THE EFfiECT OF ESOTQNC COMPO‘SE?EON 0N THE ELEQTRECAL flESESTANCE OF LITHIUM Thesis for Hie Degree of DH. D. MICHIGAN STATE UNIVERSE“ Richard Gordon Leffl'er 1961 THESIS H ‘ Im“131:11inn'fimmi‘lifliflmfllwuI * 93 017 This is to certify that the thesis entitled 3 TFv EFFECT CF ISCTCFIC CCKPGSITICN i CZ? TEE ELECTRICAL RESISI‘AL’CE CF LITHIUEZ presented by Richard Gordon Leffler has been accepted towards fulfillment of the requirements for Ph. D. degree in P11278108 3.), Man“ Majol' proeror 3 Date August 1, 1961 0-169 IJBRARY' Michigan State 1 University ABSTRACT THE EFFECT OF ISOTOPIC COMPOSITION ON THE ELECTRICAL RESISTANCE OF LITHIUM by Richard Gordon Leffler The electrical resistance of solid metallic lithium containing varying proportions of lithium-6 and lithium-7 was measured between 4. 20K and 2950K. For the isotopically—pure substances, the main features of the behavior agree with the predictions of the Bloch- Grueneisen law, the characteristic temperature being inversely pro- portional to the square root of the mass. The deviation in the details is just that found for most other metals. This portion of the work represents an extension and refinement of our earlier work, and a confirmation of subsequent work by others. For the isotopic alloys, the behavior of the resistance as a function of temperature can be described just as that of an isotopically-pure substance with a mass dependent on the isotopic composition. In fact, the temperature dependence of resistance for all compositions, including the pure isotopes, can be represented as a universal curve, by use of appro- priate scaling factors. The temperature-scaling factor is determined merely by some kind of average isotopic mass. For isotOpic alloys of lithium, there is very little numerical difference between the arithmetic mean and the harmonic mean of the isotopic masses and it is impossible to decide from the present experiments which average is preferable and thereby choose between certain theoretical proposals concerned with the effect of isotopes on lattice-vibration spectra. Abstract ' Richard Gordon Leffler ‘On the other hand, the results show clearly that there is no need at all to invoke a scattering mechanism that looks upon isotopes as impurities in the lattice. The effect of the martensitic transition of lithium at low temperatures is barely, if at all, discernible. THE EFFECT OF ISOTOPIC COMPOSITION ON THE ELECTRICAL RESISTANCE OF LITHIUM BY Richard Gordon Leffler A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Physics and Astronomy 1961 ‘ ‘ I . n w; a" t, L? g" i?" (:1 / a-.. “‘1 <4 1"" :0 fir? 4 _/ L4," 4""? ACKNOWLEDGMENTS Doctor D. J. Montgomery has suggested the problem investi- gated herein and has directed its progress. I am indebted to him for his counsel and suggestions at each stage of the investigation. His insight and broad perspective have been a source of inspiration and encouragement. The funds for this research and for a special graduate research assistantship were furnished by the United States Army Office of Ordnance Research. It has also benefited from funds for related work supplied by the Metallurgy and Materials Branch of the Research Division United States Atomic Energy Commission. Doctor P. S. Baker, director of the Isotope Sales Division, Oak Ridge National Laboratory, very kindly arranged for chemical purification and analysis of certain of the lithium samples used. He also made many valuable suggestions on handling the lithium. Doctor Harold Forstat made suggestions on the design of the cryostat. Doctor F. J. Blatt supplied the thermocouple wire for the temperature measurements. Doctor T. H. Edwards lent us the potentiometric recorder on which our data were taken. Mr. Charles Kingston and his technical staff did the major part of the building of the apparatus. Their help in many problems of the construction and operation is much appreciated. Mr. Kermit W. Richardson worked on the design and building of part of the apparatus as a Special problem. His assistance was most valuable. Union College, Lincoln, Nebraska, kindly provided financial support and arranged an extended leave of absence for the completion of the experimental work. :1: >:( s}: >:< >',< >:< 3}: >§< fl: :1: ::< ii TABLE OF CONTENTS Page IN TRODUC TION ....................... 1 THEORETICAL CONSIDERATIONS ............... 5 SAMPLES ............................ 15 A. Procurement and Analysis ............. . . 15 B. Preparation ...... . . ......... . . . . . 16 APPARATUS ........................... 18 A. Measurement of Resistance .............. 18 B. Measurement of Temperature ............. 23 C. Cryostat and Sample Holder ......... - ..... 24 D. The External Circuit . . . . . . ........... Z9 EXPERIMENTAL PROCEDURE ................ 30 RESULTS .................. . . . ....... 33 SUMMARY ............................ 53 REFERENCES CITED ...................... 55 iii LIS T OF TA BLES TABLE Page 1. Residual and Normalizing Resistances for Lithium Specimens. ....... 35 2. Normalized Net Resistances, Run No. 10 . . ..... 37 3. Normalized Net Resistances, Run No. 15 ....... 38 4. Normalized Net Resistances, Run No. 17 . . . . . . . 39 5. Normalized Net Resistances, Run No. 18 . . ..... 4O 6. Normalized Net Resistances, Run No. 20 ....... 41 7. Normalized Net Resistances, Run No. 21 ..... . . 42 8. Comparison of Data Below 800K with Dugdale €_:_t a_L_l. . . 46 9. The Data Reduced to a Universal Curve . ..... . . 51 iv LIST OF FIGURES FIGURE 10. ll. . Over-all View of the Apparatus . . . . . ........ .TheDryBox.... ..... ..... . Close-up View of the Interior of the Dry Box ...... . Block Diagram for External Circuit ......... . Cross-sectional View of Cryostat Assembly . . . . . ., Detail of Measuring Chamber . . . . . . . ...... Detail of Sample Holder ........ . . . . . . . . Representation of the Normalized Net Resistance for Li-6 and Li-7 as a Function of Temperature ..... Normalized Net Resistance of Li-6 and Li-7 Between 60°Kand800K................ ..... Comparison of Data with Dugdale <_e_t a}. above 1000K . . Comparison of the Present Results with Theory . . . 21 22 25 26 27 34 43 44 47 IN TRODUC TION Over three decades ago Dirac could write ". . . The underlying physical laws necessary for the mathematical theory for a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble. . . ."(1) In that large part of physics .is included solid-state theory, which is almost always concerned with energy changes small compared with the rest energy. The number of particles demanding simultaneous treatment is so enormous that indeed direct solution of the quantum mechanical equations appears hopelessly beyond reach. In the usual way, then, models of varying degrees of crudeness have been developed to handle different problems. For transport phenomena in crystals, Specifically for electronic conduction in metals, the standard treatment is to begin with a model of a perfect crystal lattice on which the ion cores are located, with a certain number of electrons unattached to specific atoms and free to move about among the ion cores. Thermal energy appears as vibrations of the lattice, and as kinetic energy of the electrons. For mathematical treatment, the quantized vibrations are considered as a phonon gas, and the electrons are represented by wave packets. In this treatment the individual ion cores and the individual electrons disappear, being replaced by collective models. Even after this profound simplification, the mathematical dif— ficulties in handling any realistic model remain insuperable. Further approximations must be made, and it becomes important then to test the predictions of the simplified theory to gain some idea of the over- all soundness of the simplified theory and of its region of validity. Quite soon after the birth of modern quantum theory, Bloch (1930) (2) derived an expression for the temperature dependence of the electrical resistance of a pure metal insofar as the resistance arises from thermal vibrations of an otherwise perfect lattice. The expression was decidedly successful, and with this splendid beginning, one might have hoped for rapid progress in refining this Bloch- Grueneisen formula, as it came to be called in view of Grueneisen's study and explanation of it (3). However, the mathematics remained intractable, and an absolute calculation could not be made. Hence we must say that despite the generally satisfactory picture given in the Bloch treatment, present theory does not permit an assessment of it validity for the details of electronic conduction in metals. We need to make use of experiment, therefore, to test various aspects