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Influence of interchain coupling on the one-dimensional magnon raman spectrum of CsCoBr3
S o l i d State C o u m u n i c a t i o n s , V o l . 3 6 , pp.1033-1037. Pergamon P r e s s Lid, 1980. P r i n t e d i n G r e a t B r i t a i n .
INFLUENCE OF INTERCHAIN COUPLING ON THE ONE-OIHENSIONAL MAGNON RAMAN SPECTRUH OF CsCoBr 3 I.W. Johnstone, D.J. Lockwood and M.W.C. Dharma-wardana D i v i s i o n o f Physics, National Research C o u n c i l , Ottawa, Canada KIA OR6 (Received 25 August 1980 by H. F. Collins) L i g h t s c a t t e r i n g from magnons in CsCoBr 3 has been measured f o r temperatures w e l l below the upper a n t i f e r r o m a g n e t i c ]D o r d e r i n g temperature o f T~.=. 28 K. These experiments reveal m u l t i p l e magnon f e a t u r e s o f energies in the range 90 Co 170 ~m"1 s l m i l a r to those f o . n d in CsCoC| 3 but p r e v i o u s l y unobserv=d f o r ~he hromlde. Prominent f e a t u r e s in the spectrum and t h e i r temperature dependence are described in terms o f a recent theory by Shiba. Other, weaker f e a t u r e s are e x p l a i n e d by a simple e x t t n s l o n o f the theory to Inc!ude f l u c t u a t l o n s . A new band is observed a t 178 ~m-1, whose, i n t e n s i t y drops s%harply p r i o r Lo Cl~e lower ]O o r d e r i n g t r a n s i t i o n a t T~ I - 10 K. This band is assigned to magnon-phonon combination s c a t t e r i n g .
1.
Introduction
U n t i l now, the spin-wave Roman s p e c t r a o f CsCoCI 3 and CsCoBr31"3 have bcen d i f f i c u l t to understand in terms o f c u r r e n t t h e o r i e s o f I s i n g Holsenberg chains, which are themselves the s u b j e c t o f much debate, k ' 8 Both m a t e r i a l s s i m u l a t e I-D I s l n g a n t l f e r r o m a g n e t i c chains but d i s p l a y )-D magnetic o r d e r a t s u f f i c i e n t l y low temperatures. In CsCoBr3, the l a r g e l n t r a c h a i n n e a r e s t - n e l g h b o u r exchange i n t e r a c t i o n , J, g i v e s r i s e to a t r a n s v e r s e spin e x c i t a t i o n band near 100 cm"1. For T ~ 30 K the Raman spectrum appears as an asyr~ne.tric peak cuper!mposed on a broader band o f e x c i t a t i o n s . On c o o l i n g , the spectrum is remarkable f o r the appearance o f a l a r g e number o f d i s c r e t e l i n e s whose i n t e n s i t i e s are c r i t i c a l l y dependent on the c r y s t a l temperature. Below l0 K, the spin arrangement in CsCoBr 3 comprises t h a t o f a t r i a n g u l a r (a-b plane) I s l n g lattice with Interchain antiferromagnetic n e a r e s t - n e i g h b o u r c o u p l i n g , J~, very weak f e r r o magnetic n e x t - n e a r e s t - n e i g h b o u r I n t e r c h a i n I n t e r a c t i o n s , J~, and the s t r o n g I s i n g c o u p l i n g along the chains (e a x i s ) . Between 1~ K and T~ - 28.3 K, o n e - t h i r d o f the chains become paramagnetic, w h i l e f o r 10 K < T~ I < 14 K, these two magnetic phases are thought t o c o e x i s t . Above T~ a l l the chains are d i s o r d e r e d but r e t a i n s i g n i f i c a n t spin c o r r e l a t i o n s along the e a x i s f o r temperatures beyond I00 K. Estimates o f the s t r e n g t h s o f the v a r i o u s magnetic i n t e r a c t i o n s J, J~ and J~ can be made from s u ~ c e p t i b i l i t y data 9 and the N~el temperatures T~ and T~ I. Thus J = 56 cm"1, J~/J = 0 . 0 , 3 and J~/J = -0.03. z The f i n e s t r u c t u r e In the low temperature (T < T~) Raman spectrum has been v a r i o u s l y a s c r i b e d to magnon conlbination s t a t e s o r m u l t i p l e bound s t a t e s . 1"3 Recently, the p e r t u r b a t i o n theory approach o f Ishimara and Shlba 7 f o r a s i n g l e chain has been extended by Shiba 8 to i n c l u d e i n t e r c h a i n c o u p l i n g and thereby e x p l a i n the spin e x c i t a t i o n spectrum o f CsCoCI 3
w i t h c o n s i d e r a b l e success. As a f u r t h e r t e s t o f t h i s approach to n o n l l n e a r magnon e x c l t a t i o n processes we apply Shlba's theory to new lowtemperature data f o r the bromide, where the I n t e r c h a i n c o u p l i n g is a p p r o x l m a t e l y t w i c e as s t r o n g as in the c h l o r i d e . 2.
Theory
The N a m i l t o n l a n f o r a one-dlmensional s p i n I s l n g l i k e a n t l f e r r o m a g n e t is w r i t t e n asT, 8 "o
zz xx " 2J {Sjsj÷l %sj,,
sjSj÷l)}
+yy
•
(')
Since CsCoBr 3 e x h i b i t s 3-D magnetic o r d e r a t sufficiently low temperatures the e f f e c t o f . the r e l a t l v e l y weak i n t e r c h a i n c o u p l i n g , HI(A, ~), must be i n c l u d e d so t h a t the model H3milton[an has the form:
H-2H(~)+2J ' T IS~_S~ +c'(S~.S~ +sY.s~ )] k 0 j~LJ~ ~ j~ ~ jA ~u j '
(2)
where
~ and Z denote summations o v e r n,n X,U j,~ chain p a i r s and n.n i n t r a c h a l n spin p a i r s respectively. Since ¢' ~ ¢ ~ 0 . 1 , the c ' J ' term ( i . e . the i n t e r c h a l n x - y term) may be Ignored t o a good a p p r o x i m a t i o n . The remaining I s i n g I n t e r c h a i n I n t e r a c t i o n In (2) ls t r e a t e d w i t h i n a molecular field approximation (e.g. Scalaplno, Imry and Pincus tO) to g i v e a s i n g l e - c h a i n effective Hamlltonlan: H - H0 -
hjSj
(31
where h i - ( - 1 ) J h and h 2 O. The r e s u l t i n g energy | I v e l scheme Is c o n s t r u c t e d from the Ntel s t a t e IN> which is favoured by the staggered m o l e c u l a r f i e l d h, t o g e t h e r w i t h admixtures o f the low l y i n g subspace o f e x c i t e d 1033
IO34
ONE-DIMENSIONAL MAGNON EAMAN SPECTRUM OF CsCoBr 3
s t a t e s . These c o n s i s t o f two blocks o f s t a t e s corresponding t o the Sz - 0 and Sz - 1 e x c i t a t ions. Thus f o r ik.rj sZ,
1:
+
sj I""
•
+i =,/'2"/'~
J ik-rj Z •
+3"
J J
i.e.
