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Some experimental and theoretical studies of magnetic properties.of solids Wong, Samuel Kim Po 1970

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SOME EXPERIMENTAL AND THEORETICAL STUDIES OF MAGNETIC PROPERTIES OF SOLIDS by SAMUEL KIM PO WONG B.Sc , (Sp. Hon.) Un i v e r s i t y of Hong Kong, 1965 M.Sc, Un i v e r s i t y of B r i t i s h Columbia, 1967 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of Physics We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA May, 1970 In p r e s e n t i n g t h i s t h e s i s i n 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 an advanced degree a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I ag ree t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r ag ree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my Depar tment o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l no t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Depar tment The U n i v e r s i t y o f B r i t i s h Co lumbia Vancouver 8, Canada Date Y^j, X4>; 1 1 1 * i Abstract This thesis i s concerned with three aspects of magnetic properties in solids. 1. The mean square value <I(I + 1)> of the proton angular momentum per molecule has been measured to be 3.73 ± 0.18 in solid CH^ at 4.2 K . The experiment was done by measuring the ratio of the proton magnetic 13 resonance free induction decay signal to that of C at the same frequency 13 in a sample containing 53% CH^ . The measured value of <I(I + 1)> differs appreciably from the high temperature value of <I(I + 1)> =3 and from the value of <I(I + 1)> = 6 when a l l the molecules are converted into the A spin symmetry species. Experiments at lower temperatures using a sample containing 0.002% O 2 impurity indicated that there may be a slight increase i n the value of <I(I + 1)> between 4.2 K and 2.45 K . The presence of 0.05% O 2 impurity shortened the time constant for conversion of C H 4 between different symmetry species. The interpretation of measurements of <I(I + 1)> in the presence of large amounts of 0 2 impurities is complicated by changes in the NMR line shapes. These complications are discussed i n terms of the theoretical results obtained in the second part of the thesis. 2 . The NMR line shape inhomogeneously broadened by paramagnetic impurities in solids was investigated theoretically using the s t a t i s t i c a l method of Margenau. The general expression for the Fourier transform of the line i i shape function was derived. The l a t t i c e can be divided into an inner s p h e r i c a l cut-of f region and i t s complement. Impurities i n s i d e the c u t - o f f region give r i s e to s a t e l l i t e l i n e s and impurities outside determine the shape of the main l i n e and s a t e l l i t e l i n e s . D e t ailed numerical c a l c u l a t i o n s were performed f o r a f . c . c . l a t t i c e when the impurities are (a) c l a s s i c a l magnetic dipoles, (b) spin 1/2 systems and (c) spin 1 systems. An asymmetry i n the l i n e shape i s predicted at f i n i t e temperatures . The peak i n t e n s i t y i s s h i f t e d from the centre by an amount l i n e a r i n the impurity concentration and i n the magnitude of the magnetic moment and the o v e r a l l l i n e shape adjusts i t s e l f i n such a way to give a vanishing f i r s t moment about the centre. The l i n e shape i s always Lorentzian at the centre and Gaussian i n the wings. I t depends on the impurity concentration. At low concentrations, the main l i n e and the s a t e l l i t e l i n e s have the same shape which i s predominantly Lorentzian. The shape of a powdered sample i s also discussed. 3 . General expressions are derived f o r the f i r s t two terms i n the expansion o f the s p e c i f i c heat of a paramagnetic s a l t i n powers of T . I t i s •assumed that the i n t e r a c t i o n between the paramagnetic ions consists of magnetic dipole and i s o t r o p i c exchange i n t e r a c t i o n s , and that the g-tensor o f each ion i s a x i a l and has the same o r i e n t a t i o n f o r a l l ions. The general formulae are evaluated numerically for cerium magnesium n i t r a t e . Comparison with the experimental data of Mess and coworkers i n d i c a t e d that the exchange parameter corresponds to a ferromagnetic i n t e r a c t i o n . i i i LIST OF TABLES Table Page 1. Experimental results for an oxygen-free sample of CH, 13 ' containing 53% CH^ 23 2. Experimental results for a sample of CH^ containing 53% 1 3CH 4 and 0.05% 0 2 34 3. Values of <I(I + 1)> and the residual entropy S Q for different l i m i t i n g cases of the energy levels of F i g . 13 46 4 . Values of £k , P^ , and for a face-centred cubic l a t t i c e 107 5. Values of b 2 and b 3 for single c r y s t a l 6MN . 157 iv LIST OF ILLUSTRATIONS Figure Page 1. A block diagram of the pulsed rf spectrometer 11 2. Plot of proton f.i.d. signal in arbitrary units in a sample of solid methane containing 53% 1 3CH 4 at 4.2°K 12 13 3. Plot of C f.i.d. signal in arbitrary units for the same sample as shown in Fig. 2 13 4. Circuit diagram of the transmitter 14 5. A diagram showing the setup in the helium inner Dewar 17 6. Plot of log[btV^(t)/sin bt] in arbitrary units 2 *• 0 versus t for the proton f.i.d. of Fig.2 21 7. Plot of log[V^(t)] in arbitrary units versus 2 13 t for the C f.i.d. of Fig. 3 22 8. Plot of Sp and S £ in arbitrary units versus time after an 02~free sample of CH^  containing 53% 1 3CH 4 had been cooled to 4.2°K . 26 9 . Plot of S and S in arbitrary units versus 13 time after a sample containing 53% CH^  and 0.05% 0o had been cooled to 4.2©K . 27 V 10. P l o t of Sp i n a r b i t r a r y units versus time f o r ..a sample of CH^ containing 0.002% 02 30 11. P l o t of S i n a r b i t r a r y units versus time f o r 13 a sample of CH^ containing 53% CH^ and 0.05% 0 2 . . 32 12. P l o t of cal c u l a t e d <I(I + 1)> , S , and C v versus B/kT f o r a s p h e r i c a l f r e e - r o t o r 43 13. Energy l e v e l s and degeneracies of the low-lying states of the A, T, and E spin symmetry species of CH^ perturbed by a t r i g o n a l c r y s t a l l i n e e l e c t r i c f i e l d -. •- 45 14. P l o t of ca l c u l a t e d values of • <I(I + 1)> , S , and C v versus T for CH^ i n a tetrahedral c r y s t a l f i e l d 48 15. P l o t of the calculated values of <I(I + 1)> , S , and C v versus T for CH^ i n a t r i g o n a l c r y s t a l f i e l d . 49 16a. P l o t of Real(J(u)) versus u . 77 16b. P l o t of -Imag(J(u)) versus u . 78 17. Line shape function I^(CLI) when pMp(q) =0.1 81 18. Line shape function I^(co) when pMp(q) = 0.5 82 19. Line shape function 1^ (u) when pMp(q) = 2 83 v i 20. P l o t o f R e a l ( J ( u ) ) and R e a l ( J ^ ( u ) ) v e r s u s u when n = (0,0) 86 21. P l o t o f -Imag ( J ( u ) ) and - I m a g ( J 1 ( u ) ) v e r s u s u when n = (0,0) 87 22. P l o t o f R e a l ( J ( u ) ) and R e a l d ^ u ) ) v e r s u s u when n = ( rr /18 , TT/36) 88 23. P l o t o f - I m a g ( J ( u ) ) and - I m a g ( J 1 ( u ) ) v e r s u s u when n = (TT/18, TT/36) 89 24. P l o t o f R e a l ( J ( u ) ) and R e a l ( J 1 ( u ) ) v e r s u s u when n = 2TT/9, 7TT/18) 90 25. P l o t o f -Imag ( J ( u ) ) and -Imag (J-^ (u)) v e r s u s u when n = (2TT/9, 7TT/18) . 91 26. P l o t o f f(w) and g(to) v e r s u s to when a = 1, b = 10, and c = 0.27 . .. 104 27. P l o t o f R e a l ( G ( v ) ) and R e a l ( G 1 ( v ) ) v e r s u s v when n = (0,0) 113 28. P l o t o f -Imag(G(v)) and - I m a g ( G 1 ( v ) ) v e r s u s v when n = (0,0) 114 29. P l o t o f R e a l ( G ( v ) and R e a l ( G 1 ( v ) ) v e r s u s v when n = (TT/4, 0) . 115 30. P l o t o f -Imag(G(v)) and -Imag(G- L(v)) v e r s u s v when n = (TT/4, 0) 116 P l o t of Real ( G(v)) and Real C G^v)) versus v when n = (IT/8, 0) . P l o t of -Imag(G(v)) and -Imag(G^(v)) versus v when n = (TT/8, 0) Values of -w obtained by f i t t i n g Eq. [3.19] to experimental data of Mess et a l . (59) at various temperatures S p e c i f i c heat of CMN as function of temperature v i i i Acknowledgement The work d e s c r i b e d i n t h i s t h e s i s was done under the s u p e r v i s i o n o f P r o f e s s o r M. Bloom and P r o f e s s o r W. Opechowski. To them I w i s h to e x p r e s s my s i n c e r e g r a t i t u d e f o r t h e i r c o n t i n u e d i n t e r e s t and g u i d a n c e . I am i n d e b t e d to P r o f e s s o r S. A l e x a n d e r f o r many s t i m u l a t i n g and f r u i t f u l d i s c u s s i o n s , and to Dr. J . N o b l e , Dr. S.T. Dembinski , and Mr. J . Lees f o r t h e i r a s s i s t a n c e i n t h i s work. The f i n a n c i a l s u p p o r t p r o v i d e d by the Commonwealth S c h o l a r s h i p Committee and by t h e N a t i o n a l R e s e a r c h C o u n c i l i s g r a t e f u l l y acknowledged. F i n a l l y , s p e c i a l thanks a r e due to my w i f e , E l e a n o r , f o r h e r s a c r i f i c e s and encouragement throughout the p r e p a r a t i o n of t h i s t h e s i s . TABLE OF CONTENTS Page ABSTRACT i LIST OF TABLES i i i LIST OF ILLUSTRATIONS i v ACKNOWLEDGEMENTS . . v i i i CHAPTER I NUCLEAR MAGNETIC SUSCEPTIBILITY IN SOLID CH^ AT 4.2°K . 1 1. INTRODUCTION .... '.. ... 1 2. THEORY OF THE EXPERIMENTAL METHOD 4 3. EXPERIMENTAL METHODS 9 4. DISCUSSION OF THE EXPERIMENTAL RESULTS 18 A. Measurement o f < I ( I + 1)>_ f o r Oxygen-free CH^ a t 4.2°K frV. ....... .i- 19 B. Dependence o f Sp on Time and Temperature ^ 25 5. THE MAGNETIC PROPERTIES OF 0 2 IN SOLID CH^ 33 6 . COMPARISON WITH OTHER EXPERIMENTS 40 CHAPTER 2 NUCLEAR MAGNETIC RESONANCE LINE SHAPES BROADENED BY PARAMAGNETIC IMPURITIES IN SOLIDS 54 1. INTRODUCTION 54 2. DERIVATION OF THE LINE SHAPE FUNCTION 57 3. LINE SHAPE FUNCTION DUE TO A SYSTEM OF CLASSICAL MAGNETIC DIPOLES . 70 4 . LINE SHAPE FUNCTION DUE TO A SYSTEM OF SPINS, S = 1 / 2 AND S = 1 9 5 5 . CONCLUSION 1 2 1 APPENDIX 1 2 5 CHAPTER 3 EFFECT OF THE EXCHANGE AND MAGNETIC DIPOLE-DIPOLE INTERACTIONS ON THE SPECIFIC HEAT OF PARAMAGNETIC SALTS AT VERY LOW TEMPERATURES 1 3 6 1. INTRODUCTION 1 3 6 2 . GENERAL THEORY 1 4 1 3. THE CASE OF CERIUM MAGNESIUM NITRATE 1 4 7 APPENDIX 1 5 9 BIBLIOGRAPHY 1 6 2 CHAPTER 1 NUCLEAR MAGNETIC SUSCEPTIBILITY IN SOLID CH. AT 4.2°K 4 1. INTRODUCTION Each of the f i v e i s o t o p i c modifications of methane CH7 D , 4-n n n = 0,1,2,3, and 4, e x h i b i t s two s p e c i f i c heat anomalies i n the s o l i d at low temperatures (1). With the exception of the lower s p e c i f i c heat anomaly of CH^, these anomalies have been demonstrated to be associated with phase t r a n s i t i o n s i n the s o l i d s . To understand these anomalies t h e o r e t i c a l l y , James and Keenan (2) proposed a model which assumes an e l e c t r o s t a t i c octopole octopol i n t e r a c t i o n of neighbouring molecules i n a f . c . c . l a t t i c e . Although the James-Keenan model (2) and i t s quantum mechanical extension by Yamamoto et a l . (3,4,5,6) i s suc c e s s f u l i n reproducing these double t r a n s i t i o n s and s p e c i f i c heat anomalies i n a semi-quantitative manner, the controversy developed over the i n t e r p r e t a t i o n of the s p e c i f i c heat anomaly associated with the lower of the two s p e c i f i c heat anomalies of CH^ remains unsettled (7,8,9). This s p e c i f i c heat anomaly was d i f f i c u l t to detect because of the fac t that hours are required f o r the establishment of thermal e q u i l i b r i u m i n the v i c i n i t y of the anomaly (7). The apparent zero temperature entropy of CH^, which i s based on s p e c i f i c heat measurements extending down to about 2.5°K has been in t e r p r e t e d by Colwell, G i l l , and Morrison (8) i n terms of Nagamiya's c a l c u l a t i o n (10) of the energy l e v e l s of the A, T, and E spi n symmetry species of CH^. The degeneracies of the ground states of the A, T, and E species of CH^ i n a tetragonal c r y s t a l l i n e f i e l d are i n the 2 r a t i o 5:9:2, r e s p e c t i v e l y , which are the same as the high temperature nuclear spin s t a t i s t i c a l weights (11). In a t r i g o n a l f i e l d only the degeneracy of the ground state of the T species i s a f f e c t e d , changing the above r a t i o s to 5:6:2 or 5:3:2 depending on the sign of the c r y s t a l f i e l d parameter. With the assumption that no conversion takes place between the d i f f e r e n t nuclear spin species, Morrison et a l . f i n d that t h e i r r e s u l t s are consistent with a 6-fold degeneracy of the ground state of the T species. The above i n t e r p r e t a t i o n of the zero-point entropy of CH^ has been challenged by Hopkins, Donoho, and P i t z e r (9) who suggest that when CH^ s o l i d i s i n equilibrium at 4.2°K, almost a l l the CH^ molecules are i n the state of the A spin symmetry species, but that the r e l a x a t i o n time for the establishment of e q u i l i b r i u m among the species i s extremely long f o r pure CH^. They base t h i s suggestion on measurements of the v a r i a t i o n of the proton magnetic resonance i n t e n s i t y i n s o l i d CH^ as a function of time f o r samples containing d i f f e r e n t amounts of 0 2 impurity. Such changes had been reported previously by Wolf and Whitney (12). Since the proton angular momentum per molecule i s 1 = 2 , 1 , and 0 f o r the A, T, and E spin symmetry modifications, r e s p e c t i v e l y , and S^ i s p r o p o r t i o n a l to <I(I + 1)> , measurements of Sp can be used to detect changes i n the r e l a t i v e number of CH^ molecules i n the d i f f e r e n t nuclear s p i n symmetry states. Hopkins et a l . (9) cooled samples of s o l i d CH^ to 4.2°K qui c k l y and monitored Sp as a function of time. They found that f o r moderate amounts of 0 2 , Sp increased by approximately 60% over a period of about 24 hours, while f o r samples with very l i t t l e (a few parts per m i l l i o n ) or very much (1%) 0 2 impurity no changes with time occurred. Hopkins et a l . 3 suggest that f o r very l i t t l e impurity, the r e l a x a t i o n time f o r conversion i s much longer than the time of the experiment (24 h r ) , while f o r 1% 0£ the r e l a x a t i o n time i s shorter than the time taken to make the f i r s t measure-ment (tens of minutes). The inference drawn from these r e s u l t s by Hopkins et a l . i s that the s p e c i f i c heat anomaly reported by Morrison et a l . i s associated with spin conversion induced by a small amount of 0 2 impurity (0.005%) i n t h e i r sample. A very extensive s e r i e s of nuclear magnetic resonance measurements was c a r r i e d out on CH^ and i t s deuterated modifications by G.A. de Wit (13, 14, 15). During the course of these measurements, an increase of Sp with time i n s o l i d CH^ cooled to 4.2°K was observed, i n q u a l i t a t i v e agree-ment with the "observation of Wolf and Whitney (12). However, the p o s s i b i l i t y that such changes could be due to almost complete conversion of the CH^ molecules i n t o the A spin symmetry species was never s e r i o u s l y considered. The main reason f o r adopting t h i s point of view was that the s p i n - l a t t i c e r e l a x a t i o n time T-j_ i s a very slowly varying function of temperature between 1.2° and 4.2°K and i s f a i r l y short, which seems completely i n c o n s i s t e n t with the i n t e r p r e t a t i o n of Hopkins et a l (9, 15). We s h a l l return to the s i g n i f i c a n c e of the •T-^  measurements l a t e r i n t h i s chapter. There i s another reason f o r being s k e p t i c a l about experimental demonstrations of conversion between nuclear spin symmetry species s o l e l y on the basis of a long time constant associated with a p h y s i c a l observable at low temperatures. A few years ago Borst et a l . (16) concluded that both s o l i d acetylene (C2H2) and the H2O molecules i n gumdrops converted to the paraspecies. These conclu-sions were based on the time dependence of the t o t a l neutron c r o s s - s e c t i o n 4 i n these s o l i d s a f t e r being cooled to 4.2°K. Subsequent nuclear magnetic resonance measurements (17) between 4.2°K and 1.2°K demonstrated that these conclusions were i n c o r r e c t . There i s an obvious need f o r a more d e f i n i t i v e measurement of <I(I + 1)> f o r the protons i n CH^ s o l i d at low temperatures. What i s required i s a r e l i a b l e measurement of <I(I + 1)> at a time a f t e r the system has established equilibrium. We report here the r e s u l t s of such an experiment. As w i l l be described i n more d e t a i l i n the next two sections, we used a sample 13 of CH^ containing 53% of the C isotope. By measuring the r a t i o of 13 S to the C magnetic resonance i n t e n s i t y S at the same frequency, we P t-obtain the value of <I(I + 1)> for the protons per CH^ molecule i n terms 13 of the known value of <I(I + 1)> = 3/4 f o r the C spins. The r e s u l t s of our measurements for pure methane and methane doped with 0.05% 0 2 are discussed i n Sections 4 and 5. The r e l a t i o n s h i p between the NMR measurements i n CH^ and s p e c i f i c heat and i n f r a r e d studies i s developed i n Section 6. 2. THEORY OF THE EXPERIMENTAL METHOD In t h i s s e c t i o n , we review the p r i n c i p l e of the nuclear magnetic resonance (NMR) experiment f o r determining conversion between nuclear spi n species. The s t a t i c nuclear s u s c e p t i b i l i t y x Q obeys Curie's law, which we write as follows : M 2 2 _ o _  NY ft <I(I + 1)> r , X Q R q 3 k T , L-L.-l-J where M 0 i s the equ i l i b r i u m nuclear magnetization per unit volume i n the 5 the f i e l d H Q and temperature T, N i s number of molecules per cm^, y i s the nuclear gyromagnetic r a t i o and <I(1 + 1)> i s the mean square value of the nuclear angular momentum per molecule. For the case of CH^, each molecule contains four protons of spin 1/2. If the angular momentum vectors of the four protons were uncorrelated, then <I(I + 1)> = 4(1/2)(3/2) = 3. On the other hand, i f each CH^ molecule belonged to one of the A, T, or E spin symmetry species, then we would have I = 2, 1, or 0, r e s p e c t i v e l y , and l e t t i n g P^ be the p r o b a b i l i t y that a molecule belongs to the i spin symmetry species, we obtain <I(I + 1)> = 6P A + 2P T . [1.2] At high temperatures, the A, T, and E spescies should be r e l a t i v e l y populated according to t h e i r s t a t i s t i c a l weights of 5,9, and 2, res p e c t i v e l y , giving <I(I + 1)> = 3, as expected. I f the ground molecular state of the A species were lower than the ground state of any other species by an energy much greater than kT, where k i s the Boltzmann's constant, then <I(I + 1)> = 6, leading to an increase i n x QT by a f a c t o r of 2. I t i s t h i s increase which Hopkins et a l . (9) claim to have observed using NMR techniques. I f a nuclear spin system i n i t i a l l y i n equ i l i b r i u m i s subjected to a r o t a t i n g magnetic f i e l d i n the plane perpendicular to H of magnitude H-^  and angular frequency U3Q = YH Q f o r a time T ^ , the x component of magnetization following the pulse i s given by (18, 19) : 6 M (t) = MQ sin e^osCcOot + <J))G(t) , [1.3] where 0-^  = ^xTl , <J> is a phase angle which depends on the i n i t i a l direction of the rotating field and G(t) is the free induction decay (f.i.d.) function. For a proper choice of the origin of the time parameter t, G(0) = 1. For example, in the limit that T-^  approaches zero, t = 0 corresponds to the time at which the pulse is applied. More generally for finite ^ t = t' + fx± , 0 <_ f <_ 1 , [1.4] where t' is the time measured from the end of the pulse. It has been demonstrated experimentally and theoretically (19) that under quite general conditions, f = 1/2 represents a proper choice of the origin of t. It has been found experimentally (19) that with this choice of origin, the f.i.d. shape G(t) is undistorted even for quite large values of T^, so that G(t) is the Fourier transform of the absorption line shape function g(u), where u = UJ - t o o , and we can write (18) •00 G(t) = g(u)e~ 1 U t du n=2 where Mj^  is the nth moment of the line shape. We are interested here in the experimental determination of MQ . Th e effect of M (t) is to induce a voltage V.(t)sin(co t + $) in a coil 7 having i t s axis p a r a l l e l to the x-axis. This voltage can be amplified and detected i n usual way (20) to give a f . i . d . voltage V Q ( t ) given by V Q ( t ) = A(t)BM osin0 1 G(t) = A ( t ) V i ( t ) , [1.6] where B i s a parameter which depends on L O 0 , T and the geometry of the c o i l and the sample, and A(t) i s the r a t i o of the output to the input voltages. I t i s time dependent because the detector i s nonlinear, i n general, and because of the f i n i t e time required by the re c e i v e r to recover from the influence of the large r f pulse. As w i l l be discussed i n the next s e c t i o n , the measured V ( t ) was compared d i r e c t l y with a stable reference voltage applied at the sample c o i l through a c a l i b r a t e d attenuator. Therefore, many instrumental d i s t o r t i o n s were, i n p r i n c i p l e , eliminated and we e s s e n t i a l l y measured the quantity V Q ( t ) / A ( t ) . In p r a c t i c e t h i s statement i s correct only f o r t greater than a c e r t a i n time t . For t < t , the influence J O m m ' of the pulse i s s t i l l so large that t h i s experimental procedure may introduce appreciable systematic e r r o r s . Now, i f the form of G(t) were known, the measurements of V^(t) for t > t could be exptrapolated back to the o r i g i n , where G(0) = 1. In p r a c t i c e , G(t) i s never known t h e o r e t i c a l l y i n NMR, since the line-shape problem i s s t i l l unsolved. I f t i s not too large, however, i t i s possible to make a meaningful empirical extrapolation to the o r i g i n of t, aided by the fac t that the slope of G(t) must be zero at the o r i g i n . I f 8 independent knowledge of the second moment M 2 i s a v a i l a b l e , we see from 2 2 Eq. [1.5] that the behavior of d G/dt near the o r i g i n provides a check of the consistency of the extrapolation procedure, and enables one to estimate the error involved i n such an estimate of V^(0). The e x p l i c i t empirical extrapolation procedures we have followed f o r obtaining the values of V^(0) 13 f o r the protons and the C n u c l e i , which we denote f o r s i n 0^ = 1 by Sp and S c , r e s p e c t i v e l y , w i l l be described i n Section 4. 