THE UN I V E R S I T Y OF MICHIGAN COLLEGE OF ENGINEERING Department of Nuclear Engineering Technical Report THE HYDROSTATIC PRESSURE EFFECT ON THE EPR SPECTRA OF Cr3+ AND V2+ IN SAPPHIRE Alan F. Clark Richard f. Sands Chihiro Kikuchi ORA Project 06029 under contract with: U S. ARMY MATERIEL COMMAND HARRY DIAMOND LABORATORIES CONTRACT NO. DA-49-186-AMC-80(x) WASHINGTON, D C. administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR May 1964

This report was also a dissertation submitted by the first author in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The University of Michigan, 1964.

ERRATA Page 3 line 3o lattice sites, though possibly including the displacements proposed in the literature, and are subject.... 4 line 25: spectrum of ruby. (also p. 5, line 1). 11 line 9: coulomb interactions 12 line 10: This type of analysis 18 Figure 1 upper 4F state is 4T1 28 Figure 3: Exchange coordinates x and y (Not E or H). 29 line 16: in equation 8 (also equation 12 is 129)o 45 line 20: three. The use of.00.0 48 line 16: is valid if. 54 line 8: to Stevens57...: 59 line 15: three hours at 7x106 rads/min.38 80 line 18: signal to noise' ratio, 85 line 21: misalignment-., 107 last two equationso k2 r2 d(kr) 109 line 18o the cubic field 116 line 12: are E, C3, and C2 3~ 124 line 8: where P20 - ~(3 cos2 1, - ). 145 figure: delete (D3) 155 equation F8o AV =r E 4A2(F) -4T1(F,P).0.0 160 last equation (Y3 (Y3....

TABLE OF CONTENTS Page List of Tables v List of Figures vi Abstract viii Chapter I. Introduction 1 A. Motivation, Objectives, and Results 1 Bo Literature Review 3 II, Theory 11 A. The Hamiltonian, Energy Levels, and Transitions 11 B. Sapphire Crystal Structure 33 C, The Elastic Constants for Sapphire 38 D. The Crystalline Field Model 44 E. Compressional Model and Predicted Behavior 46 IIIo Experiment 58 A o Crystals and Crystal Orientation 58 Bo Experimental Apparatus 61 C, The Two Sample Method 69 Do Experimental Procedure 74 IV, Results 78 A. Cr3+ in Sapphire 78 B V2+ in Sapphire 89 C, Summary 94 V, Conclusions 95 A. Direct Conclusions 95 Bo Analytical Conclusions 101 Co Summary 110 D. Further Studies 111 Appendices Ao. Crystalline Field for Sapphire 114 B, Change in Crystalline Field with Fressure 121

Page Appendices (contrd) C-. The Elastic Constants 126 D. Data Reduction 144 E-. Equipment List 152 F. Racah Parameters and Symmetry Tables 153 G. Matrix Elements for Axial and Zero Field Splittings and g Shifts 159 References 164

LI ST OF TABLES Page Table I. Location of the neighbor and nextnearest-neighbor ions with respect to a substitutional impurity in oc A1203 37 II. The elastic stiffness constants, cij, and compliance constants, sij,j for sapphire at room temperature. 42 III. Summary of the theoretical calculations of the change with pressure in the crystalline field parameter as a function of displacement along the trigonal axis in A1203. 52 IV. Calibration of absorption frequency meter HP X532A No. 15030 with atransfer oscillator measurement. 77 V. Comparison of crystal field parameters for V2+, Cr3+, and Mn4+ in sapphire 97

LIST OF FIGURES Page Figure 1. Energy levels of a 3d~ 4F-state ion versus cubit crystalline field 18 2. Energy levels of ruby with cubic field 20 3. Energy levels of ruby ground state versus applied magnetic field 28 4. EPR resonances in ruby for x band as a function of applied magnetic field and the field s angle to the crystalline axis. 32 5. Octahedral coordination in ruby 35 6. Crystal coordination for aluminum site in sapphire 36 7. Projection of oxygens in A1203 onto a plane perpendicular to the trigonal axis 36 8. Relative positions of the nearest and next-nearest-neighbor ions in sapphire 53 9. Crystalline field splitting of F ground state for a 3d3 ion 55 10. Spectrometer schematic 62 11. Pressure bomb and microwave cavity 63 12. High pressure system schematic 64 13. Line center shift due to overlapping lines 72 14. Bomb, coils, and flux lines schematic 72 15. Sketch of pressure effect on Mn2+ in MgO lines 75 16. Zero field splitting versus pressure for Cr3+ in A1203 81 vi

Figures (cont d) Page 17. g value versus pressure for Cr3+ in A1203 84 18. Zero field splitting versus pressure for V2+ in A1203 92 Cl Stress components 128 C2 One and two dimensional strain 131 C3 Examples of specialized strains 133 vii

Abstract The interaction of the Cr3+ and V2+ impurity ions with the c.-A120~ crystal lattice was studied by observing the hydrostatic ressure dependence of their electron paramagnetic resonance spectra to 6000 atmospheres. A two sample comparison method was used in which two identical samples are placed in the X band microwave resonant cavity. One end of the cavity and one sample are subject to hydrostatic pressure. The changes in the spin Hamiltonian parameters are obtained by measuring the difference in the magnetic field required for the same resonant condition in each sample as a function of pressure. The observed changes were: r3+o A ( 8 0 0 x / D-/?m J.'_'Y -3/ V2+o -- (, 77 ( + 71) x 10-/ k a., where D is the coefficient of the crystalline field term in the spin Hamiltonian which behaves under rotation like the spherical harmonic Y and (g-gO)z is the difference in the g value parallel to e crystalline axis and the free ion g value, From this it was concluded that the ions occupy similar lattice sites, though possibly including the displacements proposed in the literature, and are subject to the same compressibility, comparable to that of the bulk compressibilityo Comparison with the point charge model led to the conclusion that the point charge model was inadequate to predict the observed pressure changes. A molecular orbital calculation as a function of surrounding ion position and further experiments are suggested. viii

Chapter I INTRODUCTION A. MOTIVATION, OBJECTIVES, AND RESULTS Interest in solid-state systems has become widespread in recent years, particularly with the use of doped semiconductors as transistors and paramagnetically doped crystals for MASERS and LASERS. Electron paramagnetic resonance (EPR) utilizes a paramagnetic impurity ion in a host lattice as a means of studying both the ion itself and its interaction with the surrounding lattice. The iron group ions are particularly advantageous with their partially filled 3d electron shells and resulting net magnetic moment. Microwaves (ca. 3 cm. wavelength) incident upon the crystal induce magnetic dipole transitions among the magnetic sublevels of the ground state in a strong magnetic field (ca. 3000 gauss). The crystalline field at the impurity ion site due to the surrounding ions or ligands also perturbs the impurity ion states. From the resonance absorption of microwave power for transitions between various energy levels, electronic and nuclear properties of the impurity ion in its host lattice can be deduced. For texts on EPR one may consult Low,l Pake,2 Ingram,3 4 and Shlichter,5 as well as the reviews by Bleaney and Stevens,6 Bowers and Owen,7 and Jarrett.8 -1

The mechanism of the interaction between the crystalline field and the impurity ion needs further study. In order to study the interaction, the usual method is to observe the changes in spectra between similar or isoelectronic 9,10,11 ions in the same lattice 9' or the same ion in a series of lattices with similar symmetry12'13 Hydrostatic pressure (ca, 10,000 atmospheres) provides a means of linearly changing the crystalline field at the impurity ion site by compressing the host lattice, Thus it is a powerful means of studying the interaction between the two, It is the object of this study to determine the hydrostatic pressure dependencies of the EPR spectra of the triply ionized chromium ion (Cr3 ) and the doubly ionized vanadium ion (V2+), which are isoelectronic, in the c<-A1203, or sapphire, lattice, The results are interpreted utilizing the crystalline field model in the hope of further understanding the behavior of zero field splitting with a linear lattice change for isoelectronic ions, Additional objectives are to check on the substitutional positions of the impurity ions in the lattice and to determine the usefulness of the crystalline field model, This study is also another step in a thorough examination of the solid state chemistry of vanadium, It is found that the zero field splitting for both Cr3 and V2 in sapphire varies linearly with pressure to 6000

-3atmospheres, and the relative change for both ions is nearly the same. From this it is concluded that the ions occupy similar lattice sites and are subject to similar local compressibilities, comparable to that of the bulk compressibility. It is also concluded that the point charge model is inadequate to predict the observed pressure changes, and a molecular orbital calculation and further experiments are suggested. B. LITERATURE REVIEW Review of Previous High Pressure Resonance Experiments. An S-state ion, such as Mn2+ whose L = 0O should have no low order mechanism to cause zero field fine structure splitting. The observed splittingsl4 aroused a great deal of curiosity about the effect of the crystalline field as a 15,16,17 perturbation on the energy levels of an impurity ion. Among the resulting papers, Watanabe suggested that the nonzero contributions due to a crystalline field of cubic symmetry must be from matrix elements containing only even powers of the crystalline field.l8 He also suggested that this could be checked by a hydrostatic pressure experiment on the EPR of such a material. Walsh performed such an experiment and found a quartic dependence on the cubic field for both Mn2 and Fe in MgO20 assuming that local and bulk compressibilities are equal, A paper by Powell, Gabriel,

and Johnston then suggested a mechanism for ground state 21 splitting due to odd powers of the crystalline field. In another hydrostatic pressure EPR experiment, Wait found a square dependence on the cubic crystalline field for Mn2+ in calcite and a first power dependence on the axial crystalline field.22 He also pointed out that according to the theory of Powell et al. Mn2+ and Fe should not necessarily have the same pressure dependence in Mg0,21 and suggested further experiments on Fe3+ in calcite as well as studies in other host lattices, All the calcite samples which could be obtained for study contained too low a concentration of iron to allow a determination of the pressure dependence. Other hydrostatic EPR experiments have been done recently by Rimai, Deutsch and Silverman23 on Fe 3, Cr3+,, and Eu in cubic SrTiO3, They conclude a local compressibility twice that of the bulk compressibility about the impurity sites in SrTiO3, Blum and Benedek24 have also reported their results for the pressure dependence of the hydrogen atomus hyperfine interactions in the CaF2 latticeo Recently several uniaxial pressure experiments have been done on the EPR and nuclear magnetic resonance (NMR) spectra of ruby but are mostly yet unpublished, Viegele and Stettler25 have studied the effects of uniaxial stress and temperature on the NMR spectra of ruby~ Terhune26

and Donoho27 have studied the EPR spectra of ruby under uniaxial stress, and their results are correlated in Chapter V with those obtained in this study. Tucker28 has performed a spin-phonon interaction experiment on ruby whose results can also be correlated as a uniaxial pressure experiment as will be shown later. A uniaxial pressure experiment on the EPR of Nd3+ and U3+ in CaF has also been reported by Black and Donoho.29 An optical resonance experiment using uniaxial stress on MgO with Cr3+ was performed by Schawlow, 2iksis, and Sugano30 to try to understand the charge compensation required when Cr3+ substitutes for Mg2+. Sturge31 followed with a similar experiment with V2+ in MgO. These optical studies were used by Walsh in the interpretation of his EPR experiment. Drickamer et al. have also done optical studies using uniaxial pressures up to 150 kbar.32,33'34 They have found increases in the crystalline field parameter, Dq, which are consistent with the R=5 law and similar pressure dependencies for various ions in the same lattice. Also, they indicate small changes in the interelectronic repulsion Racah B parameter. Higbh Pressure Resonance Experiments on Cr3+ and V2+ in Sapphire. The pressure experiments which are compared analytically or discussed qualitatively with the present experiment are now briefly summarized. Two EPR experiments

-6on ruby (Cr3+ in sapphire) subject to uniaxial stress have been done but not reported in the literature. Terhune26 has communicated his results for the change in the zero field splitting, 2D (2D =-.38 cm.-1) for stress parallel ( //) and perpendicular ( 1 ) to the c axis of the crystal as follows3l)5, + 0,3 x c3/k krw. j A D) = 3.8 0.3 x /O1 k atm. He did not observe a change in the g value. Donoho27 has indicated that he also could not observe any changes in the g shift for uniaxial stress and his experimental limit yields ~- 8~) Z A third EPR experiment has been reported in progress35 but the nature of the stress and quantitative results are unavailable. The optical resonance study of ruby under pressure was done by Stephens and Drickamer.36 They report a change in the cubic field splitting, Dq (10 Dq = 18,000 cm.-1), of _ ) = 0. x 10-3/Q /

-7and a distinct increase in the trigonal splitting parameter, K(K ~= 350 cm,-l), above 60 katm. This change in Dq was shown to be equivalent to 30 katm. to that expected from a simple change in the metal-ligand distance, R, from the compressibility data and the relation /0.). _ C' -' r',,/'3 where q is the ligand charge and < r4 > is the integral of r4 over the radial 3d wave functions. From the change in Dq they determine a change in the interelectronic -1 repulsion parameter, B(B = 630 cm, ), of 3 2 = -O 3v X 10x / k,att. They also conclude that the decrease in B shows an increase in covalency and the change in K indicates the onset of distortion above 60 katm. The effect of uniaxial stress on the electric field gradient, q, at the impurity was reported by Viegele and Stettler25 from an NMR experiment in ruby. They report (~ ) +- 3,a xO -3/I cf-li. and but their analysis of these values has not been published.

One other experiment which can be correlated with this study as a pressure experiment is the spin-phonon interaction in ruby performed by Tucker.28 He analyzes the / perturbation, I, with the form where G is defined as the magneto-elastic coupling constant and e is the strain. He reports values26 of G2 -30 G a.s x/o5 ergOs - i!-3o GR ~ I dx i0 erO; for longitudinal waves parallel and perpendicular to the c axis. TO the author0s knowledge there are no pressureresonance experiments for V2+ in sapphire and no others for ruby. Studies of Cr3' and V2+ in Sapphire. A great deal of work has been published for ruby because of its wide application in recent LASER and MASER technology Summarized below are only those fundamental studies and related works which are used in the analysis of this problem. A summary of the crystal field parameters of the isoelectronic ions is given in Table V.

The EPR spectra of Cr3+ and V2 in sapphire are reported by E.O, Schulz-DuBois37 and Lambe and Kikuchi38 respectively. More recent evaluations of the spin Hamiltonian parameters are summarized by Laurence and Lambe39 for an ENDOR (electron nuclear double resonance) experiment to study the quadrupole interactions in sapphire. The electric field gradients determined in their experiment are also used in the discussion in Chapter V. The optical spectrum of ruby was studied by Sugano and Tanabe40 and interpreted with qualitative success (except for the zero field splitting) by crystal field theory. More recent discussions of the optical spectrum of ruby and similar structures and ions are by Wood, et a113 and Weakliem and McClure.41 From his studies of the optical spectrum McClure42 has suggested that the chromium ion is not substitutional for an aluminum. The optical spectrum of V2+ in sapphire is reported by Sturge43 along with the collected optical parameters for the other isoelectronic ions, Cr3+ and Mn4+ Molecular orbital theory was used by Lohr and Lipscomb44 3+ to study theoretically the spectra of Cr in various crystals, including sapphire. Using a semiempirical LCAO-MO method they qualitatively predict the observed spectra with some improvement over the crystal field theory in the value of the zero field splitting. Their theory

-10was used by Royce and Bloembergen45 to analyze electric field effects on the EPR spectrum of ruby with good success. The sapphire crystal structure was obtained from Wyckoff46 and the most recent compilation of elastic constants is by Bernstein 47 One other reference that should be mentioned is a discussion of the expansion of the impurity ion wave functions upon insertion of the ion into a lattice by Marshall and Stuart,48 since according to them, this would be expected with lattice compression. They show that the reduction of the spinorbit coupling parameter, X, of an ion upon insertion into a lattice is due to radial expansion of the electron wave functions and support their arguments with neutron scattering form factors and super-hyperfine experiments.

