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Measurement of nuclear magnetic moments by a magnetic resonance spectrometer Collins, Thomas LeGear 1950

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1 0 3 3 MEASUREMENT OF NUCLEAR MAGNETIC MOMENTS BY A MAGNETIC RESONANCE SPECTROMETER by Thomas LeGear C o l l i n s A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY In PHYSICS. THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1 9 5 0 . ABSTRACT A recording nuclear magnetic resonance spectrometer has been b u i l t and i s i n operation. The instrument i s based upon a simple o s c i l l a t i n g -detector of o r i g i n a l design, for which the author presents a complete analysis of s e n s i t i v i t y and s i g n a l to noise r a t i o . This analysis i s based upon the van der Pol o s c i l l a t o r and contains a theoret i c a l estimate of the modulation noise i n such an o s c i l l a t o r . Comparison with other methods shows the ultimate s e n s i t i v i t y of the o s c i l l a t i n g detector to be the same as that obtainable by bridge methods, but the former i s much more f l e x i b l e . Although s i m i l a r instruments are apparently i n use, no d e t a i l s are given i n the l i t e r a t u r e . Other features of the spectrometer are a 1200 l b . electromagnet and a l o c k - i n detector of great s t a b i l i t y . Some of the resonances recorded by the spectro-meter are discussed. These are: (a) C u ^ i n the wire of the detector c o i l . (b) Br79 and B r ^ 1 i n NaBr and KBr solutions. (c) 1127 i n Nal and KI solutions. (d) S b 1 2 1 i n SbCl^ and HSbCl£. The copper signal provides a c r i t e r i o n of the sen-s i t i v i t y of the spectrometer. The measurement of the resonant frequency agrees with values reported by Knight for me t a l l i c copper. The resonance widths of the bromine isotopes are ten-fo l d narrower than values given by Pound, an important d i s c r e -pancy since i t throws doubt upon our a b i l i t y , at present, to calculate quadrupole moments from nuclear magnetic resonance. The iodine width i n Nal agrees-with Pound's value. The antimony resonance i n HSbCT^ confirms work of Proctor (private communication). The resonant frequency i n SbCl^ i s about . 0 7 % higher. This s h i f t i s an example of the chemical effect recently discovered for f l u o r i n e and nitrogen isotopes. T H E UNIVERSITY OF BRITISH COLUMBIA F A C U L T Y OF G R A D U A T E STUDIES P R O G R A M M E O F T H E F I N A L O R A L E X A M I N A T I O N F O R T H E D E G R E E O F D O C T O R O F P H I L O S O P H Y of T H O M A S L E G E A R COLLINS B.A. (University of British Columbia) 1942 M.A. (University of British Columbia) 1943 MONDAY, M A Y 1st, 1950, A T 2 P.M. R O O M 201 PHYSICS BUILDING COMMITTEE IN CHARGE: Dean Walter H . Professor Gordon M . Shrum Professor George M . Volkoff Professor Arthur M . Crooker Professor Harry Adaskin Gage, Chairman Professor Roy Daniells Professor J. Gilbert Hooley Professor Ralph D. James Professor Kenneth C. Mann THESIS M E A S U R E M E N T OF N U C L E A R M A G N E T I C M O M E N T S BY A M A G N E T I C R E S O N A N C E S P E C T R O M E T E R A recording nuclear magnetic resonance spectrometer has been built and is in operation. The instrument is based upon a simple oscillating-detector of original design, for which the author presents a complete analysis of sensitivity and signal to noise ratio. This analysis is based upon the van der Pol oscillator and contains a theoretical estimate of the modulation noise in such an oscillator. Comparison with other methods shows the ultimate sensitivity of the oscillating detector to be the same as that obtainable by bridge methods, but the former is much more flexible. Although similar instruments are apparently in use, no details are given in the literature. Other features of the spectrometer are a 1200 lb. electromagnet and a lock-in detector of great stability. Some of the resonances recorded by the spectrometer are discussed. These are: (a) C u 6 3 in the wire of the detector coil. (b) B r 7 9 and B r M in NaBr and KBr solutions. (c) I ' 2 7 in Nal and KI solutions. (d) S b 1 2 1 in SbCl 5 and HSbCl 6 The copper signal provides a criterion of the sensitivity of the spectrometer. The measurement of the resonant frequency agrees with values reported by Knight for metallic copper. The resonance widths of the bromine isotopes are ten-fold narrower than values given by Pound, an important discrepancy since it throws doubt upon our ability, at present, to calculate quadrupole moments from nuclear magnetic resonance. The iodine width in Nal agrees with Pound's value. The antimony resonance in HSbCl 6 confirms work of Proctor (private communication) . The resonant frequency in SbCl 5 is about .07% higher. This shift is an example of the chemical effect recently discovered for fluorine and nitrogen isotopes. G R A D U A T E STUDIES Field of Study: Electromagnetic Physics. Electromagnetic Theory—Professor H . D. Smith. Quantum Mechanics—Professor G. M . Volkoff. Nuclear Physics—Professor G. M . Volkoff. Spectroscopy—Professor A. M . Crooker. X-Rays and Crystal Structure—Professor J . B. Warren. Electronics—Professor A. van der Ziel. Theory of Measurements—Professor A. M. Crooker. Electron Optics—Professor G. G. Eichholz. Chemical Physics—Professor A. J . Dekker. Other Studies: Electrical Communication—Professor H . J . MacLeod. Theory of Functions of a Complex Variable—Professor R. Hull . Ordinary and Partial Differential Equations—Professor S. A. Jennings. Physical Chemistry—Professor J. G. Hooley. Theory of the Chemical Bond—Professor C. Reid. Radiochemistry—Professor M . Kirsch. Molecular Structure—Professor B. A. Dunell. PREFACE The research described i n this thesis has been supported by grants from the National Research Council of Canada, B.C. Telephone Co., U.B.C. Research Committee, and the B.C. Academy of Sciences. Without t h e i r a i d , the magnetic resonance spectrometer could not have been b u i l t . I am personally indebted to the B. C. Telephone Company for a scholarship held during 1947-48, and to the National Research Council f o r a studentship (1948-49) and a fellowship (1949 -50) . I would l i k e -to thank Professor F e l i x Bloch and his group f o r t h e i r kindness to me during an extended v i s i t to their laboratory at Stanford University i n the spring of 1948. I also thank my professors, Dr. G. M. Volkoff and Dr. A. van der Z i e l , for t h e i r u n f a i l i n g i n t e r e s t and help, p a r t i c u l a r l y during the years when progress was slow. F i n a l l y , I wish to thank the University of B r i t i s h Columbia f o r the opportunity and honour of being one of i t s f i r s t Ph.D. candidates. T. L. C o l l i n s A p r i l , 1950. TABLE OF CONTENTS page Introduction 1 Chapter 1 . The phenomenon of nuclear induction Nuclear paramagnetism 3 The simple induction experiment 6 Blocht s equations 8 The reaction on the driving c o i l 10 Relaxation times 11 Transient phenomena - "wiggles" 13 Chapter 2 . Experimental methods L i s t of methods 1.5 Comparison of methods 16 Quantitative consideration of simple voltage measurement 17 The autodyne or o s c i l l a t i n g detector 20 Chapter 5 . The recording nuclear magnetic resonance spectrometer The o s c i l l a t i n g detector 2j? s e n s i t i v i t y 27 signal to noise r a t i o 29 d e t a i l s of construction 51 The l o c k - i n detector 32 page Rest of the apparatus 33 Chapter 4. The magnet and i t s controls Magnet design 35 . Current control 37 Proton s t a b i l i z a t i o n 38 Chapter 5» Measurements and standards Choice of sample 39 Operation procedure 40 Measurement of frequency 41 Calculation of spin 41 Calculation of magnetic dipole moments 42 Calculation of e l e c t r i c quadrupole moments 44 Chapter 6 . Results Cu ' from copper i n the driv i n g c o i l 45 Magnetic moment of S b * ^ 47 Line widths of Br and I 49 Appendix I . An estimation of noise i n a van der Pol o s c i l l a t o r 52 Appendix I I . Table of nuclear resonances 57 References 60 FULL-PAGE ILLUSTRATIONS Figures Facing Page 5 Meter recording of Na 22 resonance. 13 6 - 7 Oscilloscope pattern from protons. 13 11 Block diagram of apparatus. 25 12 O s c i l l a t i n g detector c i r c u i t . 26 14 - 16 Photographs of o s c i l l a t i n g detector. 51 18 Magnet design. 55 1? Current control for magnet. . 57 20 Proton magnet s t a b i l i z e r . 58 21 Recording of from c o i l . 4 5 22 Recording of S b 1 2 1 resonances. 48 23 Recording of B r ^ 1 , I 1 2 ? resonances. 49 INTRODUCTION The attempt by phy s i c i s t s to understand the struc-ture and binding forces of nuclei i s based on the c o r r e l a t i o n of measurable nuclear properties. The fundamental properties which have so f a r been observed are: (a; mass; (b) charge; (o) s i z e ; (dj binding energy; (e) s t a t i s t i c s ; If) spin; lg) magnetic dipole moment; (h) e l e c t r i c quadrupole moment. The subject of the research described i n t h i s thesis i s the measurement of nuclear magnetic moments; however, we s h a l l use also the spin and quadrupole moment i n our discussion. Optical spectroscopy provided us with our i n i t i a l knowledge of nuclear moments (K?) and continues to provide many useful values, but the accuracy i s inherently low and the measurements represent the culmination of decades of technical development. A more di r e c t measurement i s given by methods using the d e f l e c t i o n of atomic and molecular beams, 1 2 p a r t i c u l a r l y when combined with resonance techniques (K4). The resonance p r i n c i p l e i s not inherently l i m i t e d to molecular beams but may be used to d i r e c t l y measure the nuclear para-magnetism of material i n i t s normal state. Late i n 1945, t h i s method was successfully applied to protons almost simultan-eously by P u r c e l l , Torrey, and Pound (P7) working at Harvard, and Bloch, Hansen, and Packard (Bj?) at Stanford. I t was quickly developed and, i n 1947, yielded r e s u l t s of the highest accuracy (approaching one part i n a m i l l i o n ) f or the ra t i o s of the moments of the neutron, proton, and deuteron (see appendix I I ) . At the same time, i t was r e a l i z e d that a new t o o l had been developed f o r the in v e s t i g a t i o n of the