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X-ray induced luminescence in single crystals of pure potassium iodide Usiskin, Sidney Robert 1955

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X-RAY INDUCED LUMINESCENCE IN SINGLE CRYSTALS OF PURE POTASSIUM IODIDE by SIDNEY ROBERT USISKIN A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in PHYSICS We accept t h i s thesis as conforming to the standard required from candidates f o r the degree of DOCTOR OF PHILOSOPHY. Members of the Department of Physics. THE UNIVERSITY OF BRITISH COLUMBIA October, 1955. Faculty of Graduate Studies P R O G R A M M E O F T H E Jffittai ©ml |Lxmttitrati*m for i\\ e P g g r e * SIDNEY ROBERT USISKIN B.Sc. (Hons.) (Manitoba) 1948 T H U R S D A Y , O C T O B E R 20th, 1955, at 3:00 p.m. I N R O O M 303, PHYSICS B U I L D I N G C O M M I T T E E I N C H A R G E H . F. A N G U S , Chairman G . M . S H R U M H . M . D A G G E T T A . M . C R O O K E R M . K I R S C H K . C M A N N P I N G - T I H O G. M . V O L K O F F External Examiner—A J . D E K K E R Dept. of Electrical Engineering University of Minnesota X - R A Y I N D U C E D L U M I N E S C E N C E in S I N G L E C R Y S T A L S O F P U R E K l A B S T R A C T The luminescent decay of pure potassium iodide has been mea- sured under various conditions of temperature, annealing and x-ray irradiation time. The decay curves are found to be approximately of the form I (t) = "> p a exp (-a t) n n n 1 1 where I (t) is the luminescent intensity at the time t. Using a method that has been developed for the analysis of decay curves, the decay constants, and the trap populations p n , have been estimated. The decay constants are found to be simple temperature functions of the form a = s exp (-E / k T ) n n n with the activation energies, E , lying between 0.4 and 0.8 ev and escape probabilities, s ,̂ between 10 4 and 10 s sec.-1. The temperature dependence of the p is complicated and" indicates that radiationless transitions strongly contribute to the emptying of traps at high temperatures. A n irreversible increase in luminescent output produced by re- peated x-irradiation without intermediate annealing has been discovered; it points to considerable deviation from a first order decay mechanism in the case of slowly decaying traps. The mechanism of the irreversible effect has been discussed on a semi-quantitative basis. LIST OF PUBLICATIONS Thermal Decay of the Luminescence of K B r , G . W. Williams, S.R. Usiskin, and A . J . Dekker, Phys. Rev. 92, 1398, 1953. X-Ray Induced Luminescence in Single Crystals of Pure K l , S. R. Usiskin and O. Theimer, Phys. Rev. (in the press). GRADUATE STUDIES Field of Study: Physics Electromagnetic Theory W. Opechowski Theory of Measurements A . M . Crooker Quantum Mechanics G . M . Volkoff Special Relativity W . Opechowski v Chemical Physics A . J . Dekker Physics of the Solid State H . Koppe Other Studies: Differential Equations Advanced Differential Equations Radiochemistry T . E . H u l l T . E . H u l l M . Kirsch and K . Starke ACKNOWLEDGMENTS I am indebted to Dr. 0. Theimer for valuable discussions during the course of these experiments and for help i n interpreting the experimental r e s u l t s . I am also indebted to Dr. A.J. Dekker, under whose supervision t h i s work was started and whose enlightening discussions have given me a f u l l e r appreciation of the problems involved i n the study of luminescence. I wish to acknowledge the kind assistance of the other s t a f f members of the Physics Deoartment and the excellent laboratory f a c i l i t e s supplied by the University of B r i t i s h Columbia Physics Department. I would l i k e to express my thanks to Dr. G.W. Williams f o r . h i s assistance i n the design and construction of the apparatus. This work was made possible by research grants to Dr. A.J. Dekker and to Dr. 0. Theimer from the University of B r i t i s h Columbia. A B r i t i s h Columbia Telephone Scholarship to the author i s g r a t e f u l l y acknowledged. ABSTRACT The luminescent decay o f pure potassium i o d i d e has been measured under v a r i o u s c o n d i t i o n s o f temperature, a n n e a l i n g and x - i r r a d i a t i o n time. The decay curves are found t o be approximately of the form i where I ( t ) i s the luminescent i n t e n s i t y at the time t . Using a method t h a t has been developed f o r the a n a l y s i s o f decay curves, n u m e r i c a l v a l u e s f o r the decay c o n s t a n t s , V k , and the t r a p p o p u l a t i o n s ^; , have been est i m a t e d . The decay constants are found to be simple temperature f u n c t i o n s o f the form w i t h the a c t i v a t i o n e n e r g i e s , , l y i n g between 0.4 and O.fct ev and escape p r o b a b i l i t i e s , $t , between 10** and 10^ s e c 0 " ^ . The temperature dependence o f the i s complicated and i n d i c a t e s t h a t r a d i a t i o n l e s s t r a n s i t i o n s s t r o n g l y c o n t r i b u t e to t h e emptying of t r a p s at high temperatures. An i r r e v e r s i b l e i n c r e a s e i n luminescent output produced by repeated x - i r r a d i a t i o n without i n t e r m e d i a t e a n n e a l i n g i s d i s c o v e r e d ; i t p o i n t s to c o n s i d e r a b l e d e v i a t i o n from a f i r s t order decay mechanism i n the case o f slowly decaying t r a p s . The mechanism o f the i r r e v e r s i b l e e f f e c t i s d i s c u s s e d on a s e m i - q u a n t i t a t i v e b a s i s . TABLE OF CONTENTS. Page ACKNOWLEDGMENTS. ABSTRACT. I. INTRODUCTION. 1 1. The Band Model of Solids.... v. 1 2. Previous Work on Alkali-Halides 3 II. DESCRIPTION OF APPARATUS. 6 III. THEORY OF THE DECAY AND EXCITATION PROCESS. 11 1. The Luminescent Decay Process 11 2. The Excitation Process « 13 IV. THE ANALYSIS OF DECAY CURVES. 15 V. EXPERIMENTAL RESULTS. 24 VI. DISCUSSION OF EXPERIMENTAL RESULTS. 27 1. The Temperature Dependence of the «̂wia,jt Values 27 2. Effect of Irradiation Time on the Decay 29 3. The Temperature Dependence of the Populations „ 29 4. Irreversible Effects of X-Irradiation 31 5. Experiments at Low Temperatures 3&* VII. CONCLUSIONS. 42 APPENDIX I A Mechanical Device For Facili t a t i n g The Graphical Analysis 44 APPENDIX II Analytical Methods For Resolving A Decay Curve Into A Sum Of Exponentials..., 46 REFERENCES. LIST OF TABLES F a c i n g Page Table I . T e s t - o f G r a p h i c a l A n a l y s i s 20 Table I I . I r r e v e r s i b l e E f f e c t s o f S u c c e s s i v e X - I r r a d i a t i o n s 25 Table I I I . E ; And S; Values 27 Table IV. Temperature Dependence o f the P o p u l a t i o n and Capture C r o s s - S e c t i o n s ( i n a r b i t r a r y u n i t s ) 30 LIST OF ILLUSTRATIONS. Facing Page Fig. 1. D.C. Amplifier 6 Fig. 2. Phptomultiplier Supply...., 7 Fig. 3. Crystal Holder (High Temp.) 8 Fig. 4. Crystal Holder (Low Temp.) 9 Fig. 5. A Typical ^(E) Spectrum 24 Fig. 6. Log Versus l/T 24 Fig. 7. Decay Dependence on t* 25 Fig. 8. Glow Curve for Kl 26 Fig. 9. Dependence on t*. 28 Fig. 10. Graphical Analysis of a Decay Curve.... 44 1 I . INTRODUCTION The emission o f v i s i b l e l i g h t by a c r y s t a l a f t e r having been exposed to r a d i a t i o n i n the u l t r a v i o l e t or x-ray r e g i o n has l o n g ! been an i n t e r e s t i n g phenomenon. The a l k a l i - h a l i d e s , because o f the apparent s i m p l i c i t y o f the luminescent system, provide a good means o f i n v e s t i g a t i n g t h i s phenomenon. A l s o , s i n c e they have been the o b j e c t o f v a r i o u s o t h e r s t u d i e s , these c r y s t a l s a f f o r d the o p p o r t u n i t y , at l e a s t i n p r i n c i p l e , of c o r r e l a t i o n o f the r e s u l t s of the study o f luminescence w i t h other known p r o p e r t i e s , e.g. p h o t o c o n d u c t i v i t y , glow curves, e t c , ^ For these reasons, an i n v e s t i g a t i o n i n t o the decay o f the x-ray induced luminescence o f pure K l c r y s t a l s seemed prom i s i n g . Luminescence i s the g e n e r a l term a p p l i e d to e m i s s i o n o f l i g h t by a c r y s t a l d u r i n g and a f t e r i t s 1 e x c i t a t i o n by, e.g., u-v l i g h t or any i o n i z i n g r a d i a t i o n . That p o r t i o n o f the l i g h t which i s emitted i n a time up t o 10"^ seconds a f t e r the end o f e x c i t a t i o n i s c a l l e d f l u o r e s c e n c e . A l l subsequent l i g h t i s termed phosphorescence. T h i s demarcation at 10~& sec. i s based on the e x p e r i - mental o b s e r v a t i o n t h a t the n a t u r a l l i f e - t i m e o f the e x c i t e d s t a t e o f an i s o l a t e d non-metastable atom i s about 10~9 t o 10"^ sec. f o r d i p o l e t r a n s i t i o n s between the i n s t a n t of e x c i t a t i o n and the i n s t a n t of e m i s s i o n o f an o p t i c a l luminescence photon.3 1. The Band Model o f S o l i d s . When i n d i v i d u a l ions are brought t o g e t h e r to form a c r y s t a l , the energy l e v e l s of the i s o l a t e d i o n s become converted i n t o broad bands because of the i n t e r a c t i o n s among the i o n s . 