Download Statistical Physics Exercises

Licence 3 et Magistère de physique
Statistical Physics
Exercises
October 22, 2015
Contents
Some useful formulae
0.1 Euler’s Gamma function
0.2 Gaussian integrals . . .
0.3 Euler’s Beta function . .
0.4 Stirling’s formula . . . .
0.5 Binomial formula . . . .
0.6 Other useful integrals .
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3
3
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4
4
TD
1.1
1.2
1.3
1.4
1: Orders of magnitude – Probabilities
Gases and solids – Orders of magnitude . . . . . . . . . . . .
Tossing a coin – Binomial law . . . . . . . . . . . . . . . . . .
Rain drops – Statistics of independent events . . . . . . . . .
Maxwell’s distribution – Doppler broadening of a spectral line
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5
5
6
7
8
TD
2.1
2.2
2.3
2.4
2.A
2: Phase space, ergodicity and density of states
Phase space of a 1D harmonic oscillator . . . . . . . . . . . . . .
Volume of a hypersphere . . . . . . . . . . . . . . . . . . . . . . .
Density of states of free particles . . . . . . . . . . . . . . . . . .
Classical and quantum harmonic oscillators . . . . . . . . . . . .
Appendix: Semi-classical rule for counting states in phase space
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10
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11
11
12
TD
3.1
3.2
3.3
3: Fundamental postulate and microcanonical ensemble
The monoatomic ideal gas and the Sackur-Tetrode formula – Gibbs paradox . . .
Thermal contact between two cubic boxes . . . . . . . . . . . . . . . . . . . . . .
Paramagnetic crystal - Negative (absolute) temperatures . . . . . . . . . . . . . .
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15
TD
4.1
4.2
4.3
4.4
4.5
4: System in contact with a thermostat – Canonical ensemble
Monoatomic ideal gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Ideal, confined, nonideal, etc... gases. . . . . . . . . . . . . . . . . . . . . . .
Gases of indistinguishable particules in a harmonic well . . . . . . . . . . .
Partition function of a particule in a box – the role of boundary conditions
Diatomic ideal gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4.6
Paramagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6.1 Classical calculation: Langevin paramagnetism. . . . . . . .
4.6.2 Quantum mechanical calculation : Brillouin paramagnetism
4.A Appendix: Semiclassical summation rule in the phase space. . . .
4.B Appendix: Canonical mean of a physical quantity . . . . . . . . .
TD
5.1
5.2
5.3
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19
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20
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22
5: Thermodynamic properties of harmonic oscillators
Lattice vibrations in a solid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Thermodynamics of electromagnetic radiation . . . . . . . . . . . . . . . . . . . .
Equilibrium between matter and light , and spontaneous emission . . . . . . . . .
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24
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TD 6: Systems in contact with a thermostat and a particle
Canonical ensemble
6.1 Ideal Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Adsorption of an ideal gas on a solid interface . . . . . . . . .
6.3 Density fluctuations in a fluid – Compressibility . . . . . . . .
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reservoir – Grand
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33
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TD 8: Quantum statistics (2) – Bose-Einstein
8.1 Bose-Einstein condensation in a harmonic trap . . . . . . . . . . . . . . . . . . .
34
34
TD 9: Kinetics
9.1 Thermo-ionic effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2 Effusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
36
36
TD
7.1
7.2
7.3
7.4
7.5
7: Quantum statistics (1) – Fermi-Dirac statistics
Ideal Fermi gas . . . . . . . . . . . . . . . . . . . . . . .
Pauli paramagnetism . . . . . . . . . . . . . . . . . . . .
Intrinsic semiconductor . . . . . . . . . . . . . . . . . .
Gas of relativistic fermions . . . . . . . . . . . . . . . .
Neutron star . . . . . . . . . . . . . . . . . . . . . . . .
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Some useful formulae
0.1
Euler’s Gamma function
def
Z
∞
Γ(z) =
dt tz−1 e−t
for Re z > 0
(1)
0
Note that integrals of the form
Functional relation :
R∞
b
dx xa e−Cx may be simply related to the Gamma function.
0
Γ(z + 1) = z Γ(z)
(2)
This allows to perform an analytic continuation in order to extend the definition of the Gamma
fucntion to the other half of the complex plane, Re z 6 0.
√
Particular values Γ(1) = 1 & Γ(1/2) = π , hence, by recurrence,
Γ(n + 1) = n!
√
π
1
Γ(n + ) = n (2n − 1)!!
2
2
(2n)!
(2n)!!
def
where (2n − 1)!! = 1 × 3 × 5 × · · · × (2n − 1) =
0.2
(3)
(4)
def
and (2n)!! = 2 × 4 × · · · × (2n) = 2n n!.
Gaussian integrals
An integral related to Γ(1/2),
Z
dx e
r
− 12 ax2
=
− 12 ax2
1
=
a
R
2π
a
(5)
An integral related to Γ(3/2),
Z
dx x2 e
r
2π
a
(6)
n+1 n+1
2 2
Γ
a
2
(7)
R
More generally
Z
n − 21 ax2
dx x e
R+
Fourier transform of the Gaussian
Z
1
=
2
− 12 ax2 +ikx
dx e
r
=
R
0.3
(8)
Euler’s Beta function
Z
B(µ, ν) =
1
µ−1
dt t
ν−1
(1 − t)
0
0.4
2π − 1 k2
e 2a
a
π/2
Z
dθ sin2µ−1 θ cos2ν−1 θ =
=2
0
Γ(µ)Γ(ν)
Γ(µ + ν)
(9)
Stirling’s formula
Γ(z + 1) '
√
2πz z z e−z
i.e
ln Γ(z + 1) = z ln z − z +
which will be used frequently to express ln(n!) ' n ln n − n or
3
d
dn
1
ln(2πz) + O(1/z)
2
ln(n!) ' ln n.
(10)
0.5
Binomial formula
(p + q)N =
N
X
n n N −n
CN
p q
def
n
where CN
=
n=0
0.6
N!
n!(N − n)!
(11)
Other useful integrals
Z
∞
dx
0
xα−1
= Γ(α) ζ(α)
ex − 1
where ζ(α) =
dx
0
n−α
(12)
n=1
π2
6 ,
is the Euler Zeta function. We give ζ(2) =
Finally, we give
Z ∞
∞
X
ζ(3) ' 1.202, ζ(4) =
x4
π4
=
30
sh2 x
(related to the previous integral for α = 4).
4
π4
90 ,
etc.
(13)
TD 1: Orders of magnitude – Probabilities
1.1
Gases and solids – Orders of magnitude
The goal of this exercise is to discuss, using qualitative arguments, some orders of magnitudes
in situations that will analyzed in greater detail throughout the year. You are thus invited to
come back to these exercises when you later encounter such situations.
A. Ideal gas.– One considers a mole of an ideal gas, say oxygen, at room temperature (T =
27o C= 300 K) occupying a volume V = 24 `. The number of molecules is given by the Avogadro
number
NA ' 6.023 1023 particles/mole .
(14)
We recall from basic thermodynamics that the energy contained in a mole of gas reads, up to a
prefactor:
U ∼ RT .
(15)
1/ Density.– Compute the density of particles n (in nm−3 ). Give an estimate for the typical
distance between two particles.
2/ Energy and velocity.– What is the nature of the energy U ? Compute (in J and then in
eV) the order of magnitude of the translational kinetic energy of an oxygen molecule. Deduce
the typical velocity of a molecule in the gas.
3/ Collision against the walls of the container.– Assuming a cubic box of side L, what is
the order of magnitude of the time needed for a particle to go back and forth from a wall? Infer
the frequency fc of molecular collisions against one of the walls of the box.
4/ Pressure.– What is the momentum transferred to a wall by a single molecule undergoing
a collision. Infer the order of magnitude of the total force exerted on the wall by the molecules.
To which pressure does this correspond? Discuss the result obtained.
5/ Collisions between the gas molecules.– The mean free path (the typical distance traveled
by one particle between two successive collisions with other particles of the gas) is given by
√
` = 1/( 2 σn)
(16)
where n is the particle density and σ the scattering cross-section. Justify the order of magnitude
σ ≈ afew Å2 (we take σ = 4 Å2 ). Derive an estimate for `. Compute the corresponding mean
time between collisions τ = `/v, in which v is the typical velocity of a particle. What conclusion
can you draw concerning the nature of the motion of the gas particles?
6/ Ideal and real gases.– The preceding question has shown that the collisions between
atoms (or molecules) are highly frequent. We now wish to discuss the validity of the ideal gas
approximation, i.e., to estimate when the interactions may be neglected. We introduce the
interaction energy u0 between two atoms and the range r0 of the interaction; in monatomic
gases one typically has u0 ∼ 1 meV and r0 ∼ 5 Å. Estimate the contribution of interaction to
the total energy of the gas. Show that
Einteraction
N r03 u0
∼
.
Ekinetic
V kB T
Give the numerical value for this ratio. Your conclusion?
5
(17)
B. Vibration of atoms in a crystal.– We consider a crystal containing 1 mole of atoms.
1/ What is the typical volume occupied by the crystal? What is the density of atoms (in nm−3 )?
2/ Energy.– We will show later that the order of magnitude of the energy of the crystal is
again given by a law satisfying (15). What is the nature of this energy?
3/ Vibration of atoms.– What is the order of magnitude of the potential energy (in eV)
corresponding to a displacement δx ∼ 1 Å in the crystal? Infer the typical displacement, at
room temperature, of an atom from its equilibrium position (in Å).
Indication: One assumes that each atom is subject to a harmonic confinement whose stiffness
constant is evaluated as follows: a displacement of δx = 1 Å corresponds to δEp ≈ 10 eV
(∼Rydberg).
C. Electrons in a metal.– Due to Pauli’s principle, electrons in a metal, considered as free
2~ 2
particles, pile up in individual energy states ~k = ~2mke until an energy called the Fermi energy
F . The order of magnitude of this energy is fixed by the electron density F ∼
~2 2/3
.
me n
1/ Energy.– In gold the electron density is about n ' 55 nm−3 . The value of the Fermi energy
is F ' 5.5 eV: check that this value is compatible with the formula for F given above.
2/ Velocity.– We write F = 21 me vF2 . Compute the Fermi velocity vF . Compare with the
p
typical velocity associated with thermal energy vth ∼ kB T /me .
3/ Diffusion.– Electrons in a metal undergo many collisions (with impurities, other electrons,. . . ). The typical distance between two collisions with impurities is given by the elastic
mean-free path. For instance, gold has a residual resistivity of ρ(T → 0) = 0.022 × 10−8 Ω.m,
which corresponds to `e ' 4 µm. Infer the mean time between two collisions, τe = `e /vF , and
`2e
then the diffusion constant D = 3τ
. What is the typical distance traveled by an electron in
e
t=1s ?
4/ Drift velocity.– Consider an electrical wire of cross section s = 1mm2 that carries a current
I = 1 A. What is the mean velocity v of the electrons corresponding to this current?
1.2
Tossing a coin – Binomial law
A.– One plays heads or tails with a coin N times in a row. We write ΠN (n) for the probability
of drawing n times tails among N draws (attention: n is random and N is a sure (= nonrandom)parameter). If the coin is fixed (= fraudulent), we write p (6= 1/2) for the probability to
get tails and q = 1 − p for the probability to get heads.
