ABSTRACT SURFACE POLARIZATION EFFECTS ON THE F INFRARED ACTIVE LATTICE VIBRATICNS ' OF CRYSTALS By James T. Luxon It is widely appreciated that atomic parameters strongly affect lattice—vibration frequencies. The fact that boundary conditions—-as determined by particle Size and Shape—-strongly affect lattice-vibration frequencies in particulate polar crystals is less widely appreciated. The influence of surface polarization on lattice-vibration frequencies is investigated by examining the extent to which the infrared absorption spectra of polar crystals are influenced by particle Size and shape. The major details of the absorption spectra can be accounted for by classical.electromagnetic scattering theory and a generalized Frohlich equation. derived herein. which relates the absorption frequencies in small particles to the transverse—optical (TO) frequencies in the bulk crystal. Calculations are based on optical parameters 1' James T. Luxon and TO frequencies from i.r. reflection spectra of large TiO . BaTiO . single crystals. The spectra of MgO. SnOZ. 2 3 and SrTiO3 are correlated with particle size and shape as measured by optical and electron-microscopic methods. Frohlich type absorptions in two distinct forms % of MgO powder are found to shift with the refractive index of the surrounding medium in a predictable manner. Absorp- E tions at 400. 550. 488. and 610 cm_1 in the Spectra of three forms-of MgO powder are accounted for on the basis of particle size and shape. In SnOZ. TO bulk-crystal fre— quencies of 605. 284. 243. and 465 cm“1 are expected to shift to 670. 330. 270. and 592 cm-1; this prediction is confirmed experimentally. Similar results are found for T102. In the case of BaTiOB. the TO bulk—crystal frequen— cies of 510. 183. and 33.8 cm.1 are found experimentally to shift to 545. 400. and 180 cm-l; the agreement with calculated frequencies is good when particle shape is considered. Similar results are obtained for SrTiO3. SURFACE POLARIZATION EFFECTS ON THE INFRARED ACTIVE LATTICE VIBRATIONS OF CRYSTALS by James T. Luxon A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Metallurgy. Mechanics and Materials Science 1969 ACKNOWLEDGMENTS I wish to express my sincere appreciation to Prof. Donald J. Montgomery for his patience, constant encourage— ment and superb counsel and guidance. I want also to ex- press deep gratitude to Prof. Robert Summitt, with whom a Close association has been very enjoyable and most reward- ing. I have been extremely fortunate to have had the 0p- portunity to work Closely with two such fine people. I would like to express thanks to all the members of the MMM department, faculty and students, who have shared their knowledge and experience with me. Particu- larly I want to thank Karl Puttlitz for his assistance with the electron microscope. I wish to thank the National Science Foundation for the Science Faculty Fellowship which made it finan- cially possible for me to pursue this work. Finally, I want to say that my wife has made a major contribution to this work by virtue of her patience, understanding, and confidence in me. ii TABLE OF CONTENTS ACKNOWLEDGMENTS . . . LIST OF TABLES. . . . . . . LIST OF FIGURES . . Chapter I. II. III. IV. INTRODUCTION. . . . . . . . THEORY. . . . . . . . . . . A. Generalized Fr6hlich Equation B. Electromagnetic Scattering Theory C. Surface Modes . D. Vibration Analysis. . i. SnOz and TiOz . ii. BaTiO3 and SrTiO APPARATUS AND SAMPLE PREPARATION. EXPERIMENTAL DATA . A. B. C. Magnesium Oxide . Rutile (SnO2 Perovskites (BaTiO 3 and T102). . iii 3 and SrTiO3) Page ii vi ll 18 3O 38 41 42 44 46 52 52 68 73 TABLE OF CONTENTS (cont.) Chapter V. INTERPRETATION OF RESULTS A. Magnesium Oxide B. Stannic Oxide and Titanium Dioxide (Rutile). C. Barium Titanate and Strontium Titanate. VI. SUMMARY . . VII. LIST OF REFERENCES. iv RECOMMENDATIONS FOR FUTURE WORK Page . 79 79 89 96 . 100 . 103 . 105 LIST OF TABLES Table Page 1. Average grain size of Mgo(3) as a function of annealing time and temperature. . . . . 61 2. Observed and calculated maximum and minimum infrared-absorption frequencies for SnO . 2 and T102 powders. . . . . . . . . . . . . . 73 3. The T0 frequencies determined from single- Crystal reflection experiments. together with observed and calculated powder- absorption frequencies for BaTiO and SrTiO3 and dielectric constants . . . . . . 78 4. M90 optical parameters based on the single- crystal-reflection dispersion analysis of Jasperse et al. [32]. . . . . . . . . . 80 5. Oscillator parameters from classical disper— sion theory. and dielectric constants of SnO2 and T102 . . . . . . . . . . . . . . . 90 10. 11. LIST OF FIGURES Infrared active symmetry modes of the rutile structure . . . . . . . . . Electron micrograph of M90 (1). Magnifica- ti0n23o4OOX.............. Infrared absorption Spectra of M90 (1) in KBr and CsI pellets. . . . . . . . Variation of maximum-absorption frequency of M90 (1) with the refractive index of the surrounding medium . . . . . . . . . Electron micrograph of M90 (2) Magnification 7.000 X. . . . . . . . . . . . . . . . . . Infrared spectra of MgO (2) fumed on poly- ethylene -----~--. and dispersed in Nujol -------- . . . . . . . . . Electron micrograph of M90 (3). Grain size 60 A. Magnification 7.000 X . . . . . . Electron micrograph of-MgO (3). Grain size 100 A. Magnification 200.000 x. . . . . . Electron micrograph of M90 (3). Grain size 600 A. Magnification 120.000 X. . . . . . Infrared spectra of MgO (3) of grain sizes 55 A. -.-.-.-.. 210 A. -------- ; and 600 A. -———————. . . . . . . . . . . . . . Variation of maximum-absorption frequency with average grain size in M90 (3) spectra vi Page 43 54 55 56 58 59 62 64 65 66 67 LIST OF FIGURES (cont.) Figure Page 12. Electron micrograph of SnOz. Magnification 54.000 X . . . . . . . . . . . . . . . . . 69 13. Electron micrograph of TiO Magnification 58.000 X . . . . . . . .2. . . . . . . . . 70 14. Experimental and theoretical infrared absorption spectra of SnO . McDevitt [26] —.—.-.-.; ours -------- . heoretical . 72 15. Electron micrograph of BaTiOB. Magnifica— tion 70000 X . . . . . o . . . . . . . . . 74 16. Electron micrograph of SrTi03. Magnifica- tion 7:000 X . . . . . . . . . . . o . . . 75 17. Infrared absorption spectra of BaTiO and . 3 SrTio3 . . . . . . . . . . . . . . . . . . 77 18. Frequency of maximum infrared absorption versus shape factor for Mgo for refractive indices of 1.0 -------- . and 1.5 for the surrounding medium . . . . . . . . 83 19. Variation of the three absorption . frequencies of BaTiO and SrTiO with shape factor . . . .3. . . . . I . . . . . 98 vii SURFACE POLARIZATION EFFECTS ON THE INFRARED ACTIVE LATTICE VIBRATIONS OF CRYSTALS I. INTRODUCTION The most common approach to a discussion of crystal lattice vibrations and their interaction with electromag- netic radiation is the infinite linear diatomic chain of Born and von Karman [1. 2]. This model is a simple. but instructive. device for studying many of the phenomena associated with lattice vi- brations since it is possible to write explicit equations of motion which may be solved exactly. The solutions con- sist of two distinct branches in the Spectrum of frequency versus wave vector. The low-frequency (acoustical) branch corresponds to adjacent atoms moving in the same direction in the limit of long wavelengths. The high-frequency (0p- tical) branch correSponds to adjacent atoms moving in oppo- site directions. again in the long wavelength limit. Both branches may be further split into motions transverse and longitudinal with respect to the wave 1 propagation vector. The longitudinal branches generally are higher in frequency than the transverse branches as a result of polarization buildup in the longitudinal motion which effectively increases the force constants. In the case of polar lattices the transverse optical (TO) modes may be excited by a transverse electromagnetic wave. whereas the longitudinal Optical (LO) modes are optically inactive. Implicit in this model is the assumption of cyclic boundary conditions which mathematically is equiva- lent to assuming that the lattice is infinite. Moreover. the Born-von Karman model in three dimensions commonly assumes the crystal lattice to be diatomic and of cubic symmetry with the result that in the optical branch. there will be but a single triply-degenerate T0 mode and a single triply-degenerate L0 mode. The long-wavelength vibrations of the acoustical branch are equivalent to elastic waves and may be treated phenomenologically from a macroscopic point of view. The long-wavelength optical-branch vibrations. as one might expect. may be treated in a Similar fashion; as is dis- cussed later. this is the region in which first-order phonon-photon interactions occur. In the phenomenoloqical approach. effects of second and higher order are accounted for by the inclusion of a damping coefficient. The macro— scopic approach is taken in this work. as will be seen in section II.A. A microscopic approach to a specific crystal e.g. NaCl. leads to exactly the same conclusions as the macro- scopic approach. It does give more detailed information. but generally in terms of atomic parameters that are un— known or difficult to obtain. such as effective Charge and polarizability. The macrosc0pic approach for a three dimensional crystal involves writing equations of motion for polariza— tion waves for each of the normal modes predicted by group theory. and then solving these equations explicitly. In the case of large single crystals. this procedure leads to a sum-of-oscillators expression for the complex dielectric constant. which in turn enables one to compute the reflec- tivity or transmittance. For "small" crystals a general Frfihlich equation relating absorption frequencies and transverse-optical (T0) bulk-crystal frequencies may be deduced. In the micrOSCOpic approach equations of motion are written for individual ions. This approach necessitates evaluating the effective electric field. i.e.. the field. E seen by an individual ion. This evaluation is ex- eff' ceedingly difficult for anything but the simplest crystals of highest symmetry. In any case the microscopic equations of motion become identical with the macroscopic equations after some manipulation [2]. except that the phenomenologi- cal coefficients are identified with atomic parameters. Previous investigators have shown that a rigorous quantum—mechanical treatment leads to the same result as the classical approach. Hence. unless there is Specific interest in relating atomic parameters to lattice—vibration phenomena. the macroscopic theory is quite adequate. The earliest calculations of Born and von Karman and others assumed infinite crystal dimensions. or cyclic boundary conditions; it is important to check the validity of this assumption. and a number of investigations have been made of the magnitude of the effect of such artificial and arbi- trary boundary conditions. Ledermann [17]. who first looked at this problem. considered only short-range forces and con- cluded that. at most. the effect of free surfaces on the frequency distribution is of the order of 1/L. where L is the length of the edge of a crystal cube in units of atomic spacing. For crystals with significant long-range forces. i.e. polar crystals. the free-surface effects are consider— ably greater. In the case of infrared activity. the free- surface effects are still negligible for crystals with di- mensions ten times or more greater than the radiation wave- length. This. of course. is the case in most experiments on reflection from bulk crystals. and experiments on trans- mission of radiation incident normally on thin films. Although there has been considerable interest in the theoretical aSpects of the effect of finite boundary conditions on lattice vibrations. little work has been re- ported dealing with the interpretation of actual experi- mental data on "small" crystals. Berreman [3] has observed the L0 mode of LiF by employing radiation obliquely inci- dent on thin films. Hass [4] has deposited thin films of LiF and NaCl onto gratings in order to obtain a long. thin. cylinder-like Specimen