* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Quantum Stat Mech Primer
Bohr–Einstein debates wikipedia , lookup
Probability amplitude wikipedia , lookup
Double-slit experiment wikipedia , lookup
Scalar field theory wikipedia , lookup
Identical particles wikipedia , lookup
Ferromagnetism wikipedia , lookup
Renormalization group wikipedia , lookup
Renormalization wikipedia , lookup
Atomic theory wikipedia , lookup
Path integral formulation wikipedia , lookup
Coherent states wikipedia , lookup
Quantum field theory wikipedia , lookup
Copenhagen interpretation wikipedia , lookup
Quantum electrodynamics wikipedia , lookup
Wave–particle duality wikipedia , lookup
Quantum dot wikipedia , lookup
Quantum entanglement wikipedia , lookup
Relativistic quantum mechanics wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
Bell's theorem wikipedia , lookup
Many-worlds interpretation wikipedia , lookup
Particle in a box wikipedia , lookup
Quantum fiction wikipedia , lookup
Orchestrated objective reduction wikipedia , lookup
Quantum computing wikipedia , lookup
Hydrogen atom wikipedia , lookup
Quantum machine learning wikipedia , lookup
Quantum teleportation wikipedia , lookup
Quantum group wikipedia , lookup
Quantum key distribution wikipedia , lookup
Interpretations of quantum mechanics wikipedia , lookup
Symmetry in quantum mechanics wikipedia , lookup
EPR paradox wikipedia , lookup
History of quantum field theory wikipedia , lookup
Quantum state wikipedia , lookup
Quantum cognition wikipedia , lookup
Canonical quantization wikipedia , lookup
Concepts in Materials Science I Quantum Statistical Mechanics Primer VBS/MRC Quantum StatMech – 0 Concepts in Materials Science I Why Quantum StatMech? Nature is really quantum mechanical...classical statmech is an approximation (and typically a good one at “high temperatures”) Classical problem cases: Specific Heat of Solids - Why does it go down at lower temperatures? Drudé’s problem: Why are the electrons not contributing to specific heat? VBS/MRC Quantum StatMech – 1 Concepts in Materials Science I Quantum StatMech – Main Ideas What is different from Classical StatMech? Mainly, Two Things A description in terms of quantum mechanical states (energy levels, typically) is necessary (as opposed to the classical phase space description) Counting has to be done carefully when there are more particles than one! Fermions and Bosons have different statistics and give rise to completely different physics...such distinction does not exist in classical statmech VBS/MRC Quantum StatMech – 2 Concepts in Materials Science I Quantum StatMech – Main Ideas EAPP remains unchanged...every quantum mechanical state with the same energy is equally likely Entropy is S = kB ln Ω...Ω is the number of quantum mechanical states...microcanoical ensemble can be worked out this way... Taking a system to have levels labeled by n with P −βEn energy En , the partition function Z = n e ...the probability of state n is P (n) = A = −kB T ln Z! e−βEn Z , and The key thing is in counting! Bose is not Fermi! And, Fermi is not Bose! VBS/MRC Quantum StatMech – 3 Concepts in Materials Science I Quantum Statmech – How it works! Look at simple examples where counting will not be an issue Example: Paramagnetism Example: Thermal properties of solids (specific heat, thermal expansion etc.) VBS/MRC Quantum StatMech – 4 Concepts in Materials Science I Paramagnetism On application of magnetic field B, a magnetization M appears in the material with is in the same direction as B ∂A M = − ∂B = χB, χ is magnetic susceptibility