Chapter II THEORY Ao THE HAMILTONIAN, ENERGY LEVELS, AND TRANSITIONS A phenomemological Hamiltonian called the "spin Hamiltonian"49'50 is usually used to interpret the experimental results of current research in electron paramagnetic resonance. It arises in the following manner. A "free ion" Hamiltonian is obtained much in the manner of the helium atom51 by a summation of the electron momenta, the coulomb interactions between the nucleus and individual electrons and the coulomb interations between electrons. This Hamiltonian can be used to obtain the wave functions for the determination of the perturbation caused when this "free ion" is inserted in a crystalline field and an external magnetic field. The crystalline field is produced by the charges or dipoles from the surrounding ligands. The perturbing Hamiltonian -- due to the crystalline field, the external magnetic field, and the remaining intraionic interactions -- to which the free ion wave functions are applied becomes with simplification the spin Hamiltonian. In the free ion case the wave functions are the product or a linear combination of permutations of the product (Slater determinant) of one electron "orbitals." The equations for the one electron orbitals, Hartree or 11

-12Hartree-Fock equations respectively, are determined by applying a variational method to the Schroedinger equation of the free ion.. A more inclusive approach is to allow some "linear combination of atomic orbitals" (LCAO) with orbitals from a neighboring ligand or excited states of the impurity ion. This allows the wave functions to contain contributions of higher lying states of the ion of the proper symmetry and wave functions from neighboring ligands, as in the "molecular orbital" calculations currently in such wide use. 52'5344 This type analysis is not used here primarily because the shift in the Hamiltonian parameters is of concern rather than the exact determination. (In fact, the comparison analysis used precludes a prior knowledge of the parameters, and that their previous determination has fit the spin Hamiltonian well.) An outline of the usual derivation of the spin Hamiltonian is shown below. It is followed by the restriction of the spin Hamiltonian to a 3d3 ion in sapphire and the resulting energy levels and transitions are then derived. The General Hamiltonian. From the general Hamiltonian for a single electron moving in electric and magnetic fields obtained from Diracos theory, one can generalize further to obtain a Hamiltonian for many electrons in an atom with external perturbations by appropriate summa54 tions of interactions with the nucleus and other electrons.

-1 3= Specification of the external perturbing fields and potentials then allows for the interactions of the atomic system when placed in the crystalline environment and magnetic field, This method does not include effects of the nucleus interacting with the external fields. The interaction of the nuclear electric quadrupole moment with the crystalline electric field and the nuclear magnetic dipole interaction with the external magnetic field must be added as further perturbations. Only the electrons in the unfilled shell, such as the 3d shell for the iron group transition elements, usually are considered. Because in resonance studies only the effects of the crystalline field and the magnetic behavior of the electrons and nuclei are of primary interest this general Hamiltonian can be further simplified as will be shown. Following the above discussion one obtains the Hamiltonian shown in equation 1, where CF S LF A- e l l Ot, = i /B Pa ( So r,3

14k, # -k e -— P R ~ ( +. is.) A N PNP + 3 Ad S =aL, I -9, NH'I I3eQ =X-, +V ) a I l where Vcf is the crystalline field and the rest of the symbols are standard notation. The terms in equation 1 and the order of magnitude of the resulting energy level separations are as follows in orderg the well known Hamiltonian of a many particle system with coulomb interactions, 105 cm, l; the crystalline field term, widely variable, but in our case of iron group series 103 - 104 cm.; spin-orbit interaction 102 cm.-lo spin-spin interaction 1 cm. * the electron Zeeman interaction with external magnetic field, 1 cm.-l the interaction of the nuclear magnetic moment and the magnetic field produced by the electrons, 10 i2 cm.-l nuclear Zeeman interaction with the

external magnetic field, 10 cmo=l, and the nuclear electric quadrupole interaction with the electrons and the crystalline field, 10=4 cm.-, in which axial symmetry has been assumed. The Wave Functions. For most transition metal ions the first term, ~o, in the total Hamiltonian is used to obtain wave functions to use in determining the perturbation due to the remaining terms. The last term in 2/ the inter-electron coulomb repulsion, prevents exact solutions for the radial part of the wave functions, which are left as empirically determined parameters. Certain combinations of these radial integrals resulting from a spherical harmonically expanded interelectron potential are the empirical Racah A, B, and C parameters55 which are used experimentally later on. (See Appendix F.) The angular part of the wave functions is soluble and group theory provides a powerful means of simplifying the problem of the impurity ion in the crystal before solving by eliminating the need to consider many combinations of wave functions. This arises because the total Hamiltonian must be invariant under the coordinate transformations which preserve symmetry since a physically observable quantity must not depend upon the choice of coordinates, For a free ion any inversion or infinitesimal rotation or combination preserves the symmetry, but in a crystal it

16 is only that group of symmetry operations that preserve crystal symmetry about the impurity ion, Thus simplification arises if one uses those combinations of wave functions which transform among themselves according to the irreducible representations of the symmetry group of the impurity ion site. Other representations are just combinations of the irreducible representations. The transformation properties of these irreducible representations are represented by a matrix for each of the symmetry operations. Since only the trace, or character, is used in applications, these properties are given in a character table. (For example, see Heine.56) These group theory arguments can still be used when considering only part of the total Hamiltonian because an energy perturbation is also a physically observable quantity. In these cases the product of representations in the matrix element evaluation of 2pert. ( Ypert. also transforms as one of the irreducible representations) must contain the identity representation or the energy perturbation would not be invariant under a symmetry preserving transformation of coordinates and the matrix element would have to be zero. Examples will be shown as they arise and the character tables of the cubic Oh group and the trigonal C3 group are shown in Appendix F along with the Oh irreducible representation product table.

17 If a simple ionic model of the lattice is assumed, the wave functions, f, are made up of products of one electron impurity ion orbitals,. If the covalent nature of the lattice is to be accounted for, the orbital of an unpaired electron must contain some of the ligand orbital,, or ~, ~~=xt>.tr a~~~Pb(2) where oCtp - 2 if the overlap is small. Thus p, or the amount of mixing of the ligand orbital, is a measure of covalency. The Cubic Field. It is generally useful to decompose Vcf into a higher symmetry V1 and a lower symmetry V2 and determine the energy levels of / — eV1 with the free ion wave functions and then apply V2 plus the rest of the terms in equation 1 as a perturbation on these levels. As will b e seen later, sapphire is primarily cubic with some axial distortion so V1 and V2 become Vcub and V respectively. The result of the first part of this procedure for a 3d3 ion such as Cr3+ and V2+ is shown as a function of cubic crystalline field strength, Dq, in Figure 1.13 The free ion energy levels are indicated at Dq = 0. he notation of the levels corresponds to the transformation properties of the wave functions (i.e. the irreducible representation of the cubic group

-1825x104 2.0xlO4 1.5 /x0 2G - I 0 x 104 4p O.Sx104 4F 4 1000 2000 DO cm.l Figure 1. Energy levels of a 3d3-4F-state ion versus cubic crystalline field,

to which the wave functions belong) which result in that energy level when applied to the Hamiltonian /O -eV1 The superscript is the spin multiplicity, 2 S + 1o] The value of Dq for ruby, for example, is 1,800 cm, and the resulting energy levels are shown in Figure 2, where the usual optical transitions are indicated. As will be shown later, the additional noncubic or axial crystalline field and spin orbit coupling will cause further 4 small splitting. In the A2 ground state this becomes the "zero field splitting" of about 0038 cm-1 and for the 2E level a splitting of about 29 cm.-l results in the well known R1 and R2 red lines of ruby. The result is similar for the V2+ ion. The Spin Hamiltonian. It remains now to apply the perturbations of Vax and remaining terms of equation 1, $ s and on, to the wave functions of the 4A2 ground state. Since there is only one combination of wave functions which transforms like the A2 irreducible representation, this state is effectively an orbital singlet and thus. it has an orbital expectation value of zero. Also, since L$ is a smaller perturbation than the inter-electron repulsion term of SO an LS type coupling scheme is applicable and Ml and Ms are good quantum numbers. Assuming constant total L f and S = S, Stevens57 has shown that L 4 1 rnn I eA ytb the position coordinates can be replaced by the components of L and the individual electron spin variables by the components

-204T1 2T2 Y ( 4000o ) 4T2 B (475oA) 2 U (5560A) R (6930A) 4A2 Figure 2. Energy levels of ruby with cubic field,

-21l of S. With the further assumption that the next higher cubic field level to the 4A2 ground state is far away (optical transitions as compared to microwave), the change in energy of the ground state can be calculated to second order using only the orbital part of the wave function. Thus we have the eigenvalue problem /VI ) Mr?, )ts: (It:, s (3) where U are those combinations of the free ion wave functions, 9 which transform as the irreducible representations of the cubic point group, where _ =. ~r), -,8 m and q stands for the quantum numbers y, L, S, I, and ml, and R(r) is the radial dependence and Y9' 9 and \ are the eigenfunctions of the angular momentum operators L, S, and I, respectively. With /> —, - e Sf PElR77- (4) one obtains + on. Lal < Nr, 81'mfyn as E A M Sas

where Eq are the eigenvalues of - e - 1. Since only the small changes in the ground state due to the crystalline and magnetic fields are of interest, Bq can be dropped~ The calculation performed to second order over the orbital part of the wave function for the orbital singlet ground state yields'-"s1 $1tvsn " r"" "" (6) This eliminate the orbi tal operator and leaves only This eliminates the orbital operator and leaves only spin operators in the "spin Hamiltoniant" as follows. Considering the last six terms in equation 1 as the perturbation one obtains el - L p i L + - ) _'S Ag L * Q- where spin-spin interactions among the electrons and the electronic interaction with the nuclear quadrupole moment have been neglected. (For a more general discussion see Bleaney and Stevens 6) Since V will be shown to be of ax the form FL L L~+ t)j/ its matrix element with the

-23orbital singlet ground state, 4A2 is zero. Also, since V transforms as the irreducible representation E of the ax cubic group and the products T1 x E x A2 and T2 x E x A2 do not contain the identity representation (see Appendix F), the axial field does not connect the excited orbital states T2 and 4T1 in the second order perturbation which follows. Therefore it is dropped from the spin Hamiltonian but must be applied to the higher orbital states before the second order perturbation calculation. Since the expectation value of Lz for the orbital singlet ground state is zero the first order part of the spin Hamiltonian from equations 6 and 7 is (47L ) = p H s +pj*S I - N I [ XIt )0 The second order contributions from 2, will arise through L and will contain terms quadratic in the components of H and S or wh1Iere jSt +the iij Sf pi j Hj where the dij, fij and A ij are matrices made up of the orbital matrix elements connecting the i and j components with the excited states. (For example, the calculation of the dij is outlined in Chapter II - E and performed in Appendix Go) The last term, quadratic in H, represents a constant energy and can be dropped.

-24= Combining t he first and second order contributions in a tensor notation in which [D = d1j and 9 =; g, +- C yields pH/ = W nwMM -=Pw V. - - - (8) The g~s, D, A, and Q are tensors whose principal axes are assumed to coincide. This spin Hamiltonian when diagonalized will yield all the energies arising from the various perturbations and therefore show all the resonance properties of the system. The first, or Zeeman, term in equation 7 contains the gyromagnetic ratio or spectroscopic splitting factor, g, which will differ from the free electron value of 2.0023 due to admixing of higher orbital states, Anisotropy of g may arise through the mixing of the spin-orbit and crystalline field interactions. The second, or fine structure, term is a measure of the further splitting, D, of the ground state from asymmetric or non-cubic contributions of the crystalline field. This also arises through spin orbit coupling as detailed in Chapter IIEB, and is called the "zero field splitting." The third, or hyperfine, term gives a measure, A, of the interaction of the nuclear magnetic moment with the magnetic field of the electrons. The nuclear electric quadrupole and magnetic dipole interactions with the crystalline and magnetic fields, respectively, are expressed by Q and gn in the last two terms.

-25The Spin Hamiltonian for a 3d3 Ion in Sapphire. The spin Hamiltonian must have the same symmetry as that of the crystalline field, in this case axial symmetry. It will be seen later that the crystalline field can be expanded in the spherical harmonics, Ylm, with many restrictions on the number of terms. The expansion of the axial field about a 3d3 ion with S = 3/2 yields only the one spherical harmonic, Y20~ so the spin Hamiltonian spin operators that obey axial symmetry must also trans58 form under rotation like Y20. Racah noted that the vector operator T1, which has the same transformation properties as Ym obeys the recursion relation r TOM with the angular momentum operator S. From this one can obtain linear combinations of the spin operators which have the same transformation properties as the spherical harmonics, assuring the same symmetry for the spin Hamiltonian as the crystalline field, Examples of such linear combinations, Slm, are aSno s 3 d(s and

SSD = - o80l3 3 -3~ S(S+tl) I+25 -G sj3 S 4S+/)S Thus equation 7 with the crystalline field axis as the z axis becomes,'W-l _ - ( L +)j or7L =# ts LAH2% X(HS+HS))]+D/Vm~S(S/)2 SPIN (AX) meeP; g e9q(tI cmponYyen (10) The Energo Levels for a 3d3 Ion in remains only to apply this spin Hamiltonian to the eigenvalue problem

=27bS1kn = (11) where now. only need contain 1 and A\ This yields the eigenvalues to second order in the hyperfine interaction with the magnetic field parallel to the crystalline field axis of +Q' (rs * jfs,D (t+))LI(/) -,)(12) where ms and m:r have been replaced by M and m, respectively, to conform to standard nxotation. The gross structure of these energy levels for S = 3/2 and different M with the magnetic field parallel to the crystalline axis are shown in Figure 3. These are plotted with the coordinates and E E

-28x rE 2 13 -2 Figure 3. Energy levels of ruby ground state versus applied magnetic field.

-29This is done so that the plot is the same for Cr3+ and V2+ which both have S 3/2 but have different D values. In additions vanadium has an I = 7/2 and one isotope of chromium has an I = 3/2 which will split these levels further into 2 I+l levels as shown in the inserts in Figure 3. EPR Transitions for a 3d3 Ion in Sapphire. To induce transitions among these magnetic sub-levels with different M, a time dependent HO(t) must be added to the steady field, H. Physically, this can be thought of as adding perpendicular to the steady field, H, an oscillating magnetic field vector, H0(t), whose frequency is equal to the Larmor precessional frequency of the spin angular momentum. The transition probability is then proportional to the square of the matrix elements of the first and last operators in equation 7, with HW(t) substituted for H, These levels of different M are connected only by the S+ and S_ operators (S+ = Sx + iS ) if the hyperfine splitting is small and the field is parallel to the crystal axis. Their matrix elements are Msl M+= > -/ M F- M;0]"'1 V (12) Thus, the selection rule for EPR is AM =+ l(Am=O). With this selection rule the resonance condition for the frequency of Hi(t), which now must be perpendicular to H, becomes

in our case h (= M?,,) - E(, hi) p,Hi + D(d n-) + ), - - X 1 -B S (+ - lg~~ f s D o hl t ( - /< eH r D ( M -O ) o -t(14) This is the equation used to analyze the results of this study. (It should be noted that usually D and A are much smaller than sep! and the second order terms can be further simplified. For Cr and V2+ in sapphire D is of the same order of magnitude as ~.oH and this is not the case.) At X band microwave frequencies (9-10 Gc/sec) and with D " k, it can be seen from Figure 3 that three fine structure transitions will occur at - 100 gauss,'Y 3500 gauss, and - 7000 gauss. For ruby a check was also performed with the magnetic field not parallel to the crystalline symmetry axis. The dominant isotope of chromium in ruby has I = 0, so the spin Hamiltonian of equation 10 simplifies to 4~~~~~, HEjj / sc. asc} tZ_I chow.)

-31Going to the case where the field is no longer parallel to the quantization axis9 one can either transform the spin Hamiltonian to a coordinate system whose axis is parallel to the applied field and use combinations of the original eigenfunctions, or perform a perturbation calculation which yields second order terms insl, g where eis the angle between the field and the crystal axis. For ruby D> 1pB H and the latter doesn't converge,, The former yields the following spin Hamiltonian37 he energy s eg nc a nd ransions are The energy levels, eigenfunctions, and transitions are calculated by Schulz-DuBois37 and the X band transitions as a function of magnetic field and the angles, ~-, are shown in Figure 4. The transitions marked forbidden are forbidden only at zero degrees. The off axis check was done primarily at the angle,' = 54.740 where cos2~' - 1/3 and the second term in equation 16 is zero and the coefficients of the last two are equal. This angle was used because the transition energies are relatively simplified to3

-322 2 Z I IC -,-FOR BIDDEN L V -FORBIDDEN 30 60 90 ANGLE,e,WITH RESPECT TO CRYSTALLINE AXIS Figure 4. EPR resonances in ruby for x band as a function of applied magnetic field and the field's angle to the crystalline axis,

33-- =3b 3'~ (17) and YFO -YRZ=;j; + 8 (X X) (18) where YV _'; X =j d 94 _ X Cito (NOTEo In equation 18, x is double valuedo) To understand the source of symmetry about the ion the sapphire crystal structure is next discussed. B. SAPPHIRE CRYSTAL STRUCTURE Sapphire is the commercial name for synthetic corundum which is chemically pure aluminum oxide in the alpha phase, or just o( - A1203. The gem sapphire is naturally occurring corundum with various coloring impurities to give it gem value, such as cobalt which gives the blue sapphire, vanadium the green, and chromium the red sapphire or ruby.

