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Nuclear orientation studies at low temperatures Daly, Patrick William 1973

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NUCLEAR ORIENTATION STUDIES AT LOW TEMPERATURES by PATRICK WILLIAM DALY B.Sc. Bishop's University, 1968 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department . - of Physics We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA March, 1973 i i In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study . I further agree thatxpermission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Physics The University of British Columbia Vancouver 8, Canada Date TTUX^A 3o } / ?73 ABSTRACT The interaction between the nuclear magnetic moment and the hyperfine field in ferromagnetic and antiferromagnetic materials has been used to orient radioactive nuclei at low temperatures. The measurement of the resulting angular distribution of the gamma radiation has yielded a variety of information in both nuclear and solid state physics. 59 The Fe nucleus in an iron sample has been studied in an attempt to verify a report by Tschanz and Sapp (1970) who measure an unexpectedly large effect in the gamma ray distribution from that nucleus oriented in double nitrate salts. Since in their experiment assumptions have to be made regarding the fraction of iron atoms"in lattice sites, the well-defined environment of the iron lattice was chosen for the present experiment in order to permit an unambiguous interpretation of the results. In contradiction to predictions based on the work of Tschanz and Sapp, a null result was obtained at a temperature of 15 mK, 59 I I -indicating an upper limit for the magnetic moment of Fe of |u| < 0.9 u ^ . The anisotropy of gamma radiation emitted in the decay of 103 Ru oriented in ferromagnetic iron was measured to determine the magnetic moment of the isotope and to resolve mixing ratios in the . decay scheme. The small effect observed at 11 mK permitted a lower limit to be placed on the magnetic moment: |p| > 0.15 u . The sign of the E2/M1 mixing ratio for the 497 keV gamma transition was determined for the first time and found to be negative. iv The techniques of nuclear orientation have also been used to observe the spin-flop transition in antiferromagnetic MnCl^^H^O. By 54 observing the gamma ray anisotropy in the decay of radioactive Mn we find that at 60 mK the transition occurs for a critical field of H£ = 6.50 ± 0.05 kG applied along the crystallographic c-axis. In the spin-flop phase, the electronic spins point at an angle of 71° to this axis. These data allow values of the molecular exchange and anisotropy fields to be determined yielding respectively = 11.7 ± 0.6 kG and H = 2.0 ± 0.1 kG in agreement with the values reported by Miedema et_ al. (1965). This experiment demonstrates that nuclear orientation is a viable technique for studying spin-flop transitions and moreover allows the measurement of the directions of electronic spin magnetization in the spin-flop phase. 207 The nuclear orientation of Bi in cobalt was studied to determine the hyperfine field. An unexpectedly small gamma ray anisotropy was observed. However, saturation of the effect occurred at a relatively high temperature. These results indicate that only a fraction (^ 12%) of the Bi nuclei felt a large hyperfine field (^ 600 kG). Presumably this was the fraction of atoms in lattice sites. Comparison of the anisotropics of the 1.064 MeV and the 1.761 MeV gamma rays shows that the former effect is enhanced, possibly because of the long half-life of the 1.633 MeV intermediate state. Tschanz, J. F.5 and Sapp, R. C. 1970. Phys. Rev. C, 2_, 2168. Miedema, A. R., Wielinga, R. F. ,and Huiskamp, W. J. 1965. Physica, 31_, 835. V TABLE OF CONTENTS ABSTRACT i i i TABLE OF CONTENTS v LIST OF TABLES v i i LIST OF FIGURES v i i i ACKNOWLEDGEMENTS x CHAPTER 1 THEORY OF NUCLEAR ORIENTATION 1.1 Introduction 1 1.2 Simple Angular Distribution Scheme 2 1.3 Details of the Angular Distribution 8 1.4 Solid Angle Correction 22 1.5 Methods of Orienting Nuclei . 25 1.6 Low Temperature Production 30 1.7 Other Experimental Methods 33 CHAPTER 2 EXPERIMENTAL APPARATUS AND PROCEDURES 2.1 The Cryostat 36 2.2 Paramagnetic Salts and Thermal Contact 39 2.3 Low Temperature Procedure 41 2.4 Gamma Ray Detection 42 2.5 Paper Tape Analysis 49 2.6 Thermometry: ^Cobalt Experiments 54 2.7 The Term "Anisotropy" 63 59 CHAPTER 3 ORIENTATION OF Fe NUCLEI W IRON 3.1 Introduction 65 59 3.2 Decay Scheme of Fe 65 59 3.3 Expected Anisotropy of Fe 68 vi 3.4 The Experiment 69 3.5 Analysis 71 3.6 Results and Discussion 77 CHAPTER 4 ORIENTATION OF 103RUTHENIUM NUCLEI IN IRON 4.1 Introduction 80 4.2 Decay Scheme of 1 0 3Ru 81 4.3 Expected Anisotropy of the 497 keV Gamma Ray 83 4.4 Experiments 86 4.5 Analysis 88 4.6 Results 95 CHAPTER 5 SPIN-FLOP TRANSITION OF MnCl2-4^0-OBSERVED BY NUCLEAR ORIENTATION 5.1 Introduction 100 5.2 Molecular Field Model 100 5.3 Decay Scheme and Anisotropy 104 5.4 The Experiment 108 5.5 Analysis 111 CHAPTER 6 ORIENTATION OF 2 0 7 B i NUCLEI IN COBALT 6.1 Introduction 117 6.2 Decay Scheme 118 6.3 Expected Anisotropies 118 6.4 Sample 122 6h5 The Experiment 124 6.6 Analysis 128 6.7 Discussion 132 REFERENCES 134 v i i LIST OF TABLES TABLE 1 TABLE 2 TABLE 3 TABLE 4 TABLE 5 TABLE 6 TABLE 7 TABLE 8 Sample output of the spectrum analysis program . 59c , 60r for Fe and Co. 60, Cobalt temperatures from Co intensities during Fe runs. 59 Average for Fe for each temperature range. U and F coefficients for possible transition modes 103 in the 497 keV and 610 keV sequences in Ru. Coefficients of B2 and B4 from 6°Co background run Effect in the 497 keV gamma ray as measured by the a x i ? l detectors. Measured and fitted angles of the spin orientation above the spin-flop transition. Fitted values of attenuation and hyperfine field 207 from the gamma rays of Bi. 74 75 76 84 93 96 113 130 v i i i LIST OF FIGURES FIGURE .•1 Typical decay scheme 3 FIGURE 2 Solid angle correction 23 FIGURE 3 Level splittings and populations during 32 adiabatic demagnetization FIGURE ^ 4 Outer cryostat 37 FIGURE 5 Inner cryostat 38 FIGURE 6 Flow diagram of detector output 45 FIGURE 7 Background calculation 51 FIGURE 8 6 0Co decay scheme 55 FIGURE 9 W(0) and W(ir/2) for 6 0Co versus inverse 58 temperature FIGURE 10 Sample 6°Co spectra 60 FIGURE 11 ^Co anisotropy versus polarizing solenoid 61 current FIGURE 12 ^Co temperature versus time elapsed from ^. . t _ '62 demagnetization FIGURE 13 5 9Fe decay scheme 66 59 FIGURE 14 versus inverse temperature for Fe 70 FIGURE 15 Sample 5 9Fe/ 6°Co spectra ' 72 FIGURE 16 Ru decay scheme 82 FIGURE 17 F 2 versus mixing ratio for 497 and 610 keV 85 gamma rays 103 FIGURE 18 Sample spectra from Ru runs 89 FIGURE 19 Gamma ray scattered into detector 90 ix FIGURE 20 Background ratio of 497 keV window versus 92 '60 r . . Co intensity FIGURE J21 Mn decay scheme 105 54 FIGURE 22 Gamma ray intensity versus cos0 for Mn 107 FIGURE 23 Sample 54Mn/60Co spectra 110 FIGURE 24 Cos Versus field along c-axis 112 FIGURE 25 Cos 03 versus field along c-axis above 115 spin-flop 207 FIGURE 226 Bi decay scheme 119 207 FIGURE 27 Saturation of Bi/Co sample 125 FIGURE 28 Sample 2 0 7Bi/ 6°Co spectra 126 207 FIGURE 29 Bi gamma ray effects versus inverse temperature 131 ACKNOWLEDGEMENTS The author wishes to express his sincere gratitude to The National Research Council of Canada and the Killam Foundation of Vancouver, British Columbia for financial assistance in the forms of an NRC Post-Graduate Scholarship and a Killam Pre-Doctoral Fellowship, Dr. P.W. Martin and Dr. B.G. Turrell for their supervision and guidance and especially for their encouragement during the final weeks, R.L.A. Gorling and R. Kisser for their assistance during the experimental runs, J. Lees, glassblower, for his help in the preparation of the Bi samples, R. Weissbach, low temperature technician, for maintaining a steady supply of liquid helium, ELndLDlyDaly whose innumerable services made the whole affair run smoothly. 1 CHAPTER ONE THEORY OF NUCLEAR ORIENTATION 1.1 Introduction AOfeafaSrfehofmipadi'o'actikeod'e^  ei is that the radiation is emitted with equal intensity in all directions. However, the spherical symmetry of the radiation field is not a property of the decay process at all but a result of the random arrangement of the nuclei. For many nuclei are definitely not spherical, possessing various vector quantities, such as magnetic moments and angular momentum. In the study of oriented nuclei, a preferred direction for these vectors is introduced and the angular dependence of the radiation is observed, revealing secrets of the decay scheme or of the orienting interactions. The mathematical derivation of the angular distribution as a function of the orienting and radiation parameters is straight-forward, but does not lend itself to a feeling for what is happening. Conversely, the physical picture does not lend itself to a complete and useful formulation. However, we shall first present the physical picture to offer insight; afterwards the general result will be stated, and then derived. Much use will be made in this chapter of the quantum theory of angular momentum, references to which can be found in Rose (1967). Other general references in this field are Blin-Stoyle and Grace (1957), DeGroot, Tolhoek, and Huiskamp (1965). 2 1 . 2 S i m p l e A n g u l a r D i s t r i b u t i o n Scheme When a n u c l e u s o f s p i n 1^ a n d z c o m p o n e n t d e c a y s t o a n o t h e r s t a t e o f s p i n I^. a n d z c o m p o n e n t b y e m d s s i o n o f a gamma r a y o f t o t a l a n g u l a r momentum L , t h e ,z c o m p o n e n t o f t h e gamma r a y m u s t b e - i n o r d e r t o c o n s e r v e a n g u l a r momentum. T h e s e q u a n t u m n u m b e r s t o g e t h e r w i t h t h e a p p r o -p r i a t e p a r i t y , a r e s u f f i c i e n t t o d e t e r m i n e c o m p l e t e l y t h e s t a t e o f t h e gamma ray, . , a n d t h u s i t s a n g u l a r d i s t r i b u t i o n . F o r e x a m p l e , f o r q u a d r u p o l e r a d i a t i o n , e i t h e r e l e c t r i c (E2) o r m a g n e t i c (M2), t h e r e a r e f i v e p o s s i b l e z c o m p o n e n t v a l u e s w i t h t h e f o l l o w i n g a n g u l a r d i s t r i b u t i o n s : M = +2: W(e) = l / 4 ( l - c o s 4 6 ) 1 . 1 a M = +1: W(6) = l / 4 ( l - 3 c o s 2 6 + 4 c o s 4 6 ) 1 . 1 b M= i : 0 : W(6) = 3 / 2 ( c o s 2 9 - c o s 4 9 ) 1 . 1 c w h e r e 6 i s t h e d i r e c t i o n r e l a t i v e t o t h e z a x i s . When t h e s e f i v e f i e l d s a r e e q u a l i n s t r e n g t h , W(6) = 1, a n i s o t r o p i c d i s t r i b u t i o n . I n p r i n c i p l e , o n e c a n d e t e r m i n e t h e a n g u l a r d i s t r i b u t i o n f o r e a c h t r a n s i t i o n a n d c a l c u l a t e t h e o v e r a l l d i s t r i b u t i o n f r o m t h e p r o b a b i l i t i e s o f t h e s e t r a n s i t i o n s . T h e s e p r o b a b i l i t i e s d e p e n d o n t h e r e l a t i v e p o p u l a -t i o n s o f t h e i n i t i a l s t a t e s a n d o n t h e p r o b a b i l i t y f o r e a c h i n i t i a l s t a t e t o d e c a y t o a p a r t i c u l a r f i n a l s t a t e . F i g u r e 1 s h o w s a s i t u a t i o n i n w h i c h a p a r e n t n u c l e u s o f s p i n 3 b e t a - d e c a y s t o a s p i n 2 s t a t e ( t h e b e t a p a r t i c l e a n d n e u t r i n o o m u s t h a v e t o t a l s p i n 1 t o a l l o w t h i s ) w h i c h p r o m p t l y g a m m a - d e c a y s t o a s p i n 0 s t a t e . Shown a r e t h e p o s s i b l e t r a n s i t i o n s f r o m t h e p a r e n t Mq = 1 s t a t e . S i n c e t h e s t a t e s o f d i f f e r e n t z c o m p o n e n t s d i f f e r o n l y i n o r i e n t a t i o n a n d n o t i n a n y Figure 1 Typical decay scheme 4 intrinsic fashion, i t is possible to find the relative transition probabili-ties through geometric considerations only. The transition probabilities are proportional to the square of the Clebsch-Gordan coefficient in the angular momentum coupling of the beta particle and final state nucleus to 2 the i n i t i a l state: that is to |CCI^ LQ 1^ ; - m, m)| . For the transi-tions in the diagram, that i s , for the transitions from MQ = 1 to = 2,1,0, the probabilities are 1/15, 8/15 and 2/5 respectively. Each subsequent gamma decay has a probability of unity, since only one final state exists. How-ever, in general, one must allow for several transitions from each i n i t i a l state in the same way as was done above for the beta decay. If a l l the parent nuclei were in the = 1 state, the gamma rays would have the angular distribution w(e) = ^ 5 j(i-cos4e) + | j j(i-3cos2e + 4cos4e) + |- j(cos2e - cos4e) "I 0 A ^ = £Q(9 + 12cos 0 - 5bos?9).. .. ~ i - Sc.s 1.2 By calculating and averaging W(Q) over each of the parent states, one would be able to determine the angular distribution for any set of parent populations. Consider the case in which a parent state of spin I , with relative populations a(MO), decays to a state of spin 1^ by emitting radiation of angular momentum Lq. If this process is followed by prompt emission of radiation with angular momentum to a state of spin I^, then this latter radiation has an angular distribution 5 W ( 8 ) = m^ n ^ V ^ W V Mo~m> M ) ' 2 | C C I 2 , L 1 , I 1 ; Mo-m-n, n) |2 F ^ e ) 1.3 where F^C6) is the angular distribution of radiation with angular momentum L and z component n. It should be emphasized that the above formula requires a prompt decay from the intermediate state; that i s , the lifetime of the intermediate state must be short enough that no reorientations can occur before the _9 decay. This requirement usually demands lifetimes less than 10 sec, the typical Larmor precession time; such short lifetimes are quite normal in gamma decays. However, exceptions do exist. The angular distribution Fjv for an individual gamma ray can be calculated from the electromagnetic multipole fields. Following the phase convention used by Biedenharn and Rose (1953), we shall use the following expressions for the vector potential of an electromagnetic wave in free space with angular momentum quantum number L, z component quantum number M, and parity i r : -> L L+l AL M (m aSn e t i c) = 1 \ TLLM 11 = C _ ) 1 A A (electric) - - iL + 1 11 / 2 h T + iL + 1 M ^ - l1 / 2 h T LM 1 L2L+1J V I ' L . L + I . M 1 L2L+1J V ^ L . L - l , ! IT = (-) where h^ is a spherical Hankel function of the first kind (an outgoing wave at large distances) and T M is a vector spherical harmonic defined as TL A M= ZC(X,1,L; M - y , u ) Y | Vi 1.5 The Y,„ are spherical harmonic functions, and the E. are spherical basis AM r u r vectors, related to the cartesian unit vectors thus: t - x+iy h " " 72 1.6 5 = z o^ r - x-xy 72" In the gauge chosen for equations 1.4, the scalar potentials are zero. Hence the radiation intensity at any place is proportional to A*A*. In fact, this density is independent of the parity, so that one may use the magnetic multipole to calculate. F L M ( 6 ) - J |C(L,1,L; M-y,y)|2|Y L ) M_ y(e)| 2 . 1.7 The proportionality constant is adjusted so that W(8) = 1 for a uniform distribution of initial states. This outline of the principles behind calculating W(6) as a function of the initial state populations does not produce a useful formula. The many sums involved in equations 1.3 and 1.7 can be simplified in terms of Racah coefficients and Legendre polynomials to give a single sum: W(9) = Z B U F P (cose) . 1.8 vv v v v The sum is carried over only even values of v and the maximum value of the index is 2 I Q , 2 1^ , 2 L^, whichever is smallest. The factors in the summand are as follows: 7 The term describes the initial state populations and is a function of the initial state alone. It is given by Bv = (2v+l)1/2 Z C(I0,v,I0; M0,0)a(M0) . 1.9 The factor U describes the reorientation created by the unobserved intermediate transition. It is a function only of the transition and of the spins of the states which it connects. Its expression is I1+In-L +v Uv = (-) ° [C2I 0+lK2I1+l)]-l/'£ W(Io,I1 >I( ),I1;L 0 v) 1.10 where W is a Racah coefficient. If several transitions precede the observed one, each is represented in equation 1.8 by another U coefficient. If the transition has mixed multipolarity, the combined coefficient is U (L) + 62U (L') U = — 1.11 1 + 6 where 6 is the mixing ratio of the multipblarities; that i s , S is the amplitude for L' divided by the amplitude for L. The observed gamma ray transition is described by the factor F , which is a function of the multipolarity and spins of the states involved. Its formula is F, I .(-:). I1"I2+1 (2L1 +1)(2I1 +1)1 / 2 C(L ,L ,v;l-l) W(I ,L , I ,L ; I ,v) . 1.12 8 F o r m i x e d m u l t i p o l e s o f a n g u l a r momentum L a n d L ' , F (L) + 26F ( L , L > ) + &2 F ( L ' ) F = 1 . 1 3 v w h e r e F £ L , L » ) = f-jW1 [ ( 2 L + 1 ) ( 2 L , + 1 ) ( 2 I + 1 ) ] 1 / 2 C ( L , L ' , v ; l - l ) : : r x W C I ^ L . I ^ L ' j I ^ v ) . 1 . 1 4 T h e f i n a l t e r m P ^ f c o s O ) i s a L e g e n d r e p o l y n o m i a l , c o n t a i n i n g t h e a n g u l a r d e p e n d e n c e . T h e U and F c o e f f i c i e n t s c a n b e e v a l u a t e d f r o m t a b l e s o f C l e b s c h -G o r d a n a n d R a c a h c o e f f i c i e n t s o r t h e y may b e f o u n d f r o m t a b l e s s u c h a s t h o s e o f F e r e n t z a n d R o s e n z w e l g ( 1 9 5 5 ) (F c o e f f i c i e n t s o n l y ) o r Y a m a z a k i a n d S a t c h l e r ( 1 9 5 5 ) . O t h e r s y s t e m s f o r s p e c i f y i n g t h e s t a t e o f t h e i n i t i a l p o p u l a t i o n s e x i s t ( s e e D e G r o o t , T o l h o e k , a n d H u i s k a m p ( 1 9 6 5 ) f o r s y n o p s i s ) b u t d i f f e r o n l y i n n o r m a l i z a t i o n . 1 . 3 D e t a i l s o f t h e A n g u l a r D i s t r i b u t i o n I n t h i s s e c t i o n w i l l b e d e r i v e d t h e a n g u l a r d i s t r i b u t i o n f o r m u l a o f e q u a t i o n 1 . 8 . The e f f e c t o f m u l t i p o l e m i x i n g s w i l l n o t b e i g n o r e d , a s was d o n e i n e q u a t i o n 1 . 