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Nuclear magnetic resonance of 169Tm in thulium iron garnet
Volume 31A, number 4
PHYSICS LETTERS
i~ dTo (iE - L)p' + ~ Jx ~ Pe = 0
where G+ -= (i~ - L ) - I and s u b s t i t u t i n g this into eq. (3) we find after some i n t e g r a t i n g by p a r t s
(2)
The Liouville equation is now in a convenient f o r m for manipulation. In d e r i v i n g an e x p r e s s i o n for the t h e r m a l r e s i s t i v i t y we find it convenient to make some changes in the p r o c e d u r e suggested by Edwards. We want to split p' into the sum of two p a r t s p~ and p~. such that p~ accounts for all the c u r r e n t . A r a t h e r g e n e r a l way of doing this is to write P'l = PeJxJx ~ where 13-1 = fd~j2xPe, d~ is an e l e m e n t of phase space, a n d J x is the m a croscopic c u r r e n t . On multiplying the Liouville equation by Jx and i n t e g r a t i n g over all phase space we have
iCJx - gx f d~JxL(JxPe~) -
dT°
f d~JxLP~.= i~2 fi-1 dTo
=
dx
-
(3)
The first term in eq. (3) is vanishingly small and the second is zero because the integrand is an odd function of the momenta. W e m a y write [I]
G+[ i~2 .
NUCLEAR
dT° + L p i )
MAGNETIC
r
ikT2 ((Ljx)G+(L(JxPe)))
]j
(5)
where the s c a l a r product b r a c k e t s denote i n t e g r a tions over all phase space and the horizontal b a r s denote a v e r a g e s over equivalent d i s o r d e r e d s y s t e m s . The t e r m s m u l t i p l y i n g Jx on the right hand side of eq. (5) define the t h e r m a l r e s i s t i v i t y . The quantum m e c h a n i c a l f o r m of eq. (5) may be obtained by following well defined r u l e s . The u s e f u l n e s s of this type of e x p r e s s i o n lies in the fact that a finite r e s u l t is obtained for a t r i a l G r e e n function G+ c o n s t r u c t e d f r o m undamped propagating modes of the s y s t e m in cont r a s t to the Kubo f o r m u l a and v a r i a t i o n a l f o r m s of it. It is t h e r e f o r e a useful s t a r t i n g point for complex s y s t e m s where one has some intuitive idea about the states of the s y s t e m .
Reference
(4)
RESONANCE
23 February 1970
1. s. F. Edwards, Proc. Phys. Soc. 86 (1965) 977.
OF
169Tm
IN T H U L I U M
IRON
GARNET
R. L. STREEVER and P. J. CAPLAN
Institute for Exploratory Research, US Army Electronics Command, Fort Monmouth, New Jersey, USA Received 5 January 1970
The nuclear magnetic resonance of 169Tm has been studied in polycrystalline thulium iron garnet at 4.2°K in zero field using the spin-echo technique.
Recently we have been studying the nuclear magnetic resonance (NMR) of rare earth nuclei in the rare earth iron garnets. As far as we know. the N M R of the rare earth isotopes have been observed only for gadolinium [1] and e u r o p i u m [2] i r o n g a r n e t . In this note we report on the o b s e r v a tion of the 169Tm NMR in polycrystailine thulium i r o n g a r n e t (TmIG) powder at 4.2°K in zero external dc field u s i n g the s p i n - e c h o technique. Line shapes obtained by plotting the s p i n - e c h o a m p l i 162
tudes as a function of frequency a r e shown (fig. 1). The e x p e r i m e n t a l echo a m p l i t u d e s at each f r e quency have been divided by the frequency s q u a r e d to take into account the fact that the n u c l e a r s i g nal induced in the s a m p l e coil is p r o p o r t i o n a l to frequency and to account for the i n c r e a s e d n u c l e a r p o l a r i z a t i o n at higher f r e q u e n c i e s . The decay times, T2, were frequency dependent and in g e n e r a l n o n - e x p o n e n t i a l varying from about 10 to 60 ~sec. The b r o a d v a r a t i o n of T 2 and r e s -
Volume 31A, number 4
PHYSICS LETTERS
IO 169 8
Tm
4.2" K 6 4 2 O 2OO
k
300
400
500
600
700
FREQUENCY- MH!
