ERRATA Page Abstract line 10: having its V-0 axis inclined at an angle of Abstract second line from bottom: of the reduced vanadyl signal 13 line 1: vanadyl ion surrounded by 15 line 5: 3-4 mm long were obtainable in 16 line 4: of the atoms in the unit cell 19 line 17: coated with a thermo22 line 15: made in the white beam 23 line 4: delete "is truly phenonienological in that one can postulate it on the basis of symmetry" and replace with "can be obtained from phenomenological considerations" 28 line 7: delete "the total spin operator S and replacing" and replace with "S (op) and replacing the functions of" 31 Table 1, column 1: x not xy 39 Footnote, line 4: peak to peak height of 3/4 ~5a b 41tan; S et$l -- L6.p X8; t X ( -28

Page 42 IHC((gs)2C O-S/S -ll -tt B9-fl, (d+cPS~6-8Alcos C.0 Sln OS; v` delete "In fact.,o what is observed." +4 -2 45 line 5: the ionic radii for V and 0o 49 Table 4, column 2: 1.9717+5 77 line 5: 77 line 10: 78 line 6: 79 bottom term 62 } L S I g 91 line lO This same expression can be used to compute A g[B 95 Ref. 5: Co P. Slichter

THE UN I VE RS IT Y OF MI CHI GAN COLLEGE OF ENGINEERING Department of Nuclear Engineering Final Report - Part 1 AN EPR INVESTIGATION OF VO+2 AND X-RAY PRODUCED V+2 IN TUTTON SALT R. Ho Borcherts ORA Project 04385 under contract with: NATIONAL SCIENCE FOUNDATION GRANT NOo G-15912 WASHINGTON, D. C. administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR June 1963

This report was also a dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The University of Michigan, 1963.

TABLE OF CONTENTS Page LIST OF TABLES v LIST OF FIGURES vii ABSTRACT ix CHAPTER I. INTRODUCTION 1 Ao Background to the Present Investigation 3 B. Literature Survey of V51 EPR Investigations 4 II, CONCLUSIONS 10 A. VO+2 10 Bo Irradiated VO+2 11 C. Two Models For VO+2 in Zn(NH4)2(S04)2.6H20 11 D. Further Studies 14 III. EXPERIMENTAL METHODS 15 A. Crystal Growth 15 B. Crystal Structure and Orientation 16 C. EPR Experimental Arrangement 19 IV. THE HAMILTONIAN 23 A. The Crystalline Field 25 Bo The Spin Hamiltonian 28 Vo EXPERIMENTAL RESULTS 31 A. V0+2 31 B. Discussion of VO+2 Results 32 C. Two Models of VO+2 in Zn(NH4)2(S04) 2~ 6H20 45 D. V+2 and Irradiated VO+2 Crystals 47 Eo Discussion of V+2 Results 50 Fo Conversion of VO+2 to V+2 53 iii

TABLE OF CONTENTS (Concluded) Page APPENDICES A, X-RAY-DOSE CALCULATIONS 57 B. CRYSTAL FIELD CALCULATIONS 59 C. ANGULAR VARIATION OF ENERGY LEVELS OF THE RHOMBIC SPIN HAMILTONIAN 72 D. FINE STRUCTURE ENERGY LEVELS (RHOMBIC) 81 E. g AND A TENSOR PRINCIPAL AXES NOT COINCIDENT 84 F. ANGULAR VARIATION OF ENERGY LEVELS FOR APPENDIX E 86 G. DATA REDUCTION FOR V0+2 CRYSTALS 90 H. DATA REDUCTION FOR V+2 CRYSTALS 93 REFERENCES 95 iv

LIST OF TABLES Table Page 1. Experimental Results For VO+2 31 2. Literature Values of VO+2 Spin Hamiltonian Constants 38 3. VO+2 Magnetic Field Resonance Values 44 4. Experimental Results For V+2 49 5. Experimental Results For V+2 (Irradiated VO+2) 50 6. V+2 Magnetic Field Resonance Values 52

LIST OF FIGURES Figure Page 1. 4F (V+2) and 3F (V+3) states in crystalline fields. 5 2. 2D (Vt4 and VO+2) states in crystalline fields. 7 3. EPR spectra along K2 axis. 12 4, Unit cell of Zn(NH4) 2(S04) 2.6H20 17 5. Faces of Tutton salt, crystal orienting devices. 18 6. Schematic diagram of EPR spectrometer system. 20 7. Position of VO+2 axes in Zn(NH4)2(S04) 2 6H20. 33 8. sin G (dO/dHm)curve and VO+2 powder spectrum. 35 9. VO02 in glass, IR-4B, and crushed crystals. 37 10. X-Y plane phase shift. 40 11. Vanadyl superhyperfine structure. 48 12. V+2 spectrum along Z axis. 54 13o Conversion of VO+2 to V+2. 56 vii

ABSTRACT In these experiments the Electron Paramagnetic Resonance (EPR) spectrum of the vanadyl ion (VO+2) in single crystals of Zn(NH4)2(S04)2.6H20 is studied and comparison of the X-ray irradiated VO+2 spectrum to V+2 in single crystals of Zn(NH4)2(S04) 26H20 is made. Investigation, at 9.3 kmc, of the single crystals containing the vanadyl ion, V0+2, showed that the vanadyl ion substitutes for the divalent zinc ion with its V-0 axis oriented in one of three directions. The three directions have a population ratio of 20:5:1 with the larger two population positions having their axes at an angle of 77o48' to each other and the third position having its V-0 axis inclined an angle of 980 to the other two. The rhombic spin Hamiltonian fits the experimental data well if the x, y principal axes of the g tensor are rotated 23o020' from the x, y principal axes of the A tensor. The z axes of both tensors are assumed to coincide. The rhombic spin Hamiltonian constants (obtained at room temperature) vary slightly for each of the three positions with the larger population position having gz = 1.9331 A = 0.018281/cm gx = 1.9813 Bx = 0.007137/cm gy = 1.9801 By = 0o007256/cm QT = 0.00024/cm;'xy = -0.0000462/cm The constant xy(- Fxy+Fyx) results from the noncoincident principal axes of the g and A tensors and arises from the inclusion of off-diagonal terms of the form. XF: (Sia+ ScI) The constants A, Bx, By, and Oxy are related to Azz, Axx, and Ayy, the principal axes values of the hyperfine tensor, and were evaluated to be Azz = A Axx = 0.0071200/cm Ayy = 0.0072439/cm ix

The three positions of the V-0 axes suggested that the direction of these axes were the (111) directions of the octahedron of water molecules that surround it. Since the octahedron of water molecules is distorted each of the four positions is ordered in energy and thus in population. The lowest population position was not observed. An alternate description in which the vanadyl ion is surrounded by five water molecules (the vanadium is at the center of the octahedron of the five water molecules and the covalent bonded oxygen) is considered a possible description of this complex. Also observed was a five-line superhyperfine structure on the vanadyl resonances for a specified orientation of the magnetic field. This structure was found to be due to the protons of the surrounding water molecules. Upon X-ray irradiation of the crystals containing the vanadyl ion it was found that the X-ray, or the subsequent high energy electron it produces, breaks the V-O bond resulting in a V+2 EPR spectrum having the same rhombic spin Hamiltonian constants and z axis orientation as crystals grown with V+2 These constants, measured at 24 kmc, were calculated to be gz = 1.9717 E = 0.02280/cm gy = 1.9733 A = 0.008263/cm D = 0.15613/cm B = 0.008246/cm with A and D having the same sign. Measurements were also made on the V+2 (irradiated vo+2) spectra in Mg(NH4)2(S04) 26H20 and ZnK2(S04) 26H20. Although X-ray irradiation, carried out in a white beam from a tungsten target tube operated at 50 KVP, 50 ma, reduced the intensity of the VO+2 signal 80-90% in 10 min (-108 rad) further irradiation failed to reduce the intensity of the remaining vanadyl signals. Also, of reduced vanadyl signal, only 15-20% of it was converted to V+2. An experiment carried out at 4.2~K failed to show any V+3.

CHAPTER I INTRODUCTION In recent years the spectroscopy of the solid state has been given a great impetus by the initial successes of such devices as the transistor, the solid state radiation detector, and the ruby maser and laser, as well as by the need for fundamental knowledge of the basic properties of solids in such new areas of application of reactor technology and space exploration. Magnetic resonance is one branch of solid state spectroscopy that deals with nuclear, ferromagnetic, antiferromagnetic, and paramagnetic resonance. The first, nuclear magnetic resonance, or NMR, is concerned with the magnetic dipoles of nuclei; the other three are concerned with the magnetic dipoles resulting from electrons. In ferromagnetic and antiferromagnetic resonance the electronic dipoles are strongly coupled to each other, while in electron paramagnetic resonance, or EPR, the individual electronic dipoles are considered separate or very loosely coupled. Paramagnetic substances differ from diamagnetic ones in that the former posses permanent magnetic dipoles arising from the orbital and spin motion of the unpaired electrons. While unpaired electrons occur in the incompleted shells of the four transition series of the periodic table —iron, rare earth, platinum, palladium and uranium —unpaired electrons also occur in some donors, acceptors, and other impurities in

solids, metals, odd electron molecules, such as free radicals and those molecules damaged by radiation, and triplet state molecules, such as 02. All these may give rise to electron paramagnetic resonance. For texts on EPR one may consult Low,l Pake,2 Ingram.,4 and Shlichter,5 as well as the reviews by Bleaney and Stevens,6 Bowers and Owen,7 and Jarrett.8 The study of the resonance phenomena of paramagnetic ions began with Zavoisky,90lO in 1945. Previously the principal techniques for studying paramagnetic materials had been magnetic susceptibility and paramagnetic rotation. The classic paper by Bethell in 1929, on the prediction of the splitting of the orbitally degenerate ground state of the paramagnetic ion on the basis of the symmetry of the ligands surrounding the ion, and the resulting application by Van Vleckl2 to the explanation of paramagnetism gave rise to the "crystalline field era" in the history of paramagnetism. Soon after Zavoisky's discovery it became apparent that besides revealing a great deal of information about the paramagnetic ion, EPR could also contribute much to the then meager knowledge of the crystalline field arising from the ligands surrounding the paramagnetic ion. Generally verifying the results of crystal field theory, EPR techniques provided data precise enough that detailed examinations revealed discrepancies between.the theoretical energy.level scheme and the experimental results. Reconsiderations of the crystalline field theory showed that the more general approach is that of forming a molecular orbital model of the paramagnetic ion and the surrounding

ligands so that covalent bonding is included. In this approach crystalline field theory becomes a limiting case of the molecular orbital treatment. * A. BACKGROUND TO THE PRESENT INVESTIGATION One ion of particular interest in EPR investigations is that of vanadium. Almost isotopically pure, vanadium consists of 99.76% V51, nuclear spin 7/2, and 0.24% V50, nuclear spin:614. V51 is also an odd proton nucleus —23 protons, 28 neutrons-so that its magnetic moment is expected to be quite large; the result is that the hyperfine structure consisting of eight resonances (21+1) should be well separated. The various oxidation states of vanadium have been observed to be +5, +4, +3, and +2. Since the electronic structure of vanadium is [A]3d34s2, where [A] denotes the closed argon shell, the +4, +3, and +2 valence states have 3d1, 3d2, and 3d3 electrons respectively. By Hund's Rule** these valence states +4, +3, and +2, will have electron spins of 1/2, 1, and 3/2; respectively, and orbital angular momentum of 2, 3, and 3, respectively. These electrons are thus unpaired; as a result the EPR signature of each valence state is readily recognized by the number and angular dependence of the fine structure groups, as well as by the separation of the hyperfine structure. *For an excellent introduction to molecular orbital as well as crystal field theory see Ballhausen.13 **Hund' s Rule applied to vanadium says that the ground state has (a) maximum spin multiplicity and (b) maximum orbital angular momentum consistent with (a). See Heine,15 p.o 97.