13 Since both the proton and C NMR experiments were done at"the same frequency and temperature and with no change i n the sample geometry, the quantity B was the same f o r each and we obtain, using Eqs. [1.1] and [1.6] and the d e f i n i t i o n of S and S , P c Y S <I(I + 1)> = q I^E- [1.7] p c 13 where q i s the f r a c t i o n of CH^ molecules having the C isotope and 13 we have used the f a c t that the f i e l d at which the C resonance was observed was l a r g e r than the proton resonance f i e l d by a f a c t o r Y /Y . Using the p c 13 known values of Y f o r the proton and C , which give ^^/Y^ = 0.25, and the value q = 0.53 f o r our sample, we have <I(I + 1)> = 0.099 S / S . [1.8] \ * p ' c Comments on the use of Steady-State Techniques. Another way to measure x D using NMR i s to measure the "out of phase" component of the nuclear s u s c e p t i b i l i t y x " ( u ) a s a function of 9 u =. W - ai . I f the amplitude of the r f f i e l d i s s u f f i c i e n t l y s m a l l that x"(u) i s independent of H-^, i t may be shown r i g o r o u s l y (16) that Xo = C foo X " ( u ) du , [ 1 . 9 ] —00 where C i s a constant. The pulse experiment analog of the i n t e g r a t i o n from -oo to +oo i n Eq. [1.9], which i n v o l v e s an e x t r a p o l a t i o n of the measured value of x"(u) over a f i n i t e range of u to ±oo , i s that the d e t e c t o r i n the pulse experiment must have a s u f f i c i e n t l y wide bandwidth and the measurement must be made s u f f i c i e n t l y soon a f t e r the pulse as compared w i t h a time which i s roughly the i n v e r s e of the width of the resonance of x"(u)» Hopkins, Donoho, and P i t z e r (9) measured dx"(u)/du as a f u n c t i o n of u. They defined the i n t e n s i t y of the NMR l i n e as the product x"( u)^ > where x"(0) w a s obtained by i n t e g r a t i n g dx"(u)/du from the wings of the l i n e to u = 0, and 6 i s the s e p a r a t i o n i n frequency u n i t s between the maximum and minimum p o i n t s i n the p l o t of dx"(u)/du versus u. S t r i c t l y speaking x"(0)<5 I s p r o p o r t i o n a l to x o only i f the l i n e shape x"(u) does not change. Presumably, t h i s c o n d i t i o n was s a t i s f i e d i n the experiments of Hopkins et a l . , but they have not commented on t h i s p o i n t i n t h e i r paper. 3. EXPERIMENTAL METHODS The NMR measurements reported here were c a r r i e d out at a frequency of 22.1 MHz corresponding to a resonance f i e l d of 5180 gauss f o r protons and 13 of 20500 gauss f o r C. The magnet was a Magnion 15-inch electromagnet 10 (Model L-158) with a 2.25-inch gap. The f i e l d could be set quite close to resonance by s e t t i n g the magnetic f i e l d regulator (Magnion HS-10200) d i g i t a l l y to the estimated value. F i n a l adjustment of the f i e l d was made by monitoring the NMR s i g n a l . A 90° pulse ( 0 - ^ = TT/2) was set up by maximizing the f . i . d . s i g n a l following the pulse and noting that t h i s corresponded to a f . i . d . s i g n a l of amplitude close to zero following a pulse applied a few mi l l i s e c o n d l a t e r . The Pulse Spectrometer A block diagram of the pulse spectrometer i s shown i n F i g . 1. The timing unit consisted of a Tektronix 162 wave form generator, which c o n t r o l l e d the pulse r e p e t i t i o n rate, and a Tektronix 163 pulse generator to c o n t r o l the width of the r f pulse. Under optimal conditions of operation, a 90° pulse 13 had a width of about 5 usee f o r protons and about 13 usee f o r C. I f the 13 value of f o r the proton and C studies were the same, the r a t i o of the pulse widths would be about 4 since Y^/Yc = 4. However, the damping r e s i s t o r R^ i n the sample c o i l tuned c i r c u i t was adjusted f o r each resonance to give a n e g l i g i b l e d i s t o r t i o n of the f . i . d . shape. Since the f . i . d . s i g n a l 13 for the protons decayed more r a p i d l y than the C s i g n a l , as shown i n F i g . 2 and F i g . 3, a smaller value of R^ had to be chosen f o r the protons, thus r e s u l t i n g i n a smaller value of H-^ . The bandwidth of the p r e a m p l i f i e r (Arenberg PA620-SN) could be adjusted separately. The wide band a m p l i f i e r was an Arenberg WA-600D. The transmitter was a pulsed electron-coupled Hartley o s c i l l a t o r followed by one class A a m p l i f i e r and three class C a m p l i f i e r s . Its c i r c u i t diagram i s shown i n F i g . 4. The detector was a h a l f wave diode detector. Y o OSCILLOSCOPE SYNCH? TIMING UNITS DETECTOR WIDE BAND AMPLIFIER P R E -AMPLIFIER REFERENCE SIGNAL ATTENUATOR GENERATOR FD100 10Pf F i g . 1. A b l o c k d i a g r a m o f the p u l s e d r f s p e c t o r m e t e r . t ( u S ) 13 1 Fig. 3. Plot of C f.i.d. signal in arbitrary units for the same sample as shown in Fig. 2. The shaded area represents rf pulse. 12AU7A 6 D K 6 6 L 6 6 L 6 pulse input ->+300v 100uh 39K't 220pf . l l H J 0 0 P h ^ -02 47k r. f . 25p f output feH I—f 8 2 9 B >1-5k •05 + 1 2 0 0 V + 7 5 0 v 6 L 6 >+500v - 1 5 0 V - 5 0 v ^ Fig. 4 . Circuit diagram of the transmitter. 15 Measurement of the Signal The r e c t i f i e d f . i . d . s i g n a l was displayed on an o s c i l l o s c o p e and i t s amplitude V Q ( t ) was measured at a preselected time following the r f pulse. Immediately before and immediately a f t e r t h i s measurement a reference s i g n a l generated by a Tektronix 191 constant amplitude s i g n a l generator was measured, the average of these two measurements being used to c a l i b r a t e the f . i . d . s i g n a l . Before each set of measurements the Hewlett Packard VHF attenuator (Model 355A, B) was adjusted so that the amplified reference -s i g n a l was within a few percent of V Q ( t ) . This was done using the zero o f f s e t feature of the Type Z plug-in of the Tektonix 531A o s c i l l o s c o p e . The procedure described above enabled us to measure V^(t) i n terms of the attenuator s e t t i n g and the output voltage of the reference s i g n a l generator, which was very stable over long periods of time. Since the c o r r e c t i o n due to the d i f f e r e n c e i n the amplitudes of the reference s i g n a l and V Q ( t ) was only a few percent, the properties of the a m p l i f i e r did not enter i n t o the measurement i n an important way. I t was found e m p i r i c a l l y , however, that t h i s procedure broke down f o r t < t , as indic a t e d i n Figs. 2, and 3. For these short times, the d i s t o r t i o n of the s i g n a l by the a m p l i f i e r was found to be d i f f e r e n t f o r the reference s i g n a l , which was present both before and a f t e r the pulse, and the f . i . d . s i g n a l , which or i g i n a t e s at the time of the pulse. Because t h i s d i f f e r e n c e depended on the tuning of the c o i l and on R^, i t was decided that only data f o r t > t m were meaningful. The analysis given i n the next s e c t i o n i s based completely on the data f o r t > t , by which time the inf l u e n c e of the r f 16 pulse on the a m p l i f i e r had become n e g l i g i b l e . 13 Samples. The samples of methane containing 53% CH^ were 13 supplied by Merck, Sharpe, and Dohme of Canada, who also measured the C concentration. The oxygen-free samples were prepared using a g e t t e r i n g technique (21) and then sealed i n a glass tube of length 21 cm and diameter 0.9 cm f o r the bottom 13 cm and 1.8 cm f o r the top 8 cm. The measurement of the 0 2 concentration i n the doped samples was based on the assumption that 1/T-^  was a l i n e a r function of the 0 2 concentration (21, 22). The sample containing 0.002% 0 2 according to t h i s c r i t e r i o n had a value of T-^  = 6.3 sec at 105°K and was taken d i r e c t l y from the c y l i n d e r of research grade methane supplied by P h i l l i p s Petroleum Company. The sample containing 0.05% 0 2 had a value of T± = 0.25 sec at '77°K. Cryogenics. The low-temperature system was a conventional double Dewar system. The setup of the inner helium Dewar and the sample tube i s shown i n F i g . 5. The samples were f i r s t cooled slowly to 77°K over a period of several hours by f i l l i n g the outer Dewar with l i q u i d nitrogen. Then l i q u i d helium was transferred to the inner Dewar where i t was i n d i r e c t contact with the sample. The upper part of the sample tube was wrapped with foam rubber padding to ensure that, during the helium t r a n f e r , the s o l i d CH^ sample at the bottom of the tube was cooled f i r s t . Otherwise i f the CH^ vapour was f i r s t cooled and condensed, the high vapour pressure of the s o l i d CH^ at 77°K would cause most of the sample to be "pumped" up and to condense i n the upper region away from the sample c o i l , r e s u l t i n g i n a greatly reduced f . i . d . s i g n a l . The temperature was determined from the HE TRANFER SIPHON .15 F I T T I N G —I HE PUMPING h AND RETURN 4; LINE 17 R F INPUT TERMINAL KOVAR SEALS FOR ELECTRICAL LEADS SAMPLE TUBE HOLDER TEFLON SPACER RADIATION SHIELDS METAL SHIELD FOAM RUBBER PADDING SAMPLE COIL F i g . 5 . A d i a g r a m s h o w i n g t h e s e t u p i n t h e h e l i u m i n n e r D e w a r . 18 helium vapour pressure. Of course, an e s s e n t i a l feature of the technique used here to measure the proton <I(I + 1)> as a function of time and 13 temperature was that the C spins served as a secondary thermometer which had the same s p a t i a l d i s t r i b u t i o n i n the sample as the proton spins. 4. DISCUSSION OF THE EXPERIMENTAL RESULTS We s h a l l discuss two types of experimental r e s u l t s . The f i r s t 13 type was performed on an 02~free sample of CH^ containing 53% CH^. The other type was performed on a pure CH^ sample containing 0.002% 0 2 13 and on a CH^ sample containing 53% CH^ and 0.05% O2. From measure-ments of Sp and S^ made over periods of approximately 24 hr, i t was possible to determine <I(I + 1)> as a function of time. Measurement of <I(I + 1)> required a determination of the f . i . d . shape function G ( t ) , as w i l l be described below, which took a long time. As a consequence, r e l i a b l e measurements of <I(I + 1)> were only made f o r times greater than or of the order of a few hours a f t e r cooling the sample to 4.2°K. We found that, within experimental error, <I(I + 1)> was independent of time under these conditions. On the other hand, large increases i n S^ were normally observed during the f i r s t two or three hours. These increases were much more rapid f o r the sample containing 0.05% O2. Assuming that the l i n e shape does not change s i g n i f i c a n t l y during the course of such measurements, they can be used to determine the time-dependence of S . Occasional checks of r e l a t i v e F p values of V^(t) for a few values of t i n d i c a t e d that t h i s assumption was j u s t i f i e d . 19 A. Measurement of <I(I + 1)> f o r Oxygen-free CH^ at 4.2°K 13 We p l o t t y p i c a l measurements of V.(t) f o r the proton and C 13 resonances f o r CH^ with 53% CH^ i n F i g s . 2 and 3, r e s p e c t i v e l y . As discussed i n Sections 2 and 3, we found,that f o r t < t , the experimental c o r r e c t i o n f a c t o r A(t) could not be determined r e l i a b l y . The values of t m > which depends on the pulse width and the value of R^, are in d i c a t e d 13 i n F i g s . 2 and 3 f o r the proton and C, r e s p e c t i v e l y . Our problem i s to extrapolate V,- (t) to t = 0 for each case to obtain S and S . An v 1 p c adequate f i t of the short time behavior of the proton f . i . d . shape i s given by (18, 23) 2 2 . „, \ , a t N s i n b t r i ^  t G(t) = exp(- ~ 2 — ) — ^ — , [1.10] where b i s found from the f i r s t zero at t = t Q by pu t t i n g bt = TT . This form of G(t) has been found to y i e l d accurate values of the second moment M 2 f o r the case of CaF 2 even though the f . i . d . zeros are found experiment-a l l y to be unequally spaced, contrary to the p r e d i c t i o n s of Eq. [1.10] (23). This i s not s u r p r i s i n g as the value of M 2 i s not influenced appreciably by the behavior of G(t) at long times. We followed the procedure of Barnaal and Lowe (23) to obtain the values of a and b, which y i e l d the second moment M 2 = a 2 + b 2 / 3. [1.11] 20 The parameter a was obtained from the slope of the p l o t of log [btV^Ct).sin bt] versus t 2 as shown i n F i g . 6. The same procedure 13 was followed for the C resonance, as shown i n F i g . 7, though t h i s i s a simpler case since V^(t) i s gaussia'n, i . e . b = 0. Using the V^(t) date f o r t < t < t-, and the measured values of a and b, values of S = V.(0) m o » l 13 were found f o r the proton and C resonances, r e s p e c t i v e l y . The values of <I(I + 1)> l i s t e d i n Table 1 were obtained from the experimental r e s u l t s using Eq. [1.8]. We postpone a d e t a i l e d discussion of the s i g n i f i c a n c e of the experimental value of <I(I + 1)> = 3.73 ± 0.18 u n t i l Section 6. For the present discussion, r e c a l l that i f no conversion between the nuclear s p i n symmetry species takes place at low temperatures, then <I(I + 1)> = 3, while complete conversion to the A species gives <I(I + 1)> = 6. Both these a l t e r n a t i v e s are highly improbable, according to our r e s u l t s . Accuracy of the Measurements of <I(I + 1)> and Systematic Errors The maximum error i n each of the i n d i v i d u a l determinations of <I(I + 1)> was about ± 10%. This estimate was made by drawing extreme l i n e s through the data of the type shown i n F i g s . 6 and 7 to obtain maximum i . . i r r- i • .max . „min . and minimum values of S corresponding to S and S , r e s p e c t i v e l y . r -j i . ' max/f,min , _min._max , t The range of values between S^ /S c and /S c corresponds to an uncertainty of ± 10%. The s c a t t e r i n the most probable values obtained i n the three runs i s quite consistent with t h i s "maximum uncertainty". o o c CO O 0 m 4 6 t 2 ( x l O O ( u S ) 2 ) 2 • 13 F i g . 7. P l o t o f l o g [ V i ( t ) ] i n a r b i t r a r y u n i t s v e r s u s t f o r the C f . i . d . o f F i g . 3 8 23 Run < I ( I + 1)> M°2 M 2 / M 2 __8 -'2 1 r i 8 -2 10 sec 10 sec Nov. 14 3.75 39.1 1.21 32.4 Nov. 17 3.50 36.7 1.17 31.4 Nov. 20 3.93 31.5 0.90 35.0 Average 3.73 ± 0.18 35.8 1.09 33.0 Table 1. Experimental r e s u l t s f o r an oxygen-free sample of CH^ containing 13 53% CH 4 24 A l l of the pos s i b l e systematic errors we have so f a r considered are smaller than the s t a t i s t i c a l e r r o r of about 5% given i n Table 1 on the basis of three runs. Our experimental procedure of comparing the f . i . d . s i g n a l d i r e c t l y with a reference voltage fed through the same a m p l i f i e r serves to eliminate many t y p i c a l systematic errors associated with a m p l i f i e r n o n l i n e a r i t y and d r i f t . The remaining systematic errors to be considered involve the extrapolation of V^(t) to t = 0. The f i r s t point to note i s that the c o r r e c t i o n f a c t o r involved i n such an extrapolation i s of the order ? 13 of exp [M2t^/2] = 1.25 and 1.05 f o r protons and C, r e s p e c t i v e l y . An error i n M 2 or the o r i g i n of t of 20% gives a change i n and S c of only a few percent. There remains the p o s s i b i l i t y that the f . i . d . shape does something unexpected f o r t < t m . If t h i s occurred, the value of M 2 would have to be much greater than the value we obtained, as may be seen from Eq. [1.5]. This p o s s i b i l i t y i s pretty w e l l eliminated by the f a c t that our value of agrees with the value obtained by J.E. P i o t t of the Uni v e r s i t y of Washington (24) and i s only 15% lower than that obtained by Wolf and Whitney (12) using d i f f e r e n t techniques. Also, the r a t i o of the 13 proton and C second moments i s very close to the t h e o r e t i c a l value assuming that the dominant cont r i b u t i o n arises from intermolecular d i p o l a r i n t e r a c t i o n with protons on neighboring CH^ molecules. In th i s case (18) [1.12] which i s to be compared with the average value of 33 given i n Table 1. 25 B. Dependence of Sp on Time and Temperature The v a r i a t i o n of Sp with time a f t e r the methane had been cooled to 4.2°K was not completely reproducible, as can be seen from one strange cooling curve which w i l l be discussed l a t e r i n t h i s subsection. Apart from that strange r e s u l t , we can say that normally there was very l i t t l e change i n S or S with time a f t e r the f i r s t few hours at 4.2°K. Some t y p i c a l p c v a r i a t i o n s of Sp and S £ with time are shown i n F i g s . 8 and 9. For the oxygen-free sample Sp increased r a p i d l y during the f i r s t h a l f hour or so followed by a further slower increase of about 25% during the next two or three hours to a nearly constant value as shown i n F i g . 8. The magnitude of t h i s change and the time constant associated with i t are i n agreement with the observations of Wolf and Whitney (12). We b e l i e v e that the i n i t i a l rapid increase i n s i g n a l was associated with cooling of the sample to 4.2°K while the slower increase was p r i m a r i l y , though not n e c e s s a r i l y completely, associated with conversion between nuclear spin species. The approach of Sp to a time-independent value i s nonexponential. This i s not s u r p r i s i n g since we are concerned with conversion among three d i f f e r e n t nuclear spin symmetry species. I f the approximately 25% increase i n s i g n a l were associated with a spin conversion at 4.2°K as suggested by Hopkins et a l . (9), t h i s would correspond to an increase i n <I(I + 1)> over i t s high temperature value by a f a c t o r of about 1.25. This indicates a value of <I(I + 1)> ~ 3.75, i n agreement with our measured values of <I(I + 1)> from Sp/S c, and with measurements of J . E . P i o t t (24). The time-dependence of Sp f o r the sample containing 0.002% oxygen was s i m i l a r to that of the previous sample. As fo r the sample containing 0.05% 0 9 and 53% CH, the corresponding M 0 UJ Q Z) • 1-00 CL < < 0 0-90 to 0-80 6b C P o ° - s p A - s c 0 4 8 12 16 20 T I M E ( HOUR ) 24 2 6 Fig. 8. Plot of S and S in arbitrary units versus time after an 02-free sample of CH^  13 containing 53% CH^  had been cooled to 4.2°K 1-1 0 L i Q H 1-00 _ l Q_ < < o 0-90 o •o° A 0 oo o A l o - Sr 8 12 16 T I M E ( HOUR ) 2 0 24 13 Fig. 9. Plot of Sp and S £ in arbitrary units versus time after a sample containing 53% CH^  and 0.05% 0 2 had been cooled to 4.2°K . N5 28 increase i n Sp occurred much more r a p i d l y as shown i n F i g . 9, i n agreement with Hopkins et a l (9). Here Sp increased to i t s nearly constant value w i t h i n an hour or so. Also, the measured value of <I(I + 1)> from S /S P t-was higher than that of the C^-free sample. The s i g n i f i c a n c e of t h i s r e s u l t w i l l be discussed i n more d e t a i l i n Section 5. We have analyzed the time-dependence of the r a t i o R(x) = f ^ p / ^ c ^ T as a function of time T a f t e r cooling the sample to 4.2°K and a f t e r the increase i n s i g n a l discussed above had been completed. We f i t t e d the data of f i v e separate runs to the formula R(T) = R ( T ± ) [ 1 + r(x - T x ) ] [1.13] for x > x^, where x^ was t y p i c a l l y choosen to be about 4 hr, and a t y p i c a l s e r i e s of measurements extended over 24 hr, as shown i n F i g s . 8 and 9. The r e s u l t s are consistent with r <_ 0.0012 hr I f the s i g n a l i s p r o p o r t i o n a l to <I(I + 1)> and we assume that <I(I + 1)> i s very slowly ( i . e . r(x - x^) << 1) r e l a x i n g towards an equilibrium value of < ^ ^ + - ' - ^ > e q U i i with a time constant x-p then i t i s easy to show that <I(I+1)> .-, - <I(I+1)> e q u i l r = < I ( i + i ) > x • [ 1 > 1 4 ] x. I l I f the hypothesis i s made that <I(I + l ^ g q y - Q = >^ corresponding to complete conversion to the A species and <I(I + 1)> = 3.73, then the upper l i m i t T i of r < 0.0012 hr gives > 600 hr. In order to support the hypothesis that the e q u i l i b r i u m state at 4.2°K corresponds to most CH^ molecules 29 being i n the A spin symmetry species, one would have to f i n d a mechanism f o r allowing p a r t i a l conversion over a few hours, furt h e r conversion then being i n h i b i t e d f o r periods of more than 2 weeks. This p o s s i b i l i t y seems to be excluded by our experiments on the sample containing 0.05% O2. As w i l l be disscussed i n Section 5, we can only place an upper l i m i t on <I(I + 1)> from our experiments on that sample. This upper l i m i t i s a value of about 4.4 which seems to r u l e out the value of 6 f o r <I(I + 1)> ., since conversion i s presumably rapid i n t h i s sample e q u i l Furthermore, measurements of Sp as a function of temperature i n t h i s sample in d i c a t e that <I(I + 1)> does not change appreciably with temperature since we found that SpT was about constant between 4.2 and 1.05°K. This statement i s q u a l i f i e d i n an important way i n Section 6. We have also demonstrated that the time constant of a few hours f o r the i n i t i a l change i n S^ at 4.2°K i s not due to a v a r i a t i o n of the temperature of the sample. We cooled a sample of CH^ with 0.002% 0 2 to 4.2°K and monitored Sp as a function of time f o r a period of almost 6 hr. A f t e r t h i s time, the temperature of the system was reduced to 2.45°K. As shown i n F i g . 10, the s i g n a l then increased by a f a c t o r of 4.22/2.45 = 1.78, wit h i n experimental e r r o r . The quantity SpT i s seen to be unchanged w i t h i n the experimental accuracy when the sample i s cooled. The curve shown i n F i g . 10 corresponds to the smooth curve drawn through the data obtained by Wong et a l . (25) using the same sample. The data was f i t t e d to a s i n g l e point. The purpose of t h i s i s to demonstrate whether there i s any i n d i c a t i o n of a change of <I(I + 1)> with decreasing temperature, as revealed by an 1-1 o — 1-00 0-90 2 4 5 °K Q o Q o o 0 o o o 0 q 0 0 < 0 0 o o o o O 0 OOo° ooo o o o ° o o P o ° o 0 ° o o o v Oo° 0 0 6 8 10 12 14 16 18 2 0 22 T I M E ( HOUR ) F i g . 10. Pl o t of S p i n a r b i t r a r y units versus time f o r a sample of CH^ containing 0.002% 0 2- A f t e r almost 6 hr, the temperature was reduced from 4.22°K to 2.45°K. Then (2.45/4.22)S p i s p l o t t e d f o r the data taken at 2.45°K. The smooth curve corresponds to the data of Wong et a l . (25). The apparatus was being repaired during the time i n d i c a t e d by the shaded area. 31 increase i n the asymptotic value of Sp at long times. An increase of about 6% i s , i n f a c t , observed. I t should be emphasized, however, that the o r i g i n of time f o r the data o f " F i g . 