THE UN IV ER SIT Y OF MI CHI GAN COLLEGE OF ENGINEERING Department of Nuclear Engineering Radiation-Solid-State Physics Laboratory Final Report MICROWAVE INTERACTIONS AT CRYSTAL DEFECT CENTERS Robert Borcherts Hossein Azarbayej ani Sophocles Karavelas Chihiro Kikuchi Glenn Sherwood ORA. iProja et.04275 AIR FORCE OFFICE OF. SCIENTIFIC RESEARCH CONTRACT NO. AF 49(638)-987 WASHINGTON, D. C. administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR January 1963

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TABLE OF CONTENTS Page LIST OF TABTLE v LIST OF FIGURES vii FOREWORD xi PART I. PROPERTIES OF RUBY AND OTHER SAPPHIRES I. Introduction 3 II. Crystal Structure 5 III. Energy-Level Diagram 8 IV. Properties of Other 3d3 Ions 12 V. Trigonal Field and Spin-Orbit Coupling 20 VI. Character Table 27 VII. The 3d2 Configuration 40 VIII. The 3d3 Configuration 46 References 61 PART II. X-RAY PRODUCED V+2 FROM VO+2 IN SINGLE CRYSTALS OF ZINC AMMONIUM SULFATE —ZnSO4(NH4) 2S04- 6H2o Foreword 3 Conclusions 3 I. Background 5 II. Experimental Methods 12 A. Crystal Growth 12 B. Crystal Structure and Orientation 12 C. EPR Experimental Setup 14 III. Experimental Results 17 A. V0+2 Crystals 17 B. V0+2 Powder 23 C. V+2 and Irradiated VO+2 Crystals 26 IV. Further Study 30 Appendix A. Calculation of Angular Variation of EPR Resonance in Rhombic Field 31 Appendix B. Exact Energy Level Calculations for V+2 Along x- y-, and z-axes for Rhombic Field 37 References 41 iii

TABLE OF CONTENTS (Concluded) Page PART III. Mn+++ EPR RESULTS IN AIIBVI COIPOUNDS I. Introduction 3 II. Crystal Preparation 4 A. Preparation of Quartz Tubes 4 B. Preparation of Sample 5 III. ESR Measurements 7 A. Experimental Method 9 B. Experimental Results 12 IV. Theory 18 A. General Hamiltonian of Mn++ 18 B. Spin Hamiltonian 25 C. Crystalline Field 27 D. Hyperfine Splitting 39 E. Determination of 3a = 4r8-2r7 Parameters 47 F. Determination of Hyperfine Splitting Constant A' 47 Appendix A. Determination of Mn++ Ground State 49 Appendix B. Reduction of the ~t- 1l to a Function of L, S and I 59 References 66 PART IV. SOLID STATE INSTRUMENTATION PART V. PUBLICATIONS AND PRESENTATIONS COMPLETED UNDER CONTRACTS NO. AF-49(638)-68 AND AF-49(638) -987 I. University of Michigan Technical Reports and Internal Memorandums 3 II. Journal Articles, Papers, and Presentations 4

LIST OF TABLES Table Page Part II I. Comparison of Experimental and Calculated Values of Magnetic Field for AM = +1, Am = 0 Transitions in VOS04(NH4)2S04.6H20. 20 Part III I. Mn++ ESR Experimental Results. 7 II. Matrix Elements of 2P [T40+ 5/-i (T44+T4_4)]. 33 III. Variation of Separation of Mn++ ESR Fine-Structure Components at 0 = 0 as a Function of p = gpH/2a = E/a. 37 A-I. MSML Table for 3d5 Configuration. 50 A-II. MSML Table for 3d2 Configuration. 51 A-III. J(m,m') Integrals in Terms of A, B, and C. 53

LIST OF FIGURES Figure Page Part I 1. Coordination of Al atom in a-A1203. 6 2. Projection of oxygens on horizontal plane. 7 3. Energy-level diagram for cubic field. 9 4. Split in 2E level by trigonal field. 10 5. Split in ground level. 10 6. Properties of V2+, Cr3+, and Mn4+ 12 7. Octahedral coordination. 15 8. Cube with [111] body diagonal. 20 9. Geometric constructions of linearly independent functions. 22 10. Geometric constructions of linearly independent functions. 22 11. Split in energy levels upon introduction of perturbations. 26 12. Splitting of states of the 3d3 configuration by an octahedral field. 46 13. Splitting of 4F ground states. 49 14. Complete splitting scheme of the terms arising from the 3d3 configuration. 49 Part II 1. EPR spectra along K2 axes. 4 2. V+3 and +2 energy levels. 6 3. V+4 and VO+2 energy levels. 7 vii

LIST OF FIGURES (Continued) Figure Page 4. (a) First derivative of crushed crystal absorption spectrum; (b) Plot of sin G [dO] for 0.1% VOS04 in Zn(NH4)2(SO24)2.6H20. 11 5. Unit cell of Mg(NH4)2(S04) 2 6H20. 13 6. Faces of Tutton crystal; orienting devices. 15 7. Experimental positions of axes in VO(NH4)2(S04) 2.6H20 (1% VO/Zn). 18 8. V0+2 axes in regular and distorted octahedra. 21 9. EPR spectrum of 0.1% VOS04 in Zn(NH4)2(S04)2.6H20 with magietic field along "tetragonal" axis. 24 10. Magnetic field parallel to Z axis, 1% V2 in Zn(NH4)2(SO4)2'.6H2o. 29 Part III 1. Tube preparation. 4 2. Rotating sample holder and resonance cavity used for measurement s. 11 3. Mn++ ESR spectrum at 3000K. 13 4. Differentiation of ESR absorption in ZnTe:Mn at 780K. 14 5. Differentiation of ESR absorption in CdTe:Mn at 420~K. 14 6, Mn++ ESR spectrum in CdTe single crystals at 780K, = 0~. 15 7. Mn++ ESR spectrum in CdTe single crystals at 4.2~K, G = 30~. 16 8. Mn++ ESR spectrum in CdTe single crystals at 4.2"K, 9 = 0"~ 16 9. Schematic representation of electron-nucleus magnetic interaction. 22 10. Schematic representation of electron-nucleon coulomb interaction. 23 viii

LIST OF FIGURES (Contluded) Figure Page 11. Schematic representation of crystalline field. 28 12. A unit cell of cubic ZnTe with reflection planes a. 30 13. Theoretical prediction of Mn++ ESR spectrum. 31 14. Effects of cubic field on 6S levels. 34 15. Energy level scheme of 3d5 6S/2 (Mn++) in a tetrahedral field at G = 0~. 38 16. Mn++ ESR spectrum in ZnS (cubic) at 3000K, G = 0~. 43 17. Mn++ ESR due to a given m. 43 18. Schematic plot of fine structure of ESR spectrum of Mn++ in cubic ZnS. 46 Part IV 1. Block diagram of telescope spectrometer and necessary circuitry for coincidence counting. 5 2. Silicon wafer mounted on pyrex polishing disc, with optical glass flat and iron flat. 6 3. Response of silicon dE/dx detector to po210 alphas. 10 4. Pulse shape of large-area dE/dx detector to Po210 alphas vs. bias across detector. 11 5. Telescope spectrometer. 11 6. Response of dE/dx detector and E detector to proton recoils using Pn-Be source. 12 ix

FOREWORD During the six years that this contract with the Air Force Office of Scientific Research has been in effect, some spectacular developments in quantum electronics have taken place. In 1956, while the first draft of the technical proposal was in preparation, the term "quantum electronics" had not yet been coined. The stated purpose of the proposed program was merely to carry out some basic investigations of chemical and lattice defects in crystalline solids, using electron spin resonance as the tool. By the end of 1956, however, Bloembergen's suggestion of the solid-state maser became very widely known, and a short time later the feasibility of the suggestion was reported by Scovil, Feher, and Seidel of the Bell Telephone Laboratories. At the Lincoln Laboratory, McWhorter and Meyer discovered maser action in chrome cyanide around the middle of 1957, and on December 20, 1957, the workers at The University of Michigan Willow Run Laboratories observed maser action in pink ruby. Development of the ruby maser as a device took place very swiftly. During these months after the discovery of maser action in ruby, Townes and Giordemaine, using a tested pink ruby crystal that had been supplied by The University of Michigan workers, announced the successful operaticon of the radiotelescope maser at the Naval Research Laboratory. A similar device was built later for The University of Michigan radiotelescope. More recently, the ruby maser was used by the workers at the Jet Propulsion Laboratory for the precision determination of the Astronomical Unit (149,589,500 Kn). At xi

the same laboratory, a program evaluating the use of ruby masers for satellite communication is in progress, and it is well known, of course, that a ruby maser is part of the Telstar communication system. The pink ruby, which previously had been used primarily for phonograph needles, was very soon found to have another important application in a device now popularly called the laser. Soon after theoretical suggestions were offered by Schalow and Townes, T. H. Maiman of the Hughes Research Laboratory and Collins and his co-workers at the Bell Telephone Laboratories reported the predicted phenomenon. The program carried out under this AFOSR contract is a classical example of the importance of basic research in technological developments, As mentioned earlier, our initial objective was merely to explore the uses of electron spin resonance as a tool to study defects in hard crystalline materials, to which only limited attention had been paid at that time. It was this program that led this principal investigator to examine the merits of such mat;erials as ZnS, MsO, CaCo3, and A1203 as maser materials, and thus to the observation of maser action pink ruby. The technological developments in masers and lasers have in turn already raised a number of questions, the answers to which can come only from more intensive basic investigations. One such question is the very old one of energy transfer and transformation mechanisms in solids. We have noticed that the mechanism invoked to explain laser action in solids is different from that needed for luminescence and scintillation phenomena. A program to study these phenomena has been started, with the hope of gaining deeper insight xii

to energy conversion processes in solids. In bringing this contract to a close, I wish to thank Mr. Charles F. Yost, now of the Advanced Research Projects Agency, and the various members of the Air Force Office of Scientific Research for the generous support that made our work on the ruby maser at Willow Run possible, and for the continuation of the program to introduce these solid-state concepts and techniques into the research programs of the Department of Nuclear Engineering. Chihiro Kikuchi December, 1962 xiii

PART I PROPERTIES OF RUBY AND OTHER SAPPHIRES by C. Kikuchi and S. Karavelas

I. INTRODUC)CTION For centuries ruby has been highly prized as a gem, but it is only very recently, since the discovery of maser action in ruby, that the indust'rial importance of this material has come to be appreciated. Before 1957, the Linde Company was the principal supplier of ruby and sapphires. The socalled pink ruby was manufactured by this company to make the so-called sapphire phonograph needles. The present surge of interest in ruby and related materials stems from the fact that ruby has been shown to be useful as a maser and laser material. Maser, an acronym for Microwave Amplification by Stimulated Emission of Radiation, is a low-noise, high-gain amplifier, for which r~by is the critical material. The ruby maser was used in the Telstar commulication system, and at the California Technology Jet Propulsion Laboratory a packaged truby maser for satellite tracking stations is under evaluation. In scientific application, the ruby maser made possible the precision measureinent of the A.stronomical Unit, the average distance of the earth fro;m the sun. The precision value of 149,589,500 ~ 500 km was obtained by the workeyrs at the Jet Propulsion Laboratory, after making careful analysis of the radar echo from Venus. Ruby is also used in lasers, a device for the generation of intense coherent optical radiations, The word Laser is an acronym for Light Amplification by Stimulated Emission of Radiation. The irndustrial and tecohrological uses of this device are still under research and development, It is possible that the device will find applications in communication systems 5

operating at optical frequencies, and in certain industrial processes such as micro-cutting, micro-etching, and micro-welding. It also seems to be usefull as a medical tool-in coagulating a detached retina onto the eyeball, for example. A great deal has been said about the usefulness of ruby as a maser and laser material, but few popular expositions attempt to explain why ruby is useful. Consequently, it will be the purpose of this discussion to point out the factors that make ruby behave the way it does.

II. CRYSTAL STRUCTURE Chemically, ruby is aluminum oxide which contains a small concentration of chromium. Aluminum oxide occurs in two forms; the one that concerns us is known as o-A1203. In mineralogy, this form is called corundum, and the commercial name of synthetic corundum is sapphire. The commercial sapphire, which is clear, transparent, and colorless, should not be confused with the gem sapphire, which is blue. (In passing, perhaps it should be noted that the technological impoxrtance of sapphire is beginning to be realized. For example, according to a recent NASA report, sapphire windows will henceforth be used on satellite solar batteries. Also, sapphire has the very unusual property of high thermal conductivity but low electrical conductivity, in contradiction to the Weidemann-Franz law, which asserts that good electrical conductors are also good thermal conductors. This particular property is used in the laboratory to provide good thermal contact and good electrical insulation. ) There are several varieties of rubies, distinguished by their colors. Gem rubies are deep red, due to the high chromium concentration, whereas the rubies important for masers and lasers are pink, due to the chromium concentration of 0.1% or less. The crystal structure of sapphire is rather complicated.3'4 It can be conveniently generated by placing A1203 molecules at the corners of a cube and stretching the cube along one of the body diagonals. Another A1203, rotated 1800 about the molecular axis with respect to the first molecules, is

placed at the center of this distorted cube. Upon careful examination of this structure, it will be seen that each Al atom is sandwiched between two groups of three oxygens, as shown in Fig. 1. The three oxygens are in a 64010'0.80 A 46~27' ~0/ \ 1.37A Fig. 1. Coordination of Al atom in C-A1203. plane about 1.37A from the Al site, and the A1-0 distance is about 1.98A, making an angle of 46027' with the crystal c-axis. The other three oxygens lie in a plane o.80 away; the A1-0 distance is about 1.84A and the angle 64010'. The relative orientations of the two oxygen triangles are not quite 1800. If the oxygens are projected on a horizontal plane, the result shown in Fig. 2 is obtained. ~'!~~

The details of this quantum mechanical calculation is somewhat involved, but the general qualitative features of the energy-level diagram can be predicted readily from group theory. We are concerned with what is called the crystal-field spectra, i.e., the spectrum which arises from electrons making transitions within the 3d shell. For a cubic field, the energy-level diagram would be as shown in Fig. 3.5 4T1 2T2 (4000ooo) 4T2 B (4750A) 2T1 2E U (5560A) R (6930A) 4A2 Fig. 3. Energy-level diagram for cubic field. The numerical values given for the wavelengths of the emitted and/or absorbed radiation are for ruby. For the other two isoelectric ions, vk and In l,i 3

III. ENERGY-LEVEL DIAGRAM Our next task is to analyze the effect of the six nearby oxygens when an impurity ion such as WV2, Cr3, or Mn4 is substituted for Al. These three ions are cited because each has the 3d3 configuration, so that energylevel diagrams of all three have many qualitative features in common. In most theoretical analyses, the assumption is made that the ion consists of an inert inner ion core (the argon core) and that the electronic properties of the ions are determined by the three electrons in the outer unfilled 3d shell. The validity of this assumption has been questioned recently, but for the present discussion we shall assume the existence of such inert inner filled electron shells. Earlier, it was emphasized that the crystal structure of sapphire is complex. To make theoretical analysis somewhat tractable, it is generally assumed that the oxygens give rise to a crystalline electric field which has predominantly cubic symmetry but which also has a small component of trigonal symmetry. Such a crystalline electric field can arise if we first imagine that the oxygens are placed at the center of the faces of a cube surrounding the impurity ion, and then imagine a small distortion produced by stretching the cube along the body diagonal. If these assumptions are made, we can first take into account the effect of the cubic component of the crystalline electric field, and later consider the perturbation of the cubic-field energy levels by the small trigonal field.

The details of this quantum mechanical calculation is somewhat involved, but the general qualitative features of the energy-level diagram can be predicted readily from group theory. We are concerned with what is called the crystal-field spectra, i.e., the spectrum which arises from electrons making transitions within the 3d shell. For a cubic field, the energy-level diagram would be as shown in Fig. 3: 4T1 2T2 Y (4000A) / 4T2 B (4750A) 2T1 U (5560A) R (6930A) 4A2 Fig. 3. Energy-level diagram for cubic field. The numerical values given for the wavelengths of the emitted and/or absorbed radiation are for ruby. For the other two isoelectric ions, and Ma+ 9~~~~~~~~adP?+

the wavelengths would be slightly different. If we next take the trigonal crystalline electric field into account,6 each of the indicated levels will split into two or more levels. The 2E level will split into two levels; the separation in ruby, for example, is about 29 cm-1, as shown in Fig. 4. 2A 2E R1 R2 ( 14,418 cm-l) (14,447 cm-') 4A2 Fig. 4. Split in 2E level by trigonal field. This results in the splitting of the R line into two components, referred to in the literature as the R1 and R2 lineso The former is the ruby laser line. The ground state also splits, as indicated in Fig. 5. 4A2 < / o0.38 cm-1 4A2 Fg.5.Spiti 3(uo8 cm-1 Fig. 5. Split in ground level6 10

The separation of the levels in this case is very small, only about 1/100 that of the 2E state. But this very small splitting is important in the ruby maser. 11

IV. PROPERTIES OF OTHER 3d3 IONS In the preceding section a few of the properties of Cr3+ in sapphire were described. The next question that arises is whether or not it is possible to fabricate other materials which have similar properties. To answer this question, a brief discussion of v2 and Mn4 will be presented. The results for Mn4+ were published very recently by S. Geschwind and others at the Bell Telephone Laboratories.7 These results along with properties for V2+ and Cr3+ are summarized in Fig. 6. v2+ Cr Mn 7 (s18,000c m 24000cm /7 I OOcm' 14,800cm /I 311cm |38cm m.39 cm t?? ct= 3.4 ms t.8ms Fig. 6. Properties of V r, Cr, and Mn4m 12

The measured values of Mn4+ are: R2 = 14,866 cm-l (790K) R1 = 14,786 2D = -.3914 gll ~gl = 1.9937 iAiI =IAI = 70.0 x 10-4 cm-1 There are several interesting facts to note. In the first place, the R1 lines are very nearly equal. Another very remarkable fact is that the results for Mn4+ in sapphire are very close to those of Mn4+ in lithium titanate. In a paper by Lorenz and Prener8 the values k = 6790k (14,700 cm-l) and t = 1.1 ms are reported. The values for the ground-state splitting are not known, because the measurements were made on a powder. The very close agreement of the numerical values for sapphire and lithium titanate, despite the apparent differences in the host crystal material, stems from the fact that coordination for Mn4 is octahedral; i.e., in both cases the Mn4+ ion is bonded to six nearby oxygens. The work on V2+ in sapphire was started in our laboratory when it was noticed that V2+ is produced by X-rays.9 Our experiments showed that noimally vanadium in sapphire is present as V3, but that after irradiation, part of this converted to V2+. The chemical impurity responsible for the stabilizaa.tion of V2+ is not yet known. For Mn in both sapphire and lithium tetanate, the presence of Mg impurity is needed. Vanadium is a favorable system to investigate because it is essentially 100% V51, and because its nuclear magnetic moment is rather large, due to 13

the fact that vanadium has an odd-proton-even-neutron nucleus. The nuclear magnetic moment can then be used as a signature to identify vanadium, and for this reason vanadium can be quite easily identified in the different oxidation states in sapphire. With these preliminary remarks, let us return to the ruby energy-level diagrams given in Section III. As mentioned before, the paramagnetic ion in ruby is Cr3+ whose electron shell structure is 1522522p63523p63d 3. The three 3d electrons in the last unfilled shell are the electrons responsible for the interesting microwave and optical properties of Cr3+. (It should be noted that V+ and Mn4+ have the same electron shell structure, so that the comments about to be made are applicable to the three ions V2, Cr3, and Mn 4.) It was mentioned earlier that the chromium ions, occupying the Al substitutional sites, are surrounded by six oxygens. The question we now wish to answer is: What effect will these oxygens have upon the Cr3 energy levels? Since the actual arrangement of the oxygens is quite complex, to make the analysis more tractable we shall first assume that the effects of the oxygens come predominantly from the octahedral arrangement of the oxygens. In other words, for the first step in the analysis we shall assume that, with the Cr3+ ion at the origin of the coordinate system, the oxygens are located along the coordinate axes at the same distance from the origin. Later we shall consider the effects of a small distortion along the body diagonal. Perhaps the graphical representation in Fig. 7 will make the physical problem clearer.

