HELICON PROPAGATION IN ALUMINIUM SPHERES by Siegfried Feser B.Sc, University of Manitoba, 1967 M.Sc, University of Briti s h Columbia, 1969 A Thesis Submitted i n P a r t i a l Fulfilment - of the Requirements for the Degree of Doctor of Philosophy In the Department of Physics We accept this thesis as conforming to the required standard The University 6t British Columbia July, 1975 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h Co lumb ia , I a g ree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s tudy . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d that c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i thout my w r i t t e n p e r m i s s i o n . Depa rtment The U n i v e r s i t y o f B r i t i s h Co lumbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 11 ABSTRACT Theoretical and experimental studies have been made of the propagation c h a r a c t e r i s t i c s of h e l i c o n waves i n m e t a l l i c spheres; the hel i c o n wave i s a branch of the magnetoplasma wave spectrum i n the pres-ence of a uniform, s t a t i c magnetic f i e l d . Our t h e o r e t i c a l considerations have shown that f o r an i n f i n i t e l y conducting sphere the h e l i c o n wave i n s i d e the sphere can be described to good approximation by a si n g l e c y l i n d r i c a l wave. This model leads to a simple, approximate, a n a l y t i c understanding of the resonant mode structure and of the h e l i c o n f i e l d d i s t r i b u t i o n i n s i d e the sample. The resonant modes can be described as a doubly i n f i n i t e s e r i e s , l a b e l l e d by (m,n), where m indicates"the number (odd) of h a l f wavelengths along the magnetic f i e l d d i r e c t i o n nearly equalling the sample diameter and n ind i c a t e s the same condition perpendicular to the magnetic f i e l d . An exact numerical c a l c u l a t i o n f o r a sphere of f i n i t e s c a l a r r e s i s t i v i t y has been c a r r i e d out by representing the he l i c o n wave insi d e the sphere by a superposition of c y l i n d r i c a l waves. In t h i s way one i s able to c a l c u l a t e the absorption peak heights and widths as well as the resonant frequencies f o r the f i r s t few modes. These r e s u l t s are found to be i n good agreement with those c a l c u l a t e d independently by Ford and Werner using vector s p h e r i c a l har-monics throughout. Whereas the formalism o£ Ford and Werner i s preferred f o r computational accuracy, t h e i r approach i s p h y s i c a l l y less transparent. Experimentally, we have substantiated the r e s u l t s of the c a l -c ulations by using s i n g l e - c r y s t a l spheres of aluminium with a r e s i d u a l r e s i s t a n c e r a t i o of about 4000. The resonant modes were studied f o r two d i s t i n c t geometries of the e x c i t a t i o n and detection c o i l s : the " p a r a l l e l " geometry had the axes of the c o i l s c o l l i n e a r but at r i g h t angles to the i i i static magnetic f i e l d , whereas the "perpendicular" geometry had the excitation c o i l , the detection c o i l and the magnetic f i e l d a l l mutually perpendicular. The anisotropy of the fundamental (1,1) absorption peak • amplitude i s about 20% and that of the resonant frequency i s about 1% for an applied magnetic f i e l d of 35 kOe. The helicon data are interpreted in terms of the anisotropics of the transverse magnetoresistivity (20%) and of the Hall coefficient (1%) . For the f i r s t time i n a helicon experiment an adequate sampling of crystallographic orientations has been made using the same sample throughout. The principal feature of the (1,1) peak height anisotropy i s the presence of a narrow trough - typically 1° wide - whenever the static magnetic f i e l d l i e s in a {100} plane. This feature, which corresponds to an equally sharp ridge in the anisotropy of the transverse magnetoresis-t i v i t y , has escaped detection in a l l previous studies of the magnetoresis-t i v i t y (by helicons and otherwise), and i t i s a consequence of the magnetic breakthrough which i s required by the Ashcroft model of the Fermi surface of aluminium. The transverse magnetoresistivity oscillates (periodically in 1/B) whenever the magnetic f i e l d i s parallel to a <100> direction and this feature i s likewise consistent with the Ashcroft model. Other quantum oscillations have been observed superimposed on the helicon resonance and their periodicities are found to agree closely with those in the de Haas-van Alphen effect. TABLE OF CONTENTS Page Abstract ........... i i Table of Contents . . . . i v List of Figures . v L i s t of Tables i . . . „ v i Acknowledgment V i i Chapter I INTRODUCTION 1 I I THE THEORY OF HELICONS ........................... 6 §II-A Helicons i n an Inf i n i t e , Isotropic Medium....... 6 § I I - B Boundary-Value Problem, Semi-Infinite Slab... 1 0 §II-C Helicons i n a Sphere: An Approximate Model 1 2 I I I GALVANOMAGNETIC MEASUREMENT TECHNIQUES 3 0 I V EXPERIMENTAL APPARATUS AND PROCEDURES ............... 3 4 §IV-A Sample Preparation 3 4 §IV-B The Apparatus ... 3 6 §IV-C Coil Geometries . . . . . . . 4 4 §IV - D The Electronic Circuitry 5 4 V EXPERIMENTAL DATA 5 7 §V-A Zero-Field Resistivity. . . 5 7 §V-B Helicons i n a Sphere, Experimental Data 6 5 V I A N I N T E R P R E T A T I O N OF THE DATA 1 0 7 V I I H E L I C O N S I N A SPHERE: GENERAL I S O T R O P I C C A S E I 3 8 V I I I CONCLUSIONS 1 5 4 Bibliography I 5 9 Appendices A HELICONS IN A FERROMAGNETIC METAL . 1 6 2 B THEORY FOR HELICONS IN AN INFINITELY CONDUCTING SPHERE, AS DEVELOPED BY M.H.L. PRYCE. 168 V LIST OF FIGURES Figure Title Page II - l An Approximate Model 16 II-2 Approximate Helicon Field 21 II-3 Resonant Modes . , 23 II-4 Exact Solutions, by Pryce Formalism 27 IV-1 Sample Holder and Coil Forms 37 IV-2 X-Ray Picture 40 IV-3 Probe Assembly 42 TV-A Dewars and Magnet 45 TV-5 Perpendicular Excitation Coils 47 IV-6 Magnetic Field Calibration 49 IV-7 Coil Geometries $2 IV- 8 Electronics 55 V- l ' Instrumental Phase Shifts 59 V-2 Zero-Field Dipole Moment 61 V-3 Zero-Field Resistivity.. . . 63 V-4 Resonant Modes 66 V-5 Field Dependence of the Resonant Frequencies 69 V-6 Helicon Signal in Quadrature......... 73 V-7 Frequency Modulation 76 V-8 Rotation in {110} Plane, "j^Geometry 78 V-9 Rotation in {110} Plane, "//" Geometry............ 81 V-10 Stereographic Projection 83 V - l l (1,1) Helicon Amplitude Anisotropy 85 • V-12 (1,1) Amplitude Contours 89 V-13 Amplitude Rotation Curves, EE... 92 V-14 Amplitude Rotation Curves, BB 94 V-15 Field Dependence of (1,1) Helicon Amplitude 96 V-16 Field Dependence of (1,1) Resonant Frequency 99 V-17 Anisotropy in DD Plane 101 V-18 Quantum Oscillations on Helicon Resonance 103 V-19 1/B Periodicity of Quantum Oscillations . 10^ VI-1 Nearly-Free-Electron Fermi Surface of Aluminium 108 VI-2 Third Zone Fermi Surface 110 VI-3 Brillouin Zones 113 VI-4 Magnetic Breakthrough . . . . . . . . . . . . . . . . . 1 1 5 VI-5 Resistivity Anisotropy 121 VI-6 Magnetoresistivity ; 123 VI-7 Magnetoresistivity Comparison ..........126 VI-8 Merrill's (1968) Data for Helicons in Al Cylinders 131 VI- 9 Quantum Oscillations I36 VII- 1 Helicon Dipole Moment m,Q VII-2 Ford and Werner's Data .150 A-l Fundamental Helicon Resonance .164 A-2 Field Dependence of the Resonant Frequency .166 v i LIST OF TABLES Table Page I I - l Resonant Frequencies ....... •..••••••.••••••••«••• 19 V - l Helicon Resonant Frequencies . . 71 VII-1 Comparison of Calculations. ...... 152 v i i ACKNOWLEDGMENTS I would l i k e to express my appreciation to Dr. A. V. Gold for his guidance during the course of this work. The advice and actions on my behalf of Dr. J. F. Carolan are gratefully acknowledged. For their contributions to the theoretical aspects of this thesis, i t i s a pleasure to thank Dr. M.H.L. Pryce, U.B.C. and Dr. G.D. Mahan, University of Oregon. G. Lonzarich, a fellow graduate student, deserves special thanks for his eagerness to discuss problems and for his valuable aid regarding experimental techniques. I shall remember his motto: "The explanation has to be simple". The financial support of the National Research Council and the University of B.C. was greatly appreciated. 1 Chapter I INTRODUCTION When an electromagnetic wave impinges on a metallic surface, the wave i s generally damped after penetrating into the metal a very small distance, known as the skin depth. The damping i s caused by the mobile charge carriers absorbing the energy from the wave, and the skin depth decreases the greater the mobility of the carriers. I f , at the same time, a large, static magnetic f i e l d i s applied to a highly con-ducting metal, the carrier motion acquires a helical character since the carriers execute cyclotron rotation about the magnetic f i e l d direction. If the metal shows a substantial Hall effect, this bending of the trajectories by the static f i e l d prevents the carriers from screening out the impressed electromagnetic wave, and at low frequencies a trans-verse, circularly-polarized wave -. called a helicon wave - can be pro-pagated through the metal with only l i t t l e damping (Konstantinov and Perel (1960); Aigrain (I960)). The characateristics of this helicon wave inside the metal depend upon the magnetoresistivity tensor of the metal, and they provide an inductive (contactless) way of measuring some of the galvanomagnetic parameters of the metal. Delaney and Pippard (1972) have given a review of the various methods that are presently used to carry out r e s i s t i v i t y measurements (see also Chapter I I I of this work). This thesis i s concerned with the use of helicon waves to probe the r e s i s t i v i t y of an aluminium, single-crystal sphere. The benefits of using a spherical sample i n a helicon-wave experiment have been recognized only recently. Rose (1965) appears to have made the f i r s t observation of helicon modes in a metallic sphere. He observed the standing-wave 2 mode structure in a sodium sphere but the limited theoretical under-standing of the problem at the time did not allow any properties of sodium to be accurately determined. His was essentially a study of helicons in a known medium rather than a study using helicons to probe the galvanomagnetic characteristics of a medium. Concurrent with the work on this thesis, Ford and Werner (1973), Delaney and Pippard (1972), Simpson (1973), and Delaney (1974) have also studied helicons in spheres. Before 1973, no finite-volume, boundary-value, helicon problem had been solved exactly. The sample shapes most commonly used were either thin slabs, with the magnetic f i e l d perpendicular to the surface, or long cylinders, with the magnetic f i e l d along the axis. In either case the major information was obtained for the magnetic f i e l d oriented only along one particular crystallographic direction. If one i s interested i n the anisotropy of the galvanomagnetic properties of a crystal as the magnetic f i e l d i s rotated through various crystallographic directions, then a spherical sample i s extremely useful. A sphere allows not only the same sample to be used for a l l crystallographic directions but also a continuous, monitoring of some properties as a function of the magnetic f i e l d orientation. In this way any fine structures in the variations are more l i k e l y to be detected. The author's i n i t i a l intention was to explore the propagation of helicon waves i n a ferromagnetic material. Problems arising in interpretation of the data using a slab sample led to the use of a sphere in the work of this thesis. Since helicon waves are electromagnetic waves, their properties in a material are determined by the dielectric constant and the magnetic permeability of that material. A simple theory, combining the magnetic permeability of a ferromagnetic material due to 3 spin waves with the usual non-magnetic dielectric constant, i n order to determine the influence of spin waves on helicon wave propagation, was presented by Stern and Callen (1963). The i n i t i a l data (see Appendix A), taken with a crossed-coils experimental geometry, using a thin rectangular slab of nickel, did not allow an unambiguous interpretation. One of the main d i f f i c u l t i e s was not knowing the (spatially non-uniform) magnetization and hence the effective f i e l d B_ as a function of the applied f i e l d H for a f i n i t e , rectangular sample shape. It became clear that i t would be more suitable to study helicons i n a spherically shaped sample of a ferromagnetic material, the magnetization being best understood for a volume bounded by a second-degree surface. Although the magnetization of a sphere i s easier to consider than that of a f i n i t e slab, the normal modes of helicon propagation in a sphere were not understood, even i n a non-magnetic material. As a result, our efforts were redirected into a theoretical and an experimental study of helicons i n non-magnetic spheres. As a result of our theoretical work and that of Ford and Werner (1973) done concurrently, the boundary-value problem of helicons in an isotropic-resistivity sphere was solved prior to the completion of t|ie experiments discussed in this thesis. The opportunity to analyze the data more precisely than was previously possible,was a further reason for continuing to work with helicon waves in spheres. It was decided to carry out the experimental work on spheres of aluminium because of the av a i l a b i l i t y of single crystals of high purity and also because comparisons could then be made with published galvanomagnetic data which had been obtained in the conventional manner (Balcombe (1963); Balcombe and Parker (1970)). 4 A general description of helicon waves i s presented i n Chapter II of this thesis. An expansion in plane waves leads to the well-known solution of the boundary-value problem for a semi-infinite slab, and i t forms the basis of an approximate model which we have developed for solving the boundary-value problem of helicons in a sphere. This simplified model for a sphere i s useful because i t results in simple analytical expressions which have enabled us to interpret the observed helicon-mode structure. Chapter I I I consists of a review of the various techniques used to measure the galvanomagnetic properties of metals. Chapter IV i s a description of the experimental equipment and procedures used i n this thesis. The experimental results relating to helicon waves in aluminium spheres are presented in Chapter V. The helicon modes have been studied in some de t a i l , and the anisotropics of the resonant frequency and of the peak height and width are presented. Quantum oscillations related to the de Haas-van Alphen effect were also observed when the helicon reson-ances were studied as a function of the strength of the applied magnetic f i e l d . Chapter VI consists of an interpretation of the experimental data of the previous chapter with reference to the anisotropy of the galvanomagnetic properties and to the Fermi surface of aluminium. A comparison of the present results with those obtained from other experi-ments leads to an evaluation of the merits of the methods of this thesis. In Chapter VII we present a general analytic formalism which we have developed for helicon waves in a sphere with isotropic r e s i s t i v i t y and Hall coefficient. This formalism i s compared with that developed 5 independently by Ford and Werner (1973). Our conclusions are summarized i n Chapter VIII, which also contains suggestions for further research. \ Chapter I I THE THEORY OF HELICONS §II-A. Helicons in an Infin i t e , Isotropic Medium Helicons are a class of electromagnetic waves that propagate i n an uncompensated solid-state plasma (eg. a metal) i n the presence of a large, s t a t i c , magnetic f i e l d , B . The term 'uncompensated' means that i n the plasma o the number of carriers with hole-like properties does not equal that of carriers with electron-like properties. More precisely, what i s necessary i s that the material show a substantial Hall effect. In an isotropic medium the wave propagates with l i t t l e damping when the Hall r e s i s t i v i t y , R B q , i s much larger than the usual r e s i s t i v i t y , p. The Hall r e s i s t i v i t y , R B q , i s the voltage per unit current density which i s developed at right angles to both the applied f i e l d and the current i n the conventional Hall effect geometry. Since under these conditions a metal can effectively become transparent to a class of electromagnetic radiation, the interest that this phenomenon has generated i s easily understood. The distinguishing characteristic of a metal i s i t s high density of mobile conduction electrons. An electromagnetic wave impinging upon the surface of a metal i s screened by some of the conduction electrons as i t enters the metal. The mobile carriers thus prevent the wave from penetrating deeply into the metal. If the carriers could be constrained so as not to be able to follow the a.c. electromagnetic f i e l d (and hence not be able to screen the wave), the wave could propagate unimpeded through the sample. When a strong, s t a t i c , magnetic f i e l d i s applied to the metal, the motion of the conduction electrons i s constrained to hel i c a l paths along the f i e l d direction. As w i l l be shown presently, this constraint i n the electron's mobility permits one transverse, circularly-polarized component of an electro-magnetic wave to propagate i n the metal. At low frequencies the wave i s damped only in so far as the electrons are not constrained, that i s , do not execute uninterrupted cyclotron rotations. This may happen i f they are scattered by any of the mechanisms causing a non-zero r e s i s t i v i t y i n the metal. This i s i n marked contrast to the situation i n the absence of the magnetic f i e l d . In the common skin effect, the larger the conductivity the less the penetration into the metal. For a comprehensible and f a i r l y complete review of the state of published knowledge concerning helicons prior to the middle of 1968, the reader i s referred to review articles by Maxfield (1969) and Morgan (1967). To calculate the wave-propagation characteristics of a metal i n the presence of a uniform, static magnetic f i e l d , B^ =B^ z_, one solves Maxwell's equations i n conjunction with the constitutive relation appropriate to the presence of B^. In Gaussian units a non-magnetic material has permeability u=l, and at frequencies low enough to neglect the displacement current, Maxwell's equations are V x B = — J , (2-1) — — c — V x E = " f |f . (2-2) The Ohm-Hall constitutive relation for the case of a scalar conductivity, a, and a static magnetic f i e l d i n the z. direction giving a cyclotron frequency, V i s I ^ J + Y ( I x J ) , (2-3) where J_ i s the current density, 8 a » ne2x / m* , (2-4) o>c - (±e)BQ / (m*c) , (2-5) and T i s the mean scattering time for the carriers whose density i s n and whose effective mass i s m*. Equation 2-3 has been linearized by assuming the helicon f i e l d to be very small i n comparison to B q , and hence only the static f i e l d B q appears i n equation 2-5. An i n e r t i a l term, JWTJ_, has been ignored i n equation 2-3, the term being small when compared to J_; this approximation i s of the same order as the neglect of the displacement current i n equation 2-1 (see Klozenberg et a l , 1965). By taking the curl of equation 2-3 and substituting from equations 2-1 and 2-2, we obtain a differential equation for the total magnetic f i e l d B_: VxVxB + to x Vxfz x (VxB)) + — | J = 0 . (2-6) C — I— I ..C-.