2 These bands extend throughout the e n t i r e c r y s t a l , and e l e c t r o n s are f r e e to move i n any s i n g l e band. On the b a s i s of t h i s p i c t u r e , the d i f f e r e n c e between conductors and non-conductors may be e x p l a i n e d . I f the upper- most band i n which t h e r e are e l e c t r o n s present i s onl y p a r t i a l l y f i l l e d w i t h e l e c t r o n s , the c r y s t a l i s a conductor, f o r a net conduction of charge can occur by v i r t u e o f t r a n s i t i o n s t o unoccupied n e i g h b o r i n g l e v e l s i n the same band. In an i n s u l a t i n g c r y s t a l , the uppermost band i n which t h e r e are e l e c t r o n s i s completely f i l l e d , so that f o r every e l e c t r o n moving i n a g i v e n d i r e c t i o n t h e r e i s another o p p o s i t e l y d i r e c t e d thus p r e v e n t i n g any net t r a n s f e r of charge. For condu c t i o n t o occur i n a n ' i n s u l a t i n g c r y s t a l , an e l e c t r o n must be r a i s e d from the uppermost f i l l e d band t o the next allowed empty band, (conduction band) i n which case both the s i n g l e e l e c t r o n i n the conduction band and the " h o l e " remaining i n the f i l l e d band may c o n t r i b u t e to a t r a n s f e r of charge. In any non-conducting n a t u r a l c r y s t a l at a temperature . above T«=0 oK f d e f e c t s i n the c r y s t a l s t r u c t u r e w i l l occur.4 These d e f e c t s cause the appearance o f l o c a l i z e d energy l e v e l s between the f i l l e d and conduction bands. E l e c t r o n s which f i n d t h e i r way to these l o c a l i z e d l e v e l s are not f r e e to move about i n the c r y s t a l . D i r e c t e l e c t r o n i c t r a n s i t i o n s from these l e v e l s t o the ground s t a t e are very improbable, mainly because o f the l o c a l i z a t i o n o f the t r a p s and o f the " h o l e s " i n the f i l l e d band. Only when a hole i s a c l o s e neighbor o f an e l e c t r o n i n a l e v e l , 3 i s t h e r e a chance f o r a t r a n s i t i o n to occur. These l e v e l s , then, p r o v i d e metastable s t a t e s f o r e l e c t r o n s and are r e f e r r e d to as t r a p s . The i n t e r p r e t a t i o n o f the r e s u l t s o f the experiments d e s c r i b e d i n t h i s work i s based upon the assumption o f the e x i s t e n c e of t r a p s i n pure K l c r y s t a l s . 2. Previous Work on A l k a l i - H a l i d e s . P r e v i o u s work on luminescence has been mainly centered about m a t e r i a l s o f t e c h n i c a l importance. Of the a l k a l i - h a l i d e s , o n l y the i m p u r i t y - a c t i v a t e d c r y s t a l s have been i n v e s t i g a t e d i n any d e t a i l . Bunger and Flechsig5 were the f i r s t to observe luminescence i n t h a l l i u m - a c t i v a t e d KC1. Because o f i t s ' a p p l i c a t i o n i n s c i n t i l l a t i o n c ounters, K a l - T l has been the s u b j e c t of c o n s i d e r a b l e r e s e a r c h . As f o r the pure a l k a l i - h a l i d e s , B o s e ^ » 7 f i r s t measured the temperature dependence o f the x-ray induced luminescence. He gave a q u a l i t a t i v e e x p l a n a t i o n o f the decay process i n terms of e l e c t r o n t r a p s . Dekker and M o r r i sh ^ ' 9 measured the luminescence decay o f KBr and L i F u s i n g a new technique employing end-window p h o t o m u l t i p l i e r tubes. Because o f the p r e l i m i n a r y nature o f the experiment, no attempt was made t o e x p l a i n the temperature dependence of the decay. The phosphorescence of pure and t h a l l i u m - a c t i v a t e d a l k a l i - h a l i d e s was measured by Bonanomi and Rossel.^O They r e s o l v e d t h e i r decays i n t o a number o f e x p o n e n t i a l s e i t h e r by d i r e c t a n a l y s i s or w i t h the a i d of i n f o r m a t i o n gained by means 4 of the thermoluminescence technique f i r s t introduced by Urbach^ and t h e o r e t i c a l l y discussed by Randall and Wilkins.^^ H i l l and Schwed^ have investigated the luminescence of x-rayed NaCl. They found that d i f f e r e n t types of traps exist i n the c r y s t a l each having the same a c t i v a t i o n energy. The author, i n collaboration with G.W. Williams and A.J. Dekker, investigated the luminescent decay of pure KBr-^. It was found that a large number of traps existed i n the c r y s t a l . The d i f f e r e n t a c t i v a t i o n energies f o r 17 of these traps were determined on the assumption that the decay process i s e s s e n t i a l l y of the f i r s t order. These detailed r e s u l t s lead to the hope that the analysis of luminescent decay curves might prove to be superior to such other standard methods as the glow curve analysis, or the investigation of photoconductivity. These hopes have not been f u l l y confirmed by l a t e r investigations. Neither the monomolecular character of the decay nor the detailed numerical r e s u l t s concerning the properties of the traps can be convincingly proven. In a manner s i m i l a r to other methods, the analysis of luminescent decay curves gives only semi-quantitative information and only a combination of d i f f e r e n t experiments may f i n a l l y y i e l d a r e l i a b l e quantitative knowledge of the mechanism of luminescence. It i s the purpose of t h i s work to carry out investigations designed to discover the p o s s i b i l i t i e s and l i m i t a t i o n s of the decay curve method. 5 Potassium iodide has provided the experimental material. This substance was chosen f o r investigation because the absorption band of the x-rayed c r y s t a l l i e s i n a d i f f e r e n t spectral region than the response of the RCA 5819 photo- m u l t i p l i e r tube used as the detector of the emitted l i g h t . Also, the luminescent decay of Kl had not yet been investigated with the sensitive apparatus which had been designed for. the purpose of measuring the weak l i g h t i n t e n s i t i e s associated with the decay of the luminescence of the pure a l k a l i - h a l i d e s . A single c r y s t a l of pure Kl has been subjected to various temperatures at which the luminescent decay occurred, to d i f f e r e n t annealing times and to d i f f e r e n t times of i r r a d i a t i o n . Preamplifier FIG. 1. D.C. A m p l i f i e r Facing Page 6 II. "DESCRIPTION OF APPARATUS In order to perform the experiments proposed, an apparatus having the following characteristics was required: (a) Maintenance of the crystal at various fixed temperatures for considerable periods of time. (b) Proximity of the crystal to the exciting source, in this case, a low-voltage x-ray tube. (c) Good geometrical efficiency for the acceptance of emitted light from the crystal by the detector, an end- window RCA 56*19 photomultiplier tube. (d) Stability of the detecting system over long periods of time. (e) Linearity of the detecting system in responding to light intensities that may vary by as much as 105. It was found that the detecting device best suited for the requirements (d) and (e) indicated above, was one using a photomuliplier tube as the i n i t i a l light detecting stage and integrating i t s pulse output to get a D.C. response. The photo-tube was operated at reduced voltage to eliminate the possibility of fatigue. The D.C. amplifier used was one whose p.utput impedance was sufficiently low so that i t could operate a recording milliammeter. The circuit diagram for this amplifier i s shown i n . f i g . 1. The output of the photomultiplier was fed to the variable (105 to 10^ ohms) grid resistor of a Victoreen 5300 electrometer tetrode. The voltage developed across this resistor by the output current of the photomultiplier was amplified by a 2 stage balanced amplifier, and passed to the meter. The output signal from the amplifier was also led to FIG. 21 . Photomultiplier Supply Facing Page 7 the lower end of the grid r e s i s t o r , supplying 100$ feed-back. An i d e n t i c a l a m p l i f i e r supplied the balancing voltage f o r the meter. The large amount of feed-back u t i l i z e d i n the D.C. amplifier served a 4-fold purpose: (1) Reduction of the input time constant of the amplifier. (2) Maintenance of the c o l l e c t o r at a fixed voltage independent of signal current. (3) Reduction of output impedance (~ 1 ohm). (4) The voltage gain of the amplifier (unity) i s made independent of tube c h a r a c t e r i s t i c s and supply voltage v a r i a t i o n . An Esterline-Angus model A-W recording milliammeter was u t i l i z e d to record the decays. The chart was run at a speed of 6" per minute fo r the i n i t i a l portion of the decay curve and switched to 6" per hour to record the l a t t e r , slowly varying, portion of the decay. The response time of the meter (^ 1 sec.) meant that about 5 sec. of the recording were l o s t when the range of the amplifier was changed. This response time, however, improved the readings at the lowest i n t e n s i t i e s smoothing out the current f l u c t u a t i o n s . The photomultiplier supply shown i n f i g . 2, i s a common, regulator c i r c u i t with an output variable from 750 to 1200 v o l t s . The voltage d i v i d e r supplying the photomultiplier dynode voltages draws 1 ma. The l a s t 4 sections of the divide: consist of RCA 5651 voltage reference tubes which act as A l . cup Leads to heater and thermocouple h 6 V [ F T Myca lex insulat ion K l c r y s t a l 3mm x 2 0 m m d. -20mm hole tapered to hold c r ys ta l . copper-constantan thermocouple 8 0 0 n N i c h r o m e h e a t e r FIG. "3.. Crys t a l Holder (High Temp.) Facing Page # d voltage regulators for currents In the range from 0.5 to 1.2 ma. This reduces the source impedance to a n e g l i g i b l e value for the l a s t dynodes where the current i s appreciable and consequently the gain of the phototube>which i s sensitive to changes i n the dynode voltageSjis unaffected by even large output currents. To check that the gain of the detecting system returned to the same value a f t e r switching the system o f f , a "standard" l i g h t source was introduced. This l i g h t source consisted of a 6 volt lamp. The current was supplied by a storage battery, and the voltage across the lamp maintained at a fixed value with the use of a potentiometer. The lamp was used f o r short periods of time only,to prevent changes i n amount of radiation which might occur a f t e r long operation. The c r y s t a l was maintained i n a vacuum between the x-ray tube and the photomultiplier tube. The vacuum chamber was as narrow as the physical s i t u a t i o n would allow, and had a 3 mil thick aluminum window facing the x-ray source and a pyrex window f a c i n g the l i g h t detector. This allowed requirements (b) and (c) to be met. The thermal i n s u l a t i n g properties of the vacuum meant that the photomultiplier tube could be kept i n close proximity to the c r y s t a l even when the c r y s t a l temperature was r e l a t i v e l y high. The requirement (a) was s a t i s f i e d by having the c r y s t a l held i n the vacuum i n a brass spool which had a Nichrome wire heater wound about i t . (see f i g . 