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Contents 1 Principles of Thermodynamics 1.1 Introduction . . . . . . . . . . . . . . . 1.2 State variables and differential forms 1.3 Equation of state . . . . . . . . . . . . 1.4 Zeroth law . . . . . . . . . . . . . . . . 1.5 Internal energy . . . . . . . . . . . . . 1.6 Work . . . . . . . . . . . . . . . . . . . 1.7 First law . . . . . . . . . . . . . . . . . 1.8 Second law . . . . . . . . . . . . . . . . 1.9 Carnot’s cycle . . . . . . . . . . . . . . 1.10 Third law . . . . . . . . . . . . . . . . . 1.11 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . 1 . 2 . 4 . 6 . 6 . 6 . 8 . 9 . 10 . 12 . 12 2 Thermodynamic potentials 2.1 Fundamental equation . . . . . . . . . . . . . . . 2.2 Internal energy U and Maxwell relations . . . . 2.3 Enthalpy H . . . . . . . . . . . . . . . . . . . . . . 2.4 Free energy F . . . . . . . . . . . . . . . . . . . . 2.5 Gibbs function G . . . . . . . . . . . . . . . . . . . 2.6 Grand potential Ω . . . . . . . . . . . . . . . . . . 2.7 Thermodynamic responses . . . . . . . . . . . . . 2.8 Thermodynamic stability conditions . . . . . . . 2.9 Stability conditions of matter . . . . . . . . . . . 2.10 Thermodynamic potentials in electromagnetism 2.11 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 15 16 17 19 19 20 21 24 24 25 27 3 Applications of thermodynamics 3.1 Classic ideal gas . . . . . . . . . 3.2 Free expansion of gas . . . . . . 3.3 Mixing entropy . . . . . . . . . 3.4 Dilute solution, osmosis . . . . 3.5 Chemical reaction . . . . . . . . 3.6 Phase equilibrium . . . . . . . . 3.7 Phase transitions and diagrams 3.8 Coexistence . . . . . . . . . . . . 3.9 Van der Waals equation of state 3.10 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 29 31 32 33 36 39 40 42 44 47 v . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CONTENTS vi 4 Classical phase space 4.1 Phase space and probability density . 4.2 Flow in phase space . . . . . . . . . . . 4.3 Microcanonical ensemble and entropy 4.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 49 52 54 57 5 Quantum-mechanical ensembles 5.1 Density operator and entropy . . 5.2 Density of states . . . . . . . . . . 5.3 Energy, entropy and temperature 5.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 59 63 66 67 6 Equilibrium distributions 6.1 Canonical ensemble . . . . . . . . . 6.2 Grand canonical ensemble . . . . . 6.3 Connection with thermodynamics . 6.4 Thermodynamic fluctuation theory 6.5 Reversible minimal work. . . . . . . 6.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 69 75 77 78 82 83 7 Ideal equilibrium systems 7.1 Free spin system . . . . . . . . . . 7.2 Classical ideal gas . . . . . . . . . 7.3 Diatomic ideal gas . . . . . . . . . 7.4 Statistics of bosons and fermions 7.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 86 91 97 101 107 8 Bosonic systems 8.1 Bose gas and Bose condensation . 8.2 Black body radiation . . . . . . . 8.3 Lattice vibrations . . . . . . . . . 8.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 109 115 121 127 9 Fermionic systems 128 9.1 Conduction electrons in metals . . . . . . . . . . . . . . . . . . 128 9.2 Magnetism of degenerate electron gas . . . . . . . . . . . . . . 135 9.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 10 Phase transitions 10.1 Description of phase transitions 10.2 Landau theory . . . . . . . . . . 10.3 Ginzburg–Landau theory . . . . 10.4 Fluctuations in Landau theory 10.5 Problems . . . . . . . . . . . . . Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 144 147 151 157 160 162 1. Principles of Thermodynamics 1.1 Introduction Thermodynamics gives a phenomenological and very general description of matter – largely independent of models of microscopic structure (which where practically nonexistent at the time of foundation of thermodynamics in the 19th century). It is based on very few basic laws plus rules of calculus. Properties of matter or concrete systems are taken from outside (experiment, statistical mechanics). System. Macrophysical entity under consideration, may interact with its environment. It is often homogeneous or consists of homogeneous phases. The usual classification according to the possibility of exchange of energy and matter between the two goes as follows: Open system: both energy and matter may be exchanged. Closed system: particle number(s) fixed, energy may be exchanged. Isolated system: no exchange of matter or energy. Thermodynamic equilibrium. State of matter without any macroscopic changes or flows. A genuine equilibrium state is unambiguously determined by externally imposed state variables like pressure, volume, electric and magnetic fields. There are no memory effects like hysteresis. Traditionally, in the description of thermodynamic equilibrium there are three different equilibria: • mechanical equilibrium: no changes of form or other processes accompanied by production of macroscopic mechanical (or electromagnetic) work; • chemical equilibrium: no changes in the macroscopic chemical composition of the system; • thermal equilibrium: no macroscopic energy flows in a system in mechanical and chemical equilibrium. In plain words: no heat flows. In local thermodynamic equilibrium macroscopic subsets of the system are in equilibrium, but in neighbouring subsystems the equilibria are different, so that the system is not in equilibrium as a whole. Currents, heat flow etc. may occur; this is the realm of hydrodynamics. In most practically important cases local equilibrium is reached in macroscopically short time. State variables. State variables are parameters needed for characterization of the equilibrium state. Usually there is only a handful of them, 1 2 1. PRINCIPLES OF THERMODYNAMICS in many cases like the prototypical one-component gas two is enough to determine the equilibrium state, in which the rest are then functions of these parameters, state functions. State variables are either extensive or intensive, the former being proportional to the number of particles (the volume VR , particle number N , internal energy U , entropy S, magnetic moment m = d3 r M (r) etc.), whereas the latter (the temperature T , the pressure p, the chemical potential µ, magnetic field strength H) are independent of the number or particles. In thermodynamic differential forms these variables appear as conjugate pairs of extensive and intensive variables. For quantities like energy and entropy the extensiveness requires weakness of interaction energy (or correlations) between macroscopic subsystems of the original system in comparison with the "bulk" quantities prescribable to the subsystems themselves. Gravity might cause problems in this respect at very large scales. Electromagnetic interaction is usually screened in matter and thus of short range. Process. A change of state is called a process in thermodynamics. In a reversible process the direction may be inverted in the ”whole universe” (system plus environment). These processes are always quasistatic, i.e. so slow that the state of the system is infinitesimally close to thermodynamic equilibrium. Not all quasistatic processes are reversible, however. An irreversible process is often a sudden or spontaneous change (e.g. mixing of gases, explosion), during which the system may be far from equilibrium and the description by the state variables is no sufficient. An irreversible process may occur quasistatically, though. A cyclic process (cycle) consists of repeating periods during which the system always returns to the initial state. 1.2 State variables and differential forms State variables are macroscopic quantities related to the equilibrium. Not all of them are independent in equilibrium, though. Once independent variables are chosen, the rest are unambiguous functions of them, say p = p(T, V, N ), U = U (T, V, N ), S = S(T, V, N ) etc. y y 1 2 x 1=2 x Figure 1–1: Examples of processes leading from state 1 to state 2. When the change is infinitesimal, the rules of calculus yield the following relation between the differential of a function and the differentials of 1.2. STATE VARIABLES AND DIFFERENTIAL FORMS 3 indepent variables: ∂p ∂p ∂p dp = dT + dV + dN. ∂T V,N ∂V T,N ∂N V,T This implies that in a cyclic process the net change vanishes: I I dp = dU = · · · = 0. 1→1 1→1 Differential and differential form. Consider the differential form − dF ≡ F1 (x, y) dx + F2 (x, y) dy, (1.1) where F1 and F2 are given functions. An example familiar from mechanics is the differential form of work exerted by a force F on a body: − dW = F · dr = Fx dx + Fy dy + Fz dz . − in (1.1) means that the differential form is not necessarily The notation dF R2 − may depend on the integration path. If the a differential, therefore 1 dF − = dF (x, y) is a differential (ofcondition ∂F1 /∂y = ∂F2 /∂x holds, then dF ten referred to as the exact differential in this context). Then the integral R2 − dF = F (2) − F (1) is independent of the path and F1 (x, y) = ∂F (x, y)/∂x 1 ja F2 (x, y) = ∂F (x, y)/∂y are coordinates of the gradient of some function F . − = F1 dx + F2 dy is not a differential, Integrating factor. If the form dF in case of two variables a function integrating factor λ(x, y) may be found such that, in a vicinity of the point (x, y) the condition − ≡ λ F1 dx + λ F2 dy = df λ dF holds, which implies ∂(λF1 )/∂y = ∂(λF2 )/∂x. Both the integrating factor λ and the function f are then state variables. In case of three or more variables the integrating factor may not exist, in general. In thermodynamics, however, the integrating factor of the differential form of heat always exists, this is partially the content of the second law. Legendre transform. Legendre transform generates changes of variables between conjugate variable pairs. Consider, for simplicity, the function f (x) and define the variable conjugate to x as y≡ df (x) . dx (1.2) The Legendre transform of f is the following function of y: g(y) ≡ f (x) − yx , (1.3) 4 1. PRINCIPLES OF THERMODYNAMICS where on the right-hand side x is expressed as a function of y from relation (1.2). Direct calculation yields dg(y) = −x , dy (1.4) so that df = ydx and dg = −xdy. Mathematical identities. In thermodynamics, fixed variables are usually indicated explicitly when calculating partial derivatives. This is because several sets of independent variables are in wide use and infer different physical meaning for partial derivatives in different sets. Examples of useful relations for various changes of variables are listed below. Jacobi determinants. The use of Jacobi determinants ∂u ∂u ∂(u, v) ∂x ∂y = ∂(x, y) ∂v ∂v ∂x ∂y is often convenient when carrying out changes of variables in differential relations. This is due to the properties ∂(u, y) ∂(u, v) ∂(s, t) ∂u ∂(u, v) = = , , ∂(x, y) ∂x y ∂(x, y) ∂(s, t) ∂(x, y) valid for Jacobians of arbitrary order and easily checkable by direct calculation for 2 × 2 determinants. Example 1.1. Consider the function of two variables F (x, y). If by some reason we want to use the pair (x, z) as independent variables, we may writw F (x, y) = F x, y(x, z) . The chain rule then yields ∂F ∂x = z ∂F ∂z ∂F ∂x + y = x ∂F ∂y ∂F ∂y x x ∂y ∂z ∂y ∂x , z . x 1.3 Equation of state Equation of state expresses the relation of state variables of the system in equilibrium. It is usually written in a form involving ”mechanical” variables and the temperature. Equation of state does not usually include internal energy or other extensive variables of dimensions of energy, and in this sense the equation of state does not give a complete thermodynamic description of the system. A few widely used equations of state are listed below. 1.3. EQUATION OF STATE 5 Classical ideal gas. The equation of state of classical ideal gas is (1.5) pV = N T . Here, p = pressure, V = volume, N = number of molecules and T = absolute temperature. Mixture P pV = N T , Pof ideal gases: Equation of state remains the same with N = i Ni . The total pressure may be expressed as p = i pi , where pi = Ni T /V = partial pressure of the ith component. Virial expansion of real gas. The equation of state of the ideal gas may be amended so that the intermolecular interaction is taken into account. Denote the (particle) number density by n ≡ N/V . In the limit of small density the pressure may be expanded in powers of the density (the virial expansion p=T n + n2 B2 (T ) + n3 B3 (T ) + · · · , (1.6) where the virial coefficients Bn depend on the temperature only. Curie’s law. Magnetic field strength H, magnetic induction B and magnetization M are related as. B = µ0 (H + M ) . Further the magnetic moment of the system shall often be denoted by m; in case of homogeneous field then m = V M . The magnetic equation of state expresses the dependence of magnetization M on the field strength H. Many paramagnetic materials (no spontaneous magnetic ordering) obey Curie’s law M= C H, T (1.7) where C is a material constant proportional to the number density of paramagnetic atoms. Responses. Thermodynamic responses describe the reaction of state variables to the change of other state variables. They usually are easily measurable quantities. The equation of state determines ”mechanical” responses like thermal expansion coefficient 1 ∂V , (1.8) αp = V ∂T p,N isothermal compressibility κT = − 1 V ∂V ∂p = T,N 1 n ∂n ∂p T , (1.9) 6 1. PRINCIPLES OF THERMODYNAMICS or isothermal susceptibility χT = ∂M ∂H (1.10) T of magnetic material. Under an adiabatic (thermally isolated) change the responses are adiabatic compressibility 1 ∂n 1 ∂V = (1.11) κS = − V ∂p S,N n ∂p S and adiabatic susceptibility χS = ∂M ∂H . (1.12) S 1.4 Zeroth law Zeroth law of thermodynamics is the observation that there is a quantity called temperature characterizing the thermal equilibrium and a thermometer to measure and compare temperatures. This comparison is transitive: if two bodies are separately in equilibrium with a third one they are in equilibrium with each other. 1.5 Internal energy In thermodynamics internal energy is the total energy of the system at rest. Usually the potential energy of the system in an external field is excluded. Then internal energy consists of the kinetic energy of relative motion of particles, energy of their interaction and structural energy of the particles. Due to interaction between the system and its environment care has to taken in dividing the world in the system and the environment, especially when long-ranged interactions occur. At the present state of our knowledge of the structure of matter it is quite obvious that the internal energy of the system is a state variable and thus an unambiguous function of the state. It is also clear that internal energy may determined even in systems which are not in a state of thermodynamic equilibrium. 1.6 Work Work is energy exchange between the system and environment which may be described in terms of work of macroscopic mechanics and electromagnetic theory. 1.6. WORK 7 There are different sign conventions. Here, the elementary work (dif− is the work exerted to the environment by the ferential form of work) dW system. In this case positive work means loss of energy by the system. In the paradigmatic SVN - system1 the work related to the change of the volume is − dW = p dV. (1.13) The work related to the surface energy of a liquid may be written as (1.14) − dW = −σ dA , where σ = surface tension and A = free surface area. With positive surface energy σ > 0 and the surface tension tends to decrease the area. An elastic deformation gives rise to the work form: (1.15) − dW = −F dL , where F = the force stretching the rod and L = the rod length. The tension is σ = F/A = force/cross-section area. According to Hooke’s law σ = E(L − L0 )/L0 , where E is Young’s modulus and L0 the rest length of the rod. The general expression of the differential form of work is − dW = X i (1.16) fi dXi = f · dX , where fi are the coordinates of the generalized force and Xi the coordinates of the generalized displacement. Work in electromagnetism. Treatment of energetic quantities in a system in electromagnetic field requires considerable care in the definition of the system, because usually the introduction of polarizable or magnetizable body in an electromagnetic field changes the field everywhere, not only in the body itself. Unambiguous definition may be obtained, if the whole electromagnetic field is considered a part of the system. In this case the elementary work required for a change of fields in the form familiar from electrodynamics (with the sign corresponding to our convention) Z − dW = − d3 r (E · dD + H · dB) (1.17) may be interpreted as the work carried out by the system. With the aid of a special experimental setup in some cases it is possible to arrive at a situation in which the polarized body does not affect the fields E and H and the field energy outside the body may be dropped from the energy balance of the system so that − dW = −V0 (E · dD + H · dB) , (I) (1.18) 1 One-component isotropic homogeneous material with the state variables S, V, N (natural variables of the internal energy), often referred to as the pVT system as well. 8 1. PRINCIPLES OF THERMODYNAMICS when the volume V0 is small enough so that the fields may be regarded as uniform. If this is not possible, the energy corresponding to fields with the same sources (free charges and conducting currents) but without the polarizable body is nevertheless often subtracted from the energy related to the system including this body. The point here is that thermodynamics is brought about in the problem by the presence of polarizable material. Without it, the problem would be that of "pure" electrodynamics. For simplicity, consider still the case in which the fields E and H are the same both with and without the polarizable body. In uniform fields then the total energy might be written as 1 1 (1.19) Etot = U + V0 ε0 E 2 + µ0 H 2 2 2 thus excluding the energy of the "empty space" from the internal energy of the system considered. In this case the differential form of electromagnetic work related to the change of the internal energy defined as (1.19) assumes, according to (1.18) the form − dW = −V0 (E · dP + µ0 H · dM ) . (II) (1.20) This convention is often used in condensed matter and solid state physics. 1.7 First law The first law of thermodynamics is the law of conservation of energy. − − . − dW dE = dQ (1.21) Here, dE is the differential of the energy of the system. With the usual convention of thermodynamics, it may be identified by the differential of the internal energy: dE = dU , provided the momentum, angular momentum and the potential energy in external field of the system remain unaltered. If the particle number may change, the chemical potential µ is introduced by the definition − − + µdN . − dW dU = dQ In a general form for several particle species the first law is X − µi dNi . − f · dX + dU = dQ (1.22) i Heat capacity. The ability of a body to receive heat is described by the heat capacity ∆Q , CA = (1.23) ∆T A 1.8. SECOND LAW 9 where the subscript A refers to fixed variables, e.g.: CV , Cp . Specific heat is the heat capacity per unit mass. Heat capacities are usually easy to measure contrary to the internal energy. Cyclic process. Cyclic processes (cycles) are especially important in the theory of heat engines. In a cycle the system return to its initial state again and again after certain periodic stages. In a simple SVN system, in which − − p dV , the area enclosed by the curve describing the process in dU = dQ the (V, p) plane is I (1.24) p dV = W . H Since dU = 0, the work during a cycle is equal to the difference of the amounts of heat received and delivered by the system. The thermal efficiency of a cycle is η = ∆W/∆Q+ , where ∆Q+ is the amount of heat received by the system during a cycle. 1.8 Second law From the formal point of view the second law states two things: • for the differential form of heat there is an integrating factor (the temperature) giving rise to the extensive state variable entropy S; in a reversible process: dS = − dQ . T (1.25) dS > − dQ . T (1.26) • In an irreversible process There are several traditional equivalent formulations of the second law: (1.8a) Heat cannot be transferred from a colder heat reservoir to a warmer heat reservoir without any other changes. (Clausius) (1.8b) There is no cyclic process with the sole result of transferring the heat received to work. (Kelvin) (1.8c) Of all heat engines working between the temperatures T1 and T2 the Carnot engine has the highest efficiency. (Carnot) All these statements are equivalent in the sense that each of them yields the others. Here, we shall not dwell on demonstration of this equivalence, however. The first law in a reversible process may now be cast in the form dU = T dS − f · dX + X i µi dNi . (1.27) 10 1. PRINCIPLES OF THERMODYNAMICS 1.9 Carnot’s cycle The notion of entropy may be approached by analyzing Carnot’s cycle consisting of four reversible stages(Fig. 1–2): a) isothermal b) adiabatic c) isothermal d) adiabatic T2 T2 → T1 T1 T1 → T2 ∆Q2 > 0 ∆Q = 0 ∆Q1 > 0 ∆Q = 0 T2 p a ∆Q 2 ∆Q 2 d ∆W b ∆Q1 c ∆Q1 T1 V Figure 1–2: Carnot’s cycle. The thermal efficiency of the process is η= ∆Q1 ∆W =1− . ∆Q2 ∆Q2 (1.28) Since the cycle is reversible, it may be also used as a heat pump. The efficiency of Carnot’s cycle depends only on the temperatures T1 and T2 of the heat reservoirs but not on the details of realization. T3 ∆Q3 ∆W 23 T2 ∆Q 2 ∆W 12 T1 ∆Q 1 Figure 1–3: Determination of absolute temperature scale. 1.9. CARNOT’S CYCLE 11 Absolute temperature. An absolute temperature scale may be determined with the aid of a serial connection of Carnot’s cycles as in Fig. 1–3. The efficiency depends only on the reservoir temperatures, therefore 1−η = ∆Qout = f (Tmax , Tmin ) . ∆Qin (1.29) From relations f (T3 , T2 ) = ∆Q2 /∆Q3 , f (T2 , T1 ) = ∆Q1 /∆Q2 , f (T3 , T1 ) = ∆Q1 /∆Q3 the functional identity follows f (T3 , T2 )f (T2 , T1 ) = f (T3 , T1 ), which has to hold for all Ti . The simplest choice is f (T2 , T1 ) = T1 T2 (1.30) which defines the thermodynamic (absolute) temperature scale up to the choice of the unit. For the efficiency of Carnot’s cycle this yields η =1− Tmin . Tmax (1.31) Consider now a cyclic (quasistatic) process divided to a large number of subprocesses with temperatures Ti and the amount of heat received ∆Qi . Imagine that these portions of heat are transferred by Carnot engines working between the system a huge heat reservoir at the temperature T0 > Ti , so that the ith engine receives the heat ∆Q0i from the reservoir. Calculate now the work done in one cycle by the system and all the Carnot engines. In one cycle the work done by the system equals the heat received: X ∆Qi . (1.32) Wsystem = i The work of ith Carnot engine is WCi = ∆Q0i −∆Qi , so that the total work is equal to the heat received by our combined system from the heat reservoir: X X X ∆Q0i = Q0 ≤ 0 , (1.33) (∆Q0i − ∆Qi ) ∆Qi + Wtotal = i i i which cannot be positive according to Kelvins statement of the second law, since the combined system consisting of the original cycle and the auxiliary Carnot machines did not give any heat to a heat reservoir at a temperature lower than T0 . Now ∆Q0i /T0 = ∆Qi /Ti . Therefore, replacing the sum over subprocesses by a contour integral in the state variable space, we arrive at the Clausius inequality I − dQ ≤ 0. T (1.34) 12 1. PRINCIPLES OF THERMODYNAMICS For any reversible process this is an equality, which means that the integrand is a differential of some state variable. This state variable is the entropy S and − dQ = dS . T If a finite portion of our process is reversible, say from state 1 to state 2, the corresponding part of the contour integral in Clausius’s inequality (1.34) yields the difference between the values of entropy in these states: S2 − S1 = Z2 − dQ , T (1.35) 1 and Clausius’s inequality takes the form S2 − S1 ≥ Z2 − dQ . T (1.36) 1 In particular, in a thermally isolated system the entropy cannot decrease. The second law seems to be in contradiction with the time-reversal invariance of the basic microscopic laws of physics, since it establishes a preferred direction of processes. The origin of this time-reversal symmetry breaking in macroscopic physics remains unclear. 1.10 Third law The third law thermodynamics, Nernst’s law, states that the entropy of an equilibrium system vanishes, when the temperature approaches the absolute zero: lim S = 0 . (1.37) T →0 In classical thermodynamics the conjecture is that this limit exists, the particular value 0 is explained in quantum statistical physics. 1.11 Problems Problem 1.1. Show that ∂x ∂y z ∂y ∂z x ∂z ∂x y = −1 (1.38) 1.11. PROBLEMS 13 and that for any function F ∂F ∂y z ∂x . = ∂y z ∂F ∂x z (1.39) Problem 1.2. Which of the following differential forms are differentials? Find the integrating factor for those differential forms which are not differentials. (a) d− u = x4 y dx + y 2 dy. (b) d− u = (10x + 6y)dx + 6x dy , (c) d− u = 12y 2 dx + 18xy dy . Problem 1.3. Define the Legendre g(y) transform of the function f (x) as df g(y) = f (x) − xy , y= , dx where on the right-hand side x is assumed to be expressed as a function of y from the condition y = f ′ . (a) Show that d2 f dx2 d2 g dy 2 = −1 . (b) Construct the Legendre transform of the function f = 21 x2 . (c) Construct the Legendre transform of the function f = −ax ln x − b, where a and b are positive constants. Problem 1.4. Thermal expansivity α and isothermal compressibility κ of matter are defined as α= 1 V ∂V ∂T ; κ=− p 1 V ∂V ∂p . T Show that ∂α ∂p T =− ∂κ ∂T ; p α = κ ∂p ∂T . V Problem 1.5. Calculate the virial coefficients B2 , B3 and B4 of Clausius’ matter. Clausius’ equation of state is p+ aN 2 (V − bN ) = N T, T (V + cN )2 where a, b and c are positive experimental constants and N the total number of particles. Can you determine all the virial coefficients for this matter? 14 1. PRINCIPLES OF THERMODYNAMICS Problem 1.6. Consider a spherical capacitor with external radius b and internal radius a charged to an initial charge Q. The capacitor is halffilled by a dielectric substance of permittivity ε in such a way that the dielectric fills the space between the plates to one side of a cross-section plane dividing the spheres in two halves, while to the other side of the plane the capacitor is empty. Express the differential form of work in terms of electric induction D and electric field E. Calculate the work exerted on the capacitor, when the charge is increased by an infinitesimal amount δQ. Proceed by subtracting the differential form of work required to increase the charge by δQ from Q in an empty capacitor. Express the result in terms of the polarization vector P . Problem 1.7. Consider the same capacitor but now with an initial potential difference ∆φ between the plates. Express the differential form of work in terms of electric induction D and electric field E. Calculate the work exerted on the capacitor, when the potential difference between the plates is increased by an infinitesimal amount δφ. Proceed by subtracting the differential form of work required to increase the potential difference by δφ from ∆φ in an empty capacitor. Express the result in terms of the polarization vector P . Problem 1.8. Experimentally it has been found that a rubber band obeys: ∂F ∂L T T =a L0 " 1+2 L0 L 3 # , ∂F ∂T L L =a L0 " 1− L0 L 3 # , where F is the tension and the constant a and the rest length of the band L0 are parameters. (a) Calculate (∂L/∂T )F and give a physical interpretation. (b) Show that dF = ∂L F dL + ∂T F dT is a differential. (c) Determine the equation of state F = F (L, T ) of the band. Problem 1.9. In a perfect gas the internal energy obeys the relation dU = CV dT . Find the equation of state for such a gas in a process, in which the heat capacity C is a constant (polytropic process). Problem 1.10. Stirling’s cycle consists of two isotherms at T1 and T2 and two isochores (processes with constant volume) at V1 and V2 . Calculate the coefficient of thermal efficiency of Stirling’s cycle working on the ideal gas. Compare with the thermal efficiency of Carnot’s cycle. 2. Thermodynamic potentials 2.1 Fundamental equation Thermodynamic potentials are extensive state variables of dimensions of energy. Their purpose is to allow for simple treatment of equilibrium for systems interacting with the environment. In thermodynamics all variables are either extensive or intensive. Mathematically this may expressed in homogeneity relations with respect to the system size. Thus, extensive variables (e.g. N, V, U, S, . . .) are first-order homogeneous functions, whereas intensive variables (like p, T, µ, . . .)are independent of the size of the system. Natural variables. are those whose differentials appear in the differential form of the first law: dU = T dS − p dV + µ dN so that S, V ja N are natural variables of internal energy. With all intensive variables fixed, extensivity of all these variables means (2.1) U (λS, λV, λN ) = λU (S, V, N ). Differentiating both sides with respect to the auxiliary parameter λ and putting λ = 1 thereafter we arive at the identity (Euler equation for homogeneous functions): U =S ∂U ∂S +V V,N ∂U ∂V +N S,N ∂U ∂N . S,V From the first law it follows that ∂U ∂S V,N =T, ∂U ∂V S,N = −p , ∂U ∂N = µ. S,V Thus, we arrive at the fundamental equation U = T S − pV + µN . 15 (2.2) 16 2. THERMODYNAMIC POTENTIALS 2.2 Internal energy U and Maxwell relations The first law dU = T dS − p dV + µ dN yields ∂U , T = ∂S V,N ∂U , p = − ∂V S,N ∂U µ = . ∂N S,V From the definition of heat capacity it follows that ! − dQ ∂U . = CV = dT ∂T V,N (2.3a) (2.3b) (2.3c) (2.4) V,N Since U may be assumed to be single-valued smooth state variable, result of iterative differentiation does not depend on order ∂T /∂N = ∂(∂U/∂S)/∂N = ∂(∂U/∂N )/∂S = ∂µ/∂S. This procedure gives rise to Maxwell relations: ∂T ∂p = − , (2.5a) ∂V S,N ∂S V,N ∂T ∂µ = , (2.5b) ∂N S,V ∂S V,N ∂p ∂µ = − . (2.5c) ∂N S,V ∂V S,N These and similar relations for other thermodynamic potentials are often useful in expressing differential relations in terms of response functions and state variables. In an irreversible process T δS > δQ = δU + δW − µδN , therefore δU < T δS − p δV + µ δN. (2.6) In an irreversible process with fixed S, V and N the internal energy decreases. Thus, in equilibrium U assumes the mimimal value with S, V and N fixed (implying, of course, that something else may change). If some other work may be done in a reversible process, then ∆U = R = −∆Wfree , where the free work ∆Wfree = −R is the work the system may carry out in given circumstances. If the process is irreversible, then ∆Wfree ≤ −∆U (2.7) even if (S, V, N ) are kept fixed. Thus, the minimal work needed to bring about the change of internal energy ∆U is R = ∆U . 2.3. ENTHALPY H 17 2.3 Enthalpy H Other thermodynamic potentials are Legendre transformsof the internal energy U (S, V, N ) with respect to natural variables S, V or N . Enthalpy (or the heat function) is obtained by using p instead of V : H ≡ U + pV . (2.8) The differential follows from the definition and the first law: dH = T dS + V dp + µ dN . (2.9) Natural variables are (S, p, N ). From the definition of heat capacity it follows that ! − dQ ∂H Cp = . (2.10) = dT ∂T p,N p,N From the expression for dH three more Maxwell relations follow: ∂V ∂T = , ∂p S,N ∂S p,N ∂µ ∂T = , ∂N S,p ∂S p,N ∂V ∂µ = . ∂N S,p ∂p S,N (2.11a) (2.11b) (2.11c) Ia an irreversible change δQ = δU + δW − µ dN < T δS. Substitution of δU = δ(H − pV ) yields δH − δ(pV ) + δW − µ dN < T δS, i.e δH < T δS + V δp + µ δN . (2.12) If in the process S, p and N remain constant, spontaneous changes drive the system to the state with mimimum enthalpy. Many practically important processes (phase transitions, chemical reactions etc) take place at constant (ambient) pressure. If the conditions include also thermal isolation, the enthalpy is the natural energy quantity to use. In hydrodynamics adiabatic flow is a popular approximation. Then the specific (per unit mass) internal energy u appears in the energy equation only in the combination u + p/ρ = h, which is the specific enthalpy and thus the natural energy variable. When (S, p, N ) are fixed, the portion of the energy of the system freely exchangeable for work obeys the condition ∆Wfree ≤ −∆H . The mimimum work required to bring about ∆H is thus R = ∆H. (2.13) 18 2. THERMODYNAMIC POTENTIALS Joule–Thomson process. Consider thermally isolated forced flow of gas through a throttle valve or a porous wall. Movement of pistons is devised to keep pressures p1 > p2 fixed. Although the flow is far from equilibrium and not reversible, a hypothetical reversible process between the same states is useful, because state variables are process-independent. For the transfer of an infinitesimal quantity of matter the work by − = p2 dV2 + p1 dV1 . the system is dW Initially V1 = Vi and V2 = 0. Finally p1 p2 V1 = 0 and V2 = Vf . For constant pressures the work is Z − Figure 2–1: Flow through porous dW = p2 V f − p1 V i . W = wall. Thermal isolation means ∆Q = 0, therefore ∆U = −W . From this it follows Uf + p2 Vf = Ui + p1 Vi . Thus, the quantity U + pV , i.e. the enthalpy H remains constant: the process is isenthalpic, ∆H = Hf − Hi = 0 . (2.14) Imagine now a reversible isenthalpic process of decreasing the pressure by infinitesimal steps. The response of the temperature to this is given by the Joule–Thomson coefficient ∂T . (2.15) ∂p H To express this coefficient in terms of already introduced quantities, use the Jacobi determinant method. It is good policy to introduce variables which are the natural variables of the thermodynamic potential appearing in this definition, because then its first derivatives are state variables. Thus, ∂T ∂p ∂(p, S) ∂(T, H) 1 ∂(T, H) ∂(T, H) = = = ∂(p, H) ∂(p, H) ∂(p, S) T ∂(p, S) H " # ∂T ∂T V ∂H ∂H ∂T 1 − − = . (2.16) = T ∂p S ∂S p ∂S p ∂p S ∂p S Cp Using the Maxwell relation (2.11a) rewrite ∂T ∂S T ∂V T ∂V ∂V = = = . ∂p S ∂S p Cp ∂S p ∂T p Cp ∂T p Thus, we arrive at the expression # " V T ∂V V ∂T = (T αp − 1) . = − ∂p H Cp ∂T p T Cp (2.17) The latter form follows form the definition of the thermal expansion coefficient : αp = V −1 (∂V /∂T )p . 2.4. FREE ENERGY F 19 In the process the pressure decreases, so that the gas is cooled, if T αp > 1, or heated, if T αp < 1. For the ideal gas the Joule–Thomson coefficient vanishes, so that the temperature of an ideal gas remains the same. For real gases the coefficient is positive below a certain pressure-dependent inversion temperature, so that the gas is cooled. Thus, the Joule-Thomson process may be and is used for cooling and eventually liquifying gases. 