Note on the crystal field parameters of rare earth garnets

J. Phys.

Chem. Solids

Pergamon

Printed in Great Britain.

Press 1966. Vol. 27, pp. 1047-1051.

NOTE ON THE CRYSTAL FIELD PARAMETERS OF RARE EARTH GARNETS J. H. VAN VLECK Lyman

Laboratory

of Physics, Harvard (Received

University,

20 January

Cambridge,

Massachusetts

1966)

Abstract-The results of various investigators on the crystal field parameters of garnets of the heavy rare earths are examined and compared, including discussion of their relation on the model which we used in the preceding paper on TmGaG. A number of inconsistencies and errors in the literature are noted. The empirical ratios of sixth to fourth order terms which have been reported are practically always larger, sometimes by about a factor 10, than the values obtained by a point charge calculation with Freeman-Watson radii for the 4f electrons. INTRODUCTION

IT WASoriginally

the preceding paper on the magnetic behavior of TmGaG and TmIG with a brief examination of how our choice of crystalline field parameters compared with the results obtained by other investigators for the erbium and ytterbium garnets adjacent in the periodic table. However, as we looked into this subject, so many strange contradictions, paradoxes, and in some instances errors, came to our attention that it seemed best to incorporate the discussion of this subject in a separate paper. We will focus our attention mainly on the cubic portion of the crystalline potential. It is well known that the local field does not have cubic symmetry in the garnets. However, the cubic approximation gives the right qualitative distinction between the behavior of the different ions. Because there are six ions in the unit cell so oriented that the overall macroscopic symmetry is cubic, the magnetic response of the rare earth lattice to the applied field or to that exerted by the iron lattice is like that of a powder (neglecting small saturation corrections). In general a powder susceptibility is peculiarly insensitive to the finer details or nuances of the crystalline field, especially when the lowest ground cubic state is a singlet or a Kramers doublet well separated from the other levels. It is for this reason that the susceptibility in particular of the Tm and Yb garnets (l?s and I, ground states respectively) can be pretty adequately described intended

to terminate

by a cubic potential, even though non-cubic members must be added to interpret the spectroscopic energy levels in detail. The most general cubic crystalline potential which can act on f electrons is* I&

=

v4p[04°-5044]+

v6’sy[06°+21064]

(1)

where the O’s are the well-known STEVENS@) operator equivalents as given by BAKERet al.,@) and p, y are ion-dependent numerical parameters defined and tabulated by Stevens. The coefficients V4, Vs are dimensionally of the order es(+)4~/Z@, e2(r6>4f/R’ and should vary from ion to ion only in virtue of the gradual changes in the orbital * Like Hutchings and Wolf, we choose the axis of quantization involved in (1) along the 110 axis (or equivalent axes for the other five ions in the unit cell). This we take as the local x axis. Then to be principal axes for the magnetic g tensor at the RE site, the y and z axes must be along 110 and 001. The orientation of the nearcube of eight oxygen ions surrounding the rare earth ion is such that then, whereas the x axis is a principal axis of this cube, the y and z axes are surface diagonals. This makes the coefficients -5,21 instead of 5, -21 as they would be if axes are taken along the principal axes of the environmental octahedron. (The change in sign is a consequence of the fact that the factor eg4bin Yn4 changes sign if 4 is advanced a/4.) The fact that the principal local magnetic axes do not coincide with either the macroscopic principal axes or the principal axes of the environmental cube has caused a considerable confusion in the literature.