Fig. 1. C o r r e c t e d spin echo s p e c t r u m (see text) of 169Tm in TmIG at 4.2OK.
onance f r e q u e n c y i n d i c a t e s that the o b s e r v e d s i g n a l s a r e coming, to a l a r g e extent, from nuclei in domain walls. The situation in T m I G is s i m i l a r to that in EuIG [2]. The hyperfine fields a r e a n i s o t r o p i c and depend on the o r i e n t a t i o n of the m a g n e t i z a t i o n M with r e s p e c t to the local o r t h o r h o m b i c axes of the r a r e e a r t h site. We can w r i t e the hyperfine field at the r a r e e a r t h site as Heft
=
(/_/x2nx2 + Hy2
ny
2 +H 2
Z nZ
2)½
(1)
Here Hx, Hy and H z a r e the p r i n c i p a l values of H etf a n d n x , ny a n d n z a r e the d i r e c t i o n c o s i n e s of M with r e s p e c t to the local o r t h o r h o m b i c axes defined in ref. 2. F r o m 169Tm M 6 s s b a u e r studies at 20OK Cohen [3] has obtained hyperfine fields at the two sets of i n e q u i v a l e n t s i t e s c o r r e s p o n d i n g to M along a (111) d i r e c t i o n of 1.70 and 0.81 × 106 Oe. Since the T m ° s u b l a t t i c e m a g n e t i z a t i o n in TmIG has n e a r l y the s a m e value [4,5] at 4.2°K as at 20°K, we expect the hyperfine fields to be n e a r l y the s a m e at these t e m p e r a t u r e s . The n u c l e a r m o m e n t
23 February 1970
of 169Tm is about -0.229 n . m . [6] and its spin is ½. We show (fig. 1) by a r r o w s the f r e q u e n c i e s c o r r e s p o n d i n g to the two (111) hyperfine fields. The two (111) hyperfine fields give i n s u f f i c i e n t i n f o r m a t i o n to evaluate the t h r e e hyperfine field p a r a m e t e r s , while the NMH f r o m nuclei in domain walls is difficult to i n t e r p r e t . The c o r r e c t set of hyperfine field p a r a m e t e r s should, however, give c o r r e c t l y the (111) fields and be c o n s i s t e n t as well with the l i m i t i n g NMR f r e q u e n c i e s . If we take the m a x i m u m o b s e r v e d hyperfine field (about 1.95 x 106 Oe) c o r r e s p o n d i n g to the line at 680 MHz as Hx (or H v ) and u s e the M S s s b a u e r v a l u e s and eq. (1) we obtain H. (or Hx) = 0.67 x 106 Oe and H z = 1.04 XYl06 Oe. The m i n i m u m hyperfine field would be n e a r l y c o n s i s t e n t with the m i n i m u m o b s e r v e d f r e q u e n c y . Although the a n i s o t r o p y of the hyperfine field is of the s a m e o r d e r as that o b s e r v e d for TmGaG, [7] one does not expect an exact c o r r e s p o n d e n c e between the two c a s e s b e c a u s e of the d i f f e r e n c e in c r y s t a l fields and effect of the exchange i n t e r actions. F u r t h e r studies of T m and Eu NMR u s i n g single single c r y s t a l s in e x t e r n a l l y applied fields a r e in p r o g r e s s and will be r e p o r t e d on in the n e a r future.
R¢f~rences 1. Le Dang Khoi, Phys. Letters 28A (1969) 671. 2. R.L.Streever, Phys. Letters 29A (1969) 710. 3. R.L.Cohen, Phys. Letters 5 (1963) 177. 4. S.Geller, J.P.Remeika, R.C.Sherwood, H.J. Williams and G. P. Espinosa, Phys. Rev. 137A (1965) 1034. 5. S.M.Myers, R.Gonano and H.Meyer, Phys. Rev. 170 (1968) 513. 6. G.J.Ritter, Phys. Rev. 128 (1962) 2238. 7. E.D.Jones, J. Phys. Chem. Solids 29 (1968) 1305.
