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Optical properties of cesium iodide in the far infrared Beairsto, James Atley Bruce 1972

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OPTICAL PROPERTIES OF CESIUM IODIDE IN THE FAR INFRARED by JAMES ATLEY BRUCE BEAIRSTO B.Sc., University of Calgary, 1970 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE In the Department of Physics We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1972 In p r e sen t i ng t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the requirements fo r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e for reference and s tudy. I f u r t h e r agree t ha t pe rmiss ion for e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s understood that copying o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l ga in s h a l l not be a l lowed wi thout my w r i t t e n p e r m i s s i o n . Department o f /ZA^?£&>9 The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8. Canada Date Jo s-JbAr ( i i ) ABSTRACT The r e f l e c t i v i t y at room temperature and the absorption c o e f f i c i e n t at room temperature, 77°K and 12°K have been measured fo r C s l . These and the r e s u l t s of a Kramers-Kronig analysis of the r e f l e c t i v i t y by Vergnat et a l . (1969) have been compared to the c a l c u l a t e d o p t i c a l constants (complex d i e l e c t r i c constant and complex r e f r a c t i v e index), r e f l e c t i v i t y and absorption c o e f f i c i e n t . The c a l c u l a t i o n , using the l a t t i c e dynamical data of Karo and Hardy (1963) , i s based on the work of Wallis and Maradudin (1962) and Cowley (1963). The cubic coupling c o e f f i c i e n t has been evaluated f o r nearest neighbors with a co r r e c t i o n due to Eldridge (1973) f o r long-range Coulombic forces. The pr e d i c t e d features i n the imaginary part of the complex phase s h i f t , gamma, are a l l assigned to s p e c i f i c two-phonon processes except f o r a small feature at 91 cm \ The agreement between theory and experiment i s very poor at room temperature but improves s i g n i f i c a n t l y i n a l l cases at low temperatures. The discrepancies between theory and experiment give evidence of the fundamental importance of t h i r d and higher-order processes i n the C s l l a t t i c e and suggest that next-nearest neighbor r e p u l s i v e forces need to be included i n the c a l c u l a t i o n of the cubic coupling c o e f f i c i e n t . ( i i i ) TABLE OF CONTENTS Page Abstract i i Table of Contents i i i List of Tables v i List of Figures v i i Acknowledgements ix Chapter 1 INTRODUCTION 1.1 General Introduction 1 1.2 The Purpose and Outline of the Thesis . 2 Chapter 2 THE THEORY OF LATTICE ABSORPTION 2.1 The Macroscopic Theory of Dielectric Absorption 4 2.2 The Microscopic Theory of Lattice Absorption 7 2.2.1 Introduction 7 2.2.2 The Microscopic Theory (Undamped) 7 A. Introduction 7 B. The Phenomenological Theory of Lattice Absorption (Undamped) 7 C. The Classical Theory of Lattice Absorption (Undamped) 10 (iv) Page D. The Quantum Mechanical Theory of Lattice Absorption (Undamped) 13 E. Experimental Aspects of the Undamped Theory 13 2.2.3 The Microscopic Theory (Damped) 18 A. Introduction 18 B. The Phenomenological Theory of Lattice Absorption (Damped) 19 C. The Classical Theory of Lattice Absorption (Damped) 20 D. The Quantum Mechanical Theory of Lattice Absorption (Anharmonic) 21 E. Experimental Aspects of the Anharmonic Theory 24 Chapter 3 The Theoretical Calculations 3.1 Introduction 32 3.2 The Basic Data 32 3.2.1 Theory of the Lattice Dynamical Data 32 3.2.2 The Inter-Ionic Potential Function 39 3.3 The Basic Calculations 42 3.3.1 The Cubic Coupling Coefficient 42 3.3.2 The Complex Phase Shift 43 3.3.3 The Crystal Dipole Moment Operator 45 3.3.4 The Optical Properties 45 Chapter 4 Experimental Considerations 4.1 The Apparatus 47 4.2 The Experimental Technique 48 (v) Page 4.3 The Sample 49 4.3.1 The Sample Crystal Structure 49 4.3.2 The Motivation For The Choice of Sample 49 4.3.3 The Sample Preparation 51 Chapter 5 Experimental and Theoretical Results 5.1 Introduction 52 5.2 Presentation and Interpretation of Results .. 52 5.2.1 The Complex Phase Shift 52 5.2.2 The Optical Constants 55 5.2.3 The Reflectivity 58 5.2.4 The Absorption Coefficient 59 Chapter 6 Summary and Conclusions 74 Appendix A The Cubic Coupling Coefficient 77 Appendix B The Kramers-Kronig Relations 90 Bibliography 93 < (vi) LIST OF TABLES Table Page 3.1 Constants Used In The Calculation Of The Optical Properties 46 A . l The Generalized Force-Constant Tensors For The Eight Nearest-Neighbors In The Csl Lattice 83 (vii) LIST OF FIGURES Figure Page 2.1 Schematic representation of EMR dispersion curve superimposed on lattice dispersion curves for a polar diatomic ionic l a t t i c e .... 16 2.2 Schematic representation of the behaviour of the r e f l e c t i v i t y and the optical constants in the undamped theory 17 2.3 Schematic representation of the behaviour of the r e f l e c t i v i t y and the optical constants in the constant damping theory 25 2.4 Schematic representation of two-phonon summation (a) and difference (b) processes .... 27 3.1 Lattice dispersion curves for Csl from the DD(-) model 36 5.1 Gamma at room temperature 60 5.2 Gamma at 77°K 61 5.3 Gamma at 12°K 62 5.4 Delta at room temperature and 12°K 63 ( v i i i ) Figure Page 5.5 Epsilon real at room temperature 64 5.6 Epsilon real at 20°K 65 5.7 Epsilon imaginary 66 5.8 Extinction coefficient 67 5.9 Refractive index 68 5.10 Reflectivity at room temperature 69 5.11 Reflectivity at 20°K 70 5.12 Alpha 71 5.13 Alpha in the difference region 72 5.14 Alpha i n the summation region 73 A.l A portion of the Csl la t t i c e 79 (ix) ACKNOWLEDGEMENTS First and foremost, I wish to acknowledge the patience and enthusiasm of my supervisor, Dr. J.E. Eldridge, which has greatly enriched my masters experience. Thanks are also due to Dr. J.W. Bichard for his periodic advice and assistance with experimental matters, Mr. K. Kembry and Mr. P.R. Cullis for their friendship and moral support and Miss S. Jackson for invaluable assistance with the computing involved in the thesis. This work was supported by the National Research Council of Canada grant number 67-5653. I also wish to acknowledge personal support in the form of a National Research Council scholarship during the f i r s t year. 1 CHAPTER 1 INTRODUCTION 1.1 General Introduction. The interaction of infrared electromagnetic radiation with polar crystals leads to a resonant energy absorption by the transverse "optic" modes of the crystal. This resonant behaviour was f i r s t observed by Rubens about 1900 accompanied by an energy-selective reflection of what was termed the "Reststrahlen" rays. The use of infrared spectroscopy as a tool to probe the crystal lattice was pioneered especially by Coblentz about 1930. The a l k a l i halides have been widely used i n this work since their relatively simple diatomic lattice involves only ionic bonding, which i s the simplest and best understood type. The resonant absorption has been studied directly via the transmission of thick single crystals C50p •> 1 cm) and thin films (a few microns) and indirectly via the front-r-face reflection of a thick crystal followed by a Kramers-^Kronig analysis. More recently, the method of asymmetric interferometry has been used to yield both the amplitude reflectance and the phase change without the use of the often unreliable Kramers-Kronig analysis (Johnson and B e l l , 1969.). In the analysis of the resonance and i t s side-^bands, two-Body central forces are usually sufficient and qualitatively acceptable 2 explanations have been produced f o r the experimentally observed phenomena. However, some questions remain: (1) the structure of the theory - i . e . the form of the i n t e r - i o n i c p o t e n t i a l function and the r e l a t i v e importance of the i n t e r a c t i o n between nearest neighbours, next-nearest neighbours e t c . ; (2) the r e l a t i v e importance of the damping mechanisms T - i . e . the anharmonic terms i n the p o t e n t i a l and the non-linear terms i n the e l e c t r i c dipole moment operator; (3) the r e l a t i v e importance of cubic and q u a r t i c terms i n the p o t e n t i a l and hence the r e l a t i v e importance of t h i r d and higher-order phonon processes i n various parts of the spectrum; and (4) the c a l c u l a t i o n of o p t i c a l properties on an absolute s c a l e . 1.2 The Purpose and Outline of the Thesis In the present work, we s h a l l measure the absorption c o e f f i c i e n t at s e v e r a l temperatures over a broad s p e c t r a l range about the resonant frequency, u)a and compare t h i s with a cal c u l a t e d spectrum considering only two-phonon r e l a x a t i o n of the main resonance, v i a the third-order anharmonic coupling. In chapter 2 we present a b r i e f o u t l i n e of the theory of l a t t i c e absorption i n c l u d i n g the formulae which w i l l be used to c a l c u l a t e the o p t i c a l properties i n a manner discussed i n chapter 3. The general experimental procedure is explained in chapter 4. In chapter 5 we present the theoretical and experimental results along with a discussion of their nature and significance. In chapter 6 the findings of this work are summarized and suggestions are made for further work on this topic. 