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Licence 3 et Magistere de physique
Statistical Physics
Exercises
October 22, 2015
Contents
Some useful formulae 30.1 Euler’s Gamma function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30.2 Gaussian integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30.3 Euler’s Beta function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30.4 Stirling’s formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30.5 Binomial formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40.6 Other useful integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
TD 1: Orders of magnitude – Probabilities 51.1 Gases and solids – Orders of magnitude . . . . . . . . . . . . . . . . . . . . . . . 51.2 Tossing a coin – Binomial law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 Rain drops – Statistics of independent events . . . . . . . . . . . . . . . . . . . . 71.4 Maxwell’s distribution – Doppler broadening of a spectral line . . . . . . . . . . . 8
TD 2: Phase space, ergodicity and density of states 102.1 Phase space of a 1D harmonic oscillator . . . . . . . . . . . . . . . . . . . . . . . 102.2 Volume of a hypersphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Density of states of free particles . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4 Classical and quantum harmonic oscillators . . . . . . . . . . . . . . . . . . . . . 112.A Appendix: Semi-classical rule for counting states in phase space . . . . . . . . . 12
TD 3: Fundamental postulate and microcanonical ensemble 133.1 The monoatomic ideal gas and the Sackur-Tetrode formula – Gibbs paradox . . . 133.2 Thermal contact between two cubic boxes . . . . . . . . . . . . . . . . . . . . . . 133.3 Paramagnetic crystal - Negative (absolute) temperatures . . . . . . . . . . . . . . 15
TD 4: System in contact with a thermostat – Canonical ensemble 164.1 Monoatomic ideal gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.2 Ideal, confined, nonideal, etc... gases. . . . . . . . . . . . . . . . . . . . . . . . . . 164.3 Gases of indistinguishable particules in a harmonic well . . . . . . . . . . . . . . 174.4 Partition function of a particule in a box – the role of boundary conditions . . . 184.5 Diatomic ideal gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1
4.6 Paramagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.6.1 Classical calculation: Langevin paramagnetism. . . . . . . . . . . . . . . . 194.6.2 Quantum mechanical calculation : Brillouin paramagnetism . . . . . . . . 20
4.A Appendix: Semiclassical summation rule in the phase space. . . . . . . . . . . . 224.B Appendix: Canonical mean of a physical quantity . . . . . . . . . . . . . . . . . 22
TD 5: Thermodynamic properties of harmonic oscillators 235.1 Lattice vibrations in a solid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.2 Thermodynamics of electromagnetic radiation . . . . . . . . . . . . . . . . . . . . 245.3 Equilibrium between matter and light , and spontaneous emission . . . . . . . . . 25
TD 6: Systems in contact with a thermostat and a particle reservoir – GrandCanonical ensemble 276.1 Ideal Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276.2 Adsorption of an ideal gas on a solid interface . . . . . . . . . . . . . . . . . . . . 276.3 Density fluctuations in a fluid – Compressibility . . . . . . . . . . . . . . . . . . . 28
TD 7: Quantum statistics (1) – Fermi-Dirac statistics 307.1 Ideal Fermi gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307.2 Pauli paramagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317.3 Intrinsic semiconductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317.4 Gas of relativistic fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337.5 Neutron star . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
TD 8: Quantum statistics (2) – Bose-Einstein 348.1 Bose-Einstein condensation in a harmonic trap . . . . . . . . . . . . . . . . . . . 34
TD 9: Kinetics 369.1 Thermo-ionic effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369.2 Effusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2
Some useful formulae
0.1 Euler’s Gamma function
Γ(z)def=
∫ ∞0
dt tz−1 e−t for Re z > 0 (1)
Note that integrals of the form∫∞0 dxxa e−Cx
bmay be simply related to the Gamma function.
Functional relation :Γ(z + 1) = z Γ(z) (2)
This allows to perform an analytic continuation in order to extend the definition of the Gammafucntion 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! (3)
Γ(n+1
2) =
√π
2n(2n− 1)!! (4)
where (2n− 1)!!def= 1× 3× 5× · · · × (2n− 1) = (2n)!
(2n)!! and (2n)!!def= 2× 4× · · · × (2n) = 2nn!.
0.2 Gaussian integrals
An integral related to Γ(1/2), ∫R
dx e−12ax2 =
√2π
a(5)
An integral related to Γ(3/2), ∫R
dxx2 e−12ax2 =
1
a
√2π
a(6)
More generally ∫R+
dxxn e−12ax2 =
1
2
(2
a
)n+12
Γ
(n+ 1
2
)(7)
Fourier transform of the Gaussian∫R
dx e−12ax2+ikx =
√2π
ae−
12ak2 (8)
0.3 Euler’s Beta function
B(µ, ν) =
∫ 1
0dt tµ−1(1− t)ν−1 = 2
∫ π/2
0dθ sin2µ−1 θ cos2ν−1 θ =
Γ(µ)Γ(ν)
Γ(µ+ ν)(9)
0.4 Stirling’s formula
Γ(z + 1) '√
2πz zz e−z i.e ln Γ(z + 1) = z ln z − z +1
2ln(2πz) +O(1/z) (10)
which will be used frequently to express ln(n!) ' n lnn− n or ddn ln(n!) ' lnn.
3
0.5 Binomial formula
(p+ q)N =N∑n=0
CnN pn qN−n where CnN
def=
N !
n!(N − n)!(11)
0.6 Other useful integrals∫ ∞0
dxxα−1
ex − 1= Γ(α) ζ(α) where ζ(α) =
∞∑n=1
n−α (12)
is the Euler Zeta function. We give ζ(2) = π2
6 , ζ(3) ' 1.202, ζ(4) = π4
90 , etc.Finally, we give ∫ ∞
0dx
x4
sh2 x=π4
30(13)
(related to the previous integral for α = 4).
4
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 magnitudesin situations that will analyzed in greater detail throughout the year. You are thus invited tocome 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 =27oC= 300 K) occupying a volume V = 24 `. The number of molecules is given by the Avogadronumber
NA ' 6.023 1023 particles/mole . (14)
We recall from basic thermodynamics that the energy contained in a mole of gas reads, up to aprefactor:
U ∼ RT . (15)
1/ Density.– Compute the density of particles n (in nm−3). Give an estimate for the typicaldistance between two particles.
2/ Energy and velocity.– What is the nature of the energy U? Compute (in J and then ineV) the order of magnitude of the translational kinetic energy of an oxygen molecule. Deducethe typical velocity of a molecule in the gas.
3/ Collision against the walls of the container.– Assuming a cubic box of side L, what isthe order of magnitude of the time needed for a particle to go back and forth from a wall? Inferthe 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 undergoinga 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 traveledby one particle between two successive collisions with other particles of the gas) is given by
` = 1/(√
2σn) (16)
where n is the particle density and σ the scattering cross-section. Justify the order of magnitudeσ ≈ afew A2 (we take σ = 4 A2). Derive an estimate for `. Compute the corresponding meantime between collisions τ = `/v, in which v is the typical velocity of a particle. What conclusioncan 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 betweenatoms (or molecules) are highly frequent. We now wish to discuss the validity of the ideal gasapproximation, i.e., to estimate when the interactions may be neglected. We introduce theinteraction energy u0 between two atoms and the range r0 of the interaction; in monatomicgases one typically has u0 ∼ 1 meV and r0 ∼ 5 A. Estimate the contribution of interaction tothe total energy of the gas. Show that
Einteraction
Ekinetic∼ Nr30
V
u0kBT
. (17)
Give the numerical value for this ratio. Your conclusion?
5
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 isagain 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 A in the crystal? Infer the typical displacement, atroom temperature, of an atom from its equilibrium position (in A).
Indication: One assumes that each atom is subject to a harmonic confinement whose stiffnessconstant is evaluated as follows: a displacement of δx = 1 A corresponds to δEp ≈ 10 eV(∼Rydberg).
C. Electrons in a metal.– Due to Pauli’s principle, electrons in a metal, considered as free
particles, pile up in individual energy states ε~k = ~2~k 2
2meuntil an energy called the Fermi energy
εF . The order of magnitude of this energy is fixed by the electron density εF ∼ ~2men2/3.