structure of the l i q u i d and s o l i d state, but t h i s aspect of the subject i s not discussed i n d e t a i l i n t h i s thesis. The sharpness of the resonant conditions that allow such precision make the search for resonance very tedious even when guided by spectroscopic values for the moments. The f i r s t nuclear "spectrometer" which hunted automatically was developed by Pound (P4) who measured the r a t i o of the magnetic moments for several nuclei within the range of h i s apparatus. This work i s now being carried on by many people and a l i s t of nuclear magnetic moments measured by t h i s method i s to be found i n appendix I I . The importance of these measurements l i e s i n the p o s s i b i l i t y of systematic v a r i a t i o n of magnetic moments with nuclear species. I f the proton followed the laws developed for the electron (Dirac equation), we would expect i t to have a magnetic moment equal t o ^ ^ ^ , where the symbols have t h e i r usual meaning. This value, known as the nuclear magneton, has the r i g h t order of magnitude, but the proton has a mag-netic moment of 2.8 nuclear magnetons; while the uncharged neutron which we might expect to have no magnetic properties has a moment of -1.9 nuclear magnetons. The precision measurements of Bloch and Anderson (see appendix I I ) on the ra t i o s of the l i g h t e s t elements have provided material for much th e o r e t i c a l speculation on the nature of nuclear forces, but i t must be admitted that the success i n cor r e l a t i n g these values has been l i m i t e d . The development of the highly successful quantum theory f o r atoms was immeasurably aided by the existence of recurrent atomic properties summarized i n the well-known periodic table. No such simple arrangement exists for n u c l e i . There i s some i n d i c a t i o n of order i n the magnetic and elec-t r i c moments ( B l ) , ( F l ) , (Ml), ( N l ) , but more values are required for proof. A great need at present i s for values of moderate precision (one part i n a thousand) for the moments of as many nu c l e i as possible. We are l i m i t e d to those n u c l e i which have a spin d i f f e r e n t from zero, but t h i s amounts to about one hundred stable species of which f o r t y -two have now been measured by t h i s method. The research described i n this thesis has been con-cerned with the development of a recording nuclear spectro-meter of s u f f i c i e n t s e n s i t i v i t y to detect nuclei with small magnetic moments. The properties of nuclear paramagnetic 4 resonance w i l l be described b r i e f l y with p a r t i c u l a r reference to parameters a f f e c t i n g the size of the s i g n a l . There follows a short discussion of methods of detection. The theory of one method i s developed f u l l y by the applica t i o n of the theory of non-linear e l e c t r i c o s c i l l a t i o n s . This theory was used by the author to guide him i n the design of a simple c i r c u i t which i s the heart of the spectrometer. Following a descrip-t i o n of the apparatus some res u l t s obtained are discussed. 5 . Chapter 1. THE PHENOMENON OF NUCLEAR INDUCTION Nuclear paramagnetism I f a sample containing n u c l e i with a magnetic moment i s placed i n a strong magnetic f i e l d H Q | there w i l l appear, i n time, a small p o l a r i z a t i o n of the sample propor-t i o n a l to the f i e l d strength. This arises from the tendancy of the nuclear magnetic dipoles JA to a l i g n themselves with the f i e l d . The number of possible orientations i s l i m i t e d to 2 1 + 1 where I i s the spin quantum number. This a l i g n i n g process i s disturbed by thermal motions and, since the energy involved i s small compared to the thermal energy kT , there w i l l be only a s l i g h t l y greater number with t h e i r magnetic moment i n the d i r e c t i o n of the f i e l d than opposed to i t . The vector sum of a l l N magnetic dipoles gives the p o l a r i z a t i o n of the sample MQ -XO ^ O w n e r e s u s c e p t i b i l i t y i s given by the Curie law l±i J»L CD At room temperatures, the values of nuclear s u s c e p t i b i l i t y are of the order of lO"" 1^ c.g.s. u n i t s , which would be com-pl e t e l y masked by the atomic diamagnetism (10~^ c.g.s.) i n s t a t i c measurements. 6 The simple induction experiment The separation of nuclear s u s c e p t i b i l i t y from the atomic s u s c e p t i b i l i t y i s accomplished by making the measure-ment at a frequency corresponding to a magnetic anomalous dispersion. In the stable state, the p o l a r i z a t i o n MQ has the same d i r e c t i o n as the f i e l d HQ . On the c l a s s i c a l p icture, i f we somehow disturb the sample momentarily so that M Q makes an angle © with H Q , M 0 w i l l precess about the f i e l d d i r e c t i o n (Figure 1.) with the Larmor frequency CJC = where X i s the gyromagnetic r a t i o defined by • I t requires energy to set the p o l a r i -zation at an angle to the f i e l d and, as t h i s energy i s gradually l o s t to the atomic l a t t i c e , the precession angle w i l l decrease to zero. I f , however, we disturb the sample with a small r - f . magnetic f i e l d 2H, c o s c o t at r i g h t r P i g . I. angles to the steady f i e l d , a stable precession w i l l occur when CO i s close to CO. This i s the resonance effect and i t i s very sharp. On the microscopic picture, we are inducing magnetic dipole t r a n s i t i o n s between possible orienta-tions of the nuclear magnetic moment i n the steady f i e l d . As Bloch (BJ) points out, we can consider the c l a s s i c a l motion of the resultant p o l a r i z a t i o n of the sample because the quantum-mechanical expectation value, i n i t s time dependence, follows exactly the c l a s s i c a l laws. The resonant frequencies for nuclei i n a f i e l d of 1 0 , 0 0 0 gauss l i e i n the range from one to f i f t y megacycles. Resonant frequencies for atomic dipoles i n the same f i e l d l i e i n the microwave region. The p o l a r i z a t i o n M Q i s measured by the magnitude of the transverse rotating component which appears only at resonance. To do t h i s , we measure the r - f . voltage induced i n a c o i l surrounding the sample with i t s axis perpendicular to the f i e l d H Q. This may be the c o i l providing the r - f . f i e l d or i t may be a separate c o i l , depending upon the method of detection (see chapter 2). The measurement i s s i m p l i f i e d i f we produce a recurrent i n d i c a t i o n of resonance by sweeping the frequency or the f i e l d strength at an audio rate. Two methods are commonly used. In the f i r s t (figure 2), the f i e l d i s swept completely through resonance and the magnitude of the transverse p o l a r i z a t i o n i s observ-ed by displaying the r e c t i f i e d output from the pick-up c o i l on an oscilloscope. I f the h o r i -zontal d e f l e c t i o n on the oscilloscope follows exactly the modu-l a t i o n of the f i e l d strength, we obtain a plot of the trans-verse component against f i e l d strength. Frequency modulation may be used i n place of f i e l d modulation, but the l a t t e r i s commonly used since i t may be simply applied with a p a i r of P c a a,. 8 small c o i l s . The second method (figure 3 ) f uses a f i e l d modu-l a t i o n much smaller than the resonance width. This produces a small modulation on the induced r - f . voltage whose amplitude depends on the slope of the resonance curve. I f the frequency or f i e l d i s changed very slowly, the slope may be plotted on a recording meter by the " l o c k - i n " detector technique. Figure 5 i s a reproduction of a meter ^ 8 * 3. tracing. Blooh* s equations In order to describe the motion of M Q , Bloch (Bj5) introduced two "relaxation" times, T-j_ and Tg, which describe the i n t e r a c t i o n of the nuclear dipoles with t h e i r environment. The component of MQ p a r a l l e l to HQ determines the energy of the p o l a r i z a t i o n . This component can only vary by absorp-t i o n of energy from the dr i v i n g f i e l d % or by i n t e r a c t i o n with the atomic l a t t i c e . I f we assume that the effect of the l a t t e r i s to produce an exponential time dependence, we can describe i t by the "time constant" However, the trans-verse components have no energy associated with them and we have the added p o s s i b i l i t y of non-energetic i n t e r a c t i o n s . The f i e l d HQ i s not homogeneous and the Larmor frequency has a spread throughout the sample. Individual nuclear 9 components i n t e r f e r e and the transverse components may d i s -appear i n a time much less than T^. By again assuming an exponential time dependence, we can describe t h i s process by a transverse relaxation time T 2. The f i e l d inhomogeneity may a r i s e from the presence of atomic dipoles and neighboring nuclear dipoles as we l l as from the magnet i t s e l f * Using a model characterized by these rel a x a t i o n times, Bloch derived a set of d i f f e r e n t i a l equations which give a complete description of the motions of M Q. These have been solved only for c e r t a i n conditions, and we are interested i n the oase of "slow passage". The solutions are expressed i n terms of u and v , the amplitudes of the components of the transverse p o l a r i z a t i o n which are i n phase and out of phase with the d r i v i n g f i e l d H-^ . u = *T*H,Ato M ( 2 ) where A CO * ° ° o ~ Following Bloembergen (B 9), we w i l l f i n d i t convenient to consider the l i n e a r complex transverse s u s o e p t i b i l i t y . Thus UL=ZHX9 V*2HX' and y" = ± y 71 Ho y , s ) 10. The components u and v have the shape of d i s -persion and absorption curves for a damped harmonic o s c i l -l a t o r . This was put into the equations by the assumption of an exponential form for the rela x a t i o n terms. As Rj. i s increased, the absorption v passes through a maximum and decreases to zero. On the microscopic picture, t h i s f a l l i n g o f f i s equivalent to equalizing the populations of the energy l e v e l s by r a i s i n g n u c l e i to higher energies faster than they can lose energy to the l a t t i c e . The reaction on the d r i v i n g o o i l In most experiments we are interested i n the change i n the impedance