. . ,......: g‘.|.1ld\N-.*" .v...,.-‘,¢....... Q- . ' ’ ‘ ~ . ’ :- I > Ii. I 7 . i ' I 77 ' ‘ V 5 T . '~ mesisfer‘ihefisgs’eeéwhb. _‘ ’ gaicgééa. smfirézfi ' , . ‘ mew—isms . - a , ' -mfi. ”Mn” nu-.. , ---. - - .'r:_.","',.r wurr-n-wunnu-nl ”Inn-.2 . -_.. ..._ LEBRARY Michigan State University This is to certify that the thesis entitled THE TRANSFERRED HYPERFINE EFFECT IN ErClB presented by Dennis E. Gallus has been accepted towards fulfillment of the requirements for Ph , D . _ degree in _I§1_G.S_Ph MM; Major professor Date 30 “113le ! 0-7639 ABSTRACT THE TRANSFERRED HYPERFINE EFFECT IN ErC13 by Dennis E. Gallus An experiment was performed to indirectly measure the transferred hyperfine effect in paramagnetic ErCl3 at 1.18 K and 500 G applied field. Directions of the principal axes were determined and the applied field rotated through planes containing the principal axes. Data on transition frequencies vs. angle of applied field was taken, using a minipulser. An effective field tensor was fitted to resonance data, and the transferred hyperfine interaction tensor decoupled from it, using measured susceptibility data and evaluating the dipole-dipole interaction by a computer summation over lattice sites. The greatest contribution to the internal field by the transferred hyperfine effect occurred for a magnetic field applied in the mirror plane, near its intersection with the cleavage plane. The applied field was decreased by 70% due to the transferred hyperfine field. The transferred hyperfine interaction tensor was decomposed into an isotrOpic part As; and an anisotropic part ép' The isotropic part is AS = 3.7 x 10-4 cm_l. ép was analyzed in terms of o and n-bond contributions, and A0 - A1T = -6. x 10-4 cm-1 for both n orbitals. This indicates a greater interaction between the Cl35 nucleus and the holes in 3p electron orbitals perpendicular to the internuclear radius. THE TRANSFERRED HYPERFINE EFFECT IN ErCl3 BY Dennis E. Gallus A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Physics 1971 PLEASE NOTE: Some Pages have indistinct print. Filmed as received. UNIVERSITY MICROFILMS To Prudence ii ACKNOWLEDGMENTS I express sincere thanks to Dr. E. H. Carlson, for both suggesting this project and providing a great amount of help and encouragement during the course of the work. I also thank Dr. J. P. Hessler for many useful conversations, and Mr. D. H. Current for help in data taking. iii II. III. IV. V. VI. VII. VIII. IX. X. XI. XII. TABLE OF CONTENTS INTRODUCTION 0 C O O O C O O O O O O O O O O . THEORETICAL O O O O O O O O O O O O O O O O 0 General Theory and Hamiltonian . . . . . Theory Particular to ErCl3 . . . . . . . CRYSTAL STRUCTURE . . . . . . . . . . . . . . SAMPLE PREPARATION . . . . . . . . . . . . . . SAMPLE MOUNTING AND LAB COORDINATES . . . . . EXPERIMENTAL PROCEDURE . . . . . . . . . . . . Locating the Principal Axes . . . . . . . General Experimental Procedure . . . . . DATA DIAGRAMS AND INTERPRETATION . . . . . . . FITTING THE TENSOR . . . . . . . . . . . . . . Approximate Fit by Perturbation Theory . Computer Fit . . . . . . . . . . . . . . Precision of Measurement . . . . . . . . DECOUPLING . . . . . . . . . . . . . . . . . . MAGNITUDE OF THE INTERNAL FIELD . . . . . . . RESULTS . . . . . . . . . . . . . . . . . . . CONCLUSION . . . . . . . . . . . . . . . . . LIST OF REFERENCES . . . . . . . . . . . . . APPENDIX A DIAGONALIZING THE QUADRUPOLE I-IAMILTON IAN C O O C O C O O O O 0 APPENDIX B COMMENTS ON TEMPERATURE DEPENDENCE APPENDIX C COMMENTS ON THE THREE-FOLD AXIS OF SYMMETRY . . . . . . . . APPENDIX D DATA TABLES . . . . . . . . . . . iv 13 18 21 24 24 25 31 33 33 36 40 48 56 66 67 72 74 82 86 88 Table 10 11 12 13 14 15 LIST OF TABLES Summary of Carlson and Adams Results Summary of Data Interpretation of Strength Symbols Change in calculated frequencies per change of 0.01 in Tab Tab per one kHz change in fn along a given axis Magnitude of the internal field for 500 G applied field in xz plane of the four-fold site Magnitude of the internal field for 500 G applied field in plane perpendicular to cleavage plane and containing the y principal axis, four-fold site Magnitude of the internal field for 500 G applied field in the cleavage plane, four- fold site Magnitudes of some Transferred Hyperfine Interactions Variation of Frequency with Temperature Data for the elliptical cone method of locating the principal axes, four-fold site, f = 4515.7 kHz O H in cleavage plane H in xz plane, four-fold site H in plane 1 cleavage, containing y of four- fold site H in xz plane of eight-fold site 40 43 44 63 64 65 68 85 88 89 91 93 9S Figure 10 11 12 13 14 15 16 LIST OF FIGURES Principal axes and Er axes . . . . . . . . . ErCl3 Crystal Structure. . . . . . . . . . . Relation between principal axes and lab axes Distillation tube in furnace . . . . . . . . H in cleavage plane. . . . . . . . . . . . . H in xz principal axis plane, 4-fold site . H in plane 1 cleavage containing y, 4-fold site. H in xz principal axis plane, 8-fold site . Goodness of fit, H in xz plane, 4-fold site. Goodness of fit, H in plane l cleavage, containing y, 4-fold site. . . . . . . . . . Goodness of fit, H in cleavage plane . . . . Internal field in xz plane, 4-fold site . Internal field in planelly axis, 1 cleavage Internal field in cleavage plane . . . . . . Zeeman splitting of pure quadrupole spectrum Frequency vs. temperature for a signal . . . vi 15 l6 19 27 28 29 30 45 46 47 57 58 59 80 84 I INTRODUCTION Nuclear magnetic resonance has proven to be a particu- larly useful tool for analyzing the degree of covalency of the metallic halides. This technique is especially applica- 35 ble to a paramagnetic salt such as ErCl in which the Cl 3: nucleus undergoes both an electric quadrupole and a trans- ferred hyperfine interaction, with a very strong resonance signal. An asymmetric charge distribution around the Cl35 nu- cleus will produce an electric field gradient which will couple to the quadrupole moment of the nucleus. In forming a covalent bond, such as in ErCl the Cl- ion transfers 3: some electronic charge to the Er3+ ion. This transfer of charge produces holes in the Cl- 35 and 3p orbitals, giving rise to an electric field gradient and hence a quadrupole interaction. But since unequal amounts of spin-up and spin- down charge are transferred, a net spin density exists which interacts with the Cl35 nucleus via the transferred hyperfine interaction. The transferred hyperfine interaction is not the only perturbation to the quadrupole interaction. In the presence of a magnetic field, there are Zeeman and dipole-dipole inter- actions as well. The purpose of this thesis is to account for all these interactions in a Hamiltonian, and to 1 evaluate the transferred hyperfine interaction tensor. This tensor is then used to estimate the fractional spin density in the C1- 35 and 3p orbitals, giving an insight into bonding modes. The method employed is to fit an effective field ten- sor to the resonance data, and to decouple from it the, transferred hyperfine tensor. The fitting of the tensor to the data, and the evaluation of the various interactions, are accomplished with the aid of a computer. Results are discussed in terms of o and n bonding between the Cl- and + . Er3 lOnS. II . THEORETICAL General Theory and Hamiltonian All nuclei of spin 1 l possess an electric quadrupole moment which can interact with the gradient of any electric field present at that nucleus. The Hamiltonian for this interaction is the Quadrupole Hamiltonian, HQ=Q° (1) "<1 "[11 where g is the tensor defining the quadrupole charge distri- bution in the nucleus and 22 is the gradient of the electric field. The quadrupole moment 9 is a measure of the ellipti- city of charge distribution in the nucleus, and is defined as eQ = 35f(3z2 - r2)p(£)d3x (2) where 0(5) is the charge density. Q is positive for an egg- shaped nucleus, and negative for a saucer-shaped nucleus. Because the nucleus possesses a spin £_and a net charge, it also possesses a magnetic moment, En’ En = ykI' (3) where y is the magnetogyric ratio of the nucleus (Y = 2624. 1 -l 35 radians sec- G for C1 ). The allowable nuclear spin states are quantized, and the component m of the nuclear spin vector I in any direction can take only one of a set of discrete values +1, (1-1), ...-I; for c135, 3/2 ... -3/2. 