8.5 Paramagnetism in Spin Systems
437
For high temperatures, by which we mean temperatures such that βg I μ N B 0 1,
we may expand the exponentials and retain only the first two terms for each of them,
to give
tanh
1
2 βg I μ N B 0
1 +
1
2 βg I μ N B 0 + · · ·
−
1 −
1
2 βg I μ N B 0 + · · ·
1 +
1
2 βg I μ N B 0 + · · ·
+
1 −
1
2 βg I μ N B 0 + · · ·
1
2 βg I μ N B 0 .
As a consequence, the average magnetic moment behaves at high temperatures as
μ z high T
1
2 g I μ N
2
B 0
k B T
,
(8.5.7a)
i.e., it varies inversely with T . This is as we might anticipate intuitively in that,
the higher the temperature the more randomly oriented the magnetic moment will
become, and thus the smaller its average will be.
For temperatures sufficiently low that βμ N B 0 1, we have e
1
2 βg I μ N B 0
e
−
1
2 βg I μ N B 0 , with a corresponding average magnetic moment given by
μ z low T
1
2 g I μ N
e
1
2 βg I μ N B 0
e
1
2 βg I μ N B 0
=
1
2 g I μ N ,
(8.5.7b)
and is constant, i.e., independent of T . Thus, if T is low enough, the value of μ z
is simply μ z =
1
2 g I μ N , since the spin particle remains in the ground state E + , for
which the energy is minimal.
Before proceeding further, it will be useful to pause and consider more carefully
the connection between the statistical mechanical description of our spin system
in terms of a canonical ensemble of a macroscopically large number N of spinparticles and the thermodynamics of a magnetizable system as described in the
previous section. We saw in Chap. 2 that two pairs of conjugate variables, namely
{T , S} and {P , V }, enter into the description of a traditional thermodynamic system,
and that any pair of these variables, one chosen from each set, may be employed
to describe the system. When we established the canonical ensemble for a fixed
number N of particles in Chap. 4, we saw that the relevant variables characterizing
it were the volume V and temperature T and that the corresponding thermodynamic
potential for the system energy treated as a function of T and V is the Helmholtz
energy, A. It is thus appropriate to take some care to ensure that we make the proper
assignment of the characteristic function for a canonical ensemble consisting of
N spins so that no contradiction arises between the thermodynamic and statistical
mechanical descriptions of a macroscopic spin system. To accomplish this goal, we
begin by noting the introduction of the magnetic internal energy U mag (S, V , B 0 )
437
For high temperatures, by which we mean temperatures such that βg I μ N B 0 1,
we may expand the exponentials and retain only the first two terms for each of them,
to give
tanh
1
2 βg I μ N B 0
1 +
1
2 βg I μ N B 0 + · · ·
−
1 −
1
2 βg I μ N B 0 + · · ·
1 +
1
2 βg I μ N B 0 + · · ·
+
1 −
1
2 βg I μ N B 0 + · · ·
1
2 βg I μ N B 0 .
As a consequence, the average magnetic moment behaves at high temperatures as
μ z high T
1
2 g I μ N
2
B 0
k B T
,
(8.5.7a)
i.e., it varies inversely with T . This is as we might anticipate intuitively in that,
the higher the temperature the more randomly oriented the magnetic moment will
become, and thus the smaller its average will be.
For temperatures sufficiently low that βμ N B 0 1, we have e
1
2 βg I μ N B 0
e
−
1
2 βg I μ N B 0 , with a corresponding average magnetic moment given by
μ z low T
1
2 g I μ N
e
1
2 βg I μ N B 0
e
1
2 βg I μ N B 0
=
1
2 g I μ N ,
(8.5.7b)
and is constant, i.e., independent of T . Thus, if T is low enough, the value of μ z
is simply μ z =
1
2 g I μ N , since the spin particle remains in the ground state E + , for
which the energy is minimal.
Before proceeding further, it will be useful to pause and consider more carefully
the connection between the statistical mechanical description of our spin system
in terms of a canonical ensemble of a macroscopically large number N of spinparticles and the thermodynamics of a magnetizable system as described in the
previous section. We saw in Chap. 2 that two pairs of conjugate variables, namely
{T , S} and {P , V }, enter into the description of a traditional thermodynamic system,
and that any pair of these variables, one chosen from each set, may be employed
to describe the system. When we established the canonical ensemble for a fixed
number N of particles in Chap. 4, we saw that the relevant variables characterizing
it were the volume V and temperature T and that the corresponding thermodynamic
potential for the system energy treated as a function of T and V is the Helmholtz
energy, A. It is thus appropriate to take some care to ensure that we make the proper
assignment of the characteristic function for a canonical ensemble consisting of
N spins so that no contradiction arises between the thermodynamic and statistical
mechanical descriptions of a macroscopic spin system. To accomplish this goal, we
begin by noting the introduction of the magnetic internal energy U mag (S, V , B 0 )