4 CHAPTER 2  THE THEORY OF LATTICE ABSORPTION 2.1 The Macroscopic Theory of D i e l e c t r i c Absorption Electromagnetic waves i n a conducting d i e l e c t r i c are governed by Maxwell's equations: T7> - r. TZ + -r- E —> (J. i) VX F. = _ c <?x } 7 - w =o ; The three c o e f f i c i e n t s describing the d i e l e c t r i c are: (1) the d i e l e c t r i c constant, E ; (2) the e l e c t r i c a l c onductivity, a; and (3) the magnetic permeability , u. Since we are interested i n non-magnetic media, we set y=l. Then, elim i n a t i n g the magnetic f i e l d vector: Since our measurements are c a r r i e d out with beams and on samples of l a t e r a l dimension much greater than the wavelengths involved, we may consider the d i s s i p a t i v e plane wave s o l u t i o n to (2.2): / T ^ O ( ( f f r ' - w t ) ! ' (3.3) 5 where w i s the angular frequency of the radiation, k is a vector of magnitude 2ir/X in the direction of propogation and r is a position vector. Inserting ( 2 . 3 ) in ( 2 . 2 ) we obtain: J 6 . 4TI r*j h. Jl) The refractive index is then, by definition: /Tl - ( £ + L u> / U t i l i z i n g the definitions: fa.s) A/ * t K e 5 € + c z (3. 6 a.) (3. (A) and the equality n =E, we find: fa. 1*) 6 Prom (2.6), (2.5) and (2.4) and the d i s s i p a t i v e plane wave s o l u t i o n i s : A/r \ I f K^L^ Since the rate of production of Joule heat i s the r e a l part o f : the absorption c o e f f i c i e n t , a, i s given by (Ziman, 1969): The absorption c o e f f i c i e n t i s the quantity a c t u a l l y determined experimentally and gives the e x t i n c t i o n c o e f f i c i e n t , K, d i r e c t l y . In f a c t , a l l of the information concerning the d i e l e c t r i c i s contained i n K since we may determine N (and hence n and e) through the Kramers-Kronig r e l a t i o n s (of Appendix B). 2.2 The Microscopic Theory of L a t t i c e Dispersion 2.2.1 Introduction The l u s t r e and o p t i c a l opaqueness of many common c r y s t a l s i s evidence that they must absorb and r e f l e c t at l e a s t part of the electromagnetic spectrum. The f i r s t observation of any resonant phenomenon, however, occured near the turn of the century with the discovery of "Reststrahlen" r a d i a t i o n - the p r e f e r e n t i a l r e f l e c t i o n from some c r y s t a l s of a narrow frequency band i n the f a r i n f r a r e d . The theory of l a t t i c e d i s p e r s i o n concerns i t s e l f with p r e d i c t i n g the behaviour of the l a t t i c e - p a r t i c u l a r l y i n the neighborhood of such resonances. 2.2.2 The Microscopic Theory (Undamped) 2.2.2 (A) Introduction The v i b r a t i o n a l modes of a c r y s t a l may be e n e r g e t i c a l l y coupled and hence damped through terms of t h i r d or higher order i n the Taylor s e r i e s expansion of the p o t e n t i a l or through terms of second or higher order i n the Taylor s e r i e s expansion of the e l e c t r i c moment induced by the i o n i c motion. In the simplest and h i s t o r i c a l l y e a r l i e s t undamped theory, we consider only second-order terms i n the p o t e n t i a l (the harmonic approximation) and f i r s t order terms i n the e l e c t r i c moment. 2.2.2 (B) The Phenomenological Theory of L a t t i c e Absorption (Undamped) The macroscopic parameters n and e, determined from the spectroscopic measurement of a, may be r e l a t e d to the force and p o l a r i z a b i l i t y c o e f f i c i e n t s i n a l i n e a r continuum theory due to Born and Huang (1966). For long-wavelength l a t t i c e vibrations in diatomic ionic crystals, the macroscopic theory i s f u l l y embodied in the phenomenological equations (Huang, 1951): — LU k, ^ + ki. £ ('J./iA) On U? + L; £ O./J I) ->• where co i s a vector whose magnitude equals the displacement of the positive ions relative to the negative ions multiplied by the square root of the reduced mass m = m, m / (m,+m). + — + — Equation (2.11a) i s the equation of motion of the l a t t i c e - linear in accordance with the harmonic approximation. Equation (2.11b) expresses the local polarization arising, f i r s t l y , from a moment due to the displacement of the ions from l a t t i c e sites and, secondly, from the moments induced in the ions by the macroscopic electric f i e l d . Eliminating a> from these equations and invoking the definition of the electric displacement: they obtain the dielectric constant: V77 L i + V1A^AL' This dispersion formula may be written in terms of the directly measurable quantities: (1) a)Q, the infrared dispersion frequency (of the reststrahlen radiation) at which E tends to i n f i n i t y ; (2 ) eo, the static dielectric constant, measured at a frequency small compared to a3 Q; and (3) E ^ , the high-frequency dielectric constant, measured at a frequency large compared to U ) q but small compared with the frequencies of electronic motion i n the crystal. 2 Since e-*°° as b ^ -»• -u>o and E -> l+bnb^ as to 00 we may make the identifications: Born and Huang also show, on the basis of energy conservation, that b ^ = ^>2\ s o t n a t w e m a v ^-et uyJy° t o sb-ow that: Thus the dispersion formula may be written as: 10 2.2.2 (C) The C l a s s i c a l Theory of L a t t i c e Absorption (Undamped) Having derived a dispe r s i o n r e l a t i o n i n terms of the force constants i n the phenomenological equations (2.11), we may e s t a b l i s h the microscopic theory of dispe r s i o n simply by deducing a s i m i l a r r e l a t i o n from an atomic theory. For a diatomic i o n i c c r y s t a l , the equation of motion f or the p o s i t i v e and negative ions i n the p r i m i t i v e u n i t c e l l , i n the presence of a transverse electromagnetic f i e l d i s (Burstein, 1964): sm cT - cdl cZ - e £ v, • uli ' where m = m+m_/(m+ m_) i s the reduced i o n i c mass; u = x +-x_ represents the r e l a t i v e displacement of the ions; e i s the e l e c t r o n i c charge and ->• E c_ i s the e f f e c t i v e l o c a l f i e l d at the l a t t i c e s i t e , ef f The e f f e c t i v e f i e l d on the ions i s not simply the sum of the macroscopic e l e c t r i c f i e l d and the macroscopic p o l a r i z a t i o n as t h i s includes the f i e l d of the ion under consideration. Born and Huang (1966) have shown that the f i e l d due to the macroscopic f i e l d and the p o l a r i z a t i o n of a l l other ions i n the l a t t i c e i s merely the sum of the macroscopic f i e l d and the Lorentz f i e l d : t i where P i s the macroscopic p o l a r i z a t i o n . This i s the proper e f f e c t i v e f i e l d and i s the same f o r eit h e r ion. 11 Assuming that the Coulomb f i e l d due to each polarized ion may be represented by a 1 short'dipole localized at the equilibrium position of the ion, we may write the macroscopic polarization, including the electronic polarization of the ions and the ionic displacement polarization, as: where v = V/N is the volume of the unit c e l l and a = a++a i s the a OO CO CO " t o t a l " ionic polarizability. On combining terms, we find: —~> p* = 1 e^L^t^dL. fjp, IQ) 2 A>K and u t i l i z i n g the Clausius-Mossotti relation: e= a, -J V77" of* (2. So) we may write: where = (e -1)/4TT i s the high-frequency dielectric susceptibility. Now, inserting (2.21) and (2.17) in (2.16) and assuming an exp(itot) time dependence for the collinear vectors u, E and P, we obtain: 12 U -I Art " ) ( 3 ->> £ «V ~ IF' 3 U) Thus, the resonance frequency for the l a t t i c e vibration w i l l be: &. 23) <UJ4 = cO-Inserting (2.22) in (2.21) we obtain: and u t i l i z i n g the definition x( w) = (e(io)-l)/4ir : This dispersion relation, f i r s t obtained by Szigeti, requires only one major refinement. We have allowed for polarization resulting from a r i g i d shift of the electron shell relative to the nucleus due to a uniform effective f i e l d at the nucleus. This f i e l d i s , however, not quite uniform and does lead to a slight deformation of the shell. The correction i s effected by replacing the electronic * charge e with the Szigeti effective charge e . We may derive an * expression for e i n terms of measurable parameters from (2.21). Letting to -> 0 we find: 13 Lowndes and Martin (1968) have used this relation to calculate e for the a l k a l i halides. The value of e /e are a l l * found to be less than unity and the value of e /e = 0.78 for Csl is among the smallest values for the a l k a l i halides. Thus, our classical dispersion relation i s : 2.2.2 (D) The Quantum Mechanical Theory of Lattice Absorption (Undamped) The quantum mechanical theory of l a t t i c e absorption in the undamped approximation (Burstein, 1964) yields precisely the same form for the dielectric constant as the classical theory and w i l l consequently not be presented here. 2.2.2 (E) Experimental Aspects of the Undamped Theory The expressions (2.15) and (2.27) for the dielectric constant w i l l indeed show a resonant behaviour at u = io and this, o w i l l be reflected i n the optical properties of the crystal. The agreement with empirical data w i l l however be only qualitative. 