Science, Faculty of
Physics and Astronomy, Department of
DSpace
UBCV
Lipsicas-Lipschitz, Max Menachem Mordechai
Doctor of Philosophy - PhD
University of British Columbia
The work reported here is a proton magnetic resonance investigation of some polyatomic gases. Pulsed nuclear magnetic resonance techniques were used throughout. Little systematic work on gases using this technique has been reported hitherto, and the present study clearly indicates the potentialities of nuclear magnetic resonance experiments for the investigation of certain physical processes occurring in gases. The gases studied were hydrogen (H₂), methane (CH₄), ethane (C₂H₆) and ethylene (C₂H₄). A spectrometer and ancilliary equipment have been constructed enabling measurements to be made at 30 Mc/s, over a temperature range from 35K° to 300K° using gas pressures up to 200 atmospheres. In hydrogen, the spin-lattice relaxation time, T₁, was studied as a function of the density, temperature and ortho-para concentration of the gas. In the "dilute" gas region, T₁ was found to increase linearly with density at fixed temperature, indicating that the relaxation process is of intra-molecular origin and that studies of T₁ are able to yield valuable information on the asymmetric part of the intermolecular interaction. The temperature dependence of T₁ (at fixed density) was studied in the "dilute" gas between 35K° and 300K°, and quantum effects were discovered at low temperatures. In the dense gas, at low temperatures, T₁ was found to be a very rapidly increasing function of density at fixed temperature. In the dilute gas, T₁, at fixed density and temperature, was found to increase with increasing ortho-hydrogen concentration, indicating that the cross-section for a reorientation collision of two orthohydrogen molecules is greater than that for a collision between an ortho-molecule and a paramolecule. Diffusion measurements in hydrogen were carried out, at 78K°, as a function of density in the "dilute" gas, and it has been shown that the coefficient of self-diffusion is inversely proportional to density. The spin-spin relaxation time, T₂, for hydrogen at 78K° has been found to vary linearly with density, in the "dilute" gas, and T₂ = 0.8 T₁. In all the hydrocarbon gases studied, the relaxation process was governed by a single exponential function of time, thus enabling a unique relaxation time T₁ to be defined. In the "dilute" gas, T₁ was found to vary linearly with density. An estimate has been made of the mean square angle of rotation of a methane molecule and of an ethylene molecule per collision, with a view to examining the applicability of a rotational diffusion model for molecular reorientation in these gases. In methane, T₁ has been studied as a function of density just above the critical point for this gas and at room temperature. The effect of traces of oxygen on the relaxation process ("impurity relaxation") in the "dilute" and "dense" gas regions has been studied also. T₁ has been studied in ethane above and below the critical point, and in ethylene above the critical point, in the "dilute" and "dense" gas regions. The oxygen impurity effects have been found to be very marked indeed in the case of ethylene. In the course of this work, a n.m.r. method has also been evolved for the measurement of the compressibility of a gas. A programme of future n.m.r. work on gases, suggested by the results obtained and experience gained in this work, is outlined.
PROTON MAGNETIC RESONANCE MEASUREMENTS IN GASES by ^ MAX MENACHEM MORDECHAI LIPSICAS-LIPSCHITZ Hons, B.Sc., University of London, 1950 Hons. B.Sc.(Eng.), University of London, 1951 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of PHYSICS We accept t h i s thesis as conforming to the standard required from candidates for the degree of DOCTOR OF PHILOSOPHY THE UNIVERSITY OF BRITISH COLUMBIA October, I960 In presenting t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree th a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I f u r t h e r agree that permission f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood t h a t copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n permission. Department of The U n i v e r s i t y of B r i t i s h Columbia, Vancouver $, Canada. PUBLICATIONS 1. K. R. Allen, M . Lipsicas, W. F. Williams and D . Fotheringham. "Stereoscopic Cameras for Use with a Wilson Cloud Chamber". Rev. Sci. Insts., 24, 501, 1953. 2. K . R. Allen and M . Lipsicas. The Cloud Chamber, Research, 6, 416, 1953. 3. K . R. Allen, M . Lipsicas, D . Major and K . Phillips. " O n the Single Scattering of 13.5 MeV Electrons in Xenon." Proc. Phys. Soc, 69, 141, 1956. 4. K . R. Allen, E . A . Finlay, M . Lipsicas, D . Major and K . Phillips. "On the Single Scattering of 9.8 M e V Positrons in Xenon." Proc. Phys. Soc; 70, 355, 1957. 5. M . Lipsicas. "Nuclear Magnetic Resonance." British J. of Comms. and Electronics, 4, 686, 1957. 6. M . Lipsicas and M . Bloom. "Nuclear Spin Relaxation Measure-ments in Hydrogen Gas." Proceedings, 7th International Low Temp. Conference, 1960. Wc\& ^ttmersrig of ^British OJoIimtJhta FACULTY OF GRADUATE STUDIES PROGRAMME OF THE FINAL ORAL EXAMINATION FOR T H E D E G R E E O F DOCTOR OF PHILOSOPHY of MAX LIPSICAS B.Sc. (Hons.) London, 1950 B.Sc. (Hons. Eng.) London, 1951 TUESDAY, SEPTEMBER 27th, I960 AT 2:30 P.M. IN ROOM 301, PHYSICS BUILDING COMMITTEE IN CHARGE G . M . S H R U M , Chairman M . B L O O M E. A L L A N G . M . V O L K O F F F. A. K A E M P F F E R R. B A R R I E W. O P E C H O W S K I J. B. B R O W N J. R. C O O P E J. B. W A R R E N G . B. W A L K E R External Examiner: C . J. G O R T E R Kamerlingh Onnes Laboratorium, Leiden, Holland PROTON MAGNETIC RESONANCE MEASUREMENTS IN GASES A B S T R A C T The work reported here is a proton magnetic resonance investi-gation of some polyatomic gases. Pulsed nuclear magnetic resonance techniques were used throughout. Little systematic work on gases using this technique has been reported hitherto, and the present study clearly indicates the potentialities of nuclear magnetic resonance experiments for the investigation of certain physical processes occur-ring in gases. The gases studied were hydrogen (H.,), methane (CH 4 ) , ethane ( C , H 0 and ethylene ( C „ H 4 ) . A spectrometer and ancillary equip-ment have. been constructed enabling measurements to be made at 30 Mc/s , over a temperature range from 35K° to 300K° using gas pressures up to 200 atmospheres. In hydrogen, the spin-lattice relaxation time, T j , was studied as a function of the density, temperature and ortho-para concentration of the gas. In the "dilute" gas region, T x was found to increase linearly with density at fixed temperature, indicating that the re-laxation process is of intra-molecular origin and that studies of T t are able to yield valuable information on the asymmetric part of the intermolecular interaction. The temperature dependence of T t (at fixed density) was studied in the "dilute" gas between 35K° and 3 0 0 K ° , and quantum effects were discovered at low temperatures. In the dense gas, at low temperatures, Tl was found to be a very rapidly increasing function of density at fixed temperature. In the dilute gas, Tv affixed density and temperature, was found to in-crease with increasing ortho-hydrogen concentration, indicating that the cross-section for a reorientation collision of two orthohydrogen molecules is greater than that for a collision between an ortho-molecule and a paramolecule. Diffusion measurements in hydrogen were carried out, at 7 8 K ° , as a function of density in the "dilute" gas, and it has been shown that the coefficient of self-diffusion is inversely proportional to density. The spin-spin relaxation time, T 2 , for hydrogen at 7 8 K ° has been found to vary linearly with density, in the "dilute" gas, arid T 2 = O. 8 T j . In all the hydrocarbon gases studied, the relaxation process was governed by a single exponential function of time, thus enabling a unique relaxation time T j to be defined. In the "dilute" gas, T , was found to vary linearly with density. A n estimate has been made of the mean square angle of rotation of a methane molecule and of an ethylene molecule per collision with a view to examining the applicability of a rotational diffusion model for molecular re-orientation in these gases. In methane, T1 has been studied as a function of density just above the critical point for this gas and at room temperature. The effect of traces of oxygen on the relaxation process ("impurity relaxation") in the "dilute" and "dense" gas regions has also been studied. T1 has been studied in ethane above and below the critical point, and in ethylene above the critical point, in the "dilute" and "dense" gas regions. The oxygen impurity effects have been found to be very marked indeed in the case of ethylene. In the course of this work a n.m.r. method has also been evolved for the measurement of the compressibility of a gas. A programme of future n.m.r. work on gases, suggested by the results obtained and experience gained in. this work is outlined. V GRADUATE STUDIES Field of Study: Nuclear Magnetic Resonance Nuclear Physics J. B. Warren Quantum Mechanics F. A . Kaempffer Advanced Quantum Mechanics F. A . Kaempffer Quantum Theory of Radiation F. A . Kaempffer Group Theory Method in Quantum Mechanics W. Opechowski Related Studies: Applied Electromagnetic Theory : G . B. Walker Probability S. Nash Quantum Chemistry J. A . R. Coope ABSTRACT The work reported here i s a proton magnetic resonance investigation of some polyatomic gases. Pulsed nuclear mag-netic resonance techniques were used throughout. L i t t l e systematic work on gases using t h i s technique has been reported hitherto, and the present study c l e a r l y indicates the p o t e n t i a l i t i e s of nuclear magnetic resonance experiments for the investigation of c e r t a i n physical processes occur-r i n g i n gases. The gases studied were hydrogen (H2), methane (CH4), ethane (C^Hg) and ethylene (C2H4). A spectrometer and a n c i l l i a r y equipment have been constructed enabling measure-ments to be made at 30 Mc/s, over a temperature range from 35K° to 300K° using gas pressures up to 200 atmospheres. In hydrogen, the s p i n - l a t t i c e relaxation time, T^, was studied as a function o f the density, temperature and ortho-para concentration of the gas. In the " d i l u t e " gas region, T^ was found to increase l i n e a r l y with density at fixed temperature, in d i c a t i n g that the relaxation process i s of intra-molecular o r i g i n and that studies o f Ti are able to y i e l d valuable information on the asymmetric part of the intermolecular i n t e r a c t i o n . The temperature dependence of T^ (at f i x e d density) was studied i n the " d i l u t e " gas between 35K° and 300K°, and quantum e f f e c t s were discovered at low temperatures. In the dense gas, at low temperatures, T^ was found to be a very r a p i d l y increasing function of den-s i t y at f i x e d temperature. In the d i l u t e gas, T j , at fixed density and temperature, was found to increase with increas-ing ortho-hydrogen concentration, i n d i c a t i n g that the cross-section for a reorientation c o l l i s i o n of two orthohydrogen molecules i s greater than that for a c o l l i s i o n between an ortho-molecule and a paramolecule. D i f f u s i o n measurements in hydrogen were c a r r i e d out, at 78K°, as a function of density i n the " d i l u t e " gas, and i t has been shown that the c o e f f i c i e n t of s e l f - d i f f u s i o n i s inversely proportional to density. The spin-spin relaxation time, T£, for hydrogen at 78K° has been found to vary l i n e a r l y with density, i n the " d i l u t e " gas, and T 2 = 0.8 T j . In a l l the hydrocarbon gases studied, the relaxation process was governed by a single exponential function of time, thus enabling a unique relaxation time T^ to be defined. In the " d i l u t e " gas, T-^ was found to vary l i n e a r l y with density. An estimate has been made of the mean square angle of rota-t i o n of a methane molecule and of an ethylene molecule per c o l l i s i o n , with a view to examining the a p p l i c a b i l i t y of a ro t a t i o n a l d i f f u s i o n model for molecular reorientation i n these gases. In methane, T± has been studied as a function of density j u s t above the c r i t i c a l point for t h i s gas and at room temperature. The e f f e c t of traces of oxygen on the relaxation process ("impurity relaxation") i n the " d i l u t e " and "dense" gas regions has been studied also. Tj_ has been studied i n ethane above and below the c r i t i c a l point, and i n ethylene above the c r i t i c a l point, i n the " d i l u t e " and "dense" gas regions. The oxygen impurity e f f e c t s have been found to be very marked indeed i n the case of ethylene. In the course of t h i s work, a n.m.r. method has also been evolved for the measurement of the compressibility of a gas. A programme of future n.m.r. work on gases, suggested by the r e s u l t s obtained and experience gained i n t h i s work, i s outlined. TABLE OF CONTENTS Chapter Page I INTRODUCTION 1 II THEORY 13 2-1 Hydrogen 13 2-2 Methane 20 2-3 Ethane 24 2- 4 Ethylene 25 III PULSED N.M.R. TECHNIQUE AND EXPERIMENTAL APPARATUS 26 3- 1 N.M.R. technique 26 3- 2 Apparatus 31 A. The spectrometer 32 B. The low temperature system 38 C. Temperature measurement 43 D. The high pressure system 46 E. Ortho-para hydrogen converter 50 IV SPIN-LATTICE RELAXATION MEASUREMENTS 55 4- 1 Hydrogen 55 4-1.1 Estimation of accuracy 57 4-1.2 Estimation of ortho-para concentration 57 4-1.3 T^ as a function of density p for normal H£ at fixe d temperature 58 4-1.4 T^/p as a function of temperature i n the d i l u t e gas 63 Chapter 4-1.5 Ti as a function of ortho-para concentration 4-2 Methane 4-2.1 T± i n the d i l u t e gas as a . function of density at room temperature 4-2.2 Tj as a function of density near the c r i t i c a l point 4-3 Ethane 4- 4 Ethylene V MEASUREMENT OF SPIN-SPIN RELAXATION TIME (T 2) AND DIFFUSION CONSTANT (D) IN H 2 5- 1 T 2 i n H 2 at l i q u i d N 2 temperature 5-2 Di f f u s i o n measurement i n Hg 5-3 Density dependence of D at fixed temperature VI SUMMARY, CONCLUSIONS AND SUGGESTIONS FOR FURTHER WORK Appendix A THE PULSED N.M.R. SPECTROMETER - CIRCUIT DETAILS 1. Timing Unit 2. Pulse Generator Unit 3. Radiofrequency Unit B OPERATING INSTRUCTIONS FOR LOW TEMPERATURE SYSTEM C MEASUREMENT OF GAS COMPRESSIBILITY Appendix Page D PROCEDURE FOR MEASURING LONG SPIN-LATTICE RELAXATION TIMES 109 ( i ) P.B. Technique 110 ( i i ) C.R.R. Technique 111 REFERENCES . 112 LIST OF ILLUSTRATIONS Figure Page 1 Proton energy l e v e l s i n a magnetic f i e l d 3 2 Oscilloscope display - two pulse n.m.r. experiment 29 3 Block diagram of n.m.r. spectrometer 34 4 E l e c t r i c a l c i r c u i t of sample c o i l 37 5 Low temperature system 39 6 High pressure gas c i r c u i t 45 7 Thermometer c a l i b r a t i o n 47 8 High pressure bomb 49 9 Parahydrogen gas converter 52 10 Automatic l i q u i d nitrogen f i l l e r 53 11 Tj versus j> in hydrogen at l i q u i d a i r temperature 59 12 Tj versus |) i n hydrogen at 41K° 61 13 Tj versus JO in hydrogen at 37K° 62 14 T l / 0 versus temperature for the " d i l u t e " hydrogen gas 64 15 T l / / p versus orthohydrogen concentration 67 16 T j versus ^ i n methane at room temperature 70 17 Tj versus p i n methane near the c r i t i c a l point 74 18 Ti versus p in ethane at room temperature 76 19 Ti versus 0 in ethane at 41C° 77 Figure Page 20 versus 0 in ethylene at room temperature 78 21 Log A ( 2 T ) versus X 84 22 versus j) i n hydrogen at 78K° ^ 85 23 Log A*(2 X ) versus T 3 91 24 1/k versus ) i n hydrogen at 78K° 93 Appendix 1A Timing unit following 98 2A Pulsed 30 Mc/s radiofrequency power unit 102 3A Phase coherent R.F. power unit following 103 ACKNOWLEDGEMENTS I have benefited from many discussions with Dr. Myer Bloom, who also made numerous suggestions toward the progress of thi s work. The members of the Physics Department, who contributed their s k i l l s and knowledge at various stages of the con-struction and operation of the experimental equipment, are too numerous to be mentioned i n d i v i d u a l l y . I should l i k e to single out, for special mention, Mr. Helmut Zerbst and Mr. Rolf Weissbach of the Low Temperature Group, Mr. William Morrison and Mr. William Meyer of the Workshop and Mr. John Lees, our glassblower. Thanks are due to Dr. Garth Jones for discussions on the electronic design of the spectrometer and to Dr. Burton Muller and Mr. John Noble for assistance i n the taking of data i n the l a t t e r stages of the work. The author i s gratef u l to Dr. G. M. Volkoff for encouraging this work and the National Research Council for generous f i n a n c i a l assistance i n the form of grants i n aid of research. -1-CHAPTER I INTRODUCTION The manner i n which a nuclear spin system comes into equilibrium with the heat bath represented by i t s surroundings i s of considerable i n t e r e s t , since a study of t h i s "spin-l a t t i c e " relaxation process i s capable of providing important information about physical processes which are going on i n the "bath". The nuclear spin system referred to here i s the assembly of a l l the nuclei of a p a r t i c u l a r species, i n the present work protons, contained i n a bulk sample of the material under investigation; the other degrees of freedom of the material constitute the heat bath. We speak in terms of a physical system which i s char-acterized by a Hamiltonian of the form: W = W s p i n + ff bath +# s p i n-bath S p i n takes account of the Zeeman int e r a c t i o n of the s p i n system with external magnetic f i e l d s , and we assume the nuclei i n each atom or molecule behave independently of the other nuclei i n the sample. The energy associated with this interaction i s considered n e g l i g i b l y small as compare with the energy associated with rlj^ath* w h i c h takes account of a l l other degrees of freedom of the material. It i s these l a t t e r degrees of freedom which represent p r a c t i c a l l y the entire energy of the system and are, therefore, responsible -2-for the s p e c i f i c heat of the material; hence the use of the term "bath". There i s a very weak coupling between the spin system and the bath, represented by ll S p i n _ b a t h and i t i s thi s term which i s responsible for the fact that we can probe the bath by means of measurements on the spin system. The sample may be i n any of the three states of matter, gas, l i q u i d , or s o l i d , and the pa r t i c u l a r process which may be studied by means of observations made on the spin system w i l l vary accordingly. In the f i e l d of s o l i d state physics, the importance of nuclear magnetic resonance (n.m.r.) studies i s now well recognized, and important contributions have been made to the understanding of metals, semiconductors, insulators, ferromagnetic and anti-ferromagnetic materials among others. Many experimental studies have been made of nuclear spin relaxation i n l i q u i d s including the important low temperature l i q u i d s , helium and hydrogen. Furthermore, the n.m.r. tech-nique may be used to study the properties of the nuclear system other than those connected with the establishment of thermal equilibrium and so y i e l d further information on the material being studied. N.m.r. techniques have been used, for instance, to study s e l f - d i f f u s i o n i n l i q u i d s . The present work i s a proton magnetic resonance i n v e s t i -gation of polyatomic gases. L i t t l e systematic work on gases has been reported h i t h e r t o ^ ^. This, i n a sense, i s somewhat surprising since the measurements are able to y i e l d much valuable information r e l a t i n g to ce r t a i n types of i n t e r -molecular c o l l i s i o n s and provide data on molecular d i f f u s i o n -3-which i s not e a s i l y accessible by any other means. Before proceeding to discuss more f u l l y the a p p l i c a b i l i t y and potential usefulness of n.m.r. studies i n the gaseous state, a br i e f account w i l l be given of some aspects of the n.m.r. technique e s s e n t i a l to the discussion. We consider a spin system i n thermal equilibrium and situated i n a magnetic f i e l d H n oriented i n the z d i r e c t i o n (H = H ). We r e s t r i c t our attention to spin 1/2 p a r t i c l e s , since the present series of experiments was lim i t e d to proton-containing gases. In the magnetic f i e l d , the protons populate two Zeeman l e v e l s , a and b (Fig. 1), corresponding to protons aligned i n the + and - z directions respectively. £ -a = 2 i i H 0 (1.2) h - V F i g . 