163
PHYSICS LETTERS
i~ dTo (iE - L)p' + ~ Jx ~ Pe = 0
where G+ -= (i~ - L ) - I and s u b s t i t u t i n g this into eq. (3) we find after some i n t e g r a t i n g by p a r t s
(2)
The Liouville equation is now in a convenient f o r m for manipulation. In d e r i v i n g an e x p r e s s i o n for the t h e r m a l r e s i s t i v i t y we find it convenient to make some changes in the p r o c e d u r e suggested by Edwards. We want to split p' into the sum of two p a r t s p~ and p~. such that p~ accounts for all the c u r r e n t . A r a t h e r g e n e r a l way of doing this is to write P'l = PeJxJx ~ where 13-1 = fd~j2xPe, d~ is an e l e m e n t of phase space, a n d J x is the m a croscopic c u r r e n t . On multiplying the Liouville equation by Jx and i n t e g r a t i n g over all phase space we have
iCJx - gx f d~JxL(JxPe~) -
dT°
f d~JxLP~.= i~2 fi-1 dTo
=
dx
-
(3)
The first term in eq. (3) is vanishingly small and the second is zero because the integrand is an odd function of the momenta. W e m a y write [I]
G+[ i~2 .
NUCLEAR
dT° + L p i )
MAGNETIC
r
ikT2 ((Ljx)G+(L(JxPe)))
]j
(5)
where the s c a l a r product b r a c k e t s denote i n t e g r a tions over all phase space and the horizontal b a r s denote a v e r a g e s over equivalent d i s o r d e r e d s y s t e m s . The t e r m s m u l t i p l y i n g Jx on the right hand side of eq. (5) define the t h e r m a l r e s i s t i v i t y . The quantum m e c h a n i c a l f o r m of eq. (5) may be obtained by following well defined r u l e s . The u s e f u l n e s s of this type of e x p r e s s i o n lies in the fact that a finite r e s u l t is obtained for a t r i a l G r e e n function G+ c o n s t r u c t e d f r o m undamped propagating modes of the s y s t e m in cont r a s t to the Kubo f o r m u l a and v a r i a t i o n a l f o r m s of it. It is t h e r e f o r e a useful s t a r t i n g point for complex s y s t e m s where one has some intuitive idea about the states of the s y s t e m .
Reference
(4)
RESONANCE
23 February 1970
1. s. F. Edwards, Proc. Phys. Soc. 86 (1965) 977.
OF
169Tm
IN T H U L I U M
IRON
GARNET
R. L. STREEVER and P. J. CAPLAN
Institute for Exploratory Research, US Army Electronics Command, Fort Monmouth, New Jersey, USA Received 5 January 1970
The nuclear magnetic resonance of 169Tm has been studied in polycrystalline thulium iron garnet at 4.2°K in zero field using the spin-echo technique.
Recently we have been studying the nuclear magnetic resonance (NMR) of rare earth nuclei in the rare earth iron garnets. As far as we know. the N M R of the rare earth isotopes have been observed only for gadolinium [1] and e u r o p i u m [2] i r o n g a r n e t . In this note we report on the o b s e r v a tion of the 169Tm NMR in polycrystailine thulium i r o n g a r n e t (TmIG) powder at 4.2°K in zero external dc field u s i n g the s p i n - e c h o technique. Line shapes obtained by plotting the s p i n - e c h o a m p l i 162
tudes as a function of frequency a r e shown (fig. 1). The e x p e r i m e n t a l echo a m p l i t u d e s at each f r e quency have been divided by the frequency s q u a r e d to take into account the fact that the n u c l e a r s i g nal induced in the s a m p l e coil is p r o p o r t i o n a l to frequency and to account for the i n c r e a s e d n u c l e a r p o l a r i z a t i o n at higher f r e q u e n c i e s . The decay times, T2, were frequency dependent and in g e n e r a l n o n - e x p o n e n t i a l varying from about 10 to 60 ~sec. The b r o a d v a r a t i o n of T 2 and r e s -
Volume 31A, number 4
PHYSICS LETTERS
IO 169 8
Tm
4.2" K 6 4 2 O 2OO
k
300
400
500
600
700
FREQUENCY- MH!
