The magnetization curve calculations of the Ising ferromagnet with S = 1

Journal of Magnetism and Magnetic Materials 67 (1987) 387-394 North-Holland, Amsterdam

387

T H E M A G N E T I Z A T I O N CURVE C A L C U L A T I O N S OF T H E I S I N G F E R R O M A G N E T WITH S = 1 J. M I E L N I C K I Institute of Physics, Polish Academy of Sciences, AI. Lotnikbw 32, 02-668 Warsaw, Poland

G. W I A T R O W S K I and T. B A L C E R Z A K Institute of Physics, University of Lbd~, ul. Nowotki 149, 90.236 Lbd~, Poland

Received 15 January 1987

In the paper the general method of magnetization calculations for the Ising ferromagnet with arbitrary spin value has been developed. Detailed calculations have been made for the particular case of S = 1 and for the sc lattice, hence the critical temperature and the magnetization curve together with the ((S z)2) vs. temperature relation have been obtained.

1. Introduction In our previous papers [1-3] we presented a simple method of magnetization calculations for the case of perfect and diluted ferromagnets with S = 1/2. This method can be extended also for the case of arbitrary spin values and as an illustration the S = 1 case in the present paper is discussed in detail. It is worth while to mention here that the magnetization of ferromagnets with S > 1 / 2 was discussed already in a number of papers (see for example refs. [4-9]), however, the various methods used there were usually very complicated and as a result in most of those papers only general equations were derived. The simplicity of our method enabled us to calculate the magnetization curve of the Ising ferromagnet with S --- 1 and sc symmetry together with the ((S z)2) vs. temperature relation in the whole temperature range, although for other crystal symmetries or higher spin values the magnetization curves with the corresponding relations for ((S z)"), n = 2 . . . . . 2S, can be obtained also without great effort.

2. The general method It can be shown that the magnetization curve of an Ising ferromagnet with arbitrary spin value is given by the generalized Callen's equation (see appendix):

(s

= S(B ( SBB )),

(1)

where

8m= E j~

0304-8853/87/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

(2)

J. Mielnicki et al. / Magnetization curve ealculations of the lsing ferromagnet

388

fl = 1 / k T , and B~ is the Brillouin function. Similar equations can be also found for ((S=):}, ((S:)3}. and so on (for derivation see appendix). For instance for <(S:) 2) one can get [10]: ((S=)2} =
(3)

C~(SflBm) = S ( S + 1) + ½cth2(½flB,,) - ½(2S + 1) cth(½flB,,,) cth((2S + 1/2)fiB,,, ).

(4)

with

For further calculations it is convenient to present (1) in the following form:

or

-c d~o B , ( S f l ~ ) ~ 1 foe dt ei,~,(e irB,,,},

(S:} = S

(6)

where the integral representation of Dirac's delta function has been used. The same procedure can be applied also for the expressions of ((S:)"}. As it is seen from (6) for further calculations the therodynamic average of e x p ( - i t B m ) should be found. For this purpose we will consider further the exchange interactions between nearest neighbours only, and we will neglect the correlations between nnn and further atoms. With these approximations made the following relation is hold: ( e-- i'B'' ) -- ( e - i'J's:} z,

(7)

where Z is the coordination number. Now, in order to calculate the r.h.s, of (7) we can write: 2S

e-~AS:= E a , , ( A ) ( S : ) "

(8)

n=0

and then substituting subsequently S== - S . . . . . + S we get the set of 2S + 1 equations for coefficients

a , ( A ) . The obtained results are identical with those derived in [8]. Hence, (7) can be presented in the following form:
(,,

~ a,,(A)((S=)"}

)"

(9)

n=0

As it is seen from (9) for magnetization curve calculations it is necessary to solve the set of 2S equations for ( S ~} . . . ((S-~)2s}. These equations can be easily found from the appropriate relations presented in the form of (6) written explicitly for (S=}, because the integral: 1 /2ei,O, dt 2'rr

_

(10)

after making use of (9) takes on the form of the sum of Dirac &functions.