of the d r i v i n g o o i l caused by the l i n e a r transverse s u s c e p t i b i l i t y at resonance. The author considers the c o i l as a p a r a l l e l combination of an inductance L and a small loss 1/R (figure 4), and finds the added loss A l/R (absorption) and the change i n inductance which appears at nuclear resonance, as seen from the o o i l terminals. We may write ~"l L = (I+W<,?C)LW <?t where q i s the " f i l l i n g f a c t o r " J for the c o i l . Then PcO T l 1 8 0 O ^ 1 l^PQcL 2 1 1 1 0 6 i s given by © X - X + J -and, since the changes are very small, 11. The r e a l part of t h i s expression i s the added loss The change i n inductance i s AL U t i l i z i n g expressions (4) and (5), and The slope of the resonance absorption z^J& r ) i f l recorded when using the l o o k - i n detector technique. The measurements w i l l be made with no saturation, that i s This expression gives two peaks (figure $) when f A to = ± The value of expression (8) at i t s two maxima i s 2Wc (9) and the relaxation time T 2 i s related to the separation by = /3 7T* (separation o f peaks i n cycles per second). Relaxation times The previous development shows the strong dependence of the output on the rela x a t i o n times. Unfortunately, l i t t l e i s known about the mechanism of energy transfer between 12. n u c l e i and the atomic l a t t i c e . The coupling i s very weak and th e o r e t i c a l predictions of rel a x a t i o n times are much i n excess of measured values. Bloembergen (B?) has calculated T]_ and T 2 by considering the frequency spectrum of the fluctuating f i e l d at the nucleus due to thermal motions of neighboring nuclear dipoles. He predicts (and has v e r i f i e d ) that T i and T 2 should be equal provided the thermal motions are rapid compared to the Larmor frequency. The relaxation times for protons i n pure water are of the order of seconds. Any energetic i n t e r a c t i o n between molecules and nuclei w i l l tend to shorten relaxation times. The addition of paramagnetic ions provides a strong coupling, and even a low concentration w i l l greatly reduce T]_ and T 2 . An e l e c t r i c i n t e r a c t i o n i s also possible with many n u c l e i . I f an e l e c t r i c quadrupole moment e x i s t s , i t can react with the non-uniform e l e c t r i c f i e l d of the molecules giving a relaxa-t i o n time inversely proportional to and to the e l e c t r i c f i e l d gradient (B9). This may explain the broad resonance for some n u c l e i . As an example, 1^27 f which i s known to have a large quadrupole moment, shows a resonance width between points of maximum slope of 14 kc; on the other hand, shows a, width of a few hundred cycles determined by the magnet. The inhomogeneity i n the f i e l d of the el e c t r o -magnet used for these experiments gives an upper l i m i t to T 2 • I f the f i e l d varies over the sample by an amount ^ H Q , ace p . IS. . __ 1 3 . T 2 w i l l have a maximum value of about l / y A H 0 . The magnet must provide a f i e l d uniform to better than 1 gauss over the sample to avoid reducing the size of the s i g n a l . The author wishes to stress the importance of the homogeneity of the magnetic f i e l d to anyone contemplating construction of proton s t a b i l i z e d magnets. Transient phenomena - "wiggles" In a f i e l d of s u f f i c i e n t uniformity, a s t r i k i n g pattern appears on- the oscilloscope when we sweep through resonance i n a time short compared to T 2 (figure 6). A simple q u a l i t a t i v e explanation i s that the nuclei do not follow the rapid v a r i a t i o n and are l e f t processing f r e e l y after the f i e l d has passed conditions of forced resonance. The precession frequency i s determined by the instantaneous f i e l d which i s steadily changing. Thus the nuclei precess at a d i f f e r e n t frequency from the r - f . f i e l d and the two frequencies produce a r i s i n g beat note, which i s the "wiggle." In figure 7> the forward and back traces have been separated. The "wiggle" appears after passing through resonance, but i n t h i s picture i t has not died away completely before the approach to the next resonance. L The wiggles may be useful. Jacobsohn and V/angsness ( J l ) showed that the pattern i s perfectly symmetrical when the f i e l d i s centered on the resonance. This has been used for precision comparison of magnetic moments (B7). The size of the wiggles depends strongly on the f i e l d uniformity and 14. t h i s provides a sensitive method f o r exploring the magnetic f i e l d . 1 5 . Chapter 2. EXPERIMENTAL METHODS In t h i s chapter, we consider the methods of con-version of the rota t i n g transverse p o l a r i z a t i o n to audio signals. The difference i n technique between various experi-mental groups l i e s i n t h i s i n i t i a l stage. A description of the basic methods i s given, but, for each method, a further v a r i a t i o n exists i n the amount of am p l i f i c a t i o n used before r e c t i f i c a t i o n of the r - f . voltages. A l l groups use both the oscilloscope and recording meter for observation of the audio signals, as outlined i n chapter 1. This chapter also contains the a u t h o r s treatment of the o s c i l l a t i n g - d e t e c t o r j the actual c i r c u i t d e t a i l s are given i n the next chapter. L i s t of methods The references i n t h i s l i s t are to sources of further information and do not neoessarily r e f e r to the originator. (a) Simple measurement of the voltage changes across the d r i v i n g c o i l when i t i s fed from a constant current source. (b) The c o i l i n (a) replaced by a resonant cavity. (B9) 1 6 . (o) A detected o s c i l l a t o r signal may be balanced against the audio signal i n (a), cancelling the variations i n o s c i l l a t o r amplitude, (amplitude bridge - T2) (d) The c o i l may be incorporated i n one arm of an r - f . bridge. (BIO) (P2) (e) An i n i t i a l balance of the o s c i l l a t o r voltage may be obtained by placing a pick-up c o i l at right-angles to the d r i v i n g c o i l . (B6) (f) The driv i n g c o i l may be the o s c i l l a t o r tank c o i l . This arrangement i s s i m i l a r to the autodyne detector, (used i n t h i s research) (g) The o s c i l l a t o r i n (f) may be quenched at an audio rate, as i n the super-regenerative detector. (Rl) Comparison of methods presence of a r e l a t i v e l y large o s c i l l a t o r signal at the r - f . detector. This "unbalance" voltage i s much larger,than that a r i s i n g from the nuclear induction and i t determines the An important feature, common to a l l methods, i s the phase of the component of the transverse p o l a r i z a t i o n which i s detected. Figure 8 w i l l make t h i s clear. V 0 i s the unbalance ' voltage whose phase with respect to Ej_ i s determined by the apparatus. The signal V s (exaggerated) has a phase dependent on detuning. I t i s composed of the 1 7 . two components u and v, which are i n phase and out of phase with H 1 # The r e c t i f i e r detects only changes i n amplitude. Thus only the component of the signal voltage i n phase with Y 0 w i l l be detected and t h i s may be pure d i s -persion (u), pure absorption (v), or a mixture of both. C i r c u i t s of type a, b, and c detect almost pure absorption, while types d, and e have an adjustable unbalance phase and may detect either component. The autodyne c i r c u i t s , i n which we are p a r t i c u l a r l y interested, usually detect absorption but may be b u i l t so that they detect dispersion; the super-regenerative types are s i m i l a r . The ease with which a c i r c u i t can be adjusted to the desired conditions governs, to a great extent, the choice of method. For hunting unknown or poorly known resonances, i t i s convenient to have a c i r c u i t whose f r e -quency can be varied over a wide range by a single co n t r o l , driven by clockwork. C i r c u i t s of type (f) and (g) have t h i s property automatically, but the bridges (d) and (e) require many simultaneous adjustments. For proton s t a b i l i z a t i o n of a magnet which i s not continuously varied, the amplitude bridge (c) provides a simple arrangment. For i n v e s t i g a t i o n of l i n e shapes or precision measurement of nuclear moments where i t i s important that only pure absorption or disper-sion i s observed, the bridge methods are preferable. Quantitative consideration of simple voltage measurament Before considering the autodyne c i r c u i t i n d e t a i l , a discussion of the equations pertaining to the simple voltage measurement of type (a) w i l l be given. I t w i l l be i n t e r e s t i n g to compare these equations with s i m i l a r ones which w i l l be derived for our c i r c u i t . The method consists of measuring the r - f . voltage across the d r i v i n g c o i l , the r e c t i -f i e r usually being i s o l a t e d from the c i r c u i t by an r - f . a m p l i f i e r . The c i r c u i t , which we assume to be tuned, i s driven with a constant current i (figure 9)« This develops a voltage V*0 = t R 0 , where Rq i s the p a r a l l e l tuned impedance of the c i r c u i t . In a s i m i l a r manner to that used i n the l a s t chapter, we can write In t h i s case AV = ^ ^ A ^ which, remembering R = QLu) &\j=iQlLo>4r*(x"+iX') may be r e a d i l y evaluated as A v - <• << b W 7 » ^ i a. -rj As shown above, we w i l l only detect the r e a l part of t h i s expression, since a large voltage of t h i s phase i s present at our detector. The current cannot be made a r b i t r a r i l y large and to remove i t from the expression, we use H, = f - t i / Z 7 do) where c i s a constant depending on the shape of the c o i l . 19 o increases as the dimensions of the c o i l are increased} the size of the c o i l , however, i s l i m i t e d by the available volume of uniform f i e l d . Thus AV ^JfTQjTcoycH,?" CO V (ll) Both v and i t s derivative (used i n the recording meter method) pass through a maximum as H-|_ i s increased. For com-parison, i t w i l l be s u f f i c i e n t to write for the detected outputs where the function i s determined by the method of modulation. - For signal to noise r a t i o comparisons, i t i s con-venient to consider the detected output from (11) without reference to the method of modulation. We also detect noise e * = 4I<TRau ^UTavQLoj i n each side-band determined by the bandwidth A\> of the audio system. These side-bands add quadratically giving a signal to noise r a t i o S_ _ " q c Q 6 0 0*) which i s not dependent upon the inductance of the c o i l . The troubles with t h i s simple arrangement are other sources of noise. Modulation of the o s c i l l a t o r amplitude may be1 balanced out by the amplitude bridge. The r e c t i f i e r contributes shot noise which i s roughly proportional to the square root of the voltage detected. 