2.4 Free energy F The Legendre transform of the internal energy with respect to S yields the free energy (Helmholtz free energy): F = U − S(∂U/∂S)V,N i.e. F ≡ U − TS . (2.18) The corresponding differential is dF = −S dT − p dV + µ dN . The natural variables are T , V and N . The Maxwell relations are ∂S ∂p = , ∂V T,N ∂T V,N ∂S ∂µ = − , ∂N T,V ∂T V,N ∂p ∂µ = − . ∂N T,V ∂V T,N (2.19) (2.20a) (2.20b) (2.20c) As before, for an irreversible process δF < −S dT − p δV + µ δN . (2.21) Thus, with fixed T , V and N , the system evolves towards the minimum of the free energy. For the free work at fixed T, V, N it follows ∆Wfree ≤ −∆F . (2.22) Free energy is an extremely important tool in statistical mechanics: in many cases it is the natural macroscopic quantity to calculate for a given microscopic model. 2.5 Gibbs function G The Legendre transform of U with respect to both S and V leads to the Gibbs function (Gibbs free energy) G ≡ U − T S + pV , (2.23) 20 2. THERMODYNAMIC POTENTIALS with the differential (2.24) dG = −S dT + V dp + µ dN . The natural variables are T , p and N ∂S = ∂p T,N ∂S = ∂N T,p ∂V = ∂N T,p and the Maxwell relations ∂V − , ∂T p,N ∂µ − , ∂T p,N ∂µ . ∂p T,N (2.25a) (2.25b) (2.25c) With fixed T , p and N , a non-equilibrium system evolves towards the minimum of the Gibbs function: δG < −S δT + V δp + µ δN , (2.26) ∆Wfree ≤ −∆G . (2.27) G = µN , (2.28) and the free work is The Gibbs function is a suitable thermodynamic potential for systems which change at fixed pressure and temperature (no thermal isolation). Since these parameters are perhaps most easily of all adjustable, the Gibbs potential has a wide scope of applications both in physics and chemistry. From the fundamental equation it follows that i.e. the chemical potential is the Gibbs function per particle in a singlespecies system. Since from (2.35) it follows that dG = µdN + N dµ and, taking into account the alternative form (2.24), we arrive at the Gibbs– Duhem equation V S dµ = − dT + dp , (2.29) N N showing that the natural variables of the chemical potential are T, p. 2.6 Grand potential Ω The grand potential is also an important quantity for calculations in statistical mechanics when the number of particles cannot be fixed. The definition is Ω ≡ U − T S − µN (2.30) leading to the differential dΩ = −S dT − p dV − N dµ , (2.31) 2.7. THERMODYNAMIC RESPONSES 21 showing that the natural variables are T , V and µ. The Maxwell relations are ∂S ∂p = , (2.32a) ∂V T,µ ∂T V,µ ∂S ∂N = , (2.32b) ∂µ T,V ∂T V,µ ∂p ∂N = . (2.32c) ∂µ T,V ∂V T,µ In an irreversible change the inequality δΩ < −S δT − p δV − N δµ , (2.33) holds revealing that in a process with fixed T , V and µ the system tends to state with the minimum of Ω. The free work under these conditions is ∆Wvapaa ≤ −∆Ω . (2.34) From the fundamental equation it follows that (2.35) Ω = −pV revealing that knowledge of Ω is tantamount to knowing the equation of state (although in non-standard variables). 2.7 Thermodynamic responses Thermodynamic responses have the form of partial derivatives (∂K/∂A)B,C,... and reveal the effect of an infinitesimal change of a state variable (A) to some quantity (K) describing the system at equilibrium. Usually these are (the most) directly measurable quantities. Coefficient of (volumninal) thermal expansion. Definition 1 ∂V . αp = V ∂T p,N (2.36) In terms of number density n = N/V : αp = − 1 n ∂n ∂T . (2.37) p In isotropic substance the coefficient of linear thermal expansion is one third of this, since a small change of volume is three times the change of length. 22 2. THERMODYNAMIC POTENTIALS Isothermal compressibility. Reaction to pressure at constant temperature 1 ∂V 1 ∂n κT = − = . (2.38) V ∂p T,N n ∂p T Adiabatic compressibility. Pressure acting in thermal isolation 1 ∂n 1 ∂V = . (2.39) κS = − V ∂p S,N n ∂p S,N This quantity determines the speed of sound: cs = √ 1 . mnκS (2.40) Here, m is the particle mass and mn = mN/V is the mass density. Isochoric heat capacity. Definition by reversible process, thus ∆Q = T ∆S, and the heat capacity in general ∆Q ∆S C≡ = T . ∆T condition ∆T condition Due to the chosen scale of temperature, heat capacities are dimensionless. Specifically at constant volume we obtain ∂S . (2.41) CV = T ∂T V,N − = Since under these conditions dU = T dS −p dV +µ dN reduces to dU = dQ T dS and, on the other hand, S = −(∂F/∂T )V,N , we arrive at relations CV = ∂U ∂T V,N = −T ∂2F ∂T 2 . (2.42) V,N Isobaric heat capacity. Analogously Cp = T ∂S ∂T (2.43) , p,N − and S = −(∂G/∂T )p,N yield (dH)p,N = T dS = dQ, Cp = ∂H ∂T p,N = −T ∂2G ∂T 2 . p,N (2.44) 2.7. THERMODYNAMIC RESPONSES 23 Connections. Relations between heat capacities under different conditions are due to differences in work. For Cp , change variables ∂S ∂S(V (p, T ), T ) ∂S ∂S ∂V = = + . ∂T p ∂T ∂T V ∂V T ∂T p p Free energy Maxwell (∂S/∂V )T = (∂p/∂T )V (2.20a) yields ∂V ∂p . Cp = CV + T ∂T V ∂T p (2.45) The change of variables in (∂p/∂T )V gives rise to the result Cp = CV + V T αp2 . κT (2.46) Compressibility is positive in stable matter, therefore Cp > CV . Construction of potentials. An equation of state like p = p(T, V ) and a thermal response, say CV , are required to this end. Consider, for instance, van der Waals matter: N2 p + a 2 (V − N b) = N T . V The heat capacity CV is directly a partial derivative of the internal energy: ∂U . CV = ∂T V To calculate the other one, change variables ∂U ∂(U, T ) ∂(V, S) ∂(U, T ) = = ∂V T ∂(V, T ) ∂(V, S) ∂(V, T ) ∂S ∂U ∂T ∂T ∂U = . − ∂T V ∂V S ∂S V ∂S V ∂V S (2.47) Here, derivatives of U are −p and T , this is the point of introducing the natural variables of U . Thus ∂U ∂S ∂p = −p + T =T − p, ∂V T ∂V T ∂T V where the free-energy Maxwell (2.20a) has been used once more. This relation shows, in particular, that for the van der Waals equation of state ∂2U ∂CV = 0, = ∂V T ∂V ∂T i.e. CV is a function of temperature only! Then the integration is simple: Z Z ∂p N2 U (T, V ) = CV dT + T . − p dV = CV (T )dT − a ∂T V V 24 2. THERMODYNAMIC POTENTIALS 2.8 Thermodynamic stability conditions Let the near-equilibrium system be divided to (semi)macroscopic subsystems (labeled by index α) each in a local equilibrium, but with different pressure, temperature etc. in neighbouring subsystems. Extensive quantities remain PaddiP P tive: S = α Sα , V = α Vα , U = α Uα . Let Njα be the particle number ofP species j in the subsystem α. Then Nj = α Njα for the jth species. Due to local equilibrium For a small change of Sα ∆Sα = α pα T α V α Figure 2–2: System near equilibrium Sα = Sα (Uα , Vα , {Njα }) . X µjα pα 1 ∆Uα + ∆Vα − ∆Njα . Tα Tα Tα j Assume a system isolated as a whole, then U , V and Nj remain constant. To simplify notation, consider two subsystems: α = A, B. Conservation laws yield ∆UB = −∆UA , ∆VB = −∆VA and ∆NjB = −∆NjA . Thus, X ∆S = ∆Sα α = 1 1 − TA TB ∆UA + pA pB − TA TB ∆VA − X µjA j TA − µjB TB ∆NjA . At equilibrium ∆S = 0 identically. Since the fluctuations ∆UA , ∆VA and ∆NjA are arbitrary, the equilibrium conditions: TA = TB pA = pB (2.48) µjA = µjB follow. Thus, in equilibrium the temperature is the same everywhere, as well as the pressure (provided no external fields impose inhomogeneity) and the chemical potential for each particle species. The conditions hold also in the case system consists of different phases (constant pressure requires flat interfaces, however). 2.9 Stability conditions of matter In stable equilibrium the entropy must be at maximum. To analyze this, the second variation of the entropy with respect to {∆Uα , ∆Vα and ∆Njα } may be used. 2.10. THERMODYNAMIC POTENTIALS IN ELECTROMAGNETISM 25 Let T , p and {µj } be the common equilibrium values. For simplicity, assume one species. In Taylor expansion at the equilibrium point S = S0 + dS + 21 d(dS) the linear term vanishes. Since d2 X = 0 for any independent variable X, we obtain 1 S = S0 + d(dS) = 2 1 1X 1 1X d (dpα dVα − dµα dNα ) . (dUα + pα dVα − µα dNα ) + 2 α Tα 2 α Tα Here, dUα + pα dVα − µα dNα = Tα dSα , so that (denote dX → ∆X) 1 X ∆Stot ≡ S − S0 = − (∆Tα ∆Sα − ∆pα ∆Vα + ∆µα ∆Nα ) . 2T α (2.49) The condition of a stable equilibrium is that this expression is negative definite. Since any subsystem α is at local equilibrium, only three of fluctuations of the quantities Tα , Sα , pα , Vα , µα , Nα are independent, the rest must be expressed as functions of the chosen three. Let ∆Nα = 0. Then only two independent variables remain. Choose ∆Tα and ∆Vα and express ∆Sα as ∆pα functions thereof. Maxwell relations allow for simplification and the result is 1 1 X CV,α (2.50) (∆Tα )2 + (∆Vα )2 . ∆Stot = − 2T α T κT V α Another possibility ∆Vα = 0 with ∆Tα and ∆Nα as independent variables leads to ( ) 1 X CV,α ∂µ ∆Stot = − (∆Tα )2 + (2.51) (∆Nα )2 . 2T α T ∂Nα T,Vα From these expressions it is readily seen that the total entropy is at maximum, when the following stability conditions hold: : CV > 0 κT > 0 . (2.52) ∂µ > 0 ∂N T,V Otherwise the equilibrium is unstable and small spontaneous disturbances give rise to growing changes which lead to another state. 2.10 Thermodynamic potentials in electromagnetism The starting point here is the basic differential form of work (1.17) Z − dW = − d3 r (E · dD + H · dB) , 26 2. THERMODYNAMIC POTENTIALS whose addition to previously introduced differentials gives rise to differentials of thermodynamic potentials in electromagnetism, say (relevant parameters only explicit) Z dU = T dS + d3 r (E · dD + H · dB) . Material parameters contained in vectors E and B like the permittivity ε and permeability µ should be expressed here as functions of the entropy S. This is inconvenient, therefore a preferable choice is the free energy, for which Z dF = −SdT + d3 r (E · dD + H · dB) . (2.53) Here, ε and µ are functions of the temperature. Thermodynamics potentials assume minimum values at equilibrium, when their natural variables are fixed. Since free charges are sources of the electric induction D and the vector potential A the source of the magnetic induction B, the free energy (2.53) is the choice for problems with fixed charges of conductors and fixed vector potentials (the latter might be difficult to control in real world, though). For other cases, new potentials should be formed R by suitable Legendre transforms. For instance, the potential FeE = F − d3 r E · D gives rise to the differential Z dFeE = −SdT − d3 r D · dE , (2.54) which reveals that the natural variables are T and E. Thus, this potential minimizes at equilibrium when the field E (or the electric potential) is kept constant. R Similarly, the potential FeH = F − d3 r H · B with the differential Z dFeH = −SdT − d3 r B · dH , (2.55) and natural variables T and H is suitable for cases with fixed currents. Combinations of these transform may appear useful as well. Unfortunately, there seems to be no standard nomenclature of the different potentials in the electromagnetic case (cf. the Helmholtz free energy and the Gibbs function of an S, V , N system). Example 2.1. Consider a vertical parallel-plate capacitor in contact with a liquid reservoir. Let us calculate, how high the liquid with the dielectric constant εr rises between the vertical plates, when the capacitor is charged and disconnected from any voltage source. The potential energy of the liquid in the gravitational field is Wg = 1 gρwdy 2 , 2 where g is the acceleration of gravity, ρ the density of the liquid, y the height of the liquid slab between the plates, d the separation of the 2.11. PROBLEMS 27 plates and w the width of the plates. The energy of the electric field between the plates is WQ = Q2 Q2 d = , 2C 2wε0 [h − y + εr y] where Q is the charge of the capacitor and h the height of the plates. Minimization of the free energy F with respect to y leads to the thirdorder equation 2 Q2 h y y+ = , ε0 − 1 2ρw2 gε0 which only has one real solution most conveniently obtained by some symbolic calculation programme like Maple or Mathematica. 2.11 Problems Problem 2.1. Calculate the value of the expression ∂T ∂p V ∂S ∂V − p ∂T ∂V p ∂S ∂p . V Problem 2.2. (a) During a thermally isolated free expansion of a gas no work is carried out and no heat exchanged, thus the internal energy of the gas remains constant (the expansion is not necessarily a quasistatic process, however). Show that the Joule coefficient for a free expansion of a gas is ∂T ∂V 1 CV = U,N p− T αp κT . (b) Is the van der Waals gas heated or cooled in the free expansion? Hint: it is more convenient not to calculate αp and κT separately. The van der Waals equation of state is p+a N2 V2 (V − N b) = N T. Problem 2.3. Find out in which systems the heat capacity CV does not depend on the volume of the system. Problem 2.4. The free energy of a crystal in which the ions have just two quantum states is F = −N T ln 1 + e−ǫ/T , where ǫ is a constant. (a) Find the entropy S of the system as a function of the internal energy U and the number of particles N , i.e. the fundamental relation S = S(U, N ). 28 2. THERMODYNAMIC POTENTIALS (b) Calculate the heat capacity Cǫ as a function of temperature T . Plot it and find the position of the peak, known as the Schottky anomaly. Such a peak is characteristic of a system in which atoms have a few low-lying closely spaced energy levels, and at low temperatures may dominate all other contributions to the heat capacity of the solid. Problem 2.5. Show (N is kept fixed) that the internal energy of the Clausius gas is U (T, V ) = N u1 (T ) − 2aN 2 , T (V + N c) where u1 (T ) is a function of temperature only, whose explicit form cannot be determined thermodynamically. ZHowever, there is another relation (show this as well) U (T, V ) = CV (T, V ) , dT , establishing a connection between u1 and the isochoric heat capacity of the gas. The equation of state of Clausius’s gas is p+ aN 2 (V − bN ) = N T . T (V + cN )2 Problem 2.6. For a unit volume of dielectric at constant density find the difference cE − cD between the heat capacities of a homogeneous isotropic dielectric at constant electric field strength E and electric induction D. Problem 2.7. Show – without resorting to the connection between Cp and CV – that in a stable thermodynamic equilibrium Cp > 0 and κS > 0 Problem 2.8. By minimizing a suitable thermodynamic potential, find how high dielectric liquid rises between the vertical plates of a parallelplate capacitor connected to a voltage source with the constant electromotive force E. Express your answer in terms of the electric field in the capacitor rather than the emf. 3. Applications of thermodynamics 3.1 Classic ideal gas For a complete thermodynamic description of a system, knowledge of the equation of state and some thermodynamic potential (energy function) is required. From the equation of state mechanical responses may be inferred and vice versa: mechanical responses suffice to reconstruct the equation of state. To determine internal energy or some other thermodynamic potential a thermal response is needed, however. The equation of state of the perfect gas pV = N T immediately yields coefficient of thermal expansion 1 ∂V 1 N αp = = (3.1) = V ∂T p,N Vp T and isothermal compressibility 1 NT 1 ∂V = . = κT = − V ∂p T,N V p2 p (3.2) It is en empirical observation that in conditions in which the equation of state of a real gas coincides with that of the ideal gas (low pressure, high temperature) the heat capacity is constant. Denote CV = 1 fN . 2 (3.3) The quantity 21 f is the specific heat capacity, (specific heat), i.e. heat capacity per molecule. In chemistry heat capacity per mole and in hydrodynamics per mass unit are preferred. The factor f is the number of the effective degrees of freedom, whose classic value depends on the number of modes of translational, rotational and vibrational motion of the molecule: monatomic molecule diatomic molecule polyatomic molecule f =3 f =5 f =6 3 translations 3 transl. + 2 rotations 3 transl. + 3 rot. Each active vibrational mode adds two effective degrees of freedom (for both kinetic and potential energy of the corresponding mode of harmonic oscillation). In what follows f is assumed constant. 29 30 3. APPLICATIONS OF THERMODYNAMICS The differential of the entropy ∂S ∂S dS = dT + dV ∂T V ∂V T ∂p 1 CV dT + = dV T ∂T V (3.4) is readily integrated in the (T, V ) plane (N is fixed) with the use of the heat capacity and mechanical responses (see Fig. 3–1) Assuming constant CV and using the equation of state we obtain Z V Z T T V N CV + = S0 + CV ln + N ln . dV S = S0 + dT T V T V 0 0 V0 T0 The integration constants S0 and V0 – as extensive quantities – may be written as S0 = N s0 , V0 = N v0 , where s0 and v0 are the the specific entropy and volume at the reference point. The entropy of the ideal gas is thus " # f /2 T V S = N s0 + N ln . (3.5) T0 N v0 The specific entropy at the reference point s0 shall be defined later with the aid of statistical mechanics. The expression (3.5) for entropy does not vanish in the limit T → 0 but even diverges V in contradiction with the III law. For real gases, however, lowering the temperature leads either to phase transitions to liquid or V0 solid or the appearance of quantum correcT0 T tions. All thermodynamic information may be Integration calculated starting from the known entropy. Figure 3–1: path for entropy. The internal energy may also be calculated in the same fashion as before for the van der Waals gas by integration in the (T, V ) plane: dU S(T, V ), V = T dS(T, V ) − p dV ∂S ∂S dT + T dV − p dV = T ∂T V ∂V T ∂S = CV dT + T − p dV . ∂V T From the Maxwell (2.20a) and the equation of state it follows that (∂S/∂V )T = (∂p/∂T )V = N/V = p/T , so that the coefficient of dV vanishes. The equation of state renders the heat capacity CV independent of volume as well. Thus, the internal energy of the ideal gas is independent of the volume and may be written as 1 U = U0 + f (T − T0 )N . 2 3.2. FREE EXPANSION OF GAS 31 or, relabeling the normalization term, as 1 U = N µ0 + f T . 2 (3.6) According to 2.7 we obtain Cp = CV +V T αp2 /κT = CV +V T p/T 2 = N ( 21 f +1). The usual notation is Cp = γCV , where γ is the heat capacity ratio (adiabatic constant) γ= f +2 Cp = . CV f (3.7) 3.2 Free expansion of gas Free expansion (Joule process) takes place, when a valve is opened or a wall removed from between two chambers with different pressures. A sudden leveling of pressures is a typical irreversible process during which the system is not in equilibrium. The initial and final states, however, are equilibrium states. Let the volume grow from V1 to V2 during the expansion. Assume thermal isolation: ∆Q = 0. Since opening the valve ideally does not involve work ∆W = 0 as well. Therefore, internal energy does not change ∆U = 0; the process is isergic.. Changes in state variables may again be calculated along a hypothetic reversible path 1 → 2. Figure 3–2: Free expansion of Ideal gas. Since in the ideal gas U = gas to vacuum. the temperature must remain the same T1 = T2 . The change in entropy may be calculated with the aid of relation (3.5). 1 2fTN, ∆S = N ln V2 . V1 (3.8) It should be noted that due to thermal isolation no entropy changes in the environment occur: the entropy production is completely of internal origin. Other equations of state. In analogy with the Joule-Thomson process the following Joule coefficient may be defined and expressed in terms of mechanical responses: ∂T αp 1 p−T . (3.9) = ∂V U,N CV κT The result yields the temperature change in an infinitesimally small expansion. For finite changes integration is needed. 32 3. APPLICATIONS OF THERMODYNAMICS 3.3 Mixing entropy Consider two gases (A and B) separated by a wall. When the wall is removed, the gases mix. Let the A B temperature and the pressure in p the final state be the same as before mixing, so that the process may be thought of as isothermal and isobaric. Obviously, disorder increases Figure 3–3: Isobaric and isothermal and increase of the entropy is ex- mixing of gases. pected. Consider a mixture of ideal gases, so that partial pressures obey pj Dalton’s law pj V = Nj T , with Nj being the particle Pnumber of species j. Its concentration is xj = Nj /N = pj /p, where p = j pj is the total pressure. In the following the increase of the entropy shall be calculated in two ways, of which the latter is easier to generalize to non-ideal systems. Way 1. Both gases are imagined to freely expand in turns to the total volume and the entropy changes related to these stages are added. This is possible, because interaction between the compounds is negligible. Since pA = pB and TA = TB , the initial volumes are Vj = V xj , and the entropy change according to (3.8) is ∆S = X Nj ln j V . Vj In terms of concentrations this is ∆Ssek = −N X xj ln xj . (3.10) j This is always ≥ 0, because 0 ≤ xj ≤ 1. Way 2. Since the process takes place at constant pressure and temperature, the Gibbs function is useful. For a single-species ideal gas the Gibbs function is G(p, T, N ) = N T [φ(T ) + ln p] = N µ(p, T ) , (3.11) where the most general form of the function φ is φ(T ) = µ0 /T − ζ − ( 12 f + 1) ln T . In the case of non-interacting gases the Gibbs function of the mixture is the sum of Gibbs functions of the compounds. Prior to mixing the pressure of all compounds is p, and the Gibbs function is the sum of those of compounds: X Gi = Nj T [φj (T ) + ln p] . (3.12) j 3.4. DILUTE SOLUTION, OSMOSIS 33 After mixing the partial pressures are pj and thus the Gibbs function X Nj T [φj (T ) + ln pj ] . (3.13) Gf = j Since pj = pxj , the difference is ∆Gmixing ≡ Gf − Gi = X Nj T ln xj . j The entropy is calculated as the partial derivative S = −(∂G/∂T )p,{Nj } . Thus, the mixing entropy is X Nj ln xj . (3.14) ∆Smixing ≡ Sf − Si = − j In isobaric mixing of real gases the volume of the system is not preserved due to interactions and the Gibbs function of the mixture is not the simple sum (3.13). Gibbs paradox. What happens, if the gases are the same, A = B? If the expressions obtained are used as such, increase of entropy follows ∆S > 0, although there is no macroscopic physical change. This is the Gibbs paradox that cannot be satisfactorily explained by classical physics. In quantum statistics the solution is that microstates differing only by permutation of numbers of identical particles are considered one physical state. 3.4 Dilute solution, osmosis Consider a system consisting of a solvent and a small amount of added solute. Molecules of the solute are far away from each other and do not interact and the whole system looks like a dilute ideal gas of the solute mixed with the solvent. Thus, the mixing entropy has to be taken into account. Denote the concentration of the solute by x = N1 /N = N1 /(N0 + N1 ), where N0 is the number of molecules of the solvent and N1 that of the solute. Differentiation yields ∂x x = − ∂N0 N . (3.15) 1−x ∂x = ∂N1 N In the Gibbs function of the solution G(p, T, N0 , N1 ) = N0 µ0 (p, T, x) + N1 µ1 (p, T, x) (3.16) the chemical potentials µ0 and µ1 may depend on the particle numbers through the concentration x only, so that from the Maxwell relation ∂µ0 ∂µ1 ∂2G = = ∂N1 ∂N0 ∂N0 ∂N1 34 3. APPLICATIONS OF THERMODYNAMICS the consistency condition follows ∂µ0 ∂x ∂µ1 ∂x = , ∂x ∂N1 ∂x ∂N0 or, in view of (3.15), (1 − x) ∂µ0 ∂µ1 +x = 0. ∂x ∂x (3.17) This must hold for all 0 ≤ x ≤ 1. Obviously, for the chemical potential of the solvent µ0 the limit x → 0 must be well-behaved and the existence of a power expansion in x is a natural assumption: µ0 (p, T, x) = µ0 (p, T, 0) + xν(p, T ) + O(x2 ) . (3.18) In µ1 the logarithmic mixing entropy term must be present, therefore we write µ1 (p, T, x) = T ln x + ψ(p, T ) + O(x) . (3.19) The consistency condition (3.17) immediately yields ν(p, T ) = −T . (3.20) Taking into account that xN0 ≈ N1 we arrive at the expansion of the Gibbs function h i x G(p, T, N0 , N1 ) = N0 µ0 (p, T, 0) + N1 T ln + ψ(p, T ) + · · · . (3.21) e P Since dG = −S dT + V dp + i µi dNi , the volume is obtained as ∂G V = ∂p T,{Nj } ∂ψ(p, T ) ∂µ0 (p, T, 0) + N1 ∂p ∂p = N0 v 0 + N 1 v 1 . = N0 (3.22) Here, v0 = ∂µ0 /∂p is the specific volume of the pure solvent and v1 = ∂ψ/∂p the additional volume required by one molecule of the solute of small concentration. For the entropy we obtain ∂G S = − ∂T p,{Nj } x ∂ψ(p, T ) ∂µ0 (p, T, 0) − N1 ln − N1 ∂T e ∂T = S(0) + Ssek + S(1) , = −N0 (3.23) where S(0) = −N0 ∂µ0 /∂T is the entropy of the pure solvent; Ssek = x −N1 ln = −N0 ln(1 − x) − N1 ln x is the mixing entropy, when both come pounds are considered ideal gases; finally S(1) = −N1 ∂ψ/∂T contains the 3.4. DILUTE SOLUTION, OSMOSIS 35 entropy of the solute and additional entropy due to, say, interactions between the molecules of the solute and the solvent. Osmosis. If the free transport of the solute is blocked by a semipermeable membrane, a pressure difference, osmotic pressure is brought about between both sides of the membrane. The osmotic pressure may easily be measured by hydrostatic means (see Fig. 3–4). If the density of the solution is nearly the same as that of the pure solvent, then ∆p = ρm gh, where ρm is the density of the solvent and h is the height of the surface of the solution above the surface of the solvent. According to Chapter 2.8 at equilibrium the chemical potential of the solvent must be the same everywhere: B A h µ0 (pA , T, 0) = µ0 (pB , T, x) , pB pA where x is the concentration of the solute. For small concentrations relations (3.18) and (3.20) yield Figure 3–4: Osmosis. µ0 (pA , T, 0) = µ0 (pB , T, 0) − xT . (3.24) Due to the Maxwell relation (2.25c) ∂V V ∂µ = = = v, ∂p T,N ∂N T,p N where v is the specific volume. Therefore, we may approximate µ0 (pB , T, 0) ≈ µ0 (pA , T, 0) + v0 ∆p , where v0 is the specific volume of the solvent. Combining with relation (3.24) we arrive at the Van’t Hoff equation ∆p V = N1 T . (3.25) Here, ∆p = pB − pA is the osmotic pressure, and N1 = xN is the number of solute molecules in the volume V . The result is of the same form as the equation of state of the ideal gas. Thus, osmosis causes in a dilute solution additional pressure following exactly the ideal gas law regardless of the nature of the solute, the solvent and their interactions. Osmotic pressures calculated by (3.25) may be significant, even orders of magnitude larger (of order of a few MPa in biological applications) than the ambient pressure (in natural conditions about 0.1 Mpa). This might raise some suspicion about the applicability of relation (3.25), because its derivation was based, among other assumptions, on the expansion of the total pressure in the solution pB in powers of the osmotic pressure ∆p with 36 3. APPLICATIONS OF THERMODYNAMICS only two leading terms retained. A closer inspection reveals, however, that for liquid solutions the Van’t Hoff equation remains applicable also for osmotic pressures much larger than the ambient pressure. The point is that the comparison of the remainder of the Taylor expansion µ0 (pB , T, 0) = µ0 (pA , T, 0) + v0 ∆p + 1 ∂v0 (p∗ , T, 0) ∆p2 , 2 ∂p where pA ≤ p∗ ≤ pB , with the second term v0 ∆p contains the factor ∂v (p∗ , T, 0) , i.e. the isothermal compressibility, which for practiκT = − 0 v0 ∂p cal liquid solvents is very small (for water, e.g., about 0.5 · 10−9 Pa−1 ). Thus, the numerical smallness of the second derivative of the chemical potential overweighs the relatively large values of the osmotic pressure ensuring the applicability of Van’t Hoff equation (3.25) for fairly large osmotic pressures. 3.5 Chemical reaction In the equation of a chemical reaction 0= X νj M j (3.26) j integer-valued stoichiometric coefficients νj express the proportions in which the amount of different species of molecules Mj change in the reaction. For instance, to the burning reaction 2H2 S + 3O2 −→ ←− 2H2 O + 2SO2 (3.27) corresponds the notation −νA A − νB B −→ ←− νC C + νD D , where A = H2 S, B = O2 , C = H2 O and D = SO2 with the stoichiometric coefficients νA = −2, νB = −3, νC = 2 and νD = 2. The progress of reaction is described by the degree of reaction (degree of advancement ξ, whose differential is defined by 1 dNj = νj dξ . (3.28) Thus, when ξ is increased (decreased) by one exactly one reaction takes place to the right (left) in the equation of reaction. The usual convention is that ξ = 0, when the reaction is at its leftmost state, i.e. when one of the compounds on the right-hand side is used up completely. With this normalization ξ ≥ 0 always. 1 In chemistry the definition goes by number of moles instead of the number of molecules used here. 3.5. CHEMICAL REACTION 37 Assume constant p and T . Consider the Gibbs function X µj Nj G= j and its differential with the account of definition (3.28): X X νj µj . µj dNj = dξ dG = (3.29) Define the unit change of the Gibbs function in the reaction as X ∂G ∆r G ≡ νj µj = −A . = ∂ξ p,T j (3.30) j j With the opposite sign this is the affinity A. At constant (p, T ) the thermodynamic equilibrium corresponds to the minimum of G. Since the degree of reaction ξ is the only independent dynamic variable [apart from (p, T )], the equilibrium condition is X ∆r Geq = νj µeq (3.31) j = 0. j In a non-equilibrium state dG/dt < 0. If ∆r G > 0, then dξ/dt < 0 and the reaction proceeds to the left. In rarified gas or dilute solution in a passive solvent the ideal gas description is reasonable for the reacting compounds. Then µj = T [φj (T ) + ln p + ln xj ] , (3.32) where the most general form of the function φj is φj (T ) = µ0j /T − ζj − (1 + 1 2 fj ) ln T . Calculation yields P Y X ∆r G = T νj φj (T ) + T ln p νj xj νj . (3.33) j The equilibrium condition (∆r G = 0) may now be cast in the form of the law of mass action: P Y (3.34) xj νj = p− j νj K(T ) . j Here, the equilibrium constant K(T ) = e− P j νj φj (T ) , (3.35) of the reaction has been introduced. For reaction (3.27) the equilibrium condition is xC 2 xD 2 = pK(T ) . xA 2 xB 3 Heat of reaction. This is the heat ∆r Q acquired by the reacting system in one step to the right. Two classes of reactions are distinguished: 38 3. APPLICATIONS OF THERMODYNAMICS ∆r Q > 0: endothermic reaction, ∆r Q < 0: exothermic reaction. In an isobaric process the amount of heat is equal to the change of enthalpy, since ∆Q = ∆U + ∆W = ∆U + p ∆V = ∆(U + pV ) = ∆H. This can be straightforwardly calculated with the use of the unit change of the Gibbs function. According to relations (3.33) and (3.35) X νj ln(pxj ) . (3.36) ∆r G = −T ln K(T ) + T j If the total amount of matter is unchanged, then in a reversible process dG = −S dT + V dp. From this it follows that 1 V G G S H V G = dG − 2 dT = − dT + dp = − 2 dT + dp , + d 2 T T T T T T T T since G = H − T S. In particular, H = −T 2 ∂ ∂T G T . (3.37) p,N From expression (3.36) of ∆r G we obtain ∆r G d ∂ =− ln K(T ) ; ∂T T dT where comparison goes between the temperature dependence of the Gibbs function before and after one reaction step. The change of enthalpy, i.e. the heat of reaction, is d ln K(T ) . ∆r H = T 2 (3.38) dT The heat of reaction is often put together with the reaction formula as CH4 + 2O2 −→ CO2 + 2H2 O, ∆r H = −890.35 kJ/mol. (3.39) It is usually quoted per stoichiometric molar changes at the reference temperature 25◦ C = 298 K. Example 3.1. One mole of H2 S and 2 moles of H2 O is gaseous state are mixed at the pressure p and the temperature T , which results in the chemical reaction H2 S + 2H2 O ⇐⇒ 3H2 + SO2 . Calculate concentrations xi of all the substances as well as the standard free energy for the reaction ∆r G as functions of the degree of reaction ξ. Find also the equation for ξ and the volume of the system at equilibrium. Use the ideal gas approximation. Here, the species labels and stoichiometric coefficients are A = H2 S, B = H2 O, C = H2 , D = SO2 , νA = −1, νB = −2, νC = 3 and νD = 1. 3.6. PHASE EQUILIBRIUM 39 Since dNi = νi dξ and ξ = 0, when only the left-hand compounds are present, we may write Ni = νi + Ni0 , where Ni0 are the initial numbers of molecules. Taking into account the initial condition we arrive at relations (N0 is the Avogadro number here): NA = N0 − ξ , NB = 2N0 − 2ξ , NC = 3ξ , ND = ξ . Since the total number of particles is N = NA +NB +NC +ND = 3N0 +ξ, the sought concentrations are xA = N0 − ξ , 3N0 + ξ xB = 2N0 − 2ξ , 3N0 + ξ xC = 3ξ , 3N0 + ξ xD = ξ . 3N0 + ξ In the ideal gas approximation µi = T [φi (T ) + ln p + ln xi ]. Thus ∆r G = T X P νi φi (T ) + T ln p i + T ln N0 − ξ 3N0 + ξ −1 i νi + T ln Y i 2N0 − 2ξ 3N0 + ξ −2 xνi i = −T ln K(T ) + T ln p 3ξ 3N0 + ξ 3 ξ 3N0 + ξ . This yields ∆r G = T ln 27pξ 4 4K(T )(3N0 + ξ)(N0 − ξ)3 . In equilibrium ∆r G = 0 and the equilibrium value ξ0 is found from the relation 4K(T ) ξ04 = . 27p (3N0 + ξ0 )(N0 − ξ0 )3 Thus, in equilibrium pV = N T = (3N0 + ξ0 )T and V = (3N0 + ξ0 ) T . p 3.6 Phase equilibrium Phases are macroscopically different homogeneous equilibrium states of matter. They may coexist. If the contact surface allows free transfer of molecules in both directions, the usual equilibrium conditions (2.48) hold (provided all interfaces are flat, curvature leads to pressure differences), so that the pressure p and the temperature T are the same in all phases. Further, the chemical potential of each species is the same in all phases: for each pair of phases (A, B) µjA = µjB , (j = 1, . . . , H) (3.40) With H species taking into account the common values of the pressure and the temperature this yields µ1A (p, T, x1A , . . . , xhA ) = µ1B (p, T, x1B , . . . , xhB ) .. , (3.41) . µHA (p, T, x1A , . . . , xhA ) = µHB (p, T, x1B , . . . , xhB ) 40 3. APPLICATIONS OF THERMODYNAMICS where h = H − 1. With F coexisting phases F − 1 independent constraints for each species in different phases follow. Altogether H(F − 1) constraints are imposed. Each chemical potential in (3.40) depends, apart from (p, T ), on concentrations of the particles xjP . There are H − 1 independent concentrations in each phase. Therefore we arrive at the result variables M = (H − 1)F + 2 pieces, constraints Y = H(F − 1) pieces. To have solutions, we must have M ≥ Y . For the number of coexisting phases F this yields the Gibbs phase rule, (3.42) F ≤ H + 2, according to which, e.g., in a pure single-species substance at most three phase may coexist at the triple point (no more freedom is left by the two equilibrium conditions imposed on the two variables p and T ). The total number of phases is not restricted by the Gibbs rule, however. For instance, ice has several different phases under high pressure, all coexisting according to the Gibbs rule (see Fig. 3–6). 