‘.——_V ..... .......... THE CONSTRUCTION OF A VIBRATING SAMPLE MAGNETOMETER AND A STUDY OF MAGNETIC INTERACTIONS IN DILIJTE ALLOYS WITH ATOMIC “DER-DISORDER TRANSITIONS Dissertation for the Degree of Phs D. MICHIGAN STATE UNIVERSITY TERRI’ WAYNE McDANIEL 197 3 " Lit; RARY “1 IuIiK Egan SL128 I Umversity r] * 1 H mm This is to certify that the thesis entitled THE CONSTRUCTION OF A VIBRATING SAMPLE MAGNETOMETER AND A STUDY OF MAGNETIC INTERACTIONS IN DILUTE ALLOYS WITH ATOMIC ORDER-DISORDER TRANSITIONS presented by Terry Wayne McDaniel has been accepted towards fulfillment of the requirements for the Ph.D. degreein Physics. 0.1. 3—Way Major professor DateAflQUSt 28; T973 3-; BIN‘BING av V" “DAB 8: SONS BOOK BINIIERT INC. LIBRARY amozns «aileron. IICIIGAI ABSTRACT THE CONSTRUCTION OF A VIBRATING SAMPLE MAGNETOMETER AND A STUDY OF MAGNETIC INTERACTIONS IN DILUTE ALLOYS WITH ATOMIC ORDER- DISORDER TRANSITIONS By Terry Wayne McDaniel An experimental investigation of the influence of binary host atomic order-disorder transitions on interactions among dilute magnetic impurities has been conducted. This study has been carried out with magnetic susceptibility measurements on three host alloy systems with the emphasis on (Cu0 83PdO 17) , where X = Mn, Fe, Co, Ni, and Gd. l-cxc Susceptibility measurements were performed with a vibrating sample magnetometer that was constructed and develOped as the initial segment of this study. The details of the design, calibration, and operation of this instrument are presented. A discussion of important experimental considerations in a systematic investigation of alloys with varying impurity concentration is given. The possibility of the coexistence of several magnetic interaction mechanisms in metallic systems and the desire to experimentally distin- guish among them served as motivation for the exploitation of the atomic order-disorder transition as a potential internal control on these interactions. A consideration of the relationship between the Terry Wayne McDaniel paramagnetic Curie point 0 and magnetic interactions precedes the presentation of the experimental data. Striking effects of the order—disorder transition have been observed in (CuO 83Pd0.]7)Mn and (Cu0.83Pd0.17)Ni. Possible explanations of the observed behavior with respect to the Kondo effect, the RKKY interaction, and local environment-direct interaction effects are offered. A local environment-direct interaction picture appears to be most consistent with the data obtained. THE CONSTRUCTION OF A VIBRATING SAMPLE MAGNETOMETER AND A STUDY OF MAGNETIC INTERACTIONS IN DILUTE ALLOYS WITH ATOMIC ORDER- DISORDER TRANSITIONS By Terry Wayne McDaniel A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Physics 1973 .1. Fl ‘. ,_ T I 3‘) D '3 I 1 ‘W\ I?“ ACKNOWLEDGMENTS I would like to thank Professor Carl L. Foiles for the expert guidance he provided in this research. I hope that this work can in some small way reflect the incisiveness of his approach to experimental problems. His helpful, friendly manner and the clarity of his teaching were the major factors in making my stay at Michigan State University most worthwhile. I want to express my appreciation to the other faculty members with whom I have had the Opportunity to interact. Also deserving of thanks are the personnel of the clerical and technical staff of the Physics Department. I am particularly grateful to Mr. Boyd Shumaker for his able assistance in preparing samples for my work. The consistent financial support of the Physics Department and the National Science Foundation in a period of fiscal difficulty has been appreciated. Finally, I want to publicly express my most heartfelt thanks to my wife, Rosemary, whose hard work, sacrifice, and constant encourage- ment were essential to the success of my graduate education. ii TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES I. II. III. IV. INTRODUCTION A. Brief History of the Problem B. Preface of Thesis Content INSTRUMENTATION General Experimental Considerations in a Study of Magnetic Properties Advantages of a Vibrating Sample Magnetometer The Sample Vibrator The Sample Holder Signal Detection and Processing Signal Detection Coils Calibration and Operation 1. Sample Holder-Detection Coil Positioning 2. Sample Positioning and Geometry 3. Calibration Tests - Measurement Procedure H. The Cryostat and Temperature Control m-nmonw > SELECTED TOPICS ON MAGNETIC INTERACTIONS IN ALLOYS A. Thoughts on the Relationship of Local Moment Theory and Experiment 8. The Paramagnetic Curie Point O and Magnetic Interactions PREPARATION OF (Cuoo 83F d0 l7)l-cxc AND PRELIMINARY TESTS A. Properties of CuO 83Pd0. 17 and Alloy Preparation 8. Magnetization as a Function of Applied Magnetic Field C. Heat Treatment D. Survey of Magnetic Properties STUDIES OF CuPd§l7IMn(c) A. Detailed Sample Reliability Tests 1. Susceptibility Versus Mn Concentration at Constant Temperature 2. Electrical Resistivity-Measurement and Results 8. Temperature Dependence of Susceptibility iii Page vi 62 62 67 75 76 80 BI 85 88 Page C. Discussion of Results 99 l. The Kondo Effect 99 2. The RKKY Interaction lOl 3. Direct Interaction-Local Environment Effects 107 VI. STUDIES OF CuPd(l7)Ni(c) 113 A. Susceptibility and Resistivity Versus Ni Concentration at Constant Temperature ll3 8. Discussion of Results ll6 VII. SUMMARY l20 APPENDIX A l. Units of Magnetic Susceptibility l24 2. A General Definition of Magnetic Susceptibility l30 APPENDIX 8 Tables of Susceptibility Versus Temperature Data l33 APPENDIX C Inverse Impurity Susceptibility Versus Temperature Plots for CuPd(l7)Fe(O.29), Cu3AuMn(l.O6) and Cu3PtMn(O.8II l4l LIST OF REFERENCES l44 General Reference l48 iv LIST OF TABLES Table Page V-l p and O values for Mn and Gd in CuPd(l7) 95 AB x;(T) data for all X-1 vs. T plots AB-l CuPd(l7)Mn(O.22) 133 AB-Z CuPd(l7)Mn(O.70) l34 AB-3 CuPd§I7)Mn(I.T6) I35 AB-4 CuPd(l7)Mn(3.l5) l36 AB-S CuPd(l7)Gd(O.4) I37 AB-6 Cu3AuMn(l.O6) l38 AB-7 Cu3PtMn(O.8l) l39 AB-8 CuPdIl7)Fe(O.29) T40 Figure 11-1 11-2 11-3 11-4 11-5 11-6 11-7 11-8 IV-l IV-2 IV-3 IV-4 1v-5 IV-6 v-1 V-2 v—4 v-5 LIST OF FIGURES Schematic diagram of VSM with cryostat Sample vibrator assembly Sample holder Block diagram of electronics Detection coil geometry. a) Foner coils. b) Mallinson coil array (with coordinat system and holder position). c) Time varying field (after Foner) Saddle point plots XN/XN,O vs. ZW/RA1 Top cross-sectional view of magnet gap with cryostat Phase diagram of Cu-Pd M vs. H for CuPd§l7IGd(O.4) M vs. H for CuPd§l7IFe(c) M vs. H for CuPd§l7ICo(c) M vs. H for CuPd§l7IMn(c) M vs. H for CuPd§l7INi(c) XAIC) vs. c at T = 300 K for ordered (ORD) and dis- ordered (DO) CuPd(l7IMn(c) XAIC) vs. c at T = 80 K for ORD—DO CuPd(l7IMn(c) p(c) VS. C at T = 300 K, 78 K for ORD-DO CuPd§l7IMn(c) [)(,I4(T)]"1 vs. T for ORD-DO Cupdg172Mn(o.22) [XA(T)3-] vs. T for ORD-DO CuPd(l7)Mn(0.70) vi Page l6 2l 26 28 3l 37 42 46 64 69 71 72 73 74 83 84 89 91 92 Figure Page V-6 [x5(1)]" vs. 1 for ORD-DO Cupdg171Mn(1.16) 93 v-7 [x$(T)]'] vs. 1 for ORD-DO CuPd(l7)Mn(3.l5) 94 V-8 O(c) vs. c for ORD- DO CuPd(l7)Mn(c) and CuMn(c) 96 v-9 [x&(r)]‘% .T for ORD DO CuPd(l 7)G d(O. 4) 98 v1-1a} XA(c) at.T= 295 K for ORD 00 CuPd(l7)Ni(c ) and 114 b ggn1(c) 115 VI-2 p(c) vs. c at T = 300 K for ORD-DO CuPd(l7)Ni(c) ll7 AC-l [x§(1)]" vs. 1 for DO CuPd(l7)Fe(O.29) 141 Ac-z [xglnn‘l vs. 1 for 0RD-DO Cu3PtMn(0.81) 142 AC-3 [xg‘nn'1 vs. 1 for ono-oo Cu3AuMn(l.O6) 143 vii I. INTRODUCTION A. Brief History of the Problem The earliest systematic studies of magnetism in metals were undertaken in the first thiro of this century. At least two major devel0pments in physics during this period account for this fact. Experimental techniques at low temperatures were being rapidly developed by the mid—l920's following their first appearance at the beginning of the century. This made possible direct measurement of bulk magnetic pr0perties such as magnetization and susceptibility as a function of temperature which is essential in any attempt to understand magnetism. Low temperature results for other prOperties of metals which were potentially related to magnetism were being published with increasing frequency. The second deveTOpment was the establishment of quantum mechanics which was to provide the theoretical basis for the under- standing of magnetism in matter, even up to the present day. In retrOSpect, it seems clear that progress in providing satisfactory explanations of magnetic phenomena would have been impossible without the establishment of quantum mechanics. There is little doubt, however, that the improved experiments demonstrated the necessity of the develOpment of a new theory and thus hastened it. The experimental thrust proceeded quite predictably. The first task was to study the magnetic behavior of the pure metallic elements and these investigations exposed the broad classes of magnetism l 2 characteristic of metals. The most striking magnetic systems had long been known to be those which exhibited Spontaneous magnetic ordering, namely certain of the 3d transition metals and the lanthanides. From the outset, therefore, it was clear that a central task was to under- stand simple ferromagnetism and antiferromaqnetism, and it is remarkable that some fifty years later this objective remains to be accomplished. This is not to say that little progress has been made, but is simply an observation that magnetic phenomena have proven to be plentiful, diverse, and often exceedingly complicated. Our intent here is to present a brief historical review of one particular path which was taken in the hope of attaining a better understanding of c00perative magnetic phenomena. As commonly occurs in physics, those who followed this path found new effects which appeared to be more fundamental than the magnetic ordering that was ultimately to be explained. As we sketch the historical development of the approach we have ad0pted, it should be remembered that this discussion is obviously not intended to supplant recent, more complete reviews in the literature.(1'4) Rather, it serves as an introduction to the t0pic of this thesis. At this point we should make clear to what we refer when we speak of magnetism in metals. We are restricting our use of this phrase to those situations where a permanent (on some relative time scale) local magnetic moment exists in a metal in the absence of an applied magnetic field. Normally, a reliable experimental indicator of this situation is a strongly temperature dependent magnetization or suscep- tibility. In this definition we are excluding weak, nearly temperature independent effects such as atomic core and conduction electron contri- butions to the paramagnetic or diamagnetic susceptibility. Although the source of most of these contributions to the magnetic character of pure metals was understood shortly after the advent of the quantum (5) theory of solids, it must be emphasized that accurate quantitative calculations of the susceptibility are exceedingly difficult many-body problems.(6’7) Moreover, there still exist serious discrepancies between experimental and calculated susceptibility values for non- dilute alloys and intermetallic compounds quite apart from the usual discrepancies attributable to approximations for computational convenience. Perhaps this emphasizes the need for reliable methods of calculating the pr0perties of alloys. This is certainly a more palatable notion than the thought that all of the mechanisms of magnetic susceptibility are not yet known after nearly fifty years of quantum mechanics. In any case, it is generally true that the major Operational difficulty in dealing with the (weak) induced magnetization in metals is a correct separation of the total susceptibility into its consti- tuent parts. After the pr0perties of the pure metallic elements had been sur- veyed, the experimental attack turned toward alloy systems. The idea was that one might better approach an understanding of magnetic phenomena if their presence and magnitude could be externally controlled in systematic ways. As an example, consider the gradual dilution of a pure ferromagnet by alloying with a non-magnetic metal (one whose constituent atoms carry no permanent magnetic moment in the metal). It is clear that at some (xxnposition of this binary system the spontaneously ordered magnetic state 1ui1l cease to exist at any temperature, and one would expect the critical temperature for ordering to vary over some intermediate range of composi- tion. On the other hand, many experimentalists felt that a more fruitful utilization of alloying was to start with the comparatively uninteresting non-magnetic system and gradually add small amounts of magnetic impuri- ties. If one could gain an understanding of the magnetic behavior of the alloy system in a piecewise approach as the concentration of the impurity was incrementally increased, it was hoped that ultimately the c00perative magnetic state would come to be understood in terms of interactions among its fundamental constituents, the atoms about whose sites the permanent moment is localized. In practice, this logical plan of attack did not always yield information that could be easily interpreted. For instance, it did not at first sight seem surprising that dilute transition metal or rare earth impurities in a non-magnetic host diSplayed a permanent moment (as evidenced by a Curie-Weiss impurity susceptibility). It was widely presumed that the free atom or free ion more or less maintained its electronic configuration as it went into solid solution without really asking how exactly this happens in a metallic environment where conduction electrons might be expected to mix with the atomic states responsible for an impurity atom's magnetic moment. In fact, there was strong experimental evidence that some impurities did not carry moments in some hosts (for example, Mn in Al). Clearly, before one could proceed to an investigation of interactions among atoms carrying permanent moments, one had to understand what determined the formation of moments in the dilute alloy. The answers were slow in coming. It was not until the mid-1950's that Friedel(8) and coworkers addressed the question of moment formation in metals when he introduced the concept of a virtual bound state, a d state strongly admixed with a band of conduction electron states. This guiding work and an increasing 5 store of experimental data on dilute alloys kindled theoretical progress on this problem; the class1c papers of Anderson(9) and Wolffno) appeared shortly thereafter. Anderson solved a simple parameterized model which included s-d mixing, while Wolff treated the conduction electron-impurity system as a resonant scattering problem. These papers first considered explicitly the necessary conditions for the existence of a localized impurity moment. There exists another large set of experimental results that were accumulated between the 1920's and mid l960's which have come to be of central importance to the study of local moments in metals. This set of experiments have been referred to by van den Berg(]) as anomalies in dilute metallic solutions containing transition element impurities. These experiments consist primarily of measurements of transport and equilibrium pr0perties at low temperature in these dilute impurity systems. The "anomalies" refer to originally mysterious minima in thermal and electrical resistivity versus temperature curves, very large thermoelectric powers at low temperatures, Schottky type peaks in the Specific heat, and others. Although long su5pected, it was not until recently that the occurrence of a minimum in electrical resis- tivity versus temperature was shown to be in one-to-one correspondence with the existence of a magnetic impurity.(]]) Kondo(3) was able to explain the electrical resistivity behavior with an isotropic s-d exchange Hamiltonian by carrying the calculation to the second Born approximation. His result predicted a logarithmic increase in impurity resistivity as temperature is decreased and this agrees with experiment ir1 many systems over a wide temperature range. The perturbation calcnflation leads to a logarithmic divergence for many physical 6 properties as the temperature is lowered toward absolute zero indicating a breakdown of perturbation theory below some temperature. In the years since Kondo's remarkable result appeared, there has been much experimental and theoretical activity in the area of dilute moment systems. Kondo's original ideas have been pursued with the h0pe of removing the low temperature divergences in perturbation theory. It seems that the "Kondo system" bears some resemblance to the phenomenon of superconductivity in that one is dealing with a genuine many-body effect when a localized moment interacts with the sea of conduction electrons. Many-body techniques are being applied to the problem in the hope that a unified description of these systems for all temperatures can be develOped, but much remains to be done. Several of the non- (3) perturbative calculations and also several experiments(2’4) have suggested the existence of a many-body singlet ground state characterized by strong impurity-conduction electron spin correlation as the tempera- ture approaches absolute zero. On the other hand, impurity systems in which moments do not develOp are satisfactorily described by Hartree- Fock one-electron calculations. The question of how one can describe a Spectrum of behaviors extending between the two extremes with a single fundamental parameter has been of continuing interest. One such parameter that appears to be apprOpriate is the mean spin fluctuation lifetime which is but one of several characteristic times that are useful in a dynamical picture of dilute impurity systems. Theories dealing with so-called "localized Spin fluctuations" (LSF)(]2) have received increased attention lately. Coles(]3) has recently synthesized these various concepts in a convenient classification scheme. 