1/ Energy.– In gold the electron density is about n ' 55 nm−3. The value of the Fermi energyis εF ' 5.5 eV: check that this value is compatible with the formula for εF given above.
2/ Velocity.– We write εF = 12mev
2F . Compute the Fermi velocity vF . Compare with the
typical velocity associated with thermal energy vth ∼√kBT/me.
3/ Diffusion.– Electrons in a metal undergo many collisions (with impurities, other elec-trons,. . . ). The typical distance between two collisions with impurities is given by the elasticmean-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
then the diffusion constant D = `2e3τe
. What is the typical distance traveled by an electron int = 1 s ?
4/ Drift velocity.– Consider an electrical wire of cross section s = 1mm2 that carries a currentI = 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 probabilityof drawing n times tails among N draws (attention: n is random and N is a sure (= nonran-dom)parameter). If the coin is fixed (= fraudulent), we write p ( 6= 1/2) for the probability toget 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
to compute the average 〈n〉 and the variance ∆n2def=Var(n) = 〈n2〉 − 〈n〉2. To do so, it is
convenient to introduce the generating function
GN (s)def= 〈sn〉 , (18)
whose variable s may be any complex number. Express GN (s) as a function of ΠN (n). Giventhe function GN (s), how can you find the first two moments 〈n〉 and 〈n2〉 ? 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) isapproximately 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 conditionsof temperature and pressure, we have seen that a molecule has a typical velocity v ≈ 500 m/sand typically undergoes a collision every τ ≈ 2 ns. By assimilating the motion of the moleculeto a symmetric random walk (p = q = 1/2) of steps ` = vτ ≈ 1 µm every τ ≈ 2 ns, calculatethe typical distance traveled by a molecule after 1 s. Compare your result with the distance amolecule 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 theinterval of time dt reads λdt.
We write P (n; t) for the probability that n events occur during the interval of time t (atten-tion: 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 equationsatisfied by this probability and solve it.
2/ Arbitrary n.– Following the same procedure, determine the set of coupled differentialequations satisfied by the distribution P (n; t).
3/ Generating function.– In order to solve these differential equations, it is practical tointroduce the generating function:
G(s; t)def= 〈sn〉 (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 〈nm〉 ? (Rk: it is actually more
convenient to introduce γ(p; t)def= G(ep; t) = 〈epn〉). 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 Gaussianin the limit λt→∞.
6/ How can you explain that the binomial and the Poisson distribution coincide with a Gaussianin the limit of large numbers?
7/ (optional) We discuss the relation between the binomial and the Poisson law. Show that inthe limit N →∞ and p→ 0, keeping pN = Λ constant, the binomial law ΠN (n) (exercise 1.2)tends towards a Poisson law (indication: N !
(N−n)! ≈ Nn).
B. Application: fluctuations of the force exerted by a gas on a wall.– We start againwith exercise 1.1 on the ideal gas. We write Ft for the time average over a period t of theforce exerted by the molecules striking the wall of the container. The frequency of the impactsderived in 1.1.A is fc ∼ 1026 impacts/s. Give the expression of the relative fluctuations of theforce 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 thevelocity distribution in a classical gas follows Maxwell’s law:
f(~v) =( m
2πkBT
) 32
exp
− m~v 2
2kBT
, (20)
where m is the mass of the particle, kB is Boltzmann’s constant, T the temperature and v2 =v2x + v2y + v2z .
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: 〈vx〉, 〈vy〉, 〈vz〉, 〈v2x〉, 〈v2y〉, 〈v2z〉, 〈vxvy〉, 〈v2xv2y〉, and〈Ec〉.
3/ Marginal distribution of component vx.– Deduce from f(~v) the probability to find vxbetween vx and vx + dvx, whatever the values of (vy, vz) (marginal distribution for vx).
4/ Marginal distribution of the modulus v = ‖~v‖.– Deduce from f(~v) the probability tofind the velocity modulus v between v and v + dv (taking advantage of the isotropy of f andchanging to spherical coordinates in velocity space). Check that the distribution so obtainedis well normalized. Compare the most probable value of v with its average 〈v〉. Evaluate thevariance of v and the standard deviation σv =
√〈v2〉 − 〈v〉2.
B. Application: Doppler broadening of a spectral line.– In a spectroscopy experimenton 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) aretreated classically. We are interested in a transition between two atomic levels separated by~ω0.
x
Ω
spectrometre
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 ofmotionless 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 offrequency ω in its frame. Because of the Doppler effect, the frequency detected in the detector(=laboratory) frame will be
Ω ' ω(
1 +vxc
)in which we assumed vx c . (22)
8
c being the speed of light.
1/ Explain why the intensity of the radiation captured by the detector is I(Ω) ' i0 〈L[Ω (1− vx/c)]〉,where 〈· · ·〉 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) inthis case.
4/ What is the temperature scale T0 that allows to discriminate between the two previousregimes?
5/ We consider a gas of Rubidium atoms (87Rb of mass M ' 87mp), excited at the transitionwave-length λ = 780 nm. What is the width (in frequency) of the spectral line when the gaz isat temperature T = 190 oC? 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 oC.
RemindersAvogadro’s constant: NA ' 6.023 1023particles/mole.Boltzmann’s constant: kB ' 1.38 10−23 J K−1.Ideal gas constant: R = NAkB ' 8.31 J K−1 mol−1.
9
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
H(x, p) =p2
2m+
1
2mω2x2 (23)
in which m is the mass of the particle and ω the oscillator frequency.
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. Solvethem 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 theoscillator 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 probabilitydensity of the position.
B. Statistical physics.– We recover the previous result by a totally different method. Weconsider that the energy of the particle is known up to an uncertainty dE so that it is situatedbetween 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 equallyprobable. Next calculate the probability that the oscillator is represented by a point having anabscissa between x and x+ dx.
C. Level counting.– The classical approach must correspond to the high energy limit of thequantum approach. It is necessary, as in quantum mechanics, to be able to count the micro-states.
1/ Which rule, borrowed from quantum mechanics, allows one to perform such counting in phasespace?
2/ Calculate the number of micro-states that are classically accessible to an oscillator of energybetween E and E + dE. Compare this result with the quantum calculation, knowing that thequantum energies of a 1D harmonic oscillator are given by
εn = ~ω(n+ 1/2) avec n = 0, 1, 2, . . . . (25)
2.2 Volume of a hypersphere
A hypersphere of radius R in Rd is the domain defined by x21+x22+· · ·+x2d 6 R2. By studying the
integral∫Rd d~x e−~x
2, calculate the surface of the hypersphere Sd(R) and show that the volume
is given by
Vd(R) = VdRd where Vd =
πd/2
Γ(d2 + 1)(26)
is the volume of the sphere of unit radius (consider the cases d = 1, 2, and 3).
10
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 thesystem is
H =
N∑i=1
~pi2
2m. (27)
By (incorrectly) considering the atoms as distinguishable, show that the volume Φ(E) of phasespace occupied by states of energy less than E is
Φ(E) =1
Γ(3N2 + 1)
(E
ε0
)3N/2
where ε0def=
2π~2
mL2(28)
Find the density of states.
Numerical example: Calculate ε0 (in J and then in eV) for helium atoms in a box of sizeL = 1 m.
2.4 Classical and quantum harmonic oscillators
We consider a system of N independent identical 1D harmonic oscillators.1D, The Hamiltonianof the system is
H =N∑i=1
(p2i2m
+1
2mω2q2i
). (29)
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 (thedimension of which will be specified). Express V(E) in terms of the constant V2N , the volumeof the hypersphere of unit radius (exercise 2.2).
b/ Using the semi-classical hypothesis that a quantum state occupies a cell of volume hN inphase space, calculate the number of quantum states of energy less than E (written Φ(E)), andthen 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 number of different ways of choosing N nonnegativeintegers (n1, n2, n3...nN ) such that their sum
∑Ni=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 onebar, then n2 balls, then one bar, . . . The total number of balls is M and the total number of barsis N − 1. Permutations of balls and bars each among themselves do not count. Only matter thenumber 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 micro-states is ensured by considering that one quantum state occupies a volume hD in classicalphase space.
