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UBC Theses and Dissertations

Localized modes and the Mossbauer effect Wells, David Ernest 1965

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LOCALIZED MODES AND THE MOSSBAUER EFFECT by DAVID ERNEST WELLS B.So. Mount A l l i s o n U n i v e r s i t y , I 9 6 I B.A.Sc. U n i v e r s i t y of B r i t i s h C o lumbia, 1963 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE IN THE DEPARTMENT OF PHYSICS We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA November, I965 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of Br i t ish Columbia, I agree that the Library shall make it freely available for reference and study. I. further agree that per-mission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives„ It is understood that copying or publ i -cation of this thesis for f inancial gain shall not be allowed without my written permission. Department of Physics  The University of Br i t ish Columbia Vancouver 8, Canada Date December•1, I 9 6 5 ABSTRACT Two t y p e s of e x p e r i m e n t s i n v o l v i n g t h e Mossbauer atom as a d i l u t e i m p u r i t y i n a h o s t l a t t i c e a r e d i s c u s s e d . F o r a zero-phonon experiment w i t h Fe^? i n a t room t e m p e r a t u r e , t h e e x p e c t e d s h i f t of t h e c e n t r a l Mossbauer peak i s Q/3OO0 The minimum time r e q u i r e d t o e x p e r i m e n t a l l y d e t e r m i n e t h i s s h i f t t o w i t h i n 10$ i s f o u n d t o be 16 weeks of c o u n t i n g w i t h a 5 m i l l i c u r i e s o u r c e F o r a one-phonon experiment w i t h Fe^? i n P t ^ 9 5 f the count r a t e due t o resonance s c a t t e r i n g i s f o u n d t o be 3«i*xlC'~^/sec., and t h e count r a t e due t o R a y l e i g h s c a t t e r i n g 2 . 2 x l 0"^/sec o The case of Fe^ i n Be^ i s a l s o d i s c u s s e d . An a i r t r o u g h Mossbauer s h i f t s p e c t r o m e t e r c o n s t r u c t e d t o p e r f o r m the zero-phonon experiment i s d e s c r i b e d . V i b r a t i o n s p r e s e n t i n t h i s a p p a r a t u s , making i t i n a d e q u a t e f o r e x p e r i m e n t a l work, a r e d i s c u s s e d . TABLE OF CONTENTS , CHAPTER ONE: A SUGGESTED EXPERIMENT ON LOCALIZED MODES A. LOCALIZED MODES B. THE MOSSBAUER EFFECT C. ZERO-PHONON EXPERIMENTS CHAPTER TWO: THE DESIGN OF A SPECIFIC EXPERIMENT A. SOURCE CONSIDERATIONS B. ABSORBER CONSIDERATIONS C. COUNTING STATISTICS D. F e 5 7 IN P t 1 9 5 E. F e 5 ? IN B e 9 CHAPTER THREE: AIR-TROUGH MOSSBAUER SHIFT SPECTROMETER A. MECHANICAL DRIVE B. AIR TROUGH SUSPENSION C. VELOCITY MONITORING SYSTEM D. CALIBRATIONS OF THE AIR SUPPLY AND THE DRIVE . MECHANISM CHAPTER FOUR: PERFORMANCE OF THE AIR-TROUGH SUSPENSION A. OBSERVED PERFORMANCE OF THE AIR-TROUGH SUSPENSION B. LASER FLUCTUATIONS C. MECHANICAL VIBRATIONS D. AIR SUSPENSION INSTABILITIES E. FLUID MECHANICAL CONSIDERATIONS APPENDIX ONE A. £>0 B. £>0 c e>o D. £>0 FREQUENCY AND LIFETIME OF LOCALIZED MODES FREQUENCY OF LM IN HARMONIC APPROXIMATION FREQUENCY OF LM FOR ANHARMONIC FORCES LIFETIME OF LM DUE TO ANHARMONIC FORCES LIFETIME OF LM DUE TO IMPURITY CONCENTRATION Page 1 1 5 7 11 11 17 19 23 24-27 . 27 33 35 38 43 43 45 47 47 51 54 54 57 58 60 Page Eo 6<0; FREQUENCY AND LIFETIME OF THE LM 6 l F, NON-ISOTOPIC IMPURITIES 62 G. ENERGY AND LINEWIDTH OF LM 63 APPENDIX TWO: ONE-PHONON EXPERIMENTS 66 A. SOURCE AND ABSORBER CONSIDERATIONS 6? B. DRIVE CONSIDERATIONS 68 C. RAYLEIGH SCATTERING 71 D. F e 5 7 IN P t 1 9 5 72 E. F e 5 7 IN B e 9 75 APPENDIX THREE: DIFFUSION OF A RADIOACTIVE IMPURITY INTO A HOST 78 A. DIFFUSION INTO AN INFINITE CYLINDER 78 B. DIFFUSION INTO A NON-INFINITE CYLINDER 80 C. RESULTS FOR F e 5 7 IN P t 1 9 5 82 APPENDIX FOUR: CONSTRUCTION OF A MOSSBAUER TRANSDUCER DRIVE 84 BIBLIOGRAPHY 96 LIST OF FIGURES Page F i g u r e 2-1: DECAY SCHEME OF F e 5 7 12 F i g u r e 2-2: CALCULATION OF SMALL SHIFTS 20 F i g u r e 3-1: BLOCK DIAGRAM OF APPARATUS 28 F i g u r e 3-2: MECHANICAL DRIVE - MECHANICAL DETAILS 30 F i g u r e 3 - 3 : MECHANICAL DRIVE - CONTROL CIRCUIT 32 F i g u r e 3-4: SIGNAL GENERATOR CONTROL CIRCUIT 34 F i g u r e 3-5: AIR TROUGH SUSPENSION 36 F i g u r e 3-6: AIR SUPPLY 37 F i g u r e 3-7: VELOCITY MONITORING SYSTEM 39 F i g u r e 3 - 8 : CALIBRATION OF AIR SUPPLY 41 F i g u r e 3 - 9 : CALIBRATION OF DRIVE MECHANISM 42 F i g u r e 4 - 1 : PERFORMANCE OF AIR-TROUGH'- I 44 F i g u r e 4-2: PERFORMANCE OF AIR-TROUGH - I I 46 F i g u r e 4 - 3 : FOURIER ANALYSIS OF HARMONIC VIBRATIONS 49 F i g u r e A l - 1 : LATTICE FREQUENCY DISTRIBUTIONS 56 F i g u r e A2-1: ONE-PHONON EXPERIMENT - GEOMETRY 69 F i g u r e A2-2: LATTICE FREQUENCY FUNCTION USED TO ESTIMATE f L M 73 F i g u r e A4 - 1 : MOSSBAUER TRANSDUCER BLOCK DIAGRAM 85 F i g u r e A4-2: TRANSDUCER DRIVE - MECHANICAL 86 F i g u r e A 4 - 3 : TRANSDUCER DRIVE - ELECTRONICS 87 F i g u r e A 4 - 4 : UNREGULATED POWER SUPPLY 88 F i g u r e A4-5: POSITIVE POWER SUPPLY (PCB1) 89. F i g u r e A4 - 6 : NEGATIVE POWER SUPPLY (PCB2) 90 F i g u r e A4 -7 : DRIVE AMPLIFIER (PCB3) 91 F i g u r e A4 r8: LINEAR AMPLIFIER (PCB4) 92 F i g u r e A4 - 9 : LINEAR AMPLIFIER (PCB5) 93 F i g u r e A4-10 : OUTPUT AMPLIFIER (PCB6) 94 F i g u r e A4-11 : TRIGGER CIRCUIT (PCB7) 95 LIST OF TABLES T a b l e 2-1 T a b l e 2-2 T a b l e 2-3 Page 13 14 PROPERTIES OF F e 5 7 PROPERTIES OF P t 1 9 ^ AND B e 9 ZERO-PHONON EXPERIMENT CALCULATED RESULTS ( F e 5 7 IN P t 1 9 5 AND B e 9 ) 26 T a b l e A l - 1 : LOCALIZED MODE PARAMETERS ( F e 5 7 IN P t 1 9 ^ AND B e 9 ) 65 T a b l e A2-1: ONE-PHONON EXPERIMENT CALCULATED RESULTS ( F e ^ 7 IN P t 1 9 5 AND B e 9 ) 77 T a b l e A3-1: I ( s ) / l ( o ) AS A FUNCTION OF Y (SEE EQUATION A3-6) 81 ACKNOWLEDGEMENTS The a u t h o r would l i k e t o thank Dr. B.L„ Wh i t e , w i t h o u t whose s u p e r v i s i o n and depth of u n d e r s t a n d i n g , both p h y s i c a l and o t h e r w i s e , t h i s t h e s i s would n e v e r have been completed. F i n a n c i a l a s s i s t a n c e from t h e B r i t i s h Columbia Hydro and Fower A u t h o r i t y i n t h e form of two Graduate S c h o l a r s h i p s I s g r a t e -f u l l y acknowledged. F i n a l l y , I w i s h t o t a k e t h i s o p p o r t u n i t y t o thank my g r a n d f a t h e r , D a v i d E r n e s t B l a c k , f o r t h e u n f a i l i n g s u p p o r t he has g i v e n me th r o u g h o u t my y e a r s of u n i v e r s i t y t r a i n i n g . "«.„When you f i r s t c o n s i d e r t h e problem of making any of t h e s e c o n s t a n t v e l o c i t y o r c o n s t a n t a c c e l e r a t i o n d r i v e s , you t h i n k t h e whole problem i s t r i v i a l and can be s o l v e d w i t h $100 and a week's t i m e , so you t h i n k your b i g g e s t problem i s g o i n g t o be t o make a s o u r c e and scatterers<> Then you get your s o u r c e and s c a t t e r e r s , and s i x months l a t e r you a r e s t i l l t r y i n g t o get t h e d r i v e work-i n g , because your f i r s t a t t e m pt had 50% v i b r a t i o n s , w h i c h you would l i k e t o reduce t o 10%„ Then you r e a l l y would l i k e t o reduce them t o 1%0 When you a r e f i n i s h e d you say, "You know, i t r e a l l y was trivial"„ You d e s c r i b e how you p l a c e d t h e s o u r c e on t h e c h e s t of a g r a d u a t e s t u d e n t , h a v i n g found t h a t h i s c h e s t beat up and down w i t h c o n s t a n t a c c e l e r a t i o n , and t h e n you j u s t p ass o v e r t h e problem as i f i t were non-existant.„." (L . G r o d z l n s , p„ 52 i n Compton and Schoen, 1962) CHAPTER ONE A SUGGESTED EXPERIMENT ON LOCALIZED MODES The Mossbauer E f f e c t (Mossbauer, 1958, 1965; F r a u e n -f e l d e r , I 9 6 3 ; B o y l e and H a l l , 1962; Compton and Schoen, 1962; Bearden, 1964; Werthelm, 1964; Woodrow, 1964) c o m p r i s e s gamma r a y r e s o n a n t e m i s s i o n and a b s o r p t i o n p r o c e s s e s which i n v o l v e t h e p r o p e r t i e s o f t h e l a t t i c e i n which t h e Mossbauer atom* i s bound. These p r o p e r t i e s may be s t u d i e d by o b s e r v i n g t h e i r e f f e c t on t h e Mossbauer resonance spectrum. T h i s c h a p t e r d e a l s s p e c i f i c a l l y w i t h t h e case when t h e Mossbauer atom i s an im-p u r i t y i n t h e l a t t i c e . We b e g i n by c o n s i d e r i n g t h e p r o p e r t i e s of l o c a l i z e d modes (LM), and how t h e Mossbauer spectrum i s a f f e c t e d by them. We t h e n d i s c u s s the d e s i g n of a s p e c i f i c e x p e r i m e n t . A. LOCALIZED MODES The v i b r a t i o n s of a p e r f e c t l y r e g u l a r and harmonic c r y s t a l l a t t i c e can be r e s o l v e d i n t o t r a v e l l i n g waves ( t h e normal v i b r a t i o n s of t h e l a t t i c e ) . I n the Debye a p p r o x i m a t i o n t h e s e form a f r e q u e n c y band w i t h a maximum f r e q u e n c y CJ 0, ( t h e Debye f r e q u e n c y ) , and w i t h a d j a c e n t f r e q u e n c i e s s e p a r a t e d by o r d e r ^Jp/N (N I s t h e number of atoms i n the l a t t i c e ) . *(Terms such as "Mossbauer a t o m , - I m p u r i t y , - s o u r c e , - e m i s s i o n , -resonance,-peak,-spectrum" w i l l be used i n t h i s t h e s i s as s h o r t hand f o r "gamma r a y re s o n a n t - " ) . - 2 -C o n s i d e r a o n e - d i m e n s i o n a l c h a i n of atoms of mass M h o s t w i t h o n l y harmonic f o r c e s a c t i n g between n e a r e s t n e i g h b o u r s . L e t t h e r e be one i s o t o p i c i m p u r i t y atom of mass M j ^ p u r i t y s i t u a t e d a t t h e o r i g i n of c o o r d i n a t e s . (An i s o t o p i c i m p u r i t y i s a p u r e l y mass d e f e c t , t h e f o r c e c o n s t a n t s r e m a i n i n g unchanged). We d e f i n e t h e f r a c t i o n a l mass d e f e c t (1-1) f^MOST F o r a l i g h t i m p u r i t y (6>0 ) M o n t r o l l and P o t t s (1955) have shown t h a t t h e normal modes d i v i d e i n t o two c l a s s e s ; t h o s e whose f r e q u e n c i e s s u f f e r s l i g h t v a r i a t i o n s i n t h e band (which can be d i s c u s s e d u s i n g p e r t u r b a t i o n t h e o r y ) , and t h o s e whose f r e q u e n c i e s a r e d i s p l a c e d out of the band (and must be a n a l y z e d by o t h e r means). A l l t h e normal mode f r e q u e n c i e s except t h o s e a t t h e upper edge of the band b e l o n g t o t h e f i r s t c l a s s , and a r e r a i s e d by o r d e r (Jr/N. T h i s i s a v e r y n e g l i g i b l e change s i n c e N i s of o r d e r 1 0 2 ^ . However, the h i g h e s t normal mode f r e q u e n c y o f t h e p e r f e c t l a t t i c e , (*) D, i s r a i s e d out of t h e band t o become a new d i s c r e t e f r e q u e n c y (J L^^CJ 0. T h i s d i s c r e t e f r e q u e n c y appears o n l y f o r £ ^ 6 c r i t i c a l ( N a r d e l l i and T e t t a m a n z i , 1962 eqn 14) where £ c r i t i c a l depends on th e a t o m i c arrangement ( V i s s c h e r , 1964). As € i n c r e a s e s , Q L M i s d i s p l a c e d f u r t h e r from 6)^ . Appendix One r e v i e w s some c a l c u l a t i o n s made on Omas a f u n c t i o n of e . I f one a t t e m p t s t o d r i v e a c r y s t a l a t a f r e q u e n c y o u t -s i d e the band of normal mode f r e q u e n c i e s , as i n t h e case of the new d i s c r e t e f r e q u e n c y above, t h e d r i v i n g wave i s s h a r p l y a t t e n u a t e d w i t h i n a few l a t t i c e c o n s t a n t s from t h e source., The e f f e c t i v e d i s t a n c e I n which t h i s f r e q u e n c y i s f e l t d i m i n i s h e s as t h e d r i v i n g f r e q u e n c y i s d i s p l a c e d from t h e band edge (Mar-a d u d i n , 1958)„ T h i s new f r e q u e n c y I s thus a s s o c i a t e d w i t h a v i b r a t i o n a l mode l o c a l i z e d around the i m p u r i t y atom. We c a l l t h i s mode a l o c a l i z e d mode (LM). An atom v i b r a t i n g i n t h i s l o c a l i z e d mode can be con-s i d e r e d an E i n s t e i n o s c i l l a t o r , w i t h f i x e d f r e q u e n c y The mass and f r e q u e n c y of such an o s c i l l a t o r a r e r e l a t e d by ^ ~ (1-2) M where k i s t h e f o r c e c o n s t a n t of t h e p e r f e c t l a t t i c e , s i n c e we a r e c o n s i d e r i n g an i s o t o p i c i m p u r i t y . Because the i m p u r i t y mass i s l i g h t e r t h a n t h e h o s t mass, we a g a i n expect k)(.M t o t e h i g h e r t h a n t h e p e r f e c t l a t t i c e mode f r e q u e n c i e s , and as 6 I n c r e a s e s , we ex p e c t ( J L M t o s e p a r a t e f u r t h e r from t h e p e r f e c t l a t t i c e f r e -quency band. A n o t h e r approach t o t h e concept of l o c a l i z e d modes i s due t o Krumhansl (1962). Assume t h a t t h e wave v e c