1047

1048

J. H.

VAN

radius Y and lattice parameter R. The graphs of LEA et aZ.@) are expressed in terms of a potential

VLECK

5/1/2, 21/2/2 thereby making the symmetry tetragonal rather than cubic. Their value of Vs/V4 (corresponding to x = -0.4) is -0.14, and is in (I- 1x1) closer agreement with our choice -0.15 than one v,, =w -5[040-50441+-F@-[otio+210641) i F(4) has a right to expect. There are certain difficulties with the potential used by Dreyfus or Ayant. (2) They do not discuss the g values for the ground and plot the energy, measured in multiples of W level E = Eof +g/3H of ErGaG, but these are as a function of x. Essentially equivalent diagrams readily calculated from their wave functions. have also been published by EBINA and TsuYA.(~) They are gs = 5.4, g, = gZ = 7.2, where x, y, z The parameters x and W have a different signifiare the local principal axes directed respectively cance for different ions in terms of the basic along 110, 110, and 001, or equivalent, relative to constants V4, Vs, but the relations between x, W the macroscopic principal axes. On the other hand, and V4, Vs, follow immediately from comparison from magnetic resonanceWOLF et al.(%ndg, = 3.2, of (1) and (2) and use of Stevens’ tables of values g, = 3.2, gZ = 12.6 and g, = 4.7, gr, = 4.0, of the ion-dependent dimensionless numbers g, = 10.7 for Er doped LuGaG and YGaG F(4), F(6), /3,y. Our choice x = 0.6 for TmIG in respectively. Of course it is true that the resonance the previous paper is equivalent to taking measurements are on dilute rather than concentrated erbium gallium garnets, but since the Er ion v,3/v4 = -0.15. is intermediate in size between the Lu and Y ions, In contrast to x or W, the dimensionless ratio it is improbable that in the concentrated material Vs/V4, which measures the relative importance of the g tensor should be so different. A more the sixth and fourth order harmonics and the scale fundamental difficulty, pointed out to the writer factor V4 (or Vs) should not vary much from ion to by Prof. Werner Wolf, is that with this model the ion. Tm is bounded by Er and Yb in the Periodic axis of approximate tetragonal symmetry is 110, Table. It turns out that comparison of the thulium with the two perpendicular equal g values along garnets with the erbium ones is more informative Ii0 and 001. On the other hand, the magnetic concerning Va/V4 and with those of ytterbium resonance measurements not only on the Er doped concerning the overall scale of the crystal splitting. gallium garnets, but also on various Yb doped garnets, show that it is usually the 001 direction VALUES OF THE RATIO F's/V4 which has a g value differing markedly from the We will first compare the values of Vs/V4 other two. obtained by various investigators. An analysis of the spectrum of ErGaG has also The key to the approximate choice of Vs/V4 is been made by PAPPALARDOUUon the basis of a cubic potential, but his interpretation is questionthat the erbium garnets have many low-lying levels,@) with highly anisotropic and environmentable, for it has a positive value O-08 of the ratio sensitive g values.(‘) This must mean that there are Vs/V4, whereas with any reasonable point charge model of the surroundings of the rare earth ion, closely spaced Kramers doublets (including cubic l?s quartets split by the non-cubic portion explicit calculation of the crystalline potential gives a negative value of this ratio, as is also the of the field). Inspection of the instructive diagrams case for all the models used by other investigators. of WHITE@) shows that this requires that Vs/ V4 be (Pappalardo does not explicitly give Vs/V4, and roughly between about -0.08 and -0.15. It was the value is taken from a paper by WOLF which this fact that guided us in our choice of x for summarizes and normalizes to a uniform notation TmGaG. the results on crystalline parameters obtained by A number of papers have been written which various investigators.) aim to interpret the magnetic properties and There remains for erbium the elaborate interspectroscopic energy levels of the erbium garnets. AYA.NTand collaborators,@) also DREYFUS et aZ.(10) pretation of the spectrum of Er doped YAlG by KONINGSTEIN and GEUSIC.@~) They use a have used for ErGaG a potential similar to (1) potential similar to (1) except that 21 is replaced except that the coefficients 5, 21 are replaced by