of the derivation. The Bloch formula can be separated into a constant (calculable from certain parameters of the metal), times a function of the temperature and of another such constant usually written as a character- istic temperature. When the form of the function is tested rough agree- ment is obtained; but the function fails to describe the details of the temperature variation of resistivity for a given metal. Of course such behavior is to be expected, for, among other things, the lattice- vibration frequency spectrum and the shape of the Fermi surface are grossly simplified. Because of the complexity of this part of the calculation, the deviations of the experimental results from the theo- retical do not shed much light on the places where the derivation is the most inadequate. In particular, we cannot be sure whether the deviations are due to inadequacy of the model, or to the mathematical simplifications necessary to effect calculation. . On the other hand, an investigation of the dependence of the constants on the metal parameters can lead to more fruitful results. Specifically, if the atomic mass is varied, all other parameters remaining essentially the same, simple predictions are possible about certain features of these constants. Mathematically, we are saying simply that it is easier to study resistivity as a function of the atomic mass rather than as a functional of the atomic field, lattice vibration spectrum, and so on. The present thesis is concerned with testing the validity of the Bloch picture by using isotopic mass as a probe. The details of this analysis are given in the theory section. In the first part we shall see that in isotopically-pure crystals the effect of atomic mass is just as would be predicted from the theoretical picture, and we are thereby given additional confidence that the shortcomings in the form of the function are due simply to over simplification in the mathematical treatment. This result was foreshadowed by earlier work in our laboratory (4), and recently by results of Dugdale and his collaborators at the National Research Council at Ottawa (5). In the second part we consider the effect of. isotopic composition-- that is, the effect of varying the amount of one isotope of a given element relative to that of another isotope of the same element. This problem is much more complicated theoretically than the corresponding one for an isotopically-pure substance. One scheme proposes that mass imperfections (isotopes of different mass) act in much the same way as field imperfections (that is, as chemical impurities, which give dif- ferent atomic fields). The mass excess of each atom is considered as a deviation from the average density of the solid, and acts as a perturbation to give the basis for a correction formula. We find this picture completely unconvincing. It seems to us that the only effect of varying the isotOpic mass is to change the lattice-vibration spectrum, and not at all to introduce a new scattering mechanism. If our surmise is correct, the resistivity of isotopic alloys of Li-6 and Li-7 should lie between the resistivities of pure Li-6 and Li~7. If the other picture is correct, the resistivity of isotOpic alloys should always be higher than those of the pure isotopes. In the present experiments, a direct comparison of this sort is not possible, since resistivities were never calculated, in view of the difficulty in knowing the dimensions of the sample accurately. Never- theless, the examination of the temperature dependence of the relative resistivities failed to show any evidence of a new mechanism of scatter- ing. Hence we conclude merely that the frequency Spectrum of the isotopic alloys is intermediate between those for the pure isotopes making up the alloy. This conclusion is in keeping with the predictions of Prigogine (6) and of Pirenne (7), who showed that to the first order the frequency spectrum is modified only by scaling the frequency in the ratio of the square root of the average mass. Prigogine takes the direct mean of the masses, and Pirenne the harmonic mean. For Li-6 and Li-7, however, there is insufficient difference between the two methods of averaging to permit a choice. In summary, the present work first of all extends the range and increases the precision of the measurements of Snyder on the resistance of metals, and corroborates the measurements of Dugdale e_t a}. on the same subject. Thereby the qualitative soundness of the Bloch-Grueneisen picture of electrical conductivity in metals is con- firmed. Secondly, the present work shows that the effect of isotopic composition can be explained by merely modifying the lattice-vibration spectrum, in accordance with the theories of Prigogine and of Pirenne, without introducing any scattering through the apparent disorder of a random mixing of isotopes on lattice sites. If this interpretation is correct, serious modification of the analysis of experiments on heat conductivity may be required. THE ORE TICA L CONSIDERA TIONS To picture first in a simple way the origin of electrical resis- tivity consider a perfect three-dimensional lattice with ions stationary at the lattice sites. Valence electrons which are nearly free are moving at random about the lattice ions, and it is seen from symmetry that no net tranSport of charge takes place. - If a uniform electric field is applied, the free electrons will be accelerated and an arbitrarily large current would flow if there were no forces to obstruct the electrons. But if the ions are thermally agitated the deviation from perfect periodicity will cause collisions to occur between the ions and the electrons. The electrons are thus scattered and the current is thereby limited. Historically, Weber (8) attempted in 1875 to account for the passage of electricity through metals by assuming that a molecule was composed of a number of electrically-charged particles, some of which would break away from the molecule and be captured by neighbor- ing molecules. His goal was to account for the empirical evidence due to Wiedemann and Franz (9) (1853) that the ratio of electrical and thermal conductivities at a given temperature is the same for all metals. No real progress could be made, however, until after the discovery of the electron. Drude (10) (1900) was able to give a theo- retical derivation of the Wiedemann—Franz law by means of the simple picture of an electron gas moving among fixed ions. The electrons, upon acceleration by an external electric field, were considered to collide with the ions with a certain relaxation time between collisions. His expression for electrical conductivity is d" = nez'r/m where Uis the electrical conductivity, n is the number of charge carriers, ’7' is the relaxation time, and -e and m are the charge and the mass of an electron reSpectively. Drude considered only average velocities and average drift velocity. He predicted not merely the form of the Wiedemann-Franz law, but also an excellent value for the . numerical constant. However, the observed dependence of 0' on temperature could not be reconciled with the picture. Lorentz (ll) improved the calculation by introducing distribution functions for the electrons. A full statistical treatment, following Maxwell and Boltzmann, together with a generalization about the dependence of relaxation time on velocity enables him to get the correct temperature dependence for 0; but only at the cost of good numerical agreement. Later, another objection appeared. The Lorentz theory required also that about one electron per atom be free to move about within the metal; but these electrons should contribute to the specific heat, and hence give to metals a much higher specific heat than to insulators, in contradiction to the Law of Dulong and Petit (12.). Not until 1928 when Sommerfeld (13) applied the work of Pauli, Fermi, and Dirac to transport phenomena, was this difficulty of the Drude- Lorentz theory reconciled. The problem was now on a quantum-mechanical basis, and Bloch (1930) (2) made the next contribution by showing that electrons having energies lying in certain bands can move through a perfect periodic crystal lattice freely, and that it is imperfections in the lattice that are reSponsible for electrical resistance. The effect of thermal motion increases with temperature, of course. Impurities and lattice defects of various other types give rise to stationary imperfections whose effects are usually small. These effects are nearly independent of temperature, and they provide the major contribution to electrical reSistivity at low temperatures, since the resistance due to thermal motion falls off rapidly with temperature. The temperature-dependent part of the electrical resistivity may be approached as follows. 