3 . No a s s u m p t i o n s w i l l b e made c o n c e r n i n g t h e n a t u r e o f t h e r a d i a t i o n , o t h e r t h a n i t s a n g u l a r momentum p r o p e r t i e s , u n t i l t h e e n d . E x t e n s i o n t o o t h e r f o r m s o f r a d i a t i o n i s t h e r e f o r e e a s i l y d o n e . ( 1 9 6 7 ) . T h e B c o e f f i c i e n t s a r e t h e o n e s m o s t c o m m o n l y u s e d , f o l l o w i n g G r a y 9 We will use the notation implied in figure 1; that i s , a state of spin Ij;is to decay to a state of spin emitting radiation of angular momentum L, L+l, L+2,.... The initi a l state can be described by a (21^ +1) x (21^ +1) density matrix p . In many situations, the system possesses rotational symmetry about some axis, which we shall call the z axis. In this case, the eigenstates of the Hamiltonian are also eigen-states of the z component of angular momentum, and the density matrix is diagonal when written with these eigenstates as basis. Therefore <I1M1|p|l1M1-> = aCM^d 1.15 where a(M^ ) is the probability (or relative population) of the state After this state decays, the system contains a nucleus of spin and radiation of angular momentum L,L+l,L+2... . We shall take as basis vectors of this final state the direct product of the radiation and nuclear states. That i s , the final state basis vectors are of the form I I„M„> IAT..> where |AT.,> is the wave function of the radiation. 1 2 2 1 LM 1 LM However, the initial state |I^ M^ > cannot decay to any arbitrary combination of final state vectors; i t may decay only to those combinations which have total angular momentum 1^  and z component M^ , i f angular momentum is to be conserved. Thus |ljM^> decays to W + * \ CtI2'L'Il'M2'MrM2)lI2M2>lAL,M1-M9> 1 A 6 10 where a. is the amplitude for the emission of radiation of angular momentum L. This amplitude depends on the nuclear decay, and as far as this dis-cussion is concerned, is completely arbitrary, except that the sum of the. squares of the amplitudes must be unity (to maintain the normalization) and that with the phases chosen in equation 1.4 the amplitudes must be real.''" The final state wave functions of equation 1.16 will have the same density matrix as the original state. To find the probability that the radiation w i l l be at the point with spherical co-ordinates r , 6 , <(>, one multiplies the density matrix by the projection operator for radiation at r , 0, <f> and takes the trace of the product. W(r,e,ij>) = Tr[p|r,e,4><r,6,4>|] = E <I1M1|p|l1M1'xI1M1I |x,9,«(»<r,e,*|l Mx> M1M2 a(M,)a?.,aT A* (r, 0,<j>) «AT (r,0,<Ji) , v lJ L'5 L L'jM^-I^ L,M^-M2crj , j x C(I2,L',I1;M2,M1-M2)C(I2,L,I1;M2,M1-M2) 1.17 where use has been made of equation 1.16, to substitute for |l.|M >, and of the orthonormality of the final nuclear states. The product <r,6 ,<j> |ALM> ->-has been written as A^(r,0,(j)). The angular co-ordinates are contained entirely within the product L^'M'^ LM" T^ is function can be expanded in terms of spherical harmonics, "*"See Biedenharn and Rose (1953) regarding time reversal invariance. 11 the coefficients of which being partially determined by the Wigner-Eckart theorem. Let where AL'M'ALM = Z f(L',L,M;vp) Y (6,*) 1.18 yv f(L',L,M;vy) = /A*,M Y . ALM ^ ^  ' 1 A 9 The dependence of the right-hand side of equation 1.19 on the projection quantum number M can be determined by the Wigner-Eckart theorem, yielding f(L',L,M;vy) = 6yQ C(L,v,L';M0)(L•||v||L) 1.20 where (L'||v||L) is called a reduced matrix element and is independent of any projection quantum numbers. It will be shown that for the electro-magnetic wave functions used in this work, the reduced matrix element is real. However, this does not mean that i t is symmetric in L' and L, for it is only f that is symmetric. Instead, (-)L'(2L+1)1/2(L]|v||L') = (-)L(2L'+l)(L'{|v||L). 1.21 The Wigner-Eckart theorem restricts y to zero, which means that in the expansion in equation 1.18, only Yvo(6<)>) = ^ [2^*]P^(cos6) contributes. Tucking the 4TT into the normalization of the reduced matrix element, one can write the angular distribution function of equation 1.17 as W(9) = E a ( M 1 ) a * , a L CU^L',^; M^-M^ C (L, v.,L! ;M1-M2,0) LLi' M1 M2 1/2 v x CCI^L,!^ M2,M1-M2) (2v+l) ' (L«||v||L) P^cose). 1.22 The sum over M2 can be eliminated by using an identity involving Racah coefficients and a sum over three ClebschGGordanccoefficients. The identity is [(2e+l) (2f+l)] 1 / / 2 W(a,b,c,d;e,f) C(a,f ,c, ; a , B + S ) = E C(a,b,e;ct,B) C(e,d,c;a+B,6) C (b,d,f;B,6) 1.23 B where B+6 remains constant throughout the sum. By making appropriate sub-stitutions and transformations among the indices of the Clebsc-hrGofdan coefficients, one can show E = C(I2,L,I1;M2,M1-M2) C(I2>L» .I^M^-M^ C (L, v,L • ;MrM2,0) = M2 I1 + L _ I2 1/2 (-) [ ( 2 L ' + l ) ( 2 I 1 + l ) ] i / Z Wa^L.I^L-jI^v) CCI^v.I^M^O) 1.24 Equation 1.24 allows us to remove the sum over in equation 22 and to replace i t by a Racah and a ClebschSGordanccoefficient. The remaining sum over now contains factors which are functions only of the initial state parameters. Let B = (2v+l) 1 / 2 E apO C (I, , v, I -M. ,0) . 1.25 v „. 1 1 1 1 M l 13 With equations 1.24 and 1.25, equation 1.22 may be written as W(9) = E B F P (cosO) 1.26 V V V V where the term B^  contains only the initial state parameters and the term F^ contains the radiation and nuclear spin terms. No projection quantum numbers are contained in F . We have v I1 + L" I2 1/2 F v = E a*,aL (-) [ (2L ' +1) (21^1) ]x / Wf^ , L, ^  ,L •; 1^ v) LL ' L 1 Li x CL'||v||L). 1.27 Normally, only two multipoles contribute, and one of these is the dominant one. The lesser multipole is measured by giving the mixing ratio 6 = aT ,/a. , where the L multipole is the dominant one. After recalling J-i L i that the amplitudes are real, and that, by equation 1.21, the summand is symmetric in L and L', one can write equation 1.27 as F = aT2 F (L,L) + 2a Ta T, F (L,L') + af, F (L',L') v L v L L 1 v L ' v F (L,L) + 26F (L,L ') + 62 F (L',L ' ) — ^ - 1.28 1 + <T It remains toe evaluate the reduced matrix element to determine completely the F coefficients. However, the reduced matrix element contains many factors which are of no importance to this discussion. Such factors are radial terms, and 14 electromagnetic normalizations. We desire to normalize the electro-m agnetic wave functions in such a way that W(0) = 1 when a(M^) = (21, + l ) " ^ . When this uniform distribution occurs, all the B vanish 1 v except for Bq which goes to unity. Since Pq is also unity, we require F q to be unity as well. An explicit expression for the Racah coefficient with a zero argument is W(a,b,c,d;e,o) = (-) e- C" d 6^ [ (201) (2d+l)] ~ 1 / 2 . 1.29 Inspecting equation 1.27 in the light of this identity, one sees that the reduced matrix element-with v=0 andL=L' must, be unity. Before calculating the matrix element between electromagnetic states, i t is necessary first to calculate it between vector spherical harmonics, as defined in equation 1.5. This in turn will reduce to the matrix element of one spherical harmonic between two other spherical harmonics. The value of this matrix element is given by a standard theorem for these functions: /YA'M. Y v p YXM d f l = t ( 4 ^ 2 A ' + 2 ^ 6M',M+y " 1.30 The matrix element of a spherical harmonic between two vector spherical harmonics can then be written: ^ T L ' A - • M 'YV O I T L A M > = Z C ( A ' , l , L ' ; M - y , y ) C ( A , 1 , L ' ; M - y , y ) C ( A , v ,A ' ; M - y , 0 ) y 15 where equations 1.5 and 1.30 have been used together with the orthonorraali of the spherical basis vectors. The factor 4ir has been dropped. In equation 1.31, the sura over y contains three Clebsch-Gordan coefficients. The Racah identity of equation 1.23 can be used to replace the sum by one ClebschTGordanand one Racah coefficient; then by combining the remaining .Clebsch-Gordan coefficient with the Racah coefficient, a new sum, independent of M, can be produced. The transitions involved are: E C(A',l,L';M-y,y) C(A,l,L;M-y,y) C(A,v,A ';M-y,0) = (_) A- V- L' + 1 y [(2L+1)(2A'+1)]1/2 W(L,A,L',A';l,v) C(L,v,L';M,0) 1.32 followed by a Racah permutation W X ' L , A ; L : , , , A ' ; 1 , V ) = W ( A , L , A ' , L ' ;1 ,v) 1.33 and transition back to a sum W(A,L,A',L';l,v) C (A,v,A';0,0) = (-) 1 + L A' V [(2A+1)(2L'+ 1)] 1 / 2 x E C (L 1,1 ,A ' ;y-y) C(L,l,A;y-y) C (L,v',l< ;pV03,": • 1-34 y Together, equations 1.32, 1.33, 1.34 applied to equation 1.31 can give the reduced matrix element for vector spherical harmonics: x E C(L',l,A';y, -y) C(L,l,A;y,-y) C(L,v,L';y,0). y 1.35 16 At this point, a comment can be made about parity restrictions. The parity of the vector spherical harmonic is given by the integer A as (-O^^. The electromagnetic wave functions are constructed from those vector spherical harmonics of the same parity; thus any discussion of the parity of the vector spherical harmonics applies also to the electro-magnetic functions. In the sum in equation 1.35, changing the sign of the index \i changes the summand by, at most, a sign. The change is (-)^+* + V . Thus, i f A+A'+v is an odd integer, the sum, and of course the whole matrix element vanishes. Let us investigate the available parities in nuclear gamma decay. When the initial and final nuclear states have definite parity, as they do in most cases, a l l the allowed multipoles of the gamma ray must have the same parity, since electromagnetism conserves this quantum number. This means that A and A' have the same parity and that v must be even. The restriction to even values of v means that W(6) con-tains only even ordered Legendre polynomials; that is, only even powers of cose. As a consequence, W(ir-0) = W(0); the gamma rays exhibit reflec-tion symmetry. However, there exist a few nuclear states of mixed parity. In these situations, odd values of v can arise from the interference of multipolarities of opposite parity. See, for example, Krane et^  al. (1971). To complete the derivation of the reduced matrix element, we note that i t is zero when the parity condition is not met, (that is, when A+A'+v is an odd integer) and we shall limit our further discussion to those cases in which the parity condition holds. The summand in equation 1.35 is thus symmetric in y, so that only the u=0 and u=l terms need be considered. 17 We are now ready to look at the matrix element of a spherical harmonic between two electromagnetic wave functions. First we shall write equations 1.4 in a more compact form using the following algebraic express-ions "for certain Oehs#iQ3orctan cco;e£:'£rd.rents: C(L,0,A;0,0) = 6 X L 1.36a. C(L,1,A;0,0) = 0 ( A = L ) 1.36b = [l^-]1 7 2 (A=L+1) 1.36c = - [2LTT] 1 / 2 (X=L-LJ • 1 - 3 6 D With these expressions we may summarize equations 1.4 as + 1 L+o Ar..(p) = Z i p C(L,p,L-s;0,0) hT TT . _ 1.37 IM K^ J ._ , ^ >r> L+s L L+s M s=-l p = 0 for magnetic multipoles = +1 for electric multipoles. The spherical Hankel function, of the first kind, used in the above equation tends to an outgoing wave at distances large compared to the wave-length. The asymptotic expression is iKr \CKr) - i " w . 1.38 By including the radial dependence in the normalization, one may replace by i ^ n eq U ation 1.37. The reduced matrix element ( L ' | | v | | L ) can now be found from equations 1.35 and 1.37. It contains two sums of the form in equation 1.37, and one of the form in equation 1.35. 18 Let us now examine one of the sums over s for u=0 in order to show that i t vanishes. Consider Z ( - ) S i L + P i L S ~ 1 C(L,p,L-s;0,0) C (L, 1, L+s ; 0,0) . This sum is over s=l,0,-l. If p=0, only s=0 contributes (by equations 1.36a) and the second coefficient vanishes, by equation 1.36b. If p=l, only s=+l and -1 contribute; the summand is antisymmetric between these two values because of the factor i , causing the sum to vanish. The u=l term may now be examined. The sums over s have this form: 2 (_)s i L + P i'L-s- 1 c(L,p,L-s;0,0) C (L,l,L+s"; 1,-1) = s Z i p + S _ 1 C(L,p,L-s>;0,0) C(L,l,L+s;l,-l) . 1.39 s To evaluate this, one needs the Clebs;chsGoManeexpEessipns in equations 1.36, as well as the following expressions: C(L,1,X;1,-1) = 1//2 X=L 1.40a = t2C2lTlT]1/2 A = L + 1 1'4 0 b [2 Tj L +TT ] 1 / 2 A = L " 2 • 1 A O c In equation 1.39, i f p=0, then s=0, and the sum, by equations 1.36a and-1.40a, is -i//2. If p=l, then s=+l and -1, and equations 1.36c, d and 1.40b, c, show that the sum is s t i l l -i//2. Because the matrix element contains the product of two such sums, one a complex conjugate, and because 19 the u = -1 term equals the u = 1 term, the result of all sums is unity. The reduced matrix element can now be written. a ' lM lD - t( ^ i ) ( 2 u i ) ] 1 / 2 c ( l f V f l . . l i 0 , 1.41 ( - ) L + 1 ( 2 L + 1 ) 1 / 2 C(L,L',v;l,-l) That equation 1 .41 has the desired normalization is seen by re-calling that C(L ,0,L ;1 ,0) = 1 . The final form of the F coefficient is found by putting equation 1 .41 into equation 1 . 3 7 . FV(L,L') = *2 1 [ ( 2 L + 1 ) C 2 L ' + 1 ) ( 2 I 1 + 1 ) ] 1 / 2 C(L,L',v; 1 , - 1 ) x W C I J . L . I J . L ' ^ . V ) . 1 . 4 2 Equation 1 . 4 2 is valid for both parity conserving and parity violating situations. When the electromagnetic multipoles of angular momenta L and L' have the same parity, v must be even; i f opposite parity, v must be odd. We haye now derived the angular distribution formula, equation 1 . 8 , i n terms of the populations of the initial state of spin 1^. But it is quite normal that one does not know these populations directly, but rather those of an earlier state in the decay. In figure 1 , it is the state of spin IQ that has been oriented; the populations of the state of spin 1^ must be deduced from the populations of the first state. Let B' be the B coefficient for the spin I, state. It is this v R 1 coefficient which appears in equation 1 . 2 6 . We wish to find B^  in terms of B^  the orientation parameters for the spin IQ state. 20 Let aCM )^ be the relative populations of the ini t ia l state, and a'(M ) those of the intermediate one. The ini t ia l density matrix is now <I M'lpll M > = a(M ) 6., . . . . 1.43 o o1 1 o o o MM' o o The init ia l nuclear state 11 M > decays to a combination of 1 o o intermediate nuclear and radiation states with the same angular momentum properties as the ini t ia l state has. 11 M > Z aT C(I,,L ,1 ;M. ,M-M1) | I.M. >\$T M M > . 1.44 1 o o L ^ 1' o o' 1' o 1 1 1 1 ' L ,M -M, M1 o' o 1 E o So far, there has been no difference between what has just been stated and what was stated at the beginning of the angular distribution derivation. At this point, we ask, not for the probability that the radiation be at a certain position, but for the probability that the nucleus is in a particular state |l^M^>, regardless of the radiation. This projection operator is LM The probability of finding the nucleus in the intermediate state I1M1> is 21 a'CMj) = Tr[p ^ ( I ^ ) ] = E <I M Ipll M' >< I M' I I.M. > | <j>T ..><<))... I <I1M1 11 M > »„ o o | K | o o o o ' l l |YLM LM 1 1' o o M M' o o L M = E a C M p l a J ^ C U ^ L , I^Mj ,M Q -M 1 ) | 2 . 1.46 L M o Since equation 1.46 is a sum over pure multipoles, we may investi-gate each one separately and do the sum afterwards. For the multipole L Q , the orientation parameter^, B^ , calculated from the intermediate populations is B' = (2v+l) 1 / 2 E C C I ^ v ^ j M ^ O ) a 'CMp M l = (2v+l) 1 / 2 E C C I ^ v . I ;M ,0)|C.(I ,L , I Q ;M ,M -M ^ | 2 a(MQ) . 1.47 M..M 1 o The Racah coefficient identity in equation 1.23 may be applied to the sum over M^ . „ I..+I -L +v 1 E C(I1,v,I1;M1J0)|C(I1,L,Io;M1,Mo-M1)|Z = (-) 0 ° [ (2^+1) (2IQ+1) ] A / M l x C[Io,v,Io;Mo,0D W C I 0 , I L F I 0 , I I ; L o , v ) . 1.48 With the help of equation 1.48, B^  can be written B* = (2v+l) 1 / 2 U E C(I ,v,I ;M ,0) a(M ) v ^ J v o' ' o' o  J K o J M o = U B V V 22 I +1 -L - v ,„ where U = (-) ° 0 [(2I1+1)(2I +1)] 7 W(I ,1 -L , v ) . 1.49 V L 1 O O 1 O 1 0 If several multipoles contribute to the transition, the resultant U is an incoherent sum over their individual coefficients, by virtue of equation 1.46. This completes the derivation of the primary terms in the angular distribution formula. Of practical interest is a further term to correct for the finite solid angle subtended by the radiation detector. This geometric factor is given in the next section. 1.4 Solid Angle Correction Because the radiation detectors do not look at a single value of 8 i t is necessary to integrate W(6) over the face of the detecting surface. This process involves only the Legendre polynomials, replacing each P^(cos6) by g^P^fcose), where g^ is a function of the geometry. Figure 2 illustrates the notation to be used for a circular detector. Let 0 be the origin, the source of radiation, C the centre of the detector face and A the point on the upper circumference of the detector face such that A is in the plane containing the z axis and the line C 0. The latter has length R. The angle between C 0 and the z axis is 0 ; the half angle subtended at 0 is a. Any point P on the detector face can have the co-ordinates § and 8, where <j> is the angle between P C and C A, and 3 the angle between P 0 and C 0. The Legendre polynomials must be integrated over a l l points P. To do this, one must know the angle 6' between the line P 0 and the z axis. The length of P 0 is R/cosg, and the length of P C is R tan 3. The z component of the point P is R cos8 + (R tan3)cos<j>sine. Dividing the 23 Figure 2 Solid angle correction 24 z component by the length of the line P 0 gives the cosine of 01. cos0' = cosgcosS + sin3sin0cos(j>. 1.50 By the addition theorem of Legendre polynomials, when equation 1.50 is valid, one may write P (cose1) = P (cosg)P (cose) +2 E y ' m l ' , P1" (cosg)Pm(cose)cos m*. v v v , fv+m)! v vv J r m = 1 1.51 Averaging the Legendre polynomial over the detector face yields , 2TT a 7 a - J •' d<j> ,/ d3 siri3 P (cose1) = P (cos0) =^ j P (cos3)sin3d3 O J O J 0 *> a where Q = 2ir / sin3d3 0 " = 2Tr{l-cosa)'. 1.52 The higher terms vanish because cos m<t> integrates to zero. Equation 1.8 is corrected for solid angle by rewriting it as W(0) = E B U F g P (cos0) v ' V V V V V V 1 a g = -j-, r- / P (cos3)sin3d3. 1.53 & v (1-cosa) ' v v J For second and fourth order Legendre polynomials, g2 = j cosa (1+cosa) 1 2 g^  = g- cosa (1+cosa)(7cos a-3). 