UNCLASSIFIED ORNL ... i P 295 که .) -ا ot ا )15: . : SLra 64-57 CONF-600:) APPLICATION OF SEMI EMPIRICAL COUPLING RULES TO THE ANALYSIS OF MAGNETIC STRUCTURES* M. K. Wilkinson UM Solid State Division, Oak Ridge National Laboratory Oak Ridge, Tennessee INTRODUCTION The semiempirical rules for the ordering of ma getic moments through indirect exchange processes have been applied to relatively few magnetic materials. These are compounds with a higli degree of symmetry in which the crystal field splitting is understood and in which it is possible to trace the important magnetic exchange interactions through the d-orbitals that result from the crystal fields. In fact, av the present time, these cou- pling rules have received a thorough application in only one type of th d m yra n , 1 h rey. comie to the Co vesten, or No o destinats, ar monitor ac ore of privistely owned righta; a As we in de bboro, pero t o , of watele water at de Cocination, w www dwy teleration, suatu madhues, or een hecho n Conden, amery dwy warnaum, marad, mother, or face and Carnesto. No my porno K de tu, aty motornetta per my liabudas ou menort to the wine a. or low we mo ma contar au totala ---LEGAL NOTICE A Hiwowy nirmy w menuda.rowie. MO Inopera ho the nov. formation co The report me prepared as an acement of Goveranset opossored work, Keither the United a a la y m o weit much mor Come use i talhe Contacto o mort. d e rent, t playa or counci Unter poporos. , to do " include my rundeing trou the dar de to at . crystal symmetry. This is the symmetry in which & transition element cation is located in an octahedral interstice of a cubic anion lattice. Many perovskite-type structures and a series of trifluorides with slight distortions from cubic symmetry have been examined, and in all cases the semiempirical rules can account for the observed magnetic structures. STRUCTURES WITH OCTAHEDRAL SYMMETRY As indicated by Dunitz and Orgel, when a transition metal ion is surrounded by six negative ions in a regular octahedral arrangement, the five d-orbitals do not have the same energy. They are split into a stable triply degenerate state tze (composed of dxyo dyze dzx) and an upper doubly degenerate state ec (composed of de 2 and d 2) with the spatial dis- tributions indicated in Fig. 1. The physical basis for this splitting . B . X . is simply the electrostatic repulsion between the d-electrons and the FE ..Mi . Research sponsored by the U. S. Atomic Energy Commission under contract with the Union Carbide Corporation. - LON my -2- surrounding negative ions. As shown in the figure, the e, orbitals point directly at these ions and are destabilized, while the to, orbitals point in directions between the ions and are stabilized. The energy gap between these two sets of levels depends upon the particular anion complex and upon the charge of the cation. As successive electrons are introduced into the d-shell, they occupy the lowest available orbitals compatible with the total spin associated with the metal ion. Fur cubic symmetry it is necessary that the d-electrons be symmetrically distributed among the d-orbitals, and consequently cubic symmetry can be achieved only in the configurations dº, (tze)? (tze bles, (tag) *, (tzederlerde, (tag)"tex)" In other configurations, a Jahn-Teller effect will be operative, which will produce distortions from cubic symmetry and a corresponding split of the e, and to energy levels. Dunitz and Orgel have pointed out that the largest distortion would be expected when the eg orbitals are not symme- configu- trically populated, such as in the rations. As shown in Fig. 1, for octahedral symmetry the primary orbital overlap occurs between the e, orbitals of the cation and the p-orbitals of the intervening anion. The important magnetic exchange interactions would therefore be associated with this overlap and the to, orbitals would have very little effect. Consequently, the semi empirical rules for . - - *The (tor)o configuration is possible when the tee, splitting is large compared to the intra-atomic exchange coupling, which favors a parallel alignment of spins within the atom. magnetic coupling would depend on the occupation of the e. orbitals, which in turn would depend on the overall occupation of the d-shell. Table I shows the orbital occupation that would be expected for some iron group ions in octahedral sites when Hund's rule is applicable.