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Dispersive--reflection spectroscopy in the far infrared Staal, Philip Ralph 1979

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DISPERSIVE - REFLECTION SPECTROSCOPY IN THE FAR INFRARED by PHILIP RALPH STAAL B.Sc, University of Waterloo, 1972 M.Sc, Dalhousie University, 1973 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES THE DEPARTMENT OF PHYSICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA ©Philip Ralph Staal, 1979 In present ing t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o 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 tha t 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 f o r re ference and s tudy. I f u r t h e r agree tha t permiss ion f o r ex tens ive copying of t h i s t he s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s en t a t i v e s . I t i s understood tha t copying or p u b l i c a t i o n of 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 pe rm iss i on . Department of The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook P lace Vancouver, Canada V6T 1W5 Date >E-6 B P 75-51 1 E i i ABSTRACT A d i s p e r s i v e r e f l e c t i o n spectrometer has been b u i l t i n which the fix e d mirror of a commercial f a r - i n f r a r e d Michelson interferometer i s replaced by a polished sample. This allows one to measure simultaneously and accurately both the amplitude r e f l e c t a n c e of the sample and the phase change upon r e f l e c t i o n . From these q u a n t i t i e s , the complete o p t i c a l properties can be e a s i l y obtained without the need f o r r e s t r i c t i v e and sometimes inaccurate Kramers-Kronig analyses. An o p t i c a l system has been designed and b u i l t i n which an ei g h t - s e c t i o n switching mask i s placed i n f ront of a partly-aluminized sample, which r a t i o s out many sample surface imperfections and eliminates most asymmetries due to the spectrometer's o p t i c s . Furthermore, by a l t e r n a t e l y switching between the sample and the reference surfaces at each step of the moving mirror, simultaneous interferograms are obtained, thereby removing most of the remaining causes of phase errors which have previously l i m i t e d the accuracy of t h i s powerful technique. Although extension of these techniques to liq u i d - h e l i u m temperatures i s d i f f i c u l t , a dewar i s described which allows d i s p e r s i v e measurements on samples at temperatures at l e a s t as low as 25 K. High-resolution r e s u l t s are presented for NaCl from 25 to 500 cm *. These are compared with a b - i n i t i o c a l c u l a t i o n s of the damping of the transverse, o p t i c resonance due to cubic and quartic anharmonicity. S h e l l -model lattice-dynamical data are used as input f o r the c a l c u l a t i o n s and corrections are also made for the damping of the " f i n a l - s t a t e " phonons. The o v e r a l l agreement i s excellent at room temperature and good at 48 K. 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 x i CHAPTER I INTRODUCTION AND LITERATURE SURVEY 1.1 Far Infrared O p t i c a l Properties . . . . . 1 1.2 D i s p e r s i v e - R e f l e c t i o n and -Transmission Spectroscopy 5 1.3 Low Temperature Dispersive R e f l e c t i o n Spectroscopy 16 1.4 Thesis Outline . . . . . . . . . . . . <~ . 18 CHAPTER II THEORY OF DISPERSIVE REFLECTION SPECTROSCOPY 11.1 The General Dispersive Interferometer . . . . . . . . 19 11.2 D i s p e r s i v e - R e f l e c t i o n Measurements 25 11.3 E f f e c t s Due To The Reference Mirror . . . . . . . . . 28 11.4 Including A Sample-Arm Window 29 11.5 The Apodizing Window Function . 31 I I . 6 Frequency R e s t r i c t i o n s Due To Sampling 36 II.7 Finding Zero Path-Difference From The Interferogram 37 iv Page CHAPTER I I I EXPERIMENTAL SETUP - ROOM TEMPERATURE III.-l Optics . 39 111.2 Far Infrared Source 45 111.3 E l e c t r o n i c s 46 111.4 Computing 49 CHAPTER IV EXPERIMENTAL SETUP - LOW TEMPERATURE I V . l Optics 51 IV.2 Dewar . . . . . . . . . . 51 IV.3 Sample Mount 57 IV. 4 Vacuum 63 IV.5 Alignment Procedure 65 IV.6 Tests Of The Instrument . . . i 65 CHAPTER V ALKALI - HALIDES THEORY V . l Introduction • 68 V . 2 The D i e l e c t r i c Constant 68 V .3 Phase S h i f t Of The Transverse Optic Resonance . . . 70 V .4 I n t e r i o n i c P o t e n t i a l , . . 70 V .5 L a t t i c e Dynamics 72 V .6 Two-phonon Damping Resulting From Cubic Anharmonicity 75 V.7 Three-phonon Damping Resulting From Quartic Anharmonicity 82 V .8 Isotope-induced One-phonon Damping . . 86 V .9 Damping Of The " F i n a l - s t a t e " Phonons 87 V Page CHAPTER VI THEORETICAL AND EXPERIMENTAL RESULTS - ROOM TEMPERATURE VI. 1 The Experiment 89 VI.2 Phase S h i f t Of The Transverse Optic Resonance . . . 90 VI.3 O p t i c a l Properties . . . •. 105 CHAPTER VII THEORETICAL AND EXPERIMENTAL RESULTS - LOW TEMPERATURE V I I . l The Experiment . 113 VII.2 Phase Shift. Of The Transverse Optic Resonance . . . 117 VII.3 O p t i c a l Properties 125 V I L A Discussion of the Disagreement i n the Magnitude of the Damping 129 CHAPTER VIII CONCLUSIONS V I I I . l The Dispersive R e f l e c t i o n Spectrometer 138 VIII.2 The Far Infrared O p t i c a l Properties of NaCl . . . . 140 BIBLIOGRAPHY 142 APPENDIX A SOURCE INTENSITY CONTROLLER 150 APPENDIX B ELECTRONICS FOR ALTERNATE SAMPLING OF INTERFEROGRAMS 154 APPENDIX C EFFECT OF NON-IDEAL REFERENCE SURFACE 163 APPENDIX D RELATIONS BETWEEN OPTICAL CONSTANTS 166 LIST OF TABLES Table Page. V-I S h e l l Model Parameters . 73 V- I I Force-constant tensor f o r the two nearest neighbours l y i n g along the p o l a r i z a t i o n d i r e c t i o n x of the incident photon 84 VI- I Constants Used In The Ca l c u l a t i o n s (290 K) . . . 92 VII - I Constants Used In The Ca l c u l a t i o n s (48 K) 118 v i i LIST OF FIGURES Figure Page 1-1 O p t i c a l properties of KBr showing inaccuracies due to a Kramers-Kro'nig analysis of the power r e f l e c t a n c e , from Hadni et a l , Reference 2 . . . . 3 1-2 S i m p l i f i e d optics of a normal Michelson interferometer . . 6 1-3 Interferograms obtained with given s p e c t r a l d i s t r i b u t i o n s 7 I - 4 An asymmetric ihterferogram and i t s complex Fourier transform 9 I I - l Spectral width functions 33 I I - 2 a. The t o t a l interferogram i n t e n s i t y with slowly changing source i n t e n s i t y , b. The r e s u l t of subtracting the average i n t e n s i t y from a. c. The r e s u l t of t r i a n g u l a r apodization of b. d. The r e s u l t of "Happ and Genzel" apodization of b 35 I I I - l Optics for room-temperature sample . 40 III-2 A diagram of the partly-aluminized sample and the e i g h t -section switching mask 41 III-3 The configuration of Parker's switching mask (Ref. 79) . . 43 III-4 The upper diagram shows schematically the blocking of l i g h t by the switching mask. The lower example interferograms show the e f f e c t of a 50% l i g h t blockage which reduces the r e l a t i v e modulation to 2/3 of i t s unblocked modulation . . 44 I I I - 5 A schematic of the detection and recording e l e c t r o n i c s . . 47 IV- 1 A schematic, drawn to scale, of the RIIC FS-720 o p t i c s , modified for d i s p e r s i v e r e f l e c t i o n measurements of a liqu i d - h e l i u m temperature sample . 52 IV-2 The s i x - l i t r e sample-holding l i q u i d - h e l i u m dewar . 5 3 IV-3 Wiring layout on dewar and f i l l - t u b e i n s e r t s . 55 IV-4 O p t i c a l and vacuum arrangement 56 IV-5 The sample-holding liquid-helium dewar 58 IV-6 The l i q u i d - h e l i u m can from the adjusters side 59 IV-7 The sample cold-mount, with alignment mechanism . . . . . '60 v i i i Figure Page IVr8 A NaCl sample clamped in to the l i q u i d - h e l i u m can . . . . . 61 IV-9 The M.H.L . Pryce vacuum system 64 IV- 10 The imaged r e f l e c t i o n from the .a lumin ized sample 66 V - l Frequency d i s p e r s i o n curves along the major symmetry d i r e c t i o n s as w e l l as some other r e g i o n s , generated by the eleven-parameter model fo r NaCl at room temperature . . . 74 V - 2 I l l u s t r a t i v e frequency d i s p e r s i o n curves showing p o r t i o n s of l a rge c o n t r i b u t i o n s to the summation and d i f f e r e n c e processes 76 V I - 1 . Ref lectance amplitude and phase d i f f e r e n c e from ir of a 70 nm-thick l ayer of aluminum . 91 VI-2 Measured damping spectrum of the t ransverse o p t i c resonance i n NaCl at 290 K with 2 - c m - 1 r e s o l u t i o n , together with the c a l c u l a t e d two-phonon c o n t r i b u t i o n to the damping with equal r e s o l u t i o n 93 VI-3 Ca lcu la ted three-phonon damping of the t ransverse o p t i c resonance i n NaCl at 290 K with 2-cm *• r e s o l u t i o n . . . . 96 VI-4 Measured damping spectrum of the t ransverse o p t i c resonance i n NaCl at 290 K with 2 cm--'- r e s o l u t i o n , together wi th the two- and three-phonon c o n t r i b u t i o n s to that damping, with equal r e s o l u t i o n . . . . . . . . . . . . . . . 99 VI-5 Damping of the " f i n a l - s t a t e " two phonons, as a f u n c t i o n of wave number 100 VI-6 Ca lcu la ted three-phonon damping of the t ransverse o p t i c resonance i n NaCl a£ 290 K a f t e r convo lu t ion wi th a Loren tz ian of 9 cm h a l f - w i d t h at half-maximum 102 VI-7 Measured damping spectrum of the t ransverse o p t i c resonance i n NaCl at 290 K with 2-cm~ r e s o l u t i o n , together with the c a l c u l a t e d two- and three-phonon c o n t r i b u t i o n s to that damping, a f t e r c o r r e c t i n g f o r the l i f e t i m e of these two and three phonons 103 VI-8 Measured and c a l c u l a t e d wavenumber s h i f t of the t ransverse o p t i c resonance i n NaCl at 290 K 104 VI-9 Ca lcu la ted and measured amplitude r e f l e c t a n c e r of NaCl at 290 K with 2 c m - 1 r e s o l u t i o n 106 ix Figure Page VI-10 Calculated and measured reflectance phase angle <{> of NaCl at 290 K with 2 cm - 1 resolution . 107 VI-11 Calculated and measured absorption coefficient a of NaCl at 290 K . . . . . 108 VI-12 Calculated and measured refractive index n of NaCl at 290 K 109 VI-13 Calculated and measured real part e' of the dielectric constant of NaCl at 290 K 110 VI- 14 Calculated and measured imaginary part e" of the dielectric constant of NaCl at 290 K . . . . . . . . . . . I l l VII- 1 Polaroid photographs of the aluminum on one of the NaCl samples after being thermally cycled several times . . . . 115 VTI-2 The effect of a bubbled aluminum reference surface . . . . 116 VII-3 The calculated isotope-induced one-phonon damping of the transverse optic resonance in natural NaCl at 48 K with 2 cm resolution . . . . . . . . 119 VII-4 Measured damping spectrumof the transverse optic resonance of NaCl at 48 K with 2 cm resolution, together with the . calculated three-phonon damping with equal resolution . . 122 VII-5 Measured damping spectrum ^ f the transverse optic resonance of NaCl at 48 K with 2 cm resolution, together with the two- and three-phonon and isotope-induced one-phonon contributions to that damping, with equal resolution . . . 123 VII-6 . Measured and calculated wave-number shift of the transverse optic resonance of NaCl at 48 K 126 VII-7 Calculated and measured amplitude reflectance r of NaCl at 48 K 127 VII-8 Calculated and measured reflectance phase angle <)> of NaCl at 48 K 128 VII-9 Calculated and measured absorption coefficient a of NaCl at 48 K 130 VII-10 Calculated and measured refractive index n of NaCl at 48 K 131 VII-11 Calculated and measured real part e' of the dielectric constant of NaCl at 48 K 132 X Figure Page VII-12 . Calculated and measured imaginary part e" of the d i e l e c t r i c constant of NaCl at 48 K 133 VTI-13 Comparison of amplitude r e f l e c t a n c e measurements at various temperatures i n NaCl i n the v i c i n i t y of peaks (1) and (2) 136 A - l The i n t e n s i t y sensor c i r c u i t of the mercury-arc source i n t e n s i t y c o n t r o l l e r . . . . . . . . . . . . . . 151 A-2 The power c o n t r o l c i r c u i t of the source c o n t r o l l e r . . . . 152 B - l Integrator and a l t e r n a t i n g system.general layout 155 B-2 Integrator board c i r c u i t . . . . . . . . 157 B-3 The eight-pole a c t i v e Bessel f i l t e r , with a buffered input 158 B-4 The two-channel sample-and-hold c i r c u i t 159 B-5 A d d i t i o n a l l o g i c f o r a l t e r n a t e sampling . 160 B-6 Timing diagram for a l t e r n a t e sampling (symmetric) . . . . 161 x i ACKNOWLEDGEMENTS Most of a l l , I wish to thank Professor John E. Eldridge f o r always being a v a i l a b l e with help and advice with a l l aspects of t h i s t h e s i s . His e t e r n a l optimism ("things could be worse, you know"), h i s enthusiasm f o r physics, and h i s p r a c t i c a l way of approaching things ("buy i t , I ' l l f i g u r e out how to pay f o r i t l a t e r " ) have helped to make my stay at U.B.C. a very valuable and enjoyable experience. I wish to thank Professor H. Gush f o r many h e l p f u l discussions on f a r i n f r a r e d Fourier transform spectroscopy. His many i n s i g h t s and extensive knowledge of the f i e l d have helped to answer many questions. I am g r a t e f u l to Professor J . W. Bichard f o r discussions on low-temperature experimental techniques. I wish to thank Herman Bless f o r h i s help i n the design of the switching mask and the asymmetric dewar, and Peter Haas for b u i l d i n g the dewar and helping with the seemingly endless number of modifications required. I would also l i k e to thank the other members of the shop f o r t h e i r expert help. I wish to thank Professor M.H.L. Pryce for the funds to keep t h i s project going when a vacuum system was needed. Thanks to Cy Sedger for h i s h e l p f u l advice on such matters as how not to k i l l myself i n the student machine shop. I am g r a t e f u l to Jack Bosma f o r h i s help with the vacuum evaporator and Rolf Weissbach for h i s help with our vacuum problems. I would also l i k e to thank Wolf Breuer f o r h i s help with e l e c t r o n i c devices, and f o r bringing around bowling candies once a year. Thanks to the U.B.C. Natural, Applied and Health Sciences Comittee f o r summer support. F i n a l l y , I wish to thank the members of the S o l i d State Coffee Club and a l l the other f a c u l t y , s t a f f and students with whom I've had stimulating discussions during my stay at U.B.C. CHAPTER I 1 INTRODUCTION AND LITERATURE SURVEY 1.1 Far Infrared O p t i c a l Properties In order to determine the o p t i c a l properties of a substance, two qua n t i t i e s must be measured at each o p t i c a l frequency. I f , f o r instance, the power re f l e c t a n c e and power transmittance are known at the same frequency, the frequency dependent r e a l and imaginary parts of the complex index of r e f r a c t i o n ( n 5 n + i k ) or the d i e l e c t r i c constant ( i. .= e' + i e " ) can be obtained at that frequency. For cases such as conducting materials and d i e l e c t r i c s near a resonance however, two complementary o p t i c a l properties are almost impossible to measure with standard methods. Where the r e f l e c t i o n i s large and the absorption i s high f o r example, transmission measurements of bulk samples are very d i f f i c u l t . I t i s i n t e r e s t i n g to note, therefore, the wide v a r i e t y of techniques which have been developed to t r y to overcome t h i s problem. I t i s possib l e , f o r instance, to c a l c u l a t e the complete o p t i c a l properties of a sample i f i t s r e f l e c t a n c e i s known accurately at a l l frequencies within a wide range of the frequency region of i n t e r e s t . The Kramers-KrSnig dispersion r e l a t i o n : ^ r e l a t e s the phase of the re f l e c t a n c e $ at one frequency v to the power re f l e c t a n c e R at a l l frequencies. Since the power r e f l e c t a n c e i s i n the i n t e g r a l as l n R, a small value of R at any frequency w i l l have (I.D a large e f f e c t on <j) at almost every frequency. For a small value of R, the f r a c t i o n a l error of i t s measurement w i l l be high and the resultant phase calculated by the Kramers-KrGnig r e l a t i o n w i l l be generally 2 inaccurate. A t y p i c a l example of t h i s i s given by Hadni et a l , (see Fi g . I - l ) a n d a demonstration of the r e s t r i c t i o n s of a Kramers-Kronig 3 a n a l y s i s with L i F has also been given by Eldridge and Howard. By methods such as evaporation, films of material can be made s u f f i c i e n t l y t h i n f o r transmission measurements, but t h e i r properties often d i f f e r from those of the bulk m a t e r i a l . Interference from m u l t i p l e r e f l e c t i o n s can also be a problem with these t h i n samples. T o t a l i n t e r n a l r e f l e c t i o n can be used to study absorption, but i t i s most appropriate for studying l i q u i d s or t h i n films rather than 4-7 bulk samples. With t h i s method, l i g h t i s t o t a l l y i n t e r n a l l y r e f l e c t e d i n a dense medium bounded by a l e s s dense medium. The r e f l e c t e d r a d i a t i o n penetrates the outside medium to a depth of about one tenth of a wavelength, and absorption i n the outside medium can be seen i n the r e f l e c t e d l i g h t . Quantitative analysis of the absorption i s d i f f i c u l t with t h i s method. Normally, multiple r e f l e c t i o n c e l l s are required f o r s e n s i t i v i t y . They must be made of low loss material such as s i l i c o n , which r e s t r i c t s the index of r e f r a c t i o n of measureable materials to l e s s than about three. This method also r e s t r i c t s l i g h t throughput and r e s t r i c t s the angle of incidence to a small range not inclu d i n g normal incidence. Measuring the reflec t a n c e of a sample with and without a t h i n d i e l e c t r i c coating also allows a c a l c u l a t i o n of the o p t i c a l properties This i s a dest r u c t i v e method, which cannot be used i f the index of r e f r a c t i o n v a r i e s widely. This method takes advantage of the mult i p l e 3 200 250 van 3* FIG. 3. At an incident angle cf 15°, Frcsnel's formulas sbov,- that J?, never exceeds Kv by more than a few percent. The curve calculated for normal reflection (solid line) passes between the Ji, and Ji, curves for the incidence angle of 15'. Note the very small value of R KBr 300'K ft 250'crtV Fio. 5. Phase shift 0 in the reflection as a function of u for four values of A W : A W - 0 . 0 0 0 0 1 (curve No. 1); A W = 0.0001 (curve No. 2); A W = 0.01 (curve No. 3); A W = 0.02 (curve No. 4). t K at 200 cm which causes the phase to be inaccurate when calculated by the Kramers-Kronig di s p e r s i o n r e l a t i o n . R mm i s the value of R at =200 cm \. KBr 300"K 250 yenv* K B r 300*K 250 yem"1 FlG. 6. Refractive index n of KI!i at room temperature for lour values of A W : A W - 0 . 0 0 0 0 1 (curve No. 1); A W = 0.0001 (curve No. 2); A W - 0 . 0 1 (curve .No. 3); A W - 0 0 2 (curve No. 4). -to. 7. Absorption index k of K B r at room temperature for ee values of A W : K^o* = 0.00001 (curve No. 1); A W " 0 . 0 1 Fw three values oi A u i i n " . — • •• , , (curve No. 3); A',„; 3*^0.02 (curve No. -1). From now on >ve shall ouiy retain curve No. 3. Figure 1-1.Optical properties of KBr showing inaccuracies due to a Kramers-Kronig analysis of the power r e f l e c t a n c e , from Hadni et a l , Reference 2. 4 interference i n a t h i n f i l m . Thus, the e f f e c t of the phase of the l i g h t r e f l e c t e d from the sample into the f i l m i s measured when the r e f l e c t i o n with multiple interference i s measured. Therefore, two o p t i c a l properties are measured: the power refle c t a n c e d i r e c t l y , and the phase of the r e f l e c t a n c e i n d i r e c t l y . The o p t i c a l properties and thickness of the overcoating layer must be known accurately. Normal-incidence measurements are possible with t h i s method. I t may also be possible to measure the o p t i c a l properties by measuring s p e c i a l angles such as the c h a r a c t e r i s t i c angles (by ellipsometry, where l i n e a r l y p o l a r i z e d r a d i a t i o n becomes c i r c u l a r l y p o l a r i z e d ) , Brewsters angles (at which unpolarized l i g h t becomes l i n e a r l y p o l a r i z e d ) , and the c r i t i c a l angle for t o t a l i n t e r n a l r e f l e c t i o n . Since these angles are u s u a l l y frequency dependent, and since only one angle can be measured at a time i n p r a c t i c e (which also r e s t r i c t s l i g h t throughput), monochromatic r a d i a t i o n must be used. This eliminates the frequency multiplexing advantage of Fourier transform spectrometers normally used i n the f a r i n f r a r e d . These angular methods do have an advantage, i n that the experiment can be set up so that there i s a minimum i n t e n s i t y at the desired angle. The.minimum can be measured e a s i l y , without an absolute c a l i b r a t i o n of the i n t e n s i t y . The e m i s s i v i t y of a sample can be used to c a l c u l a t e o p t i c a l pr op er ti es , but of course the emittance of materials at low temperatures becomes extremely weak. I f several appropriate angles or p o l a r i z a t i o n s are used, complete o p t i c a l properties can be obtained from power r e f l e c t a n c e measurements. The mathematics f o r t h i s i s complicated however, and the measurements are not very accurate. One i s also r e s t r i c t e d to non-normal i n c i d e n c e , and the angular spread of the r a d i a t i o n must be kept s m a l l , which means a weak s i g n a l . F o r t u n a t e l y , there i s one method which allow s d i r e c t simultaneous measurement of two o p t i c a l p r o p e r t i e s at normal in c i d e n c e from bulk samples of moderate s i z e and any index of r e f r a c t i o n : d i s p e r s i v e F o u r i e r transform spectroscopy (DFTS). For a survey of methods f o r measuring o p t i c a l p r o p e r t i e s , i n c l u d i n g the d i s p e r s i v e methods which he 8 and J . E . Chamberlain pioneered, see E.E. B e l l ' s review. 