•
+2i-1'
ik.rj
+. + S j S j + l S j + 2 IN> ÷S"
÷
S"
+2 +4
- ~
~ e J
S+
sj 3+tsj+z j+3 j+4 IN>
i ,, 1, 2, e t c .
- /2-~ e
i.e.
~2i'
ik-r.
ik-r.
+.
JSjSj+IIN> + .
+
.
JsjSj+lSj+2sj+3JN> (4b)
I " 1, 2, e t c .
Note t h a t the x - y I n t e r a c t i o n (the c term) has converted the I s i n g s p l n - f l l p s (S~, S~S]+I, +
Sj+2, e t c . )
i n t o ~ dependent p r o p a g a t i n g modes
which correspond to the s o l l t o n p i c t u r e (moving bloch w a l l ) o f V I I l a l n 5 and the dimer gas p i c t u r e o f Hubbard I I (see a l s o Fowler and Pugat2). In the l i m i t h + O, ¢ + 0 these s t a t e s have energies E21. I - E21 " 2J w i t h respect to the N~el s t a t e IN>. When the m o l e c u l a r F i e l d is r e s t o r e d the degenerate energies s p l i t to g i v e the s e r i e s E21. j - 2 J + 2 h ( 2 1 - 1 ) , and s i m i l a r l y E i " 2J+211(21). F i n a l l y , when ¢ is nonzero t~e x - y i n t e r a c t i o n couples +i to +1+2 to mix the Sz - 1 subspace. At the zone c e n t r e (k - O) t h i s r e q u i r e s the d l a g o n a l i z a t l o n o f the trldlagonal matrix 2 <+21-11H/2Jl~2i ' - I > - 1 + } c 2 . ~c 6i , i + ( 2 i . i ) ~ for i - i' - ¢
for 1 " I'
± 1.
36, No. 12
eigen v e c t o r s and eigenvalues o f (5) to o b t a i n the r e l a t i v e s t r e n g t h o f the l i n e s , which drop o f f in i n t e n s i t y as we go to h i g h e r magnon states. Thus i f we now use k t o I d e n t i f y the d i a g o n a l i z e d l i n e a r combination s t a t e s where the p r i n c i p l e c o n t r i b u t i o n comes from +21-1 o f ( 4 a ) , then
(4a)
and Sz - O:
Vol.
(5)
Note t h a t the Sz - 0 subspace mixes w i t h IN> to g i v e a p e r t u r b e d ground s t a t e Ig> and a c l u s t e r o f e x c i t e d s t a t e s e x t e n d i n g upwards from 2J + 4h + 0(¢ 2 ) but do not mlx w i t h the Sz - i subspace which gives a s e t o f energy l e v e l s e x t e n d i n g upwards from 2J + 2h + O(cZ). However, i f the Sz - 0 e x c i t a t i o n s are r e l a t e d to the, we can use the
F o l l o w i n g Shlba, 8 the magnitude o f h f o r each o f the two low temperature phases may be e a s i l y determined by c o n s i d e r i n g the n e t t molecul a r f i e l d experienced by one chain due to i t s neighbours (see f i g u r e 1). For T < T~ I , the a chains are surrounded by s i x a n t l f e r r o m a g n e t i c a l l y coupled nelghbours so t h a t the n e t t f i e l d is 6J' This gives r i s e to the A s e r i e s o f excitations. In c o n t r a s t , the n e t t f i e l d experienced by the B and ¥ chains is z e r o , g l v l n q r i s e to an e x c i t a t i o n continuum. For T~ I < T < T ~ one t h i r d o f the chains 4Y) become paramagnetic so t h a t t h e i r spin is e i t h e r up o r down w i t h respect t o the o t h e r chains w i t h equal probability. Hence the m o l e c u l a r f i e l d s experienced by the ~ and B chains are 6 J ' , 4 J ' , 2J' and O, w i t h p r o b a b i l i t i e s o f I/8, 3/8, 3/8 and I / 8 , r e s p e c t i v e l y ( f o r y c h a i n s , h - O). Thus, in t h i s phase we expect t h r e e s e r i e s o f d i s c r e t e l i n e s comprising the A, B[ and B2 s e r i e s t o g e t h e r w i t h a continuum. 3.
Experiment and r e s u l t s
Large s i n g l e c r y s t a l s (60xlO mm2) o f CsCoBr 3 were grown by R. R l t c h i e (Physics Department, U n i v e r s i t y o f Canterbury, N.Z.) and B. B r l a t ( L a b o r a t o l r e d~Optlque Physique, P a r i s , France) using the Bridgeman technique. Raman s p e c t r a were recorded under computer c o n t r o l 13 using 50 mW o f 501.7 nm argon l a s e r e x c i t a t i o n and a s p e c t r a l r e s o l u t i o n o f 2 cm" l . Suitably cleaved samples were mounted in a Thor S500 v a r i a b l e temperature c r y o s t a t where the sample is immersed in a helium exchange gas. The sample temperature could be m a i n t a i n e d to w i t h i n tO.2 K, but because o f l o c a l l a s e r h e a t i n g (~3 K) the a c t u a l c r y s t a l temperature w i t h i n the l a s e r f i l a m e n t could o n l y be e s t i m a t e d . Spectra were recorded in the 90" s c a t t e r i n g geometry X ( . . ) Y , where X, Y, Z correspond to the c r y s t a l a, b, c axes, r e s p e c t i v e l y . Typical s p e c t r a a r e presented in f i g u r e 2 f o r T - 6 and 14 K. The peaks a t 72.8, 89.9, 118.1 and 166.1 cm" ! are due to Raman s c a t t e r i n g From the E-I , E2-, E2-~ and . A I . z o n e - c e n t r e phonons, r e s p e c t i v e l y . 3 With the e x c e p t i o n o f the Elg mode these bands are f o r b i d d e n in X(ZX)Y p o l a r i z a t i o n but appear through the d e p o l a r i z a t i o n e f f e c t s o f c r y s t a l b i r e f r i n g e n c e and s t r a i n e d c r y o s t a t windows. Other f e a t u r e s present in the range 90 tO 170 cm" | are (ZX) p o l a r i z e d and are due to zone c e n t r e magnon s c a t t e r i n g analogous to t h a t observed f o r CsCoC13,1 but e x t e n d i n g over a w i d e r range o f e n e r g i e s . Frequencies o f d i s t i n c t f e a t u r e s are l i s t e d in t a b l e 1. On warming t o 14 K, new bands appear and a l l bands grow markedly in I n t e n s i t y except those a t 105.2 and 138.4 c m ' l , whlch decrease. The behaviour o f the weak 160-cm"1 band is masked by new bands appearing nearby.
