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A nuclear magnetic resonance study of UNI₂AL₃ Gardner, Michael W. 1993

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A NUCLEAR MAGNETIC RESONANCE STUDY OF UNhAL3ByMichael W. GardnerB. Sc. Hons. (Physics) Simon Fraser University, 1991A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTERS OF SCIENCEinTHE FACULTY OF GRADUATE STUDIESDEPARTMENT OF PHYSICSWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAOctober 1993© Michael W. Gardner, 1993In presenting this thesis in partial fulfilment of the requirements for an advanced degree atthe University of British Columbia, I agree that the Library shall make it freely availablefor reference and study. I further agree that permission for extensive copying of thisthesis for scholarly purposes may be granted by the head of my department or by hisor her representatives. It is understood that copying or publication of this thesis forfinancial gain shall not be allowed without my written permission.Department of PhysicsThe University of British Columbia6224 Agricultural RoadVancouver, B.C., CanadaV6T 1Z1Date:oc-T. Cl/ lqqaAbstractThe theory for the combined effects of both a quadrupolar interaction and a Knightshift has been examined for the case of non-axial symmetry. A detailed derivation isgiven, correct to second order in perturbation theory, for the case where the quadrupoleinteraction is significantly smaller than the Zeeman interaction. These results have beenapplied to the study of the Al' nuclear magnetic resonance spectrum of a single crystalof UNi2A/3, measured using a steady state technique. A value for the asymmetry ofthe electric field gradient has also been determined for this material. In addition, thetemperature dependence of the Knight shift was observed over the range of 6 to 300K.No significant temperature dependence was found over this range.11Table of ContentsAbstractTable of Contents^ iiiList of Figures^ vList of Tables^ viiAcknowledgements^ viii12IntroductionKnight Shift142.1 Isotropic Knight Shift ^ 52.2 Anisotropic Knight Shift 102.3 Core Polarization ^ 152.4 Orbital Spin Paramagnetism ^ 163 Quadrupole Effects 173.1 Perturbation Treatment of Quadrupole Hamiltonian ^ 213.1.1^First Order Perturbation ^ 233.1.2^Second Order Perturbation 233.2 Quadrupole Frequency Shifts ^ 243.3 Angular Dependency of the Quadrupole Moment ^ 26iii4 Combined Quadrupole-Knight Shift Effects^ 374.1 Resonance Shift Effects From Inequivalent Sites. . . . . . . .^.^405 Experimental Considerations^ 495.1 General Method ^  495.2 Resonance Absorption and Dispersion Effects ^  505.3 Skin Depth Effects ^  525.4 Magnetic Field Calibration ^  535.5 Basic Experimental Procedure ..... . ....... . .^575.6 Experimental Determination of Knight and Quadrupole Shifts ^ 606 Experimental Setup^ 646.1 Apparatus and Experimental Setup ^  646.2 Sample Orientation ^  687 Experimental Data and Conclusion^ 717.1 Experimental Data ^  717.2 Experimental Results  737.3 Some Comments ^  757.4 Knight Shift Temperature Dependence ^  787.5 Conclusion ^  78A Wigner Matrix Elements^ 80Bibliography^ 82ivList of Figures2.12.2Knight Shift vs. Magnetic Field Angle 0, 0 = 0°. . . . . .^.Knight Shift vs. Magnetic Field Angle 0, 0 = 90°^14153.1 (5/2 4-* 3/2), (-3/2 4-* -5/2) transition.^........^. 323.2 (3/2 4-* -1/2, -1/2 4--> -3/2) transition. 333.3 (1/2 4-* -1/2) transition^ 333.4 Quadrupole Spectra for 0 - 0° ^ 343.5 (5/2 4-* 3/2), (-3/2 4-> -5/2) transition. ^ 343.6 (3/2 4-* 1/2), (-1/2 4-* -3/2) transition. 353.7 (1/2 4-* -1/2) transition^ 353.8 Quadrupole spectra for 0 = 90°.^.^.^.^........^.^.^.^.^.^. ^ 364.1 Spectra of Combined Quadrupole and Knight shifts, 0 - 0° ^ 384.2 (1/2 4-* -1/2) line of quadrupole-Knight shifts, 0 = 0° 394.3 (1/2 4-* -1/2) line of quadrupole-Knight shifts, 0 = 90° ^ 394.4 Spectra Of Combined Quadrupole and Knight Shifts, 0 = 90° ^ 404.5 UNi2A13 Atomic Crystal Structure^ 414.6 C-Axis View Of UNi2A13 Crystal 414.7 Inequivalent Sites In Magnetic Field Presence^ 424.8 (5/2 4-* 3/2) transition, 0' = 10° ^ 434.9 (-3/2 4-* -5/2) transition, 01= 10° 444.10 (3/2 4-* 1/2), (-1/2 4-* -3/2) transitions, 0' = 10° ^ 444.11 (1/2 4-* -1/2) transition, 0' = 10° ^ 444.12 Spectra Of Quadrupole-Knight Shift From Inequivalent Sites ^ 454.13 C-Axis View Of Crystal With Field Direction Of 100^ 454.14 C-Axis View Of Crystal With Field Direction Of 30°  464.15 (5/2 4-* 3/2), (-3/2 4--* —5/2) transitions, O' = 30°. .^ 474.16 (3/2 4-* 1/2), (-1/2 4-* —3/2) transitions, 0' = 30°  ^474.17 (1/2 4-* —1/2) transition, O' = 30°  ^475.1 Theoretical Absorption and Dispersion curves ........ .^505.2 Derivative Of Absorption Curve ^  515.3 Derivative Of Dispersion Curve^. .......... .^515.4 Mixed Absorption and Dispersion Curve ^  525.5 Aluminum Reference Signal for 10.602MHz  555.6 Aluminum Reference Signal for 10.675MHz ^  555.7 Signal From (-1/2 (--* —3/2) Line For 9852kHz  575.8 Resonance Line and Fitting Line Before Fitting Procedure ^ 585.9 Resonance Line and Fitting Line After Fitting Procedure  596.1 Schematic of Apparatus and Equipment used in the Experiment ^ 656.2 Diagram of the Tuned Circuit . . .......... . . .^.  ^666.3 Sketch of Crystal's Physical Appearance ^  686.4 (3/2 4-* 1/2) transition, 0/ = 100^ 697.1 Placement of x, y and z Axes with respect to the Crystal Structure . . . 76viList of Tables7.1 Experimental Shifts with Magnetic Field in Basal Plane. . . ..... . .^717.2 Experimental Shifts with Magnetic Field Along C-Axis ...... .^.^73viiAcknowledgementsI wish to express my sincere thanks to Dr. D. Li. Williams for the patient guidanceand invaluable instruction he provided concerning the theoretical aspects of this work. Ialso wish to thank him for his generous financial support.I am also indebted to Dr. E. Koster for his help in running the spectrometer. Histechnical abilities and amicability made a difficult experiment a more pleasant experience.I would also like to thank Dr. B. G. Turrell for his comments regarding this thesis.Finally, I would like to thank the academic staff at the U. B. C. physics departmentfor making the learning process a challenging and rewarding experience.viiiChapter 1IntroductionNuclear magnetic resonance has, over many years, proven to be an invaluable tool innumerous research fields, providing a wealth of information which ordinarily could notbe readily obtained using other techniques. The study of condensed matter, which isone such field, has benefited greatly from its application with many discoveries beingattributed to nmr research. One area of condensed matter physics where this techniquehas proven to be particularly useful involves the study of metals. This thesis is primarilyconcerned with this area of study.Part of the appeal of nmr originates from the implicitly elegant and compact theoryon which it is based. In its simplest form, the theory centers around the idea that thenuclear magnetic moments of nmr samples interact with an applied magnetic field toproduce quantized energy levels separated by an energy AE. Transitions between theselevels, which are the basis of all nmr research, are induced by the additional applicationof an external oscillating magnetic field of the appropriate frequency v = /E/h.The advantage of nmr lies in its ability to be used as a sensitive probe of the localmagnetic fields which are present in the vicinity of atomic nuclei. These local fields,which originate from the surrounding electrons and neighbouring nuclei, essentially alterthe strength of the local magnetic field experienced by the nuclei and produce a resultantshift in the the nmr frequency. This effect is observed in metals, where the metallicconduction electrons produce a local field that is noticeable in the form of frequencyshifts, commonly termed Knight shifts. In metals that have a crystal symmetry that is1Chapter 1. Introduction^ 2lower than cubic, these shifts also exhibit an orientational dependency with respect tothe applied magnetic field and are called anisotropic Knight shifts. In contrast, the shiftsof metals with crystal cubic symmetry are not affected by the magnetic field orientationand are consequently called isotropic Knight shifts.Another phenomenon which can also be a significant factor in the study of solids,for nuclear spins I > 1/2, is the nuclear quadrupole effect which is a consequence ofthe interaction between the nuclear quadrupole moment and the electric field gradientat the nuclear site. Its presence is noticeable in the form of significant frequency shiftswhich are the result of the reorientation of the nuclei relative to the applied magneticfield against local electric field forces.Although both the Knight and quadrupole effects can be observed using powderedcrystalline samples, the orientational dependencies of their shifts can only be properlyobserved using single crystals. This is principally due to the fact that the randomlyorientated crystallites of a metallic powdered sample tend to destroy any anisotropywhich may be inherently present in the metal.Beginning with a derivation of the isotropic and anisotropic Knight shifts, this thesisprovides the reader with a theoretical background so that these effects may be betterunderstood from an experimental point of view. Initially, the Knight shift effect is derivedin chapter 2 from an interaction Hamiltonian which depicts the interaction between anucleus and a conduction electron. This is followed by a thorough derivation of thesecond order quadrupole effect in chapter 3. Many papers that deal with this effecteither avoid the explicit second order derivation altogether, or simply quote the finalresult while referring the reader to another source for the actual work. As far as theauthor knows, this thesis provides the only real step-by-step derivation of the angulardependent second order quadrupole effect. The experimental part of this thesis involvesthe application of these equations to the Aln resonance spectrum in a single crystal ofChapter 1. Introduction^ 3the intermetallic compound UNi2A13, which exhibits a magnetic transition at 4.5K. Thestudy of this transition is beyond the scope of this thesis and we will confine our interestto determining the relevant parameters in the non-magnetic state. Since the Knight andquadrupole shifts coexist in this material, it is necessary to combine their effects into oneequation. This is done in chapter 4. In addition, because the crystal structure containsthree Al sites, a characteristic line splitting occurs in general. Several plots of this effectare also provided in this chapter. The basic experimental procedure, along with themethod used to determine the Knight and quadrupole shifts are given in chapter 5. Inchapter 6 a review of the apparatus is given along with the procedure used to orient thecrystal, while in chapter 7 the experimental results are quoted.Chapter 2Knight ShiftThe Knight shift which is present to some extent in all metals is largely a product of theinteraction between nuclear moments and conduction electrons near the Fermi surface.Normally, in the absence of a magnetic field, the unpaired conduction electrons of a metalhave no preferential orientation; they simply move rapidly from atom to atom withouteffecting any of the metallic nuclei. This is not the case, however, when a static magneticfield is present. In this instance the electron spins, which possess a Pauli paramagneticspin susceptibility xi,' , become polarized inducing a net magnetic field at the nucleusin addition to the applied static field. This net field, which is often referred to as the"hyperfine field" or "effective field" is largely responsible for the Knight shift effect.In more quantitative terms, the magnetic interaction which exists between a nucleusand a conduction electron in a metal may be depicted by the interaction Hamiltonian [1]—77,he f • L— ^ r32-yrtftohr^371(§ r.)^167r+^-ynyflhi.