fast thermal reversal of magnetic particles

ELSEVIER Journal of Magnetism and Magnetic Materials 171 (19971 209 217

Journal of magnetism and magnetic

J H materials

Fast thermal reversal of magnetic particles

Rodrigo Arias*, H. Neal Bertram Center/or Magnetic Recording Research, b)Tiversity o/" Cal!lbrnia, San Diego, La Jolla CA 92093-0401, USA

Received 10 February 1997

Abstract

In several physical systems the experimental thermal decay from an initial metastable state to a final stable state is frequently logarithmic over a few decades in time. The quantity plotted is the probability of not reaching the final stable state, or an equivalent variable. Logarithmic slopes locally higher than ln(10)/e ± 0.85 are shown to rule out two frequent explanations for these processes: thermal activation over a single barrier with Arrhenius exponential decay, or decay through parallel processes of Arrhenius type occurring at different rates. We present a model of thermal decay that can show these high logarithmic decay rates or fast relaxation, as here defined. A downward cascade of metastable states is assumed between the initial and final states, and the evolution of the probability distribution is described by a Master equation. 'Fast ' relaxation is obtained for consecutive states separated by equal direct and equal reverse barriers, the latter being much higher. These solutions correspond to almost degenerate decay rates, and to low relative dispersion of switching times. The results of this model may explain measurements of 'fast' thermal relaxation of the magnetization of single magnetic particles near the coercive field.

PACS: 75.60.Jp; 75.60.Lr; 05.40. + j

Kevwords. Thermal relaxation reversal magnetic particles

1. Introduction

The results presented in this paper were mot iva ted by some experimental results [1] on thermal switching of magnet ic particles, that evidence 'fast' relaxation. In the ment ioned work as in other thermal

* Corresponding author. Present address: Departamento de Fisica, Universidad de Chile, Blanco Encalada 2008, Casilla 487-3. Santiago, Chile. Tel.: (56 2) 678 4675: fax: (56 2) 696 7359; e-mail: [email protected].

relaxation processes, a logari thmic decay of a physical variable over a few decades has been observed. The peculiarity of the results of Ref. [1] is that they cannot be explained in terms of thermal activation over a single barrier or as effective single barrier processes occurring in parallel at different rates. This happens because the slopes of decay are higher than what is possible under these models. Evidence of 'fast ' relax- a t ion solutions has also been found in Monte Car lo simulat ions of simple magnet ic systems [-2-4].

In the single bar r ie r model it is assumed that the rate of switching is given by the wel l -known

0304-8853/97/$17.00 :,i~ 1997 Elsevier Science B.V. All rights reserved PII S0304-88 5 3 ( 9 7 ) 0 0 0 5 3 - X

210 R. Arias, H.,~ Bertram/Journal qf Magnetism and Magnetic Materials 171 (1997) 209 217

Arrhenius formula: F = l / r = Q exp(-EB/kT), the principal term is the exponential, EB the energy barrier, T the temperature, and f2 a frequency prefactor. It predicts for high reverse barriers a simple decaying exponential for the probability of not switching from the initial metastable state to the final stable state: PNs(t) = exp( - t / r ) , where r is the above relaxation time. Logarithmic decay over a few decades has also been attributed to several processes of thermal decay, effectively over a single barrier, occurring at different rates [5]. This corresponds mathematically to:

PNS(t) = ~, Wk e x p ( - r d ) , (1) k

where the weights Wk are all positive and sum to one (Pus(0) - 1), and the rates rk are assumed nondegenerate. This form of the probability of not switching, of which the single barrier is a special case, has a logarithmic derivative bounded in absolute value by ln(10)/e --- 0.85. According to this, we define that there is 'fast' relaxation when

L~PNs ln(10) > e -- 0.85. (2)

Consequently, within the formalism of the Master equation we became interested in answering the following question: ls it possible to have a solution like (1) where the weights are not all positive (they still have to sum to one due to PNs(0t = 1), and at the same time give rise to 'fast' relaxation'?

