Linear and nonlinear superparamagnetic relaxation at high anisotropy ...

PHYSICAL REVIEW B 66, 214406 共2002兲

Linear and nonlinear superparamagnetic relaxation at high anisotropy barriers Yu. L. Raikher* and V. I. Stepanov Institute of Continuous Media Mechanics, Ural Division of RAS, 614013, Perm, Russia 共Received 20 May 2002; revised manuscript received 17 July 2002; published 11 December 2002兲 The micromagnetic Fokker-Planck equation is solved for a uniaxial particle in the low-temperature limit. Asymptotic series in the parameter that is the inverse barrier height-to-temperature ratio are derived. With the aid of these series, the expressions for the superparamagnetic relaxation time and the odd-order dynamic susceptibilities are presented. The obtained formulas are both quite compact and practically exact in the low 共with respect to FMR兲 frequency range that is proved by comparison with the numerically exact solution of the micromagnetic equation. The susceptibility formulas contain angular dependencies that allow to consider textured as well as randomly oriented particle assemblies. Our results advance the previous two-level model for nonlinear superparamagnetic relaxation. DOI: 10.1103/PhysRevB.66.214406

PACS number共s兲: 75.30.Cr, 75.30.Gw, 75.50.Tt

I. INTRODUCTION

The problem of superparamagnetic relaxation in singledomain ferroparticles formulated, explained, and basically analyzed by Neél1 about fifty years ago, has continued to attract attention. Nowadays this interest is mainly due to the expanding number of nanometer granular magnetic media used in information storage and related high technologies. When analyzing magnetic dispersions, solid or fluid, a promising idea is to evaluate the granulometric content, particle material parameters, and relaxation rates by combining the data on linear and nonlinear dynamic susceptibilities. Recently, this approach 共it originates from the spin glass science兲 became quite feasible in experimental realization.2 However, to benefit from it, one needs an adequate model. Surprisingly, the Neél1 concept of superparamagnetic behavior of fine magnetic particles that had been substantially advanced by Brown3,4 and refined by numerous researchers 共see the review article Ref. 5 with about 400 references兲 lacks a nonlinear extension. In Ref. 6 we begun to fill up this gap and proposed a numerical procedure involving continuous fractions by means of which the linear and cubic susceptibilities for a solid system of uniaxial fine particles could be obtained. With allowance for the polydispersity of real samples, the worked out description provided a fairly good agreement with the dynamic magnetic measurements taken on Co-Cu nanocomposites.2 Recently, our approach was used successfully7 for the linear and cubic susceptibilities of the samples of randomly oriented ␥ -Fe2 O3 nanoparticles. Hereby we carry on the build up of the nonlinear superparamagnetic relaxation theory by working out a set of compact and accurate analytical expressions that considerably facilitate calculations as well as experiment interpretation. The paper is arranged in the following way. In Sec. II we discuss the problem of superparamagnetic relaxation and show the way to obtain the asymptotic solution for the micromagnetic Fokker-Planck equation in the uniaxial case. In Sec. III the perturbative expansions for the orientational distribution function are obtained, which are used in Sec. IV to construct asymptotic expressions for the nonlinear dynamic susceptibilities. The explicit forms of those expansions are 0163-1829/2002/66共21兲/214406共17兲/$20.00

given and their accuracy is proved by comparison with the results of numerical calculations. Section V contains the enveloping discussion. II. SUPERPARAMAGNETIC RELAXATION TIMES A. Uniaxially anisotropic particle

The cornerstone of the superparamagnetic relaxation theory is the Arrhenius-like law for the relaxation rate of a magnetic moment of a single domain particle predicted by Neél in 1949. The framework of this classical problem is as follows. Consider an immobile 共e.g., fixed inside a solid matrix兲 single-domain grain of a volume v . This particle possesses a uniaxial volume magnetic anisotropy, K being its energy density and n its easy axis direction. Since the temperature T is assumed to be much lower than the Curie point, the particle magnetization I, as a specific parameter, is practically constant and the magnitude of the particle magnetic moment may be written as ␮ ⫽I v . Denoting its direction by a unit vector e, one concludes that the magnetic state of such a particle is exhaustively characterized by a pair of vectors ␮⫽I v e and n. Thence, the orientation-dependent part of the particle energy 共in the absence of external magnetic fields兲 is U⫽⫺K v共 e•n兲 2 ,

共1兲

where K is assumed to be positive. As Eq. 共1兲 shows, this energy has two equal minima. They are separated by the potential barrier of the height K v and correspond to e储 ⫾n because for the magnetic moment e the directions n and Àn are equivalent. At zero temperature, the magnetic moment e, once located in a particular potential well, is confined there forever. At finite temperature, the probability of an overbarrier 共interwell兲 transition becomes nonzero. If the ratio ␴ ⬅K v /kT is high enough, the transition rate is exponential thus yielding the Neél law ␶ ⬀exp(␴) for the reference time ␶ of the particle remagnetization. Brown4 shaped up those semi-qualitative considerations into a rigorous Sturm-Liouville eigenvalue problem by deriving the micromagnetic kinetic equation

66 214406-1

©2002 The American Physical Society


YU. L. RAIKHER AND V. I. STEPANOV

2 ␶ D ⳵ W/ ⳵ t⫽JˆWJˆ共 U/kT⫹ln W 兲 ,

共2兲

where W(e,t) is the orientational distribution function of the magnetic moment, Jˆ⫽(e⫻ ⳵ / ⳵ e) is the infinitesimal rotation operator with respect to e, and the time ␶ D is introduced below by formula 共4兲. Generally speaking, Eq. 共2兲 is incomplete since a gyromagnetic term is absent there. This means that the consideration is limited by the frequency range ␻ ␶ 0 Ⰶ1, where ␶ 0 is the relaxation time of the Larmor precession of the particle magnetic moment in the internal anisotropy field H a ⬃2K/I, where K includes the possible shape contribution. Comparing this condition with the other one ␻ L ␶ 0 ⱗ1 which evidences a low-to-moderate quality factor of the Larmor precession for real nanodisperse ferrites, one estimates the allowed frequency as ␻ Ⰶ ␻ L that means, in fact, a fairly wide range.24,25 In the statistical description delivered by Eq. 共2兲, the observed 共macroscopic兲 magnetic moment per particle is given by the average m共 t 兲 ⫽ ␮ 具 e典 ⫽

冕

eW 共 e,t 兲 de.

共3兲

Note that with allowance for Eq. 共1兲 the function W has a parametric dependency on the vector n so that, in fact, the angular argument of W is (e•n). The magnetodynamic equation underlying the Brown kinetic equation 共2兲 can be either that by Landau and Lifshitz or that by Gilbert. To be specific, we adopt the former one. Thence, the reference relaxation time in Eq. 共2兲 is written

␶ D ⫽I v /2␣␥ kT,

共4兲

where ␥ is the gyromagnetic ratio for electrons and ␣ is the precession damping 共spin-lattice relaxation兲 phenomenological parameter. Assuming uniaxial symmetry of the time-dependent solution and separating the variables in Eq. 共2兲 in the form 1 W 共 e,t 兲 ⫽ 2␲

⬁

兺

ᐉ⫽0

A ᐉ ␺ ᐉ 共 e•n兲 exp共 ⫺␭ ᐉ t/2␶ D 兲 ,

共5兲

where the amplitudes A ᐉ depend on the initial perturbation, one arrives at the spectral problem Lˆ ␺ ᐉ ⫽␭ ᐉ ␺ ᐉ ,

Lˆ ⬅Jˆ关 2 ␴ 共 e•n兲共 e⫻n兲 ⫺Jˆ兴 ,

共6兲

where the non-negativity of the decrements ␭ ᐉ can be proven easily. Expanding the eigenmodes ␺ ᐉ in the Legendre polynomial series 1 ␺ ᐉ⫽ 2

⬁

兺

k⫽1

共 2k⫹1 兲 b (ᐉ) k P k 共 cos ␪ 兲 ,

k⫽1,3,5, . . . , 共7兲

where ␪ is the angle between e and n, one arrives at the homogeneous tridiagonal recurrence relation

冋

1⫺

册

冋

k⫺1 ␭ᐉ b (ᐉ) b (ᐉ) k ⫺2 ␴ k 共 k⫹1 兲共 2k⫺1 兲共 2k⫹1 兲 k⫺2 ⫹

1 共 2k⫺1 兲共 2k⫹3 兲

b (ᐉ) k ⫺

k⫹2 共 2k⫹1 兲共 2k⫹3 兲

册

(ᐉ) b k⫹2 ⫽0.

共8兲 Note that Eqs. 共5兲–共8兲 describe only the longitudinal 共with respect to the easy axis兲 relaxation of the magnetic moment. We remark that under condition ␻ Ⰶ ␻ L , i.e., far from the ferromagnetic resonance range, the transversal components of m⫽ ␮ 具 e典 are of minor importance. B. Interwell mode

Spectral equation 共6兲 describes the temperature-induced 共fluctuation兲 motions of the vector e in the orientational potential with a symmetrical profile 共1兲. With respect to the time dependence, the set of possible eigenmodes splits into two categories: interwell 共overbarrier兲 transitions and intrawell wanderings. In the spectral problem 共6兲 the interwell transitions of the magnetic moment are associated with the single eigenvalue ␭ 1 . As the rigorous analysis shows,8 it drastically differs from the others: whereas for ᐉ⬎1 all the ␭ ᐉ gradually grow with ␴ , the decrement ␭ 1 exponentially falls down proportionally to exp(⫺␴). In the opposite limit ␴ →0, all the decrements, including ␭ 1 , tend to the sequence ␭ ᐉ ⫽ᐉ(ᐉ⫹1) and thus become of the same order of magnitude. This regime corresponds to a vanishing anisotropy so that the difference between the interwell and intrawell motions disappear, and the magnetic moment diffuses almost freely over all the 4 ␲ radians with the reference time ␶ D introduced by Eq. 共4兲. From Eqs. 共3兲 and 共5兲 one finds that the longitudinal component of the magnetic moment evolves according to ⬁

m共 t 兲⫽␮

兺 A ᐉ e ⫺␭ t/2␶ ᐉ⫽1 ᐉ

D

冕

1

⫺1

x ␺ ᐉ dx,

共9兲

where x⫽cos ␪⫽(e•n). For a symmetrical potential like 共1兲 the equilibrium value m 0 of the particle magnetic moment is zero. With the abovementioned structure of the eigenvalue spectrum, the term with ᐉ⫽1 in Eq. 共9兲, being proportional to exp(⫺e⫺␴t/␶D), at ␴ ⬎1 is far more long-living than any other one. The dominating role of the decrement ␭ 1 had been proven by Brown, and for it he had derived4 the asymptotic expression ␭ B⫽ 共 4/冑␲ 兲 ␴ 3/2e ⫺ ␴ 共 ␴ Ⰷ1 兲 .

