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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 UNhAL3 By Michael W. Gardner B. Sc. Hons. (Physics) Simon Fraser University, 1991  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTERS OF SCIENCE  in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF PHYSICS  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA October 1993 © Michael W. Gardner, 1993  In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission.  Department of Physics The University of British Columbia 6224 Agricultural Road Vancouver, B.C., Canada V6T 1Z1  Date:  oc-T. Cl/ lqqa  Abstract  The theory for the combined effects of both a quadrupolar interaction and a Knight shift has been examined for the case of non-axial symmetry. A detailed derivation is given, correct to second order in perturbation theory, for the case where the quadrupole interaction is significantly smaller than the Zeeman interaction. These results have been applied to the study of the Al' nuclear magnetic resonance spectrum of a single crystal of UNi2A/3, measured using a steady state technique. A value for the asymmetry of the electric field gradient has also been determined for this material. In addition, the temperature dependence of the Knight shift was observed over the range of 6 to 300K. No significant temperature dependence was found over this range.  11  Table of Contents  Abstract Table of Contents^  iii  List of Figures^  v  List of Tables^  vii  Acknowledgements^  viii  1  Introduction  1  2  Knight Shift  4  3  2.1  Isotropic Knight Shift ^  2.2  Anisotropic Knight Shift ^  10  2.3  Core Polarization ^  15  2.4  Orbital Spin Paramagnetism ^  16  5  Quadrupole Effects  17  Perturbation Treatment of Quadrupole Hamiltonian ^  21  3.1.1^First Order Perturbation ^  23  3.1.2^Second Order Perturbation ^  23  3.2  Quadrupole Frequency Shifts ^  24  3.3  Angular Dependency of the Quadrupole Moment  3.1  iii  ^  26  4 Combined Quadrupole-Knight Shift Effects^  37  4.1 Resonance Shift Effects From Inequivalent Sites. . . . . . . . ^.^40 5 Experimental Considerations^  49  5.1 General Method ^  49  5.2 Resonance Absorption and Dispersion Effects ^  50  5.3 Skin Depth Effects ^  52  5.4 Magnetic Field Calibration ^  53  5.5 Basic Experimental Procedure ..... . ....... . . ^57 5.6 Experimental Determination of Knight and Quadrupole Shifts ^ 60  6 Experimental Setup^  64  6.1 Apparatus and Experimental Setup ^  64  6.2 Sample Orientation ^  68  7 Experimental Data and Conclusion ^  71  7.1 Experimental Data ^  71  7.2 Experimental Results ^  73  7.3 Some Comments ^  75  7.4 Knight Shift Temperature Dependence ^  78  7.5 Conclusion ^  78  A Wigner Matrix Elements^  80  Bibliography^  82  iv  List of Figures  2.1  Knight Shift vs. Magnetic Field Angle 0, 0 = 0°. . . . . .^.  14  2.2  Knight Shift vs. Magnetic Field Angle 0, 0 = 90° ^  15  3.1  (5/2 4-* 3/2), (-3/2 4-* -5/2) transition. ^........^.  32  3.2  (3/2 4-* -1/2, -1/2 4--> -3/2) transition. ^  33  3.3  (1/2 4-* -1/2) transition ^  33  3.4  Quadrupole Spectra for 0 - 0°  3.5  (5/2 4-* 3/2), (-3/2 4-> -5/2) transition. ^  34  3.6  (3/2 4-* 1/2), (-1/2 4-* -3/2) transition. ^  35  3.7  (1/2 4-* -1/2) transition ^  35  3.8  Quadrupole spectra for 0 = 90°.^.^.^.^........^.^.^.^.^.^. ^  36  4.1  Spectra of Combined Quadrupole and Knight shifts, 0 - 0°  ^  38  4.2  (1/2 4-* -1/2) line of quadrupole-Knight shifts, 0 = 0°  ^  39  4.3  (1/2 4-* -1/2) line of quadrupole-Knight shifts, 0 = 90°  ^  39  4.4  Spectra Of Combined Quadrupole and Knight Shifts, 0 = 90° ^  40  4.5  UNi2A13 Atomic Crystal Structure ^  41  4.6  C-Axis View Of UNi2A13 Crystal^  41  4.7  Inequivalent Sites In Magnetic Field Presence ^  42  4.8  (5/2 4-* 3/2) transition, 0' = 10° ^  43  4.9  (-3/2  4-*  ^  34  -5/2) transition, 01= 10° ^  44  4.10 (3/2 4-* 1/2), (-1/2 4-* -3/2) transitions, 0' = 10° ^  44  4.11 (1/2 4-* -1/2) transition, 0' = 10° ^  44  4.12 Spectra Of Quadrupole-Knight Shift From Inequivalent Sites ^ 45 4.13 C-Axis View Of Crystal With Field Direction Of 100^ 45 4.14 C-Axis View Of Crystal With Field Direction Of 30° ^ 46 4.15 (5/2 4-* 3/2), (-3/2 4--* —5/2) transitions, O' = 30°. .^ 47 4.16 (3/2 4-* 1/2), (-1/2 4-* —3/2) transitions, 0' = 30°  ^47  4.17 (1/2 4-* —1/2) transition, O' = 30°  ^47  5.1 Theoretical Absorption and Dispersion curves ........ . ^50 5.2 Derivative Of Absorption Curve ^  51  5.3 Derivative Of Dispersion Curve ^. .......... .^51 5.4 Mixed Absorption and Dispersion Curve ^  52  5.5 Aluminum Reference Signal for 10.602MHz ^  55  5.6 Aluminum Reference Signal for 10.675MHz ^  55  5.7 Signal From (-1/2 (--* —3/2) Line For 9852kHz ^ 57 5.8 Resonance Line and Fitting Line Before Fitting Procedure ^ 58 5.9 Resonance Line and Fitting Line After Fitting Procedure ^ 59 6.1 Schematic of Apparatus and Equipment used in the Experiment ^ 65 6.2 Diagram of the Tuned Circuit . . .......... . . .^. 6.3 Sketch of Crystal's Physical Appearance ^ 6.4 (3/2 4-* 1/2) transition, 0/ = 100^  ^66 68 69  7.1 Placement of x, y and z Axes with respect to the Crystal Structure . . . 76  vi  List of Tables  7.1 Experimental Shifts with Magnetic Field in Basal Plane. . . ..... . . ^71 7.2 Experimental Shifts with Magnetic Field Along C-Axis ...... . ^.^73  vii  Acknowledgements  I wish to express my sincere thanks to Dr. D. Li. Williams for the patient guidance and invaluable instruction he provided concerning the theoretical aspects of this work. I also wish to thank him for his generous financial support. I am also indebted to Dr. E. Koster for his help in running the spectrometer. His technical 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 department for making the learning process a challenging and rewarding experience.  viii  Chapter 1  Introduction  Nuclear magnetic resonance has, over many years, proven to be an invaluable tool in numerous research fields, providing a wealth of information which ordinarily could not be readily obtained using other techniques. The study of condensed matter, which is one such field, has benefited greatly from its application with many discoveries being attributed to nmr research. One area of condensed matter physics where this technique has proven to be particularly useful involves the study of metals. This thesis is primarily concerned with this area of study. Part of the appeal of nmr originates from the implicitly elegant and compact theory on which it is based. In its simplest form, the theory centers around the idea that the nuclear magnetic moments of nmr samples interact with an applied magnetic field to produce quantized energy levels separated by an energy AE. Transitions between these levels, which are the basis of all nmr research, are induced by the additional application of 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 local magnetic fields which are present in the vicinity of atomic nuclei. These local fields, which originate from the surrounding electrons and neighbouring nuclei, essentially alter the strength of the local magnetic field experienced by the nuclei and produce a resultant shift in the the nmr frequency. This effect is observed in metals, where the metallic conduction electrons produce a local field that is noticeable in the form of frequency shifts, commonly termed Knight shifts. In metals that have a crystal symmetry that is 1  Chapter 1. Introduction^  2  lower than cubic, these shifts also exhibit an orientational dependency with respect to the applied magnetic field and are called anisotropic Knight shifts. In contrast, the shifts of metals with crystal cubic symmetry are not affected by the magnetic field orientation and 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 of the interaction between the nuclear quadrupole moment and the electric field gradient at the nuclear site. Its presence is noticeable in the form of significant frequency shifts which are the result of the reorientation of the nuclei relative to the applied magnetic field against local electric field forces. Although both the Knight and quadrupole effects can be observed using powdered crystalline samples, the orientational dependencies of their shifts can only be properly observed using single crystals. This is principally due to the fact that the randomly orientated crystallites of a metallic powdered sample tend to destroy any anisotropy which may be inherently present in the metal. Beginning with a derivation of the isotropic and anisotropic Knight shifts, this thesis provides the reader with a theoretical background so that these effects may be better understood from an experimental point of view. Initially, the Knight shift effect is derived in chapter 2 from an interaction Hamiltonian which depicts the interaction between a nucleus and a conduction electron. This is followed by a thorough derivation of the second order quadrupole effect in chapter 3. Many papers that deal with this effect either avoid the explicit second order derivation altogether, or simply quote the final result while referring the reader to another source for the actual work. As far as the author knows, this thesis provides the only real step-by-step derivation of the angular dependent second order quadrupole effect. The experimental part of this thesis involves the application of these equations to the Al n resonance spectrum in a single crystal of  Chapter 1. Introduction^  3  the intermetallic compound UNi2A13, which exhibits a magnetic transition at 4.5K. The study of this transition is beyond the scope of this thesis and we will confine our interest to determining the relevant parameters in the non-magnetic state. Since the Knight and quadrupole shifts coexist in this material, it is necessary to combine their effects into one equation. This is done in chapter 4. In addition, because the crystal structure contains three Al sites, a characteristic line splitting occurs in general. Several plots of this effect are also provided in this chapter. The basic experimental procedure, along with the method used to determine the Knight and quadrupole shifts are given in chapter 5. In chapter 6 a review of the apparatus is given along with the procedure used to orient the crystal, while in chapter 7 the experimental results are quoted.  Chapter 2  Knight Shift  The Knight shift which is present to some extent in all metals is largely a product of the interaction between nuclear moments and conduction electrons near the Fermi surface. Normally, in the absence of a magnetic field, the unpaired conduction electrons of a metal have no preferential orientation; they simply move rapidly from atom to atom without effecting any of the metallic nuclei. This is not the case, however, when a static magnetic field is present. In this instance the electron spins, which possess a Pauli paramagnetic spin susceptibility xi,' , become polarized inducing a net magnetic field at the nucleus in 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 nucleus and a conduction electron in a metal may be depicted by the interaction Hamiltonian [1] —77,he f • L nen — ^  2-yrtftohr^371(§ r.)