t o r a s s o c i a t e d w i t h the wave p r o p e r t i e s of the d i s p l a c e m e n t f i e l d i s not p u r e l y r e a l , but has r e a l and i m a g i n a r y p a r t s . There t h e n e x i s t e x p o n e n t i a l l y g r owing and damping waves due t o t h e i m a g i n a r y p a r t of t h e wave v e c t o r . I n the p e r f e c t r e g i o n of the l a t t i c e , t h e s e waves a r e e x c l u d e d f o r p h y s i c a l r e a s o n s by n o r m a l i z a t i o n . However, around an i m p u r i t y , which s c a t t e r s phonons, an inc o m i n g growing wave can be s c a t t e r e d by t h e i m p e r f e c t i o n back on i t s e l f t o be-come an o u t g o i n g damped wave. T h i s i s t h e LM, and i t s f r e q u e n c y l i e s o u t s i d e t h e d i s p e r s i o n spectrum of t h e undamped phonons. Anharmonic components of t h e i n t e r a t o m i c f o r c e s l e a d t o an i n t e r c h a n g e of energy between normal modes, and between t h e LM and t h e normal modes. T h i s produces two e f f e c t s . F i r s t , t h e LM w i l l have a f i n i t e l i f e t i m e , and, from the u n c e r t a i n t y r e l a t i o n LINEWIDTH x LIFETIME f , d - 3 ) w i l l be broadened. Second, the LM w i l l be s h i f t e d downward i n energy ( V i s s c h e r , 1964). I n any r e a l experiment t h e r e i s a f i n i t e c o n c e n t r a t i o n of i m p u r i t y atoms, r a t h e r t h a n a s i n g l e i m p u r i t y a t t h e o r i g i n as we assumed above. T h i s l e a d s t o a f u r t h e r b r o a d e n i n g of t h e LM ( D i n h o f e r , 1963). As the I m p u r i t y c o n c e n t r a t i o n i s reduced t o z e r o , o r as 6 approaches £ critical» t h e l° w est v a l u e of t h e f r a c t i o n a l mass d e f e c t f o r which a LM w i l l o c c u r , the broadened spectrum s h r i n k s t o the s i n g l e f r e q u e n c y O ^ . Appendix One re v i e w s some c a l c u l a t i o n s made on t h e e f f e c t of anharmonic f o r c e components and i m p u r i t y c o n c e n t r a t i o n on t h e LM l i f e t i m e . B r o u t and V l s s c h e r (1962) have shown t h a t an a p proximate LM can e x i s t w i t h 6 ( 0 . I n t h i s case O ^ ^ O ^ a n d t h e s t a t e decays i n t o t h e normal mode f r e q u e n c y continuum. However, the decay r a t e w i l l be slow compared t o 6 J L K J i f \£\)) I . Appendix One r e v i e w s some c a l c u l a t i o n s made on the f r e q u e n c y and l i f e t i m e of t h i s v i r t u a l l o c a l i z e d mode. B. THE MOSSBAUER EFFECT When t h e i m p u r i t y atom i n t h e d i s c u s s i o n above i s a Mossbauer atom, t h e l a t t i c e v i b r a t i o n s of the i m p u r i t i e s w i l l have an e f f e c t on t h e Mossbauer resonance spectrum. A s t u d y of t h i s e f f e c t i s of i n t e r e s t f o r two r e a s o n s . F i r s t , t o i n v e s t i -g ate t h e dynamics of an i m p u r i t y atom, and t h e f o r c e s between h o s t and i m p u r i t y atoms. Second, i f the Mossbauer t e c h n i q u e i s t o be used f o r f u r t h e r s t u d y of t h e p r o p e r t i e s of t h e s o l i d , t h e above e f f e c t must be known (Maradudin e t a l , 1958). B o y l e and H a l l (1962) g i v e t h e f o l l o w i n g e x p r e s s i o n f o r the p r o b a b i l i t y of Mossbauer e m i s s i o n w i t h gamma r a y energy E ( t h e symbolism b e l o n g s t o B o y l e and H a l l ) W ( B ^ Y e * f { [ ^ * J —00 exp { i k -(R.-E-)} exp (-2W)exP{ S n m (k,t)) * (1-4) Where - 6 -S n m M = < ( k - ! i h W } { k - a h , w } > T £ s e x p ( - i u f t t ) n £ 4 +€xp(-L£.(g„-Bj]x e x p U < J f s t ) ( h f s H ) ] ( 1 " 5 ) The l a s t e x p o n e n t i a l i n e q u a t i o n 1-4 can be expanded i n powers of i t s exponent (1-6) The n term i n e q u a t i o n 1-6 i s a sum of p r o d u c t s each c o n t a i n i n g n f a c t o r s l i k e exp>(± t ( J ^ s t . ) „ We can i n t e r p r e t t h i s n - f a c t o r p r o d u c t as t h e n-phonon c o n t r i b u t i o n t o W(E), s i n c e each f a c t o r r e p r e s e n t s a change of one phonon. Thus t h e f i r s t term on t h e r i g h t s i d e o f e q u a t i o n 1-6 c o r r e s p o n d s t o r e c o i l l e s s o r z e r o -phonon e m i s s i o n , and t h e second term c o r r e s p o n d s t o gamma ray-e m i s s i o n I n which one phonon i s e i t h e r e m i t t e d o r absorbed i n the Mossbauer e m i s s i o n p r o c e s s . H i g h e r terms, c o r r e s p o n d t o m u l t i p l e phonon t r a n s i t i o n s . I f f n i s t h e n-phonon c o n t r i b u t i o n t o t h e t o t a l p r o b a -b i l i t y o f Mossbauer e m i s s i o n W(E), and we sum over a l l n, th e n from t h e above d i s c u s s i o n - 7 -V ( E . J « £ f„ = n i and f = f . { i h O / f . ) ) " ( 1 - 8 ) h ! I n f o r m a t i o n about the LM can be g a i n e d from two p o s s i b l e t y p e s of e x p e r i m e n t s ; t h o s e which a r e r e c o i l l e s s , and one-phonon experiments,, Appendix Two c o n s i d e r s one-phonon e x p e r i m e n t s . The remainder of t h i s c h a p t e r d i s c u s s e s zero-phonon experiments: i n g e n e r a l . C h a p t e r Two c o n s i d e r s a s p e c i f i c zero-phonon e x p e r i m e n t . C. ZERO-PHONON EXPERIMENTS Both t h e p o s i t i o n and a m p l i t u d e of the c e n t r a l Moss-bauer peak w i l l be a f f e c t e d by t h e f a c t t h a t t h e Mossbauer atom i s an i m p u r i t y (Maradudin and F l i n n , 1962). The p o s i t i o n w i l l be s h i f t e d - 8 -due t o t h e second o r d e r D o p p l e r e f f e c t , which depends on the mean square v e l o c i t y of t h e e m i t t i n g atoms. The a m p l i t u d e of t h e r e s o n a n t peak w i l l be a f f e c t e d by changes i n t h e Debye-Waller f a c t o r , w h i c h i n t u r n depends on t h e a m p l i t u d e o f v i b r a t i o n of the e m i t t i n g atom. Hence, zero-phonon Mossbauer e x p e r i m e n t s , w i t h t h e Mossbauer atom as an i m p u r i t y , s h o u l d g i v e i n f o r m a t i o n about t h e c o n t r i b u t i o n of t h e LM t o t h e m o t i o n of t h e i m p u r i t y , the l i f e t i m e of t h e LM, and t h e f r e q u e n c y a t t h e top of t h e band of normal modes ( B o y l e and H a l l , 1962). I n t h i s I n v e s t i g a t i o n we seek i n f o r m a t i o n about the l i f e t i m e o f t h e LM by o b s e r v i n g i t s e f f e c t on t h e p o s i t i o n o f the c e n t r a l Mossbauer peak. We d e r i v e an e x p r e s s i o n f o r t h e magnitude of t h i s s h i f t , based on t h e model of the Mossbauer I m p u r i t y atom as an E i n s t e i n o s c i l l a t o r . The ground s t a t e energy of t h i s o s c i l l a t o r i s § j~] GL) L M where C O ^ i s t h e f r e q u e n c y of t h e LM. Suppose t h a t a t t i m e t = 0 the r e c o i l from a gamma t r a n s i t i o n i n t h e Mossbauer atom, which p r e c e e d s t h e Mossbauer t r a n s i t i o n , e x c i t e s n phonons o f t h e LM. The o s c i l l a t o r w i l l t h e n have energy (n+ M . S i n c e t h e LM has a l i f e t i m e t h i s energy decays back t o t h e ground s t a t e E ( t)=h1lU L r l exp ( - t / r L M ) + { 1 ? ) Q L „ ( 1 - 9 ) The v e l o c i t y o f t h e atom c o r r e s p o n d i n g t o t h i s energy i s - 9 -v J (t) = a E (t) /M (1-10) A t time t = 0 , t h e Mossbauer atom was a l s o p l a c e d i n the n u c l e a r I s o m e r i c s t a t e w i t h l i f e t i m e V09 from which t h e Mossbauer t r a n s i t i o n o c c u r s 0 The p r o b a b i l i t y of Mossbauer gamma e m i s s i o n a t time t i s P( t ) -= exp (- t / r 0 ) T o (1-11) The second o r d e r D o p p l e r s h i f t i s g i v e n by S - < v " > (1-12) 2 C where K^r^ i s t h e mean square v e l o c i t y of the e m i t t i n g atom, and i n our case i s OO { v l (t) P(t) dt f p ( t ) d t _ ^ n ) ^ { J L M ( e x p OO - t / j _ + J _ \ dt r„ r , LM/J (1-13) - 10 -We have n e g l e c t e d t h e ground s t a t e c o n t r i b u t i o n t o ^v'y* because t h i s i s p r e s e n t a l s o i n .the a b s o r b e r , and w i l l not c o n t r i b u t e t o a shift„ Hence I + V ^ t « M c (1-14) I f we assume t h a t a l l t h e f r e e r e c o i l energy from t h e p r e c e e d i n g gamma goes toward e x c i t i n g t h e LM, and remember t h a t f o / r t M = P c ^ / P o t h e n _S_ = R K EQ (1-15) T o r c + r L M M c x As an example we c a l c u l a t e S f o r t h e case o f F e ^ 7 i n P t 1 9 5 . F o r t h e 14„4 kev t r a n s i t i o n i n Fe 5* 7, E Q/Mc 2 = 2 07xlO" 7 and To = 4.5xl0"*9 ev. F o r t h e 123 kev gamma r a y w h i c h p r e c e e d s i t , t h e f r e e r e c o i l energy R« = 0.142 ev. From Appendix One, the LM l i n e w i d t h a t tempe r a t u r e T i s PLM = 3xl0~^(l + 0.00964 T) ev. Then, a t T = 300°K, 6= 1.5X10"11 ev = To/300. - 11 -CHAPTER TWO THE DESIGN OF A SPECIFIC EXPERIMENT I n t h i s c h a p t e r we w i l l c o n s i d e r an experiment i n which t h e s o u r c e i s Co57 as a d i l u t e i m p u r i t y i n g i v i n g r i s e t o a LM Q t^6) bo A s e c t i o n i s i n c l u d e d which c o n s i d e r s Co57 as a d i l u t e i m p u r i t y i n Be^, g i v i n g r i s e t o a v i r t u a l LM^ U M^<*) 0» F i g u r e 2-1 , and T a b l e s 2-1 and 2-2 g i v e t h e s a l i e n t p r o p e r t i e s of t h e s e m a t e r i a l s . Mossbauer l i n e s h i f t s can be caused by mechanisms o t h e r t h a n t h e e x c i t a t i o n of t h e l o c a l i z e d mode. Any d i f f e r e n c e i n environment between t h e s o u r c e and a b s o r b e r c a n , i n p r i n c i p l e , cause a s h i f t . Examples a r e t h e Jos e p h s o n E f f e c t ( c a u s e d by a d i f f e r e n c e i n temperature) and t h e isomer s h i f t ( c a u s e d by a d i f f e r e n c e i n c h e m i c a l e n v i r o n m e n t ) . To a v o i d t h e isomer s h i f t we a t t e m p t t o make t h e c h e m i c a l environment of t h e a b s o r b e r t h e same as t h a t o f t h e s o u r c e . Hence t h e a b s o r b e r c o n s i s t s o f Fe57 as a d i l u t e i m p u r i t y i n the same h o s t l a t t i c e as t h e s o u r c e , i d e a l l y t o t h e same c o n c e n t r a t i o n . A. SOURCE CONSIDERATIONS L e t I ( o ) be t h e i n i t i a l a c t i v i t y of t h e Mossbauer s o u r c e . The a c t i v i t y o f the s o u r c e a t some l a t e r t i me t i s - 12 -2 7 0 D A Y C o 57 K ELECTRON CAPTURE 137 k e v 9 % 9 1 % 123 k e v I + . 3 6 k e v * F 4 - . 3 6 k e v 1 1 o< = ? . 7 -7 STiA BLE F e 5 7 FIGURE 2-1: DECAY SCHEME OF F e ^ 7 - 13 -TABLE 2-1: PROPERTIES OF F e ^ 7 H a l f l l f e o f Co5? s o u r c e 270 days D i s i n t e g r a t i o n c o n s t a n t of Co^7 A 3 x l 0 " 8 / s e c . Abundance of Fe57 i n n a t u r a l i r o n 2.17$ D e n s i t y ^ 7.85 gm/cc 22 Atomic d e n s i t y n 8 .3x10 atoms/cc Debye temperature © D 490°K E l e c t r o n i c a b s o r p t i o n c o e f f i c i e n t ix 510/cm a t 0 .862 A Energy o f Mossbauer gamma r a y E Q 14 . 3 6 kev o Wavelength o f Mossbauer gamma r a y X 0.862A X 1 . 3 8 x l 0 " 9 cm Mean l i f e of 14 . 3 6 kev l e v e l X0 1.4x10 s e c . N a t u r a l l i n e w l d t h of 14 . 3 6 kev H 4.5xlO - 9 ev l e v e l T o t a l e l e c t r o n c o n v e r s i o n c o e f f i - 0( 9«7 cent of 14 . 3 6 kev t r a n s i t i o n F r a c t i o n of 14 . 3 6 kev gamma r a y s f Q 0-.7 ( a t 300°K) e m i t t e d w i t h o u t r e c o i l , _]_o 2 Mossbauer a b s o r p t i o n c r o s s s e c t i o n OQ 22.6x10 7 cm Free r e c o i l energy of 14 .36 kev if 1.9x10 J ev F r e e r e c o i l energy of 123 kev If R1 0.142 ev - 14 -TABLE 2-2: PROPERTIES OF P t 195 AND Be' P t 195 Be-D e n s l t y A t o m i c d e n s i t y Debye t e m p e r a t u r e E l e c t r o n i c a b s o r p t i o n n M 21.37 gm/cc 1.84 gm/cc 6 . 6 x l 0 2 2 atoms/cc 1 . 