共10兲

A short time after, using a continued fraction method, Aharoni constructed9 for ␭ 1 a fairly long power series in ␴ and also showed numerically that Brown’s expression 共10兲 resembles the exact one with the accuracy of several percent for ␴ ⲏ3. In the 1990’s the eigenvalue ␭ 1 became a subject of extensive studies. Efficient numerical procedures were developed10 and a number of extrapolation formulas with a good overall accuracy were proposed.11–14

214406-2


LINEAR AND NONLINEAR SUPERPARAMAGNETIC . . . C. Asymptotic solution of the Brown equation

The study that we describe below was inspired by our work on fitting the dynamic susceptibility measurements for real assemblies of fine particles. Those data typically describe polydisperse systems in the low-frequency bandwidth ␻ /2␲ ⫽1 –103 Hz. As ␶ 0 ⬃10⫺9 s or smaller, then, using formula 共10兲 for estimations, one concludes that the mentioned frequency interval becomes a dispersion range for the interwell 共superparamagnetic兲 mode at

␻ ␶ 0 e ␴ ⲏ1,

that is,

2 ␺ 0 ⫽Z ⫺1 0 exp共 ␴ x 兲 , Z 0 ⫽2R 共 ␴ 兲 ,

冕

1

0

exp共 ␴ x 2 兲 dx,

which corresponds to ᐉ⫽0 and ␭ 0 ⫽0; note also the asymptotic expansion for the partition integral R( ␴ ) found in Ref. 17:

冕

⫺1

⫺Jˆ␺ 0 Jˆ␸ k ⫽␭ k ␺ 0 ␸ k ,

冕␺

dx ␸ j ␺ k ⫽ ␦ jk .

共14兲

0 共 J␸ j 兲共 J␸ k 兲 dx⫽␭ k ␦ jk ,

ˆ

ˆ

共15兲

where the second one follows from the first after multiplication by ␸ i and integration by parts. Note that in the second formula action of each operator reaches no farther than the nearest closing parenthesis. On rewriting Eq. 共15.1兲 in terms of a single orientational variable x⫽(e•n), the spectral problem takes the form

冋

册

d␸k d ␺ 共 1⫺x 2 兲 ⫽⫺␭ k ␺ 0 ␸ k . dx 0 dx

共16兲

In the equilibrium state Eq. 共16兲 reduces to

冋

册

d␸0 d ␺ 共 1⫺x 2 兲 ⫽0, dx 0 dx

共17兲

whose normalized solution is ␸ 0 ⫽1. This solution, being a true equilibrium one, turns the inner part of the brackets, i.e., the probability flux in the kinetic equation 共2兲, into identical zero. As remarked in Sec. II B, at ␴ Ⰷ1 the most long-living nonstationary solution of Eq. 共16兲 is the eigenfunction with ᐉ⫽1, whose eigenvalue is exponentially small, see Brown’s estimation 共10兲. We use this circumstance for approximate evaluation of ␸ 1 in the ␴ Ⰷ1 limit by neglecting the righthand side of Eq. 共16兲 for ᐉ⫽1. On doing that, the equation obtained for the function ␸ 1 formally coincides with Eq. 共17兲 for ␸ 0 . However, the essential difference is that now the content of the bracket is nonzero:

␺ 0 共 1⫺x 2 兲

R 共 ␴ 兲 ⫽e G/2␴ ,

共12兲

1

Qualitatively, from Eq. 共14兲 one may say that ␸ k are the same eigenfunctions but ‘‘stripped’’ of the exponential equilibrium solution ␺ 0 . Substituting Eq. 共14兲 in Eq. 共6兲, one gets two useful relationships

␴

1 3 15 共 2n⫺1 兲 !! ⫹ ⫹ ⫹•••⫹ ⫹••• . G 共 ␴ 兲 ⬅1⫹ 2␴ 4␴2 8␴3 2 n␴ n

共13兲

here ⫹ denotes Hermitian conjugation. The eigenfunctions of these two families are orthonormalized and related to each other in a simple way:

␺ k⫽ ␺ 0␸ k ,

共11兲

Lˆ ⫹ ␸ j ⫽␭ j ␸ j ;

Lˆ ␺ k ⫽␭ k ␺ k ,

␴ ⲏ10.

For temperatures up to 300 K this condition holds for quite a number of nanomagnetic systems. Application of the best fit procedure to a set of experimental data implies numerous recalculations of the linear and nonlinear susceptibility curves ␹ (k) of the assembly. Any such curve, due to a considerable polydispersity of the particles, is a superposition of a great number of partial curves ␹ (k) ( ␴ ) spread over a wide size 共or, in the dimensionless form, ␴ ) range. For successful processing, one needs a fast and very accurate algorithm to evaluate ␹ (k) ( ␴ ) everywhere including the domain ␴ Ⰷ1. The existing extrapolation formulas are no good for that purpose due to their illcontrollable error accumulation. A plausible way out is an asymptotic in ␴ ⫺1 solution of Eq. 共6兲. In the course of the fitting procedure, this approximation can be easily matched in the intermediate ␴ range with the well-known expansions for the small ␴ end. It is noteworthy that some 20 years ago Brown himself resumed15,16 studies on ␭ 1 and modified the preexponential factor in Eq. 共10兲 transforming it into an asymptotic series in ␴ ⫺1 . On the base of Eq. 共6兲 he had constructed an integral recurrence procedure, and evaluated ␭ 1 down to terms ⬀1/␴ 10. What we do below, is, in fact, carrying on this line of analysis that had not been touched since then. Our method advances Brown’s results in two aspects. First, for ␭ 1 it is more simple. Second, it provides not only the eigenvalue but the eigenfunction as well. Only having the latter in possession, one is able to obtain theoretical expressions for the directly measurable quantities that is the susceptibilities ␹ (k) . Taking Eq. 共6兲 as the starting point, we remark its equilibrium solution

R共 ␴ 兲⫽

The operator Lˆ in Eq. 共6兲 is not self-conjugated and thus produces two sets of eigenfunctions, which obey the respective equations

d␸1 1 ⫽ C, dx 2

共18兲

where 12 C is the integration constant. Note also that, contrary to ␸ 0 , the sought for solution ␸ 1 is odd in x. Using the explicit form of ␺ 0 from Eq. 共11兲 and integrating, one gets for x⬎0

214406-3



FIG. 1. Eigenmode ␸ 1 (x) determined with the aid of the numerical solution of Eq. 共8兲 for the dimensionless barrier height ␴ : 5 共dashed line兲, 10, 20, 25 共solid lines兲; the arrow shows the direction of ␴ growth. Thick dashes show the stepwise function that is the limiting contour for ␸ 1 at ␴ →⬁.

␸ 1 ⫽CR ⫽CR

冕冕

x

0 x

0

e

⫺␴x2

1⫺x 2

dx

2

e ⫺ ␴ x 共 1⫹x 2 ⫹x 4 ⫹x 6 ⫹••• 兲 dx.

共19兲

The integrals in expansion 共19兲 are akin. Denoting F n⫽

冕

x

2

x 2n e ⫺ ␴ x dx,

0

one can easily write for them the recurrence relation and ‘‘initial’’ condition as F n ⫽⫺

⳵ F , ⳵␴ n⫺1

F 0⫽

冑␲ 2 冑␴

erf共冑␴ x 兲 ,

F n ⫽ 关共 2n⫺1 兲 !!/2 ␴ 兴 F 0 , n

F 0 ⯝ 冑␲ /2冑␴ .

共22兲

Applying to Eq. 共22兲 the normalizing condition 共14兲, one evaluates the constant as C⫽1/RF 0 G. Therefore, from Eqs. 共20兲–共22兲 the principal relaxational eigenmode determined with the exp(⫺␴) accuracy emerges as an odd step function

␸ 1共 x 兲 ⯝

再

⫺1

for x⬍0,

1

for x⬎0.

␭ 1⫽

共23兲

In Fig. 1 the limiting contour 共23兲 is shown against the exact curves ␸ 1 (x) obtained by solving numerically Eq. 共8兲 for

冕

1

⫺1

␺ 0 共 Jˆ␸ 1 兲 2 dx⫽

1 R

冕

1

0

2

e ␴ x 共 1⫺x 2 兲

冉冊 d␸1 dx

2

dx. 共24兲

Substituting the derivative from Eq. 共18兲, one finds ␭ 1 ⫽C⫽ 共 2/冑␲ 兲 ␴ 1/2/RG,

共21兲

Comparing this with expression 共12兲 for the function G, we get the representation

␸ 1 共 x⬎0 兲 ⯝CRF 0 G.

several values of ␴ . We remark that in the statistical calculations carried out below, the typical integrals are of two kinds. In the first, the integrand consists of the product of ␸ 1 ␺ 0 and some nonexponential function. As ␺ 0 ⬀exp (␴x2), the details of behavior of ␸ 1 in the vicinity of x⫽0 are irrelevant because the approximate integral will differ but exponentially from the exact result. The integrals of the second type contain d ␸ 1 /dx in the integrand. For them a stepwise approximation 共23兲 with its derivative equal identical zero everywhere except for x⫽0 is an inadmissible choice. So, to keep the exponential accuracy in this case, one has to get back to Eq. 共18兲. The eigenvalue ␭ 1 corresponding to the approximate eigenfunction ␸ 1 from Eq. 共23兲 is evaluated via formula 共15兲 that can be rewritten as

共20兲

respectively. Using the asymptotics of the error integral, with the exponential accuracy in ␴ one finds n

FIG. 2. Asymptotic expression 共25兲 for the eigenvalue ␭ 1 with allowance for terms up to ␴ ⫺9 共solid line兲 compared to the exact numeric value 共dashed line兲.

and using expression 共12兲 for R finally arrives at ␭ 1 ⫽ 共 4/冑␲ 兲 ␴ 3/2e ⫺ ␴ /G 2 ⫽␭ B /G 2 .

共25兲

⫺1

With G expanded in powers of ␴ , see Eq. 共12兲, this formula reproduces the asymptotic expression derived by Brown in Ref. 15. At G⫽1 it reduces to his initial result,4 corresponding to the above-given Eq. 共10兲. Function ␭ 1 ( ␴ ) from Eq. 共25兲 is shown in Fig. 2 in comparison with the exact result obtained by a numerical solution. Indeed, at ␴ ⲏ3 the results virtually coincide. According to expansion 共5兲, each decrement ␭ ᐉ defines the reference relaxation time

214406-4

␶ ᐉ ⫽2 ␶ D /␭ ᐉ .

共26兲


LINEAR AND NONLINEAR SUPERPARAMAGNETIC . . .

