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OPTICAL AND PARAMAGNETIC RESONANCE ABSORPTION SPECTRA OF CHROMIUM IN YTTRIUM ALUMINUM GARNET By ROBERT WALKER McMILLAN A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1974 ACKNOWLEDGEMENTS The author acknowledges with gratitude the help and encouragement of the many people who have aided in completing this work. He is grate ful to Mr. James Gallagher for suggesting the problem which formed the basis for this work, and to Mrs. Thelma Torvik and Mrs. Dorothy Averill for carefully and patiently typing this manuscript. Messrs. Joseph Schell, James Stonebraker, and John Rawls did most of the machine work, Mr. Robert W. Phillips helped in computer programming, and illuminating technical discussions were held with Drs. Harry Bates and Norman Barnes. Also, help and encouragement have been given by Drs. Bob Landrum and Vincent Corcoran, and by Messrs. Edward Kelly, J. Ronald Thornton, Paul Rushworth and William Smith. The author is especially grateful to his committee chairman, Dr. Ralph Isler, who made several key suggestions which led to the successful completion of this work, and to the other members of his committee and his teachers at the University of Florida, who have provided challenge and given inspiration. The support of the Orlando Division of Martin Marietta Aerospace, which provided time, equip ment, and facilities, is also gratefully acknowledged. The author is deeply grateful to his wife, Ann, and to his children, Marisa, Robert, and Natalie. They have willingly sacrificed much to help him finish this study, and have never failed to give encouragement. This work is dedicated to them with love and gratitude. CONTENTS Page Acknowledgements . . . ii Abstract . . . v I. Introduction . . . 1 I.A. Brief Historical Sketch . . 1 I.B. General Crystal Field Theory . 3 I.C. General PMR Theory . .. 4 I.D. Outline of Procedure . . 5 II. Theory of the Crystalline Field in YAG Doped with Chromium 10 II.A. Expansion of Crystalline Field in Solid Harmonics 10 II.B. Determination of Correct Zero Order Wave Functions 13 II.C. Calculation of 4F Crystal Field Matrix Elements .. 16 II.D. Interaction of the 4FF4 and 4P States . .. 23 II.E. The Effect of the Trigonal Distortion in First Order 29 II.F. The Effect of the Trigonal Distortion in Second Order 33 II.G. Exact Solution of the Crystal Field Matrix Including 4F/4p Interaction and Trigonal Distortion .. 36 III. Theory of Paramagnetic Resonance in YAG Doped with Chromium 41 III.A. Perturbation of the Ground State by the Magnetic Field and Spin Orbit Coupling . . 41 III.B. Determination of the Ground State Energy Levels with i Parallel to the Symmetry Axis . ... .46 III.C. Determination of the Ground State Energy Levels for an Arbitrary Magnetic Field Orientation . .. 52 III.D. Calculation of Ground State Energy Levels Including Trigonal Distortion Effects . ... 54 IV. Experimental Procedure . . .. 56 IV.A. Sample Preparation . . ... .56 IV.B. Optical Absorption Measurements . .. 60 IV.C. Paramagnetic Resonance Measurements . .. 60 CONTENTS (Continued) Page V. Results .... . .. 69 V.A. Paramagnetic Resonance Measurements . .. 69 V.B. Optical Absorption Measurements .. 76 V.C. Determination of the Crystal Field Parameters ...... 81 V.D. Accuracy of Results .... . 85 V.E. Summary and Conclusions .... . 87 References . ... . 92 Biographical Sketch ..... . . 94 Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy OPTICAL AND PARAMAGNETIC RESONANCE ABSORPTION SPECTRA OF CHROMIUM IN YTTRIUM ALUMINUM GARNET By Robert Walker McMillan December 1974 Chairman: Ralph C. Isler Major Department: Physics Measurements of optical and paramagnetic resonance spectra made on yttrium aluminum garnet doped with 0.1 percent chromium have been suc cessfully fitted to existing theory. The locations of the 4FF and 4 4 1 F4', Pr4 levels were determined to be 16,700, 23,200, and 35,700 cm1 respectively, by optical absorption measurements. From these results, 1 the crystal field parameter Dq was found to be 1670 cm The locations of the two r4 levels were fitted within 3.9 percent to the theory of 4 4 Finkelstein and Van Vleck which predicts that the 4FF and Pr4 states are coupled by the octahedral crystal field perturbation. The results of paramagnetic resonance measurements give the gfactor = 1.980 and the l zero magnetic field ground level splitting = 0.509 cm These results were used to determine the splitting of the excited r5 level due to the i trigonal distortion, which is found to be 2140 cm The locations of the two component energy levels of the r5 level are then used to calcu late the contributions to the crystal field potential expansion caused by the trigonal distortion by a method which treats the distortion as a small perturbation of the r5 level. This perturbation calculation was carried out to second order to determine the series expansion coeffi cients of this distortion. Only the coefficient of the second order spherical harmonic y is found to contribute to the deviation from a pure octahedral cubic crystal field. The corrected potential expansion was then used to determine the splitting of the two r4 levels. Chairman SECTION I INTRODUCTION I.A. Brief Historical Sketch Paramagnetic resonance may be observed in almost any substance which has a net electron spin different from zero. Examples are: organic free radicals, some free atoms, and ions with a partly filled electron shell. In many instances, paramagnetic resonance (PMR) is observed in substances contained in host crystals. In materials of this type, the PMR absorption is actually caused by the dopant ion, but the structure of the host crys tal contributes greatly to the PMR behavior. For this reason, PMR has proven to be a valuable method for the study of the crystalline field, es pecially when complemented by optical absorption measurements. Ions of the iron and rare earth groups have been studied extensively by this technique. The first magnetic resonance measurements were performed by E. Zavoisky in Russia in 1945. He used a radio frequency source with a wavelength of 25 meters to excite resonance transitions in several substances at very low magnetic fields, at which clearly defined resonances were scarcely discern ible. The availability of sophisticated microwave sources subsequent to World War II had a great impact on this area of experimentation, and Zavoisky and other workers used these sources to expand PMR measurements into the microwave frequency region. Much of the early work was done by the groups at Oxford who have made most of the contributions to both the PMR theory and 1,2 measurement techniques. Their many publications'2 provide a valuable store of information for the study of PMR and crystal field theory. During the past few years, PMR has been observed in the optically excited states of chromium and manganese by a double resonance technique developed by Geschwind and others. In addition to the spinorbit interaction perturbation which acts upon free ions, dopant ions in crystalline hosts are also acted upon by the crystal field perturbation. For iron group ions, this perturbation is stronger than the spinorbit interaction, and the various energy levels resulting from the crystal field are usually separated by energies which may be determined by optical absorption measurements. For this reason, these measurements may be used as the basis for calculating the coefficients in the expansion of the crystal field potential. PMR measurements comple ment optical measurements in the calculation of the spinorbit interaction coefficient and smaller crystal field distortions, such as departures from high symmetry, if present. For rare earth ions in crystalline hosts, the spinorbit interaction is stronger than the crystal field perturbation be cause the incomplete inner shell of electrons is partially shielded by an outer shell. In this case, the spinorbit states are separated by energies measurable by optical absorption techniques, and results of optical measure ments may be used to calculate the spinorbit coupling parameter. The re sults of PMR measurements may then be used to calculate the crystal field parameters, as was done by Elliott and Stevens4'5 for several rare earth ethyl sulfates. Complementary optical and PMR absorption measurements therefore provide a means of characterizing the effects of crystal field and spinorbit perturbations on the energy levels of both iron group and rare earth group ions in host crystals. I.B. General Crystal Field Theory The effects of the crystal field on the energy levels of dopant ions in host crystals are evaluated by expanding the crystal field potential V(r,O6,) in a series of solid harmonics m m V(r,6,4) = E A r y (6,), A,m m m where the Am are the expansion coefficients, y are spherical harmonics, and r,98, are the coordinates at which the potential is measured. The effects of the crystal field on the energy levels of the dopant ion are determined by calculating the matrix elements of the above potential using the free ion states which are eigenfunctions of the wave equation including the coulomb interaction between electrons, but not including the spinorbit interaction. This matrix is then diagonalized to give the eigenvalues of the total Hamiltonian which are then substituted into the original crystal field matrix equation to obtain eigenfunctions diagonal with respect to this Hamiltonian. It may be convenient to perform the calculation described above for a high symmetry crystal field, and to treat slight departures from this high symmetry as perturbations of the states resulting from this cal culation. This approach is useful in the study of the trigonal distortion present in several iron group ions in various crystalline hosts. The spin orbit interaction and magnetic field perturbation are then considered as perturbations of the crystal field states determined in the calculations described above. The true crystal field energy levels are determined by optical absorption measurements, and the measured values of energy are fitted to the theoretical values by varying the expansion coefficients Am fitted to the theoretical values by varying the expansion coefficients A . The object of the crystal field calculations is therefore to obtain values of the expansion coefficients which give the best fit to measured data when substituted into the crystal field potential expansion. I.C. General PMR Theory Consider an ion of mass m possessing a resultant angular momentum J, placed in a constant external magnetic field I. The angular momentum vector of the ion then processes about the axis of S with the angular frequency = g ( )H, (1) 2mc where g is the spectroscopic splitting factor, e is electronic charge, and c is the speed of light. Now assume that a circularly polarized radio fre quency magnetic field is applied to the ion with such a sense and frequency that it rotates about H in synchronism with the angular momentum vector. This vector will then ultimately reverse its projection on H with a resultant exchange of energy with the radio frequency radiation field. This phenomenon is called paramagnetic resonance. The magnetic moment y of the ion is t = (2) in which 8 is the Bohr magnetron and ti is Planck's constant. The energy of the ion in the magnetic field is then E = 'H. (3) Taking the magnetic field direction along the Z axis and substituting (2) for t gives for the expectation value 5 where m is the magnetic quantum number. In the magnetic field, the energies of the ion corresponding to the various spatial orientations of J are then gBHm, and the selection rules for the allowed transitions, which are magnetic dipole in character, are Am = 1. Transitions between the various levels for which this selection rule holds may be induced by quanta of energy tw = gIH, (5) which is seen to be identical with Equation (1). Consider now a system of the ions discussed above situated in a crystal lattice and in thermal equilibrium with their surroundings. These ions may be considered distinguishable, and the ionic energy levels in the magnetic field will be populated according to Boltzmann statistics. At high tempera tures, the various states will be nearly equally populated, and there will be little net energy absorption at resonance, because transitions from a lower to a higher state and transitions from a higher to a lower state are almost equally probable. The strength of PMR absorptions is therefore en hanced at lower temperatures because the population difference is greater, resulting in a greater energy absorption by the ions. This resonance ab sorption can be detected by measuring the energy lost from the radio fre quency field, which causes a damping of the tuned circuit in which the para magnetic sample is placed. I.D. Outline of Procedure Both optical and paramagnetic resonance spectra of the Cr3+ ion in various crystalline hosts have been extensively studied. Much of the work has been done on the various chrome alums by Finkelstein and Van Vleck,6 Kleiner,7 Bleaney,8 and Davis and Strandberg.9 their host materials have Kleiner, Bleaney, and Davis and Strandberg. Other host materials have been studied by Lowl0 who treated MgO, Zaripov and Shamonin,11 who examined ruby, and Carson and White,12 who studied yttrium aluminum garnet and yttrium gallium garnet. Much of the interest in Cr3+ in recent years has been oc casioned by the potential of this ion as a laser material, and its application as an active ion in the ruby laser. The crystalline host yttrium aluminum garnet (YAG) has found broad ap plication as a host material for laser ions, especially the rare earth ions 3+ 3+ 3+ Nd Er and Ho Since YAG is a hard crystal with high thermal con ductivity, and is available in large crystals of excellent optical quality, it is well suited to be a laser host material.13 Although it has been deter 14 mined by Burns et al.14 that chromium doped YAG is an inferior laser material to chromium doped aluminum oxide, e.g. ruby, the material is of interest be cause of its similarity to ruby and because the basis for the crystal field and paramagnetic resonance (PMR) calculations may be used for other materials. Trivalent chromium has three d electrons and its ground state is F 2. 