10 and that represented by the f i t t e d curve i s not n e c e s s a r i l y the same because of the uncertainty i n the time at which the sample was cooled to 4.2°K. By adjusting the o r i g i n , the asymptotic value could be increased by several percent. Also, there are some po s s i b l e small systematic errors a r i s i n g i n t h i s experiment associated with the v a r i a t i o n with time of the l e v e l of l i q u i d helium, since the reference s i g n a l and the NMR s i g n a l are generated at opposite ends of the half-wave l i n e . This type \ 13 of experiment should be repeated c a r e f u l l y with a sample containing CH^ i n which such systematic errors are eliminated. In one experiment on the sample doped with 0.05% O2, we observed an anomalous time-dependence of Sp a f t e r the sample had been cooled to 4.2°K. The r e s u l t s are shown i n F i g . 11. The time required to achieve a steady value of Sp was more than 8 hr. Since t h i s behavior was not reproducible, we have no conclusive explanation of t h i s strange time-dependence. We t e n t a t i v e l y ascribe the r e s u l t to the contraction of the s o l i d methane when the sample was cooled quickly from l i q u i d nitrogen to l i q u i d helium temperature. Perhaps, i n t h i s experiment, the sample p u l l e d away from the walls of the sample tube leaving only a small area of contact f o r the t r a n s f e r of heat to the helium bath by conduction. In order to e s t a b l i s h the v a l i d i t y of t h i s i n t e r p r e t a t i o n , more experiments are required. I f the explanation i s c o r r e c t , one should be very cautious i n analyzing the implications of the time-dependence of experimental q u a n t i t i e s i n samples quickly cooled to l i q u i d helium temperatures. 1-00 0-80 0-60 U 0 4 0 0-20 P-0 o o_ o o-0 8 10 12 14 T I M E ( H O U R ) 16 18 20 2 2 13, Fig. 11. Plot of Sp in arbitrary units versus time for a sample of CH^ containing 53% CH^ and 0.05% O2. The behavior of S shown in this graph was not reproduced in subsequent experiments. u> 33 Summary. We summarize our r e s u l t s as follows : Measurements of the r a t i o Sp/S c i n d i c a t e that the e q u i l i b r i u m value of <I(I + 1)> i n pure CH 4 with 13 53% CH 4 i s w i t h i n 5% of the value of 3.73 at 4.2°K. This departure from the high temperature value of <1(I + 1)> = 3 i s consistent with the observed increase of S^ with time by about 25% f o r samples of pure CH^ and CH 4 with 0.002% 0 2 (25). There i s some experimental i n d i c a t i o n that the e q u i l i b r i u m value of <I(I + 1)> increases by a few percent as the temperature i s reduced from 4.2°K to 2.45°K, but these experiments are not conclusive. 5. THE MAGNETIC PROPERTIES OF 0 o IN SOLID CH. 2 4 The measurements of S^ and S c f o r a sample containing 0.05% 0 2 were analyzed i n the same way as the data for the oxygen-free sample, and the r e s u l t s are presented i n Table 2. The values of Sp/S c are s i g n i f i c a n t l y l a r g e r than the corresponding values f o r the oxygen-free sample. We b e l i e v e , however, that these r e s u l t s should only be used to deduce an upper l i m i t to H <1(I + 1)> . The reason for t h i s b e l i e f i s that whereas tl^ i s the same Q for t h i s sample as f o r the oxygen-free sample, the value of M^ i s s i g n i f -H C i c a n t l y greater. The o r i g i n of t h i s change i n the r a t i o M2/M2 i s almost c e r t a i n l y associated with the magnetic properties of the 0 2 molecule i n a s o l i d l a t t i c e . I f this i s so, i t can be shown i n the following paragraphs that we have s i g n i f i c a n t l y underestimated S c i n extrapolating V^(t) to values bf t < t thus r e s u l t i n g i n an overestimate of <I(I + 1)> . m & Nevertheless, as we have pointed out i n the previous s e c t i o n , the upper • 34 Run T S /S <I(I+1)> p' c M H 2 2 °K • 10 8sec~ 2 o _o 10 sec Dec. 8 4.22 42.3 4.18 33.4 2.02 16.5 Dec. 12 4.22 44.6 4.42 32.2 1.59 20.3 Jan. 18 4.22 45.7 4.52 33.6 1.56 21.6 Average 4.22 44.2 4.37 33.1 1.72 19.5 Jan. 18 1.78 35.7 Jan. 18 1.08 51.0 5.05 35'.4 1.24 28.7 Table 2. Experimental r e s u l t s f o r a sample of CH^ containing 53% 1 3 C H 4 and 0.05% 0^ 35 l i m i t s obtained f o r <I(I + 1)> f o r these samples are very u s e f u l . We can i n f a c t estimate the er r o r introduced i n t o the extrapolated value of V^(0) f o r the oxygen doped sample using the r e s u l t s of the theory developed i n chapter I I . In ad d i t i o n to using the theory of NMR l i n e broadening by paramagnetic impurities, we assume that the c o r r e l a t i o n f u nction of the 13 NMR spectrum of the resonating nuclear spins (protons or C) i n the presence of 0 2 molecules i s the product of two c o r r e l a t i o n functions, one being that obtained when 0-> molecules are absent and the other being that due to the influence of 0 2 molecules alone. This i s equivalent to assuming that the l i n e shape function i n the presence of 0 2 molecules i s the convolution of the o r i g i n a l 0 2 - f r e e l i n e shape function and the l i n e shape function due to the influence of 0 2 molecules alone. I f we now choose a set of data, V- (t) , (where V-.- (t) was defined i n page 7 ) obtained from the 1 oxygen 1 r & / sample containing 0.05% 0 2 and a corresponding set of data, ^ i ^ ^ f r e e ' obtained from the 0 2 ~ f r e e sample, and i f we p l o t the set of points V.(t) /V.(t).. , then the function F(t) so obtained i s p r o p o r t i o n a l 1 oxygen 1 -free' ^ ^ to the c o r r e l a t i o n function due to the 0 2 molecules alone. To correct f o r the error i n our extrapolation process introduced by the change of f . i . d . shape associated with 0 2 i m p u r i t i e s , we need to divide the value of a set of data ^ i ^ * - ) O X y g e n experimentally measured by the corresponding value of F(t) to give a new set of data V.(t),. , and then use V.(t),. f o r & 1 free 1 free extrapolation. Unfortunately, due to the large s c a t t e r of the experimental r e s u l t s , d i f f e r e n t p a i r s of sets of V,- (t) and V - ( t ) , y i e l d ' r 1 oxygen 1 free J d i f f e r e n t F(t) with widely d i f f e r e n t decay times, f o r c i n g us to analyze the er r o r i n the following le s s accurate way. 36 We f i r s t consider the case of the proton resonance. The cor r e l a t i o n function due to 0.05% .0^ impurities alone i s , to a very good approximation, represented by an exponential function i n time with a decay constant t ^ (chapter I I ) . At 4.2°K the value of . t ^ i s not sensitive to the value of the c r y s t a l f i e l d parameter D i n the spin hamiltonian of the 0 2 spins. This w i l l be shown i n Chapter I I . By setting D = 0, we obtain the following expression for t ^ (from Eq. [2.78.]) 9/3 a J [ l + 2 cosh(ggH/kT)] n t , = : , Ll-15 J 64TT pY3g sinh(3gh/kT) where a i s the l a t t i c e constant of the f.c.c. l a t t i c e , H i s the value of the external magnetic f i e l d , p i s the f r a c t i o n a l 0 2 concentration, and g i s the g-factor of the 0 2 molecule which i s assumed to have an i s o t r o p i c g of value 2. Substituting H = 5180 gauss and p = 0.005 into Eq. [1.15], we obtain t ^ = 301 usee. To estimate the error introduced by the extrapolation process, we note that both sets of data V.(t).. and V.(t) were v ' l free l oxygen f i t t e d to the same expression -M t 12 Ce (sin bt / bt) i n a semi-logarithmic scale, whereas V.(t) should have the form & I oxygen -M°t2/2 - t / t d C 1e (sin bt / bt) , where C and C^ are arbitrary constants, and M° i s the second moment of 37 the oxygen-free f . i . d . s i g n a l . Since the change of the value of b i s neg l i g i b l y small when 0 2 molecules are introduced, our extrapolation process i s equivalent to f i t t i n g the curve f(s) = M°s/2 + s ^ / t , [1.16] I d by the function g(s) = M2s/2 + P , [1.17] o where s = t , M 2 i s the "measured" second moment, and P i s the y-intercept determined by the extrapolation. For a set of points centering around 2 s Q = t , i f we take the slope of f(s) at- s Q as that of g(s) at s Q , and use g(s) to calculate the second moment, we get an apparent increase i n the second moment given by M 2 - M ° = l / ( s * t d ) - l / ( t D t d ) [1.18] If t =12 usee i s chosen near t , the increase i n "measured" second o m ^ g 2 moment i s M0 - M„ = 2.31 x 10 sec I f t = 28 usee i s chosen near the £ L o t a i l of the f . i . d . s i g n a l , then the increase i s lowered to M 2 - M° = 9.89 x 10 7 sec 2 These changes are consistent with the experimental values as shown i n Tables 1 and 2. The value of P i s Eq. [1.17] i s given by 38 p = t\ (M° -M 2) / 2 + t c / t d = ' ^ . a Therefore the y - i n t e r c e p t of g(s) has a p o s i t i v e value which depends on t Q . Since i n our e x t r a p o l a t i o n process, p o i n t s i n s i d e the i n t e r v a l 12 usee _< t _< 20 usee r e c e i v e d much more weight than p o i n t s o u t s i d e , the co r r e c t e d value of Sp should be increased by a f a c t o r ranging i n value from e P = 1.016 to 1.027. 13 In the case of C resonance, the decay time of the c o r r e l a t i o n f u n c t i o n due to 0 2 molecules alone i s t j = 389 usee i n an e x t e r n a l magnetic f i e l d of 20500 gauss. The corresponding i n c r e a s e s i n the o 7 ^ ~j _ 2 "measured" second moment are M 2 - M 2 = 8.6 x 10 sec and 2.4 x 10 sec when t assumes' the values of 30 usee and 110 usee, r e s p e c t i v e l y . Again, i n our e x t r a p o l a t i o n , p o i n t s i n s i d e the i n t e r v a l 30 ysec t Q <_ 110 ys re c e i v e d much more weight than p o i n t s ouside, the c o r r e c t e d value of should be inc r e a s e d by a f a c t o r ranging i n value from 1.039 to 1.152. The overestimate i n the value of <I(I + 1)> i n the oxygen doped sample can th e r e f o r e range from 1% to 14% . This i s c o n s i s t e n t w i t h the i n c r e a s e i n the "measured" value of <I(I + 1)> from 3.73 f o r the oxygen-free sample to 4.37 f o r the sample c o n t a i n i n g 0.05% oxygen. There i s a l s o a d d i t i o n a l experimental evidence t h a t our measurement of S /S i n the oxygen-containing sample i s r e a l l y an overestimate. In one experiment, measurements of Sp were made at 4.2, 1.78, and 1.08°K w h i l e measurements of S were made at 4.2 and 1.08°K. The product S T was c P independent of the temperature T to w i t h i n the experimental accuracy of 39 about 5% which i s consistent with no change i n the populations of the d i f f e r e n t nuclear spin symmetry species occurring as the temperature i s changed. The value of S CT at 1.08°K was .about 30% lower than the value of 4.2°K. Since an underestimate of S c of the,type we suggested above would be more important, the l a r g e r the induced paramagnetism of 0 2 , and since the induced paramagnetism of 0 2 increases with decreasing temperature, t h i s change i s consistent with our p i c t u r e . The induced 0^ paramagnetism also causes systematic errors i n the evaluation of Sp , though these errors are much smaller than the errors involved i n the evaluation of S c . Thus the fac t that SpT was found to be independent of temperature may pos s i b l y be due to an ac c i d e n t a l c a n c e l l a t i o n of the increase i n SpT with decreasing T by the influence of the induced 0 2 paramagnetism on our measurement, since the induced magnetic moment, and consequently the value of 1/t^ i s not l i n e a r i n 1/T at such low temperatures around 1°K. In f a c t , the value of the component of the induced magnetic moment < ^ z > i - n t n e d i r e c t i o n of the external f i e l d averaged over the o r i e n t a t i o n of the 0 2 f i g u r e - a x i s varies only s l i g h t l y with the c r y s t a l f i e l d D-parameter which i s less than about 5.7k, where k i s the Boltzmann constant. The value of < l i z > i - s approximately l i n e a r i n 1/T f o r T down to about 3°K when the external f i e l d i s greater than 5 Kilogauss. At lower temperatures, < 1 J Z > i s s t i l l approximately l i n e a r i n 1/T down to T = 1°K when D = 0. However, when D has some f i n i t e value le s s than 5.7k, <VLZ> becomes saturated at higher temperatures depending on the value of D. Due to t h i s complicated increase of 1/t^ at lower temperatures, we conclude that we cannot r u l e out a change of S T as great as 10% as AO the temperature is changed from A.2 to 1.08 K. It would be interesting to make a more thorough analysis of the influence of induced 0 2 paramagnetism on NMR second moments and signal intensities but this would be better done using data obtained from a more suitable experimental set-up; we require either a pulse spectrometer with a much shorter recovery time or a careful experiment with a steady state spectrometer. 6. COMPARISON WITH OTHER EXPERIMENTS. Heat capacity results. Colwell, G i l l , and Morrison (8) have measured the heat capacity of solid CH^ down to about 2.5°K. Extrapolation of their heat capacity data smoothly down to 0°K gave a residual entropy of S Q = A.93 ± 0.10 cal/mole/°K . The residual entropy can be expressed i n terms of the apparent degeneracies Q^ , Q^,, and and the probability P., P„, and P for the molecules being in the A, T, and E species, A 1 £J respectively, S 0 = R( P A logQ A + P T logQ T 4- P E logQ E) + S m. x , [1.19] where R = 1.987 cal/mole/°K and the entropy of mixing S . is given by mix Smix - " R ( PA l 0 § P A + PT ^ T + P E l o § P E ) ' [ 1 ' 2 ° ] When the different species are populated according to their relative 41 s t a t i s t i c a l weights, i . e . P^ : P T : P E :: Q A : Q T : Q £ , then the r e s i d u a l entropy i s simply S Q = R l o g ( Q A + Q T + Q E) . [1.21] The above condition i s c e r t a i n l y true i f the energy differences between the ground l e v e l s of the three species are much less than kT where T i s the lowest temperature reached f o r the extrapolation. Morrison et a l . (8) assumed that no conversion occurred between the d i f f e r e n t nuclear spin species during the time of t h e i r experiments (many hours) i n the l i q u i d helium temperature range. I f th i s i s true then P^ : P^ : P^ = 5 : 9 : 2 . They then took Nagamiya's values of Q , Q , and Q for d i f f e r e n t c r y s t a l l i n e f i e l d symmetries and found that within the accuracy of t h e i r determination, only a t r i g o n a l f i e l d , corresponding to Q = 5, Q = 6, and 0 = 2 gi v i n g S =5.06 cal/°K/mole, gave an adequate f i t to the experimental E O value of S Q . The analysis of Morrison et a l . (8) brings out a subtle point, that i t i s pos s i b l e to remove some of the entropy associated with nuclear spin without any change i n the r e l a t i v e population of the nuclear spin species taking place. I f the r e s i d u a l entropy was associated with complete nuclear spi n disorder i n CH^ , then we would have S 0 = R log 16 = 5.51 cal/mole/°K. To understand how t h i s decrease i n S Q can take place without conversion, we note that the high temperature r e l a t i v e s t a t i s t i c a l weights of 5 : 9 : 2 are associated with spin s t a t i s t i c a l weights only, i . e . the A spin species has one spin quintet ( 1 = 2 ) , the T spin species has 3 spin t r i p l e t s (I = 1), and the E spin species has two s i n g l e t s ( 1 = 0 ) . Now, consider the ground free r o t a t i o n a l states of d i f f e r e n t species s l i g h t l y perturbed 42 by a tetrahedral c r y s t a l l i n e f i e l d . The ground J = 0 state remains a quintet because 1 = 2 . The n i n e - f o l d degeneracy of the J = 1 state of the T species i s also unaffected. However, the n i n e - f o l d degeneracy i n t h i s case corresponds to 3 values of M combined with 3 values of M , i . e . there i s only one spin t r i p l e t , so to speak, i n t h i s case. The t e n - f o l d degeneracy of the J = 2 state i s reduced to a two-fold spi n degeneracy, the f i v e - f o l d M degeneracy being s p l i t i n t o a doublet and a t r i p l e t . I f one «J now follows Nagamiya's argument (10) to assume that the number of ground r o t a t i o n a l l e v e l s i s equal to the order of the symmetry group of the molecule or of the c r y s t a l l i n e f i e l d , whichever i s the l a r g e r , then the number of ground l e v e l s i s 12, which i s the order of the tetrahedral group of the CH^ molecule, and the E spin symmetry species has a doublet of ground l e v e l s from the J = 2 r o t a t i o n a l s t a t e s . Therefore the imposition of a t r i g o n a l f i e l d a f f e c t s only the M degeneracy of the T species g i v i n g r i s e to a t h r e e - f o l d and a s i x - f o l d degenerate s t a t e . I f only one of these states i s occupied, we see from Eqs. [1.19] and [1.20] that the r e s i d u a l entropy i s lowered below R log 16 even i f no conversion takes place. Our experiments r u l e out the hypothesis that no spi n conversion takes place since, regardless of the symmetry of the c r y s t a l l i n e e l e c t r i c f i e l d , t h i s would give <I(I + 1)> = 3.00 i n disagreement with our value of <I(I + 1)> = 3.73 ± 0.18 . The framework used by Morrison et a l . (8) can s t i l l be used to i n t e r p r e t the values of <I(I + 1)> and SQ , but i t must now be assumed that conversion can take place. The influence of the c r y s t a l f i e l d on the low-lying states must also be s i g n i f i c a n t . F i g . 12 shows the temperature dependence of the values of <I(I + 1)> , entropy S, and B / K T F i g . 12. P l o t of c a l c u l a t e d <I(I + 1)> , S, and C v e r s u s B/kT f o r a s p h e r i c a l f r e e - r o t o r . 44 s p e c i f i c heat C v associated with the r o t a t i o n a l degree of freedom f o r a l e v e l arrangement resembling that of a f r e e - r o t o r with r o t a t i o n a l constant 13 B. Although our sample contains 53% CH^ , i t has been demonstrated by McDowell (26) that the i s o t o p i c s h i f t of the value of B i s n e g l i g i b l e , and we can take B = 5.24 cm , which i s equivalent to 7.54°K, g i v i n g a value of <I(I + 1)> =5.81 i n disagreement with the experimental of 3.73 ± 0 . 1 8 . There i s a p o s s i b i l i t y that the e f f e c t of the c r y s t a l l i n e f i e l d on the low-l y i n g l e v e l s i s to leave the l e v e l arrangement unchanged, but to modify the value of B only. Then a value of B = 1.8°K gives <I(I + 1)> = 3.73. However, t h i s w i l l also give a value of <I(I + 1)> = 4.68 at 2.45°K which i s too high as compared with the observed value. The energy l e v e l s and degeneracies of the low-lying states of the A, T, and E spin symmetry species perturbed by a t r i g o n a l f i e l d are shown i n F i g . 13. For a t e t r a h e d r a l f i e l d e i = e 2 • T ^ e values of <I(I + 1)> and the r e s i d u a l entropy S Q f o r a l l the d i f f e r e n t l i m i t i n g cases of the energy l e v e l s corresponding to F i g . 13 are given i n Table 3. The three configurations : (a) e , £ 2 , e 3 << kT, (b) e^, £ 3 » kT, £ 2 << kT, and (c) e^, e^, >>kT, obviously do not f i t the experimental values of <I(I + 1)> and can therefore be eliminated. The c o n f i g u r a t i o n which f i t s best the experimental values of <I(I + 1)> and S q corresponds to e 2 , e 3 >> kT, E j << kT, where T = 4.2°K. However, the data also seem to f i t the assumption of a tetrahedral f i e l d , i n which case E i = e2 ' with E 3 >> kT, providing that the r e s t r i c t i o n £ 1 << kT was l i f t e d . Let us now consider the temperature dependence of <I(I + 1)> , S, and C v f o r these two cases : (d) e » kT, e = 0; and e = e 2 = 7.04k for the case of 45 T e, — — — - T -Energy Degeneracy Spin Species F i g . 13. Energy l e v e l s and degeneracies of the low-lying states of the A, • T, and E spin symmetry species of CH^ perturbed by a t r i g o n a l ; ' c r y s t a l l i n e e l e c t r i c f i e l d . 46 Case <I(I + 1)> S 0(cal/°K/mole) e l , e 2 , e 3 « kT 3.00 5.51 e 3 » kT; e 1 ( e 2 << kT 3.42 5.24 £ 2 , e 3 » kT; e]_ << kT 3.82 4.76 £ 1 , e 3 » kT; e 2 << kT 4.50 4.31 £ 1 ' e 2 > £ 3 > > k T 3.73 ± 0.18 4.93 ± 0.10 Table 3. Values of <I(I + 1)> and the r e s i d u a l entropy S Q f o r d i f f e r e n t l i m i t i n g cases of the energy l e v e l s of F i g . 13. 47 the t r i g o n a l f i e l d ; and (e) e 3 >> kT, e = e 1 = = 1.33k f o r the case of the tetrahedral f i e l d . The value of e i n each case i s so chosen that gives the experimental value of 3.73 to <I(I + 1)> at 4.2°K. The r e s u l t s are shown i n F i g s . 14, and 15, r e s p e c t i v e l y . The behavior of <I(I + 1)> alone indicates that the p o s s i b i l i t y of a tetrahedral f i e l d i s very u n l i k e l y , because f o r such a f i e l d the value of <I(I + 1)> = 4.08 at 2°K and increases very r a p i d l y to the value of 4.7 as the temperature decreases to 1°K, whereas the experimental value of <I(I + 1)> increases only slowly from about 3.8 at 4.2°K to about 4.0 as the temperature goes down to about 1°K as demonstrated by J.E. P i o t t (24). On the other hand, the case of t r i g o n a l f i e l d f i t s the experimental values of <I(I + 1)> very w e l l . In ad d i t i o n to the two p o s s i b i l i t i e s j u s t considered, there are numerous other p o s s i b l e , though les s l i k e l y , schemes to arrange the ground l e v e l s . In a recent paper (27), Runolfsson, Mango and Borghini quoted a value of <I(I + 1)> = 5.88 ± 0.3 from t h e i r measurement of CH^ at 1.06°K and 25 Kilogauss with 0.1% O2, suggesting that at 1.06°K the CH^ molecules belong to the A spin symmetry species with 1 = 2 . Although t h i s r e s u l t i s obtained at a temperature much lower than ours which i s at 4.2°K and the two r e s u l t s cannot be compared d i r e c t l y , there are doubts about t h e i r quoted value and conclusions. The doubts a r i s e from that using the same method of measurement they obtained a value f o r the p o l a r i s a t i o n of HD 50% higher than that expected for free protons, and t h i s anomaly cannot be explained away t h e o r e t i c a l l y . Most recently, H. G l a t t l i at Baclay, France, (private communication) obtained preliminary values of <I(I + 1)> = 4.1 and 4.7 at 4.2°K and 1.1°K , r e s p e c t i v e l y , with a nominal er r o r of T ( ° K ) F i g . 14. P l o t o f c a l c u l a t e d v a l u e s of < I ( I + 1)> , S, and C v v e r s u s T f o r CH^ i n a t e t r a h e d r a l c r y s t a l f i e l d . •—s LU 1 o 2 5.2 MOLE ; " < 1(1+1 )> / 3-8 o ^ 0-2 _J < u w 4-8 ( CAL ; S _ ^ ^ [ 3-7 7 LO > 0 . 1 U NX 4-4 0 . i 3.6 J • I I 1 I I 0 2 4 6 • T ( ° K ) F i g . 15. P l o t o f the c a l c u l a t e d v a l u e s o f <I(I + 1)> , S, and C v v e r s u s T f o r CH^ i n a t r i g o n a l c r y s t a l f i e l d . 