Paramagnetism of Caesium Titanium Alum Lucjan Dubicki and Mark J

Paramagnetism of caesium titanium alum Lucjan Dubicki and Mark J. Riley

Citation: The Journal of Chemical Physics 106, 1669 (1997); doi: 10.1063/1.473320 View online: http://dx.doi.org/10.1063/1.473320 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/106/5?ver=pdfcov Published by the AIP Publishing

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Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 130.102.82.2 On: Tue, 18 Oct 2016 06:21:12 Paramagnetism of caesium titanium alum Lucjan Dubicki Research School of Chemistry, Australian National University, Canberra, 0200, Australia Mark J. Riley Department of Chemistry, University of Queensland, St Lucia, 4072, Australia ͑Received 31 May 1996; accepted 16 October 1996͒ • Caesium titanium alum, CsTi͑SO4͒2 12H2O, is a ␤ alum and exhibits a large trigonal ﬁeld and a 2 dynamic Jahn–Teller effect. Exact calculations of the linear T2 e Jahn–Teller coupling show that in the strict S6 site symmetry the ground multiplet consists of a Kramers doublet 2⌫6 with magnetic Ϫ1 splitting factors gʈϭ1.1 and gЌϭ0, a ⌫4⌫5 doublet at ϳ60 cm with gʈϭ2.51 and gЌϭ0.06 and Ϫ1 another ⌫4⌫5 doublet at ϳ270 cm with gʈϭ1.67 and gЌϭ1.83. The controversial g values

observed below 4.2 K, gʈϭ1.25 and gЌϭ1.14, are shown to arise from low symmetry distortions. These distortions couple the vibronic levels and induce into the ground state the off-diagonal axial Zeeman interaction that exists between the ﬁrst excited and the ground vibronic levels. © 1997 American Institute of Physics. ͓S0021-9606͑97͒01804-7͔

I. INTRODUCTION Since for positive ␷ the ground state of the Ti3ϩ ion will always remain 2⌫ with g ϭ0, the early attempts to account Crystals of caesium titanium alum, CsTi͑SO ͒ •12H O 6 Ќ 4 2 2 for g ϭ1.14 in CsTiS employed a negative value of ␷. The ͑abbreviated as CsTiS͒, have the cubic space group Pa3 with Ќ 2⌫ ground state has g ϭ0 by symmetry, regardless of four equivalent Ti3ϩ ions in the unit cell. The nominally 6 Ќ whether there is Jahn–Teller coupling operating or not. The octahedral complex ion, Ti͑H O͒3ϩ , is slightly compressed 2 6 earlier work culminated in the paper by Shing and Walsh7 along the molecular trigonal axis that lies along one of the who proposed a ⌫ e Jahn–Teller coupling with a very body diagonals of the unit cell. The site symmetry at the Ti3ϩ 8 small trigonal field. This model was based on the assumption ion is S and the fractional atomic coordinates of the sulphur 6 that the spin–orbit coupling of the Ti3ϩ ion breaks the atoms identify the CsTiS alum to be of the ␤ modification.1 2T ( ϩ ) e coupling into e and quenches the The paramagnetism and, in particular, the magnetic g 2 ⌫8 ⌫7 ⌫8 pseudo Jahn–Teller coupling between ⌫8 and the higher ly- values of CsTiS have been an unresolved problem for more 2 than forty years.2–7 Unlike the electron paramagnetic reso- ing ⌫7 , where ⌫8 and ⌫7 are spin–orbit components of T2 in 3ϩ 8 the cubic limit. This analysis is not valid because the spin– nance ͑EPR͒ of Ti doped into an ␣ alum such as RbAlS Ϫ1 where twelve magnetically inequivalent sites were observed orbit constant, ␨ϳ120 cm in bound Ti͑III͒, is comparable or even less than the Jahn–Teller stabilization energy that and correspond to one chemical species with rhombic sym- 3ϩ 11,13,14 metry, the single crystal EPR of CsTiS indicated one chemi- has been observed in several Ti complexes. ϭ A new insight into the electronic structure of alums has cal species of trigonal symmetry with gʈ 1.25 and 15 g ϭ1.14.3 The large linewidth of the EPR lines, about 250 been revealed by the theoretical work of Daul and Goursot, Ќ 16 gauss, and the variation of the magnetic susceptibility at low the electronic Raman measurements of vanadium alums temperatures indicated the presence of low lying paramag- and by more recent x-ray and neutron diffraction structure 17–20 netic states.3 determinations. The work of Best, Forsyth, and Tregenna-Piggott18–20 have been particularly definitive. All If the trigonal field that lifts the degeneracy of the t2 d-orbitals of the Ti3ϩ ions has a positive sign, CsMS alums with metal ions having the electronic configu- ration, (tn) nϽ6, are ␤ alums that are characterized by ␷ϭE(t2x0)ϪE(t2xϮ) is greater than zero, then the ground 2 M͑III͒ ions coordinated by planar water molecules. The OH Kramers doublet in S6 symmetry has the symmetry classifi- 2 1 2 planes are all rotated about the M–O bonds by 20° with cation (t2xϮ) T2(2⌫6) where we use the complex trigonal ϳ 9 basis for t2 orbitals and the double group notation of Koster the sense that the oxygen p␲ lone pairs normal to the OH2 10 3ϩ et al. plane are tilted towards the trigonal z axis of the M͑OH2͒6 3 1 The 2⌫6 doublet transforms as the M JϭϮ 2 components ion. The structure of CsTiS at 100 K is shown in Fig. 1͑a͒ 3 3ϩ of angular momentum Jϭ 2 and has gЌϭ0. Indeed, for Ti displaying the S6 symmetry of the Ti͑III͒ site. The ϳ20° Ϫ1 doped in Al2O3 where ␷ϭ700 cm the observed g values twist of the planar water molecules is midway between the 11 19 are gʈϭ1.07 and gЌϳ0.0. The gʈ value differs appreciably Th and the ‘‘all horizontal’’ D3d symmetries. The angular from the prediction of the static ligand field model that gives overlap model shows that such a rotation generates a large 15 to first order, gʈ(2⌫6)ϭ2Ϫ2KzϷ0.5, where Kz is the effec- positive trigonal field, ␷. 1 2 tive orbital reduction factor for the (t2) T2 multiplet. The Consequently, the previous work on CsTiS has focused explanation of this discrepancy as well as accounting for the on the wrong part of the ligand field energy diagram. Since Ϫ1 energies of the low-lying vibronic states at 38 cm and 108 the trigonal splitting of the t2 orbitals ͑see Sec. IV͒ is much cmϪ1 was a major success for Ham’s effective Hamiltonian greater than the spin–orbit coupling of Ti3ϩ or the Jahn– 2 11,12 treatment of the T2 e Jahn–Teller problem. Teller coupling, it is the trigonal field which tends to break