-34The crystal structure46 can be generated by placing A1203 molecules at the corners of a cube and stretching the cube along one of the body diagonals. Another A1203 molecule which is rotated 1800 about the molecular axis with respect to the others is inserted in the center of the distorted cube. Without distortion each aluminum ion would thus be at the center of six oxygens in octahedral coordination as shown in Figure 5. This results in the cubic crystalline field. The distortion along the body diagonal yields the axial or trigonal crystalline field from the inequivalent sets of three adjacent oxygens as shown in Figure 6. Three of the oxygens lie in a plane 0 1.37A away from the aluminum site. The aluminum=oxygen (Al-0) distance is 1.98 i and makes an angle of 460 279 with the crystalline c axis. The other three oxygens are in a plane 0.80 A away and the A1-0 distance is 1.84 and makes an angle of 640 10Q with the c axiso In addition, the two sets of oxygens are not rotated exactly 1800 with respect to each other. A projection of one set on the plane of the adjacent set (Figure 7) shows an additional rotation of 40 220. The crystal belongs to the 3 m (or D - Shoenflies) 3d space group and the aluminum occupies a site which has the symmetry of the 3 (or C3) point group. The four degree offset is often neglected for which the symmetry becomes 3 m (or C~v)o

-35I I I 21 I 3 ~ Figure 5. Octahedral coordination in ruby,

46027' OPC/ I\.1.37A J~~~~~~~ Figure 6. Crystal coordination for aluminum site in s apphi re, AX 4~ 22' l >,7<,g Figure 76. PCro ject ion of oxygens in Al t n perpendicular to the trigonao axis.

37= In the crystalline field calculation the next-nearestneighbor aluminum ions are accounted for~ A list of the 14 next-nearest-neighbors, their distances from the substitutional site, and approximate angles (cosines are exact) with the crystalline c axis are shown in Table Io The oxygens are also included for completeness. TABLE I Location of the neighbor and next-nearest-neighbor ions with respect to a substitutional impurity in o -A1203.59 Site # of ions Distance A cos approx. Impurity 1 00.. 01 3 1.84937 0043575 640 02 3 1.98828 -0o 68899 1330 Al 1 1 2073273 Lo00000 1800 Al 2 3 2~80912 ~-02.0072- 1010 A1 3 3 3. 18581 0o50374 600 A1 4 3 3,50390 0o61893 520 Al 5 3 3 O50390 -0O61893 1280 A1 6 1 4,67959 1.00000 00 It should be noted for later reference that if one moves along the c axis of the crystal, one. adjacent octahedral site will contain an aluminum but the other is empty. It is the change of position of the surrounding ions with pressure that causes t he crystalline field to change. The displacement of the ions with hydrostatic pressure can be obtained from the following discussion of the elastic constantso

-38CO THE ELASTIC CONSTANTS FOR SAPPHIRE For any elastic deformation there is a linear relation between the stresses applied and the resulting strains. These relations are derived in Appendix C and put into the notation for which the values are usually reported in the literature. In the usual matrix notation this relation is expressed 6 - = J J (19) j-I where the c.. are the elastic "stiffness" coefficients, Pi are the stresses, and uj the resulting strains. Similarly, (20) where the sij are the elastic tcomplianceVO coefficients. The stresses, Pi, are related to the components of the stress tensor c. i as follows: 1j - - x Y y (21) /~3 a a- PG CXY YX The strains, ui, are related to the components of the strain tensor C; as follows / -v- + Gt 2 &, = A M = (22)

-39A component, a-_j, of the stress tensor can be thought of as the tensile force per unit surface area in the ith direction applied to a surface whose normal is in the jth direction. Similarly, a component, El,, of the strain tensor can be thought of as the elongation per unit length in the ith direction for j = i and the change in the angle between the ith and j th axes for i f j. It is shown in Appendix C that the symmetric matrices sij and cij generally have 21 independent components and that the crystal symmetry restricts this number further, The most general compliance tensor for the sapphire lattice which has 3 m symmetry has the six independent components shownt Li s S, S,0-S O p Us a 0 /0P3 s~ O o (23) SY1. 5'. In the case of hydrostatic pressure t' - - - =-? (24) and P c r = al = o ~ The compressional analysis done later utilizes the x and z displacements, Ax and Az, of ions neighboring the impurity

-40O ion, These are then related to the elastic compliance constants by the followingg = -- = X_= — (S,, + Sl + S,p LA ( - (25) (A = E- -- = — (3 + SP For uniaxial compression, = -e;? = P? -5 =?- = o (26) The displacements are then calculated with the elastic compliance constantst L43 = -s,3/e (27) Then for both cases the x and z displacements are simply u1 x and u3z. The most recent compilation of elastic constants for sapphire is that of Bernstein who quotes the values C /= I C-33 (28) c,, = 1. / " C = -o.a3 " "

Since the compliance constants are used to determine strain, the reciprocal matrix of these c.. values must be obtained. 13 For a crystal of 3 m symmetry the inversion is simplified to6 -- _ 5, = ~ /c3 S, =- -- C, /C (29) S =, - CC,/ C 33 where = C.. (C1, + C,~) - C, - C-, (C,, -c )-2,,, This yields the following values S-/- 0,, Q / x /o (DY'/6 ou-/ S,2 = -,&, ("Mt " ) 33 = —o3fO D (30) 5,> = - 003eo

-42These are the compliance constants used in the analysis of this experiment and are summarized with the stiffness constants in Table II. TABLE II The Elastic Stiffness Constants, C, and Compliance Constants, SJ, for Sapphire at Room Temperature. ( c;j from BoTo Bernstein, Jo APPlo hys. 34, 169 (1963); S,by inversion.) cll = 4.92 x 1012 dyne/cm2 sll = 0.241 x 10-12 cm2/dyne c33 = 4o92 s33 = 0.222 " c44 = 1.47 " s44 0.697. C12 = 1.68 $122 -0o0755 " C 13 1,16 S13 -0o0390 " c14 =-0,234 " s14 = -00503 Since these constants are for the bulk crystal and the local compression about the impurity site is of concern, the bulk compressibility must be related to the "local compressibility" about the impurity ion, The usual assumption is that they are equal.20'22 This implies that the neighboring ions will compress about the impurity ion as if they were all imbedded in a medium with the bulk elastic properties of sapphire. There will be "local relaxation" if the compliance constants of this hypothetical medium are larger than for sapphire and "local tension" if they are smaller, ie., local relaxation implies that the neighboring ions displace more than expected for a given stress,

-43Local relaxation could be expected if the atomic forces between the impurity ion and the ligands were less than for an original lattice ion. For example, this would be the case on a purely ionic b asis for a substitutional atom with a lower valence state. Since covalency effectively reduces ionic charge, it would be expected to lead to local relaxation also, but to what extent the covalent bonding strengthens the interatomic force to compensate for this is unknown. An estimate of the combined effects 61 can be obtained from bulk compressibilities. Kittel has shown that the bulk compressibility is generally inversely proportional to the interatomic force. In general, highly covalent structures such as diamond (/3= V?4Tt1 =./16) silicon (0.32), SnO2 (0.49), and germanium (1,42) have lower bulk compressibilities than highly ionic structures such as AgBr (2.74), NaCl (4.18), KCL (5.65), and RbBr (7.97). This implies that a higher degree of covalency for the impurity ion than the normal lattice ion could lead to local tension. Local tension would also result if polarization effects are significant since added distortion is necessary with compression. Now it is necessary to determine the crystalline field at the impurity ion site and predict the changes due t o the above displacements. This is done in the next two sections.

-44Do THE CRYSTALLINE FIELD MODEL The crystal field model will be used in which the crystalline field is determined by a calculation of the potential at the impurity ion site when approximating the surrounding ions as appropriate point charges at the ionic lattice sites, In view of the criticism of the crystal field model62 and the failure of the simple theory to satisfactorily predict the zero-field splitting of the 4 3 40 A2 ground term of the Cr3 ion in ruby, the use of this model must be justified. First, a more recent expansion of the crystal field analysis to include distortion of the wave functions by the trigonal field through configuration mixing and the covalent 7r bonds has shown good agreement.63 Also, the crystal field calculations to fit optical spectra show reasonable agreement when some displacement of the impurity ion is assumedA42 This assumption is supported by ENDOR (Electron Nuclear DOuble Resonance) experiments.39 Thus a model that can reasonably predict the splitting should be able to predict changes much more accurately. Finally, the point charge model has worked before for trigonal calcite,22 With these justifications the crystal field model is used with the idea that the results will give some indication of its validity, If it is assumed that the impurity ion is surrounded by point charges and that the impurity ion charge does

-45not overlap those of the surrounding ligands, then Laplaceus equation holds (31) and the crystalline field potential due to the point charges can be expanded about the impurity ion site. In this region Laplace's equation has general solutions of the form /V(r) = A ^ r Ap, 6,~ ( Zm (32) where r, 9, and 0 are the spherical coordinates of r centered at the impurity site. Equating this to the potential 2Vr _1 -'f. /_~ -_ U i(33) at the position r, due to the sum of the charges qi at the surrounding ion sites ri at the coordinates ( ri j 8 } ~af ), results in the coefficients.iv =t 1 7 t (34) (The details of t his and the following analysis are carried out in Appendix A.) The C3 symmetry of the crystal lattice restricts the m values to integral multiples of three and j values to even integers. The use of 3d wave functions for which, 2 cuts the series at = 4 and

-46allows only even values of ( The a term is just a constant and since Cr3 and V both have S = 3/2, only the term in a20 remains. This coefficient becomes, where the spherical harmonic is put in terms of Legendre polynomials, In the next section D is shown to be proportional to a20 and the change in a20 with lattice compression is also derived. Calculations of a20 are included in Appendix A and a summary of the results in Table III are also in the next section. E, COMPRESSIONAL MODEL AND PREDICTED BEHAVIOR In order to evaluate the results of this experiment with respect to the crystalline field model, some model of change of the crystal lattice with hydrostatic compression must be assumed. This, then, must be related to the changes in the c rystalline field spin Hamiltonian parameterso Assumptions. The following assumptions are made and discussed in turno 1. The local compressibility is the same as the lattice compressibility.

-47This implies that the surrounding ions will compress about the impurity ion site in the manner as if they were imbedded in a material with the same elastic constants as bulk sapphireo Local relaxation, or greater compressibility, about the impurity ion site was deduced in several other materials. There has been some evidence of local relaxation in MgO34 and SrTiO 23 but Drickamer has shown that none exists for Cr3+ or V3+ in sapphire. This is probably due to its stiffness. 2. The impurity ions are substitutionalo This assumption is under serious question both from optical studies42 and ENDOR results.39 It is also suspected that the vanadium and the chromium do not occupy the same site. It is one of the objectives of this study to check on this assumption. 3, The changes in the spin Hamiltonian parameters are due primarily to changes in the intensity of the electrostatic crystalline field~ The spin Hamiltonian parameters could also be affected by changes in covalency (or the degree to which ligand orbitals must be used in determining wave functions) and changes in the distribution of the 3d wave functions as shown later in this section. A direct measure of both ~~~of ths armtr i h achB55 64 it is of these parameters is the Racah B parameter~' It is shown in Appendix F that both an increase in the degree of

-48covalency and an expansion of the radial distribution of the wave functions, which would be expected with increasing pressure, will tend to decrease B, Drickamer s optical experiment36 shows that the change in the Racah B parameter is very small in our pressure rangeso The decrease in B observed in ruby is 0,o03% as compared to observed shifts in the crystalline field parameter of about 0~2%o Therefore this assumption seems reasonable. 4, The point charge model can be used to describe the changes in the electrostatic field. For a crystalline field which is cubic and in which the compression has the effect of simply shrinking the distance to the surrounding ions, this assumption is exact. For this reason the point charge model was used in the work by Walsh20 and Wait,22 Even for a distributed ligand charge this assumption is valid of the compression is "conformal" or angle preserving and the charge distribution lies outside the ion~ This is shown in Appendix Ao One of the reasons for choosing such a model for ruby, which has trigonal symmetry and nonconformal compression, is that it was used with success for trigonal symmetry before,22 even though the calcite compression was also far from conformal. Also, the agreement that Drickamer36 obtains with a simple point charge analysis supports this assumption, However, one of the motivations in choosing the sapphire structure was to cheek this agreement and that for calcite,

Change in the Coefficient a20. The method used will be to determine the rate of change of a20 with change in the position of a surrounding ion and simply multiply by the change in position per unit pressure as predicted by the elastic constants and the first assumption. As is easily seen from Figure 6, by taking the z axis as the axis of trigonal symmetry, the contributions to the crystalline field on the axis from each of the three oxygens in a plane are equal and any dependence on ~ can be neglected. It is easiest now to consider only one of the oxygens, namely the one lying on the positive axis. The effect of this ion will then be one third of the effect of all three for any symmetry preserving motion of the one. Thus, if one were considering only the effect of the oxygens, the sum in equation 35 would be over the two values i L / r0, S R7 6Y /0 a Ad 98 R f 6 c27 (36) where r, and 9, are obtained from Table I and then lao = 3 Q Yox. 3 p, (cs e)7) where Q is just a combination of constants, and ox. is the charge on an oxygen ion, A similar condensation can be performed for the fourteen nearest neighbor aluminumns when the sites listed in Table I contain more than one ion,

o50To get the change in a20 with pressure it is easier to transform to a rectangular coordinate system with the same z axis but with the x axis through an oxygen site. Then we can determine a change in a20 as 4tLo bc E x \{,). (38) The Axi and Azi are obtained from the elastic constants (See Section C) and are respectively u~ xi and u3Zio In this coordinate system there is no need to consider AYi since yi for the ion is zero. Thus the compression is symmetry conserving, allowing the condensation above. The rate of change of a20 with position can be obtained from equation 35. For xk = x,z one obtains OL E (COs e) Ak k' (39) Since r = (XiYxi + i2 + zi2 the following relations hold1 r = 8 = -.- -. -- /= LI One now obtains by substitution into equation 38 el` 6 -a'Y (40)

-51where _ -3(3cosoG cos ~ O Xk Jos Qc: Cos "Xk k and 4 _,, d cos, / r/, The substitution of the structure constants for the six oxygens and fourteen nearest neighbor aluminums from Table I and the elastic constants from Table II is shown in Appendix Bo Since there was suspicion that the impurity ion was not substitutional but displaced along the trio gonal z axis by about 0.1 A toward the empty octahedral site (ie.o toward the top of Figure 6), these calculations were performed at various positions in this direction. One calculation was done in the opposite direction as a check~ A summary of the values obtained are shown in Table III, The contribution of the fourteen nearest neighbor aluminums is relatively small and is therefore sometimes neglected. It is interesting to note that this contribution becomes negligible at a position 0o15 i towards the empty octahedral siteo If one plots the relative positions- of the surrounding ions from Table I as shown in Figure 8, it can be seen that t his is about where the aluminum contribution is symmetric.

-52TABLE III Summary of the Theoretical Calculations of the Change with Pressure in the Crystalline Field Parameter as a Function of Displacement Along the Trigonal Axis in Al 20 (a0 is in units of Q x 10-2 A3 where Q = 1/e o — an Aa2o0a20 is in units of 10-3/katm. Positive displacement is towards the empty octahedral site.) I n: <3 - - _ m + " co o WOQ 4 0T X t I) l f Et 0 ~~~~~~ ~ 0Y aO dc z/ v q~\9 ~C k~~~~~~) U'tZ 0 ~

-53AL 6 c- Axis (I) (3) b J, AL4 I I ~/; I(3) 0I (3) 0/ — IMPURITY.66 A 2.74A! 1.44 A I I / L AL2 (3) 1 G I \ Y I (3) 02 I (3) I AL5 (3) AL I (1) Figure 8. Relative positions of the nearest and next-nearest-neighbor ions in sapphire.

