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The nuclear spin resonance spectrum of Al²⁷ in spodumene Robinson, Lloyd Burdett 1957

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Faculty of Graduate Studies PROGRAMME OF THE F I N A L O R A L E X A M I N A T I O N FOR THE DEGREE OF D O C T O R O F . P H I L O S O P H Y of LLOYD BURDETT ROBINSON B. A. (Saskatchewan) 1953 M. A. (Saskatchewan) 1954 M O N D A Y , J U L Y 29th, 1957, at 10:30 a.m. I N R O O M 300, PHYSICS B U I L D I N G C O M M I T T E E I N C H A R G E D E A N G . M . S H R U M , Chairman M. BLOOM F. M. C. GOODSPEED A. M. CROOKER H. G. HEAL J. M. DANIELS A. D. MOORE J. B. GUNN V. J. OKULITCH External Examiner: H. E . PETCH McMaster University T H E N U C L E A R S P I N R E S O N A N C E S P E C T R U M O F A L 2 7 I N S P O D U M E N E ABSTRACT Using the techniques of radio-frequency resonance spectroscopy, experimen-tal studies of the nuclear spin resonance spectrum of A l 2 7 in single crystals of spodumene (LiAKSiOs) 2) have been carried out over a wide range of external-ly applied magnetic field. In spodumene, interactions of the magnetic dipole moment with an external magnetic field, and also interactions of the electric quadrupole mom-ent with the crystalline electrostatic field, gradient can affect the energies of Al2? nuclei. We define R as the ratio of magnetic to electrostatic interaction energies. The object of this work has been to measure experimentally a nuclear resonance spectrum over a wide enough range of magnetic field to link the regions where R is much greater than or much less than unity. Much experimental data is available in the literature for crystals where R differs appreciably from unity, but no experimental results have been given before for the intermediate region where R is of the order of unity and where the spectrum is more complex. Using data obtained from high field measurements in spodumene by Petch and Cranna, Lamarche has calculated energy levels by exact diagon-alization of the Hamiltonian over a wide range of R values for a particular orientation of the spodumene crystal in a magnetic field. Calculations for other orientations have been made using electronic computers at the University of Toronto and at the University of British Columbia. Several of the predicted resonances have been observed. One transition has been observed over a range of magnetic field covering the region from R much less than unity to R much greater than unity. Resonance frequen-cies observed have been in good agreement with calculated transition freq-uencies. A new method of using a knowledge of the spin eigenstates to predict signal voltage for an induction spectrometer has been checked at values for R of the order of unity. It gives good agreement with experimental signal voltage measurements. Pure quadrupole transitions have been observed in spodumene at 75L5 kc and 793.5 kc with an estimated probable error of 2 kc, using an induction spectrometer with Zeeman modulation. These measurements are the lowest frequency pure quadrupole resonances reported to date. PUBLICATIONS Measurements of Gamma-ray Absorption in Carbon R. N. H. Haslam, R. J. Horsley, H. E . Johns, and L. B. Robinson Canadian Journal of Physics 31, 636.(1953) Fine Structure of the gamma neutron Activation Curve in Fluorine J. V. G. Taylor, L . B. Robinson, and R. N. H. Haslam Canadian Journal of Physics 32, 238. (1954) GRADUATE STUDIES Field of Study: Physics Electromagnetic Theory J . R. H. Dempster Nuclear Physics K - c - Manr Quantum Mechanics G. M. Volkof. ' Noise in Physical Systems R. E . Burges;-Advanced Electronics R- E - Burgess Other Studies: Network Theory Servomech anisms Numerical Analysis A. D. Moore E . V. Bohn F. M. C. Goodspeed T H E N U C L E A R SPIN R E S O N A N C E S P E C T R U M O F A l 2 7 IN S P O D U M E N E by L L O Y D B U R D E T T ROBINSON A THESIS S U B M I T T E D IN P A R T I A L F U L F I L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F D O C T O R O F PHILOSOPHY i n theTJepartment of PHYSICS We accept this thesis as conforming to the standard required from candidates for the degree of D O C T O R O F PHILOSOPHY T H E U N I V E R S I T Y O F BRITISH C O L U M B I A July 1957 A B S T R A C T Using the techniques of radio-frequency resonance spectroscopy, experimental studies ;of the nuclear spin resonance spectrum of A l 2 ? i n single crystals of spodumene (LiAl(SiU3)2) have been carried out over a wide-range of externally applied magnetic f i e l d . In spodumene, interactions of the magnetic dipole moment wi th an external magnetic f i e l d , and.also interactions of the electr ic quadrupole moment wi th the crysta l l ine electrostatic f i e ld gradient can 27 affect the energies of A l n u c l e i . We define R as, the.ratio of magnetic to electrostatic interaction energies. The object of this thesis has been to measure experimentally a nuclear resonance spectrum over a wide enough range of magnetic f i e ld to l ink the regions where R i s much greater than or much l e s s than uni ty . M u c h experimental data i s avai lable in.the literature for crystals where.R differs appreciably from unity, but no experimental results have been given before for the: intermediate region where R is. of the order of unity and where the spectrum i s more complex . Us ing data obtained from high f i e ld measurements in spodumene by Petch and Cranna, Lamarche has calculated energy leve ls by exact diagonalizat ion of the Hamil tonian over a wide range of R values for a particular orientation of the spodumene crys ta l i n a magnetic field. Calculations for other orientations have been made using electronic computers at the University.of Toronto and at the University of British Columbia. Several of the predicted resonances have been observed. One transition has,been.observed over a range of magnetic field covering the region from R much less than unity to R much greater than unity. Resonance frequencies :observed have been in good agreement with calculated transition frequencies. A new method of using a.knowledge of the spin eigenstates to predict signal voltage for an induction spectrometer has been checked at values for R of the order of unity. It gives good agreement with experimental signal voltage measurements. Pure quadrupole transitions have been observed in spodumene at 751.5 kc and 793.5 kc-with an estimated probable.error of 2 kc, using an induction spectrometer with Zeeman modulation. These measurements are the lowest frequency pure quadrupole resonances reported to date. GRADUATE STUDIES Field of Study: Physics Electromagnetic Theory _.. - J. R. H. Dempster Nuclear Physics K. C. Mann Quantum Mechanics G. M. Volkoff Noise in Physical Systems R. E. Burgess Advanced Electronics R- E. Burgess Other Studies: Network Theory Servomechanisms . Numerical Analysis A. D. Moore E . V. Bohn F. M. C. Goodspeed Faculty of Graduate Studies PROGRAMME OF THE F I N A L O R A L E X A M I N A T I O N FOR THE DEGREE OF D O C T O R O F P H I L O S O P H Y of LLOYD BURDETT ROBINSON B. A. (Saskatchewan) 1953 M. A. (Saskatchewan) 1954 M O N D A Y , J U L Y 29th, 1957, at 10:30 a.m. I N R O O M 300, P H Y S I C S B U I L D I N G C O M M I T T E E I N C H A R G E D E A N G . M . S H R U M , Chairman M ! B L O O M F. M . C. G O O D S P E E D A. M . C R O O K E R H . G. H E A L J. M . DANIELS A. D. M O O R E J. B. G U N N V. J. O K U L I T C H External Examiner: H. E. PETCH McMaster University T H E N U C L E A R SPIN R E S O N A N C E S P E C T R U M OF A L 2 7 I N S P O D U M E N E ABSTRACT Using the techniques of radio-frequency resonance spectroscopy, experimen-tal studies of the nuclear spin resonance spectrum of A l 2 7 in single crystals of spodumene (LiAKSiOs) 2) have been carried out over a wide range of external-ly applied magnetic field. In spodumene, interactions of the magnetic dipole moment with an external magnetic field, and also interactions of the electric quadrupole mom-ent with the crystalline electrostatic. field gradient can affect the energies of A l 2 7 nuclei. We define R as the ratio of magnetic to electrostatic interaction energies. The object of this work has been to measure experimentally a nuclear resonance spectrum over a wide enough range of magnetic field to link the regions where R is much greater than or much less than unity. Much experimental data is available in the literature for crystals where R differs appreciably from unity, but no experimental results have been given before for the intermediate region where R is of the order of unity and where the spectrum is more complex. Using data obtained from high field measurements in spodumene by Petch and Cranna, Lamarche has calculated energy levels by exact diagon-alization of the Hamiltonian over a wide range of R values for a particular orientation of the spodumene crystal in a magnetic field. Calculations for other orientations have been made using electronic computers .at the University of Toronto and at the University of British Columbia. Several of the predicted resonances have been observed. One transition has been observed over a range of magnetic field covering the region from R much less than unity to R much greater than unity. Resonance frequen-cies observed have been in good agreement with calculated transition freq-uencies. A new method of using a knowledge of the spin eigenstates to predict signal voltage for an induction spectrometer has been checked at values for R of the order of unity. It gives good agreement with experimental signal voltage measurements. Pure quadrupole transitions have been observed in spodumene at 751.5 kc and 793.5 kc with an estimated probable error of 2 kc, using an induction spectrometer with Zeeman modulation. These measurements are the lowest frequency pure quadrupole resonances reported to date. P U B L I C A T I O N S Measurements of Gamma-ray Absorption in Carbon R. N. H. Haslam, R. J . Horsley, H. E . Johns, and L. B. Robinson Canadian Journal of Physics 31, 636.(1953) Fine Structure of the gamma neutron Activation Curve in Fluorine J. V. G. Taylor, L. B. Robinson, and R. N. H. Haslam Canadian Journal of Physics 32, 238. (1954) In presenting this thesis in partial fulfilment of the requirements for en advanced degree at the University of British Columbia,I agree that the Library shall meke i t freely nvsilpble for reference and study. I further agree thst permission for extensive copying of this thesis for scholarly purposes^  may be granted by the Head of my Department or by his representative. It is understood thp.t copying or publication of this thesis; for financial gaia shall not be allowed without my written permission. i i . T A B L E O F C O N T E N T S P A G E A C K N O W L E D G E M E N T S v A B S T R A C T v i C H A P T E R I - I N T R O D U C T I O N 1 C H A P T E R II - T H E O R Y A . Energy L e v e l s , 1 . The Hamil tonian 5 2 . Solution of the Hamiltonian.for 9=0 11 3 . Solution of the Hamil tonian for 6 / 0 12 B. Transi t ion Probabili t ies and Signal Intensities 14 C . Signal to Noise. Rat io 18 D . Choice of Crys ta l for the Experiment 21 C H A P T E R in - E X P E R I M E N T A L A P P A R A T U S A . Descript ion of Samples 24 B . Choice of-Spectrometer 26 C The Nuclear Induction Spectrometer 1. General Design 29 2 . Pure Quadrupole Induction Spectrometer 33 3 . The.Magnet 34 D . Diff icul t ies 37 C H A P T E R IV - E X P E R I M E N T A L P R O C E D U R E 41 C H A P T E R V - E X P E R I M E N T A L , R E S U L T S A . Resonance Frequencies 1. Interfering Resonances 45 2 . A l 2 7 Resonances for©* 0 46 3 . A127 Resonances f o r e / 0 49 i i i . P A G E B . Signal Intensities 51 C . Pure Quadrupole Measurements and Numer ica l Values for Quadrupole Coupl ing Constants 53 . C H A P T E R V I - DISCUSSION 57 A P P E N D I X A - Phenomenological Equations for a Two L e v e l System 64 A P P E N D I X B - Notes on Diagonal izat ion of Matr ices 67 A P P E N D I X C - Experiments on Bridge Type Spectrometers 70 R E F E R E N C E S 72 T A B L E I - Comparison of Resul ts Obtained for /^and e^qQ at Different Values of R . 56 L I S T O F I L L U S T R A T I O N S F A C I N G P A G E F i g . 1 - Energy L e v e l s for B= 0 11 F i g . 2 - Transi t ion Frequencies for 0=0 11 F i g . .3 - Energy L e v e l s for 0= 10° 13 F i g . 4 - Theoret ical Var ia t ion of Signal Voltage as a Funct ion of R 18 F i g . 5 - Ci rcu i t Diagram for Induction Spectrometer 30 F i g . 6 - Block .Diagram of Exper imental Apparatus 37 F i g . 7 - Resonance.Spectrum for 6- 0 46 F i g . 8 - Resonance Spectrum for &= 5° 50 F i g . 9 - Resonance Spectrum for 6= 1 0 ° , d = 20° 50 P L A T E 1 - Comparison of Records.Obtained Under Identical Conditions 44 II - Records of y>2> ^l* ^A> ^ 6 4 8 III - "Spl i t t ing" of V 2 a t 8 0 0 k c . 4 9 IV - Records of Pure Quadrupole Resonances i n Spodumene 54 V - A Re cord of L i 7 and A l 2 7 Resonances at 313 k c . 58 V I - Selected Runs to Demonstrate Good Signal to Noise Rat io 59 VII - A Record Obtained from the Solution of the Harnil tonian Mat r ix by the Computer Alwac. I l l E 70 V . A C K N O W L E D G E M E N T S The work described i n this . thesis was supported by the Nat ional Research Counci l of Canada, by means of research grants to Professor G . M . Volkoff and by the award of two studentships to the author i n the period 1955-57. I should l i k e to express my. appreciation to D r . Volkoff , who suggested this inves t iga t ion . H i s advice and encouragement were largely responsible for i ts successful comple t ion . D r . Bloom has been very generous wi th both h is t ime and ass is tance , and i n numerous.conversations has.contributed greatly to the author's understanding of this f ie ld of P h y s i c s . I should also l i k e to express my thanks to M i s s Jayaseetha R a u , who ass is ted i n many of the numerical calculat ions; to M r . A . J . F razer , M r . W . Morr ison and M r . J . . L e e s for their co-operation in the construction of equipment. This research has been greatly aided by the use of the electronic computers.at the .University of Toronto and at the Universi ty of Br i t i sh C o l u m b i a . The assistance and co-operation given by the. staff associated wi th these machines . is gratefully acknowledged. F i n a l l y , I should l i ke to express appreciation to m y wi fe , who has helped both by her encouragement and by her assistance i n preparing the draft of this t hes i s . C H A P T E R I I N T R O D U C T I O N In 1924 Paul i suggested that the atomic nucleus posseses a magnetic moment . It was also suggested that the.magnetic moment i s -associated wi th an intrinsic.angular momentum or " sp in" of the nuc leus . These ideas-make i t poss ible to understand the hyperfine sp l i t t ing which occurs i n atomic spectra. Quantum theory shows.that the projection of total angular momentum on any axis can only give integral or half- integral mul t ip les .of h/27T , where h i s Planck 's constant. Thus i t i s not poss ib le to observe a. continuous range, of values .of angular momentum, but only discrete, ones, and only discrete values w i l l be. obtained for the projection of the magnetic moment of the nucleus along any a x i s . T h i s leads to discrete values or l eve l s for the energy due to the interaction of the nuclear magnetic moments wi th a magnetic f i e l d . Direct evidence.of the discrete orientations.of magnetic moments was obtained for protons i n 1933, by us ing .the deflecting effect of an inhomogeneous magnetic f i e ld on the magnetic moment of the protons i n a stream Of hydrogen molecules (1). The beam of molecules .d iv ided into three, indicat ing three possible values.of the to ta l magnetic moment of the two protons present i n each m o l e c u l e . The discreteness of the energy l eve l s i n a magnetic f i e ld was shown directly i n 1937, by. caus ing transitions between levels by use of a radio-frequency magnetic f i e ld (2). 