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2 EJTϭA1/ប␻ for T2 e coupling, where ប␻ is the wave number of the Jahn–Teller active e vibration. In the complex trigonal basis the coupling matrix becomes

1 1 1 ͉Ϯ 2 Xϩ͘ ͉Ϯ 2 XϪ͘ ͉Ϯ 2 X0͘ 1 ͗Ϯ 2 Xϩ͉ 0 Qϩ ϪQϪ

1 1 ͗Ϯ 2 XϪ͉ ϪQϪ 0 ϪQϩ ϫͱ 2 A1 ͑2͒

1 ͉Ϯ 2 X ͉ ͩ Qϩ QϪ 0 ͪ 0

where QϮϭ(ϯͱ1/2)(Qx Ϯ iQy) and Qx ,Qy are the two components of the degenerate e vibration. This Jahn–Teller coupling matrix was included in the 10ϫ10 electronic matrix containing the ligand ﬁeld and spin–orbit coupling as given by Macfarlane, Wong, and Sturge.11 Each electronic basis function was then expanded in terms of nϭ0,1,2,...,nv levels of a two-dimensional harmonic oscillator vibrational basis. Matrix elements of the Jahn–Teller coupling were then evaluated and the resulting large complex matrix diagonal- ized. The total basis size without exploiting the vibronic 1 symmetries was Nϭ10ϫ 2ϫ(nvϩ1)(nvϩ2). A fast Lanc- zos diagonalization routine for real symmetric matrices was used the ﬁnd the lowest vibronic energies and wave func- tions. This meant a further doubling of the size of the origi- nal complex matrix. A typical calculation with nvϭ21 re- quired the diagonalization of a 5060ϫ5060 sparse matrix

FIG. 1. ͑a͒ The S6 symmetry of the ␤ alum structure viewed down the with 16 148 nonzero off-diagonal matrix elements. trigonal axis. ͑b͒ A schematic representation of the perturbations acting on In addition to these calculations, we also used an effec- the d electronic levels of a Ti͑III͒ ion. 1 2 1 2 tive (t2) T2 basis corrected for mixing of (e ) E to the second order. By using this electronic basis the vibronic matrix 2 2 2 2 2 1 T2( Eϩ A) e coupling into E e and A, and quenches was reduced to Nϭ3ϫ 2ϫ(nvϩ1)(nvϩ2) as only one com- the pseudo-Jahn–Teller coupling between 2E and the higher ponent of every Kramers doublet needed to be calculated. lying 2A, where E and A are the trigonal orbital species of The other Kramers component could be generated by per- the T2 parent. Figure 1͑b͒ shows a schematic hierarchy of forming the time reversal operation. This reduces the com- perturbations to the d electronic levels of a Ti͑III͒ complex. putational overhead for nvϭ21 to the diagonalization of a These perturbations act simultaneously and are treated as 1518ϫ1518 real matrix with 5544 nonzero off-diagonal ele- such here. The figure does not imply that a perturbation ments. Both types of calculation gave almost identical re- scheme has been used in this work. sults. In both cases, the vibronic levels consist of Kramers 2 doublets, calculated in the absence of an applied magnetic II. T2 e COUPLING IN TRIGONAL SYMMETRY field. The g values were then calculated from evaluating the Since the ␷ trigonal field in CsTiS is very large ͑Sec. matrix elements of the Zeeman operator from the zero field IV͒, the vibronic energy levels and the g values will no 12 eigenfunctions. longer be described accurately by Ham’s theory. Instead For specific cases of near degenerate vibronic levels ͑⌬E we diagonalize exactly the T2 e problem with any trigonal Ͻ10 cmϪ1͒, the off-diagonal Zeeman terms between vibronic field by using a sufficiently large vibronic basis so that both levels were also evaluated. The calculations were restricted the calculated energies and g values are independent of the to linear Jahn–Teller coupling and a single vibrational mode basis size. of cubic e symmetry. The T2 e Jahn–Teller coupling matrix is given by In the absence of Jahn–Teller coupling the effective 1 2 ͉␰͘ ͉␩͘ ͉␨͉͘ Hamiltonian in the (t2) T2 multiplet may be written as 1 ͱ3 ͗␰͉ Ϫ 2Qxϩ 2 Qy 00 2 2 HϭϪ⌬t͑LzϪ 3͒ϩ ␭␣L␣S␣ 1ͱ3 ͚␣ ͗␩͉ 0Ϫ2QxϪ2Qy0ϫA1͑1͒ ␨ ͩ 00Qͪ ͗ ͉ x ϩ ͑K␣L␣ϩ2S␣͒␮BB␣ . ͑3͒ ͚␣ in terms of the real d-orbitals quantized along the cubic axes. Here A1 is the linear Jahn–Teller coupling constant. It is The effective ligand field parameters to the second order related to the Jahn–Teller stabilization energy by are

