ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN ANN ARBOR TECHNICAL REPORT NO. 1 THE INFRARED ABSORPTION SPECTRA OF DIAMOND, SILICON, AND GERMANIUM By WILLIAM G. SIMERAL Project M957 SIGNAL CORPS, DEPARTMENT OF THE ARMY CONTRACT DA 36-039 sc-5581, SC PROJECT 152B-0, DA PROJECT 3-99=15=022 SQUIER SIGNAL LABORATORY, FORT MONMOUTH, N. J. June 1, 1955

PREFA CE The following text has been approved as a dissertation for the doctorate at the University of Michigan. Since the author has derived support from a Signal Corps Contract both in stipend (Sumrmers of 1951 and 1952) as well as in funds for emuipment, the full account of the work is being submitted as a report to the Signal Corps. The principal financial aid, however, was received from a University of Michigan Fellowship (1951-1952) and a National Science Foundation Fellowship (1952-1953). The author is pleased to acknowledge the assistance which he has received from the staff of the Physics Department and from his fellow members of the "Infrared Group". In particular, he is indebted to Professor G. B. B. M. Sutherland who suggested the problem and who has been generous with advice during the course of the work and in the preparation of this manuscript. In addition, grateful acknowledgment is made to the following: Dr. D. L. Wood, who has given advice pertaining to the experimental portions of this work on many occasions, particularly with regard to infrared instrumentation. Dr. S. Krimm and Dr. C. Liang, who have assisted materially in obtaining spectra in the far infrared. ii

Mr. W. Childs, who operated the cyclotron during bombardment experiments. Dr. K. N. Tanner, who helped in the vacuum ultraviolet instrumentation. Dr. G. A. Morton, of the RCA Laboratories, who supplied samples of germanium. The following firms and individuals have generously lent diamonds for use in this work: The Diamond Trading Company (Mr. Grodzinski) The Lazare Kaplan Company Triefus and Company Professor C. B. Slawson Dr. W. C. Parkinson In construction of equipment P. Weyrich, H. Roemer, and G. Kessler have contributed their not inconsiderable talents. A. Dockrill has been of great assistance in the work on the concentrated arc source. Finally, the author wishes to take this opportunity to acknowledge the many important contributions which his wife has made to the preparation of this manuscript.

ERRATA Page (1.2 read 14 for 6) iv Page number opposite(l.3 read 23 for 14) (1.4 read 30 for 23) 38 Line 19, read numerical for numberical 77 Addendum line 15. The linear.range equals the quotient of the range in mg/cm2 and the density. 113 Line 24, read numerical for numberical 127 Line 23, read (8, 0, 0) for (8, 8, 0) 131 Line 26, read (d) for (b) 144 Read "See Figures 32 and 33" for "See Figure 34" 1 In Table 1, interchange 2250 and 3000

TABLE OF CONTENTS Preface ii List of Tables v List of Illustrations vii Introduction 1 Chapter 1: REVIEW OF PREVIOUS EXPERIIIJENTAL WORK 6 1.1 X-ray 6 1.2 Ultraviolet and Visible 6 1.3 Infrared Absorption 14 1.4 Other Properties 23 1,5 Summary 33 Chapter 2: PRESENT EXPERIMENTAL WORK 2.1 Apparatus 36 2.2 Account of Experimental Work Performed 46 2.3 Summary 78 Chapter 3: REVIEW OF PREVIOUS TIEORETICAL WORK 80 3.1 Infrared 80 3.2 Ultraviolet Cutoff 92 3.3 X-ray 94 3.4 Blue Fluorescence and Absorption 95 3.5 Other Properties 95 3.6 Summary 98 Chapter 4: PRESENT THEORETICAL WORK 100 4.1 The Frequency Distribution of Lattice Vibrational Modes 100 4.2 Application of Lattice Dynamics to Infrared Absorption 137 4.3 Summary 176 Appendix A: THE VARIATION OF SPECIFIC HEAT WITH TEMPERATURE 178 Appendix B: LIST OFi DIAMOTNDS 185 Bibliography 186 iv

LIST OF TABLES Table Number 1 Properties of Tvpe I and Type II Diamonds 1 2 The Occurrence of Extra Streaks in the X-ray Diffraction Pattern of Diamond 11 3 Principal Lines and Bands Observed in the Ultraviolet and Visible Spectrum of Diamond 20 4 Maxima of Infrared Absorption Bands in the Spectra of Diamond 27 5 Impurities Found in Diamond 34 6 The Absorption Coefficient at 6,5p for 20 Diamonds 49 7 The Absorption Coefficient at 7,8,i for Various Points in Diamond M14 54 8 The Absorption Coefficients of Long Wavelength Bands in 7 Diamonds 57 9 Absorption Maxima in the Germanium Spectrum 68 10 Change in Ultraviolet Cutoff with Branching 72 11 Frequencies of Vibration of Diamond Under the Action of A. First and Second Neighbor Forces 109 B. First Neighbor Forces Only 110 12 Calculated iMaxima Allowed in Raman Scattering 112 13 Elastic and Force Constants for Germaniun and Diamond 115 14 Corresponding Maxima in igure 27 and l'igure 25B 117 15 iElastic and iForce Constants for Silicon and Germanium 119 16 Siingular Points in Branches o2 and 05 129 v

Table uNmi-b I. r 17 Locations of Singularities in the Frequency Distributions of Diamond, Silicon, and %erian iiiuim 150 18~ COimp riCon of T xiia - nd S in ul a i Po ints in Dio,c nd' s Di str ibu ti on 132 19 Comparwiso n oof Maximoa. an d Si.ngular Points in Gerrmaniumi' s 1Distribution 132 20 Calculated iaxima in tte Branches of the LFrequency Distributions of Diamond, Silicon, and Geri.,n ium 1354 21 The Stable Isotopes of Carbon, Silicon, and Germanium 141 22 2T Ic,.1axirma cof Calcul;ted Combinactions AlloJed in Infrared 1A; sorp-ttion 144 23 Tentative.ignmet iof.:'-ilation;1 ardsr on t'he BaSis of Positions of Lax.ir ima 149 24 Fincal Assignment of Combination Bands 154 25 Rel ative i-bsor:ptioni Coefficients Due to the I otope,Effect 165 26 Observed vs 1 alculated iiayirma in the iundamei ital A1sor-ption Spectra of.diaioindi, lilicon, and Ge Prna niumi 169 27'reeLnc;- of the Seeccndary L;xi.....m of ir) 181 vi

LIST OF ILLUSTRATIONS Figure Number 1 The Bragg Structure of Diamond 7 2 Covalent Bonds in Diamond 8 3 Ultraviolet and Visible Absorption Spectra of Diamonds 16 4 Frequency of Occurrence of UV Cutoff in Various Wavelength Intervals 17 5 (a) Absorption Neat 4155 A. U. in Diamond (b) Fluorescence Near 4155 A. U. in Diamond 21 6 The Infrared Absorption Spectra of Typical Diamonds 24, 25 7 The Reflecting Microscope 39 8 Cario-Schmidt-Ott Spectrograph 42 9 Hydrogen Discharge Lamp 43 10 Apparatus for Viewing Ultraviolet Transmission Patterns of Individual Diamonds 45 11 Two Spectra Showing the 6.5i5 Band 47 12 Log Log Io/Ivs Wavelength for Diamond 52 13 Transmission Pattern of Diamond M4 for 2537 A. U. Light 54 14 Spectra of 3 Diamonds from 300 to 1500 cm-1 56 15 Relations Between Absorption Coefficients of IR I Bands 58 16 The Infrared Absorption Spectrum of Diamond F-1 61 17 The Infrared Spectra of Four Powder Samples 63 vii

Figure Number 18 Transmission of Germanium from 1 to 15 Microns 65 19 Absorption Spectra for 3 Pure Germanium Crystals 66 20 Absorption Spectra for 2 Impure Germanium Crystals 67 21 The Structure of Adamantane (C10H16) 71 22 The Ultraviolet Transmission of Adamantane 74 23 Comparison of Debye and Blackman Frequency Distributions 85 24 The Four Possible Electronic Structures of Diamond 90 25 The Frequency Distribution of the Vibrational Modes in Diamond (After Smith) A. First and Second Neighbor Forces 107 B. First Neighbor Forces Only 108 26 The Second Order Raman Spectrum of Diamond (After Smith) 111 27 The Frequency Distribution of the Vibrational Modes in Germanium (After Hsieh) 114 28 Variation in Branch Contour with Numerical Integration Parameters 121 29 The Contour of the Frequency Distribution Near Singular Points 125 30 The Branches of the Distribution of Vibrational Modes in Diamond 135 31 The Branches of the Distribution of Vibrational Modes in Silicon and Germanium 136 32 The Calculated Combinations Allowed in Infrared Absorption in Diamond 145 33 The Calculated Combinations Allowed in Infrared Absorption in Silicon and Germanium 146 34 The Observed Infrared Absorption in the Combination Region for Diamond, Silicon, and Germanium 147 35 Calculated Contour If Each Allowed Combination Has the Same Intensity 148 viii

Figure Numlber 36 Calculated vs Observed Combination Bands in Diamond 151 37 Calculated vs Observed Combination Bands in Germanium 152 38 Calculated vs Observed Combination Bands in Silicon 153 39 Observed Fundamental Absorption in Diamond, Silicon, and Germanium 168 40 Calculated vs Observed Fundamental Bands in Germanium and Silicon 170 41 Calculated vs Observed Fundamental Bands in Diamond 171 42 Variation of Debye Temperature: Diamond, B4C, SiC 179 43 Variation of Debye Temperature: Si, Ge, Grey Sn 180 44 eD/m vs T/Om for Valency Crystals 183 ix

INTRODUCTION Because of its remarkable physical properties, diamond has been the subject of many experimental and theoretical investigations. Perhaps the best known properties of diamond are its great hardness, high index of refraction, and low specific heat at room temperature. Until recently, it was assumed that all diamonds were essentially identical. However, in 1934, Robertson, Fox, and Martin reported two types of diamond, based on careful investigations of infrared absorption, position of ultraviolet cutoff, photoconductivity, and birefringence. The essential properties of the two types are given in Table Io Table 1 Property Type I Type II Infrared Absorption Near 4[ and near 8{3 Near 4tonly Ultraviolet Cutoff 2250 A. Us 3000 A. U. Photoconductivity Poor Ten times better Birefringence Present Absent 2 Somewhat later Raman and Rendall detected variations between the Laue diffraction patterns of different diamonds. 1

2 They reported that diamonds of Type I had extra streaks near certain Laue spots while Ty-pe II diamonds showed no such ex3 tra streaks. Also, Lo:nsdale found that, in general, Type I diamonds had less mosaic texture than Type II diamonds. All of these workers found that Type I diamonds were more common and more perfect externally than Type II diamonds, 4,5 Later work showed that many diamonds could not be classified as either Type I or Type II since their properties were intermediate between the two types. This fact has led to some confusion in the literature. One worker may describe a diamond as Type I because its ultraviolet cutoff falls at 2900 A. U., while another worker may describe the same diamond as Type II because it shows very little absorption at 8. In order to avoid the difficulty in the notation introduced by Robertson, Fox, and Martin, we have modified the notation in the following manner: With regard to ultraviolet absorption, a diamond will be described as UV II if its cutoff occurs between 2250 A. U. and 2500 A. U. It will be described as a weak UV I diamond if the cutoff falls between 2500 A. U. and 2800 A. U.; medium UV I, between 2800 A. U. and 3000 A. U.; strong UV I, greater than 3000 A. U. With regard to X-ray diffraction, X II indicates a diamond showing no extra streaks in the Laue diffraction pattern. X I indicates a diamond showing extra streaks, while the adjectives weak, medium, and strong, indicate the intensity of the streaks. With regard to infrared absorption, IR II denotes a diamond which displays no absorption at wavelengths longer than 6p. IR I denotes a diamond which dis

3 plays absorption at wavelengths longer than 6A, while the adjectives weak, medium, and strong indicate the intensity of absorption at 8,p relative to the intensity of absorption at 4p. Raman and his coworkers have studied the properties of 6 diamond in great detail. They have found that the Raman scattering spectrum is the same for all diamonds. Variations between diamonds occur in their fluorescence spectra and their visible absorption spectra. The workers in India have made many observations of these properties. On the 7 basis of his own theory of lattice dynamics and on the basis of his experimental results, Raman has formulated a 8 theory to explain the anomalous properties of diamonds. The theory is based on the assumption that the local electronic configuration in diamond has four possible forms. This theory, as well as Raman's theory of lattice dynamics, 3, 9, 10 has received severe criticism and is not accepted by most workers in the fields of study concerned. 11 Blackwell and Sutherland have proposed a different theory to account for the anomalous properties of diamonds. Their theory is based, in large part, on the experimental 10 data obtained by Blackwell. He performed experiments on a collection of several hundred stones. He studied: (1) Spectroscopic properties in the infrared, visible, and ultraviolet (2) Raman scattering, (3) Color, (4) Crystal habit and external perfection, (5) Several other properties. This work was devoted to the correlation of the various properties in an attempt to find regularities in the variations between diamonds. The theory put forward by

4 Sutherland and Blackwell proposed that the variations between diamonds are due to structural imperfections or foreign atoms. The only subsequent work which is of importance is that of Dr. Grenville-Wells. The work was devoted to a continuation of Lonsdale's study of the X-ray diffraction properties of diamond. The work was similar to that of Blackwell in that correlations were found between mosaic texture, extra streak intensity, ultraviolet cutoff, and other properties, The present work began in 1950. Because many of the diamonds from Blackwell s collection were available, it has been possible to obtain new data for the same stones for which considerable information was already available. Initially, the object of the work was to devise two types of experiments: (1) Those which would improve and extend Blackwellts data, (2) Those which would afford a test of the impurity theory. Included in the first set of experiments were observations of the absorption spectra of diamonds in the far infrared, in the vacuum ultraviolet, and in the region of atmospheric absorption near 6t. In the second set of experiments were variations of infrared and ultraviolet absorption in individual diamonds, the effects on infrared absorption caused by deuteron and neutron bombardment, and comparisons between the spectra of diamonds and those of silicon and germanium of various states of purity. As the work progressed, it became evident that the main difficulty in the interpretation of observation lay in the fact that very little effort had been made to apply existing theory of crystal spectra to the case of diamond. On the other hand, many of the phenomena connected with the elec

5 tronic structure can be interpreted, at least in a qualitative fashion, in terms of existing theory. The band structure of the electronic energy levels has received new 13 theoretical treatment. However, the extra streaks in the Laue diffraction pattern and the varying absorption at 8P in diamond remain unexplained. Consequently, the theoretical part of this work is mainly concerned with an explanation of the infrared absorption spectrum of diamond. Our approach has been through 14 H. M. J. Smithts calculation of the frequency distribution 15 of diamond based on the Born theory of lattice dynamics. Smith was mainly concerned with the explanation of the second order Raman spectrum of diamond. Our goal has been to establish what the absorption spectrum of an ideal diamond should be, and then to examine the theoretical justification of Raman's theory and of Blackwell's and Sutherland's theory in accounting for those features of the spectrum which are not associated with the ideal case. For the latter work we have used the theory developed by 16 I. M. Lifshitz to explain the anomalous variation of intensity of certain infrared bands. Finally, it was found that the theories developed for diamond apply equally well to silicon and germanium since these substances have crystal structures and associated physical properties similar to those of diamond.

Chapter 1 REVIEW OF PREVIOUS EXPERIMENTAL WORK 1.1 X-ray 17 In 1913, the Braggs determined the structure of diamond, The Bragg structure consists of two interpenetrating face-centered cubic lattices displaced from one another one quarter of the way along the space diagonal, This structure is shown in Figure 1. Such a structure has a center of symmetry midway between each neighboring pair of carbon atoms. Each carbon atom has four symmetrically placed first neighbors located at the vertices of a tetrahedron. The most recent measurement of the uvnit cell di18 mension is by Straumanis who finds the cube side to be 3.668 A. U. which gives a nearest neighbor distance of 1L544 A. U. In Figure 2 we show the structure of diamond in terms of the bonds between neighboring atoms, The perfection of individual diamond crystals has been the subject of considerable study. Ehrenberg, Ewald, and 19 Mark measured the angular width of the Bragg reflections for certain diamonds, In the most favorable cases, the measured width approaches the theoretical width for an ideal crystal, Such narrow reflections imply regions of perfect crystal which are many thousands of layers in depth, 6

7 Fig. 1 THE BRAGG STRUCTURE FOR DIAMOND

8 I Fig. 2 COVALENT BONDS IN DIAMOND

9 2 In 1940, Raman and Nilakantan found extra reflections on Laue photographs of diamond. They proposed a new theory for interpreting these reflections. Prior to 1940, the 20 21 15 theory developed by Faxen and Waller based on Born's lattice dynamics had proved adequate to explain diffuse spots associated with reflections. Diffuse spots are normally temperature sensitive and have a natural explanation in terms of thermally excited lattice vibrations. Raman and Nilakantan found that the extra streaks in diamond were relatively sharp and temperature insensitive. They were led to propose a completely new theory which was then applied to diamond and other substances. The arguments cover9, 22 ing this new theory are in the literature. The majority of the workers in the field of X-ray diffraction are inclined to minimize the importance of Raman's theory, and, in fact, many consider it to be completely in error. In 23 any event, Lonsdale has shown that the extra streaks are separate and distinct from the thermal diffuse spots which are found in other crystals. She found that both thermal diffuse spots (primary extra reflections) and the extra streaks (secondary extra reflections) appear in the diffraction patterns of diamond. The diffuse spots occur in all diamond patterns, and they are temperature sensitive. The sharper extra streaks vary in intensity from diamond to diamond and are insensitive to temperature. In some diamonds no secondary extra reflections are detected. 12 Grenville-Wells studied the occurrence of the extra streaks in a large collection of stones. The extra streaks

10 always appear at the same positions with roughly constant re24 lative intensities. Hoerni and Wooster have verified the work concerning the position of the streaks and have made a study of the intensity distribution within the streaks as well as the relative intensity of streaks. Their results are summarized in Table II. For those streaks studied (111, 220, 311, and 331) it is found that the extra streaks have an intensity distributed according to the relation D ^lFhkli 2 R-n where D is the scattering density along the spike in reciprocal space corresponding to the extra streaky R is the distance from the corresponding reciprocal lattice point from which the spike extends, F is the structure hkl amplitude of the reciprocal lattice point as determined by 25 Brill and n = 2.2 + 0.1 for the measured points. From the relative intensities it can be determined that when "h" has a given value, independent of the value of k and 1, the quantity D/Fhkl2 for a spike parallel to (100) always has the same value. Similar statements hold for "k" and "1'? spikes parallel to (010) and (001) respectively. The results for the spikes extending from 111 are the only discrepancy this overall picture. Hoerni and Wooster indicate thet the FI value obtained by Brill may be too high. According to Hoerni and Wooster, the relations between the indices h, k, 1, and the intensity of the spikes determine certain features; of a stratification parallel to the cube faces, and the irn verse square law (n ~ 2) is to be expected if the stratification is subject to random variations.

11 Table 2 THE OCCURRENCE OF EXTRA STREAKS IN THE X-RAY DIFFRACTION PATTERN OF DIAMOND Indices of Reciprocal Zone Indices of Spike Lattice Point _.(...) (010) (001) 111 87 87 87 220 76 76 absent 113 100 100 absent 222 75 75 75 004 5 5 30 331 7 7 104 224 72 72 30 115 present present absent 333 absent absent absent Numbers indicate relative intensity. /7. Hoerni and W. A. Wooster, Experintia 8, 297 (1952)j7 I -I- -. —`J f " — 1. —- I — - I —- ---- — I -- -- ----— 1; ---- I — -- i ~ —I2.- - - _~7~~;

1 2 3 12, 26 Lonsdale and Grenville-Wells have used the method of divergent beam X-ray photography to studyI the mosaic texture of diamonds. Before giving their results, we wish to make the following point. The term mosaic texture is employed throughout X-ray literature. While the term arises from the concept of perfect crystalline blocks which forim a mosaic to make up a real crystal, the use of this picture of a real crystal is only a mathematical idealization. Structural defects will have definite effects on the X-ray pattern. For a theoretical treatment one is forced to approximate these defects by a mosaic of sm.il. perfect blocks. Wlhlen the blocks are very large, the crystal may appear to be nearly perfect, i.e. exhibit narrow Brag, g reflections, high extinction, and poor divergent beami p iiotographs. When the blocks decrease in size the result is broadened Bragg reflections, decreased extinction, and improved divergent 27 beam patterns. Finally, James has pointed.out that the width of Bragg reflections is not always correlated to the amount of extinction, t"..there is no strict correlation between the breadth of the reflection curves and the amount of small scale irregularities. Some crystal s with comparatively narrow reflection curves show quite small extinction." Consequently, in the following, it should be realized that the relative perfection of crystal as measured by X-ray techniques depends upon the technique usedo In the work on mosaic texture by Lonsdale and GrenvilleWells an attempt has been made to correlate the intensity of extra streaks with the extent of the mosaic texture. Their

13 results are not clean cut, Vhile it is goenerally true that strong extra reflections occur in stones which are the least mosaic, the correlation is by no means universal. GrenvilleWells has found some rare diamonds which are not mosaic nor do they show extra streaaks. (ch diamonr!s are ideal from the X-ray standpoint. On the otherz hand, Grenville-J-ells has increased the amount of mosaic structure by heat treatment and by bombardment without altering the intensit of the e:tra reflections. One mu'tt conclude th-:t tie o'ccurrence of the extra streaks cannot be associated with lack of mosaic,strulcture, in general, biut t1lot t'- e pecific anomalies of i itrl cture which cause the extra reflections usually do not occ.,r in nctura.ll7 mosaic diamonfs. In addition, the structuro rdefect respronsible for thre extra reflections is not seeni. as mnosaic structure in.t.Y-?livergent beam technbmiri.e. T'l;.e. roe i.ning. experi'e"rnt;al;i-ray: work has concerned the 222 reflection, toi? refletionion i forbidden if the scatt-erinlg at each lattice site ia sp'erically,;-l e-tric. The 222 reflection occurs for all diamonds. However, the 12 intensi.t of this reflection varies, and Grenville-&eells finds that it occurs with createst inten.nit"l in diard onds showing strong extra streaks,.'- occurrence of the 222 reflection is usually attributedd to the tetrahedral distribution of the outer nlectrc n in the valency bonds. 28 Heidenreich objects to t; his interpretation since the contribution of nt'oo 1ln lc -b cctterirv.;? [.-t!r " mti.. b:ol.' - ffb o i'-n!t Coul onr have shown that the X-ray diffraction pattern is consistent with

localized electronic charge extending along the bond directions. The fact that the 222 reflection varies in intensity can be explained by alteration in the electronic distribution to increase the aspharicity of scattering at lattice sites. It is clear that almost any distortion of the structure will produce such an effect at sites near the distortion. Beyond qualitative statements, no interpretation of the 30 secondary extra reflection s iin exxistence. Eorn wishes to ascribe the streaks to strain in the lattice, perhaps 23 associated with displaced atoms. Lonsdale is reticent to accept any such theory because of the small amount of mosaic tructure pre sent in ianly diamonds showing extra streaks. aio suLmiLarize the current status of X-ray data as follows: (1) Certain diamond s exhibit secondary extra reflections wh iich indi at hat te usual Bragg odel is incompletec in some respect since no other cr-lstall exhibits sim-ilar reflections. (2)'' hore is so;ue evi cenc t'hat the- e e:xtra sreas ma' be conn-cted with stratification parallel to the cube faces occurring in a random manner. (5) The e:tra streaks most frequently occur in diamonds haviing no mosaic structure, but case s occ u.r w:l ere mosaic diamonds show extra streaks and vice versa. (4) The 222 reflection is found to be stronger in diamonds showing extra streaks. 1.2 Ultraviolet and Visible In the present problem we are concerned with the absorption and fluorescence spectra of diamonds in the range from