Integrated density of states.– LetH(qi, pi) be the Hamiltonian governing the dynamicsof a system. We denote by Φ(E) the number of micro-states of energy less than E. In thesemi-classical limit, we have
Φ(E) =1
hD
∫H(qi,pi)6E
D∏i=1
dqidpi ≡1
hD
∫ D∏i=1
dqidpi θH (E −H(qi, pi)) (30)
where θH(x) is the Heaviside function.
Density of states.– The density of states is given by
ρ(E) = Φ′(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 (forinstance an ideal gas of N particles moving in three-dimensional space, D = 3N), wemust multiply by an extra factor 1/N ! to take into account the fact that the particlesare indistinguishable (i.e. that micro-states differing only by a permutation of particles areequivalent):
Φindist(E) =1
N !Φdist(E) (32)
Notice, however, that this expression accounts only partially for the symmetrization postu-late of quantum mechanics. The full consequences of the latter will be studied in detail intutorials 8 and 9.
12
TD 3: Fundamental postulate and microcanonical ensemble
3.1 The monoatomic ideal gas and the Sackur-Tetrode formula – Gibbs para-dox
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 ofthe (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 microcanon-ical entropy S∗(E, V,N) of the gas.
3/ Distinguishable atoms.– Here, we do not take into account the symmetrization postulateof quantum mechanics.
a/ Give the integral Φdisc(E) for the density of states of the gas composed of N atoms (you maywish to recall exercice 2.3). Calculate the corresponding microcanonical entropy S∗dist.
b/ Does this expression for S∗dist satisfy the extensivity property formulated above?
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 withthe wall removed.
∆Smelange = S∗disc(2E, 2V, 2N)− 2S∗disc(E, V,N) . (34)
Why do we say that this result is paradoxal?
4/ Indistinguishability.– Quantum mechanics asserts a principle of indistinguishability be-tween identical particules.
a/ Considering this principle, calculate the integral Φindisc(E) for the density of states of the gascomposed of N atoms and give the Sackur-Tetrode formula (1912).
S∗(E, V,N) = NkB
[5
2+ ln
(V
N
[mE
3π~2N
]3/2)](35)
Verify that this expression obeys the extensivity property of the entropy.
b/ Show that S∗ can be written as S∗ = 3NkB 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 temperatureand pressure. Find S∗/NkB.
3.2 Thermal contact between two cubic boxes
We consider a closed system composed of two identical cubic boxes of edge length L with oneparticle in each box. The lowest energy levels and their degeneracy for a particle in a box maybe 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 surroundedby an adiabatic wall.
13
1. We consider the wall between the two boxes as adiabatic. At time t = 0 the energy of theparticle in each box is EI = 12 ε0 and EII = 18 ε0 where ε0
def= h2
8mL2 .
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 arethere?
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 levelin a box degeneracy
3 ε0 3 = 12 + 12 + 12 16 ε0 6 = 12 + 12 + 22 39 ε0 9 = 12 + 22 + 22 311 ε0 11 = 12 + 12 + 32 312 ε0 12 = 22 + 22 + 22 114 ε0 14 = 12 + 22 + 32 617 ε0 17 = 22 + 22 + 32 318 ε0 18 = 12 + 12 + 42 319 ε0 19 = 12 + 32 + 32 321 ε0 21 = 12 + 22 + 42 622 ε0 22 = 22 + 32 + 32 324 ε0 24 = 22 + 22 + 42 326 ε0 26 = 12 + 32 + 42 627 ε0 27 = 12 + 12 + 52
27 = 32 + 32 + 32 429 ε0 29 = 22 + 32 + 42 630 ε0 30 = 12 + 22 + 52 6
TABLE 2
two particles (*) level (**) levelin a box degeneracy degeneracy
6 ε0 1 imposs.9 ε0 6 312 ε0 15 614 ε0 6 315 ε0 20 1017 ε0 30 1518 ε0 15 620 ε0 60 3021 ε0 12 622 ε0 15 923 ε0 60 3024 ε0 31 1525 ε0 60 30
(*) two distinguishable particles of the samemass.(**) 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. Eachparticle 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 thespins with the external magnetic field.We write n+ for the number of magnetic moments aligned with ~B and n− for those opposed to~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 negativevalues. 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) temper-atures. Hint: analyze what happens when you bring two systems of different microcanonicaltemperatures into contact.
What happens if we establish a contact between two identical systems with temperaturesT0 > 0 and −T0 ?
Figure 3: A typical record of nuclear magnetic inversion. The magnetization of the sample istested every 30s by NMR. Vertical bands on the graph represent 1mn. On the left is sketcheda typical signal of normal thermal equilibrium (T ≈ 300 K) revealing the magnetization ofthe 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 equilibriumvia a zero magnetization (at this point, the temperature goes from T = −∞ to T = ∞). Thisinversion is observed in lithium fluoride crystals. This behavior is possible because the relaxationtime between nuclear spins (t1 ∼ 10−5 s) is very short compared to the relaxation time betweenthe spins and the lattice (t2 ∼ 5 mn). When the field is rapidly reversed during a time betweent1 and t2, the system of nuclear spins can reach the thermal equilibrium and exhibit an absolutenegative temperature. Reference : E. M. Purcell and R. V. Pound, Phys. Rev. 81, 279 (1951).
15
TD 4: System in contact with a thermostat – Canonicalensemble
4.1 Monoatomic ideal gas
We study again the ideal gas (cf. exercice 3.1) but here we consider the gas at a fixed temperatureT , 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 Maxwell-Boltzmann approximation. Give the partition function using the de Broglie thermal wavelengthdefined by
ΛTdef=
√2π~2mkBT
. (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 thatit 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 definitionof the heat capacity,
CVdef=∂E
C(T, V,N)
∂T(i.e. CV =
(∂E∂T
)V,N
in thermodynamic notation ) (37)
Calculate the energy fluctuations of the system, Var(E)def= E2
C −(E
C)2. Compare the fluctua-
tions to the average value of the energy.
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-Tetrodeformula. 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 marginalprobability 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 givenby the Hamiltonian
H(~ri, ~pi) =N∑i=1
~pi2
2m+ U(~ri) . (38)
Give the expression for the canonical distribution, to be denoted by ρC(~ri, ~pi).b) How can we obtain the distribution function f that characterizes the position and momentumof a single particle? We define f(~r, ~p)d~rd~p as the probability for a particle to have a position ina 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 distributionof the particle velocities in the gas.
3/ Other gases.– In this question, we want to test the validity in more general cases of theresults obtained for the monoatomic gas. For each of the following situations, answer these twoquestions:
• 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. Discussthe density profile of the gas.
b) Gas of interacting particules (nonideal gas), i.e. U 6= 0.
c) Relativistic ideal gas, i.e. E =√~p 2c2 +m2c4.
d) Ultrarelativistic ideal gas i.e. E = ||~p||c.Calculate z explicitly in this case (express z as z = V/Λ3
r where Λr is the relativistic thermalwavelength). 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 willquantum 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 calculatedquite easily when the gas is enclosed in a harmonic well. Here we use this remarkable fact tocarefully 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 systemis
H =N∑i=1
(p2i2m
+1
2mω2x2i
)(40)
A. Distinguishable particules.
1/ Calculate the quantum mechanical partition function of the gas. Deduce the mean energy ofthe system.
2/ Discuss the high temperature approximation (for Zdisc and for EC
).
B. Maxwell-Boltzmann approximation. Give the expression of the partition function ofa gas of identical (and therefore indistinguishable) particles in the Maxwell-Boltzmann approx-imation. In this case, does it matter whether the particles are bosons or fermions? Give theexpression 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. Calculateexplicitly the partition function Zbosons. Deduce the high temperature limit of the partitionfunction and show that your result is identical to the Maxwell-Boltzmann approximation. Whatis the temperature T∗ that separates the quantum from the classical behavior? Discuss the originof the dependence of T∗ on N .
2/ Show that the free energy is
Fbosons(T,N) = N~ω2
+ kBTN∑n=1
ln(
1− e−n~ω/kBT). (41)
What is the physical meaning of the first term in Fbosons ?
3/ Calculate the mean energy of the gas of bosons.
4/ We define the canonical chemical potential as: µ(T,N)def= 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 usingthe de Broglie thermal wavelength ΛT (exercice 4.1).