14 For a normally incident beam, the re f l e c t i v i t y i s given by the well-known classical formula: I- -h I + 7? As co approaches a) from below, we see from (2.15) that n rises from /e to approach i n f i n i t y and thus R approaches unity. At io = u^, n jumps from +°° to -00 and remains negative according to (2.15) u n t i l : 1 u> = ( 4S Yto. = J. In this range, n i s purely imaginery and hence R = 1. Beyond W^ , which we recognize from the Lyddane-Sachs-Teller (1941) relation as the longitudinal optic frequency, R again rapidly decreases. Thus the dispersion formulae (2.15) and (2.27) give a natural explanation of the selective r e f l e c t i v i t y in the v i c i n i t y of the reststrahlen frequency. However, while the r e f l e c t i v i t y i n the range LO^ -*• OJ^ i s indeed found to be high, i t is never 100%. In transmission experiments we may expect high absorption classically when an induced polarization of the la t t i c e matches the inducing f i e l d spatially and temporally; or quantum mechanically, from the conservation of energy (frequency) and momentum (wavelength) when an incident photon creates a l a t t i c e phonon. Thus, a simple way to predict regions of l a t t i c e absorption is to draw a line 15 representing the dispersion relation for an electromagnetic wave in the l a t t i c e across the l a t t i c e dispersion curves (cf figure 2.1). The points of intersection with the transverse optic branches w i l l give us the dispersion frequency u of the dispersion relations (2.15) and (2.27). We find, that since the slope of the dispersion curve (the group velocity of the EMR) i s so great, these points of intersection occur only at very long wavelengths near the zone centre, T. Thus the dispersion frequency, u) o,,will effectively be the transverse optical frequency at r. We may then calculate the absorption coefficient: • U3o) which may be written, using the relation (2.7), as: A/c and since e i s purely re a l , a = 0 except where N = 0 and hence K = /e. This occurs when to = OJ and we may expect therefore a temperature-independent o absorption coefficient of the form a(co) = 6(a)-too). In practice, a rather broad absorption peak i s found with a temperature-dependent strength and frequency. The behaviour of the optical constants in the undamped theory i s shown schematically in figure 2.2. 16 Fig. 2.1. Schematic representation of EMR dispersion curve superimposed on lattice dispersion curves for a polar diatomic ionic l a t t i c e . 17 Fig. 2.2. Schematic representation of the behaviour of the re f l e c t i v i t y and the optical constants in the undamped theory. 18 2.2.3 The Microscopic Theory (Damped) 2.2.3 (A) Introduction The undamped theories not only show quantitative disagreement with the phenomena which they predict q u a l i t a t i v e l y , but they f a i l e n t i r e l y to p r edict any subsidiary structure away from the resonance at U) q. In the a l k a l i h a l i d e s , t h i s subsidiary structure c h a r a c t e r i s t i c a l l y takes the form of one or more "nebenmaxima" on the high frequency side of the fundamental absorption peak. These discrepancies a r i s e from the f a c t that, i n the undamped theory, the l a t t i c e modes are independent and hence energy fed s e l e c t i v e l y i n t o any one mode w i l l remain i n that mode i n d e f i n i t e l y . In r e a l i t y , the modes are coupled by either an anharmonic mechanism (higher-order terms i n the p o t e n t i a l ) or a higher-order e l e c t r i c moment mechanism or a combination of these. On the basis of t h e o r e t i c a l c a l c u l a t i o n s (Mitskevich, 1963) and the comparison of measured absorption i n various materials, (Kleinman and S p i t z e r , 1960) several authors have ascribed the coupling e n t i r e l y to the anharmonic mechanism. The most important argument for t h i s assignment i s based on the f a c t that the absorption i n non-polar c r y s t a l s such as diamond and s i l i c o n , which i s caused by the second-order moment only, i s much smaller than the side-band absorption i n i o n i c c r y s t a l s . However, the two mechanisms have the same s e l e c t i o n r u l e s and combined d e n s i t i e s of states so that i t i s not easy to discern t h e i r r e l a t i v e importance. Born and Huang (1966) have concluded that the anharmonic mechanism causes the broadening of the absorption peak 19 close to the resonance, whereas the second-order moment mechanism causes a more uniform absorption further away from the peak. Szigeti (1960) has treated both mechanisms and their interactions systematically and concludes that, far from the resonance, the second-order moment may even be the dominant mechanism. shall merely introduce an "ad hoc" damping without reference to i t s origin, and in the quantum mechanical treatment we shall consider the damping only of the anharmonic mechanism. The damping, so introduced, should bring the theory into much closer accord with the experimentally observed phenomena. 2.2.3 (B) The Phenomenological Theory of Lattice Absorption (Damped) To account for the experimentally observed energy dissipation in the v i c i n i t y of OJ q (i.e. a / o) we introduce a simple damping term into equation (2.11a) as follows (Born and Huang, 1966): For complex periodic solutions with a time dependence of the form exp(-iwt) this reduces to: In the phenomenological and classical treatments we so that the damping force opposed to the motion may be treated equivalently by replacing b_ 1 with b11+iyto and the dispersion relation (2.13) becomes: 20 auj) - I 4 </7rl3X + — iEL^Li (j.34) - 4, - i %W - LO 1 or i n terms of the p h y s i c a l l y measurable parameters: 2.2.3 (C) The C l a s s i c a l Theory of L a t t i c e Absorption (Damped) If we i n s e r t a s i m i l a r damping term i n the equation of motion (2.16) as follows (Burstein, 1964): — > •m ZC - <m ti IC - /vn u) c2 - £ <-v/ (3.36S) 7 •¥ the r e s u l t i n g expression f o r the d i e l e c t r i c constant i s : -f c 21 2.2.3 (D) The Quantum M e c h a n i c a l Theory (Anharmonic) We s h a l l u s e t h e r e s u l t s o f c a l c u l a t i o n s by W a l l i s and M a r a d u d i n ( 1 9 6 2 ) , and Cowley (1963) w h i c h use o n l y t h e f i r s t o r d e r d i p o l e moment and i n c l u d e t h e c o n t r i b u t i o n f r o m c u b i c terms i n t h e p o t e n t i a l . Under t h e s e c o n d i t i o n s , t h e t r a n s f o r m a t i o n w h i c h d i a g o n a l i z e s t h e h a r m o n i c p a r t o f t h e c r y s t a l H a m i l t o n i a n i n t o a s u p e r p o s i t i o n o f h a r m o n i c o s c i l l a t o r H a m i l t o n i a n s l e a d s t o a c o u p l i n g between t h e modes and t h e s t u d y o f t h e dynamic p r o p e r t i e s o f t h e c r y s t a l t h u s n e c e s s i t a t e s t h e s o l u t i o n o f a many-body p r o b l e m . e f f e c t s ( i . e . t h e damping e f f e c t s ) may be e x p r e s s e d a s a complex - > - - * • ' phase s h i f t Ato(kj) + ir(kj) s u f f e r e d by each o f t h e no r m a l mode t h e s y stem. The r e a l p a r t o f t h e phase s h i f t g i v e s t h e change i n f r e q u e n c y ( e n e r g y ) and t h e i m a g i n a r y p a r t i s t h e r e c i p r o c a l o f t h e now f i n i t e l i f e t i m e o f t h e phonon s t a t e (kj). B o t h p a r t s a r e t e m p e r a t u r e - d e p e n d e n t . I n t erms o f t h i s phase s h i f t , W a l l i s and M a r a d u d i n (1962) f i n d t h e d i e l e c t r i c s u s c e p t i b i l i t y : S o l u t i o n o f t h i s p r o b l e m shows t h a t t h e many-body f r e q u e n c i e s to(rcj) - t h e u n p e r t u r b e d " s i n g l e - p a r t i c l e " e n e r g i e s o f X yoLz> 22 where v is the volume of a unit c e l l and M and M are a y v components of the dipole moment operator. The frequency shifts AOJ (oj) and damping constants T(oj) were obtained from the behaviour of an analytic function of (x~ie) near the real axis by using the relation: (3. 39) where (P denotes the principal part. They are of the fo rm: -/ h. Jo) The expression (2.38) and (2.40) are useful in that they reveal the form of the solution but, in our calculations, we shall be using a form adapted from Johnson and Bell (1969): 6. (co) , + 2TT where we have incorporated the optical isotropy of the cubic l a t t i c e . In this expression, N q i s the number of unit cells in the crystal, co^  i s the angular frequency of the zero wave vector T.O. phonoh and M(oj) i s the leading term in the expansion of the crystal dipole 23 moment operator (cf section 3.3.3). Aaj(oj) and T(oj) are the components of the complex phase shift (or Hermitean and anti-Hermitean parts of the proper self-energy matrix) for the zero wave vector T.O. phonon. The lowest-order expressions for Aw(oj) and T(oj) are taken from Cowley (1963). The frequency shift i s given by: A uj (o j *) -where 4-The inverse lifetime i s given by: where 5 (oo\ + In the expressions for R(w) and S(w), n(kj) i s the occupation number for the mode (k j ) given by: The notation X -1 P indicate the principal part. The term V ( o j i k ^ ^ j k ^ j ^ ) i s the cubic coupling coefficient that connects the resonant T.O. mode (oj) to the two other modes 2^2^ a n C* 3^3^' This coefficient i s evaluated for the Csl l a t t i c e , considering nearest neighbors only (with an appropriate correction for the rest of the lattice) in appendix A and an explicit expression for i t i s given there. 2.2.3 (E) Experimental Aspects of the Anharmonic Theory The ad hoc formulae of the phenomenological and classical treatments show a qualitative agreement with the observed phenomena. The effect of a constant damping term i s shown schematically in figure 2.3. gain any physical insight into, the anharmonic crystal we require an explicit treatment of the coupling of modes as i n the quantum mechanical calculation. The three major consequences of the theory are: (1) a temperature variation of the dispersion frequency resulting from thermal expansion Can anharmonic process) and the self-energy s h i f t ; (2) a broadening of the fundamental absorption line resulting from the anharmonic coupling and consequent f i n i t e lifetime of the However, in order to get a detailed picture of, or to 2 5 Fig. 2.3. Schematic representation of the behaviour of the r e f l e c t i v i t y and the optical constants in the constant damping theory. phonon modes; and (3) the appearance of subsidiary structure resulting from multi-phonon processes. The two-phonon processes potentially contain much more information than the one-phonon processes. The incident photon excites the T.O. phonon Cserving as a v i r t u a l intermediate state) which then couples with two other phonons satisfying relaxed selection rules. While momentum must s t i l l be conserved at a l l stages of the process, energy need only be conserved between the i n i t i a l and f i n a l states. Thus, while we were previously restricted to interactions at the zone center with radiation of energy -h we may now consider the interaction of EMR with any set of phonons having zero total momentum. This leads to subsidiary structure in the absorption spectrum. There are two possible two-phonon processes of this type. In a summation process the T.O. phonon decays into two other phonons and in a difference process the T.O. phonon creates one phonon and destroys another phonon of lesser energy. These processes are represented schematically in figure 2.4. A consideration of the informal "selection rules" imposed by the form of the cubic coupling coefficient and common sense should allow us to identify the particular region of the zone responsible for any subsidiary structure. The position of the subsidiary peaks i s dominated by the consideration that the slopes of two phonon branches must be equal (for difference bands) or Fig. 2.4. Schematic representation of two-phonon summation (a) and difference (b) processes. 