7 Thus, we see from this brief sketch of developments in the dilute magnetic impurity problem how this particular approach toward under- standing cooperative magnetic phenomena has, for quite a long time, been apparently diverted from proceeding toward the original goal. But the diversion is only apparent, for in the meantime we have come to better appreciate the full complexity of problems in magnetism while much new physics has been learned. The necessary tools and concepts are being develOped for continued assaults on the puzzle of magnetism in metals. B. Preface of Thesis Content The experiments described in this thesis can now be placed into the perSpective of the continuing efforts to extend beyond the dilute magnetic system to one in which impurity-impurity interactions compete with and ultimately dominate the already complex isolated impurity- conduction electron interaction that has been the focal point of most studies through the l960's. The purpose of this study has been to carry out a systematic investigation of the influence of a particularly convenient, externally controllable parameter on magnetic impurity interactions. The parameter of interest, namely the degree of atomic order present in the host matrix, is characteristic of a very small class of non-dilute binary alloys. (The atomic order-disorder transition under consideration is distinct from the fonnation (common to many binary alloys) of an ordered atomic phase which occurs only at, or very near, simple stoichiometric compositions and which is usually referred to as an intermetallic compound.Genuine atomic order- disorder binary alloys are characterized by equilibrium phase diagrams with regions of atomic order spanning composition ranges of the order of 10 atomic percent. Sharp transition temperatures separate the ordered and disordered states. A prototypical atomic order-disorder alloy is the Cu-Au system.) As conceived, these experiments were seen as one possible way of controllably switching on or off particular types of magnetic interactions without depending upon distinctly different alloy systems to distinguish among interaction mechanisms. The Specific program, then, has been as follows: Given a single non-magnetic binary host alloy which could be ordered or disordered with a pr0per preparation procedure, one would study interactions among some Species of magnetic impurity atoms with susceptibility measurements as a function of impurity concentration and host atomic order. The study would be open- ended in that several impurities and hosts could be investigated if interesting effects were discovered. In no way was it presumed that this particular approach would render experimental interpretation to the realm of the trivial, but at least sufficient controls seemed to be built into the eXperiment to effectively reduce the complications of competing effects to a manageable level. We have restricted the admissible types of atomic order-disorder transitions fiW‘thIS study to a smaller subset of the already limited possibilities. Because one of the possible effects anticipated was the influence of the local crystalline environment on magnetic impurity behavior and interaction, it was thought essential to hold one important feature of the host crystalline structure constant in any order-disorder 'transition, namely, the lattice structure of atomic sites. For example, 'if at some fixed composition of the binary alloy the lattice of atomic sites in the disordered phase is face-centered cubic (fcc) (i.e., a random arrangement of the two species on an fcc lattice), then we insist that upon ordering the system by the appropriate means that an identical fcc lattice of atomic sites be retained and the atoms simply rearrange on that lattice to create an atomically ordered array. While crystallographers would maintain that what is described in the preceding sentence is a change in crystal structure (i.e., a new unit cell with a new basis), the essential feature for our concern is just that the lattice of atomic sites remains unchanged in the ordering. This does not always occur. It can happen that the lattice Of sites distorts upon ordering, and we have considered this to be an added complication to be avoided. Therefore, we have been doubly selective in choosing host matrices for studying interactions. (See Section A of Chapter IV for Specific details of the particular alloy systems we have investi- gated. In Chapter III, Section A, a further discussion of atomic order-disorder in regard to magnetic interactions is presented.) Before proceeding to a discussion of the details and results Of the experiments, we describe in Chapter II the considerations that went into the selection Of a particular device for performing susceptibility measurements, as well as the details of its construction and Operation. Also treated are the modifications we have incorporated into our instrument that perhaps make it unique among instruments of its type. II. INSTRUMENTATION A. General Experimental Considerations in a Study of Magnetic Properties When one decides to embark on extensive survey studies Of the magnetic character of a wide variety of physical systems with the constraint of limited resources to allocate for instrumentation, several choices as to approach are presented. Magnetism is a phenomenon which is present in nature in a multitude Of forms and which manifests itself directly or indirectly in many experimentally accessible physical prOperties. There are several reasons why one would prefer to measure a direct magnetic prOperty. Indirect methods involve measuring a property which can yield information on the magnetic prOperties of a given system under the following conditions: (1) Enough prior knowledge of the system is at hand to allow a meaningful interpretation Of the indirectly related data; (2) The indirect prOperties are present and experimentally accessible in all materials that might be chosen for study. The prOperties directly related to the magnetic state Of ainaterial fall into two general categories: (l) Bulk prOperties which reflect macro- scopic Spatial averages Of localized effects; (2) Microscopic or local properties which might be separably related to any one of the components of a bulk sample (e.g., nuclei or electrons, impurity atom or host atom). Both bulk and local prOperties are useful, but in (rifferent ways and under different circumstances. Bulk properties are 10 ll most useful when one is beginning a study in some previously unmeasured system because they allow one to determine the general type of magnetism present and some useful parameters (e.g., ordering temperature, para- magnetic Curie point, effective moments, etc.). It is perhaps simpler to determine systematic trends in magnetic behavior as some external parameter is varied with bulk measurements. Knowledge of local magnetic behavior is most valuable after the categorization and systematization acquired from bulk measurements is completed. Only a local magnetic probe can adequately resolve the details of magnetic behavior once the guidelines have been sketched with bulk measurements. TO carry out the type of survey of magnetic interactions envi- sioned in Chapter I, and keeping possible diverse future studies Of magnetism in mind, we decided to undertake direct bulk measurements of magnetic prOperties. The history Of past work of this kind indicates that the prOperty bearing most directly on bulk magnetic character is the magnetic susceptibility, or alternately, the bulk magnetization. (See Appendix A for definitions of these terms and a discussion Of the various units in use.) From Appendix A we have + _, + + M(r.t.T) = x(T)H(r.t) (2.1) where N is the magnetization, X is the magnetic susceptibility, H is the total magnetic field intensity, E denotes spatial position in the Inaterial, t is the time, and T is the temperature at F. For a uniform field H the resulting magnetization M is uniform. One can measure :rtatic or dynamic susceptibility. Static measurements are entirely satis- 'factory for the determination Of the properties of interest in this 12 thesis. Only when knowledge of fluctuations and relaxation effects are required is it essential to measure the dynamic susceptibility. Two physical effects associated with magnetic materials are commonly exploited to directly measure the static bulk magnetic susceptibility (hereafter referred to as simply susceptibility). The force methods and the induction methods are the two broad classes Of measurements derived from these effects. A recent and rather complete discussion Of the various methods within these broad classes is available.(]4) We Shall briefly review the principles involved in each method before proceeding to a detailed discussion Of the vibrating sample magnetometer (VSM). The existence of a force on a continuously magnetized material placed in an external field gradient arises in the following simple way.(15) The magnetostatic potential energy of an inductively magnetized, linear, isotropic, homogeneous Specimen in a field H is u=-% [ flfifiymfl]=-%gdnmflfifin. m2) Specimen volume Write F -- To + “F". For fixed To, d? = div“. Thus + + u(?0) = — %-x f d?‘ [H(?0+F').H(?0+?')] . (2.3) Specimen volume The force on the specimen is “*+ ++ +_’.‘a to “a F(r0) = - VU(ro) , where V = 1 ~——-+ j ———-+ k-——— . F(ro) = x f dr' [1(———5;————--H(ro+r')) + ~--- ] (2.4) 0 + For'the isotrOpic specimen, we see that F is prOportional to the sus- ceptibility x. In practice, one shapes the field such that the force T3 is non-negligible only along a single Cartesian coordinate. Furthermore, "* + the specimen volume is made small enough such that 32 r) -H(F) is essen- ' Oi tially constant over the volume. Then, for example, r0) 4 -H(r0 I). (2.5) Fz— —x(volume) (a: O ZO This result applies to the Faraday method. A related force method which relies on the same physical principle is the Gouy method. In this method, the sample is of a long cylindrical geometry, with one end in a highly uniform magnetic field and the other in negligible field. From Equation (2.3), the total force on the sample is '+-+ l “ 8 2 + F(r ) = -x f dr' [i— H 2(r) + j H 2(r) + k ———-H (r)] (2.6) o 2 Specimen 8x0 3y0 320 volume If the cylinder axis is along 20, then 5%—-H2(F) and 5§—-H2(F) are negli- O gible compared to 5%—-H2(r) over the length of the Specimen. Furthermore, the integrand -E—-H2(ro+r' ) has r' held constant. Thus d? = dfb and O + A F(Fi) s JZ—XA f dz o[k g—Hzfiw r0)] =-;- AHZCF') k (2.7) Specimeno length where A is the cross-sectional area of the cylindrical specimen. The induction methods of measuring static susceptibility all are based on Faraday's law of induced electromotive force. A magnetized Specimen produces a magnetic field in space. If the field lines are linked with a detection coil, this constitutes a magnetic flux in the coil. When some relative motion Of the coil and specimen are effected, a voltage will be induced in the coil which is equal to I%%I: where o = I: f B dAn is the flux. Thus 4 is derived from B, the magnetic turns field produced by the magnetized specimen, and the field in turn depends upon the specimen magnetization or susceptibility. SO all Of the induction methods involve the measurement of a time varying voltage l4 induced in some geometrical array Of detection coils, and the voltage is proportional to the Specimen magnetization. The proportionality factor includes the frequency of vibration and a complicated geometrical factor which must be calibrated out. 8. Advantages Of a Vibrating Sample Magnetometer A successful VSM was first described in detail by Foner(]6) in l959. A schematic diagram Of the essential features Of such a device is shown in Figure 11-1. The basic principles of Operation are as follows. A specimen whose magnetization is to be measured is posi- tioned in a sample holder between the poles Of a laboratory magnet. The sample holder is rigidly supported on a shaft which is attached to an electrical-tO-mechanical transducer which sustains a sinusoidal vibra- tion of the sample. Some geometrical array of detection coils is secured to the poles of the magnet and they sense the time varying field produced by the motion Of the uniformly magnetized specimen. This results in the induction of a sinusoidal voltage in the coils which is directly prOportional to the magnetization of the specimen. Note that measurement Of the magnetization does not require the appli- cation Of an external magnetic field to the sample if a magnetization exists without the external field. This contrasts directly with the force methods which require a field gradient and a field to create a force. When a field must be applied to induce a magnetization fOr detection with the VSM, the apprOpriate field is one that is constant in time and uniform in Space in the vicinity Of the sample. This irusures that a unifOrm magnetization is detected with the VSM and that Figure II-l. IS Schematic diagram of VSM with cryostat. (A) Support flanges (B) Support platform (C) Sample vibrator housing (D) Thermocouple vacuum gauge (E) Exchange gas valve (F) Vacuum pumping ports and hoses (G) Quick couple-capillary tube feed-through for electrical leads to sample holder (H) Connector flange (1) Vacuum tube quick couple (J) Dewar support platform (K) Dewar support flange (L) Outer vacuum jacket of Dewar (M) Inner vacuum jacket of Dewar (N) Stainless steel vacuum tube (0) Brass guide tube (P) Laboratory magnet ole pieces (Q) Detection coils in plexiglass form (R) Sample holder Figure II-l .rL / c o ffltfié : w—Q—Efiaj/ 1‘1 [1"! 1/K (i :I F L M r N O ‘i \ /Q P R/ l7 magnetization as a function of uniform applied field can be conveniently measured. This contrasts again with the force methods in which case field dependence measurements are difficult since lggi may vary when [HI does. The basic principle of operation of the VSM is seen to be quite simple and much Opportunity for flexibility exists. One can measure sample magnetization as a function Of specimen orientation or temperature (Section H, Chapter II). The sensitivity of the instrument depends upon the production Of as large a voltage in the detection coils as possible consistent with practicability and with maintenance of desired flexibility (Sections C, E, and F, Chapter II). Because one makes an analog measurement with voltage correSponding to magnetization, it is a simple matter to electrically measure magnetizations which range over all magnitudes found in atomic matter. Faraday measurements at low temperatures are known to involve complications.(]7’]8) Gerritsen and Damon(17) have discussed the effects of buoyancy and thermomolecular flow due to exchange gas and the errors that can arise. The VSM may be susceptible to induced eddy currents in metallic samples at low temperatures when sample resistivity is very low. These currents would produce unwanted signals, but experiments have shown that such signals are reduced by highly homo- geneous magnetic fie1ds.(‘9) Furthermore, such signals are rejected in the normal lock—in detection since they occur at twice the vibration frequency. We conclude that the VSM is not plagued with low tempera- ture problems as serious as those encountered with the Faraday balance. The geometry of a specimen is not as critical with the VSM as it is with the force methods. The Faraday method requires a Specimen 18 + whose volume is small if the simple approximation F2 2 x(volume)(%g¢H) is to hold. Furthermore, the linear dimensions should be comparable in all directions, e.g., Spherical samples might be desirable. On the other hand, the Gouy method requires very long samples, and this can result in large expenses for sample materials. We found it con- venient to fabricate cylindrical samples which are neither very long nor of nearly equal linear dimensions. These were apprOpriate for anticipated electrical resistivity and thermoelectric power measure- ments on the same Specimens used for susceptibility determinations. The VSM can be calibrated for any geometry as long as the entire Specimen lies within the uniform applied field. TO summarize the considerations involved in selecting an instrument for performing bulk susceptibility measurements, it was first deter- mined that the Foner VSM is clearly the best of the induction instru— ments for the many reasons discussed in Foner's paper.(16) Only the better analytical balances can match or surpass its sensitivity. How- ever, these force methods were judged less desirable than the VSM with regard to required applied magnetic field conditions during measurement, low temperature reliability, and Specimen geometry requirements. Furthermore, the force methods are mechanically more complicated, requiring specialized magnets and delicate balances, while the Foner VSM can be assembled with materials normally available in a reasonably well-equipped laboratory. Thus, one could hOpe to build a Foner type instrument at a considerable savings, and yet have a competitive piece of apparatus which is adaptable and flexible to a degree that most analytical balance magnetometers are not. In the following sections Of Chapter II, we cover in more depth the designs 19 we have adopted in the construction Of a VSM. Each section deals with a particular component of the VSM, or with testing and calibration. C. The Sample Vibrator The transducer assembly which provides the sinusoidal vibration of the Specimen under measurement is shown in Figure 11-2. The essential element is the permanent magnet and moving coil structure of a low frequency loudspeaker. The cone material and the original Speaker frame have been removed. An appropriate loudspeaker for this purpose is one that can deliver enough acoustic power (without significant distortion) at frequencies below approximately lOO Hertz (Hz) to sustain peak to peak vibrations of amplitudes up tO l millimeter (mm) while driving a long rigid shaft supporting a Specimen and holder. This requirement is apparently not so difficult to meet. We have produced sufficient vibrations while driving with less than 3 watts at 33 Hz an inexpensive, lightweight, 20.3 centimeter (cm) diameter loudSpeaker that utilizes a small ceramic permanent magnet. The permanent magnet assembly that has been used in actual Operation is a much larger, metal one Of cylindrical geometry. Its length is about 7.5 cm and the diameter is approximately 9 cm, so it is quite massive. The magnet assembly is rigidly mounted to the tOp plate of the brass vacuum housing such that the excursions Of the Speaker coil are in a vertical direction parallel to the axis Of the cylindrical vacuum housing. A lightweight rigid rod affixed to the moving coil form transmits the mechanical vibration through a small hole in the center of the Figure 11-2. 