The Crystal Field Interaction

An array of charges in a crystal produces an electric field at any one ion, the so-called crystalline electric field. The presence of this field causes a Stark splitting of the free ion energy levels which results in a substantial modification of magnetic, electrical and thermal properties of the material. The theory of the crystal field and its interpretation in terms of group theory are originally due to Bethe (16). 

If the origin of the coordinate system is taken at the nucleus of the rare earth ion, an expression for the electrostatic potential at a point (r, 6, i/ ) near the origin due to the surrounding k ions may be written as 

This perturbing potential partially lifts the (2 J + 1) degeneracy of the ground multiplet of the free rare earth ion. The resulting eigenvalues and eigenfunctions of the rare earth ions may be evaluated by the straightforward but tedious methods of perturbation theory (17). This is seldom done because of the convenience of a cal-culational method introduced by Stevens about 25 years ago (vide infra). 

For ions located at points of very high symmetry the coefficients with the same n may be related to each other. 

The B coefficients are determined in part by the surrounding ions and in part by the radial extension and the total angular momentum of the rare earth ion. B is evaluated from the expression 

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The electrostatic and spin-orbit parameters for Pu + which we have deduced are similar to those proposed by Conway some years ago (32). However, inclusion of the crystal-field interaction in the computation of the energy level structure, which was not done earlier, significantly modifies previous predictions. As an approximation, we have chosen to use the crystal-field parameters derived for CS2UCI6 (33), Table VII, which together with the free-ion parameters lead to the prediction of a distinct group of levels near 1100 cm-. Of course a weaker field would lead to crystal-field levels intermediate between 0 and 1000 cm-1. Similar model calculations have been indicated in Fig. 8 for Nplt+, Pu1 "1 and Amlt+ compared to the solution spectra of the ions. For Am t+ the reference is Am4" in 15 M NHhF solution (34). 

The crystal field interaction gives rise to an energy splitting into a number of Kramers doublets. In the case of high-spin Fe " with spin S = 5/2, there are three Kramers doublets, each of which give rise to separate contributions in the Mossbauer spectra of samples with slow paramagnetic relaxation. For 1 = 0 and a = 0, they can be labeled l/2), 3/2) and 5/2). 

When the Zeeman energy is large compared to the crystal field interaction, the electronic wave functions are approximately the 5z) states with energy... 

In amorphous frozen solutions with only one type of species (e.g. [Fe(H20)g] ) the crystal field interaction of the Fe " ions may be similar, but the orientations of the crystal field axes in general differ. When magnetic fields are applied, this... 

The electron spin resonance (ESR) technique has been extensively used to study paramagnetic species that exist on various solid surfaces. These species may be supported metal ions, surface defects, or adsorbed molecules, ions, etc. Of course, each surface entity must have one or more unpaired electrons. In addition, other factors such as spin-spin interactions, the crystal field interaction, and the relaxation time will have a significant effect upon the spectrum. The extent of information obtainable from ESR data varies from a simple confirmation that an unknown paramagnetic species is present to a detailed description of the bonding and orientation of the surface complex. Of particular importance to the catalytic chemist... 

The crystal field interacts directly only with the orbital motion of the unpaired electrons and it has an effect on the electronic spins only through the spin orbit coupling. The strongest spin-lattice interaction will therefore occur for ions with ground states having an appreciable orbital character. 

If the atom or ion is situated in an environment of different atoms or ions, as, for instance, ions in a soUd, the surrounding hgands exert on it a further interaction, which is called the crystal field interaction Hcp and enters Eq. (9). One has to compare Hcf with Hi and H2, in order to decide whether it is or not a small perturbative term. In actinide solids, it is usually found that Hcf is of the same order of magnitude as Hi and H2, so that intermediate coupling schemes are necessary which include Hcf as well. (For a more exhaustive treatment of couplings in actinides, see Chap. D.)... 

The inclusion of the crystal field destroys the rotational symmetry of the ion and lifts the degeneracy of J levels (except of course Kramer s degeneracy) the only good quantum numbers will be T s, the irreducible representations of the point-group symmetry operation. If the crystal field interaction is comparable to J-J splitting (and we see from Table 2 that this is the case of actinides) it will also cause an admixture of different J multiplets. 