123/2 In the YAG crystal, it occupies a site of octahedral symmetry2 being4sur rounded by an octahedron of six oxygen ions. Superimposed on this octahedral field is a small trigonal distortion caused by a slight displacement of the Cr3+ ion along the local and [11l crystal direction. This distortion ac counts for the interesting optical and magnetic properties of the material. Figure 1 shows the evolution of the energy levels in YAG:Cr under the in fluence of the perturbations to be considered in this work. The perturbations are shown in the order in which they will be treated. The octahedral crystal 4 field splits the F3/2 ground state of the free chromium ion into a singlet ground state r2 and two higher lying triplets (F4, 5). Each of these levels Ln 'j rN (_ z 0 H  I 1 is fourfold degenerate, including Kramers degeneracy, so that the total number of states in the 4F ground manifold is 47 = 28. Each of the higher lying triplet states is further split into two crystal field sublevels by the action of the trigonal distortion, in agreement with the predictions of group theory.5 Second order spinorbit coupling interaction between the ground (F2) level and the first excited triplet (5 ) level, together with the trigonal distortion, work to split the ground level into two Kramers doublets, characterized by m = 3/2 and m = 1/2. Upon application of s s a magnetic field H, paramagnetic resonance transitions may be observed be tween these various double levels, providing a means of determining the splitting of the ground state double levels and the spinorbit coupling parameter. The effect of the trigonal distortion on the F5 crystal field level may be determined by treating this distortion as a small perturbation on the larger octahedral cubic field. The perturbation splits the triply degenerate F5 level into a doubly degenerate level and a nondegenerate level. The magnitude of this splitting is determined by a calculation based on the PMR measurements. The knowledge of this splitting, together with the center of gravity condition for energy levels, enables one to de termine the location of the F5 sublevels, from which the constants in the crystal field potential expansion contributing to the trigonal distortion may be determined. Knowledge of these constants provides a means of deter mining the effect of the trigonal distortion on all of the crystal field sublevels. 9 The upper triplet crystal field level (F4) interacts with a higher lying 4P level through the crystal field interaction, causing the F4 level to be depressed and the 4P level to be raised in energy, with appropriate mixing of the two states. The theory of this interaction has been worked out in detail by Finkelstein and Van Vleck,6 who predict the location of all of the crystal field levels in chrome alum in terms of the crystal field parameter Dq. Optical absorption measurements made on chromium doped YAG, to be described later, agree well with this theory. SECTION II THEORY OF THE CRYSTALLINE FIELD IN YAG DOPED WITH CHROMIUM II.A. Expansion of Crystalline Field in Solid Harmonics The crystal structure of YAG, Y3A12(A104) 3 is bodycentered cubic 16 and the unit cell contains eight molecular formula units or 160 ions. Each of the metal ions is surrounded by a polyhedron of oxygens, which is a distorted cube for the yttrium ions, and is either a tetrahedron or an octahedron for the aluminum ions. All experimental evidence ob tained thus far indicates that the chromium dopant ions replace aluminum only in octahedral sites. The chromium ion may therefore be viewed as being at the origin of a cartesian coordinate system, with six oxygen ions placed at equal distances from the origin on the X, Y, and Z axes. Superimposed on the octahedral field, there is a small trigonal distortion caused by a slight displacement of the chromium ion along the local and crystalline [11) direction. It is the objective of the study of the crystal field to determine the effect of the octahedral field and trigonal distortion on the energy levels of the chromium dopant ion. Assuming that the crystal field potential V. satisfies Laplace's equation, it can be expanded in spherical harmonics Y (k6,) as follows: V = E V., with J 3 V .= AM r y (6,0), (6) 3 Z,m 9 m ,m where the A are expansion coefficients and r is the radius vector to m the point at which the potential is evaluated. The product r y (9,M ) is a solid harmonic. The V. notation is used to indicate that the potential operates only on the wave function of the jth electron, even though the potential is identical for all electrons outside closed shells. In evaluating the effects of the potential (6) on the Cr3+ ion, the fact that the crystal field energy level splitting within a given multiple may be comparable in magnitude to the splitting between multiplets caused by the coulomb interaction between electrons must be considered. In order to be strictly correct, the coulomb interaction and crystal field perturba tions must be simultaneously applied to hydrogenic wave functions, and the resulting matrix must be diagonalized numerically. This calculation has been made by Sugano and Tanabe,7 who treated several transitionmetal ions by this method. Finkelstein and Van Vleck6 have determined the energy levels of potassium chrome alum by a different method. They first determined the matrix elements of the crystal field potential within each of the multi plets using the free ion RussellSaunders states as basis functions. The resulting matrices were then diagonalized individually for each multiple level, and a new set of basis functions, simultaneously diagonal within a given multiple with respect to the free ion Hamiltonian and the crystal field perturbation, were calculated. This new set of basis functions is called the LF representation. The matrix elements of the crystal field potential were then evaluated between the various multiple levels using this LF representation as a basis. To the matrix of the crystalline poten tial calculated in this way, the diagonal matrix of the RussellSaunders energy was added. The resulting matrix was then diagonalized to give energy levels of the chromium ion in the crystal field. This procedure was followed in this work to determine the energy levels of the 4F and 4p multiplets of chromium in the YAG crystal, and will be discussed in some detail in the following sections. The number of terms in the expansion (6) which contribute to the potential can be considerably reduced. The term for k = 0, which is a constant term, can be dropped because it has no effect on the relative energies of the crystal field states. Furthermore, all terms for which > 4 will have zero matrix elements and can be omitted, because in evaluating integrals of the form fu*UvdT, where u and v denote delectron wave functions and U is a spherical harmonic, the density u*v does not contain harmonics for which > 4. Therefore, if > 4, the integral vanishes by the orthogonality of the spherical harmonics. By a similar argument, all terms for which is odd have zero matrix elements because the density u*v has inversion symmetry, and the term U in the potential expansion does not have this symmetry for odd so that the above inte grand is an odd function which vanishes upon integration. For convenience in the subsequent treatment of the trigonal distor tion, the polar axis for determining the form of the crystal field per turbation is taken along the ([111 direction. This direction is a three fold symmetry axis of the octahedron, and the potential expansion for a regular octahedron in this case is18 V 4 (, 0) + ( [y1 (9, 0 y3 (0, } (7) m where D' is the only A coefficient left in the expansion (6) because of the high symmetry. The elements in the secular determinant are the in tegrals S* Z eV (r,86,) mI dT, (8) where the sum is taken over all electrons outside closed shells, i.e., the potential function is a single electron operator. II.B. Determination of Correct Zero Order Wave Functions To evaluate the integrals (8), it is necessary to determine the correct zero order wave functions which are eigenstates of the free ion Hamiltonian, not including spinorbit coupling. They are linear combi nations of the antisymmetrized single electron wave functions. In deter mining these functions, use is made of the theorem in quantum mechanics which states that if an operator A commutes with the Hamiltonian with eigenstates nr, then A n is also an eigenstate of the Hamiltonian.19 In nnIn particular, the lowering operator L_ = L iL , x y where L and L are the x and y components of the angular momentum oper x y ator, commutes with both the basic part of the atomic Hamiltonian and the repulsion perturbation part.20 If a wave function is chosen with maximum magnetic quantum number m consistent with the orbital angular momentum quantum number then all 2+1 wave functions characterized by Iml < A can be generated by successive applications of L_. For three d electrons, a proper F state eigenfunction having = m = 3 is +++ m)> = 133> = (210), (9) where the numbers in parentheses indicate the m values for each of the three electrons and the + signs indicate that the spin of each electron is taken to be +1/2. This choice is made for ease of computation of the crystal field energy levels, and since the crystal field operator does not affect spin, no loss of generality occurs if the spins are chosen in this way. These signs may be omitted in later references to wave functions of this type, so that electronic m values shown in paren theses may also be taken to mean that the spin quantum number m = +1/2, unless otherwise noted. The required antisymmetric nature of the wave functions is implicit in the notation (210), because this expression is shorthand for the linear com bination21 7 [X2(1) X1(2) X0(3) + X2(3) X1(1) XO(2) + X2(2) X1(3) XO(1) (i X2(2) X1(1) XO(3) + X2(1) X1(3) X0(2) + X2(3) X1(2) X0(1) , in which the subscripts represent angular momentum mvalues, and the numbers in parentheses are electron coordinates. Note that an odd number of co ordinate permutations changes the sign of this wave function and an even number leaves the sign unchanged, as required of an antisymmetrized wave function. m The lowering operator L_, acting upon a wave function X gives m m1 L X = 1 [(+1) m (m1) Xm (10) The operator L_ is defined as L + L2 + L , where the subscripts refer to the three electrons outside closed shells. The lowering operator is usually defined as L_/i, and the factor t will be omitted for conciseness in future references to operators of this type. The effect of this operator on the wave function (9) is then 132 = L_(210) = iV'/(110) + 2/(200) + 1(211). (11) The first two terms on the right hand side of this equation are forbidden by the Pauli exclusion principle with the result that the m = 2 wave function, after normalization, is 132> = (211). The other 4F eigenstates of the repulsion interaction, which are the proper starting functions for the crystal field calculation, are ob tained by successive applications of the lowering operator L_. These functions are: 131> = [ (212) + 76 (201)] , 130> = [(101) + 2(202)] (12) 13> = / [2(212) + V (102)] ,(12) 132> = (112), 133> = (012). The 4P eigenstates of the repulsion interaction will be required for the determination of the matrix elements of the crystal field between 4F and 4p multiplets. These eigenstates have been determined by Theissing and Caplan20 and are given by Il> = 1 [V6(212) 2(201) , I10> = [2(101) + (202) (12a) 11>= A1 [V6(212) 2(102) . The matrix elements of the crystal field perturbation (7) between 4P states are all zero. 4 II.C. Calculation of F Crystal Field Matrix Elements The 7 x 7 4F crystal field perturbation matrix is formed by using the wave functions (12) as basis functions for the calculation of the matrix elements of the perturbation potential (7). Let Xm be a single electron wave function with magnetic quantum number m. Since the per turbation potential is a single electron operator, it affects only one of the wave functions in parentheses at a time. For example, the 3,3 matrix element of the crystal field perturbation is <33v133> = f(210)*V(210)dT = fX2*X2dT /fX*XldT fXO*VXOdT (13) + fX2*X2dT fX1*VX1dT fX0*XOdT + X2*VX2dT f/X1*X1dT /X0*X0dT. The integrals in which the potential V does not appear are all unity be cause the wave functions are orthonormal, with the result <331V133> = IX2*VX2dT + IX1*VXldT + fXo*VXodT. (14) This property of the potential aids in reducing the number of integrals which must be evaluated in determining the offdiagonal matrix elements as follows. Consider the integrals fXa*XbdT fXc*XddT fXe*VXfdT, (15) and assume that Xe *VXfdr is