UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Microwave absorption by ortho pairs in solid hydrogen Statt, Bryan Wayne 1979

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Notice for Google Chrome users:
If you are having trouble viewing or searching the PDF with Google Chrome, please download it here instead.

Item Metadata

Download

Media
831-UBC_1979_A6_7 S83.pdf [ 4.47MB ]
Metadata
JSON: 831-1.0094599.json
JSON-LD: 831-1.0094599-ld.json
RDF/XML (Pretty): 831-1.0094599-rdf.xml
RDF/JSON: 831-1.0094599-rdf.json
Turtle: 831-1.0094599-turtle.txt
N-Triples: 831-1.0094599-rdf-ntriples.txt
Original Record: 831-1.0094599-source.json
Full Text
831-1.0094599-fulltext.txt
Citation
831-1.0094599.ris

Full Text

MICROWAVE ABSORPTION BY ORTHO PAIRS IN SOLID HYDROGEN by BRYAN WAYNE STATT B.Sc. U n i v e r s i t y o f B r i t i s h Columbia, 1976 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF - * MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF PHYSICS We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA February, 1979 Bryan Wayne S t a t t , 1979 In presenting th i s thes i s in pa r t i a l fu l f i lment of the requirements for an advanced degree at the Univers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make it f ree ly ava i l ab le for reference and study. I further agree that permission for extensive copying of th is thesis for scho lar ly purposes may be granted by the Head of my Department or by his representat ives. It is understood that copying or pub l i ca t ion of th i s thes is f o r f inanc ia l gain sha l l not be allowed without my written permission. Department of The Univers i ty of B r i t i s h Columbia 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 Date fe/)- / , IV1 i i ABSTRACT The microwave a b s o r p t i o n s p e c t r a of ortho p a i r s i n low ortho c o n c e n t r a t i o n samples of s o l i d hydrogen i s s t u d i e d t h e o r e t i c a l l y and e x p e r i m e n t a l l y . A theory i s developed f o r s e v e r a l l i n e broadening mechanisms and a comparison made with experiment. Phonon induced l i f e t i m e e f f e c t s are c a l c u l a t e d to be l e s s than 2 MHz i n the temperature range T=1.2-4.2 K but have not yet been observed e x p e r i m e n t a l l y . Inhomogeneous broadening due to i s o l a t e d ortho molecules and to i s o t o p i c mass d e f e c t i m p u r i t i e s i s observed and found to be c o n s i s t e n t with t h e o r y . Several new l i n e s i n the out-of - p l a n e p a i r c o n f i g u r a t i o n are reported which confirm the assignments made p r e v i o u s l y , whereas two new l i n e s i n the next nearest neighbor c o n f i g u r a t i o n support a reassignment o f these t r a n s i t i o n s . A l s o , three e x c i t e d s t a t e t r a n s i t i o n s which f i x the p o s i t i o n of the 10 P l e v e l are reported and compared with the t h e o r e t i c a l p r e d i c t i o n s o f H a r r i s e t . a l . (1977) and L u r y i and Van Kranendonk ( 1 9 7 9 ) . i i i TABLE OF CONTENTS Page A b s t r a c t i i Table of Contents i i i L i s t o f Tables v L i s t o f F i g u r e s v i Acknowlegements v i i CHAPTER I - INTRODUCTION 1 CHAPTER II - COMPUTER CONTROLLED MICROWAVE SPECTROMETER 4 2.1 Computer System 5 2.2 Phase-Locking System 12 2.3 M u l t i p l i e r Chain 12 2.4 Phase Locked Loop 13 2.5 M i l l i m e t e r Wave Phase Lock System 19 CHAPTER I I I - THEORY 23 3.1 Ortho Broadening of the P a i r Spectrum 24 3.2 Phonon Broadening of the 0r t h o - H 2 P a i r Spectrum 44 3.3 i s o t o p i c Impurity Broadening 56 CHAPTER IV - EXPERIMENT 4.1 C r y o s t a t 4.2 Sample P r e p a r a t i o n 63 63 66 i v CHAPTER V - EXPERIMENTAL RESULTS 70 5.1 Data A n a l y s i s 70 5.2 New Lines 71 5.3 Temperature Dependence of Lineshapes 83 5.4 S t r a i n and Ortho Broadening 85 5.5 I s o t o p i c S u b s t i t u t i o n E f f e c t s 88 5.6 New Features 92 CHAPTER VI - CONCLUSIONS 100 APPENDIX A: Device Codes and I n s t r u c t i o n s 103 APPENDIX B: Assembler Subroutine D e s c r i p t i o n s 106 APPENDIX C: Ba s i c Programs 112 BIBLIOGRAPHY 119 l V LIST OF TABLES Page I Sweeper Commands 10 II Second Moments and Linewidths 38 I I I Summary o f T r a n s i t i o n Rates 54 IV Phonon Induced Linewidths f o r G, and C 3 55 V Observed Ortho P a i r T r a n s i t i o n s 73 VI High Frequency Lines 76 VII Observed Linewidths 87 VII I R e l a t i v e S t r a i n Broadening F a c t o r s 90 IX Observed Ortho Broadening 90 X Channel Spectra Peaks 98 v i LIST OF FIGURES F i g u r e Page 1 M u l t i p l i e r Chain 14 2 Phase Lock Loop 15 3 15 GHz Locked S i g n a l 18 4 M i l l i m e t e r Wave Phase Locking System 20 5 Ortho S h i f t e d P a i r L e v e l s 29 6 P a i r Molecule O r i e n t a t i o n s i n the S o l i d 33 7 f ° r the T r a n s i t i o n C 3 40 8 Lineshapes J(w) 41 9 Phonon Processes and t h e i r Feynman Diagrams 45 10 Ortho P a i r - I m p u r i t y C o n f i g u r a t i o n 58 11 Lineshape Due to I s o t o p i c Mass Defect I m p u r i t i e s 60 12 Impurity C o n c e n t r a t i o n Dependence of Linewidths 61 13 C r y o s t a t 64 14 Ortho P a i r Energy L e v e l s 72 15 High Frequency Spectra 75 16 K-Band Spectra 78 17 K-Band Spectra 79 18 X-Band Spectra 81 19 G, f o r 0.1% Ortho-H 2 Sample 86 20 G, f o r 0.35% Ortho-H 2 Sample 86 21 P-Band Spectra 89 22 Channel Spectrum from 11-24 GHz 97 v i i ACKNOWLEDGEMENTS I would l i k e to acknowledge the support o f Dr. W.N. Hardy i n the s u p e r v i s i o n of t h i s p r o j e c t . I have a l s o b e n e f i t e d g r e a t l y from numerous d i s c u s s i o n s with Dr. A.J. B e r l i n s k y . F i n a l l y I wish to thank the N a t i o n a l Sciences and Engineering Research C o u n c i l of Canada f o r a Postgraduate S c h o l a r s h i p . i i 1 CHAPTER I Introduction In t h i s thesis a study of the ortho pair spectrum in samples of s o l i d hydrogen of low ortho concentrations is presented. Hydrogen is a molecular s o l i d whose molecules are e s s e n t i a l l y free rotors. The rotational energies given by [1] E=BJ(J+1) where B=60cm_l (86.4 Kelvin) for H2 , are large compared to the anisotropic interactions in the s o l i d which are of order 7cm"' (10 K). This implies that J i s a good quantum number in the s o l i d . Since a hydrogen molecule i s composed of two nuclei of spin g" , the t o t a l wavefunction must be antisymmetric under p a r t i c l e permutation. This requires the symmetric rotational wavefunctions of even J to be associated with the antisymmetric nuclear spin state 1=0 (para species) and s i m i l a r l y odd values of J must have 1=1 (ortho species). Conversion between these species i s generally very slow and hence only the J=0 and J=l levels are populated at the temperatures of the s o l i d ( t r i p l e point 13.8 K). At the low ortho concentrations of interest here the s o l i d is known to be hexagonal close packed (hep) at a l l tempe ratures. The ortho pairs, in t h i s case nearest or next nearest neighbours in the l a t t i c e , have nine states whose le v e l s are determined by interactions between the two molecules of the pair and other two- and three-body interactions between the pair and surrounding para molecules. The dominant interaction between ortho molecules i s the e l e c t r i c quadrupole-quadrupole (EQQ) 2 i n t e r a c t i o n . T h i s s p l i t s the F=2 l e v e l s (F i s the t o t a l angu lar momentum of the p a i r ) but l e a v e s the F=0 and F=l l e v e l s d e g e n e r a t e . T h i s degeneracy i s then removed by o t h e r i n t e r a c t i o n s such as the c r y s t a l f i e l d or e l e c t r o n i c p o l a r i z a b i l i t y . These i n t e r a c t i o n s are d i s c u s s e d i n d e t a i l i n H a r r i s e t . a l (1977) (HBHII) . E l e c t r o m a g n e t i c r a d i a t i o n can be absorbed by o r t h o p a i r s v i a the quadrupo le induced d i p o l e mechanism even though the hydrogen m o l e c u l e s themselves have no permanent d i p o l e moment. T h i s e f f e c t d e r i v e s from the d i p o l e moment induced i n the s u r r o u n d i n g para m o l e c u l e s by the quadrupole moment o f an or tho m o l e c u l e . In t h i s case microwave energy can then be absorbed through p a i r t r a n s i t i o n s . These t r a n s i t i o n s have been observed by Hardy e t . a l . (1977) (HBHI) and have been d e s c r i b e d i n some d e t a i l . Some o f these t r a n s i t i o n s are s t u d i e d more e x t e n s i v e l y here and o ther p r e v i o u s l y unobserved t r a n s i t i o n s are r e p o r t e d . A major p a r t o f t h i s t h e s i s i s concerned wi th the l i n e -shapes o f these t r a n s i t i o n s . The energy l e v e l s o f an or tho p a i r i n an o therwi se pure para hydrogen l a t t i c e w i l l be homogeneously broadened by phonon p r o c e s s e s . However, i n a sample o f f i n i t e or tho c o n c e n t r a t i o n the r e l a t i v e l y more abundant i s o l a t e d or tho m o l e c u l e s g i v e r i s e to an inhomogeneous broadening o f the p a i r l e v e l s v i a the EQQ i n t e r a c t i o n . T h e o r i e s are deve loped for both phonon and or tho broadening and e x p e r i m e n t a l v e r i f i c a t i o n i s a t t e m p t e d . 3 A less i n t e r e s t i n g , but unfortunately often dominant form of broadening is caused by s t r a i n s in the sample. The EQQ interaction determining the pair l e v e l s i s proportional to r ~ 5 where r i s the separation between the pair molecules. Therefore a d i s t r i b u t i o n of pair spacings w i l l cause an inhomogenous broadening of the pair spectrum. The major source of th i s inhomogeneity i s mechanical s t r a i n on the sample. This unavoidable e f f e c t i s a result of the procedure used to grow the H2 samples. The temperature of the niobium c e l l containing l i q u i d i s slowly lowered u n t i l s o l i d i f i c a t i o n takes place. Upon further cooling to 4 K a free standing sample would contract by about 3 % . However the hydrogen sticks to the walls of the c e l l resulting in an inhomogeneous s t r a i n f i e l d . It i s not surprising that the hydrogen sticks well to the metallic walls. A molecular s o l i d is held together by Van der Waal's forces which are weaker for two hydrogen molecules than for an H2 molecule and a metal surface. Thus the s o l i d would rather sti c k to the metal walls than p u l l away when contraction occurs. This s t r a i n broadening makes observance of the previously mentioned broadening mechanisms very d i f f i c u l t because of i t s unpredictable nature and p o t e n t i a l l y large s i z e . A substantial e f f o r t has been made in the present work to make these effects as small as possible. However i t seems clear that one must develop techniques for growing free standing single c r y s t a l s before s t r a i n e f f ects can be further reduced. 4 CHAPTER II Computer Controlled Microwave Spectrometer The system to be described i s a computer controlled phase-locked microwave source with data c o l l e c t i o n , real time processing, and display f a c i l i t i e s . The Nova 2 minicomputer is interfaced to the microwave equipment through d i g i t a l to analog (D/A) converters which allow the computer to sweep the microwave source as desired. The microwave sources are backward wave o s c i l l a t o r s (BWO's) which are phase-locked to a 0.01-500 MHz frequency synthesizer. The frequency ranges covered by t h i s system at present are 8-12.4 GHz, 12.4-18 GHz, 18-26.5 GHz, 26.5-40 GHz, 60-90 GHz, and 75-120 GHz. In addition to sweeping the microwave frequency the computer c o l l e c t s data through analog to d i g i t a l (A/D) converters and can process i t in real time to display a spectrum as i t i s c o l l e c t e d . A moving head disk unit i s used for storing data as well as the operating system software. The high l e v e l languages available in the system also allow one to carry out sophisticated data processing i f desired. 5 2.1 Computer System: This section w i l l describe the hardware available in the present system and the software used to control these devices. The Nova 2 minicomputer is interfaced to several standard peripherals as well as a custom designed interface. Software supplied by Data General is used to control the standard devices whereas user defined support routines control the interface. The standard commercial equipment in thi s system is as follows: -Nova 2 minicomputer -Diablo 44 Disk Drive and co n t r o l l e r -LA36 Decwriter II -High speed paper tape punch, reader and co n t r o l l e r The Real Time Disk Operating System (RDOS) is used to provide basic communication between the user and the system. This is a f i l e orientated system and standard commands exist for contr o l l i n g the commercial peripherals. An assembler i s available for writing machine langauge programs as well as an editor and several debugging programs. These programs are useful in the development of the interface c o n t r o l l i n g software. Several high l e v e l languages are available as well. These are Basic, Fortran and Al g o l , which along with a l l the other software i s stored on disk and readily accessible. The interface was b u i l t by the UBC Physics department's electronics shop and is designed to allow the computer to have control of the experimental apparatus. The devices presently in 6 the system are: -16 b i t D/A c o n v e r t e r -Two 12 b i t D/A c o n v e r t e r s -8 input m u l t i p l e x e d A/D c o n v e r t e r -Programmable i n t e r v a l timer -Timer c l o c k - S y n t h e s i z e r c o n t r o l l a t c h e s - A u x i l a r y I/O l a t c h e s -4 input and 6 output d i g i t a l p o r t s -6 reed r e l a y s -9 d i g i t LED d i s p l a y The d e v i c e codes and i n s t r u c t i o n s f o r the use of these d e v i c e s are presented i n Appendix A. The f o l l o w i n g paragraphs d e s c r i b e how each d e v i c e i s used in the present system. The 16 b i t D/A c o n v e r t e r i s used to supply a p r e s e t t i n g v o l t a g e to the e x t e r n a l frequency c o n t r o l input o f the BWO supply. T h i s ensures that the BWO frequency w i l l be w i t h i n the capture range of the phase-lock system. Sweeping t h i s v o l t a g e i n step with the s y n t h e s i z e r maintains the phase-lock throughout the sweep. The s y n t h e s i z e r frequency can be set by a p p l y i n g the d e s i r e d b i n a r y coded decimal (BCD) s i g n a l s to i t s e x t e r n a l i n p u t s . T h i s i s done through the a i d of a group of l a t c h e s which allows the computer to set and hold the d e s i r e d frequency. 7 Event synchronization is maintained with the programmable interval timer. This device generates a pulse of specified length which i s used to control timing for the sweeper, CRT display, p l o t t i n g etc. The two 12 b i t D/A converters are used to provide a vis u a l display of data either to a monitor scope, s t r i p chart recorder or X-Y recorder. The two outputs are routed through the reed relays to the three devices in such a way that only one device is operative at any given time. One of the d i g i t a l outputs i s used to provide blanking for the CRT during retrace and when the CRT i s not in use. Data i s collected with the 14 b i t A/D converter. This device i s preceded by an eight input multiplexer which allows the A/D to measure one of the eight available inputs. The user selects which input i s to be measured and then starts the conversion. About 50 microseconds l a t e r the conversion i s completed and the d i g i t a l representation of the input voltage is ready to be read i n . The subroutines used to access the interface are called through Extended Basic. This langauge i s chosen because i t is interactive and has f a c i l i t i e s for c a l l i n g user assembler language subroutines. Although Basic is slow to execute and a cumbersome language to work with compared to Fortran, the former properties outweigh these disadvantages. Each assembler language subroutine used is described in Appendix B. The two main Basic programs used to run the spectrometer are l i s t e d in Appendix C and their use w i l l now be described. 8 The f i r s t program creates a voltage to frequency conversion table for presetting the microwave frequency of each BWO . This is done with the program CALIBRATION which sweeps the 16 bit D/A starting at a preselected voltage and speed. As an example one could c a l i b r a t e the D/A every 250 MHz throughout the band. This is done with a "pushbutton box" which c o l l e c t s the voltages desired as well as changing the speed and d i r e c t i o n of the sweep. Once th i s has been done the voltages collected are stored on disk for la t e r use. The second program is stored in SWCONTROL and consists of two parts. The f i r s t part runs the spectrometer and has f a c i l i t i e s for outputing data to the disk, CRT and p l o t t e r s . The other part i s used for data analysis. When the program is f i r s t run i t w i l l ask for the default data direc t o r y and then the c a l i b r a t i o n points filename. Once these are entered the spectrometer i s limited to the band s p e c i f i e d . The program must be run again in order to change the operating band. The program then enters the command mode and s i g n i f i e s t h i s by typing the prompt >. In the analysis mode the prompt i s #. Commands to be entered are either one or two l e t t e r s followed by a carriage return (CR). The f i r s t l e t t e r indicates the type of command and the second l e t t e r indicates a change in mode(M) or parameters(P). The mode i s associated with the sweeper and .plotting routines and s i g n i f i e s either the slow, fast or synthesized sweep or i s used to indicate the X-Y or s t r i p chart recorder when p l o t t i n g . The parameters of a command are entered upon request after the command is issued and can be used again 9 by t h e same command i f o n l y t h e command l e t t e r i s t y p e d . I f a c h a n g e i n p a r a m e t e r s i s d e s i r e d a P i s e n t e r e d a s t h e s e c o n d l e t t e r . F o r c o n v e n i e n c e t h e i n p u t r e q u e s t s a r e n o t t y p e d o u t i n t h e a n a l y s i s mode u n l e s s an S i s e n t e r e d a s t h e s e c o n d l e t t e r i n t h e command. T a b l e I l i s t s a l l o f t h e a v a i l a b l e commands . The p r o g r a m s e g r e g a t e s t h e d a t a a r r a y DATA i n t o f o u r s e c t i o n s a s f o l l o w s : A r r a y S t a r t C A L . T B L . 