• §8(7-1)^(2.1)nen ^r3 r 2where -yn is the nuclear "gyromagnetic ratio", yo is the Bohr magneton, 71 is the radiusvector to the position of the electron with the nucleus at the origin, S and f are therespective electron and nuclear spins and L is the electron orbital momentum. Thefirst term represents the interaction of the nuclear spin with the electron orbital motion.With the exception of the rare earths, this term may be generally ignored since suchan interaction is usually quenched in metals. This restriction may be lifted later on ifnecessary. The second term is the source of the anisotropic Knight shift where the spin4Chapter 2. Knight Shift^ 5dipolar interaction between the electron and nuclear spins creates an angular dependencyin the shifts of non-cubic metals. In metals with cubic symmetry, this term vanishes,leaving only the angular independent isotropic Knight shift. The isotropic Knight shiftoriginates with the third term. Here it is generally assumed that the s-wave functionsdescribe the major part of the conduction electron behaviour.We will now proceed to derive the actual terms for the isotropic and anisotropic Knightshifts in the following sections. In order to accomplish this, however, we must first assumefor simplicity that the electrons are only weakly interacting. Such an assumption, whichapplies to low energy processes has been theoretically justified by Bohm and Pines[2].Another useful approximation known as the adiabatic approximation may also be appliedin this case. In this instance, the electronic and nuclear motions may be separatedwith the nuclei being treated as stationary lattice points. The result is a completewave function that is a product of many separate wave functions from each electron andnucleus.2.1 Isotropic Knight ShiftSince a large number of particles reside within a metal we will focus our attention on theinteraction energy of a single nucleus and, using the third term of Eq.(2.1) along withthe one-electron description of conduction, sum over all the individual electrons:167iH(ein) =^ •,17211,13n E ^§ia(02=1(2.2)where j represents the §th nucleus and i represents the ith electron.Our main goal will be to find the expectation value of Eq.(2.2) with regards to themany electron wave function 0, since this will provide us with the isotropic Knight shift.Before this is done, however, we should first note that the wave functions for each electronChapter 2. Knight Shift^ 6that comprise be come in the form of Bloch functions:^= uk.(71)ei);7?^ (2.3)where eirc.f is a plane wave that is modulated by a periodic potential u(r) possessingthe periodicity of the metallic lattice. Since a spin dependency is also required of theelectron wave function, it is necessary to factor in a spin function Os as well so that theBloch function will acquire the more complete form:zkk,s(71^uk(71e4.i..03^ (2.4)Basically, the many electron wave function 0, is a product of these spin dependent Blochfunctions, properly antisymmetrized to satisfy the Pauli exclusion principle.Using 0, and assuming that the electrons are quantized along the z-direction by anexternal static magnetic field we now find that the expectation value of Eq.(2.2) is:(Oeln(ej.)10e) = 1637r^7niti3h1; • I 0:^§ibVi)Oed're^(2.5)Or\^167r^ 2(0e Iii(e3n) 10e) =^7nittOhiz3 >m lok,s(o)1 h(k, s)g,s 2is the probability of the electron with wave vector -k and spin s being in the10r,s (0 )vicinity of the ith nucleus and rns is the eigenvalue of the operator gs. h(k., s) is theprobability of occupancy by an electron of the energy level E(k,.^).We may average over this electron occupancy number for a specific temperature sothat Eq.(2.6) acquires the form:(2.6)(7ibe 7.43n) e )^1637r7,,,aohIz3 Ems 10(o)rc,,2^-■f (k, s) (2.7)where f .^) is the Fermi function:1 f (E(k , s)) = f (E) =exp[(E — EF)/ kB7-] + 1(2.8)Chapter 2. Knight Shift^ 7E F is the usual Fermi energy and kB is the Boltzmann constant.The sum over s is now readily obtainable since an electron has only two spin stateswhich are m, = 1/2 or —1/21(0e17-4371)10e) = 167-rnitohiz33^k. 210/41(0)12 f(k, 1/2) — (0) ^(1.c, —1/2). (2.9)Again assuming negligible spin-orbit interaction2^03)12 = 10(C))1we actually have:7r^1(0e In(e3n)10e) = 163 7nitonlz3 E - "1/, ( 0 ) 2 f(, 1/2) — f(k, 1/2)}.^(2.10)k. 2An approximation of the sum over k, which is now required may be found by usingthe density of states function which in its most general form is:1^d3k^1Ern^F(k) =F(k) — E F(k).^(2.11)v_,00 vThe practical application of this equation to finite yet microscopically large systemsrequires the assumption that 1/V Ek• F( /0) differs insignificantly from its infinite volumecounterpart. Using this relation for Eq.(2.10) produces:ell.din)^/e\^167T 7ntiohiz3 21 I d8:k31,(u ) {f(, 1/2) — f(fc, —1/2)1 .^(2.12)3where V is defined to be a unit volume.Knowing the energies of the two electron states which are:^E(1/2) = E(k)+H0^ (2.13)^E(-1/2) = E() — jHo (2.14)Chapter 2. Knight Shift^ 8with ±,a0H0 being the Zeeman energies, and doing a Taylor series expansion of the Fermifunctions F(ic., 1/2) and f(k, —1/2) about E(k) gives us:(0e1H22)10e) = 11 67 70-L2h-iz-Ho d3k3^13^3^873 2 ofaE0,-; ( 0 ) (2.15)where higher order terms have been neglected.SincevE(k)1where ds is an infinitesimal surface area of constant energy in k-space, we will get:e 7i(e.in)^16; 7n4hizjHo —by performing the integral over dE, where a f/aE has an action similar to that of a deltafunction.Now defining an average of 10k.I2 over the Fermi surface to be:ook(0)12)E,= ;.+-;s3( ok(o) 2 AVE(k)1) /I 4—:3 ^AVE(/*C)1) (2.17)where the density of states at the Fermi surface is:ds ^1N(EF) = 2 87r3 IvE() (2.18)and where a factor of 2 takes the two electron states into account we find:ds (10_(0\12,11\7E(k)1) =, 1\r(1)JF) (101-J0)12\J 871-3^ I^I /EF(2.19)so that Eq.(2.16) now takes the form:1)el7-1(e3n)11Pe)^—±r8 Nit2h/zjHON(EF) (10k(0)12)^.3^# EF(2.20)where N(EF) is the density of states.d3k = dEds8c1R-33 lorco)12 AvElk.)1 (2.16)Chapter 2. Knight Shift^ 9The spin susceptibility of conduction electrons, which is essentially temperature in-dependent, is given by the Pauli paramagnetic susceptibility which is:XP = 43N(EF).Substitution of this quantity into the above equation provides us with an explicit sus-ceptibility dependency so that:(zPeIn2i)10e) =^{;1-r xi, (10k(o)2)EF} (2.21)When consideration is given to the Zeemann energy which in terms of the appropriateHamiltonian 1-lz is:(Cbeinzilke) = —77ihiz2H0^ (2.22)it becomes immediately apparent that the bracketed quantity in Eq.(2.21) is in reality anextra magnetic field produced by the s-conduction electrons at the nucleus in the formof:2AH =^( 10,0)1 )3 EFWe may now define the dimensionless quantity:(2.23)AH 87rH =o —3 XP (107,(0) (2.24)which is in effect the quantity we require; the isotropic Knight shift (kiso).By examining Eq.(2.24) it becomes apparent that the Knight shift should have severaldistinct properties when observed in ordinary metals. One such property is its appar-ent temperature independence which, except for a few cases, Cd and the intermetalliccompounds AuGa2 and BiIn, is common in most metals. This property is a product ofthe largely temperature independent terms (101-:(0)12)EF and xi, that make up the aboveequation for the Knight shift. Another property, which has also been experimentallyChapter 2. Knight Shift^ 10observed and which is another feature of Eq.(2.24) is the independence of the fractionalshift AH/Ho from the static magnetic field value. Two other properties which are alsopresent experimentally and in the Knight shift relation include the predominantly pos-itive nature of the shift in most metals and its dependence on the nuclear charge Z.The origin of the positive shift is apparent since the terms that make up Eq.(2.24) areinherently positive, while the nuclear charge dependence is a product of the (10-12kEFterm which is strongly affected by the nuclear size and the Z dependence of 0.2.2 Anisotropic Knight ShiftIn the absence of cubic symmetry, the spin dipolar interaction between the nuclear andelectronic spins, which is represented by the second term of Eq.(2.1), produces an angulardependent line shift which is readily noticeable in single crystal studies [1, 3] . Thisinteraction is present also in powder studies, but not in the form of a shift. In this caseit manifests itself in the form of an added structure and increased width in the nmr linedue to the random orientation of the crystallites.In order to derive a quantitative expression for the anisotropic Knight shift, it is firstnecessary to express the shift in terms of a tensor:k =1Zso that in analogy to the isotropic case, the Zeeman Hamiltonian for the jth nucleus,along with the anisotropic Knight shift can be written as:(0e17-0 7-((e3n)kbe) = (0e17(3)10e) = --Ynhil • (T. + k) • fi^(2.25)where fio is a magnetic field vector, 1 is the angular momentum vector, 1 is the tensor:1= ii + +Chapter 2. Knight Shift^ 11and the tensor version of the Knight shift for the anisotropic case is:1Z— ksx' -i)Cxy; -i'Vxzk. Trik ^:ikyy3. fiCyzij kiCzx?7,' kzy; 1;-Kzz /C.-4^4-4^4-4In a sense, the term Ho • (1 4- k) is actually an effective magnetic field Hef f so thatEq.(2.25) may be expressed simply as a Zeeman Hamiltonian with a corresponding mag-netic field:(0e17(3)10e) = —Nnileff 1./^ (2.26)It is now necessary to determine the orientational dependency of the Knight shift4-4tensor X. In order to achieve this we must first define a set of orthogonal axes, called theprincipal axes, where the tensor IC, which is symmetric, has zero off-diagonal elements.Specifically, in the principal axis system we have:4-4k= ip ip + JP 1C JP + kp^kpNow in the general case the magnetic field is in an arbitrary direction with respectto the principal axis system. In the laboratory system fio is defined to be along theziab-axis, where Ho = Hok'hib, while in the principle axis system we have that:ii = HoxpiP HOypJP HOzpk.PFrom this we get a corresponding effective magnetic field vector:fief f = (1 + kss, )Hoxi:ip + (1 + kyy, )Hoy,:ip + (1 + kz.z, )Hoz,the magnitude of the vector being:Hef f — NI(1+ kxsp)2Igxp + (1 + kyyp)211dyp + (1 -1- r^II 2— zzp , 2 — ozp - (2.27)The direction cosines of the magnetic field Ho, in terms of the principle axis systemmay be defined as:Osp = HOzp /HO^ii3yp = 110yp / HO^= HOZ p I HOChapter 2. Knight Shift^ 12Using these quantities for Eq.(2.27) produces the result:He ff = (1 + IC= p)2 131, + + 1CYYP)2 13L + + ICZZp)2 13.?),110Realizing that the Knight shift components are very small compared to 1, this equationmay be re-expressed, using binomial expansion:He f f =__ ( 1 + kxspox2, kyyp f3y2p iczzp oz2p)H0.Using this last expression for Eq.(2.26) we now find that:(N7-1(j)kbe) == -4,A(1 -1-)Cxxpox2 p kyvp y2p kzzp 13z2p )H0^(2.28)Since we require the actual frequency shift, we can re-express Eq.(2.28) by using theenergy difference equations:hv AE^— Em = 'YnhHeffso that:V = Vo (1 + /Cs 02 + k 02 + kz 02 )P xp^yp^ZP Zpwhere vo = 7H0/27r and I z3 = m.The direction cosines for the magnetic field may be converted to the convenient spher-ical coordinate form with the corresponding angles 0 and 0:i3xp = sin 0 cos 0Ovp sin 0 sin 0= cos 0so that we get:v vo (1 + /Cssp sin2 0 cos2 + kyyp sin2 0 sin2 + kzzi, cos2 0)(3 cos2 0 — 1)2(2.29)Chapter 2. Knight Shift 13This equation will now be written in a simpler and more useful form so that it willbe notationally consistent with the first order terms of the quadrupole frequency shift inthe following chapter.V = V0 + VoiCTxp Sill2 0(1 + cos(20)) vokyyp sin2 0 (1 — cos(20)) IC„, cos2 02^ 2After some algebraic manipulation we get a final form of:(ksxV^ "P+ kyyp + ICZ2p)^(21CZZ -^KXXp)110 + VO^P^ 1)13^P3 3vo (Kyyp --)C. p) sin2 0 cos(20)x ^•2The following quantities are now defined:1(k_/Cis° — 3 kX p kyyp + ZZp)where /Cis° is the isotropic Knight shift and,= 2ICyyp — kyyp kxsp3and)C2 ==)Cyyp --)Cxxp,so that Eq.(2.29) acquires the form:voki^2 0 1) v 2oK2 V = VO VOkiso^(3 cos2 sin2 0 cos(20) (2. 30 )Experimentally, Knight shift values are generally given in terms of nonmetallic refer-ence compounds in measurements of high accuracy. Although they do not exhibit Knightshifts, these compounds, which are usually metallic salts, do have what is termed, chem-ical shifts which usually amount to no more than a few percent of any given Knight shiftvalue. These shifts are primarily the result of diamagnetic contributions from valence andChapter 2. Knight Shift^ 14closed shell electrons which produce an opposing field to the applied static magnetic field.If in contrast, a salt has a strong paramagnetic susceptibility due to unpaired molecularspin or orbital moments, then it is not suitable for use as a reference compound. It isgenerally assumed that, in addition to the Knight shift, the same degree of chemicalshifting that is present in the reference salt is also present in the corresponding metal[4] . So despite their relatively small contributions, it is generally necessary to specifythe reference compound when a Knight shift value is quoted in order to account for anydiamagnetic contributions.As a result we will define the resonance frequency with respect to a general referencecompound to be v„f, , so that Eq.