3 ) Electronic regulation of the current to t h i s heater provided s a t i s f a c t o r y temperature control. The brass spool was suspended i n the parallelepiped //,,,7////iM Pare* window >| Kl cmtdl .glass to Kovar seal rubber gasket brass Al window FIG. 4' . Crystal Holder (Low Temp.) Facing Page 9 vacuum chamber from a cover that made a vacuum-tight seal on a rubber gasket at the top of the chamber. The chamber was continually pumped from an arm at right angles to the l i n e running through the x-ray tube, c r y s t a l and photomultiplier tube. The c r y s t a l was placed i n an aluminum cup before being put into the i n t e r i o r of the brass spool. This cup, which covered the face of the c r y s t a l toward the x-ray tube served a dual purpose: (a) It maintained the face of the c r y s t a l from which the luminescence was mainly o r i g i n a t i n g , at a uniform temperature (b) It served to r e f l e c t the l i g h t into the photo- •cathode of the photomultiplier tube and thus increased the geometrical e f f i c i e n c y f o r detection of the l i g h t . For purposes of measuring the temperature of the c r y s t a l , a copper constantan thermocouple, whose e.m.f. was recorded on a potentiometer, was fastened to the inside of the brass spool. For the measurements taken at l i q u i d oxygen temperatures, the c r y s t a l holder described above was replaced by a pyrex Dewar vessel, whose inner portion was made of metal, (see f i g . 4.) This Dewar made a vacuum seal on the same rubber gasket as the one described above. The inner metal portion of the Dewar was made of Kovar and a glass-to-metal seal was thus e a s i l y effected. The c r y s t a l was held.in a brass spool and t h i s was fastened to the metal i n t e r i o r of the Dewar by means 10 of a heavy brass bar. The c r y s t a l was cooled by conduction and i t s ' temperature was measured by a thermocouple attached to the brass spool i n very close proximity to i t . I I I . THEORY OF THE DECAY AND EXCITATION PROCESS. I t i s generally'assumed t h a t the l o n g - l i v i n g phosphorescence i s due to e l e c t r o n s i n t r a p s r e t u r n i n g to the ground s t a t e v i a h i g h e r energy l e v e l s o f the c r y s t a l , most probably v i a the conduction band. In a pure c r y s t a l , e l e c t r o n s i n the ground s t a t e ( f i l l e d band) may be e x c i t e d t o the conduction band by the a b s o r p t i o n o f x - r a y s . These e l e c t r o n s i n the co n d u c t i o n band move through the c r y s t a l w i t h great speed and e v e n t u a l l y f a l l back e i t h e r t o the. f i l l e d band, with the emission o f l i g h t , or i n t o the t r a p s . Those e l e c t r o n s whose l i f e - t i m e i n the conduction band be f o r e r e t u r n i n g t o the f i l l e d band i s l e s s than 10"^ sec. g i v e r i s e to the f l u o r e s c e n c e . 1. The Luminescent Decay Process. I f ^ e l e c t r o n s are present i n the t r a p s o f depth a f t e r the e x c i t a t i o n process i s terminated, then e l e c t r o n s w i l l leave the t r a p s per u n i t time and reach the conduction band. I f thermal energy f l u c t u a t i o n s are the onl y cause f o r the discharge o f the t r a p s , the decay co n s t a n t s w i l l be temperature f u n c t i o n s o f the form A. * s ;e" VtiT i d ) where T=temperature, k=Boltzmann's constant, S ^ = t r a n s i t i o n p r o b a b i l i t y . The emitted l i g h t i n t e n s i t y i s p r o p o r t i o n a l t o the number o f r a d i a n t t r a n s i t i o n s t h a t are made per u n i t time. I f the t r a n s i t i o n p r o b a b i l i t i e s f o r the r a d i a n t t r a n s i t i o n s from the conduction band t o the f i l l e d band are very much l a r g e r 12 than for retrapping and for the thermal discharge of the traps, the l a t t e r , then, are the rate-determining process, and lead to a luminescent decay of the f i r s t order (monomolecular decay): law, as given by equation 111(2),can never be f u l l y r e a l i z e d i n nature. The reason f o r t h i s i s that the number of "holes" or empty places i n the ground state of the c r y s t a l matrix i s approximately equal to the number of electrons i n the conduction band and i n the traps. I f the l a t t e r number i s small, and i t i s necessarily so toward the end of a decay process, the number of empty electron acceptors i n the ground state becomes equally small, and the t o t a l t r a n s i t i o n p r o b a b i l i t y for radiant t r a n s i t i o n s becomes dependent not only upon the thermal discharge of the traps but upon the number of holes i n the f i l l e d band as well. . The decay constants X;, w i l l therefore become time dependent toward the end of a decay. becomes i n f l u e n t i a l , and the decay gradually becomes transformed into one of 2nd order (bimolecular decay). Although t h i s f i n a l complication i s unavoidable, i t may be n e g l i g i b l e f o r the major portion of the decay, and contribute s u b s t a n t i a l l y only to those l a s t portions of the decay which, at moderate temperatures are unobservable anyway, because of the small luminescent i n t e n s i t y . A feature which has u n t i l now been neglected, i s the p o s s i b i l i t y that r a d i a t i o n l e s s t r a n s i t i o n s from the traps, as I I K 2 ) It i s obvious that the simple monomolecular decay In t h i s s i t u a t i o n , retrapping of the electrons 13 w e l l as those g i v i n g r i s e to the observed luminescence, are allowed. I f r a d i a t i o n l e s s t r a n s i t i o n s are considered, the r a t e of discharge of a t r a p w i l l be described by where A, represents the escape p r o b a b i l i t y g i v i n g r i s e to r a d i a n t t r a n s i t i o n s , represents the escape p r o b a b i l i t y g i v i n g r i s e to r a d i a t i o n l e s s t r a n s i t i o n s and X*" i s t h e " e f f e c t i v e " escape p r o b a b i l i t y . I n t e g r a t i n g equation 111(3) gi v e s t> "= : I I I U ) The observed l i g h t i n t e n s i t y at any given time i s given by the product o f the population i n the t r a p at t h a t time and "Xt , the r a d i a n t escape p r o b a b i l i t y . I W ' fc*.* I I K 5 ) 2. The E x c i t a t i o n Process. In order t o make a study of the temperature dependence of the i n i t i a l populations i n the t r a p s , a mechanism f o r f i l l i n g the t r a p s must be assumed. The assumptions made were: Cl) X-rays l i f t great numbers of e l e c t r o n s from the f i l l e d band to the conduction band. (2) Most of these e l e c t r o n s f a l l d i r e c t l y back to the f i l l e d band g i v i n g r i s e t o the intense fluorescence observed. 14 (3) The number of electrons which are caught i n traps, per unit time, i s only a very small f r a c t i o n of the t o t a l number of electrons in the conduction band at any time during the e x c i t a t i o n . (4) The concentration of electrons i n the conduction band during the exci t a t i o n process remains constant, being determined c h i e f l y by the rate at which electrons are raised from the f i l l e d to conduction band by the x-rays }and the i fluorescent t r a n s i t i o n s . 'With these assumptions, the rate of f i l l i n g of traps may be described by U < - r ^ - ( k * + , > ; % n i ( 6 ) where = the "charging constant", which i s proportional to the i n t e n s i t y of the x-rays and i s , most probably, very s l i g h t l y temperature dependent, i f at a l l . = number of traps characterized by the subscript i ^* = discharge "constant" i . e . rate at which the x-rays eject electrons that are caught i n the traps. Most l i k e l y k L * < * K v ^ = the population i n the trap at the time t * I f equation 111(6) i s integrated between the l i m i t s \-0 and t s *t*s charging time and and ̂  one obtains 111(7) •Equation 111(7) gives the population i n the traps at the end of the i r r a d i a t i o n time "t*. 15 IV. THE ANALYSIS OF DECAY CURVES Assuming the simple monomolecular decay process t o be v a l i d , e q uation n i ( 2 ) r e q u i r e s t h a t every decay curve be r e s o l v e d into a sum of e x p o n e n t i a l s . The q u e s t i o n of the •' IK • >'.. I uniqueness, o f such a r e s o l u t i o n i s the main problem t o be i n v e s t i g a t e d . In those cases where the decay c o n s t a n t s , \ t of the components o f a decay curve are o f r a t h e r d i f f e r e n t v a l u e s , e.g. the decay of a mixture o f two r a d i o - i s o t o p e s w i t h g r e a t l y d i f f e r e n t h a l f - l i v e s , the maximum c o n t r i b u t i o n s o f the d i f f e r e n t components to the t o t a l i n t e n s i t y which i s e x p e r i m e n t a l l y measured, w i l l occur at w i d e l y separate times and they can thus be separated. I f , however, the decay c o n s t a n t s , ,have v a l u e s t h a t are not very d i f f e r e n t , the c o n t r i b u t i o n s o f the d i f f e r e n t components w i l l o v e r l a p to a g r e a t e x t e n t . At any g i v e n time the t o t a l i n t e n s i t y w i l l be composed o f the approximately equal c o n t r i b u t i o n s from two or more components. I f a p l o t o f the l o g a r i t h m o f the i n t e n s i t y o f the decay curve vs. time i s made, i n the case of the w i d e l y d i f f e r e n t decay co n s t a n t s the end o f t h i s p l o t ( i . e . t - » » ) w i l l be a s t r a i g h t l i n e , e s s e n t i a l l y produced by a s i n g l e decay co n s t a n t . In the case o f many c l o s e l y spaced decay c o n s t a n t s , the end o f the p l o t w i l l be curved s i n c e i t w i l l be produced by two or more o f the decay c o n s t a n t s . Thus, a measure o f the r e l a t i v e spacings of the decay co n s t a n t s can i n p r i n c i p l e be made from the c u r v a t u r e o f t h i s p l o t at very l a r g e v a l u e s of time. The r e s o l u t i o n o f any decay curve i n t o e x p o n e n t i a l s may be accomplished i n the f o l l o w i n g manner: Log I ( t ) i s p l o t t e d v s . t and a tangent i s drawn t o the curve at a p o i n t corresponding t o a l a r g e t v a l u e , say t ' where I(t') i s so s m a l l t h a t f o r t>t', the d i f f e r e n c e between the tangent and the curve i s s m a l l e r than the noise f l u c t u a t i o n s and, hence not observable. T h i s tangent, which may be re p r e s e n t e d f o r m a l l y by the equation: Û ItCO« lô VA-O"**)*.