28 opposite (for summation bands) over a significant range of k for strong absorption to occur. The coupling is aided by a large relative motion such as occurs near the zone boundary (notice the (3) multiplicative sine term in V which i s a maximum at the boundary). The possi b i l i t y of third and higher-order phonon processes does exist. Their smaller matrix element i s balanced by a greater number of possible phonon combinations satisfying the selection rules and the contribution from higher-order processes may be significant away from the fundamental or subsidiary peaks or at high temperatures. It i s unlikely, however, that such processes w i l l be responsible for features in the spectrum. The temperature dependence of the various processes may be inferred i f we re c a l l that the transition probabilities, W, for the creation or destruction of a phonon have the form (Ziman, 1960). Mi) Thus, for example, the net probability of creating two phonons (a summation process) i s equal to difference of the probabilities of the creation and destruction of two phonons: 29 6, </<p) The net probability of creating one phonon, (k 1j.) and destroying another, (k.^]p, (a difference process) i s similarly equal to the difference of the probability of this process and the reverse process: From C2.48) and (2.49) we can see that the difference processes w i l l completely disappear as T -*- 0 whereas the summation processes should remain. We may similarly derive the net probability of the various three-phonon processes. Let us consider, as a typical example, the probability of creating three phonons: + 30 In the high-temperature l i m i t , we may write: so that both of the two-phonon processes w i l l exhibit a linear temperature dependence and a l l of the three-phonon processes w i l l exhibit a quadratric temperature dependence. The temperature dependence of the absorption coefficient, a, may be seen i f we write a in terms of the dielectric constant given by equation (2.41), using equations (2.7) and (2.10): the calculated values of Aco(oj) and T(oj), these factors i n the denominator of (2.52) w i l l have a negligible effect except very near OJq. Thus the temperature dependence of a w i l l essentially be contained in T(oj) and hence in S(co). Below the reststrahlen o(-- f m 6(ui) As may be seen from an examination of the magnitude of 31 frequency the absorption i s due to difference processes and above the reststrahlen frequency i t is.due to summation processes. To the order of the calculation then, we may expect a linear temperature dependence in both regions with the difference absorption disappearing while the summation absorption remains at low temperatures. 32 CHAPTER 3  The Theoretical Calculations 3.1 Introduction The basic calculation required i s the total infrared absorption in a Csl crystal due to the relaxation of the main T.O. resonance near. 60 cm \ to a l l possible two-phonon processes. These lat t e r , as mentioned, may be sum or difference processes, and satisfy momentum and energy conservation with the incident infrared photon. For this calculation we need to know the functions Aa)(oj) and r(oj) (which are the frequency-dependent frequence shift and damping respectively of the T.O. mode due to the anharmonic coupling with the two phonons). The eigenfrequencies and eigenvectors of the la t t i c e are needed to calculate these functions. This l a t t i c e dynamical data was produced by a deformation dipole model due to Karo and Hardy (1963) and we are indebted to them for the use of their data. 3.2 The Basic Data 3.2.1 Theory of the Lattice Dynamical Data Whereas classically the l a t t i c e vibrations were described by a vector obeying a linear, homogeneous wave equation of second order in space and time co-ordinates, quantum mechanically the l a t t i c e vibrations may be described either as an i n f i n i t e number of distinguishable, quantized oscillations or as a gas of indistinguishable particles - the phonons. We shall adopt the description in terms of particles as i t i s particularly useful in the treatment of optical processes resulting from interaction with the photons of the quantized electromagnetic f i e l d . In the adiabatic approximation, the kinetic energy of the l a t t i c e may be written: 2. . r- ^ ^ = z> i^s^lr (3.1) where u -»• represents the displacement from equilibrium of ion s ,r "s" in the primitive c e l l " r " . In the harmonic approximation, the potential energy may be written as a truncated Taylor series in u s ,r <3V 5, r 1 J 1 ) Since we may set V q=0 and equilibrium conditions dictate that ( ^ / ^ O L ^ T ? ) = O , this simplifies to: For notational simplicity we shall drop the subscript so that: The total Hamiltonian may then be written: 3 4 fit f '-•» / \ _ ' We shall define the generalized force-constant tensor: which defines the force on an ion (s,r) due to a displacement u of the ion (s ,r ). Since the order of differentiation is unimportant for infinitesimal displacements, K w i l l be symmetric and since i t cannot be a function of the bravais c e l l index K must be of the form K '(r-r) ss Now, i f we look for normal mode solutions to (3.5) with an exp(icot) time dependence, we are confronted with the equation: which i s formidable but may be simplified since u, being a bulk excitation of the solid, must obey the Bloch theorem: - K« e — W 6 ? ) e fa?) Thus, (3.6) simplifies to: If we now express: 35 the equation (3.8) becomes: (3. io) and for the three Cartesian coordinates j , j =. x,y,z we may write the general equation for the la t t i c e vibrations as: which represents a set of uncoupled harmonic oscillators. Thus the energy associated with each bulk mode of wave vector k i s quantized in units of ft oo - the phonons. Karo and Hardy (1963) based on an analysis such as the preceding but incorporating a modification due to Szigeti. They correct for a minor distortion of the ions by assuming that only the negative ions are deformed and that the deformation dipole moment, a function only of the configuration of the eight nearest neighbours, is proportional to the overlap repulsion. They then construct a DD(-) model with the dipole on the negative ion and a DD(+) model with the dipole on the positive ion. While i t i s not clear "a p r i o r i " which model should be used, subsequent work by Karo and Hardy on the second-order Raman spectra of cesium halides and Eldridge (1970) on the thermal diffuse • / Our l a t t i c e dynamical data i s obtained from a model by 36 o CM Fig. 3.1. Lattice dispersion curves for Csl from the DD(-) model. 3 7 X-ray scattering of cesium iodide indicates that the DD(-) model i s probably more valid. This data was therefore used. There i s one significant d i f f i c u l t y involved with this type of data. The quantum mechanical theory uses the harmonic eigenfrequencies of the l a t t i c e and then corrects this data, via the complex phase s h i f t , to account for the anharmonic effects. The Karo and Hardy model however, being based on the observed macroscopic constants of Csl at room temperature, already includes these anharmonic effects. However, the actual l a t t i c e frequencies may be used i f we renormalize the expressions for Ato, I*' and'e (to) according to a scheme proposed by Cowley (1963). He has shown that i f T i s small, the harmonic frequencies may be replaced by "quasi-harmonic" frequencies given approximately by: Inverting this relationship we find that the dielectric constant, in terms of the quasi-harmonic eigendata, has a denominator of the form: 38 This expression may be simplified by noting that o> >> Ato whence As we may see, the same effect may be obtained by a r i g i d shift of the function Aio so that the frequency shift i s zero at the measured resonance frequency to^  . J Qo There remains but one note on the eigendata. Since we cannot reasonably generate a l l of the phonon modes, we sample the Bril l o u i n zone with a cubic grid of 999 points. (This i s a rather coarse grid giving only 5 points in each symmetry direction.) Furthermore, just as the translational symmetries of the l a t t i c e allow us to study the reciprocal l a t t i c e in terms of the reduced Bri l l o u i n zone scheme, so group theory shows that the rotational and reflective symmetries of the l a t t i c e allow us to consider a yet smaller 1/48 portion of the zone which uniquely defines the entire energy spectrum. Thus, in fact, only 55 sites in the 1/48 irreducible part of the zone are considered. We then generate a number of energetically equivalent points elsewhere in the zone from any site in the irreducible portion by applying the point group to that site. The "multiplicity" of the site i s 48 divided by the number of elements in the point group. The 55 unique points thus generate a grid of 999 points in the zone (excluding the origin). the dielectric constant may be written: 3 9 3 . 