NOTE

ON

THE

CRYSTAL

FIELD

PARAMETERS

by 15 -75 and a term is added in 026, so that the symmetry is tetragonal rather than cubic. The value of V6/V4 obtained from their analysis is -0.28 or -0.20, depending on whether one examines the ratios of the coefficients of 066, 046 or of 064,044. They do not calculate the g values of the ground state, and it is questionable if the proper anisotropy would be obtained. Still higher values of V6/V4are obtained by KONINGSTEINand G~usIc(I~~~5) for Nd, Eu, or Tb doped YAlG. For Tm doped YAlG, Koningstein finds (personal communication of work not yet published) that V6/V4 is -0.20 or -0.12, depending on whether one takes the ratio of the coefficients of 046 and 066, or of 064 and 064. The agreement with the value -0.15 we used for TmGaG is as good as can be expected in view of the crudeness of our cubic model. Also, as Koningstein points out to the writer, the parameter ratios are somewhat different for different garnet matrices, and so the value for TmGaG need not be the same as for Tm doped YAlG. His work on europium doped garnets seems to indicate that V6/V4 is smaller for the gallium or iron than for the aluminum garnets. In the preceding paper we assumed that this ratio had the same value in TmGaG and TmIG. This procedure was probably admissible since the cubic model is only an approximation, and especially since the ground state was well separated from the other levels. We now turn to the ytterbium garnets.* A cubic potential has been used by BRUMAGEet aZ.(r7) to describe the susceptibility of ErGaG. They suggest that the ratio V6/V4 falls between -0.06 and -t-0*02. However, this conclusion depends delicately on the interval F6-Fs between the two upper states which affects the susceptibility only slightly and is sensitive to non-cubic corrections (especially the splitting of I’s). By far the most refined calculations on the ytterbium garnets are those by * Like HUTCHINGSand WOLF, or BRUMAGEet a1.rrr) we assume that the level observed spectroscopically at about 300 cm-l in the Yb garnets is a vibrational rather than a real Stark level, as otherwise it seems impossible to explain the observed susceptibility. A model of crystalline field parameters for the ytterbium garnets proposed by KONINGSTEIN~~~) assumes that this levelis real. The values of Vs/V4 deduced from his model are lower in magnitude than those obtained by Hutchings and Wolf.

OF

RARE

EARTH

GARNETS

1049

and WOLF,@) who utilize instead of (1) a nine-parameter potential of the form

HUTCHINGS

v = Lz(v,ooas+ vsaoas> +j3(V46046+ v4204s+ V44044) + r( V66066 + v,soes + V64064 + V66066). (3) The dimensionless ratios V4’4n”/V46 and V6m/V66 are obtained from an elaborate point charge calculation in which the fields of many surrounding ions are summed. The coefficients T/is, Vss, V46, V6’ are evaluated by fitting experimental data rather than theoretically. Comparison of their value of V6’/V4’ in (3) with (1) yields V6/V4 = -0.37 and that of V64/V44 gives V6/V4 = -0-58. The coefficients of the cubically allowed harmonics in a non-cubic potential may not, however, represent the best cubic approximation. They do not try to calculate the susceptibility of YbGaG, and instead determine the sixth order part of the potential by fitting the g values of the ground state. Unpublished calculations just communicated to the writer by J. J. Pearson reveal that their model unfortunately does not yield the proper value for the part of the susceptibility which is independent of temperature at low T. COMPARISON

WITH FREEMAN-WATSON

RADII

The coefficients V46, V6’ in the potential (3) used by Hutchings and Wolf are proportional respectively to (r*1)4f and (+)4f. By combining their empirical values of V6’/V46 with their point potential expansion, one can deduce a value of (r6)4f/ (r4)4f and see how this compares with the value computed theoretically by FREEMAN and WATSON@@ from wave functions for free Ybs+ ions, which they obtain by an elaborate Hartree-Fock calculation. The Hutchings-Wolf model gives 7.2 x lo-16 cma, whereas according to Freeman and Watson (r6)4f/ (1.4)4f = O-9 x 10-16. The discrepancy by a factor 8 is discouraging. Furthermore, the crystalline potentials used by KONINGSTEIN and GEUSIC(I~-I~) forvarious rareearthdoped garnets, taken in conjunction with the point charge calculation of the crystalline field made by Hutchings and Wolf, yield values of <~6)4f/(r4)~~ from four to ten times as large as those calculated

1050

J.

H.