I 21 *- - - - - -- ~Y - Fig. 7. Octahedral coordination. Electrons 1., 2, and 3 are moving in the spherically symmetric coulomb field of +6 e. Each electron has orbital and spin angular momentum li and si respectively. There are mutual coulomb repulsive forces among the electrons, and the negative charges located along the coordinate axes will also have an important influence on the motion of the electrons. The problem indicated here is a complicated one; consequently it will be necessary for us to make a series of approximations. In order to avoid the complexities of theoretical arguments, we shall first consider the case of a single 3d electron, and shall then show how additional factors have to be brought into the analyses as we proceed to the 3d2 and the 3d3 cases. The case of the single 3d electron will be dealt with in great detail in order to bring out many features of the theoretical techniques. The single 15

3d electron is in the central spherical symmetric coulomb field, but its orbital motion is profoundly affected by the oxygens surrounding it. Furthermore, the electron has orbital and spin angular momentum, so that the coupling of these vectors will affect the electron energy level. The Schroedinger equation we need to solve is H4r = Er (4-1) with P__v 4yZ 3 4) + Ft(Xy+yZ+ZX) + %I =i - - 2- V2 + V(r) + Fc(x4+y4+z4- r s (4-2a) 10 The full trigonal part of the Hamiltonian up to fourth power is =-+Fxyz2(XY+YZ+ZX) 12 Vtr = Ft(xy+yz+zx) + F[xyz(x+y+z) - (xy+yz+zx ] it,,, "(x+y+z) it]ti + Ft(x+y+z) + Ft xyz + Ft [x3+y3+z3_ 2Xy-2b) 5+ + if the approximation is made that the symmetry of the Al or Cr3 site is C3v. This is equivalent to neglecting the angle of 4o22' shown in Fig. 2. In the following discussion we restrict ourselves to the form (4-2a) for the sake of simplicity. However, one important characteristic of (4-2b) must be kept in mind, namely, that it has no center of inversion. This is important when selection rules for electric dipole transitions are examined. There are three steps to the Schroedinger equation, as follows: I. f- V2 + V(r) + Fc(x4+y4+z4- 3 r4) 2m 5 2 II. ~2 V2 + V(r) + Fc(x4+y4+z4- - r4) + Ft(xy+yz+zx) III. 2 V2 + V(r) + Fc(x4+y4+z4- 5r4) + Ft(xy+yz+zx) + s 2m 5 16

The purpose of Step I will be to show how the symmetry properties of the Hamiltonian can be exploited to construct wave functions, and also to point out the physical reasons for the importance of group theory in attacking these problems. Consider then the first three terms in (4-2a), i.e., - - ~2 V2 + V(r) + Fc(x4+y4+z4- 3 r4) (4-3) 2m 5 We notice that the first two terms are invariant under all rotations, finite or infinitesimal. Consequently, as is well known, if the last term were absent, the solution of the Schroedinger equation could be represented by the spherical harmonics. The corresponding energy levels are called the s, p, d, etc., states. The mathematical significance of this remark is that the wave functions belonging to a particular energy level can be represented by polynomials of x, y, and z which transform among themselves under infinitesimal rotations. For example, the wave functions for the d-electron can be represented either by the set of spherical harmonics of degree 2 —Y, m = O, +1, +2-or by the polynomials xy, yz, zx, x2-y2, 3z r2. But what happens if the symmetry of the Hamiltonian is lower? For example, in (4-3), the Hamiltonian is invariant under the simple substitution of ~x, +y, +z among themselves, but not under infinitesimal rotations. In the latter case, x -) x - GY y — Y + EY (4-4) and polynomials in mixed iowers of x and y are generated. 17

However, the Hamiltonian is invariant under a group of transformations called the cubic group. This is the group of operations that carry a cube into itself. Let us see how we can make use of the Hamiltonians invariance under the cubic group to generate wave functions belonging to the same energy level. For this, we need to recall that if 4r is a wave function belonging to the energy level E of H*r = (4-1) and if Rg is one of a group of transformations that leaves the Hamiltonian unchanged, i.e., Rg(Hr) = H(Rg*) (4-5) then the set of functions RgV generated by the group of operations Rg are also solutions belonging to the same energy level E. To illustrate the physical significance of this result, let us examine the effect of the cubic crystalline electric field upon the 3d electrons. As mentioned earlier, the 3d wave functions can be represented by the five linearly independent monomials of the second degree: xy, yz, zx, X2-y2, and 3z2-r2. Consequently, let us suppose that f(r)xy (4-6a) is a solution of (4-1) belonging to E. Since f(r) is a function independent of the angle, in order to exhibit the angular dependence hereafter we shall write simply xy (4-6b) instead of f(r)xy. According to the comment made earlier, if Rg is an opera18

tion of the group then other solutions can be generated by the operation. For example, (4-3) does not change if the substitution x ~ X y ---- z z 7- -y (4-7a) is carried out. Geometrically, this corresponds to a 900 rotation about the x-axis. Carrying out this substitution on (4-6b) we obtain xz (4-6c) Another possible substitution is x z Y- Y z - x (4-7b) Geometrically, this represents a reflection on a diagonal plane passing through the y-axis. Applied to (4-6b) this operation yields yz (4-6d) which along with xy and zx belong to the same energy level E. Also, it is clear that for the rest of the substitutions that carry a cube into itself, the functions xy, yz, and zx, will transform among themselves, but that at no time will they transform into x2-y2 and 3z2-r2. But since these last two polynomials will transform among themselves, they belong to another energy level. 19

V. TRIGONAL FIELD AND SPIN-ORBIT COUPLING In the last section it was shown how symmetry properties can be used to show that the 3d electron will split into two energy levels, with 2- and 3fold degeneracy respectively, In order to bring out other properties, we shall consider the trigonal case in detail. A detailed discussion of this case is possible because there are only six operations involved. As mentioned earlier, the six oxygens in sapphire form a distorted cube, This can be realized by distorting the cube along one of the body diagonals. If this is done along the [111] body diagonal (see Fig. 8), the additional term Ft(xy+yz+zx) shown in (4-2a) appears in the Hamiltonian. What effect will this term have upon the energy levels? In particular, we shall consider the effect of such a distortion upon the energy level associated with the functions xy, yz, and zx, or some suitable linear combination of these functions. Fig. 8. Cube with [111] body diagonal. According to the above comments, we need to look for the group of substitutions or operations that leaves the Hamiltonian unchanged. Clearly the 20

Hamiltonian 4 _- -- V2 + V(r) + Fc(x4+y4+z4- 3 r4) + Ft(xy+yz+zx) (5-1) 2m 5 will be invariant for substitutions that leave Ft(xy+yz+zx) unchanged; and these will be: x x E: yr - y (5-2a) z - z x ) y x - z ~1) (2) c(l) y > Z C3 y y- x (5-2b) z - x z y x >x x -- z x > y Cil) ( z (2) y Y 3) x (5-2c) z y z —- x z >z Substitution (5-2a) is the identity operation; substitutions (5-2b) represent 120~ and 240~ rotations about the [111] body diagonal, and substitutions (5-2c) represent reflections in the three face diagonal planes. From the set of functions xy, yz, and zx we note that the function xy + yz + zx (5-3) can be constructed. Clearly, this function transforms always into itself under the operations indicated in (5-2). Physically this means that one of the three functions will split off due to this trigonal field. To construct the remaining two linearly independent functions consider the geometrical procedure shown in Fig. 9a. 21

k(z) xy Y1 + -x + xy B J(y)...zy A (x) ye (a) (b) Fig. 9. Geometric constructions of linearly independent functions (simplified). The plane normal to the vector i+j+k intersects the xy plane along AB, so that a vector in this direction can be written -i + j (5-4) Referring to Fig. 9b, this suggests the corresponding function -yz + zx (5-5) The vector normal to -i+j and i+j+k is given by I I x' ( i++i-2k) Fig. 10. Geometric constructions of linearly independent functions. 22

i j k -1 1 o = i + j - 2k (5-6) 1 1 1 so that the combination yz + zx - 2xy (5-7) is suggested. We shall now verify that the functions of (5-5) and (5-7) transform among themselves for the substitutions (5-2) and we shall furthermore construct the matrices representing the operations of (5-2). If we let fi - -yz + zx f2 - yz + zx - 2xy (5-8) then E carries fl into fl and f2 into f2. Consequently the representative matrix is 1 0o\ 0 / Consider next Ci1) In this case 1 fl -zx + xy = fl - f2 2 2 f2 zx + xy - 2yz = f 1- f2 (5-10) so that the matrix representation is 1 1 M(C)) = 2 ( 2 2 25

Similarly, 1 11 1 2 -xy+yz = 2 f + 2 f2 M 2 = 2 -2a) C) M 2\3 (5-12a) f2 ) xy+yz-2zx = - 2 2 2.2 2 2 2 (5-l2b) 2 2 1 1 1 of 2 I/+x =2 +2f2(v( 1 )(5-12c) f2 > xy+xz-2yz fl - f2 2 2 2 xy + yz + zx (2 2 2) is taken, this function is such that it always transfos into itself under or( 3) mZ2Y O 3 (5-12d) to construct some linear combination of f- and f2 of (5-8) —say gl and g2such that gl will always transform into itself and g2 into itself? The answer to this question is no; it is provided by group theory and stems from the fact that the set of numbers obtained by taking the diagonal sum of the matrices is identical to the row of numbers in the group character table. The character table for the C3V group, with which we are concerned here, is as follows: 24~~~

E 2C3 3 r A1 1 1 1 A2 1 1 -1 E 2 -1 0 The symbols appearing at the top of the column represent the symmetry operations, and the numbers in the column below give the trace or the diagonal sum of the representative matrix. The numbers under E give the dimensionality for the so-called irreducible representations A1, A2, and E. (The maxtrix representing the symmetry operations A1 and A2 are l-x-l matrices, whereas those for E are 2-x-2 matrices.) Clearly, then, xy + yz + zx (5-3) transforms like A1. If, next, the last term in (4-2a) is taken into account, the transformation properties of the spin functions have to be considered. The appropriate character table is the one for the double C3V group, which is as follows: E R 2C3 2C3R 3 5v 3 avR A1 1 1 1 1 1 A2 1 1 1 1 -1 -1 E 2 2 -1 -1 0 0 E 2 -2 1 -1 0 0 -1 -1 1 i -i 2P -1 -1 1 -i i ll~ ~~~~2

It can be shown that the spin functions for S = 1/2 transforms like E. Figure 11 shows how the energy levels split upon introducing the perturbations in (4-2a) in succession in the indicated order. Spherical Cubic Trigonal Spin-Orbit (2) (2)E - 2 / E / / / / / A1 _ (3) x (2)E E 2A Fig. 11. Split in energy levels upon introduction of perturbations. 26

VI. CHARACTER TABLE In the previous section, we mentioned such terms as character table, irreducible representations, etc., without adequate explanation. Here we shall discuss these concepts in greater detail, with particular reference to the cubic group. A clear-cut understanding of the character table is necessary before we can proceed to the two-electron 3d2 and three-electron 3d3 configurations. We shall present a discussion of what is meant by the character table, point out some of the important properties, and show how this table can be used. Perhaps it should be emphasized that the character table is somewhat like the multiplication table; if we know how to use the table, we can use it to work out multiplication problems! Consider, then, the group character table for the cubic group: E 8C3 3C2 6 6C4 A1 1 1 1 1 1 A2 1 1 1 -1 -1 E 2 -12 0 0 T1 3 0 -1 -1 1 T2 3 0 -1 1 -1 The symbols across the top of the table represent the different classes of symmetry operations and the number in front of the symbols gives the number of operations belonging to that class. A "class" of operations is a set of similar operations and can often be obtained more or less intuitively. For 27

exanple, for the group of operations that carry a sphere into a sphere, all rotations of a given angle independent of the axis of rotation belong to the same class. For the cubic group, there are the set of rotations about the body diagonals. The two rotations of 120~ and 240~ about a body diagonal will carry a cube into a cube. Since there is a total of four body diagonals, we might intuitively guess that all eight such operations belong to the same class. In the above table, the rotations about the body diagonals are in the class C3. As another example, consider C2, the class of 1800 rotations about the coordinate axes. Since there are three axes, we should expect three operations in class C2, as indicated in the table. Class C' comprises the set of 180~ rotations about the face diagonals, and C4 the class of 900 rotations about the coordinate axes. Mathematically, the set of similar operations belonging to a given class is generated by the similarity transformation Rc = RgRcRgl (6-1) If all symmetry operations Rg belonging to the group are used, a set of distinct operations Rc will be generated. This is the set belonging to the class C. For the cubic group, there are five classes, and the total number of symmetry operations is 1 + 8 + 3+ 6+ 6 = 24 (6-2) The symbols A1, A2, El, T1, T2 are the different irreducible representations of the cubic group. In order to see what is meant by this, consider a 28

set of polynomials of degree 2. There are, as is well known, 2Q+1 linearly independent such polynomials. Suppose these polynomials are represented by J1,~2... Xi...' n in which n = 22+1. If, now, some operation Rg of the cubic group-and these operations are simple linear substitutions- is applied to any one of the functions Oi(x,y,z), the new function will be a polynomial of x,y,z of the same degree 2; therefore the new function can be written as a linear combination of the original set of functions. Thus Rg(oi) = Mij(Rg)oj (6-3) and the operation Rg, operating on the set o)l, can be represented by the matrix M(Rg) (6-4) The dimension of this matrix is clearly (22+1) x (22+1). Also, since there are 24 operations in the cubic group there will be 24 such matrices, one corresponding to each operation Rg. This set of matrices is said to be a representation of the cubic group because the symmetry operations of the group and the matrix representing the operation can be set into one-to-one correspondence; i.e., if Ri, Rj, and Rk are any three operations such that RiRj = Rk (6-5a) and if Ri < M(Ri) Rj (- M(Rj) Rk e - M(Rk) (6-5b) 29

then M(Ri)M(Rj) = M(Rk) (6-5c) This group of matrices may have one of two possible forms. It is possible that all the matrices in the set will simultaneously have the form o o o 0 O O (6-6) 000 0 \ o o o o or can be put into it. In this form the non-zero blocks occur along the diagonal only. If the matrices have or can be put into this form, the group is said to be reducible; if not, it is said to be irreducible. For example, for the 3d electron case discussed earlier, I = 2, the linearly independent functions are xy, yz, zx, x2-y2, and 3z2-r2, so that 24 operations of the cubic group will generate 24 5-x-5 matrices. Furthermore, each of the 24 matrices will have the form 3o

3x3 U (6-7) group. matrices, all of which will have one of the forms in (6-8)? 2x2 0 (6-8) -.. ---— I —-- lxl tained by taking the diagonal sum of the represet xy, yz, and zx. Thative matrices will be 3-x-3, as shown earlier.ys 31

be identical to the set T2. The symbols Al, A2, E1, T1, and T2 stand for a set of matrices that are in one-to-one correspondence with the symmetry operations of the cubic group. These five sets of 24 matrices are irreducible in that the matrices of each set cannot all be put into the form (6-6) simultaneously. Furthermore, group theory assures us that whatever its dimension, any set of 24 matrices which represents the cubic group operations can be decomposed or reduced to a linear combination of the five irreducible representations. For example, for polynomials of degree 100, there will be 201 linearly independent polynomials; the cubic operations will generate a set of 24 matrices of rank 201. But the rank of the highest sub-matrix cannot be more than three. However, if we were to select a set of matrices from the original 24 such that the selected set satisfies the group property, then this set could possibly be reducible. For example, consider the rotations about a body diagonal. If the axis of rotation is the (111) direction, then x ) x x y >xy /1 0 0 \ E: y y yz yz M(E) = 0 1 0 (6-9a) z- z zx > zx 0 1O x - y xy >yz 0 1 0 C(1) Y- z yz - zx M() =3 0 0 1 0 (6-9b) z x zx - xy 1 0 0 32

x — z xy > zx /0 0 1 C3(2 y x yz - xy MK) = 1 0 O (6-9c) z )y zx -yz 0 1 0 0 0 M1)M(C(2 = 0 1 0 = M(E) (6-9gd) 0 0 1 The above symnetry operations constitute the group C3, whose character table is as follows: E 2C3 A 1 1 E 2-1 xy,yz, zx 3 0 = A+E Mathematically, this means that if an appropriate linear combination of xy, yz, and zx is taken, the three matrices can be reduced. As indicated earlier, the appropriate linear combinations are fo = xy + yz + zx fl = -yz + zx f2 = yz + zx- zxy C() fo -) fo Cl) f1 )-f _zx + xy = 2 fl - f2 (1) 3 1 C3 1)f2 zx + xy - 2yz = 2 fl - 2 f2 33

so that M(Ci0)), 12 2 (6-10) 2 2 ~~~~~~~~~~(2)~~~~~~ and for C(2) we obtain 1, 0 0 -- - - - - - - - - - - 3 = o O 2 2 (6-11) 3 1 2 2 Since 1/ i 0 0 M( E) = O 1 0 (6-9a) o 0 1 all three matrices can be partitioned as indicated by the dotted lines. The characters, or the traces, satisfy the vertical and horizontal orthogonality relations. Let x(ci, r) (6-12) represent the character, or the trace, of the matrix which represents operation Rg in class Ci for the irreducible representation r. Then the vertical orthogonality relation states that x(ci,r)x(cj,r) = G ij (6-13) n(Ci) r~~~3

in which G is the order of the group, or the number of symmetry operations of the group, and n(Ci) represents the number of symmetry operations in class Ci. For example, if the numbers in the column 3C2 are multiplied by the corresponding numbers in 6C^ and added, we find 1-1 + 1(-1) + 2(0) + (-l)(1) + (-l)(-1) = o (6-14) On the other hand, taking the sum of the squares of the characters in 3C2, we find 12 + 12 + 22 + (-1)2 + (_1)2 = 8 (6-15) which is 24/3. This orthogonality relation applied to the class E gives 12 + 12 + 22 + 32 + 32 = 24 (6-16) which states that the sum of the squares of the degrees of the irreducible representations is equal to the order of the group. The horizontal orthogonality relation states that n(Ci)X(Ci,rC)X(Ci, Fj) = GCp (6-17) i For example, if we take the representations A2 and T2 we find l(l)(3) + 8(1)(0) + 3(1)(-l) + 6 (-1)(1) + 6(-l)(-l) = 0 (6-18) On the other hand, taking the square of T2, we find 1.32 + 8.02 + 3(-1)2 + 6(1)2 + 6(-1)2 = 24 (6-19) 35

This theorem is important in determining the irreducible components of a reducible representation. Suppose that X(CiD) is the character of the matrix of an operation in class Ci for some representation D of the cubic group. If the latter is reducible, its characters are given by the sum of the irreducible components. Thus, if a(r) represents the number of times the irreducible representation r is contained in D, then X(Ci,D) = a() X(Ci,) (6-20) If we multiply this equation by n(Ci)X(Ci,rp) and sum over classes, we obtain Z n(Ci)X(Ci,D)X(Ci,r) = n i)x( ci, r) a( r) x( Ci, r) i o,i cGap a(r) = Ga(Fr) therefore a(r ) G= n(Ci)X(CD)X(Ci,D)X(i,r) (6-21) i In many instances, the irreducible representations can be determined by inspection. Another important theorem is concerned with the reduction of a product representation. For example, this problem will arise when we consider the two- and three-electron cases. Since the wave functions will be products of wave functions of the individual electrons, we shall be concerned with the transformation properties of such product functions. The theorem simply states that 36

x(ci, rrP) = x(ci,Ir')x(cCi,r) (6-22) and the reduction of this is carried out by the recipe stated earlier. This theorem is useful in determining the condition under which certain integrals can be expected to vanish. In the calculation of the energy levels, we need the integrals f t-iVa jdT (6-23) The perturbation potential is invariant under the symmetry operation, and so in the case of the cubic group it belongs to the identity representation A1. Now it can be verified that Alri = ri (6-24) where Fi is A1, A2, El, T1, or T2. Furthermore, it can be shown that rirj for i i j does not contain A1. Now V4j contains Fj but the integral will contain A1 only if i = j; furthermore, the integral will be automatically zero if A1 is not contained in the product representation. This, then, tells us that in the perturbation calculation we need to calculate only those integrals which connect states belonging to the same representation. The theorem given in (6-22) is also useful in determining the selection rules for electric dipole transitions. For this the relevant integral is 4r*rfdT (6-25 in which i and j are the two states involved in the transition. Now r transforms like T1 so that the integral will be zero, unless the product of Ali and Jj (or more precisely, FiFj) contains T1. Using the cubic group character 37

table, we find A2T1 = T2 A2E = E A2T2 = T1 ET1 = T1 + T2 ET2 = T1 + T2 = A1 E2 = A1 + A2 + E Ti = A1 + E + T1 + T2 T2 = A1 + E + T1 + T2 T1T2 = A2 + E + T1 + T2 (6-26) The allowed transitions then are A2 - >T2 E <- >T1 E > T2 T1 <-> T1 T2 < - T2 T1 > T2 (6-27) We see, then, that the levels E and T2 brought about by the splitting of the 3d levels can give rise to an allowed transition, despite the fact that AQ = O' Of course this is true for the cubic group 0, which has no center of symmetry. For the cubic group Oh, the levels derived from the 3d orbitals be38

long to the even (gerade) representations Eg and T2g, whereas r transforms according to the odd (ungerade) representation Tlu, so that integrals of the form i(Eg) r J(T2g) d vanish identically and electric dipole transitions are not allowed between states of these levels. 39

VII. THE 3d2 CONFIGURATION Earlier we indicated that the 3d-electron energy level is split into levels belonging to the cubic irreducible representations E and T2. In our notation henceforth we shall speak of the t2 or the e electron depending upon whether the electron occupies one of the T2 or E orbitals. Since there are three T2 and two E orbitals, it is clear that there can be all together 2 x 3 = 6 electrons occupying the T2 orbitals, and 2 x 2 = 4 electrons occupying the E orbitals. Consequently, we may speak of the t2 and the e subshells of the d-electron shell. As in atomic spectroscopy, we shall consider next the possible electron configuration arising from two equivalent t2 electrons. The basic Schroedinger equation Hr Elr (4-1) we wish to solve is such that e2 H H1 + H2 + —-- (7-1) r12 in which Hi 2m V2 + V(ri) + Fc(x+y+z- 5 (7-2) The method of attack is to expand the solution in terms of product functions of the two electrons. For the t2 configuration, there are nine product functions obtained by multiplying the set (xy) 1,(yz), (zx) into (xy)2,(yz)2, (zx)2. The indices 1 and 2 refer to the two t2 electrons~ Earlier, the 40

symmetry properties of the Hamiltonian were exploited to obtain information about the nature of energy levels and wave functions, The Hamiltonian (7-1) is invariant under the operations of the cubic group, so that some of the previous arguments can be used. However, we note that the Hamiltonian exhibits an additional symmetry property of being invariant when the indices 1 and 2 are permuted. This means that the solutions of (4-1) must constitute the basis functions for the symmetric group S2. For this group, there are only two operations: the identity operation, represented by the permutation (1) (2), and the operation exchanging the indices 1 and 2, represented by (12). Clearly, then, there are only two irreducible representations, and the character table is as follows: (12) (2) - Class (1) (2) (12) S 11 1 A 1 -1 9 3 = 6S + 3A This table shows that the irreducible representations are one-dimensional, i.e., the basis functions are such that each transforms into itself or into its negative upon permutation of the indices 1 and 2. These are the familiar symmetric and antisymmetric representations. Consider now the effect of these operations upon the nine product functions:

(yz) 1(yz)2 (zx) 1(yz)2 (xy) l(yz)2 (yZ) 1(zx) 2 (ZX) 1(ZX) 2 (xy) 1(zx) 2 (7-3) (yZ) 1(xy)2 (zx) l(xy)2 (xy)1 (xy)2 The two permutations are represented by 9-x-9 matrices. The character, or the trace, is simply equal to the number of functions that are not affected by the permutations. Consequently for the class (12) the character is 9, since none of the functions change when left alone. For the permutation (12) only three functions-(yz) l(yz)2, (zx)l(zx)2, and (xy) l(xy)2-transform into themselves. Consequently the character for class (2) is 3. By inspection (there is no need here to appeal to the general theorem enunciated earlier') it is seen that the reducible representation contains six symmetric and three antisymmetric irreducible representations. Since these representations are onedimensional, the nine functions of (7-3) can be grouped into six symmetric and three antisymmetric functions. The symmetric functions are (yZ) 1(yz)2 (YZ) 1(zx)2 + (YZ)2(ZX)1 (ZX)1(ZX)2 (yZ) (xy)2 + (yZ)2(xy)1 (7-4) (xy)1 (xy)2 (zx) 1(xy) 2 + (zx)2(xy)1 and the antisymmetric functions are (yZ) 1(zx)2 - (yz) 2(zx) (zx)1(xy)2 - (zx)2(xy)l (7-5) (xy) l(yz)2 - (xy) 2(yz)1 42

Next consider transformations induced by the operations of the cubic group. For (7-4) we shall obtain 24 6-x-6 matrices and for (7-5) 24 3-x-3 matrices. Our task is to decompose these into a sum of irreducible representations. The calculation of the characters would be rather tedious if we had to carry out the linear substitutions mentioned earlier. Fortunately, however, by using the result X(R,S) = 1 {[X(R) ]2 + X(R2)1 (7-6a) and (R,A) =1 [ {X(R) ]2 - X(R2) (7-6b) the necessary decomposition can be readily carried through (see, for example, Ref. 12, p. 134). From the cubic character table we have E 8C3 3C2 6C4 T2 3 0 -1 1 -1 XT 9 0 1 1 1 X(R2) 3 0 3 3 -1 To the characters of T2, we need to add or subtract those of X(R2). In so doing, we note that E2 = E C3 C3 C2 = E C2 C4 = C2 4

giving the set of numbers indicated in the row X(R2). Consequently, characters for the symmetric and antisymmetric representations are E 8C3 3C2 6C' 6C_ s 6 O 2 2 O A 3 0 -1 -1 1 so that S = A1 + E + T2 (7-7) and A = T1 (7-8) As a check, we note that (T2) = A1 + E + T1 + T2 = S + A (7-9) According to Paul:'s principle the overall wave function, including both orbit and spin, must be antisymmetric in the indices 1 and 2. Therefore if the orbital function is symmetric, the spin functions must be antisymmetric, and vice versa. The symmetric and antisymmetric spin functions have spins 1 and 0, respectively. Consequently, we obtain the allowed states 1Al +'E + 1T2 (7-10a) and 3T1 (7-11a) in agreement with the result obtained by other procedures. For future use, we shall restate our results in terms of Young tableaux (see Refs. 12 and 13). Using this notation we have shown that

= 1A + 1E + 1T2 (7-lOb) = 3T (7-llb) We shall need these results to determine the allowed states for the 3d3 configurations.