nonzero. The term (15) will then vanish unless a = b and c = d, which implies that matrix element terms of the form /(ace)V(bdf)dT, which occur only in offdiagonal elements, will be zero unless at least two sets of functions, such as a, b and c, d are equal. The other ele ments of the crystal field matrix are calculated as indicated in Equa tion (13). Since V and X are functions of r, 8, and (, the integrands of the integrals (8) will contain expressions of the form [f(r) r' where f(r) is the product of the radial dependence of the wave functions X, which are in turn products of these functions of r and the y(80, ). Since the exact form of these functions of r is not known, it is conventional to take the integral ff(r)r dr = r as part of the crystal field parameters mm to be determined. These parameters are then of the form Ak r In mak ing this assignment, the assumption is made that the functions f(r) are constant over the range of energies spanned by the 3d3 configuration, 1 6 which is about 76,500 cm This assumption will be valid as long as the 3d3 configuration does not interact appreciably with higher energy configurations, an approximation which will be accurate if the energy differences between the 3d3 and higher configurations is sufficiently large. The magnitudes of these energy differences are not known, so that the accuracy of the above approximation must be tested by agreement of measured and calculated energy levels. A method of general applicability for determining crystal field matrix elements involves the use of Wigner coefficients. In calculating matrix elements by this method, use is made of extensive tables of the Wigner 3j symbols available in the literature. The matrix elements of the crystal field perturbation (7) were determined by using Wigner coefficients in the 22 following way.22 The matrix elements of the tensor operator T(kq) between state vectors of a system which are simultaneous eigenvectors of the angular 2 momentum operators J and J of the system are given by z (Y'j'm' T(kq) lyjm) = (1)j'm'( kj) (y'' I IT(k) I Ij), (16) where y is a general quantum number, ( jk ) is a Wigner 3j symbol and the final term is the reduced matrix element. The spherical harmonic oper ator Ym (9, ) appearing in the crystal field expansion is a tensor operator with k = Z and q = m, as shown in Reference 22. Since j = Z for crystal field calculations for the chromium ion, the reduced matrix elements are given by (.y(k),,) = ( P_) (2)'+1)(2k+l) (2+1] ('k ( S4 0 00 (17) where Y(k) = YO The definition of the 3j symbols provides another method of re ducing the number of integrals which must be evaluated in determining the crystal field matrix elements. The defining equation for the 3j symbols22 contains a term 6(m'm,q) so that all of the integrals (15) for which m 3 m'q are zero. As examples of the use of Equations (16) and (17) for calculating matrix elements, both a diagonal and an offdiagonal element of the crystal 0 field matrix will be calculated. Diagonal elements are given by the Y4 term in the expansion (7) and offdiagonal elements are given by the 3 3 (Y Y ) term. Because of the 6(m'm,q) term in the definition of 4 4 the 3j symbols, the off diagonal term will couple only wave functions which differ in m values by 3. Using Equations (16) and (17), the 3,3 matrix element is determined to be 4[( 202 k 101) + kooo0j 00( <33Y403> = (21 Y(4) 2)>  242 4) 04 o (18) The doublebar term (21 Y(4) 12) is common to both diagonal and off diagonal elements, and is given by 1/2 (211Y(4)112) = [5) (19) Making use of the table of 3j symbols in Reference 22 to evaluate the 3j symbol on the righthand side gives (21 Y(4) 12) =5 (3~ ) (20) Substitution of this result into Equation (18), and evaluation of the 3j symbols inside brackets finally gives for the matrix element <331Y40133> = [ (5)1. (21) 18 The coefficient D' in the expansion (9) is given by Hutchings as 14 Y/ Ze2 r4 D' = (22) 9 5 a where Z is the atomic number, e is electronic charge, a is the distance of the octahedral charges from the chromium ion, and r is the average value of the fourth power of the distance from the chromium ion at which the potential is measured and is given by r4 = If(r)r 4T, (23) as mentioned earlier. Muliplying Equation (21) by D' gives the matrix element 35 Ze (i) (2) = 2Dq. (24) ( a /\a05) The first and second terms in parentheses in Equation (24) are conven tionally designated D and q respectively, and are the basis for specify ing energy levels in all iron group ions. The constant Dq is usually determined from optical absorption measurements as will be discussed later. As an example of the calculation of an offdiagonal matrix element, the term 3,0 will be evaluated. Making use of the wave functions of Equation 12, this matrix element is given by 1 3 3 <331 (Y3 Y3) 30> = f(210) (Y 4 Y (101)dT 2 3 3 (25) + 2 (210) (Y3 Y3) (202)dT. (25) /5 4 4 Note that in the first integral, m' = 2 and m = 1, while q may be either 3 3. The requirement stated earlier that m = m'q will then cause the Y4 term in the first integral (27), to vanish. In the same way, it is easily 3 seen that the Y term in the second integral also vanishes. In computing 3 the 0,3 matrix element, it is found that the Y4 term vanishes in each inte 4 gral, but this element is equal to the 3,0 element in accordance with the requirement that quantum mechanical operators are Hermitian. In computing the offdiagonal elements, Equation (16) is used in the form (yj'm' T(kq) yjm) = (_1)jm (j'jm'mlj'jkq) ( j), (26) 1/2 (iT(k) ), (2k+1) in which (j'jm'mlj'jkq) = (1) q(2k+l) jkj km' qm) 21 is called the vectorcoupling coefficient. Note that the reduced matrix element in Equation (26) is the same as that of Equation (16). The 3,0 element is then I T ) < 1 433> ( 1 3( 222112243) 2 (221212243) (28) Using the tables of vectorcoupling coefficients found in Condon and 23 (lO\ 1/2 Shortley23 and multiplying by 1/ D' as required by Equation (7) finally gives <331V130> = 2 V1 Dq. (29) The other matrix elements are calculated in a similar way, so that the 7 x 7 4F crystal field perturbation matrix is found to be m 0 3 3 1 2 1 2 0 4 2/1 2 10 3 2/10 2 0 3 2 /v 0 2 (30) 1 2/3 4/3/5 2 4/3/5 14/3 1 2/3 4/3/5 2 4/3/5 14/3 where all of the elements are expressed in terms of the constant Dq. Solution of the 3 x 3 submatrix gives E = 12Dq, 2Dq and 6Dq. The other two submatrices each give E = 2Dq and 6Dq so that the 4F energy levels for the chromium ion in a purely octahedral field are found to be E = 12Dq singlet, E5 = 2Dq threefold degenerate, (31) E = 6Dq threefold degenerate. The state functions for each of the levels (31) are found by sub stituting the eigenvalues into the original matrix equation, solving for the coefficients of each of the component states, and normalizing these coefficients. For example, the coefficients Cm of the P2 state are given by 4 2/41 2/ V10 CO CO 2/Vi 2 0 C = 12 C3 (32) 2 /1 0 2 C3 C giving the three equations, 4CO + 2/4 C3 2/10 C_3 = 12 C0, 2/ i C0 2 C = 12 C3, 2/1 CO 2C_3 = 12 C_3' Solving for the coefficients and normalizing gives, for E = 12Dq, 130> 33> + 33)> ,(33) where the Roman numeral is a designation for keeping track of the various functions. In a similar way, the other states are determined to be: E = 2Dq II"> 33> + 13> , 1I>/753> + /L32> . IV> = 31 32> (33a) E = 6Dq IV> =330O>+ (133> 33> ), VI> =/ 3 1> 32> , VII>= / 31> + /j32>. In Equations (33a), the first three functions are the 4FF5 states, and the last three are the 4 F4 states. In the expressions for the 6 Dq wave functions, the primes are used to distinguish these wave functions from those that will result when the 4P interaction is considered. These wave functions are the LF representation for the 4F multiple. Since the crystal field perturbation has no effect on the 4P multiple energy level, Equations (12a) give the LF representation for that multiple. The wave functions (12a), (33) and (33a) will be used as a basis for evaluating the interaction of the 4F and 4P states through the crystal field perturbation. II.D. Interaction of the FFI and 4P States In the free Cr3+ ion, the 4P level lies about 14,000 cm above the i 4F ground level, but this interval is reduced to about 10,000 cm1 when the ion is placed in a crystalline field. Owen24 has shown that this reduction ion is placed in a crystalline field. Owen has shown that this reduction in energy is caused by covalent bonding between the Cr3+ ion and the surrounding oxygen atoms, reducing the coulomb interaction between the d electrons and resulting in a reduction in the apparent energy level of the 4P state. Covalent bonding has also been considered by Sugano 25 and Peter25 in their treatment of the energy spectrum of ruby. The crystal field perturbation (7) has no effect on the P energy level be cause the matrix elements of this perturbation in the 4P state are zero. However, after the application of the perturbation (7) to the ground level and the determination of the eigenfunctions (33a), it is found that the crystal field perturbation couples the 4FF4 states and the 4p level with resultant changes in energy levels and mixing of states. 4 The P wave functions for the free chromium ion are given by Equations (12a). Each of these functions is coupled to one of the three FT wave functions by the perturbation (7). The matrix elements of V between each of the three 4P states and the corresponding 4FF4 state are determined in the same way as were the elements for the matrix (30), and are found to be in the notation of Equations (12a) and (33a). These results differ slightly from the results of Finkelstein and Van Vleck who obtained 4 Dq for each of the matrix elements (34). Note that this sign difference will not affect the energy interval between the admixed 4F and 4P levels, but it will affect the state compositions of these levels. The matrix 4 4 which accounts for the coupling between P and 4F4 levels is then, in multiples of Dq, 4FF2 4F5 4F4 4p4 22 4Fr2 12 4Fr5 2 (35) Fr4 6 4 4P4 +4 E(4p) where E(4p) is the energy of the 4P state of the ion measured from the ground state (i.e., from 4F = 0). Solution of this matrix yields a sing let FF2 at 12Dq and a triplet 4F5 at 2Dq as before, but the 4F4 and 4 4pF4 levels are shifted by the perturbation. The positions E4 and E5 of these new levels are found by solving the 2 x 2 submatrix in Equation (35) and are E = 6 Dq + E(P) E(P) 6 Dq (4 Dq) (36 E 2 2 + (4 Dq) (36) S 2 2 and 6 Dq + E(P) P)P) 6 Dq 2 ( E5 2 + 2 + (4 Dq) (37) Figure 2 shows the effects on the 4F and 4P levels of the Cr3+ ion which result from the application of the perturbation (7). The energies of the three excited levels shown in this figure may be determined from optical absorption measurements and the constants Dq and E(P) may be calculated based on these measurements. E = E(P) P 4 Energies Given by Equations (36) and (37) E = 12Dq Figure 2. 4F and 4P Energy Levels of Cr3+ in YAG It is convenient at this point to look ahead to the experimental results for a solution to Equation (36) for E(P), because the value of this parameter will be such that an exact solution of the 2 x 2 sub matrix of (35) will be obtained. In Section V, it will be shown that 4 _ 1 the FF5 and E levels lie at 16,700 cm1 and 23,300 cm respectively. The location of the 4FF level, which is 10Dq above the ground level, i gives Dq = 1670 cm Equations (31) show that the free ion level lies i 12Dq = 20,000 cm above the ground level. The E level then lies 1 4 23,300 20,000 = 3300 cm above the free ion F level and this sepa ration is equal to 2 Dq within experimental error. To summarize the above discussion, the energies of the pertinent levels relative to the free ion 4F level are 4F 2: 12Dq = 20,000 cm1 4F5: 2Dq = 3340 cm1 (38) i E : 2Dq = 3340 cm1 Equation (36) may be solved for E(P) to give (E4 8Dq) (E4 + 2Dq) E(P) = (39) E 6Dq i Substituting E4 = 2Dq into this equation gives E(P) = 6Dq = 10,000 cm , in good agreement with similar calculations performed on chromium in three different complexes as cited by Low,10 who gives F(P) = 10,000 cm1 for 3+ 3+ 1 3+ Mg:Cr and Cr(NH3 ) +, and 10,200 cm1 for Cr(H 0) 3. When substituted into Equation (37), the above values of Dq and E(P) give E5 = 10Dq = 16,700 cm. The calculated energy levels for Cr3 in a purely octahedral field, including 4F/4 P interaction, are then E = 12Dq singlet (0), E = 2Dq threefold degenerate (16,700), E 4= 2Dq threefold degenerate (23,400), (40) 41 E42 = lODq threefold degenerate (36,700), where EF41 and E 42 denote the energies of the lower and upper F4 levels i respectively. The numbers in parentheses are the energies in cm relative to the ground level, and will be compared to experimental values in Section V. The state functions for each of the energy levels (40) are found as before by substituting the eigenvalues into the original matrix equation, solving for the coefficients of each of the component states, and normal izing these coefficients to unity. The F2 and F5 states are not affected by the perturbation V because matrix elements of V between these states 4 and the P states are zero. Therefore, it is only necessary to use the 2 x 2 submatrix of (35) to determine the compositions of the admixed 4F/4 P levels. In making this calculation, the sign of the offdiagonal element is chosen to be consistent with Equation (34). For the 2Dq level, and the wave function IV> the matrix equation is 6 4 CV2 V2 = 2 (41) 4 6 Cp2 Cp2 where CV2 and Cp2 are the coefficients of the component states for 2Dq. This set of equations gives C = C = 1 so that the new wave function V2 p2 is V> = 1V V> 10 = 3 + 3 > 3 1 > (42) v2 In a similar way, the remaining wave functions for E = 2Dq (F41) are found to be iVI> = 31> /i32>+ 1 > , (42a) VII> = 2 31> + L 132> + 111> , 2/2 r2 and the states for E = 10Dq (F42) are VIII> = /30> + 33> 4133> + 10> , IX> = 31> 32> 11> (43) X> .