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Nuclear quadrupole resonance in kernite Haering, Rudolph Roland 1955

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NUCLEAR QUADRUPOLE RESONANCE IN KERNITE by RUDOLPH ROLAND HAERING A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS i n the Department of P h y s i c s We accept t h i s t h e s i s as conforming t o the standard r e q u i r e d from candidates f o r the degree of MASTER OF ARTS Members o f the Department of P h y s i c s THE UNIVERSITY OF BRITISH COLUMBIA J u l y , 1955 ABSTRACT The -pure quadrupole resonance frequencies due to B 1 1 nuclei at two different sites (denoted by E and P) i n a single crystal of kernite (Na^jjOy.ljJB^O) have been determined both by observing the B 1 1 lines i n zero external magnetic f i e l d , and also by investigating ^ h e i r Zeeman splitting in magnetic fi e l d s up to 30 gauss. The values obtained are 1281.1 * 2 Kc/sec and 1287.0 * 1 Kc/sec for the E and P sites respectively. These fre-quencies agree within experimental error with values predicted on the basis of studies i n a high magnetic f i e l d . The agreement for the P site i s much better than for the E site. Some other interaction, such as asym-metric nuclear shielding, may account for the discrepancy. Resonances also have been obtained i n zero f i e l d with a polycrystalline sample.. Indications of B 1 0 resonances were observed at the frequencies expected assuming Dehmelt's value for the ratio of the quadrupole coupling constants for B l l and B^O. A Na^ resonance line was observed at l£60 t i Kc/sec i n agreement with Proctor's result for this line. The other Na2^ resonance expected from high f i e l d work has not been observed to date. ACKNOWLEDGMENTS I am indebted to Dr. G.M. Volkoff for guiding this research and for taking a constant interest i n the work. Thanks also go to Dr. H.G. Dehmelt for his as-sistance with the experimental aspects of the work as well as for many illuminating discussions. I would also like to thank Dr. J.M. Daniels for stimulating discussions; Mr. W. Morrison and the other members of the machine shop for the machining of the nuclear induction head as well as for their aid with construction d i f f i c u l t i e s ; Mr. E. Price for his assistance with the electronic circuits and my wife for help with the diagrams in the thesis as well as for i t s editing. The support of the National Research Council i n the form of a Bursary to the author and a research grant to Dr. Volkoff i s gratefully acknowledged. TABLE OF CONTENTS Page INTRODUCTION . . 1 CHAPTER I. THEORY k Calculation of the approximate energies and wave functions for the pure quadrupole Hamiltonian k Calculation of the Zeeman perturbation of the pure quadrupole levels 5> II. APPARATUS 1 10 The transmitter and receiver . 10 The nuclear induction head 11 III. EXPERIMENTAL PROCEDURE 17 Methods for obtaining the pure quadrupole f requencie s 19 IV. RESULTS .20 Results on B 1 1 i n kernite 20 Results on B 1 0 in kernite 2k Results on Na23 ±n kernite 2k V. DISCUSSION 25 Comparison of theory and experiment . . . . 2f> Suggestions for further work 27 APPENDIX Axial and radial inhomogeneities of a Helmholtz c o i l 29 Relative concentration of K39 and B nuclei in KN02 and Na2B^0?.^H20 31 Total absorption for pure Zeeman and pure quadrupole transitions for I = 3/2 33 REFERENCES 2>k * « % LIST OF ILLUSTRATIONS Facing Page Circuit of the transmitter- 10 Circuit of the receiver 10 Block diagram of the spectrometer . . . 11 (a) Photograph of receiver and transmitter assembly, top view (b) Photograph of receiver and transmitter . assembly, bottom view 11 (a) Photograph of the induction head, . assembled (b) Photograph of the induction head, . side plate removed 11 Selected traces of B 1 1 resonance curves. 22 Low f i e l d spectrum when H 0 i s pa r a l l e l to c-axis 22 Low f i e l d spectrum when H Q i s perpendi-cular to c-axis 22 LIST OF TABLES Table Facing Page 1. Pure quadrupole frequencies and asymmetry parameters for E and F site B 1 1 nuclei 8 2. Direction cosines of the electric f i e l d gradient tensor axes (x,y,z) relative to X,Y,Z for the E and F site B l l nuclei 8 3. Predicted low f i e l d spectrum for the E and F site B l l nuclei 8 Experimental pure quadrupole frequencies for the E and F site B 1 1 nuclei . . . . - 21 * -* # INTRODUCTION The work of H.H. Waterman (1) at t h i s laboratory-has g i v e n a complete a n a l y s i s o f the quadrupole c o u p l i n g t e n s o r s f o r B n u c l e i i n NapB^O-^HpO, ( k e r n i t e ) . H i s work i s an e x t e n s i o n o f the study of n u c l e a r resonance a b s o r p t i o n s p e c t r a of s i n g l e c r y s t a l s i n h i g h e x t e r n a l magnetic f i e l d s , i n s o f a r as i t t r e a t s a spectrum compli-cated by the e x i s t e n c e of non-equivalent s i t e s f o r n u c l e i of the same s p e c i e s . I n a d d i t i o n to l e a d i n g t o inform-a t i o n r e g a r d i n g the o r i e n t a t i o n o f the p r i n c i p a l axes o f the e l e c t r i c f i e l d g r a d i e n t t e n s o r r e l a t i v e t o the c r y s t a l -l o g r a p h i c axes, a study o f the dependence on c r y s t a l o r i e n t a t i o n i n the e x t e r n a l magnetic f i e l d o f the Zeeman resonances p e r t u r b e d by the quadrupole i n t e r a c t i o n o f the n u c l e i , p r e d i c t s the value of the quadrupole c o u p l i n g constant and of the asymmetry parameter which c h a r a c t e r -i z e the quadrupole H a m i l t o n i a n . Waterman 1s a n a l y s i s (1) showed t h a t B 1 1 n u c l e i occur at f o u r d i f f e r e n t s i t e s . The p r e d i c t