V o l . 36, No. 12
O N E - D I H E N S I O NHAGNON AL R.aJ4AN SPECTRUN OF CsCoBr 3
•
•
s
(o) Figure 1
•
o
•
o
1035
•
(b)
The hexagonal arrangement o f c o b a l t ions in the c7~ plane o f CsCoBr 3 showing (a) the f u l l y ordered magnetic s t r u c t u r e f o r T < T~ I i n v o l v i n g up(+) o r down(-) s p i n s , a~d (b) the p a r t l y Ordered s t r u c t u r e For T~ I < T < T~ where the spin on atom y Is now disordered (0).
>F-Z
P
hl l-Z Z
nr
I
7o
I00
130
160
190
FREQUENCY (cm -I) Figure 2
Temperature dependence o f the 70 to 190 cm"X r e g i o n o f the Raman spectrum o f CsCoBr 3 recorded in X(ZX)Y polarization with a spectral resolution o f 2 cm" l . P denotes a phonon.
No frequency s h i f t s are o b s e r v a b l e w i t h i n t h i s temperature range. Another t e m p e r a t u r e dependent f e a t u r e occurs a t 177.7 cm-1 in both (ZZ) and (ZX) p o l a r l z a t i o n w l t h the i n t e n s i t y r a t i o I z z / I z x - 2 . 5 . This band drops even more r a p i d l y in i n t e n s i t y w i t h i n c r e a s i n ~ t e m p e r a t u r e than the magnon band a t 105.2 cm"L.
4.
Comparison w i t h theory
At low temperatures the theory p r e d i c t s a sequence o f sharp e x c i t a t i o n s (the A s e r i e s ) superimposed on a broad continuum. These bands are p r e d i c t e d to reduce In i n t e n s i t y by a f a c t o r I / 4 in the i n t e r m e d i a t e phase T~ I < T < T~. From o u r s p e c t r a a t 6 K (T
i036
Vol. 36, No. 12
ONE-DIMENSIONAL MAGNON RAMAN SPECTRUM OF CsCoBr 3
Table 1 Comparison between t h e o r y and e x p e r i m e n t f o r the f r e q u e n c i e s ( i n cm"1) and r e l a t i v e i n t e n s i t i e s ( i n b r a c k e t s ) o f peaks in the magnon Raman spectrum of. CsCoBr 3. For the c a l c u l a t i o n s J - 50.5 cm"1 and ¢ = 0 . 1 3 . The m o l e c u l a r f i e l d h - h / J .
A series (h - 0 . I I ) Expt. Theory
B! s e r i e s (~ " O.077) Expt. Theory
B2 s e r i e s ~0.045) Expt. Theory
C series (~ = 0.012) Expt. Theory
105.2 (I.00)
107 (l.O0)
I01.3
I02 (I.00)
97.1
96 (1.00)
87
86 (I.00)
138.4 (0.25)
136 (0.12)
125.6
125 (0.22)
112.1
I12 (0.42)
92.6
93 (0.74)
160
159 (0.01)
I~2.7
142 (O.03)
123.5
125 (0.15)
158
158 (0.002)
133.4
135 (0.04)
-
103 (0.47)
149
145 (0.OO4)
-
108 (0.36)
(0.08)
(T • T~ I) the sharp f e a t u r e s a t 105.2 and 138.4 cm"1 can be p o s i t i v e l y assigned t o the A s e r i e s o f bands because t h e i r I n t e n s i t y f a l l s by t h i s f a c t o r . The weak 160-cm " ! band is a l s o l i k e l y t o be an A s e r i e s band because t h e r e is no c o r r e s p o n d i n g f e a t u r e In the h i g h e r t e m p e r a t u r e phase. The f r e q u e n c i e s o f the A s e r i e s bands were c a l c u l a t e d a c c o r d i n g t o the model g i v e n in s e c t i o n 2 w i t h h - 6 J ' / J - 0 . 1 1 , J - 50.5 cm" ! and e - 0 . 1 3 . The r e s u l t s , g i v e n In t a b l e I , a r e In e x c e l l e n t agreement w i t h e x p e r i m e n t and a l s o c o n f i r m t h a t the weak band a t 160 cm"! Is Indeed an A s e r i e s peak. The r e l a t i v e I n t e n s i t i e s o f these bands are a l s o in good agreement w i t h t h e o r y as can be seen in t a b l e 1. In the I n t e r m e d i a t e phase, T~ I < T < T~, the B s e r i e s bands dominate the magnon spectrum. T h e i r f r e q u e n c i e s a r e a c c u r a t e l y reproduced by t h e o r y f o r h - 0.077 (cf. 4 J ' / J - 0.073) f o r the B! s e r i e s , and ~ - 0.045 ( o f . 2 J ' / J - 0.037) f o r the B2 s e r i e s , w i t h J and c as g i v e n above. However, because o f o v e r l a p o f bands w i t h i n B; and B2 s e r i e s i t is d i f f i c u l t t o make d e f i n i t e assignments In some cases e . g . the bands near 125 cm"1. While t h i s o v e r l a p a l s o p r e c l u d e s the e x t r a c t i o n o f a c c u r a t e i n t e n s i t y d a t a , t h e r e is a g e n e r a l q u a l i t a t i v e agreement between t h e o r y and e x p e r i m e n t a p a r t from the band near 143 cm- 1 . The t h e o r y o f Shiba as a p p l i e d h e r e does n o t p r e d i c t any m a g n e t i c s c a t t e r i n g below the B2 band a t 97.1 cm" t . However, f e a t u r e s a r e observed a t 92.6 cm- l and ~87 cm" | on e i t h e r s i d e o f the E2a phonon a t 89.9 cm" 1 . A l t h o u g h these f e a t u r e s - h a v e low i n t e n s i t y they a r e r e p r o d u c i b l e and t h e y m a i n t a i n t h e i r i n t e n s i t y w i t h I n c r e a s i n g t e m p e r a t u r e as shown In f i g u r e 2. An approach t o u n d e r s t a n d i n g these Iowi n t e n s i t y f e a t u r e s in the spectrum may be made as f o l l o w s . W l t h i n the m e a n - f i e l d scheme used here l e t us w r i t e
= ½(!
"
= ½(l
+ AS)
-½(i + a~)
Y
- ~½(I
+ ay) +
Ay)
I f o r T < TN I II f o r TN < T < T N
98 (0.58)
where the small c o r r e c t i o n terms Aa, AB, Ay a r e t e m p e r a t u r e dependent. In f a c t Ay is expected t o go t o zero a t the t r a n s i t i o n t e m p e r a t u r e . Then the m o l e c u l a r f i e l d s f e l t by the a, B and y atoms are f o r T < T~ h
- 6J' + 3J'
(AB+Ay)
hB -
0
+ 3J'
(Ay-A~)
h Y
0
+ 3J'
(AB-Aa)
and s i m i l a r l y f o r ~he 4 J ' , ! J ' and 0 f i e l d s which appear f o r T~ < T < T_I. For example, h
- 4J'
+
J'
(3AB+Ay)
h
- 2J' +
J'
(3A6-Ay)
h
=
+ 3J'
(AB-Ay)
0
.