1/ Distribution.– Give the expression for ΠN (n). Check the normalization.
2/ Generating function and moments.– In order to analyze the distribution, we are going
def
to compute the average hni and the variance ∆n2 =Var(n) = hn2 i − hni2 . To do so, it is
convenient to introduce the generating function
def
GN (s) = hsn i ,
(18)
whose variable s may be any complex number. Express GN (s) as a function of ΠN (n). Given
the function GN (s), how can you find the first two moments hni and hn2 i ? Calculate GN (s)
explicitly. Find the average and variance of n. Compare the fluctuations with the mean value.
6
3/ Limit N → ∞.– In this question we analyze the distribution directly in the limit N → ∞.
Using Stirling’s formula, expand ln ΠN (n) around its maximum n = n∗ . Explain why ΠN (n) is
approximately Gaussian when N → ∞. Carefully draw the shape of the distribution.
B. Application: molecule in a gas.– In exercise 1.1 on the gas under standard conditions
of temperature and pressure, we have seen that a molecule has a typical velocity v ≈ 500 m/s
and typically undergoes a collision every τ ≈ 2 ns. By assimilating the motion of the molecule
to a symmetric random walk (p = q = 1/2) of steps ` = vτ ≈ 1 µm every τ ≈ 2 ns, calculate
the typical distance traveled by a molecule after 1 s. Compare your result with the distance a
molecule would have traveled ballistically.
1.3
Rain drops – Statistics of independent events
A.– We study the distribution of independent events, for instance the fall of rain drops,
occurring on average with frequency λ. I.e. the probability that an event occurs during the
interval of time dt reads λdt.
We write P (n; t) for the probability that n events occur during the interval of time t (attention: n is a random variable and t a sure parameter).
1/ n = 0.– Express P (0; t + dt) as a function of P (0; t). Determine the differential equation
satisfied by this probability and solve it.
2/ Arbitrary n.– Following the same procedure, determine the set of coupled differential
equations satisfied by the distribution P (n; t).
3/ Generating function.– In order to solve these differential equations, it is practical to
introduce the generating function:
def
G(s; t) = hsn i
(19)
a) Given the generating function, how can you find the P (n; t)?
b) Poisson distribution.– Obtain from 2 a differential equation for G(s; t). What is G(s; 0)?
Infer G(s; t) and then the distribution P (n; t).
4/ Generating function and moments.–
Given G(s; t), how can you compute the moments hnm i ?
(Rk: it is actually more
def
p
pn
convenient to introduce γ(p; t) = G(e ; t) = he i). Using the result of 3, calculate the average
and the variance of the number of events.
5/ The limit λt → ∞.– Using Stirling’s formula, show that P (n; t) is approximately Gaussian
in the limit λt → ∞.
6/ How can you explain that the binomial and the Poisson distribution coincide with a Gaussian
in the limit of large numbers?
7/ (optional ) We discuss the relation between the binomial and the Poisson law. Show that in
the limit N → ∞ and p → 0, keeping pN = Λ constant, the binomial law ΠN (n) (exercise 1.2)
!
tends towards a Poisson law (indication: (NN−n)!
≈ N n ).
B. Application: fluctuations of the force exerted by a gas on a wall.– We start again
with exercise 1.1 on the ideal gas. We write Ft for the time average over a period t of the
force exerted by the molecules striking the wall of the container. The frequency of the impacts
derived in 1.1.A is fc ∼ 1026 impacts/s. Give the expression of the relative fluctuations of the
force δFt /Ft and their order of magnitude for t = 1 s, 1 ms, 1 µs,. . . . Comment.
7
1.4
Maxwell’s distribution – Doppler broadening of a spectral line
A. Maxwell’s distribution of velocities in a classical gas.– We will see later that the
velocity distribution in a classical gas follows Maxwell’s law:
m 3
m~v 2
2
,
(20)
f (~v ) =
exp −
2πkB T
2kB T
where m is the mass of the particle, kB is Boltzmann’s constant, T the temperature and v 2 =
vx2 + vy2 + vz2 .
1/ Joint distribution of velocity components.– Interpret f (~v ) in terms of probabilities.
Show that f (~v ) is well normalized.
2/ Calculate the following mean values: hvx i, hvy i, hvz i, hvx2 i, hvy2 i, hvz2 i, hvx vy i, hvx2 vy2 i, and
hEc i.
3/ Marginal distribution of component vx .– Deduce from f (~v ) the probability to find vx
between vx and vx + dvx , whatever the values of (vy , vz ) (marginal distribution for vx ).
4/ Marginal distribution of the modulus v = k~v k.– Deduce from f (~v ) the probability to
find the velocity modulus v between v and v + dv (taking advantage of the isotropy of f and
changing to spherical coordinates in velocity space). Check that the distribution so obtained
is well normalized. Compare the most probable
p value of v with its average hvi. Evaluate the
variance of v and the standard deviation σv = hv 2 i − hvi2 .
B. Application: Doppler broadening of a spectral line.– In a spectroscopy experiment
on an atomic gas, the motion of the atoms is responsible for a broadening of the spectral lines.
We study this phenomenon in a model where the atoms (assumed to be non-interacting) are
treated classically. We are interested in a transition between two atomic levels separated by
~ω0 .
Ω
spectrometre
x
Figure 1: Vapor of excited atoms emitting photons towards a detector.
Doppler effect.– If an atom at rest is in its excited state, it emits a photon of frequency ω ∼ ω0
after a typical time 1/Γ, where Γ is the intrinsic width of the excited level. A population of
motionless atoms produces a radiation of intensity i(ω) = i0 L(ω) at frequency ω, in which
L(ω) =
Γ/π
.
(ω − ω0 )2 + Γ2
(21)
We assume that the detector is placed along the x axis (figure). An atom emits a photon of
frequency ω in its frame. Because of the Doppler effect, the frequency detected in the detector
(=laboratory) frame will be
vx Ω'ω 1+
in which we assumed vx c .
(22)
c
8
c being the speed of light.
1/ Explain why the intensity of the radiation captured by the detector is I(Ω) ' i0 hL[Ω (1 − vx /c)]i,
where h· · ·i is the average over vx . Express I(Ω) as an integral.
2/ Low temperature.– What form does p(vx ) take in the limit T → 0 ? Infer the form of I(Ω)
and represent it in a figure.
3/ High temperatures.– In the limit of high temperatures, it is legitimate to substitute L(ω) →
δ(ω − ω0 ). Infer I(Ω). What is the width of the spectral line? Represent the shape of I(Ω) in
this case.
4/ What is the temperature scale T0 that allows to discriminate between the two previous
regimes?
5/ We consider a gas of Rubidium atoms (87 Rb of mass M ' 87 mp ), excited at the transition
wave-length λ = 780 nm. What is the width (in frequency) of the spectral line when the gaz is
at temperature T = 190 o C? Compare with the curves of figure 2.
6/ What could be the interest of cooling an atomic vapor in a spectroscopy experiment?
./
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Figure 2: Emission spectrum of a Rubidium vapor at temperature T = 190 o C.
Reminders
Avogadro’s constant: NA ' 6.023 1023 particles/mole.
Boltzmann’s constant: kB ' 1.38 10−23 J K−1 .
Ideal gas constant: R = NA kB ' 8.31 J K−1 mol−1 .
9
40-:'%)$;<
0)
TD 2: Phase space, ergodicity and density of states
2.1
Phase space of a 1D harmonic oscillator
A 1D harmonic oscillator has the Hamiltonian
p2
1
H(x, p) =
+ mω 2 x2
2m 2
in which m is the mass of the particle and ω the oscillator frequency.
(23)
A. Classical mechanics.– We analyze the oscillator in the spirit of classical mechanics.
1/ Check that Hamilton’s equations for this system are the expected equations of motion. Solve
them for the initial conditions
x(t = 0) = x0
and
p(t = 0) = 0
(24)
2/ Define the phase space of the system. Draw its trajectory. What is the energy E of the
oscillator for this trajectory?
3/ Calculate the fraction of time during which the particle has a position between x and x + dx.
Write the result using the notation w(x) dx, in which w(x) is interpreted as the probability
density of the position.
B. Statistical physics.– We recover the previous result by a totally different method. We
consider that the energy of the particle is known up to an uncertainty dE so that it is situated
between E and E + dE.
1/ Draw in phase space the surface in which the accessible states of the system are located.
2/ We assume that all accessible micro-states (defined in the previous question) are equally
probable. Next calculate the probability that the oscillator is represented by a point having an
abscissa between x and x + dx.
C. Level counting.– The classical approach must correspond to the high energy limit of the
quantum approach. It is necessary, as in quantum mechanics, to be able to count the microstates.
1/ Which rule, borrowed from quantum mechanics, allows one to perform such counting in phase
space?
2/ Calculate the number of micro-states that are classically accessible to an oscillator of energy
between E and E + dE. Compare this result with the quantum calculation, knowing that the
quantum energies of a 1D harmonic oscillator are given by
n = ~ω(n + 1/2)
2.2
avec n = 0, 1, 2, . . . .
(25)
Volume of a hypersphere
A hypersphere of radius R in Rd is the domain defined by x21 +x22 +· · ·+x2d 6 R2 . By studying the
R
2
integral Rd d~x e−~x , calculate the surface of the hypersphere Sd (R) and show that the volume
is given by
Vd (R) = Vd Rd
where Vd =
π d/2
Γ( d2 + 1)
is the volume of the sphere of unit radius (consider the cases d = 1, 2, and 3).
10
(26)
2.3
Density of states of free particles
We consider a gas of N free atoms in a cubic box of volume V = L3 . The Hamiltonian of the
system is
N
X
p~i 2
H=
.
(27)
2m
i=1
By (incorrectly) considering the atoms as distinguishable, show that the volume Φ(E) of phase
space occupied by states of energy less than E is
Φ(E) =
1
Γ( 3N
2 + 1)
E
0
3N/2
def
where 0 =
2π~2
mL2
(28)
Find the density of states.
Numerical example: Calculate 0 (in J and then in eV) for helium atoms in a box of size
L = 1 m.
2.4
Classical and quantum harmonic oscillators
We consider a system of N independent identical 1D harmonic oscillators.1D, The Hamiltonian
of the system is
N 2
X
pi
1
H=
+ mω 2 qi2 .
(29)
2m 2
i=1
1/ Semi-classical treatment.– We suppose that the oscillators are classical.
a/ We denote by V(E) the volume occupied by states of energy 6 E in phase space (the
dimension of which will be specified). Express V(E) in terms of the constant V2N , the volume
of the hypersphere of unit radius (exercise 2.2).
b/ Using the semi-classical hypothesis that a quantum state occupies a cell of volume hN in
phase space, calculate the number of quantum states of energy less than E (written Φ(E)), and
then the density of states ρ(E).
2/ Quantum treatment.– We now suppose that the N oscillators are quantum mechanical.
We know that the energy levels of each oscillator are nondegenerate and given by εn = (n +
1/2)~ω (where n is an integer > 0).
a/ Calculate the number of accessible states of the system when its energy is equal to E.
Indications: We wish to calculate the numberPof different ways of choosing N nonnegative
integers (n1 , n2 , n3 ...nN ) such that their sum N
i=1 ni equals a given integer M . To do so we
use the following method: each choice may be represented by a diagram of n1 balls, then one
bar, then n2 balls, then one bar, . . . The total number of balls is M and the total number of bars
is N − 1. Permutations of balls and bars each among themselves do not count. Only matter the
number of different ways of placing N − 1 bars in a linear array of M balls.
c/ Calculate the quantum density of states of the system. Show that, in the limit E N ~ω,
one recovers the semi-classical result of question (2).