1.54 25 1.5 Methods of Orienting Nuclei Having established that a collection of nuclei whose spins are not randomly oriented can produce decay products with a non-uniform angular distribution, we must now illustrate how the nuclear spins may be oriented. Before doing so, however, i t is necessary to explain some terminology. An ensemble of nuclei is said to be oriented i f the relative populations of the magnetic sublevels are not equal; that is, i f a(M) ^ (21+1) ^ . Furthermore, i f a distinction exists between up and down, that is, i f a(M) ^ a(-M) for at least one value of M, the nuclei are said to be polarized. On the other hand, i f the nuclei are oriented, but with up and down symmetry (i.e. a(M) = a(-M) for all M), then the system is said to be aligned. In the case of alignment, the odd orders of the orientation parameters vanish, since they are composed of the expectation values of odd powers of M. It is often useful to note whether a method of orientation produces polarization or only alignment. In this work static nuclear orientation is employed; that is, the nuclear orientation degeneracy of the Hamiltonian is broken and, following thermal equilibrium, the relative populations of the nuclear substates are given by the Boltzman distribution: a(M) « exp(-E(M)/KT). In this scheme, "reorientation" processes must exist to allow thermal equilibrium to arise. Nuclei caiupossess the.follgwingcelectromagneticemoments: electric charge, magnetic dipole moment (I >_ 1/2), and electric quadrupole moment (I >_ 1). The interaction between the charge and an electric field cannot orient the nucleus; the interaction between the magnetic moment and a magnetic field can polarize; and the interaction between the electric 26 quadrupole moment and an electric field gradient can align. Moments of higher order are "usually negligible. Let us examine the contribution from these moments to the Hamilton-ian. Reviews of these methods are to be found in Blin-Stoyle and Grace (1955) and DeGroot, Tolhoek and Huiskamp (1965). i . Brute Force The most direct way to orient nuclei is to apply a large magnetic field. The additional term to the Hamiltonian is ->• ->-E = -H-y = - ^  H-l = - Y HM 1.55 where H is the magnetic field, I the nuclear spin, and y the nuclear magnetic moment. The last line of equation 1.55 assumes that the quantization axis is the axis of the magnetic field. With this interaction the nuclear populations are a(M) = Z"1 exp(BM) Z = E exp(BM) M = sinh 3(1+1/2) sinh;.-> 3/2 1.56 where 3 = yH/IKT. 27 To achieve appreciable orientation B must be of the order of unity. -24 For magnetic moments of the order of the nuclear magneton (5.05 x 10 7 ergs/gauss) one requires H/T to be 3 x 10 gauss/kelvin. At room tempera-9 ture, H would have to be about 9 x 10 gauss, a rather large value. Even -2 5 at low temperatures (T ~ 10 K) the field must be about 3 x 10 gauss. Dabbs et al. (1955) used this method to orient ^^Tn. i i . SpincHaffii^tpnian -So far i t has been assumed that the atoms are free. This is not true in general, for in crystals the atoms may feel very strong forces. For a paramagnetic crystal with axial symmetry about the z axis Abragam and Pryce (1951) have derived the effective spin Hamiltonian X = [g„ H S + g,(H S + H S )] + D[S2 - is(S+l)] + AS I + B(S I + S I ) L 6 I I z z 6 i xx y y J L z 3 z z xx y y + Q[I 2 - j 1(1+1)] - ^ -H-T . 1.57 The electron states are represented by an effective spin S, such that 2S+1 equals the number of spin levels. The first term is the interaction of the electrons with the external magnetic field, which has different strengths along and perpendicular to the z axis; the D term is the splitting of the electronic levels by crystalline fields; the A and B terms are the hyperfine coupling of the electronic and nuclear spins; the Q term is the nuclear level splitting caused by the crystalline electric field gradient; the last term is the external magnetic field acting on the nuclear magnetic moment. Different methods of orienting nuclei use various terms in this Hamiltonian. 28 i i i . Gorter-Rose Method The method proposed by Gorter (1948) and Rose (1949) employs the hyperfine field in a paramagnetic material. A field of a few hundred gauss is needed to saturate the electronic magnetic moments through the g terms of equation 1.57; then, the nuclei are made to follow the electrons by the A and B hyperfine terms. The advantage of this procedure is that one can produce effective fields at the nucleus a thousand times stronger than the external field. Ambler et al_. (1953) have done the first experi-ments with this method. iv. Bleaney Method The electronic spins can be ordered by internal as well as by external means. In the method of Bleaney (1951) the D term in equation 1.57 splits the electron levels, and thus the nuclear levels through A and B. No external field is needed; only low temperatures are required. The first successes of this method were performed by Daniels et^  al_. (1951) and Gorter et_ al_. (1951). v. Pound Method Pound (1949) suggested using the nuclear quadrupole interaction. Like the Bleaney method, no external field is needed, but one does require large electric field gradients (^  10 e.s.u.). Uranium and neptunium have been so oriented by Roberts, Dabbs 131 and Parker (1958), and I has been aligned by Johnson, Schooley and Shirley (1960). 29 v i . Ferromagnets In ferromagnetic material, the strong exchange force between the electrons causes their spins to be completely polarized along some axis, say the z axis. One may then set and to zero. Marshall [1958) has shown that the hyperfine interaction behaves like an effective magnetic field at the nucleus given by AT H = - — S + H . 1.58 eff p As a result the nuclei within a domain are automatically polarized when the temperature is sufficiently low. Grace et al. (1955) proved this method by orienting ^Ce in a single crystal of cobalt metal. In multidomain samples, an external field is s t i l l necessary to magnetize the whole specimen so that a single hyperfine field is present. Fields of a few kilogauss are required to achieve this in iron, whereas over ten kilogauss may be necessary to saturate cobalt. This method, which is the one employed in the present work, is not restricted to magnetic atoms. Samoillov et al. (1960) oriented dia-magnetic atoms in ferromagnets. The nucleus of a non-magnetic atom in a ferromagnetic host sees a hyperfine field caused by the polarization of the conduction electrons. The systematic behaviour of the field for different impurities in iron has been given by Shirley and Westenbarger (1965). Calculations of the field have been done by Campbell (1969). Using this effect, the number of nuclei which can be oriented is greatly increased. One is not restricted to using the hyperfine methods only for magnetic atoms; i t may be used for any atom which can be introduced into the ferromagnetic host. 30 v i i . Antiferromagnets The case of antiferromagnetism is similar to that of ferromagneT tism, except that the exchange forces align rather than polarize the electron spins. Ear sa ssingle crystal "with;es^ mmetry.tl:ow.er than cubic no applied field is heeded; -thev electron" and nuclear spins align along a crystalline axis as the temperature is lowered. 54 Daniels, Giles, and LeBlanc (1961) observed orientation of Mn and 6 0Co in crystals of MnCl2-4H20, CoCl2-6H20, MnSiF6-6H20, and Co(NH4)2(S04)2-6H20. v i i i . Dynamic Nuclear Orientation There exist methods for producing non-uniform populations of the nuclear magnetic states without low temperatures. In dynamic nuclear orientation, optical and microwave radiations excite atomic transitions; the subsequent emission of radiation by different channels can create nuclear orientation. For a complete discussion of these methods see Jeffries (1963). 1.6 Low Temperature Production A description of the low temperature apparatus is given in a later chapter; this section will only give an outline of the principle used for cooling. Preliminary cooling down to one kelvin is accomplished by liquified gases. Liquid nitrogen can cool to 77 K and liquid helium at one atmos-phere to 4 K. Pumping the liquid helium to a pressure of 1 mm of mercury 31 further reduces the temperature to 1 K. At this point, the system is ready for the final plunge. The last step is an adiabatic demagnetization of a paramagnetic salt. At this low temperature, the entropy remaining in the salt is mainly that of the ionic spins. Since entropy is a measure of disorder, it may be reduced by ordering those spins in a magnetic field. This process can be illustrated in figure 3. Initially, with no external field, the electronic spin states are only slightly split by the inter-atomic interactions; the splitting is much less than KT, and the states are very nearly equally populated, as indicated in figure 3 by the thickening of the lines. When the external field H is applied, the levels are widely split; after thermal equilibrium is re-established, the lowest levels are the most heavily populated. Thermal contact to the 1 K temperature bath is broken and the field is removed adiabatically. In an adiabatic process, the entropy remains constant; that is, the populations do not change as the level splitting is decreased. A low temperature is-produced7: "f sp '.J picduc^c The final temperature after demagnetization is determined by the inter-atomic interactions. It is possible to reduce these forces by separating the magnetic ions by non-magnetic ions and water of hydration. Some common salts used are ferric ammonium alum [FeNIfy(SO4)2"I2H2O], manganous ammonium alum [MnCNfy)2(SO4)2"ol^O], chrome potassium a'iumi'fCrK'CSO^ ) ^ .lZH^O]and r cerium ^magnesium nitra'tecJge^g'j'CNOp • 2'4Hg'Of •:. I- C , , A 2«.:-o Further details regarding this and other methods of cooling can be found in the manual for low temperature physics by White (1968). 32 Figure 3 Level splittings and populations during adiabatic demagnetization 33 1.7 Other Experimental Methods In the standard method of orienting nuclei the anisotropy of the emitted radiation is observed as a function of temperature and/or angle. Equation 1.8 contains the temperature only in B terms, which are functions of 3 = uH^^/IkT. The hyperfine interaction yHg^^ may be so obtained. In addition, once the B terms are known i t is possible to evaluate UF, which can give information about spin assignments and gamma ray multi-polarities in the nuclear decay scheme. Other methods which can determine these quantities are: i . Angular Correlations An experiment with a basic idea similar to that of nuclear orientation is angular correlation. When two radiations are emitted in sequence the angle between them has a distribution W(0) = E F (1) F (2) P (cose) 1.59 V where the F terms are the same ones used in nuclear orientation. Spins and multipolarities can be found in this way. The advantage of this method is that i t is performed at room temperature, i r. : . i . ^ A review of this method is given by Frauenfelder, and Steffen (1965) ."v.. . .- • ' • . i i . Perturbed Angular Correlations If in a sequence of nuclear transitions the intermediate state is long-lived, the precession of the nucleus in the hyperfine field can 34 be observed in the time dependence of equation 1.59. A periodic re-orien-tation combined with the exponential decay must be included in this equation to give the correlation as a function of the time between transi-tions. The precession frequency so determined gives the hyperfine inter-action. A ful l treatment of this technique can be found in Karlsson ejt al. (1964). i i i . Internal Conversion A nuclear gamma transition can occur not only by emitting electro-magnetic radiation (the gamma ray) but also by emitting inner orbital electrons. This process is called internal electron conversion. The ratio of converted electrons to gamma rays does not depend on nuclear matrix elements but only on the transition energy and multipolarity. The theory of internal conversion is discussed by"Rose (1965) and experiments are reviewed by Ewan and Graham (1965) . iv. MBssbauer Effect The recoil-free absorption of gamma radiation (Mo'ssbauer , 1965) can be used to determine the hyperfine splitting of the nuclear levels which is seen as structure in the gamma peak. v. NMR Nuclear magnetic resonance observes the hyperfine splitting by the absorption of electromagnetic radiation at the resonant frequency •'di = yH ^/ftl. The first "bbse-r-vation'Of-.this'effect: dn ferromagnets was reported by- Portis and Gossard -(I960) whorrdejtee;teed an\ NMR. signal in cobalt.- The "signal".was- very strong duetto enhancement': in the domain walls".- -l •/rf,-.. • • - ; > . : : - - . . . 35 Impurities in ferromagnets can be studied by NMR, but the concen-tration must be > 0.1% in order to observe a measurable signal. v i . NMR/Nuclear Orientation Trace impurities in ferromagnets can be studied by a combination of NMR and nuclear orientation. A small coil is placed around the sample so that i t produces an r.f. field perpendicular to the orientation axis. The thermal equilibrium distribution of nuclear populations is destroyed by putting in radio-frequency power at the hyperfine resonant frequency. The observed effect will be a decrease in the anisotropy. This will be restored by spin-lattice relaxation upon removal of the r.f. field. This method has the advantage of the high accuracy found in resonance experiments, this accuracy being limited by the line width observed. Typically, hyperfine fields can be measured to MD.03%. The resonant destruction of nuclear orientation was first observed by Matthias and Holliday (1966). Some problems with the method are r.f. heating and the prepara-tion of samples with the thickness of the skin depth. 36 CHAPTER TWO EXPERIMENTAL APPARATUS AND PROCEDURES 2.1 The Cryostat The low temperature apparatus is described in detail by Gorling (1970). A brief description of the'cryostat which is illustrated in figures 4 and 5, is given here. Liquid nitrogen at 77 K and liquid helium at 4 K were contained in two glass dewar vessels. Immersed in the liquid helium were the superconducting magnets and the brass jackets of the inner cryostat. Two superconducting solenoids were necessary. The primary one provided the large field necessary for cooling by adiabatic demagnetiza-tion. It was made from niobium-titanium wire wound onto a solenoid of outer diameter 10 cm., inner diameter 5 cm., and length 18 cm. It provided a central field of 0.8 kG per amp and would take up to 62 amps, or 50 kG. The secondary magnet produced the field necessary to saturate the magnetization.of the ferromagnetic sample. It was constructed from superconducting niobium-titanium wire wound around a core of diameter 3.5 cm. and length 3.5 cm. The number of turns was 1700. Its maximum current was 25 amp (or 9.8kG). The inner cryostat consisted of three brass jackets the inner-most of which contained the paramagnetic salt pil l s and the radioactive specimen. The middle jacket held liquid helium which was kept at a temperature of 1 K by continuous pumping. This liquid was insulated from the 4 K helium by the vacuum (10 ^  torr.) in the outer jacket space. 37 PUMP LINES OUTER JACKET I LIQUID NITROGEN I A | i LIQUID HELIU , \ fcT] RETURN LINE f ) M S C A L E : C M I 1 L 0 5 10 15 OUTER DEWAR INNER DEWAR LIQUID HELIUM Figure 4 Outer cryostat 38 INNER JACKET PUMP LINE OUTER JACKET PUMP LINE GUARD PILL MAIN PILL COPPER HEAT CONDUCTOR BATH PUMP LINE NEEDLE VALVE OUTER JACKET SPACE 1K BATH INNER JACKET SPACE SUPER-CONDUCTING COOLING MAGNET POLARIZING MAGNET RADIO-ACTIVE SPECIMEN S C A L E : C M I 1 1 l _ l 0 2 4 6 8 Figure 5 Inner cryostat 39 High frequency vibrations could be a source of heating, either through mechanical flexing of the supports, or through currents induced by a conductor moving in a magnetic field. For this reason efforts were taken to reduce vibrations. The entire cryostat was supported on a dexion frame sunk into a cement base. The weight of the cement lowered the resonant frequency of the frame. To isolate the assembly from vibrations from the floor, the frame rested on springs or on a foam rubber pad,bothliof /whichfhad-equal effect. 2.2 Paramagnetic Salts and Thermal Contact i . Main P i l l In figure 5 i t may be seen that the innermost jacket of the inner cryostat contained two paramagnetic salt p i l l s . The primary p i l l was made of chrome potassium alum or cerous magnesium nitrate. The latter salt allowed lower final temperatures to be reached but the former had a higher heat capacity. With chrome potassium alum one expects to reach a temperature of ^12 mK, the N6el point for this salt, but with cerous magnesium nitrate one can obtain temperatures down to 2^ mK. The lowest temperature that was reached with the present system using the nitrate was 8 mK. i i . Guard P i l l and Shield The secondary paramagnetic salt p i l l consisted of manganous ammonium alum. The purpose of this p i l l was to reduce heat conduction down the supports to the primary p i l l . In later experiments a guard 40 shield (not shown in figure 5) was introduced. This shield was a thin (0.1 mm.) copper sheet wrapped around the p i l l and sample assembly such that it was in contact with the brass wall of the guard p i l l but did not touch the main p i l l or the sample. The role of the guard shield was to adsorb the helium gas remaining in the inner jacket during demagnetiza-tion and to transfer the heat of adsorption to the guard p i l l . The two pills were separated by a nylon tube. Short bits of nylon wire projecting from the guard shield kept the assembly from touching the wall of the jacket. The whole assembly was suspended from the top of the jacket by two 1.5 cm. lengths of stainless steel tubing, 0.56 mm. in diameter. i i i . Thermal Contact Because a magnetic field of several kilogauss had to be applied to the radioactive sample to achieve nuclear orientation, and because the low temperature was produced by the removal of a magnetic field from the paramagnetic salts, the sample and the salts had to be spatially separated and yet remain thermally connected. This was accomplished by a folded copper sheet with one end embedded within the main p i l l and the other end extending 9?cm. below the p i l l . The radioactive sample was attached to the lower end of the copper. If the sample was a metal f o i l , i t was soldered to the copper. If it was a crystal, i t was smothered in Apiezon N grease and wrapped within part of the copper f o i l ; needless to say, in the latter case a much poorer thermal contact was obtained. iv. P i l l Preparation The paramagnetic salt p i l l s were made by mixing finely ground salt with an equal mass of glycerine. The glycerine served to make thermal contact with the copper and to form a glass-like solidaat low temperatures. The folded copper fins filled the entire length of the main p i l l . The salt was packed between the folds to maximize the contact area, which 2 was about 900 cm . The total amount of salt and glycerine was about 120 grams. Outside the p i l l , the fins were pressed and glued together. The surfaces were previously covered with an epoxy paint to reduce heating from eddy 'Currents. 