Problems and Solutions on Thermodynamics and Statistical Mechanics

Statistical Physics 329

2143
Consider a crystalline lattice with Ising spins sc = t at each site L In
the presence of an external field H = (0, 0, Ho), the Hamiltonian of the
system may be written as

4c L

where J > 0 is a constant and the sum

only (each site has p nearest neighbors).

(a) Write an expression for the free energy of the system at temperature
T (do not try to evaluate it).

(b) Using the mean-field approximation, derive an equation for the
spontaneous magnetization rn = (so) for Ho = 0 and calculate the critical
temperature T, below which m # 0.

(1 - T/TJD as T + T,.

0), near T = T,.

Solution:

Denote by NA and NB the total numbers of particles of sc = +l and

st = -1 respectively. Also denote by NAA, NBB, and NAB the total number

of pairs of the nearest-neighbor particles that both have st = 1, that both

have sc = -1, and that have spins antiparallel to each other respectively.

The Hamiltonian can be written as

is over all nearest-neighbor sites

ZC

(c) Calculate the critical exponent /3 defined by m(T, Ho = 0) - const.

(d) Describe the behavior of the specific heat at constant Ho, C(Ho =

(Princeton)

H = -J(NAA + NBB - NAB) - POHO(NA - NB).
Considering the number of nearest-neighbor pairs with at least one s~ = +I,
we have
PNA = ~NAA $-NAB.
Similarly, PNB = ~NI~B + NAB. As N = NA + NB, among NA, NB, NAB,
NAA and NBB only two are independent. We can therefore write in terms
of NA and NAA
NB=N-NA ,
NAB = PNA - ~NAA ,
NBB = PN/2 - PNA + NAA.