4 N VD 50 5% . The value at 4.2°K i s consistent with our value of 3.73 ± 0.18, while the value at 1.1°K disagrees with that of Runolfsson et a l . However, the r e s u l t s of G l a t t l i favour a l e v e l scheme corresponding to a te t r a h e d r a l c r y s t a l f i e l d over that of a t r i g o n a l f i e l d which i s indi c a t e d by the r e s u l t s of J.E. P i o t t . In order to deduce unambiguously the correct l e v e l scheme, a more precise measurement of the temperature dependence of <I(I + 1)>, S, or at low temperatures, e s p e c i a l l y below 1°K, i s needed. Infra-red studies. The most d e f i n i t i v e study of conversion between nuclear spin symmetry species of CH^ has been done by Frayer and Ewing (28). A s i m i l a r study has also been c a r r i e d out on several other molecules (29). These experiments were not c a r r i e d out i n s o l i d methane, but rather on small concentrations of CH^ i n s o l i d Argon. Under these conditions i t proved to be possible to r e l a t e d i f f e r e n t features of the v i b r a t i o n - r o t a t i o n absorption spectra to d i f f e r e n t nuclear spin symmetry states of CH^. This enabled Frayer and Ewing to measure the dependence of the populations of the A and T spin species on time a f t e r the sample was cooled to 4.2°K. In the case of the Argon l a t t i c e , there i s a good t h e o r e c t i c a l basis f o r b e l i e v i n g that the CH^ molecules are subjected to a c r y s t a l l i n e e l e c t r i c f i e l d of cubic symmetry, and Frayer and Ewing u t i l i z e d some c a l c u l a t i o n s of the energy l e v e l s of CH^ i n a cubic f i e l d by H.F. King (unpublished) and by King and Hornig (30) to i n t e r p r e t t h e i r r e s u l t s . Some aspects of the i n f r a - r e d studies r e l a t e d to our r e s u l t s are as follows : (1) The time constant f o r spontaneous conversion of CH^ between the A and T species due to intra-molecule d i p o l a r and spin r o t a t i o n i n t e r a c t i o n s (31 , 32) was found to be 90 min. This i s very s i m i l a r to the behavior we found i n s o l i d 51 CH^, as given i n F i g . 8, but the agreement i s probably f o r t u i t o u s as the two studies were made on d i f f e r e n t s o l i d s and the d i f f e r e n c e i n energy between the A and T ground states i s apparently quite d i f f e r e n t f o r the two cases. (2) The influence of 0^ i s very s i m i l a r i n the two cases. (3) The eq u i l i b r i u m concentration of the A spin species at 4.2°K was found to be about 90%, which i s considerably greater than that f o r s o l i d CH^ according to our r e s u l t s . I t was found t h e o r e c t i c a l l y that even octahedral c r y s t a l l i n e f i e l d s as large as 10 or 20 times the r o t a t i o n a l constant B do not change the spacing between J = 0 and J = 1 states by a large amount. This f a c t enable Frayer and Ewing to understand, at le a s t semi-quantitatively, the large amount of conversion they observed. The i n s e n s i t i v i t y of the J = 0 and J = 1 energy separation to the cubic f i e l d can be e a s i l y understood i n terms of the f a c t that the eigenfunction of a state of r o t a t i o n a l angular momentum J transforms as Y ^ ( f i ) , where Y ^ ( f i ) i s a s p h e r i c a l harmonic and denotes the molecular o r i e n t a t i o n . An octahedral p o t e n t i a l can be wr i t t e n i n terms of s p h e r i c a l harmonics, the lowest term i n v o l v i n g Y^ . Therefore, the f i e l d only influences the energy l e v e l s of the J = 0 and J = 1 states i n second order, the lowest unperturbed states a f f e c t i n g the J = 0 and J = 1 states being the J = 4 and J = 3 st a t e s , r e s p e c t i v e l y . I t would be i n t e r e s t i n g to i n v e s t i g a t e the corresponding t h e o r e t i c a l problem i n s o l i d CH^ where a tetrahedral f i e l d should be present, and where we might imagine also an a x i a l or t r i g o n a l f i e l d a r i s i n g from the ordering of the or i e n t a t i o n s of the CH^ molecules associated with the low-temperature phase t r a n s i t i o n . I t would appear that i n t h i s case too the e f f e c t of the c r y s t a l f i e l d would only appear 52 i n second order. The lowest permanent e l e c t r i c m ultiple moment of a te t r a h e d r a l molecule i s an octopole moment. Therefore, the t r i g o n a l , as w e l l as the t e t r a h e d r a l , e l e c t r i c f i e l d would manifest i t s e l f i n the lowest order i n terms of a s p h e r i c a l harmonic of order Y„ i n the molecular o r i e n t a t i o n . 3m This means that the J = 0 and 1 states are influenced i n lowest order by the J = 3 and J = 2 s t a t e s , r e s p e c t i v e l y . I t would be i n t e r e s t i n g to examine t h e o r e t i c a l l y whether the f a c t that both the t r i g o n a l and t e t r a h e d r a l terms influence the J = 1 state i n second order i s compatible with the l i m i t i n g case e 2 >> kT, e^ ^ << kT, which f i t s the data so w e l l i n the v i c i n i t y of 4.2°K. Nuclear magnetic resonance. The most comprehensive NMR study of s o l i d methane has been c a r r i e d out by G. de Wit (13, 14, 15) who studied the s p i n - l a t t i c e r e l a x a t i o n time T-^  as a function of temperature down to 1.2°K fo r CH^ and i t s deuterated modifications. They represent a f r u i t f u l area for t e s t i n g of d i f f e r e n t molecular models of molecular r e o r i e n t a t i o n i n s o l i d methane. We s h a l l comment here on a few matters d i r e c t l y r e l a t e d to the problem of spin conversion i n s o l i d CH^ . I t was found that the T-^  versus temperature pl o t s f or CH^ and a l l of the deuterated modifications of methane went through a c h a r a c t e r i s t i c minimum i n the region of the low temperature s o l i d phase. For CH^ the T^ minimum occurred near 6°K and was very shallow. The value of T^ at the minimum was ^ i ^ m i n ~ -^0 msec f o r a resonance frequency of 28.5 MHz, while the value of 1.2°K was only about 10-20% higher. The conventional theory of spin r e l a x a t i o n p r e d i c t s a value of (T^) . = 8 msec (15) . The 53 f a c t that (T,) . i s a c t u a l l y l a r g e r than t h i s can be understood i n terms 1 mm J G of the f a c t that only the T spin symmetry species can rela x by i n t r a -molecular d i p o l a r i n t e r a c t i o n (32). I f the r e l a x a t i o n rate of the spins belonging to the T spin symmetry species i s 1 / (T^)^ , then the re l a x a t i o n rate of the e n t i r e system i s (15) c VP 1 T 1 T 1 -(—) = (—) Tl 221 T l CA + C T T l T 6 P A + 2 P T T l T ' where and C^ , are the heat c a p a c i t i e s associated with the Zeeman energies of the A and T spin species, r e s p e c t i v e l y ; P and P have been defined previously. The experimental f a c t that T^ increases very l i t t l e as the temperature i s reduced from G to 1.2°K ind i c a t e s that very l i t t l e change i n P T/P. takes place. This i s consistent with our observations that <I(I + 1)> does not change very much as the temperature i s v a r i e d from 4.2°K to 1.08°K . 54 CHAPTER 2 NUCLEAR MAGNETIC RESONANCE LINE SHAPES BROADENED BY PARAMAGNETIC IMPURITIES IN SOLIDS 1. INTRODUCTION I n C h a p t e r 1 we r e p o r t e d experiments performed to measure t h e n u c l e a r magnetic s u s c e p t i b i l i t y o f s o l i d CH^. I t was n o t i c e d t h a t when t h e sample o f s o l i d CH^ was doped w i t h a s m a l l c o n c e n t r a t i o n o f 0 2 m o l e c u l e s , 13 which a r e p a r a m a g n e t i c , t h e second moment o f the a b s o r p t i o n s p e c t r u m o f C res o n a n c e i n c r e a s e d s i g n i f i c a n t l y , w h i l e the second moment o f t h e p r o t o n r e s o n a n c e , b e i n g more than 20 times l a r g e r i n v a l u e , remained unchanged w i t h i n e x p e r i m e n t a l e r r o r s . The b r o a d e n i n g o f the re s o n a n c e l i n e was thought to a r i s e from the i n t e r a c t i o n between the n u c l e a r magnetic moment, which i s r e s p o n s i b l e f o r the resonance a b s o r p t i o n o f the r f energy, and t h e magnetic moment o f the 0 2 m o l e c u l e s i n d u c e d by the e x t e r n a l magnetic f i e l d . The problem o f r e l a t i n g the m i c r o s c o p i c i n t e r a c t i o n s among a system o f p a r t i c l e s to the b r o a d e n i n g o f the m a c r o s c o p i c a l l y o b s e r v e d s p e c t r u m o f i n d i v i d u a l u n c o u p l e d p a r t i c l e s i s o f g r e a t importance i n resonance s t u d i e s . I t sheds l i g h t on the n a t u r e and magnitude o f t h e s e i n t e r a c t i o n s . A l t h o u g h an e x a c t s o l u t i o n t o the g e n e r a l problem i s s t i l l l a c k i n g , v a r i o u s ways t o s o l v e i t a p p r o x i m a t e l y have been i n t r o d u c e d so t h a t f o r a g i v e n range o f p h y s i c a l c o n d i t i o n s and e x p e r i m e n t a l p r e c i s i o n , u s e f u l r e s u l t s can be d e r i v e d i n one o f t h e s e ways. B a s i c a l l y t h e r e a r e t h r e e methods o f approach t o t h i s l i n e shape problem. 55 The well-known method o f the moment e x p a n s i o n (33) i s the s i m p l e s t and most e l e g a n t . The moments a r e e s s e n t i a l l y c o e f f i c i e n t s o f t h e M a c l a u r i n e x p a n s i o n o f the F o u r i e r t r a n s f o r m o f t h e a b s o r p t i o n l i n e shape f u n c t i o n . I n p r i n c i p l e , knowing a l l the moments s h o u l d e n a b l e one t o know t h e l i n e shape f u n c t i o n c o m p l e t e l y . However, t h e r e a r e two drawbacks i n t h i s a pproach. F o r some l i n e shapes, t h e M a c l a u r i n s e r i e s may n o t converge. F o r example, a L o r e n t z i a n l i n e has a F o u r i e r t r a n s f o r m w i t h a cusp a t the o r i g i n ; i t s odd moments v a n i s h and a l l even moments, e x c e p t t h e z e r o t h moment, d i v e r g e . A l s o , s i n c e t h e b e h a v i o r o f the l i n e shape n e a r the c e n t r e i s r e f l e c t e d by the b e h a v i o r o f i t s F o u r i e r t r a n s f o r m a t v e r y l o n g t i m e s , t h i s i n f o r m a t i o n cannot be o b t a i n e d from a f i n i t e number o f moments, wh i c h a r e m a t h e m a t i c a l l y e q u i v a l e n t t o the d e r i v a t i v e s o f the t r a n s f o r m a t z e r o time. I n most c a s e s , the c a l c u l a t i o n to the s i x t h moment i s a l r e a d y v e r y l e n g t h y and cumbersome. I f the q u a l i t a t i v e f e a t u r e of the l i n e shape i s known, the n the c a l c u l a t i o n o f the f i r s t few moments i s v e r y v a l u a b l e t o f i x the l i n e shape f u n c t i o n q u a n t i t a t i v e l y . The s t o c h a s t i c t h e o r y (34, 35) aims a t c a l c u l a t i n g the a u t o c o r r e l a t i o n f u n c t i o n o f the t r a n s i t i o n o p e r a t o r S x ( t ) by t r e a t i n g the i n t e r a c t i o n s as a randomly f l u c t u a t i n g p e r t u r b a t i o n to the re s o n a n c e l i n e . As a r e s u l t , a b r o a d e n i n g i s o b s e r v e d . I t has wide a p p l i c a t i o n s , p a r t i c u l a r l y f o r i n v e s t i g a t i n g such e f f e c t s as "exchange and m o t i o n a l n a r r o w i n g s " . A l t h o u g h the d e r i v a t i o n o f the a u t o c o r r e l a t i o n f u n c t i o n i s r i g o r o u s , the f i n a l s t a g e o f s p e c i a l i z i n g the f u n c t i o n to c o n c r e t e cases i s g e n e r a l l y too c o m p l i c a t e d . One i s the n f o r c e d t o i n t r o d u c e assumptions c o n c e r n i n g the n a t u r e o f the random p r o c e s s to s i m p l i f y the c a l c u l a t i o n s . Very o f t e n t h e s e assumptions a r e s t a t i s t i c a l 5 6 i n n a t u r e and cannot be j u s t i f i e d r i g o r o u s l y . The s t a t i s t i c a l approach ( 3 6 , 3 7 ) was i n i t i a t e d by Margenau t o s t u d y p r e s s u r e e f f e c t s on s p e c t r a o f gases. E x t e n s i v e r e v i e w s on t h i s method have r e c e n t l y been g i v e n by Stoneham ( 3 8 ) and e a r l i e r by Grant and S t r a n d b e r g ( 3 9 ) . T h i s i s the most p o w e r f u l approach to c a l c u l a t e inhomogeneously broadened l i n e shapes. By assuming t h a t the c o n t r i b u t i o n s to t h e b r o a d e n i n g due to i n d i v i d u a l p a r t i c l e s a r e a d d i t i v e and t h a t the many p a r t i c l e p r o b a b i l i t y d e n s i t y f u n c t i o n can be f a c t o r i z e d i n t o a p r o d u c t o f s i n g l e p a r t i c l e p r o b a b i l i t y d e n s i t y f u n c t i o n s , the F o u r i e r t r a n s f o r m o f the l i n e shape f u n c t i o n can be e x p r e s s e d i n a c l o s e d form. I t can then be e v a l u a t e d by a continuum a p p r o x i m a t i o n ( 3 8 , 3 9 ) o r by the n u m e r i c a l e v a l u a t i o n o f t h e l a t t i c e sums i n v o l v e d . T h i s method works whenever the above two assumptions a r e v a l i d , w h i c h s h o u l d be so f o r a d i l u t e medium. The p r e s e n t work i s c o n c e r n e d w i t h the c a l c u l a t i o n of the NMR a b s o r p t i o n l i n e shape f u n c t i o n due to inhomogeneous b r o a d e n i n g by p a r a m a g n e t i c i m p u r i t i e s i n a s o l i d u s i n g the s t a t i s t i c a l approach. However, i n the f o r m u l a t i o n o f the problem, the assumption on the f a c t o r i s a b i l i t y i s r e l a x e d . The g e n e r a l e x p r e s s i o n s f o r the l i n e shape f u n c t i o n and i t s F o u r i e r t r a n s f o r m w i l l be d e r i v e d i n the f o l l o w i n g s e c t i o n . The s p e c i a l i z a t i o n o f t h e s e e x p r e s s i o n s t o a system, i n which the i m p u r i t i e s a r e c l a s s i c a l m a gnetic d i p o l e s , i s d i s c u s s e d i n d e t a i l i n S e c t i o n 3 . The cases i n which the i m p u r i t i e s a r e o f s p i n S = 1 / 2 and S = 1 a r e c o n s i d e r e d i n S e c t i o n 4 . In the c o n c l u d i n g s e c t i o n , r e l a t i o n s h i p between t h i s c a l c u l a t i o n and o t h e r works i s d i s c u s s e d . 57 2. DERIVATION OF THE LINE SHAPE FUNCTION We s h a l l make t h r e e assumptions t h a t w i l l be t h e b a s i s o f o u r c a l c u l a t i o n s . 1. The f r e q u e n c y s h i f t s p r o d u c e d by the i m p u r i t i e s a r e a d d i t i v e , i . e . the f r e q u e n c y s h i f t o f a g i v e n r e s o n a n t c e n t r e i s the sum o f f r e q u e n c y s h i f t s p r oduced by each i n d i v i d u a l i m p u r i t y i n t h e absence o f a l l o t h e r i m p u r i t i e s . T h i s i s e q u i v a l e n t t o assuming t h a t i n t e r a c t i o n s between t h e i m p u r i t i e s can be n e g l e c t e d . 2. I f q i s a s e t o f v a r i a b l e s which, t o g e t h e r w i t h the p o s i t i o n v e c t o r Y, a r e n e c e s s a r y t o s p e c i f y the s t a t e o f an i m p u r i t y a t Y, t h e n t h e p r o b a b i l i t y f o r the i m p u r i t y to assume a p a r t i c u l a r s e t o f v a l u e s o f q i s i n d ependent o f "r and of the p r e s e n c e o f a l l o t h e r i m p u r i t i e s . 3. The c o n c e n t r a t i o n o f i m p u r i t i e s i s s m a l l . The v a l i d i t y o f the second assumption w i l l become o b v i o u s l a t e r when v a r i a b l e s q a r e s p e c i f i e d . I n g e n e r a l , the f i r s t a s s umption i s no t c o r r e c t because t h e i m p u r i t i e s u s u a l l y i n t e r a c t among t h e m s e l v e s . However, i n the case o f s u f f i c i e n t l y low i m p u r i t y c o n c e n t r a t i o n s , and h i g h e x t e r n a l magnetic f i e l d (a few K i l o g a u s s ) the assumption s h o u l d h o l d t o a good a p p r o x i m a t i o n . We s h a l l now d e r i v e the l i n e shape f u n c t i o n due t o a g i v e n c o n c e n t r a t i o n p which i s e x p r e s s e d i n molar f r a c t i o n s . The p o s i t i o n o f the " r e s o n a t i n g c e n t r e " whose resonance i s to be o b s e r v e d i s chosen t o be 58 the o r i g i n of the coordinate system. The n impurities are d i s t r i b u t e d among N a v a i l a b l e l a t t i c e s i t e s . Let D denote a configuration of such d i s t r i b u t i o n s , i . e . D = ( r 1 , q±; • ..; r n , q^} , [2.1] where r_^, q^ represent r e s p e c t i v e l y the p o s i t i o n vector and the i n t e r n a l state of the i t h impurity. From the f i r s t assumption, the angular frequency s h i f t of the resonant centre due to the impurities having c o n f i g u r a t i o n D i s given by n u D = I u ( r . ,q) ' [2.2] where co(r\,q^.) i s the angular frequency s h i f t caused by the j t h impurity •when a l l other impurities are absent. From the second assumption, the p r o b a b i l i t y f o r the configuration D can be f a c t o r i s e d i n t o the following factors : n P(D) = P(d) n p(q.) , [2.3] where P(d) i s the p r o b a b i l i t y f o r the impurities to be found i n the configuration d = {r. , • • •, r } 1 n and p(q.) i s the p r o b a b i l i t y f o r the j t h impurity to be i n state q.. The 59 two p r o b a b i l i t y density functions are so defined that they are separately normalised to unity, Ip(q) = 1 , ' [2.4a] q and £p(d) = 1 . [2.4b] d From these normalisation conditions, i t follows that )>(D) = I I P ( d ) II p ( q ) D q l ' " " ' q n d J = 1 n I n P(q.) q r - " , q n j = l = n (I p ( q ) i = l q . J [2.4c] If we denote the l i n e shape function i n the absence of impurities by I 0(co), and the l i n e shape function i n the presence of impurities by I(co) , then 1(a)) = £P(D) I 0(to - u>D) . [2.5] Here we have assumed that an impurity has only the e f f e c t of s h i f t i n g a resonance l i n e along the cu-axis, without changing i t s shape. 60 Using the two following i d e n t i t i e s IQ(to - coD) = I (to - y)5(u - to )dy O Ll J —oo and 6 ( U " V = "27 i(u-co ) t e dt Eq. [2.5] can be rewritten into the following form Kco) = 2TT dy dt I (to - y ) el y t F ( t ) [2.6] where -ito t F(t) = J>(D)e D D I f we now denote i t O t T , / * . \ J 4 -e F ( t ) d t , [2.7] then we get Kco) = I Q(to - y ) I Q ( y ) d y [2.8] Eq. [2.8] shows that the t o t a l l i n e shape function I (to) i s the convolution of the " unperturbed " l i n e shape function l Q(w) , and the function I^(to). Since the function Ij(to) contains a l l the information concerning the broadening by the impurities, and I^(to) becomes i d e n t i c a l with I (to) when I 0 ( w ) i s i n f i n i t e l y sharp, we need no longer be concerned with the " unperturbed " l i n e shape function I Q(to). Henceforward, we s h a l l thus 61 assume I 0(w) to be i n f i n i t e l y sharp. We s h a l l correspondingly c a l l I^(w) the l i n e shape function and denote i t simply by I(o)), i . e . 1(0)) = 2TT e i W t F ( t ) d t [2.9a] -im t and F(t) = £P(D)e . . [2.9b] D Eq. [2.9] shows that the c o r r e l a t i o n function F ( t ) , which i s the f o u r i e r transform of the l i n e shape function I(o)) , contains a l l the information of the broadening by the imp u r i t i e s . Our task now i s to f i n d an e x p l i c i t expression f o r F ( t ) . We f i r s t denote the average £ p ( q ) e - i u ( Y ' q ) t = < e - ^ t > e . f ( ? j t ) . [ 2 . 1 0 ] q The c o r r e l a t i o n function F(t) can be rewritten i n a more convenient form n F(t) = ) > ( d ) n f ( r \ , t ) . [2.11] d j = l 2 Eq. [2.11] shows that the c o r r e l a t i o n function F(t) i s mathematically the average of a product of n functions, each of which i s chosen from the set of N functions, f ( f \ , t ) , j = 1, N. If an impurity i s found at s i t e j , the function f ( r \ , t ) i s chosen. Since no more than one impurity can occupy the same s i t e and interchanging impurities i n s i t e s i and j gives the i d e n t i c a l product of the n functions, the t o t a l number of configurations d i s 6 2 / N \ = _ NI In' n ! ( N - n)! We now assume that the d i s t r i b u t i o n i s random such that a l l configurations are equally probable. Then we have P(d) = 1 0 ' and Eq. [2.11] becomes F ( t ) = 7 N T E Acr^ t)..-- f ^ C r ^ t ) . [2.12] U n i + - - - + n N = n n.=0,l l To evaluate the summation i n the above expression, we use a method introduced by Darwin and Fowler (40). The idea i s to transform the summation in t o an i n t e g r a l which can then be evaluated by the method of steepest descent. We f i r s t define a generating function N _ cb(z) = n [1 + z f ( r . , t ) ] . [2.13] j = l 2 I f <j>(z) i s expanded as a polynomial i n z , N ,(z) = I c z% , [2.14] 1=0 then the sum i n Eq. [2.12] i s exactly equal to the c o e f f i c i e n t c n i n Eq. [2.14], i . e . 63 I f n i ( r 1 , t ) . . . . f ^ C r ^ t ) n l + * " • + n N = n ii.=0,1 1 The coefficients c n is then calculated from 1 ~n 2TT± L -n-1 0 z <j>(z)dz . [2.15 ] Since cj>(z) is an entire function, C can be any closed contour around the origin. Since both n and N are very large , the integral in Eq. [2.15 ] can be evaluated by the method of steepest descent, and the value is given by (see for example, page 437 of Ref. 41) c n = z-QU 1<J)(z0)//2Tr|g"(z0)| , [2.16] where the function g(z) is the logarithmic function of z n <^f>(z)> i.e. N ;(z) = -(n + l ) l n z + J l n [ l + zf (r. ,t) ] , [2.17] 1=1 3 and z Q is the saddle point at which g ' O O = o . Differentiating the function g(z) twice, we get 64 , N f ( r . , t ) g > ( z ) - - 1 L ± 1 + J 1 + z ; ( - > t ) . [2.18a] J = 1 J . _ N f 2 ( r \ , t ) and g " ( z ) e l L ± A . . J J . [ 2.l8b] z^ j = l [1 + z f C r j 5 t ) ] 2 The value of Z q i s then obtained by putting z = z Q into Eq. [2.18a] and then s o l v i n g the equation g'(z Q) = 0 , i . e . n + 1 N f ( r . , t ) ~z~ = E 1 + z f ( r . , t ) ' [ 2 ' 1 9 ] With the usual assumption that n and N may be regarded as tending to i n f i n i t y while the concentration p = ^ i s kept constant whatever the values of n and N are, we show i n the Appendix to t h i s chapter that when N i s s u f f i c i e n t l y large, z o = r^7 > [ 2 - 2 0 A ] n l n N and c n = I f ( r - j ^ t ) f Cr" N,t) n l + ' • ' % n.=0,l >"pN " \ (1 - P ) p N " \ J - J l + j ^ - f ( r . , t ) ] : , |2.20b ] /2TT¥ provided that the frequency s h i f t by impurity at r i s 65 u(r,q) o C T A (A 1 3 ) , [2.20c] and that the value of t l i e s w i t h i n a f i n i t e time i n t e r v a l , i . e . -T <_ t <_ T , [2.20d] for some per-assigned value of T. We note here that Eqs. [2.20c] and [2.20d] impose no r e a l r e s t r i c t i o n on our subsequent c a l c u l a t i o n s . Since we s h a l l apply our r e s u l t s to broadening by magnetic d i p o l a r i n t e r a c t i o n s , where . -3 u)(r,q) oC r ; Eq. [2.20c] w i l l be s a t i s f i e d . As to Eq.[2.20d], we can always choose a very large T, say 10"^ sec, such that F ( t ) l « F(0) = 1 f o r tI > T , then a l l the important i n f o r m a t i o n concerning F ( t ) i s contained i n the i n t e r v a l ) 66 A naive, non-rigorous and yet convenient way to see how the r e s u l t s expressed i n Eq. [2.20] are obtained from Eqs.