3.7 Phase transitions and diagrams In Figs. 3–5 typical phase diagrams of the SVN system have been plotted in (T, p) and (V, p) planes. In the (V, p) plane coexistence regions of solid, liquid and gaseous phases occupy finite area, whereas in the (T, p) plane the coexistence region is a curve. p p kiinteä C C neste kiinteä neste kylläinen höyry K isotermejä kaasu koeksistenssi kaasu K T V Figure 3–5: Phase diagrams of a liquid in (T, p) and (V, p) planes. Typical features include the triple point K, where solid, liquid and gaseous phases meet. For instance, for water (H2 O) on pK = 610 Pa, TK = 0.01◦ C. The coexistence curve of liquid and gas ends at the critical point 3.7. PHASE TRANSITIONS AND DIAGRAMS 41 C; for water pc = 22 MPa, Tc = 374.15◦ C. Gas in the coexistence limit is called saturated vapour. Typically, the chemical potential µ = ∂G/∂N is continuous at crossing of the coexistence curve, but the rest of the partial derivatives of G, S = −∂G/∂T and V = ∂G/∂p, do not necessarily share this property. If some of these derivatives are discontinuous, the phase transition is of first order. If all the first derivatives are continuous, but discontinuities (or worse singularities) appear in the second order derivatives, then we are dealing with a second order or continuous phase transition. The jumps in the entropy and volume in a first-order transition are ∆S = − ∆V = ∂G ∂T ∂G ∂p (2) + p (2) T − ∂G ∂T ∂G ∂p (1) , p (1) . T It should be borne in mind that the Gibbs functions describing different phases are separately perfectly smooth functions of state variables in a discontinuous transition, and the apparent discontinuity stems from the change in the description from the Gibbs function of one phase to that of the other phase depending on which is less at constant pressure and temperature. When the coexistence curve is crossed T and p remain constant. In such a process the amount of heat is equal to the change of the enthalpy (heat function), since ∆Q = T ∆S = ∆U + p ∆V = ∆(U + pV ) = ∆H. This is the latent heat of the phase transition (heat of fusion, vapourization etc.) p (kbar) 30 1608 atm 17.78 K VII VIII kiinteä hcp 20 kiinteät faasit 10 neste III V -50 I 0 135.4 atm 3.14 K kiinteä bcc 33.5 atm VI II kiinteä fcc p 50 T (C) neste 28.92 atm 0.32 K C kaasu 1.15 atm 3.32 K T Figure 3–6: Examples of phase diagrams of real substances: (a) water (H2 O), (b) 3 He. 42 3. APPLICATIONS OF THERMODYNAMICS 3.8 Coexistence On the coexistence curve in the (T ,p) plane µ1 (p, T ) = µ2 (p, T ). According to the Gibbs-Duhem equation dµ1 dµ2 dp dT coex = s2 − s1 1 ∆h = ., v2 − v1 T ∆v 1 2 dp S1 V1 dT + dp , N1 N1 V2 S2 dp . = − dT + N2 N2 = − dT Along the coexistence curve this yields the Clausius–Clapeyron equation p T Figure 3–7: Coexistence curve of two phases. (3.43) where s is the specific entropy, v the specific volume and h the specific enthalpy, so that ∆h is the specific latent heat of the phase transition. If the phase 2 corresponds to higher temperature, then ∆h > 0, because the discontinuous transition takes place at the crossing of the chemical potentials of different phases µ2 (T ) = µ1 (T ). If the phase 2 is stable at higher temperature, then ∂µ2 ∂µ1 < , ∂T ∂T so that s2 > s1 and ∆h > 0. For liquid-vapour transition the Clausius-Clapeyron equation may readily be integrated. In this case obviously v2 ≫ v1 . Neglecting the specific volume of the liquid and using the ideal gas equation of state for the saturated vapour we obtain ∆h p∆h dp = = 2 . dT T v2 T For constant latent heat of vapourization this yields the equation of the coexistence curve in the form 1 1 ∆h − T0 T , p = p0 e which yields the temperature dependence of the pressure of saturated vapour. This is a reasonable approximation if the temperature (and pressure) differences are small. If not, then it might be better to assume that the specific entropy in the gaseous phase is much larger than that in the liquid phase: s2 ≫ s1 , and neglect s1 . With the subsequent substitution of the perfect-gas entropy an integrable equation is obtained in this case as well. Example 3.2. Show that the boiling temperature of a liquid at constant pressure becomes higher upon dissolution in it a small amount of any 3.8. COEXISTENCE 43 non-volatile solute. Derive for the change of the boiling temperature in the limit of small concentration the result δT = x T2 , ∆h where x is the molar fraction of the solute and ∆h the specific (per molecule) heat of evaporation of the solvent. The chemical potential of the liquid solvent in the limit of small concentration of the solute is µl (p, T, x) ≈ µl (p, T, 0) − xT . In equilibrium this must be equal to the chemical potential of the solvent in the gaseous phase µg (p, T ), µl (p, T, 0) − xT = µg (p, T ) , (3.44) which determines the boiling temperature as a function of then pressure T = T (p, x). The response of the boiling temperature to a small addition of the solute is δT = ∂T ∂x x. p From condition (3.44) at constant pressure it follows −sg dT = −sl dT − xdT − T dx so that ∂T ∂x = p T T2 T ≈ , = sg − sl − x sg − sl ∆h which yields the desired relation. Example 3.3. A pot of soup boils at 103◦ C at the bottom of a hill of height 300 m and boils at 98◦ C at the top. What is the latent heat of vapourization of the soup? Small difference in boiling (coexistence) temperature, therefore the latent heat may be considered constant and the Clausius-Clapeyron equation used in the form (this is independent of the molar fraction of the soup vapour in the air and probably worth checking): p∆h dp = . dT T2 In the present problem we need the dependence of the boiling temperature T on the height rather than on the pressure. The latter are related from the mechanical equilibrium condition of a gas column as dp p dz , = −ρg = −µg dz Tair where µ is the average mass of a molecule in the air. Combining the two equations yields the relation −µg dz = ∆h Tair dT . T2 44 3. APPLICATIONS OF THERMODYNAMICS Thus, M g∆zTtop Tbottom = 8.1 kJ/mol , Tair (Tbottom − Ttop ) where NA is the Avogadro number, M the molar mass of air and the temperature of the ambient air assumed to be Tair = 20◦ C. ∆H = NA ∆h = 3.9 Van der Waals equation of state In terms of the specific volume v = V /N the van der Waals equation of state contains only intensive quantities a p + 2 (v − b) = T . (3.45) v For the pressure p we obtain p= a T − 2. v−b v (3.46) In Fig. 3–8 the phase diagram brought about by the van der Waals equation of state is sketched in the (v, p) plane (i.e. the curves depicted are isotherms). The critical point corresponds to the infliction point: ∂p = 0, ∂v ∂2p = 0. ∂v 2 The former of these equations implies, in particular, an infinite compressibility at the critical point. This is an example of singular behaviour typical of response functions at the critical point. The coordinates of the critical point are readily found: vc = 3b 8a Tc = . (3.47) 27b a pc = 27b2 In terms of dimensionless variables p = p/pc , T = T /Tc and v = v/vc the van der Waals equation assumes the form of the law of corresponding states 3 p + 2 (3v − 1) = 8T . (3.48) v without any explicit dependence on the material parameters a and b. This is an illustration of universality, which is a generic feature of (especially continuous) phase transitions: important properties may be expressed in mathematical form fairly insensitive to physical details and nature of the phase transition. Coexistence region. Isotherms in the phase diagram of the (v, p) plane are not monotonically decreasing at T < Tc . In Fig. 3–8 between the points 3.9. VAN DER WAALS EQUATION OF STATE p 45 T>Tc C T=Tc C D A T< Tc B Nb V Figure 3–8: Phase diagram of the van der Waals matter. B and C there is a rising portion with κT < 0 on. This means unstable states and in the corresponding regions the isotherms are excluded from the phase diagram. The unstable region is surrounded by regions in which the matter is in metastable states , which are stable against small fluctuations. Large enough disturbances cause phase separation to liquid and gaseous phases The region AB corresponds to superheated liquid, whereas the region CD corresponds supercooled vapour. Global thermodynamic equilibrium corresponds to states beyond these regions on the isotherm bounded by the points A and D. In coexistence pA = pD . To find this common pressure of the coexistence state the remaining equilibrium condition µA = µD . may be used. According to the Gibbs– Duhem equation dµ = −s dT + v dp , on an isotherm dµ = v dp. Integration along the isotherm yields µD − µA = Z D v dp. p A II I D (3.49) V A From the geometric construction of Figure 3–9: Fig. 3–9 we infer tion. Z D v dp = Area I - Area II Maxwell construc(3.50) A – the difference of areas. The points A and D must thus be chosen such that the areas I and II are the same. This is the Maxwell construction. Example 3.4. For the van der Waals matter in the vicinity of the critical point: 46 3. APPLICATIONS OF THERMODYNAMICS (a) Calculate the jump ∆n = n+ (p, T ) − n− (p, T ) in the number density n = N/V , where n± are the number densities of the liquid and gaseous phases on the coexistence curve. In the notation ∆n ∝ (Tc − T )β , calculate the value of the critical exponent β. Such non-analytic dependence on state variables is also a typical feature near the critical point and the corresponding non-integer powers (critical exponents) are important quantities in the description of critical phenomena. (b) Construct the phase diagram in the (T, p) plane. (c) Calculate the latent heat ∆H. (a) Expand first the law of corresponding states in small deviations from the critical values: p = 1 + π, T = 1 + τ and n = 1/v = 1 + ν to obtain 3 3 (3.51) π ≈ 4τ 1 + ν + ν 3 . 2 2 Since τ and π are equal in both phases, the equation of state (3.51) yields the connection 4τ (ν1 − ν2 ) = ν23 − ν13 = (ν2 − ν1 ) ν22 + ν2 ν1 + ν12 whose obvious solution ν1 = ν2 is physically uninteresting, so that we are left with ν22 + ν2 ν1 + ν12 + 4τ = 0 . (3.52) Another relation is obtained by the Maxwell construction, whose geometric content is easier to realize by integration over ν than over π. In these terms the condition (3.50) means Zν2 π dν ≈ 0= ν1 , ν2 4τ ν1 ν+ 3 2 ν 4 + 3 4 ν . 8 This condition is simplified by the use of the equation of state yielding 3 4 4τ ν22 − ν12 = ν1 − ν24 . 2 Excluding the uninteresting solution ν1 = ν2 we arrive at 3 4τ (ν2 + ν1 ) = − (ν2 + ν1 ) ν12 + ν22 . 2 Now the solution ν12 + ν22 = −8τ /3 with the account of equation (3.52) yields the uninteresting solution ν1 = ν2 again and we are left with ν1 = −ν2 . This finally yields the desired temperature dependence (let ν1 > 0) √ n1 − n2 = 2ν1 = 4 −τ = 4 r Tc − T , Tc withe the value of the critical exponent β = 12 . 3.10. PROBLEMS 47 (b) The result ν1 = −ν2 allows to write the explicit equation of the coexistence curve near the critical point as π = 4τ . (c) The latent heat is found from the Clapeyron-Clausius equation dp ∆h = dT T (v2 − v1 ) as ∆h = T (v2 − v1 ) dp Tc ≈ dT 2b n1 − n2 n2c r = −6Tc Tc − T . Tc 3.10 Problems Problem 3.1. Assume that Earth’s atmosphere consists of an ideal gas with a molecule’s mass m and that the acceleration of gravity g is a constant. (a) Show that as a function of altitude the pressure changes as mg dp =− dz . p T (b) For an isothermal atmosphere, derive the barometric formula: mgz p(z) = p0 exp − T . (c) Assume then that temperature changes due to adiabatic expansion and show that in such an adiabatic atmosphere p(z) = p0 1 − mg T0 1− 1 γ z γ γ−1 . Problem 3.2. Given the following expression for the chemical potential of the ideal gas µ(T, p) = µ0 + T −ζ − f + 1 ln T + ln p , 2 where µ0 and ζ are constants, and the equation of state pV = N T , calculate the thermodynamic potentials U (S, V, N ), H(S, p, N ), F (T, V, N ), G(T, p, N ) and Ω(T, V, µ) as functions of their natural variables. Problem 3.3. In two isolated tanks (of volumes V1 and V2 ) there is the same amount (N particles in each tank) of the same ideal gas at the same temperature, but at different pressures p1 and p2 . Find the entropy change, when the tanks are joined (the wall separating them is removed). 48 3. APPLICATIONS OF THERMODYNAMICS Problem 3.4. Show that in a mixture of perfect gases the heat of reaction is X X 1 ∆r H = νj µ0j + T νj 1 + fj . 2 j j Here, the term with µ0j accounts for the changes in the chemical energies (electronic binding energies) of the molecules. Problem 3.5. Consider production of atomic hydrogen in the reaction e + H+ ↔ H. Show that for the equilibrium concentrations the Saha equation I ne nH+ ≈ nQ exp − , nH T holds, where I is the ionization energy of the hydrogen and nQ a function with a weaker (than exponential) dependence on the temperature. What is this function? Problem 3.6. The humidity of the air is defined as the ratio of its steam pressure and the pressure of the saturated steam (i.e. steam in equilibrium coexistence with water). Let the humidity of the air at 20 ◦ C be 50 %. Estimate its humidity in a sauna at 80 ◦ C. Hints: The heat of vapourization of water at the pressure of 1 atm is 2.26 MJ/kg. Construct the pressure of the saturated steam p = p(T ) assuming the perfect gas laws for the steam and that both the specific volume and specific entropy are much larger than that of the coexistent water. Problem 3.7. Consider superconducting matter. In the normal phase the magnetization M is negligible, whereas in the superconducting phase the magnetic induction B = 0 in the bulk. The phase transition takes place at a constant temperature T < Tc , when the magnetic field strength becomes less than the critical value " Hc (T ) = H0 1 − T Tc 2 # . Draw the phase diagram in the (T,H) plane, calculate the change in the Gibbs function, the latent heat and the change in the specific heat CH in the phase transition. What is the order of the transition, if H = 0? Hint: For a homogeneous sample assumed here, the differential of the magnetic Gibbs function dG = −SdT − V BdH + µdN . Problem 3.8. Calculate the entropy of the van der Waals gas as a function of the temperature and volume. Problem 3.9. Analyse the behaviour of the mechanical responses αp and κT of a van der Waals gas near the critical point. Show, in particular, that on the isobar pb = p/pc = 1 αp ∼ 8 + ··· 9Tc (vb − 1)2 κT ∼ 2 + ··· 9pc (vb − 1)2 where vb = v/vc . Find also αp and κT as functions of the pressure on the isochore vb = 1 in the gaseous phase, when pb > 1. 4. Classical phase space 4.1 Phase space and probability density Phase space. The mechanical state of a classical many-particle system (point-like particles, for simplicity) is completely described by the position vectors and velocities of the particles {r i (t), v i (t)}, i = 1, . . . , N . In Hamiltonian mechanics generalized coordinates and momenta are used instead: q = (q1 , . . . , qN d ) . (4.1) p = (p1 , . . . , pN d ) N particles in d dimensional space gives rise to 2N d variables. The phase space is a 2N d dimensional space or manifold whose each point P = (q, p) corresponds to a possible mechanical (microscopic) state of the system. During the evolution of the system the image point P (t) moves in the phase space along a trajectory in a manner determined by the canonical Hamilton equations of motion: dqi dt dpi dt ∂H ∂pi ∂H = − ∂qi = (4.2) Here H ≡ H(q1 , . . . , qN d ; p1 , . . . , pN d ; t) = H(P, t) is the Hamilton function. For timeindependent H, the trajectories are stationary. Except for some special points, the trajectories cannot intersect, branch or merge. The temporal evolution of an arbitrary function F (q, p, t) related to properties of the system may be expressed using Poisson brackets X ∂F ∂G ∂G ∂F − . {F, G} ≡ ∂qi ∂pi ∂qi ∂pi i (4.3) P(t) qi pi Figure 4–1: Trajectories in a phase space. When the argument point P (t) of F is the image point of the system moving on a trajectory the total time derivative of F is dF ∂F = + {F, H} . dt ∂t 49 (4.4) 50 4. CLASSICAL PHASE SPACE Measure. For a probabilistic treatment a measure in the phase space dΓ has to be defined. With N particles in d-dimensional space the choice is: dΓ = Nd 1 Y dqi dpi 1 −N d = h dq1 . . . dqN d dp1 . . . dpN d . N ! i=1 h N! (4.5) The normalizing factor 1/N ! removes the degeneracy related to the permutation symmetry of identical particles (details in the next chapter). The introduction of the Planck constant h = 6.62607 × 10−34 Js is based on the quasi-classical correspondence of the phase space volume of h to one quantum state. Ensemble. Calculation of macroscopic quantities from "first principles" requires solution of the Hamiltonian equations of motion, which is not feasible in a large system, nor is even determination of the initial conditions for a proper setup of the problem. A statistical approach is used instead. A large number of different microscopic states give rise to the same macroscopic state. A statistical ensemble consists of different systems ("copies" of the original microscopic system in various states allowed by conditions defining the macrostate considered) in these microstates described by the image points {P j ; j = 1, . . . , n} in the phase space. In the limit n → ∞ the distribution of the image points gives rise to a probability density ̺(P ) normalized as Z dΓ ̺(P ) = 1 . (4.6) nV δS Γ0 Figure 4–2: Derivation of the continuity equation. For any function in the phase space f (P ) = f (q, p) a statistical average may be then defined as the integral Z hf i = dΓ f (P ) ̺(P ) . (4.7) Equations of motion. Image points belonging to a statistical ensemble move according to the Hamiltonian equations of motion and thus their number remains the same at all times. Introduce a velocity field in the phase space as V = (q̇, ṗ) . (4.8) R Consider the ensemble measure Γ0 ̺ dΓ of a region Γ0 of the phase space at the initial time instant. This is the number of image points in Γ0 . During the evolution of the ensemble the image points move which results in the change of both the probability density ̺ and the region Γ0 they occupy. The change of the region brings about a local change of its volume. This is given by the volume of an oblique cylinder with the basis on a surface element 4.1. PHASE SPACE AND PROBABILITY DENSITY 51 dS of Γ0 and spanned by the trajectories starting at this surface element. The direction of the axis of the cylinder formed during the movement of the image points of the surface element is thus given by V and its height during a short period ∆t by ∆h = V · n∆t, where n is the unit outward normal vector to the surface element dS. Thus, the total change of the ensemble measure of the region Γ0 may be written as a sum of contributions from the change of ̺ and the change of Γ0 as Z Z Z d ∂̺ dΓ + ̺ dΓ = V ̺ · ndS . dt Γ0 ∂Γ0 Γ0 ∂t R R Gauss’ theorem yields ∂Γ0 dS · V ̺ = Γ0 dΓ ∇ · (V ̺). Therefore, due to arbitrariness of Γ0 we arrive at the continuity equation ∂̺ + ∇ · (V ̺) = 0 , ∂t or, in more detail, ∂ ∂̺(P, t) X ∂ + (q̇i ̺) + (ṗi ̺) = 0 . ∂t ∂qi ∂pi i (4.9) From the Hamilton equations q̇i = ∂H/∂pi , ṗi = −∂H/∂qi it follows that ∂ q̇i ∂ ṗi + = 0, ∂qi ∂pi (4.10) which – summed over i – yields ∇·V = 0. Substitution of ”incompressibility condition” (4.10) in continuity equation (4.9) leads to ∂̺ X ∂̺ ∂̺ q̇i + + ṗi = 0, (4.11) ∂t ∂qi ∂pi i or in a shorthand form ∂̺/∂t + V · ∇̺ = 0. Introducing the convective time derivative d dt ≡ = ∂ +V ·∇ ∂t X ∂ ∂ ∂ q̇i + + ṗi ∂t ∂qi ∂pi i (4.12) we may now cast (4.11) in a compact form known as the Liouville theorem: d ̺ P (t), t = 0 . dt (4.13) Thus, apart from conservation of the probability measure in the phase space, the probability density is also conserved on a trajectory. 52 4. CLASSICAL PHASE SPACE Substitution of the Hamilton equations q̇i = ∂H/∂pi , ṗi = −∂H/∂qi leads to the Liouville equation ∂̺ i = L̺ , (4.14) ∂t for the probability density, where L is the Liouville operator X ∂H ∂ ∂H ∂ L = i{H, } ≡ i − . ∂qj ∂pj ∂pj ∂qj j (4.15) The Liouville equation (4.14) is not easier to solve than the Hamilton equations of motion, therefore other ideas are needed to construct methods for calculation of macroscopic properties in systems with large number of particles. What is done in practice is that a stationary solution of the Liouville equation is sought which leads to the condition {̺, H} = 0 . It is also desirable that physically independent subsystems are statistically independent, which means factorization of the probability density in the fashion ̺(1, 2) = ̺(1)̺(2), where indices are shorthand for variables corresponding to two such subsystems. From this it follows that ln ̺ should be an extensive quantity and thus a linear combination of additive integrals of motion. In mechanical systems there are seven such integrals of motion: the Hamilton function H, the total momentum P and the total angular momentum L. The last six are usually excluded from the list of integrals of motion by putting the system in a box and thus only the Hamilton function is left for the construction of the density function so that ̺ = ̺(H). Different arguments, some of which are presented below, are then used to choose a suitable and reasonable density function. 4.2 Flow in phase space Hamiltonian dynamics gives rise to flow of image points in the phase space resembling the flow of an incompressible fluid. An energy surface ΓE is a (2N d − 1) dimensional manifold determined by the condition H(q, p) = E. Its measure or surface area may be expressed as Z Z ΣE ≡ dΓ δ H(P ) − E = dΓE . (4.16) When energy is conserved, image points flow on this surface. Applicability of the statistical approach depends crucially on the character of the flow in the sense how thoroughly the whole energy surface is covered. In this respect the conventional classification includes ergodic and non-ergodic flows (or corresponding systems) illustrated in Fig. 4–4. Non-ergodic flows. An initial area element∆ΓE of the energy surface explores only a part of the energy surface ΓE . This is the case, e.g., for 4.2. FLOW IN PHASE SPACE 53 periodic motion and integrable systems allowing complete solution in terms of action-angle variables. Figure 4–4: Flow in phase space: (a) non-ergodic, (b) ergodic, not mixing, (c) ergodic and mixing. Γ E + ∆E E Ergodix flow. Almost every point of the surface ΓE approaches any other point of ΓE arbitrarily close in the long run. Mathematically this may be stated as equality of temporal and statistical averages ∆h = ∆E ∇H f = hf iE Figure 4–3: Constant energy surfaces in phase space. (4.17) the time average is 1 T →∞ T f ≡ lim (4.17) for any smooth enough function f (P ) on the energy surface ΓE and almost all starting points P (t = 0). In relation Z 0 T dt f P (t) and the statistical average on the energy surface Z 1 dΓE f (P ) . hf iE ≡ ΣE (4.18) (4.19) The microcanonical ensemble is defined by the density function ̺E (P ) = 1 δ H(P ) − E , ΣE (4.20) which allows to express expectation values on the energy surface as ensemble averages in the microcanonical ensemble: Z hf iE = dΓ f (P ) ̺E (P ) . 54 4. CLASSICAL PHASE SPACE Mixing flow. This is a special type of ergodic flow in which the image points of a small element dΓE of the energy surface are dispersed in a nearly uniform distribution over the whole energy surface. For arbitrary non-stationary density ̺E (P, t) on the energy surface ΓE and any smooth enough function f (P ) Z lim hf i ≡ lim dΓ ̺E (P, t) f (P ) t→∞ t→∞ Z 1 = dΓ δ H(P ) − E f (P ) ΣE = hf iE . (4.21) Thus, in a mixing flow an arbitrary probability density describing a nonequilibrium state evolves towards the microcanonical ensemble. Ergodic theory. Ergodic theory analyzes the character of flow in phase space. For instance, it has been rigorously proved that a two-dimensional gas of hard discs is ergodic. It is also physically plausible that dynamics of the usual three-dimensional fluids is ergodic in the framework of classical mechanics, although no rigorous proof exists. Although ergodic theory is an exiting branch of mathematics, its value for physics is rather limited. Statistical mechanics is usually applied to physical systems which rarely are truly isolated and therefore hardly exhibit non-ergodic behaviour. One could also say that statistical mechanics deals with ergodic systems only. 4.3 Microcanonical ensemble and entropy One of the basic problems in statistical physics is to find an ensemble or density function ̺ giving the correct description of a macroscopic system. If the energy of a macrosopic ergodic system is fixed, it may be described – as shown in the preceding section – by the microcanonical ensemble with the density 1 δ H(P ) − E . (4.22) ̺E (P ) = ΣE The δ function may be technically inconvenient, therefore an alternative definition may be given as a uniform distribution in an energy shell of thickness ∆E: ̺E,∆E (P ) = 1 θ E + ∆E − H(P ) − θ E − H(P ) . ZE,∆E (4.23) Here, the step function 1 1/2 θ(x) = 0 , , , x>0 x=0 x<0 (4.24) 4.3. MICROCANONICAL ENSEMBLE AND ENTROPY 55 has been used to confine the probability density to the interval E ≤ H ≤ E + ∆E. In R this region the density is 1/ZE,∆E , and from the normalization condition dΓ ̺ = 1 it follows that Z ZE,∆E = dΓ [θ(E + ∆E − H) − θ(E − H)] . (4.25) The normalizing constant ZE,∆E is the microcanonical statistical sum or the partition function expressing the number of states in the energy shell of the phase space (according to the quasiclassical normalization that ∆0 Γ = 1 corresponds to one state). Thus, in the microcanonical ensemble the probability density is distributed completely evenly in the allowed part of the phase space, i.e. between the ZE,∆E states. Example 4.1. Calculate the microcanonical partition function for a system of N free point particles of mass m. In general, the microcanonical partition function is difficult to calculate. The ideal gas is one of the rare easily calculable cases. The energy shell is defined by the inequalities E≤ N X p2i i=1 2m = H ≤ E + ∆E , √ which is geometrically a spherical shell with the radii 2mE and p 2m(E + ∆E) in the 3N dimensional momentum space of the system. The volume √ of such a (thin) shell is the surface area of the sphere of radius, say, 2mE inpa 3N dimensional space multiplied by the thickness of the shell: ∆p ≈ 2m/E∆E/2. The surface area of a sphere of radius r in a d-dimensional space is d Sd = 2π 2 rd−1 , Γ d2 where Γ(z) is the Gamma function. The coordinate part gives for each particle the volume V of the box in which the system is enclosed. Thus (remember the measure) the partition function – with the degeneracy factor 1/N ! included – is " ZE,∆E = 3 3 π 2 V (2mE) 2 h3 #N ∆E . N ! Γ 3N E 2 Entropy. In the statistical theory of Gibbs the density ̺ corresponding to a macroscopic state may be derived with the aid of a variational principle. To this end the statistical entropy S[̺] is defined as the following functional of the density, Z S = − dΓ ̺(P ) ln ̺(P ) , (4.26) 56 4. CLASSICAL PHASE SPACE and a variational principle is set requiring the physical density ̺ to maximize the entropy. For the uniform distribution (4.23) this is S = ln ZE,∆E = ln W , (4.27) which is the Boltzmann entropy,, where W is the statistical weight of the macroscopic state i.e. the number of all microscopic states compatible with the macroscopic boundary conditions. Derivation of the microcanonical ensemble. Let ∆ΓE be the part of the phase space with energy in the interval (E, E + ∆E). We seek density function ̺, which maximizes the statistical entropy (4.26). The variation of entropy Z Z δS = − dΓ(δ̺ ln ̺ + ̺ δ ln ̺) = − dΓ δ̺(ln ̺ + 1) ∆ΓE ∆ΓE must vanish at an extremum. The constraint Z δ1 = dΓ δ̺ = 0 ∆ΓE forces to look for a conditional extremum. This may be done with the use of a Lagrange multiplier λ, which leads to the condition ln ̺ + 1 + λ = 0 , with the solution ̺(P ) = ̺0 , (P ∈ ∆ΓE ). (4.28) Thus, the result is a constant probability density in the energy slice – just as in the microcanonical ensemble. Due to linearity of the normalization condition in ̺, the second variation of the entropy may be calculated as if the variation δρ were completely free, so that Z Z 1 1 ∂2 1 δ2 S = − dΓ (δ̺)2 2 (̺ ln ̺) = − dΓ (δ̺)2 ≤ 0 , 2! ∂̺ 2 ∆ΓE ̺ ∆ΓE which means that the extremum of S is indeed a maximum. The normalization condition yields ̺ = 1/ZE,∆E in the energy slice, and substitution in the expression for entropy yields Z 1 1 S=− dΓ ln = ln Z = ln W . Z Z ∆ΓE This is the Boltzmann entropy revisited. Proof of additivity. Consider a system S consisting of two statistically uncorrelated subsystems S1 and S2 . Let the corresponding phase spaces be 4.4. PROBLEMS 57 Γ1 and Γ2 . The phase space of the combined system is Γ1+2 = Γ1 ⊗ Γ2 with the volume element dΓ1+2 = dΓ1 dΓ2 . Due to statistical independence the probability density is factorized ̺1+2 = ̺1 ̺2 , and the normalization assumes the form Z Z Z dΓ1+2 ̺1+2 = dΓ1 ̺1 dΓ2 ̺2 = 1 . According to Gibbs’ formula the entropy of the combined system is Z S1+2 = − dΓ1+2 ̺1+2 ln ̺1+2 ZZ = − dΓ1 dΓ2 ̺1 ̺2 (ln ̺1 + ln ̺2 ) Z Z = − dΓ1 ̺1 ln ̺1 − dΓ2 ̺2 ln ̺2 . This is exactly the desired result: (4.29) S1+2 = S1 + S2 . Thus, entropy is an additive quantity for weakly enough interacting systems. No state of equilibrium in subsystems or between then was assumed. 4.4 Problems Problem 4.1. Write down the evolution equation for the density function ρ(t, p1 , q1 , p2 , q2 ) of coupled one-dimensional harmonic oscillators with the Hamilton function H= p2 k p21 + 2 + (q1 − q2 )2 2m 2m 2 and find the solution with the initial condition ρ(0, p1 , q1 , p2 , q2 ) = Z −1 exp −β p2 p21 + 2 2m 2m (unnormalized, Z is the normalization constant and β a positive parameter). Problem 4.2. Calculate the partition function ZE,∆E of the classical microcanonical density function for (a) N non-interacting atoms, (b) N independent harmonic oscillators. Hint. Volume of a d-dimensional ball of radius R: Vd = π d/2 Rd . Γ(d/2 + 1) 58 4. CLASSICAL PHASE SPACE Problem 4.3. Using the microcanonical definition of entropy S = ln ω(E) calculate the heat capacity of a system of N independent harmonic oscillators. Hint. Use the microcanonical definition of temperature to find E = E(T ) first. 5. Quantum-mechanical ensembles 5.1 Density operator and entropy The complete description of the state of a system in quantum mechanics is given by the wave function (i.e. the solution of the Schrödinger equation generated by the Hamilton operator of the system conforming to given initial and boundary conditions) in a suitable Hilbert space. In the case of a large system (but often not only in that case) this level of description is practically impossible. Since the quantum-mechanical description is inherently probabilistic, therefore, in contrast to classical mechanics, no new tools are actually needed for the statistical treatment of systems with large numbers of degrees of freedom, no new tools. The probabilistic measure required for the statistical description is provided by the density operator introduced in quantum mechanics for incomplete description of physical systems. Ensemble. The notion of a statistical ensemble is similar to that in classical statistics: it is formed by copies of the system in different microstates corresponding to the same macrostate. A microstate is determined by a wave function. Pure state. If the wave function Ψ is known, the density operator is the projection operator ̺ = |ΨihΨ| . (5.1) This is a pure state. It corresponds to an ensemble in which each state is the same |Ψi (modulo the phase factor). Statistical mechanics of the pure state reduces to the usual quantum mechanics, in which, e.g., expectation values are hAi = Tr ̺A = hΨ|A|Ψi. Mixed state. Basic properties of the density operator in the general case (”mixed state”) may be inferred by considering a the wave function Ψ(x, y) of a large system with a weak coupling between subsystems whose variables are labeled as x and y. Weak coupling means that Htot = H + H ′ , where H does not act on the y variables and H ′ on the x variables. Due to symmetry (antisymmetry) properties imposed on the wave functions the wave function of the large system does not factorize to a product of wave functions of the subsystems, however. In this case the density matrix (density operator in the coordinate basis) for the x-labeled subsystem may be defined as Z ̺(x, x′ ) = dy hx, y|ΨihΨ|x′ , yi, . (5.2) 59 60 5. QUANTUM-MECHANICAL ENSEMBLES It is immediately seen that the density matrix is hermitian, positive definite and normalized as Tr̺ = 1. It is often convenient to express the density matrix using the projection operators to eigenfunctions of P Cn (y)hx|ni and ̺(x, x′ ) = H: let H|ni = En |ni, then hx, y|Ψi = n PR ∗ dy Cn (y)Cm (y)hx|nihm|x′ i from which it follows that n,m ̺= XZ ∗ dy Cn (y)Cm (y)|nihm| . (5.3) n,m The expectation value of an operator A acting on the x-labeled subsystem only may be calculated as XZ ∗ dy Cn (y)Cm hAi = Tr ̺A = (y)hm|A|ni . (5.4) n,m Properties of the density operator. Obvious requirements for the density operator are positivity, normalization and self-adjointedness: ̺ = ̺† hΨ|̺|Ψi ≥ 0, ∀ |Ψi ∈ H Tr ̺ = 1 (5.5) Since the operator ̺ is Hermitian, it possesses a complete set of eigenfunctions in the basis of whose X ̺= pα |αihα| . (5.6) α Here, the standard normalization hα|βi = δα,β is implied. The eigenvalues P obey 0 ≤ pα ≤ 1 and α pα = 1. An expectation value in this basis assumes the form X pα hα|A|αi . (5.7) hAi = Tr ̺A = α Equation of motion of the density operator. Taking the time derivative of the density operator (5.2) and using the fact that hx, y|Ψi is a wave function, i.e. i~∂t hx, y|Ψi = (H + H ′ ) hx, y|Ψi we obtain i~ ∂̺(x, x′ ) X = ∂t n,m Z ∗ dy Cm (y)hm|x′ i (H + H ′ ) Cn (y)hx|ni ∗ − Cn (y)hx|ni (H + H ′ ) Cm (y)hm|x′ i , (5.8) where operators act on functions on functions to the right. Since H ′ acts only on variables y and is a hermitian operator, the contributions containing H ′ cancel. The operator H is acting on its eigenfunctions in the coordinate representation, therefore Hhx|ni = hx|H|ni and Hhm|x′ i = hm|H|x′ i. 5.1. DENSITY OPERATOR AND ENTROPY 61 Thus ∂̺(x, x′ ) X = i~ ∂t n,m Z ∗ dy Cn (y)Cm (y) hx|H|nihm|x′ i − hx|nihm|H|x′ i . In the operator form this the von Neumann equation i~ d ̺(t) = [H, ̺(t)] dt (5.9) which plays the same role in quantum statistics as the Liouville equation in classical statistics. It gives a principal possibility to find the density operator as the solution this equation together with suitable initial condition and normalization. Stationary ensemble. In a stationary ensemble all expectation values are time-independent. This implies a time-independent density operator: ̺˙ = 0. Then [̺, H] = 0, which is possible, e.g., if ̺ is a function of H: ̺ = ̺(H). In a stationary ensemble the density operator and the Hamilton operator also possess common eigenstates. Therefore, in the basis of these eigenstates |ni the density operator is diagonal X |nipn hn| (5.10) ̺= n and the eigenvalue pn may be prescribed the meaning of the probability to observe the system in the eigenstate |ni (this is a genuine wave function here, i.e. a solution of the Schrödinger equation with H). The ensemble expectation values are now X hAi = Tr ̺A = pn hn|A|ni . (5.11) n Here, hn|A|ni is the quantum-mechanical expectation value of the observable A in the quantum state |ni. It should be noted, however, that the expectation value (5.11) is different from that of in a pure state |Ψi, which P also may be expressed as a linear combination of the basis vectors |Ψi = n an |ni. Here, |an |2 is the probability to observe the system in the state |ni in the usual quantum-mechanical sense. The expectation values in the pure state are X X hΨ|A|Ψi = |an |2 hn|A|ni + a∗m an hm|A|ni (5.12) n n6=m and we see that in the density-operator description (5.11) the interference terms of the pure ensemble are absent. Many-particle systems. It is convenient to express wave functions of many-particle systems as linear combinations of products of one-particle wave functions of the form Ψℓ1 (ξ1 )Ψℓ2 (ξ2 ) · · · ΨℓN (ξN ) , 62 5. QUANTUM-MECHANICAL ENSEMBLES where ξi = (xi , si ) (si is the spin of the particle) and Ψn are eigenfunctions of the one-particle Hamiltonian enumerated by the quantum number ℓ. This allows to introduce the occupation number representation for identical particles: the number of occurrences nℓ of one-particle functions of a particular quantum number ℓ is the number of particles in this state. This is an illustrative way of speaking only: in a many-particle quantum system the very notion of one particle in a given state is actually meaningless. For identical particles symmetry conditions are imposed on the physical wave functions. The particles are either bosons of fermions. The former obey Bose–Einstein statistics and the wave function must be completely symmetric with respect to all permutations of the particle arguments. The latter obey Fermi–Dirac statistics, requiring complete antisymmetry (change of sign) in any permutation of a pair of particle arguments of the wave functions. These requirements have a profound effect on the statistical mechanics. Fock space. From the point of view of statistical physics, the quantum mechanics of many-particle systems is most conveniently formulated in the Fock space, which allows to treat systems with variable number of particles. The Fock space is a direct sum of all properly (anti)symmetrized N -particle Hilbert spaces. In the coordinate representation the elements of its basis may be written as row vectors Φ = (ζ, ψ1 (ξ1 ), ψ2 (ξ1 , ξ2 ), . . . , ψN (ξ1 , . . . , ξN ), . . .) , where ζ is a complex number, ξi = (xi , si ) and ψN (ξ1 , . . . , ξN ) is a completely (anti)symmetric function. The normalized basis elements for N -particle states may be written as |{nℓ }i ≡ |n1 , . . . , nℓ , . . .i, where n1 , . . . , nℓ are the occupation numbers P of the one-particle states {ℓ} = {1, 2, 3, . . .