* i v ( i ) can be co n s i d e r e d as forming one member o f the sum 1 1 1 ( 2 ) . In e x a c t l y the same way, a tangent t o the t a i l o f the f i r s t d i f f e r e n c e curve, logjjT(,i) - T , w Q y i e l d s and ^ k f o r a second member o f t h e sum and proceeding i n the same way, one f i n a l l y o b t a i n s a d i s c r e t e set o f |\ and va l u e s which, when s u b s t i t u t e d i n t o equation 1 11(2) has been found t o y i e l d t he observed decay curve w i t h i n the l i m i t s o f the experimental e r r o r . A ^(A) spectrum i s d i s p l a y e d , i f the ̂  v a l u e s are p l o t t e d a g a i n s t ^ . T h i s spectrum w i l l have a d i s c r e t e l i n e c h a r a c t e r i f the ^ and Ak v a l u e s o b t a i n e d i n the g r a p h i c a l >. manner d e s c r i b e d above, are u t i l i z e d . Since A i s a f u n c t i o n o f the a c t i v a t i o n energy o f the t r a p s , such a spectrum r e p r e s e n t s the e l e c t r o n i c p o p u l a t i o n d i s t r i b u t i o n i n the v a r i o u s t r a p s o f the c r y s t a l . The statement t h a t the spectrum obtained by the above method y i e l d s the observed decay curve when 17 substituted into 111(2), does not imply that i t i s the only possible spectrum. In f a c t , because the experimentally observed decay I ( t ) , becomes less r e l i a b l e as the noise background i s reached and because log I ( t ) always appears to have some curvature even when the noise background i s reached, a d i f f i c u l t y a r ises i n the drawing of the tangent to t h i s curve. I f the tangent i s drawn to the observed part of the decay curve, equation IV(1) does not yet afford a good approximation, since there i s never a straight portion of the observed curve. I f , on the other hand, one attempts to extrapolate the observed curve to reach a straight portion, the d i f f i c u l t y a r ises anew, since extra- polation i s no unique process. The same ambiguity arises when pl o t t i n g the tangents to a l l the difference curves. Further, because of the lack of knowledge of the decay curves for times 10"^ sec; < t < 2 sec., the decay curve must also be extra- polated to zero- time, and because of the exceedingly rapid rate of change of the intensity at these times, large errors may be introduced. It i s thus obvious that many d i f f e r e n t spectra may be found f o r a single experimental curve, each-of which, when substituted into equation 111(2), reproduce that curve equally well. This has the following consequences: (a) The number of l i n e s i n a spectrum i s rather meaningless, phy s i c a l l y , since t h i s number can be a r b i t r a r i l y 18 increased by superimposing n d i f f e r e n t allowed spectra: lw»£ L h . » > . i k I V ( 2 ) Only the smallest number of l i n e s s u f f i c i e n t to reproduce the decay curve i s possibly of some i n t e r e s t , as in d i c a t i n g the smallest number of traps which w i l l give r i s e to the observed luminescent decay. (b) The i n d i v i d u a l ^ w and A; values of a spectrum are rather meaningless i f considered separately. The e f f e c t s of displacements of l i n e s by amounts AX^on the reproduction of I ( t ) can e f f e c t i v e l y be compensated by suitable changes Â>; i n the magnitudes of the l i n e s . Only i f the length and pos i t i o n of the l i n e s in a spectrum are j o i n t l y considered, can the re s u l t of an analysis be given i n a manner which i s reasonably invariant against the a r b i t r a r y choice of tangents to log I ( t ) . This can be understood from the following argument: Consider two equivalent decay curves IV(3a) rZ-V^ V A ; K' I V ( 3 b ) which are i d e n t i c a l within experimental error although produced by two di f f e r e n t l i n e spectra. The notation indicates that to each l i n e k and A t of equation IV(3a), corresponds 19 a group of one or several l i n e s and which are denoted by the symbols with the double index ^ i k and X-^ - \ + Here, i s the separation of the l i n e s i n the q-spectrum from the corresponding l i n e i n the p-spectrum. Consider a cert a i n set of values as given t and attempt to determine the ^ ^ i n such a manner that, according to the required equivalence of the decay curves, I ( t ) and l i t ) , t h e i r mean square deviation i s a minimum, i . e . ' f * X -mow. IV(4) Inserting the values defined i n equations IV(3) into IV(4), d i f f e r e n t i a t i n g with respect to one of the ^CK^ay <fyi'** > integrating over the time and putting the r e s u l t equal to zero gives: r = 21 * i.v i v ( 5 ) The denominators are now expanded and the second and higher powers of A ^ are neglected. This gives: ) This r e l a t i o n s h i p is v a l i d for a l lA^ ^therefore in the approximation that ^ L V ^ c ^ » "the must s a t i s f y the following conditions: L r <T M i I V ( 7 a ) IV(7b) TABLE I (continued) Facing Page 20 H 0 M]_ M 2 M3 XL  S I ^ : sIfc>; alK\ § Analysis III p1-4d V I . 02-10-3 273 15.09 1.134 0.0929 p2-111.0 \«4.2S.10~ 2 P3=114 ^=9.04-IO-2 Analysis IV p -48" A=1.02.10-3 252 13.04 0.912 0.0691 P2=50.ct \=3.54«10-2 P3-110 V 6 ^ 3 ' 1 0 " 2 V 4 3 \ - 9 . - 5 7*lCT 2 Average over the 4 analyses. BT . MT MT MT o 1 c 3 254 13.55 0 . 9 9 3 . 0 7 9 9 TABLE I. Facing Page 20 TEST OF GRAPHICAL ANALYSIS M 0 M x M 2 M 3 ^ x ^ g K * f Actual Spectrum Px=50 A, »10-3 250 13.05 0.95 0.0766 p2=50 ^=4-10-2 p 3«50 *,«5-10- 2 p^»50 "X.,-7'10-2 P 5»50 V ' 1 0 " 1 Analysis I p1=48 \ =1.02.10-3 245 P2-S1.5 \=4.11'10- 2 P3=115.5 ^,=8.2-10-2 Analysis II p -48 Xt -1.02* IO"3 24^ p2=135.5 \-4 . 79-10- 2 p3=64.5 -A^l.05-10-1 12.82 .9139 0.0693 1 3 . 2 7 1 . 0 1 6 0 . 0 8 8 6 20 From these follow the additional conditions IV ( 7 c ) IV(7d) Conditions IV (7a) and IV (7d) together show that the "center of gravity" of the spectrum as a function of X i s invariant; and equations IV (7b) and IV ( 7 c ) together show that t h i s i s true not only f o r the spectrum as a whole, but also f o r parts of the spectrum with a width of the order of magnitude of . It must be emphasized that these r e s u l t s are v a l i d f o r the condition &\<^/x^ < < I , and strong deviations from the equations IV (7) may be expected from large values of /\\^ - From these considerations, i t can be seen that i n spite of the ambiguities involved i n the resolution of an experimental decay curve into a sum of exponentials, c e r t a i n properties of the |>fa) spectra obtained should prove to be rather invariant. An a r t i f i c i a l decay curve was mathematically constructed and then analyzed by the graphical procedure outlined above, i n order to see whether t h i s method of resolving a decay curve into exponentials i s capable of obtaining r e s u l t s that are compatible with equation I V ( 7 ) . Four d i f f e r e n t j>(>) spectra were found f o r t h i s curve, and the f i r s t four moments fVXb; i M, SI^ ; i \̂x= E ; were calculated f o r each of these spectra. The re s u l t s of t h i s procedure are given i n table 1. 21 In analyzing actual decay curves, i t was found that the t o t a l population of a l l traps,JL^i was nearly invariant when calculated from d i f f e r e n t possible b(x") spectra. By the very nature of the graphical analysis described above, the i n i t i a l i n t e n s i t y , i s always co r r e c t l y reproduced by the ̂ c and X v values of a possible spectrum i f the decay curve i s known f o r a l l times including the very short times for which t->0. This l a t t e r condition was not f u l f i l l e d in the experiments performed, and hence, the v a r i a t i o n i n Jlb-'A;, for the d i f f e r e n t spectra was greater than normally expected. It must be pointed out, however, that t h i s uncertainty does not depend upon the a n a l y t i c a l method used, but i s due to the lack of knowledge of the i n i t i a l portion of the curve. The f a c t that the invariant properties of the ind i v i d u a l ^(X) spectra are the f i r s t few moments of these spectra means that no det a i l e d knowledge of the number, depth and populations of the traps can be obtained i n t h i s way. A better physical description of the luminescent system was therefore looked f o r . I f the assumption i s made that the d i s t r i b u t i o n i s continuous rather than d i s c r e t e , one ar r i v e s at the s t a r t i n g point of the method of analysis which proved to be the most useful i n leading to rather detailed knowledge of the traps i n the c r y s t a l . I f n d i f f e r e n t l i n e spectra which are a l l compatible with a given decay are superimposed a f t e r d i v i d i n g the length 22 o f each l i n e by n i n the manner o f IV(2) one then o b t a i n s a s i n g l e many-line spectrum which i s a l s o compatible w i t h the decay. Such a many-line spectrum can be transformed i n t o an e q u i v a l e n t continuum i f the c o n d i t i o n t h a t the c e n t e r o f g r a v i t y be conserved w i t h i n s m a l l r e g i o n s o f the spectrum i s observed. T h i s may be a c h i e v e d , i n p r a c t i c e , by the f o l l o w i n g procedure: f i r s t , one c o n s t r u c t s a "sum c u r v e " . T h i s c o n s i s t s o f r e p l a c i n g t h e value o f b. at *V by the sum t- . The p o i n t s thus obtained are j o i n e d by a smooth curve and the d e r i v a t i v e o f t h i s smooth curve at each p o i n t g i v e s the magnitude o f the continuous f u n c t i o n . This l e a d s to the formulae The many-lined spectrum, IV(2), was found t o r e q u i r e more than about 25 l i n e s i n the observable A range, f o r the continuous |*(x) spectrum obtained from i t t o be i n v a r i a n t i n shape. Since a s i n g l e g r a p h i c a l a n a l y s i s o f a decay curve y i e l d e d 5 or 6 l i n e s , each decay curve had t o be analyzed i n t o at l e a s t 5 or 6 d i f f e r e n t l i n e s p e c t r a i n order to o b t a i n a s u f f i c i e n t number of l i n e s i n the many-lined spectrum. The d i f f e r e n t ^(^) c o n t i n u a obtained f o r d i f f e r e n t decay curves e x h i b i t e d d i s t i n c t maxima and minima. These continua probably g i v e the most accurate d e s c r i p t i o n o f the r e a l p h y s i c a l s i t u a t i o n , s i n c e any maxima i n d i s t r i b u t i o n s t h a t s u r v i v e repeated s u p e r p o s i t i o n i n g and averaging when another new l i n e spectrum i s added, should be r e a l and not i. due to the mode of analysis.   