[2.16], [2.18b] and [2.19] i s as follows. In the l i m i t when N -* oo , the main c o n t r i b u t i o n to the sums i n Eq. [2.19] _ -SL i s from terms i n which j i s very large. Since co(r,t) r (£ >_ 3 ) , the value of the functions f(r^.,t) i s e s s e n t i a l l y unity f o r very large j . By putting a l l f(r"^,t) = 1 and n + 1 = pN i n Eq. [2.19], and s o l v i n g the equation, we get Jo 1 - p ' Substituting t h i s value of z Q into Eq. [2.18b] and again p u t t i n g a l l f ( r \ , t ) = 1 , we obtain j " ( z Q ) = N ( l - p) 3 o 1 Substituting z Q = ^ P and g"(z 0) i n t o Eq. [2.16], we obtain again Eq. [2.20b]. Using S t e r l i n g ' s formula, f o r large N and n = pN, we have \n' n N! ! (N - n) ! MN -N N e n -n.TN-n,n ,N-n -N+n /„ . . . v n e N (1-p) e /2Trn(N-n) 'p- p N ~ \ (1 - p ) p N N ~ \ 67 From Eqs. [2.11], [2.20b] and the above r e s u l t , i t r e a d i l y follows that the c o r r e l a t i o n function F(t) i s given by F(t) = (1 - p ) N n [1 + -r-Z— f ( F , , t ) ] (N -> oo ) 1=1 1 " P J = n {(1 - p ) [ l + r - 2 - f(r".,t)]} (N + oo) j = l 1 ~ P j oo -io)("r.)t = n {1 - p[l - <e 2 >]} . [2.21] 3=1 The c o r r e l a t i o n function F ( t ) , as expressed i n Eq.[2.21], i s v a l i d f o r any value of impurity concentration p provided, of course, that- the broadening e f f e c t s produced by the impurities are a d d i t i v e . For low concentrations of p, we can approximate each f a c t o r i n Eq. [2.21] by an exponential function, -iu)(r".)t - i w ( r \ ) t -p [1 - <e 2 > ] 1 - p [1 - < e J > ] - e , and Eq. [2.21] becomes oo -ico(r.)t -p E [1 - <e >] F(t) = e J=l . [2.22] Here we note that i f one follows Grant and Strandberg (39) and Stoneham (38) by assuming that the many p a r t i c l e p r o b a b i l i t y density function i s f a c t o r i s a b l e i n t o a product of s i n g l e p a r t i c l e p r o b a b i l i t y density functions, one a r r i v e s 68 at Eq. [2.22] d i r e c t l y . In c a l c u l a t i n g the summation i n the exponent of the c o r r e l a t i o n function F ( t ) , i t i s more convenient to d i v i d e the l a t t i c e i n t o two regions, I and I I . Region I contains M s i t e s that are nearest to the o r i g i n . Region II contains the remaining s i t e s . In region I, the distance of the s i t e s to the resonating centre i s small, and the function <e ^ - a ) ^ t > changes r a p i d l y from one s i t e to another, and the summation over s i t e s i n s i d e region I i s retained. In region I I , the function <e ^ w^ r^ t> changes more smoothly over the s i t e s and, to a good approximation, the l a t t i c e can be assumed to be an i s o t r o p i c , homogeneous continuum and the summation over s i t e s i n region II can be approximated by an i n t e g r a l over region I I . Rewriting the summation i n Eq. [2.22], we have F(t) = F I ( t ) F I I ( t ) , [2.23] M - i a ) ( r . ) t -pM + p E <e > , where F-^t) = e J = 1 co -ico(r".)t -p Z [1 - <e 3 >] and F I I ( t ) = e J = M + 1 I f we now define i a ( w ) (a = I, II) as the l i n e shape function due to impurities i n region a alone, i . e . I (io) = ± -or 2TT e i t 0 t F a ( t ) d t (a = I, I I ) , [2.24] 69 then as a consequence of the f a c t o r i s a b i l i t y of the t o t a l c o r r e l a t i o n function F ( t ) , the t o t a l l i n e shape function can be expressed as the convolution of I-j.(u>) and 1^.^.(0)), K u ) = —CO I K U H J KU ) - y)dy . [2.25] This r e s u l t follows as a s p e c i a l case from a general theorem i n p r o b a b i l i t y theory that the j o i n t p r o b a b i l i t y density function of the sum of a number of independent random variables i s obtained by convoluting a l l the i n d i v i d u a l p r o b a b i l i t y density functions (42). In t h i s case, the independent v a r i a b l e s are the contributions to w due to impurities i n region I and region I I . In another case when the l i n e shape i s broadened by se v e r a l species of non-interacting impurities i n such a way that each species of impurities can be d i s t r i b u t e d as i f a l l other species are absent ( t h i s i s approximately true when the t o t a l concentration of impurities i n c l u d i n g a l l the species present i s low), then the t o t a l l i n e shape function i s simply the convolution of the i n d i v i d u a l l i n e shape functions when a p a r t i c u l a r species of impurities alone causes the broadening (38, 39). Using Eqs. [2.10] and [2.22], the c o r r e l a t i o n function F ( t ) can also be rewritten as F(t) = n F(q,t) , [2.26] q where 70 - i t o ( r . , q ) t - pp(q) mZ [1 - e J ] F ( q , t ) = e J " 1 T h e r e f o r e when t h e c o n c e n t r a t i o n i s l o w , t h e c o r r e l a t i o n f u n c t i o n F ( t ) , can be f a c t o r i s e d i n t o a p r o d u c t o f i n d i v i d u a l c o r r e l a t i o n f u n c t i o n s , F ( q , t ) ; s u c h t h a t each F ( q , t ) c o r r e s p o n d s t o a c o r r e l a t i o n f u n c t i o n when a l l t h e i m p u r i t i e s a r e i n t h e same s t a t e q. The t o t a l l i n e shape f u n c t i o n I (to) can t h u s be e x p r e s s e d as a c o n v o l u t i o n o f a l l t h e l i n e shape f u n c t i o n s I(q,to) c o r r e s p o n d i n g t o d i f f e r e n t s t a t e s q, where I(q,to) = ~ ^ e l u ) t F ( q , t ) d t . [2.27] J _o 3. LINE SHAPE FUNCTION DUE TO A SYSTEM OF CLASSICAL MAGNETIC DIPOLES We s h a l l now s p e c i a l i z e t h e e x p r e s s i o n s d e r i v e d e a r l i e r t o t h e l i n e shape b r o a d e n e d by a s y s t e m o f d i l u t e c l a s s i c a l m a g n e t i c d i p o l e s , each h a v i n g a i n t e r n a l s t a t e s . The m a g n e t i c moment o f a d i p o l e , d e n o t e d by y~(q) , i s d e t e r m i n e d u n i q u e l y by t h e s t a t e v a r i a b l e q = 1, • • • ,o . I t i s s t r e s s e d h e r e t h a t q i s c o m p l e t e l y g e n e r a l . F o r example, i t can r e p r e s e n t a quantum s t a t e o f t h e i m p u r i t y , an o r i e n t a t i o n i n s p a c e o f t h e m o l e c u l a r a x i s o f t h e i m p u r i t y , o r a s p e c i e s o f i m p u r i t i e s when s e v e r a l s p e c i e s o f i m p u r i t i e s a r e p r e s e n t . The l o c a l m a g n e t i c f i e l d p r o d u c e d by TT(q) a t t h e r e s o n a t i n g c e n t r e , w h i c h i s a t the o r i g i n , i s g i v e n by 71 H C r . q ) = - ±- [y ( q ) - • [2.29] r 3 r Since we are r e s t r i c t i n g our c a l c u l a t i o n s to the high f i e l d l i m i t when H(r,q) << H where H i s the external magnetic f i e l d and to the case i n which the impurities and the host centres are unlike spins, only the component of H(r,q) which i s p a r a l l e l to the external f i e l d H i s e f f e c t i v e i n broadening the resonance l i n e . The frequency s h i f t i s then given by r~ \ Y r—f ^  — 3(y"(q) -T) (n-7), r o oi(r,q) = -[u(q)-n 2 -] , [2.30] r • r where n i s a unit vector along the d i r e c t i o n of H and Y i s the gyromagnetic r a t i o of the resonating centre. When the external f i e l d H i s s u f f i c i e n t l y high, which i s usually true f o r most nuclear magnetic resonances, ll(q) precesses r a p i d l y about H and only the l o c a l f i e l d produced by the averaged "jT(q) i s e f f e c t i v e i n broadening the NMR l i n e shape. In other words we assume that the magnetic dipole moments ]T(q) are p a r a l l e l to the d i r e c t i o n of the external f i e l d H . Eq. [2.30] then becomes W ( r \ q ) = - (1 - 3cos 20) , [2.31] r where 9 i s the angle between H and r . A case i n which the magnetic dipole moment i s not exactly p a r a l l e l to the external f i e l d w i l l be discussed 72 i n the next sec t i o n . where From Eqs. [2.23[ and [2.24], the c o r r e l a t i o n function becomes a F(t) = n F q ( t ) , ' [2.32] q=l F q ( t ) = F ^ ( t ) F j I ( t ) , [2.32a] r \ co -, M " " i 0 0 ( R. >q) t , F?(t) = e " p M P ( q ) E £ ( p p ( q ) ? e 3 ) \ [2.32b] i k=0 K ! j = l oo • - i t j j ( r . , q ) t -pp(q). S (1 - e J ) F ^ C t ) = e J = M + 1 , [2.32c] and p(q) i s the p r o b a b i l i t y that a magnetic dipole i s found i n the state q. The t o t a l c o r r e l a t i o n function F(t) i s f a c t o r i z e d into a product of a functions, F q ( t ) , q = 1, • • • ,a , and each F q ( t ) i s the c o r r e l a t i o n function r e s u l t i n g when a l l the magnetic dipoles are found i n the state q. For the rest of t h i s s e c t i o n we s h a l l consider only the c o r r e l a t i o n function F q ( t ) and i t s Fourier transform, which i s the l i n e shape function I q(co). The t o t a l l i n e shape function i s obtained, as mentioned before, by successively convoluting I q(to) , q = 1, • • • ,a Let us f i r s t c a l c u l a t e Fj-j-(t) which i s due to impurities i n region I I . From now on we s h a l l always choose a s p h e r i c a l volume as region I. 7 3 I t w i l l be r e f e r r e d to as the c u t - o f f region and i t s radius as the c u t - o f f radius. I f the cut-of f radius r Q i s s u f f i c i e n t l y large, the v a r i a t i o n i n the frequency s h i f t s produced by impurities that are neighbours to each other w i l l be s u f f i c i e n t l y small that,the summation can be approximated by an i n t e g r a l . Denoting v as the volume of a unit c e l l i n the c r y s t a l l a t t i c e , a as one of the l a t t i c e constants and f as the number of l a t t i c e s i t e s per unit c e l l the summation i n Eq. [2.32c] can be approximated by 00 - e r 2 d r [ l [2.33] o Sub s t i t u t i n g y = cos 8 , x = ( r D / r ) 3 , a(q) = Y y ( q ) / r 0 , and u = a(q)t into Eq. [2.33] , we get = q = MJ(u), [ 2 . 3 4 ] where M = 4irfr o/(3v) i s the number of l a t t i c e s i t e s i n s i d e the c u t - o f f region. The evaluation of the i n t e g r a l has been done by Grant and Strandberg 74 (39), although one of their asymptotic series (Eq. [63a] of (39)) is incorrect. The value of the integral is given by J(u) = \ -1 - 2iu(l + — log ^ ~ 1 )/3 /T~ /T~ + 1 + e 1 U [ | + | M(l,3/2, 3iu) + | M ( l , l / 2 , 3iu) ] • /o _9- 0 0 ( l / 2 ) r ^ " T/3 6 r I 0 ( " 1 / 2 ) r r ' ( r + 1/2) M(l,-r+l/2,2iu)|, [2.35] where the function M(a,b,z) i s a confluent hypergeometric function (43), Our problem now i s to i n v e s t i g a t e the behavior of the function J(u ) . F i r s t l y we have the following important r e l a t i o n J*(u) = j(-u) , [2.36] where * represents the complex conjugation. The above r e l a t i o n follows d i r e c t l y from the d e f i n i t i o n [2.34] of J ( u ) ; or from applying the i d e n t i t y M*(a,b,ic) = M(a,b,-ic) (c r e a l ) to Eq. [2.35]. For small and large values of u, we expand J(u) i n t o a power 75 series and an asymptotic series, respectively (a). Power series expansion n=0 where P = j n k ^ Q (n-k)!k!(2k+l) (b). Asymptotic series expansion J(u) = -1 -4 iu(l + ±- l o g ^ J , ) + 2^| u| /3 /3 + 1 3/3 „ r . s+1 (1/2) _ 1 Ar IU y yn 3/16 S,. sn-% n=2 (xu) -2iu s (1/2) 1(2n+l) s (1/2) f J n - l _ + l n+1 9 1 / 0. .n ^ . n n J n=l (3iu) n=l (2iu) + 0(|u| S 1) , [2.38] where c = y ,—..,. , n ; . n k=l fa-k)!k!(k-M?) and (a) n = a(a + 1) (a+n-1) . For intermediate values of u, i t is most convenient to represent 76 J(u) g r a p h i c a l l y . We performed the i n t e g r a l i n Eq. [2.34] numerically using the IBM 360/67 computer. The function J(u) and i t s asymptotic -7/2 expansion, up to the term u , are p l o t t e d i n F i g s . 16a and 16b f o r u 210 . We denote the r e a l part of J(u) by Real (J(u)) and the imaginary part by Imag ( J ( u ) ) . For u < 0, the behavior of J(u) can be obtained from the graphs using the r e l a t i o n given Eq. [2.36]. From the graphs and s e r i e s expansions, i t i s c l e a r that : f o r |u| <_ 1 , J(u) = 2u 2/5 + 0(u 3) ; [2.39] and f o r | u| >_ 4 J(u) = — | u f - 1 - 4 i u ( l + — l o g ^ ~ 1 ) + OClul" 1). [2.40] 3/3 /3 /3 + 1 From the properties of the function J(u) , one can deduce the general behavior of the l i n e shape function 1^(10) , which i s due to impurities i n region II and being i n the magnetic state q. The f i r s t conclusion one can draw i s that the l i n e shape function i s not symmetrical about i t s centre, which by d e f i n i t i o n corresponds to no frequency s h i f t . Since the l i n e shape function I^(u>) i s obtained by Fourier transforming the corresponding c o r r e l a t i o n function F j j ( t ) , the l i n e shape near the centre i s determined by the behavior of F ^ ( t ) at large t . Since 77 F i g . 16a. Plot of Real(J(u)) versus u. The s o l i d l i n e represents Real(J(u)), while i t s asymptotic expansion up to the -7/2 term u i s represented by points 0. 79 F^Ct) = e " p P ( c l ) M J ( t a ( q ) ) , . [2.41] i t i s dete r m i n e d by the a s y m p t o t i c b e h a v i o u r o f J ( t a ( q ) ) . I t can be seen from Eq. [2.40] t h a t the a s y m p t o t i c b e h a v i o u r o f J ( t a ( q ) ) i s l i n e a r i n t . C o n s e q u e n t l y the shape n e a r the c e n t r e i s L o r e n t z i a n w i t h t h e peak i n t e n s i t y s h i t t e d by an amount A q = _ 8 T r f Y p p ( q ) p ( q ) [ 1 + 1_ /3_-± _ [ 2 > 4 2 ]  1 1 9 V /3 /3 + 1 T h i s s h i f t o f the peak i n t e n s i t y i s p r o p o r t i o n a l to b o t h t h e i m p u r i t y c o n c e n t r a t i o n p and to the magnetic moment u ( q ) , b u t i s ind e p e n d e n t o f the c u t - o f f volume. The b e h a v i o r of l!jl (u>) a t the wings i s d e t e r m i n e d by the b e h a v i o r of J ( t a ( q ) ) f o r s m a l l t . S i n c e the l e a d i n g term i n the power s e r i e s e x p a n s i o n o f J ( t a ( q ) ) i s q u a d r a t i c i n t , the wings a r e p r e d o m i n a n t l y G a u s s i a n . However, due to the i n f l u e n c e o f the term w h i c h i s c u b i c i n t , the r e g i o n o f t h e wings which i s n o t too f a r out i s asymmetric i n such a way t h a t , i n s p i t e o f the s h i f t i n the peak f r e q u e n c y , the f i r s t moment of t h e l i n e shape v a n i s h e s . T h i s asymmetry of the l i n e shape i s e x p e c t e d when t h e system has a f i n i t e r a t h e r t h a n i n f i n i t e temperature as has been demonstated by M c M i l l a n and Opechowski by the method o f moment e x p a n s i o n s (46). The l o g a r i t h m o f the c o r r e l a t i o n f u n c t i o n F q ^ ( t ) i s l i n e a r i n the p r o d u c t pMp(q). I t r e p r e s e n t s the average number o f i m p u r i t i e s which i s i n s t a t e q c o n t a i n e d i n s i d e the c u t - o f f volume. F o r l a r g e v a l u e s o f pMp(q), the c o r r e l a t i o n f u n c t i o n d e c r e a s e s v e r y r e p i d l y as t i n c r e a s e s . C o n s e q u e n t l y 80 the l i n e shape i s mainly determined by the short time behavior of J ( t a ( q ) ) , and looks l i k e a Gaussian. On the other hand, when the value of pMp(q) i s small, the l i n e shape w i l l be determined by the long time behavior of J(ta(q) ) and w i l l be predominantly Lorentzian. The parameters that influence the l i n e shape are neither simply the concentration of the impurities nor the cut- o f f volume, but involves the product of both f a c t o r s . For pMp(q) = 0.01, the l i n e shape i s p r a c t i c a l l y Lorentzian; f o r pMp(q) = 0.1, 0.5, and 2.0, the l i n e shape function and the corresponding Gaussian and Lorentzian l i n e s f i t t e d to the same peak i n t e n s i t y are shown, r e s p e c t i v e l y , i n F i g s . 17, 18, and 19. From the graphs, we see that, to a good approximation, the l i n e shape can be represented by a Lorentzian curve when pMp(q) < 0.1, and by a Gaussian one, when pMp(q) > 2. In.the Lorentzian l i m i t , the l i n e shape i s characterized by the half-width u, which i s defined as the distance from the centre of the l i n e to the point of h a l f i n t e n s i t y and i s given by This half-width i s pro p o r t i o n a l to the impurity concentration p and the magnitude of the magnetic moment u(q), but i s independent of the cu t - o f f volume. This i s i n agreement with the c a l c u l a t i o n s and observations of Anderson (44) and Grant and Strandberg (45). In the Gaussian l i m i t , the 2 l i n e shape i s characterized by the second moment <OJ (q)> which i s given by q = 8T r 2pfYy(q)p(q)  % 9/3 v [2.43] * ( q ) > = 1 6 . f P Y 2 y 2 ( q ) p ( q ) . [2.44] 1 5 r j v 81 F i g . 17. Line shape function I^(w) when pMp(q) = 0.1. The Lorentzian and Gaussian l i n e s , both f i t t e d to have the same peak i n t e n s i t y as I q ( w ) , are represented by points x and 0, r e s p e c t i v e l y . 82 F i g . 18. L i n e shape f u n c t i o n I q (UJ) when pMp(q) = 0.5. The L o r e n t z i a n and G a u s s i a n l i n e s , b o t h f i t t e d t o have the same peak i n t e n s i t y as I q ^ ( w ) , a r e r e p r e s e n t e d by p o i n t s x and o. 83 <*> / oc ( q ) F i g . 19. L i n e shape f u n c t i o n 1^(10) when pMp(q) = 2. The L o r e n t z i a n and G a u s s i a n l i n e s , b o t h f i t t e d t o have the same peak i n t e n s i t y as I q ( t o ) , a r e r e p r e s e n t e d by p o i n t s x and o, r e s p e c t i v e l y . 84 The second moment i s proportional to the impurity concentration p and the square of the magnetic moment, but i s i n v e r s e l y p r o p o r t i o n a l to the c u t - o f f volume. The same f u n c t i o n a l r e l a t i o n i s obtained i f i t i s c a l c u l a t e d by means of the method of moments (33, 4 6 ) . We s h a l l now estimate the e r r o r introduced by approximating the d i s c r e t e l a t t i c e sums i n Eq. [2.32c] by the i n t e g r a l of Eq. [2.33]. We f i r s t note that when each p o s i t i o n vector r\ changes by an amount <5r\ , the f r a c t i o n a l change i n the c o r r e l a t i o n function F q ^ ( t ) i s , from Eqs. [2.3l] and [2.32], rjl -iw (r ,q)t I I ; r e 2 -9 = 3itpYy(q) I - [(3cos 9. - l ) 6 r . + r . s i n 26.69.] F q ].(t) j=M+l r Due to destrucive i n t e r f e r e n c e , the absolute value of the i n f i n i t e sum i s assumed to be about the same or le s s than the absolute value of the extremal term i n the i n f i n i t e sum. Since S r . and r.SO . < a , we have J J 3 ~ ' 5 F l l ( t ) ' K 3 t P Y y ( q ) a | F q x ( t ) | 3 27/3 ,v ,a t , . 1 N < — 2 ~ ( — ) — — (when p M « 1) 8TT rQ o T c < r^ x Tr, O C 85 where x i s the time at which the value of the c o r r e l a t i o n function i s c decreased to — of i t s value at t = 0, i . e . e F I I ( T c ) = F q i ( 0 ) / e . When p < 1% and | t | < 5x £ , the f r a c t i o n a l change i s le s s than -—^ when the cut- o f f radius r Q i s greater than a few l a t t i c e constant i n length. To compare the continuum i n t e g r a l with the d i s c r e t e l a t t i c e sums, we c a l c u l a t e numerically the sum i u r ^ ( l - 3 c o s 2 6 ) / r 3 J x ( u ) = i I [1 - e ° ' J J ] [2.45] j=M+l fo r a face-centred cubic l a t t i c e and f o r d i f f e r e n t c u t - o f f radius : r D = > m = 1, 3, and 6. The l a t t i c e sum J-^(u) as defined here depends on the d i r e c t i o n of the external f i e l d H , because the angles 8^  depend on the l a t t e r . The behavior of J-^(u) when n l i e s i n the d i r e c t i o n (0,0) which i s one of the highest symmetry d i r e c t i o n s i n the face-centred cubic l a t t i c e i s shown i n F i g s . 20 and 21. The case when rT l i e s along two other a r b i t r a r y d i r e c t i o n s , (TT/18, TT/36) and (2TT/9, TT/18) , i s shown i n F i g s . 22 to 25. In a l l cases, J-^(u) converges quite r a p i d l y to J(u) when r Q increases from a//2 to /3a. I t can s a f e l y be concluded that f o r a face-centred cubic l a t t i c e the continuum approximation should be good whenever o a 0 10 20 U . 20. Plot of Real(J(u)) and R e a l C J ^ u ) ) versus u when n = (0, The s o l i d l i n e represents R e a l ( J ( u ) ) . RealCJ^u)) f o r r /a = 1//2 , /3j2 , /3 i s represented by points., x, • and r e s p e c t i v e l y . 87 CD < 2 0 - 2 • X K 0 10 u 20 F i g . 21. Plot of ~Imag(J(u)) and -ImagCJ^u)) versus u when rT = ( 0 , 0 ) . The s o l i d l i n e represents -Imag(J(u)). -Imag(J (u)) for r Q / a = , /372 , and / 3 i s represented by points., x, and o, r e s p e c t i v e l y . 88 F i g . 2 2 . P l o t of R e a l ( J ( u ) ) and R e a l ( J (u)) v e r s u s u when n = (ir/18, n'/36). The s o l i d l i n e r e p r e s e n t s R e a l ( J ( u ) ) . ReaKJjCu)) f o r r Q / a = l f / 2 , /3]~2 , and /3 i s r e p r e s e n t e d by p o i n t s . , x, and o, r e s p e c t i v e l y . s o l i d l i n e r e p r e s e n t s - I m a g ( J ( u ) ) . -Imag ( J ^ (u) ) f o r r Q / a = 1//2, fs~l2, and i s r e p r e s e n t e d by p o i n t s . , x, and o, r e s p e c t i v e l y . 90 R e a l ( J (u)) for r Q / a = 1 / / 2 , /3/2 , and / J i s represented by points., x, and o, r e s p e c t i v e l y . 10 20 U Fig. 25. Plot of -Imag(J(u)) and -ImagCJ^u)) versus u when n= (2TT/9, 7TT/18). The solid line represents -Imag(J(u)). -Imag (J 1 (u)) for r Q/a = l/7l, /3/2, and V3> is represented by points . , x, and o, respectively. 92 the cut- o f f radius i s as small as /Ja or l a r g e r . Since we are r e s t r i c t i n g our c a l c u l a t i o n to low impurity concentrations (p <_0.1%), we can always choose a s u i t a b l e c u t - o f f radius r Q (for example, we can always choose r Q = /3a ) such that pM << 1 i s always true and the l i n e shape i s Lorentzian i n t h i s l i m i t , i . e . pMp(q) u j I q T ( o 3 ) = - ^ - — — — , [2.46] (to|) + (to - Ato^) where to? and Aw?. are given i n Eqs. [2.42] and [2.43], r e s p e c t i v e l y . Here we note that although r i g o r o u s l y the f i r s t moment of I ^ u ) always vanishes, we i d e n t i f y the centre of the Lorentzian l i n e with the peaks of I^O*)),-not with i t s centre, since the s h i f t of the peaks from the centre i s (from Eqs. [2.42] and [2.43]). only about 0.13 of the value of the half-width of 'the l i n e . To evaluate the c o r r e l a t i o n function F q ( t ) due to impurities i n s i d e the c u t - o f f region, we note that the sum (see Eq. [2.32b]) M i Y y ( q ) t ( l - 3cos 26.)/R 3 has a magnitude of not exceeding M and we may choose pM << 1. We can then approximate Eq. [2.32b] by r e t a i n i n g only the f i r s t two terms, i . e . „ , N M i Y y ( q ) t ( l - 3 c o s 2 0 . ) / R 3 F q ( t ) = e " p M P ( q ) [ l + pp(q) I e j j ] . [2.49] 3=1 93 The l i n e shape function Iq(to) due to impurities i n the cut-off region i s thus the sum of a set of 6 - f u n c t i o n s , one represents the main l i n e and the others represent side-bands whose weights are proportional to the concentration P : M l q(to) = e ~ p M p ( q ) [ 6 ( c o ) + pp(q) I 6(10 - co.)] [2.47] 3=1 J where co. = - Y y ^ q ) (1 - 3cos 26.) . 