}. In an N -particle state obviously ℓ nℓ = N . Let ℓ1 , ℓ2 , . . . , ℓN be labels of one-particle states. The coordinate representation of a basis vector is then rQ X ℓ nℓ ! εP hξ1 |ℓ1 i · · · hξN |ℓN i , (5.13) hξ1 , . . . , ξN |{nℓ }i = N! P (ξ1 ,...,ξN ) where the summation goes over all N ! permutations of the argument permutations and εP is a sign factor, which for bosons is always 1, whereas for fermions ±1 according to the parity of the permutation (in the fermionic case this is a N × N Slater determinant). The normalization factor in (5.13) may by calculated by the traditional ”box-filling” procedure, in which ”boxes” corresponding to one-particle states are filled by enumerated balls in all possible ways to count the number of linearly independent terms in the sum on the right-hand side of 5.2. DENSITY OF STATES 63 (5.13). Let us start by putting a ball in the first box. There are N possibilities to do this. For the second ball there are N − 1 possibilities for all initial N cases, thus N (N −1). After having the required number n1 of balls in the first box, we notice that we have the counted each different term n1 ! times, since the order of casting the balls does not make any difference. Restore the correct relative weight of each term by dividing by n1 !. So far N (N − 1) . . . (N − n1 + 1)/n1 ! different terms with n1 particles have been obtained. For the second box, the starting number of possibilities is N − n1 . After completing the filling of all boxes, we arrive at the number of X N! nℓ = N , n1 !n2 ! · · · nℓ ! · · · ℓ of linearly independent terms involved in the basis vector (5.13). Entropy. Statistical entropy related to the ensemble described by ̺ is defined in analogy with the classical Gibbs entropy as a functional of ̺: S = −Tr ̺ ln ̺ . (5.14) In the basis, in which ̺ is diagonal, equation (5.6), in terms of eigenvalues pα of ̺ we may write X pα ln pα . (5.15) S=− α The relation between the statistical entropy and the thermodynamic entropy requires separate analysis, that they may be identified is not at all self-evident. 5.2 Density of states Let the spectrum and eigenfunctions of the Hamiltonian be X En |nihn| . H|ni = En |ni, H = (5.16) n In a finite volume V the spectrum of H is completely discrete. In the analysis of condensed matter or macroscopic bodies passing to thermodynamic limit V → ∞, N → ∞ with fixed the number density N/V is eventually implied leading to practically continuous spectrum. An important quantity in the analysis of the energy spectrum is the cumulative distribution function of states: X θ(E − En ) . (5.17) J(E) = n At the point E + 0+ it yields the number of states with energies ≤ E. Since dθ(x)/dx = δ(x), the density of states may be expressed as ω(E) = dJ(E) X δ(E − En ) . = dE n (5.18) 64 5. QUANTUM-MECHANICAL ENSEMBLES The difference J(E + ∆E + 0+ ) − J(E − 0+ ) ≈ ω(E) ∆E is the number of states with energies in the interval [E, E + ∆E]. In large systems the energy spectrum is dense, due to which a slight coarse-graining leads to ω(E) and J(E) which are smooth functions of E. At low temperatures small values of E are important and usually a detailed knowledge of the structure of the spectrum near the ground state is required. Example 5.1. Free particle. The Hamilton function is H = p2 /2m. Eigenstates may be conveniently enumerated with the periodic boundary conditions in a cubic box of volume V = L3 . The eigenfunctions are plane waves 1 ψk (r) = √ eik·r , V (5.19) 2π (nx , ny , nz ) ; L with integer nj . The energy of the particle is (5.20) where k= εk = p2 ~2 k 2 = . 2m 2m (5.21) Density of states in the k space is (L/2π)3 , and with the account of possible spin degeneracy g = 2S +1 in the limit of large box the ”smoothed” summing rule follows Z Z X XZ V V 3 (5.22) d k · · · = g d3 p · · · , ··· ⇒ dNk · · · = g 3 3 (2π) h σ kσ with the last expression in terms of the momentum p = ~k. The ”smoothed” cumulative distribution function of states is XZ p2 dNk θ E − J1 (E) = 2m σ Z p V 4π 3 V dp′ p′2 = g 3 p . (5.23) = g 3 4π h h 3 0 In terms of the energy E = p2 /2m in the continuum limit (V → ∞) we arrive at ) J1 (E) = 32 C1 V E 3/2 , (5.24) √ ω1 (E) = C1 V E where the constant C1 is C1 = 2πg 2m h2 3/2 . (5.25) 5.2. DENSITY OF STATES 65 kz ω 1 (E ) dk ky kx E Figure 5–1: (a) one-particle states in the wave-vector space, (b) smoothed density of states. Example 5.2. Maxwell–Boltzmann gas. Consider now a system of N free particles in a box of volume V = L3 with periodic boundary conditions. If quantum-mechanical conditions on occupation numbers are neglected, then every particle may occupy any momentum state independently of others. The energy of the system is E= N X j = p2j , 2m and the smoothed cumulative √ distribution function of states is given as the volume of a ball of radius 2mE in the 3N -dimensional momentum space √ 3N 1 N N π 3N/2 2mE g V . JN (E) = N! h3N Γ(3N/2 + 1) (5.26) The factor N ! removes approximately the permutation degeneracy of N particles. This is an acceptable approximation, when the occupation numbers of the one-particle states are small, i.e. in a dilute gas. Indeed, the counting of the number of states by the volume of a ball in the wave-vector space regards each combination of wave-numbers as the quantum numbers of Q a separate state. Physical states, however, are linear combinations of N !/ ℓ nℓ ! linearly independent products of one-particle wave functions which means a huge overcounting of physical states in the approximation we have used. This overcounting may be removed in a simple way only in the limit, when the degeneracy is the same for the overwhelming majority of all physical states, which is the case, when almost all occupation numbers are either 0 or 1. Then the overcounting is N ! fold and we arrive at the ”quasiclassical” expression (5.26) for the cumulative distribution function of states. Differentiating with respect to E we obtain the density of states ωN (E) = gN V N N !Γ( 32 N ) 2mπ h2 3N/2 E 3N/2−1 = (C2 V )N E 3N/2−1 . N !Γ( 23 N ) (5.27) 66 5. QUANTUM-MECHANICAL ENSEMBLES 5.3 Energy, entropy and temperature The density operator of a statistical ensemble allows to calculate all properties of the system, e.g. the expectation value of energy and the entropy. For a connection with thermodynamics the temperature of an equilibrium state has to be defined. To this end, introduce the microcanonical ensemble, whose density operator may be built requiring the energy of the system be in the interval [E, E + ∆E] and maximizing the statistical entropy with this constraint. This is most transparently carried out in the energyeigenfunction basis, where the quantities to be varied are functions. Therefore, the result is similar to that of the classical case 1 pn = p(En ) = 1 θ E + ∆E − En − θ E − En . ZE (5.28) The normalization factor, the microcanonical partition function is ZE = J(E + ∆E) − J(E) , i.e. (literally) the number of states with energies in the interval [E, E +∆E]. For small ∆E we may write ZE ≈ ω(E)∆E. For practical purposes SE = ln ω(E) , (5.29) because only this term is extensive in a large system. The term brought about by the energy resolution ln ∆E is not extensive and numerically negligible regardless of the macrophysical choice of ∆E. Definition of temperature. We would like to identify the statistical entropy with that of the thermodynamics. The guiding relation is the differential form of the first law dU = dE = T dS − p dV + µ dN which for constant V, N yields ∂S ∂E V,N = 1 . T (5.30) The temperature that has to be prescribed to the equilibrium system described by the microcanonical ensemble depends – according to relations (5.29) and (5.30) – unambiguously on the density of states corresponding to the energy E: 1 ∂ ≡ ln ω(E, V, N ) . (5.31) T ∂E 1 The width of the energy window ∆E is omitted here as a parameter because it turns out to be irrelevant. 5.4. PROBLEMS 67 The density of states ω depends naturally also on the variables V and N determining the Hilbert space. With the notation β= 1 T (5.32) we arrive at the result β = ∂ ln ω/∂E, i.e. β is, according to Fig. 5–2, the slope of the curve ln ω(E). ln ω(E) α tan α = β E Figure 5–2: Logarithm of the density of states as a function of energy. The slope of the curve is β = 1/T . Equilibrium thermodynamics of a macroscopic system may thus be derived from the density of states of the system ω(E, V, N ). This is tantamount to carrying out calculations in the microcanonical ensemble, which in the thermodynamic sense corresponds to the isolated system. In many cases, however, calculation of the density of states is very difficult or even impossible with fixed E and N and other ensembles are better suited for practical use. 5.4 Problems Problem 5.1. Consider a physical system described in a 3-dimensional Hilbert space. Probabilities to observe the system in the states 2 3 1 |ψ1 i = 4 0 5 , 0 2 3 0 1 |ψ2 i = √ 4 1 5 , 2 1 2 3 0 1 |ψ3 i = √ 4 −1 5 2 1 are 1/2, 1/3 and 1/6, respectively. Calculate the 3 × 3 density matrix of the system. Show that Tr ̺2 < Tr ̺ = 1. Problem 5.2. Calculate the microcanonical density matrix ρ(x, x′ ) of a one-dimensional free particle described by plane waves in the interval − L2 ≤ x ≤ L2 with periodic boundary conditions. The energy shell is defined by the inequalities 2ℏ2 π 2 (N + ∆N )2 2ℏ2 π 2 N 2 ≤ En ≤ . 2 mL mL2 68 5. QUANTUM-MECHANICAL ENSEMBLES Problem 5.3. A large quantum system consists of N non-identical independent particles each of which may occupy one of two states with energies 0 and ǫ. Energy eigenstates may thus be enumerated by the sequences ν = (n1 , n2 , . . . , nN ), where ni = 0 or 1, and Eν = N X nj ǫ. j=1 Calculate the cumulative distribution function of states, density of states, entropy, temperature and heat capacity C = dE/dT . 6. Equilibrium distributions 6.1 Canonical ensemble To the canonical ensemble corresponds the probability distribution which maximizes entropy with the constraint that the expectation value of energy hHi ≡ Tr ̺H = E (6.1) is a given constant. This constraint as well as the normalization condition Tr ̺ = 1 may be taken into account by the Lagrange-multiplier method. This leads to the variational condition 0 = δ (S − λhHi − λ′ hIi) = δTr (−̺ ln ̺ − λ̺H − λ′ ̺) = Tr δ̺(− ln ̺ − I − λH − λ′ I) . Here, λ, λ′ are the Lagrange multipliers. The last expression in brackets must vanish. This yields, with a different notation for constants ̺= 1 −βH e . Z (6.2) The normalization factor Z is the canonical partition function Z = Tr e−βH = X e−βEn = n Z dE ω(E) e−βE . (6.3) The probability distribution (6.3) is the canonical or Gibbs distribution. If the energy scale is chosen such that the ground-state energy is zero and unbounded states are possible, then the canonical partition function may be regarded as the Laplace transform of the microcanonical partition function: Z= Z∞ dE ω(E) e−βE 0 In the canonical distribution probabilities of the energy eigenstates |ni are pn = 1 −βEn e . Z (6.4) The canonical distribution of a one-particle system is also called the Boltzmann distribution. If the one-particle energies are εν , then the partition 69 70 6. EQUILIBRIUM DISTRIBUTIONS function and the probability to observe the state ν are Z= X e−βεν ; pν = ν 1 −βεν e . Z (6.5) The exponential exp (−βεν ) is the Boltzmann factor. The parameter β has the meaning of the inverse temperature of the system: β = T −1 . Entropy and temperature. Since ln ̺ = −βH − ln Z, the entropy assumes the form S = −hln ̺i = βE + ln Z , (6.6) where E = hHi = Tr (H e−βH )/Z. The parameter β determines values of both energy and entropy. To find the temperature, we relate variations of these quantities as functions of β. Calculate first δZ, δZ = Tr δ(e−βH ) = −δβ Tr He−βH = −δβ EZ ; then δS = δZ E δβ + β δE + Z = β δE . According to the definition of the temperature δE 1 T = = , δS V,N β (6.7) i.e. β = T −1 . Canonical partition function and free energy. The partition function of a distribution is a central quantity. With its aid all thermal properties of an equilibrium system may be calculated. Differentiating with respect to β:n we obtain ∂ Z = −Tr e−βH H = −ZhHi , ∂β so that the expectation value of the energy is E=− ∂ ∂ ln Z ln Z = T 2 . ∂β ∂T (6.8) Substitution in expression (6.6) for entropy yields entropy solely in terms of Z: ∂ (T ln Z) . (6.9) S= ∂T Comparison with the thermodynamic definition F = E − T S leads to an important result [which may be inferred directly from relation (6.6) as well] F = −T ln Z . (6.10) 6.1. CANONICAL ENSEMBLE 71 Since Z = Z(T, V, N ), the statistical definition of the free energy in (6.10) is given in terms of the natural variables. The density operator may now be written also as ̺ = eβ(F −H) . (6.11) is Fluctuations. The probability distribution of the energy E of the system P (E) ≡ hδ(H − E)i = Tr ̺ δ(H − E) . ω (E ) e (6.12) −βE ∆E E(β ) E Figure 6–1: Probability distribution of energy in the canonical ensemble. In the canonical ensemble this is P (E) = 1 ω(E) e−βE . Z In a large system energy integrals may be calculated with the aid of the saddle-point method, since in the weight function ω(E) exp(−βE) = exp[−βE+S(E)] the argument of the exponential is (implicitly) proportional to the number of particles N . Consider, e.g., the partition function Z Z = dE exp[−βE + ln ω(E)] The microcanonical entropy is ln ω(E) = S(E), therefore the expanding the exponential we obtain: ln ω(E) − βE = ln ω(E) − βE + ∂S(E) − β (E − E) ∂E 1 ∂ 2 S(E) (E − E)2 + · · · . + 2 ∂E 2 The stationarity condition T (E) ≡ ∂E = β −1 = T , ∂S(E) 72 6. EQUILIBRIUM DISTRIBUTIONS renders the microcanonical temperature equal to the canonical one. The second-order term is negative definite ∂2S ∂ = ∂E 2 ∂E 1 1 ∂T 1 =− 2 =− 2 , T T ∂E T CV so that the partition function is expressed in terms of a convergent integral Z 1 −βE 2 Z ≈ ω(E) e (E − E) . (6.13) dE exp − 2 2T CV This relation shows that also microcanonical and canonical entropies coincide at leading order in N . The energy distribution in relation (6.13) is normal and the variance of energy p √ ∆E = T 2 CV = O( N ) . (6.14) Thus, the relative inaccuracy of the energy is 1 ∆E ∝√ , E N (6.15) which for a large system is extremely small. For any practical purposes the canonical and microcanonical ensembles are thus p equivalent. More directly the variance of energy ∆E ≡ hH 2 i − hHi2 may be calculated by noting that hHi = − TrH e−βH ∂ ln Z = ∂β Tre−βH and ∂ ∂2 hHi = − 2 ln Z = hHi2 − hH 2 i . ∂β ∂β Thus D E ∂2 2 (∆E)2 ≡ (H − hHi) = − 2 (βF ) = T 2 CV . ∂β (6.16) Classical canonical distribution. In classical statistical mechanics the canonical distribution may be derived in the same way as above. The canonical distributionmaximizing the Gibbs entropy (4.26) turns out to be ̺(P ) ∝ exp −βH(P ) , where P is a point in the phase space. Thus, the classical canonical partition function is Z Z = dΓ e−βH , (6.17) where the volume element dΓ is given by relation (4.5). 6.1. CANONICAL ENSEMBLE 73 Example 6.1. Canonical partition function of a free point particle. The Hamilton function is H = p2 /2m. Choose plane waves as the eigenfunctions 1 (6.18) ψk (r) = √ eik·r , V where 2π (nx , ny , nz ) ; L with integer nj . The energy of the particle is (6.19) k= εk = ~2 k 2 . 2m (6.20) In the thermodynamic limit replace X nx ,ny ,nz V ··· ⇒ (2π)3 Z (6.21) d3 k · · · , to obtain b Z1 = Tre−β H = X 2 2 e−β~ nx ,ny ,nz k /2m ≈ V (2π)3 Z 2 2 d3 ke−β~ k /2m . (6.22) Due to rotational symmetry the angular integral in the spherical polar coordinates is taken immediately giving rise to the density of states of the free particle and an integral calculable as Γ( 21 ): 2πV (2m)3/2 Z1 = (2π)3 ~3 Z∞ √ V V (2πmT )3/2 = 3 , dε εe−βε = 3 h λT (6.23) 0 √ where λT = h/ 2πmT is the thermal de Broglie wavelength which may be understood as the measure of broadening of the single-particle wave packet due to thermal motion. Example 6.2. Canonical partition function of N free point particles. The wave function is a symmetrized or antisymmetrized product of N plane waves, which thus includes a large number of linearly independent terms. In the canonical ensemble, this imposes restrictions on the sum over states in the partition function which are difficult to take into account. Approximately this may be done in the case of small occupation number, in which case the number of linearly independent terms in the (anti)symmetric wave function is close to N ! and thus equal for all possible wave functions. This allows to sum over all wave vector independently and remove the overcounting introduced this way simply by dividing the result by N !. Thus ZN ZN = 1 . (6.24) N! 74 6. EQUILIBRIUM DISTRIBUTIONS Example 6.3. Canonical density operator of the harmonic oscillator in the coordinate representation. The unnormalized density operator is ∞ X 1 b hx |e−β H |x i = e−β~ω(n+ 2 ) ψ (x )ψ (x ) , 1 2 n 1 n 2 n=0 with the eigenfunctions where q = p mω ~ mω 1/4 e−q2 /2 H (q) √ n ψn (x) = π~ 2n n! x and Hn (q) are Hermite polynomials n q2 Hn (q) = (−1) e d dq n 2 −q 2 e eq =√ π Z∞ 2 du (−2iu)n e−u +2iqu . −∞ Substitution gives b |x i = 2 −β H hx1 |e r mω (q12 +q22 )/2 e π3 ~ Z∞ du −∞ Z∞ −∞ dv ∞ X (−2uv)n n! n=0 1 Here, 2 × e−β~ω(n+ 2 )−u +2iq1 u−v 2 +2iq2 v . ∞ X (−2uv)n e−β~ωn = exp −2uve−β~ω . n! n=0 The remaining Gaussian integral may be calculated with the aid of the relation (derivation see below in section 6.4): Z p −1 1 −1 1 Dx C e− 2 xsx+x·y = det (s/2π) e 2 ys y . In our case s is a 2 × 2 matrix: 2 s= 2e−β~ω 2e−β~ω 2 , and the vector y = (2iq1 , 2iq2 ). Therefore, r e−β~ω/2 mω b √ hx1 |e−β H |x2 i = π~ 1 − e−2β~ω 2 q 2 − 2q1 q2 e−β~ω + q22 q + q22 . − 1 × exp 1 2 1 − e−2β~ω With the aid of definitions and transformation rules for hyperbolic functions this expression may be cast in the more symmetric form r mω b −β H hx1 |e |x2 i = 2π~ sinh β~ω mω β~ω β~ω 2 2 × exp − (x1 + x2 ) tanh . + (x1 − x2 ) coth 4~ 2 2 6.2. GRAND CANONICAL ENSEMBLE 75 The normalization factor is Tr e b= −β H Z∞ −∞ b dx hx|e−β H |xi = q 1 β~ω 2 sinh β~ω tanh 2 so that the density matrix is ρ(x1 , x2 ) = r mω β~ω tanh π~ 2 β~ω β~ω mω 2 2 (x1 + x2 ) tanh . + (x1 − x2 ) coth × exp − 4~ 2 2 6.2 Grand canonical ensemble The grand canonical ensemble is obtained by maximization of the entropy S with the following constraints: b = E = given energy , hHi b i = N = given particle number . hN The accented quantities are operators. The Lagrange-multiplier method leads to the grand canonical distribution ̺G = 1 −β(H−µ b Nb ) . e ZG (6.25) Here, the Hamilton operator is the sum b = H (0) + H (1) + · · · + H (N ) + · · · , H where the operator H (N ) acts only on the Hilbert space of the proper N b on an N particle functions. The action of the particle-number operator N b is replaced by particle function is that of the multiplication operator, i.e. N its eigenvalue N . The grand canonical partition function is most conveniently written as the sum over all particle numbers X X (N ) ZG = eβµN Tr N e−βH = z N ZN . (6.26) N N Here, TrN is the trace in the N -particle Hilbert space and ZN ≡ Tr N exp −βH (N ) is the canonical partition function of the N -particle system. In relation (6.26) also the fugacity, z = eβµ (6.27) 76 6. EQUILIBRIUM DISTRIBUTIONS has been introduced. Grand canonical partition function and grand potential. Derivatives of the partition function 1 ZG = Tr exp[−β(H − µN )] yield expectation values. A good starting point is the extensive logarithm ln ZG . Thus, 1 ∂ ln ZG = Tr e−β(H−µN ) βN = βhN i = βN , ∂µ ZG and 1 ∂ ln ZG =− Tr e−β(H−µN ) (H − µN ) = −hHi + µhN i = −E + µN . ∂β ZG Therefore, the expectation values of the particle number and energy are ∂ ln ZG , ∂µ (6.28) ∂ ln ZG ∂ ln ZG + Tµ . ∂T ∂µ (6.29) N =T E = T2 With the use of relations (6.28)–(6.29), the entropy S = −hln ̺i now assumes the form 1 ∂ (T ln ZG ) , (6.30) E − µN + ln ZG ⇒ S ≡ −hln ̺i = T ∂T Comparison with the thermodynamic definition of the grand potential Ω = E − T S − µN leads to the identification (6.31) Ω = −T ln ZG . Here, ZG is given in terms of the natural variables T , V and µ of Ω. The density operator may be written as (6.32) ̺ = eβ(Ω−H+µN ) . Fluctuations. Differentiating the partition function twice with respect to µ we obtain ∂2 Tr e−β(H−µN ) = ZG β 2 hN 2 i . ∂µ2 Combining this result with relation (6.28) we obtain after a straightforward calculation (∆N )2 ≡ hN 2 i − hN i2 = T 2 = T 1 ∂N . ∂µ ∂ 2 ln ZG ∂µ2 (6.33) Superfluous accents denoting operators will be omitted. Operators, eigenvalues and expectation values may be distinguished by the context. 6.3. CONNECTION WITH THERMODYNAMICS 77 Since ln ZG and N are extensive quantities, the relative inaccuracy of the particle number is ∆N 1 √ , (6.34) =O N N which again is very small, for a mole of the order 10−12 . Thus, particle number, energy and other extensive quantities are determined in practically equal accuracy in all canonical ensembles provided all relevant response functions remain finite (which is typically not the case in continuous phase transitions). In many cases, however, it is easier to calculate the grand canonical partition function than the canonical or microcanonical. Example 6.4. Grand canonical partition function of a system of free point particles. The construction is readily accomplished with the use of the canonical partition function: ZG = ∞ X N =0 eβµN Z1N = exp Z1 eβµ N! and gives rise to the grand potential Ω = −T ln ZG = −T V eβµ (2πmT )3/2 . h3 6.3 Connection with thermodynamics Consider an adiabatic change in the quantum-mechanical system, i.e. a change of energy eigenfunctions and eigenvalues which does not affect the structure of energy levels: H|ni = En |ni (H + δH)(|ni + δ|ni) = (En + δEn )(|ni + δ|ni) . (6.35) Forming a scalar product of latter equation with hn|, at the linear order in the variations we obtain δEn = hn|δH|ni = hδHin . If the change of the Hamiltonian is due to small changes of parameters P δH = i ∂H δxi , then ∂xi ∂H ∂En = . (6.36) ∂xi ∂xi n If the parameters xi (t) change slowly enough as functions of time t, the system in eigenstate n remains in that eigenstate without transitions to other energy eigenstates. This is the adiabatic change in quantum mechanics. 78 6. EQUILIBRIUM DISTRIBUTIONS Thus, a change in the parameters of the Hamiltonian leads, not unexpectedly, to a change in its eigenvalues. The expectation value of the Hamilton operator depends, however, also on the probabilities to observe the system in an energy eigenstate, i.e. on the density operator. Therefore δhHi = Tr δ̺ H + Tr ̺ δH = Tr δ̺H + X δxi Tr ̺ i ∂H . ∂xi Define the generalized force ∂H Fi = −Tr ̺ =− ∂xi ∂H ∂xi , conjugate with the generalized displacement δxi . Then X Fi δxi . δhHi = Tr δ̺H − (6.37) i On the other hand, the change of the statistical entropy due to a change in the density operator is δS stat = −Tr ln ̺δ̺ , since the normalization condition requires Tr δ̺ = 0. Further steps depend on the choice of the ensemble. In case of the canonical ensemble ln ̺ = −H/T stat − ln Z and 1 (6.38) δS stat = stat Tr Hδ̺ . T Combining (6.37) and (6.38) we arrive at the relation X δE = T stat δS stat − Fi δxi (6.39) i in the form of the first law for a reversible change δU = T ter δS ter − δW . Thus, the following identifications maintain the consistency between statistical mechanics and thermodynamics hHi = E = U = internal energy T stat = T ter stat ter S P =S i Fi δxi = δW = work . (6.40) 6.4 Thermodynamic fluctuation theory A large system near equilibrium may be divided in relatively weakly interacting macroscopic subsystems. Then extensive quantities related to subsystems may be defined such that their operators in different subsystems (at least) approximately commute with each other and the Hamiltonian of the whole system. Correspondingly, the phase space may be divided in parts 6.4. THERMODYNAMIC FLUCTUATION THEORY 79 (or the Hilbert space to manifolds) characterized, in addition to the total energy E, by the values of these quantities X1 , . . . , Xn . The set of parameters (E, X1 , . . . , Xn ) determines a (close-to-equilibrium) macrostate. Let the measure of the macrostate be Σ(E, X1 , . . . , Xn ) (volume of the corresponding part of the phase space or the number of microstates in the Hilbert space) and the total measure corresponding the energy E of the system X Σ(E) = Σ(E, X1 , . . . , Xn ) . {Xi } Microstates corresponding to the set E, X1 , . . . , Xn occur at the relative probability Σ(E, X1 , . . . , Xn ) f (E, X1 , . . . , Xn ) = . Σ(E) The entropy of the macrostate (E, X1 , . . . , Xn ) is (up to non-extensive terms) S(E, X1 , . . . , Xn ) = ln Σ(E, X1 , . . . , Xn ) , (6.41) so that its probability is f (E, X1 , . . . , Xn ) = 1 exp S(E, X1 , . . . , Xn ) . Σ(E) (6.42) The equilibrium values of the parameters Xi0 maximize the entropy. If deviations from the equilibrium values xi = Xi − Xi0 are small, then a Taylor expansion of the entropy is reasonable S = S0 − 1 1X sij xi xj ≡ S 0 − xsx . 2 ij 2 (6.43) Here, the coefficients are sij = − ∂2S ∂Xi ∂Xj 0 . (6.44) The entropy matrix s is real and symmetric, and for stable equilibrium positive definite. Thus, the variables xi possess a Gaussian probability density 1 f (x) = C e− 2 xsx , (6.45) where C is a normalization factor to be calculated below. To calculate moments of the probability distribution (6.49) the generating function is useful Z 1 G(y) = Dx C e− 2 xsx ex·y , (6.46) 80 where x · y = 6. EQUILIBRIUM DISTRIBUTIONS P i xi yi . Correlation functions are calculated as Z 1 hxp · · · xr i ≡ Dx C e− 2 xsx xp · · · xr ∂ ∂ . = ··· G(y) ∂yp ∂yr y=0 (6.47) To calculate the generating function explicitly, make a shift of the integration variable x → x + a and choose the auxiliary vector a such that the linear in x term in the exponential disappears: a = s−1 y. This leads to representation Z 1 G(y) = e 2 y T s−1 y 1 Dx e− 2 xsx . C (6.48) Due to positive p definiteness of the matrix s the p following change of variables is legal z = s/(2π)x. The Jacobian is det s/(2π), therefore Z Z Z Pn 2 p −1 1 dz1 · · · dzn e−π i=1 zi . C Dx e− 2 xsx = C det s/(2π) The remaining integral factorizes to an n-fold product of well-known onedimensional integrals Z∞ 2 dz e−πz = 1 . −∞ The normalized probability density is thus r s − 1 xsx . e 2 f (x) = det 2π (6.49) Note that no explicit dependence on the space dimension n remains here. Finally, the generating functional assumes the form 1 −1 , (6.50) hxi xj i = (s−1 )ij . (6.51) G(y) = e 2 ys y Calculation of the second derivative yields Fluctuations in SVN-system. In section 2.9 the deviation of the entropy from the equilibrium value was expressed in terms of deviations of state variables to analyze stability of the extremum of the entropy. In case of equilibrium this fluctuation is negative definite and gives the probability distribution of fluctuations of state variables from their equilibrium values; according to relation (2.49) the probability density is 1 f = C exp − (∆T ∆S − ∆p ∆V + ∆µ ∆N ) . (6.52) 2T 6.4. THERMODYNAMIC FLUCTUATION THEORY 81 Fix the particle number for simplicity ∆N = 0 and express the remaining fluctuations ∆T , ∆S, ∆p and ∆V in terms of the two chosen as independent variables. Let ∆T and ∆V be the independent variables. The quadratic form of the entropy fluctuation has been expressed in terms of these variables in Chapter 2.9 in relation (2.50), therefore 1 CV 1 2 2 f (∆T, ∆V ) ∝ exp − (∆T ) + (∆V ) . (6.53) 2 T2 V T κT The elements of the entropy matrix s are thus sT T = CV 1 , sV V = , T2 V T κT sT V = sV T = 0 . Since the matrix s is diagonal, the quadratic expectation values according to relation (6.51) follow immediately h(∆T )2 i = T2 CV h∆V ∆T i = 0 , h(∆V )2 i = V T κT . (6.54) The thermodynamic theory of fluctuations and the statistical approach may lead to seemingly different results. Let us calculate the fluctuation of the internal energy in the thermodynamic theory. From the expression ∂E ∂p ∂E ∆T + ∆V = CV ∆T + T − p ∆V ∆E = ∂T V ∂V T ∂T V with the use of relations (6.54) it follows that 2 ∂p h∆E 2 i = CV T 2 + T − p T V κT , ∂T V which is apparently different from the expression h∆E 2 i = CV T 2 obtained in the canonical ensemble. The reason of this discrepancy is that in the canonical ensemble the volume V is a fixed parameter, whereas in the thermodynamic fluctuation theory fluctuations of the volume are not restricted. The statistical theory gives the same result, if an ensemble is used which allows for fluctuations of the volume (i.e. the isothermal-isobaric ensemble). Example 6.5. Calculation of the correlation function h∆V ∆pi. The most straightforward way is to express ∆p as a function of T and V and use the formulae obtained for fluctuations of the temperature and volume. Thus, ∂p ∂p ∆T + ∆V , ∆p = ∂T V ∂V T wherefrom – with the aid of (6.54) – we obtain ∂p h∆V 2 i ∂p h∆V ∆pi = h∆V ∆T i + h∆V 2 i = − = −T . ∂T V ∂V T V κT 82 6. EQUILIBRIUM DISTRIBUTIONS Example 6.6. Landau-Placzek formula. Elastic scattering of visible light in gases (Rayleigh scattering) is brought about by permittivity fluctuations. These are mainly due to density fluctuations (in gases the temperature dependence of the permittivity is negligible). The density fluctuations may be considered caused by fluctuations of the entropy and density. They have the important difference that the entropy fluctuations do not propagate but decay due to thermal conduction, whereas the pressure fluctuations propagate at the speed of sound, which gives rise to the Doppler shift of frequency of the scattered wave. Thus, scattering on the entropy fluctuations brings about scattered light at the frequency of the incident wave, whereas the pressure fluctuations give rise to the Mandelshtam-Brillouin doublet with frequencies on both sides of that of the incident wave. Let us calculate the ratio of the intensity of the doublet (the sum of both components) and the total intensity of the scattered wave. The total density Itot ∝ h∆ρ2 i. Here, ρ = m/V so that ∆ρ = −(ρ/V )∆V and Itot ∝ h∆V 2 i. Substitute ∂V ∂V ∆S + ∆p ∆V = ∂S p ∂p S to obtain 2 h∆V i = ∂V ∂S 2 p Cp − T ∂V ∂p , S ∂p . The latter term ∂V S 2 in the expression for h∆V i yields thedoublet contribution. On the other hand, according to (6.54), h∆V 2 i = −T ∂V and ∂p T ∂V −T ∂(V, S) ∂(p, T ) ∂(V, S) ∂(p, T ) CV Idoublet ∂p S = = = = . Itot ∂(p, S) ∂(V, T ) ∂(V, T ) ∂(p, S) Cp −T ∂V ∂p T because h∆S 2 i = Cp , h∆S∆pi = 0 and h∆p2 i = −T This is the Landau-Placzek formula. 6.5 Reversible minimal work and probability of fluctuations. The probability of the fluctuation ∆X = X − X 0 of the quantity X may be calculated also with the use of the notion of reversible minimal work. To this end, recall that the equilibrium entropy is a monotonically increasing function of the energy of the system. Then the decrease in the entropy ∆S corresponding to the fluctuation ∆X may be achieved also by decreasing the energy of the system reversibly in order to follow the the dependence S = S(E) of the equilibrium entropy. This energy is the reversible 6.6. PROBLEMS 83 minimal work R to be exerted to the system to cause the fluctuation ∆X. Therefore, we may write S(E − R) ≈ S(E) − ∂S 1 R = S(E) − R , ∂E T which, according to relation (6.42), yields for the probability density of the fluctuation R . (6.55) f (∆X) ∝ exp − T The nice feature here is that R is expressed in terms of mechanical work or the like. Example 6.7. Fluctuation of a string under tension. Consider a string of length L under tension caused by the longitudinal force F . For a small transversal displacement y of a string element at the point x the restoring force is y y F⊥ = F p +Fp , 2 2 (L − x)2 + y 2 x +y which gives rise to the elementary work i h p p dW = F⊥ dy = d F x2 + y 2 + F (L − x)2 + y 2 . Thus the total work to move quasistatically the string element to the displacement position is i h p p W = F F x2 + y 2 + F (L − x)2 + y 2 − L ≈ F y2 L . 