PHGN341: Thermal Physics
Quiz 6
March 29, 2013
NAME: KEY
1. Consider a spin-1 paramagnetic atom in thermal equilibrium with a reservoir at temperature, T, in an external magnetic
field, B0 . The energy of such an atom in an external magnetic field is U = −µ0 B0 mj , where mj is the quantum number
for the z-component of the angular momentum and µ0 is the Bohr magneton. (Recall that a spin-1 atom can have three
possible m-substates: mj = {+1, 0, −1}.)
a. Find the partition function for the atom.
Solution: From the definition of the partition function, we have:
X
e−βµ0 B0 mj = e−βµ0 B0 (−1) + e−βµ0 B0 (0) + e−βµ0 B0 (+1) = 1 + 2 cosh βµ0 B0 .
Z=
mj =−1, 0, +1
b. Find the average energy of the atom.
Solution: Given the partition function, the average energy, E, is found:
E=−
1 dZ
(2µ0 B0 ) sinh βµ0 B0
=−
Z dβ
1 + 2 cosh βµ0 B0
c. What is the atom’s average magnetic moment, µ = µ0 mj ?
Solution: The average value of mj is:
mj =
X
mj =−1, 0, +1
mj
e−βµ0 B0 (−1)
e−βµ0 B0 (0)
e−βµ0 B0 (+1)
sinh βµ0 B0
e−βµ0 B0 mj
= (+1)
+ (0)
+ (−1)
=2
.
Z
Z
Z
Z
Z
where Z = 1 + 2 cosh βµ0 B0 . Thus,
µ = 2µ0
sinh βµ0 B0
.
1 + 2 cosh βµ0 B0
(Note that this could have been obtained directly from the average energy since E = −µB0 .)
1