1.2 D i s p e r s i v e - R e f l e c t i o n and -Transmission Spectroscopy The Michelson i n t e r f e r o m e t e r has two important advantages as a spectrometer i n the f a r i n f r a r e d (FIR). I t has l a r g e - a p e r t u r e , h i g h -throughput o p t i c s , and i t passes many frequencies at the same time. These advantages g i v e a l a r g e output i n t e n s i t y which i s needed f o r a good s i g n a l - t o - n o i s e r a t i o (SNR) i n the f a r i n f r a r e d where sources are weak and d e t e c t o r s are poor. FigureI-2shows a normal i n t e r f e r o m e t e r i n •which the c o l l i m a t e d r a d i a t i o n from a mercury-arc source i s sent by a beam s p l i t t e r to two plane m i r r o r s , recombined at the beam s p l i t t e r , then sent to a d e t e c t o r . I f one m i r r o r i s moved towards or away from the beam s p l i t t e r , the i n t e r f e r e n c e on recombination i s a f f e c t e d by the d i f f e r e n c e i n path lengths of the two arms, and the detected i n t e n s i t y v a r i e s . T h i s v a r y i n g s i g n a l i s c a l l e d an inte r f e r o g r a m (IFG). I f the frequency spectrum of the l i g h t passing through the i n t e r f e r o m e t e r c o n s i s t s of one frequency, the in t e r f e r o g r a m w i l l be a cosine i n i n t e n s i t y w i t h respect to the zero p a t h - d i f f e r e n c e (ZPD) p o s i t i o n ( F i g . I - 3 a ) . T h i s zero p a t h - d i f f e r e n c e p o s i t i o n i s where the path lengths to the f i x e d and moving m i r r o r s are equal (x=0). The 6 Figure 1-2. S i m p l i f i e d o p t i c s of a normal Michelson interferometer. mutt: IF INTENSITY IP INTENSITY w •Wftftft. • • • » » I « J _ A U . I I • • • • • • • ""^ B f t f t . f t . f t . * . * . * * * * • • * * ! » ! » - » » — • » ' ; • * t n t l t t t m M < j i • i • • • • • • sun* I » » « ft ft ft ft , , • W f t f t - k f t . . , , , , , ^ ess*'" <Wk*.*.*.*.* » • • • • • a * A > . * . * . * . * » » • » • » • • r " • » » » » • » » • » # , • » » % s * . • • • v. W W 2 • ftftftft* ,»»»«*> Ik*.*.*.* •• *** t%** •tt**,*,*. ft ft ft ft ft *ft*ftTft*» •s»»»»i t: .•.•.•.•.•.ss* -MMftftJftft"*® ft ft ft ft ft ft ft ft ft ft,, ,»__JV »m»»*»* ffiuV/.* • N t M ftftft'' •» www ^7 Oft. • * . « ft Jj-i-jUjHt #*att 11:: 8 interferogram i s a maximum at zero path-difference because a l l frequencies i n t e r f e r e c o n s t r u c t i v e l y at t h i s p o s i t i o n . If the spectrum consists of two d i s c r e t e frequencies (delta functions), then the r e s u l t i n g interferogram i s the standard two-frequency beat-interference pattern (Fig.I-3b).If the two frequency bands are no longer i n f i n i t e l y narrow, the larger range of frequencies i n t e r f e r e at large path-d i f f e r e n c e with, on the average, random phases so that the i n t e n s i t y smooths out to h a l f that at zero path-difference ( F i g . I - 3 c ) - T h i s i s because at zero path-difference the e l e c t r i c f i e l d s add c o n s t r u c t i v e l y at the beam s p l i t t e r a f t e r being recombined, but at large path di f f e r e n c e s the e l e c t r i c f i e l d s add with random phases which means that the i n t e n s i t i e s add. If we define the e l e c t r i c f i e l d amplitudes i n the two arms to be 2E,then at zero path-difference we get the detected CZPD 2 2 i n t e n s i t y I„ p r. = ( E + E ) = 4E , whereas at large path d i f f e r e n c e s we 2 2 2 get the detected i n t e n s i t y I L p D = E + E = 2E = I z p D / 2 . For a broadband frequency spectrum t y p i c a l of a Fourier spectrometer's throughput, the interferogram becomes almost a sine function ( F i g . I-3d). In the e a r l y I960'?, E.E. B e l l and J.E. Chamberlain pioneered a 9 10 technique i n which one arm of the interferometer contains a sample. ' At each frequency, the sample reduces the amplitude and retards the phase of the electromagnetic wave i n the sample arm of the interferometer. Therefore, each frequency component of the interferogram i s reduced i n amplitude, and the e f f e c t i v e zero path-difference p o s i t i o n for each component i s s h i f t e d . Since there i s no longer a unique zero path-d i f f e r e n c e p o s i t i o n for a l l frequencies, the interferogram i s asymmetric and may look s i m i l a r to that shown i n FigureI-4.Fourier transforming t h i s interferogram gives a magnitude and a non-zero phase. 9 Figure 1-4. The upper f i g u r e i s an asymmetric interferogram. The lower figures are the magnitude and phase of the complex Fourier transform of the above interferogram. 10 This magnitude is proportional to the, amplitude of the electro-magnetic (EM) wave that passes through the sample arm of the interferometer, since i t is the amplitudes which interfere, not the intensities. The phase from the transform is proportional to the delay or extra path length introduced by the sample, i f the zero of the Fourier transform (FT) integral i s taken to be.at the position where zero path-difference would be i f the sample did not delay the electro-magnetic wave. Thus, two quantities are measured simultaneously at each frequency, giving the complete optical properties of the sample. There are two ways of placing the sample in one arm of the interferometer; either as a transmitter, in which case the mirrors remain as they are, or alternatively as a reflector, i n which case the sample replaces one of the mirrors. Dispersive transmission spectroscopy (DTS) was done f i r s t , since i t is easy to introduce and remove a sample in the fixed arm of the interferometer between the beam spl i t t e r and the fixed mirror. The position of the sample in transmission is not c r i t i c a l , whereas in reflection i t must replace the mirror exactly, i f the absolute value of the phase shift i s to be known. Solids were f i r s t to be measured in transmission. Slab samples were used which were optically f l a t for the highest frequency of interest. Comparison was then made between the two interferograms obtained with and without the sample in one beam of the interferometer. A variation on this was done by Chamberlain in 1965, who compared interferograms obtained with different angles between the beam and his sample surface.^ The effective thickness of the sample, and hence i t s effect on the beam, changes with this angle. This method, with monochromatic maser radiation, allowed index of refraction calculations 11 without-the:.necessity of Fourier transforming the interferograms. C r y s t a l quartz has been measured many times as a standard test sample for• new/methods. I", and other common f a r i n f r a r e d o p t i c a l materials (polyethylene, t e f l o n , sapphire, mylar, metal mesh, KBr, KI, S i , C i s -9—18 77 and trans- d e c a l i n , and polypropylene), have been measured. ' 19 Other s o l i d samples measured include carbon disulphide , and ordinary 20 s o d a - l i m e - s i l i c a glass. A l l measurements have been done at room temperature. Gases have also been investigated by the asymmetric transmission 21 method, s t a r t i n g with Chamberlain's measurement of a i r i n 1965. The 22 2^ 23 26 gases HCl, HBr and HI, ammonia gas, ' XCl^ molecules (X = C,Si, 27 28 128 29 Ge,Sn), N 2, 0^ and CO,, gases, '. and water vapour, have a l l been measured since. Windows of mylar or TPX have been used to confine a sample i n front of the f i x e d mirror. The o p t i c a l properties are obtained by comparing the interferograms obtained with the sample and window, and with j u s t the window. A l l these measurements were made at room temperature. Liquids were l a s t to be measured with d i s p e r s i v e transmission spectroscopy, because of the d i f f i c u l t y of making c e l l s s u i t a b l e for containing the l i q u i d s i n a vacuum while passing an undistorted f a r i n f r a r e d wavefront. The f i r s t experiments with l i q u i d s were reported i n 1967. In these, the l i q u i d was i n the form of a gravity-held pool over the f i x e d mirror. Inaccuracies due to vapour d i s p e r s i o n , and the necessity of having the f i x e d mirror arm accurately v e r t i c a l l y aligned, influenced experimenters to use c e l l s with windows of such materials as germanium, s i l i c o n or c r y s t a l quartz. The analysis for l i q u i d s can be quite d i f f i c u l t , since i t i s often necessary to unscramble several 12 overlapping interferograms resulting from different interfaces. Chamberlain's group (from 1969 to the present),~~ and later Honijk's 41-48 group (from 1972 to the present) dealt with the analysis of such interferograms. The optical properties of H^ O and have been 4 9 - 5 5 measured many times, as well as the optical properties of the aliphatic alcohols, halobenzenes, chloroform, bromoform, toluene, benzene, dimethyl- and diethyl- carbonates, aniline, decahydronapthalene, dimethyl acetylene, and XY^ molecules (X = C, S i , Ge, Sn; Y = Cl, B r ) . " ^ ~ ^ Dispersive transmission measurements on liquids have been done at room temperature with one group reporting measurements at 50 to 70 C.^^ Dispersive reflection spectroscopy (DRS) is possible with liquids, but i s primarily used for measuring the optical properties of highly absorbing solids. To measure solids, one face of the sample is polished optically f l a t . Then, one interferogram is measured with the normal fixed mirror in place, and an asymmetric interferogram is measured with the sample exactly replacing the fixed mirror (B in Fig; T _2). The ratio of the two amplitudes of the complex Fourier transforms of the interferograms gives the amplitude of the reflectance, and the difference of the two phases from the transforms gives the phase change upon reflecting from the sample. A major d i f f i c u l t y with this method is the exact replacement of the mirror by the sample. For a phase error of less than one degree at 500 cm ~, the sample surface must replace the mirror surface within half of l/360t*1 of a wavelength. This is approximately 28 nm. Starting with the symposia at Ohio State University from 1962 to 1965, Bell and co-workers described a method for dispersive reflection spectroscopy,^ ^ then used the method to examine the optical properties of KBr at room 10 12 13 temperature. ' They made a custom built interferometer, for use with small samples, in which the mercury arc source i s focussed onto 14 the fixed and moving mirrors. To precisely replace the fixed mirror with a sample, they rested them on three posts made by cutting sections from a glass lens of 20 centimetre radius of curvature. A major d i f f i c u l t y with this method i s that soft samples, such as a l k a l i halides, deform around the posts, preventing proper positioning. Bell could position a sample within 50 nm. In 1969, Johnson and Bell used the instrument to measure KC1 and KBr at room temperature, and obtained reasonable agreement with calculations based on first-order dipole interaction with the transverse optic (TO) phonon, which relaxes through cubic anharmonicity to two final-state phonons. The input lattice-dynamical data, consisting of phonon frequencies and eigenvectors, were obtained from a deformation dipole model due to Karo and H a r d y . I n the same year, Parker, Chamberlain and Burfoot reported measurements of the complex permittivity of ferroelectric KDP and antiferroelectric ADP, at room temperature, in the frequency region of the transverse optic ferroelectric mode.^ In the last reported work on Bell's instrument, Berg and Bell measured KI at room temperature, and compared i t with similar theory to Bell's KC1 paper, except that phonon frequencies and eigenvectors came from a shell model f i t t e d to neutron diffraction data.^ In 1973, Gast and Genzel reported on an instrument based on Bell's techniques, but with both arms of the interferometer focussed into a c r y o s t a t . ^ ' ^ They measured InSb and InAs at room temperature in the region of the lat t i c e vibration frequencies and free-carrier plasma edges. Their instrument allowed replacement of the fixed mirror by a sample within 300 nm. This thesis work was begun in 1974 by trying to find a way of easily modifying a commercial Michelson interferometer for dispersive reflection spectroscopy. A l l previous instruments had been custom b u i l t . The problem of mirror replacement is obviously solved by using an aluminized portion of the sample surface as the reference mirror. The small effects due to the f i n i t e aluminum thickness and less-than-perfect reflectance can easily be calculated. While this method was being explored for this thesis, papers describing the method were 80—82 published by Parker, Chambers and Angress. In their method, background and sample interferograms are taken by masking the appropriate portions of a partially aluminized sample. This positions the sample and reference surfaces at a path difference due only to the aluminum thickness as mentioned. In their configuration, circular masks illuminate either the sample or other metallized portions. Because of asymmetries, an outer ring reference mirror and four separate interferograms are required. Backlash of the moving mirror and thermal expansion or contraction of the instrument can cause large phase errors between interferograms. Also, their mask design requires an increase in path length in both arms of the interferometer in order to accomodate i t s switching. This increases the need for thermal stabilization, makes the alignment more sensitive, requires an extra module, and degrades the optical performance. In 1975, Gauss, Happ and Rother reported measurements on KDP (KH2P04) and DKDP (KfHj D J ^ ; x = 0.64, 0.86) from 120 K to room temperature with an instrument based on that of Gast and Genzel. ' In 1976, Zwick, Urslinger and Genzel also presented low temperature 85 measurements: Mg 2Si at 80 K (as well as at room temperature). Birch, P r i c e and Chamberlain published room temperature measurements on KDP and ADP (NH^H^PO^) i n 1976 with a . three post replacement method i n which most of the sample weight i s counterbalanced, l a r g e l y reducing 86 the sinkage of soft specimens onto the supports. Their replacement error was within 100 nm. Bi r c h and Murray have j u s t published (July 1978) papers describing a mode of operation of t h e i r instrument which eliminates errors due to s h i f t s i n the zero path-difference p o s i t i o n during the time between the reference interferogram and the sample i n t e r f e r o g r a m . ' ~ ~ ~ S t a r t i n g i n 1976, Parker and co-workers published a s e r i e s of measurements from 100 K to 300 K on NaCl, C s l , KBr, KDP, ADP, TIBr, KTaO^, and S r T i 0 3 ; a l l with t h e i r p a r t i a l 87—93 aluminization technique. Happ and Rother examined DKDP i n 1977 over a wider frequency range than reference 83, and from 200 K to 94 300 K. A r e f i n e d configuration of the sample aluminizing technique 95 was completed f o r t h i s t hesis and published i n 1977. In t h i s c onfiguration, simple modifications of an RIIC FS-720 Fourier spectrometer allow i t to be e a s i l y converted f o r room-temperature d i s p e r s i v e r e f l e c t i o n spectroscopy on a p a r t i a l l y aluminized sample. In a d d i t i o n , no increase i n path length i s required, and many asymmetries are s u b s t a n t i a l l y reduced, r e s u l t i n g i n greater accuracy, even with imperfect samples. Backlash has been completely eliminated by a l t e r n a t e l y recording the background and sample interferograms at each path-difference p o s i t i o n , and only these two interferograms need be recorded. These d e t a i l s w i l l be given i n Chapter I I I . This mo d i f i c a t i o n was used to obtain the o p t i c a l properties of NaCl at room temperature with high accuracy. These r e s u l t s were then compared with the r e s u l t s of the most d e t a i l e d and precise t h e o r e t i c a l c a l c u l a t i o n s yet done for the a l k a l i h a l i d e s . These included r e l a x a t i o n of the transverse optic mode r e s u l t i n g from both cubic and quartic ariharmonicity ( i n the l a t t e r case, three f i n a l - s t a t e phonons are involved), as w e l l as the l i f e t i m e s or r e l a x a t i o n of the three f i n a l -state phonons themselves. The input data, co n s i s t i n g of eigenfrequencies and eigenvectors, was generated by a shell-model program which used the parameters found by a least-squares f i t to the d i s p e r s i o n curves measured by i n e l a s t i c neutron s c a t t e r i n g . The g r i d density of points i n the B r i l l o u i n zone was v a r i a b l e . The o v e r a l l agreement was e x c e l l e n t , and represents the most accurate measurements and c a l c u l a t i o n s performed f o r t h i s m a t e r i a l . D e t a i l s of t h i s work have been r e p o r t e d ^ and 9 7 published. 1.3 Low Temperature Dispersive R e f l e c t i o n Spectroscopy Since many i n t e r e s t i n g changes i n the o p t i c a l properties of r e f l e c t i n g s o l i d s occur when the temperature i s lowered, a means of cooling the sample to liquid-helium temperatures constituted the major portion of t h i s thesis project. At the same time, Gast's, Gauss' and Parker's groups began to publish d e t a i l s of d i s p e r s i v e r e f l e c t i o n instruments capable of operating at l i q u i d - n i t r o g e n temperatures (References 80, 84, 85, 88, 91-94). Among the several considerable d i f f i c u l t i e s involved i n cooling the sample for these measurements are the following: cooling a d e l i c a t e sample which may be up to eight centimetres i n diameter and one centimetre thick to near l i q u i d helium temperature without cracking i t or warping i t by more than approximately one micron over the sample surface; maintaining the sample surface 17 within approximately one micron of the plane required for interference; aligning the sample to be within that plane while at near helium temperature; preventing contaminants from freezing onto the sample; and preventing room temperature radiation incident on the sample from warming the sample without allowing windows to disturb the interference wavefront. Any windows used have to pass the vi s i b l e and far infrared, as well as be uniform in thickness within approximately 0.1 um. An instrument which succeeds in overcoming these d i f f i c u l t i e s and allows dispersive reflection spectroscopy down to below 25 K has been bu i l t and tested for this thesis project, and described in a paper. While this thesis project was being completed, Parker's group published results of dispersive reflection measurements on KDP, KC1 and 98-101 NaCl from 105 K to 300 K, as well as a summary of refractive indices obtained with this method for NaF, NaCl,. KC1, KBr, KI, RbCl, 1 go ] 0 8 RbBr, Rbl, KTa03, S.rTi03 and TlBr at 100 K and 300 K. ' Just recently, they have modified their instrument to permit liquid-helium temperature measurements, and to incorporate the switching arrangement 147 developed for this thesis. They require long pathlengths, and require the switching mask to be at liquid-helium temperatures. They 148 also have produced a single-pass dispersive transmission set-up. Mead and Genzel have recently described modifications to their custom-103 127 built interferometer allowing measurements down to about 6 K. ' ' 146,150,151 . . i This spectrometer is vibration sensitive, has long pathlengths, radiation farther from normal incidence, and w i l l not allow the use of the partially-aluminized sample method. 18 1.4 Thesis Outline, In Chapter I I , a review of the theory of dispersive reflection spectroscopy i s presented along with the effect of a window inserted in front of the. sample. The room-temperature apparatus which was developed i s then described in Chapter I I I , followed by a description of the dewar and other modifications used.to cool the sample down to liquid-helium temperatures,' :in Chapter IV. The theory for the far infrared optical properties of the alkali-halides i s given i n Chapter V. Results of room-temperature and low-temperature dispersive reflection measurements, power, transmission measurements, and their comparison with theoretical calculations of NaCl are then given in Chapters VI and VII. These accurate theoretical calculations and their comparison with the experimental results are given in order to show how well the technique works, and to improve our understanding of anharmonic interactions which lead to the broad-band l a t t i c e absorption i n the alkali-halides. This absorption extends, in NaCl for instance, from a few wavenumbers up to the near infrared where this small residual absorption due to weak multiphonon processes limits the use of these materials as windows for high-power laser systems. 