1 The r e l a t i v e population densities i n the two le v e l s w i l l be described by a Boltzmann d i s t r i b u t i o n : n. -± = exp (2AjH_/kT) n b r o (1.3) where yu i s the proton magnetic moment, k i s Boltzmann*s constant and T i s the temperature of the sample. Since n a > n b there w i l l be a net magnetic p o l a r i z a t i o n - 4 -F= n a " n b ( 1 ' 4 ) of the spin system in the z d i r e c t i o n , the degree of polar-i z a t i o n being proportional to H Q and inversely proportional to the temperature at high temperatures (yHHc << kT). Correspond-ing to this microscopic magnetic p o l a r i z a t i o n there w i l l be a macroscopic nuclear magnetic p o l a r i z a t i o n M 0 of the whole sample. Now, i n a n.m.r. experiment we f i r s t disturb the system from thermal equilibrium by applying a r . f . magnetic f i e l d which causes t r a n s i t i o n s to occur between the Zeeman level s and so increases the spin population of l e v e l b at the expense of l e v e l a. Correspondingly, there i s a change i n the z com-ponent of the macroscopic nuclear p o l a r i z a t i o n and, further-more, there i s set up, i n general, a macroscopic nuclear magnetization vector M with non-vanishing M , M and M com-x y ^ ponents. We then study the recovery of the system towards equilibrium by examining the manner i n which the magnetization M reverts to the equilibrium condition: M z = M0, Mx = 0, My = 0. C l e a r l y , i n dealing with the approach to equilibrium we must consider both the i n t e r a c t i o n of the spins among them-selves and the int e r a c t i o n of the spin system with the thermal bath. The f i r s t i n t e r a c t i o n does not involve any change i n the t o t a l energy of the spin system and i s monitored by the M„ and M„ ("transverse") components of magnetization. The x y second i n t e r a c t i o n involves an exchange of energy with the bath and i s monitored by the " l o n g i t u d i n a l " component M*z. For -5-gases and l i q u i d s one can usually describe the time v a r i a t i o n of both longitudinal and transverse components of magnetiza-tion i n terms of an exponential rate process, but, i n general, the time constant for the longitudinal component i s not the same as that for the transverse component T 2 owing to the d i f -ferent natures of the mechanisms responsible for the change i n each component. (In the most general case there w i l l be more than one longitudinal time constant and more than one trans-verse time constant.) The recovery of the magnetization M towards the equilibrium magnetization MQ i s then governed by the well known Bloch equation ( 4 ) : dM r M x - M y - M z ~ M o dT + Tg" + J + k ~ T T - = * M x H o ( 1 * 5 ) T^ i s referred to as the " l o n g i t u d i n a l " , "thermal" or "spin-l a t t i c e " relaxation time. T 2 i s the "transverse" or "spin-spin" relaxation time. The solutions to (1.5) may be written: M z = M 0 ( l - e~V/Tl) M = (M y) e T / T 2 U . 6 ) Mx = (M x ) e" T / T 2 x X T=o The nuclear magnetic resonance technique allows one to monitor Mx, My and Mz and hence deduce the values of T^ and T 2. We now seek the mechanism through which Mz recovers to i t s equi-librium value MQ or, i n other words, the physical process which enables the spin system to relax back into thermal equilibrium with the bath. A measurement of T^ then represents a means Of -6-studying t h i s physical process. The spin system was disturbed from thermal equilibrium by the application of an external Larmor frequency magnetic f i e l d . In order to relax, the spin system must lose i t s excess energy through coupling to a s i m i l a r Larmor frequency magnetic f i e l d which, however, now has to be provided i n t e r n a l l y . This i n t e r n a l magnetic f i e l d i n a gas i s activated, so to speak, by the thermal bath and arises from two sources (neglecting quad-rupole e f f e c t s ) : 1) The dipole-dipole i n t e r a c t i o n between spins. 2) The r o t a t i o n a l magnetic moment of the molecule where i t e x i s t s . One must di s t i n g u i s h between two cases: a) A monatomic gas. b) A diatomic or polyatomic gas. In the monatomic gas the molecules are s p h e r i c a l l y sym-metric and the only i n t e r n a l magnetic f i e l d available i s the dipole-dipole f i e l d established between molecules during a c o l l i s i o n . One must then require that fluctuations i n t h i s f i e l d have an appreciable Larmor frequency component. Now, the dipole f i e l d i s of appreciable magnitude only during a very short time i n each c o l l i s i o n , of order 1 0 - * ^ seconds. The 7 Larmor frequency, by comparison, i s of order 1 0 cps. Under these conditions then, the most one could hope for to achieve relaxation i s that the dipole f i e l d be applied as often as possible, i . e . the spin system w i l l relax faster the larger the number of c o l l i s i o n s . On t h i s argument we f i n d then, -7-that: TjoC l/f> (1.7) where f> i s the gas density and where we have assumed a l l c o l -l i s i o n s are two body c o l l i s i o n s ("dilute" gas). A detailed c a l c u l a t i o n shows that t h i s relaxation process i s always a comparatively slow one (Tj i s of order IO 6 seconds for He 3 at N.T.P.) ( 2 ). Thermal relaxation of the spin system i n a diatomic or polyatomic gas, which was one of the main topics of i n v e s t i -gation i n t h i s work, i s effected lar g e l y by an intra-molecular f i e l d i n c o n t r a d i s t i n c t i o n to the inter-molecular f i e l d d i s -cussed above for the monatomic gas. This leads to a much more e f f e c t i v e relaxation process, shorter relaxation times and a t o t a l l y d i f f e r e n t dependence on the gas density. The intra-molecular f i e l d a r i ses from two sources: 1) The magnetic f i e l d H' set up by the r o t a t i o n a l mag-netic moment of the molecule i f the l a t t e r i s i n a J > 0 state. 2) The dipole f i e l d H'' of one proton at the position of the other in each molecule. In the hydrogen molecule, 1) and 2) are approximately equal i n magnitudej as i s known from Rabi's molecular beam ' • " j . "(5) . experiments '; H' = 27 oersted H " = l/2#£^g-j = 34 oersted (1.8) v = gyrbmagnetic r a t i o of the proton r = interproton distance, and i s averaged over the v i b r a t i o n spectrum of the molecule. - 8 -We s h a l l defer to chapter two a more detailed discussion of the int e r n a l f i e l d s i n the gases studied i n t h i s work. The point that should be noted at t h i s stage i s that i n these gases the inte r n a l f i e l d e x i s t s at a l l times but, i n order that i t may be e f f e c t i v e i n the relaxation process, i t must be modulated so as to give an appreciable Larmor frequency component. The modu-l a t i o n i s provided by the inter-molecular c o l l i s i o n s which are able to change the mj state of the molecule (and i n cert a i n cases the J state of the molecule too) and hence change the e f f e c t i v e value of H' and H'' at the position of the relaxing proton. For the case where the c o l l i s i o n frequency J , c i s much large r than the Larmor frequency j , Q (this condition always holds for the gas densities necessary for an observable n.m.r. s i g n a l ) , the component of f i e l d at the Larmor frequency i s increased when the molecule spends a longer time, on the average, i n each mj state. Thus, frequent c o l l i s i o n s hinder the relaxation process and we expect that T i « ' / r c C 1 - 9 * where X i s the "c o r r e l a t i o n time" describing the average time c spent by the molecule i n any mj state before c o l l i s i o n with another molecule causes a t r a n s i t i o n to another mj state. A more detailed t h e o r e t i c a l analysis of the model discussed above, ( 2 ) due to Schwinger v , i s discussed in chapter two and provides the proportionality constant i n (1.9). It i s immediately apparent from (1.9) that the density dependence of T^ for the diatomic gas i s the inverse of that for the monatomic gas (1.7) since T c °C \/0. t c i s a measure -9-of the coupling of the r o t a t i o n a l degrees of freedom of the molecule with the t r a n s l a t i o n a l degrees of freedom and, i n p r i n c i p l e , i t i s possible to calculate T c i n terms of the c o r r e l a t i o n functions of the anisotropic part of the i n t e r -molecular p o t e n t i a l . It i s , of course, only the anisotropic part of the intermolecular potential which i s responsible for reorienting a molecule during a c o l l i s i o n , and a measurement of i s p r a c t i c a l l y the only s e n s i t i v e means available to us of studying t h i s part of the intermolecular i n t e r a c t i o n . (This part of the intermolecular i n t e r a c t i o n plays a n e g l i -g i b l e role in the transport properties of most gases, which can be accounted for s a t i s f a c t o r i l y on a " b i l l i a r d b a l l " model of s p h e r i c a l l y symmetrical c o l l i d i n g molecules. ) For experimental purposes, i t may be more meaningful to write T C e x p l i c i t l y i n terms of a c o l l i s i o n cross-section, CT , where 0* i s the cross-section for those c o l l i s i o n s which reorient the molecule, i . e . for thos c o l l i s i o n s i n which mj changes. We may write: and, hence, from (1.9) oC 0 ffAr (1.11) It should be noted i n introducing the concept of a c o l -l i s i o n cross-section, we have made the i m p l i c i t assumption that we regard a l l the'molecular c o l l i s i o n s as two-body c o l l i s i o n s . This assumption w i l l c l e a r l y not be v a l i d at high densities and i t can be shown that (1.11) represents the f i r s t term of expansion of T^ written as an e x p l i c i t function of the gas - 1 0 -density, p , In analogy to the v i r i a l expansion for a gas: T j = A p + B p 2 + C p 3 + ( 1 . 1 2 ) We are now in a position to examine the expected depend-ence of T j on density, temperature and ortho-para concentra-tion i n hydrogen on the basis of equations ( 1 . 1 1 ) and ( 1 . 1 2 ) . We f i r s t consider the " d i l u t e " gas, where the t h e o r e t i c a l dependence of T j i s f u l l y accounted for by ( 1 . 1 1 ) : 1) At any fixed temperature and ortho-para concentration Tj should vary l i n e a r l y with density, i . e . ^ i / f i s a constant under these conditions. 2 ) At low temperatures CT would be expected to increase because of quantum mechanical d i f f r a c t i o n e f f e c t s when the deBroglie wavelength of the molecules becomes comparable with molecular dimensions. This quantum e f f e c t should become apparent from a study of T i / p a s a function of temperature at low temperature and fi x e d ortho-para concentration. .3) At any fixe d temperature Tj^p w i l l be a function of the ortho-para concentration of the gas sample used, since 0~ i s decreased as the para concentration i s increased. This point w i l l be discussed i n greater d e t a i l i n chapter two. .4) At higher temperatures, where higher J states of both ortho and para hydrogen are excited, CT should vary with tem-perature and one would expect to observe a corresponding e f f e c t on T l / p • -I* might be possible to correlate the values of T l / p i n t h i s range of temperature with the c o r r e l a t i o n time calculations for ortho-hydrogen of N e e d i e r ^ K In the high density region, two basic e f f e c t s may be expected to influence the behaviour of with temperature: i ) The e f f e c t of higher order c o l l i s i o n s . i i ) At low temperatures the gas behaves as a quantum gas with a momentum d i s t r i b u t i o n which d i f f e r s from the Maxwell-Boltzmann d i s t r i b u t i o n . Both these e f f e c t s w i l l appear as higher order terms i n p , i . e . they are represented by the second and higher order terms of (1.12). We conclude from the above discussion that a detailed study of the s p i n - l a t t i c e relaxation time of a gas i s able to provide much valuable information on molecular c o l l i s i o n s , r o t a t i o n a l states and the s t a t i s t i c a l mechanical properties of the gas. Furthermore, i t i s of interest to compare T^ for hydrogen with the values of T^ for several of the much more massive hydrocarbons. Differences are to be expected because of the c l a s s i c a l (high J states) nature of these gases and because of the larger number of spins per molecule and molecu-l a r symmetry considerations. The work described i n the following chapters i s an attempt to study the e f f e c t s i n hydrogen gas enumerated above and to examine the relaxation process i n some " c l a s s i c a l " gases represented by methane, ethane and ethylene. In addi-tion to the T^ measurements, some measurements were made of the phase coherence of the nuclear spins i n hydrogen with a view to determining the s e l f d i f f u s i o n constant of t h i s gas. A more detailed consideration of t h i s problem w i l l be deferred to a l a t e r chapter. In chapter two the theory of s p i n - l a t t i c e -12-relaxation i n gases i s presented in greater d e t a i l . Chapter three discusses the pulsed n.m.r. technique and some of the problems involved i n the design of the experimental equipment for t h i s work. ( F u l l d e t a i l s of the equipment w i l l be found in Appendix A,B.) in chapter four the experimental r e s u l t s for are presented and discussed. Measurements of spin-spin relaxation time and some work on d i f f u s i o n i n H 2 at l i q u i d N 2 temperature are reported i n chapter f i v e . In chapter six, we review the main r e s u l t s and conclusions from t h i s work and present a number of suggestions for further experiments. -13-CHAPTER II THEORY In t h i s chapter the theory of nuclear spin l a t t i c e relaxa-tion i n diatomic and polyatomic gases w i l l be discussed i n some d e t a i l . Before proceeding to the theory i t s e l f , i t w i l l be neces-sary to discuss some of the physical properties of the gases studied i n t h i s work insofar as they influence the relaxation phenomena. The gases i n question are: hydrogen (Hg), methane (CH 4), ethane (C 2Hg) and ethylene (C 2H 4). It i s proposed to consider each of these i n turn. 2-1 Hydrogen (H 2) C r i t i c a l temperature 33.2K°. C r i t i c a l pressure 12.8 Atm. The hydrogen molecule i s composed of two atoms and each ( atom consists of a proton and an electron. H e i t l e r and London v showed that i n the ground state o f t h e molecule the el e c t r o n spins always have a n t i p a r a l l e l o r i e n t a t i o n . On the other hand, both a n t i p a r a l l e l and p a r a l l e l orientation of the nuclear spins can occur i n the molecule and these d i f f e r e n t orientations com-bined with c e r t a i n values of the r o t a t i o n a l quantum number dist i n g u i s h the two hydrogen modifications f parahydrogen and orthohydrogen. The parahydrogen molecules have a n t i p a r a l l e l nuclear spins and even r o t a t i o n a l quantum numbers, while the -14-orthohydrogen molecules possess p a r a l l e l nuclear spins and odd r o t a t i o n a l quantum numbers. Since spontaneous conversion of these two d i f f e r e n t sets of molecules p r a c t i c a l l y does not ( 8 ) (9 ) take place either by radiat i o n or i n c o l l i s i o n processes v , the ortho and para molecules can a c t u a l l y be considered as molecules of two gases d i f f e r i n g from each other i n c e r t a i n o p t i c a l and thermal properties. As w i l l be seen l a t e r , the ortho and para molecules behave rather d i f f e r e n t l y i n proton magnetic resonance experiments too. The thermodynamical equilibrium between ortho- and para-hydrogen i s governed by Boltzmann*s d i s t r i b u t i o n law. Accord-ing to t h i s law the f r a c t i o n of the molecules Nj of the t o t a l number N Q i n the r o t a t i o n a l state J i s given by the equation: where since and g Nj o N Q P j exp ( r E j / k T j P j = 2J + 1 for J even, 3C2J+1) for J odd \j = g n u c l x 1 for para, I = 0, 3 for ortho, 1 = 1 (2.1) (2.2) (2.3) nucl (2.4) where and E t o t a l nuclear spin of molecule j = J(J+l)£^2y = B J(J+1) (2.5) -15-where $ denotes the moment of i n e r t i a of the molecule. As a l l para molecules have even and a l l ortho molecules odd r o t a t i o n a l quantum numbers, the r e l a t i o n of the concentration of the two modifications f , i s given by: n PH 2 J = e v ^ ( ^ l ) e - B J ( J t l ) / k T J - oSg = < 2- 6> J = Q ( £ 3 ( 2 j + l ) e - B J X J + l ) / k T For hydrogen B/k = 84.837K° (2.7) so that at high temperatures B « kT and f = 1/3. This value of P i s p r a c t i c a l l y attained at room temperatures and thus "normal" hydrogen i s composed of 1/4 parahydrogen and 3/4 orthohydrogen. At low temperatures B »• kT p —*-00 , i . e . i n equilibrium there i s only pure parahydrogen. This corresponds to the a n t i c i p a t i o n that as the temperature i s lowered a l l molecules gradually pass into the lowest state having the quantum number 0 which i s a para state. However, i t must be emphasized that, i n general, the equilibrium conditions r e l a t -ing to the orthohydrogen-parahydrogen conditions are not set up because of the absence of intercombination. In other words / A J = 1 the \ A l « 1 t r a n s i t i o n i s a highly forbidden one. In the present work most of the measurements were made on normal hydrogen, but some measurements were made on samples contain-ing approximately 50% ortho and 50% parahydrogen. The l a t t e r samples were obtained by c a t a l y t i c conversion of normal hydrogen at l i q u i d a i r temperature, as w i l l be described i n chapter three. The percent d i s t r i b u t i o n of the molecules i n the d i f f e r e n t J states f o r : -16-1) Equilibrium ortho-para concentration gas 2) 100% para gas 3 ) 100% ortho gas 4) "Normal" gas i s shown i n Table 1 for a number of d i f f e r e n t temperatures. Table 1 Temperature H 2 J O J=l J=2 J=3 J=4 J=5 50 K° 100 Ke 200 K° 300 K° Equilibrium 76. 