20 Sample vibrator assembly (A) Three support screws l20° apart (B) Support flanges (C) Electrical feed-throughs to Speaker and nulling coils (D) Top plate (E) O-ring in channel (F) Support platform (G) Magnet support posts (H) Speaker magnet assembly (1) Vacuum housing (J) Magnet gap and speaker drive coil (K) Drive rod guide (L) Flat disk between Sponge rubber rings (M) Vacuum pumping and monitoring ports (N) Vibration transmittance rod (0) Nulling signal coils (P) Permanent magnet (Q) Capillary tube for electrical leads (R) Quick couple with O-ring seal (S) Guide tube sup ort flange (T) Al coupling (U) Connector flange (V) Quick couple collar (W) O-ring in channel (X) Vacuum tube restraining collar (Y) Drive rod guide tube (2) Stainless steel vacuum tube (AA) Glass drive rod /// _f//// /: flmarsm \‘I \ 1 ) fiv /"’ %Z "W Figure 11-2 22 bottom plate of the vacuum housing. The motion of this rod is con- strained tO a vertical line by a cylindrical guide over about 10 cm Of the length of the rod. The contact Of the rod with the walls of the guide is made nearly frictionless with three vertical lines of ball bearings mounted in the walls Of the guide, and the guide itself is fixed with screws to the magnet assembly. The guide for the rod also serves the purpose that the Speaker frame originally served by keeping the cylindrical, single layer moving coil centered in the narrow circular gap of the speaker magnet. It iS crucial that the rod be constrained to linear motion only so that the coil and its form do not rub the magnet in the gap. At a position near the center of the rod guide, a flat disk Of 3.2 cm diameter is centered on the rod and fixed to it. The rod guide has a cavity above and below the disk which contains two readily deformable Sponge rubber rings. These serve to maintain the moving coil and rod in a vertical equilibrium position, and also provide restoring forces during the vibration. A small cylindrical permanent magnet is mounted through a hole in the transmittance rod below the rod guide. A pair Of pickup coils wired in series is mounted on the bottom of the rod guide with their axes parallel to the rod. They are positioned symmetrically on either side Of the small permanent magnet. As the speaker is driven, a voltage is induced in these coils equal to lggi where o is the total magnetic flux through the coils due to the field of the small permanent magnet. Therefore, the wave form of this voltage is an indicator Of the physical displacement with time of the driven rod. The signal is ultimately used both to monitor the sample vibration and as a nulling Signal which is 23 mixed with the signal produced by the specimen under measurement. This will be discussed further in Section E Of this chapter. All components Of the sample vibrator discussed thus far belong to a unit which is attached to the tOp plate of the vacuum housing. Electrical leads from the nulling signal pickup coils and the speaker drive coil are soldered to vacuum tight feed-throughs on the tOp plate. Three support flanges are bolted to the tOp Of the tOp plate. A heavy cylindrical brass vacuum housing for the vibrator assembly fits around the assembly and is held to the top plate with screws. A lO cm diameter rubber O-ring and silicon grease insure a vacuum seal at the tOp. A l.6 cm diameter pumping port with an O-ring valve is soldered into a side wall Of the housing. A thermocouple vacuum gauge and a needle valve are ported into the housing Opposite the pumping port to allow monitoring and control Of exchange gas pressure in the Specimen environment. The vibration transmittance rod discussed above protrudes l.3 cm through a hole in the bottom plate Of the vacuum housing. An 84 cm length of glass tubing transmits the speaker vibration to the sample holder (Section D, Chapter II). This tubing,with inside diameter 0.39 cm and outside diameter 0.59 cm, is chosen for its straightness, low mass, and low thermal conductivity. An aluminum coupling was machined to slide 3.8 cm into the glass tubing at the tOp and is held by a black wax which is easily loosened with the application Of low heat. The other end Of the coupling fits around the protruding transmittance rod and is held with three small Allen set screws. A hole is drilled through the coupling to insure that the interior of the glass tubing can be evacuated. A length Of brass tubing is soldered to a flange 24 which is held with screws to the bottom surface of the vacuum housing with an O-ring cushion. The brass tubing serves as a housing and guide for the glass drive tubing. Its inside diameter is l.l cm, the outside diameter is 1.27 cm, and the length is such that when it is secured at the tOp to the vacuum housing, 2.4 mm of glass rod clear the lower end Of the tubing. Three teflon set screws are spaced l20° apart near the bottom of the brass tube, and they center and guide the glass drive rod within the tube. The O-ring cushion between the brass tube support flange and the vacuum housing allows one to adjust the position Of the brass guide tube vertically so that the glass drive tubing and the guide tubing are coaxial, and binding Of the glass tube is avoided. Several holes are drilled through the brass tube support flange inside the cushion O-ring to transmit the vacuum from the vibrator housing to the Specimen environment. A larger flange with two O-ring quick couples is positioned and centered around the brass guide tube support flange. It also is held by screws to the bottom plate of the vibrator housing around a rubber O-ring. (For more details on the purpose of this flange, see Section H, Chapter II.) D. The Sample Holder The purpose Of the sample holder on the VSM is to secure the sample in a fixed orientation relative to the detection coils while directly transmitting sinusoidal mechanical vibration to the sample. There are three overriding considerations in the design of the holder: con- venience, magnetic prOperties, and thermal prOperties. It should be a quick and Simple Operation to change a Specimen. This not only saves 25 time, but reduces the possibility Of physically disturbing the position or delicate wiring Of the holder. This important point will be dis- cussed in more detail in Section G, Chapter II. Since the holder and its support are undergoing an identical vibration as the Specimen, and since the holder is as near the signal detection coils as the sample, any magnetization in the holder assembly will contribute a signal coherent with the specimen signal. Obviously, one wants every part Of the sample assembly tO have weak, temperature independent magnetic prOpertieS to prevent the masking of weak magnetic effects in the sample. In this regard, it is clearly advantageous to minimize the amount of material used for the holder assembly. Finally, the holder ought to have high thermal conductivity to insure a uniform Specimen temperature, but a low heat capacity to reduce thermal inertia during heating or cooling. To conform as closely as possible with the above considerations, we have used brass for the holder shown in Figure 11-3. The Specimen cavity is formed by a cylindrical tube with inside diameter 2.80 mm and length 1.27 cm. A short brass support tube, tapered at the lower end, joins the Specimen cavity perpendicular to its axis. The joint con- sists of an 0-80 filister brass screw, the head of which is soldered into the tapered support tube, and mating 0-80 threads tapped into the wall Of the specimen cavity. The support tube inserts about 2.5 cm into the glass drive rod and is bonded with a lzl mixture of GE 703l varnish and toluene-alcohol (l:l) (hereafter called GE varnish). A small O-80 set screw is threaded into the underside of the specimen cavity. Hence, a specimen can be easily removed or mounted just by turning a set screw befOre or after sliding the specimen through the 26 A BRASS GUIDE TUBE B GLASS DRIVE ROD C TEFLON GUIDE SCREWS D BRASS EXTENSION T UBE E ELECTRICAL LEADS F HOLDER O DUMMY COIL AND THERMOMETER H SET SCREW Figure 11-3. Sample holder. 27 cylinder. It should be noted that special care has been taken in all phases Of construction and in handling of the holder so as not to contaminate it unnecessarily with potentially magnetic impurities. The outer surface of the specimen cavity was tightly wound with 55 turns of #36 Belden Nyclad insulated cepper wire and this coil was cemented in place with GE varnish. This serves as a specimen nulling coil or as a dummy Specimen when a direct current is passed in the coil. A platinum resistance thermometer is formed by a 16 turn bifilar winding of 2 mil Pt wire (Sigmund COhn Corp.). The thermometer has a resistance of 12 Ohms at T = 295 Kelvin (K). (See Section H, Chapter II for more details.) The thermometer, dummy coil, and holder are electrically insulated from each other, but are in intimate thermal contact. Current and potential leads (#36 Belden Nyclad) are soldered to the ends of the Pt wire and cemented into a multilayered coating of GE varnish over the specimen cavity tube. The six electrical leads from the holder are twisted pairwise and secured by taping along the outside Of the brass guide tube up to the quick couple flange just below the vibrator housing. Care is taken to position the leads with sufficient slack between the holder and the brass guide tube so that any restraint on the vibration is avoided, even at low temperatures. E. Signal Detection and Processing In Figure 11-4 a block diagram of the electronic system associated with the VSM is shown. A 33 Hz oscillator supplies a 4 watt audio frequency power amplifier which drives the sample vibrator speaker coil. .As the drive rod and sample vibrate in phase, the magnetized sample .muwcosaumpo Lo Emcmowu xoopm .euHH ogamvm 28 ZOrhomhwo 1.426; quI:..mzwm A] mmdra mm......:1m mmdfn. zoiOmkmo 4 EOkajnzomo 105.53; NI mm man—Idm 29 and the permanent magnet in the vibrator housing produce a pair of coherent electrical signals in their respective detection coil arrays. The latter Signal is the nulling signal, and it first is amplified and then controllably phase Shifted. Next it enters a variable attenuator after which it is added in the mixer directly to the signal produced by the specimen. The sum of the two signals is fed to a high gain (A 2 106), sharply tuned preamplifier, the output of which enters a phase-lock amplifier and a null detector oscillOSCOpe. The phase shifter in the nulling circuit is adjusted so that the two Signals are 180° out of phase, and the variable attenuator is set to produce a null reading at the phase-lock amplifier. Therefore, at null, the attenuator setting is proportional to the magnetic moment of the sample, and is independent of vibration amplitude, frequency, or the characteristics of the electronics beyond the mixer. The nulling signal as produced at typical vibration amplitudes is of the order Of tenths Of millivolts with a signal-to-noise ratio of greater than 102. The amplifier which it enters is in two stages. Stage one has a gain of about 102 and drives the phase shift and attenuator networks. The signal from stage one is amplified by about 102 in stage two and the resultant voltage allows one to monitor the vibration amplitude with a 3 volt full scale a.c. voltmeter. The weakest detectable signals from the sample detection coils is put at approximately 2(10'9) volts by several independent estimates. There- fore, extraneous signals at this level at the output of the nulling signal attenuator and in the mixer must be avoided if possible. Given the geometry of the sample detection coils used, we judge the limit of electrical sensitivity to be governed by input noise in 30 the tuned preamplifier. F. Signal Detection Coils Several papers have discussed Optimum signal detection coil geometries since Foner's original treatment.(16) A closed analytical calculation Of %%-for general specimen geometry and detection coil configuration is not possible, but helpful Special cases have been treated and these serve as guideposts. At any rate, one must stay within certain practical constraints in the design of detection coil systems. Of course one wants to produce as much Signal with as high a Signal-to-noise ratio as possible, but the Space for detection coils is limited by the dual requirements of a small magnet gap (high, uniform magnetic field) and sufficient Space for a cryostat. Our earliest instrument tests were conducted with a pair of 4500 turn matched coils. They were wound with Belden #40 T-2 c0pper wire and had a d.c. resistance Of 1020 Ohms each. Following Foner, the pair was wired in series (Figure II-5(c)) and positioned as Shown in Figure II—5(a) with the coil axes parallel to the direction of Specimen vibration and perpendicular to the applied magnetic field. Although this configuration turns out not to be the most sensitive to the changing field Of the moving sample,(20) it has the distinct advantage of being largely insensitive to applied field fluctuations if the coils are nearly matched. The coils are mounted tightly into a form which is essentially a pair of circular plexiglass plates with threaded expanders around the perimeter. When in place, the circular plates press outward against the 17.8 cm diameter flat pole pieces of the XXX OOO L\\\\\\ C) C) C) \\\\ XXX XXX (D C) C) Figure 11-5. 31 / / z / o x YL+X o X/ X / (a) / / ooo / z LX xxx Y ><><>nditions was about 10'4 emu of magnetic moment. (This is approxi- mattely the moment induced in 3(10-2) cm3 Of COpper in a 4 kilogauss ap>p1ied field.) In an attempt to gain sensitivity we wanted to try tr|e~coil array described as "Optimum" by Mallinson(20) (Figure II-5(b)). Matched coils are much more crucial with this geometry since the coil axes are parallel to the applied field, and thus each coil encloses the maximum possible flux. Four coils are wired in series in the sense shown in Figure II-5(b). Note that each coil adds signal due to the V‘i brating field of the specimen (Figure II-5(c)), but that each pair or: the pole faces of the magnet is insensitive (in theory) to applied f‘i eld fluctuations. As a first approximation to Mallinson's infinitely extended coils, we used four Miller #986 air core chokes. The hOpe was that these machine-wound coils would be well matched. Each coil has 1347 turns and a d.c. resistance Of about 95 Ohms. They are 2.16 cm in diameter and 0.95 cm thick. In spite Of having fewer turns than the Foner coils, the fOur C011 array increased sensitivity by greater than a factor of five. 1Their low source impedance reduced input noise, and, surprisingly, they Seemed tO be as insensitive to the applied field as the original pair of C01 ls. When we discovered this, a second four coil array using 33 Miller #991 coils was built. Each coil has 2160 turns and a d.c. resistance Of 256 ohms. This set proved to be perhaps twice as sensi- ‘tive as the first Miller coil array, but input noise also increased so ‘Unat it was not clear that one had increased the signal-tO-noise ratio. Ftar this reason, and because coil thickness was a bit too great for OLJr forms, we put aside the second Miller coil array and concentrated OL1r efforts toward calibration of the first. G. Calibration and Operation Calibration Of the VSM has been discussed extensively else- wricere.(]6’2]’22) In addition to covering the basic points, we shall pr‘casent some considerations not dealt with adequately in the literature. We: feel that our particular approach in calibration is the correct one for- our instrument. 1. Sample Holder-Detection Coil Positioning In Figure II-5(b) we Show the sample holder-detection coil geometry used in our measurements. It is Obvious that recalibration mus t be performed for every change in the detection coils and/or sample location and geometry because the time variations of the flux CC>lagiling in the coils (hence, the voltage induced) will be a sensitive function of these factors. Our approach has been as follows. The hardened cardboard forms on which the four Miller #986 coils "are wound were turned down on a lathe to very nearly the width of the Ni nding itself. Only then could the four coils be mounted into the I-2277 cm wide plexiglass forms which were to house them. The diameter °\I€Er~ the winding of the coils is 2.16 cm and the cavity in which they 34 reside is 2.54 cm in diameter, so it was necessary to center and fix the coils coaxially in each cavity with three small styrofoam pads per coil. In addition, similar pads were placed over the coils before the thin plexiglass caps were tightened over the coil cavities. This padding insures against vibration of the coils with respect to the form and hence the magnet. The form was then positioned in the gap of the laboratory magnet and aligned such that the plane formed by the coil axes vertically bisects the round pole faces. Several constant magnetic fields were applied to the coil system to determine if it was suitably insensitive to field variations caused by the magnet power supply. After having been satisfied by this test, the coil positions were tightly secured with respect to the magnet by expanding the form as far as possible. Next, the magnet was moved on its track until the sample holder was suspended near the point Of inversion symmetry of the four coils. This position Of the laboratory magnet was carefully marked such that it could always be accurately reproduced. A plumb line was hung from the VSM support stand (fixed to the wall of the laboratory) and a point on the magnet yoke directly below the plumb was marked. Because the magnet is constrained to translation on its track plus rotation about a vertical axis, this method allows a unique and highly reproducible positioning of the magnet. Next, the sample holder was carefully positioned so as to be centered on the inversion symmetry point of the detection coils with its cylindrical axis parallel to the coil axes. The long brass guide tube was made parallel to the plumb line. These fine adjustments are made with three support screws threaded through support flanges bolted 35 to the top of the vibrator housing vacuum can. One is able to rotate, translate, or cant the entire VSM by small amounts as needed to restore the holder to a centered position in the magnet gap, and this is checked before every Operation of the instrument. 2. Sample Positioning and Geometry Once the holder and coils are correctly positioned relative to one another, the only flexibility remaining is sample positioning and geometry. Both are important and one would like to know how the output voltage Of the coils varies with certain parameters describing sample position and geometry. Let us discuss the former first. Studies of variation in coil output with sample position are normally presented as "saddle point curves". One produces a permanent magnetic moment in the holder with either a magnetized sample Of standard geometry (i.e., for our sample holder, a standard geometry would be a circular cylinder of 1.3 cm length and 2.7 mm in diameter) or a d.c. current in the dummy coil (Section D). The vibrating holder is then displaced along the x, y, and z axes respectively by small amounts and the output voltage measured. Plots of output voltage V as a function Of displacement out yield curves symmetric about the origin for symmetric detection coil geometries. At the saddle point dvout/dxi = 0. For a given sample geometry, the nature of the saddle point curves depends wholly on the coil size, shape, and orientation. From the standpoint of reproducibility Of measurements, coils with wide saddle points are preferable to those with narrow ones since precise sample positioning is less critical for the former. The width Of a saddle point is defined by the diSplace- ment allowable to maintain the output voltage within some percentage 36 of its value at the saddle point. In Figure II-6 we show saddle point curves for our four coil Mallinson array. A study Of displacements along the z axis was not performed Since sample height is highly reproducible for our VSM, and controlled displacements along 2 are difficult to effect. In can be seen that i 2 mm displacements along both the x and y directions introduce deviations of about one percent in vout' This compares favorably with saddle point analyses for other coil systems given elsewhere.(16’23) It is relatively easy to maintain the holder within t 1 nm of the center Of inversion without painstaking centering measurements before each Operation Of the instrument. As alluded to earlier, sample geometry is an important considera- tion in calibration. However, sample geometry is not important when the sample dimensions are small compared to the separation between sample and detection coils, because then a magnetized sample acts as a dipole.