configuration. He showed that the absorption of infrared radiation varied in a predictable manner for radiation polarized parallel and perpendicular to the axes of the cylinders. Axe and Pettit [5] have in- terpreted the absorption spectra of U02 and Th02 in terms of the shape of particles in the powder spectrum. Summitt interpreted the absorption spectra of SnO powder [6] and 2 fibrous SiC [7] in terms of effects of particle size and shape. One of the earliest discussions of lattice vibra— tions in finite polar crystals was given by Fr6hlich [8]. a He related the absorption frequency of a "small" spherical. isotropic polar crystal. having a single long-wavelength transverse-optic (T0) mode. to the high-frequency and low— g frequency dielectric constants and the TO frequency. Others 1 have considered the problems of lattice dynamics of finite chains [9]. thin slabs [10]. cubes [11]. and spheres [12]. but without experimental confirmation. Many of the treat- ments that have been reported consider only short-range forces; such studies cannot lead to accurate quantitative predictions concerning the interaction of radiation with finite polar crystals. Since here long—range forces. and perhaps retardation. must be considered. Nevertheless. there has been general agreement that additional modes (surface modes) will result from the imposition of finite boundary conditions. and that at least some of these should lead to measurable infrared absorption. Ruppin and Englman [13]. recently have included in calculations the long-range forces and retardation effects. concluding that all addi- tional modes due to finite boundary conditions automatically are accounted for by classical scattering theory. In other words. scattering theory is a continuum approximation to the effect of a finite number of discrete modes which arise entirely from the use of finite boundary conditions. Hence. in principle it is not difficult to account for finite- Crystal modes. The considerations discussed above are not important in intramolecular vibrations. since such vibrations are rel- atively insensitive to other parts of the molecule or its surroundings. The spectra of such materials thus may be ob- tained from powdered specimens dispersed in Nujol and other media. the absorption frequencies showing little if any ef- fect of particle size or surrounding medium. Similarly the nature of the boundary conditions for large single crystals and certain thin film or slab configurations are unimportant in that they have negligible effect on the reflection spec— tra or transmission spectra. respectively. Hence. when dealing with ”large" crystals. although lattice vibrations of ionic crystals depend on the entire array of ions because cxf long-range forces. boundary conditions can be ignored or artificial ones employed. The resulting theory of photon— phonon interaction is simplified; then optical constants can be easily interpreted in terms of lattice vibrations. and optically-active lattice-vibration frequencies can usually be accurately identified. Consequently much theoretical and experimental work has been done on thin films and large single crystals. It is not possible or always convenient to evaporate thin films of some materials. and often it is neither practical nor economical to grow suitable single crystals. It is. therefore. frequently desirable to study g ‘Fa .rn‘ materials in the form of powders. Consequently. because of the scarcity of interpretive information on the spectra of powders. we have undertaken the task of developing appropri- ate procedures for such interpretations. We have selected well-known materials for the investigation to insure ade- quate data for comparative purposes. All of the materials to be considered have been studied both in powder and single-crystal form. We have contributed additional exper- imental information where necessary. This investigation aims to establish the usefulness of powder techniques in far infrared spectroscopy as a substitute for and to comple— ment measurements on bulk single crystals or thin crystals or films. The materials selected for this study were cubic and the ferroelec- MgO. Tetragonal TiO (rutile). and SnO 2 2’ tric perovskites BaTiO3 and SrTi03. MgO was Chosen because of the ease with which powders with particles of various average grain Size could be produced and because of its simple cubic symmetry. TiO2 and SnO2 were Chosen because of their uniaxial symmetry which provides a relaxation of symmetry Conditions.and serves as a check on the applica- bility of our methods to crystals of noncubic symmetry. BaTiO3 and SrTiO3 were selected chiefly because of their interest in the theory of ferroelectricity. and the possi- bility that work on powders of these materials might shed some light on the recurring question of the low—frequency T0 mode of BaTi03. We utilize classical electromagnetic scattering theory as applied to variously shaped particles to predict absorption frequencies. The requisite optical parameters and TO bulk—crystal frequencies are obtained from single- crystal reflection results. Also included is a derivation of a generalized FrBhlich equation which relates TO bulk- crystal frequencies and powder absorption frequencies. This equation is applied to Check the consistency of scat- tering—theory results. and in some cases as a means of predicting absorption frequencies. Infrared-absorption spectra have been recorded for all the materials to Check and extend spectra reported in 10 the literature. MgO powders of various average grain Size were produced. and spectra were recorded. Particle sizes and shapes have been determined by optical and electron microscopy. I I . THEORY This investigation is concerned with the optical properties of polar crystalline materials in so far as these properties are influenced by vibrations of the atoms within the crystal lattice. Specifically. we are interested in the optical properties of crystals having at least one dimension comparable with or smaller than the wavelength of the electromagnetic radiation involved. In a crystal containing N atoms there are N coupled oscillators; according to classical vibration theory. in the harmonic approximation. these will give rise to 3N vi- bration modes. Three of these modes will be pure transla- tions. leaving 3N - 3 true vibration modes which are treated as quasiparticles (phonons) in quantum mechanics. In a polar material. changes in polarization and polarizability. which lead to infrared and Raman activity respectively. ac- company some lattice vibration modes. Phonons with which electromagnetic radiation may interact (referred to as transverse optical. T0. modes in which Oppositely charged ions are moving in opposite directions) typically have 11 12 frequencies of the order of 1013Hz corresponding to infra- red frequencies. Since the shortest wavelength that a lattice-vibration mode may have is comparable with the unit cell dimensions. the maximum value of k = l/l is of the order 108cm-l. When the photon is completely absorbed in creation of the phonon. conservation of quasiparticle momentum gives k(phonon) = k(photon) + Q where §_is the reciprocal lattice vector. which may be neg- lected for our purposes. For a frequency of the order 1013 Hz the photon wavevector l/A is of the order 103cm-l. Hence k(phonon) is of the order 103cm-l. and therefore photon-phonon interactions occur only near the center of the Brillouin zone. i.e. at long wavelengths. This is commonly referred to as the long-wavelength (k = 0) ap- proximation. In actuality these wavelengths are long com- pared with unit-cell dimensions but with respect to overall crystal dimensions they may be smaller than. comparable with. or even larger than such dimensions. The latter two cases are of primary interest in this study. Since the wavelengths in question are long compared with unit cell dimensions. the number of optical lattice- vibration modes is far less than the 3N - 3 possible l3 lattice vibration modes. Indeed. if we assume the infinite— wavelength approximation. equivalent atoms (i.e.. those atoms generated from the unit cell by any of the space sym- metry operations of the crystal lattice) will move in phase. Therefore. since equivalent atoms will have identical mo- tions. the number of possible modes may be determined by considering the number of degrees of freedom of a single unit cell. For example. in the rutile structure the unit cell contains two Ti02 (Snoz) "molecules" (Cf. Fig. 1. ru- tile structure in sec. II.D.) of three atoms each; hence eighteen degrees of freedom and a like number of lattice modes occur. Three of these are pure translational modes corresponding to the transverse acoustical (TA) branches. and the remaining fifteen are transverse optical modes. The result of group—theoretical analysis as applied to the rutile structure will be discussed later. Corresponding to each T0 or TA mode there is a longitudinal mode. L0 and LA respectively. The L0 modes do not interact directly with electromagnetic radiation. The LO-mode frequencies are higher than the corresponding TO frequencies as a consequence of an effective increase in force constant caused by the build—up of polarization along the direction of propagation of the wave. For a crystal 14 having but a single infrared active mode. these frequencies are related by 01/0 = (eo/em)%. (1) where D1 and u are reSpectively the L0 and TO frequencies. 60 is the static dielectric constant. and em is the high- frequency dielectric constant. This equation. first ob- tained by Lyddane. Sachs. and Teller (LST) [15]. has been generalized by Barker [16] for the case of lattices having more than one infrared active mode. to the form njullz/u§ = eO/em. (2) j j For non-cubic crystals this equation applies to polarization along a specific crystallOgraphic direction. The signifi- cance of the absolute value sign on uijill become apparent later. Equation 2. the general LST relation applies to the most general crystal class and includes damping to account for the anharmonic effects. The derivations of Eqs. 1 and 2 are based on the assumption that the crystal lattice is large compared with the vibration wavelength. Since bound— ary conditions are but a small perturbation in the vibra- tions of large crystals. artificial boundary conditions may 15 be imposed at the surface of the crystal without signifi- cantly affecting the validity of these equations. The infrared reflection Spectra of bulk single crys- tals may be analyzed to obtain the TO-mode and LO-mode fre- quencies of optically-active modes. This analysis is made by obtaining a fit to the reflectivity data either by means of a sum of Classical oscillators——which gives the T0 modes directly. and the LO modes from zeroes of the frequency de- pendence of the real part of the dielectric constant e’—-or by the Kramers-KrCnig method--which gives the TO modes in terms of the poles. and L0 modes in terms of the zeroes of the 6’ vs frequency curve. In addition. the TO mode can be measured directly from the absorption spectrum of electro— magnetic radiation normally incident on a thin film. These techniques for identifying the TO and LO fre- quencies all involve crystal dimensions parallel to the di- rection of polarization which are large compared with the wavelength. and hence requires no consideration of boundary effects. Fréhlich has pointed out the difference between ab- sorption frequencies of "small" polar crystals and the T0 bulk-crystal frequencies. He Showed that the absorption frequency 0’ of a spherical Specimen of a cubic polar 16 crystal is related to the TO frequency 0 by [8] u’/u = [(e0+2)//(e +e>13. (4) O 00 where e is the ratio of the major axis a to the minor axis P b of the cross-section ellipse. In the case of a thin film e 5 b/a = o. and Eq. 4 reduces to *5 L. uI/u = (EC/6w) . i.e.. the same form as Eq. 1. except that in this case. 0’ is the absorption frequency. Hass [4] deposited LiF and NaCl on gratings to achieve a configuration of long thin particles. With Eq. 4 he was able to explain the observed difference in fre- quency of the absorption bands for radiation polarized parallel and perpendicular to the grating grooves. Axe and Pettit [5] have used Eq. 3 to explain discrepancies between absorption spectra of U02 and Th02 as powders and as bulk crystals. For this purpose they modified Eq. 3 to account for the dielectric constant em of the medium in which the powder is suspended. u’/u = [(60+26m)/(€m+2€m)]%. (5) 18 Summitt has applied Eq. 3 to explain the absorption spectrum of SnO powder [6] and Eq. 4 to explain the absorp- 2 tion spectrum of fibrous SiC [7]. A. Generalized Frfihlich Equation In this section. we derive a generalized form of i Eq. 3. that is applicable to crystals of all symmetry classes. and includes damping. The development of the theory in Frahlich's treatise [8] is followed closely. generalizations being added where needed. In the absence of electromagnetic radiation. free charge. and conduction currents. Maxwell's equations are v.13 = 0. VXE = 0. (6) for polarization waves in a crystal. Since Q_= E44w2, WE = -4nv.g = -4n11<_.g. (7) for i (i.r-wt) Eager E« e The three vectors E, Q, and g are. therefore. parallel. For longitudinal waves k_is parallel to g_or 2, hence v o 2 = ih’g = 0! requires that Q.