3 If one applies a steady magnetic field H to the nucleus, there is an interaction between the applied field and the mag- netic moment En' The Hamiltonian for this interaction is Hz = -En‘H. (4) For small fields, this magnetic interaction may be viewed as a perturbation of the quadrupole interaction, one that breaks up the two-fold degeneracy of the pure quadrupole frequency. See Appendix A for the diagonalization of the quadrupole Hamiltonian in the presence of a weak magnetic field. There is also an interaction between the magnetic moments of the nucleus and the electrons of unfilled shells, or elec- trons in the unfilled shells of neighboring ions. The clas- sical interaction energy E between two magnetic moments En and He is 'u - 3(u °“1')(u "1?) E = —e —n —e ' (5) 3 r where e is the unit vector from En to He and r is the distance between the two moments. If there is a distribution of elec- trons over many lattice sites, the total dipolar Hamiltonian is the sum over all the electron magnetic moments. The crystal is paramagnetic, the Er3+ magnetic moments are lined up with the applied field, and the electron magnetic moment must be replaced by its thermal average . Thus 1 - 3 ?j ii Hdd = En Z 3 <“e> (6) J rj where £j is the vector from the nuclear magnetic moment to the jth_electron. Representing the sum by a symmetric, second rank inter- action tensor 3, and substituting the definitions En ='Y‘h’_I_ (7) (Ee>=-UB.g.<§> (8) where “B is the Bohr magneton, g is the electron g-tensorl, and S is the electron spin, one has Hdd = -quB £°2'2° - (9) According to the usual meaning of the term "hyperfine", that is, having extra-nuclear origin and coupling to the nu- clear spin, the dipole-dipole interaction is a hyperfine term. In this work, however, the term "hyperfine" will refer only to 35 nucleus the transferred hyperfine interaction between the C1 and the charge transferred in forming the Er-Cl bond, exclu- sive of the dipole-dipole coupling. The Hamiltonian for the hyperfine interaction is Hh=y IIZ’ '§ - (10) g is the second rank hyperfine interaction tensor. As will be shown later, it is the sum of an isotropic and an aniso- tropic part. The dipole-dipole and hyperfine Hamiltonians written above are not very useful, because the spin §_is not a clearly defined quantity. In a paramagnetic crystal, the electron magnetic moments align with the field, but thermal disorder resists this alignment. Due to the crystal structure, the magnetic moments are more easily aligned in certain directions than in others; the anisotropic electron g-tensor describes this. Further, there are spin relaxation effects.2 What all of this implies is that the nucleus responds only to an average electronic field, preportional to the spin average <§>. Thus, the hyperfine Hamiltonian is better written Hh = _I_-§-. (11) For a paramagnetic crystal, the magnetization M_is propor- tional to the thermal-averaged spin: L4 = N11 9' : (12) here, N is avogadro's number. Also, M = z'fi . (13) Thus, _ 1.x.H (S) = :-—————:——-: 14 This defines the thermal-averaged spin in terms of measurable quantities. The hyperfine tensor A can be determined, at least pheno- menologically, from plots of Cl35 transition frequencies. The various interactions are viewed as if they modified the applied magnetic field and produced an internal field which caused the splitting of the observed transition frequencies. Writing the total Hamiltonian as ” = ”o ' WEE-Wares-<§>+y§°<§> . (15> . -l H #1 l g g I 16 = .- o + ... o _ . O Y _ l: (g YhuB ) N 1 g , ( > one has H = HQ - th'g'fl. (17) with _ T - 1 + [D §.g ]' g (18) = _ : = Yh-UB N . The choice of signs is consistent with that of Shulman and Sugano3 in the case of KMnF3, which does not, however, have a quadrupole interaction. For purposes of analysis, the total Hamiltonian is best separated into a quadrupole part and a magnetic part; H = HQ + H . (19) The magnetic part of the Hamiltonian can be thought of as the interaction of the Cl35 nuclear spin with an effective internal magnetic field Eeff: Hm = -th-Eeff ° (20) Eeff is the sum of the applied field and the internal fields due to the spin of the electron: :13 ll :11 —eff — + Edd + Hh (21) . (22) ll Ill-3 HI: This equation defines the effective field tensor T, whose effect is to modify the applied field in such a way as to cause the Cl35 transition frequencies observed in the lab. Theory Particular to ErCl3 35 in ErCl3 has been written, the various terms must be evaluated in order to Now that the Hamiltonian for C1 uncouple the transferred hyperfine interaction tensor. Solving equation (18) for g, g = yhuB(2 - u(g-;)-x'1 )‘g . (23) The dipole-dipole term is evaluated by summing over the Er3+ lattice sites, using a computer program based on equation (6). g is fitted from the data of transition frequencies vs. angle of applied field. The values of x-1 and g have been deter- mined experimentally by Fairall, EE.EL'4 The A tensor obtained in this decoupling describes the total transferred hyperfine interaction between a Cl35 nucleus and its two nearest neighbor Er3+ ions. By means of similarity transformations and symmetry properties, one may separate out the effect of only one Er-Cl bond. Thus, one obtains éEr' the transferred hyperfine interaction tensor in the Er system 35 of axes, centered on the Cl ion, whose x' axis coincides with the Er-Cl internuclear radius. éEr may be decomposed into an isotropic component, and an anisotropic, traceless component A : (25) A = A 1 + A . — s: =p The separation of g into isotropic and anisotrOpic parts is useful because it can then be used to deduce whether the Cl- electron wave function has primarily s or p-state character. In forming the covalent bond, the charge transfer from the Cl— 3+ to the Er creates 3s and 3p holes in the Cl- electron ErBI .3 plane g, E are crystal lattice vectors xz plane is mirror plane (x,y,z) are principal axes (x',y',z') are Er system of axes 3? crystal Er3+ Figure 1: Principal axes and Er axes 10 distribution. The total Cl_ electron wave function is thus 'a sum of s and p-state wave functions. As describes a 35 nucleus and the net hyperfine interaction between the C1 spin in the 33 holes; ép describes an interaction between the nucleus and the 3p(x'), 3p(y'), and 3p(z') orbital holes. The diagonal elements of A are Br 2 A.. = A + Z A . (3cos 6.. - 1). (26) 11 S j=xl'yl’zI 13(3) 1:] ép being traceless, it has only two principal elements.5 Hence, one cannot determine the individual contributions of the three 3p orbitals. But one can deduce the isotropic term and the pairwise differences between the three p-dipolar hyperfine interactions A Writing equation (26) in 3p(i)’ explicit form, c0526ij = 0, coszeii = 1, and AX'X' = AS + 2A3p(X') " A3p(y,) - A3p(2') (27) = — + _ Ay'y' As A3p(x') 2A3p(y') A3p(z') ‘28) Az'z' = A8 - A3p(x') - A3p(y') + 2A3p(z')° (29) Adding these three equations, one sees _ 1 AS — 3 Z A.. . (30) j=x'ly.lz' 33 Subtracting equation (28) from equation (27), one sees A - A x'x' = 3( Y'Y' A3p(x') — A3p(y')) (31) hence A3p(x') — A3p(y') ') ' (32) Subtracting equation (29) from equation (27), = 3(A3p(x.) - A3p(z.)), (33) 11 (A , - Az'z') . (34) _}_ A3p(x') A3p(z') I 3 x'x Subtracting equation (29) from equation (28), Ayiyl — Azlzl = 3(A3p(yi) — A3p(zl)) (35) _ l _ A3p(y|) - A3p(z|) " 3 (Aylyt AZ'Z'). (36) The 3p(x') orbital is along the internuclear radius, the line of bonding, while the 3p(y') and 3p(z') orbitals are perpendicular to the internuclear radius. In the notation of chemical bonding, 3p(x') would be designated a o orbital, and 3p(y') and 3p(z') are designated n orbitals, fly and nz reSpectively. Symmetry does not, in the case of ErCl3 crystal structure, allow one to say that "y and nz are equal. But one can determine relative amounts of the 0, fly and n2 orbitals. From equation (32), using c and n notation, _ i - AO - A,"y — 3 (Ax'x' Ay'y') . (37) From equation (34), A - A - -1- (A - A ) (38) o “z 3 x'x' z'z' ' And from equation (36), A - A = J'- (A - A ) (39) Try “-2 3 Ylyl zlzl ' The c and n notation denotes the type of chemical bonding (or antibonding). The 3p-o electrons are those in the 0 bond, having no angular momentum about the internuclear radius. The 3p-n electrons are in n bonds, which do have angular momentum about this axis. Thus, AO interactions arise from an electron wave function in the Cl 3p orbitals which lie along the Er-Cl internuclear axis, and ATI interactions arise from an electron wave function in the Cl 3p orbitals perpendicular to this direction. 