(24) In practice, one rarely has this simple situation with the VSM since one wants to use as large a Specimen as possible positioned as closely to the detection coils as possible to achieve maximum sensitivity. Therefore, one must calibrate for sample size effects. We have combined the sample geometry calibration tests with the cali- bration of the voltage divider in the nulling signal network to arrive at self-consistent results. We review in the following subsection a typical calibration test. 3. Calibration Tests-Measurement Procedure The primary susceptibility standard we use is high purity A1. A Specimen Of nearly the volume Of the sample holder was produced by cold rolling a length Of the Al into a rod of octagonal cross-section 37 .muopa paves «Fuuom .G-HH oc=m_a 4 x E. 434,225+ 4 > o» umjéaao 86+ ‘38. W n. W . O P2m3m0(4cm.o m1 u4m3rr¢4u¢ 38 with a maximum width of the cross-section being 2.66 mm. The rod was then Spark cut to a length of 1.33 cm, both end faces being planes nearly perpendicular to the axis of the rod. This specimen was lightly etched to remove possible surface contamination from the rolling and cutting. We now recount a typical calibration run. The Al Specimen is centered and secured in the holder. We vibrate all samples at the same amplitude, one that is as large as feasible with our sample vibrator system. The phase shifter in the nulling signal circuit is tuned while supplying a d.c. current to the dummy coil on the sample holder so as to put the nulling signal 180° out of phase with the sample signal in the mixer. The phase setting Of the phase-lock amplifier is adjusted for maximum sensitivity to the signal from the mixer. Then, with the current in the dummy coil Off, we null out the signal produced by the Al standard as the applied magnetic field is held at several values from 0 to 4.7 kilogauss. The null is recorded as a setting on a precision ten-turn potentiometer in the nulling signal attenuator. An identical procedure is performed for the empty sample holder. The difference Of the null readings at given field values is then prOportional to the magnetization Of A1 at that field. A plot of the null difference versus applied field should be linear for paramagnetic or diamagnetic materials, and the SlOpe is prOportional to the susceptibility of a sample. Therefore, using the known susceptibility for Al, we have a correspondence between sample magnetization and the settings on the nulling potentiometer for the particular geometry Of the A1 sample. We know 39 V _ d¢ _ + OUt - 'd—t- " Iml (LIG (2.8) where |R| is the magnitude of the sample dipole moment, w is the frequency Of vibration, and G is a complicated composite geometrical factor for the change in flux in the detection coils as the sample iS displaced in one direction. We expect G to be factorable into a detection coil part GC and a sample geometry part GS. Therefore + + v Iml wGSGC - |M| VwGSGC , out or leI VquSGC , (2.9) vout where we have used A = NV, with V the sample volume and N = XH, the definition Of volume susceptibility (see Appendix A). Furthermore, Vout = A (AN), where A is a constant and AN is the difference in the setting on the nulling potentiometer with and without the sample in the holder. Now A, IHI, w, and Gc are fixed constants for any inter- comparison Of nulls at a given applied field. Therefore, letting the superscript x represent any specimen whose susceptibility is to be measured, we have x x T x x Vout = A (7111*) g X W" “’6ch (2 10) W A (ANN) AilfilvAleAlG ’ ° out X 5 C Al x Al ANX VA] GS or X = X E ] . (2.11) ANAT vx 62 One can Obtain the mass or molar susceptibility from this relation by using the definitions of those quantities from Appendix A. TO relate 40 A1 mass or molar susceptibilities, simple replace Vm/Vx by Dis-§-;L——Or mole fraction (A1)_ . , mass . mole fraction (x) respect1vely. It 15 clear from the above express1on that unless the geometries of the unknown sample and the standard are identical, one needs to have values fOr 621/6: . It is only in the limit Of small Specimens (both Al and x) that the dipole approximation holds, and Gél/G: = l regardless of the shapes of the Al and x. For our particular holder, sample dimensions are quite small in two directions (transverse to the holder cylindrical axis), but not in the direction of the sample length. We then expect GA] 5 = RX 212) a;“— GOIEKTJ a I - S where Go(tx/EA1) is some smooth function of the sample length ratio which must be determined empirically. In order to check all Of these assertions, we produced a series of high purity tungsten samples Of various lengths, cross-sectional areas, and irregular end face geometries which would fit into our sample holder. These were made from a long cylindrical W rod by spark cutting and chemically etching to various desired Sizes. Each Specimen was centered in the holder, and nulls were achieved for several applied fields for each. From the plots Of null setting versus applied field (linear in every case), a null value at some chosen field was taken. We calculated the molar susceptibility of each sample using Al 1 xfi=A-N-:-[Z(—M——K?—F—A—J . (2.13) MF AN sample mass sample molar mass quantity is a constant determined from the Al calibration. Note that where MF = mole fraction = , and the bracketed 41 we have effectively taken Gél/G: = l for every sample, which cannot be correct. In Figure 11-7 we show XH/XN,O plotted against tw/AA], where x:,0 is the accepted value for the molar susceptibility of W. For W samples with irregular end faces, an average length has been used. We find that the plot is linear with a small positive slope, and that the fit passes through XH/XN,O = l at Jaw/ILA1 = 1. This single plot is a strong indication of the self-consistency of the above analysis. Although samples Of various cross-section and irregular end face geometry were used, the dominant geometrical parameter is the length, for only in that dimension is there appreciable departure from the dipole approximation. From this plot, the functional form of G (ax/2A1) is deduced to be G = [0.29 “X + 0.71]‘1 for 0.5 < ix < 1.5. o o a" '7”- Furthermore, the reliability Of the Al standard is increased by this test since a correct susceptibility result for tungsten emerges. The Al standard has been further successfully cross-checked against Pd, CuPd, Cu3Au, AuGa2 and other pure samples. H. The Cryostat and Temperature Control As mentioned earlier, one of the attractive features Of the VSM is its adaptibility fer temperature dependence studies of susceptibility. Such studies are essential for understanding bulk magnetic behavior as discussed in Chapter I. At the conclusion of Section C, Chapter II, a flange with two vacuum quick couples soldered to it was briefly described. Its primary purpose is to accommodate a thin-wall (20 mil) stainless steel vacuum tube of length 85.6 cm and outside diameter 2.54 cm. This tube is 42 m._ v._ N._ 0.. . . o. .- 2.1% e, uxfix A :23: QC md v.0 Nd _ _ _ N._ 43 sealed Off with a brass plug at the lower end, and has a heavy brass restraining collar soldered around it 2.5 cm from the top Of the tube. This tube Slides up around the sample holder and brass guide tube, and mates with a 2.54 cm inside diameter vacuum quick couple on the afore- mentioned flange at the base Of the vibrator vacuum housing. The restraining collar on the tube limits its penetration into the quick couple when a partial vacuum exists inside the tube. The tube serves a dual purpose as retainer for the partial vacuum in the Space around the specimen and as a barrier against the cryogenic liquid into which it is immersed. This particular tubing was chosen for this purpose because Of its strength, low thermal conductivity, and low magnetic permeability. This latter characteristic insures that the externally applied magnetic field or the field produced by the sample is not shielded by tubing.(24) The vacuum tube is aligned coaxially with the brass guide tube by pumping out the system with a mechanical forepump so that the vacuum tube restraining collar is held tightly to the quick couple. The laboratory magnet is then rolled into position and the vacuum tube is centered between the detection coils by shimming the contact between the restraining collar and the quick couple. We find that once the shims are positioned, subsequent readjustment is unnecessary. This centering is essential SO that the vibrating sample holder and the vacuum tube surrounding it do not make physical contact, fOr any vibration transmitted to the vacuum tube results in coherent background signal in the detection coils reflecting the magnetic properties Of the stainless steel tube. Even when care is taken to eliminate direct physical contact, we fbund that the recoil vibration Of the vibrator 44 housing, small as it is for the massive housing, couples directly to the vacuum tube. This produced unwanted background Signal which we were able to reduce only by loading lead bricks onto the tOp Of the vibrator housing to increase the mass Of the recoiling unit. We believe that the appreciable recoil we find results from the heavier than necessary drive transmittance rod with ball bearings that is used in the vibrator assembly. We are in the process Of correcting this deficiency in a subsequent design. The electrical leads from the dummy coil and thermometer on the holder leave the vacuum system via a small vacuum quick couple with a 1.6 mm diameter hole which is soldered into the same flange which supports the vacuum tube quick couple (see Figure II-l). The leads are fed through a stainless steel capillary tube and sealed in the tube with molten black wax. The tube seals in the quick couple with a rubber O-ring. The vacuum system can be pumped out to a pressure of several microns of Hg with the mechanical forepump and this is sufficient for our purposes. The heat exchange gas can be controllably admitted to the sample region with a needle valve mounted on a port in the vibrator housing. A thermocouple pressure gauge monitors the pressure in the vacuum chamber. We employ a custom Kontes-Martin Dewar (Figure 11-1) for low temperature experiments. The Dewar fits around the vacuum tube and is supported by a padded flange which bolts to a plywood platform supported by the same structure which holds the VSM. The Dewar is lifted into position and secured with the laboratory magnet rolled away. When the magnet is returned to its prOper position under the VSM, the tail 45 of the Dewar is centered between the detection coils. A clamping device mounted on the magnet yoke is used to hold the Dewar securely with reSpect to the magnet. This insures that the Dewar does not touch the detection coil form as the cryogenic liquid boils in the Dewar. There is approximately 1.5 mm clearance between the coils and either side Of the Dewar tail. The Dewar tail cross-section is shown in Figure II—8. The unfilled Space along the longer diameter of the tail serves as a reservoir for cryogenic liquid. A typical temperature dependence measurement over the temperature range 78 K to 300 K is conducted as fellows. The sample region is pumped out to the capability Of the forepump. Liquid nitrogen is transferred into the open tOp of the Dewar to a height of 30 to 36 cm (about one liter). When the boiling ceases after about 2 minutes, the temperature Of the sample has fallen approximately 10 K below room temperature. With a vacuum Of several microns of Hg, the cooling rate decreases continually as the sample temperature falls. One adjusts fer a null condition as the temperature slowly drops. The applied magnetic field and the sample vibration amplitude are held constant throughout. The rate of cooling is controlled by adding He exchange gas. Of course, a convenient cooling rate depends upon the magnitude of the temperature dependence Of the specimen susceptibility. A cool down from 300 K to 78 K usually can be comfortably completed in about 60 minutes. The heat flow into the liquid nitrogen bath is reduced by the low thermal conductivity of the vacuum tube. The glass drive tube helps to thermally isolate the sample holder from the warmer environment above the holder. The sample temperature is monitored by measuring the potential drop across a Pt wire for a constant known current in the wire. As O O \\\\\\\\\\\\\\ \\\\\\\\\\\\\\ D A MAGNET POLES B DETECTION COILS C PLEXIGLASS COIL HOLDER D DEWAR E VACUUM TUBE F SAMPLE Figure 11-8. Top cross-sectional view of magnet gap with cryostat. 47 discussed in Section D Of this chapter, the Pt wire is in close thermal contact with the holder which surrounds the sample. This feature is unique to this magnetometer since the temperature of a specimen is conventionally monitored at some location whose thermal communication with the specimen is via exchange gas.(17’25’26) The platinum thermometer was calibrated against the measured temperature dependence of a very pure Pt wire carried out previously in this laboratory. The calculated temperature dependence for our thermometer was made to fit experimentally established resistances at liquid nitrogen temperature, the ice point, and at room temperature. AS a cross-check on our temperature scale, we have successfully reproduced the temperature dependence of the susceptibility of pure Pd and Al between 78 K and 300 K. Of course, the measured weak temperature dependence Of the sample holder must be subtracted out in all measure- ments. III. SELECTED TOPICS ON MAGNETIC INTERACTIONS IN ALLOYS A. Thoughts on the Relationship Of Local Moment Theory and Experiment In Chapter I we alluded to the theoretical difficulties inherent in even the simplest magnetic impurity problem imaginable, the single isolated impurity in an otherwise pure simple metallic host. As involved as the isolated impurity problem clearly is, it is just the beginning Of one approach toward an understanding of cooperative magnetic phenomena as previously discussed. In any real alloy one will surely have impurity-impurity interactions present to some extent. Of all the experimental work done in the area of dilute alloys, only a small fraction has been performed on systems approaching the dilute impurity limit. (The experimental dilute limit corresponds to impurity concentrations Of the order Of 10 to a few hundred parts per million.) Only in these dilute limit cases could one argue that isolated impurity effects dominate impurity-impurity interactions. These difficult experiments have certainly been essential to the development Of our present understanding Of moment formation in metals, but the point we make here is that impurity-impurity effects must also be understood if we are to proceed in the approach Of gradually building up the cooperative magnetic state out of known interactions. But as soon as one admits the possibility of inter- actions among impurities, the problem for the theorist has been made 48 49 (27) more complex than before. The experimentalist, in moving away from the dilute limit, has unburdened himself Of two very significant problems, but has acquired new ones which appear potentially more troublesome. This being a treatise on experimental work, let us confine our attention to the concerns of the latter worker. When working in the dilute limit Of impurity concentration and attempting to measure effects due to the presence of those impurities, it is apparent that one requires the ultimate sensitivity in these measurements that current technology can provide. This is what one 5 4 Of the constituent atoms of the would expect when only 10' to 10' sample are responsible for the effect under measurement. Of course, it is not impossible that effects might be large in this case, but in general it is not expected. The second Significant problem connected with work in the dilute limit is the metallurgical-technological one of requiring extremely pure materials. Interestingly, both of these problems are strongly coupled with financial factors, cost scaling with increasing sensitivity and/or purity of materials. As one proceeds from the dilute limit, one expects the above problems to diminish. The new troubles might be classified as metallurgical and interpretational. The metallurgical problems are largely those Of solubility. There are countless examples Of binary alloy systems in which a solute will not dissolve and remain in true solid solution beyond some small solubility limit. With the degenera- tion of a solid solution comes the formation Of solute clusters and the breakdown of the fundamental structure one hOpes to study. Metallurgical problems present insurmountable barriers fOr systematic concentration 1dependence studies in many systems Of potential interest. Another 50 metallurgical consideration that has bothered some workers follows from the Simple Observation that no two alloys of the same composition are exactly alike. The probability of producing two identical systems of 1023 particles each in a few attempts is vanishingly small. This unsettling thought can usually be put to rest by claiming that values Of 23 atoms have an -1/2 = bulk prOperties which result from an average over 10 overwhelmingly high probability of lying within a factor of N 10'23/2 of some most probable value. This means that if one strives to produce homogeneous random alloys, then there is negligible prob- ability of making an alloy with a bulk property value noticeably removed from a mean value. The point is that one should be quite certain he is producing homogeneous random alloys if a systematic concentration dependence study is contemplated. The interpretational problem is Obvious. If impurity-impurity. interactions are superimposed on impurity-host effects, the separation of the two may not be clear cut.‘ The situation is generally more complicated than a simple superposition Of effects. As the impurity concentration becomes appreciable, the host itself is altered, so the impurity-host interaction is perturbed. Ideally, one wants to seek out experimental methods which suggest that some degree of separability of effects might be possible. As pointed out in Chapter I, the specific motivation for the work described in this thesis followed directly from a desire to investigate the influence of the local crystalline environment on magnetic inter- actions and to incorporate into the investigation a possible method of distinguishing between competing interaction effects in a local moment system. With regard to the former, Jaccarino and Walker(28) in 1965 51 Offered an interesting explanation for the variation with c (atomic fraction expressed as a decimal in formulae or as a percent in the text) of the local moment in Fe impurity Sites (1 atomic percent) in the non- magnetic binary alloy hosts Nb(l-c)M°c and Rh(]_c)Pdc. They suggested that within the framework Of the Anderson model, one could explain the development of Fe moments as c increases on the basis of the local environment Of an impurity rather than some average character of the host. In fact, they proposed that moment formation might be sudden as some local condition is satisfied, and not a gradual buildup as the average prOperties Of the host change. From bulk susceptibility measurements it is impossible to distinguish between a probabilistic occurrence of Fe sites fulfilling the magnet moment existence conditions as the host composition is changed and the gradual develOpment Of a moment on all sites at a rate determined by the average properties Of the whole host crystal. These workers cited direct evidence for their model in that the NMR resonance on CO59 in Rh(]_c)PdC indicates that some CO sites remain unmagnetized up to at least c = 12.5 atomic percent, while bulk measure- ments indicate that the average moment per CO atom is near 50% Of its final value at that concentration. Further experimental support for this model has been given recently by Brog and Jones.