Thence from Eq. 共25兲 we get

␶ 1 ⫽2 ␶ D /␭ 1 ⫽ ␶ BG , 2

具具 x 共 t 兲 x 共 0 兲典典 0 ⫽

␶ B⬅2 ␶ D /␭ B ,

共27兲

where ␶ B denotes the asymptotic relaxation time obtained by Brown in Ref. 4. Substituting in Eq. 共27兲 the explicit asymptotic series 共12兲 for G, one gets

␶ 1⫽ ␶ D

冑␲ e ␴ 2␴

3/2

冉

1⫹

冊

1 7 9 ⫹ ⫹ ⫹••• . 2 ␴ 4␴ 2␴3

共28兲

⫽

⬁

␶ int⫽

冕

⬁

0

具m共 t 兲m共 0 兲典0 具m 共 0 兲典0 2

dt⫽

冕

⬁

0

具x共 t 兲x共 0 兲典0 具 x 2共 0 兲典 0

dt, 共29兲

where the angular brackets stand for the statistical ensemble averaging over the equilibrium distribution 共12兲. As follows from Eq. 共29兲, the integral relaxation time equals the area under the normalized decay of magnetization. The Green function of Eq. 共2兲, i.e., the probability density of a state (x,t), provided the initial state is (x 0 ,0), writes

␶ int⫽ 兺

W 共 x,t;x 0 ,0兲 ⫽

兺

ᐉ⫽0

␺ ᐉ共 x 兲 ␸ ᐉ共 x 0 兲 e

.

共30兲

⬁

兺

k⫽1

␸ ᐉ⫽

兺

k⫽1

␺ 1⫽

1 2

兺共 2k⫹1 兲 S k P k共 x 兲 , k⫽0 共 2k⫹1 兲 Q k P k 共 x 兲 .

共34兲

共35兲

共36兲

共32兲

The procedures to evaluate the coefficients S k and Q k and the explicit asymptotic forms for Q 1 and S 2 are given in Appendix A; note representation 共11兲 for the equilibrium function ␺ 0 . Due to Eq. 共14兲, the coefficients in formulas 共31兲 are re(ᐉ) lated to each other by b (ᐉ) k ⫽ 具 P k P k ⬘ 典 0 a k ⬘ . In those terms one gets for the correlator in Eq. 共14兲

共37兲

Substitution of Eqs. 共36兲 and 共37兲 in 共35兲 with allowance for relationships 共12兲, 共25兲, and 共27兲 gives the asymptotic representation in the form

␶ int⫽ ␶ B

⬁

兺

2 2 ␶ ᐉ 关 b (ᐉ) 1 兴 /具x 典0 .

具 x 2 典 0 ⫽ 共 1/2␴ 兲共 e ␴ ⫺1 兲 ⫽1/G⫺1/2␴ ,

⬁

k⫽0

兺

ᐉ⫽1

The equilibrium moment calculated by definition is written as

Q 1 ⯝1/G.

and introduce special notations for the first two functions 1 2

兺

ᐉ⫽1

⬁

2 关 b (ᐉ) 1 兴 ⫽

2 2 2 2 ␶ int⫽ ␶ 1 关 b (1) 1 兴 / 具 x 典 0⫽ ␶ 1Q 1/ 具 x 典 0 .

a (ᐉ) k P k共 x 兲共31兲

␺ 0⫽

2 ␶ ᐉ 关 b (ᐉ) 1 兴 /

and for ␴ Ⰷ1, using formula 共A5兲 of Appendix A we get

⬁

共 2k⫹1 兲 b (ᐉ) k P k共 x 兲 ,

共33兲

Unlike ␶ 1 , which in principle cannot be evaluated analytically18 at arbitrary ␴ , for ␶ int an exact solution is possible for arbitrary values of the anisotropy parameter. Recently two ways were proposed to obtain quadrature formulas for ␶ int . One method19 implies a direct integration of the Fokker-Planck equation. Another method20 involves solving three-term recurrence relations for the statistical moments of W. The emerging solution for ␶ int can be expressed in a finite form in terms of hypergeometric 共Kummer’s兲 functions. Equivalence of both approaches was proven in Ref. 21. In the present study, as mentioned, we are dealing in the high-barrier approximation. In this limiting case ␭ 1 is exponentially small, so that the term with ᐉ⫽1 in the numerator in Eq. 共34兲 is far greater than the others. With allowance for Eq. 共32兲 it can be written as

Similarly to Eq. 共7兲, we expand the eigenfunctions in Legendre polynomials as 1 ␺ ᐉ⫽ 2

2 ⫺␭ ᐉ t/2␶ D , 关 b (ᐉ) 1 兴 e

⬁

ᐉ⫽1

⬁

⫺␭ ᐉ t

兺

ᐉ⫽1

xx 0 ␺ 0 W 共 x,t;x 0 ,0兲 dxdx 0

where averaging over the current coordinate x is performed with the function W from Eq. 共30兲 whereas that over the initial conditions—with the equilibrium function ␺ 0 . Substituting expression 共33兲 in Eq. 共29兲 one gets the integral time in the form

D. Asymptotic integral time

The decrements ␭ ᐉ or, equivalently, relaxation times ␶ ᐉ , being the characteristics of the eigenfunctions of the distribution function, are not observable if taken as separate quantities. However, in combination they are involved in a useful directly measurable quantity, the so-called integral relaxation time. In terms of correlation functions this characteristics is defined as

冕冕

冉

冊

冑␲ e ␴ 2␴G 1 3 13 ⫽␶D 1⫹ ⫹ ⫹ ⫹••• . 3/2 2 ␴ 2␴ 共 2 ␴ ⫺G 兲 2␴ 4␴3 共38兲

As it is seen from formulas 共28兲 and 共38兲 written with the accuracy up to ␴ ⫺3 , the asymptotic expressions for the interwell and integral times deviate beginning with the term ⬀ ␴ ⫺2 . This contradicts the only known to us asymptotic expansion of ␶ int given in Eq. 共60兲 of Ref. 20 and repeated in Eq. 共7.4.3.22兲 of Ref. 22. The latter expression written with the accuracy ⬀ ␴ ⫺2 , instead of turning into Eq. 共38兲 coincides with the Brown’s expression 共28兲 for ␶ 1 . Meanwhile, as it follows from formula 共35兲, such a coincidence is impossible and therefore Eqs. 共60兲 of Ref. 20 and Eq.

214406-5



共7.4.3.22兲 of Ref. 22 are misleading. The necessity to rectify this issue made us to begin the demonstration of our approach with the case of the integral relaxation time. Further on we consistently apply our procedure to description of the nonlinear 共third- and fifth-order兲 dynamic susceptibilities of a solid superparamagnetic dispersion.

(n⫺1) ˆ ˆ Jˆ␺ 0 Jˆg (n) J共 e•h兲 , 0 ⫽J␺ 0 g 0 (n⫺1) ˆ ˆ Jˆ␺ 0 Jˆg (n) J共 e•h兲 , 1 ⫽J␺ 0 g 1

respectively. Set 共43兲 solves easily for g 0 since ⫽1. Starting with n⫽0, one gets sequentially

共43兲 g (0) 0 ⫽␸0

g (1) 0 ⫽ 共 e•h 兲 , III. PERTURBATIVE EXPANSIONS FOR THE DISTRIBUTION FUNCTION

1 2 2 g (2) 0 ⫽ 关共 eh 兲 ⫺ 具共 e•h 兲典 0 兴 , 2

A. Static probing field

1 1 3 2 g (3) 0 ⫽ 共 e•h 兲 ⫺ 共 e•h 兲具共 e•h 兲典 0 , 6 2

To find the nonlinear susceptibilities, one has to take into account the changes that the probing field induces in the basic state of the system. In the limit ␴ Ⰷ1, which we deal in, the relaxation time ␶ 1 of the interwell mode ␺ 1 is far greater than all the other relaxation times ␶ k . This means that with respect to the intrawell modes the distribution function is in equilibrium. So it suffices to determine the effect of the probing field H⫽Hh just on ␺ 0 and ␺ 1 . Assuming the energy function in the form U⫹U H ⫽⫺K v共 e•n兲 2 ⫺I v H 共 e•h兲

g (4) 0 ⫽

1 ⫺ 关共 e•h兲 2 具共 e•h兲 2 典 0 ⫺ 具共 e•h兲 2 典 20 兴 , 4 g (5) 0 ⫽

共39兲

关compare with Eq. 共1兲兴, and separating variables in Eq. 共2兲, one arrives at the eigenfunction problem Lˆ f ␤ ⫽ ␰ Vˆ f ␤ ,

共40兲

where ␰ ⫽I v H/kT and notation f ␤ refers to the distribution function modes that stem from ␺ 0 or ␺ 1 at H⫽0, i.e., ␤ ⫽0 or 1. In Eq. 共40兲 operator Lˆ is defined by Eq. 共6兲 while Vˆ ⫽⫺ ␰ Jˆ(e⫻h) is the operator caused by the energy term U H in 关Eq. 共39兲兴. As in the above, for the non-self-conjugated spectral problem 共40兲 we introduce the family of conjugated functions g ␤ and set f ␤ ⫽g ␤ ␺ 0 . Following our approach, in the low-temperature limit ( ␴ Ⰷ1) we set to zero the eigenvalues corresponding to both f 0 and f 1 ; compare with Eqs. 共17兲 and 共18兲 for ␺ 0 and ␺ 1 . Assuming the temperature-scaled magnetic field ␰ to be small, we treat U H as a perturbation Hamiltonian and expand the principal eigenfunctions as f 0⫽

兺

n⫽0

␰ n f (n) 0 ,

f 1⫽

兺

n⫽0

␰ n f (n) 1 .