24 V. EXPERIMENTAL RESULTS. Decay curves for a single crystal of pure (Harshaw) Kl were obtained for temperatures between 66°C and 200°C. Between each experimental determination of the decay curves, the crystal was annealed at a high temperature (*̂ 300°C) for about 30 hours. The crystal was always irradiated at the temperature at which the decay was observed. The continuous f>(^ distributions for the range of temperatures from 66°C to 200°C was determined from the experimental decay curves by the method described in section IV. In every case, distinct maxima and minima were found. Thus the line spectra of Williams et a l 1 ^ are here replaced by continuous spectra with the lines replaced by bands. A typical spectrum, in which s in equation III(l) has been ar b i t r a r i l y chosen as 10$, may be seen in f i g . 5. The half width of the bands may represent the uncertainty in our knowledge of the peak positions, A^v.> ° r the real physical situation: that traps of a certain class (responsible for a certain band) are not completely identical. The values of A.^^ are plotted in a logarithmic scale vs. l/T in f i g . 6. The point marked with a double arrow represents a missing point (perhaps a non-resolved broad band in the b(x) distribution), while the brackets indicate an excess point. In order to obtain a greater amount of information as to the mechanism of decay, a study of the dependence of the jit decay curve on the irradiation time t was made. Fig. 7 shows a typical result where the irradiation times for the curves  TABLE II. Facing Page 25 IRREVERSIBLE EFFECTS OF SUCCESSIVE X-IRRADIATIONS.. t(sec.) I x ( t ) I 2 ( t ) I 3 ( t ) I 4 ( t ) ^ ( t j / l i t t ) ^ ( t j / l x l t ) ^ ( t j / l i t t ) 40 269 386 412 424 1,438 1.530 l c575 100 64.4 91.6 103 '111 1.418 1.600 1.725 150 38.4 54.6 62.3 67.2 1.422 1.622 1.75 200 27.6 39.6 45.7 55.6 1.435 1.660 2.02 300 17.0 26.5 30.2 33.8 1.560 1.775 1.98 I i ( t ) are the i n t e n s i t i e s of the i t n decay at the time t„ 2'5 * . shown were t • 5, 15, 45 seconds. A series of experiments of the following nature was made: the crystal was x-rayed, and the decay recorded. Then, when the luminescent light had disappeared into the noise background, the crystal was re-irradiated with x-rays and the consequent decay of luminescence recorded. This procedure was repeated a number of times without benefit ,of intermediate annealing of the crystal. The results of this experiment are summarized in table II. It was found that this "irreversible" increase in luminescent intensity could be removed by annealing the crystal for long periods of time (up to 60 hours at~300°C). In other words, two successive decay experiments were completely reproducible i f and only i f the crystal was f u l l y annealed between experiments. Since there are arguments for believing that the decay curve i s most closely approximated by a simple mono- molecular mechanism at low temperatures, an experimental determination of the decay curve for Kl was made at liquid air temperatures. One of the features of this experiment was the fact that i t was found possible to re-excite luminescence in the crystal at this low temperature by means of visible light after the original light intensity had dropped to the noise level. After an irradiation with visible light of 1 s e c , an appreciable amount of luminescence was observed. The crystal was then exposed to white light for about 90 minutes. At the FIG. 8. Glow Curve for K l Facing Page 26 26 end of t h i s time, no luminescent l i g h t c o u l d be d e t e c t e d from the c r y s t a l . The c r y s t a l was then allowed t o warm up and the temperature and i n t e n s i t y o f emitted l i g h t recorded as a f u n c t i o n of time. The r e s u l t s of t h i s glow curve are i n d i c a t e d i n f i g . 6*. TABLE I I I Facing Page 27 E^ AND S L VALUES. i n ev log i n ev log s: 0 . 6 8 4 . 8 0 . 6 9 7 . 1 0 . 7 4 5 . 0 0 . 6 7 7 . 4 0 . 7 0 6 . 1 0 . 5 9 6 . 9 0 . 6 9 6 . 4 0 . 4 5 5 . 7 27 IV. DISCUSSION OF EXPERIMENTAL RESULTS. 1. The Temperature Dependence of the A-w,m. Values. In the re s u l t s obtained f o r Kl by the procedure of superposition of many "allowed" l i n e spectra and transformation into an equivalent continuum, no attempt was made to choose, a p r i o r i , the values of s L appearing i n equation 111(1), or to impose a temperature dependence on the A > * w The values of the ̂ *,fcK.were plotted i n a logarithmic scale against l/T, (see f i g . 6) and an attempt made to f i n d a set of straight l i n e s which would cover a l l the points so obtained i n accordance with the requirements of equation I I I ( l ) . The only p o s s i b i l i t y of doing so, i s indicated i n f i g . 6; a l l other p o s s i b i l i t i e s were excluded f o r physical reasons. The values for Ej, and S t can be calculated from equation I I I ( l ) with the help of f i g . 6, and these values are exhibited i n table I I I . • It must be emphasized that these values represent crude approximations for rather questionable physical quantities. The approximate character of the values i s revealed by the large scattering of points about the l i n e s , which indicates either large experimental and a n a l y t i c a l errors or a wrong physical i n t e r p r e t a t i o n of the points. The large scatter obviously prevents quantitative statements as to a s l i g h t curvature of the l i n e s or a common point of i n t e r - section at l/T • 0, or l/T = o» , which have been proposed by other authors.^-3 > ^ There are other arguments against an over-optimistic interpretation of f i g . 6. These include the fact that there  2$ may be e f f e c t s due to the self-absorption of the luminescence li g h t by the c r y s t a l which i s colored by x - i r r a d i a t i o n . Also, i t has been found that the X.*, ^.values corresponding to decays which occur under i d e n t i c a l conditions, vary with the i r r a d i a t i o n time (see f i g . 9). This also points to d e f i c i e n c i e s either i n the analytic method used i n f i n d i n g the Xv,a.^. values or i n the physical picture of the decay process. It should be pointed out, too, that the differences between the El̂  and S t values of individual traps as given i n the table are of the same order of magnitude as the thermal broadening of these quantities. This fact j u s t i f i e s the introduction of a continuous d i s t r i b u t i o n function but renders s l i g h t l y questionable the r e a l i t y and exact location of i n d i v i d u a l bands i n t h i s d i s t r i b u t i o n . The p o s s i b i l i t y that r a d i a t ionless t r a n s i t i o n s may help to empty f i l l e d traps introduces further complications. Equation 111(5) indicates that i f r a d i a t i o n l e s s t r a n s i t i o n s are allowed, the observed l i g h t i n t e n s i t y must be described i n terms of X* the " e f f e c t i v e " decay constant as well as X, the radiant t r a n s i t i o n p r o b a b i l i t y . Since graphical analysis y i e l d s the decay constant appearing i n the exponential, f i g . 6 may represent the temperature dependence of X*. The scatter of the points about the l i n e s i n t h i s figure may be due to t h i s f a c t , since X.* i s a rather complex quantity with an uncertain temperature dependence. 29 2. E f f e c t o f I r r a d i a t i o n Time on t h e Decay. E q u a t i o n 111(7) i n d i c a t e s t h a t i f ( b ^ r * *t) "t* i s g r e a t e r t h a n 4 , say, t h e |* become independent o f t h e t i m e o f c h a r g i n g , i . e . r e a c h s a t u r a t i o n . A study o f t h e e f f e c t o f d i f f e r e n t i r r a d i a t i o n t i m e s upon t h e l u m i n e s c e n t decay was made, a t y p i c a l r e s u l t b e i n g shown i n f i g . 7. I f t h e a s s u m p t i o n i s made t h a t a t t h e t i m e t , the decay c o n s t a n t X - /-t i s t h e main c o n t r i b u t o r t o t h e i n t e n s i t y and t h a t X(0~^>A> t h e n t h e s e e x p e r i m e n t s i n d i c a t e d t h a t even f o r t h e l a r g e s t o b s e r v a b l e X - v a l u e s , s a t u r a t i o n was not reached f o r an e x c i t a t i o n t i m e o f 30 seconds. (For a l l o t h e r e x p e r i m e n t s p e r f o r m e d , t h e i r r a d i a t i o n t i m e was 30 seconds.) T h i s t h e n a l l o w e d an upper l i m i t t o be put upon th e n u m e r i c a l v a l u e of (K* C ) such t h a t (MIC) * d s e c . - x . 3 . The Temperature Dependence o f t h e P o p u l a t i o n s Decay e x p e r i m e n t s g i v e i n f o r m a t i o n about t h e decay c o n s t a n t s A ; as w e l l as t h e p o p u l a t i o n s ^ . These p o p u l a t i o n s a r e , i n p r i n c i p l e , as c h a r a c t e r i s t i c o f t h e decay mechanism as t h e . T h e r e f o r e , the b e h a v i o u r o f t h e ^ as a f u n c t i o n o f t e m p e r a t u r e s h o u l d a l s o be t a k e n i n t o c o n s i d e r a t i o n when an e x p l a n a t i o n o f t h e decay mechanism i s a t t e m p t e d . T h i s has been done i n few p r e v i o u s i n v e s t i g a t i o n s . The t e m p e r a t u r e dependence o f t h e p o p u l a t i o n s t h a t was e x p e r i m e n t a l l y f o u n d i n t h i s work w i l l be e x p l a i n e d i n terms o f an extreme model where t h e e x c i t a t i o n and decay mechanisms a r e o f t h e f i r s t o r d e r (monomolecular). The TABLE IV. Facing Page 30 TEMPERATURE DEPENDENCE- OF THE POPULATION AND CAPTURE•CROSS-SECTIONS (in arbitrary units) • T°cN i l 2 3 4 5 6 7 8 9 2 0 0 K 1 5 4 0 . 