19 CHAPTER II THEORY OF DISPERSIVE REFLECTION SPECTROSCOPY II.1 The General Dispersive Interferometer This chapter uses thenotation of E.E. Bell and follows, in 12 places, his formulation of dispersive spectroscopy. The general form of an interferogram due to a sample in one arm of an interferometer is developed, followed by the specific case of a reflecting sample, and then including the effect of a window in the sample arm. The effects of apodization, sampling, and a partly aluminized sample are included. The following assumptions are made: (1) SUPERPOSITION: The output of the detector is a linear function of the light input to the interferometer, e.g. Different source intensities should not change the optical properties of the sample. ( i i ) TIME INDEPENDENCE: A displacement in time of the entire input . signal should shift the output signal the same amount, e.g. The optical properties should not change i f measured at different times. ( i i i ) POLARIZATION: Only one polarization of the incoming light i s considered, since the other polarization i s treated exactly the same way. (iv) CAUSALITY: No output occurs before the input excitation. We w i l l also assume that the original interferometer is totally symmetric so that the two arms have identical throughput. The superposition p r i n c i p l e then y i e l d s : v < T ) ( t ) - G ( t ' ) V < T ) ( t - t ' ) dt' (II.1) - G(t) * V ^ ( t ) ' where * denotes convolution, and: (T) Vj (t) = throughput e l e c t r i c f i e l d i f only the f i x e d mirror i s i n the interferometer. " T " indicates that the function i s zero outside t = -T to +T which i s large compared to any delays of the system. (T) (t) = throughput e l e c t r i c f i e l d of the interferometer with one arm blocked and a sample i n the other arm. G(t) = impulse response function (response to an instantaneous u n i t pulse input at time t = 0. i . e . Dlrac d e l t a function input) f o r the sample: (TV r • V j A ' ( t ) = G(t') 6(t-t') dt' = G(t) (II.2) The throughput e l e c t r i c f i e l d due to only the moving mirror i s (T) (t-x) where the s i g n a l i s delayed a time T = x / c , where c i s the speed of l i g h t i n the interferometer and x i s the extra o p t i c a l path due to the displacement of the moving mirror a distance x / 2 from the zero path-difference p o s i t i o n away from the beam s p l i t t e r . The detected e l e c t r i c f i e l d w i l l be the sum of the f i e l d (T) r e f l e c t e d through the moving mirror arm ( t - t ) and the sample arm (T) con t r i b u t i o n (t)... 21 The detected power i s therefore: • 2 2 { v | T ) ( t - T ) + V ^ T ) ( t ) } 2 = v J T ) ( t - x ) + V ^ T ) ( t ) + 2 V ^ T ) ( t - x ) V < T ) ( t ) (II.3) The second term on the r i g h t hand side i s obviously a constant, independent of x and therefore x, and the f i r s t term i s also a constant since the power detector has a response time much longer than x. This assumes, of course, that the source i n t e n s i t y does not f l u c t u a t e . The t h i r d term, however, represents the mixing of the two f i e l d s which w i l l be a function of the delay between them. The.power detected due only to the t h i r d term i s : P 1 2 ( x ) = Lim (T) 1 —CO V< T )(t-x) V < T ) ( t ) dt (II.4) i n the l i m i t of an i n f i n i t e measuring time. This i s the interferogram obtained with the sample i n one arm and i s a r e a l function of the time delay x (or o p t i c a l path d i f f e r e n c e x ) . In p r a c t i c e , the f i r s t two terms of equation (II.3) are also measured with the detector, but are l a t e r removed:computationally. Because the e l e c t r i c f i e l d s are truncated outside +T to -T, they " have f u l l range Fourier transforms: .00 v 1 T ( f ) - V ^ T ) ( t ) exp(i2Trft) dt (II.5) and inverse transforms: v<T)(t) - v l T ( f ) exp(-i27rft) df (II.6) 22 which are both symbolized by: * 1 T ( f ) * V < T )(t) ( i i . 7 ) where the subscript T indicates that v i s the Fourier transform of a truncated function and i s not a truncated function i t s e l f , and the circumflex i n d i c a t e s that v-may be complex.'Note that the values of• v f o r negative frequencies are the complex conjugates of those f o r p o s i t i v e frequencies: v,j,(-f) = v^(+f) because they are transforms of r e a l functions. Fourier transforming (II.1) gives: v 2 T ( f ) = g(f) v 1 T ( f ) (II.8) where v 2 T ( f ) # V ^ T ) ( t ) (II.9) and g(f) s> G(t) (11.10) also g(-f) = g" (f) since G(t) i s r e a l . Because the response G(t) cannot occur before the input pulse, G(t<0) = 0. As shown by T o l l , t h i s c a u s a l i t y i s equivalent to the v a l i d i t y of the Kramers-Kronig disp e r s i o n r e l a t i o n s which r e l a t e the r e a l and imaginary parts of g ( f ) ^ ^ S t e r n describes the a p p l i c a t i o n of these to o p t i c a l property measurements.^ The interferogram can be expressed i n terms of the input e l e c t r i c f i e l d and the impulse response function by s u b s t i t u t i n g (II.1) 23 into ( I I .4 ) : P (T)=Lim(T) T-x» V ^ T ) ( t - T ) { [ G ( t ' ) V < T ) ( t - t ' ) dt'} dt (11.11) J —oo —oo which may be arranged by reversing the order of i n t e g r a t i o n to: * f CO #co • P 1 9 ( x ) = 'G(t'){Lim(T) _ 1 V p ^ t + T - t ^ V p ^ t ) dt} dt' (11.12) I f the interferometer i s operated i n the symmetric fashion, with no sample i n the f i x e d arm, the | ( f ) i n (II.8) must obviously equal one. The function G(t') must therefore be the Fourier transform of one, which i s the d e l t a function 6 ( t ' ) . The r e s u l t i n g interferogram P ^ ^ ( T ) > obtained by s u b s t i t u t i n g 6(t') i n (11.12), i s : P n ( x ) = Lim ( T ) " 1 V ( T ) ( t + x ) V ( T ) ( t ) dt (11.13) T-K» J-00 This may be substituted back into (11.12): P 1 2 ( x ) = f G(t') P j ^ x - t ' ) dt' = G(x) * P ( T ) (11.14) J —00 which shows that the sample interferogram i s the convolution of the  background interferogram with the sample impulse response function. Fourier transforming (11.14) gives: £ 1 2 ( f ) = g ( f ) p n ( f ) (11.15) 24 where $ 1 2<f) * P 1 2< T> (11.16) and (11.17) From the above three equations, i t i s easy to see that by measuring interferograms with and without the sample (P^ 2(x) and (x) r e s p e c t i v e l y ) , Fourier transforming themand then c a l c u l a t i n g t h e i r complex r a t i o £ ^ 2 ( f ) ^ ( f ) , gives the frequency response function f o r the sample. This complex response function contains phase and amplitude information at each frequency, which are both required to c a l c u l a t e the complete o p t i c a l properties at that frequency. In the f a r i n f r a r e d , i t i s simpler to use the o p t i c a l path d i f f e r e n c e x instead of x, and to use the wavenumber v ( i n u n i t s r e c i p r o c a l to x) instead of f. In t h i s case, x i s measured i n cm and v i s i n cm The important r e s u l t s of t h i s s ection i n these u n i t s are therefore: P n ( x ) # 0 n ( v ) (11.18) (11.19) i ( v ) e n ( v ) (11.20) g(v) v 1 r r ( v ) (11.21) 25 II.2" Dispersive - R e f l e c t i o n Measurements, As mentioned i n Chapter I* using a polished face of a sample to replace the f i x e d mirror of the Michelson interferometer has been s u c c e s s f u l l y used to obtain both the r e a l and imaginary o p t i c a l properties by r e f l e c t i o n at normal incidence. I f the sample absorbs s u f f i c i e n t l y , or i f the sample i s wedged so that any l i g h t reaching the back face i s r e f l e c t e d away from the detector, then s i n g l e surface r e f l e c t i o n can be measured. F i r s t , a background interferogram i s measured with the interferometer i n the normal symmetric mode ( P ^ ( x ) ) . Then, a sample i n t e r f erogram i s measured .(P^OO). with the " f i x e d " mirror replaced by the sample, and with the r e f l e c t i n g surface of the sample l y i n g i n the exact plane previously occupied by the r e f l e c t i n g mirror surface. From (11.21), we know that for the mirror: v (v) = g (v) v (v) (11.22) Mirror Mirror Incident where for an i d e a l mirror: g (v) = 1 exp(iTT) (11.23) Mirror and we also know that for the sample: v (v) = g (v) v (v) (11,24) Sample Sample Incident where: g (v) = f ( v ) (11.25) Sample 26 where the complex sample reflectance: f(v) = r(v) exp(i<j>r(v)) (11.26) = (l-n)/(l+fl) (11.27) = (l-{n+ik})/(l+{n+ik}) (11.28) r = (({1-n}2 + k 2)/({l+n} 2 + k 2 ) ) H (11.29) <(>r = arctan(2k/(n +k -1)) . (11.30) The above two equations can be inverted to: n = (l-r 2)/(l+r 2+2r cos $ ) k = (-2r sin $ ) I (l+r 2+2r cos <|> ) After the background interferogram P (x) and the sample interferogram P^ 2(x) are measured, they are transformed with complex Fourier transforms, where the zero value of x for the transforms i s taken to be the position of the moveable mirror for which the maximum signal (i.e. at zero path-difference) i s obtained in the symmetric background interferogram. The resulting spectra are: P 1 2 ( v ) = P 1 2(v) exp(i<t>12(v)) (11.33) (11.31) (11.32) 27 .Pn(y) = P n ( v ) e x p d ^ v ) ) (11.34) From (11.20): ;(v) = P 1 2 ( v ) / P u ( v ) (11.35) = . P 1 2 ( v ) / p n ( v ) exp(i{<)) 1 2(v)-((, 1 1(v).}) (11.36) = g(v) exp(i<j> (v)) (11.37) The amplitude f r a c t i o n of the incident l i g h t i n the sample arm returned to the beamsplitter to i n t e r f e r e , r e l a t i v e to the background, i s g(v), and the phase of the returned l i g h t r e l a t i v e to the background i s (J> (v) . Therefore: g(v) = g (v) / g (v) (11.38) Sample Mirror Substituting (11.23) and (11.25) into the above gives: g(v) = f(v) / exp(iTr)" (11.39) so that the required sample reflectance i s : r(v) = g(v) = P 1 2 ( v ) / P u ( v ) (11.40) 4> r(v) = <j)g(v) + TT = <J>12(v) - + 1 1 ( v ) + TT ( 1 1 . 4 1 ) 28 Normally, 4>r(v)-ir i s presented, because t h i s i s zero i f the sample has no absorption, and also because t h i s i s what i s more d i r e c t l y measured. II.3 E f f e c t s Due To The Reference Mirror There are two main e f f e c t s which the reference mirror has on the measured interferograms. The - f i r s t occurs : i f the.reference mirror i s not positioned i n the exact plane previously occupied by the r e f l e c t i n g sample surface. With a p a r t i a l l y metallized sample, the r e f l e c t i n g surface i s a distance x'/2 c l o s e r to the beam s p l i t t e r than the r e f l e c t i n g sample surface, where x'/2 i s the thickness of the deposited metal f i l m . This reduces the o p t i c a l path distance, i n r e f l e c t i n g from the metal, by a distance x', which w i l l advance l i g h t of frequency v by a phase: Unless accounted f o r , t h i s phase w i l l appear to be added to the calculated phase refl e c t a n c e of the sample. By using a standard c r y s t a l -thickness monitor while evaporating the r e f l e c t i n g metal onto portions of the sample, x'/2 can be measured to s u f f i c i e n t accuracy to take complete account of <f»(v) i n the calculated phase r e f l e c t a n c e . The reference mirror has a second e f f e c t , i n as much as i t does not r e a l l y have the i d e a l r e f l e c t i o n of (11.23). I t i s a c t u a l l y of the form: <Kv) = 2 TT x' v E r r (11.42) Err g (v) = r (v) e x p f K * (v) + TT)) (11.43) Mirror Mir Mir 29 This w i l l make the calculated sample amplitude r e f l e c t a n c e appear high and the calculated sample phase appear low,, unless accounted f o r . Since f a r - i n f r a r e d frequencies are much lower than the plasma frequencies for metals* t h i s c o r r e c t i o n amounts to only a few percent. I t can be and has been ca l c u l a t e d , including multiple r e f l e c t i o n s i n the t h i n metallized f i l m , f or t h i s t h e s i s . These r e s u l t s are .presented i n Chapter VI. If we now include these two reference-mirror e f f e c t s , (11.39) becomes: g(v) = f ( v ) / (r(v) exp(i{<Kv) + ir» exp(-i<Kv))) (11.44) Mir Mir . Err so that the required sample r e f l e c t a n c e i s : r(v) = g(v) r(v) = P 1 2 ( v ) r(v) / P n ( v ) (11.45) Mir Mir •_(v) = <f>p(v) + <})(v) + TT - <J>(v) S Mir Err = ^i2^ ~ * n ( ^ + * ( ^ ) + ~ - 2TTX'V (11.46) Mir II.4 Including A Sample-Arm Window If a window i s inserted into the sample arm of the interferometer with i t s surfaces canted so that l i g h t r e f l e c t e d from i t misses the detector (>^5°), then the absorption, surface r e f l e c t i o n and r e f r a c t i v e index w i l l reduce the i n t e n s i t y and retard the phase of the sample arm beam (remember that the beam passes through the window twice). 30 If the window is of non-uniform shape or thickness, i t w i l l retard various portions of the beam by different amounts. This i s equivalent to averaging over a range of path differences, since the detector averages the power over the whole beam. Thus, the resulting interferogram w i l l be smoothed, high frequencies w i l l be lost, and i f the non-uniformity is too large, there w i l l be no measured interferogram at a l l . In order to have less than a ten percent- loss in spectral signal at the highest frequency used, the window's effect should be uniform to within approximately one tenth of the shortest wavelength used. For a maximum frequency of 500 cm \ this means a uniformity of better than y^m since the radiation passes the window twice, and since the window w i l l l i k e l y have a refractive index of approximately 1.5 to 2.5. It may be shown however, that i f the window i s of uniform thickness, then i t s effects on the sample and background interferograms completely cancel when their ratio i s taken. Using (11.20), the resultant spectrum due to a background interferogram taken with a window i n the sample arm i s : where two passes through the window w i l l have some frequency response function l w ( v ) . The resulting spectrum due to a sample interferogram taken with a window and sample in the fixed arm Is: P 11W (v) = g w(v) p n ( v ) (11.47) (11.48) Taking the ratio of these two spectra in the same fashion as for 31 spectra without a window gives: *12W^? 1 ^11W(U) = h l ^ 1 .(«w<^> * n = Pl2G) / Pn(~) = g(v) (11.49) by equation (II.38) . Thus, the >same {sample frequency-response 'function is measured, since the window's effect cancels out when ratioing the background and sample spectra. II.5 The Apodizing Window Function The typical experimental system i s limited to measureable optical path differences of up to a few centimeters. This limits the spectral resolution to values of Av of the order of X ~ or larger,- where the optical path difference i s limited to -X $ x < X. This i s equivalent to the multiplication of the i n f i n i t e l y extended interferogram by a window function which i s zero outside the range of x and one inside the range. For a window function W(x) in optical path space, there is a spectral width function w(v) in frequency space given by: w(v) £ W(x) (11.50) which, when multiplied by the interferogram gives: P(x) W(x) p(v) * w(v) (11.51) 32 Thus the spectrum i s that of i n f i n i t e l y high r e s o l u t i o n convoluted with the s p e c t r a l width function. I f the true spectrum were to consist of one s i n g l e frequency v', the resultant measured spectrum would be w(v'). For the window function which i s one from -X to +X, the s p e c t r a l width function, which corresponds to the s p e c t r a l s l i t width i n normal spectroscopy, i s the f a m i l i a r sine d i f f r a c t i o n function: w(v) = 2X (2TTVX) - 1 sin(2irvX) (11.52) which has large " f e e t " , or side lobes, as shown i n Figure .11-1. Since these feet a f f e c t the accuracy of measurements away from the c e n t r a l peak, i t i s desireable to remove these " f e e t " or apodize (from the Greek apodoi, from a - without and podoi - f e e t ) . This can be done by d e f i n i n g a s u i t a b l e W(x). In doing so however, a compromise i s made, since the width of the s p e c t r a l width function increases when the side lobes are reduced. It i s convenient to have the s p e c t r a l width function r e a l and symmetrical, which i s ensured by making W(x) r e a l and symmetrical. Another consideration i s the complexity of c a l c u l a t i o n of W(x). There are near-optimum functions for W(x) for any desired compromise between the s p e c t r a l width function's width and i t s f e e t , but they may take too 109 long f or routine computation. They may of course be precomputed and stored i n a look-up table, but t h i s often takes too much space. A good general apodizing window function, due to Happ and G e n z e l , ^ ^ which i s used f o r a l l the computations i n t h i s project, i s : W(x) = 0.54 + 0.46 c o s ( W X ) (11.53) 2X-w(v'—v) X 2X* sine function: w(v'-v) = sinf2TrX(v'-v)) TT(V'-V) 0-6034 X 1-08-1 •X Ik 33 w(v '-v) 0-54-^  w(v'-v) = sin{2TTX(v'-v)} 2TT 1.08 r— » —\"t 2X " ( V " V ) ~ 0-9076 W(x) = 0.54 A +0.46cos(irx/X) •h Figure I I - l . Spectral width functions. 34 whose s p e c t r a l width function i s also shown i n Figure IT-1./This window function does, not change much near x=0, which i s good for asymmetric interferograms where the e f f e c t i v e zero path-difference i s s h i f t e d s l i g h t l y . As mentioned i n section II.1, the measured i n t e n s i t y c o n s i s t s of the interferogram plus two added "constant" terms. The magnitude of the interferogram and these terms may d r i f t , due to f a c t o r s such as a gradual change of the source-lamp i n t e n s i t y . Such a t o t a l interferogram i n t e n s i t y i s shown i n Figure I I - 2 a , T n which the magnitude of the d r i f t i s exaggerated for i l l u s t r a t i o n . I f the two "constant" terms are treated by subtracting a constant equal to the average of the t o t a l interferogram i n t e n s i t y , then the resultant interferogram resembles that of Figure II-2b. I f the common tr i a n g u l a r apodization i s then used ( F i g . I I - 2 c ) , the r e s u l t i n g spectrum w i l l have some low frequency noise due to the slowly varying background l e v e l of the interferogram. This i s u s u a l l y unimportant, since there i s usually l i t t l e information at such low s p e c t r a l frequencies. There w i l l be a l i t t l e high-frequency noise, however, from the d i s c o n t i n u i t y i n slope at ±X. I f an apodization function, such as that of equation (11.53), i s used ( F i g . II-2d), which does not go smoothly to zero at ±X, then much larger spurious high frequencies can be created. To avoid t h i s , the f i r s t step of removing the two "constant" terms i s done d i f f e r e n t l y . The t o t a l interferogram i n t e n s i t y has subtracted from i t a l i n e a r least-squares f i t to i t s f i r s t and l a s t sections. If spectra of d i f f e r e n t r esolutions are to be compared, i t i s sCO — desireable to keep /' w(v) dv = constant. This can be enforced by having W(0) = 1 for a l l window functions. 35 A W(x) = 1-x/X Figure II-2 a. The t o t a l interferogram i n t e n s i t y with slowly changing source i n t e n s i t y , b. The r e s u l t of subtracting the average i n t e n s i t y from a. c. The r e s u l t of t r i a n g u l a r apodization of b. d. The r e s u l t of "Happ and Genzel" apodization of b. 36 The r e s o l u t i o n i n Fourier transform spectroscopy i s a rather d i f f i c u l t quantity to define. The r e s o l u t i o n from the Rayleigh c r i t e r i o n applied, to the Happ and.Genzel apodizing window function gives a r e s o l u t i o n equal to l'/X, which i s used i n t h i s t h e s i s . For more discussion see Chantry and Fleming. II.6 Frequency R e s t r i c t i o n s Due To Sampling The analysis of interferograms i s best done with a d i g i t a l computer. For most e f f i c i e n t operation, the interferogram should be sampled at equal i n t e r v a l s Ax of o p t i c a l path d i f f e r e n c e . The sampled interferogram w i l l then be the product of the continuous interferogram function and an i n f i n i t e set of Dirac d e l t a functions spaced at u n i t i n t e r v a l s along the x a x i s . This set of d e l t a functions i s represented by III(x) and c a l l e d the Shah function. The Fourier transform of t h i s function^ i s : III(x / Ax) = III(vAx) (11.54) The sampled interferogram P(x) III(x/Ax) then has the Fourier transform £(v) * III(vAx). This Fourier transform has the p e r i o d i c i t y introduced by the Shah function for unit i n t e r v a l s of vAx. Thus the p e r i o d i c i t y i s 1/Ax cm The true spectrum, however, consists of p o s i t i v e frequencies and t h e i r corresponding complex conjugate negative frequencies, so that only h a l f of the repeated frequency spectrum i s of use. Therefore, the useable frequency bandwidth i s l/(2Ax). Any s p e c t r a l structure outside t h i s bandwidth w i l l be a l i a s e d into i t , 37 possibly d i s t o r t i n g the desired spectrum. Of course, i t i s possible to a l i a s l i m i t e d bandwidths of higher frequencies into the f i r s t repeat zone while blocking out the f i r s t zone frequencies, i n order to measure high frequency spectra with fewer data points. I I . 7 .Finding Zero ,Path-Dif,f er,ence From .The In t e r f erogram Because of component p o s i t i o n - d r i f t and other t e c h n i c a l problems, the sampled interferogram w i l l u s u a l l y not be sampled r i g h t at the zero path-difference p o s i t i o n , and following interferograms may be sampled at d i f f e r e n t places. For the purposes of interferogram averaging, and phase determination i n d i s p e r s i v e Fourier transform spectroscopy, t h i s zero path-difference p o s i t i o n must be found i n r e l a t i o n to the sampled points. In the f a r i n f r a r e d , a "symmetric" background interferogram i s indeed very close to symmetric, and therefore has a pha^e spectrum of very c l o s e to zero when the Fourier transform i s computed with respect to the zero path-difference p o s i t i o n . I f the Fourier transform i s computed with respect to a point x' away from zero path-difference (such as one of the sampled interferogram data), then there i s a l i n e a r phase as a r e s u l t (see equation (11.42)). This phase i s unimportant fo r power spectrum measurements, made by using a complete interferogram, since they are analysed by taking the absolute value of t h e i r complex Fourier transforms. For s i n g l e sided interferogram a n a l y s i s , averaging of interferograms, and phase determination i n d i s p e r s i v e Fourier transform spectroscopy, i t i s necessary to include t h i s phase. For machine c a l c u l a t i o n of zero path-difference from the phase, the Fourier 38 transform should f i r s t be c a l c u l a t e d by d e f i n i n g x=0 s u f f i c i e n t l y c l o s e to zero.Lpath-difference, that t h e - r e s u l t i n g a p p r o x i m a t e l y - l i n e a r phase spectrum does not vary much more than 2TT r a d i a n s . I f the maximum i n t e n s i t y p o i n t of a sampled background spectrum i s defined to be the . x=0 p o i n t f o r the F o u r i e r transform, t h i s i s u s u a l l y s u f f i c i e n t . Then, by doing a l i n e a r l e ast-squares f i t to the phase, x' can be determined. Thus, the centre can be determined, and the F o u r i e r transform can be computed w i t h respect to t h i s c e n t r e . To determine the centre f o r an asymmetric in t e r f e r o g r a m , the assumption i s made that the sampling p o s i t i o n s w i t h respect to zero p a t h - d i f f e r e n c e do not change, which can be assured by using a p a r t i a l l y aluminized sample, and a l t e r n a t e l y sampling the background and sample interferograms at each p o s i t i o n of the m i r r o r step d r i v e . 