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Proton resonance in kernite Hedgecock, Nigel Edward 1955

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i PROTON RESONANCE IN KERNITE by NIGEL EDWARD HEDGECOCK A THESIS SUBMITTED IN PARTIAL FULFIIMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS IN THE DEPARTMENT OF PHYSICS We accept this thesis as conforming to the standard required from candidates for the degree of MASTER OF ARTS Members of the Department of Physics THE UNIVERSITY" OF BRITISH COTJJMBIA September 1955 i ABSTRACT The proton resonance line in a single crystal of NagB407'4HgO (kernite) placed in a magnetic field of 6300 gauss, was examined experimentally as the monoclinlc crystal was rotated about i ts twofold symmetry axis (cryetallographic b-axis) maintained perpendicular to the magnetic f i e ld . The observed variation with crystal position of the proton line shape was found to be consistent with that expected as a re-sult of nuclear magnetic dipole-dlpole interaction between protons in the molecules of water of crystallization* No evidence was found for any additional splitting of the proton line whioh might be correlated with the previously observed splitting of some of the components of the speotrum of B 1 1 in kernite. The expected spectrum of B^ 0 in kernite was cal-culated theoretically for a Larmor frequency of 4 M c . / s e c , corresponding to an external magnetic field of 8741 gauss, and an unsuccessful experimental search for i t was made. i i TABLE OF CONTENTS Page ABSTRACT i l CHAPTER 1: Introduction 1 CHAPTER 2: The Proton Line I Procedure ..... 9 II Results 6 CHAPTER 3: The B 1 0 Spectrum 8 CHAPTER 4: Conclusion 12 ACKNOWLEDGMENTS 13 REFERENCES 14 ILLUSTRATIONS FIG. 1 Selected traces of the recorded derivative of proton absorption lines facing page 7 TABLE 1 Fourier coefficients for the spectrum of B 1 0 i n kemite ... facing page 11 i i i CHAPTER 1 IKTROIXJCTION A detailed experimental study of the nuclear magnetic resonance ab-sorption of B 1 1 i n a single crystal of kernite (NagB^v'ffljgO) placed i n an external magnetic f i e l d of about 7000 gauss, and i t s analysis i n terms of the perturbation of the nuclear Zeeman energy levels by the interaction of the nuclear electric quadrupole moment with the crystalline electrostatic f i e l d gradient, has been carried out by Waterman and Volkoff (1)* This thesis deals with an attempt to throw some additional light on two points raised, but l e f t unanswered, i n the above work* The work of Waterman and Volkoff has shown the existence of four es-sentially non-equivalent sites for the boron atoms i n kernite. Two of these sites, denoted by C and D, have relatively weak eleetric f i e l d gradients, while the other two, denoted by 1 and F, have larger f i e l d gradients, but smaller asymmetry parameters. These results are consistent with the assign-ment of four different boron sites on the basis of X-ray data by Portoles (2). Sinee the spin of B 1 1 i s I » 3/2, three B 1 1 lines are associated with each site. The "central" line corresponding to the transitions between the Zee-man levels 1/2**--1/2 are denoted by E c , F c , while the two " s a t e l l i t e s " corres-ponding to the 1 3/2- 1 1/2 transitions are denoted by E', E", F«, F". An additional s p l i t t i n g , dependent on the crystal orientation i n the external magnetic f i e l d , but independent of the strength of this f i e l d , of a maximum amount of 11 kc./sec, was observed i n some of the B 1 1 lines associ-ated with the E and F sites characterized by the larger electric f i e l d gradi-ents. The following i s a brief summary of the information on this s p l i t t i n g 2 -from r e f . ( 1 ) . I n the r o t a t i o n o f the m o n o c l i n i c c r y s t a l about i t s b - a x i s ( t w o f o l d symmetry a x i s ) , which was l a b e l l e d as the X r o t a t i o n a x i s , s p l i t t i n g o f the l i n e s E £ , E £ and E £ was observed i n the r e g i o n 8g «• 7 5 ° t o 1 1 5 ° , w i t h a m a x i -mum s e p a r a t i o n o f the components o f Eg near 0% = 9 0 ° . T h i s l i n e was a l s o examined a t 9^ • 84° i n e x t e r n a l magnetic f i e l d s o f 6260, 5006 and 3794 gauss, e x h i b i t i n g the same degree o f s p l i t t i n g i n each case . The p a i r s o f l i n e s F £ , and F£ a re s p l i t by amounts up t o 8 k c . / s e c . i n the r e g i o n 0 Z • 4 0 ° t o 7 0 ° . The o rder o f magnitude o f t h i s s p