20 V0 = CQLoo = cH,^co i t can be seen that some improvement i n signal to noise r a t i o from t h i s source may be gained by making L large. A better method i s to provide s u f f i c i e n t a m p l i f i c a t i o n between the c i r c u i t and the detector to make the shot noise n e g l i g i b l e . The tuned c i r c u i t would then be adjusted to give the minimum possible noise factor for the r - f . a m p l i f i e r . Noise i n the r e c t i f i e r may be reduced s t i l l further by almost cancelling V 0 by means of a bridge network, but t h i s i s accompanied by a considerable reduction i n the f l e x i b i l i t y of the c i r c u i t . The autodyne or o s c i l l a t i n g detector The autodyne detector i s a weakly o s c i l l a t i n g o s c i l l a t o r which shows large changes i n amplitude whenever i t receives a voltage of frequency near to i t s own. I t was used i n the e a r l i e s t days of radio communication because of i t s extreme s e n s i t i v i t y , the name a r i s i n g from the hetero-dyne note produced when i t i s almost tuned to the incoming wave. The name o s c i l l a t i n g detector i s suggested by the author f o r an autodyne detecting i t s own frequency. There i s a p a r a l l e l with the l o c k - i n detector which i s a zero-frequency heterodyne detector; the signal i s mixed with a voltage of i t s own frequency. The use of t h i s method for the detection of nuclear induction signals was suggested by Roberts (Rl) and was used independently by Pound (P4). Many others are apparently now using the same method (HI) (P8) (EL) (A2) ( A l ) , 2 1 . but few d e t a i l s have been published. No c i r c u i t diagrams are available and the author i s not aware of any published analysis of t h i s method of detection of nuclear resonance. In order to understand the operation of the o s c i l -l a t i n g detector, we must inquire into the mechanism l i m i t i n g the amplitude of a vaeuum tube o s o i l l a t o r . I t i s w e l l known that the equation f o r a harmonic o s c i l l a t o r contains no l i m i t -ation on the amplitude of o s c i l l a t i o n , and we may expect a vacuum tube o s c i l l a t o r with more than c r i t i c a l regeneration to have an i n f i n i t e amplitude. This, of course, does not occur; the usual explanation i s the added loss i n the c i r c u i t when the amplitude i s large enough to cause a flow of g r i d current. However, i t i s possible to adjust o s c i l l a t o r s so that the amplitude i s stable when no g r i d current i s flowing (class A). This problem was p a r t l y solved by van der Pol i n 1 9 2 0 , and we s h a l l apply his r e s u l t as presented i n h i s Nonlinear Theory of E l e c t r i c O s c i l l a t i o n s ( V 2 , 1 9 3 4 ) , to our problem. Consider the simple triode o s c i l l a t o r shown i n figure 1 0 . The rate of change of the plate current i s = Q l . C t v where v i s the voltage ott 3** ott across the tuned c i r c u i t . The voltage induced i n the tank c o i l i s the sign depending on the d i r e c t i o n of f*Lo* the windings. . • . Now L L = <-R + L c or and It 1^ - oC V - £ (aV - j if V 3 which i s non-linear, then This i s the d i f f e r e n t i a l equation for the vacuum tube o s c i l l a t o r , assuming that three terms of a power series expansion are s u f f i c i e n t to describe the vacuum tube charac-t e r i s t i c . This equation may be applied to any c i r c u i t by using for — ©t the negative conductance that the tube places across the tuned c i r c u i t . Van der Pol solved t h i s equation under the assumption that j3 = 0, that i s the tube i s working about a point of i n f l e c t i o n i n i t s c h a r a c t e r i s t i c . We s h a l l see that this, condition i s s t r i c t l y s a t i s f i e d i n our c i r c u i t , but the results have some v a l i d i t y f o r other c i r c u i t s since the behaviour i s not strongly dependent on ^ 3. Van der Pol find s , for the case where the changes are slow compared to the frequency, that the amplitude V which reaches a l i m i t i n g value of V = / f 3 M This equation shows how an o s c i l l a t o r may be used as a sensitive detector. A change of i n the tank c o i l , as found at nuclear resonance, or a change of feedback oc produces a change i n the amplitude of o s c i l l a t i o n . The non-linear term X determines the magnitude of the change i n amplitude, which i s the nuclear resonance s i g n a l . This may be very large i f the c i r c u i t i s almost l i n e a r , a condition which cannot be obtained with normal g r i d leak biased o s c i l l a t o r s . The operation of class A o s c i l l a t o r s requires some form of automatic amplitude control (A.V.C.), since only a small change i n oc i s required to raise the amplitude of o s o i l l a t i o n from zero to the point where grid current start s to flow. The A.7.C. may control the trans-conductance of the tubes (and therefore oC ) with a voltage obtained by r e c t i f y i n g the r-rf. output. This w i l l be des-cribed f u l l y with reference to the p a r t i c u l a r o s c i l l a t o r used i n t h i s research. The o s c i l l a t i n g detector method i s not l i m i t e d to investigations of nuclear magnetic resonance but may perhaps be applied with p r o f i t to other physical measurements. For example, i t should be possible to f i l l the cavity of an external cavity klystron o s c i l l a t o r with a gas showing micro-wave absorption. The amplitude of o s c i l l a t i o n may be con-t r o l l e d by an A.7.C. on the gr i d c o n t r o l l i n g the beam current. I f an almost l i n e a r o s c i l l a t o r can be obtained, t h i s simple microwave spectrograph should have great s e n s i t i v i t y * Nor i s the method l i m i t e d to high frequencies. Schneider (Si) uses an audio o s c i l l a t o r as both the source of a modulating voltage and a " l o c k - i n " detector of the audio voltage after some physical measurement. He discovered that the normal l i m i t a -t i o n - the product of response time and bandwidth i s approxi-mately unity - no longer holds. This he a t t r i b u t e s to the essential non-linear form of the equations governing the amplitude of an o s c i l l a t o r . p r -o p M L O C K - I N Y . T . V . M . O-C. R-F. D E T E C T O R Recording M l l I ammeter I t . 25 Chapter 3 . THE RECORDING NUCLEAR MAGNETIC RESONANCE SPECTROMETER The spectrometer b u i l t for t h i s research i s shown i n block diagram i n figure 1 1 . Most of the parts follow standard electronic design; but two of them, the o s c i l l a t i n g detector and the l o c k - i n detector, were developed s p e c i f i -c a l l y for t h i s instrument. These w i l l be described i n d e t a i l . The magnet was also designed by the author but i t w i l l be _ • described, along with i t s controls, i n a separate chapter. The o s c i l l a t i n g detector or o s c i l l a t i n g detector used i n this apparatus. This c i r c u i t was developed with the guidance of the theory given i n the l a s t chapter. The c i r c u i t i s not a standard one, so the conditions for o s c i l l a t i o n w i l l be developed. The o s c i l l a t o r shown i n figure 12 i s the autodyne Figure 13 shows one tube of the push-pull p a i r . C]_ . The current i flowing i n t h i s condenser i s given by out of phase with the voltage The voltage applied to the g r i d i s applied to the feedback condenser 0 St l\ \ a t In g D e tec tor. Ri» tOK C? - iooo R3» l.f K Gs 3 I0OO f?5» *?K Co* . 5 * ^ . R6--U0K Cu tu»o ge*^ - 140 pf. Rs* tMe$ Ui - to*e« sr-7 -l0-*3foc. Ci 8&rb oMAmic Cft» 500 L l - tun** o.t 7 Mc uKfcK Cv 1000 cotl co^acct^ + 3 cs- . o U k R l l tubes 6 R G 5 +15© to /«C€ /»*5< 2 6 • 26 The plate load of the tube i s e f f e c t i v e l y only the capacity o2 . so v„ = gmVjc.co Thus I = V ( j C , c o - | l or the admittance looking into the feedback condenser i s C which has a negative conductance — -—L cj*, • I f the two tubes are placed across a tuned c i r c u i t , o s c i l l a t i o n s w i l l s t a r t when t h i s negative conductance com-pl e t e l y cancels the c i r c u i t l o s s . I f , as usual, l/R i s the loss and C i s the t o t a l tuning capacity • R ~ Q ~ 2 C O So for one tube n > O c i Ceo _ 0 C % I /IQ» I t i s important to note that there i s no reactive component of feedback involving , r e s u l t i n g i n frequency s t a b i l i t y against changes i n tube voltages. Since, for a given c o i l , Q i s approximately independent of frequency, the c i r c u i t may be tuned with a variable condenser over a wide range without readjustment of C^/G^ . The Van der Pol theory may be applied d i r e c t l y to t h i s o s c i l l a t o r since push-pull c i r c u i t s automatically operate about a point of i n f l e c t i o n . The d i r c u i t was designed for s t a b i l i t y , wide tuning range, and s i m p l i c i t y ; and t h i s t h e o r e t i c a l s i m p l i c i t y i s a bonus. I t i s desirable to have as nearly l i n e a r tubes as possible; 6AG5rs were the best pentodes found for t h i s purpose. 