2/ Dirichlet boundary conditions.– What is the energy spectrum of a particle if we use theDirichlet boundary condition ψ(0) = ψ(L) = 0 ? Calculate the partition function zβ expressedas a series in ΛT /L. Calculate with the aid of the Poisson formula the first terms of a hightemperature expansion for zβ.
3/ Periodic boundary conditions.– Same questions for the periodic boundary conditionsψ(0) = ψ(L) and ψ′(0) = ψ′(L).
4/ Comparaison.– Compare the partition function at high temperature obtained with the twokinds of boundary conditions. What seem to you the most convenient boundary conditions (inparticular in relation to the semiclassical analysis)?
Appendix: The Poisson formula.– Let f(x) be a function (or a distribution) defined on
R and let f(k)def=∫R dx f(x) e−ikx be its Fourier transform. It is possible to show the (very
nontrivial) identity ∑n∈Z
f(n) =∑n∈Z
f(2πn) (42)
4.5 Diatomic ideal gas
We study here the thermodynamics of a gas of diatomic molecules. Each molecule has threetranslational, two rotational, and one vibrational degree of freedom. The Hamiltonian for amolecule is
H '~P 2
2M+~2
2I+
p2r2mr
+1
2mrω
2(r − r∗)2 (43)
where ~P is the total momentum and ~ the orbital angular momentum which characterizesthe rotation of the molecule. Furthermore (r, pr) is a pair of conjugate variables that describesthe vibration of the molecule (in relative coordinates).
18
1/ Give the quantum mechanical spectrum of the translational, rotational, and vibrationalenergy. Show that the partition function for a molecule may be factorized according to z =ztranszrotzvib. 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-Boltzmannapproximation?
2/ High temperature, semiclassical approximation.– Calculate the partition function Zβof the gas in the semiclassical approximation (~ → 0) when all degrees of freedom are treatedclassically (discuss the validity of the result). Calculate the average energy of the gas and itsheat capacity.
3/ At lower temperatures it may not be possible to treat all degrees of freedom classically. Whatis the average energy of vibration when kBT ~ω ? Same question for the energy of rotationwhen kBT ~2/I. Discuss the behavior of the heat capacity as a function of temperature (take~2/I ~ω). Give your comments on the figure.
(K)
rot Tvib
0105
1
2
3
50 100 500 1000 5000
experimenttheory
T
CV /NkB
T
Figure 4: Specific heat of a gas of HD (deuterium-hydrogen). Trotdef= ~2/2kBI and Tvib
def= ~ω/kB.
Excerpt from R. Balian, ”From microscopic to macroscopic I”.
4.6 Paramagnetism
4.6.1 Classical calculation: Langevin paramagnetism.
We intend to find the equation of state of a paramagnetic material, i.e., the relation betweenthe total magnetic moment ~M of the material, the temperature T , and the external magneticfield ~B applied to the material. We consider N independent atoms, fixed at the sites of a crystallattice. 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 themagnetic moment of each atom is specified by two angles θ and ϕ. When a magnetic field ~B isapplied along the z axis, each atom acquires a potential energy
Hpot = −~µ · ~B = −µB cos θ . (44)
If each atom has a moment of inertia equal to I, then its dynamics is governed by the Hamilto-
19
nian1 :
H =1
2 I
(p2θ +
p2ϕ
sin2 θ
)+Hpot . (45)
1. Calculate the canonical partition function associated with H. Write the result in the formz = zcinzpot where zpot = 1 for B = 0. Show that zcin = Vol/Λ2
T where ΛT is a thermallength 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 pointsin the direction (θ, ϕ). Check that the probability density is normalized. Scketch w(θ, ϕ)vs. θ.
3. Calculate the average magnetic moment 〈µz〉 per atom. We will refer to M = N 〈µz〉 asthe total magnetization of the material. You may use the result given in the appendix andcalculate ∂z/∂B.
4. Discuss the behavior of 〈µz〉 as a function of magnetic field and temperature. Show thatthe 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 Hamiltonianof 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 orbitalangular momenta and the spins of the electrons for an atom in its ground state. We let J standfor the associated quantum number. The magnetic moment ~µ of an atom is related to ~J by
~µ = gµB ~J/~ , (46)
where µB = qe~2me' −9, 27.10−24 A m2 is the Bohr magneton. The Lande factor g is a dimen-
sionless constant typically of order one 2.
1. What are the eigenvalues and the eigenvectors of the Hamiltonian of an atom (rememberthat the projection Jz of the angular momentum of an atom may take the values m~ avecm ∈ −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 ?
1The kinetic energy of a moment of inertia I is Hcin = I2[θ2 + ϕ2 sin2 θ]. Then pθ = ∂Hcin/∂θ = Iθ and
pϕ = ∂Hcin/∂ϕ = Iϕ sin2 θ.2If 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 thequantum number associated with the total angular momentum ~J = ~L + ~S. The three numbers obeys at thetriangle inequality.
20
4. Calculate <µx> and <µz> for arbitrary J and for J = 1/2. Deduce the magnetizationM . Find an expression for M in terms of the Brillouin function
BJ(y) =d
dyln
J∑
m=−Je±my/J
=
1 + 12J
th[(1 + 1
2J )y] − 1
2J
th ( y2J )
.
Near the origin BJ(y) = J+13J y +O(y3). Show that the high temperature approximation
gives Curie’s Llaw. Show that at low temperature the quantum mechanical result is verydifferent from the classical result obtained in 4.6.1 (except for large values of J).
5. Determine from the experimental results sketched below the value of J for each ion.SPIN PARAMAGNETISM OF Cr+++, Fe+++, AND Gd+++ 56i
netic moments for our analysis. This analysis consistsof normalizing the calculated and experimental valuesat chosen values of H/T. Although space quantizationand the quenching of orbital angular momentum are un-mistakably indicated by the good agreement of simpletheory and experiment for the Pg2 state of the freechromium ion, there appears to be a small, second-order departure of the experimental results from theBrillouin function. In searching for the source of thesmall systematic deviation, one must consider thefollowing: (1) experiznental error in the measurementof M, II, and T, (2) dipole-dipole interaction, (3) ex-change interaction, (4) incomplete quenching, and (5)the eGect of the crystalline field splitting on the mag-netic energy levels. The diamagnetic contribution is,of course, too small to affect the results.It is felt that since the moment can be reproduced
to 0.2 percent and the magnitude of II/T is known toless than 1 percent, especially for 4.21'K, experimentalerror as a complete explanation must be discarded.It is true that the field seen by the ion is the applied
140
80
120
llo
l00
~ 90l—I 80
~ 70
g QQI—
40
20
10
4 8 l2 l6 20 24 28 52 M
/T & IO GAUSS DEG
pzo. 2. Plot of relative magnetic moment, M„vs II/T forpotassium chromium alum. -The heavy solid line is for a Brillouincurve for g= 2 (complete quenching of orbital angular momentumand J=S=3/2, fitted to the experimental data at the highestvalue 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 aLangevin curve fitted at the highest value of II/T to obtain thelower curve and fitted at a low value (slope fitting) of fI/2' toobtain the upper curve
~iii ZITI
B.OO
=-:o.oo:
e 1.30 'Ki 200 oK
~ 5.00 'K~ 4.21 'K
BRILLOUIN
oo( I
10 20 I 40"/T x 10 GAUSS /OEG
FiG. 3. Plot of average magnetic moment per ion, p 2fs H/T for(I) potassium chromium alum (J=$=3/2), (II) iron ammoniumalum (J=S=5/2, and (III) gadolinium sulfate octahydrate(J=S=7/2). g=2 in all cases, the normalizing point is at thehighest value of H/T.
