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Quadrupole transient effects and a super-regenerative spectrometer Sheikh, Aftab Ahmad 1963

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QUADRUPOLE TRANSIENT EFFECTS AND A SUPER-REGENERATIVE SPECTROMETER by AFTAB AHMAD SHEIKH A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the Department of PHYSICS We accept t h i s thesis as conforming to the required standard: The UNIVERSITY OF BRITISH COLUMBIA SEPTEMBER 1963. / In presenting th i s thesis in p a r t i a l fulf i lment of • the requirements for an advanced degree at the Univers i ty of B r i t i s h Columbia, I agree that the L ibrary shall , make i t free ly avai lable for reference and study. I further agree that per- mission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives . I t i s understood that copying, or p u b l i - cation of this thesis for f i n a n c i a l gain sha l l not be allowed without my written permission. Department of P H s / S l C S The Univers i ty of B r i t i s h Columbia,. Vancouver 8, Canada. Date I 2- Sg^XjtVvv^eyV \ ̂  k 3 i i - ABSTRACT A f i e l d modulated super-regenerative spectrometer was constructed. The theory of i t s operation was developed and put to test by observing the pure quadrupole resonance of Chlorine 3 5 i n Para —. di-chlorobenzene powder at room temper- ature. The spectrometer's operation i s c l o s e l y r e l a t e d to the quadrupole transient e f f e c t s . So another experiment was done to measure the amplitude and the decay time constant of Free Induction Signal i n Para-di-chlorobenzene as a function of pulse width and magnetic f i e l d , using a pulsed r.f. trans- mitter, constructed for t h i s purpose, The r e s u l t s of t h i s . • • • • i ' < experiment were;then applied to the theory of the spectrometer to explain i t s behaviour. - v i i - ACKNOWLEDGMENT I record my thanks to the Government of Canada for the award of the scholarship under the Colombo plan, which gave me a unique opportunity to carry out this work at the Uni- versity of British Columbia. I wish to express my sincere thanks to Professor M. Bloom for his many contributions to the understanding of the experiments described here and for his invaluable help in interpreting the data. His personal encouragement and interest contributed much to the f i n a l success of the ex- periment. Mr. J.D. Noble helped me very much in the construction of the pulsed, transmitter and in carrying out the pulse experiment for which I am very much obliged. I am also grateful to Mr. C. White for his many useful suggestions in building up the Super-regenerative Spectrometer. Finally, I wish to thank the Government of Pakistan for selecting me as a recipient of the Colombo plan scholar- ship. - i i i - TABLE OF CONTENTS Page Abstract <> i i L i s t of I l l u s t r a t i o n s — - • V Acknowledgement — — « — VII Chapter I Introduction 1 Chapter II Theory (The Pure Quadrupole Induction Signal) 5 Chapter III Apparatus and Experimental Procedure 11 A Description of the Apparatus 11 B Measurement of T^ 16 C Measurement of T^ 17 Chapter IV The Results of the Pulse Experiment 22 A Amplitude and Time Constant of the Free Induction Decay 22 B Discussion of the Observed Ch a r a c t e r i s t i c s of Free Induction Signal 29 Chapter V The Theory of Super-regenerative Spectrometer 34 Chapter VI The Description and Working of the Super-regenerative Spectrometer 49 A O s c i l l a t o r 49 B The Quenching C i r c u i t 53 C The Frequency Divider 55 D Audio Amplifier 57 E Phase Shifter 57 - i v - Page F Phase Sensitive Detector 57 G Power Amplifier 57 H Helmholtz c o i l s 59 I Working of the Spectrometer 59 J Measurements 62 Chapter VII Results obtained with the Spectrometer 63 Appendix C i r c u i t Diagrams 70 Bibliography 72 - V - LIST OF ILLUSTRATIONS Page Fig. 2.1 9 Fig. 3.1 Block Diagram of the Pulse Apparatus 12 Fig. 3.2 Block Diagram of the Pulsed Transmitter 13 Fig. 3.3 Circuit Diagram of the Transmitter 14 Fig. 3.4 Pulse Timing Circuit 15 Fig. 3.5 Plot of In A t V8;. t 2 18 Fig. 3.6 Plot of In ( A o o - A t) V s # t 21 Fig. 4.1 Plot G f T 2 * V s . H 23 Fig. 4.2 Plot of 1 V s H 2 24 Fig. 4;3 Plot of T V Q Pulse-width 26 Fig. 4.4 Plot of Signal V c Pulse-width 27 Fig. 4.5 Plot of Signal v. Magnetic Field s. 28 Fig. 5.1 - — . — ——•-- 36 t'. - Fig. 5.2 45 Fig. 6.1 Block Diagram of the Super-^regenerative Spectrometer 50 Fig. 6.2 Photograph of the Super-regenerative Spectrometer 51 Figl 6.3 Circuit Diagram of the Oscillator, .: Detector and Inegrator 52 Fig. 6.4 Circuit Diagram of the Quench Generator 54 - v i - F i g . 6.5 C i r c u i t Diagram of F i g . 6.6 C i r c u i t Diagram of F i g . 7.1 Plot of Signal V s. F i g . 7.2 Plot of Signal V S o F i g . 7.3 Plot of Signal V D S o F i g . A - l C i r c u i t Diagram of F i g . A-2 C i r c u i t Diagram of Page the Frequency Divider 56 Phase Sensitive Detector 58 Pulse-width ^ 64 Quench Frequency 66 Modulation F i e l d 68 Phase-Shifter and Amplifier 70 ' 4 Audio Amplifier 71 - 1 - I. INTRODUCTION The purpose of t h i s work i s to study the operation of a super-regenerative spectrometer by observing the pure quadrupole resonance of Chlorine 35 i n para-di-chlorobenzene i n powder form, '•6 A high inherent gain (of the order of 10 ) and the ease with which frequency can be swept over a wide range make the superregenerative spectrometer very useful i n searching for un-? known quadrupole resonance l i n e s . The o s c i l l a t o r used i n the spectrometer i s usually of the C o l p i t i s type, However, i t d i f f e r s from the standard form because i t i s turned o f f and on at a c e r t a i n rate by applying a suitable "quenching" voltage at one of the electrodes of the o s c i l l a t o r tube or the o s c i l l a t o r may be made s e l f quenching by connecting an RC network of s u f f i - c i e n t l y large time constant to the g r i d of the tube. Thus the output of the o s c i l l a t o r consists of a series of pulses, the frequency of which i s the same as that of the quench voltage. The width of the pulse, i . e . the duration of the on-time of the o s c i l l a t o r can be changed by suitable controls. The o s c i l l a t o r , i n quadrupole resonance work, i s usually operated i n the logarithmic mode i . e . the r . f . o s c i l l a t i o n s reach a steady value before they are quenched. In the absence of any s i g n a l the r . f . o s c i l l a t i o n s are i n i t i a t e d by noise. The time during which the r . f . o s c i l l a t i o n s a t t a i n t h e i r steady value depends upon the amplitude of the i n i t i a t i n g s i g n a l . When the o s c i l l a t o r ' s frequency i s swept through resonance, a nuclear s i g n a l i s induced across the c o i l . Now the r . f . o s c i l l a t i o n s s t a r t from the s i g n a l plus noise, so they a t t a i n t h e i r steady - 2 - value sooner which increases the pulse area. In the super- regenerative spectrometer the o s c i l l a t o r i s followed by a de- tector and an integrator. Thus the integrated output of the detector increases as the frequency of the o s c i l l a t o r passes through resonance, thus giving an i n d i c a t i o n of resonance. The frequency spectrum of the o s c i l l a t o r , because of quenching, consists of a main ce n t r a l frequency with side bands on either sides separated by the quench frequency. If the side bands are of s u f f i c i e n t i n t e n s i t y , they can also excite the resonance as the o s c i l l a t o r ' s frequency i s scanned. Thus the output of the spectrometer does not consist of a single l i n e but a number of l i n e s . This property of the super-regenerative method i s es- p e c i a l l y confusing when i t i s necessary to detect small s p l i t t i n g s of the quadrupole resonance. The generation of the nuclear s i g n a l by the o s c i l l a t o r at the resonance frequency i s c l o s e l y associated with the quadru- pole transient e f f e c t s (1) which can be q u a l i t a t i v e l y described as follows. The quadrupole system has no net magnetization because of the two f o l d degeneracy of the pure quadrupole l e v e l s . However, there e x i s t s a magnetization Ht p a r a l l e l to the symmetry axis of V E due to the difference i n population between the | m> and | m-l)> states and s i m i l a r l y an equal and oppo- s i t e magnetisation e x i s t s because of -| rn> and - ,|m'-'l̂ >. states. The discussion here i s only confined to I = ^/2 and a quadrupole system with a x i a l symmetry, so that the eigen- functions of I z given by I 2 | n n r v > - m | W ^ > m - are also the eigen functions of the pure quadrupole Hamiltonian. The l i n e a r l y polarized r . f . magnetic f i e l d applied during the pulse i s equivalent to two c i r c u l a r l y polarized f i e l d s , the one i n phase with the precessing magnetization vector rotates i t by angle a. The remaining component of the l i n e a r l y polarized r . f . magnetic f i e l d rotates the other magnetization vector by the same angle a. Thus both r o t a t i n g components of r . f . f i e l d are u t i l i z e d i n quadrupole resonance. After the pulse, each of the magnetization vectors begins to precess about the symmetry axis at an angle a, i n opposite di r e c t i o n s and thereby they give r i s e to an o s c i l l a t i n g magnetization p a r a l l e l to the axis of the r . f . c o i l . The voltage induced across the r . f . c o i l because of t h i s o s c i l l a t i n g magnetization forms the induction s i g n a l . Thus i n quadrupole pulse experiments the transmitting and r e - ceiving c o i l s are e s s e n t i a l l y the same. The induction s i g n a l begins to decay with a time constant T 2 because of spread i n precessional frequency of the n u c l e i . In order to observe the quadrupole resonance s i g n a l s , the super-regenerative spectrometer i s usually operated such that the time between two successive pulses i s much shorter than T 2 1 (about —±— of T 0 ) , so the induction s i g n a l following a pulse 10 i s only p a r t i a l l y decayed during the off-time. The voltage across the c o i l at the end of the o f f period i s responsible for the i n i t i a t i o n of the next burst of r . f , o s c i l l a t i o n and i t i s t h i s voltage that determines the spectrometers action. The actual s i g n a l obtained by the spectrometer i s observed by sub- j e c t i n g the r . f . c o i l containing the specimen to a sinusoidal magnetic f i e l d . The magnetic f i e l d d r a s t i c a l l y changes T 2 and thereby causes smearing of the resonance l i n e twice a cycle. Thus the si g n a l i s modulated at twice the frequency of the magnetic modulation. Thus the modulation allows the use of a narrow band-width amplifier which improves the si g n a l to noise considerably. The noise i s further reduced by the use of a phase s e n s i t i v e detector. Thus we can study the theory of the spectrometer's operation only when the value of T 2 as a function of the various parameter of the spectrometer i s known. The f i r s t part of the thesis describes the pulse experiment which was done to measure the amplitude and the decay time con- stant of the induction s i g n a l as a function of the magnetic f i e l d and pulse width with a discussion of the r e s u l t s obtained. The theory of the spectrometer i s developed i n part I I and then the r e s u l t s of part I are used to interpret the theory. I I . THEORY The Pure Quadrupole Induction Signal. As discussed i n the a r t i c l e by Cohen and Reif (2) the Hamiltonian for the in t e r a c t i o n of the quadrupole moment of a nucleus with an a x i a l l y symmetric e l e c t r i c f i e l d gradient V E at the nuclear s i t e due to surrounding changes i s given by: (2.1) where V E i s symmetric about an axis i n space , say Z - axi s . e q i s the scaler e l e c t r i c f i e l d gradient. e Q i s the scaler nuclear e l e c t r i c quadrupole moment. A S [ < HGL > I Z ] 3 8 °* * n e e i s e n * u n c * l ° n s ° * quadrupole in t e r a c t i o n ( ̂  ). are simultaneous eigenfunctions of I z • • Therefore ' ^ = ^ 0 ^ - l t t ^ = ^ . (2.2) and I N' «. = m ^ • z 1 m T m An r . f . f i e l d 2H^coswt which couples to the nuclear mag- net i c moment induces t r a n s i t i o n s between the various energy l e v e l s given by (2.2) corresponding to A m ~ ± !• Because of the degeneracy of the + m states, there are I t r a n s i t i o n f r e - quencies for i n t e g r a l spins and I - ̂  for half i n t e g r a l spins. Being concerned with Chlorine, we w i l l consider a single t r a n s i - 3 t i o n for I = /2 to i l l u s t r a t e the general properties of i n - - 6 - duction signals. In nuclear magnetic resonance experiments (3), there exists i n i t i a l l y a net magnetization MQ in the direction of the static magnetic f i e l d H Q ( Z direction) because the nuclear spins when in thermal equilibrium with their surrounding are populated among the (21 + 1) levels according to the Boltzmann distribution law. This magnetization i s rotated by an r . f . magnetic f i e l d 2H 1cosu Qt applied at right to H q by an angle ~ y \ H | ty provided 1 . y H _ *̂>S A H where t u t (j * i s the width of the applied r . f . pulse, y the gyromagnetic ratio of the nucleus, w Q y H Q the resonance frequency and AK denotes the line width. After the pulse non-vanishing x - y com- ponents of M exist and a voltage proportional to M 0Sin(^H 1tw) is developed across the sample c o i l placed at right angles to H . The signal decays in a time T 0 , the time taken for the O *i nuclear spins to "dephase". In case of quadrupole systems, i t can be shown that the expectation value of nuclear magnetization i s zero, at thermal equilibrium. Expanding in terms of the eigenfunctions of the quadrupole interaction M/ - T f M/ At thermal equilibrium, the population of the states is governed by the Boltzmann equation. - 7 - X T - Em Nm a e KT B u t Em = E-m .'. N = N. m —m so that I =0. z Thus no macroscopic nuclear magnetization i s present at thermal equilibrium; It i s therefore not obvious that free induction e f f e c t s s i m i l a r l y to those previously mentioned can be pro- duced there. However, a quantum mechanical c a l c u l a t i o n (4) shows that the e f f e c t of inducing magnetic t r a n s i t i o n s by pulses of the r . f . magnetic f i e l d i s to produce a macroscropic o s c i l l a - t i n g nuclear magnetization. For the case of a symmetric e l e c t r i c f i e l d gradient VE, t h i s magnetizan i s produced i n the plane perpendicular to the axis of symmetry of VE, and i t processio- nal motion can be given a semiclassical description. A quadrupole system has no net macroscopic magnetization due to two f o l d degeneracy of +_ ro states. However, there e x i s t s a magnetization HQ along the symmetry axis due to difference i n population between \ m)> and | m - i/> states and an equal and opposite magnetization due to - |m> , -Jro-l̂ states. We w i l l consider these two magnetization vectors separately. The population difference between | lii^ and | ro-J^ states, at thermal equilibrium and f o r E^ - E m - ^ / KT <<f 1, i s given by Boltzmann s t a t i s t i c s as N ( E m - Em-i) / KT(2I+-1) where N i s the no. of resonant n u c l e i . Therefore the magneti- zation M Q due to t h i s difference of population i s M0 = N^(r7^-E^-,)/KT(2.T+l) (2.3) The l i n e a r l y polarized r . f . magnetic f i e l d 2H^cosut applied at r i g h t angles to symmetry axis (Z axis) i s equivalent to two f i e l d s r o t a t i n g i n x y plane, one i n clockwise sense and the other i n counter-clock-wise sense. The counter-clockwise rot a t i n g magnetic f i e l d rotates the magnetization MQ by an angle ^fdyH^tu and then there e x i s t s a magnetization along each of the three axes. During the pulse the components of mag- netiza t i o n MJJ, My and M z obey equations s i m i l a r to Bloch's equation (5). A transformation i s made to a frame of reference in which the (x, y) plane rotates at the frequency of the applied r . f . f i e l d H . Therefore the vector (M x, M , M z) i n the laboratory frame transforms to the vector M (u, v, M ) i n Z the r o t a t i n g frame, where i s fixe d p a r a l l e l to the u com- ponent. The transformation i s given by (for posi t i v e e 2 q QJ M = u coscjt - v s i n wt M„ = -.(u s i n ut + v cosut) (2.4) M = M z z Si m i l a r l y the magnetization due to -m.. ? - (m - 1) states i s rotated by the clockwise ro t a t i n g magnetic f i e l d . Thus both the rot a t i n g components of the r . f . f i e l d are u t i l i z e d i n quadrupole resonance. Mx, My for t h i s case are given by MY = u coswt - v s i n ut (2.5) M = + (u s i n ut + v coswt) y \ y The observed induction due to equal contribution from both pair of m states i s obtained by (2.4) and (2.5). M = M (m) + M (-m) = 2(u cosut - v s i n ut) X X X My = My(m) + My(-m) = 0 M, = M (m) - M (-m) = 0. z z z - 9 - FIG. 2.1 - 10 - Thus no s i g n a l of induction would be detected i n a c o i l oriented at r i g h t angles to the r . f . c o i l which serves both to detect M as well as transmit 2H^. F i g . (2.1) shows a vector model of macroscopic spin pre- cession i n an a x i a l e l e c t r i c f i e l d gradient for positive signs of eVQ and y. The s p e c i a l case i s shown where the macroscopic moment M( mj and M(~m) are rotated by 90° about the r . f . f i e l d during a pulse. If a small magnetic f i e l d H 0 i s applied to the spins along Z axis for example, such that y H Q <CX e 2q Q the entire vector diagram can be thought of as precessing about H Q i n a d i r e c t i o n determined by the sign of y. Therefor the symmetry of alignment about the X axis of M ^ j and M(_ m) i s removed, which i n essence means that the degeneracy of the + m states i s removed. A low frequency modulated induction s i g n a l w i l l then appear along the y axis