0 0 3 0 5 . 5 7 . 9 185 0.0121 7 . 2 . 7 . 8 190 0.0555 12 .6 1 1 . 7 1 1 0 0 . 1 4 7 16 .6 20 .8 173 h A; 400 . 0 . 0 0 2 1 3 . 8 3.30 105 0.005 4 . 0 13 .6 3 0 0 0.015 1 2 . 4 7 . 7 4 6 0 0.055 3 1 . 0 4 . 6 6 147 *L 1 2 0 0 0.0004 40 1 4 0 0 0 . 0 0 3 6 1 3 . 5 3 . 6 4 0 0 0 . 0 0 3 6 • 1 3 . 5 5 . 8 5 6 5 0 0 . 0 1 3 5 26 .6 3 . 7 3 2 6 0 0 . 0 4 3 6 15 .5 2 . 6 6 2 1 0 0 . 1 5 3 2 5.14 117 A; A V * . 1 4 7 0 0 . 0 0 0 2 4 9 1 9 5 0 0.0009 3 1 . 7 2 . 4 5 9 6 0 0 . 0 0 2 9 3 2 2 . 6 4 6 2 0 0 . 0 1 2 8 2 5 1 . 2 4 0 0 0 . 0 4 4 4 24 3 . 3 4 2 2 0 0.164 3 6 •> 9 2 A; 1 9 0 2 0 . 0 0 0 3 6 3 . 5 1 . 3 1 1 3 2 0 . 0 0 1 2 3 7 . 7 1 . 3 6 9 0 6 0 . 0 0 4 6 3 2 0.897. 4 6 6 0 . 0 1 8 5 2 0 3 . 2 2 . 2 8 2 0.0764 2 4 1 . 9 3 3 8 6 0.27 1 0 0 6 6 i 2 3 6 4 0 . 0 0 0 0 5 7 9 1 2 5 3 5 0 . 0 0 0 3 8 4 . 5 1 800 0.0007 2 6 . 6 1 1 6 2 0 0 . 0 0 3 5 4 1 7 5 0 0.014 3 1 . 1 5 2 2 576 0.119 0 . 4 4 7 65 2 5 8 30 p o s s i b i l i t y of r a d i a t i o n l e s s e l e c t r o n i c t r a n s i t i o n s from the t r a p s t o the ground s t a t e i s , however, not excluded. if- I f i t i s assumed t h a t K<-« equation I I I ( ? ) allows the v a l u e s of k ;N; (which s h a l l be termed the capture c r o s s - s e c t i o n s ) t o be c a l c u l a t e d f o r each band t h a t occurs i n the continuous J>(A) d i s t r i b u t i o n o b t a i n e d from the a n a l y s i s o f the decay curve. In t h i s case, the Jpt must be r e p l a c e d by an i n t e g r a l JbOOdA taken over the band and ^ by • By t h i s method, which i n v o l v e s a c o n s i d e r a b l e a r b i t r a r i n e s s i n the d e f i n i t i o n o f the band l i m i t s and w i t h the assumption \ 5 ^ * ( I . e . t h a t o n l y r a d i a n t escapes from the t r a p s are p o s s i b l e ) the values o f t a b l e IV have been c a l c u l a t e d . An i n t e r e s t i n g f e a t u r e of t h i s t a b l e i s t h a t the capture c r o s s - s e c t i o n s ,K;Î ;, f o r the s l o w l y decaying t r a p s tend to have a temperature dependence such t h a t the c r o s s - s e c t i o n s f o r these t r a p s s t r o n g l y i n c r e a s e w i t h d e c r e a s i n g temperature. T h i s i s a r a t h e r unexpected r e s u l t . The capture c r o s s - s e c t i o n s f o r a l l the t r a p s should most pr o b a b l y be temperature independent. The observed temperature dependence o f these c r o s s - s e c t i o n s i s not due to the f a c t t h a t they have been c a l c u l a t e d under the assumption t h a t ( K i ^ W * ) i s z e r o . I t was found t h a t no o t h e r constant v a l u e o f the sum c o u l d remove the temperature dependence. The observed decrease i n luminescence i n t e n s i t y w i t h i n c r e a s i n g temperature i s i n d i c a t e d i n t a b l e IV as a c o r r e s p o n d i n g decrease i n the capture c r o s s - s e c t i o n s . T h i s decrease i n i n t e n s i t y c o u l d a l s o be e x p l a i n e d , i f i t were 31 assumed that radiationless t r a n s i t i o n s become increasingly important as the temperature i s r a i s e d . The p o s s i b i l i t y that r a d i a t i o n l e s s t r a n s i t i o n s may occur was neglected i n the c a l c u l a t i o n of the values of table IV and t h i s neglect lead to t'he unexpected result of the temperature dependent cross-sections. The postulate was therefore made that the capture cross-sections are constants independent of temperature and the r a t i o s A * / * , ; calculated to s a t i s f y t h i s postulate. These r a t i o s are also l i s t e d i n table IV. They were calculated on the assumption that A * / A , = I for the lowest temperature of observation. The reason f o r t h i s assumption i s that r a d i a t i o n l e s s t r a n s i t i o n s play a minor role at lower temperatures and may therefore be neglected at these temperatures. It can be seen from the r a t i o s X * / ^ , ; that capture cross-sections which are temperature independent are possible only i f radiationless t r a n s i t i o n s become increasingly prevalent with increasing temperature. This represents a strong argument for the assumption that traps are p a r t i a l l y discharged by r a d i a t i o n l e s s t r a n s i t i o n s . This assumption, though, implies a difference between \n and ^ and complicates the i n t e r p r e t a t i o n of equation I I I ( l ) which has been s * experimentally found to be v a l i d for Aj . 4. I r r e v e r s i b l e E f f e c t s of X-Irradiation. Reproducible r e s u l t s could be obtained f o r a decay curve produced under i d e n t i c a l conditions only i f the c r y s t a l 32 was annealed f o r about 30 hours at **300°C. T h i s prompted an experiment , the r e s u l t s o f which s h a l l be c a l l e d the i r r e v e r s i b l e e f f e c t . An annealed c r y s t a l was i r r a d i a t e d with x - r a y s , the decay then recorded u n t i l i t disappeared i n t o the background, the c r y s t a l immediately r e - i r r a d i a t e d , t h e decay re c o r d e d , e t c . and t h i s process repeated a number of times. The e f f e c t o f t h i s procedure i s i l l u s t r a t e d by t a b l e I I , the v a l u e s given b e i n g t y p i c a l o f a l l other s i m i l a r experiments. I t can be seen t h a t repeated x - i r r a d i a t i o n i n c r e a s e s t h e i n t e n s i t y at any time t a f t e r e x c i t a t i o n and the amount o f i n c r e a s e i s g r e a t e r at t h e end of. the decay. At the . time c o r r e s p o n d i n g to the end o f the decay o f the annealed c r y s t a l a f t e r the f i r s t e x c i t a t i o n , the i n c r e a s e i n i n t e n s i t y w i t h s u c c e s s i v e i r r a d i a t i o n s tends toward a value o f about twice the i n t e n s i t y e m i t t e d by the c r y s t a l at t h i s time a f t e r the i n i t i a l i r r a d i a t i o n . To g a i n more i n f o r m a t i o n about the decay process, attempts have been made t o e x p l a i n the e f f e c t s caused by repeated x - i r r a d i a t i o n s . Three e x p l a n a t i o n s are c o n s i d e r e d and the c o n c l u s i o n s drawn from them may be checked a g a i n s t the e x p e r i m e n t a l r e s u l t s . i ) Changes o f the i n i t i a l p o p u l a t i o n s ^>i0 produced by repeated x - i r r a d i a t i o n . I f the t r a p s are r e f i l l e d by s u c c e s s i v e i r r a d i a t i o n s b e f o r e they have had a chance to be completely emptied, the p o p u l a t i o n s found i n t h e t r a p s immediately a f t e r e x c i t a t i o n may increase with successive i r r a d i a t i o n s and thus lead to an increase i n luminescence i n t e n s i t y . The c r y s t a l was re - i r r a d i a t e d with x-rays only a f t e r the luminescence i n t e n s i t y had dropped each time to the same small value where i t became unobservable because of the back- ground noise fl u c t u a t i o n s . Thus, i t may be assumed that the t o t a l trap populations just before the 2nd and subsequent i r r a d i a t i o n s were equal to one another, and also higher than that before the i n i t i a l e x c i t a t i o n . This assumption would explain an increase i n luminescence output only f o r the 2nd e x c i t a t i o n , a l l subsequent x - i r r a d i a t i o n s occurring with the same trap populations, thus leading to decay curves of the same in t e n s i t y as that a f t e r the 2nd i r r a d i a t i o n . This i s contrary to the observed behaviour, and an alternative explanation i s required. i i ) The production of new traps by x - i r r a d i a t i o n . By combining equations 111(2) and 111(7) i t can be seen that the i n t e n s i t y of the luminescence i s proportional to the n u m b e r , , of traps present i n the c r y s t a l . Hence, the i r r e v e r s i b l e increase of luminescence output could be attributed to the production of new traps by the x - i r r a d i a t i o n . This explanation i s consistent with the suggestion that luminescence i n a pure c r y s t a l matrix i s due to the electron traps associated with l a t t i c e defects. Creation of such defects by x-rays i s suggested by various observations; 34 e.g. the volume d i l a t a t i o n that accompanies the x - i r r a d i a t i o n of a l k a l i - h a l i d e s . 2 The production of new traps by x - i r r a d i a t i o n may explain the i r r e v e r s i b l e e f f e c t s of such i r r a d i a t i o n on the luminescent decay, but i t f a i l s to explain i n a simple manner the r e l a t i v e l y quick saturation of t h i 3 e f f e c t and i t s * prevalence i n the t a i l of the decay. i i i ) - Indirect changes of the decay constants by x - i r r a d i a t i o n s . As has been mentioned in ' s e c t i o n I I I , the decay "constants" may become time dependent toward the end of a decay when deviations from a raonomolecular decay process are most l i k e l y to occur. I f the observed luminescence i s caused i n part by a decay process of the second order, then i t i s conceivable that successive x - i r r a d i a t i o n s may a f f e c t the values of the decay constants X . This may be seen from the following argument: Suppose there e x i s t i n the c r y s t a l a number of very deep traps; i . e . traps with a large a c t i v a t i o n energy and from which electrons cannot be discharged by thermal f l u c t u a t i o n s . Such deep traps w i l l be characterized by very small values of the decay constants and, consequently, by virt u e of equation 111(7), w i l l not be saturated a f t e r an x - i r r a d i a t i o n of 30 seconds (the time used i n these experiments) i f + i s also assumed to have a very small value. Thus, successive x - i r r a d i a t i o n s w i l l tend to f i l l these deep traps to a greater extent u n t i l saturation i s reached a f t e r the nP^1 i r r a d i a t i o n . 