VIII. THE 3d5 CONFIGURATION The aim of this section will be to present a number of arguments to make the Orgel-Sugano-Tanabe diagram of Cr3+ understandable. As has been pointed out already, the cubic crystalline electric field will lead to the splitting of free ion energy levels. The Orgel-Sugano-Tanabe diagrams show how the energy-level splittings are affected by the strength of the crystalline field, as shown in Fig. 12. I 3d2 3 4 y=4.50 Cr IV:3d (F)4S, F''11 B=918 f (dedy2) 2A2 70 -,E 2F E/B 50 o4F. 4 40 F2(de2dy) 2F 2 30H2 10 4A e (des) 4Fo 0 -I - 2 3 Dq/B Fig. 12. Splitting of states of the 3d5 configuration by an octahedral field (Ref. 5, Fig. 2). To develop the qualitative ideas, on the basis of symmetry arguments we shall first consider the case of weak crystalline field and then the case of the

strong crystalline electric field. To carry out this program we shall make frequent reference to the group character table and to the reduction of reducibility representations. For the sake of convenience the group character table, the characters of the reducibility representation, and their irreducible components are presented in the following cubic character table: E 8C3 3C2 6C~ 6C4 A1 1 1 1 1 A2 1 1 1 -1 -1 E 2 -1 2 0 0 T1 3 0 -1 -1 1 T2 3 0 -1 1 -1 P 3 0 -1 -1 1 T1 D 5 -1 1 1 -1 E+T2 F 7 1 -1 -1 -1 A2+T1+T2 G 9 0 1 1 1 A1+E+Tl+T2 H 11 -1 -1 -1 1 E+2T1+T2 A2 x A2 1 1 1 1 1 Al A2 x E 2 -1 2 0 0 E A2 x T1 3 0 -1 1 -1 T2 A2 x T2 3 0 -1 -1 1 T1 E x E 4 1 4 0 0 A1+A2+E E x T1 6 0 -2 0 0 T1+T2 E x T2 6 0 -2 0 0 T1+T2 T1 x T1 9 0 1 1 1 A1+E+T1+T2 T1 x T2 9 0 1 -1 -1 A2+E+T1+T2 T2 x T2 9 0 1 1 1 A1+E+T1+T2 47

The terms for the 3d3 electron configuration (see, for example, Ref. 12, p. 423) are 2p, (2D)2, 2F, 2G, 2H, 4D, and 4F. According to the Atomic Energy Tables,14 these levels are arranged as follows: 2F - - - - -36490 2H - _- - - 21078 a2D - - -20218 2G - - - 15064 2p - - - -.14185 4P - - -14072 4F - - - -O (The numerical indicates the term value for the lowest J value of the multiplet. ) When this 3-electron system is placed in an electric field having cubic symmetry, the above atomic energy levels will split, as indicated by the character table. For example, since F = A2 + T2 + T1 the 4F ground state will split into three levels, as shown in Fig. 13. The splitting will be similar for other levels. The complete splitting scheme of the terms arising from the 3d3 configuration is given in Figo 14. 48

4T1 / / 4F / 4\4A Fig. 13. Splitting of 4F ground states. 2 b2D' _ _A 2EF _PT T1,2E (Degenerate) a2D 2E 2T, a2D —-- 2T -~.~ 2T2 2p 2T1 ~4p ~~ 4T1 4T 4F_-_ 4T2 Fig. 14. Complete splitting scheme of the terms arising from the 3d3 configuration. 11

Group theory does not give the sequence of terms shown in Fig. 14. However, provided the constant b4 is positive, detailed calculations of the sequence can be obtained from pages 13 and 17 of Ref. 15. The scheme gives an explanation of the left-hand side of the OST diagram. It should perhaps be noted that Sugano and Tanabe5 have indicated the splittings only of states 4F, 4P, 2G, and 2F. Consider next the right-hand side of Fig. 14. To understand how these levels came about, let us first recall an earlier remark that the effect of the cubic crystalline field is to remove the equivalency of some of the delectrons. In a spherically symmetric electric field and d-shell can accommodate as many as ten equivalent electrons. On the other hand in a cubic field six electrons will be affected differently from the remaining four, so that often the terms t2-subshell and e-subshell are introduced, The electrons in the t2-subshell will be equivalent to one another, but not to those in the e-subshello It is easy to see, then, that the possible electron configurations are (t2)3, (t2)2e, t2(e)2, (e)3 The right superscript, as in atomic spectroscopy, represents the number of electrons with the indicated orbital. For example, t2(e)2 means that there is one electron occupying the t2 level, and two electrons occupying the e levels, We shall discuss the allowed states associated with each configurationo Consider first the configuration ta. The Hamiltonian will be invariant under the permutation of the indices 1, 2, and 3, so that the wave functions 5o

must constitute the basis for the irreducible representations of the symmetric group S3. This group consists of six permutations, belonging to three classes as indicated below: Classes ( 13) (2,1) (3) Permutation E (12) (123) (13) (132) (23) There are, accordingly, three irreducible representations of degrees 1, 2, and I, respectively. These representations will be denoted by the symbols [3], [2,1], and [13] respectively. The group character table (see, for example, Ref. 12) is (13) 3(2,1) 2(3) [3] 1 1 1 [2,1] 2 0 -1 [13] 1 -l 1 t~ 27 9 3 The characters of the reducible representation t3 given in this table are obtained by noting how the three electron wave functions transform under permutations of the group S3. As mentioned earlier, the t2 wave functions are given by = (yz) = (zx) = (xy) 51

The 3-electron wave functions are given by the product of 1-electron wave function. Since the wave function for each electron can be any one of the three functions indicated above, these are 3 = 27 product functions, so that the character for the identity operation, E, is 27, Consider next the character for the permutation of class (2,1), and in particular consider the permutation (12). The character for this operation and consequently for the class (2,1) is given by the number of functions that are not changed by the permutation. This will occur if the orbital part for electrons 1 and 2 is the same. A typical function is, say, t(1) x 5(2) x T1(3) The orbital of electrons 1 and 2 can be one of the three functions, and electron 3 can also, independently, have any one of the three orbitals. Consequently, the number of product functions remaining unchanged by the permutation (12) is 9, which means that the character for this class is 9. Finally, the character for class (3) is the number of functions that are not affected by the permutation (123) or (132). Clearly, there are only three such functions, given by t(1) x 2 x -q(x2) x(3)3 t;(l) x;(2) x Y(3) and hence the character is 3. Using the theorem enunciated earlier, we find that (t2)3 = 10 [3] +8 [2,1] + [13] 52

or, in terms of Young's tableaux, (t2)3= 1 + + (10) (8) (1) The dimensionality of the representation given by eachi shape is given by the numbers underneath. The representation [3] or, for which the orbital part is totally symmetric, is forbidden by Pauli's principle. Or to use the language of tableau calculus, in order for the total wave function to satisfy Pauli's principle, the tableau representing the spin part must be the transpose of the orbital part. Furthermore, since the electron has only two possible spin orientations, the electron spin tableau must not have more than two rows. Thus, of the three shapes the first one is forbidden for the electron spin, which leaves and as the only allowed orbital tableaux. Now belongs to 4A2 of the cubic irreducible representation. 53

This can be seen as follows. The tableau represents the totally antisymmetric space function and is given by (xy) (xy)2(xy)3 *a = N L 8p P[(xy)l(yz)2(zx)3] = N (YZ)1(YZ)2(yz)3 p (zx) 1(ZX) 2(ZX)3 where r+l for an even permutation -1 for an odd permutation Selecting now one operation out of each class of the cubic group and operating on the *a, we get: Operation Result Character (x ) y E: ( y_ — y EJVa''Va 1 E3: ( y EaC3a = *a 1 7x )y y C3:2 ( Y )-Y C3Ea ='a 1 kz C2: y- -Y C24fa ='a 1 /x Y C y y x ) C2a = *a x y) C4: ( y- x C4a = a -l These are the characters of the irreducible representation A2. 54

The irreducible cubic representations contained in the other tableaux can be readily obtained as follows. The outer product gives + Now EC belongs to T2 belongs to 3T1 and furthermore T1 + T2 = A2 + E + T1 + T2 so that 2E= E+ T+ 2T2 Collecting our results, we have shown that the allowed configurations of t2 are t3 = 4A2 + 2E + 2T1 + 2T2 This is in agreement with Bethe's result, which is obtained by a very different procedure. 55

The states for the configuration (t2)2e can be obtained easily by noting that (t2) = 3T1 +'A+ 1El + 1T2 and by making use of the results for product representations. Thus, the addition of an e-electron to the state 3T1 leads to (3T,) (E) = 2T1 + 2T2 + 4T 4T2 The others are ('A1) ( E) = E ('El) (E) = 2A1 + 2A2 + 2E (1T2) (E) = 2T, + 2T2 Carrying through a similar analysis for t2(e)2 we find that t2(e)2 = 4T1 + 2T + 2T2 + 2T1 + 2T2 And finally for e3, the single state 3 2E e = 2E In sunmary we have found that the states arising for the weak-field case are as follows:

4A2 4T1 4T2 2A1 2A2 2E 2T1 2T2 4p 1 4P 1 2p 1 2G 1 1 1 1 2(2D) 2 2 H 1 2 1 2F 1 1 1 Total 1 2 1 1 1 4 5 5 From the strong-field analysis we obtain: (t2) 1 1 1 (t2)2e 1 1 1 1 2 2 2 t2(e)2 1 2 2 (e) 1 Total 1 2 1 1 1 4 5 5 The two totals agree, as they should. Each of the states 4A2, 4T2, 2A1 and 2A2 occur only once. These are the states indicated by the straight lines connecting the left- and right-hand sides of the OST diagram. Since there are two 4T1 states, the associated energy levels are given by the solution of a 2-x-2 determinant. Furthermore, according to well known results in quantum mechanics, the energy states of different symmetry do not interact, but those of the same symmetry tend to repel. Consequently, on the left-hand side, the two T1 states should start out parallel to each other, but the separa57

tion should increase with increasing strength of the crystalline electric field. The energy levels for the 2E, T1, and T2 states are obtained by solving 4-x-4 (for 2E) and 5-x-5 (for 2T1 and 2T2) determinants. We shall now analyze the energy levels in greater detail. The term values for the d3 configuration are given in Ref. 11, pp. 206 and 233 (see also Ref. 16). These values are: 4F -15B 4p 0 2G -11B + 3C ~P.2H - 6B + 3C a2D 5B + 5C -f93B2+ 8BC+ 4C2 2F 9B + 3C b2D 5B + 5C +V193B2 + 8BC + 4C2 Comparison of these results with the experimental spectrum given earlier shows very clearly the inadequacy of even the atomic theory. If the term values for the 4F, 4P, and 2G states are used, the following numerical values are obtained: B = 917.36 cm1 C = 3678.67 cmand C 7 = = 4.01_ On the other hand, Sugano and Tanabe5 have used 58

B = 918 cmand y = C = 4.50 for their analysis. For the octahedral coordination, the crystalline field energy of the t2 and e orbitals are -4Dq and 6Dq respectively. These values are chosen by taking the separation of t2 and e to be lODq, with the center of gravity at zero (or in matrix language, the trace of the cubic field energy matrix is taken to be zero.) Thus 3E(t2) + 2E(e) = 0 E(e) - E(t2 = lODq therefore E(e) = 6Dq, E(t2) = -4Dq Then the crystalline field energies for the (t2)n (e)-n configurations are given by E(tn e3n) = (-4Dq)n + (3-n)(6Dq) so that we obtain: n Configuration Crystalline Energy 3 (t2)3 -12Dq 2 (t2)2e - 2Dq 1 t2(e)2 + 8Dq O (e) 18Dq 59

This means that for very large values of Dq (ioe., for a large crystalline field) the terms arising from the above configurations will cluster about the values indicated in the last column. 60

REFERENCES 1. R. Findley, "Telephone a Star is the Story of Communication Satellites." National Geographic Magazine 21, 638 (1962). 2. W. K. Victor, R. Stevens, and S. W. Golomb, "Radar Exploration of Venus." JPL Tech. Report No. 32-132. 3. R.W.G. Wyckoff, "Crystal Structures." Interscience Pub., Inc., New York. 4. P. P. Ewald and C. Hermann, Strukturbericht Vol. I (1913-1924), Akademische Verlogsgesellschaft M.B.H./Leipzig. 5. S. Sugano and Y. Tanabe, "Absorption Spectra of Cr3+ in A1203a." J. Phys. Soc. Japan 13, 880 (1958). 6. J. S. Griffith, "Transition Metal Ions." Cambridge Univ. Press (1961). 7. S. Geschwind, P. Kisliuk, M. P. Klein, J. P. Remeika and D. L. Wood, "Sharp-Line Fluorescence, Electron Paramagnetic Resonance, and Thermoluminesence of Mn4+ in c-A1203." Phys. Rev. 126, 1684 (1962). 8. M. R. Lorenz and T. S. Prener, "Effect of Crystal Structure Upon the Luminesence of Manganese-Activated Lithium Titanate." J. Ch. Phys. 25, 013 (1956). 9. J. Lambe and C. Kikuchi, "Spin Resonance of, +, V3 V4 in c-Al20s. Phys. Rev. 118, 71 (1960). 10. Y. Tanabe and H. Kamimura, "On the Absorption Spectra of Complex Ions." J. Phys. Soc. Japan 13, 394 (1958). 11. E. V. Condon and G. H. Shortley, "The Theory of Atomic Spectra." Cambridge Univ. Press (1959)~ 12. M. Hamermesh, "Group Theory and its Application to Physical Problems," Addison-Wesley Pub. Co., Inc., 1962. 13. D. E. Rutherford, "Substitutional Analysis." Edinburgh Univ. Press, (1948). 14. C. E. Moore, "Atomic Energy Levels." Circular No. 467 of the National Bureau of Standards.

15. Seminar notes by Dr. H. Watanabe (Fall 1962) Nuclear Engineering, The University of Michigan. 16. G. Racah, Phys. Rev. 62, 438 (1942). 62

PART II X-RAY PRODUCED V+2 FROM VO+2 IN SINGLE CRYSTALS OF ZINC AMMONIUM SULFATE —ZnSO4(NH4) 2S04 6H20 by Robert Borcherts

Foreword The purpose of this report is to present the results, to date, of a set of electron spin resonance experiments on: (a) VO+2 and V+2 in single crystals of zinc ammonium sulfate, and (b) X-ray irradiated VO+2 in single crystals of zinc ammonium sulfate, magnesium ammonium sulfate, and zinc potassium sulfate. Conclusions Upon X-ray irradiation of the single crystals containing VO+2 it was found that the X-ray, or the subsequent energetic electron that it produces, breaks the VO bond, removing the oxygen from the VO+2 site and leaving V+2 behind. Since the V+2 produced by irradiation and the crystalline field surrounding it is identical (as far as EPR measurements can determine) to that found in single crystals of "grown" V+2 in zinc ammonium sulfate, the oxygen must be at a distance so removed that the crystalline field is unaffected by its presence. Figure 1 shows the EPR spectrum along the K2 axis for the VO+2 crystal, the irradiated VO+2 crystal, and the V+2 crystal, depicting the results quite vividly.

4,44 0~~~~~~~~ CD I -t:CB 02 0" 0.... CD 0 61~~ ~~~ ~~~~~~~~~ 4 cr D06012 t ODOfO~; iiil.'~~~~~~~~~~~~~~~~~~~~~~~~~~~~ —7 ao~~~~~~~~~~~~~~~~~~~~~t ~~~~~~~-a oo~~~~~~~~~~~~~~~~~~~~~~~~~a C ct I I;~~~~~~~~~~~~~~~~~~0 I I r J. ~-il~ ~ ~ -t ~~~~~~~~~~~~~~~~ C CC CC 0 CC C Ci CC x il~~~~~~~~~~~~~~~i L o T -i —-; d~~~~~~~~~~~~~~~~Nco a.1 44 I o~~~~~~~~ i,~j PC rC~~c o~ aD F

I. BACKGROUND The first successful application of microwave techniques to the study of the centimeter and millimeter range of energy level separation was by Zavoiskyl in 1945. Since then the study of electron spin resonance spectra of various ions in crystalline and non-crystalline solids has made a major contribution to knowledge in the field of solid state physics. One ion of particular interest is that of vanadium. The +2, +3, and +4 spin states of V51 (abundance 99.76%, I = 7/2) have electronic spins of 3/2, 1, and 1/2 respectively. Consequently the electron spin resonance signature of each oxidation state is readily recognized-in non-cubic crystalline fields by the number and angular dependence of the fine structure groups as well as by the separation of the hyperfine structure, and in cubic crystalline fields by the separation of the hyperfine structure (see Figs. 2 and 3). In 1950, Bleaney, Ingram, and Scovil2 reported on the EPR spectrum of v+2 in ZnSO4(NH4)2SO4.6H20, the same host crystal used in this set of experiments. In this crystal the six water molecules form a distorted octahedron surrounding the V+2 ion, producing a rhombic crystalline field. As a result the 2S+1 degenerate ground state splits into two spin doublets, Ms = +3/2 and Ms = ~1/2 respectively, separated by 2D. Applying a magnetic field splits the remaining degeneracy and application of microwave energy at a frequency v gives rise to three fine-structure groups at g - 2, corresponding to the selection rule AM = +1. Because of the 7/2 value of the nuclear spin of V51, each of the fine-structure groups are composed of eight lines (2I+1) with a

P T1, / (I) At2 12 Dq / (3) 4 Tag \Dq (,T,g (2) E t_. \-' —' _ <) Ahvu2g FREE ION CUBIC F-P TETRAGONAL LUS COUPLING MAGNETIC FIELD FREE ION CUBIC INTERACTION (Elongated) P T, dS't+ (3) T (2) / t < (I) 6Dq 2Dq \ ()3) (2 Etg \ i i \ (I) Btosg + I Dq (I) A,I. Fig. 2. +3 and V+2 energy levels. 6