as well as the x axis due to the addi- t i o n a l precession imposed by H . - 11 - I I I . APPARATUS AND EXPERIMENTAL PROCEDURE A. Description of the Apparatus. A block diagram of the apparatus used for the pulse experi- ment i s shown i n Figure (3.1). The block and c i r c u i t diagram of the pulsed transmitter are given i n Figures (3.2) and (3.3) respectively. The f i r s t part of tube converts the po s i t i v e pulses from Tektronix Type 163 PUlse generator into negative pulses. The screen g r i d , g r i d and cathode of tube V together with the tuned c i r c u i t are con- nected as a Hartley o s c i l l a t o r which i s allowed to o s c i l l a t e only when the tube V i s switched o f f by the negative pulse at i t s g r i d . Thus acts as a gate. The s p e c i a l feature of the transmitter i s that the o s c i l l a t o r i s electron coupled to the plate of and thus to the load which makes i t quite immune to the changes of the load. acts as a tuned buffer amplifier whose output i s fed to a tuned push-pull power amplifier (V ) 6 through the gate formed by V„. The power amplifier V operates 5 6 only when the negative pulse from V switches the gate tube V 1 5 o f f , which allows the use of the 829B tube at much higher v o l - tages than i t s normal ratings. The cathode of V i s at -300 5 v o l t s so that i t s heater must also be at -300 v o l t s d.c. The 4 |if condenser connected at the output stage i s to store the pulse energy. The c o i l s of the Hartley o s c i l l a t o r , tuned buffer amplifier and that of the power amplifier are mounted mutually at r i g h t angles to avoid any d i r e c t pick-up. T E KTRONIX TYPE IG3 IPULSE GENERATOR S A M P L E HELMHOLTS COtL.% PULSE TIMER SWEEP a R.0 R.C FILTER TI6GER • <• -<- FlG3rl BLOCK DIAGRAM OF THE PULSE APPARATUS PRF- AMPLIFIER ARENBERG WIDE-BAND AMPLIFIER v DETECTOR Hartley Oscill elecb-iron Coupled to its load Amplifier •> OUT PUT FIG. 3-2 BLOCK DIAGRAM OF Pulsed Transmitter \ZAU7 >o 1 7 0 63 V-AC -.01 r - 3 0 0 V + 1 2 0 0 +750 6SN7 -300 «- C0\LS 1 OSCILLATOR ^TORNS/ , g SPACE 2. BUTTER TURKS 4\5 FIG 33 TRANSMITTER 6 S N 7 6SN7 6SN7 6SN7 C.R-0 TRiQQCR FIG3.4 PULSE TIMING- CIRCUIT - 16 - The c i r c u i t of the pulse-timer i s given i n F i g . (3.4). It consists of four multi-vibrators. The f i r s t one i s f r e e - running and i t s R-C time constant determines the r e p e t i t i o n rate. It triggers the second multi-vibrator. The R-C time constant of t h i s multirvibrator determines the delay time. The pulses from the second multi-vibrator trigger the t h i r d and fourth multivibrators. The two 6SL7 tubes mix the pulses producing a trigger for the transmitter. B. Measurement of T g. The transmitter was mounted ju s t at the top of Helmholtz c o i l s used to provide a d.c magnetic f i e l d . The c o i l con- tai n i n g the sample i . e . para-di-chlorobenzene was suspended at the centre of the Helmholtz c o i l s by s t a i n l e s s s t e e l tubing about l o " long. This was done to avoid the use of a half wave- l i n e . The Chlorine 35 pure quadrupole resonance i n p-dichloro- benzene was found to be at 34.2 MC/sec at room temperature and 34.7 MC/sec at l i q u i d nitrogen temperature as measured by a super-regenerative spectrometer. These r e s u l t s were i n agreement with the previous measurements as given i n (6) and (7). The preamplifier was tuned to 34.2 MC/sec by observing proton induction signals i n glycerine. The various stages of the transmitter were tuned to 34.2 MC/sec by a H a l l c r a f t e r * s receiver model SX-42. The pure quadrupole induction s i g n a l was seen with- out much d i f f i c u l t y . The signals under the various conditions of magnetic f i e l d and pulse-width were recorded on f i l m with a Dumont scope camera type 2620. - 17 - The l i n e shape of the induction signals can be approximated by a Gaussian function. Thus the amplitude of the induction t a i l at instant " t " a f t e r the pulse i s given by Ab = A0e (3.1) Fig.(3.5)gives two t y p i c a l decay curves (plot of log A(t) versus 2 t ). The curve A corresponds to z e r o f i e l d and B corresponds to a magnetic f i e l d due to 75 milliamperes current. T Q measurements were made as a function of magnetic f i e l d and pulse width both at room and l i q u i d nitrogen temperatures. The r e s u l t s are given i n the next chapter. C. Measurement of The presence of free induction signals i n nuclear quadrupole resonance requires that, before application of the f i r s t pulse of r . f . , there be a difference i n the populations of the energy l e v e l s between which t r a n s i t i o n s are induced. The i n i t i a l amplitude of the induction s i g n a l i s proportional to t h i s d i f f e r - ence i n population, n. For a spin 3/2 system, n = n 3 / 2 " n V 2 where n3/2 ** n+3/2 + n-3/2 = t l i e P ° P u l a t i o n °* t n ® m = | ̂  | states and n l / 2 = n+l/2 + n - l / 2 ~ t n ® population o f the m = states. We assume that t h i s difference of population has been established by the nuclei coming into thermal equilibrium with t h e i r sur-  - 19 - roundings before the f i r s t pulse has been applied. Now the effect of inducing transitions i s to change the surplus population from i t s equilibrium value. For a two energy level system the surplus population, when the system is not at thermal equilibrium, w i l l recover toward i t s equilibrium value exponentially with a time constant T^, the spin lattice relaxation time d - n » - - n ( t ) dt T ] L where n Q i s the surplus population when the spin i s at thermal equilibrium and n ^ . i s the surplus population at a time " t " . r -| (t - U ) Therefore n ( t ) = n Q - [ n Q - n ( t i ) J e ^ . Suppose we apply pulses of r - f at t=0 and at t= T , the length of the pulses being tw. V(tw) and V( T + tw) are the amplitude of the signal following the two pulses. If, makes an angle Q, with the symmetry axis of VE, the following can be shown (8) V(tu) - V( T + t w ) a f ( S, ) e ~ T1 where f (SO = Si/rv 0. ( & H.USi/rv Q.) [ l - Coo (»/5 * H,GwSA/fv6,)] Thus even in case of crystalline powder where an integration must be performed over 9| , ~ ~ i s given by the slope of log [ V(t w) - V( T + t u ) ~J vs T curve. At room temper- ature, T^ was measured by exciting the sample by two pulses of identical widths such that the signal following the f i r s t pulse is maximum. For this pulse width the signal following the second pulse was found to be close to i t s minimum value, though - 20 - not quite zero. The amplitude of the s i g n a l following the second pulse was measured as a function of the time i n t e r v a l between the two pulses. The value of T^ thus obtained i s 26.6 + 2.5 milliseconds. At l i q u i d nitrogen temperature T^ was expected to be quite long so the sample was saturated at time t=0 by a t r a i n of c l o s e l y spaced r . f . pulses which produce the i n i t i a l non- equilibrium condition of zero population d i f f e r e n c e . Then at a d e f i n i t e time " t " l a t t e r a single pulse was applied to measure the surplus of the population recovered i n time " t % The value of T^ thus obtained i s 570 + 25 milli-seconds. These values are close to those given i n references (9) and (10), A plot of the s i g n a l (at l i q u i d nitrogen temperature) versus time i s shown i n F i g . (3.6).  - 22 - IV. THE RESULTS OF THE PULSE EXPERIMENT In a super-regenerative spectrometer, the quadrupole system i s excited by a series of r . f . pulses and the s i g n a l i s observed by applying a sinusoidal magnetic f i e l d modulation. The theory of the spectrometer, as discussed i n the next chapter, requires a knowledge of T 2 and the amplitude of the induction si g n a l as a function of magnetic f i e l d and pulse width and also the value of T^, for para-di-chlorobenzene, the substance used to test the super-regenerative spectrometer. The object of the pulse experiment reported here i s to provide t h i s information so that the theory of the spectrometer may be worked out. A, Amplitude and Time Constant of the Free Induction Decay F i g . (4.1) shows T 2 , the decay time constant of the free induction s i g n a l as defined by equation (3.1), as a function of the current through the Helmholtz c o i l s producing the magnetic f i e l d for a pulse width of 60 usee. The value of the magnetic f i e l d i n gauss can be obtained by multiplying the current i n milli-amperes by ,068, as calculated by the dimensions of the Helmholtz c o i l s given i n Chapter VI. By the Figure (4.2) which shows a plot of 1 against the T*2" square of the magnetic f i e l d , i t i s seen that T 2 * i s f i t t e d quite well by 4** - A + KTfY -(4.1) where T 2 i s the value of the decay time constant of the free FIG 41 T*4H  25 - induction s i g n a l i n zero f i e l d and T 2* i s the value corresponding to f i e l d "H". K and A are constants. A discussion, of the con- stants and the equation (4.1) i s given l a t t e r . F i g . (4,3) shows T as a function of pulse width i n the 2 absence of magnetic f i e l d . The general trend i s that T 2 i n - creases with increasing pulse width. It should be emphasized that T 2 i s here the decay time parameter for a Gaussian plot of the induction s i g n a l versus time as given by equation (3,1). From figure (3.5A) which shows a representative induction decay i t i s seen that the decay i s only Gaussian for long times. A large f r a c t i o n (of the order of i ) of the s i g n a l decays i n a much shorter time than T 2 > It was d i f f i c u l t to make accurate quantitative measurements of the behaviour of the short time component as a function of pulse width because of s i g n a l to noise problems at short pulse lengths and receiver saturation e f f e c t s . In large magnetic f i e l d s (H \>> 10 gauss) the induction decay f i t t e d a Gaussian curve over the entire range within the experi- mental e r r o r . F i g . (4.4) shows the amplitude of the induction s i g n a l immediately following the pulse as a function of the pulse-width for the various values of the current producing the magnetic f i e l d . The s i g n a l seems to reach a broad maxima for pulse widths near 50 usee, for low applied f i e l d s and at lower pulse widths as the external f i e l d i s increased. F i g . (4.5) shows the induction s i g n a l as a function of the current (producing the external magnetic f i e l d ) for the various values of pulse widths. The curves corresponding to 50 and 60 u seconds are approximately the same as that of 40 u seconds FIG 4.3 Tz Vs. PULSE-WIDTH - 27 - - 28 I i i i i \ MILL! AMP o 50 loo V5o 2oo .2-50 3oo FIG 4 5 SIGNAL Vs. MAGNETIC FIELD - 29 - pulse width. The curves may be summarized q u a l i t a t i v e l y by noting that the s i g n a l tends to decrease as the applied f i e l d i s i n - creased, the rate of decrease being greater for longer pulse widths. B. Discussion of the Observed C h a r a c t e r i s t i c s of Free Induction Signal As shown i n Chapter I I , the eigenfunctions of pure quadru- pole Hamiltonian, i n case of a x i a l symmetry are the eigen- funct ions of I£ (the Z component of the s p i n ) . lZ I ^ » m | n£> where m « + 1/2, + 3/2 f o r I » 3/2, and the states |+ 3/2^> , j+ */2y> are two f o l d degenerate. When we apply a constant magnetic f i e l d H at polar angles O in a coordinate system having the symmetry axis of VE as Z axis, using f i r s t order perturbation theory i t can be shown that the e f f e c t of magnetic f i e l d for e 2 q Q >̂̂ > y\l i s to remove the degeneracy of + p states and also to mix + /2 states and yet leave a l l other m states pure (5), The c o e f f i c i e n t determining the proportion of +_ ,1/2 states i n the mixed state are functions 2 1/2 of f ( 8 ) = (1 + tan© ) . Now the time dependent density matrix "/'"can be calculated using the equation of motion where j{ i s the t o t a l Hamiltonian consisting of the pure qua- drupole and the Zeeman parts. The induction s i g n a l i s the time d e r r i v a t l v e of M (t) « y "h Trace (I f). As the sample used was i n powder form, the s i g n a l obtained by taking the time de r r i v a t i v e of M ($) w i l l have to be averaged over a l l the values of Q , <ft. The matrix elements of the density matrix contain complicated functions of f ( 9 ) , due to which t h i s averaging becomes" very d i f f i c u l t . As t h i s pulse experiment was performed only to know the values of the various parameters occuring i n the theory of the super-regenerative spectrometer, i t was not considered worthwhile to carry out the averaging due to i t s very complicated nature. Instead the r e s u l t s obtained from the pulse experiment have been interpreted semi-empirically based on a physical i n t e r - pretation of the exact programme described above. The decrease i n the value of T g with the magnetic f i e l d can be explained i n terms of l i n e broadening because of the Zeeman s p l i t t i n g of the pure quadrupole l e v e l s (11). For e q 2 Q ^ > yEt the frequency corresponding to \+3/'2/> * • J 1 ' ^ t r a n s i t i o n s p l i t s into four frequencies; the amount of s p l i t t i n g being dependent upon the applied f i e l d H and 6 (the angle between H and symmetry axis of V E ) . Both Q and H, being not the same fo r a l l n u c l e i , w i l l give r i s e to a d i s t r i b u t i o n of the resonance frequency thus broadening the l i n e width and so T* being inversely proportional to l i n e width w i l l decrease. In the absence of the external magnetic f i e l d , the decay of the induction s i g n a l being Gaussian, the amplitude of the s i g n a l at instant " t " a f t e r the pulse i s A(t) a e ^ 2 T | . When the external magnetic f i e l d i s applied, the s i g n a l i s further a t t e - nuated because of the l i n e broadening which i s of the order of yYL f o r large H ( ^ 1 0 gauss). As the decay of the signal,' i n the presence of the applied f i e l d , i s also found experimen- t a l l y to be Gaussian, i t i s reasonable to write 31 - At e ^ - e 1 - B + K Y2 H 2 where B « 1 (4.2) fWZ ' f 2 This relation Is the same as equation (4.1) which gives the it-observed dependance of T 2 upon H with the exception of constant B. For large H, the major attenuation of the induction signal w i l l be due to line broadening caused by H. S c T 2 w i l l be of the order of 1 . and then K should be of the order of one. The experimental value of K as obtained by the slope of the curve shown in Fig. (4.2) i s 1.7. This provides a f a i r l y good test of the validity of (4.1). The intercept of this curve should be of the order of 1 as predicted by the equation (4.2). However this intercept i s found to be negative. This i s because the values of T * as shown in Fig. (4.1) are only + 5% which gives quite a wide range for the values of 1 especially at high f i e l d . Thus a line can be drawn from the high f i e l d points, omitting the low f i e l d points, with a positive intercept. But we are concerned with low values of f i e l d in the spectrometer, so the curve in Fig. (4.2) has been drawn to include the low f i e l d points, even though i t has a negative intercept. This means that the constant B in equation (4.2) has been changed with another constant A in the empirical equation (4.1) which takes care of the consequences arising due to the departure from the high f i e l d approximation. - 32 - The decrease of the induction s i g n a l with the magnetic f i e l d can also be explained i n terms of the l i n e broadening. In the l i m i t i n g case when H ^ > (H^ i s half the amplitude of the r . f . magnetic f i e l d ) , only those nuclei within a frequency range Z^w <̂  y H of the frequency w of the r . f . f i e l d , w i l l — 1 be excited so as to contribute appreciably to M x and My. The e f f e c t i v e value of M Q i n equation (2.3) should, therefore, begin to decrease when H becomes of order of H^. Therefore the si g n a l , being proportional to M , also decreases. o Another factor which also contributes to the attenuation of the s i g n a l i s the relaxation during the pulse which occurs when T 2 drops to such a value, due to increase of H, that i t becomes comparable and f i n a l l y even much less than t u , the pulse width. For small pulse-widths of the order of 10 - 20 usee, the angle through which.M i s rotated during the pulse i s very small, so the e f f e c t of decrease i n the e f f e c t i v e value of M Q becomes less pronounced and also T * requires a large f i e l d ( 20 gauss) to become comparable with t w . This makes the amplitude of the si g n a l more or less independent of the f i e l d at short pulse widths, t i l l H get as large as 25 gauss when the relaxation during the pulse becomes s i g n i f i c a n t . This e f f e c t i s demon- strated by the curves of F i g . (4.5). As i n the super-regenerative spectrometer, pulse-widths are of the order of 15 ixsec and the peak value of modulation f i e l d i s about 3 gauss, we may omit the e f f e c t of the modulation f i e l d on the amplitude of the s i g n a l j u s t following a pulse. The increase of the signal's amplitude with increasing pulse width for constant can be explained by the fact (12) that the - 33 - si g n a l i s proportional to Sin ( ,̂ 3 yE^iu Sin 6^) where 0 i s the angle between and the symmetry axis of 7E, As has got a random orientation, so i s quite i n - homogeneous which explains the fa c t that the maxima observed in the curves of F i g . (4.4) are'very broad:. - 34 - V. THE THEORY OF SUPER-REGENERATIVE SPECTROMETER A super-regenerative receiver is characterized by the re- peated build-up and decay of self-oscillations in a valve o s c i l - lator, known as the super-regenerative oscillator, operating on, or near, the signal frequency. The ci r c u i t i s made alternately oscillatory and non-oscillatory by the application of a periodic voltage to one of the electrodes of the oscillator