12 To appreciate the physical significance of the constants, consider the 2P3/2 ground state of the Cl atom. The only contributions to the transferred hyperfine effect come from the single unpaired electron. Quantizing along the x axis, the asymmetry parameter n = 0, and Ay = Az’ and experimentally6 _ -4 -1 A3p(x') — 51.2 x 10 cm . (40) The isotropic hyperfine interaction constant for the Cl- ion has been calculated to be 1 A38 = 0.148 cm . (41) These numbers are presented only to give an indication of the order-of-magnitude of the hyperfine interaction tensor. III CRYSTAL STRUCTURE ErCl3 has a crystal structure isomorphic to AlC13, having a tetramolecular cell of the monoclinic space group C2/m. The ions occupy the following positions of point symmetry:7’8 4Er:(g), 2, t(0u0;%,u+%,0) 4C1(l):(i),m, i(u0v;u+k,%,v) 8C1(2):(j),l, i(xyz;x§z;x+%,y+%,z;x+%,%-y,z) The parameters have not been determined for ErCl3. They are taken as being the same as in YCl : 3 u(Y) = 0.166 u(Cl,l) = 0.211 v(Cl,l) = 0.247 x(Cl,2) = 0.229 y(Cl,2) = 0.179 z(C1,2) = -0.240 The cell dimensions for ErCl3 are (A) .... .. __= ._.. o i aO - 6.80 b0 — 11.79 cO 6.39 B 110 42 The crystal structure is a cubic close-packing of Cl- 3+ ions in one-third of ions distorted by the inclusion of Er the octahedral spaces. The ions are arranged in parallel layers as Cl-Er-Cl:Cl-Er-Cl:, with easy cleavage parallel to the plane of the layers; see Figure 2. This structure suggests an approximate three-fold axis perpendicular to the plane of the layers. Carlson and Adams7 have measured the nuclear quadrupole resonance parameters in ErC13, using the low field "elliptical 13 14 9,10 cone" method. For the four-fold Cl(l) site, they find °-3 q = :3.103 A , and the x principal axis of this site is within 3° of the Er-Cl(l)-Er angle bisector. The two bisectors of the eight-fold Cl(2) sites are within 4° of the x principal axes of the Cl(2) sites. The remainder of the Carlson and Adams results are summarized in Table l, where the asymmetry parameters ”NQR and bond angle 8N R are calculated from Q resonance data. The fQ are the observed pure chlorine quad- rupole resonance frequencies. Orientation of the principal axes of the crystal with respect to the lab axes can be seen from Figure 3. TABLE 1 Summary of Carlson and Adams Results 8 fQ(kHZ) nNQR ncalc NQR Cl(l) four-fold 4515.7:0.2 0.52:0.03 0.523 100.0°i0.6° Cl(2) eight-fold 4456.6:0.2 0.53i0.02 100.2°i0.4° Cl- ions lying in the mirror plane are designated as four— fold sites; the eight-fold site C1_ ions are situated between the mirror planes. This difference in location creates a difference in the electric field gradient at the respective sites. Hence the pure quadrupole frequency for the four-fold site differs from that of the eight-fold site. As can be seen from a crystal diagram such as Figure 2, an approximate three-fold axis of symmetry exists in the crystal. Apart from the fact that they occupy different 15 <:>| C1- I + I Er \ bottom layer top layer <3 <> <> Cleavage plane is plane of paper--Ions not drawn to size Figure 2: ErCl3 Crystal Structure [Itlllrtlll 16 cleavage plane . (x,y,z) are principal axes Z (X,Y,Z) are lab axes Y,y(h-fold) 120° 120° u-fold xz p ane x '8-roid (8 g 1 ) . xz planes y(B-fold) - o d Figure 3: Relation between principal axes and lab axes 17 crystal sites, the Cl— ions about this three-fold axis 3+ . ions. appear to be similarly situated with respect to Er This suggests that the tensor 3 describing the effective field may also have three-fold symmetry, and this will be investigated in Appendix C. This symmetry about the axis perpendicular to the cleavage plane is only approximate because the two sites have different quadrupole frequencies. The magnetic (Zeeman) splitting of the Cl35 transition frequencies appears to be three-fold symmetric about this axis, however. if! EI.‘ IV SAMPLE PREPARATION Single crystals of BrCl3 are prepared from commercial anhydrous ErCl3 powder. The powder, in a Vycor tube, is first dried by slow heating under vacuum in a vertical furnace. When waters of hydration are removed, the heat of the furnace is increased to melt the sample. The fused, but impure, sample is then placed in the open end of a Vycor distillation tube and the tube sealed. The sealed tube is placed in a horizontal furnace whose right and left sides are controlled by separate variacs. During the entire distillation operation, the tube is under a vacuum of 10_6 mm Hg. First the collecting side of the tube is heated to about 1000 C to outgas impurities from the tube. This side is then cooled to a temperature just slightly higher than the melting point of ErCl Now the sample end of the tube is 3. heated above the melting point, until the ErCl3 gradually vaporizes and distills across to the collecting side of the tube, leaving behind involatile impurities. Violent boiling of the sample must be avoided, or splashing, hence contamina- tion, will result. The pure ErCl3 condenses on the walls of the collecting side of the tube, then runs down the walls into the ampule. See Figure 4. 18 19 o psdoa pm adsoeb Hops: Mao poamom ma masses maopmhmoom poHHonusoo ma oosshsu ho spam nose coachsh i.) 388 «A! manage oases“ adsorb on — oosunsm Distillation tube in furnace Figure 4: 20 When most of the ErCl3 is distilled across, the distilla- tion tube is cooled and the ampule sealed off under vacuum. A single crystal is grown in the ampule by lowering it through a gradient furnace. Before lowering is begun, the position of the ampule is repeatedly adjusted until several crystals form in the s-shaped tail of the ampule. The sample is then molten and at a temperature only slightly above the melting point; only the tip of the tail extends from the heated zone of the furnace. The crystals which form in the tail are examined at twelve hour intervals, and the actual dropping of the ampule through the furnace not begun until a crystal is present in the tail with its cleavage plane oriented in a favorable direction. The ampule is then lowered for approximately six days, and cooled slowly. The ampule is sawed open with a diamond saw, and the crystal immersed in mineral oil. In order to prepare a spherical sample, a single crystal with no cleaving evident along its length is needed. The crystal is hand-ground with a slowly revolving grinding wheel, being continually bathed in mineral oil. Roundness is determined crudely by passing the crystal sample through holes drilled in a sheet metal strip. Only an approximately spherical shape is attainable by this method, and care must be taken to get that, as the crystal cleaves very easily when oil-soaked. V SAMPLE MOUNTING AND LAB COORDINATES For an experiment, the sample is placed in a rotatable goniometer attached to a cryostat. The goniometer can be rotated through more than 360° about its horizontal axis; this axis is taken as the lab Z direction. The lab Y axis is taken to be pointing up; lab X is then horizontal and perpendicular to Y and Z, so a right-handed system results. When the crystal is mounted on the goniometer, the cleavage plane is perpendicular to the plane of rotation of the gonio- meter. Hence, X and Y lie in the cleavage plane of the crystal. The lab angle ¢1ab is read off of the face of the goniometer by pointing a flashlight through the unsilvered portion of the dewar. The goniometer is marked in divisions of 5°, with a major division every 30°. Accuracy of aligning the goniometer is estimated at :1°. The magnet, rotating about the Y axis, defines elab; an aluminum scale running about the base of the magnet is marked at intervals of 1°. The magnet pointer indicates 0° with the field along Z. When setting up each run, care is taken so that the magnet pointer reads 270° when the magnet pole faces are parallel to the Z direction. The goniometer consists of a flat nylon disc centered on a brass bearing. The disc may be rotated from outside 21 22 the dewar by means of a rotatable rod which moves a wire and pulley. Depending upon the shape of the crystal sample, it is mounted either directly on the goniometer, or held in a detachable nylon cup. Samples shaped like sections of cylin- ders are mounted directly on the goniometer disc, being held in place with vacuum grease and a small nylon clamp, which is screwed down with nylon screws. By pressing the cleavage plane of the crystal directly onto the plane of the disc, one insures that the cleavage plane is perpendicular to the axis of rotation of the goniometer. Spherical samples are placed in the nylon cup and held in place with soft clay. In the case of cylindrical samples, a coil of #28 varnished wire could be wound directly on the sample. In the case of spherical samples, the coil must be wound around a thin paper cylinder whose radius is slightly larger than the radius of the sphere. This coil is then inserted in the nylon cup and held in place with clay. Coil leads are made long enough to avoid binding when the gonio- meter is rotated. During a long experiment, care must be taken to avoid rotating the goniometer in only one direction, to avoid the problem of snarled leads. Before placing the spherical sample in the cup, a small part is cut off in the direction of cleavage to leave a flat spot. This spot is mounted facing out of the cup, and a small, flat mirror fragment is affixed to the flat spot with grease. Thus the mirror is parallel to the cleavage plane. The goniometer is then rotated, and the reflection of a target located above the plane of the disc is observed. 