Lˆ f ␤(n) ⫽Vˆ f ␤(n⫺1) ,

共42兲

that for the particular cases ␤ ⫽0 and 1 with the aid of the identity e⫻h⫽Jˆ(e•h) takes the forms

1 1 共 e•h兲 5 ⫺ 共 e•h兲 3 具共 e•h兲 2 典 0 120 12 ⫺

1 共 e•h兲关具共 e•h兲 4 典 0 ⫺6 具共 e•h兲 2 典 20 兴 . 24

共44兲

All the obtained functions are constructed in such a way that the corresponding f ␤(n) satisfy the abovementioned zero average requirement. We remark also that there is no problem to continue the calculational procedure to any order. Evaluation of g 1 is done in two steps. At the first one, we equal to the antisymmetric stepwise function 共23兲 set g (0) 1 and its derivative equal zero. After that from the second of Eqs. 共43兲 we can express g (k) 1 in closed form. Taken up to the fourth order these ‘‘zero-derivative’’ solutions are written g (1) 1 ⫽ ␸ 1 共 e•h 兲 ⫺ 具 ␸ 1 共 e•h 兲典 0 , 1 2 g (2) 1 ⫽ ␸ 1 共 e•h 兲 ⫺ 共 e•h 兲具 ␸ 1 共 e•h 兲典 0 , 2 1 3 3 g (3) 1 ⫽ 关 ␸ 1 共 e•h 兲 ⫺ 具 ␸ 1 共 e•h 兲典 0 兴 6 1 ⫺ 具 ␸ 1 共 e•h兲典 0 关共 e•h兲 2 ⫺ 具共 e•h兲 2 典 0 兴 , 2

共41兲

Thence for the field-free (H⫽0) case one has f (0) 0 ⫽ ␺ 0 and f (0) ⫽ ␺ . The same kind of expansion is assumed for g ␤ 1 1 (0) (0) with g 0 ⫽1 and g 1 ⫽ ␸ 1 . Note also that in order to retain the normalizing condition we require that f ␤(n) have zero averages. Substituting expansion 共41兲 in Eq. 共40兲 and collecting the terms of the same order in ␰ , we arrive at the recurrence relation

1 关共 e•h兲 4 ⫺ 具共 e•h兲 4 典 0 兴 24

g (4) 1 ⫽

1 1 ␸ 共 e•h兲 4 ⫺ 具 ␸ 1 共 e•h兲典 0 关共 e•h兲 3 24 1 6 1 ⫺3 共 e•h兲具共 e•h兲 2 典 0 兴 ⫺ 共 e•h兲具 ␸ 1 共 e•h兲 3 典 0 . 共45兲 6

Note the alternating parity in e with the term order growth in both Eqs. 共44兲 and 共45兲. It is instructive to compare the approximate expressions 共45兲 with the numerical results obtained without simplification of g (0) 1 . To be specific, we consider the case when probing field is applied along the particle easy axis n. Then Eqs. 共43兲 become one dimensional and the second of them is written

214406-6



our procedure. For that we integrate Eq. 共46兲 two times by parts and substitute there the ‘‘zero-derivative’’ form of g (1) 1 from Eq. 共47兲: 1 2 1 g (2) 1 ⫽ x ␸ 1 ⫺x 具 x ␸ 1 典 0 ⫹ 2 2

冕

x2

d␸1 dx. dx

共48兲

Thus one finds that the corrected g (2) 1 differs from this of Eq. 共47兲 by adding a stepwise 关alike that of Eq. 共23兲兴 term 1 2 g (2) 1 ⫽ x ␸ 1 ⫺x 具 x ␸ 1 典 0 ⫹D 2 ␸ 1 , 2

共49兲

with the amplitude D 2⫽

FIG. 3. Function g (1) 1 found numerically 共solid兲 and evaluated in the ‘‘zero-derivative’’ approximation 共dashed兲.

dg (n) 1 dx

⫽g (n⫺1) . 1

共46兲

1 2 g (2) 1 ⫽ ␸ 1 x ⫺x 具 ␸ 1 x 典 0 . 2

共47兲

In Figs. 3 and 4 these functions are compared to the numerical solutions of Eq. 共46兲. For our calculation, the most important is the behavior of those functions near x⫽⫾1 since these regions yield the main contribution when integrated with the weight function ␺ 0 . As one can see from the figures, the ‘‘zero-derivative’’ solution g (1) 1 agrees well with the exact one, while g (2) 1 deviates significantly. This discrepancy is due to the change of the barrier height that occurs in the second order with respect to the probing field amplitude, and manifests itself in all the even orders of the perturbation expansion. Correction of solution 共47兲 makes the second step of

冕

1

x2

0

d␸1 dx. dx

共50兲

We remark that the results of evaluation of the integrals I 2k ⫽ 兰 10 x 2k (d ␸ 1 /dx)dx can be arranged in the table

Its ‘‘zero derivative’’ solutions up to the second order follow from the first two lines of Eqs. 共45兲: g (1) 1 ⫽ ␸ 1 x⫺ 具 ␸ 1 x 典 0 ,

1 2

k

0

1

2

I 2k

1

1⫺G ⫺1

1⫺(1⫹1/2 ␴ )G ⫺1

共51兲

so that Eq. 共50兲 gives 1 1 G⫺1 1 5 37 ⫽ ⫹ ⫹ ⫹ ⫹•••. D 2 ⫽ I 2k ⫽ 2 3 2 2G 4␴ 4␴ 8␴ 16␴ 4 共52兲 Function g (2) 1 corrected in such a way is shown in Fig. 4 by asterisks. It is seen that the corrected dependence with a fairly good accuracy follows the numerically obtained curve. In a similar way one can prove that the corrected function g (4) 1 has the form g (4) 1 ⫽

1 1 ␸ 1 x 4 ⫺ 关具 ␸ 1 x 典 0 x 3 ⫺3x 具 x 2 典 0 兴 24 6 1 ⫺ x 具 ␸ 1 x 3 典 0 ⫹D 2 g (2) 1 ⫹D 4 ␸ 1 , 6

共53兲

where the corrected function g (2) 1 given by Eq. 共49兲 is used and D 4⫽

1 10␴ G 2 ⫺22␴ G⫹G⫹12␴ I 4 ⫺D 22 ⫽⫺ 24 48␴ G 2

⫽⫺

FIG. 4. Function g (2) 1 found numerically 共solid兲 and evaluated in the ‘‘zero-derivative’’ approximation 共dashed兲. Asterisks show a corrected calculation with allowance for the coefficient D 2 , see Eq. 共49兲.

1 32␴

⫺ 2

1 16␴

⫺ 3

5 32␴

⫺ 4

29 64␴ 5

⫹•••.

共54兲

In the general case, when the direction of the probing field does not coincide with the particle anisotropy axis, the corrected functions g (n) 1 still can be written as

214406-7

1 2 g (2) 1 ⫽ ␸ 1 共 e•h 兲 ⫺ 共 e•h 兲具 ␸ 1 共 e•h 兲典 0 ⫹D 2 ␸ 1 , 2



Substituting Eq. 共57兲 in 共56兲 we arrive at the recurrence

1 3 3 g (3) 1 ⫽ 关 ␸ 1 共 e•h 兲 ⫺ 具 ␸ 1 共 e•h 兲典 0 兴 6

set 共 2in ␻ ␶ D ⫹Lˆ 兲 W (n) ⫽Vˆ W (n⫺1) ,

1 ⫺ 具 ␸ 1 共 e•h兲典 0 关共 e•h兲 2 ⫺ 具共 e•h兲 2 典 0 兴 ⫹D 2 g (1) 1 , 2 g (4) 1 ⫽

that we solve sequentially starting from n⫽1. At the first step the function in the right-hand side corresponds to the equilibrium case ( ␰ ⫽0). Therefore, W (0) ⫽ ␺ 0 , where the latter function is defined by Eq. 共11兲 and is frequencyindependent. Combining Eq. 共42兲 written down for ␤ ⫽0 and n⫽1 and Eq. 共58兲, we eliminate the operator Vˆ and get

1 ␸ 共 e•h兲 4 24 1 1 ⫺ 具 ␸ 1 共 e•h兲典 0 关共 e•h兲 3 ⫺3 共 e•h兲具共 e•h兲 2 典 0 兴 6 1 ⫺ 共 e•h兲具 ␸ 1 共 e•h兲 3 典 0 ⫹D 2 g (2) 1 ⫹D 4 ␸ 1 . 6

共55兲

But since Eqs. 共43兲 cannot be reduced to a form like Eq. 共46兲, the correcting coefficients D 2 and D 4 cannot be presented in a closed form. In this case the corrected solutions taking into account the behavior of function ␸ 1 around zero are built up as power series near x⫽0; such a procedure for the coefficients D 2 and D 4 is described in Appendix B. B. Dynamic probing field

To obtain the dynamic susceptibilities, one has to find the distribution function W in the oscillating probing field ␰ exp(i␻t). For this situation the kinetic equation 共2兲 takes the form

冉

2␶D

冊

⳵ ⫹Lˆ W 共 t 兲 ⫽ ␰ Vˆ e i ␻ t W 共 t 兲 , ⳵t

共56兲

where the operators Lˆ and Vˆ have been introduced in above. Assuming that the exciting field amplitude is not too high, we expand the steady-state oscillatory solution of Eq. 共56兲 in a power series with respect to ␰ : W共 t 兲⫽

兺

n⫽0

␰ n W (n) e in ␻ t .

共57兲

Note that, mathematically, representation 共57兲 is not complete. Indeed, in a general case the exact amplitude of the n ␻ mode must contain, along with the contribution ⬃ ␰ n , an infinite set of terms ⬃ ␰ n⫹2 , ␰ n⫹4 , etc. However, in a weak field limit ␰ ⬍1 the terms with higher powers are of minor importance so that the main contribution to the magnetization response signal filtered at the frequency n ␻ is proportional to ␰ n .

W (2) ⫽ f (2) 0 ⫺

共 2i ␻ ␶ D ⫹Lˆ 兲 W (1) ⫽Lˆ f (1) 0 .

冋

共59兲

Now we expand the functions subjected to operator Lˆ with respect to the set 兵 ␺ k 其 of its eigenfunctions, see Eq. 共6兲: W (1) ⫽

兺 c (1) j 共 ␻ 兲␺ j

f (1) 0 ⫽

兺共 ␸ j 兩 f (1) 0 兲␺ j ;

共60兲

here ( ␸ 兩 f ) denotes functional scalar multiplication, i.e., the integral of the product ␸ f over all the orientations of e. Substitution of Eq. 共60兲 in Eq. 共59兲, multiplication of it from the left by ␸ k , and integration render the expansion coefficient as (1) ⫺1 , c (1) k 共 ␻ 兲 ⫽ 共 ␸ k 兩 f 0 兲关 1⫹i ␻ ␶ k 兴

共61兲

where the reference relaxation times are defined by Eq. 共26兲. In the low-frequency limit only ␻ ␶ 1 is set to be nonzero while all the higher modes are taken at equilibrium ( ␻ ␶ k ⫽0). Thence, when constructing W (1) via Eq. 共60兲, by adding and subtracting a term with c (1) 1 (0), one can present the first-order solution in the form W (1) ⫽ f (1) 0 ⫺

i␻␶1 共 ␸ 兩 f (1) 兲 ␺ 1 , 1⫹i ␻ ␶ 1 1 0

共62兲

where f (1) 0 , as seen from Eq. 共59兲, is the equilibrium solution for the same value of the field amplitude ␰ . We remind the reader that the functions without upper index belong to the fundamental set defined by Eqs. 共6兲 whereas those with an upper index are evaluated in the framework of the perturbation scheme described in Sec. III A. In the next order in ␰ the function W (1) is substituted in the right-hand side of Eq. 共58兲 and through a procedure alike to that leading to Eqs. 共59兲–共61兲, the function W (2) is found. We carry on this cycle up to k⫽5. The results write

i␻␶1 共 ␸ 兩 f (1) 兲 f (1) 1 , 1⫹i ␻ ␶ 1 1 0

(1) (2) (3) (0) (1) (2) W (3) ⫽ f (3) 0 ⫹ 关共 ␸ 1 兩 f 0 兲共 ␸ 1 兩 f 1 兲 ⫺ 共 ␸ 1 兩 f 0 兲兴 f 1 ⫺ 共 ␸ 1 兩 f 0 兲 f 1 ⫹

