UBC Theses and Dissertations

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UBC Theses and Dissertations

Ordered magnetic systems studied by nuclear orientation Gorling, Robert Lloyd Albert 1976

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ORDERED MAGNETIC SYSTEMS STUDIED BY NUCLEAR ORIENTATION by LLOYD GORLING B.Sc., U n i v e r s i t y o f B r i t i s h Columbia, 1966 M.Sc., U n i v e r s i t y o f B r i t i s h Columbia, 1970 A THESIS SUBMITTED IN" PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n The Department of P h y s i c s We accept t h i s t h e s i s as conforming t o the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA Ap r i l , 1 9 7 6 0 Lloyd Gorling, 1 9 7 6 i i In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced d e g r e e ac the 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 , 1 a g r e e t h a t the 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 a g r e e 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 the 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 not 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 tmen t o f f..jL<-^ The 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 2075 W e s b r o o k P l a c e V a n c o u v e r , C a n a d a V6T 1WS Date A b s t r a c t The antiferromagnetic and spin-flop phases of MnC^^H^O have been investigated by observing the nuclear 54 orxentatxon of Mn i n that material. The s u b l a t t i c e magnetizations i n the absence of an external f i e l d were found to l i e i n a d i r e c t i o n between the a and c c r y s t a l axes at an angle of 11.5° ± 3.5° to the c axis. The f i e l d dependence of the spin configuration i n the spin-f l o p state indicates that second order anisotropy i s s i g n i f i c a n t i n t h i s system. The molecular f i e l d s were determined by combining the r e s u l t s of t h i s work with other measurements of t h e - c r i t i c a l f i e l d s (Rives and Benedict, 1975). The r e s u l t s are: the exchange f i e l d i s 11.05 ± 0.21 KOe; the b i a x i a l s i n g l e ion anisotropy f i e l d s are 0.75 ± 0.22 KOe and 2.35 ± 0.23 KOe along the a* and b axes re s p e c t i v e l y ; the second order anisotropy f i e l d i s 1.45 ± 0.19 KOe and the anisotropic exchange f i e l d i s 0.1 ± 0.3 KOe. The spin-flop t r a n s i t i o n region was found to be adequately described by a 'domain' structure i n which regions of antiferromagnetic phase and regions of spin-f l o p phase co-exist i n the c r y s t a l . i v Measurements were made of the temperature dependence of the s p i n - f l o p t r a n s i t i o n f i e l d and, c o n t r a r y t o the e x t r a p o l a t e d r e s u l t s of Rives and B e n e d i c t (1975) , the s p i n - f l o p f i e l d was found t o decrease w i t h d e c r e a s i n g temperature from 0.3K to 0.15K. I f t h e r e i s a minimum i n the t r a n s i t i o n f i e l d i t must occur a t lower temperatures. The c o o l i n g of the MnCl 2*4H 20 c r y s t a l which was h e l d i n c o n t a c t w i t h a copper heat s i n k by A p i e z o n N grease was f i t t e d t o the r e l a t i o n Q = kA (T-^-T^ 1 1) where and are the temperatures of the c r y s t a l and copper heat s i n k r e s p e c t i v e l y , and A i s the c o n t a c t a r e a . F o r n = 4 the v a l u e o b t a i n e d f o r the c o n s t a n t k i s (8.2 ± 1.9) , „3 -4 -1 -2 x 10 ergK sec cm Nuclear o r i e n t a t i o n experiments were a l s o p e r -103 59 formed on the systems Ru-Fe and Fe-Fe. The gamma-ray a n i s o t r o p i e s f o r these systems (at temperatures o f 10 and 15 mK, r e s p e c t i v e l y ) were v e r y s m a l l ; however, i t was p o s s i b l e t o determine l i m i t s f o r the magnitudes o f the n u c l e a r magnetic moments of the a c t i v e n u c l e i . The 103 Ru moment was found t o be g r e a t e r than 0.15uN and the 59 Fe moment was found t o be l e s s than 0.9uN. J.E. R i v e s and V. B e n e d i c t , Phys. Rev. B12, 1908 (1975). I V TABLE OF CONTENTS Page A b s t r a c t i i i Table o f Contents v L i s t o f Ta b l e s v i i i L i s t of F i g u r e s i x Acknowledgements x Chapter I I n t r o d u c t i o n 1 1.1 Nuclear O r i e n t a t i o n 1 1.2 Theory f o r an A n t i f e r r o m a g n e t i c I n s u l a t o r 3 1.2A Simple M o l e c u l a r F i e l d Theory 3 1.2B General Mean F i e l d Approach 7 1.2C Dynamics of the S p i n - F l o p 11 1.2D Spin-Wave Theory 12 1.3 P r o p e r t i e s o f M n C l 2 * 4 H 2 0 15 1.3A C r y s t a l l o g r a p h y 15 1.3B Phase Boundaries 16 1.3C S p e c i f i c Heat 20 1.3D Nu c l e a r O r i e n t a t i o n o f MnCl 2'4H 20 21 1.3E NMR-ON 25 1.4 I n t e r a c t i o n s i n A n t i f e r r o m a g n e t i c I n s u l a t o r s 29 1.4A The Superexchange I n t e r a c t i o n 29 1.4B The A n i s o t r o p i c I n t e r a c t i o n s 33 1.5 Contact C o o l i n g 36 v i Page C h a p t e r II The Experimental P r o c e d u r e 2.1 Growth of the Cry s t a l Specimens 40 2.2 The Cryostat and the Cry s t a l Holder 41 2.3 Gamma-Ray Detectors and Spectrum Analysis 44 2.4 Analysis of Anisotropies 46 2.5 Description of an Experimental Run .48 2.6 The Experiments 49 2.6i The Nuclear Orientation Experiments 52 2 . 6 i i The Nuclear Magnetic Resonance Experiment 54 2 . 6 i i i S u s c e p t i b i l i t y Measurements 56 2.7 The Internal F i e l d C a l c u l a t i o n 58 Chapter III Results 3.1 Spin Configurations i n the Antiferromagnetic 60 State (H < H. ,) th 3.2 The Spin-Flop State 63 3.3 The Molecular F i e l d Values for MnCl 2-4H 20 65 3.4 The Spin-Flop T r a n s i t i o n Region 71 3.5 The Temperature Dependence of the Spin-Flop 81 T r a n s i t i o n F i e l d 3.6 Cooling of the Specimens 84 v i i Page 103 Chapter IV N u c l e a r Orientation o f . Ru 4.1 Introduction 91 103 4.2 The Nuclear Orientation Parameters of Ru 94 in Iron 4.3 The Experimental Procedures ' 9 7 4.4 Analysis of the Spectra 99 4.5 Analysis of Anisotropics 102 59 Chapter V Nuclear Orientation of Fe 5.1 Introduction 105 5.2 Experiments and Analysis 108 5.3 Discussion 109 Appendix 112 References 116 v i i i L i s t of Tables Page Table 1.1 Coordinates of the Manganese Spins 17 Table 2.1 Detector Orientations 50 Table 2.2 Cr y s t a l Parameters 51 Table 3.1 Spin Configurations without Applied F i e l d 62 Table 3.2 F i e l d Dependence of the Spin Configuration 64 , i n the Antiferromagnetic Phase Table 3.3 Spin-Flop State Configuration 66 Table 3.4 Spin Configuration i n the T r a n s i t i o n 73 Region Using the Two-sublattice Model Table 3.5 Tr a n s i t i o n Region Using the Domain Model 76 Table 3.6 Temperature Dependence of the Spin-Flop 8 2 F i e l d Table 3.7 Specimen Cooling 8 6 Table 4.1 U^ and Fv C o e f f i c i e n t s for 1 0 3 R u 96 103 Table 4.2 Ru Anisotropics 101 i x L i s t of F i g u r e s P a g e F i g u r e 1.1 N o t a t i o n f o r t h e S p i n A n g l e s 5 54 F i g u r e 1.2 D e c a y Scheme o f Mn 23 54 F i g u r e 1.3 Gamma-ray a n i s o t r o p y f o r Mn i n 24 M n C l 2 - 4 H 2 0 F i g u r e . 1 . 4 S c h e m a t i c d i a g r a m o f t h e c o o l i n g o f 37 . M n C l 2 - 4 H 2 0 F i g u r e 2.1 The C r y o s t a t 42 F i g u r e 2.2 F i e l d Sweep t h r o u g h t h e S p i n - F l o p 53 T r a n s i t i o n F i g u r e 3.1 P o l a r A n g l e i n t h e S p i n - F l o p P h a s e 6 9 F i g u r e 3.2 Two S u b l a t t i c e M o d e l i n t h e T r a n s i t i o n 74 R e g i o n F i g u r e 3.3 8 i n t h e T r a n s i t i o n R e g i o n 78 F i g u r e 3.4 The M a g n e t i z a t i o n a n d 6 i n t h e T r a n s i t i o n 80 R e g i o n F i g u r e 3.5 T e m p e r a t u r e D e p e n d e n c e o f t h e S p i n - F l o p 83 T r a n s i t i o n F i e l d F i g u r e 3.6 C o o l i n g o f t h e S p e c i m e n 88 103 F i g u r e 4.1 The D e c a y Scheme o f Ru 95 59 F i g u r e 5.1 The D e c a y Scheme o f F e 106 F i g u r e A . l Gamma-ray S p e c t r u m 113 X A c k n o w l e d g e m e n t s I am v e r y g r a t e f u l t o D r . B.G. T u r r e l l who f i r s t i n t e r e s t e d me i n n u c l e a r o r i e n t a t i o n a n d who h a s d i r e c t e d my r e s e a r c h i n t h i s w o r k . I was f o r t u n a t e i n h a v i n g D r . R. K e i s e r a n d D r . P.W. D a l y a s c o l l e a g u e s . D r . K e i s e r p r o v i d e d c o n s i d e r a b l e a s s i s t a n c e t o me i n t h e l a b o r a t o r y a n d D r . D a l y a nd I c o l l a b o r a t e d o n p a r t o f t h e w o r k i n t h i s t h e s i s , n o t a b l y t h e ^ F e and "'"^Ru e x p e r i m e n t s . Many o t h e r members o f t h e P h y s i c s D e p a r t m e n t a t t h e U n i v e r s i t y o f B.C. h a v e a s s i s t e d me t h r o u g h t h e d i f f i c u l t i e s e n c o u n t e r e d i n my r e s e a r c h . I t h a n k my p a r e n t s f o r t h e i r e n c o u r a g e m e n t . I t h a n k my w i f e Norma who c o n t r i b u t e d so much t o t h i s w o r k . 1 CHAPTER I I n t r o d u c t i o n The o b s e r v a t i o n o f n u c l e a r o r i e n t a t i o n o f n u c l e i i n a magnetic s o l i d depends on the h y p e r f i n e i n t e r a c t i o n and the c h a r a c t e r of the decay scheme. Consequently the n u c l e a r o r i e n t a t i o n technique can g i v e s o l i d s t a t e or n u c l e a r p h y s i c s i n f o r m a t i o n or both. The experiments d e s c r i b e d i n t h i s t h e s i s i n v o l v e both these f i e l d s . The dominant p a r t of the work i n v o l v e s a study o f the a n t i f erromagnet MnC^ * ^ 2*3. T h i s i s d i s c u s s e d i n the f i r s t t h r e e c h a p t e r s . The remaining c h a p t e r s d e s c r i b e experiments y i e l d i n g predominantly n u c l e a r p h y s i c s 5 9 10 3 i n f o r m a t i o n c o n c e r n i n g the nuclei Fe and Ru. 1.1 N u c l e a r O r i e n t a t i o n F o l l o w i n g B l i n - S t o y l e and Grace (1957), the angular d i s t r i b u t i o n of gamma r a d i a t i o n from o r i e n t e d , n u c l e i i s g i v e n by where 8 i s the angle between the a x i s o f d e t e c t i o n o f the r a d i a t i o n and the a x i s o f q u a n t i z a t i o n and P i s the W ( 6 ) (1.1.1) K even 2 Legendre polynomial. XJ and FT. depend on the nuclear parameters and are discussed i n d e t a i l in section 4.1. The parameters completely describe the state of orien-tat i o n of the i n i t i a l system. For a n u c l e i with spin I the B (for K < 21) are given by B , „ v = ]C(2K+1) (IKI ;mO)w'(m) ; K < 21 (K) m where C i s the Clebsch-Gordon c o e f f i c i e n t and w(m) i s the normalized population of the substate with magnetic quantum number m. If the nucleus, with magnetic moment u , experiences a hyperfine f i e l d the ground state i s s p l i t into 21+1 equally spaced Zeeman l e v e l s separated i n energy by uH^/I. In thermal equilibrium the population of these l e v e l s i s given by the Boltzmann d i s t r i b u t i o n so that the normalized population of a l e v e l i s w(m) = w Qexp(-uH h fm/IKT), where w i s the normalization constant, o If the detection axis i s given by a uni t vector D and the axis of quantization by a unit vector M then the above expression for w (equation 1.1.1) becomes W ( D ) = Z) A K P K ( - " - } k where AT, = BVXJVFV. For an antiferromagnet the c r y s t a l K K K K l a t t i c e i s c o m p o s e d o f s e p a r a t e m a g n e t i c s u b l a t t i c e s . The d i r e c t i o n s o f t h e s u b l a t t i c e m a g n e t i z a t i o n s M. d e f i n e t h e — i n u c l e a r o r i e n t a t i o n a x e s t h r o u g h t h e h y p e r f i n e i n t e r a c t i o n , I f t h e s u b l a t t i c e s h a v e r e l a t i v e p o p u l a t i o n s p ^ t h e n W ( D ) = Z l P i A P (D-NL ) . i k I f u" K, F K a n d t h e h y p e r f i n e i n t e r a c t i o n a r e known, a m e a s u r e m e n t o f t h e a n g u l a r d i s t r i b u t i o n o f t h e gamma-rays a l l o w s t h e d e t e r m i n a t i o n o f t h e t e m p e r a t u r e T o f t h e s p i n s y s t e m ( t h r o u g h t h e A t e r m s ) , t h e s u b l a t t i c e m a g n e t i z a t i o n a x e s M. ( t h r o u g h t h e P t e r m s ) a n d t h e r e l a t i v e l is. p o p u l a t i o n s p^. F o r two e q u a l s u b l a t t i c e s p^ = p 2 = h-I n t h i s w o r k t h e q u a n t i t y l-W(D) i s c a l l e d t h e ' a n i s o t r o p y 1 . 1.2 T h e o r y f o r an A n t i f e r r o m a g n e t i c I n s u l a t o r 1.2A S i m p l e M o l e c u l a r F i e l d T h e o r y The g e n e r a l f i e l d d e p e n d e n t b e h a v i o r o f a n a n t i f e r r o m a g n e t c a n be e x p l a i n e d by c o n s i d e r i n g a s i m p l e m o d e l ( e . g . M a r t i n , 1 9 6 7 ) . A s s u m i n g a H e i s e n b e r g i n t e r a c t i o n b e t w e e n two s p i n s a n d S 2 , t h e i n t e r a c t i o n e n e r g y i s g i v e n by 4 where J i s the exchange co n s t a n t and i s p o s i t i v e . The p r e f e r e n c e f o r the s p i n s to l i e along a f i x e d a x i s i n the c r y s t a l , the easy a x i s , i s g i v e n by an a n i s o t r o p y term ( u n i a x i a l ) (Nagamiya e t a l . 1957) o f the form 2 F = -ijLcos ch an Y where <j> i s the angle between the easy a x i s and the s p i n a x i s . In a d d i t i o n a p p l i c a t i o n of a magnetic f i e l d produces an i n t e r a c t i o n (where y g i s Bohr magneton and g i s the gyromagnetic g - f a c t o r ) F R = -H-S ( y B g ) Using the n o t a t i o n of f i g . 1.1 the t o t a l f r e e energy F f o r N s p i n s i s g i v e n by 2 2 2 F/N = S Jcos(a-g) - % L ( c o s a+cos 0) - gu_SFf ['cos (a + t|>)+cos ( B+iJ>) ] . r> On m i n i m i z a t i o n o f the energy i t i s seen t h a t the s p i n system can e x i s t i n three d i s t i n c t phases f o r the f i e l d a p p l i e d along the easy a x i s . A t low f i e l d s the s p i n s are a l i g n e d a n t i p a r a l l e l a long the e a s y - a x i s (a=0, B = T T ) . For l a r g e f i e l d s the paramagnetic s t a t e o c c u r s i n which the s p i n s are both p a r a l l e l to the a p p l i e d f i e l d (a=B=0). For i n t e r m e d i a t e f i e l d s the s p i n - f l o p s t a t e o c c u r s f o r which a=-6=9. The t r a n s i t i o n from the a n t i f e r r o m a g n e t i c phase to the s p i n - f l o p phase o c c u r s i n thermal e q u i l i b r i u m 5 F i g u r e 1.1 N o t a t i o n f o r the Spin Angles. 6 at a f i e l d u'H t h = [ L ( 2 J - L ) ] % where u' = gy D/S. The spin-flop angle 8 i s given by-cose = y'H /(2J-L) The t r a n s i t i o n from the spin-flop to paramagnetic state takes place when cose = 1 which occurs i n a f i e l d given by u'H = 2J-L P z I f the applied f i e l d i s perpendicular to the easy-axis there i s a t r a n s i t i o n from the antiferromagnetic phase (a=7r — (3 = 9) to the paramagnetic phase (a=8=%7r) at a f i e l d y'H = 2J+L P The spin d i r e c t i o n i n the antiferromagnetic state i s given by . sine = y'H /[2J+L] The spin-flop t r a n s i t i o n (H^) i s a f i r s t order of t r a n s i t i o n because a f i n i t e change i n magnetization occurs. In fac t there e x i s t s a possible hysteresis at the t r a n s i t i o n . The antiferromagnetic state i s stable up to the f i e l d , ,. u 'H+ = [L(2J+L) Y'2 7 whereas the spin-flop state i s stable down to a minimum f i e l d given by y ' H ~ = ( 2 J - L ) [ L / ( 2 J + L ) ^ The t r a n s i t i o n to the paramagnetic state i s continuous in the magnetization. I t i s therefore second order and no hysteresis i s expected. The int e r a c t i o n s are commonly given i n terms of e f f e c t i v e f i e l d s . These are defined by the following expressions H E = JS/gy B ; = LS/gy B . In t h i s notation the energy i s F/N = gy^SH^cos(a-B) - g yB HA(cos 2a+cos 26) B E ~2T~ - gy^SH (cosa+cosB ) The c r i t i c a l f i e l d H ^ i s given by H t h = ^ H A ( 2 H E - V ^ • 1.2B General Mean F i e l d Approach The magnetic phase diagrams of antiferromagnetics have been studied i n d e t a i l i n terms of a more general molecular f i e l d approximation by several investigators including Nagamiya et a l . (1955), Rohrer (1969) , Blazey 8 et a l . (1971), Yamashita• (1972) and Morrison (1973). Following Blaze'y et a l . (1971) the free energy can be expressed by F/N = S 2 [ J 6 a S B C O S (ct-6) + 3 5 J ' ( 6 2 + 6 2 ) + K r6 J n c o s a c o s g - h (K, -K ' ) (6 2 c o s 2 a + 6 2cos 2B ) 1 Jli A B x A D - g u B S H [ S A c o s ( a - * ) + < 5 f i c o s ( ) ] - T [ s ( 6 A ) + s ( 6 B ) ] where a , B are the angles of the s u b l a t t i c e magnetization and i|> i s the angle of the applied magnetic f i e l d H with the easy axis; 6.., 5 D are the s u b l a t t i c e magnetizations A B r e l a t i v e to the T=0 value; J , are the i n t e r s u b l a t t i c e exchange and anisotropic exchange i n t e r a c t i o n ; J ' , K' are the i n t r a s u b l a t t i c e exchange and anisotropic exchange in t e r a c t i o n s ; and i s single ion anisotropy. I t i s convenient to use the molecular f i e l d s defined by: H E = J S / g y B , H K 1 = (I^-K' ) S/gy B HKE = K E S / ^ B ' V = K ' S / g i J B * The s t a b i l i t y l i m i t s at T=0 between the a n t i -ferromagnetic and spi n - f l o p phases are given by H + = t ( H K L + H K E ) ( 2 H E + H K L + H K E ) ] ^ H" = ( 2 HE-HKI + HKE ) [ ( HKAE ) / ( 2 HE + HKI + HKE ) ] ! 