THE UNIVERS ITY OF MICHIGAN COLLEGE OF ENGINEERING Department of Nuclear Engineering Technical Report AN ELECTRON SPIN RESONANCE STUDY OF VANADIUM IN CALCIUM TUNGSTATE CRYSTAL Nasser Mahootian Chihiro Kikuchi ORA Project 06029 under contract with: HARRY DIAMOND LABORATORY CONTRACT NO, DA-49-186-AMC-80(X) WASHINGTON, D.C. administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR August 1966

ERRATA Page Line Correction 2 4 spectra of 4 Add the following footnote: *K is a function of the spin-orbit coupling constant A, usually referred to as the A-value 4 20 valence states 6 4 crystal were 7 16 This result is 7 21 was made 12 2 ESR spectrum 14 Add the following footnote~ 3 *Dro Dunn has suggested that (V04) may be in the form of [(V = 0)033 where the V, 0 bond is considerably shorter than the other three and the paramagnetic electron upon irradiation comes from oxygen in the V 0 bond. 16 Table 2.1, 6.3 ~ 2.2 last line 21 2nd paragraph The ESR spectrum 23 1 1096 28 15 spectra are 29 21 (17o9 ~+ 02) 39 Table 4.2 A1, A2, and <A> are expressed in gauss 42 4 possess axial symmetry 42 Eq. (4o6) H0 - Am/g 46 15 sets in the ab-plane

ERRATA - -(CONTINUED) Page Line Correction 52 13,14 tensors 55 Tables 4.3 estimated error for gx and g is -0.005 and 4,4 and for AX and Ay is +~1 gauss. 66 20 77"K spectrum 68 7 set of lines is expected 74 4 admixture 87 lines 4 and 5 of Eqo (B.13) Ep and E, respectively 91 10 Lx = (1/2)(L+ + L_ )

This report was also a dissertation submitted by the first author in partial fulfillment of the requirement for the degree of Doctor of Philosophy in The University of Michigan, 1966.

TABLE OF CONTENTS Page LIST OF TABLES........................ iv LIST OF FIGURES........................ v ABSTRACT.................. Chapter I. INTRODUCTION..................... 1 1. Present Investigation and Motivation 2. A Brief Description of EPR Spectroscopy 3. Literature Survey II. SUMMARY OF RESULTS................ 11 III. EXPERIMENTAL PROCEDURE AND CRYSTAL STRUCTURE..... 18 1. The ESR Spectrometer 2. Sample Preparation and Composition 3. Crystal Structure and Orientation IV. EXPERIMENTAL RESULTS.................. 28 1. Group A Spectra 2. Group B Spectra 3. Group C Spectra V. DISCUSSION AND CONCLUSIONS......... 56 1. Site Occupancies of V and RE Ions in CaWO4 2. Line Width and Temperature Dependence of Vanadium Spectra 3. Splitting of the Group A Lines at 4.2~K APPENDICES............... 69 A. The Spin Hamiltonian and Position of Spectral Lines B. S4 Symmetry and Splitting of the 2D Ground State C. Relaxation and Saturation Phenomena REFERENCES................... 102 iii

LIST OF TABLES Table Page 11.o g-Values and Hyperfine Coupling Constants of Vanadium Ions in Various Host Crystals. o. o. o o 8 2.1. Experimental Versus Calculated a g for the Group C Lines.. o.................. 16 3.1. Atomic Concentration of Vanadium and Rare-Earth Ions in CaWO4 Samples................... 21 4.1. Calculated and Measured Values of the Resonant Magnetic Field for the Group A at 77~K. H \I c-Axis....... 34 4.2. Average Values of g and A of the Group A at 4.2~K in the ab-Plane..... 39 4.3. Principal Values and Directions of the g-Tensor for the Group C at 77~K.................. 55 4.4. Principal Values and Directions of the A-Tensor for the Group C at 77~K................... 55 5.1. Average Values of V4+ g- and A-Tensors in Several Host Crystals.... o............ 60 5.2. Ionic Radii of Metal Ions and Their Substitutes in CaWO4........................ 62 B.1. Polar Coordinates of the Eight Nearest Oxygen Neighbors of Ca in CaWO4. o..... o. o... o.. 83 iv

LIST OF FIGURES Figure Page 1.1. Zeeman and Nuclear Splitting of the Ground Level of V4+.......................... 5 31oo Conversion of V5 to V4+ by X-Irradiation...... 20 3.2. Metal Ions in CaWO4 Unit Cell.......... o o 24 23.3, (W04) Tetrahedron................. 25 3.4. Projection of (W04)2 Tetrahedron on the ab-Plane.. 25 3.5. (Ca, O) "Long Bond" Tetrahedron............ 26 3.6. (Ca, 0) "Short Bond" Tetrahedron.. o..... o o o 26 3.7. Projection of the (Ca, O) Double Tetrahedron on the ab-Plane....... o. o... e....... 27 4.1. ESR Spectra of X-Irradiated CaWO4:V, Tb (0.13%, 0.37) at 77 K o.......... o.......... 30 4.2. ESR Spectra of X-Irradiated CaWO4:V (0.057%) at 77~K. H 1 a-Axis and H II c-Axis o........... 31 4.3. ESR Spectrum of X-Irradiated CaWO4:V (0.057/) at 77~K. H in an Arbitrary Direction............... 32 4.4. Angular Variation of the Line Width of the Group A Spectra in the ab-Plane at 77~K........... 36 4.5. Group A Lines at 770K and 4.2~K. o.... o o o... 37 4.6. Angular Variation of the Group A g-Values at 4.2~K. H Varies in a Plane Containing the c-Axis and Making 8= 30~ with the ac-Plane.. o. o....... o 40 4.7. Angular Variation of the Group A g-Value at 4.20K. H Varies in the ab-Plane.... o o...... o... o. 41 v

Figure Page 4,8. Relative Position of the Group A, B, and the DPPH Line at 77~K....................... 43 4.9. Angular Variation of the g-Value for the Group B Lines of CaWO4:V, Tb in the ab-Plane at 770K.......... 44 4.10. Angular Variation of the g-Value for the Group B Lines of CaWO4:V, Tb at 77~K. H Varies in a Plane Containing the c-Axis and One of the g-Tensor's Principal Axes... 45 4.11. Calculated Versus Measured g-Values of the Group B in the ab-Plane at 77~K.... 47 4.12. Angular Variation of the g-Value for the Group C in the ab-Plane at 77 K................... 48 4.13. Angular Variation of the HFS for the Group C in the ab-Plane at 77~K......... 49 4o14. Angular Variation of the g-Value for the Group C in the ac-Plane at 77~K................... 50 4.15. Angular Variation of the HFS for the Group C in the ac-Plane at 77~K.......... 51 4.16. Group C, g- and A-Tensors............... 54 B.1. Point Charge Potential.................. 80 B.2. Energy Levels of V4+ in a Ca Site of CaWO4 89 C.lo Energy Absorption as a Function of the Microwave Frequency as Obtained from Bloch Equation....... 101 vi

Abstract In this study, the electron spin resonance (ESR) spectrum of vanadium (23V51, nuclear spin I = 7/2) is investigated in single crystals of calcium tungstate at microwave frequencies of about 9.5 Gc/sec. To some samples, rare-earth (RE) ions Tb3+ or Nd3+ were added as charge compensators for vanadium. To produce vanadium paramagnetic centers, the samples were irradiated with 50 kvp, 35 ma X-rays for about 15 minutes ( - 108 rad). It is believed that vanadium enters the crystal in diamagnetic pentavalent (V5+) oxidation state and the observed spectra are due to the irradiation-produced V4+ (S = 1/2). Three vanadium paramagnetic centers were detected, at 77~K and 4.2~K, with the corresponding spectra designated as Groups A, B, and C, and intensities in the ratio of 100:5:70 respectively. Each group consists of sets of eight (2 I + 1) hyperfine lines, arisen from unequivalent paramagnetic sites, which become one set when the applied static magnetic field H is parallel to the c-axis of the crystal. In general, the Group A lines, seen in the spectra of all samples, consist of four sets at 4.2~K, The maximum components of the g- and A-tensors are in the direction of the W-O bond in CaW04. Their magnitudes are gmax 2.044 + 0.003 A a (19.9 + 0.2) x 104 cm-l At 770K, one set of lines emerges at the average position of the four low-temperature sets with isotropic g- and nearly isotropic A-tensors: g = 2.0245 + 0.0005 -4 -1 All = (17.9 + 0.2) x 10 cm Al = (19.0 + 0.2) x 10 4 cm1 where All is measured with H parallel to the c-axis. This group is attributed to V4+ ions at W6 sites, i.e., to VO4complexes; the paraparamagnetic electron in this complex is supplied by one of the four oxygen ligands. The temperature dependence of the spectrum is interpreted in terms of thermal reorientation of the unpaired electron among the four possible V-0 directions. The Group B lines, shown by the (V,RE)-doped samples only, arise from two unequivalent sites with the g- and A-tensors given below: vii

A I A = Al = (8.5 + 0.2) x 104 cm1 glI = 2.068 + 0.001 glI = 2.004 + 0.001,, 90", = -lO~ el = 900~ c -10: 90~, g: 80~ where 9b and +> are the polar and azimuthal angles of the gellipsoids symmetry axes. This group is also assigned to V at a W site. The anisotropy of the spectrum is attributed to a reduction of the local symmetry due, perhaps, to crystal imperfections. The Group C lines are seen in the spectra of the samples in which vanadium is the only dopant or there is an excess of this element over the RE ions. This group arises from four nonequivalent sites: In general, four sets of eight lines are observed which become two in the ab-plane and unite to one when H Ij c-axis. Both the g- and A-tensors are anisotropic with only one common axis, taken as the z-axis, which is very close to the direction of the line joining two nearest Ca atoms of two adjacent planes in CaW04 lattice. The x- and y-axes of the g-tensor are rotated by about 30~ in c.c.w. direction with respect to the corresponding A-tensor axes. The principal values and directions for these tensors for one site are shown below. The directions for the other three sites are obtained by adding' /2, IT, and 3 Ir /2 to the azimuthal angle. x V Z g 1.990 1.965 1.901 + 0.001 e 69 51 46 + 1. 112 220 0 +1 A 34 65 165.0 + 0.3 e 90 44 46 + 1 (f 90 180 0 +1 A-values are given in 10-4 cm-1. The x and y components are approximate values calculated from measurements in the ab- and ac-planes. From the ESR data the following models are proposed for site occupancies and charge compensation in CaW04: In the presence of charge compensator (RE)3+tions, V ions occupy W6 sites and (REv ions enter Ca2' sites. In the absence of the rare-earth additives, V ions substitute both W6+ and Ca sites and charge compensation is accomplished according to the following scheme: 4 VW replace 4 W6 while 2 V5+ substitute 2 Ca2+ and one Ca vacancy is also formed in the nearest site. viii

CHAPTER I INTRODUCTION 1o Present Investigation and Motivation In recent years, calcium tungstate crystal has received great attention as an excellent host for laser-active rare-earth (RE) ions. * 1-9 Optical and paramagnetic properties of a number of these ions have 2+ been studied in this crystal lattice. Since RE ions, except Eu, enter 7-10 the crystal in their trivalent oxidation states some charge compensation mechanism must exist in order to conserve the required electrical neutrality of the sampleo Commonly, diamagnetic sodium (Na ) and niobium (Nb ) ions are used for these purposes. Effects of the charge compensation on laser activity of the crystal have been investigated by 12-14 researchers at the Bell Telephone Laboratory. There are, however, certain ambiguities concerning the lattice sites of the impurity ions. An example is the neodymium-doped crystal, CaWO4:Nd3, which is a good laser material. In general, electron spin resonance (ESR) spectroscopy (described briefly in Section 2 of this chapter) provides us with an excellent tool for determining the site of paramagnetic ions A number of articles on the optical properties of the RE-doped calcium tungstate has been recently reviewed and discussed in a threepart paper by Gorlich et al 11 1

2 in a crystal. But in this example, the ESR spectra, which vary with concentrations of Nd and the charge compensator ions, are too complex to 3,4,5 reveal decisively the site or sites occupied by the RE ions.'' In the present investigation, ESR spectra or vanadium in calcium tungstate single crystals are studied. Some of the samples were also doped with Nd or Tb in addition to the vanadium. This research was undertaken as part of a general program for studying paramagnetic properties of vanadium ions in crystalline solids. It is also an attempt to shed some light on the above-mentioned ambiguities and to obtain information on the RE laser centers in this crystal by using vanadium as an ESR probe. Vanadium is particularly well suited for this purpose because it lies immediately above niobium in the periodic table. Furthermore, as an ESR probe, it has several merits such as: (1) With an abundance of 99.76%, it is practically isotopically pure and hence its ESR spectrum does not present the complications usually encountered in the case of elements which have many isotopes; (2) due to the nuclear spin of I = 7/2, its spectrum is characterized by a hyperfine structure of 8 lines which is easily identified; (3) it shows spectrum at 77~K in contrast with accompanying Tb or Nd impurities; (4) its valence state can be changed easily by a few minutes X- or gamma-irradiation; (5) different valence states of this ion have been studied in a variety of crystalline field symmetries (see Table 1) In this program, vanadium has been studied experimentally in o. -A203, CaO, MgO powder and crystal, zinc-ammonium Tutton Salt, and Sn02 (Refo 15-19) and, theoretically, in rutile-type crystals Sn02, TiO2, and GeO2 (Ref. 21-23).