^167r +^-ynyflhi.• §8(7-1)^(2.1) r3^r3^r 2  where -yn is the nuclear "gyromagnetic ratio", yo is the Bohr magneton, 71 is the radius vector to the position of the electron with the nucleus at the origin, S and f are the respective electron and nuclear spins and L is the electron orbital momentum. The first 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 such an interaction is usually quenched in metals. This restriction may be lifted later on if necessary. The second term is the source of the anisotropic Knight shift where the spin 4  Chapter 2. Knight Shift^  5  dipolar interaction between the electron and nuclear spins creates an angular dependency in 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 shift originates with the third term. Here it is generally assumed that the s-wave functions describe the major part of the conduction electron behaviour. We will now proceed to derive the actual terms for the isotropic and anisotropic Knight shifts in the following sections. In order to accomplish this, however, we must first assume for simplicity that the electrons are only weakly interacting. Such an assumption, which applies to low energy processes has been theoretically justified by Bohm and Pines[2]. Another useful approximation known as the adiabatic approximation may also be applied in this case. In this instance, the electronic and nuclear motions may be separated with the nuclei being treated as stationary lattice points. The result is a complete wave function that is a product of many separate wave functions from each electron and nucleus.  2.1 Isotropic Knight Shift Since a large number of particles reside within a metal we will focus our attention on the interaction energy of a single nucleus and, using the third term of Eq.(2.1) along with the one-electron description of conduction, sum over all the individual electrons: 167i  E •^§ia(0  H(ein) =^ ,17211,13n 2=1  where j represents the  §th  (2.2)  nucleus and i represents the ith electron.  Our main goal will be to find the expectation value of Eq.(2.2) with regards to the many 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 electron  ^  Chapter 2. Knight Shift^  6  that 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) possessing the periodicity of the metallic lattice. Since a spin dependency is also required of the electron wave function, it is necessary to factor in a spin function Os as well so that the Bloch 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 Bloch functions, properly antisymmetrized to satisfy the Pauli exclusion principle. Using 0, and assuming that the electrons are quantized along the z-direction by an external static magnetic field we now find that the expectation value of Eq.(2.2) is:  1637r  ^7niti3h1; (Oeln(ej.)10e) =  • I 0:^§ibVi)Oed're^(2.5)  Or  \^167r^  (0e Iii(e3n) 10e) =^7nittOhiz3  2  >m lok,s(o)1 h(k, s)  (2.6)  g,s  10r,s (0  2 )  is the probability of the electron with wave vector -k and spin s being in the  vicinity of the  ith  nucleus and rns is the eigenvalue of the operator gs. h(k., s) is the  probability of occupancy by an electron of the energy level E(k,.^). We may average over this electron occupancy number for a specific temperature so that Eq.(2.6) acquires the form: (7ibe 7.43n) e )^1637r  7,,,aohIz3 Ems rc,,  2^-■  10(o) f (k, s)  (2.7)  where f .^) is the Fermi function:  f (E(k , s)) = f (E) =  1 exp[(E — EF)/ kB7-] + 1  (2.8)  ^  Chapter 2. Knight Shift^  7  E 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 states which are m, = 1/2 or —1/2 (0e17-4371)10e) =  167  1 -rnitohiz3  3^k. 2  10/41(0)12 f(k, 1/2) —  (0)  ^(1.c, —1/2). (2.9)  Again assuming negligible spin-orbit interaction 2^  03)12 = 10(C))1  we actually have: 7r^1 (0e In(e3n)10e) = 163 7nitonlz3 E - "1/, ( 0 ) 2 f(, 1/2) — f(k, 1/2)}.^(2.10) k. 2  k, which is now required may be found by using  An approximation of the sum over  the density of states function which in its most general form is: 1^d3k^1 Ern^F(k) =F(k) —  v_,00  E F(k).^(2.11)  v  The practical application of this equation to finite yet microscopically large systems requires the assumption that 1/V Ek• F( /0) differs insignificantly from its infinite volume counterpart. Using this relation for Eq.(2.10) produces: ell.din)^e\^167T 7ntiohiz3 21  /  3  I  d8:k31,  (u ) { f(, 1/2) — f(fc, —1/2)1 .^(2.12)  where V is defined to be a unit volume. Knowing the energies of the two electron states which are: E(1/2) = E(k)+H0 ^E(-1/2)  = E() — jHo  ^  ^  (2.13) (2.14)  Chapter 2. Knight Shift^  8  with ±,a0H0 being the Zeeman energies, and doing a Taylor series expansion of the Fermi functions F(ic., 1/2) and f(k, —1/2) about E(k) gives us:  d3k  (0e1H22)10e) = 11 67 70-L2h-iz-Ho 3^13^3^873  0,-; 0 (  2 of )  aE  (2.15)  where higher order terms have been neglected. Since  d3k = dEds vE(k)1 where ds is an infinitesimal surface area of constant energy in k-space, we will get: e 7i(e.in)^  16;  7n4hizjHo — 8c1R  33 -  lorco)12 AvElk.)1  (2.16)  by performing the integral over dE, where a f/aE has an action similar to that of a delta function. Now defining an average of 10k.I2 over the Fermi surface to be:  ook(0)12)E,= ;.+;s3(  ok(o) 2 AVE(k)1) 3 /I  4— : ^AVE(/*C)1)  (2.17)  where the density of states at the Fermi surface is:  ds ^1 N(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  (2.19)  /EF  so that Eq.(2.16) now takes the form:  k(0)12)^. Nit2h/zjHON(EF) (10 83^#^ 1)el7-1(e3n)11Pe)^—±r EF  where N(EF) is the density of states.  (2.20)  9  Chapter 2. Knight Shift^  The spin susceptibility of conduction electrons, which is essentially temperature independent, 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 susceptibility 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 appropriate Hamiltonian 1-lz is: (Cbeinzilke) = —77ihiz2H0^  (2.22)  it becomes immediately apparent that the bracketed quantity in Eq.(2.21) is in reality an extra magnetic field produced by the s-conduction electrons at the nucleus in the form of:  AH  =^( 10,0)1  2  (2.23)  ) 3^EF  We may now define the dimensionless quantity: AH  87r  (2.24)  Ho = —3 XP (107,(0)  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 several distinct properties when observed in ordinary metals. One such property is its apparent temperature independence which, except for a few cases, Cd and the intermetallic compounds AuGa2 and BiIn, is common in most metals. This property is a product of the largely temperature independent terms (101-:(0)12)  EF  and xi, that make up the above  equation for the Knight shift. Another property, which has also been experimentally  Chapter 2. Knight Shift^  10  observed and which is another feature of Eq.(2.24) is the independence of the fractional shift AH/Ho from the static magnetic field value. Two other properties which are also present experimentally and in the Knight shift relation include the predominantly positive 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) are inherently positive, while the nuclear charge dependence is a product of the (10-12 k  EF  term which is strongly affected by the nuclear size and the Z dependence of 0. 2.2 Anisotropic Knight Shift In the absence of cubic symmetry, the spin dipolar interaction between the nuclear and electronic spins, which is represented by the second term of Eq.(2.1), produces an angular dependent line shift which is readily noticeable in single crystal studies [1, 3] . This interaction is present also in powder studies, but not in the form of a shift. In this case it manifests itself in the form of an added structure and increased width in the nmr line due to the random orientation of the crystallites. In order to derive a quantitative expression for the anisotropic Knight shift, it is first necessary to express the shift in terms of a tensor:  k =1Z so 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 where fio is a magnetic field vector,  ^  (2.25)  1 is the angular momentum vector, 1 is the tensor: 1= ii + +  Chapter 2. Knight Shift^  11  and 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.  -4^4-4^4-4  In a sense, the term Ho • (1 4- k) is actually an effective magnetic field  Hef f  so that  Eq.(2.25) may be expressed simply as a Zeeman Hamiltonian with a corresponding magnetic field: (0e17(3)10e) = —Nnileff 1./^  (2.26)  It is now necessary to determine the orientational dependency of the Knight shift 4-4  tensor X. In order to achieve this we must first define a set of orthogonal axes, called the principal axes, where the tensor IC, which is symmetric, has zero off-diagonal elements. Specifically, in the principal axis system we have: 4-4 k= ip ip + JP 1C JP + kp^kp Now in the general case the magnetic field is in an arbitrary direction with respect to the principal axis system. In the laboratory system fio is defined to be along the ziab-axis, where Ho = Hok'hib, while in the principle axis system we have that:  ii  = HoxpiP HOypJP HOzpk.P  From 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^ zzp , 2 II — ozp 2—  (2.27)  The direction cosines of the magnetic field Ho, in terms of the principle axis system may be defined as: Osp  = HO zp /HO^ii3yp = 110yp / HO^= HOZ p I HO  Chapter 2. Knight Shift^  12  Using these quantities for Eq.(2.27) produces the result:  He ff = (1 + IC= p)2 131, + + 1CYYP)2 13L + + ICZZp)2 13.?),110  Realizing that the Knight shift components are very small compared to 1, this equation may 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 the energy difference equations: hv AE^— Em =  'YnhHeff  so that: V  2 = Vo (1 + /CsP0xp^yp^ZP + k 0 2 + kz 02Zp )  where vo = 7H0/27r and I z3 = m. The direction cosines for the magnetic field may be converted to the convenient spherical coordinate form with the corresponding angles 0 and 0: i3xp = sin 0 cos 0 Ovp  sin 0 sin 0  = cos 0 so that we get: v vo (1 + /Cssp sin2 0 cos2 + kyyp sin2 0 sin2 + kzzi, cos2 0)  Chapter 2. Knight Shift  13  This equation will now be written in a simpler and more useful form so that it will be notationally consistent with the first order terms of the quadrupole frequency shift in the following chapter. V = V0 + VoiCTxp Sill  2  0  (1 + cos(20))  (1  vokyyp sin2 0 2^  —  cos(20))  2  IC„, cos2 0  After some algebraic manipulation we get a final form of: (ksx  + kyyp +  ICZ2p)^(21CZZ  110 + VO^P^ V^  1)13^P  3^ vo  -^KXXp) "P  3  (3 c o s2 0 — 1) 2  (Kyyp --)C.xp) ^• 2 sin 0 cos(20)  2  (2.29)  The following quantities are now defined: 1(k_ /Cis° — 3  kX  p  kyyp +  ZZp)  where /Cis° is the isotropic Knight shift and, =  2 IC yyp — kyyp kxsp  3  and )C2 ==)Cyyp  --)Cxxp,  so that Eq.(2.29) acquires the form: voki  2 2 0 1) voK2 2^ sin 0 cos(20)  V = VO VOkiso^^2 (3 cos  ( 2. 