Geld with corrections due to the demagnetization factor'and the Lorentz polarization" (effect of field of neigh-boring ions). However, since the sample is spherical,these two opposing corrections cancel" each other inerst approximation. Therefore, any error thus intro-duced is a second-order correction to a second-ordereGect which is negligible. For potassium chromiumalum, the chromium ions are greatly separated, prac-tically eliminating dipole-dipole and exchange inter-actions (ignoring the possibility of superexchangebased on the existence of excited states of normallydiamagnetic atoms) .Experiments which were carried out with iron am-
monium alum" (iron in sS@s state for the free ion) andgadolinium (ssrfs state for free ion) sulfate octahydrateshow (Fig. 3) slight departures from the Brillouinfunctions for free spins. Since L is zero for both free ions,these slight departures which remain for the two ionsare not attributable to incomplete quenching. Energylevels taken from Kittel and Luttinger" and based onthe effect of a crystalline cubic 6eld through spin-orbitinteraction, ' have been used to calculate magneticmoments 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 decomposition were carefully avoided."C.Kittel and J. M. Luttinger, Phys. Rev. 73, 162 (1948)."J.H, Van Vleck and W, Q. Penney, Phil. Mag. 17,961 (1934).
Figure 5: Average magnetic moment per ion (in units of the Bohr magneton) vs. B/T for certainparamagnetic salts: (I) Cr3+, (II) Fe3+ and (III) Gd3+. Here g = 2 in all cases (since ` =0). Dots are experimental results and curves are theoretical results obtained with our quantummechanical model [from W. E. Henry, Phys. Rev. 88, 559 (1952)].
21
Appendix 4.A: Semiclassical summation rule the phase space.
In the canonical ensemble the summation rule discussed in appendix 2.A takes the followingform: for a system with D degrees of freedom and Hamiltonian H(qi, pi) the partitionfunction is given by :
Zβ =1
hD
∫ D∏i=1
dqidpi e−βH(qi,pi). (47)
If the particles are indistinguishable, the partition function is given by the Maxwell-Boltzmann approximation
Z indiscβ =
1
N !Zdiscβ . (48)
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 = −Xdφin the expression of the energy. The canonical average of X is obtained by deriving thethermodynamic potential F (T,N, · · · , φ, · · · ) with respect to the ”conjugate force” φ,
Xc
= −∂F∂φ
(49)
Example: X → M is the magnetization, φ → B the magnetic field. The mean magneti-zation is given by M
c= −∂F
∂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 quantumharmonic oscillator of frequency ω,
εn = ~ω(n+
1
2
)for n = 0, 1, 2, . . . . (50)
1/ Calculate the partition function for the quantum harmonic oscillator.
2/ Deduce the average energy εC and the average occupation number nC. Analyze the classicallimit(~→ 0) of εC. Give a physical interpretation.
3/ Express the specific heat in the form cV (T ) = kB f(~ω/kBT ), where f(x) is a dimensionlessfunction. 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 aroundits equilibrium position (a site of the crystal lattice). With the 3N degrees of freedom weassociate 3N independent one-dimensional (quantum) harmonic oscillators. We assume thatall 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 areN ! 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 partitionfunction describing lattice vibrations.
2/ Deduce the total energy of the 3N oscillators (the result for T →∞ may be obtained more
easily) and the vibrational contribution C(Einst.)V (T ) to the heat capacity of the solid.
3/ Compare the limiting behavior of the heat capacity C(Einst.)V in Einstein’s model with the
experimental results (see Figure 6):
• High temperature (T →∞) : CV → 3NkB (Dulong & Petit’s law).
• Low temperature (T → 0) : CV ' a T + b T 3 with a 6= 0 for an electric conductor anda = 0 for an insulator.
C. The Debye model (1912).– The weakness of Einstein’s model, at the origin of the dis-
crepancy between the theoretical expression C(Einst.)V (T → 0) and the experimental observations,
lies in the assumption that all oscillators have the same frequency, (i.e., that atoms are inde-pendent). A more realistic model should account for the fact that, although the atoms may bedescribed as identical quantum oscillators, they are coupled (strongly interacting due to thechemical bonds). Nevertheless, the energy may be written as a quadratic form which may inprinciple diagonalized, that is, rewritten in the form
H =
3N∑i=1
(p2i2m
+1
2mω2
i q2i
)(51)
where (qi, pi) are pairs of conjugate coordinates associated with the vibrational modes of thecrystal (like the modes of a vibrating string). The crystal is characterized by a full spectrumof distinct eigenfrequencies ωi that form a continuous spectrum. The distribution of theeigenfrequencies 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 vibrationaleigenmodes.
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 wavelengthassociated with ωD ?
b) Give a sum rule for∫ ωD0 ρ(ω)dω.
c) In the Debye model we assume that the spectral density has the simple form
ρ(ω) =3V
2π2c3sω2 for ω ∈ [0, ωD] (52)
Explain the origin of the behavior ρ(ω) ∝ ω2. 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∫ ωD0 dω ρ(ω) f(~ω/kBT ). Deduce the high tempera-
ture behavior and compare to C(Einst.)V .
b) Justify that in the T → 0 limit only the low frequency behavior of ρ(ω) is important. De-
duce the low temperature behavior of CV (T ). Compare to C(Einst.)V and explain the difference
physically. Compare to the experimental data of figure 6.
0 20 40 60 80
T 3 [ K 3 ]0
100
200
300
C
[ mJ.
mol
1 .K1 ]
0 2 4 6 80
10
20
Figure 6: Left: Specific heat of diamond (in cal.mol−1.K−1). Experimental values are comparedto 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 fitto 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 supposedin 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 av-
erage electromagnetic energy as a sum over the modes: ECe−m =
∑modes ε
Cmode. Identify the
contribution of the vacuum, Evacuum (= limT→0ECe−m).
24
3/ Radiation energy.– Radiation corresponds to excitations of electromagnetic field: ECradia =
ECe−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
frequencies, 1V E
Cradia =
∫∞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 expressionin terms of the average number of excitations in each mode (i.e., in terms of the number ofphotons).
6/ The Stefan-Boltzmann law.– Calculate the photon density nγ(T ) and the radiationenergy density utot(T ) =
∫∞0 dω u(ω;T ).
B. Cosmic Micowave Background Radiation.– About 380 000 years after the big bangatoms formed and matter became eletrically neutral, i.e., light and matter decoupled: the uni-verse became “transparent” for radiation. The period between 380 000 year and 100–200 millionyears, the time of formation of the first stars and galaxies, is referred to as the “dark ages” of theuniverse. After matter-light decoupling, the “Cosmic Microwave Background Radiation” (CMBor CMBR) has maintained its equilibrium distribution while its temperature has decreased dueto the expansion of the universe.
1/ Today, at time t0 ≈ 14 × 109 years, the CMBR temperature is T = 2.725 K. Calculate thecorresponding 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 beenmostly dominated by the energy of nonrelativistic matter, which leads to the time dependenceof the CMBR temperature according to3 T (t) ∝ t−2/3. Deduce nγ(T ) (in µm−3) and utot(T )(in eVµm−3) at time tc.
(Compare this to the temperature at the surface of the sun corresponding to the emittedradiation, 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 thatconsistency between quantum mechanics and statistical mechanics implies an imbalance be-tween 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 ofspontaneous emission, which originates in the quantum nature of the electromagnetic field.
1/ Emission and absorption.– We focus on two quantum levels |g 〉 (ground state) and |e 〉(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 〉 and state |e 〉, 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 itsground state with rate Ae→g + Be→gI(ω0) (spontaneous and stimulated emission), whereI(ω0) is the intensity of the radiation field at frequency ω0.
3Avant tc, l’expansion fut plutot dominee par l’energie du rayonnement, ce qui conduit a T (t) ∝ t−1/2.4Albert 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 etpresentes par F. Balibar, O. Darrigol & B. Jech.
25
• The transition rate between the ground state and the excited state is Bg→eI(ω0) (absorp-tion).
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 areresponsible for establishing thermal equilibrium between light and matter. Assuming that equi-
librium is described by the canonical distribution, find an expression for P(eq)g /P
(eq)e .
3/ Thermal equilibrium for radiation.– Assuming thermal equilibrium, recall the expressionfor the spectral density (Planck’s law) u(ω;T ) (i.e. Vol × u(ω;T )dω is the contribution to theenergy of radiation of the frequencies ∈ [ω, ω + dω]). Henceforth we will assume that the fieldintensity 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 thatBe→g = Bg→e.
From here on we will simply denote the Einstein coefficients that describe spontaneous andstimulated emission/absorption by A ≡ Ae→g and B ≡ Be→g = Bg→e.
b) Show that A/B ∝ ω30.