B. LITERATURE SURVEY OF V51 EPR INVESTIGATIONS _V+2 In 1951 Bleaney, Ingram, and Scovill6 reported on the EPR spectrum of V+2 in Zn(NH4)2(S04)2'6H20 —the same host crystal used in this investigation. In this crystal six water molecules form a distorted octahedron surrounding the V+2 ion, producing a rhombic crystalline field. As a result the 2S+1 degenerate ground state splits into two spin doublets, MS=+3/2 and Ms=+l/2, separated by 2D. A magnetic field splits the remaining degeneracy, and application of microwave energy at a frequency v gives rise to three fine structure groups with g Z 2(1 - 4 ) corresponding to the selection rule AM = +1 (Fig. 1). lODq Because of the 7/2 value of the nuclear spin of V51 and its large magnetic moment, each of the fine structure groups are composed of eight resonances (2I+1) with a separation of approximately 90 gauss. Their spin Hamiltonian constants are gz = 1.951, D = 0.158/cm, E = 0.049/cm, and A = 0.0088/cm. Low17 investigated V+2 substituted in the cubic site of Mg in MgO and found g = 1.980 and A = 0 0074/cm. V+3. In 1958 Zverev and Prokhorovl8 and later Lambe, Ager, and Kikuchil9 reported on V+3 in Q-A1203 (corundum or sapphire); their EPR results confirm the magnetic susceptibility measurements of Siegert20 and van den Handel and Siegert.21 Pryce and Runciman22 have performed optical investigations of V+3 in sapphire. In sapphire six oxygens surround each aluminum site in a distorted octahedron, giving rise to a trigonal crystalline electric field. If

P Tig / (I) A29 Btg (I) d2 Vi 12 Dq (3) 1 T ng Etg (2) BINTERACTION (Elongated) FREE ION CUBIC F-P TETRAGONAL L-S COUPLING MAGNETIC FIELD (A,B<O) P TMT d3 Vf2 -4rArF+ *Bia Ateg (I) (3) Tig / ~~~t Etg (2) 6Dq 4AF2& -~~~~'r~~~~+ \ Dq A Btlg (I) (,).~ Apg ZD~.35/cm Fig. 1. 4F (V+2) and 3F (V+3) states in crystalline fields.

V+3 is substituted for the A1+3 ion, the trigonal component splits the T2 energy level into a doublet E and a singlet B. This splitting acting through the LoS coupling causes the ground state to split into a lower spin singlet Ms = 0 and an upper spin doublet Ms = +1 separated by approximately 10/cm (Fig. 1). An applied magnetic field causes the Ms = +1 level to split and the application of microwave energy at a frequency v causes a "forbidden" transition, AM = +2, to occur at H = hv/2g., Since the ground state is separated from the excited state by only a few wave numbers, the relaxation time of this AM = ~2 transition is so short that the EPR experiment must be performed at very low temperatures, -4.20K, in order to observe the spectrum. If the temperature becomes too low, less than 2~K, the Ms = +1 state becomes so depopulated that no signal is observed at all. For V+3 in ic-A1203, g = 1.98, and A = 0O0102/cmo V+4. In 1960 Lambe and Kikuchi23 found a small amount of V+4 in sapphire having g l 1o97 and Al 0 0132/cm. They were also able to show that by X-r.ay or gamma irradiation that some Vf2 is produced from V+3 Also in 1.960 Gerritsen and Lewis24 and Zverev and Prokhorov25 reported on V+4 in Ti02 (rutile) In rutile six oxygen atoms surround the Ti+4 site, producing a tetragonal field with a small rhombic component. Since V+4 has only one 3d electron and thus a spin of 1/2, only one fine structure group corresponding to the AM = +1 transition is observed (Fig. 2). Theoretically, the V+4 spectrum in rutile has yet to be satisfactorily explained in terms of ordering and separation of energy levels. One notes that vana

(I) Atg FREE IO I T (2) E D' 4Dq'_I, z o3' (3) Tgg 9 / +-9Art JP2 r FREE ION CUBIC TETRAGONAL MAGNETIC (Squashed) FIELD (I) Al (2) El / 3d' (V+4) D^, }/ I ) X g, 2(1 — D ) (2) Ep < FREE ION. CYLINDRICAL TETRAGONAL MAGNETIC (Cmv) FIELD Fig. 2. 2D (V+4 and V0+2) states in crystalline fields.

dium oxide, V02, forms a "rutile-like" structure but with one short V-O bond (Andersson26) Since V+4 tends to form VO+2 ions it may well be, as Ballhausenl3 suggests, that in TiO2 V+4 tends to form a strong covalent bond with one of the oxygens. The observed hyperfine spectrum of V+4 in rutile is very anisotropic, having a separation of Az = 0.0142/ cm, Ax = 0.0031/cm, and Ay = 0.0043/cm. The g values, gx = 1.915, gy = 1.913, and gz = 1.956 also reflect the rhombic symmetry of the crystalline field. Although the valence state of vanadium in the vanadyl radical VO+2 is also +4, the strong cylindrical field formed by the vanadium and the oxygen produces an energy level structure quite different from that of V+4 in a cubic field (Jorgensen27 and Fig. 2). The first EPR spectrum of the vanadyl ion VO +2 was reported by Garif'ianov and Kozyrev28 and Kozyrev29 in both liquid and frozen aqueous and acetone solutions and later by Pake and Sands30 and Sands31 in aqueous, acetone, and ether solutions and in glass; by O'Reilly32 in vanadyl etioporphyrin I dissolved in benzene and high viscosity oil; by Faber and Rogers33 on various adsorbers (charcoal, Dowex-50, IR-4B, and IR-100); and by Roberts, Koski, and Caughey34 in some vanadyl porphyrins, As mentioned above, the vanadyl ion carries its own crystalline field so that it matters little what other kind of field surrounds the ion as evidenced by the similar spectra found by these investigators~ The results of these investigations are that the gz and gl values are 1.881o93 and 1o98-1 99, respectively, and a hyperfine splitting of 0.0147

0.0184/cm parallel to the z axis and 0.0055-0.0078/cm perpendicular to the z axis. The variation in these values is suggested to be due to the variation of the covalent bonding between the vanadyl ion and the surrounding ligands (Faber and Rogers33). Because the samples used in these investigations were either powders or solutions, the resulting EPR spectrum is either an average of the randomly oriented VO+2 ions in the frozen or powdered samples (Bleaney,35 Sands,31 Searl, Smith and Wyard36 or Ibers and Swalen37) or a time averaged spectrum in solution as a result of the motion of the V0+2 ions (McConnell38 and Rogers and Pake39). Because of the need for EPR information on oriented* VO+2 and because of the success of producing the various oxidation states of vanadium by Lambe and Kikuchi23 and Wertz, Auzins, Griffiths and Orten,40 this present work was undertaken. Underlying these immediate reasons is perhaps a deeper one-that of a systematic study of the solid state chemistry of vanadium. *There are brief comments on oriented V0+2 in VOS04o2H20 by Hutchinson and Singer,41 as well as some unpublished investigations on VO+2 in ZnK2( S04)2 6H20 by Gagero42

CHAPTER II CONCLUSIONS A. VO+2 The results of the present EPR experiments on oriented VO+2 indicate that the vanadyl ion is substituted for the divalent zinc ion in single crystals of Zn(NH4) 2(S04)2-6H20 having its V-O axis oriented in one of three directions. The three directions are found to have a population ratio of 40:10:2 with the larger two population positions having their V-O axes at an angle of 77048' to each other and the least populous position having its V-O axis inclined at 98~1' to the other two. The room temperature spin Hamiltonian constants differ slightly for each position, with the larger population position having: gz = 1.9331~2 Bx = 0.007137~4/cm gx = 1.9813+2 By = 0.007256+4/cm gy o1.9801+2 oxy = -0o0000462+2/cm A = 0.018281+5/cm lQ'| = 0.00024~2/cm The quantity0 xy('xy - Fxy+Fyx) results from the inclusion of terms when the principal axes of the g tensor do not coincide with the principal axes of the A or hyperfine tensor, The experiments indicate that the z axes of the two tensors coincide, but in the xy plane the angle between the x axes of the two tensors is 23~20'. Using the values for My, Bx, and By, we find the principal axes values of the hyperfine interaction 10

11 constants to be Axx = 0.0071200/cm, Ayy = 0.0072439/cm, and Azz - A. B. IRRADIATED V0+2 The results of the present EPR experiments on V+2 and irradiated VO+2 in single crystals of Zn(NH4)2(SO4)2o 6H20 indicate that the X-ray, or the subsequent high energy electron it produces, breaks the V-O bond, resulting in a V+2 EPR spectrum having the same spin Hamiltonian constants and z axis orientation as crystals of Zn(NH4)2(S04)26H20 grown with V+2. The measurements on the irradiated VO+2 crystals give the following room temperature spin Hamiltonian constants: gz = 1.9717~5 E = 0.02280+3/cm gy = 1.9733+5 A = 0.008263+5/cm D = 0.15613+3/cm B = 0.008246+5/cm with A and D having the same sign. Figure 3 depicts these results quite vividly. In this cry'stal the V0+2 spectrum is reduced in intensity by X-ray irradiation a factor of two for a dose of 5xlO6 rads. Although saturation occurs after 1x108 rad, resulting in a 80-90% reduction in the VO+2 signal, only 10-20% of the vanadium has been converted to V+2. An EPR experiment carried out at 4.20K failed to show any of the vanadium in trivalent form. C. TWO MODELS FOR VO+2 IN Zn(NH4)2(S04) 2'6H20 Ballhausen and Gray43 have proposed a molecular orbital description

12'T~'~.'...'~'! o~ o' ;-, Jt - fL,i....."........ i....~,......... - i1 -::i:::,:::-' ST! —I. —'lI:1-' X'. I W~~~~~~~~~~~~~~~~~~~~~~~' -.-. —. -, e A I i~ > Iq- =1.:......i!.' 1_;_i'" j ID oI i i~!_.......... o L:o o o e 1 1 1 i 4::": i 1: 1! j Ij..........-. _.o _; t a IC Q i _ai-1: —-....,_....... O''. t __.> -I 5T to~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(..,. 9 - -t 0; 1 2 -ito cd......_i................. c=. _ -:..........L, ~~~ —-- ~~-~~~~~ ~ ~ - -[ —— l?| —{- [l Lt!;|i|g1, m?, E t. Y 1 i 1. - o.:~..L1T T:T ~ T::-T - f F t W a } O l, Ol, I...... ~I ~,.........1 9 |-: - - - |- -,-' 1 i.. i................ |-;-[i l. it i-Ltl —i i L i CD~~~~~~~~~~~~~~~C ~~~~~~~~~~~~~~ —- 3~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - -_.l;-j o!-l | —' | 1' 1-a1 i r ~t - l (i,!!o 40 to m.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~..............~ x~~(O( i f ~~~~1~- I~ ~~N I rI~ ) 0 0 ~ ~ r ~~~- - ~L~+ ~~ — l~-~I~...~i-'-~~-~ ~~ —~-1 — t`1 r

.13 of the vanadyl ion currounded by five water molecules found for VO+2 in solution and VO+2 in VOS042H20 (the vanadium is located at the center and the oxygen and the five water molecules are at the vertices of an octahedron). Their description agrees so well with optical absorption data and the g values reported in the literature as well as the g values found in these experiments, that it should be considered that the VO+2 has this pentahydrate environment in the Tutton salt. One other experimental result supporting this pentahydrate model is the three orientations of the V-O axes found in this crystal. If one assumes the validity of the pentahydrate model, the results of the present experiments showing the irradiated VO+2 EPR spectrum and the V 2 EPR spectrum to be identical imply that the oxygen removed from the VO+2 ion takes the place of the missing water molecule of the octahedron, producing a crystalline field identical, as far as EPR can determine, to that for V+2 surrounded by six water molecules. An alternate "model" results if one considers the V-O axis pointing in the (111) direction of the cube where the six water molecules are placed at the centers of the faces. This arrangement predicts four possible directions of the V-O axis, but since the octahedron of water molecules in the Tutton salts is quite distorted, the four directions are ordered in energy and thus in population. The fourth direction, not found experimentally, is then expected to be so energetically unfavorable that it is not detectable by EPR techniques.