2.3 Low Temperature Procedure After the cryostat had been pre-cooled to 4 K by liquid helium the large superconducting solenoid would be charged to 50 amps (40 kilo-gauss) in five minutes. A superconducting wire across the magnet leads (the persistent switch) could be made normal (i.e. possessing resistance) by a heating wire around i t . While the magnet was being charged, the switch would be normal (an open circuit); to put the magnet into the persistent mode, the heater was turned off, the switch made superconduct-ing, and the magnet short-circuited. After the power supply was turned off the current remained in the magnet but completed the circuit through the switch rather than through the power supply. The heat of magnetization in the paramagnetic salts was conducted away by helium gas. The pumping of the outer jacket space would be stopped, and the pumping of the 1 K bath would begin. By pumping the liquid helium in this space to 0.2 torr., the temperature was reduced to 1.05 K. Helium gas was added to the inner jacket space as required to maintain thermal contact with the bath. After the paramagnetic salt p i l l s had ceased to cool, as indicated by a carbon resistor embedded in one of them, the inner jacket space would be pumped down to 10 ^  torr. When this had been achieved, the pumping was stopped. The magnet was discharged by restoring the current from the power supply to 50 amps, making the persistent switch normal, and reducing the power supply current to zero in ten minutes. The 7 - 6 pressure in the inner jacket space normally f e l l to about 10 to 10 torr. The polarizing solenoid was now charged and maintained in the persistent mode. The counting of the gamma rays could then begin. 2.4 Gamma Ray Detection Once the radioactive nuclei had been oriented one could observe the anisotropic emission of gamma rays. To do this, devices sensitive to such radiation were positioned about the source. i . Types of Detectors Two types of gamma ray detectors were used. Initially two sodium iodide detectors were employed because of their high efficiency, but as the gamma ray spectra became more intricate the higher resolution of lithium-drifted germanium detectors became necessary. The sodium iodide detector consisted of a crystal of Nal(Tl) which was 2 inches in diameter and 2 inches long. It was in optical 43 contact with a photomultiplier (RCA 6342 A) biased at 1100 volts. Because this detector was extremely sensitive to magnetic fields, i t was shielded by mur metal and an iron tube along its entire length. Such protection screened only the field of the polarizing solenoid; when the large magnet was charged these detectors were inoperable. The resolution (full width at half maximum) of the 1.33 MeV photopeak of ^Co obtained in this system was 60 keV. The separation of the two lines in ^Co is 159 keV; thus these lines are not quite fully separated. The Ge(Li) detectors consisted of a crystal of lithium-drifted germanium kept at the temperature of liquid nitrogen at all times. The first of these detectors had a trapezoidal crystal with an active area 2 of 11.2 cm and a drifted depth of 1.1 cm. The bias on the crystal was 2 1000 volts, negative. The relative photopeak efficiency for this Ge(Li) detector was 3.9%. The peak-to-Compton ratio was 9 to 1, and the resolu-tion of the 1.332 MeV peak was 5.0 keV. The second Ge(Li) detector had a cylindrical crystal. It was 3.8 cm. in diameter and 4.1 cm. long. The bias was 3000 volts, positive. In this case, the relative photopeak efficiency was 6.7% with a peak-to-Compton ratio of 25 to 1 and a FWHM of 2.4 keV. The relative photopeak efficiency of a Ge(Li) detector is defined in this work as the efficiency relative to that of a 3 in. x 3 in. Nal detector for the 1.332 MeV peak of 60co when the source is 25 cm. from the detector. 44 i i . Handling of Detector Output The pre-amplifiers of the detectors produced a voltage pulse _7 with a rise time of the order of 10 sec. and a decay time of the order of 10 ^  sec. The height of the pulse was proportional to the energy of the detected gamma ray. Sorting the pulses according to their heights therefore produced the energy spectrum of the radiation. Figure 6 shows the path of the detector signals. The pre-amplifier pulse was shaped and enhanced by a linear amplifier (Canberra Industries model 1410). The height of the output from this unit ranged from 0 to 10 volts, and was also proportional to the energy of the origi-nal gamma ray. Pulse shaping was achieved by a choice of appropriate differentiation and integration time constants to produce a bipolar out-put. For the Ge(Li) detectors, the best resolution was obtained using 2 u sec. for the first differentiation, 1 u sec. for the integration, and 7 u sec. for the second differentiation. For the Nal detectors, the bipolar pulse produced by double delay lines was used. The output from the linear amplifier was analyzed in two ways: by single channel analyzers (SCA) and scalers, and by a multichannel analyzer. The single channel analyzers produced a logic pulse of fixed height whenever the height of the input f e l l between two pre-set levels. The logic pulses were counted by a scaler (Canberra Industries model 1473). The linear amplifier signal could be sent to several SEAs, each set to select a different gamma ray. The resulting array of scalers was controlled by a timer/sealer (Canberra Industries model 1492) which stopped them after a pre-determined time. The scalers were used during an experiment to monitor the most interesting gamma rays. 45 DETECTOR PRE-AMP SIGNAL • FROM SECOND LINEAR AMP LINEAR AMP DUAL INTEGRAL DISC. MIXER/ REJECTOR TIMING SCA J T . TIMER/ SCALER SCALER ROUTER MULTI-CHANNEL ANALYZER Figure 6 Flow diagram of detector output 46 The more important path for the signals was the one leading to the Nuclear Data model 2200 multichannel analyzer. This unit contained 512 channels and had a digitizing rate of 50 MHz. With i t the entire spectrum could be recorded by pulse height analysis. Since two detectors were normally used during an experiment i t was desirable to be able to accumulate both spectra simultaneously in separate sections of the memory. Three additional units were necessary: a dual integral discriminator (Nuclear Data 531), a mixer/rejector (Nuclear Data 521) and a router (Nuclear Data 538). The analogue signals from the two linear amplifiers were presented to the dual integral discriminator. When either of the inputs exceeded a pre-set level, a corresponding logic pulse was sent to the router. The router generated signals for the analyzer to determine which half of the memory was to receive the analogue pulse. The dual discriminator also sent two signals to the mixer/rejector. One was the sum of the signals at the two analogue inputs; the other was the sum of the logic signals which went to the router. The mixer/rejector could therefore determine whether the analogue signal i t received was a coincidence between the two detectors. If so, that signal had to be suppressed. The outputs of the mixer/rejector went to the analyzer input. One Was the analogue pulse, which was de-livered to the analogue-to-digital converter, and the other was a logic pulse fed to the coincidence input of the analyzer. If this logic pulse was missing, the analogue signal was .r.ejiect'ed. The multichannel analyzer would accumulate in this way for a pre-set time. Afterwards the contents of the memory could be displayed on an oscilloscope and/or put onto paper tape in 40 sec. by a high speed Teletype punch. The memory would be erased and a new count begun. The 47 paper tape was then the only record of the last count. After the experi-ment, the contents of the paper tapes were transferred to magnetic tape and processed by the IBM 360 computer. i i i . C Counting Procedure The two most convenient angles for observing the change in gamma ray intensity when the nuclei were oriented were along the axis of orien-tation (the axial direction) and perpendicular to i t (the equatorial direction) . Detectors placed in these positions measured W(0) and W(TT/2) respectively. Since the orientation axis was a vertical one, the axial detector was placed beneath the cryostat and the equatorial detector beside i t . Since the two Nal detectors were identical it did not matter at which station either one was put. That was not true for the Ge(Li) detectors. The axial detector was 20 cm. away from the source while the equatorial was only 10 cm. away. It was therefore advantageous to put the more efficient detector at the axial position. The angular distribution If(8) is normalized to unity at high temperatures. One therefore measured W(0) and W(ir/2) by dividing the gamma ray intensity by its value when the specimen was warm. During the time preceding the demagnetization, as many counts as possible would be acquired so that after demagnetization, when the specimen was cold, the gamma ray intensities could be properly normalized. _2 After the specimen had been cooled to 10 K and the polarizing field had been applied, cold counts would be taken on the multichannel analyzer and on the single channel analyzers. The counts from the latter 48 were recorded in a log-book. A good indication of W(0) and W(TT/2) could be immediately provided by the single channel analyzers. Of special interest were the intensities of the ^ Co peaks, since they acted as an in situ thermometer, as will be described. But because the scaler counts included not only the gamma ray, but also any background of the same energy, the final results were those given by the computer analysis at a later time. The procedure followed while the specimen was cold depended upon the type of experiment being performed. When only a small anisotropy was expected counts would be acquired for as long as the temperature was below a desired level. Count times ranged from 1000 to 4000 sec. After the specimen temperature had exceeded the limit (three to six hours after demagnetization) the system would be warmed to 1 K and another magnetization would be initiated. However, i f the experiment consisted of observing changes in the anisotropy as a function of some other para-meter (applied magnetic field or temperature) several counts of 200 to 400 sec. would be taken at each setting. Counting would proceed until the specimen was too warm to be of use. Remagnetization and often a retransfer of liquid helium would be necessary. Warm counts would be acquired again during the few hours prior to the next demagnetization. They would also be accumulated at the very end of the experiment, provided that the liquid helium level was above the source. This was because the liquid absorbed a few percent of the gamma rays, and a noticeable increase in the equatorial intensity would be observed when there was no liquid between source and detector. 49 2.5 Paper Tape Analysis After the completion of a run the data consisted of a li s t of scaler counts in the log-book and a great deal of paper tape. The infor-mation stored on the tape was first converted to magnetic tape for analysis by a computer. i . Spectrum Analysis The scaler counts taken during the .experiment gave the first approximation of the observed anisotropy. As has been mentioned the background included in the count could produce errors. A constant back-ground tended to decrease the difference between the warm and the cold counts. But the background was not necessarily constant. Associated with the photopeak of a gamma ray is a broad Compton plateau of lower energy exhibiting the same anisotropy as the peak. Also, gamma rays emitted in one direction could be scattered, with reduced energy, into a differ-ent detector. This background would possess the anisotropy pertaining to the original direction of the gamma ray. The total background under a peak therefore depended in a very complicated way on the peaks of higher energy. The main component, however, was the Compton contribution. The anistropy of this unwanted segment could overwhelm that of the peak of interest, especially when only a small effect was present in that peak. In this event the scaler readings were not even a reasonable first approx-imation. The separation of the peak from the smooth background upon which i t stood was performed by our computer program SPECTAN reading the multi-channel analyzer output. The program could also detect and correct 50 displacements of the peak positions. For nuclei of short lifetimes a correction for the decay could be made. By analyzing the warm counts first, the cold counts could be immediately normalized. The statistical error was calculated assuming that the number of counts in each channel had a variance equal to that count- (the. Poissonf distribution.),. Up to five peaks from one detector could be analyzed at one time. i i . Background Calculation The assumption used to find the background beneath a peak was that the background spectrum was linear over a distance of about twice the total width of the peak. This assumption was far more realistic for peaks taken with Ge(Li) than with Nal detectors because the peaks of the former were so much narrower. Since only relative intensities were required in this work, this procedure for analyzing spectra was found to be satisfactory in most cases. The method of calculation is illustrated in figure 7 . The first step was to sum the channels within the specified peak window which pro-duced an uncorrected count equivalent to the scaler count. The second step was to skip several channels (the "gap" in figure .'7Q on either side of the peak window and to sum some more channels (the "width"). This second sum, divided by the number of channels contributing to it (that is, twice the "width") is the average background per channel be-tween the two background windows. Because the peak window is symmetrical-ly located between the background windows, the background in the peak window is the average times the number of channels in the peak window. 51 • • • e • « w 1 1 D T G H A P H WINDOW 4-* J 1 1 L O 5 10 15 20 25 30 WINDOW s 7 TO 22 GAP = 2 WIDTH = 4 UNCORRECTED COUNT 22 • I YIN) Ns7 BACKGROUND = Y(N) 4/ 28 z 25 Y(N) 22-7+1 2«4 Figure 7 Background calculation 52 The limits of the peak window, the number of channels in the "gap" and in the "width" were all variable. The corrected peak intensity is the uncorrected count less the background. The variance of this quantity is the sum of the individual variances. These variances were easily estimated since the uncorrected count and the second sum both possess Poisson statistics. The variances were taken as the sums themselves. Since the background is proportional to the second sum, its variance equals this sum times the square of the proportionality factor. If a correction for the decay was to be made it was applied to the background-free count. The time at which the spectrum had been taken was included on the magnetic tape, the half-life was given as in-put to the program, and the peak was increased to correspond to a time prior to the experiment. The variance was similarly adjusted. i i i . Normalization The program SPECTAN would be given the spectra to be analyzed and was told the number of warm counts. It analyzed these first so that i t would have the average normalization count and corresponding variance before i t started the cold counts. The intensities of the peaks in the cold spectra were produced in the same way as in the warm ones, but a further step was added. The intensity was divided by the normalization. The statistical error in the normalized count would be found by using the rule that the relative variance of a quotient is approximately the sum of the individual relative variances. This assumes that the dividend and divisor are independent (which they certainly were). 5 3 iv. Shifting Peaks An additional feature of the program SPECTAN was the detection and correction of the peak position. Adequate results were obtained only i f the peak spanned tens of channels, for the window could be shifted by only an integral number of channels. This condition was met by spectra taken on a Nal detector, but usually not by those from Ge(Li) detectors. However, i t was more often the Nal detectors that tended to shift . o r : : ^ — The program calculated the centroid of the peak window by giving to each channel a weight equal to its count. The centroid of the first spectrum was used as the standard. If a subsequent spectrum had the .centrbid--in-a;different channel, the peak window was shifted accordingly. The centroid was calculated for the new window and another shift was made i f necessary. The corrections continued until the cen-troid had the proper position within the window. There was of course the possibility that a divergent or oscillating situation would occur, thus a maximum number of attempts was specified beforehand. If centroid corrections were not desired, the maximum was put to zero. v. Program Output The computer print-out would show the results of these calcula-tions. For the warm spectra one would be given the uncorrected and corrected counts, the variance for the corrected count, the peak window, and the number of centroid corrections. After the last warm spectrum the totals and averages would be produced. For the cold spectra one would be given the same data and, in addition, the normalization and 54 their standard deviation errors. If the output was to be subjected to further computation, some of these results would be written into the computer memory. For both warm and cold spectra the program would write the uncorrected count, the decay correction factor, the corrected count, and its variance. This data could be used by programs written for specific experiments. 