15 2000 A.. 0. to 6000 A. U. The spectrum of each diamond is essentially unique in that the absorption coefficients and oemision intensities vary from diamond to diamond. We shall consider the following: (1) The ultraviolet transmisasi:n- limit or cutoff, (2) Ultraviolet and visible absorption band-s, (3) Fluorescence bands. 1.2.1 Ultraviolet Cutoff 1 Rcbertson, Fox, and Iartin showed that while some diamonds transmit to wavelengths as short as 2250 A. U. with total absorption at shorter wavelengths, many diamonds will transmit no wavelengths shorter than 3100 A. U.?Famranthan discovered that diamonds are not divided into t:ic definite classes by the positions of their ultraviolet cutoffs. He found that the position of cutoff may fall at any point between the limits of 2250 A. U. and 3100 A. U, i1 (see pIi, ure 3 and Figure 4), Blackwioll measured the position of cutoff for over one hundred stones. lie was able to correlate the position of cutoff with infrared band intensities. W.- sh.al1l discuss these correlations in the infrared section. 2 At the time Raman and I Tilakantan published their first observation of the secondary extra X-ray spots, they also noted that those diamonds which transmrit to 2250 A U. show 12 no extra spots. More recently, Grenville-Wells has made a correlation between the occurrence of the secondariy extra spots ad th;e position of the ultraviolet cutoff. The correlation is necessarily qualitative since the extra spot intensities?ere judged on a qualitative basis. Nevertheless, the

16 60 0 (a) (b) __ k UV I_______ (c) SLF /+0 - -,-... —----.. —-----------....... --- o 20 (Medium UV I) Fig, 3 ULTRA-VIOLET AND VISIBLE ABSORPTION SPECTRA OF DIAIONDS

17 160 320 — 0*,r4 0.... UV I UV II 2T UV. 200 2460 26 o0 280 3000 3200 (A.JU. Fig. 4 THE FREQUENCY OF OCCURiZENCE OF UV CUTOFF IN VARIOUS WAtELENGTH INTERfVALS

18 results generally confirm the original statement by Raman and iilakantan. However, there are some exceptions to the correlation of ultraviolet transmission and absence of extra spots. In particular, extra spots have been found in some UV II stones, whereas some medium UV I stones have no detectable extra spots. 10, 3 Various experiments have shown that many diamonds are not uniform in their ultraviolet transmission properties. Certain regions of a given diamond may be transparent to the 2537 A. U. mercury line while other areas may be opaque to this wavelength. With this fact established, it becomes possible to explain cases in which ultraviolet absorption above 2250 A. U. is measured in stones which show no extra spots. X-ray techniques utilize a relatively small volume of crystal, while standard ultraviolet absorption measurements utilize a larger volume. Ccnse1uently, transparent regions with no extra spots may be "seen" by X-rays, but masked by surrounding regions in ultraviolet absorption measurements. On the other hand, the occurrence of extra spots for ultraviolet transparent diamonds would appear to be a real exception to the usual correlation, An interesting result of the discovery that large diamonds are often non-uniform is that a large collection 26 of small,perfect octahedra assembled by Grenville-Wells contains a larger proportion of ultraviolet transparent diamonds than any similar collection hitherto reported. Grenville-Wells explains this result as a direct consequence of the fact that small crystals are less likely than are

19 large crystals to contain regions of varying properties, The existence of such a collection also tends to disprove 1, 10 the statement, often made, that UV II and IR II diamonds are externally imperfect, although the statement is surely true for large diamonds. 1.2.2 Ultraviolet and Visible Absorption Bands Table 3 lists all of the known lines and bands commonly found in diamonds in the absorption spectrum from 2250 A. U. 10 to 6000 A. U. We are not concerned with all of these lines but list them for completeness. The lines between 2300 A. U. and 3208 A. U. are, of course, only observed in diamonds whose cutoffs occur at some shorter wavelength than that of the line involved. The lines between 3034 and 3208 A. U. appear with constant relative intensities. However, their absorption coefficients vary from diamond to diamond. The strongest line in this group is at 3157 A. U. In a large collection of diamonds ranging from UV II to strong UV I, Blackwell found that the intensity of the 3157 A. U. line could be correlated with the position of cutoff. As the cutoff moves towards short wavelengths, the intensity of the 3157 A. U. line (and the intensity of each line associated with it) decreases. The 4155 A. U. line has several diffuse maxima at shorter wavelengths associated with it. The group is shown in Figure 5a as a plot of cm'l displacement from the 4155 A. U. line. The diffuse maxima follow the 4155 A. U. line in intensity. Blackwell has measured the intensity of the 4155 A, U. line in many diamonds. He finds there is no correlation

20 Table 3 RItINCI L IEJ E; AHl; BA IDS OD3E)LRVD I TEl1J ILTRAVIOLET AND VISIBLE SG'ECTRUI OF DIf -OND iWavelength Vlavelength in in An stro ___ Intensitynt s Intensit 2359 VS 4041 Maxime of 2364 VS 3951 Diffuse Region of 2399 S 3817 Medium Intensity iCeak lines 3766 some time s occur between 4155 S 2400 and 3000 4200 VVJ 3034 W 4400 T 3068 X 4530. 3157 S 4647' 3181 W 4770 SD 3208 Bla ckwell' s Data

21 l00 40.... ~4 ositive Displacemen from 155 AU. 100 co 60_\ C ) 0a u \ 0 i 0 2000 4000 (cm-0 ) Neositive Displacement from 4155 A.U. (a) (b) F. 5 (a) ABSORPTION NEAR 4155 AU. IN DIAMOND (b) FLUORESCENCE N AR 4155 A.U. IN DIAI:OND

22 between the absorption coefficients of the lines at 41-55 A. U. and 3157 A. U. Because of the correlation between the position of UV cutoff and the intensity of the 3157 A. U. line, it follows that there is no correlation between the position of cutoff and the intensity of the 4155 A. U. line. By "no correlation" we mean that while the 4155 A. U. line does not occur in UV II diamonds, it cay appear with variable intensity in UV I diamonds having the same cutoff. 12.3 Fluorescence Bands There are several colors of fluorescence which occur 10 in'iamo-nds. e're concerned'ith the moat cormmonly observed fluorescence, which is blue. When diamonds are irradiated with radiation of wavelength shorter than 4155 A. U., while man.:i- are non-fluorescent, many others fluoresce blue with varying intensities. The spectrum of the blue fluorescence consists of a sharp line at 4155 i. U. together with a group of diffuse maxima at longer rwavelengths. This spectrum is shown in Figure 5b as displacement in cm from the line at 4155 A. U. When this spectrum is compared to the similar blue absorption bands, Figure 5a, it is found that, although the maxima are displaced in opposite directions from 4155 A. U., the structure of the diffuse bands is similar for absorption and emission. Blackwell has visually estimated the intensity of the blue luo en fluorescence for many diamonds. He has also measured the intensity of the 4155 A. U. fluorescence line in some of these stones. He finds that (1) the relative intensity of the 4155 A. U. line and the diffuse bands is not constant

23 froL;. diam.on to d.am.ond. The 4155 A. U. line may be absent or very strong for a given intensity of the diffuse group. (2) The intensity of the fluorescence line at 4155 A. U. does not follow the intensity of the absorption line at 4155 f.. -U. lhle only established correlation between the blue emission and absorption is that blue fluorescence never occurs in stones which show no blue absorption. The converse statemenrt does not hold. (3) Blue fluorescent stones tend to cutoff at wavelengths below 3100 A. U. whereas most non-fluorescent UV I diamonds cutoff near 3200 A. U. Mani has shown that blue fluorescence increases in intensit-y at lo.w temperatures. It is found that all of the spectrum near 4155 A. U, becomes sharper and more intense at liquid air temperatures. 1.3 Infrared Absorption 1 Robertson, Fox, and Martin were the first to show that diamonds are not all alike in the intensity of absorption at 8p. The. classifi' dia mn:,in an -:itl -r absorn. b.rr. r* n'absorbers in the 8p region of the spectrum. Four typical infrared spectra found in diamnonds are shown in Figure 6. The spectrum is seen to consist of a group of bands near 5V, which appears in all diamonds, plus a group of bands near 8p which can either appear or be absent. Sutherland and 5 Willis showed that the absorption coefficients of all the bands at wavelengths shorter than 6& are constant for all diamonds while the absorption coefficients of the bands at

26 wavelengths longer than 6i vary from zero to values larger than the coefficients for the short wavelength bands. In addition, they showed that the relative intensities of the long wavelength bands, when they occur, vary significantly. 10 Blackwell has made an important contribution to the infrared absorption data. All of his measurements on ultraviolet and visible absorption, fluorescence, and other properties have been related to the corresponding infrared effects. Blackwell's results on properties other than infrared absorption have been discussed. In the infrared, Blackwell found several general spectral features and a group of anomalous bands. In most cases, the anomalous bands appear in diamonds of unusual color or crystal habit. We shall omit reference to these bands since they are deemed extraneous. In Table 4 are listed all of the principal Infrared bands as located by Blackwell. The bands at wavelengths longer than 6i have been divided into Group A (Blaokwell'a Group I) (7.8, 8.3, 9.2, 12,8v) and Group B (Blaakwell's Group II) (7.0, 7.3, 7.5, 8.5, 100) for the following reason: It was found that the bands in Group A follow one another in intensity and the bands in Group B follow one another in intensity, That is, Group A band intnsPtiea correlate and Group B band intenaitin scorrelate. Uowevter there 1i, in general, no correlation of intensities btwsen bands of Group A and Group B, Group B it also unique in that the 70,0 7,5, and 7.6 bands are quite narrow while 411 other bands in both groups are quite broad (see Figure 6), Although the!ntensitiea of the Groups are not correlated,

27 Table 4 IMAXIMIA OF INFRARED ABI^R T 0H B i Di II THE SPEC TRA OF DI AI'.CLDS Wi-Jave- Frequency Wave- Frequency length (Wiave- Inten- length (Wave- Inten(IKicrons) numbers) sity _ j(Iicrons) numbersr) _sity 2.8 3570 W 7.51 (B) 1332 VWT 3.2 3125 W 7.80 (A) 1282 S 4.03 2480 M 8.31 (A) 1203 S 4,59 2180 S 8,54 (B) 1171 W 4.98 2008 S 9.15 (A) 1093 M 7.01 (B) 1426 WI 9.97 (B) 1003 W 7,29 (B) 1372 W 12,8 (?) 784 VW Blackwell' s Data

29 pend on the integrated intensity of the emitted light. Consequently, the effects are determined principally by the diffuse bands near 4155 A. U. rather than the intensity of the 4155 A. U. line itself. Therefore, although it is found that the intensity of the line at 4155 A. U. cannot be correlated with any of the other spectral features, correlations with overall "blue fluorescence" are not precluded. Blackwell finds that blue fluorescence never occurs in the absence of blue absorption. This implies that Group B infrared absorption also occurs in blue fluorescent diamonds. This point is verified experimentally. In addition, it is found that, relative to bands in non-fluorescent diamonds which have Group B absorption, blue fluorescent diamonds have strong Group B absorption and weak Group A absorption. Finally, Blackwell finds that, while, in general, there is no correlation between Group A and Group B band intensities, the "blue fluorescent" diamonds form a special class for which the intensities of the two groups are correlated in the sense that their absorption coefficients follow one another in magnitude. The final correlation of importance is between infrared absorption and X-ray extra spots. Unfortunately, very few data are available on this point. Original work by Raman 2 10 and Nilakantan and a small amount of work by Blackwell confirm the fact that extra spots occur in diamonds having absorption at 8p. This fact can be inferred from the previously mentioned qualitative correlation between extra spot intensity and position of ultraviolet cutoff. However, the

30 work is not sufficiently quantitative to determine the correlation between extra spot intensity and a particular group of bands in the infrared. When diamonds are heated to 400~ C, the only detectable change in the infrared spectrum is a decrease in the intensity 10 of the 7.29k (Group B) band by a factor of 2/3, This band also shifts to longer wavelengths as the temperature increases by a factor of 1 cm-1 /50~ C. Heating to 1700~ C did not introduce any change in the room temperature infrared spectrum in specimens examined by Blackwell, 1.4 Other Properties Several other physical properties of diamond are known which exhibit anomalies similar in some ways to those already described, These will now be briefly discussed together with some important properties which show no anomalies, The latter are the Raman scattering spectrum and the density. We shall deal with them first. 1,441 Raman Effect All diamonds show a strong, narrow, weakly polarized Raman 10, 33 line at 1332 cm-1. After many studies, it is well-established that there is no effective change in this line from diamond to diamond. The line moves to lower frequencies as the temperature increases. Its displacement with temperature is 34 approximately the same as noted for the 7.29[ band in absorption, For certain large, ultraviolet transparent diamonds, 35 Krishman has succeeded in recording a weak set of Raman

51 bands extending to an upper frequency limit of 2665 cm-1. Figure 26 shows this group of bands. The 2537 A. U. line of mercury is the only exciting line which is sufficiently intense to scatter this set of bands with enough energy to record. Consequently, no UV I diamonds have been studied. It is generally assumed that the second order Raman scattering is the same for all diamonds. 1.4.2 Density The most accurate measurements of density have been 36 37 made by Tu and by Bearden. Their average results are 3,5142 and 3.51536 ~.00004 g/cc at 23.5~ C respectively. No such accurate methods have been employed in attempts to 1, 10 find density variations between diamonds. No significant density variations have been found. 1,4.3 Color, Crystal Habit, and External Crystal Perfection The three properties listed are variable between diamonds, 10 Blackwell has studied the variation of these properties and has attempted to correlate them with spectroscopic properties. There appears to be no new information to gain from introducing the highly involved relationships found in this phase of Blackwell's work. We have already mentioned 12 that Grenville-Wells assembled a collection of externally perfect, colorless octahedra. The full range of X-ray and spectroscopic anomalies is shown by her collection, Consequently, the variables of color, crystal habit, and perfection appear to be extraneous to our work. However, since some color changes have occurred in experiments performed in the present study, it is pertinent

%2 to mention that Blackwell found that blue diamonds are invariably IR II, brown are either IR II or weak IR I, and green diamonds are strong IR I. 1.4.4 Birefringence From its cubic symmetry, the diamond lattice is expected to be isotropic. However, many diamonds exhibit patterns 1 of birefringence when viewed between crossed polarizers. The birefringence may be due either to strain introduced through external distortion or to lattice imperfections. Since both factors may be in operation, and since only strain due to imperfections is of interest, the results are diffi38 cult to interpret. In favorable cases, Raman and Jayaraman have shown that sometimes there is similarity between patterns of birefringence and patterns of fluoresence, One deduces that both effects arise from a common lattice imperfection. In general, the results from the study of birefringence cannot be interpreted in such a way as to give concrete information as to the origin of anomalies in diamond properties. 39 The Indian school has placed considerable stress on the laminae seen in certain birefringence patterns. The laminae occur parallel to octahedral and dodecahedral planes, 8 According to Raman's theory for diamond, which we will discuss in a later section, the laminae occur at the boundaries between regions of differing local electronic configuration. 1.4.5 Photoconductivity Diamonds, which are normally good insulators, sometimes 1 become conductive when exposed to ultraviolet light. It is

33 found that the photoconductivity of UV II diamonds is as much as ten times the photoconductivity of UV I diamonds. 1.4.6 Impurities All, diamonds contain' detectable amounts of chemical im40 purities. Chesley made qualitative spectrographic analyses of thirty-three diamonds, listing their ultraviolet absorption properties as well. He found no correlation between the type of impurity and the position of the ultraviolet transmission limit. Table 5 lists the common impurities. 18 Straumanis lists the impurities found in two diamonds used for X-ray work. The impurities are shown in Table 5. 36 From his X-ray results and the density measurements of Tu 37 and Bearden, Straumanis concludes that his diamonds are essentially perfect with respect to spacing, vacant sites, and interstitial atoms. However, his measure of perfection depends upon atomic weight and density data which is necessarily less accurate than spectroscopic or X-ray data. 1,5 Summary From our review of previous experimental results we can draw certain conclusions about the correlations between anomalous properties. We assume that the exceptions to correlations which sometimes occur can be neglected. On this basis, we can state that whenever a diamond differs from ideal UV II, X II, IR II the following phenomena will occur, the occurrence of one inferring the occurrence of the others:

%4 Table 5 IMPURITIES FOUND IN DIAMOND (a) (b) Impurity Diamond 1 Diamond 2 Impurity Occurrence Al 3 3 Al A B 1 - Ba S Ca 1 4 Ca A Co 2 - Cr S Hf - Cu S Fe? 1 Fe S b?? Pb Pb Mg 4 2 Mg S Mn - - Na S Pt - 1 Si 4 2 Si A Ag - 2 Ag S Sn - - Sr S Ti 2 3 Ti S Zn 2 - (a) M. E. Straumanis and E. Z. Aka, J.A.C.S., 73, 5643 (1951). (b) F. G. Chesley, Amer. Min., 27, 20 (1942). 4 Major contaminant A: Occurs in all diamonds tested 1 Faint trace S: Occurs only in some diamonds' _.._.- ~:- -: ~ - -',.:.~ ~ L_, ~:,::i:_:: " — -'::.::-:- _':.:-:~'' ~ -::~::.:: --.-:.::..~~,''..'-'.

35 Class I: (1) X-ray streaks (2) Ultraviolet cutoff at wavelengths greater than 2500 A U. (3) Group A absorption in the infrared (4) Absorption at 3157 A. U. These Class I properties are correlated in the sense that the magnitudes of the effects are proportional, is e. as the intensities of effects (1), (3), and (4) increase, the ultraviolet cutoff moves towards longer wavelengths. In some of the diamonds displaying Class I properties, another group of phenomena occurs: Class II: (1) Absorption at and near 4155 A. U. (2) Group B absorption in the infrared These Class II properties are correlated in the sense that the absorption coefficient at 4155 A. U. is proportional to the Group B absorption coefficients. Finally, in some diamonds displaying both Class I and Class II properties, a third group of phenomena occurs: Class III: (1) Diffuse fluorescence near 4155 A. U. (2) Group A absorption coefficients proportional to Group B absorption coefficients (3) Group B stronger with respect to Group A than in non-class III diamonds. In this class, the correlations are not so clean-cut. The listing of properties is meant to convey that the presence of strong blue fluoresence seems to introduce a regularity between the effects of Class I and Class II, i. e. the normally independent intensities of Group A and Group B are now correlated.

Chapter 2 PRESENT EXPERIMENTAL WORK 2.1 Apparatus (2.1.1 Near Infrared) (2.1.2 Far Infrared) (2.1.3 Vacuum Ultraviolet) (2.1.4 Ultraviolet Transmission) 2.1.1 Near Infrared SPECTROMETERS: Commercial spectrometers were used to obtain spectra in the range from 2 to 33., These spectrometers are made by the Perkin-Elmer Corporation of Norwalk, Connecticut. Because the Perkin-Elmer spectrometers have been fully described elsewhere, we shall mention only the general features of the instruments. 41 The Model 21 is a double-beam recording spectrometer equipped with a sodium chloride prism. This instrument records percent transmission versus wavelength from one to fifteen microns. In regular use, a sample 10 by 25 mm is required to cover the area of the energy beam. An adapter makes it possible to use samples as small as 2 by 15 mm. Smaller samples cover only part of the entrance slit and consequently reduce the energy at the thermocouple. 36

37 42 The Model 12C is a single-beam recording spectrometer in which give different prisms (LiF, CaF2, NaC1, Cs3r, KRS-5) may be used. With these prisms the wavelength interval from one to 35 microns can be covered with resolution of the order of two wavenumbers. Sample size requirements for the 12C are the. same as those for the 21. 43 The Model 112 is a single-beam recording spectrometer which is a modified version of the 12C. The modification 44 consists of the introduction of the Walsh optical system. This system causes the radiation to be sent through the prism four times instead of twice (Littrow system monochromator). The effect of the alteration is to increase the resolving power of the spectrometer. In addition, since the radiation is chopped after it has been dispersed the effect of scattered radiation is greatly reduced. Scattered radiation beyond 25V makes the 12C spectrometer very unreliable. For example, at 25p, 25 percent of the energy detected by the thermocouple is due to scattered radiation. In the 112 instrument less than 2 percent of the energy at 25V is due to scattered radiation. REFLECTING MICROSCOPE: It has been noted that the spectrometers used in this work require samples at least 2 x 15 mm for optimum performance. A sample smaller than this will cover only part of the length of entrance slit. With such a sample, only- part of the beam enters the spectrometer. In order to bring the amount of energy at the thermocouple up to the minimum (determined by the sensitivity of the thermocouple and the gain of the amplifier) which is

38 needed for recording spectra it is necessary to increase the slit width. The spectra obtained under such conditions will show less resolution. The sample size which can be used is determined, therefore, by the resolution required. Since many of the diamonds used in this work will cover only a small fraction of the slit length, and also because there is interest in the spectra of individual portions of diamonds, a reflecting microscope has been used. 45 The microscope used is one constructed by D. L. Wood. In Figure 7 is shown a schematic drawing of the microscope and its relation to the monochromator. In all of the present work the monochromator and auxiliary equipment have been those of the 12C spectrometer. The microscope design is based on the theory of 46 47 Schwartzchild as applied by Burch, Its advantage over a refracting microscope is the fact that its properties are achromatic. Because of the obstruction of the beam by the small convex mirror and the hole in the large convex mirror, approximately 45% of the numberical aperture is lost when both the large concave and the small convex mirrors are 48 spherical. Wood ground his large spheres sufficiently aspherical to reduce the obstruction to 14% of the numerical aperture. The effective focal length of the system is 0.30 cm. The numerical aperture, not corrected for obstruction, is 0.75. Since the numerical aperture of the Model 12C monochromator is 0.12, the magnification for optimum operation is 0.75/0.12 or about 6. Wood's system is not limited to this magnification, since for a range of magnification

39 Monochrmar "Qs M7^^ 4 \ M5 > 1 4M3 MIicroscope r SM2'\.m Fig. 7 REFLECTING IMICROSCOPE

40 mirror M6 can be adjusted so that the collimator of the spectrometer is filled. In practice, with a field as small as 100 by 300p, the microscope makes it possible to record spectra using slit widths about twice the widths used in "macro" operation. The limitations of the reflecting microscope are (1) Resolution is reduced by a factor of two from "macro" operation. (2) Opening the slit of the monochromator beyond the width of the image of the source on the slit (0.6 mm) will produce no gain in energy. At long wavelengths (beyond 15p) where "macro" slit widths are greater than 0.3 mm, the microscope cannot be used. 2,1.2 Far Infrared To measure spectral absorption from 30 to 100L (330 -1 to 100 cm )a vacuum grating instrument has been used. This instrument was constructed by Randall and coworkers and 49 has been described elsewhere. The basic elements of the spectrometer are (1) an incandescent chromel metal strip which supplies the energy, (2) an off-axis parabola, (3) a plane grating, (4) a reststrahlen plate, (5) a thermocouple detector, (6) a galvanometer amplifier, (7) a tun3d electronic amplifier (13 seconds / cycle), and (8) a recording potentiometer. The reststrahlen plate and the grating are changed from one wavelength interval to the next. In obtaining spectra from 30 to 100I one is required to alter the grating-reststrahlen combination at least four times,

41 i. e. four separate runs are necessary for each sample. One is seriously limited by the small amount of energy available. For satisfactory results a sample must cover most of the exit slit. The minimum sample dimensions are about 30 x 5 mm. 2.1.3 Vacuum Ultraviolet In the current work some measurements have been made of spectral absorption in the region from 1200 to 2000 A. U. The instrument used is a Cario-Schmidt-Ott vacuum fluorite 50 spectrograph, which has been described elsewhere. A schematic diagram is shown in Figure 8. The instrumentts characteristics are below the Figure. The recorded spectrum, (Hilger Q1 Special plates were used), extends from about 1200 A. U to the red of the visible with dispersion decreasing rapidly at wavelengths longer than 2000 A. U. Atmospheric absorption below 2000 A. U. can be effectively eliminated by pumping continuously with a Welsh Duoseal vacuum pump. A hydrogen discharge lamp served as a source of continuous radiation. A drawing of the lamp with its accompanying apparatus is shown in Figure 9. In operation, the discharge tube and the lower halves of the tubes enclosing the aluminum electrodes are immersed in a tank of water. Hydrogen gas, from a cylinder, brought to atmospheric pressure by a mercury bubbler, is pulled through a capillary tube into the discharge tube. The pumping rate and the dimensions of the capillary regulated the rate of gas flow into the discharge tube and hence the pressure within the discharge