= 31> + 32>  1 . 2/35> 2 These states, along with the F2 and F5 states given by (33) and (33a) will be used to determine the effects of the trigonal distortion on each of the threefold degenerate energy levels. II.E. The Effect of the Trigonal Distortion in First Order The trigonal distortion of the crystal field in YAG:Cr, resulting from a slight displacement of the chromium ion along the crystalline [l11 direc tion, has a considerable effect on the location of the crystal field energy levels. This distortion manifests itself as a splitting of a few hundred wave numbers in the energies of the F4 and r5 states. Since these levels lie 16,000 to 20,000 wave numbers above the F2 ground level, it is convenient to consider this trigonal distortion as a perturbation of the basic states. The crystal field expansion in terms of spherical harmonics which in cludes the effects of the trigonal distortion, may be written11 0 2 0 0 4 0 3 4 3 3 V = A2r Y2 + A4 r 4 + A r4 ( Y4 ). (44) For a crystal field of octahedral symmetry, the potential (44) will approach the basic potential (7) as the coefficient of the y2 term approaches zero. It will be shown in Section V that the calculation of crystal field energy levels using (7) gives results that agree within about 3.9 percent with the measured centers of gravity of the optical absorptions. These results imply that the effects of the trigonal distortion on the locations of the centers of gravity of the various energy levels is negligible, and that the 0 contribution of the y term is small. If this contribution is small, the 0 3 3 coefficients of the y4 and (y4 y) terms are almost unaffected by the trigonal distortion. The crystal field potential expansion, including the trigonal distortion, is then given to good approximation by V r2 y20 D' r4 y0 (0 1/2 () (45) 0 2 0 . The term A2 r y2 is then treated as a perturbation of the basic states (33), (42), (42a), and (43), which are eigenstates of the basic potential (7). The states r2, F41, 42 and F5 are separated by energies on the order i of 10,000 cm and it is therefore feasible to consider the trigonal dis tortion as a perturbation of each of these states individually. To account for interaction between states, it is necessary to carry the perturbation calculation to second order. The F5 wave functions interact with the F2 ground level wave functions through second order spinorbit coupling, i splitting the F2 level into two sublevels separated by less than one cm1 This splitting can be measured by the method of PMR, and if the spinorbit coupling coefficient is known, the splitting of the r5 level due to the trigonal distortion can be calculated. The value of this splitting is then 0 2 used to evaluate the coefficient of the trigonal distortion A2 r Using this result, the effects of the trigonal distortion on the F2 and r4 levels can be determined. The matrix elements of the trigonal distortion Vig= A r 2 (46) trig 2 2 will now be evaluated using the F5 wave functions as the basis. This cal culation is made in the same way as the matrix elements of the basic poten tial were calculated, and the desired matrix elements are determined to be + 1<33V 33> () A0 2 2 trig 14 2 (47) + <32 trig32> + 2 <311Vtrig 32> 51/2 1 0 2 ()/ () AO r2 7T 28 2 0 Note that all offdiagonal elements are zero because y is a diagonal operator. Equation (47) therefore gives the energies of the singly and doubly degenerate sublevels of the F5 state, correct to first order of the trigonal distortion. The trigonal distortion also splits each of the two F4 levels into doubly degenerate and nondegenerate levels in the same way as the F5 level is split. The first order energies of the component states of the F4 level with E = 2Dq ( 41) are determined to be [2 K301y' 3> + K331y Y1 3> + 3 <33Iy133> + 10 y1>011 2< 30 yIO21 36 2 2 2 3 31 ( 140 140 \v 0 2 A r , 2 (48a) VIIA2 0r Y21VI> = S <31yO 3> + 32yO3> 2 [r 2 12 2 _023 0 +] 31 (5A02 2 ~ 21 /6 2280 lt 2 In a similar way, the energies of the F4 level with E = lODq (F42) are given by l 2 IA r2 1 < YIII A12 r y2 VIII > = 140 AO r 7T 0 2 A2) r , ~L12 2 I~= I 2 2 Z~ 280 (48b) 0 2 A2 r 2 The energy levels of all of the r4 and F5 states, 0 2 0 of the perturbation A2 r y 2' are then = 2Dq + L 14 E =E = 2Dq III IV correct to first order (7) 0 2 A r 2 2 A) r 28 Tr 2 (49) = 2Dq + (5) 0 2 A2 r , 31 (5A0 2 280 A2 r 2 2 80 7T 2 EVI = EVII = 2Dq VI VII EII II 1 5 0 02 E VIII= 10Dq 140 A2 r 1 0 2 EIX = E = 1Dq + 280 t A2 r II.F. The Effect of the Trigonal Distortion in Second Order 1 The r5 and the lower r4 states are spaced 4Dq = 6680 cm apart, so that it is reasonable to expect that second order trigonal distortion effects might influence the positions of the energy levels and affect 0 2 determination of the coefficient A r The second order correction E(2) to the energy E of a state > caused by a perturbation V is E(2) k E E k n k n where the index k refers to other states coupled to In> by the pertur bation. To apply Equation (50) to the F5 states, it is necessary to de 0 termine matrix elements of y between the r states and all of the other 2 5 states. A careful examination of the eigenfunctions (33), (42) and (43) shows that many of these matrix elements are zero. For example, the matrix element 1 <33yO 33> + <33yO33> 3 "3 But <33yO 33> = <33 yO 3> so that the above matrix element is zero. Similarly, nonvanishing matrix elements are determined to be 2<2 6 y2 2>3 yO1>5 i <32lI32> <3l 1> (51) 6 J222 3 2 20 / 2 T5) = 0 and 32 1yo1 3 > + <31 yO ll) (52) 9 5A(i> 140 A 2 f where <31yOlY l> = ( * These results show that the level of the state III is unchanged by the second order correction. The energies of IIII> and IIV> are equal, and are given by the following relation, correct to second order: 1 0 2 E = EIV =2Dq \ A r III IV 2 0 2Dq 2 (53) S160 31 5) 0 2 280 2 S2 r 81 ) S2*28*140 510 2 280 2 0 2 The term involving A2 r in the first bracketed expression in Equation (53) will not be small enough, compared to 4Dq, to be negligible, but the corresponding term in the second expression will be negligible com pared to 12Dq. The basis for this assertion will be evident when the 0 2 final values of Dq and A r are calculated. The splitting A of the F5 level, correct to the second order of the perturbation A2 r y2, is then 3 0 2 2 Y2 A =E E 3 (5 A r II III 28 0i022 + 81 228140(12Dq) * Substituting for the constants and clearing fractions gives the follow ing cubic equation for A2 r in terms of Dq and A: 2.2967 x 103 A 2 + 0.41169Dq (Ar2 (54) 02 2 + 12Dq (0.54068Dq 0.13967A) A0 r 48 (Dq) A = 0. This equation will be solved in Section V when values of Dq and A are determined. The second order trigonal distortion corrections to the r2 and r4 levels are determined in the same way as were the analogous corrections for the r5 level. For the r2 level, the corrected energy is 2 A0 2 9 2 EI = 12Dq 27 1 ( 0 ) 2 (55) 2*735 22Dq In writing Equation (55), the splitting of the levels to which the 2 level is coupled by the perturbation are considered small compared to the level separation. The energies of the P4 levels correct to second order will not be listed, but these energies will be determined in Sec 0 2 tion V after the A r coefficient has been calculated. 2 Note that the effect of the second order y2 correction on the r 2 2 ground state is to slightly depress the ground state energy. This change will have a small effect on the determination of the constant Dq as will be shown in Section V. II.G. Exact Solution of the Crystal Field Matrix Including 4F/4p Interaction and Trigonal Distortion Although the calculations of Section II.F. give the crystal field energy levels to good accuracy, it is still meaningful to solve the 10 x 10 crystal field matrix which includes all F and P states and the effects of the trigonal distortion. This calculation was carried out earlier in Section II.D. without including this distortion. In making this calculation, the trigonal distortion is included from the beginning, with the result that all of the resulting wave func tions will be diagonal in this perturbation. The calculation of Section II.F. is therefore considered more accurate than the solutions of the 10 x 10 matrix including the distortion because second order effects were shown to be significant in the last section. The most important reason for solving the 10 x 10 secular equation which includes the dis tortion is to show how the eigenstates are mixed by this perturbation. In particular, it will be shown that one of the 4p states is slightly mixed with the ground state I> given by Equation (33). Again, several new matrix elements must be evaluated before this calculation can be made, namely the elements <3m jyOlm >. These elements are evaluated as before and are given by 3+lyOj1+ > = 2i F (56  35 7T (56) <301y 01 > = 6 (5) The 10 x 10 crystal field matrix breaks down into a 4 x 4 and two 3 x 3 matrices. The 4 x 4 matrix is P,m 3,0 3,3 3,3 1,0 2 5 0 2 8 6 5 0 3,0 4Dq35 j O /Dq 2 q + q A2r 3,3 2/1iDq 2Dq+145)Ar2 0 + 2/1ODq S2 (57) 3,3 2Dq 0 q 0 2D+ Ar2 j 2Dq 1,0 + q5 2 + Dq 6Dq+ ~A x 3 matrices are, J,mR 3,1 32 1,1 2 6 5 0 2 3,1 AMr 5DMq Dq 35 AI r 32 /5Dq Dq +24 Dq (5 1,1 5)Dq 00 r2 0 2 1,1 A2 r + 2 q 6Dq( A r Zm 31 32 31 2 3 0 2 70 5 A2 r 3 5r5Dq S2 2/6 5 A0 2 11 3Dq 35 A2 r 3 35 2 The results Dq = 1670 cm and A2 r2 2 rived in Section IV, are now used to Upon making these substitutions, the A,m 3,0 3,3 3,0 3,3 3,3 1,0 7,639 10,562 10,562 1,576 3,1 3,2 1,1 10,562 2,141 0.0 3,521 1,832 4,979 5,076 3 2 11 4 2 2 /A5\) 0 2 5Dq 6Dq 5 J AO r 3 q 3 35 A 2 r 1 D 2 / Dq (58b) D 3 1 0 2  2 0Dq 6Dq 10 A2 r 1 = 13,300 cm which will be de solve the secular Equations (58). matrices (58) become 3,3 1,0 10,562 0.0 2,141 3,521 3,2 4,979 7,793 6,098 1,576 3,521 3,521 13,380 5,076 6,098 8,341 (59a) (59b) 8a) The two 3 k,m 3,1 3,2 1,1 3,1 1,832 4,979 5,076 3,2 4,979 7,793 6,098 (59c) 1,1 5,076 6,098 8,341 i where all of the above matrix elements are expressed in cm In gen eral, only three significant figure accuracy is allowed, and the solu tions to the above secular determinants must be rounded off to this ac curacy. The accuracy of the measurements leading up to the determination of the elements of the matrices (59) will be discussed in Section V. The secular determinants (59) were solved on the Martin Marietta IBM 370 computer using a matrix diagonalization program available in the program library. This program calculates eigenvalues and normalized eigenvectors for complex matrices of dimensionality up to 20 x 20. The results of this calculation are given in Table I. TABLE I Eigenvalues and Eigenvectors of the Crystal Field Matrix Including the Trigonal Distortion 4 1 E = 2.02 x 10 cm II'> = 0.761130> 0.457 ~ > + 0.45733> + 0.060110> 3 1 E = 4.54 x 10 cm IIIl> = 0.93413> + 0.255132> + 0.248 11> IIV'> = 0.93413> 0.255132> + 0.2481l> E = 2.14 x 103 cm III'> = 0.70713> + 0.707+33> 3 1 E = 6.81 x 10 cm IV'> = 0.552300> + 0.42133> 0.42113> + 0.58411> (continued) = 0.011 31 = 0.011131 = 0.355131 = 0.355131 = 0:342 13 TABLE I (continued) 3 1 E = 1.96 x 10 cm1 0.718132 + 0.696111 + 0.718132 + 0.696111 4 1 E = 1.69 x 10 cm n ca4l 2 0 74A 11 E > + 0.648 32 = 1.70 x 104 + 0.33833>  0.674111 1 cm  0.338133> 0.809110> Comparison of the coefficients of the states given in Table I with those derived without including the trigonal distortion shows that the trigonal distortion has little effect on the state compositions. As noted earlier, there is a very slight mixing of the 4P state 110> with the ground state, which will be shown to have little effect on the PMR results. lVI'> VII, > IX'> IVIII I SECTION III THEORY OF PARAMAGNETIC RESONANCE IN YAG DOPED WITH CHROMIUM III.A. Perturbation of the Ground State by the Magnetic Field and SpinOrbit Coupling Paramagnetic resonance (PMR) transitions are observed between the 3+ m = +3/2 and m = +1/2 doublet sublevels of the r2 ground state of Cr s s 2 in YAG. Second order spinorbit coupling interaction between the ground (F2) and the first excited triplet (F5) level, together with the trig onal distortion, combine to split the ground level into two Kramers doublets, m = +3/2 and m = +1/2. PMR transitions are observed between s s  these various doublet levels, providing a means of determining the split ting of the ground state and the spinorbit coupling parameter. These multiple transitions are called the PMR fine structure. The results of PMR measurements also give the splitting A of the F5 level, thus giving the other condition for calculating the energy levels E and E2. An effective perturbing Hamiltonian H can be written in terms of s the magnetic field H, spectroscopic splitting factor gs, and spinorbit coupling parameter A as follows: H s HT + g s) + ALS, (60) s s where the operators in this case are state operators instead of the single electron operators that were used in the crystal field calculations of Section II. It is convenient to evaluate the matrix elements of this Hamiltonian using the representation for L, with the mutually commuting 2 2 operators L L S and S because both the crystal field energy contri z z bution and the