= 0 which implies e = O. For transverse waves k_is perpendicular to Q'or P, so that Eq. 7 along with Eq. 6 results in §_= 0. Thus we know that Q_= O. or §_= -4n g for longitudinal waves (8) and. §_= O. or D = 4n §_for transverse waves. Consider now an ellipsoidal specimen whose dimen- sions are large compared with the lattice spacing. but small compared with the wavelength of the polarization wave. In this case the applied field causes nearly homog- eneous polarization in the Specimen. The total polarization is written as the sum of the optical and infrared polariza- tion. .g = P + P. (9) If the ellipsoid is brought into a homogeneous fieldlgo parallel to any of the main axes. the field inside is uni- form and is given by E = _§0/[(eO-1)g+1]. (10) where so is the static dielectric constant and g is a fac— tor depending on the shape of the ellipsoid [18]. From 20 the fact that g =.§(eO—l)/4n we obtain 1 (Go-l) (ll) p.= 4” (€0‘1)9 + 1 go (Static case). If E0 has a frequency which is large compared with the in- F. frared absorption frequency but small compared with the op- tical resonance frequency. then 6 = n2 = em and the equation l corresponding to Eq. 11 is. g [ 30 = (1/4”)(em-1)§o/[(€.’l>9+11~ (12) By definition. in the static case. Here a. do and air are respectively the total. Optical. and infrared polarizabilities. and V is volume. Hence. from Eqs. 12. 11. and 9. air/v = (l/4n)(GO—em)/E(€m—1)g+1][(€m+l)g+Ifl. (14) The field inside an ellipsoid located in a uniform electric field is homogeneous and parallel to the applied field if the applied field is parallel to a principal axis of polarization [18]. If it is assumed that the crystallo- graphic axes correspond approximately to the main axes of the ellipsoid. then the principal axes of polarization may 21 be taken as the ellipsoid axes. In a uniaxial crystal. such as rutile. there will then be two distinct principal polarization directions. parallel and perpendicular to the c—axis. each with a distinct set of static and high- frequency dielectric constants. For each polarization direction we can write down a scalar macroscopic equation of motion for the polarization for each allowed mode. In the case of no applied field. P. + w: P. + 7.P. = O. (15) where ms, 3 is the force-constant parameter (undamped oscil- lator frequency). 7j is the damping coefficient. and the subscript j indicates the j-th lattice-vibration mode. Note that ij/Zw would be the infrared absorption frequency in the absence of damping. It will be shown later that the effect of damping is to lower the absorption frequencies relative to the undamped case. The presence of an external time-dependent field Eo introduces a forcing term to the right side of Eq. 15. 2.. 2 ° (lflmsipir. + (Vj/ws.)Pir.+Pir. = (air./V)Eo. (16) J J J J 3 3 Ft, 3.5,“ __ 22 where the coefficient of EO has been determined by setting Pirj = Pirj = O for the static case. The polarization. Pirj' and polarizabilities. airj' are related to the total polarization. Pir' and total polarizability air' by N and N (1. =20“ I (18) where N is the number of infrared-active modes for the given crystallOgraphic axis. Equation 6 is identical to Fr5hlich's Egg 18.12 if it is restricted to Spheres with one active mode and the damping is neglected. Equation 16 applies to an ellipsoidal Specimen smaller than the wavelength in an applied external field E0. Now consider this ellipsoid to be simply a por- tion of a large crystal. and the field E0 to be due to the remainder of the crystal surrounding the region in question. The field Eo will be proportional to the total polarization which now must be considered to be a superposition of the infrared polarization. Pir' and an induced optical polariza- tion. Po' given by Eq. 12. Furthermore. the field inside 23 the small region. E. is composed of E0. the contribution of the region outside the ellipsoid. and ES. the self-field of the ellipsoid. Since P =(l/4n)(e-l) E and E==EO/[(€-l)g+l] we can write P = (l/4n)(e-l)EO/[(e-l)g+l]. (19) But the field inside the ellipsoid.E. equals the external field. E0. plus the self—field. Es' which. with the aid of the relation between E and E0. leads to ES = -(€-l)gEO/[(E-l)g+1]. (20) Substituting Eq. 20 into Eq. 19 yields E5 = —g4fiP. (21) This equation for the self—field applies to a particular principal polarization direction. and g will take on dif- ferent values for different axes. Previously we established that for longitudinal waves E = -4nP and for transverse waves E = 0. Therefore. E0 = E+g4nP = g4wP(Transverse waves) (22) -(l-g)wP(Longitudinal waves). 24 Now write a = + , Eo q4nP q47r(PO Pir) (23) where q is g for transverse waves and — (l-g) for longi- tudinal waves. Equation 23 is now substituted into Eq. 12 to eliminate E0. This yields P0 = (em-1)qPir/[(em-l)g+l-q(€m-l)]. (24) Introducing Eq. 24 into Eq. 23 produces the desired result forE : 0 E0 = q4nPir[(em-l)g+l]/[(€m-l)(g—q)+l]. (25) It is now possible to write the equations of motion in the form. (26) 2 .. 2 , q47TPiri (600-1) 9+1] (l/wsj)Pirj+(7jflwsj)Pirj+Pirj=(airj/v) (em-l)(g-q)+l Equation 26 represents the equations of motion of the polar- ization waves for a specimen which is large compared with the wavelength. Solutions of these equations yield the T0 bulk-crystal frequencies for q = g. and the LO frequencies for q = -(l-g). in terms of the force constant parameters wsj of Specimens which are small compared with the wavelength. 25 N 2 air ' the coefficient of P. on the l 3 1r -j= Since a. i right side of Eq. 25 can be replaced by Cj' where N c = E c. = (air/v)q4n[g+11/fi(em-1)(g-q)+1fl (27) q(€O-€m)/fi(€o-l)9+l][(em-l)(g-q)+lfl. ei (kx-wt) Solutions of the form Pirj = P3 are substituted into Eq. 26 yielding (28) + . . + ~= . PN) O 2 2 . 2 -(m /ws.)Pl+(1vlw/wsiPl+Pl(l—Cl)+C1(P2+P3 -(w2flm: )P2+(iyzw/w: )P 2 2 +P2(l-C2)+C2(P1+P +. . . +P ) = 2 3 —(w2/w:N)PN+(inwSN)PN+PN(l-CN)+CN(P1+P2+. . . +PN-l) = Equations 28 formaiset of N linear homogeneous equations in the N polarization amplitudes for nontrivial solutions to exist. The determinant of the coefficients must vanish. With A = -w2/w:l + iylw/wsl. and so forth. the determinant 1 becomes 26 1 1 "C1 "C -C2 A2+1-C2 -C2 —CN "CN ”ON ,.,A +1-C N (29) By subtracting column one from every other column this de- terminant may be put in the form. (Al+1)--Cl -C2 _C3 which upon evaluation leads to the result. -(Al+1) -(A1+1) . -(A2+l) O 0 (A3+1) 0 o -(Al+l) O O (AN+1) (30) 27 [(A1+1)-Cl][(A2+l)(A3+l) . . . (AN+1)] +C2 [-(A1+l)(A3+l) . . . (AN+1)] -C3 [(A1+1)(A2+2)(A4+1) . . . (AN+1)] +C3 [-Al+l)(A2+1)(A3+l)(A5+l) . . . (AN+1)] (31) -Cn [(A1+l)...(An_l+l)(An+l+l)... . (AN+1)] - A + + . . . = . cN.[< 1 1)(A2 1) (AN-1+l)] 0 Carrying out the multiplication for a given value of N . . . . N 2N gives an equation with the leading term (-1) w and a 2 2 2 y 2 constant termwsl wsz w83 . . . wsN(l - Cl - C2 - ... - CN). The coefficients of the even-power terms are real. whereas the coefficients of the odd-power terms are imaginary. Con- sequently if mj is a solution. then -w; is also a solution. For an equation of this form the product of the solutions must equal (-l)N times the constant term. Hence. nlel = mm (l—C). (32) where (-l)N has been cancelled on each side. or 2 2 h HQ) /|j | 3 = [(eo-l)g+1]/[(€m-l)9+1] (36) and the L0 modes by. I; w: /|m‘.‘°|2 -[[(eo-l)g+l]/[(e -1)g+11](e /€ ). (37) J j J . °° °° 0 Dividing Eq. 36 by Eq. 37 leads to 29 L0 T0 _ 131le l/le ] - eo/eoo. (38) which gives the LO modes in terms of the TO modes independent of shape. as would be expected. Equation 36 is the desired generalized Fr6hlich equation. and Eq. 38 is the generalized LST relation. Barker [16] has pointed out that the transverse-phonon fre- quencies. defined as poles of the dielectric constant. are related to the undamped bulk-crystal—oscillator frequencies by 03° 2 _+. Jag-73M + i'yj/Z. (39) . 2 . Noting that (w? | = wj Eq. 38 can be written. L02 2_ . g] j l flmj — eo/em. (40) and Eq. 36 becomes Hmz A02 = [(e -1>g+e 1/[g+e 1. (41) j sj j o m ‘ m m where em. the dielectric constant of the medium. has been included. Equation 40. the general LST relation. agrees with BarkerJS result derived from a purely macroscopic point of view. as we have done here. but without introduc- tion of the small-particle absorption frequencies. wsj. 30 In actuality the wsj's introduced here. aside from a factor of 2n. are the undamped small-particle absorption frequen- cies (force—constant parameters). Equation 41 relates the wsj's to the bulk-crystal undamped TO frequencies (force- constant parameters). The actual absorption frequency for the bulk crystal is given by the real part of Eq. 39. Un- less excessive damping occurs. however. as apparently is the case in the "soft" optic mode of BaTiO the damped and 3! undamped frequencies are essentially the same. With the ex- ception of BaTiO . damping in the materials considered in 3 this work is not sufficient to cause a difference of more than one or two wavenumbers between these two frequencies. B. Electromagnetic ScatteringiTheory Classical electromagnetic scattering theory allows us to compute the absorption frequencies of powdered polar crystalline materials from the complex refractive index. m = n - ik. The complex refractive index may be determined from experimental reflectance data by use of Kramers-Kranig analysis or Classical dispersion analysis. All of the re- fractive-index data used in this study has been computed by the latter method. W 31 In classical dispersion analysis. the complex di- electric constant is written in terms of a sum of harmonic oscillators N 4flp.u.2 JIJ e = e + is = em + Z 2 . (42) '-l‘0.-u +i .u u. j J 7J J for polarization parallel to a given crystallographic axis. In this equation em is the high—frequency electronic contri- bution to the dielectric constant. 7j is the dimensionless damping coefficient. 47rpj is the dimensionless mode strength. and uj is the j—th mode oscillator frequency. The damping. yj. takes into account anharmonic effects which cause line broadening. and thus is related to the line*width of the j'th oscillator mode. The mode strength. 4ij. is related to the effective Charge and polarization of the j'th mode. An equation identical to Eq. 42 applies to each unique axis of the crystal. e.g.. rutile requires three oscillators perpendicular to the c-axis. and one oscillator parallel to it. The parameters. Dj' yj. and 4Trpj are adjusted by successive approximation to give the best fit to the exper- imental reflectance data through the relation 32 2 R - [(1 -~/’€)/(ll+~fe)| . (43) 2 where 4np.u.(u.-uz) , 2 2 3 3 3 e = n -k = e +2 2 2 2 2 2. (44) W 3 ‘ - . (uj u ) +7juju and 4n 0 u ” 93'ij 6 = 2nk = Z (45) 2 2 2 2 3 (Di-u ) +7ju In the case of BaTiO3 and SrTiOB. Barker and Hopfield [19]. by use of a coupled-oscillator model. were able to obtain a better fit to the reflectivity data. The dielectric con- stant for a three-oscillator model with two interacting oscillators is given by (46) where 2 . 2 2. — + E =g4nzl+i4nzlzzuvlz/[02 D-HD(72 712)] ’ (47) 1 2 2 . 