2 7 . They have the added a t t r a c t i o n of a uniformly low noise r e -sistance (1000 ohms), which i s a great improvement over the e r r a t i c 6AKJ5. I t should be possible to use twin triodes but res u l t s with the minature 6«T6 were unsatisfactory. Careful measurements on 6AG5»s showed that i t i s possible to vary the transconductance from 3ma/v to 7 ma/v by varying the g r i d bias over a 2 v o l t range. The t h i r d d i f f e r e n t i a l c o e f f i c i e n t X"' remains p r a c t i c a l l y constant at 0.4 ma/v^ during t h i s . v a r i a t i o n . This allows us to apply the A.V.C. necessary to maintain the o s c i l l a t i o n at a low l e v e l by varying ©C without a f f e c t i n g X . To prevent loading the c i r c u i t with the non-linear c r y s t a l detector, a wide band low gain amplifier i s used. The c r y s t a l i s connected so that i t generates negative voltage which i s used to bias the 6AG5!s. A network inserted i n t h i s A.V.C. lead reduces the feedback of audio voltages 20 times for a l l frequencies above about 30 cycles. I t i s necessary that a l l audio frequencies of interest have the same feedback factor i f the detector i s to have a uniform audio frequency response. The r - f . voltage may be adjusted by varying the capacities C^. We usually work at about 1 v o l t across the whole tuned c i r c u i t : t h i s remains constant throughout the tuning range. Under these conditions °* - T ^ C|m =• 2 x 1 0 " ^ varying with tuning Y — J_ £j_ Yf - 0 . 2 x 10"*^ (constant) S e n s i t i v i t y of the o s c i l l a t i n g detector The s e n s i t i v i t y i s given by the change i n output 28. voltage for a change A l/R In the tuned c i r c u i t . We define m as the d-c. output for 1 v o l t across the tuned c i r c u i t (m=2), and b as the change i n oc for a 1 v o l t change i n output due to the A.V.C. The voltage across the tuned c i r c u i t V i s .Applying A.V.C. ^ V a V _ A ( ± ) or a\/ A ( i i r ) (19) The audio output voltage V a i s m times t h i s At small voltages For a-c. changes b has a smaller value than for d-c. changes. Above J50 cycles b=?10""^ and i s constant. has a maximum for H, near zero. Substituting from (6) with H, small , , 9-nr -T-when the d r i v i n g f i e l d H-j, i s small. For the d i f f e r e n t i a t e d signal used i n the l o c k - i h method where Sco i s the audio magnetic f i e l d sweep - <5co =• XSH0 F i n a l l y , substituting for *JC0 from ( l ) 29. Comparison of equation (21) with (12) for the out-put from the simple voltage measurement shows two important differences. F i r s t , the output i s increased i n the o s c i l l a t -ing detector by decreasing the inductance. Secondly, the output for a given material does not depend on the frequency, provided the tank capacity only i s varied. This was v e r i f i e d for proton signals, which have the same size at jmic. or lOmc. This property i s useful when comparing signal sizes with the spectrometer. The signal to noise r a t i o I t i s desirable to compare the sig n a l to noise r a t i o of th i s method with the simple measurement of voltage. This requires a knowledge of the noise present as a modulation on a van der Pol o s c i l l a t o r . This i s a very d i f f i c u l t prob-lem i n non-linear mathematics for which a solution has not been published. An attempt by the author to evaluate t h i s noise i s contained i n appendix I . The v a l i d i t y of a basic simplifying assumption could be v e r i f i e d by an exact s o l u t i o n , or by experiment. Such experiments have not been carried out, as yet, since they constitute, i n themselves, a separate research. The solution i s not inconsistent with the behaviour of the o s c i l l a t i n g detector. The audio noise which may be detected as a modu-l a t i o n on the o s c i l l a t o r i s (equation 13, appendix I) Since the A.V.C. feedback introduces no detuning, i t has no effect on the signal to noise r a t i o . 30. The amplitude change of the o s c i l l a t o r i n the absence of A.V.C. i s . _ A (is) ^ Wo X" and the signal to noise r a t i o becomes I t i s desirable to remove V so that we may compare th i s r e s u l t with the simple case. V cannot be a r b i t r a r i l y large because of saturation. We can remove i t by using V~iLu> =• c Hf^L cO where c i s the same as i n (10), This gives ^ _ V/7yc X" ' H, (V**)"* . (2*) Introducing „ s Hoc Q <*> . i r / 2 S ) which has a maximum value, since the noise i n i t i a l l y decreases faster than the signal as H]_ i s rais e d . Compare t h i s to equation (13) - they d i f f e r only by a factor of two. The necessary modification of the simple c i r c u i t by an amplitude or r - f . bridge w i l l remove t h i s small difference. Thus the o s c i l l a t i n g detector has the same u l t i -mate s e n s i t i v i t y as the simple voltage measurement across a dr i v i n g c o i l fed with a noiseless current. The o s c i l l a t i n g detector i s a much simpler c i r c u i t . For equivalent performance, the "simple" measurement requires: a separate o s c i l l a t o r , a tuned driving c o i l , an amplitude or r - f . bridge, an r - f . a m p l i f i e r , and a detector. The number of controls for continuous frequency v a r i a t i o n i s much larger than the single one required for the o s c i l l a t i n g detector. As pointed out i n the appendix, the noise factor of the tubes must be taken into account. This factor depends on 3 1 . the tuned c i r c u i t impedance and the frequency, but not strongly. I t i s also convenient to have the noise and the signal from the o s c i l l a t i n g detector large enough that no special care need be taken with subsequent audio am p l i f i e r s . These considerations, together with the p r a c t i c a l l i m i t a t i o n of available tuning condensers of sturdy construction, deter-mine the inductance of the o s c i l l a t o r tank c o i l . The induct-ance w i l l generally be smaller than average practice. In our c i r c u i t , the l i m i t a t i o n was imposed by the available tuning condenser; an increase i n s e n s i t i v i t y with a s l i g h t reduction i n signal to noise r a t i o should be possible by doubling i t s size. Details of construction The o s c i l l a t i n g detector i s housed i n a r i g i d l y constructed box to reduce microphonics (figures 14 & 1 5 ) . This box s i t s between the magnet c o i l s and the tubes are oriented so that the stray f i e l d w i l l have the l e a s t effect on t h e i r beam current. The tank c o i l i s housed i n a small box (figure 1 6 ) , connected to the main chassis by 1/2" brass tubes which carry the leads. The tank c o i l consists of 40 turns on a t h i n walled l u c i t e form, 1/2" long. Samples are placed i n 10 mm. test tubes, which f i t snugly into the c o i l form. The sides of the c o i l box carry the f i e l d modulation c o i l s . The frequency of the o s c i l l a t o r i s adjusted by a single control which can be driven by a synchronous motor and adjustable gear t r a i n . The slowest speed, which i s often used, 3 2 . i s one complete revelution i n 60 hrs. With the c o i l described, the frequency range i s 5 . 7 "to 10.4 mc. The l o c k - i n detector The block diagram shows the r e l a t i v e p o s i t i o n of the lo c k - i n detector. I t converts the amplified audio s i g n a l , which appears i n the small sweep method (figure 3 ) , into a d-c. voltage which i s recorded by the vacuum tube voltmeter and recording meter. A l o c k - i n detector i s a zero frequency heterodyne detector which has the important properties of res-ponding to only a narrow band of frequencies centered on the l o c a l o s c i l l a t o r frequency, and of responding proportional to the cosine of the phase angle between the signal and l o c a l o s c i l l a t o r voltages. For t h i s apparatus, a necessary property i s extreme s t a b i l i t y of the zero s i g n a l d-c. output voltage. The c i r c u i t i n figure (17) which was developed f o r t h i s apparatus has a stable zero signal voltage of l e s s than 0 . 0 0 5 volts independent of the size of the l o c a l o s c i l l a t o r signal. The four c r y s t a l diodes form a diode switch. When the l o c a l o s c i l l a t o r signal reaches i t s peak a l l four diodes conduct simultaneously, e f f e c t i v e l y shorting the input con-denser to ground for a f r a c t i o n of a cycle. When properly phased, t h i s switching action converts the incoming s i g n a l to a d-c. voltage with a r i p p l e of equal s i z e . The r i p p l e i s 3 3 . removed by an adjustable time constant ( 0 . 1 , 0 . 3 , 1 , 3 , 10 sec.) which also determines the bandwidth of the whole appara-tus. The d i s t i n c t i v e feature of t h i s c i r c u i t i s the use of a pen-light c e l l for diode bias; normal resistance-capacity bias has a much poorer s t a b i l i t y . Rest of the apparatus The remaining parts of the spectrometer follow conventional designs. (a) The amplifier contains two sections, each a two tube feedback amp l i f i e r with cathode follower output. Change of gain i s accomplished by feedback switching; the o v e r a l l gain i s 30 to 100 db. adjustable i n 10 db. steps. (b) The vacuum tube voltmeter i s described on page 480 of "Valley and Wallman (71). I t i s very s a t i s f a c t o r y and provides f u l l scale ranges of 0 . 1 , 0 . 3 , 1, 3 , and 10 v o l t s on the recording Esterline-Angus 1 ma meter. (c) The sweep and l o c a l o s c i l l a t o r are provided by a Hewlett-Packard audio o s c i l l a t o r with suitable matching transformers. (d) Power for the o s c i l l a t i n g detectors and the amplifier i s supplied by a vacuum tube regulated power supply. Filament power comes from storage batt e r i e s . (e) Frequency measurements are made by a General Radio type 620-A heterodyne frequency meter, often with the aid of a communications receiver. A l l apparatus i s housed i n a shielded cage. The high r . f . noise l e v e l i n t h i s b u i l d i n g makes th i s necessary. 34. The cage also provides freedom from interference by neighbor-ing spectroscopic arcs. Yoke constructed Prow hot rolled ekeel bolted together. K I T " i b o o few™ W | a - _ _ r x x x x x x x x m eottt cooling lo^e*. 2B5-*-7lfc 0.0. copper tube. C*o*v$ecUo* of coil. 1 2 0 0 l b , P 1 R G N E T . detail of fidge. pole cap- for /o /ace 3 5 . Chapter 4. THE MAGNET AND ITS CONTROLS The magnet used for nuclear spectroscopy costs as much as a l l the rest of the spectrometer. For t h i s reason alone, a description of the design deserves i t s own chapter. Magnets of this type, p a r t i c u l a r l y s t a b i l i z e d magnets, w i l l be useful i n many other experiments and the magnet b u i l t by the author f o r the nuclear spectrometer w i l l undoubtedly out-l i v e the spectrometer by many years. The descriptions i n t h i s chapter of the magnet properties and controls are intended primarily for i t s future users. Magnet design The mechanical design of the magnetic yoke i s shown i n figure 1 8 . The material i s hot r o l l e d s t e e l , no spe c i a l magnetic materials have been used. The poles are 6 inches i n diameter with a gap of 3-1/2 inches. This gap i s reduced to 1-1/2 inches by two shaped pole caps, adjusted mechanically u n t i l they are p a r a l l e l to better than . 