2.6 Thermometry: ^Cobalt Experiments In nuclear orientation experiments the problem of measuring the specimen temperature is solved by using the anisotropy of some known radioactive material. Such an isotope is ^ Co dissolved in iron. The magnetic moment and hyperfine field are known, the decay scheme has been completely determined, and a large anisotropy exists at the desired temperatures. Additional advantages are that the isotope is reasonably long-lived (half-life of 5.25 years) and that the two gamma rays have identical anisotropics. This thermometer was first used by Stone and Turrell ((M)6'2). i . Decay Scheme The decay scheme for ^ Co as given by Lederer, Hollander and Perlman (1968) is shown in figure 8. The ^ Co nucleus decays by emission of beta particles to an excited state of ^ N i . Two gamma rays are promptly emitted as the ^ N i nucleus decays to its ground state. The energies of the gamma rays are 1.173 and 1.332 MeV, and both are pure E2 transitions. 55 Figure 8 Co decay scheme 56 Using equations 1.10 and 1.12 (or the tables of Yamazaki, 1967) we find that for both gamma rays U 2F 2 = -0.42056 U 4F 4 = -0.24281. 2.1 n . Anisotropy and Temperature The temperature dependence of the anisotropy depends on the hyperfine field of Co dissolved in iron and on the magnetic moment of the ^Go nucleus. The hyperfine field is given by Koster and Shirley (1971) as -287.7 kG and the magnetic moment is given by Shirley (1971) as 3.754 nuclear magnetons. The relative intensity of the gamma rays in the axial and equatorial directions can be calculated from equations 1.8, 1.9 and 1.56: 6 uH/IkT a(M) Z exp(BM) Z sinh g (1+1/2) sinh B/2 B (2v+l) 1/2 I C(5,v,5;M,0)a(M) M v W(0) 1 + B 2U 2F 2 + B 4U 4F 4 W(TT/2) 2.2 57 In figure 9 are plotted W(0) and W(TT/2) as functions of tempera-ture. At 12 mK we have W(0) = 0.60 and W(ir/2) = 1.16. This anisotropy is large enough to serve as a good thermometer in this temperature region. i i i . Sample Preparation The first experiments with this apparatus were carried out on ^Co in iron as a test of the system. A carrier-free solution of ^CoCl,, with an activity of 1 mCi was obtained from Atomic Energy of Canada, Limited. Solution containing 2 uCi was placed on an iron f o i l of dimensions 1.0 cm x 1.0 cm x 0.03 cm. After the solution had evaporated to dryness, the iron was heated at 700°C in a hydrogen atmosphere for three days. To test that the diffus-ion of cobalt into iron had been successful, and to remove any surface activity, the specimen was immersed in HCl for one minute. The activity after this operation was seen to be slightly lower than it had been pre-viously, but after subsequent acid washes i t was constant. iv. The ^Co Experiment The objects of the experiments with ^Co were to see what temper-atures could be achieved, how long they would last, and how much current was needed in the polarizing solenoid to saturate the magnetization of the iron. These experiments were'done before the guard shield was used and before the Ge(Li) detectors were obtained. The salt used was chrome potassium alum. Two Nal detectors were used, one in the axial and one in the equatorial positions. Their distances from the source were 9.0 and 3.5 58 59 inches respectively. With a radius for the detector face of 1 inch, the solid angle factors of equation 1.54 are axial: a = 0. I l l g 2 = 0.991 g 4 = 0.970 equatorial: a = 0.286 g 2 = 0.940 g4 = 0.809 . 2.3 Counting times of 200 sec. were used for the cold counts, while the warm counts were usually accumulated for 1000 sec. An example of a 200 sec. warm count is given in figure 10. The current in the polarizing solenoid was increased from 0 to 5 amps at 1 amp intervals, and similarly decreased. Counts were taken at each current setting. In figure 11 is shown a plot of the anisotropy of the ^Co peaks as the current was increased and decreased. Saturation occurs around 3.5 amps. For this polarizing solenoid this current corres-ponds to a central field of 1.5 kilogauss. The quantity plotted in figure 14 is the anisotropy as is usually defined in the literature: _ W(TT/2) - W(0) W O / 2 ) ' Z' 4 More will be said about this function in section 2'.,7i. Figure 12 shows the plots of the temperatures of the ^Co against elapsed time since demagnetization. The temperatures were calculated by 60 2 0 0 100 h 6 0 C O SPECTRA, 2 0 0 SECONDS 60 AXIAL 4 0 2 0 J 2 Z < o CC a i a. w t-z O O 3 0 0 EQUATORIAL 4 0 8 0 1 2 0 160 CHANNEL NUMBER 2 0 0 2 4 0 Figure 10 Sample ^°Co spectra Figure 11 Co anisotropy versus polarizing solenoid current 211- 62 RUN 1 17 13 I M V) 5 z > UJ 25 UJ QC I— < CC UJ I 21 U J < S17 13 9 -I " I I . I I I RUN 2 0 3 0 100 1:30 TIME AFTER DEMAGNETIZATION (HOURS) 2:00 Figure 12 Co temperature versus time elapsed from demagnetization 63 taking a weighted average of the temperatures indicated by the axial and equatorial detectors, thus 2 2 T = °2 T L + G L 7 2 °1 + ° 2 2 2 2/ R 2 2, a = o2/ [ a1 + a2] 2.5 2 2 where a. is the variance of T. and a is the variance of T. The error l I bars in figure 12 indicate plus or minus one standard deviation. Figure 12 shows that, at this early stage, temperatures less than 20 mK could be maintained for over an hour and a half and that lower temperatures around 10 mK could last a half hour. In more recent experiments using a guard shield around a main p i l l of cerous magnesium nitrate a temperature below 12 mK was maintained for 7 hours. 2.7 The Term "Anisotropy" The usual definition of "anisotropy" is given by equation 2.4. As an average of the effect seen by the two detectors i t is undesirable because i t is not linear in both detectors. Even the difference WO/2) -W(0) places equal weight on both detectors, which may not be justified. For example, consider the case in which the fourth order term in equation 1.8 is zero. We then have W(0) = 1 + A 2 W(TT/2) = 1 - I A 2.6 6 4 where = ^ 2 ^ 2 ^ 2 * Clearly i - n this case a far better measure of the "effect" is the term k^. If both W ( 0 ) and W(TT/2) have the same error, then A^  estimated by the axial detector has half the error as A^ esti-mated by the equatorial. The axial result should therefore be weighted four times the equatorial figure. For these reasons very l i t t l e use will be made in this work of equation 2 . 4 . The word "anisotropy" will instead be used to refer to a non-uniform angular distribution. The "effect" seen by a detector at an angle 6 to the axis of orientation is e ( 0 ) = W ( 6 ) - 1. 2.7 The "effect" is the increase of the gamma ray intensity relative to its warm value. A negative effect is exhibited by ^Co in the axial direction. 65 CHAPTER THREE ORIENTATION OF 5 9Fe NUCLEI IN IRON 3.1 Introduction 59 The Fe nucleus has been oriented by Tschanz and Sapp (1970) in the hyperfine fields of paramagnetic Ce-Zn nitrate and Nd-Zn nitrate. An unexpectedly large anisotropy was observed, the temperature dependence of 59 which yielded a magnetic moment for the Fe nucleus of 1.1 +_ 0.2 u ^ . However, there are several uncertainties in this work, especially regarding the fraction of iron atoms at the double nitrate sites. A verification of their result by a different method would be useful. We 59 decided to orient Fe in iron since in this system there exist no problems about solubility. The hyperfine field is -333 kG (Hanna et al. 1960). 59 3.2 Decay Scheme of Fe 59 The decay scheme of Fe is illustrated in figure 13. The parent 59 nucleus beta-decays with a 45 day half-life to two excited statesLof Co which decays to the ground state emitting gamma rays of energies 1.292 and 1.095 MeV. The life-times of the excited states should be sufficiently short (compared to the nuclear spin precession time) to neglect intermediate state reorientation. Tschanz and Sapp (1970) consider the possibility of some reorientation but reach no definite conclusion. In any event, com-parison between our results and theirs can be made under the assumption that no reorientation is present. The main features of the decay scheme in figure 13 are obtained from Lederer, Hollander and Perlman (1968). The half-lives of the inter-66 1.292 MEV 1.095 MEV 0.0 M E V ' Figure 13 Fe decay scheme 67 mediate states, measured by Sidhu and Gupta (1967), Nordhagen, et^  al_. (1967) and Agarwal et_ al_. (1967) are 0.59 nsec. and 4-50 psec. for the 1.29 and 1.10 MeV levels respectively. Following Blair and Armstrong (1965) the 59 spins of the both excited states of Co are known to be 3/2. Internal conversion coefficients measured by Collin et^  al. (1964) indicate that both gamma transitions are predominantly E2. For the beta-decay the mix-ing of the Fermi (L 0 = 0) and Gamow-Teller (L. = 1) contributions is P P unknown. 59 An estimate of the magnetic moment of the Fe nucleus may be obtained from the shell model. The nucleus contains 26 protons and 33 neutrons. The number of neutrons is 5 in excess of the 28 necessary to close the li^j^ shell; these 5 are to be distributed among the 2p^ 2 a n <^ l f ^ 2 shells such that the total spin is 3/2. Clearly the configurations ( 2 p 3 ^ 2 ) 4 ( l f 5 ^ 2 ) 1 and (2p 3^ 2)°(lf 5^ 2) 5 have total spin 5/2. In the config-1 4 2 3 3 2 urations (2p3y2) ( l f 5/ 2) > ^3/2J (-lf5/2-) a n d (-2p3/2^) (-lf5/2") w e a s s u m e that the neutrons tend to pair within each shell such that there is no resultant contribution from shells with an even number of neutrons. It remains to find the magnetic moment of the shells with an odd number of neutrons. For a single neutron with orbital angular momentum L and total angular momentum I, the magnetic moment is p = -(1.91)uN (L = I - 2~) Y7f Ci-9i)y N (L = I + j) . 3.1 The single neutron states 2p^ 2 a n <^ ^5/2 t n e r e : L : o : r e have magnetic moments 68 -1.91 and +1.36 respectively. Chapter 12 of Rose (1967) discusses the situation of three fermions in equivalent orbits (that is, states differing only in the Iz quantum number). In order to achieve a totally anti-symmetric wave function certain limitations must be placed on the allowed values of the total spin J. The resultant magnetic moment is (J/I) times the single particle 3 3 moment. For both (2p^ 2) a n ^ ^^5/2^ a t o t a l sPi- n °f 3/2 is allowed and the magnetic moments are -1.91 and (3/5) (1.36) = 0.82 u N respectively. Although no definite value has been predicted this discussion does 59 indicate that the magnetic moment of the Fe.nucleus is likely to be small. 59 3.3 Expected Anisotropy- of Fe 59 Because the spin of the Fe nucleus is 3/2, the angular distribu-tion function of equation 1.8 need be carried only to the second order. We have W(0) = 1 + A2 W(TT/2) = 1 - j =A2 A 2 = B 2 U 2 F 2 3.2 The quantity A 2 is the most convenient measure of the anisotropy. For a pure E2 transition between states of spins 3/2 and 7/2, the coefficient F 2 is F 2 = -0.14286 3.3 69 The re-orientation following the beta-decay yields = 1.00 (pure Fermi) and = 0.2 (pure Gamow-Teller). For the present we shall assume the situation which produces the least anisotropy, that is, pure Gamow-Teller. The orientation parameter has an extreme value of 1.0 for com-plete polarization of the nuclear spins. The ultimate value for k^ is therefore -0.0286. Of course, i f there is any Fermi contribution to the beta-decay, the anisotropy will be increased. Assuming a working value for the magnetic moment as 1.0 and using the hyperfine field of -333 kG, one calculates the temperature de-pendence for k^ as in figure 14. It can be seen from this graph that the effect is small even at temperatures of 10-12 mK which we expect to achieve. 3.4 The Experiment 59 The Fe source was prepared by Atomic Energy of Canada Limited. An iron fo i l 1.0 cm. x 0.8 cm. x 0.03 cm. was neutron irradiated until 59 the activity of Fe was 20 uCi. 60 A specimen of Co dissolved in iron was prepared in the manner described in section 2..'6. The activity of ^ Co was 2 yCi. 59 Because the expected anisotropy of the Fe gamma rays is so small, i t was decided to use a salt p i l l of cerous magnesium nitrate. Lower temperatures should therefore be possible, but warm-up times would be faster since this salt has a lower heat capacity. The guard p i l l was made of ferric ammonium alum. The two gamma ray peaks of *^Fe are too close to those of ^ Co to 70 b y Figure 14 A 2 versus inverse temperature for Fe permit the use of Nal detectors. Instead, the two Ge(Li) detectors were employed. The axial detector was 25 cm. from the source and its detecting area had an effective radius of 2.3 cm. The equatorial detector was 8.5 cm. away, and its detecting area had an effective radius of 2.0 cm. The solid angle corrections were: axial: a = 0.092 g 2 = 0.994 equatorial: a = 0.236 g 2 = 0.959 3.4 Sample spectra are shown in figure 15. The count times were set at 200 sec. so that the temperature could be closely monitored. After the first demagnetization the samples warmed from 15 mK to 60 mK in one hour, but on later runs they remained below 30 mK for an hour. Afterwards counts were taken for only 20 min. before another magnetization was begun. In this way more cold counts could be obtained and less time was spent at the warmer temperatures. 3.5 Analysis After eleven demagnetizations were conducted there were 85 pairs 59 of spectra to be analyzed. It seemed that any effect present in the Fe gamma rays was indeed very small, and that i t would be necessary to reduce the statistical error as much as possible by combining spectra of the same temperature. It was hoped that a measurable temperature dependence could be found, for such a phenomenon would allow one to f i t the results to the 10005 800 72 FE 1.10 EQUATORIAL 600H FE 1.29 400 U J Z Z 2 0 0 | < X o OC U J O-V) Z300{ O o CO 1.17 CO 1.33 AXIAL 240 180 120 60 40 ~80 ••••• .«•••*• 1 160 • • • 120 200 CHANNEL NUMBER t n r i r n r i ' - T - i i 240 Figure 15 Sample Fe/ Co spectra known temperature dependence of the B2 term. If this could be done, one would have 3 = p H^^/IkT independent of the other terms in k^. An observable effect would also have made i t possible to carry out an NMR/ nuclear orientation experiment. Further, i t would even be possible, in principle, to find those other terms and determine the beta-decay mixing ratio. For these reasons the temperatures of the spectra were carefully evaluated. The first step was to analyze the peak intensities by means of the program SPECTAN, described in section 2.(5. Table 1 gives the output of this program for one pair of warm and one pair of cold spectra. The second step was to calculate the temperature of each spectrum by means of the ^Co intensities. First, a grouping of spectra was carried out. Consecutive counts with l i t t l e difference in ^ Co intensities were put together; four or five spectra would be in each group, and there were 21 groups in a l l . Then W(0) and W(TT/2) were evaluated for the ^Co peaks and a weighted average was taken of the axial and equatorial measure-ments of the temperature. For the 17 groups with a temperature below 25 mK Table 2 shows the axial and equatorial intensities and temperatures, as well as the average temperatures. With the aid of Table 2 groups which f a l l within the same tempera-ture ranges could be combined. Only then, when the temperature of each 59 spectrum had been so ascertained were the Fe peaks investigated. The standard deviation errors of the group temperatures were used as a guide to selecting the temperature ranges. Each range should be a few standard deviations wide. The ranges chosen were: 13 to 14 mK, 14 to 15 mK, 15 to 17 mK, 17 to 19 mK, 19 to 23 hiK, and 23 to 27 mK. One may 59 now find A„ for the Fe peaks. The average values within these ranges, 74 TABLE 1. Sample output of the spectrum analysis program for 59„ , 60„ Fe and Co. Uncorrected Corrected Spectrum Peak Count Norm Error Count Norm Error Window Axial 1 1450 1182 (Warm) 2 870 872 3 525 421 4 420 398 Equator. 1 7239 5162 (Warim)) 2 3899 3569 3 2432 1606 4 1614 1467 Axial 1 1436 .990 .027 1237 1.046 .034 ( 45, 56) (Cold) 2 861 .•998.9 .034 813 .983 .037 (186,197) 3 342 .6650 .036 239 .568 .050 (100,110) 4 253 .602 .038 240 .603 .051 (217,228) Equator. 1 7453 1.029 .012 5171 1.002 .018 ( 38, 55) (Cold) 2 3947 1.012 .016 3626 1.016 .019 (174,197) 3 2789 1.147 .022 1961 1.220 .036 ( 91,109) 4 1874 1.161 .027 1718 1.170 .032 (205,226) Peak identification: 1 5 9Fe, 1 .10 MeV 2 5 9Fe, 1 .29 MeV 3 60 r n , Co, 1 .17 MeV 4 Co, 1 .33 MeV 75 TABLE 2. Cobalt temperatures from ^Co intensities during ^^Fe runs, Group W(-0) W(TT/2) T(0) mK T(TT/2) mK T (Average) mK 1 .696 +_ .017 2 .815 .018 3 .880 .018 4 .667 .015 5 .666 .017 6 .772 .018 7 .758 .018 8 .797 .018 9 .726 .018 10 .798 .018 11 .728 .016 12 .727 .015 13 .635 .015 14 .694 .016 15 .8151 -016 16 .649 ..017 17 .668 .015 1.122 + .013 1.105 .013 1.017 .012 1.168 .012 1.116 .013 1.094 .012 1.121 1.092 1.132 .012 .012 .012 1.101 .012 1.114 .011 1.141 .011 1.159 .011 1.158 .011 1.076 .011 1.156 .012 1.138 .011 15.2 +_ 0.6 21.5 1.2 28.5 2.7 14.2 0.5 14.2 0.5 18.7 1.0 18.0 0.8 20.2 1.1 16.4 0.8 20.2 1.1 16.5 0.8 16.5 0.8 13.1 0.5 15.1 0.6 21.5 1.4 13.6 0.6 16.6 +_ 1.5 18.8 2.0 24.9 2.8 11.7 1.0 17.4 1.6 20.6 1.7 16.7 1.5 34.2 1.7 16.0 1.2 19.5 1.9 17.6 1.5 14.5 1.6 12.7 1.0 12.7 1.0 23.7 2.3 13.0 1.4 14.2 0.5 14.8 1.0 15.4 + 0.6 20.8 1.0 26.8 1.9 13.7 0.4 14.5 0.4 19.2 0.9 17.7 0.7 24.3 0.9 16.3 0.7 20.0 1.0 16.7 0.7 16.1 0.7 13.0 0.4 14.5 0.5 22.1 1.2 13.5 0.5 14.3 0.4 76 as measured by the axial and equatorial detectors are given in Table 3. TABLE 3. Average A„ for Fe for each temperature range. 