Zero-field-cooled magnetization and initial susceptibility of magnetic particle systems

Journal of Magnetism and Magnetic Materials 120 (1993) 203-205 North-Holland

~ l l ~ ~~/[~[I |

Zero-field-cooled magnetization and initial susceptibility of magnetic particle systems H. Pfeiffer a and R.W. Chantrell b a Institut ffir Physikalische Hochtechnologie, 0-6900 .lena, Germany b Physics Department, Keele University, Keele, Staffordshire ST5 5BG, UK General expressions for the zero-field-cooled magnetization and the initial susceptibility (g) of magnetic particle assemblies are set up. These equations are evaluated analytically by specifying the distribution of geometric and magnetic properties of the particles. We derive relations between X and the magnetic viscosity.The influence of interaction effects is examined. One of the most important characteristics of fine magnetic particle assemblies is the dependence of the zero-field-cooled magnetization (ZFCM) on the temperature and field [1]. The ZFCM is the magnetization measured at field H and temperature T after cooling the system in zero field from a high enough temperature at which all particles are superparamagnetic. Assuming a distribution of particle volumes V, the ZFCM results from magnetic stable particles ( V > Vs) as well as from superparamagnetic particles ( V < Vs), where Vs is the critical volume for superparamagnetism. Taking a typical measuring time of 100 s, we have

the volume distribution .f(V) and the H a distribution P ( H a) we have

Vs(a/t , Ha, H ) = 50 kT/M~H~r(~, H/Ha) ,

In the following, we use for f and P the normalized distribution

(1)

where H a is the anisotropy field, M s is the saturation magnetization, and 1/, is the angle between the easy axis of the particle and the field. For simplicity, we assume H a and M s to be constant within the considered temperature range. The function r can be expressed analytically for ~ = 0 and 1/, = ~r/4, whereas for general a/, only approximate expressions are given [2]. For instance, we have r = (1 - H / H a )2 for a/F= 0 and H < H a. The magnetization for particles with V > V~ is simply the virgin curve m -- m i ( ~ / H / H a) according to the Stoner-Wohlfarth theory [3]. For V < Vs we have to use the magnetization curve for superparamagnetic particles m = L ( ~ , V, Ha, H), where L is the modified Langevin function [2]. For g' = 0 we often use the approximation L = tanh(VMsH/kT ) .

(2)

The total ZFCM m is obtained by summing all contributions. Taking into account the texture function g ( ~ ) ,

m

= fo~/2d~tg(11") foXdHaP (Ha) x ( fov'('e" H*' ")dVF( V )

×L(¢, V, Ha, H) + m i ( ~ , H/Ha)

F(x)

=O~ ¢°+1

H)dVf(V) .

e-~Xx~/r(to + 1),

(3)

(4)

where F ( x ) = f ( V ) for x = V and F ( x ) = P ( H a) for x = H a ;F(a) is the F-function. The parameters a and to can be expressed by the average 2 and the width of the distribution A = (x -'~ - 2 2 ) 1/2. It holds that a = 2 / A 2 and to = ('x2/A2)-1. As an example, we consider the aligned case with = 0 and assume only one value of H a. Further we use eq. (2) for simplicity and replace tanh x by 1 + e - 3 x - 2e -2., which is a very good approximation. Under these conditions, eq. (3) can be obtained in closed form O/~o+1

re(H, T)

3

F(to + 1) •

cnY(to + 1, U~V~)/U~ +1

n=l

(5) with

c A = - [ 3 ( - 1 ) n + 1]/2, Un = a + 2 2 - ~ n ( n Vs -- 50 kT/MsHa(1 - H/Ha) 2.

1 ) M s H / k T x and

3ffa, x ) = _f o dte-tta-1 is the incomplete F-function, Correspondence to: Prof. R.W. Chantrell, Physics Department University of Keele, Keele, Staffordshire ST5 5BG, UK.

tabulated in ref. [4]. According to eq. (5), m increases monotonically with H. It follows that re(H, T --* O) -- 0

0304-8853/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved

H. Pfeiffer, R. W. Chantrell / Magnetization and susceptibilityof magneticparticle systems

204

for all H < H a. In the same way, analytical expressions for the Z F C M in the case qt = av/4 can be derived by using the results of ref. [2]. Now, we treat the aligned case (qt = 0) as above but take the volume constant and use an H a distribution of type (4). We obtain

re(H, T) --- r ( ~

1

+ 1, alia) tanh k----T-T ,

X0") = lim re(h, ~)/h, h---,0

(7)

where we use, as in fig. 1, the reduced field h = n / H a and the reduced temperature z = T / T o with the m e a n blocking temperature T n = VMsHa/50 k [1]. For the aligned system ( ~ = 0), eq. (7) gives

X = 5~Ozfo~dHaP ( Ha ) foV~°(14"")d v V f ( v ) ,

E

(9)

"c°+l

For high temperatures, eq. (8) can be rearranged to X -----50['/" + "rA(~')] -1,

where Has is the critical H a for superparamagnetism, defined analogously to eq. (1). W e have Ha, = H + (p/V)[1 + ~/1 + 2I-IV/p ], p = 2 kT/M, [5]. In fig. l(a), the field dependence of Z F C M (6) is shown for two temperatures. For high temperatures we have a uniformly increasing function, in contrast with low temperatures where we get first a strong and then a weak increase. Fig. l(b) shows the Z F C M as a function of temperature for two fields. For small fields, eq. (6) gives a sharp peak, whereas for large fields a very broad maximum follows. It is remarkable that we have m > 0 for T = 0 and H > 0, in contrast with the preceding case (5). The initial susceptibility is defined by

I

X~q

VM~H

(6)

.

where V~o = Vs(H= 0 ) = "fVI-Ia/n a. For low temperatures we obtain with eq. (4) the power-law

(8)

where

A(z)

=

f"dnaV(Ha)f" "0

dVv.