14 D..FURTHER STUDIES While the major questions of the EPR spectrum of VO+2 in Zn(NH4)2(S04)2.6H20 and the conversion of V0+2 to V+2 in this same crystal seem to be answered sufficiently, several unanswered questions suggest further work, such as; (a) an investigation of the superhyperfine structure on the vanadyl resonances that arise from the protons of the surrounding water molecules. That this structure is due to the protons is verified by its absence in crystals grown with D20. Replacement of the H20 by D20 reduces the line width of the vanadyl resonances by onehalf, and the smaller magnetic moment of the deuteron (0.85738 n.m. as contrasted to that of the proton, 2.79275 n.m.) reduces the strength of interaction so that the superhyperfine spectrum due to the deuteron is unable to be resolved. This investigation might be best pursued with ENDOR techniques and might provide information as to the correct model of V0+2 in the Tutton salt and a molecular orbital description of the complex; (b) a more accurate measurement of the strength of the quadrupole interaction also best accomplished with ENDOR techniques; (c) an investigation of a vanadyl radiation dosimeter. Vanadium in glass is observed to be in the form of the vanadyl ion and showed a decrease in signal intensity upon X-ray irradiation. Thus, the possibility of a vanadyl glass dosimeter immediately suggests itself.

CHAPTER III EXPERIMENTAL METHODS A. CRYSTAL GROWTH Single crystals of 0.05, 0.1, 0.5, and 1,0% concentrations (vanadyl to zinc) of the Tutton salt Zn(NH4)2(S04)2.6H20 were grown by evaporation from a water solution into which ZnS04'7H20, (NH4)2S04, and VOS04.2H20 were dissolved in the prescribed amounts. Translucent light blue crystals up to 3-4 mm long wer obtainable in 1-2 days, Zinc potassium and magnesium ammonium Tutton salts containing 1.0% VO+2 were also grown in this manner.* ** For the crystals containing heavy water, or D20, the water of hydration in ZnS04o7H20 and VOS04o2H20 was first removed by heating, and then the above evaporation technique was employed,*** In contrast to the relative ease of the growing of the vanadyl crystals, the growth of the 1.0% and 2.0% (V+2 to Zn) Zn(NH4)2(S04)2.6H20 crystals required a greater degree of skill because of the rapidity at which V+2 becomes oxidized in solution, Cathodic reduction of the 1.0 and 2.0% (VO+2 to Zn) zinc ammonium sulfate solutions, in which the platinum cathode and anode are separated by a porcelain cup, followed by evaporation in a carbon dioxide atmosphere yielded opaque lavender*ZnS04. 7H20 —Mallinckrodt, Analytical Reagent; (NH4)2S04 —Allied Chemical, Reagent; V0204.2H20-Fisher, Purified; MgS04 —Allied Chemical, Reagent; and K2S04-Mallinckrodt, Analytical ReagentO **Activation analysis of the 0o5% V0+2 Tutton salt showed a concentration of 0.004% V/Zno This analysis was performed by Mr. Ho Nass of the Radiation Chemistry Group under the supervision of Profo We W. Meinke of the Chemistry Department of The University of Michigan. ***The D20 was obtained from Norsk Hydro (Norway) and is 99,78% pure. 15

16 colored crystals 2-3 mm long in 3-4 days. B. CRYSTAL STRUCTURE AND ORIENTATION Figure 4 shows the unit cell of a typical Tutton salt. The location of the atoms on the unit cell is taken from the X-ray data listed in Wyckoff,44 who lists the space group of these double sulfates as Ch P21 2h la, In this crystal there are two molecules per unit cell and the water molecules surrounding the (O, 1/2, 1/2) position are derived from the (0,0,0) position by a translation to (0,1/2,1/2) followed by a reflection in the ac plane. Although the several views of the octahedron in Fig. 4i show a large deviation from cubic and tetragonal symmetry, suggesting that any EPR results from paramagnetic ions placed in the di-' valent site will need to be fitted with the rhombic spin Hamiltonian, Wyckoff44 remarks that because of the closeness of several of the atoms certain distances'can only be taken as approximate. Yet EPR results do indicate considerable rhombic symmetry (see Bleaney, Ingram and Scovill6 as well as the results obtained in Chapter V of this investigation). As shown in Fig. 5, these double sulfates or Tutton salts grow with well-recognized faces (Tutton45), so that orientation of the crystal becomes a task accomplished with relative ease. Figure 5 also shows the two devices for orienting the crystal with respect to the magnetic field. The one device positions a 22-in., 0.25-in.-diam, 0.025-in.-wall quartz rod to the crystal so that when the rod is vertical the adesired orientation is in the horizontal plane-the plane of rotation of the magnet.

6.225 A q. - 9.205' A 3 51 A92' t 4.82 A d IAxis\ 2.30A 12.475 A UNIT CELL OF MaHj(_~).Q.6__,O TWO MOLECULES PER UNIT CELL, DATA FROM WYCKOFF, CRYSTAL STRUCTURES, VOL. III jjaen QX NH4 0 s o o Mg Scale -H4 I I Angstrom Fig. 4. Unit cell of Zn(NH4)2(S04)2..6 H20

SURFACES OF A i t t A W DUAL MODE, RECTANGULAR, i ~<~~~~~~~~~ *~4.,~~~OPTICAL,CERAMIC CAVITY DEVICE FOR INITIALLY POSITIONING QUARTZ ROD IN RELATION TO THE CRYSTAL Fig. 5. Faces of Tutton salt, crystal orienting devices.

19 The position of the crystal is fixed in this device by the grooves that are machined at the same angle as the crystal growth faces. The crystal is attached to the quartz rod with Goodyear Pliobond cement. After placing the crystal, now fastened at the end of the quartz rod, in the cavity and the other end of the quartz rod in the second orienting device, one makes the final adjustments of the crystal's orientation with respect to the magnetic field by moving the quartz rod in either of the directions indicated in Fig. 5 until the desired EPR spectrum is displayed on the recorder. C. EPR EXPERIMENTAL ARRANGEMENT EPR spectra were obtained with a Varian V-4500 EPR Spectrometer, Varian V-4560 100-kc Field Modulation Unit and a Varian 4012-35 12-in. rotating electromagnet. The schematic diagram of the spectrometer system is shown in Fig. 6. X-Band System. For the VO+2 crystal measurements a Varian V-153/ 6315 Klystron operating at approximately 9.3 kmc was coupled to a dual mode (TEol) rectangular ceramic cavity internally coated with a themosetting silver paint (Hanovia 32A). The thin silvered walls of the cavity allows 100-kc modulation of the magnetic field at the sample crystal. Measurement of the klystron frequency was accomplished by means of tapping a small amount of power from the microwave circuit via a coaxial cable adapter to a Berkeley 7580 Transfer Oscillator and a Berkeley 7370 Universal EPUT and Timer. Such a measurement system is

20 K BAND SYSTEM VARIABLE ATTENUATOR SPECTROMETER ATTENUATORI~~~ I(VARIAN 4500) PHASE SHIFTER KLYSTRON POWER. SUPPLY UNIT X BAND SCRYSTEMAL TO CV-4500-AV4 M ITY SHIFTER ~~~RCATTENUATORCR V153/6315 FERRITE VARIABLE KLYSTRON ISOLATOR ATTENUATOR AND20DB CRYSTAL WAVEGUIDE TO COAXIAL CABLE 0 CONNECTOR DUAL -MODE CAVITY - X-T UNIVERSAL EPUTREOD (VARIAN G-10) AND TIMER (BECKMAN 7370).. / 12 INCH ELECTROMAGNET (VARIAN 4012-35) CEOSCILLATOR FIELD MODULATION UNIT (BECKMAN 758) (V (VARIA -N'45O) Fig. 6. Schematic diagram of EPR spectrometer system.

21 accurate to one part in 106. The system was operated at room temperature with a magnet gap of 3.25 in. K-Band System. The V+2 and the irradiated VO+2 crystals require the use of a higher frequency klystron since the zero field splitting of V+2 in the Tutton salts is of the approximate value as the X-band frequency (see Lambe and Kikuchi23). A Varian VA-98 Klystron operating at approximately 24-kmc was employed with a 2.0-in. magnet gap. The cavity for these measurements was a dual mode (TEo11) cylindrical ceramic cavity. The klystron frequency was measured with a Hewlett-Packard K-532A absorption type wavemeter having a least count of 0.010 kmc and a resettability of 0.002 kmc. Magnetic Field Measurements. Magnetic field measurements of the individual resonances were made with a Varian Model F-8 Fluxmeter connected to the Berkeley 7580 Transfer Oscillator and the 7380 Universal EPUT and Timer. For magnetic field intensities below 8000 gauss measurement of the proton magnetic resonance frequency is used (4205776 mc at 10 kilogauss). For higher fields corresponding to the V+2 measurements at Kband frequencies the resonant frequency of the deuteron is used, resulting in a slight lowering of the accuracy of the magnetic field (6~536 mc at 10 kilogauss). Measurement of the magnetic field was made directly beneath the cavity, resulting in only a -0.1 and +0.4 gauss difference between the magnetic field intensity at the position of measurement and the position of the sample for 3300 gauss (3o25-in, gap) and 8600 gauss (2o0-ino gap), respectively. For the measurements on the crystals con

22 taining D20 this value was-0O77 gauss. C Axis Position. For the Tutton salts the relation between the magnetic axes and the crystalline axes was obtained by measuring the angle of the c axis to the zy-plane of the V+2 spectrum or the c axis and'the K1K3 plane of the VO+2 spectrum. The measurement was made with a short focal length transit, With the crystal mounted so that the zy-plane was horizontal, as recognized by the EPR spectrum on the recorder, the optical axis of the transit could be aligned along the c axis of the crystal since an intersection of two of the growth planes of the crystal lay parallel to the c axis (Fig. 5). Of course, the length of the intersection of the two planes determines the accuracy with which the angle can be measured, but for a 4-mm length of intersection of the crystal growth planes such a measurement could be made to within ~0,03~ X-Ray Irradiations. X-ray irradiations of the VO+2 crystals were made in the shite beam from a tungsten-target, Machlett AEG-50S tube operated at 50 KVP-40 ma. Using cerrous-cerric dosimetry after Weiss46 we estimated the dose rate to the crystals to be 7x106 rad/min (Appendix A)o Although Weiss46 found the cerrous-cerric yield to be independent of energy from 100 KV X-rays to 2 Mev gamma rays and independent of dose rate from 1/2 Roentgen/sec to 500 Roentgen/sec, and although the above measurement falls out of both ranges, the Machlett tube sheet claims that emission may be more than 2x106 Roentgen/mmn under certain conditions so that the experimental value seems approximately correct.* *The energy absorbed by the crystal appears to be greater than the tube emission as a result of the higher absorption coefficient of the crystal compared to that of air,

CHAPTER IV THE HAMILTONIAN For electron paramagnetic resonance phenomena almost all of the experimental results can be described by a phenomenological Hamiltonian called the "spin Hamiltonian."* While the spin Hamiltonian is truly phenomenological in that one can postulate it on the basis of symmetry, we shall give an account of the "derivation" as usually seen in the literature. We consider the ion imbedded in a crystalline field produced by the charges or dipoles from the surrounding ligands** and enumerate the interaction terms of the n electron ion in a crystalline electric field subject to an external magnetic field; Interaction Energy wi — _Zn rK __ e I/ e) K WI -- Y\ **Recently Lacroix and Emch49 have generalized the spir Hamiltonian to include the effects of covalent bonding. 23