39 ..' CHAPTER I I I EXPERIMENTAL SETUP - ROOM TEMPERATURE I I I . l O ptics A.commercially a v a i l a b l e C o u r i e r - transform;spectrometer (Beckman-Research and I n d u s t r i a l Instruments Company FS-720, from Glenrothes, S c o t l a n d ) , w i t h a step d r i v e , was modified f o r t h i s p r o j e c t of measuring d i s p e r s i v e r e f l e c t i o n . This was accomplished w i t h only one major a d d i t i o n a l o p t i c a l element i n v o l v i n g no disturbance to any other p a r t of the o p t i c s . F i g ure I I I - l shows the FS-720 o p t i c s w i t h the r e l e v a n t m o d i f i c a t i o n s that have been made. F i r s t , the sample i s clamped i n the same place and manner as was the f i x e d m i r r o r , p r e v i o u s l y . Secondly, s i n c e the sample i s u s u a l l y smaller than the 8 cm diameter c o l l i m a t e d source beam, an a d j u s t a b l e - i r i s aperture stop i s placed before the beam s p l i t t e r . I t i s d e s i r e a b l e to l i m i t the beam to the same s i z e f o r both the sample arm and the moving-mirror arm, s i n c e any e x t r a non-i n t e r f e r i n g r a d i a t i o n from e i t h e r arm incr e a s e s the e f f e c t of source n o i s e . F i n a l l y , a s w i t c h i n g mask i s placed i n f r o n t of the sample i n order to a l t e r n a t e l y measure the sample and the background i n t e r f e r o g r a m s . The s w i t c h i n g mask designed by the author, shown i n Fi g u r e I I I - 2 , a l t e r n a t e l y exposes the aluminized sample surface and the exposed sample surface at each step of the moving m i r r o r . The mask i s made of aluminum and coated w i t h f l a t b l a c k p a i n t to e l i m i n a t e r e f l e c t i o n from i t s s u r f a ce. The apertures i n the s w i t c h i n g mask are made approximately Q LU GO £ o in O X ° & < p Figure I I I -2. A diagram of the p a r t l y aluminized sample and the e i g h t -section switching mask. 42 \ mm undersize to allow for s l i g h t misalignment of the sample and mask. There are. four, main reasons f o r the eight s e c t i o n mask design: ( i ) I t can be implemented with no increase i n interference-path length, since the mask rotates about an axis normal to the sample surface. This reduces the need f o r thermal s t a b i l i z a t i o n , s e n s i t i v e alignment and extra modules which are needed i n Parker's arrangement, i n which the mask rotates' about an axis -in the plane of "the-sample (see Figure TIT-3) Also, the o p t i c a l performance i s not degraded by the extra path-length which i s obviously required i n Parker's arrangement, ( i i ) The mask needs l i t t l e movement to switch, enabling f a s t switching by a small solenoid with l i t t l e v i b r a t i o n and no heating problem. This i s e s s e n t i a l i f one i s switching at each step of the moving mirror, as we are, i n order to eliminate errors i n determining zero-path^-difference and to measure both interferograms under as close to i d e n t i c a l conditions as pos s i b l e (see sections II.7 and I I I . 3 ) . The author i s the f i r s t to have recorded dual interferograms i n t h i s manner, ( i i i ) The e f f e c t s of beam non-uniformity, sample non-uniformity and other o p t i c a l asymmetries are reduced by the eight section average over the beam. Thus, four-80 interferogram r a t i o i n g i s not necessary, (iv) An outer alignment 80 r i n g i s not necessary since the aluminized segments sample the beam uniformly and reach a s u f f i c i e n t radius to be used f o r alignment. There i s , however, a disadvantage to the p a r t l y - m e t a l l i z e d sample technique, i n that the mask ne c e s s a r i l y blocks part of the l i g h t . I f the mask blocks h a l f of the l i g h t as i n Figure III-4, then B=0. The 2 i n t e n s i t y 1^ = A . For zero path-difference, the amplitudes add and 2 2 I 2 = (A + A) = 4A . For large path-differences, the i n t e n s i t i e s add, 2 2 2 and I 2 = A + A = 2A . Thus, the t o t a l detected i n t e n s i t y at zero PARKER AND CHAMBERS: DISPKHSJV E-KEFJ.KCTION SPKCTROSCOl'Y ; LAUP MOOUUkTIOM. Fig. 1. Schematic diagram of the Michelson interferometer showing the two movable screens Si and 52 in front cf the fixed reflector" and the pallia of the coherent beams of radiation using part A. Fig. 2. The geometry of the two screens and the division of the field of view at the fixed reflector. Figure I I I - 3 . The configuration of Parker's switching mask (Ref. 44 s o u r c e D e t e c t o r S w i t c h i n g M a s k A 2B A 2A M o v e a b l e M i r r o r • < J = Z A / / / F i xed M i r r o r WITHOUT SWITCHING MASK WITH SWITCHING MASK Path Difference Path Difference Figure III-4. The upper diagram shows schematically the blocking of l i g h t by the switching mask. The lower example interferograms show the e f f e c t of a 50% l i g h t blockage which reduces the r e l a t i v e modulation to 2/3 of i t s unblocked modulation. path-difference i s 1^ + 1^ - and at large path-differences 2 1^ + = 3A . I f the mask i s not used, B = A and the t o t a l i n t e n s i t y 2 2 2 at zero path-difference i s 1^ + 1^ = (A+A) + (A+A) = 8A and at large path-differences i s ^  + * 2 = ( a 2 + a 2 ) + X a 2 + a 2 ) = ^A 2. Thus, although the i n t e n s i t y d i f f e r e n c e between zero path-difference and large .path-differences i s halved due .to the mask, the modulation of the t o t a l i n t e n s i t y i s only reduced to two t h i r d s of that with no mask. This i s important when considering the e f f e c t of source noise. III.2 Far Infrared Source Since samples r e f l e c t l e s s l i g h t than the reference mirror, and since the switching mask reduces the i n t e n s i t y modulation as shown i n III.1, noise due to source i n t e n s i t y f l u c t u a t i o n s becomes more important. Even with a symmetric interferogram, the signal-to-noise r a t i o would t y p i c a l l y be only 200:1 with the o r i g i n a l P h i l i p s HPK 125W mercury vapour lamp provided with the spectrometer. This t r a n s l a t e s to a signal-to-noise r a t i o of approximately 20:1 i n a t y p i c a l spectrum. Frequently, these lamps would change t h e i r i n t e n s i t i e s i n a step-wise manner, causing much worse noise. By t r y i n g many.lamps, a reasonably quiet one could be found, and the e f f e c t of i n t e n s i t y steps could be reduced computationally. I t was c l e a r , however, that a large improvement was necessary f o r disp e r s i v e r e f l e c t i o n spectroscopy. D i f f e r e n t types of lamps were, t r i e d , without much success, and then an a c t i v e source i n t e n s i t y c o n t r o l l e r was designed and b u i l t (see Appendix A) which supplies the lamp with a DC voltage, i n part to remove 120 Hz i n t e n s i t y f l u c t u a t i o n s . The c o n t r o l l e r regulates the detected i n t e n s i t y to approximately one part i n ten thousand, eliminating the e f f e c t of l i n e voltage and power supply v a r i a t i o n s . Unfortunately, i t i s d i f f i c u l t to put a sensor into the instrument which has the same f i e l d of view and frequency response as the normal detector. This i s necessary for optimum performance, since i t i s the wandering of the arc i n the lamp, not a change i n the arc's i n t e n s i t y , which causes, most of the perceived i n t e n s i t y change. Beckman-RIIC recently replaced the HPK 125W A.C. mercury lamp with a D.C. one, which they c a l l the IR-7L. I t i s a ninety watt lamp with a short 2 cm electrode gap instead of the longer 3 cm HPK125W gap. The envelope i s a smooth tube, drawn to give a wedged cros s - s e c t i o n to reduce channel spectra, as opposed to the inwardly-dimpled envelope of the HPK 125W. These improvements reduce the e f f e c t of arc wandering by an order of magnitude. With both the new lamp and the a c t i v e c o n t r o l l e r , the source i n t e n s i t y i s stable to t y p i c a l l y better than one part i n three thousand, depending on the frequency range. III.3 E l e c t r o n i c s The e l e c t r o n i c s used to record dual-interferogram data are shown i n Figure III-5. The s i g n a l detected by the bolometer i s amplified by a low noise FET-input AC a m p l i f i e r , then detected synchronously with the chopping frequency. The chopping frequency was made v a r i a b l e , as opposed to the f i x e d frequency design that came with the FS-720, i n order to optimise the signal-to-noise r a t i o f o r each p a r t i c u l a r detector. This requires a l o c k - i n a m p l i f i e r external to the FS-720 e l e c t r o n i c s . F i f t y Hertz i s used with the bolometer. A f a s t s e t t l i n g • S T E P P I N G M I C R O M E T E R HS- - — S W I T C H I I M O M A S K S T E P ^ - D R I V E A N D S A M P L I N G C O N T R O L L O W -NOISE BOLOMETER S O H E R T Z C H O P P I N G R E F E R E N C E L O C K - I N A M P P A R !24A 8 P O L E A C T I V E F I L T E R I N F R A R E D L A B S INC. I N T E G R A T O R A N D S A M R L E - A N D - H O L D S !~ GAIN "I 8 O F F S E T / D U A L - P E N ^ ' C H A R T R E C O R D E R ± D I G I T A L V O L T -M E T E R — W P U N C H H E A T H S R 2 0 6 '.. \ .. ••. 48 Bessel f i l t e r then passes the s i g n a l to a s i g n a l - c o n d i t i o n i n g box. The r e s t of the e l e c t r o n i c s i s c o n t r o l l e d by the step-drive and sampling c o n t r o l , i n the following sequence: ( i ) The mask i s switched to i l l u m i n a t e the sample surface, while the moving mirror and the dual-pen s t r i p - c h a r t recorder are moved to t h e i r next p o s i t i o n s . At the same time, the integrator i s zeroed and the gain and o f f s e t f o r the sample i n t e r f erogram are appliecl. ( i i ) A f t e r s u f f i c i e n t time f o r everything to s e t t l e (- 0.5 seconds), the sample s i g n a l i s integrated fo r 0.5 to 32 seconds, depending on the noise present and the number of sampling points required, ( i i i ) The i n t e g r a t i o n i s stopped, the d i g i t a l voltmeter samples and punches the r e s u l t and t h i s r e s u l t i s sampled and held f o r the sample channel on the s t r i p - c h a r t recorder, (iv) The mask i s switched to i l l u m i n a t e the aluminized surface while the i n t e g r a t o r i s zeroed and the gain and o f f s e t f o r the reference interferogram are applied. Steps ( i i ) and ( i i i ) are then repeated for the reference channel. The e n t i r e sequence i s repeated s u f f i c i e n t times to produce interferograms with the required r e s o l u t i o n . The data i s punched onto paper tape which i s then fed into a Datagen Nova 1200 minicomputer. Crosstalk between adjacent datum points must be eliminated f o r t h i s switching method. Crosstalk can be t o l e r a t e d with a s i n g l e interferogram, since t h i s w i l l r e s u l t i n a predictable high-frequency attenuation and phase s h i f t i n the r e s u l t a n t spectrum. With the simultaneous recording of the two interferograms, however, the e f f e c t of c r o s s - t a l k between them i s not e a s i l y p r edictable and must be avoided. Long s e t t l i n g - t i m e s between datum points have been eliminated by using a f a s t s e t t l i n g f i l t e r followed by an integrator (see Appendix B), which replaces the simple R-C f i l t e r provided i n the FS-720 e l e c t r o n i c s . The Golay detector, 49 standard w i t h the FS-720, i s a l s o u n s u i t a b l e f o r these measurements due to i t s long time-constant of approximately one second. Thus, a bolometer, having a sho r t e r time constant of approximately two m i l l i s e c o n d s , permits f a s t e r s w i t c h i n g as w e l l as having a s u p e r i o r s i g n a l - t o - n o i s e r a t i o . I I I . 4 Computing A complete set of computer programs was w r i t t e n i n Extended B a s i c , w i t h machine language subroutines, f o r the a n a l y s i s of d i s p e r s i v e -r e f l e c t i o n interferograms. Each program s t o r e s the r e s u l t of i t s o p e r a t i o n on a d i s c . The r e s u l t of any op e r a t i o n can be d i s p l a y e d on a C.R.T., p l o t t e d w i t h a poi n t p l o t t e r , p r i n t e d , or punched w i t h a h i g h -speed punch. The f i r s t of these programs s e l e c t s a l t e r n a t e p o i n t s from the d a t a tape, to separate the background and sample.interferograms. Another program a l l o w s e d i t i n g of such f e a t u r e s as n o i s e s p i k e s i n the interferograms. A t h i r d program i s used to su b t r a c t a l i n e a r l e a s t -squares f i t to the ends of the inte r f e r o g r a m as described i n s e c t i o n I I . 5,; apodize the i n t e r f erogram as described i n the same s e c t i o n ; f i n d i t s centre by performing a lea s t - s q u a r e s l i n e a r f i t to the phase of a l o w - r e s o l u t i o n f a s t - F o u r i e r transform of the reference i n t e r f e r o g r a m as described i n s e c t i o n I I . 7 ; and f i n a l l y to F o u r i e r - t r a n s f o r m the In t e r f e r o g r a m w i t h respect to t h i s c e n t r e . Three more programs a l l o w t h e r a t i o i n g , averaging and j o i n i n g of s p e c t r a . The seventh program c o r r e c t s f o r the n o n - i d e a l reference surface (see s e c t i o n I I . 3 and Appendix C ) . When a f i n a l amplitude and phase r e f l e c t a n c e spectrum i s 50 produced, an eighth program can convert t h i s i n t o the power r e f l e c t a n c e , d i e l e c t r i c constants, r e f r a c t i v e i n d i c e s , conductivity, absorption i • • • c o e f f i c i e n t , , or the damping and wavenumber s h i f t of a resonance (see Appendix D). c 51 CHAPTER IV ! EXPERIMENTAL SETUP - LOW TEMPERATURE I " . . . . I IV.1 . Optics The primary change to the room temperature instrument, required f o r measuring the o p t i c a l properties of cold samples, i s the a d d i t i o n of a dewar to hold the sample and the consequential change i n o p t i c s . Figure IV-1 shows the FS-720 op t i c s modified f o r low-temperature measurements. A l i q u i d - n i t r o g e n temperature mylar window i s i n front of the sample held on a liquid-helium dewar. A beam-splitter type of mount, holds the mylar f l a t to prevent d i s t o r t i n g the incident wavefront. The window has more than a two degree t i l t to the beam d i r e c t i o n i n order' to prevent i t s r e f l e c t e d l i g h t from reaching the detector. Also, a spacer extends the moving mirror arm to compensate for the extra path-length introduced by the one-inch-thick gate valve and the dewar. For proper masking, the switching mask i s extended as close to the sample as p o s s i b l e . IV.2 , Dewar |The s i x - l i t r e l iquid-helium dewar shown i n Figure IV-2 holds i l i q u i d f o r about eight hours with a 5.9 cm diameter mylar window and aluminum f o i l on a l l nearby surfaces facing the sample. The u n f i l t e r e d mercury arc source increases b o i l o f f by l e s s than f i v e percent. Four 0.4 mm diameter piano wires, j o i n i n g the bottom of the helium can to outside adjusters, r i g i d l y hold the helium can to prevent sample i i 52 Figure IV -1 . A schematic, drawn to scale, of the RIIC FS-720 o p t i c s , modified f o r disp e r s i v e r e f l e c t i o n measurements of a liquid- h e l i u m temperature sample. « 2 5 T0 ; c m Moveable shield A l i g n m e n t He igh t , ad jus ter 53 72Z&. 2 — Precoo l b y p a s s / M y l a r W i n d o w G a t e valve s w i t c h i n g m a s k F i x e d a r m F . I . R . B e a m R . I . I . C . F S - 7 2 0 V a c u u m Figure IV-2. The s i x - l i t r e sample-holding liquid-helium dewar, 54 movement and to f a c i l i t a t e , h o r i z o n t a l alignment of the sample w i t h the s w i t c h i n g mask. A height a d j u s t e r , which suspends the helium can, allows, v e r t i c a l alignment and compensation f o r thermal c o n t r a c t i o n s . The sample i s held on an a l i g n e r i n s i d e a 9 cm inside-diameter copper tube which i s surrounded by l i q u i d helium. A bypass tube a l l o w s the l i q u i d n i t r o g e n used f o r p r e - c o o l i n g the dewar to be blown out, and allow s l i q u i d helium to be forced to t!he bottom of the dewar when f i l l i n g i t . The whole dewar, shown i n F i g u r e IV - 2 , i s suspended by two turnbuckles from the frame which holds the spectrometer t a b l e . The connection to the spectrometer through the gate v a l v e provides a d d i t i o n a l support. As shown i n Figure IV-3, the w i r i n g f o r the l i q u i d helium can goes from a n i n e - p i n vacuum connector, down around one of the s t a i n l e s s -s t e e l s u p p o r t / f i l l tubes, to a t e f l o n t e r m i n a l b l o c k between the two f i l l tubes. Connections on t h i s b l ock then go to a temperature sensor attached to the sample and to a temperature sensor mounted on the copper tube-. The l a t t e r i s u s e f u l f o r monitoring the cool-down. A 1 kfi heater w i r e wrapped around the copper sample-backing p l a t e i s a l s o connected to the b l o c k . This heater i s used to speed up the warming of the dewar a f t e r a run, and a l s o to keep the sample above the temperature of the r e s t of the cTewar w h i l e warming,so that nothing condenses upon the sample. F i g u r e IV-4 shows the bolometer and preamp to the l e f t , and the sample dewar to the r i g h t , both attached to the spectrometer. The pumping st a c k ; c o n s i s t i n g of gate v a l v e , spool p i e c e , l i q u i d - n i t r o g e n t r a p and o i l d i f f u s i o n pump; are shown under the dewar. The pressure obtained i s monitored w i t h i o n i z a t i o n gauges at the spool p i e c e and on IIII 55 9-Pin c o n n e c t o r I n s e r t s T fl I Li B r a s s e n d p l u g s Terminal -block He 4 Temperature s e n s o r s H o l e s S u p p o r t i n g * — pin •—Fishing line spi ra l w r a p S t a i n l e s s s tee l tube Heater 10 c m Figure IV-3. Wiring layout on dewar and f i l l - t u b e i n s e r t s . 57 top of the spectrometer. A c l o s e r view of. the dewar i s shown i n Figure IV-5. Note the sample alignment.adjusters on the large flange at l e f t , the piano-wire adjusters, and the gate valve j o i n i n g the dewar to the spectrometer. The helium can i s shown i n Figure IV-6. Note the wiring held by yellow mylar tape, the aluminized surface to reduce r a d i a t i o n heating, the t e f l o n receptacles f or the sample a l i g n e r s , and the copper-tube 'temperature sensor."The heat input to the helium can through the support tubes i s reduced by an order of magnitude by putting i n the i n s e r t s shown i n Figure IV-3. The b o i l o f f gas i s forced to s p i r a l around the surface of the tubes, which cools them. IV.3 Sample Mount The f i r s t attempts at cooling the sample were made with three phosphor-bronze spring fingers holding the sample against the copper backing p l a t e , with s i l v e r impregnated grease sandwiched i n between. The sample went to ^60K. A very t h i n layer of Apiezon N grease was then t r i e d , but since NaCl contracts on c o o l i n g much more than copper, the sample broke into two pieces. F i n a l l y , a p l e x i g l a s clamp was used to press the sample against a 50 AWG bare, s o f t , copper wire mat, on a copper plate held by the gimbals shown i n Figure IV-7. The p l e x i g l a s was used because i t contracts more than the NaCl sample on cooling, thus holding i t securely. This i s also shown i n Figure IV-8. Note the copper mat, which can be seen through the transparent s i n g l e - c r y s t a l NaCl sample, the aluminized pattern on the sample, the temperature sensor which can be seen p a r t i a l l y behind the centre of the sample, and the c l e a r p l e x i g l a s r i n g which i s held by nylon b o l t s that screw i n t o 5 c m fine A l i g n e r s coa r se P l e x i g l a s c l a m p liquid He Figure IV-7. The sample cold-mount, with alignment mechanism. 62 another" piexiglas' r i n g attached^ to-the copper sample-backing p l a t e . Four- copper braids.cool the pl a t e and allow f o r alignment movement. An expandible copper r i n g attaches the braid to the copper tube as shown i n Figure IV-7, and determines the p o s i t i o n of the sample holder i n the tube. A temperature sensing diode i s held against the back of the sample by a l i g h t s t e e l spring, and i s insulated from the copper pl a t e by a t e f l o n spacer..