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TABLE I. Matrix of the Zeeman Hamiltonian.a

͉␾1ϩ͘ ͉␾1Ϫ͘ ͉␾2ϩ͘ ͉␾2Ϫ͘ ͉␾3ϩ͘ ͉␾3Ϫ͘

͗␾1ϩ͉ z1 z1Ј x12Љ Ϫiy12Љ Ϫx12Ј Ϫiy12Ј x13Љ Ϫiy13Љ Ϫx13Ј Ϫiy13Ј ͗␾1Ϫ͉ Ϫz1 x12Ј Ϫiy12Ј x12Љ ϩiy12Љ x13Ј Ϫiy13Ј x13Љ ϩiy13Љ ͗␾2ϩ͉ z2 x2Ϫiy2 z23 x23Ϫiy23 ͗␾2Ϫ͉ Ϫz2 x23ϩiy23 Ϫz23 ͗␾3ϩ͉ z3 x3Ϫiy3 ͗␾3Ϫ͉ Ϫz3

a 1 1 The entries ␣i or ␣ij stand for the diagonal, 2 g␣(␾i)␮BB␣ , or off-diagonal, 2g␣(␾i ,␾ j)␮BB␣ , matrix elements of the Zeeman operator ͓Eq. ͑7͔͒ for the trigonal basis given in Eq. ͑5͒. The explicit forms of g␣ are given in Table II. All xiϭy i and xijϭyij .

2 ⌬tϭ␷ϩ͑␷Ј͒ /⌬, where the real cubic tensors are normalized as 2 2 ␭ϭ␨ϩ͑␨Ј͒ /⌬, V͑Eu͒ϭ1/ͱ6 ͑3Lz Ϫ2͒, 2 2 ␭zϭ␭Ϫ2ͱ2␨Ј␷Ј/⌬, V͑Ev͒ϭ1/ͱ2 ͑Lx ϪLy ͒, ͑8͒ ͑4͒ ␭xϭ␭ϩͱ2␨Ј␷Ј/⌬, V͑T2␣␤͒ϭ1/ͱ2 ͑L␣L␤ϩL␤L␣͒. The corresponding tensors for trigonal symmetry are KzϭkϪ2ͱ2kЈ␷Ј/⌬,

V͑EuϮ͒ϭϯ1/ͱ2 ͓V͑Eu͒ϮiV͑Ev͔͒, Kxϭkϩͱ2kЈ␷Ј/⌬, V͑T x ͒ϭ1/ͱ3 ͓V͑T yz͒ϩV͑T xz͒ϩV͑T xy͔͒, ͑9͒ where ⌬ is the cubic ligand field splitting of the t2 and e 2 0 2 2 2 orbitals, ␷, ␨ and k are the basic one-electron trigonal field, Ϯ ϯ V͑T2xϮ͒ϭϯ1/ͱ3 ͓␻ V͑T2yz͒ϩ␻ V͑T2xz͒ spin–orbit coupling and orbital reduction parameters for t2 orbitals, respectively, and ␷Ј, ␨Ј and kЈ are the corresponding ϩV͑T2xy͔͒, set connecting t and ed-orbitals. Ϯ 2 where ␻ ϭϪ1/2Ϯiͱ3/2. The energy levels of the (t1)2T multiplet subject to a 2 2 The matrix of the Zeeman Hamiltonian ͑Eq. 7͒ with the Jahn–Teller T e coupling and small have been discussed 2 ␷ trigonal basis ͓Eq. ͑5͔͒ is given in Table I. The explicit forms in detail by several authors and will not be reproduced of the g values in terms of the effective first and second here.11,21,22 In this paper we will examine more closely the g order g values in Eq. ͑7͒ are given in Table II. The latter can values of the lowest set of vibronic levels. be expressed in terms of the basic one-electron ligand field For S6 site symmetry, we use the complex trigonal basis: 1 ␾1Ϯ͑2⌫6͒ϭ͉Ϯ 2XϮ͘, 2 a 1 1 TABLE II. Second order g values for a T2 multiplet in trigonal symmetry. ␾2Ϯ͑⌫4⌫5͒ϭcos ␪͉Ϯ 2X0͘ϩsin ␪͉ϯ 2XϮ͘, ͑5͒

