Some properties of magnetic ions and their interaction

Introduction
The techniques of electron paramagnetic resonance (E.P.R.) have been used to investigate the interaction of rare earth ions with (a) their crystalline environment, by measuring their effective g tensor, and, (b) each other, by observing their pair spectra. The experiments have been carried out in host lattices which, when magnetically concentrated, show interesting ordering properties.
The Apparatus
Measurements have been made at both X and K Band frequencies. The X Band spectrometer, originally constructed by Drs. A. H. Cooke and J. G. Park, was modified by the addition of a 115 Ke/s, and 50 c/s, phase sensitive detection system to give greater sensitivity. The K Band spectrometer was designed with a tuneable cylindrical H011 transmission cavity, and incorporated an improved system of cryogenics.
E.P.R. of rare earth ions in diamagnetic garnets
The investigation of the properties or rare earth ions la diamagnetic garnets is of importance in connection with their profound influence on the ferrimagnetism of the iron garnets (for example see Dillon et al. 1960). The results of Ryan (1960) on the E.P.R. of rare earth lone in yttrium aluminium garnet (YAG) and yttrium gallium garnet (YGG) have shown a considerable difference of their g tensor in the two lattices, particularly for the case of Kr3+. In order to investigate further this variation, in particular the dependence on the host ion in the rare earth site, measurements have been made of the g tensors in the lutetium garnets LAG and LGG. The results, obtained mainly at X Band and 4.2°K, are given in Table 1 together with that for Dy3+ in YAG where no resonance had previously been observed. They indicate that the variation between lutetium and yttrium lattices it not so great as that between the aluminium and gallium lattices.
Table 1. Principal g values of four rare earth ions in diamagnetic garnet lattices
Ion (1%) Lattice gx gy gz
Nd3+
LGG 2.083 (±0.007) 1.323 (±0.007) 3.550 (±0.007) LAG 1.789 (±0.005) 1.237 (±0.006) 3.834 (±0.005) Dy3+ LGG 13.45 (±0.10) 0.57 (±0.010) 3.41 (±0.03) LAG 2.29 (±0.03) 0.91 (±0.05) 16.6 (±0.3) YAG 0.73 (±0.15) 0.40 (±0.20) 18.2 (±0.4) Er3+ LGG 3.183 (±0.013) 3.183 (±0.013) 12.62 (±0.10) LAG 6.93 (±0.02) 4.12 (±0.03) 8.43 (±0.04) Yb3+ LGG 3.653 (±0.013) 3.559 (±0.013) 2.994 (±0.011) LAG 3.842 (±0.006) 3.738 (±0.007) 2.594 (±0.004)
Direct use of the g values
In general the above dependence of the g tenor on the exact garnet environment precludes their direct use for the explanation of the magnetically concentrated compounds. However, in two cases they have been used successfully.
Dysprosium aluminium garnet orders antiferromagnetically at 2.5°K (Ball et al. 1962). The highly anisotropic g values found for Dy3+ in YAG suggest that the ions in DyAG, which has a similar lattice constant, may be treated to a first approximation a Ising spins with gz = 18.0, gx = gy = 0. Assuming only dipolar interaction a possible ordered state for such ions has been proposed, and the model allows an explanation of the main features of the susceptibility curve to be made.
As the g values of Yb3+ show relatively little variation with lattice, it la possible to use those found in the gallium garnets with some confidence for a discussion of YbIG. From the theory of Wolf (1962), canting of the rare earth sublattices at 20° to the Fe3+ sublattice has been predicted, and the order of magnitude of the spontaneous moment at 0°K accounted for.
Discussion of the garnet g values, cubic approximation
In order to make theoretical predictions of the g values other magnetically concentrated gamete a thorough understanding of the resonance results In terms of the local crystal field is required.
As a first approximation this crystal field has been considered to arise mainly from the nearest eight oxygen neighbours, which are arranged in the form of a distorted cube about the rare earth site. Using the general cubic field calculations of Lea et al. (1962) it has been found that the gross features of the g values can be accounted for on such a model. But only in the case of Yb3+ can the values be quantitatively explained to within a few percent.
The trends in the observed g values for Nd3+ and Yb3+ have been found to follow closely those in the structure parameters describing the positions of the oxygen neighbours. From the Kr3+ results an estimate has been made of the ratio of the parameters describing the crystal field, A04/A06, and this has been compared with other estimates. The extremely anisotropic g values found for Dy3+ in YAG suggest a ground state wavefunction of nearly pure |JZ = 15/2> . (Where Z is the [001] crystal direction).
Point charge calculations of the garnet crystal field
A general account of the method of derivation of the crystal field parameters from a point charge model of the crystal lattice has been written. In conjunction with D. K. Ray, the validity of an ionic model of the crystal lattice has been tested by considering the case of PrCl3. This indicated the inadequacies of the model, even when accents on neighbouring lane were considered, but suggested that calculated ratios of parameters may be a good approximation to the true values.
Attaining a model of point charges only, a computer has been used to calculate the contributions from (a) nearest neighbours only, and (b) all the ions in the lattice, to the crystal field parameters Amn at the rare earth site in YAG, YGG, LAG, LGG and YIG. The results of these calculations have been discussed. They indicate quite large distortions from a cubic field, and that the parameters for YGG are likely to lie closest to those in YIG.
A possible set of crystal field parameters for Yb3+ in YGG
The case of Yb3+ in the garnets is best suited to a more detailed theoretical analysis. As there are insufficient experimental data available to enable all the parameters to be found from experiment alone, ratios calculated on the point charge model have been used to reduce the number of parameters to be fitted. Treating first the J = 5/2 multiplet by itself, and then the whole 2F term, a set of parameters which give a fair agreement with experiment have been found using a computer programme. The discrepancies, which should be small, indicate the inadequacies of the approximations made. The wavefunction found for the lowest J = 5/2 doublet is similar to that found from perturbation theory, and in general the fit to the data for the J = 5/2 state alone is better than that to the whole 2F term.