3 0 )  Experimentally, Knight shift values are generally given in terms of nonmetallic reference compounds in measurements of high accuracy. Although they do not exhibit Knight shifts, these compounds, which are usually metallic salts, do have what is termed, chemical shifts which usually amount to no more than a few percent of any given Knight shift value. These shifts are primarily the result of diamagnetic contributions from valence and  Chapter 2. Knight Shift ^  14  closed 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 molecular spin or orbital moments, then it is not suitable for use as a reference compound. It is generally assumed that, in addition to the Knight shift, the same degree of chemical shifting 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 specify the reference compound when a Knight shift value is quoted in order to account for any diamagnetic contributions. As a result we will define the resonance frequency with respect to a general reference compound to be v„f, , so that Eq.(2.30) now takes the final form:  k 2Pref 1Pref 1C^ 1) sin2 0 cos(2) (3 cos2 0^ V -= (1 + kiso)V^ ref^ --r2^ 2  (2.31)  The angular dependency of this equation becomes apparent in the following figure where 0 varies from 00 to 180°, q = 00, v„f = 10MHz and the Knight shifts have been assigned the values ktso = 0.002, 1Ci = 0.001 and /C2 = 0.002 which are typical of the 35 — N x30 25 -120 >,15 0 (010 w  0  25^50^75^100 125 150 175 Field Angle 0 (Degrees)  Figure 2.1: Knight Shift vs. Magnetic Field Angle 0, cb = 00 .  Chapter 2. Knight Shift^  15  systems we are studying. In Fig.(2.2) 4 is set to 900.  35 — N x30 25 -1 20 ), 15 0 w 10 5 0  25^50^75^100 125 150 175  Field 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 same degree that it does when cb = 0° as in fig.(2.1).  2.3 Core Polarization One contribution to the Knight shift which has been ignored so far is the core polarization effect. This phenomenon, which is common in the transition metals, is the result of the inner core electron shells being distorted by the d-type conduction electrons. In the absence of any d-type interaction, the inner core electrons are normally perfectly paired with no resultant hyperfine field. Since d-type conduction electrons have a high density of states at the Fermi surface, they have a large susceptibility and as a result, have a large effect on the inner core electrons and conduction electrons. In the presence of a magnetic field, they effectively polarize the inner electrons causing an imbalance in the pairing scheme. Since there is a contact interaction between the inner s-electrons and the nucleus, this imbalance causes a magnetic field at the nuclear site. The result  Chapter 2. Knight Shift^  16  is a contribution to the Knight shift known as the core polarization shift Kcp(d). In this particular experiment, which measures the spectrum of Al27, this effect will not be significant.  2.4 Orbital Spin Paramagnetism Originally, it was assumed that the electron orbital angular momentum was quenched with the result that we could omit the first term of Eq.(2.1). While this may hold for most metals, it is not true for those metals with non-s band conduction electrons, such as in the case of transition metals. In this case, the first term of Eq.(2.1) can make a contribution to the Knight shift, but only to second order where the interaction between the magnetic field and the orbital momentum can distort the occupied states. This distortion is caused by the mixing of unoccupied states from above the Fermi level into occupied ground states that have a quenched orbital angular momentum. By acting on the first term of Eq.(2.1), these distorted states produce a non-zero contribution K to the Knight shift. Again this effect is unlikely to be significant in our experiment.  Chapter 3  Quadrupole Effects  Nuclear quadrupole moments and their interactions with local electric field gradients are an important and sensitive aid in the study of solids. These interactions which have a dependency on nuclear orientation can be readily examined using ordinary nmr techniques. Two areas of study which utilize quadrupole effects involve the use of either a high or low magnetic field. In the high field case the nuclear magnetic moment energy is assumed to be significantly greater than the interaction energy produced by quadrupole coupling. In this case the coupling simply manifests itself through the splitting of the central resonance line into several components [1] . In contrast the low field case involves quadrupole effects that are large enough to become the dominant factor in the spin orientation of a nucleus [5] . This present study of UNi2A/3 will be concerned with only high 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 to be a charge distribution with radial dependency p(i) in an electrostatic potential v(r.). The resulting electrostatic interaction energy is represented by  E  = 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 its variation over the nuclear volume is small 17  18  Chapter 3. Quadrupole Effects^  avol v(0 v(o) + E a xi  1^a2v(71  + Ex,x ^  2^3 aXiaX i f=0^ "j^  (3.2) 7;•= 0  Defining,  vi =  av(r.) OXi  = 97=0  a 2 v 77,) axiaxj 9=- 0  we get for the interaction energy  E = v(0) f p(71)d3r^  vJ  1 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 external field. Having no orientational dependency it does not contribute to the nmr signal. The second 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 the integrand in the second term is antisymmetric about the origin resulting in a vanishing integral. The third term represents the electric quadrupole interaction. Any following terms may be neglected since they either vanish for parity reasons or are outside the range of experimental detectability. We may now define an electric quadrupole moment tensor which is not only symmetric but also has a vanishing trace (v2Q = 0) and consequently only five independent components.  Qii = f[3xixj —  8i.i1712]P(71d3r  ^  (3.4)  In terms of this tensor the quadrupole energy equation (3.3) becomes: EQ^Evi,[c2i,  " i,a  + si,  'For more details see P.23 of "Nuclear-Moments"  f  1712p(f)er]^ (3.5)  Chapter 3. Quadrupole Effects^  19  The potential v(f-) satisfies Laplace's equation v2v = 0 at the origin causing the second term to vanish with the result that:  1 EQ =  E vijQii  (3.6)  where vij is also a symmetric , traceless, 2nd rank tensor. Laplace's equation applies to the potential v(71) basically because any electronic charge which penetrates the nucleus originates from the spherically symmetric s-electron states which do not provide any quadrupolar coupling. Now to get the Hamiltonian operator of the quadrupole interaction we may simply replace 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 the nucleus. With this expression the classical operator becomes:  Q ij e E(3xipxip — 6ii Mp2)^  (3.8)  producing the quadrupole Hamiltonian 7c2: 1 x--N 7-(Q =^Vi3Qi3 v  (3.9)  Since our main concern will be with high field effects we will require the matrix elements of the quadrupole operator for the perturbation calculations. Before this is done however, this operator must first be altered to an equivalent angular momentum form using the Wigner-Eckart theorem which states that the corresponding matrix elements of all second-rank, symmetric, traceless tensors are proportional[1]. The resulting form using this theorem for the quadrupole operator is: (  I, mle E (3 x ip x.ip 8i .i 1 71p2 )1 I, in ) = C (I 771q^+ j h) —^.1211 , in)^(3.10)  Chapter 3. Quadrupole Effects^  20  where the constant C is independent of m or m' and where /2 = J + 1 + I. C is determined by requiring the spin to quantized along the z-direction so that it is equal to a 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 when a nucleus is in a state of definite angular momentum its reorientation energy depends only on the difference between the charge parallel and transverse to the z-direction. In a classical sense, the nucleus has a cylindrical charge distribution about its spin which is responsible for this orientational dependency. It is for this reason that Q = 0 if i j and  Qss =  Q. Since Qxx+ Qyy Qzz = Owe get that Qss =  Qyy =  —Qzz/2 indicating  that 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 proton charge and Q is called the " Nuclear Electric Quadrupole Moment ". Since:  (II13Iz2 — 12111) = 1(21—i)^  (3.12)  C is: eQ C= ^ 1(21 — 1)  (3.13)  resulting in an angular momentum space representation of Eq. (3.9)  eQ^3 E vi,^+^-^61(21 -1)^2 -  ^  (3.14)  It will be more convenient if we express Eq.(3.14) in terms of the raising and lowering operators:  = L+ 1y^= -  21  Chapter 3. Quadrupole Effects^  so that: eQ = 41(21 — 1) [(3/:—/2)Vo-F(/./++4/z)V_,/__/„)Vk_2FV_2+P_V2] (3.15)  where:  vo^  vzz  -17±1  Vzx Vzv  v±2  _ 1 =^— Vyy)  We may now proceed to treat the quadrupole Hamiltonian with first and second order perturbation theory.  3.1 Perturbation Treatment of Quadrupole Hamiltonian In the absence of any quadrupole interaction a magnetic field will induce equally spaced levels which are characterized by the magnetic energy Hamiltonian:  = —Dio r 1-1  ^  (3.19)  Application of the field along the z-axis defines a coordinate system where this Hamiltonian 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 mechanical description of resonance. An electromagnetic field transverse to H will cause transitions between these levels producing one resonance line. The action of the quadrupole interaction shifts the Zeeman energy levels disproportionally so that they are no longer  Chapter 3. Quadrupole Effects^  22  equally spaced. Effectively, the degeneracy of the transition energy levels is lifted, splitting the magnetic resonance line into its 21 components. We may represent this action by the simple Hamiltonian : 71 == 'Ho +1--(Q^  (3.21)  E = E7(7?) +^+ E,C2)^  (3.22)  with the energies:  where E,(71,-) and EiC?) are the first and second order perturbations on the magnetic energy levels respectively. In general, the expression for the energy perturbation is to second-order:  Em = Et(,?) + (I ,^,^  E  74711  ,TTITHQ1-1,0(i,n1HQII rn)  E°  -  (3.23)  EOn  + higher order terms. where Ei(7?) is the zero-order energy of the mth quantum state (I, m11-(01I, m). It is apparent that the first-order term (/, m11-1Q1/, m) is a diagonal matrix element while the secondorder term is a sum over off-diagonal elements. In order to facilitate the perturbation calculations we must first calculate the individual diagonal and off-diagonal matrix elements using the general transforming property of the raising and lowering operators:  m)^V(1 L I I, m =  —  V( 1  m)(.1 m 1)Irri + 1)  (3.24)  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: (I,^I , m) =-  eQ  41(21 — 1)  ([3m2 — 1(1 + 1)]170  (3.26)  Chapter 3. Quadrupole Effects^  (I, m +117-(QII,m) =  23  eQ (2m + 1)(I m)(/ m 1)VT1^(3.27) 41(21 —1)  (I, m 211 (Q1I, m) = -  eQ 4/(2/— 1)  I + m)(/ m 1)(/ —1)(/ m 2)V±2. (3.28)  Matrix elements for lm — mil > 2 may be omitted since we are only considering second order perturbation. Third order effects have been given due consideration by Volkoff [7, 8]  3.1.1 First Order Perturbation The form of the first order perturbation calculation is readily apparent as it is simply Eq.