1 Consider the behavior of free electrons in a perfect lattice (thermal motion absent). Consider independently the motion of ions about their lattice sites. Superpose these two models, let them interact weakly, and determine transition probabilities; from them find the scattering term in the Boltzmann equation. With the use of appropriate statistics an expression for electrical conductivity is obtained. Let us consider this problem in more detail. To treat the lattice vibrations, take a collective model. For the crystal write a Hamiltonian which ignores electron coordinates, considering only the atomic cores. The electrons are light particles and presumably accommodate very quickly to the nuclear motion. This "adiabatic principle" enables the electronic states to be treated as a unique function of the nuclear coordinates at any instant. To a first approxi- mation this system may be analyzed into a set of independent normal vibrational modes, each equivalent to a Simple harmonic oscillator. The energy is given by E:2(;(nq+%-) fiwq the eigen states corresponding to traveling waves characterized by a wave vector 3. Because one is dealing with lattice waves and not with a continuum dispersion exists; moreover the allowed values of_q_ are discrete and finite. The maximum value of_q corresponds to a wave length of twice the lattice spacing. Larger q's correspond to nothing new physically. The transport of energy in one of the states turns out to be proportional to the group velocity of the waves, in the same way as in classical field theories. A quantum of lattice vibration may thus be defined as a phonon-~in analogy with the photon_--which travels through the lattice with the group velocity of that mode. In a sense phonons may be considered as particles, and they then satisfy Bose-Einstein statistics. Two transverse modes and one longitudinal mode are associated with each 3 vector. If there is more than one particle per unit cell, additional branches may arise in the dispersion curve, a feature of no interest for the present work. We can define g(w)dw as the number of modes with frequencies between to and wdw. A good low temperature approximation is for g(w)~w. The values of q are distributed with uniform density in reciprocal space, and the number with wave constants lying in the interval q and q + dq would be prOportional to qqu in a continuum. Then q(w) c: wz It is convenient to define a new parameter 0 such that he is the maximum allowed energy of a phonon. For the electrons, let us consider in particular a monovalent metal, and concern ourselves only with conduction electrons. For a single atom it is assumed that the entire mass is associated with the nucleus and that the inner closed shells of the electrons form a rigid halo about the nucleus, the valence electrons moving independently in the field of the core. When all the atoms are brought together to form a metallic solid, the valence electrons move in the field of all the ions combined, together with a smeared-out field due to all the other valence electrons. The wave functions for such a system are not localized, but Spread out throughout the crystal field. The group of valence electrons as a whole forms a pool from which the energy states for a crystal as a whole may be filled. At this stage of the argument assume that the lattice is perfect, that is, the ions are at rest in their equilibrium positions. The Hamiltonian for such a system contains a kinetic-energy term for the electron, a potential-energy term for the ion-electron interaction, and another potential-energy term for the electron-electron inter- action. From the mathematical point of-view the resulting Schroedinger equation is hopelessly complicated. The simplest thing to try is to neglect all but the kinetic-energy term in the Hamiltonian. Then E(k) = ’thZ/Zm where k = 231/). is the magnitude of the wave vector. The wave function is a simple product function of plane waves not containing a spin function and not antisymmetrized in the coordinates of the electrons. A func- tion incorporating these features, such as that given by a Slater determinant would be too hard to work with. A simple product function is tractable, and is kept realistic by the stipulation that not more than two of the single electron functions may be identical. This case amounts to discussing free electrons and is essentially the Sommerfeld pre- sentation. The surfaces of constant energy in k space are Spheres. This model serves as a general guide to the behavior of electrons in metals, but to get any details of electron behavior, notably band structure, the next term in the Hamiltonian must be taken into account. Considering the potential term to act as a small perturbation on the free electrons, one finds that there are values ofk for which there may be discontinuities in the energy. That is, gaps exist in the quasi.- continuous energy Spectrum, and the energy is no longer proportional to k2. The regions of allowed energy separated by energy gaps are called energy bands. The surfaces in k Space mapped out by these surfaces of energy discontinuity form the boundries of the Brillouin zones of the reciprocal Space. Surfaces of constant energy ink space tend to be Spheres for small values of 1:, but those surfaces which cross zone boundaries must cross them at right angles, and hence must be drastically distorted from spherical shape. 10 The electronic states considered so far have been stationary states, apprOpriate to equilibrium conditions. We are interested in transport problems wherein a steady state-~rather than an equilibrium state--is attained under the influence of external driving forces of some kind. For this purpose wave packets must be constructed from the simple plane waves, the motion of which then corresponds to a particle motion. In a dispersive medium wave packets travel with the group velocity, given by _l__3_§ 115‘s 35 ' The dynamics of such a wave packet may be quite different from that of free classical particles. From now on we think of this wave packet in the solid as an electron and associate with it the crystal"momentum" his. The velocity will approach zero at a zone boundary Since aE/ak goes to zero then. In general k and Xk are not parallel, as seen below. The electric current associated with an electron in state 1: is e_vk. A consequence of a theorem due to Wannier (14) is that hlé = eE for a single non-overlapping band. This equation of motion says that the electron is accelerated until it reaches the Bragg wavelength, at which time it is reflected by the lattice, to be accelerated again. The acceleration of the electron is given by s = l ———32E . r 4‘ 412 ‘95 as " where F_ is an external force, and the factor just preceding it is an inverse mass tensor. This quantity may become negative and may then be interpreted as a positive hole. A consequence of electrons obey- ing Fermi-Dirac statistics is that, practically speaking, only the electrons lying in a "thermal layer" of the order of kT about the Fermi surface can have their states altered. Thus the shape and area of the Fermi surface are determining. 11 Everything said so far has assumed that the phonons and electrons are entirely independent. That is, excitations of one are entirely independent of those of the other. With these simplified models no transport properties can be predicted. To go further one no longer neglects the anharmonicity of the lattice forces generating the phonons, the coulomb-forces between electrons, or the complicated forces which act on electrons due to thermal distortion of the lattice . Real crystals contain also imperfections in the lattice due to vacancies, dislocations, impurity sites, grain boundaries, and so on. The effect of these prOperties is treated by including in the Hamiltonian terms which allow for weak interactions between them, usually in the form of perturbations of the harmonic independent-particle states, the perturbations causing exchange of energy between the states. The effect of the perturbing term in the Hamiltonian is to give the system initially in the state”) i>, of energy Ei’ a probability of finding the system in a state I f>, of energy Ef after a time t. The time derivative of this probability is called the transition probability. 