(29) A search fOr local environment effects indicates the necessity of using binary hosts. Studying all possible interactions suggests working away from the dilute impurity concentration limit. In order to suppress as much as possible the potential problem of the uniqueness Of each alloy produced (as outlined above), we sought host systems with an internal degree of freedom not present in an ordinary binary alloy, namely the possibility 52 Of turning on or off a long range atomic order on some fixed lattice Of atomic Sites. This single feature seemed very attractive in that it ought to bear on each Of the considerations discussed above. Consider magnetic impurity atoms which dissolve in the order-disorder host matrix by substituting randomly for the two host constituents. It would seem that some average over local environments at impurity sites would be different for the ordered and disordered hosts and this might be reflected in the magnitude of the average moment develOped. (This is one measure of the impurity-host interactions.) Furthermore, it would seem plausible that impurity-impurity interactions might differ with the changing character of the host medium. For example, a dis- ordered host would increase the scattering Of conduction electrons carrying Spin polarization information between local moment sites and effectively reduce mediated impurity-impurity interactions. Finally, the production of an order-disorder transition in a single alloy avoids the uncertainties involved in drawing comparisons between distinctly different alleys. B. The Paramagnetic Curie Point O and Magnetic Interactions It has been known for a long time that many pure metals and alloys with atoms which sustain permanent magnetic moments diSplay a Curie-Weiss type Of paramagnetic susceptibility X = TE5- over some temperature range. In fact such behavior is the usual experimental criterion for the existence Of well-defined local moments. This generalization of the familiar Curie law X = E- was introduced by P. Weiss(30) in 1907. T The additional parameter came about originally as the result of relating 53 several of the prOperties of ferromagnets to the presumed existence Of a strong internal molecular field (originally called the Weiss field, 15) now referred to as the exchange field( in reference to the quantum mechanical electrostatic forces believed to be reSponSible for strong spin correlations in matter). This exchange field was postulated to be prOportional to the magnetization, Hex = AM . With this isotropic internal molecular field, it is easy to derive the Curie-Weiss law and find the conditions under which it is expected to hold. For a system of N magnetically free atoms or ions per unit volume with angular momentum quantum number J in an externally applied magnetic field, the magnetization is given(5) by QJWBH M=NgJIJB BJ(X) , XETT— , (3.1) B where g is the Landé g-factor, us is the Bohr magneton, kB is the Boltzmann constant, and 2JBJ(x) is the Brillouin function defined as (2J+l)x *) - %j-ctnh (g3) . (3.2) Suppose we let H + H + Hex = H + XM. If x = ngB(H+XM)/kBT << 1, we can approximate BJ(x) by using the series expansion 3 ‘1 u u 000 ctnh u - U—+ §-- 45’+ Then, 2J+l (2J+l)x 1 2J x lim BJ (X) = ——2—J—[ - [___+ ] x 0 (2J+1)x 3123) 2.] x STE)" 2 l'B( =—42—(2‘m x- x = 2.1 22.1 +1-1 x13 J X) 3(23) 3(23)2 3(23)2 [( )2 + ( ) J . _ x _ L 113 BJ(X) - §T§37'(2J + 2) - 3d (J + 1) Substituting back into Equation (3.1), gJu (H+AM) + M = “QJPB 33] [ 237 NgZJ(J+1)pg C M - 3k T (H+AM) : T'(H+AM) B Solving for M, M = C H , T(l - T-X) and M E XH implies that the volume susceptibility = __£_. =._9_ X T-CX - 1-o ’ where the Curie constant C NgZJ(J+l)h§ _ ”(PU312 3kB “ 3kB ' Here p E g[J(J+1)]]/2 is the paramagnetic Curie point. is the effective Bohr magneton number. (3.3) (3.5) O 5 CA We see from this derivation that the Curie-Weiss law applies to magnetically free magnetic moments in a uniform magnetic field gJuB(H+AM) H + AM when B k T’ << 1. The presence of the molecular field AM is associated with a Spatially uniform effective field at some moment site which arises from the magnetization due to all other magnetic entities of the specimen, and therefore corresponds to an effective interaction 55 among them. Thus, 0 = CA is traditionaly related to any interaction which can be phenomenologically associated with the production of an effective molecular field at a magnetic moment site. In this sense, then, one might associate a non-zero O in a metallic system with inter- actions between a localized moment and other such moments, or between localized moments and the conduction electrons. The extreme example of the former case is o z T or O = TNéel for systems diSplaying Curie spontaneous magnetic order, and the (large) interaction is assigned to exchange interactions between closely packed localized moments. The latter case is demonstrated by the non-zero O values found experimentally for extremely dilute local moment systems such as Aufe, Qufe, and CuMn (18,3l,32) where the probability of appreciable direct or even indirect impurity-impurity interactions is extremely low. Whether or not the apprOpriate interactions in this latter case can be cast into a classical effective field will be discussed below. Before proceeding too far with such an obviously oversimplified theory as an effective field model, let us make clear the context in which this presentation is made. It should be emphasized that the Curie-Weiss law is an experimental jagt_for many dilute and concen- trated magnetic systems over wide temperature ranges. In cases where it is not realized eXperimentally, it should not be invoked in analysis under any circumstances. But when it holds experimentally, the con- nection of real interactions with effective fields is justifiable as a first analysis of systematic trends provided one is aware of the shortcomings of such a model.(33’34) Dellby(35) has recently extended the effective field model by explicitly including several magnetic interactions between electronic 56 states in a metallic alloy. His particular concern was the behavior of alloy systems with exchange enhanced hosts, but some of his results are presumably applicable when enhancement is absent. Without giving the details of Dellby's derivation, let us define the relevant quantities which appear in his main result. Then we shall present his result and consider the explicit interactions introduced into the Curie-Weiss law. Dellby considers the following electronic states in his treatment: conduction electrons (s), non-conduction electrons in unfilled bands associated with a base (host) matrix (b), and impurity resonant bound electron (i). The label "dia" refers to diamagnetism of all states. Five molecular field coupling constants (c.c.) correSponding to the role played by A in Hex = AM are given as follows: c.c. between 5 and b or s and i states (a), c.c. between b and i states (B), c.c. between 5 states among themselves (Y) c.c. between i states of different impurity atoms (6), and c.c. between b states associated with different base atoms (a). The susceptibilities corrected for interactions within the same state are labeled Xs’ Xb’ or Xi’ and uncorrected susceptibilities have the superscript (o), for example, x?. The relation between corrected and uncorrected susceptibilities is (for the 5 state) X5 = x:[l-yx:]']. Dellby calculates an expression for the total alloy susceptibility (Xmeas) and fbr the pure host (xbase E [Xmeas]xi=0)' He takes the difference of these two quantities (the usual approach for isolating impurity effects) and assumes a Curie law for non-interacting impurities, x? = %, where C is the Curie constant as defined in Equation (3.6). His 57 result is a2x5+8 2 C[]+axs+(T?E§;—9(Xbase’xs'xdia)] AX Z Xmeas’xbase = 2 2 ’ (3’7) + T-C[6+azxs+(;:5§;59 (Xbase-Xs'Xdia)] This has the form of the Curie-Weiss law where the bracketed quantity in the numerator is an enhancement factor for p, the effective Bohr magneton number, and the bracketed quantity in the denominator is an explicit expression for A in Dellby's particular alloy system. Let us consider two simple cases to check the reasonableness of the expression for Ax and look at Dellby's interpretation of some of the terms. First, suppose that the host material is an insulator (a and X5 are zero). This implies that no enhancement exists (Xbase = x5 + Xdia)° The trivial result TS _£L__ AX = T-CCS : (3-8) a very simple Curie-Weiss law with an unenhanced p and only a direct interaction between impurity moments. For an unenhanced metallic host, the impurity susceptibility from Equation (3.7) is C(lmxs)2 Ax = T - C(d+a2xs) (3.9) Dellby interprets the p enhancement factor (l+axs) as due to polariza- tion of the conduction electrons near an impurity site which in effect dresses the impurity moment. Exchange scattering(3) is the mechanism presumed responsible for the additional net polarization(3) of the conduction electron-impurity moment system. The new interaction 2 between impurity moments represented by the a Xs term in the denominator 58 of Equation (3.9) can be associated with an indirect coupling via double scattering of conduction electrons. Dekker(36) obtained a result corresponding to Equation (3.8) from a simple statistical mechanical model for the behavior of the random alloy QuMn(c) in the low c range. We assumes both ferro- magnetic (FM) and antiferromagnetic (AFM) interactions for certain impurity configurations. He finds that a Curie-Weiss law for the susceptibility is valid for T >> O, and he derives a form 0 = f(c) [g(c)0f - h(c)O ] . (3.l0) a Here 6f E yf/kB and 0a E va/kB, where yf and Ya are positive exchange integrals, and thus are measures of the FM and AFM coupling strengths. f(c), g(c), and h(c) are power law functions of c reflecting the prob- ability of occurrence in a random alloy of the FM and AFM configurations he assumes. Dekker's result for O is a quite general form of Dellby's term 0 = C6 in Equation (3.8). While Dellby considers his 6 to represent a direct coupling between impurities which may assume either sign, Dekker has explicitly shown how the sign of 0 reflects the dominant interaction when there is competition. Furthermore, he has included concentration dependence in a rather rigorous way. The only explicit concentration dependence in Dellby's model is the linear dependence normally included in the Curie constant C. Dekker's G varies linearly with concentration as c approaches zero as all other less detailed effective field theories do. Owen et_al,(37) used a molecular field model to obtain a result analogous to Dellby's result Equation (3.9). Their direct interaction term showed explicit competition between FM and AFM coupling like 59 Dekker's, but without the concentration dependence other than that in the Curie constant. The indirect interaction term appeared exactly as in Dellby's result. It is important to note that term Caz X5 in Equation (3.9) yields a positive contribution to O regardless of the sign of a. The physical mechanism associated with a is exchange scatter- ing between conduction electrons and impurity atoms which gives rise to a relative Spin correlation. But this is precisely the source of the well-known Rudermann-Kittel-Kasuya-Yosida (RKKY)(38'4O) interaction (see Section C.2., Chapter V). Since the RKKY interaction can couple impurity moments ferromagnetically or antiferromagnetically depending on spatial separation of those moments, the contribution to O arising from this interaction can be of either sign. A realistic calculation of o arising from the RKKY interaction requires the performance of a complicated lattice summation procedure,(41’42) and the sign of O can be either positive or negative<43’44) depending upon the details of the host's crystalline and electronic structure. An effective field model like that of Dellby or Owen §t_al, cannot account for this behavior because it is not sensitive to the non-uniformity in Space of the magnetization of a particular species. The Spatially oscillatory conduction electron Spin density characteristic of the RKKY interaction represents magnetization of precisely this type. These effective fields take just a mean value of this oscillatory magnetization. With this Situation it is easy to see that impurity moments always couple ferromagnetically whether a is positive or negative. A Similar weakness may be present in Dellby's interaction tenn azxs + B C (—l_773&;92 (Xbase-Xs'xdia) (3'11) 60 from Equation (3.7). Equation (3.ll) represents the additional coupling between impurity moments arising from the exchange enhanced nature of the host metal and is always positive if Xbase‘xs'xdia > 0. Never- theless, Dellby's model is useful in that it suggests which types of interactions constribute to p and 0 in the Curie-Weiss law. To rigorously consider all interaction processes in a system like EdMn(l.O) (the alloy system Dellby considers) is probably far too difficult for present theoretical capabilities, so one must tolerate some of the weaknesses of the molecular or effective field model in order to proceed with a complicated magnetic interaction problem. Finally, Dellby hints that a classical effective field model would be unable to account in any way for the Kondo effect, a many- ( body phenomenon. This appears to be the case. Heeger 4) has shown (Section C.l., Chapter V) that the susceptibility of a Kondo system can approximate Curie-Weiss behavior over a certain temperature range. B where J is an effective s-d exchange integral, and p is the conduction electron density of states per atom. In Dellby's model, the term CazxS in Equation (3.9) would be most closely related to a Kondo mechanism of S-d scattering. However, if a and J are proportional, and if XS is taken as prOportional to p aS in the free electron model, it is clear that the functional dependencies of Tk and ODellby on J and p are entirely different. Furthermore, since Dellby's Cazxs is always positive and Tk must be positive, there is a sign discrepancy (see Equation 5.6). The Kondo effect, an isolated impurity moment 61 phenomenon, is not accounted for in considering the interaction between an impurity moment and conduction electrons in the effective field models of Dellby and Owen et_al, IV. PREPARATION OF (Cu XC AND PRELIMINARY TESTS 0.83Pd0.l7)l-c A. Properties of CUO.83PdO.l7 and Alloy Preparation The desire to use a non-magnetic binary alloy with an atomic order-disorder transformation as a host matrix for magnetic impurities drastically limits the possible choices. The classic atomic order- disorder system Cu-Au immediately comes to mind as a candidate. This system has been studied rather extensively, however, and at least two papers on impurity susceptibility in Cu3Au have appeared recently.(45’46) The binary alloys of Cu with Pd and Pt also have atomic order-disorder (47) Although some magnetic susceptibility studies (48-52) (48) transformations. have been done on the Cu-Pd system, only one paper relates the susceptibility to the order-disorder aspects of the alloy. Appar- ently very little work has been done on the magnetic prOperties of Cu-Pt.(53) Pd and Pt would seem to be interesting candidates as one of the components to complement the noble metal Cu in a binary host, particularly with regard to local environment effects. This Specula- tion is suggested by the exchange enhancement effects known to occur in Pd- or Pt-rich alloys. One might expect a magnetic impurity atom to behave very differently with Pd or Pt neighbors as opposed to Cu neighbors due to polarization effects that occur in Pd and Pt hosts.(54’55) Although we have carried out some preliminary measurements in systems with Cu3Pt and Cu3Au as host matrices, we focus in this thesis on 62 63 more extensive results obtained to date in CuO 83Pd0 17 (hereafter notated CuPd(l7)) based alloys. A careful study of the transformations in the Cu-Pd alloys by Jones and Sykes(56) substantiated that the maximum degree of atomic order as demonstrated by superlattice x-ray lines(57) and electrical resistivity<58) occurs at l7 atomic percent Pd (Figure IV-l). Further- more, the lattice structure of the disordered Cu3Pd type alloys is face-centered cubic (fcc) in the disordered state in analogy with Cu3Au, and below 20 atomic percent Pd, the le Cu3Au fcc structure is retained upon ordering. This is not the case for Cu3Pd type alloys of greater than 20 atomic percent Pd which devel0p a tetragonal distortion upon slow cooling below the ordering temperature.(56) Similarly, the CuPd type alloys exhibit a phase change upon ordering, becoming body-centered cubic (bcc) from fcc. AS pointed out in Chapter I, a change in the lattice of atomic sites would be undesirable in our intended search for local environment effects induced by ordering. Therefore, to realize the greatest degree of atomic order while maintaining a fixed lattice of atomic sites, it was natural to choose the composition CuPd(l7) as a host matrix. A master CuPd(l7) alloy was prepared from 6 9'5 grade Cu and 5 parts per million (PPm) impurity Pd Sponge from Johnson, Matthey, and Company. Approximately l0 grams of Cu was chemically etched with dilute nitric acid to remove surface impurities. The Cu was carefully weighed on a Mettler balance to a precision of :lxlO'5 grams and placed in a high purity alumina crucible. The apprOpriate complementary mass of Pd Sponge was weighed into the crucible. This procedure allows one to control the nominal composition before melting to within O.l atomic ‘C 64 WEIGHT PER CENT PALLADIUM 10 20 30 40 50 60 70 80 90 ‘Sool i J I 1.? I L I I ‘ I 1 15540 I O REF. I x REF. 2 l . / 1400 , /’ ‘ II 1300 /"z 1200 .5 ; ’ 1100‘ /” 1003- '— 1506 :8. a. or lCu.Pdl 900 reapenarunc. soc. PdCu i—PdCU; 700 V 600 $00 ‘2‘ E I ’I -~ \ \ ‘ 400 --.4 -— h w P-——H ug:m.a .9: I N _ 0 — _ :o»m mmpu