(3.26). We may define another quantity vQ which represents the strength of the quadrupole interaction so that: VCAVO 0 4) = ^ [m` —^+ 1)/3] 2eQ  (3.29 )  where vQ 3e2Q/21(21 — 1)h. 3.1.2 Second Order Perturbation The second order perturbation calculation of the quadrupole energy is certainly more involved than the first order case. The second order terms of Eq.(3.23) using Eqs.(3.27) and (3.28) become:  E,!)  1(771^217-(Q1702  Eon, — EOrn —2  I ( m + 111-1 Q 1m)1 2 Eom — Eo m+i  or more explicitly:  4_  1(m —11HQ177)12 Em — Eom-i l(rn+211-(Q1m)12 Em — EOrn+2  (3.30)  ^  Chapter 3. Quadrupole Effects^  24  E„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 — —2hvo  1)(I  m  + 414212}  Simplifying 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 Shifts The splitting of the magnetic resonance line by the quadrupole action into its 21 components 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 because they are only weakly allowed and do not contribute greatly to the overall resonance signal. Using Eq.(3.29) we get for the first order transition frequency change:  Vo v(1) — (E(1) — E(1))/h = vQ— (m — 1/2) m^m cq Or  74,,P = vQT0(m — 1/2)^  (3.34)  Chapter 3. Quadrupole Effects^  25  where To = Vo/eq is a dimensionless quantity. The central resonance line, which corresponds to the (-1 ^A) transition, is apparently left unaffected by the quadrupole interaction to first order since Eq.( 3.34) vanishes for m=1/2. This is not the case for the other transitions, however, where the lines called satellites, are shifted and arranged symmetrically on each side of the central line by the first order action. The second- order change in the frequency can be expressed using Eq.(3.31) as: v2 = (E1 — E7T)/ h = ^ Q 117112(24m(m — 1) — 41(1 + 1) + 9) 18voq2e2 1 —11412(12m(m —1) — 41(1 + 1) + 6)1 2 Or  v(2) = ^ {1T112(24m(m — 1) — 41(1 +1) + 9) 12vo 1, (3.35) IT212(12rn(m — 1) — 41(1 + 1) + 6)1 2 = (2/3)11412/q2e2 and 1T212 = (2/3)IV 2,2/q2e2 are dimensionless quantities. I where IT],^ Actually, because of the equalities in Eqs.(3.32) we may define that: IT112 =^= IT+112^111212 = IT-212 = 171E212 ^(3.36) where:  3 eq  eV±2 3 eq  T±1  (3.37)  T±2  (3.38)  The quantities To, T±1 and T±2 have been introduced in order to simplify some of the derivations in the following sections. It is apparent from Eq.(3.35) that the central resonance line is now shifted by the second-order quadrupole interaction. The satellite lines are also shifted in the second  Chapter 3. Quadrupole Effects^  26  order 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 of Eq.(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 be correct 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 future reference. 0 r 1^v 2 Urn= vo vQT0(m — —) + ^ illi12(24m(m —1) — 41(1 +1) + 9) 2^12v0 1 -- 171212 (12m(m — 1) — 4I(I+ 1) + 6)} 2  (3.39)  3.3 Angular Dependency of the Quadrupole Moment  Electric quadrupole effects generally exhibit an angular dependency with respect to the applied magnetic field direction. An understanding of this dependency may be achieved by examining the transforming properties of the principal axes (x',y',z°). These effectively reduce the symmetric tensor -143 to a diagonal form. The result is a tensor with only three components Vx/Ti, Vvy, and Vz,,, which are not independent since V23 is also traceless. As a result, only two parameters are needed to sufficiently define a field gradient in the principal-axis system. For future reference we may define two such useful quantities q and 77 called the field gradient and the asymmetry parameter, where:  eq  VeSi Vy,y,  (3.40)  Vz,z,  An appropriate reorientation of the principal-axis system can be performed so that:  (3.41)  Chapter 3. Quadrupole Effects^  27  where 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 axial symmetry. 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/2 with the result that:  ToP = 0 Tr2,77/v6 (3.42) In general, the magnetic field is applied along an arbitrary direction with respect to the principle z'—axis. We require a transformation that will rotate this principle axis system into one which has a z-axis that is parallel to the magnetic field direction. Such a 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 the principle z'—axis by a. This will then be followed by successive rotations about the new  y and z axes through the respective angles # and 7. Actually, only the first two rotations (a, #) are necessary to specify the transformation in this case since the quadrupole frequencies 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 do the spherical harmonics yr [9]. In other words, they will transform according to the irreducible representation of a second degree rotation group.2 More explicitly, 2  E D(2) ({a713 M)11'12Ttl:' 2See sections II and III of "Solid State Physics" vol. 5 from Cohen & Reif  (3.43)  Chapter 3. Quadrupole Effects  ^  28  where 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 matrix which combines the five equations of (3.42) to produce five more equations Till for the new 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'  (3.4 4)  2^2  In our case k = 2 and since the representation D(k) is 2k + 1 dimensional we get a 5dimensional rotation matrix. The factorials in the denominator restrict the summation over 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 is depicted 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  TH1  =  TY(2)2,_i)TP2^D((2)1,_1)TPi+1,(02),-1)T(3  D((12,)_i)  Tj-1  7)(2,_2)TT  + 1)((22?-1)TT  7,((2)2,0) TP2 +^1,((2)1,0)T131^1)(02?0)T(1)3  1:;112,)0)Tr  D((22?0)Tf  711  =  D((2)2,1)Tf2^D((-2)1,1)7T1^7)(02),1)T(1,  D((i2 ,)i)Tr  1)((22?1)7113  711  =  7:12)2,2)TP2^D((2)1,2)711'3,^1.")(023?2)T  13112 ,)2)T113  1)((22),2)TT  ^  Chapter 3. Quadrupole Effects^  29  or more specifically: 11(2) 77  1-42)^  TH -2  ^ -r^  ,-2  ,Th(2)^111(12)  TH - 1^ '-2,-1^,-1^N/6 7/^2:)(21)3 + v(2) ^TH^v(2) -2,0 v-6^2,0  I/  ,v6  I/ 2:300 + v(2) y ^T111^v(2) 2,1^ .v6 -I-^ 2,1 N/g -  ^111^ -2 2  17  ^v(21^2)(2)  ' V6^ 1/6  Since our main concern will be in finding the orientational dependencies which are present in T0,171112 and 1T212 only three of the five equations, T,, for the new coordinate system are required; this simplification being a direct result of the equalities expressed in Eqs.(3.36). The first three rows of the above matrix will be sufficient for our determination 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-Y  N/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^  30  N/6e22 cos2(0/2) sin2(0/2) / ^ (3.47)  Simplification of the above expressions yields:  TH - e2i1' -2 — ^  4  (2[cos(2a) cos2 cos(2a)] - 4i cos 0 sin(2a)) + 6  ^sin2 /31^(3.48)  TH -1 =^sin 0 [cos 0 cos(2a) - i sin(2a)]-- +16 sin2 /3^(3.49) 16 ToH  sin2 cos(2a)y 3 1 ^ + - co s 2^2^2 -  (3.50)  -  Now 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 take TH, and multiply by its complex conjugate. The resulting expressions are: 3  2 2 ITi-H12 = sin2 /3^ {^ 1/2 - cos2 /3 cos(2a)7/ -I- - cos2 /3 [COS 3 COS (2CX) + sin(2a)]6^2  (3.51)  and icos2(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^) sin /3^y 4^ 16  (3.52)  Eqs.(3.50),(3.51) and (3.52) are now substituted into Eq.(3.39) for To, ITI/12 and In/12 for the quadrupole frequency shift with the result:  r  1 3^2^1^77 sin2 cos(2a)] Y^cos 11' 13 2 +^2 vm^' 2  +  sin 2 /3 [COS 2 13 COS 2 (2a) + sin(2a)]1/ 12v0 { 6  Chapter 3. Quadrupole Effects^  31  3  — cos2 /3 cos(2a)y + — cos2,31 [24m(m — 1) — 4a + 9] 2 cos' cos2(2a) + cos2(2a) cos2 )3) ^Y 2 —1 f (cos2(2a) +^/3 + cos2 /3 si112(2a)) w 2 1^4^4 ^2^ ( (cos(2a) + cos2 ,3 cos(2a)) sin2 /3)66 sin4,3 } [12m(m — 1) — 4a + 6]} (3.53) +^ 4 16 This equation may be simplified and written in the more convenient form: Lim = Vo  -f- VQ (M, — -21)^COS2^ -  21 +  7/ sin2  /3 cos(2a)] 2  vQ2  2  (1 — cos2,3) [{102m(m, — 1) — 181(1 + 1) + 39} cos2 /3 (1 — — ri cos(2a)) 32vo 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} cos2/3 72vo  51  9  m(m — 1) —^+ 1) + 1 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 spherical coordinate angles, 0 and 0. In order to accomplish this, we must recall that a y-axis rotation was executed by the transformation matrix Eq.(3.43) and as a result a may be given 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:  ^  32  Chapter 3. Quadrupole Effects^  The result is a final form for the quadrupole frequency shift in the form of [11]: 1) [3^2^0 (^ Vnt Vo VQ M —^COS V^ 2  +  VQ  32v0  (1 — COS20){{102m(rn  cos(20) ] 2 2  - 1) - 181(1 + 1) + 39} cos2 0 (1 + - 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 0 +^ 72vo 51^9^39 - {-2-m(m - 1) -^+ 1) + — cos2(20)(cos2 0 - 1)21^(3.56) 4 The behaviour of Eq.(3.56) with respect to the angle 0 can be seen in Fig.(3.1) where the following values have been assigned to the various parameters; vo = 10MHz, vQ = 750KHz, q  = 0.500, I = 5/2 and q = 00. — N  1500 1000  w  500  tfl^  w  0  -500  0-1000 r4 -1500 ^ 80 ^20^40^60 Field Angle 0 (Degrees)  ^Figure 3.1: (5/2^3/2), (-3/2 4-> -5/2) transition.  33  Chapter 3. Quadrupole Effects^  The first three values, while somewhat arbitrary, have been chosen specifically because they are similar to the values found and used in this experimental study of UNi2A/3. In addition, since we are mainly concerned with observing the A/27 spectra I has been set equal to the aluminum nuclear spin of 5/2. The top line at 0 = 00 in Fig.(3.1) corresponds to 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. 800 600 400 4.4  200  m^0  zu -200 -400 a)  r. -600 -800  20^40^60  ^  80  Field Angle 0 (Degrees)  Figure 3.2: (3/2 4-* —1/2, —1/2 4-* —3/2) transition.  20^40^60  ^  80  Field Angle 0 (Degrees)  Figure 3.3: (1/2 4-* —1/2) transition.  34  Chapter 3. Quadrupole Effects^ 1500 1000 4_1^  w  -H  500 0  -500 w -1000  r=4  -1500  20 -  60  8-0  Field 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) transitions can be seen in Fig.(3.2) while the single line for the (-1/2  4-> 1/2)  transition is shown in  Fig.(3.3). All five spectra have been combined into a single plot in Fig.(3.4) so that the lines 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.  1500 1000 500 •  0 -500  a) -1000 -1500  20^40^60  ^  80  Field Angle 0 (Degrees)  Figure 3.5: (5/2 4-* 3/2), (-3/2 4-* —5/2) transition.  Chapter 3. Quadrupole Effects^  35  800 N x  600 400  4-1 4--1  -,1  200  4 M 0  >I O -200 z a) • -400 tr a) • -600 rx.. -800  20^40^60^80 Field Angle 0 (Degrees)  Figure 3.6: (3/2 4-4 1/2), (-1/2 4-4 —3/2) transition.  '-'  N  x x —  40  .0 20  4-1 - c-1  4 M >1^0 0  a) ty, -20 w rr. -40  20^40^60  ^  80  Field 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 purpose of making a comparison with the Fig.(3.4) spectra. Such a comparison indicates that an increase in 0 has the effect of decreasing the degree of quadrupole shifting that is present when 0 approaches 900.  