24 Interaction Energy e tv r r0 3( ( oc Wi-/31_L-La s] + -H-g 1 Wc L\l (cIrysl;leid) The energy of the crystalline field can be noted experimentally to fall into three classifications, (1) Strong fields (7104/cm) where there is considerable covalent bonding, Then (2) Medium fields (~103/cm) as in most ions in the iron series and in this *orko Then (3) Weak fields (~102/cm) as in most complexes with the rare earth elements. Then The reason for making the distinction is that our perturbation treatment of the crystalline field on the ion depends on where, in terms of interaction energy, Wcf falls in the series of X = Wc+Wso+Wss+Wn+Wqo Since the crystalline field in these experiments falls into classifi

cation (2), we assume the single electron approximation. That is, the ion is constructed of single electron orbitals such as (ls)2(2s)2(2p)6 (3s)2(3p)6(3d), which are solutions of the hydrogen-like Hamilton, P2/2m-Ze2/r. For the closed argon shell, [A] = (s)2(2s) 2(2p)6(3s)2(3p)6, we note that the mutual repulsion term 2.. merely adds to the total energy of the shell. However, in our paramagnetic ion with the unpaired 5d electrons this term gives rise to certain configurations of the 3d electrons that are lower in energy than others. Since we need to know the ground state of the "free ion" in our crystalline field one assumes Russell-Saunders coupling of the 3d electrons and constructs antisymmetrized wave functions I to calculate matrix elements of the type < T IWcfl I > so that the configurational energy can be computed. The result is a confirmation of Hund's Rule that the ground state has (1) maximum spin multiplicity and (2) maximum orbital degeneracy consistent with maximum spin multiplicity. Thus for 3d,, 3d2, and 3d3 electrons the ground states, are 2D, 3F, and 4F. A. THE CRYSTALLINE FIELD In order to calculate the splitting of the ground state due to the crystalline field, which is assumed to satisfy Laplace's equation \72C=C, we expand VK in a series of normalized spherical harmonics and precede to calculate the one electron matrix element for the 5d elec

26 trons (Appendix B). We note that (a) if inversion symmetry exists about the paramagnetic ion, all odd terms disappear since VK must have even parity and Ym has parity (-), and (b) no terms exist for I > 4 for 3d electrons since the direct product of two d orbitals span no representation higher than I = 4. Also, since VK must conform to the symmetry of the ligands producing VK, other terms may be zero. Thus we obtain for cubic fields, vc AQ 4\JO + or, Calculation of the matrix elements for this cubic field for the ground states 2D 3F, and 4F results in the splittings shown below where the numbers in parenthesis indicate the orbital dengeneracy and the letters Eg, Tg, and Ag refer to the symmetry of the state wave function. The energy level separation parameter Dq is defined as Dq r4 where 6 al5 Q is the charge on the ligand and r4 is averaged over the 3d radial function.

27 (0) Az 0!) Az. (z)z 6% 6 D 4MEU'F; Fbo- T,,~lt BXt TIa 4D \ I- - (3) 7 C?) cut:ic t-~_aq, CU6C cub IC +e+ra, cubic ~i.e cubto,,+-a't) c Cub > (convtraot d) (lo,, ae ) ri,t ) If there is an elongation (contraction) of the charges along the z direction then there is a lowering of the symmetry from Oh(cubic) to D4h(tetragonal). The crystalline field then contains the additional terms B2r2Y~+B4r4Y~ and the degenerate levels split as shown above or in Figs. 1 and 2. In calculating fields of lower symmetry (rhombic) Ballhausen (Ref. 13, p. 108) points out that it may be necessary to include other terms in the Hamiltonian such as spin orbit coupling or dynamical effects (JahnTeller) rather than augmenting the crystalline field potential with a rhombic field component. The important feature one should notice is that the lower symmetry reduces the degeneracy of the ground state to an orbital singlet. In fact, a theorem by Jahn and Teller50 states that a molecule having orbital degeneracy always distorts to remove the degeneracy. The exceptions are linear molecules and Kramers degeneracy.* *Kramers theorem states that a system containing an odd number of electrons will have even degeneracy if placed in an external electric field.51

28 B. THE SPIN HAMILTONIAN The terms in =W= Wso+Wn+Wq+Wmf which are left from the original Hamiltonian are nbw treated as a perturbation to the system. But instead of calculating the change in energy directly with this perturbation Hamiltonian Y, we use the operator equivalent method of Stevens52 to obtain a transformed Hamiltonian valid for constant L and constant So This method consists of replacing the individual electron spin variables, which transform like the components of a vector, by the components of the total spin operator S and replacing the position coordinates, which transform like second order spherical harmonics, by the components of L. This is valid providing that we remain within a manifold of constant L and constant SO With this transformed Hamiltonian the change in energy of the orbital singlet ground state is calculated to second order using only the orbital part of the wave function-now in the L,S representation. The result is the elimination of the orbital operator leaving a Hamiltonian with only spin operators-the spin Hamiltonian. X~ s.g.S tS~DS -IS_ t I'QI_5 +A a The first term contains the spectroscopic splitting factor g and differs from 20023 as a result of admixing from higher states. D is a measure of the splitting of the ground state in a noncubic field and gives rise to the fine structure, while the th third term containing A expresses the hyperfine interaction, The fourth and fifth terms express to first order the quadrupole interaction and the interaction of the magnetic

29 field with the nuclear moment, respectively. A constant factor has been dropped since it shifts all levels equally. Since the spin Hamiltonian must conform to the symmetry of the ion in the crystal we note that for axial, cubic, and rhombic symmetry surrounding the paramagnetic ion and providing the principal axes of each of the tensors coincide, we obtain the following spin Hamiltonians for cubic, axial and rhombic environments. X b - ah H ~ Is -Sags H I _i3 H3l 49eeI 0 (l4 tS~ DH( S -L (si + TjS In general the principal axes of the tensors may not coincide in which case cross terms may appear. For the example of the g and A tensors not having the same principal axes, additional hyperfine terms of the type Fi.(SiIj+IiSj) may appear~'4- 3 3X Jr 4- Q1 2:IZ Jt (M 4 The spin Hamiltonian is truly phenomenological in the sense that we could have postulated it from a posteriori arguments on the type of interactions and the symmetry of the environmento In fact for electronic spins higher than 3/2 such as Mn+2 (S = 5/2) in cubic environments, one

3o0 must include a term quartic in spins (i e, a(S4+S4+S4). Although higher r' x Y zo order terms may also be allowed it can be shown that any monomial in Jx' Jy, and Jz of order more than 2J can be reduced to a linear combination of monomials of order less than or equal to 2Jo Koster and Staatz53,54 have obtained a more general Hamiltonian that is less restrictive than the spin Hamiltonian. However, the number of constants in their Hamiltonian requires data to be obtained at high and low magnetic fields for a proper evaluation of the constants. Also the difficulty of application of this Hamiltonian increases as the symmetry surrounding the paramagnetic ion decreases. Because of this difficulty in using the Koster-Staatz Hamiltonion and that of relating the results of this work to that of other investigations, we chose the spin Hamiltonian as the method of evaluating the experimental data. The derivation of the angular variation of the energy levels from the rhombic spin Hamiltonian is done in Appendix C and an "exact" calculation of the fine structure energy level separation for the magnetic field parallel to the x, y, and z axes is carried out in Appendix D. Also, the angular variation of the energy levels for the g and A tensor not coinciding is calculated in Appendices E and F. Finally, Appendices G and H list the equations that were used to reduce the data.

CHAPTER V EXPER IMENTAL RESULTS A. VO+2 +2 From the EPR measurements at X-band frequencies of the VO ion in Zn(NH4)2(S04)2.6H20 crystals, the room temperature rhombic spin Hamiltonian constants with noncoinciding principal axes of the g and A tensor listed in Table 1 were calculated for the three observable locations of the two sites in the unit cell.* TABLE 1 EXPERIMENTAL RESULTS FOR VO+2 Location 1** Location 2 Location 3 gz = 1.9331+2 1.9316+3 1.9299+4 gx = 1.9813+2 1.9808+4 gy = 1.9801+2 1.9797~4 1.9811+10 A = 0.018281+5/cm 0.018275+5/cm 0.018441+10/cm Bx = 0.007137+4/cm 0.007104+4/cm 0.007250+10/cm By = 0.007256+4/cm 0.007255+4/cm xy = -0.0000462+2/cm IQ' | = 0.00024~2/cm 0.00024~5/cm Line width (H20) 5.0+0.4 gauss 5.0+0.4 gauss 5.0~0.4 gauss Line width (D20) 2.4+0.2 gauss Relative intensity 40+10 10+2 2+1 *The errors associated with each of the numbers in the table is the experimental error. **For this location the constants are listed for the crystals containing D20. With the exceptions of reduction of line width and the altering of superhyperfine structure, the presence of D20 or H20 produces no detectable difference on the spin Hamiltonian constants. 31

32 Figure 7 shows the positions in the crystal of the z axes of the rhombic spin Hamiltonian for the above locations. The positions are referred to the three mutually perpendicular magnetic susceptibility axes, K1, K2, and K3o The K3 axis coincides with the b axis of the crystal and the other two axes lie in the ac plane. Experimentally, these three axes show the positions of the magnetic field where the spectra from the four most populous sites coincide. These positions can be located with respect to each other within ~+0,3~ B. DISCUSSION OF VO+2 RESULTS Comparison of Spin Hamiltonian Constants. As mentioned in Chapter I, the published spin Hamiltonian constants ofthe vanadyl ion have been obtained from randomly oriented samples and that this was one reason for attempting single crystal measurements on this ion. The following will show how one can obtain the axial spin Hamiltonian parameters from a powdered spectrum before comparing the constants found in this investigation to that published in the literature. Since the spin Hamiltonian for VO+2 in Zn(NH4)2(SO4)2.6H20 shows almost axial symmetry, the EPR spectrum of a random orientation of VO+2 ions should be able to be -explained quite easily from the single crystal spin Hamiltonian constants. With.the assumption of axial symmetry we see that for a given magnetic field only those crystals or crystallites whose V-O axes are at an angle G to the magnetic field are contributing to the resonance-where 9 is related to the magnetic field by

—. —— i-. -- -i-l —--- -i A- I#. It i -.- R — i -A -:8- V-O AXIS,"Z"~~~~~~~~~~~V-0AXIS1121 v-C AXIS2"40" EPR SPECTRUM WITH MAGNETIC FIELD FARAPLLEL TO KI AIXIS \ V_ — AXIS "loO' 51.10K3b 19.10 \ V-O~~~~~~~~~~~~~~~ AXIS," IO" 98. u" XIS,"40" / ~~~~~~~~~~51.10 ~ V-OAXIS:'2' ~ 51.1" 19'1 60.50 60.5 ~~~~~~16' 16' (Wyckoffl EXPERIMENTAL RESULTS LOCATION OF V-0 AXES IN Zn(NH4)p(SO4)2 6H,O *2" INDICATES RELATIVE INTENSITY Fig. 7. Psto fV +2 axes in Zn(NH4)2(S04)2.6g20..

34 H(9) ~ y - 0p4 - +)(Cs and where,,L "l; F(T- + I I Figure 8 shows the plot of such an equation evaluated for the single crys +2 1ins d9, The number of VO axes at this angle is proportional to - sinG dG, which is just the probability that a vrO2 axis lies at an angle o to the magnetic field. Thus, the number of VOt ions contributing to paramagnetic resonance between magnetic field H and H+dH is proportional to Figure 8 shows the plot of such an equation evaluated for the single crysperimental curve in this figure is the first derivative of absorption showing the effect of the finite width of the resonances. The sharp peak

FIRST DERIVATIVE OF CRUSHED CRYSTAL ABSORPTION SPECTRUM GAUSS 271 2910 3060 3109 3181 3252 3309 3415 3512 3603 3716 3922 4130 3117 3330 3506 UJ1 ABSORPTION prbItrary units) mO 9O PLOT OF sine[d FOR 0.1% VOSO Z*H4nP0k6VO F=-o Fi g. 8. sin a (de/d8~m) curve and VO+2 powder spectrum.