e d pure quadrupole f r e q u e n c i e s due to boron-n u c l e i at the two s i t e s w i t h s t r o n g quadrupole c o u p l i n g ( l a b e l l e d E and F) are 1286 and 1287 Mc/sec. Quite s i m i -l a r f r e q u e n c i e s have been p r e d i c t e d by P r o c t o r e t a l ( 2 ) , the u n c e r t a i n t y i n each case b e i n g o f the o r d e r o f 5 Kc/sec. A pure quadrupole resonance i n k e r n i t e was f i r s t 2 observed by Cranna (3) at 1.270 - .010 Mc/sec, with a superregenerative spectrometer. A more accurate measure-ment was later made by Proctor and Kiddle (2) and the result of this measurement was 1.279 - .001 Mc/sec. In both instances, only one line was observed. In a private communication to Dr. Volkoff, Dr. Proctor has mentioned that for the Na 2 3 nuclei in kernite which on the basis of high f i e l d measurements are found to occupy two types of sites with quadrupole coupling constants differing even more widely, again only a single line was observed. If the resonances were at the predicted frequencies, then i n the case of the E and P site boron nuclei at least, one would certainly not expect the lines to be resolved i n a very small magnetic f i e l d because of the line widths. However, there is a possibility that the lines w i l l separate appreciably i n somewhat larger f i e l d s , and i t may then be possible to ob-serve two resonances. Also, since the E and F site nuclei are non-equivalent, there will.be orientations of the crystal in which any observed resonance w i l l be almost entirely due to only one of these sites. One might therefore expect to separate the contributions due to the two different sites by suitable orientation of the crystal with respect to the axes of the spectrometer and the small magnetic f i e l d . It was also f e l t that a check on Proctor's measurement might be use-f u l , i n order to establish whether or not the small difference between the observed and predicted resonance frequencies i s significant. With this experiment in mind, i t was decided to build a nuclear induction spectrometer of the Bloch type, since the group working under Proctor at Seattle had achieved considerable success with this type of spectrometer. This instrument i s of course not limited to the above experiment, and i t was hoped that the results on LiAl(SiC"3)2 (spodumene) obtained by Volkoff et a l . (lj.,5,6,7) and on HBeAlSiOcj (Euclase) obtained by Eades (8) might also be confirmed by a .measurement of the respective pure quadrupole resonances. CHAPTER I THEORY The object of the theory that follows i s to cal-culate the frequency shift of the resonance lines due to the E and P site boron nuclei, when the pure quadrupole states are perturbed by a small magnetic f i e l d . Since the i t i s convenient to use approximate expressions for the energies and wavefunctions of the unperturbed states. .The approximate expressions w i l l then be used as a starting point for the perturbation of the pure quadrupole states by a small magnetic f i e l d . 1. Calculation of the unperturbed energies and the approx- imate wavefunctions. Kriiger (9) gives the Hamiltonian for the problem asymmetry parameter, ft , i s small for both of these sites, We break up the Hamiltonian as follows: Where: and treat as a perturbation on *K A simple calculation results i n the following approximate energy eigenvalues: i,a 4-f e Q 6 x x 4 0£] so that there are two energy states, each 2-fold degenerate The wavefunctions describing the above states i n terms of the angular momentum wavefunctions are : *2 These expressions check with those given by Kruger (9)• They provide a convenient starting point for a perturbation of the pure quadrupole levels by a small magnetic f i e l d . 2. Calculation, of the Zeeman perturbation of the pure  quadrupole energy levels. The complete Hamiltonian, referred to the prin-cipal axes of the electric f i e l d gradient tensor i s now: where, H 0 i s the perturbing magnetic f i e l d , and are the direction cosines of H 0 relative to the principal axes of the electric f i e l d gradient tensor. In terms of the angular momentum eigenfunctions, the perturbation Hamiltonian i s : ( $* ) l i z o o V 2 2 O O . 2 » - * ( i Z O O J i * - 3 * where: z-«*t ; 2 * This matrix must now be represented relative to the -basis, where them's are the approximate eigenfunctions of the quadrupole Hamiltonian. For a f i r s t order perturbation calculation, not a l l the matrix elements need be known. Since the states described by vy, and y t are degenerate i n the pure quadrupole case, (as are also those described by vy5 and vj^  ), we need to diagonalize i n two by two blocks. The result-ing diagonal terms are the f i r s t order corrections to the energy. The required matrix elements are: 2 3 4 where t Z - o f - 1 Solving the two resulting equations and « o %4 44 = o results i n : Table 1 Pure quadrupole frequencies and asymmetry parameters f o r the E and P s i t e B l l n u c l e i i n k e r n i t e E. e Q 4 * * ( v + £ ) Mc/sec ah n 1.286 - .001]. .163 - .010 1.287 - .003 .117 - .010 Table 2 D i r e c t i o n cosines of the f i e l d g r a d i e n t tensor axes (x,y,z) w i t h respect to X,Y,Z at the E and P B H s i t e s i n k e r n i t e . . X y z • E, X Y Z -.76 - .ok +(.22 - .03) +(.60 - .07) -.60 ± .03 +(.05 - .01) -(.79 - .07) -.21 - .01 + (.97 - .02) + (.10 - .01) P, X Y Z -.05 - .03 -(.92 - .01|.) + (.37 ~ .02) -.91+ - .01|. -(.09 - .03) -(.33 - .02) + .3I4.O - .005 ± ( . 3 6 8 - .005) ± ( . 8 6 5 - .001) Table 3 Predicted low f i e l d spectrum, for the E and P site B-^ nuclei i n kernite. (Calculations based on Waterman's data.) H Q p a r a l l e l to c-axis H Q perpendicular to c-axis i ^ i * (weak) ^3.4 (strong). • ^ i , * . (weak) »a,4. (strong) E-sites 1286 ±"1 .575 . 