We will present a model that considers multiple inlermediate metastable states between the initial state and the final stable state: it is assumed that consecutive states are separated by equal direct energy barriers, and also by equal reverse energy barriers. The dynamics of the population of the different states is described by a Master equation [6]. Within this model one obtains solutions with a 'fast' relaxation rate when the reverse barriers are much higher than the direct barriers. For these 'fast' relaxation solutions the different relaxation rates rk become similar and there is a low relative dispersion of switching times.

In the case of a magnetic particle we assume then the existence of intermediate metastable states, and

also of stationary rates of transition between these states (of the Arrhenius type). The picture of the existence of metastable states of the magnetization has been used for example to explain hysteresis in magnetic materials [7 10]. Theoretical and numerical approaches to thermally activated magnetization reversal can be found in [11, 12], and numerical applications to systems of two three interacting particles in [13, 14].

The formalism of the Master equation is very general, and the solution presented here could be relevant to other physical systems. For example, for interacting magnetic particles or grains, one definitely expects many local energy minima or mctastable states. Other phenomena described by Master equations are gas-phase relaxation processes, chemical kinetics, polymer dynamics, etc.

We do not present a general analysis of the conditions under which a Master equation leads to 'fast' relaxation solutions (this would mean to find st{fficient conditions on given rates of transition between all states). Instead we draw attention to the fact that some of these solutions do exist, as in our model, and that their nature is physically reason- able for thermal switching of magnetic particles near the coercive field. Thus. we do not pretend to provide a complete theory of ferromagnetic relaxation. Issues like intrinsically nonexponential behavior are not addressed.

As far as the 'fast' relaxation observed in single magnetic particles, we can argue the following in favor of the applicability of our model: fast relaxation is experimentally observed when the applied field approaches the coercive field and precisely at that point our picture of a downward cascade of metastable states becomes more plausible because a "primary' energy barrier has decreased its value, indicating that secondary barriers may play an important role; also, metastable states of the magnetization can be conjectured by imperfections of the particles (surface effects, nonmagnetic inclusions, etc.); and finally, within numerical micromagnetic models of single particles, starting from arbitrary magnetization configurations, applying energy minimization codes leads to local energy minima that are generally always different [15].

R. Arias, It.N. Bertram / Journal o f Magnetisn and Magnetic Materials 171 (1997) 209 217 211

2. Master equation

In this section we review the Master equation formalism. We consider N states, with occupat ions nM), i = 0 . . . . . N - 1, and restricted to: 0~< "M) ~< 1. In our model we assume that the initial slate is i - 0, the intermediate states are i = 1 . . . . . N - 2, and correspond to a labeling in descending energy levels, and the state N - 1 corresponds to the final or ground state.

it is assumed that the rate W~j of transition from state j to state i (i # j ) is given by the following Arrhenius form:

14,.~ = Q exp( - EBji/kT), (3)

where E ~ is the energy barrier for going from state ,] t o i, EBj i =-- E lji) - - E i. E (ji) = E (i'i) is the energy at the top of the barrier and ~2 is a frequency prefactor. The equilibrium solution, n[ q, for the occupat ion of the different states is given by the Bol tzmann distribution, n~ q = exp( -E¢ /kT) /Z , with Z ~ e x p ( - E d k T ) the parti t ion function. Detailed balance holds for the rates [6]: at equilibrium dns/dt = O, and there is no net interchange of probability, Wi?~ q = W j i neq.

The Master equation in its general form is

dni dt - ~ K~ini' (4)

J

where Kij= Wi~ (i:/-j), and K i i = - ~ t ~ i W , . K , represents the rate at which state i transfers probabili ty to all the other states. These forms of the K's assure conservat ion of overall probabil i ty since ~,iKij = 0 or, equivalently, ~idni/dt = O.