3 2. A Brief Description of ESR.,Spectroscopy This technique, which is also called electron paramagnetic resonance (EPR) spectroscopy, was introduced in physics about twenty years ago for studying systems that possess unpaired electrons and hence net angular momentum and magnetic moment. Accordingly, these systems are referred to as paramagnetic or spin systems. Unpaired electrons are found in ions of transition metals. These are five groups of elements which have one or more unfilled shells underneath their valence shell. Vanadium (23V 51), for instance, with its electronic structure ls2 2s2 6 32 6 3d3 2 belongs to the i oup elmens 22T 9 where 2p 3s 3p 3d3 4s belongs to the iron-group elements (22Ti-29Cu) where the 3d shell is only partially filled. On the other hand, 60Nd and 65Tb belong to the rare-earth group (58Ce-70Yb) where the 4f shell is unfilledo The other transition groups are palladium, platinum, and actinide groups. Unpaired electrons are also found in organic free radicals, odd-electron molecules, irradiated crystals, just to mention a few examples. Usually, the paramagnetic system is composed of paramagnetic centers (eog., transition metal ions) distributed uniformly, with a concentration of 1-0.001%, in a diamagnetic substanceo All or most of the orbital degeneracies of the ion is lifted by its interaction with the electrostatic field of the surrounding ligands and by the spin-orbit coupling within the ion itself. For instances in the case of V4+:(S - 1/2) in CaWO4 crystal, the lowest energy level is an orbital singlet, but it is still two-fold degenerate in electronic spin. Now, when the sample is placed in a static magnetic field H, this level splits into two (Zeeman splitting) corresponding to the spin quantum

4 numbers Ms + 1/2. The energy separation between these levels, to first order, is g pH where g and p are the spectroscopic g-factor and Bohr magneton. If the interaction between the electronic spin (S) and nuclear spin (I) is also taken into account, each of the spin levels splits further into (21 + 1) hyperfine components characterized by the nuclear spin quantum numbers MI as depicted in Figure lo With the hyperfine correction, splitting between the spin levels to first order is now gpH + KMI where K is a constant which gives the hyperfine separations. Transitions between levels of similar nuclear orientations (\ MI = 0) can be induced by photons of energy hv o The resonance condition is thus, h g3H + KMI (1.1) In practice, v is commonly kept constant and (1.1) is satisfied by varying H slowly. Since, at thermal equilibrium, the lower energy levels are more populated, the resonant transitions are accompanied by a net absorption of energy from the radiation field. From the values of the resonant magnetic field and the number of the absorption peaks and their separations we can obtain valuable information about the nuclear and electronic spins of the paramagnetic ions, their valence state, their lattice sites, and the crystal field symmetry and strength. By measuring the intensity of the absorption lines against that of a known standard, concentration of the paramagnetic ions in the sample can be found. These ESR quantitative measurements are easily extended to minute concentrations beyond detection of standard analytical techniques (chemical or optical).

5 I..... i' - 7/ Figo l,1 oZeeman and nuclear splitting of the ground level of V4+ 3. Literature Survey This is a descriptive bibliography of ESR and some optical investigations on vanadium ions. The measured ESR parameters (g and hyperfine coupling constants) for different host crystals are shown in Table 1, to which we shall need to refer later with 3d 3d2, and 3d configurations corresponding to the spectroscopic terms 4F3/2 3F2 and 3/2 3/2s 2' 3/2citvebbigahyo'S n sm pia

6 Divalent vanadium was first investigated in 1951 by Bleaney and 24 coworkers in the rhombic crystalline field of Tutton salt, Zn(HN4)2(SO4)2.6H20. Later, accurate measurements of the ESR parameters 2+t 19 of V2+ in this crystal was carried out by Borcherts and Kikuchi. In w a s f i r s t n v e s t g a t e d b y 2 5 cubic site, V2+ was first investigated by Low who used MgO as the host 26 crystal. Later, it was investigated by Wertz and collaborators who studied electron transfer processes among the iron group transition ions 18 in MgO single crystals. Mahootian and Kikuchi studied EPR spectrum of V2+ in MgO powder samples. They have also reported an anomalous angular dependence of the line width in MgO single crystals. Similar observa27 tion is reported by Van Wieringen and Rensen. The observed anomaly is thought to be due to small tetragonal components, probably arising from 28 2+ internal strains. In calcium oxide, V was studied by Low and 29,30 17 Rubins and Azarbayejani and Kikuchi. This ion was also studied 31 32 33 15,16 in Si, ZnO, and the double fluoride KMgF3. Lambe and Kikuchi were first to study V2+ in o -A1203. They produced this ion by X-irradiation of vanadium-doped corundum. As these authors have shown, before irradiation the paramagnetic vanadium ions in A1203 are V3 and a small concentration of V4+. The effect of irradiation is to convert V3 to V2+ Later, accurate measurements of the ESR parameters of V2+ in A1203 was made by Laurance and Lambe. Sturge studied energy levels of V2+ in MgO and A1203 by means of fluorescence and excitation optical spec36 tra. Imbusch and collaborators studied temperature dependence of the optical spectrum (2E->4A2 transitions) of V2+ in MgO. Trivalent vanadium was first investigated in 1958 by Zverev and 37 16 Prokhorov in c -AlOo Lambe and Kikuchi, in a systematic study of

7 various valence states (V2+, V3+, V4+) of vanadium in sapphire, extended the measurements of the angular dependence of the V3+ spectrum to a wider range which was inaccessible to the former authors. Optical investigations of V3 in sapphire was performed by Pryce and Runciman.3 V was also investigated in CdS by Woodbury and Ludwig and in ZnS by 40 Holtono The latter observed both A Ms = 1 and A Ms = 2 transitions with isotropic g-, and A-values. The ESR of V4+ has also been investigated in rutile-type crystals Ti02, SnO2, and GeO2 (orthorhombic cation site symmetry). 41 42 Gerritsen and Lewis and Zverev and Prokhorov studied V4+ in rutile, 43 44 TiO2o Later, this ion was studied in SnO2, and GeO2o In all these crystals, V4+ showed similar ESR spectra with an ambiguity in the sequence of the energy levels. In an attempt to remove this ambiguity, Karavelas and Kikuchi21 have performed molecular orbital calculations for V4+ in TiO2, GeO2, and SnO2o Their calculations indicated that the 2 2 ground electron level was 3d(x -y ). This result was further supported 20 by the ESR investigations of Kikuchi, From and Dorain who observed a large superhyperfine shift (SHFS) in SnO2 V4+ The mechanism of this 22 SHFS was studied by Chen, Kikuchi, and Watanabeo A systematic study of the theory of SHFS interaction of the iron group elements d electrons 23 is made by Chen and Kikuchio Tetravalent vanadium, in the form of unoriented vanadyl complex 2+ VO, has been studied in various liquids, powders, and other amorphous media. o'67a 5 1 ESR spectrum of oriented VO2+ was studied 19 by Borcherts and Kikuchi in zinc-ammonium Tutton salt single crystals. They also showed that VO2+ could be reduced to V2+ by X-irradiationo

TABLE 1.1 g-VALUES AND HYPERFINE COUPLING CONSTANTS OF VANADIUM IONS IN VARIOUS HOST CRYSTALS Hyperfine* Host Frequency Coupling Ion Crystal Band T (~K) g Constants Ref. V2+ MgO X 290 1.9803(5) 74.24(2) 25 2+ MgO X 300 1.980(.5) 74.19(5) 18 2+ MgO X 290 1.9800(5) 74.1 (1) 27 V CaO X 290 1.9683(5) 76.04(5) 29,30 V2+ i X 77 1.9683(5) 76.15(5) V2+ X 20 1.9683(5) 76.22(5) V+ CaO X 300 1.9687 76.30 17 V2+ Si X 1.3 1.9892 42.10 31 ZnO+ X 1.3 (gi = 1.977(1) 46.7 V2+ ZnO X 1.3 i gl 2 -32 V2+ KMgF X 77 1.9720(2) 86.2(2) 33... 3

TABLE 1.1 —Continued Hyperfine* Host Frequency Coupling Ion Crystal Band T(~K) g Constants Refo { gZ = 1.9717(5) A = 86.63(5) sa+Tutton XK 300^ g = 1.9733(5) B = 82.46(5) salt y g g = 1.991 A = 73.538(8) v2+ o~-Al10 X 300 34 2 -2 3 XI0 g1 = 1.991 I B = 74.267(30) V3 oc -A1203 X 4.2 1.98 102 15,16 o A 63 V3+ CdS 10 1.933 39 A = 66 V3+ ZnS X 1.3 1.9433(5) 63.0(1) 40 V4+ o(-A20 3 X 300 1.97 132 16 g l0 1.956 A 142 140 9J Al 110V4+ TiO2 X,K 4.2, 78 glio 1o915 = A 0 31 41 gc = 1.913 I Ac 43

TABLE l1 —Continued Hyperfine* Host Frequency Coupling Ion Crystal Band T (~K) g Constants Ref. gll10 1. 955(1) A^0 141.5(7) V4+ TiO2 X 77 i g10 = 1.913(1) Al 1 30.9(3) 42 {g = 1.912(1) Ac = 44.1(3) 1 -10 1.943 110 144 V4+ SnO X 77 ggi10 1.939 Al =1 21 43 gc =1.903 Ac 44 gl = 1.963(.3) All = 134.36(2) 91.10=1110 V4+ OeO 2 V4+^ Ge02 g = 1.921(.6) Ail = 36.69(1) 44 gC = 1.921(.1) A = 37.54(1) (gx 1.9813(2) (Ax 71.20(4) VO2 Tutton XK 300 g y = 1.9801(2) Ay = 72.44(4) 19 gsalt =Z 1.9331(2) A = 182.8 (5) Absolute values in units of 104 cm1.