2x(L − x) This is the minimal reversible work to bring about the displacement of the string element, therefore the probability density of the corresponding fluctuation is F y2 L − 2x(L − x) f (y) ∝ e and the root-mean-square fluctuation ∆y is given by ∆y 2 = hy 2 i − hyi2 = x(L − x)T . FL 6.6 Problems Problem 6.1. Calculate the canonical partition function for a system of N identical non-interacting harmonic oscillators with an angular frequency ω. Find the internal energy and and heat capacity. Consider both classical and quantum-mechanical cases. 84 6. EQUILIBRIUM DISTRIBUTIONS Problem 6.2. Generalized equipartition theorem. Let the Hamilton function of a classical system be quadratic with respect to all canonical coordinates and momenta ξj : H= ν X αj ξj 2 , j where αj s are constants. Calculate the canonical partition function and show that E = 12 νT . Find also the density of states as the inverse Laplace transform of the canonical partition function with respect to β (to be treated as a complex variable). Problem 6.3. A large quantum system consists of N non-identical independent particles each of which may occupy one of two states with energies 0 and ǫ. Energy eigenstates may thus be enumerated by the sequences ν = (n1 , n2 , . . . , nN ), where ni = 0 or 1, and Eν = N X nj ǫ. j=1 Calculate the canonical partition function, free energy, entropy, and heat capacity C = dE/dT . Problem 6.4. Consider a gas of point-like free particles with vanishing rest mass, i.e. with the one-particle energy ǫ(p) = c|p|. Calculate the density of states of one particle, calculate then the canonical partition function of one particle and N particles. Problem 6.5. Consider a gas of point-like free particles with vanishing rest mass, i.e. with the one-particle energy ǫ(p) = c|p|. Calculate the density of states of a system of N such particles. Hint. The inverse Laplace transform of the canonical partition function might be useful. Problem 6.6. Consider a gas of point-like particles with vanishing rest mass, i.e. with the one-particle energy ǫ(p) = c|p|. Calculate the grand canonical partition function and the grand potential Ω of the gas and establish the equation of state. Calculate also the chemical potential µ = µ(T, n) of the gas. Problem 6.7. Calculate the electric dipole moment of an ideal gas consisting of linear molecules with a constant dipole moment p0 , when the gas is in an external electric field E. Use the classical canonical ensemble. Problem 6.8. The length of the arm of a classical mathematical pendulum is l, the mass attached to it is m and the acceleration of the gravity of the Earth is g. Determine the fluctuation of the swinging angle of the pendulum at rest ∆ϕ and also the heat capacity related to small oscillations of the pendulum. 6.6. PROBLEMS 85 Problem 6.9. With the use of the thermodynamic theory of fluctuations, show that h(∆S)2 i = Cp , h(∆p)2 i = T /V κS , h∆S ∆pi = 0, and also h∆S ∆V i = V T αp . Hint. Use S and p as independent variables. Problem 6.10. In the canonical ensemble, show that h(E − hEi)3 i = ∂ 2 hEi . ∂β 2 Problem 6.11. In the isobaric–isothermal ensemble only the average volume is fixed and the partition function may be written as Z Z(T, P ) = b dV TrV e−β H(V )−βP V Z = Z dV e−βP V Zcanonical (T, V ) = dV e−βP V e−βF (T,V ) . Show that in this ensemble the energy fluctuation obeys the relation h(∆E)2 i = T 2 CV − T ∂V ∂p T T ∂p ∂T V 2 −p . in accordance with the thermodynamic theory of fluctuations. 7. Ideal equilibrium systems 7.1 Free spin system In ideal systems there are no interactions between particles. If the solution of the one-particle problem is known, properties of the ideal many-particle system may be simply found by taking into account the proper statistics. The simplest example is the the system of free spins. States. Consider N particles with spin s = 12 for simplicity. No interactions between the particles. Ignore also spatial motion. This model is fairly realistic for a crystallic paramagnetic substance, in which at low temperatures site-bound atoms only have the spin degrees of freedom left. Each particle has two spin states sjz = ± 12 ~. Thus, in the N -particle system there are 2 × 2 × · · · × 2 = 2N different quantum states. Localization of host atoms allows to consider spins distinguishable. This means that the statistical ensembles constructed here are classical, although the very notion of spin is quantum-mechanical. Let the total spin be N X Sz ≡ sjz = ~m. (7.1) j=1 For simplicity, consider even N , so that the quantum number m may assume the values 1 (7.2) m = 0, ±1, ±2, · · · , ± N, 2 altogether N + 1 different values. Denote the numbers of spin-up and spindown particles by N + and N − , respectively; then 1 N+ = N + m 2 (7.3) 1 − N = N −m 2 The number of ways W (m) to obtain the total spin ~m is given by the binomial distribution N! N! N 1 W (m) = = + − = 1 (7.4) N+ N !N ! N + m ! 2N − m ! 2 Thus, this is the degeneracy of the state with the total spin Sz = ~m. 86 7.1. FREE SPIN SYSTEM 87 In the limit of large numbers an approximation with the use of the Stirling formula follows ln W (m) ≈ N ln N − N − N + ln N + + N + − N − ln N − + N − 1 1 1 + 2m/N = N ln 2 + N ln − m ln . 2 1 − 4m2 /N 2 1 − 2m/N (7.5) In particular, in the vicinity of the maximum m = 0 we obtain ln W (m) ≈ ln W (0) − N2 m2 . Thus, the binomial distribution follows approximately the Gaussian normal distribution 2 W (m) ≈ W (0)e−2m /N (7.6) √ with the standard deviation ∆m = 12 N . In case of completely random √ spins the probable values of the total spin (∝ N ) much less than the largest possible value (∝ N ). Energy. In a system of non-interacting spins only a coupling to an external field may appear in the Hamilton function. Let B = µ0 H and H = Hez . The Hamilton function describing the potential energy of the spins is X X b = −µ0 µjz , (7.7) µj · H = −µ0 H E j j where µj are magnetic moments of the particles. The magnetic moment of a particle is proportional to its spin µ = γs . (7.8) The gyromagnetic ratio γ is usually different form its classical value γ0 = q/(2m), where q is the charge of the particle and m its mass. For instance, for the electron γ ≈ 2γ0 = −e/m, where e is the elementary charge. In the following the value of the coefficient γ is left unspecified. Hence, the potential energy of the spin system in the state m is E = −µ0 γHSz = −εm (7.9) with the microscopic energy unit ε = µ0 ~γH . (7.10) Each energy level m has still the degeneracy W (m). Therefore, the density of states E 1 , (7.11) W − ω(E) = |ε| ε since according to the definition ω(E) |∆E| = W (m) |∆m|, ja |∆E| = |ε∆m|. Since in the energy E contains a coupling to the magnetic field such that the field strength H is an external parameter, the energy should be interpreted as the magnetic enthalpy. 88 7. IDEAL EQUILIBRIUM SYSTEMS Mirocanonical ensemble. It is instructive to derive the statistical mechanics of the spin system first with the aid of the microcanonical ensemble, which is rarely feasible due to difficulties in the calculation of the density of states ω(E). Denote E0 = 21 εN , so that the total energy obeys −|E0 | ≤ E ≤ |E0 |. According to relations (7.5), (7.9) and (7.11) the statistical entropy S(E) = ln ω(E) is E0 − E E0 2 E ln . (7.12) S(E) = N ln 2 + ln 2 + 2E0 E0 + E E0 − E 2 According to the statistical definition of the temperature (5.31) β(E) = 1/T (E) = ∂S/∂E. Differentiation yields β(E) = N E0 − E ln . 2E0 E0 + E (7.13) Solving for the energy we obtain the result 1 µ0 ~γH βE0 = − N µ0 ~γH tanh . E = −E0 tanh N 2 2T (7.14) Magnetization is the magnetic moment per unit volume. For it we obtain 1 X N µ0 ~γH M= µj = ez , (7.15) ~γ tanh V j 2V 2T due to direct proportionality E = −µ0 HV Mz . Canonical ensemble. Denote the values of an individual spin sjz = ~νj , then νj = ± 21 and µjz = ~γνj . Calculation of the trace yields the partition function 1 1 N 2 2 X X X µjz ··· exp βµ0 H ZN = ν1 =− 21 = 1 2 X ν=− 12 νN =− 12 j N eβµ0 ~γHν = Z1 N , where Z1 is the partition function of a single spin. 1 µ0 ~γH βµ0 ~γH − 21 βµ0 ~γH 2 +e = 2 cosh Z1 = e . 2T (7.16) (7.17) The partition function yields the free energy, which should be interpreted as the magnetic Gibbs function : G(T, H) ≡ −T ln ZN µ0 ~γH = −N T ln 2 + ln cosh . 2T (7.18) 7.1. FREE SPIN SYSTEM 89 The entropy may be calculated as the temperature derivative of the free energy: ∂G µ0 ~γH S=− = N ln 2 + ln cosh ∂T H 2T µ0 ~γH µ0 ~γH − tanh . (7.19) 2T 2T Differentiation with respect to magnetic field yields X W (m) ∂G T ∂ZN − = µ0 ~γ e−βεm · m = µ0 ~γhmi = µ0 V Mz . = ∂H T ZN ∂H Z N m Hence, the differential of the free energy is (7.20) dG = −S dT − µ0 V M · dH and the magnetization M =− 1 µ0 V ∂G ∂H = T N ~γ tanh 2V µ0 ~γH 2T , (7.21) which is the same as in the microcanonical ensemble. Susceptibility. According to the definition of the magnetic susceptibility we obtain 2 ∂ G 1 ∂M =− . (7.22) χ= ∂H T µ0 V ∂H 2 T From the expression (7.21) of M it follows 2 1 ~γ µ0 N 2 . χ= · V T cosh2 µ0 ~γH 2T (7.23) In the limit of weak field this yields Curie’s law χ= where the constant is C= N µ0 V C , T 1 ~γ 2 (7.24) 2 . Adiabatic demagnetization. The magnetic properties of a paramagnetic material may be used for cooling. The method easily allows to reach millikelvin temperatures. The lowest temperatures are reached, however, with the aid of nuclear demagnetization. Adiabatic demagnetization is based on the almost complete degeneracy of the energies of the spin states, due to which large entropy may persist 90 7. IDEAL EQUILIBRIUM SYSTEMS to extremely low temperatures. Removal of the entropy with the aid of the magnetic field then lowers the temperature dramatically. In simplified notation the entropy (7.19) may be cast in the form S = ln 2 + ln cosh x − x tanh x , N (7.25) with the dimensionless variable x x= µ0 ~γH . 2T (7.26) Entropy is a monotonically decreasing (increasing) function of x (T ). In the limit x → ∞ it follows ln cosh x = x − ln 2 + e−2x + · · · , and tanh x = 1 − 2e−2x + · · · . The limit x → 0 is trivial. The asymptotic behaviour of the entropy is: S 1 → ln 2 − x2 , x→0 (T → ∞), N 2 (7.27) S → 2xe−2x , x→∞ (T → 0). N S/N H1 < H2 ln 2 1 2 T1 T2 T Figure 7–1: Adiabatic demagnetization. In Fig. 7–1 the temperature dependence of the entropy is shown for two different field strengths. The sample is cooled first by some other effective method in very strong field (magnetic induction of several teslas). Then the sample is isolated thermally and the field strength adiabatically decreased to almost zero. Since the entropy is a function of the ratio H/T only, the sample is cooled in the ratio if the field strengths H1 T1 = . T2 H2 (7.28) Negative temperature. The temperature of the spin system may be rendered negative with the aid of adiabatic demagnetization. The system is then in a metastable state, as illustrated in Fig. 7–2. In equilibrium in a non-vanishing field the energy of the systems is negative. If the direction of the field is adiabatically reversed, occupation numbers of quantum states cannot change accordingly, although their energies change sign. The energy of the system becomes positive, and the situation formally corresponds to an equilibrium system at the negative temperature β ′ < 0. According to relation (7.12) the entropy S(E) has a maximum at E = 0, around which it symmetrically decreases to zero. Thus, S(E) is not a monotonically growing 7.2. CLASSICAL IDEAL GAS 91 ω (E) e-βE ω (E) e-β′E e-βE e-β′E -E 0 E0 E E E0 -E 0 Figure 7–2: Negative temperature. function of E. In particular, at E > 0 the derivative of the entropy is negative exhibiting negative temperature. Metastable population inversion occurs in other systems as well, e.g. in an exited laser. Note that boundedness of the energy spectrum from above is required for these phenomena. 7.2 Classical ideal gas Maxwell–Boltzmann statistics. In ideal gas interactions between the molecules are neglected, apart from collisions occuring every now and then. Their most important effect is the thermalization of the systems, i.e. the approach to the thermodynamic equilibrium describable by an equilibrium ensemble. In the ideal gas limit collisions are elastic and practically instantaneous, so that the energetics of a single molecule is completely determined by its properties as a free particle between the collisions. Thus, thermodynamically a single molecule is a closed system with only a weak coupling to the rest of the gas. Hence, the single-particle statistics is determined by the Boltzmann distribution (kappale 6.1) ̺ℓ = hℓ|̺|ℓi = 1 −βεℓ e ; Z1 Z1 = X e−βεℓ , (7.29) ℓ where εℓ are energies of the one-particle states ℓ. In a translationalinvariant system of point-like particles the one-particle energies are εℓ = pℓ 2 /(2m) = 12 mv ℓ 2 . The volume element of the velocity space is d3 v = m−3 d3 p = (~/m)3 d3 k. The probability distribution of velocity is mv 2 ∝ hk|̺1 |ki, f (v) = C exp − 2T with the normalization constant determined by the condition Z d3 v f (v) = 1. (7.30) (7.31) 92 7. IDEAL EQUILIBRIUM SYSTEMS This is a Gaussian integral, therefore Z mv 2 d v exp − 2T 3 = r 2πT m 3 ; and the Maxwell distribution in the velocity space results: f (v) = m 3/2 mv 2 exp − . 2πT 2T (7.32) It gives rise to the probability density F (v) of the speed v = |v| through the normalization condition Z ∞ Z Z ∞ 3 dv F (v) = d v f (v) = 4π dv v 2 f (|v|) = 1. (7.33) 0 0 Thus, (7.34) F (v) = 4πv 2 f (v) . It is illustrated in Fig. 7–3. F(v) From the speed distribution the following vm v v2 v Figure 7–3: Maxwell distribution for speed. characteristic quantities are readily inferred: r 2T vm = = most probable speed, m r Z ∞ 8T dv vF (v) = hvi ≡ = mean speed, πm 0 Z ∞ 3T dv v 2 F (v) = hv 2 i = = mean square of velocity. hv 2 i ≡ m 0 (7.35) From the Maxwell distribution it also follows that h 12 mvx 2 i = h 21 mvy 2 i = h 12 mvz 2 i = 12 T . This is an example of the equipartition principle of classical statistical mechanics. Example 7.1. Particle flux density of the perfect gas. Calculate the average number of particles of a Maxwell-Boltzmann gas hitting the wall 7.2. CLASSICAL IDEAL GAS 93 p Gas H2 He N2 O2 CO2 hv 2 i, m/s 1 900 1 350 510 477 407 Root-mean-square velocities of gas molecules at NTP. of the vessel containing the gas per unit time (giving the particle flux) and per unit area of the wall (which is the particle flux density). Let the z axis lie along the outward normal to the wall. Consider particles the z component of the velocity of which is in the interval (vz , vz + dvz ) contained in a straight cylinder perpendicular to the wall with the crosssection area ∆A. The number of these particles hitting the wall during the time interval ∆ is ∆N = n Z∞ dvx Z∞ dvy dvz f (v)vz ∆t∆A −∞ −∞ where f (v) is the probability density (7.32) of the Maxwell distribution and n the particle density of the gas. Thus, the contribution of these particles to the flux density ν is r m −mvz2 /2T ∆N dν = =n e vz dvz ∆A∆t 2πT and the total particle flux density r r Z∞ m T 1 −mvz2 /2T ν=n = nhvi , e vz dvz = n 2πT 2πm 4 0 where hvi is the mean speed of the particles. Example 7.2. Angular distribution in effusion. Calculate the expectation value of the cosine of the angle of flight (measured from the outward normal to the thin wall of the vessel) of molecules of a gas leaking from a vessel through a fine hole. The number of molecules moving through the hole per unit area and unit time in the solid angle determined by the azimuthal angle in the interval (θ, θ +dθ) (from the direction of the outward normal to the hole) is, by virtue of an argument similar to that of the preceding example, Z∞ 1 F (v)v cos θ sin θdv dθ , ν(θ)dθ = n 2 0 94 7. IDEAL EQUILIBRIUM SYSTEMS where F (v) is the probability density for the speed (7.33). Dividing by the flux density of the molecules moving through the hole ν we arrive at the probability density of the departure angle of the molecules f (θ) = 2 sin θ cos θ . Thus, π hcos θi = 2 Z2 sin θ cos2 θ dθ = 2 . 3 0 Example 7.3. Maxwell distribution for relative velocity. Consider a mixture of two perfect gases with molecules of masses m1 , m2 and velocities v 1 , v 2 , respectively. To calculate the probability density for v = v 1 − v 2 , it is convenient to introduce the center-of-mass variables through m1 v 1 + m2 v 2 , m1 + m2 m2 v, v1 = V + m1 + m 2 V = = v1 − v2 v v2 =V − m1 v, m1 + m 2 (7.36) (7.37) with the Jacobi determinant equal to unity. The kinetic energy in these terms is 1 1 1 1 m1 v12 + m2 v22 = (m1 + m2 )V 2 + µv 2 , 2 2 2 2 m m 1 2 where µ = m + m is the reduced mass. The probability density of the rel1 2 ative velocity is thus the Maxwell distribution of a particle with the reduced mass µ. Partition function and thermodynamics. Calculate the singleparticle partition function Z1 (β) as the Laplace transform of the singleparticle density of states ω1 (ε) (5.24). The result is Z1 (β) = Z dε ω1 (ε)e−βε = g X k ~2 k 2 V √ exp −β = g 3 2πmT 3 , 2m h where g is the degeneracy factor of internal degrees of freedom. For pointlike particles it is the spin degeneracy, for molecules it is the partition function of the degrees of freedom of the molecule in the center-of-mass frame. The partition function assumes a suggestive form with the introduction of a characteristic length scale: the thermal de Broglie wave length r h2 λT = . (7.38) 2πmT This quantity gives the spatial spread of the wave packet of the particle in the typical thermal motion. For the oxygen molecule at NTP, e.g., it is 7.2. CLASSICAL IDEAL GAS 95 λT = 0.187 Å and thus clearly less than the diameter of the molecule. The partition function is thus Z1 (β, V ) = g V . λT 3 (7.39) The canonical partition function of a many-particle system ZN (β, V ) may be correspondingly expressed as the Laplace transform of the density of states ωN (E). In the MB gas particles are completely independent, therefore ZN 1 = N! g X !N exp(−βεk ) k = 1 Z1 N , N! (7.40) where N ! removes the classical permutation degeneracy of identical particles. For the free energy FN = −T ln ZN we obtain F (T, V, N ) = N T N 3 3 h2 ln − ln T + ln − 1 − ln g V 2 2 2πm . (7.41) Here, the free energy is a thermodynamically extensive quantity. Without the degeneracy-lifting factor 1/N ! the dependence on the particle number would have been quite wrong. The usual equation of state of the perfect gas immediately follows: p = −∂F/∂V = N T /V . Further, S = −∂F/∂T = −F/T + 3N/2, so that the internal energy is U = F + T S = 32 N T confirming the result obtained in chapter 3.1. The heat capacity is determined by the degrees of freedom of the translation motion, hence f = 3. The explicit expression for the entropy of the MB gas is 3 3 2πm 5 V + ln T + ln 2 + ln g + . (7.42) S = N ln N 2 2 h 2 With the aid of the Gibbs function G = F + pV = µN and the equation of state we arrive at the expression for the chemical potential µ(p, T ) = T ln p − 5 3 h2 ln T + ln − ln g . 2 2 2πm (7.43) The chemical constant is thus 3 ζ = ln g + ln 2 2πm h2 " 3/2 # 2πm = ln g . h2 (7.44) The constant µ0 = 0, because only the kinetic energy of the translational motion is included in the energy of the molecule. 96 7. IDEAL EQUILIBRIUM SYSTEMS Grand canonical partition function. Denote the fugacity z = exp βµ and calculate according to the general rule X 1 X N N N zZ1 βµ gV z ZN = ZG (T, V, µ) = . (7.45) z Z1 = e = exp e N! λT 3 N N The grand potential is Ω(T, V, µ) = −T ln ZG = −T eβµ gV . λT 3 (7.46) The average number of particles is P N z N ZN Ω ∂Ω gV ∂ ln ZG =− = eβµ 3 = − , = N = PN N ∂ ln z ∂µ T z Z λ N T N and the pressure p = −∂Ω/∂V = −Ω/V , so that we arrive at the ideal gas equation of state once more. Validity of the MB gas law. As will be shown later, the MB approximation may be good only if the expectation values of the occupation numbers nℓ obey for all ℓ the condition nℓ ≪ 1. From the single-particle Boltzmann distribution it follows nℓ = e−β(εℓ −µ) . (7.47) The lowest (kinetic) energy is value is 0, therefore eβµ ≪ 1. We have seen that (for g = 1) exp βµ = N λT 3 /V . With the aid of the mean distance between the particles (the radius of the ball containing one particle in the average) ri the condition may be expressed as λT ≪ ri , (7.48) which allows for a simple interpretation: MB approximation is reliable, when the wave packets of the particles do not overlap. Internal degrees of freedom. Even in monatomic gases the degenerP −βε ℓ e may be thermodynamically important due to the acy factor g = ℓ possible fine and hyperfine structure of atomic energy levels. If there is no fine or hyperfine structure (practically this means that the angular momentum is zero), then only the usual temperature-independent spin degeneracy g = 2S + 1 remains. It should be noted first that only the ground state is of concern for gases, because the energy gap between the ground state and the first excited state (without spin-orbit effects) is of the same order as the ionization energy of the atom. Thus, at temperatures which give rise to appearance of atoms in excited states in an appreciable amount, the number of ionized atoms is of the same order and the system ceases to be a monatomic gas. 7.3. DIATOMIC IDEAL GAS 97 Assuming description of the fine structure in terms of the LS coupling, P the factor g may be expressed as g = J (2J + 1) e−βεJ , where the degeneracy with respect to the component of the total momentum J is explicitly taken into account. At low temperatures βεJ ≫ 1 and only the lowest order term survives giving rise to the degeneracy factor g = 2J + 1. At high temperatures βεJ ≪ 1 and the exponents are all close to the unity. Therefore, g approaches the total number of states in the fine structure with given L and S, i.e. g = (2L + 1)(2S + 1). Interaction of the electrons with the nuclear spin gives rise to the hyperfine structure, in which the differences between energy levels, however, are always small compared with the temperature and only give rise to the degeneracy factor 2I + 1, where I is the nuclear spin. This factor is often omitted. 7.3 Diatomic ideal gas Diatomic gases may be homopolar with molecules consisting of two identical atoms like H2 , N2 , O2 etc, or heteropolar like CO, NO, HCl etc. The additional constraints in the former case have to be taken into account. When the internuclear distance is close to the equilibrium distance the energy of the molecule may be expressed as a sum of several independent terms: H = H tr + H rot + H vib + H el + H yd . (7.49) Here, • H tr = p2 = translation energy; m = mass of the molecule. 2m L2 = energy of rotation; rotation about the center of mass. 2I I = the moment of inertia of the nucleai, L = the angular momentum of the molecule (electrons included, this is due to the usual approximation method). The eigenvalues of are the (2ℓ + 1)-fold degenerate energies ~2 ℓ(ℓ + 1). 2I 1 = energy of vibrations. The electronic energy has • H vib = ~ωv n b+ 2 a minimum at the equilibrium distance between the nucleai, around which the harmonic potential is a reasonable approximation. Here, n b is an operator with natural numbers 0, 1, 2,. . . as eigenvalues. Energy levels are non-degenerate. • H rot = • H el = electronic energies due to Coulomb interaction between the electrons in the electric field of the immobile nucleai. Differences be- 98 7. IDEAL EQUILIBRIUM SYSTEMS 4 tween these terms are of the order > ∼ 1 eV ≈ 10 K, so that the electronic term is that of the normal state of the molecule. However, apart from the case of vanishing molecular spin S = 0 and vanishing angular momentum with respect to the axis connecting nucleai Λ = 0 the fine structure of electronic terms must be taken into account. • H yd = energies related to nuclear degrees of freedom. Of interest here are the nuclear spins, which affect the rotational degrees of freedom in case of homopolar molecules. In case of heteropolar molecules their effect is a constant degeneracy factor due to the hyperfine structure. Denote the nuclear spins by I1 and I2 . Then the number of states of the hyperfine structure is gy = (2I1 + 1) (2I2 + 1) . As in case of monatomic gas, this factor is often omitted which is tantamount to redefining the entropy in such a way that it approaches the constant value ln gy instead of zero in the limit T → 0. Near the equilibrium distance of the nucleai there are no significant couplings between the different terms. Therefore, the partition function factorizes to a product of partition functions connected to different degrees of freedom. Calculate first for a single molecule Z1 = p2 ~2 ~ωv 1 gy (2ℓ + 1) exp − − ℓ(ℓ + 1) − n+ 2mT 2IT T 2 n=0 ∞ X ∞ XX p ℓ=0 (7.50) = Z tr Z rot Z vib Z yd . The factors are Z tr = X p Z rot = p2 exp − 2mT ∞ X ℓ=0 Z vib = V ; = λT 3 λT = Tr (2ℓ + 1) exp − ℓ(ℓ + 1) ; T r Z yd = gy = (2I1 + 1) (2I2 + 1) . (7.51) ~2 2I (7.52) Tr = −1 1 Tv ~ωv n+ = 2 sinh ; exp − T 2 2T n=0 ∞ X h2 2πmT Tv = ~ωv (7.53) (7.54) Due to the indistinguishability of the molecules, in the partition function of N the division by the factor N ! has to be introduced (Chapter 5.2), so that ZN (T, V ) = 1 Z1 (T, V )N . N! (7.55) 7.3. DIATOMIC IDEAL GAS 99 This factor is most conveniently related to the traslational motion. the free energy F = −T ln ZN is split to terms N 1 3 2πmT V F tr = −T ln + ln + 1 , Z tr = −N T ln N! N 2 h2 (∞ ) X Tr rot (2ℓ + 1) exp − ℓ(ℓ + 1) , F = −N T ln T ℓ=0 Tv F vib = N T ln 2 sinh , 2T F yd = −N T ln gy . Then (7.56) (7.57) (7.58) (7.59) With the use of relations U = F + T S = F − T ∂F/∂T = −T ∂(F/T )∂T it is readily seen that the translational degrees of freedom yield the ideal gas result 3 3 (7.60) U tr = N T → CVtr = N. 2 2 Since only F tr depends on the volume V , also for the diatomic gas the ideal gas equation of state follows p=− ∂F NT = ∂V V → 2 pV = N T . (7.61) Rotation. The temperature parameter related to rotation Tr is always clearly less than the room temperature. The partition function allows for analytic result in both limits T ≪ Tr and T ≫ Tr . r r ⇒ F rot ≈ −3N T exp − 2T ⇒ U rot ≈ T ≪ Tr : Z rot ≈ 1 + 3 exp − 2T T T r 6N Tr exp − 2T . This leads to the heat capacity T 2 2Tr Tr exp − (7.62) CVrot ≈ 12N −→ 0. T T T →0 R∞ T ≫ Tr : Z rot ≈ ℓ=0 dℓ (2ℓ + 1) exp − TTr ℓ(ℓ + 1) = ⇒ U rot ≈ N T . In this limit the heat capacity is CVrot ≈ N . T Tr ⇒ F rot ≈ −N T ln TTr (7.63) In the limit T ≫ Tr the molecule thus has two degrees of freedom: rotations about the x and y axes. Approximative calculation between the limits reveals a weak maximum of CVrot , see Fig. 7–4. Vibration. The temperature parameter of vibrations Tv is usually much higher than the room temperature. In the limit T ≪ Tv on F vib ≈ 12 N Tv − N T exp − TTv . The heat capacity becomes 2 Tv Tv vib 2 2 exp − . (7.64) CV = −T ∂ F/∂T −→ N T T T →0 100 7. IDEAL EQUILIBRIUM SYSTEMS Gas H2 N2 NO O2 HCl Cl2 Tr (K) 85.4 2.9 2.4 2.1 15.2 0.36 Tv (K) 6100 3340 2690 2230 4140 Parameters of diatomic gases In the limit T ≫ Tv in turn F vib ≈ N T ln rise to the heat capacity CVvib ≈ N . Tv T ⇒ U vib ≈ N T , which gives (7.65) Here we see two effective degrees of freedom as well, viz. translational motion and potential energy. Vibrational degrees of freedom become excited at relatively high temperatures; at low temperatures the thermal energy of order T is insufficient for this. The temperature dependence of the heat capacity is depicted in Fig. 7–4. CV/N 7/2 5/2 3/2 huoneenlämpötila Tr ionisaatio, dissosiaatio ym. Tv T Figure 7–4: Heat capacity of diatomic gas. Rotation of homopolar molecule. The rotational partition function used above is valid for heteropolar molecules only. In case of identical nucleai symmetry requirements imposed on the nuclear wave function have to be taken into account. Consider the simple (and practically most important) example of H2 . The spin of the nucleus ia I = 12 . Thus, the total spin of the protons of H2 (I1 = I2 = 12 ) may be 0 or 1; the two cases are: Ortohydrogen: I = 1, Iz = −1, 0, 1. Triplet, symmetric spin wave function. Parahydrogen: I = 0, Iz = 0. Singlet, antisymmetric spin wave function. Since protons are identical fermions, their wave function must be antisymmetric. Permutation of the coordinate variables of the protons corresponds 7.4. STATISTICS OF BOSONS AND FERMIONS 101 to the change of sign of the relative position vector r = r 1 − r 2 . With respect to this operation the parity of rotation states (described by the spherical harmonics Yℓm ) follows the parity of the quantum number ℓ. Thus, the rules of calculation are: X Tr (2ℓ + 1) exp − ℓ(ℓ + 1) , Ortohydrogen, I=1: Zorto = T ℓ=1,3,... X Tr (2ℓ + 1) exp − ℓ(ℓ + 1) , Parahydrogen, I=0: Zpara = T ℓ=0,2,... The partition function of the equilibrium system is thus Z rot−yd = 3Zorto + Zpara . (7.66) At high temperature molecular collisions lead to frequent enough conversions between the orto and parastates to bring about thermodynamic equilibrium. Then the partition function (7.66) is applicable. For T ≫ Tr we see that Zorto = Zpara and thus all four spin states are equally probable. At low temperature T ≪ Tr the parahydrogen is dominant: Zpara ≫ Zorto . In may happen, however, that in the cooling the orto-para conversion does not occur often enough, and the gas stays as a metastable mixture of orto and paragases with the spin populations in the ration 3:1. The only example of this phenomenon in practice is the hydrogen gas. Instead of the equilibrium partition function (7.66) for the N -particle system the partition function rot−yd ZN,meta = Zorto 3N/4 Zpara N/4 (7.67) must be used, since the numbers of the orto and paramolecules are 3N/4 and N/4, respectively. Due to factorization separate additive contributions to the internal energy follow: rot−yd = Umeta 3 1 Uorto + Upara ; 4 4 with the corresponding heat capacity rot−yd CV,meta = 1 3 Corto + Cpara . 