2 3 x l 0 2 ^ atoms/cc 240°K 1160°K 3950/cm 0.764/cm c o e f f i c i e n t a t 0.862A - 15 -l t = I 0 « P ( - a t ) (2-1) where A i s t h e d i s i n t e g r a t i o n c o n s t a n t . The t o t a l a c t i v i t y of the s o u r c e i s i l t 4 t - I C o ) / * = M (2-2) o T h i s means t h a t M r a d i o a c t i v e Mossbauer atoms must be d i f f u s e d i n t o the s o u r c e h o s t m a t e r i a l . must a c c e p t a c i r c u l a r a c t i v e spot of a r e a A. So t h a t t h e LM r e a l l y i s l o c a l i z e d , the Mossbauer atoms i n t h e s o u r c e must be i s o l a t e d from each o t h e r . The p r o c e s s of d i f f u s i o n a c t u a l l y produces an i m p u r i t y d i s t r i b u t i o n w hich i s a p p r o x i m a t e l y G a u s s i a n (see Appendix T h r e e ) . We assume t h a t the d i f f u s i o n produces a u n i f o r m d i s t r i b u t i o n w h i c h g r e a t l y s i m p l i f i e s c a l c u l a t i o n s and produces r e s u l t s c o r r e c t t o w i t h i n a few p e r ce n t (see Appendix T h r e e ) . We s p e c i f y a u n i f o r m c o n c e n t r a t i o n a f t e r d i f f u s i o n of I d e a l l y we would l i k e a p o i n t s o u r c e ; p r a c t i c a l l y we r (2-3) - 16 -one Mossbauer atom p e r 1000 h o s t atoms, o r 10 l a t t i c e s p a c i n g s between i m p u r i t i e s . That I s , (2-4) A d l o o o where ng I s t h e a t o m i c d e n s i t y of the h o s t . A f t e r d i f f u s i o n , t h e I n t e n s i t y of gamma r a y s e m i t t e d a t a depth x i s Ix d x = A A C(-) d x ( 2-5) The f r a c t i o n of t h e s e gamma r a y s t h a t r e a c h t h e s u r f a c e i s a f f e c t e d by e l e c t r o n i c a b s o r p t i o n and r e s o n a n t (Mossbauer) a b s o r p t i o n . S i n c e the c o n c e n t r a t i o n of i m p u r i t y atoms i s d i l u t e , t h e n t h e e l e c t r o n i c a b s o r p t i o n w i l l be due al m o s t e n t i r e l y t o the ho s t atoms. The r e s o n a n t a b s o r p t i o n i s due o n l y t o I m p u r i t y atoms w h i c h have a l r e a d y decayed, and may be r e e x c i t e d by r e s o n -a n t l y a b s o r b i n g a gamma r a y . F o r our h o s t , and d i l u t e i m p u r i t y c o n c e n t r a t i o n , we can assume t h a t the r e s o n a n t a b s o r p t i o n i s n e g l i g i b l e . M a r g u l i e s and Ehrman (196l) t r e a t t h e case when the Mossbauer atom i s a ho s t atom, and a l l atoms i n the ho s t can a b s o r b r e s o n a n t l y . The i n i t i a l i n t e n s i t y a t t h e s u r f a c e of t h e so u r c e due - 17 -t o e l e c t r o n i c a b s o r p t i o n o n l y i n t h e so u r c e i s w h e r e ^ ^ i s the gamma a b s o r p t i o n c o e f f i c i e n t of t h e h o s t . B. ABSORBER CONSIDERATIONS When the a b s o r b e r i s D o p p l e r s h i f t e d t o d e s t r o y r e s o n a n t a b s o r p t i o n , the i n t e n s i t y of gamma r a y s r e a c h i n g t h e gamma c o u n t e r i s where oi i s t h e e l e c t r o n c o n v e r s i o n c o e f f i c i e n t of t h e Mossbauer t r a n s i t i o n , f 0 i s the r e c o i l l e s s f r a c t i o n of Mossbauer gamma r a y s , A J L i s t h e s o l i d a n g l e subtended a t t h e sour c e by the c o u n t e r , /4H i s t h e gamma a b s o r p t i o n c o e f f i c i e n t of the a b s o r b e r h o s t mater-i a l , and X i s t h e a b s o r b e r t h i c k n e s s . L e t us d e f i n e d i m e n s i o n l e s s s o u r c e and a b s o r b e r t h i c k -n e sses Ts = to <X n H a s d ( 2 - 8 ) TA = to <z nrt a* x - 18 -where n^ i s t h e a t o m i c d e n s i t y of t h e h o s t m a t e r i a l , a s and a a a r e t h e i m p u r i t y / h o s t atom r a t i o s , d i s the d i f f u s i o n depth i n th e s o u r c e and x i s t h e t h i c k n e s s of the a b s o r b e r h o s t m a t e r i a l . a Q can be c a l c u l a t e d from a. D A„ X. x % A n - ^ V H (2-9) where D i s t h e mass s u r f a c e d e n s i t y of I m p u r i t y m a t e r i a l d e p o s i t e d on t h e a b s o r b e r , ^ H i s t h e mass d e n s i t y of t h e a b s o r b e r h o s t mat-e r i a l , Ag and A j a r e t h e a t o m i c numbers of t h e h o s t and i m p u r i t y . F o r 1, M a r g u l l e s and Ehrman (1961) g i v e the f o l l o w -i n g e x p r e s s i o n f o r t h e depth of the Mossbauer peak,h , h = f . [ l ~ e x p ( - r A / 2 ) I 0 ( T A / 2 ) ] (2-1°) where I Q (x) i s t h e m o d i f i e d B e s s e l f u n c t i o n of t h e f i r s t k i n d o f o r d e r z e r o , F o r T a ^ 1, we can ex p r e s s b o t h t he e x p o n e n t i a l and B e s s e l f u n c t i o n s i n e q u a t i o n 2-9 i n s e r i e s form, and o b t a i n T° 1 a n 3 0 7 2 (2-12) - 19 -Co COUNTING STATISTICS To observe s m a l l s h i f t s £ of the Mossbauer peak, counts a r e a c c u m u l a t e d a t f o u r p o i n t s on t h e peak (A, B, C, and D i n f i g u r e 2-2) w i t h r e l a t i v e v e l o c i t i e s between s o u r c e and a b s o r b e r of + V]_ and + V2- The v e l o c i t y d i s p l a c e m e n t s of the s e p o i n t s from the c e n t r e of t h e peak a r e v , - v 2 + s V& = V, +• S (2-13) v c = v, - s I n what f o l l o w s a l l v e l o c i t i e s and s h i f t s a r e e x p r e s s e d i n terms of t h e l i n e w i d t h VQ of t h e Mossbauer peak. S e v e r a l a u t h o r s (Cranshaw, p. 64 i n Compton and Schoen, 1962; B o y l e and H a l l , 1962; B o y l e e t a l , i960) have g i v e n the f o l l o w i n g e x p r e s s i o n r e l a t i n g t h e s h i f t £ t o the observ e d number of c o u n t s I n e q u a l t i m e s a t t h e p o i n t s A, B, C, D (2-14) 1 ( N A - N B W N U - NO) where AV = V2 - v l ° T h i s e x p r e s s i o n i s e x a c t o n l y i n t h e l i n e a r and q u a d r a t i c a p p r o x i m a t i o n s M i = N „ ( i - h ( i - a v ; ' ) ) (2-15) FIGURE 2-2: CALCULATION OF SMALL SHIFTS - 2 1 -where h i s t h e depth of t h e Mossbauer peak as g i v e n by e q u a t i o n 2 - 1 2 . When t h e shape of t h e Mossbauer peak i s de t e r m i n e d by l i f e t i m e e f f e c t s , i t has a B r e i t - W i g n e r l l n e s h a p e I n t h i s case e q u a t i o n 2 - 1 4 becomes 2 (NA-NB)-(NC-ND) ( 2 - 1 7 ) I f S ^ f V i , V2 t h i s reduces t o 2 (M, -MB)"(MC-N 0 ) ( 2 - 1 8 ) The s t a t i s t i c a l f l u c t u a t i o n i n bo t h (NA/ + NB) - (Nc + NQ) and ( N A - N B) - ( N C ~ N D) i s + ( N A + % + N c + K D ) ^ so t h a t t h e r i g h t s i d e of e q u a t i o n 2 - 1 8 I s m u l t i p l i e d by a f a c t o r - 22 -(MA+N&\-(Wc+ND\ > (2-19) where we have assumed the r e l a t i v e f l u c t u a t i o n s a r e much l e s s t h a n u n i t y . Assume t h a t we w i s h t o de t e r m i n e £ t o w i t h i n 10$. Then ( _ L (2-20) / O O d e t e r m i n e s t h e ti m e r e q u i r e d t o accumulate a s u f f i c i e n t number o f c o u n t s . Assume t h a t V1 = Pa /k and V 2 = 3 I"© A . R e t a i n i n g o n l y f i r s t o r d e r terms i n £, from e q u a t i o n s 2-13 and 2-16 1 3 0 / (2-21) 09 and from e q u a t i o n 2-20 - 23 -Q Q (2-22) h * 8 ' I s t h e minimum t o t a l number of counts t h a t must be a c c u m u l a t e d . I f I c i s t h e i n t e n s i t y , then + + > l 0 ° (2-23) i s t h e minimum t i m e r e q u i r e d t o d e t e r m i n e & t o w i t h i n 10$. D. P e 5 7 IN P t 1 9 5 F o r our s o u r c e we s p e c i f y 5 m i l l i c u r i e s of C o ^ 7 e l e c t r o -p l a t e d onto 0 .5 m i l P l a t i n u m f o i l i n a 1/4" d i a m e t e r a c t i v e s pot ( g i v i n g an a c t i v e a r e a A = 0 .3 c m 2 ) . I n p r a c t i c e , r a t h e r t h a n s p e c i f y an i m p u r i t y atom den-s i t y a f t e r d i f f u s i o n , we must s p e c i f y t h e d i f f u s i o n c o n d i t i o n s ( t e m p e r a t u r e , d u r a t i o n , and ambient atmosphere), from w h i c h i t i s p o s s i b l e i n p r i n c i p l e t o c a l c u l a t e t h e rms d i f f u s i o n depth and mean d e n s i t y of t h e G a u s s i a n i m p u r i t y d i s t r i b u t i o n produced by the d i f f u s i o n (see Appendix T h r e e ) . T h i s c a l c u l a t i o n r e q u i r e s a knowledge of t h e d i f f u s i o n c o e f f i c i e n t under t h e s p e c i f i e d con-d i t i o n s , w h i c h i s u n a v a i l a b l e i n our c a s e . So we assume th e u n i f o r m i m p u r i t y d i s t r i b u t i o n of e q u a t i o n 2-3» and s p e c i f y an i m p u r i t y / h o s t atom r a t i o a f t e r d i f f u s i o n of one atom of C o ^ 7 t o 1000 atoms of P t 1 9 ^ . F o r our a b s o r b e r we s p e c i f y 0.01 mg Fe-5? /cm 2 e l e c t r o -- 24 -p l a t e d onto t h e same f o i l as t h e s o u r c e , and d i f f u s e d t o g i v e u n i f o r m d e n s i t y . We assume t h a t AfL , t h e s o l i d a n g l e subtended a t t h e so u r c e by t h e c o u n t e r i s 0 .1 s t e r a d i a n s . T a b l e 2-3 l i s t s some c a l c u l a t e d v a l u e s o b t a i n e d i n t h i s c h a p t e r . The ex p e c t e d count r a t e I c - 358 c o u n t s / s e c o n d . The ex p e c t e d d i p h = 5°5$» From Ch a p t e r One, the p r e d i c t e d s h i f t & = f o / 3 0 0 . Then, from e q u a t i o n 2 -23 , t h e minimum time r e q u i r e d t o d e t e r m i n e £ t o w i t h i n 10$ i s 9.7x10^ seconds, o r 16 weeks. E. Fe^? IN B e 9 F o r our sour c e we s p e c i f y 5 m i l l i c u r i e s of Co5? e l e c t r o -p l a t e d onto a 1/4" a c t i v e s p o t of 2 m i l t h i c k B e r y l l i u m f o i l , and d i f f u s e d t o g i v e an i m p u r i t y / h o s t atom r a t i o of one atom of C o ^ 7 t o 1000 atoms of B e 9 . F o r our a b s o r b e r we s p e c i f y 0.06 mg Fe^' /cm e l e c t r o -p l a t e d onto the same f o i l as the s o u r c e , and d i f f u s e d t o g i v e u n i f o r m Fe-^ 7 d e n s i t y . As above, we assume A A - 0 .1 s t e r a d i a n s . From T a b l e 2 - 3 , t h e e x p e c t e d count r a t e I c = 9.4x10^ counts/second,. The e x p e c t e d d i p h = 25$. B e r y l l i u m f o i l h a v i n g l e s s t h a n one p a r t i n 2000 i r o n i m p u r i t i e s , and l e s s t h a n one p a r t I n 200 t o t a l i m p u r i t i e s i s - 25 -d i f f i c u l t t o o b t a i n . S u p p l i e r s of Mossbauer m a t e r i a l s * supply-P l a t i n u m backed s o u r c e s and a b s o r b e r s a t a p p r o x i m a t e l y h a l f t h e p r i c e of B e r y l l i u m - b a c k e d m a t e r i a l s . * New England N u c l e a r Corp., 575 A l b a n y S t r e e t , B o s t o n 02118 N u c l e a r S c i e n c e and E n g i n e e r i n g Corp., P.O. Box 10901, P i t t s b u r g h 15236 U.S. N u c l e a r Corp., P.O. Box 208, 801 Lake S t . , Burbank, C a l . , 91503 The R a d i o c h e m i c a l C e n t r e , Amersham, -Bucks., E n g l a n d - 26 -TABLE 2 -3; ZERO-PHONON EXPERIMENT CALCULATED RESULTS ( F e ^ 7 IN P t 1 ^ AND B e 9 ) SPECIFIED; I n i t i a l a c t i v i t y of so u r c e Host f o i l t h i c k n e s s I m p u r i t y / h o s t r a t i o i n s o u r c e I m p u r i t y s u r f a c e d e n s i t y i n a b s o r b e r S o l i d a n g l e subtended by c o u n t e r a t s o u r c e CALCULATED; I m p u r i t y d i f f u s i o n d e p t h I n s o u r c e I n i t i a l s u r f a c e a c t i v i t y o f so u r c e I n t e n s i t y a t c o u n t e r I m p u r i t y / h o s t r a t i o i n a b s o r b e r D i m e n s i o n l e s s s o u r c e t h i c k n e s s D i m e n s i o n l e s s a b s o r b e r t h i c k n e s s D i p LM s h i f t Minimum c o u n t i n g t i m e t o d e t e r m i n e g w i t h i n 10% K o ) X as D d a £ •a h g F e 5 7 IN P t 1 9 ^ F e i L l J B e 9 1 , 8 x l 0 8 / s e c o n d 1 . 8 x l 0 8 / s e c o n d 1.27xl0"^om 1/1000 5.08xl0" 3cm 1/1000 l.OxlO^gm/cm 2 6.0xl0~- 5gm/cm 2 0.1 s t e r a d i a n 3xl0~i|'cm I ( s ) °»58 I ( o ) 0.1 s t e r a d i a n 1.6xl0"^cm K o ) 358 c o u n t s / s e c . 9»4xl0^ c o u n t s / s e c . 1/790 1/1000 0.031 0.166 f^/300 16 weeks 0.031 0.99? 2.5% -"•27 -CHAPTER THREE AIR TROUGH MOSSBAUER SHIFT SPECTROMETER I n t h i s c h a p t e r we d e s c r i b e a p p a r a t u s d e s i g n e d and b u i l t t o p e r f o r m t h e experiment o f Chapter Two. The purpose of t h i s a p p a r a t u s i s t o p r o v i d e t h e s m a l l v e l o c i t i e s r e q u i r e d t o D o p p l e r s h i f t t h e s o u r c e and absorber,, These v e l o c i t i e s must be c o n s t a n t and f r e e from v i b r a t i o n o v e r the t r a v e r s e of t h e system. F i g u r e 3-1 shows a b l o c k diagram of t h e a p p a r a t u s . I t w i l l be d e s c r i b e d I n t h r e e s e c t i o n s : t h e m e c h a n i c a l d r i v e , t h e a i r t r o u g h s u s p e n s i o n , and t h e v e l o c i t y m o n i t o r i n g system. C h a p t e r F o u r d e s c r i b e s t h e performance of the a i r t r o u g h s u s p e n s i o n . A. MECHANICAL DRIVE The m e c h a n i c a l d r i v e c o n s t r u c t e d f o r t h i s a p p a r a t u s had t o meet t h e f o l l o w i n g n e c e s s a r y r e q u i r e m e n t s : 1. P r e c i s e l y c o n t r o l l a b l e and a c c u r a t e l y c o n s t a n t l i n e a r output; v e l o c i t y , 2. Output v e l o c i t y range- from 0 - 1 mm/second. • 3 . Output v e l o c i t y f r e e from v i b r a t i o n s . 