5 ' 9 whereas the thermodynamic t r a n s i t i o n , for which the energies of the two states i s equal i s given by H t h = (E+-E~)h [ ( H K 1 + H K E ) ( 2 H E - H K 1 + H K E » ^ For the. paramagnetic t r a n s i t i o n Hpy = 2 H E + HK1 + HKE ' H p z = 2 H E " HK1 + HKE ' for H applied along the y-axis and z-axis (easy axis) respectively. In the spi n - f l o p phase the spins make an angle ±9 with the applied f i e l d given by cos0 = H/(2H E+H K E-H K 1) I t i s found that there are c r i t i c a l points marking the end of the antiferromagnetic to spi n - f l o p t r a n s i t i o n as the applied magnetic f i e l d deviates from the easy axis. These c r i t i c a l points are given by the c r i t i c a l f i e l d s H — H, , , z th such that there exists a continuous t r a n s i t i o n (neither f i r s t nor second order) from the antiferromagnetic to spin-10 flop phase. The cri t i c a l condition i s a + 3 = + TT/2. The maximum angle of the applied f i e l d at which a f i r s t order t r a n s i t i o n can be observed at T=0 i s tanip = H f / H f x z For the case of orthorhombic symmetry, the 2 u n i a x i a l anisotropy term i n the free energy, ^K^(cos a+ 2 cos 8), must be replaced by a term (Nagayima et a l . , 1955) where y, and v, are the d i r e c t i o n cosines with the x A , B A , B and y axes respectively of the s u b l a t t i c e magnetizations. The z-axis w i l l be the easy axis, and the y-axis w i l l be the second easy axis i f 0 < K, < Kn l y l x If we define H K l y = - K l y S / ^ B  H K l x = " K 1 X S / ^ B then a l l the above expressions r e t a i n the same form with Hj,^ replaced by H K ^ . In addition, the paramagnetic t r a n s i t i o n with the applied f i e l d i n the x-direction i s given by H = 2H_ + HT,, + H_,^  px E Klx KE 11 C o n s i d e r the second s i n g l e i o n a n i s o t r o p y which c o n t r i b u t e s a term i n the above e x p r e s s i o n f o r the f r e e energy of the form - ( N S 2 ) ^ K 2 ( c o s 4 a + c o s 4 6 ) In t h i s case the f i e l d dependence of the s p i n o r i e n t a t i o n i n the s p i n - f l o p s t a t e i s gi v e n by (Morrison, 1973) cose = H/(H -2H _ c o s 2 9 ) , where H Q = 2H £ + H R E - H R l and H R 2 = K ^ / g ^ . The paramagnetic t r a n s i t i o n i s then ( p u t t i n g cos8=l) H = H — 2 H_ _ pz 0 K2 The paramagnetic t r a n s i t i o n f o r a f i e l d p e r p e n d i c u l a r t o the easy a x i s i s g i v e n by py 0 KI I f HT.0cos8/H_, << 1 the s p i n - f l o D t r a n s i t i o n p o i n t i s g i v e n b y: 2 H t h = ( H K E + H K 1 + H K 2 ) H 0 1.2C Dynamics o f the S p i n - f l o p The dynamics of the s p i n - f l o p t r a n s i t i o n have been c o n s i d e r e d r e c e n t l y by K e f f e r (1973) and Chow (1974). They have g i v e n a p i c t u r e o f the t r a n s i t i o n as f o l l o w s : 12 2 2 i n increasing f i e l d at a f i e l d H , = H ^ H ^ + H , which i s less 1 ki A A than the thermodynamic t r a n s i t i o n value H ^ the magnons of the surface layers of the c r y s t a l soften and surface-spin-flop regions develop. Similar regions might also develop near impurity c l u s t e r s , d i s l o c a t i o n l i n e s and other imperfections. As the f i e l d H approaches H ^ these regions develop i n three dimensions and encompass the ent i r e material. The system i s continuously i n stable equilibrium and the energy b a r r i e r i s bypassed making the t r a n s i t i o n second order. Ideally, i n a decreasing f i e l d the sequence i s reversed. The t r a n s i t i o n takes place at the same f i e l d i n increasing and decreasing f i e l d s ; that i s , there i s no hysteresis. 1.2D . Spin Wave Theory Quantum mechanical treatments of the spin-wave theory have been given by Anderson (1952), Ziitian (1952) and Kubo (1952). In p a r t i c u l a r the low temperature behaviour of an anisotropic Heisenberg antiferromagnet i n the neighbourhood of the magnetic phase boundaries has been studied by Feder and Pytte (1968) and Morozov (1972). Following Feder and Pytte we begin with the Hamiltonian = J y _ S ; • S . + K _ _ J S ' . S . f—*.— 1 — 1 • . Z l ZT - L( ? S 2. + ? S 2 .) 1 Z l ] Z ] - H(?S . + ?S . ) 1 Z l j zj 13 where J , K and L r e p r e s e n t r e s p e c t i v e l y t h e e x c h a n g e i n t e r a c t i o n , t h e a n i s o t r o p i c e x c h a n g e and t h e s i n g l e i o n u n i a x i a l a n i s o t r o p y ; and y = gy . T h e o r d e r i n g i s assumed t o be o f a s i m p l e two s u b l a t t i c e n a t u r e s u c h t h a t t h e z n e a r e s t n e i g h b o u r s o'f an i o n o f one s u b l a t t i c e a r e on t h e o t h e r s u b l a t t i c e . F o r t h e a n t i f e r r o m a g n e t i c p h a s e t h e s p i n wave ; r e l a t i o n a t T=0 (and n e g l e c t i n g z e r o p o i n t m o t i o n ) becomes i i w 0 ( k ) = S / Z J [ (1+K' + 2L' $}) 2 - Y 2 ( l i ) ] ^ ± UH where K' = K / J , L' = L / z J , £ 2 = 1 - 1/2S Y (k) = z ^ £ e — — — P_ and p_ g i v e s t h e l a t t i c e p o s i t i o n o f a s p i n . T h e t r a n s i t i o n f r o m t h e a n t i f e r r o m a g n e t i c t o t h e s p i n - f l o p p h a s e o c c u r s a t t h e f i e l d f o r w h i c h t h e f r e q u e n c y becomes z e r o . T h i s o c c u r s f o r k = 0 s o t h a t t h e t r a n s i t i o n o c c u r s f o r K" = 0 a t : H + = 2SzJ[L' £ 2 (1+L' K2) ~t2 F o r t h e s p i n - f l o p s t a t e t h e s p i n wave d i s p e r s i o n r e l a t i o n becomes -Rw°(k) = S z J [ l + y (k.)+L'u 2 ' (1-e) f'2 x [ l + Y ( k ) ( l - 2 r , 2 u 2 ) - L u 2 5 ( l + ? ) ] 3 s 14 w h e r e u = s i n 6, c o s e = u H / [ 2 S z J {1 + %K'-L^)] a n d n^ = 1 + 1 . The s p i n - f l o p t o a n t i f e r r o m a g n e t i c p h a s e b o u n d a r y i s d e t e r m i n e d by a g a i n s e t t i n g w ^ = 0. T h i s o c c u r s f o r k = 0 and g i v e s ( f o r L t y p e a n i s o t r o p y o n l y ) uH~ = 2SzJ (1-L ' ? 2 ) x { L l S ( l + C ) / [ 2 + L f C ( l + ? ) ] } J s T h e r e i s no n e e d t o i n t r o d u c e two s u b l a t t i c e s i n t h e p a r a m a g n e t i c p h a s e s o t h a t t h e a n a l y s i s i s f o r m a l l y s i m i l a r t o t h e f e r r o m a g n e t i c s y s t e m . The s p i n wave r e l a t i o n i s hw° (k) = yH - S z J { l - 2 L ' C 2 + k - y(k)} The r e s u l t i n g t r a n s i t i o n p o i n t b e t w e e n t h e p a r a m a g n e t i c s t a t e a n d t h e s p i n - f l o p s t a t e i s yH = 2 S z J (1 + %K 1 - L 1 ^ 2 ) p z I t i s e v i d e n t t h a t t h e c r i t i c a l f i e l d s d e t e r m i n e d b y s p i n wave c a l c u l a t i o n s a r e t h e same a s t h o s e c a l c u l a t e d f r o m t h e m o l e c u l a r f i e l d m o d e l p r o v i d e d o n e s i m p l y r e p l a c e s L f o r t h e m o l e c u l a r f i e l d a p p r o x i m a t i o n b y 2 f o r t h e s p i n wave c a s e . I t i s a l s o i n t e r e s t i n g t o com p a r e t h e m a g n e t i z a t i o n s i n t h e two m o d e l s . The n e t m a g n e t i z a t i o n <^z> and t h e s u b l a t t i c e m a g n e t i z a t i o n <M 2 1> i n t h e s p i n - f l o p s t a t e a r e r e l a t e d by <M2> = 2<M '>cos9 15 I n t h e m o l e c u l a r f i e l d c a s e t h e s u b l a t t i c e m a g n e t i z a t i o n <^ z'> i s e q u a l t o M Q = u gSN and i s assumed c o n s t a n t f o r T=0 i n a l l f i e l d s . H o w e v e r , i n t h e s p i n wave c a s e t h e s u b l a t t i c e m a g n e t i z a t i o n i s an i n c r e a s i n g f u n c t i o n o f H a n d e q u a l s M Q o n l y i n t h e p a r a m a g n e t i c p h a s e . As t h e f i e l d i s i n c r e a s e d a t c o n s t a n t t e m p e r a t u r e t h r o u g h t h e s p i n - f l o p r a n g e t h e n e t m a g n e t i z a t i o n < M Z > i n c r e a s e s t h r o u g h two c o n t r i b u t i o n s . F i r s t l y , cos6 i n c r e a s e s l i n e a r i l y w i t h H, a n d s e c o n d l y t h e s u b l a t t i c e m a g n e t i z a t i o n i n c r e a s e s ( b e c o m i n g e q u a l t o I- l j O n l y a t H^z. C o n s e q u e n t l y <MZ> i n c r e a s e s f a s t e r t h a n l i n e a r l y w i t h H, as c o n t r a s t e d w i t h t h e s t r i c t l y l i n e a r v a r i a t i o n p r e d i c t e d by m o l e c u l a r f i e l d t h e o r y . The t h e r m o d y n a m i c t r a n s i t i o n b e t w e e n t h e a n t i -f e r r o m a g n e t i c a n d s p i n - f l o p s t a t e s w h i c h c a n be d e t e r m i n e d by s e t t i n g t h e f r e e e n e r g i e s o f t h e two s t a t e s e q u a l h a s n o t b e e n d e t e r m i n e d . I t w o u l d be o f i n t e r e s t t o know t h i s t r a n s i t i o n p o i n t i n t h e s p i n wave t h e o r y b e c a u s e many e x p e r i m e n t s - on a n t i f e r r o m a g n e t s show no h y s t e r e s i s - a t ' t h e s p i n - f l o p t r a n s i t i o n . 1.3 P r o p e r t i e s o f MnCl 2-4H 20 1.3A C r y s t a l l o g r a p h y S e v e r a l s t u d i e s o f t h e m a g n e t i c s t r u c t u r e o f M n C l - 4 H 2 0 h a v e b e e n made. The i o n i c p o s i t i o n s a n d c o n f i g u -16 rations have been determined by.x-ray analysis (Zalkin et a l . , 1964) and neutron d i f f r a c t i o n (El Saffar and Brown, 1971). In addition the antiferromagnetic configuration has been determined without applied f i e l d by Altman et a l . (1975) using neutron d i f f r a c t i o n . It i s a monoclinic antiferromagnet (6=99.7°) with four magnetic ions per unit c e l l . The c r y s t a l l o g r a p h i c space group i s 5 P2!/a (C„, ) . A tv/o s u b l a t t i c e c o l l i n e a r model f i t s the 1 2h data for the antiferromagnetic state. The p o s i t i o n s and ++ spin assignments of the Mn ions are given i n Table 1.1. I t was formerly thought that the easy axis was along the c-axis, but Altman et a l . have determined that i t i s i n c l i n e d between the c-axis and the c*-axis. The projection of the s u b l a t t i c e magnetizations on the a-c plane was found to be 6.9° ± 1.4° from the c-axis towards the c'-axis (perpendicular to the ab plane). The projection on the b-c plane was 0.5° ± 0.5° from the c-axis towards the b-axis. 1.3B Phase Boundaries Measurement of the d i f f e r e n t i a l s u s c e p t i b i l i t y i s a p a r t i c u l a r l y s e n s i t i v e method to determine magnetic phase boundaries. This technique has been used by Rives (1967), and Rives and Benedict (1975) to determine the MnClp*4H50 phase boundaries i n temperatures down to 0.3K. TABLE 1.1 Coordinates of the Manganese Spins Unit C e l l Coordinates Cartesian Coordinates x, y, z 1.571 1.627 6.178 x, y, z .565 7.886 0.085 h+x, h-y, h+z 7.688 3.130 2.969 -h-x, -h+y, -h-z 2.448 6.383 3.134 unit c e l l parameters' 3 a = 11.186(6)A b = 9.513(5)A o o C q = 6.186(2)A B = 99.74 (4) 0 a Altman et a l . (1975) b Zalkin et a l . (1964) 18 T h e y f o u n d t h a t t h e s p i n - f l o p t r a n s i t i o n t e m p e r a t u r e d e p e n d e n c e f o r t e m p e r a t u r e s l e s s t h a n 0.6 5K c a n be f i t t e d b y a n e q u a t i o n o f t h e f o r m H p z ( T ) = H P Z ( 0 ) ( 1 - A t 3 / 2 + B t V 2 > where H p z ( 0 ) = 7.065 KOe A = 0. 066 ± 0. 007 K ~ 3 / / 2 B .= 0.129 + 0.020 K ~ 5 / / 2 T h i s f u n c t i o n has a minimum a t 0.3K which was the lowest temperature p o i n t measured. They a l s o s t u d i e d the p a r a -magnetic t r a n s i t i o n f i e l d s . E x t r a p o l a t i n g these f i e l d s to zero temperature g i v e s a measure o f the m o l e c u l a r f i e l d s u s i n g a b i a x i a l model. In p a r t i c u l a r , the zero temperature f i e l d s they o b t a i n are H =18.55 KOe pz H =22.95 KOe py and H = 2 4.55 KOe px Using the equations of S e c t i o n s 1.2B and 1.2D y i e l d s the f o l l o w i n g m o l e c u l a r f i e l d s PL = 10.375 ± .03 KOe H , = 2.20 ± .05 KOe ; H -. = 3. 80 + 0. 05 KOe K l y K l x 19 In a d d i t i o n they f i t t e d t h e i r d a t a u s i n g the s p i n - f l o p t r a n s i t i o n p o i n t d e f i n e d as H + = (2H EH A + K2A)h to o b t a i n the a n i s o t r o p i c exchange Hv„ = .00 ± .025 KOe KE w i t h the o t h e r f i e l d s unchanged. The experiments were performed on specimens which had been a l i g n e d t o w i t h i n ±2 degrees o f the easy a x i s as determined by the minimum v a l u e o f the t r a n s i t i o n p o i n t . Other methods have been used t o observe the phase boundaries. Gijsman e t a l . (1959) determined H down t h to 1.0K u s i n g zero frequency e l e c t r o n resonance (with a s i g n a l frequency o f 5 MHz) and n u c l e a r magnetic resonance o f the p r o t o n s . M. Abkowitz and A. Honig (1964) performed paramagnetic and a n t i f e r r o m a g n e t i c resonance a t a frequency o f about 24 GHz a t temperatures as low as 0.32K. They o b t a i n e d the ' c r i t i c a l f i e l d 1 resonance a t 7500 Oe w i t h a s i g n a l o f 23.5 GHz a t T^OK. In a d d i t i o n a resonance a t 6.0 KG was observed and one t e n d i n g t o 10.0 KG a t T=0 was seen. These resonances o c c u r r e d w i t h the a p p l i e d f i e l d a l o n g the c - a x i s . The l i n e w i d t h o f the c r i t i c a l f i e l d resonance was about 0.8 KG. Some h y s t e r e s i s depending on 20 the d i r e c t i o n of f i e l d sweep was observed i n t h i s resonance (as high as 0.50 KG although samples with less than 0.10 KG hysteresis were chosen to determine the c r i t i c a l f i e l d values). The other two l i n e s had a width of about 1.60 KG and the low f i e l d resonance was observed only at low temperatures (T< 0.6K). Another technique to study the phase boundaries was used i n an e a r l i e r study by McElearney et a l . (1967). They observed the magneto-caloric e f f e c t at the spin-flop t r a n s i t i o n and obtained t r a n s i t i o n f i e l d s with which the more accurate r e s u l t s of Rives and Benedict are i n agreement. 1.3C S p e c i f i c Heat S p e c i f i c heat measurements of MnCl2*4H20 have been reported by Miedema et 'al. (1965). In conjunction with other experimental data they were able to' determine the coordination number q and the size of the exchange i n t e r a c t i o n . The coordination number obtained i s q = 6 and the exchange i n t e r a c t i o n i s J/k r a = 0.057K which give the molecular f i e l d parameter H„ = 12.8 KOe h i This value of q =6 corresponds to an approximately simple cubic arrangement of the 3 pairs of nearest neighbours of the manganese spins and i s consistent with experimental 21 m e a s u r e m e n t s o f t h e C u r i e - W e i s s c o n s t a n t 9 , t h e s p e c i f i c h e a t f o r T >>T N, t h e e n e r g y g a i n e d o n t r a n s i t i o n t o t h e a n t i f e r r o m a g n e t i c s t a t e f r o m t h e p a r a m a g n e t i c s t a t e a t T=0, a n d t h e z e r o f i e l d s u s c e p t i b i l i t y . A l s o t h e n u c l e a r s p e c i f i c h e a t was d e t e r m i n e d . A s s u m i n g an i n t e r a c t i o n ' o f t h e f o r m H= A S - I t h e n u c l e a r h e a t c a p a c i t y f o r T >>A/k 0 i s g i v e n b y C M T 2 / R = I (I+D S 2 A 2 / 3 k 2 N B 2 -2 The v a l u e o f . C T = 2.6 x 10 J K / m o l e d e t e r m i n e d by t h e e x p e r i m e n t s g i v e s A / k = 0.0131 + 0.00025 K . B 1.3D N u c l e a r O r i e n t a t i o n o f M n C l 2 ' 4 H 2 0 N u c l e a r o r i e n t a t i o n i n a n a n t i f e r r o m a g n e t was f i r s t o b s e r v e d by D a n i e l s e t a l . (1961) . They a c h i e v e d n u c l e a r o r i e n t a t i o n i n a number o f a n t i f e r r o m a g n e t s i n c l u d i n g M n C l 2 - 4 H 2 0 u s i n g b o t h ^ M n a n d ^ C o i s o t o p e s f o r d e t e c t i o n . The gamma-ray a n i s o t r o p i e s o f s i n g l e c r y s t a l s w e r e m e a s u r e d w i t h no a p p l i e d f i e l d s . The maximum 54 a n i s o t r o p y o b s e r v e d was 9% ( w i t h Mn) a t a n e s t i m a t e d t e m p e r a t u r e o f 9 0-mK. The d e t e c t o r s w e r e p l a c e d i n t h e 22 d i r e c t i o n s o f the c and b axes. They found t h a t the s a l t s c o o l e d very s l o w l y ; f o r example, i t r e q u i r e d s i x hours t o ach i e v e the measured maximum a n i s o t r o p y . They suggested t h a t t h i s l o n g c o o l i n g time was due to a long n u c l e a r s p i n - l a t t i c e r e l a x a t i o n time, T^. Miedema e t a l . (1965) observed, i n the experiments d e s c r i b e d i n s u b - s e c t i o n 1.3c above t h a t the time r e q u i r e d to ac h i e v e thermal e q u i l i b r i u m a t O.IK was i n excess o f one hour. They suggested t h a t f a s t c o o l i n g o c c u r r e d f o r a s m a l l p o r t i o n o f n u c l e a r s p i n s , perhaps those w i t h a s h o r t T^ due to t h e i r p r o x i m i t y to magnetic i m p u r i t i e s , f o l l o w e d by much slower c o o l i n g of the remaining s p i n s . 