CHAPTER II SUMMARY OF RESULTS In this study, electron spin resonance spectra of vanadium are investigated in calcium tungstate single crystals at microwave frequencies of about 9.5 Gc/sec. Some of the samples were also doped with the rare-earth elements Tb and Nd in addition to vanadium. Concentrations of the additives are shown in Table 3olo Prior to X-irradiation no vanadium ESR spectrum was observed. After the irradiation, spectra were seen only at 77~K and 4.20Ko It showed, in all samples, a hyperfine structure of 8 lines corresponding to a nuclear spin of I = 7/2, and effective electron spin of S = 1/2. The observed spectra are attributed to V4+ since a cobalt-doped CaW04 sample (ICo 7/2) failed to show any ESR signalo In general, three groups of lines were identified in the vanadium spectra and labeled as Groups A, B, and Co The intensities of these groups are in the ratio of about 100:5:70 respectively. 1. Group A Spectra At 770~K this group consists of a set of 8 lines, observed in the spectra of all samples, and characterized by an isotropic g- and nearly isotropic A-values. 11

12 g = 2.0245 + 0.0005 Al l 19.1 0.2 gauss Al = 20.3 ~ 0.2 gauss At liquid helium temperature, the (V,Tb)-doped samples did not show any vanadium ESR spectra while the other samples did. This is discussed in detail in Chapter V where we have attempted to show that, due to cross-relaxation between V and Tb, such temperature dependence is expected. At this temperature, the Group A lines were seen to split into four sets of 8 lines which coalesced to two in the ab-plane and to one when the applied static magnetic field was parallel to the c-axis. It was also observed that the average position of these sets coincided with the position of the Group A at 77~K, as shown in Table 4.2. The maximum components of the g- and A-tensors at 4.2~K were found to be gmax - g = 2.044 ~ 0.003 Amax A = 21.3 + 0.2 gauss z 0 600~3, = 30 + 1 where 9 and ~ are the polar and azimuthal angles of the tensors z-axes which is very close to the W-O bond (see Figures 3.3 and 3.4). The values of g and A along the crystal axes were measured; the results are:

13 a b c g 2.033(1) 2.017(1) 2o024(1) A(gauss) 21.2(1) 19.7(1) 19.1(1) This group, with its positive ( g ( A g gg where ge g of free electron = 2,0023) and small hyperfine separation (HFS), is assigned to V4+ at the covalent tungsten site. A preliminary molecular56 orbital calculation by Karavelas and Kikuchi produced results in excellent agreement with the experimental g-value at 77~Ko For the low-temperature splitting of the Group A, we have tentatively adopted a model analogous to that proposed by Watkins and 70 Corbett for the interpretation of similar temperature dependences of the ESR spectra of the silicon E-center. Before X-irradiation, the vanadium at the W site is in the diamagnetic V04 form. Due to the irradiation, it acquires an electron from one of the oxygen ligands to become VO4, a paramagnetic complex ion with effective spin S - 1/2. The fact that gz is in the direction of W-O bond lends some support to this model. Now, suppose the unpaired electron jumps randomly from one oxygen ligand to another in V040 Thus, there are four non-equivalent sites available to it. At 4. 2~K the jumping rate is slow, the life time of the spin at each of the four sites is long, and according to Gutowsky and Saika, we expect to see distinct resonance spectra from each center. As the temperature is increased, the life time t of the unpaired electron in each orientation becomes shorter and lines become broader until eventually the resonances from the four centers are not

14 distinguishable. At still higher temperatures (near 77~K), t becomes so short that motional narrowing takes place with emergence of one set 72 of lines at the average position of the four. 2. Group B Spectra This Group was seen only in the spectra of the samples which contained a rare-earth ion besides vanadium. It is composed of two sets of 8 lines which coalesce to one when the applied static magnetic field is parallel- to the c-axis of the crystal. The ESR spectra arise from two non-equivalent sites with isotropic A- and axial g-tensors: Ajl = AL 9.1 + 0.2 gauss g! = 2.068 0.001 g, = 2.004 ~ 0.001 The symmetry axes of the two g-tensors are in the ab-plane and make angles H - -10 and = 80 with the a-axis. This group, similar to the Group A, is also attributed to V4+ at W site, but with some reduction in the site symmetry to account for the observed anisotropic spectrum. The cause of this symmetry reduction is not quite understood. It may well be due to some local imperfections in the crystal. The fact that they did not appear in the spectrum of some newly prepared samples provides some support for this view. Due to its low intensity and power saturation effect, this group was not easily observable at 4.2~K.

15 3o Group C Spectra This group was observed only in the ESR spectra of the samples which contained only vanadium or an excess of vanadium over the rareearth dopants(e.g., CaWO4:V, Nd, 1.0%, 0.1% concentrations in the melt) The spectrum at 4.2~K is similar to that at 77~K except for some line deformation and broadening due power saturation at the lower temperature. The Group C is characterized by anisotropic spectra consisting, in general, of four sets of hyperfine lines which coalesce to two in the ab-plane and to one when H is parallel to the c-axis, This shows that the ESR spectra arise from four non-equivalent sites. The principal values and directions of the g- and A-tensors for these centers are shown in Table 4.3. Because of its negative A g and large HFS, this group is assigned to V4 in the ionic site in CaW04, i.e., the calcium site. In fact, as Table 1.1 shows, the g- and A-values of V4+ and other valence states of vanadium in ionic bonds are in the range of the values we have 4+ found for the Group C. Also, the assignment of V to the Ca site is supported by the results of our crystal field calculations shown in Table 2o.1 4o Conclusions Correlating the ESR results of the three groups, A, B, and C, we have concluded that: 1) As mentioned earlier, vanadium enters the calcium tungstate lattice in pentavalent oxidation state which is diamagnetico As a

16 TABLE 2.1 EXPERIMENTAL VERSUS CALCULATED i g. FOR THE GROUP C LINES Crystal Field g-shift I Experiment Calc. a^~y 1 ~<0 <0 93, < o <o a!3 <o <o A9/a<~ >1 >1 /\9/ngy ^ >1 >1 (ks + ) 4.4~ 1.2 8 result of X-irradiation, it is converted to paramagnetic V4+ with one unpaired electron. Similar reducing effect of X-ray on vanadium ions 16,18,19,26 has also been observed by others. 2) In samples where the rare-earth ions Tb or Nd are present, V5 ions occupy the W sites while the rare-earth ions, as trivalent ions, enter the Ca sites. This scheme also satisfies the required electrical neutrality of the crystal. 3) In samples where the vanadium ions are the only intended dopants, V5+ ions occupy both the Ca and W sites. We propose the follow5+ 6+ ing model for charge compensation: Four V ions occupy four W sites while two other V ions substitute two Ca2+ ions and one calcium vacancy is also generated. According to this scheme, the intensities of

17 the Groups A and C must be in the ratio of 2:1. This is in agreement with results of our quantitative measurements (using CuSO4 as an intensity standard) within + 20% which is the usual error limit in the ESR quantitative measurements of this nature. Also, according to this scheme, every two vanadium ions at the Ca sites are coupled to a nearest calcium vacancy. The direction of this coupling makes an angle of 42~, 40' with the c-axis in the ac- or bc-plane (see Figure 3.2). This is also in good agreement with the direction of the common z-axis of the g- and A-tensors for the Group Co A number of temperature annealing and growth rate experiments were carried out in an attempt to see if there exist. some electron transfer processes between the paramagnetic centers of the A and C spectra. From the results, it appears there is no simple charge-exchange correlation between the two centers. Rate of increase of the paramagnetic vanadium due to X-irradiation as a function of the irradiation time.

CHAPTER III EXPERIMENTAL PROCEDURE AND CRYSTAL STRUCTURE 1. The ESR Spectrometer The ESR spectroscopy of the CaWO4 single crystals was achieved at 770K and 4.2~K at X-band frequencies (~9.5 Gc/sec.) using a Varian 4012-35 12" rotating electromagnet, a V-153c Klystron, a cylindrical cavity operating in the TE011 mode, and a balanced bridge homodyne detection. Data at 4.2~K were obtained at 60-70 db below klystron power level (300 rw) using superheterodyne detection. The microwave cavity was made of lavite, coated with the thermal silver paint Hanovia 32A on the inside and electroplated at 30 ma for about one hour. The static magnetic field was measured by means of a proton probe connected to a Varian Fluxmeter Model F-85. A Beckman Transfer Oscillator 7580 and Beckman Universal EPUT and Timer 7380 were also connected to the fluxmeter for more accurate measurements. Magnetic field measurements were made at a point just underneath the cavity. The measurements were corrected for the distance between the sample and the magnetic probe, using the ESR signal of a small DPHH* standard glued to o, o< -diphenyl P -picryl hydrazyl, (C6H5)2N-NC6H2(N02) 3 18

19 the sample. The klystron frequency was measured using a Hewlett-Packard K-532A absorption type wavemeter, This device, with divisions to o.010 Gc, allowed easily the frequency measurements to be extended to 1/4 of a division, i.e., to 0,0025 Gc/sec, For more accurate measurements the transfer oscillator was usedo 2. Sample Preparation and Composition Calcium tungstate crystals used in this study were grown at the Harry Diamond Laboratory, Washington, Do Co, using the Czochralski tech10 nique as employed by Nassau and Broyer. Vanadium and rare-earth dopants were added to the melt in the form of their most stable oxides, i.e., V205 and (RE)2030 To produce paramagnetic centers, the samples were X-irradiated for about fifteen minutes by a tungsten target Machlett AEG-50 S X-ray tube with beryllium window, The tube was operated at 50 Kvp x 35 mao The dose rate at the location of the sample was about 6.Ox106 rad/min. The concentration of the paramagnetic vanadium was found to reach about 95% of its maximum value after fifteen minutes of irradiation. The maximum paramagnetic conversion ratio was about 6% of the vanadium present in the sample. This was measured by comparing the vanadium ESR signal intensity to that of a standard CuSO4 single crystal. Figure 3.1 shows variation of the concentration with the irradiation time, The vanadium concentrations in the samples were determined for us, using arc discharge spectroscopy, by the Physics Instrumentation

20 100 > 80 I- z w H 60_ D / LL 40 2O 20 / 0 2 4 6 8 10 12 14 X-IRRADIATION TIME, min. Fig. 3.1. Conversion of Vto Vby X-irradiation.

21 Laboratory, Department of Industrial Health, of the University of Michigan. The results are shown in Table 31oo TABLE 31ol ATOMIC CONCENTRATIONS OF VANADIUM AND RARE-EARTH (RE) IONS IN CaWO4 SAMPLES In Melt In Crystal V/Ca RE/Ca V/Ca RE/Ca Sample (70) (70) (7o) (o) CaWO4: V 0.5 0 0.05 0 CaWO4: VTb 1 1 0.13 0.3 CaWO4 ~ VNd 1 1 0o13 0.2 The concentrations in melt were given to us by the Harry Diamond Laboratory. The last column in the table shows the rare-earth concen10 trations estimated from their distribution coefficients K C /C c m where Cc and Cm denote concentrations in the crystal and in the melt, respectively. ESR spectrum of Mn2+ was also detected in all CaWO4 samples. In some cases, the intensity of the Mn2+ signal was comparable to that of the vanadium. 3, Crystal Structure and Orientation The structure of calcium tungstate (Scheelite) is characterized by the space group C h or the tetragonal 141/a with four molecules in 53 the unit cello Recently, the dimensions of the tetragonal body

22 centered unit cell of this crystal were determined thoroughly by 54 55 neutron and X-ray diffraction techniques. The results obtained by the two methods are in excellent concordance within experimental error. In our calculations of angles and distances, we have used the X-ray diffraction data given below. Unit cell dimensions: a = b=5.243 ~ 0.002O c=11.376 ~ 0.003X Oxygen coordinates, using a W atom as the origin: x=(0.2415 + 0.0014)a y=(0.1504 + 0.0013)a z=(0.0861 ~ 0.0006)c The site symmetry of both Ca and W is S4. The structure of 9- 9-2+ CaWO4 may be viewed as composed of WO- anions ionically bonded to Ca2 cations. The metallic sites are found on planes separated by a distance of c/4 from each other and perpendicular to the c-axis, as shown in Figure 3.2. The symmetry configuration of metal ion sites on planes at c/4 and 3c/4 are mirror images of those on the ab- or c/2 planes with respect to a (001) plane. The lines joining each metal ion with the nearest metal ions in the adjacent planes make an angle 42~, 40' with the c-axis. Each tungsten ion in CaWO4 crystal is bonded to four oxygens to form a slightly squashed tetrahedron with dimensions 2.11 x 2.11A x

23 0O lo90A and W-O distance of 1o78Ao The diagonal plane of the tetrahedron makes an angle of 31054' with the a-axis. The W04 tetrahedron and its projection on the ab-plane are shown in Figures 3.3 and 3.04 The calcium atoms, on the other hand, are surrounded each by eight oxygens which form two distorted tetrahedra with Ca-O distances of 2.44A and 2,48A. These tetrahedra and their projections on the ab-plane are shown in Figures 3.5, 3o6, and 3.7. Crystal orientation and alignment were made by X-rays. To place the samples in the resonator cavity, they were glued to the end of thin wall quartz tubings.