1.10 MeV Peak Temperature Range (mK) A2 (Axial) A^  (Equatorial) A2 (Average) 13-14 + .008 + .008 -.0006 +_ .009 + .0037 +_ .006 14-15 + .005 .008 -.018 .009 -.0067 .006 15-17 + .005 .008 -.020 .008 -.0085 .005 17-19 -.009 .012 -.032 .016 .020 .010 19-23 -.013 .006 +.0015 .008 -.0056 .005 23-27 + .011 .008 -.010 .011 + .0005 .00068 1 .29 MeV Peak 13-14 +.0006 +_ .009 +0.008 +_ .010 + .0043 + .0067 14-15 +_.005 .009 + .018 .0096 + .0115 .0066 15-17 + 0006 .0075 -.011 .008 - .0025 .005 17-19 -.004 .013 + .010 .016 + .003 .010 19-23 -.0008 .00068 -.010 .008 + .0046 .005 23-27 -.003 .009 -.002 .011 .0025 .007 77 3.6 Results and Discussion For the size of the effects relative to the s t a t i s t i c a l errors no fi t t i n g to the temperature dependence can be done. Instead, one can average A2 over the lowest three temperature ranges (13 - 17 mK) with the result: Because of the beta-decay mixing there i s no reason to expect the two gamma rays to possess exactly the same value of k^, however, they both must be negative. Also the minimum effect for pure Gamow-Teller decay is the same for both peaks. We are therefore justified in averaging over the two gamma rays. The conclusion is that to within an error of 0.003, A2 is zero. 59 An upper limit may be placed upon the magnetic moment of the Fe nucleus. By assuming that the beta-decay is pure Gamow-Teller, we have at the mean temperature of 15 mK: A, (1.10 MeV) = -0.0038 + 0.0033 A2 (1.29 MeV) = +0.0044 +_ 0.0036 3.5 |B2 U2 F2| '< 0.003 B2 < 0.105 IkT y H < 0.480 y < 0.9 y. N 3.6 Although, this result i s just consistent with the result of Tschanz and Sapp (y = 1.1 +_ 0.2 y^) i t does not explain the large effect which 78 they observed. At a temperature of 10 mK they find A2 ~ -0.05. The hyper-fine field in their system is 594 kG, thus 10 mK corresponds to 6 mK in 59 our work. Furthermore, they estimate that only 30% of the Fe nuclei were at double nitrate sites and that the remaining nuclei were in pockets of frozen solution contributing nothing to the zero field anisotropy. 59 Their value of k^ z -0.05 is an average over al l Fe nuclei, hence the value of A 2 for those nuclei in the sites is much larger s t i l l . The largest possible anisotropy occurs i f the beta-transition is pure Fermi, in which case A 2 = -0.14 when nuclear spins are fully saturated. 59 In that event the upper limit imposed on the magnetic moment of Fe by our work is 0.38 u^, considerably less than that measured by Tschanz and Sapp. Since they determined the moment by the temperature dependence of A 2 their result is independent of the beta-decay mixing. Tschanz and Sapp measured the temperature from the susceptibility of the crystals. Since they used Ce-Zn nitrate for low temperatures (< 10 mK) and Nd-Zn nitrate for higher temperatures any f i t of their results to a theoretical curve requires accurate matching of the two temperature, scales. This is especially so because the overlap region contains the greatest variation in k^. Any mis-matching can greatly alter the value of the magnetic moment. Moreover the lack of agreement between the theoretical fits and their data implies that the temperature dependence is not contained solely in B 2 > The most questionable assumption in the interpretation of their results concerns the effect from nuclei not in the double nitrate sites. If this effect had a separate temperature dependence, instead of the con-stant effect assumed, the observed anisotropy would not show the B2 depen-dence . 79 These difficulties, the temperature uncertainty, the influence of the nuclei outside the sites and the non-FS^  behaviour of k^, make their experiment hard to interpret. Our measurement, however, is easy to interpret but contains a null result. Lower temperatures might yield an observable effect, which would permit an NMR/nuclear orientation experi-ment. This would best resolve the problem of the magnetic moment and would also settle the other factors in A0. 80 CHAPTER FOUR ORIENTATION OF 103RUTHENIUM NUCLEI IN IRON 4.1 Introduction 103 Several new examinations of the decay scheme of Ru (Avignone and Frey, 1971, Pettersson et_ al_. 1970) have settled one of the spin assignments, making i t possible to use the method of nuclear orientation to resolve the multipole mixing of the dominant gamma transition. In-ternal conversion studies have been unable to fix the mixing ratio because the two multipolarities involved have almost the same theoreti-cal conversion coefficients. The nuclear orientation factor F, on the other hand, could be very sensitive to the mixing. According to Hansen (1958) Ru is soluble in iron to a few atomic percent, which is quite sufficient for the trace impurity needed to 103 orient Ru in iron. A small anisotropy [e = (1. +. 0.5)%] for Ru in iron at a temperature of 40 mK has been reported by Kul'kov et al_. (1965), and thus one would expect a much larger effect at 10 mK. The hyperfine field on Ru in iron is known from Kistner (1966). Thus anisotropy of a few percent would allow the NMR/nuclear orientation technique to be used to measure the magnetic moment with great precision. The decay scheme parameters could be found from the remaining factors in the effect. 81 103 4.2 Decay Scheme of Ru 103 Figure 16 shows part of the decay scheme for Ru following Avignone and Frey (1971), Pettersson, et_ al_. (1970) and Raeside, et a l . (1969). The dominant decay mode is by beta emission to the 537 keV 103 level of Rh followed by a gamma transition (of energy 497 keV) to the 40 keV level. The 497 keV gamma is known from internal conversion data to be a mixture of E2 and Ml multipolarities. Avignone and Frey 2 report that Ml is favoured, while Pettersson et_ al_. give 6 (E2/M1) = (1-17)%. Previously the spin of the 597 keV level had been known to be either 7/2 or 5/2, but both Avignone and Frey, and Pettersson e£ al_. have found an El transition between the 537 keV level and the 295 keV level of spin 3/2, which rules out the possibility of a spin 7/2 for the higher i i 3 level. The second most dominant gamma transition is the 610 keV one between the 650 and 40 keV levels. The intensity of this gamma ray is 6.2% of the intensity of the 497 keV gamma ray. Its multipolarity is Ml or E2; no mixing ratios have been determined. 103 The Ru nucleus contains 44 protons and 59 neutrons. This number of neutrons is 9 over the magic number 50. The odd neutron is in a 2d^2 o r -^§7/2 s n eH- The Schmidt limits for the 2d^^ magnetic moment are -1.91 and +1.36 y^. For three neutrons in the l g7/2 o rb i t , such that total spin is 5/2, the magnetic moment is (5/7) times the moment for one neutron in that shell, as discussed in section 3.2. The •z Pettersson et^  al_. (1970) initially concluded that the 242 keV transition is M2 because of a plotting error. In an erratum they explain that the El assignment is correct. 82 Figure 16 Ru decay scheme 83 3 resultant moment for the (lg^2) configuration is +1.06 u^. However, these numbers can only be considered as guides because the large number of nucleons makes a proper calculation very complicated. One can only 103 conclude that the magnetic moment of the Ru nucleus would be expected to be between -1.91 and +1.36 j l,.. N 4.3 Expected Anisotropy of the 497 keV Gamma Ray 103 The spins involved in the decay of Ru through the 497 keV transition are such that the second and fourth order terms of equation 1.8 will be present. For a parent spin of 5/2 the orientation parameters B2 and have maximum values of 1.3363 and 0.4629 respectively. The value of 3 which makes B2 equal to half its maximum is 1.0. 99 The hyperfine field on Ru in iron has been found by Kistner (1966) to be 500 kG. If one assumes a working value of one nuclear 103 magneton for the magnetic moment of Ru, then B = 1.0 corresponds to T = 7.3 mK. The other unknown factors in the anisotropy calculation are the Fermi/Gamow-Teller mixing ratio and the M1/E2 mixing ratio; table 4 gives the possible U and F coefficients. Since F^ = 0 for an Ml transition (which we expect may dominate) and since ^ /B^ > 10 for 8 < 1.0, we may, at least temporarily, neglect the fourth order term of equation 1.8. The sign of the anisotropy effect is given by the sign of the F2 term, which can be negative by virtue of the large interference term. For mixed multipoles the resultant F^ is calculated by equation 1.13 and plotted as a function of the mixing ratios in figure 17. The range of F2 for the 497 keV peak is +0.93 to -0.47 84 TABLE 4. U and F coefficients for possible transition modes in the 497 keV 103 and 610 keV sequences in Ru Transition Mode 497 keV 610 keV Fermi 1.00 1.00 Gamow-Teller 0.65714 -0.14286 0.87482 0.58029 Transition Mode Ml 0.13363 0 -0.43643 0 E2 0.32453 0.11756 0.24939 -0.47755 M1/E2 Interference 0.69437 0 -0.37796 0 Figure 17 F ? versus mixing ratio for 497 and 610 keV gamma rays 86 and its zeroes occur for 6(E2/M1) = -0.1 and 6(M1/E2) = -0.24. The 2 former value is just within the range 6 (E2/M1) = ( 1-17)% given by Pettersson et_ al_. (1970) . The anisotropy of the 497 keV gamma ray can be large or small, positive or negative. Of particular interest are the sign of the effect and its temperature dependence. There is a similar lack of information about the 610 keV gamma ray. It is known to be Ml or E2, or any mixture of these. The U and F coefficients for the transitions in this sequence are given in table 4. The anisotropy of this radiation may also be large or small, positive or negative. The coefficient F^ ranges from -0.604 to +0.416 with zeroes at 6(M1/E2) =0.28 and 6(E2/M1) = -0.50, as shown in figure 17. 4.4 Experiments i . Source Preparation The sources for the experiment were made by the same method used for the ^ Co sources. To prepare a sample some ^ ^RuCl^ obtained from AmershamyiSeariee, Corp. of Arlington Heights, Illinois, was evaporated on an iron f o i l of dimensions 1.0 cm. x 0.8 cm. x 0.03 cm. Since Ru is more electronegative than Fe, the Ru atoms replace the Fe atoms in the f o i l . The sample was furnaced for 72 hours at 900°C in a hydrogen environment to clean the surface and diffuse the Ru atoms into the iron. Afterwards, surface activity was removed by etching the specimen with 103 HCl. Because of the short half-life of Ru several samples had to be made during the course of the experiments. The activities of the speci-mens ranged from 5 to 10 uCi. 87 i i . Initial Runs The *^Ru source was soldered with a separate ^ Co source to the end of the copper heat link. The main paramagnetic salt used was cerous magnesium nitrate with a guard p i l l of manganous ammonium alum. The guard shield was around the main p i l l and source. The counting in the first experiments was done by Nal and Ge(Li) detectors. These early runs showed that no large effect existed in the 497 keV peak at 10-15 mK, and that the Compton background from ^Co might mask any.small effect. The Nal detector was especially suspect of faulty background separation because the continuous spectrum on either side of the wide peak was very far from being linear. Fitting to poly-nomials was tried, but was finally abandoned in favour of a more direct way of determining the background, as discussed below. i i i . Final Runs As i t was important to estimate accurately the amount of back-ground originating from the ^Co spectrum, a separate run was undertaken using ^Co alone. This was necessary because the precise shape of the ^Co spectrum depended not only on the geometry and electronic settings but also on the temperature of the sample. The two Ge(Li) detectors were used, with the more efficient one in the more distant axial position. The run, with a 5 yCi source of ^Co in iron, covered the range of 8.5 mK to 15 mK with 18 pairs of spectra of count time 1000 seconds each. The next run was conducted under identical conditions, but with 103 the inclusion of an 8 uCi.source of Ru. The background in the 497 keV 88 window was 45% of the total count. A total of 73 pairs of spectra was obtained, at temperatures from 9 to 13 mK. The results of this run did show a very slight effect. However, 60 in order to remove any doubts about whether the Co might be causing this, a second run followed using the same geometry and apparatus, but 103 with a much larger Ru/Co ratio. The Ru source was 28 uCi and the ^Co source was 0.8 yCi. This ruthenium activity was considerably larger than would normally be considered desirable in view of heating effects in the sample but proved to be tolerable for the purpose.of these tests. The background in the 497 keV window was 1% of the total. In this run 35 pairs of spectra were obtained in the temperature region of 10-12 mK. Figure 18 shows two axial spectra, one from the preliminary ^Co run and one from the second run with the very large Ru/Co ratio. The count time for each is 1000 seconds. 4.5 Analysis The background run was analyzed fir s t . The background in the 497 keV window was found to have a temperature dependence similar to, but less than, that of the ^ Co peaks. In the ^ 3Ru/^Co runs, the number of counts in the 497 keV window was first determined knowing the temperature from the analysis of the ^ Co peaks, the corresponding background was removed and the decay correction applied. Statistical errors were also calculated. 8 r 89 6 0 C O BACKGROUND 1.173 20\ 10| 1.332 co O z" 0 z < X u cc UJ 0. t - ' z ' o :30h 40 _L 80 120 160 CHANNEL NUMBER 1 0 3 R U AND 6 0 C O 200 0.497 MEV X 30 1.173 240 1.332 40 0.610 80 120 160 200 240 Figure 18 Sample spectra from Ru runs 90 i . Background Run The analysis of the ^ Co background run determined the ratio of the continuum background within the window of the 497 keV ruthenium transition to the intensity of the ^ Co peaks. This background consisted of true Compton counts from the 6 0Co peaks plus radiation scattered from other directions. Ambient background would also be in this window. The ratio of this background within the 497 keV window to the ^Co peak intensity, denoted.as R, depends on the selected windows and is quite different for the two detectors. . It is the temperature dependence of R that is desired, but since W(0) and W(TT/2) for the ^ Co peaks are functions of temperature and since they are much simpler to measure, R was found as a function of the corresponding W. The first step in the analysis of the background spectra was to find the temperature dependence of the 497 keV background and to f i t i t as follows. Consider a gamma ray emitted in the direction 6, which is scattered into the detector at a different direction, thus recording a spurious event, as in figure 19. There is a probability f(9) that Figure 19. Gamma Ray Scattered into Detector. 91 such a scattering occurs and that the scattered radiation falls within the 497 keV window. Since the intensity of gamma radiation into the direction 9 is W(9), the number of background counts in the 497 keV window is BG = A + J'f(9) W(9) d ( c o s 9 ) 4.1 where A is a constant background. By virtue of equation 1.8, W(9i),egandbe writ.teniassaosumeof -.. Legendre polynomials of second and fourth order. The scattering func-tion f ( 9 ) may also be expanded into Legendre polynomials, but as a result of the integration in equation 4.1, only the zero, second and fourth orders survive. The temperature dependence of the background is contained in B2 and B^ , the ^ Co orientation parameters. Thus BGfJD = BGQ[1 + aB2(T) + bB4(T>] 4.2 The observed background/temperature points can be fitted to equation 4.2 to find a and b for each detector. The constant BG is found from o the warm counts. Table 5 lists the results of such a f i t for the back-ground run as well as the corresponding coefficients for the ^ Co peaks. The ratio R could now be calculated. For every temperature, the coefficients in Table 5 could be used to find the 497 keV background and intensity of the ^ Co peaks. In figure 20 are plotted the fitted R-W curves for each detector; also shown are the experimental points from which the fits were made. 92 93 TABLE 5. Coefficients of B„ and B. from Co background run Window a b Axial 497 keV Background -0. .2657 -0, .1358 Axial 6 0Co* -0, .41702 -0. .23606 Equatorial 497 keV Background 0, .1600 -0, .0706 i3 * • i 60 * Equatorial Co 0, .20356 -0. .08173 * Includes solid angle correction. i i . Ru Runs 103 The half-life of Ru is 40 days, a time short enough that during the 48 hours of the experiment the source decays by 2.5%. Since effects of less than 1% were being sought, this decay had to be carefully cor-rected. The half-life given by Flynn, Glendenin and Steinberg (1965) is 39.6 +_ 0.1 days. The program SPECTAN summed the channels in the 6 0Co and 497 keV 60 windows, and calculated the decay factor. A second program used the Co peaks to determine the temperature, the corresponding background/^Co ratio, and thus the background in the 497 keV window. This was removed and the decay correction applied. The result was the number of counts in the 497 keV gamma ray peak. 94 i i i . Statistical Error The statistical error in the background arises from the statisti-cal error in the ^ Co peaks. Let nab'e the number of counts in these peaks, N the number at high temperature, R the ratio of background and ^Co intensities. The ratio R is a function of W = n/N which is approxi-mately linear over changes in W of the order of /n/N. Therefore we may write R ~ a + b W. The background becomes BG = Rn 2 = an + bn /N = £ (n.2 + cn) 4.3 where c = aN/b. The variance in the background is then VCBG) = (b/N)2 V(n 2 + cn) . 4.4 However, the variance on the right hand side of equation 4.4 can be expanded thus: 2 2 2 2 2 V(n + cn) = <(ri + cn) > - <n + cn> 4 3 2 2 2 2 = <n > + 2c<n > + c <n > - <n > 2 2 2 - 2c<n ><n> - c <n> . 