54 The decay scheme (Lederer e t a l . , 1964) f o r Mn i s shown i n f i g u r e 1.2. XJ and Tv c o e f f i c i e n t s f o r the K K angular d i s t r i b u t i o n W (equation 1.1.1) are U 2 = 0.828 , U 4 = 0.418 ; and F 2 = -0.598 , F 4 = -1.079 . The magnetic moment has been determined by Templeton and S h i r l e y (1967) to be u = 3.302 uN. The h y p e r f i n e f i e l d o f manganese i n M n C l 2 - 4 H 2 0 was determined from the n u c l e a r s p e c i f i c heat (see s e c t i o n 1.3E f o l l o w i n g ) t o be H = 645 ± 12 KG. The a n i s o t r o p y as a f u n c t i o n o f temperature i s g i v e n i n f i g u r e 1.3 f o r a x i a l and e q u a t o r i a l d e t e c t i o n . 23 3 + 303 days 'EC •835 KeV E2 0 + 54 Mn 54 Cr Figure 1.2 Decay Scheme of Mn. Figure 1.3 Gamma-ray anisotropy for Mn i n MnCl.2* 4H20. 25 1. 3 E NMR-ON The o b s e r v a t i o n o f n u c l e a r m a g n e t i c r e s o n a n c e i n m e t a l s by t h e d e s t r u c t i o n o f n u c l e a r o r i e n t a t i o n (NMR-ON) h a s become a s t a n d a r d e x p e r i m e n t a l t e c h n i q u e . I n t h e p a r a -m a g n e t i c i n s u l a t o r L a 2 M g ^ (NO^) 1 2 " 2 4 H 2 0 t h e NMR o f o r i e n t e d 54 Mn was o b s e r v e d by N e i s e n e t a l . ( 1 9 6 8 , 1 9 7 0 ) . The o b s e r v a t i o n o f NMR-ON i n a n a n t i f e r r o m a g n e t w o u l d p r o v i d e i n f o r m a t i o n o n t h e n u c l e a r s p i n - l a t t i c e a n d s p i n - s p i n i n t e r a c t i o n s , t h e e l e c t r i c q u a d r u p o l e i n t e r a c t i o n , a n d w o u l d a l l o w a n a c c u r a t e e s t i m a t e o f t h e h y p e r f i n e i n t e r a c t i o n . o f o r i e n t e d n u c l e i c a u s e s t r a n s i t i o n s b e t w e e n t h e m a g n e t i c l e v e l s w i t h a b s o r p t i o n o f p o w e r t h e r e b y r e d u c i n g t h e n u c l e a r o r i e n t a t i o n . The e f f e c t o f r e l a x a t i o n m e c h a n i s m s on r e s o n a n t a b s o r p t i o n h a s b e e n c o n s i d e r e d i n a phenomeno-E x c i t a t i o n o f t h e n u c l e a r m a g n e t i c r e s o n a n c e l o g i c a l a p p r o a c h b y B l o c h ( 1 9 4 6 ) . C o n s i d e r a s y s t e m o f n u c l e i m a g n e t i z e d i n t h e z - d i r e c t i o n b y a c o n s t a n t f i e l d H Q . L e t T ^ a n d T 2 be t h e s p i n - l a t t i c e a n d t r a n s v e r s e r e l a x a t i o n t i m e s r e s p e c t i v e l y . I f a r a d i o - f r e q u e n c y f i e l d H^, o s c i l l a t i n g a t t h e L a r m o r f r e q u e n c y W q = y H ^ i s a p p l i e d i n t h e x - y p l a n e , t h e m a g n e t i z a t i o n M z a l o n g t h e z - a x i s i s g i v e n b y 1 + y ^ T ^ z 1 26 where i s the magnetization in the absence of and Y = u/Ih. In t h i s case the condition for saturation of the resonance i s S Q =- ( Y H 1 ) 2 T 1 T 2 ^ 1 . Consider the value of for MnCl 2*4H 20 at low temperatures. In an antiferromagnet the enhancement of the applied r . f . s i g n a l by the e l e c t r o n i c s u s c e p t i b i l i t y i s not large. In fact, i f the r . f . sig n a l H, i s applied i n lap . the x-direction and z i s the d i r e c t i o n of the s u b l a t t i c e magnetization then the magnetization rotates by an angle 6 given by sine = H l a p / ( 2 H E + HA) Since the hyperfine f i e l d follows the e l e c t r o n i c magnetization an x-component i s induced i n the hyperfine f i e l d H, . = H ..sine l i n . hf The enhancement factor, n, for the r . f . sig n a l i s then given by H. . H, _ _ l i n _ hf lap E For H, c = 600 KG and = 10 KG the enhancement i s n = 30. hf E 27 C o n t r i b u t i o n s t o t h e t r a n s v e r s e r e l a x a t i o n t i m e T 2 i n an a n t i f e r r o m a g n e t come f r o m t h e S u h l - N a k a m u r a s p i n - s p i n i n t e r a c t i o n ( S u h l , 1 9 5 9 ; N a k a m u r a , 1958) a n d n o r m a l i s o t r o p i c c o n c e n t r a t i o n s t h e S u h l - N a k a m u r a i n t e r a c t i o n i s d o m i n a n t . T h i s i n t e r a c t i o n w h i c h i s e f f e c t i v e o n l y b e t w e e n n u c l e i w i t h i d e n t i c a l g y r o m a g n e t i c r a t i o s v a r i e s w i t h t h e s e p a r a t i o n R o f t h e n u c l e i a s w h e r e a i s t h e l a t t i c e s p a c i n g . 54 F o r o u r Mn xn M n C l 2 * 4 H 2 0 s p e c i m e n s x n w h x c h t h e c o n c e n t r a t i o n o f t h e r a d i o i s o t o p e i s o n l y ^ 1 0 " ^ m o l e 54 t h e n e i g h b o u r i n g Mn s p i n s a r e e s s e n t i a l l y o u t o f t h e r a n g e o f t h i s - i n t e r a c t i o n . I n t h i s c a s e , t h e Raman p r o c e s s e s w h i c h a r e i n d e p e n d e n t o f c o n c e n t r a t i o n d o m i n a t e t h e t r a n s v e r s e r e l a x a t i o n r a t e w h i c h i s g i v e n b y d e p e n d i n g o n h y p e r f i n e a n d a t o m i c p a r a m e t e r s o f t h e s y s t e m 7 -1 C h a s b e e n c a l c u l a t e d f o r M n F 2 t o b e 3 x 10 K - s e c ( M i t c h e l l , 1957) and i t h a s a p p r o x i m a t e l y t h e same v a l u e f o r M n C l 2 ' 4 H 2 0 . We e s t i m a t e t h a t t h e v a l u e o f T 2 i s - v l s e a t a t e m p e r a t u r e o f 0.1K. Raman p r o c e s s e s ( f o r e x a m p l e s e e J a c c a r i n o , 1 9 6 5 ) . F o r T w h e r e T A E = (2EL_,HA) 2 g u B / k a n d w h e r e C i s a c o n s t a n t 28 The s p i n l a t t i c e r e l a x a t i o n t i m e f o r t h i s s y s t e m i s a l s o n o t r e a d i l y d e t e r m i n e d t h e o r e t i c a l l y . H o w e v e r , t h e s p e c i f i c h e a t m e a s u r e m e n t s o f Miedema e t a l . (1965) s u g g e s t e d a r a t h e r l o n g r e l a x a t i o n t i m e (>10 min) f o r t h e n u c l e a r s y s t e m w h e r e a s t h e r e s u l t s t o b e p r e s e n t e d l a t e r i n d i c a t e t i m e s , o f l e s s t h a n one h o u r . H e n c e , i t 3 i s r e a s o n a b l e t o e s t i m a t e T^ t o be o f t h e o r d e r o f 10 s e c . 3 A p p l y i n g t h e v a l u e s n = 30, - 1 s e c , T^ - 10 s a n d 7 H^ - 1 mG, a v a l u e o f S Q ^  10 i s o b t a i n e d so t h a t t h e r e s o n a n c e s h o u l d be r e a d i l y o b s e r v e d . The r e s o n a n t f r e q u e n c y c a n be d e t e r m i n e d b y t h e v a l u e o f t h e h y p e r f i n e i n t e r a c t i o n m e a s u r e d f r o m t h e 54 s p e c i f i c h e a t (Miedema e t a l . , 1965) o f t h e Mn n u c l e a r s y s t e m . The h y p e r f i n e i n t e r a c t i o n i s g i v e n b y H = A S Z I Z . i g n o r i n g q u a d r u p o l a r e f f e c t s . The n u c l e a r l e v e l s a r e s p l i t b y -fiv = A S O 7. The h y p e r f i n e f i e l d i s d e f i n e d b y AS I = g. Ty^ T I H, _ z z N z h f 29 The J"*Mn r e s o n a n c e i s g i v e n by 54 5 4 5 5 % ^ o = W h f S z = 5 5 - ^ - S z g N The v a l u e o f 5 5 A / k D i s 0.0131 ± 0.0025 K 54 55 and g / g = -0.83 3 Hence t h e r e s u l t s f o r 5 4 M n a r e 5 4 A / k T 5 = 0.0104 ± .0002 K H, , = 645 ± 12 KG h f a n d v Q = 5 4 2 ± 1 0 MHz 1.4 I n t e r a c t i o n s i n A n t i f e r r o m a g n e t i c I n s u l a t o r s 1.4A The S u p e r e x c h a n g e I n t e r a c t i o n S i m p l e d i r e c t e x c h a n g e i n t e r a c t i o n s b e t w e e n a t o m i c o r b i t a l s a r e n o t s u f f i c i e n t t o a c c o u n t f o r t h e o r d e r e d m a g n e t i s m f o u n d i n many i o n i c s o l i d s . The l o n g r a n g e i n t e r a c t i o n g i v i n g r i s e t o t h e m a g n e t i c o r d e r i n g i s c a l l e d s u p e r e x c h a n g e . I t i s c l e a r t h a t l i g a n d wave f u n c t i o n s a r e m o d i f i e d b y t h e p r e s e n c e o f a m a g n e t i c i o n . A n d e r s o n (1950) c o n s i d e r e d t h e s i m p l e s t s u c h m o d i f i c a t i o n i n w h i c h a n e l e c t r o n i s t r a n s f e r r e d f r o m a l i g a n d o r b i t a l ( p e r h a p s a f i l l e d 2p o r b i t a l ) t o an empty d o r b i t a l o f t h e m a g n e t i c 30 ion. The i n t e r - i o n i c exchange int e r a c t i o n s then couple the magnetic and ligand ions i n t h i s excited state so that the i o n i c spins are aligned ferromagnetically or a n t i f e r r o -magnetically. If there i s some d i s t o r t i o n of the p e r f e c t l y i o n i c configuration, due to overlap of the wavefunctions, the wavefunctions of the system are given i n perturbation theory by small admixtures of excited states. Hence, the previously mentioned magnetic states w i l l be admixed i n the ground state of the system. Conservation of angular momentum w i l l then require the ground state to have the same magnetic state (defined by the t o t a l angular momentum) as the lowest energy excited states. For example, an excited state with t o t a l spin equal to zero w i l l only couple with a ground state of spin equal to zero. By th i s process a small admixture of excited states can lead to magnetic ground states with large exchange energy. The admixture of an excited state can also be considered as a v i r t u a l t r a n s i t i o n . Such a t r a n s i t i o n i s short- l i v e d , does not conserve energy and cannot be detected by absorption or emission of r a d i a t i o n . The system i s induced by the perturbation (of the i o n i c wave-functions) to make a t r a n s i t i o n from the ground state to the excited states. The preferred spin of the excited stated w i l l then p e r s i s t i n the ground state. Only when 31 averaged over time does t h i s model give the charge d i s t r i b u t i o n of the true perturbed motion. A more general approach to superexchange has been presented by Anderson (1969). One f i r s t considers a wave function of a magnetic ion surrounded by diamagnetic groups and i n fact the whole l a t t i c e , excluding from the wavefunction the exchange e f f e c t of other magnetic ions. I t i s assumed that the exchange e f f e c t s do not s u b s t a n t i a l l y disturb the ligand f i e l d wave function. The magnetic ions are then brought together and the exchange i n t e r a c t i o n i s determined. This i n t e r a c t i o n i s now the consequence of d i r e c t overlap of longer ranging o r b i t a l s . Since these o r b i t a l s are orthogonal, ferromagnetic i n t e r a c t i o n s would be expected to r e s u l t . However, excited states can be expected i n which the magnetic electrons are on the same ligand f i e l d wavefunction. For the state with p a r a l l e l spin momenta t h i s i n t e r a c t i o n cannot occur, but f o r a n t i -p a r a l l e l spins i t gives a reduction i n the t o t a l energy. This 'configuration mixing' favours antiferromagnetic interactions and i s c a l l e d 'kinetic exchange' by Anderson. The d i r e c t exchange favouring ferromagnetic alignment i s c a l l e d p o t e n t i a l exchange. For the case of MnCl2*4H20 the manganese ions are separated by non-linear chains of oxygen, chloride, and hydrogen ions. The d e t a i l s of the ligand f i e l d s and the 32 v i r t u a l t r a n s i t i o n s a r e e x t r e m e l y c o m p l i c a t e d i n s u c h a c a s e . N e v e r t h e l e s s t h e f o r m o f t h e H a m i l t o n i a n c a n be g i v e n v e r y s i m p l y , as f o l l o w s . F o r a p a i r o f i n t e r a c t i n g i o n s e a c h w i t h a s i n g l e u n p a i r e d e l e c t r o n t h e e n e r g y o f t h e s p i n s t a t e s r e l a t i v e t o t h e mean v a l u e c a n be g i v e n by an e f f e c t i v e e x c h a n g e H a m i l t o n i a n ( s e e f o r e x a m p l e M a r t i n , 1 9 6 7 ) ~H = -J . . m + s . • s .) " e x 13 —1 —3 F o r a H a m i l t o n i a n w h i c h c o n t a i n s no s p i n d e p e n d e n t t e r m s t h e e i g e n s t a t e s o f a p a i r o f I o n s c a n be s e l e c t e d a s 2 e i g e n s t a t e s o f S a n d S w h e r e S = S. + S. a n d S. i s an z — —1 —3 1 i o n i c t o t a l s p i n . An e f f e c t i v e H a m i l t o n i a n w h i c h g e n e r a t e s s u c h e i g e n s t a t e s i s H = - J . . ( s . - s . ) + J : . ( s . • s . ) 2 + . . . e x 13 -1-1. 13 1 3 I f t h e t r u e H a m i l t o n i a n ( n o t t h e e f f e c t i v e H a m i l t o n i a n ) o f t h e s y s t e m c o n t a i n s s p i n d e p e n d e n t t e r m s s u c h as s p i n -o r b i t c o u p l i n g t h e s e c a n be t r e a t e d s e p a r a t e l y a s a n i s o t r o p i c i n t e r a c t i o n s ( s e e s e c t i o n 1 . 4 B f o l l o w i n g ) . The f i r s t t e r m i n d~t s h o u l d d o m i n a t e . T h e n t h e t o t a l e f f e c t e x f o r an i o n i c s o l i d c a n b e r e p r e s e n t e d b y t h e e x c h a n g e t e r m K e x = - S J S . - S . ( 1 . 4 . 1 ) i j . T h i s i s t h e H e i s e n b e r g H a m i l t o n i a n . 33 1.4B The Anisotropic Interactions The coupling between an electron of, spin and the o r b i t a l motion of the same electron i s given by 2m2 C2 | h ' (grad.V(r i) x p. , ) where p^ i s the momentum of the electron and V(r^) i s the •potential- I f the p o t e n t i a l derives from a central force f i e l d the energy can be written i n the form H = ^ ( r . ) l . - s . where 1 . i s the o r b i t a l angular momentum, and £ (r.. ) i s a function of the distance of the electron from the nucleus. There i s a s i m i l a r expression for the coupling of the spin of one electron and the angular momentum of another electron. For the case of states of given L and S the in t e r a c t i o n i s simply given by "H = AL-S Li o - - — This i s the spin o r b i t coupling. Generally, the c r y s t a l l i n e f i e l d 'quenches the o r b i t a l angular momentum. In the sp e c i a l case of 3d^-ions (Mn + +, F e 3 + ) the spectro-6 scopic state i s ^5/2 9 i v i n < ? L = 0* Hence, the spin o r b i t coupling vanishes. However, higher order c a l c u l a t i o n s in perturbation theory (that i s , mixing of excited states) 34 y i e l d s non-zero values for the energy. The form of the spin Hamiltonian o f an ion depends on.the symmetry of i t s surroundings as well as on the magnitude of i t s spin. For many compounds the environment of the ions i s primarily cubic consisting of 4 or 6 almost equidistant anions. The departure from cubic symmetry i s usually a x i a l in nature, although there may be terms of lower symmetry. These non-axial terms.are neglected here. With t h i s approximation the most general spin Hamiltonian for a spin S < 5/2 i s (Abragam and Pryce, 19 51) X. = gu0S-H + \ a ( S 4 + S 4 + S 4 ) + DS 2 +•fS 4 (1.4.2) d V- B 6 x y z' a a where a denotes the d i r e c t i o n of the a x i a l d i s t o r t i o n . For paramagnetic s a l t s D i s often considerably larger than a, while f i s n e g l i g i b l y small. The f i r s t term i n 3~t , which i n the paramagnetic case represents the e f f e c t of an applied f i e l d , i n the case of a magnetic material represents the exchange i n t e r a c t i o n of the ion with i t s magnetic neighbours. Van Vleck and Penney (1934) obtain the quar t i c term i n H . by f i f t h order perturbation theory involving the octahedral c r y s t a l f i e l d to the f i r s t power 2 and A' to the fourth power. The S term can be obtained a i n fourth order perturbation theory involving X, as well 35 as from configuration i n t e r a c t i o n s within t h e ion which cause small departure of the electron cloud from sp h e r i c a l symmetry (Abragam and Pryce, 1951). Note that, although 2 a term i n S i s present, g i s verv often e f f e c t i v e l y a i s o t r o p i c . I f t h e anisotropic Hamiltonian i s given by the D and a terms i n l~i denoted above, the anisotropic contribution to the free energy of the material i s (Kanamori, 1963) to f i r s t order, assuming the a x i a l anisotropy l i e s along the z-axis of the octahedral symmetry (and ignoring constant terms), F a = [ (D-2a)cos 26 + 2acos 4 e ] , (1.4.3) where 9 i s the angle the spin makes with the z-axis. 2 4 The c o e f f i c i e n t s of cos 9 and cos 9 are known resp e c t i v e l y as the f i r s t and second s i n g l e - i o n anisotropy. The i n t e r i o n i c dipolar i n t e r a c t i o n between spins S^, S_. i s given by where r . . i s the vector joing S. and S.. This term gives l j J ^ — l — j • ^ r i s e to an anisotropy energy to f i r s t order. When spin-o r b i t coupling i s not neglected i n the c a l c u l a t i o n of the exchange energy a term s i m i l a r to the dipo l a r term arises (Van Vleck, 1937; Tessman, 1954). This i n t e r a c t i o n a r i s e s 36 in second order perturbation theory and i s given by pd where A i s the s p i n - o r b i t coupling, A E i s the energy difference between the ground and excited states of the o r b i t a l l e v e l s and J i s the exchange coupling. This coupling i s c a l l e d pseudo-dipolar. The net e f f e c t of the d i p o l a r and pseudo-dipolar i n t e r a c t i o n s i s c a l l e d the anisotropic exchange and i s usually simply given by 1.5 Contact Cooling The cooling mechanisms of a specimen of MnCl2'4H.20 are shown schematically in figure 1.4. At temperatures below 200 mK the dominant heat capacity of 55 the system i s that of the n a t u r a l l y abundant Mn nuclear 55 spins. The spin l a t t i c e relaxation times for the Mn 54 and Mn systems are not known and could d i f f e r greatly (as discussed i n section 1.3E). The l a t t i c e i s cooled by contact with a copper heat sink through a grease. The heat i s conducted from the specimen to the contact material by phonons. L i t t l e (1959) has given the t h e o r e t i c a l formula 4 4 Q =• KA(T X - T^ ) (1.5.1) Copper heat sink Mn spin system Contact Medium Cr y s t a l l a t t i c e Mn spin system Figure 1.4 Schematic diagram of the cooling of MnCl 2-4H 20 38 w h e r e A i s t h e c o n t a c t a r e a a n d a n d g i v e t h e t e m p e r a -t u r e o f t h e l a t t i c e a n d c o n t a c t m a t e r i a l ) f o r a chrome a l u m t o c o p p e r i n t e r f a c e . E x p e r i m e n t a l o b s e r v a t i o n s ( A n d e r s o n e t a l . , 1961) f o r t h i s s i t u a t i o n c o n f i r m t h e 4 T r e l a t i o n a n d g i v e t h e c o n s t a n t a s 4 -i _ 4 _ 2 K = 6.2 x 10 e r g s e c K- cm E m p i r i c a l o b s e r v a t i o n s h a v e b e e n made a s s u m i n g Q - ( T n - T Q n ) . (1.5.2) f o r c o n t a c t b e t w e e n m a t e r i a l s a t l o w t e m p e r a t u r e s . Some o f t h e s e r e s u l t s h a v e b e e n g i v e n b y V i l c h e s a n d W h e a t e l y (1966). A s i m i l a r r e s u l t m i g h t be e x p e c t e d f o r t h e MnC^'^H^O t o c o p p e r c o n t a c t . A s s u m i n g e q u a l l a t t i c e a n d n u c l e a r t e m p e r a t u r e s , t h e c o o l i n g o f t h e s p e c i m e n i s g i v e n b y t h e f o l l o w i n g e q u a t i o n Q = C N T + q w h e r e Q i s t h e h e a t l o s s t h r o u g h t h e c o n t a c t , q i s t h e h e a t l e a k i n t o t h e s p e c i m e n a n d C„T i s t h e n u c l e a r h e a t c a p a c i t y . U s i n g e q u a t i o n (1.5.2) g i v e s K A ( T n - T ") = — C .T + q (1.5.3) C N The u l t i m a t e t e m p e r a t u r e o f t h e s y s t e m , T , i s g i v e n when T=0, h e n c e q = K A ( T o n - T c n ) (1.5 The n u c l e a r h e a t c a p a c i t y i s g i v e n b y C N = C Q / T 2 (1.5 C o m b i n i n g e q u a t i o n s (1.5 . 