Chapter 3. Quadrupole Effects  36  Chapter 4  Combined Quadrupole-Knight Shift Effects  The relations for the Knight shift and quadrupole effects which were derived in the last two chapters may now, in a simple way, be combined to form a more general expression for the frequency shift. In the following chapter this expression is used extensively, in conjunction with the experimental aluminum spectra, to determine the values of the asymmetry parameter, the quadrupole frequency and the Knight shifts of the UNi2A/3 single crystal sample. Using Eqs.(2.31) and(3.56) the overall frequency shift from the Knight and quadrupole effects may be expressed together as 1: ][^  lim = Lk'  kivrefl  + I-2142^-^+  2  J  (3 cos 2 0 — 1) 1^1 -- [C2vref (m — — ) vQii] sin2 0 cos(20)  2  2  2  vr)  ^ (1 - COS20) {{102711(171 - 1) - 181(1 + 1) + 39} cos2 0 (1 + — 71 cos(20)) 32 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 cos2  72vo  — {-251m(m — 1) —^+ 1) + 39 7 } 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 that reference for a comparison.  37  Chapter 4. Combined Quadrupole-Knight Shift Effects ^  — m  38  1500 1000  w .0 0  500 0 -500  w -1000 -1500  20^40^60  ^  80  Field 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 defined in the previous chapters. The one exception is in the definition of vo for the quadrupole terms 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 vo and consequently change the size of the quadrupole shift by a small degree. To be strictly correct we should also include the anisotropic Knight shift in vo, however, since we are dealing with second order effects it is reasonable to omit any such angular dependence in this term. The behaviour of Eq.(4.1) may be seen in Fig.(4.1) where it has been plotted with 0 = 00 for the five aluminum transitions which includes the four satellite lines (5/2 4-* 3/2), (3/2  4—)'  1/2), (-1/2 4-* —3/2), (-3/2 4-* —5/2) and the central  resonance line (1/2 4-* —1/2). The constants 7, vQ, //ref, and 1C,s„, ki, k2 have been assigned the same values that they were assigned in the previous chapters for the Knight and 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 of  •• Chapter 4. Combined Quadrupole-Knight Shift Effects ^  39  the (1/2 4-4 —1/2) transition, are not significantly altered by the presence of a Knight shift. 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 is more obvious which is evident if the spectrum of Eq.(4.1) is plotted again for the central line (1/2 —1/2) and compared to (1/2 —1/2) line of Fig.(3.3). ^ 40 • 4.)  20 0  (/)  -20  a) -40 o' (1) -60 -80  20^40  ^  60  ^  80  Field 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 the N  60  4-)^40 4-4 .c co >, 20  cr^0 -20  20^40^60  80  Field Angle 0 (Degrees)  Figure 4.3: (1/2 4-4 —1/2) line of quadrupole-Knight shifts, 0 = 900.  Chapter 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 lines from Eq.(4.1) for 0 = 90° are plotted where it will be noticed again that the satellite lines are not significantly altered by the Knight shift. This can be verified by a comparison with Fig.(3.8).  15 00 1000 44 .0  500 0  -500  rs, a) —1000  rx.  -1500  20^40^60  ^  80  Field 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 are noticeably more complicated than the actual theoretical spectra of Eq.(4.1). This is not a result of any problem with the equation but rather the product of the aluminum site positioning in the sample crystal structure. This effect may be understood by examining the hexagonal UNi2A13 crystal structure which is depicted in Fig.(4.5). The distance between the individual uranium atoms in the crystal basal plane is a = b = 5.207A and c, which is the perpendicular distance between these planes, is 4.018A.  Chapter 4. Combined Quadrupole-Knight Shift Effects ^  •  Uranium  ^  OAluminum ()Nickel  Figure 4.5: UNi2A13 Atomic Crystal Structure.  •  Uranium^• Aluminum 0 Nickel  Figure 4.6: C-Axis View Of UNi2A6 Crystal.  41  Chapter 4. Combined Quadrupole-Knight Shift Effects ^  42  Direction 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 viewed from the top along the c-axis. In Fig.(4.7) the three numbered aluminum sites, which are responsible for the more complicated spectral structure are pictured along with a magnetic field component parallel with the xl-axis. These sites, which are commonly termed inequivalent sites, actually produce a more complicated spectrum by splitting each transition line into three separate lines. This may be understood by noting in Fig.(4.7) that each site has a definite orientation with respect to the magnetic field direction. Normally, in the absence of a field the three numbered aluminum atoms would be indistinguishable from one another which can be verified by a simple 60° rotation of the crystal about the c-axis. By the introduction of a magnetic field, this symmetry is effectively destroyed, introducing an angular dependence in the three aluminum site  Chapter 4. Combined Quadrupole-Knight Shift Effects ^  43  1500 1000 500 0 -500 w -1000 -1500  20^40^60  ^  80  Field Angle 0 (Degrees)  Figure 4.8: (5/2^3/2) transition, 0' = 100 positions. For example, if the aluminum atom at site-2 in Fig.(4.7) is to produce the same spectrum 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 nucleus at each site will produce its own spectral shift which is dependent on the relative magnetic field direction in the basal plane. This effect can be easily reproduced from Eq.(4.1) by simply 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„f  and 1CO3 1C1, 1C2 in the last section are used here. We get similar results for the other satellite 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 plotted in 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.  Chapter 4. Combined Quadrupole-Knight Shift Effects ^ 1500 1000  500  44  -500 a) tr -1000 -1500  20^40^60^80 Field Angle  0  (Degrees)  Figure 4.9: (-3/2^—5/2) transition, 0, 100 Ts; 750  500 1_4  4-4 250 0 49 -250  w -500 -750  20^40^60 Field Angle  0  ^  80  (Degrees)  Figure 4.10: (3/2 4-+ 1/2), (-1/2 4-4 —3/2) transitions, 0' = 100 — N  60 40  >, -20  (1)., -60 -80  20^40^60^80 Field Angle  0  (Degrees)  Figure 4.11: (1/2 4-4 —1/2) transition, 0' = 100  44  Chapter 4. Combined Quadrupole-Knight Shift Effects  ^  45  — 1500 m x • 1000 — 4J^500 w —1 U)^0 >1 o • -500 w V w -1000 ri.4  -1500  20^40^60  ^  80  Field Angle 0 (Degrees)  Figure 4.12: Spectra Of Quadrupole-Knight Shift From Inequivalent Sites  Direction of Magnetic Field  Figure 4.13: C-Axis View Of Crystal With Field Direction Of 100 It will be noticed that for 0' we used 100 instead of 00 or 90°. The reason for this  Chapter 4. Combined Quadrupole-Knight Shift Effects ^  46  Direction of Magnetic Field  Figure 4.14: C-Axis View Of Crystal With Field Direction Of 300 change is that at the other two angles, two of the three lines that are produced by the inequivalent sites for each spectral transition combine to form one line. This effect can be 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 axes have the same magnetic field components because they make the same angles with respect to the field direction. As a result the corresponding aluminum nuclei at sites 2 and 3 experience the same fields and consequently produce identical spectra. Similarly, this also occurs when the magnetic field angle e in the basal plane is 30° with respect to the xl-axis. See Fig.(4.14) for the magnetic field direction with respect to the inequivalent sites and Figs.(4.15), (4.16) and (4.17)for the corresponding spectra.  •  Chapter 4. Combined Quadrupole-Knight Shift Effects ^ 1500 1000  500  4-1  C^0 i)  -500  w  ry a) —1000 —1500  20^40^60  ^  80  Field Angle 0 (Degrees)  Figure 4.15: (5/2 4-4 3/2), (-3/2 4--+ —5/2) transitions, e = 300. ▪  750 500  •  250 0  >-4  -250 w -500 -750 20^40^60^80  Field Angle 0 (Degrees)  Figure 4.16: (3/2^1/2), (-1/2  4-*  —3/2) transitions, O' = 30°  80 60 40 I-) 4-4  20 0  O-20 >,  • -40 -60 -80  20^40^60^80 Field Angle 0 (Degrees)  Figure 4.17: (1/2 4--+ —1/2) transition, 0' = 30°  47  Chapter 4. Combined Quadrupole-Knight Shift Effects^  48  In 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 effect along with the complexity of the aluminum spectra in Fig.(4.12) indicates some of the difficulties we will actually face when experimentally we attempt to observe the actual resonance lines.  Chapter 5  Experimental Considerations  The main purpose of this experiment was to investigate the Knight and quadrupole shifts produced by a single crystal of UNi2A13. This was achieved by observing the resonance spectrum 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 Knight and quadrupole shift values.  5.1 General Method In a general sense, the methods used for detecting steady state nmr signals may be separated into two broad categories. One involves detecting the change in the susceptibility of the sample associated with the onset of resonance. In this case the sample material is placed inside a coil which is part of a tuned circuit. A resonance signal is detected when the inductance of the coil is altered by the variation of the sample susceptibility. In this particular arrangement, the coil is orientated with its axis perpendicular to the magnetic field direction. The other method employs a double coil arrangement in which one coil is fed from a signal generator while the second coil, perpendicular to the transmitter coil axis and field direction, experiences an induced voltage from the forced precession of the nuclear spins. For this particular experiment the first method, commonly termed nuclear magnetic resonance absorption, was utilized.  49  Chapter 5. Experimental Considerations ^  50  Figure 5.1: Theoretical Absorption and Dispersion curves 5.2 Resonance Absorption and Dispersion Effects The change in the susceptibility of a nmr sample and its effects on the observed signal at the onset of resonance may be better understood by considering its more general complex behaviour which may be seen in Fig.(5.1) In this figure x' represents what is termed resonance dispersion while x" is called resonance absorption. Both terms comprise the complex 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. They are derived by solving the Bloch equations for the case in which the sample experiences a weak electromagnetic field from the surrounding coil. This derivation may be found in any standard nmr text. While these Lorentzian curves are typically observed in experiments conducted 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 rigid lattice are not motionally averaged by the motion that occurs in a liquid or a gas. In an actual experiment, the Gaussian resonance lines are really seen in terms of their  Chapter 5. Experimental Considerations ^  51  derivatives where the resonance absorption peak of Fig.(5.1) corresponds to the zero point in Fig.(5.2) and the zero point of the dispersion line corresponds to the peak in Fig.