11
Appendix 2.A: Semi-classical rule for counting states in phase space
For a system with D degrees of freedom the phase space of vectors (q1 , · · · , qD , p1 , · · · , pD )
has dimension 2D. The correspondence between classical and quantum counting of microstates is ensured by considering that one quantum state occupies a volume hD in classical
phase space.
Integrated density of states.– Let H({qi , pi }) be the Hamiltonian governing the dynamics
of a system. We denote by Φ(E) the number of micro-states of energy less than E. In the
semi-classical limit, we have
1
Φ(E) = D
h
Z
D
Y
H({qi ,pi })6E i=1
1
dqi dpi ≡ D
h
Z Y
D
dqi dpi θH (E − H({qi , pi }))
(30)
i=1
where θH (x) is the Heaviside function.
Density of states.– The density of states is given by
ρ(E) = Φ0 (E)
(31)
i.e. ρ(E)dE represents the number of quantum states of energy in the interval [E, E + dE).
Indistinguishable particles.– If the system contains N indistinguishable particles (for
instance an ideal gas of N particles moving in three-dimensional space, D = 3N ), we
must multiply by an extra factor 1/N ! to take into account the fact that the particles
are indistinguishable (i.e. that micro-states differing only by a permutation of particles are
equivalent):
1
Φindist (E) =
Φdist (E)
(32)
N!
Notice, however, that this expression accounts only partially for the symmetrization postulate of quantum mechanics. The full consequences of the latter will be studied in detail in
tutorials 8 and 9.
12
TD 3: Fundamental postulate and microcanonical ensemble
3.1
The monoatomic ideal gas and the Sackur-Tetrode formula – Gibbs paradox
We consider a ideal gas of N atoms confined in a box of volume V .
1/ Define the microcanonical entropy S ∗ . We call Φ(E) the integral of the density of states of
the (quantum) system. Explain why we can use the expression
S ∗ ' kB ln Φ(E)
(33)
and give its limits.
2/ Extensivity.– Formulate the extensivity property that must be satisfied by the microcanonical entropy S ∗ (E, V, N ) of the gas.
3/ Distinguishable atoms.– Here, we do not take into account the symmetrization postulate
of quantum mechanics.
a/ Give the integral Φdisc (E) for the density of states of the gas composed of N atoms (you may
∗ .
wish to recall exercice 2.3). Calculate the corresponding microcanonical entropy Sdist
∗
satisfy the extensivity property formulated above?
b/ Does this expression for Sdist
c/ Gibbs paradox.– We consider two identical volumes of the same gas separated by a wall.
Calculate the difference between the entropy of this system and the entropy of the system with
the wall removed.
∗
∗
(E, V, N ) .
(34)
(2E, 2V, 2N ) − 2Sdisc
∆S melange = Sdisc
Why do we say that this result is paradoxal?
4/ Indistinguishability.– Quantum mechanics asserts a principle of indistinguishability between identical particules.
a/ Considering this principle, calculate the integral Φindisc (E) for the density of states of the gas
composed of N atoms and give the Sackur-Tetrode formula (1912).
"
3/2 !#
5
V
mE
S ∗ (E, V, N ) = N kB
+ ln
(35)
2
N 3π~2 N
Verify that this expression obeys the extensivity property of the entropy.
b/ Show that S ∗ can be written as S ∗ = 3N kB ln a∆x∆p/h , where ∆x is a distance and ∆p
a momentum. Interpret this result.
c/ Calculate the microcanonical temperature T ∗ and pressure p∗ .
d/ Numerical example: Calculate ∆x, ∆p, and ∆x∆p/h for a helium gas at normal temperature
and pressure. Find S ∗ /N kB .
3.2
Thermal contact between two cubic boxes
We consider a closed system composed of two identical cubic boxes of edge length L with one
particle in each box. The lowest energy levels and their degeneracy for a particle in a box may
be calculated for Dirichlet boundary conditions. The results are summarized in table 1 below.
We call these two boxes I and II bring them into contact. The complete system is surrounded
by an adiabatic wall.
13
1. We consider the wall between the two boxes as adiabatic. At time t = 0 the energy of the
def
h2
particle in each box is EI = 12 ε0 and EII = 18 ε0 where ε0 = 8mL
2.
Calculate the number of microstates accessible to system I, system II and the total system.
2. We consider now that suddenly heat may be transferred by the wall between the two boxes,
so that the system evolves towards a new equilibrium state.
Which quantity is conserved during this equilibration?
What are the energy states accessible to system I and system II ?
What are the microstates accessible to the total system ? How many such microstates are
there?
Compare with the previous situation.
3. We suppose now that the total system has reached is thermal equilibrium.
What is the probability of a given microstate?
What is the probability that the energy of system I be 6 ε0 , 9 ε0 , 15 ε0 ?
Sketch the energy probability distribution for systems I and II at equilibrium.
What is the most likely energy for each system?
4. Do this exercice again considering now that each box contains two distinguishable particles.
5. Do it again considering that each box contains two indistinguishable particles of spin zero
(note that these particules are hypothetical).
TABLE 1
one particle
in a box
3 0
6 0
9 0
11 0
12 0
14 0
17 0
18 0
19 0
21 0
22 0
24 0
26 0
27 0
29 0
30 0
3 = 1 2 + 12 + 12
6 = 1 2 + 12 + 22
9 = 1 2 + 22 + 22
11 = 12 + 12 + 32
12 = 22 + 22 + 22
14 = 12 + 22 + 32
17 = 22 + 22 + 32
18 = 12 + 12 + 42
19 = 12 + 32 + 32
21 = 12 + 22 + 42
22 = 22 + 32 + 32
24 = 22 + 22 + 42
26 = 12 + 32 + 42
27 = 12 + 12 + 52
27 = 32 + 32 + 32
29 = 22 + 32 + 42
30 = 12 + 22 + 52
level
degeneracy
1
3
3
3
1
6
3
3
3
6
3
3
6
4
6
6
two particles
in a box
6 0
9 0
12 0
14 0
15 0
17 0
18 0
20 0
21 0
22 0
23 0
24 0
25 0
TABLE 2
(*) level
degeneracy
1
6
15
6
20
30
15
60
12
15
60
31
60
(**) level
degeneracy
imposs.
3
6
3
10
15
6
30
6
9
30
15
30
(*) two distinguishable particles of the same
mass.
(**) two identical fermions of spin 0.
14
3.3
Paramagnetic crystal - Negative (absolute) temperatures
We consider a system of N spin 1/2 particles located on the sites of a crystal lattice. Each
~
particle has a magnetic moment µ. This system is submitted to an uniform magnetic field B.
We suppose that the interaction between the spins is much smaller than the interaction of the
spins with the external magnetic field.
~ and n− for those opposed to
We write n+ for the number of magnetic moments aligned with B
~
B.
1/ Give the expression for n+ et n− in terms of N , B, µ, and E (total energy of the system).
2/ Calculate the number of states Ω(E, N, B) accessible to the system (calculate first Ω(n+ , n− , B)).
3/ Give an expression for the microcanonical entropy S of the system when n+ 1 and n− 1.
Sketch S vs. the energy E.
4/ Calculate the microcanonical temperature T of the system and show that T may take negative
values. Describe the state of the system when T → +∞, T → 0+ , T → 0− , and T → −∞.
5/ Show that negative (absolute) temperatures are ”hotter” than positive (absolute) temperatures. Hint: analyze what happens when you bring two systems of different microcanonical
temperatures into contact.
What happens if we establish a contact between two identical systems with temperatures
T0 > 0 and −T0 ?
Figure 3: A typical record of nuclear magnetic inversion. The magnetization of the sample is
tested every 30s by NMR. Vertical bands on the graph represent 1mn. On the left is sketched
a typical signal of normal thermal equilibrium (T ≈ 300 K) revealing the magnetization of
the sample. Subsequently, the magnetic field is reversed during a short time (T ≈ −350K).
The nuclear spins ”follow” the field and then relax toward the ”normal” thermal equilibrium
via a zero magnetization (at this point, the temperature goes from T = −∞ to T = ∞). This
inversion is observed in lithium fluoride crystals. This behavior is possible because the relaxation
time between nuclear spins (t1 ∼ 10−5 s) is very short compared to the relaxation time between
the spins and the lattice (t2 ∼ 5 mn). When the field is rapidly reversed during a time between
t1 and t2 , the system of nuclear spins can reach the thermal equilibrium and exhibit an absolute
negative temperature. Reference : E. M. Purcell and R. V. Pound, Phys. Rev. 81, 279 (1951).
15
TD 4: System in contact with a thermostat – Canonical
ensemble
4.1
Monoatomic ideal gas
We study again the ideal gas (cf. exercice 3.1) but here we consider the gas at a fixed temperature
T , i.e., we will use the canonical ensemble.
1/ Calculate the partition function of an ideal gas of N atoms in a volume V within the MaxwellBoltzmann approximation. Give the partition function using the de Broglie thermal wavelength
defined by
s
2π~2
.
mkB T
def
ΛT =
(36)
Numerical example: Calculate the numerical value of ΛT for helium at room temperature.
2/ Deduce the free energy of the gas in the thermodynamic limit. Express this result such that
it clearly exhibits the extensivity property of the free energy.
3/ Calculate the average energy of the gas as well as its heat capacity. We recall the definition
of the heat capacity,
C
∂E (T, V, N )
CV =
∂T
def
(i.e. CV =
∂E
∂T V,N
in thermodynamic notation )
def
C
Calculate the energy fluctuations of the system, Var(E) = E 2 − E
tions to the average value of the energy.
C 2
(37)
. Compare the fluctua-
4/ Calculate the canonical pressure of the system. Provide your comments.
5/ Calculate the canonical entropy of the system. Compare this result to the Sackur-Tetrode
formula. Discuss the behavior of entropy at high temperature.
6/ Calculate the chemical potential of the gas.
4.2
Ideal, confined, nonideal, etc... gases.
Recommandation: Go back to exercice 1.4 to refresh you memory about joint and marginal
probability laws.
We consider a gas of N indistinguishable particles without internal degrees of freedom,
enclosed by a box of volume V in contact with a thermostat at temperature T .
1/ Classical canonical distribution
a) Recall how microstates are described classically. The classical dynamic of the system is given
by the Hamiltonian
N
X
p~i 2
+ U ({~ri }) .
(38)
H({~ri , p~i }) =
2m
i=1
Give the expression for the canonical distribution, to be denoted by ρC ({~ri , p~i }).
b) How can we obtain the distribution function f that characterizes the position and momentum
of a single particle? We define f (~r, p~)d~rd~
p as the probability for a particle to have a position in
a volume d~r at ~r and a momentum in a volume d~
p at p~.
16
2/ Monoatomic ideal gas
a) Justify briefly that the partition fonction can be factorized according to
Z=
zN
,
N!
(39)
where z is the partition function for one particle (we recall that z = V /Λ3T ).
b) Calculate the distribution function f (~r, p~) explicitly. Derive Maxwell’s law for the distribution
of the particle velocities in the gas.