    z .m->~ acamwa 8v: : e. m N _ o + + 1 EomEvuaao ommwomoma 0.03 + 1 ommmomo mmduo o curacy—00.0 mwduo n 0N0 0¢.0 00.0 6.523 4.53 2 00.0 00.. 0N._ 72 .hchuaquuaau Lac 1 .m> x .¢->~ «cause 85 I 3.82.3330 3323.... 23 2.322.. 1.. 00;uo+ 50.0.6 0 0N0 “0.22: .3an :— 0¢.0 00.0 73 .A60czflhpwuaau to. z .m) z .m->~ «camva 8v: 1 ¢ m N _ 0 fi _ q _ .O . L 1 00.0 Lmtz: . .152. z 1 cm; L ommwomoma 0.._uo+ . 3:22.328 ommmamo onouo o- co _ 74 .AUVSz N_ case toe I .m> 2 3v: 1 AorzCLv—caao N q .m->H «guard ommmomo owmmomOmE No.3 0 ~.o_....o + 0_.0 $.22: 4.55 2 0N0 00.0 75 C. Heat Treatment Jones and Sykes(56) indicated that significant ordering was produced in CuPd(lS) by soaking the alloy at 470°C (just below the critical temperature) for l2 hours and cooling at 30°C per hour. AS a first attempt to obtain atomic ordering in our CuPd(l7) alloy, we rather arbitrarily decided to anneal at 470°C (critical temperature approximately 500°C) for 24 hours and cool to room temperature at about 20°C per hour. One Specimen of CuPd(l7) and one of CuPd(l7) Fe(0.l) were individually sealed in one-third atmosphere of approxi- mately 99 percent pure Ar gas inside clean Vycor tubes which had been previously evacuated to lO'4 mm Hg. The anneal was conducted in a Lindberg Hevi-duty furnace which has a manually controllable temperature range of about 300°C to l350°C. The furnace is adjustable in steps of 0.05°C and appears to be stable over long time periods to i O.l°C. The furnace temperature was monitored near the samples with a chromel- alumel thermocouple. Susceptibility measurement on the annealed CuPd(l7) indicated significant ordering in the heat treatment (Section 0, Chapter IV). The diamagnetic susceptibility of the sample quenched from the melt increased in magnitude by about l00% upon annealing. This is in qualitative agreement with the only other such data(48) known to us. Furthermore, the room temperature electrical resistivity (Section A.2., Chapter V) of the annealed CuPd(l7) was fOund to have decreased by a factor of about one-third from the value for the material quenched from the melt. 76 A later heat treatment of alloys with 0.70 and l.l6 atomic percent Mn in CuPd(l7) caused some surface tarnishing of the samples in the 99 percent pure Ar atmosphere. Although this surface effect did not seem to affect the magnetic prOpertieS of the alloys (susceptibility measurements before and after etching agreed), subsequent anneals were conducted using a very high purity grade of Ar (impurity concentration in ppm). There has been no further evidence of thermally induced surface effects. We have carried out a heat treatment on one Specimen from the pair of samples cut for nearly every alloy which displayed a linear M(H) in the disordered state. However, as pointed out above, only the systems CuPd(l7) Mn(c) and CuPd(lZ)_Ni(c) provided complete series of alloys of varying c with both ordered and disordered hosts available for detailed magnetic studies (Chapters V and VI). 0. Survey of Magnetic Properties We have measured the susceptibility of three different samples of CuPd(l7) quenched from the melt. The susceptibility is independent of temperature over the range 78 K to 300 K to within our experimental accuracy. This is in agreement with measurements by Ekstr6m gt_ l.(52) 3 l on CuPd(23). We measure the molar susceptibility as -6.6 i 0.7 cm mole" at T = 300 K. The magnitude of the uncertainty in this result is slightly greater than the present precision of our magnetometer, and reflects an apparent Spread of values for XM among the three samples which were produced from different master alloys. Whether the Spectrum of values is reflective of some variation in nominal composition, and/or 77 of some variation in the degree of atomic ordering unavoidably introduced in a quench from the melt is not known at this time. Ekstr6m gt__l,(52) report a Similar lack of reproducibility for CuPd(20). However, on the basis of marked changes in XM and the electrical resistivity p (Section A.2., Chapter V) induced by the heat treatment of CuPd(l7), we have tentatively concluded that the degree of unwanted atomic ordering occurring during a fast quench from the melt is slight. That is, any apparent Spread in values of XM or p (comparable to the pre- cision of any single determination) for the quenched alloy is very much less than the difference between the annealed alloy values and the mean quenched alloy values. For the ordered CuPd(l7), we find 1 = ~13.l t 0.5 cm3-mole' and this value is independent of temperature XM from 78 K to 300 K to within experimental accuracy. In light of the clear cut effects of our heat treatment procedure, we shall hereafter refer to the quenched samples as disordered and the annealed samples as ordered. We shall now survey the magnetic character of the system CuPd(l7)X(c) where X represents the 3d transition elements from Mn to Ni. Most of the details for X = Mn and X = Ni are reserved for Chapters V and VI respectively. For CuPd(l7)Mn(c) we have observed a Curie-Weiss impurity suscep- tibility superimposed on the temperature independent contribution of the host for values of c in the range 0 to 3.2 with the host both ordered and disordered. For the disordered host, the paramagnetic Curie point 0 is essentially independent of the concentration over c from 0.22 to 3.2. p, the effective Bohr magneton number, shows a Slight tendency to increase with c. The ordered system shows markedly 78 different behavior with O decreasing linearly with c. dgéc is large (2 — l0 gffg‘fi57 and of opposite sign compared to CuMn(c) and many other dilute magnetic impurity in simple host systems. Again, p appears to increase Slightly with c Similarly to the disordered case, but the magnitude is approximately 3 to 5 percent greater for the ordered host. We have pointed out above (see Figure IV—3) the solubility problems which plague CuPd(171Fe(c). Three alloys have been prepared which displayed a linear M(H) at T = 300 K. Only at c = 0.1 were we able to anneal the alloy without inducing significant Fe precipitation. A good disordered alloy at c = 0.29 was obtained upon quenching. Each of these samples exhibited a Curie—Weiss susceptibility, but unfor- tunately the Fe concentrations, and hence our experimental precision, are too low to conclusively state whether or not Significant order- disorder effects are present. It seems clear that the p and O values for X = Fe are significantly different from those for X = Mn at equi- valent c values. Figure AC-l in Appendix C shows the temperature dependence of x; for CuPd(l7)Fe(O.29). As shown in Figure IV—4, we were not able to obtain a linear M(H) relationship for CuPd(l7)Co(c), c = 0.67 and l.06, and the magnitude of the magnetization is reduced from that of comparable concentrations of Fe. The susceptibility Shows considerable temperature dependence, increasing with decreased temperature, but more slowly than a Curie- Weiss susceptibility. Hence, we have concluded that Co as an impurity in CuPd(l7) probably sustains a localized magnetic moment, but that there is a strong tendency toward clustering of Co atoms as is well- known in CuCo and AuCo.(]) 79 The contribution of the Ni in CuPd(lZ)Ni(c) to the total alloy susceptibility per atomic percent Ni is the smallest of the four 3d impurities we have considered. In fact, we find that the impurity susceptibility per atomic percent impurity falls monotonically as we proceed through the 3d transition metals from Mn to Ni just as one would expect on the basis of a filling d shell. The susceptibility of CuPd(l7)Ni(c) is temperature independent over 78 K to 300 K for c = l.23 and l0.2, ordered and disordered, and on this basis we believe that Ni atoms in CuPd(l7) do not carry local moments. This does not, however, preclude the possibility that this is an interesting system magnetically. We present evidence in Chapter VI that suggests that magnetic interactions may be important at Ni concentrations beyond approximately 1 atomic percent, particularly in the ordered alloy. In summary, we submit that compelling evidence has been found for markedly modified magnetic interactions in the system CuPd(l7)X(c), X = Mn, Fe, Co, and Ni, as the state of atomic order of the host matrix is changed. Experimentally, it has been much easier to syste- matically study these effects for X = Mn and Ni, since metallurgical difficulties intrude for X = Fe and Co. In Chapters V and VI we shall present the details of our experimental findings for X = Mn and Ni respectively, and offer suggestions directed toward understanding the physics involved. We emphasize at this juncture that order-disorder effects are clear in the magnetic behavior of both systems, but that the nature of the magnetism in each is very different. One might infer that Fe and Co would supply the smooth transition from local moment to "no moment" magnetic character in CuPd(l7)X(c) if metallur- gical difficulties could be avoided. V. STUDIES OF CuPd(l7)Mn(c) A. Detailed Sample Reliability Tests Prior to presenting the data on magnetic interactions in CuPd(l7)- Mn(c) and a discussion of the findings, we describe in the following two subsections work that was undertaken in part with the intention of further establishing, as conclusively as possible without detailed crystallographic or chemical analyses, the homogeneity and continuity of the alloys of the impurity concentration series CuPd(l7)Mn(c). Chronologically, most of the analysis reported was carried out after the interesting effects mentioned in Section D of Chapter IV had been discovered simply to cross-check some readily accessible properties of this alloy system that has at no time given any indication of metallurgical difficulties. A further intention was to empirically check a simple additivity relation for the susceptibility of dilute alloys. We maintain that the results presented, coupled with the earlier linear magnetization versus applied field data, are, beyond all reasonable doubt, supportive of the reliability of the alloys CuPd(l7)Mn(c) and justify the application of the type of analysis offered in the latter portion of this chapter. 80 Bl l. Susceptibility Versus Mn Concentration at Constant Temperature When a non-magnetic host material is alloyed with magnetic impurities, the simplest concentration dependence for the total alloy susceptibility one could expect is xA(C) = (l-C)xH + cx‘ . (5.1) A E Xalloy, XH : XhOSt, XI ; lepurIty, and c is the impurity H and XI are taken independent of c, the where X _ atomic fraction. When X validity of Equation (5.l) is restricted to the limit of c + 0 if one believes that a simple superposition of susceptibilities will break down when an appreciable host-impurity interaction exists. One might extend the validity of Equation (5.l) to larger c and account for 1+ x‘m. the breakdown of superposition by letting xH-+ xH(c) and X but then one is faced with the need for explicit expressions for xH(c) and XI(c). We have checked Equation (5.l) empirically at fixed temperatures for CuPd(l7)Mn(c), O s c s 3.2, with the host ordered and disordered. Our hOpe was that in the concentration range studied we H by the constant lim XA(c) and xI(c) by C c -O c ’ c+0 where C(c) is the normal Curie constant (i.e., independent of tempera- could represent X ture) as a function of c with its usual explicit linear c dependence placed in the coefficient of XI + XI(c) in Equation (5.1). We then have . [P(C)uB]2 3k (5.2) C(c) B Note that if we can adequately describe xA(c) for CuPd(l7)Mn(c) in this way, we will have justified in an independent manner the customary cavalier separation of XI from xA by simply subtracting out a constant 82 host susceptibility. It is not obvious a_prigrj_that such a procedure is valid as c becomes appreciable and additivity questionable. Further- more, a successful description of xA(c) by the adopted generalization of Equation (5.l) serves as a self-consistency check on the values of p(c) and C(c) determined from temperature dependence measurements of xA(C). Figures V-l and V-2 show XG(C) as a function of c at T = 300 K and T = 80 K for CuPd(l7)Mn(c), ordered and disordered. The Mn concen- trations as plotted are the nominal ones. The absolute accuracy of 3 the values of XM is put at i l x l0“6 cm -mole"]. There are at least two important points to be made with regard to these plots. (T) If XH and XI were independent of c, then the plot of xA(c) versus c would be linear. This is very nearly the case for the disordered host at both temperatures. Any slight systematic departure from linearity could be due to uncertainty in the actual Mn concentrations or to real c dependence in XI or x“. In light of the c dependence of p and O inferred from the data of the next section, one can conclude that XI for the disordered host is only very weakly concentration depen- dent, if at all. At c = O, xA(c) extrapolates to x“. Equation (5.1) works very well for the disordered alloy. For the ordered host, xA(c) departs from linearity weakly for T = 300 K and drastically for T = 80 K. This can be accounted for in a self-consistent way when one considers the concentration dependence of p and 0 found for these alloys from the temperature dependence studies. Also, cxI(c) >> (l-c)xH for c 3 0.22 and T = 300 K in CuPd(l7)Mn(c), and the temperature dependence of xI(c) insures that for T = 80 K, xI(c) completely dominates xA(c). Therefore, these plots of xA(c) versus c are rather sensitive probes of 83 4.0 + DISORDERED CuPd(l7)Mn(c) 0 ORDERED T=300K 3.0 - TI; _, 2.0 - o 2 I ”2 3 L < 8 >.< *0 LO»— 0 1 I 1 I L I O I 2 3 c (AT. 96) Figure V-l. xA(c) vs. c at T = 300 K for ordered (0RD) and dis- ordered (00) CuPd(l7)Mn(c). 84 l5.0 +D|SORDERED CuPd(lflMn‘C) +°0RDERED T=80K IZLCDP- T- LI] .1 09.0?- :5 '0' IE 8 _ < 1 >.< c 6.0- 9.. 3.0F o 1 I 1 I 1 I C) I 21 :3 c (AT. 96) Figure V-2. xA(c) vs. c at T = 80 K for 0RD-DO CuPd(l7)Mn(c). 85 the c dependence of p and 0, especially for T = 80 K. Again, for the ordered host, a linear extrapolation to c = 0 implies that lim x5(c) = xH and Equation (5.l) as generalized is adequate. c (2) Two of the nominal concentrations for the Mn series were adjusted to more reasonable values on the basis of the XA(C) plots for the disordered host. ‘From the O values deduced from the temperature dependence studies, there was reason to expect that XA(C) should be linear in c for disordered CuPd(l7)Mn(c). But the values of XA(c) for T = 300 K at c = O.l4 and c = 3.42 fell well below the linear trend established by the values at c = 0, 0.22, 0.70, and l.l6. When the assumed nominal c values for the anomalous samples yielded seemingly unreasonable p values from the temperature dependence studies of these two alloys, it became clear that the nominal c values were in error. 8y fbrcing the xA(c) values for c = O.l4 and 3.42 onto the linear plot, we could deduce revised c values, and from the revised values of c and the temperature dependence measurements, recalculate p for each. The modified p values fell smoothly and self-consistently onto the estab- lished trend for p(c) and we were able to conclude that c = 0.14 should go to c = 0.07 and c = 3.42 should be c = 3.l5. This procedure is admittedly more trustworthy for the alloy of lower c, because an extrapolation of p(c) or XA(C) from c = l.l6 to c = 3.15 is speculative at best. However, we have cross-checked our procedure with electrical resistivity measurements (see below) and they too are consistent with these adjustments. 2. Electrical Resistivity-Measurement and Results Electrical resistivity measurements of estimated i 8 percent accuracy have been carried out on each alloy of the series CuPd(l7)Mn(c). 86 Measurements were made at T = 300 K and 78 K for ordered and disordered hosts. The purpose of these determinations was (l) to check the degree and consistency of atomic ordering achieved with identical anneals for increasing impurity concentration, (2) to look for correlations between electrical resistivity and magnetic susceptibility behavior as a function of impurity concentration and host atomic order, and (3) to provide a further check on the nominal impurity concentration values. The relatively low accuracy of these measurements, although clearly sufficient to satisfy (1) and (2), and probably (3), is attributable to the method of measurement necessitated by a desire to perform the measurements on the same samples for which the susceptibility was determined. We did not want to introduce surface magnetic contamination as could readily occur if current and potential leads were soldered or arc welded directly to the sample. Instead, we felt that current and potential contacts with the sample could be best achieved by forcing appropriately shaped, non-magnetic conductors against the sample surface. Electrical leads would be soldered directly to the conducting contacts and Specimen contamination thereby minimized. Four probe resistivity measurements at room and liquid nitrogen temperatures were effected by fabricating a small sample holder which mounted on the end of a dipstick. The body of the holder consists of a cylindrical electrical insulator (teflon) through which two ports were cut to admit coolant. The ends of the teflon encasement are formed by plane-faced brass plugs, one of which is removable via threads. The brass plugs provide the current source and sink which press against the ends of a cylindrical sample. To promote uniform conductive contact over the entire end cross-section of the samples, 87 we have placed several folded layers of Pb foil between the brass plugs and the specimen end faces. We find that the malleable Pb provides a relatively high conductivity path in places that would be free of direct brass-sample contact in its absence, Since all samples do not have smooth planar end faces precisely perpendicular to their axes. Uniform conductive contact helps to insure uniform current densities throughout the specimen, a necessity for meaningful resis- tivity determinations. The potential drop along the length of the Specimen is probed by a pair of 0-80 brass screws whose ends have been filled to straight knife edges. These screws are threaded into the wall of the teflon such that the knifieedges contact anticipated equi- potentials on the surface of the specimen. The maximum possible separation of the potential probes along the direction of current flow is arranged to insure maximum sensitivity in the potential difference measurements. An Allen set screw through the wall of the teflon opposite the pair of potential contacts presses the Specimen firmly against the knife edge. We could conveniently pass a maximum current of O.l amperes through the samples. This resulted in potential drOpS over a length of 0.97 i 0.02 cm of specimen of the order of 40 uV : l%. The primary sources of error in our resistivity results are the determination of octagonal cross-sectional areas of the susceptibility samples and lack of certainty as to the degree of uniformity of current densities given the relatively large cross-sectional areas. The estimated accuracy we quote is based on a composite Of these factors and the reproducibility of any given determination. 88 We plot in Figure V-3 the electrical resistivity of CuPd(l7)Mn(c) as a function of c at T = 300 K and 78 K with the host ordered and disordered. Allowing for the assumed accuracy of the measurements, d 300K the fits of each of the four sets of data is linear in c. c uQ-Cm . pQ-cm equals l.9 Effififi'for the disordered host and 1.2 attiMn for the 300K - p78K, reflective of some ordered host. The quantity Ap g 0 portion of the lattice thermal component of p, is l.3 : 0.4 uQ-cm for the disordered host and l.8 1 0.4 pQ-cm for the ordered material. These data suggest internal self-consistency among the samples and provide rather irrefutable evidence of atomic ordering achieved in the heat treatments. We note that p(c) for the ordered host is linear, while xA(c) was not. This differs from the linear xA(c)-linear p(c) correlation for disordered CuPd(l7)Mn(c). Although the linear fits we find for p(c) are entirely consistent with the modified nominal concen- trations discussed in Section A.l., the limited accuracy of our resistivity results precludes a definitive statement as to the correct— ness of the reassignment. 8. Temperature Dependence of Susceptibility We have measured xA(c,T) for CuPd(l7)Mn(c) over the temperature range 78 K to 300 K for both ordered and disordered hosts. To isolate the temperature dependence of xI(c,T), we have made the separation xI(C.T) = c"[xA(c.T> - (l-C)xH(C)] (5.3) in accord with Equation (5.l) which appears to be valid. We have no H evidence that X depends on c for c < 3.2. In Figure;V-4 through V-7 89 25 +DISORDERED, T=300K o " T=77.4K - DOROEREO, T=3OOK c Pd 7Mn(c) _A .. T=77.4K 20.. g . O u '5- c3 __ , i 0 0. 5_ 1 I 1 I 1 I 0 I 2 3 c (AT. '36) Figure V-3. p(c) vs. c at T = 300 K, 78 K for ORD-DO CuPd(l7)Mn(c). 90 we show [XM(T)J-] as a function of temperature for c = 0.22, 0.70, l.l6, and 3.l5. Each figure includes data for each state of host order. In every case, a linear least—squares fit adequately characterizes the data. (See Appendix B for a tabular listing of all [XM(T)]-1 versus T data in this thesis.) The absolute accuracy of each value of XI is t l x l0"6 cm3-mole']. From the fit we extract the Curie-Weiss para- meters 0 and p which appear in 2 [p(C)u I I C B X (”I = T-"0L(_)—Cg = 3kBTT-o(cll ' (5'4) Table V-I lists the values of 0 and p for each of these measurements. In Figure V-8 we plot 0(c) for CuPd(l7)Mn(c) and CuMn(c) for comparison. (18) (37) The _C_uMn(c) data is from Hurd, and Mom-s £91,032) Owen et_al,, From Figure V-8 we see that 0(c) for disordered CuPd(l7)Mn(c) is nearly independent of c. In sharp contrast is the strong, linear in c behavior of 0(c) for ordered CuPd(l7)Mn(c). As quoted earlier, ggéEl-z - l0 Et_%Mn' for the ordered alloy. For CuMn, Morris and Williams(62) indicate dgéc = + 7 atfiifin" There seems to be much inconsistency with regard to the exact magnitude of the latter figure, but there is little doubt that for CuMn(c), d0(c)/dc is positive and of the order of l0 ETTMMH" Two striking features of our data that require explanation are the very different behavior of 0(c) for ordered d0(c) '7T__- and disordered CuPd(l7)Mn(c) and the opposite signs of c ‘QuMn(c) and ordered CuPd(l7)Mn(c). for As a control test on our susceptibility versus temperature results for CuPd(l7)Mn(c), we measured the temperature dependence of the susceptibility of CuPd(l7)Gd(O.4) for both states of host order. The 7 Gd impurity, being a 4f rare earth ion with tightly bound magnetic .ANN.oVezAA_Veaeu oo-oxo Lee A .m> P-Hfievqxu .e-> deemed . 