⫹

共58兲

册

1 1 (1) (2) (0) 共 ␸ 1 兩 f (3) 0 兲 ⫹ 共 ␸ 1 兩 f 0 兲共 ␸ 1 兩 f 1 兲 f 1 , 1⫹3i ␻ ␶ 1 2 214406-8

冋

1 3 (2) (1) (2) (0) 共 ␸ 1 兩 f (1) 0 兲 f 1 ⫺ 共 ␸ 1 兩 f 0 兲共 ␸ 1 兩 f 1 兲 f 1 1⫹i ␻ ␶ 1 2

共63兲

册共64兲


LINEAR AND NONLINEAR SUPERPARAMAGNETIC . . . (1) (2) (3) (1) (1) (3) W (4) ⫽ f (4) 0 ⫹ 关共 ␸ 1 兩 f 0 兲共 ␸ 1 兩 f 1 兲 ⫺ 共 ␸ 1 兩 f 0 兲兴 f 1 ⫺ 共 ␸ 1 兩 f 0 兲 f 1 ⫹

⫹

冋

册

冋

1 3 (3) (1) (2) (1) 共 ␸ 1 兩 f (1) 0 兲 f 1 ⫺ 共 ␸ 1 兩 f 0 兲共 ␸ 1 兩 f 1 兲 f 1 1⫹i ␻ ␶ 1 2

1 1 (1) (2) (1) 共 ␸ 1 兩 f (3) 0 兲 ⫹ 共 ␸ 1 兩 f 0 兲共 ␸ 1 兩 f 1 兲 f 1 , 1⫹3i ␻ ␶ 1 2

册共65兲

(5) (0) (1) (4) (0) (4) (1) (2) (3) (2) (2) (0) W (5) ⫽ f (5) 0 ⫺ 共 ␸ 1 兩 f 0 兲 f 1 ⫹ 共 ␸ 1 兩 f 0 兲关共 ␸ 1 兩 f 1 兲 f 1 ⫺ f 1 兴 ⫹ 关共 ␸ 1 兩 f 0 兲共 ␸ 1 兩 f 1 兲 ⫺ 共 ␸ 1 兩 f 0 兲兴关 f 1 ⫺ 共 ␸ 1 兩 f 1 兲 f 1 兴

冋册冋

再

册冎

⫹

15 1 3 5 (2) (2) (4) (0) 2 共 ␸ 兩 f (1) 兲 f (4) 共 ␸ 1 兩 f (2) 1 ⫺ 共 ␸ 1兩 f 1 兲 f 1 ⫹ 1 兲 ⫺ 共 ␸ 1兩 f 1 兲 f 1 1⫹i ␻ ␶ 1 1 0 2 8 4

⫹

1 1 5 (1) (2) (2) (2) (0) 共 ␸ 1 兩 f (3) 0 兲 ⫹ 共 ␸ 1 兩 f 0 兲共 ␸ 1 兩 f 1 兲 f 1 ⫺ 共 ␸ 1 兩 f 1 兲 f 1 1⫹3i ␻ ␶ 1 2 2

⫹

1 1 3 3 (1) (4) (1) (2) 2 (3) (2) (0) 共 ␸ 1 兩 f (5) 0 兲 ⫹ 共 ␸ 1 兩 f 0 兲共 ␸ 1 兩 f 1 兲 ⫹ 共 ␸ 1 兩 f 0 兲共 ␸ 1 兩 f 1 兲 ⫹ 共 ␸ 1 兩 f 0 兲共 ␸ 1 兩 f 1 兲 f 1 . 1⫹5i ␻ ␶ 1 4 8 2

冋冋

We remark an important feature of Eqs. 共63兲–共66兲: they do not contain dispersion factors of even orders. This ensures that the frequency dependence of the full distribution function W incorporates only dispersion factors with odd multiples of the basic frequency. Qualitatively, this is the result of absence of the interwell mode for the statistical moments of even orders. Technically, it is due to vanishing of the products ( ␸ 1 兩 f (ᐉ) k ) entering Eqs. 共62兲–共66兲 if the sum k⫹ᐉ is even. This rule follows immediately from combination of the oddity of ␸ 1 , see Sec. II, with the parity properties of the functions f (ᐉ) k introduced in Sec. III A. For actual calculations one needs the values of the scalar products entering Eqs. 共62兲–共66兲. In Appendix C we obtain their representations in terms of the moments Q k and S k of the functions ␺ 0 and ␺ 1 , respectively. The procedures of asymptotic expansion of Q k and S k are given in Appendix A. IV. DYNAMIC SUSCEPTIBILITIES

The set of magnetic susceptibilities of an assembly of noninteracting particles with the number density c is defined by the relation M ⫽ ␹ (1) H⫹ ␹ (3) H 3 ⫹ ␹ (5) H 5 ⫹•••

兺 Hn n⫽1

I n⫹1 v n⫹1 共 kT 兲 n

e in ␻ t

冕

册

共66兲

through the perturbation functions W (n) found in the preceding section. Therefore, evaluation of ␹ (n) becomes, although tedious, but simple procedure. Remarkably, the final expressions come out rather compact. A. Linear susceptibility

The resulting expression can be presented in the form

␹ ␻(1) ⫽ ␹ (1) 0

冉

B (1) 0 ⫹

B (1) 1 1⫹i ␻ ␶ 1

冊

,

␹ (1) 0 ⫽

cI 2 v 2 , 3kT

共69兲

which follows from substituting Eq. 共62兲 in 共68兲. Each of the two frequency-independent coefficients B (1) , being the result of statistical averaging over the orientational variable e, see Appendix C, expands into a series of Legendre polynomials with respect to ␤ , the angle between the direction h of the probing field and the particle easy axis n. This can be written as

共67兲 (1) (1) B (1) 0 ⫽b 00 ⫹b 02 P 2 共 cos ␤ 兲 ,

that describes the magnetization of the system in the direction of the probing field H⫽Hh. Therefore, of all the components of the corresponding susceptibility tensors, we retain the combinations that determine the response in the direction of the probing field. With representation 共57兲 for the distribution function, this magnetization component takes the form

M ⫽cI v 具共 e•h兲典 ⫽c

册

(1) (1) B (1) 1 ⫽b 10 ⫹b 12 P 2 共 cos ␤ 兲 ,

冉

(1) b 00

(1) b 02

(1) b 10

(1) b 12

冊冉 ⫽

1⫺Q 21

2S 2 ⫺2Q 21

Q 21

2Q 21

冊

.

共70兲

共 e•h兲 W (n) de,

共68兲

and the susceptibilities can be found by a direct comparison with Eq. 共67兲. In other words, the set of ␹ (n) is expressed

Definitions of functions Q 1 and S 2 and their explicit asymptotic representations are given in Appendix A. The (1) derived on the base asymptotic series for the coefficients b ␣␤ of expansion 共12兲 and Eq. 共37兲 are

214406-9



(1) b 02 ⫽⫺

⫹

that is the asymptotic representation of the full expression given by formula 共39兲 of Ref. 6.

1 1 13 165 ⫹ ⫹ ⫹ 3 4 ␴ 4␴ 8␴ 16␴ 5 2273

34577

32␴

64␴ 7

(1) ⫽1⫺ b 10

⫺

⫹ 6

⫹

581133 128␴ 8

B. Cubic susceptibility

⫹•••,

As follows from definitions 共67兲 and 共68兲, the third-order susceptibility is defined through the response at the triple frequency that at weak H scales as H 3 . Performing calculations along the same scheme as for ␹ (1) , one arrives at the sum of relaxators representation

1 3 2 31 153 ⫺ ⫺ ⫺ ⫺ ␴ 4␴2 ␴3 4␴4 4␴5

3629

1564

16␴

␴7

⫺ 6

⫺

785931 64␴ 8

⫹•••.

冉

(1) (1) The other components, namely, b 00 and b 12 , may be constructed straightforwardly using their relations with the given ones, see Eqs. 共70兲. For a random system, that is for an assembly of noninteracting particles with a chaotic distribution of the anisotropy axes, the average of any Legendre (1) , and the linear dypolynomial is zero, so that B (1) k ⫽b k0 namic susceptibility reduces to

␹

1⫹i ␻ ␶ 1 b 共001 兲 (1) (1) ⫽ ␹ , ␻ 0 1⫹i ␻ ␶ 1

(3) b 00 ⫽

(3) ⫽ b 02

30␴ 1

42␴

(3) ⫽⫺ b 04

(3) ⫽ b 10

(3) b 12 ⫽

␹ (3) 0 ⫽

3

⫹

⫹ 3 2

35␴

3

47 240␴ 2 21␴ ⫺

4

⫹

⫹ 4 8

35␴

4

4 7␴ ⫺

共74兲

⫹

815 96␴

6

⫹

7837 120␴

7

⫹

355391 640␴ 8

1385

11231

19083

336␴

336␴

64␴ 8

5

⫺

⫹ 6

50 7␴

6

⫺

⫹ 7

1756 35␴

7

⫺

63749 160␴ 8

⫹•••,

⫹•••,

⫹•••,

1 1 23 61 1357 235447 11962691 694849241 15133953221 ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫹•••, 15 6 ␴ 240␴ 2 192␴ 3 960␴ 4 30720␴ 5 245760␴ 6 1966080␴ 7 5242880␴ 8

65 13 25 863 3931 698911 35309123 2061480665 45071465669 ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫹•••, 2 3 4 5 84 168␴ 168␴ 1344␴ 1344␴ 43008␴ 344064␴ 6 2752512␴ 7 7340032␴ 8

(3) ⫽ b 14

1 2 1 73 17033 1007549 64390439 4493994417 1 ⫹ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫹•••, ⫺ 35 14␴ 35␴ 2 112␴ 3 560␴ 4 17920␴ 5 143360␴ 6 1146880␴ 7 9175040␴ 8

(3) ⫽⫺ b 30

(3) ⫽⫺ b 32

(3) ⫽⫺ b 34

共73兲

up to the fourth Legendre polynomial in cos ␤. (3) are The explicit expansions for the amplitudes b ␣␤

41 35␴

,

k⫽0,1,3,

5

⫹ 5

共 kT 兲 3

(3) (3) (3) B (3) k ⫽b k0 ⫹b k2 P 2 共 cos ␤ 兲 ⫹b k4 P 4 共 cos ␤ 兲 ,

49 40␴

cI 4 v 4

where the coefficients expand as

共72兲

1

冊

B (3) B (3) 1 (3) (3) 1 3 ␹ 3(3) ⫽ ␹ B ⫹ ⫹ , ␻ 0 4 0 1⫹i ␻ ␶ 1 1⫹3i ␻ ␶ 1

共71兲

3 2 1 337 499 85309 2563751 245269747 47628510799 ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ ⫹•••, 15 10␴ 16␴ 2 960␴ 3 320␴ 4 10240␴ 5 49152␴ 6 655360␴ 7 15728640␴ 8

29 11 1279 1881 320765 48133699 920146163 178560431695 43 ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ ⫹•••, ⫹ 2 3 4 5 84 56␴ 56␴ 1344␴ 448␴ 14336␴ 34406␴ 6 91750␴ 7 22020096␴ 8

47 11 2 559 2419 409499 4080395 1166954357 75334335763 ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ ⫹•••. 2 3 4 5 105 210␴ 21␴ 1680␴ 1680␴ 53760␴ 86016␴ 6 3440640␴ 7 27525120␴ 8 214406-10

共75兲



For a random system, the averages of Legendre polyno(3) mials drop out and B (3) k ⫽b k0 . With respect to formalism constructed in Ref. 6, the above expressions yield the asymptotic representations for formulas 共42兲 and 共43兲 there.