* The data in this table can be directly applied to mačnetic coupling in compounds which possess octahedral symmetry. Since no completely filled orbitals exist in these ions, only three of the six empirical rules are applicable. These are the three rules that were suggested by Wollan * to explain the ex- perimental data prior to the development of a proper theory relating to them. They are diagramatically illustrated in Fig. 2, and they may be stated as follows: 1. When half-filled orbitals (orbitals with one electron) of two mag- netic cations overlap, respectively, the two ends of a given anion p-orbital, the magnetic coupling is antiferromagnetic. 2. When empty orbitals of two inagnetic cations overlap, respectively, the two ends of a given anion p-orbital, the magnetic coupling is antiferromagnetic. When an empty orbital of one magnetic ion overlaps one end of an anion p-orbital and the other end of the same anion p-orbital over- . - laps a half-filled orbital of another magnetic ion, the magnetic - . - - .- coupling is ferromagnetic. -- : -1 . . Several neutron diffraction investigations have played a prominent . P part in the development and application of these rules, and the important L- * *Goodenough has shown that Hund's rule does not always apply for triva- lent and tetravalent cobalt. . S 2 results of these investigations will be discussed in the following sections. 1. Perovskite-Type Compounds [(1-x)La, xCa) Mnoz One of the most important and comprehensive neutron diffraction investigations, which has provided results to develop and test the semiempirical rules of magnetic ordering, was performed on a series of perovskite-type compounds ((1-x) La, xCa? Mr.0, by Wollan and Koehler. This investigation was not performed to test the rules, because no rules existed at that time. It was performed in an at- tempt to understand some of the very unusual properties of these compounds that were observed in the magnetic measurements of Jonker and Van Santen. During the course of the investigation, Goodenough' developed his ideas on semicovalent exchange and the neutron results were interpreted on this basis. The three rules, as listed previously, were later developed and shown to be directly applicable. The ideal cube of a perovskite contains one ABO, molecule, where A is a large ion such as Ca or La's located at the cube center, B is a small ion such as Mn's or co*s at the cube corners, and the oxygen ions are at the midpoint of the cube edges. Many examples of the perovskite structure are known, and most of them are slightly distorted from cubic symmetry. In this particular perovskite series, the nonmagnetic La or Ca ion does not affect the magnetic exchange interactions and the value of x merely determines the relative amounts of Mn and Mn". Since the magnetic properties were very strongly affected by the relative ionic content, this quantity was independ- ently determined in all samples by chemical analyses. Over twenty -5- different specimens were studied, with x varying between zero and one, and many different types of magnetic structures were observed and analyzed. For CaMnoz, in which only Mn++ ions are present, an antiferro- magnetic structure was observed in which the chemical unit cell is doubled along each of the three cube axes. This magnetic structure, which has been identified as the G-type, shown in Fig. 3a, is one in which each Mn ** ion is surrounded by six Mn *neighbors whose spins are antiparallel to the given ion. This is exactly the situation expected by the coupling rules, because only empty €, orbitals are involved in the coupling process through the intervening oxygen ions, S and consequently, all interactions must, ba antiferromagnetic. +4 For compositions near 20% Mri") and 80% Mn++, a magnetic struc- ture was observed which required that the chemical cell be doubled in two directions. This structure, which was designated the C-type, resembles ferromagnetic chains of moments with adjacent chains - - - - - oriented antiparallel. Such a structure, which involves the empty e, orbitals of the Mn*+ ions and both empty and half-filled e orbitals of the Mn*3 ions, also can be accounted for by application of the magnetic coupling rules. For compounds with the reverse composition, in which the Mn" content was near 25%, a ferromagnetic - - . . . - - - - - - +4 -- .. - structure was observed. Figure 36 shows how this ferromagnetism can be explained by the coupling rules when the d. 2, 2 orbitals of the Mn*3 ions are half-filled and the d, orbitals are empty. The pure Lamno, compound also became antiferromagnetic, and the structure (designated A-type) is one in which the magnetic unit cell 22 -6- requires that the chemical unit cell be doubled along