1286 - 1.10H. 1286 - 2.73H. 1286 - 1.2i|H0 P-sites 1287 - 2.68H0 1287 - .86H, 1287 - 2.12H. 1287 - .610H,, 8 The resulting f i r s t order energies are: In the experiment described in this thesis the frequency measurement i s considerably more precise than the measurement of HQ, which may be i n error by as much as Hence products of the type can be eliminated from equations (1) without seriously affecting the val-i d i t y of the resulting formulas. The simplified expres-sions for the energy levels are then: For a given orientation of the crystal i n the magnetic f i e l d , cc({£ * can be obtained from, the high-field analysis by Waterman ( 1 ) . Since r\ and e Q 4>» a r e a i s 0 known, the equations (1) w i l l predict the shift of the energy levels i n small fi e l d s , and hence the shift of the resonance absorption lines. Tables l.and 2 were taken, from Waterman's work ( 1 ) . The numerical values i n these tables were used in the calculation of the low f i e l d spectrum. The notation i n Table 2 i s as follows: the sub-scripts on the E and F refer to the two nuclear sites con-nected by a rotation about a two-fold symmetry axis. The axes labelled x, y and z refer to the principal axes of the electric f i e l d gradient tensor, while those labelled X, Y and Z refer to the crystallograph!c b, bxc and e axes res-pectively. Table 3 gives the calculated low-field spectrum for the two cases when H Q i s either p a r a l l e l or perpen-dicular to the crystallographic c-axis. The numerical values i n Table 3 are'based on the equations (1), the data in Tables 1 and 2, and the known value of (1.367 Kc/sec per gauss). Four lines result for each site i n a given orientation. Of these, two w i l l be weak, since they are essentially due to the admixture of states. This admix-ture i s entirely due to the non-zero value of tj , and i s small since i s small. The experimentally observable transitions can be identified by letting approach zero. In Table 3 they turn out to be the transitions labelled with the subscripts 3 and 1+. No account has been taken of the uncertainties in Tables 1 and 2. These errors, together with the exper-imental uncertainties, w i l l be discussed i n Chapter V. + 3 0 0 V ^ — W W W 5 K Fiq / 7 W £ TRANSMITTER CHAPTER II APPARATUS The apparatus used i n t h i s research consists of a nuclear induction spectrometer s i m i l a r to those described by Proctor (10) and Weaver (11). The o s c i l l a t o r supplying the radio frequency f i e l d to the sample Is of the push-pull type described by C o l l i n s (12). The actual c i r c u i t compon-ent values resemble c l o s e l y those used by Proctor and h i s group i n a s i m i l a r spectrometer. The c i r c u i t diagram i s given i n Pig. 1. The o s c i l l a t o r was designed to give up to 30 v o l t s across the transmitter c o i l . Usually, the voltage across the c o i l was kept at about 10 v o l t s r.m.s., r e s u l t i n g i n an r . f . f i e l d of approximately ^ gauss. The radio frequency amplifier consists of three low gain stages using SAC?^ as pentodes. The gain per stage i s approximately 5, r e s u l t i n g i n an o v e r a l l gain of 125. The band width i s about 3 Mc/sec, extending from 200 Kc/sec. to 3 Mc/sec. The o s c i l l a t o r and receiver are , b u i l t on the same chassis and are connected to the induction head by short, matched, coaxial cables. A p a r t i t i o n and base plate prevents excessive r . f . leakage, which would r e s u l t i n undesirably large r . f . voltage at the diode following the amplifier. The o s c i l l a t o r tank c i r c u i t em-ploys the outside two sections of a t r i p l e gangue, 1J.0-1J.00 JY"f- condenser, i n conjunction with a c o i l having P i g . I j . (a) Photograph of t r a n s m i t t e r and r e c e i v e r assembly top view (b) Photograph of t r a n s m i t t e r and r e c e i v e r assembly bottom view F i g . 5 (a) Photograph o f the i n d u c t i o n head, assembled (b) P h otograph o f the i n d u c t i o n head, s i d e p l a t e removed RADIO FREQUENC) RECEIVER METER AUDIO AMPLIFIER PHASE SENSITIVE DETECTOR TIME CONSTANT NETWORK DE 7ECTOR RF RECEIVER AUDIO OSCILLATOR PHASE NETWORK RECORDING MILL I AMMETER D.C. ' 4 D.C. FIELD COILS K MODULATION COILS U3 NUCLEAR INDUCTION HEAD POWER AMPLIFIER F i S 3 BLOCK DIAGRAM OF SPECTROMETER ' 200 + - . 5 i n d u c t a n c e . The tuned receiver input employs the central section of the variable condenser together with a c o i l of 100 ± p\i . This arrangement leads to a Q of about 75 for the tuned r . f . input stage, when the two cables connecting the oscillator and the receiver to the induction head are about 12-16 inches long. After the three condenser sections have been matched by suitable bending of the plates and the transmitter and receiver inductances have been equalized by means of the small tunable inductance i n the oscillator circuit, the oscillator and receiver track closely over a range of about $0 Kc/sec, allowing a sweep of 100 Kc/sec. provided the receiver and transmitter are aligned at the centre of the range. Sweeps of up to 200 Kc/sec. may be used, but the sensitivity at the end of this range becomes somewhat less because of detuning. The design of the nuclear induction head resembles closely that given by Weaver (11). Weaver's design was simplified by the omission of the internal ground planes between the oscillator and receiver coils> and the fine V-mode control. Pig. 5 shows a photograph of the induction head. Its overall size i s ij." by ij. n by 1 7/8". The U-mode control consists of