Fig. 1. C o r r e c t e d spin echo s p e c t r u m (see text) of 169Tm in TmIG at 4.2OK.
onance f r e q u e n c y i n d i c a t e s that the o b s e r v e d s i g n a l s a r e coming, to a l a r g e extent, from nuclei in domain walls. The situation in T m I G is s i m i l a r to that in EuIG [2]. The hyperfine fields a r e a n i s o t r o p i c and depend on the o r i e n t a t i o n of the m a g n e t i z a t i o n M with r e s p e c t to the local o r t h o r h o m b i c axes of the r a r e e a r t h site. We can w r i t e the hyperfine field at the r a r e e a r t h site as Heft
=
(/_/x2nx2 + Hy2
ny
2 +H 2
Z nZ
2)½
(1)
Here Hx, Hy and H z a r e the p r i n c i p a l values of H etf a n d n x , ny a n d n z a r e the d i r e c t i o n c o s i n e s of M with r e s p e c t to the local o r t h o r h o m b i c axes defined in ref. 2. F r o m 169Tm M 6 s s b a u e r studies at 20OK Cohen [3] has obtained hyperfine fields at the two sets of i n e q u i v a l e n t s i t e s c o r r e s p o n d i n g to M along a (111) d i r e c t i o n of 1.70 and 0.81 × 106 Oe. Since the T m ° s u b l a t t i c e m a g n e t i z a t i o n in TmIG has n e a r l y the s a m e value [4,5] at 4.2°K as at 20°K, we expect the hyperfine fields to be n e a r l y the s a m e at these t e m p e r a t u r e s . The n u c l e a r m o m e n t
23 February 1970
of 169Tm is about -0.229 n . m . [6] and its spin is ½. We show (fig. 1) by a r r o w s the f r e q u e n c i e s c o r r e s p o n d i n g to the two (111) hyperfine fields. The two (111) hyperfine fields give i n s u f f i c i e n t i n f o r m a t i o n to evaluate the t h r e e hyperfine field p a r a m e t e r s , while the NMH f r o m nuclei in domain walls is difficult to i n t e r p r e t . The c o r r e c t set of hyperfine field p a r a m e t e r s should, however, give c o r r e c t l y the (111) fields and be c o n s i s t e n t as well with the l i m i t i n g NMR f r e q u e n c i e s . If we take the m a x i m u m o b s e r v e d hyperfine field (about 1.95 x 106 Oe) c o r r e s p o n d i n g to the line at 680 MHz as Hx (or H v ) and u s e the M S s s b a u e r v a l u e s and eq. (1) we obtain H. (or Hx) = 0.67 x 106 Oe and H z = 1.04 XYl06 Oe. The m i n i m u m hyperfine field would be n e a r l y c o n s i s t e n t with the m i n i m u m o b s e r v e d f r e q u e n c y . Although the a n i s o t r o p y of the hyperfine field is of the s a m e o r d e r as that o b s e r v e d for TmGaG, [7] one does not expect an exact c o r r e s p o n d e n c e between the two c a s e s b e c a u s e of the d i f f e r e n c e in c r y s t a l fields and effect of the exchange i n t e r actions. F u r t h e r studies of T m and Eu NMR u s i n g single single c r y s t a l s in e x t e r n a l l y applied fields a r e in p r o g r e s s and will be r e p o r t e d on in the n e a r future.
R¢f~rences 1. Le Dang Khoi, Phys. Letters 28A (1969) 671. 2. R.L.Streever, Phys. Letters 29A (1969) 710. 3. R.L.Cohen, Phys. Letters 5 (1963) 177. 4. S.Geller, J.P.Remeika, R.C.Sherwood, H.J. Williams and G. P. Espinosa, Phys. Rev. 137A (1965) 1034. 5. S.M.Myers, R.Gonano and H.Meyer, Phys. Rev. 170 (1968) 513. 6. G.J.Ritter, Phys. Rev. 128 (1962) 2238. 7. E.D.Jones, J. Phys. Chem. Solids 29 (1968) 1305.
163