-54Predicted Behavior. Before predicting the pressure dependence of the zero field splitting due to changes in the crystalline field, the dependence of the splitting on the crystalline field must be determined, With the restriction to only the a20 term, equation 30 becomes AX - at O r o = ~lO' (41) Substitution of orbital angular momentum operators accord46 ing to Stevens yields VR (42) where g is the axial field splitting and the integral is already performed on r2. As shown in Appendix G, Vax 4 4 4 splits the excited states T2 and T1 but not the 4A2 ground state. The matrix elements are evaluated in Appendix G and result in the splittings of 15 6 ( or 3/2 K) shown in Figure 9. Since there is no splitting of the ground state due only ton the axial field, the observed splitting arises through second order spin-orbit coupling to the higher stateso Evaluation of the matrix elements of the second order perturbation X LoS, as the third term in equation 5, to the axial split excited states in Figure 9 becomes AU=- Y (<3ol)Sl lLSlo> -E - -— F -......(43) l | r _rL

-55E +10=- -K / / 4T1 +18Dq,' /''.., Z -5s= +K/2 / F / E +08 =-K \\ 0 \ \4T2 +1ODq \ -'4A, - =+K/2 \4A2 — 4A2 D FREE ION CUBIC FIELD AXIAL FIELD SPIN-ORBIT Figure 9. Crystalline field splitting of 4F ground state for a 3d3 ion.

- 56where the sum is over the axial split excited states and o refers to the unsplit 4A2 ground state. Since L transforms as the irreducible representation T1 of Oh and the product resulting from the 4T1 excited state matrix elements, T1 x T1 x A2 (see product table Appendix F), does not contain the identity representation, its matrix elements are zero and only those of the 4T2 state are considered. These matrix elements are calculated in Appendix G and the result is /O Dt ~ lo /O.D SS44 (44) where the excited states due to the excited doublet 2T2 with its larger energy denominators have been neglected. The other excited doublet states do not couple directly with the ground state as in the case of the 4T state. 1 Since i is much smaller than Dq, equation 44 can be approximated as LE^ - D- 7 ( [>(S s j ) -A) (45) or L\E -Q Ztt (46) or since Sx2 + Sy2 + Sz2 S(S+l)

57 AE -t- lot| + De QSe-5 ~ + I) (47) The first term is simply the isotropic part and adds a constant to all levelso The coefficient of the term 3S2 S(S+1), or D, is k sadf)? (48) where g contains the a20 of the previous section, Thus, for small changes, the relative change in D is:D. >A o) * (49) Similarly, the spin orbit coupling will shift the g value from the free ion values go, and an equation analogous to equation 48 is (see Appendix G) ~ot', (50) For small changes then ~(9 -9~~), A~ X~ D* (51) These equations (49 and 51) can then be used to predict the shifts in the spin Hamiltonian parameters D and (g-go) if pressure data is available for the changes in Dq and, a Only the change in Dq with uniaxial pressure is known from Drickamer s worko36 Further analysis from the available data and the data from this experiment will be performed in Chapter Vo

Chapter III EXPERIMENT For this experiment an electron paramagnetic resonance spectrometer for studying crystals under high hydrostatic pressure has been constructed65 having many distinct advantages over earlier such instruments.66,67,35 The advantages are the ease of cavity coupling adjustment under pressure, the use of a two sample comparison, and the resulting simplicity of data analysis. Also, the pressure is hydrostatic and a precise pressure measurement is possible. In Section A the crystals and crystal orientation are discussed, in Section B the principles of operation and the construction of the apparatus are described, in Section C the advantages of the two sample system and the solution of the "cross talks problem are discussed, and in Section D the method of taking data is included. A.o CRYSTALS AND CRYSTAL ORIENTATION The crystals used in this experiment were identical pairs of washer-shaped single crystals of A1203+0.1% Cr203 and A1203+0o1%V203 obtained from Linde Optical Company. The crystalline c axis was in the plane of the washer + 2.00~. Their dimensions were -58

59~ I Do -.090".005" O.Do D.240" 005'" Thickness - 0060'".005" The naturally occurring chromium has a 9o5% abundance of the Cr53 isotope whose nuclear spin is 3/2, but all other isotopes in natural chromium have a nu:clear spin of. zero. The chromium substitutes for the aluminum in a plus three valence state. The A103Cr3+ crystals were used as 20 crystals were used as obtained from Linde and gave clear strong signals. Naturally occurring vanadium is primarily the V51 isotope whose nuclear spin is 7/2. The vanadium also substitutes for the aluminum in the plus three state with a very small amount occurring in the plus four state. The plus three state is not discernible with microwave spectroscopy at room temperature and the plus four state does not interfere with the observed plus two spectra.38 To obtain the doubly ionized vanadium, the crystals were subjected to 50 kev, X=rays for a period of three hours038 Because the conversion efficiency to V2+ is small and the signal strength is spread over eight lines (I = 7/2) instead of one as in ruby, the signal strengths were considerably weaker~ The crystals were oriented spectroscopically. As will be seen in the next sections, the cylindrical axis of the washer shaped crystals is coincident with the axis of rotation of the magnet. The plane of the washers is thus

-60perpendicular to this axis and held in the plane of rotation by the sample holders to much better than the + 20 quoted by the crystal manufacturer. No further attempt was made to orient the c axis in this plane. Using Figure 4, which is a plot of the resonant magnetic field values as a function of angle, two points were used to orient the crystalline c axis of the ruby crystals with respect to the magnetic field. At - 54,740 two levels cross and can be observed by the recordings to + 0.50, This was used as a check, Principally, the crystals were oriented by using the maximum for the high field line at i- = 00, Observing the maximum of the resonances on the oscilloscope while rotating the magnet allowed orientation to the readability of the magnet scale or + 0,20. The vanadium crystals were oriented by observing the disappearance of the forbidden transitions which occur between the eight high field hyperfine lines when the c axis and the magnetic field are not exactly coincident,43 This allowed orientation by observing the spectra to about + 0O50 and there was no indication that the alignment in the plane of the washer was significantly worse than this,

B,o THE EXPERIMETAL APPARATUS The principles of operation are as follows: The two samples are placed in the atmospheric and high pressure sections of the microwave resonant cavity (Figure 11), which is located in a DC magnetic field. The magnetic field about each sample is modulated at a different frequency. The microwave power absorbed by the cavity is then fed as signal to two phase sensitive detection systems, each of which is sensitive to one of the two modulation frequencies. The signals from both samples are thus separated and plotted out on a pair of identically driven recorders. Thus the percentage change can be directly determined, and with a calibrated field sweep the relative shift can be directly measured in gauss. This not only reduces the error (formerly the difference between two large numbers) but greatly simplifies the analysis and procedure by reducing the need for considering second order corrections, hysteresis effects, and ambient changes. The high pressure is generated by a hand pump, magnified with a piston intensifier (Figure 12),and fed to a beryllium-copper bomb through thick walled stainless steel tubing. Spectrometer. The spectrometer is of the homodyne type68 and utilizes X band (8-12 kMc/sec,) frequencies and phase sensitive detectorso Reflected power from the cavity

-62KLYTRON = A FC RECORDERS ISOLATOR A B v METER PREAMP AL B MATTEN E SHIFT CRYSTAL PHASE SENSITIVE DETECTORS ATTEN 3db A B MODULATION BELOW CUTOFF ATTENUATOR WAVE GUIDE TO I I FEEDBACK COAX COUPLER SAMPLE A MAGNET MAGNET SAMPLE B CAVITY HIGH PRESSURE FEED Figure 10. Spectrometer schematic.

Section A- A Section B - B SectBon C-C 118" Stainless Steel Tubing Threaded and.2 8 " ~Y~ ~\\\\\\~i~,\\\\\\\\~j ~ Soldered into Place X-Band 2,~ ~ ~~05 5' ntetc IC Prssd ora FIa'nge2.5Daer Be Cu PsoBand B Wa ve Guide AASample B 311CBryliu Copr 12 wie(Gudt6edwalwthDc lco on lTaper 0.50 in. per in. Be Cu. 19' Svnthetic MICA (Pressed normal F U OR PH LOG OPI TE) 1113211 71 1611 5/811 Phosphor Bronze Guard Ring Rin B~rass \_ntrance for High Presur Sample Fui Beryl ium Copper /2 ie(le e wl t Dc (Beryl o 25) Extraction Collar 3/2 wd- Gue o n wl wt Dc (Brassc Figure 11. Pressure bomb and microwave cavity.

-64Charging System (Low Pressure) Relief Valve Manganin Ce I Pump I Charging Pump L 1 Cavity Pump2 I ntensifier Figure 12. High pressure system schematic.

-65is used for detection68 at present, however, and the bypass arm in the microwave system has been built in for future use. Referring to Figure 10, the microwaves are generated by the klystron and coupled through the electric field into the co-axial (Figure 11) cavity with the coupling adjusted by a below cutoff attenuator. The reflected power is then detected by a crystal detector. The microwave frequency is locked to the cavity by an automatic frequency control which uses a 10 kc frequency modulation and signal feedback. The cavity is coaxial and built into one of the plugs of the high pressure bomb as shown in Figure 11. The cavity entrance is at the wave guide to coaxial converter and about 8 wavelengths from the high pressure sample. Just the last portion of the cavity, shown in detail in Figure 11, is at high pressure; this not only allows the coupling to be changed readily with pressure but keeps the volume and consequently the stored energy and the cavity0s resonant frequency change with pressure at a minimumo This portion is sealed from the atmospheric portion by a beryllium-copper cone which is seated on an insulating cone of synthetic mica (pressed normal fluor phlogopite)o The sealing cone is part of the center conductor of the coaxial cavity.

The high pressure sample is located at the end of the cavity. It is usually in the shape of an annulus about the center conductor and surrounded by pentane, the high pressure fluid. The identically oriented low pressure sample is placed part way up the cavity, and the rest of the cavity is filled with teflon. The location of the second sample was determined by three requirementso First, it had to be beyond the cone and yet in a comparable magnetic field. The fields do not have to be exactly identical since it is the change in one absorption line with respect to the other that is measured. The magnetic field homogeneity was increased tenfold with the addition of simple ring shims which yielded a 3" diameter region in which the field was constant to within 0,1 gauss. (A 9" diameter Varian rotable magnet with a 3~25" gap is used). Second, the modulation of the magnetic field at one sample due to the modulation of the other had to be at a minimum, This was accomplished by placing each sample in the null of the modulating field of the other modulation coil. Third, the sample had to lie near a maximum of the microwave field. These requirements fixed the separation at 1,87". The frequency dependence of the cavity with pressure is small (ca, 2.5 Mc/seco per 1000 atm), If one uses a thicker and shorter mica cone than indicated in Figure 11,

-67? filling the rest of the space with Teflon, the cavity "Q" is about 500. The Teflon is primarily used to hold the long center conductor motionless to prevent modulation pickupo Teflon, however, has a broad resonance about g = 2 and for sensitivity in this region a system of washers can be usedo The absorption signal is fed into a preamplifier and then to two phase sensitive detectors whose first stages are twin-tee filter amplifiers adjusted to the two modulation frequencies, 228 cps and 800 cps. (The modulation frequencies have to be low in order to penetrate the bomb.) These are followed by the lock-in amplifiers. The outputs are then plotted by two identical recorders with synchronized charts driven by the same chart drive shaft. High.Pressure System, The components of the high pressure system are shown schematically in Figure 12, Most of the components were manufactured by Harwood Engineering. The system has a low pressure charging section and high pressure generating equipment. Hand pumps are used with a 201o,5 piston intensifier used to obtain the high pressure. The pressure is determined with a manganin cell and bridge circuit using a precision four decade resistance box and galvanometero The sensitivity of the measurement circuit has been determined to be less than 1 atmost

phere, The accuracy is thus limited by the pressure dependence of the resistance of the magnanin wire, which is linear with pressure to within 005%. The measuring system was calibrated by observing the 3% volume change as mercury freezes at 7420 atmospheres and 00~. This was done using a specially constructed Invaro oil hardening steel bomb with a much larger volume than the experimental bomb. The high pressure is transferred to the bomb through thick walled, cold worked stainless steel tubing with a 0,0075" inside diameter and 0,125" outside diameter. The bomb, shown in Figure 11, is 2.5" OD, and 11/32" I D. and open on both ends, The bomb, the two end plugs, and the sealing cone are made of Berylco 25, a nonmagnetic, high strength beryllium copper alloy which is machined in the annealed state and heat treated afterwards. The plugs are sealed with rubber, teflon, and phosphor bronze ring seals of the unsupported type69 and the unit is designed to hold a pressure of 10,000 atmospheres. Because of the seals a practical limit is about 8,000 atmospheres, Normal pentane is used as the transfer fluid and at these high pressures it becomes viscous, As much as 15 minutes is necessary for the pressure to be transmitted through the tubing, The bomb is clamped into the magnet by means of brass clamps attached to circular brass caps which are

clamped over the magnetos pole pieceso These brass caps also hold the ring shims and the modulation coils. The clamps permit motion only in the vertical direction along the two-sample axis, which is also the axis of magnet rotation. This allows precise centering and accurate repositioning. C. THE TWO SAMPLE METHOD Perhaps the most practical advantage of comparing the spectra of a sample under high pressure to one at atmospheric pressure is that during the course of the experiment one can see at a glance that a pressure shift in the spectra is indeed there, estimate its order of magnitude, and determine its direction without any calculations or analysis whatsoever. The fact that the two channels are similar reduces the need to consider electronic distortion of the line shape. It also eliminates any consideration of ambient changes since the difference is always what is measured. Hysteresis effects in the magnet, except for homogeneity, can also be ignored unless a set of lines is being considered. Placing both samples in the same cavity so that the transitions in each occur at the same microwave frequency provides many advantages. Second order frequency corrections in unshifted parameters are completely eliminated.

Also if the pressure perturbations are small, and they usually are, most second order perturbations can be neglected since these would be second order corrections to a pressure shift which is already second order. The fact that both samples are at the same frequency allows some very helpful simplifications in the analysis. For example, consider the spin -~ to ~ transition for Cr3+ in A1203 with the magnetic field along the c axis of the crystal, from equation 14 with I = 0 =tp s c u p u S n (52) where the primes indicate under pressure. Solving for Q9;F = 9t - ge is simplified by the equality above to I -H - ~-_ 9 -, H' - -~ H' (53) Thus, a precise measurement of a g shift involves only an ordinary field measurement and no frequency measurement. In a similar manner the - to 3/2 transition h, = giP H C + D = h (a /H + At) A D (54) simplifies to (since g shifts are very small) (55)

Also, if there is a close set of lines that falls within the sweep range so that the unshifted spectra can be used to calibrate the shifted spectra (See Section D), no field measurements whatsoever are needed. In this case, however, linearity of the field sweep to first order is necessary. The only disadvantage of the two sample method and the major problem of any two signal system is the channel separation or "cross talk". In our case, this is electronically quite simple and narrow band amplifiers and phase sensitive detectors assure that signals carried on each of the two modulation frequencies are definitely separated. To be certain that the signal carried on one of the modulation frequencies is purely due to one sample is not as simple. If one defines a measure of cross talks KA, as the ratio of the signal from sample A that is carried through on channel A to the signal from sample B that is carried through on channel A, then KA and KB can be measured experimentally by using two different samples. A lower limit must then be set for K, The center of the absorption line, which is what is measured in this experiment, will be shifted most by a second signal when the second signal is 2 line width away, as shown in Figure 13, so that its maximum slope occurs at the true center. The observed center of the sum (or difference, depending on polarity) of the two signals will then be shifted by 1/4K times the line width

-724K| —-__ I +I/K I4 Figure 13. Line center shift due to overlapping lines. I III'4' //Lc~ t~t\-~V-4K I'B"COILS a FLUX LINES Figure 14. Bomb, coils, and tlux lines schematic. ~" A" COIL ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~00 p ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~.00\

7 3= The choice of minimum permissible K is thus determined by the line width and the expected pressure shifts and in our case was about 10, The primary criterion in determining the location of the second sample was that it be at a maximum in the microwave fields in the coaxial cavity, yet be as close as possible to the high pressure sample so that they both remain in the homogeneous portion of the DC magnetic field~ This separation then determined the size of the modulation coils so that a pair of coils centered on one sample had a null at the other as shown in Figure 14. Without the pressure bomb and with proper orientation of the coils, KA 10 and KB 103 were obtained o However, with the A/a B bomb, K = 10 and KB 1, Consideration of these values and the bomb shape (schematically shown in Figure 14) and the fact that the skin depth at these frequencies is considerably smaller than the bomb thickness, lead to the conclusion that the modulation was being caused by induced currents, Therefore another method had to be used to eliminate the cross talk in channel B. Since this was due to the field modulation about sample B being carried up to and around sample As but not vice versa, a portion of the B modulation signal can be inverted and fed into the A modulation to exactly cancel the cross talk, Thus KB can be set by a phase and amplitude adjust

-74ment of this feedback to be as large as desired or even infinity. In this manner, cross talk was reduced to less than one part in 104. D,, EXPERIMENTAL PROCEDURE When there is more than one line falling within the sweep range of the DC magnetic field, the data taking is extremely simple. The known line separations on the atmospheric pressure chart provide a calibration for the high pressure chart. Using identical chart paper on recorders driven by the same shaft assures equality of recordings. (It was found, however, that the chart paper grid spacings and drive hole spacings are not always reliable,) Figure 15 shows a sketched pair of recordings for the first three fine structure lines for Mn++ in MgO. In actual practice the signals are driven a long way off scale in order to more accurately determine where the center of the absorption line falls (derivatives are what are plotted and constant line shape is assumed, which is supported by the work of Wait.22) In this method magnet hysteresis can be a problem unless a field sweep (as opposed to current sweep) is used or the sweep is calibrated. If no close set of lines is available, the procedure is to stop the sweep exactly when the recorder shows the

-753a a ATMOSPHERE HIGH PRESSURE Figure 15. Sketch of pressure effect on Mn2 in MgO lines.