1 For the case of electron-phonon interactions, one conceives of an electron in a particu- lar eigenstate and a lattice vibration described by its eigenstate. The phonon represents a lattice disturbance in which some of the atoms are moved out of their ideal lattice sites. An electron is affected by this disturbance and is liable to be scattered out of its eigenstate. As shown by a direct calculation, of the matrix elements, the general condition for scattering is that the total wave vector cannot be changed except by a reciprocal lattice vestor, i. e. s=E-E-s where g is a reciprocal lattice vector. When g = 0 the process is called a normal process. When _g_ 3! 2 the process is called an Umklapp proces s . 12 To take into account the other electrons, the adiabatic approxi- mation or Born-Oppenheimer approximation is adopted, wherein the motion of the ions is assumed to be so slow compared with that of the electrons that the electron configuration is always the equilibrium configuration for the particular positions of the ions at that instant. This assumption enables the eigenfunctions for the total Hamiltonian to be broken up into a product of two functions, one containing the nuclear coordinates only, the other containing electronic coordinates as variables with the nuclear coordinates entering merely as para- meters. The adiabatic principle allows uS to think of the electronic state of the system as being almost independent of the lattice vibra- tional state of the system. Each makes an independent contribution to the total energy. This separation of energies is the most valuable contribution of the principle. The small cross term previously neglected in the Hamiltonian is what gives such properties as electrical resistivity. The matrix element calculated from this cross term does not vanish, and corres- ponds to events in which both electronic and vibrational configurations are altered, as in electrical resistance. The matrix element refers to a process in which a phonon is absorbed or emitted, and an electron is scattered. Examination of this matrix element shows that in the scattering process energy must be conserved, i. e. E'zE k +‘fiwq k The change in the phonon energy is usually very small, and it is sometimes permissible to consider that the electron is scattered into a state of essentially the same energy. In evaluating the integrals in the expression for transition prob- ability, the assumption of a Debye frequency spectrum is almost the 13 only reasonable one to adopt and still be able to calculate. It turns out that the electron-lattice interaction is mainly due to longitudinal phonons. The scattering term of the Boltzmann equation is now known in principle. The final result for the ideal electrical conductivity, known as the Bloch-Grueneisen equation is l 5 g «- o;=[4A(1§-)J5(—T>] (1) where A is approximately constant, J5 is a Debye integral, and 9 is a parameter for the electrical resistivity analogous to the Debye temperature in the theory of specific heats. Define x 2.5 dz (eZ-l) (l-e”z) J5 (X)EI 0 No mention has been made of the term in the Hamiltonian for the interaction of the conduction electrons. The question arises whether electron-electron scattering contributes to electrical resistance. - AS before one may speak of normal and Umklapp scattering processes. Normal processes correspond to the interchange of momentum between the conduction electrons and do not contribute to resistance. Umklapp processes do contribute a term which is proportional to Tz’ at low temperatures. For a monovalent metal this contribution is very small, and while it is expected that this T2 term should dominate the resistance at very low temperatures this has never been observed. The static imperfections of most interest here are isolated point imperfections such as chemical impurities, vacancies, and inter- stitials. To a first approximation the contributions of these to electrical resistance are independent of each other, of temperature, and of the temperature dependent contributions, because the scattering mechanisms are assumed to be independent. Calculations (Sondheimer, 1950) (15) and experiment support this assumption. The additive 14 aspect of the separate contributions to the resistivity was determined experimentally by Matthiessen (1862) (16) and is known as the Matthiessen rule. Thus for the total resistivity p= +0 p lattice vibrations electron— electron pimpurities The adequacy of the functional form of equation (1) has been extensively investigated for natural lithium in the papers of Kelly and MacDonald (17). Results of this investigation will be discussed later. The isotopic mass dependence of the parameter 9 in equation (1) should be examined now to test the soundness of the underlying structure of the theory. According to the Bloch formula, the conductivity at any temperature T (after normalization through division by the conductivity at some standard To) should be a universal function of T/G. How well this prediction is verified will be treated in the Discussion of Results. To study the effects of isotopic composition, we make use of some work of Pirenne (7) on the change of frequency distribution when isotOpic impurities are introduced randomly in the lattice. His funda- mental treatment for monatomic crystals shows that to first-order terms in 6 M/M, where M is the average atomic mass, and (SM is the deviation from it for a given isotope, the frequency Spectrum is modified in proportion to the mass excess. (Actually Pirenne shows that the harmonic mean is a better choice than the arithmetic mean, a parameter proposed earlier by Prigogine (6) in a perturbation treatment.) For (SM/M less than say 1/5 the difference between the two means is insignificant, and it is impossible to decide between the two representations in typical experiments. SAMPLES A. Procurement and Analysis Stable isotopes of lithium having high chemical purity and high isotopic enrichment have been made available by Oak Ridge National Laboratory. . All of the lithium samples used in this experiment were made from Li-6 (99.3% Li—6; 0.7% 1.1—7), Li-7 (0.01% Li-6; 99.99% 1.1-7), and natural lithium (7. 52% Li-6; 92.48% Li-7). The natural lithium was obtained from Lithium Corporation of America, Minneapolis, as their low-sodium grade lithium, in rods of 3/8-inch diameter. The chemical specifications were the following: Na 0.005% K 0.01 Ca 0.02 N 0.06 Fe 0.001 The highly-enriched isotopes were distilled at Oak Ridge National Laboratory to remove as much of the chemical impurity as possible. The spectrosc0pic analyses before distillation are as follows: Lot No. SS5(b) Isotopic Analysis (atomic percent): Li-6, 99.3 i 0.2%; 1.1-7, 0.7 :0.1% Spectrographic Analysis (element and weight percent, presumed precision i 100%): A1 .OlT Fe .05 Pb .01 Ba .01 K .01 Sn .01 Be .001 Mg .01 Sr .01 Ca .25 Mn .01 v .01 Cr .Ol_T Na .02 Zn .25 Cu .02 Ni .01 15 16 Lot No. SS7(b) Isotopic Analysis (atomic percent): Li-6, 0.01i0.01%; 1.1-7, 99.99: 0.01% Spectrographic Analysis (element and weight percent, presumed precision i 100%): Al .05 K .01 Rb .01 Ba .02 Mg .01T Si .05 Ca .02 Mn .02 Sn .05 Cr .05 Mo .05 Sr .2 Cu .01T Na .01 V .02 Fe .05 Ni .05 No chemical analysis was made after distillation. A comparison with analyses before and after distillation on other lithium isotopes (Lot No. SS5(a)) indicates that a substantial decrease in impurities occurs in those elements which are present in amounts greater than . Ol atomic percent. (An exception is found in the case of strontium, presumably because the vapor pressure curves for strontium and lithium are very nearly identical.) It is therefore assumed that the material actually used contained impurities in amounts less than these of the above analyses. B. Preparation The samples consisted of wires of lithium about one millimeter in diameter. They were extruded from a steel die in a dry-box containing a carbon dioxide atmosphere. The samples remained in this atmosphere within the dry-box during installation into the measuring apparatus until the sample chamber was evacuated. This procedure was necessary because of the chemical reactivity of lithium with the nitrogen, oxygen, and water vapor in the air. "Alloys" of the isotopes were prepared by melting weighed amounts of the pure isotopes in mineral oil previously degassed by long 17 heating above 1000C. After thorough mixing, the molten material was drawn into a rod suitable for insertion in the steel die, and the samples of wire were extruded. Under the assumption of uniform mixing, the isotopic concentrations of the samples used are as follows: Sample No. 10-5 10.6 15-5 15-6 17-5 17-6 18-5 18-6 20-5 20-6 21-5 21—6 Li-6 99.3 0.01 99.3 7.5 .01 7.5 99.3 .01 25 25 50 75 H. :1: .2% .01 .01 [\J .01 Li—7 0.7 99.99 92.5 99.99 92.5 99.99 75 75 50 25 .2% .01 .01 APPARATUS SA. Measurement