-«dt — -If we now set B = B + B* , (2-6a) — —o — * and assume a plane-wave form for B/ B' = b exp i(qz-tot) , (2-7) equation 2-6 yields VxVxB' — ut t •—• (VxB1 ) - ^£2- B. = o ( 2_ 7 a ) C dZ C^ — — which i s separable i n Cartesian co-ordinates. The z component of equation 2-7a yields t>z = 0, that i s , the wave i s transverse. The other two equations give maximum insight when transformed by setting B! = B' ± i B' (2-7b) ± x y ' which results i n Q2 B; ± i q2 V B ; - i i t e B ; = o. ( 2 - 7 c ) The simultaneous solution of these two equations implies two possible values of q 2, denoted by q + 2 ; 2 _ ~ iATTcoa/c2 (0_fl> Q ± ~ 1 ± i u T ' ' • ( 2 8 ) c -For the case .-(i) -T.».1, which ^ experimentally implies a very pure metal at low temperature i n the presence of a large magnetic f i e l d , 2 ~ + 4TTUJCT _ + 4Tra)(nec) ^ (2-9) ±^ ~ ~ c 2 co x ~ c 2 B C O Hence, depending on the sign of coc, one of q + w i l l be real and one trans-verse, circularly-polarized component ( B + or B ) of the helicon f i e l d w i l l propagate with l i t t l e damping i f io T » 1 . Calculation from equation 2-9 shows, and experiments verify, that (for example) in an aluminium crystal of high purity i n the presence of an applied magnetic f i e l d of 35 kOe, helicons have a wavelength of about 1 cm at an excitation frequency of about 50 Hz. This corresponds to a very small phase velocity of about 5 0 cm/sec. In vacuum the corresponding wavelength would be 6 x 1 0 8 cm. A further difference between helicons and vacuum waves is that the helicon phase velocity i s dispersive, that i s , (o/q (2-10) as can easily be seen from equation 2-9. It follows that i f one could detect standing half-wavelengths in a slab, these would occur i n a pattern uniform h i n to at a fixed value of B q . The p o s s i b i l i t y of detecting half-wavelengths i n a slab w i l l be demonstrated after an implicit assumption that has been made up to this point i s c l a r i f i e d . It i s usual to ignore the displacement current i n Maxwell's equations when solving for helicon waves. This would not be permissible at very high frequencies, however another effect makes this consideration only theoretical. We have assumed a "local theory" for the conduction process, that i s , a point relationship between J_ and E_ as J(r) = _ ' _ _ ) . The "non-l o c a l " theory i s applicable when the mean free path of the carriers becomes comparable to the helicon wavelength and this regime has been studied by Cohen et a l (1960). It i s found that at a frequency of several megahertz, well below that required to make the displacement term significant, i t i s possible to have the electrons absorb energy from the wave in a resonant fashion due to a Doppler-Shifted Cyclotron Resonance. Additional damping (called Landau damping) also occurs for off-axis propagation i n the non-local l i m i t . §II-B. Boundary-Value Problem, Semi-Infinite Slab If a semi-infinite slab of material i s excited symmetrically on both sides, helicon waves are generated at each surface and these propagate inward, interfering with each other to form standing waves when the plate thickness i s an integral multiple of the helicon half-wavelength. Since the results for the semi-infinite slab of a metal w i l l be required in Chapter VI, we quote f i r s t a generalization of the dispersion relation (2-9) obtained by Penz (1967) following closely the methods of §II-A, but considering an i n f i n i t e v X ' 11 medium of arbitrary r e s i s t i v i t y tensor p_ = a - 1 . The dispersion relation found by Penz i s . ±{-p p -[(p -P ? / , ] } — - i[(P +P )/2] £ ^ ATTCO . X 1 "xx / 4 J J L V x x yy' l f p ° 2 ^ Pxx Pyy _ Pxy Pyx) ^ (2-11) In the analysis i t i s assumed that _z i s the direction of propagation rather than the direction of B^. Generally, the r e s i s t i v i t y tensor i s simplest i n the coordinate system where = B Q z^. To get the appropriate r e s i s t i v i t y tensor for the above formalism when the propagation direction z_ i s not paralle l to , the tensor i s transformed by a similarity transformation relating the two coordinate systems. To satisfy the boundary condition that the magnetic f i e l d be con-tinuous across the boundary of a semi-infinite slab of thickness d, four plane waves are needed (see Penz, 1967), i f the excitation f i e l d i s chosen to be a uniform f i e l d linearly p olarizedin the»x^direction. If a detection c o i l surrounding the slab has i t s axis along x, to be denoted the "par a l l e l " or '"//" geometry, the induced voltage i n the c o i l , ignoring the direct coupling between the excitation c o i l and the detection c o i l , w i l l be proportional to Em m2 [P /(P +P )] +iQ(to/to ) a) ,-xx -xx y y - : m_ ( 2 _ 1 2 ) mz . . 1 + iQ[U/w ) - (to /to)] m=i m m odd where = ( T ^ f - l ( P x K P y y - p x y p y x ) % , (2-13) Q « ( pxx pyy- pxy pyx ) / ( p x x + P y y ) . (2-14) If the detection c o i l has i t s axis along to be denoted the "perpendicular" or " | " geometry, the induced voltage i s proportional to 12 P / (p +p ) xy xx yy m2 1 + i Q[(w/w^ -(o> /»)] . . (2-15) m=l odd For both geometries i n the case that P = -P.. P =P,T,T=P and |P_l»|p| xy xx y y xy the resonant frequency to^ i s given as a, = f ^ f ^ P , (2-16) m V 4d z I xy and the Q of the resonance i s given by Q - P w / (2P) « ^ . (2-17) xy i. In this case the resonant frequencies correspond to an odd number of one-half wavelengths equalling the thickness of the slab. The frequencies of the resonant modes of a semi-infinite slab are i n the same ratio to each other as the squares of the odd integers m are to each other. §II-C. Helicons in a Sphere: An Approximate Model In Section II-A, the fact that the helicon wave equation was separable i n a Cartesian coordinate system allowed a solution for a semi-i n f i n i t e slab geometry to be found. As well, the wave equation i s separable and an analytic solution exists i n a cyli n d r i c a l coordinate system. However, in spherical coordinates the wave equation i s not separable and no simple solutions have been found. This i s clearly a result of the i n t r i n s i c cylin-d r i c a l symmetry when a uniform magnetic f i e l d i s applied to an isotropic medium. In this section the cylindrical solutions w i l l be employed to obtain an approximate solution for helicons i n a sphere. Because of the mathematical d i f f i c u l t i e s associated with a spherical boundary, a simplified, analytic model w i l l be presented in order to obtain an understanding of the transverse helicon modes in a sphere of i n f i n i t e conductivity. Following a suggestion by G.D. Mahan, the boundary of the sphere i s modelled by a short cylinder of dimensions comparable to exact cylindrical helicon waves are used to satisfy an approximate boundary condition. The result i s a prediction of a doubly i n f i n i t e mode structure for a sphere with resonant frequencies and the f i e l d distribution being calculation for a sphere of i n f i n i t e conductivity. The formalism for this latter calculation i s due to Pryce (1973) and i s outlined i n Appendix B. The predictions of the approximate model as regard the resonant frequencies and the f i e l d distribution inside the sample were found to be in satisfactory agreement with the results of the exact calculation. The simultaneous solution of equations 2-1, 2-2, 2-3 i s known i n cylindrical coordinates (Klozenberg et a l , 1965) and only an outline of the solution i s given here. Assuming solutions of the form where (p, z, $) are the usual cyl i n d r i c a l coordinates, yields the helicon wave equation, the sphere; the axis of the cylinder i s taken to be p a r a l l e l to 1^. The given analytically. This model i s then compared to an exact, numerical B' = b(p) exp[iW + qz - tot)] , (2-18) (2-19) where q 2 = iu 4ITO7C (2-20) Equation 2-19 can be factored as 1 ll-Cjx -Bi) (Vx -e 2) B* = 0 (2-21) and the constants (3i and 32 are.then determined as ^ j 1 ^ 1 - ^ ^ ] ^ - (2"22) The u t i l i t y of factoring the wave equation l i e s i n being able to treat each factor separately, the general solution being the sum of the two solutions. The solution i s straightforward as given by Klozenberg et a l (1965). Written i n terms of B'=(B',B!,B'), where B' = B' ± i B' , (2-23) ± x y s the solution to the helicon wave equation i s —=-Bz=__] A n J m(Y nP) exp|i[m«5 + qz - t o t ] | , n _ 1 (2-24) 2 B± =X) M n ( 3 n ± q ) J m ± l ( Y n p ) ^ P " ^ ^ 1 ) * + * z " n=l Y n where Y i s defined by n J ( Y 1 > 2 ) 2 s ( B l , 2 ) 2 -q 2 • (2-25) The value of m i s to be determined by the nature of the excitation f i e l d symmetry, and the values of A m, A m and q are to be solved for by application of the boundary conditions. For the special case of "w x-*», equation 2-3 reduces to E_ = -—- — X — v (2-26) and the helicon wave equation specifies only one non-zero value for 3i 2» namely, 15 6 «= -q2 / ( i i o j q) . (2-27) For a uniform, circularly-polarized excitation m=+l, the form of the solution i n c y l i n d r i c a l coordinates i s B ' " AiJiCYP) sin(qz) exp[i(j5-tot)] , - A i ^ y ^ J 2 ( Y P ) cos(qz) exp[i(2ftj-cot)] , (2-28) K f l ^ — ^ - Jo(YP) cos(qz) exp[i(iot)] . For the purpose of gaining some physical insight regarding the form of the helicon magnetic f i e l d at resonance, we consider the following, non-rigorous assumptions. At resonance, the B' component of the helicon f i e l d (this i s the polarization of the f i e l d which propagates i n an i n f i n i t e medium) dominates to the extent that i n matching at the boundary only B' need be considered. At resonance, i t i s assumed that B' (at the boundary) = 0, and although the shape of interest i s a sphere (radius = a), an approximate boundary i s taken to be the surface of a short cylinder with dimensions comparable to a, see Figure I I - l . In accordance with these assumptions, at resonance J Q ( Y sa) = 0 , (2-29) and cos(q sa) = 0 , (2-30) 16 Figure I I - l . An Approximate Model The scaling of the approximated cylinder to the actual sphere i s shown. If the c r i t e r i a of equal cross-sectional areas i n the plane containing i s used the relationship R=sa implies 8=1.^/2. 17 18 where s i s a factor which scales the dimensions of the cylinder to the sphere (see Figure I I - l ) . It follows that the resonant modes of this simplified model consist of a doubly i n f i n i t e series of resonant frequencies corresponding to the quantized values of y and q implied by equations 2-29 and 2-30. The resonant frequency spectrum i s found from the dispersion relation 2-27 and appears as Table I I - l . The modes have been labelled according to which zero of the cosine term and the J term i s i n effect, as follows: o (m,n) = (m extrema along z_, n extrema along p) . (2-31) This labelling bears close resemblance to counting "half-wavelengths" along the two independent directions z_ and q_. The predicted resonant frequencies are presented i n Table I I - l i n terms of a normalized frequency parameter, p , defined by The form of the helicon f i e l d at resonance can now be plotted along z_ and £_ as i n Figures II-2 and II-3; this Is accomplished by using equations 2-28. The u t i l i t y of this approximate model i s confirmed by an exact com-puter calculation (see Appendix B) appropriate to a sphere of i n f i n i t e con-ductivity. The volume average of the helicon f i e l d was calculated and i s shown as a function of the frequency i n Figure II-4. The normalized resonant frequencies have already been quoted i n Table I I - l . Figure II-4 shows the B_ f i e l d distribution inside the spherical sample and should be compared to Figure II-3 showing the f i e l d distribution for the approximate model. 19 Table I I - l . Resonant Frequencies The values of a parameter p (m,n) which i s proportional to the resonant frequency of a sphere of unit radius i s given, calculated and labelled by the approximate method described i n the text. The values i n parentheses are exact values calculated numerically using the formalism of Appendix B. 20 (m.n) 1 -> 1 3 5 7 9 1 5.74 11.5 17.6 2 3.8 30.0 1 (4.90) (10.6) (16.2) (2 2.4) 3 31.7 (30.0) 43.5 59.1 76.2 82.1 96.0 5 1 57. 7 -21 Figure II-2. Approximate Helicon Field The approximated helicon magnetic f i e l d distribution for the (1,1) mode inside the f i n i t e cylinder i s shown. The amplitude i s drawn i n units of Ai (see equation 2-28) which, at resonance, i s i n f i n i t e . The plot i s meant to show only the relative amplitudes of B^, B^, B^ along the _ axis (figure (a)) and along the £ axis (figure (b)). 22 23 Figure II-3. Resonant Modes The normalized f i e l d distributions for the f i r s t five lowest frequency modes i n the f i n i t e cylinder are sketched. The dotted line shows the spherical boundary to which this approximate solution i s supposed to apply. The relative amplitudes at (p,z) = (0,0) are determined by equation 2-28 and can be obtained using the following factors. (1,1) x 1.85 (3,1) x 4.17 (1,3) x 1.32 (3,3) x 2.17 (1,5) x 1.21 The-small circles have been calculated by the exact formalism due to Pryce and are plotted normalized at (p,z) = (0,0). 24 25 +2 J> ( 1 .5 ) - 2 26 27 Figure II-4. Exact Solutions by Pryce Formalism The exact magnetic f i e l d distribution near the resonant f r e -quencies for a metal sphere with COCT->« is given. The f i e l d s have a l l been normalized to unity at maximum amplitude. The formalism used to calculate these i s due to Pryce (1973) and i s given i n Appendix B. Table I I - l shows the comparison between the actual resonant frequencies and those calculated by the approximate theory. 28 29 Regarding the mode scheme shown i n Figure II-3, i t should be noted that actually only the f i r s t few of the (l,n) modes are 'pure'. At frequencies corresponding to the mode (3,1) and higher, one gets a superposi-tion of nearly degenerate modes of different order m, and the f i e l d distribu-tion w i l l be a weighted sum of these modes. This effect can be clearly seen by reference to the mode labelled (3,1) + (1,9) i n Figure II-4. The two modes (3,1) and (1,9) are nearly degenerate and the f i e l d distribution i s composed of a sum of two modes. The fact that one cylindrical wave i s sufficient to give the helicon wave modes i n an isotropic sphere to an accuracy of 10% for the case of CU^ T-*00 suggests generalizing this approach. In Chapter VII a superposition of cylin d r i c a l waves for an arbitrary value of OJ^ T i s used to match the vacuum f i e l d on the boundary of a sphere. 3 0 Chapter III GALVANOMAGNETIC MEASUREMENT TECHNIQUES Delaney and Pippard (1972) have given an extensive critique of the various methods which are used to measure the r e s i s t i v i t y of metals. This chapter includes a short description of these methods i n order to present a context within which to judge the value of using helicon waves i n spheres. The most common technique for measuring r e s i s t i v i t i e s i s to attach leads to the sample. This method generally involves a long, thin, rod-shaped sample. The current i s constrained to flow along the crystallographic direction determined by the long axis of the rod, and hence a different sample is needed for each new orientation to be studied. The many d i f f i c u l t i e s i n -volved i n interpreting the data gathered by this method are described by Delaney and Pippard (1972). A variety of inductive methods has been used to eliminate the need to attach leads to the sample. Most a.c. methods require the frequency to be low enough to allow complete, uniform penetration of the a.c. f i e l d into the metal sample. Generally, calculations have only been done for this l i m i t . The responses that are commonly monitored are the induced magnetic moment and the torque. Visscher and Falicov (1970) have calculated the torque when the excitation i s caused by rotating a large magnetic f i e l d about a spherical sample. An inherent d i f f i c u l t y i n interpretation i s that the magnetic moment i s proportional to a mixture of r e s i s t i v i t y tensor components and since only one torque component is measured for each orientation, analysis depends upon being able to approximate some components of the r e s i s t i v i t y tensor. These low frequency, inductive methods have been used i n the determination of open orbit directions. Using conventional electrode measurements, both the ex-istence of open orbits and the occurrence of compensation lead to a rapid increase in the r e s i s t i v i t y as the applied magnetic f i e l d i s increased. The 3 1 torque measurements, using a sphere, can resolve this ambiguity since, i f open orbits and compensation are the only two p o s s i b i l i t i e s , a compensated metal i s indicated i f the torque saturates and open orbits are indicated i f the torque increases without limit (see, for example, Datars and Cook (1969)). Several different kinds of experiments using helicon waves have been done to measure some galvanomagnetic properties of metals. As seen i n Chapter II, the frequency involved need not be kept particularly low since there i s no assumption made regarding uniform penetration of the excitation f i e l d into the sample. In fact, i t i s usual to have the sample size larger than a helicon wavelength. The experimental geometries can be categorized as follows: (i) a thin slab with J _ to the surface, ( i i ) a long cylinder with / / t o the axis, ( i i i ) a cylinder with J _ to the axis, (iv) a spherical sample. Cases (i) and ( i i ) have a serious limitation of the crystallographic orienta-tions which may be studied with a given sample. As well, the boundary-value problems relating to these two geometries have been solved exactly only for semi-infinite samples. Case ( i i i ) has the u t i l i t y of being symmetric with respect to rotation of the magnetic f i e l d , however no boundary-value problem has been worked out relating to that geometry. Three d i s t i n c t l y different kinds of helicon experiments involving spherical samples have been reported recently. Ford and Werner (1973) use what they have called a "longitudinal helicon" geometry i n a study of a sphere of potassium. The term 'longitudinal' refers to the fact that a l l three c o i l s , the solenoid producing the large magnetic f i e l d , the excitation c o i l and the detection coil(s) are colinear. It follows that for an iso-tropic material the induced magnetic moment w i l l also be i n the same direction. A major benefit to be derived from this arrangement i s that there 32 are no torques between any of the coils nor between the magnetic f i e l d and the induced magnetic moment i n the sample. This arrangement i s ideally suited to the use of a superconducting solenoid. Delaney (1974) and Simpson (1973) have used a "soft helicon" technique advanced by Delaney and Pippard (1972). The term 'soft' refers to the sample being allowed to oscillate due to the effect of the torque between the transversely-excited helicons and the magnetic f i e l d rather than being held r i g i d l y . The object i s to separate the longitudinal and the transverse parts of the r e s i s t i v i t y by using the r i g i d mounting i n alternation with the flexible mounting. Their work precedes the reporting of the solution to helicon waves i n a sphere by Ford and Werner (1973) and the analysis relies on dimensional arguments. The analysis i s complicated by the now-known ex-istence of "longitudinal helicons". Compared to the previous two methods, the method used i n this thesis i s most closely related to the traditional helicon geometry. I t was novel to use a sphere to enable the magnetic f i e l d to be aligned along an arbitrary crystallographic direction, however the geometric arrangement of the magnetic f i e l d and the excitation and detection coils i s standard. Two distinct ar-rangements of coils are reported i n the next chapter. In both cases the ex-citation c o i l i s perpendicular to the magnetic f i e l d but i n one case the detection coils are colinear with the excitation c o i l , i n the other case they are perpendicular to both the magnetic f i e l d and the excitation c o i l . The use of an iron core magnet, which generally implies that the magnetic f i e l d i s perpendicular to the helium dewar axis, makes these two geometries the favoured. There i s an ease of^realigning the magnetic f i e l d within a specified crystallographic plane which results from being able to rotate the magnetic f i e l d about the sample. This i s of great benefit when doing an anisotropy . study. 