36 ing at the 90~ end of the curves is a result of a [cos 0] 1 dependence of the equation. This can be interpreted as saying that, since there is axial symmetry, more V-O axes are oriented at 90~ to the magnetic field than at 0~O In the powder spectrum the separation of the two extreme peaks corresponds to 7A/gz5, and if one neglects the quadrupole contribution the separation of the two extreme lines in the central part of the spectrum (if they are distinguishable) is 7P/g~o Thus, if in addition to measuring the frequency of the microwave source, one measures the magnetic field at these four locations, the spin Hamiltonian constants gz' gl' A, and B can be calculated to the accuracy limited by the width of the peaks and the ability to pick out the significant lines in the central part of the spectrum. Figure 9 compares the spectrum of the crushed VO t Tutton crystal to that of vanadyl ions adsorbed on IR-4B and V205 dissolved in glass.* The latter spectrum shows that vanadium in Na20'3SiO2 glass, melted under either oxidizing or reducing conditions, is present as VO2. The larger width of the peaks in the glass and resin spectra, as contrasted to the sharpness of the peaks in the crushed crystal spectrum, reflects the nonuniformities in the crystalline field from ion to ion. Table 2 compares the literature values of the axial spin Hamiltonian constants to those found in this investigation. *The glass samples were supplied by Dro W. Nelson, Owens-Illinois, Toledo, Ohio~

V0*2 O AMBERLITE EPR SPECTRA OF RANDOMLY ORIENTED V0' 0.5% V205 IN (X BAND).tp. 510St GLASS.0% vO" IN ZnCN( VSIj-56 H.0 (crushed crystol) 7 101 _____fl~~~~~T Al __ _ Fig. 9. VO2 in glass, iR-4B, and crushed crystals.

38 TABLE 2 LITERATURE VALUES OF vo+2 SPIN HAMILTONIAN CONSTANTS gz gL IAJxlO2/cm IBIxlO2/cm Material Present work 1.9331 1.9805 1.828 0.7137 Tutton salt Faber et al.33 1.93 1.983 1.80 0.750 IR-100 1.88 1.979 1.84 0.740 Dowex-50 1.983 0.704 Charcoal 1.93 1.989 1.58 0.612 IR-4B Roberts et al.34 1.947 1.988 1.58 0.54 Vanadyl porphyrins O'Reilly32 1.948 1.987 1.59 0.52 VEP I Kozyrev29 1.92 1.96 1.78 0.7 Frozen solution Gager42 1.950 1.995 1.80 0.75 Tutton salt

39 Fit To Spin Hamiltonian. That the rhombic spin Hamiltonian with noncoinciding principal axes of the g and A tensors is a good description of the EPR spectra of the vanadyl ion in Zn(NH4)2(S04)2.6H20 is illustrated in Fig~ 10. The curves in this figure are predicted from the spin Hamiltonian with the values listed in Column 1, Table 1, and the circles represent the experimental values of tho magnetic field. The crystal for these measurements was grown from heavy water, reducing the line width of the resonances and enabling one to make more accurate measurements of the magnetic field * Noncoinciding Principal Axes of the g and A Tensors. One of the unusual results of this set of experiments is that the x and y principal axes of the g and A tensors do not coincide but in fact are separated by 23~20' If one uses the first order expression for the magnetic field in the xy plane of spin systems a reduction of 2 in the line width increases the slope of the derivative curve at x = 0 by a factor of 4-2-, increasing the accuracy with which one can measure the magnetic field at resonance. Since the magnetic moments of the surrounding nuclei produce inhomogeneities in the magnetic field at the paramagnetic ion and the deuteron has a magnetic moment less than one-third that of the proton, the line width of the resonances can be reduced by growing the crystals with heavy water. Experimentally the width is found to be reduced by a factor of 2.

40 3619 3618 3617 3616 3615 3614 3613 3522 3521 3520 3519 2 M=- 5 3518 3517 3516 - 3430 i KR KM FIELD 3427 X 23X20'~IN XY PLANE FOR VOEIN 3425 - 0 0 4 6 80LAN 3343 MAGNET ANGLE Fig. 10. X-Y plane phase shift. 334X AU K, K, (O 3128 - -80 -60 -4'0'.~o 10 ~o 2 EXPRIMNTA Fig. 10. X-Y plane phase shift.

41 where $><-C3g1tdx 2nA 91 Sc gSivlZ0 and 6 is the angle between the magnetic field and the x axis, as the major contribution to the angular variation of the magnetic field, then it can be easily shown that the maxima and minima of the resonance magnetic field occurs when n (S+- B) Y)1 +rn;3 -( X ) for this experiment * From this result one notices that if the principal axes of the two tensors coincide then&7 xy = 0 and the maxima and minima occur at 6 = 0 and 90~ for all values of m, This is the usual result encountered in EPR experiments. If one assumes that the z axes of the g and A tensors coincide but the x axes do not, then it can be shown (Appendix E) that the principal values of the hyperfine tensor, Axx and Ayy are related to ~xy EBx, and By by the following expressions *Second order terms such as (Bx-By)(g gg) are ignored.

42 =x " AyyCODSG + AcSlozg xU- (AM? Axn)s' n c~s' where 9 is the angle between the x axes of the g and A tensor. Using the experimental values for xy, Bx, and By listed in Table 1, the principal axes values of the A tensor are found to be A4X>- 0.0071/zQ /c (- o0071367 /G. ) Al\'O Oo 72z39/cM (a=o o 7zS9Lf/ ) and the angle between the x axes of the two tensors to be7 G = -233 -~0 The assumption that the z axes of the two tensors coincide may not be a valid assumption since the first order terms involving xz and.yz appearing in the expression B-s (i~coszv2s- -j+ (A 2) 4.StrtCose@ x. 8;E - disappear when G = O. In fact, aHal ( )) = sog is j ovio- erO]

showing that the maxima and minima occur at 8 = O and 90 for all values of m —exactly what is observed.* However, one may argue why the z axes of these tensors, should be expected to coincide, If one first considers an isolated vanadyl ion, the V-0 axis will be the z axis of axial spin Hamiltonian and the z axes of both the g and the A tensor since the charges on the vanadium and oxygen are the only physical reasons for producing a preferential direction. Then, for example, if one places the vanadyl ion in a weak crystalline field produced by an octahedral arrangement of water molecules, the surrounding ligands may destroy the pure axial field of the vanadyl ion, setting up an x direction for the g tensor and the A tensor. The z axes of these two tensors will remain coincident or almost coincident is to be expected since the z axis of the ligand field is weak in comparison.with the cylindrical field of the vanadyl ion. That the vanadyl crystalline field is very strong is evidenced by the similar spectra found in the investigations (28) to (33), indicating that the vanadyl ion carries along its own crystalline field. Also the fact that gL > gz can be explained if one assumes the vanadyl ion to have a very strong cylindrical field (Refo 44 and Appendix B)o *Other than including second order terms, which does not appear practical, the only way to detect the possible noncoinciding z axes of the g and A tensors would be to look for an asymmetry in the resonance magnetic field for positive and negative values of O. In this manner the term involving ~ xz, yz changes sign as ( goes through 0~o However, since O can be measured to only ~0ol~, this error can produce a ~0.7 gauss error in the measurement of magnetic field which might mask the observed asymmetry. Table 3 lists the experimental and calculated values of the magnetic field for 0 between O and 90~, The calculated values are obtained from the axial spin Hamiltoniano

TABLE 3 V0+2 MAGNETIC FIELD RESONANCE VALUES Degree Magnetic Field 7/2 5/2 3/2 1/2 -1/2 -3/2 -5/2 -7/2 0 Calculated 2714.0 2910.8 3109.5 3309.5 3512.3 3716.5 3922.6 4130.5 Experimental 2714.1 2910.9 3110.0 3310.1 3512.7 3716.8 3922.8 4130.2 Difference -.1.1 -.5.6.4.3.2.3 10 Calculated 2720.0 2914.7 3111.0 3308.8 3508.2 3709.3 3911.9 4116.1 Experimental 2719.4 3911.9 Difference -.6.0 20 Calculated 2733.3 2922.8 3113.0 3304.1 3496.0 3688.9 3882.9 4077.9 Experimental 2736. 3 2925.0 3114.8 3305.8 3496.0 3688. o 3880.8 Difference 3.0 2.2 1.8 1.7 -.0 -.9 -2.1 30 Calculated 2766.2 2944.2 3121.8 3299.3 3476.7 3654.35 3832.3 4010.8 Experimental 2764.6 4010.5 Difference -1.6 -.3 40 Calculated 2805.1 2967.2 3131.6 3292.5 3452.3 3611.4 3770.1 3929.0 Experimental 2803.8 2968.2 2130.9 3291.9 3452.2 3611.7 3769.6 3929.0 Difference 1.3 -1.0.7.6.1 -.3.5.0 50 Calculated 2854.4 3000.9 3144.2 3285.2 3424.5 3562.7 3700.7 3839.0 Experimental 2854.2 3000.5 3144.9 3283.3 3422.9 3561.7 3699.4 3837.8 Difference.2.4 -.7 1.9 1.6 1.0 1.3 1.2 60 Calculated 2912.0 3037.2 3158.7 3277.6 3395.0 3512.2 3630.2 3750.3 Experimental 2911.3 3749.2 Difference..7 1.1 70 Calculated 2974.2 3074.4 3172.1 3268.8 3365.8 3464.8 3567.2 3674.5 Experimental 2973.1 3073.7 3268.1 3365.4 3567.6 3675.4 Difference 1.0.7.7.4 -.4 1.1 80 Calculated 3032.7 3104.4 3179.4 3258.1 3341.2 3429.2 3522.7 3622.2 Experimental 3030.7 3179.8 3257.8 3523.0 3622.2 Difference 2.0 -.4 -.2 -.3.0 90 Calculated 3059.2 3115.7 3179.1 3251.9 3330.7 3416.0 3507.3 3604.2 Experimental 3059.2 3115.5 3180.1 3252.0 3330.8 3415.8 3507.1 3604.1 Difference.0.2 -1.0 -.1 -.1.2.2.1

C. TWO MODELS OF VO+2 IN Zn(NH4)2(S04)2 6H20 Model 1. If one considers a regular octahedron of water molecules located at the center of the faces of a cube whose dimension is approximately 4t (the Tutton salts) the length of the V-O bond, 1l67A,55 and the ionic radii for V+4and o+2, 06 and 0+2 060and 1. 3, respectively,56 require the direction of the V-O axis to be along the body diagonal of the cube. If this is true, there are four equivalent positions of the V-O axis in a regular or elongated octahedron of water molecules. In the Tutton salts the six water molecules do not form a regular octahedron but rather one with considerable rhombic symmetry. This is borne out by the X-ray data, Wyckoff,44 and the V+2 EPR data presented in the following section. This rhombic distortion of the water molecules makes the four positions of the V-O axes nonequivalent and ordered in energy and thus in population. Experimentally, there are only three positions with a population ratio 40:10:2. The fourth position might be expected to be so energetically unfavorable that its low population makes it unobservable. One check on this model would be a comparison of the angle between (111) and (111) directions in a cube, 70~335, with that of the angles that the three V-O axes make with each other. Figure 7 shows the angle between the z axes of the two most populous positions to be 77~48' and that between these two axes and that of the least populous position to be 98~

