PHZ7427–Solid State II
Spring 2000
Problem Set 3
Feb. 17, 2000
Due: Mar. 3, 2000
Reading: Chs. 31-33, Ashcroft & Mermin; Ch. 4, Kittel
1. Ising Model. This problem requires the use of the “ising” software from the
SSS package.
The Ising model is
H=−1
2JX
ij n.n.
SiSj−X
j
SiH, (1)
where the first sum is over nearest neighbor sites (“n.n.”), and His an external
field.
(a) Use preset 2 to obtain a quantitative plot of magnetization vs. field at
T=50J. Run until program stops with maximum number of sweeps
set to 200. Increase external field by 10J by moving field slider, and run
again. Repeat up to a field of 100J. Note since you were increasing the
field at a constant “rate”, you have a plot of magnetization vs. field at
fixed temperature. Save as postscript file and print to hand in.
(b) In the dimensionless units used by the simulation, the magnetization of a
free spin in a noninteracting paramagnet is
M(T) = tanh(H/T); (2)
note this is the same as the Brillouin function defined in the notes for the
case of spin = 1/2.
Compare the experimental values detemined in (a) with Eq. (2). Is the
magnetization you measured more or less than predicted? Why?
(c) Use preset 2 and take a data series recording T,H,hMiand h(M−
hMi)2i.ChooseHvalues large enough so that Mis easily measurable,
but small enough so that Mstays less than 0.25, so that you are in the
linear field regime, and can sensibly define χ=M/H. Start reducing the
temperature (you will find you will have to reduce Has well to satisfy the
above conditions). Be sure to reset the graph before each change in T!Try
to estimate the temperature where the susceptibility χdiverges. One way
is to plot χ−1vs. T.
(d) Convince yourself that the mean field theory result for M,
M=tanhH+4M
T(3)
1
