X-Ray Studies of the Electronic Band Structure of Metals
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X-RAY STUDIES OF THE ELECTRONIC BAND STRUCTURE OF METALS DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State U n iv e r s ity By NATHAN SPIELBERG, B .A ., M .Sc. The Ohio State University 1952 Approved by* ----- A d v iser ACKNOWLEDGEMENTS This work was done under the supervision of Professor C. H. Shaw of the Department of Physics and Astrononiy of The Ohio State University. It is a pleasure to acknowledge indebtedness to Dr. Shaw for guidance, encouragement and tutelage. Thanks are due to Mr. J. A. Soules for his fine work and aid in construction and maintenance of equipment and to Mr. Carl McWhirt and his capable machinists in the Physics Shop. I am deeply grateful to Mrs. Alice B. Spielberg for her constant encouragement and for her excellent typing of this dissertation. Gratitude is due to the Office of Naval Research for the granting of a fellowship through the Ohio State University Research Foundation, 19Jj9-.!?0, to E, I. du Pont de Nemours and Company for their grant of a fellowship for 19!?0-5l, and to the Atomic Energy Commission for supplying funds for a Research Associateship through the 0. S. U. Research Foundation. Funds for equipment were obtained from the Graduate School of The Ohio State University, from the 0. S. U. Research Fotmdation, and from the Development Fund of The Ohio State University. i 4 £00487 TABLE OF CONTENTS Page I. Introduction . 1 II. Theory li A. Bands for metals U 1 * Orxgxn • « ...... 2. Brillouin zones. »••••• 5 3. N(E) curves ...... 8 iu Calculation of N(E) curves .... 9 B. Spectra 12 1. Widths of inner levels . • • • .12 2. Shapes and widths of band spectra • 0 . 1 3 3. Perturbations to shapes of band spectra . .1 6 U. Suitable spectral regions. • • • .17 I l l * A pparatus A. Spectrometer 19 1. Design requirements . .19 2. The spectrometer shell . , .20 3. The crystal axes . .26 U. Alignment procedure ...... 36 B. X-ray tube I4.0 1. Design requirements . • UO 2. The tube ........ UO 3. The power supply . • • . U8 Evaluation of x-ray tube and auxiliary equipment . .... 32 C. Detector 53 1. Requirements . .53 2. The Allen type multiplier. • . 5U 3. Auxiliary equipment . .59 h» Operating data ....... 60 5. Evaluation of multiplier . • . .69 D. The crystals 70 1. Requirements • . .70 2. Possible crystals . .71 3. Preparation of beryl crystals . • . .7 2 U. ( 1,- 1 ) rocking curves and percent reflection • 78 5. The (1010) grating space of beryl . 8 U i i Page IV , Data 87 A. Molybdenum ..,..*•.87 B • Copper • «.».«.». 9^ C• Zinc .«..»*.«•« 97 D. Longer wavelengths ....... 102 V. Discussion of Data 103 A. Molybdenum ........ 103 B. Copper . , . loU C. Zinc .......... 109 VI. Conclusions 112 Autobiography 13.Ii i i i 1 X-RAY STUDIES OF THE ELECTRONIC BAND STRUCTURE OF METALS I . INTRODUCTION Many of the properties of solids can be explained in terms of an energy band picture. In an isolated atom, the orbital electrons may be found at certain discrete energy levels, the levels of lowest energy being associated with the innermost orbits. "When this atom is found in a solid, the innermost orbits, and hence the lowest levels, are practically unchanged from the free casej however, the outer orbits are greatly disturbed by the presence of neighboring atoms and the levels are broadened into bands of allowed energy. These bands, the degree to which they are filled and the forbidden energy ranges between bands, account for and correlate electrical and thermal conductivities, magnetic properties, radiation properties, mechanical properties, alloying properties, e t c . One of the standard techniques for studying energy levels of atomic systems is the interpretation of emission and absorption spectra. For free atoms, transitions between levels are observed as relatively sharp spectral lin es. As the atoms become more complicated, or as atoms join to form molecules, the resulting more complex network of energy levels introduces fine structure into the spectral lines, requiring more resolving power from instruments. However, so long as levels remain discrete, one can always find, in some region of the spectrum, a line corresponding to a transition between levels. This has led to the development, for example, of infrared and microwave spectroscopy. On the other hand, when there are whole bands of levels, as in solids, the line spectra become band spectra, extending over considerable energy r a n g e s . If a transition takes place between two energy bands, or within a band, any photon energy between values corresponding