Fig. 9. Plot of the crystal-field interaction strength quantities ScF = g- +2X)m>0 l ml2]) f°r...

The crystal field interaction can be treated approximately as a point charge perturbation on the free-ion energy states, which have eigenfunctions constructed with the spherical harmonic functions, therefore, the effective operators of crystal field interaction may be defined with the tensor operators of the spherical harmonics Ck). Following Wyboume s formalism (Wyboume, 1965), the crystal field potential may be defined by ... 

The 5fn electrons of the actinides represent an intermediate case where there is still shielding of the crystal fields but it is not as effective as in the lanthanides. The crystal field interactions are larger than the lanthanides but not as large as in the transition metals. The lines of most transitions are sharp and all the actinide ions could be used potentially as probes of the local environments of minerals. 

For the weak field case, we have the situation where the crystal field interaction is much weaker than the electronic repulsion. In this approximation, the Russell-Saunders terms 3F, 3P, 1G, lD, and 5 for the d2 configuration are good basis functions. When the crystal field is turned on, these terms split according to the results given in Table 8.4.2 ... 

Fig. 8.3. Energy levels of the f1 configuration as a function of the relative strengths of the spin-orbit and crystal field interactions. For chi = 0, only the spin-orbit interaction is considered for chi = 1, only the crystal field interaction is considered. The energy levels are numbered by 2 x L.

Dy +, Ho +, and others in the f-group series. Depending on the details of the crystal field interaction, these ions can be either Ising or XY. 

The secular determinantal equation is set up in the usual manner, the wavefunctions corrected for the crystal-field interaction are used in the perturbation treatment, energies are generated, and these are used in conjunction with the secular equations to generate new wavefunctions that have now been corrected for spin-orbit coupling. These corrected wavefunctions are used for the calculation of the Zeeman effect. 

In the next sections we describe briefly the main interactions, which are in charge of splitting of the 3d ions energy levels in crystals. These interactions include the Coulomb interaction, the crystal field interaction, the spin-orbit interaction and the JT interaction. As it was pointed out by Ham [13], the observed spin-orbit and trigonal field splittings of the orbital triplet states are significantly affected by the dynamic JT effect. 

In calculations involving higher J multiplets, matrix elements of the crystal field Hamiltonian between states belonging to different J multiplets are needed. Although these can be calculated by the method of operator equivalents extended to elements non-diagonal in J, it is convenient to use a more general approach, utilizing Racah s tensor operator technique (26). In this method the crystal field interaction may be written as... 

The crystal field interaction has pronounced effects on the heat capacity behavior of the system. At very low temperatures, ions occupy the lowest crystal field states. With increasing temperature, excitation within the crystal field spectrum takes place resulting in a significant contribution to the heat capacity (38). This contribution is given by the expression... 

Table 7. Characteristics of RNi2 indicating the influence of the Crystal Field Interaction (86,104).

A study of the crystal field interaction in NdAlg by heat capacity, susceptibility, and resistivity measurements over the range 4—300K has been reported. Thermal transformations in thiourea compounds of neodymium, samarium, europium, and gadolinium have been shown to take place according to the scheme M(C3H302)3, CS(NH2)2,3H20 M(C3M,02) CSfNH ) NH CNS MO-... 

Here p is the density of phonons, n(co) is the Planck number corresponding to the thermal distribution of phonon excitations and k(co) is the spin-phonon coupling constant written as a function of frequency (instead of the wave-vector star k and the phonon branch index, as previously). Only those phonons with energy equal to the Zeeman energy h coa are of interest in a direct relaxation process. This energy is characteristically 0.1 cm-1 and the relevant phonons are of the long-wave acoustic type. Their role is to modulate the crystal field interacting with the electron. 

The temperature dependence of p in the Cp Ln-R complexes appears to arise from the difference of site symmetry and the strength- of the crystal field interactions. 

In tensor operator notation the crystal-field interaction is written... 

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