1 1 gz1 g1Ϫ2KzϪ1/(3ͱ3)g3 ␾3Ϯ͑⌫4⌫5͒ϭsin ␪͉Ϯ 2X0͘Ϫcos ␪͉ϯ 2XϮ͘, gЈ 1 ͱ6 z1 Ϫ g ϩ g where XϮ and X0 stand for the degenerate and nondegenerate 3 2 9 3 orbital components of the t2 orbitals, respectively. If ␷ has a gz2 ͱ3 ͱ3 c2͑␪͒ g ϩ2 g ϩs2͑␪͒ Ϫg Ϫ2K ϩ g positive sign then the energy order of the three Kramers dou- ͩ 1 9 3ͪ ͩ 1 z 9 3ͪ blets is E1ϽE2ϽE3 for both the static ligand ﬁeld limit and 1 ) ϩs͑2␪͒ Ϫ 3g2Ϫ g3 in the presence of linear T2 e coupling. The angle ␪ is ͩ 18 ͪ given by ͱ3 1 ͱ3 gz23 s͑2␪͒ g ϩK ϩ g ϩc͑2␪͒ g ϩ g ͩ 1 z 18 3ͪ ͩ3 2 18 3ͪ 1 tan 2␪ϭͱ2␭x /͑ 2␭zϪ⌬t͒, ͑6͒ 1 gxЈ12 c(␪)(1/(3ͱ2)g2Ϫ1/(3ͱ3)g3)ϩs(␪)( 6g2Ϫ1/(3ͱ6)g3) which is independent of the Jahn–Teller coupling within 1 gxЉ12 c(␪)(Ϫ 6g2Ϫ1/(6ͱ3)g3Ϫͱ2Kx)ϩs(␪)(g1ϩ1/(6ͱ3)g3) 23 Ham’s perturbation treatment of the T2 e model. The ac- 2 2 gx2 c (␪)(g1Ϫ1/(3ͱ3)g3)ϩs (␪)(Ϫ1/(3ͱ2)g2Ϫ2/(3ͱ3)g3) curacy of Eq. ͑6͒ should deteriorate as ␷ becomes large be- 1 ͱ6 cause, as we will show, the transition energies E31ϭE3ϪE1 ϩs͑2␪͒ Ϫͱ2K ϩ g ϩ g ͩ x 6 2 36 3ͪ and E21ϭE2ϪE1 and the g values obtained from an exact calculation progressively deviate from Ham’s model. gx23 1 ͱ2 ͱ3 1 ͱ6 2 s͑2␪͒ g1ϩ g2ϩ g3 Ϫc͑2␪͒ Ϫͱ2Kxϩ g2ϩ g3 The second order Zeeman Hamiltonian for a T2 multip- ͩ 2 12 18 ͪ ͩ 6 36 ͪ let in axial symmetry may be written9

a 2 2 H͑B͒ϭ␮B͕g1S–Bϩg2͓S–V͑E͒–B͔ c͑␪͒ϭcos͑␪͒, s͑␪͒ϭsin͑␪͒, c ͑␪͒ϭcos ͑␪͒, and so on. The expressions for g and g are obtained from g and g by replacing c͑␪͒ by s͑␪͒ and – – – – ␣3 x13 ␣2 x12 ϩg3͓S V͑T2͒ B͔ϩK L B͖, ͑7͒ s͑␪͒ by Ϫc͑␪͒. For all cases gxϭgy .

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FIG. 2. Comparison between the transition energies E21 and E31 obtained FIG. 3. Comparison between the g values of the three lowest vibronic levels from exact diagonalization of T2 e coupling ͑solid lines͒ and the corre- for T2 e coupling which were obtained by exact diagonalization ͑solid sponding energies obtained from Hams perturbation model ͑dotted lines͒. lines͒ and from perturbation formulae ͑dotted lines͒ given in Eq. ͑10͒ and ͓See Eqs. ͑12a͒ and ͑12b͒ in Ref. 11.͔ The parameters used were ⌬ϭ19 500 Table II. The parameters gz2 and gx2 have negative signs. The parameters Ϫ1 Ϫ1 Ϫ1 Ϫ1 cm , ␷Јϭ0, ␨ϭ␨Јϭ120 cm , kϭkЈϭ0.8, ប␻ϭ450 cm , EJTϭ336 cm . gz2, gz3 and gx2 change sign for small values of ␷ but this behavior was omitted for clarity. The ligand ﬁeld parameters are the same as in Fig. 1.

parameters. Stevens has gone further and determined the explicit forms in the presence of T2 e coupling, but for small ␷. If we transform his results into our parametric scheme ͓Eq. III. THE gЌ VALUES FOR ␾1(2⌫6) IN TRIGONAL ͑7͔͒ then we obtain, SYMMETRY