l i t t i n g , and i t s independence o f H Q , a r e con-s i s t e n t w i t h what would be expected as a r e s u l t o f an i n t e r a c t i o n between c l o s e l y ne ighbour ing n u c l e a r magnetic d i p o l e s i n the l a t t i c e . Van V l e c k has shown (3) t h a t the i n t e r a c t i o n o f a n u c l e a r magnetic d i p o l e i n a c r y s t a l w i t h the averaged-out e f f e c t o f a l l the o the r d i p o l e s leads t o a l i n e w i d t h which agrees w i t h the observed one . I n some ca se s , however„ euen as t ha t o f two pro tons i n a molecule o f water o f c r y s t a l l i z a -t i o n , t he re are p a i r s o f i n t e r a c t i n g d i p o l e s whose members a re much c l o s e r to each o the r than t o o the r d i p o l e s i n the l a t t i c e . T h i s g i v e s r i s e t o an o r i e n t a t i o n dependent s p l i t t i n g o f the cor responding l i n e s . Analogous eases o f th ree (4) and fou r (5) c l o s e d i p o l e s are a l s o t r e a t e d i n the l i t e r a t u r e . The s p l i t t i n g o f a resonance a b s o r p t i o n l i n e due t o d i p o l e - d i p o l e i n t e r a c t i o n between two n u c l e i o f magnetic moments y\ p 2 a d i s t a n c e r apart i s g i v e n by JU,JUZ f ( y ) , where t(y) i s a f u n c t i o n o f the angle y between the s t a t i c f i e l d H Q and the l i n e j o i n i n g the two n u c l e i . Pake (6) has c a l c u -l a t e d t h i s f o r two i d e n t i c a l n u c l e i w i t h I 1 / 2 , o b t a i n i n g the r e s u l t f ( y ) -6(3 cos y - 1 ) , and has a p p l i e d i t t o s e v e r a l cases o f i n t e r a c t i o n between pro ton p a i r s i n molecules o f water o f c r y s t a l l i z a t i o n . A s u c c e s s f u l a n a l y s i s o f the angu la r dependence o f such s p l i t t i n g on the o r i e n t a t i o n o f the c r y s t a l 3 with respeet to H 0 yields the length of the line joining the two nuclei and i t s orientation i n the l a t t i c e * I f the observed s p l i t t i n g of the B 1 1 lines i s postulated to eome from such dipole-dipole interaction, then i t i s of interest to determine whether a given B 1 1 nucleus interacts with other neighbouring B 1 1 nuclei, or with N a 2 3 or nuclei, which, apart from the'less abundant B* 0, H 2 nuclei, are the only other nuclear magnetic dipoles i n the crystal. I f the B 1 1 interacts with either the N a 2 3 or the H 1 nuclei, then the other nucleus should be similarly affected. No sp l i t t i n g i n the Na 0 spectrum was observed i n a cursory examination i n ref. (1), nor i n the more detailed study by Blood and Proctor (7) • The proton lin e was not examined at a l l i n ref. (1). Blood and Proctor (7) looked at i t i n only one crystal orientation, and reported no observable s p l i t t i n g or line structure. Sinee kernite con-tains i t s protons i n molecules of water of crystallization, one would expect to observe at least the proton-proton s p l i t t i n g of the type described by Pake (6) for some crystal orientations. I t was decided to make a detailed observation of the proton resonance line i n kernite for several orientations to see f i r s t , whether the expeeted proton-proton s p l i t t i n g was vi s i b l e at least for some orientations, and second, whether there might be any sign of additional s p l i t t i n g of the proton line by an amount attributable to a pos-sible interaction between the protons and the B 1* nuclei. A second point l e f t uninvestigated In ref. (1) was the B ^ spectrum. Sinee the spin, magnetic moment and the electric quadrupole moment of the B ^ nucleus are a l l known, i t i s possible to predict the expected B 1^ spect-rum from the data for B 1 1 i n r e f . (1). Although the fourfold lower isotopie abundance and the twofold higher spin of B 1 G as compared to B 1 1 make the ex-pected signal-to-noise ratio just on the border of deteetability with the presently available spectrometer, i t was decided to make a search for the lines. This search has unfortunately not been successful to date. CHAPTER 2 THE PROTON LINE I Procedure A preliminary examination of the proton line was carried out using the apparatus described by Collins (8). This Includes an electromagnet whose f i e l d 1B normally stabilized with a proton l i n e . In this case, however in order to avoid Interaction between the two osci l l a t i n g detectors when op-erating close together i n frequency, the f i e l d was