only one cube edge. Such a magnetic structure consists of ferromagnetic layers with alternate layers oriented antiparallel, and this antiferromag- netic structure is identical to that shown in Fig. 4 for MnF,. Many other compositions of Mns and Mn *4 ions were also examined. +3 For some composition ranges both ferromagnetic and entiferromagnetic phases were present, and for other compositions, rather complicated antiferromagnetic structures were observed. However, there were no apparent discrepancies in the observed exchange interactions and those which could be expected on the basis of the coupling rules. -- It should also be mentioned that some of the effects that were , ta observed in the neutron experiments can be explained by two-sublattice canting of the magnetic moments, which de Gennes has shown to be the most usual state in systems where double exchange and superexchange interactions are in competition. In systems involving both Mn"s and Mn++, where there are empty orbitals and half-filled orbitals over- lapping opposite sides of a p-orbital, both types of interactions do indeed exist. 2. Transition Metal Trifluorides, MF, The trifluorides of transition group elements crystallize as modifications of a cubic structure in which the metal ions lie on the corners of the cube and the fluorine ions are near the centers of the cube edges. This arrangement is similar to that of the transition metal ions in the perovskite structures, except that there are no possibilities of complications due to the large ion in the center of metal ions in the perovskite struct ept that there are no the cube. A series of iron group trifluorides, including VF,, CrF,, -7- FeFz, Cof, and MnFz, was therefore investigated by neutron diffrac- tion to check the coupling rules that were suggested from the in- vestigations of the (la,Ca )Mno, perovskites. FeFz, and Cof, all had antiferromagnetic transitions to a G-type structure, in which the magnetic ions are coupled antiferromagneti- cally via the intervening anions to all six nearest neighbors. This was the same magnetic structure observed for Camno, and would be predicted for all three trifluorides on the basis of the coupling rules. Trivalent chromium has only empty e orbitals, whereas both trivalent iron and trival.ent cobalt have only half-filled orbitals; consequently, only antiferromagnetic exchange interactions are possible through the intervening fluorine p-orbitals. The distortions from cubic symmetry in these compounds made it possible to determine the orientation of the spin moments relative to the crystallographic axes. As seen in Fig. 3a, the magnetic structure actually consists of ferromagnetic sheets of moments, perpendicular to the unique axis of the rhombohedral unit cell, and adjacent sheets are oriented anti- parallel. CrF, and FeF, have the magnetic moments oriented parallel to the ferromagnetic planes, while in CoF, the moment orientation is perpendicular to those planes. In the case of CrF,, additional mea- surements were made with an external magnetic field applied to the specimen so that the antiferromagnetic domain properties could be studied. The antiferromagnetic structure that developed in MnF, at low temperatures was different from that in the other trifluorides of 26 ya this group. As mentioned previously, the structure is identical to the one observed for LaMnO2, and it is shown in Fig. 4. In this figure, it is assumed that the single electron in the e, orbitals goes into a d,, orbital, but the same structure can be developed on the assumption or an empty d 2 orbital and an electron occupying the d2 orbital. The spatial ordering of the orbitals, which is re- quired to account for this type of structure, appears reasonable on the basis of the fluorine positions. Long (2.18), medium (1.98), and short (1.88) distances to the manganese ions have been observed, and these are shown schematically in the figure. Since fluorine ions lie half-way between ferromagnetic layers, it is probable that identi- cal magnetic ion a-orbitals overlap opposite ends of the p-orbital, and this situation requires antiferromagnetic coupling. Conversely, the ordered arrangement of long and short distances within the ferro- magnetic layers suggests different types of d-orbitals on opposite ends of a p-orbital, which is the condition for a ferromagnetic interaction. A subsequent investigation was performed on trifluorides of 4d-transition elements to ascertain if similar magnetic structures and coupling rules might be applicable to these compounds. No magne- tic ordering was observed in PaF, and RuFz, but MoF, was found to order with a magnetic structure which is identical to that for CrF, as would be expected on the basis of the orbital population. The relative Néel temperatures indicate a stronger magnetic coupling in MoFg, which would be expected if the 40-orbitals have a larger over- lap with the p-orbitals than the corresponding 3d-orbitals. The -9- lack of any observable magnetic neutron scattering from Paf, and RuFz, either from an ordered magr:etic lattice or from a paramagnetic lattice, could be explained with the assumption that Hund's rule does not apply to ions in the 4d-transition series. 