2 2 2 . 01-0 +10(71+712)+u 712/[02-0 +lu(72+712)] and 62 is given by cyclic permutation of the subscripts in Eq. 47. The remaining dielectric term. 6 is given by 3! _222. E3 - 4nz3/(03-u +1073). (48) 33 In Eqs. 47 and 48. 4nzi/ui is the same as 4npl. and yl/ul the same as 71. in our previous notation. The interaction damping between oscillators l and 2 is accounted for by 712' The reflectance and the complex refractive index are still related to e by Eq. 43. e’=n2—k2 and EAEan. so that the procedure for determining the absorption frequencies of the powders remains unchanged. We have mentioned that Barker [16] has pointed out that the transverse-phonon frequency for independent. damped harmonic oscillators is given by . 2 2‘ 0'30 = ”/2: .[7/4 + u. . (49) J J J J The absolute square of the complex transverse phonon fre- quency DPj' equals the square of the undamped oscillator frequency uj) thus. '1). I (50) 3 which justifies the use of the undamped frequencies in the general Fr6hlich equation (Eq. 41) as well as the general LST relation. The actual frequency at which the phonon- photon interaction occurs in the case of absorption in thin films. thin crystal slabs. or in Raman scattering is 34 Idcyi/4 + u: . which is within one percent of uj for a value of yj/uj = 0.3. Consequently. no Significant Shift should appear unless extreme damping occurs. To estimate the effect of damping in the case of coupled oscillators. we examine the poles of the dielectric constant which are given by P 43. 222 D +D (”nwinIzVizHD ”“2”“1'(72+V12)° 2 . " 2 2 . + . . (71 712)+712]+°(1)_°2(Vi+712)+°i (51) 2 2 By virtue of the form of the coefficients. if DPi is a solution then -DPi is a solution; thus. 2 2 2 2 In I In | = u u . (52) p1 p2 l 2 and _ * _ * - u u + u u 71 + 72 + 2712 . (53) Equations 52 and 53 are based on the fact that the sum of all possible products of the roots of a polynomial taken k . k . at a time equals (-1 ak/ao)for a polynomial of the form. 35 ” Upon writing upl = D61 + iupl. etc.. Eq. 53 leads to Upl + 092 = k(71+72+2712) . (54) If we guess that A! II UPI = %(vl+712) and D92 = 5(v2+712) (55) then Eq. 52 leads to 2 2 . = + - , 0P1 - fflvlwlz) +0 + land/1+7”) and (56) '0 2 2 . + _ p2 _aJ k(72+712) +02 + ki(72+712). That these are the correct roots may be checked by applying the root-coefficient relation to the coefficients of 02 and 0. It may be observed that in the event that the analysis requires coupled harmonic oscillators. the damped and un- damped frequencies differ only slightly except in the case of extreme damping. as in the case of independent damped oscillators. Once the complex refractive index has been computed as a function of frequency. the electromagnetic extinction 36 cross-section may be determined. The extinction cross— (ext) section. C . relates to the energy loss from a radia- tion beam due to a single particle. and may be written as the sum of a pure scattering cross—section. C(sca)' and a . . (abs) . pure absorption cross—section. C . For spheres of arbi- trary size. C(eXt) may be written as a series. (ext) 2 .mz-l 4 mz-l 2 m4+27m2+38 C =-7raIm4x2 +1-ng 2 m +2 m +2 2m +3 (57) 2 2 + . + x4Re-3 m2-1 l +-§ mz-l x + .. m +2 m +2 where x e 2na/A. m a n - ik. and a is the particle radius [20]. For sufficiently small arbitrarily-shaped particles. C(eXt) and C(abs) may be expressed approximately as (sca) 8 4 2 C. = "" Wk 0.. I (58 J 3 l 3) > and C(abs) = -4flkIm(a.) J 3 I where k = 2n/A and the polarizability. dj. is given by v/4mj = gj + l/(mz-l) . (59) The subscript j in Equations 58 and 59 refers to the par- ticular direction of polarization. In the case of'a 37 uniaxial crystal. separate scattering cross-sections must be computed for polarization parallel and perpendicular to the c-axis. In Eq. 59 V is the particle volume and gj has the same identification as previously. i.e.. it is a shape factor which allows us to consider the more general case of ellipsoidal particles. In this work. calculations for various Shape factors have been made. and consequently Eqs. 58 primarily were employed. An additional term in the series expansion for spheres was included in some calcula- tions. however. to provide a basis for judging the effect of particle size. To determine the transmittance of a powder consist- ing of particles of a uniaxial crystal. the incident radia- tion intensity is averaged over all crystallographic orient— ations. The particles may be assumed to be randomly oriented in the absence of any force that might be present to align them. The desired average is determined by finding the average of the square of the electric-vector projection on a particular axis for all possible orientations of the elec- tric vector E. Since the average squared projection for each of three mutually orthogonal axes must be equal (E;2 = E?2 = Eéz). and the sums of the squares must equal 38 the square of the electric vector (E;2 + E&2 + Egz =“Fl-2). the average squared projection equals EQ/B. On the average. then. each axis will see one-third the incident energy. and the effective cross-section. C(eff) should be written as a sum of the cross-sections for polarization along the partic- ular axes. Hence. C(eff) = l C(eXt) + i C(eXt) + -l- C(eXt) 3 x 3 y 3 z _ 2 (ext) _1 (ext) -3 Cx 4-3 Cz . (60) for a uniaxial crystal. The transmittance may then be written as T = 1 - (N/3)[2Cx(eXt)+C;eXt)] (61) (ext) . . . . . where CX applies to the E i_C direction of polarization. (ext) . . . . . C applies to the E H C direction of polarization. and Z N is the total number of particles in the beam. Multiple scattering and interaction have been neglected. C. Surface Modes According to the work of Rupin and Englman [13]. the (ext) calculations of C includes contributionsto the absorption 39 spectrum by surface modes arising as a consequence of long- range forces and finite crystal dimensions. Their treatment of spheres includes electro-magnetic retardation. but neglects changes in short-range forces near the surface. Except in the case of particles whose dimensions are of the order of ten to one hundred unit cells. the Changes in short-range forces would be expected to be minor compared with the long- range electromagnetic force. This assertion is based on the fact that surface-tension effects and changes in mean- square-displacements near the surface of crystals appar— ently vanish within five to ten atomic layers [21. 22]. This contention leads us to conclude that the presence of a free surface has no effect on atoms much more than five to ten unit cells from the surface. except for the case of long-range polarization effects. (ext) Since C accounts for all of the polarization- type surface modes. it is not necessary to consider surface modes directly. It is useful. however. to compare the pre- diction of Ruppin and Englman with our scattering-theory computations. and to examine explicitly the limiting fre— quencies at which surface modes may contribute to absorp- tion. Ruppin and Englman [13] have found that the 4O electric-type modes (which they refer to as surface modes. essentially arising from long-range forces) occur at fre- quencies given by e [koan(koR)]’j£(klR)={kleL(klRfl’nm(koR) (62) and the magnetic-type modes (due to short—range forces) are lav.“ ' u-\1v’vr m r 3 given by 1PT. -. . i o o , o i . i , 3L(k R){k Rnl(k R)} unm(k R){k le(k R)] . (63) In Eqs. 62 and 63 jL and n1 are respectively Bessel and Neumann functions. kl is‘JE ZED/c. k0 is Znu/C. and R is the particle radius. The prime indicates differentiation with respect to the argument of the Bessel or Neumann func- - .+ tion. For kR much less than 1.0. na approximates (kR) (L 1). (kR) and jl approximates (kR)n. These approximations lead to E = —(£+l)/LI L = 11213 . . o (64) for the electric modes; the contribution from the magnetic modes vanishes since. in this approximation. the number of such modes-—proportional to the volume--approaches zero. For a one—oscillator model having a dielectric constant of the form 6 = €w+ (SO—em)/(l-02/u:) . (65) 41 Eq. 64 becomes. Us/ut = [€o+€(.$+l)/£]/[eoo+€m(£+l)/£] . (66) where em has been included to account for the refractive index of the medium surrounding the particles. For.£ = 1 Eq. 66 reduces to the Fr5hlich relation. and for.L = m we obtain the limit to which the surface-mode frequencies con- u /u = (60+em)/(em+em). (67) The total number of modes is not infinite. but is propor— tional to the surface area. Equation 66 suggests that ab- sorption lines should be asymmetrical. i.e.. skewed toward higher frequencies. According to Ruppin and Englman. this effect is more pronounced in larger particles. which is con- sistent with our scattering calculations based on the com- plex refractive index. D. Vibration Analysis This section presents a brief statement of the re— sults of standard group—theoretical analysis as applied to crystals to determine their first—order. long-wavelength 42 optical vibration modes. The case of MgO is excluded from this discussion since only one such mode in this structure (the rock-salt structure. two interpenetrating face-centered (cubic lattices) gives rise to a Change in polarization. This mode is a vibration in which the two lattices (Mg2+ and 02-) are each vibrating as a whole. but in Opposite directions. The number and type of fundamental T0 modes for a y" given crystal structure may be determined by the methods of group-theoretical analysis. Application of these methods to the crystal space group allows the prediction of the sym- metry Species of each of the allowable modes. and quantum— mechanical selection rules predict which modes are infrared or Raman active. Symmetry coordinates may be computed in a Straightforward manner. The symmetry coordinates provide a 'graphic description of the motions of the atoms in the var— ious modes. In the absence of degeneracy the symmetry Inodes are normal modes. (1) SnOfl and TiO2 In the case of SnO2 and TiOz. which have tetragonal Structure. Dig. group theory leads to the irreducible repre- sentation [24]. ‘+3E u u P = A +A +A +B +B +E +2B u g .1 lg 29 2 lg 29 43 The modes of symmetry species A and Eu are infrared ac— 2u tive. The symmetry coordinates of the A2u and three doubly-degenerate Eu modes are reproduced in Fig. l (d) Figure l.--Infrared-active symmetry modes of the rutile structure. Each diagram in Fig. l is a View along the C-axis . . . . + of the rutile unit cell. The small Circles represent Ti4 + . . 2- . or Sn4 ions. and the large Circles represent 0 ions. 44 Dark ions are in the t0p plane of the unit cell. Light Circles are in the mid-plane. + and - represent displace- ment in the positive and negative z- direction (c-axis). It is clear from the symmetry coordinates that the A2u mode is active for infrared radiation polarized parallel to the c—axis. and that the three Eu modes are active in radiation polarized perpendicular to the C-axis. (ii) BaTi03 and SrTiO3 The perovskites. BaTiO3 and SrTiO3. have Oh sym— metry in the cubic phase. and C symmetry in the tetrag— 4v onal (ferroelectric) phase. The ferroelectric transition temperatures for BaTiO3 and SrTiO3 are. respectively. 