42 tS E S: Entrance Slit; L: Fluorite Lenses P: Fluorite Prism; F * Plate Holder Scale: ~ size; Focal Length: 10 cm; Speed: f/12 Plate Factor: 6 A.U. at 1250 A.U. Fig. 8 CARIO-SCHIMIDT-OTT SPECTROGRAPH

Capillary Tube, —-Ground Glass Joint High Potential Lead eTrc -To hydrogen gas tank Aluminum Bubbler lectrode, _ _ _d B-e To vacuum pump / apen Blockedl. Reservoir ae l ^ 1Fluorite -Window Water Discharge Tube Fig. 9 HYDROGEN DISCHARGE LAMP

44 tube. The glass tube covering the capillary is seated on a ground glass joint and can be removed to adjust the length of the capillary by breaking off small sections. By proper regulation, a discharge of any desired character can be produced. The lamp has been operated at approximately 200 volt amperes, the hydrogen continuum can be recorded in a few seconds. For the present work no accurate calibration has been necessary. Lines in the hydrogen spectrum, the Schumann Runge bands of oxygen, and mercury vapor lines provided reference points for estimating wavelengths. Scattered light is serious at short wavelengths and accounts for 51 roughly 10% of the observed intensity at 1500 A. U. 2.1.4 Ultraviolet Transmission A technique has been developed for examining the variation in the transmission of ultraviolet light in a diamond. The apparatus is shown in Figure 10, The grating monochromator is a standard commercial instrument made by Bauch and Lomb. With a mercury vapor lamp as a source, monochromatic light is available at the exit slit. This light is reflected through a diamond mounted in a holder. An enlarged image of the diamond produced by a quartz lens, is formed at the focal plane of a camera. This image can be recorded photographically or observed visually by means of a plate covered with anthracene crystals which fluoresce in ultraviolet light. Aberrations in the system and scattering of the light make it necessary to use diamonds

46 which have flat sides. For less uniform stones, a contact print of the transmission pattern can be obtained by placing the sample directly on the photographic plate. Incident monochromatic light then produces an unmagnified pattern which can be enlarged photographically. 2.2 Account of Experimental Work Performed (2.2.1 The Near Infrared Spectrum of Diamond) (2.2.2 The Far Infrared Spectrum of Diamond) (2,2.3 Infrared Spectra of Powdered Solids) (2.2.4 The Infrared Spectrum of Germanium) (2.2.5 The Ultraviolet Absorption of Adamantane) (2.2.6 The Absorption of Diamond in the Vacuum Ultraviolet) (2.2.7 Bombardment Experiments 2.2.1 The Near Infrared Spectrum of Diamond 6.5 BAND: Twenty diamonds in the present collection were large enough to permit use of the double-beam spectrometer to record absorption spectra from one to 15p. These diamonds included 1 IR II, 3 W IR I, 2 M IR I, and 17 S IR I. Fourteen of the diamonds (1 M, 13 S IR I) showed a weak band near 6.5p. This band has not been reported by other investigators. The reason for its detection in the present work lies in the fact that the double-beam instrument automatically compensates for the intense water vapor absorption between 6 and 7~. In previous work, single-beam spectrometers were used, and observations between 6 and 7p were difficult, especially for the detection of weak bands. When the 6.5p band is strong enough for accurate observation, it appears as a doublet with maxima at 6.48L (1540 cm-1) and 6.57p (1520 cm-1). In Figure 11 the spectra of two strong IR I diamonds are shown

47 100 80..... 0 i4 60 0 3 4 5 4 7 8 (microns) ) 20 l........... _, 3 4 5 6 7 8 (microns) 100',Fig. 11 0TWO SPECTRA SHO 0. 60.............. 20 Diamond F 1 0 3 4 5 6 7 (microns) Fig. 11 TWO SPECTRA SHOWfING THE 6.5 MICRON BAND

48 in which the 6.5v band is relatively strong. This band does not appear in the IR II diamond or the weak IR I diamonds. Its intensity in IR I diamonds does not appear to be correlated with either Blackwell's Group A or Group B. In Table 6 the absorption coefficient for the 6.5k band is given for the diamonds examined. We define the absorption coefficient (here, and in the following section) as k in the expression I/Io 10kt, where I/I. 100 is the measured percent transmission, and t is the sample thickness in centimeters. We attribute the lack of correlation between 6.5p and other IR I bands to two sources: (1) The band is weak and difficult to observe with accuracy. (2) The absorption at 6,5p due to the long wavelength portion of the 5i band and the short wavelength side of Group B bands. These contributions to the absorption at 6.5V introduce errors in intensity of absorption assigned to the band. BAND INTENSITY MEASUREMENTS: Unless scattering of the light occurs, due to imperfect surfaces, the percentage of incident light transmitted by all diamonds is constant from 1 to 2.5L. The loss of energy in this wavelength 10 interval is due to reflection. In addition, as Blackwell has shown, the intensity of absorption in the near infrared spectra of diamonds is never sufficient to alter the reflection coefficient at absorption bands by a significant amount. Since one can adjust the percent transmission scale of the double-beam instrument so that the recorded transmission between 1 and 2.5p is 100%, all of the recorded

49 Table 6 THE AB3iORR-TION COEFFICIIENT OF THE 6.5~ BAND Diamond T_.e tr(mm) (65l_)-(cm ) B2 SI 1.32 0 BP2 II 1.89 0 F1 SI 3.84 0.12 F2 SI 3.96 0 F3 SI 3.56 0.10 I7 II 3.0 0.41 I20O d'fI 4.0 1M4?-o 0.79 0 S1 SI 4.67 0.34 S2 SI 4.19 0.25 s3 SI 5.36 0.57 3LF127 SI 2.55 0 SL09 I 3.10 0. 42 SLO15 SI 0.54 0.33 SL025 I 0.53 0.20 SL042 3I 3.75 0.10 3L044P2 SI 0.69. 80 SYR1 I 3.45 0.38 SYR8 SI 0.30 0.30 SYR9 SI 0.51 0.51

50 spectra can be automatically compensated for loss by reflection. The absorption coefficient at any wavelength can then be determined from the recorded spectrum without assumptions as to the position of the base line. In order to test the validity of previous experiments which showed that the absorption coefficients of the bands from 2,8 to 6.0L are constant, we have adopted the following method. Spectra are adjusted to compensate for loss by reflection as already described. The resulting spectra are plotted as log log Io/I vs wavelength, where I/Io is the measured fractional transmission. Such plots are independent of sample thickness except for a vertical displacement along the log log scale. This is shown as follows: (1) I/Io - l-kt (2) logl0I/I = kt; logl0 log10 Io/I = log k + log t Besides their independence of thickness, these log log plots have the virtue that wide variations of the absorption coefficient are represented on a logarithmic scale where they are more obvious than on a linear scale where variations in k are overexaggerated. However, regions in which k is zero or nearly zero cannot be represented on log log plots, When the double-beam spectra are plotted in this manner, it is found that a linear shift along the log log scale will superimpose (within experimental error) the bands between 2.5 and 6.0[t in 17 of the 20 diamonds examined with the double-beam spectrometer. Three of the diamonds (F2, I20, T38) were too rough to give acceptable spectra

51 in the short wavelength region. The resulting contour is shown in Figure 12. These results verify the previous 5 10 observations by Sutherland and Willis and by Blackwell that the absorption coefficients for bands from 2.5 to 6.0p are the same for all diamonds. The log log I /I presentation is also useful to show the variable absorption at wavelengths greater than 6.0~. Plots for T15 (W IR I) and SYR8 (S IR I) are shown in Figure 12, The short wavelength portions are superimposed so that the differences at long wavelength are independent of thickness. CORRELATION OF INFRARED AND ULTRAVIOLET ABSORPTION: The reflecting microscope has been used to detect variations in the spectrum between different parts of individual diamonds. Only one clear example of absorption variation within a single diamond was found. The ultraviolet transmission patterns of all suitable diamonds (i. e. flat sides) were examined using the apparatus described on page 44. The 2537 A. U. line of mercury was used for illumination. This wavelength lies in a region where UV I diamonds are opaque but UV II diamonds transmit. Patches of transmitted 2537 A. U. light indicate the presence of UV II regions, The pattern is viewed through the use of the anthracene coated plate. In 50 diamonds (listed in appendix) only two stones showed variations in transmission at 2537 A. U. One of these stones (K2) was too rough to separate the effect of scattering from the variation in absorption, Wlhen K2

Log log Io/I (Arbitrary Scale) MI -. o o C' o... -,..C)-...... trj 0 0 1-~ 01I

53 was examined with the reflecting microscope, the rough surfaces caused too much refraction of the beam to permit reliable estimates of the change in absorption coefficient between different points in the stone. Diamond M4 (W IR I) showed a clear transmission pattern which is sketched in Figure 13. Using the reflecting microscope, spectra were recorded for several points in the diamond. Using log log Io/I plots the variations in the 8p. regions were apparent. In Table 7 below Figure 13 these variations are listed in terms of the absorption coefficient at 7.8t. The correlation with the transmission pattern is obvious. This result confirms Blackwell's results obtained for the variations between different stones, i. e. the intensity of absorption at 7.8V increases as the position of the ultraviolet cutoff moves to longer wavelengths. ABSORPTION OF POLARIZED LIGHT: Because many diamonds display birefringence patterns, we have looked for dichroism in the infrared which might accompany the birefringence. On a "macro" scale, the spectrum of SLF127 (S IR I) was recorded using the double-beam instrument equipped with reflection polarizers. No effects were observed. Since the birefringence is localized, a better experiment was the recording of spectra using the reflecting microscope with a polarizer. In this case, individual regions in SLF127 and T37 ( IR II) which displayed birefringence were tested for dichroism. Again, no effects were observed.

54 0 17 1 (Lined region-opaque) (Unlined region-transparent) Fig. 13 TRANSMISSION PATTERN OF DIAMOND M4 FC7. 2537 A.U. LIGHT Table 7 The Absorption Coefficient at 7.8 Microns for Points in Diamond M4 Point K(cm-l) Point K(cm-1) Point K(cm-1) 1 4.8 7 1.2 13 0.3 2 4.0 8 20 14 0.4 3 2.8 9 0.5 15 0.5 4 1.6 10 0.6 16 0.3 5 1.4 11 0.2 17 0.3 6 1.2 12 0.3 18 0.3

55 2.2.2 Far Infrared Spectra of Diamonds 300 TO 900 cm 1 In this region, the CsBr prism was used, mounted in the Model 112 spectrometer. The reflecting microscope cannot be used at these wavelengths for reasons already discussed. Because the energy from the "blackbody" source drops off rapidly with wavelength, s amples must cover a larger portion of the slit height than in the experiments from 1 to 15p. (See page 37 ). Consequently, spectra from 300 to 600 cm-l have been obtained only for the large flat diamonds in our collection. The diamonds examined include one IR II (BP2), one weak IR I (M4) and six strong IR I (Fl, S1, S2, S3, SLF127, SL025). The spectra of Fl, S2, and SLF127 from 300 to 1400 cm-1 are shown in Figure 14. In the long wavelength region we have found no absorption in the IR II diamond. However, two bands occur in all IR I diamonds. These bands are located at 20.8p (480 cm-1) and at 30.5p (328 cm-1). The 21p band was first reported by 31 Danielson in 1951. The 30p band has not been reported previously. In Table 8 we list the absorption coefficients for the 21 and 30p bands together with other IR I bands. The bands at 21 and 30p are typical IR I bands in that their absorption coefficients vary from diamond to diamond. In order to investigate any connection between the two long wavelength bands and other IR I bands we have prepared the charts shown in Figure 15 in which are plotted the absorption coefficients of bands which may be related. Chart (a) demonstrates that

56 100 -- -_______ __________ 25 wo 1- ---- - ---- - --- 7 I h 25 ---------.......... 22 7 rm -4I -— I /1 —— 1 50 —4l 1-r —- - U —2. 300 500 700 900 1 1100 1300 1500 ~~d)~~(cm ) FIG.S 14 SPECTRA OF 3 DIAMONDS FROM 300 TO 1500 CM~1

57 Table 8 THE ABSORPTION COEFFICIENTS FOR THE LONG WAVELENGTH BANDS IN DIAMOND UndeterminedG Group A Diamond t(mnm) 21[ 305, 13M 7. 0 7.53P 10 7.8J Fl 3.84 0.86 1.1 1.4 1.0 H 2.9 H N4 0.79 0.5 0,3 0 0 0 0 3.1 1 44.67 1.0 1.25 1.5 1.05 H 3.2 H S2 4.19 1.8 0,17 0.72.10 0.57 0.95 H 33 5.36 0.93 2.4 2.0 1,75 H 5.3 H SLF127 2.55 0.46 0.87 0.97 0,81 3.2 2.2 4.9 S1025 0.53 3,7 1.0 2.5 1.0 5,2 3.2 19.6 Note: H indicates a band too intense to measure.

58 2.5 2.5 2.- 2.1. 1.K K 30 I K30 * e 0.-5-0 10. ~ ---- 0-. 0 1 2 3 4 0.5 1.0 1.5 2.0 K K o(Group B) 4 6 SLO 25"-c 3- 4. K21 21 K "21 ^- ^O^-. -10 1- 1. K0- 0 -- l l-. lI 0 5 10 15 20 0.5 1.0 1.5 2 2.5 K7 g(Group A) K13 Fig. 15 RELATIONS BETEIGEN ABSOiRPTION COEFFICIENTS OF IR I BANDS

59 the absorption coefficient of the 30p band varies independently of the absorption coefficient of 21i band. Chart (b) demonstrates that the 301 band is a Group B band in that its absorption coefficient is essentially proportional to the absorption coefficient of the 7.0p band (see page 26). Chart (c) demonstrates that the 21l band is probably a Group A band. There are only three cases in which the 7.8p band is weak enough (in intensity) to measure accurately, but the wide range of K21 and K7.8covered, gives good support to our statement. The 13p band (Blackwellts 12.75A band) has not been correlated with a particular group in the past. Blackwell made a few measurements of the band and assumed that it was a Group A band. In Chart (d), Figure 15, the correlation is shown between 10 (a Group B band) and 135. Except for SL025, the relationship appears to be quite good. We explain the deviation of SL025, by pointing to the fact the K1 is very high in SL025. The remaining Group A bands associated with 21 will also be strong and will tend to upset correlations between Group B bands lying in the wavelength interval where both Group A and Group B absorption occur, i. e. between 7.8 and 30p. We have intentionally chosen 10p rather than 7p as the representative Group B band because it lies closer to 135 than the other Group B bands and should be affected by Group A absorption in a manner similar to 13p. In fact, the position of the point for SL025 gives a good indication that the absorption band at 21p has a tail extending to at least 135 and that Group A

60 absorption at 10 is less than Group A absorption at 135. If this conjecture about SL025 is accepted, our measurements can be interpreted to mean that the 135 band is a Group B band. 30 TO 10Ox: Two samples large enough to use in the vacuum grating instrument (page 40 ) were made from (1) Four strong IR I diamonds of average thickness 0.63 mm. This sample is designated as SLO.Stones SL015, 25, 44(a) and 44(b) were used. (2) Three strong IR I diamonds of average thickness 3.74 mm. This sample is designated F since Fl, 2, and 3 were used. Neither sample shows absorption bands at wavelengths longer than 30. By comparing transmission in the overlapping regions between the records obtained in the near infrared, the CsBr region, and the far infrared it was established that the transmission measured at 35p and beyond was equal to the transmission measured between 1 and 2.5k. That is, the only losses at long wavelengths are due to reflection. The reason that one cannot rely on the absolute value of the transmission measured at long wavelengths is that the stones used are not flat or uniform in thickness. The radiation is often completely blocked by certain rough portions of the diamonds. The results are incorporated in Figure 16 in which the infrared spectrum of F1 is shown from 100 to 3800 cm1. 2.2.3 The Infrared Spectra of Powdered Solids In the course of our work in the far infrared, the possibility of using diamond dust for a sample was considered.

77 0 Cl) 75__________ _____ ______ / \/ ____________ ____ 00 \ H 100 600 1000 1400 1O00 2200 2600 3000 3400 3900 (cmr1) Fig. 16 THE INFIRARED ABSORPTION SPECTRUTIJ OF DIAPLOND F-l

62 Such a sample would be useful in that it could be made as large in area as necessary to cover the whole length of the exit slit of the spectrometer. Diamond dust obtained commercially proved to be contaminated with SiC and/or SiO2. This was established by recording spectra of powdered samples of these materials (see Figure 17). Methods of purification proved unsatisfactory. Consequently, it was not possible to use diamond dust in the experimental work in the far infrared. The spectrum of silicon powder was also recorded. (See Figure 17). It will be noted that a strong band occurs at 9t. A similar band occurs in all the other powder spectra. A similar band also occurs in the spectrum of 52 crystalline silicon. However, it seems probable that 53 this band is due to SiO2. The similarity of the powder spectra points to this fact. A further discussion of this point is given on page I 5 7 2.2.4 The Infrared Spectrum of Germanium 54 55 Lord and Briggs have published absorption spectra for crystals of germanium for the region from one to 35A. However, it was desirable to obtain additional spectra in this region for use in our calculations. In addition, the germanium absorption spectrum beyond 355 is of interest and has not been published. We have received samples of germanium crystals from Dr. G. A. Morton of the R. C. A. Laboratories, and we are indebted to Dr. Morton for supplying these samples.

64 The samples were of varying thickness and purity. Samples A1 (8.90 mm), A2 (2.54 mm), and A3 (0.99 mmn) were relatively pure (impurity concentration 1013 atoms/cc) according to their electrical properties (resistivity between 30 and 38 ohm cm). Pure germanium would have a 56 resistivity of approximately 47 ohm cm. Samples B1 (8.80 mm) and B2 (2.44 mm) had arsenic added to a con16 centration of about 101 atoms/cc (resistivity between 56 0.4 and 0.6 ohm cm). Arsenic is known to act as an elec57 tron donor in germanium. The positive arsenic ion is located at a lattice site where it is tetrahedrally bound to neighboring atoms. The extra electron is essentially free to add to the number of carriers. Consequently, germanium with arsenic impurities is "n" type since the conduction takes place through the movement of electrons (as opposed to conducting by holes). While this section was 58 in preparation, a communication by Collins has appeared in which it is stated the "n" type germanium displays absorption at long wavelengths which increases with A2. This absorption is independent of lattice absorption and is characteristic of absorption by free carriers. Absorption spectra for our samples were obtained from 1 to 100O. The spectra are shown in Figures 18, 19, and 20. In Table 9 the observed maxima are given together with the absorption coefficients at these maxima. In Figure 18 the absolute transmission from 1 to 15p is given. Beyond the sharp cutoff at 1.75p, each sample increases in transmission to a maximumn of about 40%. The surfaces of the

Pure 40 0 to or 20 (microns) Impure 40 1 —----------— j -----— i ----— I -----— 1 —---— 1 —---- rl~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~C 0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~0 20 t~o 1 3 5 ~~~~~~~~~~~~~~7 9 11 13 15 (microns) B2 "I lab Os go 0~2.r 20 ---— ^^ —- -O —---------- 2 0 CO Fig. 18 TRANJKIS3ION uF GEmC'NIAJM FROIM 1 TO 15 MICRONS

o Al - Al: Thickness-8.90 mm\ A2: Thickness-2.54 mmI A3: Thickness-0.99 mm 0 *H60 0 CO A2 ~40 a) 20 A3 0 -M - 900 800 700 600 500 400 300 20010 (cmi) Fig. 19 ABSORPTI0ON SPECTRA OF 3 PURE GERKt.'ANIUhk. CRYS-TALS

100. i| i | | Bl: Thickness- 8.80 mm B2: Thickness- 2.44 mm Bl C! l | | 1X I 1 1 I / 0 60 -.............. I, — 900 800 700 600 500 400 300 200 100 ~-I. 60 0 B2 (cm-1) Fig. 20 ABSORPTION SPECTRA OF 2 IMPURE GSE"Ii',NIJT CRYSTALS 2 0 -- - ---------— / ----- -4 ---------------------— R ---— L -

68 Table 9 ABSORPTION MAXIMA IN THE GERMAINIUIi SPECTRUMI Position Absorption Position Absorption Coefficient Coefficient.... cm," (cm-1) l's cm-l i (coe1). 11.87 845 0.10 23.8 420 3.3 13.35 750 0.10 29.0 345 25 15.6 640 0. 30 36.3 275 5.3 17.9 560 0.85 50.0 200 2.2 19.2 520 1.00 - To the base 10 samples were only rough ground, so that the departure from 52 constant transmission between 3 and 10p (observed by Briggs and others) may be attributed to scattering. In addition, the absolute transmission at wavelength where scattering is not important (near 10p) varies from sample to sample, and the loss is greater than that due to normal reflection as measured by Briggs (^, 50%). The excess loss is attributed to diffuse reflection by rough portions of the surface and is not due to absorption. A comparable situation occurs with diamonds which have been sawn but not polished. For all samples except A3, the transmission between 6 and 10 is essentially constant. For the spectra between 100 and 1000 cm1 (Figures 19 and 20) we have assumed that the loss

69 due to all types of reflection is the same as that measured between 1000 and 1600 cm'l (10 to 6p). Since in sample A3, the transmission increases continuously from 6 to 14i, we assume that the reflection loss from 100 to 700 cm1 is the same as that at 700 cml (14). In the spectra between 300 and 900 cm"1 (Figures 19 and 20) the absorption coefficients for all of the bands appear to be the same in all five samples regardless of purity. Beyond 300 cm, the impure samples (B1 and B2, Figure 20) absorb too strongly to make accurate measurements of the absorption coefficients, but there is no question that they absorb more strongly than the pure samples. On this point, 58 we agree with Collins. Concerning the magnitude of the increase of absorption between pure and impure samples, Collins states that this absorption is due to free carriers and that it increases as X2. On the other hand he finds that the amount of the observed "free carrier" absorption is several orders of magnitude greater than that predicted by theory. In the absence of published data (Collinst work appears as an abstract), it is not possible to examine these conclusions in detail. However, in all cases, Collins assumes that the absorption due to lattice vibrations is the same in all samples. It seems a remarkable coincidence that the increased absorption in impure samples sets in at precisely the place where fundamental lattice absorption occurs (Collins and 59 Fan have already proposed that the 345 cm-l band in germanium is a fundamental lattice absorption). In diamond, where no

70 free carrier absorption is expected, absorption in the fundamental region also changes from sample to sample. Since our pure germanium samples show absorption at the longest wavelengths, i. e. fundamental bands occur in the region from 100 to 300 cm'1, it is difficult to exclude the possibility that at least part of increased absorption in impure samples is due to a change in the absorption coefficients of the fundamental bands. 2.2.5 The Ultraviolet Absorption of Adamantane Adamantane is a hydrocarbon, C10H16, having the structure shown in Figure 21. If the hydrogens are omitted, the structure of adamantane is similar to the local structure of diamond (Figure 2). Since the position of the ultraviolet cutoff is determined by the electronic configuration, it is reasonable to expect that the cutoff for adamantane should be close to the cutoff of the similar structure, diamond. 60 Platt has shown that the position of the cutoff of aliphatic compounds moves to longer wavelengths as the branching around the C-C bond increases (see Table 10). The highest branched structure is diamond in which each C-C bond is surrounded by 6 branches. From Platt's data, one expects such a substance to cutoff near 1800 A. U. However, the cutoff is not sharp in all substances and can be expected to move to longer wavelengths as the concentration increases. Similarly, one expects adamantane to cut off at some wavelength longer than 1800 A. U. The cutoff of adamantane was determined by making transmission measurements on a solution of the substance in either

72 Table 10 VARIATION OF ULTRAVIOLET CUTOFF WITH BRANCHING Number of TransaBranches mission Around Limit C-C Bond..Type. Example(A. U) 15500 -C-C Ethane 15601 -C-C-C- Propane Not Measured 2 -C-C-C-C- n-pentane 17001730 C 1 17303 -C-C-C-C- 3-Methylhexane 1740 C C 1 4 -C-C-C-C- 2,3-dimethylhexane 1770 CC 1 5 -C-C-C-C- 2,2,3-trimethylpentane 1785 1 C C C 1 1 6 -C-C-C-C- 2,2,3,3-tetramethylpentane 1795 1 1 C C 60 Platt's Data

73 cyclohexane or n-hexane. The cells used were either lcm in thickness made of fused quartz or 0.1 mm in thickness made by using a lead shim between two fluorite windows. All transmission measurements were made with a Beckmann Model DU spectrophotometer. 61 In agreement with Potts we found that the purest solvents available from commercial suppliers transmitted feebly from 2500 to 2000 A. U. By shaking the solvents with sulfuric acid, the contaminants could be reduced so that transmission measurements could be made to 2000 A. U. The transmission of adamantane is shown in Figure 22. The concentration was measured for only one case, where it was found to be about 0.5 mg/cc. The cell thickness of 1 cm with such a concentration of adamantane is equivalent to -4 about 10 cm of diamond. Since the measured concentration of 0.5 mg/cc in a one cm path produces a long gradual absorption which transmits only 10% at 2300 A. U., one is led to the conclusion that either the sample of adamantane was contaminated or the absorption of adamantane is different from that of diamond which has a sharp cutoff (see Figure 3). Attempts to purify the solutions of adamantane did not alter the absorption. Another characteristic of the absorption spectrum of adamantane is the shoulder near 2250 A. U. In higher concentration, the shoulder could represent the beginning of total absorption. Since the shoulder lies close to the cutoff of UV II diamonds, it may be characteristic of the similar electronic configurations in adamantane and diamond.