to the lines F and G (figure 1) may be involved. Moreover, radiation of a given wavelength may correspond to any one of the transitions A, B, C, D since the energies involved are the same, and it is quite difficult to determine what portions of the bands are involved in a given transition. Transitions between bands usually involve optical and infrared spectra. On the other hand, if a transition takes place between one of the sharp inner levels and the outer band, each energy photon can be correlate': th a definite location in the band, as shown at A, B, C, I in figure 2. The only ultimate uncertainty in the determi n a t i o n ■ energy of the band state is that due to the natural •! inner level. In this case, the intensity of the ou- spectrum reflects rather directly the statistical weights of the various energy states making up the energy band. The wave lengths involved here fa ll in the ultraviolet and x-ray regions,the shorter wavelengths becoming more important with increasing atomic number of the material being studied. X-ray spectroscopy, therefore, is a useful tool for the study of electronic band structures. 3 A If6 Wed b o n d ABC b o n d A B CD F & F igu re 1 . Optical transitions between bands Aff owed v b o n d & £ A ! t o w e e / b o n d & £ . JL i du rrer /c**<=d AB C D Figure 2. X-ray transitions from inner level to band h I I . THEORY A. Bands for metals 1. Origin. The existence of energy bands in metals has been derived from a particular type of one-electron solution to the Schrodinger equation, known as the Hund-Mulliken-Bloch scheme.'*' When applied to solids, this is called the Bloch scheme. It states that the "Jfunction for an individual electron in the solid shall be considered as extending throughout the volume of material, being largest in the neighborhood of individual atoms and rather flat and like plane waves at large distances from the atoms. For a valence electron, in particular, this is equivalent to saying that the electron belongs to the crystals as a whole rather than to an individual atom. The Schrodinger equation, in one dimension, for example, for valence electrons is written therefore as For a one-dimensional crystal with lattice constant equal to a, V (x+a) sr V (x). It may be shown^*^ -that there are three types of solutions, one of which increases without lim it as x approaches e it h e r + 0 0 o r - 0 0 . When dealing with an infinite crystal these must be discarded. The eigenvalues of energy, E, corresponding to •^-Seitz, Frederick, The Modern Theory of Solids, Mew York: McGraw-Hill Book Company, 19lt0, p. 2^1 ^Seitz, op. cit., p. 278 ^Mott, N. F. and Jones, H., The Theory of the Properties of Metals and Alloys, London: Oxford University Press, p. 57 5 to these solutions are forbiddenj there are no electrons allowed in these energy states. The allowed solutions are of the form where k is real and Cfg(x) has th e l a t t i c e p e r io d ic it y a . The sub script k represents the momentum of the electron. 2. Brillouin zones. More detailed assumptions about the nature of the potential function V (x) lead to more information about the allowed and unallowed energy bands. If V (x) is small enough to be treated as a perturbation, then the energy is given by^ A £ / n-=£o E=o — to the second order approximation, except when f~ f l , . c„ - —-—— and represents the energy for the free electron 2,/n approximation. a ^jfC „ x /^ Vn f VCc) e d x 'd- £r„ , kn - k ~ 2JTJ2- H /f)f&qrci{ ex. f Except when /£. —£v, / is small, E follows very closely the free electron formula and is approximately equal to £&, / S0 —£ n ( i s small whenever k is close to *2J£ , and the perturbation formula ex. above for E no longer holds, but rather £ ^ -k + En */(£*-£„)*• K,* ] the negative sign holding for k <.HIT and the positive for k >!2Zf . Q CL Therefore, at there is a discontinuity in E, the energy jump ^Mott and Jones, op. cit., p. 6 being given by A E ~ ~Z IV* t • A plot of E vs, k is given in figure 3» In three dimensions, for a simple cubic lattice, the results are exactly analagous, with k and n now being treated as vectors. = h U A T * m ' _*£2m I" k— " cx I 1 Again there is a discontinuity in the energy when E = Bn , w hich occurs whenever / Ac j^~ — } k — z — J which reduces to n - k — ~~ ~ <x If coordinate axes are taken in ,!k-spaceTI and the last vector equation is plotted, it is found that a series of planes are obtained which divide the space into cells called zones or Brillouin zones Within each zone the energy changes continuously with varying k; however, whenever k crosses a zone boundary, there is a discon tinuity in the energy.