␹ 5(5) ␻⫽

冉

B (5) B (5) 1 (5) (5) 1 3 ␹0 B0 ⫹ ⫹ 16 1⫹i ␻ ␶ 1⫹3i ␻ ␶ ⫹

B (5) 5 1⫹5i ␻ ␶

冊

, ␹ (5) 0 ⫽

cI 6 v 6 共 kT 兲 5

,

共76兲

with the coefficients C. Fifth-order susceptibility

(5) (5) (5) B (5) k ⫽b k0 ⫹b k2 P 2 共 cos ␤ 兲 ⫹b k4 P 4 共 cos ␤ 兲 (5) ⫹b k6 P 6 共 cos ␤ 兲 ,

The susceptibility of the fifth order writes in an expectable way as a sum of three relaxators:

(5) ⫽ b 00

(5) ⫽ b 10

(5) b 30 ⫽⫺

(5) ⫽ b 50

80␴

5

⫹

The explicit asymptotic series are

367 2240␴

共77兲

6

⫹

123 70␴

7

⫹

41233 2240␴ 8

⫹•••,

1 1 65 79 85913 72636131 4543038053 14938598691 19 ⫹ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫹•••, ⫺ 2 3 4 5 6 96 420␴ 120␴ 1792␴ 336␴ 57344␴ 6881280␴ 55050240␴ 7 20971520␴ 8 47 29 437 5473 1046209 169435283 684614895 230861266333 11 ⫺ ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ ⫹•••, ⫹ 560 35␴ 280␴ 2 1920␴ 3 4480␴ 4 143360␴ 5 3440640␴ 6 1835008␴ 7 73400320␴ 8

311 13 5911 2141 1874309 299470403 17964831133 400677748549 137 ⫹ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫹•••, ⫺ 2 3 4 5 3360 420␴ 105␴ 26880␴ 1920␴ 286720␴ 6881280␴ 6 55050240␴ 7 146800640␴ 8 (5) ⫽ b 02

(5) b 12 ⫽

1 112␴

5

⫹

3 28␴

6

⫹

507 448␴

7

⫹

5377 448␴ 8

⫹•••,

13 23 737 2959 99733 50499149 350973527 72765921299 19 ⫹ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫹•••, ⫺ 2 3 4 5 6 504 168␴ 672␴ 8064␴ 5376␴ 28672␴ 2064384␴ 1835008␴ 7 44040192␴ 8

(5) ⫽⫺ b 32

149 5 193 5245 18677 1785635 289305193 5846947361 394448762615 ⫹ ⫺ ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ ⫹•••, 21 168␴ 672␴ 2 8064␴ 3 5376␴ 4 86016␴ 5 2064384␴ 6 5505024␴ 7 44040192␴ 8

(5) b 52 ⫽

27 139 109 1343 9203 431321 9839105 196654913 30690812563 ⫺ ⫹ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫹•••, 2 3 4 5 504 28␴ 336␴ 2016␴ 2688␴ 21504␴ 73728␴ 6 196608␴ 7 3670016␴ 8 (5) ⫽⫺ b 04

(5) ⫽ b 14

1

k⫽0,1,3,5.

3 140␴

5

⫺

1563 6160␴

6

⫺

7767 3080␴

7

⫺

613353 24640␴ 8

⫹•••,

183 15 713 433 409 319665 8222083 5744848239 403943151013 ⫺ ⫹ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫹•••, 2464 6160␴ 24640␴ 2 19712␴ 3 4928␴ 4 630784␴ 5 2293760␴ 6 201850880␴ 7 1614807040␴ 8

(5) ⫽⫺ b 34

29 47 7081 74647 7137293 385804437 4682760003 1580817298041 293 ⫺ ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ ⫹•••, ⫹ 2 3 4 5 280 770␴ 385␴ 24640␴ 49280␴ 788480␴ 6307840␴ 6 10092544␴ 7 403701760␴ 8 214406-11



(5) b 54 ⫽

2929 1713 2551 34863 92061 34432191 23756287 15647080587 7317549380671 ⫺ ⫹ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ 2 3 4 5 12320 6160␴ 24640␴ 98560␴ 49280␴ 3153920␴ 327680␴ 6 28835840␴ 7 1614807040␴ 8 ⫹•••, (5) ⫽⫺ b 06

(5) ⫽⫺ b 16

616␴

6

⫺

3 77␴

7

⫺

1467 2464␴ 8

⫹•••,

1 1 7 337 53 51433 2188103 471762913 4428495037 ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ ⫹•••, 2 3 4 5 6 7 1584 1848␴ 1056␴ 88704␴ 1848␴ 315392␴ 2064384␴ 60555264␴ 69206016␴ 8

(5) ⫽⫺ b 36

(5) b 56 ⫽

1

10 1 17 103 2489 236615 38344237 776232845 52467158027 ⫹ ⫺ ⫹ ⫹ ⫹ ⫹ ⫹ ⫹ ⫹•••, 84 231␴ 924␴ 2 3168␴ 3 14784␴ 4 236544␴ 5 5677056␴ 6 15138816␴ 7 121110528␴ 8

661 1207 17 5525 1169 9116467 131486063 1918435847 639291980689 ⫺ ⫹ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺ 55440 9240␴ 12320␴ 2 88704␴ 3 3520␴ 4 4730880␴ 5 10321920␴ 6 20185088␴ 7 807403520␴ 8 ⫹•••,

共78兲

and for a random system, as for the lower orders, (5) B (5) k ⫽b k0 . V. DISCUSSION

The above derived formulas despite their hefty look are very practical. Indeed, they present the nonlinear initial susceptibilities of a superparamagnetic particulate medium as analytical expressions of arbitrary accuracy. With respect to the frequency dependence they give the exact full structure of the susceptibility and prove that it is very simple thus putting former intuitive considerations on a solid ground. This makes our formulas a handy tool for asymptotic analysis. Yet more convenient they are for numerical work because with their use the difficult and time-consuming procedure of solving the differential equations is replaced by a plain summation of certain power series. For example, if to employ Eqs. 共72兲–共78兲, a computer code that fits simultaneously experimental data on linear and a set of nonlinear susceptibilities taking into account the particle polydispersity of any kind 共easy axes directions, activation volume, anisotropy constants兲 becomes a very fast procedure. Graphic examples justifying our claims are presented in Figs. 5 and 6, where the components of two nonlinear complex susceptibilities are plotted as the functions of the parameter ␴ . For a given sample, ␴ in a natural way serves as a dimensionless inverse temperature. In those figures, the solid lines correspond to the above-proposed asymptotic formulas where we keep the terms up to ␴ ⫺3 . The circles show the results of numerically exact solutions obtained by the method described in Ref. 6. Note that even at ␴ ⬃5 the accuracy is still rather high. The model that may be called the predecessor of the afore-derived results was proposed in Ref. 23. There, the authors calculated the initial susceptibilities up to the seventh

order having replaced a superparamagnetic assembly by a two-level macrospin system. The interrelation between the present work and Ref. 23 closely resembles the situation with the evaluation of the rate of a superparamagnetic process. First in 1949 Neél1 and then, ten years later, Brown3 evaluated the superparamagnetic time in the framework of a two-level model. In such a framework, one allows for the magnetic moment flips but totally neglects its possible diffusion over energetically less-favorable directions. In 1963 Brown4 succeeded to overcome this artificial assumption and took into account the possibility for the magnetic moment to wander over all 4 ␲ radians. In the present case, the obtained v /T dependencies of the nonlinear susceptibilities and those from Ref. 23 are qualitatively the same. Their most typical feature is the double-peak shape. Quantitatively, however, the corresponding lines differ and do not reduce to one another in any case. Indeed, as long as the temperature is finite 共whatever low兲, the configurational space for the unit vector e of the magnetic moment is the full (4 ␲ -radian兲 solid angle; its reduction to just two directions along a bidirectional axis could not be done otherwise than ‘‘by hand.’’ This is exactly what the two-level Ising-like model does: it forcibly imparts a quantum property 共discrete spin projections兲 to a macrospin assembly. From the calculational viewpoint, another essential demerit of the results23 is that the coefficients in the susceptibility formulas are not given in an analytic form. The authors propose to evaluate them by solving an infinite set of recurrence equations. Hence, the procedure23 does not provide any gain with respect to former ones neither in analytical considerations nor in constructing fitting codes. In the presented framework the results by Klik and Yao 共including the analytical formulas for them missing in Ref. 23兲 can be obtained immediately if to take the function ␸ 1 in a stepwise form 共23兲 and not to allow for the corrections

214406-12



FIG. 5. Real 共a兲 and imaginary 共b兲 components of the cubic susceptibility of a superparamagnetic assembly with coherently aligned easy axes; the direction of the probing field is tilted with respect to the alignment axis at cos ␤⫽0.5; the dimensionless frequency is ␻ ␶ 0 ⫽10⫺6 . Solid lines show the proposed asymptotic formulas taken with the accuracy ␴ ⫺3 , circles present the result of numerically exact evaluation, dashed lines correspond to the ‘‘zero-derivative’’ approximation 共45兲. The discrepancy of the curves is commented in the text after Eq. 共B12兲.

caused by the finiteness of its derivative at x⫽0. In our terms this means to stop at set 共45兲, i.e., ‘‘zero-derivative’’ solution, and not to go further. The emerging error is however, uncontrollable and not at all small. As an illustration, in Fig. 5 we show the result obtained with this model 共dashed lines兲 for the cubic susceptibility ␹ 3(3) ␻ in a textured system where the particle common axis n is tilted under the angle ␤ ⫽ ␲ /3 to the probing field. One can see that deviations are substantial. In Ref. 6 we have proposed, although without rigorous justification, a formula for the cubic susceptibility of a random assembly