100 1 1 1 (2) 4-) ( 60 ^^^'^ ^. concentration of approx.... ig ori 0.Equivalent to 10 cm of diamond. Solvent: Cyclohexane 40.... —..... concentration of approx. 0.5 2000 2200 2400 2600 2800 (A.U.) Fig. 22 THE ULTRAVIOLET ABSORPTION SPECTRUMI OF ADAMANTANE

75 2.2.6 The Absorption of Diamond in the Vacuum Ultraviolet The position of the ultraviolet cutoff is generally assumed to be the long wavelength edge of a broad continuum extending into the X-ray region. However, until recently, calculations of the electronic structure predicted a cutoff nearer 1500 A. U. In order to determine whether or not any region of transmission exists between 1200 A. U. and the cutoff of diamonds, we used the fluorite spectrograph described on page 41, The diamonds which were examined included 5 S IR I, 1 W IR I, and 2 IR II. The diamonds are listed in Appendix B together with their IR and UV characteristics. None of these diamonds showed measurable transmission at wavelengths shorter than the position of its cutoff (between 2250 and 3200 A. U., depending on the sample). We conclude that the position of the ultraviolet cutoff is a true absorption band. We discuss the recent theoretical work concerning this point in section 3.2. 2.2.7 Bombardment Experiments 52 In 1905, Crookes reported that the color of diamonds could be altered. A diamond exposed to cathode rays, in vacuum, for one year turned black on its surface. Heating to 600~ C restored the original color. Exposing a diamond to radium radiations for one year turned it blue. This color 63 could not be removed. In 1942, Cork reported that bombardment of diamonds by 10 Mev deutrons turned their surfaces green. This color could be removed by heating. Later, 64 Blackwell and Sutherland, in an attempt to test their impurity theory by introducing defects in diamonds, placed

76 diamonds in the Harwell pile. Several weeks of exposure to the pile radiations (principally fast neutrons) caused the diamonds to turn black (presumably a volume effect). No infrared energy was transmitted by the bombarded diamonds, so that the effect of bombardment on 8V absorption could not 65 be determined. Grenville-Wells performed similar experiments in which the diamonds were bombarded for as much as a month. In this case, the diamonds turned black, but it was still possible to observe the X-ray diffraction pattern. No effect was found on the intensity of the extra streaks. She found similar effects in color change on heating diamonds in 66 vacuum. In this case, the formation of graphite can be detected by X-ray. The similar effects in experiments on bombardment and heat treatment lead to the conclusion that the diamonds are changed to graphite locally. Since, in the previous experiments, pile irradiations had been carried on for several weeks before observation of the effects was made, it was considered worthwhile to examine the effects in the infrared before the diamond is rendered opaque to infrared radiation. Four diamonds, GM39 (S IR I), GM40 (S IR I), GM74 (IR II), and GM75 (W IR I), were sent to Oak Ridge for insertion in the pile. After six hours in the pile, all four diamonds showed a slight blackening, but they were still transparent in the visible and showed no changes in their infrared spectra. Additional bombardment for 24 hours rendered all four diamonds nearly opaque in the visible (a faint amount of red light was transmitted). By

77 reflected light the diamonds appeared black. In the infrared, no changes in the spectrum could be detected. We also placed two diamonds in the University of Michigan cyclotron where they were bombarded with 10 Mev deuteronas T37 (IR II) was bombarded for a total of 100 microampere minutes. It turned pale green on its surface, but no change occurred in its infrared absorption spectrum. K3 (IR I) was bombarded for a total of 250 microampere minutes. It turned pale brown on its surface, and again no change occurred in its infrared spectrum. By examining the edge of T37 under a microscope, the depth of penetration of the color change was measured to be approximately 0.2 mm. This corresponds to the range of 10 Mev deuteronin diamond as calculated from 67 curves given by Cork. For carbon, his curves indicate a range of approximately 90 mg/cm2. The density of diamond is 3.51 g/cc. Consequently, the calculated range is 0.25 mm. Since the penetrating power of fast neutrons is higher than that of deuteron, and since the flux density in the pile is higher than that in the cyclotron, it is not likely that enough defects could be introduced by bombardment with deuterons to produce changes in the infrared spectrum which were not produced by neutron bombardment. Although we have made no measurements of the ultraviolet absorption spectrum of the bombarded diamonds, the visible absorption is sufficient evidence that the electronic structure has been altered on a large scale by neutron bombardment. Since no comparable effect in the infrared spectrum is pro

78 duced by short periods of neutron bombardment, while long periods of neutrom bombardment evidently materially change the character of the local structure by turning it to graphite, we conclude that neutron bombardment of diamonds introduces defects different from those causing infrared absorption at 8C. Since vacant sites are the most probable defects introduced by bombardment, we deduce that a random distribution of vacant sites is not responsible for the anomalous absorption at 8~.. 3 Summary In our experimental work on diamonds we found an absorption band at 6.5k and another at 30, neither of which has been reported previously. While the 6.5p band appeared only in IR I diamonds, it was not associated uniquely with other IR I bands. The 135 and 30Ji were shown to be Group B bands, and the 211 band to be a Group A band. We verified the fact that the bands between 2.5 and 6.O0 occur with the same absorption coefficients in all diamonds. The variation in ultraviolet absorption (. 2537 A. U.) in diamond M4 correlated with the variation in the absorption coefficient of the 7.8V band, We found no dichroism in the infrared at those points which show birefringence in the visible. Between 33 and 100 there was no absorption, and the reflection loss proved to be essentially equal to that between 1 and 2.5k. Bombardment by neutrons and deuterons did not alter the infrared spectra of diamonds, although marked changes occurred in the visible spectrum. From these experiments, we concluded that vacant

79 sites are not the defects causing anomalous absorption in the infrared. A comparison of the spectra of powdered SiO2 and powdered Si indicated that the 91 band in silicon may be due to SiO2. 2' A comparison of the spectra of 3 pure Ge samples and 2 impure Ge samples showed that the absorption coefficients were the same for all samples at wavelengths shorter than 30p, but that the impure samples absorb more strongly at wavelengths larger than 30p. We found that adamantane absorbs strongly near the position of the ultraviolet cutoff of UV II diamonds (2250 A. U.). No diamond tested showed transmission between 1200 A. U. and the position of the ultraviolet cutoff, proving that the cutoff is the long wavelength edge of a continuum.

Chapter 3 REVIEW OF PREVIOUS THEORETICAL WORK Although a rigorous theoretical treatment of the physical phenomena which display anomalies in diamond has not been made, these are in existence well developed theories which have been successful in explaining similar phenomena in other substances. In addition, when it is said that some properties of diamonds display anomalies, it is implied that a normal behavior can be predicted. Finally, there are two theories, that proposed 8 11 by Raman, and that proposed by Blackwell and Sutherland, which try to account for the observed anomalies. In this section, we will present portions of the general theory which has been developed for several of the physical phenomena in which we are interested. From this theory, the normal behavior for an ideal diamond will be inferred, whenever possible. In those cases, where anomalous properties in other substances parallel the anomalous properties of diamond, the pertinent facts will be discussed. Finally, with this background of accepted theory, the proposed theories of Raman and of Sutherland and Blackwell will be discussed. 3.1 Infrared Absorption To explain the infrared absorption spectrum of diamonds there are two questions which must be answered. (1) What 80

81 is the absorption spectrum of an ideal diamond having the Bragg structure? (2) Wlhat causes the variation in the observed spectrum? The absorption spectrum in the infrared must arise from transitions to vibrational levels. Consequently, we must investigate the vibrational spectrum of the ideal diamond lattice. We will digress slightly to review the development of the theory of the vibrational spectrum of the crystalline lattice. As is well-known, the temperature variation of the specific heat of crystalline solids is a function of the frequency distribution of the normal vibrations of the solid. Historically, it was because some substances, (principally diamond), displayed a molar specific heat considerably lower than the classical Dulong and Petit value of 6 calories / mole0 K that the problem of the temperature coefficient of the specific heat received 68 attention. Originally, Einstein applied the concept of the Planck quantized harmonic oscillator by assuming that, to first approximation, the atoms of a monatomic crystal do not interact and that they all vibrate with the single frequency, yE, characteristic of the vibration about their equilibrium position. This approximation accounted for many of the observed features of the variation of specific heat with temperature. The total energy of the crystal is now represented as the thermal average of 3N oscillators of frequency vE. Since the average energy of a Planck oscillator at temperature T is E - he eh E/kT-l the total energy of the crystal is E = 3RT E7l where e~E/T-1

82 f h EAE/K and has the dimensions of temperature, The corresponding specific heat varies as C,= 3R (E/T) e^-/T at temperature low with respect to &E* Experimentally the specific heat at low temperatures does not approach zero exponentially, but varies more slowly with T. Consequently, while the Einstein approximation fits high temperature specific heat data, it fails at low temperatures, 69 Nernst and Lindemann made a modification of the Einstein equation. They obtained considerably better agreement with experiment by introducing a second term, 0/2, i. e. E = 3RT.1/2 /T + G/2T. However, at low e6/T- 1 eU/2T - 1 temperatures the predicted specific heat varies exponentially with temperature. The next major step in calculating the frequency distribu70 tion was performed by Debye, The Debye theory treats the crystalline solid as an elastic continuum with the provision that elastic waves shorter in wavelength than the atomic spacing are not supported by the lattice. Consequently, the Debye spectrum is characteristic of a continuum except that there is a high frequency limit. It is further assumed that the crystal is isotropic and that there is no dispersion of the elastic waves. The Debye spectrum is composed of modes whose density increases as the square of the frequency, reaching a maximum at the high frequency limit. The position of the limit is established so that there be only 3N modes. For the energy of the crystal Debye obtained E 9T (T/ i)5 JD/T9 3dx where eD - ^,x h /k j eal

83 1D is the upper frequency limit and D is called the Debye temperature. At low temperatures Cv X464 (T/9D)3 cal/mol~ K. The Debye results fit experimental data over a wide range of temperatures. In particular the low temperature values of specific heat fit the T3 variation more closly than the exponential variation. Because the Debye function for specific heat can be tabulated as molar C vs &D/T, it is a simple matter to calculate the effective OD for any substance at a given temperature when the specific heat at that temperature is known. Consequently, specific heat data are frequently represented as a plot of effective OD vs T. Deviations from constant 0D represent deviations of the solid from the ideal Debye solid. It is important to recognize that even the very successful Debye theory is only an approximation to the true frequency spectrum of crystalline solids. Through the work of Born and 71 von-Karman' a rigorous treatment of the calculation of the frequency spectrum is possible, at least in principle. Born pointed out that the problem involved is essentially a classical problem of normal vibrations. If one assumes a potential for the forces binding the atoms together, then the equations of motion and the eigenfrequencies for small vibrations can be calculated. We shall present the detail of this theory in a later section. Born calculated the frequency distribution for simple linear lattices. From his calculations, it was apparent that the principal errors in the Debye approximation are (1) the assumption that there is no dispersion of the elastic waves, (2) the neglect of the effects due to having

84 more than one atom per unit cell, Nevertheless, the Born theory predicted the same temperature variation of specific heat at very low temperatures as the Debye theory. Consequently, very little interest was displayed in making the laborous computations for the Born method until significant variations from 72 the Debye approximation were found experimentally. 73 In a series of papers, Blackman applied the Born theory to several simple lattices,in particular, the simple cubic lattice. While it was necessary to make assumptions concerning the forces between atoms, he found that certain general features of the frequency distribution were present regardless of the nature of the assumed forces. In particular, while there is a strong maximum near the upper frequency limit of the distribution, there is also a weaker secondary maximum at some lower frequency. In Figure 23 we show the relation between Blackmants results and those of the Debye theory. Blackman showed that the effect of the secondary maximum is to cause a minimum in the 0 vs T curve. Following Blackman, Fine74 calculated the frequency distribution of the normal modes of a face 75 centered cubic lattice and Leighton calculated the distribution for a body-centered cubic lattices In each calculation, special assumptions must be made concerning the nature of the forces between atoms, For real crystals three calculations have appeared in the 76 literature: (1) NaC1 by Kellerman, (2) Diamond by H. M. J. 14 77 Smith, (3) Germanium by Hsieh, We will discuss these calculations in a later section. In the calculations for

85 *4~ ~~~~~~~~~~~~~1~ ~ / / I 20 After Debye B. A fter Blackman I, /^^ I oi! 1 / I 0 2 6 8 10 Frequency A. After Debye B, After Blackman Fig. 23 COMPARISON OF DEBYE AND BLACKMAN FREQUENCY DISTRIBUTIONS

86 real crystals, only the elastic constants must be known in order to calculate the frequency distribution, if the assumed force field does not contain more constants than there are elastic constants. Consequently, when one uses the Born theory, one calculates the frequency spectrum, the specific heat, and related phenomena such as spectral absorption without recourse to experimental data other than the elastic constants. For a cubic crystal three parameters, which can be directly measured, determine the calculations. When reviewing the status of the Born theory, which is based on firm classical theory and which Born has shown fits the requirements of quantum theory also, it appears that the problem of the frequency spectrum of the crystalline solid 7 has received a satisfactory solution. However, Raman has not agreed with the Born theory and has chosen to formulate a new treatment of the frequency spectrum. Raman's theory is not supported by basic reasoning, but it is based on Raman's interpretation of the observations he has made experimentally. Before describing Raman's theory of lattice dynamics, we must point out that the Raman theory of the anomalies in the properties of diamond is a separate theory which stands apart from the theory under discussion. Raman arbitrarily divides crystal vibrations into two classes: (1) Those which are on "...a large scale and may be described without any reference to the fine structure of the solid. These are the elastic vibrations...". (2) Those which are "essentially dependent on the fine structure of the solid." According to Raman,

87 the first class is a continuous distribution of frequencies while the second class is made of "discrete and enumerable monochromatic frequencies in the infrared region of the spectrum." No proof of the preceding statements exists. In addition, Raman originally discarded the contribution of the first class to the specific heat. Here he made an error which could be shown experimentally. Since the second class can give only a sum of Einstein contributions to the specific heat, the low temperature specific heat must go to zero exponentially according to the Raman theory. Recently, however, in order to explain the specific heat of diamond, a coworker 78 of Raman has used 3 Debye functions and 9 Einstein functions. With 12 adjustable parameters, it is not surprising that the 79 specific heat curve can be matched. Blackman has compared predictions by Raman with those according to the Born theory for the specific heats of crystalline solids and shows that the Born results, which require no adjustable parameters, 80 are superior to Raman's results. MacDonald reviewed the Raman theory and concluded that the Raman theory is no more than an extension of the Nernst-Lindemann theory which achieves success through the introduction of many adjustable parameters. Let us now consider the absorption spectrum of a crystalline solid. Using Born's dynamics one predicts that the only active frequencies are those corresponding to modes which are represented by elastic waves whose wavelength is comparable to the 81 wavelength of light. In terms of atomic dimensions, the wavelengths of such elastic waves are essentially infinite,

88 Consequently, one finds that the absorption spectrum should consist of a few sharp lines whose corresponding modes involve the vibrations of atoms in the unit cell against one another. Corresponding atoms in all unit cells vibrate in phase. For all other wavelengths, the dipole moment change will cancel to zero and no absorption occurs. Similar statements hold 82 for Raman scattering. Raman has not recognized that the Born distribution has only a few allowed frequencies in absorption as well as in Raman scattering, and claims that his theory of lattice dynamics is born out by the fact that the first order Raman scattering spectra of many crystals are composed of discrete lines. On the other hand, when the overtone and combination spectra of crystals are considered Raman has difficulty in explaining the occurrence of broad bands. Since the Born spectrum is expected to have broad overtone and combination bands, a final blow is dealt Raman's theory of lattice dynamics. We shall discuss the selection rules in greater detail in a later section. However, it is now clear that we need only consider the Born theory of lattice dynamics. We can now answer the first question: "Vhat is the absorption spectrum of an ideal diamond?" It must consist of no more than discrete lines in the fundamental portion of the spectrum and should consist of some broad combination bands. We will show later that no discrete lines are allowed in the fundamental region due to symmetry selection rules. In any case, it is clear that the appearance of the broad IR I bands at wavelengths longer that 7p is in violation of the general

89 selection rules for any crystalline solid. Therefore, we can conclude that IR II diamonds display the absorption spectrum of an ideal diamond. IR I diamonds must depart, in some manner, from the Bragg structure. It is in connection with the variation of the IR I diamonds from ideal behavior that the theories of Raman and of Sutherland 8 and Blackwell have been proposed. Raman claims, without analytical support, that there are four possible configurations for the electronic structure around individual carbon atoms. His diagrams for these configurations are shown in Figure 24. It can be seen from the figure that the structures differ from one another only in the directions indicated by the arrows lying along the tetrahedral bonds. What the nature of the forces may be that support the atoms in the Td I, Td II, and OH II structures has never been explained. In addition, the electronic density derived from X-ray measurements is consistent only with OH I, i. e. the Bragg structure as represented in Figure 1. As with his lattice dynamics, the basic tenets of the Raman theory for diamond are not well-founded in theory but are only qualitative ideas. Since, in Raman's theory, there are four possible electronic configurations, varying mixtures of the four configurations are supposed to produce the observed variations in properties. In the infrared, the structures with no center of symmetry (Td) account for IR I bands while the structures with center of symmetry (Oh) account for the absence of these bands in IR II diamonds. It is assumed that all vibrations in the Oh structure are automatically centrosymmetric and those

I I TD I TD II ~~ 0 I I a I,_-.4 —s *+ —-* —*OH I OH II Fig. 24 THE FOUR POSSIBLE ELECTRONIC STRUCTURES OF DIAMOND (AFTER RAMAN)

91 in the Td structure are non-centrosymmetric. This theory is extended by the addition of Rarman's lattice dynamics. The combination of his two theories leads Raman to the conclusion that the broad absorption from 7 to 12L is due to the "resonance 83 interaction" of nine discrete frequencies. The band at 7.8, is supposed to have the same origin as the band at 7.51l 84 (1332 cm-1) in the Raman spectrum. As Sutherland has pointed out, it is not explained how the interaction can affect the energy levels involved when observing the bands in absorption while it does not affect them when observing the band in scattering, Without further ado, we state that there is no support on any theoreticalg rounds for the Raman theory for diamond. Another theory for the anomalies in the properties of dia11 monds is that proposed by Sutherland and Blackwell. They proposed that defects in the lattice structure cause the anomlies. This theory did not receive any theoretical justification when it was proposed. It was simply stated that the presence of impurities or structural defects could cause a breakdown in the local symmetry of the structure and hence break down selection rules which forbid the fundamentals, 10 Blackwell pointed out that the Born vibrational spectrum of 14 diamond, as calculated by Smithr was very similar to the observed absorption spectra of IR I diamonds. Since these bands do not appear in IR II diamonds, they must be normally forbidden. Hence some structural anomaly must occur in IR I diamonds which breaks down the selection rules. In a later section, we will amplify this theory and attempt to give it theoretical justification.

92 3.2 Ultraviolet Cutoff Clearly, the onset of absorption whether at 3200 A. U. or 2250 A. U. is associated with electronic transitions. Since diamond is an insulator we may expect that the electronic zone structure can be represented in the band approximation as a filled band separated by a relatively wide gap from an empty conduction band. In this respect, it differs from silicon and germanium only in the width of the gap separating the filled and conduction bands. These qualitative statements are supported by several calculations of the electronic structure of diamond. The first such calculation was made by 85 Kimball. He found that the eight electrons per unit cell just fill the lower zone system which exists at the observed interatomic distance. He calculated the gap between the 86 filled and unfilled bands to be 7ev (1700 A. U.), Pauling 87 and Slater discussed the form of the Heitler - London function appropriate to valence crystals such as diamond. Pauling showed that the particular linear combination of one s function and 3p functions which are orthogonal and which has maximum directional localization have maximum electron density along the tetrahedral direction, i. e. along the directions connecting nearest neighbors in the Bragg structure, 29 The calculations of electron density by Coulson from X-ray 88 data confirm the Pauling and Slater view. In 1951, Hall made a calculation of the electronic structure of diamond based on this valency description of the crystal. He obtained substantially the same energy levels as those calculated by

93 12 Kimball. In 1952, Herman made a new calculation of the electronic structure of diamond using more refined methods. He found a separation of the filled and unfilled levels equal to approximately 6ev which corresponds to a wavelength of about 2050 A. U. This value is close to the observed cutoff of UV II diamonds at 2250 A. U. There can be little doubt that the ultraviolet cutoff in UV II diamonds corresponds to transitions from the filled valency bands to the unfilled conduction bands. Now we must consider the fact that many diamonds do not transmit to as short wavelengths as 2250 A. U, Some alteration in the electronic structure must occur in UV I diamonds, In the zone structure, the longer wavelength absorption implies the existence of levels lying in the forbidden region between the filled and unfilled bands. The role of such intermediate levels in determining the electric properties of silicon and 89 germanium is well-known in the theory of semiconductors. In general, these levels are introduced by the presence of impurities, vacant sites, or structural imperfection, We can be confident that the explanation for the ultraviolet absorption in UV I diamonds lies in this same field. The theory of Blackwell and Sutherland is entirely consistent with the viewpoint that structural anomalies alter the electronic structure in UV II diamonds. The connection between the infrared and ultraviolet absorption lies in the fact that both phenomena depend on the periodicity of the lattice. In the collective 90 electron treatment each electron is described by a wave function extending throughout the lattice, and it is the

94 periodic field of the lattice which leads to a separation of the permissible states into zones. Deviations from the ideal periodic structure alter the field of the lattice and hence alter the zone structure. Similarly, in Born's lattice dynamics the elastic waves which represent the normal modes of the lattice are plane waves only so long as the lattice has its ideal periodicity. Deviations from the periodicity alter the forms of the normal modes and hence alter the phase relationships between unit cells, 3.3 X-ray In the review of experimental work we have pointed out that there is no explanation for the occurrence of secondary extra spots in X I diamonds. Therefore, we shall not attempt to review the various existing theories involved in the associated X-ray phenomena. It is certain, however, that current theories of X-ray diffraction do not predict the extra spots in any crystalline material regardless of its symmetry. Consequently, it is safe to assume that X I diamonds depart from the ideal Bragg structure in some manner. However, since the principal features of the diffraction pattern of X I diamonds do fit the Bragg structure, and since there is a marked regularity in the position of the extra spots, any departure from the ideal Bragg structure must not change the gross structure over wide regions of the crystal, and it must have some definite orientation with respect to the axes of the crystals