The general solution of the Master equation [Eq. (4)] is a linear combinat ion of exponentially decaying terms:

,Ti(t) = ~ ckul kl e x p ( - ' V ) . (5) k

The coefficients ck are determined by the initial conditions. The negative of the rates rk and the column vectors ul k) are the eigenvalues and eigen- vectors respectively of the equation obtained by substituting Eq. (5) into Eq. (4):

- (k) (Kij + rk0ii)u j = 0 . (6)

i

In some cases, the rates rk could be degenerate. If - r is one of these degenerate eigenvalues, and I is

its multiplicity, solutions of the form:

< . ) -- ce ~' y~ dlJ~t j (7) j = 0 /

are associated with the rate r. The probabili ty of not switching, PNs(t) is the

probabili ty of the system staying in the states 0 . . . . ,N -- 2:

N - 2

P~,-s(t) = ~ hi(t) = 1 - n~,-~(t) (8) i = O

(conservation of probabil i ty was used). Combining Eq. (5) into Eq. (8) (restricting ourselves to the nondegenerate case), one obtains the probabili ty of not switching as a linear combinat ion of decaying exponentials:

PNS(t) = ~ Wk exp(--rkt), (9) k

N - 3 with the weight Wk defined as Wk ~ ~ = ( 7 .(k/ Cku i .

Since initially the particle is saturated, PNS(0) = 1, and from Eq. (9) the weights Wk sum to one:

Wk = 1. (10) k

3. Fast relaxation and single barrier models

The relaxation of a metastable state into a stable one through thermal activation over a single barrier is described by the following simple Master equation:

dno , dn 1 - - = -- ano + ap-nl , - ano - ap-nl , (11) dt dt

a = (2 exp ( -Eud /kT) is the direct rate of switching, and ap 2 the reverse rate of switching Qexp(-EBr/kT), where p2 ~ e x p [ - ( E 0 - El)~ kT] represents the relative occupat ion in equilibrium of these states. EBa and Em are the direct and reverse barriers, respectively. Fig. 1 schematically represents this case:

Conservat ion of probability, no(t) + hi(t) = 1, combined with Eq. (11) determines the solution

212 R. Arias, H.N. Bertram /Journal o/' Magnetism and Magnetic Materials 171 (1997) 209-217

c~

ap 2

EO ~ EB~

p = e-(Eo-EI)/ZkT

E1 J

1 o o

8o

6o

el_ 40

2o

0

P a r t i c l e , 0 = 7 ° : ~.~I°0~:~°; I

!z ", i

I PJ I , " ' - d

o.1 1 lO T im e (s)

Fig . 1. S c h e m a t i c s o f the e n e r g y levels , e n e r g y b a r r i e r s a n d r a t e s

o f t r a n s i t i o n o f lhe s i ng l e b a r r i e r case . Fig. 2. Experimental probability of not switching as a function of log(t) for a single "/-ferric oxide particle, with an applied field near the coercive field and at 7 to the easy axis. according to Ref. [1].

n0(t), or the probability of not switching:

p 2 1 - a ( 1 +p:)t no(t) = PNs(t) -- 1 + p2 + ~ e (12)

In the limit of a higher reverse than direct barrier, or p ~ 0 , Eq. (12) becomes PNs(t)= exp(--f/r) as mentioned in the Introduction, with the relaxation time r = 1/a.

We now prove that a simple decaying exponential like Eq. (12) for a single barrier model, or a linear combination of decaying exponentials with normalized positive weights, have their logarithmic derivatives restricted in value. In the nondegenerate case the logarithmic derivative of the probability of not switching is from [Eq. (9)]:

dPNs ln(10) ~ Wk(rkt) exp(--rd). (13)

d log(t) k

The proof of relation [Eq. (14)] for a single decaying exponential is just a special case with W 1 = 1.

We are aware of two experimental publications that report magnetic thermal relaxation of single particles faster than exponential in certain regimes. One corresponds to low temperatures ( ~ 5 K) measurements made in Co particles [16]. The other, reports measurements at room temperature on elliptic 7-ferric oxide particles of 600A by 3000.4 [1]. In particular, Fig. 3 from Ref. [1], that we reproduce here as Fig. 2, shows plots of the probability of not switching for a single particle.

The absolute value of the slope of these plots, ]dPys/dlog(t)l, increases as the applied field approaches the coercive field. The highest slope shown is approximately IdPNs/d 1og(t)lMAx --~ 1.4 > 0.85.