1 TO 2002 T l 4004 TPLOT 6006 w h e r e t h e v o l t a g e - f r e q u e n c y c o n v e r s i o n t a b l e i s s t o r e d i n C A L . T B L . and t h e o t h e r t h r e e s e c t i o n s a r e u s ed f o r d a t a c o l l e c t i o n . I n t h e a n a l y s i s mode one h a s a c c e s s t o a n y p o r t i o n o f DATA and i s n o t r e s t r i c t e d b y t h e s e b o u n d a r i e s . When t h e p r o g r a m i s f i r s t r u n t h e c a l i b r a t i o n p o i n t s r e a d f r o m t h e d i s k a r e t e m p o r a r i l y s t o r e d i n DATA s t a r t i n g a t l o c a t i o n 7 5 0 0 . T h e s e p o i n t s a r e t h e n t r a n s f e r e d t o a B a s i c a r r a y and h e n c e may be o v e r w r i t t e n a t a n y t i m e . The f o l l o w i n g command and r e s p o n s e s e q u e n c e i l l u s t r a t e s t h e u se o f t h e p r o g r a m . The u n d e r l i n e d p o r t i o n s a r e t h o s e t y p e d i n b y t h e o p e r a t o r . T h e s e commands w i l l t a k e two s p e c t r a , s t o r e them on d i s k , e n t e r t h e a n a l y s i s mode , d i v i d e t h e two s p e c t r a , p l o t t h e r e s u l t o n t h e s c o p e and t h e s t r i p c h a r t r e c o r d e r , s t o r e t h e r e s u l t o n d i s k and t h e n r e t u r n t o t h e command mode . 10 TABLE I Sweeper Commands Command Mode (>) Mode/Parameter Switch R run the spectrometer (M,P) S s t r i p chart or X-Y recorder (M,P) P monitor scope plot (P) D disk read or write A go to analysis mode Analysis Mode (#) M array m u l t i p l i c a t i o n A array addition V array d i v i s i o n P monitor scope plot (P) B slope addition to array D disk read or write X find min, and max. values in array S s t r i p chart or X-Y recorder (M,P) T output array to teletype I integration R return to command mode 11 >RP FSTART.DELTA F,NO.PT,DELTA T,A/DCHAN 12..001.1001.1000.0 S=0_ SYNTH. FREQ.= 381.25 MHZ DELTA F=31.25 KHZ RUN _0_ >D_WRITE(0) OR READ (1) ? j0_ ENTER FILENAME & NO.PT A.1,1001 TO BE WRITTEN OR READ FROM CAL.PTS. (0) ,CAL. TBL. (1) ,T0 (2 ) ,T1 (3 ) ,TPLOT(4) _2_ >_R RUN _0_ >D_WRITE(0) OR READ (1) ? 0_ ENTER FILENAME & NO.PT A.2,1001 TO BE WRITTEN OR READ FROM CAL.PTS.(0),CAL.TBL.(1),T0(2),T1(3),TPLOT(4) 2 >A_ DEFAULT DIR. DATA: #D. WRITE (0) OR READ (1) ? _1_ ENTER FILENAME & N A.1,1001 TO BE WR OR READ FROM 0,1,2,3,4 JL_ #VS_ D(I)=C1*D(J)/(D(K)+C2) ENTER I, J, K, N, CI, C2 4004,2002,1,1001,16 384,0 #PP 3,1001 #S_ STRIP CHART(19) OR X-Y(18) OR POINT(21) _18_ ? 1,1001,100,6006 #D_ WRITE (0 ) OR READ (1) ? _1_ ENTER FILENAME & N A.1,1001 TO BE WR OR READ FROM 0,1,2,3,4 _1_ #R_ > The input S allows one to correct for the d r i f t in the 16 b i t D/A converter. S i s added to the 16 b i t code used to set the D/A. A number of the order of 500 can move the setting frequency completely through the capture range of the phase-lock system. This program is also l i s t e d in Appendix C. Once the data has been collected and i n i t i a l l y processed with the above program i t i s then possible to do r e l a t i v e l y sophisticated analysis of the data using programs written in Fortran. For example one can put the data through a low pass f i l t e r or perhaps f i t a spectrum with a desired lineshape. The computer can also carry out calculations used to interpret the results of the data analysis. 12 2.2 Phase-Locking System: T h i s system c o n s i s t s o f a frequency s y n t h e s i z e r , a frequency m u l t i p l i e r c h a i n , and e l e c t r o n i c s to phase lock the BWO to a harmonic of the s y n t h e s i z e r . The BWO and s y n t h e s i z e r can be swept synchronously, while m a i n t a i n i n g the l o c k , and the combination c o n s t i t u t e s a high r e s o l u t i o n microvave spectometer. The output of the s y n t h e s i z e r (frequency f 0 ) i s fed i n t o a frequency doubler c h a i n which prov i d e s a s i g n a l with a frequency e i g h t times t h a t o f the s y n t h e s i z e r at e s s e n t i a l l y the same power l e v e l . T h i s s i g n a l i s mixed with a sampled p o r t i o n of the BWO output frequency (f) in a harmonic mixer. The in t e r m o d u l a t i o n frequency (IF=8Nf 0-f) i s d i v i d e d by ten with a d i g i t a l p r e s c a l a r and then fed into a s y n c h r o n i z e r . T h i s d e v i c e p r o v i d e s a DC e r r o r s i g n a l p r o p o r t i o n a l to the d i f f e r e n c e i n phase of the IF s i g n a l and a re f e r e n c e 20 MHz o s c i l l a t o r s i g n a l . The e r r o r s i g n a l i s then fed i n t o the FM input o f the BWO, completing the phase lock l o o p . The power l e v e l s present are s u f f i c i e n t to phase l o c k the BWOs up to 40 GHz but another l o c k i n g system must be used to lock the m i l l i m e t e r wave sweeper. F i r s t the X8 m u l t i p l i e r c h a i n w i l l be d i s c u s s e d and then the phase locked l o o p . 2.3 M u l t i p l i e r Chain: The purpose of t h i s c h a i n i s to ensure t h a t the harmonic mixer produces an ac c e p t a b l e IF s i g n a l . I f the s y n t h e s i z e r frequency were mixed d i r e c t l y with the BWO s i g n a l the harmonic of f0 would normally be too high to produce an IF s i g n a l of 13 s u f f i c i e n t power. The m u l t i p l i e r c h a i n does some of the harmonic m u l t i p l y i n g and p r o v i d e s a s i g n a l of about 5 dBm ( i . e . 5 dB above 1 m i l l i w a t t ) i n the frequency range 2.8-4.0 GHz. T h i s reduces the harmonic number i n the mixer used to lock the BWO. A block diagram of the m u l t i p l i e r c h a i n i s shown i n f i g . l . The output o f the frequency d o u b l e r s c l o s e l y approximates a r e c t i f i e d s i n e wave and c o n s i s t s predominantly o f the second i harmonic of the input frequency. However the e f f i c i e n c y o f the doubling process drops i f the other harmonics are not f i l t e r e d out. The band pass f i l t e r s present serve t h i s purpose. In a d d i t i o n the broad-band a m p l i f i e r s are chosen to a m p l i f y o n l y s i g n a l s i n the frequency ranges of i n t e r e s t , i e . 0.5-1.0 GHz. f o r the f i r s t stage of m u l t i p l i c a t i o n . The f i n i t e width of the s k i r t s o f r e a l f i l t e r s r e q u i r e s one to r e s t r i c t the sweeping range to l e s s than one octave (e.g. f o=350-500 MHz). 2.4 Phase Locked Loop: The phase locked loop i s the feedback system which keeps the BWO frequency locked to a harmonic of the s y n t h e s i z e r frequency. T h i s removes the i n s t a b i l i t y o f the BWO frequency and r e s u l t s i n a very narrow s p e c t r a l width governed by the s y n t h e s i z e r , m u l t i p l i e r c h a i n and s y n c h r o n i z e r phase noi s e and s t a b i l i t y . A block diagram of the phase locked loop i n f i g . 2 shows the r e s t of the apparatus i n the spectrometer. S t a r t i n g at the BWO output the microwaves are sampled with two 10 dB d i r e c t i o n a l fo 3 5 0 - 5 0 0 MHz DOUBLER 2 f o AMPL IFIER 0 . 7 - 1. BAND FILT OGHz PASS ER V DOUBLER 4 f o 1 . 4 - 2 . 0 GHz BAND PASS FILTER V DOUBLER 8 f o A 2 . 8 - 4 BAND FI'Ll . 0 GHz PASS fER AMPLIFIER Figure 1 M u l t i p l i e r Chain MIXER LOW PASS FILTER •<—>-ISOLATOR ISOLATOR ) ] — » LEVELLER CRYSTAL DETECTOR TO CELL ERROR SIGNAL FM INPUT FROM BACKWARD WAVE OSCIL COMPUTER .ATOR i SWEEP INPUT 16 BIT D/A SYNCHRONIZER 20 MHz BAND PASS SAMPLER AMPLIFIER BAND PASS FILTER LIM ITER 200 MHz FILTER -*— LO MULTIPLIER CHAIN SYNTHESIZER FR OM COMPUTER Figure 2 Phase Locked Loop 16 c o u p l e r s . The second coupler i s connected to a c r y s t a l d e t e c t o r whose output i s used i n the l e v e l l e r c i r c u i t o f the BWO to keep the power l e v e l constant with frequency. The l e v e l l e r c i r c i u t keeps the d e t e c t o r v o l t a g e constant and hence the power spectrum e s s e n t i a l l y takes on the response f u n c t i o n of the d e t e c t o r . The second coupler i s connected to the harmonic mixer. The low pass f i l t e r prevents harmonics from e n t e r i n g the experiment and the i s o l a t o r s prevent r e f l e c t e d s i g n a l s from e n t e r i n g the BWO and l e v e l l i n g l o o p . The output o f the m u l t i p l i e r c h a i n i s fed into the harmonic mixer and the output i s a m p l i f i e d and put through a 200 MHz bandpass f i l t e r . The s i g n a l then goes through a l i m i t e r and a d i v i d e by ten p r e s c a l a r and then the s y n c h r o n i z e r . T h i s d e v i c e compares the incoming frequency to i t s r e f e r e n c e 20 MHz o s c i l l a t o r with two phase s e n s i t i v e d e t e c t o r s . A DC e r r o r s i g n a l i s then produced which i s p r o p o r t i o n a l to the phase d i f f e r e n c e measured . The s y n c h r o n i z e r w i l l o n l y l o c k onto one of the IF sidebands, determined by the s i g n of the v o l t a g e -frequency r e l a t i o n s h i p of the FM input o f the BWO. The harmonic spacing of 8 f 0 i s much g r e a t e r than the capture range of the phase l o c k loop ensuring that the BWO frequency i s u n i q u e l y determined by f c and the p r e s e t t i n g v o l t a g e fed to the BWO sweeper u n i t . The capture range of the s y n c h r o n i z e r i s increased by a f a c t o r of ten with the p r e s c a l e r which r e l a x e s the i n i t i a l c o n d i t i o n s r e q u i r e d to b r i n g the system i n t o l o c k . A l s o , the l a r g e amount of FM present i n the BWO output makes the t i g h t e r lock d i f f i c u l t to e s t a b l i s h . The l a r g e r capture range 1 7 i s a l s o u s e f u l i n overcoming the d r i f t of the 16 b i t D/A c o n v e r t e r . However the s e n s i t i v i t y of the loop i s decreased by the same amount which i n c r e a s e s the phase noise of the output over that f o r a system not using the d i v i d e by ten p r e s c a l a r . T h i s i s q u i t e a c c e p t a b l e i n the present s i t u a t i o n where the width of the l i n e s measured are g r e a t e r than one MHz. The 200 MHz narrow band f i l t e r i s necessary because of the d i g i t a l nature of the p r e s c a l e r . Without i t the s y n c h r o n i z e r could l o c k onto a harmonic of the p r e s c a l e r which would mean an IF frequency being a subharmonic of 200 MHz. A spectrum analyser can be used to monitor the output of the spectrometer when i t i s phase- l o c k e d . In the 12.4-18.0 GHz band the 3 dB p o i n t i s l e s s than 3 KHz, the s m a l l e s t bandwidth a v a i l a b l e on the spectum a n a l y s e r used. The power spectrum of the spectrometer i s shown i n f i g . 3 where f=15 GHz and the t o t a l harmonic number i s 32. The observed phase noise a p p a r e n t l y comes from the s y n t h e s i z e r . To see t h i s we note that a f t e r frequency m u l t i p l i c a t i o n by n the s i g n a l - t o - n o i s e r a t i o of the output decreases by 201og(n) dB (Kroupa, 1973). The s i g n a l to noise r a t i o of the power spectrum 100 KHz away from peak power i s 60 dB where the bandwidth i s 3 KHz. T h i s i m p l i e s an incoming S/N of 60 dB + 35 dB + 30 dB = 125 dB with 1 Hz bandwidth. T h i s agrees with the s p e c i f i c a t i o n s of the s y n t h e s i z e r g i v e n by the manufacturer.(Rohde and Schwarz model 103.9968.51) 18 Figure 3 15 GHz Locked S i g n a l 19 2.5 M i l l i m e t e r Wave Phase Lock System: The 75-120 GHz BWO cannot be phase locked d i r e c t l y from the m u l t i p l i e r c h a i n s i n c e the l a r g e harmonic number produces i n s u f f i c i e n t IF s i g n a l . For t h i s reason the m i l l i m e t e r wave BWO i s phase locked to the e x i s t i n g 12.4-18.0 GHz s y n t h e s i z e r phase locked source. T h i s u n i t p r o v i d e s more power than the m u l t i p l i e r c h a i n and a l s o ensures the use of lower harmonics. The l o c k i n g system i s s l i g h t l y d i f f e r e n t from the one used before s i n c e another s y n c h r o n i z e r i s not a v a i l a b l e . Also the phase l o c k acquired i s not as s a t i s f a c t o r y as with the previous system because of the increased t e c h n i c a l d i f f i c u l t i e s a s s o c i a t e d with the higher frequency. A block diagram of the system i n f i g . 4 shows that the sy n c h r o n i z e r has been replaced by a mixer and the frequency p r e s e t t i n g v o l t a g e i s s u p p l i e d with a wide-band frequency d i s c r i m i n a t o r r a t h e r than another D/A c o n v e r t e r . The d i s c r i m i n a t o r p r o v i d e s the rough frequency c o n t r o l f o r the BWO. I n i t i a l l y the m i l l i m e t e r BWO sweeper i s manually s e t so that the system goes i n t o l o c k . Then as the 12.4-18.0 GHz source i s swept the d i s c r i m i n a t o r p r o v i d e s an e r r o r s i g n a l .to the i n t e g r a t o r which i s then fed i n t o the FM input o f the BWO sweeper. The mixer i n the d i s c r i m i n a t o r w i l l produce a s i g n a l approximately p r o p o r t i o n a l to the d i f f e r e n c e between the IF frequency and f =c / A, where A i s the wavelength f o r which the time d e l a y corresponds to A/4. The A/4 de l a y l i n e i s preset to match the 20 MHz r e f e r e n c e s i g n a l from the phase d e t e c t o r system. T h i s d i s c r i m i n a t o r ensures that the BWO frequency i s 20 BACKWARD WAVE OSCILLATOR CRYSTAL DETECTOR kl_ u i 9 MIXER VCV 12.4-18.0 GHz SWEEPER AMPLIFIER ISOLATOR A WAVEGUIDE / \ ADAPTOR TO CELL -r 10 PRESCALER 20 200 MHz BAND PASS FILTER TO FM INPUT AMPLIFIER OF SWEEPER MHz POWER SPLITTER r MIXER 7 MHz LOW PASS FILTER INTEGRATOR | fy4 DELAY LINE DISCRIMINATOR MIXER 9 MHz LOW PASS FILTER ~1 I AMPLIFIER TO TORROID ON BWO 20 MHz REFERENCE OSCILLATOR PHASE DETECTOR J Figure 4 M i l l i m e t e r Wave Phase Locking System 21 always close enough to the desired harmonic of the sweeper signal to enable the phase detector system to lock. The phase detector produces an error signal proportional to the phase difference between the IF signal and the reference o s c i l l a t o r which i s then fed into the discriminator's integrator and a coupling toroid on one of the high voltage grids (we have used grid #2) in the BWO tube. The frequency response of the toroid is about 10 KHz to 1 MHz and the FM input response is DC to 10 KHz so that the two systems more or less overlap. The rest of the system is similar to that of the low frequency locking apparatus. Monitoring the IF spectrum when the system is phase locked reveals that the noise l e v e l is much higher than can be attributed to the synthesizer. Apparently the phase detector system does not have a high enough frequency response and cannot compensate for a l l the phase disturbances occuring in the BWO. When sweeping, the discriminator's integrator gradually supplies a larger DC voltage to the BWO sweeper in order to keep the BWO frequency in l i n e with the sweeper frequency. One disadvantage of this method is that i f for some reason the system unlocks momentarily the integrator output may move s u f f i c i e n t l y far that lock cannot be reestablished. This means the sweep must be stopped and restarted. A further d i f f i c u l t y i s that because of the various interactions between the amplitude and frequency response of the BWO g r i d s , operation of the l e v e l l i n g c i r c u i t prevents phase locking. This means that the power le v e l i s not constant with frequency. In practise the power le v e l was 22 recorded simultaneously with the spectra and l e v e l l i n g accomplished in the computer analysis. 23 CHAPTER I I I Theory In t h i s chapter some of the c o n t r i b u t i o n s to the p a i r t r a n s i t i o n l i n e w i d t h s are examined. F i r s t a theory i s developed for the inhomogeous broadening of the ortho p a i r l i n e s due to the random d i s t r i b u t i o n o f i s o l a t e d ortho molecules. T h i s i s followed by estimates o f the temperature dependent phonon induced l i f e t i m e broadening of the p a i r l e v e l s . F i n a l l y , the e f f e c t o f i s o t o p i c mass d e f e c t i m p u r i t i e s on the p a i r l i n e w i d t h s i s examined. 24 3.1 Ortho Broadening of the P a i r Spectrum: A s i n g l e ortho p a i r i n an otherwise pure para hydrogen c r y s t a l w i l l have i t s energy l e v e l s l i f e t i m e broadened by phonon proce s s e s . However, i n a sample with f i n i t e ortho c o n c e n t r a t i o n there are other p o s s i b l e mechanisms f o r broadening. In p a r t i c u l a r there are many more i s o l a t e d ortho molecules^ than there are ortho p a i r s . These ortho molecules i n t e r a c t with the p a i r molecules v i a the dominant e l e c t r i c quadrupole-quadrupole (EQQ) i n t e r a c t i o n causing a small s h i f t i n the p a i r energy l e v e l s . Because of the s t a t i s t i c a l nature of the d i s t r i b u t i o n of o r t h o s , each ortho p a i r w i l l experience a d i f f e r e n t c o n f i g u r a t i o n of ortho molecules. T h i s r e s u l t s i n an inhomogeneous broadening of the p a i r l e v e l s . The l i n e s h a p e of a p a i r t r a n s i t i o n can then be c o n s t r u c t e d by forming a d i s t r i b u t i o n f u n c t i o n of the d i f f e r e n c e s between the f i n a l and i n i t i a l p a i r l e v e l s h i f t s over a l l p o s s i b l e ortho molecule c o n f i g u r a t i o n s and s t a t e s . A s t a t i s t i c a l model d e s c r i b i n g the NMR s p e c t r a l d e n s i t y f u n c t i o n s of a system of random impurity s p i n s with i n t e r a c t i o n s having r a d i a l dependence R - n has been formulated by Hama e t . a l . and a p p l i e d to s o l i d hydrogen by F u j i o e t . a l . Since they are l o o k i n g at the i n t e r a c t i o n between a random s p i n d i s t r i b u t i o n and a s i n g l e s p i n ( r a t h e r than a p a i r of spins) t h e i r r e s u l t s are not d i r e c t l y a p p l i c a b l e here. However one i From now on i s o l a t e d ortho molecules w i l l be r e f e r e d to simply as ortho molecules, which are to be d i s t i n g u i s h e d from molecules that are part o f an ortho p a i r . 