35 The electrons caught i n these deep traps w i l l remain there u n t i l the c r y s t a l i s annealed, since at the temperature of the experiments thermal energy fluctuations are i n s u f f i c i e n t to eject them. Since these electrons w i l l originate from the f i l l e d band, t h e i r trapping i n the deep traps w i l l increase the number of holes i n the f i l l e d band at a l l times and thus allow the discharge of the shallower traps to remain mono- molecular for longer periods of time, i . e . the A^ w i l l remain time-independent for a longer time. Thus, x-rays may induce i n d i r e c t changes i n the decay constants A; , tending t o make them less time-dependent at the t a i l of the decay curve. In other words, i t i s possible that successive x-raying w i l l tend to make the decay process more nearly monomolecular i n character for a longer time. The experimental r e s u l t that successive i r r a d i a t i o n with-x-rays causes successive increases i n the luminescent decay would be explained, then, by the above arguments, i f i t could be shown that i n the relevant time i n t e r v a l s , a mono- molecular decay leads to higher i n t e n s i t i e s than a bimolecular decay. What i s meant by the relevant time i n t e r v a l i s discussed f i r s t . For a monomolecular decay process, the discharge of traps i s described by which, when integrated gives 36 The i n t e n s i t y o f t h e decay i s p r o p o r t i o n a l t o the r a t e o f di s c h a r g e of the t r a p s , i . e . I J ^ - - & « - frA**1* V I ( 1 ) F o r a b i m o l e c u l a r decay, the d i s c h a r g e of the t r a p s i s d e s c r i b e d I n t e g r a t i n g t h i s e q u a t i o n g i v e s i e 1 x M The i n t e n s i t y i s . g i v e n by I n t r o d u c i n g s |>.o ^ t h i s equation may be r e - w r i t t e n as • X It) •= a. VI(2) Equations VT(1) and VI(2), i f considered as f u n c t i o n s o f X and X' e x h i b i t maxima f o r A = 'A: X ' = '/t T h i s f a c t may be i n t e r p r e t e d as f o l l o w s : At any g i v e n time, say T, the value o f "X which w i l l . c o n t r i b u t e the g r e a t e s t amount t o t h e monomolecular i n t e n s i t y w i l l be X =  X/T and s i m i l a r l y X*Jp w i l l be the major c o n t r i b u t o r t o the b i m o l e c u l a r i n t e n s i t y . T h i s i s t r u e o n l y i f the d i f f e r e n c e s i n the i n i t i a l p o p u l a t i o n s o f v a r i o u s t r a p s are n e g l e c t e d . 37 In the case of a continuous trap d i s t r i b u t i o n , the i n t e n s i t y at the time t w i l l be mainly contributed by the value of (or A') ,which i s equal to l / t . It i s therefore possible to compare the i n t e n s i t i e s that would r e s u l t , at the time t , corresponding to a mono- or bimolecular decay, by considering A-*X" '/t. This implies comparing the i n t e n s i t i e s of the decay from a trap with a given i n i t i a l population and i n i t i a l i n t e n s i t y X V o corresponding to the two d i f f e r e n t decay mechanisms. One finds T -'At r—Ua, ^ L w * t = l VI(3) Equation VI(3) indicates that i n the relevant time i n t e r v a l a monomolecular decay leads to a higher light output than a bimolecular decay.starting from the same i n i t i a l conditions. Returning to the problem of the i r r e v e r s i b l e i r r a d i a t i o n e f f e c t s , the above findings may be summarized as follows: The luminescent decay of f i l l e d traps i s not purely monomolecular i n character, owing to the l i m i t e d number of electron acceptors (or holes in the f i l l e d band) and the , r e s u l t i n g retrapping of released electrons. The deviations from the monomolecular decay process are small f o r the quickly decaying traps which produce the i n i t i a l portion of the observed luminescence decay. However, f o r the slowly decaying traps, i . e . those that e s s e n t i a l l y produce the t a i l of the observed decay, these deviations are large. 38 Repeated x - i r r a d i a t i o n f i l l s v e ry deep t r a p s and p r o v i d e s a g r e a t e r number o f e l e c t r o n a c c e p t o r s (or h o l e s ) . T h i s i n c r e a s e s the monomolecular c h a r a c t e r o f the decay and so i n c r e a s e s the luminescent i n t e n s i t y . The e f f e c t shows a quick s a t u r a t i o n a f t e r an i n t e n s i t y i n c r e a s e o f about two, and i s l e a s t pronounced i n the i n i t i a l p o r t i o n o f the decay which i s always n e a r l y mono- molecular and cannot be made much more monomolecular i n c h a r a c t e r . These t h e o r e t i c a l p r e d i c t i o n s agree with the experimental r e s u l t s . They point t o c o n s i d e r a b l e d e v i a t i o n s from a p u r e l y monomolecular decay mechanism f o r s l o w l y decaying t r a p s . 5. Experiments at Low Temperatures. At l i q u i d a i r temperatures, the i n i t i a l i n t e n s i t y o f the decay curve was found to be about 100 times g r e a t e r than t h a t at room temperature. Consequently, i t was p o s s i b l e t o observe the decay f o r a g r e a t e r l e n g t h o f time b e f o r e i t disappeared i n t o the background n o i s e . A n a l y s i s o f the decay curve, however, d i d not g i v e any new or u s e f u l i n f o r m a t i o n . An i n t e r e s t i n g f e a t u r e o f the low temperature experiment was the f a c t t h a t a f t e r the o r i g i n a l luminescence had disappeared i n t o the background, i r r a d i a t i o n o f the c r y s t a l w i t h white l i g h t f o r 1 sec. was s u f f i c i e n t t o r e - i n d u c e luminescence. T h i s i n d i c a t e s t h a t at t h i s low temperature, thermal energy f l u c t u a t i o n s cannot r e l e a s e a l l the e l e c t r o n s caught i n a l l o f the t r a p s and t h a t t h e i r r e l e a s e r e q u i r e s the 39 g r e a t e r amount o f energy contained i n the l i g h t quanta. This r e - a c t i v a t i o n w i t h white l i g h t can be e x p l a i n e d by one of the f o l l o w i n g mechanisms: (a) The a b s o r p t i o n of an o p t i c a l quantum causes a t r a n s i t i o n o f an e l e c t r o n from, a very deep t r a p d i r e c t l y t o a s h a l l o w e r t r a p , from where the e l e c t r o n may escape to the conduction band by means of thermal energy and g i v e r i s e t o the r e - i n d u c e d luminescence. (b) The a b s o r p t i o n o f an o p t i c a l quantum causes a t r a n s i t i o n d i r e c t l y to the conduction band, from whence the e l e c t r o n may become trapped t o be r e l e a s e d l a t e r by thermal e n e r g i e s , o r f a l l back to the f i l l e d band, ( g i v i n g r i s e t o .a "secondary f l u o r e s c e n c e " ) . Since i t was i m p o s s i b l e t o view the c r y s t a l d u r i n g the i r r a d i a t i o n w i t h white l i g h t because o f t h e h i g h l i g h t i n t e n s i t y t h a t would have been r e f l e c t e d upon the photo-cathode of the p h o t o m u l t i p l i e r tube, i t was not p o s s i b l e to determine whether "secondary f l u o r e s c e n c e " o c c u r s . A f u r t h e r r e s u l t of the low temperature experiment was the f o l l o w i n g : A f t e r the o r i g i n a l luminescence had disappeared, i r r a d i a t i o n o f the c r y s t a l w i t h white l i g h t f o r sho r t p e r i o d s o f time could induce f u r t h e r luminescence. However, when the i r r a d i a t i o n with white l i g h t was continued f o r about 90 minutes, no luminescence c o u l d then be detected from t h e c r y s t a l . T h i s can be taken t o mean t h a t a l l t r a p s , f o r which thermal energy f l u c t u a t i o n s , and a l s o o p t i c a l quanta i n the v i s i b l e r e g i o n , are e f f e c t i v e i n causing a d e p l e t i o n of p o p u l a t i o n , had been emptied. A f t e r the 90 minute period of v i s i b l e l i g h t i r r a d i a t i o n , the temperature of the c r y s t a l was allowed to i n c r e a s e , the c r y s t a l being observed continuously. The r e s u l t s of t h i s experiment are shown i n f i g . This f i g u r e shows that three d i s t i n c t luminescent b u r s t s occurred i n the temperature r e g i o n shown. In a d d i t i o n , one other extremely broad, and very weak increase i n i n t e n s i t y was observed at a higher temperature, but i t appeared more as a general increase i n i n t e n s i t y r a t h e r than as a d i s t i n c t peak of luminescence and was t h e r e f o r e not shown i n t h i s f i g u r e . The phosphor operating temperature T at the peak of each glow curve band i s r e l a t e d t o the corresponding t r a p depth E by the approximation s c " E / k T -as 1 sec?1 Assuming values of s ranging from 10̂ * to 10^ as determined by the method p r e v i o u s l y d e s c r i b e d , the range i n a c t i v a t i o n energies f o r the three observed peaks i n the . luminescence output of the warming c r y s t a l are given by: f o r the peak at 142°K , E = 0.112 - 0.225 ey 154°K , E » 0.122 - 0.244 ev 172°K, E = 0.136 - 0.272 ev These values of the a c t i v a t i o n e n ergies, E i , l i e i n an e n t i r e l y d i f f e r e n t region of the energy range between f i l l e d and conduction band than the values as c a l c u l a t e d from the luminescent decay curves. It should be mentioned here that these values of E; were obtained a f t e r the c r y s t a l was allowed to de-excite i t s e l f by thermal means and also af t e r i t was completely bleached with white l i g h t . This may indicate that the traps responsible for the i n t e n s i t y observed i n the glow curve are of a d i f f e r e n t nature than those responsible f o r the luminescent decay. VII. CONCLUSIONS. U t i l i z i n g a method of analysis that has been developed for decay curves, i t has been possible to ascertain the existence of ct d i f f e r e n t types of traps i n K l whose act i v a t i o n energies l i e i n the range 0.4 to 0.8 ev and whose