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Volume A, number 4 PHYSICS LETTERS 9 September 1985

MAGNETIZATION CURVES OF A SPIN-ONE BETHE LATTICE

K.G. CHAKRABORTY ’ and J.W. TUCKER

&~~rtn~~ ~t f Ph _v . r r c . c .~ mr ~ wr i ~ f Sh e f / r r l d h e f f i e l d 3 7RH. UK

Received 12 March 1985; revised manuscript received 3 May 1985; accepted for publication 24 June 1985

and single ion anisotropy on the Bethe lattice is presentedrelative strength of the single ion anisotropy is varied.

An exact calculation of the spontaneous magnetization of a spin-one Ising model with the inclusion of biquadratic exchangerhe magnetizition curves exhibit some u&al features as the

We treat a spin-one Ising model on the Bethe latticewith the inclusion of biquadratic exchange and single-ion anisotropy in addition to the usual bilinear ex-change interactions. Our purpose is to show that thepresence of single-ion anisotropy can produce drasticchanges to the form of the magnetization curves undercertain conditions. We consider the hamiltonian

H=Ho,i+CHiii’C CHiiiikt..., 1)i ’ ii’

where

Ho,i = -JS, c Si - J’S,2 C Si2 t OS021 2

2)

Hi ii = -JSi C Sii - J’S; C Si + DS” , (3)J i

etc. The index i runs from 1 to z, andj, k, . . runfrom 1 to z - 1, z being the coordination number. Sois taken as the central spin. J and J’ are the nearest-neighbour bilinear and biquadratic exchange constantsrespectively and D is the single-ion anisotropy. Eachspin has three eigenvalues 1, 0, - 1. Since the size ofthe lattice is infinitely large, each lattice point isequivalent and so the spontaneous magnetization perspin can be calculated from

’ On leave from Department of Physics, Basirhat College,West Bengal 743412. India.

m = c c . So exp(-/3H)se S{i}

-1

X c c . . exp -PH) ,so qi}

(4)

where fl= I /k T k being the Boltzmann constant.The symbol IZsii) indicates a trace over the states ofthe appropriate spins. Using a generalisation of the

method of Katsura and Takizawa [l] as expoundedin ref. [2] we find that the magnetization can be ex-pressed in the form

m = aZ - b”)/ a” + bZ + c”) ,

with

5)

a = exp[-(L2 + L,)] + 2 exp(K’ - 0’) cosh(K + L1),

6)b=exp[-(L2 +L3)] t2exp(K’-D’)cosh(K-Ll),

(7)

c=exp[-(LZ tLj)+D’] t2cosh(Ll), (8)K=BJ,K’=PJ’,D’=PDlzandL1,L2 andLgaregiven by

exp 2L,) = a/b)“-l , 9)

exp(2L,) = (a/~)“-~ , 10)

exp(2L3) = (LI/c)~-~ 11)

Likewise, the quadrupolar moment, 4 = tSi> is found

0.375-9601/85/S 03.30 0 Elsevier Science Publishers B.V.

(North-Holland Physics Publishing Division)

205

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Volume 1 l l A , number 4 PHYSICS LETTERS 9 September 1985

t o b e

q = ( a z + b Z ) / ( a z + b z + e Z ) . ( 1 2 )

F r o m t h e se e x p r e s s i o n s , o n e c a n d e d u c e t h e f o l l o w i n ge q u a t i o n f o r t h e t r a n s i t i o n t e m p e r a t u r e T c

2 [ ( z - 1 ) s inh (Kc) ] z - 1 [ ( z - 1 ) s inh (K c) - cos h (K c) ]

X exp[ (c~ - zc g )K c]

= [( z - 1 ) s i n h ( K c ) - c o s h ( K c ) + e x p ( - a K c ) ] z - 1 ,

( 1 3 )

wi th K c =J / k B Tc , eL = Y / J a n d c t = D / J z .I n t h e a b -s e n ce o f b i q u a d r a t i c e x c h a n g e t h i s e q u a t i o n r e d u c e st o t h a t o f K a t s u r a [ 3 ] , a n d a ls o r e d u c e s t o t h a t o fO b o k a t a a n d O g u c h i [ 4 ] i f t h e si n g le - io n a n i s o t r o p yi s a l so n e g l e c t e d .