Since the function x e x p ( - x ) has a maximum value equal to 1/e at x = 1, and the Wk are assumed all positive, the absolute value of dP~,,s/d log(t) is bounded:

dPNs ln(10) ln(10) -dT-y E - ~ 0.85. (14)

e k e

4. Example of solution of a master equation corresponding to four states that shows fast relaxation

We present a solution to a Master equation corresponding to four states consecutively separated by equal direct and equal higher reverse barriers, that shows 'fast' relaxation.

R. Arias, H.N. Bertram/,h)urnal o/'Magnetism and Magnetic Materials 171 (1997) 209 217 213

~ " B d ~ [ EBr

E2

a ap2

"" E ~ , J ~

EN 1

Fig. 3. Schematics of the energy levels and barriers of a downward cascade of states with equal direct barriers, and equal reverse barriers.

Transi t ions are only allowed between nearest- neighbor states, which makes the Master equat ion tridiagonal. The direct rates are all equal to a=Qexp( -EBa /kT) , and the reverse rates to ap-' - Q e x p ( - Em/'k T), with EBa and EBr the direct and reverse energy barriers respectively. Fig. 3, with N 4 is a schematic representat ion of this case.

According to the tr idiagonal Master equat ion [Eq. (A.1)], in the N = 4 case ;to.l._, = a are the direct rates (23 = 0), and tq.2.3 = aP 2 are the reverse rates (tt4 = 0). The solution for the logari thmic derivative of the probabi l i ty of not switching, dP,~s/d log ( t )= -ln{lO)t(dn3/dt)is (see the appendix):

dPxs ln(lO)at

d log(t) - 2p 2

x [ - I + cosh(ax/2pt) ] exp [ - a ( l + p2)t]. (15)

By simple integration, and using the initial con- dition P~,s(0) = l, one can determine the probabi l - ity of not switching, PNs(t):

Pxs(t) = p2 4- p4 _t_ p("

1 + p 2 + p 4 + f ,

e - u( 1 - \ -2p + pe}t +

@2(1 - \/"2p + p2)

e - . ( l + p:)t

2p2(1 + p2)

e a ( l + ~ - p + p 2 ) t +

@2(1 + \ f 2 p + p 2 )

061

F r o m this expression, the weights Wk and rates rk corresponding to Eq. (9) can be clearly identified:

p2 + /)4 + p6

W° = 1 + p2 jr_ p4 + p6" r 0 = 0,

1 W I = 2p2( 1 + f ) ' r l = a ( 1 +p2) ,

1 U / 2 =

4P 2(1 -- X/ -P + p2)

r 2 = a(l - vf2p + p2),

1 W 3 =

4pc(1 + \/J~p + p2)

r3 = a(1 + x /2p + p2). (17)

The weight W~ is large and negative when p --+ 0, as is the case for larger reverse barriers (the weights W2, W3 are large and positive in this limit). The rates r~, r2 and r3 are closely spaced in this limit, or a lmost degenerate. To lowest order in p2 one obtains:

dPNs ln(10) -- ~__ (at)% "'. (18)

d log(t)

The function x 3 e x p ( - x ) has a m a x i m u m value equal to (3/e) 3 at x = 3. It follows that the maximum value of the logar i thmic derivative is

dENs d log(t) MAX -~ ln(lO)33/2e3

ln(10) 1.55 > -- 0.85. (19)

C

Thus, for this example, 'fast ' relaxation is evident. We want to remark that 'fast ' re laxat ion occurs for p near zero, with still nondegenera te rates, and that the limit p --+ 0 adequately reproduces the degenerate solutions of the type in Eq. (7).

5. Fast decay for states consecutively separated by equal barriers

In this section we present solutions of the Master equat ion for N states that show 'fast ' relaxation. In

2 1 4 R. Arias, [L N. Bertram / JounTal q /Magne t i sm and Magnetic ?vIateriaLs" 171 (19971 209 217

order to search for solutions of Master equat ions with fast relaxation, we make some simplifying as- sumptions. First, by choosing a tr idiagonal Master equation, we assume that only transit ions between 'nearest neighbor ' states are allowed. Also it is

N 1 assumed that the direct barriers are equal as well as

L = the reverse ones, but that the reverse barriers are ~_~ higher. Fig. 3 is a schematic representat ion of the energy levels and the equal direct and equal higher reverse energy barriers for this downward cascade of states. The direct energy barriers are denoted as EBd, and the reverse as Eu,, with EBd = p 2 E m.

p2 = exp[ - - (E i -- E i + l ) / k T ] <~ 1 is a measure of how much lower the direct barr ier is than the dn,~- 1 a(at) "~' 2

reverse one. The direct rates ,;.~ are equal to a, and dt (N - 2)! the reverse ones , /G to ap 2.