24 3c/4 i^A~~ <.f11. 376 A I ^\ I c/2 42 40' \ c/4 of 5.243A A ~ W-ion 0 Ca-ion Fig. 3.2. Metal ions in CaW04 unit cell.

25 56071 0 ~1/ —--- \. 96 A 211A " ~ 2. 11 A W Fig. 3.3. (W04)2 tetrahedron. =0 b Fi..31 54' a Fig. 3.4. Projection of (W04)2 tetrahedron on the ab-plane.

26 C 66 45.......1. 96 A Ca.. P, / _ 3.3.23 A Fig. 3.5. (Ca, O) "long bond" tetrahedron. C 3.73 A Ca ~~o /;22A --- 3.2.6 a22 A --— ~ ^>'/0 Fig. 3.6. (Ca, O) "short bond" tetrahedron.

27 * {I \ Ca 59~48', -, a o Ca-ion C0) Projection of oxygen ion on the ab-plane Fig. 3.7. Projection of the (Ca, 0) double-tetrahedron on the ab-plane.

CHAPTER IV EXPERIMENTAL RESULTS Electron spin resonance absorption spectra of vanadium-doped CaW04 single crystals, with or without the rare-earth dopant Tb or Nd, were studied at microwave frequencies of about 9.5 Gc/sec ( A 3cm). Concentrations of the additives are shown in Table 3.1. The only spectra observed, before X-irradiation, were those of Tb3+ or Nd3+ (seen only at 4o2~K) and the unintended Mn2+ impurity. After the irradiation, new groups of lines were detected at liquid air (77~K) and liquid helium (4o2~K) temperatures. The resonant magnetic fields were in the range of about 3-4 kilogauss corresponding to a spectroscopic g-factor in the vicinity of 2. The additional absorption spectrum consisted, essentially, of sets of 8 lines, of almost equal separations and intensities, thus suggesting a hyperfine structure of (21 + 1) with I = 7/2. Since, besides vanadium, cobalt has also a nuclear spin of 7/2, a sample doped with this element was examined but the additional lines failed to showo Hence, the new spectra were attributed to vanadium. In general, the position of the spectral lines showed a dependence upon the direction of the applied static magnetic field with respect to the crystal axes. From the angular dependence, we have 28

29 classified the vanadium spectra in three groups which will be referred to as the Groups A, B, and Co The intensities of the three groups are in the ratio of 100: 5: 70, respectively. The Group A, sharper than the other two, was seen in the spectra of all samples. At 77~K, it consists of 8 isotropic lines, separated by about 20 gauss with a positive A g (Figures 4,1, 4,2)o At 4.2~K, and for a general direction of the magnetic field, each line of the Group A splits into 4 lines as we shall see later. The Group B lines were observed only in the spectra of the samples which contained, in addition to vanadium, either Tb or Nd as shown in Figure 4.lo They are characterized by an isotropic hyperfine separation (HFS) of about 10 gauss and an axial g-tensor with positive A g. The Group C was seen in the spectra of CaW04 crystals which contained only vanadium or an excess of vanadium over neodymium. This group has a large HFS varying from about 36 to 176 gauss and an anisotropic g with negative A g (Figures 4.2 and 403)~ lo Group A Spectra At liquid air temperature, this group consists of 8 lines with an isotropic g = 2.0245 + 0.0005. The hyperfine coupling constant A, however, was found to be slightly anisotropic: A l = A 19ol ~ 0.2 gauss = (18o2 + 0~2) x 104 cm1 Al = B = 20.3 + 0.2 gauss = (19.0 ~ 0.2) x 10-4 cm-1 where A Il was measured along the c-axiso The experimental data at 77~K can be described by the spin Hamiltonian (see Appendix A),

................,............1................:::........... ~ i:::::.,...................................I.........i............ I..................................................... 0 I.. K..... K 2 1 I~~~09 us o I 6 oz I I i 177 LA r~~~~~~~~~~~~'':..... i......=... [~.~.i. ~.I. I~ 00 ~-.................................. C rn~~~_ "';-~ ~~ ~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~................ I.....'.~i~~~~l~~~~l-~~ ~. o n. -=-I................................,.........::, ~~ ~ c —-— ~ *I~~~~~~~~~~...... 0 001 09 v-: I0?;,J~K',~:,{7 7i.7QL 71Io 0 ~ ~.................... ~.............,....................' —~..................... 0' =............. "'. ~' S ~-~I~~~~I~ —'~....."''''"w: I,': I:~'... ":,'~;-~ I~~~- ~~~ (1~~~~~~~~~~~3 I ~~~~~~~~~~~~~~~~~PI b~~~~~~~~~~0 I or I NA ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~........ "".:'!t:'!~:, O rrr o;~........1............................ —:-........ ~.'-.'i: i:'i':i~~~~~~~~~~~~~~~~~~i ii.....~ ~~~~~~~~............ "~':'',...: c~ I 0 I I I I.................. an oy..I.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~.......

cm~~ ~~~~~~~~~~~~~ ~~~~~~..................................... I~.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~.......p~~~~~~.~ - T~ ~........" Xrl~~~~~~~~~~~~~~~~~~~~~~~ _, ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~f OE izd I ~l A..- I: ~~~~~~~~~~~~~~~~~~~~~............. 0 o I ~~~~I ),.,-'0'' I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ------,n 03 o m m c Xr1.o o L rt ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~ i-A.~) 9D: i K 0j -~~~~~ 3L~~~~06 08 GOLB 9 o'. rio.(,.4,4j

32!i..'............:,"...t-... _... I. i i4. I; — I.-.;.,,e,'.)............. I.,<..'~ _,i...........'-'..... ~,, [,,,'.... -.:.....'*'~..... *'t-.,L. T~...r..........., 10 20 30:, 1 - or.c,,. 0,4-;:. 1 1...[. *5.l-. —:*..........ze......,,~ * Cd) o...........:....,.I, - i 1 - I:' I i- * I j.....................:.....::.. 1-. 1.;.':::........... "l.....4 Q-4 -. [.....,.-,,~~~~~~~~~~~~~~~.~,::'*: "114.:; i' ~.:.','i,~ ~ ~:.....,I]i:i:'! ~i~.................:...':........',::..ii,.'i-+ L.X ~.;......

33 H3 = s+ A3 5 T 14 I ( I + ~ s ) (4.1) with S - 1/2 and I = 7/2. The position of the resonant magnetic field for transitions (M- >M-1, A m = O), obtained from (4.1), is'_ 4 j Y \ 2 2 2 - 3)7H, ( A< )51n s cos (9 (4.2) where, = Bohr magneton m = nuclear magnetic quantum number M = electron spin quantum number Ho h 2 /gp h h = microwave photon energy K = A cos + B sin e and 9 is the angle of the applied magnetic field with the symmetry axis, in our case the c-axis. For H Ij c-axis, 9 = 0 and we have, H (o,)- d- n ( )- (4.3) and in the ab-plane, H^ (rrT/a H. L _-3 i

34 TABLE 4.1 CALCULATED AND MEASURED VALUES OF THE RESONANT MAGNETIC FIELD FOR THE GROUP A. T = 77~K, H PARALLEL TO THE c-AXIS..... Resonant Magnetic Field (Gauss) m Measured Calculated - 7/2 3430.3 _ 0.5 3430.2 - 5/2 3410.8 3410.8 - 3/2 3391.6 3391.4 - 1/2 3372.5 3372.2 + 1/2 3353.5 3353.1 + 3/2 3334.0 3334.1 + 5/2 3315.7 3315.3 + 7/2 3296.8 3296.5

35 Table 4.1 shows a comparison between the measured and calculated values of the resonant magnetic field when H is parallel to the c-axis. The line width was found to depend on the direction of H. This angular dependence was more pronounced for the RE-compensated samples as shown in Figure 4.4. At liquid helium temperature, the Group A splits, in general, into four sets each of which consists of 8 lines with positive A g and hyperfine separations of about 20 gauss. They coalesce into two sets in the ab-plane and into one when H is parallel to the c-axis. Figure 4.5 shows the low-temperature splitting of this group. These sets largely superpose each other for all orientations of H: the maximum separation between the centers of the lowest and the highest lying sets in the abplane spectrum is only about 40 gauss, whereas the total separation between the first and the last lines of each set is about 140 gauss. The measurements of g and A in different planes indicated that the maximum values of these tensors occurred when H was in the direction given by the polar coordinates B 60" + 3, 30~ + 2 gmax - 2.044 + 0.005 Ama = 21.3 +- 0.5 gauss This is, within the experimental error, the same as the direction of the W-O bond in calcium tungstate (see Figures 3.3 and 3.4). The g- and Avalues along the three crystal axes are

36 CaWO4 v, Tb 1a5 and CaWO4: V, Nd ( at 1% of each ) CaWO4: V 10.0 (atO,5%) 2 I7. 5 z -I 5.0 Z51 0 10 20 30 40 50 60 70 80 90 H IlL Fig. 4.4. Angular variation of the line width of the Group A spectra in the ab-plane at 77~K.

37 0-~r - -^ -'~~~ ~ ~ i.. - & 1Ci" r~~-~- -_ ~m - ~ _-^- ~\~ - -: ~ l ~: 4 - ~-~ I L_ L T L0AwlLFF1FT 1~ ~! -' i - 133.9 gauss \ 3225.3gauss'.. 3359.2 gauss 4 t I KI t an a I:i.45 Iu I ne t 7i 7 I$ t; Ia a 4.2 Ibto i ji III - -' —- 133. 9gauss I! F —-q- - -+ -1 _Fig.~ 4.5.IGroup A i a n i C 2i 1rr rI

38 a b c g 2.033 2.017 2.024 A(gauss) 21.2 19.7 19.1 The angular- dependence of the g-factor in the ab-plane and in the plane defined by the c-axis and the azimuthal angle -= 30~ is shown in Figures 4.6 and 4.7, respectively. It was found that the average position (defined by <g> and <A>) of the four sets, which emerge from the Group A at 4.20K, was the same as the position of this group at 77~K. This is shown in Table 4.2 for H in the ab-plane. In this plane, as we have seen, the four lowtemperature sets reduce to two. Subscripts 1 and 2 are used to designate g- and A-values of these sets. We notice, in this table, that <g>2oK - 2.025 + 0.001 <A>4.2~0 20.3 + 0.1 gauss which agree well with the values found at 77~K. 2o Group B Spectra At 770~K this group is composed, in general, of two sets of 8 lines which collapse to one when H is parallel to the c-axis. The relative positions of the Groups A, B, and the DPPH absorption line are shown in Figure 4.8. The angle dependence of the g-factor for the two sets in the abplane were the same except for a 90~ phase difference as shown in

39 TABLE 4.2 AVERAGE VALUES OF g AND A OF THE TWO SETS EMERGING FROM THE GROUP A AT 4.2~K IN THE ab-PLANE IS MEASURED FROM THE a-AXIS <g> <A> CPgl g2 (g1+g2)/2 A1 A2 (A1+A)/2 0 2.007 2.033 2o025 19,7 21 20.3 10 2.04 - - 19.9 - 20 2.012 2.039 2.025 20,0 20.7 20.4 25 2.011 2,039 2.025 20.0 20.7 20.4 30 2.011 2.039 2.025 20.1 20.6 20.4 40 2.012 2.038 2.025 20.3 20.5 20.4 50 2.012 2.036 2.024 20.2 20.2 20.2 60 2.018 2.030 2.024 20.4 20.0 20.2 70 2.024 2.026 2.025 20.9 19.8 20.3 80 2.029 2.022 2.025 21.0 19.8 20.4 90 2.033 2.017 2.025 21.2 19.7 20.5

40 2.050 2.040 2.030 2.020 2.010 X x 2.000 ~I I I I I 10 30 50 70 90 8 Fig. 4.6. Angular variation of the Group A g-value at 4.20K. H varies in a plane containing the c-axis and making e = 30~ with the ac-plane.