W Z LLI (D CD) (-9 z CI)J -J o.i w w ~o -JI I I Z;:l~o cu ilO~~~~~ V i,i~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~L Cib~~~~~~~~~~i F- c aI: IZa 04 C)~~~~~~~~~~~~~Q C-)~~~~~~~~~~~~~~~~~~~Q orl Z. 0, z~~~~~~~ -zl -l~ CD I- L LL O c~~~~~~~~~~~ C-~~~~~~~~~~~~~w H C- ) -C j wCC~~~~~~~)W C z8 ~~3~ 0 co~~ ~ ~~ ~ ~~~~~;4- w1I I 0k w 0w 7

separation of approximately 90 gauss. In 1958, Zverev and Prokhorov3 and later Lambe, Ager, and Kikuchi 4 reported on V+3 in sapphire (oC-Al203). And in 1960, Lambe and Kikuchi5 found a small amount of V+4 in A1203. They were also able to show that by X-ray irradiation or gamma irradiation some V+2 is produced. Also in 1960, Gerritsen and Lewis6 and Zverev and Prokhorov7 reported on V+4 in rutile (TiO2). In sapphire six oxygens surround each aluminum site in a distorted octahedron, giving rise to a trigonal crystalline electric field. If V+3 is substituted for the A1+3 ion, the trigonal component of the crystalline field splits the T2 energy level into a doublet E and a singlet B. This splitting, acting through the L'S coupling, causes the ground state to split into a lower spin singlet Ms = 0 and an upper spin doublet Ms = +1, separated by -10/cm. Ar applied magnetic field causes the Ms = +1 level to split and as a result a forbidden electron spin resonance transition corresponding to AM = +2 can be observed at a magnetic field of H = hv/2gp. Since the ground state is separated by -10/cm the relaxation time of this transition is so short that in order to observe the spin resonance transitions the experiments must be performed at very low temperatures-approximately 4.2 K. If the temperature is too low, less than 2~K, the Ms = +1 state becomes so depopulated that no signal is observed at all,' In rutile, six oxygen atoms surround the Ti+4 site in a distorted octahedron, producing an orthorhombic crystalline electric field. Since V+4 has only one 3d electron and thus a spin of 1/2, when it is substituted for the Ti+4 ion only one fine structure group corresponding to a AM = ~1 transition

is observed. However, if tetragonal symmetry is present the relaxation time may be long or short depending on whether or not the tetragonal symmetry is "squashed" or "elongated" (see Fig. 3). For rutile the tetragonal field appears to be elongated since the experiment has to be performed at liquid nitrogen temperatures (77~K) in order to observe the EPR spectrum. The observed hyperfine spectra is very anisotropic, having a separation of 157 gauss along the z-axis and 35 gauss and 48 gauss along the x- and y-axes respectively. The g-values (gx = 1.915; gy = 1.913; gz = 1.956) reflect the rhombic symmetry of the crystalline field. Although the "valence state" of vanadium in the vanadyl radical VO2 is also +4, the strong cylindrical crystalline field formed by the vanadium and the oxygen produces an energy level structure quite different from that of V+4 in a cubic field8 (see Fig. 3). The first electron spin resonance spectrum of the vanadyl radical (VO+2) was reported by Garifianov and Kozyrev9 in a frozen aqueous solution and later by Pake and Sands10 in aqueous, acetone, and ether solutions; by O'Reilly11l in vanadyl etioporphyrin dissolved in benzene and high viscosity oil; by Faber and Rogers12 in various adsorbers (charcoal, Dowex-59, IR-4B and IR-100); and by Roberts, Koski, and Caughey13 in some vanadyl porphyrins. As mentioned above, the V0+2 radical produces a very strong cylindrical of axial field so that it matters little what other kind of field surrounds the V0O2. This is evidenced by the similar spectra found by these investigators.9-13 The results of these investigations are that the gL and g\ values are 1.98-1.99 and 1.88-1.93 respectively, and that there is a hyperfine splitting of 160

200 gauss parallel to the z-axis and of 60-85 gauss perpendicular to the zor VO+2 -axis. The variation in these values is suggested to be due to the extent of covalent bonding between the vanadyl ion and the surrounding ligands~12 Because the samples used in all these investigations were either powders or solutions, the EPR spectrum, is either an average of the randomly oriented VO+2 radicals in the frozen or powdered samples (see Fig. 4a), or a time-averaged spectrum as a result of the motion of the V0+2 in solutions. Thus, because of the need for EPR information on oriented V0+2 and because of the success of Lambe and Kikuchi5 and Wertz, Auzins, Griffiths, and Orten4 in producing the various oxidation states of vanadium, the experiments reported here were undertaken. Underlying these immediate reasons is perhaps a deeper one-that of a systematic study of the solid state chemistry of vanadium. 10

0. -7 i - - ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ - /c r/~~~~~~~~~~~~~~~~~C 0 ~~~~~00 - 0~~~~~~~~~~~4,~~d' ~. f "o /~~~~~~~~~ 4 -"I-Z CI) 0~~~~-4~I 0 0'"'-' ~~~~4 ~ - CQ 0~~~~~~ V - /~~~~Q4 ~~ /~~~~~~~`~ -U.~ ac~

II. EXPERIMENTAL METHODS Ao CRYSTAL GROWTH Single crystals of 0.05, 0.1, 0.5, and 1.0% (vanadium to zinc) concentrations of ZnSO4(NH4)2S04*6H20 were grown from a water solution into which reagent grade ZnSO4'7H20, (NH4)2'S04 and VOS04.2H20 were added in the prescribed amounts. Light blue crystals up to 3-4 mm in length were usually obtainable in 1-2 days. Zinc potassium sulfate and magnesium ammonium sulfate crystals containing 1.0% VO+2 were also grown by this method. In contrast to the ease of growing the vanadyl crystals, the growth of the 1.0% and 2.0% vanadous ammonium sulfate crystals required a greater degree of skill due to the rapidity at which V+2 becomes oxidized. Cathodic reduction of the vanadyl solution, in which the cathode and the anode are separated by a porcelain cup, followed by evaporation at 5~C in a carbon dioxide atmosphere was the method used to grow these crystals. B. CRYSTAL STRUCTURE AND ORIENTATION Figure 5 shows the two molecules contained in the unit cell of MgSO4(NH4)2S04.6H20. The position of the atoms is taken from the data in Wykcoff.15 The octahedron of water molecules surrounding the divalent metal ions forms the crystalline electric field which determines the constants in the spin Hamiltoniano That is, the data obtained from paramagnetic ions placed in these positions directly reflects the environment and the interaction of the electrons with the environment, Measurements of EPR experiments 12

w IL) I CD o (.3 C94 _ 0 WC~~~~~~~~~~~ a o ff-o o —; 0~~~~~~:D O. C cmJ,-,o wP~~~~~~ lo 0 c'J ~~~~~~~~~~~~~~~~~~~~~ \~~~~~~~~~~~~ PO~ ~~~~ z 0 0 O~~~~~~ -. ~~~~ ~O0 \ ~ ~ ~ ~ ~ ~ ~ ~ ~~~~~~~\ E0'-,-U "~I -\~~~~~~~~ -'-4 O0 CL z 0~~~~~~~~~~~~~~~~~~u z a, ~-oj — ~o uL.CD 4 c 0 O 15 cz C-i ~~ ~ ~ ~ ~ ~ ~ Z

can be made with great precision so that very minute changes in the environment can be observed. As shown in Fig. 6 these double sulfates or Tutton salts grow with well recognized facesl6 so that orientation of the crystal in the cavity becomes a task accomplished with relative ease. Figure 6 also shows the device used for positioning the 22-inch quartz rod to the crystal to within a degree or two of the desired orientation. After placing the crystal in the cavity the final adjustments of the crystal orientation are made by moving the quartz rod until the desired spin resonance spectrum is observed on the recorder. The experimental data that can be recorded are: (a) the magnetic field at resonance, (b) the angle of the magnetic field to some axis, (c) the frequency of the klystron, (d) the first derivative of the line shape as presented on the recorder, and (e) the angle that the c-axis makes with the horizontal. The angle that the c-axis of the crystal makes with the horizontal is observed through the optical hole in the side of the cavity by means of a surveyors transit o C. EPR EXPERIMENTAL SETUP Electron spin resonance spectra and magnetic field measurements were made at room temperature with a Varian V-4500 EPR Spectrometer and a Varian Model F-8 Fluxmeter connected to a Berkeley 7580 Transfer Oscillator and a Berkeley 7370 Universal Eput and Timer. Both X- and K-band klystrons (~10 kmc and -24 kmc respectively) were usedo The K-band klystron was needed for the V+2 measurements since the zero field splitting is 9,8 kmc.5 Both cavities were a silvered ceramic so that 100 kc modulation of the DC magnetic field could

J) d OOU rd','0~~~~~~~~~~~~~~~~.H O (O Y~~~~~~~~~ (%4 a) *10, ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~C.) >~~~~ 0 bDO a) U) /~~~~~~~~ cH IL C) cio.1-i ILl w~~~~~~~ L15 COO co cli CPO~~~~~~ o~~~~~~~~~~~~~~c crL 10 ~ ~ ~ 1 -t i~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~w hi c\r~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~r..._ L- -~~~~~~~~~~~~~~~~~~~~-~~~~.~~~~~:..-/ Z j (U~ ~ ~ ~~~~~~~~~~~~~~~~~- O U 9 e~~~~~~~~~~~~~~~ w._i to~~~~~~~~~~~~~~~~~~~O

be employed, X-ray irradiations were made for approximately 15 minutes in the white beam of a GE X-ray machine operated at 45 kvp-40 ma with the crystal approximately 1 inch from the tungsten target. Magnetic field measurements could be made accurate to 0,2 gauss and 0o1 gauss for K- and X-band frequencies respectively. Klystron frequency measurements of 5 Mc/sec and 005 Mc/sec accuracy for K- and X-bands respectively gives a lower limit of +00005 on the error of the g-value.

III. EXPERIMENTAL RESULTS A. VO+2 CRYSTAIS From the spin resonance measurements of the VO+2 crystals the following axial spin Hamiltonian* constants were calculated: Present Work Other Results (Powder) gll = 1.9328 + 5 1.94713 1.94811 1.88 - 1o9312 gl = 1o9802 + 5 1.988 1.987 10978 - 1.989 iAI = 0.01824 ~ 2/cm 0.0158 0,0159 0.0158 - 0.0184 IBI = 0.007162 ~ 5/cm 0.0054 0.0052 0.0061 - 0.0075 IQ I = 0.00024 ~ 5/cm The VO+2 radical enters the crystal with the VO+2 axis in one of four different directions, as shown in Fig. 7. A pair of VO+2 axes lying in the same plane belong to the same type of sites (the crystal has two molecules per unit cell), and the lower concentration position (10%) has slightly different g-values (g,, = 1.9314 + 5, gl = 1.9812 + 5) and a line width of 4.7 gauss compared with 5.7 gauss for the 40% position. No variation in intensity ratio was noted for crystals of different VO/Zn concentration. * 4axial = gll HzSz+ H g [HxSx+HySy] + AIzSz + B(IxSx+IySy) + Q' [I2 -I(I 1) 17

FOUR EQUIVALENT VO 2 POSITIONS IN A REGULAR OCTAHEDRON, THE VO+2 AXES POINT TOWARDS THE CENTROIDS OF THE TRIANGULAR FACES TWO POSITIONS OF V0O2 AXES IN ~ IN A DISTORTED OCTAHEDRON. THE MOST POPULATED POSITION IS WHERE THE VO+l AXIS POINTS TOWARD THE CENTROID OF THE TRIANGLE WITH THE LARGEST AREA. Fig. 7. Experimental positions of axes in Vo(Ni4)2(S04)26H20 (1% VO/Zn). / \~~~~a

The angle between the plane formed by the K1K3 axis (see Fig. 7) and the c-axis was found to be 0.60 + 0,.3 A similar measurement for the V+2 crys tals could not be made since it was found that the yz-plane of the one site made an angle of -6~ with the yz-axis of the other site. 1. Discussion The axial spin Hamiltonian constants (gl, gll IA|A, IBI, and IQ' 1) obtained from measurements at O and 900 fit the angular part between O and 900 to a remarkable degree, as shown in Table I. In Fig. 7 the z-axis for the analysis of the data is the VO+2 axis; in fact, it is the location of the zaxis and the result of g1 > gl (see Fig. 3) that indicates the vanadyl radical is present as an entity in the crystal and that its axis is oriented in the directions shown. One can attempt to explain why there are four directions for the V0+2 radical and only two molecules per unit cell on the following basis. The VO+2 ion is expected to enter an octahedron of water molecules in one of four equivalent directions (see Fig. 8) in order to achieve minimum energy. However, in the Tutton crystal the water molecules do not form a regular octahedron or even one with axial symmetry, but rather one with considerable rhombic asymmetry. This is seen from the X-ray data and from the V+2 EPR data presented in Section III-C. This rhombic symmetry makes the four positions of the VO+2 ion non-equivalent and ordered in energy. That is, one of the positions (40% - see Fig. 7) is energetically more favorable, another (10~ - see Fig. 7) is energetically less favorable, and the remaining two positions are so energetically unfavorable that the small population of these 19

CM _D 0I0) I. -1 a (~I o D0 "lN NI 0N1w0)0II0)OI NN1 5di'-'" 0I''d' WC n F C\ W CM C o >I 0) ) l 0) <C- ) I a 0 to E rrK )rI ) () N0K) roJ 10K) O ~ O> Z mI' %,,i (D( (c O W) C) oDNO ) I I (0]. n Z.I ~ 0 n_ W IO CO I~ I N 0 ) 1~|D0 1 03 0 ~I t o 1 I x cn cE I) ) ) or ) 10) )) 1) w0 0 to C4) II I E | O 1) NN N N On H E z 1 K 01 OD Nn" (0I oi C ( X o 0 CD If 0 - _ K) SOW aI iot >Z oDC E N N Nr N N n n.on n a~~1 ~ 0 ~olo0 1 ~ Z ~ -c d 11 00 0) 00 0 0; 0' b bb,. Eau CM ODD ie-D OD W/ -- OD:I- 0 02 E IE II IC)I r OD O

ITO l,'1 I' I l,' 1, I, I i,.,.' i,:, 0 oO ii M 5 i 2!4 i'- i: - i r. -— 7 7 ~ ii 7. K I i;~ 0F, 4 i |! i | i 4 " 1 1! |! _ TT 0' 0 - | -' | - I | i - O i # 3L,L I ~ II;I i* 4 *H L''i j I i l''i'1 1' - - - -1 1 1':,.'.:. 1,'' o, j, ll,J!'i' i i;!, i i,;!i i I 1i'! I jil Z! 1. l',';:,,!i::! I " I',c:I 211 - Ti ii,::~ ~ ~ ~ ~ ~~~~~~~~~C

positions are unobservable by EPR techniques. The assignment of a 10% position and a 40% position to one of the two types of sites can be made by looking at the positions of the VO+2 ions in the water octahedra and noting the energetically favored position. This assignment is then not arbitray to the extent that the X-ray data are assumed to be correct and that the VO+2 ion distorts the water octahedra a negligible amount. One interesting result lies in the value of IQv 1~ Though evaluated at 900 where its effect is of the order of one gauss, the term involving xQ'I increases to 4 gauss at 60~~ And since the difference between calculated and experimental values, as shown by Table I, is of the order of one gauss, this value for IQv I should be quite close to being correcte Now is related( to the quadrupole moment of the nucleus, Q, by the relation =, _ 3eQ 62V 4I(I+1) 6z2 where (~2v \ + < 1/r3 >ecto 6Z22/rystal eeto Evaluating this expression for V51, (eQ = 0.2 x 10-24 CM2)18 (~)rystal +</~electron=5x02/m The value of < 1/r3 >,, if evaluated by the expression <l/r3> __ _ b

where b =(AB) / and is 1.3 x 1025/cm3. Thus it appears as if (2V/Z2) is of opposite sign to < l/r3 > and equal in magnitude to three places Another interesting experimental result is the splitting of each of the resonances into five separate resonances when the magnetic field is in the direction of the tetragonal axis of the surrounding water molecules (see Fig. 9). This splitting, approximately 4 gauss may provide some information on the molecular orbital wave function of the vanadyl ion and the surrounding water molecules. B. VO+2 POWDER The EPR spectrum obtained from the powder formed by crushing the VO +2 crystals gives the same qualitative spectrum (see Fig. 4a) that is reported in Refs. 9-13. One should be able to explain this spectrum quantitatively from the spin Hamiltonian constants of the single crystal. Since the resulting crystal spectrum shows only axial symmetry, the spectrum is dependent only on G., the angle between the magnetic field and the VO+2 axis. Thus, for a given magnetic field, only those crystals or crystallites whose VO+2 axes are at an angle 9 to the magnetic f ield such that H = Hm(G) will contribute

c~~~~~~~c b C) I N CD~~~~~~C'El E 4-' E U) 0E a. In co 0z0 0. 0~~~~~~~~~~ o 1 0~~~~~~ Co~~~~~~~~~~~L 10~ ~ ~~~ 0 o co C b > > co~~~~~~w: ECD 4-) 00 w~~~~0 ~~~~~Ii b~~~~~~~~~c to 21i.~ ~ ~ ~

to the absorption.* Because of the axial symmetry, the number lying at this angle is proportional to the solid angle sinG dg. Thus the number of V2 ions contributing to paramagnetic resonance between magnetic field H and H+dH is -7/2 sing dg m=-7/2 | dH (G) where the sum is over the eight values of I, the nuclear spin. Figure 4b shows a plot of such an equation evaluated for the spin Hamiltonian parameters for a single crystal containing VO2. Note that in this figure the experimental curve is the first derivative of the absorption curve and that it shows the effect of the finite width of the resonances. The sharp peaking of the calculated curve corresponds to those V2 oriented at 90~ and corresponds also to a (cosg)' dependence. From the powder spectrum, H =G hv K m B[2 1H + 2K LI(+l m2 (A~2 f7g,1 si2 o2G+2(? 2sin2Gcos2g 2g2p2Ho ~K/.2I K~5g5p x A2B2g2g2m[)4i(i+l) - 8M2 - 1] -2 sin4G 0 2 K 3gp g x [2I(I+1) - 2m2 - l]m where K~2g2 -g21A~2cos2G + gj2B 2sin2G

the separation of the two extreme peaks corresponds to 71AI/g!!5, and if one neglects the quadrupole contribution, the separation of the two extreme lines in the central part of the spectrum is 71BI/g15 so that IAI, IBI, g~, and g_ can be obtained from a powder sample-to the accuracy limited by the width of the peaks, C. V+2 AND IRRADIATED V0+2 CRYSTALS Operation at K-band frequemn ies gave the following values for the rhombic spin Hamiltonian constants (see Appendix B): V+2 (Grown) V+2 (Irradiated VO+2) V+2 (Reference 2) gz = 1.9718 + 5 1.9719 ~ 5 1.951 + 2 gx = gy = 1.9750 + 5 1.9745 ~ 5 I DI = 0.15603 + 5/cm 0.15609 ~+ 5 0.158 + 10 E E I = 0. 02297 ~ 5/cm 0.02505 ~ 5 0.049 ~ 40 JIA = 0.008270 ~ 5/cm 0.008270 ~ 5/cm 0.0088 *r=20~ 3,0o 30~+1.00 20 20-50 ~ 0.50 20,50 ~ 0.50 220 For the V+2 produced by X-ray irradiation in the three different Tutton salts, the following rhombic spin Hamiltonian constans were calculated:

Zn(NH42 (S04) 2*61H20 Mg(WNH4) 2(_S04) 26H20 ZnK2(~64) 26H2o gz 1.9718 +~ 5 19720 ~ 5 1.9722 ~ 5 x= gy = 1.9745 ~+ 5 197 ~ 5 1.9750 ~ 5 ID = 0.15609 + 5/cm 0,15795 + 5 (+1.2%) 015246 ~ 5 (-2.%) | E 0.02303 + 5/cm 0.02456 + 5 (-6%) 0.02746 ~ 5 (+19.2%) Al = 0.008270 + 5/cm 0o008270 + 5/cm 0.008270 ~ 5/cm 50130 + 110 110 ~20 = 20~5~ + 0.5~ 2000~ + 0.50 14~70 50 The angle that the K1K3 plane makes with the c-axis could not be determined since the zy-plane of the one site makes an angle of 60 with the ZY-Plane of he other site. 1. Discussion As seen by the close agreement of the V+2 (grown) and V,2 (irradiated vo+2) spin Hamiltonian constants,, and the angle ce, one can assume that when the oxygen moves away from the VO+2 site leaving V+2 behind, it is removed far enough so that its influence on the crystalline field is no longer felt., This is also implied by the data on the irradiated V,0+2 in the Tutton salts where Mg is substituted for Zn and K for NH4. That is., these ions., though several angstroms away from the V2 site and'its associated octahedron of water molecules, distort the octahedron by an amount that can be measured by the changes in the values of Dl, JEl, a~, and 4, *Tn experiments on Cu+ in various Thtton salts, Bleaney, Penrose and Plump-.

If one performs a general coordinate rotation on the spin Hamiltonian for rhombic symmetry (in order to diagonalize the Zeeman term-Appendix A), it is seen that only when the magnetic field is along the x-, y-, or z-axes do the cross terms become zero, simplifying the spectrum and enabling one to make measurements that can be used to determine the spin Hamiltonian constants, For this reason no attempt was made to measure the angular part of the spectrum, An illustration of the effect of this mixing can be seen in igo 10. Note that in both the x- and y-directions the mixing and the result ing spread of the spectrum from the second site is so complete that it is almost unobservable, The DI value can be calculated from the data along the z-axis and the IEl value from that along the x- or y-axis. However, the plane of the zyaxis from the one site does not coincide with that from the second site. For this reason the center part of the spectrum in Fig. 1 showing the magnet'ic field along the K2-axis is not a "pure" spectrum, The K2-axis is approximatelY 50 from the x-axis of each of the two sites. Note that the value of JEl reported by Bleaney, Ingram and Scovil2 is almost twice the value found in these experiments, Since they used the information along the z-direction to evaluate both IDl and JEl, and JEl enters as a second-order effect along this directilon, it is thought that their value is not as correct as the value reported here,,

1 20ta.I cc I U I i L -4.1.s W V 29~~

IVe FURTHER STUDY The additional structure of the EPR absorption lines shown in Fig. 9 might yield some information about its origin if attacked with double resonance (ENDO) techniques. Furthermore, with ENDOR techniques the size of the quadrupole interaction, which may be present to some extent as suggested by EPR experiments, can be more accurately determined. Since second-order effects indicate that A and D have the same sign, a low temperature experiment to indicate the relative intensity of the high and low field transitions should determine the absolute sign of D and thus A. The growing of both V++ and VO++ in the same crystal, though the initial attempt has been unsuccessful, would serve as a very precise indicator that the environment of the V++ (grown) and the V+ (produced by X-ray irradiation) are identical. Similar experiments should be carried out to determine the optical absorption spectrum for determination of the spin orbit coupling constant ~X and to measure the production of V + (irradiated) as a function of the radiation dose to the crystal.