valve. The source of this periodic voltage is usally a separate quench os- c i l l a t o r , although self-quenching may be arranged by a suitable choice of the grid leak and grid condenser of the super- regenerative oscillator. In either case the quench frequency is necessarily much lower than the natural frequency of the super- regenerative oscillator but must be higher than that of the signal modulation. The sample i s subjected to bursts of high r . f . power by placing i t into the oscillator's c o i l . At resonance the o s c i l - lations build up from the nuclear signal voltage developed across the c o i l , otherwise they start from noise existing in the c i r c u i t . During the period of build-up the oscillations may become as much as a million times greater the signal. There are two clearly defined modes of the operation of the oscillator with separate quench, according to whether or not the oscillations are allowed to build up to an equilibrium value, as in a normal oscillator, before they are quenched. If the oscillations which build up during a single quench cycle are - 35 - quenched before they reach the l i m i t i n g equilibrium amplitude determined by the tube c h a r a c t e r i s t i c , t h e i r peak amplitude i s proportional to the si g n a l (or noise J voltage from which the o s c i l l a t i o n s grew. The o s c i l l a t o r i s then said to operate i n the l i n e a r mode. I f , however, the build-up period i s made long enough; the o s c i l l a t i o n s reach a steady value before they are quenched and the o s c i l l a t o r i s said to operate i n logarithmic mode, as the output, i n t h i s mode of operation, increases as log^(V 2/vi) as si g n a l voltage increases from to V 2. The o s c i l l a t o r was operated i n logarithmic mode by a nearly rectangular quench voltage. The osc i l l a t o r ' s action under these conditions has been shown i n F i g . ( 5 . 1 ) . During the quiescent part of the quench cycle, when the valve i s not conducting, the t o t a l e f f e c t i v e conductance i s determined by c i r c u i t loses, i . e . G = G q and the voltage across the c i r c u i t i s that due to noise or nuclear s i g n a l . As soon as the quench voltage gets higher than the voltage at which the o s c i l l a t i o n s s t a r t , the conductance G becomes negative, s e l f o s c i l l a t i o n begin to b u i l d up at the resonant frequency of the c i r c u i t formed by L and C and thus the super-regenerative period s t a r t s . In the beginning, the o s c i l l a t i o n s b u i l d up l i n e a r l y t i l l the time T L when the amplitude of the o s c i l l a t i o n i s large enough for l i m i t a t i o n to begin due to curvature of the tubes c h a r a c t e r i s t i c . Thus the mode of the o s c i l l a t o r ' s operation t i l l T_ i s e s s e n t i a l l y l i n e a r a f t e r which i t changes to logarithmic. As the o s c i l l a - t ion amplitude further increases, the conductance s t e a d i l y decreases to zero ( 1 3 ) . On reaching zero i t remains there u n t i l the end of the super-regenerative period, during which - 36 0 S C I I A T I O N S T A R T ? H | C U T - O F F (-̂ r) M E A M fi-R\D ftUENCH W A V E - F O R M TIME VOLT ME 4 CONDUCTANCE TIME j Sl)PER-RE.GENE|RATWe PERIOD V" DAMPING PERIOD SENSITIVITY T SAflPLINQ PERIOD -TIME Q $ C l L f t T | Q N | AMPLITUDE | BUILD-UP STARTS FIG 5-1 TIME <> - 37 - time the o s c i l l a t i o n s are maintained at t h e i r equilibrium am- pl i t u d e . At the end of the super-regenerative period, the conductance becomes posit i v e and the o s c i l l a t i o n s are damped out t i l l the next quench cycle. The damping during the quench cycle should be s u f f i c i e n t to reduce the s e l f o s c i l l a t i o n s below noise l e v e l before the b u i l d up s t a r t s again otherwise the o s c i l l a t o r s e t t l e s down i n the coherent state. The maximum s e n s i t i v i t y occurs exactly at the time when the t o t a l conductance G goes from p o s i t i v e to negative at t = 0. The s i g n a l voltage across the c r i c u i t at t h i s instant plays the greatest r o l e i n the determination of the time at which the amplitude of the s e l f - o s c i l l a t i o n s attains i t s equi- li b r i u m value. An element of s i g n a l occurring before time t = 0 has time to decay before b u i l d up starts and, consequently, has less e f f e c t than a s i m i l a r s i g n a l occurring at t = 0. An element of s i g n a l , a r r i v i n g l a t e r than t h i s instant, again has less e f f e c t because part of the build-up has expired before i t s a r r i v a l . The s i g n a l i s , therefore, sampled for a short period i n each quench c y c l e . Over the greater part of the cyc l e , the s i g n a l has a n e g l i g i b l e e f f e c t . The way i n which the s e n s i t i - v i t y varies with time depends upon the nature of quench and upon the tube's and c i r c u i t parameter's. In the logarithmic mode the amplitude of the o s c i l l a t i o n envelope i s always the same. The s i g n a l advances the apparant s t a r t i n g time of the o s c i l l a t i o n s thus increasing the area under the envelope as shown i n F i g . (5,1 d). Thus the area of the envelope changes with the amplitude of the s i g n a l . A suitable detector c i r c u i t converts t h i s once more into a change of amplitude. - 38 - Now we calculate the nuclear signal induced in the c o i l . The nuclear spin i s subjected to bursts of high r . f . power which brings about,, several effects. I. The nuclear absorption during the presence of r . f . power reduces the Q of the c o i l and thus reduces the integrated pulse energy. II. Any coherent nuclear precession, which may occur when T 2 is comparable to or greater than the quench period, w i l l cause the r . f . bursts to be initiated earlier by nuclear signals rather than by noise. This latter effect gives rise to an increased integrated signal response. Dean (14) suggests that the second mechanism i s the dominant one for quadrupole resonance detection by the super-regenerative method. A calculation of the signal, based upon the second mechanism, has been made. During one quench period T ,*the oscillator i s on for time "tw" and i s off for the rest; of the quench period. Thus the output of the super-regenerative oscillator can be supposed to consist of a series of equally spaced pulses of equal widths. The quadrupole system i s excited by the pulses and i t relaxes both longitudinally as well as transversely between the time interval during successive pulses t i l l a steady state i s reached. The quadrupole system has a magnetization if along the symmetry axis (Z axis) due to difference in population between - 39 - respectively, then - a . Usually i s much longer than the time interval between two consecutive pulses and T 2 is also longer or comparable with the latter. Therefore the longitudinal component of M w i l l be in process of relaxing towards i t s —=7 —"> equilibrium value M and the transverse component of M o w i l l be decaying during the time between the pulses. Let the angle between ~ll and Z axis just before the second pulse be a . Further rotation of magnetization w i l l depend upon the phase of the r . f . oscillations during the pulse relative to the phase of ~M? This phase difference depends upon the conductance of the ci r c u i t during the on and off periods, the departure of the frequencies of the pulse from the resonance frequency and the past history as well. As the various parameters specifying the phase may vary from pulse to pulse, this phase difference may be assumed to be random. So the angle between 1? and Z axis after the second pulse w i l l be a* ̂  « 2 + a* T h e s a m e e v e n t s take place when other pulses arrive. Thus the problem of finding out the angle between ~M and Z axis after the quadrupole system has been excited by a train of pulses i s similar to that of a random walk in which there i s a certain amount of relaxation during the period of two successive steps. In actual practice we get the Integrated effect of the pulses in a certain time determined by the time constant of the phase sensitive detector. The detected output of the super-regenerative oscillator-detector V( t) w i l l consist of a steady value V and a fluctuating compo- nent A V^ tj •'• V ( t ) = * + A V ( t ) - 40 - We assume that A V ( t j is a Gaussian variable i.e. at any time i t has an apriori probability of being between V and V-fdV given by P <JLvwhere P ( v ) - A & ( V ~ V ) / j 2 6 + cx> and PWjdlV = I and J^\)^ - & -co The signal that we measure when the time constant of the phase- sensitive-detector is T* i s ayprox. given b-y r ' c r Vo)^b = V + J L JAV<t) d t o 0 The square mean variance of the signal i s given by / re flit' J d l b ' AMtf) AV(t"; We now assume that the fluctuations are governed by a corre- lation time which in our case i s the same as quench period T -r^/r z -Wr and t = t' - t" , It i s easily shown that Therefore the R.M.S. fractional noise = Tc V A typical value of T and Tc used in the spectrometer are T = 20 usee and T c = 2.5 sec which even for 8 = V gives a signal to noise as high as 62500 . Thus we see that the randomness in the phase-difference between M and the r . f . oscillations during the pulse has got practically no effect upon the signal under the practical operating conditions. Hence we can assume that in the steady state the angle between the - 41 - Z axis and magnetization, after a pulse, i s always a_ which s relaxes to a because of T, t i l l the advent of next pulse which s 1 rotates i t to a Q again. If M~ i s the magnitude of the mag- netization before a pulse, then the amplitude of the signal just after the pulse w i l l be proportional to M~ Sin o + . Now to s s calculate M~ we make the following assumptions and approxi- s mations. I. It is assumed that a l l pulses are of identical widths. The rise time of a pulse depends upon the amplitude of the signal which initiates the pulse. The signal starting the n pulse i s proportional to magnetization before the (n - 1) pulse. As the magnetizations before the I, II, III, ... pulses are different, so the rise time of the corresponding pulses w i l l also be different. In practice, the rise time of the pulse is only a very small fraction of even the shortest pulse. So we can, to a good approximation, neglect the difference in rise times as compared to the width of the pulses and assume that a l l pulses are of equal width. II. It is assumed that pulses are very short as compared to T 2. This corresponds to the effective f i e l d in the rotating refer- ence frame being governed only by H^,so that the precession during the time r . f . pulse i s applied may be referred to as pure nutation. In the spectrometer, the value of pulse-width i s abbut one tenth of T 2 even in the presence of the modulation f i e l d , so this approximation i s well j u s t i f i e d . 42 - III. The relaxation during the pulse i s neglected. The results of the pulse experiment show that the signal i s independent of * the external magnetic f i e l d i.e. T 0 for short pulse width ( ex. 10 usee) and for low values of f i e l d ( — 5 gauss) thus showing that the relaxation during the pulse, under these conditions can be neglected. The spectrometer was operated to satisfy these conditions. IV. The effect of transverse components of magnetization in establishing the value of steady state magnetization i s neglected. • —± If the magnetization i s rotated by the f i r s t pulse by an angle a , the x, y components of magnetization after the pulse w i l l be proportional to M 1 Sin a. As a ^ .05 radian, X these components w i l l be negligible as compared to Z component = cos a . Thus the main contribution to the steady state magnetization w i l l come from the rotation and relaxation of the Z component of magnetization. V. i s supposed to be uniform. Note that H i s never uniform for a powder since Hj e f f = H 1 Sin 8^ where 0^ i s the angle between and the symmetry axis of "VE. VI. Special effects due to tendency of spin systems to establish a "spin temperature" in the rotating reference frame are neglected (15), Now we calculate the value of the magnetization in the steady state within the frame-work of these approximations. - 43 - Suppose a pulse of width t w Is applied at the time n r Let the Z component of magnetization M before the pulse be z represented by M~ z^ nj and after the pulse by M*( n). If Mz obeys the equation dMz . MQ - M2 dt T x then i t can be shown that = Mo- { M 0 - M2(<YO] e C 5 - 1 ) and M +,„, = M , . cos a (5.2) z^ n* z\«i where a i s the angle through which M ( n) is rotated by z the n t n pulse. Now we consider the value of IT" (n) where n = 0, 1, 2, 3 ... z to form a suitable expression for M~(n). Z st I pulse:- M (1) = M_ z u M+(l) = M cos a. z o nd II pulse: - By putting n = 1 in (5.1) M2 = M© (i - e. T ,j + Mocooo<eT* ... By (5.2) M o D O o C (,_ ̂  + I I I r d pulse:- _ - V T M 2 (3)=M 0 o - f i ) + M 0 Coo^C»-e J e 1 + M0CoOo( -e. and + -TV . -TZ . \ + M0 ODU e 44 I V t h pulse:- _ . "r/\ + . T M2C^ - Mo - CW0tW ) e - r / T -T/T ~Vr ^ M0(\-e /T ,) + M0Cci0c(i-€.7 ;)e T l Thus the expression for M (n) forms a geometric progression with f i r s t term as M (1 - e 1 ) and common ratio as cos a e ' , .". M (steady state) = sum of geometric series as n-» «» Due to assumption IV, M = M . z s .*. the amplitude of the induction signal just following the pulse i s "V- This signal decays with a time constant T 2* during the time T - t w and initiates the r . f . oscillations of the next pulse at the end of the quench period T . However the effect of the assumption VI is to introduce an uncertainty in the effective zero of time for the Gaussian decay of the induction signal (16). Thus the time interval T - t w i s replaced by T where T l i e s in between T* - t w and T . So the nuclear signal that initiates the r . f . oscillations i s VS(W,T) = M ° ° ' f c l*?-** e. 2 (5.3) Now, knowing the nuclear signal that initiates the r . f . o s c i l l a - tions, we can calculate the voltage of the signal given by the spectrometer. Figure (5.2) shows the build up envelope, after time T^ (Fig. 5.1), in greater det a i l . The two curves represent - 45 - - 46 - the build up from two voltages and Vg. i s the voltage reached at time T L by the oscillations building up from a sample of noise in the sensitive period, earlier in the cycle. V i s due to signal plus noise. For the time being, we are considering the events in a single quench cycle. Although the noise voltage i s indeterminate for a single cycle, i t s r.m.s. value i s given by . - 2 where V* i s the mean square noise at input^V^ i s the signal input, and u L i s the voltage gain during the interval of linear build-up i.e. from the instant build-up starts to T^. For simplicity, l e t us take the build up curves to be truely ex- ponential in form u n t i l they are suddenly limited at the equi- librium value V m. As we are starting with the values of voltages at t = T L, we shall hence-forth measure time from this point and not from the beginning of the true build-up period. As V n i s even less than one-hundreth of V . the z, m difference in area to the l e f t of vertical axis is negligible. Thus the incremental area to the right of axis (shaded in Fig. 4.2) represents the increase in output due to signal during a single cycle of quench. Referring to Fig. (5.2) Vm = v » e - -Me - 47 - where a is the time constant of build-up « 2 c , where Gl i s the capacitance of the tuned c i r c u i t and i s the con- ductance of the c i r c u i t during the on period (13). .*.. t i = a 1 n V m and t« = a 1 n V m . Vi V 2 Hence the incremental area i s given by (5.4) For signals greater than noise, the f i r s t term in the bracket is of the order of V whilst the second is much smaller. m Thus we can write As the 1 . 5 cycle narrow-band width amplifier and the phase- sensitive detector used in the output stage of the spectrometer, allow only a very narrow spectrum of noise to effect the f i n a l signal, so We can assume that V 2 V"n . Thus The f i n a l signal voltage as recorded by the spectrometer w i l l be V = a , V ^ 2 ^ V s / ( ^ ~ ^ ( 5 < 5 ) where T i s the quench period and K i s a constant of the spectrometer determined by the detector, amplifiers, etc. The presence of T in the denominator of 5 . 5 i s because of the fact that A A i s the increase in area of a single pulse and the voltage V i s the integrated effect of pulses where 7c is the time constant of the phase-sensitive-detector. - 48 - As f is a variable quantity, i t has been separated from the constant K. Substituting the value of V from (5.3) into ( 5 . 5 ) , s V - A t ^ r ) - - L J £ ( 5 . 