The e f f e c t o f these small c o r r e c t i o n s Is to make the e f f e c t i v e m o l e c u l a r f i e l d s d i f f e r somewhat from the 6 : 4 : 2 r a t i o s , as was a l r e a d y seen In our a n a l y s l s o f the A, B1, B2 s e r i e s . These d i f f e r e n c e s can be used to e s t i m a t e v a l u e s f o r A=, AB and Ay = 10.002/J'l and thus the u n c e r t a i n t y i n the "hy - O" m o l e c u l a r f i e l d is o f the o r d e r o f J ' . This p h y s i c a l p i c t u r e can thus e x p l a i n the e x t r a f e a t u r e s o f the spectrum in terms o f weak t r a n s i t i o n s c o r r e s p o n d i n g to h = J ' , g i v e n in t a b l e ] as the C s e r i e s . A more d e t a i l e d t h e o r y 10 may be c o n s t r u c t e d by c o n s i d e r i n g the l i n e a r response o f the chains to e v b l u a t e and so on, but Is n o t w a r r a n t e d f o r the p r e s e n t s p e c t r a . 5.
Spin e x c i t a t i o n s
in CsCoCl 3
In view o f the s a t i s f a c t o r y p i c t u r e o f t h e CsCoBr 3 spectrum o b t a i n e d w l t h l n the a n a l y s i s o f the p r e v i o u s s e c t i o n s , we have r e - e x a m i n e d the data f o r CsCoCI 3. Here a g a i n we use the w e l l d e f i n e d and unambiguous A s e r i e s t o f i x a s u i t a b l e c h o i c e o f J, ¢, and J' w i t h l n the neighbourhood o f the v a l u e s (J-52 an " l , ¢ - 0 . 1 3 , J~/J = 0.0056) g i v e n i n the l i t e r a t u r e . 8 These v a l u e s a r e then used t o p r e d i c t the B! and B2 series. Since J~ in CsCoCI 3 Is c o n s i d e r a b l y
ONE-DIMENSIONAL MAGNON RAMAN SPECTRUM OF CsCoBr 3
V o l . 36, No. 12
less than that o f the bromide we do not expect the C-serles type o f features to be so important h e r e . The r e s u l t s are given in t a b l e 2 and hence the spectrum is c o n s i s t e n t w i t h the parameters (J, J') to be expected from susceptib i l i t y data. 1~ 6.
Combination band at 178 cm-1
As mentioned p r e v i o u s l y , the i n t e n s i t y o f the 178-cm -1 band has a marked temperature .. dependence, and has very low i n t e n s i t y by T~ I , which would e x p l a l n why t h i s band has not been observed p r e v i o u s l y . The extremely r a p i d decrease in i n t e n s i t y when the temperature is increased from 6 to 14 K i n d i c a t e s that the band is associated in some way w i t h the magnetic o r d e r i n g g i v i n g r i s e to the A series o f bands. However, no A-series magnetic s c a t t e r i n g is expected o f t h i s i n t e n s i t y at 178 cm-1 and the i n t e n s i t y decreases too r a p i d l y . The band is assigned to Eig phonon-magnon combination s c a t t e r i n g for the f o l l o w i n g reasons. F i r s t l y ,
the intense Elg phonon at 72.8 cm"1 and the strongest A s e r i e s magnon at 105.2 cm"1 can combine to g i v e a combination mode at 178 cm"1. Secondly, the Ela phonon i n t e n s i t y is s p i n dependent 3 and t ~ e r e f o r e the combination band w i l l have a stronger temperature dependence than e i t h e r parent alone in agreement w i t h experiment. T h i r d l y , the combination has + AZg + E2~ symmetry in the O6h E19 X Elg t h e r e f o r e ,s , l l . e d in (ZZ) p o i n t group polarization. The f a c t that the band is a l s o observed in (ZX) p o l a r i z a t i o n can o n l y be e x p l a i n e d f o r such a combination by invoking the c r y s t a l maqnetic space group symmetry, Cm'c2~ f o r T < T~T. 9 For then the combination has 2A 1 + 2B l"symmetry in the Czv p o i n t group and i t is now a c t i v e in both diagonal and (ZX) polarizatlons exclusively. Acknowledgements - We wish to thank Dr. G.D. Jones and Dr. B. B r i a t for p r o v i d i n g the c r y s t a l s o f CsCoBr 3 and Dr. H. Shiba for p r o v i d i n g p r e p r i n t s o f his work.
Table 2 Comparison between the experimental values I for frequencies (in cm"1) o f peaks in the magnon Raman spectrum o f CsCoCI 3 and the t h e o r e t i c a l values computed using J = 49.2 cm-! and e = 0.13. The p r e d i c t e d r e l a t i v e i n t e n s i t i e s are given in brackets beside the Frequencies.
A series (h - 0.037) Expt. Theory
B] series (~ = 0.026) Expt. Theory
B2 series (~ = O.017) Expt. Theory
90.5
91.3 (1.00)
88.8
88.4 (!.00)
86.5
85.5 (1.00)
106.6
105.5 (0.48)
100.1
99.8 (0.59)
94.3
9h.3 (0.68)
116.3
116.5 (0.21)
109.0
108.8 (0.35)
101.0
101.2 (0.48)
-
125.6 (0.07)
116.3
116.3 (0.18)
105.0
107.1 (0.32)
-
122.9 (0.09)
112.9
112.5 (0.24)
116.3
117.2 (o.16)
References 1. 2. 3. 4. 5. 6. 7.
W. B r e l t l l n g , W. Lehmann, T.P. S r l n i v a s a n and R. Weber, S o l i d State Commun. 24, 267 (1977). loW. Johnstone and D.J. Lockwood, S o l i d State Commun. 32, 285 (1979). I.W. Johnstone and L. Oublckl, J. Phys. C: S o l i d State Phys. 13, 4531 (1980). J. des Clolzeaux and H. Gaudin, J. Math. Phys. Z, 1384 (1966). J. V i l l a i n , Physlca 790, I (1975). H. Fowler, Phys. Rev. BIT, 2989 (1978). H. Ishimura and H. $hlba, Prog. Theor. Phys. 63, 743 (1980).
1037
8. 9. 10. I1. 12. 13. 14.
H. Shiba, Prog. Theor. Phys. to be published. W.B. Yelon, D.E. Cox and H. Eibschutz, Phys. Rev. B1...22, 5007 (1975). D.J. Scalaplno, Y. Imry and P. Pincus, Phys. Rev. BI_II, 2042 (1975). J. Hubbard, Phys. Rev. B17, 494 (1978). H. Fowler and H. Puga, Phys, Rev. B1_88, 421 (1978). N.L. Rowell. D.J, Lockwood and P. Grant. J. Raman Spectroscopy in press. N. Achiwa, J. Phys. Soc. Japan 27, 561 (1969).