of Resistance The resistance was measured potentiometrically. In this labor~ atory previous measurements down to liquid-air temperatures were made with a Rubicon Standard Kelvin Bridge. For measurements down to liquid-helium temperatures however, this method is no longer practical, because the design of the bridge demands that one of the current leads to the unknown sample have very small electrical re«- sistance and therefore correspondingly large heat conductance. The large heat leak thereby introduced into the cryostat hinders cooling and. renders the achievement of thermal equilibrium impossible. . Moreover, even when the Kelvin bridge is in balance a small current flow in the potential leads from the sample. Hence these leads cannot be of arbitrarily small diameter. The potentiometric method, on the other hand, allows use of both current and potential leads of small diameter, and accurate results may be obtained with small currents. The choice for the optimum lead sizes was made according to the suggestions of McFee (18). Figures 1, 2, and 3 are photographs of the experimental setup. Figure 4 is a schematic diagram of the system. A potential ofabout 6 volts was supplied to the external circuit by a leaduacid storage battery. - A decade resistance box controlled the current. A reversing switch was placed across the battery. The current, whose value was kept at about . O36 ampere for most of the measurements, was determined by noting the potential drop across a ten-ohm standard resistor. This drOp was measured by a Leeds and Northrup K-3 Potentiometer used in conjunction with a high-sensitivity dwc galvanometer (7 x 10‘“8 v/mm). 18 Figure l. Over-all view of apparatus. To the left of center is the metal Dewar. From its mouth projects the top of the glass Dewar covered by the brass top plate. In the center at the top is the control and switching panel. Below it is the Speedomax recorder whose input consists of the output from the d-c microvolt amplifier just beneath it. The Mueller bridge is placed on a shelf in front of the amplifier. At the bottom are the storage batteries and current-control circuit. On the table to the right is the K-3 potentiometer with its galvanometer. 20 Figure 2. View of internal parts of apparatus positioned in dry box and ready for sample placement. At the top of the picture is the conduit through the center of brass top plate. The conduit projects through the balloonbuoyed up by carbon dioxide from the cylinder to the left. Within the dry box may be seen in the foreground a vise holding the extrusion apparatus, and at its end the aluminum radiation shield. Behind the vise is the brass tank. 21 Figure 3. Close-up View of the interior of the dry box. At the top is the brass plate holding the aluminum doughnut with openings through which the leads have been wrapped. Three Teflon stand-off insulators carry an aluminum plate, which in turn supports the temperature-equalizing ring around which the leads are wrapped several times. Beneath it is the Teflon sample holder with the lithium samples (covered by polyethylene envelopes) looping down from it. The aluminum radiation shield hangs in the middle of picture, just in front of the extrusion apparatus. 22 «DE—1524 .330an Hmcaouxo no“ Ednmdflp 03on .v madman h40>0mu_2 .u d 20.. .0233“ oEoEofaSa onovounm reassure m -x Z d A 03:32:55 use .2228 o .o 5:25.. I], 5 #5050 35.3th muhquz mm4a2 “\Ub//' / F j ,, SAMPLE HOLDER ‘//// (TEFLON) LITHIUM SAMPLE THERMCCOUPLE x/ Figure 7. Detail of sample holder. To avoid confusion the leads are not Shown. 28 of the sample container. The sample container was an aluminum cylinder surrounding the sample holder and samples. The sides and bottom of this cylinder were made from aluminum foil of double thick- ness, and the top plate was made from 1/8-inch aluminum plate. A radiation shield of aluminum foil surrounded this cylinder. The assem- bly was enclosed in a brass cylinder with side thickness of about 1/24 inch, a bottom plate about 1/8 inch thick, and a top plate consisting of an outer brass ring soldered to a l/l6-inch brass disc. This ring is 1/8 inch thick, with outer diameter 5 inches and inner diameter about 4 inches. A similar ring is soldered to the tOp of the cylinder. Matching holes were drilled in both rings, so that a vacuum seal could be formed between the top plate and the rest of the cylinder by a lead fuse-wire gasket. A stainless-steel tube, 1/2 inch outside diameter, and about 2 1/2 feet long, was soldered inside a 1/2 inch hole in the middle of the t0p plate. This tube served both for evacuation of the cylinder and for taking the electrical leads out of the cryostat. The leads were sealed at the t0p of the stainless-steel tube by imbedding in a disc of cast plastic 1/2 inch thick and 2 inches in diameter (Liquid Casting Plastic, Castolite Casting Company, Woodstock, Illinois). A vacuum seal could then be obtained between the plastic and a small plate soldered to the stainless- steel tube. The assembly was placed in acylindrical glass Dewar flask of height 24 inches and inside diameter 6 inches. The topmost five inches of the Dewar consisted of glass pipe with a ground collar at the top. This flat surface permitted a 3/8-inch brass plate to be clamped over the top of the plate, to allow partial evacuation of this Dewar. The glass Dewar itself was placed in a stainless-steel Dewar, as shown in Figure 5. When measurments were being made below liquid-air temperature, this outer Dewar permitted precooling samples, and served as a radiation shield. 29 D. The External Circuit The electrical leads consisted of seven Teflonwcovered No. 27 c0pper wires for the current leads to the six samples, seven Formvar- covered No. 30 copper wires for potential leads to them, three Teflon- covered No. 27 wires for two heaters, and the six thermocouple wires mentioned earlier. The leads enter the assembly through the plastic disc, descend through the stainless-steel tube, and wind four times around an aluminum torus serving as a heat exchanger. The torus is placed in good thermal contact with the underside of the brass plate to which the tube is silver soldered (that is, the top plate of the vacuum chamber). The leads are fastened to the torus with insulating varnish. They then descend to the top plate of the aluminum cylinder surrounding the sample holder, where they are wrapped several times around spokes in the aluminum plate and secured with varnish. The leads go next to the copper bolts on the sample holder where they are secured with copper nuts. Thermal emf's are minimized by making the circuit almost entirely of c0pper. After leaving the cryostat, the potential leads and the thermocouple wires go to a twelve-point thermal-free switch. The current leads to to a rotary switch which connects each sample in turn with the external circuit. The differential thermocouple wires go directly to a galvanometer. Each heater circuit contains an ammeter, a lOOO-ohm potential divider, and a Heath-kit Battery Eliminator as a power supply. EX PERIMEN TA L PR OC ED URE Alloys of the isotopes were prepared by melting weighed quanti- ties of the high-purity lithium isotopes in mineral oil. Before the lithium was melted it was cleaned of all reaction products by scraping the surface with a steel knife blade to remove the oxide and nitride coating. Cleaning, as well as weighing, was done in a dry box with a dry carbon-dioxide atmosphere. The two chunks of lithium were pressed together before melting and stayed together during melting, to form a single globule of molten metal floating in the oil. During the melting the surface of the melted lithium did not stay Shiny. Apparently a surface reaction had taken place,.to form a very thin surface layer. This layer actually served as a protective surface, and was pliable enough that mixing by mechanical deformation could be carried out. The globule was kneaded by means of a stainless steel rod for several minutes to insure a homo- geneous mixture. From this globule some of the metal was drawn up into a glass tube coated with hot oil, and immediately extruded into a beaker of cooler oil. In preparation for extruding the lithium wires, the surface of the alloy was again cleaned with a knife in the carbon-dioxide atmosphere. The extruding device consisted of a steel barrel with a plunger threaded to advance along threads tapped in the barrel. The plunger forced the lithium through a 1-mm circular hole in a steel die screwed into the end of the barrel. The samples were extruded and placed in the sample holder while kept in the dry box. Then the brass chamber and stainless-steel tube, complete with all of the apparatus inside it, were placed in the dry box. The samples were extruded from the die by placing it in a vise and turning 30 31 the plunger with a wrench. Care was taken during the mounting not to deform the wires in any way except where the wire was attached to the current junctions. After the sample holder had been loaded, the potential contacts were made, the sample container and radiation shield were put- in place, and the brass cylinder was assembled with the lead gasket in place. The double-Dewar apparatus containing the brass cylinder was then evacuated, in order to prevent any surface reaction due to slight amounts of impurities in the dry-box atmosphere, or to slow reaction with the carbon dioxide. . At temperatures below 800K the apparatus was pre-cooled to liquid- air temperature by introducing helium gas inside the glass Dewar and into its vacuum jacket. The transfer tube from the helium liquefier was put into place, the helium-gas return to the liquefier was connected, and the vacuum jacket of the glass Dewar was then pumped out. Liquid helium was then brought in from a Collins helium liquefier. Six to eight liters introduced into the glass Dewar lowered the temperature to about 4. 