3 R3. (It may happen that some of the d e l t a functions i n the second term may coincide, hence the number of side-bands may be smaller than M). Now the l i n e shape due to contributions of impurities i n both region I and II when the impurities are i n state q i s , from Eq. [2.32], 1 % ) = r°° lq(co - y ) I q I ( y ) d y In the Lorentz l i m i t , from Eqs. [2.46] and [2.47], we have * L (a,J)2 + (to - Aco^) 2 M + Pp(q) I ~ — 0 ~ ~ — r \ • [2.48] j = l (toj) 2 + (w - (o. - Ato q T) 2 *2 3 94 Therefore, the line shape due to contribution of impurities in both regions consists of a main line centred at Aw!* and side-bands at co. + Aw!* > II J I I and a l l the lines, main line and side-bands, are Lorentzian, since pM << 1. The above considerations should also be valid for other l a t t i c e systems. For lattices which are less symmetric than the face-centred cubic l a t t i c e , the value of r Q / a may have to be somewhat larger for the continuum approximation to be sufficiently close to the discrete l a t t i c e summation. However, in these cases, the number of sites in a unit c e l l is smaller, and a cut-off radius r Q can usually be chosen such that pM << 1 and the la t t i c e sums can be approximated by the continuum integral. Therefore, we can conclude that the correlation function of a system of classical magnetic dipoles u (q), q = 1, . . . ,0 , can be expressed, to a good approximation, as a product of a functions ™ / W I - L T / ^ t \ \ M iYy (q)t(l-3cos 26 .)/R3 F q ( t ) = e - p M p ( q ) ( l + J ( t a ( q ) ) [ 1 + p p ( q ) ^ e j 3 ] > [ 2 5 Q ] 3=1 The Fourier transform lq(co) of F q(t) consists of a main line at the centre and side-bands at - (1 - 3cos 2 G . ) . As before, a l l the lines , RT 3 3 main line and side-bands, are Lorentzian, since pM << 1. The total line shape function is the convolution of the functions l q(co), q = 1, • • • ,o . 95 4. LINE SHAPE FUNCTION DUE TO A SYSTEM'OF SPINS, S = 1/2 AND S = 1 We s h a l l now consider two examples. We s h a l l c a l c u l a t e the l i n e shape function of a powdered sample : one due to a system of spins S = 1/2, and the other S = 1. SPIN 1/2. The d e r i v a t i o n of the frequency s h i f t Or,q) due to a spin which i s located at r" and i n the magnetic state q i s simple and straight-forward. For s i m p l i c i t y we s h a l l assume that the spins have an i s o t r o p i c g-factor and that they i n t e r a c t with the n u c l e i by means of the d i p o l a r i n t e r a c t i o n . The p a i r hamiltonian f o r a nuclear spin I at the o r i g i n and the spin S = 1/2 at Y then consists of three terms : H = H + H + H , [2.51] z e d ' H = - YI-H , z ' H& = 3gS-H , and H = M . [s.I - 3 ( g-r)(I-r) d 3 2 J r r Here represents the Zeeman energy of the nuclear s p i n I, which has gyromagnetic r a t i o Y, i n an external magnetic f i e l d H. represents the Zeeman energy of the impurity spin S. H represents the magnetic 96 dipolar i n t e r a c t i o n between the nuclear spin I and the impurity spin S. f3 i s the Bohr magneton. I f the magnitude of H i s s u f f i c i e n t l y large . ( >_ 1 kilogauss), as i s generally found i n most of the resonance experiments, then the following inequality holds : tf >> tf » tf, . [2.52] e z d We can evaluate the frequency s h i f t by means of the perturbation method. We treat tf + tf as the zeroth order term and tf , as the perturbation, e z d r In a coordinate system i n which the z-axis i s i n the d i r e c t i o n of the external f i e l d H, we have tf = -YHI , z z and tf = ggHS . e z The eigenstates of tf and tf are denoted by <J> (m ) and ij> (m ) (m = ± 1/2) corresponding to eigenvalues E(m ) = - m'YH and E (ra ) o JL J. 6 L> = ggm H , respectively. The eigenstates of tf + tf are then given by (j)(mT)^(m ) corresponding to eigenvalues (-m Y + m Bg)H . The f i r s t order dipolar energy i s then E (m s,r; ir^) = ^  m ^ d - 3cos 26), [2.53] r where 0 i s the angle between r and the z-axis. Since nuclear magnetic 97 resonance i s observed for the t r a n s i t i o n m^. > m^. - 1 , the frequency s h i f t io("r,m ) of the resonance i s then given by co(r,ms) = ^ | m s ( l - 3cos 2 e ) . [2.54] r This equation i s the same as Eq. [2.31] i n the case of c l a s s i c a l magnetic dipoles. In f a c t , the induced magnetic moment of the spin S i n state m^  i s -Bg<iKms) |s|ir(ms)> = -egmgn , where n i s a unit vector along the di r e c t i o n of H . With u(q) = -$gq , q = ± 1/2 , the case of S = 1/2 i s the same as the previously discussed case of c l a s s i c a l dipoles each of which has two states. The p r o b a b i l i t y with which a spin S i s found i n state q i s given by the Boltzmann factor -BgqH/kT p ( q ) 2cosh(BgH/2kT) where T i s the temperature and k i s the Boltzmann's constant. The l i n e shape function i s the convolution of two l i n e s , Iq(o>), q = ± 1/2, each of which i s a Lorentzian l i n e with side-bands i n the wings. Since the convolution of two Lorentzian l i n e s i s again a Lorentzian l i n e s which has a s h i f t and half-width equal to the sum of the s h i f t s and half-widths, respectively, by the two convoluted l i n e s , neglecting the side-bands, from Eqs. [2.42] and [2.43] the l i n e shape I(o>) i s again Lorentzian with h a l f width 98 2 = 4TT pfYBg t a n h ( g g H / ( 2 k T ) ) > [2.55] 2 9/3v and s h i f t AM = - — co, [1 + — log ^  " 1 ] [2.56] 17 2 /3 /3 + 1 In the case when the temperature i s i n f i n i t e l y high and the side-bands can be neglected, which i s equivalent to putting p(q) = 1/2 and making the cut-off radius r Q shrink to zero, the c o r r e l a t i o n function becomes 2 7 T f r o P 3 3 r — - [ J ( Y 3 g t / ( 2 r 3 ) ) + J(-Yegt/(2r 3))] F(t) = l i m e J V  ro-K) From the assymptotic expansion, Eq.[2.38], of the function J ( u ) , we have J ( u ) + J ( - u ) = -2 + 4TT | u ] / (3/3) + O d u l " 1 ) , and F(t) = e - ^2 f p Y 3 g | t | / ( 9 / 3 v ) m [ 2 > 5 8 ] The l i n e shape function i s therefore a Lorentzian with a half-width = 4 T r 2fpY3g / ( 9/3v) This r e s u l t i s the same as that obtained by Abragam (18) 99 In the above considerations, i t has been assumed that the impurity-states have a l i f t time which i s long compared to the precession period of the nuclear spins. This happens when the coupling between the impurity spins and thermal bath i s much smaller than the Zeeman energy of the nuclear spins. In this "long l i f e - t i m e c o r r e l a t i o n l i m i t " , the general expression fo r the c o r r e l a t i o n function i s represented by Eq. [2.26], i n which £-E(q)/kT P ( q ) = Z e - E ( q ) / k T ' [ 2 ' 5 7 ] where E(q) i s the energy of the impurity i n state q. In the case of the "short l i f e - t i m e c o r r e l a t i o n l i m i t " , when the t r a n s i t i o n rates between d i f f e r e n t states of the impurity spins are very much l a r g e r than the precession frequency of the nuclear spins i n the l o c a l f i e l d produced by the impurity spins, the l o c a l f i e l d s f l u c t u a t e s so r a p i d l y that, i n e f f e c t , the nuclear spin experiences an averaged l o c a l f i e l d produced by an averaged magnetic moment -tf /kT ^ true <y> = -tf /kT • [ 2' 5 9 ] t r e 6 For S = 1/2, the observed s h i f t i n resonance frequency i s then w(r) = -Yu(l - 3 c o s 2 9 ) / r 3 , [2.60) where u = -pg tanh(Y3gH/(2kt))/2 100 In the Lorentzian l i m i t , s u b s t i t u t i n g the value of magnetic moment u in t o Eqs. [2.42] and [2.43], and comparing the s h i f t and half-width with those i n the case of the "long l i f e - t i m e c o r r e l a t i o n l i m i t " , i t can be seen that both cases give the same s h i f t and half-width. This r e s u l t i s not only true f o r S = -|- , but i s true generally i n the Lorentzian l i m i t . This can be shown i n a s t r a i g h t forward fashion. From Eqs. [2.42] and [2.43], i n the "long l i f e - t i m e c o r r e l a t i o n l i m i t " , the t o t a l half-width and the t o t a l s h i f t are given by, r e s p e c t i v e l y , 8Tr 2pfY „ . . •• ' ^ = Ep(q)y(q) , 2 9/3v q and Aco = - Sp(q)y (q) . 9v q But then E p(q)p(q) i s the average magnetic moment of the impurity i n the q case of the "short l i f t - t i m e c o r r e l a t i o n l i m i t " . Therefore, both l i m i t s give the same l i n e shape. However, when the impurity concentration i s high , the Lorentzian l i m i t cannot be used, and the l i n e shape i s d i f f e r e n t f o r the two cases. The "short l i f e - t i m e c o r r e l a t i o n l i m i t " then usually gives a narrower l i n e . We s h a l l now inve s t i g a t e the l i n e shape due to impurities of spin i n a powder sample. From Eq. [2.50], the c o r r e l a t i o n function, f o r a p a r t i c u l a r • o r i e n t a t i o n of H, can be expressed as 101 102 where 8TT fp . u > = Z ~ 5 » 2 9/3 P . / J • /3 + 1 v R. J and 6 = Yu/v . The l i n e shape function consists of a c e n t r a l main l i n e I . (to) = main r 2 TT [ t O , + ( t O + CO ) ] P which i s Lorentzian with half-width SrTfpYu 9/3v and s side-bands pN(k)owl T ( U ) =: 2 k TT .dv_ o t o 2 + (to + (Op + t 0 j ( l - 3 y 2 ) ] 2 k = 1,•••,s, each of which corresponds to the c o n t r i b u t i o n due to impurities i n a nearby s h e l l . The shape of the side-band has a general form given by the i n t e g r a l 103 f.(o>) = o a 2 + [to + b ( l - 3 y 2 ) ] 2 [2.64] To see how f ( t o ) behave, the f u n c t i o n s f ( t o ) and g ( t o ) , which i s d e f i n e d a s , g(co) = c(to + b) 2 2b > to > -b o t h e r w i s e are p l o t t e d i n F i g . 26 for the following values : a = 1, b = 10, and c = 0.27. From the p l o t i t i s seen that the behavior of f(to) i s s i m i l a r to the smearing out of the well-known l i n e shape g(to) of a p o l y c r y s t a l l i n e sample due to a n i s o t r o p i c Knight s h i f t (47, 48). The i n t e g r a l i n Eq. [2.64] can be evaluated i n a straight-forward manner, gi v i n g f(u>) 4a/3bri cos (7- + <f>) log n + 3b + n/6b( coscj) + sin<|>) n 2 + 3b - r|/6b"(cos<j> + sincj>) , 0 . , I T , ,. r , . -l/6b~ n (sind) - cbscb) -, \ + 2 s i n ( r + <j)) [ TT + tan — ^ ] V , 3b - n 2 ' [2.65] where ri = [ a 2 + (to + b ) 2 ] 1 / 4 o 104 OJ 26. Plot of f ( c o ) and g (co ) versus co when a = 1, b and c = 0.2 7. . -= 10, 105 , 1 _ -l,co •+ b N and n = — tan ( ) z. a It would be us e f u l to know the p o s i t i o n and peak i n t e n s i t y of these s i d e -bands. To do t h i s , we f i r s t define the function f°° A o a2 + [co + b ( l - 3 y 2 ) ] 2 [2.66] Then for a l l values of co, we have f^co) > f(co) > 0 . Th e i n t e g r a l i n f (to) can be evaluated e a s i l y , g i v i n g f (co) = — sinCr + <j>) 4a/3bn When co = com = - b + a//3 , f (co) attai n s i t s absolute maximum value max . . (3) 4a/2ab We now expand the shape function f(co) of the side-band about the point co„ . Let e be an i n f i n i t e s i m a l deviation from ui . Then m m f ( % + e ) = [ 1 - 1 0 v ? , , # ) 3 / 2 + 0(e 2) + . , . ] . [2.67] 4a/2a¥ 2 7 T T 3 1 / 4 b 106 Therefore, whenever the maximum of f(to) i s located f a r t h e r away from the centre than the half-width of the main l i n e , i . e . b > a , the maximum of f(to) occurs at about the same p o s i t i o n to , and the maximum 3 l / 4 can be approximated by the expression — — . For a face-centred cubic 4a/2ab l a t t i c e and for low impurity concentration (p <_0.1%), i f the c u t - o f f region includes a few nearest s h e l l s , then the above approximation i s indeed nearly exact as the side-bands are r e a l l y f a r away. I t i s u s e f u l to r e l a t e the p o s i t i o n and peak i n t e n s i t y of the side-bands to the line-width to, and the peak i n t e n s i t y I Q of the main l i n e . Denoting to ^ . as the p o s i t i o n and 1^ . as the peak i n t e n s i t y of the side-band Ik(u) corresponding to the kth s h e l l , we define the following q u a n t i t i e s : / 9/3v k mk % „ 2 D3 ' 32TT pRT P ' - 1 / i - 3l/W>u vh P k V X o - 4 / 2 - I5 kl a , n - T / T ( . ~ P | 3 1 / 4 p N ( k ) . r 13/4 a n d Q k " Ik / Imain ( wmk ) ~ P k + l ?k> The values of E , , P. , and Q, for the f i r s t 20 s h e l l s f o r a face-centred k k k cubic l a t t i c e with l a t t i c e constant a are l i s t e d i n Table 4. From the table, we see that f o r low impurity concentrations, the side-bands are d i s t i n c t i v e l y r e s o l v a b l e . For example, when p = 0.05%, the p o s i t i o n of the peak i n t e n s i t y 107 S h e l l k N(k) i y a - p § k P k P _ 3 / 2 Q k p l / 2 1 12 0.707 1.396 X i o " 1 23.47 4.575 X i o - 1 2 6 1.000 4.936 X i o " 2 19.74 4.809 X i o " 2 3 24 1.225 2.687 X I O " 2 107.0 7.725 X i o " 2 4 12 1.414 1.745 X i o " 2 66.39 2.022 X i o " 2 5 24 1.581 1.249 X i o " 2 157.0 2.448 X i o " 2 6 8 1.732 9.499 X i o " 3 59.99 5.413 X i o " 3 7 48 1.871 7.538 X i o " 3 404.1 2.296 X i o " 2 8 6 2.000 6.170 X i o " 3 55.83 2.125 X i o " 3 9 36 2.121 5.171 X i o " 3 365.9 9.783 X _3 10 10 24 2.236 4.415 X i o " 3 264.0 5.145 X i o " 3 11 24 2.345 3.827 X i o " 3 283.6 4.152 X i o " 3 12 24 2.449 3.358 X i o " 3 302.7 3.414 X i o " 3 13 72 2.550 2.978 X i o " 3 964.3 8.554 X i o " 3 14 48 2.739 2.403 X i o " 3 715.7 4.133 X i o " 3 15 12 2.828 2.181 X i o " 3 187.8 8.935 X i o " 4 16 48 2.915 1.992 X i o " 3 786.1 3.118 X i o " 3 17 30 3.000 1.828 X i o " 3 512.8 1.714 X i o " 3 18 72 3.082 1.686 X i o " 3 1282 3.642 X i o " 3 19 24 3.162 1.561 X i o " 3 444.0 1.082 X i o " 3 20 48 3.240 1.451 X i o " 3 921.1 1.938 X i o " 3 T a b l e 4. V a l u e s o f £, , P, , and 0. f o r a f a c e - c e n t r e d c u b i c l a t t i c e . 108 of the side-band due to the s h e l l as f a r away as the 9th s h e l l i s s t i l l about 10 times the half-width of the main l i n e , while t h i s peak i n t e n s i t y i s about 0.4 of the i n t e n s i t y of the main l i n e there. Spin S = 1 The p a i r hamiltonian f o r spin S = 1 i s the same as that of spin S = 1/2 (Eq. [2.51]), except that terms due to the c r y s t a l f i e l d are to be added to H . These a d d i t i o n a l terms make the c a l c u l a t i o n of the e magnetic moment and consequently the expression f o r the di p o l a r broadening of the resonating centre very complicated. Except f o r some s p e c i a l d i r e c t i o n s of the external magnetic f i e l d , one has to apply numerical s o l u t i o n s to i n d i v i d u a l cases. Once the magnetic moment i s known, the re s t of the c a l c u l a t i o n and discussions of the shape function w i l l be the same as that treated i n the previous s e c t i o n and i n the case when S = 1/2 of t h i s s e c t i o n . For our example, we s h a l l consider the broadening by oxygen 3 molecules at low temperatures. The molecule i s i n the ground e l e c t r o n i c state with no net e l e c t r o n i c o r b i t a l angular momentum. The l a t t i c e i s again face-centred cubic. We s h a l l consider only the ."short l i f e - t i m e c o r r e l a t i o n l i m i t " . In order to reduce the mathematics of the problem but to r e t a i n the e s s e n t i a l physics, we assume that the molecule has an i s o t r o p i c g-factor equal to that of a free electron and that the spin hamiltonian i s a x i a l l y symmetric (49, 50,51) : ' H = ggS-H + D[(S.n, ) 2 - S(S + l ) / 3 ] , [2.68] 109 where D i s the c r y s t a l f i e l d parameter and n^ i s a unit vector d i r e c t e d along the oxygen molecule f i g u r e a x i s . When D = 0, the r e s u l t is' exactly the same as the case of spin S = 1/2, except that the value of the magnetic moment i s now given by y ( q ) = - g g q E q = 0, ± 1 , [2.69] where n i s the unit vector i n the d i r e c t i o n of H . In the "short l i f e - t i m e c o r r e l a t i o n l i m i t " , the average induced magnetic moment becomes _ _ 2gg sinh(ggH/kT)n r„ n 1 y 1 + 2 cosh(ggH/"kT) * U , / U J When D ^ 0, the eigenvalue problem f o r i s much more involved. Except for the two cases when n^ i s e i t h e r p a r a l l e l or perpendicular to n, the expressions for the energy and the magnetic moment are too complicated to be u s e f u l i n actual c a l c u l a t i o n s . We s h a l l solve the problem numerically. To express H i n a more convenient form, we choose a coordinate e system that i s f i x e d with respect to the c r y s t a l l a t t i c e . Let n = (0,<f>) and n^ = (a,g). Eq. [ 2 . 6 8 ] can be rewritten as H = BgH[ sine (S cosd) + S sin*) + S cos0 ] e x T y 2 + D{[ sina(S cosg + S sing) + S cosa ] 2 - 2/3} . [2.71] x y z 110 At t h i s point, we note that i s not known a p r i o r i although there must be some more favorable o r i e n t a t i o n s i n which the energy of the molecule i s at a minimum. One expects that these o r i e n t a t i o n s to be along some symmetry di r e c t i o n s of the c r y s t a l l a t t i c e , e.g. (1,0,0), (1,1,0), (1,1,1) etc f o r a face-centred cubic l a t t i c e , and that the p r o b a b i l i t i e s f o r the molecule to be found along equivalent d i r e c t i o n s are equal. Since the (1,0,0) di r e c t i o n s i n the face-centred cubic l a t t i c e have the highest symmetry, we s h a l l f i r s t assume that the molecule i s equally l i k e l y to be found i n any of the s i x (1,0,0) d i r e c t i o n s . Then f o r a given set of values of (D, H, n", nj^) > 1 = 1> ""*> 6, i s diagonalised numerically to obtain the induced magnetic moment ^ ( n k ^ ) • The frequency s h i f t produced i s then _ _ _ 3(,T(E k i).r)(n.r) w(r, n k ± , n) = - — [ y (n" k ±)-n - ] . [2.72] r r The c o r r e l a t i o n function F(t,rT) and l i n e shape function I(co,n) f o r a given external f i e l d o r i e n t a t i o n i s then calculated by means of the average value . , . , 6 iw(r,n^ ,n)t < e i W ( r , n ) t > = 1 £ e \p p=l 6 L - C2.73] and Eq. [2.26]. Since the hamiltomian i s i n v a r i a n t under the transformation Ti^ — > - rTk , i n actual c a l c u l a t i o n s Eq. [2.73] can be replaced by • / \». i 3 icoCFjn, ,n)t < e l w ( F ' n ) t > =\l e kP , [2.74] p=l I l l where i s one of the following o r i e n t a t i o n s : (1,0,0), (0,1,0), and (0,0,1). The c o r r e l a t i o n function Fjj(t,n") i s then simply given by • 3 » - r i v r 3 [S(p)-n-3(S-(p).r ) ( n . r . ) / r 2 / r 3 . ] >i F I l ( t , H ) = e P = 1 J = M + l t \ [2.75] 3 where v = g Y g t / r 0 and S(p) = y ( i ^ )/(Bg) For s u f f i c i e n t l y large r Q (/3a), the summation involved i n the logarithm of F(t,n) can again be approximated by a sum of i n t e g r a l s . To do t h i s , we again assume that the average spin S(p) points i n the d i r e c t i o n of ri approximately, i . e . S ( p ) - n - 3(S(p) -T) (rT.r)/r 2 * S ( p ) - n ( l - 3cos 26) where 8 i s the angle between r and n . This approximation i s good only i f the c r y s t a l f i e l d term i n the hamiltonian i s not too much greater than the Zeeman term. However, as w i l l be shown l a t e r , t h i s i s a good approximation f o r D = 5 . 6 k , H = 2 0 kilogauss, and T = 4.2°K . Since we want to apply the t h e o r e t i c a l r e s u l t s to i n t e r p r e t 13 the broadening of the C and proton resonances l i n e s by oxygen i m p u r i t i e s , the above p a r t i c u l a r values of H and T are chosen so that they correspond 112 to experimental values already discussed i n chapter I. The value of D = 5.6 k i s chosen using some p l a u s i b i l i t y considerations (50, 51) as i t s actual value f o r oxygen molecules i n s o l i d methane l a t t i c e i s unknown. We now denote the summation i n Eq. [2.75] by , 3 » , ivr 3 [ " s(p).n-3(S(p ) . T.)(rI . T.)/r 2]/r 3. p=l j=M+l From Eqs. [2.33], and [2.34], the summation can be represented by the average of three i n t e g r a l s 1 3 - • G(v) = ±. I J((S(p)-n)v) . [2.77] When T = 4.2°K and D = 5.6 k , the behavior of G (v) and G(v) are represented i n Fig s . 27 to 32. The functions are p l o t t e d f o r three d i f f e r e n t d i r e c t i o n s of n , (0,0), O|-50), and (|">0); a n d i n each d i r e c t i o n of n , three d i f f e r e n t values of r Q are chosen, r Q = (p/2) 5 , p = 1, 2, and 4. For a l l three d i f f e r e n t d i r e c t i o n s of n", G(v) i s already quite a good approximation of G^(v) when r Q = /2a . Also i f one compares the numerical values of G(v) for the f i v e d i f f e r e n t d i r e c t i o n of n : (0,0), (77,0), (T-,0), &,\), and ), one finds very l i t t l e v a r i a t i o n i n G(v) o 4 o o 4 4 when n changes. When n^ i s along some other sets of equivalent d i r e c t i o n s , such as the ( l , l , 0 ) ' s and the (1,1,1)'s, the v a r i a t i o n of G(v) i s again n e g l i g i b l e . This i s consistent with the high symmetry of the face-centred cubic l a t t i c e . Therefore, f o r low concentrations of i m p u r i t i e s , [2.76] 27. Plot of Real(G(v)) and RealCG^v)) versus v when n = (0,0). The solid line represents Real(G(v)). ReaKG^v)) for r Q/a = 1//2, 1, and /l is represented by points o, ., and x, respectively. > CD b < ~ 0 -1 -2 e e 9 0 10 20 30 U0 F i g . 28. P l o t of -Im(G(v)) and -ImCG^v)) versus v when n = (0,0) The s o l i d l i n e represents -Im(G(v)). -InKG^v)) for r D / a = 1//2, 1, and /2 i s represented by points o, ., and x, r e s p e c t i v e l y . Fig. 29. Plot of Real(G(v)) and RealCG^v)) represents Real(G(v)). RealCG^v)) points o, and x, -respectively. versus v when n = (TT/4, 0). The solid line for r D/a = 1//2, 1, and Jl is represented by V Fig. 30. Plot of -Imag(G(v)) and -ImagCG^v)) versus v when n = (TT/4, 0). The solid line represents -Imag(G(v)). -ImagCG^v)) for r Q/a = 1//2, 1, and /2 is represented by points o, ., and x, respectively. 2 0 -> < LY. 10 0 o O o O v*"® O X''© o _* O X^o o S-"© o 0 10 20 30 V 50 Fig. 31. Plot of Real(G(v)) and RealCG^v)) versus v when n = (TT/8, 0). The solid line represents Real(G(v)). RealCG^v)) for r Q/a = 1//2, 1, and /2 is represented by points o, and x, respectively. r-1 F i g . 32. P l o t o f -Imag(G(v)) and -Imag(G 1 ( v ) ) v e r s u s v when n = (ir/8, 0). The s o l i d l i n e r e p r e s e n t s - I m a g ( G ( v ) ) . '-Imag (Gl ( v ) ) f o r r D / a = 1//2, 1, and /2~ i s r e p r e s e n t e d by p o i n t s o, ., and x, r e s p e c t i v e l y . oo 119 when the side-bands corresponding to the f i r s t few s h e l l s are sp f a r away at the wings and can be neglected, the l i n e shape function of a powder sample broadened by oxygen impurities can again be approximated by a Lorent-zian l i n e with half-width given by pK where K i s the c o e f f i c i e n t of the l i n e a r term i n t of the asymptotic expansion of G(v). K can be c a l c u l a t e d i n the following way. We take the external f i e l d to be along the z-axis, i . e . n" = (0,0). The average spin value can be c a l c u l a t e d f o r the three d i f f e r e n t o r i e n t a t i o n s of the oxygen molecular axis. Along the (0,0,1) d i r e c t i o n , we have S(l) = (0, 0, - 2sinh(BgH/kT) eD/kT + 2 cosh(3gH/kT) Along the (0,1,0) and (1,0,0) d i r e c t i o n s , we have S(2) = S(3) = (0, 0, 2w(w + A + w2 ) s i n h ( D / l + w2 /(2kT)) ^ (1 + w2 + w/l + w 2 ) ( e _ D / ( 2 k T ) + 2cosh(D/l + w2 /(2kT)) where w = 2ggH/D From Eqs. [2.75] and [2.43], the half-width i s given by co, = pK = pC(S (1) + 2S (2))/3 ^2 Z Z where ? = 8 T r 2fYBg / ( 9 / 3 a 3 ) 120 At T = 4.2°K , the half-width when D = 5 .6 k i s to, = 0.359p<; • At t h i s temperature, the half-width i s rather i n s e n s i t i v e to the change i n the D-parameter. When D = 0 , Eqs. [2.43] and [2.70] give = 16TTP f Ygg s inh (3 gH/kT) ^ 9 / 3 a 3 ( l + 2cosh(3gH/kT)) = 0.399p? , which i s only about 11% higher than that when D = 5.6 k . As the l i n e shape i s rather i n s e n s i t i v e to the change of the para-meter D, Eq. [2.78] i s used i n chapter I to estimate the broadening of the-13 C and proton resonance l i n e s i n s o l i d methane by 0 2 i m p u r i t i e s . 