an inductance loop and a 10 ohm series resistance mounted at the end of a lucite rod. The as-sembly i s inserted into an accurately machined hole d r i l l e d axially into the stationary part of the transmitter c o i l . Its operation i s described by Weaver (11). This arrange-ment allows one to reduce the r.f . leakage voltage i n the receiver input to about 2 m i l l i v o l t s when the voltage across the transmitter coils i s of the order of 20 volts. This de-coupling i s essential for proper operation of the spectro-meter, since the voltage on the grid of the third 6AC7 must not exceed 1 volt i n order to ensure essentially linear operation. For larger leakage voltages, the gain of the r.f. amplifier would have to be reduced. It was found, how-ever, that optimum operation i s achieved with minimum leakage voltage. When the U-mode i s observed, both controls are i n i t i a l l y used to balance out the leakage voltage. A small amount of U-mode leakage i s then induced with the U-mode control. Similarly, for observation of the V-mode, a small leakage i s induced with the V-mode control. The detected receiver output i s fed into a high gain, tuned audio amplifier. The gain of the amplifier i s variable i n steps of 3 from 103 to about 10?. The amplifier is tuned to the modulation frequency, 99.5 cps., and has a of 1+0. The modulation frequency was chosen so that stray pickup at 60 and 120 cps. was minimized. The. audio output of this amplifier i s then detected by a phase sensitive detector of the Schuster (13) type. The circuit was slightly modified by Waterman and the actual circuit i s given i n his thesis (1). The output of the phase sensitive detector i s fed onto an Esterline-Angus chart Recorder after passing through a variable time-constant network. The modulation f i e l d to the sample i s supplied by a pair of Helmholtz coils mounted directly on the sides of the nuclear induction head. The coils consist of 30 turns of #21+ magnet wire and supply, a f i e l d of about 12 gauss per ampere. The current flowing through these coils i s supplied by a power amplifier capable of producing 16 watts output into a matched load. Suitable phase shift networks are in-corporated in the input to the power amplifier as well as In the reference grid of the phase sensitive detector. This arrangement allows a 360 degree phase shift and i s used to optimize the phase of the two voltages on the phase detector. The block diagram of the entire spectrometer is given i n Fig. 3 . When the spectrometer i s in operation, the t r i p l e gangue condenser i s driven by a variable speed drive at rates between 8 Kc/min. and 100 cycles per min. The time-constant network following the phase detector can be adjusted to these speeds over a range of from 1 to 80 seconds; the optimum value for a given speed being determined by the line width of the resonance absorption. The spectrometer was primarily built for the pur-pose of studying pure quadrupole resonances i n the 1 mega-cycle region. In this application, the pure quadrupole resonance frequency i s perturbed by a small magnetic f i e l d applied perpendicularly to the axes of the receiver and transmitter coils i n the nuclear induction head. The small magnetic f i e l d i s supplied by a second set of Helmholtz coils consisting of $00 turns each of #18 magnet wire and capable of producing 30 gauss per ampere. However, the spectrometer is not restricted to this application. The nuclear induction head was designed so that i t would f i t into the 2 n gap of the permanent magnet des-cribed by Waterman (1). This feature was found to be use-f u l in the preliminary tests made before the spectrometer was operating satisfactorily. Since the f i e l d of the above mentioned permanent magnet is approximately 7000 gauss, the nuclear resonance of K 3 9 occurs at about 1,L\. Mc/sec. Thus, a saturated solution of KN02 or any other f a i r l y soluble potassium salt provides a convenient test signal in the 1 megacycle region. Since the resonance obtained from potas-sium in solution i s comparatively sharp, unnecessarily long searching i s hereby eliminated. After some adjustments, the potassium resonance was observed with a signal to noise ratio of 100:1, using a fast sweep and a 5 second time constant. A search was then made for the B 1 1 pure quadrupole resonances i n a single crystal of kernite, (Ha2B^0y. LLE^)), predicted by Waterman (1) at 1.286 and 1.287 Mc/sec. and by Proctor et a l . (2) at 1.285 and 1.290 Mc/sec. One or both of these reson-ances due to the E and P sites boron nuclei described by Waterman (1) have been observed by Cranna (3) and by Proctor and Kiddle (2). After several unsuccessful searches, some reasons for possible failure were investigated i n some de-t a i l . These may be li s t e d under: 15 1. Insufficient modulation due to the fact that the resonances due to the two different sites are spaced i n such a way as to make the overall line width abnormally large. 2. Lower concentration of nuclei i n the case of the kernite crystal due to a poorer f i l l i n g factor and fewer nuclei per cc. within the crystal as compared to the potassium in solution. 3. Possible smearing" of the resonance lines be-cause of unduly large inhomogeneities i n the f i e l d obtained from the Helmholtz coils. 1+. Some fundamental difference between the amount of absorption in the case of a quadrupole resonance and a dipole resonance at the same frequency. 