-76derivative to be zero and measure the field with a nuclear magnetic resonance probe. Similarly, the field is measured for the shifted line. The difference between the field at the probe and that at the sample is of no concern as long as it is constant since the difference between resonance conditions is what is measured. The nuclear magnetic resonance probe used the proton resonance of a 0.2 molar solution of CuCl2 in water for the low field lines and the deuteron resonance of a 1.0 molar solution of CuC12 in 99.9% pure heavy water for the high field lines. The only other necessary measurement is that of the microwave frequency. The frequency does not appear in equations 40 and 42 but it is necessary to correct for any residual field difference between the two samples at atmospheric pressure. This difference must be subtracted from the pressure shifts and since the frequency changes a little with pressure, this zero pressure difference must be corrected with frequency as is explained in the data reduction in Appendix D. Since accuracy to better than one part in 104 was not necessary a previously calibrated absorption type wavemeter was used. This allowed frequency measurements to 0.001 Gc/sec out of about 9.4 Gc/sec. Table IV shows the calibration of the frequency meter when a transfer oscillator was used to measure the microwave frequency to better than one part in a million.

-77TABLE IV Calibration of Absorption Frequency Meter HP X532A #15030 with a Transfer Oscillator Measurement. Actual Frequency Gc/sec. Frequency Meter Reading Gc/sec. 9,0468 9.0470 9.1792 9.1790 9.2157 9,2160 9.2234 9,2230 9.2345 9,2350 9.2614 9,2620 9,2951 9,2950 9,3433 9,3430 9,4003 9,4000 9.5073 9, 5070 A list of all the electronic equipment used is given in Appendix Eo,

Chapter IV RESULTS A o Cr 3 IN SAPPHIRE Data for the resonant field conditions as a function of hydrostatic pressure were taken for the single resonance line of the I - 0 Cr3+ ion at five positions of angle and field. These were the low field and midfield lines with the magnetic field parallel to the crystalline axis, the higher two lines at 54,70, and one midfield line at 300. The last three "off-axis" resonances were used primarily for a check, For the principle chromium isotope no hyperfine interaction exists and the resonance condition for the magnetic field parallel to the c axis shown as equation 14 in Chapter II becomes /v -g, IH +s duaMI] (56) The most significant pressure dependence was observed for D,70 the crystalline field parameter in the spin Hamiltonian. It will be discussed first followed by the results for gz and A. All shifts were determined relative to the parameters obtained by Schulz-DuBois37 and Laurence and Lambe,39 g= 1o9840 +.0006 2D = -11 493 + 6Mc/sec. (57) A =B = 48o5 Mc/seco -78

which agreed with those obtained here, gz = 1.9850 +.001 and 2D = -11,492 + 10 Mc/sec. D Shift. The pressure dependence of the zero field splitting, D, was determined with the low field line with the magnetic field parallel to the crystal axis and checked at three non-parallel positions. The low field line is due to the M = +3/2 to +1/2 transition for which equation 56 becomes (58) Because the change in gz was expected to be very small compared to that in D from previous pressure experiments and, indeed, showed to be small (5 x 10-5/katm as opposed to 2 x 10-3/katm.) from this study, it was neglected in obtaining the pressure shifts in D. Since both samples were subject to the same microwave frequency, the above resonant condition held for both the high pressure and reference samples so that (59) where the primes indicate the high pressure condition. Solving for the change in 2D gives na a = - 9, p PH w (60)

The analysis of the experimental data is explained in Appendix D, The results are shown in Figure 16, Note that since D is negative, the results were plotted with increasing magnitude downward so that the negative slope implies a relative increase in magnitude. The A2D obtained from this experiment was added to the value of 2D obtained from Schulz-DuBoiso37 A least squares fit to the straight line yi = a + bxi gives a slope b = -21.15 Mc/sec. katm. (61) The error in the plotted values can be due to the field measurement, the pressure measurement, and the location of the zero condition of the absorption derivative. The field was measured to + 0,15 kcP ("Proton" kilocycles/sec. for proton NMR probe, see Appendix D), or + 0,1 Mc/sec. The pressure measurement was to +1 atmosphere in several thousand and is therefore neglected. The accuracy of the absorption line zero determination can be estimated from the line width, about 10 gauss, and the signal to noise ration, which was about 400:1, (The signals were driven off the recorders in order to enhance this accuracy in the zero determination,) Assuming constant line shape and linearity of the second derivative between the two points of maximum derivative yields an accuracy of + 0,025 gauss, better than the accuracy of the field measurement, This was experimentally confirmed. Thus the error in A2D becomes

Shift in 2 D in Ruby from Ms = 1/2 to 3/2 transition at 8 0; g,, assumed constant -11,493 Mc A2D - -g,,, AH - 11,500 - - 11,520 - 11, 540/ Slope = -21.15 Mclk Atm. - 11,560 2D Megacycles -11,580- - 11,600 -11,620 0 1 2 3 4 5 6 Pressure 1000 Atmospheres Figure 16. Zero field splitting versus pressure for Cr3+ in A1203.

-82E~= fox+ 0 a./sc Mcv'/ -e = o_.2, P / ~c. (62) This yields a relative change in 2D of 2 D.O.OR vD k attl. (63) The method used to check this value for other resonant conditions was to use the pressure dependence determined above and check for agreement in the transition energies, such as equations 17 and 18 in Chapter II, with the experimental datao This was done because of the extreme complexity in solving for D from these equations. In all cases the agreement to the second significant figure in the pressure shift was excellent. g Shift. The pressure dependence of gz, the impurity ion's gyromagnetic ratio, can be obtained directly from the midfield line with the magnetic field parallel to the c axis. The midfield line is the M = -~ to M = 2 transition and equation 56 simplifies to (64) Again, since both samples were in the same cavity and subject to the same microwave frequency, the resonant condition for both samples was the same and

-83~ ~P~H -= D~S~H~~ ~ (65) where the primes indicate pressure. This leads immediately to the result / H-I bH =a =9R9 = 9 H = 9 H' (66) Analyzing the results of the experiment as detailed in Appendix D led to the plot shown in Figure 17. In this plot the Agz determined experimentally was added to the gz value from Shulz-DuBoiso37 As can be seen from Figure 17 there is an initial first step of about 1 x 10-4 and the remaining points fall on an approximate line that has no slope outside of the experimental error but has a slight general upward trend. In the following discussion it will be shown that neither of these is significant. The initial step between one and a few hundred atmospheres was reproducible and occurs in the region of the largest physical changes in the microwave cavity as indicated by the changes in the cavity frequency. For example, the total change in a 905 Gc/seco frequency between 1 and 7000 atmospheres is about 120 Mc/sec, Of this, about 60 Mc/sec. occur in the first 100 atmospheres and 100 Mc/sec in the first few hundred atmospheres Because

Shift in gl,,for Ruby from Ms = -1/2 to 1/ 2 transition at 00 H-H' AH Ag,, H' g,1 H' g1, value 1.9844 109843 o 0~~~~~~ 0 0 0 0 1.9842 109841 1.9840 0 1 2 3 4 5 6 Pressure 1000 Atmospheres Figure 17. g value versus pressure for Cr3+ in A1203

of this and the fact that the step maintained the relative sign and proportional magnitude of the remaining field difference when both samples were at atmospheric pressure, it is felt that this step was due to either rotational or vertical motion of the high pressure sample with respect to the reference sample. The rotational motion could arise from a circular motion of the pentane as it enters the sample holder in the initial compression but is not likely to be reproducibleo The vertical motion arises from compression of the high pressure sealing cone shown in Figure 11 to which the sample holder is attached. This would be readily reproducible. An order of magnitude of the effect can be estimated as follows. The sealing cone of mica has a thickness of 0,030" and a very small conical angle, ~ o A compression of Ax in thickness will result in a vertical mot ion of Ax/sin $. Since the angle is small, a vertical motion of about 0,o030" can be taken as a reasonable upper limit. The samples are 1N3/4V" apart so this is a 3% changeo The zero pressure field difference due to slight misalighment or inequivalent fields at the two samples was 30 kcP, Assuming that this difference is linear a 3% change would allow a 1 kcP change which is equivalent to 2x10-4 in Agz, larger than the observed step.

=86Rotational motion would make the magnetic field no longer parallel to the crystalline axis and would allow mixing of the very large D term, This would add a correction term of the order of DR,'/ P/ H/ where - is now the small rotation, From the magnitude of the step, about one part in 2 x 104, an estimate of the necessary rotation can be found, Both gz H and D are on the order of 104 Mc/sec. Thus, one part in 2 x 104 of gzp H is 0,5 Mc/sec., or - M-):/I. a.'D'-' YS~ ~ (67) Thus a rotation of a few tenths degree could introduce such a step. A check for rotation of the sample with cycling of the pressure was performed. The change of crystal orientation was observed by the motion of resonance lines as explained in Chapter III, Part Ao This could be done to +0,20 and no motion was observed, This does not preclude, however, that this effect does not contribute, but is an indication that it is less likely,

-87The slight slope shown in Figure 17 is well inside the experimental error. In spite of this an effort was made to see if it could be significant. As shown in Appendix D the zero pressure field difference correction was normalized to the zero pressure condition. This correction is not the same when normalized to a different pressure. Renormalization to the next lowest pressure decreased the slope somewhat. This, coupled with the fact that in rechecks of the g shift the slope disappeared and in one case was slightly negative corresponding to a larger negative initial step, led to the conclusion that the two effects are interconnected and the slope is not significant. In any case, an upper limit can be put on the change in the gz value or change in the difference in gz from the free ion value of go = O 200230 By far the largest error occurs in AH since g is fixed and HI is measured to one part in 105o From Appendix D (. k'0 _ O -H V (68) where all the fields are measured to +0,15 kcP in 1.5 x 104 kck and ) to five parts in 105l It is assumed that the field inhomogeneity does not change. The factsthat the step was reproducible within other experimental errors and the field was always cycled repeatedly before measurements support this assumption,

-88= Therefore an error estimate for AHwould be E~ = i lo o..+ /x /o ~ o (69) E e =o, D-Y Thus & 1 v /, - 19 0*;7. 0 t(70) or since go-gz is about 2 x 10-2 I9 KffS~i| < S<~ = X /0j,, Q(71) over a range of several thousand atmospheres. Therefore it can be said that the change in the difference between gz and the free ion value of go is certainly less than 2.5 x 10-3 and probably on the order of 1 x 10-3/katm. or less. A Shift. An attempt was made to observe a shift in the hyperfine splitting due to the Cr53 isotope which is 9.5% abundant and has an I = 3/2. Only the two outermost hyperfine lines, m = + 3/2, were observable and only at the midfield line with the magnetic field parallel to the crystalline axis~ They were observed as two "pips" or "'wings" on the,sides of the very broad I = 0 resonance

Because there was no clear cut zero to the derivative of these absorption lines, a determination of the A value could be made to only two parts in one thousand. A slight decrease in A, on the order of 0,5 x 10-3/katm., was noticed. This shift was the same order of magnitude as observed elsewhere (0,6, 0,44, and 0.035 x 10-3/katm.20,22) but since A could only be determined to 2 x 10,it was inconclusive. B, V2+ IN SAPPHIRE The spectra for the V2+ ion in sapphire is more complex than that for ruby, This is primarily due to the addition of the hyperfine interaction with the vanadium nucleus which has a nuclear spin of 7/2. With small (2 ) misalignments, forbidden hyperfine transitions are allowed so that only when the magnetic field is parallel to the crystalline axis is the eight line "signature" of vanadium undistorted, Further complications arise because at X band frequencies the midfield transition occurs very near the energy level crossing indicated as a dotted enclosure in Figure 3, Both here and at the low field transitions the mixing of states has been observed39 resulting in a very complicated spectrum, Since the ruby data indicated that g shifts could not be observed, g was assumed constant as before, and the necessary data was

90O taken from the high field lines, the only usable spectrum at X band frequencies, for which M = + 1/2 to +3/2. For the above case equation 14 in Chapter II becomes (72) where f is the second order hyperfine correction terms and is a function of A, G, gz, H, D, and m. The concentration of the V3 ions in the vanadium impurity samples was the same as that for the Cr3+ ions in the ruby sampleso Because the conversion of the V ions to V2+ ions by X irradiation is not very high and the same signal strength is now spread over eight lines instead of one, the signal to noise ratio was considerably reduced. The cross talk bucking circuit described in Chapter III, which eliminated the sample cross talk between channels, is obviously dependent on the modulation amplitude. Because the modulation negative feedback is dependent on both modulation amplitudes, maximizing the high pressure sample signal while still eliminating the cross talk resulted in insufficient signal to noise for the reference sample signals. No reasonable balance which maintained the required accuracy was possible so a straightforward determination of the parameters A and D as a function of pressure was done using only the high pressure sampleo In order to overdetermine the values, the resonant

_91l conditions were observed for the first three (most clear) hyperfine lines at each pressure, for which m takes the values -7/2, =5/2, and -3/2. The data was analyzed as detailed in Appendix D by an iterative process. Since the value of A was expected to change very little compared to the change in D, it was assumed constant with pressure. The zero pressure parameters 39 were taken from the work of Laurence and Lambe and are: gz = 1 991 2D = 69 607.2+2 OMc/sec. A = -220,o 615+0o 025 Mc/sec B = -222,890+0o100 Mc/sec, Once D as a function of pressure was determined, its dependence was used to determine the dependence of Ao From these the second order correction was corrected for pressure and the process repeated. ID Shift. The results of this analysis for the pressure dependence of the zero field splitting are shown in Figure 18. Again, the D value increases in magnitude and is plotted downward as an increase in a negative numbero The slope by a least squares fit to the straight line Yi = a + bxi is b = -17.0 Mc/sec. katm, (74)

-9,600 SHIFT IN 2D WITH HYDROSTATIC PRESSURE -V2 IN A1203 TRANSITION M= 1/2 — 3/2; m= -7/2, -5/2,-3/2; e=00 -9,620 SLOPE = -17.0 Mc / k ATM. RELATIVE CHANGE=+ 1.77 x 103/k ATM - ERROR I -9,640 - 2D Mc -9,660 -9,680 0 PRESSURE k ATM 0 I I I I I 0 1 2 3 4 5 Figure 18. Zero field splitting versus pressure for V2+ in A1203

The error is now larger because there are additional differences in the calculation and the signal to noise ratio was reducedo Also, the resonance probe for field measurements used a deuteron resonance so the +0,15 kcD becomes about +0,7 Mc/sec. The uncertainty in this value due to the reduced signal to noise ratio (5021) raises this to +1,0 Mc/seco Since A2D is now the error in A2D becomes E - + o, 5 + 05- 1 i, o ",o _ ~ 3. 0 Mc /.r Ec. 6A~D (76) This yields a relative change in 2D of -— z — = (/77 o. ) x to/ k'tm. (777 A Shift. The larger errors here were expected to mask any pressure shifts in A unless they were unusually large. Expected shifts from other experiments would be on the order of 10=3/katmo With the value A = 220 Mc/sec., this would be shifts of about 0,2 Mc/sec, katmo which is well inside the experimental error, No large shifts were observed. However, the data showed again a slight decrease, as in the case for ruby, of the expected order of magnitude, But this also was termed inconclusive.