121 5. CONCLUSION. The l i n e shape function f o r the resonance absorption of a resonating centre due to inhomogeneous broadening by a d i l u t e system of perturbing impurities was ca l c u l a t e d as the Fourier transform of a c o r r e l a t i o n function F(t) by the s t a t i s t i c a l method. I f the i n t e r a c t i o n responsible for the broadening i s made up of a number of i n t e r a c t i o n s , each of which broadens the l i n e and i s uncorrelated with the others, then F(t) i s the product of c o r r e l a t i o n functions corresponding to the i n d i v i d u a l i n t e r a c t i o n s . In the frequency space, the t o t a l l i n e shape function i s obtained by convoluting l i n e shape functions a r i s i n g from the uncorrelated broadening i n t e r a c t i o n s . This i s a u s e f u l well-known r e s u l t . In general, one can separate the broadening e f f e c t i n t o a set of uncorrelated e f f e c t s which are convenient for the p a r t i c u l a r problem under i n v e s t i g a t i o n . The l i n e shape function due to two or more d i f f e r e n t species of impurities can be obtained by convoluting the l i n e shape functions due to each species of i m p u r i t i e s , as pointed out by Grant and Strandberg (39), and by Stoneham (38). For a given species of impu r i t i e s , the t o t a l l i n e shape function can be expressed as the convolution of functions, each representing the l i n e shape when the impurities are a l l i n a p a r t i c u l a r i n t e r n a l state or when the impurities are d i s t r i b u t e d i n one of the s p a t i a l p a r t i t i o n s of the l a t t i c e . In general, i t i s more convenient to p a r t i t i o n the l a t t i c e i n t o two regions, an inner "cut-off region", which i s a sphere around the resonating centre, and i t s complement. We have shown that impurities i n s i d e the c u t - o f f region give r i s e to a side-band structure to the resonance l i n e 122 and impurities outside give r i s e to a general broadening of the l i n e shape function. This has also been pointed out by Grant and Strandberg (39) and by Stoneham (38). We have performed d e t a i l e d numerical c a l c u l a t i o n s f o r the case when the perturbation i s the magnetic d i p o l a r i n t e r a c t i o n and when a l l the perturbing dipoles are time-independent with the same magnetic moment and are d i s t r i b u t e d randomly i n the l a t t i c e . The value of the magnetic moment here would correspond to the time averaged moment of the impurities i n the short c o r r e l a t i o n time l i m i t as w i l l be discussed l a t e r . The numerical c a l c u l a t i o n s using a f . c . c . l a t t i c e show that i n t e g r a l s are i n t h i s case a reasonable approximation to the l a t t i c e sums when the cu t - o f f radius i s s u f f i c i e n t l y large f o r the l a t t i c e anisotropy to be averaged out. I t turns out that the continuum approximation i s very good even when the cu t - o f f radius i s only a few l a t t i c e constants. As a consequence, while the p o s i t i o n of the side-bands changes as the f i e l d o r i e n t a t i o n varies with respect to the c r y s t a l a x i s , the shape of the main l i n e remains the same. The l i n e shape due to impurities outside the cu t - o f f volume depends on the product of impurity concentration and the volume i n s i d e the cut-off region, i . e . depends on the average number of impurities i n s i d e the cu t - o f f volume. When the number i s large, the shape i s predominantly Gaussian; but when i t i s small, the shape i s a truncated Lorentzian. This dependence has already been obtained by Grant and Strandberg (39). For a given c u t - o f f volume, the l i n e shape i s therefore Lorentzian when the impurity concentration i s very low and becomes more and more Gaussian when the concentration increases. This i s i n agreement with the r e s u l t obtained by K i t t e l and Abrahams (52) using the method of moment expansions. However, f o r the s t a t i s t i c a l method to be v a l i d , the concentration cannot be too large and only the case when 123 the Lorentzian shape predominates i s important. In t h i s case, the side-bands of a s i n g l e c r y s t a l have i n t e n s i t y p r o p o r t i o n a l to the impurity concentration and have the same shape as the main l i n e . In general, the l i n e shape i s Lorentzian at the centre and i s Gaussian at the wings. As can be a n t i c i p a t e d from the r e s u l t s obtained by McMillan and Opechowski (46) from moment expansions at f i n i t e temperatures, the l i n e shape i s not symmetrical about the centre of the unperturbed l i n e . Rather, the p o s i t i o n of i t s peak i n t e n s i t y i s s h i f t e d by an amount that i s l i n e a r i n the impurity concentration and i n the magnetic moment of the perturbing i m p u r i t i e s . The whole l i n e shape i s asymmetrical i n such a way as to make the f i r s t moment about the centre of the unperturbed l i n e vanish. The s h i f t of the peak i n t e n s i t y can be observed experimentally f o r systems i n which i t i s not too much broadened by the unperturbed l i n e . One system which has been studied experimentally by 13 us i s 0 2 impurities i n s o l i d CH^. From Chapter 1, the r.m.s. width of 13 ' 13 C resonance f o r an 02~free sample of CH^ containing 53% CH4 at T = 4.2°K and H = 20 Kilogauss i s < M > = 1.6 gauss. The s h i f t due to 0 2 impurities i s 6H = 2.5 x 102 p<S z> gauss, where p i s the molar concentration of 0 2 . Since < S z > has a magnitude of about unity, the s h i f t i s comparable to the r.m.s. width of the unperturbed l i n e when p > 0.2% , and should be observable. The case i n which the perturbing spins have value 1/2 was i n v e s t i -gated by numerical c a l c u l a t i o n s . In the "long l i f e - t i m e c o r r e l a t i o n l i m i t " , a broadened l i n e with a s h i f t e d peak r e s u l t s at f i n i t e temperatures. However, at i n f i n i t e temperature, the s h i f t vanishes, and a Lorentzian shape appears 124 when the cut-off radius shrinks to zero. This l i m i t i n g r e s u l t i s the same as that of Abragam (ref.(18), page 126). In the "short l i f e - t i m e c o r r e l a t i o n l i m i t " , each paramagnetic impurity has a time-independent magnetic moment equal to the average magnetic moment per spin.- The r e s u l t s obtained f o r the c l a s s i c a l dipoles are thus d i r e c t l y applicable to t h i s case. When the sample i s i n the form of a powder, the side-bands are smeared out into a shape s i m i l a r to that due to the an i s o t r o p i c K n i g h t - s h i f t of a p o l y c r y s t a l l i n e sample. I f the unperturbed l i n e i s s u f f i c i e n t l y narrow, these side-bands should be resolvable and t h e i r i n t e n s i t i e s and po s i t i o n s give information on the impurtity concentration and value of the induced magnetic moment. An example i n which the broadening-is due to 0 2 impurities at low temperatures was also investigated numerically. In t h i s example, we have S = 1. As expected, the general features of the broadening are the same as that i n the S = 1/2 case. At T = 4.2°K the l i n e shape function varies only s l i g h t l y with the an t i c i p a t e d c r y s t a l f i e l d D-parameter f o r 0 2 i n CH^ , as the average magnetic moment varies only s l i g h t l y with the D-parameter at 4.2°K . At lower temperatures, the e f f e c t of the D-parameter would be more pronounced. 125 APPENDIX We s h a l l now show that when n = p N , - T ^  t T f o r some f i x e d T , and to x~l where I > 3 , Eqs. [2.16], [2.18b] and [2.19] lead to z - - P -o 1 - p ' p - p N - l / 2 ( 1 _ pN-1/2 N [ 1 + _ J ^ f ( Y i j t ) ] i = l x p . j and c n = i n the l i m i t when N — > oo . /2irN To t h i s end we f i r s t write f ( r . ,t) = 1 - e ( r . , t ) . [A2.1] 3 3 Eq. [2.19], can then be rewritten as N z n [ l - e ( r \ , t ) ] pN + 1 = I j=l 1 + z Q - z 0 e ( r j , t ) Nz Q z Q N e ( r \ , t ) rr-v— ^ r —7-— I  2 [A2.2] 1 + z Q + z Q j = 1 1 + ^ _ v ( 7 . > t ) From Eq.[A2.2], we obtain a f t e r some straightforward algebraic manipulations, 126 z Q = x Q + n , [A2.3] where " x Q = ° , [A2.4] 1 ( N E ( r t) Substituting Eqs. [A2.1], [A2.4] and [A2.5] into Eq. [2.18b], we have M j . i N f 1 - e(r. ,t) w G „ ( Z O ) = . P N ± J L _ £ I J 2. [ A 2 . 6 ] <*o + ?N> j - 1 1 1 + ( x Q + ? N ) [ 1 " ^ j . t ) ] j p-N+1 P N , 1 P ^ N ( 2 X Q + ^ Now, — * = + 1 — <xo + ?N> Xo ( xo + V xo<xo + CN> N i 1 - e(F , t ) > and £ i J = [ j = l I 1 + (x Q + ? N ) [ 1 - s ( r t ) ] ) N (1 + g N ) £ ( r . , t)[2 + 2xp + E;n - s ( r . , t ) - 2x0e(r~ , t ) - ^ e C ^ . t ) ] (1 + x Q ) 2 j = r (1 + x Q ) 2 \1 + (x G + c N ) [ l - e ( r . , t ) ] We have g"(z0) = N { -P- - 1 + P(N) ) , [A2.7] 1 2 7 where P(N) = 1 2-N ( * o + ?N) V x o + 4> 1 N (1 + c N ) £ ( r . > t ) [ 2 + 2x Q + c N - e C ^ . t ) - 2 x ^ ( 7 . , t ) - ^ ( r . ,  N j = l (1 + x c ) 2 { 1 + (x Q + C N ) [ 1 - t(J.,t)] }2 t ) ] From Eqs. [2.13], [A2.1] and [A2.3], we have C ~ V z 0 ) = ( xo + ? N > " P N _ 1 | I { 1 + ( x o + ? N ) f ( Y j ' t } } = R(N)S(N)x Q n ^ ( X Q ) , [A2.8] ^ _ P ^ N -nN-1 N where R(N) = [1 + - ] p W X [ l + (1 - P ) C J N , [A2.9] p IN N r C i f C r .t) and S(N) = n' 1 J \ . [A2.10] j - 1 I [1 + x Q - x 0 e ( r j , t ) ] ( l + x Q + ? N ) Before we proceed to prove that, i n the l i m i t N — > oo , (1) £^ = 0, (2) P(N) = 0 , (3) R(N) = 1 and (4) S(N) = 1 , we s h a l l e s t a b l i s h the following i n e q u a l i t y : -Jl If - T <_ t <_ T f o r some T , co(r,q) r (£ ^ 3) , and N i s s u f f i c i e n t l y large, then N I | e ( r . , t ) | <_ M log N , [A2.ll] j = l 3 . 128 where M i s a constant independent of N. To prove i n e q u a l i t y [ A 2 . l l ] , we f i r s t l a b e l the l a t t i c e s i t e s according to the following scheme : i > j implies that r ^ > r.. . From Eqs. [2.10] and [A2.1], we have iio(r ,q ) t | e ( r ,t)| < I p(q )|1 - e I . ioo ( r . ,q .)t Since I 1 - e 3 3 I \l ^ [ i c o C T q ) t ]] k=l * 3 3 l k I = iirr | c o(r.,q.)t| k | U ( T j 5 q . ) t | < e |u(r ,q ) t | < [wCr , q ) t [ e J J aT a T / r l r .-J where a = max. { r^ „ m . { r . | a ) ( 7 . , q . ) | } aT/r^" we have |slr..t>|<I p C q ) 2 | e 1 = \ 1 M J R 5 129 a T / r * where D = aTe and N j = l J < D N I N r . J = l [A2.12] We s h a l l now denote by d the maximum d i s t a n c e between two p o i n t s i n a u n i t c e l l w hich has volume v and c o n t a i n s f s i t e s . L e t S be a a f a m i l y o f c o n c e n t r i c s p h e r i c a l r e g i o n s c e n t r e d a t the o r i g i n : S a = < T I r 1 r N } a = 1 » - - ' » k a s u ch t h a t (a) r N - r N <_ d (a = l , - . . , k - l ) , d+1 a (b) number o f s i t e s c o n t a i n e d i n S i s N , and (c) N = N f c . For each a, we d e f i n e r ( a ) such t h a t fa) _ ™ Now, as k — > oo , we may assume t h a t — > v . k Comparing t h i s l i m i t w i t h = v , we have k & r N = r ( k ) [A2.13] k 130 Let e a = r(a + 1) - r(a) a = l,...,k - 1 . Since lim e, .. = lim [r(k) - r(k - 1)] = lim (r„ - r „ ) < d , k-^ 0 k-1 k"*00 \ \ - l ~ the value of e a is bounded . Let e = max {e } From [A2.13], there exists an integer m > 0 , such that | r N - rfa)| <_y whenever a >_ m [A2.14] a We can always choose an integer p such that p > m and r(p) >_ 4e N 1 k N« 1 3=1 a=p+l j=N^_j+l r where Q = I ~ + I I j = l r. a = 2 j=N a_ 1 +l r. Now, for each a > p + 1 , we have N a N - N y _1_ < a l V l j=N +1 r 3 r 3  J a-1 j rN . a-1 131 N a - N a - l [ r(a - 1) - e] 4irf — v r.(a) 2 A r ar r ( a - l ) [r(a - 1) - e ] ; 4TT f r(a) 2 , r dr r ( a - l ) (r - 2e)' Therefore, we have N f r ( a ) 3-1 r V a=p+l r dr r ( a - l ) (r - 2e) < Q + 4lTf ^r ( k ) r 2 d r r(p) (r - 2e) < Q + 16irf r(k) r(p)-2e dr r < Q + MIL! { l o g ( r ( k ) ) _ l o g [ r ( p ) _ 2 e ] } ^ n . 16TTf ^T 16lTf I . V I , 16lTf | n r / \ < Q + -^7- log N + | log ^ | + — — |log[r(p) 3v - 2e] 64ir f <_ l°g N f ° r s u f f i c i e n t l y large N [A2.15] 132 From Eqs. [A2.12] and [A2.15], we obtain the i n e q u a l i t y represented by Eq. [A2.ll] with M = D . We can now prove the four l i m i t s . (1) l i m r = 0 Denoting y = max j <• | l + z Q - z 0 e ( r \ , t ) | from Eq. [A'2.5], we have UNI i a T ^ { l + l*ol + 1 * ^ , t ) | } -TT "T7)N { 1 + |z0| + yM log N } i ^ f 1 [A2.16] for some K which i s independent of N . Since l i m l o ^ N = 0 , we have l i m r = 0 . [A2.17] (2) The proof of l|m P(N) = 0 i s s i m i l a r to that i n (1) . (3) l i m R(N) = 1 . From Eq. [A2.9], we have 133 f U - p k >, (1 - p)r log R(N)= N \ log [1 + (1 - p)e ] - p log [1 + - log [1 + 51 j ~- I. P J P Expanding the f i r s t two logarithmic functions i n t o power s e r i e s , and subtracting one from the other, we get log R(N) = N ( l - p ) 2 4 j J (-if k + 1 N | - l o g [ 1 + — »] . Then oo k-1 k-1 -k | log R(N)| < N ( l - P ) 2 | g 2 | I (-Dk ( 1 ~ P ) h >, . , ( 1 - P ^ N k=l k + 1 1 | i o g 1 1 p < na - P)2ig2{i fig1-1} + i^u + j , - N ( 1 - p ) 2 u Ni1rnr j - + r i i T ^ r } + ( 1 " P P H " " tA2-181 Now, we have implying that l i m log R(N) = 0 and l^m R(N) = 1 [A2.19] 134 (4) lim S(N) = 1 . Expanding the product i n Eq. [A2.10], we have S(N) = i + I <-cNr I J jn TT £=1 1 j61+-..+JlN=£ [1 + x Q - x 0 e ( r j S t ) ] 1 (1 + x Q + ? N) £.=0,1 e ( r N , t ) " £ N % [1 + x Q - x 0 e ( r N , t ) ] (1 + x Q + ? N) N>j | l + x Q - x 0 e ( r t ) | | l + x Q + ? N Using V = max •( : ) , the above expression f. J leads to |S(N) " 1| = I I <-?/{ I } £.=0,1 < ~~ £ IV*|C NI*{ I | E ( r t ) | \ . . . * | s ( ? N , t ) | l N } .=1 I £ 1+. • -+£ N=£ J J £.=0,1 x N N 1 I [V|C N| I |s(r t ) . | ] A £=1 j=l J K(N)[K(N) - 1] - K(N) - 1 N where K(N) = v|?„| I | e ( F t ) | . .1=1 3 From Eqs. [ A 2 . l l ] and [A2.16], we have 2 l im K ( N ) = VKM l i m ^ l o j | N ) = 0 Therefore, l im |s(N) - l l = 0 and l i m S(N) = 1 F i n a l l y , from Eqs. [A2.3] , [A2.4] and [A2.17], we have l im z = x Q = - r — 2 — [A2 ^xo o o ]_ - p From Eqs . [16] , [A2.6] , [A2.8] , [A2.17] - [A2.29] , we have -0N -1 N xo i S l ^ 1 + x 0 f ( r ,t)] l im c_ = l im J J  ^ n J^OD / 2TTN j P 1 x o ( 1 + x o > 2 P-PN-1/2 pN-1/2 J [ 1 + ^ f ( F f t - ) ] j=l 1 - p J = l i m /2W 136 CHAPTER 3 EFFECT OF THE EXCHANGE AND MAGNETIC DIPOLE-DIPOLE INTERACTIONS ON THE SPECIFIC HEAT OF PARAMAGNETIC SALTS AT VERY LOW TEMPERATURES 1. INTRODUCTION. The method of adiabatic demagnetization of paramagnetic s a l t s proposed o r i g i n a l l y by Debye (53) and Giauque (54) and f i r s t s u c c e s s f u l l y applied by de Haas, Wiersma and Kramers (55) has made i t poss i b l e to carry out experiments i n the m i l l i k e l l v i n range oi : temperatures. For t h i s reason, the temperature dependence of the s p e c i f i c heat of paramagnetic s a l t s at thi s temperature range has become important f o r at l e a s t two reasons. F i r s t , i t determines the e f f i c i e n c y of the s a l t as a coolant, and the lower l i m i t of temperature reached by demagnetization. Second, the s p e c i f i c heat i t s e l f i s s t i l l one of the primary thermometric parameters i n the m i l l i k e l v i n range. For temperatures below 1 Kel v i n , the l a t t i c e s p e c i f i c heat i s n e g l i g i b l e . The contribution to the s p e c i f i c heat i n the absence of external magnetic f i e l d i s then mainly due to the following e f f e c t s : 1. The i n t e r a c t i o n of the i n d i v i d u a l paramagnetic ions with the c r y s t a l f i e l d ; 2. The i n t e r a c t i o n of the i n d i v i d u a l paramagnetic ions with t h e i r nuclear magnetic dipole and e l e c t r i c quadrupole moments ("hyperfine i n t e r a c t i o n " ) ; 1 3 7 3 . The exchange i n t e r a c t i o n of paramagnetic ions ; 4. The magnetic d i p o l e - d i p o l e i n t e r a c t i o n of paramagnetic i o n s . ( We d i s r e g a r d some other much s m a l l e r e f f e c t s ). In many s a l t s , the paramagnetic ions have a degenerate ground s t a t e when they are f r e e and the s p l i t t i n g due to the c r y s t a l f i e l d i s much l a r g e r than the thermal energy of the ions at T = 1 K e l v i n . The e f f e c t of the c r y s t a l f i e l d on the s p e c i f i c heat i s then i n d i r e c t and only gives r i s e to an a n i s o t r o p i c g-tensor. We s h a l l r e s t r i c t our c a l c u l a t i o n s o nly to such cases. I t has been accepted from both t h e o r e t i c a l c o n s i d e r a t i o n s and experimental evidence (56) t h a t , f o r s a l t s i n which the i n t e r a c t i o n s ( i n c l u d i n g exchange, magnetic d i p o l e - d i p o l e and hy p e r f i n e ) are s m a l l compared to kT where T i s the lower l i m i t of the temperature under i n v e s t i g a t i o n and k i s the Boltzmann constant, the s p e c i f i c heat C v can be represented by the expression : which i s obtained by r e t a i n i n g the f i r s t term i n the 'high temperature' expansion s e r i e s of the s p e c i f i c heat, constant [ 3 . 1 ] [ 3 . 2 ] 138 Here, as i s well-known (57, 58), A n = c o e f f i c i e n t o f N i n <Hn> = T r a c e H n/(2J + 1) N ; N i s the number of p a r a m a g n e t i c i o n s i n the volume V , and H i s the H a m i l t o n i a n which i n c l u d e s the exchange, m a g n e t i c d i p o l e - d i p o l e , and h y p e r f i n e i n t e r a c t i o n s , and J i s the e f f e c t i v e s p i n quantum number o f the p a r a m a g n e t i c i o n s . One would e x p e c t t h a t a t temperatures T such t h a t kT i s comparable to the energy r e p r e s e n t e d by H , the b e h a v i o u r o f C v w i l l d e v i a t e from -2 the T law. Recent e x p e r i m e n t s on the s p e c i f i c h e a t o f a number o f p a r a m a g n e t i c s a l t s a t m i l l i k e l v i n temperatures i n f a c t show c o n s i d e r a b l e d e v i a t i o n s (59). To see what s o r t o f d e v i a t i o n s can be e x p e c t e d from the t h e o r e t i c a l p o i n t o f view, we s h a l l c a l c u l a t e the c o e f f i c i e n t s o f the f i r s t two terms i n the e x p r e s s i o n r e p r e s e n t e d by Eq. [3.2]. I n our c a l c u l a t i o n s , we s h a l l d i s r e g a r d c o m p l e t e l y the h y p e r f i n e i n t e r a c t i o n , s i n c e e v e n t u a l l y we s h a l l a p p l y our t h e o r e t i c a l r e s u l t s o n l y t o the case of c e r i u m magnesium n i t r a t e i n which the h y p e r f i n e i n t e r a c t i o n o f the c e r i u m i o n s i s a b s e n t . F o r i o n s w i t h h y p e r f i n e i n t e r a c t i o n s , i t has been p o i n t e d out by D a n i e l s (60) t h a t the c o n t r i b u t i o n s to the s p e c i f i c h e a t C v due to the h y p e r f i n e i n t e r a c t i o n and t h a t due to the exchange and magnetic d i p o l e - d i p o l e i n t e r a c t i o n s _3 a r e a d d i t i v e up to the term T i n Eq. [3.2]. I n the f o l l o w i n g c a l c u l a t i o n s , we s h a l l assume t h a t the p a r a m a g n e t i c i o n s a r e a l l a l i k e and a r e i n the same quantum s t a t e whose a n g u l a r momentum i s J . They a r e assumed t o form a 139 Bravais lattice with an axial symmetry and the interaction consists of exchange and magnetic dipole-dipole interactions, both of which are anisotropic. The coefficients X2 » A3 s and X^ have been calculated by Van Vleck (61) and Joseph and Van Vleck (62) for the less general case of isotropic magnetic g-factor, isotropic nearest neighbour exchange interaction, and a simple cubic lattice. Opechowski (63) obtained expressions for A2 with the most general anisotropic interactions. Daniels (60) ignored exchange interaction and obtained general expression up to X^ for a pure magnetic dipolar interaction, but he gave numerical values only for the X^ coefficient. Marquard (64), in a recent paper which we saw only after completing our calculations, gave general formulae for the coefficients X^ , X^ , and X^ . However, when evaluating them numerically (for the case of GdCl3), he assumed magnetic isotropy. _3 After obtaining a general expression for C y to the term T in the next section, we shall specialize the expression to the case of cerium magnesium nitrate (CMN) in Section 3. All the general expressions given below have already been published by us (65) together with the numerical computations of the coefficients Aj_ (i =2, 3). However, these numerical computations have been based on the values of the lattice constants of CMN which, according to the latest crystallographic data, are not quite correct. In this thesis the numerical computations have been redone using the new more accurate lattice constants (66). It should be added that, very recently but quite indepen- . dently, D.J. O'Keeffe (67) has calculated X2 , A3 and X^ for magnetic 140 dlpole-dipole i n t e r a c t i o n without taking i n t o account exchange i n t e r a c t i o n and has computed them numerically f o r the case of CMN. The s a l t CMN i s chosen, because (59, 68) : (1) i t i s used most extensively, among paramagnetic s a l t s , as a thermometer i n the m i l l i k e l v i n range, (2) large amount of s p e c i f i c heat data have been c o l l e c t e d down to about 0.003K, (3) deviation of the s p e c i f i c heat from the T law has been observed, and (4) the con t r i b u t i o n to the s p e c i f i c heat comes predominantly from magnetic di p o l e - d i p o l e i n t e r a c t i o n rather than exchange i n t e r a c t i o n (60, 68). The l a t t i c e sums of the rhombohedral l a t t i c e , which i s the l a t t i c e of CMN, are c a l c u l a t e d numerically by means of a d i g i t a l computer. The r e s u l t s agree with experiment (59) down to about 0.004K i f the exchange constant i s assigned a value which corresponds to a ferromagnetic i n t e r a c t i o n . At lower temperatures, higher order terms i n the expansion would have to be taken into account. 