20K. The brass cylinder was evacuated, and the heaters were adjusted manually so as to approach thermal equilibrium, as indicated by the differential thermocouple. In several trial runs it was found that adjust- ment of the heaters did not significantly affect the measurements. Hence for most of the measurements between 4. 20K and liquid-air temperature, natural warming, due to the normal leak of heat into the sample container, was sufficiently gradual that near thermal equilibrium was almost always achieved. The consistency of the experimental results, as well as the agreement with the work of Kelly and MacDonald (l7) lends confidence to this assumption. Above 800K, essentially the same pro- cedure was followed, with liquid air replacing liquid helium as the cooling medium. AS mentioned earlier, the thermocouple reference-junction temperatures were monitored with a platinum resistance thermometer. 32 Periodic checks were made on the reference—junction temperature. Simultaneous readings of temperature and resistance were not possible, but nearly continuous readings for temperature and resistance could be obtained by sampling the two specimens and the two thermocouples in order. The breaks in the curves on the recorder chart were then filled in. RESULTS The direct experimental results for the resistance consist of voltage readings which are translated into resistance values. Since the dimensions of the Specimens are not known precisely, the actual values of resistance are of little interest. Instead we consider the ratio of these resistances to that at some reference temperature, To, taken as 293. 10K. Moreover since we are concerned primarily with the phonon-electron interaction, we need to subtract the temperature- independent residual resistance due to chemical impurities and other imperfections. At 4. 20K the temperature-dependent part of the total resistance is negligibly small. We compute then the ratio of net resistance R(T) at temperature T, defined as the total resistance R'(T)- R' (4. 20), divided by the net resistance R(To) at reference temperature To. The behavior of this normalized net resistance is Shown schematic- ally in Figure 8 for isotopically—pure Li-6 and Li-7. The numerical values will be given later. Residual Re sistances The actual values of the residual resistance give an excellent indication of the chemical purity of the material. At room temperature our samples had resistances of about 20 milohms, decreasing to about 30 microhms at liquid helium temperature, to give a ratio of about one or two parts per thousand. These values are summarized in Table '1. The results give confidence that possible failure of Matthiessen's :rule cannot affect the conclusions very much Since the residual resistances are small and of uniform magnitude for all samples. The values agree very well with those found by Snyder and by Dugdale (it 2:1. 33 34 .DHENHDQEB m0 «.530de .m we bugwfifl pad oufidwfifl MOM mocmumwmou uoc ponflmfiuoc 0:» mo Goflducomohmmm .w madman OE wmah<¢wa2ub oon ecu oo. o A l 0.0 l EE\E m Table 1. Residual and Normalizing Resistances for Lithium Specimens Total Total Resistance Resistance Sample at 4. 2°K at 293.10K R'(4. 2) Material Number R'(4. 2) R'(293. 1) R' (293. 1) "131.6" 10.5 .0412 my, 23.63 ij, 1.74 x10"3 (99. 3%) “Li-7" 10-6 .0350 23.44 1.49 (99.99%) "Li-6" 15-5 .0416 23.82 1.75 Li-Nat. 15-6 .0280 24. 20 1.16 (92. 5% Li-7) Li-Nat. 17-5 .0233 23. 29 1.00 "Li-7" 17-6 .0318 23.68 1. 34 "Li-6" 18-5 .0385 23.90 1.61 "Li-7" 18-6 .0206 16.37 1.26 25% “Li-6" 20—5 .0355 15.94 2.23 75% "L1,?" 25% "Li-6" 20-6 .0345 15.03 2. 30 75% "Li-7" 50% "Li-6" 21-5 .0350 19.44 1.80 50% "Li-7" 75% "Li-6” 21-6 .0378 21.70 1.74 25% "Li-7” 36 'For the isotopically-pure samples the residual resistivity is somewhat higher than that of the natural lithium. This result is not surprising in view of the techniques which must be used for preparation of the separated isotopes in metallic form. Only small quantities of starting material are available, and the distillation technique is special. For the isotopic alloys the residual resistivities are only a little higher than those for the separated isotopes. This increase is to be expected, for the manipulations involved in their preparation can only introduce additional impurities. Nonetheless the smallness of the increase is gratifying, and lends confidence to the assumption that no spurious. effects have been introduced. Effect of Temperature The effect of temperature on the normalized net resistance of lithium metal of varying isot0pic composition is presented in Tables 2, 3, 4, 5, 6, and 7. Graphical presentation of this body of data in full is not feasible. To give an idea of the reproducibility of the data, Figure 9 shows a section of the data for lithium-6 and lithium-7 between the temperatures 600K and 800K. The reproducibility from one sample to another of the same material is usually better than one part per thousand. . Comparison with Other Workers The present results, which are considerably more precise than those of Snyder, are not in contradiction with his. Our values agree well with those of Dugdale e_t a_.l. , who claim somewhat greater precision than we do. To give an idea of the agreement, Figure 10 shows results of calculation from a portion of the data, with the ratio of resistivities for lithium-6 and lithium-natural plotted as a function of temperature for our work and for that of Dugdale e_t a_.l. Since the dimensional data for our samples are not available we do not have direct knowledge of the 37 Table 2. Normalized Net Resistances, Run No. 10 Sample Number 5, L16; Sample Number 6, Li7 No.5”. No.6- Tenux No.5 No.6 Tenux .00000 .00000 4.2 .04680 .05363 63.0 .000017 .000013 8.8 .04680 .05363 63.1 .000046 .000043 11.3 .05958 .06839 68.1 .00005 .00005 12.2 .05958 .06839 68.1 .000085 .000085 12.7 .07050 .08025 71.8 .00013 .00017 14.1 .07050 .08025 71.9 .00086 .00093 20.6 .08138 .09168 75.2 .00086 .00093 20.6 .08138 .09168 75.3 .0012 .0017 22.9 .08625 .09787 76.8 .00163 .00176 24.1 .08625 .09787 76.9 .00234 .00258 27.0 .3143 .3304 133.2 .00362 .00402 30.4 .3701 .3856 145.7 .00431 .00483 31.9 .3990 .4142 152.3 .00431 .00483 31.9 .4195 .4350 157.0 .00512 .00580 33.4 .4643 .4786 167.0 .00630 .00712 35.4 .6428 .6535 207.9 .00724 .00823 37.0 .7541 .7633 233.8 .00944 .01088 39.7 .8003 .8108 245.2 .01198 .01378 42.3 .8567 .8672 258.7 .01198 .01378 42.3 .9005 .9129 269.2 .01587 .01830 45.7 .9433 .9590 279.4 .01587 .01830 45.7 .9805 .9982 288.4 .02069 .02409 49.4 .9816 .9828 288.7 .02069 .02409 49.6 1.0000 1.0000 293.1 .02818 .03310 54.2 1.0014 1.0015 293.5 .02818 .03310 54.2 1.0285 1.0263 299.9 38 Table 3. Normalized Net Resistances, Run No. 15 Sample Number 5, LiNat; Sample Number 6, Li6 No.5 No.6 Temp. No.5 No.6 Tmnp. .0000 .0000 4.2 .2523 .2361 115.6 .00004 .00004 11.0 .3090 .2949 128.7 .00014 .00014 14.2 .3451 .3318 136.8 .00036 .00036 17.2 .3664 .3530 141.6 .00098 .00094 20.7 .4095 .3965 151.7 .00127 .00122 21.8 .4304 .4171 156.6 .00182 .00174 24.3 .4457 .4325 160.1 .00179 .00174 24.6 .4471 .4345 160.1 .00221 .00213 26.0 .4744 .4630 166.5 .00289 .00277 28.0 .5145 .5030 176.2 .00326 .00303 29.1 .5256 .5147 178.2 .00379 .00348 30.1 .5521 .5416 184.4 .00464 .00428 31.9 .5661 .5567 187.8 .00531 .00477 32.9 .5767 .5663 190.5 .00672 .00603 35.1 .5835 .5739 191.8 .00917 .00812 38.5 .6091 .6003 197.9 .01254 .01100 41.7 .6413 .6322 205.8 .01477 .01295 43.5 .6430 .6356 205.9 .01798 .01550 45.9 .6459 .6381 206.5 .02198 .01898 48.5 .6483 .6406 207.1 .02561 .02235 50.7 .7298 .7238 226.8 .03486 .03027 55.3 .7426 .7372 229.6 .04387 .03759 59.0 .7554 .7494 233.1 .05231 .04442 62.4 .7876 .7821 241.1 .06310 .05542 66.4 .8264 .8224 250.4 .06909 .06079 68.6 .8264 .8228 250.5 .07640 .06767 71.0 .8719 .8690 261.8 .08417 .07464 73.3 .8913 .8883 266.4 .08876 .07884 74.8 .9860 .9857 289.8 .09645 .08631 77.0 .9909 .9908 291.2 .10690 .09622 80.0 .9921 .9916 291.5 .1315 .1203 86.7 .9959 .9958 292.5 . 