4 4 (7.68) This is an example of non-ergodic behaviour of a macroscopic system, when the approach to the global thermodynamic equilibrium is hindered by dynamic reasons or the relaxation time to arrive at equilibrium is too long. 7.4 Statistics of bosons and fermions Bose–Einstein and Fermi–Dirac statistics. Relativistic quantum mechanics shows that there is a deep connection between the spin os a particle and its statistics. For identical particles with an integer spin s = 0, 1, 2, . . . the many-particle wave function must be symmetric under permutations. In case of a half-integer spin s = 21 , 32 , 25 , . . . an antisymmetric 102 7. IDEAL EQUILIBRIUM SYSTEMS wave function is required. In the former case the particles are bosons, obeying Bose–Einstein statistics; in the latter fermions, obeying Fermi–Dirac statistics. Further, the abbreviations BE- and FD-statistics will be used. Enumerate the one-particle states with the label ℓ. For non-interacting particles many-particle wave functions are constructed as symmetrized (bosons) or antisymmetrized (fermions) products of one-particle wave functions. Due to the (anti)symmetrization it does not matter which arguments are prescribed to each one-particle wave function initially. What matters is the number of wave functions corresponding to a given state in the set of one-particle functions with the aid of which the many-particle wave function is constructed. This set is thus unambiguously described by the occupation numbers nℓ of the one-particle states. The distribution of particles to these states is illustrated in Fig. 7–5. By construction of the wave function, to each set of occupation numbers n1 , n2 , n3 . . . , nℓ , it corresponds exactly one (and thus non-degenerate) many-particle wave function. Therefore, summation over all microstates in the partition function is tantamount to summing over all possible occupation numbers of one-particle states, which are 0 and 1 for fermions and any non-negative integer for bosons. ε ε ε3 ε4 ε1 ε0 ε1 ε0 Figure 7–5: Occupation scheme of one-particle states for (a) bosons, and (b) fermions. The energy eigenvalue in the corresponding many-particle basis state |{nℓ }i = | n1 , n2 , n3 . . . , nℓ , i of the Fock space is E= X εℓ nℓ . (7.69) ℓ Only the grand canonical partition function may be readily calculated. Since no constraints are imposed on the occupation numbers in this case, all combinations of allowed one-particle state occupations are included in the partition function. This means that the sum over the occupation numbers of each state ℓ is independent of others. For a system of bosons we 7.4. STATISTICS OF BOSONS AND FERMIONS 103 obtain ZG,BE = X {nℓ } = " exp −β "∞ Y X ℓ e X ℓ # nℓ (εℓ − µ) = −βn(εℓ −µ) n=0 # = Y ℓ ∞ ∞ X X n1 =0 n2 =0 ··· 1 1− e−β(εℓ −µ) Y ℓ exp [−βnℓ (εℓ − µ)] (7.70) , where µ is the chemical potential. In a fermionic system the Pauli exclusion rule allows for no more than a single-particle occupancy, therefore " # " 1 # 1 1 X X X Y X −βn(εℓ −µ) ZG,F D = · · · exp −β nℓ (εℓ − µ) = e n1 =0 n2 =0 = Yh 1+e −β(εℓ −µ) ℓ i ℓ ℓ n=0 (7.71) . In either case the probability in the ensemble of a many-particle quantum state |{nℓ }i is # " X 1 nℓ (εℓ − µ) . (7.72) exp −β P ({nℓ }) = ZG ℓ From now on both the bosonic and fermionic results are exposed parallelly, because they differ in a couple of signs only. In the formulae the upper signs refer to the boson and lower signs to the fermion statistics. The grand canonical partition function gives rise to the grand potential BE FD Ω(T, V, µ) = ±T X ℓ n o ln 1 ∓ e−β(εℓ −µ) , (7.73) which depends, apart from the explicitly indicated variables T, V, µ, on the one-particle energy spectrum εℓ . The derivative with respect to εℓ is the expectation value of the occupation number ∂Ω ∂εℓ ∂ ln ZG ∂εℓ # " X 1 X nk (εk − µ) = hnℓ i = nℓ nℓ exp −β ZG = −T = (7.74) k {nk } according to the definition of the probability density (7.72). Calculation of this derivative from the expression (7.73) for Ω yields BE FD nℓ = 1 eβ(εℓ −µ) ∓1 . (7.75) The occupation number of a Fermi gas fulfils the condition 0 ≤ nℓ ≤ 1. Distribution of the occupation numbers heavily depend on the chemical potential µ. With the usual normalization of the one-particle energy levels 104 7. IDEAL EQUILIBRIUM SYSTEMS with vanishing ground-state energy ε0 = 0 in an ideal boson system µ ≤ 0. The chemical potential of a fermion system may have both signs, but it is often positive and even µ ≫ T , and then the degenerate fermion system exhibits strong quantum effects. Thermodynamic variables. According to section 2.6 the differential of the grand potential is dΩ = −S dT − p dV − N dµ. Thus, the entropy is i 1X X h ∂Ω S=− ln 1 ∓ e−β(εℓ −µ) + =∓ nℓ (εℓ − µ) . ∂T µ T ℓ ℓ This result may be written completely in terms of the occupation numbers. Since 1 ± nℓ 1 ± 1 ⇒ β (εℓ − µ) = ln , eβ(εℓ −µ) = nℓ nℓ the entropy assumes the form X BE {± (1 ± nℓ ) ln (1 ± nℓ ) − nℓ ln nℓ } . S= (7.76) FD ℓ The average particle number N and the energy E (= the internal energy U ) may be obtained from the thermodynamic relations N = −∂Ω/∂µ and E = Ω + T S + µN = Ω − T (∂Ω/∂T ) − µ(∂Ω/∂µ), or from the definition of expectation values as X N =N = nℓ , (7.77) E= X ℓ (7.78) n ℓ εℓ . ℓ Further results depend on the structure of the one-particle spectrum. Translation-invariant ideal quantum gas. In translation-invariant case plane waves in a box may be used as the spatial wave functions. For non-relativistic particles the spectrum is εℓ = p2 /2m and in the thermodynamic limit the sum over one-particle states may be replaced by the integral (see section 5.2) Z ∞ X X ··· = dε ω1 (ε) · · · , (7.79) ··· = g ℓ where ω1 (ε) = C1 V √ 0 k ε and C1 = 2πg 2m h2 32 . (7.80) Here, g is the spin degeneracy factor (or the partition function of internal degrees of freedom). The particle number, energy and grand potential then are, according to relations (7.77), (7.78) and (7.73), Z ∞ √ 1 BE N = C V dε ε β(ε−µ) , (7.81) 1 FD e ∓1 0 7.4. STATISTICS OF BOSONS AND FERMIONS BE FD BE FD E = C1 V Z ∞ dε 0 Ω = ±C1 V T Z ∞ dε √ 0 105 ε3/2 eβ(ε−µ) ∓1 , h i ε ln 1 ∓ e−β(ε−µ) . (7.82) (7.83) Integration by parts in the last relation and comparison with (8.4) allows to relate the grand potential and the internal energy as 2 Ω=− E, 3 (7.84) which immediately gives rise to the equation of state in the form pV = 2 E. 3 (7.85) Here, the energy is expressed as a function of V , T and µ. Therefore, to arrive at the equation of state in the usual variables p = p(N/V, T ), the chemical potential µ = µ(N/V, T ) has to be solved from the ”normalization condition” (7.81). For small occupation numbers nℓ ≈ e−β(εℓ −µ) ≪ 1 the grand potential may be calculated explicitly with the result Ω ≈ −eµ/T gV T 5/2 (2πm)3/2 , h3 which is exactly the same expression as that was obtained for the MB gas (7.46) and confirms most directly and irrevocably the correctness of the counting of states with the aid of the normalization factor N ! for the MB gas. Adiabatic equation of state of ideal quantum gas. With the aid of the change of variables ε = T z it follows from the integral representations (7.81) - (7.83) that µ , (7.86) Ω = −pV = V T 5/2 f T µ ∂Ω , (7.87) = V T 3/2 g S=− ∂T T ∂Ω µ N =− , (7.88) = −V T 3/2 f ′ ∂µ T where the explicit form of the functions f and g is unimportant, although integral representations may be constructed in an obvious way, e.g., from relation (7.83). Consider fixed N and a reversible adiabatic process. Then the entropy S must be constant as well. From relations (7.87) and (7.88) it then follows that the ratio S/N is a function of the ratio µ/T only. Hence, in a reversible adiabatic process the ratio µ/T is also a constant. From equation (7.86) we 106 7. IDEAL EQUILIBRIUM SYSTEMS then infer that in the adiabatic process of a quantum ideal gas the expression pT −5/2 remains constant and thus arrive at the adiabatic equations of state for translation-invariant ideal BE and FE gas: −5/2 pT −5/2 = p0 T0 , 3/2 pV 3/2 = p0 V0 3/2 V T 3/2 = V0 T0 , . Example 7.4. Fluctuation of the occupation number and the particle number of boson gas. Let us calculate ∆n2ℓ = h(nℓ − nℓ )2 i in a boson gas. By definition P P ℓ nℓ (εℓ − µ)] {nℓ } nℓ exp [−β P . (7.89) nℓ = P ℓ nℓ (εℓ − µ)] {nℓ } exp [−β Differentiating with respect to εℓ we obtain ∂nℓ = β n2ℓ − hn2ℓ i . ∂εℓ On the other hand we have seen that nℓ = 1 , eβ(εℓ −µ) − 1 (7.90) whose derivative is ) ( 1 1 ∂nℓ 2 = −β + 2 = −β nℓ + nℓ . β(ε −µ) ∂εℓ eβ(εℓ −µ) − 1 e ℓ −1 Thus, the fluctuation of the occupation number is given by the relation h(nℓ − nℓ )2 i = nℓ + n2ℓ . Fluctuation of the particle number is !2 + * X X 2 2 ∆N = h N − N i = [hni nj i − ni nj ] . (ni − ni i,j i To calculate the correlation function of the occupation number, differentiate (7.89) with respect to εm : ∂nℓ = β (nℓ nm − hnℓ nm i) . ∂εm On the other hand, the derivative of (7.90) yields therefore ∂nℓ = −βδℓm nℓ + n2ℓ , ∂εm h(ni − ni ) (nj − nj )i = δij ni + n2i and the fluctuation of the particle number obeys D 2 E X nℓ + n2ℓ . = N −N ℓ 7.5. PROBLEMS 107 7.5 Problems Problem 7.1. Show that for the heat capacity of a system of noninteracting magnetic moments with the total angular momentum J the relation CH ≡ T ∂S ∂T = H J(J + 1) V µ0 H 2 χH = N T 3 µB gB T 2 holds in the weak-field limit (µB gB ≪ T ). Here, µB is the Bohr magneton and g the Lande factor. Problem 7.2. A paramagnet in one dimension can be modelled as a linear chain of N + 1 spins. Each spin interacts with its neighbours in such a way that the energy is E = nǫ, where n is the number of domain walls separating regions of up spins from regions of down spins. Calculate the entropy S(E) with the use of the statistical weight of the macrostate with the energy E = nǫ, and show that in the limit of both n and N large the energy may be expressed as E= Nǫ . eǫ/T + 1 Problem 7.3. Consider the model of one-dimensional paramagnet as a linear chain of N + 1 spins interacting in such a way that the energy is E = nǫ, where n is the number of domain walls separating regions of up spins from regions of down spins. Calculate the partition function, free energy and entropy of this system in the canonical ensemble. Problem 7.4. A monatomic gas at the temperature T is contained in a vessel from which it leaks through a fine hole. Show that the average kinetic energy of the molecules leaving through the hole is 2T . Problem 7.5. In a thin-walled vessel of volume V there are N0 molecules of a perfect gas. At the time instant t = 0 the gas begins to leak out through a tiny hole of area A. Assuming that the outside pressure is negligible, calculate the number of molecules in the vessel as a function of time t. Problem 7.6. Calculate hv 2 i and the average speed h|v|i in a gas whose center of mass is moving at the velocity V , i.e. the velocity distribution is 1 2 f (v) ∝ exp − m(v − V ) . 2T Problem 7.7. a) From spectroscopy it is known that the nitrogen molecule has vibrational excited states with energy levels En = ~ω(n + 12 ). If the level separation ~ω = 0.3 eV, what is then the ratio of the number of molecules in the first excited state (n = 1) and in the ground state (n = 0), when the gas is in thermodynamic equilibrium at the temperature 1000◦ K? 108 7. IDEAL EQUILIBRIUM SYSTEMS b) The energy gap between the ground state and the first excited state of a helium atom is 19.82 eV. Assuming nondegenerate ground state and degenerate excited state with g = 3 estimate the relative frequency to find atoms in these states in a gas at the temperature 10 000 K? Problem 7.8. Derive the following results for the fluctuations of a perfect FD gas: h(nl − nl )2 i = h(N − N ) i = 2 nl (1 − nl ) X l nl (1 − nl ) . 8. Bosonic systems 8.1 Bose gas and Bose condensation Perfect Bose-Einstein gas. In case of translational invariant ideal boson system with the non-relativistic dispersion law εℓ = p2 /2m in the thermodynamic limit the sum over one-particle states may be replaced by the integral Z ∞ X X ··· = dε ω1 (ε) · · · , (8.1) ··· = g ℓ √ where ω1 (ε) = C1 V ε and 0 k C1 = 2πg 2m h2 32 . (8.2) The particle number, energy and the grand potential are then Z ∞ √ 1 , (8.3) N = C1 V dε ε β(ε−µ) e −1 0 Z ∞ ε3/2 , (8.4) E = C1 V dε β(ε−µ) e −1 0 Z ∞ Z ∞ h i √ 2 ε3/2 dε ε ln 1 − e−β(ε−µ) = − C1 V Ω = C1 V T dε β(ε−µ) . (8.5) 3 e −1 0 0 In the limit of dilute gas (n = N/V → 0, when also z = eβµ → 0) these quantities coincide with those of the Maxwell–Boltzmann ideal gas so that in this region the ideal Bose gas obeys the equation of state of the classical ideal gas. Corrections to the MB limit may be conveniently obtained, when the functions (8.3), (8.4) and (8.5) are expressed as series in the fugacity z. Consider, for instance, the integral for the grand potential: Z ∞ Z ∞ ε3/2 2 ε3/2 ze−βε 2 dε βε −1 = − C1 V dε . Ω = − C1 V 3 e z −1 3 1 − e−βε z 0 0 The denominator of the integrand gives rise to geometric series, and having changed the order of integration and summation we arrive at readily calculable integrals of the form Z ∞ T 5/2 Γ(5/2) . dε ε3/2 e−βε(n+1) = (n + 1)5/2 0 109 110 8. BOSONIC SYSTEMS We thus obtain 2 Ω = − C1 Γ 3 ∞ X 5 zn V zT 5/2 . 2 (n + 1)5/2 n=0 The series here is one definition of a less known special function, the polylogarithm Liν : ∞ X zn Liν (z) = . (8.6) nν n=1 From this series of unit radius of convergence we immediately see the connection with the Riemann ζ function: (8.7) Liν (1) = ζ(ν) . From the calculation of the integral for the grand potential the following generalization is readily inferred: Liν (z) = 1 Γ(ν) Z ∞ 0 dt tν−1 , −1 et z −1 (8.8) which allows for analytic continuation to values Re z < 1. The polylogarithm Liν (z) has a branching point at z = 1, and the corresponding branch cut is usually put on the real axis from z = 1 to z = ∞. Thus, the fugacity expansion of the grand potential may be compactly written as 5 2 V T 5/2 Li 52 (z) . (8.9) Ω = − C1 Γ 3 2 Similarly 3 V T 3/2 Li 32 (z) . N = C1 Γ 2 (8.10) Quantum corrections to the MB limit of the boson gas may be expressed in more transparent variables by solving for the chemical potential µ = µ(n, T ) from (8.10) and substituting in (8.9) to write the grand potential as a function of the volume, particle density and the temperature. As a matter of fact, this is the procedure for construction of the virial expansion for the BE gas, since Ω = −pV . Bose–Einstein condensation. In the limit of dense matter properties of the ideal BE gas differ dramatically from those of the classical ideal gas. Since in stable matter ∂N /∂µ > 0 and µ ≤ 0, the particle number N at given temperature approaches its maximum value in the limit µ → 0− . Denote this maximum value N1 (T ): N1 (T ) = C1 V Z 0 ∞ √ ε = C1 V T 3/2 Γ dε βε e −1 √ 3 3 3 π ζ = ζ C1 V T 3/2 , 2 2 2 2 8.1. BOSE GAS AND BOSE CONDENSATION 111 where ζ(1.5) ≈ 2.612. The maximum particle density is thus √ N1 (T ) π 3 n1 (T ) = = AT 3/2 ; A = ζ C1 . V 2 2 (8.11) For fixed particle density n direct calculation yields ∂µ ∂T n =− 1 T Z ∞ ε1/2 (ε − µ)eβ(ε−µ) (eβ(ε−µ) − 1)2 0 < 0. Z ∞ ε1/2 eβ(ε−µ) dε β(ε−µ) (e − 1)2 0 dε revealing that the condition n < n1 (T ) may be fullfilled only above the 3/2 critical temperature Tc determined by the condition ATc = n as h2 Tc = 2πm n gζ 3 2 ! 32 . (8.12) When the temperature is further lowered, the chemical potential cannot grow any more. In the region T < Tc the Bose condensation takes place meaning that a macroscopic portion of the particles occupies the lowestenergy one-particle state ε0 = 0. In this region in the sum over oneparticle states (8.1) the continuum approximation becomes inapplicable. The lowest-energy state must be left outside the integration as a separate term. The correct rule is Z ∞ X dε ω1 (ε) · · · . (8.13) · · · = (term ℓ = 0) + 0 ℓ In a system quantized in a finite volume the energy levels εℓ are discrete and differences between low-lying levels ∝ V −2/3 . When µ approaches the energy ε0 = 0 from below, the occupation number n0 = 1 e−βµ −1 ⇒ N0 (T ) becomes arbitrarily large. The continuum integral (8.3) represents – up to extensive terms – correctly the number of the excited particles (ℓ 6= 0), so that in the region T < Tc on N = N0 + AV T 3/2 . With the use of the result (8.11) we obtain 23 T N1 =n V Tc " (8.14) 32 # N0 T =n 1− V Tc These functions are illustrated in Fig. 8–1. Bose–Einstein gas is an ex- 112 8. BOSONIC SYSTEMS N N1 N 0 µ (< 0) N0 T µ Figure 8–1: Particle numbers and chemical potential of ideal Bose gas as functions of temperature in a fixed volume. ceptional example of a system, in which the grand canonical ensemble is not statistically equivalent to the canonical ensemble. In the condensate phase the grand-canonical fluctuation of the particle number is macroscopic ∆N/N ∝ 1. This is clearly unphysical, so that in principle it would be more correct to use the canonical ensemble. This may practically effected – up to extensive terms – by putting the chemical potential equal to zero in the mean occupation number of the excited states and replacing thereafter the sum over excited states by the continuum integral. Direct contributions of the condensate drop out from the thermodynamic potentials. For instance, the internal energy is determined by the excited particles only Z ∞ Z ∞ 5 5 x3/2 ε3/2 = C1 V T 5/2 = C1 V T 5/2 Γ ζ . dx x E = C1 V dε βε e − 1 e − 1 2 2 0 0 From here the heat capacity immediately follows 5 5 ∂E 5 3/2 = C1 V T Γ ζ . CV = ∂T V 2 2 2 Integration of the expression for the heat capacity as the derivative of the entropy allows, together with the normalization due to the third law, to find the entropy as well: ∂E 5 3 5 3/2 S= ζ . = C1 V T Γ ∂T V 5 2 2 Finally, the equation of states follows from the observation that in the extensive approximation F = Ω = − 32 E and ∂F 5 5 2 p=− ζ . = C1 T 5/2 Γ ∂T V 3 2 2 The pressure is thus independent of the volume! Physically this means that any change in the volume is accompanied by such a change in the number of 8.1. BOSE GAS AND BOSE CONDENSATION 113 particles in the condensate that the pressure remains the same (at constant temperature). For instance, an isothermal compression would lead to the growth of N0 , but N1 /V would remain constant together as the pressure p. There is no energy, entropy or pressure related to the particles in the condensate state ε0 = 0. The phase diagrams of the matter in the Vp and Tp planes are depicted in Fig. 8–2. Due to infinite compressibility a part of the Tp plane is completely excluded. p normaali faasi p T3 > T2 > T1 poissuljettu alue kondensoitunut normaali faasi kondensoitunut faasi V T Figure 8–2: Phase diagrams of the ideal Bose gas. Bose condensation in alkali vapours. The Bose-Einstein condensation was experimentally observed in 1995 in alkali vapours (87 Rb, 23 Na, 7 Li) laser cooled below microkelvin temperatures. The cooling took place in magneto-optical traps, in which the potential energy of the magnetic moments of the atoms in a radio-frequency alternating magnetic field may be described by a harmonic potential V (x, y, z) = 1 1 1 k1 x2 + k2 y 2 + k3 z 2 . 2 2 2 (8.15) The system is thus inhomogeneous and the Bose condensation shows not only in the momentum space but also in the real space. The genuine Bose condensation requires a continuum spectrum of the one-particle states starting from the ground state. In the boson gas in a box this is achieved in the usual thermodynamic limit: N, V → ∞ with constant N/V . In the harmonic potential the corresponding limit is N → ∞, ωi → 0 with N ω1 ω2 ω3 fixed, i.e. the harmonic potential becomes flat. However, differences in the approach to the flat full space lead to physical results different from those obtained in the thermodynamic limit of a gas initially enclosed in a rectangular box, which emphasizes the sensitivity of the thermodynamic limit to the way it is accomplished. For instance, relations (8.14) are replaced by 114 8. BOSONIC SYSTEMS N1 = N " T Tc 3 N0 = N 1 − T Tc (8.16) 3 # with different powers of the temperature. To see this, let us calculate the density of states of a boson in the harmonic potential (8.15). The eigenvalues of the Hamilton operator are given by 1 1 1 εn1 ,n2 ,n3 = ~ω1 n1 + + ~ω2 n2 + + ~ω3 n3 + , (8.17) 2 2 2 where the ni = 0, 1, 2 . . . are integers labeling the eigenfunctions. Expectation values are calculated over all values of ni weighed by the mean occupation number. In the thermodynamic limit it would be desirable to arrive at an integral over the one-particle energies. Let us first introduce the variables u = ~ω1 n1 , v = ~ω2 n2 , w = ~ω3 n3 , for which the difference between adjacent values ~ωi becomes small in the limit ωi → 0, thus allowing for representation of the sum over ni as an integral sum: X n1 ,n2 ,n3 Z∞ Z∞ Z∞ X 1 1 ∆u∆v∆w −→ 3 du dv dw . = 3 ~ ω1 ω2 ω3 u,v,w ~ ω1 ω2 ω3 0 0 0 The average number of particles, for instance, may be expressed as N= X n1 ,n2 g gn (εn1 ,n2 ,n3 ) −→ 3 ~ ω 1 ω2 ω3 ,n 3 Z∞ Z∞ Z∞ du dv dw 0 0 1 eβ(u+v+w+ε0 −µ) 0 −1 , (8.18) where ε0 = ~2 (ω1 + ω2 + ω3 ) is the energy of zero-point oscillations. This relation suggests that the thermodynamic limit – in analogy with the gas in the box – corresponds to N → ∞, ωi → 0 such that the product N ω1 ω2 ω3 is kept constant. Introduce then the variables u = x2 , v = y 2 , w = z 2 in which it is evident that the the integral over one-particle excitation energies may be obtained in the spherical polar coordinates, since the mean occupation number depends on x2 + y 2 + z 2 = r2 only: 8g N= 3 ~ ω1 ω2 ω3 Z∞ Z∞ Z∞ dx dy dz 0 0 0 xyz . eβ(x2 +y2 +z2 +ε0 −µ) − 1 8.2. BLACK BODY RADIATION 115 After calculation of the angular integrals and introduction of the excitation energy ε′ = r2 as the new radial integration variable we arrive at the expression Z∞ g 1 N= 3 (8.19) dε′ ε′2 β(ε′ +ε −µ) 0 2~ ω1 ω2 ω3 e −1 0 showing that the density of states of the BE gas in the harmonic potential well is 2 g (ε − ε0 ) . (8.20) ω1 (ε) = 3 2~ ω1 ω2 ω3 The condensation temperature is now determined by the condition µ = ε0 as 1 N ω 1 ω2 ω3 3 . (8.21) Tc = ~ gζ(3) As for the gas in the box, from relations (8.19), (8.21) the temperature dependence of the number of particles is inferred in the form (8.16) quoted above. The thermodynamics of the BE gas in the harmonic well below Tc is also determined by particles above the condensate only. For instance, the internal energy assumes the form g E = N0 ε 0 + 3 2~ ω1 ω2 ω3 Z∞ ε′ + ε 0 π4 T 4 dε′ ε′2 βε′ = N ε0 + . e −1 30~3 ω1 ω2 ω3 (8.22) 0 8.2 Black body radiation Quantization of the energy of electromagnetic radiation. Consider a free electromagnetic field with vanishing scalar potential φ = 0 and the vector potential A subject to the transversality condition ∇ · A = 0. Then the fields are E=− ∂A , ∂t B = ∇ × A, (8.23) and the Maxwell equations give rise to the wave equation for A ∇2 A − 1 ∂2A = 0. c2 ∂t2 Put the system in a box of volume V and express the vector potential as the linear combination of plane waves (normalized to unity in the volume V ) X e−iωt+ik·r eiωt−ik·r √ , (8.24) + a∗k √ ak A= V V k where k · ak = 0 and ak ∝ e−iωt with the dispersion law ω(k) = c|k|. The sum in (8.24) is taken over all integers labeling the components of the wave 116 8. BOSONIC SYSTEMS 1 , 2πn2 , 2πn3 . Introduce the canonical vector in the box, where k = 2πn L L L x variables y z ε0 ak e−iωt + a∗k eiωt , √ P k = Q̇k = −iω ε0 ak e−iωt − a∗k eiωt . Qk = √ (8.25) Expressing the coefficients of the Fourier expansion of the vector potential in terms of the canonical variables 1 Pk −iωt ak e = √ Qk + i 2 ε0 ω 1 P k a∗k eiωt = √ , (8.26) Qk − i 2 ε0 ω substituting in the expression for the Hamilton function of the electromagnetic field becomes Z ε0 E 2 B2 H = d3 r + , 2 2µ0 resolving the wave-vector sums with the aid of the identity Z ′ d3 reir·(k−k ) = V δn1 n′1 δn2 n′2 δn3 n′3 and the transversality condition k · ak = 0 we arrive at the the Hamilton function of the electromagnetic field in the form of the sum of Hamilton functions of independent harmonic oscillators: Z B2 1X ε0 E 2 P 2k + ω 2 Q2k . + = H = d3 r 2 2µ0 2 k Due to transversality k·ak = 0 the vectors P k and Qk have only two components corresponding the two independent polarization modes. Therefore, in the box the Hamilton function of the electromagnetic field in the canonical variables may be written as 1X 2 H= Pk,λ + ω 2 Q2k,λ , (8.27) 2 k,λ where λ labels the polarization components. This representation suggests quantization with the introduction of the momentum and coordinate operators with the standard commutation rules i h b k′ ,λ′ = −iℏδλ,λ′ δk,k′ . Pbk,λ , Q (8.28) Due to the orthogonality of plane waves with different wave vectors the Hamilton operator assumes exactly the same form as the Hamilton function with canonical variables replaced by operators: X 2 b2 b =1 Pbk,λ + ω2 Q (8.29) H k,λ . 2 k,λ 8.2. BLACK BODY RADIATION 117 The energy eigenvalues are thus E= X k,λ 1 ℏω nk,λ + , 2 (8.30) where nk,λ = 0, 1, 2, . . . for each set k, λ. Similarly, the expression for the momentum of the electromagnetic field Z 1 P = d3 rE × B µ0 c in terms of the canonical variables (8.25) gives rise to the momentum operator in the form X k 2 b =1 b 2k,λ , P (8.31) Pbk,λ + ω2 Q 2 ck k,λ whose eigenvalues are also expressed with the aid of the occupation numbers nk,λ as X 1 ℏk nk,λ + P = . (8.32) 2 k,λ The representations (8.30) and (8.32) allow for the interpretation of the excitations of the quantized electromagnetic field as photons of energy ε = ℏω and momentum p = ℏk with the dispersion law ε = cp. The quantity nk,λ then is the number of photons of frequency ω, direction of propagation k/k and polarization state λ. Note that this involves a new physical interpretation of the excited states of the harmonic oscillator, each of which now represents nk,λ particles with the energy ε = ℏω, momentum p = ℏk and the polarization vector eλ . Since the occupation number nk,λ = 0, 1, 2, . . . may be any non-negative integer, the photons obey Bose-Einstein statistics. The polarization state is a degree of freedom corresponding to spin, whose value is actually 1, but the longitudinal component is absent in the radiation field and the spin degeneracy factor is g = 2. In the description of radiation the zero-frequency energy and momentum are usually subtracted from expressions (8.30) and (8.32) and deferred to the definition of the ”vacuum”. The energy of photons may be arbitrarily small, therefore it is not possible to fix their number or even its average as the condition for the construction of a statistical ensemble. Rather, the number of photons is determined as an equilibrium condition for the free energy as ∂F = µ = 0. Therefore, ∂N the mean ”occupation number” of the photons corresponding to an oscillator mode is 1 n(ω) = β~ω . (8.33) e −1 This is the Planck distribution.. The density of states is often determined with respect to the angular frequency. In the thermodynamic limit the sum 118 8. BOSONIC SYSTEMS over wave vector components is first replaced by the integral over the wavevector space with account of the isotropy of the mean occupation number dNk = g(L/2π)3 d3 k = gV /(2π 2 ) k 2 dk . When the condition dNk = dNω = f (ω)dω is then imposed, the density of states with respect to ω is found as: Z ∞ X ω2 dω f (ω) · · · ; f (ω)dω = V 2 3 dω ··· = (8.34) π c 0 ℓ The full energy density of the radiation field now assumes the form Z ∞ E E(T ) ≡ = dω E(ω, T ) , (8.35) V 0 where the spectral energy density E(ω, T ) is given by the Planck radiation law ~ω 3 . E(ω, T ) = 2 3 β~ω (8.36) π c (e − 1) ε(ω,T) ω max ω max ω max T3 T2 T1 ω Figure 8–3: Spectral energy density of the radiation field according to Planck’s radiation law. In Fig. 8–3 the form of the intensity distribution of the radiation according to Planck’s law is depicted. In the limit of large frequency or low temperature this law tends to Wien’s law E(ω, T ) ∝ ν 3 exp(−hν/T ), where ν = ω/2π and h is a fitting parameter. In the limit of small frequency the Rayleigh–Jeans law is recovered E(ω, T ) ∝ ω 2 T , which corresponds to the classical equipartition theorem. In Planck’s radiation law the frequency of the maximum energy density ωmax ∝ T thus explaining Wien’s law for the peak of the spectrum. 8.2. BLACK BODY RADIATION 119 Integration over the frequency in Planck’s radiation law yields Z ∞ Z ∞ x3 ~ω 3 T4 dx x dω 2 3 β~ω = 2 3 3 . E(T ) = π c (e − 1) π c ~ 0 e −1 0 The remaining integral is of the familiar type, thus Z ∞ x3 π4 dx x = Γ(4)ζ(4) = e −1 15 0 leading to the energy density E(T ) = 4 4 σT , c (8.37) where σ is the Stefan–Boltzmann constant σ= π2 . 60~3 c2 (8.38) In the ordinary units σ = π 2 kB 4 /(60~3 c2 ) ≈ 5.671 × 10−8 W/(m2 K4 ). Thermodynamics. Thermodynamics of the photon gas may be deduced from the expression for the energy density (8.37), since the internal energy U = EV , i.e. 4σ U= V T 4. (8.39) c Therefore, the heat capacity is 16 3 ∂S ∂U = . (8.40) σT = T CV = ∂T V c ∂T V Integrating the last equation with the account of the condition S → 0, T → 0 we arrive at the expression for the entropy S= 16σ V T3 . 3c (8.41) Therefore, the free energy F = U − T S is F =− 4σ V T4 . 3c (8.42) Calculation of the pressure p = −∂F/∂V yields the equation of state as p= 1 4σ 4 T = E(T ). 3c 3 (8.43) It is worth noting that the pressure is independent of the volume leading to infinite isothermal compressibility as in the Bose condensate. 120 8. BOSONIC SYSTEMS It should be borne in mind that as a many-particle system photons constitute an extremely ideal gas, because even scattering events between photons are very rare. Therefore, a pure photon gas is a non-ergodic system that cannot thermalize due to intrinsic dynamics. If the photon gas is in equilibrium, which strictly speaking hardly ever is the case, then this is solely due to interaction with a suitable environment, a black body,. The black body is an equilibrium system acting as a heat bath by emitting and absorbing photons without reflection, isotropically in spatial directions and within a continuous spectrum. I cτ θ A Figure 8–4: Black-body radiation and absorption. Emission and absorption of a black body. A black body may be modelled as a hole in the wall of a hollow container with isotropic black-body radiation in it. Then the radiation power in a fixed direction may be calculated with the aid of the geometric construction depicted in Fig. 8–4b. During the time τ the radiation in the direction of the angle θ from the outward normal of the surface comes from the region, whose depth is ℓ(θ) = cτ cos θ and the volume Aℓ(θ). The total energy of the photons propagating from this region in the solid angle element dΩ in the direction θ is therefore E(T )Acτ cos θ dΩ , 4π since of all the photons in a volume element the part dΩ/(4π) is propagating in the required direction. The radiance L is the radiation power in a given direction in a unit solid angle per visible surface area of the radiating body. Since in the direction θ the visible emitting area is A⊥ = A cos θ, the radiance of the black body is (to obtain power, divide energy by τ ) L= c E(T ) . 4π Since L is independent of direction (i.e., the black-body radiation is diffuse), the radiating surface appears equally bright from any oblique direction as from the direction of the normal. Radiant excitance M is the radiant power to the half-space above the surface per surface area. Thus, it may be written as the integral of the 8.3. LATTICE VIBRATIONS 121 radiance multiplied by the ratio cos θ = A⊥ /A over the corresponding solid angle: Zπ/2 Zπ/2 Z2π 1 c E(T ) Mb = dθ sin θ dϕL cos θ = 2π dθ sin θ cos θ = E(T )c . (8.44) 4π 4 0 0 0 Thus, black-body radiation obeys Lambert’s law M = πL,˛. Due to relation (8.37), the Stefan–Boltzmann law for the radiant excitance Mb = σT 4 (8.45) follows from equation (8.44). A black body absorbs all the incident radiation, emits diffuse radiation and its spectral excitance (or spectral radiance) is a universal function depending on the temperature and the angular frequency of the radiation only. Irradiance E is the radiant power incident to the surface per surface area. It may be absorbed (denote the corresponding portion by Ea ), reflected (Er )or transmitted (Et ). Thus, radiant properties of the surface are characterized by the (spectral) absorption ratio (Ea (ω) = α(ω)E(ω)) Ea = αE, reflection ratio Er = ρE and transmission ratio Et = τ E. Energy conservation implies that α + ρ + τ = 1. For a black body α = 1, ρ = τ = 0. Emission of a generic radiating body is characterized by its emissivity (or spectral emissivity) ǫ: M (ω, T ) = ǫ(ω, T )Mb (ω, T ) , where M (ω, T ) is the excitance of the body at hand. The spectral emissivity of a grey body is independent of the frequency. If the body is at equilibrium with the radiation field, then Kirchhoff’s radiation law ǫ(ω, T ) = α(ω, T ) holds. 8.3 Lattice vibrations The relative motion of nucleai in molecules and solids is so slow in comparison with the motion of electrons that these degrees of freedom may be separated in the Born–Oppenheimer or adiabatic approximation. The system of nucleai is then described the Hamilton operator on which the influence of electrons appears as a potential energy depending on the nuclear configuration N X pj 2 + V (r 1 , r 2 , . . . , r N ) , (8.46) H= 2Mj j=1 122 8. BOSONIC SYSTEMS where Mj are the masses of the nucleai (atoms). Denote the equilibrium positions by Rj and displacements thereof uj = r j − Rj . Expanding with respect to the latter we obtain V = V0 + X Vjx ujx + jx 1 X xy Vjk ujx uky + · · · , 2! jk,xy because the linear in displacements terms vanish due to the equilibrium condition. In the harmonic approximation the Hamilton function is H= N X pj 2 1 X X xy Vjk ujx uky . + 2Mj 2! xy j=1 (8.47) jk A Hamilton function of this form may always be diagonalized with the aid of a canonical transformation which leads to the representation in the normal coordinates as a system of independent harmonic oscillators: H= 1X Pℓ 2 + ωℓ 2 Qℓ 2 , 2 (8.48) ℓ where the angular frequencies ωℓ are obtained from the characteristic equation h i xy det Vjk − ω 2 Mj δjk δxy = 0 . In the quantized theory the canonical variables Pℓ , Qℓ are replaced by operators obeying the usual commutation rules (8.28) so that the Hamilton b ℓ in (8.48). Thus, the operator is obtained by the substitution Pℓ , Qℓ → Pbℓ , Q energy eigenvalues may be written as X 1 (8.49) E= ℏωℓ nℓ + 2 ℓ giving rise, as in the case of electromagnetic radiation, to the interpretation of the excited states of the lattice vibrations as a system of particle-like entities, phonons. Exact solution. The set of harmonic oscillators (8.48) with the energy eigenvalues (8.49) gives rise – in analogy with the quantized electromagnetic field – to statistical description in terms of grand canonical ensemble with vanishing chemical potential. Thus, F = Ω and we arrive at the formally exact relation X (8.50) ln 1 − e−βℏωℓ . F =T ℓ This expression gives rise to a simple analytic form in the limits βℏωℓ ≫ 1 and βℏωℓ ≪ 1. 8.3. LATTICE VIBRATIONS 123 In a box with periodic boundary conditions the wave functions may be labeled by the components of the wave vector k and the polarization index λ. Then (8.49) is replaced by X 1 ℏωλ (k) nλ (k) + E= . (8.51) 2 k,λ In a lattice no continuum limit is taken, so the number of components of k remains finite and equal to the number N of elementary cells in the specimen. Roughly speaking, the elementary cell is the smallest volume element of the solid whose periodic continuation to fill the space reproduces the lattice at hand. The elementary cell does not always convey all the symmetry properties of the lattice, therefore it is often replaced by the Wigner-Seitz cell, which obeys the lattice symmetries. The corresponding entity in the wave-vector space is the first Brillouin zone. The number of components of the wave vector is usually restricted to the value N by requiring that all k belong to the first Brillouin zone in the wave-vector space. It should be borne in mind, although it is not used here, that the components of k are not necessarily orthogonal. The elementary cell may contain several atoms. Let the number of atoms in the elementary cell be ν, then there are, in general, 3ν polarization states of the lattice vibrations. Due to translational invariance, there are always three polarization states with the dispersion law ω ∝ k. These are sound waves or acoustic modes of lattice vibrations. If the lattice is not simple, i.e. ν > 1, then optical modes of lattice vibrations appear whose frequency has finite limit, when k → 0. Fig. 8–5 shows typical dispersion laws for a lattice with two atoms in the elementary cell. ω(k) optiset akustiset k 0 Figure 8–5: Dispersion curves of phonons. At low temperatures (βℏωℓ ≫ 1) the exponential cuts off the largeenergy vibrations. The vibration modes with the lowest energy are sound waves. In an isotropic substance – which is assumed for simplicity – the sound waves have three independent vibration modes: one longitudinal and 124 8. BOSONIC SYSTEMS two transversal modes, for which the dispersion relations may be written as ωt = ct k , ωl = cl k , (8.52) which allow for transformation from the wave-vector sum to wave-vector integral in a fashion similar to that used for the photon gas. Since the speed of sound for the transversal components is the same, the energies have a twofold degeneracy with respect to polarization and we obtain, at T → 0: TV F = 2π 2 Z∞ dk k 2 2 ln 1 − e−βℏωt + ln 1 − e−βℏωl 0 TV = 2π 2 2 1 + 3 c3t cl Z∞ dω ω 2 ln 1 − e−βℏω 0 = 3T V 2π 2 c3 Z∞ dω ω 2 ln 1 − e−βℏω , (8.53) 0 where c is the geometric-like mean speed of sound. We see that the free energy is formally the same as for the photon gas, therefore the substitution c → c and g = 3 immediately yields the basic thermodynamic quantities of the phonon gas: F − F0 = − π2 V T 4 , 30(ℏc)3 S= 2π 2 V T 3 , 15(ℏc)3 CV = 2π 2 V T 3 . 