satisfactorily stabilized using a signal from F * In hydrofluoric acid. The proton line was examined In several crystal orientations with the b-axis of the crystal perpendicular to the static magnetic f i e l d of 6300 gauss, and was found to be s p l i t i n some crystal positions. I t was then decided to perform a complete rotation of the crystal. Since the physi-cal construction of the available Collins type osc i l l a t i n g detector i s not suitable for precise specifications of the orientation of the crystal with respect to HQ, for greater accuracy the modified Pound and Knight eireuit built by Waterman and used i n his original work (1) on B 1 1 was employed. This apparatus has the additional advantage that the level of r . f• o s c i l l a -tion i s easier to control, thus avoiding saturation of the sample. The actual sensitivity of the two circuite i s about the same. A kernite crystal approximately 2.5" x 0.5W x 0.5" In size was used throughout, the b-axls corresponding to the longest axis. For the prelimin-ary examination of the proton l i n e , the crystal was supported i n the magnetic f i e l d by the r . f . c o l l I t s e l f , which was wound directly on the erystal. In the more detailed study, the crystal was carefully aligned with a machined brass j i g , and one end was fixed to a Incite block with hard wax. The r . f . 6 c o l l was then wound about the other end of the crystal. A careful comparison of signals corresponding to the same crystal orientation i n the two cases showed no difference i n the l i n e , indicating that the hydrogen in the lucite support, which was entirely outside the c o i l , made no appreciable contribu-tion to the signal* Since the electromagnet gap i s only 1*5" compared to the 2" gap i n the permanent magnet used by Waterman, the lucite support and the brass tube enclosing both the lucite and the crystal were replaced by ones of similar design but reduced i n size compared to those constructed by Waterman* A complete 360° rotation of the crystal about i t s b-axis was made, with orientations at 20° intervals. Half the readings were made with a peak-to-peak modulation amplitude of approximately 3 kc./sec. (0.7 gauss), and the rest with about 1 kc./sec. (0.24 gauss), a l l at a frequency of 220c./see. No improvement i n resolution was observed i n the second set of readings, but the deterioration i n signal-to-noise r a t i o compared with the f i r s t set would tend to annul any such improvement. As expected from theoretical considerations, the line shapes for c r y s t a l ^ positions separated by 180° were the same within the accuracy of observation. The signal was observed at some additional orientations i n order to determine the position and magnitude of the maximum s p l i t t i n g . A set of the best experimental traces covering a half rotation has been assembled from the complete set, and i s reproduced i n f i g . 1. II Results These results on the variation of the shape of the proton resonance in the course of the b-rotation turned out to yield very l i t t l e information of a quantitative nature. The line possesses fine structure, but resolution i s so poor that a definite separation into more than one component occurs only i n the region G - 75° to 90°, where the integral of the observed F i g . 1. Selected traces of the recorded derivative of proton absorption lines, in the neigh-bourhood of 27 Mc./sec, in a single crystal of kernite. The angular position of the crystal i s measured from the % = 0° posi-tion in which the crystal e-axis i s per-pendicular to H 0. The sweep speed i s ap-proximately 60 kc./sec. per chart d i v i -sion. The traces at = 120° and 140° were recorded with a higher gain setting than the others. 7 derivative signal shows two peaks, separated by approximately 40 kc./sec, as compared with a total line width of 60 kc./sec* Between 0^ = 120° and 20°, only a single component appears, with a minimum line width of 35 kc./sec. near 9j « 160°. Presumably the single observation by Blood and Proctor (7) must have been made i n this region* The maximum line width of 75 kc* see. was observed at 6j • 40°, where the integrated signal consists of a central peak with small shoulders at each end. A l l line widths were measured between the outermost peaks on the derivative lines. The maximum spli t t i n g of 40 kc./sec. i n the proton line i s of the same order of magnitude as found by Pake (6) for proton-proton interaction in gypsum, and