2 . 2 The Inter-Ionic Potential Function As the a l k a l i halides are so highly ionic, their wavefunctions are minimally distorted by inclusion i n a l a t t i c e and simple perturbation theory is adequate for a consideration of the l a t t i c e forces. The f i r s t -order perturbation leads to the classical long-range Coulombic attraction between ions of opposite charge and repulsion between ions of l i k e charge. To this level of approximation, therefore, we may treat the ions as point electronic charges. However, Earnshaw's theorem t e l l s us that a system of stationary charges cannot be i n equilibrium just under their own electrostatic Coulombic forces of attraction and repulsion. If we r e c a l l , however, that each ion in fact consists of a nucleus surrounded by an electron cloud we w i l l see that when the ions are close enough together that these charge clouds begin to overlap there w i l l be an additional repulsive force between neighbouring ions as a result of the Pauli exclusion principle. There are also higher-order forces in the l a t t i c e but these are generally ignored. The potential energy of one ion due to the presence of any other ion may be written as: where r i s the distance between the ions, 6XTkT i s a Kronecker delta NN type function which i s one for nearest neighbors and zero for other ion pairs, and X and p are two constants characterizing the strength and range of the repulsive force. Here we have used an exponential form for the repulsive force f i r s t suggested by Born and Mayer in view of the known exponential decay of the mutually repelling radial (3. /£) 40 wavefunctions. The constants X and p must be determined from compressibility and l a t t i c e constant and we must therefore sum (3.4) over the entire l a t t i c e to determine the total energy per unit c e l l . The summation encounters d i f f i c u l t y due to the long-range nature of the Coulombic force (« 1/r) and the rapidly increasing number of contributing ion 3 site (« r ). Madelung f i r s t performed the summation using a theta function transformation to approximate the summation over l a t t i c e sites beyond a certain distance with an integral (cf. Born and Huang, 1966). We may then write the l a t t i c e energy per unit c e l l as: where a i s a pure number, known as the Madelung's constant, dependent only on the l a t t i c e structure, M i s the co-ordination number of the l a t t i c e (equaling the number of nearest neighbors) and r^ i s the nearest neighbour distance. As stated, we may use this form to determine X and p from the inter-ionic separation and the compressibility at static equilibrium. Following Born and Huang (1966) we note that the volume of the unit c e l l , v , i s proportional to the third power of r , so that: a o -- 3 (3./?) 41 Then the pressure and the compressibility are: Po = du. i z . i l ) At static equilibrium, of course, pQ= 0 so that = 1 * 5 ) * r j. -ct + — P i and the constants X and p may be evaluated. In the evaluation of the cubic coupling coefficient, we require the f i r s t three spatial derivatives of the inter-ionic potential. From (3.4) we can see that: (r) - -a •3 - c ; ^ v ¥ e 42 t We note that (f> (r ) 0 since i t i s the total l a t t i c e potential i which has a minimum i.e. u = 0. More rigorous theories of the potential function may include the repulsive forces between next-nearest neighbours. Born and Huang 1966) have shown however that this term i s generally less than one-fifth of the nearest neighbour force and i s the least important in the cesium salts of a l l the a l k a l i halides. 3.3 The Basic Calculations 3.3.1 The Cubic Coupling Coefficient In order to calculate Aio and T, we need to know the cubic coupling coefficient (o^^lk^^''^-^ 3^ given in appendix A equation (A.7). The physical significance of this coefficient may be seen in the t r i p l e summation terms. The tensor <J> i s a generalized force-a l a 2 a 3 constant tensor relating the force experienced by an ion due to the relative motion of any two other ions i n the l a t t i c e . The factors i , , -A (0K;L K ) represent the relative displacement of the two ions at 1 , 1 -»• (3) (OK) and (L K ) in the mode ( k ^ ) . Thus V expresses the la t t i c e energy due to the motion of an ion (in the resonant T.O. mode) in the force f i e l d resulting from the motion of a l l other pairs of ions in the l a t t i c e . 4 3 In practice we must restrict this summation in some manner as a summation over a l l ion pairs in a f i n i t e l a t t i c e w i l l clearly be impractical. Since any ion w i l l be coupled most strongly to those ions nearest i t , we have made the most natural restriction and considered only the coupling coefficient due to the ion's nearest t neighbours. Thus the summation E i s reduced to eight terms in the L'K* Csl structure. This truncated coefficient w i l l certainly be quite valid for a treatment of the short-range repulsive forces but w i l l probably be quite invalid for a treatment of the long-range Coulombic forces. In an attempt to minimize this error without making the calculation prohibitively long and complicated a procedure due to Eldridge (1973) has been adopted, whereby the long-range Coulombic forces between far-neighbours have been considered in an appropriate fashion. 3.3.2 The Complex Phase Shift. (3) Knowing V we may calculate Au> and T by means of a simple summation over the Brillouin zone. In fact we have calculated T according to equation (2.44) and then calculated Aco from T in the following way. We write the relations (2.40) in a short-hand way as: - £ ^ <Pf «>-*>']'' (3.31°) 44 » where to = co + to and each of the summations extends over a l l of the m n phonons. Then i t may be written: Eldridge (1973) has shown this to be the equivalent of a Kramers-Kronig relation between Aco and T which is a consequence of the causality of the function from which they are derived. In our case the summations over m and n were actually over the i phonons generated by the model and the frequencies co were grouped in one wave number intervals. The analytic representation of the principal function was: 7%+ Y (3.33) o. ol 45 The value of Ato, once calculated, was adjusted so that the frequency shift was zero at the measured resonance frequency. This accomplishes the same thing as the renormalization of the eigendata so that the original expression (2.41) could be used rather than the renormalized expression (3.14) 3.3.3 The Crystal Dipole Moment Operator The only undefined term i s now M(oj). Following Wallis et a l (1962) we have taken: where e i s the Szigeti effective charge and a and a are the ionic p o l a r i z a b i l i t i e s . 3.3.4 The Optical Properties The calculation of the dielectric function (2.41) i s straightforward once AOJ and T are known. Subsequently a l l of the optical parameters are easily calculated by means of the relations derived in section (2.1) One f i n a l adjustment was made in the calculations. The value of the Szigeti effective charge was adjusted so that our calculated value of the low-frequency dielectric constant was i n agreement with the tabulated values of Lowndes and Martin (1968). TABLE 3.1. Constants Used In The Calculation Of The Optical Properties Resonant absorption frequency c l Static dielectric constant High-frequency dielectric constant' , , b Madelung constant "Eldridge" constant 0 Nearest-neighbour distance^ Screening radius^ Overlap energy Szigeti effective charge** Ionic polarizability^ Ionic mass Second potential derivative Third potential derivative 62.0 cm_?" at room temperature 65.5 cm at 12°K 6.54 at room temperature 6.29 at 12°K 3.02 at room temperature 3.09 at 12°K 1.7627 4.0356 —8 3.95x10 _ cm at room temperature 3.90x10 cm at 12°K —8 0.318x10 Rcm at room temperature 0.291x10 cm at 12°K —8 2.60xlO_R ergs at room temperature 6.62x10 ergs at 12°K 0.67 at room temperature 0.75 at 12°K -24 3 3.137x10-, cm-5.982x10 cm 132.9 a.m.u. 126.9 a.m.u. 12 -3 0.492x10.-erg cm 0.582x10 erg cm at room temperature at 12°K 12 -3 2.924x1019 erg cm _ at room temperature 3.634x10 erg cm" at 12°K (c) See reference (Eldridge, 1973) (d) See reference (Karo and Hardy, 1963) (a) See reference (Lowndes and Martin, 1969) (b) See reference (Born and Huang, 1966) CHAPTER 4  EXPERIMENTAL CONSIDERATIONS 4.1 The Apparatus A Beckman R.IIC FS-720 Fourier spectrophotometer (essentially a Michelson interferometer) with step drive was used. The infrared radiation was produced by a high-pressure mercury arc lamp and measured by a Golay detector. The principle advantage of such an instrument i s the so-called Fellgett or multiplex advantage - the a b i l i t y to sample a l l spectral elements simultaneously and hence to significantly increase the signal-to-noise ratio. However, i n the region between 60 cm ^ and 90 cm the absorption was so high that the noise from the detector s t i l l prevented measurement of the transmitted intensity through the thinnest crystal we could polish (74u). The i n s t a b i l i t i e s of the source have been discussed by Dowling (1967). They take the form of slow uniform d r i f t s and step-function discontinuities. The slow d r i f t s were seen to create noise at very long wavelengths (<5 cm "*") while the discontinuities, ascribed by Dowling to fluctuations in the plasma rather than the quartz envelope, give rise to higher frequency noise (Dowling, 1967). Whenever.possible, the experimental runs were repeated i f either of these i n s t a b i l i t i e s were observed. For low-temperature work, the samples were cooled by being pressure mounted at one end against a brass tail-piece i n thermal contact with a reservoir of liquid nitrogen or helium. At nitrogen temperatures 48 the thermal contact was assumed perfect between the reservoir and the tail-piece and between the tail-piece and the samples. At helium temperatures the temperature of the tail-piece was measured, with a carbon resistance thermometer, to be 11°K. Other workers (Kuwahara, 1971) have measured the temperature, with a germanium resistance thermometer as 12±2°K. The pressure mount (with silver grease) has been assumed to create perfect thermal contact between the samples and the tail-piece. 