88 23 .11 0. 01 Para 99. 98 - 0. 02 Ortho r 100 .0 Normal 24. 995 75 .0 0. 005 Equilibrium 37. 36 61 .46 1. 15 0.02 Para 97. 02 - 2. 98 • . _ -Ortho - • 99 .95 - 0.05 Normal 2$. 25 74 .97 0- 74 0.03 Equilibrium 18. 65 71 .67 7. 27 2.38 0, .13 0 .002 Para 71. 82 - 28. 05 0, .13 -:-Ortho -, 96 .78 -. 3.21 - 0 .003 Normal 17. 95 72 .59 7. 01 2.41 0, .03 0 .002 Equilibrium 12. 89 65 .78 11. 78 9.07 0. .40 0 .08 Para 51. 40 47. 0 - 1 1, .59 -Ortho - 87 .80 - . 12.10 - 0 .10 Normal 12. 85 65 .84 11. 75 9.08 1. .40 0 .08 Since the t o t a l nuclear spin of a parahydrogen molecule i s zero, the protons i n parahydrogen molecules w i l l not give r i s e to a resonance s i g n a l . The spin system on which measurements were made i n t h i s work was thus the assembly of protons of the orthohydrogen molecules alone. However, i t w i l l be shown below that the c o l l i s i o n s of parahydrogen molecules with orthohydrogen molecules s t i l l have some e f f e c t on the orthohydrogen spin -17-system, e.g. they w i l l play a role in the establishment of thermal equilibrium of the spin system. We may now develop a physical picture of the nuclear spin l a t t i c e or thermal relaxation process i n hydrogen i n the f o l -lowing terms: For the ortho-para mixtures used i n t h i s work over the range of temperatures studied here the molecules were predominantly i n the lowest r o t a t i o n a l state (J=0 para and J»i ortho) and we assume that a molecule i s unable to change i t s J state by c o l l i s i o n because of the high a c t i v a -tion energy (10B = 848.37k for ortho-molecule and 5B = 424.19k for para-molecule). However, during a c o l l i s i o n the mj state of the molecule may well change. The int e r n a l f i e l d exper-ienced by a proton i n the orthomolecule w i l l be very strongly dependent on the orientation of the proton-proton axis i n the molecule and the r o t a t i o n a l axis of the molecule r e l a t i v e to the d i r e c t i o n of the applied f i e l d H Q. It i s the f l u c t u a -tions i n the i n t e r n a l f i e l d caused by changes i n the mj state of the molecule which provide the Larmor frequency magnetic f i e l d necessary for the relaxation process. Now, a change in the mj state of the molecule during a c o l l i s i o n can only be caused by an anisotropic i n t e r a c t i o n potential between the c o l l i d i n g molecules. The anisotropic part of the interaction i s much more pronounced i n the c o l l i s i o n of two J = 1 molecules (ortho-ortho c o l l i s i o n ) than i n the c o l l i s i o n of a J = 1 molecule with a J = 0 molecule (ortho-para c o l l i s i o n ) , because the J = 0 molecule has spherical symmetry. There are three types of interactions between two hydrogen -18-molecules and, in general, each of these w i l l contribute to the anisotropic part of the i n t e r a c t i o n ^ ^ ) . The interactions are: 1) The valence i n t e r a c t i o n due to the overlap of the electron wave functions of the two molecules. The r a d i a l —1 87v/a dependence of this interaction i s of form e * o ( a 0 = Bohr radius) and i t i s thus a short range int e r a c t i o n i n terms of molecular dimensions. 2) The van der Waals inte r a c t i o n , with a r a d i a l depend-ence of r~®. 3) The quadrupole-quadrupole i n t e r a c t i o n , which only exi s t s i f each molecule i s i n a J = 1 or higher state. This term has r ^ r a d i a l dependence and i s probably the dominant term i n the ortho-ortho int e r a c t i o n , but i s completely absent i n the ortho-para int e r a c t i o n because of the spherical symmetry (J = 0) of the para molecule. (Measurements in s o l i d hydrogen^!) indicate that t h i s i n t e r a c t i o n plays an e s s e n t i a l r o l e i n the anomalous s p e c i f i c heat (A. t r a n s i t i o n ) of s o l i d hydrogen at about 2K°, which i s supposedly due to the co-operative quenching of molecular rotation i n orthohydrogen. The t r a n s i t i o n temperature i s very strongly dependent on ortho concentration, decreasing with decreasing ortho concen-t r a t i o n , and, i n fact, below 60% ortho concentration the t r a n s i t i o n changes i t s nature.) It follows from t h i s discussion that the s p i n - l a t t i c e relaxation time, T^, i n hydrogen gas i s a function of the strength of the inte r n a l f i e l d experienced by the relaxing -19-proton and of the mean l i f e t i m e of the molecule i n an mj state, also referred to as the "c o r r e l a t i o n time". The calcu-l a t i o n of T j for hydrogen gas was f i r s t performed by Schwinger ( 2 ) . The term ^ s p i n - b a t h i n C 1- 1) i s i n two parts: 1) One considers the inte r a c t i o n of the nuclear spin with the r o t a t i o n a l moment of the molecule, using a Hamiltonian of form: = H ' l j (2.8) where T — T j + T g = vector sum of the nuclear spins 7 = ro t a t i o n a l moment of the molecule H 1 = magnetic f i e l d at the position of the proton due to the molecular rotation = 27 o e r s t e d ^ ) . 2) One then adds the contribution of the spin-spin i n t e r -action (dipole-dipole interaction) using a Hamiltonian of form: H" = - —-j- [ 3 ( T r I T ) ( T 2.n) - T j . T ^ (2.9) where n i s the unit vector pointing from one proton to the other and r i s the distance between them. Schwinger finds : Tj" = 2 T c X 2 [| H' 2J(J +1) +3H"2 ( 2 j - l K i j + 3 ) ] = 2.74 x 1 0 1 2 T c (2.10) where H'' = x ^ = 34 o e r s t e d ^ ^ i s the e f f e c t i v e f i e l d due to one proton at the position of the other. The Schwinger n-1 1 theory assumes that T c -— which condition, as noted i n yo the previous chapter i s always well f u l f i l l e d experimentally. The Schwinger theory i s only concerned with the i n t r a --20-molecular interactions and makes no attempt to calculate t c, which i s a function of the inter-molecular interactions. A more general theory, i n which both i n t r a - and inter-molecular interactions are taken into account, has been developed by (12 ) Bloom and Oppenheim '. This theory e f f e c t i v e l y deduces the l i f e t i m e of the r o t a t i o n a l states of the molecule ( T c or, more generally a number of f _'s) from considerations of the pair d i s t r i b u t i o n function of the molecules and the i n t e r -molecular p o t e n t i a l . Hence, i t i s able to predict exact values for T-^ and to take account of the temperature depend-ence of T^ through the temperature dependence of the molecular pair d i s t r i b u t i o n function. Numerical values based on t h i s theory are not yet available. 2-2 Methane (CH 4) C r i t i c a l temperature 191.0K°. C r i t i c a l pressure 45.8 Atm. The methane molecule i s a spherical top molecule ( i . e . i t has three equivalent axes of symmetry) and i s composed of one atom of carbon and four, atoms of hydrogen arranged in a tetrahedral array about the carbon atom. There are thus four protons per molecule and there i s a correspondingly larger intra-molecular dipole-dipole i n t e r a c t i o n i n methane than i n hydrogen. The interaction of the nuclear spin with the r o t a t i o n a l moment of the molecule i s much weaker than in hydrogen, however, because of the considerably smaller rota-t i o n a l moment of the heavier molecule. Consequently, there i s some s i m p l i f i c a t i o n i n the c a l c u l a t i o n of TT for methane -21-because i t i s possible to neglect the I.J term, but th i s i s more than made up for in the added complications which ari s e because of the large number of dipole-dipole interactions per molecule and because of the large number of r o t a t i o n a l states which the molecule may occupy. The large number of J states available to the molecule implies that there i s almost a con-tinuum of mj states and consequently each molecular c o l l i s i o n r e s u l t s i n a molecular reorientation, which, however, may correspond to mj = 0,1,2, The Schwinger theory cannot be applied to t h i s case and a d i f f e r e n t approach due to Hubbard^ 1 3), which makes use of R e d f i e l d ' s ^ 1 4 ^ semiclassical theory of relaxation and F u r r y ' s ^ * ^ r o t a t i o n a l d i f f u s i o n equation, has been developed. In th i s approach, one describes the molecular reorientation i n terms of a large number of small "random" rotations given by a r o t a t i o n a l d i f f u s i o n equation. The random rotations here correspond to random changes of mj i n successive c o l l i s i o n s . The Redfield density matrix formulation enables one to take account of the degener-acy of the spin energy l e v e l s for a four spin 1/2 system (I = 2). An important point to consider i n the case of methane i s that there are three types of modification of the molecule: meta (I = 2), ortho (I = 1J and para (I =0), with s t a t i s t i -c a l weights 5:9:2, and both the meta and para give r i s e to a n.m.r. s i g n a l . Since the r o t a t i o n a l states allowed for each modification are determined by symmetry considerations and since t r a n s i t i o n s between d i f f e r e n t modifications are -22-highly forbidden, we might expect, i n general, to observe a relaxation process described by two d i f f e r e n t time constants a r i s i n g from the ortho and meta molecules respectively. However, on closer examination of the allowed J states for meta and ortho-me thane, we find(*®) Meta: J = 0 , 3 , 4 . . . . (2 .11) Ortho: J = 1 , 2 , 3 , 4 , 5 , . . . and the temperatures of our experiments are s u f f i c i e n t l y high for the ortho and meta modifications to share the same r o t a -t i o n a l states. (At room temperature, the value of J obtained by equating the r o t a t i o n a l energy of the J state, J(J+1) $ 2 / 2 I Q , to kT i s approximately J = 6 .) Thus, although there w i l l be two time constants, the difference between them w i l l be too small to resolve. Similar considerations w i l l apply to other polyatomic gases. Hubbard's analysis of the spin l a t t i c e relaxation process i n methane shows that, even apart from consideration of the modifications of the molecule, one would expect more than one time constant for a four spin 1/2 molecular system. Hubbard finds that the approach of the z component of magnetization to equilibrium may be written i n the form: 3 M M ^ 92-13 J46 M z " M e q ^ T§3 e x P 92+13 >f46 + —3 exp ~i~(73+2 /46) "|^(73-2 746) i f " (2 .12) r 6 where T D = °^ ^ lV*c (2 .13) -23-r Q = proton-proton separation = 1.78 x 10~° cm and T _ = 1/6 D (2.14) c where D i s the r o t a t i o n a l d i f f u s i o n constant. The two exponential decays i n (2.12) have r e l a t i v e amplitudes of 56:1 and time constants which d i f f e r by only about 25%. Differences of t h i s order are not measurable experimentally, mainly because of signal/noise considerations. From (2.12) and (2.13) we see that we may write: T l - 7 . 8 x'lQlHc ( 2 ' 1 5 ) Equation (2.15) i s formally very s i m i l a r to (2.10 for hydro-gen. However, the c o r r e l a t i o n time T c i s defined rather d i f f e r e n t l y i n the two cases. In Hubbard's theory, one considers a r o t a t i o n a l random walk and i f one then postulates that the mean time between c o l l i s i o n s , X , as given D y the transport properties of the gas, i s also the mean time between successive rotations i n the r o t a t i o n a l random walk, one finds, using (2.14), that: T / ~ = <Tc^ 2> (2.16) c o where < > i s the mean square angle of rot a t i o n per c o l l i s i o n . This interpretation i s based on the assumption that T T / t « 1, i.e.o^ should be very small, which i s c inherent i n a simple random walk formulation. It i s possible to extend the theory to the opposite case of random rot a t i o n by very large steps, i . e . T" / T „ ^> 1. c In t h i s case, the molecule loses a l l memory of i t s orientation -24-after each c o l l i s i o n and i t may be shown that Hubbard's equation (2.15) i s s t i l l v a l i d provided one makes f = t . c The experimental values of T j enable one to make an estimate of T c on the basis of (2.15). The value of T can be calculated on the basis of a hard sphere model for c o l l i d -ing molecules^1^^: i = 4 d 2 ( ^ ) ' l 7 x ! 0 8 A T 1 / 2 ( 2 . 1 7 ) where d = diameter of CH^ molecule = 4 x 10 cm m = mass of CH4 molecule = 2.62 x I O - 2 3 gm A = density i n Amagats. One may then evaluate i n (2.16) and check which of our assumptions as regards T c f i t s the experimental s i t u a t i o n . If one extended the Hubbard theory to other molecules, i t may turn out that for some molecules the Furry r o t a t i o n a l walk model i s a good approximation, while for others random rotation by very large steps i s more plausible. In the former case, one may also obtain a value for the r o t a t i o n a l d i f f u s i o n constant D from (2.14). 2-3 Ethane (C 2Hg) C r i t i c a l temperature 305.4K°. C r i t i c a l pressure 48.2 Atm. Ethane i s an asymmetric top molecule with 6 protons per molecule. The theore t i c a l evaluation of T^ i n terms of T c has not been c a r r i e d out, but i t should be possible to extend the Hubbard theory to t h i s molecule, taking account of the d i f f e r e n t molecular configuration as compared with the methane -25-molecule. 2-4 Ethylene (C2^4 ) C r i t i c a l temperature 282.7K°. C r i t i c a l pressure 50.6 Atm. Ethylene i s a symmetric top molecule with four protons per molecule. Here again there i s no th e o r e t i c a l c a l c u l a t i o n as yet for T. i n terms of X , though the Hubbard theory i c could be extended to t h i s geometry. -26-CHAPTER III PULSED NUCLEAR MAGNETIC RESONANCE TECHNIQUE AND EXPERIMENTAL APPARATUS 3-1 N.M.R. Technique The method used to study the magnetization of the nuclear spin system i n t h i s work was the pulsed n.m.r. "spin echo" technique of Hahn^ 1®^. Since t h i s i s a now well established technique, i t i s only intended to give here a b r i e f account of the theory of pulsed n.m.r., the basic ideas for which were given by Bloch^®). We consider a nuclear spin system i n a magnetic f i e l d H Z = H o a n < * i n thermal equilibrium with a "bath". As d i s -cussed i n chapter 1, there w i l l be a corresponding macro-scopic moment, which can be represented by a vector M = MQ i n the z d i r e c t i o n . We now examine the behaviour of the system when a radiofrequency magnetic f i e l d at the Larmor frequency, ^ Q , i s applied at ri g h t angles to the d i r e c t i o n of the fixed f i e l d H Q, i . e . H x = 2Ht c o s 4 S 0 t , H y = 0 H z = H Q ^ Q = 2 TT \ Q (3.1) In practice, the sample i s surrounded by a c o i l having i t s axis i n the x d i r e c t i o n and energized by a voltage pulse from a Larmor frequency o s c i l l a t o r . The plane polarized Larmor frequency magnetic f i e l d represented by H x = 2H^ * c o s ^ Q t i s equivalent to the sum of two f i e l d s rotating -27-around the z d i r e c t i o n , namely: (1) H = Hi c o s f i ) n t H„ = H-, s i n ^ _ t x x o y x o ( 3 2 ) (2) H X = H j c o s ^ 0 t H Y = -Hj s i n ^ Q t It i s convenient to go over to a frame of reference which i s rotating at the Larmor frequency. Since the gyromagnetic r a t i o of the proton i s po s i t i v e , f i e l d (1) w i l l be synchro-nous with the gyromagnetic precession of the protons about the fi x e d f i e l d H Q. Thus, on our macroscopic picture, f i e l d (1) w i l l exert a constant torque on the nuclei of magnitude T = M x H j (3.3 ) F i e l d ( 2 ) , being far from synchronous with the gyromagnetic precession of the protons, w i l l have very l i t t l e e f f e c t and may be n e g l e c t e d ^ ^ . Now, the torque represented by equation (3 .3 ) w i l l lead to the macroscopic magnetization M rot a t i n g about the H Q d i r e c t i o n , which, in the rotating frame of reference, may be represented by the vector M being tipped away from the z axis. The angle 6 between the vector M and the z axis i s a function of the energy supplied by the r . f f i e l d , i . e . i t w i l l depend upon the magnitude of H j and the length of time i t i s applied, and i s given by: XHI t, = 6 where 6 i s i n radians (3 .4 ) and t w i s the duration of the r . f . f i e l d pulse 0 = IT/2 corresponds to the nuclear magnetization being wholly i n the transverse (x-y) plane, which, on the energy l e v e l picture of chapter 1, i s equivalent to equalizing the -28-l e v e l populations. The component of magnetization i n the transverse plane w i l l induce a voltage i n the c o i l surrounding the sample, and the magnitude of t h i s voltage w i l l be a maximum for a Tf/2 ro t a t i o n . The induction signal observed on an oscilloscope af t e r amplification and detection has a c h a r a c t e r i s t i c t a i l -l i k e appearance (Fig. 2). The t a i l indicates that immediately afte r the TT/2 rotation the transverse magnetization dies away because of the dephasing action of the magnetic f i e l d inhomogeneity and of the spin-spin i n t e r a c t i o n . However, the i n i t i a l height of the TT/2 t a i l (assuming fast amplifier response) i s a d i r e c t measure of the z component of magneti-zation immediately prior to the 7T/2 rotation pulse. Now, i n the measurement of the s p i n - l a t t i c e relaxation time, we are concerned with a measurement of the regrowth of the z component of magnetization after disturbing the system from thermal equilibrium, e.g. by applying a TT/2 pulse. Thus, a simple method of measuring T^ i s to apply two TT/2 pulses separated by a known time T , the f i r s t pulse disturbing the system from equilibrium and the second monitoring the regrowth of the z component of magnetization. By repeating the pulse sequence of two TT/2 pulses at a r e p e t i t i o n rate which allows the system to recover to equilibrium before each sequence but varying the separation T between pulses, one may deduce the value of the s p i n - l a t t i c e relaxation time T^ from the height A ( T ) of the second t a i l , which w i l l obey -29-the law: ( 3 . 