25 would expect the gen e r a l f e a t u r e s o f the broadening to be the same i n both cases and t h i s does seem to be borne out. T h i s i s an N-body problem i n v o l v i n g a random d i s t r i b u t i o n of the i n t e r a c t i n g s p e c i e s , and even though the i n t e r a c t i o n Hamiltonian i s we l l known the problem cannot be solved in g e n e r a l . The approximations made here e s s e n t i a l l y reduces the c a l c u l a t i o n to a two-body problem. For convenience the ortho p a i r w i l l be placed at the c o o r d i n a t e o r i g i n and the r e s t o f the l a t t i c e d i v i d e d i n t o d i s c r e t e s p h e r i c a l s h e l l s o f r a d i u s r . In the l i m i t o f low ortho c o n c e n t r a t i o n the set of ortho molecule c o n f i g u r a t i o n s c o n s i s t i n g of o n l y para molecules i n s i d e a sphere of r a d i u s r (except the ortho p a i r ) , one ortho molecule i n the s h e l l r and a random d i s t r i b u t i o n of o r t h o s beyond w i l l approximate the a c t u a l set of a l l p o s s i b l e c o n f i g u r a t i o n s . In a d d i t i o n the e f f e c t on the p a i r l e v e l s o f the ortho molecules beyond the f i r s t one w i l l be replaced by t h e i r average, which i s zero, as w i l l be shown l a t e r . T h i s i s probably a c c e p t a b l e s i n c e - 5 the EQQ i n t e r a c t i o n i s r e l a t i v e l y s h o r t ranged (R ). The model d e s c r i b e d so f a r i n c l u d e s the s t a t i c s h i f t s i n the p a i r l e v e l s assuming no i n t e r a c t i o n between the ortho molecules o u t s i d e the f i r s t ortho s h e l l . T h i s ignores the c o r r e l a t i o n s between the ortho molecules. For example i f two ortho molecules have equal c r y s t a l f i e l d s p l i t t i n g s then each molecule can " f l i p " i t s s t a t e and s t i l l conserve energy. T h i s would lead to a type of motional narrowing observed i n nucl e a r s p i n systems. In t h e i r approach Hama e t . a l . have t r i e d to take t h i s i n t o account by e x c l u d i n g c o n t r i b u t i o n s to the s p e c t r a l 26 d e n s i t y coming from an ortho molecule which has an ortho neighbour c l o s e r than some s p e c i f i e d d i s t a n c e . I n c l u s i o n of t h i s e f f e c t does narrow the c e n t r a l p o r t i o n o f the l i n e as expected. In the present work the c o r r e l a t i o n e f f e c t i s ignored. Now the l i n e s h a p e f u n c t i o n J(<u) can be w r i t t e n i n the form oo [2 ] J(w)=2 d3r P(r) E ^ n ) ) where d r p(r) i s the p r o b a b i l i t y o f having the f i r s t ortho molecule i n the s h e l l o f r a d i u s r and Ej;(rfn) i s the p a i r t r a n s i t i o n frequency s h i f t caused by an ortho molecule at a d i s t a n c e r and o r i e n t a t i o n fl=(6,0) with r e s p e c t to the p a i r a x i s . The a p p r o p r i a t e l a t t i c e sum i s r e p l a c e d by an i n t e g r a l s t a r t i n g at some c u t - o f f r a d i u s r c , chosen to be a few l a t t i c e s p a c i n g s . I f N r i s the number of l a t t i c e s i t e s i n s h e l l r then [ 3 ] dV p r r > * (l-X) ' " -N r X( l-Xy where (1-X) i s the p r o b a b i l i t y o f having no ortho molecules i n s h e l l s 1 through r - 1 and N f X ( l - X ) ^ ' i s the p r o b a b i l i t y o f having one ortho molecule i n s h e l l r . Now l e t r N = ]> whence [4] dV pOO - e ~ N X N r X 3 I f ~3~^x = T^Ty * s t n e average volume per ortho molecule then ts] <jv p(x) = 3 e ^ f -X T h e r e f o r e 27 [6] T(to) zlaf lldr^_ fa^si)) and the n o r m a l i z a t i o n constant is f7T -rVr/* [7] a " T~ An ortho molecule i n an otherwise pure para H z l a t t i c e experiences a c r y s t a l f i e l d with 3 - f o l d symmetry about the c-a x i s which s p l i t s the J=l s t a t e i n t o l e v e l s with mj=±l and nij- = 0. The s i g n of the c r y s t a l f i e l d s p l i t t i n g V c i s not known but i t w i l l be assumed that the irij = 0 l e v e l i s lower i n energy than the doubly degenerate mj= ±1 l e v e l . These l e v e l s are a s s o c i a t e d with the s t a t e s |l,mj> = Y ™ T ( oJ3) where 0J3 i s the angle between the ortho molecular a x i s and the q u a n t i z a t i o n a x i s (which i s the c r y s t a l c - a x i s ) . In the presence of other ortho molecules the EQQ i n t e r a c t i o n s p l i t s the two degenerate l e v e l s an amount o f -3 the order of a few MHz f o r X-2xl0 . T h i s i s much l e s s than the c r y s t a l f i e l d s p l i t t i n g of 300 MHz and hence o n l y the m^  = ±1 s t a t e s are mixed. Diagonal i z i n g the matrix elements < 1, m;j-| K^oQ l 1 ' m T > with r e s p e c t to m.j=±l leads to the non-degenerate s t a t e s [8] I W = y f and as before Jl,z>=Jl,0>. These "dumbell o r b i t a l s " are o r i e n t e d with the x and y o r b i t a l s each r o t a t e d by an angle Y 28 from the x and y q u a n t i z a t i o n axes r e s p e c t i v e l y . The angle f i s determined by the c o n f i g u r a t i o n and s t a t e s o f o r t h o s which surround the ortho molecule of i n t e r e s t . In g e n e r a l f i s time dependent s i n c e the ortho molecules can change s t a t e s as d e s c r i b e d e a r l i e r . However t h i s e f f e c t i s being ignored here. The c o n f i g u r a t i o n of ortho molecules surrounding the one c l o s e s t to the p a i r i s not completely random s i n c e there can be no other ortho molecules i n s i d e the f i r s t ortho s h e l l . However, i t w i l l be assumed that V i s a random v a r i a b l e . Hence the f i n a l l i n e s h a p e w i l l c o n s i s t of an average of the f-dependent l i n e s h a p e s . In order to f u r t h e r s i m p l i f y the c a l c u l a t i o n i t w i l l be assumed that as the p a i r molecules undergo a t r a n s i t i o n , the e i g e n s t a t e s of the c l o s e s t ortho molecule w i l l not change. T h i s i s not p r e c i s e l y c o r r e c t , s i n c e the EQQ i n t e r a c t i o n with the ortho depends on the s t a t e of the p a i r molecules. However, s i n c e the degenerate l e v e l i s s p l i t by the presence of many ortho molecules i t i s not unreasonable to assume that i n most cases the two s p l i t s t a t e s w i l l remain e s s e n t i a l l y the same. The l e v e l diagram i n f i g . 5 i l l u s t r a t e s the e f f e c t an ortho molecule has on the p a i r l e v e l s , and on the t r a n s i t i o n s allowed between these l e v e l s . The o n l y t r a n s i t i o n s c o nsidered here are those where the p a i r s t a t e changes but the ortho s t a t e remains the same. The f o l l o w i n g arguement w i l l show t h a t i n t e n s i t i e s f o r other types of t r a n s i t i o n s are much l e s s than the kind considered here. (a) EQQ S p l i t P air Levels and Transitions |,,o> i Lone Ortho E Q Q S p l i t (b) EQQ S p l i t Ortho Molecule Levels Figure 5 Ortho Shifted Pair Levels 30 The integrated intensity associated with a pair t r a n s i t i o n is proportional to a sum of terms containing <f |P < n ) | i X i \ p ( m ) * f f> (HBHII) where |i> and | f> are the i n i t i a l and f i n a l pair states and i s a component of the dipole moment operator. This operator i s s p l i t into a scalar and tensor term P =PS + Pt Considering f i r s t the scalar term where (*), and a)2 are the orientation angles of the pair molecules and C 0 3 is that of the ortho molecule a l l with respect to the pair axis. The product states are |i>= | F , M > |1,X> and •P f |f>= | F , M> [l,X> . Hence the integrated intensity i s proportional to terms and <TFM| r;V) t o-y;v(u>,) | F,M> c<i,x | i,x> For a pair t r a n s i t i o n the l a t t e r term drops out and the former term contributes to the int e n s i t y i f |l,X> = |l,X> . The tensor term P^  contains a sum of terms YZ(VI) / 2 m + V j ) ^ ^ ( - f l i j ) R.^  * where i and j label the ortho molecules as before ( i , j = l , 2 ; 2,3; 3,1). The intensity therefore contains the i,j=l,2 term multiplied by L<l,x|l,X> -4-and the other two terms contain R-L3 where i = l , 2 . This l a t t e r contribution w i l l be ne g l i g i b l e compared to a l l the others in 31 the low or tho c o n c e n t r a t i o n l i m i t where the average o r t h o - p a i r d i s t a n c e i s many t imes R 0 . I t then f o l l o w s t h a t the s t r o n g e s t p a i r t r a n s i t i o n s are those i n which the or tho molecu le does not change s t a t e . The o t h e r s are too weak to be c o n s i d e r e d . The EQQ i n t e r a c t i o n H a m i l t o n i a n i s g i v e n by ( H a r r i s , 1970) »5 where i and j l a b e l the i n t e r a c t i n g or tho m o l e c u l e s . The p a i r s t a t e s |F,M> are g i v e n by (HBHII) [12] | F , M ) = ;> co i f > ,M -mj x'k) y,M(uM where F i s the t o t a l angu lar momentum of the p a i r and M i s i t s p r o j e c t i o n a long the p a i r a x i s . These s t a t e s g i v e r i s e to a two f o l d degeneracy f o r the |2,±2>, | l , t l > and |2,±1> s t a t e s which i s l i f t e d by the n o n - c y l i n d r i c a l l y symmetric i n t e r a c t i o n s d i s c u s s e d in HBHII. The non-degenerate s t a t e s are then where the x - a x i s i s chosen so tha t the x -z p lane i s a r e f l e c t i o n p lane o f the c r y s t a l . The s t a t e s o f the or tho molecu le are c o n s t r u c t e d wi th j 1 ,m-r>=Yf T(ui^ where (O3 i s measured wi th r e s p e c t to the c r y s t a l c -a x i s . S ince i t i s more conven ien t to work i n the p a i r frame t h i s q u a n t i z a t i o n a x i s must be a l i g n e d with the p a i r a x i s by the r o t a t i o n [13] ,., . J f <_ „ m where are the E u l e r ang les d e s c r i b i n g the r o t a t i o n o f the c r y s t a l c - a x i s to the p a i r a x i s . The y - a x i s o f the c r y s t a l 32 f i e l d frame can be a r b i t r a r i l y chosen so that °<=0 and f o r the three cases of i n t e r e s t jS and X are g i v e n i n f i g . 6 . E v a l u a t i o n of E(r,_a) w i l l r e q u i r e the matrix elements <l,mg| < F , M | X[3 | F , ± M > |l,m3'>. With the h e l p o f Rose (1952) [15] < ' ^ 3 | ( F , M | M U | F , M > | I ^ ^ ^ R(TLO C F I W and [16] where C ' zl/r[3(2F+l)j,/2W(2IF(3lF)c(2FF;ZAArM) 33 J A k A 1 + 1 + 2 0) Nearest neighbor i n - p l ane sin/9 A k J r « o * i b) Nearest neighbor out-of-plane A i A J r - o -6 A k C) Next nearest neighb or crystal field axes tl • + +2 I A A k, A 2+ • A — — L + + ' local pair axes Note that i £ x ; f k & and only the i £ or J which l i e s i n the basal plane, represented by dots, i s shown. The pluses represent the next plane up. Figure 6 P a i r Molecule Orientations i n the S o l i d 3 4 and l - « l 3 Note that C ( 2 2 4 ; 2 M , y ) = 0 i f M = 2 and hence S ,(m 3,m 3 /) does not enter i n t o the matrix elements where F = M = 2 . Using the forms o f |l,X L> in [8] and [9] [19] < U ^ R M | } ^ R A / i ; > |l,X<> - R(ri3) S(^f»i) where |F,M>„< i s e i t h e r | F , M > ( ° ( = 0 ) . or | F , |M] >± (<*=±l) and the v a r i a b l e s i n v o l v i n g i are d e f i n e d by i x t m{ 1 x -1 1 2 y +1 1 3 z 0 0 Now t h a t the matrix elements have been determined one can 35 proceed with the c a l c u l a t i o n of J (oj) . Assuming that r / ? =r 2-j=r and J l ( 3 = i l 2 3 =Jl which i s a c c e p t a b l e f o r l a r g e r , [20] EL(r>fL)= zR(r) AC { S(mi,»>.) + ^ & e S(»i > -m;)]} where and I F ^ M ^ is the i n i t a l p a i r s t a t e and JF',*!'^/ the f i n a l p a i r s t a t e . Averaging over a l l ^  and i [21] * x V r * Replacing ^-Zjr.fl)) with JoV $( W - (£)V) S(w' - ( \ f ^ > A)) g i v e s t 2 2 1 TM = ™j"r. ^ r e J ^ Vj o V ; where [ 2 3 ] T ^ ; = ^ I ^ v d a S (<o - ( y 5 E j ; r , A ) ) i s the s p e c t r a l d i s t r i b u t i o n f u n c t i o n a s s o c i a t e d with an ortho molecule i n a s h e l l of r a d i u s r about the p a i r . T h i s f u n c t i o n i s independent o f r . 36 (o)) i s now c a l c u l a t e d n u m e r i c a l l y by a computer program which f o r each p o i n t on a g r i d of r1 and A c a l c u l a t e s (^) J l ) and adds one to the a p p r o p r i a t e "energy b i n " forming a histogram. Two of the checks performed on the r e s u l t s o f t h i s program are the comparison of the f i r s t and second moments of ^(co) which can be c a l c u l a t e d a n a l y t i c a l l y . The f i r s t moment i s e a s i l y c a l c u l a t e d and i s found to be zero. [24] Jdu> ojf(u)) = ~ ^ f ^ d A ^ f E / ^ ) = 0 *-v ( sf * f -> s i n c e E ^ r , i l ) i s a sum of (PL) S and JolA. (XL) ~Q . T h i s a l s o i m p l i e s t h a t the f i r s t moment of J (<s>) i s zero l e a v i n g the average value of the p a i r f r e q u e n c i e s unchanged. C a l c u l a t i n g the second moment r e q u i r e s a l i t t l e more algebra r e s u l t i n g i n [25] JVeo cu* JVU>,= + J _ (ucosH) +£2- cosxf[ + 2(AC AC')[srff(i<*sx? ~i) ivo I S O ' • V-sin*? ces1-f + — s i n 29 f l 37 The computer calculated f i r s t moment i s always less than one half the "bin size" which compares well with the a n a l y t i c a l result of zero. A l i s t of calculated second moments and l i n e -widths appear in table II. To the accuracy given in the table, the computer results agreed per f e c t l y with [25]. The lineshape is also calculated by computer but instead of computing the integral from r=r a to oo the change of variable i s made which results in the f i n i t e integration range of 0 to d}/. Now where W„ i s the maximum or minimum value of to depending on whether 61 i s positi v e or negative. In the l i m i t of large OJ , [26] y i e l d s J(o))^ OJ in agreement with Hama's spectral d i s t r i b u t i o n function. [26] also gives J (to)~ (J^ iy) as Before discussing the computer results we note that E(r,/1) given in [20] contains two terms with c o e f f i c i e n t s A c and A C . These two c o e f f i c i e n t s are the only expressions containing F,M and c< so that an examination of &C and A C * allows one to determine which tr a n s i t i o n s have i d e n t i c a l lineshapes. Also i f AC=AC'=0 then there i s no broadening by t h i s mechanism. Transitions A2,A^,G( ,G5,G^ and G3 have zero linewidth and the following pairs of tra n s i t i o n s have the same lineshape: (B~,C3) ( B * ,C, ) (G M-H 3) (A,,-A5) (G 2 ,-G3') 38 Table II Second Moments and Linewidths T r a n s i t i o n s Second Moment Width (FWHM) In-plane t r a n s i t i o n s C, ,B?; ( 2 , 1 ) + - » ( 1 , 0 ) 0.347 16.1 C3 ,B" : (2,1)_ - » (1,0) 0. 344 11.9 B 3 0.115 6.7 G f,H 3 0.115 6.7 (2,2) + - » ( 2 , 0 ) 0.230 13.4 Out o f plane t r a n s i t i o n s A, , A ? 0.344 16.1 A i , A 3 0.000 0.0 B, ' ,K2 . ,Kv. 0.342 11.9 B, ,K, ,K 3 0. 345 16.1 G, ,G 5 0.000 0.0 H i 0.115 6.7 H, 0.114 10.3 Next nearest neighbour t r a n s i t i o n s G 2,G 3 0.000 0.0 G/,G3' 0. 345 16.1 G 6 0.114 10.3 G 7 0.115 6.7 G s 0.342 11.9 G,, 0. 344 16.1 See Table V (pg.73) f o r t r a n s i t i o n l a b e l l i n g * i n u n i t s o f [ ~?as/lW[K0/R)? ] * 10~ Z i n MHz with X=0.2% and (7=20.15 GHz 39 The l a s t three p a i r s have J A (co) =JS (-cd); i . e . they are m i r r o r r e f l e c t i o n s about the l i n e c e n t e r , and hence the minus s i g n . The computer produced (J1 (ca) have been p l o t t e d f o r each t r a n s i t i o n l i s t e d i n t a b l e I I . The apparent noise i s due to the JT- g r i d not having enough p o i n t s to evenly f i l l the energy b i n s of 7^(60). When the g r i d s i z e i n J (^co) i s increased the n o i s e i s c o n s i d e r a b l y reduced. However, the r e s u l t i n g l i n e s h a p e J (OJ) i s v i r t u a l l y unchanged. Fig.7 shows an example of T'(^) and f i g . 8 the J ( c o)'s. In a d d i t i o n to the d u p l i c a t e l i n e s h a p e s a l r e a d y mentioned the f o l l o w i n g t r a n s i t i o n s , upon i n s p e c t i o n of 7^(0)), have e s s e n t i a l l y the same shapes: a) ,Gj, H i , - B 3 ,-H3 b) B, , G 2 , -G 3 ; K i , K 3 c) A, ,-A 5,-Gq d) B / ,-Gy 1 K z 1 fCif. e) H, ,G 6 In the f i r s t four groups each l i n e has the same second moment and lineshape but i n the l a s t group the ^(aO's as w e l l as the second moments are not i n exact agreement. Also Tfw) i n groups b) and c) are q u i t e s i m i l a r as are t h e i r second moments. These r e s u l t s show t h a t t h e i r i s l i t t l e d i f f e r e n c e i n (^co) f o r nn out-o f - p l a n e and nnn t r a n s i t i o n s between l i k e s t a t e s , whereas there i s l i t t l e s i m i l a r i t y between nn i n - p l a n e t r a n s i t i o n s and i d e n t i c a l t r a n s i t i o n s of the other two p a i r t ypes. T h i s i s not t r a n s p a r e n t upon examination of E(r,il) i n [17], [18] and [20]. 