i s so much greater than the 11 kc./sec. maximum splitting observed In the B* 1 spectrum, that the former cannot be due to Interaction between protons and B^ - nuclei. These observations do not, however, preclude the existence of such an effect, since a sp l i t t i n g of only 11 kc./sec. would be obscured i n the gross structure of the l i n e . The overall effect i s most probably due to proton-proton interaction as described previously. On this basis, since there are four water molecules per unit c e l l i n kernite, up to four pairs of lines might be expected to make up the proton signal. The poor resolution, however, makes i t Impossible to state definitely the number of components present, though the observed line shapes seem to indicate the presence of at least two such pairs of lines. The situation here i s similar to that reported by Pake (ref. 6, p. 331) i n connection with the proton line In a decahydrate, where the large number of essentially non-equivalent pro-ton pairs per unit c e l l combine to produce a wide "smeared-out" l i n e . CHAPTER 3  THE B 1 0 SPECTKPM Before the search for the B 1 0 l i n e , the spectrometer was adjusted to give maximum signal-to-noise ratio to the B 1 1 lines c£ and E0.. In a f i e l d of 5000 gauss, corresponding to a Larmor frequency of approximately 6.8 Mc./sec. for the B ^ nucleus, the best signal-to-noise ratio obtained was about 20:1. This figure was reduced to somewhat less than 10:1 at a Larmor frequency of 5.2 Mc./sec. (H 0 • 3800 gauss). The isotopic abundance of B 1^ i s roughly 20# compared to 80$ for B 1 1 , and i t s spin of 3 gives r i s e to twice as many lines i n the spectrum as i n the case of B 1 1 , so that, assuming the same line width, the expected signal-to-noise ratio at a given frequency i s a factor of eight lower for B*-0 than for B 1 1, and hence just on the limit of detectability. This estimate i s con-firmed by preliminary results on the pure quadrupole spectrum of B 1 0 i n this laboratory, with a Bloch type apparatus, which indicate that the B 1 0 line i s roughly one tenth as strong as the B ^ li n e , and about the same width. The search for the B^° lines was made i n a frequency region centred at 4.1 Mc./sec, which i s the B*° Larmor frequency for the f i e l d of about 9000 gauss used. As would be expected from the preceding discussion, no con-clusive sign of any line has been found to date, even when sweeping slowly with a long time constant i n the detector. Since B 1 0 has I » 3, for a particular B 1 0 s i t e , i t s nuclear resonance spectrum i n a high external magnetic f i e l d should consist of three pairs of lines, one pair corresponding to each of the three possible pairs of |A mj • 1 transitions characterized by 0*-* ±1, +2 and ± 2«-» ±3. The frequency 8 9 difference between the lines i n such a pair of satellites i s dependent on crystal orientation, and i n accordance with the theory given by Volkoff (9) i t should be possible to choose an orientation i n which the two members of each of three pairs of s a t e l l i t e s coincide, thus increasing the signal i n -tensity by a factor of two. With this i n view, the spectrum of for the E and F sites i n kernite has been calculated for the X rotation. In this rotation the C and D lines w i l l not f a l l on one another i n any position (ref. 1: Waterman's thesis). The relevant theory has been worked out i n detail for the general case i n ref. (9), and i s outlined here for 6a® 1 - 3 . The B 1 0 nuclei find themselves i n the same environment In the crys-t a l as those of B 1 1, so that the components (f>^ of the electric f i e l d gradi-ent tensor available i n ref. (1) are applicable i n both eases. The frequency difference between the components of a pair of satel-l i t e s with the quantum number m (m-. m-1 and -m+l«- -m) i s given by (V' - y " ) m = (m - &)(a + b cos28 + e sin 26) (1) where for 1 = 3 the coefficients are defined by ajj. - 1/10 (YYY + Tzs)-*X " 1 / 1 0 ^YY " YZZ ) (2) w l t h f i j " ± * P i j These f i r s t order coefficients a, b, e for B*° are obtained from the corres-ponding values for the B 1 1 spectrum i n ref. (1) through multiplication by a by a factor 1/5 Q 1 0/^ 1 1; where 1/5 i s the ratio of the factors i 1(21 - 1) TABLE I Fourier coefficients for the spectrum of at sites E and F i n kernite, calculated from experimental Fourier Coefficients for B"^, and known relative nuclear moments of and B^. A l l values are in Mc./sec. The coefficients n, p, r, u, v are calculated for a Larmor Frequency v Q = 4 Mc./sec. corresponding to H Q = 8741 gauss. Site E F Axis X 2 2 X Yj , 2 2 a 0.224 - 0 . 