-94CO SUMMARY Listed below are the changes in the spin Hamiltonian parameters with hydrostatic pressure determined in this experiment on Cr3+ and ions in -Al 0 2 3+ Cr3+ &,,2 = (], 8 + 002)x /O-yk o..O. ( 3 _ * I u a. x / k k v2+ i1: n - (i. 77 o.~ ) /0-3/k*m

Chapter V CONCLUSIONS A. DIRECT CONCLUSIONS Hydrostatic compression of the sapphire lattice con2+ 3+ taining isoelectronic V and Cr3+ impurities has shown that the crystalline field parameter, D, in the spin Hamiltonian varies linearly with pressure to about 7000 atmospheres for both ions. In addition, the relative changes, AD/D, for both ions are very close in value and indicate an increase in magnitude. Similar results have been obtained for the cubic field splitting of other isoelectronic ions. Mn2+ and Fe3+ in Mg020 and Gd3+ and 23 Eu+ in SrTi03 show this same identity in behavior with pressure. With no other considerations it can be concluded 2+ 3+ that both the V2+ and Cr3+ ions occupy the same or very closely the same lattice position, though not necessarily the aluminum site, and that the compressibility about each ion is nearly identical~ The implication that the ions occupy similar sites in the same lattice is supported by the identical pressure dependence of Dq found by Drickamer for five different ions (Ti3+ V3, Ni2, Cr3, and Ni3+) in sapphire. 3436 It is in opposition to the substitutions proposed by Weakliem and McClure,41 who postulated a varying displacement along.95

the c axis to explain the differences in the trigonal field parameter calculated from optical spectra for various 3+ 3+ 3+ 3+ 3+ 3+ triply ionized ions (Ti3+ V Cr3+ Mn3 Co3+ and Ni in sapphire. However, equal trigonal field parameters were found by Geschwind and Remeika71 who carried out precise measurements of the anisotropy of the g shifts from EPR spectra. The anisotropy in g-go arises because different excited states are coupled to the ground state for (g-go), and (ggo)_L,(see Appendix G). Since the excited states are split by Vax, this anisotropy can be used to determine the splitting. They observed the 3d ions Co2 V, Cu3+, and Ni2+ and the 4d ion Ru3 in sapphire and determined nearly equal trigonal fields for all but Cu3 and Ni2+, Kamimura63 has recently explained this behavior and by invoking distortion of the covalent bonds of the impurity has determined equal trigonal fields for all of the above ions implying similar positionso Another experiment discussing ion position is by Laurence and Lambe39 who observe widely different values of the electric field gradient at the impurity nucleus in ENDOR experiments for the isoelectronic ions, Cr and Mn 4 in sapphire. These values and other data are collected in Table Vo They point out, however, that the variation in the electric field gradient (non-monotomic and opposing signs) shown in Table V cannot be quantitatively explained by simple point charge model displacemento

-97TABLE V Comparison of crystal field parameters for V2+, Cr3+ and Mn4+ in sapphire. Va- C 3+ M3-+ 13 79 I 0 (c~ ~. 13 13,6 00 J 63 - (20 k a(e 6_,0 B| 3 0f 75 3 OG K ~c-3 U13,s, &0o 40ki (X0 k)O 3 3o I FR, T 0o qOkA, 3_ 37 7' D ~a 0.383 O. 3 I,a -D*8 O,01 35 j'Vto7 _7__ 37 "-!'700'370 PACFTR 6I 3'6 /\ 13 ~x (cm~-) 0 X CO-3/'-' I ~' —-- Cv

=98= V3* C i Mtn R t V 1 * Q (b) o.3 - 0.03 o.3ss I.'? -CIL - 1.87 c. 7Ya I (1-,) | 12.7 q3 g (g urxs) | -140 I9, 00 -2. Lco -197,9 00,, N.u. 0,53 ]? G 1 3 0 ~8 o, TI < r> (a. I8 1.8 O 7 * No independent determination of Q has been done. These values were arrived at by assuming the field gradient was the same as for A13+o ** Calculated by Watson from Hartree-Fock wave functions. Those in parantheses determined experimentally from 10tQ = "t'f.,'/3, and linear scaling to < r2 >.43

It should be pointed out that all of the above determinations except the field gradient have involved the spin-orbit coupling whose anisotropy63 must also be accounted for as still another parameter, Examining the previous experiments as to the location of the ions, only two imply varying displacements. Laurence and Lambe calculate a displacement for the Cr3+ ion and propose other distortions to explain the other electric field gradientso Weakliem and McClure41 originally proposed displacement from varying axial field parameters determined from optical spectra. The axial field parameters from EPR spectra71 do not show this variation, even though both were determined from the many parameter crystal field theory. In a molecular orbital calculation Lohr and Lipscomb show that the zero field splitting, 2D, is a more sensitive function of position than is the trigonal field splitting, 3/2 K, which is more sensitive to configuration mixing. From the similarity of the values of 2D from Table V and the above discussion, it seems reasonable to conclude in agreement with Drickamer34,36 and Geschwind and Remeika71 and Kamimura63 that the Cr3+ and V2+ ions substitute in similar sites in the sapphire lattice though not necessarily the aluminum site~ Further analysis at the end of the next section supports this conclusiono

-100The difference in charge of the two ions, V2+ and Cr3+, suggests that the local compressibilities would be different. The local ionic binding energy would be ex2+ pected to be significantly lower for V leading to a 20 higher local compressibility. Also, the polarization of the 02- ions72 would be less, leading to a higher local compressibility. But the similar pressure dependence implies that these charge dependent effects are not noticeably different for the two ions. Though the V2+ ion is larger and higher covalency would be probable, the values of B/Bo and /A,, from Table V indicated no significant difference, so one would expect no difference in compressibility due to a different degree of covalency. The fact that no anomalously large shifts in the difference of the g value from that of the free ion, g-go, or in the hyperfine splitting term, A, are observed implies that the local compressibility is the same or smaller than the bulk compressibility for sapphire~ Thus it can be concluded, in agreement with Drickamer34 36 that the compressibility about the different ions is the same and comparable to that of the bulk compressibility of sapphire. A further test of this discussion would, of course be the pressure dependence of Mn4+ in sapphire.,

-101l B. ANALYTICAL CONCLUSIONS It is worthwhile to compare the results obtained in this experiment with those from a spin-phonon interaction experiment by Tucker28 and a uniaxial stress EPR experiment on ruby performed by Terhune26. Tucker analyzed his experiment for longitudinal acoustic waves along the c axis with the interaction represented by a perturbation of the form Ge, = Gte,2lF:i (78) where G is defined as the magnetoelastic coupling constant and ezz the elastic strain amplitude The product Gezz is the change in the coefficient of Sz with uniaxial displacement and should therefore yield the A2D for a uniaxial pressure experiment. He obtains a value of G2 = 2.8x10-30 (ergs)2 for longitudinal acoustic waves along the c axis. (Note. The reference 28 reads 1.5 x 10o29 but this has been corrected by private communication26 to the quoted value.) This yields a A2D/2D = 4.5 x 10=3/katm. From this and the data perpendicular to the c axis a hydrostatic pressure shift of A2D/2D L= L58 x 10 3/katm. is predicted which has an error of about 50%. In the uniaxial stress experiment Terhune obtained a value of A2D/2D-(5.2 + 0.3) x 10=3/katm, for a stress parallel to the c axis. From this and the value obtained perpendicular to the c axis,

.102he predicted a hydrostatic pressure shift of A2D/2D = 1.41 x 10-3/katm. also with an error of 50%. Both of these predictions include the observed value for hydrostatic compression of A2D/2D = 1.84 x 10-3/katm. The observed values of changes in D with hydrostatic and uniaxial compression can now be compared to those changes predicted for the a20 term in the crystalline field expansion shown in Table III. This can be done using equations 49 and 51 (Chapter II) which are Das L 6,) s (49) and (9 9O d.Dc. (51) The parameters on the right are proportional to the position variables as follows. % o(la() 0. (79) X c r c (80) (81)

-103where c rn > indicates the average value of rn over the radial part of the 3d wave functions. There is a reasonable amount of evidence that g-go varies with pressure primarily as Dq-1. Drickamer's experiment36 showed that for ruby Dq behaves like R to about 50,000 atm., implying that < r4. changes very little in our pressure range. In Walsh"s experiment20 on Cr3+ and Ni2+ in MgO, he obtains values of'(9V~) 2s 0 + 0.2 and 1.9 + 0.3 respectively. The corresponding values for from Minomura and Drickamer34 are 1.84 and 2.0. This is an indication that both < r4and r-3> also change very little. In the same experiment Minomura and Drickamer observe (as in ruby36) that the Racah B parameter changes very little,.026%/katm. From the above discussion and the small change in B for ruby it seems reasonable to conclude that to a first approximation < rn changes insignificantly, and the parameters in equations 79-81 become 6 aao (82) = O I - (83) 7 Rt (a4)

Substitution of a value for ADq/Dq calculated from equation 84 and the elastic constants and of the point charge model values of Aa20/a20 from Table III for a substitutional 0 position and +0.15 A removed yields equation 85 and 85w, respectively. These are (in units of 10-3/katm.) aX = O.,f t,. 0._ o= I -.7.; 0o 03 (85) =7 o,'18 ~o10 # -o.O 0o'03 which are compared to the experimental value of 1.84 + 0.02. Similarly the uniaxial pressure values give = 3 +D OI —. (- o.6 "'~0~).o0 2. L'0.03 a.D (86) aD = I,3O_.O/ l. (-_o.,0,o) o,/+io~ 0.03 (86g) which are compared to the experimental value of 5,0 + 0.2, where 860 is for the removed position and ADq/Dq is now Drickamer's experimental value. If relocation of the impurity ion is used to look for agreement, in both pressure cases the ion position would have to be far removed in the opposite direction from that proposed by McClure,42 Laurence and Lambe,39 and Lohr and Lipscombe.44 There is also no position at which both equations would agree, so sub

-105stitution of the impurity ion at a site other than the aluminum site cannot account for the results without relaxing the other basic assumptions. The three other basic assumptions from Chapter II-E which were used to look for agreement can now be questioned. These were, 1. The change in the spin Hamiltonian parameters is due to changes in the crystalline field. 2. The local compressibility is the same as the bulk compressibility. 3. The point charge model can be used to describe the changes. With regard to the first assumption, the optical experiment using uniaxial pressure on ruby by Drickamer36 showed that equation 84 appears to be applicable. This and the small decrease in B were used (see Chapter II-E) to assure this assumption. The same experiment was also used to assure the second assumption. Since it is one of the objectives of this study to check on the last assumption and since the other assumptions are experimentally supported, it can be concluded that the point charge model is inadequate to describe the observed pressure effects. Disregarding any other experiments and relaxing all the assumptions except that of the point charge model, a check

: X06 on the D shift, equation 49, can be done using the g shift, equation 51. Substitution of the latter into the former yields (87) or substituting the Table III values for a20/a20 as before and the experimental upper limit for A(g-go)/ (gago) yields as -- pr (88) r >. f O0qa wO 01 ~ rt ~ (88') which are compared to the experimental value 1.84 + 002. Thus, this experiment alone does not eliminate possible agreement and an accurate value of the g shift is desired, However, arguments will now be given that any expected value of a A(g=go)/(g-go) would be negative and stronger than <r >,/' ro, again leading to disagreement. Stuart and Marshall have argued that the decrease in the spin-orbit coupling parameter, X, for a free ion upon substitution into a crystal is primarily due to expansion of the impurity ion wave functions. This arises from screening of the 3d electrons by overlap of the ligand charge clouds. They postulate that the

-107radial wave function expansion can be represented to a first approximation by a simple scaling factor with appropriate renormalization, ioe. Tohe, hv'sed-(89) They have quantitatively supported this approximation with neutron scattering form factors- for Mn2+ ions and superhyperfine splittings in MnF2 with k A 0.9 going from the free ion to the solid. With this approximation the variables C 9 X 9 and Dq will now have a k dependence from equations 79-81 with k decreasing further with compression. For example, consider the integral of r2 < r J fr r $= r'dr 50L IL1 SOL D doL-. Substituting the scaled free ion wave function yields < rr, Sr 3i r) r) r cl r S of.,t D F F E I or'.LI9 ke, J FRE F,~EE' But since a change of variable doesn0t affect the integral S' Rkr) k~r f(kr) kr ) &c j; fr r~ and one obtains

=108-.S O ki F (I CSimilarly the k dependence of all the parameters can be shown to be X N< k3 ~D "< k`- R - (90) For small decreases in k (Ak/k is a negative number) equation 87 becomes tR.D R k0, k IY OL;I (91) or a -D A Ma0 k R (92) or substituting values as before f R () (93) (93~)

-109Since Ak/k and AR/R are both negative numbers for compression, agreement still is not possible with the point charge model. Now it is possible to look analytically at the ea 2+ 13+ equality of AD/D for V2 and Cr o Equation 49 with the variation of all the parameters becomes a> -.. r>> — - o (94) or in terms of the expansion scaling factor, k, a D ea._ a'd R.%~a /if R. (95) The expansion factors, k, would not be expected to vary significantly between V2+ and Cr. In fact, /4"free ion in this analysis equals k3. From Table V the values of f.i. are similar so the factors k=:are quite similar and Ak/k in equation 95 for Cr and V should thus be nearly identical. Disregarding the expansion one can argue similarly for equation 94 from the equality of the pressure dependence of the c.ubid field splitting, a, for isoelectronic S-state ions, which was indicated earlier20,23. The cubic field splitting for S-state ionsfdepends in some manner on A and Dq and therefore on < r-3 >, r4 >, and R. The fact that the pressure dependencies are equal in the same lattice implies that either r r and are -< r-'> and lr r' are

-110the same for isoelectronic ions or small compared to zR/1. In either case one could expect the same to hold true for z~'rTa/~ r, Assuming, therefore, that the n. (or the - ) are the same for both ions and knowing the experimental equivalence of A2D/2D one can write (LsL26Cr3* (A D) J/w(96) or Since a20 and R are parameters that depend only upon the lattice and not the ions, it can be concluded that the ions occupy similar lattice sites and are subject to similar compressibilities. C. SUMMARY In summary, it can be concluded that: 1. The V2+ and Cr3+ ions occupy similar lattice sites, though possibly including the displacements proposed in the literature. 2. The local compressibilities about the ions are similar. 3, The local compressibility is approximately the same as the bulk compressibility. 4. The point charge model is inadequate to predict the pressure changes.

111D. FURTHER STUDIES The most immediately useful experiment which would complement the work reported here would be a determination of the changes in Dq and, with hydrostatic pressure for ruby. The former can be done optically but a direct determination of the latter is currently unknown. Since the pressure dependence of the g shift involves A and Dq in the same manner as in the pressure dependence of D, a complete quantitative analysis according to equations 49 and 51 for ruby could also be done with a better determination of the change in the g shift and a more realistic prediction of the change in The latter could be done by a molecular orbital calculation, such as done by Lohr and Lipscomb,44 as a function of displacement of the surrounding ions, and the former is discussed in detail below. A measurement of the change in (g-go)z and a more precise determination of the D shift for uniaxial pressure would similarly provide all the data necessary. it would also be interesting to see if the D shift for a third isoelectronic ion, Mn, would agree with those of the Cr3+ and V2. An optical pressure experiment on V2+ in sapphire could also confirm this agreement. Other EPR experiments which would mutually complement this experiment and that by Walsh20 would be the two S state

-112ions, Fe3 and Mn2, in sapphire and V in the cubic MgO. It also comes to mind that resonance shifts with pressure, more practically optical resonances, could be used as a pressure measurement with a potential in accuracy on the order of a per cent or less to pressures of several hundred thousand atmospheres. Further measurement of the g shift. From the preceding study and previous high pressure EPR experiments20'22, an expected value of A(g-go)z/(g-go)z would be 1 x 10-3/katm. Since (g-go)z for ruby is about 0.02, this is a shift in g of 2 x 10-5/katm. For 10% accuracy a g shift experiment should thus be able to see aAg of 2 x 10-6/katm. Since this can be observed over several thousand atmospheres, a reasonable g shift experiment should at least be accurate to one part in 105 in the measurements of the g value. The necessary restrictions for such accuracy can now be determined from the g shift analysis in Chapter IV, The frequency and the field must be measured to about 5 x 10 6, which requires a factor of two better field measurement. A constant inhomogeneity in the field of 0.1 gauss per inch would be sufficient to permit the suspected motion of 0~030"w, This is readily obtainable without the bomb, but inhomogeneities inside the bomb would have to be determined. Rotation would have to be held to less than 1/60. The mixing of the change in D with

-113pressure becomes too large with a misalignment of + 3~0 Assuming constant line shape, the determination of the center of the 12 gauss wide absorption line requires a signal to noise ratio of about l000lo If all of the above requirements can be met, a A(g-go)z/(g~go)z 1 x 10-3/katm. could be observed.