(2.30) now takes the final form:1Pref1C^kV -= (1 + kiso)V^--r-ref (3 cos2 0^2Pref1)   sin2 0 cos(2)2 2(2.31)The angular dependency of this equation becomes apparent in the following figurewhere 0 varies from 00 to 180°, q = 00, v„f = 10MHz and the Knight shifts have beenassigned the values ktso = 0.002, 1Ci = 0.001 and /C2 = 0.002 which are typical of the35—Nx3025-120>,150(010w0 25^50^75^100 125 150 175Field Angle 0 (Degrees)Figure 2.1: Knight Shift vs. Magnetic Field Angle 0, cb = 00 .Chapter 2. Knight Shift^ 15systems we are studying. In Fig.(2.2) 4 is set to 900.25^50^75^100 125 150 17535—Nx3025-1 20), 150w 1050Field Angle 0 (Degrees)Figure 2.2: Knight Shift vs. Magnetic Field Angle 0, 0 = 90°.It will be noticed in this last figure that the Knight shift does not drop to the samedegree that it does when cb = 0° as in fig.(2.1).2.3 Core PolarizationOne contribution to the Knight shift which has been ignored so far is the core polarizationeffect. This phenomenon, which is common in the transition metals, is the result of theinner core electron shells being distorted by the d-type conduction electrons. In theabsence of any d-type interaction, the inner core electrons are normally perfectly pairedwith no resultant hyperfine field. Since d-type conduction electrons have a high densityof states at the Fermi surface, they have a large susceptibility and as a result, have alarge effect on the inner core electrons and conduction electrons. In the presence ofa magnetic field, they effectively polarize the inner electrons causing an imbalance inthe pairing scheme. Since there is a contact interaction between the inner s-electronsand the nucleus, this imbalance causes a magnetic field at the nuclear site. The resultChapter 2. Knight Shift^ 16is a contribution to the Knight shift known as the core polarization shift Kcp(d). Inthis particular experiment, which measures the spectrum of Al27, this effect will not besignificant.2.4 Orbital Spin ParamagnetismOriginally, it was assumed that the electron orbital angular momentum was quenchedwith the result that we could omit the first term of Eq.(2.1). While this may hold formost metals, it is not true for those metals with non-s band conduction electrons, suchas in the case of transition metals. In this case, the first term of Eq.(2.1) can make acontribution to the Knight shift, but only to second order where the interaction betweenthe magnetic field and the orbital momentum can distort the occupied states. Thisdistortion is caused by the mixing of unoccupied states from above the Fermi level intooccupied ground states that have a quenched orbital angular momentum. By acting onthe first term of Eq.(2.1), these distorted states produce a non-zero contribution K tothe Knight shift. Again this effect is unlikely to be significant in our experiment.Chapter 3Quadrupole EffectsNuclear quadrupole moments and their interactions with local electric field gradientsare an important and sensitive aid in the study of solids. These interactions whichhave a dependency on nuclear orientation can be readily examined using ordinary nmrtechniques.Two areas of study which utilize quadrupole effects involve the use of either a highor low magnetic field. In the high field case the nuclear magnetic moment energy isassumed to be significantly greater than the interaction energy produced by quadrupolecoupling. In this case the coupling simply manifests itself through the splitting of thecentral resonance line into several components [1] . In contrast the low field case involvesquadrupole effects that are large enough to become the dominant factor in the spinorientation of a nucleus [5] . This present study of UNi2A/3 will be concerned with onlyhigh field effects where the quadrupole coupling is treated with perturbation theory.The coupling term can be derived in a classical manner by considering the nucleus tobe a charge distribution with radial dependency p(i) in an electrostatic potential v(r.).The resulting electrostatic interaction energy is represented byE = f p(ilv(fl)d3r (3.1)where the integral is taken over the volume of the nucleus.The scalar potential v(r.) can be expanded in a Taylor series about = 0 since itsvariation over the nuclear volume is small17Chapter 3. Quadrupole Effects^ 18avolv(0 v(o) + E  a xiDefining,av(r.)vi =OXiwe get for the interaction energy1^a2v(71+f=0^Ex,x ^2^3 aXiaX"jia 2 v 77,)axiaxj(3.2)7;•=0=97=0 9=-01E = v(0) f p(71)d3r ^ v J xip(r.)d3r^vij f xixip(F)d3r (3.3)where the subscripts i, j range from 1 to 3.The first term represents the interaction energy of a nuclear point charge in an externalfield. Having no orientational dependency it does not contribute to the nmr signal. Thesecond term vanishes because the nuclear ground state wave function has a definite parity[6] resulting in p(-71) = 1. Obviously, this consideration clearly indicates that theintegrand in the second term is antisymmetric about the origin resulting in a vanishingintegral. The third term represents the electric quadrupole interaction. Any followingterms may be neglected since they either vanish for parity reasons or are outside therange of experimental detectability.We may now define an electric quadrupole moment tensor which is not only symmet-ric but also has a vanishing trace (v2Q = 0) and consequently only five independentcomponents.Qii = f[3xixj — 8i.i1712]P(71d3r^ (3.4)In terms of this tensor the quadrupole energy equation (3.3) becomes:EQ^Evi,[c2i, + si, f 1712p(f)er]^ (3.5)" i,a'For more details see P.23 of "Nuclear-Moments"Chapter 3. Quadrupole Effects^ 19The potential v(f-) satisfies Laplace's equation v2v = 0 at the origin causing thesecond term to vanish with the result that:1EQ = E vijQiiwhere vij is also a symmetric , traceless, 2nd rank tensor.Laplace's equation applies to the potential v(71) basically because any electronic chargewhich penetrates the nucleus originates from the spherically symmetric s-electron stateswhich do not provide any quadrupolar coupling.Now to get the Hamiltonian operator of the quadrupole interaction we may simplyreplace the classical density p() by the operator:P(op)(0^e E^77.p)^ (3.7)where the sum runs over the number of protons of charge e which reside within thenucleus. With this expression the classical operator becomes:Q ij e E(3xipxip — 6ii Mp2)^ (3.8)producing the quadrupole Hamiltonian 7c2:1 x--N7-(Q =^Vi3Qi3v(3.9)Since our main concern will be with high field effects we will require the matrixelements of the quadrupole operator for the perturbation calculations. Before this is donehowever, this operator must first be altered to an equivalent angular momentum formusing the Wigner-Eckart theorem which states that the corresponding matrix elementsof all second-rank, symmetric, traceless tensors are proportional[1]. The resulting formusing this theorem for the quadrupole operator is:(3.6)( I, mle E (3x ipx.ip 8i.i 1 71p2 )1 I, in ) = C (I 771q^+ j h) —^.1211 , in)^(3.10)Chapter 3. Quadrupole Effects^ 20where the constant C is independent of m or m' and where /2 = J + 1 + I. C isdetermined by requiring the spin to quantized along the z-direction so that it is equal toa single tensor element where m = m' = I and i = j = z.This requirement which is not an arbitrary definition is based on the fact that whena nucleus is in a state of definite angular momentum its reorientation energy dependsonly on the difference between the charge parallel and transverse to the z-direction. In aclassical sense, the nucleus has a cylindrical charge distribution about its spin which isresponsible for this orientational dependency. It is for this reason that Q = 0 if i jand Qss = Q. Since Qxx+ Qyy Qzz = Owe get that Qss = Qyy = —Qzz/2 indicatingthat all the components of the tensor Qij may be expressed in terms of a single element.Eq.(3.10) now becomes:(Hie E(34, — 1711))1//) = C(//13/ —^ (3.11)The left term is defined to be equivalent to the single quantity eQ where e is the protoncharge and Q is called the " Nuclear Electric Quadrupole Moment ". Since:(II13Iz2 — 12111) = 1(21—i)^ (3.12)C is:eQC =^1(21 — 1)resulting in an angular momentum space representation of Eq. (3.9)eQ^3- ^ E vi,^+^--^61(21 -1)^2(3.13)(3.14)It will be more convenient if we express Eq.(3.14) in terms of the raising and loweringoperators:= L+ 1y^= -vo^vzzVzx Vzv_ 1=^— Vyy)-17±1v±2Chapter 3. Quadrupole Effects^ 21so that:eQ= 41(21 — 1) [(3/:—/2)Vo-F(/./++4/z)V_,/__/„)Vk_2FV_2+P_V2] (3.15)where:We may now proceed to treat the quadrupole Hamiltonian with first and second orderperturbation theory.3.1 Perturbation Treatment of Quadrupole HamiltonianIn the absence of any quadrupole interaction a magnetic field will induce equally spacedlevels which are characterized by the magnetic energy Hamiltonian:= —Dior 1-1^(3.19)Application of the field along the z-axis defines a coordinate system where thisHamiltonian is diagonal and has eigenvalues:=^= — hvo m^ (3.20)where m =^+ 1, ... I — 1,1 with 21 + 1 energy levels separated by hvo.The Hamiltonian which is called the Zeeman energy is the simplest quantum mechan-ical description of resonance. An electromagnetic field transverse to H will cause tran-sitions between these levels producing one resonance line. The action of the quadrupoleinteraction shifts the Zeeman energy levels disproportionally so that they are no longerChapter 3. Quadrupole Effects^ 22equally spaced. Effectively, the degeneracy of the transition energy levels is lifted, split-ting the magnetic resonance line into its 21 components. We may represent this actionby the simple Hamiltonian :71 == 'Ho +1--(Q^ (3.21)with the energies:E = E7(7?) +^+ E,C2)^ (3.22)where E,(71,-) and EiC?) are the first and second order perturbations on the magnetic energylevels respectively.In general, the expression for the energy perturbation is to second-order:,TTITHQ1-1,0(i,n1HQII rn) Em = Et(,?) + (I ,^,^E74711 E° - EOn (3.23)+ higher order terms.where Ei(7?) is the zero-order energy of the mth quantum state (I, m11-(01I, m). It is apparentthat the first-order term (/, m11-1Q1/, m) is a diagonal matrix element while the second-order term is a sum over off-diagonal elements.In order to facilitate the perturbation calculations we must first calculate the individ-ual diagonal and off-diagonal matrix elements using the general transforming propertyof the raising and lowering operators:m)^V(1 — m)(.1 m 1)Irri + 1) (3.24) L I I, m = V(1 m)(I — m — 1)1ni — 1) .^(3.25)Using Eqs. (3.23), (3.24) and (3.25) and the fact that(mIrni) = mm, we get:eQ(I,^I , m) =- 41(21 — 1) ([3m2 — 1(1 + 1)]170 (3.26)Chapter 3. Quadrupole Effects^ 23eQ(I, m +117-(QII,m) = 41(21 —1)(2m + 1)(I m)(/ m 1)VT1^(3.27)(I, m 211-(Q1I, m) =4/(2/ 1)eQI + m)(/ m 1)(/ —1)(/ m 2)V±2. (3.28)— Matrix elements for lm — mil > 2 may be omitted since we are only considering secondorder perturbation. Third order effects have been given due consideration by Volkoff [7, 8]3.1.1 First Order PerturbationThe form of the first order perturbation calculation is readily apparent as it is simplyEq.(3.26). We may define another quantity vQ which represents the strength of thequadrupole interaction so that:VCAVO 04) =^ [m` —^+ 1)/3]2eQwhere vQ 3e2Q/21(21 — 1)h.(3.29 )3.1.2 Second Order PerturbationThe second order perturbation calculation of the quadrupole energy is certainly moreinvolved than the first order case. The second order terms of Eq.(3.23) using Eqs.(3.27)and (3.28) become:E,!)or more explicitly:1(771^217-(Q1702 4_Eon, — EOrn —2I ( m + 111-1Q 1m)1 2 Eom — Eom+i1(m —11HQ177)12 Em — Eom-il(rn+211-(Q1m)12Em — EOrn+2(3.30)Chapter 3. Quadrupole Effects^ 24E„2,)e2Q 2^f (i — m ) (I + rn 1)(/^1)(-1^+ 2)1V-212 ^1612(21 — 1)2 1 +2hvo(277/^1)2(/— 7/1,)(/ +^1)IV_112^(2771— 1)2(1 M)(/ — TT/^1)1V+112 hVo —hVo(1 + n)(' rn 1)(I rn — 1)(I m + 414212}—2hvoSimplifying the above expression we get:E(2) — 18v0q2e2 {11412(8m2 — 4I(I +1) +1) + IV212(-2m2 2I(I +1) — 1)1 (3.31)where^1V-212 = 114212 = 1V212^1V-112 = 114112 = 11412°^(3.32)3.2 Quadrupole Frequency ShiftsThe splitting of the magnetic resonance line by the quadrupole action into its 21 com-ponents can be qualitatively expressed as:vin = (Em_i — Eni)/h = vo 147,1) +^ (3.33)where h =Planck's constant.Transitions between levels where 'And > 1 have been omitted principally becausethey are only weakly allowed and do not contribute greatly to the overall resonancesignal.Using Eq.(3.29) we get for the first order transition frequency change:Vov(1)  — (E(1) — E(1))/h = vQ—cq(m — 1/2)m^mOr74,,P = vQT0(m — 1/2)^ (3.34)Chapter 3. Quadrupole Effects^ 25where To = Vo/eq is a dimensionless quantity.The central resonance line, which corresponds to the (-1^A) transition, is appar-ently left unaffected by the quadrupole interaction to first order since Eq.