Model 2. Ballhausen and Gray43 have constructed a molecular orbital description of the vanadyl ion in which the vanadyl ion is surrounded by five water molecules in the form of a pyramid-the vanadium at the center of the base and the covalent bonded oxygen sticking out perpendicular to the base a distance of 1o67o. Ho h a po67A Such a description is found in V0204.2H20 and is also supposed to be the description of the vanadyl ion in solution —see Ballhausen and Gray.43 With this molecular orbital description of this model they found gz = 1.94Q, g1 = 1.983 and < g >=1o969. They also obtained good agreement with optical absorption spectra and magnetic susceptibility measurements. Although the g values found in this work differ slightly (this work: gz = 1.9330, g1 = 1.9805 and < g > = 109647) from that of Ballhausen and Gray43 it should be considered possible that their pentahydrate model of V0+2 may be that found in the Tutton crystal, particularly since this pentahydrate model willshow only three orientations of the V-0 axis when the model is viewed as an octahedron. However, if

47 this model is to be correct, the result of producing a V+2 spectrum from VO+2 by X-rays identical to that of V+2 surrounded by six water molecules implies that the oxygen released from the vanadyl ion by Xray irradiation must take the place of the missing water molecule producing a crystalline field identical to that of V+2 surrounded by six water molecules. Vanadyl Superhyperfine Structure. An interesting experimental result is the additional structure found on the resonances of the vanadyl ion for a particular orientation of the magnetic field (Fig. 11). The five resonances are each separated by approximately four gauss and result from overlap of the wave function onto the protons of the surrounding water molecules (i.e., superhyperfine structure). That these additional resonances arise from the protons is verified by the fact that they are unobservable in the crystals grown with heavy water. Since these additional resonances are superhyperfine structure a measurement of.the strength of this interaction may provide some information on the molecular orbital description of the vanadyl ion and the surrounding water molecules and the validity of one model or the other. D. V+2 AND IRRADIATED V0+2 CRYSTALS From the EPR measurements at K-band frequencies (-24 kmc) of the vanadous ion in Zn(NH4)2(S04)2'6H20 crystals, the room temperature rhombic spin Hamiltonian constants shown in Table 3 were calculated

56.3-ft.- 64.6- -7 2.0- -78.8 85.0- 92.3- 97.1 —V EPR SPECTRUM OF.1% VOSO4in Zn(NI14)e(SO4)6H2O WITH!MAGNETIC FIELD ALONG "TETRAGONAL" AXIS Fig. 11. Vanadyl superhyperfine structure.

49 by means of the method in Appendix H. TABLE 4 EXPERIMENTAL RESULTS FOR V+2 V+2 (Grown) V+2 (:Irradiated VO+2) v+2 (Bleaneyl6) gz 1. 9717+* 1.9718+5 1.951 gy 1.9733+5 1.9733+5 |DI 0.15613+3/cm 0.15609+3/cm 0.158/cm I E 0.02290+3/cm 0.02297+3/cm 0.049/cm A O. 008263+5/cm 0.008267+5/cm 0.0088/cm BI O. 008246+5/cm 0.008249+5/cm 20.5 ~0.50 20.5 +0.50 220 r 2~+5~ 3~+1~ 2~ Line width 6.0+0.6 gauss 6.0+0.6 gauss 6.0+0. 6 gauss *The errors associated with the values in Tables 4 and 5 are the experimental errors and do not include the 0.o08 error on the wave meter calibration. **Oa and Flocate the z axis of the spin Hamiltonian with respect to the ac plane and the c axis. For the V+2 produced by X-ray irradiation in the three different vanadyl Tutton salts, the following room temperature rhombic spin Hamiltonian constants in Table 5 were calculated~

50 TABLE 5 EXPERIMENTAL RESULTS FOR V+2 (IRRADIATED VO+2) Zn(NH4) 2( S4) 2 6H20 Mg( NH4) 2(SO4) 2.6H20 ZnK2( S04) 2 o 6H2 gz 1o.9718 1o 9720 1.9722 gy 1o 9733 1.9723 1.9741 JDI 0.15609 0.15793 (1.2%) 0.15244 (-2.3%) I E 0.02297 0.02452 (+6.6%) 0.02742 (19.2%) IAI 0o.008267 0.008262 0.008253 I B 0.008249 0.008253 0.008212 Y/ 3~+10 1~+10 110+20 20 5~+0.5~ 20.00+0o5~ 147~0+0,5~ E. DISCUSSION OF V'2 RESULTS As seen by the close agreement of the V+2 (grown) and V+2 (irradiated VO+2), spin Hamiltonian constants and the identical (within experimental error) location of the z axes for both cases leads to the following conclusions. Either, (1) the VO+2 ion existing in the Tutton salt before irradiation is surrounded by six water molecules, and after irradiation the released oxygen moves far enough away from the divalent site so that its influence on the crystalline field is no longer felt; or, (2) the vanadyl ion in the Tutton salts is the pentahydrate model of Ballhausen and Gray,43 and after irradiation the released oxygen takes the place of the missing water molecule, completing the octahe

dron and producing a crystalline field identical to the one with the V+2 surrounded by six water molecules. That the crystalline field must be identical for Vt2 and irradiated VO+2 is also supported by comparing the spin Hamiltonian constants and z axis positions for V+2 (irradiated VO+2) in the three Tutton salts listed in Table 5. These results indicate that the substitution of different host ions in the monovalent and divalent positions distort the octahedron an amount that is measured by the changes of D, E, a, and ~.* Note that the value of E reported by Bleaney, Ingram and Scovil,l6 and listed in Table 4, is almost twice the value found in these experiments. Since they used the EPR data along the z direction to evaluate both D and E, and since E enters the equation in this direction as a second order correction (Appendix H), we expect that their value is not as correct. Fit to Spin Hamiltonian. Table 6 compares the experimental and calculated values of the resonance magnetic field for the z and y axes for each of the eight resonances in the three fine structure groups. Note that although the spin Hamiltonian is fitted to an average of the magnetic field values in a fine structure group (Appendix H) the individual values of the calculated resonances deviate in a consistent *The result that changes in the monovalent ion influence the position of the z axis in the crystal to a great extent was noticed by Bleaney, Penrose, and Plumpton57 in their EPR investigations of Cu+2. They also noticed large changes in the g values, l1%, which were not observed in this work.

TABLE 6 v+2 MAGNETIC FIELD RESONANCE VALUES (GAUSS) Z Axis (v = 24.118 kmc) Y Axis (v = 24.112 kmc) Calc. Diff. Exper. Calc. Diff. Exper. 12439.6 (-.7) 12440.3 11468.1 ( -4 3) 11472 4 12348.5 (.9) 12347.6 11376.8 (-.6) 11377.4 12258.1 ( 1.8) 12256 3 11285.0 -(.9) 11284.01 12168.3 ( 1.9) 12166.4 11194.1 ( 2.1) 11192.0 12079.2 ( 1.3) 12077.9 11103.9 ( 2.7) 11101.2 11990.8 (. 1) 11990.9 11014.4 ( 2.4) 11012.00 11903 0 (-2.2) 11905.2 10925.0 ( 1.4) 10924.2 11815.9 (-5.1) 11821.0 108375 ( o 1) 10827.6 9029:6 ( 107) 9027.9 8854o9 ( o3) 8853.6 8937.0 ( 3) 8936.7 8762.6 (. 3) 8762.3 8845.4 (.6) 8846.o0 8671.2 (-.6) 8671.8 8754.7 (-1.0) 8755.7 8580o0 (-1.0) 8581.8 8665.0 (-1.4) 8666.4 84913 (-1o 0) 8492.3 8576.2 (-.7) 8576.9 8402.7 (-.7) 8403.4 8488.2 ( o2) 8488.o 8315.1 ( 1) 8315 0 8401o3 ( 1.7) 8399.6 8228,4 ( 1,5) 8226,9 5676.0 ( 7.3) 5668.7 6644.2 ( 2.0) 6642.2 5580.2 ( 1.5) 5578.7 6552.2 (-.6) 6552o8 5486. o (-2.3) 5488.3 6461.4 ( -1o 4) 6462.8 5393.2 (-5.2) 5398.4 6371.9 (-2.0) 6373.9 5301.9 (-5.7) 5307.6 6283~7 ( -14) 6285.o 1 5212.2 (-4.6) 5216.8 6196,7 ( 3) 6196,4 512359 (-2.0) 5125.9 61110o ( 2.9) 6108 1 5037.2 ( 2.5) 5034.7 6026.5 ( 6.2) 6020.3 manner that corresponds to second-order hyperfine correction that is too large by a factor of two or three. No explanation suggests itself unless it is that the experiment is performed at a microwave frequency insufficiently removed from the energy level crossover which mixes the levels M, m-l and M-l, m (Lambe and Kikuchi23). It should also be noted

53 that, if the experiment is performed at a frequency too close to the zero field splitting value 2D, the measured value for 2D will be smaller than the correct value. X Axis Data. If one performs a general coordinate rotation on the rhombic spin Hamiltonian in order to diagonalize the Zeeman term (Appendix C), only when the magnetic field is along the x, y, or z axes do the cross terms become zero, simplifying the spectrum and thus enabling one to make precise measurements for the spin Hamiltonian constants. The effect of this mixing is illustrated in Fig. 12, which shows the EPR spectrum with the magnetic field parallel to the z axis of one of the two sites in the crystal. Note that, as a result of this mixing, the spectrum from the second site is so spread as to make it almost unobservable. Figurel? shows the effect of not having the magnetic field precisely parallel to either the x, y, or z axes. In this figure the magnetic field bisects the angle made by the x axes of each of the two sitesapproximately 10~. Since the spectrum from the one site could not be differentiated from that of the second site, the magnetic field measurements for the x direction could not be made to any precision; hence, no gx value is listed in Tables 4 and 5. F. CONVERSION OF VO+2 TO V+2 Because irradiation of the vanadyl crystal results in the breaking of the V-O bond and the production of V+2, it might be expected that all the vanadyl ions might be able to be converted to V+2 in accordance with

::~ T —— i';:-'. —_-s —t!-.-tt.-:. ~ =:.-:-_'f-"t-'1 — - - -: --- — I — - m:_ _ m _ _ _..... LL LOW FIELD CENTER FIELD HIGH FIELD MAGNETIC FIELD PARALLEL TO Z AXIS 1% V In ZnS04(NH4S04-6HO, V=24 KMc Fig. 12. V+2 spectrum along Z axis.

55 the exponential relation N(V+2) = No(VO+2)(l-e-R), where No(V0+2) is the number of VO+2 ions originally present in the crystal and R is the energy dose absorbed by the crystal. While the initial conversion of VO+2 to V+2 by irradiation may follow such a law, some mechanism interferes to prevent more than a reduction of 80-90% in the vanadyl signal intensity and only 10-20% of this amount is converted to V+2. These results are illustrated in Fig. 13, which shows the conversion of VO+2 to V+2 as a function of irradiation time.* The degree of saturation appeared to vary slightly for the three different host crystals with the same initial concentration of VO+2, but remained constant for different sizes of the same host crystal, grown from the same solution, indicating that the X-rays penetrated the entire crystal. Also a slight annealing affect was noticed. These same effects, saturation, variation in saturation for "different" crystals, slight annealing, and loss of the reduced signal were also observed in the experiments by Lambe and Kikuchi23 and Sturge58 in the conversion of V+3 to V+2 in sapphire. *While a graph of the absolute intensity of V+2 and V0+2 versus dose might be more meaningful in light of the observation that the decrease in VO+2 signal intensity is not entirely accounted for in the increase of the V+2 signal, such measurements were a practical impossibility with the apparatus used in this set of experiments. Also, comparison to a standard such as hydrazyl was impossible since an irradiation center was produced in the crystal masking the hydrazyl resonance.

10 CONVERSION OF VO TO V\4BY X-RAYS (DOSE RATE 7x 106 RADS/MIN.) 8 6 4 RELATIVE INTENSITY I I 20_ _ I 1 10 20 30 160 IRRADIATION TIME (MINUTES) Fig. 13. Conversion of Vo+2 to V+2.

