DYNAMICAL CRITICAL BEHAVIOUR OF HEISENBERG FERROMAGNETS FOR T ≥ Tc

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DYNAMICAL CRITICAL BEHAVIOUR OF

HEISENBERG FERROMAGNETS FOR T

≥ Tc

H. Iro

To cite this version:

JOURNAL DE PHYSIQUE

Colloque C8, Suppl6ment a u no 12, Tome 49, decembre 1988

DYNAMICAL CRITICAL BEHAVIOUR OF HEISENBERG FERROMAGNETS FOR

T

1

T c

H. Iro

Institut fiir theoretische Physiiz, Universittit Linz, A-4040 Linz, Austria

Abstract.

-

From an 0 (E)-calculation (E = 6

-

d) of the dynamical spin correlation function in a Heisenberg system for T 2 T, an expression for that function applicable in dimension d= 3 is deduced. It comprises the expected limiting behaviour in different sectors of the critical region.

It has been shown that the MaiMazenko model of a Heisenberg ferromagnet [I], evaluated within an &-expansion, E = 6

-

d [2, 31, accounts very well for certain features found by neutron scattering in Fe, Ni and EuO [P6] at T,. To investigate the applicability of the model also for T

>

T,, one has to calculate the spin correlation function for arbitrary values of q, w, and the correlation length J. Since the calculation involves an expansion around d= 6 [7], the result must be processed further in order to apply in d= 3 [8].

The equation of motion for the order parameter S in a Heisenberg system with effective Hamiltonian

is given by [I]

with the Gaussian distributed random noise

5.

For the spin correlation function C(q, w ,

0 ,

expressed via the fluctuation dissipation theorem by the self energy

II

(q[,

$)

of the response function, which can be more conveAiently calculated, one has [7] (x = qJ, w = w/XqZ)

1

q - 2 ~ (2) -Re 2

h " -iw

+

[X ( x )

II

(x, iw)]-' ' (2) Here X (x) = (1

+

1/z2) is the scaling function for the static susceptibility (i.e. (-2 = r ) and the dy-

d+2 E

namical critical exponent z equals -=4-- [I]. (In

2 2

X (x) and in z we have neglected q ; X has been renor- malized.) The self energy

II

has been calculated t o first order in E (E is related to the fixed point value of f * by f * = ( 9 6 1 ~ ~ ~ ) " ~ [I])

11 (x, iw) = 1

-

EF (z, iw) ₍₃₎ where F (x, iw) is a rather involved expression, which can be found explicitly in reference [7]. We quote only

the two limits of F (x, iw) used in the following. At T, one finds for large values of w

1 1

F ( a , ' iw) = --In (-iw)+ -In 2 + 0 (I/&)

,

(4a)

8 8

and in the hydrodynamic region, i.e. x

<<

1, for w = 0 the limit is

This asymptotic behaviour of F(S, iw) has to match the known asymptotic power law behaviour of 11 (x, iw) [8], namely

II

(co, iw) N w(~-")'" for w -r co (5a)

and

II

(x, 0) N x " - ~ for z -+ 0. (5b)

In the same spirit as in reference [3] this can be achieved by replacing F (x, iw) by a simpler function showing the logarithmic behaviour of equations (4a, b)

where

a = 0.46 and b = 3.16. (7) By exponentiating the right hand side of equation (3) with F given by (6) and observing conditions (5a, b) one arrives at the final expression for

II

(x, iw)

2-e/4

I

6 / ( 8 - & )

n

(z, iw) = [(I

+

f )

-

aiw (8) Insertion into equation (2) and setting E = 3 gives the

final form for the correlation function C (q, w, J) of isotropic Heisenberg ferrornagnets [9].