33 As was shown i n Chapter II-B, the helicon resonance i n a f i n i t e sample allows the determination of three quantities, the peak height, the resonant frequency, and the peak width at each orientation and magnetic f i e l d strength. In so far as these quantities can be related successfully to the re s i s t i v i t y tensor, more information than that achieved by the torque experiments should be possible. 34 Chapter IV EXPERIMENTAL APPARATUS AND PROCEDURES §IV-A. Sample Preparation The starting material for a l l the aluminium samples that were used was a large zone-refined aluminium ingot* consisting of several large grains of typical dimensions 1 to 2 cm. The ingot had a nominal purity rating for the bulk material of 99.9999%. Three aluminium, single-crystal spheres of typical dimension 0.5 cm were prepared from this ingot by spark erosion under kerosene using an Agietron Spark-Cutter as follows. Using low spark current and voltage with a rotating, hollow, copper electrode, a solid cylinder of aluminium was shaped. The cylinder was then mounted on a slowly rotating 'lathe' with i t s axis horizontal. A rotating, hollow, copper electrode with i t s axis about 5° off v e r t i c a l was slowly lowered to f i n a l l y intersect with the rotating cylinder. In this way an accurate sphere i s shaped with only one small area (less than 1% of the total surface area) s t i l l attached to the original cylinder. This small bridge was then removed through slow spark erosion. After being thus spark-cut, two of the three spheres were mechan-i c a l l y polished**. After being handled i n this manner, X-ray photographs re-vealed a polycrystalline surface layer. Before being chemically etched (see W.J. McG. Tegart, 1959), the sphericity was measured** to be about one part i n 10 5. * purchased from Cominco Ltd., Montreal, Quebec, Canada. by Joseph A. Corriveau, 18421 Ardmore Avenue Detroit, Michigan, U.S.A. 1 35 We distinguish the spheres as follows: LABEL DIAMETER POLISHED CHEMICALLY ETCHED Al-1 0.430 ± .002 cm yes 2 minutes Al-2 0.439 ± .005 cm no 2 minutes Al-3 - 0.25 cm yes 10 minutes Most of the data presented in this thesis was gathered from sample • Al-1. Amundsen and Jerstad (1972) have noticed changes of a factor of 2 i n the magnetoresistivity of a pure aluminium single crystal, depending upon annealing. For the sake of completeness two sets of experiments were done on Al-1, one before annealing and the other after annealing at 500°C for 3 days, followed by a slow cooling. No differences i n the Q of the helicon resonance or i n the anisotropy features were noticed after annealing. Possible effects due to the mechanical polishing of Al-1 were checked by using Al-2 i n a repeat of some experiments done with Al-1. In a l l features, the two samples gave consistent results. Sample Al-3 had a rather thick polycrystalline layer after being mechanically polished and had to be rather heavily chemically etched (about 10 minutes) to give an identifiably single-crystal X-ray picture. Even after extensive etching the X-ray spots were much more diffuse than those obtained from Al-1. Because of the smaller volume of Al-3, background signals prevented making as clear an interpretation of the helicon results as when using Al-1. Nevertheless, results were, as with Al-2, consistent with the more extensive study done with Al-1. The low temperature r e s i s t i v i t i e s of Al-1 (after annealing) and Al-2 (unannealed) were measured at 4.2°K by an inductive method (see Chapter V). The residual resistance ratio (RRR) i s defined to be the resistance at room temperature compared to the resistance at 4.2°K, and was measured to be 4000 ± 100 for Al-1 and 3300 ± 200 for Al-2. 36 The mounting and orienting of the samples are described i n the next section. §TV-B. The Apparatus , The experiments of this thesis were designed around a rotatable, 15 inch, iron core Varian electromagnet. The geometric arrangement used to measure the zero-field r e s i s t i v i t y of the samples as well as some of the helicon data to be presented i n Chapter V has the excitation c o i l and the de-tection coils colinear and both perpendicular to the static magnetic f i e l d . This arrangement i s labelled the " p a r a l l e l " or "//" geometry. In this case i t was most convenient to wind a l l three coils on the Kel-F p l a s t i c c o i l form supporting the sample-holder insert as shown i n Figure IV-1. The detection coils were about 1000 turns of #46 insulated copper wire each. The "//" excitation c o i l was about 200 turns of #42 insulated copper wire. There were no connections made between any of the coils-inside the dewar. G.E. adhesive #7031 was used to glue adjacent layers of windings and adhesive-impregnated paper was used to separate the coils from each other. Two detection c o i l s , wound i n opposition, were used i n order to buck out the direct coupling between the excitation c o i l and the detection system. The balancing of these co i l s i s discussed i n §IV-C. The other geometric arrangement of c o i l s , labelled the "perpendicular" or " | " geometry, Involved a set of heavy c o i l forms bolted to the magnet at right angles to the detection c o i l s ' axis and w i l l be discussed shortly. The spherical sample was placed i n the plastic sample holder (see Figure IV-1) and cushioned by a small amount of cotton hatting. The sample-holder insert could be removed from the co i l s and mounted on a Philips X-ray machine. A small stick glued to the sample allowed i t to be aligned relative to the horizontal plane and to the locating pin axis (see Figure IV-1) to better than 1°. The sample was then held i n position by a slight tightening 37 Figure IV-1. Sample Holder and Coil Forms The c o i l forms and the sample holder insert are shown. The two detection coils were about 1000 turns each of #46 insulated, copper wire. The "//" excitation c o i l was about 200 turns of #42 wire. A l l coils were carefully wound on a Meteor Coil Winder and were separated from each other by glue-impregnated paper spacers. V . 38 temperature — sensing resistor locating pin — excitation coil -for//geometry G.E. adhesive 7031 cotton nylon screw. 4 W'W$Wl/t:ii mmmm, warn •"-".•'.•'•A'in i III::: 'iiliii:'" ' ' I ^iiiiiiiiii non-magnetic stainless steel tubing •Kel-.F plastic •nylon screw hole inner detection coil •SAMPLE -outer detection coil KehF plastic phosphor - bronze spring 39 of the cotton batting and plunger onto the sample and by impregnating the cotton batting cushioning with adhesive. After the adhesive had hardened and the stick was unglued from the sample, the crystal was re-X-rayed. If the alignment was satisfactory, the holder was transferred to the c o i l form and probe assembly. A typical X-ray picture with the crystal aligned with a four-fold axis near center i s shown as Figure IV-2. Experimentally there has been some confusion i n distinguishing a <110> direction from a <100> direction i n aluminium (see Chapter VI). To avoid the possibility of any such confusion i n this work, the orientation of the crystal was determined by an analysis of the X-ray "spots" and then checked by physically rotating the sample a known amount to locate another high symmetry direction, generally the three-fold <111> direction. I t was necessary to glue the sample after being aligned; otherwise during the helicon experiment the torque between the anisotropic induced magnetic moment and the applied magnetic f i e l d would cause the sample to rotate. In the only case clearly observed, the four-fold axis tended to line up with the magnetic f i e l d . As a result of this motion these data were not reproducible and several discontinuities were observed as the magnetic f i e l d was rotated. Gluing the sample i n the manner mentioned prevented these d i f f i c u l t i e s . As precaution, the samples were re-X-rayed to ensure that the sample had not moved after each series of experiments. After the sample holder was transferred from the X-ray machine to the probe assembly, and that into the dewar, the horizontal plane was s t i l l accurately known but the orientation within the plane was known to only about 5° . However, i n a l l cases the symmetry of the helicon data allowed the orientation within the plane to be determined to ± h degree. The probe assembly with the sample holder and coils at i t s bottom end i s shown i n Figure IV-3. The copper baffles and the three small-diameter, stainless steel tubes served to make the assembly more r i g i d and thus to 40 Figure IV-2. X-Ray Picture A typical X-ray picture of Al-1 i s shown. A four-fold axis i s nearly centered, and the horizontal plane i s near one of the {110} - planes. The film to sample distance was approximately 3 cm. The film used was Polaroid, 3000 speed, 4 x 5 Land film, type 57. 42 Figure IV-3. Probe Assembly The probe assembly with the sample holder and coils at i t s bottom end i s sketched. The overall length was about 1 meter. The coaxial feed-throughs were connected to the two detection c o i l s ; the multi-prong elec-t r i c a l feed-through was connected to the "//" excitation c o i l and the temperature sensing resistors. •43; coaxial feed-thru-brass flange oYing seat twisted leads holes in tubing non-magnetic stainless steel phosphor-bronze springs helium transfer electrical feed-thru' non-magnetic stainless steel tubing copper baffies temperature sensing resistors sample holder reduce the voltage pickup due to mechanical vibrations. Twelve phosphor-bronze springs were attached near the bottom end of the probe also to help reduce vibrations by holding the probe tightly against the inner dewar wall. The e l e c t r i c a l leads from the coils were brought up to the top flange as tightly, uniformly - twisted, insulated copper wire (#42). The wires were held against the non-magnetic stainless steel tubing by G.E. adhesive and/or teflon tape. The temperature sensing resistors were used to monitor the liquid helium level i n the dewar. The liquid helium dewars were made of non-magnetic stainless steel by Oxford Instruments. The dewars were mounted on a heavily weighted gantry which was isolated from the floor of the electromagnetically shielded room by shock-absorbing springs and padding. The magnet used was a 15", iron core, electromagnetic Varian magnet. A schematic drawing of the dewars and the , magnet appears as Figure IV-4. Because of severe space limitations inside the dewar, the " | " excitation coils had to be mounted outside the dewar. In order to hold these coils securely as the a.c. torque acted on them, the coi l s were wound on solid non-magnetic stainless steel c o i l forms which were held tightly i n an aluminium yoke which i n turn was solidly bolted to the magnet. This arrangement i s illustrated by Figure IV-5. Because the excitation c o i l s were located centrally i n the magnetic f i e l d , the Hall probe for the Varian magnet had to be displaced outside the high-field pole tip s . As a result the magnetic f i e l d had to be calibrated. A rotating c o i l magnetometer was used to measure the true central f i e l d ; the magnetic f i e l d calibration i s shown as Figure IV-6. SIV-C. Coil Geometries For the helicon experiments here envisioned three arrangements of coils are commonly in use. Defining z_ as the direction of B^, the large, static magnetic f i e l d , one could have the excitation c o i l with i t s axis parallel to z. For the case of an isotropic metal sphere, symmetry requires 45 Figure IV-4. Dewars and Magnet The Oxford Instruments non-magnetic stainless steel dewars and the rotatable, 15'', electromagnetic Varian magnet are shown. The probe f i t inside the dewars with the sample centered i n the magnetic f i e l d . 46 47 Figure IV-5• Perpendicular Excitation Coils This drawing shows the perpendicular excitation coils wound on their non-magnetic stainless steel c o i l forms. The c o i l forms were mounted i n the aluminium yoke which i n turn was bolted to the magnet. Note the position of the Hall probe. Because of i t s non-central position the magnetic f i e l d had to be calibrated using a rotating c o i l magnetometer. 4-9 Figure IV-6. Magnetic Field Calibration The graph shows the true central f i e l d as measured by a rotating-coil Rawson-Lush magnetometer (0.1% accuracy) plotted against the nominal f i e l d derived from the Hall probe i n i t s non-central position. 50 NOMINAL FIELD FROM HALL PROBE (kOe) 51 a l l Induced moments to be parallel to z_ and thus the detection c o i l would op-timally be parallel to z. This i s the geometry (m=0) used by Werner, Hunt and Ford (1973) in their study of potassium; i t i s the most natural geometry to be used with a superconducting magnet and has the advantage that there i s no torque on the excitation c o i l or on the isotropic sample. The two c o i l geometries used for this study have been designed around an iron-core, rotatable electromagnet and have the axes of both the excitation c o i l and the detection coils perpendicular to B Q Z . In the geometry which we label " p a r a l l e l " or "//", the excitation and detection coils are collinear and perpendicular to the plane of rotation (ie., p a r a l l e l to the axis of rotation) of B' z. As the direction z changes relative to the o — — crystallographic directions, the excitation f i e l d and the detection coils re-main fixed relative to the crystal. In the geometry which we label "perpendicular" or " [ ", a l l three axes are mutually perpendicular. As the magnetic f i e l d i s rotated w.r.t. the crystal, the excitation f i e l d also rotates, remaining at right angles to both z_ and the axis of the detection c o i l , which remains fixed w.r.t. the crystal. Figure IV-7 i l l u s t r a t e s the "pa r a l l e l " and "perpendicular" arrangements of the excitation c o i l , the detection c o i l , and the static f i e l d B z. o — The torque caused by the alternating current through the excitation c o i l interacting with the static magnetic f i e l d caused some experimental d i f -f i c u l t i e s . The " p a r a l l e l " geometry with a l l the coils compactly mounted on the probe did not give rise to as great a vibration problem as did the "perpendicular" geometry. To mount the " | " excitation coils sufficiently r i g i d l y , a solid aluminium yoke was bolted to the magnet and s o l i d , non-magnetic stainless steel c o i l forms were tightly bolted to this yoke (see §IV-B). Because of i t s larger physical dimensions, even with this heavy duty mount, vibrational problems were serious i f the frequency was greater than about 2. kHz. The smaller structure associated with the'"/'/" coils suggests a 52 Figure IV-7. Coil Geometries Illustrated i s the geometric arrangement of the excitation and detection coils and the magnetic f i e l d for the "p a r a l l e l " and the "perpendicular" geometries. Sketches of the actual devices are shown i n Figures IV-1 and IV-5. 53 X higher acoustical resonant frequency and did not cause as great a vibrational problem at low frequencies. §IV-D. The Electronic Circuitry The circuitry to drive the excitation coils and that to detect the response Is indicated in Figure IV-8. A voltage-controlled oscillator (VCO) feeding a constant current amplifier provided an alternating current of constant amplitude through either the "//" or the "_J_" excitations c o i l s . The frequency was controlled by the linear voltage ramp shown in the figure and was measured with an electronic frequency counter. The voltages induced i n the two detection coils were added 180° out of phase with each other. R c was outside the dewar and was set to give a minimal direct coupling between the excitation c o i l and the detection c o i l s , with the sample removed. The direct coupling was i n this way reduced to 1% of that which would result i f one detection c o i l had been used. Since one of the detection coils was substantially closer to the sample, any non-uniform a.c. response due to the sample would induce a net voltage i n the detection c o i l system. This v o l -tage was detected using a lock-in amplifier, the reference phase being given by the current through the excitation c o i l s as determined by the voltage developed across the series resistor R (see Figure IV-8). The two-phase s (quadrature) output of the phase-sensitive detector was recorded on either a two-pen chart recorder or x-y recorder as a function of frequency, magnetic f i e l d , or orientation. 