2. When.transitions occurred, a change in the number of molecules in one of the beams was observed. The magnetic moment of. nuclei should lead to a weak paramagnetic effect. The paramagnetism was first observed in 1937 in solid hydrogen (3). This observation showed that the effects of the nuclear magnetic moment could be observed in bulk material as well as in molecular beams. It was pointed out by Gorter (4) in 1936 that in principle at least, transitions between the nuclear magnetic energy, levels should be observable in bulk material. In l946,two groups, working independently, succeeded in observing the transitions between nuclear magnetic energy levels. HLoch, Hansen and Packard (5) measured induction signals from transitions between proton levels in water, while Purcell, Torrey and Pound (6) detected the absorption of energy from a resonant electronic circuit by proton transitions, in paraffin wax. The energy difference.between nuclear magnetic energy levels is such that transitions between the levels give rise to radiation in the radio-frequency range for magnetic fields obtainable in the laboratory. The branch of radio-frequency; spectroscopy dealing with such transitions is called Nuclear Magnetic Resonance, often abbreviated to NMR. Atomic spectroscopy led to another piece of information about the nucleus. Deviations from an interval rule in the spectrum of Europium, observed in 1935, were explained by Casimir (7), who suggested that the nucleus had a non-spherical charge, distribution which would lead to 3 a quadrupole moment. The interaction of the nucleus with the .electrostatic field gradient produced by the surrounding electron cloud is.different for different orientations of the nucleus. This effect perturbs the.electronic . energy levels; it also leads to.a. change in the energy levels of the nucleus. In many nuclear magnetic resonance experiments the electro-static field gradient averagesjout to zero, so that the quadrupole effect can be neglected so far as the energy levels are.concerned. However, in the case of some molecules, and in crystals, it is possible for a nucleus to be placed at- a position where the electrostatic field gradient does not average to zero, and the resulting interaction will change the energy levels of the nucleus. In 1948 Bloembergen (8) reported.effects, of quadrupole interactions with the deuteron. Pound (9) found nuclear magnetic resonances which were split into several lines by the quadrupole interaction in single crystals. In 1951 Dehmelt and Kruger (10) found.resonances due to transitions between nuclear quadrupole energy levels, using the techniques of nuclear magnetic resonance but with no applied magnetic field. This is termed Nuclear Quadrupole Resonance, sometimes.abbreviated to NQR. hi the six years which have elap sed. since pure quadrupole resonances were first observed, many experiments have been performed to study various.aspects .of quadrupole and magnetic interactions of nuclei. Observations have been made of pure quadrupole resonances for many nuclei in a variety of materials. The perturbing effect of a small magnetic interaction on a large quadrupole interaction has been studied, and the effect 4. of a s m a l l quadrupole interaction on a large magnetic interaction has a lso been observed. Wertz (11) has used the name Nuclear Spin Resonance to point out the essent ia l characterist ic of these transitions and this term w i l l be used in the rest of the thes i s . Up to the present t i m e , no experiment has been reported where a transition has been observed over a sufficient range of magnetic f ields to permit a study of the gradual change from a quadrupole interaction perturbed by a magnetic interaction, to a magnetic interaction perturbed by a.quadrupole interact ion. T h i s thesis describes an experiment where transitions were observed between l eve l s of the a luminum nucleus, i n a c rys ta l of spodumene, from zero and low magnetic f i e ld where the quadrupole; interaction i s dominant, through the intermediate f ie ld region where electr ic quadrupole and magnetic dipole. interaction energies, are of the same order, and up to higher f ie lds where the magnetic interaction i s most important. The emphasis i n this study i s not on the investigations of the properties of the spodumene crys ta l or of the a luminum nuc leus . The work was done rather to provide the first example . in the literature on Nuclear Spin Resonance of a radio-frequency spectrum which covers a complete range of ratios of the magnitudes of e lect r ic quadrupole and nuclear dipole interactions to which a nucleus i s subjected. Particular attention i s pa id to the transition region not previously investigated elsewhere, in which this ratio i s c lose to uni ty, and i n wh ich the observed spectrum exhibits certain characteristic differences from the previously studied l i m i t i n g cases where the ratio i s either much greater than or much less than uni ty . 5. C H A P T E R II T H E O R Y A ) Energy, L e v e l s 1 . The Hamil tonian In any branch of spectroscopy, the observation of transitions between discrete energy l eve l s is of primary interest . W h i l e a n most branches of spectroscopy, many effects determine the values ;of the energy l e v e l s , and numerical predictions of the energy leve ls may be very difficult or imposs ib l e , i n radio-frequency spectroscopy the energy levels are often due to only a few interactions and their values may be calculated us ing quite s imple , ideas . The energy leve ls of an isola ted nucleus i n a magnetic f i e ld are known i n terms of the magnetic f i e l d , and the spin and magnetic moment of the nuc leus . They are given by E m = m / t H Q / I where the spin I i s given in units of h , m gives the magnetic substate, taking on integral or half integral values i n unit steps from -I to I . If the magnetic f i e ld Ho i s . i n gauss , and i f the magnetic dipole moment JJL i s i n c . g . s . units, E i s i n ergs. The energy leve ls of a nucleus whose energy i s due only to quadrupole interactions may alsO be written down immediate ly if;.the electro-static gradient i s ax i a l ly symmet r i c a l , and the-axisx>f quantization i s also the axis of symmetry . The energy leve ls are given by E m = e 2 q Q ( 3 m 2 - I 2 ) 4 I(21 - 1) eq i s the value of the electrostatic f i e ld gradient along the axis of symmetry , and eQ i s the electr ic quadrupole moment of the nuc leus . If, however, the electrostatic . f ield gradient i s not ax i a l ly symmet r i ca l , the Hamil tonian of the system i s more complex . In this case i f the Hamil tonian i s referred to the pr inc ipa l axes x , y , z , of the electrostatic f i e ld gradient tensor defined i n reference (12), then Q - ^ S Q ( ' 3 I z 2 - l 2 + n ( i x 2 . I y 2 ) ^ e 2 c 41(21 - 1) where 1 i s the asymmetry parameter of the above tensor, defined by and i s the electrostatic potent ia l . I x , Iy and I x are the spin angular momentum.operators. The terms i n I x 2 and Iy2 lead to non-zero values for the elements ^ m > m - 2 0 * t r i e Hamil tonian in. the representation which diagonalizes I z . and so to find the energy l eve l s we must diagonalize the Hami l ton i an . Fortunately, i t turns out that for spin of odd half- integral values , the matr ix .of order 21 +. 1 may be broken up into two submatrices of order I + 1 / 2 . When diagonal ized, each sub-matrix leads to the same set of eigenvalues,due to the symmetry of the Hamil tonian for £ m states . In the general ca se , we have both magnetic and quadrupole interact ions. It i s convenient to define a ratio R of magnetic interaction energy ju H 0 , to quadrupole interaction energy e 2 q Q 4 We can discuss three general regions: R » l , R « l , and R ~ l . A well explored region in nuclear spin.resonance experiments is that where R >> 1. The eigenvalue energies in this case can be found by time independent perturbation theory, and several authors have treated different aspects of this problem. In particular, a calculation was made by Volkoff (13) using perturbation theory up to third order. This, showed how one might find the quadrupole interaction energy of a nucleus in a crystal, as well as the electric field gradient asymmetry parameter at the site of the nucleus, and the spin of the nucleus itself, by rotating the crystal about two arbitrary axes and measuring the variation of resonant frequencies with angular position. The theory was applied by Fetch, Cranna.and Volkoff (13) to an experiment on the. crystal Spodumene (LiAl ( S i 0 3 ) 2 ) . They obtained complete information on the quadrupole coupling of aluminum in the crystal as well as the. orientation of the field gradient and its asymmetry at the site of the nucleus. Perturbation theory may also be used for the case where.R«l. This has been treated by Kruger (14), Bersohn (15), Cohen (16), and is discussed in a forthcoming, re view by, Das and Hahn (17). A calculation and experiment was also carried out in this laboratory by Haering (18) for the case of B1* nuclei in kernite. Apure quadrupole line predicted from high field.measurements by Waterman (19) was found, and perturbation theory was.used to calculate the Zeeman splitting for R « 1. Measurements were made, of the Zeeman splitting of the line up to R •= 0.2. For the region where R is of the order of unity, no perturbation calculation will work. The Hamiltonian must be diagonalized exactly if the energy levels are to be predicted. Various methods.of diagonalizing a matrix are available but in general, matrices for spin greater than unity must be diagonalized by numerical methods. Thus, no general solution can be given for the energy levels, and the best procedure is.often different for different cases. We now limit the discussion to the case of spin 5/2, which fits the experiment to be described. For spin 5/2 the quadrupole.interaction Hamiltonian has the following form: W O S O O O O T O V O O / S O U O V o O V o u o s O O V O T O o o o s o w W - 1 L A 4 A n 4 vio T = -A 20 v = 3 A y\ 2 o v r ,u 5 A = e2qQ This matrix may be separated into two 3x3 sub-matrices, each of which yields the same third order secular equation for the energy levels; E 3 .- 7E (3+)\2)+20 (1 - )^2) =0 Where, the units of E are e2qQ/20, this agrees with the equation given by Cohen (16). This cubic equation can be solved for any particular value of K ^ , and then use of the quadrupole interaction constant e 2 q Q leads directly to the.expected pure quadrupole transition frequencies. In order to calculate the energy, l eve ls related to the experimental condit ions, we consider the Zeeman Hamil tonian for the case that the magnetic f i e ld l i e s i n the plane of the x z pr incipal axes of the electr ic f ie ld gradient tensor, making an arbitrary angle wi th the z a x i s , which i s chosen as the axis .of quantizat ion. Note that the x and y directions of the sys tem are not equivalent because of the asymmetry of the crysta l l ine f ie ld gradient. L F 0 O 0 °\ F M G O 0 0 G N K 0 0 0 0 K - N G 0 0 0 0 G - M F / 0 0 0 0 F J •( - 1 ) L = B cos & M - i ? B c o s 0 5 N = JL B cos& 5 F « I B s i n ^ G .- 2 ^ B s i n & K = 1 _ B s i n £ VT 5 5 B = / > C H 0 The total Hamil tonian 7f of the system i s given by The energy levels of the sys tem are found by d iagonal iz ing the matr ix form of the Hami l ton ian , which is exact ly equivalent to so lv ing the eigenvalue problem: - E . TA. I . i - i T h i s leads to a secular determinant wh ich may be written.as fol lows: a g h o 0 b g b i j . o 0 h i c k j o o j k d i h o o j i e g o o o h g f 0 a = - E i - L + W = -Ei - B :cos^-+-A 1 T b = - E i - M + T = - E i - 3 E'CQBd - A 5 20 - E i - N + U E i " _L-B-;cos# - A d = - E i + N + U = - E i -t l B c o s d - A 5. 5 e = - E i +• M i- T = - E i + 3 B c o s 5 - A 5 20 f = g h = E i + L + W = - E i + B c o s 0 + A 4 F = - B s in 8 S - A^V 4 VTo" 0 .4 .8 \2 1.6 2.0 2.4 2B 3.2 36 4.0 R Fig. 1 - Energy Levels for 9 - 0 Facing page 11 Fig. 2 - Transition Frequencies for 6 = 0 Facing page 11 i = - G = " 2 sva.0 5 V = 3 A V\ 20 VT k .= K - 3 8 sin & These values agree with the ones calculated by Lamarche (20). 