VBS/MRC Curie’s Law χ ∼ 1 T (Gd salt, Kittel) Quantum StatMech – 5 Concepts in Materials Science I Paramagnetism Simplest model...Lattice containing N sites each with one unpaired s-electron (spin 12 , µ = µb , no net orbital angular momentum) P i i tot tot Hamiltonian of the system −µSz B; Sz = i Sz , Sz is the z-component of spin operator at site i (energy states from quantum mechanics!) Each site can have spin pointing along the magnetic field Szi = 1 or Szi = −1...Thus there are a total of 2N configurations! Since spins at different site are independent QN Z = i=1 Zi , with Zi = e−β(−µB) + e−β(+µB) = 2 cosh(βµB) and Z = (2 cosh(βµB))N VBS/MRC Quantum StatMech – 6 Concepts in Materials Science I Paramagnetism A = −kB T ln Z = −N kB T ln (2 cosh(βµB))...Thus ∂A = N µ tanh βµB M = − ∂B For “small” B, M = herself! N µ2 kB T B or χ = N µ2 kB T , ...its Curie Physics...spins want to align with magnetic field, but they get “kicked” around by temperature Most paramagnetic materials will not be spin 12 ...could be larger...but same theory can be applied...and works wonders! Take metals, they should also obey this, right? VBS/MRC Quantum StatMech – 7 Concepts in Materials Science I Paramagnetic Salts Works wonders! (Kittel) VBS/MRC Quantum StatMech – 8 Concepts in Materials Science I And, Metals Works not wonders! (Kittel) Why? Its Pauli, not Curie! VBS/MRC Quantum StatMech – 9 Concepts in Materials Science I Thermal Properties of Solids Specific heat...why it falls with temperature? Whence thermal expansion? Strategy...compute free energy A via partition function Z...everything else falls through Work in 1D, model illustrates basic ideas... VBS/MRC Quantum StatMech – 10 Concepts in Materials Science I Simple 1D Model of Solid A chain of N atoms (mass m) interacting with their neighbors via a potential energy φ(|xi −xj |) (φ is given [how?]) ξ a a Model assumption...fix the neighboring atoms at an average distance a (which depends on temperature) and imagine the central atom to be “jiggling” described by ξ VBS/MRC Quantum StatMech – 11 Concepts in Materials Science I Simple 1D Model of Solid The potential energy of the atom = 12 (φ(a − ξ) + φ(a + ξ)) ≈ φ(a) + 12 φ00 (a)ξ 2 (call φ00 (a) = mω 2 (a)), note ω in general depends on a) Hamiltonian of atom : mω 2 ξ 2 P2 H = 2m + 2 + φ(a)...harmonic oscillator! There are N of them (treated independently as an approximation)...treat these quantum mechanically (energy states from quantum mechanics) Energy states En = (n + 12 )~ω(a) + φ(a) VBS/MRC Quantum StatMech – 12 Concepts in Materials Science I 1D Solid: Specific Heat The probability of state n ∼ e−βEn , or partition P∞ −βEn function is Z = n=0 e Sum can be evaluated easily : Z = e−βφ(a) 2 sinh1 β~ω ( 2 ) ln Z Internal energy U = − ∂∂∂β = φ(a) + ~ω 2 + 2 ~ω Specific Heat : Cv = ∂U = k B ∂T kB T ~ω eβ~ω −1 ~ω kB T e ~ω kB T e −1 2 kB T ~ω =⇒ Cv → kB ...nice! 2 − k~ωT ~ω kB T ~ω =⇒ Cv → kB T e B → 0...Nicer! VBS/MRC Quantum StatMech – 13 Concepts in Materials Science I Specific Heat 1D and 3D solids 1 shcv.nb Cv kB 1 0.8 0.6 0.4 0.2 1 2 3 T kB 4 ÑΩ The specific heat of an oscillator vanishes exponentially when kB T ~ω...”quantum alacrity” The above model was proposed by Einstein for 3D solid (just 3N oscillators instead of N)! So why do we see T 3 specific heat in insulating solids? VBS/MRC Quantum StatMech – 14 Concepts in Materials Science I Specific Heat, Debye Theory In real solids, the phonon frequencies are not equal VBS/MRC In particular long wavelength phonons (elastic waves) have very low freqency...Thus, for