55 Figure IV-8. Electronics The excitation c o i l was powered by a Kepco BOP operating i n a constant current mode. The reference signal to the lock-in amplifier was derived from the voltage across the series resistance, Rg. One of two lock-in amplifiers was used, either a PAR 124 or a PAR 129. The PAR 129 gave a quadrature output which was recorded on a two-pen recorder, either an x-y recorder or a chart recorder. The excitation frequency was supplied by using a Wavetek VCO (model 142). The voltage ramp (a homemade integrator) was used to sweep the frequency slowly and uniformly over the desired range. In the " [ " geometry the excitation f i e l d was typically 5 Oe; i n the "//" geometry the value was about 50 Oe. 56 voltage controlled oscillator -if frequency counter chart recorder constant .current amplifier 57 Chapter V EXPERIMENTAL DATA §V-A. Zero-Field Resistivity The zero-field r e s i s t i v i t i e s of the aluminium spheres designated Al-1 and Al-2 i n Chapter IV were measured inductively and the results were interpreted making use of the theory for a spherical sample with a scalar r e s i s t i v i t y as given by Smythe (1950). The r e s i s t i v i t y of any cubic crystal (eg. aluminium) i s scalar i n the absence of an applied magnetic f i e l d , and thus the theory can be applied exactly to the spheres of aluminium. The method i s essentially the same as that used by Ford and Werner (1973) to de-termine the r e s i s t i v i t y of a spherical crystal of potassium. The metal sphere of radius 'a' i s excited by means of a uniform, a . c , magnetic f i e l d , b_, and the induced magnetic dipole moment, M, i s parallel to b_- with i t s .magnitude and phase given by M = | b a 3 j 2 ( q o a ) / j ^ a ) , (5-1) where q o - (1 + i) tZmaa/.c2)** , (5-2) and j Q ( x ) and J2OO are the complex spherical Bessel functions of order zero and two, respectively. Measurements of the dipole moment of Al-1 (annealed) and Al-2 (unannealed) were done at 4.2°K i n the absence of the sta t i c , magnetic f i e l d . In order to determine both the phase characteristics of the co i l s and of the electronics as a function of frequency and also the magnitude and phase of the direct coupling between the excitation and detection c o i l s , dummy runs 58 were done without the sample i n the c o i l s . It was found that at the low frequencies involved (1 - 10 Hz), because of the high conductivity, the only significant frequency-dependent phase sh i f t was that introduced by the PAR 124 lock-in amplifier. Although the instrumental phase sh i f t became large at these low frequencies, i t was easily calibrated (see Figure V-l) and was allowed for by making an adjustment of the phase setting of the lock-in for each frequency. The second correction was to subtract from the total signal the direct coupling between the excitation c o i l and the detection c o i l s ; this correction was of the order of 5% of the t o t a l signal and was measured with the sample removed from the c o i l s . Figure V-2 gives the cor-rected in-phase and out-of-phase signal for Al-1. A third effect which has not been allowed for i s that i n practice the amplitude of the oscillator decreased by about 10% at the lowest frequencies. Figure V-2 i s presented for i t s qualitative features only; these conform well to theory. The quantitative comparison to theory i s done for the ratio of the in-phase to out-of-phase response which i s independent of the excitation amplitude. This information i s presented as Figure V-3, i n which the data for Al-1 i s pre-sented as circles with a best lin e drawn through the data and the four +'s mark the theoretical predictions with the conductivity being chosen to f i t theory to experiment at 2.1 Hz as follows: Equation 5-1 predicts an in-phase to out-of-phase ratio of unity for 4inoo"a2/c2 = 11.5. Knowing the values of co, a, and c allows an accurate determination of a to be made. For Al-1 at 4.2°K, a was determined to be 1.4 x 10 1 1 (fi-cm) - 1. This result i s more usefully given i n terms of the residual r e s i s t i v i t y ratio (RRR) by using the value of 0.35 x 10® (ft•cm) - 1 for the conductivity at room temperature as given by the Handbook of Chemistry and Physics (1971). For Al-1 the RRR was 4000 ± 100. A similar set of measurements and calculations for Al-2 determined i t s RRR to be 3300 ±200. These figures typify the material as being of relatively low residual r e s i s t i v i t y i n comparison with the aluminium 59 Figure V - l . Instrumental Phase Shifts The graph shows the phase setting of a PAR 124 lock-in amplifier needed to nu l l the output when the same low frequency signal was connected to the signal and reference channels. The lock-in was set to operate at the lOx frequency scale. This phase sh i f t i s similar to that shown as Figure 111-21 i n the PAR 124A Lock-In Amplifier Operating and Service Manual. 60 0-| P — , , , , — r 1 2 3 4 5 6 7 FREQUENCY (Hz) 61 Figure V-2. Zero-Field Dipole Moment The in-phase and the negative of the out-of-phase components of the zero-field dipole moment are shown for Al-1. The direct coupling between the "//" excitation c o i l and the de-tection coils as well as the instrumental phase sh i f t (see Figure V-l) have been allowed for. There i s a decrease i n the excitation amplitude of about 10% at the lowest frequencies and thus comparison to theory i s done for the ratio of the two components (see Figure V-3). 62 ro to ARB. UNITS + O + G „ + O + o + o +o o+ o + o + o + i o + © + o + o + o + o + m O + O £ O + m • Z o o + o • • • • ; . + o .. + o o + + H 0 ? _ = 0) 09 O »• W 1 CD O I T J 3 -0 ) CD to O i l 1 63 Figure V-3. Zero-Field Resistivity The comparison of the experimental data using Al-1 to theory i s shown. The ratio of the negative of the in-phase to the out-of-phase component i s excitation-amplitude independent and this ratio has been f i t t e d to theory using the conductivity as the free parameter. Theoretically, the ratio i s unity when 4ircooa2/c2 - 11.5. Knowing a and c and the frequency at which the experimental data shows a ratio of unity, allows a to be determined. The four +'s show the comparison between the data (the solid line i s a best f i t to the data) and theory. To be noticed i s that the conductivity chosen to f i t at one frequency allows an accurate matching at a l l frequencies. 64 R A T I O 5 -( '"-Phase) ro T CO (Out-of-Phase) ro -n •— m O 01 G m Z o < X N CO 0 - f m — m 73 m Z x m O •< ID 65 used i n other experiments. With further efforts i n refining i t i s now possible to attain a RRR of greater than 20,000 for an aluminium crystal. Chiang et a l (1970), Amundsen and Jerstad (1972) and Delaney (1974) a l l report galvanomagnetic experiments on aluminium single crystals with RRR values lying i n the range 20,000 to 25,000. §V-B. Helicons in a Sphere, Experimental Data Helicon experiments using spheres can be divided conveniently into two classes. The f i r s t consists of experiments on materials with an isotropic r e s i s t i v i t y . Experiments on potassium by Werner et a l (1973) f a l l into this class and they confirm the predictions of the theory of helicons i n isotropic-r e s i s t i v i t y spheres as developed i n this thesis and by Ford and Werner (1973). The second class of experiments are helicon experiments i n anisotropic-r e s i s t i v i t y spheres and these form the major part of this chapter. The theory as developed for isotropic samples i s used to give an approximate analysis of the anisotropic data. The Mode Structure In this section, data for helicon modes w i l l be presented for the two geometries, " J _ " and "//", as discussed i n Chapter IV. Figure V-4 i s representative of the experimental resonant mode structure of transversely excited helicons i n a sphere for an arbitrary orientation of the magnetic f i e l d . In the figure the magnetic f i e l d strength and direction were held constant as the frequency was varied. The slowly varying background i s due to the slight, direct coupling between the excitation and the detection coils i n this " J _ " geometry. The remaining, resonant structure i s a measure of the volume average of the strength of the helicon f i e l d inside the sphere. As discussed i n §II-C, the mode structure of helicons i n a sphere i s an i n f i n i t e series of primary modes (m,l) each with an i n f i n i t e series of sub-sidiary modes (m,n). For a f i n i t e value of to T, because of the f i n i t e width 66 Figure V-4. Resonant Modes This frequency sweep at 35 kOe i s representative of the helicon resonant modes i n the aluminium sphere Al-1. The labell i n g , (m,n), of the various modes i s i n conformity to the predictions of the approximate theory given i n §II-C. The time taken to vary the frequency linearly from 30 Hz to 3000 Hz was 6 minutes. The time constant was 1 second. The slowly varying background signal i s due to the direct coupling between the excitation c o i l and the detection coils i n this " | " geometry. 68 of the modes, only the f i r s t few; modes w i l l be relatively unmixed. In these experiments, given that IO CT^20, there i s even a large admixture between the f i r s t resonance peak ((1,1) i n the terminology of §II-C) and i t s f i r s t sub-sidiary, peak (1,3) (see also Figure V-4). It i s not straightforward to measure the characteristics (eg., anisotropy of the amplitude) of the sub-sidiary peak because of this large admixture of the variation of the main peak which i s superimposed. Hence, unless otherwise specified, a l l the sub-sequent data w i l l refer to the fundamental resonant mode (1,1) of the sphere. By analogy with other helicon experiments i t might be expected that for a particular resonant mode the ratio, B /co , of the f i e l d and r ' o m' frequency at which the resonance occurs be a constant. A plot of tom versus B q for the f i r s t few modes i s presented as Figure V-5. Table V - l shows a comparison between the predicted resonant frequencies (see Table II-l) and the experimentally observed frequencies. The'mode structure of a sphere with <^(J^co was predicted to be a doubly i n -f i n i t e series. This result i s better confirmed by the results of Werner et a l (1973) using a high purity potassium crystal. Anisotropy Data In order to become familiar with the nature of the anisotropy data to be presented, i t i s useful to study Figure V-6. The figure sketches the two-phase (quadrature) output of the PAR 129 lock-in amplifier as the fre -quency i s continuously varied at a fixed B q value and orientation. Conform-ing to usual designations, curve (a) i s called the "absorptive" component "A" and curve (b) i s called the "dispersive" component "D". The three quantities of greatest interest are the absorptive peak amplitude and width, and the resonant frequency; these are indicated i n the figure. The data to be presented relating to the anisotropy of peak am-plitude and resonant frequency were collected both point-by-point and 69 Figure V-5. Field Dependence of the Resonant Frequencies Plotted are the resonant frequencies of the f i r s t few modes of helicons i n a sphere as a function of the magnetic f i e l d . A straight-line through the origin i s characteristic of helicon waves i n other sample shapes of non-magnetic materials. 71 Table V - l . Helicon Frequencies A comparison between the predicted frequencies and the experimentally observed frequencies i s shown. The underlined values are experimental. Refer to Table I I - l for the two theoretical numbers. The experimental f r e -quencies are those associated with the maxima of B 1 *v» to B_' rather than B_' which i s calculated theoretically, and hence one expects the experimental values to be somewhat too large. In fact, because the resonances are quite narrow, this effect results i n a sh i f t of only about 1% for the fundamental mode and less for the other modes. This effect i s discussed by M e r r i l l (1970). 72 (m,n) 1 -> 1 3 5 7 9 I 5.74 11.5 17.6 23.8 30.0 1 4.8 + .1 11.+.3 (4.90) (10.6) (16.2) (2 2.4) 3 31.7 30.0+.2 (30.0) 43.5 48. 59.1 76.2 82.1 96.0 5 7 5. 100. 157. 7 140. --73 Figure V-6. Helicon Signal i n Quadrature , The absorptive, A, and the dispersive, D, components of the detected helicon signal are ill u s t r a t e d . The curves are representative of the " [ " geometry. The amplitude and the half-width of the absorptive peak and the resonant frequency are indicated. 74 75 continuously with respect to orientation of the magnetic f i e l d . The point-by-point procedure i s obvious and needs no elaboration. Experience indicated that the anisotropy of the resonant frequency was very slight for these experiments. In order to measure the peak amplitude anisotropy con-tinuously i t was found to be sufficient to tune (frequency or magnetic f i e l d ) at some arbitrary orientation and then to rotate the magnetic f i e l d slowly to change the orientation, confident that the mode remained very nearly tuned to i t s resonant value. The actual, though slig h t , anisotropy of the resonant frequency could also be measured continuously and with accuracy by monitoring the dispersive signal. It i s clear from Figure V-6 that the dispersive signal near resonance i s very sensitive to small shifts from the resonant frequency. Precautions were taken to validate the data. Spot checks were done at many orientations to check being tuned to the absorptive peak. Data were collected point-by-point and cross-checked with that of the continuous procedure. A third method, which ensured recording the true absorptive peak amplitude, was to modulate the frequency with a low-frequency, sinusoidal variation. Experimentally this was achieved by tuning the frequency to the absorptive peak maximum and then frequency modulating the voltage controlled oscillator at 0.5 Hz. Figure V-7 shows the response when the frequency sweeps through the resonant value and when i t does not. Since the absorptive signal as a function of frequency i s peaked, the response i s at twice the modulation frequency i f the modulation frequency caused one to sweep through the peak. If the resonant peak i s not i n the range of modulation the response i s at the modulation frequency. If properly tuned, the envelope of the resulting variation of the absorptive signal with orientation guaranteed the true peak height as the magnetic f i e l d was rotated. A l l these methods gave consistent results. Figure V-8 shows the absorptive peak amplitude and the dispersive signal as a function of the magnetic f i e l d orientation at a fixed frequency 76 Figure V-7. Frequency Modulation The response due to modulating the frequency at a low frequency, f m , has a component at 2 f m when the peak amplitude i s within the range of modulation. If the range of modulation does not include the peak, the response i s at f_. 77/ HELICON AMPLITUDE near ly t u n e d m o d u l a t i o n r a n g e 78 Figure V-8. Rotation i n {110} Plane, "_L" Geometry Data of the absorptive peak amplitude, A, and the dispersive component, D, as i s rotated i n a {110} plane are shown The magnetic f i e l d strength i s 35 KOe and the frequency i s tuned to the (1,1) mode (at f = 45 Hz). These anisotropy data are representative of the results for the " | " geometry. The ordinates for the A curve are on the l e f t , for the D curve on the right; and both curves are drawn to the same scale. .!• 80' near the resonant condition (1.1) for the " | " geometry. The fast, small-amplitude variation of the absorptive peak amplitude i s due to slight de-tuning from the peak as the orientation i s changed. Figure V-14 which shows the true amplitude versus orientation curve (using the frequency modulation method described previously) differs from Figure V-8, only by the absence of these fast oscillations. Fieure V-9 shows the analogous curves for the "//" geometry. Here the dispersive signal i s shown on a greatly expanded scale relative to the absorptive component. In both geometries, the anisotropy of the absorptive peak amplitude i n a {110} plane is about 20% and the two geometries give very nearly the same shaped curve. In the " | " geometry, the dispersive curve shows large amplitude, fast oscillations as a function of orientation. These oscillations depended on the amplitude of the excitation i n a complex way and no systematic study was made of this behavior. The fast oscillations were greatly de-creased in the "//"geometry.; -the -excitation-amplitude being ten times that i n the " | " geometry. In the "//" geometry the dispersive signal contains a slower variation which varies i n step with the absorptive peak amplitude. Possible interpretations w i l l be presented i n Chapter VI. In order to attempt a more complete understanding of the helicon wave propagation characteristics for an arbitrary orientation of the magnetic f i e l d , the plane of rotation was changed five times, by rotating the crystal in i t s holder. A l l the directions investigated i n this series of experiments are indicated in Figure V-10. The planes of rotation have been remapped onto the 1/48 irreducible part of the stereographic projection. A l l points of intersection have been labelled and these same labels reappear i n the presentation of the data in Figure V - l l . The labelled points of intersection i n Figure V-10 indicate that the s t a t i c , magnetic f i e l d was i n an equivalent crystallographic direction for the two distinct planes. However, the excitation f i e l d was not i n the same 81 Figure V-9. Rotation i n {110} Plane, "//" Geometry Data equivalent to Figure V-8 for the "//" geometry are presented. The absorptive peak amplitude, A, and the dis-persive component, D, are shown as B_ ( = 35 KOe) i s rotated i n a {110} plane. Since the fast dispersive o s c i l -lations seen i n Figure V-8 are absent here, the dispersive signal has been amplified 10 times relative to the absorptive component. The ordinates for the A curve are on the l e f t , for the D curve on the right; the scale for D has been expanded to ten times that of A. PEAK AMPLITUDE ( m V ) DISPERSIVE S IGNAL (mV) 83 Figure V-10. Stereographic Projection A stereographic projection of the five crystallographic planes, AA to EE, i n which the magnetic f i e l d was rotated i n these experiments i s shown. The planes have been remapped onto a 1/48 zone of the stereographic net as allowed by the symmetry of the crystal. The various directions which are common to more than one plane are labelled and these labels reappear on the data to be presented i n Figure V - l l . <|> i s the angle of tip of each of the planes from the {100} plane. 