2. 0400 2. 0300 - /~ ^ 2. 0200 2. 0100 I I I I I I~~~1 L I 0 10 20 30 40 50 60 70 80 90 Fig. 4.7. Angular variation of the Group A g-value at 4.20K. H varies in the ab-plane.

42 Figure 4.9. The g-value remained unchanged in the planes which contain the c-axis and make an angle c = -10~ or - = 80~ with the a-axis (Figure 4.10). This suggested that the g-tensors for the two sites possessed axial symmetry. The symmetry axes are given by the polar coordinates ( -1 = 90~, i = -10~) and ( l0 = 90~, 8 = 80~). The principal values for the g-tensors are, g = - 2.068 ~ 0.001 g -2 2.004 + 0.001 The hyperfine separation was found to be isotropic within experimental error. Its value is A - A1 9.1 + 0.2 gauss Hence, the experimental data can be described by an axial spin Hamiltonian with, A =A =A A x y z Thus, =( H$S-) + I(4.5) with an effective spin S = 1/2 and nuclear spin I = 7/2. The position of the resonance absorption lines, obtained from (4.5), is given by 2 H= H-l~A A _ (G 2- ) (4.6) a i(gp^ 4

DPPH I - - low field lines 141.4 3298. 7 gauss Intense lines 3440. 1 gauss high field lines H Fig. 4.8. Relative position of the Group A, B, and the DPPH line at 770K.

9 2.068 2.064 2.060 / low field set 2.056 - \ high field set 2. 052 2.048 2. 044 z 040 - 2. 036 2.032 2.028 2. 024 / 2.020 2.016 2.012 2.008 2.004 t 0 10 20 30 40 50 60 70 80 90 100 110 (, measured from the a-axis Fig. 4.9. Angular variation of the g-value for the Group B lines _-C T.T. r T N.'TT + 1.^ --- 1, +- -7-70 7

2. 068 - 2. 064 K 2. 060 2.056 - 2.~/~ 056 *~~ low field set 2. 052 / a~~~~~~/ 052x high field set 2. 048 2. 044 2. 040 2. 0362. 032 - v 2. 028 - 2. 024 2 020 2.0162. 012 2 008 2. 004. ^ K > x x -10 0 20 40 60 80 90 100 105 G measured from the c-axis Fig. 4.10. Angular variation of the g-value for the Group B lines of CaWO4:V, Tb at 77~K. H varies in a plane containing the c-axis and one of the g-tensor's principal axes.

46 where, Ho j -j/0y, and 2 9 4l eas @ + gL S _ (4.7) where 9 is the angle of the static magnetic field with the symmetry axis. Figure 4.11 shows that the angular variation of the g-value measured in the ab-plane are in good agreement, within the experimental error, with the values calculated from (4.7). At 4.2~, the spectra of the B and C groups showed some line broadening and deformation due to power saturation. This made the identification of the low-intensity Group B very difficult. 3. Group C Spectra The spectrum at 4.2~ was similar to that at 77~ except for the above-mentioned saturation effect at the lower temperature. All the measurements were made at 77~Ko The Group C, in general, consists of four sets of 8 lines which unite to two sets of the ab-plane and to one when H is parallel to the c-axis as shown in Figures 4.2 and 4.3. The spectrum, arising from four non-equivalent sites, is characterized by anisotropic g- and A-values, with a negative A g and a relatively large HFS varying from about 36 to 176 gauss. Figures 4.12 through 4.15 show the angular variation of g- and A-values in the ab- and ac-planes. The two tensors have rhombic symmetry with only one coinciding principal axes chosen as the z-axis; x- and y-axes of the g-tensor seem to be rotated by about 30~ in the counter clockwise direction with respect to the corresponding axes of

47 2.080 0 Measured x Calculated 2.060 2 040 2.020 2.000' I I I I 0 20 40 60 80 100 Fig. 4.11. Calculated versus measured g-values of the Group B in the ab-plane at 770K.

2.000 1. 990 1. 980 1. 970 1,920 - 1.910 - 1.900 -45 -35 -25 -15 -5.0 5 15 25 35 45 55 65 75 85 95 105 115 125 135 Fig. 4.12. Angular variation of the g-value for the Group C in the ab-plane at 770K.

140 130 120 110 100 90 80 70 60 50 40 j 30 -45 -35 -25 -15 -5 0 5 15 25 35 45 55 65 75. 85 95 105 115 125 135 0, MEASURED FROM [100] DIRECTION. Fig. 4.13. Angular variation of the HFS for the Group C in the ab-plane at 770K.

2. 000 1. 990 1.980 1. 970 1.960 1. 950 1. 940 1. 930 1. 920 1. 910 1. 900 0 10 20 30 40 50 60 70 80 90 OMEASUREDFROM [001] DIRECTION Fig. 4.14. Angular variation of the g-value for the Group C in the ac-plane at 770K.

51 180 170 160 150 140 130 110 100 90 \ 80 70 60 50 40 3 0 10 20 30 40 5 60 70 80 90 0, MEASURE FROM THE (001 ) DIRECTION Fig. 4.15. Angular variation of the HFS for the Group C in the ac-plane at 770K.

52 the A-tensor, as shown in Figure 4.16. o and A are the minimum and maximum values of these tensors. The large difference between the magnitude of these and the x, y components made the accurate measurements of the former possible. In contrast, the x and y components were difficult to determine experimentally due to the admixture of the four sets of lines and strong second order effect in the xy-plane. Approximate values and directions of these components were calculated (by computer) from the measurements in the ab- and ac-planes, using relationships g = 9 ) i ( nh + V 3 A = A1 -+AYZ 1 + 4n2 =A Ym. +'IJ Y\3 where ni and mi are direction cosines of g and A with respect to the tensor's axes. The results are shown in Tables 4,3 and 4.4. In the tables, 9 and < are polar and azimuthal angles of the tensor's axes. Indices 1 to 4 are used to designate the ~ coordinates of the tensors for the four unequivalent sites. The Group C lines can be described by a modified rhombic spin Hamiltonian of the form, x 3 H,= 3 LHet'H+9 HX s + YYSyl+Ail x x x x by S+ Fa jC I (4.8) where the last term is added to the usual rhombic spin Hamiltonian to account for the non-coinciding x and y axes of the g- and A-tensors, 19 The constants Bx and By and Fij are given by

53 2 2 B = A cos 2 + A sin 2 x x y 2 2 B = A sin + A cos 2 y x y F = F = 1/2(A -A ) sin S cos (4.9) xy yx x y where 3 = 30~ is the angle between x- or y-axes of the g- and Atensors and A, Ay, and A are the principal values of the hyperfine x y z coupling constants. Putting experimental data into (4.9), we obtain for the constants of the spin Hamiltonian (4.8) B = 42 x 104 cm -4 -1 B = 57 x 104 cm y Fij = F.. = 0.217(A.-A), i,j x,y,z ij 3z z

54 To a nearest C2 Ca 42 69 / a Az=165 / 220 = 18002 Pa46 \Gru C g I \ r\ \ I I] - i^ /^^ ^x = I 0 0 0 = 51 6 = 44" p= 2200 = 180' Fig.4.16. Group C, g- and A-tensors.

55 TABLE 4.3 PRINCIPAL VALUES AND DIRECTIONS OF THE g-TENSOR FOR THE GROUP C LINES T=77 K gx gy gz 1.990 1,965 1.901 + 0.001 S 690 51 46 + 1 t 112" 220 0+1 203 310 90 + 1 P3 ~293 40 180 + 1 +c 23 130 270 + 1 TABLE 4.4 PRINCIPAL VALUES* AND DIRECTIONS OF THE A-TENSOR FOR THE GROUP C LINES T. 77~K Ax- Ay Az 34 65 165 + 0.3,I.....,.. 0 6 90 44 46 + 1 <,, 90 180 0+ 1 tc 180 270 90 + 1 3 270 0 180 + 1 0 90 270 + 1 In units of 10' 4 em1.

CHAPTER V DISCUSSION AND CONCLUSIONS 1o Site Occupancies of V and Re Ions in CaW04 In the previous chapter we found the following correspondance between the doped calcium tungstate samples and the observed ESR spectra: Sample Spectra CaWO4: V Group A (60%) + Group C (40%) CaWO4: V, RE Group A (9570) + Group B,(57o) where RE represents Tb or Nd ions and the % shows the percentage of the total spectral intensity. From the ESR results, presented in Chapter IV, we draw the following conclusions concerning the paramagnetic centers of these groups. a) It was seen that before X-irradiation the samples did not show any vanadium ESR spectrum. On the other hand, after the irradiation spectra arising from V4+ ions were detected in all samples. Hence, it is reasonable to assume that before the irradiation the vanadium in CaW04 is in the diamagnetic pentavalent oxidation state V5+ Then, as a 56

57 result of the irradiation, it is reduced to paramagnetic V4 ion with S - 1/2. In fact, the reducing effect of X- or -rays on vanadium 18,26 ions has been observed in vanadium-doped magnesium oxide, aluminum 16 19 oxide, and Tutton salt. Also, we could not detect V3+ ions in the samples before the irradiation. b) The Group C lines, in the spectrum of the uncompensated CaWO4:V samples, arise from the V4 ions at ionic Ca sites. One evidence for this assignment is the negative A g of this group: This is generally observed for ions at ionic sites as shown in Table 1.1. Crystal field calculations for V4+ in a Ca site result in the following energy levels (see Appendix B). 3____ > 5.2 Dq?//- a; ~3.6 Dq 2D(5) -3 Dq -11.1 Dq where, r3 4t';Lo r/ ~~

58 a) + 3 ),_ (,_ 3 ) P ^T ^~^+ + f Using this energy scheme, we have calculated the shift of the g-factor. In the z-direction, we have n,8 A gz = 8 8.1 Dq and for the x or y components, -2 -- _ A gx=-a gy 16.3 Dq where the spin-orbit coupling coefficient A is positive (less than half filled shell). The calculations not only predict a g <o but also that Or, in fact, A9P/ 98 Experimentally, we have found Z g = - 0.1013 + 0.0005 z g = -0.0123 + 0.005 x - 0.033 a g~ = - 0.0373 + 0.005

59 Thus, a 93 /Z 9, = 10 + 4 9/ =2.7 + 0.4 The agreement between the calculated and experimental results is good when we consider the approximate nature of the former and the relatively high error in a g measurements for g-values close to ge' The reason why the theoretical calculations make no distinction between gx and gy is that we have assumed perfect S4 symmetry at the paramagnetic ion site in our crystal field calculations. In fact, one would expect the site symmetry to be disturbed by charge compensation mechanisms specially in the case in point (the Group C lines) where vacancy compensation is operative. Such a perturbation can cause splitting of the degenerate 13,4 level. It is easy to show that if we assume new wave functions for these levels such as, then in general Z/ gx A gy. 4+ Another evidence for associating the Group C with V4+ in calcium site is the large HFS of this group. In Table 5.1 average values of the g- and A-tensors for V4+ ions in several host crystals are shown. As we observe, <A> for the Group C lies between those of V4+ in the rutile structure crystals and in the zinc-ammonium Tutton salt which

60 TABLE 5.1 AVERAGE VALUES OF V4+ g- AND A-TENSORS IN SEVERAL CRYSTALS Ion Host Crystal < R > < A >* V4+ TiO2 1.928 72 V4+ Sn2 1.928 69.7 V4+ GeO2 1.935 69.5 VO2+ Tutton 1.9648 108.8 Salt V4+ CaW04 1:952 88 (Group C) In units of 104 cm1. is a strongly ionic crystal. Hence, we conclude that the C spectrum originates from V4+ at ionic calcium sites. c) Judging by the spectrum alone, the Group A may be associated with V4+ at either a W or a Ca site both of which possess a S4 symmetry. But this group, in contrast with the Group C, has positive L g and small HFS, and thus it cannot be related to a Ca site. Also, Karavelas 56 and Kikuchi have made a preliminary molecular orbital calculation for vanadium orthovanadate V0O which is converted to VO4- upon X-irradia4. 4 tion. Allowing for a charge compensation correction (for the extra electron at the orthovanadate complex), they anticipate g II = 2.0268 and gL = 2.0231 which give <g> = 2.024 in excellent agreement with the experimental value g = 2.0245. Hence, the Group A is assigned to V4+ ions in W sites.