D e t a i l e d c a l c u l a t i o n s o f t h e m a g n e t i z a t i o n a s g i ve nb y e q . ( 5 ) h a v e b e e n m a d e f o r v a ri o u s s t re n g t h s o f t h ei n t e r a c t i o n s J , J ' a n d D a n d f o r s e v e ra l c o o r d i n a t i o nn u m b e r s z . H e r e o n l y t h e r e s u l t s f o r z = 6 a n d J ' / J =

1 . 0 5 a r e p r e s e n t e d s i n c e t h e s e a r e m o s t i n t e r e s t i n ga n d c o n t a c t w i t h t h e w o r k o f C h a k r a b o r t y a n dM o r i t a [ 5 ] c a n b e m a d e . I n fi g. 1 t h e m a g n e t i z a t i o na s a f u n c t i o n o f1 / I J J i s s h o w n f o r v a r i o u s v a l u e s o f c t'.I t i s s e e n t h a t f o r ~ ' = 0 t h e m a g n e t i z a t i o n e x h i b i t s a na n t i -C u r i e t e m p e r a t u r e Ta c i n a d d i t i o n t o t h e C u r i et e m p e r a t u r e T C . A s - a ' i n c r e a s e s , t h e a n t i - C u r i e t e m -

p e r a t u r e m o v e s t o l o w e r t e m p e r a t u r e s a n d t h e m a g n e -t i z a t i o n c u r v e g r a d u a l l y st a r ts t o p e a k o n t h e l o w t e m -pe ra tu re s ide . A t a c r i t i ca l va lue C~'c,w h i c h c a n b es h o w n t o b e - 0 . 0 5 f o r o u r c h o i c e o f c~, t h e a n t i - C u r i et e m p e r a t u r e i s p r e d i c t e d t o b e z e r o . In t h i s c a se t h em a g n e t i z a t i o n i n c r ea s e s s t e a d i ly w i t h d e c r e a s in g t e m -p e r a t u r e f r o m t h e C u r ie p o i n t , b u t i n a t e m p e r a t u r er a n g e e x t r e m e l y c l o s e to z e r o i t d r o p s r a p i d l y t o z e r oa t z e r o t e m p e r a t u r e . F o r v a lu e s o f - t ~ ' g r e a t e r t h a nt h i s c ri t ic a l v a l u e t h e m a g n e t i z a t i o n a p p r o a c h e s u n i t ya t z e r o te m p e r a t u r e . A l t h o u g h f o r - a ' o n l y s l i g h tl y

g r e a t e r t h a n - ~ c t h e m a g n e t i z a t i o n c u rv e e x h i b i t s as o m e w h a t u n f a m i l i a r f o r m , t h e c o n v e n t i o n a l s h a p e i sr e a c h e d f o r r e l a t i v e l y sm a l l v a l u e s o f t h e s i n g l e - io na n i s o t r o p y ( s e e t h e u p p e r c u r v e i n f ig . 1 ) . We h a v e a l -s o c a rr i e d o u t e x t e n s i v e c o m p u t a t i o n s f o r o t h e r v a l-u e s o f t~ a n d o t h e r c o o r d i n a t i o n n u m b e r s . A f u l l ac -c o u n t o f t h e s e r e su l t s w i l l b e r e p o r t e d i n d u e c o u r sew h e n o t h e r a s p e c t s o f o u r w o r k , i n c l u d i n g a d i sc u s -s i o n o f t h e q u a d r u p o l a r m o m e n t a n d t h e p h a s e tr a n -s i t i o n a r e p u b l i s h e d .

1 0

m

O 5

3

k a T J

Fig. 1. The m agnetization, m, as a functio n o fk B T / J for J / J= - 1 .05 and z = 6. The successive curves from (a) to (b) aref o r D / J z = 0 , - 0 . 0 1 , - 0 . 0 3 , - 0 . 0 5 , - 0 . 0 8 , - 0 . 2 a nd - 0 . 6respectively. The curve for - 0.05 falls sharply to the originat T = 0 from its max imum value shown.

I t i s t i m e l y t o c o m m e n t o n t h e v e r y r e c e n t w o r k o fC h a k r a b o r t y a n d M o r i t a [ 5 ] . T h e l o w e r m a g n e t i z a -t i o n c u r v e i n f ig . 1 c o r r e s p o n d i n g t o t h e a b s e n c e o fs i ng l e -i o n a n i s o t r o p y c a n b e c o m p a r e d d i r e c t l y w i t ht h e r e s u l t s h o w n i n t h a t p a p e r. ( N o t e t h a t t h e r e t h eb i q u a d r a t i c e x c h a n g e i s d e f i n e d w i t h t h e o p p o s i t es ig n .) A l t h o u g h t h e C u r i e a n d a n t i -C u r i e t e m p e r a t u r e sa g r e e , t h e m o s t n o t i c e a b l e d i f f e r e n c e i s t h a t t h e r e t h em a g n e t i z a t i o n i s a f a c t o r " 6 s m a l l e r t h a n o u r e x a c tr e s u l t . We c a n s h o w t h a t t h e i r t h e o r y l e a d s t o t h e r e -s u lt o f eq . ( 5 ) b u t w i t h t h e e x p o n e n t s z r e p l a c e d b y