( 3.34) vanishesfor m=1/2. This is not the case for the other transitions, however, where the linescalled satellites, are shifted and arranged symmetrically on each side of the central lineby the first order action.The second- order change in the frequency can be expressed using Eq.(3.31) as:v2= (E1 — E7T)/ h = Q^18voq2e2 117112(24m(m — 1) — 41(1 + 1) + 9)1—11412(12m(m —1) — 41(1 + 1) + 6)12Orv(2) = ^ {1T112(24m(m — 1) — 41(1 +1) + 9)12vo1 ,2 IT212(12rn(m — 1) — 41(1 + 1) + 6)1 (3.35)= (2/3)11412/q2e2where IT],^ and 1T212 = (2/3)IV I2,2/q2e2 are dimensionless quantities.Actually, because of the equalities in Eqs.(3.32) we may define that:IT112 =^= IT+112^111212 = IT-212 = 171E212^(3.36)where:3 eqeV±23 eqT±1T±2(3.37)(3.38)The quantities To, T±1 and T±2 have been introduced in order to simplify some of thederivations in the following sections.It is apparent from Eq.(3.35) that the central resonance line is now shifted by thesecond-order quadrupole interaction. The satellite lines are also shifted in the secondChapter 3. Quadrupole Effects^ 26order case, but in a uniform manner, so that their frequency differences Av(2) =^—about the central resonance line actually vanish. This effect, which is a direct result ofEq.(3.35) being an odd function of m indicates that if we use the first-order equation(3.34) to express the frequency difference between corresponding satellites it will becorrect in the second-order case as well.Eq.(3.33) will now be rewritten in terms of Eqs.(3.34) and (3.35) for clarity and futurereference.1^v 02 rUrn= vo vQT0(m — —) + ^ illi12(24m(m —1) — 41(1 +1) + 9)2^12v01--2 171212 (12m(m — 1) — 4I(I+ 1) + 6)} (3.39)3.3 Angular Dependency of the Quadrupole MomentElectric quadrupole effects generally exhibit an angular dependency with respect to theapplied magnetic field direction. An understanding of this dependency may be achievedby examining the transforming properties of the principal axes (x',y',z°). These effec-tively reduce the symmetric tensor -143 to a diagonal form. The result is a tensor withonly three components Vx/Ti, Vvy, and Vz,,, which are not independent since V23 is alsotraceless. As a result, only two parameters are needed to sufficiently define a field gra-dient in the principal-axis system. For future reference we may define two such usefulquantities q and 77 called the field gradient and the asymmetry parameter, where:eq VeSi Vy,y,Vz,z,(3.40)An appropriate reorientation of the principal-axis system can be performed so that:(3.41)Chapter 3. Quadrupole Effects^ 27where using Vxx Vyy Vzz = 0 we get 0 < 7), < 1.The asymmetry parameter /7 is a measure of the departure of a field gradient from axialsymmetry. In cases where the field gradient is cylindrically symmetric, 117y,y, = Vez'and 7/ vanishes. This is also true for the spherically symmetric and cubic cases where== ==In the principal axis system Eqs. (3.16), (3.17) and (3.18) reduce to the simpler forms:VO =^141 = 0 V±2 = riVezi/2with the result that:ToP = 0 Tr2,77/v6 (3.42)In general, the magnetic field is applied along an arbitrary direction with respect tothe principle z'—axis. We require a transformation that will rotate this principle axissystem into one which has a z-axis that is parallel to the magnetic field direction. Sucha transformation can be specified in a general way by the consecutive Euler rotations(a, /3, -y). Initially, this transformation will rotate the principal axis system about theprinciple z'—axis by a. This will then be followed by successive rotations about the newy and z axes through the respective angles # and 7. Actually, only the first two rota-tions (a, #) are necessary to specify the transformation in this case since the quadrupolefrequencies are independent of 7 as will be verified by the following derivations.In order to acquire the form of the transformation, we must realize that Eqs.(3.42),which are the quantities we are interested in, will transform in the same manner as dothe spherical harmonics yr [9]. In other words, they will transform according to theirreducible representation of a second degree rotation group.2 More explicitly, 2E D(2) ({a713 M)11'12Ttl:' (3.43)2See sections II and III of "Solid State Physics" vol. 5 from Cohen & ReifChapter 3. Quadrupole Effects^ 28where TI; are Eqs.(3.42) for the principle axis system.D(2)({ce, 0, 7}) is the transformation matrix we require. It is a 5-dimensional matrixwhich combines the five equations of (3.42) to produce five more equations Till for thenew coordinate system where the z-axis lies along the direction of the magnetic field.The explicit form of this matrix can be found using the general relation from Wigner[10]:D(k)( cf ,^7 )til ti E( —^V(k it)!(k — it)!(k ft')!(k(k 11' 0!(k + — 0!V^— it)! f x {eit''' cos2k+12-p'-2 11 sin2+P'-'a L3 eit1.1'2^2 (3.44)In our case k = 2 and since the representation D(k) is 2k + 1 dimensional we get a 5-dimensional rotation matrix. The factorials in the denominator restrict the summationover to only a few integers between the larger of 0 or tt — /2 and the smaller of k —or k ,a. The actual elements that comprise this matrix may be found in appendix A.Knowing the form of the matrix we may now carry out the transformation which isdepicted in Eq.(3.43). We have:T-112 = TY(2)2,_2)TP2^1)((2)1,-2)T131+ 7)(02?-2)71,1) 7)( i2 _2)Tr 7)(2,_2)TTTH1 = TY(2)2,_i)TP2^D((2)1,_1)TPi+1,(02),-1)T(3 D((12,)_i) + 1)((22?-1)TTTj-1 7,(2)2,0) TP2 +^1,((2)1,0)T131^1)(02?0)T(1)3 1:;112,)0)Tr D((22?0)Tf711 = D((2)2,1)Tf2^D((-2)1,1)7T1^7)(02),1)T(1, D((i2 ,)i)Tr 1)((22?1)7113711 = 7:12)2,2)TP2^D((2)1,2)711'3,^1.")(023?2)T 13112 ,)2)T113 1)((22),2)TTChapter 3. Quadrupole Effects^ 29or more specifically:TH 1-42)^-r^11(2) 77-2 ,-2TH- 1^,Th(2)^111(12)'-2,-1 ,-1^N/6^TH^v(2) 7/^2:)(21)3 + v(2) I/-2,0 v-6 2,0 ,v6^T111^v(2) I/- 2,1^2:300 + v(2) y.v6 -I-^ 2,1 N/g^111^17 ^v(21^2)(2)-2'2 V6 1/6Since our main concern will be in finding the orientational dependencies which arepresent in T0,171112 and 1T212 only three of the five equations, T,, for the new coordinatesystem are required; this simplification being a direct result of the equalities expressedin Eqs.(3.36). The first three rows of the above matrix will be sufficient for our determi-nation of the explicit forms of these three quantities.Using the elements for 7)(2)(a, 3 , -y) from appendix A, we get for TI12:^TI12 = G-2ia cos4(/3/2)e-2i1(—T)= + Vcos2(/9/2) sin2(02)e-2"^sin4(0/2)e-2i-YN/6(3.45)Similarly for TH3t:= —2e-2 cos3(0/2) sinG13/2)e-"_36 \/6 cos3 (3 / 2) sin(0/2)e-"— -V6 cos(0/2) sin3(/3/2)e-il' + 2e' cos(3/2) sin3(f3/2)e-i-Y-1—^(3.46)and for 71,11:TOW = "V6e-21° cos2(3/2) sin2(3/2)i cos4(0/2) — 4 cos2(3/2)sin2(0/2) sin4(0/2)Chapter 3. Quadrupole Effects^ 30N/6e22 cos2(0/2) sin2(0/2) /^ (3.47)Simplification of the above expressions yields:TH - e-42i1'-2 — ^ (2[cos(2a) cos2 cos(2a)] - 4i cos 0 sin(2a)) + ^sin2 /31^(3.48)6TH =^sin 0 [cos 0 cos(2a) - i sin(2a)]-- +16 sin2 /3^(3.49)-1 16sin2 cos(2a)y + -3 cos  - -1ToH  ^ (3.50)2^2^2Now for the second order perturbation we need 171112 = ITN2 and 1T21112 = in-1212,so we take T-1/2 and multiply by its complex conjugate and do the same for THi; we takeTH, and multiply by its complex conjugate. The resulting expressions are:ITi-H12 = sin2 /3^[COS2 3 COS 2 (2CX) + sin(2a)]-1/2 - cos2 /3 cos(2a)7/ -I- -3 cos2 /3{6^2(3.51)andicos2(2a)^cos4^cos2(2a)^cos2(2a) cos2 13^y2=^ COS2 /3 sin2(2a))4 4 2 6(cos(2a)^cos2^cos(2a))^6 sin4.^2^) (3.52)sin /3^y4 16Eqs.(3.50),(3.51) and (3.52) are now substituted into Eq.(3.39) for To, ITI/12 and In/12for the quadrupole frequency shift with the result:1 r 3^2^1^77 sin2 cos(2a)]vm^'11'Y^cos 13 2 +^2+12v0 {sin2 /3 [COS 2 13 COS 2 (2a) + sin(2a)]-1/26Chapter 3. Quadrupole Effects^ 31— cos2 /3 cos(2a)y + —3 cos2,31 [24m(m — 1) — 4a + 9]22—1 f (cos2(2a) +^/3cos' cos2(2a) + cos2(2a) cos2 )3)^Y+ cos2 /3 si112(2a)) w2 1^4^4 ^2^(  (cos(2a) + cos2 ,3 cos(2a)) sin2 /3)6 166 sin4,3 }4+^[12m(m — 1) — 4a + 6]} (3.53)This equation may be simplified and written in the more convenient form:Lim = Vo -f- VQ (M, — -2-1)^COS2^21 + 7/ sin2 /3 cos(2a)]2vQ232vo (1 — cos2,3) [{102m(m, — 1) — 181(1 + 1) + 39} cos2 /3 (1 — —2ri cos(2a))3— {6m(m — 1) — 21(1 +1) + 3} (1 +^cos(2a))]n2v2,^ [24M(M — 1) — 41(1 +1) + 9 — {30m(m — 1) — 6/(/ + 1) + 12} cos2/372vo51m(m — 1) —^+ 1) +91 cos2(2a)(cos2 /3 — 1)2]2^2 4(3.54)To get this equation into its final form, we must re-express it in terms of the sphericalcoordinate angles, 0 and 0. In order to accomplish this, we must recall that a y-axisrotation was executed by the transformation matrix Eq.(3.43) and as a result a may begiven in terms of 0 as:(3.55 )As a result there are some sign changes associated with the cos(2a) terms of Eq.(3.54)since:cos(2a) = cos(7 + 20) = — cos(20)Incidentally, the relation between the angles /3 and 0 may be expressed simply as:Chapter 3. Quadrupole Effects^ 32The result is a final form for the quadrupole frequency shift in the form of [11]:(^1) [3^2 0 cos(20)Vnt Vo VQ M —^COS V^]22VQ+32v0 (1 — COS20){{102m(rn - 1) - 181(1 + 1) + 39} cos2 0 (1 + -2 n cos(20))3- {6m(m - 1) - 21(1 + 1) + 3} (1 - iicos(2))]7720, [+ (^-1 24m(m - 1) - 41(1+1) + 9- 130m(m - 1) - 61(1 +1) +12} cos2 072vo51^9^39^- {-2-m(m - 1) - + 1) + —4 cos2(20)(cos2 0 - 1)21^(3.56)The behaviour of Eq.(3.56) with respect to the angle 0 can be seen in Fig.(3.1) wherethe following values have been assigned to the various parameters; vo = 10MHz, vQ =750KHz, q = 0.500, I = 5/2 and q = 00.1500—N1000500wtfl^0-500w0-1000r4-1500 ^2 ^40^60^80Field Angle 0 (Degrees)^Figure 3.1: (5/2^3/2), (-3/2 4-> -5/2) transition.8006004004.4 200m^0u -200z-400a) -600r.Chapter 3. Quadrupole Effects^ 33The first three values, while somewhat arbitrary, have been chosen specifically becausethey are similar to the values found and used in this experimental study of UNi2A/3. Inaddition, since we are mainly concerned with observing the A/27 spectra I has been setequal to the aluminum nuclear spin of 5/2. The top line at 0 = 00 in Fig.(3.1) correspondsto the (5/2 4-4 3/2) transition, where m = 5/2, while the bottom line represents the(-3/2 4-* —5/2) transition where m = —3/2.-80020^40^60^80Field Angle 0 (Degrees)Figure 3.2: (3/2 4-* —1/2, —1/2 4-* —3/2) transition.20^40^60^80Field Angle 0 (Degrees)Figure 3.3: (1/2 4-* —1/2) transition.150010004_1^500w-H0-500w -1000r=4-15008-0602-0Chapter 3. Quadrupole Effects^ 34Field Angle 0 (Degrees)Figure 3.4: Quadrupole Spectra for 0 = 00.Similarly the two spectral lines for the (3/2 4-* 1/2) and (-1/2 4-* —3/2) transitionscan be seen in Fig.(3.2) while the single line for the (-1/2 4-> 1/2) transition is shown inFig.(3.3). All five spectra have been combined into a single plot in Fig.(3.4) so that thelines for the various integer spins, m, may be seen relative to one another. Figures (3.5),(3.6) and (3.7) represent the same transitions that were depicted by figures (3.1), (3.2)and (3.3) only in this case 0 has been increased to 900.15001000500• 0-500a) -1000-1500 20^40^60^80Field Angle 0 (Degrees)Figure 3.5: (5/2 4-* 3/2), (-3/2 4-* —5/2) transition.Chapter 3. Quadrupole Effects^ 35800N 600x4004-14--1 200-,14M 0>IO -200za)• -400tra)• -600rx.. 20^40^60^80Field Angle 0 (Degrees)Figure 3.6: (3/2 4-4 1/2), (-1/2 4-4 —3/2) transition.-800'-'N 40xx—.0 204-1- c-14M>1^00a)ty, -20wrr.-4020^40^60^80Field Angle 0 (Degrees)Figure 3.7: (1/2 4-4 —1/2) transition.All five spectral lines for this new angle are combined into Fig.(3.8) for the purposeof making a comparison with the Fig.(3.4) spectra. Such a comparison indicates that anincrease in 0 has the effect of decreasing the degree of quadrupole shifting that is presentwhen 0 approaches 900.36Chapter 3. Quadrupole EffectsChapter 4Combined Quadrupole-Knight Shift EffectsThe relations for the Knight shift and quadrupole effects which were derived in the lasttwo chapters may now, in a simple way, be combined to form a more general expressionfor the frequency shift. In the following chapter this expression is used extensively, inconjunction with the experimental aluminum spectra, to determine the values of theasymmetry parameter, the quadrupole frequency and the Knight shifts of the UNi2A/3single crystal sample.Using Eqs.(2.31) and(3.56) the overall frequency shift from the Knight and quadrupoleeffects may be expressed together as 1:][^kivrefllim = Lk' + I-2142^-^+ 2 J (3 cos2 0 — 1)1^1--2 [C2vref (m — —2) vQii] sin2 0 cos(20)vr)32^ (1 - COS20) {{102711(171 - 1) - 181(1 + 1) + 39} cos2 0 (1 + —271 cos(20))vo 3— {6m(m — 1) — 21(1 +1) + 3} (1 —^cos(20))]7720,^ [24rn(m — 1) — 41(1 +1) + 9 — {30m(m — 1) — 61(1 +1) +121 cos272vo— {-2-51m(m — 1) —^+ 1) + 739 } cos2(20)(cos2 0 — 1)21^(4.1)where: Urn = vo Av(Knight shift) + Av(quadrupole shift).1-There is an error in [12], " Metallic Shifts in NMR" concerning this formula, see eq. 6.18 of thatreference for a comparison.37Chapter 4. Combined Quadrupole-Knight Shift Effects^ 38—m15001000w 500.0 00 -500w -1000-150020^40^60^80Field Angle 8 (Degrees)Figure 4.1: Spectra of Combined Quadrupole and Knight shifts, 0 = 00.Most of the parameters and variables in this equation are the same as those definedin the previous chapters. The one exception is in the definition of vo for the quadrupoleterms where it is now equal to the first term of Eq.