3/ Other gases.– In this question, we want to test the validity in more general cases of the
results obtained for the monoatomic gas. For each of the following situations, answer these two
questions:
• Does the factorization (39) still hold?
• Is the velocity distribution given by Maxwell’s law?
a) Gas confined by an external potential Uext (~r).
Application: a rubidium gas is trapped in a harmonic potential created by several lasers. Discuss
the density profile of the gas.
b) Gas of interacting particules (nonideal gas), i.e. U 6= 0.
p
c) Relativistic ideal gas, i.e. E = p~ 2 c2 + m2 c4 .
d) Ultrarelativistic ideal gas i.e. E = ||~
p||c.
Calculate z explicitly in this case (express z as z = V /Λ3r where Λr is the relativistic thermal
wavelength). Derive the energy of the gas and its equation of state.
4/ What is the limit of the classical approximation, i.e. of equation (39) ? In which cases will
quantum effects will (if T % or & ? n = N/V % or & ?)
4.3
Gases of indistinguishable particules in a harmonic well
In spite of the indistinguishability (i.e. the symmetrization postulate in quantum mechanics),
the exact partition fonction of a gas composed of indistinguishable particles may be calculated
quite easily when the gas is enclosed in a harmonic well. Here we use this remarkable fact to
carefully discuss the limits of the semi-classical approximation of Maxwell-Boltzmann.
We consider N particles in a one-dimensional harmonic well. The Hamiltonian of the system
is
N 2
X
pi
1
2 2
H=
+ mω xi
(40)
2m 2
i=1
A. Distinguishable particules.
1/ Calculate the quantum mechanical partition function of the gas. Deduce the mean energy of
the system.
C
2/ Discuss the high temperature approximation (for Zdisc and for E ).
B. Maxwell-Boltzmann approximation. Give the expression of the partition function of
a gas of identical (and therefore indistinguishable) particles in the Maxwell-Boltzmann approximation. In this case, does it matter whether the particles are bosons or fermions? Give the
expression of the chemical potential µMB (T, N ) of the system.
C. Bosons.– We now consider the particules as identical bosons.
17
1/ What are the quantum states? Discuss the difference with distinguishable particles. Calculate
explicitly the partition function Zbosons . Deduce the high temperature limit of the partition
function and show that your result is identical to the Maxwell-Boltzmann approximation. What
is the temperature T∗ that separates the quantum from the classical behavior? Discuss the origin
of the dependence of T∗ on N .
2/ Show that the free energy is
N
X
~ω
Fbosons (T, N ) = N
+ kB T
ln 1 − e−n~ω/kB T .
2
(41)
n=1
What is the physical meaning of the first term in Fbosons ?
3/ Calculate the mean energy of the gas of bosons.
def
4/ We define the canonical chemical potential as: µ(T, N ) = F (T, N ) − F (T, N − 1). Analyze
the high and low temperature limit of µ. Sketch carefully µ vs. T and compare with µMB (T, N )
obtained in B.
4.4
Partition function of a particule in a box – the role of boundary conditions
We consider a particle of mass m moving freely in a 1D box of size L.
1/ Semiclassical calculation.– Give the semiclassical partition function and express it using
the de Broglie thermal wavelength ΛT (exercice 4.1).
2/ Dirichlet boundary conditions.– What is the energy spectrum of a particle if we use the
Dirichlet boundary condition ψ(0) = ψ(L) = 0 ? Calculate the partition function zβ expressed
as a series in ΛT /L. Calculate with the aid of the Poisson formula the first terms of a high
temperature expansion for zβ .
3/ Periodic boundary conditions.– Same questions for the periodic boundary conditions
ψ(0) = ψ(L) and ψ 0 (0) = ψ 0 (L).
4/ Comparaison.– Compare the partition function at high temperature obtained with the two
kinds of boundary conditions. What seem to you the most convenient boundary conditions (in
particular in relation to the semiclassical analysis)?
Appendix: The Poisson formula.– Let f (x) be a function (or a distribution) defined on
def R
R and let fˆ(k) = R dx f (x) e−ikx be its Fourier transform. It is possible to show the (very
nontrivial) identity
X
X
f (n) =
fˆ(2πn)
(42)
n∈Z
4.5
n∈Z
Diatomic ideal gas
We study here the thermodynamics of a gas of diatomic molecules. Each molecule has three
translational, two rotational, and one vibrational degree of freedom. The Hamiltonian for a
molecule is
~` 2
P~ 2
p2
1
H'
+
+ r + mr ω 2 (r − r∗ )2
(43)
2M
2I
2mr
2
where P~ is the total momentum and ~` the orbital angular momentum which characterizes
the rotation of the molecule. Furthermore (r, pr ) is a pair of conjugate variables that describes
the vibration of the molecule (in relative coordinates).
18
1/ Give the quantum mechanical spectrum of the translational, rotational, and vibrational
energy. Show that the partition function for a molecule may be factorized according to z =
ztrans zrot zvib . Calculate the expression for the partition function for these three types of motion.
What is the relation between z and the partition function of the gas in the Maxwell-Boltzmann
approximation?
2/ High temperature, semiclassical approximation.– Calculate the partition function Zβ
of the gas in the semiclassical approximation (~ → 0) when all degrees of freedom are treated
classically (discuss the validity of the result). Calculate the average energy of the gas and its
heat capacity.
3/ At lower temperatures it may not be possible to treat all degrees of freedom classically. What
is the average energy of vibration when kB T ~ω ? Same question for the energy of rotation
when kB T ~2 /I. Discuss the behavior of the heat capacity as a function of temperature (take
~2 /I ~ω). Give your comments on the figure.
CV /Nk B
experiment
theory
3
2
1
0
Tvib
Trot
T (K)
5
10
50 100
500 1000
5000
def
def
Figure 4: Specific heat of a gas of HD (deuterium-hydrogen). Trot = ~2 /2kB I and Tvib = ~ω/kB .
Excerpt from R. Balian, ”From microscopic to macroscopic I”.
4.6
4.6.1
Paramagnetism
Classical calculation: Langevin paramagnetism.
We intend to find the equation of state of a paramagnetic material, i.e., the relation between
~ of the material, the temperature T , and the external magnetic
the total magnetic moment M
~
field B applied to the material. We consider N independent atoms, fixed at the sites of a crystal
lattice. Each atom has a magnetic moment µ
~ of constant modulus.
In this first exercice we consider µ
~ as a classical vector. The spatial orientation of the
~ is
magnetic moment of each atom is specified by two angles θ and ϕ. When a magnetic field B
applied along the z axis, each atom acquires a potential energy
~ = −µ B cos θ .
Hpot = −~
µ·B
(44)
If each atom has a moment of inertia equal to I, then its dynamics is governed by the Hamilto-
19
nian1 :
1
H=
2I
p2θ
+
p2ϕ
!
sin2 θ
+ Hpot .
(45)
1. Calculate the canonical partition function associated with H. Write the result in the form
e 2 where Λ
e T is a thermal
z = zcin zpot where zpot = 1 for B = 0. Show that zcin = Vol/Λ
T
length and Vol an accessible volume. Express zpot as a function of x = βµB.
2. Give the expression for the probability density w(θ, ϕ) that the magnetic moment points
in the direction (θ, ϕ). Check that the probability density is normalized. Scketch w(θ, ϕ)
vs. θ.
3. Calculate the average magnetic moment hµz i per atom. We will refer to M = N hµz i as
the total magnetization of the material. You may use the result given in the appendix and
calculate ∂z/∂B.
4. Discuss the behavior of hµz i as a function of magnetic field and temperature. Show that
the high temperature approximation gives the Curie law M ∝ B/T .
4.6.2
Quantum mechanical calculation : Brillouin paramagnetism
We now consider a system of N quamntum mechanical magnetic moments. The Hamiltonian
of a particle is given by Hpot [Eq. (44)]. The magnetic moment µ
~ is now an operator that acts
~
on the quantum states. We call J the total angular momentum, which is the sum of the orbital
angular momenta and the spins of the electrons for an atom in its ground state. We let J stand
for the associated quantum number. The magnetic moment µ
~ of an atom is related to J~ by
~ ,
µ
~ = gµB J/~
(46)
qe ~
where µB = 2m
' −9, 27.10−24 A m2 is the Bohr magneton. The Landé factor g is a dimene
sionless constant typically of order one 2 .
1. What are the eigenvalues and the eigenvectors of the Hamiltonian of an atom (remember
that the projection Jz of the angular momentum of an atom may take the values m~ avec
m ∈ {−J, −J + 1, .., J}) ?
2. Calculate the partition function Z of an atom as a function of J and y = βg |µB | JB.
Determine Z for the special case J = 1/2.
3. What is the probability for an atom to be in the quantum state of quantum number m ?
1
The kinetic energy of a moment of inertia I is Hcin = I2 [θ̇2 + ϕ̇2 sin2 θ]. Then pθ = ∂Hcin /∂ θ̇ = I θ̇ and
pϕ = ∂Hcin /∂ ϕ̇ = I ϕ̇ sin2 θ.
2
If the angular momentum is due only to the electron spins, we have g = 2. If it is due only to the orbital
angular momentum, we have g = 1, and if it is of mixed origin, then g = 3/2 + [S(S + 1) − L(L + 1)]/[2J(J + 1)],
where S and L are the quantum nimbers of the spin and the orbital angular momentum, respectively. J is the
~ + S.
~ The three numbers obeys at the
quantum number associated with the total angular momentum J~ = L
triangle inequality.
20
4. Calculate <µx > and <µz > for arbitrary J and for J = 1/2. Deduce the magnetization
M . Find an expression for M in terms of the Brillouin function
( J
)
1
1
X
1 + 2J
d
±my/J
2J
BJ (y) =
ln
−
e
= .
y
1
dy
th ( 2J
)
)y
th (1 + 2J
m=−J
3
Near the origin BJ (y) = J+1
3J y + O(y ). Show that the high temperature approximation
gives Curie’s Llaw. Show that at low temperature the quantum mechanical result is very
different from the classical result obtained in 4.6.1 (except for large values of J).
Determine from the
results
Gd+++ below the
Cr+++, Fe+++,
SPIN 5.
PARAMAGNETISM
56ivalue of J for each ion.
OFexperimental
ANDsketched
~iii
netic moments for our analysis. This analysis consists
of normalizing the calculated and experimental values
at chosen values of H /T. Although space quantization
and the quenching of orbital angular momentum are unmistakably indicated by the good agreement of simple
heory and experiment for the Pg2 state of the free
chromium ion, there appears to be a small, secondorder departure of the experimental results from the
Brillouin function. In searching for the source of the
small systematic deviation, one must consider the
ollowing: (1) experiznental error in the measurement
of M, II, and T, (2) dipole-dipole interaction, (3) exchange interaction, (4) incomplete quenching, and (5)
he eGect of the crystalline field splitting on the magnetic energy levels. The diamagnetic contribution is,
of course, too small to affect the results.
It is felt that since the moment can be reproduced
o 0.2 percent and the magnitude of II/T is known to
ess than 1 percent, especially for 4.21'K, experimental
error as a complete explanation must be discarded.