3. The Ising ferromagnet with S = 1

In order to illustrate the above method we will discuss further in detail the S = 1 case. In this case the set of two eqs. (1) and (3) for (S=) and ( ( S : ) 2 } should be solved and for S = 1 they can be presented in

J. Mielnicki et al. / Magnetization curve calculations of the Ising ferromagnet

389

the form:

(S z ) = 4

a~m

(11)

-

tgh2( l jSjj~mSf ) + 3 and

tgh2(½/7J X S f ) + l } (($2)2) = 2

x

jErtl

(12)

Now, according to (6) and (7) one can write: .,.-.1 m = f_~ d~o fl(½floo)-4£ 1 f~

ei.,,(e_i.~s~,)z '

(13)

dt ei~t(e-ivs~') z,

(14)

dt

where the following notation has been introduced: m ~ (SZ),

(15)

x --- ((s=)2),

(16)

fl(a)

= 4 tgh

(17)

a/(tgh2a + 3),

(is)

f2(a) = 2(tghZa + 1)/(tgh2a + 3), and

~, = tJ.

Now, from (8) the following relation for ( e x p ( -

( e x p ( - i ' / S Z ) ) = x(cos 7 - 1) + 1 - i m sin 7.

iySZ))

results for S = 1: (19)

For further calculations we will assume below the particular case of a sc lattice with Z = 6. Then, from (19), one can get: 6

6

(exp(-i~,SZ)) 6 = ~ A, cos n~, - i Y'~ n=0

C,

sin yn,

(20)

n=l

where A, and C, are the polynomials with respect to xkm ~. It can be noted that the first sum in (20) is the even function of 7 while the second sum is the odd function. This statement is important because as fl is the odd function, then the even part of (20) does not contribute to (13). Similarly, the second sum in (20) can be neglected in the calculations of (14). Now, taking into account the relations:

f ~ doa f,(½flo~) ~---4 f~_ dt

ei'°'i sin n7 =

-fl(½1~n7)

(21)

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390

and dodf2(½flo9

f;

oo d t e i C ° t c ° s n y = f 2 ( i f l n Y )

1

oo

(22)

'

oo

we get on the basis of (13), (14) and (20) the following set of equations for m and x: 6

(23a)

m = E C, f l ( n a ) , n=l 6

x = E A,f2(na)

(23b)

n=0

which, after making order with respect to x~m; takes on the form: 5

k+l

4

k+2

6 E xk E a(k'z)fl(la) + 20m2 E xh 1 E b2ll)fl(hx) k=l

/=1

k=l

2

l=l

k+4

+ 6 m 4 E xk-1 E C~1z)f,(la) = 32(1 - - 6 f l ( a ) ) , k=l 7

(24a)

/=1

k

5

k+l

Y~ x k - ' ~ a~az)f2((l- 1 ) a ) + 15m 2 ~ x k 1 ~ b22)f2((l_ 1 ) a ) k=l

l=l

k=l

l=1

k+4

3

6

+ 15m 4 ~ x k-1 ~2 ~(2)r "k/J2 ( ( l - 1 ) a ) k=l

+ m 6

l=1

E d~2)f2((l- 1)a) = O,

(24b)

l=l

where we have denoted:

a = (J/ZkBT~)(l/y),

(25)

y = T/T~

(26)

and T~ is the Curie temperature. Moreover, a ~ . . . 8O - 320, 560, - 480, 165,

-160,

a[)l) =

400, - 560, 420, -132,

24, 72, - 84, 36,

0,

-

4? =

(2)

akl =

20, - 20,

32 - 240, 720, - 1600, 2100, -1512,

462,

80 - 240, 270, -110,

- 24,

8 24, 18,

-

48, -27, 0, 5,

12 - 24,

-2, -10, 10,

192 - 960, 2400, - 3360, 2520, - 792,

40 - 80, 44,

12, 0, - 4,

2 - 2,

240 - 960, 1680, -1440, 495,

d} 2) are numerical coefficients and are listed below:

(27) 10 -10,

1

(28)

6 -6,

1

] 1 ]'

160 - 480, 540, - 220,

(29)

(30) 60 - 120, 66,

12 -12,

J. Mielnicki et al. / Magnetization curve calculations of the Ising ferromagnet

391

1.(

• .2

i A

.5

.8

. . . .

1.2

1.0

TITt

1.4

Fig. 1. The magnetization curve (solid line) and the ((SZ) 2) vs. temperature relation (dashed-dotted line) for an Ising ferromagnet with S = 1. ( J / k T c = 0.2842). The dashed line - the magnetization curve of an Ising ferromagnet with S = 1 / 2 . ( J / k T c = 0.7884) [2].

b~2) =

0, - 32, 96, -112, 48,

-16, 64, - 120, 112, - 42,

12, c(2) kt = , _ 24, 14,

0, 8, -8,

d}2)=

0,

-10,

16 -64, 96, -64, 15,

-16, 32, -17, 15,

0,

] 32 -96, 104, -40,

0, -12, 12, -6,

4 -8, 2, 1].