4 C Output v e l o c i t y r e v e r s i b l e ' upon c l o s i n g m l c r o s w i t c h e s . W h i l e not a b s o l u t e l y n e c e s s a r y , t h e f o l l o w i n g r e q u i r e -ments make t h e a p p a r a t u s more c o n v e n i e n t t o use: R7 " MICR05W|TC«£S S I G N A L G E N E R A T O R C O N T R O L C I R C U I T F I G U R E 3 - 4 -H E W L E T T - P A C K A R D 2 4 - I A O S C I L L A T O R T f O O K 11 r—rW'' £ > F 5 / x F I 0 O K yAA/W 'WW L/VJ -Z7V 5 > F L E F T R l & H T H E A T H K I T A A - 2 1 S T E R E O A M P L I F I E R M E C H A N I C A L D R I V E C O N T R O L C I R C U I T F I G U R E 3 - 3 I M O T O R A N D M E C H A N I C A L D R I V E F l f f O R E " 3 - 2 A I R T R O U G H S U S P E N S I O N F I G U R E 3 - 5 I 1 / E L O C l T r M O N I T O R I N G S T S T E M FiGURe 3 - 7 S C A L E R S A - l R SOPPLY F I 6 0 R E 3 - 6 FIGURE 3 - l t BLOCK DIAGRAM OF APPARATUS - 29 -5. Output v e l o c i t y c a p a b l e of b e i n g a u t o m a t i c a l l y changed between two d i s c r e t e v a l u e s . 6. When system e n e r g i z e d , i n i t i a l o u t put v e l o c i t y a lways i n t h e same d i r e c t i o n . 7. C o n t r o l of s c a l e r s . P u l s e s r o u t e d t o one of f o u r s e t s of s c a l e r s by v e l o c i t y c o n t r o l system, depending on d i r e c t i o n and magnitude of v e l o c i t y . F i g u r e s 3-2, 3-3» and 3-4 show components of t h e mechan-i c a l d r i v e system. The remainder of t h i s s e c t i o n d i s c u s s e s t h e s e f i g u r e s . F i g u r e 3-2; The synchronous motor has a v e r y c o n s t a n t v e l o c i t y c h a r a c t e r i s t i c , and may be r e v e r s e d d u r i n g r o t a t i o n . A l t h o u g h t h e motor i s b u i l t t o be synchronous a t 60 c p s , o p e r a t i n g a t from 60 t o 100 cps s h o u l d not a p p r e c i a b l y a l t e r i t s c o n s t a n t v e l o c i t y c h a r a c t e r i s t i c . The b a l l d i s c i n t e g r a t o r g i v e s f l e x i b i l i t y i n t h e range of o u t p u t v e l o c i t i e s a v a i l a b l e , A m i c r o m e t e r head s e t t i n g con-t r o l s i t s o u t p u t . W i t h a moment of i n e r t i a of 248 l b i n 2 , t h e f l y w h e e l f i l t e r s out n o i s e ( v i b r a t i o n s ) g e n e r a t e d by t h e motor and b a l l d i s c i n t e g r a t o r . The ou t p u t v e l o c i t y i s f u r t h e r reduced by two p u l l e y s . On t h e o u t p u t s i d e of t h e f l y w h e e l , o n l y f a b r i c t a p e s and s l e e v e b e a r i n g s a r e us e d , so as t o m i n i m i z e n o i s e . The p u l l e y arrangement - 30 -T A P E E T A P E D • T A P E : C 2 ] P U L L E Y B - A C P O W E R M O T O R : B O D l N E E L E C T R I C C O . C A T . N O . 2 2 7 0 4- L E A D S Y N C H R O N O U S C A P A C I T O R A C M O T O R M^V , ' 6 0 C P S , I / 7 5 T H P , 1 8 0 0 RPM> O U T PUT DIA. 0 . 5 I N T E G R A T O R : L I B R A S C O P E I W C . P A R T N O . 8 7 9 2 5 © - / B A L L - D I S C I N T E G R A T O R , O U T P U T 0 - 2 X I N P U T I N P U T A N D O U T P U T D l A . 0 . 5 " F L Y W H E E L : R E K - O - K U T M O D E L £ 1 B R O A D C A S T T U R N T A B L E I N P U T D I A . 1 5 . 5 " , O U T P U T D I A . 0-5* PULLEY A : INPUT D I A . - 2 . 9 7 5 " , O U T P U T D l A . O - Z " P U L L E r 6 : I N P U T D I A . 3 " , OUTPUT D l A . 0 . 3 7 5 " T A P E A , D : F A B R I C IT" X 1 / 2 " * 12. M I L T A P E B : M Y L A R 5 7 " ^ l " * J M I L T A P E C : F A B R I C * 1/X" x 18 M I L T A P E E : S T E E L 5 4 - " x 5 0 MIL x 3 M I L FIGURE 3-2: MECHANICAL DRIVE - MECHANICAL DETAILS shown I n F i g u r e 3-2, g i v e s a maximum ou t p u t v e l o c i t y of 0.64 mm/ second. F i g u r e 3-3: R l and R2 form a b i s t a b l e system. W i t h t h e motor r o t a t i n g i n one d i r e c t i o n , R l i s on and R2 o f f . D e p r e s s i n g t h e a p p r o p r i a t e m i c r o s w i t c h , S I o r S2, changes t h e s t a t e of b o t h R l and R2, and a l s o R3, the r e p e a t e r of R l . When t h e r e l a y system has power a p p l i e d , R l i s always i n i t i a l l y on. The a p p r o p r i a t e s c a l e r c o n t r o l r e l a y , RIO, R l l , R12 o r R13, must be on f o r t h e c o r r e s p o n d i n g s c a l e r t o r e g i s t e r c o u n t s . The c i r c u i t r y f o r t h i s i s not shown. D e p r e s s i n g t h e a p p r o p r i a t e m i c r o s w i t c h i n i t i a t e s t h e f o l l o w i n g sequence: e i t h e r R4 o r R5 i s t u r n e d o f f ; R l and R2 change s t a t e , R6 i s t u r n e d o f f , and T l and T2 a r e e n e r g i z e d ; R3 changes s t a t e , power i s c u t o f f t o t h e motor and s c a l e r c o n t r o l r e l a y s ; t h e motor c o n t a c t s a r e r e v e r s e d , and the s c a l e r l o g i c r e -r o u t e d . When t h e t h e r m a l d e l a y r e l a y s T l and T2 c l o s e , power i s r e s t o r e d t o the motor which now o p e r a t e s i n r e v e r s e . When t h e motor has moved t h e system enough t o r e l e a s e t h e m i c r o s w i t c h , power i s r e s t o r e d t o the s c a l e r c o n t r o l r e l a y s . Changing t h e s t a t e o f R? (see d i s c u s s i o n of F i g u r e 3-"+ below) changes t h e s t a t e of R9, which r e r o u t e s t h e s c a l e r l o g i c , and t u r n s on R8 f o r t h e t h e r m a l d e l a y time of T3 and T4, x«rtiich c u t s o f f t h e s c a l e r s w h i l e t h e motor changes v e l o c i t y . - 32 -S I 51 R 2 Rl ] r R2. w r ] c R 3 i r R 4 - R 5 R 4 R51. T l R 6 R 9 R9 ] C T 4 - 1 1 T 3 S C A L E R C O N T R O L R E L A Y ' S R 8 R6 R 3 R I O R II L R 6 — » — IIOV n R6 — 9 -R3 BLUE R 3 A 1 B L A i K •——AAAA-B L A C K M O T O R —AAAAr—j 3 .7S /«F -9> >• B L U E T 2 R l , 2 ; ^ , 5 , 8 , 1 0 , 1 1 , 1 2 , 1 3 D P D T ( K R P i l A ) R 3 , 6 , 9 3 P D T ( K R P l * A ) T l , 2 5 S E C . N J . O . T H E R M A L D E L A V T 3 . 4 10 S E C . I M . C . T H E R M A L D E L A Y S l , 2 S P D T M I C R . O S f c / ) T C H € S R 7 I N S I G N A L G E N E R A T O R C O N T R O L C I R C U I T ( F l C U R E 3 - 4 - ) A L L R E L A Y C O I L S 6 - 3 V A C - 33 -F i g u r e 3-4: The u n i j u n c t i o n t r a n s i s t o r r e l a x a t i o n o s c i l l a t o r , Q1/Q2, g i v e s a p o s i t i v e p u l s e a t t h e c o l l e c t o r of Q2 once e v e r y 5 t o JO m i n u t e s , depending on the v a l u e of t h e v a r i a b l e r e s i s t a n c e . T h i s p u l s e changes the s t a t e of b i s t a b l e m u l t i v i b r a t o r Q3/Q4, and of r e l a y R7„ Q6 i s b i a s e d o f f by the z e n e r d i o d e when Q4 i s i n the "on" s t a t e . As R? changes s t a t e , one of t h e 4 0 y u F c a p a c i t o r s d i s c h a r g e s t h r o u g h s o l e n o i d s A and B a l t e r n a t e l y , c h a n g i n g t h e f r e q u e n c y of o s c i l l a t i o n of t h e H e w l e t t - P a c k a r d P u s h - B u t t o n O s c i l l a t o r . B. AIR TROUGH SUSPENSION An a i r - t r o u g h s u s p e n s i o n was chosen f o r t h i s a p p a r a t u s because of the low f r i c t i o n c h a r a c t e r of the a i r b e a r i n g . I t was hoped t h a t v i b r a t i o n - f r e e movement c o u l d be e a s i l y a t t a i n e d w i t h such a s u s p e n s i o n . T h i s s e c t i o n d e s c r i b e s t h e s u s p e n s i o n system c o n s t r u c t e d . C h a p t e r F o u r d i s c u s s e s the performance of t h i s s u s p e n s i o n . The s u s p e n s i o n was i n s p i r e d by t h e l i n e a r a i r t r a c k d e v e l o p e d by Neher and L e i g h t o n (I963) and S t u l l (1962). L i k e t h e a i r t r a c k , our s u s p e n s i o n c o n s i s t s of a r i g h t a n g l e r a i l , i n whic h many s m a l l h o l e s have been d r i l l e d , and a r i g h t a n g l e r i d e r w h i c h f l o a t s on t h e r a i l , s u p p o r t e d by a i r f o r c e d t h r o u g h t h e h o l e s . I n t h e a i r t r a c k system, the r i d e r i s f r e e t o move many FIGURE 3-4: SIGNAL GENERATOR CONTROL CIRCUIT - - 35 -meters a l o n g t h e r a i l , c o v e r i n g o n l y a s m a l l f r a c t i o n o f t h e a i r h o l e s a t any t i m e . I n our system, t h e movement of t h e r i d e r i s c o n t r o l l e d by t h e m e c h a n i c a l d r i v e d e s c r i b e d above; the r i d e r i s f r e e t o move o n l y about one i n c h a l o n g t h e r a i l b e f o r e d e p r e s s i n g a r e v e r s i n g m i c r o s w i t c h ; and a l l t h e a i r h o l e s a r e always c o v e r e d by t h e r i d e r . F i g u r e 3-5 i s a photograph of t h e system. The a i r t r o u g h i s s u p p o r t e d on a heavy base t o damp out v i b r a t i o n s t r a n s m i t t e d t h r o u g h t h e b u i l d i n g frame. From top t o bottom t h i s base c o n s i s t s of two s t e e l p l a t e s b o l t e d t o g e t h e r and w e i g h i n g a p p r o x i m a t e l y 300 pounds; a sheet of r u b b e r v i b r a t i o n -damping m a t e r i a l ; a 1/8" s h e e t of l e a d ; and a p p r o x i m a t e l y one t o n of c o n c r e t e b l o c k s . The a i r t r o u g h I s mounted i n s i d e an a c o u s t i c -t i l e d box t o damp out v i b r a t i o n s t r a n s m i t t e d t h r o u g h the a i r . The a i r s u p p l y f o r t h e a i r t r o u g h i s shown i n F i g u r e 3 - 6 . The a i r f l o w can be v a r i e d from about 150 t o 300 c u b i c f e e t p e r hour, a t a few o u n c e s / i n c h ^ p r e s s u r e i n s i d e the r a i l , w i t h t h e p r e s s u r e r e g u l a t i n g v a l v e . V i b r a t i o n s t r a n s m i t t e d t h r o u g h t h e a i r s u p p l y a r e damped out by t h e o i l drum r e s e r v o i r . To use the s u s p e n s i o n i n a Mossbauer e x p e r i m e n t , the so u r c e i s mounted on t h e r a i l , and the a b s o r b e r on t h e r i d e r . A gamma d e t e c t o r i s p l a c e d b e h i n d t h e a b s o r b e r . Co VELOCITY MONITORING SYSTEM A v e l o c i t y m o n i t o r i n g system was c o n s t r u c t e d which can FIGURE 3-5: AIR TROUGH SUSPENSION - 37 -B U I L D I N G A I R S U P P L Y FILTER AND DESSICANT P R E S S U R E R E G U L A T O R P R E S S U R E G A U G E p| -F L O V / M E T E R n o H O L E S D I A . r / » 6 " R E S E R V O I R f S 5 G A L D R U M ) 3 / 4 - " - " D I A H O S E - I Q " L O N G A I R T R O U G H R A I L ( C A P A C I T Y " 2 S A L . ) C O N T R O L L E D L E A K PRE SSU RE GAUGE P2. FIGURE 3-6; AIR SUPPLY - 38 -measure b o t h t h e f r e q u e n c y and a m p l i t u d e of v i b r a t i o n s w i t h amp-l i t u d e s as s m a l l as 10"" ^  m i l l i m e t e r s . A s c h e m a t i c diagram i s shown i n F i g u r e 3-7. The f r i n g e s produced by d i r e c t i n g t h e l a s e r beam i n t o the M i c h e l s o n i n t e r f e r o m e t e r w i l l sweep a c r o s s the p h o t o c e l l a t a f r e q u e n c y p r o p o r t i o n a l t o t h e v e l o c i t y of t h e a i r t r o u g h r i d e r 0 A smooth m o t i o n w i l l produce a tone a t t h e speaker,, V i b r a t i o n s i n t h e m o t i o n produce a b u b b l i n g sound a t the s p e a k e r and d i s t i n c -t i v e waveforms on t h e o s c i l l o s c o p e . A F o u r i e r a n a l y s i s o f t h e v i b r a t i o n s may be o b t a i n e d on t h e s t r i p - c h a r t r e c o r d e r by sweeping t h r o u g h t h e f r e q u e n c y spectrum w i t h t h e t u n e d a m p l i f i e r . The system p r o v e d t o be e x t r e m e l y s e n s i t i v e t o v i b r a -t i o n s . Heavy f o o t s t e p s twenty f e e t away d e f l e c t e d the m i r r o r s enough t o be e a s i l y d e t e c t e d . B r o a d c a s t i n g t h e f r i n g e p a t t e r n o v e r t h e l o u d s p e a k e r produced sound waves wh i c h d e f l e c t e d t h e m i r r o r s enough t o g e n e r a t e p o s i t i v e f e e d b a c k , even when the m i r r o r s were s u r r o u n d e d by t h e a c o u s t i c - t i l e d box. The p h o t o c e l l has an adequate response up t o about 5 k i l o c y c l e s , c o r r e s p o n d i n g t o an a i r t r o u g h r i d e r v e l o c i t y of 1.5 mm/second. D ° CALIBBATIONS OF THE AIR SUPPLY AND THE DRIVE MECHANISM I t I s d e s i r a b l e t o d e t e r m i n e t h e p r e s s u r e / f l o w r a t e c h a r a c t e r i s t i c s o f t h e a i r s u p p l y of F i g u r e 3-6. E x p e r i m e n t a l d e t e r m i n a t i o n of t h e f l o w r a t e Q and p r e s s u r e i n s i d e the a i r t r o u g h , P2, f o r v a r i o u s v a l u e s of t h e p r e s s u r e a t t h e r e g u l a t i n g P O L A . R O I P C A M E R A M o G CO W i <J O o M K 3 O M 3^ O W (H O CO t< CO + 2 . 