5. Unfortunate orientation of the crystal i n the small magnetic f i e l d , leading to low transition probabi-l i t i e s . Some of the above p o s s i b i l i t i e s , (2, 3 and Ij.) were inves-tigated i n the Appendix. The pure quadrupole resonance i n kernite due to the E and F site boron nuclei was f i r s t observed i n a f i e l d of about 12 gauss. In this f i e l d , the lines are just re-solved, the line widths of the two components being about 10 Kc/sec. In order to determine the pure quadrupole fre-quency, the magnetic f i e l d was then varied over a wide range and the resulting frequency shifts were noted. Two straight l i n e p l o t s r e s u l t when frequency i s p l o t t e d versus magnetic f i e l d . The common i n t e r s e c t i o n o f these two l i n e s with the l i n e H 0 = 0 g i v e s the pure quadrupole frequency. A b e t t e r way to measure the pure quadrupole frequency i s to make use of the symmetry of the low f i e l d spectrum. A d i r e c t mea-surement i n zero f i e l d i s a l s o p o s s i b l e . These methods w i l l be d i s c u s s e d i n Chapter I I I . CHAPTER III EXPERIMENTAL PROCEDURE The object of this section i s to c l a r i f y and justify the various techniques for obtaining the two pure quadrupole frequencies due to the E and P site borons in kernite. Since the E and P site pure quadrupole frequen-cies cannot be expected to be resolved due to line widths, one must make use of the crystal orientation i n order to untangle the low-field spectrum of these two sites. Under the action of a small magnetic f i e l d , the pure quadrupole resonances s p l i t up into two strong and two weak resonances, provided n=^o . Therefore, at least four lines should be observed in the low f i e l d spectrum of the E and P site boron nuclei. However, the intensity of these lines is strongly dependent on crystal orientation, (cf. ref.(6)) being a maximum when the "long" axis of the elec-t r i c f i e l d gradient tensor i s perpendicular to the r . f . f i e l d and a minimum when the two are p a r a l l e l . Making use of the data in Tables 2 and 3, i t i s apparent that choosing HQ p a r a l l e l to the crystallographic c-axis strongly favours the P sites, while the E sites w i l l be very e f f i c i e n t l y suppressed. (For a l l measurements, the b-axis was along the receiver c o i l axis.) However, i f HQ i s chosen perpendicular to the c-axis, the E sites w i l l be favoured. The P sites are i n this case not very e f f i c i e n t l y sup-pressed, since their long axis does not l i e para l l e l to the r . f . f i e l d in this orientation. In fact, because the P sites f a l l into two categories, Pi and P£, and because the angle between the long axis of the Pi and P2 i s ap-preciably different from 180°, i t i s impossible to balance out the P sites completely.in any orientation. The reason is simply that even i f the r.f. f i e l d were chosen to l i e exactly along one of the long axes, say that of Pi, the P2 site long axis would not be simultaneously p a r a l l e l to the r . f . f i e l d , and hence a resonance could be expected. Hence, an P site contribution to the E site resonances may be ex-pected. On the basis of Waterman's data, these lines should be resolved i n a f i e l d of about 30 gauss, so that Ij. lines should result. Experimentally, only 2 lines were observed, even i n fields of up to 80 gauss. This discrepancy i s dis-cussed i n Chapter IV. The above argument also shows that i f HQ were chosen at about 1+5° to the c-axis, both sites should be almost equally favoured, and Jj. lines should be observed in f i e l d s of 30 gauss or more. This was i n fact found to be the case. Three methods of measuring the pure quadrupole frequencies now suggest themselves. A l l of these methods depend upon crystal orientation for separating the E and P site resonances. 1. A p l o t of the low f i e l d spectrum f o r each  o r i e n t a t i o n . For s m a l l f i e l d s , the Zeeman p e r t u r b a t i o n i s l i n e a r , and a l e a s t square f i t t o the experimental p o i n t s can be made. The common i n t e r s e c t i o n of the best s t r a i g h t l i n e s through the p o i n t s w i t h the l i n e HQ. = 0 gives the pure quadrupole frequency. 2. A p l o t o f the low f i e l d spectrum f o r each  o r i e n t a t i o n . The theory r e v e a l s that f o r a given s i t e , the low f i e l d spectrum i s symmetric about the l i n e P = v£ Hence, i n any given small f i e l d the pure, quadrupole f r e -quency i s given by: A s e r i e s of measurements of \7X and i n d i f f e r e n t f i e l d s can be made, and the r e s u l t i n g estimates of \?a are then averaged. 3. A d i r e c t measurement of the pure quadrupole  frequencies i n zero f i e l d . At f i r s t s i g h t , t h i s seems impossible because of the opposite p o l a r i z a t i o n s of the two (overlapping) components. Hence, one might expect complete c a n c e l l a t i o n . This i s , however, not so because of the a d d i t -i o n a l phase s h i f t introduced by the modulation f i e l d . Thus, when the modulation amplitude i s i n c r e a s i n g , one of the com-ponents w i l l move up and the other down i n frequency, caus-i n g the l i n e s to add. CHAPTER IV RESULTS The low f i e l d spectrum due to the E and P s i t e B 1 1 n u c l e i i n NagB^Oj.L\IL^) ( k e r n i t e ) was observed i n order t o e s t a b l i s h the pure quadrupole resonance f r e q u e n c i e s due to n u c l e i at these two s i t e s . I n o r d e r t o o b t a i n b o t h of these f r e q u e n c i e s , two e a s i l y r e p r o d u c i b l e o r i e n t a t i o n s o f the c r y s t a l were used. I n b o t h o r i e n t a t i o n s , the c r y s t a l -l o g r a p h i c b - a x i s was chosen along the a x i s o f the r e c e i v e r c o i l . However, the two o r i e n t a t i o n s are d i f f e r e n t i n s o f a r t h a t one time the r a d i o frequency f i e l d i s chosen perpend-i c u l a r t o the c - a x i s , while the other time i t i s chosen p a r a l l e l t o the c - a x i s . F o r a l l measurements the s m a l l s t a t i o n a r y magnetic f i e l d was p e r p e n d i c u l a r t o the plane c o n t a i n i n g the axes o f the r e c e i v e r and t r a n s m i t t e r c o i l s . P i g s . 