The solution to a Master equat ion cor respond- ing to states consecutively separated by equal direct and equal reverse energy barr iers can be obta ined following the results of the appendix (for arbi t rary values of p between 0 and 1). Cont inued fraction expressions for the Laplace t ransforms of the occupat ion of the different states are obta ined by plac- ing the appropr ia te direct and reverse rates ).~ = a, tq ap 2. After inversion of these Laplace transforms, exact solutions for the occupat ion of the different states corresponding to any value of the reverse barr ier have been obta ined for an arbi t rary number of states, N.

The most interesting quant i ty is dn~,_~/dt = - d P x s / d t , since it is the rate of switching at t ime t: (dn~. ~/dt) d t is the probabi l i ty that the switching will occur between t and t + dr. The exact expres-

rate of switching was obtained sion for this in Eq. (A.22):

dnN 1 2a a( 1 + p2)t - - e

d t N p x - 2

X ~ (--1)Js in 2 e 2apc°s(rq:;O' (20) j = 1

We are interested in the low p (or higher reverse barriers) behavior of the rate of switching dnu- ~/dt:

we expand the second exponential in a power series in p:

dt lN 1 _ 2a a l l +p'~)t

dt NF~ ~ 2 e

.... (2pat), .,',, - ~ ( ~ ) ( ~ ) x ~, 1! y~ ( - -1)Js in 2 cos I . (21)

l = 0 j = 1

Interestingly enough, it can be proven that the sum:

( - , ) J s i n 2 ( ~ ) cos@ZJ~ . , \ N /

(22)

is exactly zero for 1 < N - 2, meaning that there are no divergent terms in dnN_ 1/dt as p--+ 0, as expected, dns 1~dr to order p2 becomes (the exponential factor is not expanded in p2):

e a(l+pZ)t

x [1 + p2(at)2 ( N -- 2) + ... ]. (23) (N -- ])N

F rom this expression one can easily calculate the average switching time, ( t ) , and its variance, 0.2 __ ( / . 2 ) __ ( t ) 2 :

f f . dn,~- 1 1 ( t ) = dz ~ t -~ - (N - 1 - 2p 2 -F ... ), (24) ) ( 1 T £1

1 G 2 = ~ [ N - - I + p 2 ( N 2 + N - 8 ) + " " ] . (25)

a -

The lowest-order term of the average switching time is easily interpretable since it is equal to the number of barriers to be traversed, N - 1, times the essential t ime scale 1/a.

The m a x i m u m value of the logar i thmic derivative can also be obta ined to order p2:

dPss MAX d log(t) = ln(l 0)e IN- 1~

(N _ lye 1 ( x [ ] ~ - - ~ . 1 - 2p a -

N-, ) N + ' (26}

The logari thmic derivative of the probabi l i ty of not switching, shows 'fast ' relaxation behavior in the limit of higher reverse barriers, or low values of p. This m a x i m u m logari thmic slope, from Eq. (26) and at p --+ 0, is an ever increasing function of N from its lowest value at N = 2, ln(10)/e -~ 0.85.

R. Arias, H.N. Ber t ram/Journa l q/3/[agnetism and Magnetic Materials 171 (1997) 209 217 215

A high N limit value of Eq. (26) at p --+ 0 is found /_

by using Stirling's formula, n! ~ n" exp( nlw~2n, valid for large n:

dPxs / N - 1 d log(r) MAX ~ ln(10} X / 2x (27)

This approx imat ion is quite accurate for N ~> 6. Eq. {27} shows that for a large number of states the m a x i m u m logari thmic derivative grows as the square root of the number of barriers to be traversed.