2 g1ϭ2Ϫ8kЈ␨Ј/͑3⌬͒Ϫ4k␨ f b /͑3ប␻͒, If ␷ has a positive sign then the lowest level of the T2 multiplet will always be ␾1͑2⌫6͒ with gЌ͑2⌫6͒ϭ0, in the g2ϭϪ8ͱ2kЈ␨Ј/⌬ϩ2ͱ2k␨fb /ប␻, presence or absence of Jahn–Teller coupling. However, it is possible to generate a nonvanishing gЌ in the limiting case g3ϭ4ͱ3kЈ␨Ј␥/⌬ϩ2ͱ3k␨fa /ប␻, ͑10͒ where the energy gap, E21 , is very small and comparable to Kzϭk␥Ϫ2ͱ2␷ЈkЈ␥/⌬ϩ2␷kfa /͑3ប␻͒, the Zeeman energy in an EPR experiment. This limit can be achieved if the Jahn–Teller coupling is sufficiently large. Kxϭk␥ϩͱ2␷ЈkЈ␥/⌬Ϫ␷kfa /͑3ប␻͒, We have investigated this problem for the case of Ti3ϩ:Al O by varying the vibronic coupling while keeping where ␥,f , f are the standard Ham reduction factors11,12 for 2 3 a b all other parameters constant. Increasing the vibronic cou- Jahn–Teller coupling to a single mode of e symmetry and frequency ប␻. Equation ͑10͒ reduces to the static ligand field pling constant, A1 , is equivalent to increasing the Jahn– Teller stabilization energy through the relation model by setting ␥ϭ1, f aϭ0, f bϭ0. 1/2 Figure 2 compares the transition energies obtained from A1ϭ͓EJTប␻͔ . We found that even if EJT was raised to 600 Ϫ1 Ϫ1 the Ham model and from the exact calculation as ␷ increases. cm the calculated energy gap, E21ϭ3.2 cm , was still too large to generate any significant gЌ in the ground state. Similarly, the approximate gʈ and gЌ values obtained from Table II and Eq. ͑10͒ are compared with the exact values in The magnitude of EJT is restricted by the weak antibond- 24 Fig. 3. ing energy of the t2 orbitals and the typical values of EJT 3ϩ Ϫ1 The agreement between the two calculations is almost that have been used for the Ti ion are EJTϭ200 cm and Ϫ1 Ϫ1 3ϩ 11 Ϫ1 exact for ␷ in the range 0 to 400 cm and the differences ប␻ϭ200 cm for Ti :Al2O3 , and EJTϭ320 cm and Ϫ1 Ϫ1 3ϩ 14 remain small for ␷ approaching 800 cm . Beyond ␷ϭ1000 ប␻ϭ258 cm for Ti :LiNbO3 . Similar results are ex- Ϫ1 cm the errors in the approximate gz1 and E21 values in- pected for CsTiS alum and we conclude that for any realistic Ϫ1 crease almost linearly with ␷. It is therefore not surprising value of Jahn–Teller coupling, EJTϽ600 cm , it is impos- 3ϩ that the Ham model works very well for Ti :Al2O3 where sible to account for the observed gЌϭ1.14 if the symmetry is ␷ϭ700 cmϪ1. On the other hand, for ␤ alums where ␷Ͼ1500 trigonal and ␷ is positive. Therefore, the trigonal symmetry Ϫ1 3ϩ cm it is essential to diagonalize the T2 e coupling exactly of the Ti͑H2O͒6 ions must be removed in the low tempera- in order to obtain accurate values for E21 and gz1. ture form of CsTiS.

FIG. 4. The dependence of the transition energies E31 and E21 on distortions FIG. 5. The effect of distortions on the g values of the ground Kramers with symmetry Eu, Ev, T2yz, T2xz, and T2xy, abbreviated as u, v, ␰, ␩ and doublet ␾1͑2⌫6͒. The parameters are the same as in Fig. 1 with ␷ ϭ1800 ␨, respectively. The parameters are the same as in Fig. 1 with ␷ ϭ1800 cmϪ1. cmϪ1.

