Chapter 5 - WebRing

CHAPTER 5. MAGNETIC SYSTEMS 231
5.2 Noninteracting Magnetic Moments
We first review the behavior of a system of noninteracting magnetic moments with spin 1/2 in
equilibrium with a heat bath at temperature T. We discussed this system in Section 4.3.1 and in
Example 4.1 using the microcanonical ensemble.
The energy of interaction of a magnetic moment µ in a magnetic field B is given by
E = −µ·B = −µzB, (5.1)
where µz is the component of the magnetic moment in the direction of the magnetic field B.
Because the magnetic moment has spin 1/2, it has two possible orientations. We write µz = sµ,
where s = ±1. The association of the magnetic moment of a particle with its spin is an intrinsic
quantum mechanical effect (see Section 5.10.1). We will refer to the magnetic moment or the spin
of a particle interchangeably.
What would we like to know about the properties of a system of noninteracting spins? In the
absence of an external magnetic field, there is little of interest. The spins point randomly up or
down because there is no preferred direction, and the mean internal energy is zero. In contrast, in
the presence of an external magnetic field, the net magnetic moment and the energy of the system
are nonzero. In the following we will calculate their mean values as a function of the external
magnetic field B and the temperature T.
We assume that the spins are fixed on a lattice so that they are distinguishable even though
the spins are intrinsically quantum mechanical. Hence the only quantum mechanical property of
the system is that the spins are restricted to two values. As we will learn, the usual choice for
determining the thermal properties of systems defined on a lattice is the canonical ensemble.
Because each spin is independent of the others and distinguishable, we can find the partition
function for one spin, Z1, and use the relation ZN = ZN 1 to obtain ZN, the partition function for
N spins. (We reached a similar conclusion in Example 4.2.) We can derive this relation between
Z1 and ZN by writing the energy of the N spins as E = −µB N i=1si and expressing the partition
function ZN for the N-spin system as
To find Z1 we write
ZN =
s1=±1 s2=±1
=
s1=±1 s2=±1
=
s1=±1
e βµBs1
= e βµBs1
Z1 =
s=±1
s1=±1
...
sN=±1
...
sN=±1
s2=±1
N = Z N 1
e βµBΣN
i=1 si (5.2a)
e βµBs1 e βµBs2 ...e βµBsN (5.2b)
e βµBs2 ...
sN=±1
e βµBsN (5.2c)
. (5.2d)
e βµBs = e βµB(−1) +e βµB(+1) = 2coshβµB. (5.3)