61 d) The Group B lines, with positive A g and small HFS, are also associated with a W site with some reduction of the local symmetry. Since the g-tensor principal axes do not bear any relationship with bonds directionsin the crystal and also because of the low intensity of its spectrum, this group is believed to arise from local imperfections in the crystal. This view is further supported by the fact that crystals prepared at some later time did not show the Group B lines. From the above conclusions we now propose a model for the charge compensation in the impurity-doped CaWO4o It was seen that in the RE-compensated samples, vanadium (as V5+) occupies the W6 site. Thus, a simple and reasonable scheme for the charge compensation is for the RE ions to enter Ca sites in trivalent oxidation state. This agrees well with the substitution model of 13 5+ 5+ Nassau and Loiacono who used Nb, instead of V for charge compensator in their Nd-doped calcium tungstate. Also, substitution of W6 (in WO ) by V5 is chemically possible for VO' is a complex vanadium O 4 4 69 ion, actually known to exist. Finally, in occupation of lattice sites by impurity ions, the match between the sizes of the substituent and the 13 substituted ions is an important factor. Table 5.2 shows how well the ionic radii of V5+ and Nd3+ (or Tb3+) compare with those of W6 and Ca2+ 57 which they replace. The case of CaWO4:V samples, however, is differento Here, vanadium is the only impurity ion in the lattice, and thus contributions to the charge compensation must come from the vanadium ions itself; vacancies might also be generated. In fact, the anisotropic spectrum of

62 TABLE 5.2 IONIC RADII OF METAL IONS AND THEIR SUBSTITUTES IN CaWO4 Ion W6+ Ca2+ V5+ Nd3+ Tb3+ Ion W Ca V Nd Tb Ionic radius (A) 0.62 0.99 0.59 1.03 1.00 the Group C is a strong evidence for possible existence of vacancies in this crystal. Hence, we suggest the following model for the charge compensation in this sample. Four V ions enter four W sites while two other V ions occupy two Ca2+ sites and one calcium vacancy is also formed. Then, upon X-irradiation, V5+ ions at Wand Ca sites are converted to V4+ which give rise to the Groups A and C spectra, respectively. According to this model the intensities of the Groups A and C must be in the ratio of 2:1. This is in agreement with the experimental results within 20%o, the error limit of ESR quantitative measurements. Also, the model suggests that each pair of vanadium ions at calcium sites are coupled to a nearest calcium vacancy. The direction of this coupling, as shown in Figure 3.2, makes an angle of 42~ 40' with the caxis in the ac- or bc-plane. This is in good agreement with the direction of the common z-axis of the g- and A-tensors of the Group C (see Figure 4.16). A number of temperature annealing and growth rate experiments *Conversion rate of V5+ — V4+ as a function of X-irradiation dose.

63 were carried out in an attempt to see if an electron transfer process existed between the ESR centers of A and Co From the results, it appears that there is no simple correlation between the two centers. The above observations lead us to the following conclusions on site occupancies of the vanadium and RE ions in CaW04. i) In the charge-compensated CaW04~V, Tb and CaWO4~V, Nd samples, vanadium ions, in V5 oxidation state, occupy W6 sites while the trivalent rare-earth ions enter Ca2+ sites. ii) In the uncompensated CaWO4:V sample, V+ ions occupy both W6 and Ca2+ sites. In this case, each pair of vanadium ions at calcium sites are coupled to a nearest calcium vacancy. 2. Line Width and Temperature Dependence of Vanadium Spectra For many years, line shape and relaxation of paramagnetic signals have been the subjects of extensive experimental and theoretical studies. In spite of this, still in most cases only a qualitative analysis of experimental data is possible. In this section, we too. will attempt to interpret qualitatively our observations of the line width and the temperature dependence of the Group A spectra on the basis of relaxation phenomena. A brief description of these processes and the related formulas, which are going to be used in our description, are presented in Appendix Co For convenience, the following notations will be used in reference to the samples under consideration:

64 S1 for CaW04 V S2 for CaWO4:V, Nd S3 for CaW4:V, Tb We shall also express the full width (at half-maximum) of the absorption line in terms of A c, i.e., the deviation from the resonance frequency so as shown in Figure Colo To first order, AZ w is proportional to A H, the measured line width in gauss through at gWaLH a) The observed angular variation of the line width of the Group A at 77~K suggests that a sizable contribution to the line broadening comes from the dipolar spin-spin coupling. Perhaps there are small contributions from other sources of line broadening such as mosaic spread of the crystal and inhomogeneity in the static magnetic field across the sample. Also, the electron-nucleus dipolar interactions may not be quite negligible in our case. Experimentally, we have observed that the ESR absorption lines of S2 and S3 are much broader than that of S1 (see Figure 4.4). This can be explained by spin-spin interactions (dipolar broadening) between vanadium ions as well as between vanadium and the RE ions. According to Equations (C.3) and (Co4) of Appendix C, the dipolar interactions tend to increase the line width as the concentrations of the impurity ions increase. The concentration of vanadium in S1 is almost half as much as that in S2 and S3 (see Table 3.1). Furthermore, the last two samples

65 contain RE ions which make substantial contribution to cX through Equation (C,4). b) As we recall, at liquid helium temperature samples S1 and S2 showed similar ESR spectra whereas S31 which contains V and Tb, did not show any vanadium spectra even at 60-70 db below the 300 mw power level of the microwave source, The Tb3+ ESR lines, however, could be observed at very low magnetic field intensities Similar phenomena have been 66,67 reported by others, and it can be explained in terms of crossrelaxation mechanism: In S1 samples, vanadium ions occupy both Ca and W sites with two different but close g-valueso At liquid helium temperature, these two spin species couple magnetically and the cross-relaxation takes place with T2 < T12 < T1 where T1, T2, and T12 are the spinlattice, the spin-spin, and the cross-relaxation times. Since the resonance frequencies of the two centers are not much different from 64 each other, we expect T12 to be closer to T2. The fact that we see the ESR spectra of S1 at 4.2~K implies that the non-saturation condition (Co6) is satisfied. That is, 2 2 Y Hr, T,Pz. where ~ is a constant and (2H1) is the amplitude of the magnetic component of the radiation field. Now, in the (VgNd)-doped samples S2$ almost all vanadium ions go to W sites and Nd3+ occupy Ca siteso The g-values of Nd+ ions in S2 are close to that of vanadium; in fact, the "For Tb3+ in CaWO4, g, 17,777 and g. < 0,15 (Ref. 2)o

66 two sets of spectra overlap in a wide region in the ab-planeo This is a favorable condition for establishment of the cross-relaxationo Again, since the g-values are similar to the preceding case SI, we would expect to see the vanadium spectrum, in concordance with the experimental results The case of the S3 samples, however, is differento Here, we have V4+ and Tb3+ in W and Ca sites, respectivelyo Due to the large difference between the resonance frequencies of these ions we expect 64 T12 to be very largeo This makes 12 > and hence-the saturation condition prevails. The reason we are able to see the Tb spectrum is that the spin-lattice relaxation time T1 for Tb3+ is smaller than that for V4+ (Tb+ does not show spectrum at 77~K while V4+ does)o Apparently the smallness of T1 compensates for large T12 so that the saturation condition is removed 3o Splitting of the Group A Lines at 402~K As we have seen in Chapter IV, at liquid helium temperature, the Group A spectrum splits, in general, into four sets of lines0 The maximum component of the g-tensor was found to be in the direction of\ the 2W-O bond in W04 Also, the 77~K spectra was found to be at the average position of the four low-temperature sets (Table 42) o This behavior cannot be accounted for only by temperature dependence of the line width: the separation between the centers of the

67 lowest lying and the highest lying sets at 4.20 reaches a maximum of about 40 gauss while the maximum line width of the Group A at 77~K, for 70 CaWO4:V samples, is only about 4 gauss. Watkins and Corbett have observed similar phenomena in the ESR of the silicon E-centers and attributed it to "thermal reorientation" of the centers. We shall tentatively apply their method to our case. Before irradiation, vanadium in W+ site forms the diamagnetic 3+ complex V04 After the irradiation, it acquires one electron and 4becomes paramagnetic V04 If we assume that the unpaired electron comes from one of the oxygen atoms, then there are four possible V-O orientations differing in the spin resonance frequencies by = 2.i" 2 77 S o This is in agreement with the ESR spectra observed at 44o2~K. Now, suppose the unpaired electron in VO4 jumps randomly from one V-O direction to another with an average life time of T in each direction. At low temperatures the jumping rate is slow and, as Gutowsky and Saika have shown, if T > / o then we expect one distinct resonance absorption for each orientation with an intensity proportional to the probability of finding the electron in that orientationo We can identify this case with the Group A at 4.2~K assuming the electron executes random jumps among the four V-0 orientations with the same probability. Thus, we expect to see four lines with equal intensities as it is actually observed in experiment. As the temperature is increased, the jumping rate increases (Z decreases) and the resonance lines undergo life time broadening. Experimentally, we have observed this when the liquid helium in the system is

68 all evaporated and the temperature of the sample is rapidly approaching that of the liquid nitrogen coolant in the outer dewar. Analogous 70,73 observations have been reported by others.' As the temperature is further increased, T becomes smaller until eventually q 1 1^ which is the condition for "fast 72 exchange" and manifestation of motional narrowing. In this case, one set of spectrum is expected to emerge at the average position of the four low-temperature lines with a line width proportional to. We have associated this case with the 77~K spectra of the Group A (see Table 4.2).

APPENDIX A THE SPIN HAMILTONIAN AND POSITION OF THE SPECTRAL LINES 1. THE HAMILTONIAN In EPR spectroscopy usually a phenomenological Hamiltonian, the "spin Hamiltonian," is employed to interpret experimental results. It is phenomenological in this sense that its form can be anticipated on the basis of physical considerations and the crystal field symmetry. It can also be derived from a general Hamiltonian for a paramagnetic ion in the electric field of the host material (a crystal, in our case) and the externally applied static magnetic field. We shall outline this derivation. The Hamiltonian for the paramagnetic ion can be written as, I =f ff+ -s + m }+ l-e + K+ (A.1) where various interactions and their order of magnitude for the iron group elements are as follows i3, the coulomb interaction between the electrons and the nucleus - )e (z 105 cm- 1- _ the crystal field iteraction Its magnitude depends on the, the crystal field interaction. Its magnitude depends on the Cf 69

70 ion. For most iron group elements, e ^3^ <^ e The explicit form of c depends, of course, on the crystal field symmetry. It will be treated in detail in Appendix B.,the spin orbit coupling interaction (O 103 cm1) - Ls A L.S, assuming Russel-Saunders coupling. 3, the spin-spin interaction (1 cm'l) ='~S Jssk Jk 1, interaction of the unpaired electrons with the applied static magnetic field (.1 cm ) = pH (L+ ) ( e = Bohr magneton) It produces the splitting of the electronic levels between which the transitions are observed., the effect of the nuclear magnetic moment on the magnetic field of electrons 10 of electrons (10-2 cm-1) = 25' C l S. 3 (rC- s: )(:.) 1) t: t L., u Ll\ L. L where gN pN and I are nuclear g factor, nuclear magneton and nuclear spin, respectively.

