8.4 Magnetism, I. Paramagnetism
427
Although we shall investigate paramagnetism is some detail in this and the next
section, we shall only provide an introduction to ferromagnetism in Sect. 8.6 via a
discussion of the one-dimensional Ising-chain model plus a short summary of some
properties of the two-dimensional Ising model solution, as its analytical solution is
not mathematically tractable at a typical undergraduate level.
8.4.1 Thermodynamics of Magnetic Dipolar Media
The present treatment of the thermodynamics of a magnetic dipolar medium
assumes that the magnetizable material is uniformly distributed over the volume V
of the thermodynamic system and magnetized by an externally-maintained constant
magnetic field H 0 . The corresponding free-space magnetic induction is given by
B 0 = μ 0 H 0 , with μ 0 the permeability of free space defined (in SI units), as precisely
μ 0 ≡ 4π × 10 −7 H m −1 . The magnetizing field H 0 infuses the thermodynamic
system with a field energy, U field , given [4] by
U field =
1
2 V B · H 0 ,
(8.4.1)
with V the volume of the thermodynamic system and B the magnetic induction,
B = μ 0 (H 0 + M) ≡ μH 0 ,
(8.4.2)
that depends upon both the external field H 0 and the internal magnetic field
associated with the bulk magnetic moment M. However, because the differential
work done by current sources that are part of the surroundings in magnetizing a
volume V of a magnetizable substance is given (see Problem 8.7) by V H 0 · dB 0 ,
it is convenient to treat B 0 as the more fundamental quantity for dealing with the
thermodynamics of a magnetizable medium. A careful and thorough analysis of the
nature and role played by an external magnetic field has also been given by Callen
[5]. The appropriate form giving the incremental energy that is transferred reversibly
into a volume d V of a magnetizable substance upon quasistatically immersing
it into a magnetic induction field B 0 is thus d U field , which can be shown (see
Problem 8.9) to be given by
d U field = B 0 · dM.
(8.4.3)
8.4.2 Magnetic Susceptibility and Adiabatic Demagnetization
In general, the macroscopic magnetic moment of a thermodynamic system is given
in terms of the magnetizing field B 0 as
M = χ
(m) (T , ρ, B 0 ) · B 0 ,
(8.4.4a)
427
Although we shall investigate paramagnetism is some detail in this and the next
section, we shall only provide an introduction to ferromagnetism in Sect. 8.6 via a
discussion of the one-dimensional Ising-chain model plus a short summary of some
properties of the two-dimensional Ising model solution, as its analytical solution is
not mathematically tractable at a typical undergraduate level.
8.4.1 Thermodynamics of Magnetic Dipolar Media
The present treatment of the thermodynamics of a magnetic dipolar medium
assumes that the magnetizable material is uniformly distributed over the volume V
of the thermodynamic system and magnetized by an externally-maintained constant
magnetic field H 0 . The corresponding free-space magnetic induction is given by
B 0 = μ 0 H 0 , with μ 0 the permeability of free space defined (in SI units), as precisely
μ 0 ≡ 4π × 10 −7 H m −1 . The magnetizing field H 0 infuses the thermodynamic
system with a field energy, U field , given [4] by
U field =
1
2 V B · H 0 ,
(8.4.1)
with V the volume of the thermodynamic system and B the magnetic induction,
B = μ 0 (H 0 + M) ≡ μH 0 ,
(8.4.2)
that depends upon both the external field H 0 and the internal magnetic field
associated with the bulk magnetic moment M. However, because the differential
work done by current sources that are part of the surroundings in magnetizing a
volume V of a magnetizable substance is given (see Problem 8.7) by V H 0 · dB 0 ,
it is convenient to treat B 0 as the more fundamental quantity for dealing with the
thermodynamics of a magnetizable medium. A careful and thorough analysis of the
nature and role played by an external magnetic field has also been given by Callen
[5]. The appropriate form giving the incremental energy that is transferred reversibly
into a volume d V of a magnetizable substance upon quasistatically immersing
it into a magnetic induction field B 0 is thus d U field , which can be shown (see
Problem 8.9) to be given by
d U field = B 0 · dM.
(8.4.3)
8.4.2 Magnetic Susceptibility and Adiabatic Demagnetization
In general, the macroscopic magnetic moment of a thermodynamic system is given
in terms of the magnetizing field B 0 as
M = χ
(m) (T , ρ, B 0 ) · B 0 ,
(8.4.4a)