z ( z - 1 ) . T h i s d i f f e r e n c e c a n b e t r a c e d t o t h e s i n g l e -s it e d i s t ri b u t i o n f u n c t i o n i m p l i c i t i n t h e t h e o r y u s e di n r e f . [ 5 ] t o c a l c u l a t e t h e m a g n e t i z a t i o n . O n t h e o t h -e r h a n d , i n t h e a b s e n c e o f b i q u a d r a t i c e x c h a n g e a n ds i ng l e -i o n a n i s o t r o p y, o u r e x p r e s s i o n s a r e c o n s i s t e n tw i t h t h o s e o f Ta n a k a a n d U r y f l [ 2 ] . We h a v e a l so s h o w nt h a t o u r r e s u l t s f o r t h e C u r i e t e m p e r a t u r e , t h e m a g n e -t i z a t i o n a n d th e q u a d r u p o l a r m o m e n t a g r ee w i t ht h o s e o b t a i n e d b y Ta k a h a s h i a n d Ta n a k a [ 8 ] w h ou s e d a v a r i a t i o n a l c o n s t a n t - c o u p l i n g a p p r o x i m a t i o n

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Volum e 111A, num ber 4 PHYSICS LETTERS 9 Septem ber 1985

w i t h t w o p a r a m e t e r s t o t r e a t t h e s a m e sp i n- 1 I s in gs y s t e m o n a s t a n d a r d l a t t i c e .

I n o r d e r t o s t u d y t h e s t a b i l i t y o f t h e s o l u t i o n s ak n o w l e d g e o f t h e f r e e e n e r g y i s r e q u i r e d . M o r i t a [ 6]h a s d e d u c e d t h a t t h e f re e e n e r g y p e r s i te f o r th e

B e t h e l a t t i c e i s e q u i v a l e n t t o t h e o n e g i v e n b y t h ec l u s t e r v a r i a t i o n m e t h o d . H i s g e n e r a l m e t h o d [ 7] o fc a l c u l a t i o n o f t h e f r e e e n e r g y h a s a ls o b e e n a p p l i e db y s e v e r a l a u t h o r s t o a v a r i e t y o f p r o b l e m s , s e e e .g .r e f s. [ 3 , 9 , 1 0 ] , a n d t h e e q u i v a l e n c e o f t h e c l u s t e r v a r i a-t i o n m e t h o d w i t h t h e c o n s t a n t -c o u p l i n g a p p ro x i m a -t i o n u n d e r c e r t a i n c o n d i t i o n s h a s b e e n s h o w n [ 11 ] .H o w e v e r , a d i r e c t c a l c u l a ti o n o f th e f re e e n e r g y f r o mt h e p a r t i t i o n f u n c t i o n f o r o u r B e t h e l a t ti c e w h e r e t h es u r fa c e e f f e c t s h a v e b e e n n e g l e c t e d f r o m t h e o u t s e ti s n o t p o s s i b l e a s i t is n o l c l e ar h o w t h e t h e r m o d y n a m -

i c l i m i t is t o b e t a k e n [ 1 2 ] . H o w e v e r , f r o m t h e a b o v eo b s e r v a t i o n s w e i n t u i t i v e l y e x p e c t t h e e q u i v a l e n c e o fo u r r e s u lt s f o r t h e C u r ie t e m p e r a t u r e a n d t h e o r d e rp a r a m e t e r s w i t h t h o s e o f th e c o n s t a n t c o u p l i n g ap -p r o x i m a t i o n a p p l i e d t o t h e I si n g m o d e l o n a n o r d i -n a r y l a t t i c e t o e x t e n d a l so t o t h e f re e e n e r g y. We h a v et h u s a d o p t e d t h e e x p r e s s i o n f o r t h e l a t t e r t o c a lc u l a t e

t h e f r e e e n e rg y o f th e m a g n e t i z a t i o n c u r v e s o f f ig . 1 .I t w a s f o u n d t h a t t h e y h a v e a lo w e r f r ee e n e r g y t h a nt h e c o r r e s p o n d i n g p a r a m a g n e t i c s o l u t i o n s - a n e c es -s a r y c o n d i t i o n f o r t h e i r s t a b i l it y.

References

[1] S. Ka tsura and M. Takizawa, Prog. Theor. Ph ys. 51(1974) 82.

[2] Y. Tanaka and N. Ury~, J. Phys. Soc. Japan 50 (1981)1140.

[3] S. Katsu ra, J. Phys. A12 (1979) 2087.[4] T. Obo kata and T. Ogucbi, J. Phys. Soc. Japan 25

(1968) 322.[5] K.G. Chakrab orty and T. M orita, Physica 129A (1985)

415.[6] T. Morita, Physica 83A (1976) 411.[7] T. Morita, J. Math. Phys. 13 (1972 ) 115 .

[8 ] K. Takahashi and M . Tan aka, J. Phys. Soc. Japa n 46(1979) 1428.

[9] S. Ohkuro and S. Katsura, J. Phys. A13 (1980) 1501.[10] I. Nagahara, S. Fu jiki and S. Kats ura, J. Phys. C14

(1981) 3781.[11] T. Morita and T. Tanaka, Phys. Rev. 145 (196 6) 288.[12] F. Peruggi, J . Phys. A16 (1983) L 713.

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