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 confirmedonly at the end of the 1920s with the development of quantum electrodynamics; in the frameworkof this theory Dirac proposed the first microscopic theory for spontaneous emission. 7
Appendix: ∫ ∞0
dxxα−1
ex − 1= Γ(α) ζ(α)
∫ ∞0
dxx4
sh2 x=π4
30(53)
(you may deduce the second integral from the firat one for α = 4). We have ζ(3) ' 1.202 and
ζ(4) = π4
90 .
5Microwave Amplification by Stimulated Emmission of Radiation6The 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 1964.7P. A. M. Dirac, The quantum theory of the emission and absorption of radiation, Proc. Roy. Soc. London
A114, 243 (1927).
26
TD 6: Systems in contact with a thermostat and a particlereservoir – Grand Canonical ensemble
6.1 Ideal Gas
We consider an ideal gas at thermodynamic equilibrium in a volume V . We fix the temperatureT 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 assumethat the Maxwell-Boltzmann approximation is justified. For this question no supplementaryhypothesis (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
Ξ = exp[eβµ z
]. (55)
Show that, under this minimal hypothesis, it is possible to derive the equation of state of theideal gas, pV = NkBT .
3/ Monatomic classical ideal gas.– Give the explicit expression for Ξ and J for the monatomic
classical perfect gas in the Maxwell-Boltzman regime. Derive NG
(T, µ, V ) and EG
(T, µ, V ), andfrom 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 amonatomic ideal gas of indistinguishable atoms.This gas is in contact with a solid interface thatmay adsorb (trap) the gas atoms. We model theinterface as an ensemble of A adsorption sites.Each site can adsorb only one atom, which thenhas an energy −ε0
The system is in equilibrium at a temperature T and we model the adsorbed atoms, i.e. theadsorbed phase, as a system with a fluctuating number of particles at fixed chemical potentialµ and temperature T . The gas acts as a reservoir.
1/ Derive the grand-canonical partition function ξtrap for a single adsorption site. Deduce thegrand-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 smallerthan the number of sites A). Rederive the results for Ξ(T,A, µ) obtained in the precedingquestion.
27
3/ Calculate the average number of adsorbed atoms N as a function of ε0, µ, A, and T . Fromthis, 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 expressionfor the site occupation probability θ as a function of the gas pressure P temperature T (notethat the number of atoms N is much smaller than the number Ngas of gas atoms).We define a parameter
P0(T ) = kBT
(2πmkBT
h2
)3/2
exp
− ε0kBT
,
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 ofN around its average value. Remember that
σ2N = (N −N)2 = N2 −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 thetotal box of volume Vtot (figure). The number N of particles in the box fluctuates with time.
Vtot
N, V
tot ,N
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 particlesN in a volume V = 1 cm3.
b) The typical particle velocity is v ≈ 500 m/s and the average collision time (the typical timebetween two successive collisions) is τ ≈ 2 ns (See Exercise 1.1). What is the typical time thata 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 numberδNτ 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 withinthe framework of the grand-canonical ensemble.
2/ Recall the derivation of the average NG
and the variance ∆N2 def= Var(N) from the grand-
canonical partition function. Derive the relation
∆N2 = kBT∂N
G
∂µ. (56)
28
3/ A thermodynamic identity.– In this question, we identify N with its average in order tore-derive a thermodynamic identity. Starting from µ = f(N/V, T ) and p = g(N/V, T ) where fand g are two functions (justify their form), show that(
∂µ
∂N
)T,V
= −VN
(∂µ
∂V
)T,N
et
(∂p
∂N
)T,V
= −VN
(∂p
∂V
)T,N
. (57)
Derive the Maxwell relation (∂p
∂N
)T,V
= −(∂µ
∂V
)T,N
(58)
Suggestion: Use the differential of the Helmholtz free energy dF = −S dT − p dV + µdN .
4/ Compressibility.– Derive the relation between the fluctuations and the isothermal com-
pressibility κTdef= − 1
V
(∂V∂p
)T,N
,
∆N2
NG
= nkBT κT (59)
where n = NG/V .
Remark: This is an example of a fluctuation-dissipation relation, 8 i.e., a relation betweenthe response function (the compressibility) and the fluctuations. Another example of thisfundamental relation is CV = 1
kBT 2 ∆E2 (Exercise 4.1).
5/ Classical ideal gas– Write down the expression for the compressibility of a classical idealgas. Deduce from it the expression for the fluctuations.
6/ (Optional) Pair correlation function.– In this question we are going to establish therelation between (59) and the fluctuations in the fluid. To this end we introduce the local densityn(~r) =
∑Ni=1 δ(~r − ~ri) where ~ri is the position of the ith particle.
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,
∆N2 =
∫V
d~rd~r ′(n(~r)n(~r ′)− n(~r)× n(~r ′)
). (60)
In a homogeneous fluid the two-point correlation takes a simpler form,
n(~r)n(~r ′) = n δ(~r − ~r ′) + n2 g(~r − ~r ′) , (61)
where g(~r) is the pair correlation function, that characterizes the distribution of the distancesbetween 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
nkBT κT = 1 + n
∫ ∞0
4πr2dr [g(r)− 1] , (62)
where we have taken advantage of the rotational symmetry. Sketch the qualitative shape of g(r)in a fluid.
8Strictly 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(ε) of one of the gas particles. What is theaverage number of particles in a state with energy ε?
2/ Derive the (implicit) equation that allows us to compute the chemical potential µ, as well asan equation for the total energy of the gas. It will be useful to introduce the Fermi integrals
Iν(α) =
∫ +∞
0
xν dx
exp(x− α) + 1.
3/ We first consider the situation of zero temperature (degenerate Fermi gas). Compute thechemical potential (Fermi energy) of the gas as well as its energy. Compare the behavior ofµ(T ) at zero temperature of an ideal fermion gas with that of a classical ideal gas. Numericalexample: copper has n ' 1029 conduction electrons per m3. Compute the Fermi energy EF andthe 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 PS0.5 1.0 IS
r 1 1 I ~ I I I I I I I ~
Potassium
TmLE IV. The heat capacity of rubidium; measurements inthe liquid-helium temperature cryostat. The units of heat ca-pacity are mJ mole ' deg '. Temperatures are based on the 1958He4 scale.'
cu 5'Cl 4IE 2
o 25
3,0
2.0O.I 0.2
T PK)
2.003
Pro. 1. C/T versus T' for potassium. Q: liquid-helium cryostat;o: adiabatic demagnetization cryostat.
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III. EXPERIMENTAL RESULTS
The heat capacity points obtained in the six experi-ments are presented in Tables I to VI.If the experimental heat capacity points were all
of equal accuracy the best method of analysis of thedata would be to plot C/T versus T' and to find y andA of Eq. (4) from the intercept and limiting slope atT'=0. In this work the measurements above 1'K areexpected to be appreciably more accurate than thosebelow 1'K, and it is therefore appropriate to givesome weight to the points above 1'K even in thedetermination of y. This can be done through the re-quirement that Eq. (5) represents the data at tem-peratures at which deviations from Eq. (4) first becameimportant. This requirement can have an appreciableaAect on the assignment of values of y and A.There is a further complication in, the analysis of
these experiments. The experimental points for po-
TABLE III. The heat capacity of rubidium: measurements inthe adiabatic demagnetization cryostat. The units of heat ca-pacity are mJ mole ' deg '. Temperatures are based on the 1958He4 scale a
a See Ref. 10.
0.5 1.0 l.5
20
tassium and cesium taken in the adiabatic demagneti-zation cryostat and those taken in the liquid-heliumcryostat do not join smoothly in the region of overlap.In each case the points obtained in the adiabatic de-magnetization cryostat are high. The discrepancy inheat capacity at 1.2'K, after the heat capacity of theempty calorimeter is subtracted, is 1.3/0 for potassiumand 3% for cesium, as will be seen in Figs. 1 and 3.This discrepancy has not been observed in other ex-periments in this apparatus, and is apparently associ-ated with the fact that in these experiments heat isintroduced to the surface of the calorimeter insteadof directly to the sample. This produces a superheatingof the calorimeter during the heating periods, with theconsequence that the heat loss from the calorimeter tothe surroundings during the heating period is greaterthan that estimated from the drift rates before andafter the heating period. This effect is not important in
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a See Ref. 10.Pre. 2. C/T versus T' for rubidium. g: liquid-helium cryostat;
~: adiabatic demagnetization cryostat.
