The magnetic transition in finite systems

Volume 35A, number 2

PHYSICS LETTERS

THE MAGNETIC

TRANSITION

1: ~

IN FINITE

19:1

SYSTEMS

W. F. HALL, R. E. DE WAMES and T. WOLFRAM North American Rockwell Science Center, Thousand Oaks, California 91360

USA

Received 10 March 1971

The spatial behavior of the magnetization in finite systems near their transition tempernture is derived for one—dimensional mean—field models.

Magnetic critical phenomena and the effects of surfaces on the static and dynamic properties of magnetic materials are subjects of much current interest [1-3]. It is the purpose of this note to point out certain simple properties of finite magnetic systems just below the critical point which follow solely from the assumptions inherent in any one-dimensional mean-field model. Such a model relates the magnitude ~iof the sublattice magnetization at the site x to the values of i~in the vicinity of x in the following manner: ~(x)

=

1 F(~-Z a(x, x’)ai(x’))

* .

X

Here F is the mean-field function (e.g., the Brillouin function), T is the absolute temperature, and a(x, x’) is the influence function, assumed to have a definite range - x’ 0 and g(z) is a positive, monotonically increasing function vanishing at z = 0. In what follows, we shall for simplicity tai.ceg(z) j3z~ as z —. 0, so that the magnetization of the unbounded system will tend to zero as c4a(1 - ‘r)L,s]1/V(y> 0), where we have put T = T/aA, A= 2~~~à(x’). However, when the systern has boundaries oeyond which the magnetization must vanish, ~ will depend upon the distance from the boundaries, and the manner in which i~(x)approaches zero will no longer be given by 71~. tThe symbol ~ integration,

stands either for summation or

To derive the qualitative behavior of ~ for the finite system it is sufficient to approximate i~in the argument of F by the first three terms in a Taylor expansion, leading to the differential equation at points further than Ra from the boundaries -1 ~ “(x) + - crrF (ii) = 0 . (1) A 2ã(x), andF~ is the inverse where a = +)2~x function 2for F. Near enough to the boundaries ij will increase roughly linearly withx. giving the boundary conditions at x = 0 and x = d. with a 1 =~x>0xd(x): a ±~-77’(~)

1 =

aTF

(?7(~j))-

~1i(2j)

The nonlinear differential eq. (1) for the magnetization is identical with Newton’s second law for the motion of a particle in a potential field, and consequently one can deduce the qualitative behavior of the solutions from its energy integral. Putting m = a2,’A, 1(ii) v i~’(x), ti/a and + (i3~a)(~i using the a)~ as ~ —, 0, this integral becomes asymptotic form FE ~mv2 + 4(1-T)772(x) - a~[i7(x):a]2~/(2+~) as T —, 1. The potential energy V(rj) defined by this energy integral consists of a parabolic well with a lip at ~ = i~, beyond which the potential falls away to -~. Depending upon the initial conditions and the value of E, the solution will fall into one of three categories: Oscillation within the well, divergent motion starting above or outside the well, or motion tending asymptotically to the lip of the well. The solution for i~appropriate to a system of finite width clearly corresponds to a positive segment of an oscillatory trajectory, moving from some a~out to the edge of the well and returning to 77ci~ Similarly, the solution 93

Volume

35A, number 2

PHYSICS

LIYI’T

ERS

appropriate to a semi-infinite sample must be

systems [4] (x

that with F = V(i7~),for only in this case does 7) increase asymptotically to ~ as x . The divergent solutions are necessarily unphysical. For the semi-infinite sample, the energy integral combined with the boundary condition 1(2) in A the limit?) 2 0 gives ~2 [77(0)] ± V(q(0)) (a])

(1~ is

2 (57

(1- ~)[a(l 2( ±~)

-

T)(3]

2)

(3)

17

May 197!

na

0. for n 1,2. . . . N, where the lattice parameter.) In summary, one finds that for points near the surface of a finite sample the ferromagnetic transition occurs in two stages: when T lies in the range 1 (1- T)~(1 7’ ~1follows i~behaves as ~“bulk”. but with different transition temperature. while fora(Td_ T)~V(1 d When the mean-field function F is the Brilbum function, so that g(z) ~z2. the solution to eq. (1) for the semi-infinite case can be ohtamed in closed form for 1: 77B(x)~[a3(l - T) ~] ~~tanh{[(l-T)A/2a 2]1~(x~2a1rA

which for — 1 has the solution 77(0) K(1 - T)1/27)~ 77g. Therefore, in the neighborhood of the surface of a semi-infinite sample the magnetization decreases more rapidly to zero than in the bulk by the factor (1 - r)1/2. Further, from (1) it can be shown that the linear region for q(x), over which 170: ~ extends over a distance l~[a2/A(l_T)]h/2, the “coherence length”, which tends toward infinity as 1, so that every finite point in a semi-infinite sample eventually follows rather than ?7~. This result does not, however, imply that the magnetization infinite samples tends eventually to zero as i~. So long as the thickness d 1, the surface of the sample will follow while the bulk will follow n~ when 1 becomes cornparable to or larger than d, however, the oscillatory behavior of the solution must be taken into account. ~e finds that in this case the boundary conditions force i~(x)to take roughly the form ,k(r)sinirx/d, which Itfrom gives that 2/Ad2]l/)~. will (1) be seen this 77= K’[l form- for T_a27r 17 vanishes not at r = 1, but at 2 2 the smaller value Td = 1 - a 2r Ad , an effect which has been discussed for finite discrete

94

)},

a result entirely consistent with the discussion following (3). The qualitative features of this result at x 0 were found recently by Mills [2], although he did not obtain the above solution. For a discrete system, this solution holds with x replaced by na0. It should be mentioned that computer solution for a nearest-neighbor-coupled discrete system using the Brilbouin function bears out the behavior obtained here from continuum arguments. Results for such a system, including the effect of changes in the exchange constants for the surface layers, will be published elsewhere.

References [1) M. E. Fisher and \V.J. Camp. 26 (1971) 565. [2] D.L.Mills, A.I’.S. [3) [4]

Phvs.

Rev. Letters

Bulletin CAl, p. 345 (March, H. D.De Wames and 1’. Wolfram, Phys. Rev. Letters 22 (1969) 137. See, for example, L. Valenta, Phys. Stat. Solids 2 (1962) 112; W.Doring, Z. Naturforsch. 16a (1961) 1008, 1146; and R. J. Jelitto, ibid., 19a 11964) 1581,