84 85 Figure V - l l . (1,1) Helicon Amplitude Anisotropy A composite of the data of the absorptive peak amplitude taken with B = 35 KOe for a l l the planes and directions o r indicated i n Figure V-10 i s presented. The labelling i s the same as i n Figure V-10. 86 1 T 1 1 1 1——i \ 1—~r-—r B B CC »15 I I DD l18 <ioo> '15 O R I E N T A T I O N ( d e g r e e s ) 0 10 20 30 40 SO 60 70 80 90 100 J L I 1_ I • 1 I I » t t 87 crystallographic direction i n each case. It would be interesting to be able to plot the helicon absorptive peak amplitude from the five different planes on the same diagram. To this end i t must be established that the direction of the transverse, excitation f i e l d i s not an important factor i n determining the anisotropy features i n these experiments. Two separate indications show this to be the case. The strongest argument i s that the "//" and the " | " geometries give essentially the same anisotropy features for the absorptive peak amplitude for every plane of rotation studied (compare curves A i n Figures V-8 and V-9). In the "//" geometry the excitation f i e l d was fixed with respect to the crystal as the magnetic f i e l d rotated and i n the " | " geometry i t rotated with the magnetic f i e l d , and since for a l l orientations of the magnetic f i e l d the results for the two cases agreed, the contention i s 6trongly confirmed. The other indication i s that for both geometries at the points of intersection between two different planes the absorptive peak amplitude was the same to within .1% or better. Thus to a high accuracy the anisotropy of the absorptive peak amplitude i s independent of the orientation of the ex-citation f i e l d , and only the orientation of the steady f i e l d B need be —o considered. Given these arguments, in principle the data from the different planes of rotation can be correlated and drawn on the same diagram. The data of Figure V - l l were collected over a period of about six weeks. Although the raw data are already self-consistent to within 1%, self-consistency of about 0.2% can be achieved by allowing for a d r i f t i n gain from run to run by uniformly scaling the data from each plane to best correlate with the data of plane AA i n Figure V - l l . The scale factors that were needed to obtain self-consistency were of the order of 1.0 ± 0.010 and i t was found that scaling a l l the data from a particular plane by the same amount enabled self-consistent results to be obtained. The effectiveness of this scaling tends to confirm that the 1% variation was indeed due to an 88 electronic d r i f t i n the gain from run to run. Results from the plane EE were not scaled since i t was not possible to orient the crystal accurately enough to place i t , with any confidence, i n a known relationship to the sharp minima of the other curves. The sharp minima occur when i s in a {100} plane and the orientation of EE i s very close to a {100} plane. Scaling the data of Figure V - l l as explained allows a comprehensive picture of the peak amplitude variation over the 1/48 zone to be made. A contour map of the peak amplitude at 35 kOe appears as Figure V-12. Linear interpolations were made between known values for points not lying on one of the four planes AA to DD and the free-hand contours have been drawn to intersect the edges of the c e l l at 90° (as required by crystal symmetry). There are three features of especial interest. 1. There i s a maximum at <111>. This i s the absolute maximum of the peak height. 2. A slow and rather slight variation over the rest of the zone i s noted, with a subsidiary maximum about 10° from <100> i n the plane about 25° from {100}. 3. Of particular interest i s a very sharp minimum whenever the magnetic f i e l d direction crosses a {100} plane and i t appears as a trough i n the contour diagram. For B near <100> the minimum has an angular width -o of about 5°. Away from <100> the width i s about 2° i n the "//" geometry whereas i n the " J _ " geometry i t i s only about 1°. Minima due to the trough appear at 0° and at 90° i n Figure V-12 for the curves marked AA, BB, CC, and DD. 89 Figure V-12. (1,1) Amplitude Contours This contour map of the absorptive (1,1) peak amplitude over the 1/48 stereographic projection of orientations of has been constructed from the data of Figure V-12. The data have been normalized to the maximum amplitude at <111>. 91 Figures V-13 and V-14 show the anisotropy of the absorptive peak amplitude of the (1,1) mode for various values of B q (and hence also the resonant frequency) for the two geometries of c o i l s . Figure V-13 shows the rotation of the magnetic f i e l d i n the plane EE for the "//" geometry and Figure V-14 shows the rotation i n the plane BB for the " | " geometry. To be noted i s the monotonic behavior of most of the features except for B^ near a <100> direction. For the <100> orientation a l l pertinent data show an oscillatory behavior with magnetic f i e l d strength. This oscillatory behavior can better be seen i n Figure V-15 which shows the amplitude versus B q for various orientations including <100>, <110>, and <111>. The oscillations when present are periodic i n 1/B Q, with period 2.2 x 10 - 6 Oe - 1, and their magnitude f a l l s off rapidly as i s moved away from <100>. For a l l other orientations which were checked (only some are indicated in Figure V-15) the variation was essentially monotonic. The data of Figures V-13, V-14, and V-15 were collected using the frequency modulation technique. In recording Figures V-13 and V-14 the center frequency did not have to be reset since the anisotropy was small. However, to remain tuned to the peak value of the amplitude as B q i s changed, required a corresponding change i n the frequency. Staying tuned while two parameters were changing was achieved by a sinusoidal modulation of the frequency i n order to sweep repeatedly (about once a second) over the peak. The magnetic f i e l d was then varied slowly enough to be able to periodically reset by hand the center frequency to ensure sweeping over the peak. The envelope of the resulting curve gives the actual amplitude without worries about becoming detuned. It i s the envelope of each curve which i s shown i n Figure V-15. It i s of interest to see how the resonant frequency of the (1,1) mode varies with B q when i t i s aligned with a <100> direction. To this end, since the frequency shifts only a small amount, the amplifying effect of the 92 Figure V-13. Amplitude Rotation Curves, EE Shown i s the anisotropy of the peak amplitude i n the plane EE using the "//" geometry, parametrically with the strength of B Q. The only orientation for which the variation i s not monotonic i s for near <100>; see also Figures V-15 and V-17. 94 Figure V-14. Amplitude Rotation Curves, BB Shown is the anisotropy of the peak amplitude i n the plane BB using the " } " geometry, parametrically with the strength of B Q. As i n Figure V-13, the only orientation of B^ showing an oscillatory behavior i s near <100>. 96 Figure V-15. Field Dependence of (1,1) Helicon Amplitude This i s a composite of many curves showing the peak amplitude as a function of the,magnetic f i e l d B q for different orientations as indicated. A l l other orientations checked showed a behavior bounded by the extremes of these curves. 97 98 dispersive signal for frequencies near the resonant value was calibrated and used. What i s actually plotted i n Figure V-16 i s the difference between the dispersive signal (calibrated as a resonant frequency shift) at <100> and at 15° from <100> with the same frequency and magnetic f i e l d . In the figure the oscillatory behavior of the frequency s h i f t i s compared with that oof the absorptive peak amplitude at <100>. Figure V-17 shows a point-by-point collection of data corresponding to the half-width, the amplitude and the resonant frequency as a function of orientation for crystal orientation DD. Of interest i s the inverse behavior or the half-width and the amplitude, the near constancy of the resonant frequency except at <100>, and the increase i n the resonant frequency at <100>. There i s no correspondingly large increase i n the resonant frequency for ^ in a {100} plane but away from <100>. This behavior of the frequency i s also indicated by the dispersive output (see Figure V-9), since any shift in the resonant frequency i s amplified by the rapid change of the dispersive signal near resonance. Quantum Oscillations The resonant mode structure of helicon waves in a sphere can also be seen by fixing the frequency and varying the strength of the magnetic f i e l d . This was not the usual procedure since O^T i s proportional to B q and hence the quality, Q, of the resonance changes as one sweeps through the resonance. Nevertheless, i t i s of interest to do this and the results of varying B q to sweep through the (1,1) resonance is shown in Figure V-18. The different curves refer to different orientations of B as indicated. —o The rapid oscillations superimposed on the helicon resonance are periodic in l/B o (see Figure V-19) and are analyzed i n Chapter VI. Of special interest in Figure V-18 is the observation that the rapid oscillations have a n u l l i n their amplitude at or near the maximum of the helicon resonance. 99 figure V-16. Field Dependence of (1,1) Resonant Frequency^ The difference i n the resonant frequency of the (1,1) peak with B^ . // to <100> and 15° away from <100> as a function of the magnetic f i e l d strength B q i s shown. For comparison the f i e l d dependence of the absorptive peak at <100> i s included. 100 ARB. UNITS 1 0 1 Figure V-17. Anisotropy i n DD Plane Shown is a point-by-point plot of the anisotropy of the absorptive peak amplitude and half-width and the resonant frequency i n the DD plane. ,To be noted i s the mirror-image behavior of the amplitude and half-width, the near constancy of the resonant frequency except for a slight increase near <;100>. . 102 1 0 3 Figure V-18. Quantum Oscillations on Helicon Resonance This i s a composite of the absorptive component of the (1,1) resonance for various values of the orientation of B as i n --o dicated. The frequency was kept constant at 36 Hz and the magnetic f i e l d was varied. The rapid oscillations are periodic i n 1/BQ and their amplitude i s zero at the maximum of the helicon resonance. 1 0 5 Figure V-19. 1/B Periodicity of Quantum Oscillations The quantum oscillations 30° from <100> i n a {110} plane are shown to be periodic i n 1/B with period 3.0 x 10 7 Oe - 1. i 106 1 0 ? Chapter VI AN INTERPRETATION OF THE DATA Aluminium Fermi Surface In order to give an interpretation of the results of the experiments presented i n Chapter V, i t is necessary to include a brief review of the topological features of the aluminium Fermi surface as i t i s presently agreed upon by Ashcroft (1963) and Anderson and Lane (1970). As far as an inter-pretation of the experimental results i s concerned, of greatest interest i s the possibility of open orbits, since dramatic changes in the r e s i s t i v i t y properties of a metal are then possible. When open orbits exist i n the plane perpendicular to the applied magnetic f i e l d i t has been shown (see Fawcett, 1964) that the transverse magnetoresistivity increases without limit as the strength of the magnetic f i e l d increases. It w i l l be shown that an increase i n the magnetoresistivity implies a decrease.in the.helicon amplitude and hence an anisotropic magnetoresistivity could account for the observed ani-sotropy of the helicon data. The actual Fermi surface of aluminium differs only s l i g h t l y from the nearly-free-electron surface appropriate to aluminium. Harrison (1960) has calculated the nearly-free-electron surface for aluminium, a valence three metal with face-centered cubic symmetry; the results are shown as Figure VI-1. The small corrections to the Harrison model which are needed to f i t the available de Haas-van Alphen data have been calculated by Ashcroft (1963) and confirmed by Anderson and Lane (1970). The fourth zone pockets of electrons disappear and the third zone becomes dismembered into separate toroidal surfaces on the square faces of the B r i l l o u i n zone; see Figure VI—2. Ashcroft (1963) concludes as follows regarding the Fermi surface of aluminium. 108 Figure VI-1. Nearly-Free-Electron Fermi Surface of Aluminium After Harrison (1960). 109 3 f « ZCHE-REGICNS Of a'NS 4th ZONt-POCKETS OF EL'NS 1 1 0 Figure VI-2. Third Zone Fermi Surface After Ashcroft (1963), as drawn by Anderson and Lane (1970). I l l 1 1 2 "The presence of the 24 points of contact onto the third zone from the second zone surface must lead to the po s s i b i l i t y of some curious open or extended orbits. The effect of breakthrough i s d i f f i c u l t to estimate for an arbitrary f i e l d direction. For fields i n the (100) plane, however, extended i f not open orbits w i l l be very probable i n the present model." Page 2081. It i s easier to see the p o s s i b i l i t i e s for open orbits i f the Fermi surface can be visualized i n an extended zone scheme. To f a c i l i t a t e this visualization, Figure VI-3 shows how the B r i l l o u i n zones for a face-centered cubic crystal stack to form an extended zone scheme. Balcombe and Parker (1970) have done a model calculation relating to the magnetoresistivity of aluminium, taking into account the conclusions of the Ashcroft model. They conclude as follows: "The model does, however, provide a qualitative explanation of the fact that the resistance tends to be higher when the f i e l d i s i n a {100} plane than otherwise. Points of con-tact between second and third zones of the Fermi surface occur i n pairs near corners W and the pairs are aligned along [100], [010], and [001] directions; thus, whenever the f i e l d i s i n a {100} plane there i s a third zone orbit which can provide a link between two second zone orbits. In other f i e l d orientations, breakdown can cause a jump from a second zone to a third zone orbit, but since there i s only one point of contact on the l a t t e r , a particle must either remain on the third zone orbit or eventually return to i t s original orbit and continue along i t . So, i n the general case, there w i l l be no extensive network of trajectories and the effects of breakdown on the resistance w i l l be relatively i n s i g n i f i -cant." Page 554. According to Ashcroft (1963) and Balcombe and Parker (1970) the Fermi surface of aluminium may support open orbits i n the presence of a magnetic f i e l d i n the {100} plane. Although the Fermi surface i s not multiply-connected, magnetically-induced interband transitions (called magnetic breakthrough or magnetic breakdown) can cause extended or open orbits to occur. Figure VI-4 shows a central (001) section through the a l -uminium Fermi surface. When the magnetic f i e l d i s aligned with [001], i n the absence of breakthrough, a hole-like orbit FABH.... on the second-zone Fermi surface w i l l occur. With breakthrough, a carrier at B may proceed 113 Figure VI-3. Br i l l o u i n Zones The Br i l l o u i n zones for a face-centered cubic crystal can be stacked to form an extended zone scheme. The region formed by the center,.T, of each zone to the principal points of high symmetry as shown define the l / 4 8 t n symmetry section. The second zone hole surface (Figure VI-1) can be visualized as nearly the same shape as the Br i l l o u i n zone but with a l l the f l a t sides scooped out so that the surface does not touch any zone boundaries. The third zone electron surface (Figure VI-2) can be visualized as separate toroidal surface centered on each of the square faces of the B r i l l o u i n zone. 114 115 Figure VI-4. Magnetic Breakthrough Figure (a) shows a central <100> s l i c e through the Fermi surface (from Balcombe and Parker (1970)). A variety of new orbits are possible when magnetic breakthrough occurs, i n particular the electron-like orbit FABCDEF... is important. Four open-orbit directions are shown as dotted lines. Figure (b) shows a non-central s l i c e for B^ tipped in a {100} plane. The electron-like orbit i s not possible, but two open-orbit directions, one shown as a dotted l i n e and the other i t s mirror image in a parallel s l i c e , are s t i l l possible. 