Ϫ1 E31ϭ273 cm . The transition energies increase slowly with 2 IV. T2 COUPLING AND LOW SYMMETRY increasing strain. In this work we will examine more closely POTENTIALS the effect of strain on the g values, illustrated in Figs. 5 and 6. The direct product t2ϫt2ϭa1ϩeϩ[t1]ϩt2 shows that 2 Figure 5 shows that a very small strain of any symmetry all the low symmetry distortions of the T2 multiplet can be induces into the ground state level a gЌ that varies linearly described by five real cubic tensors V(⌫M) ͓see Eq. ͑8͔͒. with ⌬(⌫M) and is described by the second order perturba- The matrices of V(⌫M) with the complex trigonal basis tion given in Eq. ͑5͒ were incorporated into the dynamic Jahn– Teller calculation. The effect of the low symmetry distor- gЌ͑␾1͒Х2͓⌬͑⌫M ͒/E21͔͗␾1ϩx͉V͑⌫M ͉͒␾2Ϫ͘ tions on the transition energies and the g values of the lowest ϫ͗␾ ͉xЉ ͉␾ ͘. ͑11͒ set of vibronic levels are displayed in Figs. 4–6, where 2Ϫ 12 1Ϫ ⌬(⌫M) is the coefficient of the operator V(⌫M) and gives a As ⌬(⌫M) becomes comparable to E21 , Eq. ͑11͒ is no measure of the one-electron splitting of the t2 orbitals. For longer accurate. For the case ⌫MϭEu and T2xy the varia- example, ⌬(Eu)ϭ(Ϫͱ6/3)⌬(t2), where ⌬(t2)ϭE(t2xy) tion of gЌ͑␾1͒ can be very accurately described by a 2ϫ2 ϪE(t2xz,yz) is the standard one-electron tetragonal matrix involving ␾1 and ␾2 levels. This perturbation model is splitting.25 less accurate for strains of other symmetry where evidently The T2 e calculations were made with the following the strain coupling to ␾3 is larger. The basic mechanism for Ϫ1 Ϫ1 ligand field parameters: ⌬ϭ19 500 cm , ␨ϭ120 cm , ␷ inducing gЌ͑␾1͒ remains essentially the same for all cases ϭ1800 cmϪ1, ␷Јϭ250 cmϪ1, kϭ0.75, kЈϭk and ␨Јϭ␨. The and we consider in more detail the simple case of strain with ␷Ј parameter was obtained by fitting the ligand field model to Eu symmetry. the zero field splitting of the ground state of CsCrS ͑see Ref. The invariance of g͑␾3͒ values to ⌬(Eu) ͑Fig. 6͒ indi- 26͒ and ␷ was deduced from the effective trigonal field ob- cates that the strain mixing of ␾3 with ␾1 and ␾2 is very served in the electronic Raman of CsVS ͑see Ref. 16͒ where small. Consider the 2ϫ2 matrix for the ␾1Ϯ and ␾2ϯ basis Ϫ1 ␷ϩ2/3␷Јϳ2000 cm . The active Jahn–Teller mode was as- with diagonal energies 0 and E21 , respectively. By applying sumed to be the skeletal Q(E) vibration with an energy of this model to the numerical output we deduce, gxЉ12ϭ1.951 Ϫ1 450 cm ͑see Refs. 27 and 28͒ and EJT was varied to give a and ͗␾1Ϯ͉V(Eu)͉␾2ϯ͘ϭ0.35 in units of ⌬(Eu).

rough agreement with the gʈ value for CsTiS and was ﬁnally The calculated g values in strict trigonal symmetry are Ϫ1 ﬁxed at EJTϭ336 cm . gz1ϭ1.108, gz2ϭϪ2.505, gz3ϭ1.665, gx2ϭϪ0.057, gx3 2 Figure 4 shows that the T2 multiplet with T2 e cou- ϭ1.828. These can be compared with the approximate values pling and large positive ␷ is characterized by a low lying obtained from Eq. ͑10͒ and Table II by taking ␪ϭ87.2° ͓Eq. Ϫ1 excited state at E21ϭ56 cm and a higher lying state at ͑6͔͒: gz1ϭ0.71, gzЈ1ϭϪ0.04, gz2ϭϪ2.96, gz3ϭ1.90, gx2

J. Chem. Phys., Vol. 106, No. 5, 1 February 1997 Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 130.102.82.2 On: Tue, 18 Oct 2016 06:21:12 1674 L. Dubicki and M. J. Riley: Paramagnetism of caesium titanium alum

2 2 ¯gz1ϭc2gz1Ϫc1gz2 , 2 2 ¯gz2ϭc1gz1Ϫc2gz2 , 2 ¯gx1ϭϪ2c1c2gx12ϩc1gx2 , 2 ͑13͒ ¯gy1ϭϪ2c1c2gx12Ϫc1gx2 , 2 ¯gx2ϭ2c1c2gx12ϩc2gx2 , 2 ¯gy2ϭ2c1c2gx12Ϫc2gx2 ,