2. Solution of the Hamiltonian for B = 0. First, consider the case 0. Then g •» i = k = 0. We can break up the determinant into two 3 x 3.determinants by.rearranging rows and columns so that the order instead of 1, 2, 3, 4, 5,6 is 1, 3, 5, 2, 4, 6. One thus obtains, two third order equations for the energy . levels, which can be solved. Lamarche has carried out the solution for spodumene over the range, of values from R • 0 to R =4 using the values for e2qQ and ^obtained by Petch et al (13). The energy levels for the case 0 = 0 are. plotted in Fig. (1). The energiesare in units of e2qQ. The energy levels fall into two classes; class L states have mixtures of spin substates m = 5/2, 1/2, -3/2, while class M states are mixtures of m = 3/2, -1/2, -5/2. Due to the selection rule A m *> - 1 only transition between levels of different classes can occur so that for this.orientation a total of nine transitions.are possible. These nine transitions, denoted by 2) 1, l)2, ...2)9, are indicated by arrows.in Fig. 1. The expected transition frequencies are plotted versus R in Fig. 2. The class L and class M energy levels of Fig. 1 are numbered according to decreasing energy. Transitions between energy levels for the B = 0° orientation are indicated by a single subscript, for example 2- The region for R > 3.2.in Fig. 1 can be described by the perturbation theory treatment applicable when R » 1 (12) and experimental results for this region have already been obtained for spodumene by Petch and Cranna (13) who studied the Zeeman lines ^) j to The intensities.of ^ 6 to ^ 9 become vanishingly small for R> 3.2. The three pairs of lines ^4, -^7; -^2» -^6) ~^8' ^9* in the range R < 0.3 can be described by perturbation theory applicable to R «, 1. .Although some experimental results were available in this region for other crystals with higher values of e2qQ than in spodumene, none were available for spodumene. In the intermediate region 0.3 < R< 3.2 in which perturbation theory is not applicable, no experimental results had been reported for spodumene or any other crystal. One of the objects of this thesis was to close this experimental gap in the radio-frequency spectroscopy of nuclei in crystals for at least one specific example, by demonstrating the gradual transition from the pure quadrupole spectrum slightly perturbed by a magnetic field (R«l) to the Zeeman spectrum perturbed by, the quadrupole interatinn ( R » l ) . 3. Solution of the Hamiltonian for 6 j 0. The case where B f 0 is considerably more.complicated. Here the determinant cannot be broken down, so.that one is left with a sixth order algebraic equatinn to solve. While in principle, numerical methods for the solutions of such equations exist, the computation over a reasonable range of R and for several angles is a task which one would not undertake if it could be avoided. Fortunately, the task can be avoided. Electronic computers have become almost commonplace aids to scientific research in the past ten years, and the work of diagonalizing a 6 x.6 matrix can be performed in a matter of minutes on one of them. Diagonalization of matrices using electronic computers is discussed in Appendix B. After it became apparent that the transitions in spodumene could be observed down to below 600 kc/sec the Hamiltonian was calculated for several values of the angle 6 and values of R from 0 to 3.2. The matrices for the Hamiltonian were submitted to the University of Toronto computer, FERUT, and in a few weeks the diagonalized matrices were returned. Some of the matrices, diagonalized at Toronto were purposely made to duplicate some of Lamarche's calculations. The agreement between the duplicate calculations was quite satisfactory. Calculations were made for both f{= 0.95 and^r 0.93. In March 1957, a computer ALWAC HIE was installed by the University of British Columbia. At several crystal positions of special interest, matrices were prepared and diagonalized on this machine. As an example of the behaviour of energy levels for B i 0°, the energy levels for 9 = 10°, Y[ = 0.93 are shown in Fig. 3. It is particularly noticeable that the energy levels do not cross. The reason for this.is that unlike the case 9 = 0°, in which the levels may be grouped into two sets of spin substates, no such grouping is possible when 0 j. 0°. The off-diagonal elements of the Zeeman Hamiltonian lead to mixing of all the spin substates, so that every eigenstate of the Hamiltonian will be a mixture of all six possible spin states and the energy levels.no longer cross one another. A total of fifteen transitions will be possible in principle for & / 0, since there will be a non-zero probability of transition between any two levels. A second object of this work was to detect some of these fifteen possible transitions. The system of numbering the energy states and transitions between them is shown in Fig. 3. The energy levels are numbered in order of decreasing energy and the transition between the i and j level is indicated by - ^ i j . The double subscript always indicates.a transition for the case B i 0. B. Transition Probabilities and Signal Intensities Nuclear resonance transitions are usually caused by a ..radio-frequency magnetic.field produced by rf current in a.coil. The resonance condition is often detected by absorption of energy from the coil by the nuclear spins. In such a case, the signal intensity is proportional to the transition probability, other things being constant. In the experiment to be described, an induction type spectrometer was used, in which a signal voltage is 'induced by the rf nuclear magneti-zation at right angles to the rf magnetic field which produces transitions. As will be shown, the signal voltage in an induction spectrometer will not necessarily be simply.related to the transition probability. We will now consider some of the factors which determine the signal strength in a nuclear induction spectrometer for transitions involving,mixed m states;. We will always use the cartesian co-ordinate system where Z is parallel to magnetic field H0, X is.parallel to the transmitter coils and to the radio-frequency magnetic.field, and Y is parallel to the receiver coil. For transitions between pure m states (such as we find for instance when looking at a proton resonance), the same signal voltage would be produced in a coil parallel to the Y axis as in one parallel to the Xaxis. This follows directly from the Bloch equations (21). Thus for transitions between pure spin states, an induction spectrometer and an absorption spectrometer will give the same signal intensity. For transitions between mixed m states the signal intensity may not be the same for the two types of spectrometer. This has been pointed out for the case of pure quadrupole resonance (22) where if no steady magnetic field is applied, no induction signal.can be obtained because.of the degeneracy of i m states for quadrupole interactions. Pure quadrupole frequencies may of course be measured with an induction spectrometer by using Zeeman modulation, and measuring the derivative of the resonance curves with respect to magnetic field. Then the signals from the transitions between plus and between minus spin states add, since the frequency of one transition increases with increasing magnetic field and the frequency of the other decreases with increasing field (23). To calculate explicitly the variation with R of received voltage in an induction spectrometer for transitions between mixed m states, we first note that the voltage detected will be due to an.oscillating component of magnetization along the . Y axis. Since magnetization M is proportional to the vector sum of the spins, the expectation value of dM/dt will go linearly with the expectation value of dl/dt. It is then sufficient to calculate the expectation value of dly/dt. Calculations of this nature for special situations have been reported in-the past by Bloom et al (24) and more recently by Feynman et al (25). Recently a calculation of expectation values of I x and Iy for a two state system where.the eigenstates are known has been carried out by Dr. Bloom. The results of this calculation can be applied to the present problem. The calculations are shown in detail in Appendix A, but will be briefly discussed here. For a many level system, one considers.only two levels, whose eigenstates are and between which a transition is to be induced. The eigenstate of the two level system is Y' = a Vi + b ^2 The difference in population between the two states is n = a*a - b*b The expectation value of Iy is:(>//*|Iy| y / ) . A bit of algebra and use of the time dependent Schroedinger equation leads to differential equations for the expectation values of dlx/dt and dly/dt. In order to take into account the interactions between the nuclei and their surroundings, the relaxation times T i and T2 used by Bloch are phenomenologically introduced into the differential equations. Then taking the ma gnetic fields as: H x - 2Hi cos cot H y = 0 H z - H0, the differential equation can be solved when n is assumed to have a constant value. Where the principal axes of the electrostatic gradient coincide with the external co-ordinate axes, the solution is dt lf-(^-^o) 2T2 2 I " u / J P = (Yl*/lx| f2) iS = ( f i * / I y/ f 2) u)0 - E l - E 2 Ej and E2 are the energies of the states between which transitions are to be induced. The resemblance to the Bloch equations is noticeable and when P = S the solutions, reduce exactly to the solution obtained by Bloch. The important aspect of this, solution with regard to the present experiment is the fact that the voltage detected will vary as SP. This is in contrast to the case when other types of spectrometers are used where the voltage detected?-just depends on the transition probability, For certain transitions in spodumene, Volkoff and Lamarche have calculated values proportional to P 2 which are proportional to transition probabilities for the rf magnetic field linearly polarized in the x direction and thus to the signal intensity of an absorption spectrometer for that case. These values show quite strong variation with R over the region where R is about unity (22). However, the variation of S is even more pronounced. The value of S goes through zero where R is near unity. This makes a great, deal of difference to an experimenter, for consider the strong line called ^2 m Fig* 2. The transition probability which is proportional to P 2 goes through a maximum at R = 0.8. However, S goes to zero near this point. Thus just where the transition probability is a maximum, no signal will be observed if an induction spectrometer is used. Relative values of SP have been plotted in Fig. 4 for some of the stronger transitions in spodumene when & - 0 and the x and y principal.axes of the electro-static field gradient tensor are parallel to the X;, Y directions respectively. C. Signal-to-Noise Ratio At this, point it seems worthwhile to consider the factors which determine whether nuclear resonance signals will be strong enough to be detected in any particular situation. It is necessary to .compare the voltages from the nuclear spin transitions to the other voltages present, which are grouped together under the> general heading of noise. The nuclear signai arises from the. radio-frequency oscillation of a net magnetization in the sample. For pure magnetic resonance, the Bloch equations (21) for radio-frequency susceptibility allow us to calculate the signal voltage which can be produced. The.rf susceptibility,depends on the number of nuclei present, their magnetic moment^ the interactions which maintain the thermal equilibrium with the surroundings, and the intensity of the radio-frequency magnetic field which causes radio-frequency variation in the, net magnetization. .An important relation which may be derived from the Bloch equations gives the value of the optimum rf magnetic field 2Hi-cos cjt in terms of the longtitudinai and transverse relaxation times and and the nuclear magnetogyric ratio % . The relation for maximum nuclear signal is ^ 2 H 1 2 T 1 T 2 = 1 Noise voltages.arise from several sources. The Nyquist noise is present in any system. For a resistive impedance R and a bandwidth for detected signals of B, the noise voltage V is given by V 2 .= 4kTRB k is Boltzman's constant T is absolute temperature in degrees Kelvin. Often, larger noise voltages-arise from other sources. Amplifiers invariably contribute some noise , and generally the oscillator which provides the rf magnetic field also produces noise. Electrical machinery and fluorescent lights cause fluctuations, not all of which can be eliminated from the output of the system. It is clear however, that a calculation of signal-to-noise ratio based on only thermal noise gives us the best possible ratio. If this calculation gives a low ratio, the experiment must be redesigned or abandoned. Andrew ( 27 ) has derived a relation for the signal-to-noise voltage ratio based on Nyquist noise and the best possible nuclear resonance signal voltage which may be calculated from the Bloch equations. Andrew's result for the ratio of signal voltage to noise voltage is Vg = K/)fNI(I-»-i)h2\ /VcQf n 3T 7 )J V n \ 48kT / I kTBTj / f Q - resonant frequency Vc - volume of sample coil Q - quality factor of the coil N - number of interacting nuclei K - includes filling factor of sample coil and efficiency of the detection apparatus. It is never better than about 0.5 To - transverse relaxation «* _J: AH- line width T'i - spin-lattice relaxation time In the case of aluminum nuclei, for instance, a line width of 5 gauss gives T2 = 3.10"^  seconds. T\ must be calculated as well, but if for an aluminum resonance of 5 gauss in width, saturation does not occur for H^ as large, say, as ,0.3 gauss, then T i is shorter than 0.01 seconds. With these estimates for Tj and T2-and taking N = 3 x l 0 2 2 Q = 50 V c = 5cc. f 0 = 500kc. and using the nuclear constants of aluminum, we get 30 for the best possible signal-to-noise voltage with a bandwidth of 1.cycle per second. The constants were chosen to fit the case of a 10 gram sample of spodumene, although the calculation does not strictly apply in this case, since the energy, of the interaction in spodumene does not arise only from magnetic effects. We certainly must divide by three since only two out of six levels enter into any one transition. This.is just an order of magnitude calculation presented to show that it is not unreasonable to expect to see some, of the resonances predicted by Lamarche and Volkoff for spodumene at low magnetic field. D. Choice of Crystal for the Experiment If the experiment was to be done at all, it was necessary to use a single crystal of some material, since the quadrupole effect, when mixed with magnetic interaction depends on the angle made by the magnetic field with the principal axes of the crystalline electrostatic field gradient tensor. Several-types of crystals were immediately available, but all had serious drawbacks. Spodumene, for example, on which the most work had been done, has pure quadrupole frequencies below one megacycle, and early attempts to observe.them had failed. Kernite, with a pure quadrupole frequency which is a bit above a megacycle and has been observed with good signal to noise ratio (18), has four different nuclear sites in each unit cell, each giving rise to a different resonance frequency for B1!, to say nothing of possible resonances from B*0 and Na2^ as well as from the protons contained in the crystal. Euclase, with pure quadrupole frequencies above a megacycle, also has two non-equivalent positions for nuclei in the unit cell (28) and attempts by Haering to detect the predicted pure quadrupole frequency met wi th fa i lure . Other c rys ta l s , on which measurements have been reported i n the.literature, appeared to suffer from one or more .of