116 117 v i a a hole-like orbit to H or, depending on the transition probability, hop onto an electron-like beta orbit and proceed towards C. At C the orbit may remain electron-like and proceed back to B or i t may make the transition to a hole-like orbit and proceed to D.... A hole at H has a similar set of possible orbits. Thus, depending on the transition probabilities hole-like orbits FABH... may occur or. electron-like orbits FABCDEF... may occur. A network of connected hole-like and electron-l i k e orbits can result in extended or open orbits. The points of contact between second and third zones on the Fermi surface (as discussed by Ashcroft (1963)) are removed when the spin-orbit coupling i s calculated. However, the s p l i t t i n g i s small and can be represented by a breakthrough f i e l d of typical value <v2 kOe (Holtham and Lonzarich, private communication). The data being presented i n this thesis w i l l be interpreted to support the Ashcroft model of the Fermi surface of aluminium. This model has been called into question by Chiang et a l (1970) and by Amundsen and Seeberg (1969). The argument of Chiang et a l depends upon the following statement. "Inasmuch as one can speak only of narrow layers of open trajectories i n the corners W ( i t i s immaterial whether they are the consequence of the presence of bridges or magnetic breakdown), there should be ob-served an anomalously sharp anisotropy of the resistance, the existence of which i s not indicated i n any of the known experiments." Page 1043. This apparent lack of a "sharp anisotropy" i s confirmed by the helicon experi-ments of Amundsen and Seeberg (1969). They use single crystal plates with a RRR of 17,100 and report seeing no anomalous anisotropy. Absorptive Peak Amplitude and Magnetoresistivity To use the statements regarding open orbits quoted from Ashcroft and from Balcombe and Parker to interpret helicon data, i t i s necessary to know the effect of open orbits on helicon propagation. Buchsbaum and Wolff 118 (1965) have shown that a "single band of open orbits causes strong damping" of a helicon wave. This effect has been experimentally confirmed by Grimes et a l (1965) i n a single crystal of s i l v e r . Thus, the sharp minima i n the helicon amplitude that are reported i n Chapter V may be interpreted as caused by the Ashcroft open orbits. The existence of the sharp trough i n the (1,1) peak height when the direction of B is close to a <100> plane has been associated with —o magnetic breakthrough. So far the oscillations of the helicon absorptive peak amplitude as a function of the strength of B q when B^ i s par a l l e l to <100> (see Figure V-16) has not been considered. Oscillations of the trans-verse magnetoresistance i n a four-probe measurement have previously been ob-served by Parker and Balcombe (1968) when B^ was near a <100> direction. These oscillations have been explained semi-quantitatively by Balcombe and Parker (1970) i n a model calculation allowing for magnetic breakthrough effects. In order to,attempt a comparison between the results of Parker and Balcombe and those of this thesis, the helicon data must be converted into quantitative information about the r e s i s t i v i t y tensor. Helicon propagation i n a sphere with isotropic zero-field re-s i s t i v i t y i s accurately accounted for by the theory of Ford and Werner (1973) and the results of Chapter VII. The behavior of the resonant frequency, tom n > the absorptive peak amplitude and the Q of the absorptive peak are well understood. Calculations by Ford and Werner give the following numerical result. The resonant frequency, wi i , of the fundamental, (1,1), helicon mode of a sphere of radius a i s found numerically to be 4.902 c 2R B o on On the other hand, the fundamental mode for an isotropic, semi-infinite slab of thickness 2t has i t resonant frequency given by 119 c 2R B % 011 -TTT*-^* ( 1 +orT) 2 ) <6"2) as can be easily shown using Equation 2-13. For t o c x » l , the square root can be approximated by using the binomial expansion and hence, c 2R B . W l " 4~x~4 x - E F ^ Cl + - ^ f r ) , ( V » 1) . (6-3) c Thus, for a sphere and a semi-infinite slab the qualitative (and semi-quantitative) behavior of the resonant frequency i s the same for both i t s B and to x dependence, o c r For a semi-infinite slab the Q of the resonance i s given by V 1 h c (see Equation 2-14). For t o c x » l , to x Q--f- • (6-5) The comparable formula for an isotropic sphere, derived numerically by the author using the formalism of Ford and Werner, i s to x Q - ~ f ~ * (6-6) For a slab as well as a sphere, the amplitude can be shown to be proportional to Q for Q>>1. Hence, the amplitude of the absorptive peak i s proportional to to cx for large to c x . A l l the above arguments and formulae are va l i d for isotropic systems only. A discussion of the behavior of a sphere with an anisotropic r e s i s -t i v i t y i n a helicon experiment can, at present, be an approximate one only. 1 2 0 We shall make the following assumption. For any given orientation of the anisotropic sphere, i t i s assumed that the wave characteristics are qualitatively and semi-quantitatively the same as for an isotropic sphere. To account for anisotropic behavior, i t i s assumed that to a f i r s t approx-imation only the values of the r e s i s t i v i t y components change, the form of the equation remaining the same as for the isotropic case. : Assuming that the results of Chapter V can be described by equations 6-1 and 6-6, some of that data shall now be translated into i t s equivalent i n r e s i s t i v i t y tensor component variations. It has been noted before (see, for example, Figure V-17) that the resonant frequency i s very nearly isotropic, which implies that p i s essentially isotropic (cf. equation 2-16 for the xy case of a slab). Given this, the Q and hence the absorptive peak amplitude both hold a reciprocal relation to the average of p and p which w i l l be xx yy denoted by p: ( pxx + p y y ) / 2 • <6"7> A l l the amplitude data gathered at a fixed frequency and magnetic f i e l d can now be inverted to give an average transverse magentoresistivity. As an example, the reciprocal of the contour map data, Figure V-12, appears as a contour map of the r e s i s t i v i t y , p , i n Figure VI-5. A l l the comments which were made regarding the helicon amplitude now hold true for the average transverse magnetoresistivity i f maximum and minimum concepts are exchanged. A fixed frequency has been specified since the induced voltage i n a c o i l i s proportional to the frequency of the magnetic moment causing i t . For data such as Figure V-15, showing the oscillatory behavior of the helicon ampli-tude as the magnetic f i e l d , and hence the resonant frequency, i s varied, this frequency dependence must f i r s t be divided out before inverting the data. The data for B along the three symmetry directions <111>, <100>, and <110> 1 2 1 Figure VI-5. Resistivity Anisotropy Shown i s the variation of the effective r e s i s t i v i t y , p, over the entire zone, as a contour map. These data are derived from the helicon amplitude contour map, Figure V-12, where B = 35 kOe and have been normalized to the peak value. 122 123 Figure VI-6. Magnetoresistivity Shown is the magnetic f i e l d dependence of the effective r e s i s t i v i t y , p, for the three high-symmetry directions, <100>, <110>, and <111>. The data for a l l curves are continuous. The error bars are not an indication of the scatter i n the data but rather show the uncertainty associated with inverting data when the zero i s not located exactly. 124 125 has been converted in Figure VI-6 to a plot of a quantity proportional to p versus the strength of B . o For an isotropic metal, u> T can be written as c W T = » R B a , (6-8) c o where R i s the Hall constant and cr the conductivity of the metal at a magnetic f i e l d strength B q. Using the known value of R (see, for example, Amundsen and Jerstad, 1972) and the value of a obtained with the zero-field r e s i s t i v i t y measurement described i n §V-A for Al-1 yields to T - 1.45 B , (B i n kOe) (6-Q) c o o . . \*» •> / At the maximum f i e l d value of 35 kOe, this implies u_x = 50.5. Using equation 6-6 for the Q of the helicon resonance, the experimentally obtained value of a) x for B = 35 kOe and aligned along <111> i s 20, along <100> c o about 16. This difference of a factor of 2.5 for B // <111> or 3.2 for B —o —o // <100> is attributed to the magnetoresistivity, p ( B Q ) / p o , of the sphere. Feder and Lothe (1965) have calculated the transverse magnetoresistance for aluminium assuming a constant relaxation time and a l l orbits closed. Their calculation i s for B // <100> and p(B )/p i s about 4.0 for u> T 10. This —o o o c magnitude of the transverse magnetoresistivity i s typical of an aluminium sample with O^T-20. Balcombe (1963) reports values of 2.5 - 3.0 and Balcombe and Parker (1970) report values around 4.0 for p(B { j ) / p Q with B^ // <100>. At this point a comparison between the data of this thesis and that of Balcombe and Parker (1970) can be made. Figure VI-7 shows the transverse magnetoresistivity, p'(B )./p for // <100> as reported by Balcombe and Parker and that obtained from the helicon amplitude anisotropy as shown in Figure VI-6 and converted into magnetoresistivity as described 126 Figure VI-7. Magnetoresistivity Comparison This i s a comparison between the four-terminal measurement of the magnetoresistivity, p(B Q)/p o, for // <100> by Balcombe and Parker (1970) and that obtained from these helicon experiments. 127 128 previously. Of interest are the observations that the oscillations match, that the amplitude of the oscillations are relatively larger for the data of Balcombe and Parker, and that the value and the slope of the magnetoresisti-vi t y i s less for the data of this thesis. Regarding the oscillations l i t t l e can be said i n addition since the only well-known numerical quantity c a l c u l -able i s the period, and there i s agreement on this point. The lower slope for the magnetoresistivity using the helicon data as compared to that obtained from a four-probe measurement i s worthy of note. Amundsen and Seeberg (1969) comment that the various experiments with aluminium "...show a great variation which cannot solely be attributed to the difference in sample purity. It i s interesting to note that the helicon measurements ' give a magnetoresistance which shows signs of satura-tion i n high fi e l d s .•.."" ...It i s probably easier to prepare and mount samples with a small number of de-fects i n a helicon experiment than in a direct current experiment due to the considerable difference i n sample size". Page 702. Discrepancies i n the slope of the magnetoresistivity have been observed i n potassium; see for example Penz and Bowers (1967). It i s of interest that the recent work of Werner et a l (1973) using a potassium sphere i n a helicon experiment shows complete saturation. Amundsen and Jerstad (1974) have i n -vestigated the slope (Kohler slope) of the transverse magnetoresistivity i n aluminium and found that the slope was particularly sensitive to strain. The greater the strain the greater was the slope. This confirms the results of this thesis. The spherical sample shape i s more easily mounted with minimal strain than i s a wire-like sample with leads attached. Amundsen and Jerstad (1974) also found that removing (by electropolishing) dislocations near the surface (presumably caused by spark cutting) gave a lower slope. A sphere has the lowest surface area to volume ratio and might be expected to show a lower slope i n the magnetoresistivity. 129 Dispersive Signal and Hall Constant So far only the absorptive peak amplitude anisotropy has been analyzed to give a measure of the transverse magnetoresistivity and i t s ani-sotropy. Figures V-9, 16, and 17 show some sharp anisotropics (of magnitude 1%) and some f i e l d dependent features of' the resonant frequency and of the dispersive signal which can be related to the resonant frequency. Equation 6-1 shows that an anisotropy of the resonant frequency can be due to a variation i n one or a l l of a, RB , or to x. The variation i n the radius, a, o c of the sphere should be very slight given that before a light chemical etch, the sphericity was measured to be 1 part i n 10 5. One might expect a small, slow change in the "effective" radius with rotation due to misalignments and this could account for the lack of perfect symmetry seen i n Figure V-10 (note that the dispersive signal was amplified lOx). Discounting a variation i n the radius, the effect of the two quantities R B q and u^x must be determined. As shown earlier (Figure VI-5), the anisotropy i n to^ x (~20) i s of the order of 10%. Using equation 6-1, this would imply an anisotropy of ± 0.07 Hz for to.. - 50 Hz i f RB remained constant. It i s clear from the data that this l.l o i s almost an order of magnitude too small to account for the observed resonant frequency anisotropy, especially for B^ near <100>, see Figure V-18. The anisotropy must then be due to an anisotropy of RB Q« It remains to at-tribute the major anisotropy to R or B . Since B =H +=4TT M and H was kept o o o 3 o constant, a variation i n B q would be due to a variation i n M. Quantum os-c i l l a t i o n s i n M would be expected from the dHvA effect and these would appear as an os c i l l a t i o n in B q and hence i n the dispersive signal or the resonant frequency. Oscillations of the resonant frequency have been observed for B^ p a r a l l e l to <Clt)0> and the dHvA effect may be the cause. This component (20%) of the total frequency anisotropy i s of the order of magnitude that i s also explicable by increased damping which causes an oscillatory behavior i n the 130 absorptive amplitude. A component (80%) of the increase i n the resonant frequency at O-00)> i s non-oscillatory and i s interpreted as representing an increase i n R and not due to the dHvA effect or to increased damping. Extending the arguments related to breakthrough i n the section discussing the Fermi surface of aluminium, i t i s an increase of R that would be expected for B^ // <100>. A simple formula for the Hall coefficient i s given by R - 1 / ( i ^ - n e) e c (6-10) where n^ and n g represent the densities of carriers with hole-like and electron-like properties respectively. When there i s no breakthrough the hole-like carriers on the second zone Fermi surface dominate. With break-through, as described previously, some hole-like carriers change to electron-l i k e carriers. This change would effectively decrease the number of carriers and hence increase R and so also the resonant frequency. The mean frequency shift near <100> i s consistent with roughly 0.01 carriers per atom under-going magnetic breakthrough at 35 kOe. Comparison with Other Experiments M e r r i l l (1968) was the f i r s t to attempt an extensive study of the anisotropy properties of various metals using helicon waves. The method involved using a cylindrically-shaped, single crystal of metal and applying transverse to the axis. By rotating the crystal, the amplitude, frequency and half-width of the helicon resonance (standing-wave) could be obtained for oriented along different crystallographic directions i n the plane determined by the geometric axis of the cylinder. The analysis of the results i s done by s t r i c t analogue with the formalism of Penz (1967) for a semi-i n f i n i t e slab of anisotropic r e s i s t i v i t y . A comparison between the helicon 131 Figure VI-8. Merrill's (1968) Data for Helicons i n Al Cylinders 132 BOL ... - ..„.•..•..—••••••..... [lOO] Al 25HG onplriuda E t fxl] . 24 leta) . 20 90 $<deg> ISO 133 amplitude anisotropy and the known magnetoresistivity of s i l v e r allows M e r r i l l to claim that a "... comparison can be made between the angular widths and separations of the interesting features of helicon-anisotropy curves and stereograms of the magnetoresistance. The agreement on angular widths and separation i s very good." Page 719. Figure VI-8 shows the aluminium data from M e r r i l l (1968) as pre-sented i n that paper (Figures"7 and 8 i n the original, p. 721). The results are rather disappointing; about a l l that can be concluded i s that there i s anisotropy. In particular, M e r r i l l notes the lack of "perfect rotational symmetry" and the lack of perfect reproducibility. In each of the figures an important observation can be added i n the lig h t of the more extensive study of this thesis. F i r s t , the symmetry-of the anisotropy data indeed does not correspond to a cylinder with i t s axis along <100>; i t does correspond to a cylinder with i t s axis along <110> (cf. our Figure V-14). Secondly, when presenting the BQ-dependence of the helicon amplitude for B^ p a r a l l e l to <100>, attention i s not drawn to the oscillations of the amplitude, which i n a l l fairness are not well-defined i n the data. It i s these oscillations which have been analysed previously i n this thesis and compared to the data of Balcombe and Parker (1970). Amundsen and Seeberg (1969) have studied helicon propagation i n single-crystal plates (20 x 20 x 1 mm) of aluminium (RRR~17,100) whose crystal directions perpendicular to the surface area [3,1,1] , [1,1,0] ± V , [1,1,1] ± 1°, and [1,0,0] ± 2°. Contrary to expectation, these samples showed a behavior typical of an isotropic-resistivity slab when the magnetic f i e l d ^ was rotated away from the normal. These results are to be contrasted to those of M e r r i l l (1968) and this thesis. Amundsen and Seeberg conclude as follows regarding the discrepency between their results and those of M e r r i l l . "We are therefore inclined to think that the reported anisotropy is due to a side effect and i s not a true property of aluminium." Page 701. 134 The existence of a large anisotropy of the absorptive peak ampli-tude and of a small anisotropy of the resonant frequency are confirmed by the experiments of this thesis. As argued e a r l i e r , this anisotropy of the resonant frequency, especially near <100> i s too large to be accounted for by increased damping only. Our results thus tend to confirm the v a l i d i t y of an anisotropy of the Hall constant; a result also indicated by Merrill's ex-periments. The magnitude of the anisotropy of the frequency i s about 1% i n our experiments and has been measured with reproducibility and correct crystal symmetry. M e r r i l l remarks that his results were not "perfectly re^ producible" and did not show the expected crystal symmetry. He observed an anisotropy of the resonant frequency of about 5%. To conclude this section, some basic results w i l l be restated. The anisotropy of the helicon absorptive peak amplitude has been interpreted as magnetoresistivity data and as such strongly supports the Ashcroft model of the Fermi surface of aluminium. Magnetic f i e l d dependent oscillations of the magnetoresistivity for parallel to <100> compares favourably with the four-probe measurement by Balcombe and Parker (1970). This oscillatory magnetoresistivity has been interpreted by Balcombe and Parker as due to magnetic breakthrough. If this i s a correct interpretation then i t would be expected that there be a small anisotropy of the Hall constant (increasing for B_ parallel to <100>)and an oscillatory dependence of i t on f i e l d strength as the change from hole-like to electron-like orbits are made possible by breakthrough. The breakthrough interpretation has been strengthened by the observations i n this thesis that the resonant frequency behaves i n conformity to the expected behavior of the Hall constant. It has been argued that the most probable influence on the frequency i s the Hall constant. It i s not presently understood why Amundsen and Seeberg (1969) and Amundsen and Jerstad (1972) have not observed either the sharp anisotropy at <100> or the o s c i l -latory dependence of the absorptive peak amplitude i n their helicon experi-ments using thin slabs of aluminium. 135 Quantum Oscillations: Comparison with the de Haas-van Alphen Data Besides the quantum oscillations shown i n Figure V-16 and inter-preted as oscillations i n the magnetoresistivity due to magnetic breakthrough in Figure VI-6 and the oscillations of the resonant frequency shown i n Figure V-17 and interpreted as a variation i n R, there i s another mani-festation of these oscillations. Figure V—19 shows oscillations superimposed on the helicon resonance obtained by fixing the frequency and varying B Q . These oscillations have been reported previously by Grimes (1964) and Vol ' s k i i and Petrashov (1968). V o l ' s k i i and Petrashov (1968, 1971) show that the oscillations are not due to oscillations i n the magnetoresistivity but are a direct manifestation of the dHvA effect showing i t s e l f as a variation i n the helicon wavelength due to the oscillatory term M ( B q ) i n ' B =H -H^ ATT M(B ) . Figure V-19 clearly shows that these oscillations are o o 3 o minimal at the helicon amplitude and hence are not due to the damping of the helicon wave. Amundsen and Jerstad (1972) using single-crystal aluminium plates (10 x 10 x ~ 1 mm) oriented along <111> ± 2° and <100> ± 2° observe quantum oscillations superimposed on the helicon resonance curve as B q i s varied. The measured frequencies of the oscillations i n the orientations <111> and <100> agree well with Larson and Gordon (1967). Grimes (1964) and V o l ' s k i i and Petrashov (1968, 1971) have previously studied these oscillations using helicons i n aluminium slabs. Figures V-9, V-10, and V-19 are meant to show the u t i l i t y of using a spherical sample to obtain the dependence of the frequencies on orientation. Figure VI-9 shows the analysis of the period (Oe - 1) of the fast oscillations shown i n Figure V-19. A comparison with the data of Gordon and Larson as presented by Ashcroft (1963) and the Fermi surface calculations of Ashcroft (1963) i s provided. 