133°C [23] and -l63°C [34]. Consequently at room tempera— ture BaTiO3 is tetragonal and SrTiO3 is cubic. All of the measurements Of this study were made at room temperature. The unit cell of this structure includes five atoms and hence there are 15 degrees of freedom. For Oh symmetry these divide into representations 4Flu + qu [23] where in each F mode is triply degenerate. One of the modes of Flu symmetry is a pure translational mode (acoustical branch). and the remaining three are infrared active. In the tetrag- onal phase each of the F modes splits into modes of sym- lu metry species Al + E. and the qu mode Splits into Bl + E. 45 Modes of symmetry Species E are doubly degenerate. Thus the Oh——-vC4v transition may be represented as. 3 Flu + 1F2u ———) 3Al + 1B1 + 4E. The Al modes are infrared active for polarization parallel to the c-axis. and the E modes are infrared active for polarization perpendicular to the c-axis. Since the Split- ting of the Flu and qu modes is not expected to be very great. we do not distinguish between Oh and C4v symmetry in this work. Separation of these modes in infrared ab- sorption spectra of these powders was not possible because of extensive broadening of the bands due to variations in particle shape. III. APPARATUS AND SAMPLE PREPARATION The spectra obtained by this writer were recorded on two instruments to cover a frequency range from 167 -l -1 -l -1 cm to 800 cm . In the range from 167 cm to 667 cm a Perkin-Elmer Model 301 double-beam Spectrophotometer was used. To extend certain Spectra to 800 cm-1. a Perkin- Elmer Model 137 double-beam potassium-bromide prism spec- trophotometer was used. Many of the MgO spectra reported herein had been recorded earlier by D. J. Montgomery and K. F. Yeung on the Model 137 Spectrophotometer mentioned above and on a Perkin-Elmer Model 21 spectrophotometer equipped with a cesium bromide prism to cover the region 270-1000 cm'l. The Model 301 spectrophotometer is a double-beam grating instrument which contains a combination of variable- transmission filters. choppers. and reflection elements to isolate a narrow band of the electromagnetic Spectrum and to decrease the influence of stray light. The radiation source in this work was a globar. A Golay detector served to determine the intensity of the transmitted beams and the 46 47 ratio of the two signals is plotted on a strip-chart re- corder. Slit width is varied automatically to maintain a nearly constant energy level over the entire frequency range. The overall performance is very good. with a sig- nal-to-noise ratio of better than 25 to l. and scattered light less than 2%. The resolution is 0.7 cm"1 at 214 cm-1. Spectra could be reproduced within less than 1.0 Possible difficulties may occur from water-vapor absorption below 400 cm-1. and excessive absorption by the polyethylene windows above 600 cm-1. The water vapor prob- lem was minimized by flushing with dry nitrogen. Results obtained with polyethylene windows were Checked with KBr windows to ascertain the effect of the loss of energy. NO significant difference was discernible in the spectra run with polyethylene and KBr windows in the 400 to 667 cm"1 region. Three instrument change points occur in the 667 cm.1 to 167 cm-1 region. Between 667 cm-1 and 320 cm.1 a filter change is required; the reproducibility at this filter change is excellent. At 320 cm-1 the choppers must be changed from opaque to KBr; the reproducibility at this point is satisfactory. A filter change at 210 cm.1 is a 48 little more troublesome for some undetermined reason. but no difficulty has been encountered in interpreting the spectra. If absolute intensities were a matter Of concern. we should have a problem here. The Perkin-Elmer Model 137 spectrophotometer. though a much less sophisticated instrument than the Model 301. yields excellent reproducibility of spectra. Repeated runs give maximum-absorption wavelengths within less than 0.1u. The source of radiation in this instrument is a globar. and the detector is a thermocouple. The spectra are recorded continuously on a drum chart. Slit width is controlled manually. All powder samples studied in this work were pre- pared as Nujol mulls. with the exception in some cases where smoke was fumed directly onto polyethylene or KBr plates. Since absolute intensities were not of particular interest in this work. most samples were prepared without weighing the oil or powder. A sample of a few milligrams of powder was added to five to ten drOps of Nujol in a plastic vial containing a plastic ball of diameter slightly less than that of the vial. The mixture was then agitated by means of a Wig-L-Bug vibrator. This operation broke up loosely aggregated particles and dispersed them throughout 49 the Nujol. The mull was then placed between polyethylene or KBr plates. the Choice depending on the Spectral range. Variations in powder concentration affected only intensi- ties. not absorption frequencies. A pair of plates was always placed in the reference beam to compensate for ex- tinction by the plates themselves. In the case of KBr plates. compensation is excellent. Differences in thick- ness of the polyethylene plates makes compensation less complete but nevertheless satisfactory. A large fraction of the M90 data quoted herein were recorded previously by D. J. Montgomery and K. F. Yeung who used a Perkin-Elmer Model 21 spectrophotometer equipped with a cesium bromide prism to cover the region 270-1000 cm_l. as well as the previously-mentioned Perkin- Elmer Model 137 Spectrophotometer. The quality of the Model 21 spectra exceeds that of the Model 137. but the data are comparable. The samples were prepared in the form of alkali halide pressed pellets. and MgO in the form of smoke deposited directly on KBr plates. Particle size and shape were observed in this work by both the optical microscope and the electron micrOSCOpe. Optical observations were made with a Unitron MPS micro- scope equipped with a Cooke AEI image—splitting eyepiece. 50 This eyepiece consists of a system of prisms linked to a micrometer screw by which the angular relation between the prisms may be varied. When the prism faces are parallel the images are superimposed. When the micrometer screw is turned the images shear across each other. Filters are in- E serted to make one image appear red and the other green. The amount of shear required to make the edges of the images touch is equal to the dimension of the Object. The microm- ; eter is calibrated with a substage micrometer. Measuring accuracy for the image-splitting eyepiece is 0.325u.for a 40X objective lens. Electron micrographs were obtained on a Hitachi HU-ll-A electron microscope. Since great precision in particle—size measurement was not necessary magnification was determined from charts of magnification versus lens current for various pole-piece settings which are supplied by the manufacturer. The negative images are recorded on Kodak electron-image plates. and additional magnification may be achieved when these negatives are printed. Support films for the powder particles were made by coating 400-mesh copper grids with formvar or carbon. gThe carbon—coated grids were prepared by vacuum-depositing carbon vapor on a microscope slide. With a scribe the 51 film is cut into squares approximately 3 mm on a side. The film is then floated off the slide by slowly lowering the slide into distilled water with an angle of about 30“ be- tween the slide and water. The squares of carbon film are picked up on the grids with a pair of pointed tweezers to hold the grid. and bringing it up underneath the film to lift the square clear of the water° The coated grids are then stored by sticking them to two-sided plastic tape which has been placed on the edge of a microscope slide. The formvar film was prepared by placing a few drops of 0.I% solution of formvar in ethylene dichloride on a Clean microscope slide and dragging a second slide across it to achieve a thin coating. After the ethylene dichloride evaporates. the film is cut and placed on the grids in the same manner as the carbon film. The powder particles were deposited on the film- covered grids by dispersing a minute quantity of the powder in distilled water. A specimen is prepared by placing a drop of this mixture on a coated grid. The powder par- ticles stick to the film after the water evaporates. Specimens of MgO smoke particles were prepared by the previous method and also by fuming the smoke particles directly onto the coated grids. IV. EXPERIMENTAL DATA This section contains pertinent infrared spectra obtained from the literature as well as those recorded by this writer and by Montgomery and co-workers. The section is divided into three main parts. according to crystal structure: the first deals with MgO (rock-salt structure). the second with TiO and SnO (rutile structure). and the 2 2 third with BaTiO3 and SrTiO3 (perovskite structure). Each part describes the salient features of the various spectra. and gives the results of optical-microscope and electron- microSCOpe investigations. A. Magnesium Oxide Three forms of particulate MgO were studied in this investigation. The first. to be referred to as MgO(l). consisted of powders Obtained from various commercial sources. such as K and K Laboratories and Harshaw Chemical Company. Although these powders varied somewhat in purity. all were 0.99+ pure and no effects were observed in the 52 53 infrared absorption spectrum which could be attributed to impurities. The optical micrOSCOpe shows that MgO(l) powder contains particles as large as several microns in diameter. After dry grinding in the Wig-L-Bug. the large particles break up into smaller ones. of diameter 0.5u or less. the proportion of larger particles remaining negligible. Grind- ing in Nujol has essentially the same effect. Figure 2 is an electron micrograph of a commercial MgO powder. This photograph confirms the optical-micrOSCOpic findings and enables a more precise estimate to be made of particle size and shape. It can be seen in this micrograph that the average particle size is indeed less than 0.5u. and that the shape is fairly irregular though devoid of flat faces or sharp corners and edges. In Fig. 3 are illustrated two typical absorption spectra Obtained by Montgomery and co-workers. One spec- trum is Of MgO dispersed in KBr and pressed into a pellet. 'the other is of MgO dispersed in CsI and pressed into a pellet. There are two strong. broad absorption bands centered near 550 cm-1. The maximum-absorption frequencies are slightly different for the KBr and CsI pellets. Figurel4 is a plot Of observed frequency of maximum abSOrption 54 Figure 2.-—Electron micrograph of M30 (1). Magnification 23.400 X. 55 .mcsmw .m .M can wwwEomucoz .b .Q “muuommm mo mousomx OGOHUHE CH nuwomam>mz mm mm Hm ma NH ma ma _ _ q . a _ OH om om ov 0m 00 on om OOH R.mumaamm HmO pom me CH Adv om: mo muuommm COHDQHOQO pmumuwcHlu.m musmam quaozad uI aoueqarmsuexm 56 xwvofl T>Huomum0m oa.H 66.H om.a oa.a . _ _ . .omm 1 0mm cum HmuooEAHmmxm 0mm HMOfluouomna 1 00m Ohm .Eoflpoa Oswpcsouusm 0:» mo xwpcfl m>flu Iomummu may QDHS Adv om: mo mocmsvmuw coHumHOmQMIEDEHxCE mo coflumflum>ll.¢ mnsmflm mo) Kouenbexg uorqdzosqe-mnmeew (I- 57 versus refractive index of pellet material for KBr. KCl. NaCl. CsBr. and C31. The absorption frequencies for each pellet material are averages of the values obtained from several Spectra of different pellets. The second form of MgO. to be referred to as MGO(2). is obtained from burning magnesium ribbon in air to form-MgO smoke. An electron micrograph of this smoke deposited directly from the burning ribbon onto a formvar- coated COpper grid is given in Fig. 5. The appearance Of micrographs of MgO(2). first agitated in water then evap- orated onto formvar-coated grids. is essentially the same. The particles appear to be very nearly perfect cubes. ex- cept for numerous chain-like particles composed of well- defined cubes stacked in a staggered fashion. Many of the single cubes are 1.50 or larger on an edge. whereas the chains mostly consist of small cubes..0.3u or less. stacked into chains as long as l.5u.* A reproduction of the absorption spectra of MgO(Z) fumed directly onto polyethylene plates. and also of the same material dispersed in a Nujol mull is given in Fig. 6. *These thin Chains are easily seen on the negative but are not distinct on the print presented in this thesis. 58 Figure 5.——Electron micrograph of MgO (2) Magnification 7000 X. 59 AHIEUV aocmsvmum oom 00¢ com com con - u _ \l/ \ /.I \ I \ l./ \ I [om \\./ \\ \II /III /\ o a/ I/ x .\x .1/. /.// es \\ //. / 1 . . III.ILV 1.) \ z” e . . I \ / .\ .. x . . ll..\\ on \ \ -om . IIIIIIII HOnsz aw pomnmm ImHO pom ..|.|.I.