3. Other Perovskite-Type Compounds After the original experiments on the perovskite-type compounds of the (La,ca)Mnosystem had indicated the applicability of specific rules for magnetic coupling, many other similar compounds were in- 71 vestigated to test these rules. Koehler and Wollant extended their original investigations to a series of compounds LaBO., in which B is a trivalent ion of one of the 3d transition elements. They found that both Lacro, and LaFeO, ordered with the G-type structure as would be expected, but that neither Lacoo, nor Lanio, exhibited nag- netic ordering at temperatures down to 4.2°K. Analysis of the LaNio, data was somewhat handicapped by an oxide impurity, but there was no indication of any type of magnetic scattering from Lacoo, that would be consistent with the expected atomic magnetic moments of the Co *3 ion. Goodenough? explained this anomaly and other results from magnetic measurements on the basis that it is possible for Co ions (and possibly also Ni ions) to have a low spin state, in which Hund's rule does not apply. A series of rare-earth-iron perovskites was also investigated by neutron diffraction, and since the iron mo- S.Si: -- > ments ordered at a much higher temperature than the rare-earth moments, the latter did not strongly influence the type of ordering which occurred within the lattice of Fe's ions. In all cases, the expected G-type lattice was found to develop. -10- An investigation by Scatturin, Corliss, Elliott, and llastings) of the magnetic ordering in a series of 3d transition metal double fluorides KBF, with the perovskite-type structure added additional confirmetion to the applicability of the coupling rules for magnetic ions in octahedral symmetry. Since the 3d ions in this series are divalent, additional evidence was provided to show that the magnetic structures were dependent on the electron configurations of the ions and not on the specific elements involved. KMnFz, KFeFz, KCoFz, and KNiF, all ordered with the G-type structure, whereas KrF, developed the A-type structure, which had previously been observed only in LaMno, and MnFz. No magnetic order was observed in KCuFz, and the authors infer that the sensitivity of the experiments should have been sufficient to detect such a structure. This is unfortunate, because magnetic coupling in this compound would have involved orbi- tals containing two electrons and orbitals containing one electron, and the coupling rules have not been verified for these conditions. If the orbital arrangement allowed completely filled d-orbitals to overlap the opposite ends of the fluorine p-orbital, then no coupling would have been expected. However, if the arrangement had involved only coupling between half-filled orbitals and coupling between a half-filled and a completely filled orbital, then an A-type structure might have been expected. This structure would require a ferronag- netic coupling through the process of virtual double exchange as indicated by the semiempirical rules. It is possible, of course, that such an interaction was too weak to produce magnetic order. : :: * -ll- OTHER TYPES OF STRUCTURES Although the indirect coupling in simple compounds with transition metal ions in octahedral sites appears to be completely explained by the semiempirical coupling rules, these rules have not received adequate ap- plication in other types of compounds. This is perhaps not too surprising, because many magnetic compounds do not have simple crystal structures. Consequently, the crystal field effects are unknown, and there are many possible paths through which magnetic interactions can occur. Wollan* has considered the case of magnetic coupling between tran- sition metal ions in tetrahedral sites, and he has shown that it is quite possible that the coupling rules can apply for magnetic compounds with this symmetry. Dunitz and Orgelt have pointed out that for this case, the e doublet