(5.3)  Figure 5.2: Derivative Of Absorption Curve  Figure 5.3: Derivative Of Dispersion Curve  Chapter 5. Experimental Considerations ^  52  Figure 5.4: Mixed Absorption and Dispersion Curve 5.3 Skin Depth Effects The skin depth effect [14] is a phenomenon which limits the degree to which an alternating magnetic field can penetrate a metal. Its presence produces a reduction in the amplitude of 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 caused by the metallic sample which induces a phase shift in the resonance signal [13] . This effect may be overcome by using a powdered sample where the metal has been ground down to a particle size that is less then the skin depth. A powdered sample was not used in this experiment because clear and detailed signals were required for the Knight and quadrupole shifts. As has been mentioned previously, a powdered sample tends to obscure the individual signals produced by these shifts. A theoretical curve of a mixed signal 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 this  Chapter 5. Experimental Considerations ^  53  effect is that we are no longer sure exactly where resonance absorption occurs. This is primarily because the central zero point of the mixed signal does not correspond to the frequency at which maximum absorption occurs as it does in the pure absorption case. In this particular experiment this problem was largely overcome by fitting the appropriately mixed theoretical curves to the actual experimental curves on a computer. Once the proper absorption and dispersion mixture was found it was then possible to determine the actual resonance frequency by simply computing where the zero point would have been if an absorption signal had only been present. The most commonly observed signal in this experiment was comprised of an equal mixture of absorption and dispersion similar to the theoretical curve of Fig.(5.4). In this figure the theoretical absorption point is indicated by the position of the vertical line.  5.4 Magnetic Field Calibration In this experiment the resonance signals were produced by linearly sweeping the magnetic 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 for the actual field strengths. These were found by observing at two different frequencies the absorption spectrum of a powdered aluminum reference sample and noting the relative signal positions. A powdered sample was used in place of a single crystal for the calibration process to avoid the problems mentioned in the previous section. Once the signal positions were determined it was then simply a matter of calculating the field intensities at these positions using the standard isotropic Knight shift equation of the previous chapters:  v = 7ito(1+ kiso)H  ^  (5.1)  Chapter 5. Experimental Considerations ^  54  and solving for H. While the reference sample was composed of only a simple metal powder which did not exhibit the usual anisotropic Knight and quadrupole shifts that were common in the UNi2A/3 sample, it was still necessary to compensate for its isotropic Knight shift in order to calculate accurately the actual magnetic field strength. It was for this reason that the above equation was used to determine H. Accurate values for the 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/10KGauss and = 0.00161 with 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 of 10.602MHz and 10.675MHz. The actual frequency ranges used to calibrate the magnetic field in this experiment generally depended on the size of the field sweep used with the common sweep ranges being 50Gauss or 100Gauss. In Figs.(5.5) and (5.6) the range was approximately 100Gauss. The signals in these figures along with all the other data were plotted with a computer. Actually, the data collecting process involved taking several sweeps of the resonance signal for signal averaging, digitizing it with an analog-to-digital converter and then displaying it with 512 separate data points on a graph.  Chapter 5. Experimental Considerations ^  55  Figure 5.5: Aluminum Reference Signal for 10.602MHz  Figure 5.6: Aluminum Reference Signal for 10.675MHz It will be noticed that two spectra are actually displayed in both figures instead of one. This is simply the result of scanning the aluminum reference signal twice with the top line in both figures corresponding to an increasing sweep of the magnetic field and the bottom line corresponding to a decreasing sweep. The reason for sweeping the spectrum twice in this manner was for the sole purpose of eliminating an RC time constant effect  Chapter 5. Experimental Considerations ^  56  which was introduced into the spectrometer to suppress some of the spurious noise that accompanied many of the spectra. While this effect was useful for noise reduction, it did tend to delay the onset of the actual resonance signals. This is noticeable in both of the above figures where it can be seen that the peaks of both spectral lines are slightly displaced horizontally from one another. By sweeping the spectrum in both directions it was possible to compute an average line position and thus determine the real onset of resonance. 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 with data point numbers. In doing so it was found that the line in Fig.(5.5) was positioned at the 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 to outline the method by using the spectra of Figs.(5.5) and (5.6). In practice this process was done automatically on a computer when fitting the theoretical mixed absorption and dispersion curves to the experimental resonance signals. The procedure began by finding the magnetic field at the positions of the pure aluminum resonance signals in both figures using Eq.(5.1). The intensity of the field at the line position of Fig.(5.5) was found 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 of data point numbers, it was then possible to calculate the magnetic field strength increase per data point which was 0.1991Gaussfpoint. Knowing this and the number of points that actually comprised the plot, the end point fields were then easily calculated. In this case they were found to be 9523.8Gauss and 9625.8Gauss. It will be noticed that the entire field sweep was actually 102Gauss.  Chapter 5. Experimental Considerations ^  57  5.5 Basic Experimental Procedure Each experiment was preceded by a cooling of the UNi2A/3 sample down to approximately the 11Kelvin range using a variable flow cryostat and a calibration of the magnetic field with the aluminum reference sample. Once these processes were completed the aluminum sample was then replaced with the UNi2A/3 specimen and the spectrometer frequency varied until an appropriate signal was found. Observing a resonance signal from this sample usually required several sweeps of the magnetic field in order to statistically enhance the signal-to-noise ratio. When a sufficient number of sweeps were completed for a specific run the entire spectrum, if satisfactory, was then stored digitally for later analysis. Following this process, it was common practice to either search for more signals by varying the spectrometer's frequency again or to simply change the temperature 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 9852kHz  Chapter 5. Experimental Considerations ^  58  After an experiment was completed the data were then analysed in order to determine the frequency shifts produced by the UNi2A/3 sample. This analysis, which required finding the correct magnetic field value at which resonance occurred was completed by fitting 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 to compensate for the distortion of the resonance curve by the mixing of the absorption and dispersion 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 along with the signal itself. Incidentally, the resonance line position on this plot (see vertical line) 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 we simply perform an actual calculation of the frequency shift for the Fig.(5.7) spectra. This can be easily achieved since the magnetic field was already calibrated for this particular data in the previous section where the increase in field strength per data point was found to be 0.1991Gauss/point.  Resonance Lines /**\  Figure 5.8: Resonance Line and Fitting Line Before Fitting Procedure  Chapter 5. Experimental Considerations^  59  Figure 5.9: Resonance Line and Fitting Line After Fitting Procedure Since we know that the resonance line position is represented by the 288th data point of Fig.(5.9) and we know the minimum and maximum field values are 9523.8Gauss and 9625.8Gauss respectively, we can immediately calculate the corresponding field strength at that point. In doing so we find a value of 9581.1Gauss. Determining the resonance frequency 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 previous section. Using this equation actually gives us the reference frequency value which in this case is 10.6293MHz. In other words, this is the frequency at which resonance would occur in the absence of any Knight or quadrupole effects. To now find the degree of frequency shifting present simply requires the subtraction of this reference frequency from the applied frequency of 9.852MHz. From this the corresponding shift value is found to be Ay = —777.3KHz. This particular method, which was performed on a computer, was used throughout the experiment in order to determine the Knight and quadrupole shifts of the UNi2A13 single crystal sample.  ^  Chapter 5. Experimental Considerations^  60  5.6 Experimental Determination of Knight and Quadrupole Shifts We 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 of Eq.(4.1) using experimental shift values with different magnetic field orientations. This modified equation was actually an approximation of Eq.(4.1) where vo in the second order quadrupole terms, in this case, represented the reference frequency without any Knight shift where: = (1 + ki„)v„f [-1^— —1) + illref I (3 cos20 — 1) 2^2^2 1^v. — — k2Vref 2 v2 Q  32vref  (1  COS20) [{102771(172  —  16m(m  7.120, + ^ [24M(M — 1) —  72vref  —  (rn  — 1)^181(1 —^+  1) — 21(1 +1)  + 3}  —  1 2  —)  vol sin2 0 cos(20)  39}^ +^ cos2^(1 0  (1  —7/ cos(20)) 3  — -d277 cos(20))]  41(1+1) + 9— {30m(m — 1) — 61(1+1) +12} cos2 0  9 ^15 1 ^— 1) —^+ 1) + 1- 1 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 order it was considered appropriate to drop this particular Knight shift presence to simplify the process used in finding the parameters. In the actual experiment an iterative method was 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 corresponded to field directions which were along the crystal c-axis, 0 = 00, or in the basal plane, 0 = 90°. The reason for this becomes immediately apparent by examining Fig.