where gxiϭgyi and gx12 is to be identified with gxЉ12 . Actu- ally, gx12 includes the small contribution of gxЈ12 because the full 4ϫ4 matrix for the ␾1 and ␾2 subspace as given in Table I can always be transformed into 2ϫ2 matrices. We now see that the anisotropy in the induced gЌ of the 2 2 ground state is 2c1gx2 and is very small because c1 is very small. The condition for inducing an apparent axial gЌ into a Kramers doublet is that gЌϳ0 in exact axial symmetry and that the doublet is coupled to another Kramers doublet by both strain and by an axial off-diagonal Zeeman term. The ␾2 level has gЌϭgx2 in exact trigonal symmetry. In 2 this case, the anisotropy in the apparent gЌ͑␾2͒ is 2c2gx2 that 2 FIG. 6. The effect of V(Eu) distortion on the g values of the three lowest is approximately constant since c2 varies from 1.00 to 0.811 Ϫ1 vibronic states in the T2 e model. The parameters are the same as in Fig. 1 as ⌬(Eu) varies from 0 to 100 cm ͑Fig. 6͒. with ␷ϭ1800 cmϪ1. V. SOME APPLICATIONS Figure 5 shows that g ͑␾ ͒ is a sensitive detector of ϭϪ0.10, gx3ϭ1.84 and gxЉ12ϭ1.86. Furthermore, Eq. ͑10͒ Ќ 1 strain of any symmetry. The previous analysis ͑Sec. IV͒ can and Table II give gxЈ12ϭ0.02, gxЈ13ϭ0.03, gxЉ13ϭϪ0.26, gx23 be applied with very little modification to all (t1) 2T mul- ϭ0.02 and gz23ϭ0.15. The second order equations evidently 2 2 give the correct signs and even useful estimates of the g tiplets that have a positive axial field, ␷ or ⌬͑t2͒,ofany values. Clearly, the most important Zeeman matrix element magnitude. Table III lists a number of examples which dis- play a range of low symmetry distortions. is the off-diagonal gxЉ12 which is always large because it con- The nonvanishing g for Ti3ϩ:Al O has already been tains a large contribution from g1 that is not quenched by the Ќ 2 3 Jahn–Teller coupling. attributed to lattice strains but a detailed investigation of a 21 If the low symmetry perturbation is diagonalised with microscopic mechanism was not pursued. If we represent ¯ ¯ the distortion by V(Eu) then the observed g ϳ0.14 requires the basis ␾1Ϯ and ␾2Ϯ Ќ ⌬(Eu)ϳ4cmϪ1. ¯ ␾1Ϯϭc2␾1ϮϪc1␾2ϯ , The g values of CsTiS can be fitted with ⌬(Eu)ϳ56 ͑12͒ cmϪ1. In Fig. 6 the curves for g ͑␾ ͒ and g ͑␾ ͒ cross at ␾¯ ϭc ␾ ϩc ␾ , z 1 Ќ 1 2Ϯ 1 1Ϯ 2 2ϯ ⌬(Eu)ϳ70 cmϪ1 where the g values are predicted to have where c2Ͼc1 and c2 is given a positive sign then for the two the value 1.31, in good agreement with the isotropic value lowest levels we obtain observed for deuterated29 CsTiS ͑Table III͒.

2 a TABLE III. Strain induced gЌ for the T2 ͑␾1͒ Kramers doublet with positive ␷.

Experimental Calculated

b b gʈ gЌ Ref. ⌬(Eu) gʈ

3ϩ Ti :Al2O3 1.07 ϳ0.14 21 4.1 1.07 3ϩ • Ti :CsAl͑SO4͒2 12H2O 1.17 0.23 33 10 1.11 1.19 0.70 30 1.16 1.24 0.93 43 1.20 • CsTi͑SO4͒2 12H2O 1.25 1.14 3 56 1.25 • CsTi͑SO4͒2 12D2O 1.31 1.31 29 70 1.31 3ϩ • Ti :NH3CH3Al͑SO4͒2 12H2O 1.40 1.61 32 110 1.44

a 3 3 ␾1 transforms as M JϭϮ 2 for Jϭ 2. b Ϫ1 The observed gЌ was used to estimate ⌬(Eu)incm ͑Fig. 5͒, which was then used to predict gʈ , given in the 3ϩ last column. Separate calculations were made for Ti :Al2O3 .