136 Figure VI-9. Quantum Oscillations This i s the analysis of the fast oscillations shown in Figure V-19. The triangles are the data of this thesis, the circles that of Gordon and Larson as quoted by Ashcroft (1963) and the dashed lin e represents the calcula-tions of Ashcroft (1963). 137 138 Chapter VII HELICONS IN A SPHERE-GENERAL ISOTROPIC CASE In this chapter helicon propagation i n a non-magnetic sphere of -material described by a scalar conductivity, c, and an arbitrary value of to cT i s discussed. We start with a solution of Maxwell's equations i n terms of cylindrical functions. These functions are decomposed into an i n f i n i t e sum of known spherical functions which match i n form the multipole decompo-si t i o n of the vacuum f i e l d s . A general superposition of the cylindrical waves constitutes the solution for a spherical boundary and the weight function (or alternatively, the response function) i s written as a ratio of two i n f i n i t e determinants as shall be shown. The numerical solution leads to some round-off error d i f f i c u l t i e s . Concurrent with the work of this thesis, Ford and Werner (1973) have also solved the boundary-value problem of helicons i n an isotropic-r e s i s t i v i t y sphere. They use only spherical coordinates and vector spherical harmonics. To satisfy Maxwell's equations plus the boundary conditions, their method requires a doubly i n f i n i t e sum of functions. The set of functions which solve Maxwell's equations are obtained via an eigenvalue problem. The second set of functions, the weight functions, i s involved in the superposition of these solutions i n order to satisfy the boundary conditions. Their weight function (or alternatively, their response function) is likewise written as a ratio of two i n f i n i t e deter-minants, the elements of which involve the previously-mentioned eigenfunc-tions. A numerical solution i s obtained by truncating the determinants. Their method is highly mathematical and does not lend i t s e l f to an easy qualitative understanding of the resonant mode structure. A more detailed comparison of our formalism with that of Ford and Werner w i l l be given at the end of this chapter. 139 Somewhat earlier, Francey and Gates (1968) presented an approx-imate solution to the zero-resistivity case of helicons i n spheres. They treat the "longitudinal", m = o, geometry discussed by Ford and Werner (1973). Of interest i s the fact that l i k e this thesis they also start with a superposition of cylindrical waves. Their results differ substantially from the exact results given by Ford and Werner. Unlike the other methods, our approach has the feature of allowing a straight-forward, analytic, albeit approximate, physical interpretation of the resonant mode structure of helicons i n a sphere. This approximate model has been presented i n §II-C. Equations 2-22, 2-24 and 2-25 of §II-C constitute the general solution i n cylindrical coordinates of the helicon f i e l d inside the con-ducting medium. To f a c i l i t a t e matching the magnetic fields at the boundary, r=a, of the sphere, the f i e l d inside the sphere i s now written as a sum of functions of spherical coordinates. The formula for the decomposition i s obtained by expanding an appropriately weighted sum of plane waves f i r s t In cylindrical functions and then i n spherical functions as follows. A A Let the polar angles for r and 3 be r - (6,« ; 6 -• (o,*) . (7-1) Thus, 8_«r_ = gr[cos 0«cos a + sin 0-sin a.cos (^ -<J>)] . (7-2) Consider the following expansion in cylindrical functions 140 2TT y^*exp(i£.r).exp(-is(^ -(|.)) d ( 7 - 3 ) o 2ir +«» = 27 exp(igr«cos e^ cos-a) y* E i P J ( 8 f s i n 6.sin a) o p=» P (7-4) exp(i(p-s)0M>)) d(iM) = exp(iBr.cos 0«cos a) i J (Br-sin 0«sin a). (7-5) The f i r s t step follows from the generating function for c y l i n d r i c a l Bessel functions, J^, and the second step follows from the requirement that p=s for non-zero contribution upon doing the integration. The same function (7-3) expanded i n terms of spherical functions i s ^y"eacp(lB_.r). exp (-is d(*-$) (7-6) o 2ir - 2 f E i ^ * ( 3 ) Y J (r) j £(er).exp(-is(^-<j))) d(*-*) (7-7) o £,m = ^ i Z (2-£+l) 3 £(gr) p | ( c o 8 a) P|(cos 0), . (7-8) where Y^ i s a complex spherical harmonic function, j ^ i s a spherical Bessel s function, P^ i s a Legendre Polynomial, and * represents complex conjugation. Combining these two results gives, for integer s, exp(iqz) J (yP) -E/" s 1 "(M <*-s).'(2£KL) p s , . ' _ S , FL, < 7 ~ 9 > 141 where X = cos a, z = r cos 6 , p 2 r sin 6 , (7-10) 1= M , The identity (7-9) i s valid for complex values of the arguments. The cylindrical solutions, equations 2-28 of §II-C, of the helicon wave equation can now be written i n terms of the same spherical functions that appear i n the vacuum f i e l d s . We now l e t q,@, and y be the quantities introduced i n equations 2-18 to 2-25 (the solution of the helicon wave equation i n c y l i n d r i c a l coordinates). The interrelations (7-10) are consistent with those defined i n Chapter II. The general solution of the wave equation consists of an arbitrary superposition of the cylindrical solutions (2-24), which we re-produce here for convenience. 2 B; = ^ A™ J m(Y nP) exp|i[mj{ + qz - cot]} , n = 1 (2-24)' 2 B±=/C M n ( S n ± q > J m ± l ( Y n P ) C"*1) 4 +'~ «•":]} , n=l Y n Experimentally one fixes 10, B q, and o and therefore q Q (=. i4ircoa/c2) and Li_x (= BQ0/nec) are fixed. In the dispersion relation (2-22) (reproduced here for convenience) there i s one free parameter which we choose to be X. « v j T % \ . (2-22) 142 A l l other parameters 3, y» a i*d q are specified by equations 7-10 and 2-22. Hence, consider an i n f i n i t e sum over values, of \ for the f i e l d inside the sphere as S ^ ^ n X J q r, 6, •) ' . ' (7-11) n=l o, n — n, o, ^ where 6 i s eliminated by using n fj2 = q 2 / (1 - ia) TX ) . (7-12) n o c n The solution to the vacuum wave equation i s well-known and, i n component form, i s given by Bvac z •P^(cos 6) exp{ilm<j»-cut]} , •P^ (cos 8) exp{i[(m+l)<J)-o)t]} , vac _ _m ,a.l+l. ,^m-lf^-m) (l-m+1) (l-m+1) fl3* •^"^•(cos 8) exp{i[(m-l) <J)-a)t] >^ , „m where F^_^ i s the multipole strength to be determined by matching at the boundary. The constant 'a' i s taken to be the radius of the sphere and the solution i s valid for r->a. 143 For the case of a nonferromagnetic metal of f i n i t e conductivity there can be no surface currents and hence the appropriate boundary condition i s that a l l components of B_ are continuous; since we are con-cerned with a quasistatic problem, the boundary conditions on the electric f i e l d s are automatically satisfied (see Legendy, 1964). With the c y l i n d r i -c a l waves written i n terms of spherical functions, the coefficients of each P™ (cos 0) exp i(m<J>-cot) term can be equated independently for each -value of Z and m. This gives Z W X . V Vena> &Xn> -S-ifco' 3 - $ (7-14) w h e r e b m i s t h e a p p r o p r i a t e c o m p o n e n t o f t h e a . c . e x c i t a t i o n f i e l d a n d m - l-m (21+1) (l-m) I J I rm _ (21+1) q - m - 1 ) ! 1 ~ X n jn+l« . 1 C , " 1 M ! P£ ( V * ( 7~ 1 5 ) n m £ - m ( 2 £ + l ) C £ - m + l ) I 1 + X n _m-l,, , n for i - 0, 1, 2 a n d „ m _ m r ( £ - m ) ( £ - m ) . ' ~|% ^e.z _ b C _ 1 ) [ 4 ^ ( £ - m - l ) i j v * = + w i < > m | " ( £ - m > tf-^D ( l - m + D f f t e^,- b i ; L8^ c-fcha-i): ~ J for £=1,2,3........ 144 Experimentally one i s interested i n an excitation f i e l d which i s uniform over the sample volume. This allows three independent excitations a l l of which have & =.0, as follows: 1. m = 0 and the external excitation couples to B inside, z 2. m = ±1 and the external excitation couples to B inside. + In a l l cases the uniform external f i e l d implies, i n our notation, that only L even modes are excited. The usual transverse excitation f i e l d i s linearly polarized and hence i s a superposition of the two modes, m = ± 1. The i n f i n i t e set of equations implied by equation 7-14 has, for each m, one non-homogeneous equation, -£ = 0, and three homogeneous equations for each of Z- 2, 4, 6, ... . The strength of the vacuum multipole can be eliminated (temporarily) for each £ £2 and this leaves, for each m, a set of equations of the form E w X , V V 6 n a ) Go C V - b > o-m for L - 0, and £ W m(q o/ n) J £(3 na) K» (X n) = 0 , (7-18) 00 n=l o, n n -c n for £ = 2, 4, 6, ... . The functions and L^m are obtained from equation 7-14 by the elimination of for each £. A possible definition of K£ m and L £ m i s G,m •?.c\.> ~=^F - ^ V (7-19) L™ (X ) = = ^ - = ^ . *. • n „ m „ m z By Cramer's Rule J 0 ( e ia)G m a i ) , ... . J 0 ( 6 n _ I a ) G m ( X n _ 1 ) , , • V*n+la>GX+l>> ' J 2 (B ia)I^(X 1), j^C^a^CXj) , • J a ^ l - ^ n - l ) . 0 , 3 2 ( ^ ) ^ 1 ) . • J 2 C e n_ 1a)L m(X n_ 1) . . . » j (6 a)G™(X n), ... o n o n , j 2(3 a)K m(X ), ... 1 / n *• n (7-20) £ 146 Experimentally, one usually uses a c o i l wrapped around the sample to measure the response to the excitation. This c o i l w i l l measure a combina-tion (depending on the c o i l geometry) of the vacuum multipole strengths To solve for °ne can use any of the three equations for each £. Using the B component (q Q) l>~ n l f ( q X_) o, n This can alternatively be written as a ratio of two i n f i n i t e determinants, the numerator of which differs from equation 7-20 as follows: j £ ( 3 i a ) j 2 ( 3 i a ) ( X X ) , . J2(Bia) L m ( X X ) , , j 2 ( 3 n a ) ( X Q ) , j 2 ( 3 n a ) L m (X Q) , (7-21) The i n f i n i t e determinants must be truncated i n order to solve them numerically. This was done by choosing a f i n i t e (~ 15), uniform sampling of the * n's and retaining that same number of rows in the determinant. In this regard the procedure is similar to that of Ford and Werner (1973) . Our formalism i s numerically more complicated since our parameter, X, i s complex (see equations 2-22 and 7-10) although there i s a constraint linking the real and imaginary parts. The corresponding parameter used by Ford and Werner is purely real. Serious numerical d i f f i c u l t i e s were encountered i n the computerized calculations. These problems stem from the conflicting symmetries, cylindrical waves and a spherical boundary. What was observed from the calculations was that the weight function Wm(^n)' w a s typically of opposite sign for adjacent 147 c y l i n d r i c a l waves (ie. adjacent values of X ). If many ^ n's were used the magnitude of the Wm(^n) increased rapidly and the resulting round-off errors due to subtraction of large numbers to yield small ones proved too great to counter even with double-precision subroutines. A typical calculation of the dipole moment using eleven values of X^ i s shown as Figure VII-1. The value wcT = 60 was chosen in order to compare with the results of Ford and Werner. The f i r s t mode (1,1) and i t s subsidiary (1,3) are f a i r l y well defined. The mode (1,5) is very broad and needs more terms to define i t . The (3,1) mode i s also shown, f a i r l y well defined. The comparable calculation by Ford and Werner (1973) i s shown as Figure VII-2. Note the difference i n naming of the modes, Ford and Werner count nodes, we count half-wavelengths. An indication of numerical "noise" i s indicated i n Figure VII-1 by an arrow. This particular 'unwanted' bump results from round-off errors because the subroutine calcula-ting the spherical Bessel functions changed algorithm at this point and changed the accuracy of the last few decimals. Table VI-1 gives a comparison of the results of the two Figures VII-1 and VII-2. To within the accuracy allowed by the round-off errors, the results agree. 148 .Figure VII-1. Helicon Dipole Moment The solid curves represent the results of the calculation using cylindrical waves. The resonances are labelled according to the scheme outlined i n II-C. The amplitude i s given in units of Jfibja3, where bj is the excitation f i e l d , as a function of 4iru>a a 2 / c 2 for to x = 60. Eleven terms were used in the c calculation. The circled dots and crosses are results from Ford and Werner (1973) (sketched by a dashed line) and are presented for comparison. A complete diagram of their results appears as Figure VII-2. 149 h-7 —I I - I— 600 1200 1800 Arrtocr a2/c2 150 Figure VII-2. Ford and Werner's Data This figure i s the same as that labelled Figure 3 i n Ford and Werner (1973), p. 3710. 151 c d c r » 6 0 N = I5 D IPOLE •4.0 Real -6.0 •8.0 J L j L J L I 1 J L 1 J L 0 1000 2 0 0 0 Airwa a 2 /c 2,^2 3 0 0 0 152 Table VII-1. Comparison of Calculations The results of the calculations of Ford and Werner (1973) and those described i n this chapter are compared. The resonant frequencies and the width are given in units of 4ir wo a 2/c 2. 153 RESONANT FREQUENCIES F & W THIS WORK (1,1) 300 30 0 ± 5 (1,3) 64 0 650 ± 20 (3,1) 18 20 1 830 ± 6 0 A M P L I T U D E RAT IOS t 1 f 1 ) \ ( 1 , 3 ) 9.6 9.0 ± .5 d , 1 ) \ ( 3 , 1 ) 4.35 4.3 ± .2 W I D T H • (1,1) 60±10 70±10 154 Chapter VIII CONCLUSIONS This thesis deals mainly with two aspects of helicon waves in metals. Chapter II and VII present a new model and the related calculations for solving the boundary-value problem of helicons in a spherically-shaped sample of an isotropic metal. Chapter IV, V and VI deal with helicon ex-periments using single-crystal spheres of aluminium and with interpretations of the data collected. A third concern, a minor one from the point of view of this thesis, i s that of helicons propagating i n a ferromagnetic material such as nickel. The results of this investigation have been b r i e f l y sum-marized in Appendix A. Previous to this thesis, Rose (1965) had observed helicon resonances in spheres of sodium. However, no attempt was made to solve the related boundary-value problem. Concurrent with the work leading to this thesis, but unknown to us at the time, Ford and Werner (1973) found a numerical solution to the problem of helicons i n a sphere of an isotropic-r e s i s t i v i t y metal. The approach taken i n this thesis differs substantially from their approach. In Chapter II, Section C, a model i s presented for an approximate solution to this boundary-value problem for the case of i n f i n i t e conductivity. It was discovered that a single cylindrical wave, character-ized by two parameters related to the helicon wavelength along the static magnetic f i e l d , E L , and perpendicular to the f i e l d , could describe the resonant frequencies and the helicon f i e l d distribution to within about 10%. The model predicted a doubly-infinite set of resonances. The major series of resonances i s characterized by having one-half of the wavelength perpendicu-lar to the magnetic f i e l d approximately equal to the diameter of the sphere. Members of the series di f f e r from each other by the wavelength along the 1 5 5 f i e l d direction. With respect to the behaviour of the helicon f i e l d along Bg, the major series of resonances strongly resembles the series of modes i n a slab with B^ normal. In the model, associated with each of these major resonances i s an i n f i n i t e set of subsidiary resonances differing from each other by the wavelength perpendicular to B q. The experiments reported i n this thesis confirm some aspects of this model. The major series of resonances was observed and one or more subsidiary resonances were observed related (from the point of view of the model) to each major one. Werner et a l (1973), using a very pure sample of potassium have been able to confirm that for the f i r s t major resonance i n their experiments there are at least five subsidiary ones. Chapter VII deals i n an exact manner with the more general case of a f i n i t e scalar value for the r e s i s t i v i t y . A superposition of c y l i n d r i c a l waves was used to match the helicon f i e l d with the excitation and the vacuum fiel d s at the boundary of the sphere. The calculation was done via computer and numerical d i f f i c u l t i e s were encountered. Nevertheless, the calculations showed quite clearly the major resonance and i t s f i r s t subsidiary resonance with the half-width, frequency and peak amplitude being accurately defined. The experiments which were done provided a check on the u t i l i t y of the isotropic model. The sample used showed some anisotropy. Even so, the predictions of the approximate model were validated and the calculation for a general scalar r e s i s t i v i t y correctly predicted the resonant frequencies, the peak shape and the relative amplitude of the major resonance and i t s subsidiary. Of greater importance i s the study of the anisotropy of the single-crystal aluminium sphere using helicon waves. These experimental studies resulted in several interesting findings. As a whole the results tended to confirm the open-orbit and magnetic-breakthrough predictions of the Ashcroft 156 (1963) model of the Fermi surface of aluminium. This i s of especial im-portance given that, as explained i n Chapter VI, recent helicon experiments using slab-shaped samples have not detected the expected anisotropic features and for this reason Amundsen and Seeberg (1969) and Chiang et a l (1970) have cast doubt on the v a l i d i t y of Ashcroft's model. For the f i r s t time a spherical sample was used to study i n detail the helicon characteristics for the f i e l d , B^, adequately sampling a l l i n -dependent crystallographic directions i n a metal. Given the present understanding of helicon waves in isotropic media, the helicon characteris-t i c s were interpreted as magnetoresistivity data. The results compared favourably with the four-probe measurement done by Balcombe and Parker (1970) dealing with an oscillatory magnetoresistivity due to magnetic breakthrough when B^ was para l l e l to <100>. As well the anisotropy of the Hall constant was noted for // <100>. A contour map showing the variation of the transverse magnetoresistivity over the l/48th stereographic projection of orientations of the magnetic f i e l d was obtained. This i s the f i r s t time, known to us, that the same sample was used to study a l l crystallographic directions in a magnetoresistivity experiment using helicons and the results obtained i n these experiments were used to reinterpret seemingly erratic results obtained by M e r r i l l (1968) for a study of helicons in aluminium cylinders. An effect which had been predicted by the Ashcroft (1963) model of the Fermi surface of aluminium has been observed for the f i r s t time. The observation was that the absorptive peak amplitude was sharply decreased everytime B^ was i n a {100} plane. This corresponds to the Ashcroft predic-tion that open or extended orbits should be possible whenever B^ i s i n a {100} plane. Our experiments indicate that this band of open or extended orbits i s no more than 1° wide, and i t would be easy to miss detecting them 157 except when i t i s possible to collect data continuously. In these experi-ments this was possible by rotating the magnetic f i e l d about the spherical sample. The experiments and theory presented i n this thesis suggest several possible areas for further investigation. A most obvious unfinished area i s that of helicons in ferromagnetic materials. With the present understanding of helicons in spheres i t should now be possible to use and interpret helicon experiments using spherical samples of a ferromagnetic material. The results reported in Appendix A indicate that some interesting findings may result from a thorough investigation of the interaction of helicons and magnons. Of consequence might be a more complete understanding of the nature of spin waves in a f i n i t e sample as well as a recording of the magnetoresistivity of the crystal and the associated concerns of explicating the various re-s i s t i v i t y mechanisms and Fermi surface models. The difference between helicon experiments using an itinerant-spin material such as nickel and a more localized-spin material such as gadolinium would be of interest to i n -vestigate. The helicon experiments using a sphere of aluminium are of impor-tance in so far as they help explain the r e s i s t i v i t y properties of aluminium. However, any metal of sufficiently high purity which can be grown as or shaped into a single-crystal sphere can now be studied using the methods and interpretations explained in this thesis. For instance, a much purer sample of aluminium would be of interest to use to extend the results of this thesis. 