95 3.4 Blue Fluorescence and Absorption The electronic transitions associated with absorption and fluorescence near 4155 A. U. in diamond are similar to those observed in ionic crystals where the transitions are associated with energy levels introduced by imperfections such as vacant sites or impurity atoms. However, it is not certain what the 37 origin of the levels in diamond may be. Raman argues that 91 the fluorescence is fundamental to the lattice while Bishui and others argue that the fluorescence is dependent upon structural imperfections. We shall not attempt to present the arguments in detail. However, since blue fluoresence and absorption occur with variable intensity and are sometimes absent, it is certain that if the levels involved in the transitions are fundamental to the lattice, the transitions must be normally forbidden, and they occur when some modification of the structure occurs which breaks down the selection rules. Consequently, it is surely valid to assume that blue fluorescence and absorption are characteristic of a lattice imperfection. 3.5 Other Properties (3.5.1 Conductivity) (3.5.2 Particle Counting) (3.5.3 Thermal Conductivity) 3.5.1 Photoconductivity The property of photoconductivity is intimately related 1 to the electronic structure. It is known that while UV II diamonds normally conduct electricity when exposed to ultra

96 violet light, UV I diamonds exhibit much lower conductivity under similar conditions. This means that in UV II diamonds, absorption of light quanta can raise electrons to the conduction band, and while some of these electrons are trapped or scattered, many migrate to the anode. The problem is reduced to considering (1) from which levels are the electrons initially excited by the light quanta, (2) the nature of the electron traps. In our discussion of the ultraviolet cutoff, we have established that UV II diamonds are essentially ideal in their electronic structure. This implies that the photoconduction electrons in UV II diamonds are excited from the tail of the fundamental absorption band, i. e. the edge of the filled band, to the conduction band. If the structure is ideal, only lattice scattering of the electrons deters their movement through the solid. However, the absorption linesbetween 2250 A. U. and 2500 A. U. in many UV II diamonds indicate that there are energy levels in the forbidden zone near the conduction band. These levels may be associated with electron traps. However, the density of these levels in UV II diamonds must be low compared to UV I diamonds. In UV I diamonds there are many levels (4155 A. U. to 2250 A. U.) in the forbidden zone. These levels are associated with imperfections some of which may serve as electron traps. Consequently, the photoconductivity 90 of UV I is very small. As Seitz points out, excitations from wavelengths well within the absorption band cannot be expected since the reflectivity will be high and few quanta will penetrate into the body of the solid. Similarly, in UV I diamonds, few quanta of sufficient energy to raise electrons

97 from the edge of the filled band to the conduction band will penetrate the solid, since the absorption coefficient (and hence the reflectivity) is large in this wavelength region. 3.5.2 Particle Counting The variation of the ability of diamonds to act as particle counters has recently received new qualitative theo92 retical consideration. Ahearn has pointed out that the counting properties of the diamonds can be related to the type and configuration of the imperfections. He proposes that the imperfections may occur in clusters so that a diamond may be composed of conductive channels next to insulating regions. We shall not present further details of this proposal, but while Ahearn's proposal is not rigorously proved, it shows that the variations in counting properties are consistent with a model of clusters of imperfections, the precise arrangement of which will influence the counting properties. His proposal is especially important in view of the variation of counting 93 properties within a given diamond. The fact that there appears to be no unique correlation between particle counting 30 ability and the UV and IR properties is probably related to the fact that in particle counting, the existence of many electron traps will deter the counting ability. For example, 94 fluorescent diamonds are usually poor counters. One infers that the levels associated with fluorescence at 4155 A. U. are connected with imperfections which act as electron traps. On the other hand, UV II diamonds are presumably ideal in struc

98 ture and hence contain few of Ahearn's clusters of imperfections. Consequently, a good particle counter can be expected to have UV and IR properties intermediate between our II and strong I. 3.5.3 Thermal Conductivity The experimental values for the thermal conductivity of diamond at low temperatures has been said to be consistent 95, 96 with Ahearn's proposal of clusters of imperfections. Klevens has applied his theory of the effect of particle size on thermal conductivity to show that the experimental values for diamond can be explained in terms of imperfections of the order of 20 to 60 interatomic distances. "Either the imperfections form linear arrays of length large and diameter small compared with these wavelengths or they form clusters whose diameters are of the same order as the size of the wavelengths." Since the thermal resistance depends upon both scattering by the lattice and scattering by imperfections, there is clearly a relationship between this deduction by Klevens and other properties which we have discussed, all of which depend upon lattice periodicity and departures from ideal periodicity. Unfortunately, no data are available on the variation of thermal conductivity between diamonds or its correlation to other physical properties, 3, 6 Summary We have seen that all of the important anomalies in the structure sensitive properties of diamond are consistent with

99 the existence of imperfections in the structures of UV I, IR I, and X I diamonds. In each case, the discussion has been qualitative in the sense that no specific imperfection has been proposed, and no quantitative calculations have been made. In addition, while each of the electronic phenomena such as ultraviolet cutoff, fluorescence and absorption in the visible, etc. has some counterpart in the observed phenomena in other crystals, i. e. the introduction of energy levels in the forbidden zone, the infrared absorption spectrum and the X-ray diffraction pattern have no such counterpart since only diamond shows the observed anomalies. In fact, since this work has started it has been shown that absorption bands occur in the spectra of silicon and germanium which are comparable to 81 bands in diamond in position, although their intensities appear to be constant. We shall discuss this later. But these two points remain unique, (1) Variation in intensity of absorption near 8p and (2) Occurrence of extra streaks in the X-ray pattern. No other substance is known to show these properties. In the following sections we will attempt to show how far current theory can go in explaining the positions of the bands in the infrared spectrum of diamond, and we will also show that the same theory applies to the spectra of silicon and germanium. Finally, we shall attempt to account for the anomalous variation in intensity of the 8V bands in diamond.

Chapter 4 PRESENT THEORETICAL WOORK 4.1 The Frequency Distribution of Lattice Vibrational Modes (4.1.1 Diamond) (4.1.2 Germanium) (4.1.3 Silicon) (4.1.4 Analyses of the Fundamental Branches) 4.1.1 Diamond The broad absorption bands observed in the infrared spectrum of the diamond crystal must have their origin in the vibrational modes associated with the diamond lattice. The 15 theory of the crystalline lattice as developed by Born states that the frequency distribution of the normal modes will be characterized by 3p analytically distinct functions (normally called "branches"), where p is the number of atoms per unit cell in the crystal. Diamond is made up of two interpenetrating face-centered lattices, displaced one quarter of the way along the space diagonal. Such a lattice can be generated from a unit cell containing two atoms, one belonging to each of the face-centered lattices. Consequently, for diamond, p = 2, and the frequency distribution will be made up of 6 branches. In order to calculate the form of the distribution of modes in a given branch, one must set up the so-called 100

101 97 "Dynamical Matrix". This matrix is formed in the following manner: (1) Assumptions are made as to the form of the forces between atoms. (2) The three equations of motion for each atom are calculated on the basis of classical mechanics. (3) Solutions in the form of triply periodic functions are substituted in the equations of motion. The periodicity is expressed through three phases, i. The space is equivalent to the well-known reciprocal lattice. (4) After substitution, one has 3p equations in the 3p amplitudes of the assumed solutions. One now demands that these 3p equations be satisfied simultaneously. A matrix of the coefficients of the amplitudes is formed. This is the dynamical matrix D (hi) with added terms on the diagonal, i. e. (D(4i) - a2I) where X a 2ST, = the frequency of the periodic solution. The condition that all 3p equations have a non-trivial solution requires that the determinant ID(i) - S2II vanish. This determinant is 3p square. Consequently, for each point in? space (t1, 2, (3) there are 3p corresponding values of 3. These 3p solutions are the values of the frequency for the 3p branches at that point in 4 space. The number of modes having frequencies between ) and i +I is the distribution function N(K). Each branch can be formed separately in the following manner. (1) A number of uniformly spaced points in d space are chosenr (2) The value of $ for the branch at each point is calculated using the dynamical matrix. (3) A small interval of frequency, A, is chosen. (4) The number of calculated )'s falling in each interval is plotted vs the average frequency of the interval, (5) n such counts are made, shifting the boundaries of each

102 interval by /n each time. (6) Ni( ) for the ith branch is a curve through the plotted points. Clearly, the accuracy of such a calculation is increased by increasing the number of points used ink space. We have, so far, neglected the calculation of the force constants entering into the dynamical matrix. In general, it is found that force constants corresponding to the assumed force field can be expressed in terms of elastic constants, Cij. In the case where a solid has n independent elastic constants, n independent force constants can be determined by the elastic constants. 14 In the particular case of diamond, H. M. J. Smith performed a calculation following the method outlined above, For diamond there are three independent elastic constants, 98 C1 C12, C44, which have been determined experimentally. Consequently, 3 independent force constants can be determined. Smith chose a force field which involved a completely general first neiglibr interaction, demanding only that the corresponding potential be expandable in a Taylor series in the displacements. Only terms corresponding to small displacements from equilibrium are retained. Because of the symmetry of the lattice, such a field requires only two constants for specification (Smith's a and ). Since there are three elastic constants, the field is overspecified and one identity results between the Cii. In addition, the general theory predicts that the upper limiting frequency of the total distribution N (i) will be Raman active. This frequency, )3, is observed. )R can be expressed in terms of a and hence in terms of the Cij. H'

103 Consequently, a second identity is obtained: a = unit cell side; m = mass of carbon atom. 4 C 11 (C C) 42 C2 m The measured values are: C 9.5 x 1012 dynes/cm C12 3.9 x 1012 C44 ~ 4.3 x 1012 I. = 1332 cm"l 2a = 3.56 x 10'8 cm m = 1.995 x 1023g Using these values, identity 1 gives the result 1.11 vs 1; identity 2 gives the result 0.46 vs 1. Since the second identity is so poorly satisfied, Smith added an additional force field in the calculation. This field is central between second neighbors and characterized by a single force constant "t". Using 4R, one identity can be established between the Cij. 8A (A + 8011 - 16044) Identity 3: 2 -2 2 Identity 53: ______- 1; A = 4 2 C2m 2 2a R (3A - 8C11 + 16C12) Using the listed values of the constants, identity 3 gives 99 1.4 vs. 1. On the basis of these results Born has pointed out that there are three possibilities: (1) The elastic constants are in error, (2) The second neighbor forces are not central, (3) More distant neighbors interact. If either of the latter two possibilities is true, Smith's formulation is inadequate. Since no better solution was available, Smith

104 chose to examine the elastic constants. There is one experimental check of Cll and C12 in the value of the bulk modulus, 1/K: 1/K a 1/3(Cll + 2 C12). The observed values of K 101 are 0.16 and 0.18 x 101 /megabar. The average, 0.17 x 10'6 /megabar, gives 1/K = 5.9 x 1012 dynes/cm2 The observed values of Cll and C12 give 1/K 5.77 x 1012 dynes/cm2 The discrepancy is 2% whereas the deviation of the observed bulk modulus is 6%, so that Cll and C12 must be essentially correct, On this basis, Smith calculated as follows: 1 13I2 cm1 = 1 XC =: 1332 cm: a = 0.157 x 106 dynes/cm R WTC m a + 82 = 2aCll: i = 0.0226 x 10 = 1/4 / a (C11 + 2 C12) + a-16 7: o 0.104 x 10 C44 1 Z: /- /a + 47: C44 = 5.0 x 1012 2a The observed value of C44 is 4,3 x 1012 The alteration by Smith is 14%. With the revised value of C44, identity 3 now gives the result 1.04 vs 1. This represents improved agreement over the previous result of 1.4 vs 1, In order to clarify the concepts involved in these calculations, it is worthwhile to consider the physical interpretation of the three force constants a, 6, and L. The expression for the highest frequency of the lattice, R' is = 1 a. The corresponding mode is the vibration of one face-centered lattice against the other along the bond direction. Since a simple diatomic C-C stretching mode would give ) = 1 r2K ~, it is clear that a can be interpre2r \ m

105 as a bond stretching force constant. The constant A does not, by itself, determine a mode, but it is analogous to the coefficients of the cross terms in the potential function of a polyatomic molecule. The constant "II" can be interpreted as a constant between second neighbors comparable to the force constants used in the central force field approximation 102 for polyatomic molecules since the corresponding potential is central, To this point, the only justification for altering the measured value of C44 is that the identities which follow from Smith's force field are not satisfied by the unaltered C44t Predictions based on the altered C44 can be used to test its validity. One such prediction is the variation of specific heat with temperature. This variation is determined by N (Q )d. Smith used her calculated N( ) to determine CT vs T and obtained excellent agreement with experiment. Smith also calculated a different N(o ) using the a and g given on page 101, but setting L - o. This calculation does not correspond to the true case for diamond since the values of a and 8 used were originally calculated on the assumption that p t o, However, we shall make use of both calculations in later sections. In order to avoid confusion in nomenclature we will name the calculations as follows: Calculation A: a 0.157 x 106, 0.104 x 106,' 0 0.0226 x 106 Calculation B: a = 0-157 x 106, = 0.104 x 106,.L 0 Calculation A is based on a force field which includes both

106 first and second neighbor interaction and is valid for diamond if the alteration of C44 is accepted. Calculation B is based on a force field which includes only first neighbor interaction but is not valid for diamond for the reason given above. The plots of N (a) vs ) obtained by Smith for the two calculations are shown in Figure 25 (A, B). Her data are given in Table 11 (A, B). Another check of the altered value of C44 is offered by Smith's analysis of the observed second order Raman effect, Rather than assuming that the potential includes anharmonic 102 terms, Smith assumed that there is "electrical anharmonicity." Because of this anharmonicity certain overtones and combinations are allowed. Without considering the theory in detail, it is sufficient to state that she determined that the following bands are allowed in second order Raman scattering: (1) The first overtones of all branches, (2) The sum and difference bands involving branches 1 and 2, 1 and 3, 2 and 3, 4 and 5, 4 and 6, 5 and 6. In calculations A, used here, branches 1 and 2 are degenerate. The contours of the combination and overtone distributions were obtained by numerical integration in the manner described for Ni()). In this case, one combines the two frequencies of the branches involved at each point in p space and then obtains the distribution of the combined values of 9. Smith's results are shown in Figure 26. She found that the observed spectrum could be matched if the relative intensities of the allowed branches was I(33): I(44): I(11): 1(13) 40:12:1:1;

107 1280 35 -...... CALCULATION A FIRST AND SECOND NEIGHBOR FORCES 30 - - ----. 25.' 0 2 J UO I 10 ___ _ ____ 667 ____ _ 5 I ) ~< p N I tg i',,i 0 200 400 600 00 1000 1200 1400 (c"m-) Fig. 25A THE FREQUENCY DISTRIBUTION OF VIBRATIONAL MODES IN DIAMOND (AFTER SMITH)

108 35 24e 30 LL CALCULATION B FIRST NEIGHBOR FORCES ONLY 25 - I.,467 \\~~ ~~~~~~~~i I _ -/J.,f/ I,,///'y-Its 0 ----—... -— [,. 0 200, 400 600 800 1000 1200 1400 (cm'1) Fig. 25B THE FRUQUEUCY DISTRIBUTION OF VIBRATIONAL MODES IN DIAMOND (AFTIER S1IITH)

L a C — ach 11~~~4 w.-rib 1C-r h2CJlra~'b. 1,-7Db __________.^r:nch Niiunber __ Pr z co 2~ 5W4 W5 " ^;__ y ""'""l ^__ _ _ _ _ _ _ _^ - -^ 8 4 0 2.51 2.51 2.12 2.12 1.48 1.48 81 2 2 2.48 2.48 2.27 2.12 1.53 1.30 8 2 0 2.51 2.51 2.17 2.17 1.44z 1.44 3 0 0 2.48 2.48 2.23 2.23 1.40 1.40 7 3 1 2.50 2.50 2.24 2.07 1.55 1.33 7 1 1 2.48 2,48 2.30 2.06 1.43 1.35 6 6 0 2.48 2.48- 2.27 2.12 1.535 1.30 4 2 2, 51 2.51 2.28 2.07 1.35 1.16 6 4 0 2.51 2.51 2.54 2.01 1.47 1.25 O 2 2 2.49 2.49 2.33 1.96 1.37 1.21 G 2 0 2.51 2.51 2.40 1.90 1.34 1.23 0 0 2.49 2.49 2.43 1.97 1.28 1.28 5 5 1 2.49 2.49 2.27 2.00 1.46 1.15 3 3 2.49 2.49 2.32 2.06 1.13 1.06 5 3 1 2.51 2.51 2.35 1.88 1.30 1.05 5 1 1 2.50 2.50 2.42 1.68 1.17 1.14 4 4 4 2.50 2.50 2.34 2.11 0.99 0.99 4 4 2 2.50 2.50 2.38 1.96 J1.14 0.99 4 4 0 2.50 2.50 2.44 1.78 1.38 0.99 4 2 2 2.50 2.50 2.49 1.71 0.98 0.98 4 2 0 2.51 2.51 2.49 1.54 1.09 0.88 4 0 0 2.51 2.51 2.51 1.35 0.97 0.97 3 3 3 2.50 2.50 2.45 1.84 0.91 0.91 3 3 1 2.50 2.50 2.43 1.55 1.01 0.s80 5 1 1 2.51 2.51 2.51 1.21 0.77 0.74 2 2 2. 2.51 2.51 2.51 1,34 0.69 0.69 2 2 0 2.51 2.51 2.51 1.07 0.73 0.54 2 0 0 2.51 2.51 2.51 0.71 0.52 0.52 1 1 1 2.51 2.51 2.51 0.71 0.37 0.37 I. K. J. Smith's Date Units of co are 1014 sec1

. able 17 1i F";-Jt's':.-; OF@ _-'' " L V TIO]f;.F;:Z-F::';.l Tni }-;- -LTIO O F.-.I RSl Tt7.EI3).O.i.;CS ONLY H ec ipi':? t.. 1 Lattic PoiNunt _mbr 2 n ch Numb r Px P P 1 2 3 4 5 6 8 4 0 2.29 2*29 I K78 1.78 1.03 1. 03 8 2 2 2.32 2 27 ].84 1.71 1.06 0.95 8 2 2. 29 2.29 9 1.78.8 1.05 1.03 03 O 0 2.2 9,29 1.78 1.78 8 103 1.03 7 3 1 2.33 2 19 1.66 1.05 0.95 7 1 1 2.31 2 30 1. 93 1.6 1.02 0. 99 6 6 0 2.32 2.27 1.84 1.71 1.06 0.95 6 4 2 2.36 2 33 1.92 1.62 0.94 0. 85 6 4 0 2.35 2.29 1.94 1.59 1.03 0.90 6 2 2 2.35 2.33 1.98 1.54 0.94 0.88 6 2 0 2.5 2.32 2.04 1.46 0.97 0.91 3 0 0 2.33 2533 2.09 1.40 0.94 0.94 5 5 1 2.37 2.31 1.94 1.60 0.98 0.85 5 3 3 2.39 2.38 1.95 1.58 0.80 0.78 5 3 1 2.38 2.35 2.07 1.42 0.89 0.79 5 1 1 2 37 2.37 2.18 1.25 0.83 0.82 4 4 4 2.40 2.40 1.91 1.63 0.73 0.73 4 4 2 2.40 2.38 2.02 1.49 0.81 0.73 4 4 0 2.40 2.34 2.11 1.37 0.90 0.73 4 2 2 2.41 2.41 2.17 1.26 0.71 0. 69 4 2 0 2.43 2.38 2.25 1.11 O.79 0.65 4 0 0 2.41 2.41 2.32 0.96 0.70 0.70 3 3 3 2.42 2.42 2.11 1.36 0.67 0.67 3 3 1 2.44 2.41 2.24 1.13 0.70 0.59 3 1 1 2.46 2.45 2 36 0.86 0.55 0.52 2 2 2 2.46 2.46 2.33 0.95 0.50 0.50 2 2 0 2.48 2.46 2.40 0.75 0.50 0.39 2 0 0 2.48 2.48 2.46 0.49 0.37 0.37 1 1 1 2.50 2.50 2.46 0.48 0.27 0.27 H. M. J. Smith's Data Units of14 -1 Units of ) are 10 sec.,

I1 Calculated, allowed combinations and overtones o 19 ho~~~~~~~~~~~1 0 0 400 800 1200 1600 2000 2400 28 0 (cm-l) (2W1 2. 2J3 5. 2(&6 7. +(4+(, 10.1-J3 12. 3-CJ6 1.l2 3. 4 6 (W1+. 4+ - 42 ^lt+u2 4* 5 12+W3 9. 5+(g 11.t- t) 13 -* - Match of Observed Bands by Selected Calculated Bands Ii a / 1700 1900 2100 2300 2500 2 00 (cmn1) A. Calculated B. Observed Fig. 26 THE SECOND ORDER RAMAN SCATTERING OF DIAMOND (AFTER SMITH)

112 all other branches having zero intensity. Smrith1s c.lulfted positions for the maxima of individcual combinations and overtones are given below. Liaxima in thte observed Raman spectrum occur at 2460 cm'1 and 2176 cm1. Table 12 CALCULA.TED MAXIMA ALLO JWLD II R PlVIAAN SCATITERI-NG Combination MIaximum Com-bination IMaximum Combination i.Maximum ~2oj 1, + o) co1 - c.,3 2602 3 2522 106 2w 0 3 4 co ~~2 (2+ (o3 L2't3 2o3 2469 ^ + o5 1805 o4 - o5 345 2o4 2177 +o4 6 1683 o4 - o, 409 2to5 1487 )5 + 06 1434 35 - o: 133 2co6 1258 The important question is whether or not Smith's calculation of the second order Raman effect is an adequate test of her calculated frequency distribution. Out of thirteen allowed bands only four are used to determine the contours of the observed bands. In addition, no allowance has been made for the effect of anharmonicity in the potential on the frequencies of combinations and overtones. Finally, there are several branches whose maxima fall in the region of the maximum of the second order Raman scattering.'iWe conclude that while Smith's method of interpreting the observed spectrum is correct, in principal, it does not afford

113 a rigorous test of the precise positions of the individual frequencies. 4.1.2 Germanium The methods already discussed also apply to the lattice of germanium since it has the same crystal structure as diamond. By the introduction of the measured elastic constants, the frequency distribution for germanium can be obtained. A 77 calculation has been made by Hsieh in which he considers only first neighbor interaction. His result for N( ) is shown in Figure 27. We will now examine the identities calculated by Smith to determine the validity of Hsieh's assumption that second neighbor interaction is negligible. The elastic constants 103 used are those recently measured by W. L. Bond and differ only slightly from Hsiehts values. Our Calculation Hsiehts Calculation 011 ~ 1.298 x 1012 dynes/cm 1.29 x 1012 dynes/cm C01 0.488 x 1012 0.48 x 1012 012 0,488 x 10I 44 0.673 x 1012 0.67 x 1012 m = 72.6 atomic units 2a = 5.62 x 10-8 cm With these constants identity 1 gives 1.017 vs 1; identity 2 gives R' 369 cml1. Because is not known for germanium, the second identity gives a numberical value for R vwhich has no direct experimental check. If we include second neighbor interaction for germanium, identity 3 gives )p = 396 cm-1. R

114 WI, 30 i - ----- Ct 25. — co.o 20.......... 15 0 4(o --.6- 03) 10 - A/ IW L~234 0 50 100 150 200 250 300 350 400 (cm-1) Fig. 27 THE FREQUENCY DISTaIBUTION OF VIBRATIOIP.AL.MODES IN ilER ANI T. (AFTER HSIEH)

115 We will now calculate the force constants which are determined by the measured elastic constants of germanium. We also make a similar calculation for diamond in which elastic constants (using Smith's altered C44) are used. This calculation differs from Smith's calculation B in that the measured value of R is not included as a parameter. In the following table those quantities with subscript "1" are calculated assumling first neighbor interaction alone and the listed values of the elastic constants are the only quantities used in determining the values. We shall call this "Calculation C". The quantities labelled with subscript "2" are calculated assuming both first and second neighbor Interaction. For germanium the calculation depends only on the measured elastic constants, while for diamond, since we use the altered C44, the calculation is equivalent to Smith's calculation A. Table 13 ELASTIC AND FORCE CONSTANTS FOR DIAMOND AND GERMANIUM 11 12 44 al 1 1 Germanium 1.298 0.488 0.673.0725.0497 Diamond 9.5 3.9 (5.0).338.239 Diamond Ratio Germanium 7.32 7.99 7.43 4.66 4.81 a2?2 _ 2 PR Rg Germanium.0839.0582 -.0014 369 396 Diamond.157.104 +.0226 1951 1332 Diamond Ratio Germanium 1.87 1.79 -16.14 5.30 3.36 -- -—'",- ":i ~,,-~-:::'':'. -''.~:-='-:-=:. =~=:":''"..'='':':''": -;":..=:-':,,,'"...:' "-=- - --,,~ =: -'.-= -=j2'U'-~; —=u= "-,,,,_ -_ r ____ y _ __ _ ~