0 0 0 1 inches across the diameter. The pole caps, which were c a r e f u l l y machined, have a rectangular ridge turned at the outside edge i n order to produce a uniform f i e l d at the center of the magnet. The shape of the ridges was determined by extending the c a l c u l a -tions of Rose (R2) to include cases where the f i e l d r i s e s 3 6 . s l i g h t l y towards the edge of the gap. With the dimensions used, the f i e l d should r i s e 1 part i n 1 0 , 0 0 0 one inch inside the edge. The actual f i e l d i s homogeneous to 1 part i n 2 0 , 0 0 0 over the central two. inches, but a v a r i a t i o n of this amount may occur i n a much shorter distance. The inhomogeneity re-s u l t s i n a reduction of Tg (broadening of the signals) f o r the narrowest signals, but the l i n e width i s narrow enough f o r a l l but the most precise measurements. The c o i l s consist of four one inch layers each separated by cooling c o i l s . Each layer has approximately 800 turns of #12 Formal wire. The cooling c o i l s are single layers of 3 / l 6 O.D. copper tubing, f i v e layers for each c o i l . By connecting the f i r s t and l a s t layers i n series and the second and t h i r d layers i n series, each c o i l can be s p l i t into two sections having equal resistances of 7 ohms. Thus the whole magnet can have resistances of I f , 7 , or 28 ohms, depending on the connection of the four sections. The maximum current per section i s 12 amps. Each section has a selenium r e c t i f i e r connected backwards across i t , which d i s -charges the f i e l d very e f f e c t i v e l y when the switch i s opened. The voltage reverses across the c o i l terminals when the current i s broken, but i t r i s e s to a value less than the forward v o l -tage. Since uniformity i s more important than high f i e l d strength, there i s no choking of the magnetic c i r c u i t at the gap. The core saturates inside the c o i l s before the pole caps are saturated. With the 1-1/2 inch gap, the f i e l d i s l i n e a r l -m a z O-C. Aw\p. ftAaxnenfc* of \9-6AS7'5 09K w\ *e*Ce* cund fun ov> 115 Va-c M R G N E T T C u R R E N T . C ONTROL. r- WOV. 2K. d*c. level o u i i u t t wevib. Tube 6 S C 7 •S" r (single oxtV\od«.) eeplafee I voltage. s^s^x v» enput stage, used m d-c. anjp. Fcg. 19. 5 7 . with current up to about 3 amps which gives 6000 gauss. 7-1/2 amps gives 10 ,000 gauss while the maximum current of 12 amps gives only 14 ,000 gauss. Current control The magnet i s usually operated with a l l the c o i l sec-tions i n series (27 ohms), the current being controlled by a bank of 19 twin triode 6AS7 ts. The power i s supplied by the bu i l d i n g 1 s motor generator sets which can d e l i v e r up to 270 v o l t s d-c. ^he control c i r c u i t (figure 19) measures the mag-net current by comparing the voltage developed across part of a 1 ohm potentiometer with 3 v o l t s from dry c e l l s . The d i f -ference i s applied as a correction to the grids of the" 6AS7!s af t e r an amplification of 5 0 , 0 0 0 . A d-c. l e v e l control i n the amplifier adjusts the current so that the difference i s near zero. Since most of the disturbance comes from the motor generators, the magnet time constant should be as large as possible. The addition of a feedback path i n which the a-c. voltages appearing across the magnet are fed into the control amplifier near i t s output restores the natural time constant of the magnet (several seconds). The amount of a-c. feedback i s l i m i t e d by the appearance of o s c i l l a t i o n . This c i r c u i t reduces the fluctuations of the magnet current to about 1 i n 10,000; however, slow d r i f t s may be several Itimes greater. The source of the d r i f t i s v a r i a t i o n i n the heater current of the f i r s t tube of the d-c. a m p l i f i e r , which i s considerably improved by the use of a M i l l e r (M2) low d r i f t c i r c u i t . sweep I Q&CIUJWM ©ST. TD.C. OUTPUT Photon *fcabUl , *er - b>!&cfc "o > a. o u , s t a b i l i z e r output 1 1 \ 1 1 - U 1 1 1 - i 1 wicthce1 tatevfclnj fi£cA»Ulse* ^ollro^e. PROTON STW&*UZRT\ON. fig- a o . , /© /ace /». 39. 38. Proton s t a b i l i z a t i o n (PI) The magnetic f i e l d i t s e l f i s further s t a b i l i z e d by a control voltage derived from a proton resonance (figure 1 9 ) . The proton signal i s generated i n an o s c i l l a t i n g detector^ which i s r e a d i l y tunable from 20 to 4 5 inc. The 500 cycle audio sweep covers a small p o r t i o n of the signal which i s am-p l i f i e d and detected by a l o c k - i n detector s i m i l a r to the one described e a r l i e r . The d-c. output of this c i r c u i t i s plotted i n figure 2 0 , which shows that a f i e l d control voltage i s , r e a d i l y available over the c ent.ral r egion. This sig n a l : i s fed into the d-c. amplifier i n series with the comparison voltage. The c i r c u i t would o s c i l l a t e i f the gain were not reduced greatly. This i s conveniently done by throwing away voltage across a simple phase advance network (figure 20) whose use permits an increase i n the maximum loop gain. Although t h i s c i r c u i t makes the normal current control superfluous - i t has been inadvertantly turned o f f for several hours - i t i s s t i l l necessary i f a large disturbance should throw the proton s i g -nal out of i t s small control range. S t a b i l i z a t i o n better than 1 part i n 1 0 0 , 0 0 0 i s achieved by use of the proton resonance. Older version, to be replaced by the c i r c u i t described i n t h i s t h e s i s . 29. Chapter 5 . MEASUREMENTS AND STANDARDS This chapter discusses the operation procedure f o r the spectrometer and the methods of computing spins, magnetic dipole moments, and e l e c t r i c quadrupole moments from the i n -formation i t provides. Summary of important equations gyromagnetic r a t i o Larmor frequency cO = ){(-{< maximum d e f l e c t i o n w - i-STTqNT* I(T+l) j ^ 8 - ^ , on recording meter v*»«y bl_ 3 k T "=p = , / 3 7 / * ( c y c l Q S between peaks)  l i n e width audio sweep width Svo«Vo"Ho, ©"H^sweep current, i Choice of sample The nuclear species i n which we are interested i s placed i n the apparatus as a f r a c t i o n of the n u c l e i of an element i n a compound. For large signals, we require as many of the p a r t i c u l a r nuclei a s possible. We also require a long relaxation time T£, but there i s no point i n having i t longer than the l i m i t set by the magnet. A long relaxation time i s usually found i n l i q u i d s or solutions containing the element, and when i t i s possible to use these with high concentrations, 4G they are used i n preference to s o l i d s . For nuclei w i t h low r e l a t i v e abundance, enriched samples have been used but with the exception of deuterium, these are not r e a d i l y a v a i l a b l e . The s i z e of sample used i n t h i s spectrometer i s about 1 ml. With samples having a relaxation time l i m i t e d by the magnet, there i s some advantage i n adding a paramagnetic catalys t . Such samples have a long T i and there i s consider-able p o s s i b i l i t y of saturation. This applies to protons i n water and may be expected f o r n u c l e i w i t h spin 1/2 which w i l l not have a quadrupole moment. Operation procedure Using any available spectroscopic data or t h e o r e t i -c a l predictions, the magnetic f i e l d required to bring the resonance within the range of the instrument i s calculated. I f t h i s f i e l d i s f e a s i b l e , i t i s set up i n the magnet and s t a b i l i z e d by the proton resonance. The sweep f i e l d current and the l o c k - i n detector time constant are set at medium values and the frequency drive i s turned on at i t s lowest : speed. The spectrometer i s l e f t to hunt automatically f o r the resonance. I t i s possible to pass through resonance without „•.  detecting i t . The optimum conditions for recording a wide weak l i n e are quite different from those f o r a narrow strong l i n e and, i n the absence of any clue (such as a large quadrupole moment), several runs have to be taken before i t can be s a i d that the resonance i s i n another^region or i s too weak to detect. Once the resonance i s found, i t can be improved by 41. adjusting the parameters of the spectrometer. A broad reson-ance allows us to increase the audio f i e l d sweep proportionally. Also, since the meter records the si g n a l i n a longer time, the time constant of the detector can be increased, reducing the bandwidth and noise. The dependence of the signal on T2 can be almost removed i n t h i s readjustment. Measurement of frequency The magnetic f i e l d at the sample cannot be measured d i r e c t l y with the accuracy required. The measurement of the f i e l d i s made by adding an i n t e r n a l standard such as Na whose gyromagnetic r a t i o i s known. The measurement reduces to a com-parison of the frequencies of the unknown and standard reson-ances, ^his i s measured w i t h a heterodyne frequency meter having an accuracy of at l e a s t 1 part i n 10,000. The exact method of comparison has many variati o n s . The reader i s r e f e r -red to methods i l l u s t r a t e d by the measurements of Sb!21 and 0u^3 given i n the next chapter. Calculation of spin Values for the spin of most nuclei are ava i l a b l e from spectroscopic and other measurements. Where the spin i s i n doubt, i t can be checked from the observed maximum meter def-l e c t i o n s , l i n e widths, and audio sweep currents f o r the unknown and standard signals. We can eliminate a l l factors common to the two resonances and obtain where W i s the distance between the peaks and i i s the sweep 42.. current, x refers to the unknown and s to the standard. In t h i s measurement, one must be c a r e f u l that the conditions of small sweep and no saturation s t i l l pertain. Calculation of magnetic