APPENDIX A X-RAY-DOSE CALCULATIONS In this appendix we wish to compute the X-ray dose given to the crystals, using the data obtained from the cerric-cerrous sulfate dosimetry of the Machlelett AEG-50S operated at 50 KVP 40 ma. The experimental setup for the irradiations is shown below ~2~crystal i/ 3.57cn pos~zon 4'costn~yv~oco kT, rce The dose rate measured by the dosimeter is 0.73x106 rep/min. If we can assume that the X-rays are monoenergetic at an energy of their maximum intensity* (33 KV for 50 KVP), and that the absorption coefficient of the crystal is that of aluminum, then the dose rate absorbed by the crystal is *Kulenkampff lists an empirical law for the white beam of an X-ray tube to be I(X) = CZ(l/\2)(l/ho-l/2)+BZ2(l/\2) where B ~ C and can be ignored for practical purposes.59 Z is the atomic number of the target, X the wavelength, and o the cutoff wavelength. 57

e (tT d 09 S@e.ai) STU'eA3 Uioje aG S [uaioejjGoo uo-qdIosqe asaTl*,KJ ~ A,'e-j......Z-C W-Pl-9' * he? / /aJg6 X 7)/xit

APPENDIX B CRYSTAL FIELD CALCULATIONS In this appendix we wish to calculate the splitting of the ground state 2D, 3F, and 4F in cubic and tetragonal fields and then calculate the effective g values for such environments. We assume that the paramagnetic ion is surrounded by point charges and that there is no overlap of charges of the paramagnetic ion and the surrounding ligands. Then the crystalline electric field in this region has a potential satisfying Laplace's equation VVO which has the general solution Y)=( XA \(gB Assuming that we have a point charge Q on each of the j surrounding ligands leads us to express V(; ) as V( ) A fre where the vectors r~, ~, and t form the triangle shown below. 59

6o From the law of cosines we may express V( U ) as ) ~}- Cr2 Z+ra - aero ap M 7 i O ( +iFit By expanding %CosoA.) by the spherical harmonic addition theorem, V( r, ) becomes:r) — > { ( YK ( Ad (XXd) (B) Then, comparing like terms in Eqs. (B-1) and (B-2), we find Athi-ant Q ) tu ( )) (B-3) 1. SYMMETRY CONSIDERATIONS By exploiting the symmetry of the locations of the ligands surrounding the paramagnetic ion, we can find that many of the Aen vanish. For example: (a) If the polar axis (z axis) possesses p-fold symmetry, the coefficients Am vanish unless m is zero or an integral multiple of 2X. That is, so that which is satisfied only when m = O, +p, +2p,... (b) Since V( ) isto be real, r =c] implies f A-.)~= A o

61 (c) If there is a reflection plane through the polar axis, and if this plane is the x-O-z plane, then V(t) = V(-5) which implies This together with (b) gives the result that the An are real. (d) If there is inversion symmetry about the paramagnetic ion, then and IYI Since the Ym have parity (-l)n we obtain the result 4%(> - 4r foo(e) For d electrons we see that the matrix elements <391 V(frV)- > will span no Am with n > 4. Thus for 4-fold symmetry about the z axis we need to consider only those terms involving AX,A,4,ACubic Fields. For cubic fields we see that from Eq. (B-5) Performing the summation for the ligands located at ~a1 along each of

62 the coordinate axes, we find that Now dropping the term involving AS since it shifts all levels equally and dropping the subscript i, we find that V( vl ) becomes 4 0A~ Ys 0 ( 4- X4 Note that V(r>Q O)- /rt2:) ) so that A~ is related to A4 by the following /"4-'l r4- 4Thus, for cubic fields " = - ~, - 4~ + (.,~ )(~ t i-';)~ or, -evbc - A ) y <x+y4* 4 - + r4] where from (B-3) Tetragonal Fields. If we lengthen (shorten) the distance along the z axis so that the charges are located at a distance +a2 along the z axis and +al along the x and y axes, then

63 V( })-V+etr Vc.obc V$*+r/'~ =- 2tY where in Vcubic ~; TT~- 7 as before. 2. 2D( dl) Cubic Field. Group theoretical considerations can show that the 2D state splits into a three-fold degenerate level, T2g, and a two-fold level, Eg, for cubic fields. If we anticipate the results and write the orbital parts of the wave function of the five-fold degenerate level as T.> 17( z+ v 5 Zr - ) E7> j ( I t'> Y)= ()- ( —;)/r Z 1 0 I- ~) _12 (

64 then the matrix elements, > become diagonal in this "cubic harmonic" representation. The results are that the three-fold T2g level is depressed by an amount 4Dq and the two-fold E2g level is raised 6Dq, where Dq - F2 + and r4 is averaged over the 5d radial function. Tetragonal Field. Since the cubic part of the tetragonal field has already been evaluated above, we need only to calculate the matrix elements of /,~\ / V te+r Q ( () y Q As z (3?2t-r) + 3 (5 3r 3 rt) where I Note that both A and B have the same sign and are negative (positive) if the octahedron is lengthened (shortened) along the z axis. Also _AX ea for I1- Ql<. Since it is easier to calculate the matrix elements in the operator equivalent form,52 we write V+4r, A 3 L2 LL(L+). B S I5 -L- -30. L(L+ ) _ZLZ-. /X &AX1 i X /tl)1l;

where G<(-al and 23 - are found by evaluating a matrix element of Vtetrag and V (OP) for 2D. Using this operator to calcutetrag tetrag late the matrix elements of the tetragonal crystal field yields the following energy level changes* AE(C) _ a +a 1 7 awhich are shown graphically below. "D cub',c -ctbLs rD' " (elonIa*Ballhausenl3 defines and r D = Dt

66 3. 3d2 (3F) Cubic Field. Group theoretical methods can shown that the sevenfold orbitally degenerate F state splits into two orbital triplets and one singlet. Again, we anticipate the results and write down the linear combination of state functions that diagonalize the matrix elements of the cubic field 8 - 4 ~3

67`4T= (~;, Z)- 4 -- ( ) -4) If we use the operator equivalent form for the cubic field VC9lI (Bl7b a3L5 [35Lf -30 L(L-t) L2 a5 Lt -oL(L+t ) +3LZ(L+')*4 ( Ltd LL4) ] where f can be evaluated to be - 2/315 for 3d2(3F), it is found that the seven-fold degenerate level splitting is T — (4 2 ) Tetragonal Field. From the operator equivalent form for V+e,+ra \/I+,_,(,oe = o Aa 3 -L(L-I-'] -,..35 L4 -30 5OLL(L+ I) +25 L -6L(L0)*3Ld(U I)~ where 01 = - 2/105, p = - 2/315 for 3d2(3F), we find the following energy-level splittings for the A2, T2, and T1 levels due to a tetragonal

68 field: -35A~- v TAe: AS~w L~a~tz1)~ 3 6 r4 /\L(IL / -%U3)S=A v21r These results are shown graphically below: Ar (I) 6Dt t LT (3) (j) cubic tetrF. (eionga )d 4. 3d3 (4F) Cubic Field; Tetragonal Field. For the 3d3(4F) ground state it can be shown that CZ = + 2/105, 5 = 2/315, in contrast to o = - 2/105, = - 2/315 for the 3d2(3F) state. Thus the energy-level splitting for

69 this 4F state, including the tetragonal field, is obtained by inverting the 3F levels. This results in the energy diagram below:* Th (3) (1) 6D8 4F(7) D_ ()?2D T2(3) ODi A' ( /) cubic.tefrad (elornaj d) 2D 5. D g VALUES The results from crystal field calculations of the 2D state in a squashed tetragonal field is summarized below: 2D (') /D' cubic _ear__ (s uashec) *If the octahedron is squashed, the tetragonal levels are inverted since A and B change signs.

70 However, if we consider the vanadyl ion to produce a strong cylindrical crystal field as a result of the covalent bond between the oxygen and the vanadium, the 2D state splits in the following manner: (1) 3?2-r rz 4 XL cyit n cai tTc To compute the g values, we first calculate the correct wave functions for the ground state when the spin orbit interaction \L~S is included. The corrected wave functions, to first order, become hr h ( X- o the a w fantions sAl that where o, B are the spin parts of the product wave functions such that S+-' is /3 3o Stn) (-<

71 The ground state is still degenerate but lowered an amount Applying a magnetic field results in an interaction of the form causing the energy levels to separate by an amount Using degenerate perturbation theory to first order, first with H parallel to z, and then with H parallel to the x axis results in Now if g~ > g1, which happens to be the case for the vanadyl ion, then 4D > Dr, which is very easily satisfied when the crystal field is assumed to have strong cylindrical symmetry. 4F g Values. For the 3 electrons in the 4F ground state in cubic or tetragonal field the first order g value is calculated in a similar manner and is found to be /I \ \

APPENDIX C ANGUIAR VARIATION OF ENERGY LEVELS OF THE RHOMBIC SPIN HAMILTONIAN We start with the rhombic spin Hamiltonian with the principal axes of the several tensors coincident (Chapter III) Y-3 p Llsqs, +XX, lfS%+ 5 + DLS -`s(s~ 0i]S #E (is> 3)_'A;IS,+ RXISX~By4"'9 A, StAQl+ 3 + rtl)J ( C-1.) t Q" Ll'-LtSH.I|~4 {2txIx H I- L H, 1. ZEEMAN TERM Since the Zeeman term, has the largest interaction by at least one order of magnitude, we proceed to make a coordinate transformation in order to diagonalize this term and then treat the additional terms by perturbation theory. The coordinate system x', y', z' in which the Zeeman term is diagonal is related to the x, y, z coordinate system by the Eulerian angles 9, ~ and V as shown below: 72

73 st & Then, since A/, - (e, S_)e, + e( ~ e' (~3 ~s 8 where el, e2, e3 are unit vectors in the x, y, z directions respectively, we find that the components of S in the x, y, z coordinate system are related to those in the xt, y', z' coordinate system by the following equations: X(iose<cos<4-5Pi ncod'R5 n A-SS (-COss vP-slrcoSP sCo Y) + SE si n G S n s s5 -- S O( coS$YcosGcosh f\IN+) + S (-C$: S I cSt)CoSiOCOS(6/) k(SV (-coS + S lc i(P)) Now, if

k-ix = \J- srn x cosS 1 S - I:1 LS\n S\ YAt' J-~ - S cos ~ and we require COS c- -- 0S Sln where and sZ _ 0ecoS~ + (Yfz (C-3) then (i.e., the Zeeman term is diagonalized) 2. FINE STRUCTURE TERMS With this transformation the axial and rhombic fine structure terms become

'-"~~~~~~~~~~C t o K CA Fri ~~~~~~~~~~~~~~~~~~~~c+ U T_, 0 o ~L c~ ~ ~-~ za H Crs —, CE t~~~~~~~~~._- 5~ (A O M ITI' -P O oI (Ifl~~~~~~~~~~~~~~~~~~~~~~~~~~O 00( -t ~~~~~~~~~~~~~~~~~0 -'qhx ImI~" ci -I Fj 6d - U~~~~~~I Nie- r~ P_ (A~~~~~~~~~ v, Q 5 C CA 9nla rj; ~ 1 O 0 I P- -- $~~~~~O r~ c )C- (D i- VI sri 3 4~~~~~~~~~~~~~~~~~~~~~~~A -AC C~~~O D COO3 ~ r-rg ~~~~~~~~~~r~ (03 -t~~~~~~~~.~(A ( ED ~~~~~~4 Jrn 4flp r\~~~~~P D Z ) r\~~ wr CA r CS-~~~~~~~ ca ~~~~~k r ru n~~~~~~~A CAV Ln o~~~~~~~~~~~j c~~~~~~~~a3Z x s~~~~~~~~~L__

the first being the diagonal contribution, the latter two resulting from second order contributions. 3. HYPERFINE TERM For the hyperfine term AIzSz+BxIxSx+ByIySy the coordinate rotation matrix for the spin operators Ix, Iy, Iz is | o blC~SW WlS~tt ~\/ i ~ O C05& Sjn~g O -.cosin cosC 9 KC where the determinant of each matrix set equal to unity defines B1 and K; that is ACos2 + BtsL = K X cosz~, By S,' - K The resulting equations for Iz, I+, and I_ become CinO / c 4 L (- cosN -i3sin)IT'_@ (83X LC+-i........ (c-6) 4- (XCOSn (I A COSY2 )I- ____, a, ~~~~~~~~~ K K ~~~~~~~~~~~ Z~~~,",~ _