40 Figure 7 C f (w) for the T r a n s i t i o n C3 The lineshapes above correspond to A: ; B: C3 ; C: G5 ; and D: G7 . See t e x t f o r a complete l i s t of i d e n t i c a l lineshapes (pg. 43). Here we have taken X=0.2%. Figure 8 Lineshapes J(w) 42 When c o n s i d e r i n g two d i f f e r e n t t r a n s i t i o n s one expects i d e n t i c a l J^(co)'s i f they i n v o l v e the same p a i r type and have the same c o e f f i c i e n t s AC and AC'. However, i f two t r a n s i t i o n s i n v o l v i n g d i f f e r e n t p a i r types are considered i t i s not so c l e a r why i d e n t i c a l J^(co) 's can r e s u l t even i f the va l u e s o f AC and A C are i d e n t i c a l f o r the two p a i r types. The value of r e i s chosen to be about two l a t t i c e s i t e s away from the p a i r . A comparison of the li n e s h a p e generated with t h i s value o f t0 and one with r c/2 shows l i t t l e change as would be expected. The numerical i n t e g r a t i o n method used, which i s poor f o r small to, causes a d i p i n the center of J (co) . However, f o r small OJ ) j (co) can be replaced with C^(^) and t h i s w i l l g i v e the peak h e i g h t . U n f o r t u n a t e l y the noise present i n j'(a)) makes the value of j"(0) d i f f i c u l t to determine and hence the l i n e w i d t h s w i l l have an e x t r a u n c e r t a i n t y a s s o c i a t e d with them. Also note that each l i n e i s e s s e n t i a l l y symmetric except near 6j=0. As expected the t r a n s i t i o n s with s i m i l a r ^ ( o ^ ' s lead to s i m i l a r J (co) l i n e s h a p e s . There are b a s i c a l l y three v a l u e s f o r the second moments o f 'f(u)) (see Table II) but there are four d i s s i m i l a r l i n e w i d t h s ( f u l l width at h a l f maximum). A narrow J^(w) w i l l produce a small l i n e w i d t h and a broad $ (co) w i l l produce a l a r g e l i n e w i d t h even though the second moments are i d e n t i c a l . The second moment o f J(w) i s d e f i n e d by [27] Tz = ru*JMdw = r ( - f . • ^ j J » , 7 J M i ' * 43 At X=0.2% this gives for a value of 445 MHz for (J 2 ) whereas the linewidth i s only 6.7 MHz. This is due to the large contribution the wings make to the second moment. For large w, -8/5 w which would lead to an i n f i n i t e second moment i f not for the f i n i t e extent of J(W) set by r0 .. This asymptotic behavior i s in fact observed in the wings of the computer generated J («) . Upon inspection of J(eo) the following tr a n s i t i o n s have the same lineshape: a) B, , G'z, G^  , K. f K 3 r A j , A^, Gq 1 b) B,' , Kz , Bi r c) G^ . f H 3 ' B 3, H^, G 7 d) H| . G 6 The appearance of OJ 'x in the expression for J(u>), [26] implies that the lineshape and l i n e width scale in frequency by 5/3 X again in agreement with the results of Hama e t . a l . Since J j ((*)) doo has been normalized to unity then the amplitude of J (w) -S /3 as defined w i l l scale as X . Since the integrated intensity must be proportional to the number of ortho pairs, i . e . X , 2 then the observed lineshape is proportional to X J (0J») whose amplitude therefore scales as X In the model we have just treated the e f f e c t of the ortho molecules beyond the closest one to the pair are ignored. This tends to underestimate the linewidths. A more serious omission is the motional narrowing caused by co r r e l a t i o n e f f e c t s . In the work of Fujio e t . a l . i t i s found that the linewidth decreases by about 60% when these effects are included. 44 3.2 Phonon Broadening of the 0rtho - H 2 . P a i r Spectrum Phonon broadening of the ortho-H 2 p a i r l e v e l s becomes important at low ortho c o n c e n t r a t i o n s (X-$0.05%) where the inhomogeneous broadening due to i s o l a t e d ortho molecules i s l e s s than the homogeneous phonon broadening. C a l c u l a t i o n s done by C u r r i e and Van Kranendonk have s t i m u l a t e d our i n t e r e s t i n pursuing t h i s matter f u r t h e r . The phonons couple to the p a i r l e v e l s through the EQQ and c r y s t a l f i e l d (CF) i n t e r a c t i o n s which dominate the p a i r Hamiltonian. The four phonon processes considered here are the f o l l o w i n g : a) D i r e c t D b) F i r s t Order Raman 1R c) Second Order Raman 2R d) Anharmonic Raman aR The Feynman diagrams i n f i g . 9 i l l u s t r a t e these p r o c e s s e s . Other processes i n v o l v i n g two or more i n c i d e n t or s c a t t e r e d phonons w i l l be ignored. These phonon processes w i l l i n gen e r a l be temperature dependent and hence can be d i s t i n g u i s h e d from the temperature independent i s o l a t e d ortho broadening e f f e c t . The l i n e broadening i s gi v e n by the t r a n s i t i o n r a t e ( H e i t l e r , 1949) c a l c u l a t e d using the Golden Rule. For s i m p l i c i t y the t r a n s i t i o n r a t e i s c a l c u l a t e d f o r each process assuming i t i s the o n l y one in e f f e c t . c) 2R This f i g u r e i l l u s t r a t e s the following a) Direct b) F i r s t Order Raman c) d) Anharmonic Raman. The s o l i d l i n e s the dashed l i n e s represent phonons. 45 b) IR d) aR phonon processes: Second Order Raman represent p a i r states and Figure 9 Phonon Processes and Their Feynman Diagrams 46 The interaction Hamiltonian i s selected from an expansion of either the EQQ or CF Hamiltonian in terms of the molecular displacements about the r i g i d l a t t i c e . The EQQ Hamiltonian i s given by Harris (1970) [ 2 8 ] XmQ = ^ 5 C(224-i^)X/k) X (%) \ W ) where C0>, , Wz are the angles between the ortho molecular axes and the quantization axis , and r ( 1 J J l u describe the position of one ortho molecule with respect to the other r e l a t i v e to the quantization axis. The c r y s t a l f i e l d interaction w i l l be described by a l o c a l l a t t i c e d i s t o r t i o n model where the surrounding para molecules deform about the ortho molecule. The CF Hamiltonian i s given by Raich and Kanney (1977) [29] K;; =y^  Bwjy ;^ ) R C where B (R) = A t R 6 and R-Lj i s the distance between the ortho molecule i and a para molecule j and ilcjis the angle between the ortho molecular axis and R;j . The t o t a l CF Hamiltonian i s obtained by summing over i = l , 2 and j over the hep l a t t i c e (excluding i ) . The states used here are |F,M> |n> where |F,M> is the pair state given in HBHII by [30] \FM) ~ 2 C ( H F j m M - * ) I («*»,) 7, M m and |n>= |{n(k ,A ) }> are the phonon states. In order to simplify the calculations the isotropic Debye model w i l l be used. Hence an 47 3 2 / a v e r a g e v e l o c i t y o f s o u n d w i l l be u s e d d e f i n e d b y —} ~ —? + ~i, a s i s d o n e i n t h e p h o n o n d e n s i t y o f s t a t e s , w h e r e v t and v^ a r e t h e t r a n s v e r s e and l o n g i t u d i n a l v e l o c i t i e s o f s o u n d . D i r e c t P r o c e s s : T h i s p r o c e s s i n v o l v e s t h e e m i s s i o n o f a p h o n o n a c c o m p a n i e d b y a p a i r t r a n s i t i o n . The i n t e r a c t i o n H a m i l t o n i a n i s g i v e n b y ~u • v H a n d h e n c e t h e t r a n s i t i o n r a t e i s n 27T — . ' K n,K ,A [31] W D ^ ^ 2 PnK <Ff^l<V ,|**Hk>|FiM v > The B o l t z m a n n d i s t r i b u t i o n , Pn^ , d e s c r i b e s t h e p h o n o n p o p u l a t i o n s and €- t and 6f a r e t h e e n e r g i e s o f t h e i n i t i a l and f i n a l p a i r s t a t e s . The l a t t i c e d i s p l a c e m e n t i s q u a n t i z e d i n t e r m s o f t h e p h o n o n c r e a t i o n and a n n i h i l a t i o n o p e r a t o r s a ^ A and a ^ w h e r e k i s t h e w a v e v e c t o r and A i s t h e p o l a r i z a t i o n o f t h e p h o n o n c o n c e r n e d . Now t h e l a t t i c e d i s p l a c e m e n t a t s i t e i i s g i v e n b y t32] uL = ( a u + <i-u) £ U e A w h e r e ^ft\ i s t h e p o l a r i z a t i o n v e c t o r . The f o l l o w i n g f o r m u l a w i l l be u s e f u l i n e v a l u a t i n g t h e d e r i v a t i v e s o f H E C ^ and H C F ( H a r r i s , 1970) [33] ftV?V(R) x/Yfi) ~- (i&fcc^j ^) c^) [R£-«7 i / w E v a l u a t i n g W0 f o r t h e EQQ p r o c e s s g i v e s 48 t 34, w 0 6 O Q = ^ {rff(¥f %P f Co^x,M)\rX^ where AM=Mf -Mj_ and [ 3 5 ] S(r~it/\L - 2 w - f o ^ - M i ' - " ) c r ^ ; ^ ^ - « f j The high temperature expansion f o r n(k) has been used s i n c e —-<1 at T=4K where 6<== ^ - e > <L 3K. T h i s g i v e s r i s e to the l i n e a r temperature dependence of W0. Upon c a l c u l a t i o n i t i s noted that S (F^ ,ML , F + ,Vi^.) i s non-zero o n l y i f Fl=Fj:=2. T h i s i s to be expected s i n c e the EQQ i n t e r a c t i o n o n l y l i f t s the degeneracy of the F=2 s t a t e s o f the p a i r . Hence the EQQ processes o n l y broaden the t r a n s i t i o n s i n v o l v i n g F-t=F^=2. Also x note the 4£ dependence of W/>. Next e v a l u a t i n g f o r the CF process g i v e s [36] < ^ § (f)\2.F,H)Wy2lFflj/^Y2f,^jm^)SC/>^ where S(dM) i n v o l v e s a l a t t i c e sum which w i l l be r e s t r i c t e d to the nearest neighbours o f the ortho molecule i o f the p a i r . Because of the symmetry of the l a t t i c e with r e s p e c t to the p a i r molecules one o n l y needs to perform the l a t t i c e sum about one ortho molecule and then m u l t i p l y the r e s u l t by 2 (see HBHII). [ 3 7 ] s(m) = ? S £-f«rw*J « M aiZ •A J J and [ 3 8 ] a ] X = ifr c(tx3jA.*«\) j) C*.B'-ZB] 49 where U),^  = (j^y ) i s give n by the Euler angles r o t a t i n g the ortho (1)-para (3) frame i n t o the p a i r frame and Hjj/ : ^ . For comparison the t r a n s i t i o n C, 3 ; |2,2>-»J2,1> w i l l be considered at T=4K. The two processes d i s c u s s e d here y i e l d WbEQQ ~ z * lo6 rad/jec W / ^ \.H- * io" rad /sec It should be noted that other t r a n s i t i o n s with s i m i l a r v a l u e s f o r &£ r e s u l t i n e s s e n t i a l l y the same t r a n s i t i o n r a t e s . F i r s t Order Raman Process: T h i s process i n v o l v e s the s c a t t e r i n g of a phonon accompanied by a p a i r t r a n s i t i o n . It w i l l be assumed that f i r s t the phonon i s absorbed by the p a i r that then undergoes a t r a n s i t i o n and f i n a l l y emits another phonon with random o r i e n t a t i o n and p o l a r i z a t i o n with r e s p e c t to the i n i t i a l phonon. The i n t e r a c t i o n Hamiltonian i s H'~j\ ? M C C ' ' ) r ~ T ~ where o n l y the a^a term i s r e t a i n e d . The t r a n s i t i o n r a t e i s now gi v e n by ,2 [39] Wn " T 1 2 Pnk <V^ -'|<^ %I X"l^/Wi> K"«> M^ + t^-ta^ i,f The sum over k i s converted into an i n t e g r a l from k=0 to k D , the Debye wave v e c t o r . A f t e r i n t e g r a t i n g over the d e l t a f u n c t i o n the other i n t e g r a l i s c a r r i e d out i n the low temperature l i m i t MAX K, KB approximation i n v o l v e s i n t e g r a l s o f the form -'o s i n c e S^ /T>>1 where — — i s the Debye temperature. T h i s 5 0 Upon ca l c u l a t i o n 1..E<a<5_ 7.0x10* rx/r , /K*T\ < C <^ H [40] W„ - T j r - 5 ( ^ F f M f ) ( i - / ^ ^ — j -where [41] - i l>L ^ T S r / l 1 C ( ' « ; A f , ^ c f / a ? j , ^ and where [43] sW;= £ 2. D-C^rrj ^ f and f(T f C( 111) k > -A*-*J C(m jAfj/iM +Ai)fa*B «+Z*.*'~ 3S)]K f*>tj) 7 This process has a strong temperature dependence of T and does not depend on as i s to be expected of a Raman process. Again note that the EQQ process only broadens t r a n s i t i o n s with 51 Second Order Raman Process: T h i s process i s the same as the f i r s t order Raman process except t h a t now the p a i r changes to a v i r t u a l s t a t e before going to the f i n a l s t a t e . Energy i s conserved o n l y between the i n i t i a l and f i n a l s t a t e s . The v i r t u a l s t a t e s used here w i l l be the p a i r s t a t e s and w i l l not i n c l u d e higher r o t a t i o n a l and r o t a t i o n a l - v i b r a t i o n a l s t a t e s o f the ^ m o l e c u l e s . These l a t t e r s t a t e s l i e above the phonon band and w i l l c o n t r i b u t e to W2.R as —2, and hence can be ignored. The i n t e r a c t i o n Hamiltonian i s giv e n by ( H e i t l e r 1949) where |l> i s the v i r t u a l i ntermediate s t a t e . The two terms correspond to e i t h e r a phonon f i r s t being absorbed and then one being emitted or a phonon being emitted f i r s t and then one being absorbed* I n s e r t i n g the two i n t e r a c t i o n terms y i e l d s [44] X + [45] S where [46] 3 = Z TL ^ ^ z M I F f ^ ) y ( f ^ F z ^ x ) [ c O ^ A f j m f ) c ( m ; A ^ M x ) and d M f a M f - M -[1 ' 4Mr=Mx-Mi A l s o [47] W where 1 ^ J 52 [48] sr ? K R ^ a^ a'^x a ^ and Rjj' are as before and The sum over I i s the sum over intermediate s t a t e s . T h i s process i s a l s o evaluated i n the low temperature l i m i t and 5 7 y i e l d s a T dependence, to be c o n t r a s t e d with T f o r the f i r s t o rder Raman proc e s s . Anharmonic Raman Process: T h i s process has been d i s c u s s e d by Van Kranendonk and Walker (1968) and i s a second order process i n v o l v i n g the d i r e c t process and the s c a t t e r i n g of the r e s u l t i n g phonon with another one. T h i s phonon s c a t t e r i n g does not occur i n a harmonic l a t t i c e . The intermediate phonon does not have to conserve energy and hence does not c o n t r i b u t e to the temperature dependence of W ^ R . Hence t h i s process w i l l have the temperature dependence of W,„ . Van Kranendonk and Walker (1968) estimate that WaR<= 100 W,R . Of the two i n t e r a c t i o n s c o n s i d e r e d the c r y s t a l f i e l d i n t e r a c t i o n i s dominant. The d i r e c t process and the anharmonic Raman c r y s t a l f i e l d process are of the same order of magnitude at 4K. However most of the experimental data i s taken at T=1-2K where the d i r e c t CF process i s dominant. T h i s process i s p r o p o r t i o n a l to A&*T and r e s u l t s i n a l i n e w i d t h o f about 1 MHz at T=2K f o r the C,/3 t r a n s i t i o n s . In summary, t a b l e s I I I and IV l i s t the e f f e c t these processes have on the l i n e w i d t h o f two l i n e s at various temperatures. 54 TABLE I I I Summary o f T r a n s i t i o n Rates Process T r a n s i t i o n Rate M f^ (C3) 37 -z -1 - 3 ' 1 4.51x10 erg sec Mif. 3.16x10 erg 5 - z 8.06x10 sec Mt-r 2.65 sec 32- _2. - i - z ? 2 9.90x10 erg sec M^ -f 1.00x10 erg 1.77x10* sec~ ' M i f 1.26x10*" 3f -2 -1 - i i - 2 4.27x10 erg sec M ; f 1.35x10 erg _ i 3 1.03 sec Mif 2.35x10 Values of c o n s t a n t s used are as f o l l o w s : T=4 K A£ =87 GHz E =20.15 GHz R0=3.784 A (HBHI) f=0.08868 g/cc (V.V. Goldman) 5 5 v A=2.19x10 cm/sec ; v t =1.16x10 cm/sec (Bezuglyi and Minyfaev) A=7.247xl0~" erg of=3.425 A~' C=5.970x10~'3 ergA* (Raich and Kanney) X 7 55 TABLE IV Phonon Induced Linewidths f o r G, and C_j Process Temperature G, FWHM C 3 FWHM (MHz) (MHz) 1. 2 K 0. 02 0. 7 2. 1 K 0. 04 1. 2 4. 0 K 0. 08 2. 2 w £4Q W (ft w w 'R CF W £<Q<3 CF 1. 2 K 0.002 0.10 2. 1 K 0. 004 0.17 4. 0 K 0.008 0.33 1. 2 K _ 2. 1 K <0.001 <0.001 4. 0 K 0.016 0. 016 1. 2 K _ 2. 1 K - -4. 0 K <0.001 <0.001 1. 2 K 0.002 0.002 2. 1 K 0. 04 0.04 4. 0 K 0.92 0.92 1. 2 K 2. 1 K - -4. 0 K <0.001 <0.001 1. 2 K <0.001 <0.001 2. 1 K 0.02 0.02 4. 0 K 1.60 1.60 1. 2 K _ 2. 1 K <0.001 <0.001 4. 0 K 0.035 0.03 5. Note that r e s u l t s quoted f o r G, are obtained by s c a l i n g those for C 3 a p p r o p i a t e l y . 56 3.3 I s o t o p i c Impurity Broadening: The presence of the i s o t o p i c i m p u r i t i e s of HD and D 2 i n s o l i d Hj. can cause an inhomogeneous broadening of the p a i r l i n e s v i a the s t r a i n f i e l d surrounding the i m p u r i t y s i t e . Since HD and Dz have v i r t u a l l y the same i n t e r m o l e c u l a r p o t e n t i a l as H 2, the e f f e c t i s due e n t i r e l y to the d i f f e r e n t mass. In p a r t i c u l a r , the heavier molecules have l e s s zero p o i n t motion and t h i s tends to p u l l the l a t t i c e i n towards the mass d e f e c t . For an i s o t r o p i c s o l i d the s t r a i n f i e l d at r i s p r o p o r t i o n a l to r _ i where r i s the v e c t o r from the i m p u r i t y s i t e to a p o i n t i n the l a t t i c e . T herefore the s h i f t i n the p a i r f r e q u e n c i e s f o r a _ 3 s i n g l e d e f e c t i s p r o p o r t i o n a l to r . At f i n i t e c o n c e n t r a t i o n s , one has to f i n d the d i s t r i b u t i o n of s t r a i n s produced by the randomly s i t u a t e d i m p u r i t i e s . The c a l c u l a t i o n of the l i n e s h a p e proceeds as f o l l o w s . A random d i s t r i b u t i o n of i m p u r i t i e s with c o n c e n t r a t i o n c i s arranged about a s i n g l e ortho p a i r i n an otherwise pure para H 2 l a t t i c e . One f i r s t c a l c u l a t e s the s t r a i n at the p a i r s i t e due to each impu r i t y and then adds these s t r a i n s to o b t a i n a t o t a l frequency s h i f t . T h i s i s repeated f o r a l l p o s s i b l e c o n f i g u r a t i o n s and a s p e c t r a l d i s t r i b u t i o n f u n c t i o n i s formed from the r e s u l t s . A L o r e n t z i a n l i n e shape i s obtained which might be expected of a r i n t e r a c t i o n ( c f . the case of d i p o l a r broadening in NMR (Abragam, 1961)). 57 T h i s problem i s solved on a computer using a Monte C a r l o program. A random c o n f i g u r a t i o n of i m p u r i t i e s i n an hep l a t t i c e i s formed out to a predetermined r a d i u s from the p a i r . The frequency s h i f t i s then c a l c u l a t e d assuming the t r a n s i t i o n f r e q u e n c i e s s c a l e as |r, -r* 2 s i n c e the EQQ i n t e r a c t i o n makes by f a r the l a r g e s t c o n t r i b u t i o n to the s e p a r a t i o n of the p a i r l e v e l s . The s h i f t i s then recorded and a histogram of s h i f t s i s formed from N c o n f i g u r a t i o n s . The l i n e s h a p e s obtained are then f i t t e d to a L o r e n t z i a n and a Gaussian and the r e s u l t s compared. The ortho s e p a r a t i o n displacement i u i s assumed to be g i v e n by [49] where A= the f r a c t i o n a l s h i f t o f the nearest neighbor d i s t a n c e . In a d i s c r e t e l a t t i c e the a c t u a l motion c l o s e to the impur i t y may be f a r from i s o t r o p i c . Here we are mainly concerned with the asymptotic behaviour away from the i m p u r i t y which we assume to be i s o t r o p i c . Fig.10 i l l u s t r a t e s the r e l a t i o n s h i p s between i m p u r i t y and p a i r molecules. Expanding r, and ri i n terms of r and & g i v e s [50] U a + AU 2 so that [51] f «n Since J'c/jl.(3co51e -/) ~0 then the s p e c t r a l d i s t r i b u t i o n of an 58 Figure 10 Ortho Pair-Impurity Configuration 59 i m p u r i t y molecule s i t u a t e d on a s h e l l of r a d i u s r has an average value of zero. T h i s i m p l i e s t h a t f o r t h i s model the average value of the p a i r f r e q u e n c i e s w i l l not change. One can see immediately from [51] that the f u l l width at h a l f maximum, i s p r o p o r t i o n a l to f 0 A . Furthermore one expects a l i n e a r dependence on the impuri t y c o n c e n t r a t i o n as f o r the d i p o l a r broadening case (see Abragam, 1961 and Hama e t . a l . , 1972 ). - f A t y p i c a l l i n eshape i s shown i n fi g . 1 1 where c=3X10 , the n a t u r a l abundance of HD, along with the f i t t e d L o r e n z t i a n . Attempts to f i t a Gaussian to the lin e s h a p e g i v e v e r y poor agreement whereas the L o r e n z t i a n g i v e s an e x c e l l e n t f i t . A p l o t of f i t t e d l i n e w i d t h s v s . c i s shown i n fi g . 