(2.31) for the Knight shift:1/0 = ( 1 + kiso)Vref •This new definition is necessary because the presence of a Knight shift will alter voand consequently change the size of the quadrupole shift by a small degree. To be strictlycorrect we should also include the anisotropic Knight shift in vo, however, since we aredealing with second order effects it is reasonable to omit any such angular dependencein this term. The behaviour of Eq.(4.1) may be seen in Fig.(4.1) where it has beenplotted with 0 = 00 for the five aluminum transitions which includes the four satellitelines (5/2 4-* 3/2), (3/2 4—)' 1/2), (-1/2 4-* —3/2), (-3/2 4-* —5/2) and the centralresonance line (1/2 4-* —1/2). The constants 7, vQ, //ref, and 1C,s„, ki, k2 have beenassigned the same values that they were assigned in the previous chapters for the Knightand quadrupole shifts. Comparing the spectra in this figure with those of Fig.(3.4),where 0 = 00, it is apparent that the quadrupole transitions, with the exception ofChapter 4. Combined Quadrupole-Knight Shift Effects^ 39the (1/2 4-4 —1/2) transition, are not significantly altered by the presence of a Knightshift. This is reasonable since the relative magnitude of the satellite quadrupole shifts,in this case, is significantly greater than the Knight shift effect. In contrast, since the(1/2 —1/2) quadrupole shift is relatively much smaller, the Knight shift presence ismore obvious which is evident if the spectrum of Eq.(4.1) is plotted again for the centralline (1/2 —1/2) and compared to (1/2 —1/2) line of Fig.(3.3).^ 40• 204.)0(/)-20a)• -40o'(1)• -60-80 20^40^60^80Field Angle 0 (Degrees)Figure 4.2: (1/2^—1/2) line of quadrupole-Knight shifts, 0 = 00 .This is also the case when 0 is increased to 90° where a comparison of Fig.(4.3) for theN 604-)^404-4.cco>, 20cr^0-2020^40^60Field Angle 0 (Degrees)Figure 4.3: (1/2 4-4 —1/2) line of quadrupole-Knight shifts, 0 = 900.80Chapter 4. Combined Quadrupole-Knight Shift Effects^ 40(1/2^—1/2) transition with the corresponding pure quadrupole transition in Fig.(3.7)definitely indicates a significant Knight shift presence. In Fig.(4.4) all five spectral linesfrom Eq.(4.1) for 0 = 90° are plotted where it will be noticed again that the satellite linesare not significantly altered by the Knight shift. This can be verified by a comparisonwith Fig.(3.8).15 00100050044.00-500rs,a) —1000rx.-150020^40^60^80Field Angle 0 (Degrees)Figure 4.4: Spectra Of Combined Quadrupole and Knight Shifts, 0 = 90°.4.1 Resonance Shift Effects From Inequivalent Sites.The actual spectra that are produced by the Aln nuclei of the UNi2A/3 sample arenoticeably more complicated than the actual theoretical spectra of Eq.(4.1). This is nota result of any problem with the equation but rather the product of the aluminum sitepositioning in the sample crystal structure. This effect may be understood by examiningthe hexagonal UNi2A13 crystal structure which is depicted in Fig.(4.5). The distancebetween the individual uranium atoms in the crystal basal plane is a = b = 5.207A andc, which is the perpendicular distance between these planes, is 4.018A.Chapter 4. Combined Quadrupole-Knight Shift Effects^ 41•Uranium^OAluminum ()NickelFigure 4.5: UNi2A13 Atomic Crystal Structure.• Uranium^• Aluminum 0 NickelFigure 4.6: C-Axis View Of UNi2A6 Crystal.Chapter 4. Combined Quadrupole-Knight Shift Effects^ 42Direction of Magnetic Field•••••••"*"Figure 4.7: Inequivalent Sites In Magnetic Field Presence.The A/2" positions may be seen more clearly in Fig.(4.6), where the crystal is viewedfrom the top along the c-axis. In Fig.(4.7) the three numbered aluminum sites, whichare responsible for the more complicated spectral structure are pictured along with amagnetic field component parallel with the xl-axis. These sites, which are commonlytermed inequivalent sites, actually produce a more complicated spectrum by splittingeach transition line into three separate lines. This may be understood by noting inFig.(4.7) that each site has a definite orientation with respect to the magnetic fielddirection. Normally, in the absence of a field the three numbered aluminum atoms wouldbe indistinguishable from one another which can be verified by a simple 60° rotationof the crystal about the c-axis. By the introduction of a magnetic field, this symmetryis effectively destroyed, introducing an angular dependence in the three aluminum siteChapter 4. Combined Quadrupole-Knight Shift Effects^ 43150010005000-500w -1000-150020^40^60^80Field Angle 0 (Degrees)Figure 4.8: (5/2^3/2) transition, 0' = 100positions. For example, if the aluminum atom at site-2 in Fig.(4.7) is to produce the samespectrum as the atom at site-1, a rotation of the field by 60° would be necessary. Similarly,a rotation of 120° would be required for the site-3 atom. In short, each aluminum nucleusat each site will produce its own spectral shift which is dependent on the relative magneticfield direction in the basal plane. This effect can be easily reproduced from Eq.(4.1) bysimply plotting the equation three times for the three distinct angles 0 = 41, 4 = 41+ 60°and 0 = + 120°. The result for the satellite line (5/2 4-4 3/2) may be seen in Fig.(4.8)where 01 = 100, m = 5/2 and where the same values that were used for 77, vQ, v„fand 1CO3 1C1, 1C2 in the last section are used here. We get similar results for the othersatellite lines which are plotted in Figs.(4.9) for the (-3/2 —5/2) transition and(4.10) for the (3/2 1/2) and (-1/2 —3/2) transitions. The central line is plottedin Fig.(4.11). All fifteen lines can be observed together in Fig.(4.12) where in Fig.(4.13)the field direction is indicated in the basal plane with respect to xl.60—N 40>, -20(1)., -60-80Chapter 4. Combined Quadrupole-Knight Shift Effects^ 441500100050044-500a)tr -1000-150020^40^60^80Field Angle 0 (Degrees)Figure 4.9: (-3/2^—5/2) transition, 0, 100Ts; 7505001_44-4 250049 -250w -500-75020^40^60^80Field Angle 0 (Degrees)Figure 4.10: (3/2 4-+ 1/2), (-1/2 4-4 —3/2) transitions, 0' = 10020^40^60^80Field Angle 0 (Degrees)Figure 4.11: (1/2 4-4 —1/2) transition, 0' = 100Chapter 4. Combined Quadrupole-Knight Shift Effects^ 451500—mx• 1000—4J^500w—1U)^0>1o• -500wVw -1000ri.4-150020^40^60^80Field Angle 0 (Degrees)Figure 4.12: Spectra Of Quadrupole-Knight Shift From Inequivalent SitesDirection of Magnetic FieldFigure 4.13: C-Axis View Of Crystal With Field Direction Of 100It will be noticed that for 0' we used 100 instead of 00 or 90°. The reason for thisChapter 4. Combined Quadrupole-Knight Shift Effects^ 46Direction of Magnetic FieldFigure 4.14: C-Axis View Of Crystal With Field Direction Of 300change is that at the other two angles, two of the three lines that are produced by theinequivalent sites for each spectral transition combine to form one line. This effect canbe understood by examining the positions of the numbered aluminum sites in Fig.(4.7)where the magnetic field is parallel to the xl-axis. In this instance the x2 and x3 axeshave the same magnetic field components because they make the same angles with respectto the field direction. As a result the corresponding aluminum nuclei at sites 2 and 3experience the same fields and consequently produce identical spectra. Similarly, thisalso occurs when the magnetic field angle e in the basal plane is 30° with respect to thexl-axis. See Fig.(4.14) for the magnetic field direction with respect to the inequivalentsites and Figs.(4.15), (4.16) and (4.17)for the corresponding spectra.Chapter 4. Combined Quadrupole-Knight Shift Effects^ 4715001000i)5004-1C^0-500wrya) —1000—150020^40^60^80Field Angle 0 (Degrees)Figure 4.15: (5/2 4-4 3/2), (-3/2 4--+ —5/2) transitions, e = 300.▪ 750500• 2500>-4-250w -500-75020^40^60^80Field Angle 0 (Degrees)Figure 4.16: (3/2^1/2), (-1/2 4-* —3/2) transitions, O' = 30°806040I-)4-4 200>,O• -20• -40-60-80 20^40^60^80Field Angle 0 (Degrees)Figure 4.17: (1/2 4--+ —1/2) transition, 0' = 30°Chapter 4. Combined Quadrupole-Knight Shift Effects^ 48In this instance the nuclei at sites 1 and 2 now produce the same spectra. Actually,this spectral degeneracy is present when 0' is 00, 30°, 60°, 90° and 120°. This effectalong with the complexity of the aluminum spectra in Fig.(4.12) indicates some of thedifficulties we will actually face when experimentally we attempt to observe the actualresonance lines.Chapter 5Experimental ConsiderationsThe main purpose of this experiment was to investigate the Knight and quadrupole shiftsproduced by a single crystal of UNi2A13. This was achieved by observing the resonancespectrum of the associated Aln nuclei for various magnetic field orientations. Eq.(4.1)was then used, along with the resulting experimental data, to solve for the actual Knightand quadrupole shift values.5.1 General MethodIn a general sense, the methods used for detecting steady state nmr signals may be sep-arated into two broad categories. One involves detecting the change in the susceptibilityof the sample associated with the onset of resonance. In this case the sample material isplaced inside a coil which is part of a tuned circuit. A resonance signal is detected whenthe inductance of the coil is altered by the variation of the sample susceptibility. In thisparticular arrangement, the coil is orientated with its axis perpendicular to the magneticfield direction. The other method employs a double coil arrangement in which one coilis fed from a signal generator while the second coil, perpendicular to the transmitter coilaxis and field direction, experiences an induced voltage from the forced precession of thenuclear spins. For this particular experiment the first method, commonly termed nuclearmagnetic resonance absorption, was utilized.49Chapter 5. Experimental Considerations^ 50Figure 5.1: Theoretical Absorption and Dispersion curves5.2 Resonance Absorption and Dispersion EffectsThe change in the susceptibility of a nmr sample and its effects on the observed signal atthe onset of resonance may be better understood by considering its more general complexbehaviour which may be seen in Fig.(5.1) In this figure x' represents what is termedresonance dispersion while x" is called resonance absorption. Both terms comprise thecomplex susceptibility which is defined to be:x(w) = xi(w) - ix"( w).The dispersion and absorption curves in the simplest case are Lorentzian in nature. Theyare derived by solving the Bloch equations for the case in which the sample experiences aweak electromagnetic field from the surrounding coil. This derivation may be found in anystandard nmr text. While these Lorentzian curves are typically observed in experimentsconducted on liquids and gases, Gaussian like curves are more typical of the solid state[3, 13] . This is because the magnetic dipolar interactions between nuclei on a rigidlattice are not motionally averaged by the motion that occurs in a liquid or a gas. Inan actual experiment, the Gaussian resonance lines are really seen in terms of theirChapter 5. Experimental Considerations^ 51derivatives where the resonance absorption peak of Fig.(5.1) corresponds to the zeropoint in Fig.(5.2) and the zero point of the dispersion line corresponds to the peak inFig.(5.3)Figure 5.2: Derivative Of Absorption CurveFigure 5.3: Derivative Of Dispersion CurveChapter 5. Experimental Considerations^ 52Figure 5.4: Mixed Absorption and Dispersion Curve5.3 Skin Depth EffectsThe skin depth effect [14] is a phenomenon which limits the degree to which an alternatingmagnetic field can penetrate a metal. Its presence produces a reduction in the amplitudeof the resonance signal and a causes distortion in the signal shape. This distortion,which is the result of the mixing of the absorption and dispersion signals, is causedby the metallic sample which induces a phase shift in the resonance signal [13] . Thiseffect may be overcome by using a powdered sample where the metal has been grounddown to a particle size that is less then the skin depth. A powdered sample was notused in this experiment because clear and detailed signals were required for the Knightand quadrupole shifts. As has been mentioned previously, a powdered sample tends toobscure the individual signals produced by these shifts. A theoretical curve of a mixedsignal can be seen in Fig.