APPENDIX A CALCULATION OF ANGULAR VARIATION OF EPR RESONANCE IN RHOMBIC FIELD In this appendix we wish to obtain formulae so that evaluation of the constants in the rhombic spin Hamiltonian can be made from experimental data. We start with the general rhombic spin Hamiltonian iI ~~~~~~~~1 ~rhombic [gzHzSz + gxHxSx + gyHySy] + D[Sz - S(S+1)] + E(S2-Sy) + AIzSz + BSxIx + CSyIy + Q' [I2 1 1( +) + Q"f[I2 - I2] and proceed to diagonalize the Haniltonian in the Zeeman term and subsequently treat the off-diagonal elements as perturbations. 1. FINE STRUCTURE rl TEM Since we would like to have gxHxSx+gyHySy+gzHzSz = gS'H we proceed to make a coordinate transformation with the new set of coordinates xT y? z' related to x y z by the Eulerian angles G,, and, EL~~~~

then Sx, Sy, and Sz are related by the following: Sx = (cos G cos 1 - sin G sin 4 cos O)Sx - (cos G sin x~~~~~~~~~~~~~~~~~~~~~~~! + sin G cos r cos +)Sy + sin G sin Sz Sy = (sin 9 cos * + cos 9 sin r cos O)S' - (sin 9 sin + cos 9 C cos cos 9 sin Sz Sz = sin $ sin Sx + sin cos S+ cos Sy Sz Now if Hix = H sin ( cos 5 Hy = H sin ( sin 5 Hz= H cos ( and if we require cos 9 = -(gy/g) sin 8 sin 9 = (gx/gL)cos 6 sin 0 = (gj/g)sin ( Cos 0 = (gz/g)cos 0 where g2 = g2Cos28 + g2sin26 g2 = o 2 S + 2 in2S g g11|CosOS + gsi20S12 then ~zHzSz + gxHxSx + gyHySy = gHSz Thus the Zeeman term is diagonalizedo The axial and rhombic field terms become, respectively, 32

U2 U) -+ U) > ~~ ~ ~~- N - N V:)s *Hr U) U) ~~C\Jtav~~~~ + + loI~~~~~ 0 I -- -N1 N U __ U)~V U) V2 0 + -- ~+ - I 0 c U) U) U) C) co CO coC) %-~~~ ~ ~~~~~~~~..D -A %- o._~ -- f —f [ o (D D(o (M U) c- N. Ua) 0 -I CM CM 0 CV o) C) _. +.. (.O V2 U) U) _ _*r - -4 b.) +~ U) rH U) coU]~ cm~~C) C) — -+ I -M ICM (]*1) U]() UX)d*r tq ~ _ iic -+ C) -+C) V -4 -+ b H o.- ~ ~ + *rl % Q < 14 X X~~~~~~~~~Cl WSA w + C c) EQ~~ ~ o o -D D + CU U) -I ~ + 0 0 -+ -i U). H U)U () 0'.H co co oQ ~ H lU + IH U) _ U) U2I *r rn 2 N OC U) U)r UD ) -DC~ IfICb e., CM + O 1."" C., -I IOJ I HI [J I CM -ji~. bO -> -'- CM bolGi H-N + i U 0 co + U) 0 C T~~~~~~~~~~~~~~~~~~~ + L U) U)U)C)T -UN''c\cm U)j I-'H +C co) U Hc.o U r~~ ~ ~ b. H *H 0~ C) + ~~~~~~~~~~~~~~~% —I% - o~~~~~~~~~~~c M b\J /-')% cCm co CO ~ gi m CM~~~~~~~~~~~~~~~~ CM I rI * _>. H CT-r - H.H H%3 co H~ ~ ~ ~ ~ ~ ~~~~~Izt HI 0 C\j N =I C~~~~~~'HC\ cm co~ ~ ~~~~~~) i b_ to 1 ( D( + v ~~~~~~~~~~~~~~~~~~~~II r..D ~ ~ ~ ~ ~ ~~b U)~~~~~~~~~~~~~~~~~a I U).D rn. CMN_ _.I U)::: U)= 55

The diagonal terms of the fine structure part involving Sz, Sz, and S+S_+ SS'S evaluated for the states IS Ms > with the selection rule M + M-1 yields AE g0H o (M ) Cos - - 1 Dg-3Eg ( cos2 - -- sin2) The non-diagonal terms, i e., the terms involving (S)2, (, SzS) and (S+Sz+SzS+), can be evaluated by second-order perturbation theory (there are no first-order terms), give the following correction to the rhombic fine structure term: 4s(s+l)-24M(M-1)-9 [ E c __ D -g (g2cos2-2gsin2) g — - sin2 cos2O 2gH 2D g4 -I-~ 2~y 2r il3 2S(S+1)-6M(M-1)-3 f g2 E + nE2-sn 2 in22F + D f sin 282 (gocos26 2 2 2 2 2 2 f COS~l~ i r:2E gxgyg!t cos 8 sin - gsin25) (1 + g Cos a + 2E Cos 3 sin 2d 2. HYPERFINE TERM In attempting to carry out a similar procedure on the rhombic hyperfine terms the results become extremely complicated. Since the experimental results indicate little, if any, rhombic symmetry in the hyperfine terms for VO++ and none for V++, we will assume that B = C and Q" = O; io e, we have AIzSz + B(IxSx+IySy) + Q'[I - I(I+1) ] + yn_H I Here we perform a rotation on the nuclear spin as we did previously on the electron spin, with the difference that where the rotation matrix for the angle I6 was 34

1 0 0 O cos > -sin O sin cos it now becomes (for Ix, Iy and Iz): 1 0 0 A B o K cos - K sin K sin - cos K K where K is defined by setting the dterminant of this matrix equal to unity, i.e., K2 = A2cosS20+B2sin2. Since we have chosen axial symmetry in the hyperfine term we may arbitrarily choose ~ = O. The hyperfine term then becomes!!! AB I I AIzSz + B(IxSx+IySy) - KIzSz + BSxIx +- S Iy + IzSy Kg2 sin ( cos ( The first term is the only one which is diagonal in Mi, m, and with the section rules M + M-l, Am = 0 its contribution to the separation of energy levels is Km. Evaluating the other terms by second-order perturbation theory (neglecting the terms of order Km with respect to gpH), we obtain B2 (A2+K [I(I+l)-m2] 2 ((M 1 m 4 K/ gSH +B \ 2 gpH 3. QUADRUPOLE TERM Since the quadrupole term Qt [IT2_ T I(I+1) ] transforms similarly to the fine structure part, one can quickly obtain the quadrupole contribution to the 35

difference in energy levels as (A)2 g sin2cos2() [4I(I+l) - 8m2 - 1] T2 g2 2KM(M-l) (Q,)2 ( g2 sin40 [2I(I+l) - 2m2 1] 9, 8KM(M-1) These are also obtained by second-order perturbation theory and then by applying the selection rules M - M-l, Am = 0. Although there is a diagonal contribution of the quadrupole part to the energy level of the form (` B2 (I+1i Q 2 os2 m2 + --- sin2 [I(I+l) - m2] - I(I+l) it is the same for each value of M, so that the selection rule Am = 0 results in cancellation of this term.

APPENDIX B EXACT ENERGY LEVEL CALCULATIONS FOR V+2 ALONG x-, y-, AND z-AXES FOR RHOMBIC FIELD Given the rhombic spin Hamiltonian with axial hyperfine term Ihombic = _H.g*S + D[S2 - I S(S+) ] + E(S2-S) + I.A.S, we obtain for the fine structure term for V+2 (S = 3/2) and for parallel to z-axis z = gzHSz + D[S2 - S(s+1)] + [S + s2] Degenerate perturbation calculations lead to the following secular determinant: M = -3/2 -1/2 1/2 3/2 3. _ gzH+D 3 E 2 2 z E 1 2 gl,H-D 5E 2 -% 1 1 N5E 2 gzH-D 2 -k 3 gzpH+D This determinant may be solved for the following energy levels: 37

+ 1 s 3 gz3H + \/3E2+(gzH+-D)2 5 -; _ 1 g,BH + \/3E2+(g,fH D)2 Sz --.; - = - gziH - JE(gzH+~D) 2' 2 For the transitions M + M-l the change in energy is: E3/2,1/2 - 2 -:1 = gZPH3/2 + l/3E2+(gzPH3/2+D)2 - 3E2+(gzPH3/2-D) E1/2,-1/2 - X:-2 = -gZfH1/2 + Nf3E2+(gzpHl/2-D)2 + \3E2+(gzfHl/2+D)2 E-1/21-3/2 - A-A1 = gZH-l/2 - I/3E2+(gzIH-1/2+D)2 + 13E2+(gziH-l/2-D)2 Making use of a Taylor's expansion for each of the square roots and solving for the magnetic field (in the major term) yields the following equation (to fourth order): (A) H3/2 hv 2( (D + 3E2/2 7 D g9zH3/2 27 E4D 1 4 (gZ)5 (H3/2)4 hv 3E2 1 (B) H1/2 ='g(H/2 gzp (gZp)2'Hl/2 1 D 2 ( gZ) 5Hi/ 2 L_ gzlH l /2 9 D F) z54() 7> gF H-'

hv 2 3E2D 1 (C) HL1/2 =2 - D )2j gzp gzJ) H_ 1/2 1 D 2 gzPH-1/2 27' E4D H 1 4 (g )5 For the magnetic field parallel to the x-direction, the Zeeman term is not diagonal. To diagonalize, a coordinate rotation about the y-axis by 900 is performed, resulting in the fine structure spin Hamiltonian = gxdz D [(s)2 + [(s+) + ( + SS + SIS] - + E (Sz)2 + [(S+)2 + (S3)2 - S S+ s- +s which leads to the following secular determinant: M = -3/2 -1/2 1/2 3/2 3 — 3H D+E 2 D-3E -2/ 1 gxD+H D+E 2 2 2 D-3E 99 2 ~~2' ~2 ~+(D3E).. _ ggxH 3 j D+E 2 2 2 39

This is the same secular determinant for kparallel to the z-axis with gx - gz -(D-3E)/2 + D, and(d+E)/2 - E. Thus with these changes, Equations (A), (B), and (C) also hold for the x-axis. Similarly, for the magnetic field parallel to the y-axis the following substitutions are made: gx +g, gz,(P) D, and D2E * E 2 2 4o

REFERENCES 1. E. J. Zavoisky, J. Phys. (USSR) 9, 211 (1945). 2. B. Bleaney, J.D. E Ingram, and H E.Do Scovil, Proc. Physical Soc. 64A, 6o0 (1951). 3. G. M. Zverev and A. M. Prokhorov, Soviet Phys. (JETP) 7, 1023 (1958). 4. J. Lambe, R. Ager, and CO Kikuchi, Bull. Am. Phys. Soc. 4, 261 (1959) 5. J. Lambe and C. Kikuchi, Physo Rev. 118, 71 (1960) o 6. H. J. Gerritsen and H. R. Lewis, Physo Rev, 119, 1010 (1960). 7. G. M. Zverev and A. M. Prokhorov, Soviet Phys. (JETP) 12, 160 (1960). 8. M. B. Palma-Vittorelli, M. U. Palma, D. Palumbo, Nuovo Cimento 3, 718 (1956). 9. N. S. Garifianov and B. M. Kozyrev, Soviet Phys. (Doklady) 98, 929 (1954); see also B. M. Kozyrev, Disc. Faraday Soc. 19, 135 (1955). 10. G. E. Pake and R. H. Sands, Phys. Rev. 98m, 266A (1955). 11. D. E. O'Reilly, J. Chem. Phys. 29, 1188 (1958); 30, 591 (1959). 12. R. J. Faber and M. T. Rogers, J. Am. Chem. Soc. 81, 1849 (1959). 13. C. M. Roberts, W. S. Koski, and W. S. Caughey, JO Chem. Phys. 34, 591 (1961). 14. J. E. Wertz, D. Avzins, J.H.E. Griffiths, and J. W. Orten, Disc. Faraday Soc. 26, 66 (1958). 15. R. Wyckoff, Crystal Structures, 3, Table XF15, Interscience, New York. 16. A.E.H. Tutton, Crystalline Structure and Chemical Constitution, Macmillan and Co, London (1910). 17. C. N. Ballhausen and H. B. Gray, J. Inorg. Chem.,, 111 (1962). 18. K. Murakawa, J. Physical Soco of Japan 11, 422 (1956). 19. B. Bleaney, R. P. Penrose, and Bo.TP. Plumpton, Proc. Royal Society, 198 A, 406 (1949). 41

PART III Mn++ EPR RESULTS IN AIIBVI COMPOUNDS by G. H. Azarbayejani

I. INTRODUCTION In earlier reports on ESR measurements in AIIBVI compounds doped with Mn1,2,j some interesting results on the ground state splitting factor 3a have been obtained. Further experiments which have been carried out pertaining to the cubic ground state splitting will be described here, In Section II crystal preparation and doping with impurities of CdTe and ZnTe will be considered. In Section III ESR measurements will be reported.

II. CRYSTAL PREPARATION4 CdTe and ZnTe crystals, which we have used in our experiments, are made from the elements in fused silica quartz tubes under vacuum and heated in R-F furnaces. The following steps are taken in preparing these crystals. A. PREPARATION OF QUARTZ TUBES All sample tubes are made of 10 mm (ID) tubing with 4-10 in. of added tubing. In one end, these tubes are closed in regular round shape, and in the other are connected to a quartz tube stub (see Fig. 1). The empty part is 4-10 in. long. SLAB - c - SAMPLE Fig. 1. Tube preparation. The tubes are checked for leaks by using the vacuum station and Tesla coil. Good tubes are cleaned in the following sequence: (1) Benzene rinse (2) Acetone rinse (3) Tapwater rinse with brushing at each stage (4) Warm concentrated nitric acid bath for 2-4 hr (5) Deionized water rinse (20 times) (6) Drip drying by holding the tubes vertically with the open end down

(7) Heating the tube at quartz fusing temperature so that any foreign impurity remaining on the walls of the quartz tube will be fused into the quartz (8) Cooling by air flow (9) Closing the tube with a cork wrapped with glassine powder paper (10) Labeling the tube prepared in this fashion with the letter "C," indicating that it is clean for use in crystal preparation. B. PREPARATION OF SAMPLE The following steps are taken for weighing the sample. (1) Recording. —On a special data sheet the elements, properties, and weight of each are recorded. The base elements such as Cd and Te in CdTe should have a purity of 99.999%, whereas in the elements of intended impurity such as Mn or Cu it can be 99.99%. (2) Treatment of the Elements. -Before weighing Cd and Zn, the metal bars are cut into pieces less than 10 mm long and are etched with concentrated nitric acid. Then they are rinsed thoroughly with deionized water, covered with high-purity metanol, agitated, drained, and allowed to dry on paper towels. Tellurium is usually used without etching but always in pieces so that its oxide content will be small. (3) Weighing of the Elements. —The amount necessary for each element is calculated by desk calculator up to 7 figures and then, with use of the elements etched as described in paragraph (2), the weighing is carried out. The elements belonging to Group IIb (Cd, Zn) must be weighed before those belonging to Group IIIb and Group VI; therefore for CdTe, Mn Cd is weighed first

and immediately added to the tube, with tellurium and Mn then weighed and added to the tube. (4) Evacuation and Seal-Off of Sample.-The stube in Fig. 1 connects the tube to a vacuum station which has a fore pump for initial evacuation and a diffusion pump for further evacuation. After *the desired vacuum is achieved the tube is heated in the "c" region to a length of 1-1-1/2 in. with about 2 in. above the top of the sample so that the walls of the tube collapse completely for a satisfactory seal-off. (5) Reaction of Elements in R-F Furnaces.-To ensure a uniform distribution of the impurity ions in the compound the tube is placed vertically in a graphite cylinder. This graphite cylinder is placed along the axis of a vertical solenoid of 1/8-in. (ID) copper tube connected to a 20-KW Lepel R-F generator. The temperature is raised above the crystalline melting point and held there for several minutes; then the power is turned off, which in 5-10 minutes lowers the temperature to room temperature. The quartz tube of the sample is then inspected for crackso If single crystals are desired, the polycr,ystalline obtained from the R-F furnace is annealed for several weeks under a temperature slightly lower than the melting point. The same technique is used for both ZnTe and CdTe doped with Mn.

III. ESR MEASUREMENTS The first experiment on AIIBVi compounds, performed by Van Wieringen5 on ZnS:Mn, raised a great demand for similar ESR experiments. Since then many other materials with structures close to cubic have been investigated. In Table I some of the recent ESR results on Mn++ which have been obtained in our laboratories or by other investigators are given. TABLE I Mn++ ESR EXPERIMENTAL RESULTS Crystalline ao Co 3a D Material Structure* Coordination (A~) (A) g 10-4cml 104cm0 Ref. MgO R 6 4.24 2.0014 55 0 a CaO R 6 4.81 2.0009 17.7 0 b aZnS Z 4 5.43 2.0025 27.7 0 a 5ZnO W 4 3.24 5.18 2.0016 6 -216.9 c ZnSe(P) Z 4 5.67 2.010 - - d ZnTe Z 4 6.12 2.005 88.9 - a CdS W 4 4.13 6.64 2.003 4.2 8.2 c CdSe W 4 4.30 7.01 2.005 47? a CdTe Z 4 6.88 2.007 83.1 - a *W stands for wurtzite (BZnS), R for rock salt (NaCl), and Z for zincblende (ZnS) structure. Refs. a. Results of our laboratories. b. Shuskes, Phys. Rev. 127, 1529 (1962). c. Dorain, Phys. Rev. 112, 1058 (1958). d. Matsumara, J. Phys. Soc. Japan 14, 108 (1959).

According to Watanabe6 3a c(Dq) 2 (1) Dq oc r4/ (2) where r is the position of an electron with respect to a paramagnetic ion and R is the position of the nearest neighbors of paramagnetic ions. This theory was originally developed for ions in an octahedral field produced in crystals with the structure of NaCl. In Table I, MgO and CaO have this structure, and if we assume that r4 of Eq. (2) remains the same for both materials we have: (aMgO F:ao =caO.8 aO) = (RCa0O)o a= CaOo (4<81410) 3~.4 (3a) aCaO T aoo, = 24 The experimental results (Table I) gives a= 55 7 = 35.1 (3b) Also 0t = 2.0 (4a) where (aiC) = 3.03 (4b) and E = experimental T = theoretical

Comparison of aZnTe and aZnS gives aZnTeE = 0.312 (5a) where /:ZnS = 3 (5b) Also ZnTe = 1.07 (6a) where (aznTe) = 1.75 (6b) aCdTe 1.75 Equations (4a) and (4b) reveal a sharp discrepancy (as much as 35%) between experimental result and theoretical prediction and indicate a great need for testing other materials having a cubic field. This is why many studies have been carried out so far. As reflected by Eqs. (5a), (5b), (6a), and (6b) the discrepancy of 35% in MgO and ZnS is raised by 1000% in the case of ZnS and ZnTe, and by 75% in the case of ZnTe and CdTe. This demonstrates that although crystalline field theory gives a satisfactory result in light AIIBVI compounds and especially in octahedral coordination such as CaO and MgO, it fails to account for heavier compounds belonging to this group, especially those in tetrahedral coordination. In the following sections the experiments carried out on ZnTe and CdTe doped with Mn will be considered. A. EXPERIMENTAL METHOD The measurements at 300, 78, and 4.2~K were carried out in an X-band

(9,2-9.4 kmc/sec) magnetic resonance spectrometer. The ZnTe single crystals containing 0.1% MnTe were prepared as discussed in Section II-B. Being brittle and of low resistivity, the ZnTe single crystals are very difficult to cleave and orient properly. Therefore we had to select many samples and design a new cavity which permitted the simultaneous and orthogonal rotation of both DC magnetic field and crystal in the cavity. This cavity is shown in Fig. 2. The teflon pulley and nylon string are used particularly to facilitate the rotation of the crystal at low temperatures. With this device we were able to orient the crystal as well as find the temperature dependence of the ESR parameters in ZnTe and other similar materials. Considering that cubic ZnS and ZnTe have exactly the same crystalline structure and that the maximum splitting between the fine structure components of Mn++ ESR spectrum in the cubic ZnS occurs when the magnetic field and [001] axis are coincident,7 one simply looks for the maximum separation in the spectrum of Mn++ as a function of angles G' and ~'. These are the angles through which the DC magnetic field and crystal in the cavity are rotated from arbitrary positions-. After maximum splitting is achieved and hence the location of one of the crystalline axes is determined further rotations regarding this axis are made to ensure the correct assignment of the axis. Before making an ESR measurement, it is of great importance to ensure that the piece of crystal is not polycrystalline or twin. Metallurgical microscopes with magnification ranging from 3 to 100 have been used for closer inspection of the faces, which are not shiny. After this stage X-ray photographs from both powdered and solid pieces of crystal are made to identify the structure as well as the singularity of the 10