1 2 c o n f i r m i n g the l i n e a r dependence of Av on c. Note that the l i n e a r i t y breaks down at higher c o n c e n t r a t i o n s . Each o f the p o i n t s at c=0.03% were computed with a d i f f e r e n t s et of random c o n f i g u r a t i o n s o f imp u r i t y s i t e s as w e l l as v a r y i n g maximum r a d i i . The value used for A i s 0.01. From t h i s graph one gets [52] = fo Ac The average frequency s h i f t s from zero are n e g l i g i b l e as expected. In the c a l c u l a t i o n so f a r i t has been assumed that the l a t t i c e constant remains f i x e d at i n f i n i t y and hence the molar volume does not change. T h i s i s not the case f o r a r e a l c r y s t a l where one f i n d s f o r an e l a s t i c a l l y i s o t r o p i c c r y s t a l ( K r i v o g l a s , 1969) that the s t r a i n at a po i n t r from an impuri t y s i t e i s gi v e n by 60 f (MHz) Figure 11 Lineshape Due To I s o t r o p i c Mass Defect I m p u r i t i e s 61 Figure 12 Impurity Concentration Dependence of Linewidths 62 I ( l+V ) 6AS ~t [53] = 1TT7 where V i s Poisson's r a t i o o f the host and i s the change i n u n i t volume due to a c o n c e n t r a t i o n c of i m p u r i t i e s . In t h i s way one can r e l a t e the change i n molar volume to the asymptotic s t r a i n f i e l d s and hence determine the value of A. One way to o b t a i n i s to c o n s i d e r the s h i f t i n the average frequency, d f , of the p a i r l i n e s due to the l a t t i c e c o n t r a c t i o n . A-f _ _ SAX - - 5 c_ IAT_ [54] f g r 3 x r t)c Now, with [52] and [53], the f u l l width at h a l f maximum (FWHM) of the l i n e s h a p e i s giv e n by [55 , AV-I.S! f.c There i s another c o m p l i c a t i o n when c o n s i d e r i n g HD as the im p u r i t y . The center of mass i s not c o i n c i d e n t with the geometric center of the molecule and hence i t r o t a t e s o f f c e n t e r . T h i s would have the tendency to expand the l a t t i c e , opposing the zero p o i n t motion e f f e c t . Since the rms zero p o i n t motion i s comparable to the H-D bond le n g t h then these two e f f e c t s are of the same order and would lead to a sm a l l e r d i s t o r t i o n e f f e c t . One could perhaps determine t h i s e x p e r i m e n t a l l y by comparing the s t r a i n f i e l d s f o r HD to that o f Di_ ( c o r r e c t i n g f o r the f a c t t h a t the mass d e f e c t f o r D^ i s l a r g e r ) . CHAPTER IV Expe r iment 63 4.1 Cryostat: The ortho pair absorption i n t e n s i t i e s are expected to be quite weak (HBHI). Therefore transmission spectroscopy would involve detecting very small changes in the transmitted power. Instead the power absorbed is measured d i r e c t l y using a calorimetric technique. The s o l i d H 2 s i t s in a niobium c e l l forming the bottom of a shorted waveguide (see fig.13). After the microwave energy i s absorbed by the pairs i t is released to the s o l i d causing a temperature r i s e in the sample. The temperature change is detected with a 31 element thermopile whose output i s measured with a nanovoltmeter. The niobium c e l l (described in HBHI) contains the s o l i d H 2 in i t s bottom ha l f . The c e l l is f i l l e d with H 2 up to the step. This step i s constructed with the dimensions of the waveguide reduced by VeT =1.14 where £ is the d i e l e c t r i c constant of s o l i d H^. This i s done to avoid mismatch in the c h a r a c t e r i s t i c impedance of the waveguide due to the hydrogen. When working with frequencies well above the fundamental band of the waveguide trapped modes can exist in the unstepped waveguide. These modes then resonate at certain frequencies causing a multitude of absorption peaks, the energy being dissipated by absorption in the Nb. The step eliminates most of these resonances, the rest of which can then be distinguished from the H~ absorptions because of the lack of temperature dependence of INCIDENT MICROWAVES STYROFOAM RADIATION TRAP REFERENCE ARM OF THERMOPILE] THERMOPILE LIQUID HELIUM RADIATION TRAP THERMO-METERS SUPER-CONDUCTING NIOBIUM WAVEGUIDE SOLID H 2 COLD FINGER Figure 13 Cryostat 65 t h e i r i n t e n s i t i e s . The niobium c e l l is superconductiong below 9 K and therefore has a low thermal conductivity which thermally isola t e s the H 2 sample from the thermal bath. The thermopile consists of 31 Au(0.07% Fe)-Nb elements with a s e n s i t i v i t y of 5 jjK at T=l. 2 K. One side of the thermopile i s connected to a copper assembly clamped to the waveguide above the sample serving as the reference arm. The other side of the thermopile is connected to a copper sheet wrapped around and bonded to the c e l l to provide good thermal contact between the c e l l and the thermopile. The c e l l walls are thin enough (0.4mm) that the temperature drop between the H and the copper sheet is small. Mounted on t h i s copper sheet i s a heater r e s i s t o r used to ca l i b r a t e the s e n s i t i v i t y of the system thereby allowing quantitative measurments of the absorptions. The large thermal mass of copper and hydrogen leads to a c e l l response time of about 2 seconds. This puts an upper l i m i t on the microwave scanning rate. The source of microwaves is the computer controlled spectrometer. The spectral width of the source i s much less than the smallest absorption linewidth. Therefore i f the scanning rate i s chosen slow enough the lineshape measured w i l l represent the absorption lineshape. Standing waves between the source and the bottom of the c e l l cause the power incident on the c e l l to change with frequency and thus a f f e c t s the background absorption. The background absorption also varies across the band since the delivered power i s not uniform even 66 when the microwave source i s l e v e l l e d . T h i s n e c e s s i t a t e s the d i v i s i o n of two s p e c t r a with d i f f e r e n t l i n e i n t e n s i t i e s i n order to e l i m i n a t e the background. T h i s process w i l l be d e s c r i b e d l a t e r . 4.2 Sample P r e p a r a t i o n : At room temperature the e q u i l i b r i u m c o n c e n t r a t i o n of the ortho to para s p e c i e s i s 3:1. In order to reduce the ortho c o n c e n t r a t i o n , the l i q u i d H2. sample i s i s placed i n c o n t a c t with a paramagnetic c a t a l y s t (Fe(OH) 3 powder). The e q u i l i b r i u m ortho c o n c e n t r a t i o n i s g i v e n by n0 - ^ e ~ ^ € . T h i s allows one to o b t a i n ortho c o n c e n t r a t i o n s down to 0.004% at T =13.8 K, the t r i p l e p o i n t . The l i q u i d i s c i r c u l a t e d through the c a t a l y s t by c o n v e c t i o n heating to ensure an even c o n c e n t r a t i o n d i s t r i b u t i o n . The gas above the l i q u i d and i n the connecting tubes i s f l u s h e d out r e g u l a r l y with converted vapour i n an attempt to reduce the regions of high ortho c o n c e n t r a t i o n . The c o n v e r s i o n process can be monitered down to n 4 - l % with a standard thermal c o n d u c t i v i t y b r i d g e . One can then estimate the time needed to complete the c o n v e r s i o n . A f t e r the e q u i l i b r i u m ortho c o n c e n t r a t i o n has been reached the sample i s t r a n s f e r r e d to a metering bulb and then immediately condensed i n t o the c e l l to keep any back c o n v e r s i o n at room temperature to a minimum. For the purposes o f condensation and growth a small amount o f l i q u i d He i s t r a n s f e r r e d i n t o the dewar, up to the f l a n g e on the bottom vacuum can. Once i n the c e l l the s o l i d i s melted with the 67 waveguide heater and with He exchange gas in the stainless steel can surrounding the c e l l a positive temperature gradient is maintained with respect to the c e l l bottom. The c e l l temperature i s then slowly lowered using a temperature con t r o l l e r and the s o l i d starts growing from the bottom of the c e l l . The cold finger mounted on the bottom of the c e l l should induce nucleation to occur f i r s t at the center of the c e l l bottom. If the s o l i d then grows from th i s nucleation center i t should be possible to obtain a single c r y s t a l . The growing time is about one-half hour and is monitored by recording the temperature gradient in the c e l l with the thermopile. After the sample has s o l i d i f i e d i t i s slowly cooled to about T=10 K. By the time this temperature is reached most of the thermal contraction in the s o l i d has taken place. Since the H^sticks to the c e l l walls a large amount of s t r a i n exists which must then be removed. By pumping on the s o l i d and applying heat pulses to the c e l l walls with one of the two heaters, one attempts to break the sample away from the walls. After this procedure the sample i s annealed for about an hour in order to remove as many internal strains as possible. The t o t a l amount of time to grow and anneal the sample is limited by the length of time the l i q u i d helium l a s t s . The f i n a l step i s to gradually cool the sample to 4 K and then f i l l the dewar with l i q u i d heluim, taking care not to heat up the c e l l with warm gas. O r i g i n a l l y only the waveguide heater was present and this was used to break the sample away from the walls. In an attempt to obtain a better success rate in breaking away samples, 68 another heater was i n s t a l l e d on the copper sheet at the bottom of the c e l l . This was done in order to supply a uniform heat pulse to a l l portions of the c e l l walls. The ef f e c t d i f f e r s from that of the waveguide heater, where a heat pulse would travel down the waveguide and reach the top portion of the sample f i r s t . After preparing many samples i t became apparent that the new heater caused more problems than i t solved. Less power was needed than for the waveguide heater and before the correct amount was discovered, several samples had been popped out of the c e l l . Apparently the combination of a large heat pulse to the c e l l walls and the pumping forced the Uz s o l i d out of the c e l l and up the waveguide where i t then attatched i t s e l f . This fact was deduced from the absence of Hj. vapour pressure upon low le v e l heating using the new heater but the presence of vapour when the waveguide heater was used. Confirming this was the thermopile response to a pulse given the c e l l heater which indicated the absense of any thermal mass associated with the H2. sample. In an attempt to reduce the amount of hydrogen melting at the walls during each heat pulse, the pulse width was reduced from 5 seconds to 0.5 seconds. While this was enough to relieve some of the stress caused by sti c k i n g to the walls i t also caused s h i f t s in the l i n e centers. One sample showed three l i n e s near G 5, a narrow one in the usual position and two broader components shifted downwards in frequency, suggesting the presence of at least three f a i r l y uniformly strained 69 c r y s t a l l i t e s . Upon regrowing this sample and applying long heat pulses to the new heater these duplicate l i n e s did not appear. Subsequent regrowths with heat pulses to the waveguide heater also f a i l e d to cause the unusual shapes to materialize. This appears to eliminate one of our concerns, namely the p o s s i b i l i t y of impurities affecting the pair frequencies. These experiences show that the improvements to the c e l l and growing procedures have probably resulted in larger and fewer c r y s t a l l i t e s being produced. Unfortunately t h i s was accompanied by l i n e replicas which severely complicated the data analysis. Subsequently the waveguide heater was used exclusively to apply heat pulses to the sample. 70 CHAPTER V Experimental R e s u l t s 5.1 Data A n a l y s i s : In order to o b t a i n the a b s o r p t i o n l i n e s h a p e <T(w) , one must c o r r e c t the raw data f o r v a r i a t i o n s i n the i n c i d e n t power P c (co) . T h i s can be done o n l y i f the hydrogen a b s o r p t i o n changes r e l a t i v e to the a b s o r p t i o n i n the niobium w a l l s . T h i s can happen i n two d i f f e r e n t ways. One i s due to the change i n Boltzmann f a c t o r s when the temperature i s changed and the other i s due to ortho molecule c l u s t e r i n g which causes the number of p a i r s to change i n time. The l a t t e r occurs' through the ortho-para exchange mechanism (Amstutz e t . a l . 1968; Oyarzun and Van Kranendonk 1971,1972). Follo w i n g HBHI the i n i t i a l spectrum i s d e s c r i b e d by [56] S 0 (0))=Po (to) [1 + n w H where £ (oo) i s the hydrogen a b s o r p t i o n r e l a t i v e to that o f the niobium. The enhanced spectrum i s g i v e n by [57] S t ( a ) ) ^ P , ( u ) [ l + ( l + f ) ^ ( w ) ] where |? r e f l e c t s the change i n the enhancement f a c t o r . From t h i s one can o b t a i n the i n c i d e n t power spectrum as f o l l o w s : [58] Pa (w)=S d(«)+ f[Sa ( u ) - ^ S t («)] The constant << can be determined p r e c i s e l y s i n c e i t i s e i t h e r due to changes i n the nanovoltmeter s c a l e or changes i n the 71 thermopile s e n s i t i v i t y with temperature. However f i s not so w e l l determined. It can be c a l c u l a t e d i n p r i n c i p a l i f temperature changes and c l u s t e r i n g are taken i n t o account but the l a t t e r i s e s p e c i a l l y d i f f i c u l t to determine p r e c i s e l y when i t o ccurs s i m u l t a n e o u s l y with a gradual temperature change. F o r t u n a t e l y there i s another way to estimate jS . The l i n e s h a p e s are i n g e n e r a l much narrower than the standing wave p a t t e r n . Therefore one can vary |3 i n such a way as to s u b t r a c t the sharp hydrogen a b s o r p t i o n s i g n a l from the raw data l e a v i n g o n l y the broad standing wave p a t t e r n . Once P6 (oj) i s determined then [59] S(co) = I f there i s a l a r g e noise component to then the spectrum can be passed through, f o r example, a low pass gaussian f i l t e r to remove the n o i s e . T h i s broadens the l i n e s but i s u s e f u l i n b r i n g i n g out the weak ones. 5.2 New L i n e s : The spectroscopy reported i n HBHI covers most o f the t r a n s i t i o n s one i s able to observe with t h i s technique. A few more l i n e s have been observed with the a d d i t i o n of the higher frequency m i l l i m e t e r wave source, and s i x low i n t e n s i t y l i n e s have been observed with the h e l p of computer r a t i o i n g . In a d d i t i o n , the l i n e s G 6 and G 7 have been r e a s s i g n e d . Energy l e v e l diagrams f o r the three p a i r types are shown i n fig.14 along with the observed t r a n s i t i o n s . Table V c o n t a i n s a l i s t o f a l l observed t r a n s i t i o n s with some of t h e i r p r o p e r t i e s . NEAREST NEIGHBOR I N - P L A N E 2,0 NEAREST NEIGHBOR OUT-OF-PLANE 2,0 2,2 + I I, I I I I B| B, A| A 2 A 3 A 5 * 1 * * 1 1 |.| + E (GHz) NEXT NEAREST NEIGHBOR 20 15 10 5 0 - 5 -10 i r G 7 G € - I , l + 1.0 G 9 2 , l + ^ -•2,0 G 2 G 3 U A. .2.2. •0.0 G 8 •Jt— 2,l_ F i g u r e 14 Ortho P a i r Energy Levels 7 3 Table V Observed Ortho P a i r T r a n s i t i o n s Line Assignment Frequency (GHz) Nearest neighbour o u t - o f - p l a n e A, (2 f l) t-»(l,l) t 64.110 Aa. (2, 1)+ -»(1,1)_ 64.947 A 3 (2,1)- -»(1,1) + 65.430 A 5 ( 2 , 1 ) , - M l , l ) _ 66.280 B, (2,1)+ -» (1,0) 69.834 B, ' (2,1). -»(1,0) G, (l,0)-» (2,2)+ 13.650 G 5 (1,0)-* (2,2)_ 16.617 K, (1,1). -»(2,2)+ 18.534 Kt (1,1)+ (2,2)+ 19.375 K3 (1,1)- -» (2,2)_ 21.489 Kit. (1,1)+ ->(2,2)_ 22. 353 H, (1,1)_ -»(2,0) 98.875 H2 d , l ) f -» (2,0) 99.716 Nearest neighbour i n - p l a n e Bj. (2,1)*-»(1,0) 70.228 B 3 ( 2 , l ) - - * ( 0 , 0 ) 71.280 C, (2,1)+ -» (2,2)_ 84.113 C 3 (2,1)_ -» (2,2)+ 86.595 G^ (0,0)->(2,2)+. 15.327 H 3 (0,0)-»(2,0) 93.26 Next nearest neighbour G 4 (2,1)+-»(1,1)_ 14.018 G3 (2,1)_-»(1,1) + 14.051 G* (1,1)- -» (2,0) 17.000 G 7 (1,1)+-» (2,0) 17.074 G, (2,1). -»(1,0) 12.058 G, (2,1)+ -»(1,0) 11.956 G3' (2,1). -» (1,1). 14.125 Those f r e q u e n c i e s not observed i n t h i s experiment are taken from HBHI. 74 Perhaps the most u s e f u l set o f l i n e s d i s c o v e r e d are those connecting to the |2,0> l e v e l f o r i n - and out- of- p l a n e p a i r s . The theory o f HBHII and of L u r y i and Van Kranendonk (1979) both p r e d i c t the p o s i t i o n of the |2,0> l e v e l . Table VI l i s t s these p r e d i c t i o n s along with the e x p e r i m e n t a l l y determined v a l u e . Fig.15 shows the e x p e r i m e n t a l l y observed l i n e shapes. The r e l a t i v e i n t e n s i t i e s of H, to Hz observed i s 2.4 i n reasonable agreement with theory (3.4) c o n s i d e r i n g the d i f f i c u l t y i n measuring these l i n e s with poor s i g n a l to noise r a t i o s . I n t e n s i t y comparisons are made at T=2.1 K, r e l a t i v e to the unc l u s t e r e d v a l u e . By comparing the a b s o r p t i o n i n the niobium between H, and G , and the i n t e g r a t e d i n t e n s i t i e s the r a t i o of the i n t e n s i t i e s o f H, to G, i s 15 in good agreement with the t h e o r e t i c a l r e s u l t o f 14.6 . E x c e l l e n t agreement i s obtained f o r the out - o f - p l a n e l i n e s and HBHII, however there i s a s l i g h t d i s c r e p a n c y between experiment and theory with the i n - p l a n e l i n e . The two s e t s o f va l u e s g i v e n by L u r y i and Van Kranendonk do not agree v e r y w e l l with experiment. A f i t o f the parameter 4^ to the out - o f - p l a n e l i n e s would be ve r y i n t e r e s t i n g e s p e c i a l l y i f i t then p r e d i c t e d the c o r r e c t p o s i t i o n f o r the in-plan e l i n e . The parameter 4^ i s a s s o c i a t e d with the d i f f e r e n c e between the ob l a t e n e s s o f the zero p o i n t motion o f p a i r molecules i n the in-pla n e and out - o f - p l a n e c o n f i g u r a t i o n s . It i s a l s o i n t e r e s t i n g to note that the frequency f o r the ground s t a t e to 1 0 r l e v e l t r a n s i t i o n of in-pla n e p a i r s , i n f e r r e d from the experimental v a l u e s o f H 3 and B 3, i s c o n s i s t e n t with the r e s u l t obtained by Raman s c a t t e r i n g ( S i l v e r a e t . a l . 1971). 75 Figure 15 High Frequency Spectra Table VI High Frequency Lines Line H, H3 (2,0)-» (2,1) HBHI I LVK ( A J =0) LVK (dj=0. Experiment 001) 98.898 97. 54 96.6 98.875±0 99.742 98. 38 97. 5 99.716±0 93.509 91.99 93. 4 93.26t0. 164.79 163.27 164.7 164.54* 165*3 A l l f r e q u e n c i e s i n GHz. *as obtained from H3 and B 3 (see text) LVK r e f e r s to L u y r i and Van Kranendonk (1979) Raman s c a t t e r i n g ; S i l v e r a e t . a l . (1971). 