INFLUENCE OF INTERCHAIN COUPLING ON THE ONE-OIHENSIONAL MAGNON RAMAN SPECTRUH OF CsCoBr 3 I.W. Johnstone, D.J. Lockwood and M.W.C. Dharma-wardana D i v i s i o n o f Physics, National Research C o u n c i l , Ottawa, Canada KIA OR6 (Received 25 August 1980 by H. F. Collins) L i g h t s c a t t e r i n g from magnons in CsCoBr 3 has been measured f o r temperatures w e l l below the upper a n t i f e r r o m a g n e t i c ]D o r d e r i n g temperature o f T~.=. 28 K. These experiments reveal m u l t i p l e magnon f e a t u r e s o f energies in the range 90 Co 170 ~m"1 s l m i l a r to those f o . n d in CsCoC| 3 but p r e v i o u s l y unobserv=d f o r ~he hromlde. Prominent f e a t u r e s in the spectrum and t h e i r temperature dependence are described in terms o f a recent theory by Shiba. Other, weaker f e a t u r e s are e x p l a i n e d by a simple e x t t n s l o n o f the theory to Inc!ude f l u c t u a t l o n s . A new band is observed a t 178 ~m-1, whose, i n t e n s i t y drops s%harply p r i o r Lo Cl~e lower ]O o r d e r i n g t r a n s i t i o n a t T~ I - 10 K. This band is assigned to magnon-phonon combination s c a t t e r i n g .
1.
Introduction
U n t i l now, the spin-wave Roman s p e c t r a o f CsCoCI 3 and CsCoBr31"3 have bcen d i f f i c u l t to understand in terms o f c u r r e n t t h e o r i e s o f I s i n g Holsenberg chains, which are themselves the s u b j e c t o f much debate, k ' 8 Both m a t e r i a l s s i m u l a t e I-D I s l n g a n t l f e r r o m a g n e t i c chains but d i s p l a y )-D magnetic o r d e r a t s u f f i c i e n t l y low temperatures. In CsCoBr3, the l a r g e l n t r a c h a i n n e a r e s t - n e l g h b o u r exchange i n t e r a c t i o n , J, g i v e s r i s e to a t r a n s v e r s e spin e x c i t a t i o n band near 100 cm"1. For T ~ 30 K the Raman spectrum appears as an asyr~ne.tric peak cuper!mposed on a broader band o f e x c i t a t i o n s . On c o o l i n g , the spectrum is remarkable f o r the appearance o f a l a r g e number o f d i s c r e t e l i n e s whose i n t e n s i t i e s are c r i t i c a l l y dependent on the c r y s t a l temperature. Below l0 K, the spin arrangement in CsCoBr 3 comprises t h a t o f a t r i a n g u l a r (a-b plane) I s l n g lattice with Interchain antiferromagnetic n e a r e s t - n e i g h b o u r c o u p l i n g , J~, very weak f e r r o magnetic n e x t - n e a r e s t - n e i g h b o u r I n t e r c h a i n I n t e r a c t i o n s , J~, and the s t r o n g I s i n g c o u p l i n g along the chains (e a x i s ) . Between 1~ K and T~ - 28.3 K, o n e - t h i r d o f the chains become paramagnetic, w h i l e f o r 10 K < T~ I < 14 K, these two magnetic phases are thought t o c o e x i s t . Above T~ a l l the chains are d i s o r d e r e d but r e t a i n s i g n i f i c a n t spin c o r r e l a t i o n s along the e a x i s f o r temperatures beyond I00 K. Estimates o f the s t r e n g t h s o f the v a r i o u s magnetic i n t e r a c t i o n s J, J~ and J~ can be made from s u ~ c e p t i b i l i t y data 9 and the N~el temperatures T~ and T~ I. Thus J = 56 cm"1, J~/J = 0 . 0 , 3 and J~/J = -0.03. z The f i n e s t r u c t u r e In the low temperature (T < T~) Raman spectrum has been v a r i o u s l y a s c r i b e d to magnon conlbination s t a t e s o r m u l t i p l e bound s t a t e s . 1"3 Recently, the p e r t u r b a t i o n theory approach o f Ishimara and Shlba 7 f o r a s i n g l e chain has been extended by Shiba 8 to i n c l u d e i n t e r c h a i n c o u p l i n g and thereby e x p l a i n the spin e x c i t a t i o n spectrum o f CsCoCI 3
w i t h c o n s i d e r a b l e success. As a f u r t h e r t e s t o f t h i s approach to n o n l l n e a r magnon e x c l t a t i o n processes we apply Shlba's theory to new lowtemperature data f o r the bromide, where the I n t e r c h a i n c o u p l i n g is a p p r o x l m a t e l y t w i c e as s t r o n g as in the c h l o r i d e . 2.
Theory
The N a m i l t o n l a n f o r a one-dlmensional s p i n I s l n g l i k e a n t l f e r r o m a g n e t is w r i t t e n asT, 8 "o
zz xx " 2J {Sjsj÷l %sj,,
sjSj÷l)}
+yy
•
(')
Since CsCoBr 3 e x h i b i t s 3-D magnetic o r d e r a t sufficiently low temperatures the e f f e c t o f . the r e l a t l v e l y weak i n t e r c h a i n c o u p l i n g , HI(A, ~), must be i n c l u d e d so t h a t the model H3milton[an has the form:
H-2H(~)+2J ' T IS~_S~ +c'(S~.S~ +sY.s~ )] k 0 j~LJ~ ~ j~ ~ jA ~u j '
(2)
where
~ and Z denote summations o v e r n,n X,U j,~ chain p a i r s and n.n i n t r a c h a l n spin p a i r s respectively. Since ¢' ~ ¢ ~ 0 . 1 , the c ' J ' term ( i . e . the i n t e r c h a l n x - y term) may be Ignored t o a good a p p r o x i m a t i o n . The remaining I s i n g I n t e r c h a i n I n t e r a c t i o n In (2) ls t r e a t e d w i t h i n a molecular field approximation (e.g. Scalaplno, Imry and Pincus tO) to g i v e a s i n g l e - c h a i n effective Hamlltonlan: H - H0 -
hjSj
(31
where h i - ( - 1 ) J h and h 2 O. The r e s u l t i n g energy | I v e l scheme Is c o n s t r u c t e d from the Ntel s t a t e IN> which is favoured by the staggered m o l e c u l a r f i e l d h, t o g e t h e r w i t h admixtures o f the low l y i n g subspace o f e x c i t e d 1033
IO34
ONE-DIMENSIONAL MAGNON EAMAN SPECTRUM OF CsCoBr 3
s t a t e s . These c o n s i s t o f two blocks o f s t a t e s corresponding t o the Sz - 0 and Sz - 1 e x c i t a t ions. Thus f o r ik.rj sZ,
1:
+
sj I""
•
+i =,/'2"/'~
J ik-rj Z •
+3"
J J
i.e.