-The 'diode i s a Lake Shore Cryotronics DT500 KL, operated with a 10 yA bias current. With t h i s method of sample mounting, successful runs were obtained,at sample temperatures of ^ 25 K. A compromise, however, must be made between temperature and sample f l a t n e s s . I f a large pressure i s applied v i a the p l e x i g l a s r i n g , the sample d i s t o r t s . This, i s minimized by tightening the p l e x i g l a s clamp while monitoring the fl a t n e s s with an o p t i c a l f l a t and monochromatic l i g h t , to keep the d i s t o r t i o n acceptably low. This i s one area where future improvements, such as d i r e c t vapour or l i q u i d cooling of the sample, would be u s e f u l . . Four sample-alignment knobs turn thin-walled s t a i n l e s s - s t e e l - t u b e two-pronged forks through a moveable p l e x i g l a s cover which i s p a r t l y aluminized and p a r t l y c l e a r f o r v i s u a l alignment. A moveable l i q u i d -nitrogen temperature s h i e l d portion i s set low at room temperature to compensate f o r thermal contraction at the a l i g n e r s . Figures IV-2 and IV-7 show i n cross-section that the forks can be inserted into two-holed t e f l o n receptacles which turn two coarse and two f i n e a l i g n e r s . The coarse a l i g n e r s push d i r e c t l y on the sample-holding gimbals, while the f i n e a l i g n e r s move the coarse a l i g n e r s through a d i f f e r e n t i a l thread and a lev e r . Various designs were t r i e d at f i r s t . A hex key guided by a cone to a hex receptacle was too r 63 c r i t i c a l to align, and very s t i f f . A T-barv pushed' into a wide- slot was better, but stuck in certain positions i f not aligned perfectly. The two prongs into two holes method allows for mis-alignment, and a support to hold the receptacles at the opposite end of the copper tube from the sample improves the alignment and allows flexible connections to be used on the more c r i t i c a l fine aligner.s (Figure IV-7) - Mounting the sample holder on an aligner of orthogonal spring "bars, which was originally designed but discarded when being built because i t was easier to machine gimbals, would probably have been better. The fine alignment i s not too smooth at liquid helium temperature, but is adequate. IV.4 Vacuum A cryotrapped Varian VES-4 diffusion pump evacuates the spectrometer directly under the beam spl i t t e r at approximately 400 l i t r e s per second for air (Figures IV-4 and IV-9). It can also evacuate the dewar directly, through the bottom port shown in Figure IV-2. The spectrometer, which can be isolated from the dewar by a gate valve for beam split t e r and f i l t e r changes, is normally evacuated to ^10 ^ Torr. The liquid-helium boiloff rate i s 10% lower when the aluminized gate valve is closed. The mylar window mentioned previously reduces sample contamination, by preventing the sample from cryopumping the spectrometer. Without this, there are considerable effects evident in the reflection spectra after an hour or so at pressures of y^lO ^ Torr. The Welch 1376-B80 300 1/m forepump is mechanically isolated by a dual bellows to prevent i t from vibrating the spectrometer, dewar and detector. 64 To A i r Coax, Trap: Veeco VS-160 Dual-bellows V i b r a t i o n I s o l a t o r Fore- pump Welch 1376 B80 300 1/m Rotary FS - 720 3" : Gate .Dewar Valve Spectro-meter T.C. Gauge T.C.,Ion Gauges I L i q u i d -1 nitrogen Trap Varian 0362 - 4" O i l \ iffusion) Varian Pump j VHS 4 Figure IV-9. The M.H.L. Pryce vacuum system. Funds for t h i s equipment were kind l y donated by M.H.L. Pryce from h i s NRC operating grant. 65 IV.5 Alignment Procedure" O p t i c a l alignment of a sample i s done as follows. At room temperature, the sample; i s mounted so that the aluminized : pattern on i t i s symmetrically aligned with the symmetry introduced by the l i n e a r arc of the source. The switching mask i s then r o t a t i o n a l l y aligned with the aluminized pattern on the sample. Also, the'sample is-centred behind the switching mask and aligned f o r interference while exposing the aluminum reference surface. A room-temperature run can then be taken as a check. A mirror i n the normal sample-module images the beam through the module's window i n order to v i s u a l l y centre the sample behind the switching mask while the moving mirror beam i s blocked. This i s done a f t e r the sample i s cold (and the interferometer i s evacuated, of course) and i s followed by realignment f o r inte r f e r e n c e . A t y p i c a l image seen when checking a helium temperature sample f or alignment i s shown i n Figure IV-10. IV.6 Tests Of The Instrument The t h i r t y - f i v e low-temperature-sample interferograms measured so f a r i n d i c a t e that the p o s i t i o n of the sample i s stable, with no v i b r a t i o n noticeable, and i n s i g n i f i c a n t zero-path-difference p o s i t i o n a l s h i f t during a low temperature run. Adequate averaging of the non-uniform beam by the switching mask was checked by switching i n front of a plane mirror i n the dewar to produce two interferograms. In the measured 30 to 500 wavenumber range, the resultant amplitude r a t i o i s unity within the noise of l e s s than one percent, and the phase i s zero within the noise of l e s s than one 67 degree-. The. interf-e-rogram*modulation i s ^-30% i n . t h i s frequency range, due.to incomplete, i n t e r f e r e n c e of h i g h f r e q u e n c i e s , due to the s w i t c h i n g mask and a l s o due to the longer pathlength. Note that the i n t e r f e r o g r a m modulation i s E ( I - I ) / I where I , I are the 0 00 CO Q O  i n t e n s i t i e s at zero path d i f f e r e n c e and l a r g e path d i f f e r e n c e r e s p e c t i v e l y . A ;76 mm diameter sample of 'NaCl ;8';mm'thick, clamped i n p l a c e and s t i l l f l a t to b e t t e r than l y over the r e g i o n i l l u m i n a t e d by the 55 mm diameter i r i s - a p e r t u r e stop, was run at room temperature. This agreed 97 w i t h previous accurate measurements t o w i t h i n the n o i s e l e v e l s of ^1% amplitude and 1° phase. 68 . CHAPTER V ALKALI - HALIDES THEORY V.l Introduction In this chapter, the theoretical... calculations of the optical properties of alkali-halides are discussed. J.E. ^ Eldridge et a l have previously performed power-reflection and -transmission measurements with several (alkali-halides and compared these f a i r l y successfully with 3 112—116 detailed calculations based on cubic anharmonicity alone. ' With the improved method used i n this project for measuring optical properties, however, refinements of this theory have become necessary ,for an accurate comparison with the experimental results. In particular, three-phonon processes, resulting from quartic anharmonicity, have been included, and furthermore, the damping of a l l the final'-state phonons has been considered. These are described in sections V.7 and V.9 respectively. V.2 The Dielectric Constant The optical absorption of the alkali-halides in the far infrared is due primarily to the "reststrahlen" absorption of the degenerate transverse optical modes. Considering only the first-order electronic dipole moment, the complex dielectric constant, due to Wallis and Maradudin^'' and Cowley^^, as expressed in reference 97, i s : e» + i c " = e + e* 2(M + + M~) OO 4- — „ — — « _ f _ _ * V * ^ r ' * r r TTVC 2 M M~{V2-V 2+2V Q (A' (0,J o;v)-ir(0,j Q;v)} } 69 wtter'e'- e i s the r e a l d i e l e c t r i c constant" i n the high frequency l i m i t , * e i s the macroscopic e f f e c t i v e charge a s s o c i a t e d w i t h the tra n s v e r s e o p t i c l a t t i c e resonance, M + and M are the masses of the a l k a l i and h a l i d e i o n s , v i s the volume of the u n i t c e l l , c i s the v e l o c i t y of l i g h t , v i s the observed wavenumber of the transverse o p t i c mode where the o wave-vector k i s e f f e c t i v e l y zero and the branch index i s 1 . A' (0,j o;v) and T ( 0 , j Q ; v ) are the r e a l arid imaginary p a r t s of the phase s h i f t , which represent the frequency s h i f t and damping (both i n wavenumbers), r e s p e c t i v e l y , of the transverse o p t i c mode when e x c i t e d by r a d i a t i o n of wavenumber v. * Since the s t a t i c d i e l e c t r i c constant E i s e a s i l y measured, e o can be c a l c u l a t e d by t a k i n g the zero frequency case of ( V . l ) : e = £ r o + : | ! V j i i Q _ ( v . 2 ) 0 TTVC 2 M M v 2{l+2A'(0)/v } 0 0 so that: * e f(e o-e o o)Trvc 2 M + M v 2{ 1+2A 1 (0) /V Q } }h M + + M " (V .3) The bulk of the d i e l e c t r i c c a l c u l a t i o n i s then the determination of the complex phase s h i f t , A'(v) and.T(v), which have been abbreviated s i n c e we w i l l assume that these r e f e r to the tra n s v e r s e o p t i c resonance at k = 0. 70 V.3 Phase- Shift' Of The: Transverse Optic Resonance Both the real and imaginary parts of the phase shift do not need to be directly calculated, since they are related through the dispersion relations. The wavenumber shift may be written, according to reference 3 as: A(v) = - m — T(v') dv' + constant (V.4) * J0 v' 2 - v 2 + e 2 where is taken beyond the three-phonon limit, and e i s a small -4 number (10 in these calculations) to avoid a singularity in the integral. The constant i s unimportant because we know that A ' ( V Q ) = 0 at the experimentally observed resonance frequency V Q , S O that we set: A'(v)'-= A(v) - A(v ) (V.5) o Thus only the damping T(v) need be calculated. This depends on the . interionic potential, the coefficient which couples the various phonons, and the phonon data i t s e l f . This latter must f i r s t be generated by a shell-model calculation using parameters determined from a f i t to inelastic-neutron-scattering data. V.4 Interionic Potential Assuming that the potential between any two ions can be expressed as the sum of a long range coulombic term and a short range quantum-mechanical repulsion, then the l a t t i c e energy per unit c e l l i s : 71 U'-- =f£ •+ n C e - r o / p (V.6) o where only nearest neighbours were considered i n the r e p u l s i v e term. Here, a i s the Madelung constant (- 1.74756, 1.76267, and 1.63805 f o r the N a C l , C s l , and ZnS s t r u c t u r e s r e s p e c t i v e l y ) , e i s the e l e c t r o n i c c h a r g e , , r Q ; is the nearest -neighbour .separa t ion j 'and n i s the number of nearest -neighbours (6, 8, and A f o r the N a C l , C s l , and ZnS s t r u c t u r e s r e s p e c t i v e l y ) . The r e p u l s i v e parameters can then be determined from the experimental va lues of the c o m p r e s s i b i l i t y 3 and the nearest neighbour separat ion XQ as f o l l o w s : r gcte2 P = (V . 7 ) 2gcte2 + 9vr o 2 r /P a n d c - ° e P e ° . (V.8) where v i s the u n i t - c e l l volume (2r^, 8 r ^ / ( 3 / 3 ) , and 16r^/(3/3) f o r the N a C l , C s l , and ZnS s t ruc tu res r e s p e c t i v e l y ) . The i n t e r i o n i c p o t e n t i a l used i n c a l c u l a t i n g T(v) i s the p o t e n t i a l <j> between j u s t two i o n s : B = T ~ + n C e ~ r / p (V.9) and the d e r i v a t i v e s used fo r the c a l c u l a t i o n are : r^ r H r^p o o o 72 ! ^ r ^ + j L . - V P •.' (v.n> i U 0 o r H r o o r " ( r ) = ^ . - M e - r 0 / P (v.12) 0 r1* p 3 o r ..... ( r ) = Z ^ l i + V ^ e - r 0 / P (V.13) p 4 V.5 Lattice Dynamics The l a t t i c e dynamical input data are generated by a shell model which was kindly supplied by G. Dolling and modified by R. Howard. This fifteen-parameter model was used as an eleven parameter model, since there were only eleven parameters used by Schmunk and Winder in their 119 shell-model f i t to inelastic neutron-scattering data. These parameters are given in Table- V-I. The basic theory of the-shell model 120 i s described by Cochran et a l . Eigendata were calculated with this model for 2792 values of in t i l the irreducible 1/48 of the Brillouin zone, which is equivalent to 108,000 wave-vector points in the f i r s t B rillouin zone, or 30 evenly spaced wave-vectors from the zone centre to the <100> boundary. These were a l l used for calculating two-phonon damping, but for the much more lengthy three-phonon calculation, 28 irreducible points or 500 points per zone were used. Some of the dispersion curves given by this model are shown in Figure V - l . 73 TABLE.V-I Shell Model Parameters Parameter Value Units A 1 2 or A 10.264 e2/2v B 1 2 or B -0.971 e 2/2v Aj j or A' -0.427 e 2/2v B 1 1 or B' 0.022 e2/2v A 2 2 or A" 0.597 e 2/2v B22 o r B " -0.025 e 2/2v Z 0.968 e a l 0.011 1/v d l -0.035 e a2 0.058 1/v d2 0.194 e Figure V - l . Frequency dispersion curves along the major symmetry d i r e c t i o n s as w e l l as some other regions, generated by the eleven-parameter model for NaCl at room temperature. 75 V .6 Two-phonon Damping Resulting From Cubic Anharmonicity Since the momentum of the far i n f r a r e d photon i s very small, only the approximately-zero-moraentum transverse o p t i c phonon can be d i r e c t l y excited. This phonon can however serve as a v i r t u a l intermediate state i n which energy i s not conserved, which then couples with other phonons ,which.conserve both?energy.and momentum. Thus, absorption-at frequencies away from V q can occur. There are two possible two-phonon processes of t h i s type. In the summation process, the transverse o p t i c phonon decays into two phonons with equal and opposite momenta, whose combined energy equals that of the incident photon. In the d i f f e r e n c e process, the transverse o p t i c phonon creates one phonon and destroys another phonon of lower energy with equal momentum and an energy d i f f e r e n c e equal to that of the incident photon. As shown i n Figure V-2, there can be a large c o n t r i b u t i o n to the summation process where the phonon branches have equal and opposite slopes, so that many possible summations S^-K^ equal the same energy. Likewise, f o r d i f f e r e n c e processes, large contributions can occur f o r equal slopes such that there are many equal di f f e r e n c e s ^ " ^ l • This two phonon damping i s given (from Cowley, Reference 118) by: r 2- Ph^> = T i b i L Iv^Vv^^-tVl 2 S(v) (cm" 1) (V.IA) /.tic n kj j —kj 2 (3) where V i s the "c u b i c - c o u p l i n g - c o e f f i c i e n t " , which couples the transverse-optic resonance mode to the two phonons, and S(v) i s a function of temperature and wavenumber r e s u l t i n g from the t r a n s i t i o n 76 .'Transverse Optic Figure V-2. I l l u s t r a t i v e frequency d i s p e r s i o n curves showing darkened portions of large contributions to the summation (SJ+ S 2 ) and d i f f e r e n c e (I^-D.) processes. Note that for the summation case, S^nd S„ are r e a l l y of opposite momentum. 77 p r o b a b i l i t i e s f o r phonbn creation and destruction, together with the conservation of energy between the, i n i t i a l photon state (wavenumber v) and the f i n a l two-phonon state. Note that the re s u l t a n t phonons must be from d i f f e r e n t p o l a r i z a t i o n branches f o r c r y s t a l s such as these, which have centre-of-inversion symmetry. From quantum theory (Ziman, Reference 121), the p r o b a b i l i t i e s f o r •creation and a n n i h i l a t i o n of a phonon are: P n + n + l * n + 1 <V'15> W l « » ' ' • (V.16) Thus, the t o t a l p r o b a b i l i t y f o r the creation of two phonons P T T (summation process) i s equal to the p r o b a b i l i t y of creating two phonons minus the p r o b a b i l i t y of a n n i h i l a t i n g two phonons: . p++ re {n(k,j 1)+l}{n(-k',j 2)+l} - n ( ^ , j 1 ) n ( 4 , j 2 ) cc nCt.jj) + n ( - k , j 2 ) + 1 (V.17) and the t o t a l p r o b a b i l i t y P + for the d i f f e r e n c e process of creat i n g one phonon and a n n i h i l a t i n g another phonon i s : P +_ «' { n ( ^ , j 1 ) + l } n ( k , j 2 ) - n ( k , j 1 ) { n ( k , j 2 ) + l } « n ( t j j ) - n ( k , j 2 ) (V.18) Here, the Bose-Einstein occupation number, n(k,j) i s given by: 78 n(k,j) - (exp{hcv(k,j)/(k_T)} - l ) -1 (V.1.9.) where kg i s Boltzmann's constant. Expanding the exponential i n a power ser i e s gives: n(k,j) = nv,(k,,j) ,1 Kv(k,j)  :kgT + 2!( k^T + • • • - 1 -1 (V.20) which, for large temperatures, gives: n(k,j) = ? » 1 Kv(£,j) (V.21) so that both summation and di f f e r e n c e processes w i l l be l i n e a r l y proportional to temperature: + + oc — — HIGH TEMP. 1 - + -v(k,j.) v(-k,j_) k„T P. "B HIGH TEMP. K vCk.j.) v(k,j_) (V.22) (V.23) At very low temperatures, n(k,j) becomes very close to zero, so that: ++ « 1 LOW TEMP. p and +- = 0 LOW TEMP. (V.24) Energy conservation i s incorporated into the S(v) term by means of a Dirac d e l t a function, as shown below i n the f i n a l forms of S(v): S(v) = (n(k,j1)+n(-k,j2)+l) 6 {v-v(^ , j 1 ) -v(4 , j 2 ) } (V.25) 79 fo r summation processes where two phonons are created, and: S(v) = ( n ( k , j 1 ) - n ( k , j 2 ) ) (6{v+v(k,j 1)-v(k,j 2)} -Stv - v C^jp+ v d , ^ ) } ) (V.26) for d i f f e r e n c e processes. The theory to 'this point i s : I d e r i t i c a l f o r the d i f f e r e n t c r y s t a l (3) structures. In i t s f i n a l form, however, V I t s general form i s : depends on the structure. v ( 3 ) i •£ 1 -t i ) = - ± v v^-j > J j » " ^ 2 2 ' 3' 3 96 [ ( T r c N ) 3 v ( k 1 , J 1 ) v ( k 2 , J 2 ) v ( k 3 , J 3 ) T 2 3 , - l " 2 -3 3 x n £=1 ^ exp(2iTik £'x(L,K)) \ exp(2TTik -x(L',K')) (V.27) where N i s the number of unit c e l l s , x(L,K) i s the p o s i t i o n vector of the K*"*1 type of ion i n the I,*"*1 unit c e l l , i s the mass of the K1"*1 type of ion, and $ ala2a3 (L,K;L'K') i s the t h i r d c a r t e s i a n d e r i v a t i v e of the p o t e n t i a l energy between the two ions at LK and L'K' (the f i r s t anharmonic term i n the potential-energy Taylor expansion), m 0 ( k . , j . ) i s the a^ t^ 1 component of the eigenvector for the type ion when disturbed by the mode lc ,j . The d e l t a function A(k.+^+k-) conserves momentum by being set equal to one when the wave-vector sum equals zero or a r e c i p r o c a l - l a t t i c e vector, and by being set equal to zero otherwise, • 8 0 The i n c l u s i o n of the r e c i p r o c a l - l a t t i c e vectors does not a l t e r the f i n a l form of T(v) and A(v), and therefore, with only the transverse o p t i c resonance (k^=0) being excited, then k2=-k2 and becomes ( 0 ,JQJ^, J J ; —^»j2^ • Since the f i r s t summation over LK gives N times the summation over K, one of the ions may be placed i n the zeroth u n i t c e l l and (V.27) m u l t i p l i e d by N. The second summation over L'K' (with the prime 'on 'the sigma excluding the term with L'K'=LK) has to be r e s t r i c t e d to c e r t a i n neighbours i f the evaluation of (V.27) i s to be t r a c t a b l e . The summation has been r e s t r i c t e d to the s i x nearest neighbours f o r the NaCl structure by Johnson and B e l l , ^ " ' and E l d r i d g e . This was reasonable since the major component of <f> (LK;L'K') comes C t l 0 l 2 a 3 from the short range overlap repulsion, which, f o r third-nearest neighbours, i s only about 0.15% of the repulsion f o r nearest neighbours (second nearest neighbours do not contribute, since they are l i k e ions, and l i k e ions i n the lc=0 transverse optic'mode maintain the same separation). Due to the cubic symmetry of the c r y s t a l , T(v) w i l l be independent of the p o l a r i z a t i o n of the incident photon. Assuming i t i s p o l a r i z e d with i t s wave-vector along the x d i r e c t i o n , therefore, cc^  i n (V.27) may be set equal to x (Johnson and Bell^"*) . Converting the e f f e c t i v e force constants <t> (LK:L'K') to r a d i a l d e r i v a t i v e s and considering nearest T x a 2 c t 3 v neighbours only, one a r r i v e s at the expression: 81 V ( 3 ) ( 0 , j 0 ; k , j i ; - k , j 2 ) . = ^ | (M++M~)n3 ^ [8N(M +M~)V o(2TTC) 3J (v(k,j 1)v(-k,j 2)) { * , " ( r o ) K ( ^ ' j l ) m x ( ~ ^ j 2 ) " ^~(k,J 1)ni+(-k,J 2)) »»(r o) <^'(ro) r r o ,2 1 fm+(t,j.)m~(-k.10) - m~(^,j 1)m^(-^,j 2))} sin(2Trr ok x) <j>"(r) <J>'(r) x sin(2-rrr k.) o o (V.28) where M" and M are the masses of the positive and negative ions respectively (Na + and Cl for NaCl), and the derivatives of <)>(r) are given in (V.10) through (V.12). Thus, the two phonon damping can be obtained by substituting (V.25), (V.26) and (V.28) into (V.14). The damping, therefore, depends on the strength of the coupling of the two phonon modes to the transverse optic resonance, and the probability of creation or destruction of the phonons involved. 82 V.7 Three-phonon Damping Resulting From Quartic Anharmonicity Around and beyond approximately the results of these dispersive measurements show discrepancies when compared with theory based on two-phonon damping alone. The extra damping due to the next higher order process was therefore calculated by J.E. Eldridge. The simplest and-most "important -three'phononicontrlbution ,is given by: T (Z) ^—'1 I I |v ( 4 ) (o i -k i -k i -k i ) l 3 " P H 2 T C ¥ ^ .