It is true that the field seen by the ion is the applied
B.OO
oo:
=-:o.
e
i
oo(
llo
gI—
20
10
x 10
40
GAUSS /OEG
Geld with corrections due
to the demagnetization
factor'
Figure 5: Average magneticand
moment
per polarization"
ion (in units
of the
Bohr
magneton) vs. B/T for certain
the Lorentz
(effect
of field
of neigh3+ , (II)
3+ and since
3+ . isHere
ions).FeHowever,
sample
spherical,
paramagnetic salts: (I) Crboring
(III)theGd
g = 2 in all cases (since ` =
these two opposing corrections cancel" each other in
0). Dots are experimental results
and curves are theoretical results obtained with our quantum
erst approximation. Therefore, any error thus intromechanical model [from W.duced
E. Henry,
Phys. Rev.
88, to559
(1952)].
is a second-order
correction
a second-order
l00
~
I
I
FiG. 3. Plot of average magnetic moment per ion, p 2fs H/T for
(I) potassium chromium alum (J=$=3/2), (II) iron ammonium
alum (J=S=5/2}, and (III) gadolinium
sulfate octahydrate
(J=S=7/2). g=2 in all cases, the normalizing point is at the
highest value of H/T.
120
I
'K
BRILLOUIN
"/T
80
l—
30
1.
200 oK
~ 5.00 'K
~ 4.21 'K
140
~
ZITI
90
80
70
QQ
40
20
10
4
8
l6
l2
/T
& IO
20
24
28
52
M
GAUSS DEG
pzo. 2. Plot of relative magnetic moment, M„vs II/T fo r
potassium chromium alum. The heavy solid line is for a Brillouin
curve for g= 2 (complete quenching of orbital angular momentum}
and J=S=3/2, fitted to the experimental data at the highest
value of II/T. The thin solid line is a Brillouin curve for g=2/5,
J=3/2 and L=3 (no quenching). The broken lines are for a
Langevin curve fitted at the highest value of II/T to obtain the
ower curve and fitted at a low value (slope fitting) of fI/2' to
obtain the upper curve
-
eGect which is negligible. For potassium chromium
alum, the chromium ions are greatly separated, practically eliminating dipole-dipole and exchange interactions (ignoring the possibility of superexchange
based on the existence of excited states of normally
diamagnetic atoms) .
Experiments which were carried out with iron ammonium alum" (iron in sS@s state for the free ion) and
gadolinium (ssrfs state for free ion) sulfate octahydrate
show (Fig. 3) slight departures from the Brillouin
functions for free spins. Since L is zero for both free ions,
these slight departures which remain for the two ions
are not attributable to incomplete quenching. Energy
levels taken from Kittel and Luttinger" and based on
the effect of a crystalline cubic 6eld through spin-orbit
interaction, ' have been used to calculate magnetic
moments at a few points for iron ammonium alum in
' C. Breit, Amsterdam Acad. Sci. 25, 293 (1922).
'oH. A. Lorentz, Theory of Electrons (G. E. Stechert and
Company, New York, 1909}.
"C. J. Gorter, Arch. du Musee Teyler 7, 183 (1932).
i2
Contamination
and
"C. Kittel
and decomposition were carefully avoided.
M. Luttinger, Phys. Rev. 73, 162 (1948).
"J.H, Van VleckJ.and W, Q. 21
Penney,
Phil. Mag. 17, 961 (1934).
Appendix 4.A: Semiclassical summation rule the phase space.
In the canonical ensemble the summation rule discussed in appendix 2.A takes the following
form: for a system with D degrees of freedom and Hamiltonian H({qi , pi }) the partition
function is given by :
Z Y
D
1
Zβ = D
dqi dpi e−βH({qi ,pi }) .
(47)
h
i=1
If the particles are indistinguishable, the partition function is given by the MaxwellBoltzmann approximation
1 disc
Zβindisc =
Z
.
(48)
N! β
Appendix 4.B: Canonical average of a physical quantity
Let X be a physical quantity with conjugate parameter φ, i.e., there is a term dE = −Xdφ
in the expression of the energy. The canonical average of X is obtained by deriving the
thermodynamic potential F (T, N, · · · , φ, · · · ) with respect to the ”conjugate force” φ,
c
X =−
∂F
∂φ
(49)
Example: X → M is the magnetization, φ → B the magnetic field. The mean magnetic
∂F
zation is given by M = − ∂B
.
22
TD 5: Thermodynamic properties of harmonic oscillators
5.1
Lattice vibrations in a solid
A. Preliminary: a single harmonic oscillator.– We recall the spectrum of the quantum
harmonic oscillator of frequency ω,
1
n = ~ω n +
for n = 0, 1, 2, . . . .
(50)
2
1/ Calculate the partition function for the quantum harmonic oscillator.
2/ Deduce the average energy C and the average occupation number nC . Analyze the classical
limit(~ → 0) of C . Give a physical interpretation.
3/ Express the specific heat in the form cV (T ) = kB f (~ω/kB T ), where f (x) is a dimensionless
function. Interpret physically the limit behavior for T → 0 and for T → ∞.
B. The Einstein model (1907).– We consider a solid of N atoms, each vibrating around
its equilibrium position (a site of the crystal lattice). With the 3N degrees of freedom we
associate 3N independent one-dimensional (quantum) harmonic oscillators. We assume that
all oscillators have the same frequency ω and that they may be considered as discernable.
They are discernable because each one is attached to a specific lattice site (however there are
N ! different ways to attach the indiscernable atoms to the discernable sites).
1/ Using the results of part A, give (without further calculation) the expression for the partition
function describing lattice vibrations.
2/ Deduce the total energy of the 3N oscillators (the result for T → ∞ may be obtained more
(Einst.)
easily) and the vibrational contribution CV
(T ) to the heat capacity of the solid.
(Einst.)
3/ Compare the limiting behavior of the heat capacity CV
experimental results (see Figure 6):
in Einstein’s model with the
• High temperature (T → ∞) : CV → 3N kB (Dulong & Petit’s law).
• Low temperature (T → 0) : CV ' a T + b T 3 with a 6= 0 for an electric conductor and
a = 0 for an insulator.
C. The Debye model (1912).– The weakness of Einstein’s model, at the origin of the dis(Einst.)
crepancy between the theoretical expression CV
(T → 0) and the experimental observations,
lies in the assumption that all oscillators have the same frequency, (i.e., that atoms are independent). A more realistic model should account for the fact that, although the atoms may be
described as identical quantum oscillators, they are coupled (strongly interacting due to the
chemical bonds). Nevertheless, the energy may be written as a quadratic form which may in
principle diagonalized, that is, rewritten in the form
3N 2
X
pi
1
2 2
H=
+ mωi qi
2m 2
(51)
i=1
where (qi , pi ) are pairs of conjugate coordinates associated with the vibrational modes of the
crystal (like the modes of a vibrating string). The crystal is characterized by a full spectrum
of distinct eigenfrequencies {ωi } that form a continuous
P spectrum. The distribution of the
eigenfrequencies is called the spectral density ρ(ω) = i δ(ω − ωi ).
23
1/ Specific heat.– Express the specific heat formally as a sum of contributions of vibrational
eigenmodes.
2/ A few properties of the spectral density.– The spectral density ρ(ω) has a finite support
[0, ωD ], where ωD is the Debye frequency.
a) What is the origin of the upper cutoff and what is the order of magnitude of the wavelength
associated with ωD ?
Rω
b) Give a sum rule for 0 D ρ(ω)dω.
c) In the Debye model we assume that the spectral density has the simple form
ρ(ω) =
3V
ω2
2π 2 c3s
for ω ∈ [0, ωD ]
(52)
Explain the origin of the behavior ρ(ω) ∝ ω 2 . Applying the sum rule, find a relation between ωD ,
the mean atomic density N/V , and the sound velocity cs (compare to the result of question a).
3/ Limiting behavior of CV (T ).
a) Justify the representation CV (T ) = kB
ture behavior and compare to
(Einst.)
CV
.
R ωD
0
dω ρ(ω) f (~ω/kB T ). Deduce the high tempera-
b) Justify that in the T → 0 limit only the low frequency behavior of ρ(ω) is important. De(Einst.)
duce the low temperature behavior of CV (T ). Compare to CV
and explain the difference
physically. Compare to the experimental data of figure 6.
300
ï1
C [ mJ.mol .K ]
20
10
ï1
200
0
0
2
4
6
8
100
3
0
3
T [K ]
0
20
40
60
80
Figure 6: Left: Specific heat of diamond (in cal.mol−1 .K −1 ). Experimental values are compared
to the curve resulting from the Einstein model by setting θE = ~ω/kB = 1320 K (from A.
Einstein, Ann. Physik 22, 180 (1907)). Right: Specific heat of solid argon as a function of T 3
(from L. Finegold and N.E. Phillips, Phys. Rev. 177, 1383 (1969)). The straight line is a fit
to the experimental data. Insert: zoom onto the low temperature region.
5.2
Thermodynamics of electromagnetic radiation
We consider a cubic box of volume V containing electromagnetic energy. The system is supposed
in thermodynamic equilibrium.
A. General.
1/ Recall how the eigenmodes of the electromagnetic field in vacuum are labeled.
2/ Electromagnetic energy.– Using the results of part A of Exercice 5.1, express the avP
C
C
erage electromagnetic energy as a sum over the modes: E e−m =
modes mode . Identify the
C
contribution of the vacuum, Evacuum (= limT →0 E e−m ).
24
C
3/ Radiation energy.– Radiation corresponds to excitations of electromagnetic field: E radia =
C
E e−m − Evacuum . Identify the contribution of each mode.
4/ Eigenmode density.– Calculate the spectral density ρ(ω) of eigenfrequencies in the box.
5/ Planck’s law.– Writing the energy density (per unit of volume) as an integral over the
R∞
C
frequencies, V1 E radia = 0 dω u(ω; T ), recover Planck’s law for the spectral energy density
u(ω; T ). Plot u(ω; T ) as a function of ω for two temperatures. Interpret physically the expression
in terms of the average number of excitations in each mode (i.e., in terms of the number of
photons).
6/ The Stefan-Boltzmann
R ∞ law.– Calculate the photon density nγ (T ) and the radiation
energy density utot (T ) = 0 dω u(ω; T ).
B. Cosmic Micowave Background Radiation.– About 380 000 years after the big bang
atoms formed and matter became eletrically neutral, i.e., light and matter decoupled: the universe became “transparent” for radiation. The period between 380 000 year and 100–200 million
years, the time of formation of the first stars and galaxies, is referred to as the “dark ages” of the
universe. After matter-light decoupling, the “Cosmic Microwave Background Radiation” (CMB
or CMBR) has maintained its equilibrium distribution while its temperature has decreased due
to the expansion of the universe.
1/ Today, at time t0 ≈ 14 × 109 years, the CMBR temperature is T = 2.725 K. Calculate the
corresponding photon density nγ (T ) (in mm−3 ) and the energy density utot (T ) (in eV.cm−3 ).
2/ “Dark ages” The expansion of the universe between tc ≈ 380 000 years and today has been
mostly dominated by the energy of nonrelativistic matter, which leads to the time dependence
of the CMBR temperature according to3 T (t) ∝ t−2/3 . Deduce nγ (T ) (in µm−3 ) and utot (T )
(in eV µm−3 ) at time tc .
(Compare this to the temperature at the surface of the sun corresponding to the emitted
radiation, i.e. T = 5700 K).
5.3
Equilibrium between matter and light , and spontaneous emission
In a famous article few years before the birth of quantum mechanics, 4 Einstein showed that
consistency between quantum mechanics and statistical mechanics implies an imbalance between the absorption and emission probability of light between two atomic (or molecular) levels.