| OCOHMSDONHOQ no ocean ANV om: mo muuoomm UwumumsHll.o wusmflm (%) aoueqqrmsuex; 60 A very strong absorption Occurs at 550 cm.1 and 488 cm.1 re- spectively. for the MgO(2) fumed onto a polyethylene plate and dispersed in a Nujol mull. Another fairly strong. sharp absorption band occurs near 400 cm-1 in both spectra. A weak absorption at 665 cm.1 in the spectrum of MgO(2) fumed on polyethylene shifts to 610 cm"1 for MgO(2) in Nu— jol. This value is near the short-wavelength limit of the Perkin-Elmer Model 301 spectrophotometer. and is in a re— gion where polyethylene absorbs strongly and thereby limits the available energy. Nevertheless. the Shift of this weak absorption with variation in the medium surrounding the particle is evident in work carried out in Montgomery's laboratory. This work was carried out with KBr plates and KBr pellets. the Perkin-Elmer Model 21 and Perkin-Elmer Model 137 spectrophotometers. whose ranges extend. respec- tively. to 1000 cm-1 and 800 cm-1. The third form of MgO. to be referred to as MgO(3). is obtained from the thermal decomposition of MgCOB. The grain size of the resulting particles can be controlled by varying the temperature and the duration of heating. Aver- age grain diameters from 50 to 1000 A are easily Obtained. In Table l are listed the various average grain sizes that 61 are Obtained by annealing for various combinations of time period and temperature according to Birks and Friedman [25]. TABLE l.—~Grain Size in angstroms as a function of annealing time and temperature. Number of Hours Temperature 2 3 4 6 8 12 400°C 55 60 65 70 600°C 100 110 125 140 800°C 210 240 260 300 900°C 380 500 1000°C 600 700 800 900 The results shown in Table 1 agree with our electron- microscope findings within a few angstroms for annealing time—temperature combinations of 2 hours at 400°C. 2 hours at 800°C. 3 hours at 800°C. 2 hours at 1000°C. (C.f. Figs. 8. 9.) Hence. we have accepted the values in Table 1 as be- ing reasonably accurate for all the powders which were pre- pared in this manner for our infrared absorption studies. Optical-microscope and low-magnification electron-microscope observations (Fig. 7) show the MgO(2) particles to be fairly large (up to five microns across). and somewhat platelike. 62 g! - . F. -. . ,9 . '4 . . ‘ ‘ 4:21. I ,~ [x I ‘r ,au -11, $11. 9'" Figure 7.-—Electron micrograph of M90 (3). Grain size 60 A. Magnification 7.000 X. 63 The higher-magnification micrographs (Figs. 8. 9) reveal the grain structure of these particles. A reproduction of the spectra Obtained from MgO(3) having average particle diameters Of approximately 55.210 T7 and 600 angstroms is presented in Fig. 10. Spectra re- corded for particles of larger grain sizes appear the same as the spectrum of the 600 A grain-size particles. The specimens were all prepared in Nujol mulls and the spectra - recorded on the Perkin-Elmer Model 301 spectrophotometer. The chief characteristics of the spectra are a very strong. distinct absorption band near 400 cm-1. and quite strong absorption to near 730 cm.1 with no definite cutoff fre- quency. The latter Characteristic has been confirmed by extending the spectra to 800 cm.1 on the Perkin—Elmer Model 137 Spectrophotometer; .As grain Size increases. the 400 cm.1 band narrows and shifts toward lower frequency. The curve in Figure 11 represents the variation of the maximum-absorption frequency with average grain diameter for grain diameters of 55. 100. 210. 380. 600 and 900 A“ Although relative intensities are difficult to compare. it is apparent that the absorption away from the 400 cm_1 band lessens with increasing grain size. and that a distinct 64 Figure 8.-—Electron micrograph of MgO (3). Grain size 100 A. Magnification 200.000 X. 65 Figure 9.—-Electron micrograph of M90 (3). Grain size 600 A. Magnification 120.000 X. 66 con AHIEOV hocmsvmum coo 00m a _ .m 000 0cm “IIIIIIII .w cam “.I.I.I.I .m mm mmufim Camum mo Amv Om: mo muuommm CmumnwcHll.oa musmflh OH ON om ov 0m 00 on quaoxed uI aouequmSUEIL 67 com mEouummcm a“ wufim samuo com com com OOH q _ q — _ . .muuommm Amv Om: CH muwm Camum mmmum>m £ua3 hocmsvmnm COHDQHOTQMIEDEHxCE mo ooflumfium>ll.aa musmflm com 00? Gav (T_mo) Kouenbeig uoquzosqe-mnmrxew 68 absorption band appears at about 500 cm.1 with increasing grain size. There are several significant similarities and dif— ferences in the spectra of the three forms of MgO powder. R“ All of the spectra exhibit considerable absorption from roughly 400 cm-1 to 730 cm-1. The MgO(2) and MgO(3) Spec- tra have in common a distinct band at approximately 400 E cm_l. No such band appears in the MgO(l) spectrum. The .L MgO(l) and MgO(2) Spectra have intermediate bands. i.e.. between 500—600 cm-l. and the MgO(3) spectra of larger— grain-size particles apparently begin to develop a distinct band in this region. The MgO(l) and MgO(2) spectra contain distinct. weaker absorptions above the intermediate range (500-600 cm-l) for MgO(2). and below this range for MgO(l). There are no clearly—defined absorptions of this type in the MgO(3) spectra. B. Rutile (SnOn and TiOz) 4 Examples of the electron micrOgraphs of SnOz and TiO2 powders are shown respectively in Figs. 12 and 13. Generally the largest dimensions of the particles do not 69 L ) e Figure 12.——Electron micrograph of SnO 54.000 X. 2. Magnification 70 a ' ‘\ L [2" g E Figure l3.--Electron micrograph of T102. Magnification 58.000 X. 71 exceed 0.50 for T10 and 1.5u for SnO In both powders 2 2' relatively few particles exceed 0.5u. The particle shape tends to be rather random. though the particles of both powders tend to exhibit flat faces and fairly sharp edges. 95 especially in the larger particles. SnO2 particles exhibit more faces per particle. up to twelve. whereas TiO par— 2 ticles seem to have no more than six to eight faces. In Fig. 14 is given the experimental absorption ' spectrum of SnO powder as obtained in the present work. 2 and in the work of [26]. These spectra. which are in fairly close agreement. indicate the existence of four absorption bands. with considerable broadening in the higher-frequency regions. Similarly. four absorption bands are obtained for TiO2 by McDevitt and Baun [27]. and by Afremow and Vandeberg [28]. We have found. how— ever. that in contrast with the findings in SnO2 the ab- sorption frequencies and band shape vary substantially with the impurity content and the method of preparation of the TiO2 powder. Rutile powders of various impurity content (95% T10 to 99% T102) were studied. and all of 2 our Spectra differ in some detail from those of McDevitt and Baun and of Afremow. To some extent the Spectra of these investigations disagree with each other. The Transmittance in percent 72 80 20 _ l l l l l l 800 600 . 400 200 Frequency (cm-1) Figure l4.--Experimental and theoretical infrared absorption spectra of SnOz. .McDevitt [26] —-—-----; ours -------- ; theoretical 73 observed frequencies of maximum and minimum absorption for TiO2 and SnO2 powders are contained in Table 2. TABLE 2.--Observed and calculated maximum and minimum infra- red-absorption frequencies for SnO2 and TiO2 powders. n1 Material FrequenCies of FrequenCies of absorption maXima absorption minima v1 v2 v3 V4 Sno2 obs* 670 325 270 610 283 295 635 EJ: SnO2 calc 670 330 270 592 290 310 630 TiO2 obs+ 610 425 350 680 365 460 650 TiO2 calc 660 440 360 700 400 500 690 . + ' . *Data due to McDev1tt[26]. Data due to McDeVitt & Baun[27]. C. Perovskites (BaTiO, and SrTiOB) Electron-microscope observations of BaTiO3 and SrTi03(Figs. 15 and 16) indicate that maximum particle dimensions seldom exceed 1.5u and that the particles tend to have nearly cubic shape. with some of the larger par- ticles exhibiting very flat faces and extremely sharp edges, Perry. Khanna and Rupprecht [29] recorded the ab- sorption spectrum of SrTiO powder from 1 to 1200 cm—1 3 74 l/l I—-| Figure 15.——Electron micrograph of BaTiO tion 7.000 X. 3. Magnifica- 75 Figure l6.——Electron micrograph Of SrTiO . . . 3' Magnification 7.000 X. (C.f. Fig. 17) 76 They observed three distinct absorption bands. In our laboratory we have verified the essential features of this Spectrum in the region 167 to 800 cm- 1 Numerous people have recorded the Spectrum of BaTiO . but 3 in the literature we could find no Spectra which extended below 300 cm—1 In higher frequency regions. Last [30] and Spitzer et a1. [31] found absorptions at 545 and 400 cm“1 We have extended the spectrum of BaTiO 3 powder down to 167 cm-1. and thereupon verified the previously mentioned absorptions and revealed a third weak absorption in the neighborhood of 180 cm-1. Figure 17 contains Last's spec- trum for BaTiO listed the observed absorption frequencies for both SrTiO and BaTiO3. 3 extended to 167 cm-1. In Table 3 are 3 Transmittance in percent 100 77 80 . 6O 40 20 100 _ 80 60 - 40 h 20 _ 1 (a) (b) BaTiO SrTiO l l 700 600 500 400 Frequency (cm-1) 300 200 100 Figure l7.--Infrared absorption spectra of BaTiO and SrTiO3. (a) (b) 3 700 to 300 cm‘l. Last [30] 300 to 167 cm-1 our work Spectrum due to Perry et al.[29] '78 F .. {I'll “liliflsild .Hmau vamflmmom can umxumm Eonm who mocam> penumcoo oeuuomamwo++ .m .nmNH .HC um muuom ou ofiaum How .HOmH anon on map mum newamm mom moaooosumum coeumuomn¢+ .HHmH .Hm um Houuwmw scum mud nonau> Ononau a voooaooum com maa med m.pm n.m u a o .axu oov o~¢ cow and moaaum ass a 6 .mx« new new new man couoaooua mom 66H m.nm O .u u m u .axo n a 664 may cow and Oaacm ooo~.u a .awo new com mam H...-0 can moconvouu coaumuounu +NQEKE aucmavoum 09 ++muadumcoo mucosvoum :oflumuonnu mocoauoum coaumHOOQO «Hauuauouoamaam bauuooaowo cowumHaoaoOInostnuh sowuuasuHMOIucwuouumom .musmumcoo owuuooamao vow modaum can Oflamm Mom moaocmsvmum COMDQHOOAMIuopsom OODMHDOHMO pom pm>u0mno £uw3 umnuomou .mucosaummxo sowuomawmu Hmumhuoloamcflm Bonn pocfleumump mowocmsvmnm OB onall.m mqm<9 m V. INTERPRETATION OF RESULTS The major features of the infrared spectra of MgO. SnO . TiO . SrTiO . BaTiO . we submit. can be explained as 2 2 3 3 a consequence of particle shape and particle size. This portion of the thesis is concerned in part with applying the FrOhlich theory and electromagnetic scattering theory to explain the spectra of materials consisting of "small" particles of various shapes. Minor details not accounted for in this manner are attributed to grain-boundary effects. and to the fact that real powder specimens do not adhere strictly to the small-particle assumption. As in the pre- vious section. the discussion will be divided into three parts. viz. one for each crystal structure studied. A. Magnesium Oxide In principle. the absorption frequency of powdered MgO may be calculated by either the FrOhlich relation. Eq. 5. for small spheres. or Eq. 41 for other shapes. or alter— natively by means of electromagnetic scattering theory. 79 80 For small spheres imbedded in a matrix of refractive index 1.5. calculation of the transmittance from Eq. 61 now re- (ext) . . . duces to T = l—NC . for MgO which absorbs isotropically. Computation of the complex refractive index of MgO required for the above calcualtion was based on parameters given by Jasperse et a1 [32]. who took a two-oscillator sum to fit the reflectivity data of MgO. The apprOpriate parameters are listed in Table 4. TABLE 4.--MgO optical parameters based on the single-crystal-reflection dispersion analysis of Jasperse et a1. [32]. Frequency(cm‘1) Strength(4np) . Damping 7 Utl 401 6.60 0.019 Dt2 640 0.45 0.160 e = 3.01 e = 9.66 00 0 The low-strength mode at 640 cm-1 required to Obtain a good fit to the reflectivity data has little effect on the calculated transmittance. and no effect on the position of the absorption maximum. Equation 5. as well as Eq. 61 yields a value Of 550 cm”1 for the absorption maximum under the assumption of spheres dispersed in a matrix of refractive 81 index 1.5. Figure 4 contains plots of frequencies of maxi— mum absorption as Obtained experimentally and frequencies of maximum absorption as obtained from Eq. 5 as a function of the refractive index of the surrounding medium. The vertical separation of these curves is l to 2%. which is '3 well within the uncertainty in ut. the TO frequency. In addition. em is probably not known to within better than 1 or 2%. Values of em from 2.95 to 3.05 are found in the L literature [32. 