(5.4) where:X = ax' bX"has been plotted for an equal mixture where a = b = 0.5. The main problem with thisChapter 5. Experimental Considerations^ 53effect is that we are no longer sure exactly where resonance absorption occurs. Thisis primarily because the central zero point of the mixed signal does not correspond tothe frequency at which maximum absorption occurs as it does in the pure absorptioncase. In this particular experiment this problem was largely overcome by fitting theappropriately mixed theoretical curves to the actual experimental curves on a computer.Once the proper absorption and dispersion mixture was found it was then possible todetermine the actual resonance frequency by simply computing where the zero pointwould have been if an absorption signal had only been present. The most commonlyobserved signal in this experiment was comprised of an equal mixture of absorption anddispersion similar to the theoretical curve of Fig.(5.4). In this figure the theoreticalabsorption point is indicated by the position of the vertical line.5.4 Magnetic Field CalibrationIn this experiment the resonance signals were produced by linearly sweeping the mag-netic field slowly over a small range while maintaining a constant spectrometer frequency.Such a technique required not only a very stable frequency but also accurate values forthe actual field strengths. These were found by observing at two different frequencies theabsorption spectrum of a powdered aluminum reference sample and noting the relativesignal positions. A powdered sample was used in place of a single crystal for the calibra-tion process to avoid the problems mentioned in the previous section. Once the signalpositions were determined it was then simply a matter of calculating the field intensi-ties at these positions using the standard isotropic Knight shift equation of the previouschapters:v = 7ito(1+ kiso)H^(5.1)Chapter 5. Experimental Considerations^ 54and solving for H. While the reference sample was composed of only a simple metalpowder which did not exhibit the usual anisotropic Knight and quadrupole shifts thatwere common in the UNi2A/3 sample, it was still necessary to compensate for its isotropicKnight shift in order to calculate accurately the actual magnetic field strength. It wasfor this reason that the above equation was used to determine H. Accurate values forthe constants 71Lbeta and ki„ were found in " Nuclear Magnetic Resonance in Metals"by T. J. Rowland [4] . The values listed for pure aluminum in that reference were:-ytto = 11.094MHz/10KGaussand= 0.00161with respect to the nonmetallic reference compound A1C13. In Figs.(5.5) and (5.6)the signal of the actual aluminum reference may be seen for the relative frequencies of10.602MHz and 10.675MHz. The actual frequency ranges used to calibrate the magneticfield in this experiment generally depended on the size of the field sweep used with thecommon sweep ranges being 50Gauss or 100Gauss. In Figs.(5.5) and (5.6) the range wasapproximately 100Gauss. The signals in these figures along with all the other data wereplotted with a computer. Actually, the data collecting process involved taking severalsweeps of the resonance signal for signal averaging, digitizing it with an analog-to-digitalconverter and then displaying it with 512 separate data points on a graph.Chapter 5. Experimental Considerations^ 55Figure 5.5: Aluminum Reference Signal for 10.602MHzFigure 5.6: Aluminum Reference Signal for 10.675MHzIt will be noticed that two spectra are actually displayed in both figures instead ofone. This is simply the result of scanning the aluminum reference signal twice with thetop line in both figures corresponding to an increasing sweep of the magnetic field and thebottom line corresponding to a decreasing sweep. The reason for sweeping the spectrumtwice in this manner was for the sole purpose of eliminating an RC time constant effectChapter 5. Experimental Considerations^ 56which was introduced into the spectrometer to suppress some of the spurious noise thataccompanied many of the spectra. While this effect was useful for noise reduction, itdid tend to delay the onset of the actual resonance signals. This is noticeable in both ofthe above figures where it can be seen that the peaks of both spectral lines are slightlydisplaced horizontally from one another. By sweeping the spectrum in both directionsit was possible to compute an average line position and thus determine the real onset ofresonance. The true resonance positions are indicated in both figures by the vertical lines.Since each plot consisted of 512 data points it was convenient to index their positions withdata point numbers. In doing so it was found that the line in Fig.(5.5) was positioned atthe 87th data point while the line in Fig.(5.6) was situated at the 417th point.To get a clearer idea of how the calibration process actually worked we will proceed tooutline the method by using the spectra of Figs.(5.5) and (5.6). In practice this processwas done automatically on a computer when fitting the theoretical mixed absorptionand dispersion curves to the experimental resonance signals. The procedure began byfinding the magnetic field at the positions of the pure aluminum resonance signals inboth figures using Eq.(5.1). The intensity of the field at the line position of Fig.(5.5) wasfound to be 9557.4Gauss while the field intensity at the other line position in Fig.(5.6)was determined to be 9606.9Gauss. Since the line positions were already given in terms ofdata point numbers, it was then possible to calculate the magnetic field strength increaseper data point which was 0.1991Gaussfpoint. Knowing this and the number of pointsthat actually comprised the plot, the end point fields were then easily calculated. In thiscase they were found to be 9523.8Gauss and 9625.8Gauss. It will be noticed that theentire field sweep was actually 102Gauss.Chapter 5. Experimental Considerations^ 575.5 Basic Experimental ProcedureEach experiment was preceded by a cooling of the UNi2A/3 sample down to approxi-mately the 11Kelvin range using a variable flow cryostat and a calibration of the mag-netic field with the aluminum reference sample. Once these processes were completed thealuminum sample was then replaced with the UNi2A/3 specimen and the spectrometerfrequency varied until an appropriate signal was found. Observing a resonance signalfrom this sample usually required several sweeps of the magnetic field in order to sta-tistically enhance the signal-to-noise ratio. When a sufficient number of sweeps werecompleted for a specific run the entire spectrum, if satisfactory, was then stored digi-tally for later analysis. Following this process, it was common practice to either searchfor more signals by varying the spectrometer's frequency again or to simply change thetemperature of the sample or reorientate the magnetic field. An actual signal from the(-1/2 4-+ —3/2) resonance line can be seen in Fig.(5.7) for a frequency of 9852kHz.Figure 5.7: Signal From (-1/2^—3/2) Line For 9852kHzResonance Lines/**\Chapter 5. Experimental Considerations^ 58After an experiment was completed the data were then analysed in order to determinethe frequency shifts produced by the UNi2A/3 sample. This analysis, which requiredfinding the correct magnetic field value at which resonance occurred was completed byfitting theoretical absorption and dispersion curves to the actual experimental curves.As mentioned in the previous section, it was necessary to use such a process in order tocompensate for the distortion of the resonance curve by the mixing of the absorption anddispersion signals. An actual fitting of the Fig.(5.7) data may be seen in Figs.(5.8) and(5.9). In Fig.(5.9), the fit which most closely approximated the signal may be seen alongwith the signal itself. Incidentally, the resonance line position on this plot (see verticalline) which was given in terms of a data point index was 288.The process of determining the shift from this fitting routine will be made clearer if wesimply perform an actual calculation of the frequency shift for the Fig.(5.7) spectra. Thiscan be easily achieved since the magnetic field was already calibrated for this particulardata in the previous section where the increase in field strength per data point was foundto be 0.1991Gauss/point.Figure 5.8: Resonance Line and Fitting Line Before Fitting ProcedureChapter 5. Experimental Considerations^ 59Figure 5.9: Resonance Line and Fitting Line After Fitting ProcedureSince we know that the resonance line position is represented by the 288th data pointof Fig.(5.9) and we know the minimum and maximum field values are 9523.8Gauss and9625.8Gauss respectively, we can immediately calculate the corresponding field strengthat that point. In doing so we find a value of 9581.1Gauss. Determining the resonancefrequency at that point simply requires the use of the simple Zeeman equation:vref =where the value of the coefficient 7ito is 11.094Gauss/10MHz as given in the previoussection. Using this equation actually gives us the reference frequency value which inthis case is 10.6293MHz. In other words, this is the frequency at which resonance wouldoccur in the absence of any Knight or quadrupole effects. To now find the degree offrequency shifting present simply requires the subtraction of this reference frequencyfrom the applied frequency of 9.852MHz. From this the corresponding shift value isfound to be Ay = —777.3KHz. This particular method, which was performed on acomputer, was used throughout the experiment in order to determine the Knight andquadrupole shifts of the UNi2A13 single crystal sample.Chapter 5. Experimental Considerations^ 605.6 Experimental Determination of Knight and Quadrupole ShiftsWe will now proceed to outline the method used to find the Knight and quadrupole shifts.The actual process used to find their values required solving a slightly modified form ofEq.(4.1) using experimental shift values with different magnetic field orientations. Thismodified equation was actually an approximation of Eq.(4.1) where vo in the second orderquadrupole terms, in this case, represented the reference frequency without any Knightshift where:^= (1 + ki„)v„f [-1^— —1) +  illref I (3 cos20 — 1)2 2^2v2Q1^v.— —2 k2VrefCOS20) [{102771(172 —1(rn — —2) vol sin2 0 cos(20)+^cos2^(1 —7/ cos(20))(132vref1)^181(1—^+ 39}^03— 16m(m — 1) — 21(1 +1) + 3} (1 — -d277 cos(20))]7.120,+^ [24M(M — 1) — 41(1+1) + 9— {30m(m — 1) — 61(1+1) +12} cos2 072vref^15 1 ^— 1) —^+ 1) + 1-91 cos2(20)(cos2 0 —1)2] .2^2 4(5.2)See Eq.(4.1) for a comparison with this equation. Since these terms were of second orderit was considered appropriate to drop this particular Knight shift presence to simplifythe process used in finding the parameters. In the actual experiment an iterative methodwas used to check if this omission had any real significance on the results.The most common shift signals used for the data analysis in this experiment cor-responded to field directions which were along the crystal c-axis, 0 = 00, or in thebasal plane, 0 = 90°. The reason for this becomes immediately apparent by examiningFig.(4.12) where it can be seen that a significant degree of line interference can occurChapter 5. Experimental Considerations^ 61theoretically in the angular region of 0 = 40° to 800. By using only the two field direc-tions of 0 = 00 and 0 = 90° the chance of mistaking one line for another was reduced.It is apparent also in Fig.(4.12) that when the magnetic field direction is in the basalplane, that is when 0 = 90°, some of the (3/2 4-* 1/2), (-1/2 4-* -3/2) and (5/2 4--> 3/2),(-3/2 4-* -5/2) spectra can mix. This however, was not a real problem because thelatter spectra were significantly weaker than any of the other spectra at this angle. In re-ality it was actually quite difficult to properly observe the (5/2 4-* 3/2), (-3/2 4-)' -5/2)transitions experimentally in the basal plane.The basic procedure used in finding the parameters began with determining vQ. Thiswas done by subtracting the experimental satellite signals, which corresponded to a fielddirection of 0 = 00, from each other and using Eq.(5.2) to solve for vQ. Specifically, theexperimental resonance frequency that corresponded to the (5/2 4-* 3/2) transition wasset equal to Eq.