operators S and S are diagonal in this representation. z The basis functions for this representation are the eigenfunctions of the crystal field perturbation with the spin quantum number taken into account. In particular, since PMR is observed only in the ground state at room temperature without optical excitation, only the ground state wave functions need be considered in the first order perturbation calculation. The four ground state basis functions, including the spin quantum numbers S and m are then s IISm>= FSm> 5 1303ms> (333ms> 1333ms>). (61) 3 1 In this equation, m takes on the values + and giving the four basis functions. The matrix elements of the first order perturbation are <2Sm Hs= 1 <2Sms / BH I 2 Sms> r2sinjHr2s> = + <2Sm sg *S .SF2Sm + A <2Sm Ls.Sr Sm> + s s O2 S> The components of H may be written H = HcosO, z (63) H = HsinO e , + where 6 and 4 are polar angles referred to the symmetry axis of the crystal, which is the [111i direction for YAG, and is the direction around which the crystal field potential was expanded in Section II.A. Using the above ex pression for H+, the operator HL becomes 1 (64) *T. = H L + 1(HL_ + H_L) (64) zz 2 + z+ To evaluate the effect of this operator on the wave functions (61), it is necessary to recall that the action of the operators Lz, L+ and L on a wave function Jim > give Lz ImZ> = mJIm , (65) L+ Em> = [(+) m (m+lj m+1 > . Similarly, for later reference, the spin operators S S and S_, acting on a function ISm s> result in S zSm> = ms Sms> (66) + Sm s = IS(S+1) m (m +l jSms +1> . Operating with L on (61) gives L  r2Sm> = /2 (33Im> + 33m . The matrix element of L using this basis is then z 2 <3 2 33 3 3 The matrix elements of L and L_ using the Equations (61) as basis functions are also zero, since the function (61) does not contain terms which have m 1. Therefore the matrix elements vanish by orthogonality. The first term of (62) is therefore zero. The third term of (62) is evaluated by using the identity Ti = L S + (L + LS+). (67) z 2 2 The effect of operating on IF2S> with LS, for example, is then P2S> = 5 31 ( 8 33 3 + 32 > (68) r2 3': 2 2 Operating from the left on Equation (68) with each of the four wave func 3 1 tions (61) formed by using m = , 5 gives zero in every case. In a similar way, it is easy to show that the matrix elements of LS are all zero for the other basis functions (61). The third term of Equation (62) is therefore zero, showing that the fourfold spin degeneracy of the ground state is not lifted by the first order spinorbit coupling perturbation. The only nonzero matrix elements of the first order perturbation are then The energy contribution from the second order perturbation is SI < F2Sms (t ) + g A "Sms 2s (70) 4 5 E(F) E(F2) r = 2,15 where F2 represents the ground state and F represents the excited states 4 r5. The energies of these states are E(F2) and E(F), respectively. The term < 2Sms Sr FSms> of the expression (70) is zero because the operator S affects only the Sm parts of the wave functions, leaving the orbital term vanishes because F2 and r are orthogonal. The second order perturbation then becomes <(2Sms' 6 L + ALS Sm > (71) r E(F) E(F2) The numerator of the above expression is expanded to give j S The last term of (72) does not depend on spin, therefore it shifts each of the four ground state levels by the same amount. Since this term causes no change in relative ground state energy, it is usually ignored.26 Combining Equations (69), (70) and (72) gives the ground state energies correct to second order of the perturbation (60) as E F = 2 S1 2sms AL Sms> E(F) E(F2) [I I' AL I Sm + 2 As noted earlier, the first term of (73) is the only contribution to the first order energy of the F2 states. If all other terms were zero, this term alone would cause an isotropic splitting, linear with magnetic field, of the fourfold spin degenerate r2 level into four separate equally spaced 3 1 sublevels characterized by m = and m = +. The second term in the s 2 s 2 summation accounts for the anisotropy of the PMR spectrum, and shows that this anistropy is due to second order spinorbit coupling between the F2 and F states. The first term in the summation gives the splitting of the ground state in the absence of a magnetic field. When a magnetic field is applied, each of the two doublet levels is further split into two sub levels. Transitions between these various levels are called the PMR fine structure. A graph of this splitting for YAG:Cr is included in Section V. Another contribution to the energy (73) which has been ignored is the contribution due to the nuclear magnetic moment of the active ion. If one of the isotopes of this ion has halfodd nuclear spin I, the pos sible spin states of the nucleus will cause 21 + 1 incremental values of magnetic field to be superimposed upon the external magnetic field, thus splitting the PMR absorption for this isotope into 21 + 1 separate ab sorptions. This phenomenon is called the nuclear hyperfine interaction, and its effect on the PMR absorption is called the hyperfine structure. 3 Chromium has one isotope (53) with nuclear spin and 9.5 percent abun dance, and one isotope (54) with integral nuclear spin. Because of the low percentage of chromium 53 in ordinary samples, the hyperfine struc ture is difficult to observe and has not been observed in this work. For this reason, the nuclear spin contributions to the energy (73) have been 27 ignored. Bleaney and Bowers7 have observed chromium hyperfine structure in dilute chrome alums enriched with chromium 53. III.B. Determination of the Ground State Energy Levels with H Parallel to the Symmetry Axis If the external magnetic field is applied parallel to the crystal line symmetry axis, the four ground state levels will diverge linearly, and the gfactor, which is defined for this orientation, may be found by measuring the splitting of the m = + levels. The zero field splitting s 2 is also found in this orientation by extrapolating the splitting measured at high fields to zero field. If the magnetic field is applied at an angle to the symmetry axis, there is competition between the crystal field and the magnetic field to determine the axis of precession of the spinning electrons, and the energy levels do not diverge linearly. This competition results from the fact that the ground state has a small angular momentum due to the combined action of the trigonal distortion and the second order spinorbit interaction. Values of both the gfactor and the zero field splitting are required to complete the characterization of the crystalline field begun in the last section. For this reason, Equation (73) will be used to calculate the energy levels with the magnetic field and symmetry axes aligned, because this approach provides insight into the way the ground state behaves when perturbed. Other orientations will be treated by using the spin Hamiltonian formalism devised by Pryce8 and Abragam and Pryce9 as applied to chromium 9 in octahedral symmetry by Davis and Strandberg. This treatment was origi nally applied to ammonium chrome alum, but it also accurately predicts the locations of PMR transitions in YAG:Cr. The basis functions which will be used for the calculation of ground state energies are those given by Equations (33) and (33a) for E = 12Dq and E = 2Dq and (42), (43) for E = 2Dq and E = 10Dq. The result of this calculation will then be used to determine the F5 level splitting which 0 2 will yield the trigonal distortion parameter A2 r By including the 0 2 effects of A2 r a new set of crystal field eigenvalues and eigenfunctions can be calculated, as was done in Section II, the results of which are given in Table I. These new eigenfunctions will then be used to reevaluate the ground state splitting to include any effects of the trigonal dis tortion. In particular, it will be of interest to evaluate the effects of the admixed 4P states. If the magnetic field is aligned with the crystalline symmetry axis, Equation (73) becomes E2 = <2Smsgs HzSz 2Sms> 1 1 [ Ii r E(P) E(T2) < 2SmsALSIrSm> 12 + 2 1 3 The matrix elements of g 8H S are simply g BH and Zg sH .The operator S z 2s z 2 s z 8H L couples only states II> and III> and its matrix element is indepen z z dent of spin. Operating on the state II> with L gives LII> = 1 L130> L 133> + L33> zL 3 z L 3 z > 3 zl = F2133> V'33> Operating with 4IIL LI I = 2, (75) and all of the other I I + > with L S gives 2 2 z z 3 3 3 3 3 3 Szli + LS 30 + L zS 33 + z 3 z + L S 133 +  3 z z 2 2 33 3 + 3 33 + 3 r2 2 / 22 Operating again with T + 3. (76) In a similar way, the other matrix elements of L S are found to be The matrix elements of LS are found by operating on each of the states r 2Sms with LS as was done in arriving at Equation (76). The result ing functions are then operated on with all of the bra vectors<4Smsl and TABLE II Matrix Elements 2 = (continued) < II I > = > (continued) TABLE II (continued) < IIjtIIj> =3 Note that all matrix elements between F4 and r2 states are zero. This result is also obtained if the 4F/ P interaction is not considered in deriving the eigenvalues and eigenvectors of the crystal field. The results of Equations (75), (76) and Table II are now substituted into Equation (74) to obtain the ground state energies. These energies are: 3 2 II3 3 2 E g3 H A 2 .>12 r 3 2 s 2 ] E 2 I EI II I 31 2 z 2 2 2ASH I I E E EE E gH A2 1 8 + 6 I II I (77a) 3 2 9 6' 12AgH E E II I E1 H A2 1 8 + 6 r 12 s E II EI E E E E 2II EIII EIV 4ASH + E (77c) II I E 3 2 A2 [E9 6 E E 3 g SH A + 2 3 2 s E EI EI EI 12ASH + (77d) EII E II I The zerofield splitting is obtained by setting H = 0 in Equations (77), and results in 2 1 1 E E = = 8A E (78) E 3 r 1 E E= E E 2+ 2+ LIII I II I where the fact that EIII = EIV is used. This energy difference is given to good approximation by 8A(E E ) II III 6 2(E5 E2)2 (79) r 5 2r where E is the average value of the energy of the F5 level and E = E 2. The effective gfactor is obtained by subtracting (77c) from (77b) to get 8ASH E E = gs H a E 1 F 1 E E 2 2 F r2 2 2 5 2 Using the condition for observation of PMR gives AE = hv = g3H = H 8A s E E 8A 5 2 so that 8A 9g (80) s E E 5 2 This equation shows that the free electron gfactor is modified by second order spinorbit interaction between r2 and r5 states. It will be shown in Section V that Equation (80), together with the measured values of g and E5 E give a value of A that yields poor results 5 2 when used in the subsequent crystal field calculation. For this reason, i 1 the free ion value 91 cm1 quoted by Lowl0 will be used for determi nation of the trigonal distortion coefficient. If the proper value of A is used, Equation (79) yields the r5 level splitting in terms of the ground state splitting 6, determined from PMR measurements, and the position of the F5 level E5 E2 determined from optical absorption measurements. III.C. Determination of the Ground State Energy Levels for an Arbitrary Magnetic Field Orientation The spin Hamiltonian is widely used to determine the ground state energy levels of paramagnetic ions. This approach reduces the orbital de pendance of the perturbation (60) to constant numbers that have only the function of parameters, and it is then only necessary to evaluate the matrix elements of the spin operators to obtain the spinHamiltonian matrix ele ments. The spin Hamiltonian therefore operates only on spin states, and its matrix elements and energy levels are evaluated by using these states. Pryce28 has expressed the spin Hamiltnian in the elegant form Pryce has expressed the spin Hamiltonian in the elegant form H = B(g 6. 2AA..) S.H. A2A..S.S. (81) s lS 13 1 13 i ' in which the summation convention is employed. In Equation (81), S repre sents spin, H is magnetic field, and the quantity A.. is the tensor 13 Aij E(r) E (r2) F 2 Note the similarity of Equation (81) to the analogous equation (73). 