APPENDIX A CRYSTALLINE FIELD FOR SAPPHIRE The crystalline electric field at the impurity ion site can be calculated by approximating the surrounding ligands by point charges and assuming that there is no overlap of the impurity ion charge and these point charges. (The validity of these assumptions are discussed in the text (Chapter II Part D) and will not be repeated here.) Then the crystalline electric field in this region has a potential which obeys Laplace~s equation (Al) Therefore, the crystalline field potential can be expanded about the impurity ion site and Laplace0s equation has solutions of the form \/(t) nn a~g~t r By (' )(A2) where r,'i, and P are the spherical coordinates of r centered at the impurity ion site and Ylm ( -J ~ ) are the spherical harmonics. The z axis will be taken as the axis of highest symmetry, the C3 or three.fold axis in the crystal. In order to evaluate the coefficient a )ms the potential in equation A2 is equated to the potential at the position r Q 1 14.

-115YV_(r) = irr. /Lr-r/ (A3) due to charges, qi, at the surrounding ion sites, ri, whose coordinates are ri, I i, and a i. From the law of cosines this potential becomes YedL E /r'r;_ 0r(A4) which in terms of the Legendre polynomials becomes \/)-4e gi r(*) o~ (cos ) (A5) where o< is the angle between r and ri. The Legendre polynomials are related to the spherical harmonics by \;oigp,~ (=j~~ S )(A6) so that equation A5 is now V(r) = y (i) C ( ) o ( (A7) The use of the spherical harmonic addition theorem (A8) gives for equation A7 a)- U: $ ~62, ) - A9 A~~~~~~~~~(9

-116Comparison of equations A9 and A2 show that the coefficients a Om are c5.~ LX (A10) where (All) The expansion in equation A2 with the coefficients in equation A10 can be restricted now due to the symmetry of the impurity ion site and many of the a /m vanish. The symmetry of the sapphire lattice at the impurity site is 3 (C 3) This means that the potential is invariant under 3 any of the symmetry operations of the C3 point group. These are E, C3, and 2C3. Application of the C3 operation demands that v (~r,, V, -,-')= V(r~ W) (A12) The spherical harmonics, however, obey the relation y (-,, —) - e (A13) So that e is unity, the product mp is restricted to integral miltiples of 21v or, since in this case p = 2lr/3, values of m to

-117 The basis functions (equation 3) used to evaluate any matrix elements for the crystalline field potential will consist of 3d wave functions for which 1 = 2. This will lead to integrals of the form 18 Ma M ( )cF 1m, (A15) In order for this integral not to vanish 1 is limited to values of 4 or less by the triangle theorem of the spherical harmonics. Also, the parity of the two basis functions are even, their product is even, and therefore the parity of the crystalline field term must be even. This limits 1 to the values 0, 2, and 4. Thus through symmetry restrictions and the evaluation of matrix elements which do not vanish when the 3d wave functions are used, the crystalline field expansion, equation A2, becomes V(rC) a %~ o V(L) 0 t0 t t~3r t3)'(A16) The aoo term will just add a constant to all energy levels and can be eliminated from consideration. The symmetry restrictions of equation A16 are for the orbital angular momentum operators since they enter through position coordinates. The spin operators in the crystalline field term of the spin Hamiltonian, however, must obey the same symmetry restrictions because they enter only through the

-118spin orbit coupling. It can also be shown that any polynomial in an arbitrary angular momentum operator, Ji, of degree 2J + 1 can be reduced to a polynomial of degree 2J-l, or ~a+ I - I a,-..' J - --- (A17) Both Cr3+ and V2+ have an S = 3/2 so the polynomials of the spin angular momentum Operators of the form S. 2S+ 1 (or Si ) allowed by the symmetry restrictions in equation A16 can be expressed as polynomials in Si (or s.2S-1). This removes the spin terms which transform like Y40 and Y43 from consideration. Thus for a 3d3 ion whose S = 3/2 only the term with the coefficient L, o (A18) need be considered. The spherical harmonics in terms of Legendre polynomials are Y*,; (- us = t (1 )OYOi' (A19) Substituting this in a20 yields the a20 given as equation 35 in Chapter II or (A20)

where (A21) It can now be shown that the point charge model can be used to describe the changes in the electrostatic field for an arbitrary ligand charge distribution if the compression is conformal or angle preserving. For an arbitrary ligand charge distribution the coefficients a,m in equation A10 becomes ~ = 6(pJ*/ ) Z (, @ / a )c - (A22) for a distributed ligand charge density P. (This can be seen by replacing the point charge distribution, qi at (ri f:i whi)e in equation A3 by an arbitrary nfl (R) p, = i,/./ where the ionaligand and ligand-ligand overlap is zero. The potential becomes V(r)-=Y-. and results in the coefficients in equation A22 in a completely analogous manner.) For the special angle preserving compression (A23) the new coefficients a m" can be expressed in terms of the old by equation A22 as

-120and Since AR/R is a constant independent of R, this model should accurately preduct any changes due to a conformal compression as in a cubic structure.

APPENDIX B CHANGE IN CRYSTALLINE FIELD WITH PRESSURE The change in the crystalline field potential with the application of pressure can be determined from the elastic constants, the lattice constants, and the compressional model derived in Chapter II. The model leads to the following equation (equation 38) 1tAO )-~A aO d ~O, (Bl) where aa ar3io 0(B2) and for xk = x, z xd x sok; * ) r L3 4 Xk (B3) and Q is just a combination of constants. The Axi and Azi for hydrostatic pressure are obtained from equations 25 U LA E = = ( x I (4) P{ (ee = (c S,3 + S33)1 by multiplying by the appropriate ion coordinate as follows -121

-122 A XL = t XS _ ST XS (B5) From Table II the values of U1 and U3 are -3 A, 3 LA, --,1~ xto /.'f> (B6) where 1 atmosphere 1.01325 bars (106 dyne/cm.2) = 1.0332 kg/cm.2 The calculations were also performed as a function of uniaxial compression. In all cases considered here the compression was taken along the z axis. For uniaxial compression (see Appendix C) the stresses of equation 24 become = -?P; P. = = a / = ps -?s ~ o (B7) From the matrix multiplication shown as equation 23 the values U1 and U3 in terms of the elastic compliance coefficients become Liz = - 5/3 p, = - s3/. P (B8) Substituting the compliance coefficients from Table II yield the values

-123U1 0.0395 x 10-3/katm. U 3 Uniaxial U3 = -0.225 x 10-3/katm. The sums in equations B1 and B2 run over the six oxygen and the fourteen nearest aluminum ions. Since many of these ions have the same effect, the sum can be reduced to eight terms with the coordinates for the eight sites listed in Table II. Each term must then be multiplited by a weighting factor, Ni, which is just the number of ions which have those particular coordinates. Also, since the constant Q appears in both the numerator and denominator, it can be dropped. With these simplifications, the substitution of equations B2, 13, and B5 into B1 yields L =I, ( (B9) -3 // For calculational purposes the numerator can be simplified using the following relations: ro -o=;.s (BO) \ D _I_. m m m ~x~ ~ O' SVri (30

-124as shown in the following steps, N1,,rR1 X; ( Z/EY s/,t<R>P; ~+ t X ~iY rr,..: r-3 Lu IVl (t, >+ Io l')+ U oS; ( -,/' Equation B10 is now;:,,. _ r, —'3 IV;~x~ (o (o;3.~t, st cC: CC ~ - ~SP' O _. _. -'~~ C 6eL t, p(Bll) where P20 -= (3 cos2 l1). This is the equation that was used to numerically calculate the values in Table III. The change in the crystalline field potential with both hydrostatic and uniaxial pressure was also done as a function of position along the z axis. To do this the values of zi were obtained from the constants ri and cos' ii in Table I. For example, zi for the oxygen site 01 is

-125r cc8 $ /i ( [i )..... 204,:. g..:7/ ij~I I i 0 If the impurity ion substitutes 0.1 A toward the empty octahedral site along the z axis, this value will be lower O O by 0.1 A or.706 A since the motion is toward the plane of these ions (i.e., up in Figure 6). With the new values zi the constants of Table II were redetermined assuming no change in xi. The calculations for hydrostatic and uniaxial pressure were then performed again. As indicated in Table III, this was done for the non-substitutional O O o O positions of -0.1A, 0,iA, 0.12A, and 0.15A where the positive number indicates a position closer to the empty octahedral site. It was seen early in the calculational process that the contribution of the fourteen aluminums was usually quite small. Therefore, their contribution was neglected in some of the calculations as shown in Table IIIo In the remaining values the aluminum contribution and the oxygen contribution were calculated separately. The table then shows the values for only the oxygens and for the combined effect of the oxygens and the aluminums. All the numerical calculations were checked by Miss P. Van Alstine by an independent determination.

APPENDIX C ELASTIC CONSTANTS In order to be able to use the elastic constants properly, it is necessary to understand their origin and meaning. Since the elastic constants are relations between stress and strain, these will first be discussed in sections A and Bo After the elastic constants and their special notation are discussed in section C, the special form that they take for isotropic or "Engineering" materials is indicated in section D. Techniques of measurement are briefly reviewed in section E. A. THE STRESS TENSOR Two assumptions are first necessary to describe the "stress" or combination of forces applied to a deformable body. First, all parts of the body are in equilibrium. Since the interaction forces in a body are molecular and therefore very short range, the second assumption is that the forces acting on any part of the body by surrounding parts act only on its surface. (This does not hold when macroscopic fields are produced such as in the piezoelectric effect.) -126

127Consider the forces acting in or upon the small differential cube in Figure Cl. By the second assumption these forces can be resolved into the forces, Cj, acting on the faces of the cube.,j can be interpreted as the force in the L direction acting on the surface, Sj whose normal is in the j direction. If i = j, it is a "normal" force, and if L j, it is a "shear" force. By convention a positive aOlp is a tensile force and negative a compressive force, It can be shown73 that the c,. are symmetric. This can be easily seen from Figure C1 and the first assumption. Since the portion of the body represented by the cube must be in equilibrium, its moments about any axis must be zero. For example, the moments about the X axis are - or and Do whose sum must equal zero. Therefore AC = O and similarly %.. = r, From the second assumption, the sum of the forces on the surface of the cube, Ol 5J ~ is equivalent to the sum of the forces within the cube, 5 F dV where Fk is the vector force per unit volume on the arbitrarily small volume element d V. From vector analysis the two can only be equal if Fz is the divergence of 0 -. or o j Le.w (Cl)

-128Xm SI aI Figure C1. Stress components..

-129Since F' is a vector, then cr-. must be a tensor of rank two. Also, the equation of equilibrium for a deformable body is then cz )e t (C2) Because it is a tensor, it can be transformed to its principle axes so that only diagonal elements exist. Some examples for uniaxial, hydrostatic, and pure shear stresses are shown below. o o 0 o -? o o o o/ )o 0P Uniaxial Tension Hydrostatic Pressure /- cr o /r o 0 \ o C/ o o Pure Shear It must be emphasized that the stress tensor is not a "matter" or crystal property tensor such as permittivity or magnetic susceptibility. It is "field" tensor and can have any arbitrary direction for its principle axes and can exist for isotropic and amorphous bodies such as glass.

-130= B. THE STRAIN TENSOR Since it is not as simple to picture strain geometrically it is easiest to understand by starting with one dimension and generalize to two and later three dimensions. A stretchable one dimensional string is pictured in Figure C2a. Consider point P a distance x from the origin and a nearby point Q at x +tAX. After stretching, P is at P a distance X - A from the origin, and Q is at a distance a + L-k from P. Since strain is an indication only in change of length, the transition from the origin is of no concern, and strain can be defined as the increase in length over the original length. Therefore increase in length _pQ &(C3 strain - -(C3) original length P More precisely, the strain at the point P is (C4) If is a constant for all X, then it is said to be "homogeneous" strain. Only homogeneous strain will be considered from here on. (For inhomogeneous strain see, for example, Reference 74.) In two dimensions the string becomes the stretchable sheet pictured in Figure C2b. There are now four e0s where

- 131X AX o P Q a.) X +u A+ AU O P O' ONE DIMENSIONAL STRAIN Xt QI b.) A QX AX. W. X.x TWO DIMENSIONAL STRAIN Figure C2. One and two dimensional strain

-132- aA, (C5) Geometrically the change A Cwould be, >eRK KJ I' ^<; = e~j AX(C6) Since ant~ and 4)j are vectors, e, is a tensor of rank 2. The special case of a4, parallel to an axis, say X? a is helpful to understand the meaning of Ce, and is shown in Figure C3a. The point cI will now not only have a component of linear extension, L~ Mt, along the axis as in the one dimensional case but shift off the axis by n 4; as well. Thus aM, ~, (C7) and e is the linear or extension component. Also from Figure C3a and from the fact that only small changes are considered TR AJu 19 d, - (C8) and e is the rotational component. Now if e is purely a strain tensor, then a transformation which allows no deformation should have all e,; = o. But a pure rotation of an angle 9 shown in Figure C3b has no deformation but

-133I a0~~~/',i o.) P AXI Q, PURE SHEAR b.) ROTATION S IMPLE SHEAR PURE SHEAR + ROTATION SIMPLE SHEAR PURE SHEAlR + ROTATION Figure C3. Examples ot specialized strains.

:134,~'i ~ o( / 1 (C9) This rotation is antisymmetric and it can be proven in general that a pure rotation is always antisymmetric as follows:-,Choose the origin as the axis of rotation and the I64Z will always be perpendicular to L xZ and (C10) Substitution from equationC6 gives Qe j, Ole LX 0 (Cll1) or jet cP; - - eje4' + (C12) The conditions (C12) are exactly those of an antisymmetric tensorQ Since any tensor can be defined as the sum of a symmetric tensor and an antisymmetric tensor, et can be defined as e. G l j a e,)+ C e ) (C13)

-1l35Aid is the symmetric part of the strain tensor and is commonly called just the strain tensor. A is the same as e_~ but tl is now (e, + e, ) and equivalent to ~ the change in angle between two axes. A simple example of this separation is shown in Figure C3c where a "simple" shear is equivalent to the sum of a "pure" shear and a rotation. Extension to three dimensions is completely general and easily done by simply letting i and j run over 1, 2, 3. Again it must be emphasized that the strain tensor will only conform to matter or crystal properties when the causing stress does. C. ELASTIC CONSTANTS AND MATRIX NOTATION Experimentally it has been known for a long time that a stress will produce a strain and that below the elastic limit, where irreversible changes start to occur, the two are proportional. This is Hooke's Law and is written 6 — Sc (C14) where S is a constant. The higher the value of S, the larger the strain produced for a given stress or the more compliant the material. The S is thus called "elastic compliance" or just compliance. Equation C14 is also written

1 36 C (C15) where now the higher the value of C, the less the strain or the stiffer the material. The C is therefore called "elastic stiffness' or just stiffness. These equations are generalized in tensor notation to (C16) and similarly (C17) where the indices range over 1, 2, 3 and the S and c now have 81 components. It can be proven that and C,:hL are tensors of fourth rank 60 Because - and oEl are symmetric, not all'of the 81 components are independent. For example, if we apply shear stress O —, t there must also be of; and,;L -O-,,2 C Iot a = (S6a (C18) It is impossible to devise an experiment to separate..;la from 5.., therefore, an arbitrary constant can be used to relate them; the easiest being equality, or "; jkz = 54, ik (C19)

-137In a similar argument, a uniaxial stress along say the X3 axis will produce the strains r, j i 3 3 Y. 4 $ (C20) But since die = CJ we can write BSjut - /.' k. (C21) Thus the independent number of components is reduced to 36. Now a change in notation is necessary before the elastic constants given in the literature are obtained. Since there are six independent components in each of the stress and strain tensors, the double indices are reduced to a single index ranging from 1 to 6, as follows-4 Matrix Notation 1 2 3 4 5 6 (C22) Tensor notation 11 22 33 (23,32)(31,13)(12,21) or in the tensor arrays. P, crsty Gu/ V:I \ aand similarly the.,j becomes i&. Thus we now have the equations used in actual practice

-138uA; = ~jpj (C23) and (C24) For example; hydrostatic pressure would be, = = - and uy / P ^ P6 = ~; uniaxial tension is o -= p and 90; pure shear is p, =-. and,4O = O or p,5. O and pl = p. Since the arrays Sj and C., are selected components of their respective fourth rank tensors, they are not tensors of rank two and do not transform like one. If it is desired to transform to a new set of axes, it is necessary to return to the four index notation. However, the arrays are still symmetric and only 21 independent components exist for an anisotropic material with no symmetry. Crystal symmetry will reduce the number of constants still further. For example, in the 4 class with the 4 (gg) axis parallel to X3 the axes transform as follows. Therefore, in the tensor notation