the. same drawbacks. From these considerations, several possible procedures present themselves . A search could be made for some crys ta l wi th an asymmetr ic f ie ld gradient and quadrupole.interaction energy of about 10 M c . / S e c . , preferably a crystal i n which a l l the nuclear si tes for the nucleus in question are equivalent * A search through the l i terature might turn up something useful , or measurements on untried crystals might be carried out at high f i e lds , i n hopes .of finding a suitable quadrupole. coupl ing constant. The.latter procedure would involve .a large amount of prel iminary work before the ma in experiment could be t r ied . Another more appealing solution would be to try to measure the resonance for Euclase .or Kerni te , since the frequencies i nvo lved for them are not too l o w , and high f i e ld data i s read i ly ava i lab le . However, the existence of more.than one nuclear site for the nuc le i i n question would lead to many l ines at intermediate magnetic f i e lds , . and untangling them might be a difficult p roblem. Except for the rather low frequency of the pure quadrupole resonance, spodumene appeared to be. an ideal, c r y s t a l . It has only one nuclear site for each of A l 2 7 and L i 7 , and a luminum has only one isotope. Thus l ines from other nuc le i than that being studied should not interfere. Complete experimental data for high magnetic f ie ld had been taken and several good clear crystals of fair ly large.volume were ava i l ab le . Furthermore, the intermediate f ie ld region calculat ions had already been done for spodumene and the predicted resonant frequencies and relative intensities showed an interesting behaviour. Finally, it seemed worthwhile to try and demonstrate that nuclear resonance measurements could be carried but at frequencies-below a megacycle, even for the. relatively broad lines found in crystals. Spodumene was chosen as the most suitable crystal for the experiment. CHAPTER IH -EXPERIMENTAL APPARATUS A. Description of Samples Spodumene is a silicate of lithium and aluminum. Its structure has been described elsewhere (29). The crystal has two good cleavage planes which intersect at an angle of 94°. Because of the symmetry of the crystal structure, the bisector of the 94° angle is parallel toone of the principal axes of the electrostatic field gradient tensor at both the aluminum and.lithium sites. This, turns out to be the x. principal .axis for aluminum:... The c axis of the crystal is parallel to the line of intersection of the two cleavage planes. Measurements at high magnetic field (13) have shown that the z principal axis of the. electric field gradient tensor at the aluminum site makes an angle of 34° with the c axis. Only,two of the three axes of the crystal can be easily identified visually so that the .orientation of the z principal axis can be only located to within.two possible positions by a simple examination of the crystal. In order to find, the third axis, part of a single .rotation of the crystal at high field.about the x principal axis was made. The identification of the third axis.of any of the samples used was then made by comparison with the results at high field which had.already been published (13). Three different samples were used in all. The first was cut from a perfectly clear piece of spodumene so that it could be rotated about the y principal axis in the 5/8" hole of the.receiver coil. The second was cut for rotation about the x axis. It also was perfectly clear. The final crystal used was cut to fit into a 3/4" hole with either the x or y axis along the receiver coil. The first two crystals had a volume of about 1 cc. and the second had a volume of about 3 cc. The larger crystal had a slightly greenish coloration and possibly contained some ferromagnetic impurity, as this crystal caused very bad pickup of the ma gnetic modulation in the pure quadrupole spectrometer, at low magnetic fields. Whatever impurity was present did not measurably affect the frequency of the resonance lines however, as some of them could be compared in all three crystals. Using sealing wax, the crystals were mounted on a lucite rod.and then aligned in the magnetic field. . The alignment in the magnetic field could be checked at 600 kc. by looking at the splitting of the lithium line. Finally the crystal could be aligned to within one half of one degree by examination of a double line which appears at about 800 kc. at the field where the transition ^  i s predicted. The choice of sample has a great deal to do with the success of.an experiment to measure quadrupole interactions. Impurities or imper-fections in a crystal may cause broadening of the resonance lines to such an extent that they areunobservable. Measurements made at high field in various spodumene samples by Petchet al (13), showed that in some crystals, the satellite lines were weak.or unobservable, while in other crystals the same resonance lines were quite strong. Even using results from a good clear crystal, the outer satellite lines, which are most dependent on quadrupole interaction, are considerably wider than.the central line which is only slightly perturbed by quadrupole interactions. This is clearly indicated by the record shown in Fig. 2 of reference (13) where the.line width of the outer satellite appears.to be about 15 kc. B. Choice of Spectrometer Considerable difficulty was.anticipated in observing the spodumene.transitions in the region below one megacycle. Several factors in the experiment tend to reduce the intensities.of signals. Firstly, the population difference between the energy levels is directly proportional to resonant frequency, so that in going from 7 to 0.7 megacycles per second the number of nuclei which can give a useful signal is reduced by a factor of ten. . Secondly, the line s in solids :may be quite wide and this further reduces the maximum intensity of the lines. Not only are the lines broad, but due to the mixed quadrupole. and magnetic splitting, instead of observing transitions between all the.energy levels at a single frequency, we only observe them between pairs of levels. For a nucleus of spin 5/2, this means that ionly 1 /3 of the nuclei interact with the exciting radiation of any one frequency to produce a signal. The disadvantages of broad lines are partially compensated by the short relaxation tirnescaused by the quadrupole interactions. This means that relatively high radio-frequency fields can be used to excite transitions. It might also be suggested that use of liquid air or even liquid helium temperatures would increase the signal intensity by increasing the population.difference.between states. Unfortunately, the quadrupole interaction is somewhat temperature dependent, (30) and so the high field measurements would have.to be done at the same temperature to get a consistent picture . Also, at low temperatures, the relaxation time is liable to be increased, so that the possibility of using high radio-frequency power levels-would be lost, thust defeating the purpose of the low temperatures. One method of making use of high radio-frequency power is by means of a.superregenerative spectrometer (31) (32). This makes use.of an oscillator which is periodically, turned off, usually by pulses from a low frequency generator. .During the periods of oscillation, the oscillator coil subjects a sample to an intense pulse of a radio-frequency magnetic, field. The oscillator is then shut off or "quenched" for a fixed period during which oscillations in the tuned circuit die.away. When the quench pulse is.removed, the precessing nuclei in the sample may induce sufficient voltage in the oscillator coil to influence the restarting of the oscillator. This will happen if the oscillator frequency,corresponds to a resonance frequency for nuclear transitions. Using a spectrometer of this type, Cranna was able to detect aluminum resonances in spodumene down to at least 1200 kc. and found the pure quadrupole line in kernite at about 1280 kc. (31). He felt that better signal to noise ratios could be achieved with more experience. Unfortunately, it is difficult to get a pure dispersion or absorption signal with this type of spectrometer, and also the quench frequency produces sidebands Jon the central frequency so that frequency measurements, of resonances are. difficult. High radio-frequency power has also been used with steady state methods. The chief difficulty here is with the added noise introduced by the oscillator. Bridge methods are often used.to balance out most of the oscillator voltage. However, most bridges must be balanced at.each frequency and are inconvenient where-measurementsare to be made.over a wide range of frequency. . Some experimental, attempts.to use bridges are. reported in Appendix C. A very satisfactory method of avoiding the oscillator noise while obtaining high rf power is provided by the induction spectrometer first developed by Bloch and co-workers. Using this spectrometer, only a small fraction of the total oscillator voltage is detected by the receiver coil, while for transitions between pure m states, the.nuclear signal.detected is as strong;as for an ordinary absorption type spectrometer. The problem of sensitivity for mixed transitions has been discussed earlier. Using a spectrometer of this type, Haering was able.to detect the kernite pure quadrupole resonance with very good signal-to-noise ratio (18). This type of spectrometer is also currently used in the commercial nuclear magnetic resonance spectrometers produced by Varian Associates. C. . The Nuclear Induction Spectrometer 1. General Design The general design of the spectrometer used for the experiment is.similar to that described by Weaver (33). Two identical transmitter coils of diameter are mounted co-axially in a rigid brass box. A similar coil is wound on a hollow lucite spool and placed between .the. two transmitter coils and at right angles to them. A hole in the brass box permits samples to be inserted into the "receiver" coil where they may be subjected to a radio-frequency magnetic field from the transmitter coils. One transmitter coil is mounted on runners and maybe moved back and forth by a.screw, so as to reduce the coupling between the transmitter and receiver. Some out-of-phase voltage is induced in the receiver coil, chiefly by eddy,currents in.the brass box. To control this, a small inductance loop similar to that described by Weaver was used. This consists of two turnsof wire wound co-axially, on a lucite rod 3/8" in diameter. One more turn is placed in series across, the end of the.rod, to form a loop with axis at right angles to the axis of the rod. A 22.ohm resistor is placed in series with the inductance loops, so that the impedance of the whole is.almost purely resistive. The lucite rod is inserted into a hole.drilled co-axially into one;of the transmitter coils. The lucite rod may be rotated without affecting the coupling of the loops to the transmitter coil but the coupling to the receiver coil may be adjusted-over a.continuous range of values. It is very important that the lucite rod fit snugly, otherwise it becomes almost impossible to get either a pure absorbtion or pure dispersion INDUCTION SPECTROMETER -••300 volts. Transmitter coils Receiver coil V Shield 25pfX 500 K 400 Pf 400 400 pf Pf -> To rf amplifier Fig. 5 - Circuit Diagram for Induction Spectrometer Facing page JO s i g n a l . Ia pract ice , the l i ic i te rod was made to fit quite snugly in i ts ho le , after a l i t t l e stopcock grease had been applied to prevent b ind ing . The transmitter co i l s were made to be part of the resonant c i rcu i t of a push-pul l osc i l l a tor ; see F i g . 5 . The capacity of the tank c i rcu i t was two gangs i n series of a three gang condenser. The.third part of the condenser formed.a resonant c i rcui t wi th the receiver c o i l . By making the inductance of the receiver c o i l and of each of the transmitter c o i l s to be the same, both transmitter and.receiver could be tuned over a range of from 500 k c / s e c to 1000 k c / s e c by adjusting the three gang condenser. T r i m m i n g condensers were provided for fine adjustment. The osci l la tor tube found to be most satisfactory was a 5692 double tr iode. By control l ing the plate, supply voltage of the osci l la tor tube, radio-frequency voltages of from 2.to 100 volts r . m . s . . c o u l d be produced across the transmitter c o i l s . The voltage coupled into the receiver c o i l from the transmitter could be reduced to belOw a m i l l i v o l t , but i n operation, a voltage of about 15 m i l l i v o l t s r . m . s . was induced for reasons given be low. The tuned receiver c o i l was coupled to a three stage, rf amplif ier by a coupling condenser s m a l l enough to severely attenuate any audio voltages p icked up by the receiver c o i l . In the sys tem used , the magnetic f ie ld was modulated at 228 cyc les per second, so that nuclear resonance signals resulted i n an audio -modulated radio -frequency voltage on the receiver c o i l . The rf amplif ier had a gain of about 100, and the output was detected by a 6 A L 5 vacuum diode. The detected audio s ignal was ampl i f ied by an audio amplif ier ^  tuned at the modulation frequency^whose gain could be increased in steps from 2.x 10^ up to 107. The output from the audio.amplifier was fed to a phase-sensitive detector of the type described by Schuster (34), and the resulting derivative of the resonance was recorded on an Esterline Angus chart recorder. The time constant on the phase sensitive detector could be varied from 2 to 200 seconds. The first spectrometer built was 1" wide and 5" square, so as to fit between the poles of the electromagnet. At first the sides of the box were made of 1 /16" brass, but this was found to be very unsatisfactory. In order to get a good nuclear spin resonance signal from the solid sample, an audio-frequency magnetic field modulation of up to 15 gauss was superimposed on the d-c magnetic field. A sheet of brass, when introduced into the combined audio-frequency and d-c field, vibrated strongly enough to be detected by touching one's finger to it. Such vibration leads to modulation of the voltage picked up by the receiver coil, and audio -frequency modulation far greater than any due to nuclear resonance is produced. Almost all of the vibration was eliminated by using sides made of 1/16" lucite, to which a.sheet of 2.or 3 mil brass - shimstock had been cemented. The spectrometer is built in as symmetrical a.manner as possible, in order that fluctuations in the level of oscillation and other instabilities should have a minimum effect on the receiver coil. The two sides :of the transmitter coils are identical,. and the centre of the tank capacitance is grounded. With the receiver coil centred between the two transmitter coils, electrostatic coupling to the.receiver should be almost completely/eliminated. A faraday shield around the receiver coil should give additional shielding from electrostatic coupling, but in practice, no notable improvement resulted from such a.shield when the transmitter coils were properly balanced. Rough measurements of the noise voltage with the d-c magnetic field held constant gave an overall noise figure for the system of less than 2, indicating that very little noise is picked up from the oscillator. The induction type spectrometer can produce either dispersion or absorption signals, depending on whether one detects the component of magnetization which is out of phase or in phase respectively with-the exciting rf magnetic field. In order to get a pure signal mode,an "autodyne" signal voltage, which is either in phase or in quadrature is; induced into the . receiver coil. The autodyne signal is much larger than any nuclear signal. Only the nuclear signal which is in phase with the autodyne signal will be detected as amplitude modulation. It turns out in practice that the,in-phase authodyne signal voltage is very easily affected by vibration, while-the.out-of-phase autodyne signal is almost unaffected. For this reason one generally chooses to observe dispersion signals with the induction spectrometer. The desired level of quadrature voltage was induced by means of the inductance coupling loop described above. 