Spin-lattice-relaxation-like model for superparamagnetic particles under an external magnetic field

PHYSICAL REVIEW B

VOLUME 62, NUMBER 5

1 AUGUST 2000-I

Spin-lattice-relaxation-like model for superparamagnetic particles under an external magnetic field ˜es-Paniago, and R. Paniago H.-D. Pfannes, A. Mijovilovich, R. Magalha Departamento de Fı´sica, Universidade Federal de Minas Gerais, CP 702, 30123-970 Belo Horizonte, Brazil 共Received 22 November 1999兲 Small monodomain magnetic particles of uniaxial magnetic anisotropy are considered. The superparamagnetic relaxation is assumed to happen by coherent rotation of the spins. The results of existing models for the calculation of superparamagnetic relaxation times are shortly reviewed. A different model based on phonon interaction with the total spin of the particle under the presence of an external magnetic field is presented and corresponding relaxation times are calculated and compared with the existing theories. The calculation of Mo¨ssbauer superparamagnetic relaxation spectra in the low-temperature limit and for higher temperatures is discussed.

I. INTRODUCTION

Ultrafine single domain magnetic particles are often found in soils, rocks, and living organisms or in artificially manufactured materials. They are of great interest in fundamental science and technology because of their nanometric size and magnetic properties that differ from the corresponding bulk materials. Many studies, like, e.g., of catalysts, high-density recording media, ferrofluids, argiles, ceramics, paintings, magnetic bacteria, ferritin, and others, in which nanosized magnetic particles are important, have been reported in the literature. The magnetization at low temperatures in such particles is oriented near easy magnetization directions which correspond to minima of the magnetic anisotropy energy, separated by energy barriers of certain heights. An external field changes the relative depths of the minima. As the temperature increases, the magnetization can overcome this barrier and turn over to near another easy direction with a certain magnetization reversal rate ␶ ⫺1 . The particle is said to exhibit a superparamagnetic behavior if the characteristic time of measurement 共time window兲 of the method used for the determination of the magnetization is greater than ␶ . Among various methods, Mo¨ssbauer spectroscopy has been widely used for the investigation of iron-containing superparamagnets since its characteristic time of approximately 10⫺8 s lies in the range of superparamagnetic relaxation times at 100– 300 K of many species of magnetic nanosized particles. The overturn of the magnetization to a different orientation happens by rotation of the spins. In the rest of the paper we consider only the case of ‘‘coherent rotation,’’ i.e., all spins remain in the initial mutual 共parallel兲 configuration throughout the rotation.1 This mode minimizes the exchange interaction and is prevalent for sufficiently small particles. For bigger particles, where the anisotropy energy is minimized, other modes 共‘‘buckling’’ or ‘‘curling’’兲 are possible 共see, e.g., Ref. 1兲. For simplicity, we presume the existence of uniaxial anisotropy, no size dispersion 共all particles have the same size and shape兲, and no interparticle interaction. We assume also that the orientation of the easy direction of all particles is along the z direction, i.e., when the anisotropy is due, e.g., to uniaxial form anisotropy, all particles are equally oriented in 0163-1829/2000/62共5兲/3372共9兲/$15.00

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space. With these assumptions and under the presence of an external field with induction B applied along the ⫹z direction, the anisotropy energy E is given by1–4 E 共 ⌰ 兲 ⫽KV sin2 共 ⌰ 兲 ⫺M S VB cos共 ⌰ 兲 ,