has a lower energy than the tztriplet, and the corresponding orbital population is given in Table II. Furthermore, in this symmetry the e, orbitals do not point toward the anion, and the indirect magnetic exchange appears to occur through an overlap of the the orbitals with the p-orbitals as shown in Fig. 5a. If the important magnetic interactions should occur through the paths shown in Fig. 56, the magnetic structured found for B-Mns would be consistent with the ccupling rules. All of the to orbitals would be half-filled so that all indirect interactions would be antiferromagnetic, and these interactions would cause the moment ar- rangement indicated in the figure. Wollan has also pointed out that application of these rules can account for the strong antiferromagnetic interaction observed in many spinel-type structures between ions on A sites and ions on B sites. However, if these rules do apply for the spinels, then this A-B interaction could be ferromagnetic in certain compounds. In .* * * -12- * particular, in spinels containing trivalent chromium with empty e, orbi. tals in octahedral sites and ions with halt-filled to orbitals, such as 3 divalent iron, in tetrahedral sites, the A-B coupling would be ferromag- netic. It is of interest to point out that mixed spinels involving chro- mium ions have been found to have unusual magnetic properties. More Information on spinels and on other compounds with magnetic ions in tetra- hedral symmetry would be very desirable in determining the applicability of these coupling rules. - - - - - . .. 13- Table I. d-Shell Configurations for Iron Group Ions in Octahedral Sites Mn +4 143 Cr 2 not us 12 out 2 T1*3 T4*2 en 00 tag too 00 110 00 111 to 11 11 11 11 111111111111111 111111 ** * ** -14. Table II. d-Shell Configuratione for Iron Group Ions in Tetrahedrai Sites Mn fe #26 000 000 100 110 111 1111111111 -15- REFERENCES 1. J. D. Dunitz and L. E. Orgel, J. Phys. Chem. Solids 3, 20 (1957). 2. J. B. Goodenough, J. Phys. Chem. Solids 6, 287 (1958). 3. Wollan, Child, Koehler, and Wilkinson, Phys. Rev. 112, 1132 (1958). 4. E. O. Wollan, Phys. Rev. 117, 387 (1960). 5. E. O. Wollan and W. C. Koehler, Phys. Rev. 100, 545 (1955). 6. G. H. Jonker and J. H. Van Santen, Physica 16, 337 (1950); Physica 19, 120 (1953). 7. J. B. Goodenough, Phys. Rev. 100, 564 (1955). 8. P. G. de Gennes, Phys. Rev. 118, 141 (1960). 9. M. A. Hepworth and K. H. Jack, Acta. Cryst. 10, 345 (1957). 10. Wilkinson, Wollan, Child, and Cable, Phys. Rev. 121, 74 (1961). 11. W. C. Koehler and E. O. Wollan, J. Phys. Chem. Solids 2, 100 (1957). Koehler, Wollan, and Wilkinson, Phys. Rev. 118, 58 (1960). 13. Scatturin, Corliss, Elliott, and Hastings, Acta. Cryst. 14, 19 (1961). 14. Corliss, Elliott, and Hastings, Phys. Rev. 104, 924 (1956). 15. E. W. Gorter, Philips Research Repts. 2, 295 (1954). જ - - - -. * a t e . SA . .." 5.IX NAT. UNCLASSIFIED ORNL-LR-DWG 68214 R THE O-ORBITAL @g-ORBITALS ( dy2-y2 720-ORBITALS ORBITAL OVERLAPS UCATION CATION - - - ANION -ORBITAL O - P - • - * : to i TE Fig. 1. Crystal Field Splitting of d-Orbitols for Iron Group Ions in Octahedral Sites. UNCLASSIFIED ORNL-LR-DWG 20121R CATION ANION CATION (9) ANTIFERROMAGNETIC (0) ANTIFERROMAGNETIC 10) ANTIFERROMAGNETIC (c) FERROMAGNETIC Cup B D HALF-FILLED ORBITAL EMPTY ORBITAL ION SPIN Fig. 2. Schematic Representatation of Indirect Exchange Process. UNCLASSIFIED ORNL-DWG 64-6783 INUM, SZCZ 111... FERROMAGNETIC (0) COOL25L00.75 MnO3 (a) COMnO3 Fig. 3. Magnetic Structures in Cox L0,-x MnO3 Compounds UNCLASSIFIED ORNL-LR-DWG 68215R ülinal.. - Fig. 4. Magnetic Structure (Half- Cell) of Mn F3 with Suggested d-Orbital Magnetic Coupling. UNCLASSIFIED ORNL-LR-DWG 28841R3 lo) Fig. 5. Possible Indirect Coupling Between lons in Tetrahedral Sites. (6) . . - - - lle - DATE FILMED 11/ 19/64 K iling . . vill, IME ! ! .MMM. i * 12 ar * . ST RE i ma he BY -LEGAL NOTICE This report was propered as an account of Goveramont sponsored work. Neithor the United Suatos, nor the Commission, nor any person acting on behalf of the Commission: A. Makes any warranty or ropresentation, expressod or implied, with respect to the accu- racy, completeness, or usefulness of the information contained in this report, or that the use of any information, apparatus, mothod, or process disclouod in this roport may not infringo privately owned rights; or B. Assumos any liabilities with respect to the use of, or for damages resulting from the uso of any information, apparatus, method, or process disclosed in this report. 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