33. .2. Pure.Quadrupole,Induction Spectrometer In order to measure pure quadrupole resonances in an induction spectrometer one must sweep the frequency across the resonance, and also have a perturbing magnetic field, since at zero magnetic field, the symmetry of the quadrupole coupling leads to plus and minus m states falling ;at the same energy, so that the induction signals cancel. Unfortunately, the phase of the autodyne signal induced in the receiver for any given adjustment is somewhat dependent on frequency. The chief cause of this frequency dependence seems to be that the eddy currents in the shielding depend somewhat on frequency. Thus it can be quite difficult to keep the spectrometer adjusted to observe a pure absorption or dispersion mode over a wide range of frequency. For the experiments at several hundred gauss, space.is at a premium so that the shielding must be quite.close to the rf coils. However, for weak fields, the spectrometer may have, almost any size, since small magnetic fields can easily be produced over a large volume. This reasoning led to the construction of an induction type spectrometer similar to the one previously,described except that the shielding was at least one inch from the rf coils. The frequency of this instrument could be swept through 20% of the total value without serious change in the phase of the autodyne signal. Using field modulation at 228 cycles per second of about 6 gauss, the pure quadrupole lines were observed in all three spodumene samples with this spectrometer. Radio-frequency magnetic fields of about 0.1 gauss peak-to-peak from the transmitter and time constants of up to 80 seconds on the phase sensitive detector were used. 3. The Magnet For convenience in magnetic field-control, measurements of weak resonance signals are often made by setting the magnetic field at a predetermined level and sweeping the radio-frequency across.the resonance region. This has been the.customary procedure in our nuclear resonance laboratory in the past. When using the induction spectrometer, the phase of the autodyne signal picked up in the receiver coil is dependent on frequency. Thus the autodyne signal may at times contain a little bit of in-phase component, which as stated earlier, is susceptible to vibration. This,means that if any.vibration at the modulation frequency is present, one gets drifting of the output to the chart recorder as the amount of in-phase, component of the autodyne signal changes. Even with the small amount of vibration still present in the spectrometer after all improvements had been made, the drift could be bad enough to cause.the recorder to go .completely, off scale.over a sweep of 100 kc. In addition, the mixture.of modes caused by the variation in-phase, of the autodyne signal as the frequency, is varied causes the shape of the resonance to become uncertain, so that to pick a weak line out of the noise becomes very difficult. It was. therefore decided that instead of changing the frequency while holding the magnetic field strength constant, the magnetic field would be varied while holding the radio-frequency constant. In order to avoid the use.of d-c amplification, the magnetic field would be modulated as before. It was relatively easy to vary the field slowly . over a wide range, since the.current through the electromagnet already in use was controlled by a servo-system which amplifies an error signal, thereby controlling the grids of a bank of 6AS7 triodes which supply the magnet current. To get a magnetic.field sweep which was linear in time, it was.only.necessary to add a linearly varying voltage to the error signal. Owing to the hysteresis.of the iron in the magnet, it is impossible-to predict the magnetic field-accurately, from a knowledge of the current in the coils . The field is normally held steady by a proton resonance field control system (35); onexould vary the field by slowly changing the frequency of the proton spectrometer. However, this procedure could become more and more .difficult as lower fields.were needed, since the proton resonance would finally become too weak to .control the field, due. to the low resonance frequency. In order to have a field.calibration for each run, a small probe was mounted in the brass.box containing the nuclear induction system. The probe was as near as possible to the. sample, and contained.the coil from an oscillating detector of the type designed by Collins (35). The coil was immersed in mineral oil, and provideda proton signal for frequencies .0.8 to 3 megacycles per second with a large signal-to-noise ratio. The output from the oscillating detector was attenuated and led into the.same.audio amplifier which amplified the.detected signals from the nucleax induction apparatus. Thus by choosing a suitable frequency for the oscillating detector, a calibration resonance line could be put on the chart at any desired field as the field swept slowly through the proton resonance. In practice, the variation of field with current was found to be linear enough to that interpolation over 20 or 30 gauss was accurate to about 1 gauss. The difference of field between the position of the proton probe and the sample coil of the nuclear induction head was measured using a sample of lithium acetate in water. A difference of from 1/2 to 3/4 gauss was found between the two positions. Since measurements were only made to an accuracy of about one gauss, because of line widths, a correction could easily be applied. A slightly more satisfactory calibration point was provided by the resonance.in the copper wire of the.receiver coils. Also, the L i 7 in the spodumene crystal gives a resonance which may be used for field calibration. Using the iron electromagnet which has been described earlier (35), measurements of aluminum resonances in spodumene were made down to about 600 kc and it was apparent that sufficiently good signals could be obtained to measure some of the spodumene transitions where R is of the order of unity. In order to simplify the measurements of magnetic field, an air.core electromagnet was constructed, free from ferromagnetic materials. The magnet was. designed to approximate to a Helmholtz coil and. consisted of two large coils, each with about 3000 turns of number 18. wire. The inner SPECTROMETER RF OSCILLATOR MODULATION COILS RF AMPLIFIER AND DETECTOR AUDIO AMPLIFIER PHASE SENSITIVE DETECTOR CHART RECORDER MONITORING OSCILLOSCOPE AUDIO OSCILLATOR ^IPHASE SHIFTER 3_ POWER AMPLIFIER M A G N E T CONTROL FROM GENERATORS MAGNET COILS "Xv POTENTIOMETER •— t9 6AS7 DOUBLE TRIODES < A.C. AMPLIFIER */ 2OHM. 2 OHM D.C. AMPLIFIER X HELIPOT-500 OHM j 6V0LTS OH  Fig. 6 - Block Diagram of Experimental Apparatus Facing page 57 diameter of each coil is about 6" and the outer diameter 10V. Each coil had.a width of 3". The coil forms used were made of a brass cylinder, and aluminum sheets of 3/8" thickness. Three, layers.of copper tubing were placed in each coil, and water could be passed through the tubing at the rate of 2 litres per minute. The magnet temperature.remained quite constant at 5° to 10° below room temperature. The current for the magnet was controlled by.the same servo-system whichwas used to control, current in the iron electromagnet. Switches were put in the system.so that one could switch from one magnet to.the other. For a. useful gap between the coils of 2", and using 300 volts d-c from the main d-c generators of the Physics Building, it was possible to get magnetic fields up to 480 gauss. Hie homogeneity of the field was estimated to be about 1/2 gauss at 200 gauss over a volume of 5 cc. from the. line width of a proton signal in water. This at first appears to be quite inhomogeneous, but the widths of the lines to be measured are all greater - than 5,gauss, so that the field homogeneity is quite satisfactory. Fig. 6 is a block diamgram showing the experimental apparatus. D. Difficulties At the highest levels of sensitivity used, some difficulty was experienced with instability of the apparatus. The instability became larger as rf power was increased and was the chief limitation on the sensitivity of the apparatus. It was only observed while the magnetic field was being varied and at times effects were produced which could have been mistaken for weak resonance signals. That the signals did not arise from nuclear spin resonance was evident from the fact that they were independent of the radio-frequency used in the spectrometer. The output from the spectrometer also showed a tendency to drift steadily as the field was swept over a wide range. During a brief visit to this laboratory, Dr. Packard of Varian Associates mentioned similar difficulties which had been experienced by other workers. He felt that most of these troubles could be traced to vibrational effects of one kind or another, particularly with the type of spectrometer where metal sides are fastened to the. box containing the receiver and transmitter coils. The amount and phase, of the. rf voltage picked up by the receiver coil is extremely, dependent on the contact resistances between 4he. sides of the box and the main part. Any vibration tends to change these resistances, and thus could produce spurious signal voltages. Experimentally, the author has often noticed that although the sides of the box are fastened down with about 40 screws, the. removal of one screw can have quite a pronounced effect on the.rf voltage picked up by the receiver coil. This effect is much less noticeable when lucite sheets with a thin brass covering;oii the outer surface are used for the sides instead of 1/16" sheets of brass. In order to eliminate possible changes in contact resistances between the walls of the box, a new spectrometer similar in principle > to the earlier spectrometer was; built from a single block.of aluminum. Holes were bored for the receiver and transmitter coils, leaving a 1/8" clearance for coils 1" in diameter. This spectrometer would take samples up to 3/4" in diameter and was used for measurements with the large greenish crystal described earlier. However, although the larger crystal gave larger resonance signals, no significant decrease in the. instability/was noticed with the spectrometer made from a single block:of aluminum. Tests showed also that the. instability was not due. to the sample. It is felt that the chief advantage of a spectrometer made froma single block is that some simpli-fication in construction is achieved. It was possible to get sufficient signal-to-noise ratio,even with the instability, by decreasing the bandwidth of the system and using lower levels of rf power. Careful adjustment to give pure quadrature phase for the rf voltage in the receiver coil also leads to considerable decrease in instability. This of course is.a strong argument for believing the difficulty is connected with some vibrational effect in the instrument. In any case, the spectrometer even.