TENSIONING MECHANISM TEFLON RACE SILK THREAD TEFLON SAMPLE PULLEY L Enlarged view of pulley assembly, showing somple in place Fig. 2. Rotating sample holder and resonance cavity used for measurements. 11

crystal. These steps are very important in the measurement of cubic field parameters because the structure of the crystal depends a lot on the method by which they are prepared. Fortunately, in the case of the tellurides of Zn and Cd the structure is zincblende alone. For the selenides of these elements, however, structures of sphalerite (cubic ZnS) and of wurtzite (hexagonal ZnS) are observed, and sometimes it is possible for both structures to be present. A proton magnetometer and Hillard-Packard frequency counter have been used to obtain the line positions. A hydrazyl marker is used for g measurements due to the fact that in each measurement there might be slight shifts in klystron frequency, which is an especially important factor for temperaturedependence measurements. B. EXPERIMENTAL RESULTS 1. e ZnTe: Mn In Fig. 3 the spectrum at 3000K is shown, At the top is the spectrum corresponding to G = 0, and at the bottom the spectrum corresponding to G = 30~~ (G is the angle between [001] and the DC magnetic field H.) This phenomenon of dependence of intensity on G is more pronounced in the materials with larger a (cubic field splitting parameter) because the farther the finestructure components a and P are from the main line y (see Fig. 17 in Section IV-D) and the higher the temperature, the more destructively they tend to combine. At 9 = 30~ the separation of the fine-structure components reduces nearly to zero; therefore the five transitions corresponding to MS = -3/2, -1/2, 1/j2, 3/2, and 5/2 occur at about the same magnetic field. Figures 3a and 3b clearly manifest the correctness of orientation assignment given to 12

the crystal. Zn Te; Mn 300~ K (a) 8= 00 z o o 0 0 uU4 0 CI N: (b) 8 = 30o MAGNETIC FIELD, H Fig. 3. Mn* ESR spectrum at 300oK: (a) G = 00; (b) 3 = 30. The spectra for T = 780 and 4.20K at G = 00 are given in Figs. 4 and 5 respectively. 2. CdTe:Mn Measurements were made at 300, 78, and 4.20K; the corresponding ESR absorption spectrum for 780K is shown in Fig. 6, and the spectra for 4.2~K ( 300~ and 0 = 0~) are shown in Figs. 7 and 8 respectively. 13

0 L ~li'9Z~~1 pm z I ICc _ =|, e -9LO2: ~ —- =9'~ 66Z x NOlid JOS8V =V'HP/VP C- _ __ o........ X oE.. = iic~___ IL ~ 9 09 N. o Iz,0 Z ~ D 9'S90~ NOIidOSStV - V'HP/VP 14

I J~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ J~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ I I I I tr I I I i I ((6 i Fig. 6. Mnn ESR spectrum in CdTe single crystals at 78~K, 9 = 00. 15

Fig. 7. Mn++ ESR spectrum in CdTe single crystals at 4.20K, G = 30~.! j'2't~f Fig. 8. n ESR spectrum in CdTe single crystals at 4.20K, 9 = 0~. 16

The superhyperfine structure corresponding to Cd ions was observed both in CdS and CdTe by other investigators. 1 17

IV. THEORY A. GENERAL HAMILTONIAN OF Mn++ To explain the spectra of Mn++ shown in Figs. 3-8 we should obtain a relation between (1) the DC magnetic field which is measured experimentally, and (2) the electron spin magnetic quantum number M, the. nuclear magnetic quantum number m, the polar angle 9, and the azimuthal angle o. Such a relation is self-evident only if one derives it directly from the following total Hamiltonian of Mn++: 8 t (M++) = Z (7) G=l where t1 = (P~/2mi-Ze7'ri) + e2/rij (8) i=l i>j =1 3 = - X 4 i32r5'.s J- 3rij 3osirij.sJ) (10) i>j 4 = (et/2mc)(i+2s'i).H = (L+2S)-H (11) X=dgiN prI[r(~i-s).+risiri I] + - (ri)si4 (12) i=l 18

2I(2I-1) ri5I*2r2 - 3(riI) ] with I* = /I(I+l) (13)? = V V(rj,ri,Gi,i) (14) j=l i=l 8 = -gN-H.I (15) -~l, [Eq. (8)] is the sum of the kinetic energies of electrons; the coulomb interaction of the electrons with the nucleus of the- ion and each other, the' is the number of electrons around the Mn++ nucleus; Z = Z-2 = 23. /a 2 [Eq. (9) ] is the spin orbit interaction. The proportionality constants aij, bij, and cij are in general functions of quantum numbers (ni,Pi, mi) The expression aiji-sj, for instance, refers to the energy associated with the force which the orbital angular momentum of the ith electron exerts on the spin of the jth electron. In general laiiI >aij It can be shown that8 2 Zeffe2 aii =2. (16) 2m2c2 <r> where r is the position of the electron with respect to the nucleus. c' s are primarily due to Heisenberg's exchange effect, which appear in the interaction terms between the two electrons i and j: 19

Vij = Kij - (1/2-2si.s)Ji (17) where Kij= = < i ii > < jTj > (18) and ij - < ilij (19) Though the dependence of Jijsi-sj on the angle between the spins is similar to that of the mutual potential energy of two magnetic dipoles r-5( r2_i. _ - 3-_i r_j ~r), they are two entirely different things. [3~ [Eq. (10)] is the magnetic dipole-dipole interaction and it can be simply derived by finding the magnetic field Hii produced by ith dipole at the position of jth dipole: Hij = VixA(rij) = -vix[4ixvi(l/rij)] i ixriJ\ vix (. 1 i iij) 3rij - i X( xrx X( ixri j rij rij r_ i-_r2 j _i+3riri. ~[irij) and = -J._Hij = rj(r2ji.j_i.rijj.rij) 20

Let _i = 22si and J = 2psJ Then 3 42r-5(r.sisj - 3risirij.sirii.sj) - ij 13-.. and = I = ij 4 r7(r2 si.j - 3rij.siiriJ.s) (20) i>j=l i>j =l (4 [Eq. (11)] is again the interaction of the magnetic moments of all ion electrons with the external DC magnetic field, and it is evident that if the DC magnetic field is not strong enough to decouple the angular momenta of the electrons then the result of interaction on the closed shells is zero. For the last shell this result is as shown in Eq. (11). ~5s [Eq. (12)] is exactly like 33, the only difference being that the total number of Z protons and A-Z neutrons is assumed to be a particle with angular momentum I at the origin; therefore the contribution of the orbital angular momentum 1i of the electron is just Xi i./r (21) where the contribution of the ith electron spin ai is ai = -2gNN r5 Er5lsi.I - 3risiri-I - 8r(ri)Ksi.I/3] (22) The first part of Eq. (22) is due to dipole-dipole interaction and the second to the electrons present at origin. Therefore Eq. (23) results (see Fig. 9): 21

5 =+a)i LI 2g(N N) ri5[r(Ii-si).I+3ri'siri.I]+81r5(ri)si.IJ (23) I NUCLEAR th ri E LECTANGULAR MOM. i ELECTRON NUCLEUS Fig. 9. Schematic representation of electronnucleus magnetic interaction. Q 6 [Eq. (15)] is the electric quadrupole moment. Consider that the nth nucleon potential on the ith electron is (see Fig. 10): 00 vni = eien a ri ni P1(ni) 1=0 and n~ A A A o VNE= E Vni = I eien Pni (ni) V:NE L _ r —-pi P ~ ( ~ ni ) i=l n=l i=l n=l Q=0 ereni [Pnio(ni ni) + Pji~(Pni) niP2(ni)+-] i,n V + V1 + V2 (24) where Sni = cos ni = (XiXn+YiYn+ziZn)/riRn (25) and p = (Rn/ri)2 (26) 22

= ) ^ rl = Ze/ri (27) i,n i =, ee 2eien8) r3Rn (XiXn+YiYn+ziZn) (28) i nriRn.th ELECTRON Mth NUCLEON y NUCLEUS Fig. 10. Schematic representation of electronnucleon coulomb interaction. The matrix elements of V1 correspond to the nuclear electric dipole moment which vanishes, and 23 v- e~ e 2 riSeie2n23 _1ni /2 (29) i9n i9n Substituting Eqs. (25) and (26) we have: 23

V 2 r3Rneien [3ri2R2( nin+ZZ+2xyiXnYn+2yi ziynZn+2zix iZnXn) i,n 5 7,ri 5rei 2nX[ X2n+y2222XY +2yi Zn+2ziXiZnXn i,n - (X? +Y2+Z2) (X2+Y2+2z) /3] — 3 ~ r.~Se ZnZiX = ri5eie n[(x2iXn2) + 18(xiYiXnYn+ 2i) 3riRn] i,n ri5 eien[(ri-3x2i)(R -3) + (r-3Y2) ( R-3Y) + (]r 3z?)(-3Zn) i,n + 18(xiyiXnYn+ ) 1 = 6 I ri5ei e en[(r2 2 2 - 2X + (r?-3y?) (R2-3Y2) + (r2-3z2)(R2-35Z) i n + 18(xiYiXnYn+') ]0) Consider now9 L en(Rn-3Xn) = eQ (- - 2) (31) nI(2I-1) x enXnyn eQ 1 (I I +I I (32) I(2-1) y y Thus we obtain: 1 ei eQ i2(3 r 3r)3r2 +I +I2 +9(Ix )18(xiIxI.) V2 = r I(2I-1) Lx (33) For all electrons the ei's are the same, ei = -e; therefore Eq. (33) reduces to: ~%.

6 V2 V2 + r5[I(I+l)r- 3(Qr' I (34) 2I(2I-1) -7 [Eq. (14)] describes the effect of the sum of the coulomb potentials produced by the neighboring ions at the ith electron. In the case of approximation of the closest neighbors in ZnTe and CdTe, this is the sum of the potentials produced by the four closest Te ions. f7 is discussed in Section IV-C. 8a [Eq. (15)] is the direct interaction of the DC magnetic field H and nuclear spin I with the gyromagnetic ration gN' gNPNI-_H is considered as a constant and neglected. B. SPIN HAMILTONIAN In case of II-VI compounds and for 3dn ions 1 > 7 > /2 +1S; therefore the ground-state electronic configuration of MI++ is still 3d5. The terms associated with this configuration are: S,P,D,F,G,H, and I. (35) One way of studying the effect of different Hamiltonians on these terms is to transform them into functions of angular momentum operators, L, S, and I, which facilitates the theoretical calculations considerably and does not affect the ESR experimental calculations. This transformation is carried out in Appendix B. The result [Eq. (B-14) of Appendix B] is: 25

{ (L,S,tI) + C = KL.S - M[(L.S)2 + 1 LS - L*2S*2/3] C + P(L+2S) *H - gN-I + N{a[L*2S-I - 3(L.SL I+L-IL.S)/2] + L.I - -.KSI + p[(L.I) /2 - L-I/2 - *2/3] + 7 (6) where L* = L(-L+); S = IS( S+); I* = VI(i+l) and K, M, N, p, and K are independent of L, S I, ML, MS or MI. The energy of each of the terms of Eq. (35) is calculated in Appendix A. Equations (A-28) of this appendix give for Mn++ (3d5) + 0.5 for 4F - 4.0 for 4D E(3d5 L2S+1 B/C= 5,,,C)-10A,E(3d5,L28_B/c-f 5,)-bA - 7 0 for 4P (37) - 7.50 for 4G -17.50 for 6S where A,B,C are Racah's coefficients. Thus the 6S state lies the lowest; considering Eqs. (36) and (37) one finds that in the case of Mn++ the eigenvalues for operators including L will be zero. Thus: < LMLI LML > = = 2S.H + S I +7(S) - gNNH (38) 26

where A = -NK Therefore, for Mn++ in the intermediate crystalline field appropriate to IIVI compounds such as MgO, ZnS, ZnTe, and CdTe, the spin Hamiltonian-which is derived from the total Hamiltonian after the latter has been transformed into functions of orbital and spin angular momenta and then integrated over orbital angular momenta L —is: (S,) + gN I = = 2PS.H + AS.I + 7(S) (59) where A = -NK and gNDNH-I is assumed to remain constant. On the basis of the assumptions made to derive in Eqs. (7)-(15), one findsthat: (1) The gyromagnetic ratio of Mn++ outer shell electronic cloud should be the -same as that of free electrons. (2) Any difference between electronic spin states Mi(-5/2,... 5/2) should result from 47(S) and AS*I in Eq. (39). C. CRYSTALLINE FIELD The crystalline field in II-VI compounds of zincblende structure can be obtained from Fig. 11 and Eq. (40). V(r) = j AnmrnYnm( G,) (40) n=O m=-n Symmetry considerations will reduce the number of terms in Eq. (40). 27

ec y Fig. 11. Schematic representation of crystalline field. (A is the ion, B is its closest neighbor, and C is its electron.) For d electrons one has to consider only expansions up to n = 4. No terms with odd n in a given configuration will contribute in ground-state splitting, regardless of whether the center of symmetry is present or not. The term A; is a constant shifting all of the levels of a given configuration by the same amount. Therefore: Veff = AO2r2Y20 + A22r2Y22 + A40r4Y40 + A44Y44 + A4_4Y4_4 The choice of z axis parallel to the fourfold rotary reflection axis S4 facilitates the determination of the coefficients in the above relations. Furthermore consider that V(r,G,) = Ej mtn-meim' nm where anm = f(r,n,m) only and = cos G 28

Thus S4V(r,i,S) = V(rG+t, ~ + 2) = V(r,r, ) results in [cos(OG+i) ]n-me im(O+T/2) n-m im( I~c os(e+n)] = (cos G) e or m = O, +4. Now consider that reality condition requires that V* = V or each term having the same condition, i.e., (A44Y44) = A44Y44 = A4-4Y4-4 since Y4 = (-1)4Y4_4 = Y4-4 9A44 A4-4 The presence of reflection planes a requires that Anm = An-m (41b) Conditions (41a) and (41b) result in: (Anm)* = An-m = Anm, (41c) namely, Anm's are real. In our experiments no axial dependence of the field on G was observed; therefore the field set up by four tellurium ions around Mn++ electronic cloud is 29

V = A40r4[Y40 + (A44/A40)(Y44+Y4-4) ] A40r4(35 cos4G-50 cos29+3) + r4A44 5 sin4A cos44 (42) Now consider Fig. 12. We have VA(r,G = 0, = O) = VB(r,G = i/2,O = O) or 8r4A40 = 3A4or4 + r4 ~ A44 or (A44/A40) = f7/ B y Mn (not in scale) O Te (not in scole) Fig. 12. A unit cell of cubic ZnTe single crystal showing a Mn++ ion surrounded by four Te ions and the reflection planes a. Therefore 1/2 7(r,G,1) = V = A40r4 [Y40+(5/14) (Y44+Y44) ] (43) and Eqs. (B-28) of Appendix B give: 3o

fr7[(r,r')n,Lx,W, Lz] = B[1 + 14 + Lz4 - (3L* -1)] (44) In the 6S state of Mn++, for H parallel to z axis, one finds from Eqs. (39) and (44) that < M,n l SIM,m > - <,m |SiM-l,m > = 2PH(M,m) + mA = hvo = 2PHo (45) and H(M,m) = Ho - mA' A' = A/2P (46) where m = nuclear magnetic quantum number. Equations (445) and (46) indicate that there should be six absorptions only (see Fig. 13). -5/2 IAI-3/2 IAI-I/21AI I/21AI 3/2 AIA 5/2 AI HH Fig. 13. Theoretical prediction of Mn++ ESR spectrum. These absorptions are observed only in powders and we realize that to describe the spectra shown in Figs. 3-8 a further refinement of theory is necessary. Bleaney and Stevens,10 after a phenomological argument, conclude that a more realistic representation of the crystal potential for Mn++ in II-VI compounds may be written as: 31

47,[(r,r')n,L,S] = B[L4 + T4 + L4 - L*2(3L* 21)/5] + P[S4 + S4 + S4 - S*2(3s*2-1)/5] (47) Therefore cubic(Mn++) L < t tlo > + I < oli In >|2/[E(O)-E(n) ] n g_H-.S + AS.I + [S + S + - S* (3S* -1)/5] (48) - s Owing to the fact that P is a measure of bi when H = 0 it is called the cubic zero field splitting factor. Bethell shows that the state 6S5/2 will be split into 2F7 and 4F8 under the effect of a tetrahedral field created by the four closest tellurium ions of Zn, Cd, and Te. To correlate P and AE = 4r 82r = 8 7 use is made of T tensors given as:12'13 -1/2 T =+ [Q(1+1) - q(q+l)] (S+,*Tq) (49) S+ = Sx ~ iSy T4+4 == 0/(16) S+. (50) Sx = (S+ + SO)/2 Sy = (S+ + S_)/2i (51) Consider that x (S+ + S )4/16 = (s+ + S4)/16 + f(s + * *S3) Sy = (S+ + S)4/16 = (s + S4)/16 + h(S+... S3) 32

Then P(Sx4 + sy = ~ (s+ + S4) + f' + h' = (T44 + T4_4) + f" + hi" X Y70 S ~(52) Evaluating S+ from Eq. (49) and substituting in Eq. (52) one gets: cui (Mn++) = gpSH + AI*S + 2 [T4o + /7511 (T44 + T4_4) (53) cubic 5 where T40= 8 (5Sz - 3S + 25Sz - 6s* + *) * = +) (54) The matrix elements of the 2P[T40+ J75/1 (T44+T44) ]/5 can be found very easily, and the result is as shown in Table II. TABLE II MATRIX ELEMENTS OF 2 [T40 (T44+T4.4) 1 5.. M' =+ - + 3 + 5 11= + — +_5 2 2 2 M= +- 6P 0 2 _+ 3 O -gP + 5 3 25P 3P In the secular equation iij -ij~l[ = (6P-E) [(-9P-E)(3P-E) - 45P2] = O 33

giving el(+ 2) = 6P, C2 = 6P, E3 = -12P (55) e1 and E2 correspond to four eigen functions with energy 6P above the ground state of the free ion. Thus the situation shown in Fig. 14 results: 4 I a 6P 8 eS5/2,(n++ free ion)/ 2a 12P "t + z 2r Fig. 14. Effects of cubic field on eS5/2 levels. The splitting of 6S5/2 state in a tetrahedral field is given by 4r 2r7 = 18P = (56) 8 7 where a = 6P Substituting for Pin Eqs. (52) and (53), one finds expressions similar to those given by Bleaney and Stevensl0 and by Lambe et al.l' t c gPS-H + a [S4+Sy+S4-S*2(3S*2-1)/5]/6 -/cubic - cubic g + A - gNN-H I (57) 34

or cubic-= gPS.H + AI-S + a[T40+ 17 (T44+T4_4) ]/15 (58) To find the energy corresponding to electronic and nuclear spin levels we set up the secular equation for fine-structure Hamiltonian: fs G ~~~~~~~~f.s.~fs. S8 =,- AI.S =grfSH + a(55S 4 - 30S*2S2 + 25Sz - 6S*2 + 3S*4)/120 + a(S + S4)/48 (59) For the DC magnetic field H parallel to one of the principal axes of the cube, namely the z axis, we will find the matrix elements of X (HIIZIIS4) = g=HSz + H a(354 -6S4+3S*46/120 + a( S++S)/48 f.s. (60) The first and second term are diagonal while the third term is off-diagonal; therefore < MlgHSz + a(35S4 -30S*2S2z+25S2-6S*2+3S*4)/120 + a(S++S_)/481M' > = < MlgHSz + a(35S4-30S*2S2z+25S-6S *2+3S*4)/120IM > + < M1a(S4+S4_)/h48jM' > = A + B (61) MM' M M'M'+4 with A = [2Me + a(14M4-95#+189)/48] B = 4 a/2 e = gSH/2 35

The secular equations (determinant) can be constructed from fs E)MM E - MMI= 0 (62) Thus \' 5/2 3/2 1/2 -1/2 -3/2 -5/2 5/2 5e+a/2-E 0 0 0 Jf a/2 0 3/2 0 3e-3a/2-E 0 0 0 V a/2 1/2 0 0 c+a-E 0 O O A = = -1/2 0 0 0 -+a-E O O -3/2 I a/2 0 0 0 -3c-3a/2-E O -5/2 0 5 a/2 0 0 0 -5e+a/2-E (63) The determinant is an equation of 6th power in E. To solve consider 0 0 0 7 0 o p o o o y O O 5 O O O A = 8\(0a - y2) (rl 72) = 0 (64) O O O XO O y o o o 0 o 0 7 0 0 0 giving E1 = a+e E2 = a-E or, provided e > a,

a a 5a2 E3 = E - + (+ 4E + a + 8 ec 2 24 E4= d -6P4++a 4h - a- -/ E4 = - /(4e+a)+5a2/4 2 - 2e+a -e 5a2 (65) E5 = -- 2 + (J(4e -a)2+5a2/4 +he -E -a + 4 - a +/e 2E a 5a2 6E =' 2 5 a2/h4 2 + a / It is obvious that the states M = +1/2 do not mix and that E1 and E2 therefore belong to M = 1/2 and M = -1/2. For other M's one has,provided a/E << 1. Ef.s.+ 1 a+ Ef (M _ 5 a + ~ a = 3a ~ 5a2/32E E's'(M + ~ 2 4T8 2 ~a(66) Ef- S, (M + + 5 = 2 /8E 32 Thus, the energy diagram of spin levels for H||Z and in the absence of nuclear spin effect is as shown in Fig. 15 and Table III. TABLE III VARIATION OF SEPARATION OF Mn4 ESR FINE-STRUCTURE COMPONETS AT 9 = 0 AS A FUNCTION OF p = gpH/2a = e/a p E-5/2-EF E-3/2-EF E-1/2-EF El/2-EF E3S /2-EF 0 - 2.00 - 2.00 1.00 1.00 1.00 1.00 1 - 4.71 - 4.62 0.00 2.00 1.71 5.62 2 - 9.59 - 7.57 - 1 3 4.59 10.57 3 - 14.56 - 10.55 - 2 4 7.05 15.55 4 - 19.54 - 13.54 - 3 5 10.54 20.54 5 - 24.53 - 16.53 - 4 6 13.53 25.54 6 - 29.52 - 19.53 - 5 7 16.53 30.52 7 - 34.52 - 22.52 - 6 8 19.52 35.52 10 - 49.52 - 31.51 - 9 11 28.51 50.5 100 -499.5 -301.50 -99 101 298.5 500.5 ~..~ ~ ~ ~ ~ ~ ~ 3