Figure 8: Heat capacity of potassium (W.H. Lien and N.E. Phillips, Phys. Rev. 133, A1370(1964)) in a figure showing C/T as a function of T 2. The linear part of the dependence of Con T corresponds to an effective electron mass of m∗ = 1, 25m. Do you know where the cubicterm comes from?
4/ We now consider the situation of low but nonzero temperature. Calculate the chemicalpotential of the gas and its energy. Deduce from it the heat capacity at low temperature.Provide your comments. The following second-order approximations in 1/α will be useful:
I 12(α) =
2
3α3/2
(1 +
π2
8α2+ O(α−4)
),
I 32(α) =
2
5α5/2
(1 +
5π2
8α2+ O(α−4)
).
30
7.2 Pauli paramagnetism
We consider an ideal gas of electrons in a uniform and constant magnetic field ~B = B~uz. Wedefine ~m as the magnetic moment of an electron, given by ~m = gemB
~S/~, where mB = qe~2me
< 0is the Bohr magneton and ge ' 2 for a free electron.
1/ Show that the density of states on the energy axis, ρ±(ε), of the electrons with spin up andspin down (Sz = ±~/2) is given by
ρ±(ε) =1
2ρ(ε∓mB) with ε > ±mB ,
where m = |gemB/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 zerotemperature. Determine the Fermi energy εF .
b. Give the value of the Bohr magneton mB in eV/Tesla. Justify the validity of the approximationmB εF (B = 0).
c. Show in particular that, in this limit, εF is independent of B.
3/ Derive from the previous results that the magnetization M at zero temperature can bewritten in the following form:
M =3
2Nm× mB
εF.
Compare with the equivalent result for distinguishable particles and explain in particular theorigin of the enormous reduction in the magnetization.
4/ Write down the (formal) expressions for N± for arbitrary temperature as well as the expres-sion for the magnetization M . Discuss the behavior of M when T →∞ (or more precisely whenT TF ). Show that in this limit the chemical potential µch is given by
expβ µch =4(β εF )3/2
3√π ch(βmB)
,
and that the magnetization may be written as M = N m th (βmB), as expected (cf. exer-cises 4.6.2 and ??). It should be noted that in this question the electron spins are treatedquantum mechanically (we have made no assumption on the relative values of kBT and mB),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 sepa-rated by gaps (Bloch’s theory of the electronic band structure of a solid). 9 In certain crystalsthe 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 calledinsulators, because no electrical conduction is possible unless some valence electrons are pro-moted to the conduction band, which lies Eg higher, under the effect of thermal fluctuations. Ifthe gap Eg is not too large (not more than 1 eV), the crystal is called a semiconductor, becauseelectrical conduction becomes (reasonably) efficient at room temperature.
If we are interested in the low-energy physics of a semiconductor, only the energy states atthe top of the (full) valence band and those at the bottom of the (empty) conduction band play
9Cf. solid-state physics or quantum mechanics course (e.g. chapter 6 of C. Texier, Mecanique quantique, Dunod,2011).
31
a role, because of the form of the Fermi-Dirac distribution. We will model the energy spectrumas two half-infinite bands,
εcond~k= Eg +
~2~k 2
2m∗eand εval~k
= −~2~k 2
2m∗t,
where the curvatures of the parabolas define the “effective masses” m∗t and m∗e (which dependon the crystal structure).
E
metal
E
quantiquesetats
etatsquantiques
occupee
vide
isolant
E
k
BV
BC
Eg
Figure 9: The energy spectrum of a crystal is a continuum of states, with gaps. Its conducting orinsulating nature is determined by how the single-particle energy states are filled by the electrons.
1/ Give a quick estimate of the Fermi energy εF .
2/ Write down the density of states associated with the states of the conduction band, ρcond(ε),and the one associated with the states of the valence band, ρval(ε).
3/ Give the expression for the average number of electrons, separating the contributions of thevalence and the conduction electrons: N(T ) = Nval(T ) + N cond(T ). Show that the averagenumber of valence electrons may be written as
Nval(T ) = Nval(0)−∫ ∞0
dε ρval(−ε)nt(ε;T, µ)︸ ︷︷ ︸Nt
.
Give the expression for nt(ε;T, µ) (establish the relation with the occupation number of theelectrons). 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 andholes) with positive energy. Using the notation ρt(ε) ≡ ρval(−ε), give the general expressions forthe number of electrons Ne ≡ N cond(T ) and the number of holes Nt ≡ Nval(0)−Nval(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 neutralityimposes that Ne = Nt. Derive from this the density n = Ne/V of charged particles in the con-duction 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, whereasfor germanium Eg ' 0.67eV, m∗e ' 0.55me and m∗t ' 0.29me. We take T = 300K. Calculate thechemical potential µ(T ) as well as the density n for each material. Compare nSi and nGe witheach other and with the free electron density in e.g. copper. Verify that the approximations ofquestion 5 are well justified.
32
7.4 Gas of relativistic fermions
We consider a gas of relativistic spin-1/2 fermions with energy ε~p =√~p 2c2 +m2c4 which is
completely degenerate (T = 0 K).
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 thenonrelativistic case and in the ultrarelativistic one.
2/ Calculate the energy of the gas in the general relativistic case. What is its limit in thenon-relativistic (pF mc) and in the ultra-relativistic case? We give the integral
∫ X
0x2√
1 + x2 dx =1
8
X(2X2 + 1)
√1 +X2 − ln(X +
√1 +X2)
=
X3
3 + X5
10 + ... X 1
X4
4 + X2
4 + ... X 1
3/ Determine the pressure of the gas and its equation of state in the ultrarelativistic and in thenonrelativistic limit.
7.5 Neutron star
A neutron star is the remainder of a supernova, a phenomenon that occurs when the iron coreof a massive star collapses after having exhausted all its nuclear fuel. If the initial star is not toomassive, 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. Itstemperature is at most of the order of 108 K in equilibrium. We assume first that the neutronstar 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 anonrelativistic and completely degenerate Fermi gas.
3/ Calculate the energy and the grand potential of the star. Derive from it the pressure of theneutron gas.
4/ A neutron is in fact an unstable particle that disintegrates according to the reaction n →p+e− + νe. 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 of N bosonic particles of mass m confined by a harmonic potential. TheHamiltonian is Htot =
∑Ni=1H(~ri, ~pi ), where the single-particle energy is H(~r, ~p ) = ~p 2/(2m) +
mω2 ~r 2/2 − 3 ~ω/2 (the zero of energy has been chosen such as to coincide with the groundstate 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 nνBE of a single-
particle 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 thermodynamiclimit 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, atleast in principle, the chemical potential µ(T,N).
(b) We introduce the function
g3(ϕ) =1
2
∫ ∞0
x2
ex/ϕ− 1dx 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) =
∞∑n=1
ϕn
n3and
∞∑n=1
n−3 = ζ(3) = 1, 202.. . (64)
Plot the function g3(ϕ) for ϕ ∈ [0, 1]. By which graphical method can you determine thefugacity ϕ = eβ µ(T,N) as a function of temperature?
(d) Show that the high temperature expansion of (64) corresponds to the classical resultobtained within the Maxwell-Boltzmann approximation: µclass(T ) = −3 kBT ln(T/T ∗). Isthe “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 whereµ = 0. Comment on this. Give the expression for TBE . Calculate its value for a typicalexperiment: 10 N = 106 and ω = 2π × 100 Hz. Compare TBE to the temperature be-low which (noninteracting) discernable particles would all enter the single-particle groundstate.
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-particleground 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
for µ→ 0). Hence, we keep n0 ≡ N0(T ) as a parameter and split the total number of particlesaccording to
N = N0(T ) +∑
ν 6=(0,0,0)
1
eβεν − 1. (65)
(a) Justify that the second term on the RHS of Eq. (65) can be rewritten as
N ′(T ) =
∫ +∞
0dε ρ(ε)
1
eβε − 1. (66)
What does N ′(T ) represent physically?
(b) Show that N ′(T )/N = (T/TBE)3. Deduce an expression for N0(T ). Plot it. Compare tothe experimental data.