any given temperature there are many phonons which satisfy kB T ~ω Quantum StatMech – 15 Concepts in Materials Science I Specific Heat, Debye Theory It can be shown that the fraction of these phonons is 3 kB T where ωD is the Debye frequency ~ωD (corresponding temperature ΘD )...hence T 3 specific heat VBS/MRC Quantum StatMech – 16 Concepts in Materials Science I Thermal Expansion, 1D Math easier at high temperatures A(a, T ) = −kB T ln Z ≈ φ(a) + kB T ln ~ω(a) kB T To find equilibrium lattice parameter set ∂A 0 (a) + k T 1 ∂ω = 0 =⇒ φ B ω(a) ∂a ∂a Say φ(a) = φ(a0 ) + Then α = φ000 (a0 ) − φ00 (a0 ) = 1 a−a0 T a0 ln ω − ∂ ∂a = φ00 (a0 ) 2 (a − a ) 0 2! kB a0 φ00 (a0 ) × + φ000 (a0 ) 3 (a − a ) 0 3! φ000 (a0 ) − φ00 (a0 ) ...note that typically ω decreses with increase in a Thermal expansion is due to anharmonicity! Harmonic crystals will not expand! VBS/MRC Quantum StatMech – 17 Concepts in Materials Science I Thermal Expansion in 3D Solids General results α = phonons γCv 3B , γ=− P ∂ ln ωi i ∂V ,i over all Grüneisen parameter γ, roughly 2 in insulators Do we expect all crystals to expand on heating? VBS/MRC Quantum StatMech – 18 Concepts in Materials Science I Negative Thermal Expansion! Crystalline materials with -ve α!! Why? VBS/MRC Quantum StatMech – 19 Concepts in Materials Science I Now for some Quantum Counting! When there is more than one particle in a quantum system, one has to be careful regarding counting Identical particles are indistinguishable Fermions: Only one particle is allowed in a quantum state Bosons: Any number of particles can occupy a state Assume that the energy levels εi are solved for and the state i is gi fold degenerate There are a total of N particles at temperature T ; what fraction of the gi states at energy level i will be occupied? VBS/MRC Quantum StatMech – 20 Concepts in Materials Science I Fermi-Dirac Statistics for Fermions Strategy: Find Entropy as a function of ni (number occupying level i), then, A; Minimize A with respect to ni P Note i ni = N ...this is a tough condition to enforce directory...so we will do the following trick.. Assume that there are ni particles occupying level i which is gi fold degenerate...How many ways are there ! to do this?... ni !(ggii−n i )! The total number of ways is just the product of Q ! number of ways at each level Ω = i ni !(ggii−n i )! VBS/MRC S =PkB ln Ω = kB i (gi ln (gi ) − ni ln(ni ) − (gi − ni ) ln(gi − ni )) Quantum StatMech – 21 Concepts in Materials Science I Fermi-Dirac Statistics for Fermions P Internal energy U = i ni εi P A = j n j εj − P kB T j (gj ln (gj ) − nj ln(nj ) − (gj − nj ) ln(gj − nj )) − P µ( j nj − N ) where µ is the chemical potential (ensure #particles is N via a Lagrange multiplier trick!) ∂A ∂ni = 0 =⇒ ni = gi eβ(εi −µ) +1 Determine µ from As T → 0, ni gi =1 P (Fermi-Dirac) gj j eβ(εj −µ) +1 = (εi ≤ µ), ngii = 0 N ,µ = µ(T, N ) (εi > µ) Chemical potential µ at T = 0 is called Fermi energy εF , and at T = 0 all levels below Fermi level are fully occupied! VBS/MRC Quantum StatMech – 22 Concepts in Materials Science I Fermi-Dirac Statistics for Fermions fd.nb n H¶L g H¶L 1 1 kB T=0 0.8 0.6 0.4 0.2 0.5 kB T =0.03 ¶ Μ 1 1.5 2 Μ In the thermodynamic limit, energy levels become very closely spaced thus gi → g(ε)dε...to get n(ε)dε = g(ε)dε eβ(ε−µ) +1 VBS/MRC Quantum StatMech – 23 Concepts in Materials Science I Fermi-Dirac Statistics for Fermions g(ε)dε Chemical potential obtained from 0 eβ(ε−µ) =N +1 R ∞ ε g(ε)dε Internal energy U = 0 eβ(ε−µ) +1 , etc.. R∞ Fermi-Dirac Strategy for Fermions: 1. Obtain density of