␹ 3(3) ␻ ⫽⫺

1 (3) 共 1⫹2S 22 兲共 1⫺i ␻ ␶ 1 兲 ␹ , 4 0 45共 1⫹i ␻ ␶ 1 兲共 1⫹3i ␻ ␶ 1 兲

␹ 5(5) ␻⫽

3 21 i ␻ ␶ 1 ⫺ ␻ 2 ␶ 21 8 4 ⫻ 共 1⫹i ␻ ␶ 1 兲共 1⫹3i ␻ ␶ 1 兲共 1⫹5i ␻ ␶ 1 兲 1⫺

共80兲

that, following the example of the already tested Eq. 共79兲, has high chances to be a good approximation for ␹ 5(5) ␻ in the whole temperature interval. As we have already ascertained in Ref. 6, the best interpolation expression for the relaxation time in the susceptibility formulas is

␶ 1⫽ ␶ D

共79兲

that proved to be well adjusted for approximating the results of numerical calculations in all the temperature interval and also appeared to be good for fitting experimental data.7 Now we see that this very expression follows from Eqs. 共73兲–共75兲 (3) up to the zeroth order with if to expand the coefficients b i0 ⫺1 respect to ␴ . This justifies Eq. 共79兲 as a formula yielding a correct frequency dispersion of the cubic susceptibility of a random assembly at low temperatures. The cause of its applicability at high temperatures is the exponential dependence of ␶ 1 on ␴ . Indeed, in the frequencies range ␻ ␶ 0 Ⰶ1, where we work, the condition ␴ ⱗ1 means ␶ 1 → ␶ 0 , and all the dispersion factors in Eq. 共79兲 drop out. This transforms expression 共79兲 in a correct static susceptibility that is also a true result. To avoid any confusion we remark that Eq. 共79兲 differs from formula for ␹ 3(3) ␻ given in Ref. 6 by the coefficient (⫺1/45) due to the difference in definitions: in Ref. 6 it was included in ␹ (3) 0 . Applying the similar procedure to Eqs. 共76兲–共78兲 we get the expression for the fifth-order susceptibility

1 (5) 共 2⫹12S 22 ⫹4S 32 兲 ␹ 16 0 945

冋冑

e ␴ ⫺1 ␴ 2 ␴ 1⫹ ␴

␴ ⫹2 ⫺ ␴ ⫺1 ␲

册

⫺1

,

proposed in Refs. 13,14. VI. CONCLUSIONS

A consistent procedure yielding the integral relaxation time and initial nonlinear susceptibilities for an assembly of noninteracting superparamagnetic particles is constructed in the low-to-moderate temperature range. Starting from the micromagnetic kinetic equation that describes intrinsic rotary diffusion of the particle magnetic moment, we obtain the results in an analytical form. They are presented as asymptotic series with respect to the dimensionless parameter ␴ that is the uniaxial anisotropy barrier height scaled with temperature. High-order expansion terms are easily accessible that allows to achieve any desirable extent of accuracy. This is proven by comparison of the proposed approximation with the numerically exact results. The susceptibilities contain angular dependencies that allow one to consider the particle assemblies with any extent of orientational texture—from perfectly aligned to random. The new

214406-13



APPENDIX A: EVALUATION OF THE EXPANSION COEFFICIENTS FOR EIGENFUNCTIONS ␺ 0 AND ␺ 1

Both functions ␺ 0 and ␺ 1 are uniaxially symmetrical about the anisotropy axis n and can be expanded in the Legendre polynomial series, see Eq. 共32兲: 1 2

k⫽0

1 2

k⫽1

␺ 0⫽ ␺ 1⫽

兺

共 2k⫹1 兲 S k P k 共 x 兲 ,

k⫽0,2,4, . . . ,

兺

共 2k⫹1 兲 Q k P k 共 x 兲 ,

k⫽1,3,5, . . . ,

共A1兲

where in accordance with the parity properties of the eigenfunctions nonzero terms are S 0 ⫽1, S k ⫽ 共 P k 共 x 兲兩 ␺ 0 兲 , Q k⫽ 共 P k共 x 兲兩 ␺ 1 兲 ,

k⫽2,4, . . . ,

k⫽1,3,5, . . . .

共A2兲

Taking into account that ␺ 1 ⫽ ␺ 0 ␸ 1 , where ␺ 0 in a finite form is given by Eq. 共11兲, one arrives at the general formula Fk ⫽ 共 1/R 兲

冕

1

0

2

P k 共 x 兲 e ␴ x dx,

共A3兲

where F is S k for even and is Q k for odd values of the index, and the function R( ␴ ) is defined by Eq. 共11兲. In particular Q 1 ⫽ 共 1/R 兲

冕

1

0

1 2 xe ␴ x dx⫽ 共 e ␴ ⫺1 兲 / ␴ R. 2

共A4兲

Using asymptotic expansion 共12兲 for R, one gets FIG. 6. Real 共a兲 and imaginary 共b兲 components of the fifth-order susceptibility of a random superparamagnetic assembly; the dimensionless frequency is ␻ ␶ 0 ⫽10⫺6 . Solid lines show the proposed asymptotic formulas with the accuracy ␴ ⫺3 , circles present the result of a numerical evaluation.

formulas stand closer to reality than those for a two-level system and are to facilitate considerably both analytical and numerical calculations in the theory of superparamagnetic relaxation in single-domain particles.

Q 1 ⫽1/G⫽1⫺ ⫺

1 1 5 37 353 4881 ⫺ ⫺ ⫺ ⫺ ⫺ 2 ␴ 2 ␴ 2 4 ␴ 3 8 ␴ 4 16␴ 5 32␴ 6

55205 64␴

7

⫺

854197 128␴ 8

共A5兲

Knowing Q 1 , one can derive all the other moments Q k with the aid of the three-term recurrence relation obtained from Eq. 共8兲 by setting there b k ⫽Q k and ␭⫽0. The same relation can be used to find the equilibrium order parameters S k . This is a head-to-tail procedure, where S 0 ⫽1 and S 2 is determined by the integral

ACKNOWLEDGMENTS

Partial financial support from the International Association for the Promotion of Cooperation with Scientists from the New Independent States of the Former Soviet Union 共INTAS兲 under Grant No. 01–2341 and by Grant No. PE– 009–0 of the U.S. Civilian Research & Development Foundation for the Independent States of the Former Soviet Union 共CRDF兲 is gratefully acknowledged. The package Maple V for PC used in the calculational work was obtained by Institute of Continuous Media Mechanics in the framework of the EuroMath Network and Services for the New Independent States - Phase II 共EmNet/NIS/II兲 project funded by INTAS under Grant No. IA-003.

⫹•••.

S 2 ⫽ 共 1/2R 兲

冕

1

0

2

共 3x 2 ⫺1 兲 e ␴ x dx.

共A6兲

Taking the latter by parts one gets 3 S 2 ⫽ 关 e ␴ ⫺R 兴 / ␴ R. 4 On comparison with Eq. 共A4兲, we find 3 3 S 2 ⫽ Q 1 ⫺ 共 3⫺2 ␴ 兲 / ␴ , 2 4 that upon substituting asymptotic series 共A5兲, transforms into

214406-14



3 3 15 111 1059 12243 ⫺ ⫺ ⫺ ⫺ ⫺ 2 3 4 2␴ 4␴ 8␴ 16␴ 32␴ 5 64␴ 6

In the left part we make use of the fact that ␸ 1 is the left eigenfunction of the operator Lˆ , in the right part the integrals are taken by parts and yield

S 2 ⫽1⫺ ⫺

165615 128␴

7

⫺

2562591 256␴

8

⫹•••.

共A7兲

␭ 1 D n ⫽2

APPENDIX B: EVALUATION OF THE CORRECTING COEFFICIENTS D n IN A GENERAL CASE

共B1兲

where the functions g (n) 1 are rendered by formulas 共45兲 and are not corrected with respect to the derivative d ␸ 1 /dx. Substituting Eq. 共B1兲 in 共42兲 and taking into account Eqs. 共45兲, we get a recurrence sequence of equations for the corrections u (n) : Lˆ u (n) ⫽Vˆ u (n⫺1) ⫹Jˆ␺ 0

共 e•h兲 n Jˆ␸ 1 . n!

共B2兲

With allowance for the fact that function ␸ (0) 1 depends only on x, Eq. 共B2兲 rewrites as

冋

册

d 共 e•h兲 n d ␸ 1 ␺ 0 共 1⫺x 2 兲 . Lˆ u (n) ⫽Vˆ u (n⫺1) ⫹ dx n! dx

冕

D n⫽

1 u (n⫺1)

␺0

0

0

d␸1 d 共 e•h兲 dx dx dx

共 e•h兲 n dx. dx n! 1

共B7兲

d 共 e•h兲 dx⫺ dx

冕

1d␸

共 e•h兲 n dx. 共B8兲 dx n!

0

1

Since ␺ 0 ⬀exp(␴x2), the first integral in Eq. 共B8兲 can be presented as an asymptotic series if the power expansion of the function u (n⫺1) in the vicinity of x⫽0 is known. A closed form for the second integral can be found with the aid of the table given in Eq. 共51兲, see Sec. III A. As an example, we calculate the coefficient D 2 . Since from the addition theorem 共 e•h兲 ⫽cos ␪ cos ␤ ⫹sin ␪ sin ␤ cos ␸ ,

u (1) ⫽cos ␤

k i␸ 2 1/2 k 兺k C (0) 兺k C (1) k x ⫹sin ␤ e 共 1⫺x 兲 k x .

共B9兲共B3兲

that follows from Eq. 共18兲, we get

冋册

␭ 1 d 共 e•h兲 n . 2 dx n!

Here the upper index of the C coefficients corresponds to the azimuthal number m of the spherical harmonic e im ␸ . Operator Lˆ now includes the azimuthal coordinate and takes the form

共B4兲 ⫺Lˆ ⫽ 共 1⫺x 2 兲

In particular, at n⫽1 Eq. 共B4兲 takes the form ␭1 d Lˆ u (1) ⫽ 共 e•h兲 . 2 dx

冋

d dx

2

⫺ 关 2 ␴ x 共 1⫺x 2 兲 ⫹2x 兴

⫹ 2 ␴ 共 3x 2 ⫺1 兲 ⫺

共B5兲

Equations 共B4兲 are solved sequentially beginning from Eq. 共B5兲 by expanding in a power series with respect to x. The right-hand sides of Eqs. 共B4兲 and 共B5兲 are proportional to an exponentially small parameter ␭ 1 . Just due to that we did not take into account the corrections of the order u (n) when deriving Eqs. 共45兲. However, the quantities

m2 1⫺x 2

册

d dx

.

