Lecture 7: Canonical Ensemble

Physics 127a: Class Notes

Lecture 7: Canonical Ensemble – Simple Examples The canonical partition function provides the standard route to calculating the thermodynamic properties of macroscopic systems—one of the important tasks of statistical mechanics X −βEj Hamiltonian → QN = e → free energy A(T,V,N)→ etc. (1) j

A number of simple examples illustrate this type of calculation, and provide useful physical insight into the behavior of more realistic systems. The following are simple because they are a collection of noninteracting objects, which makes the enumeration of states easy.

Harmonic Oscillators Classical The Hamiltonian for one oscillator in one space dimension is

p2 1 H(x,p) = + mω2x2 (2) 2m 2 0 with m the mass of the particle and ω0 the frequency of the oscillator. The partition function for one oscillator is Z ∞ p2 1 dx dp Q = −β + mω2x2 . 1 exp 0 (3) −∞ 2m 2 h The integrations over the Gaussian functions are precisely as in the ideal gas, so that

/ / 1 2πm 1 2 2π 1 2 kT Q = = , (4) 1 h β βmω2 hω 0 ¯ 0 introducing h¯ = h/2π for convenience. For N independent oscillators N N kT QN = (Q1) = (5) hω¯ 0 and then hω¯ A = NkT 0 , ln kT (6) U = NkT, (7) kT S = Nk ln + 1 , (8) hω¯ 0 hω¯ µ = kT 0 . ln kT (9)

Equation (7) is an example of the general equipartition theorem: each coordinate or momentum appearing p2/ m Kx2/ 1 kT as a quadratic term in the Hamiltonian (e.g. 2 , 2) contributes 2 to the average energy in the classical limit. The proof is an obvious generalization of the integrations done in Eq. (4)—see Pathria §3.7 for a more formal proof. For oscillators in 3 space dimensions, replace N by 3N in the above expressions.

1 Quantum The quantum calculation is very easy in this case. The energy levels of a single, one dimensional harmonic oscillator are 1 Ej = (j + )hω¯ 2 0 (10) so that X −β(j+1/2)hω¯ 0 Q1 = e (11) j −βhω / e ¯ 0 2 = = 1 . −βhω¯ (12) 1 − e 0 2 sinh(βhω¯ 0/2)

N For N one dimensional oscillators QN = (Q1) from which the thermodynamic behavior follows 1 −βhω¯ 0 A = NkT ln [2 sinh(βhω¯ 0/2)] = N hω¯ 0 + kT ln 1 − e , (13) 2 U = 1Nhω (βhω / ) = Nhω 1 + 1 . ¯ 0 coth ¯ 0 2 ¯ 0 βhω (14) 2 2 e ¯ 0 − 1

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0.4 specific heat

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0.5 1 1.5 T 2 Speciﬁc heat of N one dimensional harmonic oscillators scaled to Nk as a function of temperature (scaled to hω¯ 0/k). It is interesting to focus on the speciﬁc heat

βhω ∂U e ¯ 0 2 C = = Nk(βhω¯ 0) . (15) ∂T βhω 2 e ¯ 0 − 1

For T →∞, β → 0 and the exponentials can be expanded, and we find the classical, equipartition result C = Nk. For any finite temperature the specific heat is reduced below the classical result, and for low 2 −hω¯ 0/kT temperatures kT hω¯ 0 the exponentials are large and C ' Nk(hω¯ 0/kT) e so that the specific heat is exponentially small. The results are plotted above. These results, with N → 3N, are the Einstein model for the specific heat of the phonons in a solid.

Paramagnetism Consider N magnetic moments µ in an applied magnetic field B. There is competition between the magnetic energy of size µB which tends to align the moments along the field, and the thermal fluctuations.

2 Classical vector spins The Hamiltonian is XN XN E H =− µE · B =−µB cos θi (16) i=1 i E taking the ﬁeld B to be in the z direction and θi is the polar angle of the ith moment. This gives the partition N function QN = (Q1) with Z βµB cos θ Q1 = d e (17) with Z Z Z 2π π d = dφ sin θdθ (18) 0 0 the integral over all angles. The average z magnetic moment is hMzi = N hµzi R βµB cos θ d µcos θe ∂ ln Q1 hµzi = R = kT . (19) d eβµB cos θ ∂B

The one-moment partition function is easily evaluated Z 1 (βµB) βµB cos θ sinh Q1 = 2π d(cos θ)e = 4π , (20) −1 βµB so that hµzi = µ L(βµB), (21) with L the Langevin function L(x) = x − 1 . coth x (22) Note that L(x →∞) → 1, and for small x

x x3 L(x) ' − +··· . (23) 3 45 For large temperatures or small ﬁelds (small βµB)

Nµ2B hMzi ' . (24) 3kT χ = ∂ hM i /∂B| The linear increase with a small applied ﬁeld is known as the magnetic susceptibility z B→0, so that Nµ2 χ = . (25) 3kT This T −1 susceptibility is known as a Curie susceptibility.

3 1 S = 1 Ising model This might correspond to a quantum spin- 2 ( 2 ) system in which each spin has only two possible orientations, or a classical spin with strong, uniaxial, crystalline anisotropy. The Hamiltonian is X H =− µiB (26) i with µi =µ the magnetic moment of the ith spin. We assume there is no interaction between different spins. There are only two states for a single spin so the calculation Q1 is very easy

βµB −βµB Q1 = e + e = 2 cosh(µB/kT ). (27)

N Since the spins are non-interacting QN = (Q1) , and so

A(N, T ) =−NkT ln [2 cosh(µB/kT )] , (28) U =−NµB tanh(µB/kT ). (29)

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1 2 3 4 T 5 Speciﬁc heat scaled to Nk as a function of temperature scaled to µB/k for N moments µ in a ﬁeld B.

The specific heat is µB 2 1 C = Nk (30) kT 2 µB cosh kT which is plotted in the figure. The specific heat is proportional to T −2 at high temperatures and exponentially small at low temperatures. In between is a peak at kT ' µB known as a Schottky anomaly. Since we can also understand the specific heat as C = T∂S/∂T, we identify the anomaly with the decrease in entropy as the moments become ordered along the field.

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1 2 3 4 T 5 Nµ µB/k N 1 Magnetization scaled to as a function of temperature scaled to for spin- 2 objects.

4 We are also interested in the average magnetic moment P −βµi B i µie hMzi = P . (31) −βµi B i e Just as in calculating the average energy we see this is conveniently written ∂ ∂A hM i = kT Q =− z ∂B ln N ∂B (32) (with partials at constant N,T). This gives µB hM i = Nµ . z tanh kT (33)

At large field or low temperature we get saturation hMzi ' Nµ, whereas at small fields or high temperature 2 the behavior is linear in the field hMzi ' Nµ B/kT. The susceptibility is

Nµ2 χ = . kT (34) again of Curie form. Pathria §3.9 studies the case of arbitrary S quantum spins.