PHY 4523 Spring 2000 – Homework 3
Due by 5:00 p.m. on Friday, February 11.
Answer both questions. To receive full credit, you should explain your reasoning and show
all working. Please write neatly and remember to include your name.
1. A magnetic lattice gas.
A “lattice gas” consists of L independent sites, each of which can be either empty (in
which case its energy is 0) or occupied by one particle (with energy ε > 0). Each
particle has a magnetic moment of magnitude µ which, in the presence of an applied
magnetic field H, orients itself either up or down (parallel or antiparallel to the field)
with energy −µH or +µH, respectively.
(a) Find the canonical partition function for this system.
(b) Evaluate the average energy Ē, the average particle number N̄ , and the average
magnetization M̄ = (N̄up − N̄down )µ as functions of the temperature T and the
field H.
(c) Expand the results of (b) in the limit ε kB T µH. Sketch the variation of
Ē, N̄ , and M̄ with H at fixed T .
Note: You are being asked to calculate the partition function for a fixed number of
sites. As part (b) implies, the number of particles is not fixed.
2. Clusters on a 1D surface.
Adsorption of atoms onto the surface of a two-dimensional solid can be modeled by
considering an infinite one-dimensional sequence of surface sites. Each site is either
empty (energy 0) or occupied by one particle (energy ε). Each atom can be said to
belong to a “cluster” of adjacent atoms. The size of the cluster is the number of
consecutive occupied sites. For example, an isolated atom forms a cluster of size 1,
while two atoms on adjacent sites with an empty site on either side form a cluster of
size 2. Assume that the solid is held at a temperature T .
(a) Calculate the probability P that a randomly chosen site is occupied by an atom.
(b) Calculate the probability Pn that a randomly chosen site is occupied by an atom
P
belonging to a cluster of size n. To check your answer, verify that n Pn = P .
(c) Calculate the probability Qn that a randomly chosen atom belongs to a cluster
of size n.
(d) Calculate n̄, the mean size of the cluster to which a randomly chosen atom belongs.
(e) Calculate hni, the mean size of a randomly chosen cluster.
Hint 1: Before getting involved in any calculation, check that you fully understand the
difference between Pn and Qn , and between n̄ and hni.
Hint 2: You may find it helpful to use the identity
P∞
n=1
nxn = x (∂/∂x)
P∞
n=1
xn .