5 ) F i g . 2 Figure 2 i s a drawing of an oscilloscope display showing the amplitude of the detected r . f . signal as a function of time for the two pulse n.m.r. experiment. Following the pulses at times 0 and V , indicated by shaded regions, are the induction signals ( " t a i l s " ) . The echo signal at 2T i s discussed below. The value of i s obtained from the height of A ( T ) at X = oo , i . e . i t i s the height of the second t a i l when the f i r s t pulse i s not applied. For the measurement of T 2 a 71/2, 7T pulse t r a i n i s used. The 7T pulse i n t h i s case ' acts as a "rephasing" pulse which cancels out the dephasing of the ind i v i d u a l nuclear spin precessions r e l a t i v e to each other caused by the inhomogeneity of the magnetic f i e l d . (We assume for the moment that there i s no t r a n s l a t i o n a l motion of the spins i n the magnetic f i e l d , i . e . we are neglecting d i f f u s i o n e f f e c t s . ) The 71 pulse does not have any e f f e c t on the dephasing action of the spin-spin i n t e r -action, however, where the dephasing i s completely random. The TI pulse, as shown by Carr and P u r c e l l ^ ^ gives r i s e to an echo signal at time 2 T after the 7T/2 pulse, provided -30-the time T between the 1^/2 and TT pulses i s short compared with T j and T®. The echo height i s a function of the spin-spin relaxation time: A ( 2 T ) = A Q e " 2 T / T 2 (3.6) Equation (3.6) has to be modified i f d i f f u s i o n e f f e c t s are to be taken into account. These e f f e c t s w i l l depend on the v a r i a t i o n of the "f i x e d " magnetic f i e l d H Q over the sample. Experimentally, one i s usually interested i n the case of a homogeneous f i e l d H Q with a uniform f i e l d gradient dH G superposed i n one d i r e c t i o n (e.g. 2. = G), and the motion dy of the nuclei through G i s described by the tr a n s l a t i o n a l d i f f u s i o n equation. The height of the echo i s then given by: A(2T)!= A Q exp (-2T/T - 2/3# 2G 2DT 3) (3.7) 2 where D i s the t r a n s l a t i o n a l d i f f u s i o n c o e f f i c i e n t . In practice the uniform gradient G i s either superposed by a suitable array of current carrying c o i l s or by shimming the magnet gap. From equation (3.7), i t follows that measure-ments of the echo decay allow one, i n p r i n c i p l e , to measure the d i f f u s i o n constant, D, (in the present case, the s e l f d i f f u s i o n constant of a gas) provided cer t a i n other constants, including the spin-spin relaxation time, are known. In chapter 5 the application of thi s technique to the measure-rs) To be more exact, any two pulse sequence i n which the f i r s t pulse i s not a TT pulse w i l l give r i s e to an echo, but the echo amplitude i s a maximum for the case described here. -31-ment of the s e l f - d i f f u s i o n constant of hydrogen w i l l be d i s -cussed i n greater d e t a i l . It i s important to note that i n these measurements the spin system i s used to " l a b e l " the (21 \ molecules. As remarked by Carr and P u r c e l l v ; - "To the extent that the 'lab e l ' has a ne g l i g i b l e e f f e c t on the d i f f u s i o n process i t s e l f , a method may be said to measure t r u l y the 'self d i f f u s i o n ' constant. In the method here described a molecule i s i n e f f e c t l a b e l l e d by the d i r e c t i o n of the nuclear magnetic moment i t c a r r i e s ; a more innocuous l a b e l would be d i f f i c u l t to imagine". 3-2 Apparatus The basic equipment required for t h i s work comprised: a) A pulsed nuclear magnetic resonance spectrometer. b) A low temperature system capable of maintaining the gas sample at any temperature between 35°K and room temperature. c) A high pressure system, including the sample "bomb" i t s e l f , designed for operation up to maximum pressures of order 200 atmospheres. Each of these three major items of equipment was s p e c i a l l y constructed for t h i s work, but, where possible, considerable f l e x i b i l i t y was allowed for i n the design to make the equipment suitable for other nuclear magnetic resonance work i n the future. The factors influencing the design of each unit w i l l now be discussed and a general description w i l l be given of the operation of the equipment. -32-Further d e t a i l s , including technical data and detailed c i r -c u i t analysis w i l l be relegated to the appendix. A. The Spectrometer Various aspects of pulsed n.m.r. spectrometers have (18 1 (22 2*3 1 already been discussed by Hahnv ; and others v ' ;. The spectrometer used here was designed to the following rough s p e c i f i c a t i o n : 1) Operating frequency approximately 30Mc/s. 2) Measurement of relaxation times i n the range lOOu sees to several seconds. 3) P o s s i b i l i t y of detecting weak signals by pulsed r . f . phase sens i t i v e detection methods. 4) Pre-set pulse sequence timing accurate to within a few micro-seconds. (2 5) Multiple sequences, including a Carr-Purcell t r a i n v " 6) Continuously variable pulse length Lu sec-lmsec. These factors w i l l now be amplified i n greater d e t a i l and the extent and manner to which they were incorporated i n the f i n a l design of the spectrometer w i l l be underlined. The magnet made available for t h i s work was the 7000 (24 ) gauss permanent magnet already described by Waterman The frequency of the spectrometer had to be matched to t h i s magnetic f i e l d , leading to the choice of a frequency of approximately 30Mc/s. Phase sens i t i v e detection e n t a i l s the use of a continuously running o s c i l l a t o r so that the design was o r i g i n a l l y based on a 15.085 Mc/s continuously -33-running c r y s t a l o s c i l l a t o r feeding pulsed amplifiers and a f i n a l frequency doubling push-push power amplifier. How-ever, d i f f i c u l t i e s were encountered due to stray leakage of 30Mc/s power during the o f f pulse periods. Since the signal strength i n the present set of experiments was expected to be quite high, i t was decided to temporarily abandon the idea of phase se n s i t i v e detection and to replace the c r y s t a l o s c i l l a t o r with a pulsed L-C o s c i l l a t o r . F u l l d e t a i l s of a continuously running R.F. power system with f a c i l i t i e s for (24) both phase sensi t i v e detection and Meiboom-Gill type pulse phasing are given in F i g . 3 A. Requirements (2), (4), (5) and (6) above were met by the development of a special timing unit capable of t r i g g e r -ing a pulsing unit comprising Textronix type 163 Pulse Generators. In t h i s way the timing sequence and duration of the R.F. pulses are controlled by the timing unit and pulsing unit respectively. A block diagram of the spectrometer i s given i n F i g . 3. The timing unit i s designed to produce tr i g g e r i n g pulses of a few v o l t s amplitude i n a predetermined time sequence. It consists of a Tektronix type RM181 Time Mark generator feed-ing voltage pulses or pips at one of f i v e f i x e d frequencies (IMc/s, lOOkc/s, lOOc/s con t r o l l e d by a f i v e position switch) into a pulse sequence sub-unit. The sub-unit con-s i s t s of a scale of 1000 Magnetron Beam Switching tube (MBS tube) counting c i r c u i t with a double channel output, giving a p o s i t i v e output voltage of several v o l t s which i s Pulsed 30 Mc/s power unit "Bomb" Sample Chamber LEL amplifier Jj$ Sample jl c o i l Pulsing unit Oscilloscope Timing unit Trigger Fig. 3. Block diagram of n.m.r. spectrometer -35-capable of trigg e r i n g a Tektronix type 163 Pulse Generator. The counting c i r c u i t i s controlled by a multivibrator gate which makes the c i r c u i t inoperative for a predetermined length of time after each count of 1000 uni t s . This gate thus controls the r e p e t i t i o n rate of the pulse sequence which can be varied continuously be adjustment of the gate duration. During each counting cycle the f i r s t output channel i s open only during the i n t e r v a l extending over the f i r s t ten counts i n the cycle and pulses corresponding to any three or less of these counts can be brought out through t h i s channel. These pulses, to be c a l l e d A pulses, may then be applied to trigger a type 163 Textronix pulse generator. The second output channel i s opened by a t r i p l e coincidence c i r c u i t , so that any count between 1 and 999, to be c a l l e d a B pulse, can be brought out through th i s channel and made to trigger a second 163 pulse generator. In the most common application, one A pulse, normally No. 5, i s brought out through channel 1 and one B pulse, at a predetermined time corresponding to the setting of the coincidence c i r c u i t , i s brought out through channel 2. Each of these pulses gives r i s e to a to v o l t p o s i t i v e going pulse from i t s corresponding 163 pulse generator which i n turn triggers the pulsed R.F. o s c i l l a t o r giving r i s e to a peak to peak 30 Mc/s R.F. voltage of about 400 vo l t s across the sample c o i l . The length of the R.F. pulse may be v a i r i e d by means of a potentiometer control on the pulse generator and thus the rotation angle of the nuclear magnetization cor--36-responding to the A and B pulses may be varied independently. Provision i s also made for more complex pulse sequences -_e.g. the coincidence c i r c u i t of channel 2 can be arranged i n double coincidence so as to give a t r a i n of 9 B pulses equally spaced i n time and of equal width. The e l e c t r i c a l c i r c u i t of the sample c o i l ±^ shown i n F i g . 4. The half wave l i n e i s a length of RG59U cable approximately 3 meters in length. The length of the half wave l i n e i s adjusted to make the value of C, when maximum signal i s observed on the oscilloscopej approximately eqUal to the value of capaci-tance required to tune the sample c o i l L at 30Mc/s when the Q of the l a t t e r i s measured on a Marconi Q meter. Cj and C2 are small capacities, 22pF and lOpF respectively, de-coupling the R.F. power and receiver c i r c u i t s from each 6 other. The LEL amplifier has a maximum gain of order 10 and a band-width of approximately 2Mc/s. The f i n a l stage of the amplifier contains a 1N295 diode detector. Pro-v i s i o n i s also made, however, for a d i r e c t R.F. output when required. The oscilloscope trigger may be taken either from the timing unit or from the A or B pulse generators as desired. The nuclear induction signals normally observed on the Tektronix 545 Oscilloscope are of order 1 v o l t amplitude. The signal/noise r a t i o was improved considerably, (by a factor of order 5) with a s l i g h t loss of signal amplitude 9 by a simple R.C. f i l t e r at the input to the oscilloscope. This improvement was p a r t i c u l a r l y useful for measurements -37-->• To LEL Amplifier Cs l i n e L Sample C o i l Fig.: 4. E l e c t r i c a l C i r c u i t of Sample C o i l -38-on the echo si g n a l , since the l a t t e r i s usually weaker than the t a i l and yet the frequency bandwidth required for the echo i s much less than that for the t a i l . B. The Low Temperature System The basic problem i n designing t h i s system was to f i n d the most e f f e c t i v e way of maintaining a high pressure gas sample, contained i n a thick-walled copper bomb of 1-inch outer diameter, at any temperature between 30°K and room temperature. Ideally one would l i k e to maintain the gas temperature constant to within ± 0.25°K for periods of at least 20 minutes. The degree of control actually achieved with the present system was considerably i n f e r i o r to t h i s ; The magnet gap length of 2 inches imposes a severe r e s t r i c t i o n on f l e x i b i l i t y i n designing the low temperature system. A "blow cryostat" system which w i l l be described below was f i n a l l y decided upon, but experience gained i n th i s work indicated that a variable heat leak system would be far superior. However, some very t r i c k y design problems would have to be solved before the l a t t e r system could be adapted to t h i s application. A schematic diagram of the low temperature system i s shown i n F i g . 5. The system i s designed to maintain the temperature of the gas i n the "bomb" at any desired tem-perature between35°K and room temperature. For temperatures between 35°K and 55°K helium gas i s used as the coolant, although, i d e a l l y , hydrogen would be preferable. The -40-- * l i q u i d helium'dewar vessel A i s surrounded by a l i q u i d nitrogen f i l l e d dewar B. Dewar A i s precooled to l i q u i d N 2 temperature'by the thermal" conduction of the residual a i r (approximately 1 m/m Hg pressure) i n i t s vacuum jacket. When l i q u i d helium i s transferred into A, the residual aid i n the jacket i s frozen out and a perfect vacuum main-tained. The gas sample "bomb" i s enclosed by a double dewar system the inner dewar C i s evacuated by a rotary pump and the outer dewar D i s f i l l e d with l i q u i d Ng. The vacuum jacket of C i s of the permanently evacuated variety. The copper bomb i s jacketed by a concentric brass cylinder F through which cold helira gas i s made to flow. Liquid helium at 4.2°K contained i n dewar vessel A vapourises and flows through the vacuum jacketed transfer pipe a into a bomb jacket F. After cooling the bomb the gas flows v i a the rad i a t i o n s h i e l d R out to the return He l i n e and the central helium recovery tank. The flow of helium gas i s cont r o l l e d by the valves (1) and (2) i n the chimney by-pass; (1) and (2) provide coarse and fin e flow controls respectively. To provide for a larger gas flow than i s attainable by the natural vapourization of the l i q u i d helium i n A additional " b o i l o f f " i s provided, when required, by means of a small heater (a 1 i i . r e s i s t o r ) immersed i n the l i q u i d helium and energized by a Variac and 6V filament transformer. The flow rate of the helium gas i s monitored by an o i l manometer attached to an o r i f i c e place flowmeter P inserted i n the return l i n e . Valve (3) i s provided to -41-f a c i l i t a t e flushing of the helium flow system with cylinder helium and to allow pressure measurements on the flow system to be made with the mercury manometer Mg. It was found i n practice that i t was not possible to maintain a constant flow of helium gas i n the flow system i f the inner tube of the transfer tube "a" was i n contact with l i q u i d helium. This i s a consequence of the pressure i n s t a b i l i t y generated by liquud helium b o i l i n g within the transfer tube. Numerous unsuccessful attempts were made to improve the s t a b i l i t y under these conditions by careful shaping of the i n l e t to t h i s tube and by introducing one or more holes i n the tube i n an attempt to provide some pressure r e l i e f . When the inner tube of "a" was just above the l i q u i d helium l e v e l i n A, the gas flow, even with the maximum amount of additional " b o i l o f f " provided by R, was only s u f f i c i e n t to maintain the bomb temperature at about 4o°K. (The " b o i l o f f " i s li m i t e d by the excess pres-sure allowable i n "A", i . e . approximately 10 cm.of Hg.) The experimental procedure for measurements i n the tempera-ture range 35°K to 50°K was therefore modified to the f o l -lowing: A wide bore tube, extending to within 1 inch of the bottom dewar A, was attached to the inner tube of "a". The bomb was precooled to near l i q u i d nitrogen temperature (78°K) by forcing l i q u i d Ng through the cooling c i r c u i t , using dewar A as a l i q u i d Ng storage dewar. Dewar A was then removed and a l l l i q u i d a i r removed from the cooling system. Dewars A and B were then mounted i n position and -42-the cooling c i r c u i t was pumped out and flushed with helium gas i n preparation for a l i q u i d helium transfer operation. Dewar A was then f i l l e d with l i q u i d helium using the con-ventional procedure for transferring l i q u i d helium into a storage dewar. The bomb was cooled down rapidl y to approximately 30°K by closing (1) and (2) completely, f u l l y c l o s i n g (4) and allowing maximum helium gas flow through