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Temperature dependence of spin-phonon relaxation time in holmium
Volume 32A, number 7
PHYSICS LETTERS
TEMPERATURE RELAXATION
DEI~ENDENCE TIME
IN
7 September 1970
OF SPIN-PHONON HOLMIUM *
M. C. LEE and M. LEVY
Department of Physics, University of California, Los Angeles, California
90024, USA
Received 20 July 1970
A phenomenological calculation is given for the temparature dependence of the spin-phonon relaxation time in the spin-spiral phase of the heavy rare earth metals. It is found that, for a single crystal of holmium, the relaxation time decreases as T -~ when the temperature is increased.
A p h e n o m e n o l o g i c a l equation for the spin phonon i n t e r a c t i o n in the s p i n - s p i r a l phase of the heavy r a r e e a r t h m e t a l s has b e e n d e r i v e d by Tachiki et al. [1]. They obtain, for the longitud i n a l u l t r a s o n i c a t t e n u a t i o n coefficient along the c-axis,
At~
n-s2 [a2j(q)/a¢zzaq]2
pv3
a2j(q)/aq2
ar2r 1+ 2 2
) p-
(1)
where n is the n u m b e r of s p i n s p e r unit volume; the t h e r m a l a v e r a g e of the spin a n g u l a r m o m e n t u m p e r t r i p o s i t i v e ion, which is d i r e c t l y p r o p o r t i o n a l to the s u b l a t t i c e m a g n e t i z a t i o n ; v l the longitudinal sound velocity along the c - a x i s ; p the density; J(q) the F o u r i e r t r a n s f o r m of the exchange i n t e g r a l ; q the s p i n wave v e c t o r along the c - a x i s ; ~zz the s t r a i n produced by the sound wave along the c - a x i s ; co the sound wave f r e quency; • the r e l a x a t i o n t i m e needed by the spin s y s t e m to r e a c h its new e q u i l i b r i u m c o n f i g u r a tion u n d e r the s t r a i n field produced by the sound wave. A s is c a l i b r a t e d a g a i n s t the background a t t e n u a t i o n in the p a r a m a g n e t i c state above the N6el t e m p e r a t u r e . In fig. 1 we show, for a single c r y s t a l of holm i u m , Air v e r s u s t e m p e r a t u r e data obtained u s i n g 15 MHz l o n g i t u d i n a l sound waves along the c - a x i s . T h e s e data a r e r e p r o d u c e d f r o m Levy and Lee [2]. T h e r e a r e two s h a r p peaks at TN and T c r e s p e c t i v e l y , which a r e due to the fluctuations in the o r d e r p a r a m e t e r [3]. A broad a t t e n u a t i o n m a x i m u m i s a l s o p r e s e n t at about 97.5OK which is p r e s u m a b l y due to the c o m p e t i tion between v a r i o u s t e m p e r a t u r e - d e p e n d e n t * Research sponsored by the Air Force Office of Scientific Research under AFOSR Grant No. 70-1847.
(IX 0
T (oK)
Fig. 1. Attenuation of longitudinal ultrasonic wave along the c-axis in Ho. Frequency = 15 M H z .
p a r a m e t e r s , n a m e l y ~, Ja and 7. We a r e a s s u m !ng that.the quantity n / p ~l s t a y s c o n s t a n t through m e WhoLe t e m p e r a t u r e region. F u r t h e r m o r e it is r e a s o n a b l e to a s s u m e that ~" is a d e c r e a s i n g f u n c tion of t e m p e r a t u r e . Let us suppose that at t e m p e r a t u r e T = To, W~o = 1; then f r o m eq. (1) s o l v i n g for ~- we obtain
.-, o
(2)
where
[~2j(q) /aC zz~ q]2 /[ O2j(q) /aq2 ] = [a2j(q)/~zzaq]2/[a2j(q)/Sq2] 0 F o r T < To we have s > So, and in o r d e r to have 0 ~ > 1 we m u s t choose the plus sign. In a f i r s t o r d e r a p p r o x i m a t i o n which i g n o r e s second o r d e r effects, J(q) can be e x p r e s s e d a s [4]
505
V o l u m e 32A
number 7
PHYSICS
100
LETTERS
7 S e p t e m b e r 1970
B 2 = - / ~ H c o / 4 G o ( 1 - c o s ~qoC) 2 .