141 2. GENERAL THEORY In the absence of external magnetic f i e l d s , the Hamiltonian of a system of N i n t e r a c t i n g paramagnetic ions i s given by the following expression : H = I H [3.3a] H.. '= VV.J'J. + D.. , [3.3b] r 3(y*. . r . .) (y".-r. .) > and D. . = — \ y,-y. ±—±3 J — i 3 — I . [3.3c] Here the terms V.. J . - J . are the i s o t r o p i c exchange i n t e r a c t i o n between i j x j paramagnetic ions i and j with exchange constant a n d t' i e t e r m s are t h e i r magnetic d i p o l a r i n t e r a c t i o n . J\ and y"j_ are r e s p e c t i v e l y the t o t a l angular momentum and the magnetic moment associated with ion i , and 7 ^ i s the vector j o i n i n g ion i to ion j . From the p a r t i t i o n function Z of the system with Z = t r e - « / k T where k i s the Boltzmann constant, the s p e c i f i c heat C v at temperature T can be cal c u l a t e d by means of the following r e l a t i o n (61) : 142 c_v = i-a_ T 2 9 i o ^ F o r m a l l y Z can be expanded i n powers o f T ^ , t h a t i s , Z = ( t r l ) I ( - l ) n ^  ( ^ ) n < H n > , [3.5] n= 0 where <H > = — — t r l N and I i s the i d e n t i t y o p e r a t o r w i t h t r l = ( 2 J + 1) I f we s b s t i t u t e Eq. [3.4] to Eq. [3.5], u s i n g the f a c t t h a t <H> v a n i s h e s i d e n t i c a l l y (which we s h a l l show l a t e r ) , we g e t ^- .<»6. I-2.<H!» I-3 + ....-; [ 3. 6 ] N k Nk 2 Hk 3 2 3 Our t a s k i s t h e n to e v a l u a t e <H > and <H > . The magnetic moment and the g - t e n s o r o f an i o n i a r e r e l a t e d by the e q u a t i o n , y i a = 3 E 8 i « Y J Y fa' Y = x»-y»' z ) > [ 3 ' 7 ] Y where g i s the Bohr magneton. U s i n g Eq. [3.7], we e x p r e s s Eq. [3.3b] i n 143 a form which i s more convenient f o r l a t e r c a l c u l a t i o n s H i j = £ K i j a B J i a J j @ > -.. .... [3.8a] . 2 3 a n d K i j a B = V i j a g + 3 ( l g i s a g j e & " ~T~ I 8 i a a 8 j p B r i j a r i j p ) [ 3 ' 8 b ] IJ i j (Here we note that K±.^ i s symmetrical with respect to i , j and with respect to a, g ) . In terms of the K±.^ 's, <W> , <tf2> , and <H3> can be rewritten as : c , 'e ' [3.10] < H > = i " > j " J>y i j a j , a . KiJ«3 Ki'j'a'g' Ki"j"a"g" t r ( Jia JjB Ji'a' Jj'g' Ji"a" g V,'g" Jj„g„)/(2J + 1 ) N . [3. From the i d e n t i t y t r J i a = 0 f o r a 1 1 i and a , i t i s obvious that f o r i ^ j , <H..> = 0 , implying <H> = 0 . S i m i l a r l y 144 2 2 we can see that only one kind of term H.. contributes to <H > and two i j 3 3 kinds of terms, H. . and H..H..H. . , contribute to <H > . Using the i d e n t i t i e s given by Rushbrooke and Wood (58) : t r ( J i a V ^ J ( J + 1 ) ( 2 J + 1 ) N + 1 *ag and t r ( J . J.„J. ) = \ J ( j + 1)(2J + 1 ) N + 1 e „ , ia ig iY 6 J y agY where e' i s defined by agY 'agY 1 i f (agY) i s an even permutation of (xyz) , -1 i f (agY) i s an odd permutation of (xyz) , 0 otherwise ; we get <H2> = ± J 2 ( J + 1 ) ^ I K 2 i>J a,( i j a g ' [3.12] <H3> - I J 3 ( j 4" i ) y y K K K. 9 " i>j>k a,g,Y 1 J a B J ^ ^ k i Y a — J 2 ( J + l ) 2 I I ( I K.. ) 3 + 2 J K.. K . . 0 K..V i>j i a l j a a a,g,Y 1 3 a 3 l j 3 Y l j Y a 36 - 3( y K.. ) y (K..FL ) 2 I a 1 J a a g,Y j B Y / [3.13] Eqs. [3.6], [3.12], and [3.13] then give a general expression for C y up to 145 _3 the term T for a general l a t t i c e . We s h a l l now assume that the paramagnetic ions form a Bravais l a t t i c e with a x i a l symmetry. We also choose a coordinate system which coincides with the d i r e c t i o n s of the p r i n c i p a l axes of the c r y s t a l . Then for a l l ions, we have 8 i a 3 < Sag 8a ' 8 1 8 2 8 x ' 8 3 8 i | Introducing the abbreviations, We obtain > 1 . . = g r. . i i a r. . °a i i a 1 2 and Q. . = g r. . i i a r. . a i i a i j J 2 2 K. . . = ( V - 8 2 + V. . ) 6 0 - — P. . P. r 3 6 a i j a3 r 3 l j a l j f i j i j Sub s t i t u t i n g t h i s expression i n t o Eqs. [3.12] and [3.13] and a f t e r a long but straight-forward algebraic manipulation, we f i n a l l y obtain the expressions f o r <H > and <H > i n terms of e x p l i c i t l a t t i c e sums : 146 <H2> = | e V (J + i ) 2 ^ "I > -1 TT 4 ^ 2i. i>i r. 2) I z2. + 9(g2 2 s2 | e 2 j 2 (J + i ) 2 ( g 2 - 4 ) I v. 3z. (1 -i>j r 2 r. . ) + ^  J 2 ( J + l ) 2 i > 3 [3.14] 2 <« 3> = \ ^ + ^ { i - f + ^ i ^ [ 2 £ + g2 - (gi - g 2)] ± > j. r i j ± > j r i j . . r i j V2. , 3Z 2.. ^ + 6 2(gl -g 2) I d - ^ 1 ) - I V 3 } i>i r . . r . . i>i J ' + f j> <j + D'{ ? 6 Z T J T T [ ~ 7 4 + 4 + 9<<£ - 8|>-T <• , i>i>k r. .r., r; . r. . J i j j k k i k i + 2 7 ^ j k ' ? k i ) ( Q j k " \ l ) 2 7 ' V V ^ k l ' V ' V V 1 " + ^ i '-r^r E - 4 8i + g1; + ecgi - g ^ ) % + 9 ( ? \ . f ] ± > 2 > k rjk rk± r j k V. .V., . 3z 2 • ) 2,2 2. r - i L J ^ (1 - - ^ i ) + 3 . £ V V V V t>3>k r k i L k i 147 3. THE CASE OF CERIUM MAGNESIUM NITRATE In s p e c i a l i z i n g the Eqs. [3.14] and [3.15] to the case of cerium magnesium n i t r a t e , CMN, (Ce 2Mg 3(N0 3) 1 2•24 H 20), we s h a l l make the s i m p l i f y i n g assumption, which i n t h i s case i s probably j u s t i f i e d , that the exchange i n t e r a c t i o n i s p r i m a r i l y important only between nearest neighbours, that i s , we s h a l l put V.. = V when ions i and i are nearest neighbours and V.. = 0 otherwise. The cerium ions i n CMN form a rhombohedral l a t t i c e with the long diagonal of each rhombohedron being p a r a l l e l to the t r i g o n a l axis of the c r y s t a l . The rhombohedral u n i t c e l l can be i n s c r i b e d i n t o a nexagonal unit c e l l whose dimensions are (66) The distance d between the nearest neighbours, i . e . the l a t t i c e constant of the rhombonedral l a t t i c e , i s then a = 11.004 A o and c = 17.296 A = 0.780a If we measure the magnitude and components of vector r „ taking a as unit and use the following abbreviations i n Eqs. [3.14] and [3.15] : 148 - g x J ( J + 1) T = a 3k w VaJ n 2 2 H a 3 9a 2 and Y = we get <H2> - 2 Nk 2 c j (5 + Y^)S 1 - 6(1 - Y 2 ) ( 2 - Y 2 ) S 2 + 9(1 - Y 2 ) 2 S ; - 12(1 - Y 2) 4 (1 ~ — ) w + 18w2 d 3 d 2 [3.16] <W3> T / 2 3 a 6 r 2 3 Z 2 2 NkJ 6J(JT1T Y S ^ + 7 [ 2 + Y - ^ - ( 1 - Y ) ] w + + " Y 2 ) d - ^ ) w 2 - 3w3 d 3 d 2 + T - j C - 7 + Y 6 ) S 5 + 3(1 - Y 6 ) S 6 + 9[S y - S 1 Q + Y 2(S 8 " 3 S 11> + + Y \ S 8 - 3S 1 2) + Y 6 ( S 9 - S 1 3 ) ] 149 + w[( - 4 + YIF)SLLF. + 6(1 - lk)S15 + 9(S16 + 2Y 2S 1 ? + Y \ 8 ) ] 3w2 (1 - Y 2 ) [4 + — ( 1 ' - — ) + — ( 1 - '—')•] } , 4d 3 d 2 p 3 p 2 [3.17] where z i s the z-component of d. The l a t t i c e sums (k = 18) and the i r numerical va lues, which were calculated by means of an IBM 7044 computer, are l i s t e d below : S = J = 39.334 1 H x; J 1 J S = I z 2 . / r ? . = 14.911 2 J l j ' 1 J S q = I z ? . / r = 7.303 S = J l / r 9 . = 64.785 J S = J = 710.540 5 , L . 3 3 3 j ,k r. . r., r. . xj j k kx S 6 = ^ 3 7 C = 2 4 8 - 6 2 9 j ,k .rf . r f . r f . xj j k kx 150 S, = t j , k (x.ik3,ki-+-».ikyki) 3 5 5 r. . r ., r. .. xj jk. kx = 187.138 (x.. x, . + y ..y,. . ) z . , z, . s = i ^ \ > ; k i i k k i = n . 7 1 1 j , k 3 5 5 r r r i j j k k i z 2 z 2  j k k i . j , k r 3 . r ^ r f . x j j k kx = 93.999 10 y ( x i . i x . i k + y j i y i k ^ ^ k ^ k i + y i k y k i ) ( x k i x i j + y k i y i i ) L> i - i - r-j,k r ^ r~" r 5 i j j k k i = 70.634 (x. .x... + y. .y ) (x.. x, . + y . , y 1 . ) s = V xj j k ' l r j i k j k k x '.ik Jki ; ** "* " c c ir 11 j k ^ k i / Z k i z i j j , k 5 5 5r r r i j j k k i = - 7 . 3 2 4 12 2 (x. . x M + y . .y ) z M z, . z. . £ i - l -Ik ' l i ' i k j k kx x j j , k r ? . r ? . r . 5 . x j j k kx = 3.901 '13 = I „ 2 n , 2 ^2 z . . z .. z, . ^1 -Ik kx j , k r ? . r ? . r, 5. xj j k kx = 36.867 S = y y = 128.899 151 z 2 'S,c = M = 43.585 k ra5 k r k i s = j £ ( * , i k * k i + - y , 1 ky k i ) = 4 5 > 7 2 3 16 ,.>. f 5 5 ^ k r j k r k i (x.. x, . + y.. y. . ) z . . z. . S - I l-^-^ -1k k l -1k k l • 7.587 ( j ) k r ? , r. 5. J j k k i , 2 ~ 2 z.. z, s i a = ; U H T T - «•<>«« ( j ) k r r. . J j k k i Where the symbol £ means t h a t summation i s t a k e n o n l y o v e r t h o s e v a l u e s o f (J) j such t h a t i o n j i s a n e a r e s t n e i g h b o u r o f i o n i . The sums S-^  (k = 1, ...,18) a r e e v a l u a t e d by t a k i n g the i o n i to be a t the o r i g i n . A l l o t h e r i o n s have t h e i r p o s i t i o n s g i v e n by r = r ^ d j + n 2 d 2 + n 3 d 3 , where d m (m = 1, 2, 3) a r e the b a s i s v e c t o r s o f the rhombohedral l a t t i c e w i t h d m = d, and n m a r e i n t e g e r s . The number of terms t a k e n i n t o a c c o u n t i n each sum S k i s c h a r a c t e r i z e d by an i n t e g e r such t h a t a l l t h o s e terms f o r w hich 152 - n ( k ) l n m l n C k ) (m = 1, 2, 3) are taken into account. An accuracy s u f f i c i e n t f o r our purposes was obtained by taking n ( k ) =40 f o r 1 < k < 4 , n ( k ) = 4 f o r 5 < k < 13 , and n ^ = 20 f o r 14 < k < 18 . The sums S-^  , S 2 , and S3 h a v e . a l r e a d y been computed b e f o r e by D a n i e l s (60), and much more a c c u r a t e l y by P e v e r l e y and M e i j e r (69). P e v e r l e y o and M e i j e r have used the o l d v a l u e s o f l a t t i c e c o n s t a n t s (a = 10.92 A , o c = 17.22 A ). Our r e s u l t s f o r S-^  , S 2 , and S3 agree almost e x a c t l y w i t h those o f P e v e r l e y and M e i j e r i f we a l s o use the o l d v a l u e s o f l a t t i c e c o n s t a n t s (see r e f e r e n c e 65). In the above n u m e r i c a l c o m p u t a t i o n s , we have o o used the most r e c e n t v a l u e s o f a = 11.004 A and c = 17.296 A , and t h e r e s u l t s f o r Sj , S 2 , and S^ agree w i t h i n 1% w i t h those o b t a i n e d u s i n g the o l d v a l u e s o f l a t t i c e c o n s t a n t s (60, 65). T h i s shows t h a t the v a l u e s o f S-L , S 2 , and S3 a r e n o t s e n s i t i v e t o s m a l l change i n the v a l u e s o f a and c . We now sub s t i t u t e the values J = 1/2 , g^ = 1.84 , g/( = 0.1 , T = 1.187 x 1 0 _ 3 K , and the values of S k into Eqs. [3.6], [3.16], and [3.17]. Neglecting terms containing Y n ( n > 2), we obtain the numerical 153 expression f o r the s p e c i f i c heat of CMN i n CGS units : [3.18] where t>2 = 6.533 + 10.720y 2 + 0.703w(l - y2) + 1.409w2 , b 3 = -25.366 - 95.466Y2 - (32.457 + 37.070Y2)w + 1.266(1 - Y 2)w 2 + 1.115w3 , and T i s expressed i n m i l l i k e l v i n . ' I f Y = g^/gji < 0.1/1.84 , the c o n t r i b u t i o n of terms containing Y to b 2 and b 3 , as given by Eq. [3.18], can be neglected. Since, experimentally (from paramagnetic resonance data) (60, 70), gj( seems to be smaller than 0.1, we put Y = 0 i n these equations. The values of b 2 and b 3 given by equation [3.18] for w = 0 do not exactly agree with those given by O'Keeffe (67). The discrepency i s of about 3% f o r b 2 and much la r g e r f o r b 3 . The o r i g i n of t h i s discrepency i s not easy to f i n d because i n O'Keeffe's c a l c u l a t i o n s the sums S^ to S 1 8 defined above do not occur e x p l i c i t l y . We s h a l l next determine the value of w by comparing our t h e o r e t i c a l expression of Eq. [3.18] ( i n which we put Y = 0 ), i . e . , 154 C v b 2 b Nk = ^ + ^I ( T I N U N I T ° F ^ [ 3 - 1 9 J b 2 = 6.533 + 0.703w + 1.409w2 and b 3 = -25.366 - 32.457w + 1.266w2 + 1.115w3 , w i t h the e x p e r i m e n t a l d a t a o f Mess e t a l . (59). We f i r s t s u b s t i t u t e the e x p e r i m e n t a l v a l u e o f the s p e c i f i c h e a t a t each te m p e r a t u r e T = (4 + n)mK n = 0, 1, 16 i n t o Eq. [3.19], and we then s o l v e Eq. [3.19] f o r w. The v a l u e s o f w o b t a i n e d by thus f i t t i n g Eq. [3.19] a t v a r i o u s temperatures a r e p l o t t e d v e r s u s -4 the temperature T i n F i g . 33. Due to e i t h e r the e f f e c t s o f t h e T term o r the u n c e r t a i n t y o f the e x p e r i m e n t a l p o i n t s o r b o t h , the v a l u e o f w v a r i e s c o n s i d e r a b l y o v e r the i n t e r v a l o f T from 4 mK to 20 mK. The maximum and minimum v a l u e s o f w a r e w = -0.122 and w = -0.438 r e s p e c t i v e l y . By p u t t i n g w = -0.122 and w = -0.438 i n t o Eq. [3.19], we o b t a i n the two cu r v e s r e p r e s e n t e d by the p o i n t s marked i n F i g . 34 by • and o , r e s p e c t i v e l y . I n a d d i t i o n , we g i v e i n F i g . 34 the two c u r v e s which c o r r e s p o n d t o w = 0 and w = -0.249 and which a r e marked by A and x , r e s p e c t i v e l y . The c h o i c e o f w = -0.249 g i v e s the s m a l l e s t v a l u e o f b 2 . The v a l u e s o f t h e c o e f f i c i e n t s b 2 and b 3 f o r the f o u r d i f f e r e n t v a l u e s o f w a r e l i s t e d i n T a b l e 5. 0 . 4 -0 . 2 0 -1 I I l _ ' < I I I L 1 0 T ( m K ) 15 2 0 F i g . 33. Values of -w obtained by f i t t i n g Eq. [3.19] to experimental data of Mess et a l . (59) at various temperatures. . 0 0 5 b . 0 0 2 . 0 0 1 F i g . 34. 1 0 0 S p e c i f i c heat of CMN as function of - temperature. The s o l i d curve represents experimental data of mess et a l . (59). Calculated values are in d i c a t e d by P A ^ O O ' *' X ' a n d 0 > c o r r e s p o n d i n g to w = 0, -0.1222, -0.249, and -0.438, r e s p e c t i v e l y . w b 2 0 6.533 -25.366 -0.122 6.468 -21.381 -0.249 6.445 -17.223 -0.438 6.495 -11.003 Table 5. Values of b 2 and b 3 f o r s i n g l e c r y s t a l CMN. (T i s i n unit of mK) 158 -4 S i n c e we do n o t know, o f c o u r s e , where t h e e f f e c t o f t h e T term becomes l a r g e r than t h e u n c e r t a i n t y o f e x p e r i m e n t a l p o i n t s , we l e a v e the q u e s t i o n open as to what v a l u e o f w s h o u l d be r e g a r d e d as the b e s t . In any c a s e , w i s c e r t a i n l y n e g a t i v e , which would i n d i c a t e a f e r r o m a g n e t i c i n t e r a c t i o n . I t may be p o i n t e d out t h a t the c u r v e i n F i g . 34. c o r r e s p o n d i n g t o w = -0.417 i s g i v e n by the f o r m u l a = 6.495 _ 11.003 Nk T 2 T 3 which i s n o t v e r y much d i f f e r e n t from the f o r m u l a v 6.4 12 _ , _ „ = T > 6.5 mK Nk 2 T 3 -g i v e n by Abraham and E c k s t e i n (71) i n the c o n c l u d i n g p a r a g r a p h o f a r e c e n t paper. I n t h a t paper, the a u t h o r s used t h e i r own e x p e r i m e n t a l d a t a which however do n o t seem to be d i f f e r e n t from t h o s e o f Mess e t a l . (59) above 6.5 mK. 159 APPENDIX The rather vague statement on p.152 "An accuracy sufficient for our purposes etc. ..." is not meant to imply that the numerical values of the l a t t i c e sums S , S., 0 as given on pages 149-151 are correct to 1 l o the third decimal digit. It only implies the author's belief that these values when substituted into the f i n a l formulae for the specific heat w i l l give values whose uncertainty is less than that of the experimental values. Dr. J.R.H. Dempster has pointed out to the author that in the case of some of the l a t t i c e sums one can make more precise statements about the uncertainty of these numerical values by considering the rounding-off errors in some detail. Let us f i r s t consider .'"-Since all^jterms are positive, the contribution due to terms outside a sphere with radius h can be approximated by an integral T - — h " V 4ir dr h r 3Vh 3 9.22 3 where a = 1 and V is the volume of a unit c e l l of the rhombohedral la t t i c e . The accuracy of the computer is 7 digits. If the summation is done in which terms are arranged in descending order of their magnitudes, 160 then a l l terms in with value less than S± x 10 7 - 39.334 x 10 7 » 4 x 10 6 are automatically disregarded by the computer. This implies that terms outside a sphere with radius h, such that ±- = 4 x I O " 6 , h 6 were discarded, resulting in an underestimate of by about 9.22 x / 4 x 10 6 . 2 x 10 2 (In the actual summation, not a l l terms are in descending order, and the error may be less). Since the sphere with radius h is contained inside the region of summation defined by -40 _< n-^ , n 2, 113 <_ 40 (page 151), we have not introduced any additional error (which would be the case i f n^"^ had been taken too small). We thus have finally 39.33 < S± - 39.35 161 S i m i l a r analysis can give corrected values of S 2, S 3 , and S^ . As to the double sums S^, S^, errors introduced by neglecting terms outside the region, which i s defined by -4 <^  n i > n2» n 3 — > may be more important than the rounding-off e r r o r s . In p r i n c i p l e , both errors can be estimated by evaluating a double i n t e g r a l of the form 16ir 2 V 2 f ( r l S r 2 ) d r x d r 2 over the domain D : r , r 2 >_ h ; |Tj - r~21 >_ 1 . Due to the d i f f i c u l t y i n evaluating the i n t e g r a l , we have note., attempted^estimating the errors i n these double sums. The errors may be one or two orders of magnitude l a r g e r than that of S^ . L a t t i c e sums S^g were evaluated using double p r e c i s i o n , which has 15 d i g i t s of accuracy. Rounding-off errors here are n e g l i g i b l e as compared to errors a r i s i n g from neglecting terms outside the region defined by -20 <^  n-^ , n 2 , n^ £_ 20 We have reason to b e l i e v e that the errors are about a few tenth of 1% f o r S14' S18 * 162 -The rounding-off errors in the lattice sums of Chapter II (Eqs. [2.45] and 1*2.76] ) are much less important. In fact they are 3 negligible since only about 5 x 10 terms corresponding to lattice sites in the 90 or so nearest shells were added and the remaining terms were approximated by an integral. •t 163 BIBLIOGRAPHY 1. J.H. Colwell, E.K. G i l l , and J.A. Morrison, J . Chem. Phys. 42, 3144 (1965). In t h i s paper, the low temperature heat 'capacities of the deuterated methanes are analyzed and a d e t a i l e d review of the l i t e r a t u r e i s presented. 2. H.M. James and T.A. Keenan, J . Chem. Phys. 31, 12 (1959). 3. T. Yamamoto and Y. Kataoka, J . Chem. Phys. 48, 3199 (1968). 4. Phys. Rev. Letters 20, 1 (1968). 5. Prog. Theor. Phys. Suppl., Extra number, 426 (1968). 6. H. Yasuda, T. Yamamoto, and Y'. Kataoka, Prog.^Theor. Phys. 41, 859 (1969). 7. J.H. Colwell, E.K. G i l l , and J.A. Morrison, J . Chem. Phys. 36, 2223 (1962). 8. J . Chem. Phys. 39, 635 (1963). 9. . H.P. Hopkins, P.L. Donoho, and K.S. P i t z e r , J . Chem. Phys. 47, 864 (1967). 10. T. Nagamiya, Prog. Theor. Phys. _6, 702 (1951). 11. E.B. Wilson, J . Chem. Phys. 3, 2 7 6 (1935). 12. R.P. Wolf and W.M. Whitney, Proceedings of the Ninth Conference on low Temperature Physics, 1118. Plenum Press (1965). 13. G.A. de Wit, Ph.D. Thesis, U n i v e r s i t y of B r i t i s h Columbia (1966). 164 14. G.A. de Wit and M. Bloom, Phys. Letters 21, 39 (1966). 15. Can. J . of Phys. _47, 1195 (1969). 16. L.B. Borst, S.L. Borst, L. Koysooko, H. P a t e l , and E. Stusnick, Phys. Rev. Letters ]_, 343 (1963). 17. M. Bloom and E.P. Jones, Phys. Rev. Letters 8, 170 (1962). 18. A. Abragam, The P r i n c i p l e of Nuclear Magnetism. Oxford Un i v e r s i t y Press (1961). 19. D. Barnaal and I.J. Lowe, Phys. Rev. Letters '11, 258 (1963). 20. E.L. Hahn, Phys. Rev. 80, 980 (1950). 21. H.S. Sandhu, J . Lees, and M. Bloom, can. J. Chem. ^8, 493 (1960). 22. H.S. Sandhu, J . Chem. Phys. 44, 2320 (1966) ?H.S. Sandhu, Ph.D. Thesis, U n i v e r s i t y of B r i t i s h Columbia (1964). 23. D. Barnaal and I . J . Lowe, Phys. Rev. 148, 328 (1966). 24. J.E. P i o t t , W.D. McCormick, B u l l e t i n of the Am. Phy. Soc. 14, 334 (1969); and p r i v a t e communication. 25. K.P. Wong, J.D. Noble, M. Bloom, and S. Alexander, J . of Magnetic Resonance 1, 55 (1969). 26. R.S. McDowell, J.Mol. Spectry. 21, 280 (1966). 27. (5. Runolfsson, S. Mango, and M. Borghini, Physica _4j4, 494 (1969). 28. F.H. Frayer and G.E. Ewing, J . Chem. Phys. 48, 781 (1968). 165 29. H.P. Hopkins, R.F. C u r l , and K.S. P i t z e r , J. Chem. Phys.'48, 2959 (1968). 30. H.F. King, D.F. Hornig, J . Chem. Phys. 44, 4520 (1966). 31. R.F. C u r l , J.V.V. Kaspar, and K.S. P i t z e r , J. Chem. Phys. 46, 3220 (1967) 32. C H . Anderson and N.F. Ramsey, Phys. Rev. 149, 19 (1966). 33. J.H. Van Vleck, Phys. Rev. 74> 1168 (1948). 34. N. Bloembergen, E.M. P u r c e l l , and R.V. Pound, Phys. Rev. TS, 679 (1948). 35. R. Kubo, Fl u c t u a t i o n Relaxation and Resonance i n Magnetic Systems, D. ter Haar Ed., page 23, O l i v e r and Boyd Ltd., Edinburgh (1961). 36. H. Margenau, Phys. Rev. 48, 755 (1935). 37. H. Margenau and J . Watson, Rev. Mod. Phys. 8, 22 (1936). 38. A.M. Stoneham, Rev. Mod. Phys. 41, 82 (1969). 39. W.J.C. Grant and M.W.P. Strandberg, Phys. Rev. 135A, 715 (1964). 40. R.H. Fowler, S t a t i s t i c a l Mechanics, Cambridge U n i v e r s i t y Press (1966). 41. P.M. Morse, H. Feshbach, Methods of Th e o r e t i c a l Physics. McGraw-Hill Book Co. (1953). 42. W. F e l l e r , An Introduction to P r o b a b i l i t y Theory and I t s A p p l i c a t i o n s , Vol I I . Page 142. • John Wiley and Sons, Inc (1966). 43. M. Abramowitz and I.A. Stegun, E d i t o r s , Handbook of Mathematical Functions, Page 504, Dover Publications (1965). 166 44. P.W. Anderson, Phys. Rev. 82, 342 (1951). 45. W.J.C. Grant and M.W.P. Strandberg, Phys. Rev. 135A, 727 (1964). 46. M. McMillan and W. Opechowski, Can. J. Phys. 38, 1168 (1960). 47. N. Bloembergen, T.S. Rowland, Acta M e t a l l u r g i c a 1, 731 (1953). 48. . Phys. Rev. 97, 1679 (1955). 49. M. Tinkham and M.W.P. Strandberg, Phys. Rev. 9*4, 937 (1954). 50. H. Meyer, M.C.M. O'Brien, J.H. Van Vleck, Proc. Royal Soc. (London) 243A, 414 (1958). 51. J.C. Burford and G.M. Graham, J . Chem. Phys 49, 763 (1968). 52. C. K i t t e l , E. Abrahams, Phys. Rev. 90, 238 (1953). 53. P. Debye, Ann. der Physik 81, 1154 (1926). 54. W.F. Giauque, J.Am. Chem. Soc. 49, 1864 (1927). 55. W.J. de Haas, E.C. Wiersma, H.A. Kramers, Physica 1_, 1 (1934). 56. D. De Klerk, Handbuch der Physik XV, 119 (1956). 57. W. Opechowski, Physica _4, 181 (1937). 58. G.S. Rushbrooke, P.J. Wood, Mol. Phys. JL, 257 (1958). 59. K.W. Mess, J . Lubbers, L. Niesen, W.J. Huiskamp, Physica 41_, 260 (1969); Eleventh I n t e r n a t i o n a l Conf. on Low Temp. Phys. Vol I, 489 (St. Andrews 1968). 167 60. J.M. Daniels, Proc. Phys. Soc. (London) 66, 673 (1953). 61. J.H. Van Vleck, J . Chem. Phys. 5_> 322 (1937). 62. R.I. Joseph, J.H. Van Vleck, J . Chem. Phys. 32, 1537 (1960). 63. W. Opechowski, Physica 14, 237 (1949). 64. CD. Marquard, Proc. Phys. Soc. (London) 92, 650 (1967). 65. S. Wong, S.T. Dembinski, W. Opechowski, Physica 42_, 565 (1969). 66. A. Z a l k i n , J.D. Forrester, D.H. Templeton, J . Chem. Phys. 39, 2881 (1963) 67. D.J. O'Keeffe, Ph.D. Thesis, The C a t h o l i c Univ. of America (1969). 68. R.P. Hudson, Cryogenics 9_, 76 (1969). . 69. J.R. Peverley, P.H.E. Meijer, Status S o l i d i 13, 353 (1967). 70. A.H. Cooke, J.H. Duffus, W.P. Wolf, P h i l . Mag. 44, 623 (1953). 71. B.M. Abraham, Y. Eckstein, Phys. Rev. Let t e r s 24, 663 (1970). 

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