71,the interaction between the nuclear magnetic moment and the D.C. magnetic field ( 10-3 cm-) -1)N -, the interaction of the nuclear quadrupole moment with the electrons (4104 cm ) = -eQ 1 (1i)~ - 3 ) r; I c^- L (.f.:)- c ] The effect of the crystalline field on the nuclear quadrupole moment is negligible. Since in our case the crystal field energy H is smaller cf than the coulomb interaction energy 3, but greater than all other energies in Eq. (A-l), the state of the free ion is usually described by H3. Then, the effects of the H c and the other interaction energies are taken into account by the application of two successive perturbations. When the state of a free ion is described by the coulomb energy only, it is implied that the electrons move independently from each other while obeying the Pauli exclusion principle. The total orbital and spin angular momenta are then given by L_1 L - 2I and J = L + S. Thus, in the absence of any external electric or magnetic field, each energy level is (2J+ 1) - fold degenerate. The

72 single electron orbitals are the usual Is 2s 2p etc. V, for instance, can be represented by [A]3d where [A] stands for the closed argon shell. The ground state of the free ion depends on its unpaired electrons, i.e., those which are not comprised in the closed shell. For 4+ the V ion, with just one unpaired 3d electron, the ground state can be 2 r readily written as D/2 (using the usual spectroscopic notation L2J where, r = 2 S+1 denotes the spin multiplicity of the state and J = L-S. 4+ The minus sign is used here because 3d shell of the V ion is less than half-full). The effect of the crystal field energy M f on splitting of the ground state of the free ion shall be discussed in Appendix B. 2. THE SPIN HAMILTONIAN So far, we have considered only the first two terms of the Hamiltonian (A.1). The remaining perturbations will now be treated collectively to derive the spin Hamiltonian. Denoting these terms by H we have Y -LS S -s me N Uv i a (A.2) Using the operator equivalent method on (A.2) and neglecting terms of second order in operators L, S, and I, we obtain A/_ A L.- p RX /1. P r\ & e' (A^.^^^^^1^)13~~I

73 where Q is related to the quadrupole moment Q by / eQ AV 8 4 X (OI -I ) )o In deriving (A.3), the spin-spin interaction term 3H1 and the first term of J; which expresses the interaction between the nuclear quadrupole moment and the electric field of electrons, have been dropped because of their negligible contributions. When, as in our case, the lowest energy level of the paramagnetic ion in the crystal field is an orbital singlet, the perturbation calculations with i/ are carried out in two steps. First, operators S and I are regarded as non-commuting algebraic quantities and the ordinary perturbation calculations are done only with the orbital part of the wave functions. The result is an expression in I and S which is called the spin Hamiltonian. Next, this new Hamiltonian is used to find the actual energy levels. Thus, to second order, for instance.5 <ol:'ox- 1!<~l'lE where, 0> and In> denote the ground state and the nth orbital excited state wave functions. The spin Hamiltonian obtained in this fashion has the general form -p H-9 - 5 + S.^ s +:-A. S S + z ~rtP,*r H,9~ N + PK H. l ^H(A.4)

74 where g, D, A, Q, gM, and \ are tensors whose principal axes are assumed to coincide. The first term in (A.4) is the Zeeman splitting term, The spectroscopic factor, g, may be anisotropic and differ from 2.0023 due to the admixing of higher orbitals. The fine structure term, SSDS, represents the splitting of the ground state in the absence of external magnetic field and the nuclear interaction. This term originates from the second order effects of the crystalline field and spinorbit coupling. The hyperfine structure term, IoAoS, describes the interaction between the nuclear magnetic dipole and the magnetic field due to the electronic spino The term Io,.oI shows the contribution of the nuclear quadrupole moment due to its interaction with the crystalline electric field, The direct interaction between the externally applied magnetic field and the nuclear magnetic dipole moment is expressed by the term H.oN.Io Finally, the last term, ~oHo/A.H can be neglected because it is independent of nuclear and electronic variables and thus shifts all energy levels by an equal amount. The eigenvalues of (Ao4) cannot be obtained in a general manner: In each specific problem, the general spin Hamiltonian 3-. must be modified to conform to the symmetry of the crystal field. For instance, for cubic symmetry the tensor quantities of (A.4) are all isotropic, i.e. x = gy C z - g Ax = Axy Az A. etc.

75 Hence, we have ( 4)cu cja9P H5+D( Sx+S + S+)+ A 1Es +(Iti 2Iy'1s)_ 9 I N v Observe furthermore, (])( (x + + S. ) - D S(+s ) = cons+, and can be neglected because it shifts all energy levels by the same amount. Thus, assuming the principal axes of the various tensors coincide, we find wbi Sp H +S + ^ I *S- +T- T(A.5) Similarly, for the axial field, (No ) 5 ^ Hb + 5( H l5s,) ~ 3. 4 B SX + IY) +Q L z 3(A.6) -P ^ R I- z N1 I )(I+ H:1 J)l and for the more genera case, the rhombic symetry, and for the more general case, the rhombic symmetry,

76 P ZZ (A.7) +e - p a I ( 9o d ge d When the principal axes of A- and g-tensors do not coincide,. cross-terms of the form must be added to the rhombic spin Hamiltonian (Ao7)o Note further that for paramagnetic systems with S - 1/2 the effective axial or rhombic spin Hamiltonians do not contain terms in D or Eo 3. ENERGY LEVELS AND LINE POSITIONS FOR S - 1/2 SYSTEMS In general, the energy levels for the state I M. m> are eigenvalues of -~ and can be obtained by the usual degenerate perturbation calculationse The results for V4 ions (S = 1/2), to second order and neglecting the nuc~lear and quadrupole interactions are as followso 9P I ) (-(A 8) where M and m denote the electronic and nuclear spin quantum numbersrespectively o The energy for A 8 1 9 A m O resonance transitions is given by

77 6 (, ~)- E ( -\ t)= h ^ from which the resonant magnetic field H (i.e., the position of the resonance absorption lines in the magnetic field) is obtained as H) Ho - A where, Ho ='/9 Similarly, for the axial spin Hamiltonian we have, E (-,,) _ E (_-\ _) _=_ h ~ z 2 _ (.I(-) _B ) co24 + K n 1 (9+ P (+) o where, 2 2 4e c2os.2 22 ^ ^ 2 -— 2.1o 22j g2. K g^ A G 0 9 S^ ~

78 g Ii and g_ are g-values measured parallel and perpendicular to the symmetry axis, respectively; 9 is the angle between the magnetic field and the symmetry axis.

APPENDIX B S4 SYMMETRY AND SPLITTING OF THE 2D GROUND STATE 4 In this appendix we shall apply crystal field calculations to V4+(3d ion, 2D ground state) at the substitutional calcium site in CaWO4 single crystal. The purpose of these calculations is to find the energies associated with splitting of the ion's ground state in the crystal field and, subsequently, to estimate L g. lo THE CRYSTAL FIELD POTENTIAL In these calculations we shall make the customary assumptions in the crystal field calculations, namely, (1) the impurity ion (the paramagnetic transition metal ion of interest) occupies a substitutional metallic site in the host crystal without distorting the field, (2) negative and positive charges in the lattice are non-overlapping point charges. If we choose the paramagnetic ion as the origin of polar coordinates, Fig. B.1, then the electrostatic potential at a space point r, due to negative charges Qi located at Ri, can be written as V(r)- = 2 -- Or,using the law of cosines, V( r) - e S; ( - v.7 ) 79

80 This, in turn, can be rewritten in terms of the Legendre polynomials as, V (r) l. ( s ^ )\ Cc i \\ e r L Fig. B.1. Point charge potential. where )i denote angles between directionsr and Ri. Polynomials Pt (cos c i) can be expressed in terms of (&,4) and (, c,), which define directions r and Ri, by making use of the spherical harmonics addition theorem. Thus, 4 r 2- r 12 ((; C - I\ (B.2) Using this in (B.1) we obtain v(r)= -e2 I O_ - E( ((6,) y ((B.3) _ = -O 2, L 4 =4

81 Eq. (B.3) can be rewritten as i=o YV I= (B.4) where A m are given by A _= e __- - 7 ( A- =) e /4..T ( _ l)) (&,) (B.5) L The number of terms in (B.4) which need to be considered is greatly reduced in each particular problem as follows. First of all, the final goal in these calculations is to find the energy levels of the paramagnetic ion in the crystal field. This is done, as we shall see later, by computing matrix elements of V(g) with respect to the wave functions associated with these levels. The paramagnetic ion in our case is V4+ with electronic configuration [A] 3dl were [A] stands for the closed argon shell. Thus, the paramagnetic electron in this case is just one 3d electron. Accordingly, the wave functions do not contain terms with t > 2. Therefore, terms with t > 4 in V(r) can be eliminated as they result in zero matrix elements. Similarly, the terms of odd J (odd parity) are excluded from the crystal potential. Further reduction in the number of terms in V(r) can be obtained by symmetry considerations. The calcium site in CaWO4 has S4 symmetry with elements E, C2, and S4. The last element, for instance, requires that the crystal potential, anywhere in the lattice, remain invariant

82 under a rotation of 900 about the c-axis followed by a reflection in a plane perpendicular to this axis. Thus, applying S4 to r, we get but, V(+,s) V( - 7 - 9) which, by virtue of (B.4), requires Yp (62 + ) _- YQ (9+ + 2 - 9 ) or, p (c5s e) e = (_co')e (B.6) Noting that P m (cos 6 ) has the parity of I-1|1 with respect to the change in the sign of its argument, the only acceptable values of t and m that can satisfy (B.6) are 0 2 4 m 0 0 +4 The term Y00 introduces just a constant shift in all energy levels and thus it can be neglected from the sum in (B.4). Hence, Y20, Y40, Y44, and Y4-4 are the only terms to be considered in the calculation of V(g), which now can be written as, Vtr) = r aoy' +. ( 4 AYt Q+^-4 - \f) (B.7) 40 LoL4 4

83 The coefficients A tm must be calculated from (B.5) by carrying out the indicated summation over the ligand charges. Taking Ca as the origin, the polar coordinates of the eight ligand oxygen ions, computed 55 from the recent crystal structure data, are shown in Table Bo.1 TABLE B.1 POLAR COORDINATES OF THE EIGHT NEAREST OXYGEN NEIGHBORS OF Ca IN CaWO4 Oxygen Number R(in A) i I 1 2.481 66,45 53,31 2 2.481 - 66,45 +WT 53,31 + 1I/2 3 2.481 66,45 53,31 + 2 r /2 4 2.481 - 66,45 + r 53,31 + 3'r /2 0o, 5 2.436 - 40,4 +1T 59,48 6 2.436 40,4 59,48 + T/2 o 7 2.436 - 40,4 +IT 59,48 + 2 I /2 8 2.436 40,4 59,48 + 3 r /2 The angles ei and; in the table are the polar and azimuthal angles of the ith oxygen, respectively. From Eq. (B.5) the expression for A20 can be written as, LTr 8 Tr A 4o — eZ_ A;,0= He S R3 O(6 S8r^< > / > (3 Cos 6-1)(B.8) -___,2iV^T.