77 One problem with the assignment of H 3 i s the l a c k o f a temperature dependence of i t s i n t e n s i t y . T h i s w i l l be d i s c u s s e d l a t e r . Confirming the previous o u t - o f - p l a n e assignments i s the d i s c o v e r y o f the l i n e s K, through . The l o c a t i o n s o f these l i n e s were determined by the a l r e a d y observed t r a n s i t i o n s . U n l i k e most o f the other l i n e s these are q u i t e weak and are not v i s i b l e i n the raw d a t a . Only a f t e r computer p r o c e s s i n g d i d these l i n e s appear, showing the value of p r e c i s e l y reproduceable frequency scanning (see fi g . 1 6 and 17). When one compares the r e s u l t s obtained with a temperature r a t i o and a time r a t i o the reproduceable f e a t u r e s are assigned to the a p p r o p r i a t e t r a n s i t i o n s . The other f e a t u r e s o c c u r i n g i n one spectrum and not the other are considered to be n o i s e . The noise l e v e l i s r e l a t i v e l y high because of the weak a b s o r p t i o n i n t e n s i t i e s . There i s good agreement between the f r e q u e n c i e s o f these l i n e s and those determined p r e v i o u s l y . The r e l a t i v e i n t e n s i t i e s observed are about the same f o r K 2, K 3, and whereas K, has about four times the i n t e n s i t y o f the o t h e r s . The r e l a t i v e i n t e n s i t y o f K 2, K 3, and compared to G, i s about 0.17 at 2.1 K whereas theory p r e d i c t s 0.25. T h i s d i s c r e p e n c y may be due p a r t l y to the ortho broadening of the K l i n e s smearing part o f the l i n e w e l l i n t o the noise (which does not occur with G, ), as we l l as the f a c t t h a t an u n c a l i b r a t e d power meter was used f o r the K-band measurement. A companion to K, i s a f e a t u r e at 18.66 GHz which does not seem to f i t any p o s s i b l e t r a n s i t i o n i n the p a i r s p e c t r a . T h i s w i l l be d i s c u s s e d l a t e r . 'E o I O hi L. UJ O U Q_ cr o co GO < 1.0 K, 0.5 19 T-I.2K (TEMPERATURE RATIO) X-0.35 % 20 21 FREQUENCY (GHz) Figure 16 K-Band Spectra CO 80 A c l o s e look at the next nearest neighbour spectrum r e v e a l s the need to r e a s s i g n and G 7. If G^ and G 7 are now taken to be the t r a n s i t i o n s from | l f l > ~ and |l r 1 to | 2 , 0 > r e s p e c t i v e l y , then the t h e o r e t i c a l i n t e n s i t y r a t i o of G 6 to G 7 of 2 . 5 0 agrees with the e x p e r i m e n t a l l y observed r a t i o of 2 . 5 . Now by using the t h e o r e t i c a l s p l i t t i n g between the | l , 0 > and | l , l->-± s t a t e s of 2 . 3 5 GHz and the other known s p l i t t i n g s one gets a value of 1 1 . 5 9 and 1 1 . 7 0 GHz f o r the | 2 , 1 > + and | 2 , 1 > - to | l , 0 > t r a n s i t i o n s r e s p e c t i v e l y . T h i s i s c o n s i s t e n t with two f e a t u r e s observed i n the X-band search of the 0 . 1 % 0 -H2 sample at 1 1 . 9 6 and 1 2 . 0 6 GHz. S i g n a l averaging the r e s u l t s i n the 0 . 3 5 % o-H2 sample shows two l i n e s at 1 1 . 9 5 6 and 1 2 . 0 5 8 GHz with approximately the r i g h t r e l a t i v e i n t e n s i t y r a t i o to confirm t h i s assignment (see f i g . 1 8 ) . The o r i g i n of the f e a t u r e at 1 1 . 9 7 9 GHz i s p r e s e n t l y unknown. The r e l a t i v e i n t e n s i t y of G s to G 2 observed i s about 0 . 2 5 , which i s i n good agreement with the t h e o r e t i c a l i n t e n s i t y r a t i o of 0 . 2 1 c o n s i d e r i n g the poor s i g n a l - t o - n o i s e r a t i o . A l s o , the observed thermal enhancement between T = l . 2 K and 2 . 1 K i s 1 . 7 compared to the p r e d i c t e d value of 1 . 9 . The thermal f a c t o r of G 6 and G 7 between T = l . 2 K and 2 . 1 K i s found to be 1 . 2 whereas theory p r e d i c t s 1 . 4 3 . T h i s g i v e s a d d i t i o n a l support f o r the new assignment s i n c e the thermal f a c t o r f o r the p r e v i o u s l y assigned t r a n s i t i o n s | 2 , l > t —» | l . 0 > would be 1 . 9 0 . We now compare these r e s u l t s to the analogous r e s u l t s f o r G z. The thermal f a c t o r of G2. between T = 1 . 2 K and 2 . 1 K i s found to be 1 . 4 , compared with 1 . 8 8 p r e d i c t e d with 81 FREQUENCY (GHz) Figure 18 X-Band Spectra 82 Boltzmann f a c t o r s . The c l u s t e r i n g enhancement f a c t o r s f o r G 6 and G 7 are 1.3 and 1.2 at T=2.1 K and T=l.2 K r e s p e c t i v e l y which i s to be compared with 1.4 and 1.3 with G 2, i n d i c a t i n g good agreement. However the r e l a t i v e i n t e n s i t y of G6 and G7 to G z at T=2.1 K i s 0.98 which i s not i n good agreement with the t h e o r e t i c a l value of 0.73. Note that the d i s c r e p a n c y would be s u b s t a n t i a l l y l a r g e r i f the p r e v i o u s assignments o f G 6 and G 7 where taken. The two l i n e s , G i and G), a s s o c i a t e d with G^and G3 are the |2 rl> + —> |1,1>+ and |2,1>_ —> | l , l > - t r a n s i t i o n s . T h e i r f r e q u e n c i e s , 13.944 and 14.125 GHz r e s p e c t i v e l y , are determined once Gz, G 3 , Ge and G 7 are s p e c i f i e d . The |l,l>± s p l i t t i n g i s set by the assignments of G 6 and G 7 . T h e r f o r e when Gz and G3 are chosen, the |2,1>± s p l i t t i n g i s determined, f i x i n g the f r e q u e n c i e s o f G'z and G3'. In the 0.1% o-H z sample a small f e a t u r e was seen at 14.122 GHz and i n the 0.35% 0-H2. sample two small f e a t u r e s are seen i n the a p p r o p r i a t e p l a c e s but could a l s o be a noise f l u c t u a t i o n . T h e r e f o r e one can say t h a t G 3 has been observed e x p e r i m e n t a l l y whereas the presence of Gi i s u n c e r t a i n . These l i n e s are much harder to see s i n c e they are ortho-broadened, i n c o n t r a s t to Gz and Gj which are not. T h i s w i l l be d i s c u s s e d l a t e r i n d e t a i l . As mentioned before i t i s suspected that some of the i n t e n s i t y of these l i n e s i s s t r a i n broadened out f a r i n t o the wings. The absolute i n t e g r a t e d i n t e n s i t y f o r G| of t h i s l a s t sample i s 0.24 * 10 GHz cm i n comparison with 1.35*10 GHz cm -' p r e d i c t e d t h e o r e t i c a l l y . T h i s d i s c r e p a n c y i s apparent i n 83 many of the samples except for the f i r s t 0.35% sample (on i t s l a s t regrowth) which showed an integrated intensity for G, of - f f - i 0.89 * 10 GHz cm , much closer to the predicted value. The v a r i a b i l i t y of integrated intensity with regrowth of the same sample supports the claim that the int e n s i t y i s being p a r t i a l l y smeared out into the wings of the l i n e s . 5.3 Temperature Dependence of Lineshapes: The most accurate lineshapes are obtained in the low frequency bands where the microwave power level i s high, and where the linewidths are much less than the period of the stand-ing wave pattern. For these reasons and because of their large i n t e n s i t i e s , G, and G5 are the best candidates. Plots of G, and G s were made for runs at T=1.2, 2.1, 3.0, 4.2 K on the 0.1% o-H2 sample (see fig.19). These runs were not under synthesizer control and hence are not as accurate as other runs. However, to within experimental error the lineshape and linewidth i s temperature independent. These results put an upper l i m i t on the change in linewidth of 0.18 MHz. Results from the 0.35% 0-H2. sample put an even lower l i m i t on the phonon broadening. Plotted in fig.20 i s G, at T=2.1 K and 1.2 K. There i s v i r t u a l l y no difference between the two plots at the d i f f e r e n t temperature. It i s also noted that G 5 has a lineshape very similar to G, . An upper l i m i t of 100 KHz w i l l then be put on this e f f e c t between T=l.2 and 2.1 K (due to the f i n i t e frequency steps at which the data is c o l l e c t e d ) . This i s consistent with the theory developed e a r l i e r which suggests a difference of 84 about 100 KHz between the l i n e w i d t h s at the two temperatures due to the c r y s t a l f i e l d p r o c e s s e s . In some recent experiments we have looked very c l o s e l y at the l i n e s h a p e s of G , , G5 and Gz at T=4.2 K f o r the X=0.35% c r y s t a l . In c o n t r a s t to e a r l i e r r e s u l t s f o r the X=0.1% sample where the p o s i t i o n of the l i n e s d i d not change with temperature, G ( and G 5 were found to decrease i n frequency from the low temperature v a l u e s by 2.7 MHz, and Gz by 2.0 MHz. At present there i s no p l a u s i b l e e x p l a n a t i o n f o r these s u r p r i s i n g l y l a r g e s h i f t s . Because of the s h i f t one cannot o b t a i n the li n e s h a p e i n the usual manner. Instead, we used the background obtained p r e v i o u s l y a t T=2.1 K from time r a t i o e d s p e c t r a . T h i s l e a d s to the r e s u l t shown i n f i g . 2 0 . The l i n e w i d t h i s 2.9'MHz, s l i g h t l y l e s s than the value 3.1 MHz obtained at T=l.2 K or T=2.1 K. However, the background i n the v i c i n i t y o f the l i n e appears to have changed which causes some d i s t o r t i o n o f the l i n e s h a p e . T h i s d i p a l s o occurs at the other two l i n e s , G5 and Gz. Since i t i s u n l i k e l y t h a t the standing wave p a t t e r n has changed d r a s t i c a l l y i n temperature there must be some broad temperature dependent a b s o r p t i o n a s s o c i a t e d with the hydrogen. The s l i g h t d i f f e r e n c e i n the l i n e w i d t h at the two temperatures c o u l d probably be accounted f o r by t h i s change i n background. We t h e r e f o r e conclude that to w i t h i n 0.2 MHz the l i n e width i s temperature independent. T h i s i s s u b s t a n t i a l l y l e s s than the c a l c u l a t e d width at 4.2 K of about 2 MHz (see Table I V ) . However the c r y s t a l f i e l d c o u p l i n g c o e f f i c i e n t i s somewhat u n c e r t a i n and i n a d d i t i o n a small change i n the e f f e c t i v e value 85 of the sound v e l o c i t y (which occurs to high powers in the formulae) could make a s i g n i f i c a n t change in the predicted width. 5.4 Strain and Ortho Broadening: As mentioned e a r l i e r mechanical strains contribute a large portion to the linewidth of the pair t r a n s i t i o n s . This component, because of i t s nature, cannot be predicted in advance and with the present apparatus has not been eliminated. If one r e c a l l s the three other contributions to the linewidth of G, discussed e a r l i e r , one notes they are n e g l i g i b l e compared to the t y p i c a l l y observed linewidth of a few MHz. A glance at table VII, which contains the smallest linewidths observed at a parti c u l a r 0-H2. concentration, shows that the s t r a i n broadening has no pa r t i c u l a r c o r r e l a t i o n with concentration. However by using G, as a standard, one can extract the inhomogeneous ortho broadening contribution to the linewidths. In general the s t r a i n broadening w i l l have a d i f f e r e n t e f f e c t on each t r a n s i t i o n due to the complicated dependence of the l e v e l s p l i t t i n g s on pair separation. Therefore one cannot take the l i n e width of G, as the s t r a i n broadening e f f e c t but must scale i t by some factor. These factors w i l l be determined by data collected from samples of 0.03% 0-H2. concentration where the ortho broadening should be n e g l i g i b l e . These factors are l i s t e d in table VIII. Now using these factors and assuming a l l the lineshapes concerned are Lorenztian (in order that one can add and subtract linewidths) the ortho broadening contribution 8 6 1.0 o_ CC ° 05 CD < X»O. I% F i g . 19 f0- 13.650 GHz r \ T-2.1 K_ — / \ T -4 .2K-s/ — <S 1 l ^ - = ^ l -10 1.0-Q. cr o to cn 0.5 < F i g . 20 -5 f -O 5 f 0 (MHz) 10 X»0 .35% f0= 13.650 GHz - T-1.2 a 2.1 K T=4.2 K H -10 -5 0 5 f - f 0 (MHz) Figure 19 G x f o r 0.1% o-H 2 Sample Figure 20 Gj f o r 0.35% o-H 2 Sample 87 Table VII Observed Linewidths Narrowest l i n e w i d t h s (MHz) vs. O-H- c o n c e n t r a t i o n Line G, Ga. O f G 5 G* G-, 0. 03% (a) (b) 2.1 2.67 3.8 3.9 2.0 3.3 2.1 2.55 - 8.4 - 8.0 0.1% 3.83 9.77 8.46 4. 40 20.6 15.9 0.2% (a) (b) 5.0 4.9 8.26 -- 4.6 0. 35% 3.09 5.20 18 3.09 25.2 25.5 Appeared as an p a r t i a l l y unresolved doublet i n t h i s sample o n l y . In each case (a) and (b) r e f e r to two d i f f e r e n t samples o f the same c o n c e n t r a t i o n . 88 to the l i n e w i d t h s at 0.1% and 0.35% o-Hz are found and l i s t e d i n t a b l e IX. As one can see there i s g e n e r a l agreement with the r e s u l t s p r e d i c t e d e a r l i e r . The l i n e s G, , G 5 and G ± (and G3 by l o o k i n g at the s p e c t r a d i r e c t l y i n fig.21) are not broadened whereas G^, G 6 and G 7 a r e . R e c a l l i n g the l i m i t a t i o n s o f the theory presented e a r l i e r the agreement of the p r e d i c t e d widths with experiment i s good at 0.35% 0-H2.. There i s however a problem with the 0.1% o-H a widths i n that they should be 5/3 (0.1/0.35) =0.12 times s m a l l e r than those f o r a 0.35% sample. The measured r a t i o i s about 0.29. T h i s d i s c r e p e n c y could be due to an e r r o r i n the c o n c e n t r a t i o n (which cannot be c o n v e n i e n t l y measured), or due to the f a c t t h a t the l i n e s h a p e s are not L o r e n t z i a n and one should not simply add the l i n e w i d t h s together but should do a proper c o n v o l u t i o n . 5.5 I s o t o p i c S u b s t i t u t i o n E f f e c t s : In order to i n v e s t i g a t e the e f f e c t on the l i n e w i d t h s o f s u b s t i t u t i o n a l i s o t o p i c mass d e f e c t s i n the l a t t i c e , s e v e r a l 0.2% o-Hi samples where doped with about 0.2% o-Dz (3=0 s p e c i e s ) . Deuterium was used rather than HD because i t should have the l a r g e s t e f f e c t of the two and i n a d d i t i o n does not have the added c o m p l i c a t i o n of o f f c e n t e r r o t a t i o n (see s e c t i o n 3.3). The changes i n frequency and l i n e w i d t h are now d i s c u s s e d . In order to e s t a b l i s h the mechanical s t r a i n broadening component of the l i n e w i d t h the sample i s grown undoped and the l i n e w i d t h s observed. Then the sample i s evapourated i n t o a bulb c o n t a i n i n g a small amount of 0-D2. and q u i c k l y condensed back i n t o the c e l l Table VIII R e l a t i v e S t r a i n Broadening F a c t o r s Linewidths at 0.03% r e l a t i v e to G, G 5 1.06±0.08 Gz 1.57*0.25 G ^  1.08^0.22 Q 6 3.15 G 7 3.00 (Average v a l u e s taken over many samples) Table IX Observed Ortho Broadening Linewidths (MHz) with s t r a i n component removed Line 0.1% 0. 35% Theory G, 0.00 0. 00 0.00 G a 3.76 0. 35 0. 00 G*. 4. 32 14.7 17.0 G 5 0. 34 0. 00 0.00 G^  8. 54 15.5 26. 2 G 7 4.41 16.2 17.0 91 and regrown. Considering G, and G 5 , the undoped linewidth i s 5 MHz and with 0.2 % o-D2 i s 13 MHz. The average l i n e position of G, and G 5 has increased by Af=6 MHz, the sign of which i s consistent with the l a t t i c e contraction expected due to the reduced zero point motion of the impurities r e l a t i v e to the host. Again assuming one can add linewidths the mass defect s t r a i n broadending linewidth i s 8 MHz. Relating Af to 4V gives = 0.75 which i s consistent with the result predicted e a r l i e r ° f ^ = 1 - 0 4 = 0 - 6 ' If one takes the measured value of Af and uses [54], one finds the change in molar volume with o-Dj. concentration to be -7^^-0.12. This i s consistent with the linear approximation applied to results for the pure species AS-4j~a c which gives 1 ^ ^ = -0.125 . The value of =8 MHz y i e l d s , from [52], a nearest neighbor d i s t o r t i o n of A=-0.0034 for o-b2 impurities (cf. A=-0.03 for fHe in 5He, P.G. Klemens e t . a l . (1964)). From this we can predict the linewidth for a Di concentration of 0.03%, the natural abundance of HD in hydrogen, to be 1.2 MHz. Therefore since the broadening effect of HD i s expected to be less than that for D2, we can conclude that the lineshape of the pair t r a n s i t i o n s in "natural" hydrogen contains a Lorentzian component with a width less than 1.2 MHz. 1using V(o-Di) = 19.88 cc/mole and V(p-H 2)= 22.73 cc/mole and U=0.269. Bostanjoglo and Kleinschmidt (1967) and Goldman (1977) 92 5.6 New Features: While studying the many samples grown in th i s experiment several unexpected phenomena were observed. The f i r s t of these is the apparent temperature independence of the i n t e n s i t i e s of in-plane t r a n s i t i o n s while out-of-plane and next nearest neighbour t r a n s i t i o n s display the expected temperature dependence. In addition to this a new spectral feature has been observed at 17.5 GHz which does not display the same ch a r a c t e r i s t i c s of the ortho pair l i n e s . Also there are some very broad background l i n e s observed in the 11.0 to 24.0 GHz reg ion. A temperature independence of the i n t e n s i t i e s of C (, C 3 and H 3 was found between T=l.2 K and T=2.1 K in two samples containing 0.35% o-H2. This e f f e c t was not observed by Hardy e t . a l . where the ortho concentrations were 0.2% or l e s s . Time ratios also show nothing, indicating either a very fast or very slow clustering rate, whereas temperature ra t i o s show only a broad background feature about 500 MHz wide and no sign of the l i n e s which are obviously there in the raw data. In order to gain more information a set of scans was taken at T=4.2 K These spectra are much noisier than those taken below the lambda point of l i q u i d He but are s t i l l useful. When rati o s are taken between the T=4.2 K and 1.2 K data a sharp l i n e of low in t e n s i t y about 20 MHz wide s i t s on top of the broad background peak at the expected positions of the three l i n e s . This indicates the p o s s i b i l i t y of two temperature dependent and competing processes which happen to cancel out when comparing T=1.2 K to 93 T=2.1 K. Recalling from HBHI that in-plane l i n e s do not show the same eff e c t s of clustering as do out-of-plane l i n e s , one might suggest that there i s a temperature dependent ef f e c t on the clustering which i s counteracting the Boltzmann population factors. Also the fact that t h i s e f f e c t has not been observed in samples of 0.2% o-H2 or less suggests that the process may be concentration dependent. The temperature independence could also be due to microwave