the cooling c i r c u i t . (1) and (2) were then opened f u l l y and (4) closed completely so as to prevent any gas flow through the cooling c i r c u i t . N.m.r. measurements were made during the subsequent warm up cycle of the bomb, which had a warm up rate of approximately 0.7°K per minute. Some improvements to the cooling system could s t i l l be made, even i f the above procedure i s to be continued i n future. The r a d i a t i o n s h i e l d R i s not e f f e c t i v e under these condi-tions and i t would be advantageous to incorporate a l i q u i d N 2 container i n the dewar cap so as to provide a surface at around 80°K at the top of the dewar. Bomb temperatures i n the range 55°K to 78°K were obtained by f i l l i n g dewar C with l i q u i d N 2 and immersing the bomb whilst pumping on the N 2 v i a 9, so as to maintain a constant vapour pressure corresponding to the required temperature. For these measurements, dewar D was f i l l e d with N 2 i n the normal manner whilst dewar A was attached to the system and evacuated together with the whole of the "cooling c i r c u i t " . Temperatures in the range 78°K - room temperature --43-were obtained by cooling the bomb down to l i q u i d N 2 tem-perature and then, having evacuated C, r e l y i n g on the slow warm-up rate of the bomb against the ra d i a t i o n s h i e l d action of N 2 f i l l e d dewar D. For certai n temperatures, e.g. 193°K, where l i q u i d or s o l i d baths are available, the bomb was immersed i n a smaller dewar f i l l e d with the appropriate coolant. The temperature s t a b i l i t y obtained over the temperature range 35°K to 300°K was as follows: 35 - 40°K 0.7°K r i s e per min. 40 - 50°K 0.5°K r i s e per min. 55 - 78°K constant to 0.5°K over periods of 20 mins. 78°K - room temperature - less than 1°K r i s e per 15 mins. C. Temperature Measurement A l l temperature measurements were made using a platinum resistance thermometer (H & B W4871) and a Wheatstone Bridge. This was found to be the most convenient arrangement, since gas thermometers become unwieldy above about 20°K. I n i t i a l l y , an attempt was made to construct a platinum resistance thermometer ^ i n the laboratory. Later, on advice from Dr. J. S. Dugdale, N.R.C., Ottawa, the Hartmann-Braun thermometer mentioned above, suitably modified to N.R.C. s p e c i f i c a t i o n , was purchased. It proved to be no more expensive (#42) than the thermometer constructed i n the -44-laboratory and was c e r t a i n l y more robust and capable of greater s e n s i t i v i t y . An indium thermometer^ 2®^ may, how-ever, have provided an alternative and possibly cheaper thermometer. The thermometer was mounted i n a copper tube soldered deep into the main body of the bomb; the copper tube contained some glycerine into which the thermometer was immersed to ensure good thermal contact at low tem-peratures. For accurate measurements in the range 20°K to 200°K a Rubicon Mueller Bridge was used. For higher tem-peratures, a Post O f f i c e Box type bridge was found s a t i s -factory, The thermometer was c a l i b r a t e d i n s i t u against the vapour pressure of pure gases. Research grade hydrogen, nitrogen, metahane and ethylene were used to provide vapour reference points. The bomb was f i r s t maintained at a temperature below the b o i l i n g point of the p a r t i c u l a r c a l i b r a t i o n sample used . A large gas sample was p a r t i a l l y l i q u e f i e d i n the bomb. The valves Vj and Vg, F i g . 6, were then opened s l i g h t l y so as to allow the l i q u i d to come into equilibrium with i t s vapour. Equilibrium was indicated by the fact that the pressure reading on gauge G was observed to remain constant despite the fact that gas was s t i l l flowing through V^? This reading of pressure was taken to be the vapour pressure of the l i q u i d at the p a r t i c u l a r resistance value measured on the bridge. Gas t a b l e s K ' were then used to convert vapour pressures to degrees Kelvin and a number of points on a resistance-- 4 5 -Gauge 60 PSI Exhaust Regulator •O 3000PSI [ j * To Rotary Pump BOMB H, Hydrocarbon H 2 Converter F i g . 6. High Pressure Gas C i r c u i t -46-temperature plot were thus established. The c a l i b r a t i o n obtained, F i g . 7, i s estimated to be absolutely accurate to ± 0.3°K within t 5°K of the c a l i b r a t i o n points and to be within - 1°K over the whole range of temperature. D. The High Pressure System The experiments envisaged when designing t h i s equipment were expected to require maximum gas pressures of order 3000 PSI. The problems involved i n designing a high pres-sure system for nuclear magnetic resoance work at these pressures are e s s e n t i a l l y twofold: 1) R e s t r i c t i o n s imposed by the magnetic f i e l d . The s p a t i a l l i m i t a t i o n here i s not too serious; however, the use of non-magnetic materials does complicate matters some-what. 2) D i f f i c u l t i e s due to the fact that the high pressure system has to withstand very large temperature changes. High pressure systems and techniques suitable for the range of temperatures encountered in t h i s work have been discussed^®). In the present work i t was decided to make the sample chamber ("bomb") out of copper i n preference to s t a i n l e s s s t e e l . This decision was based on a number of factors: 1) A v a i l a b i l i t y of material. 2) It was desired to have a non-magnetic bomb and th i s would have necessitated the use of type 321 s t a i n l e s s s t e e l which i s rather d i f f i c u l t to machine. i -48-3 ) It was important that the bomb should be at a uniform temperature during measurements and that the thermal time constant of the bomb should be small. The high thermal conductivity of copper i s very advantageous i n t h i s respect. The f i n a l design evolved i s shown i n F i g . 8. The sample c o i l , wound on a thin glass former, i s normally "immersed" i n the gas being studied. One side of the c o i l i s anchored (and grounded) to a copper plate held i n position by the pressure screw P. The seam between P and the bomb casing i s sealed by a f i l l e t of soft solder. The other end of the c o i l i s brought out through a Kovar glass seal soft soldered to the bomb casing. I n i t i a l attempts to seal t h i s end of the bomb with a commercial high pressure cone j o i n t f i t t i n g were barely s a t i s f a c t o r y at room temperature. At low temperatures, however, i t was found impossible to make such a f i t t i n g leak-tight, even after Dow Corning s i l i c o n e f l u i d had been applied. The Kovar seal, on the other hand, has been found to be very s a t i s f a c t o r y . For measurements of the d i f f u s i o n constant of gases a smaller, s p a t i a l l y well defined, gas sample must be used. : In t h i s case, the i n t e r i o r of the bomb i s f i l l e d with a Teflon plug which con-fine s the gas to the volume enclosed i n the glass former. The Teflon i s suitably shaped to allow the gas to leak into t h i s volume (Fig. 8). The bomb casing has been con-siderably overdesigned, with a very large factor of safety. The inner diameter i s approximately one half of the outer diameter, which corresponds to a bursting pressure of -50-18000 P S I ^ 9 ) . Some additional protection i s also provided by the Teflon cap placed over the Kovar seal as a pre-caution against the glass of the seal shattering. Commer-c i a l s t a i n l e s s s t e e l pressure tubing and Ermeto type high pressure f i t t i n g s and valves were used for the gas transfer and handling system connecting the gas cylinder to the bomb. Sample pressures above those by l i q u e f y i n g a sample i n the bomb. The f i n a l pressure achieved depends, however, on the r a t i o of the volumes either side of the i s o l a t i n g valve V^. When the sample volume was small, as i n the " d i f f u s i o n " set-up, t h i s was somewhat of a l i m i t a t i o n and i t would have been desirable to have had an i s o l a t i n g valve closer to the bomb i n l e t . The gas pressure i n the bomb was measured on a 6-inch diameter Budenberg Bourdon type gauge ca l i b r a t e d against a Barnett dead weight tester. Four alternate gauges were available - 0-200 p . s . i . , 0-600 p . s . i . , 0-1500 p . s . i . and 0-3000 p . s . i . The scale c a l i b r a -tion of each was found to be within 1% of the stated values. E. Ortho-para Hydrogen Converter The conversion of ortho-Hg into para-Hg i s not a spontaneous process and requires a c a t a l y s t , as has been discussed above. The gas obtained from a hydrogen cylinder at room temperature ("normal Hg") i s a 3:1 ortho-para mixture. In the present work i t was desired to carry out experiments on both normal H 2 and on a Hg mixture of -51-higher para content. The % para content i n an equilibrium mixture of H_ gas i s a function of the temperature of the gas as discussed in chapter 2. Thus the conversion of normal Hg has to be catalysed at a low temperature. (30 Ideally c a t a l y s i s i n the l i q u i d phase would be d e s i r a b l e v However, t h i s would have required a l i q u i d helium cryostat and i t was, therefore, decided to simplify matters somewhat by catalyzing the reaction in the gas, using a charcoal c a t a l y s e r ^ at l i q u i d Ng temperatures. (The equilibrium concentration of ortho-para at this temperature i s about 50%. ) The conversion i s a rather slow process, however, at the gas pressures of order 100 atomospheres used i n the present experiments, both because of the f i n i t e rate of c a t a l y s i s and because of the slow rate of gas d i f f u s i o n at high pressures and low temperatures. A converter (Fig. 9) was attached to the high pressure gas c i r c u i t (Fig. 6). It consists'of a s t a i n l e s s s t e e l (Type 303) pressure vessel with a demountable plug which housed the charcoal catalyser. The converted gas i s drawn o f f , when required, from the volume nearest the charcoal. (This ensures that the most highly converted gas i s removed f i r s t . ) The converter i s surrounded by a l i q u i d Ng f i l l e d dewar. This dewar i s r e f i l l e d with l i q u i d Ng automatically by a f i l l i n g c i r c u i t (Fig. 10) and the converter i s thus kept at constant temperature (78°K) for i n d e f i n i t e periods. The pressure seal, which has proved s a t i s f a c t o r y , u t i l i z e s a copper gasket and a -52-F i g . 9. Parahydrogen Gas Converter - 5 3 --oo 110 V 60c/s ® Indicator G.E. Carbaloy Thermistors ompressed a i r valve F i g . 10. Automatic Liquid Nitrogen F i l l e r -54-t e f l o n gasket, as shown. Conversion to 50% para completed i n about 5 days. -55-CHAPTER IV SPIN LATTICE RELAXATION MEASUREMENTS A serie s of measurements of the s p i n - l a t t i c e relaxation time T j i n hydrogen, methane, ethane and ethylene was per-formed over a wide range of temperatures and gas densities. It i s proposed i n thi s chapter to discuss these measure-ments and present the r e s u l t s for each of these gases. Although the measuring technique used for a l l these gases was e s s e n t i a l l y the same, some v a r i a t i o n i n procedure was found necessary or convenient to allow for the wide range of relaxation times encountered (approximately 10 m i l l i -seconds to 20 seconds} and the range of temperatures studied. The measurements on each gas w i l l be presented separately, together with some discussion of the experi-mental procedure adopted i n each case. The r e s u l t s on each gas w i l l be interpreted i n terms of the theory out-l i n e d i n the previous chapters and some discussion given of further work which i s suggested from the data obtained thus f a r . 4-1 Hydrogen Measurements of T j i n hydrogen gas were ca r r i e d out over a range of densities up to 700 Amagat and temperatures between 35K° and 300K°. The values of T, were found to -56-l i e i n the very convenient experimental range of approxi-mately 10 milliseconds to 140 milliseconds. The major ex-perimental d i f f i c u l t y i n these measurements was to maintain a fi x e d temperature i n the range 35K° to 50K° during the period of a measurement. It was found that the best one could do with the present apparatus was to work on the warm-up cycle of the bomb after the l a t t e r had been cooled down to about 3QK°, and thi s provided temperature s t a b i l i t y of about 0.5K° over the period of measurement of approximately one minute. In t h i s connection, the preset timing f a c i l i -t i e s of the spectrometer coupled with the use of printed log sheets were of great value i n reducing the period of time necessary for a measurement to a minimum. The uncertainty of the temperature measurement due to the f i n i t e time required for the thermometer and the gas sample to follow the changes i n temperature of the bomb body have been calculated to be of order 0.3K°. T^ was measured using a r e p e t i t i o n rate of approximately one per second and either a) a 7T , 71/2. pulse sequence, or b) a 7T/2, 7T/2 pulse sequence In either case, the f i r s t pulse disturbs the system from equilibrium, i n a) by inverting the nuclear magnetization (M = -M0) and i n b) by making the magnetization wholly trans-verse (M z =0). The second pulse i s a searching pulse, the amplitude A( T ) of i t s induction t a i l being a measure of the recovery of the z component of magnetization. In either case, TT i s determined from the slope of the plot of log A as a function of t , as discussed i n chapter -57-4-1.1 Estimation of Accuracy The experimental accuracy i n the determination of T j was generally of order ± 5%, although at low temperatures and nigh density (max. signal/noise) t h i s figure was improved to ± 3%, whilst at high temperatures and low density the accuracy was only of order ± 8%. The errors i n estimating the density were mainly due to the inaccuracy i n the tem-perature measurements. The gas pressure readings were correct absolutely to ± 0.5% and t h i s inaccuracy would have l i t t l e e f f e c t on the density determination. In the low temperature high density region, the density estimates are correct to ± 2%, at higher temperatures to ± 0.5%. 4-1.2. Estimation of Ortho-para Concentration The estimation of ortho-para concentration for "con-verted" samples ( i . e . samples prepared i n the c a t a l y t i c coaverter unit described i n chapter 3) was based on n.m.r. signal amplitude measurements. The procedure adopted was as follows: A sample of converted gas was admitted into the pre-viously evacuated bomb. Let the gas pressure and temperature be Pj P.S.I, and K° respectively. A pulse $ of fi x e d length and r e p e t i t i o n rate (usually a 7V/2 pulse) was 'applied and the amplitude A of the induction t a i l measured. The bomb was then evacuated and a.fresh sample of normal H2, di r e c t from a room temperature gas cylinder, was introduced into the bomb at temperature K° and at a pressure Pg such - 5 8 -that when an i d e n t i c a l S pulse i s applied, with the receiver amplifier bias se t t i n g remaining unchanged, the amplitude of the induction t a i l was again A. Since the n.m.r. signal i s proportional to the number of ortho-molecules i n the sample, the % ortho content G of the converted sample i s : Z 2 G = S— x 75% (4.1) Z l where Zg i s the compressibility of H2 at pressure Pg and temperature K° and Z-^ i s the compressibility of Hg at pressure P^ and temperature K°. Usually these measurements were made at l i q u i d a i r tem-perature when, to a l l intents and purposes, i t may be assumed that = Z 2 = 1. The accuracy of these estimates of ortho-para concentra-ti o n i s of order 1 3%. The experimental r e s u l t s for Hg w i l l now be presented and discussed i n terms of the considerations enumerated i n chapter 1. 4-1.3 T i as a function of d e n s i t y ^ for normal H 2 at fixed temperature a) At l i q u i d a i r temperature A plot of T^ as a function of p i s shown i n F i g . 11. It w i l l be noted that, in the density region studied here, T^/p i s a constant, i . e . this region corresponds to the " d i l u t e " -60-gas. Since we were r e s t r i c t e d to a maximum pressure of about 1200 p . s . i . i n these experiments, i t was not possible to study the dense gas at and above l i q u i d a i r temperature. F i g . 11 confirms that for low densities T^/p i s a constant, equal to 0.12 msecs/Amagat, and hence from 2.10 we derive a value for T c at th i s temperature: T c = 3 x 10~ /j) sees, where p i s i n Amagats. b) At 41K° A plot of Tj as a function of j> i s shown i n F i g . 12. It w i l l be noted that Tj / p i s again constant at low density, though the value has increased to approximately 0.14, but for densities i n excess of about 500 Amagats.the v a r i a t i o n of T j with p i s of the more general form expressed by (1.12). The data for the high density region i s inadequate for a det a i l e d quantitative analysis of the dependence of Tj on density and temperature i n t h i s region of density. However, comparing F i g . 12 with F i g . 13, we see that at 37K° Tj i s c e r t a i n l y a more ra p i d l y varying function of p than at 41K°. It would c l e a r l y be of intere s t to extend the measurements to higher densities and a new cryostat and high pressure system i s how being designed to f a c i l i t a t e these measurements. Since we have no measurements of T j i n the dense gas at high temperatures, we are unable to separate out the e f f e c t s of higher order c o l l i s i o n s from any other e f f e c t s (e.g. of a quantum mechanical nature) at low temperatures. - 6 3 -4-1.4 Tj/p as a function of temperature i n the d i l u t e gas From the measurements of T j , a series of plots were obtained for T j as a function of p. at a number of d i f f e r e n t temperatures between.35K° and room temperature. We now replot the slope of each of these plots ( i . e . T^ /p ) i n the low density ("dilute gas" ) region as a function of tempera-ture. The r e s u l t i n g plot, F i g . 