33]. We assert that the results illustrated in Fig. 3 establish the nature of the strongest absorption in the M90 (1) Spectrum (Fig. 3) as being of the FrOhlich type. i.e. at a frequency predicted by Eq. 5. Further support for this assertion is obtained by an inspection of the electron micrograph of MgO (1). Fig. 2. which shows that y a large proportion of the particles do indeed approximate spheres. The secondary absorption in the MgO (1) Spectrum also is observed to shift toward lower frequency with in- creasing refractive index of the matrix. although no de— tailed analysis Of this shift has been made. A possible explanation of this absorption may be sought in a consid— eration of the presence of a fairly large incidence of 82 double particles; axial ratio 2:1. Figure 18 shows plots of maximum absorption frequency versus shape factor 9 for matrix refractive indices of 1.0 and 1.5 for MgO. as calcu- lated from Eq. 41. If it is assumed that these particles approximate prolate spheroids. the shape factors for polar- fl ization parallel and perpendicular respectively to the long axis are gl|= 0.172 and g _L and n = 1.5 yields an absorption frequency of 450 cm-l. = 0.414. From Fig.18. gll= 0.172 which agrees closely with the experimental value of 460 cm-1. The absorption frequency corresponding to g = 0.414 1 occurs at 540 cm.1 which is too Close to the spherical- particle absorption of 550 cm”1 to be resolved but which contributes to the strength of absorption in this region. The strongest absorption in the MgO(2) spectrum is interpreted in a fashion similar to the interpretation of the strongest absorption in MgO(l). It is Clear. however. that this absorption. which occurs at 546 cm-1 for smoke fumed on polyethylene and 490 cm-1 for smoke in a Nujol mull. is not due to spherical particles. for these. accord- ing to Eq. 5. would absorb at about 605 cm.1 when deposited on polyethylene and at 550 cm—1 in a Nujol mull. The cub- ical character of the smoke particles. Fig. 5. leads one to expect absorption at lower frequencies than for spheres. 83 uouomm wmmnm o.H m.o 0.0 ¢.o N.o 4.. _ . j j . . A a q _ .EDHUOE mcflpcsouusm mnu How Illlllll.m.H Ucm . IIIIIIII o.H mo mmOflOCH O>Huomum0u mom Omz How Houomm wmmnm msmum> COHDQHOQO Consumcfi Esaflxme mo hocmsvmumll.ma musmflm 00¢ 000 com (I_mo) uoquiosqe-mnmrxem go Xouenbexg 84 This shift is a consequence of the lower overall depolariza- tion effect which gives rise to effective force constants that are less than would be obtained with spheres. No at- tempt has been made to solve the quite difficult problem of computing the depolarization (shape) factor for cubes. Ii It will be shown that it may be estimated. however. with the aid of Fig. 18. From the curve for a matrix refractive index of 1.0 an absorption frequency of 546 cm-1 (Observed L’ for MgO smoke fumed on polyethylene) yields a shape factor of 0.21. This shape factor leads to absorption at 500 cm”1 for a refractive index of 1.5. according to the lower curve of Fig. 18. in good agreement with 490 cm_1 observed for MgO smoke in a Nujol mull. Hence. it is reasonable to con— clude that the strongest absorption in the MgO(2) spectrum is of the FrOhlich type. Eq. 41. for cubes. and that the appropriate shape factor for cubes is approximately 0.21. The weak absorption that appears as a Shoulder in the MgO(2) spectrum may also be interpreted as a particle- shape effect. Again from Fig. 18. an absorption at 665 cm-1 yields a shape factor of approximately 0.5 for a matrix refractive index of 1.0. This shape factor in turn leads to an absorption at 610 cm.1 according to the lower curve of Fig. 18 (matrix refractive index 1.5). which agrees 85 closely with the value of 607 cm-1 Obtained for MgO smoke in a Nujol mull. A shape factor of 0.5 corresponds to ab- sorption due to polarization perpendicular to long thin. rod-like particles. Polarization parallel to these rod- like particles would cause absorption at the TO bulk-crystal frequency. which incidendally. is present in this spectrum. although it is not believed that this is the primary cause for absorption at the TO frequency. Such rod-like (or Chain-like) particles do occur in the MgO smoke. Fig. 5. The regularity of the stacking of the cubes in these part— icles indicates that they are single crystals whose growth has followed a characteristic pattern; hence. intimate con- tact is realized between the stacked cubes along the chain. The primary cause for the fairly strong. sharp ab- sorption at the TO bulk-crystal absorption frequency (~1400 cm_l) is the presence of large MgO cubes for which the small- particle assumption is invalid. The real part of the com~ plex refractive index of MgO is approximately 15.0 at 400 cm-l. This high value produces an internal wavelength of 25u/15 = 1.67u which is comparable with the particle size of the larger particles. A few wavenumbers above 400 cm-1 the real part of the MgO refractive index is reduced by at least an order of magnitude. and the small-particle 86 assumption is again valid. .As pointed out previously. the chain or rod-like particles. as well as the large cubes. will absorb at the TO frequency. and neither will exhibit shifts with variation of the refractive index of the sur- rounding medium (matrix). The sharp low-frequency cutoff of this absorption is consistent with this interpretation. Since in the harmonic approximation under no circumstances can the TO frequency lead to absorption below the bulk- crystal value. Furthermore. the half-width of the absorp- tion is comparable with the half-width of the peak in the real part of the complex refractive index 25-30 cm-l. which lends additional credence to the large-particle explanation. The strong absorption around 400 cm_1 in the MgO (3) spectrum (Fig. 11) as in the previous case of M90 (2). results from the presence of particles which are not small compared with the relevant wavelength of radiation. that is. the internal wavelength. AS optical microscopy and electron micrographs (Figs. 7. 8. 9) indicate. M90 (3) consists of large flaky particles (a large percentage are several microns across). with a fine-grain structure. Be— cause of the cubic structure of MgO. the grain structure does not alter the general properties with respect to 87 lattice-vibration absorption. In other words the composite of small grains agglomerated into a single particle behaves optically as a single crystal except for grain-boundary perturbations. The effect of the grain boundaries is mani- fested both in broadening of the absorption with decreasing grain size-(C.f. Figs. 8. 9) and in shift of the abSorption frequency to higher frequencies with decreasing grain size (cf. Fig. 10). Although a complete analysis of the grain-boundary effects has not been attempted. some general observations may be made. Both effects correlate Closely with the variation in percent of surface atoms of the grains. For a grain size of 55 A. more than 50% of the atoms are surface or grain-boundary atoms. At a grain size of 380 A. the percentage of surface atoms is less than xx. Up to particles with average grain size of 380 A the variation in line-width and maximum-absorption frequency is appreciable; beyond this grain size. there is no percep- tible variation in either of these spectrum properties. In- creasing line—width with decreasing particle size is un- doubtedly a result. in large part. of the inefficient (ire regular);packing in the grain boundaries. An additional effect. particularly with respect to the skewing of the ab- sorption line toward higher frequencies and the resultant 88 shift in maximum-absorption frequency. arises from the overall shape of the particles (not to be confused with the grains which make up the particles). The fact that the smaller grain-size particles (cf. Fig. 10) exhibit relatively stronger absorption at both the high-frequency limit. the LO bulk-crystal frequency. and at frequencies just above the TO bulk-crystal mode-—as well as in between-- suggests the presence of particles of highly irregular shapes. As discussed previously this conjecture has been confirmed by the low-magnification electron micrograph Fig. 7 of M90 (3) (approximately 60 A grain size). The platelet-like particles will absorb near the LO mode for polarization perpendicular to the large surface and near the TO bulk-crystal frequency (on the high side. Since the particles are not infinite) for polarization parallel to this surface. Other irregularities in shape will cause absorption at intermediate frequencies. The two extreme cases cited above may be confirmed by substituting the appropriate shape factors into Eq. 41. For E parallel to the large surface. g «'0.0+. for E perpendicular to the large surface g = 1.0 -. For other shapes 9 lies between zero and unity. 89 It has also been pointed out in Sec. IV that as the average grain size increases. the grains become more clearly defined and begin to separate. that is. strong intimate con— tact is reduced (Cf. Figs. 8. 9). Consequently. the par- ticles will become less irregular. as sharp edges are more easily knocked off and thin platelet-like particles are broken up in the grinding operation. This interpretation is consistent with the relatively diminishing absorption at the L0 and just above the TO bulk-crystal frequencies. It also accounts for the appearance of the distinct inter— mediate absorption for larger grain sizes. The latter ab- sorption is a result of polarization-induced (FrOhlich type) absorption of the grains themselves acting as inde- pendent particles wherever the grain boundaries between individual grains have broken down. B. Stannic Oxide and Titanium Dioxide (Rutile) To analyze the absorption spectra of powdered SnO2 and TiO2 we first interpret them in terms of the generalized FrOhlich relation. Eq. 41. The particles are assumed to be approximately spherical (g = 1/3). 90 Electron micrographs. Figs. 12. 13. indicate that this assumption is more accurate in the case of SnO2 than for TiO2 as discussed in Sec. IV. For the single mode that is active in polarization parallel to the c-axis (E No). Eq. 41 becomes: ug/u4 = [(60 + 26m)/(e>0 j-Zém)]%. (68) ‘H H With the appropriate parameters from Table 5 the frequen— cies for SnO2 and TiO2 are found to be respectively 592 cm-1 and 700 cm-1. TABLE 5.--Oscillator parameters from Classical diSpersion theory. and dielectric constants of SnO and TiO . 2 2 Frequency(cm’l) Strengths 01 02 O3 O4 4flp1 4flp2 4fip3 4fip4 Sn02* 605 284 243 465 1.64 1.25 5.8 5.4 Ti02+ 479 386 189 189 2.0 2.2 78.5 165.0 Damping Dielectric Constants 'Y 'y 'y 'y 6 £0 E 6 1 2 3 4 °ll _L °°l| °°_L SnO 0.034 0.022 0.032 0.040 9.6 12.5 4.17 3.78 2 T102 0.025 0.032 0.101 0.040 173.0 89.0 8.4 6.0 *SnO2 data due to Summitt [6]. +TiO2 data due to Spitzer et al. [31]. 91 Referring to the absorption frequencies listed in Table 2. we may see that absorption bands occur at 700 cm"1 for TiO2 and 610 cm.1 for SnOZ. We therefore assign these bands to 0;. thereby identifying the frequencies corre5pond— ing to the E||c mode. The remaining three bands then are u’ and 0’. without attempt- assigned to the E-Lc modes. 0’ 2. 3 1' ing exact identification. To check this assignment. we write Eq. 41 for E||c again taking 9’ = 1/3 in the spherical approximation: ’/0 u 03 = [(eo.