-1393 3 3 -3 3- 9 or in matrix notation I/ 3 ->; -- - The array of indices becomes in the matrix notation la -' _r' -/ -t?, /I 1 3 -0i -7A 33? -3 Y -3 fir -rq - rs SY ~ m ~FTR" SC Y 11 and when equated with the original, results in the following relations between constants for this particular classx * S,~~~~~

-140This is a common means of displaying these relations (Reference 60, p. 140) and a heavy dot implies a non-zero value, a line adjoining two dots ism equivalence, and a circle is negative equivalence. Further examples would be triclinic, which is simply 21 dots and cubic, which is D. ISOTROPIC STRESS STRAIN RELATIONS For isotropic materials the constants are reduced to two. Its relations are like cubic above except S,~ = a(,,- I,;). The constant relating pure tensile stress and pure extensive strain is called "Young's Modulus", E, where (C25) and can be thought of as a coefficient of extension. The ratio of transverse compression to longitudinal extension is

-141-,t 54(C26) and is called "Poisson's Ratio",. Substitution into equation C23 gives, for example, (A - e p -E (p; tp,)}. (C27) The "Rigidity Modulus",', comes from the shear components of equation C23, or (C28) or (C29) and can be thought of as a resistance to shear. The volume change or "dilation", A, is (l%2~~~~~ t,(+ ) t-/ A(C30) which for small strains becomes A- I - U* /,A Us A3 U(C31) For hydrostatic pressure where A = A- = = -P and P = 0 -/\ /p just becomes the sum of the nine components in the upper left-hand corner of the. matrix and for cubic and isotropic is

-142_ =3 (5s, I.2,4) =V - (C32) where K is the "Bulk modulus". E. MEASUREMENTS If the medium is now subjected to an elastic wave, equation C2 has an additional term so that the balance of forces is still maintained and the equation of equilibrium becomes Pr~~ ib~~ tox;~ ~(C33) where is the density and the left-hand side is a mass times acceleration. Since th'e "oc. s can be related to the hi 5r (Reference 73, p. 98), equation C33 can be reduced to a set of equations of the form,,~~ _O d~c4 3ja:Q i;(C34) which is now in matrix notation and Cj contains only combinations of the elastic constants and is the "velocity of sound" in the material in the j direction. Most measurements of the elastic constants use some form of equation C34 and a determination of the, elastic wave velocities in various directions.47,75-77 There are two principle methods of this type of measurements, pulse echo

-143and resonance, both of which measure the adiabatic constants. The pulse echo method utilizes a determination of the time delay for a single pulse to echo a given number of times in a crystal of known length. The resonance technique involves a variation of input frequency to a crystal and observation of resonances for a known length in the crystal. Static methods are simpler in theory but more difficult in practice and measure the isothermal constants.78 This method just measures a known strain for a known stress. The isothermal constants, however, can be obtained from the adiabatic ones by the relation F H Lj] =..........(C35) Where ol is the c hange in strain} us, with temperature and C, the heat capacity at constant stress. These differences are usually quite small.

APPENDIX D DATA REDUCTION A CHROMIUM IMPURITY For the Cr3+ ion the data obtained from the experimental procedure were the pressure, the microwave frequency, and the magnetic field measurements at the two resonant conditions for the two samples. The frequency measurement is direct in gigacycles from a calibrated absorption type frequency meter (See Chapter III). The field measurements were in "proton" kilocycles (kcP) for a proton nuclear resonance condition and were converted directly into pressure shifts as will be shown later. The pressure readings were in 0O01 ohms resistance to balance the bridge circuit including the manganin wire whose resistance changes with pressure. By judicious choice of bridge components this reading was almost equivalent to atmospheres pressure. Calibration of the pressure system by observing the freezing point of mercury showed a correction to the resistance reading as (ate^ = c. 905 R (o0,l onus) g(D1) For the determination of any shifts in the gz value the equation =144=

-145w92~ = wgb H' (D2) is used. The value of gz is obtained from Schulz-DuBois37 and is 9 = i. 98t0 -t.0006 (D3) The ratio AH/HQ needs no conversion of units and "proton" kilocycles/sec. can be used directly. In the g shift determinations there often remained a field difference when both samples were at atmospheric pressure. This zero pressure difference which is subtracted from high pressure differences must be corrected for the frequency changes in the cavity. Consider the following diagrams: H (D3) p - o; v = Ho p; When no pressure is applied the resonance field condition for the high pressure sample, Hbo, is different from that for the reference sample, Hao. Applying pressure to the system changes the frequency so that these two new resonant conditions are

-1461/4 -6 ta*^ = Hae (D4) because (D5) In addition, the high pressure samples resonant field value is shifted because of a change in the spin Hamiltonian parameter to the value H9bx. The value that is desired (D6) thus becomes in terms of the measured parameters This - equation x and equao D2 we (Dusedts7) This equation and equation D2 were used to evaluate shifts in gz with pressure. The shift in the spin Hamiltonian parameter, 2D, was evaluated from experimental data by the equation 2 X _~ g~ p~~ aH ~ (D8)

-147The value gz is the unshifted value of equation D3 and is the Bohr magneton )e = 9 3, 732( xl G erg L (D9) Converting the ergs to megacycles and the gauss to "proton" kilocycles/sec. with %/Pf' =4.25770 kcP/gauss yields = C.3.72 3, Pc /4sec kcP (D) (D10) and equation D8 is A; D = - o. 65 3 (3 c/, c c,~P) kc (k P). (Dll) Since for these data runs there was n egligible or no zero pressure field difference, this was the equation used to evaluate the data. The values obtained were fit by the least squares method to the equation Yi = a + b SX (D12) where Yi is the parameter A2D; xi, which has essentially no error (+ 1 atm. in thousands), is the pressure; and a and b are ~ - (D3) (D13) ^ w A

=148The standard error of the fit is the rootsmean-square of the y derivations (D14) For the check of the pressure shift in 2D when the magnetic field was not parallel to the crystalline field axis, equations 17 and 18 (Chapter II) were used. These equations involve the values x and y where H /(-D t h I -D - (D15) The values of gz and P from equations D2 and D10 were used with the measured frequency and magnetic field values and the D(P) obtained from the zero angle determination. A check for equality to three significant figures in the pressure shift was then determinedo B. VANADIUM IMPURITY The same data was obtained for the'V ion as for the Cr3+ ion except that it was only for the high pressure sample, Also, the resonant magnetic field conditions for the first (in increasing field) three hyperfine lines were obtained for each pressure. This allowed two separate

-149determinations of the two spin Hamiltonian parameters involved, D and A (See Chapter IV). Because there was no two sample comparison in this case, equation 14 was used directly and is shown below where the second order hyperfine correction is simplified to f(B,A,gz, D,H,m) and the transition M = +~ to +3/2 is used; hY - 9Bz39 + as +, ten (D16) The frequency, the pressure, and the magnetic field were measured as before; however, the field probe contained heavy water and the deuteron resonance was used. Therefore, "deuteron" kilocycles/sec. were measured and with ~= /ar = 0.65356 kcD/gauss the value for P becomes (D17) Since for V2+ in A012U3 the values gz H and 2D are of the same order of magnitude, the second order correction, f, is quite significant as can be seen from equation 14. An iterative process was used to determine the p ressure dependence of this correction. From the first part of this experiment and previous high pressure EPR experiments, the shift in D with pressure was expected to be much larger than that in A and gz. (The change in B was assumed to be the same as A.) Thus to first order in

-150pressure shifts gz, A, and f were assumed constant and the change in D with pressure was calculated from equation D16 as (D18) where the primes indicate pressure. This was done for each of the three hyperfine lines at a given pressure. From the pressure dependence of D = D(P) the pressure dependence of the second order correction, f~(..,D,*..) was calculated. This was at most a 2% change in a term which was at most a 0.4% correction. iHowever, this was used to attempt to detect shifts in A. For a given pressure the resonant field conditions for two hyperfine lines are ti dAU I t I he)31~~~~~~~~ e / a +t(D19) and h/':, = ttP Al, aD Subtracting the former from the latter yields I 40p1 4 I / I R -7Jk45*3tIM' t(ri. () * (D20) (Note= In this iterative process it is impossible to calculate a AA = Al - A analogous to equation D18. Since it would take form

-151l (D21) where the Dg and f were calculated from the same data ( v, Hg, F, and H) assuming a constant A, it will just return a constant A.) Evaluation of equation 020 between two independent pairs of hyperfine lines for each pressure indicated no shift in A outside of the experimental error. Thus, A was assumed constant, and the calculation of equation D18 was redone using a pressure corrected second order hyperfine correction, f9, or yel r f h s * (D22) This then yielded the results for the pressure dependence of the crystalline field term in the spin Hamiltonian.

APP1:NDIX E EQUIPPE~NT The following is a list of the commercial electronic equipment used in the experiment and shown in the schematic in Figure 9. The high pressure generating equipment was manufactured by Harwood Engineering. The pressure bomb and cavity were fabricated under the direction of Mr. Herman Roemer in the University of Michigan Physics Department's Instrument Shop. All other components except standard waveguide were constructed i n the laboratory. A.o SPECTROMETER Klystron - Varian V153C Wavemeter - Hewlett Packard X532A Isolator - Kearfott 177=2c=1 20 db Attenuator - Hewlett Packard X375A Phase Shifter - Hewlett Packard X885A 50 db Attenuator FXR - X164A Detector - Microwave Associates IN23E Power Supply, Preamp, and AFC - Varian V4500-10 Signal Generator - Heathkit AG8 Modulation Amplifier ~ Heathkit AA121 Recorders Varian G10 B. MAGNET Varian 3400 90"D. 31 " gap rotable magnet with Varian V2503- 7 kva power supply C. PRESSURE MEASUREMENT Galvanometer Leeds and Northrup 2430D Decade Resistance - General Radio 1432U -152=

APPENDIX F RACAH PARAMETERS AND SYMMETRY TABLES The Racah parameters arise from the two electron matrix elements of the interelectron Coulomb repulsion integral, ioe., matrix elements of the last term in / and are all of the form c4 tie*| e &6 - (F1) (With i = m and j = t this is just a Coulomb integral, J, or with i =t a nd j = m an exchange integral, K.) The potential is expanded as r< (r, r- a~-; r,r~,, A')'~ - r, r Or - 00 k (F2) E rk kk (= O) where r< and r> are the lesser and greater of the two electron coordinates r1 and r2. The integral can now be done over the angular part of the wave functions leaving C<(e l C Rh X 4 k (F3) where the angular integration is lumped in the constant, ak -153

1 54One defines now Slater Condon parameters (F4) =.e sic a4 A IF5) (It is easy to see that kF and / akC L ) commo denominator r o k - A common denominator I /# exists in the ak and is usually lumped with Fk for a new Fk F. The sum is cut off at k = 2. because of the angular part and for d electrons Racah introduced the three linear combinations of Fk as basic parameters: C - F, - 5 F, C = 3r-F (F6) The first order electrostatic energies are linear combinations of these parameters. In particular for 3d3 free ions they are F = 3g-1~8.p = 3,q (F7) 14H P = 3P -G + 3

-1 55 a6 = 3g - ii B.3C F = 3R +8 B 3c X = 3R + B- +3 S (C3 + FeB + Cc) Note that A enters the same in all levels and is usually just dropped. B and C for a given ion are thus not independent but connected by a ratio C/Bo In a crystalline field these levels will be split and related by a function of the field and these parameters. Thus by observing transitions between levels of the same electronic origin, one determines the field splitting. Combining this with transitions between levels of different electronic origin, one determines the Racah parameters. It is in this manner that Drickamer34 determined the change in B with pressure. For example, for Cr3+ in sapphire 7 = a F (F) e /T;F)S -):De -DIS -o(-8aDI 18 =/ODt hvA = E A4(F) T;fJ =D? - jS B DfSB -F8Dri /;o BDltr8) From the frequency dependence of the 4T2 - 4A2 transition he determined the pressure dependence of Dq. From this and the pressure dependence of the 4T 4A transition he 1 2 calculated the pressure dependence of B.

-156That B decreases with increasing covalency as indicated in Chapter II-E can be seen as follows. B is made of integrals of the following form55 h)e(R) s I ra l av1? rfio(rn dr (F9) where R is a radial wave function, r. and r> are the lesser and greater coordinates of the two repelling electrons and l is an integer. With covalency R becomes R(r) = o K(r) + Y(r) where e and r are the impurity and ligand orbitals and P< <.. Thus the integrals above are primarily proportional to E< which decreases with covalency. That B decreases with radial expansion can be seen from the wave function expansion discussion in Chapter V (equation 89) and the observation that the dominant r dependence in the operator is a negative power. Therefore B will be proportional to some positive power of k, the radial expansion scaling factor, which decreases with increasing pressure. The irreducible representations of the Oh and C3 point groups are given below along with the product table for the Oh group.

-1570 E C3 3CR 6 C., 9c I I I -I -I E a -I a o O x-y 3 -- T, 3 o -I -I 0 x, y>,;, IY, I T, 3 0 - - xy, y, 3x 0 = 0 x ITversLo n C3 C~- C3 C ____ El C3 _ __X y; _ _-_ _, _Y _ ~E J wt~L ~,; LtJI ~~~~~~A) ~ ~ ~ ~ ~ j.

-158Products of the Irreducible Representations of Oh E T T. A, a Ft, E ~, T. E E F7-,7 * T 7 T. T TA T | T% T7T,+++ R2I.T-'r,T. T + O a +,F+T-.T puE+T,.+TT F.~~ ~~~ % -. % &er.~p..,r

APPENDIX G MATRIX ELEM2ENTS FOR AXIAL AND ZERO FIELD SPLITTINGS AND g SHIFTS Before the orbital matrix elements of the spin orbit coupling between the 4A2 ground state and the 4T2 excited state can be calculated, the splitting of the excited state due to the axial field must be determined. First the orbital wave functions will be determined. The cubic symmetry operations on the spherical harmonics 3t - r 3 -3 __ T TY~ (G1) >/ - 2jJi X+ly iL^ r y3 t3 = g; | 2 E (X + y) r -33 = g S3 <-353 Y3D in Oh symmetry with the four fold axis as the z axis53 ~y3o19 -15 9

-160{ Y -I8 Y. 4 (Y - )~) where they have been classified according to their transformation properties. These must be transformed to the 111 trigonal axis by the Eulerian rotation R(8 M o ) where =( 450 is the rotation about the z axis, P = crccos 1/j is the following rotation about the new axis, and = O is the following rotation aboUt the newest z axis. This transformation yields 3.3,+1,'6 3} (G3) a / fY3 J+ Y3j Ta, 2 2 { t3 ff y3 } it in Y,? i-3) _ i0

-161Since the 4T state does not couple directly to the ground state it will no longer be considered. Equation 42 (Chapter II) becomes for L = 3 V - -r g -L(LtI)= + l (42) A~X ~ ~ ~ -'~ (42) 4 and the matrix elements for the T2 state are < 7;+,|V"X| 7;v = iu;x 0(8 ) (} )} (G4) < %.,., s'O ) 1 -YJ7 - r,~ To prove that the ground state doesngt split under Vax I f|Vl > > = a - f(9'-i) (i 8 = C < (G5) Now equation 43 can be evaluated to obtain the ground state splitting O E -X >< ~1 L.S5+i(LL sL_S.)I ><.I__-_. I9, > -, =.~ E (Y%, ) -. E(':.) (43) To see which states are connected by which operators the following operations are evaluated; L- |. - 3 ( J/ } (G6) Y Ifi/7Y'j

-162Because the Y!s are orthogonal only the following matrix elements need to be calculated; < T L. /7> = 3 - 2 (G7) < T.,, L:7 > =.r- 3 C lL 3 { I I - I =(r Similarly < a/LI = > < 1l.T (G8) Therefore for L S one obtains Z Z LZ. /"9 t 1) S f9 i /ID (G9) and for ~(L+S + L S+ ) one obtains E,.., = -A L (G10) Ss, EL.Z = s_. 57 but(G (Gil)

163so =Lt A iof. - 5 -Dt (G12) Combining AEL and AEL, yields E 4_, [(S 1 )1 i9 F 19Sl 1p~e2 o o2 - 5E 5 (44) which is equation 44 (Chapter II). The change in the g value from the free ion value, go0 is for gz from equations 8, G6 and G7 L 1 7I <% ol'2 |L~ I > I D I + I (G13) or ~( - 2. - /o9D +/0d (50) which is equation 50 (Chapter II). For gL the L+ connect to the 4T2+1 states and from equations 8, G6 and G7 (G14) or LoDe -ss 1 9f(- (Gl) (G15)

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