1784 .1653 98.5 1.0000 1.0000 293.15 . 2357 .2214 111.8 1.0054 1.0059 294.44 . 2355 .2215 111.9 k 39 Table 4. Normalized Net Resistances, Run No. 17 Sample Number 5, LiNat; Sample Number 6, Li7 No.5 No.6 Tmnp. No.5 No.6 Tmnp. 0.000073 0.000076 8.8 0.3731 0.3733 139.0 0.000056 0.000076 10.8 0.3982 0.3983 148.8 0.000322 0.000304 14.8 0.4500 0.4506 160.3 0.00128 0.00125 21.9 0.4581 0.4590 162.7 0.00248 0.00247 26.4 0.4848 0.4852 169.0 0.00833 0.00842 37.0 0.4899 0.4907 169.8 0.01114 0.01119 40.0 0.4968 0.4975 171.9 0.01458 0.01465 43.2 0.6028 0.6030 197.6 0.01832 0.01835 45.8 0.6556 0.6558 210.5 0.22117 0.02155 48.0 0.6690 0.6698 211.9 0.03922 0.03926 57.4 0.7316 0.7318 227.9 0.04873 0.04873 61.3 0.8922 0.8936 266.9 0.06123 0.06111 65.6 0.9133 0.9134 271.7 0.07097 0.07128 69.1 0.9506 0.9514 281.2 0.08162 0.08205 72.7 0.9584 0.9590 283.1 0.09004 0.09024 75.0 0.9785 0.9797 288.2 0.09906 0.09916 77.6 0.9742 0.9742 286.9 0.1069 0.1077 79.8 0.9863 0.9869 290.0 0.1663 0.1676 95.6 1.0008 1.0017 293.5 0.2481 0.2483 114.6 1.0030 1.0042 294.3 0.2797 0.2802 121.9 1.0232 1.0245 299.4 40 Table 5. Normalized Net Resistances, Run No. 18 Sample Number 5, Li6; Sample Number 6, Li7 No. 5 No. 6 Temp. No. 5 No. 6 Temp. .000473 .000654 18.4 .08565 .09664 76.9 .000992 .00117 21.7 .0918 .1032 78.7 .00254 .00304 27.9 .09762 .1094 80.4 .00456 .00555 32.7 .2655 .2833 122.5 .00595 .00712 35.1 .2972 .3150 129.8 .00852 .01036 38.9 .3318 .3494 137.9 .01172 .01410 42.4 .3729 .3892 146.7 .01492 .01809 45.3 .4003 .4165 152.8 .01907 .02267 48.8 .4460 .4625 162.9 .02286 .02716 50.9 .4753 .4917 169.3 .02881 .03366 54.9 .5289 .5415 181.4 .03703 .04290 59.0 .5598 .5721 188.8 .04106 .04743 60.6 .5941 .6042 196.7 .04837 .05558 63.8 .6448 .6536 208.6 .05490 .06286 66.3 .8536 .8570 257.8 .06046 .06909 68.4 .9134 .9151 272.2 .06506 .07446 70.2 .9481 .9487 280.5 .07096 .08100 72.2 .9762 .9768 287.4 .08025 .09114 75.2 41 Table 6. Normalized Net Resistances, Run No. 20 Sample Number 5, 25% L516, 75% Li7; Sample Number 6, 25% L16, 75% 147 No. 5 No. 6 Temp. No. 5 No. 6 Temp. .000659 .000625 16.4 .4747 .4740 166.9 .000922 .000858 18.3 .4881 .4876 170.1 .00149 .00150 21.9 .5065 .5059 174.4 .00315 .00327 27.6 .5356 .5323 180.5 .00529 .00536 32.1 .5524 .5506 185.2 .00756 .00767 35.7 .5681 .5675 188.9 .01093 .01092 39.6 .5792 .5782 191.8 .01347 .01347 41.9 .6065 .6054 198.5 .01792 .01796 45.7 .6306 .6296 204.5 .02264 .02271 48.7 .6440 .6423 207.8 .03136 .03137 53.5 .6582 .6568 211.1 .03871 .03852 57.1 .6724 .6710 214.3 .04623 .04631 60.3 .6769 .6760 215.5 .05270 .05281 62.8 .6913 .6906 218.9 .06437 .06454 67.1 .7064 .7052 222.5 .07089 .07106 71.2 .7302 .7285 227.9 .08375 .08343 73.4 .7547 .7545 233.9 .08940 .8922 75.1 .7666 .7665 236.6 .09586 .09594 77.1 .7735 .7738 238.4 .1062 .1062 80.0 .7880 .7871 241.9 .1144 .1142 82.2 .8005 .7997 244.9 .1421 .1420 .89.8 .8124 .8117 247.75 .2009 .2013 104.5 .8237 .8230 250.5 .2261 .2263 110.1 .8344 .8337 ‘ 253.0 .2644 2649 118.6 .8469 .8463 256.0 .2668 .2666 119.6 .8582 .8570 258.8 .2845 .2845 123.3 .8708 .8696 261.7 .3037 .3032 128.2 .8934 .8916 267.0 .3458 .3449 129.0 .9021 .9009 269.3 .3970 .3971 149.2 .9122 .9108 271.6 .4053 4054 151.2 .9197 .9188 273.6 .4104 .4107 151.9 .9348 .9348 277.25 .4161 .4259 156.0 .9467 .9468 280.2 .4422 .4419 159.6 .9768 .9767 287.55 .4553 .4548 162.6 k. 42 Table 7. Normalized Net Resistances, Run No. 21 Sample Number 5, 50% L516, 50% L17; Sample Number 6, 75% L16, 25% L17 \ No. 5 No. 6 Temp. No. 5 No. 6 Temp. .00024 .00029 15.7 .3824 .3778 146.6 .00053 .00055 17.5 .4002 .3974 150.8 .00151 .00152 22.9 .4180 .4148 154.9 .00225 .00219 25.3 .4422 .4382 160.2 .00335 .00335 28.2 .4565 .4530 163.5 .00407 .00416 30.2 .4814 .4780 169.6 .00558 .00551 32.8 .4928 .4896 172.1 .00740 .00728 35.6 .4983 .4955 173.4 .00979 .00958 38.6 .5130 .5101 176.7 .01214 .01187 40.8 .5190 .5161 178.0 .01506 .01463 43.6 .5301 .5276 180.2 .01848 .01802 46.4 .5522 .5502 ‘185.8 .02295 .02225 48.9 .6008 .5982 197.8 .02709 .02586 51.6 .6148 .6129 201.3 .03404 .03315 55.3 .6269 .6244 204.1 .04211 .04030 59.1 .6349 .6327 206.1 .05100 .04912 62.7 .6560 .6553 211.4 .05928 .05705 65.8 .6700 .6687 214.35 .06640 .06401 68.4 .6785 .6774 216.45 .07237 .06982 70.4 .6921 .6903 219.6 .07979 .07687 72.8 .7056 .7037 222.8 .08571 .08276 74.7 .7186 .7166 225.7 .09413 .09097 77.3 .7312 .7290 228.8 .10005 .09696 78.9 .7432 .7419 231.6 .1058 .1026 80.5 .7548 .7539 234.5 .1324 .1298 88.1 .7663 .7654 237.2 .1457 .1422 91.6 .7844 .7839 241.6 .1617 .1581 95.8 .8230 .8221 250.5 .1801 .1762 100.4 .8390 .8378 254.25 .1978 .1943 104.8 .8516 .8475 256.55 .2185 .2146 109.6 .8796 .8788 264.15 . 2396 .2357 114.5 .8887 .8876 266.25 .2572 .2537 118.6 .8982 .8972 268.55 .2859 .2818 124.7 .9283 .9286 276.0 .3228 .3191 130.8 .9423 .9424 279.35 .3359 .3319 136.4 .9669 .9668 285.3 .3474 .3435 139.0 .9809 .9811 288.6 .3483 .3442 139.2 .l3652 .3613 142.9 -:3696 .3662 144.2 43 TEMPERATURE (K') 0.150)" +/ f” .. / 0100 ‘Fg 19 R(T) J/ A/ R(T) / 8/ / 1 xtHSU 03H. .VHO .xomam 6e 622:8 oomuoooH owns.“ ongmuomgou 05 GM Hengmcuggfifl mo 2.65 o» 0185.303 mo 3933mm.” mo 03mm .2 656E O... u¢3buao 2.3 :83“: uwmop >m .nncfidflfifl 90m 0 nouogmfimm 05 mo m0.3.m> 03¢ no“ 3.2 nonwoamdnu £3on 05 o» wnwpuooow popmfidofido mo>ufio mum mocfl USOm 08H. .mpcwom Housman—€098 05 gmdoufi 85me o>ufiu ommuokm Gm ma 9H3 poammp 09H .ougduomgmu 330m“: ma mmmwomad 93. 5153:“: mo 35 o» pugsmfifi n3 monoumwmou um: 66338.8: 05 m0 036.“ 05. mo >399 no>o 90.038 300qu on» 3 muddwpuo 9;. 6:005 5%? 33mm.” «Cowman 65 mo GOmManEoU .: ondmfim on: m¢3h4¢wm1uh OOn 0mm com on. O0. On 0 o _ l 2 ("")OOI 43 48 of lithium-6 at temperatures between 500K and 3000K. The dashed line represents our experimental data, the solid curves the predictions of the Bloch-Grueneisen formula for two values of the parameter 9 for the heavier isotope. It is clear that adjustment of this parameter cannot much improve the agreement over the temperature range shown. At lower temperatures, the theoretical curve continues to rise, but flattens off to approach the vertical axis at about 38 percent. The experi- mental points begin to scatter badly, since the absolute values are only a few percent of the room temperature values, and differences between small quantities are involved. We believe that the scatter is due to experimental uncertainties inherent in the present method and apparatus, but we cannot rule out possible effects of the martensitic transition, or even of departures from Matthiessen's rule. With reSpect to the behavior of the parameters occurring in the Bloch- Grueneisen formula, only the parameter 9 will yield much of interest. All the other quantities appearing in the formula (lattice con- stant, atomic field, and so on) are virtually identical between isotopes of lithium (see, e. g.- , Covington and Montgomery, 1957) (19). In our own experiments the lack of knowledge of the sample dimensions pre- vents our verifying these identities; but the work of Dugdale and collaborators however shows indeed that the limiting resistivities of the two isotopes at high temperatures are the same. So far as the behavior of the parameter 6 is concerned, it is easier to describe the results for the isotOpic alloys and the pure iso- topessimultaneously. Accordingly we postpone this discussion. Dependence on Isotopic Composition (including pure isotOpes) For the isotopically-pure metals, it is clear from the equations of motion that the lattice-vibration frequency Spectrum must be identical between isotopes, after application to the frequency of a scaling factor 49 proportional to the square root of the isotopic mass. For in the equation of motion the mass of each particle is multiplied by the second derivative 1 nor time appears elsewhere with respect to time. Since neither mass in the equations, the solutions can contain mass and frequency only in the combination mass times frequency squared. A dimensional-analysis argument next suggests that the only way the frequency can enter is through the combination ’hto/kT, where the characteristic frequency w is proportional to the square root of some