5(ℏc)3 (8.54) Here, F0 is the free energy of zero-mode vibrations, which is independent of the temperature. The physically most important conclusion is the temperature dependence CV ∝ T 3 of lattice vibrations, so that the vibrational contribution at low temperatures vanishes faster than the electronic contribution. At high temperatures (βℏωℓ ≪ 1) we approximate 1 − e−βℏωℓ ≈ βℏωℓ and write 3N ν X ℏω F = F0 + ln βℏωℓ = F0 + ln , (8.55) T ℓ where ω is the geometric mean frequency and 3N ν is the number of vibration modes (N is the number of elementary cells in the lattice and ν the number of atoms in the elementary cell). Relation (8.55) gives rise the the heat capacity CV = 3N ν , ℏωℓ ≪ T in accordance with the Dulong-Petit law conforming to the classical equipartition theorem (the average energy of each vibration mode is T ). Debye model. A fairly realistic analytic approximation to the lattice heat capacity of solids is provided by the Debye model, whose basic assumptions are: 8.3. LATTICE VIBRATIONS CV/N 125 Dulong-Petit 3 Debye Einstein θ T Figure 8–6: Lattice heat capacity of a solid. • Take into account only acoustic modes: one longitudinal and two transversal modes. • Assume linear dispersion relations: ( ωl (k) = cl k (8.56) ωt (k) = ct k • Cut off the spectra at the Debye frequency ωD = θD /~ (where θD is the Debye temperature) chosen such that the number of vibrational modes is 3N ν. Density of states in each mode is dNj = L 2π 3 4πk 2 dk = V ω 2 dω , 2π 2 c3j so that the total density of states is 2 1 V + 3 ω 2 dω . dN = 2π 2 c3t cl Z ωD dN , yielding The total number of modes is 3N ν = ω=0 3 ωD Nν = 18π 2 V 2 1 + 3 c3t cl −1 . (8.57) The density of states is thus dN (ω) = 9N ν 2 3 ω dω. ωD (ω < ωD ) (8.58) 126 8. BOSONIC SYSTEMS Substance Li Na K Be Mg Ca * diamond θD (K) 400 150 100 1000 318 230 Substance B Al Ga C* Si Ge θD (K) 1250 394 240 1860 625 360 Substance Cu Ag Au Zn Cr Fe θD (K) 315 215 170 234 460 420 Debye temperatures of some substances In the Debye model the free energy is 9N ν F = F0 + ωD 3 ZωD dω ω 2 ln 1 − e−βℏω . (8.59) 0 Stretching integration variable we obtain F = F0 + 9N νT T θD 3 θZD /T dz z 2 ln 1 − e−z . (8.60) 0 From this expression the internal energy follows in the form E = F + T S = E0 + 3N νT T θD 3 θZD /T 0 z 3 dz = E0 + 3N νT D ez − 1 T θD , (8.61) where the last equality defines the Debye function D(x). In the Debye model 12π 4 CV −→ N 5 T →0 T θD 3 . (8.62) which is an exact result, because the model is tailored to exact for lowenergy phonons. In Fig. 8–6 the specific heat of the Debye model is compared with results of other models. The best fit is given by the Debye model. For solids with complex elementary cells the agreement with the experiment is worse. 8.4. PROBLEMS 127 8.4 Problems Problem 8.1. Derive the following expressions for the fugacity, chemical potential and internal energy of the ideal Bose gas: η2 + 23/2 z = η− µ = T ln η − E = 1 1 − 3/2 4 3 η 23/2 +3 η 3 N T 1 − 5/2 − 2 2 2 η 3 + O(η 4 ) 1 1 − 5/2 24 3 1 35/2 − 1 24 η 2 + O(η 3 ) η 2 + O(η 3 ) where η = N λ3T /gV . Problem 8.2. Consider an ideal BE gas in the harmonic potential 1 1 1 V (x, y, z) = k1 x2 + k2 y 2 + k3 z 2 . 2 2 2 This is the effective potential energy of alkali atoms in the experimental observations of the BE condensation in alkali vapours. The BE condensation takes place in the limit ki → 0, N → ∞, which replaces the usual thermodynamic limit. In this limit, the density of states of nonrelativistic boson gas may be written as ε2 dN , = 3 dε 2~ ω1 ω2 ω3 where ωi are the angular frequencies of the harmonic oscillations in the potential V . Calculate the internal energy U , heat capacity CN and entropy S of this gas below the condensation temperature, and find also the equation of state. Problem 8.3. Is the BE condensation possible in a two-dimensional perfect boson gas? Consider the gas both in a box and harmonic potential well. Problem 8.4. Calculate the radiant energy of the Sun emitted in the microwave band of width 1,0 MHz centered at the wavelength 3,0 cm. Consider the Sun a black body at the temperature 5800 K. Problem 8.5. Show that in an adiabatic expansion of isolated photon gas the wavelengths of the photons grow proportionally to the diameter of the space they occupy. What was the radius of the Universe in the end of the radiation era (i.e. when the ions formed atoms and radiation and matter decoupled), if it is now about 15 · 109 light years? Problem 8.6. Show that, according to the Debye theory, the heat capacity of lattice vibrations at T ≪ θD (=Debye temperature) is CV = and at T ≫ θD " 12π 4 N 5 1 CV = 3N 1 − 20 T θD θD T 3 2 , # + ··· . 9. Fermionic systems Perfect Fermi gas. The ideal translation invariant Fermi gas is a realistic starting point for many weakly interacting fermionic subtances, e.g. the system of conduction electrons of a metal. The non-relativistic density of states in a three-dimensional space is ω1 (ε) = V 2πg 2m h2 23 √ (9.1) ε. The spin degeneracy is now g = 2. Since the mean occupation number is n(ε) = 1 eβ(ε−µ) +1 (9.2) , the particle and energy densities are N = 4π V E = 4π V 2m h2 2m h2 23 Z dε 0 23 Z √ ∞ ∞ dε 0 ε eβ(ε−µ) (9.3) +1 ε3/2 eβ(ε−µ) + 1 . (9.4) They may be expressed in terms of the polylogarithm as N = −2 V 2πmT h2 3Ω E =− = −3 V 2V 32 Li 23 (−z) , (9.5) 32 (9.6) 2πm h2 5 T 2 Li 25 (−z) , where z = eβµ is the fugacity. 9.1 Conduction electrons in metals Degenerate Fermi gas. The most important Fermi systems are far from the Maxwell-Boltzmann limit: they are dense and heavily degenerate, since µ > 0 and T /µ ≪ 1. The average occupation number (9.2) is then nearly a step function and its derivative the δ function with the minus sign. (see Fig. 9–1). 128 9.1. CONDUCTION ELECTRONS IN METALS 129 (∆n)2 = – T dn dε 1/4 n 1 T 0 ε µ µ ε Figure 9–1: Average occupation number of the degenerate Fermi gas. Completely degenerate Fermi gas (T = 0). In this limit the mean occupation number is the step function n(ε) = θ(µ − ε). The chemical potential at T = 0 is called the Fermi energy (which in the adopted system of units is the Fermi temperature as well) µ = εF = ~2 kF 2 ; 2m (9.7) where ~kF = pF is the Fermi momentum. In the momentum space all states up to the Fermi momentum are occupied. In the free Fermi gas the Fermi surface is a spherical surface of radius kF in the wave-vector space dividing the occupied and empty states. The particle density is calculated from the normalization condition N = 2πgV 2m h2 3/2 ZεF 3/2 √ 2m 4 3/2 dε ε = πgV εF 3 h2 (9.8) 0 as n= gkF 3 6π 2 ⇒ kF 3 , 3π 2 (9.9) where the latter form applies to spin- 21 particles like electrons. The energy per particle is R1 dx x3/2 3 = εF . ε = εF R01 5 dx x1/2 0 From the general non-relativistic relation pV = 2 E. 3 (9.10) the equation of state follows in the form p= g ℏ2 15π 2 2m 5/3 6π 2 n . g (9.11) The density of the electron gas of metals is so high that the ordinary room temperature T is always much lower than the Fermi temperature (or the 130 9. FERMIONIC SYSTEMS degeneration temperature, T ≪ TF = εF . It should be borne in mind, however, that this is not the case for conduction electrons in insulators and semiconductors. Most properties of the degenerate electron gas may be calculated in the limit of zero temperature. The parameters ri and rs describing the density of the electron gas are defined as V 1 4 = = πri 3 ; N n 3 rs = ri , a0 (9.12) where a0 is the first Bohr radius a0 = In metals 1.9 < ∼ rs below. < ∼ Metal Be Al Ga Zn Mg Cu Li Na K Rb Cs 4πε0 ~2 = 0.529 Å. me2 5.6. Some properties of metals are quoted in the table rs 1.88 2.07 2.19 2.30 2.65 2.67 3.25 3.93 4.86 5.20 5.63 Valence 2 3 3 2 2 1 1 1 1 1 1 n (1028 m−3 ) 24.3 18.2 15.4 13.3 8.67 8.47 4.70 2.66 1.41 1.15 0.904 TF (eV) 14.2 11.7 10.5 9.48 7.14 7.03 4.75 3.25 2.12 1.85 1.58 Parameters of metals The low-temperature asymptotic behaviour of integrals like (9.3) and (9.4) whose integrand contains the mean occupation number may obtained in the following way. Introduce in the generic integral a suitable change of variable Z∞ Z∞ f (µ + T z) dz f (ε) dε I= =T , ez + 1 eβ(ε−µ) + 1 0 −µ T and divide the integration interval at the origin with the subsequent change of sign of the integration variable to obtain µ ZT Z∞ f (µ − T z) dz f (µ + T z) dz + T . I=T ez + 1 e−z + 1 0 0 (9.13) 9.1. CONDUCTION ELECTRONS IN METALS 131 In the latter integral the identity 1 1 =1− z e−z + 1 e +1 allows to single out a term similar to the first integral in decomposition (9.13) so that µ I=T ZT 0 µ ZT Z∞ f (µ − T z) dz f (µ + T z) dz −T . f (µ − T z) dz + T ez + 1 ez + 1 (9.14) 0 0 So far no approximations have been made. Further we use the observation that in the limit T → 0 the upper limit of the last integral in relation (9.15) grows without limit (because µ has the finite limit εF ), and – with the exponential accuracy of order ∼ e−µ/T – we send this limit to infinity. Thus µ I≈T ZT 0 Z∞ [f (µ + T z) − f (µ − T z)] dz f (µ − T z) dz + T ez + 1 0 The change of variable µ − T z = ε in the first integral together with the Taylor expansion in T z of the numerator of the integrand in the second yield i h Zµ Z∞ 2T zf ′ (µ) + 1 (T z)3 f ′′′ (µ) dz 3 + ... (9.15) I = f (ε) dε + T ez + 1 0 0 Integrals appearing here may be calculated in a fashion analogous to that used in the calculation of the similar bosonic integral: Z∞ Z∞ Z∞ ∞ X z x−1 e−z z x−1 x−1 = dz = dz z (−1)n e−z(n+1) dz z e +1 1 + e−z n=0 0 0 0 ∞ X n+1 (−1) = nx n=1 Z∞ ∞ X (−1)n+1 . dz z x−1 e−z = Γ(x) nx n=1 (9.16) 0 Here, the alternating series does not give the anticipated ζ function directly. The following trick, however, allows to reorganize it to a difference of easily identifiable positive-term series: ∞ ∞ ∞ X X X (−1)n+1 1 1 = − = x x x n (2k + 1) (2l) n=1 k=0 l=1 ∞ X k=0 ∞ ∞ l=1 l=1 X 1 X 1 1 + − 2 (2k + 1)x (2l)x (2l)x 132 9. FERMIONIC SYSTEMS Combination of the two first series on the right-hand side is the desired series of Riemann’s ζ function, so is the third series from the terms of which the common factor 2−x is easily extracted, thus we arrive at the conclusion Z∞ dz 0 z x−1 = Γ(x)ζ(x) 1 − 21−x . z e +1 (9.17) Taking into account the numerical results ζ(2) = π2 ; 6 ζ(4) = π4 90 we arrive at the low-temperature expansion for the typical Fermi-gas integral (the Sommerfeld expansion) Z∞ dε 0 f (ε) ≈ β(ε−µ) e +1 Zµ 7π 4 4 ′′′ π2 2 ′ T f (µ) + T f (µ) + · · · dεf (ε) + 6 360 (9.18) 0 Heat capacity. The particle density n = N/V of the electron gas in metals is fixed by the charge neutrality. Expressing the same particle density at low temperatures with the aid of the Sommerfeld expansion of the normalization condition (9.3) on one hand and at zero temperature in terms of εF [see relation (9.8)] on the other we arrive at the relation √ Z ∞ 2 π2 2 1 2 ε dε β(ε−µ) = µ3/2 + T √ + · · · = εF 3/2 . 3 12 µ 3 e + 1 0 The solution is the low-temperature expansion of the chemical potential: # " 2 π2 T + ··· . (9.19) µ(T ) = εF 1 − 12 TF To calculate the energy according to (9.4) the following integral is needed Z∞ dε ε3/2 eβ(ε−µ) +1 = 0 = 2 5/2 π 2 2 √ µ + T µ + ··· 5 4 " 2 # 5π 2 T 2 5/2 εF 1+ + ··· . 5 12 TF Here, expansion (9.18) has been used first, after which the expression of µ from equation (9.19) has been substituted. The energy per particle is now ε(T ) = π2 T 2 3 εF + . 5 4 TF (9.20) 9.1. CONDUCTION ELECTRONS IN METALS 133 Further, the heat capacity is CV = ∂ (N ε) π2 T =N . ∂T 2 TF (9.21) The heat capacity of the electron gas vanishes as a linear function of T at low temperatures. Numerically it is small, however, because the degeneracy temperature TF is very high. The heat capacity of lattice vibrations of a solid vanishes as the third power of the temperature at low temperatures (8.62), therefore in metals the heat capacity of the electron gas becomes dominant at low temperatures. The crossover takes place at temperatures of the order of a few kelvins. Electrons and holes in semiconductors. In the band theory of electrical conductivity of a crystalline solid the periodic structure of positive ions brings about the band structure of one-particle electron states in the crystal. The bands – densely filled by one-particle states – are separated by relatively wide gaps in the electron energy spectrum. From the point of view of electrical conductivity, the two highest-energy bands with occupied states are of particular interest. At T = 0 the valence band is completely filled, whereas the conduction band above it is partly filled in conductors and empty in insulators and semiconductors, the difference between the latter being in the magnitude of the energy gap between the bottom of the conduction band and the top of the valence band. The electron states within bands correspond to almost freely moving electrons with a rather complicated dispersion relation, though. Near the bottom of the conduction band, for instance, the dispersion relation – with the one-particle energy measured from the bottom of the band – is typically that of a non-relativistic electron with an effective mass me of the order of the physical mass of the electron. Near the top of the conduction band the dispersion relation gives rise to the description of conductivity in terms of moving empty electron states called holes, which behave as positively charged particles with an effective mass mh of the order of the physical 2 2 electron mass and the dispersion relation εh (k) ≈ − ~2mk , where the origin h of the energy scale is at the top of the valence band, hence the negative energy of the hole. Since the hole is an electron absent at a state in the valence band, the mean occupation number of a hole state is the mean occupation number of the corresponding electron state subtracted from the maximum occupation number one: 1 1 = β(−ε +µ) . (9.22) nh (ε) = 1 − β(ε −µ) h e h +1 e +1 Thus, the chemical potential of holes is the chemical potential of electrons with the minus sign. From the point of view of the use of the perfect Fermi gas in the description of the electron gas in insulators and semiconductors it should be 134 9. FERMIONIC SYSTEMS noted that the number density and thus the Fermi temperature of currentcarrying electrons in these materials is much less than in metals. In most cases the electron gas is actually far from degeneracy and rather accurately described by the Maxwell-Boltzmann distribution. Example 9.1. Chemical potential in a semiconductor. Consider the model of an intrinsic semiconductor corresponding to Fig. 9–2. Assume that electrons and holes behave as free particles of masses me and mh , respectively, and that Eg ≫ T . Show that the densities of electrons particle E and holes obey the relation ne = nh ∝ exp − 2Tg , calculate the coefficient of proportionality and the chemical potential at low temperatures. Here, Eg is the energy gap between the conduction and valence bands. elektronitilat Eg aukkotilat Figure 9–2: Creation of a hole in the valence band through the excitation of an electron to the conduction band. In the MB approximation the number of electrons in the conduction band may be calculated as 3 Z∞ Z∞ p 2me 2 −β(ε−µ) dε ε − Eg e−β(ε−µ) , Ne = dε ω1e (ε − Eg ) e = V 2πge 2 h Eg Eg where the origin of the energy scale is chosen at the top of the valence band, and the one-particle energies of electrons in the conduction band are 2 2 ε = Eg + ~2mk . Calculation yields e 3 2me π 2 3 −β(Eg −µ) Ne = V g e T2e . h2 On the other hand, in the MB limit the number of holes is Nh = V 2πgh 2mh h2 23 Z0 −∞ √ dε −ε e −β(−ε+µ) = V gh 2mh π h2 32 3 T 2 e−βµ . In an intrinsic semiconductor at T = 0 the conduction band is empty and the valence band completely filled, therefore Ne = Nh . Moreover, in the 9.2. MAGNETISM OF DEGENERATE ELECTRON GAS 135 product of the expressions for Ne and Nh the chemical potential is cancelled, so that √ 2 me mh π 3 3 −βEg ne nh = n2 = ge gh T e h2 and ne = nh = √ 3 √ 2 me mh π 2 3 −βEg /2 ge gh . T2e h2 The chemical potential is readily found from the ratio Ne /Nh as 3 ! gh mh2 1 1 µ = Eg + T ln , 3 2 2 g me2 e which corroborates the MB approximation by producing small occupation numbers both for electrons in the conduction band and holes in the valence band. At T = 0 the chemical potential assumes the constant value µ(0) = 21 Eg . This is sometimes also called the Fermi energy, but it should be borne in mind that other authors reserve the name Fermi energy to the energy of the highest occupied electron state at T = 0. In the present model this is at the top of the valence band, while the zero-temperature value of the chemical potential does not correspond to any electron state at all. 9.2 Magnetism of degenerate electron gas Pauli paramagnetism. The magnetic moment of the electron is µe = γe s ≈ 2γ0 s, where γ0 = −e/(2m) is the classic gyromagnetic ratio. Introducing Bohr’s magneton µB = e~ eV = 5.66 × 10−5 2m T we obtain µz = −µB σz , where σz a Pauli spin matrix. In an external magnetic field the energy of the electron is εpσ = p2 − µz B = εp + µB Bσ . 2m (9.23) (σ = σz = ±1) The spin degeneracy is lifted, but otherwise the effect is a shift of the chemical potential, so that Ω = −T = X k ( " ln 1 + e −β # ℏ2 k2 +µ B−µ B 2m " + ln 1 + e −β #) ℏ2 k2 +µ B−µ B 2m 2 1 1 1 2 ∂ Ω0 Ω0 (µ − µB B) + Ω0 (µ − µB B) = Ω0 (µ) + (µB B) + ... , 2 2 2 ∂µ2 (9.24) 136 9. FERMIONIC SYSTEMS where Ω0 is the grand potential of the electron gas, when B = 0. Taking into account that ∂Ω0 = −N we see that ∂µ ∂N 1 2 + ... , Ω = Ω0 (µ) − (µB B) 2 ∂µ T,V so that the magnetization is 1 M =− V ∂Ω ∂B T,V = µ2B B T,V ∂n ∂µ ∂n ∂µ , T,V and the susceptibility χ= ∂M ∂H = µ0 µ2B B=0 . T,V In the degenerate electron gas, in which the average occupation number is nearly a step function Zµ ω1 (ε) dε N =V 0 so that ∂n ∂µ = ω1 (µ) = D(µ) T,V is the density of states (per unit volume) at the Fermi surface (in this limit µ = εF ) so that M = µB 2 D(µ)B and Pauli’s paramagnetic susceptibility assumes the form χpara = µ0 µB 2 D(µ) , (9.25) which remains valid also in the case, when the dispersion law is different from the usual isotropic parabolic form (in solids the effective dispersion law is, as a rule, anisotropic, which leads to a more complex density of states). Landau levels. The magnetic field affects the electron energy also through the orbital motion. This is also a pure quantum effect – in classical statistics there is no diamagnetism of the electron gas. The effect on the orbital motion is described by the minimal coupling, in which the kinetic momentum p is replaced by the difference p − qA in the Hamilton function (A is the vector potential and q the charge of the particle). In case of the electron q = −e and the Hamilton function of a free electron (without the account of spin) in the magnetic field is 2 H= (p + eA) . 2m 9.2. MAGNETISM OF DEGENERATE ELECTRON GAS 137 For the uniform magnetic field B = Bk choose the vector potential in the Landau gauge: Ax = −By, Ay = Az = 0. The choice of the gauge does not affect the energy eigenvalues, but it does affect the functional form of the eigenfunctions, and the Landau gauge turns out to be the simplest to solve. It should also be borne in mind that the choice of the vector potential reflects the physical boundary conditions of the problem. The Hamilton operator corresponding to the Landau gauge is ie~ −~2 e2 B 2 2 2 b = 1 (b H p + eA) = ∂x 2 + ∂ y 2 + ∂ z 2 + By∂x + y . (9.26) 2m 2m m 2m b commute, solutions of the Schrödinger Since the operators pbx2 , pbz2 and H equation with the Hamiltonian (9.26) may be sought in the form ψ(x, y, z) = eikx+ikz χ(y). For the function χ the differential equation follows: ℏkx p2 ~2 ′′ e2 B 2 2 yk = χ + (y − yk ) χ = ε − z χ ; , − 2m 2m 2m eB (9.27) where ε is the energy eigenvalue of the electron. The equation (9.27) is of the same form as that of the quantum harmonic oscillator with the spring p constant K = e2 B 2 /m and angular frequency ω = K/m = eB/m = ωc . The energy eigenvalues are thus grouped into Landau levels with 1 p2 p2z = z + µB B (2n + 1) . + ~ωc n + (9.28) εn = 2m 2 2m The wave functions are localized in the y directions with the width parameter r ~ . b0 = eB In the x direction they are extended (plane waves) (see Fig. 9–4b). Each Landau level is heavily degenerate, because the energy eigenvalues (9.28) are independent of the quantum number kx labeling the energy eigenfunctions. Consider boundary conditions for a rectangular slab with dimensions Lx × Ly × Lz . Impose periodic boundary conditions in the x direction, then kx = kj = 2π j, Lx (j = 0, ±1, ±2, . . .) ⇒ ∆kx = 2π . Lx (9.29) Wave functions corresponding to consecutive values of j are – according to (9.27) – shifted with respect to each other in the y direction by the distance ∆y = ~ h ∆k = . eB eBLx 138 9. FERMIONIC SYSTEMS In the area Lx × Ly there is room for Ly eB = Lx Ly ∆y h eigenfunctions. In case of a thin slab it may be said that each Landau level has the surface density of states nL = eB . h (9.30) εn 0 x Figure 9–3: Landau levels and the degeneracy of the electron states, schematic illustration corresponding to the symmetric gauge. Landau diamagnetism. Thus, the grand potential of the electron gas in the uniform magnetic field may be written as ℏ2 k2 ) ( ∞ −β 2mz +µB B(2n+1)−µ gLx Ly eBT X X . ln 1 + e Ω=− h n=0 (9.31) kz In the z direction – for a slab of macroscopic thickness – the usual trick applies: Z∞ X Lz −→ dkz . 2π kz −∞ ℏ2 k 2 After the change of variable 2mTz = x and integration by parts we arrive at the representation √ ∞ gV eB 2πm T 3/2 X β[µ−µB B(2n+1)] 3 Li Ω= −e . 2 h2 n=0 (9.32) The desired limit B → 0 cannot be taken directly, since the sum over Landau levels diverges in this limit and the ambiguity of the multiplication of B and the divergent series must be resolved. To this end, expression (9.32) 9.2. MAGNETISM OF DEGENERATE ELECTRON GAS 139 is traditionally approximated with the aid of the Euler-Maclaurin formula Zb n X 1 f [a + (k − 1)h] = h ∞ X Li 32 −eβ[µ−µB B(2n+1)] = k=1 f (x)dx a h 1 [f ′ (b) − f ′ (a)] + O h2 , (9.33) − [f (b) − f (a)] + 2 12 where b = a + (n − 1)h. Substituting f (x) → Li 23 −eβ(µ−x) , h → 2µB B, a → µB B and b → ∞ we arrive at the asymptotic relation n=0 + 1 2µB B Z∞ µB B Li 32 −eβ(µ−x) dx µB Bβ β(µ−µB B) ′ β(µ−µB B) 1 Li 23 −eβ(µ−µB B) − e Li 3 −e + O B2 . 2 2 6 Expanding the right-hand side to the linear order in B, we obtain ∞ X n=0 Li 32 −eβ[µ−µB B(2n+1)] 1 = 2µB B Z∞ µB Bβ βµ ′ Li 23 −eβ(µ−x) dx + e Li 3 −eβµ + O B 2 , 2 12 (9.34) 0 in which the limit B → 0 is straightforward. It is clear that the first term on the right-hand side of (9.34) gives rise to the grand potential without the magnetic field Ω0 , but this is readily seen by direct calculation as well: n Z∞ X Z∞ ∞ −eβ(µ−x) β(µ−x) dx = dx Li 32 −e n3/2 n=1 0 0 = ∞ X n=1 n Z∞ n ∞ X −eβµ −eβµ −βx = T Li 52 −eβµ , dx e = T 3/2 5/2 n n n=1 (9.35) 0 which gives rise to the expression (9.6) for the grand potential. To identify the second term on the right-hand side of (9.34), use the important property of the polylogarithm ∞ ∞ d X zn 1 X zn 1 d Liν (z) = = = Liν−1 (z) , dz dz n=1 nν z n=1 nν−1 z which allows to estimate the grand potential of the electron gas in the mag- 140 9. FERMIONIC SYSTEMS netic field (9.32) as Ω≈ gV (2πm)3/2 T 5/2 Li 25 −eβµ − 3 h 2 (µB B) ∂ 2 gV (2πm)3/2 T 5/2 βµ Li 52 −e 6 ∂µ2 h3 = Ω0 − 2 (µB B) ∂ 2 Ω0 . (9.36) 6 ∂µ2 Since N = − ∂Ω0 , we immediately see that in the diamagnetic case ∂µ 2 (µB B) Ω = Ω0 (µ) + 6 ∂N ∂µ + ... , T,V so that Landau’s diamagnetic susceptibility may be written as 1 1 χdia = − µ0 µB 2 D(µ) = − χpara , 3 3 (9.37) and the total susceptibility of the electron gas is χ= 2 µ0 µB 2 D(µ) . 3 (9.38) Quantum Hall effect. In semiconductor interfaces by electric means conditions may be created such that the conduction electrons are bound to the interface but free to move along it. This gives rise to an effectively two-dimensional electron gas with arbitrarily adjustable surface density. In a current-carrying conductor located in a transversal magnetic field the Hall effect may be observed, which means induction of a voltage between the faces of the conductor corresponding to an electric field perpendicular to both the current density and the magnetic field induction. The basic setup is depicted in Fig. 9–4a. To maintain a steady current in the x direction, the Lorentz force acting on the conduction electrons (or holes which may also carry current in a semicponductor) −e [v × B]y = evx B must be compensated by the electric field Ey = vx B. The ratio ρxy ≡ vx B B Ey = = jx −envx −en (9.39) is the Hall resistivity. In relation (9.39) n is the number density of the electrons and jx the current density. In the two-dimensional electron gas the Hall resistance may be correspondingly defined as Vy B Rxy ≡ = , (9.40) Ix −en2 9.2. MAGNETISM OF DEGENERATE ELECTRON GAS y y z B 141 b0 Ey I Vy ∆y yj +1 yj x x Figure 9–4: (a) Motion of electrons on a two-dimensional surface and the Hall effect, (b) transversal wave functions in the Landau gauge. where n2 is the surface density of electrons (the subscript 2 reminds of the spatial dimension). The resistance in this case is the same quantity as the resistivity. The Hall resistance (9.40) is a linear function of the magnetic field. If the experimental setup is such that the surface density of electrons n2 is independent of the magnetic field, the classic argument yields the dashed straight line in Fig. 9–5. Experimentally it has been observed (Klaus von Klitzing 1980), however, that at low temperature in the limit of strong magnetic field and small electron density the dependence is not linear, but with the growth of B the Hall resistance exhibits also constant plateau values h 1 · , (9.41) e2 ν where ν = 1, 2, . . .. This is the integer quantum Hall effect. The experimental accuracy 3 is high: 10−5 . . . 10−7 . Later plateaus at fractional values of ν were observed as well. The 4 integer effect may be explained by and large 5 6 with the aid of Landau levels in the effective two-dimensional electron gas and is thus a B one-particle problem. In the fractional effect correlations due to the Coulomb interacFigure 9–5: Hall resistance tion are important and thus this is a genuine in the quantum Hall effect. many-particle problem. Here, only the integer quantum Hall effect will be considered. If exactly ν lowest Landau levels are filled, the surface density of electrons is n2 = νnL . Substitution of this value in relation (9.40) yields exactly the observed plateau value (9.41). These are pointwise values, however, therefore they does no explain, where the plateaus come from. It is also not clear that all electron states contribute to the current (in three dimensional electron gas of solids only states in the conduction band do). The long-distance phase coherence turns out to be important. R xy n=2 Rxy = 142 9. FERMIONIC SYSTEMS The locking to the plateau values may be explained as follows. In the real matter the Landau levels are broadened to energy bands, with a large number density at the Landau energy, but very small between the bands. In solids smallest impurities are able to bring about localized electron states in the region of low density of states. Electrons in such states do not carry current, thus forming mobility gaps. When the electron states are filled up to the Fermi energy εF , the latter may occur in a mobility gap. In this case, the conduction bands are either completely filled or empty. This brings about a plateau in the Hall resistance, because the number of extended states in the conduction bands – and, correspondingly, the number density of current carriers – changes proportionally to the magnetic induction thus rendering the Hall resistivity intact. With the further increase (decrease) of the magnetic induction a conduction band adjacent to the mobility gap moves on the Fermi energy causing a decrease (increase) of the occupation of the electron states of this conduction band. This, however, is compensated by the increase (decrease) of the number of the extended states in all conduction bands, so that the number density of current-carrying electrons remains constant and the Hall resistivity increases (decreases) with the magnetic induction. 9.3 Problems Problem 9.1. It has been observed that the radius R of a large nucleus depends on its mass number A according to the relation R = 1.1A1/3 fm, (1 fm = 10−15 m) . Using this, calculate the density of saturated nuclear matter, its Fermi momentum and Fermi energy. Note that the degeneracy factor gs = 4 accounting for both the isospin (proton, neutron) and spin degrees of freedom (the quantum number of both of them is 1/2). How high is the degeneracy pressure? Is the matter relativistic? What is its energy density (including the rest energy of the nucleons mp ≈ mn ≈ 940 MeV /c2 ). Problem 9.2. Calculate the compressibility of a highly relativistic (ε = cp) degenerate electron gas at the accuracy of T 2 . Problem 9.3. Calculate the total magnetic susceptibility of the free electron gas in a homogeneous magnetic field B in the Landau gauge without the decomposition to paramagnetic and diamagnetic susceptibilities, i.e. start with the Hamilton operator b = H pb2y (pbx − eyB)2 pb2 e b + + z + s·B. 2m 2m 2m m Problem 9.4. In a semiconductor there are n0 bound electron states per unit volume (donor states) with the energy −ε0 < 0 all filled at the temperature T = 0 (see Fig 9–6). When the temperature rises, part of 9.3. PROBLEMS 143 the donor electrons may be transferred to the conduction band, in which the one-particle density of states is √ ω1 (E)dE = V A E dE (A is a constant). Show that at the temperature T the number density of the conduction electrons is n = n0 e−(ε0 +µ)/T , +1 e−(ε0 +µ)/T where the chemical potential µ at low temperatures is 1 2n0 1 µ ≈ − ε0 + T ln √ 3/2 . 2 2 A πT Assume the valence band located so low that its hole excitations need not to be taken into account. E johtavuusvyö 0 -ε0 donoritilat Figure 9–6: Energy level structure of a simple model of a doped n type semiconductor. 10. Phase transitions 10.1 Description of phase transitions Classification of phase transitions. The basic thermodynamic potential in the description of phase transitions is the Gibbs potential G = µN , because one of the equilibrium conditions – in addition to that the temperature and the pressure are uniform – is that the chemical potential coincides in all coexisting phases. Phase transitions are traditionally classified by the singular behaviour of the derivatives of the Gibbs potential. If some of these derivatives are discontinuous, the phase transition is of first order. If all the first derivatives are continuous, but discontinuities (or worse singularities) appear in the second order derivatives, then we are dealing with a second order phase transition. These names suggest generalization according to the behaviour of higher derivatives, but this line of classification could not be consistently extended. Order parameter. Another popular classification scheme emphasizes the behaviour of the central quantity in the modern theory of phase transition, viz. the order parameter. This is a quantity, which by definition is zero in one phase and assumes finite values in the other. In spite of the loose definition, in most cases there is a natural choice for the order parameter. In the gas-liquid transition, for instance, such a natural choice is the difference between the densities of liquid and gas, whereas in the paradigmatic ferromagnetic transition the order parameter is the magnetization. If the change of the order parameter in the phase transition is finite, then we are dealing with a discontinuous phase transition. If the order parameter tends to the zero value or departs from it in a continuous fashion, then a continuous phase transition takes place. It is typical of the continuous phase transitions that the order parameter, various response functions and correlation functions exhibit nonanalytic behaviour as functions of thermodynamic variables in the vicinity of the critical point, i.e. the point of the continuous phase transition in the space spanned by the thermodynamic variables. This non-analytic behaviour is usually a powerlike dependence in deviations from the criticalpoint values of variables such as the temperature and some ”external field” like the pressure in the gas-liquid transition or the magnetic field strength in the ferromagnetic transition. The exponents appearing in such asymptotic relations are the critical exponents (or critical indices) of the transition. 144 10.1. DESCRIPTION OF PHASE TRANSITIONS 145 The most common critical exponents will be defined and calculated below within the Landau theory of phase transitions. In many cases phase transitions involve changes in symmetries of the system, which are called symmetry breaking. In the ferromagnetic transition, for instance, in the paramagnetic phase the material is often macroscopically isotropic and thus possesses the three-dimensional rotational symmetry. In the ferromagnetic phase the direction of the macroscopic magnetization establishes a preferred direction and the rotational symmetry remains at most in the plane perpendicular to the direction of the magnetization. Symmetry breaking and order parameter. It is typical of a second order (or continuous) phase transition that some symmetry of the (usually) high-temperature phase is spontaneously broken in the low-temperature phase. The degree of symmetry breaking may be described by an order parameter vanishing in the symmetric (usually) high-temperature phase, but finite in the ordered phase with the broken symmetry. Examples: • Structural transformation of a crystal lattice. In barium titanate (Fig.) electric polarization is brought about, this is ferroelectricity. The polarization P is the natural order parameter. P Ba, Ti, O Figure 10–1: Structural transformation of BaTiO3 . The polarization P is the order parameter. • Ferromagnetic phase transition. The broken symmetry is the spin rotation symmetry. Below the critical temperature magnetization M 6= 0 appears; this is the order parameter of the system. • Superconducting transition of electron system and superfluidity transition of 4 He. Here, a gauge symmetry related with the particle number conservation is broken. • Symmetry breaking of the electroweak interaction in particle physics. A gauge symmetry is broken here as well. 146 10. PHASE TRANSITIONS Figure 10–2: Magnetic ordering. Universality. It is a remarkable feature of the continuous phase transitions that they are largely insensitive to material properties of the system apart from such global features like rotational or other symmetries. Physically different phase transitions may be described in a unified fashion as soon as important global properties such as the number of components and the tensor character of the order parameter, symmetries of the system and the space dimension are found to be the same. This is the universality of continuous phase transitions, and identification of the universality class of the particular phase transition is one of the important tasks of its analysis. Most importantly, the values of critical exponents turn out to coincide fairly accurately in physically different systems belonging to the same universality class. Singularities in the thermodynamic limit. Mathematical description of continuous phase transitions is much more difficult than that of the discontinuous transitions. In the latter case different phases have chemical potentials of their own and at the phase transition the coexistence condition µ1 = µ2 holds. On the other hand, by definition of the first-order transition, e.g., ∂µ1 ∂µ2 6= ∂T ∂T so that rising or lowering the temperature across the transition temperature necessarily involves a reversal in the ordering of the chemical potentials and in equilibrium the phase corresponding to the minimum of the chemical potential prevails. Chemical potentials of different phases are smooth functions of the state variables. In case of continuous phase transition, on the contrary, the number of state of equilibria changes: a single chemical potential corresponds to the symmetric phase, whereas several (even infinitely many in case of breaking of a continuous symmetry) chemical potentials of equal value but different values of the order parameter are available for the ordered phase. This is a singularity which is hard to find in the statistical ensembles, whose partib tion functions, say Z = Tr e−β H are smooth functions of parameters, at least in a finite system with the physically prevailing effectively short-range interactions (Coulomb force is screened is electrically neutral systems, and gravity is small). In the thermodynamic limit, however, singular behaviour may follow 10.2. LANDAU THEORY 147 for suitable values of parameters, as was seen in the case of Bose condensation. Other possible sources of singularity are long-range interactions and zero-temperature limit, on which, however, we shall not dwell here. Exact results for physically interesting interacting systems are rare, therefore more or less phenomenological approaches are popular in description of continuous phase transitions. The most general approach, based on a variational principle, is that of the Landau theory. 