158 In studying the anisotropy of the r e s i s t i v i t y of a material, one technique which has not yet been tried may be of interest. In the geometry used by Ford and Werner (1973), the magnetic field,-B^, and a l l excitation and detection coils were colinear. If a material with scalar r e s i s t i v i t y i s used a l l induced moments w i l l also be i n the same direction. A material which does not have a scalar r e s i s t i v i t y might be expected to show a response normal to the direction of B . To excite par a l l e l to B and detect normal -o —o to could possibly allow a purely scalar r e s i s t i v i t y material to be dis-tinguished from an anisotropic one. This technique might profitably be ap-plied to a sphere of sodium to unravel the d i f f i c u l t i e s associated with detecting an anisotropic Fermi surface, the anisotropic part of which i s locked to the magnetic f i e l d direction. One aspect of the theoretical considerations dealing with the boundary-value problem is completed. The case of helicons propagating i n a non-magnetic, metallic sphere of scalar r e s i s t i v i t y has been sat i s f a c t o r i l y solved and understood by the work of Ford and Werner (1973) and this thesis. The case of an anisotropic r e s i s t i v i t y has not been solved. This i s an important problem since the greatest benefit derived from using a sphere i s to study the anisotropy. In this thesis the approximation of treating helicon waves with i n a l l orientations as being described by the equations derived for an isotropic material with only the r e s i s t i v i t y para-meters varying was used. Some of the results obtained via this interpretation were substantiated by direct magnetoresistivity measurements. It would be significant to be able to interpret the data more exactly and to that end the anisotropic r e s i s t i v i t y case must be solved. 159 BIBLIOGRAPHY Aigrain, P. Proc. Intern. Conf. Semicond. Phys. Prague, 224 (1960) Amundsen, T. and Jerstad, P. J. Phys. F 2, 657 (1972) J. of Low Temp. Phys. 15, 459 (1974) Amundsen, T. and Seeberg, P. " ' "~ J. Phys. C 2, 694 (1969) Anderson, J.R. and Lane, S.A. Phys. Rev. B 2, 298 (1970) Ashcroft, N.W. Phi l . Mag. 8^, 2055 (1963) Balcombe, R.J. Proc. Roy. Soc. A 275, 113 (1963) Balcombe, R.J. and Parker, R.A. Phi l . Mag. 21, 533 (1970) Buchsbaum, S.J. and Wolff, P.A. PRL 15, #9, 406 (1965) Chiang, Yu.N., Eremenko, V.V. and Shevchenko, O.G. S.P. JETP 30, 1040 (1970) Cohen, M.H., Harrison, M.J. and Harrison, W.A. Phys. Rev. 117, 937 (1960) Datars, W.R. and Cook, J.R. Phys. Rev. 187, 769 (1969) Delaney, J.A. and Pippard, A.B. Rep. Prog. Phys. 35, 667 (1972) Delaney, J.A. J. Phys. F 2, 657 (1972) Feder, J. and Lothe, J. Ph i l . Mag. 12, 107 (1965) Francey, J.L.A. and Gates, D.J. J. Phys. A 2, 710 (1968) Fawcett, E. Advances i n Phys. XIII #49-52, 139 (1964) Ford, G.W. and Werner, S.A. Phys. Rev. B8, 3702 (1973) (see also Werner and Ford (1975)) 160 Grimes, C.C. and Kip, A.F. Phys. Rev. 132, 1991 (1964) Grimes, C.C., Adams, A. and Schmidt, P.H. PRL 15, #9, 409 (1965) Handbook of Chemistry and Physics The Chemical Rubber Company (1971) Harrison, W.A. Phys. Rev. 118, 1182 (1960) Klozenberg, J.P., McNamara, B. and Thonemann, P.C. J. Fluid Mech. 21, 545 (1965) Konstantinov, O.V. and Perel, V.I. S.P. JETP 11, 117 (1960) Larson, CO. and Gordon, W.L. Phys. Rev. 156, (1967) Legendy, CR. Phys. Rev. 135, A1713 (1964) Mahan G.D. private communication Maxfield, B.W. Amer. J. of Phys. 37, 241 (1969) McG.Tegart, W.J. Electrolytic and Chemical Polishing of Metals i n Research and Industry, Pergamon Press, New York (1959) M e r r i l l , J.R. Phys. Rev. 166, 716 (1968) ; Amer. J. of Phys. 38_, 417 C197Q) Morimoto, T. and Ueda, S. Physics Letters 42A, 457 (1973) Morgan, D.P. Phys. Stat. Sol. 24, 9 (1967) Parker, R.A. and Balcombe, R.J. Phys. Letters 27A, 197 (1968) Penz, P.A. J. of Applied Phys. 38, 4047 (1967) Pehz, P.A. and. Bowers, R. Solid State Commun. 5^ > 341 (1967) Pryce, M.H.L. private communication (1973) 161 Rose, F.E. Cornell Univ. Ph.D. Thesis (1965) Magnetoresistance and Helicon Resonances i n Sodium, Potassium, and Lithium Simpson, A.M. J. Phys. F 3, 1471 (1973) Smythe, W.R. Static and Dynamic E l e c t r i c i t y McGraw-Hill Book Co., Inc., New York (1950) Stern, E.A. and Callen, E.R. Phys. Rev. 131 (2), 512 (1963) Visscher, P.B. and Falicov, L.M. Phys. Rev. B 2_, 1522 (1970) V o l ' s k i i , E.P. and Petrashov, V.T. S.P. JETP Letters 7_, 335 (1968) Werner, S.A., Hunt, T.K. and Ford, G.W. Solid State Comm. 14, 1217 (1974) Werner, S.A. and Ford, G.W. Phys. Rev. B 11, 1772 (1975) 162 Appendix A HELICONS IN A FERROMAGNETIC METAL The i n i t i a l intent that led to the work of this thesis was to study the interaction of helicon waves with spin waves (magnons) i n a sample of ferromagnetic metal. To this end a slab of nickel (approximately 10 mm x 5 mm x 1 mm) with RRR - 4000 and the perpendicular close to <110^ was * used. A crossed-coils geometry with B^ directed along the normal enabled us to locate and study the f i r s t three resonant modes. In a non-magnetic material these modes correspond to one-half, three-halves, and five-halves of the helicon wavelength matching the sample thickness. In these experiments the frequency was held constant and the magnetic f i e l d was varied. Figure A - l shows the fundamental resonance observed i n this manner. For helicons i n a non-magnetic medium a plot of the resonant f r e -quency versus the applied, static magnetic f i e l d i s a straight l i n e going through the origin. Stern and Callen (1963) used a simplified theory of the effect of a ferromagnetic material on helicon propagation. Essentially they calculated the magnetic permeability, u, and used i t i n Maxwell's equations i n l i e u of u=l, appropriate to a non-magnetic material. For an i n f i n i t e medium they calculated 4TT M " = 1 + H - T H — o anis where Mg i s the saturation magnetization, H q i s the applied f i e l d and H A N £ G i s the anisotropy f i e l d . With these assumptions a plot of the resonant f r e -quency versus magnetic f i e l d has an intercept at - H a n £ S « If the sample i s f i n i t e , usually one lets H q -»• H - DMg, where D i s the demagnetizing factor, and one gets an intercept at (-H . + DM ) . 3X11S 8 * Kindly supplied by W.A. Reed, Bell Telephone Laboratories, N.J. 163 Neither the non-magnetic nor the Stern and Callen calculation matches our experimental results. A plot of the resonant frequency of each of the three modes as a function of magnetic f i e l d i s shown i n Figure A-2. Indeed, none of the curves go through the origin as was correctly predicted by Stern and Callen (1963). However, each mode has a different value for the intercept, i n contradiction to the Stern and Callen prediction. What possible interpretations are there? Since the sample was necessarily f i n i t e and the theory applied only to i n f i n i t e media, this could be one possible factor needed to explain the discrepancy between theory and experiment. For a slab, the demagnetizing f i e l d i s a re l a t i v e l y complicated function of the applied f i e l d and i t is simplistic to write i t as DM . Because of sharp corners the crystal may not be a single domain s at the low fi e l d s and the effect of domains was not considered in the theory. In the theory the coupling between helicon waves and spin waves Is very strong. As a result i t must be calculated what influence some dynamic effects have. For instance, what are the proper boundary con-ditions for magnons at the surface of the nickel slab? How w i l l this influence the helicon resonance condition? These problems were l e f t unsolved since the work was redirected to study the p o s s i b i l i t y of using a sphere of nickel as the sample, the de-magnetizing effect being best understood for this shape of sample. However, the boundary value problem of helicons i n even a non-magnetic sphere had not been solved then and hence we directed ourselves to the experiments and theory which make up this thesis. Morimoto and Ueda (1973) have also reported an observation of helicon standing waves in a slab of nickel. Their sample has the normal // <111> and the results are not directly comparable to ours (normal // <110>) since the anisotropy f i e l d varies greatly for changes in orientation. 164 'Figure A - l . Fundamental Helicon Resonances The absorptive peak of the fundamental helicon resonance in a slab of nickel i s shown. The magnetic f i e l d was directed along the geometric normal to the surface which was near <110>. The frequency was 90 Hz. 165 MAGNETIC FIELD (kOe) 166 .Figure A-2. Field Dependence of the Resonant Frequency The resonant frequencies of the three modes observed are plotted as a function of the magnetic f i e l d . For a non-magnetic material a l l curves would pass through the origin. 167 168 Appendix B THEORY FOR HELICONS IN AN INFINITELY CONDUCTING SPHERE The formalism presented here i s due to Pryce (1973). It was made available to the author who used i t as the basis for a computerized calcula-tion of the resonant frequencies and the helicon f i e l d distribution inside an i n f i n i t e l y conducting non-magnetic sphere • The equations are written for a medium with a scalar r e s i s t i v i t y , p , and the parameter O J C T i s taken to be °°. The results are presented i n Chapter II of this thesis." The f i e l d equations are written as follows: p2 B ± = -4TT13zJ± (B-l) p B = 2iTiO J,_ + 3 J ) z - + + -where p 2 = 4iru>/pc and 4fr J = ± 13 B,. + iS^B ± z ± ± z 4* J = - | (3_B + - 3+B_) with the divergence condition given by (B-2) 3,B + 3 B + 23 B =0 (B-3) + - - + Z Z • \ -»/ The magnetic f i e l d , B, and the current density, J, are expanded as a power series i n r and an appropriate series of Legendre polynomials, P n > and their f i r s t two derivatives, P' and P 1 1, as follows: n n ' ; 169 B - ' X/ ( 2 n + 1 ) ( p r ) S U ( n , s ) P n » n - ° n,.s B + •- ^ (2n+l) (pr) S v(n,s) sin 20 e 2 1 * , n^2 (B-4) n,s B z (2n+l) (pr) S w(n,s) sin 6 e 1* P q , n^2 n,s with s>nand n and s both even. 4irJ_ (2n+l) ip (pr) S f (n,s) P q , n > l n,s + =S ( 2 n + 1 ) ± P ( p r ) S S ( n' s ) s i n 2 e « 2 ±* P n ' ' n - 3 ( B - 5 ) 4TTJ n,s 4 l f J z =X) ( 2 n + 1^ ( P r ) S h(n,s) s i n 9 e 1* P' , n i l ' n n.s with s>n and.n.and s both odd, «• where u(n,s), v(n,s), w(n,s), f(n,s), g(n,s) and h(n,s) are unknown quantities depending on n and s; they w i l l be determined by applying the boundary conditions. Using the recurrence relations relating P , P', P " to n n n Pn + 1 Pn±l a n <* Pn±l a ^ o w s o n e t o S U D S t i t u t e the expansions (equations B-4 and B-5) into equations B-l, B-2 and B-3 i n such a way that the equations become relations between u, v, w, f, g and h without any r or 9 dependence. From the six independent equations one can solve for recurrence relations involving only u and v. Defining 1?0 5 (n,s) = u(n,s) + (n)(n-l) v(n,s) n (n,s) = u(n,s) + (n+l)(n+2) v(n,s) allows one to present the recurrence relations as follows: e(n ^ = (s-n+2)(s+n+3) -(n+2)(2n2+n+l) f f . «n,s) ( 2 n + i ) (2n+3) £(n,s+2) (nxn-1) r,(n,s+2)j- ^ ^ f f i * 5 * n(n+2,s+2) , n>0 , „,„ v _ (s-n+2)(s-n+4) . (s-n+2)(s+n+3) n(n,s) j ^ i j ) S(n-2,s+2) + ( 2 q + 1 ) [^-(n+l)(n+2) 5 ( n,s+2) + - ^ g ^ ^ n ( n , s + 2 ) J n>2 . If one defines w(n,s) = r(n,s) + (n+2) g (n,s) + (n-1) n (n.s) 2 (2n + 1) and uses the divergence condition, one gets (s-n+1) In 5 (n-l,s) + 2(n-1) C(n-l,s)] +(s+n+2) [(n+1) n (n+l,s) + 2 (n+2) r( n+l,s)] = 0. 171 One can now compute a l l the £'s, n's and r»s In terms of 5(0,s) and £ (n,n). Outside the sphere, the vacuum equations yield _ . , E n(n-l) A P B = b + n n n " 0 n+T r _ A s i n 2 9 e 2 ± * P" B = - E _n n_ (B-10) n n+1 r _ • ' (n-1) A sin 6 e 1* P ? i i _ = - L n n , z n n+1 r where b i s the applied excitation f i e l d strength and A are the unknown o n vacuum multipole f i e l d strengths. Continuity.at the surface, r=a, implies E (pa) 8 u(0,s) = bn o (2n+l) E (pa) S u(n,s) = n(n-l) A / a 1 ^ 1 (2n+l) E (pa) S v(n,s) = -A /a s n (2n+l) E (pa) 8 w(n,s) = -(n-1)A / a n + 1 n+10 (B" 1 1) The A 's can be eliminated to give n (pa) 8 S (0,s) = b o (pa) 8 I (n,s) = 0 (B-12) (pa) 8 C (n,s) = 0 . This i n f i n i t e set of linear equations can be truncated for some large value of n=N and can be solved. As N+», the values of the coefficients converge to the coefficients describing the f i e l d .
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Helicon propagation in aluminium spheres Feser, Siegfried 1975
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Title | Helicon propagation in aluminium spheres |
Creator |
Feser, Siegfried |
Date Issued | 1975 |
Description | Theoretical and experimental studies have been made of the propagation characteristics of helicon waves in metallic spheres; the helicon wave is a branch of the magnetoplasma wave spectrum in the presence of a uniform, static magnetic field. Our theoretical considerations have shown that for an infinitely conducting sphere the helicon wave inside the sphere can be described to good approximation by a single cylindrical wave. This model leads to a simple, approximate, analytic understanding of the resonant mode structure and of the helicon field distribution inside the sample. The resonant modes can be described as a doubly infinite series, labelled by (m,n), where m indicates the number (odd) of half wavelengths along the magnetic field direction nearly equalling the sample diameter and n indicates the same condition perpendicular to the magnetic field. An exact numerical calculation for a sphere of finite scalar resistivity has been carried out by representing the helicon wave inside the sphere by a superposition of cylindrical waves. In this way one is able to calculate the absorption peak heights and widths as well as the resonant frequencies for the first few modes. These results are found to be in good agreement with those calculated independently by Ford and Werner using vector spherical harmonics throughout. Whereas the formalism of Ford and Werner is preferred for computational accuracy, their approach is physically less transparent. Experimentally, we have substantiated the results of the calculations by using single-crystal spheres of aluminium with a residual resistance ratio of about 4000. The resonant modes were studied for two distinct geometries of the excitation and detection coils: the "parallel" geometry had the axes of the coils collinear but at right angles to the static magnetic field, whereas the "perpendicular" geometry had the excitation coil, the detection coil and the magnetic field all mutually perpendicular. The anisotropy of the fundamental (1,1) absorption peak amplitude is about 20% and that of the resonant frequency is about 1% for an applied magnetic field of 35 k0e. The helicon data are interpreted in terms of the anisotropics of the transverse magnetoresistivity (20%) and of the Hall coefficient (1%). For the first time in a helicon experiment an adequate sampling of crystallographic orientations has been made using the same sample throughout. The principal feature of the (1,1) peak height anisotropy is the presence of a narrow trough - typically 1° wide - whenever the static magnetic field lies in a {100} plane. This feature, which corresponds to an equally sharp ridge in the anisotropy of the transverse magnetoresistivity, has escaped detection in all previous studies of the magnetoresistivity (by helicons and otherwise), and it is a consequence of the magnetic breakthrough which is required by the Ashcroft model of the Fermi surface of aluminium. The transverse magnetoresistivity oscillates (periodically in 1/B) whenever the magnetic field is parallel to a <100> direction and this feature is likewise consistent with the Ashcroft model. Other quantum oscillations have been observed superimposed on the helicon resonance and their periodicities are found to agree closely with those in the de Haas-van Alphen effect. |
Subject |
Plasma waves |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-02-05 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
IsShownAt | 10.14288/1.0084782 |
URI | http://hdl.handle.net/2429/19627 |
Degree |
Doctor of Philosophy - PhD |
Program |
Physics |
Affiliation |
Science, Faculty of Physics and Astronomy, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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