{ 3 ' C O . J o ' k l ^ l » k 2 ^ 2 ' k 3 ^ 3 ) | ± J X ± J 2 ^ 3 x S(v) (cm - 1) (V.29) where: S(v) = (( n i+l)(n 2+l)(n 3+l) - n ^ n j 6 ( v - v ^ - v ^ (V.30) for a summation case, and: S(v) = (( n i+l)(n 2+l)n 3 - n in 2(n 3+l)} 6 ( v - V j - ^ + v ^ (V.31) for a difference case in which, for example, the third phonon is destroyed. These are developed in the same manner as equations (V.25) 122 and (V.26). The notation of Wallis et a l has been used, i n which the sum over ±j_j, indicates a sum over ± v^» and of course the appropriate expression for S(v) must be used i n each case. Only a few authors have calculated this contribution, and in each 83 case c e r t a i n approximations have been made to reduce the computing time and complexity. These have included working i n the high-temperature -v (A) (4) l i m i t , ignoring the k dependence of V and by t r e a t i n g V as a constant. We thought i t would be worthwhile to c a l c u l a t e t h i s three- . phonon damping as accurately as p o s s i b l e , since i t i s the predominant term around V q at room .temperature, .and j u s t above .the. two-phonon .l i m i t at most temperatures. There i s also evidence of broad structure i n t h i s 123 l a s t region f or c e r t a i n compounds. Consequently we have made none of the above approximations. The general expression f o r the V c o e f f i c i e n t s i s given i n Equation (2.2) of Reference 122. We have evaluated from t h i s f o r the roc k s a l t structure i n a manner s i m i l a r to that outlined i n Reference 115 (3) f o r V f o r G s l . This involved considering nearest neighbours only (which i s the usual approximation and a f a i r l y good one since the second-nearest neighbours are l i k e ions and do not contribute at a l l ) , and also that the incident photon i s p o l a r i z e d with the e l e c t r i c f i e l d along the x a x i s . When one converts the fourth Cartesian d e r i v a t i v e of the p o t e n t i a l <j) to r a d i a l d e r i v a t i v e s , and r e s t r i c t s a i to x, then the r e s u l t f o r the neighbours i n the x d i r e c t i o n may be seen i n Table V-II. Here <f>"(ro) and <f>'(ro) have been neglected since they are so small (see Table v l - I ) • Only i> contains <f>'1 1 (r ) , which i s xxxx T o e a s i l y the l a r g e s t term (see Table VI-I again). However, 6<j>' 1 1 (r )/r i s j u s t over h a l f of <j>' ' 1 ' (r ) and one would not expect to neglect i t . The c a l c u l a t i o n of any of the general terms <J> , however, takes at xct2Ct3ai t l e a s t N/N as much computing time as the c a l c u l a t i o n of <J> , where N i s the number of wave-vector points i n the f i r s t B r i l l o u i n zone and N^ i s the number of i r r e d u c i b l e points i n the same zone. In our case, 84 TABLE V-II Force-constant, tensor f o r the two nearest neighbours l y i n g along the p o l a r i z a t i o n d i r e c t i o n x of the incident photon: Jxxxx 4 xxxy > XXX z n 0 0 <(. xxyx + xxyy xxyz 0 T 0 xxzx i> xxzy <i> xxzz 0 0 T xyxx 4> xyxy <J> xyxz 0 T 0 <J> xyyx xyyy xyyz = r " T 0 0 xyzx xyzy xyzz 0 0 0 <(> xzxx <f> xzxy xzxz 0 0 X • xzyx xzyy xzyz 0 0 0 xzzx xzzy xzzz T 0 0 < n » <j,""(r ) - ( 6 / r . ) • ' " ( r ) x = (1/r )<|»,,,(r ) 85 we chose an N^ of 28, which gave N equal to 500. The central-processing-u n i t time needed to compute the f u l l threerphonon spectrum using.just ct was 24 minutes. I t can be seen, therefore, that t h i s predominant xxxx r term i s the only one which can be calculated exactly, but that the s i x other non-zero terms i n Table V-II w i l l cancel f a i r l y w e l l with the .<}>'' \(r ) term i n .tj> . The f i n a l .expression, therefore, contained only <J>" " ( r '). (Neighb ours l y i n g along the y. and z d i r e c t i o n s contribute no term with f ' " ' ( r ) and only three with <f>'' ' (r ), which have also o o nece s s a r i l y been neglected.) The expression derived i s : v I . U , J 0 , K 1 , J 1 , K 2 , J 2 , K 2 J . , J 3 ; - 4 8 N (2irc ) t f v 0 v 1 v 2 v 3 •»»»'(r o) M++M" M+M ( A x l A x 2 A x 3 ^ A x l A x 2 A ^ 3 ) + ^ I A x 2 A x 3 ^ r i A x 2 A r f ) c ° s < 2 ^ 0 k l x ) — + — + — + + (A ,A 0A _-A ,A „A „) cos(2irr k„ ) x l x2 x3 x l x2 x3 o 2x - - + + + -+ (A ,A 0A _-A .A „A _) cos(2nr k„ ) x l x2 x3 x l x2 x3 o 3x A ^ t ^ ) (V.32) where: A x l ' < ^ r J 1 ) / ( M + ) J 5 (V.33) i n which m x ( k ^ , j ^ ) ^ s t n e x component of the eigenvector f o r the Na ion when disturbed by the mode ^j>J^» a n d i s t n e x component of the wave-vector of the f i r s t phonon. The A function conserves momentum and has the property that: A(k1+k2+k3) = \ 86 ^ -»- -V -V -V 1 i f k.+k9+k_ =0 or G (umklapp process) (V.34) 0 otherwise where G is a reciprocal-lattice vector. The potential derivative <t>'(r ) i s determined from (V.13). o .Thus., -the .three-phonon. damping can .be ^ .obtained ,by .substituting (V.30), (V.31) and (V.32) into (V.29). V.8 Isotope-induced One-phonon Damping This contribution i s almost negligible, even at low temperatures for NaCl, but has nevertheless been included. It i s the absorption of the photon by a single phonon with any wave-vector or momentum. Momentum need not be conserved, since the crystal symmetry i s destroyed by the isotopic impurities. It i s s t i l l , however, a result of anharmonicity, since the transverse optic resonance i s f i r s t excited and then decays to the single phonon. In natural NaCl, the chlorine i s 3 5 C l (75.53%) and 3 7C1 (24.47%), and the appropriate expression for r , the isotope induced damping, is given by: 3 r fn A ^ - *v3M+<(AM-)2>p-(v) rT C n(.U,j n;v) = (V.35) I S ° ° 4v (M-)2(M++M~) o where AM is the deviation at a particular site,of the M ion mass from the average M , and p (v) i s the phonon density of states for the M sublattice, given by: 87 I Jm"(k,j)| 2~*<V£^ ~ ~) p"(v).= k^ : 1 (V.36) I | m " ( k , j ) | 2 k,j where m (k,j) i s the eigenvector of the M ion associated with the phonon of wave-vector IT and branch index j . P ( V ) is normalized ^such that: / o p-(v) dv = 1 (V.37) The isptope-induced damping Is then added to the two and three phonon dampings and then inserted i n t o (V.l) and (V.4) f o r the c a l c u l a t i o n of the d i e l e c t r i c constant. V .9 Damping Of The " F i n a l - s t a t e " Phonons The t h e o r e t i c a l c a l c u l a t i o n s described i n t h i s chapter have considerably sharper features than i s experimentally observed (see sections VI.2, VII.2 ). C l e a r l y , the l i f e t i m e s of the two or three phonons themselves, which are involved i n the r e l a x a t i o n of the X 2 A 123 transverse o p t i c mode, have to be included. ' Fischer has done 126 t h i s i n a s e l f - c o n s i s t e n t manner. In t h i s t h e s i s , the t o t a l damping, calculated i n previous sections of t h i s chapter, i s convoluted with a normalized Lorentzian function of varying width. The width i s determined by the sum of the widths of the phonons involved i n the damping. These widths can be obtained to s u f f i c i e n t accuracy, f or t h i s refinement of the c a l c u l a t e d damping, from the widths of the phonon branches measured i n the neutron d i f f r a c t i o n experiments used to obtain the l a t t i c e dynamical input to the c a l c u l a t i o n s . The width of the transverse o p t i c a l mode at k-0 can of course be obtained from t h i s experiment. 89 CHAPTER* VI THEORETICAL AND EXPERIMENTAL RESULTS - ROOM TEMPERATURE VI. 1 The Experiment The modification for room-temperature dispersive reflection 95 spectroscopy was tested f i r s t with KBr samples, and then with NaCl 97 samples. NaCl was investigated because i t had not previously been accurately measured, because i t s optical properties could be accurately calculated, and because large f l a t laser windows of NaCl were commercially available. The NaCl samples used were 76 mm-diameter 8 mm-thick laser windows from Harshaw Chemical Co., which .we polished to within 0.3 ym over our working area. An i r i s diaphragm limited the working area to a diameter of 55 mm. The thickness was sufficient to reduce any reflected intensity from the back face to a negligible quantity. Vacuum-evaporated aluminum coated the samples.through a mask cut out of steel shim stock. The thickness was given by a d i g i t a l thickness monitor to be 70 nm ± 5%. The reflection results shown in this chapter are the average of four runs, each of f i f t y minutes duration. The resolution obtained, after apodization, was 2 cm ^ (the maximum path difference was 0.5 cm). The r e f l e c t i v i t y i s sufficiently strong to give good amplitude results from 25 to 500 cm ^. The phase becomes very small below 110 cm * and above 340 cm ^. In these regions i t was obtained more accurately Indirectly from transmission measurements on cleaved crystals, varying 90 in thickness from 0.03 to 0.2 cm, which yielded directly the absorption coefficient. The results shown have been corrected for the fact that the aluminum is not a perfect reflector and. that the phase change i s not exactly TT radians. This correction i s shown in Figure VI-1, and the calculation to obtain this i s given in Appendix C. VI.2 Phase Shift Of The Transverse Optic Resonance As mentioned in Section V.3, only the damping T(v) need be directly calculated. The two-phonon, three-phonon, isotope-induced, and final-state phonon damping have been calculated. These are compared with the experimental damping spectrum obtained from the measured optical properties, which are shown in Section VI.3. The various parameters used to calculate the damping may be found i n Table VI-I. The results of the two-phonon damping calculation may be seen in Figure VI-2 together with the experimental spectrum. Both the calculated summation and difference damping, as well as the total, have been indicated. Only five prominent features occur in the calculated spectrum and these have been labelled in Figure VI-2. They are a l l summation features. There are three main c r i t e r i a for establishing the combination 1 1 2 strength of any two phonons. F i r s t , for a summation band, the two contributing phonon branches should have equal and opposite slopes. This gives a maximum in the combined density of states (see Section V..6 and Equations (V.14), (V.25) and (V.26)) . Second, the s i n ( 2 T T r Q k a ) - type terms in Equation (V.28) are zero whenever k =0 or (2r Q) ~ t e.g. k at the point X equals (2r^) ~. Finally, from the eigenvectors in the ^ ' " ( r j 91 I E c r LU CD UJ > < 5 (S39j5ap) 3 S V H d Figure VI -1 . R e f l e c t a n c e amplitude and phase d i f f e r e n c e from IT of a 70 nm-thick l a y e r of aluminum. The phase d i f f e r e n c e (a) i s due to the o p t i c a l path d i f f e r e n c e w h i l e (b) i s due t o the s k i n depth and f i n i t e t h i c k n e s s of the aluminum. 92 TABLE VI-I Constants Used In The Calculations (290 K). Observed TO resonance wave, number V 0 (cm 1) 164 Static dielectric constant e 0 5.90 High-frequency dielectric constant z oo 2.33 Madelung constant Nearest-neighbour distance*5 a r o (IO - 8 cm) 1.74756 2.8138 Sodium-ion mass M + (amu) 22.9898 Chloririe-ion mass (amu) 35.453 Compressibility Repulsive-overlap-3 C (10 - 1 2/bar) ( I O - 1 0 erg) ..... 4.17 16.64 potential parameters Fourth potential derivative P r o (10~ 8 cm) <J>"'»(ro) (10 1 2 erg* cm ' 3 ) 0.323 62.03 Third potential derivative " ( r o ) . ( i o 1 2 erg* cm ' 3> -5.926 Second potential derivative <j>"(r )/r o o ( i o1 2 erg * cm • 3> 0.198 First potential derivative O 0 (101 2 erg * cm 3 ) 0.261 See Reference 134 k See Reference 135 c See Reference 136 93 O (,_UK>) (/L ! °f ' 0 ) J Figure VI-2. Measured damping spectrum o| the tra n s v e r s e o p t i c resonance i n NaCl at 290 K w i t h 2-cm r e s o l u t i o n , together w i t h the c a l c u l a t e d two-phonon c o n t r i b u t i o n to the damping w i t h equal r e s o l u t i o n . The large-dashed l i n e s show the separate c o n t r i b u t i o n s from the d i f f e r e n c e and summation processes. 94 term, i t may be seen that combining modes must have at l e a s t one e igenvector component i n the same d i r e c t i o n ( i . e . the x d i r e c t i o n i n Equation (V.28)) f o r the r o c k s a l t s t r u c t u r e . The i n t e n s i t y i s enhanced i f one of the e igenvectors i s of opposi te s ign to the other th ree , producing a strong coupl ing between "normal" a c o u s t i c and o p t i c modes. It has been seen, with both KI and L i F , that few of the strong fea tures are due to phonon p a i r s at the highest-symmetry p o i n t s , and indeed on ly at L are they p o s s i b l e (see F igure V - l f o r d i s p e r s i o n c u r v e s ) . The next most favourable reg ion i s u s u a l l y hal f -way along T to K where £E p a i r s are found, but then the remainder come from low-symmetry reg ions i n the centre of the zone. The s i t u a t i o n i s found to be s i m i l a r here fo r N a C l , except that combinations from branches running from V- to Q are even more predominant. While severa l regions cont r ibu te at the wave number (234 cm *) of fea ture (1) i n F igure V I -2 , the peak i t s e l f i s due e n t i r e l y to the sum of the two branches l a b e l l e d 1' and 2 i n F igure V - l at Q. I t may seem strange at f i r s t that a strong combination would be obtained between two a c o u s t i c modes, s ince i n the c r i t e r i a a l ready mentioned, one would expect strong coupl ing between "normal" a c o u s t i c a l and o p t i c a l modes. These two branches at Q, however, s a t i s f y very s t rong ly a l l the remaining c r i t e r i a . They have equal and opposi te s l o p e s , being near ly f l a t (see from L to W i n F igure V - l ) . The s i n ( 2 T r r k ) term i n the o a coupl ing c o e f f i c i e n t i s l a rge f o r a l l three wave-vector components. A l l modes at Q have e igenvector components i n the same d i r e c t i o n , and f i n a l l y the m u l t i p l i c i t y of 24 at Q i s the h ighest p o s s i b l e f o r any zone-boundary p o i n t . Feature (2) at 251 cm * i s due to L^L^, which has been the strongest i n a l l of the compounds invest iga ted so f a r , p lus Z^(TA)Z 4 (TO) . The c a l c u l a t e d feature (3) i s a c t u a l l y a p a i r of ? 95 combinations: branches 3 and 4 near Q, at 286 cm ~ t and 2 and 5 near Q, at 295 cm ~. Experimentally, these do not resolve. Feature (4) is from E^(TA)Ej(LO),and branches 1 plus 6, and 4 plus 5, two-thirds of the way from V to Q. The last feature, (5) i s partially E^E^ and partially from a region near W. It i s clear from Figure VI-2 that the two-phonon damping i s insufficient around V q and that the damping beyond 400 cm 1 must come from higher-order terms. The result of the three-phonon damping calculation may be seen in Figure VI-3, where the summation and difference processes have been drawn separately where they overlap. In this regard, difference may be taken to mean the destruction of only one phonon, since the damping due to processes involving the destruction of two phonons i s so small that i t i s hardly v i s i b l e on the graph. The form and magnitude of Figure VI-3 is. similar to that obtained by Bruce 124 (4) for KBr, who, i t i s f e l t , used a comparable expression for V (see for example Reference 129) although this is not clear. It i s interesting to assign the phonons responsible for the structure seen in the spectrum, and the four main features have been indicated. (The structure around 300 cm 1 is mainly grid noise from the calculation.) The predominant phonon involved in a l l of the features i s the transverse optic mode, which i s so f l a t across most of the zone (see Figure V - l ) . It has an average wavenumber of 170 cm ~, and so peak (6) at 170 cm 1 is the creation of two and the destruction of one, while peak (9) at 510 cm 1 is the creation of three. Peak (8) at 485 cm 1 is the creation of two transverse optic modes together with a mode around 140 cm 1 (e.g. between X and K, and X and L), while peak (7) at 444 cm 1 is two transverse optic modes plus an acoustical mode around 100 cm ~. Also o Figure VI-3. Calculated three-phonon damping of the transverse o p t i resonance i n NaCl at 2 9 0 K with 2-cm r e s o l u t i o n . The .dashed l i n e s show the separate contributions of the summation and di f f e r e n c e processes. 97 contributing- here* is* the combination o f transverse optic plus two -1 (4) modes around 140 cm . One can see by inspection of V (Equation V.32) that once again the coupling i s enhanced i f one or a l l three of the modes have an optical nature, i.e. the eigenvectors are of opposite sign. Note, however, that the eigenvectors for the longitudinal optical mode between T and X, for example, change halfway across the zone to having the same sign, in order to preserve the optical nature of the mode, in which the nearest neighbours are displaced in opposite directions. This i s of course accompanied by a similar change in the longitudinal acoustical mode eigenvectors, so that the two-phonon enhancement remains strong. This enhancement in the three-phonon case i s not strongly evident in the present spectrum for NaCl, since there i s no energy gap in the phonon spectrum, so that many modes cross and change nature throughout the zone. Of greatest importance i s the high density of states of the transverse optic mode, as already mentioned. Subsequent to Reference 112 in which the two-phonon enhancement was 130 pointed out, Duthler and Sparks reported i t as a quasiselection rule. Recently, Duthler has extended this rule to multiphonon processes and theoretically demonstrated in a much more instructive manner the enhancement mentioned above, namely, that obtained where an.odd number 125 of optical phonons i s involved. In particular, a distinction i s drawn between materials.without a gap, such as NaCl and LiF, and those with a gap, such as Nal and KI in which the modes tend to maintain a constant nature across the zone. In these materials then, the combination of two optic modes and an acoustic mode is obviously unfavourable, and accentuated by the gap i t s e l f , the feature due to the sum of three optic modes stands out clearly from the rest of the spectrum. This 98 123 feature has been experimentally observed. We have calculated the three-phonon.. spectrum for KI at 300 K and 80 K and demonstrated, this 132 behaviour in agreement with experiment. Two further points may be mentioned. The f i r s t is the relatively equal magnitude of the difference damping in Figure VI-3 compared with the summation damping, as opposed to the two-phonon case of Figure VI-2. This is to be expected, since there are three times as many ways to destroy one transverse optic mode and create two, as there are ways of creating three transverse optic modes. The second is the wavenumber at which the summation processes 114 start, which as expected is approximately three times the wave number, 80 cm \ of the lowest acoustical modes halfway across the zone. Figure VI-4 shows the effect of adding in the above three-phonon contribution.There i s an improvement, but there remains an obvious fault in the computed spectrum, namely the.excessive sharpness of some of the features (e.g. at v and 350 cm ^ ) . Previously, this was attributed 3 to the neglect of further neighbours in the coupling coefficients, but since like ions do not contribute at a l l , this i s clearly not the main reason. As mentioned in V.9, the damping of the "final-state" phonons must be included. Figure VI-5 shows the half-width at half-maximum of the Lorentzian with which the calculated spectrum had to be convoluted in order to agree with experiment. In particular, the widths were matched for the four sharp features in Figure VI-2. The values obtained are reasonable, when one considers which phonons are involved, what their widths are l i k e l y to be (see for example Reference 126) and that the sum of the individual widths has to be used. Due to the predominant role of the high-k transverse optic phonons in the three-phonon contribution, the damping of the three-phonon states i s a l i t t l e easier ( i - ' - U o ) ( ^ '• ° f * 0 ) J Figure VI-4. Measured damping spectrum of the transverse o p t i c resonance i n NaCl at 290 K with 2 cm r e s o l u t i o n , together with the two- and three-phonon contributions to that damping, with equal r e s o l u t i o n . 100 i—r T— i — r i — i — I — r T — i — i — r O CD CM CO O O 10 O o I £ LU ZD LU > < ( , - W 0 ) OMldlAIVa Figure VI-5. Damping of the " f i n a l - s t a t e " two phonons, as a f u n c t i o n of wavenumber. The damping was exp e r i m e n t a l l y determined at the four f e a t u r e s i n d i c a t e d above and i n F i g u r e V I - 2 . A c u b i c polynomial was used to j o i n the four p o i n t s above, w i t h zero slope at each p o i n t . The c a l c u l a t e d two-phonon spectrum was convoluted w i t h a L o r e n t z i a n of the v a r y i n g h a l f - w i d t h at half-maximum given above. 101 to estimate. The damping of the zero-lc transverse o p t i c mode i s what has been measured i n t h i s experiment, and may be seen to be about 3 cm * at V q i n Figure. VI-2. This would i n d i c a t e that a Lorentzian with a constant halfwidth of 9 cm * should be used to convolute the spectrum of Figure VI-3, and the. r e s u l t may be seen i n Figure VI-6. The previous summation features of Figure VI-3 are now seen to be unresolved. The f i n a l r e s u l t i s seen i n Figure VI-7, where the o v e r a l l agreement i s very good. In p a r t i c u l a r , the agreement at i s excellent and shows the proportion of two- to three-phonon damping responsible (both of the d i f f e r e n c e type). The remaining discrepancies may be due to one or more of the f o l l o w i n g : errors i n the neutron data, s p e c i f i c a l l y the eigenvectors; neglect of further neighbours i n the coupling; neglect of the second-order dipole moment above 350 cm ^; i n c o r r e c t three-phonon damping above 350 cm ^; neglect of multiphonon damping above 350 cm ^; i n c o r r e c t width of " f i n a l - s t a t e " phonons above 350 cm ^; neglect of other contributions to the damping from d i f f e r e n t diagrams (see 350 cm would be the most important, and s p e c i f i c a l l y at 400 cm 1 a strong broad c o n t r i b u t i o n from the four-phonon process i n v o l v i n g the c r e a t i o n of three transverse o p t i c modes (510 cm and the d e s t r u c t i o n of an a c o u s t i c a l mode (110 cm ^) was expected. Since t h i s should decrease quickly with decreasing temperature, the low-temperature experiment was performed to i n d i c a t e whether the supposition was correct or not. The wavenumber s h i f t i s f i n a l l y c a l c u l a t e d as outlined i n V.3 from the t o t a l damping spectrum. This i s shown i n Figure VI-8 together with the spectrum obtained from the measurements. I t may be seen from o multiphonon processes above -1 Figure VI-6. Calculated three-phonon damping of the transverse o p t i c resonance i n NaCl a| 290 K a f t e r convolution with a Lorentzian of 9 cm half-width at half-maximum. 103 o (,-LU3) ( / L t 0 f < 0 ) J Figure VT-7. Measured damping spectrum of the transverse o p t i c resonance i n NaCl at 290 K w i t h 2-cm r e s o l u t i o n , . together w i t h the c a l c u l a t e d two- and three-phonon • c o n t r i b u t i o n s to that damping, a f t e r c o r r e c t i n g f o r the l i f e t i m e of these two and three phonons. 