23 Position of the sphere is adjusted until the goniometer can be rotated without causing the image of the target as seen in the mirror to move. The cleavage plane of the crystal is then perpendicular to the axis of rotation of the goniometer disc. When the cryostat is inserted in the dewar, the plane of rotation of the goniometer must be parallel to the field direction when the magnet pointer is set at 270°. To this end, a laser beam is reflected from a small mirror, mounted parallel to the cleavage plane of the crystal, and the rod supporting the goniometer turned. When the reflections from this mirror and a mirror perpendicular to the pole faces of the magnet coincide with the laser source (the laser being situated about ten feet away), the apparatus is in good adjustment, and a cartesian system established. VI EXPERIMENTAL PROCEDURE Locating the Principal Axes All frequencies referred to are Cl35 37 transition frequen- cies. The signals of C1 , in addition to being only 25% as strong, are lower in frequency by a factor of Q37/Q35 = 0.785; Q is the nuclear quadrupole moment. The C137 frequencies were not sought. Carlson and Adams7 give the orientation of the x and z principal axes with respect to the cleavage plane. But the orientation of the y axis in the cleavage plane must be determined every time a crystal is mounted on the goniometer. Thus, the first thing done in each experiment was to deter- mine the location of the principal axes of the crystal. At 77 K, observing free induction decay on an oscillo- scope and using a minipulser11 , the four-fold site pure quadrupole frequency was traced by rotating the magnet through elab to find the signal for every 5° change in position of the goniometer. The applied field was 500 G. Refer to Data Table 11 in Appendix D. These loci of zero-splitting were plotted on a stereographic net, and the resulting closed figure rotated up to become an ellipse centered on the pole. Using the "elliptical cone" method9 for the low field case, the orientations of the principal axes were determined. 24 25 The orientation of the x and z principal axes with respect to the cleavage plane was found to agree with that of Carlson and Adams. See Figure 3. General Experimental Procedure An applied magnetic field of 500 G was rotated through the cleavage plane at a temperature of 1.18 K. Transition frequencies are plotted in Figure 5 and the data is listed in Data Table 12 in Appendix D. From the graph, it was concluded that an error of 10° in the assigned direction of the y principal axis had occurred previously in rotating up the ellipse. It was found that the orientation of the y principal axis could be located much more precisely and easily by sweeping the frequency while rotating the magnetic field through the cleavage plane of the crystal. The determination of the direction of the y axis by this method should be accurate to within one or two degrees. Succeeding runs followed the same format: after aligning the apparatus, the crystal was cooled to 1.18 K. The applied field was first rotated through the cleavage plane to determine the locations of the y principal axes of the four-fold and eight-fold sites. Orientations of the x and z principal axes were then taken to be as in Figure 3. Knowing the locations of the y principal axes, the goniometer could be set so that the plane of rotation of the magnet included one or two princi- pal axes of either the four-fold or eight-fold site. Representa- tive data and figures are summarized in Table 2. An interpreta- tion of the various figures follows. TABLE 2 Summary of Data Figure Data Table* Plane of Rotation of H 5 12 Cleavage 6 13 xz of four-fold 7 l4 1_cleavage, containing y of four-fold 8 15 xz of eight-fold *In Appendix D 26 27 and ems L . . _ xeaoeumv» III‘ '1', I‘ ‘3’ oeeaode co names .naHu 8 S _ [J H _ 4 + cacao Apaouiwvh Hounds a ma.“ u a 0 com I m spam UHOMI: P r (r r b 1? N :1 m .3 m.: m.: an ‘Kouanbsag H in cleavage plane Figure 5: 28 .632 so 398 53¢ om as on 0 can con cam _ . . A . J1 _ 4 . _ . . _ - i L 4 A . _ Ilu. 'O'O‘O'O'O' I III C 77". .I‘.‘ 8‘. 'O'O'O'n IOIIC‘I'IIIOa‘ I. No: a mH.H u a 6 com a m 09am vacuum IIIII ri ouam paouié.lllll .Irm.e O I‘ |IIOIIIOIIIOII.'O'.’O'O O 'O'O'olol' O ' 'O'OIIO-IO‘I ’O'OICIIO'OIIO r: . _ _ . _ r - _ _ r _ - . _ . . _ . . elm: zuu ‘Kouanbsxg H in xz principal axis plane, u-fold site Figure 6: 29 ..eamae ac cameo .pmao om on on o omn con cam _ - . _ 4‘ -. _ x, . _ a . H|.+ i q _» Ill N‘\O\O\O|‘IIC'"|‘I‘IJJI"I “J No: I l \‘ a ’0’, [IO]. .‘C‘. \\O\ J : a 4" \ IVQu’IIi'EIQ’ o‘.\ \O\ \.\/Inl I I .HV. ‘HVAuHQ\ Ve/I/ 1 Avah \\ iol...‘ . I I . II. \\I 0‘ ole / l W 3 \ ‘0‘ 1': ’0’ I, I. a O\\ \\\o ole/AI // adv H 'ln‘llll“ "',|*\A ti :1 w.: :1..-) a on; a a \«idni 4 I If 6 com a m \ \.. rll dI‘IJ.’ ”pd" “Hog-” ..... \ \ I 5 O A.) undo afloat: (.xx 2 ICIOIU'ID'IIOIOIDIOX ZHR “louanbead H in plane ; cleavage containing y, h-fold Bite Figure 7: 30 neaoae co mamas .naao cm 00 on 0 0mm 00m CNN . _ a _ _ I 1 NI N III-bull... I I- I. 5' “|.‘v‘ [OI ll ’ I \‘ ,"\\ x IL ‘\\ I’D ‘ l '1, 'I‘P‘\|'Iu ’Ir'r Ilf‘IO‘O Ill 1' ' .Il r. l haze moaocosaoam Heuusoo II a mu.” I B I] 0 com I m oval uncut: zun ‘Kouanbaxg ms .s.oe 0.: 5.: m.: H in xz principal axis plane, 8-fold site Figure 8: VII DATA DIAGRAMS AND INTERPRETATION As datum was recorded in the data book, it was also plotted on graph paper, frequency vs. lab angle. The first step in the analysis consisted in connecting the data points, usually at 5° intervals, by curves. Next, approximate sym- metries of these curves about the two pure quadrupole fre- quencies were noted, and the curves thus identified by site. For purposes of fitting, data that had been organized by site was transferred to computer punch cards, which contained the two lab angles, the four recorded frequencies belonging to that particular site at this orientation, signal strength symbols and site number. There now follows a description of some representative data plots: Figure 5: H in cleavage plane of the four-fold site. The four-fold and eight-fold sites are easily distinguished by noting their symmetries about the respective pure quadrupole frequencies f Maximum splitting of lines occurs in the y 0' principal axis direction. It is seen that the y axes of the four-fold site and the two eight-fold sites are separated by 60°, indicating the approximate three-fold axis of symmetry (since the y axis is two-fold). Figure 6: H in xz principal axis plane, four-fold site. Principal axes of the crystal are as indicated. The two 31 32 crossing points of the two inner four-fold site lines, used in hand-fitting the g tensor, are obvious. Frequencies of the two eight-fold sites coalesce in this plane. Figure 7: H in plane JL cleavage, containing the y principal axis of the four-fold site. The two eight-fold site lines are again distinguishable. Figure 8: H in the xz plane of the eight-fold site. Central lines only are shown. Now the eight-fold curves strongly resemble the four-fold curves of Figure 6, and the four-fold curves resemble the eight-fold curves of that same figure. This is with the goniometer rotated 60° from the position it was in when the data shown in Figure 6 was recorded, thus further indicating the existence of an approximate three-fold axis of symmetry perpendicular to the cleavage plane. VIII FITTING THE TENSOR Approximate Fit by Perturbation Theory Since the data at 500 G shows only a small deviation from symmetry about the pure quadrupole frequency, it was decided that the simpler equations of perturbation theory would yield sufficient information for a first approxima- tion to g. The results from this approximation could then be fitted to the data using more powerful methods. The equations of perturbation theory are given in the second part of Appendix A. One modification must be made in attempting to describe the transition frequencies by perturbation theory. This theory assumes that the nuclei in a crystal see a magnetic field which is equal in magnitude and direction to the applied field. The inclusion of the dipole-dipole interaction and transferred hyperfine effect changes this picture. With the inclusion of these effects, the nuclei in the crystal see a magnetic field which differs in magnitude and direction from the applied field; it is this internal magnetic field Hi that must be used in the equations of perturbation theory. The data from the rotation of the applied field H in the xz plane of the four-fold site (see Figure 6 and Data Table 13, Appendix D) indicates two positions where the two middle transition frequencies f2 and f3 are equal. Denoting 33 34 these two angles by 0 and 8L2, one obtains values for f L1 1' 3, and f4 at each of these angles (f1 > f2 > f3 > £4). Defining 0 and ¢ as the spherical polar angles between f f 2! the internal field and the principal axes of the crystal, one sees that, because the xz plane is a mirror plane, ¢ = 0 for H applied in the xz plane. One may express the 0 dependence of Hi in terms of the applied field a and the magnet angle 6L by means of the tensor 2: -1 = E'H (42) or r . ‘ ' l ’ . IHiI Sln 8 Tll 0 Tl3 Sln 8L) [HI 0 = o T22 o 0 (43) {cos 8» LT31 0 T33 I Lcos 6L; O = = = = ne assumes that T12 T21 T32 T23 0 because of the fact that the xz plane of the crystal is a mirror plane. Matrix multiplication yields H. E— Sin 8 = TllSIn 8L + T13 cos BL (44) and H. _ . Hi cos 8 - T1351n 8L + T33 cos BL (45) Note that data from the xz plane does not allow a determination of T22. From Appendix A, equations (ASS-A58), f1 = to + (kai/Zh){[3/2] + [1/21} (46) f2 = £0 + (kai/Zh){[3/2] - [1/21} (47) f3 = fQ - (thi/Zh){[3/2] - (1/21} (48) f4 = f0 - (thi/Zh){[3/2] + [1/21} (49) 35 where fQ is the pure quadrupole frequency. Here, as in equation (A40). _ 2 2 2 2 . 