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Submitted on 1 Jan 1978
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THE LIFETIME OF PHONONS IN THE ORBACH
RELAXATION PROCESS OF CMN
A. van der Bilt, A. van Duyneveldt
To cite this version:
JOURNAL DE PHYSIQUE
Colloque C6, supplément au n° 8, Tome 39, août 1978, page C6-1002
THE LIFETIME OF PHONONS IN THE ORBACH RELAXATION PROCESS OF CMN
A. Van der Bilt and A.J.van Duyneveldt
Kamerlingh Onnes Laboratory, University of Leiden, The Netherlands.
Résumé.- Le présent article décrit la relaxation Orbach avec "phonon bottleneck" en fonction du champ et de la concentration en ions Ce dans LaMN. Les résultats prouvent que le temps de vie des phonons est% 10-8s,indépendamment de la température entre 1,5 et 4,2 K.
Abstract.- The bottlenecked Orbach relaxation process was studied as a function of field and concentration of Ce ions in LaMN. It is shown that the lifetime of the phonons in the relaxation equals nJ 10~°s, independent of temperature between 1.5 and 4.2 K.
In the Orbach relaxation process the energy from the spin system is transferred via two phonon bands, which are centered around energy A of the excited doublet. The width . of these phonon bands is of the order of the width of the excited doublet, and is in fact the reason for observing three dis-tinct regions in the x(H) curves if the relaxation is bottle-necked /l/. At small magnetic fields the phonon bands overlap, so no net effect on the relaxa-tion time x is seen. At intermediate fields the overlap reduces upon increasing H, which causes an increasing bottleneck. Explicite calculation /l/ for this field region shows that x will vary as H z c * . A t strong fields the phonon bands are se-parated, the bottleneck being at maximum and inde-pendent of the actual field.
Lichti and Culvahouse 111 determined x for (Ce :La, )MN at 4.2 K as a function of the cerium
c i—c
concentration c. From analyzing their results and other data available from the literature, they con-cluded that the phonon relaxation time x , which is the essential factor in the bottleneck, is of the order of 10~° to 10""s. In this paper we report on non-resonance studies of the bottlenecked Orbach process, thus allowing T,H and c to be varied 131.
In this way we determine x and show that the pho-non lifetime does not change upon varying T.
Measurements of the differential suscepti-bility were performed on a series of samples (Ce : La )MN with c in the range 0.06 - 1.0. We deter-mined . x(H) curves at various temperatures between 4.2 and 0.36 K, in the usual way /3/. A typical set of such curves at 2.1 K is given in the figure. At the weakest fields x becomes independent of field, then x increases with H and is described
satisfac-I
torily by x °= H . The region of separated phonon bands, corresponding to field-independent
relaxa-tion times is missing in the figure.
Fig. 1 : Relaxation time at T : 2.10 K as a func-tion of external magnetic field for CMN samples of different magnetic dilution.
The curves tend downward above 1 kOe, an effect arising due to the onset of the direct relaxation process, which happens to be the more effective way of energy exchange at strong fields. The
r e l a x a t i o n times i n t h e i n t e r m e d i a t e f i e l d r e g i o n were found t o follow t h e p r e d i c t e d c concentra- t i o n dependence of T.
We determined T(T) curves i n t h e f i e l d r e g i o n of 100 Oe
-
1 kOe. These d a t a can b e analy- sed i f one c o n s i d e r s n o t only t h e e x p o n e n t i a l temperature dependence of t h e Orbach r e l a x a t i o n time, b u t a l s o t h e p a r a l l e l o p e r a t i n g p r o c e s s e s . The d e t a i l s w i l l be p u b l i s h e d elsewhere, f o r our p r e s e n t aim i t s u f f i c e s t o mention t h a t a l l our d a t a i n t h i s f i e l d r e g i o n were d e s c r i b e d byThe f i r s t term r e p r e s e n t s t h e b o t t l e n e c k e d Orbach p r o c e s s , dominant above 1.5 K. The second one i s t h e Raman p r o c e s s , which i s n o t dominant between 4.2 and 0.36 K, b u t on t h e o t h e r hand i n f l u e n c e s t h e observed T(T) curves j u s t enough t o be essen- t i a l f o r t h e a n a l y s i s . The t h i r d term i s not t h e d i r e c t p r o c e s s , though i t s temperature dependence would s u g g e s t i t , b u t t h e o r d e r of magnitude of
t h e so-called i m p u r i t y r e l a x a t i o n , e f f e c t i v e below 1 K ~ ) . We w i l l not d i s c u s s t h i s term now, b u t i t i s worth mentioning t h a t i t i s fundamental f o r allowing CMN t o be u s e f u l f o r low-temperature t h e r - mometry.