7 and 8 are p l o t s o f the ex p e r i m e n t a l and t h e o r e t i c a l low f i e l d spectrum o f the E and P s i t e B 1 1 n u c l e i i n k e r n i t e . P i g . 1 e x h i b i t s the r e s u l t s o b t a i n e d w i t h the c r y s t a l l o g r a p h i c c - a x i s p a r a l l e l the s m a l l s t a t i o n -ary magnetic afield H Q and p e r p e n d i c u l a r t o the plane con-t a i n i n g the r e c e i v e r and t r a n s m i t t e r c o i l axes. •The theo-r e t i c a l curves are based on a p e r t u r b a t i o n c a l c u l a t i o n u s i n g Waterman's d a t a i n Tables 1 and 2 f o r t h i s o r i e n t -a t i o n . S i m i l a r l y , P i g . 8 e x h i b i t s the r e s u l t s o b t a i n e d w i t h the c r y s t a l l o g r a p h i c c - a x i s p e r p e n d i c u l a r t o H 0 and Table 1+ Measured pure quadrupole frequencies of the E and F site B l l nuclei i n kernite. Method of determination Si E te F Least square f i t to low f i e l d spectrum Symmetry of low f i e l d spectrum Direct measurement i n zero f i e l d 1282.0 - 2 1281.1+ - 2 1280.0 t 2 1286.7 - 1 1287.1+ - 1 1286.8 ± 1 Average pure quadrupole frequency 1281.1 - 2 1287.0 - 1 A l l entries are i n Kc/sec. p a r a l l e l t o the t r a n s m i t t e r c o i l a x i s . The t h e o r e t i c a l curves i n P i g . 8 are based on a s i m i l a r c a l c u l a t i o n , em-p l o y i n g the data i n Tables 1 and 2. The pure quadrupole f r e q u e n c i e s of the E and P s i t e B 1 1 n u c l e i i n k e r n i t e were measured by each of the t h r e e methods d i s c u s s e d i n Chapter I I I . The r e s u l t s of these measurements, to g e t h e r w i t h the averaged pure quad-r u p o l e f r e q u e n c i e s , are g i v e n i n Table ij.. A l l f r e q u e n c i e s were measured w i t h a BC-221A type frequency meter which had been p r e v i o u s l y c a l i b r a t e d a g a i n s t WWV. The quoted u n c e r t a i n t i e s i n the measurements are r a t h e r l a r g e r t h a n the u n c e r t a i n t y i n the a c t u a l frequency measurement, i n order to i n c l u d e s y s t e m a t i c e r r o r due to impure modes and u n r e s o l v e d l i n e s t r u c t u r e . When the s m a l l s t a t i o n a r y magnetic f i e l d H Q i s p a r a l l e l t o the c r y s t a l l o g r a p h i c c - a x i s one i s o b s e r v i n g p r i m a r i l y the P s i t e n u c l e i s i n c e the E s i t e s are e f f i -c i e n t l y suppressed i n t h i s o r i e n t a t i o n . Hence o n l y two l i n e s may be expected i n the low f i e l d spectrum when t h i s o r i e n t a t i o n i s employed. T h i s was i n f a c t found to be the case. From P i g . 7 i t can be seen t h a t the agreement be-tween theory and experiment i s v e r y good i n t h i s case. The incomplete s u p p r e s s i o n of the P s i t e s has ben mentioned p r e v i o u s l y . I t was s t a t e d t h a t t h i s should l e a d to f o u r r e s o l v e d resonance l i n e s i n f i e l d s exceeding 30 gauss when H Q i s p e r p e n d i c u l a r t o the c r y s t a l l o g r a p h i c F i g . 6 S e l e c t e d t r a c e s o f B l l r e s onanc e cur v e s f o r t h e F s i t e . The time c o n s t a n t was 10 s e e s , and t h e m o d u l a t i o n a m p l i t u d e about 1 gauss. (a) H Q i s z e r o . One c h a r t d i v i s i o n i s a p p r o x i m a t e l y l± K c / s e c . (b) Eg i s about 1$ gauss. One c h a r t d i v i s i o n i s a p p r o x i m a t e l y 8 K c / s e c . (c) H 0 i s about 20 gauss. One c h a r t d i v i s i o n i s a p p r o x i m a t e l y 6 K c / s e c . I n a l l t r a c e s the c - a x i s was p a r a l l e l t o H 0 and p e r p e n d i c u l a r t o the t r a n s -m i t t e r a x i s . F I G 7 LOW FIELD SPECTRUM OF B" IN KERNITE Ho PARALLEL TO C-AXIS 1310 - -EXPERIMENTAL POINTS THEORETICAL CURVE FOR E-SITE B" NUCLEI THE OR E TIC A L C URVE FOR F-SITE B" NUCLEI F i s 8 10 zo FIELD (GAUSS) LOW FIELD SPECTRUM H. PERPENDICULAR TO 30 II OF B IN C - A X I S KERNITE c-axis. Only two lines were observed experimentally, even in fields of 80 gauss or more. A slight asymmetry of the observed lines was observed, which may indicate unresolved line structure. Furthermore, the experimental low f i e l d spectrum does not agree with the theoretical spectrum very well. This discrepancy may be due to poor crystal align-ment or errors in the measurement of the magnetic f i e l d . However, i t seems more l i k e l y that the theoretical pre-dictions for the E sites are i n error, since the predicted pure quadrupole frequency for the E site B 1 1 nuclei also exhibits a discrepancy. In an orientation intermediate to those mentioned above, four resonances of approximately equal intensity were observed i n field s of 30 gauss or more. The centres of the two pairs of resonance lines agreed within experi-mental error with the pure quadrupole frequencies given i n Table ij.. This orientation was not used to obtain quanti-tative results, but confirms the presence of two non-equivalent nuclear sites in a more striking manner. An additional check on the presence of the two j nonequivalent nuclear sites can be obtained by using a polycrystalline sample. A measurement i n zero f i e l d with this sample resulted in