3 ) t o (1.5.5) y i e l d s , - T / T 2 = ( T n - T o n ) (1.5 o T h i s e q u a t i o n c a n be u s e d t o d e t e r m i n e t h e t h e r m a l c o n d u c t i v i t y c o n s t a n t K o f t h e c o n t a c t i f t h e s p e c i m e n c o o l i n g h a s b e e n d e t e r m i n e d . 40 CHAPTER II The E x p e r i m e n t a l Procedures 2.1 Growth of the C r y s t a l Specimens C r y s t a l specimens of MnCl 2"4H 20 were grown by e v a p o r a t i o n from the s a t u r a t e d s o l u t i o n . I n i t i a l l y they were grown on h a i r s i n a c o n s t a n t temperature bath. However, the f o l l o w i n g s i m p l e r method produced c r y s t a l s of good q u a l i t y . A s a t u r a t e d s o l u t i o n of 10 ml of MnCl 2*4H 20 54 c o n t a i n i n g about 200 y C i of c a r r i e r - f r e e Mn was p r e p a r e d . T h i s s o l u t i o n was poured i n t o a warmed e v a p o r a t i n g d i s h t o a depth of about 1 cm. A drop of hot water was added t o the s o l u t i o n and s t i r r e d i n so t h a t a s l i g h t l y under-s a t u r a t e d s o l u t i o n , w i t h no seeds, was o b t a i n e d . A s m a l l 3 seed c r y s t a l (of about 3 mm volume) was p l a c e d i n the s o l u t i o n on the bottom of the e v a p o r a t i n g d i s h . The c o n t a i n e r was then covered w i t h a f i l t e r paper and l e f t a t room temperature f o r about two days i n which time the c r y s t a l had grown to about 1 cm i n l e n g t h . The c r y s t a l was then removed from the s o l u t i o n and s t o r e d . C r y s t a l s of MnCl 2*4H 20 of dimensions of about 1 cm x 1 cm, x 0.3 cm and having an a c t i v i t y of about 25 y C i were o b t a i n e d . The 41 c r y s t a l l o g r a p h i c a x e s o f t h e s p e c i m e n s w e r e r e a d i l y d e t e r m i n e d by i n s p e c t i o n o f t h e c r y s t a l f a c e s ( G r o t h , 1 9 0 8 ) . 2.2 The C r y o s t a t a n d t h e C r y s t a l H o l d e r The low t e m p e r a t u r e s r e q u i r e d f o r t h e e x p e r i m e n t s w e r e a c h i e v e d b y d e m a g n e t i z a t i o n o f a p a r a m a g n e t i c s a l t . A s c h e m a t i c d i a g r a m o f t h e c r y o s t a t i s shown i n f i g u r e 2 . 1 . The p a r a m a g n e t i c s a l t u s e d was chrome p o t a s s i u m a l u m . Embedded i n t h e a l u m w e r e a p p r o x i m a t e l y 5000 c o p p e r w i r e s 2 h a v x n g a c o m b i n e d s u r f a c e a r e a o f a b o u t 2000 cm . T h e s e w i r e s w e r e s o l d e r e d i n t o a c o p p e r c o l d f i n g e r t o w h i c h t h e c r y s t a l h o l d e r ( d e s c r i b e d b e l o w ) was a t t a c h e d . A 6 0 C o - F e p l a t e was a l s o s o l d e r e d t o t h e c o p p e r c o l d f i n g e r t o p r o v i d e a l o w t e m p e r a t u r e n u c l e a r o r i e n t a t i o n t h e r m o m e t e r ( C a m p b e l l e t a l . ' 1965) . A c o p p e r h e a t s h i e l d e n c l o s i n g t h e c h r o m e a l u m p i l l a n d c o l d f i n g e r was c o o l e d b y a manganous ammonium s u l p h a t e . ' g u a r d ' p i l l . A j a c k e t c o o l e d by a IK pumped l i q u i d h e l i u m b a t h e n c l o s e d the e n t i r e s a l t p i l l a s s e m b l y . A n o u t e r 4K l i q u i d h e l i u m b a t h c o n t a i n e d t h r e e s u p e r c o n d u c t i n g s o l e n o i d s . Two o f t h e s u p e r c o n d u c t i n g s o l e n o i d s , p r o d u c i n g ' f i e l d s o f 35 K G - a n d 10 KG, w e r e u s e d t o d e m a g n e t i z e t h e chrome a l u m p i l l a n d t h e g u a r d p i l l r e s p e c t i v e l y . W i t h t h i s c r y o s t a t t h e t e m p e r a t u r e o f t h e c o l d f i n g e r , a s m e a s u r e d by t h e n u c l e a r o r i e n t a t i o n t h e r m o -42 A u x i l i a r y cooling s a l t Demagnetizing solenoids Main cooling s a l t Heat s h i e l d A u x i l i a r y cooling s a l t Copper cold finger C r y s t a l holder P o l a r i z i n g solenoid The Cryostat. 43 m e t e r , was l e s s t h a n 20 mK f o r a p e r i o d e x c e e d i n g 48 h o u r s . M a g n e t i c f i e l d s w e r e a p p l i e d t o t h e s p e c i m e n by a s u p e r - ' c o n d u c t i n g s o l e n o i d h a v i n g a maximum f i e l d o f . 1 5 KG a n d a n i n h o m o g e n e i t y o f l e s s t h a n 1% o v e r a s p h e r e o f 1.5 cm d i a m e t e r . To a c h i e v e o p t i m u m h e a t t r a n s f e r f r o m t h e c r y s t a l s p e c i m e n t o t h e c o p p e r c o l d f i n g e r by c o n t a c t c o o l i n g i t was n e c e s s a r y t o u t i l i z e a s much c o n t a c t a r e a a s p o s s i b l e . F u r t h e r m o r e , a c o v e r i n g o v e r t h e c r y s t a l w o u l d p r e v e n t h e a t i n g c a u s e d b y t h e a d s o r p t i o n o f a n y r e s i d u a l e x c h a n g e g a s . T h e r e f o r e , t h e c r y s t a l was m o u n t e d i n a c o p p e r h o l d e r w h i c h h a d b e e n s o f t - s o l d e r e d t o t h e c o p p e r c o l d f i n g e r . The c r y s t a l h o l d e r was made o f c o p p e r f o i l (1/64 i n c h e s t h i c k ) f o l d e d t o f i t t h e c r y s t a l . The c r y s t a l was m o u n t e d i n t h e h o l d e r u s i n g A p i e z o n N g r e a s e f o r t h e r m a l c o n t a c t . The c r y s t a l p l a n e s and e d g e s w e r e i n c o n t a c t w i t h t h e p l a n e s a n d e d g e s o f t h e h o l d e r s o t h a t t h e l a t t e r s e r v e d t o o r i e n t t h e c r y s t a l . Good c o n t a c t t o t h e c r y s t a l was o b t a i n e d b y t y i n g t h r e a d t i g h t l y a r o u n d t h e h o l d e r . A c r y s t a l h o l d e r was n o t u s e d f o r t h e NMR-ON e x p e r i m e n t s b e c a u s e i t w o u l d h a v e a t t e n u a t e d t h e a p p l i e d r a d i o - f r e q u e n c y f i e l d . I n t h i s c a s e , t h e c r y s t a l was s i m p l y f a s t e n e d t o t h e h e a t s i n k w i t h A p i e z o n N g r e a s e a n d 44 t h r e a d . F o r t h e s u s c e p t i b i l i t y m e a s u r e m e n t s t h e c r y s t a l was i n s e r t e d i n a s e c o n d a r y c o i l w h i c h s e r v e d a s t h e c r y s t a l h o l d e r a n d w h i c h was g l u e d w i t h GE 7031 v a r n i s h t o t h e c o p p e r c o l d f i n g e r . I n o r d e r t o d e t e r m i n e t h e d i r e c t i o n s o f t h e s u b -l a t t i c e m a g n e t i z a t i o n s r e l a t i v e t o t h e c r y s t a l a x e s o f t h e m o u n t e d s p e c i m e n i t was n e c e s s a r y t o know t h e c r y s t a l o r i e n t a t i o n i n t h e f r a m e o f r e f e r e n c e o f t h e d e t e c t o r s , t h a t i s , t h e l a b o r a t o r y f r a m e . T h i s m e a s u r e m e n t was f a c i l i t a t e d by t e m p o r a r i l y m o u n t i n g a s m a l l m i r r o r o n t h e c r y s t a l o r c r y s t a l h o l d e r . A l a s e r l i g h t a n d m i r r o r s m o u n t e d on t h e l a b o r a t o r y w a l l s w e r e t h e n u s e d t o d e t e r m i n e t h e s p e c i m e n o r i e n t a t i o n i n t h e l a b o r a t o r y f r a m e . 2.3 Gamma-Ray D e t e c t o r s a n d S p e c t r u m A n a l y s i s T he n u c l e a r o r i e n t a t i o n e x p e r i m e n t s w e r e p e r f o r m e d u s i n g f r o m two t o f o u r d e t e c t o r s a t a t i m e . T h e s e d e t e c t o r s w e r e p o s i t i o n e d t o o p t i m i z e t h e i n f o r m a t i o n o b t a i n e d . The d e t e c t o r o r i e n t a t i o n s r e l a t i v e t o t h e c r y s t a l p o s i t i o n w e r e d e t e r m i n e d i n t h e l a b o r a t o r y f r a m e o f r e f e r e n c e . The gamma r a d i a t i o n i n t e n s i t y was m e a s u r e d w i t h b o t h N a l a n d G e ( L i ) d e t e c t o r s . The N a l c r y s t a l s w e r e 2" d i a m e t e r b y 2" l e n g t h . T h e two G e ( L i ) d e t e c t o r s u s e d i n t h e e x p e r i m e n t s had e f f i c i e n c i e s o f 6% a n d 1 3 % c o m p a r e d t o a 3" x 3" N a l c r y s t a l . The o u t p u t s o f t h e d e t e c t o r s w e r e a m p l i f i e d a n d r o u t e d t o a m u l t i - c h a n n e l a n a l y s e r w h i c h a c c u m u l a t e d t h e s p e c t r a . I n a d d i t i o n s e l e c t e d gamma-ray p e a k s w e r e m o n i t o r e d t h r o u g h s i n g l e c h a n n e l • a n a l y s e r s a n d s c a l e r s . A f t e r e a c h c o u n t i n g p e r i o d t h e a n a l y s e r s p e c t r a w e r e p u n c h e d o u t o n p a p e r t a p e w h i c h was r e a d i n t o t h e U n i v e r s i t y ' s IBM 370/168 c o m p u t e r f o r a n a l y s i s . 54 The gamma-ray s p e c t r a c o n t a i n e d one Mn 6 0 p e a k a t 0.8 37 MeV and two Co p e a k s a t 1.17 a n d 1.33 MeV. The p h o t o p e a k i n t e n s i t i e s w e r e d e t e r m i n e d by s u b t r a c t i n g a b a c k g r o u n d c o u n t from, t h e t o t a l number o f c o u n t s i n a 'window' s e t o n t h e p h o t o - p e a k . The ' b a c k g r o u n d c o u n t ' was d e t e r m i n e d by m e a s u r i n g t h e c o u n t s on e a c h s i d e o f t h e p e a k and i n t e r p o l a t i n g . I n t h i s m a n n e r , t r u e b a c k -6 0 54 g r o u n d f r o m d e g r a d e d Co r a d i a t i o n a n d some a c t u a l Mn p e a k r a d i a t i o n was i n c l u d e d i n t h e b a c k g r o u n d c o r r e c t i o n . The a s s u m p t i o n t h a t t h e b a c k g r o u n d i s l i n e a r i s b e t t e r f o r a G e ( L i ) d e t e c t o r t h a n a N a l d e t e c t o r . N e v e r t h e l e s s , s i n c e o n l y r e l a t i v e i n t e n s i t i e s a r e r e q u i r e d i n t h i s w o r k , t h i s b a c k g r o u n d c o r r e c t i o n was f o u n d t o be s a t i s f a c t o r y . I n d e e d , i n t h e c a s e o f t h e N a l d e t e c t o r s , i t was f o u n d t h a t t h e 54 a n i s o t r o p y o f t h e e s t i m a t e d b a c k g r o u n d o n t h e Mn p e a k was 54 a l m o s t e q u a l t o t h e Mn p e a k a n i s o t r o p y i n d i c a t i n g t h a t t h e t r u e b a c k g r o u n d made o n l y a s m a l l c o n t r i b u t i o n . T h e a n a l y s i s o f t h e a p p r o x i m a t e l y 100 s p e c t r a p e r 46 r u n was done u s i n g c o m p u t e r p r o g r a m s . A l l o f t h e s p e c t r a i n t h e r u n were a n a l y s e d u s i n g t h e same window w h i c h was c o r r e c t e d f o r any s h i f t s i n peak p o s i t i o n . The n o r m a l i z e d i n t e n s i t y W ( e q u a t i o n 1.1.1) f o r e a c h d e t e c t o r was o b t a i n e d by d i v i d i n g t h e c o l d c o u n t by t h e a v e r a g e 'warm c o u n t ' f o r t h a t d e t e c t o r . The warm' c o u n t s were o b t a i n e d w i t h t h e s p e c i m e n a t s u f f i c i e n t l y h i g h t e m p e r a t u r e (>1K) t h a t t h e a n i s o t r o p y o f t h e r a d i a t i o n was n e g l i g i b l e . F o r some o f t h e s h o r t e r c o u n t i n g p e r i o d s s c a l e r c o u n t s were t a k e n w i t h o u t t h e a n a l y s e r c o u n t s ( b e c a u s e o f t h e l o n g p r i n t o u t t i m e r e q u i r e d by t h e a n a l y s e r ) . I n t h i s c a s e , t h e s c a l e r windows and c o u n t s were p e r i o d i c a l l y c h e c k e d a g a i n s t a n a l y s e r s p e c t r a t o d e t e r m i n e w h e t h e r any s h i f t s h a d o c c u r r e d . I f r e q u i r e d , a s u b s e q u e n t r e n o r m a l i z a t i o n o f s c a l e r c o u n t s c o u l d be p e r f o r m e d . 2.4 A n a l y s i s o f A n i s o t r o p i e s A s d i s c u s s e d i n s e c t i o n 1.1, t h e a n g u l a r d i s t r i -b u t i o n c a n be a n a l y s e d t o d e t e r m i n e t h e o r i e n t a t i o n o f t h e s p i n s y s t e m s and t h e i r t e m p e r a t u r e . I t r e q u i r e s two a n g l e s t o l o c a t e a d i r e c t i o n i n s p a c e so t h a t i n o r d e r t o d e t e r m i n e two s u b l a t t i c e d i r e c t i o n s and t h e t e m p e r a t u r e a s w e l l , i t i s n e c e s s a r y t o have a t l e a s t f i v e i n d e p e n d e n t m e a s urements o f t h e a n i s o t r o p y , t h a t i s , f i v e m e a s u r e m e n t s 4 7 a t d i f f e r e n t a n g u l a r p o s i t i o n s . F o u r d e t e c t o r s w e r e a v a i l a b l e f o r t h e s e e x p e r i m e n t s a n d t h i s was t h e maximum .number u s e d . A l s o , f o u r d e t e c t o r s g a v e a c o u n t i n g s y s t e m w h i c h was n o t o v e r l y c o m p l e x . I n p r a c t i c e , b e c a u s e c e r t a i n p h y s i c a l a s s u m p t i o n s c o u l d be made, a d e q u a t e i n f o r m a t i o n was O b t a i n e d w i t h t h i s number o f d e t e c t o r s . A l l t h e o b s e r v e d a n i s o t r o p i e s f o r a g i v e n a p p l i e d f i e l d w i l l r e p r e s e n t t h e same m a g n e t i c c o n f i g u r a t i o n i n t h e s p e c i m e n . A gamma-ray i n t e n s i t y m e a s u r e d o n a d e t e c t o r l o c a t e d i n d i r e c t i o n EK f o r a s p e c i m e n t e m p e r a t u r e T_. i s ex t h d e s i g n a t e d W. .. T h e o r e t i c a l i n t e n s i t i e s W. . ({V, }) c a n b e I D 1 j k d e t e r m i n e d f o r a n y s e t o f v a r i a b l e s {V^.} s u c h a s t h e t IT s u b l a t t i c e o r i e n t a t i o n s a n d t h e t e m p e r a t u r e . The W.. i l e x i n t e n s i t i e s c a n t h e r e f o r e be f i t t e d t o t h e W.. t o d e t e r m i n e 13 t h e d e s i r e d v a r i a b l e s . The b e s t f i t was d e t e r m i n e d b y m i n i m i z i n g c h i s q u a r e d : w h e r e e.. i s t h e e s t i m a t e d s t a n d a r d d e v i a t i o n i n t h e 13 a n i s o t r o p i e s . I n c l u d e d i n t h e s t a n d a r d d e v i a t i o n e.. i s t h e I D u n c e r t a i n t y i n t h e d e t e c t o r p o s i t i o n . I f t h e d e t e c t o r i h a s i t s p o s i t i o n s p e c i f i e d b y t h e a n g l e s 6^, <j>^ , t h e n e 2 . = (,3Wth/9e • ) e 2 (e . ) + ( 9 W t h / 3 $ . ) e 2 U • ) + S 2 I D ij i i l y y i yx 48 w h e r e e ( x ) i n d i c a t e s t h e u n c e r t a i n t y i n Xi a n < 3 5 i s t h e s t a n d a r d d e v i a t i o n i n t h e c o u n t r a t e . The a c c u r a c y o f t h e s o l u t i o n was d e t e r m i n e d n u m e r i c a l l y . F o r any v a r i a b l e o f t h e s o l u t i o n , V^., t h e e r r o r i s g i v e n b y • • e 2 ( V k ) . = 2 O V /9P.) 2 e 2 (P,) , i / j J J w h e r e t h e P. a r e a l l t h e i n o u t p a r a m e t e r s s u c h a s t h e o b s e r v e d a n i s o t r o p i e s a n d t h e d e t e c t o r p o s i t i o n s . T h e p a r t i a l d e r i v a t i v e s ( 3 V ^ / 3 P j ) w e r e c a l c u l a t e d b y i n c r e m e n t i e a c h p a r a m e t e r a n d d e t e r m i n i n g t h e c h a n g e i n t h e v a r i a b l e s i n t h e r e s u l t a n t b e s t f i t . 