24 -48, 26, 4 -4,

, 8 - 8, ] ],

(31)

1 (32)

1 (33)

Thus, we get the set of two eqs. (24a) and (24b) for m and x with the coefficients depending on the temperature. However, before the calculations of m and x, the J / k B T ~ value should be first found. For this purpose eqs. (24) should be solved for the case of y = 1 and m = 0. With these values eqs. (24) become the equations for J / k B T ~ and x for T = Tc and after computer calculations the following values have been obtained: J / k B T c = 0.2842

(34)

and 2

Xrc = ( ( S z ) >r=rc = 0.6976.

(35)

Now, taking into account (34) the magnetization curve ( S z ) = re(T) together with the ((SZ) 2 ) = x ( T ) relation have been obtained on the basis of (24a, b) and the results are presented in fig. 1, where for the comparison also the magnetization curve for S = 1 / 2 is given.

4. Discussion In the paper we have discussed the general method of magnetization calculations of Ising ferromagnet with arbitrary spin value which was then applied for the detailed calculations of ferromagnet with S = 1. As a result we have obtained the coupled equations for ( S z) and ((SZ)2). The same method can be used for the case of higher spin values, however, according to eq. (8) we will obtain then the set of 2S coupled equations for ( ( S Z ) ' ) , n = 1 . . . . . 2S, what is the only complication of the problem. On the basis of numerical calculations made for crystals with sc symmetry first of all the J / k T ¢ value equal to 0.2842 has been obtained. This can be compared with the value 0.338 resulting from the

J. Mielnicki et al. / Magnetization curue calculations of the lsing ferromagnet

392

Bethe-Peierls Weiss method. The difference between these two results is due to the inaccuracy of the present method in comparison with the much more accurate B - P - W approach. Then, the ( S ~) and ( ( S : ) 2 ) vs. temperature curves have been obtained in the whole temperature range. As it is seen from fig. 1, the ( ( S : ) 2 ) tends to 2 / 3 for T--+ zc, what could be expected because the energy levels corresponding to S: = _+1,0 become equally populated in the limit of infinite temperature.

Appendix. The derivation of generalized Callen's equation The derivation of generalized Callen's equation (1) together with (3) is based on the approach first applied by Ferchmin [10]. We start with the correlator {[S,+,(t), exp(aS,,)S,,,]+) together with its Fourier transform:

([ S£+,(t),

f£p,",,(~

e"S~'s£] + ) =

)e

i~ dco,

(A.1)

] +) e i , ~ d t

(A.2)

where

1 £ ~ ([S+(t).e.S.,,Sm 0,,,(a) = ~-~v S,,+ ( t ) = exp( -

itB m)S,,+

(A.3)

and

B~ = Y'~ J,,iS/"

(A.4)

j'Em

Then, making use of the representation of correlation functions [11]:,


~c

I:(~o) e i~, do:,

(A.5)

"am ) = j_,I",,,(~0)e/~'~ e ix,, d~0

(A.6)

we get: a 60 ) / ( e ,.",(~o) = o,,,(

/3~

+ 1),

{A.7)

hence, taking into account (A.4), (A.5) and (A.6), the relation for ([S+(t), exp( aS~,,)S,,]_ the form:

<[S£(t), e"Si"S,,]_ )= f. I <' b(Boa)O~,,(co) e

)

can be found in

~'do:

(A.8)

~c

with b(flao) = tgh(flo0/2). Making now the inverse Fourier transform of (A.8) and then substituting p~, by the r.h.s, of (A.2), we get: 1 2~

~c e-''Bm . [s£ '

e"S~"Sm]-)

ei'~' a t = ~1

itB"[S,,+,,e"S2Sm] + ) e i ~ ' d t

(A.9)

In the derivation of (A.9) also eq. (A.3) has been used. Now, taking into account the following commutation relations:

[S+,eaS's-]±=e-aeuS:[s(s+I)+Se-(sz)e]+_[S(S+I)-Se-(S:)2]e"X:

(A.10)

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393

and introducing the notation: ~2(a) = e as~',

(a.ll)

A(fl~o) = [1 + b(flco)] e " - [1 - b(fl~o)], B(B~o) = [1 + b(/3o:)] e a + [1 - b(fl~o)],

(A.12) (A.13)

we get after simple calculations the following equation for I2(a):

)

da2, 2--~f_~ (e-i'BmI2(a)) ei'~t dt + A(fl~o---~da ~ f ' - s e-i'Bm~2(a)) ei~t dt -S(S+l){2-~L(e-itSmI2(a

1

) ei'~tdt}=0.

