8.6 Magnetism, II. Ferromagnetism
463
Note that the value μ eff = 4.6553 given in Table 8.3 for the ground level 2ndorder Zeeman contribution to the effective magnetic moment for Eu 3+ is actually
a considerable overestimate of μ eff , as it assumes a fractional population 1 for the
ground term, rather than only about 0.46, which is the relevant fractional population
for the Eu 3+ ground level at temperature T = 293 K. A more realistic estimate,
μ eff (Eu
3+ ) = 3.1524, for the 2nd-order Zeeman contribution is obtained upon
employing the full z so (T ) value of 2.1808 for Eu 3+ at T = 293 K.
8.6 Magnetism, II. Ferromagnetism
8.6.1 Setting the Stage
It has long been known in thermodynamics that changes in phase of a substance,
such as the gas-liquid or liquid-solid phase transitions that occur at specific
temperatures (commonly referred to as the boiling point and the freezing point) for
the substance and at which two or more phases co-exist typically give rise to analytic
discontinuities in some of the thermodynamic functions for the system, especially
in the neighbourhood of what is called the critical point. These discontinuities
may become quite distinct in the vicinity of a critical point. No such non-analytic
behaviour is seen for the ideal gas, whose constituent particles do not interact, and
whose success is in principle limited to those ranges of temperature and pressure in
which the actual interparticle interactions play only minor roles. Interparticle interactions play essential roles, however, in phase transitions, such as gas condensation,
melting of solids, or phase co-existence phenomena. A characteristic feature of the
ideal gas model is that energy levels of a macroscopic collection of noninteracting
particles (atoms, ions, molecules) can be expressed directly in terms of the energy
levels of the individual particles themselves. For macroscopic systems of interacting
particles, however, it happens that under certain circumstances, large numbers of the
constituent particles are able to interact with one another in a cooperative manner
that assumes a macroscopic significance at a particular temperature (referred to as
a critical temperature, T c , for the system) that corresponds to the onset of a change
of phase for the macroscopic substance. A proper treatment of such cooperative
phenomena can prove to be formidable, so that it is often necessary to invent simple,
mathematically tractable, models in order to obtain an understanding of them. We
shall be interested here specifically in attempting to reach some understanding of
the phenomenon of ferromagnetism and the ferromagnetic to paramagnetic phase
transition. For the interested reader, more complete and detailed expositions of these
interesting phenomena can be found in [17–21].
When we examined monatomic solids in Chap. 5 and the paramagnetism of solid
inorganic salts in previous sections in the present chapter, we treated the atoms/ions
as sets of noninteracting particles located at fixed lattice sites defining the solid
crystal. We have also examined the role of interatomic/intermolecular interactions
463
Note that the value μ eff = 4.6553 given in Table 8.3 for the ground level 2ndorder Zeeman contribution to the effective magnetic moment for Eu 3+ is actually
a considerable overestimate of μ eff , as it assumes a fractional population 1 for the
ground term, rather than only about 0.46, which is the relevant fractional population
for the Eu 3+ ground level at temperature T = 293 K. A more realistic estimate,
μ eff (Eu
3+ ) = 3.1524, for the 2nd-order Zeeman contribution is obtained upon
employing the full z so (T ) value of 2.1808 for Eu 3+ at T = 293 K.
8.6 Magnetism, II. Ferromagnetism
8.6.1 Setting the Stage
It has long been known in thermodynamics that changes in phase of a substance,
such as the gas-liquid or liquid-solid phase transitions that occur at specific
temperatures (commonly referred to as the boiling point and the freezing point) for
the substance and at which two or more phases co-exist typically give rise to analytic
discontinuities in some of the thermodynamic functions for the system, especially
in the neighbourhood of what is called the critical point. These discontinuities
may become quite distinct in the vicinity of a critical point. No such non-analytic
behaviour is seen for the ideal gas, whose constituent particles do not interact, and
whose success is in principle limited to those ranges of temperature and pressure in
which the actual interparticle interactions play only minor roles. Interparticle interactions play essential roles, however, in phase transitions, such as gas condensation,
melting of solids, or phase co-existence phenomena. A characteristic feature of the
ideal gas model is that energy levels of a macroscopic collection of noninteracting
particles (atoms, ions, molecules) can be expressed directly in terms of the energy
levels of the individual particles themselves. For macroscopic systems of interacting
particles, however, it happens that under certain circumstances, large numbers of the
constituent particles are able to interact with one another in a cooperative manner
that assumes a macroscopic significance at a particular temperature (referred to as
a critical temperature, T c , for the system) that corresponds to the onset of a change
of phase for the macroscopic substance. A proper treatment of such cooperative
phenomena can prove to be formidable, so that it is often necessary to invent simple,
mathematically tractable, models in order to obtain an understanding of them. We
shall be interested here specifically in attempting to reach some understanding of
the phenomenon of ferromagnetism and the ferromagnetic to paramagnetic phase
transition. For the interested reader, more complete and detailed expositions of these
interesting phenomena can be found in [17–21].
When we examined monatomic solids in Chap. 5 and the paramagnetism of solid
inorganic salts in previous sections in the present chapter, we treated the atoms/ions
as sets of noninteracting particles located at fixed lattice sites defining the solid
crystal. We have also examined the role of interatomic/intermolecular interactions