共1.1兲

where K is the anisotropy constant, V the volume, M S the 共saturation兲 magnetization per volume of the particles, and ⌰ represents the angle between the magnetization and the ⫹z direction. E(⌰) exhibits one minimum at ⌰⫽0 共magnetization parallel to the external field兲 with energy E(0) ⫽⫺M S VB and another at ⌰⫽␲, with higher energy E( ␲ ) ⫽⫹M S VB 共magnetization antiparallel to the external field兲. Above a limit field B lim ⫽2K/M S only one minimum exists. For B⫽0 both minima are separated by an energy barrier of height E( ␲ /2)⫽KV. The probability f (⌰)d⌰ to find the magnetization oriented towards ⌰ is given by the thermal with g(⌰)⫽exp average f (⌰)⫽g(⌰)/ 兰 0␲ g(⌰)d⌰ 关⫺E(⌰)/(kBT)兴sin ⌰.1,2 For high temperature and low fields this is different from zero in the whole range 0⭐⌰⭐ ␲ , meaning that the magnetization can fluctuate between different directions. It is our objective to attribute relaxation rates to these fluctuations of the magnetization direction and to derive explicit expressions for them. II. CLASSICAL MODELS

The classical theories of superparamagnetism are the models of Ne´el4 and Brown.5 In both models coherent rotation of the spins is assumed. Ne´el notes that at low temperature and in the presence of an external field directed along the ⫹z direction, the magnetization of P particles of a total of N identical particles present, at an instant t, is found near the ⫹z direction and that of the remaining (N⫺ P) particles near the ⫺z direction. Corresponding time constants ␶ ⫹ and ␶ ⫺ are introduced by writing the balance equation P˙ ⫽⫺ P/ ␶ ⫹ ⫹ 共 N⫺ P 兲 / ␶ ⫺ ,

共2.1兲

where ␶ ⫹ and ␶ ⫺ correspond to the passage of magnetizations from ⌰⫽0 关lower minimum of E(⌰)] to ⌰⫽ ␲ 关higher minimum of E(⌰)] or vice versa, respectively. With 3372

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SPIN-LATTICE-RELAXATION-LIKE MODEL FOR . . .

the solution of Eq. 共2.1兲 the total relative magnetization M (t), which is proportional to P⫺(N⫺ P)⫽2 P⫺N, can be written as M 共 t 兲 /M S ⫽N 共 ␶ ⫹ ⫺ ␶ ⫺ 兲 / 共 ␶ ⫹ ⫹ ␶ ⫺ 兲 ⫹2 关 P 共 0 兲 ⫺N ␶ ⫹ / 共 ␶ ⫹ ⫹ ␶ ⫺ 兲兴 exp共 ⫺t/ ␶ 兲

ries and derived an analytical expression for the smallest positive eigenvalue. An external field was not included in their calculations. By comparing with numerical solutions an agreement within 1% in the range 0.5⭐ ␣ ⭐50 was established. The corresponding relaxation rate, assuming again ␩ ⫽( ␥ 0 M S ) ⫺1 , is ⫺1 ⫺1 ␶ BBD ⫽ ␶ 0BBD exp共 ⫺ ␣ 兲

共2.7兲

⫺1 ␶ 0BBD ⫽K ␥ 0 M S 共 1⫹ ␣ /4兲 5/2␣ ⫺1 .

共2.8兲

共2.2兲 ⫺1

⫺1 ⫺1 ⫽␶⫹ ⫹␶⫺ .

with ␶ For the calculation of ␶ ⫾ Ne´el relates the vibrational strain energy of the particles to the thermal energy, calculates the corresponding magnetostrictive energy, and from this the magnetic anisotropy field in which the magnetization pre⫺1 are then cesses. The superparamagnetic relaxation rates ␶ ⫾ calculated by multiplying the precession rates by the population of the states near the top of the barrier and by an Arrhenius factor.4,6 Including also demagnetization effects Ne´el obtains 关 ␣ ⬅ ␤ KV⫽KV/(k B T) 兴 ⫺1 2 1/2 2 1/2 ␶ ⫾Neel ⫽ 共 2K/ ␲ G兲 1/2␥ 0 M ⫺1 S 兩 3G ␮ ⫹DM S 兩 ␣ 共 1⫺h 兲