a doubly peaked resonance l i n e . The line peaks were measured to be at 1280 and 1286 Kc/sec. Again, the measurement served only as a, qualitative check on the results in Table 1±. An accurate measurement would 23 be d i f f i c u l t i n this case because of the poor signal to noise ratio. Most of the measurements were taken with a 10 second time constant at a sweep rate of approximately 2 Kc/sec. per minute. This ratio was chosen in-order to avoid lengthy sweep times and undesirable phase lags. In a l l cases, the modulation amplitude was about 1 gauss, which for the B 1 1 resonances i n kernite i s approximately 1/8 of the half-line width. This low modulation amplitude reduces the signal amplitude considerably but ensures f a i t h f u l line reproduction. With the above settings of time constant, sweep rate and modulation amplitude the signal to noise ratio of the B 1 1 lines from kernite was approximately 10:1. In the case of overnight searching a sweep rate of ^ Kc/sec. per minute was used with a time constant of 60 seconds. The signal to noise ratio of the B-*-l lines i n kernite was then approximately 5>0:1, provided a modulation amplitude of about 6 gauss was employed. Waterman (1) states that a B 1 0 resonance i n kernite should be observed at 1333 Kc/sec. This resonance i s due to + + the - 3—*• - 2 transition. Two further lines should be ob-served as a result of the - 2 —*• - 1 transitions, since the - 1 level i s the only level that i s spl i t by the small i n f i r s t order. Dehmelt (31}.) has calculated the splitting as: AV= f-hv> A simple calculation for the P site B^u nuclei results in two resonances at 734 and 862 Ec/sec. Several searches through the 860 and 1330 Kc/sec. region have revealed in-dications of resonances. In each case the signal to noise ratio was only about 2t l . An accurate measurement of the frequency has not yet been carried out but both lines ap-pear to be within £ Kc/sec. of the predicted values. The line widths appear to be about 1/3 the line width of the corresponding B 1 1 lines. A measurement of the Ha23 resonances was also attempted. In order to interpret any results, one would require data on the orientation of the electric f i e l d gradient tensor at the two sites similar to the data given for B 1 1 i n Table 2. A pure quadrupole resonance due to Wa^ 3 w a s measured at l£60 t i Kc/sec. Wo change i n this frequency was noticed as the crystal was rotated about the b-axis. A second line due to has not been observed to date. CHAPTER V DISCUSSION This experiment was designed to verify the work done by Waterman (1) at this laboratory. Both of the measured B 1 1 resonances due to the E and P site nuclei l i e within experimental error of the predicted frequencies. In the case of the P sites, the agreement seems to be excel-lent. In fact, the average value for the P site pure quad-rupole frequency of B 1 1 agrees exactly with the value pre-dicted by Waterman (1). Furthermore, the experimental low f i e l d spectrum follows the predicted curves very closely throughout the entire range of field s investigated. Slight deviations from the theoretical spectrum are easily explained by errors in the measurement of H Q and i n the crystal align-ment. The magnetic f i e l d was measured to an accuracy of about 5% by a simple measurement of the current flowing in the Helmholtz coils . No accurate method of crystal alignment was employed and the orientations used are uncer-tain within about % degrees. The effect of these errors i s an uncertainty i n the slopes of the linear experimental spectrum. The pure quadrupole frequencies are not affected by these errors. It i s possible to explain the poor agree-ment between theory and experiment in the E site case partly by a consideration of these errors. However, the deviation of the experimental pure quadrupole frequency from the theoretical value cannot be explained on this basis. It i s true that the E site pure quadrupole frequency l i e s within the uncertainty range of the theoretical value, but i n the opinion of the author the deviation i s significant. This belief is based of the knowledge that both Waterman and the author have been somewhat generous with estimates, of uncertainty. One might attempt to explain the deviation due to the presence of the incompletely suppressed F sites which have been mentioned earlier. However, i f one accepts the apparent fact that the F site frequency l i e s higher than the corresponding E site frequency, this attempt i s n u l l i -fied since incomplete F site suppression would tend to raise and not to lower the E site frequency. If one is willing to accept the deviation as being real, an explanation could perhaps s t i l l be found in the suggestion made by Dr. Daniels and quoted in Waterman's thesis (1). Dr. Daniels suggested that the deviation may be due to asymmetric shielding of the nueleus by i t s surroundings. If such an effect i s pre-sent, i t may be that i t i s more pronounced i n the case of the E sites. Certainly, such shielding could be different for the E and F sites, since these two sites are non-equivalent. The preliminary results on B 1 0 are somewhat un-certain. If the measurements to date are reliable, It would seem that the ratio for the quadrupole coupling con-stants given by Dehmelt (l£) i s verified. His observation of the relative line width of B and B x w nuclei i n similar surroundings also seems to be confirmed, although in the case of kernite this ratio i s rather smaller than 6, the figure given by Dehmelt (llj.). 