•
+2i-1'
ik.rj
+. + S j S j + l S j + 2 IN> ÷S"
÷
S"
+2 +4
- ~
~ e J
S+
sj 3+tsj+z j+3 j+4 IN>
i ,, 1, 2, e t c .
- /2-~ e
i.e.
~2i'
ik-r.
ik-r.
+.
JSjSj+IIN> + .
+
.
JsjSj+lSj+2sj+3JN> (4b)
I " 1, 2, e t c .
Note t h a t the x - y I n t e r a c t i o n (the c term) has converted the I s i n g s p l n - f l l p s (S~, S~S]+I, +
Sj+2, e t c . )
i n t o ~ dependent p r o p a g a t i n g modes
which correspond to the s o l l t o n p i c t u r e (moving bloch w a l l ) o f V I I l a l n 5 and the dimer gas p i c t u r e o f Hubbard I I (see a l s o Fowler and Pugat2). In the l i m i t h + O, ¢ + 0 these s t a t e s have energies E21. I - E21 " 2J w i t h respect to the N~el s t a t e IN>. When the m o l e c u l a r F i e l d is r e s t o r e d the degenerate energies s p l i t to g i v e the s e r i e s E21. j - 2 J + 2 h ( 2 1 - 1 ) , and s i m i l a r l y E i " 2J+211(21). F i n a l l y , when ¢ is nonzero t~e x - y i n t e r a c t i o n couples +i to +1+2 to mix the Sz - 1 subspace. At the zone c e n t r e (k - O) t h i s r e q u i r e s the d l a g o n a l i z a t l o n o f the trldlagonal matrix 2 <+21-11H/2Jl~2i ' - I > - 1 + } c 2 . ~c 6i , i + ( 2 i . i ) ~ for i - i' - ¢
for 1 " I'
± 1.
36, No. 12
eigen v e c t o r s and eigenvalues o f (5) to o b t a i n the r e l a t i v e s t r e n g t h o f the l i n e s , which drop o f f in i n t e n s i t y as we go to h i g h e r magnon states. Thus i f we now use k t o I d e n t i f y the d i a g o n a l i z e d l i n e a r combination s t a t e s where the p r i n c i p l e c o n t r i b u t i o n comes from +21-1 o f ( 4 a ) , then
(4a)
and Sz - O:
Vol.
(5)
Note t h a t the Sz - 0 subspace mixes w i t h IN> to g i v e a p e r t u r b e d ground s t a t e Ig> and a c l u s t e r o f e x c i t e d s t a t e s e x t e n d i n g upwards from 2J + 4h + 0(¢ 2 ) but do not mlx w i t h the Sz - i subspace which gives a s e t o f energy l e v e l s e x t e n d i n g upwards from 2J + 2h + O(cZ). However, i f the Sz - 0 e x c i t a t i o n s are r e l a t e d to the
F o l l o w i n g Shlba, 8 the magnitude o f h f o r each o f the two low temperature phases may be e a s i l y determined by c o n s i d e r i n g the n e t t molecul a r f i e l d experienced by one chain due to i t s neighbours (see f i g u r e 1). For T < T~ I , the a chains are surrounded by s i x a n t l f e r r o m a g n e t i c a l l y coupled nelghbours so t h a t the n e t t f i e l d is 6J' This gives r i s e to the A s e r i e s o f excitations. In c o n t r a s t , the n e t t f i e l d experienced by the B and ¥ chains is z e r o , g l v l n q r i s e to an e x c i t a t i o n continuum. For T~ I < T < T ~ one t h i r d o f the chains 4Y) become paramagnetic so t h a t t h e i r spin is e i t h e r up o r down w i t h respect t o the o t h e r chains w i t h equal probability. Hence the m o l e c u l a r f i e l d s experienced by the ~ and B chains are 6 J ' , 4 J ' , 2J' and O, w i t h p r o b a b i l i t i e s o f I/8, 3/8, 3/8 and I / 8 , r e s p e c t i v e l y ( f o r y c h a i n s , h - O). Thus, in t h i s phase we expect t h r e e s e r i e s o f d i s c r e t e l i n e s comprising the A, B[ and B2 s e r i e s t o g e t h e r w i t h a continuum. 3.
Experiment and r e s u l t s
Large s i n g l e c r y s t a l s (60xlO mm2) o f CsCoBr 3 were grown by R. R l t c h i e (Physics Department, U n i v e r s i t y o f Canterbury, N.Z.) and B. B r l a t ( L a b o r a t o l r e d~Optlque Physique, P a r i s , France) using the Bridgeman technique. Raman s p e c t r a were recorded under computer c o n t r o l 13 using 50 mW o f 501.7 nm argon l a s e r e x c i t a t i o n and a s p e c t r a l r e s o l u t i o n o f 2 cm" l . Suitably cleaved samples were mounted in a Thor S500 v a r i a b l e temperature c r y o s t a t where the sample is immersed in a helium exchange gas. The sample temperature could be m a i n t a i n e d to w i t h i n tO.2 K, but because o f l o c a l l a s e r h e a t i n g (~3 K) the a c t u a l c r y s t a l temperature w i t h i n the l a s e r f i l a m e n t could o n l y be e s t i m a t e d . Spectra were recorded in the 90" s c a t t e r i n g geometry X ( . . ) Y , where X, Y, Z correspond to the c r y s t a l a, b, c axes, r e s p e c t i v e l y . Typical s p e c t r a a r e presented in f i g u r e 2 f o r T - 6 and 14 K. The peaks a t 72.8, 89.9, 118.1 and 166.1 cm" ! are due to Raman s c a t t e r i n g From the E-I , E2-, E2-~ and . A I . z o n e - c e n t r e phonons, r e s p e c t i v e l y . 3 With the e x c e p t i o n o f the Elg mode these bands are f o r b i d d e n in X(ZX)Y p o l a r i z a t i o n but appear through the d e p o l a r i z a t i o n e f f e c t s o f c r y s t a l b i r e f r i n g e n c e and s t r a i n e d c r y o s t a t windows. Other f e a t u r e s present in the range 90 tO 170 cm" | are (ZX) p o l a r i z e d and are due to zone c e n t r e magnon s c a t t e r i n g analogous to t h a t observed f o r CsCoC13,1 but e x t e n d i n g over a w i d e r range o f e n e r g i e s . Frequencies o f d i s t i n c t f e a t u r e s are l i s t e d in t a b l e 1. On warming t o 14 K, new bands appear and a l l bands grow markedly in I n t e n s i t y except those a t 105.2 and 138.4 c m ' l , whlch decrease. The behaviour o f the weak 160-cm"1 band is masked by new bands appearing nearby.