6 ) where _ Mo(t-€. )Si/Yvc£s and B i s the constant of the spectrometer. In Chapter IV the following emperical formula was obeyed 1 2 2 T ^ 2 - A + K r H 2 Here H i s the modulation magnetic f i e l d given by H m cos Unft, substituting the value of T 2 in (5.6) V = 1 U ( T ) - 1 ( T ' \ KJL^WT7-)-!. tfJ^^c^ic^C (5.7) A strong signal was observed when the phase-sensitive detector was tuned to 2u/m. According to equation (5.7) i t i s given by The results obtained with the spectrometer have been discussed in Chapter VII on the basis of equation (5.8) and the data of Chapter IV. - 49 - VI. THE DESCRIPTION AND WORKING OF THE "SUPER-REGENERATIVE SPECTROMETER" A block diagram of the super-regenerative spectrometer is given in Fig. (6.1) and i t s photograph has been shown in Fig. ( 6.2). A. The Oscillator The c i r c u i t diagram of the oscillator, detector and inte- grator together with the c i r c u i t that allows the quench voltage applied to the oscillator has been shown in Fig. (6.3). The basic ci r c u i t was designed by Dean (14) for use in the region of 30 MC/sec. The present c i r c u i t has been taken from McCall's thesis (17) after suitable modification. The oscillator i s the grounded-plate version of the Colpitt's oscillator. The quenching action is as follows: The two halves of the 6J6(V 1), in parallel, have their cathodes tied to the cathode of the oscillator tube V2. The plates of V^ are by- passed for r . f . When a positive pulse from the external quenching c i r c u i t i s applied to the grid of Vji, the additional cathode bias shuts off V 2. Thus the oscillator is operating only during that part of the quench during which V^ i s no con- ducting. At the time V^ i s cut off, the cathode impedance i s shunted across part of the tuned c i r c u i t . This large positive conductance shortens the decay time-constant of the tank ci r c u i t and aids in quickly damping out the free oscillations. _ C. R . O AUDIO GENERATOR INTEGRATOR £ DETECTOR QUENCH GENERATOR MOTOR DRIVE OSCILLATOR A SAMPLE HELMHOLTS COIl AWMETER -0" 6-1 THE SUPER-REGENER/ AUDIO AMPLIFIER RECORDER NARROW BAND-WIDTH AMPLIFIER FREQUENCY DIVIDER POWER A M P L I F I E R PHASE SENSITIVE DETECTOR x PHASE SHIFTER A U D I O G E N E R A T O R cn O SPECTROMETER - 51 - FIG 6-3 OSCILLATOR , DETECTOR AND INTEGRATOR - 5 3 - It is important that this impedance should not have any damping effect during the on-period of the oscillator and to ensure this, the grid of Vj must be very negative (of the order of - 120 volts). A fixed bias that can be set from 0>- 90 volts i s provided by a battery and potentiometer (shown in the quench c i r c u i t ) . The over-all positive to negative swing of the quench voltage i s about 120 volts. The frequency of the oscillator i s changed by driving the ) tuning condenser slowly by a motor. The motor drive can be replaced by connecting a variable-capactiy diode in parallel with the tuning condenser as shown by dotted line in the cir c u i t diagram. The sawtooth has a period of about 5 - 1 0 minutes. A variable capacity-divider, following the tank c i r c u i t , allows some control of the fraction of the tank voltage applied to the grid of the i n f i n i t e impedance detector V . The output from the cathode of V_ is fed into a four section, RC f i l t e r network which integrates the pulse area. Part of the detector's output i s fed to a cathode follower whose output i s dis- played on an oscilloscope. The on - off time of the oscillator is measured from this wave-form. B. The Quenching Circuit The c i r c u i t diagram of the quenching ci r c u i t i s given in Fig. ( 6 . 4 ) . The input to the ci r c u i t is provided by a Hewlett-Packard audio oscillator (Model 200 DR). The frequency of the input varried from 10 K.C. /sec to 70 K.C./sec. The tubes and V"2 square the sine-wave input. They are followed by a differentiating network. The resulting positive FIG 6-M Q U E N C H G E N E R A T O R - 55 - and negative pips appear at the grid of Vg which has a fixed negative bias, well below the cut-off. Therefore only the positive pip affects the tube current. The output pip from V^ triggers the "one-shot" multi- vibrator formed by V 4 and Vg. With no input, V 4 is cut off and V i s conducting V and V have a common plate load. Thus 5 4 3 the positive pip at the grid of Vg appears as a negative pip at the plate of V 4 and the grid of Vgj in i t i a t i n g the change to metastable state. The duration of this state is determined by the R and c values in the grid of V and the setting of the 4 potentiometer in the grid of V4. The output, taken from either plate, is a square wave in which the duration of the positive and negative portions may be varied. The multi-vibrator's out- put, after having been amplified by Vg is biased by a 90 volt battery and then applied to the grid of the quenching tube. The potentiometer in the bias battery c i r c u i t is found to have a fine control over the on - off time of the oscillator, C , The Frequency-Divider The c i r c u i t diagram of the frequency-divider is shown in Fig, (6.5)- The tubes Vj and V 2 convert the input sine wave to a square wave which triggers a bistable multi-vibrator formed by tubes V3 and V4, The output of the multi-vibrator i s a square wave of half the frequency of the input. The tubes V5 and V7 together with the L,C, f i l t e r s allow only the fundamental sine wave to pass to the output stage which i s a cathode follower formed by tube Vg, Thus the output of the cathode-follower is a sine wave of half the frequency of the input sine wave. The FIG 6.5 FREQUENCY DIVIDER 57 - tube V i so l a t e s the mul t i -v ib ra to r from the f i l t e r stages. 5 D. The Audio Ampl i f ie r The audio ampl i f ie r i s a modified vers ion of Metropoli tan V i c k e r ' s Main A m p l i f i e r . I t s c i r c u i t diagram has been given i n 3 the Appendix, I t has an over a l l gain = 3 . 7 x 10 . I t has a bandwidth from 10 cycles /sec to 300 cyc les / sec . This band- width i s because of the capaci tors c,, and c 2 . E, The Phase Shi f te r The c i r c u i t diagram of the phase sh i f t e r i s given i n the Appendix. I t changes the phase from 0 to 320° . However, the output s i g n a l changes with the se t t ing of phase. Thus with each se t t i ng of phase, the input to the phase sh i f t e r has to be adjusted for constant output. F . The Phase-Sensitive Detector The c i r c u i t diagram of the phase-sensit ive detector as shown i n F i g . (6.6) i s due to Schuster (18). The c i r c u i t con- s i s t s of a pentode tube Vj the plate load of which i s switched a l t e rna te ly to and R 2 by the reference s i g n a l . The input s i g n a l at the g r i d of the pentode establ ishes a plate current i — gm eg and the switching tube (V<j ) determines which of the two r e s i s t o r s Rj or R w i l l be traversed by the current . This i s accomplished by a reference voltage about 30 vo l t s i n the secondary of T^ appl ied to gr ids of V^., a l t e rna te ly cu t t ing off one-half of the tube and causing the other ha l f to conduct. The switch tubes ( V 9 , V 2 K ) have a resis tance of about 7000 ohms FIG 6.6 PHASE SENSITIVE DETECTOR - 59 - where as the pentode has a resis tance of 1.5 Meg ohms. This feature assures the balance s t a b i l i t y of the c i r c u i t r e l a t i v e to va r ia t ions i n switching tube cha rac t e r i s t i c s or reference vol tage. Long time constants fo l lowing the detector are obtained by feeding the d .c . output to a pa i r of cathode followers ( V 3 , V 4 ) through the appropriate R-C elements, G. The power ampl i f ie r used i s Bogon's Model M 0 3 0 boaster ampl i f i e r . The narrow bandwidth ampl i f ie r i s White's Operational Selec t ive Ampl i f i e r Model 236A together with White 's twin T of type 546. , H. The Helmholtz C o i l s Each c o i l has 1350 turns and a resis tance of 30 ohms. Their mean radius as w e l l as the distance between the i r centres i s 7 inches. A c a l c u l a t i o n shows that the peak value of the magnetic f i e l d at the centre of the c o i l s i s given by .096 x i gauss where i i s the r .m.s . value of the a l t e rna t ing current f o l - lowing through the c o i l i n mil l i -amperes , I . Working.of the Spectrometer To achieve a good f i l l i n g fac tor , the c o i l of the o s c i l l a t o r i s placed d i r e c t l y in to the sample (para-di-chloro-benzene) kept at the centre of the Helmholtz c o i l s . The o s c i l l a t o r s frequency i s s lowly changed by der iv ing the tuning condenser by a motor. The passage of the de r iv ing frequency through the resonance should be "slow ad iaba t ic" . This requires ~ ^< - 60 - where i s the l i n e width i n frequency u n i t s . During the experiment, o s c i l l a t o r ' s frequency was driven at the rate 3.290 K C / s e c 2 . Thus the aforesaid condi t ion i s we l l s a t i s f i e d as the l i n e width i s about 50 KC. Also the t o t a l time taken by the der iv ing frequency to cross the l i n e should be at leas t several times the time constant of the phase-sensitive detector . At the same time a good s igna l - to-noise r a t i o requires the time constant to be as large as poss ib le . A time constant of 5 sec was found to be a good compromise between these requirements. As the frequency passes through the resonance, the nuclear s igna l i s induced across the c o i l . To record the r e - sonance the s igna l i s modulated by a s inuso ida l magnetic f i e l d of 82.5 c p . s . and having a peak value about 5 gauss. As the Zeeman s p l i t t i n g of the pure quadrupole l eve l s depends upon the o r i en ta t ion of the e l e c t r i c f i e l d gradient axis with respect to the magnetic f i e l d and th i s o r ien ta t ion being random i n case of powder sample, the resonance i s smeared twice a cycle by the modulating magnetic f i e l d . Thus the s igna l appears at the second harmonic of the modulation frequency i . e . at 165 c . p . s . as the output of the in tegra tor . This s igna l after being amplif ied by the audio-ampli f ier and the narrow-band-width ampl i f ie r appears at the input of phase-sensit ive