0 V SHASTA 854A AMPLIFIER l o o K S I G M A 4-HI8 C A D M I U M S U L F 0 S E L 6 M I D E P H O T O -C O N D U C T O R C E L L 1—yfK. S P E C T R A P H Y S I C S 1 3 0 H E L I U M -N E O N L A S E - R 6328A 2 . N 1 3 0 * O N E F R I N G E E V E R r A / 2 * 3 . l 6 * x I 0 _ 4 " M M . H A L F - ^ f L V E R E D M I R R O R M O V I N G M I R R O R M O O W T E b - o K j * A I R T R O O G H filDFR O S C I L L O S C O P E T H E A T H K I T A A - 2 2 . A M P L I F I E R SPEAKER H E W L E T T - P A C K A R D i - I O C V O L T M E T E R . V~0 FIXED MIRROR G E N E R A L R A D I O 12 3 2 - A T U N F D F I L T E R M O S E L E T 6 8 0 S T R I P - C H A R T R E C O R D E R - 40 -v a l v e , - Pl$ is."shown I n F i g u r e 3-8 f o r two l o a d i n g c o n d i t i o n s . F i g u r e 3~8(a) shows the case w i t h no e x t r a l o a d i n g on the r i d e r ( w hich weighs 8.5 l b ) ; F i g u r e 3-8(b) shows t h e case w i t h 1.5 l b w e i g h t s p l a c e d on each arm of t h e r i d e r . We i n t e r p r e t the l e v e l l i n g o f f of t h e p r e s s u r e c u r v e s t o mean t h a t t h e a i r p r e s s u r e i n t h e a i r t r o u g h i n c r e a s e s o n l y u n t i l i t i s s u f f i c i e n t t o f l o a t t he r i d e r . F u r t h e r i n c r e a s e of the p r e s s u r e a t t h e r e g u l a t i n g v a l v e causes i n c r e a s e d f l o w . , We assume optimum c o n d i t i o n s t o be when t h e r i d e r j u s t f l o a t s , i n d i c a t e d by p o i n t s A and B i n F i g u r e 3-8. Waveforms produced by t h e v e l o c i t y m o n i t o r i n g system when the r i d e r i s moved a t c o n s t a n t v e l o c i t y , w h i l e P I i s v a r i e d , g i v e s rough sup-p o r t t o t h i s a s s u m p t i o n , a l t h o u g h t h e r e appears some l a t i t u d e (about one ounce/square i n c h e i t h e r way) i n c h o o s i n g the optimum v a l u e of P I . The d r i v e mechanism of F i g u r e 3-2 was c a l i b r a t e d by me a s u r i n g t h e average f r e q u e n c y o u t p u t of t h e v e l o c i t y m e a s u r i n g system a t v a r i o u s m i c r o m e t e r head s e t t i n g s . Two methods were used; c o u n t i n g t h e f r e q u e n c y peaks on o s c i l l o s c o p e t r a c e s , and f i n d i n g t h e c e n t r e o f t h e F o u r i e r a n a l y s i s peak on t h e s t r i p -c h a r t r e c o r d e r . The r e s u l t s were c o n s i s t e n t and a r e shown i n F i g u r e 3 - 9 . - 41 -o 5" ft) F / S U R F 3-8 a) /5T FIGURE 3-8; CALIBRATION OF AIR SUPPLY - 42-FIGURE 3-9t CALIBRATION OF DRIVE MECHANISM _ 43 -CHAPTER FOUR PERFORMANCE OF THE AIR TROUGH SUSPENSION Ch a p t e r Three d e s c r i b e s a p p a r a t u s c o n s t r u c t e d as a Moss-bauer S h i f t Spectrometer,, Many months have been spent a t t e m p t i n g t o make t h i s a p p a r a t u s f u n c t i o n a d e q u a t e l y enough t o p e r f o r m t h e experiment of C h a p t e r Two e From Ch a p t e r One 9 t h e e x p e c t e d s h i f t of t h e Mossbauer l i n e i s of t h e o r d e r of 1% of t h e Mossbauer l i n e w i d t h . I f t h e v e l o c i t y produced by the s p e c t r o m e t e r a t each c o n s t a n t - v e l o c i t y s e t t i n g i s a d e l t a - f u n c t i o n , t h e n from Chapter Two i t w i l l t a k e 16 weeks of c o u n t i n g t o d e t e r m i n e t h e s h i f t t o w i t h i n 10$. I f t h e v e l o c i t y i s some o t h e r f u n c t i o n t h a n a d e l t a - f u n c t i o n , t h i s f u n -c t i o n must r e p l a c e t h e d e l t a - f u n c t i o n i n t h e e q u a t i o n s f o r t h e shift„ Not o n l y w i l l t h i s c o m p l i c a t e t h e d e r i v a t i o n of t h e ex-p e c t e d s h i f t , i t w i l l a l s o i n c r e a s e the time r e q u i r e d t o e x p e r i -m e n t a l l y d e t e r m i n e the shift„ I n a d d i t i o n a knowledge o f t h e v e l o c i t y f u n c t i o n i s r e q u i r e d . I n our case v i b r a t i o n s a r e p r e s e n t , t h e p a t t e r n and causes of w h i c h have not been c o m p l e t e l y d e t e r m i n e d . Hence th e a p p a r a t u s remains u n u s a b l e . T h i s c h a p t e r d e s c r i b e s t h e p r e s e n t performance of t h e apparatus.. A. OBSERVED PERFORMANCE OF THE AIR-TROUGH SUSPENSION F i g u r e 4-1 shows o u t p u t waveforms from t h e v e l o c i t y m o n i t o r i n g system f o r f o u r v e l o c i t i e s of t h e a i r t r o u g h r i d e r . IWi 1 mmmmmmwkmmmwkmwi A B c D V O L T S / C M 5 V 5 V 5 V T I M E / C M 5 0 MS ^ o MS 2 0 MS 10 M S A I R P R E S S U R E ( P I ) 6 O F / I N 2 6 O Z / | N * 7 . 5 O t / | NJ 7 5 0 £ / l M l M I C R O M E T E R S E T T I N G 0.975 1 . 0 0 0 1 . 0 2 5 " I . 0 5 0 R I D E R V E L O C I T Y " .022 M r t / s . 0 5 0 M f l £ . 0 7 5 " MM/S , 1 0 5 M r t / s T O P T R A C E S - R l D l s R M O V I N G W E S T B O T T O M T R A C E S - R I D C R M O V I N G E A S T F I G U R E 4 - 1 P E R F O R M A N C E O F A I R T R O U G H - I - 45 -These waveforms c o n t a i n f r e q u e n c y m o d u l a t i o n s w h i c h , as n o t e d above, i t i s n e c e s s a r y t o e l i m i n a t e i n o r d e r t o use the a p p a r a t u s . There a r e f i v e o b v i o u s p o s s i b l e s o u r c e s of n o i s e a t the ou t p u t of t h e v e l o c i t y m o n i t o r i n g system; 1. i n t e n s i t y f l u c t u a t i o n s i n t h e l a s e r beam 2. d e f l e c t i o n of t h e m i r r o r s due t o m e c h a n i c a l v i b r a t i o n s o r i g i n a t i n g o u t s i d e t h e a p p a r a t u s 3 . d e f l e c t i o n o f the m i r r o r s due t o m e c h a n i c a l v i b r a t i o n s i n t h e d r i v e mechanism 4 . d e f l e c t i o n o f t h e m i r r o r s due t o f l u i d d y n a m i c a l i n s t a b -i l i t i e s i n t h e a i r s u s p e n s i o n 5. f r i c t i o n due t o i m p e r f e c t i o n s i n t h e d r i v e and s u s p e n s i o n . The f o l l o w i n g s e c t i o n s c o n s i d e r t h e s e n o i s e s o u r c e s . B. LASER FLUCTUATIONS F i g u r e 4 - 2(a) shows waveforms produced by t h e i n t e n s i t y f l u c t u a t i o n s i n t h e l a s e r beam when no i n t e r f e r e n c e o c c u r s (one of t h e m i r r o r s i n the M i c h e l s o n i n t e r f e r o m e t e r was b l o c k e d ) . These waveforms have f r e q u e n c y about 30 cps and a m p l i t u d e 1/2 v o l t . They may be caused by t h e power s u p p l y i n t h e l a s e r . The s p e c t r a l p u r i t y of the l a s e r was v e r i f i e d by d i r -e c t i n g t h e beam i n t o a c o n v e n t i o n a l M i c h e l s o n i n t e r f e r o m e t e r , moving one o f t h e m i r r o r s a t c o n s t a n t v e l o c i t y , and r e c o r d i n g the f r i n g e system w i t h the p h o t o c o n d u c t o r and s t r i p - c h a r t r e c o r d e r . No b e a t s were found i n a p p r o x i m a t e l y 1000 f r i n g e s . A B C D V O L T S / C M 0.5 V below/ 10V 5V T / M E / C M 20 MS 10 MS 20 MS 50 MS AIR PRESSURE (PI) 0 below 6 Ofr/lN2 be levy RIDER IN MOTION NO NO YES Y E S MICROMETER SETTING - — 1.000 1.000 3 . T O P T O B O T T O M P I » O i + ) 6 , 1 7 O " * / | N * £ V / C M * 5 > * * i 2 0 V C . T O P 4 T R A C E 4 M O V I N G W E S T , B O T T O M 2 . E A S T O . T O P - 9 O l / l N ^ W T S O N E N D S . M I D D L E - 9 0 ? / l N * W T S O N A R M S . ' B O T T O M - 6 O j / | N l , N O W T S . F I G U R E 4 - 2 * P E R F O R M A N C E OF A l R - T R p O G H - 3 1 47 -C. MECHANICAL VIBRATIONS W i t h no a i r f l o w i n g t h r o u g h t h e a i r t r o u g h and t h e d r i v e mechanism o f f , 120 c p s , 5 v o l t waveforms were o b s e r v e d . W i t h the a i r s u p p l y t u r n e d on but t h e r i d e r s t i l l a t r e s t , t h e s e waveforms remained. I f the a i r p r e s s u r e ( P I ) was s e t much h i g h e r t h a n t h e optimum s e t t i n g , t h e r i d e r v i s i b l y and a u d i b l y v i b r a t e d , and t h e waveform a m p l i t u d e s were up t o 20 v o l t s (see F i g u r e 4 - 2 ( b ) ) . An i n t e r p r e t a t i o n of t h e s e waveforms i s t h a t t h e r i d e r i s v i b r a t i n g so t h a t one o r two f r i n g e s a r e sweeping back and f o r t h a c r o s s t h e p h o t o c o n d u c t o r . T h i s means t h a t e i t h e r the amp-l i t u d e o f v i b r a t i o n of t h e r i d e r i n t h e d i r e c t i o n of t h e l a s e r beam i s one h a l f t h e l a s e r w a v e l e n g t h , i n whi c h case t h e f r e q u e n c y of t h e v i b r a t i o n i s 120 c p s , o r e l s e t h e a m p l i t u d e i s one l a s e r w a v e l e n g t h and t h e f r e q u e n c y 60 c p s . I n e i t h e r case we assume the o r i g i n of t h e v i b r a t i o n s t o be e l e c t r i c a l m a c h i n e r y elsewhere i n the b u i l d i n g , and t h a t v i b r a t i o n s from t h i s m a c h i n e r y i s t r a n s -m i t t e d t h r o u g h t h e base o f t h e a p p a r a t u s . T h i s s o u r c e c o u l d not be l o c a t e d . L a y e r s of foam r u b b e r p l a c e d between t h e l a y e r s of c o n c r e t e b l o c k s I n t h e base might e f f e c t i v e l y damp out t h e s e v i b r a t i o n s . D, AIR SUSPENSION INSTABILITIES W i t h t h e a i r s u p p l y s e t a t the optimum s e t t i n g ( p o i n t A i n F i g u r e 3-8) and t h e r i d e r i n m o t i o n , waveforms w i t h f r e q u e n c y m o d u l a t i o n s of 20 - 25 cps o c c u r (see F i g u r e s 4-1 and 4 - 2 ( c ) ) . P o s s i b l e causes a r e m e c h a n i c a l v i b r a t i o n s i n t h e d r i v e mechanism - 48 -o r f l u i d d y n a m i c a l i n s t a b i l i t i e s i n t h e a i r s u s p e n s i o n . As t h e r i d e r v e l o c i t y i s v a r i e d , t he f r e q u e n c y o f t h e s e v i b r a t i o n s appears t o remain c o n s t a n t . Hence we assume t h a t t h e y a r i s e from t h e a i r s u s p e n s i o n . I f t h e a i r s u s p e n s i o n i s c a u s i n g t h e s e v i b r a t i o n s we would expect t h e i r f r e q u e n c y t o be a f u n c t i o n of t h e moment of I n e r t i a of the r i d e r . To a l t e r t h i s moment of i n e r t i a , 1.5 l b w e i g h t s were p l a c e d on each arm, and the n on each end of t h e r i d e r . F i g u r e 4-2(d) shows t h a t t h e v i b r a t i o n f r e q u e n c y appears u n a l t e r e d . I f we assume t h a t t h e v i b r a t i o n s of t h e r i d e r a r e harmonic, X ( t ) ^ X 0 S I N p t (4-1) t h e n when t h e r i d e r moves a t a c o n s t a n t v e l o c i t y v 0 , and w i t h t h e s e v i b r a t i o n s superimposed, t h e i n s t a n t a n e o u s r i d e r v e l o c i t y i s Vft)= V 0 + X 0P COS pt ( 4 " 2 ) The f r e q u e n c y produced by t h e M i c h e l s o n i n t e r f e r o m e t e r by moving . one of t h e m i r r o r s a t a v e l o c i t y v i s f ( t ) = f V(t) = f c t A f cos pt <*-3) F i g u r e 4-3(a) i s a p l o t of 4-3. A F o u r i e r a n a l y s i s of 4-3 d e t e r m i n e s t h e f r a c t i o n of time t h e r i d e r spends moving a t a v e l o c i t y v, as a f u n c t i o n of v. - 49 -I n v e r t i n g 4-3 - 50 -R e f e r r i n g t o F i g u r e 4 - 3(a) (4-5) F i g u r e 4-3(t>) i s a p l o t of 4~5» The c u r v e i s peaked a t t h e ends (-f = f<>- ). A F o u r i e r a n a l y s i s was performed on t h e waveforms of F i g u r e 4 - 1 , and i n a l l c a s es t h e c u r v e o b t a i n e d was peaked a t t h e c e n t r e ( T~ - f 0 ) • The a s s u m p t i o n of harmonic o s c i l l a t i o n s appears i n v a l i d . I n F i g u r e 4 - 2(c) i t i s seen t h a t t h e waveform a m p l i t u d e s a r e a f u n c t i o n of t h e d i r e c t i o n i n w h i c h the r i d e r i s moving. The i n t e n s i t y of t h e l i g h t f a l l i n g on the p h o t o c o n d u c t o r depends on the amount of i n t e r f e r e n c e produced by t h e M i c h e l s o n I n t e r f e r o m e t e r , w h i c h depends on t h e a l i g n m e n t of t h e m i r r o r s . We assume the m i r r o r a t t a c h e d t o t h e r i d e r i s a t d i f f e r e n t a n g l e s t o t h e v e r t i c a l when moving i n o p p o s i t e d i r e c t i o n s . To v e r i f y t h i s t h e p o s i t i o n of t h e r e f l e c t i o n of t h e l a s e r beam from the moving m i r r o r was o b s e r v e d as a f u n c t i o n of t h e d i r e c t i o n of m o t i o n . W i t h t h e l a s e r and r e -f l e c t e d s p o t b o t h 194" away from the m i r r o r , a d e f l e c t i o n of 0 . 0 5 " i n t h e p o s i t i o n o f t h e r e f l e c t e d s p o t was o b s e r v e d as t h e r i d e r changed d i r e c t i o n . T h i s c o r r e s p o n d s t o an a n g u l a r change a t t h e -4 m i r r o r o f 1.3x10 r a d i a n s , o r a d i f f e r e n c e i n h e i g h t of t h e ends - 51 -of t h e r i d e r of 1„5 m i l . S i n c e t h e r i d e r i s suspended o n l y t h e o r d e r of a few m i l above t h e r a i l , i t i s p o s s i b l e t h a t f r i c t i o n between t h e r i d e r and r a i l i s r e s p o n s i b l e f o r t h e ob s e r v e d v i b -r a t i o n s . E. FLUID MECHANICAL CONSIDERATIONS Two d i f f e r e n t i a l e q u a t i o n s d e s c r i b e t h e dynamics of f l o w of a v i s c o u s f l u i d : t h e e q u a t i o n of c o n t i n u i t y | £ + J i / ( * v ) = O ( 4 . 