29 Abragam and Pryce have expanded the spin Hamiltonian formalism to in clude the effects of the nuclear hyperfine interaction, which need not be considered for YAG:Cr. The spin Hamiltonian approach has been applied to chromium in octa 9 hedral symmetry by Davis and Strandberg, who used it to explain PMR in ammonium chrome alum. They have solved the 4 x 4 secular determinant, 1 3 corresponding to m = +~, +, in terms of a normalized energy and mag s 2 2 netic field W = 2E/6 and X = 2g6H/6, where 6 is the ground state split ting. Their solution to the secular determinant is W4 W2 2 + 2 X2 + 2X2W (3cos2 1) + 1 2 2 1 2 9 4 3X cos 2 + X + X =0, (82) 2 16 where 6 is the angle which the magnetic field makes with the crystalline symmetry axis. This equation is solved in Section V, using measured values of g and 6, for several different orientations, and the energy levels will be plotted as a function of magnetic field. III.D. Calculation of Ground State Energy Levels Including Trigonal Distortion Effects The calculation of energy levels carried out in Section III.B. was repeated using the wave functions of Table I, which include the trigonal distortion effects. To perform this calculation, the corrected wave functions were substituted into Equation (74) to give expressions analo gous to Equations (77) for the ground state energies. As before, the matrix elements of g sH S are simply g SH and s 2 z 2 s z 3 gs H The matrix elements of L are obtained by first operating on the wave function II''> with L to obtain LJzI'> = (3)(0.457) 33> (3)(0.457) 3> . The only wave function which couples to the above expression is the func tion III''> so that The matrix elements of L S are given by z z < IIL S lI >= 2.91 2' z z 2 (84) The other matrix elements of LTS are determined in the same way as the matrix elements given in Table II, but these new elements will not be given here. Using the above results, the parameter 6 is given to good approximation by 7.54A2(E E II III 2 (85) 6 2 ' (E E ) 5 2 and the corrected gfactor is 7.78A g = gs (86) s E E2 5 2 These results show that the trigonal distortion has little effect on the parameters 6 and g, as might be expected in view of the slight changes in state function coefficients caused by the distortion. Of more significance is the fact that all matrix elements of TS between F2 and r4 states no longer vanish. Most of these elements remain small enough to have negli gible effects on the energy levels, but the LS interaction will still mix small amounts of higher lying states with the r2 ground state. This mixing probably accounts for the observation of PMR transitions which do not obey the magnetic dipole transition selection rule, both in this work and in the 9 "' 0 work of Davis and Strandberg. The LS perturbation simply mixes a higher lying state of the proper spin with the ground state in such a way that the selection rule is satisfied for a small component state of the wave function and a weak transition is therefore observed. SECTION IV EXPERIMENTAL PROCEDURE IV.A. Sample Preparation The sample of YAG:Cr studied during this work was obtained from the Airtron Division of Litton Industries. The sample was finished in the form of a 6 mm x 12 mm cylinder and was specified to have 0.1 percent of the aluminum ions replaced by chromium ions. At this dopant level, the crystal is very pale green  almost colorless. This green color is caused by broad optical absorption bands centered at wavelengths of 431 and 595 nanometers. Although the dopant level was specified to be 0.1 percent, no attempt was made to verify this concentration. The sample was mounted in a goniometer, which was then placed in a back reflection Laue Xray diffraction camera. A Laue photograph was made of the crystal which showed that the cylindrical axis was very nearly parallel to the [l111 direction. The goniometer was adjusted to precisely align the Xray beam with the [ill direction, and another photograph was made to verify this alignment. The goniometer was then removed from the camera, with this alignment preserved, and a face was ground on the crystal, parallel to the (111) planes, with a diamond abra sive wheel. This procedure was repeated for the E[10o and [001o direc tions, resulting in a crystal with faces ground parallel to the (111), (110) and (001) sets of planes. Figures 3a, 3b, and 3c show Laue photo graphs made along each of these directions. Note that the spot pattern on each photograph displays the characteristic symmetry of the axis along which the crystal was oriented for the photograph. Each of these pic tures was made using a molybdenum Xray tube operated at 19.5 kilovolts and 30 milliamperes, and was exposed for 30 minutes. Figure 3a. Laue Xray Diffraction Pattern of YAG:Cr Sample in the [111] Direction Figure 3b. Laue Xray Diffraction Pattern of YAG:Cr Sample in the [110] Direction Figure 3c. Laue Xray Diffraction Pattern of YAG:Cr Sample in the [001] Direction The end of the crystal on which the faces were ground was cut off and retained for PMR measurements. The remaining piece, a cylinder of dimensions about 6 mm x 8 mm, was polished on both ends for optical ab sorption measurements. IV.B. Optical Absorption Measurements The optical absorption spectrum of the YAG:Cr sample was measured on a Beckman Model DK2 spectrophotometer. This instrument is used in many laboratories to measure optical transmission of filters, laser mirrors, chemicals, and other optical materials in applications where extreme accuracy is not required. It is suitable for measurements on YAG:Cr because the absorption bands are broad and illdefined, so that use of a more sophisticated instrument is not required. The Beckman DK2 is a dual channel prism spectrometer which uses the second channel for reference in cancelling the effects on the spectrum due to source and detector variations with wavelength. It uses a tung steniodide quartz lamp as a source, and a photomultiplier tube or lead sulfide cell as a detector, depending on the wavelengths to be studied. The DK2 covers a wavelength range of 200 to 4000 nanometers. The re sults of optical absorption measurements made on YAG:Cr, using the DK2, will be discussed in Section V. IV.C. Paramagnetic Resonance Measurements Microwave spectrometers for the detection of PMR absorptions were designed and built in the course of this work for measurements at fre quencies near 10 Hz (Xband) and 3.5 x 10 Hz (Kaband). These 30,31,32 spectrometers are of conventional design, except that they do not use the superheterodyne detection technique used in the most sensitive spectrometers. The sensitivity of these spectrometers is adequate for the observation of the relatively strong absorptions which occur in YAG:Cr. Figure 4 is a block diagram of the 10 GHz system. Microwave energy,_ obtained from an Xband klystron, is coupled to the system through an isolator and EH tuner. A small amount of the energy is picked off at the cross guide coupler and mixed with a harmonic from a 1 MHz frequency standard and fed back to lock the klystron to the standard for stable operation. The radio frequency oscillator mixes with the frequency multiplier output to provide frequencies between the harmonics of the standard. The majority of the klystron energy is coupled through an attenuator to one port of a magic tee. This device divides the energy equally between two outputs, one of which goes to the tuned microwave cavity between the poles of an electromagnet, and the other goes to an amplitude and phase altering system consisting of an attenuator and a slotted guide tuner. This system is adjusted until the energy reflected from it just balances the energy reflected from the cavity in which the sample is mounted. Therefore, little or no energy leaves the fourth port of the magic tee unless an absorption occurs in the cavity. This bridge arrangement minimizes the quiescent signal incident on the detec tor and, therefore, reduces detector noise. The sample cavity is tuned to resonate at the applied microwave frequency, with a cavity mode con figuration chosen such that the microwave and external magnetic fields are orthogonal. This resonant condition is characterized by a high cavity "Q" or quality factor, and a sharply defined absorption of micro wave energy at the resonant frequency. When a PMR absorption occurs in the sample, microwave energy is absorbed and the Q of the cavity changes, 0 OH 41 C 0 U thus increasing the amount of signal incident on the fourth port of the magic tee. Energy from this fourth port is then detected and fed into a lockin amplifier, which drives a strip chart recorder on which the PMR spectrum is recorded. The reference output of the lockin is used to drive modulation coils mounted on the pole pieces of the magnet. Fre quency measurement is accomplished by picking off part of the energy from the adjustable arm of the magic tee, zerobeating it with a harmonic of the transfer oscillator, and counting the transfer oscillator fre quency. Precise magnetic field measurements are made with a nuclear magnetic resonance gaussmeter. The cavity is suspended in a double dewar and may be cooled if necessary, but cooling was not required to observe the strong resonances in YAG:Cr. In this system, the microwave frequency is held constant while the magnetic field is swept slowly. PMR absorptions appear as differentiated lines because of the method of modulation and detection used. Figure 5 is a block diagram of the 35 GHz PMR spectrometer. The operation of this system is almost identical to the operation of the 10 GHz system, and will not be discussed in detail except for the few important differences between the two systems. The 10 GHz phase locked source discussed earlier is reduced in frequency to less than 9 GHz and is used as a stable reference for the 35 GHz spectrometer, which is phase locked to the fourth harmonic of this source. The output frequency of the 35 GHz system is then a factor of four higher than the frequency of the 9 GHz input, plus or minus the 10 MHz reference frequency, and can be precisely determined by measuring the frequency of the 9 GHz source with the transfer oscillator and frequency counter. A 10 MHz ambiguity in frequency, introduced by the reference, can be easily resolved by 64 O 0 o~ ~ o ] l 0 oc 3r 0 0 0 O O 0 U)0 BO a  ) 0 m u 4 uJ 1 90 o , . 4 40 r1 4 U $4I 4O O . U \ S0 w  x4 a10 1r l 0 a) u a44 0 o0 > I, I X C r, o( 1) 1 0 0 (0 l ua l o o *^3 LJ H 0 0  fo >1 WII 0L using the wave meter to give a coarse frequency measurement. A duplexer is used in place of the magic tee for coupling energy to the sample cavity. This device is a ferrite coupler used for directing transmitted and received energy to and from antennas in microwave radars and com munications systems. The sample cavities for both spectrometers were fabricated by using an electroforming process in which a stainless steel mandrel of the de sired dimensions was machined and polished to a very smooth finish. A thin layer of gold was then deposited on the mandrel, followed by the main cavity deposition which was about 2 mm of copper. The mandrel was then pulled from the electroformed deposition, leaving a cavity with inner surfaces gold plated and with very precisely defined corners. Flanges were then soldered to the cavity and the proper microwave coup ling iris sizes were determined by increasing the hole size until the desired degree of coupling was obtained. The cavity used for 10 GHz measurements was rectangular and the 35 GHz cavity was circular in cross section. Because of space limitations in the dewar, the 10 GHz cavity was fixtuned and would support only one resonant mode. The 35 GHz cavity was tunable and was large enough, be cause of the higher frequency, to support several modes. Tuning was ac complished by a movable plunger coupled by stainless steel rods to a micrometer screw mounted on the dewar flange. For observation of PMR absorptions, the external magnetic field and the microwave magnetic field must be orthogonal. In a single mode rect angular cavity, this orthogonality may be achieved simply by mounting the sample centered in the bottom of the cavity. However, if the external magnetic field is rotated to obtain data on other orientations, this orthogonality is lost with a resultant reduction in signal strength. This problem is solved by mounting the sample on the side wall of the cavity as shown in Figure 6a, with the result that the two fields are orthogonal regardless of the orientation of the electromagnet. The sample may be mounted centered on the bottom of the cylindrical cavity, because a cavity mode can be found in which the magnetic field is axial, as shown in Figure 6b. Figure 7 shows how PMR in several crystal orientations of interest can be observed in a cubic crystal without the necessity for removing and remounting the sample. Faces are ground on the crystal parallel to (111), (110), and (001) planes as described in Section IV.A. The cry stal is then mounted in the cavity with its (110) planes horizontal, so that the external field is always parallel to these planes when the field is rotated. The external field may then be aligned with the [11" , [110], and 001 directions, as well as many others, as the external electromagnet is rotated, without remounting the sample. 4ir^ fIt (a) Rectangular Magnetic Field Lines Magnetic Field Lines (b) Circular Figure 6. Methods of Mounting Samples in Rectangular and Circular Cavities so That the External and Microwave Magnetic Fields Are Always Orthogonal. The External Field May Be Oriented in Any Direction in a Horizontal Plane Perpendicular to the Plane of the Page. Sample Sample Y~I~ (111) Plane  Y Rotation Axis Orientation and Rotation of Sample for PMR Measurements. (110) Plane Figure 7. SECTION V RESULTS V.A. Paramagnetic Resonance Measurements PMR absorption spectra were measured at both Xband and Kaband. The Xband measurements were made as the magnetic field was rotated in 100 steps relative to the crystalline symmetry axis, beginning with the field parallel to the [i10] direction and ending with the field parallel to the [o00 direction, as shown in Figure 7. The Kaband measurements were made with the magnetic field parallel to the [110], [111], and [0013 directions only, because these directions are considered most important. Actually, only the measurements made in the ["i direction are neces sary for characterization of the trigonal distortion of the crystal field. Figures 8, 9, and 10 show the results of Kaband measurements made for the three orientations of interest. The Xband measurements are not shown because they are not as easily interpreted as the Kaband results. These lower frequency results are complicated by the fact that the energy of the zerofield splitting is greater than the energy of an X band photon, resulting in a large grouping of transitions at lower fields where lines are weaker and may be broadened by illdefined energy levels. In a cubic crystal, there are four possible equivalent [ll direc tions. If an external magnetic field is aligned with one of these direc tions, the field direction will make an angle 00 with that direction and an angle 70.60 with the other three directions. Measurement of PMR ab sorptions along this particular direction should then show contributions from the Cr3+ sites displaced parallel to the field as well as those displaced at angles of 70.60 to the field direction. This assertion is C14 0 w 4 tC n M m m I H ('4 U)OV 0 0) rrf m .1TTTTr' T TI l r 4 4Tri4il T 4 t*t4> I H+l 1 '~ J t It~tm I I II i 11 1 '!"