This result can be unders tood in a simple, ap- proximate, way. First, conservat ion of probabi l i ty can be written as: o-ldn,~,_ ~/drlMAx~ 1. The maximum of the absolute value of the logari thmic derivative IdPys/d log(t)l - ln(10)t dnx_ ~/dr can be approx ima ted by (t)ldn~,_ 1/dt[MAX. Finally, by using our previous expression for conservat ion of probabil i ty:

dPr, s - ( @ ) ~ X/,'NT {28)

which is the behavior shown by Eq. (27). In the limit p --+ 0, where the fastest relaxation is

obtained, the rates of decay of the solution to the Master equat ion become degenerate, and produce a low relative dispersion of switching times, a / ( t ) .

Acknowledgements

The authors wish to acknowledge the invaluable assistance of Dr. Har ry Suhl, through useful con- versations and suggestions.

Appendix A. Solution by Laplace transform methods of tridiagonal mas(er equations

An alternative method for solving tr idiagonal Master equat ions is to use Laplace transforms. This me thod has been effectively applied to t r idiagonal matrices Kii [171, and also to pentadiagonal matrices [181. In these cases it is possible to obtain cont inued fraction expressions for the Laplace t ransforms of the occupat ion of the different states.

The general t r idiagonal Master equat ion can be written as

tio(t) = - 2ono(t) + ~qnl(r),

lii(t) = f~i ltli 1(/) - - (•i -~- [2i)~1i([) -IF Ill+ 1/7i+ 1(/)

(A.I)

with i = 1 , 2 , . . . , N - - 1. If N - 1 is the last state considered, we utilize ).~,_ 1 = 0 and t~.~' = 0. The Laplace t ransform of the occupat ion of the state i, ni(t), is defined as

hi(z) = dr exp(-zt)ni(r) (A.2) )

with z = x + i y , x > 0 . The appl icat ion of the me thod starts by applying the Laplace t ransform opera to r to the full Master equation:

I t l n l ( z ) = - - I 4- (.'~o 4- Z)no(Z),

Ill+ l n i+ l(Z) = - - 2 i_ i t / i_ i(,7) 4- (2i 4- Iti 4- z)rli(2).

(A.3)

A cont inued fraction solution can be obtained for no(z):

1 no(z) = . (A.4)

/~o + z - t/120/(21 + ~tl + Z -- ...)"

The following convenient functions are defined, A,(z) and B,,(z), th rough the recursion relations:

A, , . l ( z ) = (2. 4- I*. 4- z)A,,(z) - ~t,,Z,, 1A . - l ( z ) . (A.5)

B,,+l(z) = (2, + It,, + z)B,,(z) - It,,).,-1B,, l(Z), (A.6)

with, A o ( z ) = O, A l ( z ) = 1, B o ( z )= 1 and B l ( z ) = ) . 0 + 2

Specifically thinking in the case of N states, and remember ing to use 24r-- 1 = 0, t~,~' = 0, no(Z) can be writ ten as

A,~,(z) no(z) - (A.7}

B~.(z}"

Making use of the results of Ref. [17], one can also derive that

B i ( z ) A N ( z ) - Ai(z)B~-(z) <{z) = (A.8)

/V[i B ~,,( z )

with i = 1 , . . . ,N - 1, and Mi ~ 1(1 ... lli.

216 R. Arias, H,N. Bertram/.Journal ol Magnetism and Magnetic Materials 171 (1997) 209-217

To ob ta in hi(t) one has to invert these Laplace t ransforms. The invers ion gives con t r ibu t ions from all poles of n~(z), or equivalent ly from all zeros of B~,(z). One has to calculate the sum of the residues at these poles. The final express ion for n~(t) is

hi(t) = n7 q + ~ e--" Resz=~]li(z ) J

= n~ q + ~, e --~' lim (z - zj)ni(z) (A.9) .I z ~ z i

with zj all zeros of Bx(z), except z i = 0. ( - z j) becomes the rate of decay rj.

In the fol lowing we app ly this me thod to states consecut ively separa ted by equal direct and equal reverse energy barriers . The direct rates between consecut ive states are cons idered equal, Wlo = 20 = W21 = ;-1 = -.. -= a, as well as the reverse rates, Wo~ = ]/1 = W 1 2 = ] /2 = . . - ~ aP 2, with p ~ e x p ( - ( C o - ~:1)/2) = exp(-(~:~ - ~:2)/2) . . . . ; Ci =-- Ei /k T.