104 £ cr LU LU > < (^UJO) ( /L < , ° f < 0 ) V F i g u r e V I - 8 . M e a s u r e d and c a l c u l a t e d wavenumber s h i f t o f t h e t r a n s v e r s e o p t i c r e s o n a n c e i n N a C l a t 290 K . The p r i m e on A ' ( 0 , j ; v ) i n d i c a t e s t h a t t h e s h i f t h a s b e e n s e t t o z e r o a t t h e ° o b s e r v e d r e s o n a n c e wavenumber v . 105 Equation V.l that the optical properties are f a i r l y insensitive to the wavenumber shift except near V q , and so the measurements quickly become noisy away from V q . It i s the damping spectrum which is important. VI.3 Optical Properties The direct measurements were of reflectance amplitude and phase, as outlined in Chapters II and III, and these may be seen together with the spectra calculated using the damping of Figure VI.7 in Figures VI-9 and VI-10 respectively. The experimental error of the reflectance ranged from 0.2% at the maximum to 2% at the minimum and 4% at 500 cm ^. The discrepancy just after 200 cm ^ in the damping of Figure VI-7 shows up in both the reflectance and phase, but to a smaller degree than might be expected. The agreement in the remainder of these two figures is excellent. It should be noted that in a l l of the figures, from Figure VI-9 onwards, the measured properties are plotted a l l the way from 25 to 500 cm ^. If the dashed line i s not v i s i b l e , i t i s because i t i s completely overlapped by the calculated f u l l line.. As mentioned in VI.1, the phase i s too close to zero to measure accurately directly, below 110 cm ^ and above 340 cm \ so that direct absorption-coefficient measurements were performed in these regions. These are shown in Figure VI-11 together with that obtained from the phase, and the measurements from Reference 123. The discrepancy in Figure VI-7 at 400 cm * is evident here together with the obvious need for higher order processes beyond. For completeness, the refractive index i s shown in Figure VI-12, and the real and imaginary dielectric constants in Figures VI-13 and VI-14, respectively. From Figure VI-13, v T n is equal 106 J ' 3 .Dr\IVi03 " U 3 c J ' . 3 a n i l " i d l / \ I V Figure VI-9. C a l c u l a t e d and measured amplitude r e f l e c t a n c e r of NaCl at 290^K w i t h 2-cm r e s o l u t i o n . The power r e f l e c t i v i t y R = f f •, where f = r exp(i<f>). 107 I £ cr LU CO ZD LU > < Figure VI-10. Calculated and measured_reflectance phase angle <f> of. NaCl at 290 K with 2,-cm r e s o l u t i o n . The power ' r e f l e c t i v i t y R = f f , where f = r exp(i<J>). > L _ 1_ 1 1 1 — j — - _ — I ( LO ro C\J —• o CM .((,-WO) T f O 3 L W 3 0 0 MOI idHOSaV ) 0 , 5o| Figure VI-11. Calculated and measured absorption .c o e f f i c i e n t a of NaCl at £90 K. The measurements below 110 cm and above 340 cm were from power-transmission experiments. Good overlap with the d i s p e r s i v e - r e f l e c t i o n measurements was obtained. The crosses are from Harrington et a l . (Ref. 123). 109 £ cr LU co LU > < ( u ' X 3 C 1 N I 3 A ! 1 0 V c J J 3 c J ) 0 , & o | Figure VI-12. Calculated and measured r e f r a c t i v e index n of NaCl at 290 K. The discrepancy j u s t a f t e r 200 cm i s again evident. 110 ( , 3 ' I N V I S N C O . 0 1 ^ 1 0 3 1 3 1 0 1 V 3 H ) 0 , , 6 o | Figure VI-13. Calculated and measured r e a l part e' of the d i e l e c t r i c constant of NaCl at 290 K. I l l o ( „ 3 ' I N V I S N C O Dldl03"13ia AUVNI9VI/MJ) O I 6 o | Figure VI-14. Calculated and measured imaginary part E" of the d i e l e c t r i c constant of NaCl at 290 K. 112 to 265 ± 0.5 cm 1 as opposed to the neutron result of 262 ± 7 cm"1.11^ The; agreement with the low-frequency real dielectric constant gives a value of 1.080 for the effective charge e . 113 CHAPTER.VII THEORETICAL AND EXPERIMENTAL RESULTS - LOW TEMPERATURE VII.1 The Experiment The modifications, for low-temperature., dispersive reflection 138 spectroscopy were tested with KC1 and NaCl samples. The NaCl was investigated to try to resolve some of the questions raised by the room-temperature investigation in Chapter VI. The sample size and preparation was the same as for the room-temperature experiment in Section VI.1. The reflection results shown in this chapter are the average of two runs., each of f i f t y minutes duration. The resolution obtained, after apodization, was 2 cm ^ (the maximum path difference was 0.5 cm). The r e f l e c t i v i t y i s sufficiently strong to give good amplitude results from 30 cm ^ to 500 cm The phase becomes very small below 167 cm ^ and above 340 cm In these regions i t was obtained more accurately indirectly from transmission measurements on cleaved crystals, varying in thickness from 0.0955 to 0.548 cm, which yielded directly the absorption coefficient. When the low-temperature runs on NaCl were done, the measured sample temperature stabilized at 48;K. After the experiments were completed, i t was discovered that the sample-holding plexiglas clamp had become loose, reducing the thermal contact between the sample and i t s backing plate. The transmission measurements and theoretical calculations were therefore also done with a temperature of 48 K. V 114 An unforseeh problem was< encountered'with the aluminum-reference surface. A f t e r a low-temperature run, the evaporated aluminum on some of the samples seemed a l i t t l e cloudy. This was not o i l or other condensed impurities. Investigation with a microscope revealed that the aluminum was breaking away from the s a l t and forming bubbles, as i t was copied. An. extreme example of t h i s ,may be seen i n Figure VII-1. The aluminum was approximately 70 nm t h i c k . The e f f e c t of th i s , bubbling was determined by comparing the room-temperature spectra of NaCl with both smooth and bubbled reference surfaces. The r e s u l t s are shown i n Figure VII-2. This type of c o r r e c t i o n was done to the r e f l e c t i o n measurements made at 48 K. This c o r r e c t i o n brings the measured amplitude re f l e c t a n c e to within 1% of the expected values (see Figure VII-7) and brings the phase angle within 1° of the phase calculated from independent transmission measurements. Some samples were much l e s s a f f e c t e d than others, but a systematic study of the optimum thickness or evaporation conditions or a l t e r n a t i v e metal coatings has not yet been performed. In Figure VII-2, above 100 cm ~, there i s an approximately l i n e a r phase lead, and an approximately l i n e a r r e f l e c t a n c e drop. The phase slope of 0.036 degrees per wave number corresponds to a 0.5 um average s h i f t of the e f f e c t i v e r e f l e c t i o n surface. This seems consistent with the bubble sizes and the percentage of the surface covered by them. One would also expect, from these bubble c h a r a c t e r i s t i c s , a drop i n r e f l e c t i o n as the wavelength becomes smaller than the bubble s i z e , approaching a l i m i t around f i v e micrometers or so. This l i m i t would correspond to r e f l e c t i o n only from the undisturbed surface and the top c e n t r a l portions of the bubbles. The behaviour below 100 cm 1 i s not so easy to i n t e r p r e t . One p o s s i b i l i t y i s cracking of the aluminum coating, producing lower Figure VII-1. Polaroid photographs of the aluminum on one of the NaCl samples a f t e r being thermally cycled several times. The areas seen i n each photograph are 230 u x 300 um. This i s the worst region of bubbling on a sample which was the most a f f e c t e d . © O ( s o © j 6 e p ) 3 S V H d Figure VII -2 . The e f f e c t of a bubbled aluminum reference surface. The average bubble s i z e i s roughly 30 um. 117 conductivity-over? longer: distances'; and attendant reduction i n the long wavelength reflectance.. I f these breaks were spaced approximately on the scale of the bubbles, the r e f l e c t a n c e would be s i m i l a r to that of a c a p a c i t i v e mesh, which would give r e s u l t s s i m i l a r to what i s 144 observed. VII. 2 Phase S h i f t Of: The Transverse Optic Resonance' As mentioned i n Section V.3, only the damping T(v) need be d i r e c t l y c a l c u l a t e d . The two-phonon, three-phonon and isotope-induced damping have been c a l c u l a t e d . These are compared with the measured o p t i c a l properties, which are shown i n Section VII.3. The various parameters used to c a l c u l a t e the damping may be found i n Table VII-I. I t should-be stressed that the damping has a very small e f f e c t on the resonance, p a r t i c u l a r l y at low -temperatures. Hence, although T(v) may not appear to agree too w e l l , the agreement of the calculated and measured amplitude and phase i s e x c e l l e n t . As an example, compare the accuracy of the damping measurement i n t h i s s ection with that of Reference 100. The l a t t i c e dynamical data for the 48 K t h e o r e t i c a l c a l c u l a t i o n s were produced as described i n Section V.5. Since no low-temperature parameters were a v a i l a b l e f or the s h e l l model, room-temperature parameters were used, and then the frequencies were a l l increased by a f a c t o r of 1.05. This i s the average s h i f t observed both i n t h i s 131 experiment and i n the 80 K neutron d i f f r a c t i o n experiment. The isotope-induced one-phonon damping was calculated as described i n Section V.8, and i s shown i n Figure VII-3. I t accounts for 118 TABLE VII- I Constants Used In The Calculations (48 K). Observed TO resonance wave number v (cm *) 0 173.4 cl S t a t i c d i e l e c t r i c constant £ 0 5.50 High-frequency d i e l e c t r i c constant e 00 2.35 Madelung constant a 1.7475( Nearest-neighbour d is tance ' 3 r Q (10~ 8 cm) -2.7940 Sodium-ion mass M + (amu) 22.9898 C h l o r i n e - i o n mass M (amu) 35.453 C o m p r e s s i b i l i t y 3 ( 1 0 ~ 1 2 / b a r ) 3.885 R e p u l s i v e - o v e r l a p - C ( 1 0 ~ 1 0 erg) 21.71 p o t e n t i a l parameters p (10~ 8 cm) 0.310 Fourth p o t e n t i a l d e r i v a t i v e r o < f r » " » ( r o ) . ( 1 0 1 2 erg'cm 71.38 Th i rd p o t e n t i a l d e r i v a t i v e • , , f ( r o ) ( 1 0 1 2 erg•cm • 3 ) -6 .666 Second p o t e n t i a l d e r i v a t i v e <|>"(r)/r ( 1 0 1 2 o o erg•cm '3> 0.236 F i r s t p o t e n t i a l d e r i v a t i v e • • ( O / r J - ( 1 0 1 2 o 0 erg* cm 'V 0.268 See Reference 134 k See Reference 135 See Reference 139 119 • et o to 1-4 0.18 0.16 0.14 0:12 0.10 O Q Q.Oo 0 . 0 4 0 . 0 2 0 1 \ 1 1 1' ( r e x p t " Q 0 9 ) < (14) V 1 X 1 \! 1 : s » ( I 3 H t 4 8 K — • f — 1 — 1 \f 1 / \ J (10) * / 1 A/ A / v o ^ 1 i (16) Mb) 0 . 100 N U M B E R 2 0 0 3 0 0 ( c m " ' ) Figure VII-3. The calculated isotope-induced one-phonon damping of the transyerse o p t i c resonance i n natural NaCl at 48 K with 2 cm r e s o l u t i o n . The long dashed l i n e s are merely f o r guidance i n seeing the experimental peaks. 120 approximately 45% of the damping at v^, but i s almost negligible beyond 10 cm.* from v . The observed structure in r___ reflects the density of o ISO J states of the Cl ion. Despite their small relative magnitude at 48 K, peaks (10) and (11) are observed in the transmission experiment, which is also plotted in Figure VII—3 on the same scale as ^  n, but lowered .Lois by 0.09 cm * so as to f i t on the graph. A lower temperature (7 K) transmission-experiment, shows these peaks at 123.cm * and 145.5 cm ^  140 141 -1 more clearly. ' The frequencies measured at 48 K are 123 ± 2 cm and 144 ± 2 cm ^. The theoretical calculation gives peaks at 123 cm * and 146 cm ^. Peak (10) i s due to transverse acoustic phonons between and at L and W.(See Figure V.l for the NaCl dispersion curves). The transverse acoustic mode at L is 121.6 cm ^ from the l a t t i c e dynamical data used,-but only 120.5 cm ^ in the 80 K neutron experiment. Therefore, the theoretical peak (10) should be 122 cm ^. Similarly, peak (11) should be at 144 cm *, according to the 80 K neutron experiment. Peak (11) i s due to longitudinal acoustic phonons near and in the faces and th corners,of the irreducible 1/48 of the Brill o u i n zone, which do not have T as a corner. Peaks (12) and (13), at 167 cm * and 177 cm are due to transverse optical phonons at and between Q and W, and T and X, respectively. Peak (14) at 182 cm ^ i s due to transverse optical phonons at the L W K face, and longitudinal optical phonons from W to K inclusive. The sharp cutoffs at (15) and (16) correspond to the longitudinal optical phonons at X and T, at 190 cm ^ and 270 cm * respectively. The fast rise in from low wavenumbers up to peak (14). comes largely from the v 3 term in (V.35). From (15) to (16), i s low because the contributing longitudinal optical mode changes frequency rapidly throughout the Brillouin zone, so that the density of 121 states? is;-very,? low?;,. The result of the three-phonon damping calculation may be seen in Figure VII-4, where the difference process has been drawn, separately in the region in which the summation and difference processes overlap. Here, as with Figure VI-3, difference processes refer to the destruction of only one phonon, since the damping, due to processes involving the destruction of two phonons i s at most a negligible 0.0001 cm \ Comparing Figures VI-3 and VII-4, i t is obvious that the damping due to the difference process has gone down much more swiftly with a reduction in temperature than has the summation process. This i s to be expected from the temperature dependence of Equations (V.30) and (V.31). Although the difference damping is small, i t accounts for approximately 15% of the damping at V q . The feature assignments are the same as for the room-temperature calculation in Section VI.2, just shifted in wave number by the factor of 1.05. The individual peaks (7), (8), and (9) do not show up experimentally, but the drop of the three-phonon damping at higher wavenumbers than peak (9) at 538 cm * is observed. The measured damping spectrum and the spectrum calculated with two- and three-phonon and one-phonon isotope-induced processes are compared in...Figure VII-5. The feature assignments are the same as for the room-temperature calculation in Section VI.2, just shifted in wave number by the factor of 1.05. F i r s t , note that the wavenumber positions of the peaks do not agree precisely. This i s obviously due to the room-temperature neutron diffraction data used in the calculation. The transverse acoustic mode at L i s at 120.5 cm ^ i n the 80 K data of Reference 131. This implies a shift by the factor 1.04, since the room--1 temperature mode is at 115.8 cm . Because peak (1) is due to transverse 122 Figure VII-4. Measured damping spectrum of the transverse o p t i c resonance of NaCl at 48 K with 2 cm - 1 r e s o l u t i o n , together with the calculated three-phonon damping with equal r e s o l u t i o n . The long-dashed l i n e shows the separate , c o n t r i b u t i o n of the di f f e r e n c e process. The measured damping i s not shown below 300 wavenumbers. 123 E NUMBER ( c m " 1 ) Figure VII-5. Measured damping spectrum of the transverse o p t i c resonance of NaCl at A8 K with 2 cm - r e s o l u t i o n , together with the two- and three-phonon and isotope-induced one-phonon contributions to that damping, with equal r e s o l u t i o n . 124 acoustic phonons around.Q, near L, one expects this shift to produce peak (1) at 233.5 x 1.04 = 243 cm - 1. This agrees well.with the experimental value of 240 cm ~. The wave number of peak (2) can be deduced from the 80 K values of L 3 and L^: 120.5 + 142.3 = 262.8 ± 1 cm - 1, which agrees with the experimental value of 264 ± 0.5 cm ~. It i s very d i f f i c u l t to say what should happen to peak (3) from the 80 K neutron -data, and peak, (4); is too imprecise forfany meaningful; comparison. The 80 K values of at 0.4 and 0.6 in tiie (1,1,0) direction give: ((137.0 + 224.6) + (161.4 + 186.3))/2 = 355 ± 5 cm - 1, in agreement with the experimental value for peak (5) of 355 ± 1 cm 1. Thus, the frequencies of the peaks in the theoretical damping spectrum agree very well with the experimentally observed damping. The absolute magnitudes of these peaks, however, do not agree as well. This is at least partly to be expected, and is discussed further in Section VII.4. The resolution of the theory is made compatible with the experiment by convoluting the theoretical result with the spectral width function corresponding to the apodizing window function of Equation (11.53). The peak widths ((1) to (5)) agree well when limited only by this instrumental resolution. This implies l i t t l e damping of the "final-state" phonons at 48 K in the two-phonon damping region, yet there may be a f a i r amount of "final-state" damping in the three-phonon region, since peaks (7), (8), and (9) are not resolved experimentally. The lack of damping, theoretically, around 420 cm 1, is s t i l l seen at 48 K. If this were due to a four-phonon process involving the destruction of an acoustical mode, this discrepancy should have disappeared at 48 K. Therefore, this discrepancy may be due to one or more of the following: errors due to the neutron data; neglect of further neighbours in the coupling; neglect 125 of th'ev second-order dipblev moment; rieglect' of other contributions to 128 the,damping, from different diagrams (see Bruce ). Finally, the wavenumber shift i s calculated as outlined in Section V.3 from.the total damping spectrum. This is shown in Figure VII-6 together with the spectrum obtained from the measurements. As mentioned in Section VI.2, the optical properties are f a i r l y insensitive, to the wavenumber shift away from v , hence the measured spectrum i s quite noisy. It is.the damping spectrum which is important. VII.3 Optical Properties The directly measured amplitude and phase reflectance may be seen, together with the spectra calculated using the damping of Figure VII-5, i n Figures VIT-7 and VII-^8, respectively. The experimental error of the reflectance ranged from 2% at the maximum to 10% at the minimum and higher wavenumbers. The experimental error of the phase is approximately 1% in the transmission region below 167 cm * and above 340 cm *. It i s approximately 2% where the amplitude reflectance i s high (167 cm * to 270 cm and grows much larger around 320 cm ^ where the amplitude reflectance is very small. The overall agreement is excellent, yet the small discrepancies, particularly those corresponding to peaks (1) and (2) at 240 cm ^ and 264 cm ^ respectively, remain outside the experimental errors considered. It should be noted that in a l l of the figures, from Figure VII-7 onwards, the measured properties are plotted at least a l l the way from 25 cm * to 500 cm If the dashed line i s not v i s i b l e , i t is because i t completely overlaps the calculated f u l l l i n e . As mentioned in VII.l, the phase i s too close to zero to measure 126 00 {Q CM O CM • ST CD CO I I I I . ( . U P ) (A !°j*0) V 8 I Figure VII-6. Measured and calculated wave-number shift of the transverse optic resonance of NaCl at 48 K. The prime on A'(0,j ; v ) indicates that the theoretical shift has been set ?o zero at the observed resonance wave number v . o 127 0 100 2 0 0 3 0 0 4 0 0 500 W A V E N U M B E R ( c r n ~ ! ) Figure VII-7. Calculated and measured amplitude reflectance r of NaCl at 48 K. The power r e f l e c t i v i t y R = f f , where f = r exp(i(j)) . Figure VII-8. Calculated and measured r e f l e c t a n c e phase angle <j> of NaCl at 48 K, where the re f l e c t a n c e i s f = r exp(icf>). 