I
a 2 j ( q ) / a ¢ z z a q c a n b e c o m p u t e d in the f o l l o w i n g 50
way:
a2 j ( q ) / a ¢ z z a q = [~la T(aJ(q)/~q)] a T l ( a ill) = (~2j(q)/~q2)(Oq/a T) a-T1
2.0
s i n c e aEzz = Ezz = a l / l . l i s t h e l e n g t h of the s a m p l e , ot T is the t h e r m a l e x p a n s i o n c o e f f i c i e n t w h i c h we a s s u m e c o n s t a n t , a q / a T i s the c h a n g e of the t u r n i n g a n g l e w i t h t e m p e r a t u r e and i s a c o n s t a n t in a f i r s t o r d e r a p p r o x i m a t i o n . T h e n f r o m eq. (3) 1 y = ( c o s ½qo c c o s -~qc - c o s q c ) / s i n 2 ~qo i c "
x
0.5
0.2 0.1 10
20
50 T (°K)
100
E0
Fig. 2. Relaxation time v e r s u s temperature. Dots are data points and solid line is the plot of a T -3.
J(q) s 2 = { B 0 + 2B 1 c o s (qc/2) + 2B2 c o s (qe)} (r2 (3) w h e r e B0, B1, and B 2 a r e c o n s t a n t s * , c i s the l a t t i c e c o n s t a n t , and a = ~ / s . s i s t h e s p i n a n g u l a r m o m e n t u m p e r ion at T = 0°K. We e v a l u a t e B 1 and B 2 at T = T O u s i n g the f o l l o w i n g two c r i t e r i a : (a) ~J(q)/~q = 0 f o r the s t a b i l i t y of the s p i n - - s p i r a l p h a s e , and (b) at T = To, j ( q ) ~ 2 _ J ( 0 ) s Z = ao P/'/co w h e r e ~ is the m a g n e t i c m o m e n t p e r ion at T = 0OK. / / c o is the c r i t i c a l m a g n e t i c f i e l d n e c e s s a r y to c h a n g e a s p i n s p i r a l p h a s e to the f e r r o m a g n e t i c p h a s e at T o . T h e n we o b t a i n B 1 = U H c o c o s ½qoC/ao(1 - c o s ½qoc) 2 , * I g n o r i n g the t e m p e r a t u r e d e p e n d e n c e of B 2 i n t r o d u c e s about 1% e r r o r in J(q).
506
In the c a s e of h o l m i u m , we c h o o s e q u i t e a r b i t r a r i l y T o = 120OK and i n v e s t i g a t e the r e l a t i v e c h a n g e of 7 b e l o w t h i s t e m p e r a t u r e . We u s e K o e h l e r et a l . ' s q v a l u e s [5] and n o r m a l i z e L i u et a l . ' s Dy s u b l a t t i c e m a g n e t i z a t i o n c u r v e [6] b e t w e e n 132°K and 20°K. F i g . 2 s h o w s ~ ' v e r s u s t e m p e r a t u r e a s c a l c u l a t e d f r o m eq. (2). The s o l i d c u r v e i s a plot of a T -3 v e r s u s T w i t h a = 2.16 x 10 -2 s e c °K3. T h u s , the l o n g i t u d i n a l s p i n phonon r e l a x a t i o n t i m e a l o n g the e - a x i s in h o l m i u m a p p e a r s to d e c r e a s e a s T -3 a s the t e m p e r a t u r e i s i n c r e a s e d . T h e r o l l - o f f f o r the low t e m p e r a t u r e r e g i o n is p r o b a b l y due to the p r e s e n c e of a n i s o t r o p y and m a g n e t o s t r i c t i o n w h i c h we h a v e not i n c l u d e d in our calculation.
Ref e~'ences [: ] M. Tachiki, M. Levy, R. Kagiwada and M. C. Lee, Phys. Rev. L e t t e r s 21 (1968) 1193. [Z] M. Levy and M. C. Lee, Phys. L e t t e r s 32A (1970) 294. [3] R.J. Pollina and B. LUthi, Phys. Rev. 177 (1969) 841. [4] R . J . Elliott, Phys. Rev. 124 (1961) 346. [5] W. C. Koehler et al., Phys. Rev. 151 (1966) 414. [6] S. H. Liu et al., Phys. Rev. 116 (1959) 1464.