(5.2) with m 5/2 and 0 = 0° while in a similar manner the frequencyfrom the (-3/2 4-* -5/2) transition was set equal to the same equation only in this casewith m = -3/2. Subtracting the latter relation from the former resulted in a simpleequation of the form:(11512,312 -/IQ =^ •4(5.3)Alternatively, this method was also used for the (3/2 4-> 1/2) and (-1/2 4-* -3/2) spectraproducing the slightly different relation:(v312,112 - 1/--1/2,-3/2) vQ2^(5.4)Both equations provided similar values for vQ. Next it was necessary to determine theasymmetry parameter 7/ and 0. These equations were formulated using resonance fre-quency data that corresponded to a field direction of 0 = 90° with respect to the crystalc-axis. It has been already demonstrated theoretically in the last chapter that a multitudeof line signals should be present for this particular angle due to the three inequivalentChapter 5. Experimental Considerations^ 62sites in the UNi2A/3 crystal. As a consequence of this we were able to use the six ex-perimental spectra associated with the (3/2 4-* 1/2) and (-1/2 —3/2) transitions tofind 77 and vQ. The actual process initially involved the same method used in derivingEq.(5.4) except in this case 0 = 900 resulting in the relation:1/1 = V3/2,1/2 — 11-1/2,-3/2^—742 [1 + i COS(201)].^(5.5)^Using the signals associated with the other inequivalent sites with 0 = ^+ 60° and= + 120° provided two more equations:Av2 = ("3/2012 — v_i 2—'7/ 3/2)600 = —//Q [1 + n cos 2(0' + 600)]^(5.6)Av3 = (v3/2,1/2 — '1/2,-3/2)1200 = —742[1 + 77 cos 2(0' + 1200)]^(5.7)Eqs.(5.5), (5.6) and (5.7) were then solved for 77 and 7iQ resulting in:AV1 + V Q=COS(2011Qtan-1 [2Lu2 + 3vg Avi]2^0(,A7/1, vQ)9Y = tan-1 [2Av3 3vQ +2^-\13-(Av1 +7N)(5.10)Actually, the last two equations are equivalent since they should, in theory, provide thesame angles for 0 and the same asymmetry parameter value when substituted in Eq.(5.8).Once 77 and 0' were determined it was only a matter of establishing the Knight shiftvalues. This was accomplished by solving three sets of equations in the form of Eq.(5.2)for /Ciso, k1 and /C2. The equations used were:(5.8)(5.9)V-112,-3I2 = kisoVrefo^1C2Vrefo + A^(5.11)Chapter 5. Experimental Considerations^ 63Vrch vi/2,-1/2 = /Cis° Vreh^k21ireh COS (201 ^ in„^B^(5.12)21'ref2^K211„f2 COS 2(01 + Oineq) CV1/2,-1/2 1= KisolireJ2^2(5.13)where Oineq 0° in Eq.(5.12) and 60° or 120° in Eq.(5.13) depending on the particularlines used. A, B and C were constants that represented the remaining terms of Eq.(5.2).In Eq.(5.11) we used the signal from the (-1/2 4-* -3/2) transition where 0 = 00 withrespect to the crystal c-axis and set it equal to Eq(5.2) for m -1/2 and the sameangle. Eqs.(5.12) and (5.13) involved two of the three signals associated with the central(1/2 4-> -1/2) line for the field angle of 0 = 90° with m = 1/2 in Eq.(5.2). The referencefrequencies are numbered for the purpose of indicating that the same reference frequencyis not required for all three equations.Chapter 6Experimental SetupIn this chapter we are mainly concerned with the experimental setup and the method usedto orient the crystal. In the first section a review of the basic experimental configurationand the equipment used will be given.Since much of the equipment was a standard type found in many laboratories itwill not be necessary to provide an exhaustive description of the functioning of everyapparatus used in the experiment.6.1 Apparatus and Experimental SetupThe experimental setup may be seen in Fig.(6.1). It is indicated in this figure, that thesample was positioned in a cryostat between the poles of an electromagnet. Actually,the sample was fixed firmly in place on the end of a 1 meter probe between the poles ofa 7400 Varian rotatable electromagnet. The magnetic field direction could be changed,by rotating the entire electromagnet about the crystal. Typical field values used in thisexperiment generally ranged from 5 to 13KGauss. The field strength was monitored atthe computer from the lock-in analyzer through the 16-bit analog-to-digital converter;see Fig.(6.1).The sample coil was made by wrapping the crystal in 25cm of No.37 coated copperwire. It was connected to the rest of the apparatus through a coaxial cable and was partof the tuned circuit shown in Fig.(6.2). The circuit's frequency was monitored by thePhilips frequency counter and sent directly to the computer.64Figure 6.1: Schematic of Apparatus and Equipment used in the ExperimentChapter 6. Experimental Setup^ 66Figure 6.2: Diagram of the Tuned CircuitThe Pound-Knight-Watkins (P.K.W.) box, phase lock frequency stabiliser (p.l.f.^),Dynatrac3 lock-in analyzer and Kepco operational amplifier (OP/Amp) were all partof the actual spectrometer. The resonance frequency was changed by adjusting thecapacitance of the P.K.W. box, which is represented by the variable capacitor in Fig.(6.2),and varying the voltage across the voltage variable diode. This voltage, which hadthe effect of altering the diodes capacitance, was supplied by the phase lock frequencystabiliser. In this case the p.l.f.s was used as a fine adjustment which could change thefrequency over a limited range in 1KHz increments. Large frequency changes could onlybe produced by the P.K.W. box.It was mentioned in the chapter concerned with experimental considerations that anRC time constant effect was needed in order to eliminate some of the noise associatedwith the signal. This was supplied by the lock-in analyzer where an RC constant of1.25sec, which corresponded to a spectrometer bandwidth of 0.1Hz, was generally used.The analyzer also modulated the magnetic field over a 10Gauss range with a 37Hz sinewave. This particular frequency was found to be the best in eliminating most of theelectronic interference that was present in the laboratory.Chapter 6. Experimental Setup^ 67The temperature of the specimen was reduced by exposing it to a continuous heliumflow in the cryostat. The temperatures used were generally in the range of 6 to 11K. Itwas necessary to reduce the sample temperature down to this range in order to enhancethe signal-to-noise ratio. A moderation of the temperature was usually achieved throughthe use of a small heating element placed near the sample. This heater was regulated bythe Lakeshore monitor and controller. The equipment was set up so that the temperaturein the vicinity of the sample could be conveniently monitored through the monitor andcontroller at the computer. When needed, the sample's temperature could be reducedbelow the helium boiling point of 4.2K by filling the cryostat with liquid helium andpumping on it. In this particular case, the helium level was monitored by a helium levelindicator which was a device that consisted of a niobium-titanium alloy wire strung alongthe length of the probe and connected to a power supply and voltage indicator.The most unique feature of this experimental setup involved the use of a computerinstead of a wave form generator to sweep the magnetic field over a specific range; usually50 or 100Gauss. The scanning of the field was controlled from the computer, through thegeneral purpose interface (G.P.I.) bus and through the 16-bit digital-to-analog converterat the magnet power supply; see Fig(6.1). The field was changed in a linear manner whilethe spectrometer's frequency was held at a constant value by the lock-in analyzer. Eachsweep of the magnetic field required 51.2 seconds to complete. Since the field was sweptin both directions by the computer in order to eliminate the signal displacement causedby the RC time constant, a full cycle actually required 102.4 seconds. Generally it wasnecessary to sweep the field repeatedly in order to statistically enhance the signal-to-noiseratio. The usual number of double scans used ranged from 1 to 25. Approximately 43minutes were required to complete 25 full cycle sweeps.Oc\sc'.c.Figure 6.3: Sketch of Crystal's Physical AppearanceChapter 6. Experimental Setup^ 686.2 Sample OrientationA sketch of the UNi2A13 sample may be seen in Fig.(6.3) where it is indicated that thecrystal had an approximately hexagonal shape and relatively small size. Its small sizealong with its fragility required that special care was taken in its handling. This wasespecially true during the mounting process which involved securely fastening the sampleon the end of the probe to a small teflon slab with teflon screws.Due to its shape it was initially thought that the crystal c-axis was positioned perpen-dicular to the samples surface. This, however, turned out not to be the case. By viewingthe (3/2 4-4 1/2) spectra it became apparent that the closest the c-axis came to beingaligned along the magnetic field direction corresponded to an angle of approximately14.5°. This was the case even though the samples surface was perpendicular to the fielddirection. Finding this angle required rotating the magnetic field in small incrementsand noting the position and relative spacing of the three spectral lines produced by thethree inequivalent sites. The idea behind this process may be understood by referringto Fig.(6.4) where, for clarity, the theoretical spectra of the (3/2 1/2) transition fromChapter 6. Experimental Setup^ 6910007505004-1250C)a) -250t:ra) - 500._750:20^40^60^80Field Angle 0 (Degrees)Figure 6.4: (3/2 4-4 1/2) transition, 0' = 100 .Fig.(4.10) have been replotted. It will be noticed in this figure that as the angle 0 of mag-netic field decreases with respect to the c-axis, the three spectral lines tend to convergeto a single maximum frequency shift. The point at which they converge corresponds tothe alignment of the field along the axis; that is when 0 = 00.Using this principle to determine the c-axis direction, it was observed that the threespectra did tend to converge up to a certain point, however, a further change in the fielddirection beyond this point resulted in a reversal of this tendency. The angle at whichthis occurred corresponded to the above field angle of 0 = 14.50. This value was laterconfirmed when the crystal was realigned using x-rays. A reorientation of the crystal wasnecessary in order to ensure that the magnetic field was properly aligned along the c-axiswhen 0 was thought to be 00.The determination of the c-axis direction required the removal of the sample from itsmount and its placement in a special holder that enabled the user to accurately observethe crystal orientation. Several x-ray photographs of the crystal diffraction pattern werenecessary before a symmetric pattern was observed. The onset of symmetry indicatedan alignment of the x-ray beam, along the crystal c-axis. After recording the crystalChapter 6. Experimental Setup^ 70orientation at which this occurred, specifically 14.5° + 0.3°, the sample was carefullyremounted on the teflon holder and placed in its new orientation on the end of thesample probe to a small brass plate. The plate on which the sample holder was placedwas bent to the above angle so that the magnetic field could then be easily aligned alongthe crystal c-axis with the appropriate rotation. This alignment was later checked byagain observing the converging behaviour of the (3/2 4- 1/2) spectra as the magneticfield direction was altered. This time it was observed that the three spectra convergedto a single frequency shift which indicated that the c-axis was finally properly alignedin the field direction. The error in this alignment, after mounting, was considered to be+0.5°. The geometry of the sample prevented the easy x-ray observation of any of theother crystal axes.Chapter 7Experimental Data and ConclusionThe experimental frequency shifts found when the magnetic field was applied along thecrystal c-axis and in the basal plane are now used in conjunction with the equations givenin chapter 5 to find vQ, the asymmetry parameter II, O' and the isotropic and anisotropicKnight shifts /Cis°, K1 and /C2.7.1 Experimental DataLines Frequency Shifts (kHz) Oineg(3/2 4-* 1/2)—477+1 00—263 ± 1 60°—282+1 120°(1/2 4-* 1/2)53 ± 1 0°71 + 1 60°67 ± 1 120°(-1/2 4--> —3/2)566 ± 1 0°380 + 1 60°396 + 1^._ 120°Table 7.1: Experimental Shifts with Magnetic Field in Basal PlaneThe experimental frequency shifts and their lines are tabulated in Table(7.1). As isindicated in this table, three frequency shifts were observed for each line resulting ina total of nine shifts for all three lines. This particular signal distribution, which wasanticipated in chapter 4, is the result of the aluminum nuclei at the three inequivalentsites experiencing Knight and quadrupole effects for a magnetic field direction in the71Chapter 7. Experimental Data and Conclusion^ 72crystal basal plane. It will also be noticed in the table that each experimental shift hasbeen assigned to a particular angle 0,,,eq. These angles, which appeared in Eqs.(5.12)and(5.13), represent the angular positions of the inequivalent sites in the crystal.The assignment of each shift to a particular angle involved using the experimentalspectra from the (3/1 4-4 1/2) and (-1/2 <-4 -3/2) lines for Avi, Av2, and Av3 inEqs.