For a comparison with neutron scattering results one has to express C(q, w, J) in terms of the measured quantities

C8 - 1564 JOURN4L D E PHYSIQUE

Here w, (q, f ) is the half width a t half maximum of the scattering intensity when q is kept constant

point. In reference [18] it was shown that dipolar inter- actions can account for the scaling function

n

found experimentally. In the shape function cP their influ- ence becomes evident at T, only at very small values of the wave vector q [19]. In conclllsion one can say that equation (2) together with (8) describes the criti- cal behaviour very well but very close to the dynamical critical point the spin nonconserving interactions be- come important. The onset of this influence depends on the quantity considered.

Acknowledgment

This work was supported by the Fonds zur Forderung der wissenschaftlichen F~~rschung.

Fig. 1. - Constant w cuts of C (q, w , E ) at 2 w / r = 0.05 for

e

ranging from far from criticality ([ = 1) to close to criticality (E = 100)

.

The solution w, is a homogeneous function, so that

where R is fixed at T, to be R ( 0 0 ) = 1. The homo- geneity of the dynamic shape function @ was already anticipated in equation (9). In reference [8] 0 and @

are given in terms of 11.

At

Tc@

reduces to the form used in reference [4], thus leaving unchanged the conclusions in references [5, 61. For T

2

T,

R (z) is similar to a mode coupling result obtained in reference [lo]. Especially for small val- ues of x there is less agreement with results found in Fe [ll] or EuO [12], but his is due to dipolar forces (see below). In the shape @, particularly at T,, the small w behaviour is not as flat as found in mode cou- pling calculations [lo]. A feature that is very sensi- tive to the shape is a constant w scan. An example is shown in the figure for different values of

<.

By in- creasing the distance from

T,

(decreasing <) the peak position is shifted very slightly to higher values of q. This weak variation, already expected in [13], matches very well to neutron scattering results as e.g. in Ni [14, 151 and EuS [16]. Further the equal intensity con- tours of C (q, w , f ) are confirmed by an experiment on FesSi [17]: in the (q, w ) domain considered a dynamic Lorentzian shape function is clearly excluded.

So the results presented above lead directly to the limitations of the model. Though dipolar interactions are small, due to their spin nonconserving property they are ultimately relevant very close to the critical

[I] Ma, S. and Mazenko, G. F., Phys. Rev. B 11

(1975) 4077.

[2] Dohm, V., Solid State Commurz. 20 (1976) 657. [3] Bhattacharjee, J. K. and Fen-ell, R. A., Phys.

Rev. B 24 (1981) 6480.

[4] Folk, R. and Iro, H., Phys. Rev. B 32 (1985) 1880.

[5] Shirane, G., Boni, P. and Martinez, J. L., Phys.

Rev. B 36 (1987) 881.

[6] Boni, P., Chen, M. E. and Shirane, G., Phys. Rev. B 35 (1987) 8449.

[7] Iro, H., 2. Phys. B 68 (1987) 485.

[8] Iro, H., J. Magn. Magn. Maten 73 (1988) 175. [9] The condition

00

J__

$C (9, w ,

0

=:

x

(a,

0

is satisfied by

II

(z, iw) as given by equation (8) in a nontrivial manner.

[lo] Aberger, C. and Folk, R., Phys. Rev. B 38 (1988) 6693.

[ l l ] Mezei, F., Phys. Rev. Lett. 49 (1982) 1096. [12] Mezei, F., Physica 136B (1986) 417.

[13] Folk, R. and Iro, H., Phys. Rev. B 34 (1986) 6571.

[14] Mook, H. A., Lynn, J. W. ancl Nicklow, R. M., Phys. Rev. Lett. 30 (1973) 556.

[15] Bijni, P. Shirane, G., J. A p p l . Phys. 57 (1985)

3012.

[16] Boni, P., Shirane, G., Bohn, H. G. and Zinn, W.,

J. A p p l . Phys. 63 (1988) 3069.

[17] Pintschovius, L., Phys. Rev. B 35 (1987) 5175. [18] Frey, E. and Schwabl, F., Phlls. Lett. A 123

(1987) 49.