The emission probability is larger than the absorption probability due to the phenomenon of
spontaneous emission, which originates in the quantum nature of the electromagnetic field.
1/ Emission and absorption.– We focus on two quantum levels | g i (ground state) and | e i
(excited state) of an atom (or a molecule). The energy gap is ~ω0 . Denote by Pg (t) and Pe (t),
the probability at time t for the atom to be in state | g i and state | e i, respectively.
We consider three processes:
• In vacuum the atom in its excited state falls back to its ground state at a rate Ae→g
(spontaneous emission).
• When submitted to monochromatic radiation, the atom in its excited state falls back to its
ground state with rate Ae→g + Be→g I(ω0 ) (spontaneous and stimulated emission), where
I(ω0 ) is the intensity of the radiation field at frequency ω0 .
Avant tc , l’expansion fût plutôt dominée par l’énergie du rayonnement, ce qui conduit à T (t) ∝ t−1/2 .
Albert Einstein, “Zur Quantentheorie der Strahlung”, Physikalische Zeitschrift 18, 121–128 (1917).
The article has been reproduced in: A. Einstein, Œuvres choisies. 1. Quanta, Seuil (1989), textes choisis et
présentés par F. Balibar, O. Darrigol & B. Jech.
3
4
25
• The transition rate between the ground state and the excited state is Bg→e I(ω0 ) (absorption).
a) Write down the pair of coupled differential equations for Pg (t) and Pe (t).
b) Derive the equilibrium condition.
2/ Thermal equilibrium for matter.– The multiple absorption and emission processes are
responsible for establishing thermal equilibrium between light and matter. Assuming that equi(eq)
(eq)
librium is described by the canonical distribution, find an expression for Pg /Pe .
3/ Thermal equilibrium for radiation.– Assuming thermal equilibrium, recall the expression
for the spectral density (Planck’s law) u(ω; T ) (i.e. Vol × u(ω; T )dω is the contribution to the
energy of radiation of the frequencies ∈ [ω, ω + dω]). Henceforth we will assume that the field
intensity is given by Planck’s law, I(ω0 ) = u(ω0 ; T ).
4/ Relation between spontaneous emission and stimulated emission/absorption.–
a) Analyze the hight temperature behavior of the equation obtained in 1.b and show that
Be→g = Bg→e .
From here on we will simply denote the Einstein coefficients that describe spontaneous and
stimulated emission/absorption by A ≡ Ae→g and B ≡ Be→g = Bg→e .
b) Show that A/B ∝ ω03 .
c) Why is it easier to make a MASER
5
than a LASER ?
6
This first prediction by Einstein (1917) on the spontaneous emission rate A was confirmed
only at the end of the 1920s with the development of quantum electrodynamics; in the framework
of this theory Dirac proposed the first microscopic theory for spontaneous emission. 7
Appendix:
Z
∞
dx
0
xα−1
= Γ(α) ζ(α)
ex − 1
Z
∞
dx
0
x4
π4
=
30
sh2 x
(53)
(you may deduce the second integral from the firat one for α = 4). We have ζ(3) ' 1.202 and
4
ζ(4) = π90 .
5
Microwave Amplification by Stimulated Emmission of Radiation
The first ammonia MASER was built in 1953 by Charles H. Townes, who adapted the techniques to light in
1962 and received the Nobel prize in 1964.
7
P. A. M. Dirac, The quantum theory of the emission and absorption of radiation, Proc. Roy. Soc. London
A114, 243 (1927).
6
26
TD 6: Systems in contact with a thermostat and a particle
reservoir – Grand Canonical ensemble
6.1
Ideal Gas
We consider an ideal gas at thermodynamic equilibrium in a volume V . We fix the temperature
T and chemical potential µ.
1/ Extensivity.– Show that the grand potential may be written as
J(T, µ, V ) = V × j(T, µ) .
(54)
Discuss the physical interpretation of the “volumetric density of the grand potential” j.
2/ Classical ideal gas.– We consider a dilute gas of particles, for which we may assume
that the Maxwell-Boltzmann approximation is justified. For this question no supplementary
hypothesis (the number of the degrees of freedom, their relativistic or nonrelativistic nature,
their dynamics, etc.) will be needed.
We introduce z, the single particle partition function. Justify that z ∝ V .
Show that the grand canonical partition function is
h
i
Ξ = exp eβµ z .
(55)
Show that, under this minimal hypothesis, it is possible to derive the equation of state of the
ideal gas, pV = N kB T .
3/ Monatomic classical ideal gas.– Give the explicit expression for Ξ and J for the monatomic
G
G
classical perfect gas in the Maxwell-Boltzman regime. Derive N (T, µ, V ) and E (T, µ, V ), and
from these the pressure pG (T, µ).
6.2
Adsorption of an ideal gas on a solid interface
We consider a container of volume V filled with a
monatomic ideal gas of indistinguishable atoms.
This gas is in contact with a solid interface that
may adsorb (trap) the gas atoms. We model the
interface as an ensemble of A adsorption sites.
Each site can adsorb only one atom, which then
has an energy −0
The system is in equilibrium at a temperature T and we model the adsorbed atoms, i.e. the
adsorbed phase, as a system with a fluctuating number of particles at fixed chemical potential
µ and temperature T . The gas acts as a reservoir.
1/ Derive the grand-canonical partition function ξtrap for a single adsorption site. Deduce the
grand-canonical partition function Ξ(T, A, µ) for all atoms adsorbed on the surface.
2/ We will now explore an alternative route. Derive the canonical partition function Z(T, A, N )
of a collection of N adsorbed atoms (Note: the number of adsorbed atoms N is much smaller
than the number of sites A). Rederive the results for Ξ(T, A, µ) obtained in the preceding
question.
27
3/ Calculate the average number of adsorbed atoms N as a function of 0 , µ, A, and T . From
this, derive the occupation probability θ = N /A of an adsorption site.
4/ The chemical potential µ is fixed by the ideal gas. This may be used to deduce an expression
for the site occupation probability θ as a function of the gas pressure P temperature T (note
that the number of atoms N is much smaller than the number Ngas of gas atoms).
We define a parameter
2πmkB T 3/2
0
P0 (T ) = kB T
exp −
,
h2
kB T
and will express θ as a function of P and P0 (T ).
5/ Langmuir isotherm.– How does the curve θ(P ) behave for different temperatures?
6/ (A question for the brave) Calculate the variance σN that characterizes the fluctuations of
N around its average value. Remember that
2
σN
= (N − N )2 = N 2 − N
2
Comment on this result.
6.3
Density fluctuations in a fluid – Compressibility
Let a fluid be thermalized in a box at temperature T . We consider a small volume V inside the
total box of volume Vtot (figure). The number N of particles in the box fluctuates with time.
Ntot , Vtot
N, V
Figure 7: We consider N particles in the small volume V of a fluid.
1/ Order of magnitude for the gas
a) If the gas is at normal temperature and pressure, calculate the average number of particles
N in a volume V = 1 cm3 .
b) The typical particle velocity is v ≈ 500 m/s and the average collision time (the typical time
between two successive collisions) is τ ≈ 2 ns (See Exercise 1.1). What is the typical time that
a particle spends inside the volume V ? (Reminder: the diffusion constant is D = `2 /3τ where
` = vτ ) How many collisions will the particle typically experience during this time?
c) We wish to estimate the particle renewal in volume V . Derive the expression for the number
δNτ of particles entering/exiting the volume in a time τ . Show that δNτ /N ∼ `/L.
d) Justify that, under these conditions, the gas inside the volume V may be described within
the framework of the grand-canonical ensemble.
G
def
2/ Recall the derivation of the average N and the variance ∆N 2 = Var(N ) from the grandcanonical partition function. Derive the relation
G
∂N
∆N = kB T
.
∂µ
2
28
(56)
3/ A thermodynamic identity.– In this question, we identify N with its average in order to
re-derive a thermodynamic identity. Starting from µ = f (N/V, T ) and p = g(N/V, T ) where f
and g are two functions (justify their form), show that
∂p
∂µ
V ∂µ
∂p
V
=−
et
=−
.
(57)
∂N T,V
N ∂V T,N
∂N T,V
N ∂V T,N
Derive the Maxwell relation
∂p
∂N
=−
T,V
∂µ
∂V
(58)
T,N
Suggestion: Use the differential of the Helmholtz free energy dF = −S dT − p dV + µ dN .
4/ Compressibility.–
the relation between the fluctuations and the isothermal com Derive
def
1
∂V
pressibility κT = − V ∂p
,
T,N
∆N 2
N
G
= n kB T κT
(59)
G
where n = N /V .
Remark: This is an example of a fluctuation-dissipation relation, 8 i.e., a relation between
the response function (the compressibility) and the fluctuations. Another example of this
fundamental relation is CV = kB1T 2 ∆E 2 (Exercise 4.1).
5/ Classical ideal gas– Write down the expression for the compressibility of a classical ideal
gas. Deduce from it the expression for the fluctuations.
6/ (Optional ) Pair correlation function.– In this question we are going to establish the
relationP
between (59) and the fluctuations in the fluid. To this end we introduce the local density
r − ~ri ) where ~ri is the position of the ith particle.
n(~r) = N
i=1 δ(~
R
a) Justify that N = V d~r n(~r) and show that the variance may be expressed as an integral of
the density correlation function,
Z
2
0
0
0
∆N =
d~rd~r n(~r)n(~r ) − n(~r) × n(~r ) .
(60)
V
In a homogeneous fluid the two-point correlation takes a simpler form,
n(~r)n(~r 0 ) = n δ(~r − ~r 0 ) + n2 g(~r − ~r 0 ) ,
(61)
where g(~r) is the pair correlation function, that characterizes the distribution of the distances
between the particles.
b) If the correlations are negligible at large distances, what is the limiting value of g(~r) when
||~r|| → ∞ ?
c) Derive from (59) the Ornstein-Zernike relation
Z ∞
n kB T κT = 1 + n
4πr2 dr [g(r) − 1] ,
(62)
0
where we have taken advantage of the rotational symmetry. Sketch the qualitative shape of g(r)
in a fluid.
8
Strictly speaking, “fluctuation-dissipation relations” usually concern dynamical response functions.
29
TD 7: Quantum statistics (1) – Fermi-Dirac statistics
7.1
Ideal Fermi gas
We consider a gas of “free” electrons (i.e. without interactions) in a container of volume V .
(This ideal fermion gas could for example represent the conduction electrons in a metal.)
1/ Give the expression for the density of states ρ() of one of the gas particles. What is the
average number of particles in a state with energy ?
2/ Derive the (implicit) equation that allows us to compute the chemical potential µ, as well as
an equation for the total energy of the gas. It will be useful to introduce the Fermi integrals
Z +∞
xν dx
Iν (α) =
.
exp(x − α) + 1
0
3/ We first consider the situation of zero temperature (degenerate Fermi gas). Compute the
chemical potential (Fermi energy) of the gas as well as its energy. Compare the behavior of
µ(T ) at zero temperature of an ideal fermion gas with that of a classical ideal gas. Numerical
example: copper has n ' 1029 conduction electrons per m3 . Compute the Fermi energy EF and
the order of magnitude of the velocity of an electron at T = 0 K.