The l i f e t i m e of phonons i n t h e two bands can b e determined from t h e f i r s t term i n e q u a t i o n
( 1 ) . The important parameter i s t h e b o t t l e n e c k f a c - t o r , Q, which f o r t h e c a s e of CMN can be given a s Q = 2 N r p / ( h p ( h ) ) , where p(A) i s t h e d e n s i t y of s t a - t e s of t h e phonon spectrum / 2 / . We used s i n g l e c r y s t a l s a s w e l l a s powdered samples, t h e numerical r e s u l t s f o r t h e b o t t l e n e c k e d Orbach p r o c e s s being i d e n t i c a l . T h i s means t h a t t h e phonon l i f e t i m e i s n o t determined by t h e time needed f o r a phonon t o
-
t r a v e r s e t h e c r y s t a l , s o t h a t we can c a l c u l a t e T = 4 x 1 0 - ~ s a s an upper l i m i t . I n t h e f o l l o w i n g
P
we use T = 10-~s f o r an o r d e r of magnitude calcu-
P
l a t i o n . The f a c t o r p(A) i n Q i s p r o p o r t i o n a l t oN/V
and t h u s t o c . Using t h e r e l e v a n t v a l u e s f o r t h e p a r a m e t e r s , we a r r i v e a t Q = 104c, from which eq.(8) i n r e f . 2 f o l l o w s a s 1 T-'= 6.7 x I O - ~ T ; ~ c- (rCH)-,
(2)where
ro
i s t h e Orbach r e l a x a t i o n time w i t h o u t a b o t t l e n e c k and r c r e p r e s e n t s t h e f i n i t e l i f e t i m e of t h e e x c i t e d d o u b l e t . I t was argued i n / 2 / t h a t T should b e e q u a l t o h a l f t h e c o e f f i c i e n t of r o .From our experiments a t weak f i e l d s
we obtain.^^
= 10-llexp(36.25/kT), SO1 1
T-l= 1 0 1 2 ~ - ? c- C' exp(-36.25/kT). (3) T h i s r e s u l t i s i n good agreement w i t h t h e e x p e r i - mental r e s u l t s ( e q u a t i o n ( 1 ) ) .
The time c o n s t a n t s a t s t r o n g f i e l d s can be c a l c u l a t e d / 2 / , a s rmC r,(l+~):. C o n s i d e r a t i o n s about T- can be used t o r e c o n s i d e r our e s t i m a t e
of t h e phonon l i f e t i m e . It t u r n s o u t t h a t a lower l i m i t & o r T may be obtained from t h e d a t a i n t h e
P
f i g u r e . Each T(H) measurement g i v e s an experimental lower l i m i t f o r t h e r a t i o T m / ~ O , s o one o b t a i n s
.A
(l+Q) 2 and t h u s Q. We considered a l l T(H) c u r v e s and t h e r e s u l t was averaged t o a v a l u e of Q , being a f a c t o r of 10 s m a l l e r t h a n t h e v k l u e used above. A s Q i s p r o p o r t i o n a l t o T t h i s means t h a t t h e
P'
lower l i m i t of T i s 10-gs, s o t h e v a l u e of 10-8s P
we used above i s a r e a s o n a b l e v a l u e indeed. Orbach and Vredevoe / 4 / suggested t h a t t r a n s v e r s e and l o n g i t u d i n a l phonons should show r e l a x a t i o n times t h a t d i f f e r by 4 o r d e r s of magni- tude. I f phonons w i t h l i f e t i m e s much longer than 10-8s were o p e r a t i v e i n t h e b o t t l e n e c k , t h e y would t r a v e r s e t h e c r y s t a l and l e a d t o s u r f a c e e f f e c t s . On t h e o t h e r hand a much s h o r t e r l i f e t i m e would g i v e an o p p o r t u n i t y t o overcome t h e b o t t l e n e c k . There i s no i n d i c a t i o n from our experiments t h a t e i t h e r one of t h e s e s i t u a t i o n s occurs.
Summarizing one may conclude t h a t t h e bot- tlenecked Orbach r e l a x a t i o n i s d e s c r i b e d adequately w i t h one
-
temperature independent-
phonon l i f e - time of about 1 0 - ~ s . Numerically t h i s r e s u l t a g r e e s w i t h t h a t i n r e f e r e n c e 2, b u t t h e p r e s e n t a n a l y s i s shows t h a t t h i s number can be used a s an average over t h e whole temperature r e g i o n of 1.5 t o 4.2 K. T h i s means t h a t t h e anharmonic i n t e r a c t i o n / 4 / can n o t b e t h e p r i n c i p a l phonon decay mechanism, a s t h i s mechanism p r e d i c t s a temperature dependence f o r T p r o p o r t i o n a l t o exp(36.25/kT). F u r t h e r s t u -P
References
/ I / G i l l , J.C., J . Phys. C? (1973) 109.
/ 2 / L i c h t i , R.L.,and Culvahouse,