14, shows two int e r e s t i n g features: i ) At temperatures below 100K° there i s a sharp i n -crease i n Tj/p as the temperature i s lowered and a value of Tj/p at 20.4K° obtained from Bloom's data^ 1^ i s found to f i t our extrapolated experimental curve. We interpret t h i s increase i n Tj /p as being due to quantum mechanical d i f f r a c -t ion e f f e c t s which cause the e f f e c t i v e value of X^" to increase. It i s in t e r e s t i n g to make a comparison of Tj/p oc CT V* at 40K° and 20K°: (Tj/p )40K° Experimentally we f i n d ( T /p )20KU ^ °' 7' Theoretically, i f we assume (Toe ~\2, where A i s the -\ a i deBroglie wavelength and ... A = — °c — we obtain; _ _ P ^ < 0 " ^ )40K° = V20K° = T - l / 2 1 ^ Q ? ( ^ ^ K 0 V40K° " Thus, we conclude that at low temperatures T = k/U 2, where k i s a constant. It i s in t e r e s t i n g that a c a l c u l a t i o n made on the basis of such a crude d i f f r a c t i o n model should give such close quantitative agreement with the experimental r e s u l t s . - 6 4 -I I F i g . 14. T]/p versus temperature f o r the d i l u t e hydrogen gas -65-i i ) At high temperatures, above about 100K°, we f i n d T ^ / p i s s u b s t a n t i a l l y independent of temperature. In this region of temperature the possible e f f e c t of the higher r o t a t i o n a l states of the molecule has to be considered. Since precise values of are not yet available from the Bloom and (12) Oppenheimv ' theory, the best one can do i s to make use of a c a l c u l a t i o n due to Needier, which extends the Schwinger c a l c u l a t i o n to take account of both the J = 1 and J = 3 ro t a t i o n a l states of orthohydrogen. Needier finds : 1 T l = c x p ! T ± + c 3 p 3 T o (4.2) where C-^ and are known c o n s t a n t s ^ ^ , which are of com-parable magnitude, f l* ^ 1 a n c i ^3* ^ 3 a r e t n e r e l a t i v e p r o b a b i l i t i e s and c o r r e l a t i o n times of the J = 1 and J = 3 states respectively. Below 100K°, the f i r s t term i n (4.2), (2) which i s Schwinger's o r i g i n a l term , should account for completely, since T~3 i s i n s i g n i f i c a n t i n t h i s range of temperature. Above 100K0., we may consider two p o s s i b i l i t i e s : i ) T x £ , T 3 For 1 1 > T 3 we may neglect the second term in (4.2), because ^ 3 ^ J° 1 over our temperature range and, i n fa c t , f 3 / p i i s o n l y s i g n i f i c a n t above about 250K°. (At 300K° 1 \ 1 ~ 7' ]For TT^ = HTg, the i n s e n s i t i v i t y of T ^ / p to tempera-ture indicates that the temperature dependence of ^ 3 i s very - 6 6 -s i m i l a r to that of T^, as would be expected i n t u i t i v e l y . Thus, on t h i s assumption, i t would seem that we may deduce from our T i measurements a temperature dependence for the cross-section over the entire range of temperatures from 20K° to 300K°. i i ) T * i «Xz This assumption i s untenable since i t would lead to the deduction that p 3 T 3 i s sub s t a n t i a l l y independent of temperature and thus that X \ and T 3 behave very d i f f e r e n t l y with temperature, 4-1.5 Tj as a function of ortho-para concentration Some measurements of T j were made on an approximately 50% ortho-para sample of Hg at a number of temperatures between 35K° and 78K° i n the d i l u t e gas region. At a l l temperatures i t was found that at constant temperature ( T 1 , 5 0 % ortho H 2 < ( T 1 f o r m a l H 2 i n ae r e e i n e nt with our remark i n chapter 2 that we expect to f i n d 0~" <T 0~ o-p 0-0 In F i g . 15 we plot Tj/p as a function of ortho-para concentration at three d i f f e r e n t temperatures. The data at 20.4K° are taken from Bloom.^ \ but corrected for an error i n the % ortho concentration^ 3 2^. The data presented here should be regarded as a preliminary to a more det a i l e d study of the dependence of Tj/p on ortho-para concentration, which i s planned for the near future. We note that Tj/p becomes more,strongly dependent on ortho-- 6 7 -- 6 8 -para concentration as the temperature i s lowered and t h i s type of dependence i s also found in l i q u i d Hg^**^. It i s possible that t h i s e f f e c t i s due, at least i n part, to a difference i n the magnitude of the quantum d i f f r a c t i o n e f f e c t s i n ortho-ortho interactions as compared with ortho-para interactions. 4-2 Methane Measurements of i n methane were made at two d i f f e r e n t temperatures: 193K°, which i s only 2K° above the c r i t i c a l point, and room temperature. i n methane i s found to be very much longer than that i n hydrogen, several seconds as compared with milliseconds. The measurement technique for T^ was modified somewhat from that described above for greater convenience and i n order to ensure better accuracy with such long relaxation times. The procedure used was s i m i l a r , in p r i n c i p l e , to that (34 ) used by B i l l i n g s & Nolle . A succession or t r a i n of Tf /2, o^ . pulse sequences was applied i n a time t, short compared with where TT/2 < °< < TV and the separation of the TT/2,0^ pulses i s very short compared with both T^ and Tg. This t r a i n ensured that the magnetization at time t was very close to zero. At a time T later, a si m i l a r t r a i n of pulses was applied and the height A ( T ) of the f i r s t echo due to th i s t r a i n was measured. We may then deduce T^ from the usual r e l a t i o n : A(t) = A^ ( l - e - t / T l ) where A ^ i s the value of A ( T ) for t > ST^. We have used -69-echo amplitude measurements rather than measurements of the amplitude of the f i r s t induction t a i l , because of the improved signal/noise r a t i o obtainable by use of a low pass f i l t e r . We have used two methods of generating the pulse t r a i n required and these are referred to as P.B. and C.R.R. respectively, and are described i n the appendix. The main difference between these two methods i s that i n the P.B. method the time t was measured by stop watch while i n the C.R.R. method the MBS timing c i r c u i t was used. The P.B. method was convenient and s u f f i c i e n t l y accurate for T j values above about 5 seconds. 4-2.1 T j i n the d i l u t e gas as a function of density at room temperature Figure 16 shows a plot of T j as a function of density up to about 100 Amagats, at 293K°. Two facts are immediately apparent; 1) T j / p i s a constant at constant temperature, as i n H 2. 2) T j / p i s roughly 200 times as long i n methane as i n — 2 4 hydrogen: 2 x 10 seconds/Amagat, as compared with 10 seconds/Amagat for Hg. Inserting t h i s value, of Tj/p i n equation (2,15) and evaluating T i n equation (2.17), (2.16) becomes: Thus i t i s not clear that the random walk assumption used i n Hubbard's theory i s r e a l l y well j u s t i f i e d i n the gas, since -71-the mean square angle of rot a t i o n i s rather large, « 0.11 o (radian) . Since the measured relaxation times were of order several seconds, one i s led to consider whether (1) relaxation by c o l l i s i o n with the bomb walls, or (2) relaxation by oxygen impurity, play any role here. We can immediately rule out relaxation by wall e f f e c t s on the basis of a simple c a l c u l a t i o n , as follows: The mean square distance t r a v e l l e d by a molecule per second i s , by d e f i n i t i o n , D,the t r a n s l a t i o n a l s e l f d i f f u s i o n constant of the gas. We can estimate D for methane from the following considerations: For any gas^ 1 7* D p = 1.4] (4.3) where the gas v i s c o s i t y , *| , i s independent of density up to (3 5) quite high densities. Now, for hydrogen v ' at N.T.P. 2 D = 1 cm /sec and the v i s c o s i t y of hydrogen and methane at N.T.P. are roughly the same. Hence, at N.T.P. Drn = 1 x ( r^- = T c n , 2 / s e c C H 4 f c H 4 8 . . the root mean square distance t r a v e l l e d by a CH 4 molecule in one relaxation time, at N.T.P., i s : V 2 1/2 1 _ 2 - 3 £r (DT X ) = g x 2 x 10 ^ 2 x 10 ° cm Now , at constant temperature, T^ i s proportional to p and D 1 ...... ( p ) l / 2 i s p r o p o r t i o n a l to — , so t h a t V.x j i s d e n s i t y i n d e p e n d e n t and ( x2~Y"A i s always v e r y s m a l l compared w i t h the i n t e r n a l -72-dimensions of the bomb i n this range of temperature. Also, the v a r i a t i o n of DTj with temperature i s very slow so that, over the whole range of temperature studied i n t h i s work, wall relaxation may be neglected because most of the molecules do not get a chance to h i t the walls during a time of the order of the relaxation time. Relaxation by oxygen impurity should follow a Tj oC i density dependence, since the i n t e r -action between the methane molecule and the oxygen molecule i s of the intermolecular dipole-dipole type. The methane gas samples used in t h i s work were obtained from commerical ( P h i l l i p s Petroleum Co.) gas cylinders having an oxygen impurity content of approximately 5 x 10~ 4%. An experiment was performed i n which atmospheric oxygen was del i b e r a t e l y introduced into the methane sample i n the bomb, so as to give an approximately 1% oxygen impurity. No difference i n Tj due to thi s impurity was observed at room temperature i n the range of density studied ( i . e . 100 Amagats max.) and i t was, therefore, assumed that our Tj measurements at room temperature are not affected by oxygen impurity. 4^2.2 T j as a function of density near the c r i t i c a l point Measurements of Tj were made at 193K°, i . e . about 2° above the c r i t i c a l point for methane. It was f e l t that the temperature s t a b i l i t y of our apparatus was inadequate to allow r e l i a b l e measurements to be made any closer to the c r i t i c a l temperature, where large changes of density with temperature occur. Measurements at the c r i t i c a l point are -73-of p a r t i c u l a r i n t e r e s t since, at high densities, i t may be possible to observe the e f f e c t of molecular clu s t e r s on Nevertheless, even at 193K°, i t i s possible to observe i n both the " d i l u t e gas" region and i n the "dense gas"at pressures not i n excess of 80 Atmospheres. Figure 17 shows the v a r i a t i o n of Tj with density for a sample obtained d i r e c t l y from the cylinder and for a d e l i -berately oxygenated sample. The density v. pressure curve for methane at t h i s temperature i s also shown. (The l a t t e r data was obtained by a n.m.r; method discussed in the appendix.) The most s t r i k i n g features of the plot of T^ versus density at t h i s temperature are: 1) In the d i l u t e gas, T-^/p i s again a constant and <Tl/p ) 1 9 3 Ko = 1.7 so that, assuming T-i/p i s decreasing (T- . /0 ) 1 I 1 T 293K° monatonically with temperature, there i s almost a 1/T tempera-ture dependence. 2) In the dense gas (above 100 Amagats, as indicated by the density v. pressure plot) T j / p appears to be a decreas-ing function with density up to about 200 Amagats. The behaviour of T j / p above 200 Amagats i s not quite clear and the measurements need repeating to substantiate the r e s u l t s shown here. The measurements at the very highest densities are most probably influenced by oxygen impurities. The oxygen content of the cylinder methane sample used i n this experiment was estimated from a measurement of Tj i n the -75-/og \ l i q u i d . A c a l c u l a t i o n based on Sandhu et a l gave the oxygen content as 2.5 x 10~ 3%. If there i s indeed no ef f e c t due to oxygen i n the 100-200 Amagat region, then our experi-mental curve shows that, 2 + BP then some of the c o e f f i c i e n t s of the higher order terms i n i f T 1 = A p + B p + are negative. The y dependence of the relaxation by oxygen impurity i s i l l u s t r a t e d by the plot of Tj v. ^ for the oxygenated sample. The oxygen content of t h i s sample was estimated to be 0.1%. In the dense gas, the relaxation time for thi s sample i s dominated by oxygen impurity relaxation, as evidenced by the fact that the T-^ v. p> curve i s a close f i t to the curve ^ T^ ~ 350 Amagat sees. For comparison, a ca l c u l a t i o n of T^ based on a dipole-dipole i n t e r a c t i o n of an Og molecule with a CH^ molecule, assuming 0.1% oxygen impurity, gives a value of p T^ ~ 12 Amagat sees. However, th i s c a l -c u l a t i o n must be regarded as very approximate i n view of the large errors introduced by inaccuracies i n estimating r, the dipole-dipole separation, ( r 4 dependence) and V the r e l a t i v e v e l o c i t y of the molecules ( I T ^ dependence). 4-3 Ethane Measurements of T^ i n the gas were made at room tempera-ture and at 41C°, i . e . both below and above the c r i t i c a l point. The measurement techniques were sim i l a r to those used with methane. The r e s u l t s are shown i n Figs. 18 and 19. Since a -78-t h e o r e t i c a l analysis s i m i l a r to the work of Hubbard on methane i s not available, i t i s not possible, at t h i s stage, to give a detailed discussion of the r e s u l t s i n terms of the r o t a t i o n a l states of the molecule. We again f i n d only one relaxation time and t h i s varies l i n e a r l y with p at fixed temperature i n the d i l u t e gas region, but T j i s about 10 times as great as i n methane. T j / p i s s u b s t a n t i a l l y independent of temperature between 20C° and 4lc° to the accuracy of the experimental data (about ± 10%) . At 41C °, i n the dense gas region, a 0 .5% oxygenated sample showed impurity relaxation e f f e c t s of the same nature as those discussed i n methane. The dense gas region at room temperature would also appear to be effected by oxygen impurity i n the cylinder sample, as evidenced by the scatter at 70 Amagats. 4-4 Ethylene Measurements of T j i n ethylene gas at room temperature are shown i n F i g . 2 0 . The technique of measurement was the same as that used for methane and ethane. The most s t r i k i n g feature of these r e s u l t s i s the very pronounced impurity relaxation e f f e c t of oxygen even at low d e n s i t i e s . A plot of 1/Tj as a function of p for a 0 .35% oxygenaged sample shows a l i n e a r r e l a t i o n s h i p up to 60 Amagats and indicates that the relaxation can be accounted for com- pl e t e l y on the basis of an impurity Og relaxation process. This strong dependence on oxygen impurity i s probably c l o s e l y related to the unsaturated nature of the ethylene -80-molecule. It i s very probable that a weak bonding between oxygen and ethylene i s formed, which lengthens the mean time that an oxygen molecule i s i n close proximity to the protons of the ethylene molecule. The planer structure of the ethylene molecule may also allow the oxygen molecule to approach much closer in to the protons than i n ethane or methane. A further point of considerable inte r e s t i s the fact that T^/p for ethylene i s 10 times as great as for methane. It i s not possible at thi s stage to make a detailed analysis (in the absence of a c a l c u l a t i o n similar to that of Hubbard for methane) but assuming that the constant pr o p o r t i o n a l i t y between T^ and V c i n C2H4 i s not very d i f -ferent to that for CH* we f i n d that ( f / ' T ) « 1 C 2H 4 i n d i c a t i n g that i n ethylene the molecule i s given a very large rotation.at each c o l l i s i o n . •81-CHAPTER V MEASUREMENT OF SPIN-SPIN RELAXATION TIME (T 2) AND DIFFUSION CONSTANT (D) IN H 2 "The simplest formula to test numerically i s that for s e l f - d i f f u s i o n , but the c o e f f i c i e n t of s e l f - d i f f u s i o n of a gas into i t s e l f i s not a quantity which admits of d i r e c t experimental determination." — S i r James Jeans, "Dynamical Theory of Gases", 1925. The purpose of t h i s chapter i s , in part, to show that the n.m.r. technique allows a d i r e c t experimental determina-ti o n to be made of the s e l f - d i f f u s i o n c o e f f i c i e n t of a gas. As was mentioned i n chapter 3, i n a two pulse experiment (e.g. "IT/2, IT) the decay of the echo amplitude i s a function of the spin-spin relaxation time, T 2, and the tr a n s l a t i o n a l d i f f u s i o n c o e f f i c i e n t of the gas, D, through equation (3.7): A(2 X) = A Q exp where V denotes the time separation of the two pulses and G i s the uniform gradient of the magnetic f i e l d . (We assume d H o dH Q dH Q (20) = G } d y = . ) Carr and Pu r c e l l have shown that, by using a somewhat more complicated pulse t r a i n , i t i s possible to p r a c t i c a l l y eliminate the e f f e c t of d i f f u s i o n - 8 2 -on the echo decay. The Carr-Purcell technique thus allows one to determine both Tg and D, and i n th i s chapter some measurements of spin-spin relaxation time and of s e l f -d i f f u s i o n i n Hg gas at l i q u i d Ng temperature w i l l be d i s -cussed. 5-1 Spin-Spin Relaxation Time i n Hg at Liquid Ng Temperature The spin-spin relaxation time i s a measure of the time for a group of spins i n i t i a l l y precessing about H Q i n phase to dephase as a r e s u l t of th e i r s l i g h t l y d i f f e r e n t pre-cession frequencies. The reason for the differences i n the precession frequency of i n d i v i d u a l spins i s due to the fact that each spin i s acted upon by the l o c a l magnetic f i e l d of i t s environment. In the general case of a time dependent environment, the l o c a l f i e l d may be considered to give r i s e to a dephasing, A'ft , i n the spin precessional frequency during a " c o r r e l a t i o n time" T c , after which time the l o c a l f i e l d of the environment w i l l change and give r i s e to a d i f f e r e n t dephasing which may d i f f e r i n both magnitude and sign. In a gas the changes i n the l o c a l f i e l d are due to the random motion and c o l l i s i o n processes of the gas mole-cules. The c o r r e l a t i o n time i n a gas i s thus very short compared with a Larmor period and, under these circumstances, one usually expects that T^ = Tg^ 3 7^. Our measurements of T 2 i n H 2 were undertaken with a view to v e r i f y i n g t h i s equality. In the Carr-Purcell technique the d i f f u s i o n e f f e c t s are eliminated by applying a single TT/2 pulse followed by a -83-t r a i n of equally spaced TT pulses. The "rephasing" pulses then act to cancel out the dephaslng action of the d i f f u s i o n process, and, i f a large number, n, of TT pulses i s applied i n a time short compared with T 2, the echo amplitude decay equation w i l l read: A(2T ) = A Q exp In our experiments, n i s always large enough (n = 15 per i relaxation period) to make the d i f f u s i o n term of the order of a few percent of the 1/T 2 term. We have also checked the e f f e c t of the n 7T pulses from the plot of log A(2T ) as a function of T , which i s a straight l i n e i n d i c a t i n g that d i f f u s i o n e f f e c t s are n e g l i g i b l e , whereas, when we use a two pulse, ( T/2, IT), t r a i n and vary T , we f i n d a pro-nounced non-linearity i n the plot (Fig. 21). The accuracy of our T 2 measurements i s of order - 10%, and, to t h i s accuracy we f i n d : 1. There are two spin-spin relaxation times, T 2 and T 2 S , with T 2 S a£0.5 T 2. 