6 ) and t h e N a v i e r - S t o k e s e q u a t i o n + ^ ( 7 » g r a d j v s - g r d p + i f 7 2 v + ^ S ™ d <Jiv 7 where ^ i s t h e f l u i d d e n s i t y , ^  t h e dynamic c o e f f i c i e n t of v i s -c o s i t y ( w h i c h we assume t o be independent of v e l o c i t y ) , P t h e exc e s s p r e s s u r e o v e r a t m o s p h e r i c , and V t h e stream v e l o c i t y a t p o i n t ( x , y , z ) . F o r gas f l o w s below Mach Number 0 . 3 (gas v e l o c i t y l e s s t h a n 0 .3 sound v e l o c i t y ) , t h e f l o w can be c o n s i d e r e d i n c o m p r e s s -i b l e ( O l s o n , 1961). I n t h i s case t h e d e n s i t y i s c o n s t a n t and from e q u a t i o n 4-6 dlv* V - O (4-8) - 52 -Combining e q u a t i o n s 4-7 and 4-8 (4-9) Few s o l u t i o n s of e q u a t i o n 4-9 a r e known. An a p p a r a t u s c l o s e l y r e l a t e d t o the a i r t r o u g h i s the a i r s u p p o r t e d puck. I n a paper d i s c u s s i n g the t h e o r y of the a i r s u p p o r t e d puck, Whitney (1964) makes a p o i n t from which i n f e r e n c e s about t h e a i r t r o u g h can be drawn. f l o w i n t h e s u p p o r t i n g f i l m i s v i s c o u s I n c h a r a c t e r . I f i t i s suppo|?^d t h a t t h e gas has z e r o v i s c o s i t y , Whitney shows t h a t t h e puck w i l l o s c i l l a t e up and down. S i m i l a r l y i f t h e f l o w i s such t h a t l n e r t i a l e f f e c t s mask v i s c o u s e f f e c t s , e r r a t i c puck b e h a v i o r r e s u l t s . S t a b l e v i s c o u s f l o w o c c u r s a t stream v e l o c i t i e s s m a l l enough so t h a t t h e l e f t hand s i d e of e q u a t i o n 4-9 may be n e g l e c t e d , l e a v i n g S t a b l e b e h a v i o r of t h e a i r puck o c c u r s when t h e gas (4-10) T a k i n g t h e d i v e r g e n c e of b o t h s i d e s (4-11) Whitney shows t h a t t h e s e e q u a t i o n s can be s o l v e d t o g i v e a s t a b l e - 53 -s o l u t i o n . To make an e s t i m a t e of t h e stream v e l o c i t y we a p p l y the or d e r - o f - m a g n i t u d e r e l a t i o n Q r v = - r - ( 4 ~ 1 2 ) h t o Whitney's s t a b l e puck, and t o our a p p a r e n t l y u n s t a b l e a i r t r o u g h . F o r Whitney's puck, t h e v o l u m e t r i c f l o w r a t e Q i s 200 cmVsec. o r 0 . 5 cfm, and t h e t h i c k n e s s o f the gas f i l m h i s 0.01 cm o r 4 m i l . Then t h e p r o d u c t of stream v e l o c i t y and r a d i a l d i s -t a n c e from t h e gas i n l e t r v i s 30 f t / s e c . I n our case Q = 1.6 cfm, h = 5 m i l g i v i n g r v = 80 f t / s e c . Because t h e s e stream v e l o c i t i e s a r e s i m i l a r , t h e g e o m e t r i c d i f f e r e n c e s between t h e two systems must be c o n s i d e r e d i f one i s t o draw any c o n c l u s i o n s about the f l o w c h a r a c t e r o f the a i r t r o u g h . - 54 -APPENDIX ONE FREQUENCY AND LIFETIME OF LOCALIZED MODES C a l c u l a t i o n s of LM f r e q u e n c y ( O c M a n d l i f e t i m e ^ L M a r e e x t r e m e l y c o m p l i c a t e d . To keep t h e mathematics manageable, s i m p l e models must be assumed. I n g e n e r a l t h e s e models a r e not r e a l i s t i c , and t h e c a l c u l a t i o n s o n l y q u a l i t a t i v e . I n t h i s a p p e n dix we s u r v e y t h e c a l c u l a t i o n s w h i c h have been made. We c o n s i d e r f i r s t t h e case when t h e f r a c t i o n a l mass d e f e c t £ (see e q u a t i o n 1-1) i s p o s i t i v e ( i . e . a l i g h t I m p u r i t y ) . We l a t e r i c o n s i d e r t h e s i m p l e r case when £ i s n e g a t i v e ( a heavy i m p u r i t y ) . We a p p l y t h e c a l c u l a t i o n s t o t h e examples of Fe57 m P t W ( £ = 0.708) and F e 5 7 i n B e 9 (£ = -5.3). A. £ > Q : FREQUENCY OF LM IN HARMONIC APPROXIMATION Most c a l c u l a t i o n s of the f r e q u e n c y of the LM g i v e C J L M / t J f t as a f u n c t i o n of t h e f r a c t i o n a l mass d e f e c t S» I n our example (Fe5? i n P t 1 9 5 ) £ = 0 . 7 0 8 . F o r a s i n g l e s u b s t i t u t i o n a l i s o t o p i c i m p u r i t y a t t h e o r i g i n i n a c u b i c B r a v a l s c r y s t a l w i t h harmonic f o r c e s , s e v e r a l a u t h o r s ( e . g . Maradudln, 1964a, e q u a t i o n 41) have shown t h a t WLM/GW)D as a f u n c t i o n of 6 i s de t e r m i n e d by I - e o* GJw») = 0 ( A l - l ) - 55 -where G^(6) z) i s the H i l b e r t t r a n s f o r m of t h e d i s t r i b u t i o n f u n c t i o n G0(t»)*) f o r t h e squares o f t h e normal mode f r e q u e n c i e s of t h e p e r -f e c t h o s t c r y s t a l . 0)L^/a)D ±s t h e n dependent on t h e c r y s t a l model ( i . e . t h e f r e q u e n c y d i s t r i b u t i o n ) chosen. We c o n s i d e r f o u r models (see F i g u r e A l - 1 ) . Model 1. F o r a l i n e a r c h a i n G 0 ^ V - { < t f u ( ^ - " f 2 f ( A 1 - 2 ) and we o b t a i n ( M o n t r o l l and P o t t s , 1955* e q u a t i o n 3.20) F o r 6 = 0.?08 t h e n 6 J t | A / ( J 0 = 1 .49 . ( A l - 3 ) (J, / F o r £ m 0.708 t h e n <A*/<»)b = 1.41. Model 2 . F o r t h e p a r t i c u l a r l y s i m p l e Debye spectrum we o b t a i n (Dawber and E l l i o t , 1963 and Maradudin, I963 ) ( A l - 5 ) Mpdel 3. F o r t h e case of a n e a r e s t - n e i g h b o u r s i m p l e c u b i c l a t t i c e FIGURE A l - 1 ; LATTICE FREQUENCY DISTRIBUTIONS - 57 -i n w h i c h t h e c e n t r a l and n o n - c e n t r a l f o r c e c o n s t a n t s a r e e q u a l , M o n t r o l l and P o t t s (1955) o b t a i n e d t h e f r e q u e n c y d i s t r i b u t i o n U s i n g t h i s d i s t r i b u t i o n , V i s s c h e r (1963) c a l c u l a t e d &)LMA>D from e q u a t i o n A l - 1 , and f i t t e d t h e f o l l o w i n g e q u a t i o n t o h i s c a l c u l -a t i o n s 2(1-€) U 1 " 7 ) F o r 6 = 0.708 t h e n (^LH/OD = 1 .36. Model 4. F o r th e case o f a n e a r e s t - n e i g h b o u r c e n t r a l f o r c e f a c e -c e n t r e - c u b i c l a t t i c e O v e r t o n and Dent ( i960) o b t a i n e d t h e f r e q u e n c y d i s t r i b u t i o n ( A l - 8 ) U s i n g t h i s d i s t r i b u t i o n N a r d e l l l and T e t t a m a n z i (1962) d i s c o v e r e d t h a t a d i s c r e t e LM appears o n l y f o r S > 0 .215, and f o r £ = 0.708 t h e n (A.M/QD - I . 3 8 . We a c c e p t t h i s model as most r e a l i s t i c , and use t h i s v a l u e of 0LM/O)D h e n c e f o r t h i n our c a l c u l a t i o n s . B. 6^0; FREQUENCY OF LM FOR ANHARMONIC FORCES I f we c o n s i d e r t h e I n t e r a t o m i c f o r c e s t o have anharmonic terms, t h e n CJL«A)0 s h o u l d have a s m a l l e r v a l u e t h a n i n t h e harmonic - 58 -a p p r o x i m a t i o n ( V i s s c h e r , 1964)* The s t r e n g t h o f the anharmonic i n t e r a c t i o n i n c r e a s e s w i t h t e m p e r a t u r e . C o n s i d e r i n g c u b i c and q u a r t i c a n h a r m o n i c i t i e s a t the h i g h t e m p e r a t u r e l i m i t i n a l i n e a r c h a i n w i t h f o r c e c o n s t a n t s chosen t o e q u a l t h o s e o f l e a d , Maradudin (1964b) has c a l c u l a t e d t h e s h i f t i n f r e q u e n c y o f th e LM from i t s v a l u e i n t h e harmonic a p p r o x i m a t i o n . F o r QL* foo = 1.38, A(ULM) _ T (Al-9) 6)D IS 6+ I f we assume t h a t t h e l i m i t i n g h i g h t e m p e r a t u r e v a l u e of th e e q u i v a l e n t Debye t e m p e r a t u r e Q ^ s T , t h e n &(QLm) a.05Ob and f o r our purposes can be n e g l e c t e d . C. €>0; LIFETIME OF LM DUE TO ANHARMONIC FORCES The i n t r o d u c t i o n o f c u b i c a n h a r m o n i c i t i e s i n t o our c r y -s t a l model w i l l r e s u l t i n f i n i t e l i f e t i m e s . Q u a r t i c a n h a r m o n i c i -t i e s do not have any e f f e c t upon the l i f e t i m e (Maradudin and F e i n , 1962). The l i f e t i m e depends upon the s t r e n g t h of th e c u b i c an-harmonic i n t e r a c t i o n , and e x p r e s s i o n s f o r t h e l i f e t i m e I n c l u d e s uch a s t r e n g t h f a c t o r . We r e v i e w f o u r c a l c u l a t i o n s below. A l l but t h e f i r s t use a s i m p l e c u b i c B r a v a i s l a t t i c e w i t h a s i n g l e s u b s t i t u t i o n a l i s o t o p i c i m p u r i t y a t t h e o r i g i n , and c a l c u l a t e t h e l i f e t i m e a t a b s o l u t e z e r o of t e m p e r a t u r e . C a l c u l a t i o n 1. Maradudin (1964b) has found an e x a c t s o l u t i o n f o r - 59 -t h e case o f a monatomic l i n e a r c h a i n a t th e h i g h t e m p e r a t u r e l i m i t . C h o o s i n g f o r c e c o n s t a n t s t o r e p r e s e n t t h o s e of l e a d , and f o r (x)LM /O)Q = 1 .38, he c a l c u l a t e s 1 a 0.1+ifcT CJ, ( A 1 _ 1 0 ) I f we can assume t h a t T ~ 9^,then \ /*X — ^b/l C a l c u l a t i o n 2. U s i n g t h e s i m p l e f r e q u e n c y d i s t r i b u t i o n of Model 2 ( e q u a t i o n A l - 4 ) Maradudin (1963 ) a r r i v e s a t t h e e q u a t i o n ! - J U - 3 i r u D * * I m ( ^ M W ( A i - i i ) r i M v v where t = G r u n e i s e n ' s c o n s t a n t = i m p u r i t y mass V = v e l o c i t y of sound X^&IMAO = 1.6 x 10"5 f o r W t M/w» = I . 3 8 F o r JfO: 2, a n d t i t ^ ^ 1/100 t h e n l / t = ^ t > / 2 0 * 7 C a l c u l a t i o n 3 . U s i n g t h e f r e q u e n c y d i s t r i b u t i o n o f Model 3 (equa-t i o n A l - 6 ) , Klemens ( I96I) a r r i v e s a t th e e q u a t i o n where |< = an a t t e n u a t i o n l e n g t h |^  » d i a m e t e r o f a sphere which a p p r o x i m a t e s t h e B r i l l o u i n Zone. X^LMM = 0.025 f o r (AU/ W o = I . 3 8 . 60 F o r |<, and tfc: 2 , ^ 1/100 as above^ l / ^ r t)b/2SOt C a l c u l a t i o n 4 . V i s s c h e r (1964) p o i n t s out t h a t o v e r s i g h t s by Klemens i n C a l c u l a t i o n 3 make t h a t e s t i m a t e of t h e l i f e t i m e t o o s h o r t . U s i n g t h e same f r e q u e n c y d i s t r i b u t i o n V i s s c h e r a r r i v e s a t the e q u a t i o n l^usiuor7- (i-e)* /QLM? Jv (oWk) (Ai - 1 3 ) where T = r a t i o of anharmonic/harmonic p o t e n t i a l energy when LM i s e x c i t e d , and l V i s s c h e r has e v a l u a t e d Xy; = 5 x 10~3 f o r t O L M / [ J 0 = 1.12 and assumes t h a t r ^  1/100, I n t h a t case | / f a c J D / a 3 S 5 A t t e m p e r a t u r e s above a b s o l u t e z e r o , t h e s t r e n g t h of the anharmonic i n t e r a c t i o n w i l l i n c r e a s e . Klemens (1961) has e s t i -mated t h a t a t tem p e r a t u r e T, (Al - 1 5 ) D. S^O; LIFETIME OF LM DUE TO IMPURITY CONCENTRATION I n any r e a l experiment t h e r e w i l l be a c o n c e n t r a t i o n of i m p u r i t y atoms, r a t h e r t h a n one s i n g l e i m p u r i t y a t t h e o r i g i n as - 61 -assumed so f a r . W i t h z e r o c o n c e n t r a t i o n t h e LM i s h i g h l y degener-a t e , and w i l l be broadened i n t o an i m p u r i t y band as t h e con c e n t -r a t i o n i n c r e a s e s , , By w o r k i n g a t low enough c o n c e n t r a t i o n s , i t s h o u l d be p o s s i b l e t o m i n i m i z e t h i s e f f e c t . U s i n g t h e f r e q u e n c y d i s t r i b u t i o n of Model 3 ( e q u a t i o n A l - 6 ) , D i n h o f e r (1963) has c a l c u l a t e d t h a t T 281 f o r t h e case £ = 0 ,4 and t h e i m p u r i t y / h o s t atom r a t i o i s 1/100, I n our case £ = 0.708 and t h e i m p u r i t y / h o s t atom r a t i o i s 1/1000, so we would expect a l o n g e r l i f e t i m e t h a n D i n h o f e r ' s . E. £ { 0 ; FREQUENCY AND LIFETIME OF LM The s o l u t i o n o f e q u a t i o n A l - 1 , based on Model 2 ( e q u a t i o n A l - 4 ) , f o r € < 0 and whi c h v a n i s h e s as °° i s (Maradudin, 1963 ) I 7 ' • ' - "•• (Al - 1 7 ) 3|€| 9ler e T I F I f |6| i s so l a r g e t h a t we can n e g l e c t a l l but t h e f i r s t term we o b t a i n t h e e x p r e s s i o n g i v e n by B r o u t and V i s s c h e r (I96I) = (- \ - V 2 ( A l - 1 8 ) Maradudin g i v e s t h e l i f e t i m e as - 62 r (Al - 1 9 ) w h i c h f o r Model 2 i s U s i n g e q u a t i o n A l - 1 , and f o r l a r g e |£| t h i s reduces t o t h e e x p r e s s i o n g i v e n by B r o u t and V i s s c h e r ( A l - 2 0 ( A l - 2 1 ) F o r £ = -5.3, from e q u a t i o n s A l - 1 8 and A l - 2 1 = 0 .251 and l / f =Uo/%0>%. From e q u a t i o n s A l - 1 7 and A l - 2 0 U)irM/&Jb= 0 .259 and ' / t = W D / / 7 .7 . F. N0M-IS0T0PIC IMPURITIES We have not t a k e n a c c o u n t of the f a c t t h a t t h e i m p u r i t y may not be i s o t o p l c . The i n t e r a t o m i c f o r c e c o n s t a n t s a f f e c t i n g t h e i m p u r i t y may be d i f f e r e n t from t h o s e a c t i n g on t h e p e r f e c t l a t t i c e . I t i s d i f f i c u l t t o p r e d i c t whether o r not t h i s w i l l be t h e c a s e , and i f s o , t o e s t i m a t e t h e magnitude of t h e d i f f e r e n c e . V i s s c h e r (1963) has assumed f o r c e c o n s t a n t changes, and has a n a l y z e d t h e i r e f f e c t on t h e LM f r e q u e n c y i n t h e harmonic a p p r o x i -- 63 -E l a t i o n . We must, from i g n o r a n c e , assume t h a t o u r i m p u r i t i e s a r e i s o t o p i c . G, ENERGY AND LINEwflDTH OF LOCALIZED MODES I n t h i s s e c t i o n we make some n u m e r i c a l c a l c u l a t i o n s of LM energy and l i n e w l d t h f o r t h e examples F e ^ 7 i n P t 1 9 ^ a n a F e ^ 7 i n Be?. We assume t h e f o l l o w i n g f r e q u e n c y and l i f e t i m e e s t i m a t e s , F o r F e ^ 7 i n P t 1 9 5 : CJLM/^D= 1.38 and • = 6 J D f I + 3 T Tl — —— s —~(~\—i \\ (Ai - 1 5 ) T L M 2J53 t <9 D (^ - *A)D- ' )J F o r F e 5 7 i n B e 9 : CJLM/cjb= 0.259 and * / V t M =• ^ J p / l 7 . 