*! f r1 Mf 0 r N M oM r co o' or 0y z .o 0o r0 [ : i t Magnetic Fields Noted in Kilogauss Lock in Amplifier Sensitivities in :Microv.olts Lmtg Tm~ ""~ "T'~v'i ~4~ i.I 4. ...4.4~4 4 A44+ 4+,, .4 rl H4$ +tt 44 ttt'7rn1.t;it tff T~ T arrw rz 1.. ta .... ... . . ::I::. ...... ... ' ii~i10 InE It Figure 5. Paramagnetic Resonance in iAG:Cr 11j Direction, 35.73 x 10 Hz 0) I 0 N 0. M Ln W OO o0d Slir (4 ( r01 (1 HHo a ooo o "I ' . I *t^2:4ttTn: ML: I * 9~J rrL 1.: ~FL Ln _ Rt~~F~iFFLtc;F~~~~fFF~tf ~ffif ititr~tttbi rrrrrrrrmrrlirrll r;rrllrir~lllll;: ii : I.lr ^ j=4^":: 1 L." V +1 AT ~C 1:fir AT. I .1: F:4 rtGIl3'! ~Lmrr F4  1 tt f H '*.hl;' : "*) *' .: ",. jwAi '.'*"t:lr*.". ; !! :_+, .% t rrl tc~ytr~rt Irtttrtl ,,mri,4^ '*f t~ T~:'~tlr* rr:^3 ; l a: :! ;4 ~i~'~f~,"tF~'Jftft~ii i'~:7~t~f~;~;"r~'d ~'~~~:~t~';;;;~~~;~ ~.~f~i~~t~;~~ ~F~~ri 2 It'i,Eiff i~iiiili;~liii :t:'l~.tiltifiF,';~F~f~~;;liiiF ~ tt 1 tO On n< oo m N m m .D N in fn N' IV LA LA I I I I rai i i i i H o (> o O 1. . .ll~m . ... .. .. .. . . 4+ 1++1++4+ + .flffff i + .. .. . 0W N0 rn^ N~r I II I I I Figure 9. Paramagnetic Resonance Spectrum of YAG:Cr Li10 Direction, v = 35.73 x 10 Hz I r LO O (N wl wr) " ~wrrn ,4.JJ4r.44a 4 4 I is +4ii, I ! +H+.t .H 4 t+ttti "N L flJt In n 17 T1W 0 I I IJ 7r 1 Flffl#Efl~"t~iiiflS;flg44itg:44444..B4ii t444 "co M o" ON 00 N0 4 r4 4 M4 1 i e`4  ;_tt i; ; :.+tCT t f IL RT.t__i . 1~t *imrumimimrn I + t iRtd.. +t km t!i... .JEHUttitrLbttitttt in wn rHI 0 M0111; M W Ln ryj r}ITtyi h. 4 t II Ak) V i: Pj9 jp' Eili' __ l 7: 4I 41  1K) Jl11j r194 MI.+T_ 1;11! (A 'r i __ ir fO VI t tp4! 1. If  III 1 4141ttl t4 410 4 4 1, 11 1;4 i_: O~w I i" I T I MWO.W:1 fi 19 HII +44 +4 fii I+ IHH4 *l! t. t 4i 41i1t1:1 Ttrti  ~t;r'Jt~rt4~ 4;rrttr;fl{4tg4~44 HEtHfftHEfflrdfEtftdt .il' 4 4t t t li Mff: 4ffIi.ff LI 1.ia f.. .. ......i1 ir. U .,I ,ftrir I t.r: .. I. 1tW = tCa;4=+;1 Figure 10. Paramagnetic Resonance Spectrum of YAG:Cr [OO] Direction, v = 35.73 GHz 0 N^'x .1112nq 4b }4tt. thh4N thH+Rit m iittlt ~';~;~tr;ft~tf~t~t~t~tr~~i~ L1 rill. _tt rt4 tl 4,4 tUTI 4a: qUrm40i  RI'11 111 2it+r ~ i ~ ~ Ii ~ ~  )ml.,lm+ I LIl;LlYI rrrrm ^n.~~~!1 41T.fgf^mmfl r[ r? t i iI^^E F; ': "" [WI+5 i f MMM M ,++ ++,P + *i r4"t 3 tiu7 TTTI:T . i t TT] :J 21J Ilil [Z LIJ. iL+_ J. I 2.i i +. ":m;+' L.U:TT, U(CrTI +R;W 4r H +1H1 P"+i f [ I I I I I i confirmed by the PMR spectrum for the [113 direction shown in Figure 8. Note that the lines are generally grouped into triplets and singlets. The singlet lines are due to ions displaced parallel to the field, and the triplets are contributed by ions displaced at an angle of 70.60 to the field. Ideally, the triplets should be degenerate, but it is pos sible that some unexplained crystal field anomaly, or a slight misorien tation of the crystal, has caused them to be split. The center line of the triplet located at 12.895 kilogauss is actually degenerate with the 00 line which results from the transition m = + r.. s 2 2 If the external magnetic field is aligned parallel to the crystal line [110] direction, two of the trigonal displacement directions make an angle of 900 with the field and the remaining two make an angle of 35.30 with the field. One would then expect the lines to occur in doub lets, an assertion which is confirmed in some measure by Figure 9. Note that some of the doublets are degenerate while others are split by more than 100 gauss. When the external field is aligned with the [0oo1 direc tion, it makes an angle of 54.70 with each of the trigonal displacements, so that the absorption lines should occur in quartets, which is confirmed by Figure 10. The zero field splitting is calculated by using the PMR transitions measured in the [111] direction which are attributed to ions displaced parallel to the magnetic field. A more accurate calculation of this parameter is possible if transitions are used which have high field values, because the locations of these transitions are known to better accuracy. This higher accuracy is a result of the splitting varying almost linearly with magnetic field at the higher field levels. To make this calculation, the locations of two Kaband and one Xband transition 74 were used, as shown in the energy level schematic of Figure 11. 3 s 2 Ek 1 m s 2 6 3 m =  s 2 H  Figure 11. Ground State Energy Level Schematic with External Magnetic Field By using this schematic, it is not difficult to show that the zero field splitting 6 is H2Ek HEx 6 1 + H + H 1 2 (87) where Ek and E are energies of Ka and Xband photons, respectively, k. x and the magnetic field intensities HI, H2 are defined by the figure. The desired Kaband resonance occurs at a frequency of 35.73 GHz and a magnetic field of 7.377 kilogauss, while the Xband resonance occurs a 10.071 GHz and 9.354 kilogauss. Substituting these results into (87) i gives 6 = 0.509 + 0.001 cm where the basis for the accuracy will be discussed later. The effective gfactor is calculated from hvk = Ek = gBH, (88) where the variables have been defined earlier. For this calculation, it is necessary to use the parameters corresponding to the transition 1 1 9 ms =2 +2', namely H = 12.895 kilogauss, and vk = 35.73 x 10 Hz. This calculation gives g = 1.980 0.002. These results agree well with those of Carson and Whitel2 who found 6 = 0.510 cm1 and g = 1.98, for YAG doped with chromium. Using the above values for 6 and g, it is now possible to solve Equation (82) for the normalized energy level splitting as a function of angle and magnetic field. This equation was solved for the 11] io] , and 0013 directions and the results are plotted in Figures 12, 13, and 14. The PMR transitions which have been observed in the course of this experiment, both at Xband and at Kaband, are designated by arrows. The long arrows indicate Kaband transitions and the short arrows represent Xband tran sitions. Equation (82) was used to compare the locations of measured tran sitions observed in the [11i direction to the locations predicted by the theory of Davis and Strandberg. The observed values of magnetic field were substituted into (82) and this equation was solved for the energies, which were then compared to the microwave energy measured for the transitions. The results of these calculations are given in Table III. TABLE III PMR Line Positions in the [111 Direction at 35.73 GHz 3+ Cr Ion CrAn ur Magnetic Field in Gauss Angular Displacement Calculated Measured Percent Difference 00 3,445 3,533 2.55 00 7,371 7,377 0.08 00 12,895 12,895 0.00 70.60 5,238 5,239 0.02 (continued) TABLE III (continued) Cr3+ Ion Magnetic Field in Gauss Angular Displacement Calculated Measured Percent Difference 70.60 6,954 6,933 0.30 70.60 10,637 10,691 0.51 70.60 12,913 12,895 0.14 70.60 14,373 14,369 0.03 V.B. Optical Absorption Measurements The optical absorption spectrum of YAG:Cr was measured over the spectral range 250 to 900 nanometers, using the Beckman DK2 spectro photometer, and the results are shown in Figure 15. No additional ab sorptions were observed when the measurement range was extended from 200 to 2800 nanometers. This spectrum shows three broad absorptions, whose positions and qualitative widths are predicted with good accuracy by crystal field theory. The accuracy of the measured energies shown i in the figure is estimated to be + 100 cm Although it is possible to achieve much higher accuracy by using a more sophisticated spectro meter, the uncertainty in absorption position caused by the widths of the absorptions does not justify this additional accuracy. The absorptions shown in Figure 15 are broadened by the trigonal distortion splitting, spinorbit coupling within the manifold, and by thermal vibration of the ions which give rise to the crystal field. Un fortunately, the latter two effects effectively mask the trigonal dis tortion splitting, so that this effect could not be measured by optical absorption techniques, and the value of the splitting calculated in the 77 8 S0 Ions 70.60 Ions 6 3 m= s 2 / 3 4.4 ms m = .22 0 3 2 s 2 6  mS =  N 8 I I I I I 2 4 6 8 10 12 Magnetic Field in Kilogauss Figure 12. Normalized Ground State Splitting Versus Magnetic Field for jtl Direction N 900 Ions / 35.3 Ions 4 2 s 2  0 2  s 8 8S I I I I I I I 0 2 4 6 8 10 12 14 Magnetic Field in Kilogauss Figure 13. Normalized Ground State Splitting Versus Magnetic Field for [110] Direction 6 4 o "s ~~ 1  Sms 4  6 8 I I I I I I I 0 2 4 6 8 10 12 14 Magnetic Field in Kilogauss Figure 14. Normalized Ground State Splitting Versus Magnetic Field for 0o1] Direction 100% Transmission 550 600 400 Wavelength in Nanometers Figure 15. Optical Absorption Spectrum of YAG Doped with 0.1% Chromium 111. IIIlt II 0 lLA r'v .. ..I U ... o :: :: : .. : : : r 0 0 4 o N^^ 1i 14 1 I Ilr llIse I IU L _L I I I l l i t t i l l m lI1l~ 14411 IEE~f HIM111~ ,I I I I I I I I I I Ir~ next section cannot be verified experimentally. Some degree of experimental verification is possible by comparing the calculated splitting to the measured halfwidth of the optical absorption, a comparison which is made in Figure 17. V.C. Determination of the Crystal Field Parameters In Section II, it was shown that the crystal field potential, including the trigonal distortion, may be written in the form 0 2 0 4 y 10/2 3 3 V = A2r 2 + D' r ( (4 y (89) 0 2 4 where A r and D' r are parameters to be determined by PMR and optical absorption measurements. It was proven in Section III that the splitting of the r5 crystal field level, in terms of parameters defined earlier, is 5E2)2 E E = A = 2 (90) II III 8A2 8A This equation gives A in terms of measurable quantities if the spinorbit coupling constant A is known. It is tempting to use Equation (80), which gives g in terms of A and E5 E to calculate A based on PMR measure 5 2 ments. Unfortunately, this procedure gives a value for A, which when sub stituted into (90), results in a value of A that is not physically reason able. Using the values gs = 2.0023, g = 1.980 from PMR measurements, and i A = 16,700 cm1 from optical absorption measurements, Equation (80) gives 1 A = 46 cm Substitution of this value of A into (90) results in A = i 8400 cm which is more than onehalf of the separation of the F2 and r5 states. This value of A does not agree with the halfwidth of the F5 energy level determined by optical absorption measurements and is therefore not physically reasonable. This lack of agreement can probably be blamed 24 on covalency effects, studied by Owen, and the need for correcting the free electron gfactor for the atomic environment, discussed in detail by Abragam and Van Vleck.33 The problem of choosing the proper value of A for this work has been solved by using the free ion value. Kleiner gives theoretical and experi mental evidence that the value of the spinorbit coupling parameter A has a negligible dependence on the crystalline field. He shows that A may have a contribution AO, valid for the free ion, and a contribution A1 due to the n 4 crystalline field. He then shows that AO/A1 10 so that the free ion value A0 is a good approximation to A in a crystalline field. The fact that magnetic susceptibility data for several iron group ions agree well with theory based on the free ion spinorbit coupling parameter also supports this contention. The appropriate free ion value, consistent with treating LS as a state operator, is given by Low, and is 91.0 cm Substituting this value of A and the previously derived value of 6 into the equation for A gives A = (2.14 0.03) x 10 cm . It must be noted that the sign of A, the r5 level splitting, is ambiguous. Equation (79) shows that if A = E E is positive, then the ground level splitting 6 is positive in the sense that E 3 is greater 2 2 than E 1 so that A and 6 have the same sign. By using optical and PMR 2 2 absorption measurements, it is unfortunately not possible to determine the sign of either 6 or A. It is, however, possible to show that A must be positive by solving the cubic equation (55) for AO r using both positive and negative values of A, and thereby showing that a positive value for A makes the most physical sense. Substituting the negative value for A, together with the value Dq = 1670 cm 1, into (55) gives A r = 2.61 x 5 4 4  10 cm while the positive value of A gives 1.33 x 104 cm1, 3.38 x 104 cm 5 1 and 2.79 x 10 cm .In the former case, the other two roots of the cubic equation are complex, and therefore not physically meaningful. The proper 0 2  value of A2 r is chosen by substituting A = 2140 cm1 into the equation 2 for the r5 level splitting correct to first order, and obtaining A2 r = 15,900 cm. The root of Equation (55) nearest this value is 1.33 x 104 obtained by substituting the positive value of A into (55). Considering the indicated tolerances for Dq and A, it is therefore concluded that 0 2 4 1 A2 r = 1.33 x 10 200 cm The crystal field parameter Dq is determined from the location of the r5 level as given by optical absorption measurements. Figure 15 shows the optical absorption spectrum of YAG:Cr. The energy of the F5 level is simply lODq as shown