2. T 2 / p i s constant (at l i q u i d N 2 temperature). (Fig. 22.) 3. T 2 = 0.8 T r We are unable, at present, to give any det a i l e d arguments to explain either 1 or 3. It would appear that the spin-spin c o r r e l a t i o n times are d i f f e r e n t to the inter-molecular c o r r e l a t i o n times, but, i n any case, 1 and 3 do not contra-d i c t the general requirement that T 2 ^ T j ( i . e . there can be no transverse magnetization for the equilibrium spin 2T K 2 G 2 D ( 2 T ) 3 12n' ~ 2T ~ A Q exp(-f-) (5.1) msecs -86-system). The lin e a r dependence of Tg on density would be expected to follow from s i m i l a r considerations to the T-^ dependence on density. 5-2 Di f f u s i o n Measurements i n Hydrogen A l l measurements of d i f f u s i o n depend on fin d i n g a suitable unambiguous method of " l a b e l l i n g " molecules. The n.m.r. technique allows one to lab e l the molecules without making any changes at a l l i n the composition or physical conditions of the molecule, and i t thus o f f e r s a considerable advantage over a l l other methods of studying d i f f u s i o n , i n that true s e l f - d i f f u s i o n measurements are possible, i n p r i n c i p l e , However, i n diatomic and polyatomic gases the molecules are not completely indistinguishable, because of the fact that they e x i s t i n the several modifications ortho, para, and meta, i n co n t r a d i s t i n c t i o n to monatomic gases where the molecules are completely indistinguishable. Thus the n.m.r. technique would be capable of measuring true 3 s e l f - d i f f u s i o n i n He , but i n the gas samples used i n the present work, the d i f f u s i o n constant, D, measured i s r e a l l y that of a "binary mixture" of ortho and para hydrogen gases; An ortho molecule i n the (J = 1 I =1) state has, i n f a c t , 9 quantum states and so i n an ortho-ortho c o l l i s i o n 2 there are 9 = 81 possible in t e r n a l states for the system. The e f f e c t of the m u l t i p l i c i t y of quantum states on the transport properties of gases has been treated i n d e t a i l by j several authors^ 3 8)(39)(4p)^ taking into account symmetry -87-e f f e c t s which are the r e s u l t of the necessary symmetrization of the wave function for a system of i d e n t i c a l p a r t i c l e s . Cohen et a l ( 3 8 ) . f i n d the t h e o r e t i c a l s e l f - d i f f u s i o n c o e f f i -cients for 100% parahydrogen and for 100% orthohydrogen are equal to a l l intents and purposes, but that there w i l l be a strong temperature and ortho-para concentration dependence for H 2 gas at low temperatures. The measurement of D i n hydrogen gas as a function of both temperature and ortho-para concentration i s thus a very worthwhile project. The only measurements of d i f f u s i o n i n hydrogen reported i n the (35) l i t e r a t u r e are those of Harteck and Schmidt for para-H^ d i f f u s i n g into ortho-H 2 at 273K°, 85K° and 20.4K° at 1 atmosphere pressure. The measurements to be discussed here are i n the nature of preliminary measurements for a detailed study of d i f f u s i o n i n gases as a function of density and temperature. It would be of considerable inte r e s t to measure the d i f f u s i o n co-e f f i c i e n t i n the dense gas region because of the possible i n t e r p r e t a t i o n of the r e s u l t s for T j i n th i s region on the basis of a d i f f u s i o n model of molecular c o r r e l a t i o n . The scope of the present measurements was to examine the f e a s i -b i l i t y of doing d i f f u s i o n measurements with the present apparatus and to examine the density dependence of D i n the " d i l u t e " normal hydrogen gas at l i q u i d N 2 temperature. For the d i l u t e gas, a rigorous r e l a t i o n e x i s t s between s e l f - d i f f u s i o n and v i s c o s i t y , which, expressed i n terms of the respective c o e f f i c i e n t s , may be written: - 8 8 -D = f -L (5.2) where D i s the c o e f f i c i e n t of s e l f - d i f f u s i o n of the gas, i . e . the mean square distance t r a v e l l e d by the molecules per second, *j i s the c o e f f i c i e n t of v i s c o s i t y for the gas and p i s the gas density, f i s a constant of order unity. The exact value of f depends on the choice of gas model and intermolecular potential employed i n the theory. For Hg, (17) the c a l c u l a t i o n s of Hirschfelder et alV ' based on a 6-12 p o t e n t i a l , give f = 1.37. Since the v i s c o s i t y of a gas at fixed temperature i s independent of i t s density over a wide range of densities, one would expect to f i n d : D = constant (5.3) I Our technique for determining D i s to examine the decay, of the echo amplitude i n a two pulse ( TT/2, Tf ) experiment and correct for the Tg term i n equation (3.7), using the measured values of T 2 discussed i n the f i r s t part of t h i s chapter. Multiplying (3.7) by e + ^ / T 2 one obtains: A* (2T ). = A Q exp | * 2 G 2 DT 3"] (5.4) = A Q exp - k T 3 The slope of the plot of log A * ( 2 T ) versus T 3 i s k and, i f the value of the gradient G i s known, one may immediately derive the value of D. Unfortunately, the natural inhomogeneity of the magnet -89-used for thi s work was so large ( ~ 1.5 gauss/cm) as to make i t extremely d i f f i c u l t to es t a b l i s h a known uniform uni-d i r e c t i o n a l gradient i n H Q. A further point which had to be considered was the e f f e c t of any inhomogeneity of the r . f . f i e l d and departure from TT/2, TT conditions on the echo decay. A t h e o r e t i c a l analysis, due to Muller and B l o o m s h o w s that the exponential describing the echo decay i s independent of the angles through which the magnetization i s rotated by the r . f . pulses, and is- independent of the in t e n s i t y and homogeneity of the r . f . f i e l d and the duration of the r . f . pulse. In a l l our d i f f u s i o n measurements the in t e r n a l volume of the sample bomb was reduced to about 0.2 cc by a Teflon i n s e r t . This helped to ensure a more uniform magnetic f i e l d gradient. (For a sample of known geometry situate%'^I* :a v >v-magnetic f i e l d with a uniform gradient, G, the magnitude of G may be determined from the shape of the induction t a i l following a 7T/2 pulse ^ 2 0 ^ . ) An attempt was made to produce a uniform gradient by means of two sets of current carrying wires (max. current 85 amps) placed 3 inches apart and sym-met r i c a l l y disposed on either side of the bomb, with the current flowing i n the same d i r e c t i o n i n each so as to pro-duce f i e l d s i n opposition. By t h i s means the t o t a l mean gradient of H Q was increased to 2.8 gauss/cm for current flow i n one d i r e c t i o n and reduced to 0.5 gauss/cm for current flow i n the reverse d i r e c t i o n . The difference between these two gradients indicates the non-linear e f f e c t s introduced by the -90-magnet pole faces. This change i n gradient was i n s u f f i c i e n t to ensure the necessary uniformity and accurate knowledge of G required for absolute determinations of D. It would, how-ever, have been possible to " c a l i b r a t e " the gradient by making a measurement on a sample, such as water, where D i s very well known^^. The experimental r e s u l t s to be d i s -cussed here were obtained with no externally applied gradient and were r e s t r i c t e d to measurement of " r e l a t i v e D", i . e . to measurements of k i n (5.4). The accuracy of the values of k obtained i n t h i s work i s of the order of 10%. The experimental procedure consisted of measuring the amplitude of the echo A(2T ) following a 7V/2, Tf sequence of pulses as a function of T . The amplifier non-linearity was corrected for by a c a l i b r a t i o n experiment i n which the induction t a i l amplitude for a single TT/2 pulse was measured as a function of the gas density at fi x e d tem-perature. Some measurements have been made of the density and temperature dependence of k i n the range 60K° to 120K°, but i t i s only proposed to discuss here the density dependence of k (and hence of D) at fixe d temperature (78K°). 5-3 Density dependence of D at fixe d temperature For t h i s measurement only the r e l a t i v e values of D are required, provided G remains the same for a l l measurements. A t y p i c a l plot of log A ( 2 T ) as a function of T i s shown i n Fig . 21, in d i c a t i n g the non-linearity introduced by the T 3 d i f f u s i o n term. Figure 23 shows the corresponding plot of log A*(2T ) versus T 3 , the slope of which i s k. Since D«C k, -92-a plot of 1/k as a function of j> should give a straight l i n e , i f (5.3) i s v a l i d . Figure 24 shows th i s plot which indeed v e r i f i e s ( 5.3). The d i f f u s i o n c o e f f i c i e n t i n normal Hg i s thus inversely proportional to density at f i x e d temperature. - 9 4 -CHAPTER VI SUMMARY, CONCLUSIONS AND SUGGESTIONS FOR FURTHER WORK The work described i n the preceding chapters may be re-garded as the f i r s t part of an extensive study of the physics of gases by n.m.r. techniques. Our p r i n c i p l e interest i s to obtain information on the asymmetrical part of the i n t e r -molecular int e r a c t i o n , from measurements of the s p i n - l a t t i c e relaxation time. However, several other aspects of i n t e r -molecular interactions and of the properties of gases may be investigated by n.m.r. methods and the present work should be regarded, to some extent, as an exploratory investigation of the p o t e n t i a l i t i e s of the n.m.r. technique and of the possible d i f f i c u l t i e s or li m i t a t i o n s which have to be considered. Most of the work reported here has been concerned with H 2 . When numerical values become available from the Bloom-(12) Oppenheim theory v ' i t w i l l be possible to derive, d i r e c t l y from our Tj measurements, the form of the asymmetrical part of the intermolecular potential. Meanwhile, we have been able to show that: at fixed temperature, i n agreement with the Schwinger theory 2. T j / p i n the d i l u t e gas, increases with decreasing temperature below 100K° and i s p r a c t i c a l l y temperature i n -dependent from 100K° to 300K°. The increase i n T,/p at the 1. In the d i l u t e gas, T j i s proportional - 9 5 -lowest temperatures i s interpreted i n terms of quantum e f f e c t s . 3. The dependence of T^/p on ortho-para concentration i s greatest at the lowest temperatures. 4. In the dense gas, at low temperatures, T^ i s a r a p i d l y increasing function of ^ . It would, c l e a r l y , be most valuable to extend the present work on hydrogen i n several d i r e c t i o n s : 1. One would l i k e to have data on T^ i n the dense gas over a much wider range of temperature, but t h i s would neces-s i t a t e the use of a gas compressor capable of providing pressures up to at least 400 Atmospheres. It would be par-t i c u l a r l y i n t e r e s t i n g to compare T^ i n the dense gas with T-^ i n the l i q u i d . 2. It would be in t e r e s t i n g to extend the Tj measurements to the high temperature range where the higher J states of the Hg molecule (J = 3 ) are well populated. (This again would require gas pressures above those now available to us.) One could then examine the intermolecular interaction as a function of the J state of the c o l l i d i n g molecules. 3. More accurate information i s needed on the difference between the ortho-ortho int e r a c t i o n and the ortho-para i n t e r -action. This, however, would require high percent para samples which have to be produced by c a t a l y t i c conversion of normal H 2 i n the l i q u i d phase. 4. At, low temperatures, i t should be possible, by comparing T^ for HD and Hg, to examine the r o l e played by the d i f f e r e n t quantum s t a t i s t i c s of these two gases. - 9 6 -Our measurements on some of the l i g h t e r hydrocarbon gases have shown that, here again, i s proportional to ^ at f i x e d temperature i n the d i l u t e gas, but that the relaxation times are considerably longer than i n H 2 . On the basis of Hubbard's theory for methane^ 1 3^, i t i s possible, from measurements i n th i s gas, to estimate the mean square angle of rotation per c o l l i s i o n . It appears very l i k e l y , from a com-parison of T^ i n methane and ethylene, that the planar con-fi g u r a t i o n of the l a t t e r gas molecule allows a much larger rotation angle per c o l l i s i o n than in the more spherical methane molecule. By extending these measurements to other gases with d i f f e r e n t molecular geometries, i t should be possible to obtain information on the mean square angle of rot a t i o n of a molecule as a function of i t s geometry'. The measurements i n the dense gas are considerably harder to interpret because of the marked e f f e c t of O2 impurity i n t h i s region. A l l the hydrocarbons studied show a d i f f e r e n t density dependence of T^ in the dense gas region, as compared with H 2 . Further work has to be done i n t h i s region, but once again t h i s requires the use of a gas compressor. The oxygen impurity e f f e c t s , i n themselves, would provide an i n t e r e s t i n g topic for further study, and, i n p a r t i c u l a r , valuable information may be obtained on "meta-stable states" of oxygen with cert a i n molecules. A further topic for study i s the dependence of T^ on changes i n molecular symmetry introduced by p a r t i a l deutration. It may prove easier to study these e f f e c t s i n the gas phase, -97-where Og e f f e c t s are not usually as dominant as i n the l i q u i d phase. In addition to making measurements of T j we have also explored the p o s s i b i l i t y of making other measurements on gases using the n.m.r. technique. We have shown that i t i s possible to measure the s e l f - d i f f u s i o n c o e f f i c i e n t D of a gas as a function of i t s temperature and density, and have v e r i f i e d that D for normal Hg i s inversely proportional to j3 It would be p a r t i c u l a r l y i n t e r e s t i n g to make measurements of D at high, l i q u i d l i k e , d e n s i t i e s . Our magnet has, however, proved very i l l - s u i t e d for d i f f u s i o n measurements on gases. In the course of t h i s work, we have developed a method for estimating the compressibility of a gas, based on the fact that the amplitude of the n.m.r. signal at f i x e d tem-perature i s proportional to the density of the gas. We discuss t h i s technique i n Appendix C . The fact that we have found Tg < T^ i n Hg at l i q u i d Ng temperature suggests that i t would be very worthwhile to study the r e l a t i o n between T j and Tg over a wide range of temperature and i n a number of gases. We have shown that", with the equipment constructed for n.m.r. work on gases, some aspects of gas physics, not accessible by the more conventional techniques, have been investigated. Further work remains to be done, but much of thi s requires a higher pressure than that accessible to us at present. -98-APPENDIX A The Pulsed N.M.R. Spectrometer - C i r c u i t Details The spectrometer comprises the following basic units: 1. A timing unit 2. A pulse generator unit 3. A pulsed radiofrequency unit. 1. Timing Unit Figure 1A i s a schematic diagram of the timing unit. Voltage pulses or pips from a Tektronix TM181 generator at one of f i v e fixed frequencies (selected by a f i v e p osition switch) are applied to the g r i d of the pulse amplifier V^. A diode clamping c i r c u i t controls, or "gates", the input to the second amplifier stage of V^, the clamping potential being controlled by the cathode of the gate driver stage Vg- The input pulses to the M.B.S. counting tube c i r c u i t , V3, Vg and Vy, w i l l thus be shut off when the gate driver stage i s operated. During the period i n which the gated amplifier i s "open", a t r a i n of pips i s amplified and then subdivided into an "odd" t r a i n and an "even" t r a i n by means of the binary c i r c u i t V3, and the pulses of each t r a i n are fed through a cathode follower Vg to the grids of the MBS counting tube. Special care i s required i n the design of the binary c i r c u i t , V3, to allow for the wide range of r e p e t i t i o n frequencies (1 Mc/s to 100 kc/s i n f i v e steps) of 22K' 22K I 2 A T 7 22 £ 7 0 G-ATE DKlVER .300V INPUT 0-22^< 5563 6-1 10J3r»| 22 0 - 3 3 M GATE BISTABLE 300V W > - L j oo Wt M —© Selectc Of • ZISV nss nr Dtcadf Cottar ^ CR»w.