129 accurately directly', below 167 cm 1 and above 340 cm ~\ so* that d i r e c t absorption c o e f f i c i e n t measurements were performed i n these regions. These are shown i n Figure VII-9 together with that obtained from di s p e r s i v e r e f l e c t i o n , and the measurements from Reference 123. The discrepancy i n Figure VII-5 at 420 cm Vis evident here. For completeness, the r e f r a c t i v e index i s shown i n Figure VII-10, and.j'the r e a l and-imaginary d i e l e c t r i c - constants, i n Figures VII-1'1 -: and VII-12, r e s p e c t i v e l y . Note that the measured r e f r a c t i v e index and imaginary d i e l e c t r i c constant become very inaccurate from 185 cm 1 to 235 cm ~, where the measured re f l e c t a n c e i s 100% w i t h i n the noise l e v e l . From Figure VII-11, v i s equal to 270 ± 1 cm 1 as opposed to the neutron -1 131 r e s u l t of 264 ± 3 cm . The agreement with the low-frequency r e a l * d i e l e c t r i c constant gives a value of 1.1)87 f o r the e f f e c t i v e charge e . VII.4 Discussion of the Disagreement i n the Magnitude of the Damping The discrepancies that seem to be outside the random and systematic errors already discussed, are mainly the magnitudes of the damping peaks (1), (2), and (3) i n Figure VII-5. There are many po s s i b l e problems with the t h e o r e t i c a l c a l c u l a t i o n , but there i s only one experimental point that should be considered ^ c a r e f u l l y : was the sample temperature r e a l l y 48 K? The sample and reference surface conditions and the p o s s i b i l i t y of a s i g n i f i c a n t s t r e s s e f f e c t a l l made l i t t l e e f f e c t , when considered. The change of o p t i c a l properties with sample temperature was the reason for low temperatures i n the f i r s t place, so n a t u r a l l y the temperature has a considerable e f f e c t on the r e s ultant damping. By 130 Figure VII-9. Calculated and measured absorption coefficient a of NaCl at 48 K. The measurements below 167 cm and above 340 cm are from transmission experiments. Good overlap with the dispersive reflection measurements was obtained. The circles are from Harrington et a l (Reference 123). Figure VII-10. Calculated and measured r e f r a c t i v e index n of NaCl at 48 K. -Figure VII-11. Calculated and measured r e a l part e 1 of the d i e l e c t r i c constant of NaCl at 48 K. 133 Figure VII-12. Calculated and measured imaginary part e " of the d i e l e c t r i c constant of NaCl at 48 K. 134 doing a series of theoretical calculations for different input temperatures, i t was found that the theoretical and experimental damping magnitudes would agree f a i r l y well i f the sample temperature was really"approximately 150 K. F i r s t of a l l , what i s the heat input to the sample? From the average liquid helium boiloff of ^ 0.8 l i t e r s per hour, we can set an extreme upper limit of ^ 0.6 Watts. From the fact that the boiloff i s ^  10% lower when the sample is shielded by the gate valve, we can set a very rough amount of 60 mW. Secondly, since the temperature sensor reads the sample's back face, could there be a temperature gradient with the front surface of the sample at 150 K, and the back surface at 48 K? Even taking the lowest thermal conductivity of NaCl in this temperature range, ^ 0.1 W«cm **deg \ the heat input would have to be an incredible 500 Watts. This implies that the sample must be sufficiently uniform in temperature for our purposes. Thirdly, could the thermometer be thermally isolated from the sample and thermally sunk to the copper backing plate? The thermometer diode leads are thermally isolated from the diode case by design, and approximately 20 cm of 32 AWG brass wire leads also isolate the diode before the leads are f i r s t taped to the helium can. This gives a very high thermal resistance of greater than 10"* deg'W ^. The diode is held against the sample by a steel spring, and i t s thermal resistance w i l l be more important. Taking a large upper limit of 0.4 W*cm ''•deg ^ for the spring's thermal conductivity, and assuming the worst case of perfect thermal contact a l l along the end co i l s , the length to cross-sectional area ratio i s ^  2000, and the thermal resistance i s much greater than 5000 deg*W ^. The thermal resistance between the temperature sensor and the sample was calculated assuming a 0.005 cm thick layer of grease covering half the 135 surface of the sensor pressed, against- the sample. This light grease layer was compacted by rotating the sample onto the sensor during the mounting of the sample. A lower limit thermal conductivity of 0.001 W*cm k*deg ^ was taken for the- grease (extrapolated from Reference 142), since the grease would be at least 48 K. Since the diode surface is 3/16" in diameter, the thermal resistance must be less than 50 deg'W Even these extreme' limits imply that* the;, measured temperature should be within half a degree of the actual sample temperature. One f i n a l check of the sample's temperature can be made by comparing the measured reflectance with published power reflectance measurements. This i s done in Figure VII-13. The results from Reference 143 were obtained with a grating instrument at 12° incidence, with a temperature accuracy of about five degrees and a power reflectance accuracy of about one percent. Comparing the temperature dependence of the reflectance at feature (1) and the frequency corresponding to a reflectance of 0.8, i t i s obvious that the sample temperature i s below 100 K. The accuracies involved in the two experiments and the transposition of the data, however, prohibit inferring the temperature more precisely by this method. The possible errors in the theoretical calculations include the omissions of further neighbours i n the coupling, the second-order dipole moment, damping of "final-state" phonons, and other contributions to the damping. There are also possible errors in the input parameters used. The transverse optic phonon frequency used was 173.4 ± 0.5 cm * from e" measured in this experiment. This agrees well with the neutron value of 172.5 ± 1 cm * from Reference 131, and 174 cm ^ from Reference 141. The real dielectric constant at low and high frequencies, and respectively, was taken from the accurate data of Reference 134. 136 UJ < o UJ LL, UJ Q ZD -1 0 .3 a . 0.2 0 PRESENT EXPERIMENTS REF. 143 era tw esse tsa <s$ 3 0 0 K C E S S K E O i X S S C m «ss 2 0 0 K 100 K _l _1 2 1 0 W A V E N U 2 5 0 2 7 0 ER ( c m - 1 ) Figure VII-13., Comparison of amplitude refl e c t a n c e measurements at various temperatures i n NaCl i n the v i c i n i t y of peaks (1) and (2). The data from Reference 143 are from power re f l e c t a n c e measurements at 12 incidence. 137 These values of 5.500' ± 0.025. and 2.350 ± 0.001 respectively, agreed with the experimental values of- 5.5.0 + 0.15 and 2.35 ± 0.02. By computing the changes in the parameters V Q , E q and required to make the damping spectra of Figure VII-5 agree, i t is clear that they would have to have values far outside the error limits, just discussed. Of the other parameters in Table. VII-I, only the compressibility and the potential parameters are: likely>to 1 "be: causing problems. Since the 2 damping is approximately proportional to ((•'''.(r ) , only a small error in this parameter w i l l cause a big effect on the damping, and complete agreement with the experiment would require that ^'''(r ) be unchanged from i t s room temperature value. The f i n a l , and possibly most serious deficiency in the present calculation, i s the l a t t i c e dynamical data used. It has already b een noted that scaling the room temperature frequencies does not give agreement of the frequencies of the features in the damping spectrum. Likewise, the eigenvectors, which have not been corrected at a l l for low temperature, could easily be sufficiently in error to cause significant errors in the damping magnitude. 138 CHAPTER VIII CONCLUSIONS V I I I . l The Dispersive R e f l e c t i o n Spectrometer A d i s p e r s i v e r e f l e c t i o n spectrometer f o r the f a r i n f r a r e d has been designed, b u i l t and tested f o r t h i s t h e s i s . I t consists of a modified.commercially-available Michelson interferometer, and enables o p t i c a l property measurements from a few wave numbers to 1000 wave numbers, at sample temperatures at l e a s t as low as 25 K and as high as 300 K, with no need for the often inaccurate Kramers-Kronig a n a l y s i s . As t h i s l a s t chapter i s being written, other approaches f o r producing such an instrument are appearing i n the l i t e r a t u r e ^ " 4 ^ and therefore these conclusions on the instrumental performance are wr i t t e n i n comparison with these other approaches. One basic consideration i s the convergence of the beams i n the interferometer arms. The standard commercially-available instruments use collimated beams, but some experimenters have custom-built interferometers with converging beams, to i r r a d i a t e small samples. In p r a c t i c e , t h i s does not increase the throughput a l l that s i g n i f i c a n t l y , except f o r very small samples. Furthermore, t h i s gives r a d i a t i o n that i s farthe r from normal incidence, longer pathlengths and lower r e s o l u t i o n . For re s o l u t i o n s of about one part i n a thousand or better, the collimated-beam approach, used i n t h i s t h e s i s i s at l e a s t as good f o r any s i z e sample. 139 A second-major consideration' Is" the method for obtaining an instrumental background spectrum. The method employed i n t h i s t hesis involves using aluminized portions of the sample surface as a reference surface, but i t should be noted that t h i s does not preclude also making measurements with the other method of completely r e p l a c i n g the sample with a separate r e f l e c t o r . The p a r t i a l l y - a l u m i n i z e d sample method, as implemented i n t h i s t h e s i s , -has: the advantages of: absolute -phase determination to high accuracy; the ratioing-out of many o p t i c a l asymmetries i n the instrument and sample surface conditions such as f l a t n e s s and scratches; the r a t i o i n g - o u t of long term d r i f t s i n components of the system, eliminated, by the switching method of obtaining two interferograms. almost simultaneously. It.has been shown to be an excellent method for measuring.-optical properties of samples down to approximately 2 cm i n diameter, below which mask alignment becomes d i f f i c u l t and i n e f f i c i e n t . For smaller samples, the mirror s u b s t i t u t i o n method can be f a i r l y e a s i l y employed, since i t i s easier to preserve o p t i c a l q u a l i t y over a smaller area. There are more chances of systematic errors than with the switching method, and the r e l a t i v e p o s i t i o n s of the sample and reference surfaces must be obtained by l e s s r e l i a b l e methods. An a d d i t i o n a l f a c t o r at low temperatures i s the adherence of the evaporated aluminum f i l m to the sample with the switching method. Since on some samples the aluminum does not come o f f , i t seems that with proper precautions t h i s problem can be avoided. T h i r d l y , a choice of down-looking or side-looking dewar o p t i c s must be made. The side-looking o p t i c s i n t h i s t hesis allow much more compact optics than those using an extra mirror for down-looking o p t i c s . The only advantage to the l a t t e r i s for very small samples where no 140 i n d i r e c t method of measuring.the absolute phase at some higher frequency point i s p o s s i b l e . In. that, case, g r a v i t y held,samples are used, r e q u i r i n g the down-looking con f i g u r a t i o n . A fourth important consideration i s the choice of a sample mount for low temperatures. The simplest approach was used for t h i s t h e s i s , with mechanical-contact cooling, a mylar 77 K filter-window only, and a large-apferture of 5V9 cm. This allows/measuremerits down to -at l e a s t 25 K, with easy alignment, and with the highly, .accurate switching arrangement. Compromising the signal-to-noise r a t i o s l i g h t l y , with smaller samples, and apertures, and using e l e c t r i c a l alignment (maximizing the detected output) with black polyethylene windows can be used f o r lower temperatures. VIII... 2 The Far Infrared O p t i c a l Properties of NaCl I t has been w e l l recognized that quartic anharmonicity must be included i n the room-temperature damping of the transverse o p t i c resonance i n the a l k a l i - h a l i d e s . This i s done for the f i r s t time i n t h i s study, and the d e t a i l e d form of the quartic-coupling c o e f f i c i e n t that i s derived i s presented. The strong r o l e of the transverse o p t i c mode i n t h i s three-phonon damping i s observed. The advantage of obtaining good high-resolution experimental data throughout the spectrum i s that the l i f e t i m e broadening of the two or three phonons to which the transverse o p t i c mode relaxes then becomes evident. This broadening'can be determined from the experimental data and incorporated i n t o the c a l c u l a t i o n s . The end r e s u l t at room temperature i s excellent o v e r a l l agreement and we be l i e v e represents the best agreement between 141 experimental, measurements, and- ab i n i t i o t h e o r e t i c a l c a l c u l a t i o n s of the f a r - i n f r a r e d o p t i c a l properties of any m a t e r i a l . 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The f i r s t section of the c i r c u i t , shown in Figure A - l , produces an output which is an amplified function of the sensed intensity. The photodarlington detector presently used is biased at a constant 3 Volts. At the desired intensity level, the current sunk by this detector creates an additional 3 Volt drop -across the 330 ti resistor and the level potentiometer, producing a 6 Volt output of amplifier A l . The 6 Volt reference voltage i s doubly regulated and thermally stable. Amplifier A2 has a variable gain, which allows an overall controller gain from 25 to 10. 000. Its output drives an optoisolator at a nominal 5 Volts seen at the "monitor" output. The second section of the c i r c u i t , shown in Figure A-2, provides and controls the power to the IR-7L mercury-vapour lamp. The opto-isolator isolates the line voltage circuitry from the intensity sensor circuitry. The line i s full-wave rect i f i e d and capacitively f i l t e r e d to provide a rough DC supply (330 VDC open c i r c u i t ) . The 100 ft 100 W resistor adjusts to provide an average current of 0.95 Amps. The other part of the lamp's ballast consists of the incandescent light bulbs and the parallel path through the 300 ti 20 W resistor and the TIP 30B power Figure A - l . The i n t e n s i t y sensor c i r c u i t of the mercury-arc source i n t e n s i t y c o n t r o l l e r . Figure A-2. The power control circuit of the source controller. 153 t r a n s i s t o r . The l a t t e r gives, a 10% c o n t r o l on the source i n t e n s i t y , and i s i t s e l f c o n t r o l l e d by the 2N 4410 t r a n s i s t o r and the o p t o - i s o l a t o r . The 75 Volt zener diode prevents the TIP 30B from exceeding i t s voltage c a p a b i l i t y . The meter i s set up to read the current through the lamp, or the con t r o l voltage. When preparing to s t a r t a run, the c o n t r o l voltage i s normally set with the l e v e l c o n t r o l to be i n the centre of i t s range. A voltage doubler.is also provided, which w i l l supply approximately 600 V o l t s , t o t a l , to the lamp, to help s t a r t the arc. 154 APPENDIX B ELECTRONICS FOR ALTERNATE SAMPLING OF INTERFEROGRAMS As described in Section III.3, crosstalk, between adjacent data points, must be eliminated for the switching; method. The circuits... described in,this appendix help avoid crosstalk, as well as providing f i l t e r e d and demodulated outputs, and different gains for the two channels. The general signal flow is shown in the block diagram of Figure B-l. The signal proportional to the detected intensity comes in from the lock-in amplifier to a buffer amplifier on the integrator board at I/B. F6r channel B, this signal i s passed straight to I/H and the.A or B output. For channel A, the signal i s f i r s t offset by up to ±10 Volts, then amplified by a factor between 0 and 20, before appearing at I/H. The potentiometer before F/13 then allows an overall gain between 0 and 2.5. The signal then passes through a Bessel f i l t e r of 48 dB/octave, with a delay of approximately 0.1 seconds, which reduces i t s output to 1% of the input at 15 Hertz. This low-pass f i l t e r i s primarily for direct signal monitoring at the f i l t e r output. From the f i l t e r , the signal i s passed through a resistor which determines the integration time, into the integrator at i/E. The integrated output at I/D passes to an output for the -7.5 V full-scale FS-720 voltmeter, and also through a -4/3 amplifier to 1/15. This gives a 10 V full-scale output which goes to S/7. Then, on the sample-and-hold board, the two channels are separated and held for the two outputs S/A and S/B. i 155 Figure B - l . Integrator and a l t e r n a t i n g system general layout. 156 The details of the integrator, f i l t e r , and sample-and-hold boards are shown in Figures B-2, B-3, and B-4 respectively. The analog switching i s done by AD7511 CMOS switches controlled by logic from the FS-720's step drive and sampling control. The logic of Figure B-5 produces the A or B channel control signals, inhibits the step drive every other sampling period, and produces drive voltages for the switching solenoid. The. timing diagram \for 'this i s shown in Figure B-6. Starting from the bottom, an internal f l i p - f l o p produces the signal to SK3/B in the FS-720. This gives a dead time of 63/120 seconds, and an integration time or sampling time (S.T.) of : S.T. = 8a / 15 - 1 / 120 seconds (B.l) where a = 1, 2, 4, 8, 16, 32, or .64. The integrator i s zeroed by the signal at SK3/A, after the d i g i t a l voltmeter has sampled and punched the result. The signal i s also sampled and held by either channel A or B before the integrator i s zeroed. The sampling time i s set at ^  0.09 seconds by the MC14528 monostables shown in Figure B-4. The alternating from sample to reference and back i s controlled by the CD4013 D-flops . of Figure B-5. They can be used in either a symmetric mode in which the channel order for sampling stays the same after each step of the moving mirror, or in an asymmetric mode in which the switching mask only has to move half as often. In this work, the symmetric mode was always used. The buffered D-flop outputs then pass through SK3/D and SK3/E to the reverse switch in Figure B-l, which allows either the sample or the background to be defined as channel A. A voltage-doubling power supply and some power transistors drive the switching solenoid through Figure B-2. Integrator board. Includes the integrator, a 4/3 analog scaler, an offset and gain for channel A, and analog switches. Figure B-3. The eight-pole a c t i v e Bessel f i l t e r , with a buffered input. 3OOK £ O-HT/iF —> 001 sec. Figure B-4. The two-channel sample-and-hold c i r c u i t . •r i l l - j ^ to 160 If •4 Figure B-5. A d d i t i o n a l l o g i c f or a l t e r n a t e sampling. 161 5KZA1 SK3A i£ no r~~ B m e Pvcses TR.t<S6Efi-—4>JL U_ H no n Sot-EM>rt OFF INT-ZERO /AfT So u£A/o I PULL. SACK & b-Vrf. SAMPLE in K I u: lA)T£6Mr£- & 7tAo >A>T-W T £ 0>K*T£ l*/T£&A./1T£. H >k IZO  1 5.77 6 3 1X0 5 . T . ~TtME. (seconds) 0 15 0 15 0 0 -24 -50 ]o 11 0 11 0 11 0 11 0 11 0 r-3 o > Figure B-6. Timing diagram f o r al t e r n a t e sampling (symmetric). 162 SK2/H. The XR320 monostable h o l d s 5 0 Volts across the solenoid for the, f i r s t ^ Jg. second in order to pul l in the solenoid's plunger. The plunger i s then held in position for the rest of the sampling interval By 24 Volts. 163 APPENDIX C EFFECT OF NON-IDEAL REFERENCE SURFACE The reference surface, as mentioned i n Section VI.1, i s a vacuum-evaporated aluminum coating, of at l e a s t 50 nm thickness. As shown i n Section I I . 3 , / t h i s thickness (x'/2) as w e l l as the, aluminum's reflec t a n c e g (v) from (11.43) must be taken into account i n order Mirror to c a l c u l a t e properly the sample's r e f l e c t a n c e . This r e f l e c t a n c e can be calculated f o r such a p a r a l l e l metal slab i n which m u l t i p l e r e f l e c t i o n s can occur, i f the thickness and the complex conduc t i v i t y 6 are known f o r the bulk metal. The Drude theory has been shown to be s u f f i c i e n t l y accurate for 133 predicting.power r e f l e c t a n c e i n the mid- to f a r - i n f r a r e d . From t h i s theory: n 2 - k 2 = 1 -.4iTcr T / (1 + OJ 2T 2) ( C l ) 2nk = 4 T r 0 o / ( a i ( l + o) 2x 2)) (C.2) where a i s the DC conductivity, taken as 3.18 x 1 0 1 7 s _ 1 f o r aluminum -14 at room temperature, x i s the s c a t t e r i n g time taken as 0.801 x 10 s, and to = 2TTCV. AS shown i n Appendix D, n and k can then be c a l c u l a t e d from ( C l ) and (C.2). The r a t i o of the r e f l e c t e d f i e l d to the incident 137 f i e l d i s given by: 164 E , ict 6 - i a -BN  r _ (e e - e e ) (1 - n^) E ± e i a e B ( l + f i ) 2 - e " " i a e - 3 ( l - n ) 2 (C.3) where: a = imx v (C.4) 8 = i rkx 'v (C.5) where x' i s i n centimetres. The required amplitude r e f l e c t a n c e f o r (11.43) i s : r (v) = r o Mir, ( e e - e e ) 2 + 4sln 2c l(e e-r 2e V + 4 r 2 s i n 2 (a+<j> ) (C.6) where the amplitude and phase r e f l e c t i o n of the bulk metal are r e s p e c t i v e l y : ( r = 00 ( n - l ) 2 + k 2 (n+1) 2 + k 2 <(> = arctan 2k n 2 + k 2 - 1 ( C 7 ) (C.8) The required phase r e f l e c t a n c e f o r (11.43) i s : <|> (v) = arctan Mir xm - yl [pil- - ymj (C.9) 165 where: 8 -6 x = e (s cos a - t s i n a) - e (u cos a + v s i n a) (CIO) 6 —8 y = e (t cos a + s s i n a) - e (v cos a - u s i n a) ( C . l l ) I = (l-n 2H-k 2).cos(a) ( e S - e _ B ) - 2nk siri(a) (e 6+e - 6) (C.12) m = 2nk c o s ( a ) ( e B - e _ B ) + ( l - n 2 + k 2 ) s i n ( a ) ( e ^ + e - 6 ) (C.13) s = n 2 - k 2 + 2n (C.14) t = -2k - 2nk (C.15) u = n 2 - k 2 - 2n (C.16) v = 2k - 2nk (C.17) 166 APPENDIX D RELATIONS BETWEEN OPTICAL CONSTANTS S t a r t i n g from the complex sample r e f l e c t a n c e : f ( v ) = r ( v ) e x p ( i 9 r ( v ) ) (11.26) and o m i t t i n g the (v) f o r s i m p l i c i t y but keeping the wavenumber dependence i m p l i c i t , the v a r i o u s o p t i c a l constants f o r non-magnetic m a t e r i a l s (u r=l) can be r e l a t e d as f o l l o w s . For the complex r e f r a c t i v e index n: fi- = n + i k (D.l) n = (1 - r 2 ) / ( l + r 2 + 2r c o s > r ) (11.31) k = (-2r s i n <j>r)/(l + r 2 + 2r cos 4>^) (11.32) and the in v e r s e r e l a t i o n s : r = (({1-n} 2 + k 2 ) / ( { l + n } 2 + k 2 ) ) ^ (11.29) <f>r » a r c t a n ( 2 k / ( n 2 + k 2 - 1)) (11.30) The complex d i e l e c t r i c constant t and the complex c o n d u c t i v i t y 6 are r e l a t e d to the above by: 167 ft2 = t +• ±26/ (cv) (D.2) where: fi • = e' + i e " (D.3) For non-conductors, such -as t h e ' a l k a l i - h a l i d e s , 6. -= 0 and -therefore: e ' - n 2 - k 2 (D.4) e" = 2nk , " (D.5) and the inverse r e l a t i o n s : n = ((e' + {e' 2+e , , 2} J 5)/2) 1' S (D.6) k = ((-£' + W2+e"2}h)/2)h (D.7) Since the samples used are much la r g e r than the wavelength, consider a plane wave i n the x d i r e c t i o n : E = E o exp(i2-rrv(nx-ct)) exp (-2iTkvx) (D.8) The inte n s i t y , w i l l be: 1 = < $ > t = 8T7 R e ( ^ * (D.9) 168 (D.10) and the absorption, c o e f f i c i e n t a comes from the exponential decay term: a = 4irkv (D.ll) and'the i n v e r s e * r e l a t i o n i s : k = a / ( 4T r v ) (D.12) 1 = 1 7 E o e x P ( ~ A l T k ^ x ) For a resonance such as the transverse o p t i c a l resonance i n NaGl, the complex phase s h i f t i s r e l a t e d . t o the d i e l e c t r i c constants by: g = em + ( e Q - e o o)/(l - ( v / v n ) 2 + 2 ( A , - i r ) / v J , CD.. 13) where e and e are the r e a l d i e l e c t r i c constant i n the low and hi g h frequency l i m i t s r e s p e c t i v e l y . From t h i s : r = V"<Eo - O 2(e" 2 + ( e ' - O 2 ) (e -e )(£'-£)• _ — o o + (V /V )• e + (E -E ) - 1 (D.14) (D.15) and the i n v e r s e r e l a t i o n s : E ' = £ + (2r/v )2,+ (l-(v/v )2+2A'/v I 2 (D.16) 169 (e -e )2T/v 0 oo 0 ( 2 r / v J 2 + ( l-(v/v o) 2+2A'/vJ 0J ( D . 1 7 ) 

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