Patterson, Bailey - Solid State Physics Introduction to theory

When we include a relaxation time (or an interaction process), we find that the time rate of change of the magnetization (along the field) is proportional to the deviation of the magnetization from its equilibrium value. This guarantees a relaxation of magnetization along the field. If we add an alternating magnetic field along the x- or y-axes, it is also necessary to add a term (M × H)z that is proportional to the torque. Thus for the component of magnetization along the constant external magnetic field, it is reasonable to write

As noted, (7.265) has a built-in relaxation process of Mz to M0, the spin-lattice relaxation time T1. However, as we approach equilibrium in a static magnetic field H0kˆ, we will want both Mx and My to tend to zero. For this purpose, a new term with a relaxation time T2 is often introduced. We write

Equations (7.265), (7.266), and (7.267) are called the Bloch equations. T2 is often called the spin-spin relaxation time. The idea is that the term involving T1 is caused by the interaction of the spin system with the lattice or phonons, while the term involving T2 is caused by something else. The physical origin of T2 is somewhat complicated. Consider, for example, two nuclei precessing in an external static magnetic field. The precession of one nucleus produces a varying magnetic field at the second nucleus and hence tends to “flip” the spin of the second nucleus (and vice versa). Waller27 first pointed out that there are two different types of spin relaxation processes.

Ferromagnetic Resonance (B)

Using a simple quantum picture, for an atomic system, we have already argued (see (7.258))

dt a

where Ba = μ0H. This implies a precession of μ and M about the constant magnetic field Ba with frequency ω = γBa the Larmor frequency, as already noted. For

27See Waller [7.66]. Discussion of ways to calculate T1 and T2 is contained in White [7.68 p124ff and 135ff].

Fig. 7.30. (a) Thin film with magnetic field, (b) “Unpinned” spin waves, (c) “Pinned” spin waves

which as expected is just the Larmor precessional frequency. In actual situations we also need to include demagnetization fields and hence shape effects, which will alter the resonant frequencies. FMR typically occurs at microwave frequencies. Antiferromagnetic resonance (AFMR) has also been studied as a way to determine anisotropy fields.

Spin-Wave Resonance (A)

Spin-wave resonance is a direct way to experimentally prove the existence of spin waves (as is inelastic neutron scattering – see Kittel [7.39 pp456-458]). Consider a thin film with a magnetic field B0 perpendicular to the film (Fig. 7.30a). In the simplest picture, we view the spin waves as “vibrations” in the spin between the surfaces of the film. Plotting the amplitude versus position, Fig. 7.30b is obtained for unpinned spins. Except for the uniform mode, these have no net interaction (absorption) with the electromagnetic field. The pinned case is a little different (Fig. 7.30c). Here only waves with an even number of half-wavelengths will show no net interaction energy with the field while the ones with an odd number of halfwavelengths (n = 1, 3, etc.) will absorb energy. (Otherwise the induced spin flippings will absorb and emit equal amounts of energy).

We get absorption when

Fig. 7.31. Spin wave resonance spectrum for Ni film, room temperature, 17 GHz. After Puszharski H, “Spin Wave Resonance”, Magnetism in Solids Some Current Topics, Scottish Universities Summer School in Physics, 1981, p. 287, by permission of SUSSP. Original data in Mitra DP and Whiting JSS, J Phys F: Metal Physics, 8, 2401 (1978)

Assuming ∫mx mx·dA etc = 0 for a large surface we can also recast the above as

which is the same as we obtained before, with a slightly different analysis. The αi are of course the direction cosines.

The anisotropy energy and effective field can be written in the same way as before, and no further comments need be made about it.

When one generalizes the equation for the time development of M, one has the Landau–Lifshitz equations. Damping causes broadening of the absorption lines. Then

rest of the terms are usually thought of as perturbations. In the discussion that follows, the hyperfine interaction will be neglected.

To avoid complicated many-body effects, we will assume that the sources of the crystal field (Ec ≡ − φc) are external to the paramagnetic ion. Thus in the vicinity of the paramagnetic ion, it can be assumed that 2φc = 0.

Weak, Medium, and Strong Crystal Fields (B)

In discussing the effect of the crystal field on the energy levels, which is important to EPR, three cases can be distinguished [47].