2.5 D e s c r i p t i o n o f a n E x p e r i m e n t a l Run A f t e r t h e M n C l 2 ' 4 H 2 0 s p e c i m e n was m o u n t e d a n d t h e d e m a g n e t i z a t i o n c r y o s t a t a s s e m b l e d , t h e c r y o s t a t was p r e c o o l e d w i t h l i q u i d n i t r o g e n c o n t a i n e d i n a n o u t e r d e w a r . L i q u i d h e l i u m was t r a n s f e r r e d i n t o t h e c r y o s t a t a n d t h e I K b a t h . The s a l t p i l l a s s e m b l y was k e p t i n t h e r m a l c o n t a c t w i t h t h e l i q u i d h e l i u m b y h e l i u m e x c h a n g e g a s . When t h e s y s t e m h a d c o o l e d t o l i q u i d h e l i u m t e m p e r a t u r e t h e i n i t i a l 'warm' ( n o r m a l i z a t i o n ) gamma-ray c o u n t s w e r e t a k e n a f t e r w h i c h t h e s u p e r c o n d u c t i n g d e m a g n e t i z a t i o n s o l e n o i d s w e r e c h a r g e d t o m a g n e t i z e t h e s a l t p i l l s . T h e l i q u i d h e l i u m i n 49 the IK bath was pumped to cool the bath. During t h i s period thermal contact was maintained between the s a l t p i l l s and the IK bath by exchange gas at a pressure of 20 mtorr. When the temperature of the s a l t p i l l assembly and IK bath had f a l l e n to approximately 1.IK the s a l t p i l l assembly was thermally i s o l a t e d by pumping away the exchange gas. The superconducting solenoids were demagnetized over about 5 minutes thereby cooling the s a l t p i l l assembly. The MnCl2"4H20 specimen cooled to very low temperatures (<150 mK) within one hour. The nuclear o r i e n t a t i o n experiments were performed by applying magnetic f i e l d s to the specimen while accumulating data from the gamma-ray detectors. A f t e r these experiments were completed the specimen and s a l t p i l l assembly were warmed to the temperature of the IK bath and addi t i o n a l warm counts were taken. 2.6 The Experiments In t h i s section a b r i e f d e s c r i p t i o n of the various runs w i l l be given. The detector and c r y s t a l o r i e n t a t i o n s are given i n table 2.1. The c r y s t a l parameters are given i n table 2.2. 50 TABLE 2.1 D e t e c t o r O r i e n t a t i o n s The d e t e c t o r and a p p l i e d f i e l d s are. g i v e n i n t h e c r y s t a l ( a * , b, c) c o o r d i n a t e s y s t e m u s i n g p o l a r c o o r d i n a t e s i n d e g r e e s Run D e t e c t o r D e t e c t o r P o s i t i o n A p p l i e d F i e l d number 6 <j> 8 cj> ' A 1 8 5 - 20 5 0 2 1 0 7 99 3 1 7 9 76 4 96 1 9 5 B 1 89 1 7 0 0 0 2 87 - 48 3 1 7 9 95 4 90 14 C 1 90 1 2 5 0 0 2 90 - 17 3 1 8 0 0 D 1 1 6 0 ° 0 0 0 2 1 9 0 ° 0 The a c c u r a c y o f a l l a n g l e s q u o t e d i s t y p i c a l l y ± 3 ° . 51 TABLE 2.2 C r y s t a l P a r a m e t e r s Run A a n d B C D d i m e n s i o n s (cm) 1.5x.89x.34 1.6 l o n g . 1.21x.70x.30 ( a l o n g a * , b , c a x e s ) a c t i v i t y ( y C i ) 24 ± 1 15 ± 1 24 ± 1 mass (gm) 0.39. ± .005 0.3 + 0.1 0.39 ± .005 c o n t a c t a r e a (cm 2) 1.7 ± 0.1 3.0 ± .5 1.1 ± 0.1 52 2.6i The Nuclear Orientation Experiments Runs A and B were performed on the same specimen of MnCl2*4H.20. However, the magnetic f i e l d o r i e n t a t i o n and the detector positions r e l a t i v e to the c r y s t a l axes were d i f f e r e n t i n each run. For run A the c r y s t a l was aligned so that the magnetic f i e l d was 5 degrees from the c-axis towards the c*-axis; for run B the c r y s t a l was aligned with the c-axis along the applied f i e l d . Comparison of the r e s u l t s of these runs for applied f i e l d s i n the spin-f l o p t r a n s i t i o n region would give information on the dependence of the t r a n s i t i o n to small changes i n magnetic f i e l d alignment. The t y p i c a l v a r i a t i o n of the normalized i n t e n s i t i e s for applied f i e l d s i n the spin-flop t r a n s i t i o n region i s shown i n figu r e 2.2 for run B. Data from two detectors i s shown. One detector was aligned along the c r y s t a l c-axis, the other was i n the a*-b plane. The f i e l d was f i r s t swept up, then down through the t r a n s i t i o n . I t may be noted that the f i n a l anisotropies at the end of the sweep down are greater i n magnitude than the i n i t i a l values at the beginning of the sweep up. This e f f e c t r e s u l t s from the cooling of the specimen during the period of measurement (about two hours). Run C was performed with the c r y s t a l c-axis aligned 1.2 W 1.1 1.0 0.9-0.8 J T y p i c a l e r r o r 7. 00 7. 25 7. 50 A p p l i e d F i e l d (kG) F i g u r e 2.2 F i e l d Sweep t h r o u g h t h e S p i n - F l o p T r a n s i t i o n , s m o o t h c u r v e s drawn' t h r o u g h t h e d a t a p o i n t s , a x i a l a n d e q u a t o r i a l d e t e c t o r s r e s p e c t i v e l y . The l i n e s shown a r e W , W a r e d a t a f o r a x ^-q un + f i e l d s w e e p i n g up. 0 f i e l d s w e e p i n g down. 54 along the a p p l i e d f i e l d . Three d e t e c t o r s were used i n the run. A d i f f e r e n t , s o l e n o i d was used f o r the a p p l i e d f i e l d i n t h i s run. I t s f i e l d was " c a l i b r a t e d by comparing the s p i n - f l o p t r a n s i t i o n w i t h the other runs. Run D was performed s p e c i f i c a l l y t o determine the d i r e c t i o n of the s u b l a t t i c e m a g n e t i z a t i o n i n the a-c pla n e w i t h no a p p l i e d f i e l d : Only two d e t e c t o r s were needed; one was a t 9 = +10 and the oth e r a t 6 '= -10 i n the a-c p l a n e . Because an attempt was a l s o t o be made t o 54 observe NMR-ON of the Mn n u c l e i no c r y s t a l h o l d e r was used i n t h i s run, the c r y s t a l b e i n g f i x e d d i r e c t l y t o the copper c o l d f i n g e r . Consequently, the c o o l i n g of t h i s specimen was not so e f f i c i e n t . In these n u c l e a r o r i e n t a t i o n runs about 350 gamma ray s p e c t r a were taken and about 10 magnetic f i e l d sweeps through the s p i n f l o p t r a n s i t i o n were made. The time r e q u i r e d t o perform a s i n g l e low temperature run was about 36 hours. 2 . 6 i i The N u c l e a r Magnetic Resonance Experiment The r a d i o - f r e q u e n c y ( r . f . ) f i e l d was a p p l i e d t o the c r y s t a l by a s i n g l e t u r n c o i l . The c o i l was a t t a c h e d o u t s i d e the -IK j a c k e t which was f i t t e d w i t h a g l a s s t a i l . The r . f . f i e l d s t r e n g t h a t the c r y s t a l was r a t h e r inhomogeneous v a r y i n g from 10 to 30 mG. The c o i l was o r i e n t e d t o d i r e c t 55 the f i e l d along the a * - a x i s . The r . f . f i e l d was frequency modulated a t 1 MHz bandwidth w i t h a 10 Hz modulation frequency. The frequency of the r . f . f i e l d was swept over a range encompassing the 54 expected resonance frequency of the Mn n u c l e a r s p i n s i n MnCl2"4H20 which had been determined from a measurement of the h y p e r f i n e i n t e r a c t i o n (Miedema e t a l . , 1965) t o be 542 ± 10 MHz. The f r e q u e n c i e s from 530 MHz t o 550 MHz were swept a t about 0.3 MHz/min. A broader range of 510 t o 53 0 MHz and 550 t o 57 0 MHz was swept at 1 MHz/min. Two gamma-ray d e t e c t o r s were used. The 6 0 temperature of the c r y s t a l and the Co-Fe p l a t e on the 54 c o l d f i n g e r were deduced from the a n i s o t r o p i e s of the Mn and ^ C o gamma^rays. A G e ( L i ) d e t e c t o r p o s i t i o n e d i n the e q u a t o r i a l p l a ne along the a * - a x i s was employed f o r these measurements. In order to d e t e c t the n u c l e a r magnetic resonance, a l a r g e d e t e c t o r e f f i c i e n c y was d e s i r e d . To t h i s end a Nal d e t e c t o r w i t h a l a r g e c r y s t a l of s i z e 5 i n c h e s by 4 i n c h e s was used to monitor the i n t e n s i t y of 54 the Mn r a d i a t i o n . I t was mounted a x i a l l y along the c r y s t a l c - a x i s . The s i z e of the e q u i l i b r i u m a n i s o t r o p y observed 3 on the Nal d e t e c t o r was 7%. The count r a t e was 2 x 10 counts per second g i v i n g an accuracy of ±0.2% i n a 100 second i n t e r v a l . The count r a t e was monitored w i t h a r a t e -meter and w i t h a m u l t i c h a n n e l a n a l y s e r i n the m u l t i s c a l i n g 56 mode. Sur p r i s i n g l y , no s i g n i f i c a n t v a r i a t i o n s from equilibrium were observed i n t h i s experiment. 2 . 6 i i i S u s c e p t i b i l i t y Measurements The spin-flop t r a n s i t i o n i n MnC^MH^ was also determined from the f i e l d dependence of the p a r a l l e l s u s c e p t i b i l i t y . A mutual inductance technique was used f o r these measurements. One half of the secondary pick-up c o i l was wound on the specimen with i t s axis p a r a l l e l to the c-axis of the c r y s t a l . The c o i l was mounted on the heat sink p a r a l l e l to the applied f i e l d . ' The second half of the secondary was an i d e n t i c a l c o i l wound i n the opposite sense and mounted on the other side of the heat sink. Each c o i l had 500 turns of number 40 AWS copper wire and dimensions of 1.0 cm x 0.94 cm x 0.3 cm. The c r y s t a l was inserted i n i t s c o i l with Apiezon N grease and both c o i l s were glued to the heat sink with GE 7031 varnish. The primary c o i l was wound on a nylon t i p at the bottom end of the heat s h i e l d . The sig n a l frequency used was 42 KHz. The secondary voltage was monitored by a l o c k - i n amplifier with a 10 Mfi input impedance preamplifier. The s u s c e p t i b i l i t y was measured for applied f i e l d s from 0 to 10 KGauss. 57 The s u s c e p t i b i l i t y of t h e specimen was determined as a f u n c t i o n of temperature. For temperatures i n the r e g i o n a c c e s s i b l e by pumping l i q u i d helium the temperature was g i v e n by the helium vapour p r e s s u r e . For temperatures of about 1 /3 K an incomplete demagnetization was performed. For an a d i a b a t i c d e m a g n e t i z a t i o n the f i n a l and i n i t i a l temperature T, T^ r e s p e c t i v e l y and the f i n a l and i n i t i a l f i e l d s H, are r e l a t e d by . sr- = t r - f o r H>>100 G i i H i T h e r e f o r e , the f i n a l e q u i l i b r i u m temperature i s simply dependent on the f i n a l f i e l d and the i n i t i a l c o n d i t i o n s . In the a c t u a l experiment the i n i t i a l temperature and f i e l d were 1 . 0 9 K and 3 2 . 3 KG whereas the f i n a l f i e l d was 8 . 8 0 KG r e s u l t i n g i n a f i n a l temperature of 0 . 3 0 K. The specimen f e l t some ' s t r a y ' f i e l d from the dem a g n e t i z a t i o n s o l e n o i d . The s i z e of t h i s s t r a y f i e l d was determined by the change i n the s p i n - f l o p t r a n s i t i o n p o i n t i n f i e l d s a p p l i e d i n o p p o s i t e d i r e c t i o n s . I t s magnitude was 0 . 5 0 6 KG. Very low temperatures (^100 mK) were a c h i e v e d by complete d e m a g n e t i z a t i o n . In t h i s case the temperature was measured 54 by the n u c l e a r o r i e n t a t i o n of the Mn i n the specimen. 58 2.7 The I n t e r n a l F i e l d C a l c u l a t i o n The i n t e r n a l f i e l d i n an e l l i p s o i d magnetized along one of i t s axes i s g i v e n by H. = H , . , - DM , i n t a p p l i e d where D ' i s the demagnetizing f a c t o r which i s dependent on the specimen geometry, and M i s the m a g n e t i z a t i o n . DM i s c a l l e d the demagnetization f i e l d . Chikazumi (1964) g i v e s an e q u a t i o n f o r the demagnetization f a c t o r of an e l l i p s o i d having dimensions a, b, c along i t s axes w i t h a>b>>c and w i t h the f i e l d a p p l i e d along t h e " c - a x i s . D = (4TT) TTC j 1-1 a-b _[_3_ a-b I 2 ] 4a [ 4 a 110 a I j where D i s i n cgs u n i t s . In g e n e r a l , D i s a t e n s o r . In order to o b t a i n an e s t i m a t e of the average i n t e r n a l f i e l d , the c r y s t a l shape can be approximated to an e l l i p s o i d and the above e q u a t i o n used t o determine D. For the s u s c e p t i b i l i t y specimen (run D) the r e s u l t a n t v a l u e of the estimated d e m a g n e t i z a t i o n f a c t o r i s D = 2.3 dgs u n i t s . A c r y s t a l which i s not e l l i p s o i d a l w i l l have a s p a c i a l v a r i a t i o n of demagnetizing f i e l d s r e s u l t i n g i n an i n c r e a s e d width of the s p i n - f l o p t r a n s i t i o n . In o r d e r t o determine the i n t e r n a l f i e l d i t i s 5 9 necessary to know .the m a g n e t i z a t i o n M. T h i s can be determined by i n t e g r a t i n g the s u s c e p t i b i l i t y . However, the s u s c e p t i b i l i t y measured i n the experiments was i n a r b i t r a r y u n i t s so t h a t i n t e g r a t i o n g i v e s M(H) = k'o H*(H) d H ( 2 ' 7 - 1 } where k i s a n o r m a l i z a t i o n c o n s t a n t which must be determined. The n u c l e a r o r i e n t a t i o n measurement of the s p i n d i r e c t i o n i n the s p i n - f l o p phase can a l s o be used t o determine the m a g n e t i z a t i o n : M ( H ) = 2 M o c o s e ( H ) where M Q i s the s u b l a t t i c e m a g n e t i z a t i o n M Q = h gyB SN , . and 6 i s the angle between the s p i n d i r e c t i o n s and the f i e l d d i r e c t i o n . The m a g n e t i z a t i o n was determined from the -1 -3 r e s u l t s of s e c t i o n 3.2 to be 96 ± 6 erg G cm f o r ^ a p p l i e d = 7 • 7 K O e « T h i s i s i n good agreement w i t h the . - 1 - 3 v a l u e of 100 ± 2 erg G cm deduced from the r e s u l t s o f Rives and B e n e d i c t (1975). The n o r m a l i z a t i o n k i n . e q u a t i o n 2.7.1 can t h e r e f o r e be determined and the m a g n e t i z a t i o n f o r any f i e l d can be c a l c u l a t e d from the i n t e g r a l of the s u s c e p t i b i l i t y curve. 60 CHAPTER I I I Results The nuclear o r i e n t a t i o n experiments have been analysed to determine the magnetic structure of the c r y s t a l . Models have been used which assume that the l a t t i c e of magnetic ion spins can be divided into a few sub l a t t i c e s i n each of which the i o n i c spins are oriented i n the same d i r e c t i o n . The o r i e n t a t i o n of a s u b l a t t i c e spin r e l a t i v e to the orthogonal c r y s t a l l a t t i c e axes a*, b, c i s defined by the polar coordinates 8 , <j> where the polar angle 8 i s the angle between the spins and the c-axis, and the angle <j> i s the angle between the proje c t i o n of the spins on the a*-b plane and the a*-axis. The spin configuration i s then s p e c i f i e d by the polar coordinates 8, <J> for a l l the s u b l a t t i c e s . This s p e c i f i c a t i o n of the spin configuration i s used throughout t h i s chapter. 3.1 Spin Configurations i n the An t i f erromagnetic State (H<H-_n) The nuclear o r i e n t a t i o n experiments i n which the applied f i e l d was less than the spin f l o p t r a n s i t i o n f i e l d have been analysed using a model for the spins, appropriate to T=0 i n which there are two equal a n t i p a r a l l e l s u b l a t t i c e s . 61 In t h i s case the t o t a l spin configuration i s given by only two angles 6 = 6! = 6 2 - 1 8 0 ° <t> = 4> i = 4> 2 In runs A and B, i n which four detectors were used, 9, cj> and the temperature T were determined by solving (using computer analysis) the simultaneous equations for the anisotropy at each detector. For run C, i n which only two detectors were used only the one angle 9 and the temperature were determined. The r e s u l t s of the analysis of data taken with no applied f i e l d on the specimen are given in table 3 . 1 . The quoted error includes the s t a t i s t i c a l error i n the ra d i a t i o n i n t e n s i t y and the error i n the alignment of the c r y s t a l and detectors. The average value for the spin d i r e c t i o n i s 6 = 1 1 . 5 ± 3 . 5 deg.; <j> = 1 0 . 5 + 5 . 0 deg. These r e s u l t s agree quite well with measurements of Altman et a l . ( 1 9 7 5 ) who obtained the more accurate r e s u l t of 6 = 6 . 9 ± 2.0 deg.; <j> = 2 ± .2 deg. by three dimensional neutron s c a t t e r i n g . The combined r e s u l t gives 9 = 7 . 6 ± 1.6 deg.; cf> = 3.2 ± 1 . 9 deg. TABLE 3.1 Spin Configurations Without Applied F i e l d Run 0 i A 9.0 ± 6.4 7.5 + 7.2 B 14.5 ± 6.4 18.0 ± 7.2 C 12.3 ± 8.0 -30 ± 20 D 11.015.0 0.03 Averages 9 =11.5° ± 3.5° <{> = 10. 5 ±5.0° run D measured proj e c t i o n of spin i n a-c plane 63 The spin configurations for applied f i e l d s less than the spin f l o p t r a n s i t i o n f i e l d • (H^.