dipole moments We have from our apparatus the r a t i o of the resonant frequencies of the unknown and the standard. u s i n g published values, i t i s a simple matter to a r r i v e at the r a t i o of the resonant frequency of the unknown i n our sample to the proton i n water. This should be the r a t i o of the gyromagnetic r a t i o s and from the known value of the proton moment, we can calculate the magnetic moment of the unknown. The proton moment i s known from two separate methods, molecular beam measurements with considerable correction, and measurements at the National Bureau of Standards of proton resonances i n a magnet calibrated by the force on a conductor carrying a known current ( T l ) . These agree very well and the best value at present i s 2.7935 + .0004 nuclear magnetons (P3). The simple c a l c u l a t i o n outlined above i s probably quite safe for accuracies of a one part i n a thousand, but i t assumes that the f i e l d s at the nuclei are not disturbed by the atomic electrons. For free atoms, there i s an Internal d i a -magnetism a r i s i n g i n the electrons. The magnitude has been calculated by Lamb (LI) who finds a correction of 0.319 Z4/3 must be applied. This correction i s applied so that i t i n -creases the calculated magnetic moment. Since uncorrected frequency r a t i o s are usually quoted, t h i s correction i s applied 43 i n the l a s t step only. We are s t i l l assuming that every compound, at l e a s t i n solution, w i l l give the same resonant frequency. Recently, Proctor (P12) and Dickinson (D2) have shown that t h i s i s not true. P r o c t o r s i l l u s t r a t i o n i s most s t r i k i n g . A solution of NB4NO3 gives two Nl4 resonances several k i l o c y c l e s apart. Pre-sumably one arises i n the NH4 ion and the other i n NO^. A s i m i l a r e f f e c t discovered by the author for Sbl21 w i l l be. d i s -cussed i n the next chapter. I f this difference arises from the diamagnetism of the molecular electrons, i t i s d i f f i c u l t i f not impossible to calculate. The lack of central symmetry i n many molecules and ions further complicates the problem. Indeed, t h i s author feels that the differences observed w i l l be u t i l i z e d as an a i d i n the determination of molecular struc?-ture rather than the use of molecular structure to determine the true magnetic moment. These discoveries show that much of the extreme accuracy quoted i n the l i t e r a t u r e for frequency r a t i o s i s of dubious value, p a r t i c u l a r l y where the exact con-s t i t u t i o n of the sample i s not given. These considerations also apply to the i n t e r n a l standard used. I t i s hoped that s i m i l a r work w i l l be done on sodium compounds since i t lias been widely used as a standard. No differences have been found f o r proton signals i n a wide variety of samples and the precise measurements of the very l i g h t n u c lei must s t i l l be accepted. Frequency comparisons d i r e c t l y with proton signals have only the possible error i n the "unknown" resonance. 44. Calculation of quadrupole moments Pound has calculated a few quadrupole moments by-comparing l i n e widths for nuclei with known and unknown quadrupole moments (P5), (P6). I f Q, i s the only cause, the relaxation time I/T2 should vary as Qp, where Q, i s the quad-rupole moment i n cm?, and as the e l e c t r i c f i e l d gradient. The l a t t e r may be neglected when both n u c l e i are the same element, but when they d i f f e r , Pound calculates the r a t i o of the two gradients using ionic r a d i i . The author's own work (next chapter) throws some doubt upon these calculations, and more experimental and th e o r e t i c a l work i s necessary before r e l i a b l e 3 can be obtained from nuclear induction measurements. FIGURE. S i . to face page 45 4 5 . Chapter 6. RESULTS The nuclear spectrometer has been i n operation for only a few weeks. Some res u l t s of significance have been obtained already and are reported i n t h i s chapter. I t i s intended to exploit the excellent performance of t h i s spectro-meter i n the immediate future by extending the measurements to other nuclear species. Cu6? from copper i n the d r i v i n g c o i l Weak signals apparently a r i s i n g from the metallic copper i n the wire of the driving c o i l were f i r s t reported by Pound. We have found signals which have been i d e n t i f i e d as ar i s i n g from t h i s source (figure 21). The frequency r a t i o to sodium was c a r e f u l l y measured because the resonance was not immediately i d e n t i f i e d as Cu^5. The r e s u l t s are summarized here since they constitute a demonstration of the r e l i a b i l i t y of the frequency measurements. The sample contained a small amount of NaCl, which gave sodium resonances very close to the c o i l resonance. The following method was used to f i n d the frequency r a t i o : (a) The frequency was swept slowly down through the Cu resonance. After t h i s was complete, the gain, detector time constant, and audio sweep were reduced so the Na signal 46., would be c l e a r l y recorded. The frequency of the Na signal was observed by noting zero beat on the heterodyne frequency meter as the recording meter passed through zero. This procedure was repeated f i v e times. (b) The rate of sweep i n the region of the signals was obtained by sweeping the frequency over the same range and marking zero beats for a series of settings of the f r e -quency meter. (c) The Cu frequency was calculated by adding the small measured differences to the Na frequency. The whole procedure was repeated by rot a t i n g the tuning condenser 180° and sweeping the frequency up through the Na and then the Gu resonances. Since rather long time constants were used, the recording meter needle may.have lagged behind the true reading. This effect w i l l cancel out by averaging the two sets of meter readings. The least squares solution for the rate of sweep , with decreasing frequency was 19 .9 kc/inch, with a paper speed of 6 n/hr. The average of the f i v e tracings gives l^Cu^/V^Na 2 3 = 1.00425 + . 0 0 0 0 5 . Similar measurements with increasing frequency gives ^Ou65/V>Na2? = 1.00402 + .00004. The difference between the two groups of readings i s r e a l and can be attributed to the long time constant f o r the Cu traces. The average of the two groups i s 1.00415 +. .00006. The precision does not take into account systematic errors. An asymmetry i s e a s i l y v i s i b l e i n the traces (figure 21) which forced the author to reduce the precision considerably. A 47. r e l i a b l e estimate gives VCu6Vl^Na23 = 1.0041 + .0002. This value i s not i n agreement with the r e s u l t s of measurements on a cuprous s a l t . B i t t e r 1 s values (B2) give for t h i s r a t i o VGu&/V*Na23 = 1.0021 + .0002. However, a f t e r i t was known that the signals came from the c o i l , the value was checked against the work of Knight ( K l ) . He compared the f r e -quency of Cu&3 i n f i n e l y powdered metallic copper and i n cup-rous chloride, and found the metallic resonance to be 23.7 kc higher than the resonance from the salt which was at 10.123 mc. Using these values, we calculate J^Cu^3/l>'Na23= 1.0044 + .0002 for metallic copper. Our result i s i n good agreement with t h i s value. Knight also publishes a tracing of the copper s i g -nals which may be compared with f i g u r e 21. The signal to noise r a t i o i s about the same, although he had much more cop-per e f f e c t i v e l y i n the r-f. f i e l d . Knowing the o r i g i n of the si g n a l , i t i s possible to explain the asymmetry noted above. The f i e l d H]_ penetrates only a short distance into the copper wire and i s s h i f t e d i n phase from the f i e l d i n free space. We are thus detecting a mixture of absorption and dispersion giving a disto r t e d s i g n a l . Magnetic moment of Sbi21 Sbl21 resonances have been recorded with samples of HSbClg and SbCl£. The SbCl^, which i s a fuming l i q u i d , was used d i r e c t l y from the re-agent b o t t l e . The HSbCl^ was pre-pared by adding the SbCl^ to concentrated hydrochloric acid i n approximately equal amounts. Reproductions of the traces are FIGURE 22 to face page 48 48 are given i n figure 2 2 . No signals were found with an HC1 solution of SbCl^. Measurements were taken by placing one test-tube each of HSbCl^, SbCl^, and saturated solution of NaBr i n the c o i l of the o s c i l l a t i n g detector one af t e r the other. The center of each resonance was measured by the heterodyne frequency meter as the recording meter needle crossed the zero l i n e . The large signals made t h i s spot measurement possible, but the accuracy depends on the s t a b i l i t y of the magnet proton co n t r o l . The measurements were repeated three times. The frequencies quoted below are twice the actual value, because the frequency meter was measuring the 2nd harmonic of the o s c i l l a t o r . 1 s t run 2nd run 3rd run N a 2 5 i n NaBr 15.084 15.084 15.084 S b 1 2 1 HSbCl 6 15 .648 15 .648 15 .648 Sb ^ S b c i ^ 15,657 1J.657' 15 .656 r a t i o B B b 0 1 6 . 9 0 4 8 . 9 0 4 8 . 9 0 4 8 r a t i o SbCl c . 9 0 5 4 . 9 0 5 4 .9055 5 S h i f t i n g the p o s i t i o n of the NaBr sample i n the c o i l gave a difference of less than 1/2 i n the l a s t place. The values are: ^ S b 1 2 1 / V N a 2 5 - . 