VOLUME 77, NUMBER 25 P HY S I CA L REV I EW LE T T ER S 16 DECEMBER 1996
(dotted line, Fig. 1). Mean-field [21,23] and many-body[24] interaction effects may also shift TcN a few percent.The second result we present is a measurement of the
energy and specific heat. Ballistic expansion, which facili-tates quantitative imaging, also provides a way to mea-sure the energy of a Bose gas [6,7,18]. The totalenergy of the trapped cloud consists of harmonic potential,kinetic, and interaction potential energy contributions, orEpot, Ekin, and Eint, respectively. As the trapping field isnonadiabatically turned-off to initiate the expansion, Epotsuddenly vanishes. During the ensuing expansion, theremaining components of the energy, Ekin and Eint, arethen transformed into purely kinetic energy E of the ex-panding cloud: Ekin 1 Eint ! E, where E is the quantitywe actually measure. According to the virial theorem,if the particles are ideal (Eint 0), E will equal half thetotal energy, i.e., E 1
2Eidealtot . However, for a system
with interparticle interactions the energy per particle dueto Eint can be non-negligible and then E aEtot, wherea is not necessarily 1
2 .The scaled energy per particle, ENkBTo, is plotted
versus the scaled temperature TTo in Fig. 2. EN isnormalized by the characteristic energy of the transitionkBToN just as the temperature is normalized by To . Thedata shown are extracted from the same cloud images asthose analyzed for the ground-state fraction. Above To ,the data tend to the straight solid line which correspondsto the classical MB limit for the kinetic energy. Mostinteresting is the behavior of the gas at the transition. By
FIG. 2. The scaled energy per particle ENkBTo of the Bosegas is plotted vs scaled temperature TTo . The straight, solidline is the energy for a classical, ideal gas, and the dashed lineis the predicted energy for a finite number of noninteractingbosons [22]. The solid, curved lines are separate polynomialfits to the data above and below the empirical transitiontemperature of 0.94To . (inset) The difference D between thedata and the classical energy emphasizes the change in slopeof the measured energy-temperature curve near 0.94To (verticaldashed line).
examining the deviation D of the data from the classicalline we see (Fig. 2, inset) that the energy curve clearlychanges slope near the empirical transition temperature0.94To obtained from the ground-state fraction analysisdiscussed above.The specific heat is usually defined as the temperature
derivative of the energy per particle, taken with eitherpressure or volume held constant. In our case thederivative is the slope of the scaled energy vs temperatureplot (Fig. 2), with neither pressure nor volume, butrather confining potential held constant. To place ourmeasurement in context, it is instructive to look at theexpected behavior of related specific heat vs temperatureplots (Fig. 3). The specific heat of an ideal classical gas(MB statistics), displayed as a dashed line, is independentof temperature all the way to zero temperature. Idealbosons confined in a 3D box have a cusp in their specificheat at the critical temperature (dotted line) [9]. Liquid4He can be modeled as bosons in a 3D box, but the truebehavior is quite different from an ideal gas, as illustratedby the specific heat data [25] (dot-dashed line): Thecritical (or lambda) temperature is too low, and the gentleideal gas cusp is replaced by a logarithmic divergence.We can compare our data with the calculated specificheat of ideal bosons in a 3D anisotropic simple harmonicoscillator (SHO) potential [20] (solid line). Note thatbecause we do not measure Epot, we must divide theSHO theory values by two to compare with our measuredexpansion energies. The specific heat of the ideal gas isdiscontinuous and finite at the transition.In order to extract a specific heat from our noisy data,
we assume that, as predicted, there is a discontinuity in the
FIG. 3. Specific heat, at constant external potential, vs scaledtemperature TTo is plotted for various theories and experi-ment: theoretical curves for bosons in a anisotropic 3D har-monic oscillator and a 3D square well potential, and the datacurve for liquid 4He [25]. The flat dashed line is the specificheat for a classical ideal gas. (inset) The derivative (bold line)of the polynomial fits to our energy data is compared to thepredicted specific heat (fine line) for a finite number of idealbosons in a harmonic potential.
4986
Figure 10: Results of the JILA group for 40000 atoms of 87Rb : Phys. Rev. Lett. 77, 4984(1996).
5/ Assume (or show)
1
6
∫ ∞0
x3 dx
ex − 1=
∞∑n=1
n−4 = ζ(4) =π4
90= 1, 082.. . (67)
Show that the average energy per boson for T 6 TBE is given by
E
N kBTBE=
3 ζ(4)
ζ(3)
(T
TBE
)4
. (68)
Compare with the experimental data shown in Figure 10. 11 Compute the specific heat atconstant volume as a function of N , T , and TBE . Compare to the specific heat of a classical gas.
11Figure: Circles are experimental data of the JILA group. The straight line is the classical result. The dottedline corresponds to (68) and to another formula describing the gas for T > TBE . The continuous line is anexperimental fit. The inset shows the difference ∆ between the classical result and the experimental data. Theexperiment 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. thediscussion in the article of the JILA group].
35
TD 9: Kinetics
9.1 Thermo-ionic effect
We aim to describe electron emission by a hot metallic cathod. Consider N noninteractingelectrons in a volume V at temperature T .
1/ Show that the average number of electrons in an infinitesimal volume of phase space is givenby
w(~r ; ~p )d3rd3p = 2f
(~p 2
2m− µ
)d3rd3p
h3,
where the Fermi factor is f(ε) =1
eβε + 1. How is the probability density w(~r ; ~p ) normalized?.
2/ We consider the flow through an elementary surface dS during a time interval dt of thoseeletrons that have a momentum ~p up to d3p. In what volume are these electrons ?
3/ Calculate the emitted current densitydjzdpz
as a func-
tion of the normal component pz of the momentum andthe other parameters (use polar coordinates in order toperform the integration over px and py).
4/ We assume that the only those electrons can escapefrom the metal that have a kinetic energy along the z axissatisfying εz = 1
2mep2z > εW > εF where εF is the Fermi
energy. We consider a temperature T much smaller thatTF = εF /kB and an extraction energy W = εW − εF kBT . Deduce Richardson’s law for the current J ,
J = AT 2 exp(−B/T ) (69)
Discuss the various hypotheses in relation with the ex-perimental observations.
9.2 Effusion
A monoatomic gas is confined to a volume V . Atoms may escape from the box through a smallhole of area δS. The hole is sufficiently small so that we can assume that the gas is at all timesin a state of thermal equilibrium (this hypothesis will be discussed later). We may then, inparticular, introduce a time-dependent temperature.
1/ The gas has temperature T and the atomic velocities are described by the Maxwell distri-bution. Give the number of atoms dN exiting the volume during time interval dt. Discuss thedependence of dN/dt on N/V and T
2/ Effusion from a volume with fixed temperature. Assume that the volume is in contactwith 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 areaδS = 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 UF6 in a box of volumeV pierced by a little hole. The two isotopes 238U and 235U are present in nature in proportions99.3% and 0.7%. Denote by N1 and N2 the number of molecules of isotope 238U and 235U ,respectively.
(a) Show that the proportion of isotopes 238U and 235U in the gas that has left the box is notthe same.
(b) How many times should the effusion process be repeated in order to reach a proportion of2.5% of 235U ? The molar mass of fluor is 19 g.
4/ Adiabatic effusion.– We now analyse the effusion problem in a situation where there is noheat exchange between the box and the external world (adiabatic walls). We will see that thenthe 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 energycarried away by the atoms exiting the box). Deduce from it the loss of energy per atomdE/dN . Give your comments.
(b) Set λ = δSV
√kB/(2πm). Show that the evolution of the temperature and the number of
particles in the box is controlled by the two differential equations
dT
dt= −λ
3T 3/2 ,
dN
dt= −λN T 1/2 . (70)
(c) Solve this system of coupled nonlinear differential equations. Compare with the solutionof question 2.
5/ Effusion between two thermostated boxes. Two boxes (1 and 2) contain a monatomicgas of same nature. They do not exchange heat and energy transfer may only occur throughparticle exchange through a little hole connecting the two boxes. The temperature T1 of box 1is maintained by a thermostat, and similarly for temperature T2 of box 2. We denote by P1 andP2 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