(4.12) where it can be seen that a significant degree of line interference can occur  Chapter 5. Experimental Considerations ^  61  theoretically in the angular region of 0 = 40° to 800. By using only the two field directions 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 basal plane, 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 the latter spectra were significantly weaker than any of the other spectra at this angle. In reality 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. This was done by subtracting the experimental satellite signals, which corresponded to a field direction of 0 = 00, from each other and using Eq.(5.2) to solve for vQ. Specifically, the experimental resonance frequency that corresponded to the (5/2 4-* 3/2) transition was set equal to Eq.(5.2) with m 5/2 and 0 = 0° while in a similar manner the frequency from the (-3/2 4-* -5/2) transition was set equal to the same equation only in this case with m = -3/2. Subtracting the latter relation from the former resulted in a simple equation of the form: /IQ  (11512,312 =^  4  •  (5.3)  Alternatively, this method was also used for the (3/2 4-> 1/2) and (-1/2 4-* -3/2) spectra producing the slightly different relation: vQ  (v312,112 - 1/--1/2,-3/2) 2^  (5.4)  Both equations provided similar values for vQ. Next it was necessary to determine the asymmetry parameter 7/ and 0. These equations were formulated using resonance frequency data that corresponded to a field direction of 0 = 90° with respect to the crystal c-axis. It has been already demonstrated theoretically in the last chapter that a multitude of line signals should be present for this particular angle due to the three inequivalent  ^ ^  62  Chapter 5. Experimental Considerations ^  sites in the UNi2A/3 crystal. As a consequence of this we were able to use the six experimental spectra associated with the (3/2 4-* 1/2) and (-1/2 —3/2) transitions to find 77 and vQ. The actual process initially involved the same method used in deriving Eq.(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)].  ^  Using the signals associated with the other inequivalent sites with 0 =  (5.5)  ^+ 60° and  = + 120° provided two more equations: Av2 = ("3/2012 — v_i /'72— —//Q [1 + n cos 2(0' + 600)]^(5.6) 3/2)600 = 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(2011Q  tan-1 [2Lu2 + 3vg Avi] 2^0(,A7/1, vQ) 9Y  = tan-1 [2Av3 3vQ + 2^-\13-(Av1 +7N)  (5.8) (5.9)  (5.10)  Actually, the last two equations are equivalent since they should, in theory, provide the same 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 shift values. 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: V-112,-3I2 = kisoVrefo^1C2Vrefo +  A  ^  (5.11)  Chapter 5. Experimental Considerations^  vi/2,-1/2 =  Vrch  /Cis° Vreh^k21ireh  2  1'ref2^K211„f2  V1/2,-1/2 1= KisolireJ2^  2  COS (201^in„^  63  B^(5.12)  COS 2(01 + Oineq) C  (5.13)  where Oineq 0° in Eq.(5.12) and 60° or 120° in Eq.(5.13) depending on the particular lines 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 with respect to the crystal c-axis and set it equal to Eq(5.2) for m -1/2 and the same angle. 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 reference frequencies are numbered for the purpose of indicating that the same reference frequency is not required for all three equations.  Chapter 6  Experimental Setup  In this chapter we are mainly concerned with the experimental setup and the method used to orient the crystal. In the first section a review of the basic experimental configuration and the equipment used will be given. Since much of the equipment was a standard type found in many laboratories it will not be necessary to provide an exhaustive description of the functioning of every apparatus used in the experiment.  6.1 Apparatus and Experimental Setup The experimental setup may be seen in Fig.(6.1). It is indicated in this figure, that the sample 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 of a 7400 Varian rotatable electromagnet. The magnetic field direction could be changed, by rotating the entire electromagnet about the crystal. Typical field values used in this experiment generally ranged from 5 to 13KGauss. The field strength was monitored at the 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 copper wire. It was connected to the rest of the apparatus through a coaxial cable and was part of the tuned circuit shown in Fig.(6.2). The circuit's frequency was monitored by the Philips frequency counter and sent directly to the computer. 64  Figure 6.1: Schematic of Apparatus and Equipment used in the Experiment  Chapter 6. Experimental Setup^  66  Figure 6.2: Diagram of the Tuned Circuit The 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 part of the actual spectrometer. The resonance frequency was changed by adjusting the capacitance 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 had the effect of altering the diodes capacitance, was supplied by the phase lock frequency stabiliser. In this case the p.l.f.s was used as a fine adjustment which could change the frequency over a limited range in 1KHz increments. Large frequency changes could only be produced by the P.K.W. box. It was mentioned in the chapter concerned with experimental considerations that an RC time constant effect was needed in order to eliminate some of the noise associated with the signal. This was supplied by the lock-in analyzer where an RC constant of 1.25sec, 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 sine wave. This particular frequency was found to be the best in eliminating most of the electronic interference that was present in the laboratory.  Chapter 6. Experimental Setup^  67  The temperature of the specimen was reduced by exposing it to a continuous helium flow in the cryostat. The temperatures used were generally in the range of 6 to 11K. It was necessary to reduce the sample temperature down to this range in order to enhance the signal-to-noise ratio. A moderation of the temperature was usually achieved through the use of a small heating element placed near the sample. This heater was regulated by the Lakeshore monitor and controller. The equipment was set up so that the temperature in the vicinity of the sample could be conveniently monitored through the monitor and controller at the computer. When needed, the sample's temperature could be reduced below the helium boiling point of 4.2K by filling the cryostat with liquid helium and pumping on it. In this particular case, the helium level was monitored by a helium level indicator which was a device that consisted of a niobium-titanium alloy wire strung along the 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 computer instead of a wave form generator to sweep the magnetic field over a specific range; usually 50 or 100Gauss. The scanning of the field was controlled from the computer, through the general purpose interface (G.P.I.) bus and through the 16-bit digital-to-analog converter at the magnet power supply; see Fig(6.1). The field was changed in a linear manner while the spectrometer's frequency was held at a constant value by the lock-in analyzer. Each sweep of the magnetic field required 51.2 seconds to complete. Since the field was swept in both directions by the computer in order to eliminate the signal displacement caused by the RC time constant, a full cycle actually required 102.4 seconds. Generally it was necessary to sweep the field repeatedly in order to statistically enhance the signal-to-noise ratio. The usual number of double scans used ranged from 1 to 25. Approximately 43 minutes were required to complete 25 full cycle sweeps.  Chapter 6. Experimental Setup^  68  6.2 Sample Orientation  Oc\sc'. c. Figure 6.3: Sketch of Crystal's Physical Appearance A sketch of the UNi2A13 sample may be seen in Fig.(6.3) where it is indicated that the crystal had an approximately hexagonal shape and relatively small size. Its small size along with its fragility required that special care was taken in its handling. This was especially true during the mounting process which involved securely fastening the sample on 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 perpendicular to the samples surface. This, however, turned out not to be the case. By viewing the (3/2  4-4  1/2) spectra it became apparent that the closest the c-axis came to being  aligned along the magnetic field direction corresponded to an angle of approximately 14.5°. This was the case even though the samples surface was perpendicular to the field direction. Finding this angle required rotating the magnetic field in small increments and noting the position and relative spacing of the three spectral lines produced by the three inequivalent sites. The idea behind this process may be understood by referring to Fig.(6.4) where, for clarity, the theoretical spectra of the (3/2 1/2) transition from  Chapter 6. Experimental Setup^  69  1000 750 500 4-1  250  C) a) -250  t:r  a) - 500. _750:  20^40^60^80 Field 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 magnetic field decreases with respect to the c-axis, the three spectral lines tend to converge to a single maximum frequency shift. The point at which they converge corresponds to the 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 three spectra did tend to converge up to a certain point, however, a further change in the field direction beyond this point resulted in a reversal of this tendency. The angle at which this occurred corresponded to the above field angle of 0 = 14.50. This value was later confirmed when the crystal was realigned using x-rays. A reorientation of the crystal was necessary in order to ensure that the magnetic field was properly aligned along the c-axis when 0 was thought to be 00. The determination of the c-axis direction required the removal of the sample from its mount and its placement in a special holder that enabled the user to accurately observe the crystal orientation. Several x-ray photographs of the crystal diffraction pattern were necessary before a symmetric pattern was observed. The onset of symmetry indicated an alignment of the x-ray beam, along the crystal c-axis. After recording the crystal  Chapter 6. Experimental Setup^  70  orientation at which this occurred, specifically 14.5° + 0.3°, the sample was carefully remounted on the teflon holder and placed in its new orientation on the end of the sample probe to a small brass plate. The plate on which the sample holder was placed was bent to the above angle so that the magnetic field could then be easily aligned along the crystal c-axis with the appropriate rotation. This alignment was later checked by again observing the converging behaviour of the (3/2 4- 1/2) spectra as the magnetic field direction was altered. This time it was observed that the three spectra converged to a single frequency shift which indicated that the c-axis was finally properly aligned in 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 the other crystal axes.  Chapter 7  Experimental Data and Conclusion  The experimental frequency shifts found when the magnetic field was applied along the crystal c-axis and in the basal plane are now used in conjunction with the equations given in chapter 5 to find vQ, the asymmetry parameter II, O' and the isotropic and anisotropic Knight shifts /Cis°, K1 and /C2. 