4.5 For a Poisson distribution of mean A 95 <n> A 3 2 A + 3A + A 4 3 2 A + 6A° + 7AZ + A. 4.6 The mean A may be approximated by n; since this is a large number 3 (MO ) only the highest order need be retained. (Recall that c is of the order of N). Then 4.6 Results The corrected normalization counts should differ only by statistics. A test of the detector stability throughout the run could be provided by comparing the sample variance of the 497 keV warm counts to the variance expected from equation 4.7. These quantities agreed to within 1.5 standard deviations. It was convenient to divide the cold counts from the first run into two temperature regions: 9 - 10 mK and 10 - 13 mK. For the second 103 run, which because of the stronger Ru activity did not achieve for long V(BG) ~ (b/N)2 [4n3 + 4cn2 + c2n] = 4 b W Rn + na 4.7 The variance in the background is estimated by equation 4.7. l . The 497 keV Transition 96 a temperature below 10 mK, the counts in the one temperature region, 10 - 12 mK, were selected. The average effect of the 497 keV gamma ray in the detectors is shown in Table 6. TABLE 6. Effect in the 497 keV gamma ray Temperature (mK) 1 - W(0) WO/2) - 1 9-10 0.0101 +_ 0.0016 1st run 10 - 13 0.0059 +_ 0.0018 1st run 10-12 0.0048 +_ 0.0007 0.0025 + 0.0005 2nd run The results in Table 6 do not include measurements from the first run in the equatorial direction. The equatorial detector was found to have a small attenuation (>1%) when the large cooling magnetic field was on. No such effect was seen on the axial detector. Since in the first run all normalization counts were taken with this field on its results are suspect. As a further test for errors in the background correction a com-parisom.'was made between the normalized count of the 497 keV peak corrected 97 by the R-W method and the same count corrected by the linear background method. For the first run there was agreement to within 0.008 with neither method producing consistently higher numbers. The errors for individual spectra were 0.011 for the R-W method and 0.015 for the linear method. For the second run, agreement was within 0.001 with individual errors of 0.004 for the R-W method and 0.005 for the linear method. The confirmation of the background correction by a different method of calculation indicates that both should be reliable. We there-fore conclude from the axial detector that the 497 keV gamma ray exhibit an effect of -0.004 to -0.005 in the axial direction at a temperature of 11 mK. The first run also indicated an even greater effect in the 9-1 mK region. However, because this effect was not confirmed in the second run and because the first and second runs did not agree entirely in the 10 - 13 mK region, the lower temperature result was not accepted as proven. Only the result of the second run with its large Ru/Co ratio was considered conclusive. i i . The 610 keV Gamma Ray The smaller intensity of the 610 keV gamma ray makes the statis-tical error in its analysis much larger. Also, the ratio of background to peak is larger, creating more difficulties in the background removal. The results of analysis from the first run, where the background is 90% of the total, are too uncertain to be reliable, but the second run, where the background is only 18% of the total, does yield some useful data. 98 After correcting the counts in the 610 keV window using the R-W relations for this energy the 610 keV gamma ray shows the normalized intensities in the 10 - 12 mK region to be W(O) = 0.996 + 0.003 WO/2) = 0.999 + 0.002 . 4.8 With the possibility of systematic errors in equation 4.8 we conclude thaitethe1: effectsiselesssthan 0.005;;" i i i . Magnetic Moment and Mixing Ratio With the negative effect seen in the 497 keV gamma ray in the axial direction one concludes that F 2 is negative, and hence that the mixing ratio 6 must be negative, as in figure 17. 103 A lower limit on the magnetic moment of the Ru nucleus can be found by assuming the beta-decay and gamma-decay mixings are such that 2 ^2^2 ^ S a m a x i m u I I K Using the extreme value 6 (E2/M1) = 0.17 given by Pettersson et al. (1970), or 6 = -0.41, and the extreme case of pure Fermi beta-decay, |U2F2| < 0.3263 . 4.9 In order to observe W(0) = 0.996 at 11 mK, the value of B2 is 0.004 2 0.3263 = 0.012. 4.10 99 For spin 5/2, has this value when y H e f f ^ > 0.098 or y > 0.15 y N 4.11 where equation 4.11 uses the hyperfine field value of 500 kilogauss and a temperature of 11 mK. For the 610 keV gamma ray, the observation of an effect less than 0.005 when B2 = 0.012 indicates B2 U 2 F 2| < 0.005 |F2| < 0.416. 4.12 The E2/M1 mixing ratio for the 610 keV gamma ray is limited as follows: 6(E2/M1) > 1.15 or S(E2/M1) < -0.03. 4.13 iv. Summary 103 Although i t had looked possible that nuclear orientation of Ru in iron might resolve uncertainties in its decay and its magnetic moment, the small effect observed does not allow this to be done, other than to say that for the 497 keV gamma ray the E2/M1 mixing ratio is negative and that the magnetic moment is greater than 0.15 nuclear magnetons. Some limitations may also be put onto the mixing of the 610 keV gamma ray. 100 CHAPTER FIVE SPIN-FLOP TRANSITION OF Mn C12-4H20 OBSERVED BY NUCLEAR ORIENTATION 5.1 Introduction 54 That the Mn nucleus in a crystal of antiferromagnetic MnCl2 : 4H20 can be oriented at low temperatures was demonstrated by Daniels et al. (1961). The hyperfine field direction is defined by that of the electronic spins so that at low temperature the nuclear spins point along the axis of electronic magnetization. Turrell, Johnston and Stone (1972) have used the angular dependence 54 of the gamma radiation from Mn in CoCl2 • ^H^ O to observe the change in direction of the electronic spins as the crystal goes through the spin-flop transition. It should be possible to study the spin-flop transition in MnCl2 * 4^0 by the same method. This experiment was carried out in collaboration with R.L.A. Gorling. 5.2 Molecular Field Model The dynamics of the electronic spins of an antiferromagnet can be described by the molecular field approximation of the Ne"el (1932, 1936), Bitter (1937) and van Vleck (1941). A thorough treatment of the model is given by Nagamija et al. (1955). 101 The forces acting on the electronic spins are the exchange force, the anisotropy energy, and the external field. The exchange energy between spins ^  and is E. . = - h J. . Q + 4 • £.) 5.1 I J i j v l y A simple model is made by letting J„ = J when spins i and j are nearest neighbours and J\ ^  = 0 otherwise. The spins may be considered to be members of two intersecting lattices A and B such that all the nearest neighbours of a spin in lattice A are in lattice B, and vice versa. For antiferromagnetism, J is negative. Equation 5.1 is minimized when the spins in lattice A are antiparallel to those in lattice B. The anisotropy energy arises from the crystalline fields, and is often of the form E. = - D[S2 - \ S(S + 1)1 5.2 l L I Z 3 J J For positive values of D the spins tend to align along the z axis, the easy axis of magnetization. The applied magnetic field adds the Zeeman energy E. = - * gB 5.3 l l & where B is the Bohr magneton and g the electronic g-factor. 102 Equations 5.1, 5.2 and 5.3 can be combined and summed over all the spins. Letting N be the number of spins in each lattice and q be the number of nearest neighbours, the total Hamiltonian, after dropping constant terms, is X = ~£ J. -. s\ • 3. - D I S2 - gg H • Z t. = - 2 Nq J \ • 3B - ND[S2z + S^] - gBM • f§A + . 5.4 In equation 5.4 all spins in lattices A are and all spins in lattice B are \ . Equation 5.4 can be re-written by introducing the molecular field HE = -2JqS/g6, the exchange field. The exchange field H~E = ^ A H E / S is the field produced by the spins in lattice A acting on the spins in lattice B . The anisotropy field, = 2DS/gB, is a fictitious field. Equation 5.4 is written as X = ^ A * ^ B -I C 4 + s 2 z ) --15 • c£A + 5.5 Let H be along the z axis and consider two possible spin configurations. In the first case, the spins are antiparallel and the energy is E(l) = - NgBS[HE + HA] 5.6 103 In the second case, the spins are at an angle 6 to the z axis but have opposite projections in the xy plane. In this situation S • tn = S2 cos 2 0 A B = S2 (2 cos20 - 1) 5.7a S. + Sn = 2S cos 0. 5.7b Az Bz The energy of this configuration is E(2) = NgB[HE(2 cos20 - 1) - H cos2© - 2H cos0]. 5.8 We choose the angle which minimizes E(2). This occurs when C 0 S 8 = 2f^H" 5 ' 9 and the minimum value of E(2) is E(2) = - NgBS[HE + H2/(2HE - H^)]. 5.10 The second case corresponds to the spin-flop configuration which occurs when the field applied along the easy axis exceeds a critical value Hc = [HAC2HE - HA)]h 5.11 When H > H , the energy E(2) < E(l). 104 The critical field in MnCl^ ' 4H2O has been measured with susceptibility and nuclear resonance studies by Gijsman et al. (1959) to be 7.87 kG at 1.04 K. A linear extrapolation to zero temperature gives 7.50 kG. Metcalfe (1971) observes a spike in the thermal resistivity at 0.132 K when the field along the crystalline c-axis (the easy axis of magnetization) is about 7.0 kilogauss. Miedema et al. (1965) using heat capacity measurements found their data consistent with the exchange field Hg = 12.8 kG and the anisotropy field H A = 2.2 kG. The advantage of the nuclear orientation experiment is that the spin-flop is directly observed (assuming the nueleif-followttheeelectron spins). The interpretation in terms of H^  and H^  is made through equations-5.9 and 5.11. 5.3 Decay Scheme and Anisotropy 54 The gamma radiation from oriented Mn nuclei is a suitable probe because its decay scheme is well established. The decay scheme:in figure 21 is from Lederer, Hollander and Perlman. The magnetic moment is 3.302y^ (Templeton and Shirley, 1967) and the hyperfine field on Mn in MnCl^ * 4^0 is deduced to be 645 ± 12 kG from the thermal capacity data of Miedema et al. (1965). 54 The coefficients in equation 1.8 for Mn are U2 = 0.82808 U4 = 0.41785 F 2 = -0.59761 F 4 = -1.06904 5.12 105 303 DAYS Figure 21 Mn decay scheme 106 The detectors used in the experiment subtended a half-angle at the source of 10°. The solid angle corrections are g 2 = 0.974 and g^ = 54 0.916. The angular distribution of the gamma ray from Mn, with this solid angle correction, is W(6) = 1 - (0.482) B2 P2 - (0.409) B4 P4 . 5.13 Equation 5.13 is plotted for several temperatures as a function of cos0 in figure 22. As the electronic spins change direction, the nuclear spins follow by means of the hyperfine interaction. The direction of the spins can be found by locating the orientation axis relative to the axes of three gamma ray detectors. For each detector equation 5.13 finds the angle between the orientation axis and the line joining the detector to the radioactive source. It should be noted that in the model in section 5.2 there are two spin lattices each with its own orientation axis. In the non-flop situation these axes coincide but are of opposite sense; the gamma radi-ation, depending on only the square of the cosine, cannot distinguish the two senses. In the flop phase, i t is not immediately obvious that the same symmetry exists. The two orientation axes have spherical polar co-ordinates (0,<J)<) and (0,ir +_<(>). Consider a detector along the axis with polar co-ordinates (6 ,<|> ). The angle a between this axis and the axis '(9,40 is cosa = cos0 cos6 + sin0 sin0 cos(d>-d> ) o o V T Yo 5.14 108 For the detector along the z axis (the axial detector in the normal phase), 6Q = 0 and a = 6 for both orientation axes. For a detector in the xy plane 8Q = TT/2 and cosa = sine cos (<j>-<|> ) . The two orientation axes therefore differ only in the sign of cosa, and thus the gamma ray intensities from them are identical. For detectors in other positions this would not be so. 5.4 The Experiment i . Source The experiment was conducted upon a crystal of MnC^ • 4^0 which 54 had been grown from a solution containing some MnC^. The activity of the crystal was 15 yCi. It was covered in Apiezon N grease and fixed to the copper foil at the end of the heat link. Also soldered to the heat link was a ^ Co-Fe fo i l of activity 2 yCi. The paramagnetic salt used was chrome potassium alum. i i . Detectors Three gamma ray detectors were employed. In the axial direction a 5 in. x 4 in. Nal(Tl) detector (denoted C) was positioned 14 in. from the source; the two Ge(Li) detectors were set up in the equatorial plane, both 4 in. from the source. The crystals had been aligned so that the crystallographic c-axis (the .easy axis."of magnetization) was vertical; the axes of the Ge(Li) detectors were mutually perpendicular, with one (denoted detector B) 35° from the crystallographic b-axis, and the other (denoted detector A) at 125°. All angles quoted here have an error of 5°. 109 The analysis of the orientation axis is simplified by using three mutually perpendicular detection axes, but because each detector cannot distinguish the sign of cos0 an ambiguity exists in the location of the orientation axis. This ambiguity may be resolved by moving the equatorial detectors so that the angle between them is 45° (or 135°). In a subse-quent experiment the A detector was moved to make an angle of 125° with the B detector. The ambiguity was settled conclusively by the results of this run. Since the multichannel analyzer is able to route the signals from only two detectors at one time i t was necessary to alternate the second input between the A and B detectors. The C detector was always the first input. Sample B and C spectra are shown in figure 23. i i i . Procedure Several normalization counts of 1000 sec. were taken before and after the experiment. After the demagnetization the current in the polarizing solenoid was increased to provide an external field along the c-axis of the MnC^^I-^O crystal?.; Several counts of 400 sec. were acquired at each current setting. The field in the centre of the solenoid had been calculated and measured to be 390 gauss per amp. The current was measured by the voltage across a resistor in the power supply for the superconducting magnets. 110 EQUATORIAL 80. 60] 40 "o 201 Z < I o cc UJ Q. v> 4 0 p Z 3 o o 0.835 MEV 50 1.173 X10 1.332 _L 100 150 CHANNEL NUMBER 0.835 MEV 1.173 200 AXIAL 250 30h 20 1 0 h X10 1.332 250 50 100 150 200 f i g u r e 23 S a m p l e 5 4 M n / ^ ° C o s p e c t r a I l l 5.5 Analysis i . Temperatures The program SPECTAN was used in the normal way to find the rela-tive intensities of the ^Co and ^4Mn peaks. The temperature of the copper heat link, as given by the ^Co thermometer, was .15 mK, but the MnC^^rL^O crystal was at a temperature of 65 to 55 mK, as measured by 54 the Mn anisotropy below the spin-flop transition. This higher temper-ature was due to the poor thermal contact and the high heat capacity of the crystal.?. The crystal continued to cool throughout the several hours of the experiment. i i . Spin-Flop Transition The spin-flop transition was analyzed by finding the orientation axis relative to each detector as a function of f i e l d along the c-axis. These angles were determined by locating the angle 6 corresponding to the observed relative intensity W(0) on the appropriate temperature curve in figure 22. In the antiferromagnetic phase the crystal tempera-ture could be found from the observed anisotropy. Because the crystal continued to cool and because the temperature could not be measured directly in the flop phase, the temperature was interpolated from those measured in the antiferromagnetic phase before and after the crystal was in the flop phase. The spin-flop effect i s shown in figure 24 where cosa for each detector is plotted against f i e l d applied along the c-axis. Through the transition the cosine i s an average over a l l the spins. The centre of the transition i s seen from figure 24 to be at H = 6.50 + 0.05 kG. 112 •8r .4* O .21 < Z Q Z o CL CO UJ OC OC o CJ UJ z 00 O O .81 .6 O .4 DETECTOR A JL DETECTOR B • * 1.0r DETECTOR C •8h .6 .4 J 9 7 8 FIELD ALONG C AXIS IN KG Figure 24 Cosa versus field along c-axis i i i . Orientation Axis The orientation axis of the spins can be found from the angles it makes with the three detector axes. The three angles are not indepen-2 2 2 dent and can be fitted to the relation cos a. + cos a„ + cos a„ = 1 to A rs L yield the fitted values of these angles in Table 7. From these spherical polar angles for the orientation axis may be found as follows: 6 = a c tan<j> = cosa^/coscig 5.15 where <j> is the azimuthal angle measured from the B detection axis. TABLE 7. Measured and fitted angles of the spin orientation above the spin-flop transition Measured Angles Fitted Angles Polar Angles* Field Applied —-, r • c t . . a_ „ a a. a_, a 6 <p along c-axis AA B B c A B c • (kG) (degrees) (degrees) (degrees) 7.01 58 + 3 42 + 2 73+2 56.0 40.4 70.8 70.8 -36.3 8.97 60+2 42+1 67+2 59.5 41.1 65.6 65.6 -34.7 0=a ; <|> is relative to axis of the B detector. The b-axis in this system is at <j) = (-35+5)°. The error in fitted and polar angles is +1° . 114 Without knowing the signs of the cosines in equation 5.15 one can-not determine the sign of <f>. In Table 7 the negative sign has been chosen because of the result of a separate experiment in which the A detector was at <j>^  = +125° relative to the B detector. With this geometry + cosa. - cosa,, costb. ± ~ A H A tan<j) = . 5.16 cosag sintf)^ A typical set of values from the scaler counts of this experiment, above the spin-flop, is cosaA = 0.71 cosaB = 0.68 5.17 Solving for. cf) by equation 5.16 yields = 62.5° or -29.5°. The negative sign for <j> in Table 7 is therefore justified. Thus the orientation axis is within the be plane, to within the accuracy of the detector settings. iv. Molecular Fields As the applied field along the c-axis is increased beyond the transition the spins tend to be pulled back to the c-axis according to equation 5.9. The molecular fields and can be found from this effect. From equation 5.9 one would expect that in the spin-flop phase cos8 is proportional to the applied field. By fitting a line through the two points above the transition and through the origin (figure 25) the value of 2Hg - is calculated from the slope. We have 2H^  - = 21.4 +_ 1.0 kG. From equation 5.11 and the transition field of 115 FIELD ALONG C AXIS IN KG Figure 25 Cos0 versus field along c-axis above spin-flop 116 Hc = 6.50 +_ 0.05 kG, HA(2H£ - H^ ) = H,2 = 42.25 +_ 0.65 (kG)2. As a result, the fields H„ and H. are E A HA = 2.0+0.1 kG PL = 11.7 + 0.6 kG . 5.18 E — These values are in excellent agreement with the results of Miedema et al. v. Summary Nuclear orientation provides a very direct means of observing the spin-flop transition in antiferromagnets. The transition in MnC^^r^O is seen to occur for an applied field of Hc = 6.5 +_ 0.05 kG at 50 mK. There is good agreement with the simple molecular field description of the spin orientation above the transition. The molecular fields deduced from the experiments concur with previous measurements. The spins are seen to lie in the be plane of the crystal. 