^) and i n the a-c plane were also determined using the same model. The r e s u l t s are given i n table 3.2. In run A the f i e l d d i r e c t i o n was between the c and c*-axis; i n run B the f i e l d was along the c-axis. For both runs the analysis shows that within experimental error the'spin configuration remained unchanged from the zero field configuration for f i e l d s up to 97% of the spin f l o p t r a n s i t i o n . In molecular f i e l d theory the, spin configuration does not change for f i e l d s l e s s than the spin-flop f i e l d i f the applied f i e l d i s along the easy axis. For a f i e l d i n c l i n e d at an angle of 7° to the easy axis the theory predicts a maximum change AO i n the polar angles of the s u b l a t t i c e s of l e s s than 3°. This value of AG was l e s s than the uncertainties in the observed polar angles so that no changes i n the spin configuration were expected. The spin configurations for applied f i e l d s i n the region of the spin f l o p t r a n s i t i o n f i e l d w i l l be discussed i n section 3.4. 3.2 The Spin-Flop State The nuclear o r i e n t a t i o n anisotropics for applied f i e l d s greater than the t r a n s i t i o n f i e l d were f i t t e d to a model i n which two equal s u b l a t t i c e s were again assumed. If the f i e l d H i s applied along the anisotropy axis both the 64 TABLE 3.2 F i e l d D e p e n d e n c e o f t h e S p i n C o n f i g u r a t i o n i n t h e A n t i f e r r o m a g n e t i c P h a s e H a p p l i e d 6 * R u n 3.19 3 . 0 ± 6 . 5 20 ± 40 A 10.0 ± 7.0 2 2 + 1 5 B 6.061 11.0 ± 6.5 12 ± 10 A 11.0 ± 7.0 19 ± 15 B 6.79 16 ± 10 -20 + 22 C 7. 018 5.0 ± 5.0 0 + 25 A 65 mo l e c u l a r f i e l d and the spin-wave models g i v e a s p i n c o n f i g u r a t i o n i n which the angles between H and the s u b l a t t i c e d i r e c t i o n s are equal and the p r o j e c t i o n s of the s u b l a t t i c e d i r e c t i o n s i n the plane p e r p e n d i c u l a r t o H are a n t i p a r a l l e l . Even i f the a p p l i e d f i e l d makes a s m a l l angle w i t h r e s p e c t to the a n i s o t r o p y a x i s t h i s d e s c r i p t i o n of the s p i n c o n f i g u r a t i o n i s a good approximation to the m o l e c u l a r f i e l d r e s u l t because of the s m a l l i n f l u e n c e of the c r y s t a l l i n e a n i s o t r o p y terms i n t h i s r e g i o n . . The gamma r a y a n i s o t r o p y d a t a has been a n a l y s e d u s i n g t h i s model of the s p i n c o n f i g u r a t i o n t o o b t a i n the p o l a r angle 8, and the angle cj> of the s p i n p r o j e c t i o n s i n the a*-b plane w i t h r e s p e c t t o the a * - a x i s , as w e l l as the temperature of the s p i n system. The r e s u l t a n t s p i n c o n f i g u r a t i o n s a re g i v e n i n t a b l e 3.3. The angle <j> which i s not expected t o depend on the a p p l i e d f i e l d i s § = 94 + 4 deg., i n d i c a t i n g t h a t the s p i n s ' f l o p ' i n t o a plane v e r y c l o s e t o the b-c pl a n e . These e x p e r i m e n t a l r e s u l t s w i l l be d i s c u s s e d i n r e l a t i o n t o the t h e o r e t i c a l models i n s e c t i o n 3.3. 3.3 • The M o l e c u l a r F i e l d V a l u e s f o r MnCl 2*4H 20 The m o l e c u l a r f i e l d s can be determined from the t r a n s i t i o n p o i n t s . R i v es and B e n e d i c t (1975) used t h e i r measured v a l u e s of the paramagnetic t r a n s i t i o n f i e l d s H^„, TABLE 3 . 3 Spin-Flop State Configuration H(KG) 6(deg) ' . *(deg) 7.656 70.0 ± 1.2 9 4 . 0 ± 3.5 10.53 61.0 ± 1.2 95.0 ± 3.5 12.44 54.0 ± 1.6 96.0 ± 3.5 14 . 36 46 . 0. ± 1 . 2 95. 0 1 3 . 5 7.656 70.0 ± 1 . 1 93.8 ± 3.5 10.53 60.0 ± 1.9 93.8 ± 3.5 13.80 49.0 ± 3.8 103.8 ± 3.5 7.89 72 ± 6 89 ± 5 10. 08 62 + 3 81 ± 5 67 H , Fl to obtain the values P Y P z H E = 10.375 ± 0.03 KOe H K l y = 2 , 2 0 1 ° - 0 5 K O e  H K l x = 3 ' 8 0 ± ° * 0 5 K O e These molecular f i e l d s can be used to determine the thermo-dynamic spin f l o p t r a n s i t i o n f i e l d H h from the molecular f i e l d theory equation (section 1.2B): 2 _ 2 th E Klx Kly giving H^-6.410.1 K O e - This value of H ^ i s i n poor agreement with the observed value for the spin f l o p t r a n s i t i o n f i e l d of 7.0 KOe. Our nuclear o r i e n t a t i o n r e s u l t s for the f i e l d dependence of cose i n the spin f l o p state and the spin f l o p t r a n s i t i o n f i e l d can be used to determine H_ and PL., provided ti Kl second order anisotropy and anisotropic exchange are both n e g l i g i b l e . Using the r e l a t i o n s (section 1.2B) 2 2 H t h " 2 H E H K 1 " HK1 and cos8 = H/(2H_ - KL,.. ) fti K l the r e s u l t s are H„ =11 .88 ± 0 . 4 2 KOe E H R 1 = 2.31 ± 0.04 KOe 68 These values are i n poor agreement with those deduced from "the observed paramagnetic t r a n s i t i o n f i e l d s . For example, the calculated value of H i s 21.45 ± 0'.42 KOe whereas the value pz observed by Rives and Benedict i s 18.55 ± 0.05 KOe. Agreement between the t r a n s i t i o n points and the f i e l d dependence of cose i n the spin flop state can.be obtained i n the molecular f i e l d model i f second order aniso-tropy i s included. Anisotropic exchange i s also included i n the analysis. In t h i s case the f i e l d dependence of the spin configuration cos8 . . i s given by (section 1.2B) cos9 = H/(H Q - 2H R 2 cos 26) (3.2.1) where H Q = 2H E + H K E - H R l . The values H. = 21.45 ± 0.42 KOe and H _ = 1.45 ± 0.19 KOe U i\Z were obtained from 3.2.1 by a f i t t i n g procedure which u t i l i z e d both the nuclear o r i e n t a t i o n data and the paramagnetic t r a n s i t i o n point H . The data and the f i t t e d curve are pz shown i n fi g u r e 3.1. The maximum deviation from l i n e a r i t y of cos8 , . i s 5.6 ± 0.8 percent of the value of cos6 T T at H . A deviation ^ H pz from l i n e a r i t y for f i e l d s applied along the easy axis i s also indicated i n the magnetization measurements of Rives and Benedict, The maximum deviation for t h e i r r e s u l t s was 6.4 ± 0.5 percent of the magnetization at H • However, the no n - l i n e a r i t y 7 0 i n the magnetization also includes a contribution from the f i e l d dependence of the sub l a t t i c e magnetization a r i s i n g from spin wave e f f e c t s (section 1.2D). This contribution has been estimated to be only 2.4% (Miedema et a l . , 1965), leaving about 4.4% no n - l i n e a r i t y which i s i n agreement with the nuclear o r i e n t a t i o n r e s u l t . The molecular f i e l d s can be determined from the t r a n s i t i o n f i e l d s given i n section 1.2B: H = 2H_ + H , + HTrT,; H = 2BL, + H , + HT,_ py E Kly KE' px E Klx KE H t h = t H 0 ( H K l y + H K 2 + HKE>^ The following f i e l d s are obtained: = 1 1 . 0 5 ± 0 .21 KOe, Hv„ = 0 .1 ± 0 . 3 KOe, K E HT,-. = 0 .75 ± 0 .22 KOe, HT,, = 2 .35 ± 0 .23 KOe Kly Klx H K 2 = 1.45 ± 0 .19 KOe, as well as H R E + H K l y = 0 . 8 6 ± 0 .20 KOe. These r e s u l t s indicate that the anisotropic exchange f i e l d i s n e g l i g i b l e which i s not sur p r i s i n g since the K E g-factor f o r the 6S state i s i s o t r o p i c (g = 2 . 0 ) ' . The r e s u l t s also suggest that the second order anisotropy f i e l d i s approximately twice the f i r s t order f i e l d . The second order anisotropy can a r i s e from the (nearly regular) octahedral 71 arrangement of the two chlorine ions and four water molecule around the manganese ion. A small a x i a l d i s t o r t i o n of the regular octahedron can contribute an a d d i t i o n a l f i r s t order anisotropy. Using equation 1.4.3 the a x i a l symmetry c o e f f i c i e n t D and the octahedral symmetry c o e f f i c i e n t a can be calculated for t h i s system: D = 0.96 ± 0.15 cm - 1; a = 0.45 ± 0.06 cm"1 These values are rather larger than those estimated from paramagnetic resonance, of Mn + + i n paramagnetic materials. T y p i c a l values given are D ^ +0.2 cm - 1, a ^ 0.0 2 cm"1 Unfortunately, because the anisotropic i n t e r a c t i o n a r i s e s from the c r y s t a l f i e l d through spin- o r b i t coupling only i n fourth or f i f t h order of perturbation theory (for the s t a t e ) , even rough estimates of the expected e f f e c t are d i f f i c u l t to obtain. 3.4 The Spin-Flop T r a n s i t i o n Region For applied f i e l d s i n the s p i n - f l o p t r a n s i t i o n region, the molecular f i e l d model for two s u b l a t t i c e s gives solutions for the spin configurations which allow the 72 polar angles e l f e 2 of each s u b l a t t i c e to assume d i f f e r e n t , independent values while the projections' of the s u b l a t t i c e spins i n the a*-b plane remain a n t i p a r a l l e l . The angles 9 l 7 6 2 and c|> = <j> x = cj> 2 for the spin configuration i n the t r a n s i t i o n region were f i t t e d to the nuclear o r i e n t a t i o n data from runs A and B. The specimen temperatures used for f i t t i n g the anisotropies were obtained from the cooling curve (figure 3.6). In f a c t , the temperatures could have been determined i n the actual f i t t i n g procedure, but using the cooling curve data reduced the number of parameters and thereby increased the accuracy of the r e s u l t . The spin configurations obtained by the analysis are given i n table 3.4 and the polar angles are plotted i n figure 3.2. The r e s u l t s indicate that i n increasing f i e l d the spin configuration goes from an antiferromagnetic phase (9 2 - 9j + 180°) to a s p i n - f l o p phase ( e 2 - -9i) with no detectable intermediate phase. However, i t i s also apparent that within the spin-flop phase between f i e l d s of 7.25 and 7.45 KG the value of cos9 decreases with increasing applied f i e l d contrary to the t h e o r e t i c a l r e s u l t that cos9 i s proportional to H. I t was concluded that the two s u b l a t t i c e model was not appropriate i n the t r a n s i t i o n region. It has been proposed by Keffer (1973) and Chow (1974) that a mechanism exists i n the spin-flop t r a n s i t i o n which gives r i s e i n the c r y s t a l to regions ('domains') of a n t i -ferromagnetic phase and regions of spin-flop phase. Consequently, 73 TABLE 3.4 i n Configuration i n the Tra n s i t i o n Region Using the Two-sublattice Model applied (kG) degrees degrees degrees 7.12 4 -176 90 7.21 53 - 15 105 7.273 -62 44 99 7.43 -78 71 95 7.21 -36 160 96 7.273 -49 35 105 7.34 -61 51 100 7.43 -59 82 , 98 + 1 . 0 -CosQ + 0 . 5 " O 5 o o o 0 . 0 -- 0 . 5 + - 1 . 0 -o 7.0. 7.1 7.2 7.3 7.4 7 . 5 Applied F i e l d (KG) Figure 3.2 Two Sublattice Model in the T r a n s i t i o n Region, x Run A 0 Run B 75 the nuclear o r i e n t a t i o n data was f i t t e d to a model i n which a f r a c t i o n 5 of the specimen i s i n the spin-flop configuration and a f r a c t i o n (1-6) i s i n the antiferromagnetic configuration. I t i s assumed that a n e g l i g i b l e part of the specimen i s i n a t r a n s i t i o n a l phase between these two configurations. The i n t e n s i t y of r a d i a t i o n from the specimen at a detector located i n a d i r e c t i o n D i s given by W(D) = (1- 6)W a f(D) + (5)W s f (D) where W - and W _ are the angular d i s t r i b u t i o n s from the at s i pure antiferromagnetic and spin-flop phases respectively. The data from runs A and B were f i t t e d to the model to obtain the f r a c t i o n 6 of spins i n the spin-flop state. The i n d i v i d u a l spin configurations i n the mixed phase specimen were assumed to be the appropriate spin configurations from just below and just above the t r a n s i t i o n region. The values of 6 which . were obtained from the analysis and t h e i r applied f i e l d s are given i n table 3.5. The random error i n 6 a r i s i n g from uncertainty i n the ansiotropies, the detector positions and the temperatures i s +0.03 for a l l values of 6. The error i n 6 a r i s i n g from the uncertainty i n the spin configuration of the domains (as determined above and.below the t r a n s i t i o n region) i s ±0.03 for a l l 5. Therefore the t o t a l expected uncertainty i n 6 i s ±0.04. The v a r i a t i o n of 6 across .TABLE 3 . 5 T r a n s i t i o n Region Using the Domain Model, i s the Fraction of Spins i n Spin-flop Phase applied (kG) 7 . 1 2 7 . 2 1 7 . 2 7 3 7 . 3 4 7 . 4 3 7 . 5 3 Run A 0 . 0 0 . 4 5 0 . 7 2 5 0 . 9 0 1 . 0 0 . 0 4 ) Run B 0 . 0 3 0 . 2 5 0 . 5 0 0 . 7 7 5 0 . 9 6 0 . 96 77 the t r a n s i t i o n i s p l o t t e d i n f i g u r e 3.3 f o r the two runs. In run B i n which the a p p l i e d f i e l d was i n c l i n e d at 7° to the easy a x i s the t r a n s i t i o n occurs at a s l i g h t l y (0.7%) higher f i e l d than i n run A i n which the i n c l i n a t i o n was only 2°. In f a c t , molecular f i e l d theory p r e d i c t s a s h i f t f o r the two s u b l a t t i c e models i n agreement w i t h the observed v a l u e . Using 80% of the range of 6 as an e m p i r i c a l measure of the width of the t r a n s i t i o n , the observed width i s 0.19 ± 0.02 KOe i n run A and 0.24 ± 0.02 KOe i n run B. The inhomogeneity of the a p p l i e d f i e l d on the specimen was estimated t o be 1% or 0.07 KOe. There i s an a d d i t i o n a l inhomogeneity i n the i n t e r n a l f i e l d a r i s i n g from the inhomogeneity of the demagnetization f i e l d of the specimen. The average demagnetizing f i e l d j u s t above the t r a n s i t i o n r e g i o n was 0.23 KOe. E s t i m a t i n g a 20% v a r i a t i o n i n t h i s f i e l d g i v e s a t o t a l inhomogeneity of the i n t e r n a l f i e l d of only 0.11 KOe which i n d i c a t e s t h a t the observed width does not f u l l y a r i s e from f i e l d inhomogeneity. K e f f e r and Chow d i d not determine the r a t i o of the phase p o p u l a t i o n s i n the t r a n s i t i o n r e g i o n ; however, they d i d determine the s t a b i l i t y boundaries. The lower boundary i n - 2 2 i n c r e a s i n g f i e l d i s (H, ) = H„H, + H,. , and i n decreasing J. Ja A A f i e l d i s H ~ = H_,3/H.. The upper boundary i s H + = 2KLH- + H 2 Z, o 4 Ji A A These r e s u l t s apply t o s i n g l e - i o n a n i s o t r o p y at T=0 w i t h the f i e l d , a p p l i e d along the easy a x i s . When the molecular f i e l d 79 values given i n section 3 . 3 are used i n the above expressions, the s t a b i l i t y boundaries are given by H r / H t h = °-83' v H t h = °- 9°' H + / H t h = i - 1 2 where H ^ i s the thermodynamic t r a n s i t i o n f i e l d . Thus the observed width of the spin-flop t r a n s i t i o n l i e s well within the t h e o r e t i c a l l y allowed boundaries. If the specimen i s i n the t r a n s i t i o n region of f i e l d the magnetization i s given by M = 6M cose r-s sf where 6 ^ i s the polar angle i n the spin-flop phase and Mg i s i s the saturation magnetization. Hence 6 and the magnetization should be proportional. 6 can be determined from the nuclear o r i e n t a t i o n data and M can be determined from the i n t e g r a l of the s u s c e p t i b i l i t y x- In figure 3.4 the i n t e g r a l of the s u s c e p t i b i l i t y and 5 are plotted against the applied f i e l d . The same c r y s t a l alignment was used for the measurements of X and <5. x was determined at temperatures between 0.3 and 0.15 K whereas 6 was determined at temperatures between 0.06 and 0.08 K. The two graphs (figure 3.4) are s i m i l a r although the s u s c e p t i b i l i t y gives a wider t r a n s i t i o n of 0.35 KOe. This increased width of the t r a n s i t i o n i s possibly caused by v a r i a t i o n i n 6 £ i n the t r a n s i t i o n region which was not sf ^ considered i n the nuclear o r i e n t a t i o n a n a l y s i s . 1. Cn O 7.0 Figure 3.4 7.2 7.4 Applied F i e l d (KG) T?6 The Magnetization and 6 i n the T r a n s i t i o n Region. The i n t e g r a l of the s u s c e p t i b i l i t y , M (in a r b i t r a r y u n i t s ) , i s shown by the s o l i d curve. The c i r c l e s give the experimental values of 6 . co o 81 I t i s concluded that whereas the two su b l a t t i c e molecular f i e l d models f a i l to describe adequately the nuclear o r i e n t a t i o n data, the. domain model using mixed phases i n the spin-flop t r a n s i t i o n region does give s a t i s f a c t o r y agreement with experiment. 3.5 The Temperature Dependence of the Spin-Flop T r a n s i t i o n F i e l d The temperature dependence of the spin-flop t r a n s i t i o n at temperatures between 0.1 and IK was also obtained from the a.c. s u s c e p t i b i l i t y measurements. The values obtained from these measurements for the c r i t i c a l f i e l d s at temperatures i n the region of 0.1K, 0.3K and 1.0K are given i n table 3.6 and figure 3.5. I t i s apparent from figu r e 3.5 that the t r a n s i t i o n f i e l d continues to decrease as the temperature i s lowered from 0.3K to 0.1K. The lowest f i e l d for the spin-flop t r a n s i t i o n occurred at the i n t e r n a l f i e l d H., = 6.98 ± 0.03 at tn the temperature T=0.12K. Rives and Benedict (1975) also determined the temperature, dependence of the spi n - f l o p t r a n s i t i o n f i e l d . They f i t t e d t h e i r r e s u l t s to the upper boundary of the a n t i -ferromagnetic phase (H+) using a r e l a t i o n c a l c u l a t e d from spin-wave theory by Feder and Pytte (1968). The f i t t e d temperature dependence of the t r a n s i t i o n f i e l d had a minimum at 0.3K, the lowest temperature they reached i n t h e i r measurements. In 8 2 TABLE 3.6 T e m p e r a t u r e D e p e n d e n c e o f t h e S p i n - F l o p F i e l d T e m p e r a t u r e A p p l i e d F i e l d (K) (kG) 1.115 ± .01 7.630 ± .002 1.09 ± .01 7.587 ± .002 0.9 ± .05 7.579 ± .002 0.8 ± .05 7.551 ± .007 0.30 ± .01 7.30 ± .01 0.15 ± .02 7.252 ±. .002 0.12 ± .01 7.25 ± .01 7 . 