A1872
W. H. LI EN AN 0 N. E. PH ILL I PS
r
0.5
1
1
I
1.0
I
~
I
I
I
TmLE IV. The heat capacity of rubidium; measurem
the liquid-helium temperature cryostat. The units of
pacity are mJ mole ' deg '. Temperatures are based on
He4 scale.
IS
I
I
I
~
Potassium
cu
'Cl
'
5
4
I
E
3,0
2
o 25
2.0
T
2.0
03
0.2
O.I
PK)
Pro. 1. C/T versus T' for potassium. Q: liquid-helium
o: adiabatic demagnetization cryostat.
cryostat;
1.1991
1.3249
1.4630
1.6163
1.7867
1.9718
24.07
32.22
43.46
59.22
81.31
111.4
149.7
200.3
269.1
3.1428
3.4535
3.7762
486.5
644.5
827.6
2. 1610
2.3660
2.5961
2.8521
a
III. EXPERIMENTAL RESULTS
361.7
4.0713
1.1952
1.3076
1.4365
1.5882
1.7523
1.9314
2.1287
2.3342
2.5471
2.7886
1009.
23.83
30.99
41.12
56.00
76.35
104.3
142.6
191.4
251.8
336.9
445. 1
581.4
3.0530
3.3361
3.6579
4.0081
1.5881
1.7361
1.9055
2.0910
2.2887
2.5015
2.7254
2.9798
3.2642
3.5784
3.9231
See Ref. 10.
Figure 8: Heat capacity of potassium (W.H. Lien and N.E. Phillips, Phys. Rev. 133, A1370
TheC/T
heat as
obtained
in thelinear
six experipoints of
capacity
and cesium of
taken
(1964)) in a figure showing
a function
T 2 . The
part oftassium
the dependence
C in the adiabatic dem
ments are presented in Tables I to
VI.
∗
zation
those taken in the liquidandcubic
cryostat the
on T corresponds to an effective
electron
mass
of
m
=
1,
25
m.
Do
you
know
where
If the experimental heat capacity points were all cryostat do not join smoothly in the region of
term comes from?
of equal accuracy the best method of analysis of the
In each case the
points obtained in the adiaba
data would be to plot C/T versus T' and to find y and
magnetization cryostat are high. The discrepa
of Eq. (4) of
the but
fromlow
and limiting
intercept
slope at Calculate
4/ We now consider theA situation
nonzero
temperature.
the atchemical
1.2'K, after the heat capacity
heat capacity
T'=0.
In
this work the measurements above 1'K are
calorimeter
is subtracted, is 1.3/0 for po
potential of the gas and its energy. Deduce from it the heat capacityempty
at low
temperature.
expected to be appreciably more accurate than those
and 3% for cesium, as will be seen in Figs. 1
Provide your comments. below
The following
in 1/α will be useful:
1'K, and second-order
it is therefore approximations
appropriate to give
This discrepancy has not been observed in ot
1'K
some weight to the points
even
in
the
above
in this apparatus, and is apparently
periments
2
2 y.3/2
determination of
This can π
the re- ated with the fact that in these experiments
be done through
−4
α
1
+
+
O(α
)
,
I
1 (α) =
that 3 Eq. (5) represents
quirement
2
8α2 the data at tem- introduced to the surface of the calorimeter
peratures at which deviations from Eq. (4) first became
of directly to the sample. This produces a superh
can2 have an appreciable
important. This requirement
of the calorimeter during the heating periods, w
2
5π of y and−4A.
aAect on the assignment
consequence that the heat loss from the calorim
I 3 (α) = α5/2 1of+values2 +
O(α ) .
There
is a further
2
complication
5
8α in, the analysis of the surroundings during the heating period is
these experiments. The experimental points for pothan that estimated from the drift rates befo
after the heating period. This effect is not impor
TABLE III. The heat capacity of rubidium: measurements in
the adiabatic demagnetization
cryostat. The units of heat capacity are mJ mole ' deg '. Temperatures are based on the 1958
He4 scale a
30
T
C
T
C
0.5
20
0
1.
l.5
7.2
Pauli paramagnetism
~ = B~uz . We
We consider an ideal gas of electrons in a uniform and constant magnetic field B
qe ~
~
~ as the magnetic moment of an electron, given by m
~ = ge mB S/~, where mB = 2m
define m
<0
e
is the Bohr magneton and ge ' 2 for a free electron.
1/ Show that the density of states on the energy axis, ρ± (), of the electrons with spin up and
spin down (Sz = ±~/2) is given by
1
ρ± () = ρ( ∓ mB) with > ±mB ,
2
where m = |ge mB /2| and ρ(E) is the density of states calculated in exercise 1.
2/ a. Calculate the number of electrons N± with spin up and spin down, respectively, at zero
temperature. Determine the Fermi energy F .
b. Give the value of the Bohr magneton mB in eV/Tesla. Justify the validity of the approximation
mB F (B = 0).
c. Show in particular that, in this limit, F is independent of B.
3/ Derive from the previous results that the magnetization M at zero temperature can be
written in the following form:
3
mB
M = Nm ×
.
2
F
Compare with the equivalent result for distinguishable particles and explain in particular the
origin of the enormous reduction in the magnetization.
4/ Write down the (formal) expressions for N± for arbitrary temperature as well as the expression for the magnetization M . Discuss the behavior of M when T → ∞ (or more precisely when
T TF ). Show that in this limit the chemical potential µch is given by
4(β F )3/2
exp{β µch } = √
,
3 π ch(β m B)
and that the magnetization may be written as M = N m th (β m B), as expected (cf. exercises 4.6.2 and ??). It should be noted that in this question the electron spins are treated
quantum mechanically (we have made no assumption on the relative values of kB T and m B),
but the Pauli exclusion principle is not taken into account (T TF ).
7.3
Intrinsic semiconductor
The spectrum of the single-particle energies of an electron in a crystal consists of bands separated by gaps (Bloch’s theory of the electronic band structure of a solid). 9 In certain crystals
the single-particle energy states are filled in such a way that the last band is completely occupied
(valence band), while the next one remains empty (conduction band). Such crystals are called
insulators, because no electrical conduction is possible unless some valence electrons are promoted to the conduction band, which lies Eg higher, under the effect of thermal fluctuations. If
the gap Eg is not too large (not more than 1 eV), the crystal is called a semiconductor, because
electrical conduction becomes (reasonably) efficient at room temperature.
If we are interested in the low-energy physics of a semiconductor, only the energy states at
the top of the (full) valence band and those at the bottom of the (empty) conduction band play
9
Cf. solid-state physics or quantum mechanics course (e.g. chapter 6 of C. Texier, Mécanique quantique, Dunod,
2011).
31
a role, because of the form of the Fermi-Dirac distribution. We will model the energy spectrum
as two half-infinite bands,
~cond
= Eg +
k
~2~k 2
~2~k 2
val
and
=
−
,
~
k
2m∗e
2m∗t
where the curvatures of the parabolas define the “effective masses” m∗t and m∗e (which depend
on the crystal structure).
vide
E
E
E
etats
quantiques
etats
quantiques
BC
Eg
occupee
BV
metal
k
isolant
Figure 9: The energy spectrum of a crystal is a continuum of states, with gaps. Its conducting or
insulating nature is determined by how the single-particle energy states are filled by the electrons.
1/ Give a quick estimate of the Fermi energy F .
2/ Write down the density of states associated with the states of the conduction band, ρcond (),
and the one associated with the states of the valence band, ρval ().
3/ Give the expression for the average number of electrons, separating the contributions of the
valence and the conduction electrons: N (T ) = N val (T ) + N cond (T ). Show that the average
number of valence electrons may be written as
Z ∞
N val (T ) = N val (0) −
d ρval (−) nt (; T, µ) .
0
|
{z
}
Nt
Give the expression for nt (; T, µ) (establish the relation with the occupation number of the
electrons). Justify that nt measures the average number of holes (“anti-electrons”).
4/ Anti-electrons.– The notion of a hole allows us to only consider “particles” (electrons and
holes) with positive energy. Using the notation ρt () ≡ ρval (−), give the general expressions for
the number of electrons Ne ≡ N cond (T ) and the number of holes Nt ≡ N val (0) − N val (T ).
5/ If βµ 1 and β(Eg − µ) 1, determine Ne and Nt explicitly.
6/ In an intrinsic semiconductor (without acceptor or donor impurities) the electrical neutrality
imposes that Ne = Nt . Derive from this the density n = Ne /V of charged particles in the conduction band, as well as the chemical potential µ(T ). Given the constraints on the temperature
(approximations of question 5), where is µ(T ) precisely located?
7/ Numerical example: For silicium Eg ' 1.12eV, m∗e ' 1.13me and m∗t ' 0.55me , whereas
for germanium Eg ' 0.67eV, m∗e ' 0.55me and m∗t ' 0.29me . We take T = 300K. Calculate the
chemical potential µ(T ) as well as the density n for each material. Compare nSi and nGe with
each other and with the free electron density in e.g. copper. Verify that the approximations of
question 5 are well justified.
32
7.4
Gas of relativistic fermions
We consider a gas of relativistic spin-1/2 fermions with energy p~ =
completely degenerate (T = 0 K).
p
p~ 2 c2 + m2 c4 which is
1/ Calculate the Fermi momentum pF of the gas within the semi-classical approximation (cf.
TD2). What is the domain of validity of that approximation? Find the Fermi energy in the
nonrelativistic case and in the ultrarelativistic one.
2/ Calculate the energy of the gas in the general relativistic case. What is its limit in the
non-relativistic (pF mc) and in the ultra-relativistic case? We give the integral
 3
5
X

Z X p
+ X10 + ... X 1

o
n
p
p
3
1
x2 1 + x2 dx =
X(2X 2 + 1) 1 + X 2 − ln(X + 1 + X 2 ) =

8
0
 X 4 + X 2 + ... X 1
4
4
3/ Determine the pressure of the gas and its equation of state in the ultrarelativistic and in the
nonrelativistic limit.
7.5
Neutron star
A neutron star is the remainder of a supernova, a phenomenon that occurs when the iron core
of a massive star collapses after having exhausted all its nuclear fuel. If the initial star is not too
massive, the collapse stops when the pressure of the gas of neutrons, formed by electron capture,
can offset the gravitational attraction. Here we consider a neutron star of mass M ' 1.4M ,
where M ' 2 × 1030 kg is the mass of the sun, and with a radius of the order of 10 km. Its
temperature is at most of the order of 108 K in equilibrium. We assume first that the neutron
star effectively consists of neutrons exclusively. A reminder: neutron mass ' 940 MeV/c2 ,
electron mass ' 0.5 MeV/c2 and ~c ' 200 MeV.fm.
1/ Calculate the number of neutrons N in the star.
2/ Using the results of the previous exercise, show that we may consider the neutron gas as a
nonrelativistic and completely degenerate Fermi gas.
3/ Calculate the energy and the grand potential of the star. Derive from it the pressure of the
neutron gas.
4/ A neutron is in fact an unstable particle that disintegrates according to the reaction n →
p+e− + ν̄e . This reaction is exothermic and frees approximately 0.8 MeV.
-a- If all the neutrons had disintegrated, we would have a gas of N electrons in the star.
Which would in that case be the Fermi energy of that gas, and the average energy per electron?
-b- Derive from this that the neutrons cannot all disintegrate.
-c- Calculate the maximum number of electrons that can be present in the neutron star and
derive from this the degree of disintegration of the neutrons. Conclusion?