power saturation, i . e . thermal equilibrium is not being maintained in the system. This p o s s i b i l i t y has been checked many times for G| and now with Ci and C 3 by either comparing the lineshapes at two d i f f e r e n t incident power le v e l s or by observing the time dependence of the absorbed power at the absorption peak. These tests were always negative and the rough c a l c u l a t i o n done in HBHI indicates that a much higher power lev e l would be needed to saturate these l e v e l s . Now that the p o s s i b i l i t y of saturation has been eliminated there i s an inconsistency between experiment and the compensating clustering e f f e c t suggested in the l a s t paragraph. The Boltzmann factors for the ground state t r a n s i t i o n s C, and C 3 decrease as the temperature i s raised whereas those for excited state t r a n s i t i o n s such as H3 increase . This means that any e f f e c t cancelling out the Boltzmann factor between T=l.2 K and 2.1 K would have to be l e v e l dependent. There i s a d i s t i n c t feature occuring near 17.5 GHz in many of the samples. This feature shows an increase in intensity when the temperature is lowered from 2.1 K to 1.2 K and shows no 94 change with time. It was not observed i n the 0.1% 0-H2. sample nor the 0.03% 0-H2. samples but appears at 17.24 GHz i n the 0.2% samples with or without o-D z and at 17.4 GHz i n one 0.35% sample and at 17.49 GHz i n the other 0.35% sample. T h i s l i n e d i s p l a y s the unusual behaviour of l o s i n g i n t e n s i t y upon repeated annealing whereas the p a i r l i n e s g e n e r a l l y i n c r e a s e i n i n t e n s i t y . T h i s i n d i c a t e s t h a t t h i s a b s o r p t i o n i s not due to the usual p a i r t r a n s i t i o n s . There are two other weaker t r a n s i t i o n s observed at 18.66 GHz and 11.90 GHz t h a t one would be tempted to a s s o c i a t e with t h i s f e a t u r e . The l i n e at 11.90 GHz d i s p l a y s the same c h a r a c t e r i s t i c s as the 17.5 GHz l i n e , except that the e f f e c t of annealing was not monitored. Therefore these two l i n e s c o u l d be r e l a t e d . However the l i n e at 18.66 GHz changes i n time because of c l u s t e r i n g , and a l s o grows in i n t e n s i t y as the temperature i s increased from .1.2 K to 2.1 K and hence i t does not seem l i k e l y t h a t t h i s l i n e i s r e l a t e d to the other two. The source of these l i n e s i s unknown as i s the reason they appear i n some samples and not o t h e r s . One p o s s i b i l i t y to c o n s i d e r i s that the 11.90 GHz and 17.49 GHz l i n e s are the ground s t a t e to |2,0> l e v e l t r a n s i t i o n s f o r the two f u r t h e r neighbour p a i r s , i e . r / r D =V8/3' and ST. The EQQ s p l i t t i n g s f o r these t r a n s i t i o n s are 17.35 GHz and 12.93 GHz r e s p e c t i v e l y and i f one were to i n c l u d e 3-body p o l a r i z a t i o n and c r y s t a l f i e l d e f f e c t s these could e a s i l y s h i f t a few hundred MHz i n t o p l a c e . The temperature dependence i s c o n s i s t e n t with t h i s h y p o thesis as i s the f a c t t h at both of these p a i r types have the same symmetry 95 as the in-pla n e p a i r s , and are t h e r e f o r e allowed t r a n s i t i o n s . Scanning from 12.5 to 18.0 GHz has revealed two l a r g e peaks i n the background which are about 0.6 to 1.0 GHz wide. These peaks are observed i n a l l the samples which where scanned across t h i s band. In a l l cases these peaks are temperature independent and show no c l u s t e r i n g . T h e i r c e n t e r s are l o c a t e d at about 13.7 GHz and 16.1 GHz. Since the i n t e g r a t e d i n t e n s i t i e s o f the p a i r l i n e s i n t h i s r e g i o n are u s u a l l y about an order o f magnitude s m a l l e r than theory p r e d i c t s i t i s assumed that a l a r g e p o r t i o n of the sample i s s t r a i n broadened and that a l a r g e p o r t i o n of the i n t e n s i t y r e s i d e s i n the wings. One might then suggest that these l a r g e background peaks are the s t r a i n broadened p a r t s o f the p a i r l i n e s . In order to check t h i s h ypothesis the i n t e g r a t e d i n t e n s i t y o f the lower frequency peak was measured i n a 0.03% 0 - H 2 sample and was found to be about 7 x 10 6 GHz cm-1 whereas the t h e o r e t i c a l i n t e g r a t e d ' i n t e n s i t y f o r - 6 - - ( G, p l u s G2 i s 0.013 x 10 GHz cm . In a d d i t i o n , these peaks do not change i n temperature or time. We conclude that these peaks are not due to ortho p a i r t r a n s i t i o n s . However the f a c t that these peaks are not seen when there i s no sample i n the c e l l i m p l i e s t h a t they are somehow connected with the presence o f hydrogen. Channel s p e c t r a due to the H^ f i t a l l the above c r i t e r i a and in order to confirm t h i s p o s s i b i l i t y , data from 11 to 24 GHz i s examined i n (fig . 2 2 ) and the center p o s i t i o n s o f the broad l i n e s are ta b u l a t e d i n t a b l e X. One c a l c u l a t e s the expected f r e q u e n c i e s from the waveguide wavelength A<j as f o l l o w s . 96 [60] S T X c 2 ~ y\ [61] for the TE ) c ? mode. Here a i s the l a r g e r t r a n s v e r s e d imens ion o f the waveguide and X i s the l e n g t h o f the hydrogen sample . These r e s u l t s are t a b u l a t e d i n t a b l e X and show good agreement f o r n=2-6. A l t h o u g h channe l s p e c t r a u s u a l l y take on a s i n u s o i d a l shape the geometry o f the c e l l may change t h i s s i t u a t i o n . The c e l l f l ange c o n n e c t i o n to the waveguide taper i s 2.55 t imes the sample l e n g t h away from the bottom of the c e l l and r e f l e c t i o n s from t h i s boundary may superimpose another s i n u s o i d o f a d i f f e r e n t wavelength on top o f the f i r s t one . A l s o there w i l l be some r e f l e c t i o n s from the c e l l - t a p e r f l ange which w i l l change the s t a n d i n g wave p a t t e r n . In p r i n c i p l e the s t ep i n the waveguide shou ld e l i m i n a t e any r e f l e c t i o n s . However i f the sample volume i s not p r e c i s e l y c o r r e c t there w i l l be an impedance mismatch near the s t ep and hydrogen s u r f a c e c a u s i n g some r e f l e c t i o n . The peak to t rough r a t i o o f the s imple s i n u s o i d i s g i v e n by where the r e f l e c t a n c e o f the hydrogen-vacuum s u r f a c e has been taken to be The observed v a l u e tends to decrease wi th f requency and has an [62] [63] r = i - yg1 - o.o65 1.0 T = 2.1 K X = 0 .35% FREQUENCY (GHz) Figure 22 Channel Spectrum From 11-24 GHz VD Table X Channel Spectra Peaks Expected Observed 21/zla Frequency Frequency -1 (GHz) (GHz) 2 11.8 11.4 3 14.0 13.7 4 16.5 16.1 5 19.4 19.9 6 22.3 22.6 l=hydrogen sample length=3.92 cm. <A<j =waveguide wavelength average value of about 1.5 which i s in reasonable agreement. 100 CHAPTER VI C o n c l u s i o n s To conclude we l i s t the main r e s u l t s obtained i n t h i s work as w e l l as some suggestions f o r f u r t h e r developement. (1) Out-of-plane p a i r s : The |2,0> l e v e l has now been e s t a b l i s h e d with the d i s c o v e r y of the two t r a n s i t i o n s H, and H2 . In a d d i t i o n , the four r e c e n t l y observed l i n e s K, through have confirmed the assignments o f t r a n s i t i o n s connecting to the |l,l>± and \2,2>± s t a t e s . A l l the l e v e l s have now been determined except that o f the |0,0> s t a t e . Most processes do not couple to t h i s s t a t e , making an experimental d e t e r m i n a t i o n o f i t very d i f f i c u l t . (2) In-plane p a i r s : The |2,0> l e v e l has a l s o been e s t a b l i s h e d f o r the in - p l a n e p a i r s with the measurement o f H 3. T h i s l e a v e s o n l y the |1,1>± l e v e l to be determined. The l a c k o f temperature dependence of the in- p l a n e p a i r t r a n s i t i o n s i s cause f o r concern and f u r t h e r e f f o r t w i l l be necessary to provide an e x p l a n a t i o n f o r t h i s e f f e c t . 101 (3) Next-nearest neighbor p a i r s : C o n s i d e r a b l e progress has been made i n the d e t e r m i n a t i o n of the next nearest neighbor p a i r l e v e l s . The recent d i s c o v e r y of the two l i n e s G f and , as w e l l as the reassignment o f G6 and G 7, have e s t a b l i s h e d the p o s i t i o n of most of the l e v e l s . (4) Inhomogeneous broadening: A f i r s t order theory f o r the broadening of the p a i r l i n e s due to the f i n i t e c o n c e n t r a t i o n of ortho molecules has been developed. Experimental v e r i f i c a t i o n has been obtained f o r t h i s theory with the low frequency l i n e s . A more q u a n t i t a t i v e comparison of experimental l i n e s h a p e s with theory awaits improvement i n the s t r a i n broadening. (5) Homogeneous broadening: The e f f e c t o f phonon induced l i f e t i m e broadening on the p a i r l i n e s has been c a l c u l a t e d . Although an upper l i m i t has been placed on t h i s e f f e c t , experimental v e r i f i c a t i o n of the temperature dependence of the l i n e w i d t h s i s needed. Again, i t i s important that s t r a i n e f f e c t s be reduced. (6) Further ortho p a i r s : The o b s e r v a t i o n of the two l i n e s at 11.90 and 17.49 GHz and the p o s s i b i l i t y t h a t these are t r a n s i t i o n s o f the t h i r d and f o u r t h nearest neighbor p a i r s w i l l h o p e f u l l y s t i m u l a t e i n t e r e s t i n the c a l c u l a t i o n of t h e i r expected f r e q u e n c i e s and i n t e n s i t i e s . The r e s u l t s o f these c a l c u l a t i o n s w i l l be awaited 102 with a great deal of i n t e r e s t . 103 APPENDIX A: Device Codes and I n s t r u c t i o n s The f o l l o w i n g p r o v i d e s a d e s c r i p t i o n of the custom p e r i p h e r a l s a v a i l a b l e i n the system. Each item c o n t a i n s the o c t a l d e v i ce code (DC) and o p e r a t i n g i n s t r u c t i o n s . 1) 16 b i t D/A Conve r t e r : DC 75 DOA AC,75 s e t s a v o l t a g e p r o p o r t i o n a l to the contents o f accumulator AC. DOB AC,75 b i t s 14,15 of accumulator AC t u r n on or o f f the two d i g i t a l outputs on the c o n v e r t e r . 2) S y n t h e s i z e r C o n t r o l : DC 2 T h i s d e v i c e r e q u i r e s the input to be i n 1 BCD. DOA AC,2 b i t s 8-11 set the 100 MHz d ig i t b i t s 12-15 set the 10 MHz d ig i t DOB AC, 2 b i t s 0-3 set the 1 MHz d i g i t b i t s 4-7 set the 100 KHz d i g i t b i t s 8-11 set the 10 KHz d ig i t b i t s 12-15 set the 1 KHz d ig i t DOC AC, 2 b i t s 4-7 set the 100 Hz d ig i t b i t s 8-11 set the 10 Hz d ig i t b i t s 12-15 set the 1 Hz d i g i t 3) 12 b i t D/A c o n v e r t e r s : DC 4 The 12 b i t twos complement contents o f accumulator AC 104 ( b i t s 4-15) s e t s the output v o l t a g e of the c o n v e r t e r , i n the range 0 to 10 v o l t s . DOA AC,4 D/A #1 DOB AC,4 D/A #2 4) 14 b i t A/D c o n v e r t e r : DC 25 The input channel (0 through 7) i s s e l e c t e d and con v e r s i o n i n i t i a l i z e d with DOAS AC,25 where AC c o n t a i n s the channel number. A f t e r c o n v e r s i o n , s i g n i f i e d by the i n t e r u p t or busy and done l o g i c , the 14 b i t twos complement r e p r e s e n t a t i o n of the analog input i s read i n with the i n s t r u c t i o n DIC AC,25 5) Programmable i n t e r v a l t i m er: DC 5 The timer increment s i z e of 10 microseconds (0<a<6) i s s e l e c t e d with the i n s t r u c t i o n DOA AC,5 AC c o n t a i n s the number a. The number of increments counted by the timer i s loaded i n and the timer s t a r t e d with the i n s t r u c t i o n : DOBS AC,5 AC c o n t a i n s the number of increments The output pulse generated by t h i s d e v i c e i s a v a i l a b l e through a set of complementary (TTL compatible) outputs or through the i n t e r r u p t or busy and done l o g i c . 6) D i g i t a l i n p u t s , outputs and r e l a y c o n t r o l s : DC 31 105 DOA AC,31 b i t s 10-15 of accumulator AC s e t the r e l a y s #6 to #1. DOB AC,31 b i t s 10-15 of accumulator AC s e t the la t c h e d output p o r t s #6 to #1 DIA AC,31 b i t s 12-15 i n d i c a t e the s t a t e of the input p o r t s #4 to #1 DIB AC,31 b i t s 12-15 i n d i c a t e the l a t c h e d s t a t e of these input p o r t s . These inputs are TTL compatible and are normaly h i g h . 7) Timer c l o c k : DC 55 The f o l l o w i n g i n s t r u c t i o n s : NIOC AC,55 w i l l r e s e t the timer. DIA AC,55 reads i n the most s i g n i f i c a n t d i g i t s DIB AC,55 reads i n the l e a s t s i g n i f i c a n t d i g i t s The 32 b i t b i n a r y number read i n c o n t a i n s the number of m i l l i s e c o n d i n t e r v a l s s i n c e the l a s t r e s e t i n s t r u c t i o n . 8) LED d i s p l a y : DC 45 DOB AC,45 s e t s the lower four d i g i t s where AC c o n t a i n s the four BCD numbers to be d i s p l a y e d . DOA AC,45 s e t s the upper four d i g i t s i n the same manner. DOC AC,45 s e t s the lone decade with b i t s 12-15 of accumulator AC and the decimal p o i n t i s set with b i t s 4-7. 106 APPENDIX B: Assembler Subroutine D e s c r i p t i o n s Some of the assembler language subrou t i n e s used to c o n t r o l the microwave spectrometer are d e s c r i b e d i n t h i s appendix. Each heading w i l l c o n t a i n the name of the program as w e l l as the Basic i n s t r u c t i o n used to c a l l the s u b r o u t i n e . SYNSW: CALL 16,FSTART,AF,START,N,AT,SYNTH,DUMMY,A/DCHAN T h i s i s the r o u t i n e which sweeps the BWo' sweeper and s y n t h e s i z e r . The s y n t h e s i z e r frequency i s set with SYNTH but the 16 b i t D/A p r e s e t t i n g v o l t a g e i s set from a frequency-voltage c o n v e r s i o n t a b l e l o c a t e d i n DATA at l o c a t i o n FSTART. T h i s t a b l e i s c o n s t r u c t e d before the program i s c a l l e d and r e q u i r e s t h a t the D/A be c a l i b r a t e d . T h i s i s done with the r o u t i n e CALIB which i s to be d e s c r i b e d l a t e r . Every AF p o i n t i n t h i s t a b l e i s read. SYNTH i s a three element a r r a y c o n t a i n i n g the s t a r t n g frequency o f the s y n t h e s i z e r AAA,ABB,BBO Hz and i t s increment frequency i n Hz. AAA,A i s stored i n SYNTH(1), BB,BB i n SYNTH(2) and SYNTH(3) c o n t a i n s the increment frequency. The sweeper i s stepped N times with a time i n t e r v a l o f A T ( i n 1 m i l l i s e c o n d u n i t s ) . The data i s c o l l e c t e d at the end of each time i n t e r v a l before the frequency i s stepped. Data from A/D channel A/DCHAN i s st o r e d i n DATA s t a r t i n g at l o c a t i o n START and data c o l l e c t e d from A/D channel #1 i s stored i n DATA s t a r t i n g at l o c a t i o n 6006. DUMMY i s a "dummy" v a r i a b l e not used at p r e s e n t . 107 CALSW: CALL 12,FSTART, F,START,N, T,POPT,PARR,A/DCHAN FSTSW: CALL 13,FSTART, F,START,N, T,POPT,PARR,A/DCHAN These two programs sweep the BWO sweeper with the 16 b i t D/A but not the s y n t h e s i z e r . CALSW i s f o r slow s i n g l e sweeps, T i n m i l l i s e c o n d u n i t s and FSTSW f o r f a s t r e p e t i t i v e sweeps, T i n 1 microsecond u n i t s . The v a r i a b l e s FSTART, F,START,N,A/DCHAN are d e f i n e d as b e f o r e . POPT i s the CRT p l o t o p t i o n . POPT=0 no p l o t POPT=l s t o r e data (T) i n s p e c i f i e d block P0PT=2 c a l c u l a t e T/TO and s t o r e P0PT=3 c a l c u l a t e 1-T/T0 and s t o r e The Basic program w i l l request the s t a r t i n g addresses o f bl o c k s T, TO and TPLOT. The data i s stored i n T and can be d i v i d e d by a background spectrum i n TO. I f t h i s i s done the r e s u l t i s stored i n TPLOT. In FSTSW, the contents o f T w i l l be p l o t t e d for P0PT=1 and the contents o f TPLOT f o r P0PT=2 or 3. For CALSW the block p l o t t e d w i l l be that g i v e n i n response to a request from the Basic program. The a r r a y PARR s p e c i f i e s the s t a r t i n g p o s i t i o n of TO and of TPLOT i n i t s f i r s t two elements and i n FSTSW the p l o t g a i n can be changed by t y p i n g i n the d e s i r e d g a i n change on the t e l e t y p e as f o l l o w s : 108 TTY k e y G a i n c h a n g e 1 1 2 2 3 3 8 8 H 1/2 T h i s g a i n f a c t o r i s u s e d t o s c a l e t h e d a t a p l o t t e d . CALSW p e r f o r m s o n l y o n e s w e e p w h e r e a s FSTSW s w e e p s r e p e t i t i v e l y and t y p i n g a c a r r i a g e r e t u r n (CR) on t h e t e l e t y p e w i l l r e t u r n c o n t r o l t o B a s i c . FSTSW i s u s e f u l f o r a d j u s t i n g e x p e r i m e n t a l p a r a m e t e r s and o b s e r v i n g t h e e f f e c t t h i s h a s o n t h e s p e c t r u m i n r e a l t i m e . C A L I B : C A L L 1 1 , S T A R T , N , V S T A R T , T T h i s r o u t i n e a l l o w s o n e t o c a l i b r a t e t h e 16 b i t D / A c o n v e r t e r . When t h i s r o u t i n e i s c a l l e d i t s t a r t s s w e e p i n g a t V S T A R T ( v o l t s ) and i n c r e m e n t s t h e v o l t a g e s e t t i n g r e g i s t e r b y 1 b i t e v e r y T m i l l i s e c o n d s . A " b l a c k b o x " a t t a t c h e d t o d i g i t a l i n p u t s 1 and 2 i s u s e d t o c o n t r o l t h i s p r o g r a m . N o r m a l l y t h e t h r e e p u s h b u t t o n s a r e o p e n ( b o t h i n p u t s l o w ) and t h e s w e e p i n g c o n t i n u e s . A t t h e d e s i r e d f r e q u e n c y , w h i c h i s d e t e c t e d w i t h a m i c r o w a v e f r e q u e n c y c o u n t e r o r w a v e m e t e r , t h e r e d b u t t o n i s p u s h e d ( b o t h i n p u t s h i g h ) and t h a t v o l t a g e i s r e c o r d e d i n DATA s t a r t i n g a t S T A R T . T h i s c o n t i n u e s u n t i l t h e m i c r o w a v e band h a s b e e n c a l i b r a t e d . P r e s s i n g e i t h e r o f t h e b l a c k b u t t o n s (one i n p u t h i g h , one l o w ) w i l l e i t h e r r e v e r s e t h e d i r e c t i o n o f t h e 109 sweep or a l t e r n a t e the ra t e of sweep by a f a c t o r o f 8. Typing a c a r r i a g e r e t u r n w i l l terminate the sweep and the number of c a l i b r a t i o n p o i n t s taken i s returned i n N. These c a l i b r a t i o