7 . There i s no r e a s o n t o assume t h a t t h e Debye temperature a t t h e i m p u r i t y w i l l be t h e same as f o r t h e h o s t . The Debye temp-e r a t u r e i s p r o p o r t i o n a l t o t h e phonon c u t o f f f r e q u e n c y , 6 J 0, and from e q u a t i o n 1-2 Hence, a somewhat l e s s crude a s s u m p t i o n ( S c h i f f e r e t a l , 1964) i s IMPURITY - 6 4 -F o r P l a t i n u m , t h e Debye tem p e r a t u r e i s 240°K, so t h a t 0^F F m 820°K. F o r B e r y l l i u m , t h e Debye tem p e r a t u r e i s ll60°K, so t h a t Q e p p = l62°K, T a b l e A l - 1 p r e s e n t s t h e r e s u l t s of c a l c u l a t i o n s on Qb L^ci» ^ ty* ^ H.Mt tiu*i» r*LM f o r t h e two ca s e s above. - 65 -TABLE A l - 1 : LOCALIZED MODE PARAMETERS ( F e ^ 7 IN P t 1 9 ^ AND B e 9 ) F e 5 7 IN P t 1 9 5 F e ^ 7 IN B e 9 Debye tem p e r a t u r e a t i m p u r i t y s i t e 820°K 162°K Debye f r e q u e n c y O p l.OSxlO^-Vseo. 2 . 1 2 x l 0 1 Vsec . R a t i o 1.38 0.259 LM f r e q u e n c y L ^ l O ^ / s e c . 5 . 5 x l 0 1 2 / s e c . Debye energy i i 6 ) D 0.0708 ev 0.014 ev LM energy 0.0975 ev O.OO36 ev LM l i f e t i m e ( a t 300°K) f LM 5 . 6 5 x l 0 " 1 2 s e c . 8.35xlO~ 1 3sec„ LM l i n e w i d t h fL M 1 . 1 7 x 1 0 " " ^ 7.9x10"^ ev - 66 -APPENDIX TWO ONE-PHONON EXPERIMENTS I n t h e case of e m i s s i o n of a Mossbauer gamma r a y w i t h a s s o c i a t e d e m i s s i o n o r a b s o r p t i o n of one phonon, t h e gamma r a y energy w i l l d i f f e r from i t s r e c o i l l e s s v a l u e by t h e phonon e x c i t -a t i o n energy. I f t h e Mossbauer atom i s one of the atoms of t h e h o s t c r y s t a l ( i . e . i s not an i m p u r i t y ) , t h e n t h e one-phonon Mossbauer a b s o r p t i o n c r o s s - s e c t i o n s i m p l y maps out t h e f r e q u e n c y spectrum of t h e h o s t c r y s t a l ( V i s s c h e r , i 9 6 0 ). However, t h i s f r e q u e n c y spectrum may a l s o be o b t a i n e d from c o h e r e n t i n e l a s t i c n e u t r o n s c a t t e r i n g e x p e r i m e n t s (Brockhouse, i 9 6 0 ), a t e c h n i q u e which i s w e l l e s t a b l i s h e d , and en c o u n t e r s fewer e x p e r i m e n t a l d i f f i c u l t i e s ( B o y l e and H a l l , 1962; V i s s c h e r , 1963)0 I f t h e Mossbauer atom i s an i m p u r i t y i n a h e a v i e r h o s t l a t t i c e , t h e n t h e f r a c t i o n a l mass d e f e c t £ ) 0 . A l o c a l i z e d mode a r i s e s as seen i n Cha p t e r One, and t h e one-phonon Mossbauer a b s o r p -t i o n c r o s s - s e c t i o n s e p a r a t e s i n t o normal mode and l o c a l i z e d mode c o n t r i b u t i o n s . The LM c o n t r i b u t i o n g i v e s r i s e t o peaks i n the Mossbauer spectrum d i s p l a c e d by t h e LM e n e r g y , t f r o m t h e c e n t r a l (zero-phonon) Mossbauer peak. A v e r y d i r e c t measurement of t h e LM f r e q u e n c y i s t h u s p o s s i b l e by i n t r o d u c i n g a v e l o c i t y between t h e s o u r c e and a b s o r b e r such t h a t t h e one-phonon LM Moss-bauer peak of t h e so u r c e i s D o p p l e r s h i f t e d t o c o i n c i d e w i t h t h e - 67 -c e n t r a l (zero-phonon) Mossbauer peak of t h e a b s o r b e r (Maleev, I960; Mozer and V i n e y a r d , 1961; V i s s c h e r , p. 30 i n Compton and Schoen, 1962). I f t h e Mossbauer atom i s an i m p u r i t y i n a l i g h t e r h o s t l a t t i c e , t h e n G ^ O a n d the v i r t u a l LM d i s c u s s e d by B r o u t and V i s s c h e r (1962) a p p e a r s . The one-phonon Mossbauer a b s o r p t i o n c r o s s - s e c t i o n t h e n c o n s i s t s a l m o s t e n t i r e l y of a resonance peak a t the low end of t h e f r e q u e n c y spectrum. An experiment s i m i l a r t o t h e one above may be performed t o measure t h i s r esonance. I n t h i s a p p e ndix we c o n s i d e r examples of e x p e r i m e n t s of b o t h t y p e s . F o r t h e case € > 0 , we use Fe57 i n P t 1 9 5 ( g = 0.708) as our example. F o r £ < 0 , we use Fe^7 i n B e 9 ( £ = - 5 . 3 ) as our example. A. SOUBCE AND ABSORBER CONSIDERATIONS L e t I(o), t h e a c t i v i t y o f t h e s o u r c e , be 5 m i l l i c u r i e s . L e t t h e s o u r c e be Fe^? a s an i s o t o p i c i m p u r i t y i n and i n Be". Assume the I m p u r i t y c o n c e n t r a t i o n i s d i l u t e enough not t o i n t r o -duce any b r o a d e n i n g e f f e c t on t h e LM peak. We conduct our e x p e r i -ment a t room t e m p e r a t u r e . From T a b l e A l - 1 of Appendix One, f o r Fe^? m P t 1 ^ , f) O t M = 0 .0975ev and 1"^ = I , l 6 x l 0" i |'ev. F o r F e ^ 7 i n B e 9 , "fi U t M = O.OO36 ev and P L K t = 7 . 9 x 1 0 " * ^ . The l i n e w i d t h of t h e c e n t r a l (zero-phonon) Mossbauer - 68 -peak i n Fe57 i s P 0 = 4.5xlO~ 9 e v , Then P 0 ^ ( 1"^, and o n l y a f r a c t i o n P D / r ^ ^ l y i n g a t t h e c e n t r e of t h e LM peak w i l l be a bsorbed when the LM peak of t h e s o u r c e o v e r l a p s t h e c e n t r a l peak of t h e a b s o r b e r . I t would be advantageous t o f i n d an a b s o r b e r w i t h a l i n e w i d t h much b r o a d e r t h a n . Such an a b s o r b e r i s t h e " b l a c k a b s o r b e r " d e s c r i b e d by H o u s l e y e t a l (1964). T h i s c o n s i s t s of a m i x t u r e of ammonium f l u o r o f e r r a t e (which has two s p l i t c e n t r a l peaks) and l i t h i u m f l u o r o f e r r a t e ( w h i c h has a s i n g l e wide c e n t r a l p e a k ) . When mixed i n t h e r i g h t p r o p o r t i o n s , t h e r e s u l t a n t Mossbauer spectrum a p p r o x i -mates a f l a t - t o p p e d l i n e of w i d t h = 15 1"^  = 6 , 8 x l 0 " ^ e v o We assume t h a t t h e a b s o r b e r s u r f a c e d e n s i t y of Fe57 i s 40 mg Fe57/cm 2. T h i s c o r r e s p o n d s t o 4 . 2 2 x l 0 2 0 atoms/cm 2. B. DRIVE CONSIDERATIONS The D o p p l e r s h i f t , t l CdL^, r e q u i r e d t o o v e r l a p the z e r o -phonon l i n e of t h e a b s o r b e r w i t h the LM peak of the s o u r c e , nec-e s s i t a t e s v e l o c i t i e s v = c r\e»)Urt/E0 = 2030 metres/second f o r Fe-57 i n P t 1 9 5 and 75 metres/second f o r Fe57 i n B e 9 . One way of a c h i e v i n g such r e l a t i v e v e l o c i t i e s i s t o employ a r o t o r d r i v e . A s c a t t e r i n g experiment can be performed by d e p o s i t i n g t h e a b s o r b e r on t h e o u t s i d e r i m of t h e r o t o r , and u s i n g t h e e x t r e m e l y s t a b l e r o t o r s w h i c h a r e a v a i l a b l e ( E g e l s t a f f , 1961). F i g u r e A2-1 shows th e g e o m e t r i c a l arrangement of such an e x p e r i m e n t . - 69 -FIGURE A2-1: ONE-PHONON EXPERIMENT - GEOMETRY - 70 -The r e l a t i v e v e l o c i t y i s VR= r o cos( 0 + e ) (A2-1) where r i s c o n s t a n t and CO can be c o n t r o l l e d t o one p a r t i n 10D O n l y t h e a n g l e of i n c i d e n c e can cause g e o m e t r i c a l v e l o c i t y b r o a d e n i n g . The i n t e n s i t y of LM gamma r a y s e m i t t e d by t h e so u r c e and f a l l i n g on t h e s o l i d a n g l e subtended by t h e r o t o r a t the s o u r c e i s where fjjyj i s t h e f r a c t i o n of gamma r a y s from t h e Mossbauer t r a n -s i t i o n i n t h e s o u r c e w h i c h a r e e m i t t e d w i t h energy E c . The f a c t o r 1/2 i s i n c l u d e d because t h e D o p p l e r s h i f t i s i n o n l y one d i r e c t i o n A J 1 i s d e t e r m i n e d by t h e D o p p l e r v e l o c i t y r e s o l u t i o n r e q u i r e d . T h i s i s t h e r a t i o of t h e a b s o r b e r l i n e w i d t h t o t h e LM energy: (A2-2) (A2-3) F o r F e 5 ? i n P t 1 9 ^ t h i s i s 7x10"? s t e r a d i a n s . F o r Fe^? i n B e 9 t h i s i s 1.9x10"^ s t e r a d i a n s . Assume t h a t a l l of I]_ i s a b s o r b e d , and t h a t t h e c o u n t e r subtends one s t e r a d i a n a t t h e a b s o r b e r . Then t h e i n t e n s i t y „ 71 counted i s X i Kl ~ — — * — — * 1 1 ( A 2 ~ 4 ) C. RAYLEIGH SCATTERING Competing i n t e r a c t i o n s o f gamma r a y s w i t h the a b s o r b e r w h i c h might produce s p u r i o u s c o u n t s may be i g n o r e d , w i t h t h e ex-c e p t i o n o f R a y l e i g h s c a t t e r i n g . I F o r R a y l e i g h s c a t t e r i n g (A2 -5) where r 0 = e 2 / m e c 2 = 2 . 8 x 1 0 " c m < > i n o u r case Qsft/z and Nelms and Oppenheim {1955, T a b l e 2) g i v e f/Z as a f u n c t i o n o f k = 2 a Q s i n (6/2) / H where a Q = fiVrnge2 = 5 . 2 9 x l 0 " " 9 cm. and /( = f i c / E 0 = 1 . 3 8 x l 0 " 9 cm. i n our c a s e . Thus k = 5*^5 and from Nelms and Oppenheim, f o r Z = 26, f = 7 . 4 7 . From e q u a t i o n A2 - 5 t h e n 6o\j6fL = 2.2 b a r n s / s t e r a d i a n . We have assumed t h a t t h e c o u n t e r subtends one s t e r a d i a n a t t h e a b s o r b e r , so 0*R = 2.2 b a r n s . The i n t e n s i t y c o u n t e d due t o R a y l e i g h s c a t t e r i n g i s (A2-6) - 72 -where Q / \ i s t h e a t o m i c s u r f a c e d e n s i t y of i n the a b s o r b e r . D. F e 5 ? IN P t 1 9 ^ We r e q u i r e some e s t i m a t e of "f^ j^ * t h e f r a c t i o n of gamma r a y s from t h e Mossbauer t r a n s i t i o n w h i c h a r e e m i t t e d from t h e sou r c e w i t h energy E + 1")CA>LM» W e f i r s t e s t i m a t e f ] _ , t h e one-phonon c o n t r i b u t i o n t o t h e t o t a l p r o b a b i l i t y of Mo s s b a u e r . e m i s s i o n . We t h e n e s t i m a t e what f r a c t i o n f L M i s of f ] _ . From e q u a t i o n 1-8, f ^ = f Q l n ( l / f Q ) . We assume t h a t f Q = 0 . 7 . Then f i = 0 . 2 5 . We now e s t i m a t e fjji, u s i n g F i g u r e A2-2 ( o r i g i n a l l y p l o t t e d by Maradudin, 1964a, f i g u r e 1). F i r s t we d i s c u s s t h e f u n c t i o n p l o t t e d I n F i g u r e A2-2. Assuming a s i n g l e s u b s t i t u t i o n a l I m p u r i t y a t t h e o r i g i n i n a c u b i c B r a v a i s c r y s t a l , Maradudin g i v e s t h e one-phonon Moss-bauer a b s o r p t i o n c r o s s - s e c t i o n as <r,= TTo*0r0 e " 2 V / I 0 ( a b » ) SGM CO ^ G»( <f)  e x p ( - ^ r > G ) t M / 2 ) etpiptjcj^h) C ^ u J . (A2 - 7 ) FIGURE A2-2: LATTICE FREQUENCY USED ,T0 ESTIMATE f i n - 74 -where G 0 ( Q ) i s the d i s t r i b u t i o n f u n c t i o n of the squared f r e -quencies of the normal modes of the p e r f e c t host l a t t i c e ; G 0(G J ) 2 i s the H i l b e r t transform of GQ(l) ) ; M i s the mass of a host atom; and 2 V = d & _ \ do) COTHOW*) G B « o a ) (A2-8) x C O T H ( y S n o L M / 2 ) 2 l \ = i_ r l t I X E * p ( / 3 f t ^ * / 2 ) (A2 - 9 ) where B ( W 2 ) i s a f u n c t i o n of the squared frequency d i s t r i b u t i o n , , F igure A2-2 shows the f u n c t i o n - 75 -G 0 C c o a ) (A2-10) p l o t t e d a g a i n s t W/00, f o r the f r e q u e n c y d i s t r i b u t i o n G 0(&) 2) of of Model k ( e q u a t i o n A i - 8 ) . From e q u a t i o n A2-7 i t can be seen t h a t a t the a b s o l u t e z e r o of temp e r a t u r e t h e shape of t h e one-phonon Mossbauer spectrum i s g i v e n by t h e f u n c t i o n A2-10. F o r t h e case 6 = 0 ( t h e Mossbauer atom i s a h o s t atom), we have c u r v e A i n F i g u r e A2-2„ Curve B I s f o r t h e case e = 0,5 (a LM peak has s e p a r a t e d from t h e band). The r a t i o of t h e a r e a under Curve B t o t h a t under Curve A i s , from t h e f i g u r e , 0 . 