10.2 Landau theory Effective thermodynamic potential. At the critical point (point of the continuous phase transition in the parameter space) the order parameter vanishes in a continuous manner, when the symmetric phase is approached. In the Landau theory the order parameter ϕ is considered a macroscopic variable describing an incomplete equilibrium, whose equilibrium value is found by minimizing the proper thermodynamic potential. The system is considered to be so close to the critical point that the order parameter is already small but still so far from the critical point that the system may be assumed to homogeneous and the thermodynamic potential a smooth function of the order parameter and state variables. The thermodynamic potential is then expanded in powers of the order parameter and the leading terms retained. Thus G(p, T, ϕ) = G0 (p, T ) + αϕ + Aϕ2 + Cϕ3 + Bϕ4 + . . . (10.1) The linear term must vanish in case of vector order parameter for a rotation-invariant Gibbs potential. The third-order term gives rise to a discontinuous transition, so that in case of continuous transition only the second and fourth order terms remain. To guarantee existence of equilibrium, the coefficient B >, and, although a function of p and T may usually be considered constant. For A > 0 the only minimum of (10.1) is ϕ = 0, whereas in case A < 0 a twofold degenerate nonvanishing solution for the minimum exists. The borderline value A = 0 then corresponds to the point of phase transition. The simplest smooth temperature dependence is A = a(T − Tc ). Ferromagnetic ordering. Here, the basic idea of Landau theory is demonstrated in the example of the prototypical ferromagnetic transition. Denote magnetization by m to emphasize that it is an order parameter assuming values different from the equilibrium magnetization. In relations dUsys dFsys = T dS + µ0 V h · dm , = −SdT + µ0 V h · dm , (10.2a) (10.2b) the quantity h ≡ h(T, m) is the derivative of the thermodynamic potential with respect to the order parameter. In equilibrium, however, it must be equal to the magnetic field strength H. The natural parameters of the free energy are T and m, i.e. Fsys = Fsys (T, m). If the system is coupled 148 10. PHASE TRANSITIONS to a magnetic field with the fixed field strength H, then the equilibrium value of the magnetization m is determined by the condition that the Gibbs function has a minimum. The magnetic Gibbs potential now becomes an order-parameter dependent function Gsys (T, H ; m) = Fsys (T, m) − µ0 V H · m . (10.3) dGsys = −SdT − µ0 V M · dH. (10.4) 1 F (T, m) = F0 (T ) + α2 (T )m2 + α4 (T )m4 + · · · . 2 (10.5) Choose m to minimize G: δG/δm = δFsys /δm−µ0 V H = µ0 V h−µ0 V H → 0. In equilibrium we must have h(T, m) = H. In other words, m = m(T, H) ≡ M is the equilibrium magnetization and the familiar result holds In an isotropic system the free energy F depends on the magnitude of the order parameter m = |m|. Expand F as power series Assume simplest possible smooth dependencies to provide stable minima in the vicinity of Tc : α2 (T ) = a · (T − Tc ) ; a>0 (10.6) α4 (T ) = b = const > 0 F F M T > Tc M0 M T < Tc Figure 10–3: Free energy as a function of the order parameter in Landau theory. Let first H = 0. Then the minimum of G is also the minimum of F . From the condition ∂F = 2a(T − Tc )m + 2bm3 = 0 ∂m the equilibrium magnetization is found as (cf. Fig. 10–3) M0 (T ) = 0, T > Tc r a M0 (T ) = ± (Tc − T ) . b T < Tc (10.7) 10.2. LANDAU THEORY 149 The latter relation determines the value of the critical exponent β, which describes the non-analytic behaviour of the order parameter ϕ as a function of the temperature near the critical point as β ϕ(T ) ∼ (Tc − T ) , T < Tc , (10.8) H = 0. Thus, in the Landau theory β = 12 . The spontaneous magnetization M0 (T ) is depicted in Fig. 10–4. The minimum value of the zero-field free energy is thus F (T, M0 ) = F0 (T ) ; F (T, M0 ) = F0 (T ) − M0 ( T ) T > Tc (10.9) a2 (Tc − T )2 ; 2b T < Tc C (T) Tc T Tc T Figure 10–4: Magnetization and heat capacity. Heat capacity. According to the definition 2 ∂S ∂ G CH = T = −T . ∂T H ∂T 2 H In zero field G = F , more accurately G(T, H = 0) = F (T, M0 (T )), yielding CH (H = 0) = −T (d2 F/dT 2 ). From relations (10.9) we obtain d 2 F0 dT 2 T > Tc : CH (T ) = C0 (T ) ≡ −T T < Tc : a2 CH (T ) = C0 (T ) + T b (10.10) The specific heat has a jump CH (Tc− ) − CH (Tc+ ) = a2 Tc b at the critical temperature (Fig. 10–4). In general, however, the singular behaviour of the heat capacity in a continuous transition is characterized by the critical exponents α and α′ 150 10. PHASE TRANSITIONS with the definition CH (T ) ∼ ( −α (T − Tc ) , T > Tc , −α′ (Tc − T ) , T < Tc , H = 0, (10.11) where usually both α and α′ are numerically small. The finite discontinuity in the Landau theory in these terms is described by putting α = α′ = 0. Susceptibility. If the external field H 6= 0, we arrive at the equilibrium condition H H M0 M M T < Tc T > Tc Figure 10–5: Determination of the equilibrium magnetization, relation (10.13). ∂F ∂G =0 ⇔ = µ0 V H , ∂m ∂m i.e., according to relations (10.5) and (10.6), 2a(T − Tc )M + 2bM 3 = µ0 V H . (10.12) (10.13) In case T > Tc (Fig. 10–5) in the limit of small field H the magnetization M = χH + O(H 3 ), with the susceptibility χ χ= µ0 V . 2a(T − Tc ) (10.14) In case T < Tc in the limit of small H obviiously (cf. Fig. 10–5) M = M0 + δM , where δM ∝ δH is small. The following relation holds, µ0 V δH 2a(T − Tc )δM + 6bM0 2 δM a = 2a(T − Tc )δM + 6b (Tc − T )δM b = 4a(Tc − T )δM . = This yields for the susceptibility the result χ= µ0 V δM = . δH 4a(Tc − T ) (10.15) 10.3. GINZBURG–LANDAU THEORY 151 From relations (10.14) and (10.15) it is seen that the critical exponents γ and γ ′ of the susceptibility, defined as ( −γ (T − Tc ) , T > Tc , χ(T ) ∼ H = 0, (10.16) −γ ′ (Tc − T ) , T < Tc , in the Landau theory assume the value γ = γ ′ = 1. M H <H 1 2 χ = (∂Μ / ∂Η)Η=0 H =0 Tc Tc T T Figure 10–6: Magnetization and susceptibility. The susceptibility is depicted in Fig. 10–6. In the same plot dependence of the magnetization M on the temperature and field strength H has been sketched. In particular, at the critical temperature T = Tc , from the condition µ0 V H = 2bM 3 it follows M (Tc , H) = const × H 1/3 . (10.17) From this relation it follows that the critical exponent δ, which describes the non-analytic dependence of the order parameter ϕ of the external field h at the critical temperature as ϕ(h) ∼ h1/δ , T = Tc , (10.18) in the Landau theory assumes the values δ = 3. 10.3 Ginzburg–Landau theory of superconductivity In a large system it is possible that the order parameter is not constant throughout the whole volume, especially, when the broken symmetry is continuous. Deviations from homogeneous order parameter are also necessary to describe fluctuations near the critical point. If local variability is allowed, then a field theory is obtained with the order parameter m(r) as a function of position. Such a generalization was put forward in V. Ginzburg in application to superconductivity and turned out to be very successful explanation of this phenomenon discovered already in 1911 by H. Kamerlingh– Onnes. The Ginzburg-Landau theory has proved a fruitful starting point for description of several other ordering phenomena as well. 152 10. PHASE TRANSITIONS Free energy. The order parameter is assumed to be a complex-valued function Ψ(r) called macroscopic wave function in this context. Physically, this is quantity describing the correlated electron pairs (Cooper pairs), whose motion gives rise to the phenomenon of superconductivity in metals. The ”genuine” wave function of such a pair of electrons depends on variables of both electrons, of course. The macroscopic wave function here is related to the motion of the center of mass of the Cooper pair, which is thus considered a pointlike object in the Ginzburg-Landau theory. The typical length scale up to which the electrons of the pair remain correlated, the coherence length ξ0 (usually ξ0 > ∼ 1000 Å), must therefore be much less than the typical spatial scale of the macroscopic wave function for the GinzburgLandau theory to be consistent. The free energy is written as local functional functional of the macroscopic wave function, where the spatial dependence is taken into account by the leading term of the gradient expansion: Z ~2 b 2 2 4 Fsys (T, [Ψ]) = d3 r f0 + . (10.19) |∇Ψ| + a(T − T ) |Ψ| + |Ψ| c 2m∗ 2 Here, m∗ is a parameter of dimension of mass. In a system with charged particles in an magnetic field gauge invariance is imposed by replacing the gradient in the canonical momentum operator −i~∇ by the covariant derivative to obtain −i~∇ − e∗ A(r), where A is the vector potential and e∗ the charge. With this substitution the Ginzburg-Landau free energy (10.19) remains invariant under the transformation Ψ(r) → Ψ(r)eiα(r) even in case of position-dependent α(r), when it is accompanied by the proper change of the vector potential. When the energy of the magnetic field is added, the total Ginzburg-Landau free energy is obtained in the form F = Z 2 ie∗ b B2 ~2 2 4 ∇− A Ψ + aτ |Ψ| + |Ψ| + , d r f0 + 2m∗ ~ 2 2µ0 3 (10.20) where for brevity the notation τ = T − Tc has been introduced. Thus, we are dealing with a 4-parameter (m∗ , e∗ , a, b) phenomenological theory. Variational conditions. It is convenient to write down the stationarity equations for the functional (10.20) with the aid of the functional derivative, whose definition for an arbitrary functional F [f ] of the function f (r) is Z δF δF [f ] ≡ d3 r δf (r) . δf (r) In practice, the usual chain rule together with partial integration is sufficient to arrive at expressions containing δf (r ′ ) = δ(r ′ − r) δf (r) which allows to resolve one spatial integral. 10.3. GINZBURG–LANDAU THEORY 153 Since the electromagnetic field interacts with charged matter, the vector potential A is a variable quantity as well. Consider first variation of the magnetic field energy with fixed boundary conditions for the fields varied. Then Z Z Z B · (∇ × δA) B 2 = δ (∇ × A) · (∇ × A) = 2 δ V V V Z Z = −2 ∇ · (B × δA) + 2 ∇ · (B × δAc ) ZV ZV = −2 n · (B × δA) + 2 δA · (∇ × B) , (10.21) ∂V V where the notation δAc means, that the derivatives in the nabla do not act on this factor. Taking into account the fixed boundary condition, we arrive at the result Z δ 1 1 ∇×B =∇×H. B2 = δA(r) 2µ0 V µ0 Thus example illustrates sufficiently the calculation of the functional derivatives also for the case of the order parameter Ψ. Therefore, we quote only the final equilibrium conditions. Superconductivity. From the requirement of stationarity with respect to variations of Ψ(r)∗ and δA(r) the Ginzburg–Landau equations for superconductivity follow 2 ie∗ ~2 2 A Ψ + aτ + b |Ψ| Ψ = 0 , (10.22a) − ∗ ∇− 2m ~ e∗ ~ (e∗ )2 2 ∗ ∗ J≡ [Ψ (∇Ψ) − (∇Ψ )Ψ] − A |Ψ| = ∇ × H . (10.22b) 2im∗ m∗ The former is a nonlinear Schrödinger equation for superconducting particles with the mass m∗ and charge e∗ . The latter equation, which determines the supercurrent density J , is Ampère’s law ∇ × H = J for static fields. The first term of the current density J is the canonical current, which is not gauge invariant. Only the account of the second term gives rise to a gauge invariant current density. Temperature-dependent coherence length. The Ginzburg–Landau equations are nonlinear, therefore an exact solution is possible only in special cases. If there is no magnetic field, it is consistent to put A = 0 everywhere. Relation (10.22b) is then automatically fulfilled, if Ψ is real. Consider a superconductor filling the half-space x > 0. Then Ψ is a function of x only and the equation (10.22a) assumes the form − ~2 d2 Ψ(x) + aτ Ψ(x) + bΨ(x)3 = 0 . 2m∗ dx2 (10.23) As a boundary condition, impose Ψ(0) = 0. The equation may be solved by multiplying by the factor Ψ′ (x) and constructing a first integral. The firstorder differential equation obtained is also solvable. It turns out that a 154 10. PHASE TRANSITIONS meaningful solution may only be found for temperatures τ < 0, i.e. T < Tc . For the order parameter the expression Ψ(x) = √ x ; ns tanh 2ξ follows, where ns = − and ξ=p (x > 0), aτ b ~ 2am∗ (Tc − T ) (10.24) (10.25) . (10.26) The constant ns = |Ψ(∞)|2 is the density of superconducting particles. The quantity ξ, which describes the thickness of the surface layer, is the temperature-dependent coherence length.. This is the typical length scale of the Ginzburg-Landau model. It is approximately equal to the coherence length √ ξ0 of the correlated electron pairs far from Tc , but since it diverges as Tc − T near Tc , it is bound to become much larger than ξ0 (which is independent of the temperature) close enough to the critical point. Meissner effect. In a weak magnetic field the vector potential A and the field strength H are small and the wave function may be formally expanded as Ψ = Ψ0 + Ψ1 + Ψ2 + . . . , where Ψ0 is the zero-field solution obtained above, and Ψn ∝ |A|n . Let the region x > 0 be superconducting, and the magnetic field directed along the y axis. Then A = A(x)ez ; B = ∇ × A = −A′ (x)ey . Rewrite equation (10.22a) in more detail, − ~2 2 ie∗ (e∗ )2 2 ∇ Ψ + A · ∇Ψ + A Ψ + aτ + b|Ψ|2 Ψ = 0. 2m∗ m∗ 2m∗ The zeroth order yields the previous equation (10.23). In the first order the vector potential is absent, because the vectors A and ∇Ψ0 are orthogonal. Thus, Ψ1 = 0, and the change of the wave function is of second order in A. Equation (10.22b) then implies that in the first-order accuracy − 1 1 (e∗ )2 A|Ψ0 |2 = ∇ × H = ∇ × (∇ × A) = − ∇2 A , m∗ µ0 µ0 (10.27) where the last from is a consequence of the relation ∇ · A = 0. Equation (10.27) is readily solved deep in the superconductor, where Ψ0 = cons. For the function A(x) with the account of relations (10.24) and (10.26) the following equation is obtained µ0 (e∗ )2 aτ d2 A(x) =− A(x) . 2 dx m∗ b (10.28) 10.3. GINZBURG–LANDAU THEORY 155 The physically meaningful solution is the exponentially falling off function x A(x) −→ const × exp − , (10.29) λ x→∞ where the parameter λ is the penetration depth s bm∗ . λ= aµ0 (e∗ )2 (Tc − T ) (10.30) Usually, λ ≫ 100 Å. Since deep in the superconductor A = 0, the magnetic field does not penetrate the matter. The superconductor is thus a perfect diamagnet. This is the Meissner effect. In the Ginzburg-Landau theory both the temperature-dependent coherence length (10.26)and the penetration depth (10.30) diverge in the same way, when the critical temperature is approached. Their temperatureindependent ratio s κ= λ m∗ = ∗ ξ ~e 2b , µ0 (10.31) the Ginzburg-Landau parameter is an important parameter of the theory, since its value determines the sign of the surface tension between the superconducting and normal phases and thus the character of the phase transition between them in strong magnetic fields. Critical field. Superconductors thus expel magnetic field. A strong enough magnetic field, however, destroys the superconducting state. The borderline value, the critical field Hc may be determined thermodynamically as follows. A superconducting body possesses a magnetic moment m = V M = −V H. The potential energy of this magnetic moment in the RH external field W = −µ0 0 m·dH = −µ0 12 m·H = 21 V µ0 H 2 is the energy in excess to the free energy of the superconductor and the energy of the magnetic field in the absence of the superconductor (which occupies the volume of the superconductor as well). The corresponding interaction energy between the magnetic moment of the same body in the normal state and the external field is negligible due to the small numerical value of the dia- and paramagnetic susceptibilities. Therefore, the difference between the energies of the body in the magnetic field in the homogeneous superconducting state and in the normal state is – up to surface effects – Fs − Fn = −V a2 (Tc − T )2 1 + V µ0 H 2 . 2b 2 (10.32) Just at the the critical field this difference vanishes, and near the critical temperature the thermodynamic critical field is Hc = a(Tc − T ) √ . µ0 b (10.33) 156 10. PHASE TRANSITIONS Surface effects, however, turn out to be of paramount significance in many practical superconductors. The point is that the transition from, say, the superconducting to the normal state requires initial nucleation of small normal state formations, and the appearance of these is hindered by the energy cost to build up the surface, when the surface tension between the normal and superconducting states is positive. In such a case of a superconductor of the I type the phase transition takes place in fields larger than the thermodynamic critical field so that the magnetic field penetrates a large volume at once (the surface area between the ordinary and superconducting phases is minimized). In a superconductor of the II type the surface tension of the interface between the superconducting and normal states is negative, which leads to the nucleation of the normal phase in a superconducting bulk at field strengths less than the thermodynamic critical field (the borderline value is the lower critical field Hc1 ) and to the nucleation of superconducting phase at field strength larger than the thermodynamic critical field (the borderline value is the upper critical field Hc2 ). In these materials nothing remarkable happens at Hc . In superconductors of the II type the magnetic flux penetrates the superconducting bulk as thin filaments (vortices) forming a vortex lattice (Abrikosov lattice). The flux in the filaments is quantized with the flux quantum h/e∗ . Observations on this phenomenon as well as on quantization of magnetic flux through a superconducting ring have shown that the charge of the superconducting particle e∗ = −2e, where −e is the electron charge. The supercurrent is carried by bound pairs of electrons. These Cooper pairs are loose formations with the diameter of the of the coherence length ξ0 and thus much larger than the distances between the conduction electrons. The quantization of the magnetic flux may be readily demonstrated in geometries of a superconducting ring in a magnetic field and for a normalstate filament aligned with an external magnetic field in the superconducting bulk. Imagine a closed contour around the filament or along the ring deep in the bulk superconductor so that on the loop the magnetic induction vanishes and the modulus of the order parameter is constant. From relation (10.22b) and the representation Ψ = |Ψ|eiφ it then follows that on the contour e∗ ∇φ = A. (10.34) ~ Integrating over the closed contour we obtain, by virtue of the Stokes theorem, I Z I e∗ e∗ e∗ A · dl = n · ∇ × A dS = ΦB , (10.35) ∆φ = ∇φ · dl = ~ ~ ~ where ∆φ is the change of the phase of the wave function after traversing over the contour and ΦB is the magnetic flux through a surface, whose boundary is the closed contour at hand. The wave function must, however, be a single-valued function of the position, which imposes the condition 10.4. FLUCTUATIONS IN LANDAU THEORY 157 ∆φ = 2πn with an integer n. Thus, the magnetic flux through a superconducting ring or normal-state filament is quantized as Z h ΦB = n · B dS = ∗ n . (10.36) e In particular, this condition imposes restrictions on the appearance of normal-state vortices in the phase transition to the normal state in a type II superconductor in magnetic field. 10.4 Fluctuations in Landau theory Landau theory is based on an effective thermodynamic potential describing the system in incomplete equilibrium described by the order parameter ϕ. The probability of such a state may be estimated in a manner similar to that used in the Einstein theory of fluctuations. Consider the classical canonical ensemble (for simplicity of notation). The partition function is the measure of the phase state with the weight e−βH : Z ′ dΓ e−βH(p,q) . (10.37) Z= The measure of the part of the phase state corresponding to the incomplete equilibrium may be written in a similar form by imposing the condition ϕ = ϕ(p, q), where ϕ(p, q) is the order parameter expressed as function of the variables of the phase space. Formally this is effected as Z ′ dΓ δ (ϕ − ϕ(p, q)) e−βH(p,q) . (10.38) Z(ϕ) = Obviously Z = Z dϕ Z(ϕ) and the relative frequency at which the incom- plete equilibrium occurs in the phase space is Pr(ϕ) = Z(ϕ) = eβ[F −F (ϕ)] , Z (10.39) where the effective free energy is F (ϕ) = −T ln Z(ϕ) . (10.40) Substituting the expression for F (ϕ) in the Landau theory (inhomogeneous system) Z F (ϕ) = F0 + d3 r g(∇ϕ)2 + a(T − Tc )ϕ2 + Bϕ4 (10.41) we arrive at the probability density for the order parameter in the form Pr(ϕ) ∝ e−β R d3 r [g(∇ϕ)2 +a(T −Tc )ϕ2 +Bϕ4 ] . (10.42) 158 10. PHASE TRANSITIONS Calculations with such a weight are only possible in the form of an expansion in B. The leading order is given by the Gaussian distribution corresponding to B = 0. Already in this approximation problems in definition of the mathematical quantities involved appear. For instance, the correlation function of the order parameter is Z Y R 3 2 2 dϕ(r) ϕ(r)ϕ(r ′ ) e−β d r [g(∇ϕ) +a(T −Tc )ϕ ] hϕ(r)ϕ(r ′ )i = r Z Y dϕ(r) e−β R d3 r [g(∇ϕ)2 +a(T −Tc )ϕ2 ] . (10.43) r In continuum space both the denominator and the numerator consist of a formally infinite-fold integral, to which some meaning should be prescribed. The simplest thing to do is to put the system on a lattice in a finite box, which resricts the integrals over values of the order parameter at different positions to a finite number. In case of a Gaussian integral for a correlation function it is also possible to use the expression (6.51), in which the dimension of the space of integration does not appear explicitly. This means that hϕ(r)ϕ(r ′ )i = −1 T T (r, r ′ ) = G(r − r ′ ) , −g∇2 + a(T − Tc ) 2 2 (10.44) where G(r − r ′ ) is the Green function of the operator −g∇2 + a(T − Tc ), i.e. the solution of the equation −g∇2 + a(T − Tc ) G(r) = δ(r) (10.45) with vanishing boundary condition at infinity. More constructively calculation of the correlation function is convenient to carry out in the wave-vector space. Put the system in a, say, cubic box and define the coefficients of the Fourier series as Z 1 d3 r e−ik·r ϕ(r) , (10.46) ϕ(k) = V and calculate the Fourier coefficients of the correlation function Z Z ′ ′ 1 3 d r d3 r ′ e−ik·r−ik ·r hϕ(r)ϕ(r ′ )i = hϕ(k)ϕ(k′ )i . 2 V (10.47) On the other hand, assuming the usual translation invariance we may write hϕ(r)ϕ(r ′ )i = C(r − r ′ ) (10.48) and 1 V2 Z d3 r Z ′ ′ d3 r ′ e−ik·r−ik ·r C(r − r ′ ) Z Z ′ 1 3 ′ −i(k+k′ )·r ′ 1 d r e d3 r e−ik·(r−r ) C(r − r ′ ) = δk′ ,−k C(k) . = V V (10.49) 10.4. FLUCTUATIONS IN LANDAU THEORY 159 Comparison of relations (10.47) and (10.49) yields (10.50) hϕ(k)ϕ(k′ )i = δk′ ,−k hϕ(k)ϕ(−k)i = δk′ ,−k h|ϕ(k)|2 i , since for a real ϕ(r) from (10.46) it follows that ϕ∗ (k) = ϕ(−k). Express now the Gaussian weight in terms of ϕ(k). Substitution of the Fourier series of ϕ(r) yields Z d3 r g(∇ϕ)2 + a(T − Tc )ϕ2 Z XX ′ = d3 r −gk · k′ + a(T − Tc ) ϕ(k)ϕ(k′ ) eik·r+ik ·r k k′ =V X gk 2 + a(T − Tc ) |ϕ(k)|2 . (10.51) k In view of expressions (10.50) and (10.51) it appears convenient to carry out the integration over the real and imaginary parts of ϕ(k). These are not all independent variables, since (10.52) ϕ(k) = ϕR (k) + iϕI (k) = ϕ∗ (−k) = ϕR (−k) − iϕI (−k) . Therefore, it is sufficient to integrate over values of ϕR (k) and ϕI (k) in a "half space" of wave vectors, chosen, for instance, by the condition k1 ≥ 0. In calculation of the correlation function h|ϕ(k)|2 i this feature is unimportant, however, because all integrals over values of ϕR (k′ ) and ϕI (k′ ) with k′ 6= k cancel in the expression P Z Y 2 −βV [gk2 +a(T −Tc )] |ϕ(k)|2 2 k dϕR (k)dϕI (k) ϕR (k) + ϕI (k) e h|ϕ(k)|2 i = k ,k1 ≥0 Z Y dϕR (k)dϕI (k) e −βV P gk2 +a(T −T k [ c) . ] |ϕ(k)|2 k ,k1 ≥0 (10.53) Thus, we are left with the following ratio of twofold Gaussian integrals h|ϕ(k)|2 i = Z∞ dϕR (k) −∞ Z∞ −∞ Z∞ −∞ 2 2 dϕI (k) ϕ2R (k) + ϕ2I (k) e−βV [gk +a(T −Tc )] |ϕ(k)| dϕR (k) Z∞ dϕI (k) e−βV [gk 2 +a(T −Tc )] |ϕ(k)| −∞ . 2 (10.54) Calculation yields h|ϕ(k)|2 i = therefore hϕ(k)ϕ(k′ )i = T 1 , 2 V gk + a(T − Tc ) 1 T δk′ ,−k 2 . V gk + a(T − Tc ) (10.55) (10.56) 160 10. PHASE TRANSITIONS We see that the length scale of the correlation function, the correlation length, is r g ξ= , T > Tc . (10.57) a(T − Tc ) Below Tc a similar relation follows: r g ξ= , 2a(Tc − T ) T < Tc . (10.58) Therefore, in Landau theory values of the critical exponents of the correlation length ( −ν (T − Tc ) , T > Tc , ξ(T ) ∼ H = 0, (10.59) −ν ′ (Tc − T ) , T < Tc , are equal and ν = ν ′ = 12 . Expression for the correlation function as a function of the position vector may now be calculated as the Fourier series ′ T X eik·(r−r ) . V gk 2 + a(T − Tc ) k k′ k (10.60) Further, it is customary to pass to the thermodynamic limit, which produces the familiar integral sum and the correlation function may be calculated as the inverse Fourier transform ′ Z ′ eik·(r−r ) T e−|r−r |/ξ d3 k = . (10.61) hϕ(r)ϕ(r ′ )i = T (2π)3 gk 2 + a(T − Tc ) 4πg |r − r ′ | hϕ(r)ϕ(r ′ )i = XX ′ ′ hϕ(k)ϕ(k′ )i eik·r+ik ·r = At the critical temperature ξ → ∞ and hϕ(r)ϕ(r ′ )i = T , 4πg|r − r ′ | T = Tc , 1 , |r − r ′ |1+η T = Tc (10.62) which fixes the value of the critical exponent η of the correlation function hϕ(r)ϕ(r ′ )i ∼ in the Landau theory as η = 0. (10.63) 10.5 Problems Problem 10.1. Consider the following expansion in the order parameter φ of the Gibbs free energy G(p, T, φ) = G0 (p, T ) + a(T − T0 )φ2 − Cφ3 + Bφ4 , where a, B and C are positive constants. Find the equilibrium value of the order parameter, show that there is a first order phase transition in this system and find the transition temperature. 10.5. PROBLEMS Problem 10.2. Calculate the latent heat of the phase transition in the model of the preceding problem. Problem 10.3. There are systems in which (on the p,T plane) a line of second-order transitions changes into a line of first-order transitions at the tricritical point. Near the tricritical point the Landau expansion of the Gibbs free energy may be written as G(φ, p, T ) = G0 (p, T ) + Aφ2 + Bφ4 + Dφ6 , where D > 0. The line of second-order transitions is determined by the conditions A(TC (p)) = 0, B > 0. On the line of first-order transitions B < 0 so that at the tricritical point A = B = 0. Find the value of the order parameter on the line of first-order transitions in the ordered phase and establish a connection between A, B and D (which is the equation of the line of first-order transitions). 161 Index absorption ratio, 121 additivity, 56 adiabatic approximation, 121 adiabatic change, 77 adiabatic constant, 31 adiabatic demagnetization, 89 critical exponents, 144, 149, 151 critical point, 144 Curie’s law, 5, 89 De Broglie wave length thermal, 94 Debye model, 124 degeneracy, 86 degeneration temperature, 130 degree of advancement, 36 degree of freedom, 29 degree of reaction, 36 density of states, 63 density operator, 59 diamagnetism of Landau, 138 of Meissner, 155 differential, 3 differential form, 3 binomial distribution, 86 black body, 115, 120 Bohr’s magneton, 135 Boltzmann distribution, 69 of MB gas, 91 Born–Oppenheimer approximation, 121 Bose condensation, 110, 113 Bose–Einstein statistics, 62, 101 boson, 102 canonical distribution, 69 classical, 72 canonical ensemble, 69 canonical equations of motion, 49 Carnot’s process, 10 chemical potential, 8 of electron gas, 132 of MB gas, 95 chemical reaction, 36 Clausius–Clapeyron equation, 42 coefficient of thermal expansion, 21 coexistence, 42, 44 coherence length, 152 temperature-dependent, 154 compressibility, 5, 22 conduction electrons, 128 continuity equation, 51 convective time derivative, 51 Cooper pairs, 152, 156 correlation length, 160 critical point, 44 temperature, 111 efficiency, 9, 11 effusion, 93 electron gas magnetism, 135 endothermic reaction, 38 energy free, see free energy free spin system, 87 internal, 6, 15, 30 of black-body radiation, 118, 119 energy band, 142 energy surface, 52 ensemble, 50 canonical, 69 grand canonical, 75 microcanonical, 53, 54, 66 enthalpy, 17, 18, 38, 87 entropy, 9, 12, 24, 30–32, 63, 70 Boltzmann, 56 of BE and FD gases, 104 of black-body radiation, 119 162 INDEX of free spin system, 88 of MB gas, 95 statistical, 55 equation of state, 4 of black-body radiation, 119 equilibrium conditions, 24 equilibrium constant, 37 equilibrium distributions, 69 equipartition principle, 92 ergodic flow, 53 ergodic theory, 54 exothermic reaction, 38 extensive variables, 2 factorizing, 98 Fermi energy, 129 Fermi gas, 128 degenerate, 128 ideal, 128 Fermi surface, 129 Fermi temperature, 129 Fermi–Dirac statistics, 62, 102 fermion, 102 Fermmi momentum, 129 ferroelectricity, 145 ferromagnetic ordering, 147 flow, 52 fluctuations, 71, 76, 78, 80, 82 flux quantum, 156 Fock space, 62 free energy, 19, 70 Gibbs, 19 Helmholtz, 19 of black-body radiation, 119 of diatomic ideal gas, 99 of MB gas, 95 free expansion, 31 fugacity, 75 functional, 152 functional derivative, 152 gauge invariance, 152 generalized displacement, 7 generalized force, 7 generating function, 79 Gibbs distribution, 69 Gibbs function, 19, 32, 33, 37 163 of free spin system, 88 Gibbs paradox, 33 Gibbs phase rule, 40 Gibbs–Duhem equation, 20 Ginzburg–Landau equations, 153 Ginzburg–Landau theory, 151 Ginzburg-Landau free energy, 152 Ginzburg-Landau parameter, 155 grand canonical distribution, 75 grand canonical ensemble, 75 grand potential, 20, 76 of BE-gas, 103 of FD gas, 103 of MB gas, 96 gyromagnetic ratio, 87 Hall effect, 140 quantized, 140 Hall resistivity, 140 Hamilton function, 49 harmonic approximation, 122 heat capacity, 8, 22, 29 of electron gas, 133 heat capacity ratio, 31 heat of reaction, 37 heat pump, 10 heteropolar, 97 hole, 140 homopolar, 97, 100 Hooke’s law, 7 hysteresis, 1 ideal gas, 5 classic, 29 classical, 5, 91 diatomic, 97 ideal system, 86 integrating factor, 3 intensive variables, 2 internal energy, 6 inversion temperature, 19 irradiance, 121 Joule process, 31 Joule–Thomson process, 18 Lagrange multiplier, 69, 75 INDEX 164 Landau free energy, 152 Landau levels, 136 Landau theory, 147 latent heat, 41 law of mass action, 37 Legendre transform, 3, 19 Legendre transforms, 17 Liouville equation, 52 Liouville operator, 52 Liouville theorem, 51 Lorentz force, 140 macroscopic wave function, 152 macrostate, 79 magnetic moment, 87 magnetization, 5 of free spin system, 88, 89 Maxwell construction, 45 Maxwell distribution, 92 Maxwell relations, 16, 17, 19–21 Maxwell–Boltzmann gas, 65 Meissner effect, 155 metastable, 45, 101 minimal work reversible, 83 mixed state, 59 mixing entropy, 32 mixing flow, 54 natural variables, 15 Nernst’s law, 12 non-ergodic, 101, 120 non-ergodic flow, 52 normal coordinates, 122 normal distribution, 87 nuclear demagnetization, 89 nuclear spin, 98 number operator, 75 occupation number, 103 order parameter, 144 ortohydrogen, 100 osmosis, 33, 35 osmotic pressure, 35 parahydrogen, 100 paramagnetism, 5 of Curie, 89 Pauli, 135 particle flux, 93 particle flux density, 93 partition function canonical, 69 diatomic ideal gas, 98 grand canonical, 75, 96, 102 microcanonical, 66 of BE gas, 103 of FD gas, 103 of free spin system, 88 of MB gas, 95, 96 penetration depth, 155 perfect Bose-Einstein gas, 109 phase diagram, 40 phase equilibrium, 39 phase separation, 45 phase space, 49 measure, 50 phase transition, 40 continuous, 41, 144 discontinuous, 144 ferromagnetic, 145 first order, 41, 144 of second order, 147 second order, 41, 144 phase transtion ferromagnetic, 147 phonon, 122 photon, 117 Planck distribution, 117 Planck’s radiation law, 118 Poisson brackets, 49 probability density, 50 process, 2 Carnot’s, 10 cyclic, 2, 9 irreversible, 2 isenthalpic, 18 isergic, 31 reversible, 2 pure state, 59 quasistatic, 2 radiance, 120 radianssi, 120 INDEX Radiant excitance, 120 Rayleigh–Jeans law, 118 real gas, 5 reflection ratio, 121 resistivity, 141 response thermodynamic, 5, 21 rotation, 97, 99 saturated vapour, 41 solution, 33 Sommerfeld expansion, 132 specific heat, 9 specific heat capacity, 29 speed of sound, 22 spin system free, 86 stability, 24, 25 stability conditions, 24 state variables, 1 statistical sum microcanonical, 55 Stefan–Boltzmann constant, 119 Stefan–Boltzmann law, 121 stoichiometric coefficient, 36 superconductivity, 145, 151 superconductor, 153 superfluidity, 145 surface density, 138 surface tension, 7 susceptibility, 6 Landau diamagnetic, 138 magnetic, 89 Pauli paramagnetic, 135 SVN-system, 7 symmetry breaking spontaneous, 145 system, 1 closed, 1 isolated, 1 open, 1 temperature, 6, 70 absolute, 11 negative, 90 thermal efficiency, 10 165 thermal expansion coefficient, 5, 18 thermal wavelength, 73 thermalization, 91 thermodynamic equilibrium, 1 local, 1 thermodynamic limit, 63 thermodynamic potential, 15 transmission ratio, 121 triple point, 40 työ vapaa, 19 universality class, 146 Van der Waals equation of state, 44 Van’t Hoff equation, 35 vibration, 97, 99 virial coefficient, 5 virial expansion, 5 Wien’s law, 118 work, 6 electromagnetic, 7 free, 16, 17, 20, 21 Young’s modulus, 7 zeroth law, 6