7.1 Experimental Data  Lines (3/2 4-* 1/2)  (1/2 4-* 1/2)  (-1/2 4--> —3/2)  Frequency Shifts (kHz) —477+1 —263 ± 1 —282+1 53 ± 1 71 + 1 67 ± 1 566 ± 1 380 + 1 396 + 1^._  Oineg  00 60° 120° 0° 60° 120° 0° 60° 120°  Table 7.1: Experimental Shifts with Magnetic Field in Basal Plane The experimental frequency shifts and their lines are tabulated in Table(7.1). As is indicated in this table, three frequency shifts were observed for each line resulting in a total of nine shifts for all three lines. This particular signal distribution, which was anticipated in chapter 4, is the result of the aluminum nuclei at the three inequivalent sites experiencing Knight and quadrupole effects for a magnetic field direction in the 71  Chapter 7. Experimental Data and Conclusion^  72  crystal basal plane. It will also be noticed in the table that each experimental shift has been 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 experimental spectra from the (3/1 4-4 1/2) and (-1/2 <-4 -3/2) lines for Avi, Av2, and Av3 in Eqs.(5.8), (5.9) and (5.10) and determining which assignment produced the most reasonable values for 0' and 17. To be more specific, the actual process first involved subtracting the experimental shifts associated with the (-1/2 <-* -3/2) line from the corresponding shifts 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 theoretical plots 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 inequivalent sites and are therefore represented by the same angles qne q, it was necessary to subtract 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, Av2 and 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 was then 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 ascertained that assigning the angles in the particular order shown in Table(7.1) produced such a result. In other words, Avi = (-477 - +566)001(Hz, Av2 = (-263 - +380)600kHz and Av3 = (-282 - +396)12ookHz. It was discovered, using various theoretical plots similar 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 in  Chapter 7. Experimental Data and Conclusion^  Lines (-1/2 4.- —3/2) (-3/2 4-÷ —5/2) (5/2 4-* 3/2)  73  Frequency Shifts (kHz) —788 + 1 —1565 + 1 1580 + 1  Table 7.2: Experimental Shifts with Magnetic Field Along C-Axis Table(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 previously mentioned 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 either Eq.(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 observed when the field direction was along the c-axis, the inequivalent site positions were not relevant and consequently their angles not required in this case. The experimental data, along with the errors listed in both tables were determined by repeatedly using the fitting procedure mentioned in the last chapter to find an average shift 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 Results The value of xicj using Eq.(5.4) was determined to be:  1 vQ = 786.3 + 0.3  The reason for using Eq.(5.3) instead of Eq.(5.4) was primarily due to the fact that the denominator in this equation was twice as large as the number used in the denominator  Chapter 7. Experimental Data and Conclusion ^  74  of Eq.(5.4) which resulted in a smaller calculated error in N. Using this value for N in Eqs.(5.8), (5.9) and (5.10), along with the (3/2 4-* 1/2) and (-1/2 4-4 —3/2) experimental shifts for Av1, Av2 and Av3 cited in the previous section, it was found that: = 2.3 ± 0.5  and ii = 0.327 ± 0.003  The error in 0' was calculated by finding the difference between the values given by Eqs.(5.9) and (5.10). It was mentioned in chapter 5 that both equations should in theory produce identical angles for 0'. This, however, was not true experimentally since we were using experimental data for these equations, not theoretical values, as a result they produced two slightly different values for this angle. The error in the asymmetry parameter was found to depend mainly on the errors in Avi 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 their values 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 to a 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 were substituted into Eqs.(5.12) and(5.13). The third (1/2 4-* —1/2) line was used to check the accuracy of the resulting Knight shift values. The reference frequency values used were 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 shifts  ^  ^_L  Chapter 7. Experimental Data and Conclusion ^  75  produced the three values:  %  %  1C2,90 %  (0.23 + 0.02)  (-0.25 + 0.02)  (0.14 ± 0.03)  It was established that the errors in these values depended mainly on the experimental uncertainties 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 inserting the above Knight shift values into Eq.(4.1) and calculating the frequency shifts. It was discovered that these calculated shifts were not significantly different from the original experimental shifts. In other words, the above Knight shift values were not affected by using an approximate form of Eq.(4.1).  7.3 Some Comments The asymmetry of the ionic contribution to the electric field gradient was calculated and compared to the experimental asymmetry parameter found in the last section. Its value was determined by considering the following ionic forms of a large number of uranium(U+), nickel(Ni2+) and aluminum(A13+) atoms and their influence on a single aluminum atom. The actual process required a computer that summed the contribution of each ion using the equations: 17xx^= {3y2  _  VYY  e r5  [ 3x2 —r 2]^  7,2]^ =  r5  Vzz = e — [42 r 5^  -  r2]^  (7.1) (7.2) (7.3)  The variable r was the distance from the single aluminum atom to an ion in the crystal lattice and x, y and z were its components with respect to the coordinate system depicted  Chapter 7. Experimental Data and Conclusion^  76  V-Axis  • Z-Axis Out Of Page X-Axis  Figure 7.1: Placement of x, y and z Axes with respect to the Crystal Structure in Fig.(7.1) relative to the crystal structure. After summing over a sufficiently large number of contributions the following values for Vsx, V  ^Vz, were found:  1/zx = —1.070 x 10mesuicm3 Vyy^  9.017 x 1013esu/cm3  Vzz = 1.676 x 1013esu/cm3  It is immediately apparent that this result indicates that the largest component of the electric field gradient is in the x direction, in complete disagreement with the experimental observations. Thus, the ionic contribution cannot be the dominant source of the electric field gradient. Using these values the asymmetry parameter was calculated to be:  ii  = 0.686  Chapter 7. Experimental Data and Conclusion^  77  The contribution from the conduction electrons can be probed through the Knight shift anisotropy tensor since it reflects the symmetry of the Fermi surface electron states which is also proportional to their contribution to the electric field gradient. This tensor takes the form: 0 Kaniso=^0  ^  0  —(k1 k2)/2^0  0^0^/C2)/2 which was easily derived using Eq.(2.31). From this equation we have that: /C2 - 2 = /C2 s + — ( 3 cos2 0 — 1) — — sin 0 cos(20). 2 2  (7. 4)  where v (1 + k)vref•  Since kzz  =  ICSX  = —(1C1 + 1C2) / 2  (7.6)  = (—K:1 + K: 2)/ 2  (7.7)  (7.5)  kiso  where 0 = = 00 for Eq.(7.5), 0 = 90° and 0 -,--- 00 for Eq.(7.6) and 0 = = 90° for Eq.(7.7), we have: k„  0  0  K80  0  0  0  /Cxr  0  0  1C,50  0  0  0  iCyy  0  0  kiso  —(1C1 + k 2 )/2  +  (—)c1 +K:2)/2  where: 4-1^4-1  K = Kis° +  'Canis° •  _  Chapter 7. Experimental Data and Conclusion ^  78  From the experimental Knight shift values we infer that this contribution would have the experimentally observed principal axes and an asymmetry parameter of: ii = 0.6 + 0.2  where  0<n<  1  and 1=  kaniso,xx  —  kaniso,yy  IC aniso,zz  Thus, at least these states appear to have the correct symmetry, and we can only assume that the remaining states not near the Fermi surface are responsible for the remaining significant contribution to the electric field gradient.  7.4 Knight Shift Temperature Dependence The temperature dependence of the Knight shift was observed over the range of 6 to 300K by measuring the frequency shifts of various resonance signals. No significant temperature dependence was observed over this range outside of the experimental uncertainty of +1kHz. This result is consistent with the theoretical derivation of the Knight shift given in chapter 2, (see Eq.(2.24)), where a temperature independent behaviour was one of the many properties found to be associated with this effect.  7.5 Conclusion We have shown how the theory of the Knight shift and quadrupolar interactions for a non-axial field gradient can be applied to the determination of both the Aln Knight shift tensor and the electric field gradient tensor in the case of UNi2A13. The experimental  Chapter 7. Experimental Data and Conclusion^  79  results 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 reasonable accuracy.  Kis° %  (0.23 + 0.02)  % (-0.25 + 0.02)  k2 % (0.14 + 0.03)  The electric field gradient is not predominantly influenced by the ionic contribution according to a lattice sum calculation, but appears to be more strongly influenced by the conduction electrons. In the temperature range studied, no temperature dependence of these parameters has been observed.  ^  Appendix A  Wigner Matrix Elements  The following terms are the matrix elements that comprise the 5-dimensional rotation matrix. In all, there are 25 elements.  = e2 cos4(3/2)e-2i1  ^  1)(2,-1 =^cos3(3/2) sin(3/2)e-1'Y  DL2L =^cos2(3/2) sin2(3/2)^_ _2e-2ice cos(3/2) sin3(i3/2)01'  L-'2,2 = C2  D(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-Y  DL2L0 = 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-Y  80  Appendix A. Wigner Matrix Elements^  7)02?-2 = \ COS2  81  (/3/ 2) sin2(P/2)e-2"^DV,_) = N/6 cos3(0/2) sin(0/2)e—" —^cos(P/ 2) sin3(0/ Di:(2 „1)  2-)322 = cos4(j3/2) — 4 cos2(P/2) sin2(0/2) sin4(0/2)  —.V6cos3(3/2) sin(P/2)ei-Y +^cos(0/2) sin3 (0/2)e"  =^cos2(0/2) sin2(/3/2)e22i7  7)122_2^2eic'  cos(3/2) sin3(0/2)e-2i1  ^  (2) -- 3e cos2(0/2) sin2(P/2)e—i'Y D1,— — 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'Y  Di22 =^cos3(0/2) sin(P/2)e2il'  'T)2) 2 = e2ic  sin4(0/2)e-2i1'  ^  1:)2()) = '16-e2ja cos2 (P/2) sin2(3/2) s4 (3/2) e2i-y  ^  DV) = 2e2' cos(P/2) sin3(P/2)e—i-Y DV1, =  2e2i' cos3(0/2) sin(i3/2)eil  Bibliography  [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, New York (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 Atomic Spectra, 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 the Theory 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 Science Pub., (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) 82  Bibliography^  83  [18] R. T. Schumacher, Introduction to Magnetic Resonance: Principles and Applications, Modern Physics Monograph Series, New York (1970) [19] M. I. Valic, A Nuclear Magnetic Resonance Study of Single Crystals of Gallium Metal, U.B.C. Thesis (1970)  


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