464
8 Electric and Magnetic Phenomena
briefly in Chap. 7 to reach some understanding of corrections to the thermodynamic
state functions when binary interactions between the atoms or molecules can no
longer be neglected. In Sect. 7.4, we also examined briefly the concept of the critical
point, particularly the concept of a critical temperature and phase co-existence
within a Van der Waals model fluid. In this section, we shall examine the role played
by nearest-neighbour interactions amongst the particles occupying lattice sites of a
crystal.
The best-known examples of ferromagnetic materials involve atoms or ions of
the ferrous metals iron, cobalt, and nickel. The relevant atoms, Fe, Co, Ni, are all
transition metals with electron configurations [Ar](4s) 2 (3d) n electronic configurations, with n = 6, 7, 8, and ground electronic terms 5 D, 4 F , and 3 F , respectively.
They thus possess both spin and orbital electronic angular momenta, coupled to give
total electronic angular momenta J = L + S, L + S − 1, · · · , |L − S|, in which
L designates the total electronic orbital angular momentum quantum number and
S the total electronic spin angular momentum quantum number. Nonetheless, it has
been experimentally well established that ferromagnetism appears to be associated
only with the electronic spins.
We have seen that a collection of N noninteracting spins gives rise to a
macroscopic magnetization M in the presence of a constant external magnetic
field. From Eq. (8.5.8b) for M z , for example, it is clear that for B = 0, the
magnetization will vanish. It is then reasonable to ask what happens when we
consider a collection of N interacting spins. This question turns out to be difficult
to answer in general, as the mathematics becomes virtually intractable for the
most relevant case, namely that of a three-dimensional solid. Although the twodimensional case with nearest- and next-nearest-neighbour remains intractable,
that with nearest-neighbour plus magnetic field interactions becomes numerically
accessible, while the two-dimensional case with nearest-neighbour interactions in
zero magnetic field is analytically solvable [22], though essentially beyond the level
of the present treatment. 9
A solution of the simplest model, namely that of a one-dimensional chain of
magnetic dipole moments, each of which may be either aligned in the direction
established by an applied magnetic field of strength B or aligned opposite to that
direction, and interacts with its nearest-neighbours, thereby establishing a linking
of the chain elements, can be attained relatively straightforwardly. This rather
important, though simple, model is known as the one-dimensional Ising model.
A history of the Ising model has been given by Brush [23], and a more detailed
treatment of the Ising model, its generalizations, and its many applications has
been given by Pathria [24]. In its simplest version, it may be represented physically
through expressing the Hamiltonian H N of the collection of N ‘spins’ as
9 An exact analytical solution to the two-dimensional nearest-neighbour interaction in the absence
of an external magnetic field was first given in 1944 by Onsager [22].
8 Electric and Magnetic Phenomena
briefly in Chap. 7 to reach some understanding of corrections to the thermodynamic
state functions when binary interactions between the atoms or molecules can no
longer be neglected. In Sect. 7.4, we also examined briefly the concept of the critical
point, particularly the concept of a critical temperature and phase co-existence
within a Van der Waals model fluid. In this section, we shall examine the role played
by nearest-neighbour interactions amongst the particles occupying lattice sites of a
crystal.
The best-known examples of ferromagnetic materials involve atoms or ions of
the ferrous metals iron, cobalt, and nickel. The relevant atoms, Fe, Co, Ni, are all
transition metals with electron configurations [Ar](4s) 2 (3d) n electronic configurations, with n = 6, 7, 8, and ground electronic terms 5 D, 4 F , and 3 F , respectively.
They thus possess both spin and orbital electronic angular momenta, coupled to give
total electronic angular momenta J = L + S, L + S − 1, · · · , |L − S|, in which
L designates the total electronic orbital angular momentum quantum number and
S the total electronic spin angular momentum quantum number. Nonetheless, it has
been experimentally well established that ferromagnetism appears to be associated
only with the electronic spins.
We have seen that a collection of N noninteracting spins gives rise to a
macroscopic magnetization M in the presence of a constant external magnetic
field. From Eq. (8.5.8b) for M z , for example, it is clear that for B = 0, the
magnetization will vanish. It is then reasonable to ask what happens when we
consider a collection of N interacting spins. This question turns out to be difficult
to answer in general, as the mathematics becomes virtually intractable for the
most relevant case, namely that of a three-dimensional solid. Although the twodimensional case with nearest- and next-nearest-neighbour remains intractable,
that with nearest-neighbour plus magnetic field interactions becomes numerically
accessible, while the two-dimensional case with nearest-neighbour interactions in
zero magnetic field is analytically solvable [22], though essentially beyond the level
of the present treatment. 9
A solution of the simplest model, namely that of a one-dimensional chain of
magnetic dipole moments, each of which may be either aligned in the direction
established by an applied magnetic field of strength B or aligned opposite to that
direction, and interacts with its nearest-neighbours, thereby establishing a linking
of the chain elements, can be attained relatively straightforwardly. This rather
important, though simple, model is known as the one-dimensional Ising model.
A history of the Ising model has been given by Brush [23], and a more detailed
treatment of the Ising model, its generalizations, and its many applications has
been given by Pathria [24]. In its simplest version, it may be represented physically
through expressing the Hamiltonian H N of the collection of N ‘spins’ as
9 An exact analytical solution to the two-dimensional nearest-neighbour interaction in the absence
of an external magnetic field was first given in 1944 by Onsager [22].