117 CHAPTER SIX ORIENTATION OF 2 0 7 B i NUCLEI IN COBALT 6.1 Introduction Nuclear orientation experiments have been carried out recently on bismuth dissolved in nickel and iron. Bacon et al_. (1972) have oriented 2 0 4 B i in iron and nickel and Kaplan et^  al_. (1972) have studied 2 0 6 B i in the same hosts. Bismuth is soluble in nickel to a few atomic percent (Hansen, 1958) and only slightly soluble in iron (yQ.l' atomic percent). It has been reported that Bi in cobalt may be soluble to 1%, but this is unconfirmed, [(Vol, Ll'96j7). Since the solubility need only be a small fraction of a percent to prepare a source for nuclear orientation experiments this may not be a problem. The purpose in studying Bi in cobalt is to compare the hyper-fine measurements for the ferromagnet hosts Ni, Fe and Co. The tendency of other non-magnetic atoms dissolved in these metals is to exhibit a hyperfine field approximately proportional to the host, electronic magnetization. This is consistent with the theory that the field arises from conduction electrons polarized by the host magnetization (Campbell, 1969) . Bacon e^t al_. (1972) measure the hyperfine field for Bi in Ni and Fe to be 325 +_ 35 and 1180 +_ 130 kG, respectively. Kaplan et al_. (1972) found the field in Ni to be 390 +_ 15 kG but could conclude for Bi in Fe only that the field was between 800 and 1000 kG. The field in cobalt is expected to be between those of Ni and Fe. 207 The magnetic moment of Bi has been measured using nuclear orientation by Kaplan et al_. (1973) and they estimate a value of 4.69 +_ 0.25p. 118 The hyperfine interaction then is quite huge. We might expect a very large anisotropy in the dominant gamma ray, which has M4 multipolarity. Problems with the experiment might.be the low solubility of Bi in Co and the reorien-tation of the nuclei in the intermediate isomeric state in the dominant decay sequence. 6.2 Decay Scheme 207 The Bi nucleus decays by electron capture with a half-life of 207 30 years to excited states of the Pb nucleus. The decay scheme is shown in figure 26 following Lederer, Hollander and Perlman (1968). The primary decay is by first forbidden electron capture to the 1.633 MeV level of 207 Pb. This state has a half-life of 0.8 seconds, probably too long to ignore reorientation effects, but also too short to assume complete re-orientation. It emits an M4 gamma ray of energy 10064 MeV. The secondary decay sequence is by electron capture to the 2.330 207 MeV level of Pb followed by prompt gamma-decay to the 0.569 MeV level. The 1.761 MeV gamma ray is 94% Ml and 6% E2 (Alburger and Gunyar, 1955). 207 The 0.569 MeV level of Pb is populated by both these gamma 207 transitions and by a small amount of electron capture from Bi. The interpretation of the anisotropy of this gamma ray, which is dominated by the influence of the 1.064 MeV transition, is rather complicated. 6.3 Expected Anisotropies The effect of the long half=life of the 1.633 MeV state on the anisotropy can be measured by observing the deviation in the measured effect from the one calculated assuming no reorientation. 119 Figure 26 Bi decay scheme 120 For the electron capture with angular momentum change L g c = 2 between the states of spins 9/2 and 13/2 and for the M4 gamma transition between levels of spins 13/2 and 5/2 we have the coefficients U 2 = 0.91606 U 4 = 0.73396 F 2 = -0.85726 F 4 = 0.39060 . 6.1 Higher order terms are present but their effect is much smaller (^0.01). The maximum values of B2 and B4 are 1.6514 and 1.0644 respectively. The term B2 is half its maximum for 8 = 0.7; using the magnetic moment of 207 4.69 y^ for the Bi nucleus and a working value for the hyperfine field in cobalt as 500 kG, then 3=0.7 corresponds to a temperature of 25 mK. At this point, W(0) = 0.37 and W(TT/2) = 1.34, easily observable effects. However, the relatively long half-life of the intermediate state could cause a change in the magnitude of the anisotropy calculated above. Two types of processes may operate (de Groot et al. 1965). First, there are sudden violent disturbances due to nuclear recoil and electronic transitions in the atom after the electron capture. The effect of these disturbances is hard to estimate theoretically but phenomenologically we can describe i t by introducing attenuation coefficients which may diminish (or enhance) the undisturbed anisotropy. Thus W(6) = E G B U P (cosO) 6.2 V V V V V where G^  allow for these changes. Secondly there will be relaxation due to spin-lattice interaction from an initial population characteristic of the oriented Bi nucleus (but possibly attenuated by the preceding effects) to one characteristic of the 121 nuclei in the 1.633 MeV state oriented by the hyperfine interaction in the Pb atom. This type of process is well known from nuclear magnetic resonance theory. It is described by a relaxation time, T^, which is typically the order of seconds or more at low temperatures. This process has been observed in nuclear orientation, for example by Stone et^  al. (1971) who measured the anisotropy of the gamma ray from the 40 sec. isomeric state 109 of Ag. An important feature of the spin-lattice relaxation is that T^  is inversely proportional to temperature (except at the very lowest temper-atures) . The 1.761 MeV gamma ray does not contain uncertainties due to reorientation in the intermediate state, but it is a much less intense transition. It is 10% as strong as the 1.064 MeV gamma ray and, being of higher energy, has a lower detection efficiency. The anisotropy parameters for this gamma ray are calculated using L g c = 1 for the electron capture reorientation, and 6(M1/E2) = -0.252 for the gamma transition. This mixing ratio is chosen because i t 2 yields 6 = 0.064, consistent with Alburger and Gunyar (1955). The nega-tive sign is selected because, as is discussed in sections 6.5 and 6.6, we observed a positive effect from this gamma ray in the axial direction, which result is compatible only with the negative value. The anisotropy parameters are therefore U 2 = 0.92496 U 4 = 0.74948 F 2 = 0.39406 F 4 = 0.59871 6.3 The orientation terms and B4 are the same as those for the 1.064 MeV gamma ray. For & = 0.7 we have W(0) = 1.38 and W(TT/2) = 0.87. 122 The anisotropy of the 0.569 MeV gamma ray is difficult to esti-mate accurately because i t contains uncertainty not only from the reorien-tation of the isomer but also from the different feeding transitions. Nevertheless, because i t is such an intense peak its anisotropy can be easily measured to give information on sample saturation, as is discussed in section 6.4 6.4 Sample 207 The Bi activity was obtained from Amersham/Searle Corp. in 207 the form of carrier-free ^ ^ l ^ * n solution.. Since BiCl^ melts at 260°C and boils at 400°C it would be impossible to prepare the sample by the regular method of evaporating the chloride on to a cobalt fo i l and furnacing at 700 - 900°C. The Bi ions, being more electronegative than Co, were allowed to plate spontaneously on to the surface of 0.2 g. of cobalt powder (99.99% Co) which had been introduced into the solution. The powder was used rather than a fo i l in order to increase the surface area. About 95% of the activity (^ 7 yCi) would be on the powder after 30 minutes. This method was suggested by R. Butters of the Metallurgy Department of the University of British Columbia. 207 The Co powder with the Bi was sealed in a quartz capsule 0.9 cm. in diameter and 4 cm. long. This length was necessary to allow the capsule to be sealed without heating the powder over 100°C. Hydrogen at 13 cm. of Hg was also in the ampoule. The Co and Bi were melted together by heating the powder to the melting point of cobalt at 1500°C. This temperature was achieved by encasing the quartz capsule within a block of graphite of length 5 cm. and 123 diameter 2 cm. and by heating the graphite to white heat by an induction furnace. The minimum power necessary to melt cobalt was determined before-hand with a non-active sample of cobalt powder. The entire length of the quartz tube had to be heated to prevent Bi from condensing on colder parts of the wall. Since quartz glass softens at 1500°C the internal pressure had to be properly adjusted. The original pressure of 13 cm. became 78 cm. at this temperature. The slight over-pressure pushed the glass walls away from the sample preventing the glass from encircling the cobalt. The graphite was maintained at 1500°C for several minutes, and was quenched to 100°C in 30 sec. by plunging it into water. The bead of cobalt so produced had an activity of 3.7 yCi and was roughly cylindrical 207 in shape, 0.4 cm. long and 0.2 cm. in diameter. A Bi activity of 3.7 yCi in 0.2 g of cobalt corresponds to a Bi-Co alloy containing 10 ^ at.r% bismuth. No further annealing nor flattening was carried out because of the results of a study by Feldman et^  al_. (1971) which showed that for Bi in iron annealing at temperatures over 300°C tended to decrease the number of Bi atoms at lattice sites. The only further procedure applied to the sample was to remove surface activity by sandpapering. No noticeable decrease in activity was found. It was necessary that the specimen should be magnetically saturated in the field of the polarizing solenoid. Cobalt alloys generally require large fields for saturation and, in this case, there was a sizable demagne-tization factor (^ 3.5) because of the shape of the specimen., The magnetiza-tion curve was determined by measuring the self-inductance of a small coil, wound around the specimen. The inductance was plotted against the field applied along the axis of the coil. Since the sample was approximately 124 cylindrical i t was oriented with the long dimension parallel to the coil axis and the applied field. Figure 27 illustrates the saturation curve so found. Also shown for comparison are W(ir/2) - W(0) for the 0.569 and 1.064 MeV gamma rays as a function of applied field, taken during the nuclear orientation experiment. All three curves are consistent with saturation at 14 kG. The superconducting polarizing solenoid in the de-magnetization cryostat could produce fields up to 16.5 kG and therefore would be adequate. 6.5 The Experiment A 1.3 uCi source of ^ Co in iron was soldered to the copper link 207 next to the bead of Bi in cobalt. The paramagnetic salt was chrome potassium alum. Initially two Ge(Li) detectors were used and figure 28 shows a sample of the spectrum taken with a Ge(Li) detector. The initial runs were performed to check the saturation of the sample and to measure the size of the anisotropy of the 1.064 MeV gamma ray. Also of interest was the sign of the effect on the 1.761 MeV gamma ray, since this determined the sign of the mixing ratio. The 1.064 MeV gamma ray showed a surprisingly small effect [W(0) = .82] at 11 mK. The 1.761 MeV gamma ray had a positive axial effect and although the statistical errors were quite large in this weak gamma ray, it was obvious that i t too had a smaller effect than expected [W(0) = 1.1]. The reduction in the observed effects compared to the expected values could be explained by assuming that the sample was not a good solu-207 tion of Bi in cobalt and that only a small fraction (M0%) of the Bi were in the lattice sites. To confirm this view the temperature dependence 125 Ar GAMMA RAY ANISOTROPY AT 11 MK 1.064 MEV o s £ I i * 0.569 MEV 1 2 , MAGNETIC SATURATION AT 300 K 1.0| i LU.8f 1.6| o UJ O . z ,< t-o . Q J L_ L 6 8 10 12 14 APPLIED FIELD - KILOGAUSS 207 Figure 27 Saturation of Bi/Co sample 8r CO o UJ z z < I o cr, UJ CL to Z3 o o 40 30 20 10 0.569 MEV 126 GE(LI) DETECTOR 1.064 • • • 1.173 1.332 X10 1.761 569 MEV NA I DETECTOR 1.064 X10 1.332 1.761 50 100 150 200. 250 Figure 28 Sample 2^7Bi/^°Co spectra 127 of the anisotropics would be needed. Using the Ge(Li) detectors the 1.064 MeV gamma ray was observed within a temperature range of 11 to 70 mK. After demagnetization the temperature was 11 mK. Higher temperatures were produced in a reversible manner by restoring a small current (up to 2 amp) to the large magnet. At each temperature four 1000 sec. counts were acquired. Ten temperature points were taken, including two at the lowest temperature (zero field from the large magnet) taken before and after the others. The study of the temperature dependence of the 1.761 MeV gamma ray was conducted with a 5 in. x 5 in. Nal(Tl) detector in the axial direction at a distance of 25 cm. from the source. The increase in the count rate over that of the Ge(Li) detector was a hundredfold producing a tenfold decrease in the statistical error. Because of the poorer resolu-tion of this detector it was necessary to reduce the activity of ^ Co to 0.2 yCi so that a cleaner spectrum could be obtained for analysis of the 1.761 MeV peak. A sample spectrum from this detector is given in figure 28. The run to determine the temperature dependence of the 1.761 MeV gamma ray was carried out with the same procedure as was usedlfoflthe •!.061MeV study. Four 1000 sec. counts at each of ten temperature points were acquired. The temperature was increased by putting some current into the large magnet. A Ge(Li) detector was placed in the equatorial direction to observe the ^ Co gamma rays for temperature measurement. These peaks could not be resolved by the Nal detector. 128 6.6 Analysis The gamma rays were analyzed by assuming the extremely simplified 207 model that a fraction f of Bi nuclei were in lattice sites of the cobalt host, feeling the fu l l hyperfine field, and that the remaining nuclei felt an average of zero field. Thus [W(6) - l ] Q b s = f[W(0) - 1] 6.4 that is, the observed effect is reduced by the factor f. In addition the 1.064 MeV gamma ray may be altered by the reorientation as discussed in section 6.3. In non-magnetic metals there exists a relationship between the Knight shift and the spin-lattice relaxation time, T^ . This is the 'Korringa relation' (Korringa, 1950) o AH V (TjT) YfjFjj) = constant . 6.5 Here is the nuclear gyromagnetic ratio and the constant involves funda-mental atomic constants. The Knight shift depends on the polarization of conduction electrons by the applied magnetic field, and we might expect a similar relation to apply for hyperfine interactions in ferromagnetic metals. Thus we assume 2 2 (T-jT) Y n H e f£ soeonstant . 6.6 Data supporting this assumption has been given by Stone (1971) and theoretical 129 arguments supporting it have been discussed by Campbellf(1969). Experi-18 2 mental evidence indicates that the constant has a value 2^ x 10 G sec K. Most of the data is relevant to impurities dissolved in iron, butOCampbell predicts should be independent of the host material. If we apply the relation to the case of Pb in cobalt we obtain T1T = 7 sec K, 6.7 where we have used the H £^ value for Pb in Co of 262 kG (Koster and Shirley, 1971) and assumed a magnetic moment of 2 y^ for the isomeric state. The hafftlife of this state is 0.8 sec. so that for the temperature range investigated spin-lattice effects may be ignored. In equation 6.2, i f the reorientation factors a n ^ G4 a re gqual (to G, say) then [W(0) - l ] o b s = fG[W(6) - 1] . 6.8 With this very simple model the temperature dependence of the gamma rays could be fitted to the expected anisotropy to find the hyperfine field and the attenuation^fG. For the 1.761 MeV gamma ray the attenuation is the fraction of nuclei in lattice sites, and for the 1.064 MeV transition, i t is that fraction times the reorientation factor. The 1.064 MeV peaks were analyzed by the program SPECTAN in the normal way. Temperatures were measured by the ^Co thermometer. The 1.761 MeV gamma ray was studied using a Nal detector with such a wide FWHM that the linear approximation for the background was not considered suitable. 130 Instead all spectra of the same temperature were added and plots were made of the summed spectra. These were analyzed by fitting the continuous background on both sides of the peak with a flexible curve, drawing in the portion under the peak, and evaluating the remaining area under the peak by a planimeter. We estimate the total error from statistics, background fitting and planimeter reading to be vl.5% for each of the cold spectra. In figure 29 are shown the plots of the axial and equatorial effects from the 1.064 MeV peak and the axial effect from the 1.761 MeV transition. The equatorial effect for the latter is not shown because it was small (^  1-2% at 11 mK), a result which is consistent with the value of f = 12% found from the axial effect. These data were fitted to the expected anisotropies (with solid angle corrections) with the attenuation and the hyperfine field as unknown parameters. The fitted values are given in Table 8 and the curves are shown in figure 29. TABLE 8. Fitted values of attenuation and hyperfine from the gamma rays t 207p. of Bi. Attenuation Hyperfine field ^ Deg. of kG Freedom .176 +_ .005 560 + 30 6.1 8 .186 +_ .007 880 +_ 65 13.5 8 .12 + .025 590 + 200 10.2 7 1.064 MeV Axial 1.064 MeV Equat. 1.761 MeV Axial 132 6.7 Discussion For the interpretation of the effect from the 1.761 MeV gamma ray, the model which assumes that there is no contribution from nuclei not in lattice sites is a very extreme one. However, our analysis indicates that this approximates to the actual situation. Of course, more likely there ise a large fraction of nuclei which feel weak fields (< 100 kG) and a smaller fraction ( M2%) which feel a much larger field of ^600 kG, of perhaps a distribution of fields of this order of magnitude. Thus we conclude from the f i t for the 1.761 MeV gamma ray that 12% of the nucleiaare in fields in the range 400 to 800 kG. For the 1.064 MeV gamma ray the interpretation is more difficult. The significance of the fitted parameters in table 8 is:.as follows: the attenuation factor determines the ratio of the observed and expected effects at saturation, whereas the hyperfine field determines the temperature range in which the saturation begins to occur. The near equality of the attenuation coefficients for the 1.064 MeV axial and equatorial effects indicates that the simple model in which and G^  were taken as equal is reasonable. We see that G = 1.6 ± 0.4, andenhancement. However, the discrepancy in the fitted values of the hyperfine field suggests that the previously ignored spin-lattice relaxation effects are important at the higher temperatures, altering the values of B^ and B^ . The axial and equatorial effects would not be changed to the same degree because of the difference in the relative signs of and in the axial and equatorial directions. The observed discrepancy could be explained by a value of T^ T of approximately 0.2 sec K, a much lower value than that estimated in section 6.6. 133 206 The nuclear orientation of Bi in cobalt would have several advantages over the present experiment. Because the half-life is 6 207 days instead of the 30 years for Bi one can achieve the same activity 207 with 1/1500 the number of atoms needed for. Bi. This reduction in concentration may allow for a larger fraction, perhaps 100%, of nuclei at lattice sites. Secondly, there are no long-lived intermediate states 206 0^207 in the decay of Bi. 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