^dth its faults gave sufficient sensitivity,to detect the resonances under investigation. One.other possible explanation may be offered for the instability. Some modulation voltage is induced into the coils of the electromagnet, which can be fed back to the servo-system controlling the d-c current to the magnet. Changes in magnet current are associated with changes in the loop gain of the system, so that as the magnetic field is gradually varied there may be variations.of small currents in the system which are. in phase with the modulation field and with the reference voltage .to the phase-sensitive detector. This could lead to drifting.of the output level of the spectrometer. Complete elimination of these effectsjis not a simple.problem, and since the purpose.of this research was to observe a certain resonance spectrum, not to develop a spectrometer, no all-out attack.has been made on the problem. These difficulties are pointed out to assure the reader that the limits of sensitivity have not been reached, and there isreason to believe that even weaker resonances than those studied here could.be observed if the apparatus was perfected. CHAPTER TV EXPERIMENTAL PROCEDURE The measurements reported in this thesis were obtained for the most part using the iron-free electromagnet where measurements of magnetic field could be obtained by, a measurement of magnet current. The current through the magnet was measured by meansof a potentiometer which measured the voltage developed across a 1/2 ohm manganin resistor in series with the magnet. The manganin resistor was wound on a bakelite form andxarefully annealed before being put into the system. The potentio-meter was fed by several number six dry, cells which were placed in a closed box to prevent rapid fluctuations in their temperature. It was calibrated regularly with a standard voltage cell. The field versus the potentiometer reading was measured from 450 gauss down to 100 gauss using Li and proton resonances from samples placed at'the position normally, occupied by the crystal. No deviation.from linearity was found. The spectrometer could be removed and replaced in the field within a few millimeters:of its former position, but the field calibration was checked before and after each set of measurements as a precautionary measure. All frequency measurements were made using a BC 221 -A frequency meter. The crystal oscillator frequency of the meter was checked against WWV at 20 megacycles. The spectrometer oscillator frequency remained stable to much better than 0.1 kc over periods of hours/ provided the voltages on the plate or the heater of the oscillator tube were held constant. Using the new electromagnet the spectrum of Al 2? in spodumene was measured dawn to about 20 gauss. Time constants between 2 and 200 seconds were used at various times. Most of the work was done with a time constant of 20 seconds, and a field sweep rate of 5 gauss per minute. Some, checks with lower time constants:of the L i 7 resonances showed that the fast sweep rate led.to a systematic error of about 0.5 gauss in measuring position of a resonance. However, the narrowest line width found for A l ^ 7 resonance was 5 gauss, and the precision of measurement was.limited by signal to noise ratio to ab out 0.5 gaus s. Thus it was. advantageous to; apply the correction rather than to reduce the time:;constant or use slower field sweep. Field modulation up to 15 gauss peak-to-peak has been used. As a general rule, the broader the modulation, the easier it is to detect a line, but the poorer is the precision of measurements. Thus for some of the measure-ments made of 2, modulation widths down to 1 gauss peak-to-peak were used, but higher modulation strength was used depending on how difficult a transition was .to observe. Rf field strengths up to 0.5 gauss.r.m.s. could be produced, but the increase in instability, for high power made 0.1 gauss r.m.s. the most practical rf field strength. The magnet current was supplied from either a stack of accumulators which produce 110 volts d-c or two motor generators which can produce up to 300 volts d-c when in series, or a transformer and bank of selenium rectifiers which would provide two amps at 250 volts d-c with considerable ripple. The current to the magnet was passed through a bank of 19 6AS7 double triodes. The grids of the 6AS7's were controlled by a voltage derived from a 2 ohm resistor in series with the magnet. This voltage is balanced by a bucking voltage, and the difference goes to a d-c amplifier which controls the grids of the 6AS7.triodes.in such a way as to keep the error voltage very small. The bucking voltage was changed linearly in time using the voltage from four No. 6 dry cells across a 500 ohm, 15 turn helipot. This produced a linear change in the magnet current and in magnetic field. An a-c feedback.loop reduced the amount of ripple on the m agriet current. To makea measurement, the oscillator was.first set accurately at the desired frequency. Then the - magnetic field was set at some point above the position where: a resonance was.expected, and a synchronous motor driving the bucking-voltage helipot gave, a field sweep which was linear in time. The output from the detection system was recorded on an Esterline Angus chart recorder. The.chart of the E .A. recorder is also driven by a synchronous motor. As the. reading on the field, calibration potentiometer passed through a given value, a calibration mark could be put on the chart record. The potentiometer setting for which a resonance occurred ^ as found by linear interpolation between these calibration marks. For some of the measurements, the.signals were very weak, and in order to get reasonable.accuracy, it was necessary, to go over the P L A T E I Facing page 44 weak resonances several times and compare the signal obtained in several runs. As an example to show clearly.the advantages of comparing results of several runs, two typical recordings taken with exactly the same conditions are shown in Plate I. The runs were made with 63 = 5° at 830 kc. The two lines are and ~i)24 and by comparing the two records, it is clear that no other resonances are present. The signal-to-noise ratio is somewhat lower than_usual in these two runs.because a small field modulation was being used. CHAPTER V EXPERIMENTAL RESULTS A. Resonance Frequencies 1. Interfering Resonances The theoretically expected transitions ^1 to %>9 for O = 0, and the 15 transitions i?^ for Of 0 have been discussed in Chapter II. We now describe what has been observed experimentally, and some of the difficulties involved. One difficulty with measurements at low frequency is that for a given frequency, the magnetic resonances of many different nuclei are found within a small range of magnetic field. When one is observing broad lines, and is consequently using broad field modulation, there is liable to be interference by unwanted signals frOm nearby resonances. In this experiment, nuclear magnetic resonances from three different sources have given some interference. The Cu^3 i m e and the }?s ^ n e cross at about 1 megacycle/sec. Thus measurements of ^3 could only be made well above and well below one megacycle/sec. The )) 7 transition runs almost on top of the L i 7 resonance from about 700 kcand -up. This is particularly unfortunate as y) 7 is rather broad and weak.and the L i 7 line.is very strong. It is fortunate, of course, that the orientations of the electrostatic gradients at the aluminum, and lithium sites are such that the lithium line has almost no splitting where the z principal axis at the aluminum site is parallel to the magnetic field. Otherwise three interfering resonances due to L i 7 would have been present. The other interfering resonance encountered was due to protons in the spectrometer. The first spectrometer had coils wound on lucite, and the protons in the lucite gave a line much larger and broader than any oflthe resonances in the crystal being studied. A second spectrometer was built using all glass coil forms. Even with this one, a strong proton signal (although much smaller than in the first spectrometer) appeared. The signal was probably due to the insulation on the receiver coil wire. In any case, this spectrometer was much more sensitive to vibrational effects, so that, a spectrometer with lucite;,coil forms was used for all measurements. The.wide.proton line prevents observation of )?2 ^ m e region from R = 0.4 to R = 0.6. 2. A l 2 7 Resonances for 0=0 In Fig. 7 some of the.observed resonances for 0= 0 have been plotted. The lines are the theoretically predicted resonance frequencies and the circles represent experimental measurements. The dot-dashed and the dashed line show the position of the. interfering proton and L i 7 resonances respectively. Only the measurements.taken in the region for R less.than 2 are shown as this is.the interesting region for which no previous measurements were available. In order to give a better picture of the progress of this thesis, the historical order will be followed in reporting the results. Transitions were first observed in the large iron-core electromagnet with a spodumene sample placed so that the . z principal axis was within about 5° of being parallel to the magnetic ..field. ^3 was first observed at about 1200 kc, and its position could be checked easily by comparison with the nearby Cu°3 resonance from the copper wire in the spectrometer. It was also observed at around 800 kc but could not be found at lower frequencies. The resonant frequencies observed for V3 were in good agreement with prediction. Next the signal from }>2 w a s observed at 1200 kc and traced down to 600 kc at which point the. field in the large electromagnet could no longer be reduced linearly. Measurements were then commenced in the air core electro-magnet. By means of measurements made on the crystal at high field, the z principal axis was aligned to within 1° with H Q. l) 2 was. now traced down to 525 kc and back to about 750 kc at around 20 gauss. Weak resonances were also observed in the vicinity of. R = 1.2 at the frequencies predicted for y)j. The V7 resonances were at first scarcely larger than the noise, but by increasing the depth of field modulation fairly creditable records could be produced. A search was now made for the other Zeeman split quadrupole lines at fields below one hundred gauss. In this region, y 2> >*4» 2^ 6 were detected down to fields of about 20 gauss, with a signal-to-noise ratio of only about 2. The use of magnetic field sweep prevented searching below 20 gauss. No clear indication of }) 7 could be found in the low field region, P L A T E I Facing page 48 as is to be expected from the curves of Fig. 4. The transitions and -)) 9 predicted by Lamarche and Volkoff (36) are shown by them to have a considerably lower transition probability than the transitions ~))\ to y)^. As the resonances observed at low field for > J^ 4» l)t w e r e a t about the limit of sensitivity, only a short search was made for >*8 a n c* ^9 • As expected, there was no detectable indication of a resonance at the fields and frequencies where they would.be found. y)\ was.also observed at frequencies between 1.2 and 1.5 megacycles per second. was never clearly observed.above 150 gauss. No search was made for ^3 at low fields, as low frequency combined with low transition possibility made such a search seem useless. Plate II shows photographs of records.obtained for a few of the.resonances at Q - 0. The upper photograph shows a run at 675 kc/sec. The large resonance at right is due to L i 7 . At left is the resonance from V 2 and in between is a smaller resonance from Pj. A time constant of 80 seconds was used with a modulation depth of the order of 10 gauss. The field has been swept from 450 gauss to 240 gauss. The lower picture shows the sort of signal-to-noise ratio which was obtained in measurements at low field. This.is a run at 900 kc showing J^ g and 1^ 4. In order to emphasize, the way in which the.record is interpreted, a drawing has been made below the record to show what would be observed if the system had no noise.or instability. It is possible to make such a drawing with confidence by comparing the results froma number of such runs. The records shown.on P L A T E HI Facing page 49 Plate I make the procedure clear. In the record shown of ~))^ and j)^, the magnetic field was swept from about 150 gauss to about 40 gauss. For all records shown, the highest magnetic fields correspond to the right hand side of the record. 3. A l 2 7 Resonances for Q j 0 In the vicinity of 800 kc and R about 1.4, }?2 was observed to apparently split into 2 equally strong lines, if the crystal was rotated by a degree or so from the 0° setting. The splitting became greater for a larger deviation from alignment, but for frequencies of about fifty kc either higher or lower one or other of the lines would disappear. In order to understand this effect, we must refer to the energy level diagrams of Figs, land 3,. At & - 0°, the energy levels L-2 and M2 cross. The transition })2 goes from to M2, and for 6= 0, no transition is possible from Li-to L2. If &4 -0> levels no longer cross, all six spin substates mix into each eigenstate, as has been discussed in Chapter 13., and transitions are possible between any pair of levels. For only a slight deviation from &= 0, the transitions equivalent to }?2 will still have greatest probability, but at the value of R where.the levels would cross, both transitions will have about the same probability. Plate III shows resonances at 800 kc for 0 •«= 0°, 1° and 5° in the region of R = 1.4. Values of 0 are accurate to about one half of one degree. The field has been swept over about 60 gauss for each record. R 1 4 Fig. 8 - Resonance Spectrum for&= 5° . R Fig. 9 - Resonance Spectrum for 9 - 10°, 9= 20° Facing page 50 Partly because of these experimental observations, the Hamiltonianwas written down-.for other orientations than £=0°, and solved by the Toronto University computer, FERUT. The transition frequencies calculated from these solutions agreed with the measured frequencies of the double lines for several orientations. This agreement was especially satisfying as it showed that even with the low signal to noise ratios obtained, measurements of resonance frequencies could be made with good accuracy. It was also nice to know for sure that the transition, frequencies could be measured before they were predicted. Some of the results for 0 / 0 have been presented in graphical form in Fig. 8 and Fig. 9. For comparison, some points for Q = 0 are also plotted. The solid curves shown give the theoretically predicted frequencies as a function of R for the cases shown. The slight deviation between theory and prediction measurements (which in no case is greater than the width of the resonance) is discussed later. The insert of Fig. 8 shows the detailed behaviour of the resonances around R = 1.4, with 0 = 0 and frequency near 800 kc. No position for the crystal was found where only a single line was observed for ^ 2 m this.region. This will also be mentioned later. B. .Signal Intensities The calculation of the variation of signal voltage with R which is discussed in Chapter II and in Appendix A resulted from an apparent discrepancy between predicted and observed transition probabilities. When measurements were first made of 3^ 2 in the region around R = 0.8, it was noticed that the signal fell off sharply in intensity and became.almost unobservable just in the region where the transition probability was calculated to be a maximum by Lamarche. This did not indicate an error in either calculations or measurements. The signal intensity is proportional to the transition probability only when using an absorption type spectrometer where the nuclear resonance signal is detected in a coil parallel to the linearly polarized radio-frequency magnetic field which causes the transitions. The signal voltage detected in an induction spectrometer is not simply related to the transition probability. In Chapter II and in Appendix A, the procedure for calculating the variation of signal voltage with R is given. The expected variation of signal voltage versus R is shown.in Fig..4, and we note.that in the region around R = 0.8, the signal voltage should go to zero just where it was in fact observed to pass through a minimum. The observation of weak resonances in this region is due to the fact that the rf magnetic field, at the sample position is. not completely polarized in any one direction, but has components in different directions in different parts of the sample. Clarification of this point will be given in Chapter VI. Signal intensity depends on line width, width of magnetic modulation, frequency of the transition, relaxation times and radio-frequency power. Measurements also depend on instrument stability. Thus any quantitative measurements of signal strength would be suspect unless all these factors were taken into account. However, the signal strength of p 2 is predicted to approach zero very sharply in the vicinity of R = 0.8 when an induction spectrometer is used, so some measurements of relative signal voltage were made in this region, holding the modulation width and radio frequency pOwer constant. The variation of overall spectrometer sensitivity was checked over the frequency region involved by measuring the amplitude of the central L i 7 line at 500 kc and 600 kc. The intensity of ^2 w a s found to decrease roughly as predicted, and semirquantitative agreement with the predictions shown in Fig. 4 was.obtained. Another check on the theory can be made by comparing signal strengths of the transitions V 2 a n c * Pi' The predicted signal voltage is.only one half as large for as for p2> near R = 1.2. The observed )) j signals were only about 1/3 as strong as for )} 2 .