6 + I X; 1 I X 1 « 7, 4 L 7. 2f 7 . C J 1 . 1 , , 0.0 0.2 0.4 0.6 0.8 1 Temperature (K) Figure 3.5 Temperature Dependence of the Spin-Flop T r a n s i t i o n F i e l d . 84 f a c t , i n an antiferromagnet the actual t r a n s i t i o n does not occur at H + but at H ^ (the thermodynamic t r a n s i t i o n point at which the free energies of the phases,are equal). I t i s + u n l i k e l y that H and would have the same temperature dependence so that the use of the Feder and Pytte theory by Rives and Benedict to extrapolate t h e i r data was not j u s t i f i e d . We conclude that any minimum i n the temperature dependence of the spin-flop t r a n s i t i o n f i e l d must occur below 0. 15K. 3.6 Cooling of the Specimens Measurement of the gamma-ray anisotropy allows the determination of the temperature of the spin systems (in addition to the spin quantization d i r e c t i o n s ) . The actual parameters determined i n the temperature analysis are the B c o e f f i c i e n t s i n the expression for the anisotropy (equ. 1.1.1). The B c o e f f i c i e n t s are a function of 3 = yjH^ f/IKT. Hence, determination of the B c o e f f i c i e n t s and previous knowledge of 1, y.j. and H ^ y i e l d s the temperature of the system. The value of H ^ used i n the c a l c u l a t i o n was determined by nuclear s p e c i f i c heat measurements. Spin wave e f f e c t s r e s u l t i n a f i e l d dependence of the s u b l a t t i c e magnetization even at low temperatures. This f i e l d dependence i s r e f l e c t e d i n the average hyperfine f i e l d of the s u b l a t t i c e . However, the t o t a l change i n 85 t h e s u b l a t t i c e m a g n e t i z a t i o n has b e e n e s t i m a t e d t o be o n l y 2 . 4 % (Miedema e t a l . , 1 9 6 5 ) . T h e r e was an a d d i t i o n a l c h a n g e i n o f n o t g r e a t e r t h a n 3% a r i s i n g f r o m t h e p r e s e n c e o f t h e a p p l i e d f i e l d . T h e s e v a r i a t i o n s i n were n o t s i g n i f i c a n t i n t h e d e t e r m i n a t i o n o f T and were t h e r e f o r e i g n o r e d i n t h e a n a l y s i s . I n t h e e x p e r i m e n t s t h e s p e c i m e n was c o o l e d by t h e c o p p e r c o l d f i n g e r t h r o u g h A p i e z o n N g r e a s e . The t e m p e r a t u r e o f t h e s p e c i m e n as a f u n c t i o n o f t i m e c a n be u s e d t o d e t e r m i n e t h e t h e r m a l c o n d u c t a n c e Q / A T o f t h e c o n t a c t . The t e m p e r a t u r e d a t a o f r u n s A and B, g i v e n i n t a b l e 3.7, has b e e n f i t t e d t o a r e l a t i o n o f t h e f o r m ( s e c t i o n 1.5) Q = k A ( T n - T Q n ) (3.6.1) where Q i s t h e h e a t f l u x , T q i s t h e u l t i m a t e t e m p e r a t u r e , A i s t h e c o n t a c t s u r f a c e a r e a and k i s a t h e r m a l c o n d u c t i v i t y c o n s t a n t f o r t h e c o n t a c t . A s s u m i n g n=4, t h e b e s t f i t o f t h e d a t a g i v e s t h e t h e r m a l c o n d u c t i v i t y c o n s t a n t : k = (8.2 ± 1.9) x 10 3 e r g K " 4 s e c - 1 c m ~ 2 , and T = (47 ± 5) mK. o. The d a t a f i t t e d c u r v e a r e shown i n f i g u r e 3.6. The d a t a was a l s o f i t t e d - t o t h e e q u a t i o n w i t h n=3. I n t h i s c a s e t h e c o n d u c t i v i t y c o n s t a n t i s k = (6.3 ± 1.9) x 10 2 e r g K - 3 s e c _ 1 c m _ 2 . 86 TABLE 3.7 Specimen Cooling-Run A end of demagnetization: to = 1600 hours T (mK) time hours/min ± 1 min T (mK) time hours/; ± 1 m 150 ± 30 1602 51 2218 130 1610 53 2300 123 1614 53 2343 112 ± 20 1617 49 2422 145 ± 30 1621 51 ± 1.5 2551 102 1626 52- 2634 102 ± 15 16 34 47% 2714 73 ± 6 1734 48 2753 65 ± 5 1813 47 2833 . 59 1921 50 ± 1 . 5 2913 58% + 3 1942 50 2954 53 2038 48% 3 035 • 50 2140 47 3123 cont 1d. 87 TABLE 3.7 (continued) Run B end of demagnetization: t = 16 00 hours T (mK) time hours/min T (mK) time hours/min 115 ± 20 113 137 100 ± 15 100 110 86 80 75 ± 4 67 1608 1612 1616 1623 16 27 1631 1636 1641 1745 1824 62 63% 58 51 57 54% 53 49 + 2 52% 1908 1949 2119 2207 2247 2342 2420 2500 2541 0 .5 1.0 2 .0 Time (hours) Figure 3.6 Cooling of the Specimen. x Run A 0 Run B 5.0 10 20 03 CO 89 with T = (46 ± 5) mK. o This f i t t e d curve i s also shown i n figure 3.6. Both curves f i t the data reasonably wel l . According to the theory of L i t t l e (1959), the boundary resistance depends on the acoustic mismatch between the materials and i s a function of t h e i r d e n s i t i e s and acoustic v e l o c i t i e s . I t has been shown (Peterson et a l . , 1973; A l l e n , 1974) that, i n addition to the sca t t e r i n g of phonons at the surface considered by L i t t l e , the scattering of phonons within the material should be considered i n the analysis of the boundary resistance. Peterson et a l . obtain good agreement with theory f o r a number of d i f f e r e n t contacts". Metcalfe (1971) has observed e f f e c t s i n the thermal conductivity of MnC^^H^O which he associated with magnon-phonon scattering and which could be s i g n i f i c a n t i n the boundary resistance. Unfortunately, i t i s not possible with the present t h e o r e t i c a l knowledge and the l i m i t e d experimental data to determine q u a n t i t a t i v e l y to e f f e c t s of the various processes. Furthermore, i n any i n d i v i d u a l measurement the contact resistance may be strongly effected by factors such as imperfect surface contact, imperfections i n the surface layers at the in t e r f a c e , and surface s t r a i n s due to d i f f e r e n t i a l contraction, a l l of which tend to increase the contact resistance. It i s hot s u r p r i s i n g , therefore, that the conductivity constant k measured i n the present case for 90 M n C ^ ^ B ^ O i s an o r d e r ' of m a g n i t u d e d i f f e r e n t t h a n t h a t . o b s e r v e d f o r a c o n t a c t of chrome p o t a s s i u m a l u m t o c o p p e r w i t h A p i e z o n N g r e a s e . The s p i n - l a t t i c e r e l a x a t i o n t i m e T^ o f t h e manganese n u c l e a r s p i n s y s t e m h a s so f a r b e e n i g n o r e d i n t h e a n a l y s i s . I n e a r l i e r w o r k o n •MnC^MH^O a t l o w t e m p e r a t u r e s ( D a n i e l s e t a l . , 1961 a n d Miedema e t a l . , 1965). t h e s u g g e s t i o n was made, t h a t t h e c o o l i n g r a t e was l i m i t e d by t h e s p i n - l a t t i c e r e l a x a t i o n . U s i n g t h e r e l a t i o n 3 . 6 . 1 f o r n = 4 w i t h t h e f i t t e d v a l u e s f o r k a n d T , a n d a s s u m i n g a t e m p e r a t u r e d e p e n d e n t t i m e c o n s t a n t T^ (T) d e f i n e d b y , dT _ T - T 0 d t T 1 ( T ) 3 g i v e s r e l a x a t i o n t i m e s T^ = 3 x 10 s a t T = 100 mK a n d 3 T^ = 40 x 10 s a t T = 50 mK. T h e s e v a l u e s f o r T^ r e p r e s e n t t h e u p p e r l i m i t o f t h e n u c l e a r s p i n - l a t t i c e r e l a x a t i o n , a s s u m i n g t h i s c o n t r i b u t i o n d o m i n a t e s t h e t h e r m a l r e s i s t a n c e . P e r h a p s i n M n C ^ M E ^ O w h e r e t h e d o m i n a n t c o n t r i b u t i o n t o t h e h e a t c a p a c i t y i s t h a t o f t h e n u c l e a r s p i n s y s t e m , t h e b o u n d a r y r e s i s t a n c e s h o u l d i d e a l l y be t r e a t e d i n t e r m s o f t h e c o u p l e d s y s t e m s o f l a t t i c e , e l e c t r o n i c a n d n u c l e a r s p i n s . 91 CHAPTER IV 10 3 Nuclear Orientation of Ru 4.1 Introduction Anisotropic radiations from oriented nuclei can be used to determine information about the nuclear system being observed. As discussed i n section 1.1, Blin-Stoyle and Grace (1957) show that the gamma-ray i n t e n s i t y , observed at angle 8 to the axis of quantization, i s given by W(6) = ' £ B U7F_P„(cos6.) (4.1) K even K K K K The c o e f f i c i e n t s B , which are discussed i n d e t a i l i n section 1.1, describe the nuclear o r i e n t a t i o n of the i n i t i a l system (preceding any nuclear decay). They are functions only of the i n i t i a l nuclear spin 1^ and, i n the case the hyperfine i n t e r a c t i o n can be described by a hyperfine f i e l d H^f, 6 = uH^/IKT where p i s the nuclear magnetic moment and T i s the temperature of the system. The c o e f f i c i e n t , F.,, depends on the angular momentum properties of the observed t r a n s i t i o n . If a state of angular momentum 1-^  decays to a state of angular momentum 1^ by emission of angular momentum L the c o e f f i c i e n t F R i s 92 given by F (LLI 0I,) f o r w h i c h t h e expression i s F K ( L L ' I 2 I 1 ) = (-1) Z 1 [ (2L+1) (2L'+1) (2I 1+1)] 2 -• C(LL'K; 1-1) W(I 1I 1LL' ; ¥.1^) (4.2) where W.is a Racah c o e f f i c i e n t . • If the observed gamma t r a n s i t i o n has mixed m u l t i p o l a r i t y then the c o e f f i c i e n t F^ i s given by F R = [ F K ( L L I 2 I 1 ) + 6 2 F K ( L , L ' I 2 I 1 ) + 2 F ( L L ' I 2 I 1 ) ] / [ 1 + 6 2 ] ( 4 . 3 ) for a 2 L pole t r a n s i t i o n i n which i s admixed an amplitude 6 of 2 L + ^ pole r a d i a t i o n . C l e a r l y t h i s expression takes account of the coherent r e l a t i o n s h i p between the admixed components. The c o e f f i c i e n t U i s a function of the angular momenta of a l l t r a n s i t i o n s preceding that observed. For the unobserved t r a n s i t i o n the Uv i s given by J- I r r I i ~ L ' U R = [ ( 2 I Q+1) ( 2 I J + 1)] 2 ( - l ) U X • W ( I 0 I . 0 I 1 I 1 ; KL') When unobserved t r a n s i t i o n s with mixed angular momenta L', and L" precede that which i s being studied, account 93 m u s t a l s o be t a k e n o f a n y a d m i x t u r e o f t h e modes o f d e c a y . I n t h i s c a s e , t h e c r o s s t e r m s due t o i n t e r f e r e n c e o f t h e a d m i x e d c o m p o n e n t s v a n i s h a n d one h a s o n l y t o r e p l a c e U~K ( L - ) by [ U R ( L ' ) + S 2 U K ( L " ) ] / ( l + 6 2 ) .• I f t h e r e i s a s e r i e s o f u n o b s e r v e d p r e c e d i n g t r a n s i t i o n s t h e v a l u e o f U v i s s i m p l y t h e p r o d u c t o f t h e UT.'s f o r e a c h K K t r a n s i t i o n . T h a t t h e p r e c e d i n g t r a n s i t i o n s i n v o l v e b e t a -d e c a y o r gamma-decay i s o f no s i g n i f i c a n c e i n t h e d e t e r m i n a t i o n o f t h e e x p r e s s i o n s . The number o f t e r m s i n e q u a t i o n (4.1) i s l i m i t e d by K < 2 I Q , 2 1 ^ 2L I t h a s b e e n a s s u m e d i n t h e a b o v e d i s c u s s i o n ' t h a t t h e f i n a l s t a t e o f t h e s y s t e m i s a f u n c t i o n o n l y o f t h e i n i t i a l n u c l e a r o r i e n t a t i o n a n d t h e s u c c e e d i n g r a d i a t i v e d e c a y s . I f i n t e r m e d i a t e s t a t e r e o r i e n t a t i o n o c c u r s t h e n e x p r e s s i o n ( 4 . 1 ) m u s t be r e p l a c e d b y -'w(e) = ILJ Q , . B u ^ p j c o s e ) ( 4 . 4 ) K e v e n K K.K K K w h e r e Q d e s c r i b e s t h e a t t e n t i o n o r e n h a n c e m e n t o f t h e is. a n i s o t r o p y due t o t h e i n t e r m e d i a t e s t a t e r e o r i e n t a t i o n . H o w e v e r , no s i g n i f i c a n t r e o r i e n t a t i o n w i l l o c c u r i f t h e 94 l i f e t i m e o f t h e i n t e r m e d i a t e s t a t e i s s m a l l c o m p a r e d t o t h e L a r m o r p r e c e s s i o n t i m e o f t h a t s t a t e . 103 4.2 The N u c l e a r O r i e n t a t i o n P a r a m e t e r s o f Ru i n I r o n Shown i n f i g u r e 4.1 i s p a r t o f t h e d e c a y scheme 103 o f Ru b a s e d o n t h e a n a l y s i s o f A v i g n o n e a n d F r a y ( 1 9 6 7 ) , R a e s i d e e t a l . (1969) a n d P e t t e r s s o n e t a l . ( 1 9 7 0 ) . The gamma-ray s p e c t r u m i s d o m i n a t e d by two p e a k s : a s t r o n g p e a k a t 497 KeV and a w e a k e r p e a k a t 610 KeV o f o n l y 6.2% t h e i n t e n s i t y o f t h e 497 KeV p e a k . F o r t h e 497 KeV t r a n s i t i o n t h e Fermi/Gamow - T e l l e r m i x i n g r a t i o o f t h e p r e c e d i n g b e t a d e c a y h a s n o t b e e n d e t e r m i n e d . F o r t h e s u c c e e d i n g gamma r a d i a t i o n , t h e E2/M1 m i x i n g r a t i o h a s b e e n d e t e r m i n e d ( P e t t e r s s o n e t a l . , (1970) o n l y t o t h e e x t e n t t h a t <5 2(E2/M1) i s b e t w e e n .01 a n d .17. F o r t h e 610 KeV e m i s s i o n t h e p r e c e d i n g b e t a t r a n s i t i o n i s p u r e Gamow - T e l l e r . T h e gamma-ray m u l t i p o l a r i t y c a n be M l o r E 2 , b u t t h e m i x i n g r a t i o i s n o t known. The n u c l e a r o r i e n t a t i o n c o e f f i c i e n t s F T. and U v K IN. f o r t h e p o s s i b l e d e c a y modes a r e g i v e n i n t a b l e 4.1. I t s h o u l d be n o t e d i n p a r t i c u l a r t h a t t h e c r o s s t e r m g i v i n g t h e m i x i n g o f M l and E2 f o r t h e 497 KeV t r a n s i t i o n i s 95 39.6 days 650 KeV 537 KeV M 1 + E 2 497 KeV 40 KeV Figure 4.1 The Decay Scheme of 1 0 3 R u . TABLE .4 . 1 U R and F R C o e f f i c i e n t s for Ru -Transition 4 97 KeV 610 KeV 4 97 KeV 610 KeV Beta Decay (LT ) K Fermi U, 2 1. 00 U4 1.00 Gamow-Tel'ler 0.657 0. 875 -0.143 0.580 Gamma Decay (F^) M l F2 0.134 •0.436 F4 F2 0 . 325 0 . 249 4 .118 -.478 M x / E 2 694 .37 8 97 large giving a very s e n s i t i v e measure of t h i s mixing. Using the Mossbauer technique, Kistner (1969) has determined the hyperfine f i e l d of ruthenium i n ir o n to be H, = 500 ± 10 KG hf 103 The nuclear magnetic moment of Ru i s not known, The nucleus has 44 protons and 59 neutrons. There are 9 neutrons above the magic number 50. Using the s h e l l model for neutrons, these 9 neutrons w i l l be d i s t r i b u t e d 5 7 between a 2d x2 l e v e l and a l g ^2 l e v e l which are c l o s e l y spaced i n energy. Because of the large number of nucleons involved the c a l c u l a t i o n of the magnetic moment would be complicated. The Schmidt l i m i t s f or t h i s nucleus (I= 5/2) are -1.91u N < u < l-36y N 4.3 The Experimental Procedures Preparation of the specimen, of 1 (^Ru i n i r o n was straightforward. Some 1 ( ^ R u C l ^ i n HCl was evaporated to dryness on an i r o n f o i l of dimensions 1.0 cm x 0.8 cm x 0.03 cm. The ruthenium replaces some i r o n which i s less electronegative. The f o i l was heated at about 900°C for 72 hours to d i f f u s e the ruthenium; the surface layers 98 were then etched o f f with HC1 to remove r e s i d u a l surface a c t i v i t y . The r e s u l t i n g a c t i v i t y of the specimen was 5 to 10 yCi. Several specimens were prepared for the experiments. The specimen was soldered to one side of the 6 0 copper heat sink. A Co-Fe_ thermometer was soldered to the other side. The demagnetization assembly used a cerous magnesium n i t r a t e s a l t p i l l with a manganous ammonium guard p i l l . The lowest temperatures recorded with t h i s assembly were about 9 mK. I n i t i a l experiments indicated only a small anisotropy was present i n the "^3Ru 497 and 610 KeV t r a n s i t i o n s . Because i t would be necessary to correct the ruthenium counts very c a r e f u l l y for any anisotropic ^ C o background, a separate ^Co run was performed i n order to determine the ^Co spectrum i n the experimental configuration. 103 Two Ru xn iron experxments were performed. In the f i r s t "^3Ru experiment the ^^3Ru source had an a c t i v i t y of 8 yCi and the 6<^Co-Fe thermometer had'a ^Co a c t i v i t y of 5 yCi. , To improve the background c o r r e c t i o n further the second "*"^ 3Ru experiment was performed with a ~*"^ 3Ru a c t i v i t y of 28 yCi and a ^Co a c t i v i t y of only 0.8 yCi. For t h i s second experiment the ^°Co background i n the ~^3Ru peak was about 1%. The temperatures obtained i n the f i r s t experiment were i n the region of 9 mK to 13 mK whereas i n 99 t h e s e c o n d e x p e r i m e n t w i t h t h e l a r g e r "^^Ru s o u r c e t h e l o w e s t t e m p e r a t u r e r e c o r d e d was o n l y 10 mK p r e s u m a b l y b e c a u s e o f t h e i n c r e a s e d r a d i o a c t i v e h e a t i n g . The g a m m a - r a d i a t i o n was m e a s u r e d by two G e ( L i ) d e t e c t o r s , p l a c e d i n t h e a x i a l and e q u a t o r i a l d i r e c t i o n s . I t was d i s c o v e r e d t h a t t h e e q u a t o r i a l d e t e c t o r was a f f e c t e d (^1%) by t h e m a g n e t i c f i e l d o f t h e m a i n magnet. T h i s g ave c a u s e f o r some u n c e r t a i n t y i n t h e e q u a t o r i a l c o u n t s m e a s u r e d i n t h e f i r s t e x p e r i m e n t i n w h i c h t h e n o r m a l i z a t i o n c o u n t s were t a k e n w i t h t h e m a i n magnet on. • The e q u a t o r i a l c o u n t s were t h e r e f o r e d i s c a r d e d f o r t h a t e x p e r i m e n t . 4 . 