E.P.R. of rare earth ions in weakly paramagnetic garnets. Interaction shift
It has been shown that the resonance from Kramers ions in a lattice containing host ions in a singlet ground state should be shifted but not broadened by th&e; interactions. This shift may be expressed in the same form as the magnetic g tensor. The effective principal g values of Yb3+ and Er3+ have been measured at X Band frequencies and 4.2°K in such lattices, end the results are given in Table 2. The case of Yb3+ in TmAG is believed to allow unambiguously such an interaction shift.
Table 2. Effective principal g values of Er3+ and Yb3+ in weakly paramagnetic garnets
Ion (~0.005%) Lattice gx gy gz
Er3+
TmGG 3.76 (±0.02) 3.48 (±0.02) 11.9 (±0.1) TmAG 7.89 (±0.03) 4.25 (±0.02) 7.65 (±0.03) Yb3+ EuGG 3.805 (±0.003) 3.664 (±0.003) 2.670 (±0.007) TmGG 3.724 (±0.008) 3.557 (±0.007) 2.944 (±0.010) TmAG 4.020 (±0.007) 3.763 (±0.008) 2.554 (±0.009)
E.P.R. of Gd3+ in hexagonal lattices
Measurements have been made at X Band frequencies of the crystal field parameters bmn for Gd3+ in LaCl3 and LaBr3.
Table 3
  b02 × 104 cm-1 b04 × 104 cm-1 b06 × 104 cm-1 |b66| × 104 cm-1 g
LaCl3 (20°K) 18.0 (±0.2) 2.4 (±0.05) 0.23 (±0.05) 0.13 (±0.07) 1.992 (±0.001)
LaBr3 (20°K) 67.6 (±0.3) 11.11 (±0.06) 3.46 (±0.06) - 1.991 (±0.001)
The results tabulated above emphasize their dependence on the lattice spacing in this structure found by Hutchison et al. (1957).
E.P.R. from pairs of Gd3+ ions in LaCl3
In order to obtain information on the interactions of a pair of Gd3+ ions in LaCl3 which is of interest in connection with the properties of the isostructural ionic ferromagnet GdCl3, their pair spectrum has been investigated at X and K Band frequencies and at 20°K. In such a case of weak isotropic exchange interaction it is possible to obtain measurements of the exchange constants directly from the resonance line position.
The nearest neighbour (n.n.) pair spectrum, with the magnetic field H parallel to the c axis, is symmetric about the central main line and extends over ~ ± 2,600 gauss. It consists of four outer groups of lines, of overall width ~130 gauss and separation ~370 gauss, and three apparently well resolved inner lines. The spectrum due to next nearest neighbour (n.n.n.} pairs, with H along their symmetry axis, shows no such grouping. There are very many lines within 600 gauss of the central main line, due to more distant neighbours.
The theoretical analysis of the spectrum has been undertaken from the start using a computer. It has not been possible to identify any of the pair transitions with certainty, and because of the large variation of line position with the values of all five parameters in the pair Hamiltonian the interpretation has proved difficult. However if the crystal field parameters for a member of the n.n. pair are assumed not to differ considerably from those of the isolated ions, the grouping of the outer n.n. lines can be explained. It indicates a value of Jox = 0.011 (±0.005) cm-1 and α = g2μ2B/r3nn = 0.022 (±0.002) cm-1 (where ℋox = Jox s(1)⋅s(2)).
E.P.R. of rare earth atoms in YCl3 and LuCl3
The principal g values of rare earth lone in YCl3 and LuCl3 have been measured at X Band frequencies and 4.2°K. The results are listed in Table 4. The cases of Py3+, Er3+, and Yb3+ show little relative variation with host lattice, and it should be possible to use the values directly for the interpretation of the bulk magnetic properties of the isostructural concentrated compounds. It has been shown that as a first approximation, the crystal field at the rare earth site may be considered to be sixfold cubic, as expected from the location of th nearest Cl- ions.
Table 4. Principal g values of rare earth ions in monochromic YCl3 and LuCl3; gb is measured along the b [010] axis, and g1 and g2 lie in the (010) plane, g1 is at an angle α to the a [100] axis.
Ion (1%) Lattice g1 g2 gb α
Dy3+ YCl3 6.86 (±0.03) 3.02 (±0.06) 9.34 (±0.03) ±9° (±5°)
LuCl3
7.21 (±0.02) 3.16 (±0.05) 8.84 (±0.02) ±7° (±6°) Er3+ YCl3 8.27 (±0.05) 3.78 (±0.05) 7.70 (±0.03) 0 (±8°) LuCl3 8.34 (±0.02) 3.51 (±0.03) 7.92 (±0.02) 0 (±5°) Yb3+ YCl3 2.99 (±0.02) 1.53 (±0.02) 2.957 (±0.006) 0 (±9°) LuCl3 2.95 (±0.01) 1.58 (±0.01) 2.96 (±0.01) 0 (±5°) Ce3+ YCl3 1.60 (±0.01) 0.66 (±0.02) 1.708 (±0.003) 0 (±10°) Nd3+ YCl3 1.16 (±0.03) 2.63 (±0.02) 1.653 (±0.005) 0 (±8°)
Miscellaneous E.P.R. experiments
Experiments which have given inconclusive or ambiguous results include measurements on Ca3+ pairs in LaCl3, and U4+ in ThO2.