V o l . 36, No. 12
O N E - D I H E N S I O NHAGNON AL R.aJ4AN SPECTRUN OF CsCoBr 3
•
•
s
(o) Figure 1
•
o
•
o
1035
•
(b)
The hexagonal arrangement o f c o b a l t ions in the c7~ plane o f CsCoBr 3 showing (a) the f u l l y ordered magnetic s t r u c t u r e f o r T < T~ I i n v o l v i n g up(+) o r down(-) s p i n s , a~d (b) the p a r t l y Ordered s t r u c t u r e For T~ I < T < T~ where the spin on atom y Is now disordered (0).
>F-Z
P
hl l-Z Z
nr
I
7o
I00
130
160
190
FREQUENCY (cm -I) Figure 2
Temperature dependence o f the 70 to 190 cm"X r e g i o n o f the Raman spectrum o f CsCoBr 3 recorded in X(ZX)Y polarization with a spectral resolution o f 2 cm" l . P denotes a phonon.
No frequency s h i f t s are o b s e r v a b l e w i t h i n t h i s temperature range. Another t e m p e r a t u r e dependent f e a t u r e occurs a t 177.7 cm-1 in both (ZZ) and (ZX) p o l a r l z a t i o n w l t h the i n t e n s i t y r a t i o I z z / I z x - 2 . 5 . This band drops even more r a p i d l y in i n t e n s i t y w i t h i n c r e a s i n ~ t e m p e r a t u r e than the magnon band a t 105.2 cm"L.
4.
Comparison w i t h theory
At low temperatures the theory p r e d i c t s a sequence o f sharp e x c i t a t i o n s (the A s e r i e s ) superimposed on a broad continuum. These bands are p r e d i c t e d to reduce In i n t e n s i t y by a f a c t o r I / 4 in the i n t e r m e d i a t e phase T~ I < T < T~. From o u r s p e c t r a a t 6 K (T
i036
Vol. 36, No. 12
ONE-DIMENSIONAL MAGNON RAMAN SPECTRUM OF CsCoBr 3
Table 1 Comparison between t h e o r y and e x p e r i m e n t f o r the f r e q u e n c i e s ( i n cm"1) and r e l a t i v e i n t e n s i t i e s ( i n b r a c k e t s ) o f peaks in the magnon Raman spectrum of. CsCoBr 3. For the c a l c u l a t i o n s J - 50.5 cm"1 and ¢ = 0 . 1 3 . The m o l e c u l a r f i e l d h - h / J .
A series (h - 0 . I I ) Expt. Theory
B! s e r i e s (~ " O.077) Expt. Theory
B2 s e r i e s ~0.045) Expt. Theory
C series (~ = 0.012) Expt. Theory
105.2 (I.00)
107 (l.O0)
I01.3
I02 (I.00)
97.1
96 (1.00)
87
86 (I.00)
138.4 (0.25)
136 (0.12)
125.6
125 (0.22)
112.1
I12 (0.42)
92.6
93 (0.74)
160
159 (0.01)
I~2.7
142 (O.03)
123.5
125 (0.15)
158
158 (0.002)
133.4
135 (0.04)
-
103 (0.47)
149
145 (0.OO4)
-
108 (0.36)
(0.08)
(T • T~ I) the sharp f e a t u r e s a t 105.2 and 138.4 cm"1 can be p o s i t i v e l y assigned t o the A s e r i e s o f bands because t h e i r I n t e n s i t y f a l l s by t h i s f a c t o r . The weak 160-cm " ! band is a l s o l i k e l y t o be an A s e r i e s band because t h e r e is no c o r r e s p o n d i n g f e a t u r e In the h i g h e r t e m p e r a t u r e phase. The f r e q u e n c i e s o f the A s e r i e s bands were c a l c u l a t e d a c c o r d i n g t o the model g i v e n in s e c t i o n 2 w i t h h - 6 J ' / J - 0 . 1 1 , J - 50.5 cm" ! and e - 0 . 1 3 . The r e s u l t s , g i v e n In t a b l e I , a r e In e x c e l l e n t agreement w i t h e x p e r i m e n t and a l s o c o n f i r m t h a t the weak band a t 160 cm"! Is Indeed an A s e r i e s peak. The r e l a t i v e I n t e n s i t i e s o f these bands are a l s o in good agreement w i t h t h e o r y as can be seen in t a b l e 1. In the I n t e r m e d i a t e phase, T~ I < T < T~, the B s e r i e s bands dominate the magnon spectrum. T h e i r f r e q u e n c i e s a r e a c c u r a t e l y reproduced by t h e o r y f o r h - 0.077 (cf. 4 J ' / J - 0.073) f o r the B! s e r i e s , and ~ - 0.045 ( o f . 2 J ' / J - 0.037) f o r the B2 s e r i e s , w i t h J and c as g i v e n above. However, because o f o v e r l a p o f bands w i t h i n B; and B2 s e r i e s i t is d i f f i c u l t t o make d e f i n i t e assignments In some cases e . g . the bands near 125 cm"1. While t h i s o v e r l a p a l s o p r e c l u d e s the e x t r a c t i o n o f a c c u r a t e i n t e n s i t y d a t a , t h e r e is a g e n e r a l q u a l i t a t i v e agreement between t h e o r y and e x p e r i m e n t a p a r t from the band near 143 cm- 1 . The t h e o r y o f Shiba as a p p l i e d h e r e does n o t p r e d i c t any m a g n e t i c s c a t t e r i n g below the B2 band a t 97.1 cm" t . However, f e a t u r e s a r e observed a t 92.6 cm- l and ~87 cm" | on e i t h e r s i d e o f the E2a phonon a t 89.9 cm" 1 . A l t h o u g h these f e a t u r e s - h a v e low i n t e n s i t y they a r e r e p r o d u c i b l e and t h e y m a i n t a i n t h e i r i n t e n s i t y w i t h I n c r e a s i n g t e m p e r a t u r e as shown In f i g u r e 2. An approach t o u n d e r s t a n d i n g these Iowi n t e n s i t y f e a t u r e s in the spectrum may be made as f o l l o w s . W l t h i n the m e a n - f i e l d scheme used here l e t us w r i t e
= ½(!
"
= ½(l
+ AS)
-½(i + a~)
Y
- ~½(I
+ ay) +
Ay)
I f o r T < TN I II f o r TN < T < T N
98 (0.58)
where the small c o r r e c t i o n terms Aa, AB, Ay a r e t e m p e r a t u r e dependent. In f a c t Ay is expected t o go t o zero a t the t r a n s i t i o n t e m p e r a t u r e . Then the m o l e c u l a r f i e l d s f e l t by the a, B and y atoms are f o r T < T~ h
- 6J' + 3J'
(AB+Ay)
hB -
0
+ 3J'
(Ay-A~)
h Y
0
+ 3J'
(AB-Aa)
and s i m i l a r l y f o r ~he 4 J ' , ! J ' and 0 f i e l d s which appear f o r T~ < T < T_I. For example, h
- 4J'
+
J'
(3AB+Ay)
h
- 2J' +
J'
(3A6-Ay)
h
=
+ 3J'
(AB-Ay)
0
.