2 k [m] — [am cos 8 + (bm + cm + mecmc052¢)51n 0] . (50) The constants are defined in Appendix A, equations (A44-A48). At either of the angles 6L1 or 8L2, where f2 = f3, one sees by equating equation (47) and (48), that [3/2] = [1/2] (51) Using equation (50), and solving for the angle 0 in terms of the constants, one has 2 2 (a ) - (a ) tanze = 112 3/2 (b 2 . (52) + c 1/2) +C 2 (b3/2 3/2) ’ 1/2 Evaluating the constants yields 0, the azimuthal angle of the internal field Hi in the principal axes system of the crystal. Subtracting equations (46) and (49), and solving for Hi' one finds Hi = (fl-f4)/(yk([3/21+ [1/21>>. (53) By inserting the above calculated value of 8 into the bracketed expressions in the denominator of equation (53), and evaluating f f4, and 6L1 from the data, one obtains a value for Hi' 1’ Using the other crossing point of f and f at a magnet angle 2 3 8L2, one obtains a second value of the internal field, Hi' One may insert each of the two sets of values of BL and Hi into equations (44) and (45) to obtain four equations in the four unknowns T11, T13, T31, T33. The applied field H is known. Using data from Data Table 13,one obtains the solutions 36 T = 0.91 T = -0.04 11 13 (54) T31 = -0.29 T33 = 0.26 , or T = -0.91 T = 0.04 11 13 (55) T31 = -0.29 T33 = 0.26 The first of these solutions was taken to be the correct one. In equation (43), the direction of Hi is expressed in angles measured in the principal axis system of the crystal; hence, the tensor as derived is also expressed in the principal axis system, and may be interpreted as bringing about a transformation of the applied field from the lab to the principal axis system. T22 may be determined from the data of either the xy or yz four-fold site planes. The previously described method is again used, with only one unknown. One obtains T22 = 0.968. Thus, a first approximation to g is _ ’ _ 1 EPA - 0.91 0 0.04 0 0.968 0 (56) ~-0.29 0 0.259 Computer Fit Because the perturbation theory used to approximate the g tensor cannot describe the asymmetric frequency splitting observed in the laboratory, an analytical method more powerful than perturbation theory is needed. The availability of computer program subroutines which calculate nuclear quadrupole resonance frequencies exactly by finding the 37 eigenvalues of the Hamiltonian permits a much more accurate fitting of the tensor to the data.12 A computer program EYEBAL was written for the CDC 6500 computer. The main function of this program is, briefly, to rotate the applied magnetic field through 180° (in steps of 5°) through some plane in the crystal whose orientation with the principal axes of the crystal is given by the angles Eulerl and Euler2. At each 5° point: 1.) The applied field is expressed in a lab-centered cartesian coordinate system. 2.) This lab field is expressed in the principal axis frame of the crystal. 3.) This applied field is then multiplied by 2, also in the principal axis system. 4.) The product, the internal field, is expressed in spheri- cal polar coordinates. 5.) The four highest nuclear quadrupole resonance frequencies for I = 3/2 are then calculated using subroutines ZEEMAN and HERDAG. These frequencies are indexed and stored in an array. When all frequencies are stored for a 180° rotation of the field through some plane of the crystal, a plotting sub- routine plots the four frequencies versus the lab angle of the magnet. Additional output includes lists of the four frequencies and the lab angle of the applied field, and the components of the internal field expressed in both lab and principal axis systems. Typically, the applied field is rotated through the xz and xy planes, and through the plane containing y and perpen- 38 dicular to the cleavage plane, for a given site in the crystal. By changing Eulerl and Euler2, any orientation is possible. The tensor g as originally derived was used as input for EYEBAL, and the output frequencies for the rotation in the xz plane compared to corresponding data. To obtain an idea of how each component of g influenced the spectrum, 0.05 was and T added to each component T in turn, and 11' T13' T31' 33 that modified tensor used as input for EYEBAL. Corrections to the tensor 2 were made in the following manner: Let f(n,0L) denote the frequency of the nth line fn (n = 1...4) at the magnet angle 8L, as calculated by EYEBAL. Let df(n,8L) denote the change in frequency necessary to make the calculated frequency equal the experimental frequency. Further, let dgab(n,8L) denote the change in the nth frequency fn ate: brought about by adding .05 to the tensor element Tab' Then df(n,0L) should equal a linear combination of the four values of dgab(n,0L) produced by incrementing each of T11, T and T by 0.05. That is, 13' T31' 33 df(n,8L) = k'dgll(n,8L) + l'dgl3(n,8L) + m'dg31(n,8L) + p-dg33(n,8L) . (57) By choosing four different frequencies (at either the same, or different angles 8L) one obtains four equations in the four unknowns k, 1, m, and p. By choosing four strong, unblended data points in regions of the spectrum where the calculated frequencies diverge most from the data, calculating the df(n,0L) for these points, and treating the matrix of coefficients as a Cramer's rule problem, one easily obtains 39 solutions on a HP9100B calculator. Solutions k, l, m, and p are multiplied by 0.05 to yield the number 6a which must b be added to Tab to produce the desired correction in the spectrum. The reliability of this method having been tested, program EYEBAL was modified to read data cards which contain lab angles of the applied field, frequencies, observed strengths, and the site number (four-fold or eight-fold). These frequencies were labelled and stored in a matrix, and then recalled and subtracted from the calculated frequencies to which they correspond. This modified program, called COMPAR, provides a rapid test for the fitting of a given tensor, and its output indicates which regions of the calculated spectrum are in least agreement with data for all three rotations of the field in the crystal. Up to this point, the choice of data points used in fitting the tensor was based upon the assigned strength symbol, as well as agreement with neighboring data points. An additional feature of program COMPAR was a confidence test which made use of the recorded strength symbols attached to each frequency when it was recorded as datum. The meaning of these symbols is given in Table 3. The confidence test recorded 2 (f(n,6L)data - f(n,0L) ) SUMSQ(n) = Z calc (58) 2 6L (24f) for each of the four transition frequencies of the spectrum. Here, Af is the observational error recorded in Table 3. 