50 s=-5/2 40 30 5/2 Ms = 3/2 20 3//2 Ms = 1/2 E(a) -10- \ Ms= —/2 free Cubic Magnetic \/2 ion Field Field 20 30 \ |Ms =-3/2 -3/2 -40 1/2 1/2 -3/2 Ms=-5/2 5/2 3/2 -50 I [ 1I I I I 0 2 4 6 8 10 Fig. 15. ~Energy level scheme of 3d5 ess/2 (Nn+F) in a tetrahedral field at 0 = 0~. 38

For the case where H has direction cosines a, D, and 7 with coordinate axes, gIH.-S = g[HzSz + (H+S_+H_S+)] = 2e(ySz+/%+S_+\S+) (67) where 2E = g3H and _ = (c f i5)/2 (68) and the energy levels arel4 E_+1/2 = pa + E + rla3/e2 + qla2/e E+3/2 = -3pa/2 + 3c + r2a3/62 + q2a2/e (69) E+5/2 = 5pa/2 + 5e + r3a3/C2 + qs3a2/e where =[1 - ((c2+p2,/2+y2C)] = ql = -55(7-258)/6 (70) = 5[1 + 6(22.755) ]/32 = 5[1 + b(50-1138)/3]/32 and rl = -58(196-16355+ 312582) /144 r2 = 5[1 + 35(79-6155+11252) ]/128 (71) r3 = -5[1 - 58(113-7058+10752) /9]/128 D. HYPERFINE SPLITTING The hyperfine splitting Hamiltonian for a cubic field is: h = S- AI = A-I = ASzIz + A(S+I_+SI+)/2 (72) 39

The exact energy to be assigned to each state M,m requires solution of a 36 x 36 secular equation. An approximate method is to use time-independent perturbation theory up to a sufficiently high order with respect to the diagonalized Hamiltonian for HIIZIIS4. Let li = i y +$l (73) 0o = gOIHSz (74) ASzIZ + A(S+I_+S_I+)/2 + crystalline(75) Then, taking free ion energy as zero, one has: E E(O) + (2) (76) M,m M,mm+,m+ E,m (76) where E(O) = <M,ml o IM,m > (77) 1)m = <M,m1lJ(1M,m> (78) EM2)m I < Mm'I'M',m'>' /[FM) - 0)M ] (79) Then the energy associated with each level M,m is given by =f) = s+ hf.s. EM, m( = 0) = E.s. + EM, s where hf- So_-= E(1). + E(2) eM,m M,m M,m

The values of EM; are given by Eq. (66) and El') and E(2) are: (2) (0) E(2) I < M,m A(S+I I +) /2 |M',mt>l2 E{' ml] M',m' =4 (/< M, mlS+I_ IM',m'><M,S_I+M',m'>) (x) < M',mt SS+I + S_I+IM,m >/[E ( 0) ) ] (82) A M11 > < I,(o) (0) 4 < M,mS+I M-l,m+l > < M-l,m+l1S_I+IM,m >/[EMm EM) m+l A+ < M,m|S_I+|M+l,m-l > < M+l,m-olS+I_|M,m >[ (o ) 1(o) Consider S+I_ M-l,m+l> = [S*2 - (M-1)M]1/2[I*2 _ (m+l)m]l/2M,m > S_I+lM,m > = [S*2 _ M(M-1) ]1/2[I*2 - m(m+l) ]1/21M-l,m+l > and (83) S_I+IM+l,m-1 >= _[S*2 - (M+1)M]1/2[I*2 (m-l)m]l/2IM,m > S+I_ JM,m > = [S*2 - M(M+1) ]1/2[I2 - m(m-l) ]1/2 M+l,m- > Consider also E,m) - gHHM,(M~), E(O) -E(O) +gpH (84) mIt~m ~l,mt9mTl M,m Mil,m~l Substituting Eqs. (82) and (83) in Eq. (84), one finds: 41

(2) = A.(S* M2+M) ( I*2-m2-m) (S*2_M2_M)(I*2_m2+m) EM,m 4 L gH gH =A [M(I*2-m2) - m(S*2-M2) + M(I*2-m2) - m(S*2-M2)] (85) 4gpH -eA M[i(I+l) - m ] - m[S(S+l) _ M2] 2gfH Considering Eqs. (67), (75), (80), (81), and (85) one finds for (HIIZIIS4): E5/Z m t e - + I(4e+a)2+5a2/4 + 5Am/2 + A2[5(35/4-m2)/2-5m/2]/2gPH 5/2,m 2 Es/2,m - E 2 + J(4E-a)2+5a2/4 + 3Am/2 + A2[3(35/4-m2)/2-13m/2]/2gPH E1/2,m e + a + Am/2 + A2[(35/4-m2)/2-17m/2]/2gPH El/2'm - (86) -e + a - Am/2 + A2[-(35/4-m2)/2-17m/2 ]/2g H E_ 3/2 m e - - /(4e+a)2+5a2/4 - 3Am/2 + A2[-3(35/4-m2/2-13m/2]/2gPH E-s/2,m 2 E 5/2 @ e 21(4E- a)2+5a2/4 - 5Am/2 + A2[-5(35/4-m )/2-5m/2]/2gpH For a given m there are six energy levels; therefore five absorption transitions will occur, giving rise to five resonance lines corresponding to a given m and to different M values. The ESR spectrum of Mn++ clearly shows this feature (see Fig. 16). Considering Fig. 16 one notes that associated with each m the spectrum at g = 0 has the form shown in Fig. 17. To identify the M values to which the lines ao, y,, 6, and X are associated we must find the intensity IM of each line. This is proportional to magnetic dipole absorption probability; 42

Fig. 16. Mn++ ESR spectrum in ZnS (cubic) at 300~K, @ = 00. a r E A Fig. 17. Mn++ ESR due to a given m. and the ratio of intensities of two lines is as follows:15 IM I< M-llrKr|lM >12 =-' (87) IMt | f<M'-1 rK. r IM' >12 But, from replacement theory, vectors of position and linear momentum can be substituted by angular momentum operators L or S, namely

A dA AA A e2 A *2 rA = r- P.r oc LL.L = LL or SS (88) where S - Sz + _S+ + g+S + = (l~i)/2 i-= f and IM,< M-11+S IM > < M I ES+jM-l > S(S+l) - M(M-1) IM' < -M'-11+S_ M'> < M I eS+IM'-l> S(S+l) - M'(M' -1) Thus I5/2:I3/2:Il/2:I1/2:I_3/2 5:8:9:8:5 (89) Note also that the separation of the measured magnetic field corresponding to each line with intensity IM, aMM,, p for G = 0 can be obtained as follows: EM,m -l,m gM + = gHo = hvo for all M,m (90) Thus EM,m-EM-l,m EM',m -EM' -l,m HgM gm M,m +,m,m (91) or rMM' M',m,m M,m,m f = /g (92) where f' (A,a)'s are given in Eq. (86) and one considers the fact that in M,m AIIBVI compounds la ++ |< IA ++l| fhMm must be found up to a2, i.e., 44

sJ(4e+a)2+5a2/4 A 4 + a + 5a2/16gH gH = 2 (93) Substituting Eq. (93) into Eq. (86) one finds a,'s, e.g., /2,5/2 - %5/2 5= H/2 - 1/2 f1/2 - f5/2 (94) where f E 1/2; E 1/2 _H 1/2 Equations (86) give E-/2 - 7_ = I2c+A+A2(55/4-m2)/2gH; 2e/gp H1/2 (95) gc_ g3 Thus f~/ = mA + A' [(35/4-m2) ]/2H A' = A/gP, a = a/gp and fs/ = mA' + 2a' + At2(35/4-m2+8m) Therefore CT5/2 -2a' - 2mA'2/H Similarly'3/2 = 5a'/2 - 5a'2/16H - mA'2/H a1/2 = 0 c 1/=2 -5a'/2 - 5a'2/16H + mA'2/H a 3/2 = 2a' + 2mA'2/H (96) Considering Eqs. (89) and (96), and also the fact that the two fine-structure components tend to merge (see Figs. 17 and 18) at higher field one can deter4 5

_ 1 rI 2 7:2 Z~~1 25 Fig. 18. Schematic plot of fine structure of ESR spectrum of Mn++ in cubic ZnS for m = -5/2 and 5/2. mine that A and a have opposite signs. According to Watanabe5 a cc (Dq)2 and thus A < O. Since (15a'/21)/(15A'2/HI) is 8, 28, and 29 for ZnS, CdTe, and ZnTe respectively, the measured DC magnetic field determines the M to be assigned to the lines a, P, y, r, X; e.g., cx corresponds to the M giving maximum separation, etc. The result (see Fig. 18) is -+ M=:-l/2; -+ 5/2; y + 1/2; +- -3/2; + 3/2 (97) Consider Eqs. (91) and (95), as well as the fact that A' < 0 and H - H = f f, M, m M, m M, m M,m Then H1/2,m - H/2m =A(m )+ A'(m-m')+ At2(m'2-m2)/2Ho (98) <0 m' <m Therefore, the lowest group of five fine-structure components a,, y, y, and x (see Fig. 18) belongs to m = -5/2, where the group at the high-side

field belongs to m = +5/2. E. DETERMINATION OF 3a = 4r8- 2E PARAMETERS The 6S/ electronic level of the Mn++ ion is split into two levels, 48 and 2r7, due to the presence of a cubic field. This energy difference is easily obtained from Eq. (96): 5 [ja(M = -1/2,m = -5/2) 1 + I (M = 3/2,m = 5/2) i] = [I-5a'/2 - 5a'2/16H - 5A'2/2HI + 15a'/2 - 5a'2/16H - 5A'2/2HI] 5 = 3[15a'/2 + 5a'2/16H + 5A'2/2HI + 15a'/2 - 5a'2/16H - 5A 2/2H1]/5 = 3a' and 3a = g5 3a'(erg), 3a'(gauss), or 3ag(CM (99) hc F. DETERMINATION OF HYPERFINE SPLITTING CONSTANT A' Equation (98) gives H1/2y,5/2 - H1/2_5/2 -5A = -1/5 (H1/2,5/2-H1/2_5/2) 100) A = gA' (erg) or A'(gauss) or gPA'/hc in (cm-l) 47

APPENDIX A DETERMINATION OF Mn++ GROUND STATE The electronic configuration of Mn++ is ls2 2s2 2p6 3s2 3P6 3d5 (A-l) According to Griffithl6 the relative energy levels of the ion can be obtained by considering only the last five electrons in the partially filled shell of 3d. The energy terms corresponding to these electrons can be found simply by the "counting method." This method consists of setting up a table with possible positive MS and ML values in rows and columns, respectively, with determining the number of times that each MS occurs for a given ML by constructing all possible wavefunctions corresponding to ML. In Eq. (A-2) a set of wave functions of ML = 2 is given. 12+ 2- 1+ -1+ -2+> with ML =2 and MS 3/2 12+ 2- 1+ -1+ -2> with ML = 2 and MS = 1/2 (A-2) 12+ 2- 1+ -1- -2+> with ML = 2 and MS = 1/2 12+ 2- 1- -1+ -2+> with ML = 2 and MS = 1/2 One can, with the same procedure, find that there are nine other wave functions with MS = 1/2 and two more with MS = 3/2. The result for other ML values can be obtained much easier; they are given in Table A-I. Among the sixteen terms in Table A-I the sextet 6S and quartets 4G, 4F, 4D and 4p are the most responsible terms which account for the observed optical and EPR 49

TABLE A-I MSML TABLE FOR 3d5 CONFIGURATION MS Total Number Difference ML' | 1/2 3/2 5/2 1/2 3/2 5/2 Energy Terms 6 1 0 0 1 0 1 I 5 2 0 0 1 0 0 2H 4 5 1 0 3 1 0 2G,2G and 4G 3 8 2 0 3 1 0 2F,2F and 4F 2 12 3 O 4 1 0 2D 2D 2D and 4D 1 14 4 0 2 1 0 2p and 4P 0 16 5 1 2 1 1 2S and 6S experimental results. According to Racah12 these terms can be calculated from the terms of 3d2 because 3d5 is an nQ2Q+1 configuration and because the sextet and quartet terms have the spin S = 2+1/2 and S = 2-1/2. The 3d2 MSML table (Table A-II) reveals that for 3d2 there are five energy terms, 1G, 3F, 1D 3P and 1S, (A-3) whose energies can be obtained by using the diagonal-sum rule. In this way one finds the sum of the energies of several terms from the relation: - ij -5E ij1 0 2 P (A-4) where Pa is a polynomial of m,th power provided p Ima m = a=l 5O

TABLE A-II MSML TABLE FOR 3d2 CONFIGURATION MS Total Number Difference Energy Terms 4 0 1 0 1 1G 4 0 1 0 1 1G 3 1 2 1 1 3F 2 1 3 0 1 1D 1 2 4 1 1 3P 0 2 5 0 1 1S with n' the number of electrons in the partially filled shell (n' < 2Q+1). Each Pa satisfies the relation P -m El +.. = O (A-5) i=1 The coefficient of -E in Eq. (A-5) is the desired diagonal-sum which we must find. It can be obtained from the relation ma Xii E (K%) (A-6) i=l Kx where J(,2) is the coulomb integral defined as 00 J(i,j) = ak(i,j)Fk(i,j) (A-7) k=O Fk is the so-called Slater-Condon parameter which for a pair of electrons i and j with the same n and ~ takes the form 51

00 00 Fk (i,j) = e2 (rk/rk+l) [R(i) ]2 [n(j) 2r2 dridrj (A-8) ni,j) e (r</r> j and ak(i,j) in this case (same n and ~) can be given as: ak( i,j) = ck(mi,mi) ck(mj,mj) (A-9) with 2r I c(m,m') Y*2k+l)_ m( 1() Ykm-m' (g,) Ym' (G,)sin 1 dd ) 2k+l) o o (A-b) From the diagonal-sum rule which is described in Eqs. (A-4), (A-5), and (A-6) we can write the energies of the terms of 3d2 as follows: E(1G) = J(2,2) (A-lla) E(3F) +E( G) = J(2,1)+J(1,2) (A-llb) E( 1D) +E( 3F) +E( G) = J(2,0) +J(1,1) +J(0,2) (A-llc) E( 3P) +E( D) +E( 3F) +E( G) = J(2,-1) +J(1,) +J(O, 1) +J( -1,2) (A-lld) and E( S) +E( 3P) +E( D) +E( 3F) +E( G) = J(2,-2) +J(l,-1) +J( 0,0) +J( -l l) +J( -2,2) (A-lle) Equations (A-6)-(A-10) reveal that J(i,j)'s of Eqs. (A-11) can be obtained from the much simpler relation Q ~ J(i,j) = j akFk = -a2kF2k = aoFo + a2F2 + a4F4 (A-12) k=O k=0 52

In Table A-III the integrals J(i,j) of Eqs. (A-11) are given in terms of the coefficients A, B, and C defined as A = Fo - 49F2 (A-13a) B = F2 - 5F4 (A-13b) C = 35F4 (A-13c) TABLE A-III J(m,m') INTEGRALS IN TERMS OF A, B, AND C m mI' J(m,m ) 2 +2 A + 4B + 2C 2 +1 A - 2B + C 2 0 A- 4B + C 1 +1 A + B + 2C 1 0 A + 2B + C 0 0 A + 4B + 3C Substituting for J(i,j) in Eqs. (A-ll) one finds the relative energy levels as follows: E(1G) = A + 4B + 2C (A-14a) E(3F) = A - 8B (A-14b) E( 1D) = A- 3B + 2C (A-14c)

E(3P) = A + 7B (A-14d) E(1S) = A + 14B + 7C (A-14e) Equations (A-14) reveal that the relative energy levels for the terms with S = n'/2 (n' being the number of electrons) can be express as16 E[dn',n'+lT(L)] = n'(n'-l)(A-8B)/2 + 3[6n' - L(L+l) ]B/2 (A-15) and therefore the energy of the triplets of 3d 2, quartets of 3d3, quintets of 3d4 and sextets of 3d5 can be found easily, e.g., E(d5, 6S) = 10A - 35B (A-16) The energies of the quartets of 3d5 cannot be obtained as easily, it can, however, be shown that they may be derived from Eqs. (A-14). Consider that each quartet of d5 has four electrons with a spin and one with f spin. We construct the wave functions ~l(m,m') of the quartet 4F of d5 so that, compared with the wave functions *l(ml..m6) of the quintet 5D of d6 *l(m.. m6,d6,5D) = 12+ 2- 1+ 0+ -1+ -2+>, (A-17) it will have the form ~l(m...m5,d5, 4F) = 1(2,) = 12+ 2- 1+ 0+ -2+>. (A-18) In Eq. (A-18), m(2) stands for the m~ of one-electron wave function with D spin and m'(l) stands for the -m~ of the extra electron in the quintet D of d6 configuration. The diagonal elements of ~(m,m') can be obtained by relating

them to the corresponding elements of the 5D terms of d4 and d6 configurations. These terms can be found from the relationl6 6 E(d6,5D) = [J(m',m) - K(m',m%) ] K<%=2 5 5 = [J(mK,mN) - K(m,mk)] + [[J(mK,m6) - K(mK,m6)] Kc<=2 K=O (A-19) 5 E[((m,mt) + J(m,m') - K(m,m') + f(mI,mN) K=l where 2 K(m,m') = 3(ss') b kFnk 0 s ~ s' (A-20a) k=0 and f(mK,mk) = J(mK,mX) - K(m,mK)j. (A-20b) Thus Eq. (A-19) can be rewritten as 5 E(d6,5D) = E[O(m,m') ] + J(m,m') + f(m,-m'). (A-21) K=l1 We have also

5 E(dS, S) = 2 [J(mK,mk) - K(mK,m) ] K<%=2 5 5 = 2 f(m,m2) + f(mK,m\) K=1 K<\=3 (A-22) 5 4 = 21 f(m)K-m) f(mK,mk) K=1 K<%=2 5 f(mK,-m m) + E(d4,5D) A comparison of Eqs. (A-21) and (A-22) gives: E[O(m,m')] = E(d6,5D) + E(d4 5D) - E(d6,6S) - J(m,m') (A-23) Equation (A-23) gives the energy levels associated with the sextet and the quartets of Mn++ and Fe+++ within a constant which can be found by considering Eqs. (A-14), (A-16), and (A-23) for the case where ML = m+m' = 0: E[O(m,m'), ML = 0] = E(d5,6S) = C' - E(d2, S) (A-24) where C' = E(d5, 6S) + E(d2,1S) = 10A- 35B + A + 14B + 7C 11A- 21B + 7C. (A-25) Equations (A-24) and (A-25) reveal that El[d5,4T(L) ] = 11A - 21B + 7C - E[d (L)] (A-26)

Substituting from Eqs. (A-14) for E[d2, T(L)] we find the relative energies of the quartets 4G, 4F, 4D and 4P of 3d5 configuration E(4G) = 10A - 25B + 5C (A-27a) E(4F) = 10A - 13B + 7C (A-27b) E(4D) = 10A - 18B + 5C (A-27c) E(4P) = 10A - 28B + 7C (A-27d) The relative energies of these levels for the case B/C =.5 and in units of C can be computed with the result as 0.5 for 4F (A-28a) - 4.0 for 4D (A-28b) E[d5, ST(L); B/C =.5] - OA - 7.0 for 4p (A-28c) C - 7.5 for 4G (A-28d) -17.5 for 6S (A-28e) A careful study of Fig. 4.4, Ref. 16, indicates that only at B/C <.08 do the quartets lie lower compared with the doublets; but even at this region Hund's Rule does not hold completely, and the 4D is lower than 4F. 57

APPENDIX B REDUCTION OF THE jft-J4l TO A FUNCTION OF L, S AND I To derive the spin Hamiltonian Eq. (35), use is made of the replacement theory:17 (Ce KJIL- IteaK) ((YtKi' |_ICIKm) = (Kin| ILIKin) L(L+l) (B-1) = B(C',oa)(Kmn (KLm) where I can represent such vectors as r or p. Thus we can replace the position coordinates with angular momentum components as follows: x+ LX, xy = (LLy+yx)/2, x + n (B-2) ki2 [Eq. (9)] is already independent of position vector and assuming that Russel-Saunders coupling holds and that bii >> bij, it will take the form: K L-S (B-3) where K is a constant, L is total angular momentum and S is the total spin. agg [Eq. (10) ] is one of the Hamiltonians containing position vectors rij in conjunction with si and sj. To replace position coordinates with the components of angular momentum we expand as follows: "hL r i[( i i i ~ ~i i = Rx + Ry + Rz 59

where ~, ~ and 5 are the Cartesian coordinates of rij and Rx = 4I 2rij rj sx - 3[ [( +zSy) ] >-5 i>j Recalling that m 0 =-cZiS XajSX = 5ijS2 S ijxS x = S (B-6) by substituting for sisj in Eq. (B-5) one finds: Rx= Cl(i,j)ij L(L+l)s - 3[ + 2 (LyLz+LzLy)(SySz+SzSy) ] i>j = r3 p[3 x"4x + 2 (LyLz+Lz)(SySz+SzSy) - L*2Sx](B-8) = - r-5 p[(34-L*2)S2 + 2(iLx+2LzL)(-iSx+2SySz) ] = r-5 p[(I3L-L* 2)S2 + 3 (LxSx+6LzLySySz) ] + o The result for Ry and Rz can be obtained from Rx [Eq. (B-8) ] and 3 can be expressed as 3 (LIS) = -M[(LS) 2+ 2 (LS) - L(L+1)S(S+1)/3] (B-9) Since [Eq. (11)] is already given in terms of L and S, [Eq. (12)] will be considered now~ Each of the three parts rij( i-si).I, 3risiri I and 8e5(ri) siI/ 3 will be treated separately as follows: 60

= N')3 zDN (L-I - KS-I) (B-10) 31 i i N 3ri si ri /r = N2 [2 (LxSxLxIx+LxxLxSx+LXLS+xSxLzIz+LzIzLxSx+LxSxLyIy+LyIyLzSz) +2 (x + y,y + x,z + z) + 2 (x + z,z + x,y + y) ] 3 N"[(L SX+yySy+IzXLXIx+LyIY+LzIz) + (LxIx+WIy+LzIz) (LxXx+LySy+LzSz) ] 3 N [L.S L.I + L.I L.S] (B-1l) = N3 -j3 (ri)s.I = N L S-I (B-12) i and finally we have N aN[L*2S-I 2 (L.S L-I + L.I L.S)] + L-I - K S-I (B-13) where N and a are constants. In the same way, 7 [Eq. (13)] can be changed into a function of L, S and I. The final result is 61