" l ' s o ( H a , "r)

(11)

I/

It holds that A ( r ~ oo) = 0. If A(~') tends to zero more strongly than 1 / % we obtain r >> 1 at the Curie law X = 50/~. For instance, this is the case for the distribution (4). If only one distribution (V or H a distribution) in eq. (8) is present, simple relations between X and the magnetic viscosity S for a zero field can be derived. Using the expressions for S [2,5] we have ( d / d ~ X ~ ' x ) = 1250 S(z)q(r), where q -- 1 for H a --- const, and q = 1/~- for V = const. For H a = const, and using the volume distribution (4), X takes the simple form 50

X = ~-F(~o + 1) 3,(~o + 2, [~ + 1]~').

(12)

There appears a maximum of X at ~'M = TM/TB, where ~'M is given by

(13)

T(w + 2, ZM) = e-ZMz~ +2,

lI)

Ii0

0.8

0.8

0.6

0.6

0.4

(10)

0.4

(b)

(a) 0.2

0.2

~-~s

I

i

I

i

t

J

i

i

i

i

i

t

0.1

0.2

0.3

0.4

2

~

6

8

I0

12

I~

16

h

='-

Fig. 1. (a) ZFCM (6) for ~o= 1 versus the reduced field h = H / ~ a for two reduced temperatures ~-= 50 kT/VMs~a; (b) ZFCM versus z for two h.

I-I. Pfeiffer, R.W. Chantrell / Magnetization and susceptibility of magnetic particle systems with z M =(to + 1)r M. It follows from eq. (13) that ~'M> 1. In particular, we have r M ~ 2 for broad V distributions and r M converges to 1 if f ( V ) becomes sharper. Analogous to eq. (12), a simple expression for X in the random case can be obtained ( g 0 F ) = sin in eq. (3)). Using eqs. (3) and (7) and the results for L and m i [2,3], we arrive at 1 2[ X=~X0+~ 1

1 F(to+l)

] 3'(to + 1, [~o + l l r ) [ , (14)

where Co = X( if' -- 0) is given by eq. (12). The condition for ~'M (13) is only slightly changed (the rhs is to be multiplied by 0.96). In contrast with eq. (12), we have x(~" = 0) ~ 0. So far, we have neglected interactions between the particles. This effect can be easily included within a mean-field approximation, replacing h in eq. (3) by h + hi, where the interaction field h i is proportional to the magnetization (h i = tim). fl > 0 (or/3 < O) means positive (or negative) interaction, which stabilizes the magnetized (or demagnetized) state. This procedure leads to the susceptibility

205

With eq. (15) and the Curie law given above we arrive for high temperatures at the Curie-Weiss law [6]. ; = 5 0 / ( ~ - ~0),

(16)

i.e. interaction gives rise to a finite ordering temperature ~'0 = 50/3. We remark that a negative ordering temperature may also occur without interaction (/3 = 0), if A ( r ) in eq. (11) vanishes with 1/r leading to r o = - l i r a pA('r). This follows for distributions P and f obeying

fo~'dVP(y/V)f(V) ~y-2

for Y - - oo

which is realized if P ( H a) ~ H a 2 or f ( V ) ~ V -3 for large H a or V, respectively. The latter case has already been treated in ref. [7]. Thus, a negative ordering temperature does not necessarily mean the presence of negative interaction. This work was supported by the CAMST (Community Action on Magnetic Storage Technology) project of the EC science programme. References

= [(I/x)

_/3]--1.

(15)

The maximum value of ,f is enhanced (or lowered) by positive (or negative) interaction. The temperature ~'M remains unchanged, if /3 does not depend on ~'. For d/3/d~-> 0 (or < 0), ~'M increases (or decreases) by interaction. The susceptibility without interaction takes for small ~'(~" 0 and 3' < 1 or /3 < 0 and 3'> 1. The inflection point occurs at ~'i --- [(1 - 3")/B/3(1 + 3')] 1/7.

[1] R.W. Chantrell, M. EI-Hilo and K. O'Grady, IEEE Trans. Magn. MAG-27 (1991) 3570. [2] H. Pfeiffer, Phys. Stat. Sol. (a) 122 (1990) 377. [3] E.C. Stoner and E.P. Wohlfarth, Phil. Trans. R. Soc. A240 (1948) 599. [4] K. Pearson, Tables of the Incomplete F-Function (Cambridge University Press, 1957). [5] M. EI-Hilo, K. O'Grady, H. Pfeiffer, R.W. Chantrell and R.J. Veitch, tobe published in IEEE Trans. Magn. [6] M. Holmes, K. O'Grady and J. Popplewell, J. Magn. Magn., Mater. 85 (1990) 47. [7] M. EI-Hilo, K. O'Grady and R.W. Chantrell, J. Magn. Magn. Mater. 117 (1992) 21.