33
TD 8: Quantum statistics (2) – Bose-Einstein
8.1
Bose-Einstein condensation in a harmonic trap
We study an ideal gas P
of N bosonic particles of mass m confined by a harmonic potential. The
Hamiltonian is Htot = N
ri , p~i ), where the single-particle energy is H(~r, p~ ) = p~ 2 /(2m) +
i=1 H(~
m ω 2 ~r 2 /2 − 3 ~ ω/2 (the zero of energy has been chosen such as to coincide with the ground
state energy).
1/ Calculate the density of states ρ(ε) within the semiclassical approximation described in TD2,
i.e. considering energy scales ε ~ ω.
2/ Recall (without demonstration) the expression for the average occupation nν BE of a singleparticle state of energy εν (the notation ν represents the set of quantum numbers (ν1 , ν2 , ν3 )
where νi = 0, 1, 2, . . .). Show that the chemical potential is negative.
3/ We first consider a high temperature, such that µ < 0 is satisfied as in the classical (dilute)
regime. We use the grand-canonical formulae (T and µ fixed) and consider the thermodynamic
limit in order to describe the canonical situation (T and N fixed).
(a) Write down the implicit relation between µ, T , and N that allows you to calculate, at
least in principle, the chemical potential µ(T, N ).
(b) We introduce the function
1
g3 (ϕ) =
2
Z
∞
0
x2
dx
ex /ϕ − 1
for ϕ ∈ [0, 1] .
(63)
Rewrite the previous relation in terms of g3 (eβµ ). Why are we restricted to ϕ ∈ [0, 1] ?
(c) Justify that
g3 (ϕ) =
∞
X
ϕn
n=1
n3
and
∞
X
n−3 = ζ(3) = 1, 202.. .
(64)
n=1
Plot the function g3 (ϕ) for ϕ ∈ [0, 1]. By which graphical method can you determine the
fugacity ϕ = eβ µ(T,N ) as a function of temperature?
(d) Show that the high temperature expansion of (64) corresponds to the classical result
obtained within the Maxwell-Boltzmann approximation: µclass (T ) = −3 kB T ln(T /T ∗ ). Is
the “correct” chemical potential above or below µclass (T )? Plot its shape.
(e) Show that, as the temperature is decreased, one reaches a critical temperature TBE where
µ = 0. Comment on this. Give the expression for TBE . Calculate its value for a typical
experiment: 10 N = 106 and ω = 2π × 100 Hz. Compare TBE to the temperature below which (noninteracting) discernable particles would all enter the single-particle ground
state.
4/ For temperatures T 6 TBE we make the two ad hoc assumptions:
(i) µ(T, N ) remains fixed at µ = 0.
(ii) if µ = 0, we cannot find an expression for the number of particles in the single-particle
ground state starting from the formula for the occupation number (the expression is divergent
10
~ = 1, 05 × 10−34 J.s and kB = 1, 38 × 10−23 J/K.
34
energy and specific heat. Ballistic expansion, which facilitates quantitative imaging, also provides a way to measure the energy of a Bose gas [6,7,18]. The total
energy of the trapped cloud consists of harmonic potential,
kinetic, and interaction potential energy contributions, or
Epot , Ekin , and Eint , respectively. As the trapping field is
for µ → 0). Hence, we keep n0 ≡ N0 (T ) as a parameter
and splitturned-off
the total
number
of particles
nonadiabatically
to initiate
the expansion,
Epot
suddenly
vanishes.
During
the
ensuing
expansion, the
according to
X remaining
1 components of the energy, Ekin and Eint , are
N = N0 (T ) +
. into purely kinetic energy E of the
(65)
thenβεtransformed
exν
e
−
1 Ekin 1 Eint ! E, where E is the quantity
panding
cloud:
ν6=(0,0,0)
we actually measure. According to the virial theorem,
if the particles are ideal (Eint � 0), E will equal half the
1 ideal
. However, for a system
total energy, i.e., E � 2 Etot
interparticle interactions the energy per particle due
(a) Justify that the second term on the RHS of Eq.with
(65)
can
be
rewritten
as
to Eint can be non-negligible and then E � aEtot , where
1
Z +∞ a is not necessarily 2 .
1 energy per particle, E�NkB To , is plotted
0
The scaled
.
(66)
N (T ) =
dε ρ(ε)
βε
versus
the
e −scaled
1 temperature T �To in Fig. 2. E�N is
0
normalized by the characteristic energy of the transition
kB To �N� just as the temperature is normalized by To . The
What does N 0 (T ) represent physically?
data shown are extracted from the same cloud images as
those analyzed for the ground-state fraction. Above To ,
(b) Show that N 0 (T )/N = (T /TBE )3 . Deduce an expression
for N0 (T ). solid
Plotlineit.which
Compare
to
the data tend to the straight
corresponds
the experimental data.
to the classical MB limit for the kinetic energy. Most
interesting is the behavior of the gas at the transition. By
FIG. 2.
The scaled energy per particle E�NkB To of the Bose
gas is plotted
vs scaled temperature T �To . The straight, solid
87 Rb
Figure 10: Results of the JILA group for 40000 atoms
: Phys.
Lett.
77,
4984
line isof
the energy
for
a classical,Rev.
ideal gas,
and the
dashed
line
is the predicted energy for a finite number of noninteracting
(1996).
bosons [22]. The solid, curved lines are separate polynomial
fits to the data above and below the empirical transition
temperature of 0.94To . (inset) The difference D between the
data and the classical energy emphasizes the change in slope
of the measured energy-temperature curve near 0.94To (vertical
dashed line).
π4
5/ Assume (or show)
1
6
Z
0
∞
∞
X
x3 dx
=
n−4 = ζ(4) =
= 1, 082.. .
x
e −1
4986 90
(67)
n=1
Show that the average energy per boson for T 6 TBE is given by
4
E
3 ζ(4)
T
.
=
N kB TBE
ζ(3)
TBE
(68)
Compare with the experimental data shown in Figure 10. 11 Compute the specific heat at
constant volume as a function of N , T , and TBE . Compare to the specific heat of a classical gas.
11
Figure: Circles are experimental data of the JILA group. The straight line is the classical result. The dotted
line corresponds to (68) and to another formula describing the gas for T > TBE . The continuous line is an
experimental fit. The inset shows the difference ∆ between the classical result and the experimental data. The
experiment seems to suggest a transition temperature 0, 94 TBE . What are the possible reasons for this difference?
[It is largely explained by the argument given by Ketterle and van Druten, Phys. Rev. A 54, 656 (1996), cf. the
discussion in the article of the JILA group].
35
0.94To obtained fr
discussed above.
The specific heat
derivative of the e
pressure or volum
derivative is the slo
plot (Fig. 2), with
rather confining po
measurement in co
expected behavior o
plots (Fig. 3). The
(MB statistics), disp
of temperature all
bosons confined in
heat at the critical
4
He can be modele
behavior is quite dif
by the specific he
critical (or lambda)
ideal gas cusp is r
We can compare o
heat of ideal bosons
oscillator (SHO) p
because we do no
SHO theory values
expansion energies.
discontinuous and fi
In order to extrac
we assume that, as p
FIG. 3. Specific hea
temperature T �To is
ment: theoretical cur
monic oscillator and
curve for liquid 4 He
heat for a classical id
of the polynomial fit
predicted specific hea
bosons in a harmonic
TD 9: Kinetics
9.1
Thermo-ionic effect
We aim to describe electron emission by a hot metallic cathod. Consider N noninteracting
electrons in a volume V at temperature T .
1/ Show that the average number of electrons in an infinitesimal volume of phase space is given
by
2
3 3
p~
d rd p
3 3
w(~r ; p~ )d rd p = 2f
−µ
,
2m
h3
where the Fermi factor is f (ε) =
1
. How is the probability density w(~r ; p~ ) normalized?.
eβε + 1
2/ We consider the flow through an elementary surface dS during a time interval dt of those
eletrons that have a momentum p~ up to d3 p. In what volume are these electrons ?
djz
as a funcdpz
tion of the normal component pz of the momentum and
the other parameters (use polar coordinates in order to
perform the integration over px and py ).
3/ Calculate the emitted current density
4/ We assume that the only those electrons can escape
from the metal that have a kinetic energy along the z axis
1
satisfying εz = 2m
p2 > εW > εF where εF is the Fermi
e z
energy. We consider a temperature T much smaller that
TF = εF /kB and an extraction energy W = εW − εF kB T . Deduce Richardson’s law for the current J,
J = A T 2 exp(−B/T )
(69)
Discuss the various hypotheses in relation with the experimental observations.
9.2
Effusion
A monoatomic gas is confined to a volume V . Atoms may escape from the box through a small
hole of area δS. The hole is sufficiently small so that we can assume that the gas is at all times
in a state of thermal equilibrium (this hypothesis will be discussed later). We may then, in
particular, introduce a time-dependent temperature.
1/ The gas has temperature T and the atomic velocities are described by the Maxwell distribution. Give the number of atoms dN exiting the volume during time interval dt. Discuss the
dependence of dN/dt on N/V and T
2/ Effusion from a volume with fixed temperature. Assume that the volume is in contact
with a thermostat which maintains its temperature. Atoms escape from the volume into vacuum.
(a) Discuss the time dependence of the number of atoms N left in the volume.
(b) Consider a gas of helium in a volume V = 1 ` at room temperature. The hole has an area
δS = 1 µm2 . How long does it take for half of the atoms to leave the box?
36
3/ Uranium enrichment. We consider a gas of uranium hexafluoride U F6 in a box of volume
V pierced by a little hole. The two isotopes 238 U and 235 U are present in nature in proportions
99.3% and 0.7%. Denote by N1 and N2 the number of molecules of isotope 238 U and 235 U ,
respectively.
(a) Show that the proportion of isotopes
the same.
238 U
and
235 U
in the gas that has left the box is not
(b) How many times should the effusion process be repeated in order to reach a proportion of
2.5% of 235 U ? The molar mass of fluor is 19 g.
4/ Adiabatic effusion.– We now analyse the effusion problem in a situation where there is no
heat exchange between the box and the external world (adiabatic walls). We will see that then
the temperature of the gas in the box will decrease with time.
(a) Calculate the energy loss dE of the gas in the box during time interval dt (i.e. the energy
carried away by the atoms exiting the box). Deduce from it the loss of energy per atom
dE/dN . Give your comments.
p
(b) Set λ = δS
kB /(2πm). Show that the evolution of the temperature and the number of
V
particles in the box is controlled by the two differential equations
dT
λ
= − T 3/2 ,
dt
3
dN
= −λ N T 1/2 .
dt
(70)
(c) Solve this system of coupled nonlinear differential equations. Compare with the solution
of question 2.
5/ Effusion between two thermostated boxes. Two boxes (1 and 2) contain a monatomic
gas of same nature. They do not exchange heat and energy transfer may only occur through
particle exchange through a little hole connecting the two boxes. The temperature T1 of box 1
is maintained by a thermostat, and similarly for temperature T2 of box 2. We denote by P1 and
P2 the pressures in the two boxes.
(a) Find a relation between temperatures and pressures expressing stationarity (i.e. dN1→2 =
dN2→1 ).
(b) Calculate the energy flow between the two boxes as a function of T2 − T1 .
Remark: we are describing here a nonequilibrium situation.
37