20 Quad MMDFQMMdsz .omm .omm .o:d .oom .oms .omz .om .o: .pu _ p z Hi _ _ _ _ . fi 8:2: m 3 $32.: 833de <53 8539 I IS 9l Qmmmgmo . Qmmwgmema q flmmdszvfitgdju I ‘ ('2', f: [DQ/BWUNT \__, a) ' 3.; /. (77 ‘ AJU 92 .Aok.00ezfim_vedeo oo-eeo Lee A .m> P-Haevqxu .m-> deemwd Q own: waspmmmmzwp .Omm .omm .Oflm .me .OwH .Omfi .00 .0: .PU 3:2: m S 243% . Dwo<4mm_o <._. _-Hfievqu .o-> deemed 3. Own: MMDFEMMnEwH .Qmm .Omm .ij .OON .Omf .ONH .0m .0: .00 ._ _ _ e _ r _ _ . Qummamo x i.S QmMMQMQmHQ .. Co flwflQVZZKSQmDQ imum mm. 8 3 \\ fimmx \. m .\ m \ 9w \\ 1.00 A 1.9 6 94 .AmP.mveznkpvea=u oo-amo Lee E .m> P-Hfievde .A-> weaned Ax owe“ mmaqummzmH .Qmm .omm .Odm .omm .owfi .omfi .00 .0: fl ammmamo. we Qummamomse.. ...t m: mczz essadao .. .. Tom 95 TABLE Y—I p.® VALUES FOR Mn,Gd IN CuPd(l7) SAMPLE p ® 00 can 00 0RD CuPd(mMn(o.22) 5.3 5.5 o -3.6 1:02 .2 CuPd(InMMOJO) 5.5 5.6 -2.5 -9.0 CuPd(l7)Mn(l.|6) 5.5 5.6 -2.5 -M.: CuPd(l7)Mn(3.2) 5.5 5.5 -3.2 -35.5 CuPd(l7)Gd(O.4) 6.2* 6.2* -3.0 -2.7 *SEE PAGE 68. IO 0- + I 1' 'IO— 2 ® ~20~ '30- +DISORDERED CuPd(l7)Mn(c) ° ORDERED CuPd(lDMMCI AggMMc), REFERENCES: IS, 37, 62 .. L I l I 1 I 400 I 2 3 6 (AT. %) Figure V-8. 0(c) vs. c for 0RD-DO CuPd(l7)Mn(c) and CuMn(c). 96 97 electrons, was expected to carry a moment in CuPd(l7) of a different type than the Mn. This expectation is based upon the experimental fact that the effective moment of Gd atoms in dilute solid solution is essentially the free ion moment. This is partially because the Gd+++ ion is in an 5 state, and thus Gd+++ does not interact with crystalline electric fields, and partially due to the fact that the occupied f states lie somewhat below the Fermi level so that the degree of s-f mixing(63) is considerably less than the correSponding s-d mixing associated with 3d virtual bound states. In short, the Gd should carry a "clean" magnetic moment as compared to Mn, and differences in the magnetic behavior as the host matrix atomic order changes could offer clues in the interpretation of the observed effects in CuPd(l7)Mn(c). In Figure V-9 we plot [XM(T)]-] against temperature for CuPd(l7)Gd(O.4). Least-squares fits for the host ordered and disordered indicate that p and O are the same for each (see Table V-l). Electrical resistivity measurements indicate that ordering comparable to that for all of the CuPd(l7) anoys was achieved. We find resistivities of 13.7 and 12.0 uQ—cm at T = 300 K and 78 K respectively for disordered CuPd(l7)Gd(O.4). For the ordered alloy, 6 = 8.8 and 7.0 uD—cm at T = 300 K and 78 K. Each resistivity determination carries the i 8 percent accuracy limit. Thus, it is clear that Gd and Mn in CuPd(l7) do behave differently magnetically. Any interpretation of the effects observed in CuPd(l7)Mn(c) must be consistent with their apparent absence in CuPd(l7)Gd(O.4). 98 .Ae.oveofle_vea=u oe-emo Lee 0 .m> P-HfihvdxH .m-> deemed nu owe. wmzpamwmzmh .owm .omm .Odm .omm .omH .omH .mm .m: .QU Amtz: m >m om<3d2 . omoqddmzu < . (D. '4 2 o— + O -5 + 0 -IO- = . I I I I I o 2 4 6 8 IO 6 (AT. %) Figure VI-la. XA(C) at T = 295 K for 0RD-DO CuPd(l7)Ni(c) and CuNi(c). 115 13(3 +DISORDERED CuPd(l7)Ni(c) ° C’ORDERED CuPd(l7)Ni(c) ~AQgNiIc) T-295K + 24~ A I8“ T" l.l.) 5‘ _ A z o ”I :5 § 12— : 2 ’5 < a __ . 25 CD 2 6L 0 " o ' I I l I I C) 22 ‘1 £5 £3 1(3 6 (AT. %) Figure VI-lb. xA(c) at T = 295 K for ORD-DO CuPd(l7)Ni(c) and ggfli(c). 116 for c 5 5.0, is valid for disordered CuPd(l7)Ni(c) and CuNi(c). For ordered CuPd(l7)Ni(c), xA(c) appears to be linear through c = 10.2. This is somewhat surprising. Equation (5.1) would seem to be valid for c 5 10.2 with an increased value of XI. We see that ngéEl-evaluated near c = 0 increases as we proceed from a pure Cu host through ordered I increases. CuPd(l7), perhaps indicating that x The electrical resistivity of the ordered and disordered alloys of the system CuPd(l7)Ni(c) at T = 300 K is Shown in Figure VI-2. While p(c) is linear for the disordered host with an impurity resistivity of about 1.0 §%T§ENT" the data for the ordered host is highly non-linear. The measurements cannot be well fit with any simple power law in c. It appears that the values of p at the three highest concentrations have defined a linear behavior with nearly the same slope as the disordered alloy plus an added constant. 8. Discussion of Results The electrical resistivity results for ordered CuPd(l7)Ni(c) cast some uncertainty on the interpretation of the results for this impurity series. The data indicates that the resistivity of the alloys with the highest Ni concentration has actually increased after a heat treatment intended to atomically order the host matrix. Although this is a surprising result, there is an obvious possible cause. Metallurgists have proven that the binary alloy CuPd(l7) orders atomically, but to our knowledge, no systematic experiments have checked the extent to which the ternary System CuPd(l7)X(c) orders. Our previous resistivity results for X = Mn and c 5 3.2 indicate ordering is not diminished in 117 25 b+D|SORDEFQED CgPd(|Z)Ni(C) OORDERED CuPd(l7)Ni(C) T'3OOK 20— SE c ‘3 C} :L l5F- Q )- IO— )- 55 I I I J C) 2! ‘1 £5 £3 6 (AT. %) Figure VI-2. p(c) vs. c at T = 300 K for 0RD-DO CuPd(l7)Ni(c). 118 that ternary alloy. However, it could well be that for X = Ni and c z 1.0 ordering is impeded by the impurity. At c 2 10 for nearly any impurity one would expect some perturbation of the host's normal behavior, including atomic ordering tendencies. Yet, our suscepti- bility results for annealed CuPd(l7)Ni(c) appear to obey a simple, unique functional dependence on c (linear) for O 5 c 5 10.2. If the resistivity data is interpreted as reflecting the reduction of induced atomic order as the Ni concentration is increased, it seems surprising that xA(c) behaves so simply. Of course, a linear XA(c) could be ”( reflective of a fortuitous combination of X c) and XI(c) (see Equation (5.1)), the concentration dependence of XH being due to the gradual reduction of the presence of order with increasing c. Detailed resis- tivity measurements over the temperature range 4 K to 300 K are now planned for this alloy series to attempt to answer the question of the possible diminution of atomic order with the addition of Ni. The similarity of the behavior of xA(c) for CuNi(c) and CuPd(l7)Ni(c) is noteworthy. As pointed out above, there is a indication that the impurity susceptibility XI is larger in CuPd(l7). Equation (5.1) 3 1 I yields x; = 130 (10'6) cm -mole' for Ni in Cu, and XM = 205 (10'5) cmB-mole'1 for Ni in disordered CuPd(l7). This is possibly reflective of exchange enhancement due to the presence of Pd. Until the question of the extent of atomic order achieved in CuPd(l7)Ni(c) is settled, any analysis of the susceptibility on the basis of atomic order must be tentative. However, since the suscepti- bility XA(c) for the heat treated alloys is linear within experimental accuracy, evaluating dxA(c)/dc near c = O and applying Equation (5.1) indicates that X1 is greater for the heat treated alloythan for the 119 disordered system. It then becomes very tempting to speculate that a local environment effect is becoming increasingly manifest as atomic ordering occurs. (Experimentally, there is no doubt that ordering occurs at low c.) Presumably there is no competing RKKY interaction to produce a long range Ni-Ni interaction in this system since there is no permanent Ni moment. Again, we emphasize that the apparent increase of dxA(c)/dc|C=0 upon ordering may be an artifact resulting from the behavior of xH(c) if ordering diminished with increased c. VII. SUMMARY A brief review of the history of studies of dilute magnetic alloys was presented. The emphasis was on the experimental approach of attempting to understand COOperative magnetic phenomena in metals by utilizing alloys to systematically construct the magnetic state in terms of the fundamental interactions among the constituent magnetic atoms. This approach forced physicists to first answer basic questions about local moment formation in metals, and then to explain the striking changes in the prOperties Of metallic systems induced by the presence of dilute magnetic impurities. Only presently are we beginning to take the next logical step of considering the basic magnetic inter- actions present in a moderately dilute magnetic system. The work described in this thesis was undertaken in the hOpe of contributing some small part to the resolution of the difficult problem of magnetic interactions in metals. A vibrating sample magnetometer has been constructed and cali- brated. This instrument has been used to perform reliable magnetic susceptibility measurements on a wide variety of metals, alloys, and insulators, including the alloys discussed in this thesis. At present, the Signal detection coils have = 5.4 (103) turns and the resolution of the magnetometer is 2 2 (10-5) emu of magnetic moment. For a Cu Specimen of 0.5 gm in an applied magnetic field of 4700 gauss, we would measure the susceptibility with a precision of the order of t 10 percent or 120 121 better. The sensitivity could rather easily be improved by an order of magnitude or greater by increasing the number of turns in the detection coils, the applied field, and/or the specimen size. However, the present sensitivity was entirely adequate for the work described in this thesis. An apprOpriate cryostat allows continuous susceptibility determinations below room temperature. Routine measurements to T = 77 K are reported, and measurements to 4.2 K are possible with the current design. A discussion of the relevant aspects of a systematic experimental study of the magnetic properties of dilute alloys was given. The particular approach we have adopted of using binary alloys with atomic order-disorder transitions as host matrices for magnetic impurities was given justification. The most important points presented include (1) the possibility of inducing local environment effects, (2) the capability of altering long range interactions, and (3) the avoidance of the uncertainties of depending entirely upon the reliability and continuity of distinctly different alloys. Because most of our susceptibility measurements indicated the applicability of the Curie-Weiss law, we discussed the relation of the paramagnetic Curie point 0 to magnetic interactions. The effective field models of Dellby,(35) Dekker,(36) and Owen gt_al,(37) were seen to be helpful in incorporating several types of interactions into 0. One must not take these models too seriously, however, since these Simple effective field models cannot, for example, account for the RKKY interaction or the Kondo effect. The selection of the alloy system CuPd(l7)X(c) was considered, and the preparation and initial reliability testing of the alloys was 122 explained. We found that a simple heat treatment process to induce atomic ordering was effective as confirmed by susceptibility and elec- trical resistivity measurements. Of the four transition metal impurities X = Mn, Fe, Co, and Hi, only Mn and Ni were sufficiently soluble in CuPd(l7) to withstand heat treatment and remain in solution up to moderate concentrations. Thus, more detailed descriptions of experimental results and their Significance could be given fOr the impurity concentration series CuPd(l7)Mn(c) and CuPd(l7)Ni(c). Rather strikingly different magnetic susceptibility behavior was reported for the ordered and disordered states of CuPd(l7)Mn(c). I These results were offered after a careful justification of the reliability of the various alloys that were measured. Possible explana- tions of the impurity concentration dependence of 0 were distilled from a consideration of three types of interactions. The Kondo effect was ruled out as a source of the effect because it is a concentration independent phenomena characteristic of very dilute alloys. It was suggested that the Kondo effect might account for the apparent non-zero O values obtained by extrapolating to zero impurity concentration or for the non-zero O values of the disordered alloy where atomic disorder effectively isolates impurities at low concen- trations. At first Sight, the RKKY interaction seems a plausible mechanism to explain our results. One is tempted to believe that the change in atomic order simply turns on or off this long range coupling of impurity moments. The failure of the free electron RKKY model to account for the observed Sign of O is worrisome, but not implausible given a binary alloy host containing Pd. The observed absence of any 123 significant shift in O for the order-disorder systems Cu3AuMn(l.O), Cu3PtMn(0.8), and CuPd(l7)Gd(O.4) is extremely difficult to reconcile if RKKY is indeed the dominant interaction in CuPd(l7)Mn(c). Although the RKKY interaction cannot be totally ruled out on the basis of our results, it does not appear to be the most likely possibility. Local environment-direct interaction effects are still sufficiently mysterious that one can attribute much of his ignorance to them. We have offered some intuitive plausibility arguments as to why direct interactions (direct exchange, superexchange, short range RKKY) might be more important in CuPd(l7)Mn(c) than in Cu3AuMn(c) or Cu3PtMn(c). Until detailed microsc0pic measurements are made on these atomic order-disorder systems, we believe that considerable speculation is involved in attempting to differentiate among them. The CuPd(l7)Ni(c) system is still a puzzle. While susceptibility measurements Show a systematic distinction between the disordered and the heat treated alloys, the accompanying resistivity measurements raise questions about the extent of atomic ordering achieved at the higher Ni concentrations. Several features of this system are clear. Ni does not sustain a localized magnetic moment at any concentration up to 10.2 atomic percent for either state of host order. The impurity susceptibility of Ni in disordered CuPd(l7) is enhanced over that in pure Cu by almost 60 percent and for Ni in "ordered" CuPd(l7), the apparent enhancement is still larger. The anomalous resistivity results for the heat treated alloys must be explained before the susceptibility effects in the "ordered" alloy become more than tentative. APPENDICES APPENDIX A 1. UNITS OF MAGNETIC SUSCEPTIBILITY 2. A GENERAL DEFINITION OF MAGNETIC SUSCEPTIBILITY APPENDIX A 1. UNITS 0F MAGNETIC SUSCEPTIBILITY There appears to be considerable variance and ambiguity in the literature of magnetism concerning the units of magnetic susceptibility. Most of the uncertainty stems from the use of the so-called emu, or electromagnetic unit, in CGS unit systems. When it appears, it is usually not defined and hence can lead to incorrect dimensionality for the susceptibility. We hOpe to alleviate some of this confusion with the following treatment. Let us first clarify the relationship between susceptibilities in the two most commonly used systems of units, the Gaussian system and the rationalized MKS system. The Gaussian system is a CGS system which combines the older electrostatic and electro- magnetic systems. Often the terms Gaussian and CGS are used inter- changeably, so we shall denote Gaussian susceptibilities with the subscript CGS. Susceptibilities in the rationalized MKS system are labelled with the subscript MKS. The relation among magnetic induction 8, magnetic intensity R, + and magnetization M is E = n (A + M) [MKS] (A.l) or E A + 4th [CGS] . (A.2) 124 125 In both systems, the volume magnetic susceptibility is defined for linear, isotrOpic materials and we call this susceptibility x. + M ; XMKS H [MKS] or M 3 Xces A . [CGS] (A.3) It follows that in linear, isotropic materials M, H, and B are all prOportional. Therefore, we have E = po(l + XMKS) M 5 ah , [MKS] (A.4) and thus 0 = 00(1 + XMKS)’ or XMKS = E;-- 1 . (A.5) Similarly, E = (l + 4n X663) 11: all, [CGS] (A.6) and thus p = (l + 4p XCGS) , or XCGS = %%l- . (A.7) We see that X is dimensionless in both systems of units. To see how the magnitude of x characteristic of a given physical system transforms between the two unit systems, consider the following. The quantity 8 describes the same physical field regardless of the system of units. Only the magnitude (not the dimensionality) of the unit of B differs from the MKS to the Gaussian (CGS) systems. In fact, the equivalence relation is the familiar l tesla [MKS] = 104 gauss [CGS]. Therefore, let us hypothesize a universal definition of volume magnetic susceptibility x'. Let X' be the ratio of the portion of 8 due to the + magnetization of matter to the remainder of 8. Therefore 26 . _ 4WM _ and XCGS - —H—-- 4n XCGS . (A.9) By the nature of the definition of X', it must be true that XMKS = Xcos , (A‘IO) or using Equations (A.8) and (A.9), XMKS = 4n XCGS (A.l1) This is the transformation of volume susceptibility values between the unit systems. Two commonly derived magnetic susceptibilities are (l) the mass susceptibility Xg and (2) the atomic or molar susceptibility XA = XM' These susceptibilities are defined in the same way in the MKS and Gaussian unit systems. 