23 The results so far obtained on the Na spectrum agree with Proctor's (2) results within experimental error. Only one site has been found so far. Investigation of the other site may require rotation of the crystal about the a or c-axis. Suggestions for further experiments. This experiment suggests some possible further investigations, which are l i s t e d below. 1. The reason for the apparent deviation of the observed pure quadrupole frequency might be thoroughly i n -vestigated. It may be useful to use a crystal in which a l l the observed nuclei are i n equivalent positions i n order to simplify the spectrum. Spodumene might be a good crystal to work with, since i t s spectrum has been f u l l y analysed pre-viously by Volkoff et a l . (4,5,6,7). 2. Measurements of the magnetic f i e l d and crystal orientation might be refined. The magnetic f i e l d could be accurately calibrated with a paramagnetic resonance signal which occurs at approximately 30 Mc/sec. i n a f i e l d of 10 gauss. The current i n the Helmholtz coils could then be controlled by controlling the voltage across a standard resistance. A proper crystal mount similar to the one d e s c r i b e d by Waterman (1) c o u l d be used to improve the mea surement of the c r y s t a l o r i e n t a t i o n and c o u l d e a s i l y be i n s t a l l e d w i t h the B l o c h type I n d u c t i o n head. APPENDIX 1. C a l c u l a t i o n s o f the a x i a l and r a d i a l inhomogeneities of  a Helmholtz c o i l . The magnetic f i e l d near the c e n t r a l p o i n t on the a x i s of the two c o i l s may be expanded i n a T a y l o r s e r i e s about t h i s p o i n t . L e t A be the a x i a l and B the r a d i a l f i e l d com-ponent. Then, near the c e n t r e : A = A.+ ( x A . + f A ^ + . j L WA„<rZ%P\t+f> zAtf.) + # ( » \ , + 4x«p + 6 y y A . , r f +... V+... where: x = a x i a l displacement from c e n t r e . p = r a d i a l displacement from c e n t r e . \ , A , e t c . = (^ ~) , (>^ *p) e t c . . . A s i m i l a r e x p r e s s i o n i s v a l i d f o r B. The f i e l d e q u a t i ons, — » -> div H = O ^ c u r | H = O y i e l d , i n the above n o t a t i o n : and, dA I B _ c ( 2 ) Substituting the above expansions f o r A and B i n (1) and (2) and equating c o e f f i c i e n t s of equal powers of X and f» , re s u l t s i n : A - A. + £ x<A„„ O) In a r r i v i n g at the above expressions, use has also been made of the a x i a l symmetry of the f i e l d . Now, where: r = c o i l radius N = number of turns per c o i l I = current i n amperes Evaluating CA„ M1 X my y i e l d s , B - Ha 2±-*4 " ( 6 ) Clearly, the inhomogeneities become very small, provided m » *f p . In the experiment described in this thesis, r was l£ cm. Since x and p are of the order of 1 cm. for the sample boundaries, the inhomogeneities may be expected to be completely negligible. II. Calculation of the relative concentration of Potassium  nuclei i n saturated KMO? solution to Boron nuclei i n  Kernite. (Na2BkP7. UE-2Q). Using the solubility of KNO2 as 3.13 gm/cc, the number of potassium nuclei per cc. i s 2.28 x 1022. Using the density of kernite as 1.953 gm/cc, the number of boron nuclei per cc. i s 1.70 x 10^2. Hence the concentrations of the nuclei whose resonances are observed are very nearly equal. After iso-topic abundance and nonequivalent sites have been accounted for, we have: K39 = B 1 1 6.16 Hence, provided the f i l l i n g factors are comparable for these two samples, the resonances obtained can also be expected to be comparable with regards to intensity. III. Calculation of the total absorption for pure quad- rupole and pure Zeeman transitions for the case  I = 3/2. The absorption at an angular frequency w i s : oo lr E* a = ^  , y H f j?(6)f(E')[eZ ]^(E'-E^<6|llt\E'>l,aE ( 1 ) where: pfe), ^(ef) = density of states with energy Z = partition function E' = E+-noJ The area under a resonance peak represents the total absorption over a l l frequencies and hence i s the integral of the expression ( 1 ) over a l l frequencies. Or: TOTAL ABSORPTION A = J C U U J -oo For discrete levels we have: in the representation in which $C i s diagonal. Rewriting the above sum of matrix elements, we get: A - s p u, f.-fe J X X I , - ( a ) This expression may now be evaluated for any given Hamilton-ian i f the spin.of the nucleus i s also known. 33 1. Pure Zeeman a b s o r p t i o n ( I = 3/2) For t h i s case Making use of the r e l a t i o n one e a s i l y o b t a i n s A 2 - W ( f s.'nh(lfe)+iri«W (3) 2. Pure quadrupole a b s o r p t i o n ( I - 3/2, H = 0) For t h i s case U s i n g «.Q<1>S 2. a s i m i l a r c a l c u l a t i o n y i e l d s A , - « ! i a { 3 ^ S , n n i ^ } (5) ~ 4**VH*f 3 (wf i (6) 1fZ. l a ~KT j Comparing (Li) and (6), we get the r e s u l t : A s 3 p r o v i d e d the two resonances are observed at the same frequency. REFERENCES (1) Waterman, H.H., Can. J. Phys. 12, 156, 1955, Ph.D. Thesis, University of Bri t i s h Columbia, Vancouver, B. C. (1954). Proctor, W.G. et a l . , Private Communication. Cranna, N.G., Ph.D. Thesis, University of Br i t i s h Columbia, Vancouver, B.C. (195^ 4-) • Volkoff, G.M., Can. J. Phys. ^1, 820, 1 9 5 3 . Lamarche, G. and G.M. Volkoff, Can. J. Phys. J l , 1010, 1953. Lamarche, G. and G.M. Volkoff, Can. J. Phys. j[2, i+93, 1954. Petch, Cranna and Volkoff, Can. J. Phys. ^1, 837, 1953. Eades, R.G., Can. J. Phys. J£, 286, 1955. Kruger, H., Zeitschrif.t fur Physik l^O, 371, 1950. Proctor, W.G., Phys. Rev. Jl> 35, 1950. Weaver, H.E., Jr., Phys. Rev. 89_, 923, 1953. Collins, T.L., Ph.D. Thesis, University of Br i t i s h Columbia, Vancouver, B.C. (1950). Schuster, N;.A., R.S.I. 22., 2$k> 1951. Dehmelt, H.G., Zeitschrift fur Physik l^l* 528, 1952. Dehmelt, H.G., Am. Jour. Phys. 22, 110, 1951}-. 

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