Weak crystal fields are by definition those for which the spin-orbit interaction is stronger than the crystal field interaction. This is often realized when the electrons of the paramagnetic shell of the ion lie “fairly deep” within the ion, and hence are shielded from the crystalline field by the outer electrons. This may happen in ionic compounds of the rare earths. Rare earths have atomic numbers (Z) from 58 to 71. Examples are Ce, Pr, and Ne, which have incomplete 4f shells.

By a medium crystal field we mean that the crystal field is stronger than the spin-orbit interaction. This happens when the paramagnetic electrons of the ion are mainly distributed over the outer portions of the ion and hence are not well shielded. In this situation something else may occur. The potential that the paramagnetic ions move in is no longer even approximately spherically symmetric, and hence the orbital angular momentum is not conserved. We say that the orbital angular momentum is (at least partially) “quenched” (this means ψ|L|ψ = 0, ψ|L2|ψ ≠ 0). Paramagnetic crystals that have iron group elements (Z = 21 to 29, e.g., Cr, Mn, and Fe that have an incomplete 3d shell) are typical examples of the medium-field case.

Strong crystal field by definition means covalent bonding. In this situation, the wave functions for the paramagnetic ion electrons overlap considerably with the wave functions of the other electrons of the crystal. Crystal field theory does not work here. This type of situation will not be discussed in this chapter.

As we will see, group theory can be an aid in understanding how energy levels are split by perturbations.

Miscellaneous Theorems and Facts (In Relation to Crystal Field Theory) (B)

The theorems below will not be proved. They are stated because they are useful in carrying out actual crystal field calculations.

The Equivalent Operator Theorem. This theorem is used in calculating needed matrix elements in crystal field calculations. The theorem states that within a manifold of states for which l is constant, there are simple relations between the matrix elements of the crystal-field potential and appropriate angular momentum operators. For constant l, the rule says to replace the x by Lx (operator, in this case Lx is the x operator equivalent) and so forth for other coordinates. If the result is a product in which the order of the factors is important, then we must use all possible different permutations. There is a similar rule for manifolds of constant J

frequently slightly different from what one might expect. Of course, the Jahn– Teller effect does not remove the fundamental Kramers’ degeneracy.

Hund’s rules. Assuming Russel–Sanders coupling, these rules tell us what the ground state of an atomic system is. Hund’s rules were originally obtained from spectroscopic evidence, but they have been confirmed by atomic calculations that include the Coulomb interactions between electrons. The rules state that in figuring out how electrons fill a shell in the ground state we should (1) assign a maximum S allowed by the Pauli principle, (2) assign maximum L allowed by S, (3) assign J = L − S when the shell is not half-full, and J = L + S when the shell is over half-full. See Problems 7.17 and 7.18. Results from the use of Hund’s rules are shown in Tables 7.9 and 7.10.

Energy-Level Splitting in Crystal Fields by Group Theory (A)

In this Section we introduce enough group theory to be able to discuss the relation between degeneracies (in the energies of atoms) and symmetries (of the environment of the atoms). The fundamental work in the field was done by H. A. Bethe (see, e.g., Von der Lage and Bethe [7.64]). For additional material see Knox and Gold [61, in particular see Table 1-2 pp. 5-8 for definitions].

We have already discussed some of the more elementary ideas related to groups in Chap. 1 (see Sect. 1.2.1). The most important new concept that we will introduce here is the concept of group representations. A group representation starts with a set of nonsingular square matrices. For each group element gi there is a matrix Ri such that gigj = gk implies that RiRj = Rk. Briefly stated, a representation of a group is a set of matrices with the same multiplication table as the original group.

Two representations (R′, R) of g that are related by

In (7.286) we say that the representation R(g) is reducible because it can be reduced to the direct sum of at least two representations. If R(g) is of the form (7.286), it is said to be in block diagonal form. If a matrix representation can be brought into block diagonal form by a similarity transformation, then the representation is reducible. If no matrix representation reduces the representation to block diagonal form, then the matrix representation is irreducible. In considering any representation that is reducible, the most interesting information is to find out what irreducible representations are contained in the given reducible representation. We should emphasize that when we say a given representation R(g) is re-