(A.14)

We introduce now the function f(~o, a) defined by the equation: 1 2---~

(e-itBmI2(a))

(e_i,Bm) ei~tdt.

i ei'~t dt=f(o:, a)~--~

OO

(A.15)

OO

Then, on the basis of (A.14) one can get the following relation:

--A(~) ~ f (

' a) + S(S + alf(~0, a) = 0.

(A.16)

The solution of this equation should fulfil the following condition resulting from (A.15): f(~0, 0) = 1.

(A.17)

The second condition results from the identity for spin operators: S

I--I ( S Z - p ) = 0

(A.18)

p=-S

hence, from (A.15) we get:

~ -p)f(~o, Hs ( 7a

a) a = o = 0 .

(A.19)

p=-S

Now, the solution of (A.16) can be found in the same manner as it was done in ref. [13], and the result is: e~S+l)°[a + b(fl,0)] ~ + '

f(~0, a) = 2b(flo:)

{[1 + b(flo:)] e ~ - 1 +

b(flo:)}

-s°[1 b (B,~)] ~+~ - e {[1 + b(flo:)] 2s+1 - 1[1 - b(fl~o)] 2s+~" (A.20)

For further considerations we can present (A.15) in the form:

~(a)~f e~

Bm~,at = f(~o, a)~--~f~_ e l(~°-B'')t at ,

(A.21)

where the integrals are simply the Dirac delta functions. Integrating now the above equation over ~o, we simply get:

(~2(a)) = (f(Bm, a)),

(A.22)

J. Mielnicki et al. / Magnetization curvecalculationsof the lsing ferromagnel

394 hence

((S:)")

:

/ -~a,f(B,,, d

a)a:0

/

"

(A.23)

I n p a r t i c u l a r cases of n = 1 a n d 2 we get f r o m (A.23) the eqs. (1) a n d (3). W e have p r e s e n t e d a b o v e the d e r i v a t i o n of g e n e r a l r e l a t i o n s for ( ( S : ) " ) . H o w e v e r , for a given spin v a l u e the e x p r e s s i o n s for ( ( S = ' ) " ) c a n be f o u n d as u s u a l (see, e.g., ref. [12]) f r o m the r e l a t i o n : ( ( S , ~ ) " ) = ( T r m ( S ; - ' ) " e /~'~/ ~r,,, 7 ~ V /,

(A.24)

where ~ is the Hamiltonian which, for exchange interactions only, can be presented in the form:

JF---(-J Y'~ S,")S:, j ~ r¢l

"

(1.25)

/

a n d the trace is t a k e n with respect to the S,, (S,~, = - S . . . . . + S ).

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]

J. Mielnicki, T. Balcerzak, V.H. Truong and G. Wiatrowski, J. Magn. Magn. Mat. 51 (1985) 151. J. Mielnicki, T. Balcerzak, V.H. Truong, G. Wiatrowski and L. Wojtczak, ibid. 58 (1986) 325. J. Mielnicki, T. Balcerzak, V.H. Truong, G. Wiatrowski and L. Wojtczak, ibid. 61 (1986) 198. N. Benayad, A. Benyoussef and N. Boccara, J. Phys. C 18 (1985) 1899. D.A. Garanin and V.S. Lutovinov, J. de Phys. 47 (1986) 767. D.A. Garanin and V.S. Lutovinov, Teoret. Mat. Fiz. 62 (1985) 263 (in Russian). D.A. Garanin and V.S. Lutovinov, Solid State Commun. 49 (1984) 1049. A. Zag6rski, Phys. Lett. A 99 (1983) 247. A. Zag6rski and W. Nazarewicz, Acta Phys. Polon. A60 (1981) 697. A.R. Ferchmin, Magnetyki o nieuporzBdkowanej strukturze (The magnetics with disordered structure). The papers of Institute of Physics, Polish Academy of Sciences, no. 71 (Ossolineum, Wrodaw, Warszawa, Krak6w. Gdahsk, 1978) (in Polish). [11] D.N. Zubarev, Usp. Fiz. Nauk 71 (1960) 71. [12] N. Boccara, Phys. Let[ A 94 (1983) 185.