4.2 The Experimental Technique The absorption coefficient was obtained from the measured intensities transmitted through vacuum, I q , and through the sample I , by using a form of Airy's formula corrected for the slight wedge shape of the samples: _ o(o( Here d i s the sample thickness and R i s the power reflectance. On the low-frequency side of the resonance the r e f l e c t i v i t y was calculated using the classical formula (2.24) where the refractive index, n, was calculated from the dispersion formula (2.15): 49 On the high-frequency side of the resonance the measured r e f l e c t i v i t y was used (but was negligible beyond 100 cm . Samples of several thickness were used at a l l temperatures so that the ratio i / l was between one-fifth and four-fifths over the o entire range of measurement. 4.3 The Sample 4.3.1 The Sample Crystal Structure A l l the a l k a l i halides have a rock-salt structure except for CsCl, CsBr, and Csl which have a cubic structure. Csl was chosen for this work for reasons set forth in the following section. The cubic structure with dual basis may be considered as two inter-penetrating cubic sub-lattices displaced by (d/2, d/2, d/2) where d i s the l a t t i c e paramter (sub-lattice cube edge). The nearest neighbour distance in Csl i s the largest for the a l k a l i halides and cesium and iodine are respectively the largest a l k a l i and halogen ions. 4.3.2 The Motivation For the Choice of Sample This investigation was undertaken as part of a continuing study of the l a t t i c e absorption of a l k a l i halides by Eldridge (see bibliography). Csl, being the "softest" a l k a l i halide, w i l l have the lowest characteristic l a t t i c e frequencies and should therefore more 5 0 rapidly show the third and higher-order processes as the temperature i s raised (cf section 2.2.3(E)). The second-order processes have a probability which varies linearly with the population numbers of the contributing phonons whereas the probability of third-order processes varies as the product of two population numbers. Thus, the third-order processes should begin to dominate when the population numbers become greater than unity. This occurs when the exponential factor i n : AI rjy) = { j y j o l i w(ty/j?B r]-1 j Us) becomes less than two. At any given temperature the factors to(kj) w i l l be the lowest, the population numbers the greatest, and the third-order processes the most prominent in Csl of a l l the a l k a l i halides. The transition from dominant third-order to dominant second-order processes should occur approximately when the exponential terms are two or when the exponent i s unity. That i s , when: Taking to(kj) to be of the order of 60cm this equality w i l l be satisfied very near the temperature of liquid nitrogen, 77°K. The third-order processes w i l l therefore clearly be dominant at room temperatures and the second-order processes at helium temperatures. 51 A further advantage of Csl i s the absence of natural 122 127 isotopes of 55 Cs and 53 I which precludes the possibility of isotope-induced absorption. As an analysis of the sample purity i s not available the question of mass defect modes does exist but common impurities are much lighter than the constituent ions so that any such effect should occur on the high-energy side of the main resonance. 4.3.3 The Sample Preparation As Csl does not cleave, the samples were cut with a wire saw and then polished. Ethylene glycol was used as a lubricant for the cutting and as a base for the grinding compound (jeweller's rouge on f e l t ) as Csl i s highly soluble i n water. The "polish" was probably largely an etch as the ethylene glycol did slowly dissolve the sample but the combination provided an optically f l a t surface very quickly. The sample was then immediately cleaned in acetone (approximately 30 seconds) to remove the ethylene glycol and then i n trichloroethylene (indefinitely) to remove the acetone which clouded the sample in time. This order of preparation was used to minimize the experimenter's contact with the more noxious solvents. The samples were stored in a dessicator and showed l i t t l e deterioration over long periods. 52 CHAPTER 5 EXPERIMENTAL AND THEORETICAL RESULTS 5.1. Introduction. The o p t i c a l constants of Csl have been ca l c u l a t e d from the l a t t i c e dynamical data as o u t l i n e d i n chapter 3. These have been compared with the measured r e f l e c t i v i t y and absorption c o e f f i c i e n t obtained as o u t l i n e d i n chapter 4. The only other experimental data a v a i l a b l e i n the l i t e r a t u r e comes from a Kramers-Kronig analysis of r e f l e c t i v i t y measurements (at room temperature and l i q u i d hydrogen temperatures) by Vergnat e t a l . (1969). The o p t i c a l constants i n t h i s work are obtained from a s i n g l e harmonic o s c i l l a t o r model with constant damping and consequently show no subsidiary s t r u c t u r e . They do, however, give a reasonable measure of the heights and widths of the main resonances. 5.2. Presentation and I n t e r p r e t a t i o n of Results. 5.2.1. The Complex Phase S h i f t . The contributions to gamma from summation and difference processes have been computed separately at one wave number i n t e r v a l s on e i t h e r side of the resonant frequency and smoothed with a 25-point polynomial.A This smoothing introduces spurious behaviour near the main resonance (e.g. negative summation values) and consequently, a l l of the c a l c u l a t i o n s have also been performed with a 9-point polynomial smoothing. Near 70 wave numbers the more reasonable r e s u l t has been used. 53 From the form of gamma, i t appears that two-phonon summation processes do not occur below 70 cm ^  so that the damping of the main resonance w i l l be due solely to difference processes. As these essentially disappear at low temperatures, there should be l i t t l e two-phonon and certainly no three-phonon effect on the main resonance at li q u i d hydrogen or helium temperatures. The calculated function shows peaks (more prominent with a 9-point smooth) at 15 cm \ 24 cm \ 83 cm \ 91 cm \ 104 cm ^  and 116 cm \ Most of these may be tentatively assigned to specific processes at certain points i n the zone by examining the specific (3) form of V , the magnitude of the population numbers (most important for difference processes), the necessity for two branches to have properly matched slopes and the eigenvector data. At X, the sine terms peak. Consequently, the L.O.-T.O. difference processes w i l l be strong (despite the relatively high energy and low population number) at 30 cm ^  as a result of the fortuitously matched slopes. The L.O. and T.A. modes w i l l not interact as the I ion i s at rest in a l l three modes. The T.O. + an d L.A. modes w i l l not interact as the Cs ion i s at rest. The T.O. + T.A. summation process w i l l be strong at 82 cm 1 as a result of the opposite slopes but this effect and the high energy w i l l severly limit the difference process. Similarly, there w i l l be a strong L.O. + L.A. summation process at 134 cm ^ . This w i l l , 54 however, be smaller than the dominant T.O. + T.A. combination as (3) a result of the frequency factors in the denominatior of V (a factor of 4 smaller), the lower population numbers as a result of the higher energy (a factor of 2 smaller) and the fact that neither mode is degenerate (a factor of 4 smaller). The L.A. -T.A. diffe rence process w i l l be strong at 22 cm as a result of the low energy of the modes and the equal slopes. At M, the sine terms are zero. At R, the sine terms peak. The L.O. and T.O. modes w i l l not interact as the I ion i s at rest in both modes. The L.A. + and T.A. modes w i l l not interact as the Cs ion i s at rest. However, the L.O. + L.A., L.O. + T.A., T.O. + T.A. and T.O. + L.A. modes w i l l a l l admit summation processes at 107 cm ^  and below as a result of the equal and opposite slopes. Along the symmetry directions, only two combinations stand out. In the (1,0,0) direction any possible combination should peak at X. In the (1,1,0) direction, the L.O. and L.A. modes should produce a strong summation process near data point (3,3,3) at an energy of 113 cm \ In the (1,1,1) direction the L.O. and L.A. modes have opposite and nearly equal slopes a l l the way to R according to Eldridge (1970). Despite the high energy of these -1 modes, summation processes may be expected at approximately 112 cm The L.O. and T.A. modes w i l l combine strongly as the T.A. mode i s lower in energy and degenerate. The average energy difference 55 here i s 16 cm \ On the basis of these considerations, we shall tentatively assign the peak at 15 cm ^  to the L.O.-T.A. difference process along (1,1,1); the peak at 24 cm ^  to the L.A.-T.A. difference process at X; the peak at 82 cm ^  to the T.O.+T.A. summation process at X; the peak at 104 cm '''to the L.O.+L.A., L.O.+T.A., T.O.+T.A. and T.O.+L.A. summation processes at R and the peak at 116 cm ^  to the L.O.+L.A. summation processes along (1,1,1) and (1,1,0). No assignment i s evident for the peak at 91 cm \ The two-phonon limit i s determined by the maximum total energy of 134 cm for the strong L.O.+L.A. summation process at X. The limit i s , however, not abrupt since the T.O.+L.O. summation processes on (1,0,0) may contribute near X at an increasing energy but with a decreasing intensity as we move towards the zone center. 