77 Putting these equations, as well as those for Sx, Sy, and Sz (C-2), in the hyperfine term and evaluating the resulting expression for the state IS.t1.n> yields the following addition to the energy level +' s,~- -Co~ 5Z0 I(I $-'5ryl2~1+h:M OO)c B i A U l X( (+ H~oss-,a,.)] s'~,.- os- (.,-a,A) BZ + AK...... %of x (c-71 ) - + C. C!Eas cc +~ coscc ( 743 where the first term is the diagonal contribution and the other terms are evaluated from second order corrections CO

78 4. — QUADRUPOLE TERMS The quadrupole term QI _- 3(I+ ] +[ Q Ix -] rotates like the fine structure term so that its contribution to the energy level can be written immediately as (I I cos, cosLL' sn os 3 W ~n(8H s, n os'~ + -QS _' ( I e4-T Ks oz l)- t sln') cs5S 42 ~ 5. NUCLEAR INTERACTION TERM For the term involving the interaction of the magnetic field with the nucleus(S. N~ s.-;T H>1, j tthe same straight forthe nucleusl the same straightforA + si H Yn K _BiS~Ot Cos s i r 8 8) slh'gNgBy*C 5~5Z-S 1 3 CZn A~ _ lncs OgN952- Ka-xcS6)(c9

79 Thus, to second order the angular variation in the energy level EM from the rhombic spin Hamiltonian is given by In EPH work the energy for resonance is given by the selection rule AM~l, Am = 0. Evaluating the expression Emhjfor this transition we obtain: 4-/ 0,- Lo X~~r, —- _ L,,oI l ~~r r!~ o'"O ~ VT) ci 4l + __ V + ~ ~.(+ L~RJ ~ ~ (4V" U r16 VI:5u LI)~~~~V NX 4-4~~"~u r- VID - 0;o 00 -IN rl 4 +, O, - V.s (J-JJ I 13 T S'I)r ~- W 11 3 010 + U~ v, Wd 7~~J ~* ti **i J4J< ri -4 Lii + nl ~ I 4r- II V) rdY V) 44 CrC) ~~L~ 6b 9in — II

+I- +. — + + H~~~~~~~~ t~Ht i~, t — c~.H (/I I-I "I xS%~~~~~~~~~~~~c c 0~~~~~-0I x rtri3 80 -f C (A Il ~ I-' ~ ~~ -I- Io 0U''t 3 ~ ~ Is -, cn VI ~~~~~~~~~~~~~~-L~~~_S 9 (A rr~~~~~~~~~~~Jr OP c aim 9 VI s OD! I I o ~ ~ n a - LA ~A - v p ci ~ ON CD co IA~ plC~1.h z I bI Cs -1 IV 0 CA~ ~~~~~~~~~~~~~0C Pi RA~~~~~~~~~~~~~~~~~~~L a tr' pl~~(A10 Nt t z Is0a I z CIP APO:' (AC ~ rQn~cl 0.0 IA~~~~~~~~~~~~~~~r Jr 0,~~~~ x'VA On x CZ0 I Cj\ crj I I~~~~~~~~~~~~~~~~~~~~~n/3~~~r rJ C~~~~-3 O~ N 3 17N

APPENDIX D FINE STRUCTURE ENERGY LEVELS (RHOMBIC) Given the rhombic spin Hamiltonian we wish to calculate the energy levels for the magnetic field in the x, y, and z directions for only the Zeeman and fine structure terms. 1. H PARALLEL TO z AXIS Degenerate perturbation calculations leads to the following secular determinant: -z _~ xl'~ -. 2. _ H4 2 xi N 12t i 3 Z)%t which may be solved for the following energy levels,X#\ —. 2-../ \,;' - 81

82 _ s A = — to - #3t'" (go2 —) Zb - For the transitions AM = +1 the change in energy is t.Pi...... -I al w p.... not diagonalTo diagonalize a Taycoordinate rotation about the square roots and solving by 0for the magnetic field (tin the major term) yields the following equfine structure spin Hati-on to fnianourth order) (A) 3/2 34 -ED --- Q 27E4 (B) L,71 D tonian

83 which leads to the following secular determinant: -3 gx + gz;-3 D d+EE X (DtE) C\5Dt 3 L13 | C (t E) _ E 3 L IIL,~~~D-sE~Z

APPENDIX E g AND A TENSOR PRINCIPAL AXES NOT COINCIDENT In this section we would like to note the effect on the hyperfine term I.A*S when the principal axes of the A tensor do not coincide with the principal axes of the g tensor. We denote by x, y, and z and x', y', and z' the principal axes of the g and A tensors, respectively. Then, performing a coordinate rotation R on the term I.A.S, we obtain in the unprimed coordinate system I AS IR-R/AR-KRS where 0 0 A( Ac;I (Ix,;e5 K and cscP sinY) o I\ C \(cosG snG 0Q R = (-ank COSLP 0 0 |c CcsQ 5n 0 o-S COSOg 0 0 /0- oS/ O O 0 where 9, I, L are the Eulerian angles relating the primed to the unprimed coordinate system as shown below.

.0IO, For the special case when the z axes of the two tensors are coincident, V = 0, and we may choose ~ = 0 also. Then IR-1 RAR-1 RS becomes, dropping the primes on the Aii, (TxIX i, )C0o510tA SlSILO Slag 0 rCOS 9(A-Hx7A 0 \ s svi Gcos (AA'-Ax AX5S@ + ACc6S z I S0 sjIz ((AixoyztvAG5 */%)+SMc oszx5i tO) tA,,S T C 1C jA jA \C,3~la~ IA,/\\,lay.a~

APPENDIX F ANGULAR VARIATION OF ENERGY LEVELS FOR APPENDIX E As shown in Appendix E, when the principal axes of the g and A tensor do not coincide, cross terms arise so that the hyperfine interaction term in the spin Hamiltonian ATSet +,IySX B tIs (E-1) has to be supplemented with terms of the form'F tSa t E G, (LsTJ- Sd i )I (E-2) where the prime on the sum denotes iVj and we have written the offdiagonal terms in a symmetric and antisymmetric form. However, the hyperfine tensor may be written in a diagonal form (E-l) so that any orthogonal transformation will leave the A tensor symmetric —thus Gij - 0 Writing 2 ij - Fij+Fji, we obtain for the form (E-2) the following terms: ai;t~~~~~~~~~~~~~ T I (E-3) Since we want the angular variation of the energy levels due to the hyperfine term, we perform a coordinate rotation on the spin operators as in Appendix C and obtain for the transformed equation of (E-3) 86

+ I e -C~~~~~~~~- CA r i miA r In2 H H t70 X~~~O 03 < rz It/% t C-~~~~~~~~~~~~~bCA a. ~n 9L)~~~~~, c- x-!~r'7~~~~~~~~ —?~~~~~~~~~~~~~~~~~~~~-''r'..'/c n ~ ~ OC (A~( 1 X- C-Z -2 CA 5 -t n a, 1Lit 42M C, cU~Q I )C I~~~ica OO~ C cn~~~~~~~~~( IA 5 fS~~~7K(A(, VI, D 3 Cc ~~Sc VP c 8~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~I IT - a 1 CW 4 - LEN

88 Thes-e terms arising from the off-diagonal contributions of IoPA S when the principal axes of g and A do not coincide must be added to the diagonal contributions of Appendix C before matrix elements of this part of the Hamiltonian can be evaluated for the state ISM,Im >. Since the resulting equation would be extremely complicated in general, and since we are primarily interested in this approach where the phase shift is the largest, i.e., the xy plane, we set p = O. The hyperfine terms then become: + I -q1 - a5wY)' 0.- +- Ci Dv os-ugl6- 6 41ItI 1 c pl) -l' r BxI lf2L+ F 3 C At -9 sost -- sG) ~ -. 4- cE-9'~) ~rlIl - titSt'-fs31: u IS'I I't (E-5)

89 Applying perturbation theory to second order on (E-5) gives the variation of the energy levels in the xy plane. Using the selection rules AM = +1, Am = 0 and solving for the magnetic field as in Appendix C results in the following equation describing the variation of the resonance magnetic field for S= 2 -- if we neglect the quadrupole terms: _h Lm Fx+g~3 a- t13 ___ ____ _z(_ 4 CA + S - En S2(S ale ~atlS S 8Q04 ~ 2. -L 4-b ~l~~s~boS i ~,(S+,j-v~2 iA U~'~U~ Uu o~ (n~cos26a~n s~n~di t-t~f~ b~OVj + 4 - Z. n "alp H at~~~~~g + I W,6b C~] Sims Z-

APPENDIX G DATA REDUCTION FOR V0+2 CRYSTALS From Appendices C and F the resonance magnetic field can be evaluated along the x(O = 6/2, 5 = O), y(O = t/2, 6 = i/2), and z(i = 0) axes to give the following equations:'4y.,_'r5JH,A...M(M-I.) El. Lofiap 8'~' -'(1_ ~r. -I/-';' (s-i4 M(M( -M +-i) xFOx (~~ —'- ~~ + 4. A e) I7+3Z 7- Q Q~~~9

91 Since the magnetic field for each of the 2I+1 resonances is measured, we choose to evaluate the spin Hamiltonian constants in the following manner. Z Axis. For S = 1/2, I = 7/2, the above equation for Hz reduces to 48L" -1~~- A By/rL (G-2) IAI and IQ". To evaluate QIT | one can look at' W -h-1-FA A +41?t \% 6 _M (G-3) for the various values of m. This same expression can also be used to compute H/gzo. ogBo To evaluate gz we use 8v~ X e Se 3Z g,2>+~- (G-4) X Axis. Similarly, for the x axis Hx in (G-l) reduces to the following for S = 1/2, I = 7/2: __ /3~v#1( (Q'4(slYmy.{LL9_ A G-5) Forming 1/2m [Hx(-m) -Hx(m) ] one can evaluate Bx/gxP and ( Q'+Q") 2. And with 1/8 [~jHx(m) ], gx can be computed. Y Axis. For the y axis a similar treatment on the data is performed by using

92 A m' 61 YY IPnF 2. (G-6) -~xy'. In the xy plane the term involving the quantity 8 xy was evaluated at _ 45~ ~. At this angle.dl: t0-H A+( k~t

APPENDIX H DATA REDUCTION FOR v2 CRYSTALS From Appendix D the rhombic spin Hamiltonian gives the following equation for the fine structure part with the magnetic field parallel to the z axis (we also add the hyperfine terms evaluated by perturbation theory from Appendix C): a H 3/2 KSJ Similarly for the magnetic field along the y axis: H - ~/h D+3E _ ~. f (-)lD~3)34 _ _ e CLH')-ktSA)24 K_ h-I A3/ 1 9 D -3L — A,, a7 A- B 11 ila8- e/tne8i3 H33ld aH7 i \3;t, I D32__,_,r-E B * - -2 gD- 4Dt). (D. ) [I t 3 Z gy| 9 I) (

94 Since the magnetic field for each of the 21+1 resonances is measured, we choose to evaluate the data in the following manner: for the z axis and the value for D, M Z -77/2~-C1 L "' + H3~L- \Li - Sf25C3 5%F ls 1 for gz, Similarly for the y axis and the value for E, X 72 )=D F3 Eta o: (~ HD+3E A 1 ( l "' S / Jt D) 3'- -3 S. 2S2+)-S i X / D t-._,, Using the experimental data for the sums on the lefthand side of the equations and evaluating the equations in a self-consistent manner yields the data listed in Tables 2 and 3, Chapter IV.

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