PHYSICS LETTERS
TEMPERATURE RELAXATION
DEI~ENDENCE TIME
IN
7 September 1970
OF SPIN-PHONON HOLMIUM *
M. C. LEE and M. LEVY
Department of Physics, University of California, Los Angeles, California
90024, USA
Received 20 July 1970
A phenomenological calculation is given for the temparature dependence of the spin-phonon relaxation time in the spin-spiral phase of the heavy rare earth metals. It is found that, for a single crystal of holmium, the relaxation time decreases as T -~ when the temperature is increased.
A p h e n o m e n o l o g i c a l equation for the spin phonon i n t e r a c t i o n in the s p i n - s p i r a l phase of the heavy r a r e e a r t h m e t a l s has b e e n d e r i v e d by Tachiki et al. [1]. They obtain, for the longitud i n a l u l t r a s o n i c a t t e n u a t i o n coefficient along the c-axis,
At~
n-s2 [a2j(q)/a¢zzaq]2
pv3
a2j(q)/aq2
ar2r 1+ 2 2
) p-
(1)
where n is the n u m b e r of s p i n s p e r unit volume; the t h e r m a l a v e r a g e of the spin a n g u l a r m o m e n t u m p e r t r i p o s i t i v e ion, which is d i r e c t l y p r o p o r t i o n a l to the s u b l a t t i c e m a g n e t i z a t i o n ; v l the longitudinal sound velocity along the c - a x i s ; p the density; J(q) the F o u r i e r t r a n s f o r m of the exchange i n t e g r a l ; q the s p i n wave v e c t o r along the c - a x i s ; ~zz the s t r a i n produced by the sound wave along the c - a x i s ; co the sound wave f r e quency; • the r e l a x a t i o n t i m e needed by the spin s y s t e m to r e a c h its new e q u i l i b r i u m c o n f i g u r a tion u n d e r the s t r a i n field produced by the sound wave. A s is c a l i b r a t e d a g a i n s t the background a t t e n u a t i o n in the p a r a m a g n e t i c state above the N6el t e m p e r a t u r e . In fig. 1 we show, for a single c r y s t a l of holm i u m , Air v e r s u s t e m p e r a t u r e data obtained u s i n g 15 MHz l o n g i t u d i n a l sound waves along the c - a x i s . T h e s e data a r e r e p r o d u c e d f r o m Levy and Lee [2]. T h e r e a r e two s h a r p peaks at TN and T c r e s p e c t i v e l y , which a r e due to the fluctuations in the o r d e r p a r a m e t e r [3]. A broad a t t e n u a t i o n m a x i m u m i s a l s o p r e s e n t at about 97.5OK which is p r e s u m a b l y due to the c o m p e t i tion between v a r i o u s t e m p e r a t u r e - d e p e n d e n t * Research sponsored by the Air Force Office of Scientific Research under AFOSR Grant No. 70-1847.
(IX 0
T (oK)
Fig. 1. Attenuation of longitudinal ultrasonic wave along the c-axis in Ho. Frequency = 15 M H z .
p a r a m e t e r s , n a m e l y ~, Ja and 7. We a r e a s s u m !ng that.the quantity n / p ~l s t a y s c o n s t a n t through m e WhoLe t e m p e r a t u r e region. F u r t h e r m o r e it is r e a s o n a b l e to a s s u m e that ~" is a d e c r e a s i n g f u n c tion of t e m p e r a t u r e . Let us suppose that at t e m p e r a t u r e T = To, W~o = 1; then f r o m eq. (1) s o l v i n g for ~- we obtain
.-, o
(2)
where
[~2j(q) /aC zz~ q]2 /[ O2j(q) /aq2 ] = [a2j(q)/~zzaq]2/[a2j(q)/Sq2] 0 F o r T < To we have s > So, and in o r d e r to have 0 ~ > 1 we m u s t choose the plus sign. In a f i r s t o r d e r a p p r o x i m a t i o n which i g n o r e s second o r d e r effects, J(q) can be e x p r e s s e d a s [4]
505
V o l u m e 32A
number 7
PHYSICS
100
LETTERS
7 S e p t e m b e r 1970
B 2 = - / ~ H c o / 4 G o ( 1 - c o s ~qoC) 2 .