The e f f e c t o f these small c o r r e c t i o n s Is to make the e f f e c t i v e m o l e c u l a r f i e l d s d i f f e r somewhat from the 6 : 4 : 2 r a t i o s , as was a l r e a d y seen In our a n a l y s l s o f the A, B1, B2 s e r i e s . These d i f f e r e n c e s can be used to e s t i m a t e v a l u e s f o r A=, AB and Ay = 10.002/J'l and thus the u n c e r t a i n t y i n the "hy - O" m o l e c u l a r f i e l d is o f the o r d e r o f J ' . This p h y s i c a l p i c t u r e can thus e x p l a i n the e x t r a f e a t u r e s o f the spectrum in terms o f weak t r a n s i t i o n s c o r r e s p o n d i n g to h = J ' , g i v e n in t a b l e ] as the C s e r i e s . A more d e t a i l e d t h e o r y 10 may be c o n s t r u c t e d by c o n s i d e r i n g the l i n e a r response o f the chains to e v b l u a t e
Spin e x c i t a t i o n s
in CsCoCl 3
In view o f the s a t i s f a c t o r y p i c t u r e o f t h e CsCoBr 3 spectrum o b t a i n e d w l t h l n the a n a l y s i s o f the p r e v i o u s s e c t i o n s , we have r e - e x a m i n e d the data f o r CsCoCI 3. Here a g a i n we use the w e l l d e f i n e d and unambiguous A s e r i e s t o f i x a s u i t a b l e c h o i c e o f J, ¢, and J' w i t h l n the neighbourhood o f the v a l u e s (J-52 an " l , ¢ - 0 . 1 3 , J~/J = 0.0056) g i v e n i n the l i t e r a t u r e . 8 These v a l u e s a r e then used t o p r e d i c t the B! and B2 series. Since J~ in CsCoCI 3 Is c o n s i d e r a b l y
ONE-DIMENSIONAL MAGNON RAMAN SPECTRUM OF CsCoBr 3
V o l . 36, No. 12
less than that o f the bromide we do not expect the C-serles type o f features to be so important h e r e . The r e s u l t s are given in t a b l e 2 and hence the spectrum is c o n s i s t e n t w i t h the parameters (J, J') to be expected from susceptib i l i t y data. 1~ 6.
Combination band at 178 cm-1
As mentioned p r e v i o u s l y , the i n t e n s i t y o f the 178-cm -1 band has a marked temperature .. dependence, and has very low i n t e n s i t y by T~ I , which would e x p l a l n why t h i s band has not been observed p r e v i o u s l y . The extremely r a p i d decrease in i n t e n s i t y when the temperature is increased from 6 to 14 K i n d i c a t e s that the band is associated in some way w i t h the magnetic o r d e r i n g g i v i n g r i s e to the A series o f bands. However, no A-series magnetic s c a t t e r i n g is expected o f t h i s i n t e n s i t y at 178 cm-1 and the i n t e n s i t y decreases too r a p i d l y . The band is assigned to Eig phonon-magnon combination s c a t t e r i n g for the f o l l o w i n g reasons. F i r s t l y ,
the intense Elg phonon at 72.8 cm"1 and the strongest A s e r i e s magnon at 105.2 cm"1 can combine to g i v e a combination mode at 178 cm"1. Secondly, the Ela phonon i n t e n s i t y is s p i n dependent 3 and t ~ e r e f o r e the combination band w i l l have a stronger temperature dependence than e i t h e r parent alone in agreement w i t h experiment. T h i r d l y , the combination has + AZg + E2~ symmetry in the O6h E19 X Elg t h e r e f o r e ,s , l l . e d in (ZZ) p o i n t group polarization. The f a c t that the band is a l s o observed in (ZX) p o l a r i z a t i o n can o n l y be e x p l a i n e d f o r such a combination by invoking the c r y s t a l maqnetic space group symmetry, Cm'c2~ f o r T < T~T. 9 For then the combination has 2A 1 + 2B l"symmetry in the Czv p o i n t group and i t is now a c t i v e in both diagonal and (ZX) polarizatlons exclusively. Acknowledgements - We wish to thank Dr. G.D. Jones and Dr. B. B r i a t for p r o v i d i n g the c r y s t a l s o f CsCoBr 3 and Dr. H. Shiba for p r o v i d i n g p r e p r i n t s o f his work.
Table 2 Comparison between the experimental values I for frequencies (in cm"1) o f peaks in the magnon Raman spectrum o f CsCoCI 3 and the t h e o r e t i c a l values computed using J = 49.2 cm-! and e = 0.13. The p r e d i c t e d r e l a t i v e i n t e n s i t i e s are given in brackets beside the Frequencies.
A series (h - 0.037) Expt. Theory
B] series (~ = 0.026) Expt. Theory
B2 series (~ = O.017) Expt. Theory
90.5
91.3 (1.00)
88.8
88.4 (!.00)
86.5
85.5 (1.00)
106.6
105.5 (0.48)
100.1
99.8 (0.59)
94.3
9h.3 (0.68)
116.3
116.5 (0.21)
109.0
108.8 (0.35)
101.0
101.2 (0.48)
-
125.6 (0.07)
116.3
116.3 (0.18)
105.0
107.1 (0.32)
-
122.9 (0.09)
112.9
112.5 (0.24)
116.3
117.2 (o.16)
References 1. 2. 3. 4. 5. 6. 7.
W. B r e l t l l n g , W. Lehmann, T.P. S r l n i v a s a n and R. Weber, S o l i d State Commun. 24, 267 (1977). loW. Johnstone and D.J. Lockwood, S o l i d State Commun. 32, 285 (1979). I.W. Johnstone and L. Oublckl, J. Phys. C: S o l i d State Phys. 13, 4531 (1980). J. des Clolzeaux and H. Gaudin, J. Math. Phys. Z, 1384 (1966). J. V i l l a i n , Physlca 790, I (1975). H. Fowler, Phys. Rev. BIT, 2989 (1978). H. Ishimura and H. $hlba, Prog. Theor. Phys. 63, 743 (1980).
1037
8. 9. 10. I1. 12. 13. 14.
H. Shiba, Prog. Theor. Phys. to be published. W.B. Yelon, D.E. Cox and H. Eibschutz, Phys. Rev. B1...22, 5007 (1975). D.J. Scalaplno, Y. Imry and P. Pincus, Phys. Rev. BI_II, 2042 (1975). J. Hubbard, Phys. Rev. B17, 494 (1978). H. Fowler and H. Puga, Phys, Rev. B1_88, 421 (1978). N.L. Rowell. D.J, Lockwood and P. Grant. J. Raman Spectroscopy in press. N. Achiwa, J. Phys. Soc. Japan 27, 561 (1969).