5.2.2. The Optical Constants. The real and imaginary parts of the complex dielectric constant were calculated f i r s t from equation (3.14) with the aid of (3.24) and (A.24).°/\ Thus: 6U\- ^ +• 4n6.xK(^ilth ) 1 j-, (fj) where: 56 Aco = AcO - AiO, C£a.&) Q O The value of the Szigeti effective charge, e*, has been adjusted so that our calculated value of the low-frequency dielectric constant agrees with the measured values of Lowndes and Martin (1969). This requires a ratio of e*/e = 0.89 at room temperature and e*/e =0.78 at 12°K compared with the measured values of e*/e = 0.78 at room temperature and e*/e = 0.77 at 12°K. The L.O. frequency predicted by the real part of the dielectric constant is 93 cm ^ at room temperature and 94 cm ^ at 12°K. That these values do not agree with the measured value of 91.5 cm ^ (Karo and Hardy, 1963) i s probably attributable to poor values of delta. From (5.1): ^ ^ + 1 u.3) except very near the resonant frequency so that: 600 - — - =2 coqo A (a) Thus, the disagreement with the L.O. frequency casts doubt on delta - particularly at zero wave numbers and the L.O. frequency. 57 As may be seen from (5.3), the minor amount of structure in epsilon real is due mainly to delta. The imaginary part of epsilon, however, contains a l l of the gamma structure since: Using equation (2.7) we may show that: X1' Id*'- 6') tf.t) Between u)0 and cJ^ however, epsilon real i s negative so that: ft a. 4. ? Thus, most of the structure in K/lcomes from 6. rather than c and the peak at 82 cm i s largely washed out. It appears again in the index of refraction, N ,/(since: fat) In both epsilon imaginary and N, the peak at 82 cm ^  w i l l be unrealistically prominent as a result of the assumption of a coupling coefficient for the entire l a t t i c e of the same form as for nearest neighbors only. While this is also true of the other features i n gamma, i t i s most important here as this feature represents the only real subsidiary structure i n the optical properties. Outside of the region between u)Q and UJg 58 this function i s quite smooth since the extinction coefficient has essentially the same structure as £ '. The peak heights and widths of a l l the room temperature optical constants exhibit significantly too l i t t l e damping when compared with the data from Vergnat et a l . (1969). This i s direct evidence for the need to include the third and higher-order processes in our calculations. The low temperature peak heights and widths are however i n reasonable agreement with the data. This encourages us to believe that the damping of the main resonance is due only to second and higher-order difference processes (which w i l l disappear at low temperatures). The r e f l e c t i v i t y has been calculated using the relation (2.28) :/| The peak r e f l e c t i v i t y calculated at room temperature i s far too high since the third and higher-order processes are prominent at this temperature and constitute the primary damping. However, we find once again at 20°K that our calculated re f l e c t i v i t y i s in reasonable agreement with the data. Because the feature i n N at 82 cm ^  i s too large, the re f l e c t i v i t y w i l l be too low i n this region and i t w i l l be too large around 70 cm ^  at room temperature as a result of 5.2.3. The Reflectivity tijt: f.10, cn 59 the inadequate damping i n the theory. This i s probably the origin of the apparent side-peak at 87 cm in the calculated r e f l e c t i v i t y which does not show up in the data. At 12°K the agreement i s much better except for the dip at 82 cm ^ produced by the unrealistically prominent feature in N. 5.2.4 The Absorption Coefficient The magnitude of the calculated absorption coefficient appears to be approximately twice as large as i t should be over the entire range. Whether this i s a result of sparse or inaccurate la t t i c e dynamical data or the effect of neglecting next-nearest-neighbour repulsive forces i s not obvious. The high wave number difference region/)is however encouraging as i t i s evident that the additional damping of higher-order terms must be included here. These higher-order effects appear to be increasing with temperature faster than the two-phonon effects as was expected and as was observed i n the peak heights and widths of the room temperature optical constants. The calculated absorption coefficient i n the summation region/Jdoes not show the feature at 115 cm . It i s possible that the slight feature at 95 cm ^ w i l l s h i f t over to this position when more accurate lat t i c e dynamical data is used for this calculation. The high absorption beyond the two-phonon lim i t i s clear evidence of significant higher-order processes in the Csl l a t t i c e . 60 Fig. 5.1. Gamma at room temperature. Summation and di f f e r e n c e processes are shown separately. 61 62 F i g . 5.3. Gamma at 12 K. Summation and difference (X100) are shown separately. 63 Fig. 5.4. Delta at room temperature and 12°K. 64 in c\i o C M tn < Q. UJ <-cc. 1. Q-UJ UJ 3 o UJ o o-00 < CD O 9 T" in Q C \ J -i in OJ-i , -RT-y © POSITIVE DATA' POINTS 5 A NEGATIVE DATA POINTS 20 40 ~60~ 80 100 WAVE NUMBER (cm ') 120 1> 140 160 180 200 Fig. 5.5. Epsilon real at room temperature, taken from Vergnat et a l . (1969). The data points are 65 •' i, • i ' \ i i j Q POSITIVE DATA POINTS ? A NEGATIVE DATA POINTS ~ l i 1 1 1 1 I I l 40 60 80 100 120 140 160 180 200 WAVE NUMBER (cm - 1 ) E p s i l o n r e a l at 20°K. The data points are taken from Vergnat et a l . (1969). 66 in-C \ | ->-DC 9 00 Q_ LU o Y Q CM. I I I 9-point smooth \ • ROOM TEMPERATURE DATA If). I • H 2 TEMPERATURE DATA 0 2 0 4 0 6 0 8 0 •1. 100 WAVE NUMBER (cm ') 120 140 F i g . 5.7. Eps i l o n imaginary, et a l . (1969). The data points are taken from Vergnat 67 C O O J (D O -7 I y LL O • i CO CD oj - i I i OJ I O in-i • ROOM TEMPERATURE DATA • H 2 TEMPERATURE DATA 0 I 1 1 1 1 2 0 4 0 6 0 8 0 1 0 0 WAVE NUMBER (cm - 1) 120 140 Fig. 5.8. Extinction coefficient. The data points are taken from Vergnat et a l . (1969). 68 25-point smooth 9-point smooth • ROOM TEMPERATURE DATA • H 2 TEMPERATURE DATA 1 1 1 : 1 1 1 n O 20 40 60 80 100 120 140 WAVE NUMBER (cm - 1) F i g . 5.9. Refractive index. The data points are taken from "Vergnat et a l . (1969). 69 Fig. 5.10. Reflectivity at room temperature. Vergnat et a l . (1969). The data points are from Fig. 5.11. Reflectivity at 20°K. The data points are taken from Vergnat et a l . (1969). 71 72 Fig. 5.13. Alpha i n the difference region. Fig. 5.14. Alpha in the summation region. CHAPTER 6 SUMMARY AND CONCLUSIONS The cubic coupling coefficient for the Csl lattice has been calculated considering the effect of nearest neighbors only with a correction due to Eldridge (1970) for the long-range Coulombic forces but no correction for the more distant repulsive forces. The complex self-energy phase s h i f t describing the many-body effects introduced by two-phonon damping of the modes was then calculated. The latt i c e dynamical data of Karo and Hardy (1963) and the results of calculations by Wallis and Maradudin (1962) and Cowley (1963) using only the first-order dipole moment and including the contribution from cubic terms in the potential were used. Using the complex phase s h i f t , the optical constants (complex dielectric constant and complex refractive index), r e f l e c t i v i t y and absorption coefficient of Csl have been calculated in the vi c i n i t y of the primary infrared resonance. These have been compared with the results of a Kramers-Kronig analysis of the ref l e c t i v i t y of Csl by Vergnat et a l . (1969), the measured re f l e c t i v i t y at room temperature and the measured absorption coefficient at room temperature, 77°K and 12°K. The imaginary part of the complex phase s h i f t contains features at 15 cm \ 24 cm \ 82 cm \ 104 cm and 116 cm ^  which are assigned to specific two-phonon processes and a smaller feature at 91 cm ^ for which no assignment i s given. A l l of these features are enhanced by the form of the correction (3) in V for long-range Coulombic forces. The peak heights and widths of the calculated optical properties show general agreement with experiment at low temperatures but very poor agreement at room temperature. The calculated re f l e c t i v i t y shows adequate agreement with the measured values at low temperatures (except for a spurious -1 feature at 87 cm ) but i s significantly too high at room temperature. The calculated absorption coefficient is approximately twice the measured value over the entire range except in the high wave number difference region where the calculation is far too low at room temperature. The calculation predicts no absorption beyond the two-phonon limit of 134 cm ^ but there i s actually significant absorption to 190 cm ^ at room temperature. These results give clear evidence that third and higher-order processes are extremely important i n the Csl l a t t i c e . The fact that the calculated absorption coefficient i s too high indicates that the complex part of the self-energy sh i f t i s too large. This i s also indicated by the low-temperature peak heights of some of the optical constants and suggests that the next-nearest-neighbor repulsive forces (which (3) would diminish V and gamma overall) should be included. While these conclusions are strongly suggested, a proper follow-up investigation of them requires the recalculation of the optical properties with higher density lattice dynamical data. At the time of writing, this work i s being ini t i a t e d by Eldridge. 77 APPENDIX A.  THE CUBIC COUPLING COEFFICIENT The third order coupling coefficient which couples the resonant transverse optical phonon to two other phonons i s : i 3 3 I*' L where m (k.j.|K) i s the a.