NH3CH3AlS is a ␤ alum that has a ferroelectric transi- prising that the ⌬(Eu) parameters in Table III are very tion at 177 K where the cubic space group changes to Pca21 . small. Furthermore, Fig. 4 suggests that the dependence of Low symmetry ligand fields are indicated by the EPR of gx,y͑␾1͒ on T2yz, T2xz, T2xy and hence on T2xϮ potentials 3ϩ 31 Cr :NH3CH3AlS, measured below 177 K. All ␤ alums ͓Eq. ͑9͔͒, may be represented empirically by an effective or have large and positive ␷. Accordingly, the previous average operator V(Eu). analysis23,32 of the EPR and spin-lattice relaxation rates of 3ϩ Ti :NH3CH3AlS have to be corrected. VI. CONCLUSIONS The g values of Ti3ϩ:NH CH AlS measured below 5 K 3 3 The low temperature paramagnetic properties of CsTiS ͑Table III͒ can be fitted to an arbitrary V(Eu) strain. The Ϫ1 are largely determined by the two lowest vibronic levels that observed gЌϭ1.61 requires ⌬(Eu)ϳ110 cm and gʈ is predicted to be 1.44, in fair agreement with the observed are coupled by a large trigonal Zeeman term that operates in the direction perpendicular to the molecular trigonal axis. gʈϭ1.40. 3ϩ This off-diagonal Zeeman term should make the spin-lattice Two features in the earlier work on Ti :NH3CH3AlS may be important. Firstly, the spin-relaxation time has a relaxation time sharply dependent on the direction of the large dependence on the direction of the applied magnetic applied magnetic field and can be easily induced into both field.23 The relaxation time is a maximum when B lies along the ground and excited vibronic levels by small low- symmetry distortions. the ͓111͔ direction and drops sharply as B deviates away from the ͓111͔ direction. In contrast to the earlier analysis, we suggest that the orbit-lattice interaction and the off- 1 J. Sygusch, Acta Crystallogr. B30, 662 ͑1974͒. 2 D. Bijl, Proc. Phys. Soc. London A63, 405 ͑1950͒. diagonal Zeeman term ͑x12Љ in Table I͒ provide the most ef- 3 B. Bleaney et al., Proc. Phys. Soc. London A68,57͑1955͒. ficient second-order amplitude for the angular dependence of 4 A. Bose, A. S. Chakravarty, and R. Chatterjee, Proc. R. Soc. London Ser. the spin-lattice relaxation time. A 255, 145 ͑1960͒. 5 A. Manoogian, Can. J. Phys. 48, 2577 ͑1970͒. This suggestion lends support to the earlier 6 33 3ϩ G. F. Dionne, Can. J. Phys. 50, 2232 ͑1972͒. interpretation of the EPR of Ti :CsAlS. In this case the 7 Y. H. Shing and D. Walsh, Phys. Rev. Lett. 33, 1067 ͑1974͒. EPR shows fine structure spread over ϳ300 gauss and has 8 G. F. Dionne, Can. J. Phys. 42, 2419 ͑1964͒. been interpreted to consist of at least three distinct chemical 9 S. Sugano, Y. Tanabe, and H. Kamimura, Multiplets of Transition Metal species with apparent axial symmetry. ‘‘The intensity of the Ions in Crystals ͑Academic, New York, 1970͒. 10 G. F. Koster et al., Properties of the Thirty-two Point Groups ͑MIT, Cam- lines decreased very rapidly as the magnetic field diverges bridge, MA, 1963͒. 33 from the direction of the cube diagonal.’’ If we use the 11 R. M. Macfarlane, J. Y. Wong, and M. D. Sturge, Phys. Rev. 166, 250 ͑1968͒. reported gЌ to fix the value of ⌬(Eu) then the predicted gʈ 12 F. S. Ham, Phys. Rev. 138, A1727 ͑1965͒. will correspond closely to the empirical gʈ values ͑Table III͒. 13 3ϩ R. Ameis, S. Kremerand, and D. Reinen, Inorg. Chem. 24, 2751 ͑1985͒. The second feature of the EPR of Ti :NH3CH3AlS is 14 O. Thiemann et al., Phys. Rev. B 49, 5845 ͑1994͒. 32 that the reported line width of ϳ20 gauss is much smaller 15 C. Daul and A. Goursot, Inorg. Chem. 24, 3554 ͑1985͒. than the value of 250 gauss observed for CsTiS. The latter 16 S. P. Best and R. J. H. Clark, Chem. Phys. Lett. 122, 401 ͑1985͒. 17 M. Brorson and M. Gajhede, Inorg. Chem. 26, 2109 ͑1987͒. value is much larger than that expected for pure magnetic 18 3 S. P. Best and J. B. Forsyth, J. Chem. Soc. Dalton Trans. 3507 ͑1990͒. dipole interaction. It seems possible that just as for 19 S. P. Best and J. B. Forsyth, J. Chem. Soc. Dalton Trans. 1721 ͑1991͒. Ti3ϩ:CsAlS, the EPR of CsTiS may consist of several unre- 20 S. P. Best, J. B. Forsyth, and P. L. Tregenna-Piggott, J. Chem. Soc. Dalton solved chemical species with different low symmetry fields. Trans. 2711 ͑1993͒. 21 C. A. Bates and J. P. Bentley, J. Phys. C 2, 1947 ͑1969͒. Such details will require precise information on the crystal 22 M. Abou-Ghantous, C. A. Bates, and K. W. H. Stevens, J. Phys. C 7, 325 structure at low temperatures. ͑1974͒. In the analysis of the data in Table III we have used an 23 Y. H. Shing, C. Vincent, and D. Walsh, Phys. Rev. Lett. 31, 1036 ͑1973͒. 24 M. Bacci, Chem. Phys. 40, 237 ͑1979͒. arbitrary distortion of Eu symmetry. In all cases the ground 25 2 2 J. Glerup and C. E. Schaeffer, in Progress in Coordination Chemistry, state is T2 ( E) and is subject to a Jahn–Teller distortion. edited by M. Cais ͑Elsevier, Amsterdam, 1968͒, p. 500. The direct product of the trigonal species E requires the dis- 26 A. G. Danilov, J. C. Vial, and A. Manoogian, Phys. Rev. B 8, 3124 torting potentials to be of Eu and T x symmetry in the ͑1973͒. Ϯ 2 Ϯ 27 parent cubic symmetry ͓Eq. ͑7͔͒. The large trigonal field ␷ S. P. Best, R. S. Armstrong, and J. K. Beattie, J. Chem. Soc. Dalton Trans. 1 2 1655 ͑1982͒. stabilises the (t2) E state and it is unlikely that the distor- 28 S. P. Best, J. K. Beattie, and R. S. Armstrong, J. Chem. Soc. Dalton Trans. tions will be of EuϮ type, which involve compression and 2611 ͑1984͒. elongation of the strong sigma M–O bonds and tend to de- 29 B. V. Harrowfield, Phys. Abstr. 75, 594 ͑1972͒. 30 stroy the ␷ trigonal field. Rather, the distortions should be of R. O. W. Fletcher and H. Steeple, Acta Crystallogr. 17, 290 ͑1964͒. 31 D. W. O’Reilly and Tung Tsang, Phys. Rev. 157, 417 ͑1967͒. T2xϮ symmetry, which involve the lower-energy bending or 32 Y. H. Shing, C. Vincent, and D. Walsh, Phys. Rev. B 9, 340 ͑1974͒. angular distortions of M–O bonds. Therefore, it is not sur- 33 G. A. Wooton and J. A. MacKinnon, Can. J. Phys. 46,59͑1968͒.