+2€m)/(e0° +2€m)]% (69) and substitute appropriate data from the oscillator parame— ters in Table 5 into this equation. This procedure gives for the left-hand and right—hand sides. respectively. for SnO2 1.40 and 1.47. and for TiO2 2.45 and 2.98. The agree- ment in the case of SnO clearly is better than for TiO 2 2' The refractive index n and extinction coefficient k have been calculated as a function of wavelength from the dispersion parameters listed in Table 5 for both SnO2 and TiOz. With these values of n and k. C(eXt') was calculated _ point-by-point as a function of wavelength for E||c and Bio. Then from Eq. 61 the transmittance may be calculated. The 92 predicted and experimental frequencies of maximum and mini- mum absorption are listed in Table 2. The agreement is ex— cellent for SnO . but not quite so good for TiO 2 Figure 14 2. is a plot of the calculated transmittance for SnOz. The transmittance T has been adjusted so that the maximum ab- sorption of the experimental curve of McDevitt agrees with the theoretical one. Hence. absolute intensities in the plot are meaningless. though a rough estimate of the number of particles in the beam does lead to the correct order Of magnitude for T. The relative intensities of the absorp- tion bands in the theoretical curve agree somewhat better with the spectrum of SnO recorded by the writer for sub- 2 stantially lower Overall absorption. The better agreement between the experimental and theoretical curves for rela- tively low absorption may indicate that interaction and multiple—scattering effects alter the relative intensities when the particle density in the mull or pellet is high. C(ext.) As previously pointed out. may be expanded in a series for greater accuracy. For particles of radius as large as 0.511.. however. no significant contribution is made by the third and higher term in this series. Moreover. (abs). C(sca) is small compared with C , yet it is useful to (sca) retain C in order to determine the form of the 93 scattering. Its magnitude provides an estimate of the ef- fect of multiple scattering when it is suspected to be (abs) sca) present. It turns out. however. that C C( ’ and both have maxima at the same frequencies. Accordingly. multiple scattering cannot change the positions of the 5 absorption maxima. As a further check on the consistency of the re- a sults we substitute the frequencies of maximum absorption E. as determined from scattering theory into Eq. 41 for Bic. and also compare the absorption frequencies for E||c de- termined by both the FrOhlich and the extinction-cross- section method. ForIEHc we see that for SnO2 both methods give 592 cm-l. but for TiO2 the extinction calculation gives 700 cm"1 and FrOhlich‘s method 705 cm_l. The left- hand and right-hand sides of Eq. 41 with 01 0;. and 0; obtained from the extinction calculation are respectively 1.43 and 1.43 for Sn02. and 2.98 and 2.98 for TiOZ. It is Clear that the two methods are consistent with regard to maximum-absorption frequencies. On the basis of the success of the calculation of the extinction cross SectiOn. it appears that the n and k values determined from single-crystal reflection spectra of SnO2 and TiO2 are essentially valid for the powder form 94 of these crystals. especially SnO As has been pointed 2. out. there is however a lack of correlation for band width. and some disagreement in relative absorption strength. This lack of band shape correlation is true for all mater- ials studied in this work. The observed bands are consid- erably broader than the calculated ones. and much stronger absorption occurs in some cases than is indicated by the calculation. Because of the accuracy with which absorption frequencies are predicted. it appears likely that the dis- crepancies do not result from fundamental differences in n and k between the single crystal and the powder. but rather as a consequence of other properties of the particles. including the surface as itself constituting a defect in the periodic structure of the lattice. The disagreement in rel— ative intensities is to some extent a consequence of broad- ening. but in some cases it is too severe to be explained by that alone. The variation Of intensities apparently is a more subtle problem involving the direct effect of finite boundary conditions on lattice—vibration modes. and perhaps multiple scattering and interaction. Deviation from Spher- icity. such as prolateness and oblateness as we have seen before. will cause shifts in absorption frequency. thus re- ducing the absorption at the fundamental frequencies and 95 increasing it elsewhere. A quantitative estimate of the effect of such deviations would require detailed knowledge of crystallographic orientation of the particles as well as of their Shape distributions. Such information is neither available nor is it readily obtainable. In addi- tion. defects introduce additional absorption frequencies. thus increasing absorption away from the fundamentals. Ac- cording to Ruppin and Englman all surface—mode contributions to the extinction cross section are accounted for by the electromagnetic scattering theory. Their finding would seem to make further Consideration of surface modes un- necesSary unless explicit values Of surface—mode frequen- cies are desired. The complex refractive index determined from class- ical dispersion theory is a meaningful physical quantity. and is essentially the same for both single crystal and powder in the case of TiO2 and SnOZ. This contention is based on the success of extinction theory in predicting absorption frequencies from the complex refractive index. and on the agreement obtained with Eq. 41. The failure to achieve complete agreement between the observed and Calculated transmittance versus frequency curves is due 96 to the inherent nature of the surface of the particles such as shape. and concentration of defects. C. Barium Titanate and Strontium Titanate The analysis of BaTiO3 and SrTiO3 powder absorption spectra is basically similar to the analysis of SnO2 and TiO2 spectra. Table 3 contains a list of both the theoret— ical and experimental absorption frequencies for the two perovskites. The theoretical frequencies listed there were determined by the extinction—theory method on the assumption of small spherical particles. The calculated low-frequency mode is in excellent agreement with the experimental low- frequency mode in the case of both BaTiO3 and SrTiO3. The high-frequency modes. however. are 7.0 - 10.0%»higher than their experimental counterparts. the highest-frequency mode being shifted the most in each case. This point will be discussed later. Substitution of the experimental absorption fre— quencies into the generalized FrOhlich relation with g = 1/3 for spheres (Eq. 41). along with the dispersion parameters listed in Table 3. yields respectively for the left-hand and 97 right-hand sides 4.97 and 5.7 for SrTiO and 11.7 and 3. 14.3 for BaTiO3. The differences are 14% and 18%.respec— tively. On the other hand. use of the calculated frequen- cies instead of the experimental ones yields 5.8 versus 5.7 for SrTiOB. and 14.6 versus 14.3 for BaTiO3. yielding errors of about 2% in each case. This set of results in- dicates that the calculated absorption frequencies are consistent with the dielectric constants of the materials. It is asserted. therefore. that the frequency assignments in Table 3 are valid. Since the electron micrOgraphs of BaTiO3 and SrTiO3. Figs. 15 and 16. indicate some tendency toward the formation of cubical particles. a check of the effect of particle-shape deviations from spherical was made. Figure 19 represents the variation of absorption frequency with shape factor for each of the three modes of BaTiO 3 and SrTiO3. Since the shape factor for cubes was deduced to be approximately 0.21 from measurements onngO. it is reasonable to expect that better agreement for all three modes in the perovskites would be obtained for a shape factor near 0.21. Such indeed is the case. as may be ob- served from inspection of Fig. 19. For example. a Shape factor of 0.24 yields absorption frequencies of 607. 393. .aOBUmm mmmrm rea3 mOaSHm can mOeBmm mo wmflocmsvmum coflumnomnm 008:» 030 MO coflumeum>ln.ma musmflh moflaum Anv Houomm Tawnm moHBmm Amy Houomm wmwcm 98 6.6 «.6 6.6 6.6 «.6 6.6 _ _ \ \- 66H - 66H v. o. lllll .. _ a \ 1 \ I 66m w 66m To 0 U I. l I 666 m 666 \ D \ \. 666 A \ I. \.I \.\\ 6 66¢ m. um I 666 .. 666 \ \ \ \ \ mo) Xouenbezg uorqdzosqv (I_ 99 1 and 175 cm-1 for SrTiOB. and 555. 375. and 180 cm- for BaTiO3. all of which are in reasonably close agreement with the observed absorption frequencies. It is asserted. therefore. that the absorption spectra of SrTiO3 and BaTiO3 powders are characteristic of I"semi-cubical" particles with a shape factor of approximately 0.24. VI. SUMMARY In this study we have interpreted the infrared ab- sorption spectra of particulate polar crystalline materials through the use of electromagnetic scattering theory and a generalized FrOhlich relation. This scheme suffices to in- terpret virtually all Of the details of the spectra of the materials studied. which were MgO (cubic. rock salt struc- ture). SnO and TiO (tetragonal rutile structure). and 2 2 BaTiO and SrTiO (SrTiO 3 3 is cubic at room temperature. 3 BaTiO3 is Slightly distorted cubic or tetragonal structure at room temperature). In the case of MgO three distinct particulate forms were studied. and in all cases the spec- tral details were found to be consistent with theoretical computations based on size and shape of the particles and refractive index of the medium surrounding the particles. For the rutile and perovskite—structure materials. the shift of the bulk—crystal TO frequencies due to surface polarization is adequately accounted for by scattering theory. These results are consistent with the generalized FrOhliCh relation. 100 101 Portions of the interpretations presented in Sec. V should not be considered independently. particularly the in- terpretations of the secondary absorptions in MgO (double particles and long chain-like particles). The overall suc- cess. however. of the scattering theory and FrOhlich methods in predicting the positions of the major absorptions lends credence to the interpretations of the secondary absorptions. The inability of the approach taken in this work to obtain a good fit of the theoretical transmittance curves to the experimental transmittance curves. besides predicting the frequencies of maximum absorption. is not unexpected. In all powdered materials the particles may be expected to vary in shape about some average. with some average shape factor 9. Variations in shape cause line broadening which would be extremely difficult to take into account quantitatively. and the effort would probably not be justified. Size varia- tions may result in the occurrence of absorptions of both the surface polarization and bulk—crystal type in the same spectrum. We have semi-quantitatively accounted for the effect of shape variation on line width by virtue of the fact that both scattering theory and the FrOhlich relation predict the presence of absorption at frequencies all the 102 way from the bulk-crystal TO frequencies to the LO frequen- cies for extreme variations in particle shapes. The success of our effort to interpret absorption spectra of particulate materials in terms of size and shape makes abundantly clear the danger of reporting absorption spectra Of powdered polar materials without taking into ac- count the effects of particle size and shape. On the other hand we have shown that with care such spectra may be inter- preted and then used in support of other data. In some cases powder spectra may provide supplementary information when it is not readily available by some other means. For example. when sufficiently large single crystals are not available for low—frequency reflection studies. or in the event that one or more frequencies are too low to measure with available equipment. the polarization-shifted frequency may be measurable. (i) (ii) (iii) (iV) VII. RECOMMENDATIONS FOR FUTURE WORK Characterization of various polar crystalline powders according to their average shape factor 9. as has been done here in the case of MgO (2). BaTiO and 3' SrTiO3. Correlation of these shape factors with op- tical properties in the visible part Of the spectrum would be Of interest. A detailed theoretical study of the effects of multiple scattering and electromagnetic interaction. although expected to be minor. might clarify some of the discrepancies between the theoretical and ob- served transmittance curves. A thorough experimental study of the effect of particle dispersion within pellets or mulls on the infrared absorption spectrum. This work should be carried out in conjunction with the work suggested in (ii). A more complete theoretical investigation of the effect of particle size variation on the infrared 103 (v) 104 absorption spectrum. particularly in the case of MgO. would provide additional information on the shape of the transmittance curve near the TO mode frequency. A careful investigation into the exact nature of the variation of absorption frequency with shape factor in crystals with more than one TO mode as in the case of BaTiO3 and SrTiO3 (Sec V). It may be noted in Sec. V. 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