116:e..... o. if ie c ife s c- Ih ^ c oe dln ar,mond. This indicates a repulsion between second neighbors. Since the value i, small, it seems likely that secnd neighbor interaction in germanium is negligible. It will be noticed that the effect of the introduction of second neighbor interaction is to reduce R in diamond and to raise R in germanium. Since our calculation C for germanium and Smith's calculation B for diamond are both on the assumption that second neighbor interaction is negligible we can compare the two calculations by comparing the force constants used: a p Our calculation C for germanium:.0725.0497 Smith's calculation B for diamond: 0.157 0.104 Ratio B/C: 2.165 2.092 Average: 2.129 Deviation: + 2% Whether the agreement of the ratios of the force constants is fortuitous or not, the fact that the two ratios are closely the same can be used to develop relations between the frequency distribution from Smith's calculation B and Hsieh's calculation for germanium which is equivalent to our calculation C. In order to find the desired relations, we must return to Smith's formulation of the dynamical matrix. It is found that each element of that matrix contains a factor a or. Since the a's and fts for the calculations B and C are mutually proportional, the proportionality factor may be removed to the front of the matrix. In addition, a factor 1/m occurs in front m of the matrix. ge/mc i 6.05. Therefore, the eigenvalues of

1]7 the matrix, V2, for calcul:ti.n B,Jill be 2.129 x 6.05 times the correporn -ing eigenvalues of calcul- tion C. Finrally, the frequency: scales for calculati:n B will be 2.129 x 6.05 - 3.58 times the frequency scale for Calculation C. In order to check the results of these calculations we compare results obtained by Hsieh for germanium (calculation C) and by Smith in her calculation B for diamond. (Figure 27 and Figure 25B). Using the positions indicated we obtain: Diamond Germanium Ratio First Maximum 1248 cm 348 cm- 3.59 Second Maximum 467 132 3.55 Minimum 695 179 3.88 The variation in the ratio indicates that Smith and Hsieh did not perform their calculations in precisely the same manner. If we use individual maxima in Hsieh's curve arising from branch maxima and compare with Smith's branch maxima we obtain: Table 14 CORRESP 1NDITG iMAXIMA IN FIGURE 27 1ATD FIGURE 25B Germanium Maxima Diamond Maxima (Calculation (Calculation B) Ratio o1o2 3 e348 cm1 1248 cm1 359 )3 284 1007 3.55 (4 234 830 3.55 W5o6 132 467 3.55 The agreement of these ratios with the predicted value of 3.58 is very satisfactory. We have ignored the low frequency maximum

118 at 66 cm-l in Hsieh's calculation. Calculations in a later section show that it is not a principal maximum of a branch. It is clear that the lack of agreement in the first comparison is due to a difference in the position of the minimum. Evidently, Smith and Hsieh used slightly different methods of numerical integration. From the results of the preceding section we are assured that a scale factor of approximately 3.58 relates the numerical values obtained for frequencies in Smith's Calculation B and the frequencies for germanium. Hsieh showed that his frequency distribution gives correct results for the variation of the specific heat of germanium with temperature. This indicates that the Calculation C for germanium corresponds to the true distribution. However, for diamond it is not the Calculation B which fits observed specific heat data. Instead, it is calculation A. Consequently, we can predict that the true distribution for germanium, while it is simply related to Smith's Calculation B for diamond, is not simply related to the true distribution for diamond, i. e. Calculation A. This prediction follows from the fact that there is no simple relation between the Calculation A and B made by Smith. Physically, the difference between the true distributions for germanium and diamond lies in the second neighbor interaction which is important in diamond, but negligible in germanium. We shall return to this point when we attempt to correlate the infrared spectra of diamond and germanium. 4.1.3 Silicon For silicon we proceed in the same manner as with germanium.

.t.ere i: rio caickuiation comar,,Pabl e< <.r t..i... 1:;<:'}.for f3:vrsani:;.;io7::V;v-r, i, ti: -i!1e c.,n f. 1oc e ".rc c":a a. tn are r-lt.ed,te o those of g'.r(an;iLiu_2j b, a c.on.an;e:"t fac.tor:., S:.n;. th' Calc:ul at! oni can 104 be used for silicon. Fo: i1ii0Lon the elastic constants and corroes-pondiiL- force con;stant: ar r aa s follow: Table 15 ELAST'IC: A D FORCE C' 7SITA'7 IL' SILICCU AID GE. J B..^ FC'I -'L III 12 IC 1 JPI Si 1, 6740.6523 7 957.0909.0765 Ge 1.293.488.6.0725.04.7 Si/Ge 1'..O 0 1.337 1.1 254 1. 54 S i. 1537.1101.00"78 663 862 Ge. 0 39.0582 -. 36 569 iO6 Si/Gc 1. 3 52 1.892 5.i 1.80 J./ The second neighbor force constant, l, is negative for silicon as for germaniuml and also relatively 1small. The identities for sil]icon give: Identity 1: 1.087 vs 1; Identity 2: RI 663 cm1; Identity 3: R2 = 862 cm-i'. The results aro very similar to those obtained for germanium. e therefore compare the force cons-,tants for the calculations -where second neighbor interaction is neglected (Caiculat ion C). These values are listed above under a.and B|. The ratios are 1.254

120 for a and 1.354 for'. The average value is 1.304; deviation is 5%. To find the scale factor for N( ) we have mGe/msi 72. 6 258 j Si/s)e = 1.304 x 2.58 1.83 +.04. The relation between the frequencies for silicon and Smith's values in Calculation B is given by 3.58 = 1.95. Therefore, 1.83 for silicon we can expect that the maxima of the distribution of normal modes will lie at frequencies 1.83 times higher than for germanium, and 1.95 times lower than for Smithis Calca-. tion B. One check of our calculation is offered by the specific 105 heat data. Keesom and Pearlman have shown that a scale factor of 1.8 will superimpose the variation of D vs T (8D = Debye temperature) for silicon and germanium. Since the Debye temperature is directly proportional to the frequency, it is expected that the ratio of 1.83 should appear. We discuss this point in Appendix A. 4.1.4 Analysis of the Branches of the Frequency Distribution In the preceding treatment we have shown that the calculations made by H. iM. J. Smith can be used to determine the frequency distribution of the normal modes for diamond, silicon, and germanium. Before applying Smith's results to the infrared absorption spectra of these substances, we will examine the contours of the individual branches. This procedure is necessary because we are not employing an analytical method but are forced to use the method of numerical integration. Figure 28 shows the contour of branch 3 (3g) from Smith's Calculation A using three methods of numerical integration.

121 25 7 20 oB 20 L........... / \ _ 15 \ 10 -4 — * A, *< / \ A | \ 11't /'J I 0 \ 2.0 2. 1 2.2 2.3 2.4 2.5 2.6 1100 1150 1200 1250 1300 1350 (ccmRl) * Scale units are 2oi)*10-14/sec. A B C 4 19 38 57 (cm-1) n 5 4 3 Fig. 28 VAFJIATION IN BRAUCH CONJTOUBR JITH fU!E1XICAL IT3GRBATIOI PARAK CARS

122 The positions of the maxima as well as the upper and lower frequency limits are clearly functions of the parameter A) introduced on page 101. In this example, we are already aware that the upper frequency limit of the branch is 1332 cm-1 (2.51 in Smith's units), because it is possible to solve for the roots at < - (0,0,0), and general theory predicts that these roots will give the limiting frequencies of the branches. However, the lower limiting frequency of an optical branch and the upper limiting frecuency of an acoustical branch is not given by the roots at p = (0,0,0). Without introducing further theory, we would be compelled to leave these frequency limits as unknowns. This difficulty, in combination with the approximation inherent in numerical integration, gives poor prospects for establishing the contours of the individual branches. In addition, it tells us that we cannot accept Smith's contours of the branches without careful scrutiny. We will first examine Smith's calculation of branch contours and then try to refine the methods she used. Examination of the branches Smith uses indicates that she has not been consistent in her representation of branch 4, Calculation A. Sho'wn as a fundamental, this branch extends to an upper limit of 2.51 (in her units) = 1332 cm'"" Shown as an overtone, it extends to approximately 4.7. 4.7/2 = 2.35 2 2.51 Since an overtone branch is the same as the corresponding fundamental except for a scale factor of 2, these results are inconsistent, On the general theory, it can be predicted that only 5 branches are "optical", i. e. extend to 1332 cm-1. Therefore,

it 1~ certain Ih!at an error \was m ade in dr aw.:,lc.e Smith did not calculate each fundamental branch:-n;a:,irt-1 eI.y, thi eJror is ex z;:: a le. iCo Sv- E, rin, e Smith c-1cul..-tc each.J-parate di ributio'n for- tb. overto f,.' br'ianc!., th re uls fci th- over-tones can be carri,.vro directly to the fut;damnrr al:..,Sith'; 1st Overtone und en ta:r anch Iaxii -.'1':; 2, 2,6 02 C3?'-.i l. c,: - 4JL (-^~~~~~~~~~~~~~~~~~~~J C.,2469 1-5. Ae 2177 i089 "ir50~ ~1487 744 (Ag 01258 629 In order to exami;,e t+heose. results,,wo. -;il go to hcory 106 developed by van Hove, He has sho.wn that si igularities will occur in Ili( 3 ), i, e. the distribution of modes in thne ih branch, wth.en-evr thL three cerva ties a, / / 3 /'-3 vanish simiultaneously. t will be recalled that the solution of the determinental equation in the dyaiamical calculation gave as a function of the ( i Van "iove sho!i fur;;.Ter thoat tlere are for-r ty-;e e cf.:r.ngul arit ties wbchl can ap.-ear. Ji-ing lhs notation the singularities in =. ()i) are (1) Mfaxima, (2) Saddle points of type 1, (3) Saddle points of type 2, (4) Non-vanishing minima. The nature of these singularities in Nfi( ) is as follows: g() ) = i,()/N, -o - volumes cf ulit cell, a > 0,' = tot^l. nurmb;eri f i.n;t'-; -nl,.

g(0) go'N) + f JfT 7 fov. 9< c (2) Sad:le pt. of irt type: Sg0)= g0)) _ 0- for 9 a3/? In Agure 29 the behavior of the function g(~) at each type o) Saddle pt of 2nd Te:analytic cont ri is discontinuous at ence'0 for )< c g(p) = go() + 2 i Aro |_ for3> 2/2 In TFigure 29 the behaviror of the function g(o) at each type of singularity is shown. The function g;() is analytic and hence smoothly continuous at the critical fre uency. The contribution of J 9- - is discontinullou at "c Ience, even though go( ) is not n-own, it is necessary that the total function g(O) follow the general patterns shown in the Figure. The following points are made by van Hove. (1) Singularities of two or more types can occur at the same ). c In this case,

(a ) (c){L,' NT( ) C I( ) -N.,T ( ) l l I I; J: ( )!__ * __ _- -_ - - -L_ C c Pig'. 29 fT,Il- COiK7TO_ Li- DI 11 T }iPX PE 0'. DIS~RIS 1IHi 7X-.' D -I UL 7-L O' _i T.

126 the resultant is the sum of the singularities. (2) At least one maximum, one saddle point of type 1, and one saddle point of type 2 must occur in every acoustical branch. (5) At least one non-vanishing minimum must occur in each optical branch. It should be noted that since the goi) are analytic functions, the singularity at is always in the slope, i. e. ^g(^ -> + 00 In order to apply this theory to a particular case, it is necessary to examine the analytic expressions for each branch: ~ $) (P) to determine points at which L * 9 0. O *~ ~ ~ ~~~~~~~ l 21 I %W Using the diamond lattice as our example, we first examine the shape of the reciprocal lattice. The first zone of a face centered lattice is described by Smith as an octahedron with its vertices cut off. Its analytic description is qx = t 1 q a 1x 1 qy ~ z c 3/2 We have adopted Smith's notation where our i = her qi. The qj are the variables of the reciprocal space and, as mentioned before, represent the phases of the triply periodic solutions to the dynamical problem. To simplify, Smith introduces variables Pi a 8qi. The first zone boundary is now P + + + ~~~~+ + 2 x t 8, Py 8, P 8, P y Py 12. Due to the symmetry of the reciprocal lattice, it is sufficient to consider that portion of the first zone bounded by O P P P e 1 P + PY + P P 12 x` y z f= We know that wApi will be zero at any zone boundary which is perpendicular to a reciprocal lattice axis. Therefore, the following point (Px, Py, PZ) will have ^P ^ ~

127 (8,0,0). It can also be so'en that the point (4,4,4) will satisfy the condition for the occurrence of a singularity. The procedure is to obtain the derivatives of the determinental equation with respect to qx, qy, qz. All factors 3/3qi are set equal to zero. The coordinates of a given point (Px, Pys Pz) are substituted, and the result must vanish identically. In this procedure, we do not use numerical values for the force constants, but demand that the functions of qx, qy, qz must cancel. In this way, we have shown that the points (8,0,0) and (4,4,4) identically satisfy the conditions for all branches. The method is rather laborious, and a return to symmetry considerations produces the result that the point (8,4,0) also satisfies conditions for the occurrence of singularities. This is shown as follows: The derivative of y with respect to qx is zero for any point (8,a,b) from general rules of zone theory; the derivative with respect to qz is zero for any point (a,b,O) since the x,y plane is a plane of symmetry of the reciprocal lattice. It remains to show that ~S)/Dqy is zero at (8,4,0). The direction parallel to the y axis through (8,4,0) passes through the following points: (8,0,0), (8,2,0) (8,4,0), (8,6,0), (8,8,0). Now the points (8,6,0) and (8,8,0) lie outside the first zone. However, it is found by transformation that they correspond to the points (8,2,0) and (8,8,0), respectively. That is, the function V (q) must be symmetrical about (8,4,0) along the above line parallel to the y axis. Consequently, the contention that (8,4,0) represents a point at which singularities in the branches occur is proved.

128 The two points (6,6,0) and (8,2,2) can be shown to be equivalent by the allowed transformations of the reciprocal lattice, Due to their positions on the boundaries of the first zone, it was suspected that either maxima or minima in a= (q) would occur for every branch at these points. However, along the line (8,2,0), (8,2,2), (8,2,4), (8,2,6), it is found that the point (8,2,4) is equivalent to the point (0,4,6) and (8,2,6) is equivalent to (6,2,0). Therefore, one can only calculate the functions p = ) (q) along this line to see if a maximum or minimum occurs in a particular branch since the symmetry does not indicate that this will occur for all branches. Table 16 shows the values of v. Since (8,2,2) is symmetrically placed with respect to the y and z axes, and since it lies on the boundary (8,a,b), the demonstration of a maximum or minimum in the Table is sufficient to guarantee the condition'/a a 3/qy /qz -0. Calculations for both the "tlst Neighbor" (Calculation B) and "2nd Neighbor" (Calculation A) cases are shown. In each case, branches two and five demonstrate horizontal slopes. For branch two, A reaches a minimum. For branch five, y reaches a maximum. Consequently, we anticipate that the corresponding frequencies will denote singular points in Ni(1) for these two branches. The total result of the considerations of van Hovets theory is given in Table 17, It must be pointed out that there may be additional singular points. The four types of singularities can be distinguished from one another by the examination of the aQ/A; at each point. If all three aC/a) are maxima, the singular point

129 Table 16 SINGULAR POINTS IN BRANCHES o AND c5 AT POINT 8,2,2 = 0,6,6 IN THE RECIPROCAL LATTICE DIRECTION PARALLEL TO EITHER y OR z AXES Point in Reciprocal1 3 24 25 Lattice 14 5 6 First Neighbor Calculation (B) 8,2,0 * 0,6,8 2.29 2.29 1.78 1.78 1.03 1.03 8,2,2 = 0,6,6 2.32 2.27 1.84 1.71 1.06 0.95 8,2,4 " 0,6,4 2.35 2.29 1.94 1.59 1.03 0.90 8,2,6 a 0,6,2 2.34 2.32 2.04 1.46 0.97 0.91 Second Neighbor Calculation (A) 8,2,0 = 0,6,8 2.51 51 21 217 2.17 1.44 1.44 8,2,2 " 0,6,6 2.48 2.48 2.27 2.12 1.53 1.30 8,2,4 - 0,6,4 2.51 2.51 2.34 2.01 1.47 1.25 8,2,6 = 0,6,2 2.51 2.51 2.40 1.90 1.34 1.23 Units: 2.51 = 1332 cm-1

130 Table 17 LOCATIONS OF SINGULARITIES IN THE FREQUENCY DISTRIBUTIONS OF DIAMOND, SILICON, AND GERMANIUM 01 ~2 "3 04 05 c6 Diamond 8,0,0 1316(d) 1316(d) 1185(b) 1185(a) 743(c) 743(c) 4,4,4 1326 1326 1240(d) 1120(a) 525(d) 525 8,4,0 1332 1332 1125(d) 1125(b) 786(a) 786(a) 8,2,2 = 6,6,0 1316(d) 812(a) Silicon 8,0,0 624(d) 624 485 485 281 281 4,4,4 654 654 520(d) 444(a) 198 198 8,4,0 624(d) 624 485 485 281 281 8,2,2 = 6,6,0 618(d) 284(a) Germanium 8,0,0 340(d) 340 264 264 153 153 4,4,4 356 356 283(d) 242(a) 108 108 8,4,0 340(d) 340 264 264 153 153 8,2,2 = 6,6,0 337(d) 155(a)

131 is a "maximum" as described by equation (1) page 124. Similarly two maxima and one minimum give a saddle point of the first type; two minima and one maximum give a saddle point of the second type; and three minima give a non-vanishing minimum. Using these criteria we have labelled each singular point in Table 17 with letters (a) Maximum, (b) Saddle point of the First Type, (c) Saddle point of the Second Type, (d) Non-vanishing minimum. Many of the points are not labelled, particularly for the calculation in which no second neighbor interaction is considered. This arises from the fact that along at least one direction parallel to an axis through the point in question, the values of ) are constant. We are not aware of the significance of this point since we have not examined the situation analytically. However, there is probably some rela107 tion to the fact that in the simple cubic lattice Newell finds that only when second neighbor interaction is introduced do the van Hove singularities appear. In the second neighbor calculations, only a few cases occur in which the singular points cannot be identified. For branches 1 and 2 the values of 9 are very close together, so that the accuracy of the calculation is insufficient to define the singularities. The point (4,4,4) in branch 6 for the second neighbor calculation is the only point in the four remaining branches which cannot be identified. Since (4,4,4) is symmetrically located with respect to the three axes, this point must be either an (a) or (b) type singular point. In order to utilize the singularities it is necessary to consider each branch in detail. Let us compare Smithts

132 maxima with singular points for diamond. (See Table 18). Table 18 COMPARISON OF MAXIMA AND SINGULAR POINTS IN DIAMOND'S DISTRIBUTION Branch.........Smith Maximum Singular Points 1 - 2 1301 1316, 1326, 1332 3 1235 1240, 1185, 1125 4 1089 1120, 1125, 1185 5 744 743, 525, 786, 812 6 629 743, 525, 786 In the same way, Hsieh's results for germanium can be compared to the results above. Table 19 COMPARTSON OF fMXIMA AND S.TrNTLAR POINTS IN GERMANIUM'S DISTRIBUTION Hsieh Maximum Singular Points 348 340, p56, 337 2 3 284 283, 264 4 234 242, 264 5 132 108, 153, 155 6

133 It is seen that some of the various singular points lie near the maxima obtained by Smith and Hsieh. Since the calculated singularities are associated with infinites in the slope, we do not expect to find distribution maxima at precisely the frequencies of the singularities. However, the nature of a particular singularity indicates the general behavior of N(]) in the immediate vicinity of the singularity. Consequently, we can hope to define the interval of frequency in which maxima may occur, To facilitate the use of the calculated singularities, we have calculated all of the fundamental branches using Smithts data and our own numerical integration. For diamond, in branches 3,4,5,6, the singular points are identified except in the single case already mentioned. Consequently, we have altered the contour obtained by numerical integration to incorporate the effect of these singularities. This procedure is purely qualitative since we do not know the magnitude of the effect of the singularities. For silicon and germanium, the number of singular points which are identified is not so great. Rather than assuming that these points will correspond to the similar points for diamond, we have used only those points which are identified. In all three distributions, the most useful part of this work with singular points is the identification of the upper limiting frequencies of the acoustical branches and the lower limiting frequencies of the optical branches. This identification allows the numerical integration to be applied with more certainty since points falling beyond the limits are known to

134 be spurious and are neglected. The results for this procedure are shown in Figure 30 and Figure 31. Maxima are given in Table 20. The principal differences between our results and those obtained by Smith lie in the location of the limiting frequencies, as already mentioned. The most significant case occurs for branches 1 and 2 whose maxima lie above the top of Figure 30, In these branches the lower limiting frequency occurs at 1316 cm-1 Smith places the maximum of the branches at 1301 cmn1 The error arises from Smithts numerical integration in which she uses 4 = 160 cmt1 while the spread of the branch is only 16 cm-. This is equivalent in spectroscopy to using a spectral Table 20 THE CALCULATED MAXIMA OF THE BRANCHES OF THE FREQUENCY DISTRIBUTIONS OF DIAiVirOND, SILICON, AND GERMiANIUM Branch Di aond Silicon Germaniumn 1 1326 654 356 2 1326 625 341 3 1285, 1180 520 284 4 1080, 930, 680, 370 444 241 5 736, 570, 330 281, 230, 122 153, 126, 67 6 700, 525 230, 122 126, 67 Note: Some of the secondary maxima listed for branches 4, 5, and 6 may be spurious due to the uncertainty in the numerical integration.