VAN

by Freeman and Watson for the respective ions.* It is perhaps significant that HUTCHINGS and RAY(~O@also find a similar type of discrepancy in the fields for PrCla and PrBrs. To explain the glaring disagreement with the Freeman-Watson radii one may make any one of four possible assumptions, viz. (a) that the crystalline field parameters of the writers we have cited are wrong, (b) that the point charge calculations are meaningless because of screening effects, (c) that the Freeman and Watson radii are inapplicable in solids, because of configuration mixing, or something of the kind, by interatomic forces or, (d) as suggested by JORGENSONet aZ.,(21) the covalent effects are so strong even in the rare earths that the method of crystalline potentials cannot be used, and instead the whole ligand must be treated as a unit. It would be sheer speculation for us to try to decide in the present state of our knowledge which is the correct way out. The ratio Va/V4 = -0.15 of sixth to fourth degree terms which we used in the preceding paper is about one third as great as for the Hutchings and Wolf model, but about three times as large as to be expected from the Freeman-Watson radii and point charge calculations. It is thus a sort of compromise between the two approaches.

VLECK

the absolute value of the Stark splitting, or, in other words, the scale factor in the crystalline potential, as it is as regards the relative size of the sixth and fourth degree terms. The empirical value of the proportionality constant in the fourth degree part of the crystalline potential employed by Hutchings and Wolf is only twice as large as one obtains from their point charge calculation and the Freeman-Watson value of (r4)4~ The scale factor which we used to interpret the susceptibility of TmGaG in the preceding paper agrees within 15% or so with that found by Koningstein in unpublished spectroscopic investigations on TmAlG. Exact agreement cannot be expected, since gallium and aluminum garnets are not the same. Also, this scale factor for TmGaG is reasonably consistent with that employed by other authors for the gallium garnets of adjacent ions. In YbGaG, experimental determinations of the temperature independent term in the paramagnetism at low temperatures show that the I’s state is about 800” above the ground level (820” and 740” according to two different sources).(ssJ’) This fact fixes the scale factor once the value of ve/v4 has been decided on. WHITE’S diagrams(s) show that for fixed I’4 and for va/v4 in the range -0.10 to -0.15 the interval Is-l?4 in Ybsf is almost exactly the same (within 2 or 3%) as the overall VALUES OF THE SCALE FACTOR Stark splitting in Tmsf, which we took to be about The situation is not as perplexing as regards 930” in the preceding paper.(l) The reduction by a little more than 10% in the scale factor in going * KONINGSTEIN and GEUSIC(~~) overlooked the fact that from Tm to Yb is readily understandable, because the Freeman and Watson values of 4f are measured in atomic units rather than angstroms. Consequently the orbital radius contracts somewhat when the the values of Bss, &s, Bss given in Table V, page 728 atomic number increases. A decrease of almost this should be increased by factors (1.9)2, (1*9)4, (1.9)6 reamount is yielded by the calculations of Freeman spectively. Also, there is a typographical error in this and Watson. If vs/v4 = -0.15, the White diatable: exponents should be 18,34,48 rather than 16,32, grams show that the overall splitting should be 64. The entries in Table IV on page 1425 of Koningstein’s paper on EuIGtrs) relate to his A,m (the same as the vnrn about 85% as great in EI?+ as in Tms+. The splitin our Eq. 3) and not his Bnm and the calculated values tings deduced by DREYFUS et al. in ErGaGo@ of the ratios A40/Aa0,AsO/A4°,AsO/Aa0 in this table should be increased bv factors (1-9)2. , ,_. (1*9P. . (1~9)~ for and by KONINGSTEIN and GEUSIC(~~)in ErAlG are about 60-65% as great as the value we used the reasons we have stated. The change spoils the near agreement between the observed parameter ratios and in TmGaG. Corrections for change of radius those calculated from theory. A notational correction with atomic number here work in the wrong should also be noted. His Ad4, As4 are respectively codirection. efficients of 8044, ~0~4, not of 5,9044-21yOs4in the Probably the reason that there is greater consistexpansion of the potential. There are also some inconsisency in the scale factor than in va/v4 is that the tencies or misprints in the signs of A44,Ae4,but they are readily straightened out from the fact that A44/A40> 0, overall splitting is a rather definite quantity As4/Aso< 0 if the axes are along the environmental which can be inferred from experimental data, cube, with the reverse true if they are local axes for the whereas more involved analysis is required to bring g tensor, as explained in a previous footnote of the present out the relative importance of the different terms. paper.