84 3 where Q is the effective charge of the oxygen ions and <1/R3 > is defined by 3 >- 2 ( R 3 _ 3 ) o O with R1 = 2.481 A, and R2 2.486 A. From Table B.1, we obtain 2 2 (,3 cos 6.- -) = 0. s6 a (B.9) Thus, combining (B.8) and (B.9), we find A = o,&~ e, "r< ^, S/ =V 3 <R3 > (B. 10) Similarly, A — \. o 30\/ < ^-S >) e 40' and,' —Y44' — Y' - = — < >e 74- 1(4i -4)-(>'094 ( Y. e where, -(- >L+ 2-) Hence, according to (B.7), we finally have

85 IIT 2. 8)0 \Tr < S Eor Y0 1 4er r ^(^Y ) _ i Y - Y )J (YY -) (B.11) For calculations of matrix elements of V(r) it is more convenient to express V in the operator equivalent form as follows, V(r), 0~,,4 Sa eQ<-'-I-LL IL op1 -pe1^113-^ 30> sL z (B.12) (,L' ) }- -- ~.l'(L ++ L)-; I.Oq,4 (tL,+L) where, 4 a~~~r f tt >~Snalin> <rn> - average of rn over 3d radial wave function - O E )'

86 2. ENERGY LEVELS The energy levels of the paramagnetic ion in the crystal field are the roots of the secular determinant L where ~ are appropriate basis functions for the S4 symmetry. These functions are chosen, by considering the symmetry operations of the S4 group, to diagonalize V or to reduce the number of the off-diagonal elements to a minimum. Thus we find, r - z ) F4 = t;-1 The non-zero matrix elements of V(r) are: r,\ I vl > 0.128 A - 0.483 B - W < r IV lqr> -0.128 A -.0.580 B W2 / I \/1 rJ > -0.128 A + 0.418B W < IV j\/ lb > -0.456 i B 4 w

87 <r I\ 0.456 i B W4 I~r3v 3>= 0.064 A + 0.322 B _W5 < ll V r,> 0.064 A + 0.322 B W5 where, A eQ<r/ R3> and B = eQ<r4/ R5> The secular determinant is set up as follows: _________) 1> r4> Lr> Lbr2) <rl1 W1-E 0 0 0 0 < 131 0 W5-E 0 0 0 &1 0 0 W -E 0 0 -5l 0 0 0 W2-E W w4 <1 ) 0 0 0 W -E From this determinant, the energy levels are found to be: E = 0.128 A - 00483 B E3 0.064 A + 0322 B E4 = 0.064 A + 0o322 B (B.13) E = -0,128 A + 0.595 B E = -0.128 A - 0.757 B In order to determine the sequence of energy levels we need to know, at least approximately, the ratio A/Bo Accurate determination of

88 this ratio which involves integrals of the radial wave function is not 68 possible. Pappalardo and Wood, using the results of Hartree-Fock calculations by Watson, have estimated* < r4 > / < r2 / R> 0.118 for Cu2+ (d9) which is conjugate of V4+ (dl). We shall use this value in our approximate calculations. Hence, employing the conventional Dq notation where, Dq = eQ < r4 / R5 > / 6 we have B = 6Dq and A= 51Dq. Using these parameters in B.13, we find the energy levels as shown in Figure B.2. The corresponding eigenfunctions, which now diagonalize the perturbation Hamiltonian V(r), are found to be 13^1= t 01 Ta-/ 2 9 3^. i) 036 t + t ) 0~'6 3 p; ( -,Fk *Private communication with the former authors. *The final result of Pappalardo and Wood on energy levels of Cu2+ in CaW04 is not correct due to some calculation errors.

89 ____________5.2 Dq /~o~ g ~3.6 Dq 2D(5) -3 Dq ______________ -11.1 Dq Fig. B.2. Energy levels of V4+ in a Ca site of CaWO4. 3. SHIFT OF THE g-FACTOR The g-factor for a free electron is ge 2.0023. However, the paramagnetic electron is not free and consequently its g-value differ from ge by an amount A g = g - g. Contributions to / g come from coupling of the orbital angular momentum of the electron with its spin angular momentum and with the externally applied magnetic field. Denoting these perturbations by H pert9 we have -AverT ~ L. A L -~ A g is proportional to matrix elements of pert in the crystal field. For orbitally non-degenerate ground state (our case),

90 <0I 1 1 |10 >. where (0 > denotes the orbital part of the ground state wave function. Hence, to second order, 19 }_lperlo> F E -FE - f where f denotes "final states." Let us choose H in the direction of the z-axis taken as the quantization axis. Thus, carrying out the perturbation calculations and collecting terms of first order in S Hz, we find W E10 LIo^ *F 0 f f o comparing this with the corresponding term in the spin Hamiltonian s = 9 H2 we find _g =f o Similar relationships can be obtained for x and y directions too. Hence, in general A 1 I E - 1f L. IoI (B.14) i = 3c,,3

91 Applying (B.14) to the energy levels of V4+ we find, 2 A S _ _ r KS |- ILzi>I~ l+ -<'LEzi >I + vr.I4|L&>1 E -E E -_ -& ^P Q1 r3F, L rFL where, Kr/. ^IVl~' - o Thus g c-y 5.1 Dq Similarly, for gx, using the relationship Lx = (1/2)(L+ + L) we have, fl -3 +where, i 4' Lt L + k ~ = o 1 4 r13 i t L 1 o I C 1 L. +L li>l — * - t./F,,i..,_ i i>{^o,.,

92 Thus,'A - -^ 2 - -- E -E 146 3 DJq 3,4 aForAgy, using the relationship L = (L+ + L) / 2 L in (B.14), we obtain o 1r.31 or A9 - g x 3

APPENDIX C RELAXATION AND SATURATION PHENOMENA In this appendix we shall describe briefly these processes and introduce the formulas which were used in our discussion, in Section 2 of Chapter V, on the line width and temperature dependence of the spectra. No effort, however, will be made here to present lengthy proofs or derivations since the subject is fully treated in a number of papers and books i. Relaxation Processes Consider a system of two spin states (Ms = + 1/2) in a crystal lattice in a static magnetic field. Let us suppose that the spins are in thermal equilibrium with each other and with the lattice. That is, there are no interactions to change the total population of each level, The steady state condition is, then, characterized by N W12 2 N W21 where N? are the equilibrium populations and W.. are transition probabilities per unit time via spin-spin and spin-lattice interactions. On the other hand, assuming Boltzman distribution, the static equilibrium is governed by o.^,, -L /l<.. K~ /r H.-I e9 93

94 \A/12 A/t g i/ N.H, E\ where A E - E2 - E1> 0. Hence, we have - A E / kT Wi12 / 21 W e Now we turn the microwave field on and disturb the equilibrium by inducing transitions between the spin levels. In this case the spins, by interacting with each other and with the lattice, try to reestablish the thermal equilibrium. The transient condition of the system can be expressed by d / dt (N - N) = 2 (N2W2 - N1W12) or (C.1) dn/dt = N (W21 - W12) - n (W21 + W12) where N N1 + N2, and n s N1 - N2. Solution of Equation (C.1) is n - no + Ae't/

95 where, n _ N W21 12 o N W21 + W12 -x W21 W12 and A is a constant of integration. The approach of the spin system toward a thermal equilibrium is called "relaxation" and the associated characteristic time T is defined as the "relaxation time." There are three major relaxation processes, namely, spin-lattice, general spin-spin interactions, and cross-relaxation. i) Spin-Lattice Relaxation This is the mechanism by which the spin system exchanges energy with the lattice and "cools-off." Here, the relaxation time (T1), in general, is a function of temperature (T); the applied magnetic field H; and parameters L and X, the crystal field strength and the spin-orbit coupling constant, respectively. The explicit form of T is one of the most disputed problems in relaxation, especially at low 59 temperatures. Since the classic work of Casimir and du Pre, many investigators have studied this problem both theoretically and experimentally. The agreement among their results in most cases is only qualitative. However, they all agree that T1 has (1/T) - dependence. For recent literature on this subject see references 60 and 61.

96 For the purpose of our discussion we will use the recent formula of Van 60 Vleck for the low temperature region, T, + T (C.2) + O_ H' O. Ht+ T o A, where a, ao, a2, a4, b, and c are constants and H is the magnetic field strength. ii) Spin-Spin Interactions These are interactions which tend to establish a thermal equilibrium in the spin system via coupling of the spins with each other and with the nuclei of the host medium. These are: (1) magnetic dipole-dipole coupling; (2) electrostatic interactions between electrons which is usually called exchange effects; and (3) hyperfine coupling between the spin of the paramagnetic ion and the nuclear spin of the host material. Of these, the second one is important only in systems with high spin densities, e.g., undiluted paramagnetic salts, and the third one is usually negligible. The theory of the magnetic dipole-dipole interaction was first 62 developed by Van Vleck who used moments method to calculate dipolar broadening of ESR lines. The spin-spin interaction was also recently studied by statistical approach which extended Van Vleck's results to very dilute spin systems. The contribution of the magnetic dipole coupling to the line width stems from the fact that if two paramagnetic ions i and j are located at a distance r.. from each other, then the classical magnetostatic energy between the two is given by

97 L I J. where Si and S are the two spin angular momenta and r. is the radius -i j1 -Ji-j vector from i to j. The energy for a system of N spin is then, N L1 Using this perturbation Hamiltonian, Van Vleck calculated the mean square derivation <( A )2>, of the resonance frequency, defined by 23 J (Wf)d where f ( - ) is a line-shape factor and In most spin resonance experiments, the frequency cw is held constant and the magnetic field H is varied. To first order, < > L o CiJ3 JH~ where A H is the line width measured in gauss. Taking the static magnetic field in the z direction, chosen as the quantization axis, Van Vleck has obtained <,A ) S ( 2 r (1 3os,) (C.3) \ 4-' ^0 2 s^ (~,J~ I~gl PL r I- 3cos

98 where r. and o are the radius vector and polar angle of the jth ion with respect to one of the ions taken as the origin of coordinates. Now suppose the system contains a second kind of paramagnetic ions whose resonance frequency is different from the first one, eg^ CaWO4V, Nd. system. In this case, if the resonance condition is set for the first ions, the contribution of the second ions to the spread of the resonance frequency, in the same coordinate system, is given by 3h3 ( >- 24 (5 +1)9 5 P 2 r k (- 3 coS ) 92 1. okk (c.4) In other words, in such a system, the total mean square deviation of is given by The characteristic tie constant f the sin-sin intera ons is shown by T2. iii) Cross-Relaxation The theory of cross-relaxation was first presented by 64 Bloembergen et al. Since then, it has been applied successfully to interpretation of a number relaxation phenomena which could not be accounted for by considering only T1 and T2 time constants. According to this theory, the dipolar interaction between the ions may induce transitions in which the total Zeeman and crystalline field energy is conserved. Consider two different species of ions oc and P with corresponding resonance frequencies d0 and Ad, and energies E

99 and EP. Now we ask what is the probability that oc "flips up" in spin while p flips down? This probability is proportional to where ap is dipole-dipole interaction Hamiltonian. The characteristic time, which is inversely proportional to the transition probability, is shown by T12 due to the fact that its magnitude is commonly between the spin-lattice relaxation time T1 and the dipolar interaction time T2. T12 increases rapidly as the difference between the resonance frequencies of the two spin systems increases. 2. Saturation Phenomena If we use only the spin-lattice and spin-spin relaxation times T1 and T2 to describe the transient state of the spin system, then the average rate of energy absorption per unit volume of the sample is given by A T. )-H(wL) H TT2 ( which is obtained from the Bloch phenomenological equation. Here magnetostatic (Curie) susceptibility H- 1/2 of the amplitude of the microwave field's magnetic component =AE../t, where A Eij splitting of energy levels between which transition takes place oo = fixed frequency of the microwave field gyromagnetic ratio

100 2 2 Under normal operational condition, the term y HT T1 T2 which will be denoted by CT satisfies the condition (T - H TT T I (Co6) and hence,'< 27 e ~t | Tt(% o)I (C. 7) The resonance condition is W=wo for which the absorption reaches its peak 2. 2 A, 3C w H T; Peadk o o I We notice that by increasing microwave power, A increases. Figure C.1 shows the shape of absorption line obtained by plotting A vso T2 (% -o) using Equation (5.7). The width of this curve between the points of half-maximum is Thw codi- (C.8) The condition (C.6) can be maintained by decreasing the microwave power. However, a limitation is imposed on us by T1 and T2. When, for any reason, (H increases so that t H, TTa ~> (C.9) then near resonance. Equation (Ce5) takes the following form A - r

101 Amo x ____ _/i_ Fi. E y an as a f n of (Wo- @) Tz Fig. C.1. Energy absorption as a function of the microwave frequency. Under this condition, as we note, the absorption curve levels off and no resonance peak is observed. This situation is called saturation of the ESR signal. It implies that relaxation mechanisms are not powerful enough to carry off the energy put in by the electromagnetic field.

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