«LT \ 300V • f V i 8 . .300V ->/50V §0-3J M o-i 3-3K • V V - ' 4-7K I>ecoie Countr 9 o-ooi-l-l3h MBS T„U 0.6«M T~ 0-OOI CfttC. Followv J)Y-I\/CW MBS Decade Counter ^ »» ^ ^ 6 A H 6 I 2 3 4 J C 7 g O l 2 3 A 5 6 7 8 9 T«M-.e!s c £ M . B.f. I ' C H . I Cointlj«r\ce CirCwif" CH H Coin t i J t n c e Circuit F I Q IA T I M I N G U N I T D r i v e r Ai*>p. M B S D r iyer Circuit" 0-33M 150V 00/ ZZK Ctf. IT o/p 22K ~T(> follow Paje. 98 -99-the input pulses. The grids of the MBS tube require a nega ti v e pulse of about 100 vol t s for s a t i s f a c t o r y operation. In an e a r l i e r version of the c i r c u i t , the pips from the Tektronix generator were f i r s t amplified by means of a f e r r i t e loaded 5:1 step-up pulse transformer (constructed i n the laboratory), i n place of Vj, and then fed into a Schmidt trigger c i r c u i t . The use of such a triggering c i r -c u i t s i m p l i f i e s the design of the following binary c i r c u i t ; the shape of the pips from the Tektronix generator d i f f e r s for each r e p e t i t i o n frequency and this makes for some d i f -f i c u l t y i n the s a t i s f a c t o r y operation of the binary c i r c u i t (This Tektronix unit i s designed for oscilloscope sweep c a l i b r a t i o n and the pips at low r e p e t i t i o n frequencies are del i b e r a t e l y lengthened to give reasonable pulse brightness The output from target 0 of the f i r s t MBS tube i s d i f -ferentia-ted and :amplifled and used to drive a second binary c i r c u i t , which i n turn produces two pulse trains which are fed into the following MBS tube. The negative going output from target 0 of the f i n a l MBS tube, generated at the end o one complete counting cycle of the MBS tube counter c i r c u i t i s fed v i a cathode follower Vs to the gate driver V2 which then actuates the diode clamp on V\ and cuts o f f the count-ing c i r c u i t from the input pulses generated by the Tektronix generator. V 2 holds the diode clamp of Vj i n the "shut o f f " position u n t i l the r e p e t i t i o n rate multivibrator Vg releases the clamp v i a the gate bistable multivibrator V 4. -100-The outputs from the targets of the f i r s t MBS tube are brought out to the 10 position "Selector". Jumper wires allow one or more target outputs to be brought out to channel I v i a the Rossi coincidence c i r c u i t V^g, ^ 3 , V\4-This l a t t e r c i r c u i t ensures that no output w i l l be available at channel I when the second or l a t e r MBS tubes are counting, 1. e. after the tenth pip of the counting sequence. The targets of each of the MBS tubes are brought out to the channel II coincidence c i r c u i t . For a two pulse sequence, jumper 5 i s connected up on the channel I selector and the channel II selector i s set at m,n,5. The "Repetition Selector" and "Coincidence Selector" switches are set at position III. The two pulses w i l l then be separated by a time T equal to (100m + lOn) x 1/f seconds, where f i s the frequency of the pips generated by the Tektronix generator. For a Carr-P u r c e l l type sequence, the "Coincidence Selector" switch i s set at II and the channel II coincidence c i r c u i t i s set at 55. One then obtains a channel I pulse at time 5 and channel II pulses at 55, 155, 255,.. . . * 955. A minimum voltage output from channels I and II of approximately 5 vol t s i s required for s a t i s f a c t o r y operation of the pulse generator unit. 2. Pulse Generator Unit This consists of two type 163 Tektronix Pulse Generator units modified to give a variable length, 90 volt positive going pulse on a steady posi t i v e or negative bias. The bias -101-i s continuously adjustable between approximately -25V and +30V. 3. Radiofrequency Unit The unit used in the present work, F i g . 2A, consisted of a 30 Mc/s pulsed o s c i l l a t o r followed by a tuned 30 Mc/s amplifier stage and a class C push-pull power stage. A unit designed for use with a continuously running o s c i l l a t o r and pulsed amplifiers i s shown schematically i n F i g . 3A. Both units w i l l now be discussed b r i e f l y . Referring to Fig . 2A, the channel I Tektronix 163 pulse generator feeds a +90 vo l t input pulse and a -20V bias through a 330/L r e s i s t o r to the grids of the pulsed 30 Mc/s o s c i l l a t o r V j . The channel II pulse generator provides a +90V input pulse but no bias voltage. Pulses of R.F. at 30 Mc/s, corresponding to the input pulses provided by the channel I and II Tektronix 163 pulse generators, are inductively coupled to the input of the amplifier stage V 2. The R.F. output of V 2, of order 300 vol t s peak to peak, i s applied to V 3, which i s running i n class C, and approximately 400 vo l t s peak to peak at 30 Mc/s are available at the input to 'the half-wave l i n e which feeds the sample c o i l . -' The use of a continuously running o s c i l l a t o r and pulsed amplifier i s advantageous because of the p o s s i b i l i t y of using phase sensitive detection and the convenience of cor-re c t i n g for errors i n the magnetization rotation angle by the +90V input pulse +90V input pulse -20V bias -170V 1000V r. f. choke fioMc/s « — , O/P 30 Mc/s o s c i l l a t o r F i g . 2A. Pulsed 30 Mc/s radiofrequency power unit -103-method suggested by Meiboom and G i l l . Figure 3A i s the c i r c u i t diagram for a unit which has been constructed but not yet tested out. The continuously running o s c i l l a t o r i s c r y s t a l controlled at 15,045 Mc/s and amplitude s t a b i l i t y i s (Art \ provided by a Gourier v 'feedback loop. Vg i s a series tuned buffer amplifier. The f i r s t gated amplifier V 4 , V 5 i s screen gated, using the pulse amplifiers Vg and Vy. The gating pulses are provided by the Tektronix 163 channel I and II pulse generators. V 7 and VQ are likewise gated on the grids. The f i n a l power stage i s a push-push class C frequency doubler stage tuned to 30.09 Mc/s. P a r t i c u l a r care was taken with the layout of the unit to reduce stray 30 Mc/s r . f . radiati o n from the unit to a minimum; the use of 15 Mc/s stages i n a l l but the f i n a l stage was dictated by the same considerations. 6 T 6 Osc. 6AK5" A m p 6CB6 Ar*p. A 150 V CHI PULSE 6CB6 2 . 2 K J • 0 I « > 2 2 K V, Z28 J o i , -orj •5--0K 33 K G-dted Amps 5763 2 2 5 V Gate</ i f i y e r i RFC -Of, 1 5 . 0 4 5 • MC/S - T — £ 47 X IK 100 K.FC. i47 150 > _L 180 •01. IOOK;: 68 2 2 K 01 •01 4 — * 4-7 = 4*7 I00K 150 'r A 300V 10 OK' -w-INIOO A 225V 6 T € CH. I FULSt C H . H PULSE - " O h " INiOO r 6DX8 "PuUe A m p . > 2,20 H' 22S1/ O l y u ^ -I7DV /5K IKV •01 •Hi-' •01* Z2a 22 .a p 3 £ 2 9 35" "Powger D o u L l e r £ C ° / P -£ JO. 09 M C / S - 1 7 0 V C i n pF 1? i n -A. f o U o h i "Pye. I03 F.g. 3A PHASE C O H E R E N T R . F . P O W E R U N I T . -104-APPENDIX B Operating Instructions for Low Temperature System Figure 5 i s a schematic diagram of the low temperature system. The complete system i s f i r s t pumped out by means of the rotary pump: (a) The gas flow c i r c u i t comprising dewar A, vacuum jacketed l i n e a, the bomb jacket and the return line r i s pumped out v i a 6, 1 and 8, with 5 closed. (b) The jacket of a v i a 7." (c) Dewar C v i a 9. (d) The vacuum jacket of dewar A v i a 10 and 12. Liquid a i r i s poured into dewar D and the gas flow c i r -c u i t i s opened up to the atmosphere by opening 1, 8 and 10, with 10 open to the atmosphere and 5 closed. Liquid a i r i s then poured into A and the vapour made to flow through the gas flow c i r c u i t and out to the atmosphere by clo s i n g 1 and using 2 as a t h r o t t l e control with 6 open. The bomb i s cooled down to l i q u i d a i r temperature by t h i s means i n about 30 minutes. Dewar A i s then removed, i t s l i q u i d a i r contents poured out and the empty dewar replaced i n position. The helium transfer syphon i s inserted into dewar A and i t s vacuum jacket evacuated through 1 1 . Dewar B, f i l l e d with l i q u i d a i r i s then placed round dewar A and 1 and 8 are closed. The cooling c i r c u i t i s then flushed with cylinder -105-helium gas v i a 3 and the whole c i r c u i t then evacuated v i a 8 and r e f l u s h e d with helium; the c i r c u i t i s l e f t p r e s s u r i z e d w i t h helium to a few cm of Hg above atmospheric pressure. L i q u i d helium t r a n s f e r v i a the syphon tube i n t o dewar A i s e f f e c t e d by p r e s s u r i z i n g the helium r e s e r v o i r container v i a 4 and a l l o w i n g the " b o i l o f f " helium to be recovered v i a 1 and 5. -106 -APPENDIX C Measurement of Gas Compressibility Since the N.M.R. signal amplitude i s proportional to the equilibrium nuclear magnetization, the magnitude of the proton magnetic resonance signal from any gas sample i s given by Sx =Knp,xfx =Knfe 5*x"K (1A» where K i s a constant of the system, which includes the "e f f e c t i v e volume" of the sample and the s e n s i t i v i t y of the system, n i s the number of protons per molecule, p x i s the number of molecules per cc, <TX i s the f r a c t i o n a l excess population of protons i n the lower energy state, Z x i s the P V compressibility of the gas defined as equal to * at the R i x temperature T x and pressure P x, and R i s the gas constant per mole. Therefore, i f the signal S x i s measured as a function of the pressure P x at a constant temperature, T x, the r e l a t i v e c o m p r e s s i b i l i t i e s , Z x, as a function of pressure may be computed from the r a t i o of P x to S x. A l t e r n a t i v e l y the r e l a t i v e c o m p r e s s i b i l i t i e s can be exhibited on a plot of S x vs P x. In order to measure the absolute compressibility of any gas, i t i s necessary to measure the magnitude of the proton signal from a gas of known compressibility i n the same apparatus described by the same K. If i t i s convenient to use hydrogen as1., the reference gas for t h i s purpose, since -107-accurate values of the compressibility of H 2 are re a d i l y ( 27 ) available and H 2 i s gaseous oyer the entire range of tem-perature of a l l gases other than He, From equation (1A), the r a t i o of the signal from gas x to that f^om hydrogen i s any gas x at temperature T x and pressure P x may be determined from the known compressibility of hydrogen Zg^ at temperature TJJ and pressure P H and from the r a t i o of the two pulsed 2 2 signals. Several experimental precautions must be taken i f one i s to obtain r e s u l t s of reasonable accuracy. In order to insure that K i s indeed a constant for a given gas, the gain of the system must be constant during the time of measurement. I f , as was the case i n the present experiments, the detected R.F. signa l i s measured, i t i s necessary to correct for the non-l i n e a r i t y of the detecting diode. If one were to observe the r - f signal d i r e c t l y on an oscilloscope or with phase sensi t i v e detection, there would be no need for thi s correction. The " e f f e c t i v e volume" of the sample i s determined completely by the dimensions of the chamber (the sample "bomb") and the r - f c o i l configuration and neither of these i s varied. Between observations one must wait several Tj's to ensure that the inspected magnetization i s the equilibrium magnetiza-t i o n . In order to ensure that K i s the same constant for two d i f f e r e n t gases, i t i s advisable to make measurements of 2 (2A) Using t h i s equation, the absolute compressibility Z x of -108-the induction signal following a TV/2 pulse rather than on the echo following a 7T/2, 7\ sequence of pulses. D i f f u s i o n decay of the echo, i f appreciable, w i l l confuse the measure-ment. The time needed to make a measurement i s li m i t e d only by T i which i s of the order of seconds for the hydrocarbon gases and milliseconds for hydrogen gas. This method pro-vides compressibility values of l i m i t e d accuracy compared to other methods but r e a d i l y and r a p i d l y . - 1 0 9 -APPENDIX D Procedure for measuring long s p i n - l a t t i c e relaxation times When the s p i n - l a t t i c e relaxation time i s rather long (of order 2 seconds or more), i t i s convenient to measure T-^ by means of a "variable r e p e t i t i o n rate" technique. In p r i n c i p l e , may be measured as follows: A t r a i n of TT/2 pulses i s applied, separated by equal time i n t e r v a l s , X, i . e . 7T/2 pulses are applied at a r e p e t i t i o n rate of l/X per second. The amplitude A ( T ) of the induction t a i l following these pulses i s then a measure of the regrowth, in a time T , of the z component of magnetization, s t a r t i n g from M_ = 0 . In practice, i t i s d i f f i c u l t , with t h i s pro-cedure, to ensure that the pulses are true 7T/2 pulses, and thus that the regrowth of magnetization s t a r t s , indeed, from zero. To ensure t h i s , a simple modification i n technique may be made which, i n addition, has the advantage of pro-viding improved signal/noise. This modification consists of replacing each single pulse of the "r e p e t i t i o n rate technique" by a t r a i n of pulses. Each t r a i n consists of one (j) pulse followed by a large number (e.g. 1 0 ) of o(. pulses, where ^ = T T / 2 ±£ and 6 i s made as small as possible, TT/2 ^ < 7T and the time t between (j) and the f i r s t o£ pulse i s very short, e.g. 1 msec or less, and the separation of the oC pulses i s likewise very short. The o£ pulses ensure that there i s no z component of magnetiza--no-tion l e f t after the l a s t pulse, and thus that the recovery of the magnetization towards equilibrium in time T s t a r t s from Mz = 0. ( T i s now the time between the l a s t pulse of a t r a i n and the ft pulse of the following t r a i n . ) The f i r s t U. pulse of each t r a i n w i l l be followed by an echo A(2t) at time 2t, which w i l l serve as the regrowth of the z component of magnetization i n the time i n t e r v a l T . It i s more convenient to measure A(2t) than A(T'), because the bandwidth required for A(2t) i s considerably less than that required for A ( T ) , and a low pass f i l t e r may be used to improve -the signal/noise of A(2t) r e l a t i v e to that of A( T ). We obtain Tj from: A ( 2 t ) r = A ( 2 t ) e o * ( l - e ~ r / T l j where A ( 2 t L i s given by A(2t ) for T 2> 5Tn . In the present work, two schemes were used for s e l e c t -ing and measuring the r e p e t i t i o n i n t e r v a l , T , denoted as the P.B. ("push button") technique and the C.R.R. (''counter r e p e t i t i o n rate") technique respectively. The former was used for T^ values above 5 seconds and the C.R.R. technique for T-^ values down to about 1/2 second. ( i ) P.B. Technique The timing was controlled by the use of a Tektronix type 162 waveform generator operated i n the "manual tri g g e r " position. The two type 163 pulse generators and the wave-form generator were set to give a pair of pulses with a - I l l -mutual time separation of approximately 1 msec and a large number (e.g. 10 or more) of such pairs was applied by depressing the "continuous tr i g g e r " push button on the waveform generator. The time i n t e r v a l between removing the continuous trigger and applying i t once again was measured by a stopwatch c a l i b r a t e d i n 1/5 second d i v i s i o n s . ( i i ) C.R.R. Technique The timing i n thi s technique was controlled by the main timing unit operating on the 10 kc/s scale. Channel I was set at 5 and channel II at 15, so as to give a 1 msec in t e r v a l between the (j) pulse and f i r s t o( pulse. The coincidence selector was set at 2, so that 9 further pulses were generated, mutually separated by 10 msecs. t r o l l e d by the r e p e t i t i o n rate control of the timing unit, and a greater degree of accuracy i n determining T was thus obtained i n the C.R.R. method as compared with the P.B. method. The r e p e t i t i o n rate, 1/f pulses was con-- 1 1 2 -REFERENCES CHAPTER I ( 1 ] M. Bloom, Physica ( 2 3 ) , 2 3 7 , 1 9 5 7 . ( 2 ] N. Bloembergen, Thesis, L e i d e n , i 1 9 4 8 . ( 3 j H. Y. Carr, Private Communication. ( 4 , F. Bloch, Physical Review 7 0 , 4 6 0 , 1 9 4 6 . (5 J. M. B. Kellog, I. I. Rabi, N. F. Ramsey and J. R. Zacharias, Phys. Rev. 5 6 , 7 2 8 , 1 9 3 9 . ( 6 J G. Needier, Thesis, U.B.C., 1 9 5 9 . CHAPTER II ( 7 , -w. H e i t l e r and F. Landau, Z.f. Physik, 4 4 _ , 4 5 7 , 1 9 2 7 . ( 8 ] E. Wigner, Z.f. Physikal. Chemie B, 4 , 1 2 6 , 1 9 2 9 . ( 9 ; H. H a l l and J. R. Oppenheimer, Phys. Rev. 35, 1 3 2 1 9 3 0 . do T. Nakamura, Prog. Theor. Phys., 1 4 _ , 1 3 5 , 1 9 5 5 . ( 1 1 J R. W. H i l l and B. W. A. Ricketson, P h i l . Mag. 4 5 , 2 7 7 , 1 9 5 4 . ( 1 2 , M. Bloom and I. Oppenheimer - to be published. ( 1 3 , P. S. Hubbard, Phys. Rev., 1 0 9 , 1 1 5 3 , 1 9 5 8 . ( 1 4 , A. G. Redfield, I.B.M. Journal of Research and Development 1 , No. 1 , 1 9 5 1 . (15 j W. H. Furry, Phys. Rev., 1 0 7 , 7 , 1 9 5 7 . ( 1 6 ] MaUe, Ann. de Physik, 3 0 , 5 5 5 , 1 9 3 7 . ( 1 7 , Hirschfelder et al., Molecular Theory of Gases and Liquids. -113-CHAPTER III (18) E. L. Hahn, Phys. Rev. 80, 580, 1950. (19) F. Bloch, Phys. Rev., 70, 460, 1946. (20) F. Bloch and A. Siegert, Phys. Rev., 57, 522, 1940. (21) H. Y. Carr and E. H . P u r c e l l , Phys. Rev., 94, 630, 1954. (22) M. Bloom, E. L. Hahn and B. Herzog, Phys. Rev., 97, 1699, 1955. (23) J. Schwarz, Rev. S c i . Inst., 28, 780, 1957. (24) H. Waterman, Thesis, U.B.C., 1954. (25) S. Meiboom and D. G i l l , Rev. S c i . Inst., 29, 688, 1958. (26) G. K. White and S. B. Woods, R.S.I., 28, 638, 1957, Ibid 29, 181, 1958. (27) Int. C r i t . Tables. (28) W. Paul, G. D. Benedek and D. M. Warschauer, Rev. S c i . Inst., 30, 874, 1959. (29) P. W. Bridgman, The Physics of High Pressure, London, 1949. (30) A. T. Stewart and G. L. Squires, J.S.I., 32, 26, 1955. (31) K. F. Bonhoffer and P. Harteck, Naturwiss, 17, 182, 1929. CHAPTER IV (32) M. Bloom - Private Communication. -114-(33) G. Seidel, W. P. A. Haas and N. Poulis, 7th International Low Temperature Conference Proceedings, I960. (34) J. J. B i l l i n g s and A. W. Nolle, J. Chem. Phys. 29, 214,, 1958. (35) Harteck & Schmidt, Z.f. Physikal. Chemie B, 21, 447, 1933, (36) H. S. Sandhu, J. Lees and M„ Bloom, Can. J. of Chem., 38, 493, I960. CHAPTER V (37) D. Pines and C. P. S l i c h t e r , Phys. Rev., 100, 1014, 1955. (38) E. G. D. Cohen, M. J. Offerhaus, J. M. J. van Leeuwen, B. W. Roos and J. deBoer, Physica 22, 791, 1956. (39) E. G. D. Cohen, M. J. Offerhaus, Physica, 24, 742, 1958. (40) R. A. Buckingham, A. R. Davies and D. C. G i l l e s , Proc. Phys. S o c , 71, 457, 1958. (41) B. H. Muller and M. Bloom, to be published, Can. J. of Physics, October I960. (42) J. H. Simpson and H. Y. Carr, Phys. Rev., I l l , 1201, 1950. APPENDIX (43) G. G. Gourier, Wireless Engineer, 27, 105, 1950. .01 i V, / U -2,00V w M —© Fn», '(F^J- > MSS nr ~5tca3e Counf?r| M8S Lnygr MBS I J>ecgjt Counbrl =.0.001 I 2. 3 + 5" 6 T 8 » e » » « » « » 0 I 2 3 4 5 - 6 7 8 9 7»r«eh; M. 8.f. T '' CH. I e. T»Iji PIBSI T&vjei> PWI T ^ t h MBfl ' CH II C°iwciJence Circuit Fig IA TIMING UNIT -• CH. 7T o/p "^o follow Pye. 38 CH I PULSE A 300V F , - 8 . SA PHASE COHERENT R.F. POWER UNIT.
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Physics
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Proton magnetic resonance measurements in gases
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