NOTE

ON

THE

CRYSTAL

FIELD

PARAMETERS

OF

RARE

EARTH

GARNETS

1051

11. PAPPALARDOR., 2. Phys. 173,374 (1963). Acknowledgements-The writer is indebted to Professor 12. WOLF W. P.. Proc. Int. Conf. on Ma.enetism in WERNER P. WOLF for helpful discussions and comment, Nottingham’p. 555 (1964). and to Professor J. A. KONINGSTEINand Dr. J. J. PEARSON 13. KONINGSTEINJ. A. and GEUSICJ. E., Phys. Rev. 136A, for communication of results in advance of publication. 726 (1964). REFERENCES 14. KONINGSTEINJ. A. and GEUSICJ. E., Phys. Rew. 136A, 1. SCHIEBER M., LIN C. c. and VAN VLECK J. H., 711 (1964). J. Phys. Chem. Solids, 27, 1041 (1966). 15. KONINGSTEINJ. A., Phys. Rev. 136A, 717 (1964). 2. STEVENSK. W. H., Proc. Phys. Sot. Lond. 65A, 209 16. KONINGSTEINJ. A., J. Chem. Phys. 42, 1423 and (1952). 3195 (1965). 3. BAKER J. M., BLEANEYB. and HAYES W., Proc. R. 17. BRUMAGEW. H., LIN C. C. and VAN VLECK J. H., Sot. 247A, 141 (1958). Phys. Rev. 132, 608 (1963). The value of Ve/V4 4. LEA K. R., LF.ASKM. J. M. and WOLF W. P., J. Phys. _ which we quote from this reference is obtained by Chem. solids 23, 1381 (1962). transcribing its equation (3) into our system of 5. EBINA Y. and TSUYA N., Rep. Res. Inst. Elect. units rather than that of Pappalardo and Wood. Commun. Tohohu Univ. B12, 1, 167, 183 (1960); 18. HUTCHINGSM. T. and WOLF W. P., J. Chem. Phys. ibid. B13, 1, 25, 43 (1961). 41, 617 (1964). 6. DIEKE G. H.. PYOC. First Int. Conf. on Puramuznetic 19. KONINGSTEINJ. A., Theor. Chim. Acta 3,271 (1965). Resonance ‘Vol. 1, p. 237. Academic Press,N.Y. 20. FREEMANA. J. and WATSON R. E., Phys. Rev. 127, (1963). 2058 (1962). 7. WOLF W. P., BALL M., HUTCHINGSM. T., LEA~K 20a. HUTCHINGSM. T. and RAY D. K., Proc. Phys. Sot. M. J. M. and WYATT A. F. G., J. Phys. Sot. Japan Lond. 81, 663 (1963). 17, suppl. B-l, 443 (1962). 21. JORGENSONC. K., PAPPALAR~OR. and SCHMIDTKE 8. WHITE J. A., J. Phys. Chem. Solids 23, 1787 (1962). H., J. Chem. Phys. 39, 1422 (1963). 9. AYANT Y., ROSSETJ. and VEY~SIEC. R., C.R. Acad. 22. BALL M., GARTON G., LEASK M. J. M. and WOLF Paris 259, 1698 (1964). W. P., Proc. 7th Int. Conf. on Low Temp. Phys. 10. DREYFUS B., VEYSSIE M. and VERDONEJ., J. Phys. p. 34 Univ. of Toronto Press (1960). Chem. Solids 26, 107 (1965).

This paper is dedicated to Professor MARIO KOTANI on the occasion of his sixtieth birthday. The writer congratulates him on this milestone in his distinguished career.