6 2 , Hence 0.38 of t h e a r e a under c u r v e A must appear under t h e LM peak i n Curve B. We i n t e r p r e t t h i s t o mean fLM = °'38 f]_ f o r t h e case 6 = 0.5 and a t T = 0°K. We make t h e as s u m p t i o n t h a t f j ^ = O.38 f x f o r e = 0.708 and T = 300°K. Then fLM = 0-09. T a b l e A2-1 l i s t s t h e c a l c u l a t e d v a l u e s o b t a i n e d i n t h i s a p p e n d i x . I n summary, t h e e x p e c t e d count r a t e due t o e x c i t a t i o n of t h e LM i s 3»l4xl0"Vs econd, and t h e e x p e c t e d count r a t e due t o R a y l e i g h s c a t t e r i n g i s 2,2xl0" 2/second. E. F e ^ 7 IN B e 9 I n t h i s case a resonance peak a r i s e s a t t h e low f r e q u e n c y - 76 -end of the one-phonon c r o s s - s e c t i o n which comprises p r a c t i c a l l y a l l of the one-phonon c r o s s - s e c t i o n . We assume = f]_ = 0.25» Table A2-1 l i s t s the c a l c u l a t e d values obtained i n t h i s appendix. In summary the expected count r a t e due to e x c i t a t i o n of the LM Is 2.35xl0~ 2/second and the expected count r a t e due to Rayleigh s c a t t e r i n g i s 6xlO~3/second. - 77 TABLE A2-1: ONE-PHONON EXPERIMENT CALCULATED RESULTS ( F e ^ 7 IN  P t 1 9 ^ AND B e 9 ) F e 5 7 IN P t 1 9 ^ F e 5 7 IN B e 9 SPECIFIED: A c t i v i t y of so u r c e A b s o r b e r l i n e w i d t h A t o m i c s u r f a c e den-s i t y of a b s o r b e r LM energy LM l i n e w i d t h CALCULATED: D o p p l e r v e l o c i t y S o l i d a n g l e subtended by r o t o r a t sour c e LM f r a c t i o n LM i n t e n s i t y R a y l e i g h i n t e n s i t y X6>) 1 . 8 x l 0 8 / s e c . 1.8xl08/sec„ P A 6 . 8 x l 0 ~ 8 e v 6 . 8 x l 0 ~ 8 e v 9 A 4 . 2 2 x l 0 2 0 a t o m s / c m 2 4 . 2 2 x l 0 2 0 a t o m s / c m 2 f)U T M 0.0975 ev p.^ 1.16X10" 2 1 , ev O.OO36 ev 7.9x10 ^ ev V 2030 m/sec. 75 m/sec. ^ I c 7xl0"°7steradians 1.9x10"^ s t e r a d i a n s 0.09 I L f 1 3 , 1 2 x l 0 " Y s e c , T » 2 . 2 x l 0'~ 2/sec. 0.25 2 . 3 5 x l 0" 2/sec, 6 x l 0 ~ 3/sec. - 78 -APPENDIX THREE DIFFUSION OF A RADIOACTIVE IMPURITY INTO A HOST I n t h i s a ppendix we d e r i v e an e x p r e s s i o n e q u i v a l e n t t o e q u a t i o n 2-6 f o r t h e r a t i o of the i n t e n s i t y of t h e s o u r c e a f t e r d i f f u s i o n I ( s ) t o t h e I n t e n s i t y b e f o r e d i f f u s i o n I ( o ) . T h i s i s a measurable q u a n t i t y , b e i n g a l s o t h e r a t i o of count r a t e s a f t e r and b e f o r e d i f f u s i o n , geometry h e l d c o n s t a n t . T h i s i s one method used t o f i n d t h e d i f f u s i o n c o e f f i c i e n t . I n our d e r i v a t i o n we assume an i n f i n i t e c y l i n d e r as ho s t m a t e r i a l . We t h e n quote r e s u l t s f o r t h e case when the h o s t f o i l i s not i n f i n i t e l y t h i c k . F i n a l l y , we e v a l u a t e I ( s ) / l ( o ) f o r the case of F e57 m P t W c o n s i d e r e d i n Chapter Two, and show t h a t t h e u n i f o r m d i s t r i b u t i o n we assumed I n Chapter Two g i v e s r e s u l t s which agree w e l l w i t h t h o s e of t h i s a p p e n d i x . A. DIFFUSION INTO AN INFINITE CYLINDER The d i f f u s i o n e q u a t i o n i n one d i m e n s i o n ( P i c k ' s Second Law) i s A_C(x,t) = K *L_ C(x.t) , where C ( x , t ) i s t h e c o n c e n t r a t i o n a t depth x, a f t e r a d i f f u s i o n of d u r a t i o n t , and we have assumed t h a t t h e d i f f u s i o n c o e f f i c i e n t K i s independent of c o n c e n t r a t i o n . - 79 -Assume t h a t we d e p o s i t M gamma-ray r a d i o a c t i v e atoms i n a t h i n u n i f o r m l a y e r on one end of an i n f i n i t e l y l o n g c y l i n d e r of c r o s s - s e c t i o n a l a r e a A. We heat t h e c y l i n d e r t o a temperature such t h a t d i f f u s i o n o c c u r s f o r a d u r a t i o n t . The boundary con-d i t i o n s on our s o l u t i o n t o e q u a t i o n A3-1 a r e 1. C(x,0) = 0 except a t x = 0 2. C(0,0) = CO oo 3. f C ( x , t ) dx = M/A The a p p r o p r i a t e s o l u t i o n i s The r o o t mean square d i f f u s i o n depth d, i s ^ x a C ( x / O d x a - _ 2 = 2 K t .The-mean d e n s i t y i s ^ C ( x / 0 * d x ^ (A3-3) \ C(x,t) Hence - 80 -I f A i s t h e r a d i o a c t i v e decay c o n s t a n t , t h e n t h e a c t i v i t y a t t he c y l i n d e r s u r f a c e b e f o r e d i f f u s i o n i s I ( o ) = A M. Assume t h a t t h e r a d i o a c t i v e mean l i f e , l / A , I s v e r y l o n g compared t o the d i f f u s i o n d u r a t i o n t . Analogous t o e q u a t i o n 2-6, the gamma r a y i n t e n s i t y a t t h e c y l i n d e r s u r f a c e a f t e r d i f f u s i o n i s (A3-6) = X(o) EXP(rV2) ERFC ( TVV?) where /A^ i s t h e gamma r a y a b s o r p t i o n c o e f f i c i e n t of t h e c y l i n d e r m a t e r i a l , and y =/4^i« T a b l e A3-1 g i v e s I ( s ) / l ( o ) as a f u n c t i o n o f y. B. DIFFUSION INTO A NON-INFINITE CYLINDER The above r e s u l t I s v a l i d f o r an I n f i n i t e l y l o n g c y l i n d e r ( i . e . t h e h o s t f o i l t h i c k n e s s x D ^ > d ) . F o r f i n i t e h o s t f o i l t h i c k -ness x 0 , and d not much l a r g e r than x G , Steigman e t a l (1939) have shown t h a t TABLE A 3 - 1 ; I ( s ) / l ( o ) AS A FUNCTION OF Y (SEE EQUATION A 3 - 6 ) Y I ( s ) / l ( o ) Y I ( s ) / l ( o ) 0.0 1.0 1.2 0.4725 0.02 0.9843 1 , 4 0.4294 0.04 O.9688 1.6 0.3942 0.06 0.9537 1.8 0.3603 0.08 0.9389 2.0 0.3362 0.1 0.9250 2.5 0.2788 0.15 0.8862 3.0 0.2430 0.2 0,8586 3.5 0.2127 0.25 0.8242 4 . 0 0.1888 0.3 0.7994 5.0 0.1538 0.4 0.7468 6 .0 0.1296 0.5 0.6992 7.0 0.1118 0.6 O.6563 8.0 0.0982 0.7 0.6182 9 .0 O.O876 0.8 0.5839 10.0 0.0790 0.9 0.5512 2 0 . 0 0.0398 1.0 0.5233 50.0 0.0159 100.0 0.0080 For Y< 0 . 1 ; I ( s ) / I ( o ) = 1 - £ Y + Y_f_ - 0 ( 1 0 - 4 ) For Y ^ 2 0 . 0 ; I ( s ) / I ( o ) * -L - O d O ^ ) - 82 -I(s) T(oT (A3-7) where V =/HX'O° A S V - * < » , e q u a t i o n A3-7 reduces t o e q u a t i o n A3-6, F o r d » x^ Steigman e t a l show t h a t I(s) I(o) i - e (A3-8) as d-*»ao , t h e n e q u a t i o n A3-8 reduces t o e q u a t i o n 2 -5 , C. RESULTS FOR Fe^? IN P t 1 9 ^ 57 195* F o r Fe i n P t we want (ana l o g o u s t o e q u a t i o n 2-4) C(x) = 1_ n H = 6 . 6 x l 0 1 9 atoms/cm3 (A3-9) 1000 From e q u a t i o n A 3 - 4 , t h i s g i v e s d = 1,7x10"*^ cm f o r our case (M = I ( o ) / A = 6 x l 0 1 5 ; A = 0 .3 cm 2). F o r P l a t i n u m , /UH = 4l60/cm, so - 8 3 -y =>ttHd = 0 . 7 , and from equation A3-6 and Table A3-1 I ( s ) / l ( o ) = 0 . 6 2 . From Chapter Two, our r e s u l t was I ( s ) / l ( o ) = 0 . 5 8 , assuming a uniform i m p u r i t y d i s t r i b u t i o n C(x) = M/Ad. In Chapter Two we assumed a f o i l t h ickness x Q = 1.27x10'" 3 cm, which gives x 0 / d = 7 . 5 . We might expect equation A3-7 to give a r e s u l t d i f f e r e n t from equation A3 -6 . However, upon computation of the f i r s t few terms i n A3 - 7 , we f i n d that the d i f f e r e n c e between the two equations i s n e g l i g i b l e . - 84 -APPENDIX FOUR CONSTRUCTION OF A MOSSBAUER TRANSDUCER DRIVE The a c c u r a t e measurement of s m a l l s h i f t s of t h e Mossbauer l i n e r e q u i r e s a c o n s t a n t - v e l o c i t y Mossbauer s p e c t r o m e t e r , as t h a t d e s c r i b e d i n C h a p t e r Three. However, i n c a s e s where z e r o r e l a t i v e v e l o c i t y need not be p r e c i s e l y d e t e r m i n e d , a v e l o c i t y - s w e e p s p e c t r o m e t e r i s more c o n v e n i e n t . T h i s appendix g i v e s t h e c i r c u i t r y f o r one such s p e c t r o m e t e r . A b l o c k diagram of t h e s p e c t r o m e t e r c o n s t r u c t e d i s shown i n F i g u r e A4-1. T h i s s p e c t r o m e t e r was t a k e n , w i t h some changes of components and c o n s t r u c t i o n d e t a i l s , from Lynch and Baumgardner (I96I). The o p e r a t i o n ) o f t h e s p e c t r o m e t e r i s d e s c r i b e d i n t h e i r p a p e r . A photograph of t h e t r a n s d u c e r appears i n F i g u r e A4-2. The c i r c u i t r y i s shown i n F i g u r e A4 - 3 . T h i s c i r c u i t r y was b r o k e n up i n t o t h e modules shown i n F i g u r e s A 4 - 4 t o A4-11, and p l a c e d on p r i n t e d c i r c t i l t boards (PCB). Three p o t e n t i o m e t e r c o n t r o l s a r e p r o v i d e d : t h e V e l o c i t y Range C o n t r o l i s s e t t o sweep the t r a n s d u c e r t h r o u g h the Mossbauer s o u r c e v e l o c i t y range of i n t e r e s t ; t h e Channel Span C o n t r o l i s s e t f o r t h e d e s i r e d v e l o c i t y i ncrement p e r c h a n n e l o f t h e m u l t i c h a n n e l a n a l y z e r ; t h e Z e r o V e l o c i t y C o n t r o l i s . s e t so t h a t p u l s e s accumu-l a t e d w h i l e t h e Mossbauer s o u r c e i s a t r e s t w i l l f a l l i n t o t h e c h a n n e l s e l e c t e d t o c o r r e s p o n d t o z e r o v e l o c i t y . - 85 -T R I A N G L E W A V E G E N E R A T O R N [ _ s S [ N l _ _ SOURCE D R I V E " AMP. ABSORBER G A M M A D E T E C T O R A M P. CHANNEL ANALYZER. i T R I G G E R C I R C U I T F I G , A« ~ 9 T T T T S T F C H A N N E L A W / ^ L r Z E - R P O W E R S U P P L I E S F i f i S A 4 - 3 , * CHASSIS F I G A 4 - I O F f c O N T P A N E L F i ( 5 A + - I I v T 3 F I G U R E A4-1; ftKSSS B A U E R T R A N S D U C E R B L O C K D I A G R A M Fl SURE A4- - 2 : T R A N S D U C E R DRIVE- M g C H A M I C A L - 87 -F I G U R E A 4 - 3 : T R A N S D U C E R D R I V E - E L E C T R O N I C S FIGURE UNREGULATED POWER SUPPLY - 89 -F I G U R E P O S I T I V E P O W E R S U P P L Y - ( P C B | ) - 90 -> I /f\—f~\ 1^ O <2T \ 2 * > In o 7 A G O o to o ID h Vn O O — -««. p 55- CO -J- vi> 0 0 r w W FIGURE A4-6; NEGATIVE POWER SUPPLY (PCB2) - 91 -o o H l — <c <c o o b h Q - O -£ 5 § 3 2 2. Z 2 OS n ! o( oi SB <£ « <~> ^ ^ — — Of Qf <zf or FIGURE A 4 - 7 : DRIVE AMPLIFIER (PCB3) - 92 -FIGURE A f r - 8 ; ' LINEAR AMPLIFIER (PCB4) - 93 -o <v> — cr> ^ * 5 $ C O <J- — or <y or FIGURE A4-9: LINEAR AMPLIFIER (PCB5) - 94 -2k i o =F f CO Q. v0 0 f\—f\ / 00 V<7 o O to 60 <9 I «0 en so £ 0 0 tli V -j < V) •J >u * t ^ o < o a: j b > £ 0 o UJ i- w FIGURE A4-10: OUTPUT ' AMPLIFIER (PCB6) - 95 -UJ J 1U z 2 < •x o Ul 51 o ti-ro o h FIGURE A 4-11; TRIGGER CIRCUIT (PCB7) - 96 -BIBLIOGRAPHY A . J . Bearden, 1964: ed. " T h i r d I n t e r n a t i o n a l Conference on the Mossbauer E f f e c t " Rev. Mod. Phys. 2k*333(1964) A.J.F. B o y l e , D.St.P. Bunbury, C. Edwards and H.E. H a l l , I960; P r o c . Phys. Soc. 76.165(1960) A.J.F. B o y l e and H.E. H a l l , 1962; Rep. P r o g . Phys. XXV,441(1962) R. B r o u t and W.M. V i s s c h e r , 1962; Phys. Rev. L e t . £,54(1962) B.N. Brockhouse, i 9 6 0 ; I n "Symposium on I n e l a s t i c S c a t t e r i n g i n S o l i d s and L i q u i d s " IAEA, V i e n n a , i 9 6 0 . D.M.J. Compton and A.H, Schoen, 1962; ed. "The Mossbauer E f f e c t : P r o c e e d i n g s of th e Second I n t e r n a t i o n a l C o n f e r e n c e " W i l e y , 1962 P.G. Dawber and R.J. E l l i o t , I963; P r o c . R o y a l Soc. 222,222(1963) A.D. D i n h o f e r , I 9 6 3 ; Phys. Rev. 121,535(1963) P„A. E g e l s t a f f , H.J. Hay, G. H o l t , J . F . R a f f l e and J.R. P i c k l e s , 1961; P r o c . I n s t . E l e c . E n g r s . (London), P a r t B, 108,26(1961) H. F r a u e n f e l d e r , 1963; "The Mossbauer E f f e c t " 2nd p r i n t i n g w i t h c o r r e c t i o n s and a d d i t i o n s , B e njamin, 1963. P.G. Klemens, 1961; Phys. Rev. 122,443(1961) J.A. Krumhansl, 1962; J . A p p l . Phys. S u p p l . 21*307(1962) F . J . Lynch and J.B. 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Mossbauer, 1958; Z. Phys, 151.124(1958) R.L. Mossbauer, 1965; Chap XXI(B) i n K. Siegbahn " A l p h a - , B e t a - , and Gamma-Ray S p e c t r o s c o p y , V o l . 2" N o r t h - H o l l a n d , I 9 6 5 . B. Mozer and G.H. V i n e y a r d , I 9 6 I ; B u l l . Am. Phys. Soc. 6,135(1961) G. F. N a r d e l l l and N. T e t t a m a n z i , 1962; Phys. Rev. 126.1283(1962) H. V. Neher and R.B. L e i g h t o n , I 9 6 3 ; Am. J . Phys. 21,255(1963) A.T. Nelms and I . Oppenheim, 1955; U.S. N a t i o n a l Bureau of S t a n d -a r d s J . of Research ii,5 3 ( 1 9 5 5 ) R.M. O l s o n , 1961; " E s s e n t i a l s of E n g i n e e r i n g F l u i d M e c h a n i c s " , I n t e r n a t i o n a l Textbook Co., I96I. W.C. O v e r t o n J r . and E. Dent, i 9 6 0 ; U.S. N a v a l R e s e a r c h L a b o r a t o r y Report 5252, Washington 25 . - 98 -J.P. S c h i f f e r , P.N. P a r k s and J . H e b e r l e , 1964; Phys. Rev. 133.A1553 (1964) J . Steigman, W. S h o c k l e y and P.C. N i x , 1939; Phys. Rev. £6,13(1939) J . L . S t u l l , 1962; Am. J . Phys. ^0,839(1962) W.M. V i s s c h e r , i 9 6 0 ; Ann, Phys. L p z . <?,194(1960) W.M. V i s s c h e r , 1963; Phys. Rev. 129 .28(1963) W.M, V i s s c h e r , 1964; Phys. Rev. 134.A965(1964) G.K. Wertheim, 1964; "The Mossbauer E f f e c t : P r i n c i p l e s and A p p l i c -a t i o n s : Academic, 1964. W.H. Whitney, 1964; Am. J . Phys. 21*306(1964) J . E . J . Woodrow, 1964; M.Sc. T h e s i s , U.B.C. 

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