detector. The band-width of the ampl i f ie r i s only 2 c . p . s . , so i t cuts down the noise consider- ab ly . The noise i s further cut down by the phase-sensitive detector . The reference s igna l for the phase-sensitive detector i s provided by a Hewlett Packard Audio o s c i l l a t o r Model 200 AB through a phase s h i f t e r . To keep the phase difference between the two s ignals feeding the phase-sensit ive detector constant, - 61 - the same audio oscillator provides the modulation signal. As the frequency of the modulation signal is half of the reference signal, so the audio oscillator feeds a frequency divider which halves the frequency and then drives a power amplifier. The power amplifier supplies current to the modulation Helmholtz c o i l s . The maximum current obtained was 400 milli-amperes, The value of the current usually used for modulation was 3 0 - 5 0 milliamperes. The output of the phase-sensitive detector i s recorded by E sterline-Angles D.C. mllliaAmeter Model AW. The signal as recorded by the ameter is maximum when the nuclear signal i s in phase with the reference signal. This is done by setting the phase shifter. The f i e l d modulation has been used to avoid the response of the spectrometer to spurious signals which were previously observed in a self quenched frequency modulated oscillator. The frequency spectrum of the oscillator is very complicated. Because of quenching, i t consists of a central frequency with side bands on either sides separated by the quench frequency. The best way to locate the central fre- quency i s to switch off the audio oscillator feeding the quench generator so that the oscillator is pperating continuously. Then i t s frequency can be easily measured by a receiver. The central frequency of the quenched oscillator, which is very close to this frequency is f i n a l l y determined by switching on the audio- oscillator and changing the quench frequency. It is only the central frequency that does not shift by changing the quench frequency. This fact locates the central frequency. - 62 - J. : Measurements The amplitude of the signal was measured as a function of the quench frequency and the modulation current. Then, for a particular quench frequency and.modulation current, the signal was measured as a function of the on-off time of the oscillator. The results have been given in the next section. These measure- ments were taken at room temperature. VII. RESULTS OBTAINED WITH THE SPECTROMETER The spectrometer was operated with low modulation f i e l d ( ~ 3 gauss) so as to satisfy most of the conditions under which the theory for the spectrometer was developed in Chapter V, Figure (7.1) shows the amplitude of the signal as a function of the pulse width (on-time of the oscillator) for the various values of the quench frequency. Three effects are quite prominent. I. The amplitude of the signal goes through a broad maxima as the pulse width is increased. II. The maxima becomes broader as the quench frequency i s decreased. ' III. In the regions of maxima, the amplitude of the signal increases with the quench frequency. The following explanation for the result can be given. The theory of the spectrometer was based on the assumption that the mode of the oscillator's operation i s always logarith- mic and the pulses are rectangular i.e. the rise and f a l l time of the pulse is negligible as compared to i t s f l a t portion. For short pulse widths ( — 5 usee), neither of these approximations hold and in the limit of very short pulses the spectrometer should be operating in the linear mode. In the linear mode the spectro- meter output w i l l be proportional to V g instead of lnV s as in logarithmic mode and increasing pulse width should give i n - creasing signal. As the pulse width i s further increased we 64 - FIG-7.1 SIGNAL Vs. PULSE-WIDTH — . P U L S E - W I D T H \h SEC 13! L. 1 _ 4 -I 5 — 1 \ 1 1 -AO 1 1 1 J L5 1 - 65 - come across such values of pulse width for which the spectro- meter operates between the linear and logarithmic modes and the signal goes through a broad maxima over a certain range of pulse width. In this plateau, the signal as predicted by the equation (5.8) should vary as ~ independent of T or tw. Note that the dependence of signal upon f ' holds only in the logarithmic mode. Figure (7.2) shows a plot of the signal versus ^ for the various values of pulse width, conforming this prediction. A plot of the signal against for different values of. ( T - to) also, gives the same result. On the right side of the plateau, the spectrometer operates in the logarithmic mode because now the pulse width i s quite large as compared with the rise time and the pulses can be assumed to be rectangular. In this region the theory predicts that the signal i s proportional to ' . The exact value of T* i s not known but i t should lie between T and T -tw. i As the value of T decreases by increasing the pulse-width, therefore the signal drops for longer values of pulse-width. However, the decrease in signal amplitude is faster than pre- dicted by the theory-. This is most probably because of the changes in the tube parameters with the increase of the on-time of the oscillator. This factor may also contribute to the existance of a definite plateau for long T , rather than a curve with a simple maximum. In addition to this, the condition that the time decay constant of the r . f . oscillations should be quite large as compared with the off-time of the oscillator, is also not satisfied for long pulses. - 66 - FIG 1.2 SIGNAL VS. OiUENCH FREQUENCY - 67 - Figure (7.3) shows the signal against the r.m.s. value of the modulation current. The peak value of the modulation f i e l d in gauss can be obtained by multiplying the r.m.s. value of the current in milli-amperes by .097. At low fields ( ^ 2 . 5 gauss) the signal is proportional to the square of the f i e l d as pre- dicted by the equation (5.8). As the f i e l d i s further increased, the signal increases less rapidly than that given by H m and i t f i n a l l y begins to drop after passing through a broad maxima at about 10 gauss. The decrease of the signal at the high values of f i e l d can be explained by considering the effects which attenuate the amplitude of the free induction signal as dis- cussed in Chapter IV. The best operating conditions of the spectrometer can be summarized as follows. I. It was found experimentally that the spectrometer is most sensitive when the noise in the detector's output, at the instant when the oscillations are building up, can be seen on the oscilloscope. This can be achieved by adjusting the controls of the quench generator to a proper on-time of the oscillator. This adjustment is quite easy to make because a part of the detector's output i s always displayed on the oscil l o s - cope. II. The frequency of the quench generator should be high (50 - 100 KC/sec), At the high quench frequency, the region in which the spectrometer can be operated for maximum signal i s very narrow. So care should be taken in selecting a proper on- time of the oscillator. A good indication of the proper time is obtained by step I. - 68 - FIG 7-3 SIGNAL Vs. MODULATION FIELD. • - 69 - I I I , The peak value of the modulation magnetic f i e l d should be about 5 - 7 g a u s 3 for ch lor ine 35 nuc l e i i n P a r a - d i - chlorobenzene. For other systems, the optimum modulating f i e l d w i l l depend on Tg and on the nuclear gyromagnetic r a t i o y. The other operating condit ions of the spectrometer have been described i n Chapter VI while discussing the working of the spectrometer. - 70 - APPENDIX HG.A2 AUD\0 AMPLIFIER - 72 - BIBLIOGRAPHY 1. T.P. Das and E.L. Hahn, Nuclear Quadrupole Resonance Spectro- scopy, Solid State Physics; Supplement I (1958), p. 70 - 80. 2. M,H, Cohen and F. Reif, Nuclear Quadrupole effects in nuclear magnetic resonance; Solid State Physics 5, 321 (1957). 3. E.L. Hahn, Phys,Rev. 80, 980 (1950). 4. M. Bloom, E.L. Hahn, and B. Herzog, Phys,Rev. 97, 1700 (1955). 5. M. Bloom, E.L. Hahn, and B. Herzog, Phys.Rev. 97, 1702 (1955). 6. Wang, Towens, Schawlow, Holden, Phys.Rev. 86, 809 (1952). 7. Meal, J.Am.Chemical Soc. 74, 6121 (1952). 8. M. Bloom, Ph.D. Thesis, University of I l l i n o i s (1954). 9. B. Herzog and E.L. Hahn, Phys.Rev. 103, 148 (1956). 10. P.A. Bender, D.A, Jennings and W.H. Tantillo, J,Chem.Physics 32, 499 (1960). 11. T.P. Das and E.L. Hahn, Nuclear Quadrupole Resonance Spectro- scopy!s(1958), p, 7 - 12. 12. M. Bloom, E.L. Hahn and B. Herzog, Phys,Rev. 97, 1704 (1955) - 73 - 13, J.R, Whitehead, Super-regenerative Receivers, Cambridge University Press. 14. C. Dean, Ph.D. Thesis, Harvard University (1952). 15, Abraghamj The Principles of Nuclear Magnetism, p. 545. 16. W.I, Goldberg, Private communications with M. Bloom, 17, D. McCall, Ph.D. Thesis, Department of Chemistry, University of I l l i n o i s (1954). 18. N.A. Schuster; Rev. of Scientific Ins. 22, 254 (1951).

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