I
a 2 j ( q ) / a ¢ z z a q c a n b e c o m p u t e d in the f o l l o w i n g 50
way:
a2 j ( q ) / a ¢ z z a q = [~la T(aJ(q)/~q)] a T l ( a ill) = (~2j(q)/~q2)(Oq/a T) a-T1
2.0
s i n c e aEzz = Ezz = a l / l . l i s t h e l e n g t h of the s a m p l e , ot T is the t h e r m a l e x p a n s i o n c o e f f i c i e n t w h i c h we a s s u m e c o n s t a n t , a q / a T i s the c h a n g e of the t u r n i n g a n g l e w i t h t e m p e r a t u r e and i s a c o n s t a n t in a f i r s t o r d e r a p p r o x i m a t i o n . T h e n f r o m eq. (3) 1 y = ( c o s ½qo c c o s -~qc - c o s q c ) / s i n 2 ~qo i c "
x
0.5
0.2 0.1 10
20
50 T (°K)
100
E0
Fig. 2. Relaxation time v e r s u s temperature. Dots are data points and solid line is the plot of a T -3.
J(q) s 2 = { B 0 + 2B 1 c o s (qc/2) + 2B2 c o s (qe)} (r2 (3) w h e r e B0, B1, and B 2 a r e c o n s t a n t s * , c i s the l a t t i c e c o n s t a n t , and a = ~ / s . s i s t h e s p i n a n g u l a r m o m e n t u m p e r ion at T = 0°K. We e v a l u a t e B 1 and B 2 at T = T O u s i n g the f o l l o w i n g two c r i t e r i a : (a) ~J(q)/~q = 0 f o r the s t a b i l i t y of the s p i n - - s p i r a l p h a s e , and (b) at T = To, j ( q ) ~ 2 _ J ( 0 ) s Z = ao P/'/co w h e r e ~ is the m a g n e t i c m o m e n t p e r ion at T = 0OK. / / c o is the c r i t i c a l m a g n e t i c f i e l d n e c e s s a r y to c h a n g e a s p i n s p i r a l p h a s e to the f e r r o m a g n e t i c p h a s e at T o . T h e n we o b t a i n B 1 = U H c o c o s ½qoC/ao(1 - c o s ½qoc) 2 , * I g n o r i n g the t e m p e r a t u r e d e p e n d e n c e of B 2 i n t r o d u c e s about 1% e r r o r in J(q).
506
In the c a s e of h o l m i u m , we c h o o s e q u i t e a r b i t r a r i l y T o = 120OK and i n v e s t i g a t e the r e l a t i v e c h a n g e of 7 b e l o w t h i s t e m p e r a t u r e . We u s e K o e h l e r et a l . ' s q v a l u e s [5] and n o r m a l i z e L i u et a l . ' s Dy s u b l a t t i c e m a g n e t i z a t i o n c u r v e [6] b e t w e e n 132°K and 20°K. F i g . 2 s h o w s ~ ' v e r s u s t e m p e r a t u r e a s c a l c u l a t e d f r o m eq. (2). The s o l i d c u r v e i s a plot of a T -3 v e r s u s T w i t h a = 2.16 x 10 -2 s e c °K3. T h u s , the l o n g i t u d i n a l s p i n phonon r e l a x a t i o n t i m e a l o n g the e - a x i s in h o l m i u m a p p e a r s to d e c r e a s e a s T -3 a s the t e m p e r a t u r e i s i n c r e a s e d . T h e r o l l - o f f f o r the low t e m p e r a t u r e r e g i o n is p r o b a b l y due to the p r e s e n c e of a n i s o t r o p y and m a g n e t o s t r i c t i o n w h i c h we h a v e not i n c l u d e d in our calculation.
Ref e~'ences [: ] M. Tachiki, M. Levy, R. Kagiwada and M. C. Lee, Phys. Rev. L e t t e r s 21 (1968) 1193. [Z] M. Levy and M. C. Lee, Phys. L e t t e r s 32A (1970) 294. [3] R.J. Pollina and B. LUthi, Phys. Rev. 177 (1969) 841. [4] R . J . Elliott, Phys. Rev. 124 (1961) 346. [5] W. C. Koehler et al., Phys. Rev. 151 (1966) 414. [6] S. H. Liu et al., Phys. Rev. 116 (1959) 1464.