in Section II, so that Dq = 1670 + 10 cm1 It is reasonable to ask whether the second order trigonal distortion perturbation will affect the F2 and r5 levels enough to make it neces sary to correct Dq for this effect. Equations (49), (54) and (56) are used to evaluate these corrections. Substituting the above value of 0 2 A2 r into Equation (56) gives for the corrected r2 energy E = 12Dq 160 cm1 Similarly, Equation (56) gives for the energy EII of the r5 level i E = 2Dq + 1200 cm1, and Equation 54 gives for the EIII and EIV levels i EIII = EIV = 2Dq 950 cm Considering the twofold degeneracy of the EIII and EIV states, the aver age energy of the r5 level is then 1 Ep = 2Dq 235 cm. 1 These results show that the energy difference E E is lODq 75 cm1 5 2 which is essentially 10Dq within experimental error. Based on these calculations, it is concluded that the second order trigonal distortion correction has negligible effect on the determination of Dq. The coefficient D'r of the fourth order term in the crystal field expansion may now be calculated using the above determined value of Dq and one of the diagonal crystal field matrix elements determined in Section II. Using the 3,3 matrix element, this calculation gives < 33y4I3> D'r = 2Dq, or 2 45 2 2 73 r \ / 05J D'r = 2Dq, which gives 4 4 1 D'r 4.14 x 10 + 250 cm. 2 The factor in the above equation results from expressing the crystal field potential in a form such that the [111] direction is the symmetry axis, as noted in Section II. Substituting for A2 r and D' r in the 2 crystal field perturbation expansion gives finally V = 13,300 y2 41,100 y + ( )/2 3 (91) Y413,300 0 [0 y4 (91) i expressed in cm . V.D. Accuracy of Results Errors in frequency measurement, magnetic field measurement, and crystal orientation are the types of errors which must be considered when evaluating the accuracy of PMR measurements. A study of these 3 1 errors was made in measuring the parameters of the > transition 2 2 in YAG:Cr at Xband. Both frequency and magnetic field for this tran sition were measured a total of 10 times while keeping the magnetic field centered on the absorption line. After each measurement, the field was readjusted to the center of the line and the frequency and magnetic field were measured again. During this series of measurements, the frequency remained stable at 10.7067 GHz, while the magnetic field varied from 9353.7 to 9354.6 gauss. The mean of the magnetic field measurements was 9354.1 gauss and the standard deviation was 0.28 gauss. Assuming that a random field measurement will fall within 3 standard deviations (3a) of the mean, the magnetic field for this transition is 9354.1 + 0.8 gauss or the accuracy is + 0.009 percent. This accuracy percentage is assumed for all of the magnetic field measurements, and the frequency measurement errors are considered negligible, based on the measured stability of the frequency source. Perhaps the largest error contribution in the PMR measurements is caused by a slight misorientation of the crystal relative to the mag netic field direction. This misorientation may be caused by errors in Xray orientation, crystal mounting, or spectrometer alignment. The Laue patterns shown in Figures 3a, 3b, and 3c show that the Xray orien tation is probably accurate to a few tenths of a degree because a 10 error in direction will cause about a 2 mm deflection of the central X ray spot. The photographs show that the central spots are centered in the pattern within less than about 0.5 mm. The standard deviation of alignment error caused by Xray orientation is then estimated to be 0.3 degrees. The crystal was oriented in the PMR spectrometer by aligning the intersections of the faces ground on the crystal with the microwave cavity geometry using a surveyor's telescope. The standard deviation of this error is estimated to be 0.5 degrees. Finally, the spectrometer was aligned with the electromagnet axis by again using the surveyor's telescope. Because of the large sizes of the magnet and spectrometer, this error is considered negligible compared to the other two error sources. Assuming that the alignment errors are normally distributed, the standard deviation of the total alignment error is E0.5)2 + (0.3)] =0.58 degrees. One would then expect the alignment error to be almost always less than (3)(0.58) = 1.74 degrees. To evaluate the effect of the above error on the location of the PMR resonances, Equation (82) was solved using the parameters determined 1 1 for the m = transition at 35 GHz for values of 0 ranging from s 2 2 50 to +50. The results of this calculation are plotted in Figure 16. Note that an alignment error of 1.7 degrees corresponds to an error in magnetic field determination of 13 gauss, which is large compared to the field measurement error and is therefore the dominant error source. This magnetic field error was substituted into Equations (87) and (88) for 6 and g to give the error limits indicated in the evaluation of these parameters. The error in 6, together with the error in optical absorp tion measurements, was then used to estimate the uncertainty in the r5 level splitting, A. The 30 error for the optical absorption measurements is estimated i to be 100 cm. Error contributions in these measurements come from spectrometer calibration alignment as well as uncertainty in position of the absorption maximum due to the width of the absorption itself. This error in the location of the optical absorption maxima was used to esti mate the error in determination of the crystal field parameters Dq and 4 D' r V.E. Summary and Conclusions The results of this study are summarized in Table IV. In those in stances where previously known results are available, they are given for comparison. Figure 17 is a series of energy level diagrams based on the calculations and measurements of the parameters given in Table IV. This Figure shows the effect of the first and second order trigonal distor tion perturbation on the energy levels, as well as the results obtained by solving the complete 10 x 10 crystal field matrix by computer. The first energy level diagram shown is that measured by optical absorption, and the shaded regions are intended to represent the absorption half widths. Absorption peaks are indicated in these shaded regions. The 88 13.02 13.00  En " 12.98 rd 0 P O r( Cu g 12.96  a 12.94 C, S12.92 12.90 12.88 I I I I I 0 +1 +2 +3 +4 +5 Angular Deviation from [11ii1 Direction in Degrees Figure 16. Magnetic Field of Resonance as a Function of Orientation Error for the Transition ms = > ms = 2 U > 12 8 ________________I_____I 0 u ^ ^ ^ Anua eitonfo il iecini ere Fiue1.MgeicFedo eoacea ucino Oretto ro frteTasto m = percentage errors between measured and calculated centers of gravity are indicated on the figure. Degeneracies are noted in parentheses. TABLE IV Summary of Results Value Parameter Crystal Field Parameter, Dq (Measured) Crystal Field Expansion (Calculated) Ground State Splitting, 6 (Measured) r5 State Splitting, A (Calculated) Spectroscopic Splitting Factor g (Measured) Location of r4 Levels (Measured) Location of F4 Levels (Calculated) This Work 1670 + 10 cm1 13,300y041,400 Oy +10 3 3] +(7 4 4 0.509 0.001 cm1 0.509 + 0.001 cm Carson and White 0.510 2.14 x 103 1.980 + 0.002 1.98 2.32 x 104 + 100 cm1 3.57 x 104 + 100 cm1 2.37 x 104 cm1 3.71 x 104 cm1 The most important result of this study is the use of complementary optical and paramagnetic resonance spectroscopy to characterize the crys talline field in YAG:Cr. The link between these two different spectro scopic methods is the connection between the trigonal distortion split ting of optical levels and the slight splitting of the ground level which has such a large effect on the PMR spectrum. Measured  Optical Absorptions Pure Octahedral Crystal Field (3) +2.8% First Order Trigonal Distortion (2) +2.8% (1) Second Order Trigonal Distortion (1) +3.9% (2) Computer Solution (1)+3.9% (2) 30 L 25 L (3) +1.2% 20 I (3) 0.0% (1) C.G. +1.2% (2) (1) C.G. 0.0% (2) (1) C.G. +2.1% (2) (1) C.G. 0.5% (2) (1) C.G. +2.5% (2) (1) C.G. 1.4% (2) 10  Figure 17. Calculated and Measured Energy Level Diagrams Peak Peak \\\\\\\X The locations of the optical absorption bands due to 4F and 4p multiple states have been successfully predicted by using crystal field theory. The widths of these bands have been qualitatively accounted for by treating the trigonal distortion as a perturbation of the octahedral crystal field energy levels. The locations of PMR absorptions have been found to agree well with the theory of Davis and Strandberg. The method of treating the trigonal distortion as a perturbation of the octahedral crystal field has application to other ions in other cry stalline complexes. For example, the neodymium ion in YAG occupies a site of orthorhombic but near cubic symmetry. It is possible that the deviation from cubic symmetry could be treated as a perturbation of the cubic states to better explain the energy levels of this important laser material. In addition, there are many materials containing iron group ions to which this procedure might be applied to better explain observed magnetic resonance and optical absorption results. REFERENCES 1. B. Bleaney and K.W.H. Stevens, Repts. Prog. in Phys., 16, 108 (1953). 2. K.D. Bowers and J. Owen, Repts. Prog. in Phys., 18 304 (1955). 3. S. Geschwind, G.E. Devlin, R.L. Cohen, and S.R. Chinn, Phys. Rev., 137, A1087 (1965). 4. R.J. Elliott and K.W.H. Stevens, Proc. Roy. Soc. (London) A215, 437 (1952). 5. R.J. Elliott and K.W.H. Stevens, Proc. Roy. Soc. (London) A218, 553 (1953). 6. R. Finkelstein and J.H. Van Vleck, J. Chem. Phys., 8, 790 (1940). 7. W.H. Kleiner, J. Chem. Phys., 20, 1784 (1952). 8. B. Bleaney, Proc. Roy. Soc. (London) A204, 203 (1950). 9. C.F. Davis, Jr. and M.W.P. Strandberg, Phys. Rev., 105, 447 (1957). 10. W. Low, Phys. Rev. 105, 801 (1957). 11. M.M. Zaripov and Iu. Ia. Shamonin, Soviet Phys. JETP, 3, 171 (1956). 12. J.W. Carson and R.L. White, J. Appl. Phys. 32, 1787 (1961). 13. J.A. Koningstein and J.E. Geusic, Phys. Rev., 136, A711 (1964). 14. G. Burns, E.A. Geiss, B.A. Jenkins, M.I. Nathan, Phys. Rev, 139, A1687 (1965). 15. M. Tinkham, Group Theory and Quantum Mechanics, McGrawHill Book Company, New York, 1964, pp. 6975. 16. K.H.G. Ashbee and G. Thomas, J. Appl. Phys., 39 3778 (1968). 17. S. Sugano and Y. Tanabe, J. Phys. Soc. Japan, 13, 880 (1958). 18. M.T. Hutchings, Solid State Physics, Vol. 16, Academic Press, New York, 1964, pp. 227273. REFERENCES (Continued) 19. E. Merzbacher, Quantum Mechanics, John Wiley & Sons, New York, 1961, p. 153. 20. H.H. Theissing and P.J. Caplan, Spectroscopic Calculations for a Multielectron Ion, Interscience Publishers, New York, 1966, Chapters 10, 12, 14, and 15. 21. J.C. Slater, Quantum Theory of Atomic Structure, McGrawHill Book Company, New York, 1960, Chapter 14. 22. A.R. Edmonds, Angular Momentum in Quantum Mechanics, 2nd Ed., Princeton University Press, Princeton, 1960, pp. 7577. 23. E.U. Condon and G.H. Shortley, The Theory of Atomic Spectra, Cambridge University Press, Cambridge, 1957 Chapter 14. 24. J. Owen, Proc. Roy. Soc. (London) A205, 136 (1951). 25. S. Sugano and M. Peter, Phys. Rev. 122, 381 (1961). 26. G.E. Pake, Paramagnetic Resonance, W.A. Benjamin, Inc., New York, 1962,Chapter 3. 27. B. Bleaney and K.D. Bowers, Proc. Phys. Soc. (London) A64 1135 (1951). 28. M.H.L. Pryce, Proc. Phys. Soc. (London) A63, 25 (1950). 29. A. Abragam and M.H.L. Pryce, Proc. Phys. Soc. (London) A205, 135 (1951). 30. W. Low, Paramagnetic Resonance in Solids, Academic Press, New York, 1960, pp 1125. 31. D.J.E. Ingram, Spectroscopy at Radio and Microwave Frequencies, Butterworths Scientific Publications, London, 1955, Chapter 4.1. 32. R.S. Alger, Electron Paramagnetic Resonance: Techniques and Appli cations, Interscience Publishers, New York, 1968, Chapters II, III, and IV. 33. A. Abragam and J.H. Van Vleck, Phys. Rev., 92, 1448 (1953). BIOGRAPHICAL SKETCH Robert Walker McMillan was born on April 18, 1935 in Sylacauga, Alabama, the son of Robert T. and Alma B. McMillan. He has one sister, Martha, now Mrs. John McDermott of Ann Arbor, Michigan. At the outbreak of World War II, his family moved to Charleston, South Carolina, where his father worked in the shipyard for the duration of the war. In 1948, the family moved back to Alabama, and Robert was graduated from Sylacauga High School in 1953. In 1957, he received a Bachelor of Engineering Physics Degree from the Alabama Polytechnic Institute, now Auburn Univer sity. Following two years' active duty as a communications officer in the Air Force, during which he served in Illinois and Japan, Robert was em ployed as an engineer by the Baltimore Division of the Westinghouse Elec tric Corporation. After two years at Westinghouse, he moved to Orlando, Florida, where he was employed as an engineer by the Orlando Division of Martin Marietta Aerospace, a position which he still holds. In 1966, Robert earned a master's degree in Physics from Rollins College, and in 1967 he began work on a PhD. degree at the University of Florida. Robert McMillan is married to the former Ann Simmons of Sylacauga, Alabama, and is the father of three children: Marisa, Robert, and Natalie. He has been granted two patents and is coauthor of three technical pub lications. He is a member of Pi Mu Epsilon, Sigma Pi Sigma, and the Arnold Air Society. He is also a member of the First Baptist Church of Orlando, where he has served as a Sunday School teacher. 