(see Plate II). Thus, the relative intensities are. in qualitative agreement with the predictions. The extra difference can be attributed to difference in line width. A comparison may also be made between signal strengths for and p(>. The signal-to-noise ratio obtained for 3^ 4 and 3^ 5 is not very good because they are quite wide. However, the records which have, been obtained show that tends to give a bigger signal than which is in qialitative agreement with the predictions shown in Fig. 4. (See Plate II for an example of 3)4 and "pg resonances.) C. Pure Quadrupole Measurements and Numerical Values.for Quadrupole Interaction Constants Measurements of transition frequencies versus field for •$2> >^4 a n d 3^ 6 f°r values of R below 0.4 when extrapolated to R - 0 indicated that the pure quadrupole frequencies were about 5 kc away from the frequencies predicted by Lamarche. For this reason a pure quadrupole induction spectrometer was built and a search was made for the pure quadrupole resonances. Unfortunately, the.large spodumene sample had. some, impurity which caused a great deal of modulation pick-up. The pick-up was dependent on orientation which probably indicated that the impurity was localized in one portion of the crystal. In any case, only some orientations of the crystal could be used. A pure quadrupole resonance about 20 kc wide was detected at 751.5 - 2 kc, with the.x principal axis parallel to the receiver coil and the z principal axis parallel to the transmitter coil. The resonance was 'checked by putting on a d-c perturbing field of a few gauss which made the resonance split into two lines. Runs were made at several values .of low magnetic field, and the Zeeman split lines, converged at 751 kc. The predicted line near 789 kc was not seen clearly with this orientation. P L A T E IV 8 5 6 k c 7 0 9 kc 818 k c 6 9 2 k c Facing page 54 Using another orientation, where the z principal axis made an angle of roughly 45° in the yz plane with the transmitter coil, and the x axis was still parallel to the receiver, both the pure quadrupole.lines were detected. The upper one was found at 793.5 ± 2 kc. Attempts to make measurements of Zeeman splitting with the z axis parallel to a small d-c magnetic field gave no useful resultsas the split lines were then.too weak to detect, because;of the low signal-to-noise ratio. Plate.IV shows two runs in which both of the pure quadrupole resonances were.observed. The disagreement with the pure quadrupole frequencies calculated by Lamarche is beyond the estimated experimental error. However, the theoretical ratio: of the frequencies was calculated for values of the asymmetry parameter r\ from .0.90 to 0.98 using the equation given on page 8. The ratio of 751.5 to 793.5 occurs when r\ has the value 0.933. The value of e2qQ then comes out to be 2.965 megacycles per second, compared to the value of 2.960 megacycles per second used by Lamarche. The estimated probable error of 2 kc for the pure quadrupole measurements gives a probable error of .007 for Y{ and 10 kc for e2qQ. The above values are in excellent agreement with the. results of Petch et al. The slight disagreement with the Lamarche calculation arises chiefly from his use of a preliminary.value for j^of 0.95, as is pointed out in the paper by Lamarche and Volkoff (3.6), although the final value obtained by Petch and Cranna was 0.94 ± 0.01. Values for e-^ qQ and f^may also be calculated from measurements in the region wh ere R ~ 1. The predicted frequency for i s almost independent of r\ for R = 0.8, so that with only an approximate value of *\ , measurements of j)2 maybe used to calculate e2qQ. The value obtained from this measurement is e2qQ = 2,975 kc. which gives an average of 2,970 kc. for the two different determinations of e2qQ. A determination of i\ may also be obtained using phenomena associated with the crossing of the energy levels L2 and M 2 . The transition j)2> between L i and M 2 has.a high probability of being induced, while the transition Lj[ to L 2 should be forbidden for & = 0. Nevertheless, just in the region of magnetic .field where lu2 and M 2 cross, resonances are observed which are due to transitions between L]^ and ~L2 • (See insert of Fig. 8). The resonance frequency at which the two transitions are predicted to have the same, frequency at the same magnetic field is dependent on f| . Using the average value, e2qQ = 2,970 k c , the cross-over frequency should be 798 kc. for ^  = 0.930 and 794 kc. for f[ - 0.950. From measurements like those shown in the insert of Fig. 8, the cross-over frequency is about 797 kc. This gives us the value l\ = 0.935. Although the extremely, close agreement between the two determinations of >\ is certainly fortuitous, the above calculation indicates, that all the determinations of e2qQ and f\ are. in good agreement. Table I compares the results obtained from all the different measurements of e2qQ and ^ for A l 2 7 in spodumene. The errors quoted for results obtained in this work are estimated probable errors based on the precision of the experimental easurements Involved. TABLE I e2qQ Mc./sec n R = 0 2.965 ±.010 0.933t.007 This work R~ 1 2.975+ .015 0.935 + .010 This work 2.950 £.020 0.94 r .01 Reference (13) CHAPTER VI DISCUSSION The main reason for undertaking the experiment was to check that the nuclear resonance spectrum for spodumene in the intermediate field region behaved according to the predictions made by Lamarche and Volkoff. The. agreement with the frequency predictions indicates that the energy levels behave as predicted. The new method for calculating signal voltage for an induction spectrometer, which has been discussed in Chapter II, gives good agreement with experimentally measured resonance voltages. , One of the interesting predictions of the calculation is that in the intermediate field region,. "extra" transitions can occur which would be.impossible at higher magnetic fields. One of these, the transition pj, between non-adjacent energy levels has been observed in the region between R = 1.1 and R = 1.4. At higher values of R it is masked by the Li resonance, and at lower values of R, the combination of low signal voltage and low value of df0/dH makes it impossible to detect with present techniques. The records.reproduced throughout this report tehow that measurements of nuclear resonance may be undertaken at frequencies below one megacycle even in the unfavourable case of broad lines and diluted samples. As an example of the sensitivity of the induction type of P L A T E V SPODUMENE RESONANCES-313 kc 12 5 255 GAUSS GAUSS Facing page 58 spectrometer, Plate IV shows a record taken at 313 kc/sec at &- 90° using the largest crystal of spodumene and a time constant of 10 seconds. In this orientation, the L i 7 line is split in three by the quadrupole interaction. The smaller resonance corresponds.to an aluminum resonance which would be the "central" aluminum line at high magnetic fields. Even this low frequency is certainly not the lower limit: for detection of resonances affected by quadrupole interactions . It just happened to be- a frequency which could be easily obtained by putting an extra condenser across the tuned circuit of the oscillator. The magnetic field has been swept from 250 to 120 gauss in the record shown. After looking at the record of )>6 and ))4 shown in Plate II, some readers may question the reliability, of measurements made with such a low signal-to-noise ratio. When the positions of the resonances are already known, it might be possible to measure some spurious.resonance or a bump in the noise as a signal. This possibility,was kept in mind throughout the measurements . Since spurious signals had been found which were independent of frequency, no measurements were accepted unless the resonance had the expected variation in magnetic field as the frequency was shifted through five or ten kc. The only resonances which were so weak as to be suspect are those for fields below 100 gauss. In order to be sure of these, it was sometimes necessary to make two or three runs, and then only by knowing the shape of the resonance could a good measurement be made. To an inexperienced observer, many of the records from which measurements P L A T E VT Facing page 59 were obtained look like nothing but noise. The most convincing proof that these measurements are indeed correct lies in the fact that they were used to predict a disagreement of the pure quadrupole frequencies with those predicted by Lamarche. When the quadrupole frequencies were observed in the new pure quadrupole spectrometer, the disagreement was found to be aspredicted, and as mentioned earlier, further investigation of the constants involved showed that the. disagreement was due only to the use by Lamarche of an inaccurate preliminary result from the high field measurements of Petch and Cranna. The signal-to-noise ratio obtained for measurements at fields ;above 100 gauss is such that there can be no suggestion that any of the resonances reported are not real. Furthermore, measurements made of the transitions ^23 a n c* ^24 f o r ^ 0 were made with no prior knowledge of where these.resonances should occur, and when the transition frequencies were finally calculated, agreement with experiment was obtained. Since many of the records shown in the thesis.were taken to demonstrate other points than a good signal-to-noise ratio, two selected runs are shown in Plate VI. The upper record shows y^23 2^ 24 resonances at 800 kc for 6=5, while the lower one shows )^2 and 1)7 resonances at 660 kc for B = 0. Tirre constants of 20 and 80 seconds were used for the first and second run respectively. The curves of Fig. 4 indicate that no signal voltage should be observed in the vicinity of R = 0.8. The curves are only exactly applicable when the radio-frequency magnetic field is parallel to either one of the x or y principal axes.of the electric field gradient tensor, and the receiver coil is parallel to the. other one. In the spectrometer which has been used, the radio-frequency magnetic field is produced by two transmitter coils of 1" diameter and 1/2" length. The two coils are separated by a distance of 1", so that components of magnetic field in all directions .will, be produced in the sample. The component of magnetization parallel to the y principal axis of the electric field gradient tensor is .the one which becomes zero around R = 0.8. Thus, if the y axis is parallel to the receiver coil-axis, no signal will be detected for one particular value of R. However, if the y axis is parallel to the transmitter coil axis, non-parallel components of the rf magnetic .field will still cause transitions, so that weak resonances may be detected for all values of R. As.may be seen by comparing.Fig. 7 and .4, measurements were actually made.close to where ^Iy^ goes.to zero. These were obtained with the y principal, axis parallel to the transmitter coils and the x principal axis parallel to the receiver coil. Some mention must also be made about the. line widths of the resonances. The line width measured is almost certainly due to quadrupole effects. The minimum width of )) 2 around 800 kc is about 5 gauss, which corresponds to.a width of about 8 kc, while the width obtained for the pure quadrupole. lines is about 20 kc. It is instructive to compare these widths with the width of the central aluminum line at high magnetic fields for which quadrupole interactions only appear as second order effects and the line width is about 5 kc or less. We also note, that at high field the outer satellites for which the quadrupole interaction is more important have a width of about 15kc . See Fig.. 2.of Reference (13). It seems clear that as the quadrupole interactions become more important, the lines become broader. This explains the.low intensities obtained for the resonances below 100 gauss since in this region the important . interaction is due to the quadrupole moment. The splitting of the y)^ resonance around 800 kc as a function of magnetic field is an interesting example of the behaviour of the angular momentum wave functions. If the z principal axis of the. field gradient tensor is accurately parallel to the net magnetic field, the eigenstates M2 and L,2'-of Fig. 1 are made up of different sets of wave functions and so they cannot interact and the energy levels can cross. If, however, the parallel condition is not satisfied, the wave functions of the two states can mix and the levels ."repel" each other. Now consider the transitions V2> ^23' a n c* 3^ 24- (See. Figs. 1 and 3). If we go off from the & - 0° position by say 10°, the states should be still about the same as at B = 0°, except that energy levels 3 and 4 no longer cross. The transition >*2 is similar to the transition 2^ 23 a t fields above the cross-over point and to the transition y>24 a t fields below the cross-over point. Since at B- 0°, y>2 was allowed and. the other transitions forbidden, then since for small deviations from B = 0°, the eigenstates will be only slightly changed, we can expect V23 to be strong at high fields and ^24 t o ^ e strong at low fields. At the field where the two lines would cross for a 0° setting, both lines.are strong. Calculation of signal voltage by the methods of Appendix A give approximately equal values for the strength of the two lines for this region in good agreement with experimental observations. With reference to Fig. 1 and 3, only one of the transitions corresponding to >^ 23 o r 3-^24 should, occur at any given value of R, for C 7 - 0 . This allowed transition at B= 0 is denoted by V 2 « In early work