25 (93, I 15 I I I I \ I \ S~~ ~~~~~~~~~~~~~~~~ )6^ 15 _ _ _ _ _ _ _ _ _ _ _ _ I r_ _ / I 5 0 200 400 600 800 1000 1200 1400 (cm-f) Fig. 30 THEl BRANCHES OF TITE DISTRIBUTION OF VIBRATIONAL K;ODES IN DIAIIOND

25 - ------— i Fig. 31 20 THE BRANCHES OF TI-E DISTRIBUTION OF VIlRATIONAL INODES IN (A) SILICON, (B) GERI, iNIUI I I \ od I I 1 11 - I A: 100 300 400 500 600oo 700 B:0 50 100 150 200 250 300 350 (cm-l)

137 slit width ten times the width of a line one is trying to resolve. This means that the principal maximum of the total dis-1 tribution cannot lie at 1280 cm as Smith calculates, but it -1 must lie closer to 1332 cm * The precise height of the maximum is difficult to determine because slight variations in the position of the lower limiting value of 1316 cm'1 will alter the height of the branches, since the area under the branches is fixed. Because of the high concentration of normal modes in such a narrow frequency interval, it is likely that the second neighbor calculation has its principal maximum at about 1320 cmi. e. the maximum will be determined by branches 1 and 2 with little contribution from branch 3. 4.2 Application of Lattice Dynamics to the Infrared Absorption Spectra o Diamond, Silicon, and Germanium (4.2.1 Selection Rules) (4.2,2. Analysis of the Observed Binary Combination Bands) (4.2.3 Assignment of the Second Overtone Bands) (4,2.4 Analysis of the Observed Fundamental Bands) 4.2.1 Selection RSules The selection rules for the normal modes of a crystalline 81 16 90 lattice are given by Teller, Lifshitz, and others. The selection rules are as follows: (1) Of all those frequencies occurring in the branches of the frequency distribution of a crystal, only those frequencies corresponding to ( = (0,0,0) can be active in infrared absorption. The modes corresponding to these frequencies are elastic waves of infinite wavelength, i. eo the phase change is zero between similar atoms in different unit cells. This selection

138 rule follows from the fact that for all other modes, corresponding atoms in neighboring unit cells vibrate out of phase with one another, so that over a region comparable in size to the wavelength of the incident radiation a dipole moment change existing in one unit cell is cancelled by summing over many cells. For any mode at a (0,0,0), the symmetry of the mode determines whether or not it is infrared active. (2) Combinations and overtones of branches can be infrared active. In this case, it is possible to pick certain combinations for which the phase change between unit cells is zero. This is possible because for each frequency in one branch there is one corresponding frequency in every other branch for which the phase )i = - Tj. That is, since the frequency is independent of the sign of (, one adds the frequencies of each branch occurring at the same reciprocal lattice point. The distribution of the resultant combination frequencies can then be active in the infrared if the modes involved have the proper s ymme trry On the basis of these selection rules, we can state the following general rules which are independent of syummetry considerations: (1) The fundamental absorption spectrum of an ideal crystal is composed of narrow lines associated with limiting frequencies 4 (0,0,0). (2) The overtone and combination absorption spectrum of an ideal crystal is composed of broad bands associated with the combinations of the branches which make up the distribution, N($). WIhether or not any of the above lines and bands appear in absorption is governed by the symmetry of the modes involved. Symmetry selection rules can forbid

139 transitions allowed under (1) and (2) but cannot allow any transitions forbidden under (1) and (2). Similar selection rules hold for the Raman effect with the same conclusions concerning the nature of the fundamental and overtone regions. The Raman effect of diamond corresponds to these rules. The single sharp line at 1332 cml- is the only feature of the fundamental region. The second order spectrum is composed of broad bands whose analysis by Smith has already been discussed. Since the ideal diamond lattice has a center of symmetry at the center of each C-C bond, one may expect that the infrared 102 and Raman spectra will be mutually exclusive. There is only one limiting frequency, i. e. the three optical branches have a common root at a 0,0,0 and since this frequency is Raman active, no fundamental bands are expected to appear in infrared absorption. Further, Smith has calculated the selection rules for combinations and overtones in the Raman effect. If the rule of mutual exclusion holds, none of these combinations or overtones is allowed in the infrared, On this basis we would predict that the maximum number of allowed infrared combinations and overtones is (1) No first overtones, (2) + + i + +3 ++ Ll W 15- 4 l - 5 1 5(3 t- 4, -3 -t 5, w3 - w6. Branches one and two are degenerate in Smithts calculation for diamond, so that all statements for 0l also hold for o2. The degeneracy is a consequence of the central force field which is assumed to exist between second neighbors. This force is not used in the calculation for silicon and germanium. Therefore, for silicon and germanium, these branches are not degenerate, and

140 separate combination maxima occur. The above statements are based on the assumption that the diamond lattice is centrosymmetric. However, a real diamond having the ideal Bragg structure is not strictly centrosymmetric due to the presence of more than one isotope of carbon. Similarly, silicon and germanium, to which the selection rules apply, have more than one stable isotope. In Table 21, the stable isotopes of the three substances are listed together 67 with their abundances. In a later section we will discuss the effect of a random distribution of isotopes on the selection rule = (0,0,0) for fundamental modes. At this point, we shall assume that the selection rule requiring = (0,0,0) still holds. However, it is clear that at least some sites of each crystal will contain atoms of weight different from thlt of neighboring atoms. Consequently, the lattice is not strictly centrosymmetric. For diamond and silicon the abundances of the less predominant isotopes are relatively small. In germanium, however, there is a distribution over several isotopes having comparable abundances. In diamond, silicon, and germanium, the masses of the various isotopes are not greatly different. The limiting mode involves the motion of one face centered lattice against the other. In such a mode, the amplitudes of two different isotopes will be only slightly different and the dipole moment change will be small. At least on this qualitative basis, no fundamental absorption is anticipated at the upper limiting frequency because of this isotope effect. Similar reasoning leads us to the conclusion that the selec

141 Table 21 THE STABLE ISOTOPES OF CARBON, SILICON, AND GERMANIUM I sotope Substance (Mass No.) Abundance Carbon 12 98.9% 13 1.1 Silicon 28 92.3 29 4.7 30 3.0 Germanium 70 20.6 72 27.4 73 7.6 74 36.8 76 7.6

142 tion rules involving mutual exclusion can be expected to hold for the combination and overtone spectra. However, due to the presence of isotopes, particularly in the case of germanium the combinations allowed in the Raman effect may appear in absorption without violating symmetry selection rules. These combinations are +1 + w2, W1 + W3, 02 - 3 4 - 5, -4 - w6, W5 - )64.2.2 Analysis of the Observed Binary Combination Bands In general, the maximum of a combination branch does not lie at the sum of the frequencies of the maxima of the two branches forming the combination. The variation of frequency with ( is different for each branch (see Tables llAand B ). In particular acoustic branches tend to low frequencies and optical branches tend to high frequencies as ( approaches zero. Consequently, when combination branches are formed by adding the frequencies for two branches at each point Q, the resulting variation of frequency with 0 will be different, in general, from either of the contributing fundamental branches. Similarly, the distribution of combination frequencies, formed by numerical integration, will, in general, have its maximum at a frequency not equal to the sum of the frequencies of the 81 maxima of the contributing fundamental branches. Teller discusses this point. It is necessary to calculate the distribution for each allowed combination. We have done this using Smith's data and employing numerical integration. Singular points were utilized to identify the frequency limits of the combinations. Diamond combinations are obtained using Smith's

143 Calculation A. Silicon and germanium branches are obtained using Smithts Calculation B and then dividing the frequency scale by 1.95 and 3.58 respectively. The maxima of the combination branches are listed in Table 22. The calculated distributions are shown in Figure 32 and Figure 33. The anticipated shift of the combination maxima with respect to the individual fundamental maxima occurs (see Figures 30 and 31 and Table 20). From our calculations we are certain thu t a freqiuency scale factor of approxirmately 1.83 connects the distributions of silicon and germaniium. ilowever, since second neighbor interaction is used for the diamond calculction, there is not a simple scale factor between tho distributions of diamond and the ot:her substances (see page 118). Consequently, while observed bands due to the same combinations in silicon and germaniuin should be recognizable from their positions and general similarity, the corresponding observed diamond bands can be identified only through the assignment of observed combinations in terms of calculated branches. 52, 54 The observed infrared spectra of diamond, silicon, and gerranium in the combination region are shown in Figure 34. The diamond frequency scale is 3.6 timles tlhe germanitum1 scale and twice the silicon scale. In iFigure 35, the coltours for the whole combination region are calculated on the assumption that every branch is equally intense. Comparing with tFigure 34, it is clear that such an assumption does not correspond to the facts. In Table 23, the observed maxima are listed together with the calculated maxima lying closest to them. This table

144 Table 22 THE MAXIMA OF CALCULATED COMIBINiATIONS ALLOWYED IN INFRARED ABSORPTION Combination Diamond Silicon Germanium co1+ 2400 to 2450 cm- 1 1100 cm'1 598 cm'1 2 + 4 2400 to 2450 cm'1 1100 598 03 + 04 2260 960 520 1 + W5 2040 905 492 a + o5 2040 880 480 W3 + 5 1920 800 440 3 5 I + W6 1940 875 478 O + )6 -1940 870 475 03 + 6 1860 760 415 Total"* 1900, 2350, 2450 790, 870 430, 475, 1090 600 * See Figure 34 $, See Figure 35

145 30 —-. F 25 20 20 D I - A \ 20to~ e / IT I?~;s I= v1 Ad A A'I R H15 ~~~~5~~~~~~~~~~~~~~ B:W 4Wt 1 6; E: 3 6 Fig. 32 CALCULATED COM1BINATIONS ALLOWED IN INFRARED ABSORPTION IN DIAMOND CALCULATGD CO~~~~~IBINI1T~IOPSALL~~3)I MFAE BO"ON IjDIA~

146 30 - - G I: 700 100900 100 1100 I e I I II: 40D 450 500 550 600 C2 E:4 G:_-1 3 ABSORPTION I ILICON (ANI 9 I I / i I / I I / I\ I / 10 i - I - I ~!' I I L B:W- WW-: G:IA)- i C:W3-W4 W2-O6 Fig. 33 CALCULATEI3D COM~BINATIONS ALLO0JED IN INFRARED ABSORPTION IN (I) SILICON, (II)GJiS-R1VNIUI4

147 10 Diamond-2.62 rnm B - C 6 2 100 Silicon-1 imm i 10 0 60 r X B ll 40 pi - 675 800 900 1000 1100 1200 1300 100 -1........... A 0 60 400 450 500 550 600 650 700 FIG, 34

148 80 C,) 1400 4 0 1 5 0 5 0:6 0 1 50 0 60 ----— IS — ^- f -------— r --------- Sir 4c~~~~~~ / ~II...... 20 / 0 A: 1300w 1500, 1700 g 19001 21001 2300 2500 130 1 0'. 7.......... B: 1400'40 50.... 1590'6o 150 C: 7009 idoo iio 12b0 (cm ) A: Diamond; B: Germanium; C: Silicon Fig. 35 CALCULATED CONTOUR IF EACH ALLOW1ED COA;1BIN0ATION HAS THE SA1IE INT2E30SITY B: ^00'; W \W ^5Q [600 [W~~~~~~~~~~~~~~~~~~ G: 00^O 00 idO l~j> 1J30(c[; A: Diamond; B: Germanium; C~~~~~~~~~~~~~~~~~~~~~~~~~~:Sico Fig. 35~~~.....ALULTE COT:-I AH LO COMBINATIO IA iii SA......NI

149 Table 23 TENTATIVE ASSIGNMENT OF COMBINATION BANDS ON THE BASIS OF POSITIONS OF MAXIMA Observed Assignment Calculated Diamond 2400-2500 1 + 4 2400-2450 1 + ~4 } 02 + W4 2165 t3 + (4 2260 1960-2020 + & 0 2040 "2 + w5 J 1 + < 6 ) 1940 se + 6 Silicon 1100-1150 ol + O4\ 1100 o2 + W4 J 900-960 o + 4 960 co1 + co5 905 736 W3 + c6 760 -Germanium 640 wo + ) 598 G2 + )4 J 520-560 (3 + W4 520 W1 + 05 492 420 3 + 05 440 03 + 06 415 * See text....._::':-':...:! Z.'.~:' ~!.!-'?- -,~'""'";-'::....;.:..~..." /.'........~',..;.'...il:: ~: J" "....

150 represents a tentative assignment on the basis of the positions of the maxima, but without consideration of the shapes of the bands. In Figures 36, 37 and 38 we have plotted the observed bands in terms of absorption coefficients. Also shown are calculated combinations whose component branches have been adjusted in relative intensity to produce the best fit with the observed bands. In the following paragraphs we discuss each case separately. In Table 24, the final assignments are listed together with the relative intensities of the individual bands. The individual combination maxima shift slightly when the various bands are superimposed. The maxima of the combined distribution are given in the table. All combinations not listed are assigned zero intensity. DIAMOND: Several calculated maxima fall near the observed doublet at 2000 cm-1. In our fit of the observed spectrum, we have included all of the calculated bands which fall close to 2000 cm"1, i. e. those listed as tentative assignments. However, both l + 06 and g2 + g6 could be deleted without greatly altering the band shape, simply by increasing the intensities of 1l + 5 and p2 + c5. Since the observed maxinum is a doublet, at least two combination bands must contribute to it. Our calculations are too approximate to justify showing fine structure in the calculated contour at 2000 cm'1, although since several bands contribute to the calculated maximum, the observed doublet is essentially explained. The second main maximum in the observed spectrum at 2165 cm1 lies approximately 100 cm'l below the closest maximium, )3 + +4, at 2260 cm"'. However, the observed band shape is similar to

151 o L=106cm-1 i j s ( 4-),p! / "' * 240 ----- / —- D 0. *~r 1 / I,, /',, 0/, / I, d I / / I\ - o _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __ 3 C C/ I V 0 ( v 1300 1500 1700 1900 2100 2300 2500 Branch:~1+4',l1~'~5:~Jl+~4:~2 +~:W2 +~5:~2 +~:~3,W$ Intensity: ~: 1: 1: ~: 1: 1: 2 Fig. 36 CALCULATED (B) VS OBSERVED (A) COMBINATION BANDS IN DIAMOND

l~I $ c | \ |....2=30 cmlW O 6 |ll \|Allowed Br cW4 W J2+W:4 4W 3+Wl+J5:W3+W W3*W6 a) e Branches' 1 4243 153 36 o6 o | II \ LIntensity: 1:1:: 2/3: 1 3 4 4 \ +-Forbidden W+W Intensity: 1/6:1/6 350 40_______ ________ Branche s 3 o 4 = JI' \ (cm-) 2d A o 350 400 450 500 550 600 650 700 Fig. 37 CALCULATED (B) VS OBSERVED (A) COMBINATION BANDS IN GERMANIUM

ago,) =54 cm-1 O IBranches:'.1~+:+2w:N3 +' 5i_ 35 6 |: +l Intensity: 1:1: 1:2:3 (I) o 650 750 850 950 1050 1150 1250 (cm-1) Fig. 38 CALCULATED (B) VS OBSERVED (A) COIBINATION BANDS IN SILICON

154 Table 24 FINAL ASSIGNMENT OF COMBINATION BANDS Observed Calculatsed (cm-1) Assignment Intensity (cm"1) Diamond 2400-2500 0 + o4 1 2400-2450 W2 + 04 1 2165 W3 + t4 2 2270 1960-2020 01 + w5 1 0 2 + 05 1 2000-2050 01 + 06 1 l "2 + 06 1 Silicon ^1100-1150 ( 1 + 04 1 100 02 + 04 1 J 960 03 + 04 1 950 900 A + o 5 2 900 736 (3 + W6 3 770 Germanium 640 ( 0 + 03 1/6 640 2 + )3 1/6 560 ( + 04 1 590 + 04 1 520 (03 + 04 1 520 0 1 + 5 2/3 420 (03 + 5 1 ) 420 3 + 6 03 l See text -...... z..' -:: ~F -:r ~ — L! -..;.-~i,~;]: [ -' ~:L Tf ~" "i ~!,ii:~:~',f- -:i ~:~'.'-' —'-~-:'~,:'::):;,..:..:.:..z-i:.~ ~:~:.u7: "-'........ -!:.- ~'

155 the calculated shape. The inaccuracy of the numerical integration (a3 " 106 cm1l) in addition to the effect of anharmonicity contribute to the discrepancy between observation and calculation. The third observed maximum at 2480 cm1' extends from 2400 to 2500 cm1l and is well approximated by the calculated +1 + 04, ~2 + (4' The fact that the calculated value of 3 + 04 lies at too high a frequency accounts for the overlapping of the calculated bands at 2325 cm'l, The calculated intensity at 2450 cm'1 is below the observed intensity because it is found that further increase of the 2450 cm1l band intensity moves the maximum at 2260 cm'l to higher frequencies. Clearly, if (3 + 04 fell closer to the observed frequency, the difficulty with w1 + W4, W2 + 04 would not occur, and the observed spectrum could be more accurately reproduced. GERMANIUM: In Table 23 it will be seen that there is a discrepancy between the observed maximum at 640 cm and the calculated position of the assigned maximum at 598 cmn1 which is in the wrong direction to arise from anharmonicity. If, however, we consider the "forbidden" combinations, which we have stated may appear in germanium, it is found that the combinations l1 + co and O2 + (3 fall precisely at the location of the 640 cm'l band, and have a similar contour (Table 24, Figure 37). Since absorption exists at 560 cm-1, and since the 1l + 04, c2 + 04 maxima occur at 598 cm1l, it is likely that they contribute to the absorption at 560 cm71 No other allowed or forbidden combination branch in the calculated set, has a non-zero density at 560 cm-1. The only other possibility

156 is an overtone, which falls in the "forbidden" class. When overtones are considered it is found that 2 o3 falls at 568 cm-1. However, its contour does not fit the observed band, especially the sharp drop at 600 cm'1. Since the contour of 1 + i4' s2 + i4 similar to the 560 cm-1, although the maximum is displaced by 30 cm'1, we assign the 560 cm-l band to 1 + G4' W2 + 04' Although 1w + w5 does not have a corresponding maximum in the observed spectrum, its inclusion improves the agreement between the contours of the calculated and observed bands. The remaining observed bands are well explained by the calculated spectrum. It should be noticed that the assigned bands in germanium differ from those for diamond. Aside from the inclusion of two weak "forbidden" bands germanium also has a strong 03 + W6 band and a somewhat weaker W3 + c5 band, neither of which occur in diamond. (Note: The Germanium contour can be reproduced in all important aspects by setting the intensity of Ac + W5 equal to zero and increasing the relative intensity of W3 + o6 from 3 to 4). Finally, neither 1 + g6 nor 02 + 06 occurs in germanium. As we have mentioned above, these last two bands are not vital to the explanation of the diamond spectrum and may not be present. The different intensities for various bands indicate that the anharmonic constants differ between germanium and diamond. SILICON: We have made no observations of the silicon spectrum. Unfortunately, the published spectra do not overlap, 52 54 i. e. Briggs published the spectrum from 1 to 12p and Lord published the spectrum beyond 12po. However, near 12, there

157 are no data available. In addition, the absorption coefficient of the 1100 cm1l band appears to be too strong to be comparable with the combination bands near it nor is it similar to any band in germanium or diamond. In the experimental section we have shown that in powder form silicon displays this band, and we have also shown (page 62 ) that SiO2 has a similar strong band near 1100 cm-. For these reasons we consider that the 53 band at 1100 cm1 in silicon is due to SiO2 impurities. Lord holds this same opinion. The remainder of the spectrum can be explained in terms of the combinations listed in Figure 38. The inclusion of Ao + 04, o2 + )4 is arbitrary since we are excluding the 1100 cm'l band from consideration. However, the level of general absorption between 1000 and 1050 cm-1 is consistent with the inclusion of these combinations. In addition, these two bands have appeared in our assignments for both germanium and diamond, and are included to show that their presence in silicon is not inconsistent with the experimental data. Except for the assigned "forbidden" band in germanium, the assigned bands and their relative intensities are very similar for the two substances, espcially since o3 + 5 is not essential to the explanation of the germanium spectrum. This similarity points to the similarity of the force fields existing in these substances. Diamond clearly differs, as we have already anticipated in our original calculations. 4.2,3 Second Overtone Bands We have not considered the two highest frequency maxima that occur in diamond (see Figure 6), silicon (see reference

158 52), and germanium (see Figure 19), These bands are weak and broad in all three substances and lie above the upper frequency limit for binary combinations and first overtones. We have not investigated the selection rules for higher order combinations and overtones, and will merely list second overtones falling close to the observed maxima. Substance Observed (cm-1) Assignment Calculated (cm-l) Diamond 3125 3 04 3240 3570 3 03 3755 Silicon 1300 3 04 1332 1470 3 3 1560 Germanium 750 3 04 722 845 3 03 852 Considering the possible effects due to anharmonicity as well as the accuracy of our calculations, the agreement can be considered satisfactory. WVe shall find that 03 and 04 occur in the fundamental spectra. In a centrosymmetric system, one expects that the second overtone of an active fundamental will 102 be allowed. 4.2o3 Analysis of the Observed Fundamental Absorption Bands It has been stated in section 4.2.1 that the general selection rules for ideal crystals forbid infrared absorption due to all fundamental modes except those corresponding to limiting frequencies of branches. In the diamond type structure there is only one limiting frequency. It is triply degenerate This frequency is forbidden in absorption by symmetry selection rules.

159 However, the absorption spectra of many diamond crystals and of all those silicon and germanium crystals which have been studied show broad absorption bands in the wavelength region where fundamental vibrations may be expected to occur. These broad bands cannot be explained as combination or difference bands. It will be shown that many of the observed maxima do correspond to the calculated maxima of the frequency distribution. Broad bands are also observed in the fundamental absorption 120 121, 122 spectrum of NaCl. Born and Blackman have shown that anharmonic terms in the potential can cause normally forbidden fundamentals to appear. According to this theory the appearance of forbidden transitions depends on the fact that the upper 90 limiting frequency in NaC1 is allowed in absorption, even in the harmonic approximation. The upper limiting frequency is forbidden in diamond, silicon, and germanium. Consequently, the Born, Blackman theory does not apply. It is necessary to explain, therefore, how the selection rules for fundamentals can be broken. For the explanation we 16 shall consider the work by I, M. Lifshitz concerning the effect of imperfections on crystal selection rules. Since Lifshitz's work is a mathematical treatment with very little interpretation, it is useful to discuss a model of an imperfect crystal on general terms to provide some connection between physical phenomena and the mathematical results. Accordingly, we will present a few general considerations based on qualitative reasoning.