(L, S,I) = l K LS - M[(L~S)2 + L.S/2 - L*2S*2/31] + (L+2S).H - yNH (B-14) + N{a[L*2S I - 3(L.S LoI+L.I L.S)/2] + L.I - K S.I] + p[(L.I)2 + _LI/2 L*2I*2/3 ] 7 where K, M, N, p, a, and y are independent of operators L, S, and I, and ~,~ [Eq. (14)] is to be obtained for each particular crystalline field. ZnTe 7 and CdTe single crystals have cubic structure with each Zn or Cd surrounded by four tetrahedrally positioned Te ions. The crystalline field appropriate to this structure is [see Eq. (43)]: Ie (r,9,O) = A40r4{Y40(9,1) + (5/14) 1/2[Y44(9,Y) + Y4-4(9,0)]} [A40[c4o/8(35z44- _30r2z+3r4) + 5 4 105 (r4 (B-15) -4r4cos 2G+r4cos 49)cos 4 ] = A40[40o/8 (35 4-30r2z2+3r4) + 5i/14 105r4sin4G cos 4 ]. ra.cullliilg that cos 40 = 8 cos4 -~ 8 cos2~ + 1 r4sin49 = (X2+y2)2 = r4 - 2r2z2 + z4 62

one finds that r4sin40 cos 4~ = X4 + y4 - 6x2y2 (B-16) Equations (B-15) and (B-16) reveal that = (A40/8)[(4o(35z4-30r2z2+3r4) + 1680 5wf/14 a44(x4+y4-6x2y2) ] (B-17) where a = ) 1/(2n+l)(n _(ml 1 n,m= ( -1) _ 2(n+m)! Thus 44 = (9/2(8)!2r) 1/2 (B-18) and (40 = (9/2=) 1/2 (B-19) Substituting for %n,m in Eq. (B-17) one finds: J o= (3A40/8 /T)[(35z4-30r2z2+3r4) + 5(x4+y4-6x2y2)] (B-20) (15A4o/ /) (x4+y4+z4 - 3r4/5). The next step is to transform r4 into a function of angular momentum components. Consider that r4 = x4 + y4 + z4 + 2(X2y2+y2z2+z2x2) (B-21) and x2y2 > 6 ( L~L+~LL+'LxLy L+IyLLxL+ LLLL+Iy5L) (B-22) 65

where LxL5rLyLx + LLXLXy = 2Lz + 2L L: - (B-23) and LxLyrLx5yT + LyLxx LX = L2 + 2LL - L2 - (B-24) Equations (B-22), (B-23), and (B-24) reveal that x2y2 _(2L+4II+3IL2z262q) = (6LftL+5L2-2LI-2Lq) (B-25) The expressions corresponding to y2z2 and z2x2 can be obtained from Eq. (B-25) by proper substitution of x, y, and Zo The sum of these expressions which is needed for evaluating Eq. (B-21) is: 6(x2y2+y2z2+z2X2) J -6( 2 2 2 2 2L+ -t 3(L+LX+L+) - 2(L 4++Lz) (3B-26) 2(L +Lf+).... 6(I T+TL2z+L2z) - (L+L+Lz) Now note that I; + gL2 + L2L Z- L*4 -( L4+1+L4) ( LLzL or 6(rL + + LL + 2Lz4) 3[L - (L-+L+L4) ] (B-27) Equations (B-20) through (B-27) reveal that

J/ (L) = B[ + + L4 3 (L*4 1 L*2 = B[ZL + L4 + L( - 1) L]2(3L*2-1~ (B-28) = B[IT + L4 - L(L+1) (3L*2+3L-1)] Equation (B-28) is similar to the Bleaney'slO results for this case and B here is a constant obtained as a result of transforming j (r,G,O) into J(L).

REFERENCES 1. Lambe et al., Phys. Rev. 119, 1256 (1960). 2. Azarbayejani et al., Bull. of Am. Phys. Soc. 6, 117 (1961). 3. Kikuchi et al., Suppl. Phys. Soc. Japan 17, 435 (B-l, 1962). 4o Semiconductor Laboratory Procedural Manual (DRM60-00169). 5. Van Wieringen, Physica 19, 397 (1953). 6. Watanabe, Prog. of Theoret. Phys. 18, 405 (1957). 7. Matarrese et al., J. Chem. Phys. of Solids 1, 117 (1956). 8. Thomas, Nature 117, 514 (1926). 9. Mack, Rev. of Mod. Phys. 22, 64 (1950). 10. Bleaney and Stevens, Repts. Prog. Phys. 16, 108 (1953). 11. Bethe, Ann, Physik 3, 133 (1929). 12, Racah, Phys. Rev. 62, 438 (1942). 13. AFOSR TN59-220 (1959). 14. deKronig et al., Physica 6, 290 (1939) o 15. Hitler, Quantum Theory of Radiation, Oxford, Clarendon Press, p. 180 (1954). 16. Griffith, The Theory of Transition Metal Ions, Cambridge (1961). 17. Feenberg, notes on Quantum Theory of Angular Momentum, p. 34, Stanford University Press (1959). 66

PART IV SOLID STATE INSTRUMENTATION by Glenn G. Sherwood* *Captain, USAF, in Civilian Institutions Program, AFIT, at The University of Michigan.

This research project is concerned with the design and development of a solid state charged particle dE/dx detector to replace the traditional gaseous ion chamber. The advantages of the solid state counter over the gas counter are: (a) Resolving times of less than 0.1 microsecond due to short collection times in the semiconductor. (b) Less scattering of particles out of the beam by multiple coulomb interactions, and hence greater counting efficiency. (c) Ionization efficiency which is 10 times greater than that of the gas counter, and hence some probable improvement in resolution. (d) Fast rise time, which permits high counting rates and the possibility of using pulse shaping techniques to discriminate against background gamma radiation. Some work on semiconductor dE/dx detectors was done by H. E. Wegner of Los Alamos Laboratory in 1960. He developed a diffused junction dE/dx detector with a thickness of 50 microns and a diameter of 3/16-in. As Wegner reported out of six original detectors this was the only one which survived the difficult grinding and etching process to give a workable thin detector. However, this one was enough to demonstrate the feasibility and advantages of semiconductor dE/dx detectors. Our research project, wherein we are attempting to measure the spatially dependent fast neutron fission spectra through several types of semi-infinite media, requires a proton recoil telescope spectrometer which has a resolving

time of less than 1 microsecond and is relatively insensitive to background neutrons and gammas. Previous telescopes with gaseous counters had extreme limitations imposed by the large resolving times, and this restricted their performance so greatly that experiments with fission sources were virtually impossible, as explained by Johnson.1 Our prescription for a possible method of measuring the spectra is shown in Fig. 1. Here is depicted a proton recoil telescope with a semiconductor detector preceded by a semiconductor dE/dx detector. The proton recoils in energy range 1 to 12 Mev must pass the dE/dx detector and stop in the thick E detector. Only those pulses which pass through both detectors and are hence in time coincidence are allowed to enter the pulse height analyzer. This reduces the large neutron background produced by the Si28 (n,p) and Si28 (n,a) reactions as well as betas from the walls of the instrument. The portion of this telescope spectrometer project covered herein involves the development of the dE/dx detector. Three factors dictate the size of the thin detector: noise, counting statistics and thickness. The energy of the protons require a detector of about 25 microns so that 1.5 Mev protons will pass into the second detector. So that the efficiency can be kept in the order of magnitude of 10-6 or greater (corresponding to an energy resolution of 15-20%) for reasonable counting statistics, the diameter of the finished detector should be as close to 2 cm as possible. Noise will limit the achievement of the ultimate resolution and efficiency. Theoretically, the noise of this detector should be about 85 Kev if predicted by the noise analysis of Goulding.2 This would be acceptable because the 12-Mev

Telescope Housing Proton Beam -dE/dx Detector E Detector ~~~~~~~~Beam Pre amplif iers Amplif iers 0r r Beom Double l IIl|Scope Coincidence -cng Unite iGate Pu Ise Multi eChnnel rSignal Pulse Analyser Oscilloscope Printout Record Fig. 1. Block diagram of telescope spectrometer and necessary circuitry for coincidence counting. proton deposits about 150 Kev in 25 microns of silicon. In attempting to develop high quality semiconductor detectors the surface barrier technique appears superior to the diffused junction method as used by Wegner because: (a) The surface barrier method does not require excessive heat which degrades the performance of the completed diode. (b) The p-n junction diode has a "window" of several microns which degrades the energy of slower protons without collecting charge.

With these advantages in mind work was started in February 1962 to develop these solid state devices. A 2000 ohm-cm, 500-microsecond, vacuum zone floated N-type silicon, 24 mm in diameter, is waxed into a circular pyrex boat which is in turn waxed to a flat aluminum plate. The silicon is then cut into discs.055-in. thick with a high speed diamond impregnated blade on a Di Met No. 120BQ cutoff machine. The silicon disc is then waxed to a pyrex optical disc 3 in. in diameter (see Fig. 2). Three small blocks of silicon are waxed 120 degrees apart, as shown in the figure, and it is the purpose of these blocks to maintain the parallelism between the face of the silicon crystal and the pyrex blocks..;....... -..;:-........ Fig. 2. The silicon wafer is shown mounted on a pyrex polishing disc. The optical glass flat and the iron flat are used in the lapping process. Two types of optical lapping flats are used in this process, a 12-in. Fig:2. T l w

polishing is done with Linde A alumina on Buehler AB Metcloth with the Buehler polishing wheel. Both iron and pyrex flats are checked periodically to ensure that they do not loose their flatness. The silicon crystal is first lapped with No. 400 silicon carbide in water on the metal flat to remove the silicon damaged by the saw. The crystal is then lapped with No. 600 silicon carbide and then with No. 1200 aluminum oxide in water. The polishing blocks are measured regularly during this process with a micrometer to maintain them at an even height with respect to the back of the polishing block. In this way the removal of silicon by lapping can be controlled so that the crystal is ground with parallel surfaces. When the pits have all been removed by the No. 1200 abrasive, as observed with a metallurgical microscope of power 110, the crystal is transferred to the glass optical flat and lapped with No. 3200 aluminum oxide. The crystal is then polished to a mirror finish with the Linde A until:-all pits and.scratches are removed. At this point the surface should have minimum damage (about 1 micron) but should still exhibit a relatively high surface recombination velocity. The crystal is then turned over on the 3-in, pyrex flat, polished side down, and the reverse side of the crystal is lapped and polished in the manner explained above. The crystal is polished down to 45-micron thickness, with particular care being taken to insure that the crystal is planar. This can be controlled by measuring the edges with a microscope. The crystal is removed from polishing by gentle heating. It is washed in successive baths of trichloroethylene, alcohol, and distilled water, dried in a clean glove box, and finally cemented to a lavite type A ring with

Araldite No. 951 amine-type epoxy (see Fig. 2). The back of the crystal is protected by sax while the entire crystal and holder is immersed in an etch bath (nitric acid, hydrofluoric acid, and acetic acid in volume proportions 2:1:1) for two minutes. After washing in distilled water, the wax is removed from the back. After 24 hours, 75 micrograms/cm2 of gold is evaporated on the front and the same quantity of aluminum is evaporated on the back. The semiconductor dE/dx detector is allowed to age for several days and is then completed. Experimentation with several etches showed that the most controlled etch producing a chemical polish on the 1/2-micron polished surface of the silicon was of the above noted composition, maintained at O~C for about 2 minutes. This procedure removes evenly about 10 microns, leaving a polished surface which appears to produce the surface barrier required. The important aspect of this technique is that we have apparently been able to control the etch rate without any preferential action on the part of the etch solution. Using this procedure on the face of the silicon crystal apparently provides the density of fast states necessary for the inversion layer. With evaporation of a gold layer, the rectifying junction appears to be complete. We are still experimenting with back contacts to achieve a stable or ohmic contact which is free from injection. We have only partially achieved this because the back contacts still provide some rectifying action. The tests to show the amount of minority carrier injection arer stlll inconclusive.

The latest dE/dx detector$ have progressed to the point where we are now producing detectors 3-cm2 in finished area, and 30 microns thick, with 210 stable characteristics over periods of weeks. Resolution for Po is 5.8% (see Fig. 3) and the peak noise extends to channels corresponding to about 280 Kevy. Tmprovement in the back contact should help to decrease the noise to the acceptable level of 150 Kevo Figure 4 shows the pulse shape versus bias for Po210 alphas. The dE/dx detector in conjunction with a thick detector is now operating in a proton recoil test spectrometer with coincidence equipment. This telescope spectrometer, shown in Fig. 5, is being used to help find those sources of noise which can be removed. (Since tests have just started there are no conclusive results as yet.) The mechanism shown in Fig. 5 is the telescope, which operates in a vacuum housing. The front wheel holds the polyethylene radiators, which are of different thickness. Behind the front wheel is the dE/dx detector, and behind this detector is the E detector. The rear of the device holds the motor for positioning the different radiators corresponding to various parts of the neutron energy spectruum being sampled. The instrument has been inritially tested with a weak Pu-Be source. Figure 6 shows the response of the dE/dx detector and E detector to recoil protons. The picture is poor because the count rate is low and a 30-45 second exposure was necessary. The coincidence counting rate is about 1 per minute. In January the instrument will be tested with the Michigan Van de Graaff. Initial results indicate at least partial success with the new semiconductor dE/dx detectors and some success with the telescope. As far as we can determine no other group has achieved success as yet in producing these solid state counters.

,0. * Pi __ 0 o.1 i *r C\J 0 PHi * 0.0 I X * Q X 0 _ 1 0o l z H'0 O 0 — O 0 C\ r i, 0 0 0 0 0 0 13NNVHD i3d SINoQo 10

25 75 150 Fig. 4. Pulse shape of large area dE/dx detector to po210 alphas vs. bias across the detector and 22-meg load resistor. Voltage is 0, 25, 75, and 150 volts; the leakage current is over 1 microampere. Fig. 5. The telescope spectrometer. The detector nearest to the proton radiator wheel is the dE/dx detector. The second detector is a semi-conductor E detector. 11

Fig. 6. The top line shows the response of the dE/dx detector and the bottom the E detector to proton recoils using a Pn-Be source. 12

REFERENCES 1. In Marion,J. B. and J. L. Fowler, Fast Neutron Physics, Vol. IV, Interscience Publishers, N. Y. (1960). 2. Goulding, F. S. and W. L. Hansen, Leakage Current in Semiconductor Junction Radiation Detectors and Its Influence on Energy Resolution Characteristics, UCRL 9436, November 1960. 13

PART V PUBLICATIONS AND PRESENTATIONS COTPLEED UNDER CONTRACTS NO. AF-49(638) -68 AND AF-49(638)-987

I. UNIVERSITY OF MICHIGAN TECHNICAL REPORTS AND INTERNAL MEMORANDUMS Baker, J., and Lambe, J., Luminescence of Color Centers in KC1, Technical Report 2616-5-T, January, 1959. Chen, S. H., Evaluation of Matrix Elements in Crystalline Field Theory, Technical Report 04275-1-T, October, 1962. Kikuchi, C., Makhov, G., Lambe, J., and Terhune, R., Ruby as a Maser Material, Technical Report 2616-6-R, May, 1959. Kikuchi, C., Sum Rules and Relative Intensities for Paramagnetic Ions of Spin 3/2, Technical Report 2616-8-R, June, 1959. Kikuchi, C., Resonance Absorption of Paramagnetic Ions with in 5/2: CaCO3: Mn, Technical Report 2616-10-R, August, 1959. Kikuchi, C., and King, J. E., Azimuthal Dependence of Spin Resonance Spectrum for S = 5/2, Internal Memorandum Z-1214, April 14, 1959. Kikuchi, C., and Lambe, J., Spin Resonance of V2, V3_, V4+ in a-A.1203, Technical Report 2616-12-R, November, 1959. Kikuchi, C., and Sims, C., Resonance Absorption of Ruby at Low Magnetic Fields, Internal Memorandum, Z-1101, August 25, 1958. Kikuchi, C., and Sims, C., Relative Intensities of Ruby Resonance Lines, Internal Memorandum, Z-1102, August 27, 1958. Makhov, G., Kikuchi, C., Lambe, J., and Terhune, R. W., Maser Action in Ruby, Technical Report 2616-1-T, June, 1958.

II. JOURNAL ARTICLES, PAPERS, AND PRESENTATIONS Azarbayejani, G. H., Kikuchi, C., and Mason, D., "Paramagnetic Resonance of ZnTe:Mn, " Bull. Am. Phys. Soc. 6, 117 (1961). Baker, J,, Lambe, J., and Kikuchi, C., "Photosensitive Spin Resonance in CdS,," Phys. Rev, Ltrs. 3, 270 (1959). Borcherts, R., Wepfer, G., and Kikuchi, C., "Paramagnetic Resonance Spectrum of Vanadyl Ammonium Sulfate," Bull. Am. Phys. Soc. 7, 118 (1962). Kikuchi, C., "Certain Sum Rules Applicable to Paramagnetic Ions of Spin 3/2," Bull. Am. Phys. Soc. 4, 261 (1959). Kikuchi, C., "MASER," presentation at Sylvania Products, Bayside, New York (April, 1957). Kikuchi, C., "Ruby Maser," Proceedings of NSIA-ARDC Conference on Molecular Electronics (1958). Kikuchi, C., "Experimental Work on Ruby Masers," invited paper at American Physical Society Meeting (November, 1958), Chicago, Illinois. Kikuchi, C., and Ager, R., "'Doublets in the Electron Spin Resonance Spectrum of Mn++ in Calcite," Bull. Am. Phys. Soc. 3 (march, 1958). Kikuchi, C., Ager, R., and Matarrese, L., "Doublets in the Electron Spin Resonance Spectrum of Mn++ in Calcite," presentation at American Physical Society Meeting, Chicago, Illinois (March, 1958).* Kikuchi, C,, and Azarbayejani, G. H., "Spin Resonance Properties of ZnTe:Mn and of Other AIIBVI Compounds," JO Phys. Soc. Japan 17, Suppl. B-I, 453 (1962). Kikuchi, C., and Lambe, J., "Spin Resonance Investigation of Certain Sapphires," invited paper at American Physical Society Meeting, Honolulu, Hawaii (August, 1959). Kikuchi, C., and Lambe, J., "Spin Resonance of V2+, V3+, V4+, in z-Al203,+ Phys. Rev. 118, 71 (1960). *This is essentially the same paper as Technical Report 2616-12-R, November, 1959.

Kikuchi, C., Lambe, J., Makhov, G., and Terhune, R. W., "Development of a Ruby Maser at Willow Run Laboratories," presentation at Fort Monmouth, New Jersey (1958). Kikuchi, C., Lanbe, J., Makhov, G., and Terhune, R. W., "Ruby Maser," presentation at Electron Rube Research Conference, Laval Univ., Quebec, Canada (June, 1958). Kikuchi, C., Lambe, J., Makhov, G., and Terhune, R. W., "Induced Microwave Emission in Ruby," Solid State Physics in Electronics and Tele-Communications, Vol. IV: Magnetic and Optical Properties, Part 2, Academic Press, london (1960). Kikuchi, C., Lambe, J., and Terhune, R., "Maser Action in Ruby,t" Phys. Rev. 109, 1399 (February, 1958). Kikuchi, C., Makhov, G., Lambe, J., and Terhune, R., "Ruby as a Maser Material," J. Appl. Phys. 20, 1061 (July, 1959). Kikuchi, C., and Matarrese, L. M., "Paramagnetic Resonance Absorption of Ions with Spin 5/2: Mn++ in Calcite," J. Chem. Phys. 33, 601 (1960). Lambe, J., and Ager, R., "Microwave Cavities for Magnetic Resonance Spectrometers,t" Rev. Sci. Inst. 30, 599 (July, 1959). Lambe, J., Ager, R., and Kikuchi, C., "Electron Spin Resonance of V2+ and V3+ in Corundum," Bull. Am. Phys. Soc. 4, 261 (1959). Lambe, J., and Baker, J., "Effect of Bleaching on F-Center Paramagnetic Resonance," Bull. Am. Phys. Soc. 3 (March, 1958). Lambe, J., and Baker, J., "Optical Effects on F-Center Spin Resonance at Low Temperatures," presentation at Quantum Mechanics Conference, High View, New York (Sept., 1959). Lambe, J., Baker, J., and Scarisbrick, I., "Spin Resonance of Atomic Tritium at 4.2K,t" Bull. Am. Phys. Soc. 4, 418 (1959). Lambe, J., and Kikuchi, C., "Spin Resonance of Donors in CdS," J. Phys. Chem. Solids 8, 492 (January, 1958). Lambe, J., and Kikuchi, C., "Spin Resonance Mn++ in CdTe," Bull. Am. Phys. Soc. 5, 158 (1960). Lambe, J., and Kikuchi, C., "Paramagnetic Resonance of CdTe; Mn and CdS:Mn," Phys. Rev. 119, 1256 (1960).

Mathews, J. H., and Lambe, J., "X-Ray Coloration of Ruby," Bull. Am. Phys. Soc. 4, 284 (1959). Scarisbrick, I., "Spin Resonance of Gamma Irradiated Alkali Hydrides," Bull. Am. Phys. Soc. 4 (June, 1959). Terhune, R. W., Kikuchi, C., Lambe, J., and Baker, J., "Hyperfine Structure of the (Cr53)+++ Ion in Ruby by Double Resonance," Bull. Am. Phys. Soc. 5, 157 (1960). Terhune, R. W., Lambe, J., Kikuchi, C., and Baker, J., "Hyperfine Spectrum of Chronium 53 in A1203," Phys. Rev. 123, 1265 (1961).

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