40 Table 3 Interpretation of Strength Symbols Symbol Strength df, Observational error, kHz A very strong :1 B strong, slight blend :2 C strong, blend :3 D medium :5 E medium, blend :7 F weak :10 G weak, blend :12 X,Q No data recorded Precision of the Measurement Before the final value of the tensor is presented, it is pertinent to ask what the theoretical limits of precision are for this fitting procedure. Since a given datum point could be in general a function of all five elements of 2, it is difficult to relate the precision of g to observational error Af. Fortunately, the effects of each of the diagonal elements of g are separable from one another. By examining the four transition frequencies along the x, y, and z principal axes of the crystal, one may observe the effect of T T 11' T13' 31' and T33, respectively, and separate it from the effects of the other components of the tensor. Hence, one may express the uncertainty in a particular element of g as a function of the uncertainty in measuring a particular frequency. The following method was used: once 2 had been fitted to nearly its final form (its goodness of fit being judged by the 41 credibility test previously described), the tensor was again deliberately altered by adding 0.01 to each of its components, in turn, and used as input for EYEBAL. The change in the calculated frequencies from their values with the unaltered tensor were calculated at angles corresponding to the four-fold site principal axes of the crystal. The results are summarized in Table 4. Table 4, showing the change in calculated frequen- cies along the principal axes per change of 0.01 in the elements of 2, indicates the sensitivity of this fitting procedure. Assuming the variation in frequency to be linear with variation in Tab' one may express the information in Table 4 as the change in Tab necessary to change the frequency along a given axis by one kHz. Since one kHz is the Optimal resolution of the spectrometer, this will give Optimal error bars for Tab' This is done in Table 5. The best possible determination of each element Tab’ assuming strong frequencies measured along the principal axes, predicts the following uncertainties: Element Uncertainty Tll :0.01 T22 :0.005 T33 :0.005 T13 :0.0l T31 :0.004 The best fit obtained yields: 42 , \ o.920:o.01 o -0.067:0.01 EPA = o 0.968:0.005 o (59) -0.284:0.004 o 0.262:0.005J This is g for the four-fold site, fitted to data at 1.18 K. Goodness of fit is illustrated in Figures 9, 10, and 11. Differences between calculated and experimental frequencies vs. lab angle are plotted for four-fold site data, with H applied in the indicated planes. Change in calculated frequencies per change of 0.01 in Ta Element T11 22 33 13 31 Axis X 9:, 2.1 0 Table 4 4:, 1.5 0 43 21:3 -0.9 0 b Table 5 Tab per one kHz change in fn along a given axis Element Axi§_ £1 £2 £3 £4 Tll x 0.005 0.007 -0.01 -0.007 T22 y 0.003 0.004 -0.005 -0.004 T33 2 0.003 0.005 -0.005 -0.003 Tl3 2 -0.014 0.017 -0.014 0.014 T31 x -0.004 0.004 -0.004 0.004 44 45 s QGHQ 0 co 3 on 0 can con 0%. m a d _ _ _ _ II- . J I O O O 1 0 e e D e o Will-III)! a O 0 3 w e . l 1 1| ll '0 O Y i O 0L O O O O O O O O o O 0 O O o 0 O O l T . nu I rl I fill.“ ( o ) O O O O O O F D W P D D O O l v we L I I I I Y H D If 0 Y a O I o O i o e H.“ I l O l p b _ r p _ P out oa+ Oat ou+ out cfr oat ou+ 23‘ .otnoJ _ queJ Goodness of fit, H in xz plane, h-fold site Figure 9: 46 I 1 T I l T T T __ I I I I I I I I .40 O\ I e e e e o d I-- —-O ‘0 q {I 0 I e {I 4 o L o I— 0 -—G P ' 0 P e P I {I e e e e (I h—-— 0 —-10 4 '- e . '3 e e I 6“ e e O —- 4 0 e 0 fig 0 e e 0 e e a e —- O 4 --I8 M e (I e e e q e H Ne 0“ 3 0 $1 “-0 $4" $4 . fig 14 11 In 11 IL .1 IL 41 O O O O O O O 0 PI H H H H H H H + I + I + I + I aux .otaoJ _ :dan Figure 10: Goodness of fit, H in plane‘L cleavage, conteining y, h-fold site 47 1 180 150 3 __ . d ,. O O — O O Ox 0 O O O O I... C O ~o d 1 e c> — 0 O O m d .I «x .3 _ «a “N u I... o l l l l l 1 l J I ll 1_I j l O O O C. O O O O H H H H H H H H + I + I + I + I 38X .otsoJ _ quaJ Figure 11: Goodness of fit, H in cleavage plane ”lab, IX DECOUPLING From the theory, equation (23), g = YhuBlg 4 N(3 - ;I;‘11 g (60) Here, y = 2624. radians/sec h = 1.05 x 10-2'7 erg-sec “B = 0.927 x 10.20 erg/gauss N = 6.022 x 1023 /mole g = the effective field tensor is the inverse susceptibility tensor is the electron g-tensor "Q X is the dipole-dipole interaction tensor I "U x 1 and g have been measured by Fairall, 2E.2l'4 g is evaluated by direct summation over lattice sites within a sphere of radius 100 A. Computer program YDIP evaluates g by evaluating fl 2 1 - 3( 'f.) g = Z 3N -1* (61) J rj by components, where the summation was over all Er lattice sites within the sphere of radius 100 A. Q ’—0.019091 0 0.000379 24 -3 g = 0 -0.019580 0 x 10 cm . (62) L 0.000879 0 0.0386714 Fairall, gt al., have measured 3 and x—1 for ErCl3 in directions perpendicular and parallel to the cleavage plane, 48 49 with zero applied field, from temperatures of 4 K to 0.03 K. At 1.18 K, the parallel component -1 Xi = 1.15 mole/cm3. = 0.40 mole/cm3, and For purposes of decoupling the trans- ferred hyperfine tensor, the inverse susceptibility must be expressed as a tensor in lab coordinates (X,Y,Z), with Z being the axis perpendicular to the cleavage plane. Fairall, 33 El- do not specify the orientation of the y principal axis of the crystal, which lies somewhere in the cleavage plane. Hence, one may not assign exact values to the X and Y compo~ nents of -1 1 . But if one assumes that the value of X"- is the average of the X and Y components of g‘l, one may write -1 zlab _ This assumption is a reasonable one. 0 0 W 0.40 0 0 1.15 J (63) 13 Garton, 33 al., measure 3 values for ErCl3 in host lattices of YCl3 and LuCl3 and find that the g values in the X and Y directions are within 5% of each other. Because x is a function of 92, it follows that the X and Y components of ;, hence z-l, may be assumed equal as an approximation. quoted by Fairall, 23 31., are used. ll-Q ll r 8.36 0 L O 0 8.36 0 o I 0 4.77J For consistency, however, g values (64) Taking the best fitted value of 3 and transforming into the lab system via the similarity transformation T =LAB §(SG°) -1 o gpAg (56 ). (65) where g is given by 50 ' cos(a) 0 sin(a)) §(0I) = 0 l 0 . (66) L—sin(a) 0 cos(a) I One obtains Elab' USing equation (60), one obtains élab' in units of ergs. In this work, energy units of cm-1 will be used. Hence, using a conversion factor of l erg = 5.035 x 1015 -1 cm , ’15.9 0 5.61) _ _ - -1 élab - 0 1.28 0 x 10 cm . (67) L 9.10 0 7.50) Transforming to the principal axis system, '3.39 0 -0.637) A = 0 -1 28 0 x 10’4cm'l (68) =PA ' ' k2.86 0 20.15 Results are quoted to three significant figures. The main limitation here is that x.1 and in the case of x -1 beyond two significant figures. must be read from a graph, , it is quite difficult to do this Hence, the numbers quoted here are accurate to three significant figures at best. Recalling that each Cl- sees two nearest neighbor Er3+ ions, and since each Er 3+ creates a hole in the Cl- 35 and 3p orbitals, the g tensor above must be separated into two tensors, g1 and £2, one for the contribution from each Er ion. "3’ ll "3’ PA' Because the xz 3+ (69) plane is a mirror plane, these two tensors are equal in the Er system of axes (x', y', z'), with the x' axis 51 lying along the Er-Cl internuclear radius: élEr = ézsr = éEr' (70) Finally, one sees that for a bond angle 0, _ 0 -l 0 91.. sees (:4 <71) and _ .9 ’1 -9 éZEr §( 2)§2§ ( 2) . (72) Here, §(a) is given by cos(d) sin(a) 0 ) §(0I) = -sin(0I) cos(0I) 0 (73) 0 0 l k A Inserting élEr and éZEr from equations (71) and (72) into equation (70), and solving for by matrix multiplication, ész . . . 0 — (u51ng the identity §(§) = g I(-%)), one finds -l l = §(%)§(%)§ § I%)§‘ (g), (74) _ 0 e g g éPA _ élPA + §(§)§(§)§1PA§ (2); (2). (75) Multiplying matrices and equating elements of the matrices on the two sides of the equation, one obtains nine equations in nine unknowns, the unknowns being the nine elements of élPA’ The algebraic solutions are: (A1PA)11 = (APA)ll/2 (76) (AlPA)22 = (APA)22/2 (77) (AlPA)33 = (APA)33/2 (78) (AlPA)13 = (APA)13/2 (79) (AlPA)3l = (APA)3l/2 (8°) 52 _ 8 (AlPA)23 - [(APA)23tan(§0]/2 (81) _ 8 (AlPA)32 — [(APA)32tan(§)]/2 (82) sin(%)cos(%) (AlPA)12 = (AlPA)21 = [(APA)11'(APA)22] 2 8 . 2 0 ' 2(cos (§)-Sln (5)) (83) One thus solves for élPA' élPA must now be rotated into élEr’ using equation (71). The result is '-8.41 0 -0.483 A = A = 0 9 47 x 10"4cm'l . (84) =Er =lEr ° L 2.17 0 10.1 ‘ This is the transferred hyperfine tensor at 1.18 K of ErCl 3! measured in the coordinate system centered on the Cl-ion, whose x' axis lies along the internuclear radius. system of axes, ' 7.97 é11ab ‘ "5°26 L 4.55 -3.60 -0.64 8.26 this tensor measures 2.80) 7.14 3.80 ‘) 10 In the lab cm- . (85) There now follows a summary of all tensors used in calculations, given in both the lab and principal axis frames. Lab Coordinate System 0.305 0 -0.131 g = 0 0.968 0 \-0.348 0 0.877 \ d (dimensionless) (86) 53 Lab Coordinate System, Continued. IIU ll ll>< |b< ll ILQ élEr — '—0.0191 0 0.000879) 0 -0.01958 0 x 1024cm"3 0.000879 0 0.03867 '0.40 0 0 ‘ 0 0.40 0 mole-cm-3 L 0 0 1.15J 2.5 0 0 ‘ 0 2.5 0 cm3-mole-l L 0 0 0.87] '8.36 0 0 0 8.36 0 L 0 0 4.77) ’0.120 0 0 ‘ 0 0.120 0 I 0 0 0.210J 15.9 0 5.61) 0 —1.28 0 x 10’4c1m'1 9.10 0 7.60 ‘~ J ’ 7.97 -3.60 2.80‘ -4 -1 - -5.26 -0.638 7.14 x 10 cm L 4.55 8.26 3.80 (87) (88) (89) (90) (91) (92) (93) 54 Principal Axis Coordinate System 0.920 0 —0.067) g = 0 0.968 0 (dimensionless) (94) L-0.284 0 0.262) 0.0198 0 -0.0271 1 p = 0 -0.0196 0 x 1024cm’3 (95) -0.0271 0 -0.000214 ’ 0.92 0 -0.35‘ -1 -3 g = 0 0.40 0 mole—cm (96) L—0.35 0 0.631 ”1.4 0 0.76) _ 3 -1 z — 0 2.5 0 cm -mole (97) L0.76 0 2.0 .I ”5.89 0 1 66 ‘ g = 0 8.36 0 (98) _1.66 0 7.24 J ( 0.182 0 -0.042‘ g‘1 = 0 0.120 0 (99) (‘0-042 0 0.148J (3.39 0 -0.637 _ -4 -1 g - 0 -l.28 0 x 10 cm (100) 55 Principal Axis System Continued. élEr - PA F 1.70 -8.86 -8.86 -0.638 -0.318 -O.363 10.1 Again, in the Er system of axes, T-8.41 -0.483 ‘ 0 10.1 J u (101) (102) X MAGNITUDE OF THE INTERNAL FIELD The result of the transferred hyperfine effect and dipole- dipole interaction altering the applied field at the Cl- ion can be best shown by graphs of the internal magnetic field vs. orientation with the crystal principal axes. To this end, program EYEBAL was used with the best-fitted effective field tensor 3 as input, with an applied field of 500 G being rotated through the principal axis planes of the four-fold site. The magnitude of the internal field was then plotted vs. the angle of the applied field, and the positions of the principal axes indicated. This output is tabulated in Tables 6, 7, and 8, and plotted in Figures 12, 13, and 14. The forms of these curves were checked against output of a previous program, HINT, which used actual data taken at 1.18 K and 500 G applied field. This program used the Parker and Spence method of moments14 and the equations of perturba- tion theory to calculate the internal field. Results are in complete agreement. The results show that the internal field nearly equals the applied field along the x and y principal axes, but is reduced to nearly 25% of the applied field strength along the z axis. The contribution of the transferred hyperfine effect to this change in internal field can be found by considering the effective field tensor 2. By equation (18), the effective 56 57 l l T l l l 500 _ .- 1 e ’ e . Happlied ' 500 G ,' .. 400 . ° ... r- . . ’PA 0 O 300 III- . . .1 I- ' ' 200 ... . . ... . O h ‘ . I—i . O O 100 _. _ o. 2m p t: d B 1 1 I I I l l 0 30 60 90 120 150 180 a 0 13b) Figure 12: Internal field in xz plane, u-fold site lililll) 58 r T I I l l 500 L— l-# ee"......‘000. ... 0.. .00..... ..'Oeoo- noo.__ ’PA h-fold —1 _ Happlied ' 500 G d r- .4 I— u—i 0 2. ‘1 H a: 1 l I 1L 1 J l 0 30 6O 90 120 150 180 0 6lab) Figure 13: Internal field in plane ll y axis, .1. cleavage Hint’ G 59 I I I I I I I 500.— .... e ' e 0 O O C e O -— e e ‘4 O O #00 —- o O —‘ . yPA III-{01d . e O 300 ' . _— ‘ Happlied ' 500 G ' —‘ 100... ... l l l l l l I 105 135 165 15 45 75 105 O ‘1.» . Figure 14: Internal field in cleavage plane field tensor is 60 O -1 1 Q g E 103 = + - — g = (g “”13 )N ( ) 2'1 4'9' '1 " L + T ' 440314 - (104) I 0.0198 0 -0.0271' 1.4 0 0.76 g = g + fi 0 -0.0196 0 x1024- 0 2.5 0 -0.284 0 —0.00028 £0.76 0 2.0 S I I '3.39 0 —0.637‘ ' 0.182 0 -0.042 - 0 -1.28 0 x 10'4 . 0 0.120 0 . YFuBN _2.86 0 20.2 I -0.042 0 0.148J r s 1.4 0 0.76 0 2.5 0 (105) 0.76 0 2.0 J 0.011 0 -0.065) I-0.091 0 —0.002 g = g + o -0.081 0 + 0 0.049 0 . (106) I—0.062 0 -0'035) -0.222 0 -0.703J d-d trans. hyp. Therefore if H is applied along the x principal axis, ’ 1 -0.011 -0.091‘ EINT = g-g = 0 IR] . (107) g -0.062 —0.2221 The transferred hyperfine field Opposes the applied field, as does the dipole-dipole field. toward the -z direction. The internal field is bent II' I: .l l 61 If g is applied along the y principal axis, 0 glNT = g-g = 1 — 0.081 + 0.049 IHI . (108) 0 The transferred hyperfine field would add to the applied field, but is Opposed by the dipole-dipole field. If g is applied along the z principal axis, - 0.065 - 0.002 EINT = 23-8 0 [HI . (109) l - 0.035 - 0.703 Here, the transferred hyperfine field dominates even the applied field along the z principal axis direction, and is primarily responsible for the large observed decrease in the internal field observed along the z principal axis. Similarly, in the lab system, substituting the tensors into equation (104), r-0.0191 0 0.000879‘ g = ; + § 0 -0.01958 0 x 1024 - L 0.000879 0 0.03867 . r2.5 0 0 ‘ 15.9 0 5.61 0 2.5 0 - 0 -1.28 0 x 10"4 , YEMEN . 0 0 0 87 I 9.10 0 7.60‘ '0.120 0 0 ' r2.5 0 0 ‘ 0 0.120 0 ' 0 2.5 0 . (110) 0 0 0.210 0 0 0.87 b J 5 J 62 It should be remarked that g was tested to determine if it correctly predicted the splitting observed at magnetic fields higher and lower than 500 G. In one experiment, the frequencies of several strong four and eight-fold site lines were measured as the applied magnetic field was varied from 250 G to 7000 G. The magnetic field was along the line of intersection of the cleavage and xz planes of the crystal. 3 and the various magnitudes of H were used as input for program EYEBAL, and the calculated frequencies agreed in all cases with the data, within the accuracy of measurement for these frequencies. This justifies the fact that most data for the fitting of g was taken at 500 G, and points to the fact that, at least at this orientation of the field, E is field- independent. TABLE 6 Magnitude of the internal field for 500 G applied field in xz plane of the four-fold site. 0 0 6lab' Hint'G elab’ Hint'G 0 443.1 90 231.2 5 424.6 95 263.6 10 403.2 100 295.3 15 379.1 105 325.7 20 352.5 110 354.3 25 323.8 115 380.7 30 293.4 120 404.7 35 261.6 125 425.9 40 229.2 130 444.2 45 197.0 135 459.3 50 166.5 140 471.3 55 139.9 145 479.8 60 120.9 150 484.9 65 114.2 155 486.6 70 121.8 160 484.7 75 141.4 165 479.4 80 168.3 170 470.6 85 199.0 175 458.5 180 443.1 63 TABLE 7 Magnitude of the internal field for 500 G applied field in plane perpendicular to cleavage plane and containing the y principal axis, four-fold site. 0 0 6lab' Hint'G elab’ Hint'G 0 443.1 90 484.0 5 443.4 95 483.7 10 444.4 100 482.8 15 445.9 105 481.4 20 448.1 110 479.4 25 450.7 115 477.0 30 453.7 120 474.1 35 457.0 125 470.9 40 460.4 130 467.5 45 464.0 135 464.0 50 467.5 140 460.4 55 470.9 145 457.0 60 474.1 150 453.7 65 477.0 155 450.7 70 479.4 160 448.1 75 481.4 165 445.9 80 482.8 170 444.4 35 483.7 175 443.4 180 443.1 64 TABLE 8 Magnitude of the internal field for 500 G applied field in the cleavage plane, four-fold site. 0 o ¢1ab' Hint’G cblab' Hint’G 0 231.2 90 484.0 5 234.2 95 482.6 10 242.7 100 478.3 15 256.1 105 471.3 20 273.2 110 461.6 25 292.8 115 449.4 30 314.1 120 434.8 35 336.1 125 418.1 40 358.0 130 399.5 45 379.3 135 379.3 50 399.5 140 358.0 55 418.1 145 336.1 60 434.8 150 314.1 65 449.4 155 292.8 70 461.6 160 273.2 75 471.3 165 256.1 80 478.3 170 242.7 85 482.6 175 234.2 180 231.2 65 XI RESULTS From equation (84), r-8.41 0 -0.483 = 0 9 47 0 x 10'4cnm'1 (111) =Er O O 2.17 0 10.1 L J By equation (30), A = l 2 A.. = 3.71 x 10"4cm'l (112) S 3 XI I I 3:) - IY '2 And from the requirement that A be traceless, =P , Apx'x' O Apx'z') =Er = AS; + O Apy'y' 0 (113) ~Apzlxl O Apzlz!’ I-12.1 0 -0.483‘ -4 -4 -1 = 3.71 x 10 1 + 0 5.76 0 x 10 ,cm (114) ~ 2.17 0 6.37 to three significant figures. Hence, by equation (37), _ 1 _ _- -4 -1 A0 ATry — 3(Ax'x' Ay'y') - 5.96 x 10 cm (115) By equation (38), A - A = l(A -A ) = -6 16 x 10‘ cm"1 (116) o nz 3 x'x' z'z' ' And by equation (39), A - A = l(A -A ) = -0 233 x 10‘4cm"1 (117) ny 82 3 y'y' z'z' ' 66 XII CONCLUSION The transferred hyperfine interaction tensor has been measured at 1.18 K and 500 G. It has been found that the tensor as measured is field-independent over a range of 250- 7000 G. Strong temperature dependence has been suggested (Appendix B). Plots of internal field vs. angle of applied field show the greatest modification of the applied field occurs close to the z principal axis of the crystal, where the transferred hyperfine interaction causes the internal magnetic field to drop to only 30% of the applied field value. The transferred hyperfine interaction tensor derived (see equation (112), Section XI) has an isotropic part, 4 -1 AS = 3.71 x 10' cm (118) and an anisotropic part, ’-12.1 0 -0.483‘ _ -4 -1 AP — 0 5.76 0 x 10 cm (119) L 2.17 0 6.37 j The pairwise differences in ép between 0 and n orbitals are, by equations (37), (38), (39): _ _ -4 -1 AC ATry — 5.96 x 10 cm (120) _ _ -4 -1 A0 - A,"z — 6.16 x 10 cm (121) _ _ -4 -l ATry - Anz — 0.61 x 10 cm (122) 67 68 The interaction between the Cl35 nucleus and holes in both 8 orbitals is nearly equal. The pairwise difference AO - An = -6. x 10.4 cm-1 indicates that n bonding dominates a bonding in ErC13. The relative magnitudes of As and the diagonal elements of ép indicates that holes in the Cl- 3p orbitals have a greater interaction with the Cl35 nucleus than do holes in the 35 orbitals. This implies that the C1- wave function is primarily p-state in ErC13. For a comparison between the hyperfine interaction of ErCl3 and other transition metal halides, see Table 9. TABLE 9 Magnitudes of Some Transferred Hyperfine Interactions Compound As (lo—4 cm-l) A0 - An (10-4 cm-l) Ref. ErCl3 3.71 -6. CuClz'ZHZO 7.8 5.0 6 Cdc12':Cu++ 9.5 4.5 6 C0C12'2H20 5.6 0.8 6 Agc1:Fe3+ 2.8 0.5 6 CsMnCl3 19 C111(1inear bridge) 1.7 0.0 ClI (bent bridge) 1.46 0.06 CsCoCl (bent bridge) 2.68 1.03 19 3 69 The small relative magnitude of the off-diagonal elements of ép' as well as a look at the crystal structure, suggests that they may be due to errors in evaluating certain contributions to the Hamiltonian. The location of the Cl- ion in the ErCl3 crystal does not suggest that there would be any x'z' or z'x‘ elements in the tensor. The two~ fold symmetry about the y principal axis, and the existence of the mirror plane argue against any off-diagonal elements. 