4 9 4 0.270 0.203 0.128 0.334 b 0.766 -0.045 -0 .719 -0.469 0 .537 0 . 0 7 5 c 0.127 +0.116 ±0.306 -0.481 +0.484 - 0 . 2 0 3 m 1 2 3 1 2 3 nx -0.0304 0.0187 0.1168 -0.0282 0.0203 0.1173 Px - 0 . 0 2 7 1 -0.0081 0.0301 0.0305 0 . 0 0 4 5 - 0 . 0 4 7 6 *x - 0 . 0 1 0 5 -0.0013 0.0170 -0.0097 0 . 0 0 4 6 0 . 0 3 3 2 u x 0.0439 - 0 . 0 0 6 7 - 0 . 1 0 7 8 -0.0009 ,0.0001 0 . 0 0 2 2 v x 0.0149 - 0 . C 0 2 3 -0.0367 0.0346 - 0 . 0 0 5 3 -0.0850 for the two nuclei. Dehmelt (10) gives for the ratio of the quadrupole moments of B 1 0 and B 1 1 the value Q 1 0/^ 1 1 • 2.084 ± 0.002, which makes this conversion factor 0.4168. Second order effects give rise to a shift of the centre of gravity 4m of each pair of sate l l i t e s from the Larmor frequency v Q given by 9 - v • n + p cos 29 + r sin 20 + u cos 40 + v sin 49 (3) m o The coefficients here depend on the coefficients a, b, c and on m i n a way given e x p l i c i t l y i n ref. (9), and are inversely proportional to v 0 . These coefficients calculated for v 0 » 4 He. sec. are presented i n Table 1. The orientations at which the members of each pair of sate l l i t e s coincide, to-gether with the frequency interval from v 0 at which these double lines are expected to appear are shown i n Table 2: TABLE 2 Predicted crystal orientations and frequency intervals i n kc./sec. from v 0 - 4 Me. sec. at which s a t e l l i t e pairs coincide. Subscripts re-fer to the m value of the transition. m *-» m-1 0«-»l 1 — 2 2*-> 3 % E l -*o E2 " •o E„ - w 3 vO 58° 6» - 70.3 + 27.5 +223.0 131° 24* - 55.4 + 26.9 +191.5 y i - * o F 8 - * o »»-'. 59° 3* - 88.0 + 22.8 +244.3 166° 42* - 24.8 + 26.6 +129.3 The double lines E x at • 58° 6* and at = 59° 3» l i e close c l a — together both in frequency and orientation, and the results of a de-tailed calculation of the frequencies of the B 1 0 lines E^, EJ, F* and FJ at these two orientations show that at « 58° 6* these lines should appear at 70.3, 70.3, 78.3 and 67.8 kc./sec. below v Q - 4Mc./sec, respectively. This means that the line FJ coincides with the double line with F^ entering Into the edge of this composite line. This should result in a threefold increase In signal intensity over a single B*° line, and an In-tensity of roughly 0.4 times that for a single B 1 1 line. CONCnJSION The variation, of the proton line shape*, while not very informa-tive, is what would be expected from the combined effect of four closely spaced and differently oriented proton pairs in each unit ce l l of the crystal to produce a wide line with poorly resolved fine structure* A further search for B 1 0 lines in kernite should be facilitated by the re-sults of the calculations presented here, though a complete experimental analysis of the B 1 0 spectrum cannot be made without using a stronger magnetic f ie ld , or a more sensitive spectrometer* 12 ACKNOWLEDGMENTS I am indebted to my research supervisor, Dr. G. M. Volkoff, for suggesting the problem, and guiding me during the course of the work. I should also like to express my appreciation to Drs. H. H. Waterman and H. G. Dehmelt, for helpful advice i n regard to the experimental aspects of the work; to Mr. W. Morrison, of the Physios Department workshop, for machining part of a new crystal holder; and to Mr. R. J. Grant, who confirmed some of the data on the pro-ton l i n e , using different apparatus. Financial aid In the form of asslstantshlps for the summers of 1954 and 1955 as part of a continued grant-in-aid to Dr. Volkoff from the National Research Council i s gratefully acknowledged. 13 REFERENCES 1. WATERMAN, H. H., VOLKOFF, C M . Can. J . Phys'. 33: 156, 1955 see also WATERMAN, H. H. Ph.D. Thesis, University of Br i t i s h Columbia, 1954 2. PGRTOLES, L., Estud. geol. Inst. Mallada 5: 3, 1947 and 7: 21, 1948 3. VAN VLECK, J . H., Phys. Rev. 74: 1168, 1948 4. ANDREW, E. R., BERSOHN, R., J . Cham. Phys. 18: 159, 1950 5. ITOH, J . , et a l . J . Ghem. Phys. 20: 1503, 1952; j , Phys. Soc. Japan 8: 287 and 293, 1953 6. PAKE, G. E«, J . Chem. Phys. 16: 327, 1948 7. BLOOD, H., PROCTOR, W. 0., Private communication 8. COLLINS, T. L., Ph. D. Thesis, University of B r i t i s h Columbia, 1950 9. VOLKOFF, G. M., Can. J . Phys. 31: 820, 1953 10. DEBMELT, H. G., Z. Physik 133: 528, 1952 14,. 


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