111 -§ , (A.12) (1) x9 where p is the mass density of the material. The units of Xg are 3-kilogram']) in MKS units and (centimeter3-gram'1) 1 = 103 cm3-g'] must be taken into therefore (meters in cos units. Note that l m3-kg‘ account in deriving a transformation for Xg between unit systems. XMKS 4" Xcss -3 = = = 4h(10 ) x (A.l3) Xg,MKs p 3 g,CGS MKS 10 “cos = :XA (2) xA xM — EF‘ . (A.l4) where A is the atomic mass of an element (i.e., the mass of 1 mole of the element) or the formula weight of a compound. The units of XA or XM 3 are therefOre (meters -mole']) in MKS units and (centimeter3-mole'1) in 127 3 1 6 3 1 CGS units. Since 1 m -mole- = 10 cm -mole- , we have - XMKS _ 4" XCGS -3 XA.MKS ‘ pMKg MKS ‘ 163—;—_-(10 Aces) . (A 13) CGS or = 4n(io'°) (A 14) XA,MKS xA,CGS ° - Of course, the transformation for XM is identical since XA = XM' Gaussian units are more commonly used for susceptibility. The remaining discussion pertains to the Gaussian, or CGS, system of units. The confusing use of the emu seems to arise in the following way. + Magnetization M is defined as the magnetic dipole moment 5 per unit + + volume, or M = The Gaussian unit of m’is the same as the old __fll___ volume ' electromagnetic unit, namely, 1 gm1/2-cm5/‘2-sec'1 = 55§E§-= l gauss-cm3 E l emu of magnetic moment. Then, the magnetization M has dimensions of 3 gauss = l emu-cm- . One can define o E Mp-1 which is magnetic moment 1 is the per unit mass and has dimensions 1 emu-gm']. Also, MA 5 MAp' magnetic moment per mole with dimensions 1 emu-mole']. These are well- defined quantities with definite dimensionality. However, many authors assign the units of these various magnetizations directly to the corres- ponding susceptibilities derived above. Hence, one finds units of emu-cm'3, emu-gm'], and emu-mole'] for the volume, mass, and atomic (molar) susceptibility respectively. This usage is incorrect. The error in the dimensionality is obvious from the relations M x = fi' 5 x = fi- . and XA = (A.lS) 9 Since experimental determinations of susceptibility probe the existence of magnetic dipoles, it is useful to see how the fundamental 128 quantity m depends on X: Xg’ or XA' From the definitions of M, o, and MA’ and Equation (A.15), we have _, + + m = M-(volume) = x(volume)-H , (A.16) or a = 6-(mass) = xg-(mass).fi , (A.l7) + "* + or m = MA-(mole fraction) = xA-(mole fraction)-H . (A.18) The quantity A is well-defined for pure elements or molecular materials. There is often an ambiguity as to the definition of A for alloys (AA) and intermetallic compounds (AIM). For binary systems A . . X(l-c)Yc’ A is usually defined as AA : (l-c) AX + cAY , (A.19) where c is the atomic fraction of component Y. For an intermetallic compound, the ratio ng-is a ratio of small integers. That is c I X Y __=_ I l-c IX ’ customary to designate and I of order 1 to 10. For intermetallics, it is AIM s IXAX + IYAY . (A.20) IM(IX,IY) represent discontinuities in AA(C)- Note that the values A The Curie law and the Curie-Weiss law for the temperature dependence of an array of weakly coupled spins is so common in the literature of magnetism that a Simple connection between a typical collection of spins and the magnitude of its associated susceptibility has proven helpful. Consider the expression for a Curie law volume susceptibility in CGS units 129 N (ppB)2 X =‘—§EE—T——' a (A.21) where N is the number per unit volume of magnetic moments of magnitude puB, that is, we express the magnitude of the magnetic moment in terms of the fundamental atomic magnetic moment, the Bohr magneton. p is therefore a dimensionless constant. kB is the Boltzmann constant and T is the absolute temperature of the environment of the spin array. If only the atomic fraction c of the total number of atoms of a specimen are the magnetic ones, then we can generalize Equation (A.21) by putting N equal to cNT where NT is the total number of atoms per unit volume. Since the atomic or molar susceptibility is most commonly used for alloy systems, we convert Equation (A.21) as generalized to an expression for the Curie law molar susceptibility by multiplying both members by A/p, which is a volume per mole for the mateiral that has NT atoms per unit volume. The result is N cegf-MnBiz xA = BkB T . V (A.22) But NTA/p is the number of atoms per mole which is a universal constant, Avogadro's number ”0' Therefore, the Curie law molar susceptibility is _ cNo(puB)2 XA - -—§EE—T—-' (A.23) Of course, we could equally well obtain an expression for the Curie law mass susceptibility by multiplying both members of Equation (A.23) by A']. However, let us instead evaluate Equation (A.23) for c = l, p l, and T = l K . 130 2 [6.02(1023)mole']][9.27(10-2])ergs:g_auss']]2 3[i.38(io‘1°)ergs-K'1] (1K) 1 3 -l g-cm —m01e 12 XA (A.24) Now one can quickly calculate the molar susceptibility arising from the Curie (b = O) or Curie-Weiss law for any value of c, p, T, or O by the prescription 2 x, =1, (ans—mam pg, , (A.25) where c is the atomic fraction of magnetic species, and T and O are in Kelvin. To illustrate, let us calculate the molar susceptibility of 1 atomic percent Mn in Cu at T = 300 K if we know that the Mn moments exhibit a Curie-Weiss susceptibility with p = 5.0 and O = 5 K. Then, substituting into Equation (A.25), 2 XA 2 %_(Cm3_mo]e-i) (0.01)(5.0)_ (300 - 5) XA z 1.06 (10‘4) cm3-moie'1 2. A GENERAL DEFINITION OF MAGNETIC SUSCEPTIBILITY It is difficult to find a fully general definition of the concept of magnetic susceptibility in the literature of magnetism. We provide such a definition by writing down a representation of the familiar macroscopic vector quantity M(F,t), the magnetization in a material at time t at Spatial position F relative to some origin, as a function of (resulting causally from) the magnetic field intensity R at all points 131 in the material. + _ —)- 1 ~r. I + +. +' . Mi(r,t) - ai(r) + g V-éf dr dt b1j(r,r ,t,t)Hj(r ,t ) + + .2 lg-fjff dF'dF"dt'dt"ci.k(F,F',F",t,t',t"). J,k V J HJ(F',t')Hk(F",t") + .... . (A 26) + + V is the Specimen volume. This is a power series expansion of M in H and is valid for non-linear, inhomogeneous, anisotrOpic materials. It is traditional to define the magnetic susceptibility as the kernel in the tenh linear in H. That is, 1.J.('F,‘f',t,t') s hij(F, F',t,t') . (A.27) X Clearly, the susceptibility as defined here is an important function, particularly for "linear" materials, those for which a1, cijk’ --~ are zero.‘ Linear materials form a large and important class of matter. Only materials which exhibit spontaneous magnetic order are excluded. However, linearity can break down at large values of H (the precise value usually depends on the temperature and possibly other factors). Some Special cases of xij(F,F',t,t') are so commonly valid as to be widely and incorrectly accepted as the general case. When isotropic materials are to be described, xij includes 6ij’ the Kronecker symbol. To describe spatially and temporally homogeneous systems (hence iso- tropic), x (F,F',t,t') = éij x(F-F‘,t-t') . (A.28) 13 Expressing this susceptibility in terms of its Fourier transform we 132 have (apart from oij) . + + +1 I X(T-F',t-t') = V 4 ff dqdwx+ e‘[q°(r‘r )+°(t't )] (A.29) qaw (217) If X3 w is independent of m, then x(F—F',t-t') = 6(t-t') V [ dq x+ e q'(r‘r') (A 30) (203 q Kittel(66) treats several Special cases of x3. He shows that X3 = 5Tb? in general for the X3 of Equation (A.30). Here m3 and ha are the Fourier transforms of M(F) and H(F) respectively. One Special choice for x3 leads to a common (but, we now see, non- general) representation of the susceptibility. Suppose X3 = X0: independent of 6. Then from Equation (A.30) X(F-F',t-t') = 6(t-t')vxoa(F-F') . (A.3l) By Equations (A.26) and (A.28), the resulting magnetization is Mi(T,t) = g %-6ij ff dF'dt'vxod(t-t')a(T-F')Hj(r',t') M1.(-F,t) = oni(-r,t) or M(FJ) = XOH(?.t) , (A.32) which is the classical result for a linear, homogeneous, isotropic —). magnetic material. One might represent M and x0 as functions of the temperature T at point F to further generalize this familiar form. APPENDIX B TABLES OF SUSCEPTIBILITY VERSUS TEMPERATURE DATA TEMP. (DEG 300.0 263.6 213.5 175.8 156.7 145.9 134.8 127.3 120.9 115.0 108.2 101.8 96.9 91.2 35.9 82.9 80.5 79.0 300.0 278.6 236.3 219.7 190.6 178.0 107.0 157.7 149.5 141.5 135.3 126.4 118.2 111.9 100.8 95.6 88.4 84.3 81.8 80.2 79.0 133 Table A8- . K) DISORDERED ORDERED l IMPURITY SUS. 00115955 .0131158 60136418 .0164053 .0200080 .0209961 60822797 60238413 .0253833 .0271385 .0283447 .0300120 .0316897 .0335661 .0354308 .0375150 .0401735 .0418200 004287QB 60439831 .0125290 .0134934 .0159407 .0169365 .0196464 .0206369 .0221243 .0232777 00248060 .0257151 .0209868 .0290627 .0306974 .0323131 .0336410 .0372090 .0430841 .0442486 .0454777 .0456892 (CC/MOLE) CuPd(l7)Mn(O.22) TEMP. (DEG 206.0 256.2 239.5 2?3.7 215.3 206.2 199.2 101.h 184.3 177.5 165.1 148.9 135.6 126.8 119.0 113.2 108.2 102.8 07.8 03.3 88.9 84.2 80.8 79.0 206.0 2H0.b 267.6 250.0 246.7 239.2 223.9 205.3 191.3 179.3 169.3 160.6 153.4 146.5 138.1 128.5 l?0.7 114.5 139.3 103.3 07.9 01.8 e8.0 H3,“ AU.9 79.0 134 Table AB-2 . K) DISORDERED (WHEHEI) [MDHRIIY BUS. .0126243 .0146490 .0157644 .0167688 .0173110 .0180139 .0185741 .0192654 .0200361 .0209837 .0222783 .0246867 .0270994 00290246 .0308094 .0323499 .0339038 .0356146 .0374167 .0391132 .0410792 .0434957 .0451394 .0462141 00129874 0013b534 .0141367 oOlQQZQQ .0151600 .0156612 .0109221 .0180911 00194330 .020638H .0220337 .0230845 .0240217 .0249601 00261870 .0282733 .0299147 .0316326 .0330050 .0348787 .0366386 .0386790 .0406474 .0425985 .0440067 .0447463 (CC/MOLE) CuPd(l7)Mn(0.70) TEMP. (UEU 296.0 270.0 284.6 236.8 228.2 270.0 212.0 204.7 197.8 100.7 194.4 172.1 15b.5 145.1 132.4 l?b.3 120.1 110.0 105.3 101.5 04.8 99.9 Rb.8 R3.7 90.« 79.0 298.8 291.4 275.8 2h8.7 283.1 296.7 Zu9.8 235.6 292.5 209.2 107.3 1R8.4 176.0 106.6 155.9 lab.9 135.4 127.“ 119.6 111.3 105.2 97.7 01.1 96.3 Al.b 90.5 . V) 135 Table AB-3 IMDHRITY bus. r11§tN¥HElQEL) ORDERED .0126981 .014001b .0148207 .0159025 .0183562 .010999] .0177994 .0183080 .0180448 .0195570 .0202977 .0214u34 .02337bl .028274u .0275082 .0291283 .0305133 .0332111 .0346919 .0360704 .0386288 .0906464 00950559 .0435730 .0952002 .0459631 .0126991 .0132866 .0136582 .0139709 .0194036 .0146631 .0158533 .0166934 .0177235 .0186012 .0196888 .0206026 .0217497 .0227108 .0241723 .0276697 .0293262 .0310020 .032770b .0350022 .0372734 .0393775 .0410320 .0917339 (CC/MULF) CuPd(l7)Mn(1.16) TEN”. (DEG 209 . 0 277.9 270.9 258.8 249.7 242.4 239.6 275.8 215.1 203.8 192.9 1M2.H 173.2 155.0 157.0 151.2 142.3 111.2 128.0 119.7 108.6 100.9 05.5 92.8 H5.9 R1.0 79.0 205.7 284.3 27b.5 270.4 259.7 248.4 232.5 270.7 209.0 198.0 186.0 178.9 170.3 160.9 147.0 138.2 128.8 122.2 116.2 110.9 103.3 Qh.b 02.0 95.2 R1.C '10 "1 . K) 0150911111351 nHDERFn 136 Tab1e AB-4 IMPUHIIY 5US. .0125213 .0133740 .0136879 .0142418 .0148089 0015292“ .0158108 .0182941 .0170381 .0180258 .0189908 .CI99537 .0209620 .0220790 .0230330 .0240732 .0253833 .0274383 .0291511 .0318218 .0342325 .0359821 .0378432 .0393802 .0425952 .0948479 .0481229 .0112469 .0117977 .0120739 .0123834 .0127118 .0131758 .0138850 .0145589 .0151885 .0158338 .0185801 .0171095 .0178718 .0188028 .0201882 .0213818 .0224230 00(34483 oUdQZZHg .0250702 .0285449 .0279853 .0291138 .0508556 .0322331 _n1/n107 (CC/MOLE) CuPd§171Mn(3.15) TEMP. (DEL) 298.0 249.7 237.9 214.2 202.8 101.9 181.3 172.5 184.1 158.9 150.0 139.3 128.5 120.4 113.5 108.8 108.13 07.2 03.2 Rd.9 95.8 42.9 78.8 298.0 259.8 292.2 246.1 232.8 218.9 202.1 187.7 178.1 165.3 156.b 148.4 139.3 129.8 122.8 118.5 108.3 1000“ 45.1 H9.9 44.2 41.3 78.4 137 Tab1e AB-S . K) DISORDFPFD DPHFRFH INDURITY SUS. .0158922 .0177120 .0198788 .0220038 .0232828 00247087 .0282303 .0275217 .0285788 .0295199 .0308985 .0331279 .0355848 .0178205 .0399310 .0420920 .0443898 .0472008 .0491458 .0515538 .0535801 00553114? .0582708 .0181183 .0182915 0015970“ .0192770 .0205388 .0219498 .0232931 .0249848 .0287801 .0283954 .0299850 .0317182 .0334375 .0354942 .0378205 .0395778 .0431082 .0481059 .0484799 .0514054 .0552132 .0573388 .0592354 (CC/MOLE) CuPd§17zGd(0.4) 138 Tab1e AB-6 TEMP. (DEG. K) IWPHRI1Y 50$. (CC/MOLE) DISOHDFPFD 204.0 .0112208 277.5 .0119288 25109 .01.}138‘0 241.0 .0136327 231.1 .0142738 222.0 .0149330 212.7 .0155349 203.3 .0181875 194.0 .0170120 185.0 .0177348 158.8 .0211393 C_u_3Ay_Mn(1.06) 148.5 .0227212 129.3 .0255813 116.7 .0286287 107.9 .0307383 07.2 .0344331 91.2 .0388384 87.4 .0388874 42.5 .0415789 00.2 .0428207 78.4 .0439475 ORDEPED 290.2 .0111183 253.3 .0132042 238.5 .0139839 221.8 .0149907 207.3 .0181038 103.5 .0171900 180.1 .0184001 170.8 .0197170 1171202 913206768 153.3 .0221121 147.2 .0230828 137.5 .0243868 128.9 .0280708 122.2 .0277002 115.2 .0292922 108.7 .0311733 103.3 .0330984 95.7 .0358932 89.0 .0390582 85.1 .0409385 R2.b .0424737 80.8 .0437497 79.0 .0445139 TEMP. (DEG 295.5 279.5 232.4 220.7 208.7 107.4 188.7 187.5 142.1 139.8 127.8 115.2 06.9 08.5 03.9 80.0 79.0 208.2 281.4 245.3 224.7 207.8 194.8 173.8 164.1 155.4 147.7 141.0 131.5 124.0 117.4 110.8 104.2 09.4 01.4 95.6 02.2 80.2 78.4 139 Tab1e AB-7 . K) DISORDFRED OPDEHFW IMPURITY SUS. .0117385 .0124388 .0149055 .0155183 0010450.?) .0173281 00183535 .0203297 .0221890 .0241835 .0281058 .0288788 .0340848 .0371449 .0389038 .0409574 .0413209 .0122480 .0127900 .0143955 .0158287 .0189997 .0188992 .0198705 .0211785 .0222582 .0234114 .0244258 .0273923 .0288891 .0305458 .0323727 .0335777 .0382783 .0388845 .0401537 .0413817 .0424880 (CC/MOLE) fl3flMn(o.81) TEMP. (DEC) 205.8 200.0 294.8 2?1.7 180.3 165.4 138.7 130.5 112.2 107.1 102.4 05.1 08.9 05.0 80.8 79.7 . K) DISORDERED 140 Tab1e AB-8 IMDURITY SUS. .0085028 .0089234 .0097883 .0113371 .0138045 .0148821 .0173814 .0182792 .0213338 .0220712 .0230521 .0244081 .0255189 .0288870 .0281123 .0287154 (CC/MOLE) CuPd§171Fe(0.29) APPENDIX C INVERSE IMPURITY SUSCEPTIBILITY VERSUS TEMPERATURE PLOTS FOR CuPd(17)Fe(0.29). Cu3AuMn(1.06) AND Cu3PtMn(0.81) 141 .Amm.ovmmflu_vumau on 2.4 4 .m) .-Hahvux2 ._-8< mL=m_2 3. 0mm: 52$”..ng .omm .omm .orm .oom .02 .og .0m .0: .o _ _ _ _ 2 _ _ _ .0 Amznjmoma Z :35383 QMMMQm—O 1.0 8.938 3:. emmmomoma 4 Lulu ENQELKSEB 1.0m . mu. 8 3 rwx m 0 .1 3 / 83 1.00 1m .0 142 m .fipm.ov=zg¢ nu oo-omo Lo; H .m> P-Hflpvqu .~-u< mgzmwm 3. Own: MMDH¢MmmzmH .Omm .Omm .03N .00N .Om: .ONH .0m .O: .00 _ _ _ L _ _ L _ . 6.52: m E 843% omo<4ama «Fae ommmomov «53¢ II va DMMMQMD x B @me mn&.QmmmamomHQ q Cc HH®.QHZZHmNHHmDQ Om mm 8 3 m 0 .nl 3 / 93 NOD A l.9 6 143 .omm .omm .o:m .omm .owH .omfi .0m .0: .o .Aoo._vcz= P-Hflpvqu .m-o< «gnaw; Ax away manpcmmmzmh P _ b TMU Amtz: m >m om<3aa omo<4am6 «~40 owmmomov T. ”Wu ammwemo x 19 ”Mu ammwemomHe q Hoe fiwzzflmmggqsu 10m LIST OF REFERENCES 10. 11. 12. 13. 14. 15. LIST OF REFERENCES G. J. van den Berg, jn_"Progress in Low Temperature Physics" (G. J. Gorter, ed.), V01. 4, Chapter 4. North-Ho11and Pub1., Amsterdam, 1964. M. D. Daybe11 and N. A. Steyert, Rev. Mod. Phys. 49, 380 (1968). J. Kondo, ig_"So1id State Physics" (F. Seitz, D. Turnbu11, and H. E. Ehrenreich, eds.), V01. 23. Academic Press, New York, 1969. A150: Progr. Theoret. Phys. (Kyoto) 32, 37 (1964). A. J. Heeger, in "So1id State Physics" (F. Seitz, D. Turnbu11, and H. E. EhrenreiEh, eds.), V01. 23. Academic Press, New York, 1969. J. H. Van V1eck, "The Theory of E1ectric and Magnetic Suscepti- bi1ities". Oxford, 1932. E. I. B1ount, Phys. Rev. 126, 1636 (1961). L. M. Roth, g;_Phys. Chem. So1ids 23, 433 (1962). J. Friede1, Can. 9, Ph 5. 343 1190 (1956); Nuovo Cimento (Supp1.) Z, 287 (195833-1n_"Meta 1ic So1id So1ution" (J. Friede1 and A. Guinier, eds.). Benjamin, New York, 1963. P. w. Anderson, Phys. Rev. 124, 41 (1961). P. A. No1ff, Phys. Rev. 124, 1030 (1961). M. P. Sarachik, E. Corenzwit, and L. D. Longinotti, Phys. Rev. 135, A 1041 (1964). N. Rivier and V. Z1atic, J, Phys. E; Meta1 Phys. 2, L87 and L99 1972 . B. R. Co1es, Review presented at the 1972 Michigan State University Summer Schoo1 on A11oys (C. L. Foi1es, ed.). T. R. McGuire and P. J. F1anders, ig_"Meta11urgy and Magnetism" (A. E. Berkowitz and E. Kne11er, eds.), V01. 1, Chap. 4. Academic Press, New York and London, 1969. E. C. Stoner, ”Magnetism and Matter," Methuen and Co., London, 1934. 144 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 145 S. Foner, Rev. Sci. Inst. 39, 548 (1959). A. N. Gerritsen and D. H. Damon, Rev. Sci. Instrum. 33, 301 (1962); Proceedings of Low Temperature Conference VII, p. 131, (1962). c. M.‘ Hurd, .1_. Phys. Chem. So1ids go, 539 (1969). Robert C. Wright, Princeton App1ied Research Corp., private communication. J. Ma11inson, 3;_App1. Phys. 32, 2514 (1966). D. Fe1dmann and R. P. Hunt, g;_Instrkde. 72(9), 259 and 313 (1964). N. E.)Case and R. D. Harrington, 3;_Res. Nat. Bur. Stand. 70C, 255 1966 . G. J. Bowden, 3, Phys. E; Sci. Instrum. 3, 1115 (1972). J. D. Jackson, "C1assica1 E1ectrodynamics," John Hi1ey and Sons, New York, 1962. C. M. Hurd, Cryogenics 3, 264 (1966). w . De11by and H. E. Ekstr6m, 33_Phys. §_4, 342 (1971). C3 . .1. Kim, Phys. _R_e_v. _13_]_, 3725 (1970). < . Jaccarino and L. R. Ha1ker, Phys. Rev. Letters 13, 258 (1965). . C. Brog and Wm. H. Jones, Jr., Phys. Rev. Letters 34, 58 (1970) . Weiss, ._1_. d_e Physigue g, 667 (1907);1_, 166 (1930). D'UX . Hurd, 3, Phys. Chem. So1ids 33, 1345 (1967). . E. Ekstrdm and H. P. Myers, Phys. kondens. Materie 14, 265 (1972). . Sago, A. Arrott, and R. Kikuchi, 3, Phys. Chem. So1ids 19, 19 1959 . Ai: . S. Smart, 3, Phys. Chem. So1ids 13, 169 (1960). . De11by, Physica Scripta 3, 187 (1971). . J. Dekker, Physica 33, 697 (1958). . Owen, M. E. Browne, V. Arp, and A. F. Kip, 3, Phys. Chem. So1ids , 85 (1957). (mac. 1: a: c. M. A. Rudermann and C. Kitte1, Phys. Rey, 33, 99 (1954). 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 146 T. Kasuya, in ”Magnetism” (G. T. Rado and H. Suh1, eds.), V01. 2B, p. 215. Acaaemic Press, New York, 1966; Progr. Theoret. Phys. (Kyoto) 43, 45, (1956). K. Yosida, Phys. Rev. 106, 893 (1957). P. G. DeGennes, 3, Phys. Radium 33, 510 and 630 (1962). S. Doniach, in "Theory of Magnetism in Transition Meta1s" (w. Marsha11, edTT' Academic Press, New York, (1967). Danie1 C. Mattis, “The Tneory cf Magnetism," Harper and Row, New York, 1965. w. C. Kok and P. w. Anderson, Phi1. 339, 34, 1141 (1971). G. Airo1di and M. Drosi, Phi1. 339, 43, 349 (1969). P. P. Kuz'menko, P. A. Suprunenko, and G. I. Ka1'naya, Fiz. meta1. meta110ved. 33, 751 (1971). M. Hansen, ed., "Constitution of Binary A110ys," McGraw-H111, New York, 1958. B. Svensson, Ann. Physik 14, 699 (1932). L. Andersson, B. De11by, and H. P. Myers, So1id State Comm. 3, 319 (1969). L. A. Ugodnikova and Yu. N. Tsiovkin, Fiz. meta1. meta110ved. 33, 223 (1969). B. De11by and H. P. Myers, 33_App1. Phys. 44, 1010 (1970). H. E. Ekstrfim, H. P. Myers, and B. De11by, Gfiteborg Institute of Physics Reports 093, October 1972. D. J. Lam and K. M. My1es, 3, Phys. Soc. Japan 34, 1503 (1966). . G. Low and T. M. Ho1den, Proc. Phys. Soc. 33, 119 (1966). G R. Segnan, Phys. Rev. 160, 406 (1967). F. H. Jones and C. Sykes, 3, Inst. Meta1s 33, 419 (1939). E. Boge1ius, C. H. Johansson, and J. O. Linde, Ann. Physik 33, 291 1928 . R. Tay1or, 3, Inst. Meta1s 34, 255 (1934). F. Bitter, A. R. Kaufmann, C. Starr, and S. 1. Pan, Phys. 33y, 33, 134 (1941). J. S. Kouve1, 3, Phys. Chem. So1ids 31, 57 (1961); 34, 795 (1963). 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 147 R. P. E11iott, "Constitution of Binary A11oys. First Supp1ement," McGraw-Hi11, New York, 1965. D. P. Morris and I. Wi11iams, Proc. Phys. Soc. Lond. 33, 422 (1959). R. E. Watson, S. Koide, M. Peter, and A. J. Freeman, Phys. Rev. 139, A167 (1965). K. Yosida and A. Okiji, Progr. Theoret. Phys. (Kyoto) 34,505 (1965). . Bense1 and J. A. Gardner, 3, App1. Phys. 41, 1157 (1970). J C. Kitte1, in "So1id State Physics" (F. Seitz, D. Turnbu11, and H. E. Ehrenreich, eds. ), V01. 22. Academic Press, New York, 1968. A . J. Heeger, A. P. K1ein, and P. Tu, Phys. Rev. Letters 13, 803 (1966). L. Roth, H. J. Zeiger, and T. A. Kap1an, Phys. Rev. 149, 519 (1966). R. E. Watson and A. J. Freeman, Phys. Rev. Letters 14, 695 (1965). Cu3Au: B. R. C01es, Physica 26,143 (1960), M. Hirabayashi and Y. Muto, Acta Met. 9, 497 (1961); A. Koman and S. Sidorov, J. Tecfl _(USSR) 11,711 (1941). Cu Pd: Reference P535. G. Reno11§t and H. J. Seemann, Z. angew Phys. 3 28,148 (1969). Cu3Pt: AT.Schneider and U. Esch, 3, E1ektrochem. 33, 291 (1944). Cu3Au: H. J. Seemann and E. Vogt, Ann. Physik 3, 976 (1929). Cu3Pd: Reference 48. Cu3Pt: T. W. McDanie1, unpub1ished. Cu3Au: P. O. Ni1sson and C. Norris, Phys. Letters 29A, 22 (1969). Cu3Pd: H. J. Spranger and H. P. Aubauer, J. Phys. Chem. So1ids 33, 2113 (1972). G. Low, 43y, jh_Phys. 1g, 371 (1969). . Moriya, Progh, Theoret. Phys. (Kyoto) 33, 157 (1965). G. T S. A1exander and P. W. Anderson, Phys. Rev. 133, A1594 (1964). B. Caro1i, 33_Phys. Chem. So1ids 33, 1427 (1967). B. Giovannini, M. Peter, J. R. Schrieffer, Phys. Rev. Letters 13, 736 (1964). G. Fischer, A. Herr, and A. J. P. Meyer, J. App1. Phys. 39, 545 (1968); G. Fischer and M. J. Besnus, So1ia‘State Comm. 7’T1527 (1969). E. W. Pugh and F. M. Ryan, Phys. Rev. 111, 1038 (1958). 148 GENERAL REFERENCE C. Herring, ih_"Magnetism" (G. T. Rado and H. Suh1, eds.), V01. 4, Academic Press, New York, 1966. L. F. Bates, "Modern Magnetism," 4th edition, Cambridge University Press, 1961.