interatomic force constant divided by the atomic mass. Now, as we have just seen, any frequency appearing in the derivation will have this same dependence on mass. Hence it is strongly indicated that a universal curve with abscissa T/9, where 9 varies inversely as the square root of the atomic mass, will describe the dependence of electrical resistance on temperature. The arguments given in Dugdale e_t a1. (5), based on derivations in Ziman (20), amount to illustrations of this conclusion for specialized models. The Bloch-Grueneisen equation itself is a very Specific illustration of this argument. For the isotOpic alloys it appears that the only effect of introducing isotopes is to modify the frequency vibration spectrum. Prigogine (1954) (6) has shown that to first-order perturbation terms the Spectrum is modified merely by scaling the frequency in inverse proportion to the square root of the arithmetic mean of the mass; Pirenne (7) has shown that the first order correction in an exact treatment gives a spectrum modified merely by scaling the frequency in inverse proportion to the square root of the harmonic mean. For isot0pic alloys of lithium-6 and lithium-7 the difference between the two means is a few parts per 1The electron orbits are modified a small amount (actually about one part in ten thousand) by the difference in reduced mass for electron- nucleus between isotOpes, so that the atomic fields are not exactly identical; but this difference may be ignored in the present work. 50 thousand at most, and we need not differentiate between them in this treatment. We shall try then to account for the effect of isotopic composition, from 100 percent lithium—6 to 100 percent lithium-7, simply by scaling the temperature in proportion to the square root of the average mass of the isotOpic mixture. We arbitrarily take lithium-natural as the standard. Then the normalized net resistance is adjusted by the scaling factors and divided by that for natural lithium. If the theory is correct this ratio should be unity. The results of this procedure are‘given in Table 9. Each entry in the table is of the form W 31(ij )/Rx(To) X RNat(T)/RNat( T0) where x denotes the isotOpic content of the sample, RX(T)/RX(T0) is the net normalized resistance of the sample x, y is defined as «I Mx/Mnat, and WK is defined as 1.0000 - (293.1 - 293.1FMx/Mnat') N, where N (equal numerically to 0. 00407) is the slope of the lithium- natural curve at 293. 10K; Mx is the average mass of the sample of isotOpic composition x. WX is the scaling factor by which the point Rx(293. 1/y)/RX(Z93OK) on the RX curve is made to lie on the Rnat curve. . In the range from Z90°K to 800K the disagreement is at most a few parts per thousand. From 809K to about 300K the disagreement is a few parts per hundred. Below 300K the absolute resistances are so small that the effects of any slight absolute errors are greatly magnified, and the numerical comparison becomes meaningless. The last rows in the table reflect the onset of this trend. The agreement then is! to be considered excellent, the variability at low temperatures reflecting simply the difficulties in the experimental work there. It is particularly significant that no strong trend appears in the deviations of the ratios from unity. Thus it would seem that there are no scattering mechanisms unaccounted for. 51 Table 9. The Data Reduced to a Universal Curve T 100% Li-6 75% Li-6 50% Li-6 25% Li-6 0% Li-6 (0K) 0% Li-7 25% Li-7 50% Li-7 75% Li-7 100% Li-7 290.0 -- -— -- 1.001 1.001 280.0 1.001 1.001 1.000 1.006 1.001 270.0 0.999 0.998 0.999 0.998 1.000 260.0 1.001 1.000 0.994 0.999 1.002 250.0 0.999 0.997 1.001 0.997 1.006 240.0 1.000 0.998 0.997 0.996 1.007 230.0 0.996 0.993 0.992 0.993 1.004 220.0 0.998 0.995 0.994 0.993 1.002 210.0 0.997 0.990 0.990 0.988 1.001 200.0 1.010 1.004 1.004 1.003 1.003 190.0 1.000 0.994 0.995 0.996 1.003 180.0 0.997 0.997 0.997 0.999 1.004 170.0 1.002 1.000 0.998 1.001 1.004 160.0 0.998 0.996 0.999 0.997 1.004 150.0 1.000 0.998 0.999 0.999 1.004 140.0 1.000 1.000 0.996 1.007 0.997 130.0 1.004 0.998 1.001 0.996 1.002 120.0 0.997 0.997 1.000 0.995 1.000 110.0 0.992 0.992 0.994 1.000 1.001 100.0 1.009 0.998 0.998 1.009 1.001 90.0 1.003 1.002 1.008 1.004 1.001 80.0 1.008 1.018 1.015 1.009 1.008 70.0 0.988 1.016 1.025 1.022 1.019 60.0 0.988 1.014 1.018 1.005 0.971 50.0 1.029 1.069 1.072 1.048 1.003 40.0 1.012 1.135 1.101 1.042 0.994 30.0 1.088 1.069 1.133 1.208 1.008 20.0 1.355 1.171 1.013 1.290 0.960 52 Effect of the Martensitic Transition Although a transformation from one crystal structure to another would be expected to affect the lattice-electron interaction, and thereby the electrical resistivity, we have been unable to locate definitely any ' such change due to the well-known martensitic transformation of lithium . from its normal bcc structure to hcp at low temperatures. , In all our measurements the material was first cooled to liquid-helium temperatures, and then kept there for some time before warming. - Consequently we were not in position to detect any hysteresis in the transition. Still, we should have noticed any irregularity in the resistance curve when passing through the transition region during warming. We have not yet scrutinized our data specifically to locate this effect, but routine examination has failed to pick it up. It is to be remembered that the resistances actually measured are about 2 milohms, and that changes of only a few percent of this value are to be expected. The slightest error in the measurement of resistance or of temperature, or the introduction of any spurious effect, would render detection of such irregularity very difficult. As pointed out earlier, the data for temperatures much below 800K show considerable scatter. Although it is possible to attribute this scatter to the martensitic transition, the absence of systematic trends in this scatter works against this interpretation. SUMMARY 1. A method was developed for measuring the resistance of small samples of metallic lithium from room temperature (2950K) to liquid helium temperature (4. 20K). 2.- Measurements were made on natural lithium, and on essentially isotopically-pure lithium-6 and lithium-7. The low values of the residual resistance, less than 2 x 10'3 times the room temperature resistance, showed that the samples were of high chemical purity. The course of the temperature dependence of resistance was in good agreement with results obtained earlier in work at our laboratory and elsewhere. 3. Measurements were made on isotopic alloys of lithium, prepared in our laboratory from the same batches of separated isotOpes used in our other measurements. The low values of the residual resistance, less than 3 x 10'3 times the room temperature resistance, showed that only slight chemical impurity was introduced in the preparation. The course of the temperature dependence of resistance was of the same type as that obtained for the natural lithium and for the separated isotopes. . 4. For all the specimens, the course of the temperature dependence showed that the Bloch—Grueneisen formula is a good first approximation to describe the variation of resistance with temperature, but that--as with most metals--the formula fails in its details. 5. For all the specimens, the effect of isotopic mass on the para- mater 9 appearing in the Bloch-Grueneisen formula could be treated simply by scaling the resistance and the temperature by appropriate factors derived from the theory. ‘ A universal curve can thus be used to represent the behavior of lithium of any isotopic composition. 53 54 6. From the previous finding it appears that the introduction of isotopes may be explained simply by their effect on the lattice-vibration frequency spectrum, and that no new mechanism of impurity scattering need be invoked when an atom of a given atomic number and atomic mass is replaced by one of the same atomic number but of different atomic mass. 7. The low-temperature martensitic transition in lithium produces little if any irregularity in the dependence of the electrical resistance on temperature. thIJN 13. 14. 15. *16. 17. 18. 19. 20. REFERENCES CITED . A. M. Dirac, Proc. Roy. Soc. A123, 714 (1929). . Grueneisen, Ann. Phys., Lpz. (5) E, 530 (1933). . D. Snyder,» R. G. Leffler, and D.J. Montgomery, Bull.‘Am. Phys. Soc.~II, _2_, 299 (1957); D. D. Snyder and D. J. Montgomery, Phys. Rev. 109, 222 (1958). P . F. Bloch, z. Phys. _5__9, 208 (1930). E D . J. S. Dugdale, D. Gugan, and K. Okumura, Submitted to Proc. Roy. Soc. , (1961). . I. Prigogine, Physica 29, 383, 516 (1954). J. 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