with the smaller crystals, an angular position was found where only one line was. found where V 2 was predicted. However, in order to separate the lines from noise, a modulation of about 4 gauss peak to peak had to be used, and lines separated by 4 gauss or less would appear as one, especially with a low signal-to-noise ratio. With the larger crystal used in the final experiments, a modulation width of about one gauss peak to peak gave s ufficient signal to noise. With this lower modulation, no po sition was found where the two lines merged into one narrow line. Even in the best position, two peaks could still be.resolved, apparently separated by three to four gauss (see Plate III). The line.width for a single line was 5 gauss. No satisfactory explanation'has been found for the fact that the lines do not merge. It may well be that the crystal was not properly aligned in the field although a careful search was made to find the point of merging if it existed. It is felt that the deviation from alignment was less than one half of one degree. There also appears to be a slight discrepancy between experiment and theory for ^23 a t ^=5°. The deviation is only about one line width or less, nevertheless, it was observed consistently for two different crystals. It seems likely that the two effects are connected, but for the present, they must remain an unsolved problem. The pure quadrupole transitions measured in spodumene appear to be the lowest in frequency which have been observed by direct methods to date. It seems that the useful-range of frequencies for crystal spectroscopy,extends to lower frequencies than has been generally assumed. One interesting experiment is.being planned with Dr. Nt Bloom to take advantage of the possibilities.of low frequency measurements in crystals. Crystalline (Cu CI2) 2H2O becomes antiferromagnetic at temperatures below 4°K. This should lead to a non-zero magnetic field at the position of protons in the crystal. The magnitude of the field.is estimated to be of about the order which would give a proton resonance somewhere between 500 and 1000 kc. If the resonance can be detected with no external field, a. method of measuring the variation of the internal field with temperature will be available. Numerous pure quadrupole transitions have been predicted at frequencies well below a megacycle per second from work at high magnetic fields. Until now, scarcely any measurements of these low frequency resonances at zero magnetic field have.been reported. It would be interesting to measure a large number of these low-frequency pure quadrupole resonances to see if any discrepancies arise between measurements-at high and low magnetic fields, such as has been reported in reference (18) for Kemite. APPENDIX A Phenomenological Equations for a Two Level System In order to determine the signal to be. expected from a radio-frequency transition using an induction spectrometer, it is necessary to calculate the component of radio-frequency magnetization parallel to the receiver coil. The magnetization will be proportional to the expectation value of the appropriate spin operator. In a many level system it is convenient to treat only,those two levels between which transitions, are to be observed. Then we may write the eigenfunction of the two level system as f.. a(t)ti + b(t) *f2 where ^fj and ^2-are the eigenfunctioUsof the steady-state Hamiltonian Mo due t» the time independent electric and magnetic interactions. The total Hamiltonian of the system will be given by where /V represents the time-dependent interaction between the magnetic dipole moment of the.nucleus and the applied rf magnetic field. The following relations may be written % 'fx " E - l f i : %>f% = E 2 ^ 2 ; ^o " El - B 2 . The.magnitude.of the rf magnetic field is given by H = 2HX xos^t Thus ?/ = y^I-H^ = YJ\(IXHJ + IyHy^-IzHz') We also define the quantities P, S and M as follows (Yi*/ ixlfi) =Of2* | k \ f i ) = p (fl*l Iy|f2) ="(t2*l lyiTl) = i S < f l * l I z l f 2 ) = <f2*l i z l f l ) ^ Ix, Iy and I z are the spin angular momentum operators when z is the direction of quantization. Diagonal elements of /¥~ will not be effective in producing transitions, so we may assume that all other matrix elements, than the ones included in P, S and T produce negligable effects. We write the time dependent part of the expectation value of operator:. 0 as. 0. Then f x = P(a* b b* a) Iy = iS(a* b - b* a) and we note that the population difference between the two states is given by n = a* a - b* b The. time derivatives of a and b may be obtained in terms of P, S, T, E^, E'2, a and b by use of Schroedingers equation iHf-.Tff since the eigenstate.of the two level system must be a solution of Schroedingers equation^. Using the above mentioned time derivatives of a and b, we obtain expressions for the time derivatives of I x, Iy ,and n, which reduce to the equation dl_ = ^  I XH for nuclei of spin 1/2. dt dlx/dt = u)0Ty P/S.- 2PS;n fiiy - \Ix/T2 ] dfy/dt - - L0oTx S/P +2n ^S(PHX + THZ) -{Ty/T2] dn/dt = 2 3T[Hyrx S/P - (PHX + THZ) Iy/S - ^ n - n0)/T1 j The terms in curly brackets have been added phenomenologically following Bloch. The number n 0 represents, the population difference in the absence of the radio-frequency field. We will consider the particular case where the rf magnetic field, is parallel to the x axis. H z = 0 H y = 0 H x = 2^003001 For steady state conditions, n* is constant in time. The experimentally observable component of magnetization will be parallel to the y axis, so we are interested in Iy. Solving the equations we obtain I = 2T0 PS. n y Hi ( _ t . , i y > , , 0 1 n \ T 2 ( ^ ~ Hi) sin^t - cbs^t ( l±(u)-u)0pT22 I 2 V ° ' J Using the approximations ' A2 {i0o2 + & -2td0{«J-*0)+l_ 2u?0(uJ-us0) T 2 2 T 2 2 A similar solution may be obtained using H z ;= 0 H x - 0 and H^ = 2H^  cos^.t. Other cases,lead to more complex solutions which will not be reproduced here. For transitions between pure spin states for P = S, the solution reduces to the result obtained from the Bloch equations. APPENDIX B Notes on Diagonalization of Matrices The most efficient procedure for obtaining eigenvalues of a symmetric matrix is not to solve the sixth order secular equation, but to diagonalize the matrix directly, using the Jacobian method. The Jacobian method is an iterative procedure which diagonalizes one 2X2 sub-matrix at a time. For a symmetric matrix, this is particularly easy, for consider the matrix A To see how the method is applied to a larger matrix, suppose we wanted to do one step in the diagonalization of the matrix for /T^ on page 9. We choose the largest off-diagonal element which in this case is K and call it w. Then: Then U*AU is the diagonalized matrix, where x is N and y is -N • Tan 2 (9 = — > ^ 2N o< = cos (f /3- sin The process always.converges, since the spur of the matrix, and the spur of the matrix squared are conserved, and each step shifts one of the off-diagonal elements into the spur. The procedure is ideal for a machine, since it may be repeated-over and.over againwith no new instruction. The. machine has only to decide which off-diagonal, element is largest at each step. It must also stop when the off-diagonal.elements have been reduced to a sufficiently, small.value. Now computors generally have a quite limited number of operations which they can perform. The computer recently installed by the University.of British Columbia, ALWAC n i v e a u add-or subtract, multiply, divide, compare numbers, look up numbers in its magnetic memory drum on being given the proper command, besides, being able to type out numbers which are stored in certain parts of its memory. It can-also choose between certain alternate operations depending on whether certain numbers are positive or negative. Information and commands are used by the machine in. "hexadecimal" code . The machine can be made to translate from the hexadecimal code to ordinary decimal 1 0 0 0 0 0 0 1 0 0 0 0 U = 0 0 o< -/3 0 0 0 0 ^  <=* 0 0 0 0 0 0 1 0 0 0 .0 0 0 1 numbers and vice versa. In order for the machine to do a useful computation, a program of the steps to be performed must be prepared. The programs for many of the standard calculations, such as the calculation of exponential and trigonometric functions are stored in the machine's memory and are available at any time. . A program for diagonalizing matrices up to order 8X8 by the Jacobian method was written by Dr. J. M. Daniels, and stored in the memory of the computer. The program consists of over one hundred eight-digit "words" in hexadecimal code. In the course:of a computation, the machine does many more than one hundred steps, as the program is so written that many/of the steps are repeated over and over. Once the program is correctly stored in the memory (one always checks this by solution of a simple standard matrix), the. procedure for diagonalizing a matrix is extremely simple. The program is moved from the main memory, storage.to the working storage.by simply typingout on.the control typewriter the command 4412. After a space, the order of the matrix is typed in, 6 in our case. Then the 36 numbers composing the matrix are fed into the machine, and last, a number is typed in which will be the maximum allowed size of off-diagonal elements, in the diagonal matrix. One then types in a space and the.calculation starts. After from three to five minutes, the elements of the diagonalized matrix.are typed out in order by,the machine and it is ready for another calculation. Plate VII shows the record obtained for the diagonalization of a particular matrix. The case shown is for d= 5°, R » 1.52 andH-= 0.933. P L A T E VII v kkU 6 -12855 7376 0 -1*81 0 0 7376 -27571 9896 -2150 -1987 0 0 9896 17713 0 -2I5O -1*81 -1H81 -2150 0 -27713 9896 0 0 -1987 -2150 9896 -12*29 7376 0 0 - l * 8 l 0 7376 6285510 -10772 1 -10 -1 -31965 -10 1997* 1 -33301 -1 1 -753* -1, 63631! Facing page 70 The block.of numbers at top is the matrix which is to be diagonalized and the column below contains the elements of the diagonalized matrix as they were typed out by the machine. The actual position of each term in the column may be determined by noting that the machine types the elements.of each row of the matrix in turn. It will be noted that the off-diagonal elements are no longer completely symmetric. This is merely due to round-off errors and does not indicate that anything.has gone wrong. APPENDIX C Experiments on Bridge Type Spectrometers After it was realized that the induction signal could disappear for transitions.involving quadrupole interactions, some experiments were undertaken to try and find resonances using the absorption of energy from one arm of a radio-frequency bridge. It seems reasonable.to expect that the oscillator noise should.be balanced.out.here and most of the . advantage of a crossed-coil spectrometer would be obtained without the disadvantage of low signal voltage for ^2 at R = 0.7. Since field sweep was being used instead of frequency sweep the chief argument against bridges, that a bridge is hard to keep in balance over a range of frequency, becomes unimportant. Several different types of bridges were built and tested including those described by Soutif (37), Anderson (3%) and Torrey (39), as well as some modifications that seemed to be worth trying. In every ease, the noise from the bridge was at least five times higher than would be obtained from pure Nyquist noise. The nOise was lowest when bridges of a symmetrical type were used, and also quadrature phase unbalance gave better results than a slight in-phase unbalance, but when the oscillator power was increased to a power level comparable to that used in the inductinn spectrometer, the signal to noise ratio obtainable was such that L i 7 signals in spodumene could scarecely be detected at 700 kc. The attempt to see resonances using a bridge was finally given up as. a waste of time. Probably the chief cause of the noise is.that high currents in the tank circuits of the bridge lead to noise, which cannot be balanced out. REFERENCES 1) Frisch, R. and Stern, O., Z. Phys. 85, 4 (1933) 2) Rabi, I. I., Millman, S., Kusch, P. and Zacharias, J. R., Phys. Rev. 55, 526 (1939) 3) Lasarew, B..G. and Schubnikow, L...W., Phys. Z. Sowjet, 11_, 445 (1937) 4) Gorter, C. J., Physica, 3_, 995 (1936) 5) Bloch, F., Hansen, W. W.. and Packard, M. E., Phys. Rev. 69, 127,(1946) 6) Purcell, E . M., Torrey, H. C. and Pound, R. V., Phys. Rev._69, 37 (1946) 7) Casimer, H. B. G., Physica 2, 719 (1935) 8) Bloembergen, N., Nuclear Magnetic Resonance, Martinus Nijhdff, The Hague (1948) 9) Pound, R. V., Phys. Rev. 79, 685 (1950) 10) Dehmelt, H. G. and Kruger, H., Z. Physik, 129, 401 (1950) 11) Wertz, J. E., Chemical Reviews,,Vol. 55, No. 5 (Oct. 1955) 12) Volkoff, G. M., Can. J. Phys., 31, 820 (1953) 13) Petch, H. E., Cranna, N. G. and Volkoff, G. M., Can. J. Phys. 31, 837 (1953) 14) Kruger, H., Z. Physik,130, 371 (1951) 15) Bersohn, R., J.Chem. Phys. 20, 1505 (1952) 16) Cohen, M. H., Phys. Rev. 96, 1278 (1954) 17) Das and Hahn, Nuclear Quadrupole Resonance Spectroscopy. Preprint to Dr. M. Bloom 18) Haering, R. R.and Volkoff, G. M., Can. J. Phys. 34, 577 (1956) 19) Waterman, H..H. and Volkoff, G. M., Can. J. Phys. 33, 156 (1955) 20) Lamarche, G., Master's Thesis, University of British Columbia, Vancouver, B.C. (1953) 73. 21) Bloch,. F., Phys. Rev. 70, 460, (1946) 22) Ingram, Spectroscopy at Radio and Microwave Frequencies, Butterworths Scientific Publications, London, (1955) 23) Haering, R.R., Master's Thesis, University of British Columbia, Vancouver, B.C., (1955) 24) Bloom, M., Hahn, E. L. and Herzpg, B., Phys. Rev ._97, 1699, (1955) 25) Feynman, R. P., Vernon, F.L., Jr. and Hellworth, R. W., Jour.App. Phys. 28, 49, (1957) 26) Volkoff, G. M. and Lamarche, G.,.Can. J. Phys.^2, 493, (1954) 27) Andrew, E. R., Nuclear Magnetic Resonance, Cambridge University Press (1955) 28) Eades, R. G., Can. J. Phys._33, 286, (1955) 29) Volkoff, G. M., Petch, H. E .and Smellie, D. W...L., Can. J. Phys. 30_, 270, (1952) 30) Bayer, H., Z. Phys. 130, 227, (1951) 31) Cranna, N. G., Ph.D. Thesis, University of British Columbia, Vancouver, B .C. (1954) 32) Dehmelt, H. G., Am. Jour. Phys. 22, 110, (1954) 33) Weaver, H. E., Jr., Phys. Rev._89, 923, (1953) 34) Schuster, N.A., R.S.I., 22, 254, (1951) 35) Collins, T. L., Ph.D. Thesis, University of British Columbia, Vancouver, B.C., (1950) 36) Lamarche, G. and Volkoff, G. M., Can. J. Phys. 31, 1010, (1953) 37) Soutif, M. Nuclear Paramagnetism, La Revue Scientifique, P. 203, (1951) 38) Anderson, H. L., Phys. Rev. 76, 1460, (1949) 39) Torrey, H. C ., Phys. Rev. 76, 1059, (1949) 

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