432
8 Electric and Magnetic Phenomena
example of an ideal gas, in which there is a mechanism, namely collisions, for
redistributing the energy (heat) transferred to the ideal gas from the surroundings
or the work done on the system by the surroundings into other dynamical modes
(such as the translational motion or internal states) of the constituent particles
of the thermodynamic system, no mechanism exists for transferring the stored
(magnetic) energy into other dynamical modes of the thermodynamic system. That
the sole outcome arising from placing a magnetizable thermodynamic system into a
magnetic field is a reduction in entropy due to the creation of preferential alignment
of spins (or other angular momenta) along the field direction (thereby increasing the
order of the system) is a potential source of confusion about the proper analogy to
be drawn between the intensive-extensive B 0 -M pair of thermodynamic variables
and the P -V or T -S pairs employed to generate the enthalpy or Helmholtz energy.
The most common analogy made between traditional thermodynamic systems and
magnetizable thermodynamic systems is that between pressure–volume work, and
the field-moment ‘work’, B 0 dM, in which B 0 plays the role of pressure, P , and −M
plays the role of volume, V [4–7]. Consistent with this manner of looking upon the
magnetizable system in terms of the internal energy U mag as a function of S, V , M
is the introduction of a magnetic enthalpy, H mag (S, P , B 0 ) ≡ U(S, V , M) + P V −
B 0 · M, a magnetic Helmholtz energy A mag (T , V , M) ≡ U(S, V , M) − T S, and a
magnetic Gibbs energy G mag (T , P , B 0 ) ≡ U(S, V , M) − T S + P V − B 0 · M. Of
course, the internal energy U remains exclusively a function of the three extensive
thermodynamic variables S, V , M, while the Gibbs energy remains exclusively a
function of the three intensive thermodynamic variables T , P , and B 0 .
The analogy between the B–M pair of complementary variables and the corresponding T -S pair would leave the enthalpy unchanged as H
mag (S, P , M) ≡
U(S, V , M) + P V , would introduce a new Helmholtz energy A
mag (T , V , B 0 ) ≡
U(S, V , M) − T S − B 0 · M, and would give the same Gibbs energy G mag (T , P , B 0 )
as that obtained using the P -V analogy. However, as neither of these two interpretations leads to full compatibility between the thermodynamic and the statistical
mechanical description of a magnetic system given in the following subsection, we
shall follow a description provided in §5.8 of [2].
8.4.3 Thermodynamic Potentials for Magnetizable Systems
Because it is experimentally very difficult to control the magnetization M (equivalently, the magnetic moment M) of a macroscopic system, while it is relatively easy
to control an external magnetic field H 0 (equivalently, a magnetic induction field
B 0 ≡ μ 0 H 0 ), it is useful to work with the Legendre-transformed internal energy
U mag , defined as
U mag (S, V , B 0 ) ≡ U(S, V , M) − B 0 · M,
(8.4.19a)
8 Electric and Magnetic Phenomena
example of an ideal gas, in which there is a mechanism, namely collisions, for
redistributing the energy (heat) transferred to the ideal gas from the surroundings
or the work done on the system by the surroundings into other dynamical modes
(such as the translational motion or internal states) of the constituent particles
of the thermodynamic system, no mechanism exists for transferring the stored
(magnetic) energy into other dynamical modes of the thermodynamic system. That
the sole outcome arising from placing a magnetizable thermodynamic system into a
magnetic field is a reduction in entropy due to the creation of preferential alignment
of spins (or other angular momenta) along the field direction (thereby increasing the
order of the system) is a potential source of confusion about the proper analogy to
be drawn between the intensive-extensive B 0 -M pair of thermodynamic variables
and the P -V or T -S pairs employed to generate the enthalpy or Helmholtz energy.
The most common analogy made between traditional thermodynamic systems and
magnetizable thermodynamic systems is that between pressure–volume work, and
the field-moment ‘work’, B 0 dM, in which B 0 plays the role of pressure, P , and −M
plays the role of volume, V [4–7]. Consistent with this manner of looking upon the
magnetizable system in terms of the internal energy U mag as a function of S, V , M
is the introduction of a magnetic enthalpy, H mag (S, P , B 0 ) ≡ U(S, V , M) + P V −
B 0 · M, a magnetic Helmholtz energy A mag (T , V , M) ≡ U(S, V , M) − T S, and a
magnetic Gibbs energy G mag (T , P , B 0 ) ≡ U(S, V , M) − T S + P V − B 0 · M. Of
course, the internal energy U remains exclusively a function of the three extensive
thermodynamic variables S, V , M, while the Gibbs energy remains exclusively a
function of the three intensive thermodynamic variables T , P , and B 0 .
The analogy between the B–M pair of complementary variables and the corresponding T -S pair would leave the enthalpy unchanged as H
mag (S, P , M) ≡
U(S, V , M) + P V , would introduce a new Helmholtz energy A
mag (T , V , B 0 ) ≡
U(S, V , M) − T S − B 0 · M, and would give the same Gibbs energy G mag (T , P , B 0 )
as that obtained using the P -V analogy. However, as neither of these two interpretations leads to full compatibility between the thermodynamic and the statistical
mechanical description of a magnetic system given in the following subsection, we
shall follow a description provided in §5.8 of [2].
8.4.3 Thermodynamic Potentials for Magnetizable Systems
Because it is experimentally very difficult to control the magnetization M (equivalently, the magnetic moment M) of a macroscopic system, while it is relatively easy
to control an external magnetic field H 0 (equivalently, a magnetic induction field
B 0 ≡ μ 0 H 0 ), it is useful to work with the Legendre-transformed internal energy
U mag , defined as
U mag (S, V , B 0 ) ≡ U(S, V , M) − B 0 · M,
(8.4.19a)