-component of the eigenvector or unit U ± 1 1 -*• 1 th polarization vector, n(k j ), associated with the K ion in the unit c e l l and the mode given by: frtf1%(Iijl Io) /Mil i , j ' t / >) (Az) 78 according to the notation of Karo and Hardy (1963). For notational simplicity, we shall define a reduced displacement: X(LK) i s the position vector of the K1"*1 basis element in the i / * 1 unit c e l l . We shall define the origin of the l a t t i c e to be the position of the K = 0 basis element in the unit c e l l under consideration (L = 0). Thus: X* (oo) = (o>0) o) ) T (o>) = (a)ai a) where 2a i s the length of one side of the unit c e l l , as shown in figure A.l. The summation 2 extends over the basis elements in K one unit c e l l and, since we shall restrict ourselves to a consideration i of nearest neighbours, the summation E w i l l be over the eight nearest L'K' neighbours only. Since the Csl crystal l a t t i c e has cubic symmetry we may consider only one Cartesian component df the transverse optic phonon, say x, and later derive the complete coefficient by invoking the , symmetry of the crystal. Thus the force constant tensor (j) i s alaZa3 -: -reduced to the matrix <f> , -. , ^ . , xa„a and the f i r s t bracket xn the product 3 6 term ir^ w i l l always be: Fig. A. l . A portion of the Csl l a t t i c e . 80 2 o + ' yV©*) - /Y)/]x - syrtf (i >(') — r — J e (A.f) In accordance with this definition ZTTI A' 7 (V) K ) (M) Thus, our truncated coupling coefficient may be written: J The required derivative of the two-body potential is 81 + / ^ — - + y c , ^ ^ j * 3 /- 1 ^ . ^ 3 >' where the spatial derivatives of the potential are given in equation (3.20). We note that (A.8) contains three types of terms : 'P"/1"". (p/f0a . From the calculated values of the spatial derivatives given in (3.20) we can see that the relative magnitudes are (p '• (y{'r<>\ (p/r<,*~ '. ', /ooo : 3b : 3>f. However, as mentioned in section 3.3.1 we are assuming a cubic coupling coefficient for the entire l a t t i c e of the same form as that for nearest neighbours only. Consequently, the potential derivatives appear in V as Y J i <P/rb £ <p /I'S . . These sums require /h* i n order to evaluate the effect of the Coulombic term in each case. The summation may be performed by a procedure due to Eldridge (1973) whence jj- , 'h - ' '1> where a i s a constant depending on the la t t i c e structure. After summation, we find: I, f \ (A 1) 82 The relative magnitudes of these terms are ^ , $ •' £ <p /r ; £ (y/)-1 ;; /ooo ; o(, : ? so that we shall ignore the terms in the f i r s t spatial derivative in subsequent calculations. The other two spatial derivatives w i l l be: <p/, >(Y>e • ( where a = /, 03£t> The force-constant matrices for the eight nearest neighbours to any ion in terms of the second and third spatial derivatives of <f> (r) are list e d in table A . l . We note here the symmetry in <b J J Txa 2a 3: 4 c ^ 3 (l ™> - - H ™, -<») lA'") This leads us to consider the contributions to the summation over nearest neighbours in pairs of symmetrically displaced neighbours. The following i s a step-by-step account of the summation for the nearest neighbours i n the (1,1,1) and (1,1,1) directions of both basis elements. 83 r/1 a \ ? -r if \->f -x p > • J T fat, oti ' ' Table. A . l . The generalized force-constant tensors for the eight nearest-neighbors i n the C s l l a t t i c e . 84 t t (i) Term 1: K = 0, (0K,L K ) = (a,a,a) Using the definition: and summing the appropriate terms from Table A.l we may write: ' A 2 .3 c4 • [(M'P'-jif'l iW + W] o n/. 3a. 7/ Cii) Term 11: K = 0, (0K,L K ) = (-a,-a,-a) Using the symmetry relation (A. 10) for <j> and the x 0 1 2 a 3 definition: A t ' [(0,0,0) : (~ar^-a)] = lj. 85 we may immediately write: * />>• /y •+v4 ( i i i ) Summation: Term 1 + Term 11 Using the fact that the delta function w i l l only admit contributions to such that lL, = -k^ and writing out the factors A* and B* e x p l i c i t l y , i t i s a simple i f tedious matter to show that: a. a. i l 3 a Arlj (a>) /r/\y(so)-/nAj(jto) ,w\u (si) -86 A precisely analogous consideration of the corresponding nearest neighbours for the other ion i n the basis (K=l) u t i l i z i n g the definitions: (in) shows that the exponential terms exp 2irik (a,a,a) which appear outside of the brackets in the definition of A^ j(a,a,a): (0K;.L K )J a l l appear in the summation in cancelling pairs. In fact, the summation analogous to the above but for the K = 1 ion i s precisely the same as the summation for the K = 0 ion except for an overall change of'sign and the fact that 1 1 A i s replaced by .A . However, from (A.5) i o x r 1 x so that the summations for K = 0 and K = 1 are equal. We may, of course, now write the summation C f o r both basis elements) of the contributions from nearest neighbours i n the '(1,1,1) and (1,1,1) directions. However, we shall pause to make the following definitions in order to express the summation more compactly: 1 M ) + at. C , £ ^J; M 1 AyVi (21) - £ 0 J d <\ 3 J'J'syYl? (m) <-ir)-y (3c) + j J * syrfyho) /yY)y (31) +• -f- yyrlc, hi) ^ - ' h i , / j o ) ~~ srrl s)tf u /Ja) srt srrf* hi) d 0 d- 0 /rA / n ^0 - W j ^30) - -rV)j0c)^y(3i)J -~, 3 f srY)7(?>) /Wl-yho) - s r y l r ho) wr7, h>)l + d + s ) Y l s r \ [ s i Y l t j h >) (j*) - sfrfyhc) h i ) 88 In this notation we may write the summation just calculated as: The factor A"*" may be written: o x J L ^IM, J Maradudin et at. (1963) have shown that: stflflcj, I.) " M and using the orthogonality condition: we may show that: i r , . A ~] I M, Mo -1/2 We also note that there i s a factor M^ i n our definition (A.3) of m (iK). We shall now redefine m (iK) as the actual rather than 8 9 reduced displacement, i.e.: An examination of (A.19) w i l l show that this redefinition -1/2 necessitates the inclusion of a multiplicative factor ( M . M ) 1 o in the summation. The analogous summations for the other three diagonal directions may be similarly derived. The general form i s : • \ (A The cubic coupling coefficient may now be e x p l i c i t l y written as: i r r 43 -•4-9 0 APPENDIX B  THE KRAMERS-KRONIG RELATIONS The real and imaginary parts of many linear causal response functions are related by the fact that causality requires the function to be analytic in some portion of a complex frequency plane so that the Cauchy Integral Theorem applies. Applying these relations to a physical system S with a time-dependent excitation £(t) and a response p(t) leads to the Kramers-Kronig relations between the real and imaginary parts of the "generalized susceptibility", ^Cco) . The validity of the Kramers-Kronig relations rests on four general assumptions concerning the system S CAbragams, 1961): (i) The system S i s linear. i.e. If ^ 1 -•• P 1 and %2 •+ p 2 then + c ^ -* V l + C 2 P 2 -( i i ) The system S i s stationary. i.e. If £(t) •* p(t) then £(t - t ) p (t - t ) . o o The condition of a monochromatic response to a monochromatic excitation follows from the conditions (i) and ( i i ) . ( i i i ) The system S obeys the principle of causality. i.e. If ?(t) = 0 for t < t , p(t) = 0 for t < t . o o 91 (iv) The system S shows a f i n i t e t o t a l response to a f i n i t e t o t a l e x c i t a t i o n . i . e . I f E f t ) = | U ( t ' ) | dt' i s f i n i t e then « o JL ' r i t J ILjCt) = |p(t ) | dt i s f i n i te. The r e l a t i o n s derived from these assumptions may be w r i t t e n f o r the d i e l e c t r i c constant as: = ^ •> ± P i " ' t ^ t t ) Mo' ' . (B.,a) 77 J (co'*-As a s p e c i a l case, consider a system with a monochromatic absorption as i n the undamped approximation. 6 "fa) = £>(*)-uJ<) (3.a) The r e l a t i o n (B.la) gives: 6'fa) - 6* - rr (to:-to") (2.3) which i s the response of a s i n g l e undamped mode. The analogous r e l a t i o n s to QB.l) e x i s t between the magnitude, p, of the conventionally measured r e f l e c t i v e t y and the phase angle, <j>, where: 9 2 as well as the real and imaginary parts of the impedance functions in classical electrodynamics. Their importance derives from the fact that the restrictions of linearity and causality allow one to completely define these complex functions by measuring but one of their components. 93 BIBLIOGRAPHY ABRAGAM, A., (1961), The Principles of Nuclear Magnetism (Clarendon Press, Oxford) Born, M. and Huang K., (1966), Dynamical Theory of Crystal Lattices (Clarendon Press, Oxford). Burstein, E., (1964), in Phonons and Phonon Interactions. Edited by T.A. Bak (W.A. Benjamin, Inc., New York). Cowley, R.A., (1963), Advan. Phys., 12, 421. Dowling, J.M., (1967), Investigations in the Far-Infrared With A  Lamellar Grating Interferometer, Air Force Report No. SSD-TR-67-30 (Aerospace Corporation, El Segundo, California). Eldridge, J.E., (1970), J. Phys. C: Solid St. Phys., 3, 1527. Eldridge, J.E. (1972), Phys. Rev. B, Vol. 6, No. 4, 1510. Eldridge, J.E., (1973), To Be Published Huang, K., (1951), Proc. Roy. Soc. A, 208, 352. Johnson, K.W. and B e l l , E.E., (1969), Phys. Rev., 187, 1044. Karo, A.M. and Hardy, J.R., (1963), Phys. Rev., 129, 2024 Kleinman, D.A. and Spitzer, W.G.,(1960), Phys. Rev., 118, 110. Kuwahara, R.H., (1971), Ph.D. Thesis, University of Brit i s h Columbia. Lowndes, R.P. and Martin, D.H., (1969), Proc. Roy. Soc. A, 308, 473. Lyddane, R.H., Sachs, R.G. and Teller, E., (1941), Phys. Rev.,'59, 673. Maradudin, A.A., Montroll, E.W. and Weiss, G.H., (1963), Theory of Lattice Dynamics i n the Harmonic Approximation (Academic Press, New York). Mitskevich, V.V., (1963), Soviet Physics - Solid State, h, 2224. Szigeti, B., (1960), Proc. Roy. Soc. A, 258, 377. Vergnat, P., Claudel, J., Hadni, A., Strimer, P. and Vermillard, F., (1969), J. Phys. (France), 30, 723. Wallis, R.F. and Maradudin, A.A., (1962), Phys. Rev., 125, 1277. Ziman, J.M., (1960), Electrons and Phonons (Clarendon Press, Oxford). Ziman, J.M. (1969), Principles of the Theory of Solids (Cambridge University Press, Cambridge). 

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