160 If the periodicity of an ideal lattice is interrupted by the presence of randomly distributed imperfections which are separated by an average distance which is short compared to the wavelength of the incident radiation, then for at least some modes, for which the dipole moment change averages to zero in the ideal case, the averaging will now take place over a shorter distance (fewer number of unit cells), and a net dipole moment change may result. It appears that a net dipole moment change will occur only for those modes whose wavelengths are comparable to or longer than the average spacing of the imperfections. For modes of much shorter wavelength, the averaging over unit cells will produce a net dipole moment change of zero, just as in the ideal lattice. The number and frequency distribution of the modes which will become active in an imperfect crystal will be determined by (1) the average separation of the imperfections (2) the distribution of vibrational frequencies with respect to the wavelengths of the corresponding modes, (3) the dipole moment change in a single unit cell associated with each particular mode, The normal modes of a crystalline lattice are uniformly distributed in phase space. If we measure phase (the phase of an elastic wave equals the reciprocal of the wavelength times the unit cell dimension) in terms of the radial distance, r, of a point in phase space from the origin, then the number of modes between r and c r + dar is 4p 2d r where is the number of modes per unit volume of phase space. Since is constant the number of modes increases as 4 2* This means that there are many more modes of short wavelength than of long

161 wavelength. In turn, we infer that imperfections must be spaced, relatively close together if many frequencies are to be active in absorption. Since optical branches tend to high frequencies near: (0,0,0), and acoustical branches tend to low frequencies near f= (0,0,0) 90, (see Table 11), imperfections will cause selection rules to break down for the higher frequencies of optical branches and the lower frequencies of acoustical branches. In the preceding qualitative discussion it has been assumed that the normal modes can be represented as plane waves even in the imperfect crystal. In addition, no consideration has been given to the effect of imperfections on the normal frequencies and their distribution. To test the validity of our qualitative 16 deductions we will now consider the work by Lifshitz. Lifshitz has written a series of three articles on the general subject "Optical Behavior of Non-Ideal Crystal Lattices in the Infrared." He shows that, in general, without reference to the nature of imperfections, the presence of inmperfections in crystals can be expected to cause normally inactive frequencies to be allowed in absorption1 In contrast to the qualitative results given above, Lifshitz finds that not just some of the modes but all of the modes are allowed in absorption in imperfect crystals. According to Lifshitz, this occurs because the normal modes are no longer plane waves. Returning to our qualitative model, if one says that the frequencies are primarily determined by the nature of the interaction forces, while the phase relationships between unit cells (which depend on the wave forms of the normal modes) are determined by the long range order of the crystal, then it is possible to understand how the frequency

162 distribution of the normal modes can remain essentially unchanged even though the selection rules are broken for all of the modes. Nevertheless, the extent to which the dipole moment change is preserved over several unit cells may vary between different modes, Lifshitz has considered two special cases for which he can obtain a solution: (1) arbitrary concentration of small distortions, randomly distributed, (2) small concentration of centers with an arbitrary distortion. Case (1) corresponds to the effect due to the presence of more than one stable isotope. Case (2) corresponds to the effect due to the presence of impurity atonis or vacant sites. Case (1): The isotope effect varies according to the frequency under consideration. We shall consider two ranges of frequency: (a) Frequencies comparable in magnitude to the upper limiting frequency. In diamond, silicon, and germanium the upper limiting frequency is normally inactive. For a case in which only two isotopes occur, Lifshitz finds that there will be weak, uniform absorption with the absorption coefficient proportional to C(1-C) /mY) where m z mI _ m, mi mass of ith isotope, m* mIIC + mI (1-C), C m concentration of one isotope, 1-C = concentration of other isotope. "Uniform absorption" for all frequencies implies absorption bands which follow the contours of the branches of the frequency distribution. (b) Frequencies very small with respect to the upper limiting frequency. In this case, the absorption decreases with increasing frequency as the fifth power of the frequency. Since this range starts from 0 = 0, the modes involved will be primarily sonic

1533 and ver- long infrared wi'vTs. Pres umably, the stron decr.ease in absorption with frequency- will render this effect unobservable in the infrared region. (2) The impurity effect depends on the nature of the impurity. In order to simplify the problem Lifshitz treated the problem in which the iipurity atom differs from the atoms of the parent crystal in mass, but does not alter the force field. The case is highly idealized, and probably only applies to ionic mixed crystals. However, the results are instructive. For example, when the mass of the impurity atom approaches the mass of the atoms of the parent crystal, Lifshitz obtains results which agree with the results obtained for the isotope effect, i. e. his treatment is consistent. For masses much greater or much less than the atoms of the parent crystal, absorption for high frequencies is uniform as in the isotope effect, but it depends only on the concentration of the impurity atoms and is independent of their mass. Also, the impurity atoms may cause new frequencies to appear which may lie either within or outside the frequency intervals of the ideal crystal branches. These new frequencies will, in general, form a set of continous bands whose positions and intensities depend on the impurity concentration. We shall now consider how Lifshitz's results can be applied to explain the absorption in the fundamental region by diamond, silicon, and germanium in terms of the presence of imperfections in the lattice. The appearance of bands characteristic of the ideal lattice can be due either to an isotope effect or to the presence of impurity atoms, or perhaps to other imperfections

164 which have not been investigated specifically. If frequencies not characteristic of the lattice appear in absorption, these may be attributed to the effect of impurity atoms. Before we consider the observed spectra we will make some estimates of the average distance between isotopes of the same mass. For diamond, C13 is present as 1.1% of the atoms while C12 makes up the remaining 98.9%. Therefore, 1 atom in 90 is C13. Since each unit cell contains two atoms, 90 atoms fill 45 unit cells, If the isotopes are uniformly distributed (an approximation to the mean spacing of a random distribution), then the distance between unit cells containing C13 atoms is 3\J 45 3.8 unit cell sides. That is, approximately 4 unit cells separate C13 atoms, on the average. Such a separation is small enough to affect even the shortest mode, which has a wavelength comparable to one unit cell side. In silicon and germanium the various isotopes are present to an even greater extent. (See Table 21). We conclude that the imperfections introduced by isotopes are close enough together to break down selection rules for all elastic waves. The question now becomes one of the magnitude of the absorption coefficient. Since the mechanism of absorption in valency crystals is not clear, 108 (see, for example, Matossi ), we will attempt no quantitative estimate of the absorption coefficient. One can calculate the factors which depend on the mass and concentration of the isotopes. In Table 25 we have calculated the quantity K' (m - mi ) Ci i-... (1-Ci) where m' is the atomic weight of the substance when the isotopes are present in their normal proportions, mi is the mass of the

165 ith isotope, and Ci is the concentration of the ith isotope, This expression for K' is the extension of Lifshitz's expression for two isotopes (page 162) to the case of more than two isotopes. Table 25 RELATIVE ABSORPTION COEFFICIENTS K' Substance Mass No Ci m K'.105 Diamond 12.989 12.01 7.7 13.011 Silicon 28.923 28*06 25.6 29.047 30.030 Germanium 70.206 72 6 71.5 72.274 73.076 74.368 76 *076 The absorption coefficient is proportional to Kt. However, we have no good estimate of the relative effects of the remaining terms in the total expression for the absorption coefficient. These terms depend on the charge distribution and the force field. Under these conditions, we cannot compare the observed absorption coefficients with calculated expressions. Table 25 does show that the contribution of the terms depending on isotope concentration and mass becomes progressively greater as we go from diamond to silicon and germanium, At the strongest absorption band in the fundamental spectra the observed absorption coefficients are

166 KDia: KSi: KGe (0-20): 5.5: 13.5 These observed coefficients also incr-ease from diamrond through germanium, if we use the IR II diamonds (K = 0) for the comparison, The fact that diamond someti-mes exhibits no fundamental absorption indicates that either the isotope C13 is absent from IR II diamonds, or the isotope effect in diamond is not sufficient by itself to bring about observable absorption. In order to test this point, measurements of the C12/C13 ratio in diamonds of varying IR properties were planned. Unfortunately, while tentative arrangements were made with thie University of Chicago to do this work, no measurements have bean made. However, results 109 on other material containing carbon as well as some diamonds, which were not classified as to Type)indicate that the maximum variation in the ratio C12/013 in naturally occurring substances is of the order of 5A of the ratio. The corresponding variation in the total number of C13 atoms, i. e. from 1 part in 90 to about 1,05 parts in 90, is insignificant from the standpoint of infrared absorption. Also, since isotope concentration can have at most very minor effects on the electronic properties whose variations are correlated with the variation in infrared absorption, it follows that the isotope effect by itself, is not responsible for infrared absorption. However, in silicon and germanium, the observed infrared absorption coefficients in pure samples do not vary from sample to sample. It seems reasonable to conclude that the absorption in silicon and germanium may be due to the isotope effect. If this is true, one must explain the fact that diamond evidently has no observable isotope effect. Such an explanation can be obtained only by a consideration of

167 the contribution to the absorption coefficient by the factors depending on charge distribution and the force field. We do know that the force field in diamond is different from that in silicon and germanium (see section 4.1). Consequently, there is no reason to expect that the neglected factors will be the same order of magnitude for all three substances. We will now consider the observed spectra, shown in Figure 39. The frequency scales have been adjusted to the calculated factor of 1.95 (Dia/Si) and 3.58 (Dia/Ge). The principal maxima of all three substances fall in the same general region. The narrow diamond band at 1372 cm1 has no counterpart in the silicon spectrum. The shoulders in the germanium spectrum at 360 and 375 cm-l are similar to the 1372 cml band. We will consider this point later. In Table 26 are shown the calculated maxima which fall closest to the observed maxima. In Figures 40 and 41 the contours of the observed spectra are compared with the calculated branches which most nearly fit the observed bands. Intensity factors used for this comparison are shown in Table 26. We will now consider the fit between observation and calculation. Since the results for germanium and silicon are less complex than those for diamond, we will consider them first. GERMANIUM: The contour of the observed spectrum is well matched in the high frequency portion by w1, p2' and g3. The maximum at 200 cm-1 is not so well explained by 04. The true shape of 04 may be somewhat different from our calculated result, and our observed spectrum i>i, this region is not of high accuracy, in any event, the calculated ~bnds appear to follow the contour

4+CO 2U0 do 9oO l~oO SQ~f lj OtO2 100il --.... M... Silicon: t= 2rOr 50.. —--------------..... - so ^=^-=-im im 104 —-----— p3 —---- Silicon: t=t2ra V ^ ~ f 5 Q _2 7 C) r\ J \ ~zCi o> "_75,- I I I... I,, 506 25 0 ___",___'0 r __,.'."+. 0 0 5 O0____ / O0 oil- IJ -i,{. I,;i - -—. —1; —?ijSiJ A q.:3. - U ~onu, n.'7 ]/' ~...,, i. L"r,.-:.,,..1-.*,'i.' ^ v-.!- -j ~,_'1'

169 Table 26 OBSERVED VS CALCULATED MAXIMA IN THE FLITDiAM NTAL ABSORPTION SPECTRA OF DIAMOND, SILICON AID GERMANIUM Observed Calculated Substance (cm-1) (cm"1) Branch Intensit Diamond A: 1280 1285 W3 3 1203 1180 W3 3 1088 1080 04 1 480 (680) W4 1 (370) B: 1400 None 1372 None 1332 1326 &1s, L2 0.1, 0.1 1170 None 1004 None 770 736 5 1 368 330 W5 1 Silicon 610 654, 625 l1, W2 1, 1 520 520 (% 2 Germanium 345 356, 541 Co ~P I, 1 275 284 0 2 200 241 w 1.3

5L5~~ ~170 15 n —- \-\-i-\_________~- i Germa niun 12 (1:W2:W3:Wd4 U 9__12- ___ _____f, 1 1 2::r l f I.w 2 9 ------- __ ____ __-_ —---------— ____ __ __ -U _ 0~~~~~~~~~~~01 C>4 ~~~~~~~~~~~~~~~~~I M = 0~~~~~~~~~~~~~~~~ g 0 2 - ______ _______' \ CT~~~~~~~~~~~~~~~~~~~~~~~~~~% 100 150 2 2 O 350 Cm ~ —S ~ WI~~ ~ ~ ~ ~ ~ ~ ~ ~~ c r - I - I: 2~~~~~~~~~~~~~~~~~~~~~~ 4 0 Silicon5 C), 0 O iO | i ~ z~L~ 1~ = >"'I 1: 2I I 200 3.00 AOO. 500 600 700

171 Group A 33:^4.3: 1 3; *H I / | ~3I ol 0 o 300 500 700 900 1100 1300 1500 (cm-1).. l. ^ ^ Kai \j* ~. i Ul —* rn 4 0 o'l.................... 30o0 500 700 900 1100 1300 1500; A M 13 s c;+. i ig. I 1 i "-4 ~ ~..~ L V -,-.-I. I ll I I~ -.

172 of the observed bands if the relative intensities of the branches -1 are o1: =2::: 04 1:12:1:3. The shoulder at 360 cm falls within the calculated interval of fundamental frequencies and could be due to a shoulder on 1, or 2.' The shoulder at 375 cmn1 falls just at the upper frequency limit of the calculated interval. However, we have no exact data on the position of this upper frequency limit, and it is possible that the 375 cm1 absorption is also a shoulder on c1 or 02 rather than a non-lattice band. SILICON: The calculated position of'1,'o2 fa:lls at a higher frequency than the observed max:ilmum at 610 cm. Tis indicates that our calculations for silicon are less accurate than those for germanium. The difference between the observed and calculated maxiila is 640-610 = 30 cm-l, or about 5o of the observed frequency. The original calculation (section 4,13) showed a deviation of 5% in the ratios of the firce constants, Therefore, the observed error of 5` is compatib- i;aih the.,prrvT a'ff <''2 calculation. rTle calculated,3 rmatches the observed spec trln near 520 cmn, altlcgh the detail of the spectru is somewhat more compl ex than that of our calc ulatoed branch. If the observed peak is as sharp as it appears to be fr-om Lord's data, then singulariti s rmay be responsible. lTe calculation (section 4.1.4) of the fu.r.:niame.ntal branches using n.o econd neigLhboar interaction (B) gave an unidentified singular point near 520 cm-. (see Table 17). If wve assu;e that the nature of the singularities in our Calculation 3 is the same as in the alterna tive calcu!l?tio~.- which does include!second. neighbor xfore s (/.), thnc the pelr obsrved at 520 cmi -1 can be epla-imed in. terml: of a sing ul r

173 point of type (b) at the lower frequency edge of the branch (see Table 17 and Figure 28). The relative intensities 1: op:' 1:1:2 are the same as those used to match the germanium spectrum. DIAMOND: In diamond, the situation is complicated by the fact that the observed absorption coefficients are not constant. In addition, the relative intensities of the bands vary. Ve have constructed separate approximate contours for Group A and Group B type bands where the relative magnitudes of the absorption coefficients have been obtained from Blackwell's correlation charts, and, in the case of the long wavelength bands, from our experimental results. Group A: The observed bands at 1280, 1203, and 1088 cm-1 are reasonably well explained by the calculated contours of 3 and 04 with a relative intensity of 3:1. The band at 21l (480 cm- ) is not well explained. Because the methods of calculation are quite inaccurate, away from the principal maxima of the branches, it is quite possible that the 480 cml1 band is due to a secondary maximum on *4. In Figure 40 we show two weak maxima on o4, both of which may be spurious; however, their presence in the calculated contour indicates the manner in which a secondary maximum can occur in the region where the calculations are inaccurate. The interpretation of the band at 480 cm1 is more fully discussed below. It will be noted that 1 and p2 are not used to account for Group A absorption, although those branches are responsible for the strongest absorption in silicon and germaniun. Analogy with silicon and germanium c"ould hbve led to the assignment of -the 1280 cm!1 band to a1, co and 1203 cm-l to 3. However, we

174 have shown (section 4.1.4) that the calculated position of the maxima of c1 and o2 lies near 1323 cm-l. In addition, the calculated primary maxim:um of 3 lies at too high a frequency to be responsible for the 1203 cm' band. However, it is possible that the force field used for diamond is in error, especially since one elastic constant was altered to fit relations arising from the nature of the assumed force field (section 4.1.1). The principal argument in favor of retaining the present calculation is that no change can be made in the predicted values of o1, O2, and 3 without a corresponding change in 4 05, 6* The maxima of 05 and w6 determine the position of the secondary maximium of the total frequency distribution. It is shown in Appendix A that the specific heat results at low temperatures are sufficiently s ensitive to establish the position of the secondary maximum close to the position obtained with the present force field. Another argument in favor of our representation of 01 and c2 is that the calculated maxima of combinations involving 01 and w2 give good agreement with the observed combination bands, Group B: In this group there are at least two bands (1372, 1400 cm"1),which lie beyond the upper limit of the frequency distribution of diamond. It will be recalled that the Raman line at 1332 cm1' fixes this limit. The weak band at 1540 cm1l which has not been correlated to either Group A or B probably belongs to Group B because of its position with respect to the upper frequency limit. The remaining bands in Group B fall itil'in the frequency distribution of the Ideal diamond. Of these bands, the maxima at 1332 and 770 cm-l correspond to

175 the calculated 1',,2' and 5. The absorption band at 1332 cm-1 is wider than the Raman line at the same frequency. For this reason we assign it to the ol, e2 bands rather than to the single upper frequency. The bands at 1170 and 1004 cm1' fall in the vicinity of strong Group A absor-tion and near the dense portions of 3 and (4. However, their mr.,axima do not correspond to calculated naxima, rind the.y c.n b, rc'.grdod as new frequencies caused by a perturbeaion of the dialmoLnd lattice. The 328 cm'l band (like the 480 cm' band) falls in the region where our calculations are too inaccurate to:maki:e an assignmlent. W-ie will add a note on the long javelength. balnds. If difference bands are considered, it is foun..d cthat the maximum.1 of W3 - c5 falls at the position of the 480 cm band. iowever, that this assignment is unlikely is proved by the fact that the corresponding su mmation band does not occur in absorption (section 4,2.2). In addition, the absorption coefficient at 480 cm'l is variable and appears to be associated with the,iroup A bands. The only othlr ma:Limum calculated to be near 480 cm'" is the rmaximum at 525 cm-1 in 6. The reaaon that t i.'9 blnd h.?9s -ct been used in assignment, i. thr t c6m has another calcula.ted mnax-rimum near 740 cm -. However, the absorption observed near 770 cm-1 belongs to Group B while that at 480 cm 1 belongs to Group A, so that they can have their origins in the same branch only if the branch is not uniformly excited. While the assignment of the other branches has been made on the assumption that they are uniformlly excited, it is not unlikely, that the low freq uc.:n:cies -f acoustical'branches may be non-uniformly excited (see page 151), however, if this were

176 true, it would be difficult to assign bands, simply because the observed maxima might fall anywhere within the interval of frequencies of the branch under consideration. For this reason, we shall not speculate further on this point. ve see from the above discussion, that the absorption bands in the fundamental frequency range of diamond follow the general behavior predicted by Lifshitz. That is, frequencies which can be associoted with the ideal frequency distribution appear in absorption with variable intensity, and, in addition, certain diamonds show additional absorption which cannot be associated with the ideal frequency distribution. It should be emphasized that the non-ideal bands alwvays appear in the saiLe positions although their intensities vary. This intric tes that if the bands are due to impurities, the impurities do not correspond to Lifshitz's idealized case (2), page 163, in which the impurity atom does not alter the force field, for in that case the positions of non-ideal frequencies vary.vith concentration. One might infer th: t the primrary effect of the imperfections responsible for Group B absorption is through a change in the force field. 4. 3 Summiary he have calculated thhe branches of the frequency distributions for diamond, silicon, and: gercmaniu u, using certain of z K. H. J. Srmithts numerical results on diamo 1nd. For silicon and germanium we have assumed thl t there is a constant proportion between the force co;?t~sants of silicon La:n:.d germanium and

177 the constants used in one of Smithts calculations (B) in connection with diamond. The validity of this assumption has been discussed and the proportionality factors evaluated. The observed combination spectra of diamond, silicon, and germanium have been explained in terms of allowed combinations, with the exception of cne we.ak band in germanium which has been shown to agree with a combination band which could become active t'->rough an isotope effect. The observed fundamental spectr-a. silicon and gerrmanium and Group A of dianmnd can be accounted for in terrms of calculated branches of the freu.enc7 distributic-ns c;- tU.e Sustances. The diamocnd Group B absorption has tw-o weal;k maxima lying near calculated branch maxi tc, but it is chiefly composed of bands which cannot be associated T:1ith calcula.ted maxima for the ideal lattice. These results, coupled with the fact that the a bbsorpition coefficien t for ban' in -ure si. ion and,erai re t t.i.l. i n. - Vi iy a -re consistent ifcont1G. -..' o1 i t. chat; isp..>. effect i. as p on p sble for absrpt-ion ira.':ur s0' ilico au. e nLt;iu, bu t.'t impuriies;,OO_. o I~'.1 J..L IL m..: >.,E c;.c..._e;o.. t (1) jie infrared,l::oi'l;? -.i:, on spectra of uiamond, silicon a.nd ger —man.ium can be ac Oounted for in ter'is of the::orn th'ieor? of l.ttice dyinailmics. (7') >*arlysis of the inf rarecd aborp in pe-t;u of diamo.nd in te-auys of the Born inf:,: b u. i. in' t. orn,theory;s l-~,. the:peri- enta:-, fcts 1<..."h htd "lackwel" and "".St to po'oe 1lG)'':.i.:;- e-ory are. in.', c': th on ZoI tl c' L lL ]_ a;z':

Appendixi A THE VARIATION OF SPECIFIC HEAT WiTH TEIIMPERATURE In section 3.1, the effective Debye temperature has been defined as e in Cv = D ( B/T) where D ( 9/T) is the Debye expression for specific heat as a function of temperature. The variation of specific heat with temperature can be expressed as g vs T where deviation from constant 3 denotes deviation from an ideal Debye solid. In Figures 42 and 43 e vs T is 110 111 111 plotted for diamond, boron carbide, silicon carbide 112 113, 117 113 silicon, germanium, and grey tin. Where necessary specific heat data in the literature have been reduced to 6 data. All of the substances except boron carbide have the 114 diamond type crystal structure. Boron carbide has a complex 115 structure not comparable to that of diamond. It is included because it is normally considered to be a "valency" crystal like diamond and it has properties such as melting point and hardness which are intermediate between silicon carbide and diamond, All of the substances display a minimum in ( vs T. This minimum is a consequence of the secondary maximum of the fre118 quency distribution of vibrational modes. Nakamura has shown that for T ce, 4 (T/9 )= / L where L the upper limiting frequency of the distribution and V denotes the frequency region making the dominant contribution to the specific heat at temperature T. In Table 27, the frequency1)l evaluated from Nakamura's approximation, corresponding to the 178

179 1940.-....... Diamond 1900...... 186 182G.. 0 100 200 300 400 T(~ 3L) 1340 43 1300 1260.~ 1220 120 50 100 150 200 T CK) 1080 Silicon Carbide 1040 ---- 1000 - I-\I — 960 920 0 50 100 150 200 T(~ K) Fig, 42 VARIATION OF DEBYE TEIiPrERiATURE: DIA;iOND, B4C, SIC

180 700 - 600 500 \_ —--------------------- Silicon 400 -- 0 100 200 300 400 T (0K) 400 36o -4 320 d) 280 --- E —4-^~ 20lllGermaniumr: 240 d) 0 50 100 150 200 T (OK) 225 175 150.I_ Grey Tin 125 0 25 50 75 100 T (~Ki) Fig. 43 VARIXATIOIN OF DEDYE TEI-PERATUIRE: SI, GE, GREY SN

181 minii mum in 9 vs T is given for each,of:e six subostnces. or diamond, si li Co and germaniun, our alculated values for the frequency of the secondary i1a-ax imu are ",so l isted. Table 27 F RE O z Y.::z..L D..._ _...:) Po S! ic of:c 0. ^.C t y - oi i i.oi.inimun in r om a! cu-1 - 1 Di amond 01820~K 220~K 1o335 x 5.:80 3 ^6 b:.e 12.5r0 ore --....... - r:?b' J 946 80 -- -- --- iLi!icorn 460 40 665 231 2:S ~G.;-i.u: rrman i tmil 25 6 22 37 128 13 Greyr T'in I142 11 -- The a.e nt i -te last to co.um s is very good. It explains. e suy its'; b- Si an > r e cih in cacul b -.- specific hea t uccess b n L v- e -.... fro. their distibutions for diaio;n:: ad.:geri:ioniusi, 1.. the secondary maxima fell at the correct position to account it: the observe6 minriium in vs T. On the othor hand, it -.. c ear that e tt i t th low re119 Quenc..' potio of the dis:i.bution (see e a. ) r;:,..~ il) JO _, (-7 i"e skS t *st X 1.! 7 S-l /- w f' -;;,;:, CJI' 1 cl P.~ i.; (at:1;- I.'. i7 e *t( te**10,. &t' {' -.. - -' t s ci- f t. Ci

182 for silicon, and germanium (Calculation B). Calculation B for diamond places the secondary maximum at 470 cm-1 which is considerably removed from the 645 cm-1 predicted from specific heat data. This is evidence that the 480 cm 1 band observed in absorption is not due to absorption at the secondary maximumi It is also evidence favoring the "second neighbor" calculation for diamond. In Figure 44 are plotted curves for 0/Um vs T/pm for the six substances. m is the average high temperature value of G. The curves for silicon and germanium superimpose over the range of temperatures considered. The scale factor between corresponding 9's and T's is 1.8. In section 4.1, we stated that this result follows from our result that the frequency scales of the distributions for silicon and germanium are related by a factor of 1.83. From the definition of 9 3 h max and the Nakamura approximation one sees that this statement is valid. The remaining curves in Figure 44 show two trends. There is a displacement to lower values of both /m and T/mm as we go from diamond to grey tim. The minima of all the diamond like substances fall on a smooth curve while the boron carbide minimum is displaced from that curve. One sees that some of the properties of the substances such as melting point and hardness can be correlated with the variation in these curves i. e. diamond at one extreme and grey tin at the other. The deviation of boron carbide is to be expected since its crystal asymmetry differs from the other substances. The fact that the

D iara \S / iC l t I\ / ^ / | | Substance,6 -- - \ — / l | Diamond 1930 OK /\ I Boron Carbide 1600 | / A rey Silicon Carbidell50 l / Tin Silicon 685 Germanium 380 Ik\1 1 / | Grey Tin 240 0.04.08.12 416,20 *24.28.32 T/n'm

184 curves for all the diamond-like substances do not superimpose indicates a difference in the forces between atoms in these crystals, i. e. their frequency distributions are not related by simple frequency scale factors.

Appendix B LIST OF DIAMONDS USED IN EXPERIMENTAL WORK Diamond IR DI2AMOND IR K 1 II GM 12 MI 2 WI 38 MI 3 SI 39 SI 4 SI 40 SI 5 Si 41 SI 6 WI 74 II 7 WI 75 WI'GR 1 MI M 4 WI 2 SI 3 SI S I SI I 7 MI 2 SI 13 WI 3 SI 20 WI 25 II T 15 WI 30 II 22 II SLF127 SIK 37 II * SLO 9 SI 38 WI 15 SI 15P SI B 2 SI 25 SI BP 2 II 25P SI 3 II 42 SI CR 1 SI 44P1 SI F 1 SI 44P2 SI 2 SI SYR 1 SI 3 SI 8 SI G 1 MiI 8P SI 2 SI 9 SI 3 SI Note: All diamonds listed were examined for uniformity of transmission at 2537 A. U. $* Stones tested in the vacuum ultraviolet. Code: K - Kaplan; M Parkinson; S - Slawson; T * Triefus; All others - Diamond Trading Company. 185

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