Initially the rate a will be considered equal to one, and will be reinstated at the end. Defining y -= 1 + p2 + z, the recursion relations satisfied by A,, are

A,,+ ~(y) = yA , (y ) - p 2 A , I(Y) (A.10)

and similarly for B,,. A closed-form solution of the recursion relations

[Eq. (A,10)J can be obtained for An(y) and B,(y):

z+'" - ),"_ A,(y) 2+ - 2_ ' (A.11)

[ ( y - p ~ - 2 _ ) ) ? + - () , - p 2 _ ) . + ) 2 ' ~ ] B,,{y) = /~+ --,4_

(A.12)

where 2+(y) are defined by

2±(3,) = 1 0' -+ ~//3,2 _ 4p2). (A.13)

Returning to n~(z), it can be written in this case, by taking into considerat ion that 2x ~ = 0, ]/N = 0:

hi(y) =

B,(y)(ANO') - AN ~0')) -- A,(y)(B,,,O') -- B a, I(Y))

M,(BNO') -- B N - fly))

(The A, and B, of this formula do not have a limiting value of n.) It can be shown that B , ( y ) -

B,_ I(Y) = - (1 + p2 _ y)A,(y) ~ zA,(y) , so a pole of n~0') is y = 1 + p2 or equivalently z = 0 corresponding to the equilibrium solution; or it can be a zero Yi of AN(y), j = 1 . . . . . N -- 1. All the zeros yj have the form .D = PZi, where the :~j satisfy the equations 0 = A u ( p T y p N - l , or they are the roots of a poly-

nomial in ~ of degree N - 1. These roots can be obtained by making use of expression [Eq. (A.I1)] for A,(y). A zero yj of Adv) satisfies "" = 2" z + _, or

F 0'j + \/Y.7 -- 4p2) n = 0'J -- ~ - @2), . (A. 15)

To ob ta in these roots , we mul t ip ly both sides

of Eq. (A.15) by ( y j - x/ /ry~j- 4p2) ", o r : ( 4 p 2 ) n =

(}'j - - x//V 2 -- 4 p 2 ) 2n. Then

/" 2 Y~ _ v / y j _ @2

= 2p( 1, e i2~/2", e i4~/2n, . . . . ei(2n - 1)2x/2n). (A . 16)

After some algebra, the n - 1 roo ts yj of A, (y ) are

)'j = 2p cos (A. 17)

with j = 1 , . . . ,N - 1. F o r example , for n = N = 4

these roo t s are: y l = x//2p, y2 = O, y3 = - -x / /2p . Re - tu rn ing to the or iginal variables, the rates of decay become

The expression for dn; /d t is given by

(A.18)

dni E czjt d t (t) = z; Re s,=:,ni(z)

J

= ~ e-'" lira zj(z - &i)ni(z) (A.19) j z ~ z ;

with

(y - y j )Xx- l(yj)Bi(yj) lira zj(z -- zflni(z) = -- lira . . . . ~,. ~, M i A N(Y)

A u - ~ ( y , ~ ) B i ( y j )

(A. 14) p2i(OAN/~y)O'j)" (A.20)

R. Arias. H.N. Bertram /Journal of'Magnetism and Magnetic Materials 171 ?1997) 209 217 217

The rate of switching dnN_l/dt can now be determined. Using expressions Eqs. (A.I1), (A.12) and (A.17) one determines that AN-I(ya) = (__ 1)j+ lpU-2 , B,,v 1{3'j) = ( - - 1)JP N a n d

~A,~- ( - - 1 ) J + l p N 2N ~ ' ( ) ' i )= (A.21) ey 2 sin2(~j/N)

The final expression for the rate of switching dn~,_ 1~dr according to (Eq. (A.19)) and (Eq. (A.20)) is (in the original variables):

dnN- 1 _ _ 2a e_.{ 1+,,:), dt Np~" 2

× ~, ( - 1)J sin 2 e2ap co~ C~j,'.','lt (A.22) j= 1

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fast thermal reversal of magnetic particles

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