Thermodynamics, Statistical Physics, and Quantum Mechanics

42 PROBLEMS

where B is an external field, is the magnetic moment, and the prime

indicates that the summation is only over the nearest neighbors. Each spin

has nearest neighbors. The spins are restricted to equal The

coupling constantJ is positive. Following Weiss, represent the effect on

of the spin–spin interaction in (P.4.87.1) by the mean field set up by the

neighboringspins Calculate the linear spin susceptibility using

this mean field approximation. Yourexpression should diverge at some

temperature What is the physical significance of this divergence?

What is happening to the spin lattice at

4.88 Spin Waves in Ferromagnets (Princeton,

Colorado)

Consider the quantum mechanical spin-1/2 system with Hamiltonian

where the summation is over nearest-neighbor pairs in threedimensions.

a)

b)

c)

d)

Fluctuations

4.89 Magnetization Fluctuation (Stony Brook)

Consider N moments with two allowed orientations in an external
field H at temperature Calculate the fluctuation of magnetization M,
i.e.,

Derive the equation of motion for the spin at site of the lattice.

Convert the model to a classical microscopic model by inserting the

classical spin field into theequation of motion. Express

to lowestorder in its gradients, considering asimplecubic latticewith

lattice constant

Consider the ferromagneticcase with uniform magnetization

Derive the frequency-versus-wave vector relation of a small spin-

wave fluctuation

Quantize the spin waves in terms of magnons which are bosons. De-

rive thetemperaturedependence of theheatcapacity.