Substitution of expansion 共B9兲 in Eq. 共B5兲 leads to the set of equations (m) ⫺ 关 k 共 k⫹1⫹2m⫹2 ␴ 兲 ⫹m 共 m⫹1 兲 2 ␴ 共 k⫹m⫹1 兲 C k⫺2 (m) (m) ⫹2 ␴ 兴 C (m) k ⫹ 共 k⫹1 兲共 k⫹2 兲 C k⫹2 ⫽N k ,

共B10兲

where m⫽0,1 and the numbers in the right-hand side are n⫽2,4,.., N (0) k ⫽

have finite values. To show that, let us multiply Eq. 共B4兲 by ␸ 1 and integrate. This yields 共 ␸ 1 兩 Lˆ u (n) 兲 ⫽ 共 ␸ 1 兩 Vˆ u (n⫺1) 兲 ⫹

冕

1d␸

冋册

we seek the solution of Eq. 共B5兲 the sum

␭1 d␸1 ⫽ , dx 2 ␺ 0 共 1⫺x 2 兲

D n ⫽ 共 ␸ 1 兩 u (n) 兲 ,

0

共 1⫺x 2 兲 u (n⫺1)

Replacing the derivative d ␸ 1 /dx in the first term of the right-hand side with the aid of Eq. 共B3兲, we arrive at the representation of the coefficient D n as

Finally, making use of the relation

Lˆ u (n) ⫽Vˆ u (n⫺1) ⫹

1

⫺␭ 1

Let us present the solution of Eq. 共42兲 in the form (n) (n) , f (n) 1 ⫽ ␺ 0 g 1 ⫹u

冕

冉冏冋册冊

d 共 e•h兲 n ␭1 ␸1 2 dx n!

. 共B6兲

再

⫺1

for

k⫽0,

0

for

k⫽0,

N (1) k ⫽

再

1,

for k odd,

0,

for k even.

In reality, one retains in expansion 共B9兲 only a finite number of terms so that Eqs. 共B10兲 could be easily solved analytically by any computer algebra solver. In terms of expansion 共B9兲, expression 共B8兲 at n⫽2 is written

214406-15



D 2 ⫽cos2 ␤

兺 C 2k(0) k⫽0

共 2k⫺1 兲 !!

X 6⫽

2 k␴ kG

⫹110S 2 P 2 共 cos ␤ 兲 ⫹33兴 ,

1 共 2k⫺1 兲 !! (1) C 2k⫺1 ⫺ sin2 ␤ 2 k⫽1 2 k␴ kG

兺

1 2G⫺3 P 2 共 cos ␤ 兲 . ⫺ ⫺ 6 6G

1 1 5 37 ⫹ ⫹ ⫹ ⫹••• 4 ␴ 4 ␴ 2 8 ␴ 3 16␴ 4 ⫺sin2 ␤

冉

共B11兲

冊

1 1 1 7 19 ⫹ ⫹ ⫹ ⫹ ⫹••• . 2 3 4 8 ␴ 16␴ 64␴ 64␴ 4 共B12兲

Y 1 ⫽Q 1 cos ␤ ,

Y 5⫽

1 Y 3 ⫽ 关 2Q 3 P 3 共 cos ␤ 兲 ⫹3Q 1 cos ␤ 兴 , 5

1 关 8Q 5 P 5 共 cos ␤ 兲 ⫹28Q 3 P 3 共 cos ␤ 兲 ⫹27Q 1 cos ␤ 兴 , 63 共C2兲

where cos ␤⫽(n•h) and the parameters S k and Q k are the expansion coefficients introduced by formulas 共32兲. (n) Now using the expressions for functions f (n) 0 and f 1 derived in Sec. III A one sees that the relevant integrals of Eqs. 共63兲–共68兲 are expressed in terms of X k and Y k as

As it should be, at ␤ ⫽0 this formula reduces to Eq. 共52兲 that was obtained for a one-dimensional case. We remark, however, that in a tilted situation ( ␤ ⫽0) the coefficient D 2 acquires a contribution independent of ␴ that assumes the leading role. This effect is clearly due to admixing of transverse modes to the set of eigenfunctions of the system, and it is just it that causes so a significant discrepancy between the ‘‘zero-derivative’’ approximation and the correct asymptotic expansion for ␹ (3) curves in Fig. 5. Evaluation of the coefficient D 4 is done according to the same scheme and requires taking into account a number of the perturbation terms that makes it rather cumbersome.

1 1 2 (3) „共 e•h兲兩 f (1) 0 …⫽X 2 , „共 e•h 兲兩 f 0 …⫽ X 4 ⫺ X 2 , 6 2 „共 e•h兲兩 f (5) 0 …⫽

1 1 1 X 6 ⫺ X 4 X 2 ⫹ X 32 ; 120 8 4

共 ␸ 1 兩 f (5) 0 兲⫽

Before proceeding to the integrals 共scalar products兲 in Eqs. 共62兲–共66兲 and 共68兲, let us consider the ‘‘primitive’’ ones

1 1 1 1 Y 5 ⫺ Y 3 X 2 ⫹ X 22 Y 1 ⫺ X 4 Y 1 ; 120 12 4 24

Y n ⫽„共 e•h兲 n 兩 ␺ 1 ….

1 „共 e•h兲兩 f (2) 1 …⫽ Y 3 ⫺Y 1 X 2 ⫹D 2 Y 1 , 2 共 ␸ 1 兩 f (4) 1 兲⫽

1 X 2 ⫽ 关 2S 2 P 2 共 cos ␤ 兲 ⫹1 兴 , 3

1 1 1 X 4 ⫺ Y 3 Y 1 ⫹ X 2 Y 21 ⫹D 4 ⫹D 2 共 ␸ 1 兩 f (2) 1 兲, 24 3 2

„共 e•h兲兩 f (4) 1 …⫽

1 关 8S 4 P 4 共 cos ␤ 兲 ⫹20S 2 P 2 共 cos ␤ 兲 ⫹7 兴 , 35

1 1 1 1 Y ⫺ Y X ⫺ X Y ⫹ X 2Y 24 5 6 3 2 6 4 1 2 2 1 ⫹D 4 Y 1 ⫹D 2 „共 e•h兲兩 f (2) 1 ….

共C5兲

W.F. Brown, Jr., J. Appl. Phys. 30, 130S 共1959兲. W.F. Brown, Jr., Phys. Rev. 130, 1677 共1963兲. 5 J.-L. Dormann, D. Fiorani, and E. Tronc, Adv. Chem. Phys. 98, 283 共1997兲. 6 Yu.L. Raikher and V.I. Stepanov, Phys. Rev. B 55, 15 005 共1997兲. 7 L. Spinu, D. Fiorani, H. Srikanth, F. Lucari, F. D’Orazio, E. Tronc, and M. Nogue´s, J. Magn. Magn. Mater. 226–230, 1927

*Email address: raikher@icmm. ru

3

L. Neél, Ann. Geophys. 5, 99 共1949兲; C. R. Hebd. Seances Acad. Sci. 228, 664 共1949兲. 2 T. Bitoh, K. Ohba, M. Takamatsu, T. Shirane, and S. Chikazawa, J. Phys. Soc. Jpn. 64, 1311 共1995兲; J. Magn. Magn. Mater. 154, 59 共1996兲.

4

1

共C4兲

1 2 共 ␸ 1 兩 f (2) 1 兲 ⫽ X 2 ⫺Y 1 ⫹D 2 , 2

The functions ␺ 0 and ␺ 1 are originally defined in terms of the angle ␪ ⫽arccos(e•n). Thus, before performing integration one needs to transform both integrands to the same set of angles. Doing this with the aid of the addition theorem for Legendre polynomials, one finds

X 4⫽

共C3兲

1 1 (3) 共 ␸ 1 兩 f (1) 0 兲 ⫽Y 1 , 共 ␸ 1 兩 f 0 兲 ⫽ Y 3 ⫺ X 2 Y 1 , 6 2

APPENDIX C: EVALUATION OF INTEGRALS

X n ⫽„共 e•h兲 n 兩 ␺ 0 …,

共C1兲

and

Since the coefficients C found from Eq. 共B10兲 are functions of ␴ , one has to perform in Eq. 共B11兲 asymptotic expansion. This gives finally D 2⫽

1 关 16S 6 P 6 共 cos ␤ 兲 ⫹72S 4 P 4 共 cos ␤ 兲 231

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LINEAR AND NONLINEAR SUPERPARAMAGNETIC . . . 共2001兲. B.A. Storonkin, Kristallografiya 30, 841 共1985兲关Sov. Phys. Crystallogr. 30, 489 共1985兲兴. 9 A. Aharoni, Phys. Rev. 135, A447 共1964兲. 10 Yu.P. Kalmykov, Phys. Rev. E 61, 6320 共2000兲. 11 L. Bessias, L. BenJaffel, and J.-L. Dormann, Phys. Rev. B 45, 7805 共1992兲. 12 A. Aharoni, Phys. Rev. B 46, 5434 共1992兲. 13 W.T. Coffey, P.J. Cregg, D.S.F. Crothers, J.T. Waldron, and A.W. Wickstead, J. Magn. Magn. Mater. 131, L301 共1994兲. 14 P.G. Cregg, D.S.F. Crothers, and A.W. Weakstead, J. Appl. Phys. 76, 4900 共1994兲. 15 W.F. Brown, Jr., Physica B 86–88, 1423 共1977兲. 16 W.F. Brown, Jr., IEEE Trans. Magn. 15, 1196 共1979兲. 17 Yu.L. Raikher and M.I. Shliomis, Zh. E´ksp. Teor. Fiz. 67, 1060 共1974兲关Sov. Phys. JETP 40, 526 共1974兲兴. 18 W.T. Coffey, Adv. Chem. Phys. 103, 259 共1998兲. 8

19

D.A. Garanin, V.V. Ishchenko, and L.V. Panina, Teor. Mat. Fiz. 82, 242 共1990兲关Theor. Math. Phys. 82, 169 共1990兲兴. 20 W.T. Coffey, D.S.F. Crothers, Yu.P. Kalmykov, E.S. Massawe, and J.T. Waldron, Phys. Rev. E 49, 1869 共1994兲. 21 W.T. Coffey and D.S.F. Crothers, Phys. Rev. E 54, 4768 共1996兲. 22 W. T. Coffey, Yu. P. Kalmykov, and J. T. Waldron, The Langevin Equation 共World Scientific, Singapore, 1996兲. 23 I. Klik and Y.D. Yao, J. Magn. Magn. Mater. 186, 233 共1998兲. 24 J. Garcıá-Palacios and P. Svedlidh, Phys. Rev. Lett. 85, 3724 共2000兲. 25 We remark that, in principle, there might occur a situation, where ␶ is very long, i. e., the quality factor of precession is very high. Then the gyromagnetic term in the kinetic equation must be retained so that the Larmor precession begins to interact with the superparamagnetic 共longitudinal兲 relaxation. An example of such a situation is considered in Ref. 24 where some interesting nonlinear effects are found.

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