3'15 the existence of In crystals such as cubic KNiF3, the four-fold axis of symmetry, the x principal axis, necessitates that there be no off-diagonal elements in ép' Further, hence the interactions in the APY'Y' = APZ'Z" Cl- 3p(y') and 3p(z') orbitals are equal. Although one cannot invoke this four-fold symmetry in the case of ErCl3, neither can one adequately explain the existence of the off-diagonal elements. Imprecision in analysis leading to off-diagonal elements could have several causes. The quantity z-l used to decouple g is the most doubtful. First of all, g-1 was interpolated from a steeply changing curve. Secondly, the two components of l-1 in the cleavage plane were presumed equal, and the results of Garton, gt 31.,13 dispute this. Thirdly, there is enough doubt about susceptibility measurements on ErCl3 in general to cast doubt on the results of Fairall, et 31., as will now be discussed. In discussing susceptibility measurements on ErClB, four points should be mentioned. First of all, a spherical sample 70 Should be used in order to control demagnetizing effects. Secondly, the crystal should not be stored in oil, because it readily absorbs oil between its cleavage planes, and this oil makes an accurate determination of the mass impossible. Thirdly, the temperature control must be well calibrated, and the sample in good thermal contact with the bath. This demands a small sample. Fourthly, one would ideally like to measure the susceptibility of the crystal with a magnetic field applied simultaneously in the same direction as the measurement. The first and the second requirements are almost incom- patible. It is difficult to obtain spherically shaped samples, and the only way attempted was to slowly grind the crystal under oil, which both aids the grinding and prevents the absorption of water vapor from the air. The Fairall measurement was made with a cylindrical sample, and with zero applied field. Further, there are theoretical complications in fitting the measured susceptibility to a Curie-Weiss law. Since the electron g-value is determined from the Curie constant, this casts doubt upon the g tensor used in decoupling A. Wolf and Heintz16 show that for concentrated materials with appreciable interactions between the magnetic moments, there exist contributions to the g-value from cross-terms between the off-diagonal part of the magnetization and the interactions. They also suggest that one must not neglect higher order 17). terms in the Curie-Weiss law (the second-order g-shift 71 Their measurements on a powdered sample of BrCl3 suggest that the Curie constant may change by as much as 35% when higher-ordered terms are included. The results of this analysis could be most improved by better susceptibility data. The fitting of the effective field tensor to the data could, however, be slightly improved by a Chi-squared fit computer program which would vary all five elements of the effective field tensor, as well as the applied field angles. Thus, the values of the transferred hyperfine interaction tensor reported here represent an estimate of a quantity that has not been previously reported on for this crystal, and give an insight into the C1- electronic state in ErC13. LIST OF REFERENCES 10. 11. 12. 13. 14. 15. LIST OF REFERENCES A. Abragam and B. Bleaney, Paramagnetic Resonance 2: Transition Ions (Clarendon Press, Oxford, 1970), Chap. 1, pp. 13-15. . A. Narath in Hyperfine Interactions (Academic Press, 5 New York, 1967), ed. A. Freeman and R. Frankel, Chap. 7, p. 301. R. G. Shulman and S. Sugano, Phys. Rev. 130, 506 (1963). C. W. Fairall, J. A. Cowen, and E. Grabowski, (to be I published). Narath, p. 304. R. Bersohn and R. G. Shulman, J. Chem. Phys. 4g, 2298 (1966). E. H. Carlson and H. S. Adams, J. Chem. Phys. 54, 388 (1969). R. Wyckoff, Crystal Structures (Interscience Publishers, New York, 1964), 2nd ed., Vol. 2, Chap. VB, pp. 56-57. W. J. M. DeJonge, G. N. Srivastava, and Paul M. Parker, J. Chem. Phys. 42, 2843 (1968). C. Dean, Phys. Rev. 24, 1053 (1954). S. I. Parks, "NMR in Paramagnetic NdBr3 and UI ", dissertation, Florida State University, 1967, B. 103. E. H. Carlson, "Computer Program for Nuclear Magnetic and Quadrupole Resonance Frequencies and Intensities, I = 3/2 to 9", University of Alabama, 1965. G. Garton, M. T. Hutchings, R. Shore, and W. P. Wolf, J. Chem. Phys. 44, 1970 (1964). Paul M. Parker and R. D. Spence, Phys. Rev. 160, 383 (1967). Narath, p. 304. 72 16. 17. 18. 19. 73 W. P. Welf and J. F. Heintz, J. Appl. Phys. 44, 1127 (1965). M. T. Hutchings and W. P. Wolf, Phys. Rev. Letters 44, 187 (1963). T. P. Das and E. L. Hahn, Nuclear Quadrupole Resonance Spectroscopy (Academic Press Inc., New York, 1958), Chap. 1, p. l. H. Rinneberg and H. Hartmann, J. Chem. Phys. 24, 5814 , (1970). APPENDICES APPENDIX A Diagonalizing the Quadrupole Hamiltonian The Hamiltonian for the interaction of the quadrupole moment of a nucleus with the electric field gradient at its position due to surrounding charges is given by the tensor- scalar product18 _ . = m -m m where Q defines the quadrupole charge distribution in the nucleus. In terms of cartesian coordinates X, Y, Z, its irreducible components are 2 03 = eQ (31z - £2)/(21(ZI - 1)) (A2) 0:1 = eQ/6 (Iin + IiIz)/(4I(ZI -1)) (A3) 0:2 = eQ/6 (Ii)2/(4I(ZI -1)) (A4) where (A5) The scalar quadrupole moment Q of the nucleus is given by eQ = fp.r? (3 c0528. - l) dT. (A6) i 1 il 1 where pi is the charge density in a small volume element dTi inside the nucleus at a distance ri from the center, and 811 is the angle between the radius vector £1 and the nuclear spin axis. The field gradient at the nucleus is 74 75 ' I defined by the tensor 2;, which has nine components -Vij which in cartesian coordinates are 2 3 V (X., xj = X, Y, Z) (A7) ij = EEETEIE i and V is the electrostatic potential at the nucleus due to the surrounding charges. If Laplace's equation holds, meaning that all charges producing the electric field are external to the nucleus, then Z;' is symmetric and traceless, and its five irreducible components are , _ 1 (VB )0 — 2 sz (A8) (VE')+1 - (vXZ _ ivYZ)//6 (A9) (VE')12 - (vXX - vYY i 2ivXY)/(2/6). (A10) Transforming to a set of principal axes x,y,z, the irreducible components become _ 1 _ 1 (VE)O - E’sz— 5 eq (All) (VE)i2 = (vxx - vyy)/(2/6) = neq/(z/E) (A13) and neq is called the quadrupole coupling constant. The asymmetry parameter n is defined by vxx - v n = V *Yy (A14) 22 and be convention, one states zz| (A15) |V|<|V| = A(3m2 - I(I + 1)) . (A22) For a system of nuclear spin I = 3/2, in a representation in which the eigenvalues of Iz are diagonal, r ‘ = A 0 0 E0 3 0 0 -3A 0 0 (A23) 0 0 -3A 0 0 O 0 3A There are two doubly degenerate energy levels +3A and -3A. The transition frequency between these levels is f0 = 6A/h. (A24) When n # 0, the off-diagonal matrix elements are = = /((I+m)(I-m+1)(I+m-1)(I-m+2)) An/Z. (A25) For an I = 3/2 system, HQ = ('3A 0 /3n 0 ) 0 -3A 0 /3n (A26) fin 0 -3A 0 77 Rearranging rows and columns and block-diagonalizing, one obtains the eigenvalues E = I A o ‘ Q 3 p 0 O 0 -3Ap 0 0 (A27) 0 0 -3Ap 0 ‘ 0 0 0 3AOJ where p = (1 + n2/3)% . (A28) As before, there are two doubly degenerate energy levels. The transition frequency between the two levels is fQ = 6Ap/h . (A29) The transitions between the energy levels may be produced by the application of an oscillating magnetic field which interacts with the magnetic dipole moment of the nucleus, producing a time-dependent perturbation. A constant magnetic field H may also be applied to break up the degeneracy of the pure quadrupole levels. If the magnetic interaction is small compared to the quadrupole interaction (ka< = -yHmcosB , (A34) and = :%§.((I-m)(l+m+l))k sine (cos¢ + isin¢ ) (A35) where 0 and 0 are the usual spherical polar angles between the applied field and the crystal principal axes. For an I = 3/2 system, = 3%E-((3/2 - m)(5/2 + m))LS * *sin0 (c050 + isin ¢). (A36) 12 were used to Computer subroutines ZEEMAN and HERDAG Obtain the transition frequencies in all programs used to fit the internal field tensor 3. However, for the initial fit of the tensor, the equations of perturbation theory were used. Perturbation Theory for I=3/2 Case For a system of nuclei of spin I=3/2 in a constant magnetic field H, the energy eigenvalues Em + where m ,_ represents the limiting values of mz, are:10 H- = E (ka/2)[3/2] (A37) E3/2,: 0 H- E 81/2,i = 0 (th/2)[1/2] (A38) correct to first order in H. Here, 2 2 2 2 2 k [m] = [am cos 0 + (bm + cm + 2bmcmc052¢)Sin 8] (A39) and again, it is presumed that ka<