states g(ε) by solving the Quantum Mechanics problem 2. Obtain chemical potential 3. Use F-D distribution above to obtain required thermodynamic potentials VBS/MRC Quantum StatMech – 24 Concepts in Materials Science I Free Electron Gas Step 1: Obtain g(ε)...g(ε) = V 2π 2 2m 3/2 √ ε ~ Step 2: Obtain chemical potential ( n(ε) @T = 0, µ = εF ; n(ε) = 1 (ε ≤ ε ), F g(ε) = 0 (ε > εF )) g(ε) h 2 i2/3 R εF 2 ~ 3π N Thus, 0 g(ε)dε = N =⇒ εF = 2m V R εF Step 3: T = 0, U = 0 g(ε)dε = 35 N εF How about specific heat ∂U ∂T ? For that we have to calculate U as a function of T ...can do this rigorously, but we will do it in an approximate way Note, for free electron gas, g(εF ) = VBS/MRC 3N 2 εF Quantum StatMech – 25 Concepts in Materials Science I Fermi Energies in Metals VBS/MRC Fermi temperatures (kB TF = εF ) are ∼ 105 K, i. e. kB TROOM /εF ∼ 10−3 (key point!) Quantum StatMech – 26 Concepts in Materials Science I Specific Heat of Free Electron Gas 1.00 ni/gi 0.75 kBT 2 0.50 0.25 0.00 0.0 VBS/MRC 0.5 1.0 ε/µ 1.5 2.0 On increasing temperatures electrons occupying states just below the Fermi level are excited to states just above the Fermi level Quantum StatMech – 27 Concepts in Materials Science I Specific Heat of Free Electron Gas Number of electrons whose states are changed ∼ g(εF ) kB4T (average increases of energy ∼ kB T 4 )...Total energy increase ∼ 2 kB T2 g(εF ) 4 2 2 kB T2 kB U (T ) ≈ ∼ g(εF ) 8 =⇒ CV = g(εF ) 4T Noting g(εF ) = 32 εNF , 1 T 3 CV = 12 TF 2 N kB 32 N kB @TROOM ...Drudé 3 5 N εF problem resolved!! Party starts again! Physics: Due to Pauli, only a fraction of electrons can participate in the thermal dance...the others continue their “worker bee” existence! VBS/MRC Quantum StatMech – 28 Concepts in Materials Science I Specific Heat of Free Electron Gas (Kittel) Physics: Due to Pauli, only a fraction of electrons can participate in the thermal dance...the others continue their “worker bee” existence! VBS/MRC Quantum StatMech – 29 Concepts in Materials Science I Bose-Einstein Statistics Many systems of interest...light, phonons, He4 , Alkali metal atoms etc.. Number of ways of distributing ni particles becomes Q (gi +ni )! gi ...resulting in n = (BE Distribution) i k gi !ni ! eβ(ε−µ) −1 Strategy is same as that of FD case 1. Obtain density of states g(ε) by solving the Quantum Mechanics problem 2. Obtain chemical potential 3. Use B-E distribution above to obtain required thermodynamic potentials VBS/MRC Quantum StatMech – 30 Concepts in Materials Science I Bose-Einstein Condensation A very interesting thing happens (in systems where particle number is conserved µ 6= 0)...Bose-Einstein condensation As temperature is lowered a finite fraction of particles N occupy the lowest energy state...temperature required are 10−9 K!!!! This is a “new state of matter”! (Colorado Group) VBS/MRC Quantum StatMech – 31 Concepts in Materials Science I Well, Whats left to do? A LOT! Phase Transitions (Order-disorder, Magnetism, Ferroelectricity) need to consider interactions...mean field theory, renormalization group etc.. Transport Properties (“Dynamics”) ... Many challenges remain! VBS/MRC Quantum StatMech – 32 Concepts in Materials Science I Materials Science – The Grand Scheme Material: Atoms in a particular configuration Properties: Which atoms? What configuration? The Scheme of Materials Design Quantum Mechanics How does energy depend on configuration? g(ε) Materials Questions What configuration gives desired properties? Statistical Mechanics "Favoured" configuration P Z= e−βε A = −kB T ln Z VBS/MRC Quantum StatMech – 33