9 0 4 8 + .0001 for a sample of HSbCl^. This confirms exactly a value obtained by Proctor (private communication)11 S b 1 2 1 / ^ N a 2 ^ 0 , 9 0 5 4 + .0flD2 for a sample of SbCl^. The difference i s quite large, but the almost exact r e p e t i t i o n of the resonant frequencies indicates that i t i s r e a l . This i s a further example of the 1The author i s indebted to Professor Bloch for t h i s information. FIGURE 23. to face page 49 4 9 . chemical s h i f t found for f l u o r i n e and nitrogen isotopes by Dickinson (D2) and Proctor (P12). I f the r a t i o i s rounded o f f to .9051 the value for the magnetic moment becomes 3 .3 6 nuclear magnetons which i s to be compared with the spectro-scopic value obtained by Crawford (C3) of 3 .7 nuclear magnetons. Line widths of bromine and iodine isotopes Pound has reported resonances observed for I 1 2 ^ B r 7 ? , and B r 8 1 (P5) (P6). He found the iodine resonance to be 14 kc. wide and the two bromine resonances to be 10 kc. wide, B r 7 9 being s l i g h t l y wider than B r 8 1 . Iodine i s known to have a large quadrupole moment and, assuming the l i n e widths to ar i s e only from quadrupole coupling, Pound calcu-l a t e d the quadrupole moment of bromine by comparing the resonance widths. The e l e c t r i c f i e l d gradient i s assumed to arise from the water molecules and Pound allowed for the greater distance of the iodine nuclei from the e l e c t r i c dipoles i n the water molecules. These calculations give a value for the quadrupole moment of bromine which has been v e r i f i e d by microwave spectroscopy ( G l ) . The author has recorded resonances for these isotopes using saturated aqueous solutions of Nal, KI, NaBr, KBr. Some of the traces are reproduced i n figure 23. This figure shows c l e a r l y that there i s a marked discrepancy f o r B r 8 1 width i n NaBr solution, but the iodine width i n Nal i s approxi-mately i n agreement with Pound's value. Widths measured on tracings taken with a higher paper speed are: 50. Sample Nucleus Freq. F i e l d Width NaBr Na 2? 7.00 mc. 6220 .55 + .05 kc. u B r 8 1 7.15 6220 .95 + .05 kc. n Br 7? 6.64 6220 1.25 + .10 kc. Nal -j-127 6.92 8110 10 - 14 kc. The bromine isotopes have s l i g h t l y narrower reson-ances i n KBr solution. Of more i n t e r e s t i s the width of about 2 kc for I 1 2 7 i n KL (figure 2 5 ) . I f Pound's assumptions are correct, one would expect the width to be the same order as found for Nal solution. A further p e c u l i a r i t y of the bromide solutions i s the extreme s e n s i t i v i t y of the bromine resonances to the addition of the paramagnetic s a l t MnCl 2.4H 20. The addition of a minute c r y s t a l to samples of KBr and NaBr caused a marked reduction i n the relaxation time. The s a l t was added to the sample as i t sat i n the apparatus, and tracings were taken without varying any parameters. The amplitude of the signal " for B r 8 1 i n KBr was reduced by a factor of four, and the l i n e width increased by a factor of two. This i n consistent with a reduction of T 2 by a factor of two. The concentration of the paramagnetic s a l t was of the order of .001 M. Similar r e s u l t s were obtained using NaBr solution, but the sodium resonance was not affected. I t i s d i f f i c u l t to imagine any experimental error which would make the bromine signals ten times too narrow. The conclusion appears to be that quadrupole moments cannot be r e l i a b l y estimated from nuclear magnetic resonance without further experimental and t h e o r e t i c a l work. 52. APPENDIX I . An Estimation of Noise i n a Van der Pol O s c i l l a t o r The noise appearing as an audio modulation of a van der Pol o s c i l l a t o r can be approximately calculated simply i n the following manner. I t i s assumed that the noise a r i s i n g i n the tuned c i r c u i t can be separated into Fourier components i n the usual way. The response of the c i r c u i t to these r - f . f r e -quencies w i l l be calculated using equations provided by van der Pol (72) and treating each component independently. The resultant noise amplitudes w i l l be detected by a l i n e a r detec-tor and the audio noise (20 to several thousand cycles) calcu-lated. Where i t i s necessary to i n s e r t values for approxima-t i o n , those found for the c i r c u i t described i n chapter 3 of this thesis w i l l be used. To set up the d i f f e r e n t i a l equation, we consider the triode c i r c u i t used i n chapter 2, with the addition of a noise current = 4k7~ALf/R flowing into the tuned c i r c u i t whose p a r a l l e l r e s i s -i 4 \ tance i s R (figure 24). The current equation gives 53. We are now ready to apply van der Pol's solution for the problem of forced o s c i l l a t i o n . He starts with the d i f f e r -e n t i a l equation c^lv _ ± U'v- x'xr*) •+• cojrv^ co^Esincjt ta) which can be found on page 1068 of his a r t i c l e (72). Comparison with our equation shows that * ' = "c * ( it) ~ (~ —O ( c h a r a c t e r i s t i c of van der Pol o s c i l l a t o r ) . We now quote r e s u l t s from his paper and modify them to our purpose. So that comparison with the o r i g i n a l work w i l l he d i r e c t , we s h a l l drop the primes i n the constants just l i s t e d and s h a l l continue f or the present i n the nomenclature used i n the relevant section of van der Pol's paper. He finds the concepts of free and forced o s c i l l a t i o n s have l o s t t h e i r rigorous meaning, but he defines a as the amplitude of o s c i l l a t i o n w i t h frequency COQ ("free"), and b as the amplitude of o s c i l l a t i o n of frequency CO, ("forced"). Both these frequencies can exist simultaneously, and he finds a 1 = <C - * Is) where Q - 0 ~ i / 3 T p " ^ s ' t l l e a J 3 1 P l i ' b u ( 3 - e of the o s c i l l a t i o n i n the absence of a forcing voltage, and fc=2tfUo-wJ Equation J? shows that when b*" > the amplitude of the free o s c i l l a t i o n i s completely suppressed. Van der Pol finds for the case where a i s zero b-{zv^i-^f} = ^ ^ (7) Equations 6 and 7 are cubic i n b 2 and give three r e a l positive solutions when z/oC i s near zero. The l a r g e r value of b 2 i s the stable one and i f E i s small b 2 / a Q 2 i s s l i g h t l y greater than unity. In this region, the free os-c i l l a t i o n i s completelysuppressed, a condition often found i n practice and known as "locking." Using the approximation b 2/ao 2==l> i t can be shown that the width of the region of three solutions: i s (f) s t("So~) for equation (6) ^ (J) =i(^7j for equation 17) W) The two solutions obtained by van der Pol do not agree upon the extent of the region of locking with small d r i v i n g v o l -tages. Substitution of numerical values for our o s c i l l a t o r into equations 8 and 9 shows that the width of t h i s region i s much l e s s than one; cycle f o r noise e.m.f.fs, and we are only interested i n those r - f . frequencies which d i f f e r from C0 0 by more than 20 cycles. For these frequencies with small E, equation 6 i s applicable and i t has only one solution for b 2 / a 0 2 which i s much l e s s than unity. We can si m p l i f y the equation to 55. A further s i m p l i f i c a t i o n r e s u l t s from our upper audio l i m i t , oc as here defined i s about 4 x 10 4 cycles when our o s c i l l a t o r i s operating at 1 v o l t r - f . o**" i s much larger than z 2 for z i n the audio range and (10) becomes x. o(. , and remembering C\0 « —— We now revert to our o r i g i n a l nomenclature. Equa-t i o n (11) becomes 7~l 4kTR&L> k - ftYV*® ( ) This i s the amplitude of r - f . noise components i n the o s c i l -l a t o r , which d i f f e r from the o s c i l l a t o r frequency by an audio frequency. These frequencies on each side of the central f r e -quency w i l l appear as an audio modulation a f t e r detection. Because of the presence of the large central frequency voltage, the side bands simply add quadrat.ically and cross products are n e g l i g i b l e . The audio noise as a modulation i s No allowance has been made f o r noise o r i g i n a t i n g i n the tubes themselves. I t i s probable that t h i s can be done by introducing the noise factor F for the tubes used as an amplifier. F m:- noise power from input c i r c u i t + amplifier noise power from input c i r c u i t alone F i s dependent upon the frequency and the impedance of the input c i r c u i t (VI). For a given tube type, F has a minimum value which l i e s between 1 and 2 for the l e s s noisy tube types. The equations above are modified by replacing R by FR J>6. wherever we are concerned with the generation of noise. dence of noise components when applied to a non-linear tube i s not s t r i c t l y .correct. However, the transconductance at any instant is almost completely determined by the large o s c i l l a t -ing voltage; the small noise voltages produce very l i t t l e change. This s i t u a t i o n i s s i m i l a r to the well-known problem of the detection of signals i n the presence of a large voltage of central frequency; as already noted, cross-modulation pro-ducts do not appear. The author feels that equation 14 must represent closely the modulation noise of a van der Pol o s c i l -l a t o r . (It) I t i s appreciated that the assumption of indepen 57-APPENDIX I I . A Table of Nuclear Magnetic Resonances This table contains values for the r a t i o of resonant frequencies to the frequency of the proton. No diamagnetic correction i s included. This table contains only those nuclei which have been measured by paramagnetic resonance methods, and only one reference i s given for each nucleus; for further information, r e f e r to the tables of Chambers and Williams (02), and Poss (P3). ^nO/^Hl = .685001 + .000030 (B8) I^j2/1^H1 = .1535063 + .000002 (B7) VeI/Vk1 = 1.066636 + .000010 (B4) A>He5/l/B2 m .7617866 + .0000012 (A2) The accuracy quoted for many nuclei i n the remaining part of the table i s j u s t i f i e d only for the p a r t i c u l a r sample used i n the measurement (P12) , ( D 2 ) . Since the exact c o n s t i -tution of the sample i s r a r e l y given, accuracy greater than 1 part i n a few thousand i s suspect. Nucleus V/iSr1 Error x 10-5 Ref. Li7 .388625 4 . 0 (B2) Be? .1405187 .02 (D3) B 1 0 .10745 11 (B2) B 1 1 .320827 .04 (Al) C13 .25143 5 (P2) Nucleus Error x 10-5 Ref. .072255 .8 ( P l l ) N15 .10132 1 ( P l l ) Fl9 .94077 10 (P2) .26450 3 (B2) Al27 .26056 3 (B2) p51 .40481 4 (B2) Cl35 .09799 7 (CI) Ci57 .11773 8 ( P l l ) v 5 l .26290 (K2) Mn55 .24789 4 ( P l l ) Co59 .23728 4 ( P l l ) Cu63 .26506 5 (12) Cu65 .28391 6 (B2) Ga69 .24009 80 (C2) Ga71 .30494 40 102) Br79 .25059 5 (Zl) B r 8 1 .27014 5 (21) Rb 8^ .09661 4 (01)... Rb87 .32718 7 (B2) C d l H .21202 4 (P10 OdH5 .22181 4 (P10 .32697 5 (P10 Snll7 .35612 11 (P9) Snll9 .37268 14 (P9) 59. cleus Error x 10"^ Ref. Sl>121 .2394 20 (t h i s thesis) 1127 .20003 7 (ZD Csl33 .13093 14 (CD Lal59 .141231 1.4 (Dl) P t W .21496 4 (P10) Hgl99 .17879 2 (P10) T1203 .571499 5 (P2) T 1 2 0 5 .577135 5 (P2) Pb207 .20898 6 (P9) 60* REFERENCES Those marked .§• provide an introduction to the whole subject. Al Anderson, D.A., Phys. Rev. 2£> 434 U949). # A2 Anderson, H.L., Phys. 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