4 A n a l y s i s o f t h e S p e c t r a The ^ C o b a c k g r o u n d , w h i c h i s s c a t t e r e d r a d i a t i o n , h as an a n i s o t r o p y w h i c h h a s t h e same t e m p e r a t u r e d e p e n d e n c e 60 a s t h e Co p e a k s b u t n o t t h e same a n g u l a r d e p e n d e n c e . Hence, t h e b a c k g r o u n d r u n (no "'"^Ru sp e c i m e n ) was a n a l y s e d 103 t o f i t t h e b a c k g r o u n d i n t h e window o f e a c h o f t h e Ru 6 0 p e a k s t o t h e a n i s o t r o p y o f t h e Co, f o r e a c h d e t e c t o r . U s i n g t h i s f i t t e d b a c k g r o u n d i n t h e "'"^Ru r u n s t h e p e a k s c o u l d be c o r r e c t e d f o r t h e ^ C o b a c k g r o u n d . A n o t h e r method u s e d t o d e t e r m i n e t h e b a c k g r o u n d 103 i n t h e Ru p e a k s assumed t h a t t h e b a c k g r o u n d was l i n e a r i n t h e r e g i o n o f t h e p e a k . The b a c k g r o u n d i n t h e p e a k was 100 then estimated by measuring t h e background above and belov; the' peak. The background-corrected counts determined by the l i n e a r method were compared to those determined by the f i t t e d method. For the f i r s t run the agreement of i n d i v i d u a l counts was within 0.8% with no systematic va r i a t i o n s . For the second run the agreement was within 0.1%. The agreement of the l i n e a r method and the f i t t e d method indicates that the background correction i s r e l i a b l e . 103 In the actual analysis of the spectra the Ru peak i n t e n s i t y was determined by subtracting the background 6 0 which could be calculated from the Co anisotropies and i n t e n s i t i e s . The ^ 3 R u i n t e n s i t y was then corrected for the decay of the source during the experiment (about 2%) and s t a t i s t i c a l errors were determined. The f i n a l anisotropies are given i n table 4.2. Evaluation of the temperature dependence of 497 KeV t r a n s i t i o n would allow the determination of the magnetic moment of "*"^ 3Ru independently of the decay parameters. Hence, the anisotropies have been grouped by temperature. Unfortunately, i n the f i r s t run the error was too large to determine the temperature dep'endence accurately enough, whereas i n the second run s u f f i c i e n t l y low temperatures were not achieved. The 610 KeV t r a n s i t i o n was too weak to give s i g n i f i c a n t r e s u l t s i n the f i r s t run. In the second run 101 TABLE 4.2 103 Ru A n i s o t r o p i e s 497 MeV T r a n s i t i o n Run 1 1 2 Temperature Range (mK) 9 - 1 0 10 - 13 10 - 12 610 MeV T r a n s i t i o n 10 - 12 W(0) 0.010 ± .0016 0.0059 ± .0018 0.0048 ± .0007 0.004 ± .003 W(|) - 1 0025 ± .0005 001 ± .002 102 the r e s u l t s are s u f f i c i e n t only to conclude that the anisotropy, 1-W(0), i s less than +0.005. 4.5 Analysis of Anisotropies Because of the small anisotropies observed, the fourth order term i n the expression for W(0) can be ignored giving the s i m p l i f i e d equation W(O) = 1 + B 2U 2F 2 The f a c t that for the 497 KeV t r a n s i t i o n W(O) < 1.0; W(|-) > 1.0 c l e a r l y indicates that F 2 i s negative for t h i s t r a n s i t i o n . This means 6 must be negative and, since i t i s established 2 that 6 (E2/M1) i s between .01 and .17, we have the r e s u l t -.41 < 5 < -.10 . Choosing the mixing r a t i o s for the beta and gamma decays which give the largest possible e f f e c t y i e l d s the maximum value for U 2 F 2 : U 2 F 2 < .33 Allowing only the smallest value for the observed anisotropy, 1 - W(O) = 0.004 , 103 at 11 mK gives the lower l i m i t of B 2 > .004/.33 = 0.012 and therefore, for a hyperfine f i e l d of 500 KG and spin 5/2, the nuclear magnetic moment i s given by U l > 0 . 1 5 u N The observation that the e f f e c t for the 610 KeV t r a n s i t i o n was less than ±.005 means B 2 U 2 F 2 < ' ° 0 5 Using the above r e s u l t , > .012, y i e l d s F 2| < .42 From equation (4.3), the mixing r a t i o for the 610 KeV t r a n s i t i o n i s therefore l i m i t e d to 5 (E2/M1) < 0.0 or 6 (E2/M1)' > 1.1 Since t h i s work was completed measurements on the same system have been made by Krane and co-workers (Krane, 1976) at temperatures as low as 5 mK. A value of B 2 U 2 F 2 = ~ 0 - 0 2 0 1 0.001 was obtained for the 497 KeV ra d i a t i o n . The temperature dependence of the anisotropy i n the temperature range 5 mK - 8 mK was used to determine 104 t h e n u c l e a r m a g n e t i c moment, t h e v a l u e o b t a i n e d b e i n g v = 0.67 ± O . l l y Our r e s u l t i s c o n s i s t e n t w i t h t h i s more a c c u r a t e v a l u e . 105 CHAPTER V 59 Nuclear Orientation of Fe 5.1 Introduction 59 Nuclear o r i e n t a t i o n of Fe i n the paramagnets Ce-Zn n i t r a t e and Nd-Zn n i t r a t e was observed by.Tschanz and Sapp (1970). However, t h i s was a d i f f i c u l t system f o r i n t e r p r e t a t i o n l a r g e l y because only a f r a c t i o n of the Fe ions were i n l a t t i c e s i t e s i n the c r y s t a l structure. Nuclear o r i e n t a t i o n in i r o n i s a more d i r e c t experiment which could clear up some of the uncertainties i n t h e i r work. 59 The decay scheme of Fe i s given i n the handbook of Lederer, et a l . (1968). The relevant features are given i n figure 5.1. There are two beta emissions with unknown Fermi/Gamow-Teller mixing r a t i o s . The two r e s u l t i n g gamma-rays each have predominantly E2 character. The l i f e -time of the 1.292 MeV l e v e l i s rather long being 0.59 nsec. A possible r e o r i e n t a t i o n could occur for t h i s state for very large hyperfine i n t e r a c t i o n s where the nuclear precession i s not slow compared to the l i f e t i m e of the 59 state. The spin of the Fe nucleus i s 3/2. The Schmidt l i m i t s for the nuclear magnetic moment, u, for t h i s spin 106 Figure 5.1 The Decay Scheme of Fe. 107 are -1.91u < u < 1.14 The nucleus has 26 protons and 33 neutrons. Using the s h e l l model, there are 5 excess neutrons to f i l l two c l o s e l y 3 5 spaced l e v e l s , the 2p x 2 and If y2 l e v e l s . A higher l e v e l 2p^2 i s u n l i k e l y to contribute. Assuming that s h e l l s with even numbers of neutrons do not contribute to the magnetic moment, only the configurations 3 1 5 4 3 9 5 3 3 3 5 ? (2p J2) L (If°2) \ (2p J2) Z (If°2) , ( 2 p J 2 ) J ( I f 3 2 ) (5.1) need be considered. The resultant magnetic moment for three p a r t i c l e s i n one angular momentum state i s given by (1/1^) times the single p a r t i c l e value where I i s the spin of the resultant state and 1^ i s the t o t a l spin of each p a r t i c l e i n the state (see for example Rose, 1967). The resultant .moments of the three states given above are -1.91u N, 0.82pN and -1.91n N r e s p e c t i v e l y . It can be seen that t h i s simple model gives a large range of values of the magnetic moment. The hyperfine f i e l d of i r o n i n ir o n metal has been measured by Hanna et a l . (1960) to be 333 KG. The expression for the i n t e n s i t y of the aniso-t r o p i c gamma-radiation (4.1) includes only terms .as high 108 as the second order since 21 = 3, so that W(9) = 1 + B 2 U 2 F 2 P 2 ^ c o s e ) The c o e f f i c i e n t s for each gamma-ray have the same value of 0.143. For the same Fermi/Gamow-Teller mixing r a t i o the U 2 c o e f f i c i e n t s would be equal also. For pure Fermi U 2 equals 1.00; for pure Gamow-Teller U"2 equals 0.20. At saturation the anisotropy, l-W(O), i s 0.143 for pure Fermi and .029 f o r pure Gamow-Teller. At 10 mK assuming u = 1.0 N the anisotropy would be 0.04 for pure Fermi and 0.008 for pure Gamow-Teller. 5.2 Experiments and Analysis 59 The Fe source was prepared by neutron i r r a d i a t i o n of an i r o n f o i l . The i n i t i a l source strength was 20 uCi. The dimensions of the source were 6 0 1.0 cm x 0.8 cm x 0.03 cm. A source of Co i n i r o n (2 uCi strength) was used for thermometry. The sources were soldered to the copper heat sink. A demagnetization p i l l made of cerous magnesium n i t r a t e was used to achieve low temperatures. A temperature of about 13 mK was achieved a f t e r demagnetization with warming to about 27 mK i n one hour. 109 Two Ge(Li) detectors were used to detect the gamma-radiation. One detector was a x i a l , while the other was i n the equatorial plane. The.spectra peaks were analysed by subtracting a l i n e a r background from each peak as discussed i n section 4.4. A l l anisotropies from the temperature range 13 mK - 14 mK were obtained for each 59 peak m the Fe gamma-ray spectrum. The r e s u l t , using A2 = B 2 U 2 F 2 ' w a s A 2 (l.lO.MeV) = -0.004 ± .003 A 2 (1.29 MeV) = +0.0044 ± .004 5.3 Discussion 59 Both Fe gamma-rays would be expected to have negative values for A 2 regardless of the mixing i n the beta-decays. Assume that the mixing i s pure Gamow-Teller for both emissions so that a minimum e f f e c t occurs. In that case the maximum size of the magnetic moment can be determined from the r e s u l t s . The average value of the e f f e c t i n the experiments i s A 2 = 0.00 ± .003 for an average temperature of 15 mK. Therefore, the upper l i m i t on B„ i s 0.105 giving 110 Tschanz and Sapp (1970) observed a value of 59 A 2 = .05 at T < 10 mK for Fe i n Ce-Zn n i t r a t e and Nd-Zn n i t r a t e . They estimated that 30% of the nuc l e i were i n c r y s t a l s i t e s and they assumed the other nuclei saw a zero hyperfine f i e l d and would not contribute to the anisotropy. In this.case, the anisotropy for the c r y s t a l s i t e s would be given by - .17. They f i t t e d the temperature dependence of the anisotropy to obtain a rough estimate of the nuclear 59 magnetic moment of the Fe ground state. Assuming the temperature dependence of A 2 a r i s e s s o l e l y from B^(T) of nuc l e i i n c r y s t a l s i t e s they obtained y = 1.1 ± 0.2yN. 59 . Our measurement for Fe-Fe gives A 2 < 0.003 at 15 mK which y i e l d s y < 0.9yN for pure Gamow-Teller beta decay and y < 0.4yN for pure Fermi decay. The largest anisotropy occurs i f a l l nuclear spins are i n the lowest Zeeman l e v e l and i f the beta decay occurs with complete Fermi mixing. In t h i s case A 2 = -0.14. This i s approximately the size of the e f f e c t seen by Tschanz and Sapp. If i t i s assumed that t h e i r r e s u l t indicates complete Fermi mixing, then by our r e s u l t s the maximum nuclear magnetic moment i s only y = 0.4yN which i s then i n disagreement with t h e i r value based on the tempera-ture f i t of the data. I t seems reasonable to question the temperature f i t of the double n i t r a t e data. Possibly the nuclei not i n c r y s t a l s i t e s contribute appreciably to the I l l anisotropy. This e f f e c t could change the shape of the temperature curve thereby a f f e c t i n g the derived value of the magnetic moment. A d i f f i c u l t y encountered by Tschanz and Sapp was the measurement of the temperature which was determined from the s u s c e p t i b i l i t y of the double n i t r a t e s a l t s themselves: Ce-Zn n i t r a t e was used below 10 mK; Nd-Zn n i t r a t e was used above 10 mK. Any inconsistency i n the temperature would serio u s l y have affected the f i t t i n g process. In order to explain t h e i r large experimental value of T s c n a n z a n < ^ Sapp have determined an enhancement factor (see equation 4.4) of about 5 to 7 for both y-rays. The enhancement i s assumed to a r i s e from intermediate state r e o r i e n t a t i o n . Agarwal et a l . (1967) noted an attenuation of the y -Y c o r r e l a t i o n f o r the 59 1.29 MeV l e v e l of Co for a me t a l l i c iron source as compared to f e r r i c and ferrous chloride sources. I t i s ' of course possible that the i o n i c environment gives an enhancement not found i n the metal. Since the completion of t h i s work Krane et a l . (1976) have measured the anisotropies of gamma-radiation 59 from the decay of Fe i n i r o n at a temperature of 3.5 ± 0.0 3 mK. They determined the nuclear magnetic moment to be y = 0.29 ± 0.03yN which i s consistent with our less accurate r e s u l t . 1 1 2 A p p e n d ! x Gamma-Ray Spectrum Analysis The gamma-ray spectra obtained i n the experiments were analysed to determine the r e l a t i v e i n t e n s i t y of the gamma-radiation f o r the t r a n s i t i o n of i n t e r e s t emitted i n the d i r e c t i o n s of the detectors. The general procedure employed for the spectral analysis can be i l l u s t r a t e d by considering a t y p i c a l spectrum obtained i n the MnCl2*4H.20 experiments from a Nal detector. This i s shown i n fi g u r e 54 A l . The spectrum has one peak from the Mn decay at 0.835 6 0 MeV and peaks from the Co decay at 1.17 and 1.33 MeV. The gamma-transition photopeaks are superimposed on a 'background' of counts a r i s i n g from several contributions. Gamma-rays which have been only p a r t i a l l y absorbed i n the detector constitute the p r i n c i p a l source of the background. The anisotropy of these counts i s the same as the photopeak anisotropy for that p a r t i c u l a r gamma-transition. '. Another source of the background i s r a d i a t i o n which has been scattered into the detector ('degraded r a d i a t i o n ' ) . These counts do not have the same anisotropy as the unscattered counts because they were a c t u a l l y emitted i n a d i f f e r e n t d i r e c t i o n . In addition, there are i s o t r o p i c background contributions 40 from the environment due mainly to degraded K r a d i a t i o n from the concrete of the b u i l d i n g . The t o t a l background i s 1 13 4 0-3 0-Mn Ui -P C 3 O CJ 201 10H1 x 10 25 b bw I 2 i n 60 Co ° • o • O 0 wb. b_ 50 Channel 3 <v 75 — r 10 Figure A l . Gamma-ray Spectrum. 114 t h e sum o f t h e c o n t r i b u t i o n s f r o m a l l t h e gamma-ray 54 t r a n s i t i o n s p r e s e n t . Hence t h e b a c k g r o u n d u n d e r t h e • Mn p h o t o p e a k i s d i f f e r e n t t h a n t h e a n i s o t r o p y o f t h e p h o t o p e a k . To d e t e r m i n e t h e p h o t o p e a k c o u n t , t h e b a c k g r o u n d c o u n t u n d e r t h e p e a k c a n be e s t i m a t e d a n d s u b t r a c t e d f r o m t h e t o t a l peak c o u n t . A s i m p l e t e c h n i q u e was u s e d i n t h e e x p e r i m e n t s i n w h i c h i t was assumed t h a t t h e b a c k g r o u n d u n d e r t h e peak was l i n e a r . C o u n t s i n a 'window' c e n t e r e d on t h e peak were d e t e r m i n e d as w e l l a s c o u n t s c h o s e n s y m e t r i c a l l y above and b e l o w t h e window (as shown i n f i g u r e A l ) . .The b a c k g r o u n d i n t h e window was t h e n e s t i m a t e d by i n t e r p o l a t i n g t h e c o u n t s B f r o m a b o v e t h e window ( c h a n n e l s b 0 t o b.) and u 3 4 t h e c o u n t s B^ f r o m b e l o w i t ( i n c h a n n e l s b ^ t o b 2 ) • I f t h e c o u n t i n t h e window ( c h a n n e l s w^ t o w 2) i s C, t h e n t h e ' b a c k g r o u n d c o r r e c t e d c o u n t P i s g i v e n by B - B P = C - — (w„-w,+l) 2 ( b 2 - b 1 + l ) The b a c k g r o u n d - c o r r e c t e d c o u n t P i s t h e n t h e e s t i m a t e d v a l u e o f t h e p h o t o p e a k i n t e n s i t y . The o n l y s o u r c e o f e r r o r i n t h e e s t i m a t e o f P t h a t a f f e c t s t h e v a l u e o f n o r m a l i z e d i n t e n s i t y W i s t h e b a c k g r o u n d w h i c h d o e s n o t have t h e same a n i s o t r o p y 54 a s t h e p h o t o p e a k and w h i c h i s n o n l i n e a r . F o r t h e Mn peak, t h i s b a c k g r o u n d i s m a i n l y t h e ^ C o d e g r a d e d r a d i a t i o n . I t 54 c o n t r i b u t e s a n e s t i m a t e d e r r o r , f o r t h e Mn d e c a y , o f ^2% 115 o f t h e v a l u e o f P. F o r t h e n o r m a l i z e d i n t e n s i t y W t h i s e r r o r i s r e d u c e d . I n t h e w o r s t c a s e , t h a t i s t h e l a r g e s t m e a s u r e d a n i s o t r o p y o f 2 0 % , t h e e r r o r i n W i s +0.4%, g i v i n g a n e r r o r i n t h e a n i s o t r o p y o f 2%. T h i s c o n t r i b u t i o n t o t h e t o t a l e r r o r i s l e s s t h a n t h e e r r o r i n t h e a n g u l a r l o c a t i o n o f t h e d e t e c t o r a n d t h e s t a t i s t i c a l e r r o r i n P. I n f a c t , t h e o t h e r N a l a n d G e ( L i ) d e t e c t o r s u s e d i n t h e e x p e r i m e n t s h a d b e t t e r r e s o l u t i o n r e s u l t i n g i n s m a l l e r b a c k g r o u n d c o r r e c t i o n s , s o t h a t f o r t h e s e d e t e c t o r s t h e r e l a t i v e e r r o r was e v e n l e s s . A l t h o u g h t h i s m e t h o d o f s p e c t r a l a n a l y s i s d o e s n o t make f u l l u s e o f a l l t h e gamma-ray i n f o r m a t i o n a v a i l a b l e i n t h e s p e c t r u m , i t i s v e r y s i m p l e a n d d i r e c t , a n d w o r k s s a t i s f a c t o r i l y f o r p e a k s t h a t a r e w e l l r e s o l v e d . 116 REFERENCES Abkowitz, M., and A. Honig, Phys. Rev. 136, A1003 (1964). Abragam, A., and M.H.L. 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