n p o i n t s can then be used to c o n s t r u c t a frequency-voltage t a b l e by i n t e r p o l a t i o n . CRT: CALL 8,START,N,DUMMY,MARK CRT: CALL 9,START,N,DUMMY,MARK,DUMMY CRT1: CALL 10 These r o u t i n e s are used to d i s p l a y the contents o f DATA on the monitor scope (CRT). C a l l i n g r o u t i n e 8 w i l l p l o t N p o i n t s i n DATA s t a r t i n g at START. The v o l t a g e a p p l i e d to A/D channel 7 i s read i n and the po i n t with the X-value corresponding to t h i s v o l t a g e i s i n t e n s i f i e d . If an M i s typed i t i s echoed on the t e l e t y p e and t h i s p o i n t i s stored i n the a r r a y 1 MARK. Th i s f e a t u r e can be u s e f u l when s e l e c t i n g p o r t i o n s o f a spectrum f o r a n a l y s i s . To e x i t t h i s r o u t i n e type a c a r r i a g e r e t u r n and c o n t r o l w i l l be t r a n s f e r e d back to B a s i c . Routine 9 only s e t s up the parameters f o r the p l o t . T h i s i s necessary before c a l l i n g the sweeper programs.' A c a l l to ro u t i n e 10 w i l l p l o t the data once and must be c a l l e d r e p e t i t i v e l y i n order to o b t a i n a v i s u a l d i s p l a y . DISKW CALL 14,FILENAME,START,N,ERR DISKR CALL 15,FILENAME,START,N,ERR These two r o u t i n e s t r a n s f e r data between core and the d i s k . DISKR w i l l read N words from the f i l e FILENAME and place them i n 110 DATA s t a r t i n g at START. The other r o u t i n e DISKW w i l l w r i t e data from DATA to the d i s k with the c a l l i n g parameters d e f i n e d above. The RDOS e r r o r code f o r the d i s k o p e r a t i o n i s returned i n ERR. XYPLT: CALL 18,YSTART,XSTART,N,AT STRIP: CALL 19,YSTART,XSTART,N,6T POINT: CALL 20,YSTART,XSTART,N These three e n t r i e s to XYPLOT w i l l p l o t a p o r t i o n of DATA on e i t h e r an X-Y r e c o r d e r , s t r i p c h a r t recorder or p o i n t p l o t t e r . The X and Y v a l u e s p l o t t e d come from DATA s t a r t i n g at l o c a t i o n s XSTART and YSTART r e s p e c t i v e l y . N p o i n t s are p l o t t e d at AT m i l l i s e c o n d i n t e r v a l s . I f XYPLT or POINT are c a l l e d then the X and Y v o l t a g e s are a p p l i e d to the X and Y in p u t s o f the p l o t t e r . In t h i s case the X input i s u s u a l l y a l i n e a r ramp from 0 to 10 v o l t s . However i f STRIP i s c a l l e d the Y value i s a p p l i e d to the c h a r t recorder input and the X vaiue i s a p p l i e d to the marker i n p u t . When one of these programs i s c a l l e d the f o l l o w i n g keyboard commands can be executed: CR - s t a r t p l o t t i n g 1 - p l o t 0V,0V 2 - p l o t 10V,10V 3 - p l o t f i r s t p o i n t These commands are e s p e c i a l l y u s e f u l i n s e t t i n g up the X-Y r e c o r d e r . During the p l o t t y p i n g a CR w i l l terminate the p l o t . A f t e r p l o t t i n g , c o n t r o l i s returned to Basic a u t o m a t i c a l l y by I l l the STRIP r o u t i n e whereas a CR i s necessary with the other two r o u t i n e s . If one then l i f t s the pen from the p l o t t e r , t h i s w i l l then e l i m i n a t e the f l y b a c k to (0,0) from being p l o t t e d . FUNC: CALL 22,F,I,J,K,N,C(5) T h i s program c o n t a i n s a number of r o u t i n e s which can be used to manipulate the a r r a y s i n DATA. F s p e c i f i e s the f u n c t i o n to be executed as f o l l o w s : F=l Add two a r r a y s F=2 B a s e l i n e a d j u s t F=3 D i v i d e two a r r a y s F=4 M u l t i p l y two a r r a y s F=5 I n t e g r a t e over an a r r a y F=6 Find the minimum and maximum p o i n t s i n an a r r a y The d e t a i l s on how to s p e c i f y the other parameters are a v a i l a b l e i n the l i s t i n g of the program ANALYSIS, i n appendix C. 112 APPENDIX C: Basic Program 00.10 P R I N T " S W E E P E R C O N T R O L P R O G R A M - - " 00 :i. 5 L E T E = 0 0016 L E T E = 0 0 0 2 0 D I M C * C 2 3 > B $ C 6 3 » M C 2 3 » N C 3 3 » F $ C 2 5 3 F D $ C 1 2 3 » G * C 1 2 3 » L C 1 0 3 R C C 1 0 5 3 » 6 C 3 3 » F C 5 0 3 0021 L E T B $ = " N N N N N N " 0025 I N P U T " I N P U T D E F A U L T D I R E C T O R Y F O R D I S K O P E R A T I O N S M I $ 0030 I N P U T " E N T E R C A L I B R A T I O N P O I N T S F R O M D I S K ( I N P U T F I L E N A M E ) OR T T I ( I N P U T T ) " . F $ ; 0040 L E T F $ = F $ v " " 0050 I N P U T • E N T E R F S T A R T » D E L T A F » N O . P T S " » F 1 » F ? » N 0060 I F F * = " T " T H E N G O T O 0 1 1 0 0070 C A L L 1 5 » F * > 7 S 0 0 » N » £ O O S O I F E = 0 THEN G O T O 0105 0081 PRINT N5" P O I N T S T A K E N " t 0090 PRINT " R E A D ERROR " yE 0100 STOP 0105 C A L L 5 r F C 1 3 » 7 5 0 0 » N 0106 GOTO 0150 . . . 0110 PRINT "ENTER P O I N T S " 0120 FOR 1=1 T O N 0130 INPUT F L I J O 0140 NEXT 1 0150 FOR 1 = 1 TO H 0151 I F FI::I::I<O T H E N L E T FI:I:I=FI:I::LI65536 0152 NEXT I 0155 PRINT 0160 O N ESC THEN S T O P 0165 INPUT ">"yC$y 0170 L E T P=0 0175 L E T M = 0 0130 IF C$ 1:2,2::!=" P" THEN L E T P = l 01S5 IF C$!::2»2:J = " M " THEM L E T H = l 01S7 O N ESC T H E N GOTO 0155 01 90 I F C$ E 1 y 1 :i = • R " T H E N G0T0 1000 0210 If CSL1 y i : i = " P " T H E N G O T O 6000 0220 I F C$L1*13="D" T H E N G O T O 3000 0225 I F C$i::i y 13="A" T H E N G O T O 9000 0230 PRINT " I L L E G A L C O M M A N D ° 0240 GOTO 0160 0630 INPUT " E N T E R F I L E N A M E " , F $ 064 0 L E T F:<; = F$y " " 0660 C A L L 14yF$y7500vNyE 0750 IF E = 0 T H E N STOP 0760 PRINT " W R I T E E R R O R "5E 0770 STOP 1000 R E M « R U N « 1010 IF M=0 T H E N G O T O 1031 1020 L E T P=l . . . 1025 LET B$C1 y i:i = " P " 1030 INPUT " I N P U T SLQUI( 1 2 ) > F A S T ( 1 3 ) y S Y N T H . < 1 6 ) S C A N • , Z 1 1031 IF BiSCl y :L:i = " N " T H E N G O T O 1020 1032 IF P=0 T H E N G O T O 1060 10 40 GOSUB 2000 1045 C A L L 9y2002yNly()y0y0 1060 IF Z 1 0 1 6 T H E N L E T Gi: i3~l 1070 L E T NC .1.3 ==2002 :l. 075 C A!... i... Z1 •/ 1 y 1 y 2 0 0 2 v 1 y T1 y G C 1 3 y N C 1 3 y C1 ' 1076 INPUT " R U N "yZ85 1080 C A L L Z l y :l. y 1 y 2002 i N1 , T1 y GII 13 •> NC13 ? C1 1090 C A L L 8 y 2 0 0 2 y N l » 0 » 0 1095 PRINT 1100 GOTO 0160 2000 REM * * S W E E P P A R A M E T E R S t% 113 2010 INPUT " F S T A R T , DELTfit F « N O . P T r D E L T A T CI. MSEC / 1 MMSEC) y A / D CH ANNEI... " y F4 yF7>'N:l. y T l y C l 2015 L E T G4 = F4 2020 L E T F2 = F 4 f F 7 * ( N l - - l > 2030 LET' V9 :=INT( (F2--F1 ) / F 9 ) + l 2040 L E T V 2 = ( F C V ? + 1 3 - I - C V 9 3 ) * ( F 2 - F 1 - ( V 9 - 1 ) * F 9 ) / F 9 + F C V 9 . 1 . 2050 L E T V 9 = INT< (F4--F1 ) /F9)+.t 2060 L E T V 1 = ( F C V"9 +11 - F i. 09 ::i ) * ( F 4 - F 1 - ( V9 -1 ) * F 9 ) / F 9 + F C V 9 3 2070 L E T V 3 = V 2 2090 L E T F3 = F2 2100 IF F L V 9 r i . .5>V2 THEN GOTO 2205 2110 L E T V5 = Fi:V9 + 13 2120 L E T F3=09*F9+F1 2205 L E T J9 = l 2206 INPUT " S="y3 2207 L E T V 8 - V 1 2210 F O R 1=1 T O 20 2215 L E T N 2 = I N T < < F 3 ~ F 4 ) / F 7 ) 2217 IF N2=0 THEN GOTO 2450 2220 IF V3<0 THEN L E T U3 = V3t-65536 2240 L E T V7=<U3~01>/N2 2241 L E T V4=V3+V7 2245 L E T N2=N2+1 2250 FOR J = l TO 101 2260 L E T CI::J:I=V3+S 2270 L E T Y 8 = V 8 + V 7 2280 NEXT J 2290 C A L L 1 y C C 1 3 y 1 i J > J 9 2300 L E T J9=J9+J 2310 L E T N2=N2-101 2340 IF N2>0 THEN GOTO 2250 2345 L E T J9 == I N T ( C F 3 - 0 4 ) / F 7 ) f 1 2347 L E T 03=V3 2350 L E T U j . = U3 2360 L E T V9 = V9-f l 2370 L E T V3 = FCv>9 + 13 2330 L E T F4=F3 2390 L E T F3=V9*F9+F1 2400 IF F3<=F2 THEN G O T O 2440 2 410 L E T F3=F2 2420 L E T V3 = U2 2424 L E T Y 9 = V 8 + V 7 + S 2 425 I F ,J9 = N1 THEN C A L L 1 y Y9 y :l. y 1 y J 9 2430 IF ,J9> = N1 THEN GOTO 2450 2440 NEXT I 2450 I F Z l O - 1 6 THEN RETURN 2460 L E T N=4 2470 IF F4>15 THEN L E T N=5 2430 L E T G:l.== v . 2 ; / ( N*S > *.1.000 2490 L E T 62 = F 7 / ( N * 8 ) * 1 E K ) 6 2500 L E T GLJ.:I = INT(G1>K:!.0) 2510 L E T G9=G1-G[: 1 3 / 1 0 2511 IF A B 8 ( G 9 X . 0 0 1 THEN L E T 39 = 0 2512 L E T G!::2:i = I N T < G 9 * 1 0 0 0 0 0 i , 5 ) 2520 L E T Gi::3:i = I N T ( G 2 * 1 0 0 + . 5 ) 2530 PRINT " S Y N T H , F R E Q . = " 5 G :L ?" MHZ DELTA F ~ *5 02 5 » KHZ 2ND HARMONIC= ° 5 G 1 * 2 5" MHZ"5 2550 RETURN 3000 REM « DISK READ OR W R I T E * * 3010 INPUT " WRITECO) OR READ<1) ? ' y X 3 i 3020 INPUT " ENTER F I L E N A M E X N O . P T "yG*yN3 3030 L E T F$ = D*yGity " " 3035 L E T M3=N3 114 3040 INPUT "TO BE WRITTEN OR READ FROM CAL. PTS. (0) rCAL. T B I . . ( J. ) » TO (2 ) y T1 < 3 ) , TPL 0 T ( 4 ) " ? Y 3 3050 LET X3=X3+14 2060 LET E3=0 3070 :VM Y3 + 1 THEN GOTO 3 0 8 0 , 3 1 0 0 , 3 1 2 0 , 3 1 4 0 , 3160 3030 CALL X3rF* f7500»N3»E3 3090 GOTO 3170 3100 CALL X3»F$»1>N3>E3 3110 GOTO 3170 3120 CALL X 3 , F $ » 2 0 0 2 , N 3 , E 3 3130 GOTO 3170 3.1.40 CALL X 3 , F S , 4 0 0 4 , N 3 , E 3 3150 GOTO 3170 3160 CALL X 3 , F $ , 6 0 0 6 , N 3 , E 3 3170 IF X3 = 14 THEN GOTO 3190 3190 IF M 3 0 M 3 THEN PRINT N3 5 • POINTS READ INSTEAD OF " JM3 3190 IF E3=0 THEN GOTO 0160 3200 PRINT "ERROR CODE "5E3 3210 GOTO 0160 6000 REM * * SCOPE PLOT * * 6010 IF P = 0 THEN GOTO 6035 6 020 LET B*t':A,6J=-"P" 6 030 GOSUB 6500 6035 IF B * t 6 t 6 2 - ' H ' THEN GOTO 6020 6036 PRINT 6040 C A L L S , S 6 v N 6 , 0 , L . i : : i : i 6050 IF I...!." :L 3~-0 THEN GOTO 0160 6 060 FOR 1=1 TO 10 6070 I F L C I J ^ O THEN GOTO 0160 6080 PRINT " M i "515" )~" 5 L C I 3 , " F R E Q . = " 5 F1+ < Ll_ I -1-1 ) *FS 6 090 NEXT I 6100 GOTO 0160 6500 INPUT " PLOT CAL .OTS . (0 )>CAL .TBL .<1 )»T0 ( 2 ) »T1 (3 )>TPL0T < 4 > • ,X6> 6 501 L E T Li::i:i=o .6510 INPUT " NO.PT TO BE PLOTTED= " ,N65 6515 LET S6=<X6--1)*2002 6520 IF X6=0 THEN LET 56=7500 6530 IF X6 = l THEN LET S6= l 6540 RETURN 9000 REM * * TRANSFER CONTROL T l A N A L Y S I S PROGRAM * * 9010 SAVE "SWCNTRL" 9020 CHAIN " A N A L Y S I S " 115 0010 RErt ** ANALYSIS ROUTINES FOR -SWCONTROL" ** 0015 DIM C*i:2:i yCC27:iyB$C63yF$C253 F D $ L " : I . 2 3 F G*C 1 2 3 y PC . 1 0 3 0016 LET B$==" NNNNNN " 0020 PRINT 0022 INPUT " DEFAULT DIR. " ,m 0025 PRINT 0030 ON ESC THEN STOP 0032 INPUT " *"tC%i 0033 LET S==l 0034 LET M=0 0 035 LET P~0 0 0 3 7 ON E S C T H E N G O T O 0 0 2 3 0 0 3 8 I F C $ i : 2 y 2 3 = " M " T H E N L E T M 0 0 3 9 I F C $ i : 2 y 2 3== " P " T H E N L E T 1" = 1 0 0 4 0 I F C $ i : 2 '23== " S " T H E N L E T i ==0 0 0 4 5 I F c$i:i y 1 3 = " A " T H E N G O T O 2 0 0 0 0 0 5 0 I F c$i::i y 13== » V " T H E N G O T O 7 0 0 0 0 0 5 2 I F c*t:i y!3== „ j » T H E N euro 7 5 0 0 0 0 5 5 I F C $ E 1 , .1.3== " P " T H E N G O T O 2 5 0 0 0 0 5 8 I F C $ C 1 y 1 3== " X " T H E N G O T O 5 5 0 0 0 0 6 0 I F c$i::i y .1.3== " B " T H E N G O T O 3 0 0 0 0 0 6 5 I F c$i:i y 13== » n 1, T H E N G O T O 3 5 0 0 0 0 7 0 I F C$C:l. y 1 3 == " M " T H E N G O T O 5 0 0 0 0 0 7 5 I F c$ci y 13== " S " T H E N G O T O 4 5 0 0 0 0 8 0 I F c$i:i yJ.3== n J „ T H E N G O T O 6 0 0 0 0 0 8 5 I F C $ E 1 y 1 J == " R " T H E N G O T O 1 0 0 0 0 0 8 7 I F c$i:i y 1 3 == " F " T H E N G O T O 8 0 0 0 0 090 PRINT " ILLEGAL COMMAND" 0160 GOTO 0030 0500 ON N THEN GOTO 0030y OSlOy 0530y 0550y 0570F 0590y 0610 0510 INPUT PC13-yPi::23 0520 RETURN 0530 INPUT !-'C 1 3 y PC23 y PZ33 0540 RETURN 0550 INPUT Pi:i3yP[:23yPE33ypi:43 0560 RETURN . . . 0570 INPUT - '.' 1 3 ? ?£2 J y F\".3» F L 43 > i~ 153 0580 RETURN 0590 INPUT p L 1 3 ? PC23 ? PC33 ? pL 1 3 .» P153 y PC6 3 0600 RETURN 0610 INPUT P C1 3 y P i: 2 :i y P C 3 3 y P C4 3 y P i: 5 3 , P C 6 3 , P C 7 3 0620 RETURN ... 1000 SAVE "ANALYSIS" 1010 PRINT 1020 PRINT 1030 CHAIN "SWCNTRL" THEN GOTO 0160 2000 REM 2010 IF S==l THEN GOTO 2040 . . 2020 INPUT " D < I ) == CI*D ( J ) + C2*D ( K ) ENTER I > J 7 K F N , CI F C2 " , P1.1 3 y PC23 y PL'33 ? PC 4 3 yPC53?PC63 2030 GOTO 2060 2040 LET N==6 2050 G03UB 0500 2060. CALL 22»1yPC13yPC23yPC33yPC43yPC53 2070 GOTO 0030 2500 REM 2505 IF •-:"•:!. THEN GOTO 2800 2510 IF P==0 THEN GOTO 2540 2520 LET B$i:6y6 3==!!P" 2530 GOSUB 2630 . . . . ... . . . ... 2540 IF B$i:6y63=="N" THEN GOTO 2520 2560 CALL 3,36yN6y0yLC13 116 2 5 6 5 IF P = 0 THEN PRINT 2570 IF LCI. .1 = 0 THEN GOTO 0030 2580 FOR 1=1 TO 10 2590 IF LCi:i = 0 THEN GOTO 0030 2600 PR I NT " H ( ' r l r ' ) ~ " y L. C1.1 J " F R E Q , •= " i F .1 f (I. C I 3 -1 ) * F 8 2610 NEXT I 2620 GOTO 0030 2630 INPUT " PLOT CAL . QTS.(0)>CAL.TBL.(1) tTO<2) ,Tl<3)»TPLOT(4) "yX6y 2650 INPUT " NO.PT TO BE PL01'TED = "rN6 2660 LET 36=(X6-1)*2002 2 6 7 0 IF X 6=0 THEN LET S6=7500 2 6 3 0 IF X6=l THEN LET 3 6=1 2635 LET L C i J » 0 2690 RETURN 2800 LET N=2 2805 IF P=0 THEN GOTO 2560 2810 GOSUB 0500 2820 LET X6 = PCi::i 2830 LET N6=PC2:]-2840 GOSUB 2660 2850 GOTO 2560 3000 REM 3010 IF S=l THEN GOTO 3040 3020 INPUT " B(I>=C1MKJ)iC2+C3*IP+C4*IP*IP ENTER I,JyN , C 1 > C 2 > C 3 yC 4 "yPCll y P i: 2 :i y P i:: 3 :.i > P C 41 •> P C 5.1, P c 6 1 , P C 71 3030 GOTO 3060 3040 LET N=7 3050 GOSUB 0500 3060 C A L L 22y2ypci::i ypc2i y 1 ypi::3:j yPL4::i 3070 GOTO 0030 3 5 0 0 REM 3 510 INPUT " WRITE(0) OR READ<1) ? "yX35 3520 INPUT " ENTER FILENAME % N " »G$*.N3 35 30 LET F$=D$yG$y" " 3540 LET M3 = N3 3550 INPUT "TO BE WR OR READ FROM 0,1v2y3>4 "yY3 3560 LET X3=X3il4 3570 LET E3=0 3530 ON Y3+1 THEN GOTO 3590; 3610y 3630y 3650> 3670 3590 CALL X3yF*y7500 ,N3yE3 3600 GOTO 3680 3610 CALL X3yF$ylyN3yE3 3620 GOTO 3630 3630 CALL X3,F$y2002yN3yE3 3640 GOTO 3630 3650 CALL X3yF$y4004yN3yE3 3660 GOTO 3630 3670 CALL X3 y F$ y 6006 y.N3 y E3 . . 3680 IF X3=14 THEN GOTO 3700 3690 IF M30N3 THEN PRINT US', " POINTS READ INSTEAD OF ' 5M3 3700 IF E3=0 THEN GOTO 0030 3710 PRINT "ERROR CODE "5E3 3720 GOTO 0030 4500 REM 4510 IF M=0 THEN GOTO,4550 1520 LET P=:L 4530 LET B$[:5y5:i = " P" 4540 I N U T " 3TRI p CH ART (19) 0R X - Y (18) 0R P01 NT ( 21 ) " y Z2 4550 IF B$C5y5::i = "N" THEN GOTO 4520 4555 IF S=l THEN GOTO 4800 . _ 4560 IF P=0 THEN GOTO 4630 4570 GOSUB 2630 4573 LET T5=l 117 4575 IF Z2 = 21 THEN GOTO 4590 4580 INPUT " TIME IN MSEC. PER POINT ",T5 4590 INPUT " START OF RAMP IN DATA ",S7r 4600 LET PCI 3=0 4610 LET PC23=0 4620 LET PC33=32767/<N6-1 )-.0001 4630 LET PC 4 3 =0 4640 CALL 22y2yS7,S7yS7,N6yPC 13 4630 CALL Z2yS6y37yN6,T5 4690 PRINT 4700 LET P=0 4710 LET M = 0 4720 GOTO 0030 4300 LET N = 4 4820 GOSUB 0500 4822 LET S6=PC13 4824 LET N6 = PC2::i 4826 LET T5=PC33 4S28 LET S7=PC4 3 4830 GOTO 4600 5000 REM 5010 IF S •• 1 THEN GOTO 5040 5020 INPUT ' D(I)=C1*D(J)*DCK) ENTER I •> J y K > N , C1 " y P C 13 y P C23 » P C 33 i PC 43 y PC53 5030 GOTO 5060 5040 LET N=5 5050 GOSUB 0500 5060 CALL 22y4yPC13yPC23yPC33»PC43yPC53 5070 GOTO 0030 5500 REM 5510 IF S = l THEN GOTO 5540 5520 INPUT " FIND MINiMAX FROM I TO I+N ENTER I , H "yPC13»PC23 5530 GOTO 5560 5540 LET N = 2 5550 GOSUB 0500 5560 CALL 22y6yPC 13y1y1tPC23,PC33 5570 PRINT " MAX="5PC33,"MIN="5PC43 5530 PRINT "LOCATION^"5PC535" n?PC63 5590 GOTO 0030 6000 REM 6010 INPUT " ENTER IyNO.PT "y!yN2 6020 LET Il:=INT<N2/2S+.5> 6030 LET I2=N2-I1*25 . 6040 PRINT 6050 FOR J l = l TO I I 6060 CALL 5»CC.13yI+<Ji-l>*25»25 6070 FOR K = l TO 25 6030 PRINT CCK3y 6090 NEXT K 6100 NEXT J l 6110 IF 12=0 THEN GOTO 6200 6120 CALL 5yCC13yfttl*25yI2 6130 FOR K=l TO 12 6140 PRINT CCK3y 6150 NEXT K 6200 PRINT 6210 GOTO 0030 7000 REM 7010 IF 3=1 THEN GOTO 7040 7020 INPUT " D<:n=Cl*DU)/(D<K)-fC2) ENTER I»J, K»N»Ci »C2 0 > PC 1 3 y PC23 > PC33 » P C43yPC53yPC63 7030 GOTO 7060 118 70 40 LET N = 6 7050 GOSUB 0500 7060 CALL 22,3.?:'i:::n»PC23.vPi:33yp£:4:i»pf:53 7070 GOTO 0030 7500 REM 7510 IF S = l THEN GOTO 7540 . .. 7520 INPUT " INTEGRATE FROM I FOR N POINTS ENTER I?N "yPC 13»PC23 7530 GOTO 7560 7540 LET H 2 7550- GOSUB 0500 7560 CALL 22y5yPC 13«PC 13,PC 13tPC23,PC33 7570 PRINT " INTEGRAL FROM "JPC13}" TO " ii PC 13+PC23 j " =•  "5PC33 7520 GOTO 0030 SOOO REM • 3010 INPUT " l'iAT A v I ) ->DATA ( J ) y SM00THED M TIMES ENTER I ? J , N0 . PT > M - , I» J , N 3yHS S020 CALL 20yIyJyN8 8030 LET M3 = M3--1 . . ' . 3040 IF M8 = 0 THEN GOTO 0030 8041 CALL 9yJyN3y0y0y0 3043 FOR 12=1 TO 25 3044 CALL 10 3045 NEXT 12 3050 FOR 11=1 TO MS 3060 CALL. 20yJyJyN8 8062 FOR 12=1 TO 25 3063 CALL 10 8064 NEXT 12 8070 NEXT I I 8030 GOTO 0030 >;< 0010 PRINT "'--CALIBRATION ROUTINE—-" • 0015 DIM F$C20 3 0016 LET E=0 0020 PRINT . 0030 INPUT "USTART X DELTA T IN MSEC. ..•>V»T 0040 LET N=-l 0050 CALL 11y7500yN?y,T 0060 IF N>1 THEN GOTO 0081 0070 PRINT "N<=1 y ERROR" 0080 GOTO 0020 ... 0031 PRINT Ny" P01 N T S T A K E N " t 0630 INPUT "ENTER FILENAME "yF$ 06 40 LET FS=F$y" " 0660 CALL 14yF$y7S00yNyE 0750 IF E=0 THEN STOP 0760 PRINT "WRITE ERROR ",E 0770 STOP 119 BIBLIOGRAPHY Abragam, A., The P r i n c i p l e s of Nuclear Magnetism , Oxford U n i v e r s i t y Press, London (1961) Amstutz, L . I . , Thompson, J.R., and Meyer, H., Phys. Rev. L e t t . 21 ,1175(1968) Bezuglyi and Minyfaev, Sov. Ph y s . - S o l i d State 9_ ,480 (1967) Bostanj og l o , 0. and K l e i n s c h m i d t , R. , J . Chem. Phys. 46_ ,2004(1967) C u r r i e , J . and Van Kranendonk, J . , p r i v a t e communication (1974) F u j i o , M., Hama, J . and Nakamura, T., Prog. Theor. Phys. 5£ ,293(1975) Goldman, V.V., J . Low Temp. Phys. 26. f203 (1977) Hama, J . , Inuzuka, T. and Nakamura, T., Prog. Theor. Phys. 48 ,1769(1972) Hardy, W.N., B e r l i n s k y , A.J. and H a r r i s , A.B., Can. J . Phys. 55_ ,1150 (1977) H a r r i s , A.B., Phys. Rev. B_l ,1881 (1970) H a r r i s , A.B., B e r l i n s k y , A.J and Hardy, W.N., Can. J . Phys. 55 ,1180(1977) H e i t l e r , W., The Quantum Theory of R a d i a t i o n ,2nd. e d i t i o n , Oxford U n i v e r s i t y Press, London (1949) Klemens, P.G., De Bruyn Ouboter, R. and Le P a i r , C., P h y s i c a , 30 ,1863(1964) K r i v o g l a s , M.A., Theory of X-ray and Thermal Neutron S c a t t e r i n g  by Real C r y s t a l s ,Plenum P r e s s , New York (1969) Kroupa, V.F., Frequency S y n t h e s i s ; Theory, Design and A p p l i c a t i o n , Halsted Press, John Wiley and Sons, Inc., Ne& York, Toronto (1973) L u r y i , S. and Van Kranendonk, J . , Can. J . Phys. _57_ ,(1979), i n press Oyarzun, R. and Van Kranendonk, J . , Phys. Rev. L e t t . 26_ ,646 (1971) Oyarzun, R. and Van Kranendonk, J . , Can. J . Phys. 5fJ ,1494 (1972) Raich, J.C. and Kanney, L.B., J . Low Temp. Phys. 28 ,95 (1977) 120 Rose, M.E., Elementary Theory of Angular Momentum ,John Wiley and Sons, Inc., New York, Toronto (1957) S i l v e r a , I.F., Hardy, W.N. and McTague, J.P., Phys. Rev. B4 ,2724 (1971) Van Kranendonk, J . and Walker, M.B., Can. J . Phys. 46_ ,2441(1968) 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            data-media="{[{embed.selectedMedia}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
https://iiif.library.ubc.ca/presentation/dsp.831.1-0094599/manifest

Comment

Related Items