(5.8), (5.9) and (5.10) and determining which assignment produced the most reason-able values for 0' and 17. To be more specific, the actual process first involved subtractingthe experimental shifts associated with the (-1/2 <-* -3/2) line from the correspondingshifts related to the (3/2 4-4 1/2) line. Since the (3/2 4-> 1/2) and (-1/2 -3/2)lines are symmetrically related to each other, which is apparent from the theoreticalplots in Fig.(4.10), this involved subtracting the two largest shifts from each other,(-477 - +566)kHz, then the two smallest, (-263 - +380)kHz and finally the last two,(-282- +396)kHz. Since symmetrically related shifts are produced by the same inequiv-alent sites and are therefore represented by the same angles qne q, it was necessary tosubtract them in this fashion so that they could be used in Eqs.(5.8), (5.9) and (5.10)for Av1, Av2 and Av3. It will be noticed in Eqs.(5.5), (5.6) and (5.7) that Av1, Av2and Av3 are defined to be related to the respective inequivalent site angles Oineq 00 ,60° and 120°. Once the differences between the experimental shifts were found, it wasthen simply a matter of substituting their values into Eqs.(5.8), (5.9) and(5.10) for Avi,Av2 and Av3 in several different ways until these equations produced a unique angle,that is one less than 60°, and a non-negative asymmetry parameter. It was ascertainedthat assigning the angles in the particular order shown in Table(7.1) produced such aresult. In other words, Avi = (-477 - +566)001(Hz, Av2 = (-263 - +380)600kHz andAv3 = (-282 - +396)12ookHz. It was discovered, using various theoretical plots simi-lar to those displayed in chapter 4, that the order of the angular assignments for the(3/2 4-> 1/2) and (-1/2 4-> -3/2) lines also produced the particular order indicated inChapter 7. Experimental Data and Conclusion^ 73Lines Frequency Shifts (kHz)(-1/2 4.- —3/2) —788 + 1(-3/2 4-÷ —5/2) —1565 + 1(5/2 4-* 3/2) 1580 + 1Table 7.2: Experimental Shifts with Magnetic Field Along C-AxisTable(7.1) for the (1/2 4-), —1/2) shifts.In order to do all this, however, we first required the value of /N. As previouslymentioned in chapter 5 this was done by subtracting the experimental satellite signals,which corresponded to field direction along the crystal c-axis, from each other in eitherEq.(5.3) or (5.4). In this particular experiment Eq.(5.3) was used with the (5/2 4- 3/2)and (-3/2 4-4 —5/2) shifts listed in Table(7.2). Since the shifts in this table were observedwhen the field direction was along the c-axis, the inequivalent site positions were notrelevant and consequently their angles not required in this case.The experimental data, along with the errors listed in both tables were determined byrepeatedly using the fitting procedure mentioned in the last chapter to find an averageshift value along with a standard error for each resonance signal.All the shifts were observed for a temperature range of 6 to 11K.7.2 Experimental ResultsThe value of xicj using Eq.(5.4) was determined to be:1 vQ = 786.3 + 0.3The reason for using Eq.(5.3) instead of Eq.(5.4) was primarily due to the fact that thedenominator in this equation was twice as large as the number used in the denominatorChapter 7. Experimental Data and Conclusion^ 74of Eq.(5.4) which resulted in a smaller calculated error in N. Using this value for N inEqs.(5.8), (5.9) and (5.10), along with the (3/2 4-* 1/2) and (-1/2 4-4 —3/2) experimentalshifts for Av1, Av2 and Av3 cited in the previous section, it was found that:= 2.3 ± 0.5andii = 0.327 ± 0.003The error in 0' was calculated by finding the difference between the values given byEqs.(5.9) and (5.10). It was mentioned in chapter 5 that both equations should in theoryproduce identical angles for 0'. This, however, was not true experimentally since wewere using experimental data for these equations, not theoretical values, as a result theyproduced two slightly different values for this angle.The error in the asymmetry parameter was found to depend mainly on the errors inAvi and AN and not to a large extent on 0'.Once vQ, 0' and 77 were determined, it was then simply a matter of substituting theirvalues into Eqs.(5.11), (5.12) and (5.13) along with the experimental (-1/2 4-* —3/2)and (1/2 4-- —1/2) shifts. The (-1/2 4-* —3/2) shift in Table(7.2) which corresponded toa field angle of 0 = 00 was set equal to Eq.(5.11), while two of the three (1/2 4- —1/2)shifts from Table(7.1), which were observed for a field direction in the basal plane weresubstituted into Eqs.(5.12) and(5.13). The third (1/2 4-* —1/2) line was used to checkthe accuracy of the resulting Knight shift values. The reference frequency values usedwere v„k = (10630.0 + 0.5)kHz for Eq.(5.11), vrek -,--  (9039.0 + 0.5)kHz for Eq.(5.12)and //ref, = (9020.0 +0.5)kHz for Eq.(5.13). Solving these equations for the Knight shifts1C2,90 % %%(0.23 + 0.02) (-0.25 + 0.02) (0.14 ± 0.03)Chapter 7. Experimental Data and Conclusion^ 75produced the three values:It was established that the errors in these values depended mainly on the experimentaluncertainties associated with the (-1/2 4-4 3/2) and (1/2 4-* —1/2) frequency shifts.The effect of using Eq.(5.2) as an approximation to Eq.(4.1) was checked by insertingthe above Knight shift values into Eq.(4.1) and calculating the frequency shifts. It wasdiscovered that these calculated shifts were not significantly different from the originalexperimental shifts. In other words, the above Knight shift values were not affected byusing an approximate form of Eq.(4.1).7.3 Some CommentsThe asymmetry of the ionic contribution to the electric field gradient was calculatedand compared to the experimental asymmetry parameter found in the last section. Itsvalue was determined by considering the following ionic forms of a large number ofuranium(U+), nickel(Ni2+) and aluminum(A13+) atoms and their influence on a singlealuminum atom. The actual process required a computer that summed the contributionof each ion using the equations:e17xx^ [=r5 3x2 —r^2]^ (7.1)^_L {3y2 _ 7,2] (7.2)=VYY r5Vzz = e— [42r5^- r2]^ (7.3)The variable r was the distance from the single aluminum atom to an ion in the crystallattice and x, y and z were its components with respect to the coordinate system depictedV-Axis• Z-AxisOut Of PageChapter 7. Experimental Data and Conclusion^ 76X-AxisFigure 7.1: Placement of x, y and z Axes with respect to the Crystal Structurein Fig.(7.1) relative to the crystal structure. After summing over a sufficiently largenumber of contributions the following values for Vsx, V ^Vz, were found:1/zx = —1.070 x 10mesuicm3Vyy^9.017 x 1013esu/cm3Vzz = 1.676 x 1013esu/cm3It is immediately apparent that this result indicates that the largest component of theelectric field gradient is in the x direction, in complete disagreement with the experimentalobservations. Thus, the ionic contribution cannot be the dominant source of the electricfield gradient. Using these values the asymmetry parameter was calculated to be:ii = 0.686Chapter 7. Experimental Data and Conclusion^ 77The contribution from the conduction electrons can be probed through the Knightshift anisotropy tensor since it reflects the symmetry of the Fermi surface electron stateswhich is also proportional to their contribution to the electric field gradient. This tensortakes the form:0^0Kaniso=^0 —(k1 k2)/2^00^0^/C2)/2which was easily derived using Eq.(2.31). From this equation we have that:/C2 - 2= /C2 s + —2( 3 cos2 0 — 1) — —2 sin 0 cos(20).wherev (1 + k)vref•Sincekzz = kisoICSX = —(1C1 + 1C2) /2= (—K:1 + K:2)/2where 0 = = 00 for Eq.(7.5), 0 = 90° and 0 -,-- 00 for Eq.(7.6) and 0 = = 90° forEq.(7.7), we have:k„ 0 0 K80 0 00 /Cxr 0 0 1C,50 0 + —(1C1 + k2 )/20 0 iCyy 0 0 kiso (—)c1 +K:2)/2 _where:4-1^4-1K = Kis° + 'Canis° •(7.4)(7.5)(7.6)(7.7)Chapter 7. Experimental Data and Conclusion^ 78From the experimental Knight shift values we infer that this contribution would havethe experimentally observed principal axes and an asymmetry parameter of:ii = 0.6 + 0.2where0 < n < 1andkaniso,xx — kaniso,yyIC aniso,zzThus, at least these states appear to have the correct symmetry, and we can only assumethat the remaining states not near the Fermi surface are responsible for the remainingsignificant contribution to the electric field gradient.7.4 Knight Shift Temperature DependenceThe temperature dependence of the Knight shift was observed over the range of 6 to 300Kby measuring the frequency shifts of various resonance signals. No significant temperaturedependence was observed over this range outside of the experimental uncertainty of+1kHz. This result is consistent with the theoretical derivation of the Knight shiftgiven in chapter 2, (see Eq.(2.24)), where a temperature independent behaviour was oneof the many properties found to be associated with this effect.7.5 ConclusionWe have shown how the theory of the Knight shift and quadrupolar interactions for anon-axial field gradient can be applied to the determination of both the Aln Knight shifttensor and the electric field gradient tensor in the case of UNi2A13. The experimental1=Kis° % % k2 %(0.23 + 0.02) (-0.25 + 0.02) (0.14 + 0.03)Chapter 7. Experimental Data and Conclusion^ 79results yield very accurate values for the electric field gradient (vc? = 786.3 ± 0.3, 77 =-0.327+0.003), which in turn allow the determination of the Knight shifts with reasonableaccuracy.The electric field gradient is not predominantly influenced by the ionic contributionaccording to a lattice sum calculation, but appears to be more strongly influenced by theconduction electrons. In the temperature range studied, no temperature dependence ofthese parameters has been observed.Appendix AWigner Matrix ElementsThe following terms are the matrix elements that comprise the 5-dimensional rotationmatrix. In all, there are 25 elements.= e2 cos4(3/2)e-2i1^1)(2,-1 =^cos3(3/2) sin(3/2)e-1'YDL2L =^cos2(3/2) sin2(3/2)^_ _2e-2ice cos(3/2) sin3(i3/2)01'L-'2,2 = C2D(2),_2 = 2e-ai cos3(3/2) sin(3/2)e-2-Yi^= e-ia cos4(3/2)e-i'(— 3e-ja sin2(/3/2) cos2(3/2)e-i-YDL2L0 = V6e-2a cos3(3/2) sin(3/2)^= 3e-ia cos2(/2) sin2(3/2)e'Y^V6e-2a cos(3/2) sin3(3/2) — e-ja sin4(i3/2)eil',Th(2)^2e-ja cos (3/2) sin3(/3/2)e2i-Y80Appendix A. Wigner Matrix Elements^ 817)02?-2 = \ COS2 (/3/ 2) sin2(P/2)e-2"^DV,_) = N/6 cos3(0/2) sin(0/2)e—"—^cos(P/ 2) sin3(0/2-)322 = cos4(j3/2) — 4 cos2(P/2) sin2(0/2)sin4(0/2)Di:(2„1) —.V6cos3(3/2) sin(P/2)ei-Y+^cos(0/2) sin3 (0/2)e"=^cos2(0/2) sin2(/3/2)e22i77)122_2^2eic' cos(3/2) sin3(0/2)e-2i1^D1,—(2) -- 3e cos2(0/2) sin2(P/2)e—i'Y— ei' sin4(13/2)e—il'DV2 = V6e cos3(P/2) sin(0/2)^/X1 = e cos4(0/2)e6'—^cos(P/2) sin3(/3/2) — 3e ^(/3/2) sin2(P/2)e'YDi22 =^cos3(0/2) sin(P/2)e2il''T)2) 2 = e2ic sin4(0/2)e-2i1'^DV) = 2e2' cos(P/2) sin3(P/2)e—i-Y1:)2()) = '16-e2ja cos2 (P/2) sin2(3/2)^DV1, = 2e2i' cos3(0/2) sin(i3/2)eils4 (3/2) e2i-yBibliography[1] C. P. Slichter, Principles of Magnetic Resonance, Spinger-Verlag, Berlin; New York(1990)[2] D. Pines, Solid State Phys., 1:38 (1957)[3] A. Abragam, Principles of Nuclear Magnetism, u. press, Oxford (1961)[4] T. J. Rowland, Nuclear Magnetic Resonace in Metals: vol. 9, Pergamon Press, NewYork (1961)[5] T. P. Das, Nuclear Quadrupole Resonance Spectroscopy, Academic Press, New York(1958)[6] N. F. Ramsey, Nuclear Moments, Wiley, New York (1953)[7] G. M. Volkoff, Can. J. Phys., 31:820 (1953)[8] G. M. Volkoff, H.E. Petch, D. W. L. Smellie, Can. J. Phys., 30:270 (1952)[9] M. H. Cohen, F. Reif, Solid State Phys., 5:345 (1957)[10] E. P. Wigner, Group Theory and its application to the Quantum Mechanics of AtomicSpectra, Academic Press, p.167 (1959)[11] H. Stauss, J. Phys. Chem., 40:1988 (1964)[12] G. C. Carter, L. H. Bennett, D. J. Kahan , Metallic Shifts in NMR: A Review of theTheory and Comprehensive Critical Data Compilation of Metallic Materials: part I,Pergamon Press, P. 64[13] F. A. Rushworth, Theory of Magnetic Resonance, London, Gordon 86 Breach SciencePub., (1973)[14] A. C. Chapman, P. Rhodes, E. F. W. Seymour, Proc. Phys. Soc. B, 70:345 (1951)[15] R. Bersohn, J. Phys. Chem., 20:1505 (1956)[16] S. N. Sharma, Nuclear Magnetic Resonance in Single Crystals of Tin and Cadmium,U.B.C. Thesis (1966)[17] J. Winter, Magnetic Resonance in Metals, Claredon Press, Oxford (1971)82Bibliography^ 83[18] R. T. Schumacher, Introduction to Magnetic Resonance: Principles and Applica-tions, Modern Physics Monograph Series, New York (1970)[19] M. I. Valic, A Nuclear Magnetic Resonance Study of Single Crystals of GalliumMetal, U.B.C. Thesis (1970)

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