escape p r o b a b i l i t y , s, has been found to vary from 10^ to 10^. These values for and S-v were calculated on the basis of the extremely simple monomolecular decay process i n which a l l t r a n s i t i o n s of electrons lead to observable r a d i a t i o n . That t h i s simple assumption i s only a f i r s t approximation i s indicated by the fact that i t f a i l s to explain the temperature behaviour of the populations i n the traps immediately a f t e r x - i r r a d i a t i o n i n a manner which i s compatible with temperature independent cross-sections for the traps. Only by introducing the p o s s i b i l i t y that radiationless t r a n s i t i o n s f o r electrons may occur and that they become of increasing r e l a t i v e importance as the temperature i s raised, can the behaviour of the i n i t i a l trap populations be s a t i s f a c t o r i l y explained. Further arguments against the simple monomolecular decay process are found i n the explanation of the i r r e v e r s i b l e e f f e c t s of x - i r r a d i a t i o n . This explanation i s based on the assumption that successive x - i r r a d i a t i o n s cause a part of the decay process to become transformed from one of higher order to a monomolecular one. 43 In s p i t e o f these f a c t s , the determined E-v and S; v a l u e s represent acceptable approximations and c o n t a i n u s e f u l i n f o r m a t i o n about t h e luminescent system. The experiments a t low temperatues have enabled the det e r m i n a t i o n o f va l u e s which l i e i n a d i f f e r e n t energy r e g i o n than those determined from the luminescent decay. The work d e s c r i b e d above i n d i c a t e s t h a t experiments on the d e t e r m i n a t i o n o f the decay o f observed luminescence cannot, alone, g i v e d e t a i l e d i n f o r m a t i o n about the decay pro c e s s . Some o f the p o s s i b i l i t i e s f o r t h i s process have been d i s c u s s e d and when c o n s i d e r e d w i t h r e s u l t s o f oth e r experiments should f a c i l i t a t e the d e s c r i p t i o n of the mechanism i n v o l v e d when 'a c r y s t a l emits v i s i b l e r a d i a t i o n a f t e r e x c i t a t i o n w i t h x - r a y s . time (sec) FIG. 10. Graphical Analysis of a Decay Curve Facing Page 44 APPENDIX I 44 A MECHANICAL DEVICE FOR FACILITATING ...THE GRAPHICAL ANALYSIS. In order t o o b t a i n a s u f f i c i e n t number of l i n e s i n the many-lined spectrum, g i v e n by e q u a t i o n IV(2),so t h a t the d e r i v e d continuous |j>(>̂  d i s t r i b u t i o n be i n v a r i a n t , i t was necessary t o r e s o l v e a s i n g l e experimental decay curve i n t o a sum of e x p o n e n t i a l s i n a l a r g e v a r i e t y o f ways. To f a c i l i t a t e the t e d i o u s g r a p h i c a l procedure, a mechanical device f o r c o n s t r u c t i n g the l o g a r i t h m o f a d i f f e r e n c e on a l o g a r i t h m i c graph, was i n t r o d u c e d . T h i s d e v i c e i s based upon the r e l a t i o n F i g u r e 10 i l l u s t r a t e s i t s ' use f o r the f i r s t t h r e e l i n e s I, 1^ Lg. A tangent 1^ i s drawn t o the curve I i n the l o g a r i t h m i c s c a l e by means o f a l o n g r u l e r which may be f i x e d t o t h i s p o s i t i o n . The i n n e r p a r t o f a s l i d e - r u l e , having the same s c a l e as the l o g paper used, g l i d e s along t h i s l o n g r u l e r , w ith the p o i n t "1.0" always kept i n c o i n c i d e n c e w i t h i t . The s l i d e - r u l e i s kept i n a v e r t i c a l p o s i t i o n as i t s l i d e s along the t a n g e n t i a l r u l e r by a T-bar attache d to i t s outer frame. T h i s T-bar g l i d e s a l o n g a h o r i z o n t a l guide at the upper edge of the graph. With t h i s d e v i c e , I2, which c o i n c i d e s w i t h the p o i n t (x-1) on the s l i d e r u l e , can be immediately p l o t t e d i n the l o g a r i t h m i c s c a l e without any i n t e r n s d i a t e numerical work, as 4 5 indicated i n f i g . 1 0 . I'and l " represent possible tangents to the curve I, i n d i c a t i n g why d i f f e r e n t analyses are possible. APPENDIX II 46 ANALYTICAL METHODS FOR RESOLVING A DECAY CURVE INTO A SUM OF EXPONENTIALS. Assuming that the trap d i s t r i b u t i o n i n the A -scale i s a continuous one, the analysis of a decay curve reduces to the problem of inverting a Laplace transform: J - f r ^ - 1 b'(x>Ae~ A(l) ^© Several methods have been attempted for achieving the inversion: i ) Introducing co (^= too* where C= a constant. equation A(l) may be re-written as and the problem reduces to one of determining the function ^ ( * ) I f one defines then, I(t> = \\ix> e * * * * = f j * C » * V where A * i s the n t n moment of X with respect to the function jWft^ , and I ( t ) , i ( t ) , i'(t) etc. can i n p r i n c i p l e be found graphically from the decay curves. Any function may be reoresented ^5 as 0 x i r where the Hj.*) are the Hermite polynomials, and 47 C«»»s\ H^OO^*)^*-. The C ^ c a n thus be rep r e s e n t e d as f u n c t i o n s o f the moments o f x w i t h r e s p e c t t o ̂ >Cx). Thus (A) can be r e p r e s e n t e d as a power s e r i e s , i n ^ where the are f u n c t i o n s o f the moments A ( ^ . T h e r e f o r e . t h e p o p u l a t i o n f u n c t i o n , b(A) may be expressed as i i ) S i m i l a r l y , the f u n c t i o n may be c o n s t r u c t e d from the numerical v a l u e s o f the moments which can be obtained from the h i g h e r d e r i v a t i v e s o f I ( t ) at the time t=0: T h i s method r e q u i r e s an accurate knowledge o f I ( t ) i n the r e g i o n at t » 0 . i i i ) I f I ( t ) i s expressed as Then Ĉ>) may be expressed a s ^ ^ ^2. + O M a + a x ̂  + ... 0 O! I! XT I n t r o d u c i n g t * '/^ Thus A l l the d e r i v a t i v e s o f Î t} with r e s p e c t t o ̂  may be obtained from t h e d e r i v a t i v e s o f I ( t ) with r e s p e c t t o t ; i . e . from the h i g h e r d e r i v a t i v e s o f the experimental decay curve at any g i v e n value o f t . I f (n+1) terms are used i n the power s e r i e s for 'X('f) ,• then the f i r s t (n+1) d e r i v a t i v e s at any g i v e n timet»VV g i v e s u f f i c i e n t i n f o r m a t i o n t o s e t up (n+1) l i n e a r equations i n the (n+1) unknowns, . Note, too, t h a t Lim - o . 0 and i f a o i s known, *r--*»o K then Lim e t c . The v a l u e s o f a l l the 0 - ^ may, i n p r i n c i p l e , be obtained by the e x t r a p o l a t i o n o f s i m i l a r function's to HzsO. i i i i ) Another method attempted i n the a n a l y s i s o f the decay curves i s based upon the formula The f u n c t i o n A-̂ C*) i s g i v e n i n t a b u l a r form i n a paper by N. R o s e n . 1 7 Expressed i n terms o f ( f o r p r a c t i c a l purposes) t h i s formula i s and the procedure o f a n a l y s i s i s as f o l l o w s : The e n t i r e A -range o f i n t e r e s t (from o to %- X ) i s d i v i d e d i n t o a g i v e n number o f i n t e r v a l s . W i t h i n each i n t e r v a l i t i s assumed t h a t may be r e p r e s e n t e d by a power s e r i e s of 3 terms, say. The t a i l o f the I ( t ) may then be represented by the power s e r i e s 3 The l e n g t h o f the t i n t e r v a l must be chosen such t h a t the c o e f f i c i e n t s jf; remain i n v a r i a n t t o a decrease o f the f i n t e r v a l used. Then, i s c o n s t r u c t e d f o r a l l times (up t o t = « o i n p r i n c i p l e ) ^Lijt) i s then s u b s t r a c t e d from and the d i f f e r e n c e analyzed by the formula TCr> - I -11U. t*' (>./r) - ^ (V*)] T h i s procedure i s then repeated u n t i l the e n t i r e X-range i s covered. Thus ^ ( ^ ) may be re p r e s e n t e d as a sum o f the power s e r i e s f o r each X r e g i o n and the p o p u l a t i o n d i s t r i b u t i o n f u n c t i o n J>(>) r e p r e s e n t e d as = C $(>)/A where C i s a c o n s t a n t . A l l these methods o f a n a l y s i s were attempted. I t was found that the l a c k o f knowledge o f the i n t e n s i t y curve f o r decay times extremely short and a l s o extremely l o n g , i n t r o d u c e d v a r i o u s d i f f i c u l t i e s . Thus, the p o p u l a t i o n d i s t r i b u t i o n was found to have some negative v a l u e s when determined by these a n a l y t i c methods. F u r t h e r , the r e p r o d u c t i o n o f the decay curve u s i n g the ^OA} d i s t r i b u t i o n determined by these a n a l y t i c methods always l e f t something t o be d e s i r e d . 50 For these reasons, the g r a p h i c a l method was found to. be most s a t i s f a c t o r y , s i n c e i t a v o i d s , from the s t a r t , the p o s s i b i l i t y o f n e g a t i v e p - v a l u e s . A l s o , the r e p r o d u c t i o n o f the decay curves was found to be much b e t t e r than t h a t obtained by use o f the a n a l y t i c methods. REFERENCES 1. Becquerel, E. 2 . . Seitz', F. 3.. Leverenz, H.W. 4. Mott, N.F. and Gurney, R.W. 5. Bunger, W. and Flechsig, Z. 6. Bose, H.N. 7. Bose, H.N. 8. • Dekker, A.J. and Morrish, A.H. 9. Dekker, A.J. and Morrish, A.H. 10. Bonanomi, J . And Rossell, J. 11. Urbach, F. 12. Randall., J.T. ani • Wilkins, M.H.F. 13. H i l l , J . J . and Schwed, P 14. Williams,G.W., Usiskin, S.R. and Dekker, A-.J. La Lumiere (Paris 1867) Rev. Mod. Phys. 26 7, (1954). Luminescence of Solids, John Wiley and Sons, Inc., New York, (1950). Electronic Processes i n Ionic C r y s t a l s . (Clarendon Press,) Oxford, (1940). Z. Phys. 62 42, (1941T7 Ind. J. Phys. 20 2 1 , (1946). Ind. J." Phys. 21 29, (1947). -~ Phys. Rev. 78 3 0 1 , (1950). Phys. Rev. 80 1030, (195077 Helv. Phys. Acta 2j> 725, (1952). Wien Ber. (IIA) 132 363, (1930). Proc. Roy. Soc. A 184 366, (1945). J . Chem. Phys. 23 652, (1955). Phys. Rev. 92 1398, (1953T7 15. Zernicke, F. 16. C h u r c h i l l , R.V. 17. Rosen, N. Handbuch der Physik, Vol. I l l , pp.. 448 Modern Operational Mathematics i n Engineering, McGraw H i l l Book Co. Inc. New York, 1944. Phys. Rev. 2 | 255-276, (1931).

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