⫻ 共 1⫾h 兲 exp关 ⫺ ␣ 共 1⫾h 兲 2 兴 ,

共2.3兲

where ␥ 0 is the gyromagnetic ratio ( ␥ 0 ⫽g␮ B /ប⬇ge/m, ␮ B ⫽Bohr magneton ⬇9.27⫻10⫺24 J/T, g⫽electronic g factor, e/m specific charge of the electron兲, G is Young’s modulus, ␮ is the longitudinal magnetostriction constant, D the demagnetization factor, and h is the reduced field h ⫽M S B/2K. Brown5 considers a random walk of the magnetization direction similar to normal Brownian motion. From Gilbert’s equation for the movement of the magnetization, in which a ជ /dt and a random-field term dhជ (t)/dt dissipative term ␩ dM is included, a Fokker-Planck-type equation is derived, whose solution can be written as a series containing exponentially decaying time-dependent terms. The corresponding decay rate constants are the eigenvalues of the Fokker-Planck equation. In the case of uniaxial symmetry the smallest positive eigenvalue ␭, which determines the dominant 共longest兲 relaxation time constant ␶ , can then be calculated from the solution of the following differential equation:

冉

d⌽ 共 x 兲 d exp关 ⫺E 共 ⌰ 兲 / 共 k B T 兲兴 共 1⫺x 2 兲 dt dx

冊

⫹␭⌽ 共 x 兲 exp关 ⫺E 共 ⌰ 兲 / 共 k B T 兲兴 ⫽0

共2.4兲

with ␭⫽2 ␣ M S /( ␶␥ 0 K), where for the dissipation constant ␩ ⫽( ␥ 0 M S ) ⫺1 is assumed. From this equation Brown deduces assymptotic values of the relaxation rate for high ( ␣ Ⰶ1) and low ( ␣ Ⰷ1) temperature: ⫺1 ␣ Ⰶ1: ␶ Br ⬇ 共 ␥ 0 K/M S 兲 ␣ ⫺1 关 1⫺ 共 2/5兲 ␣ ⫹ 共 48/875兲 ␣ 2

⫹ 共 2/5兲 h 2 ␣ 2 兴

共2.5兲

and

with

In the limit of high temperature ␶ 0BBD approaches zero as does ␶ Brown from Eq. 共2.5兲. A different approach to superparamagnetic relaxation, especially adapted to the calculation of Mo¨ssbauer spectra of superparamagnetic particles, has been proposed by Jones and Srivastava 共‘‘many states’’ model8兲. With this model also ’’collective excitations’’1 can be taken into account in a correct way. The Mo¨ssbauer spectrum 共diagonal hyperfine interaction兲 I( ␻ ) for a pair of lines, 共e.g., lines 1 and 6兲 of a six-line spectrum with splitting proportional to the z component of the magnetization is

ជ M⫺1 1ᠬ 兲 I 共 ␻ 兲 ⬀⫺2 Re共 W

共2.9兲

M⫽ 关 i 共 ␻ ⫺ ␻ i 兲 ⫹⌫ 兴 1⫺R,

共2.10兲

with the matrix

ជ consists of probabilities 共Boltzmann where the row vector W factors兲 of the different magnetization directions 共attributing ‘‘states’’ S z to them兲, the diagonal matrix 关 i( ␻ ⫺ ␻ i )⫹⌫ 兴 1 contains the line positions ␻ i according to the hyperfine splitting, ⌫ is the natural linewidth of the Mo¨ssbauer isotope, and 1ᠬ is a column vector with all elements equal to 1. R is a relaxation matrix whose off-diagonal elements r i j are the transition probabilities per time w ij between the states i and j (i⫽ j) and whose diagonal elements are r ii ⫽⫺ 兺 j w ij (i⫽ j). The complete Mo¨ssbauer spectrum is obtained by summing up the separately calculated three two-line spectra from Eq. 共2.9兲, weighted with the adequate intensity factors, in our case, e.g., 3:0:1 when the ␥ direction is parallel to the z direction. A more exact expression, valid when off-diagonal hyperfine interactions are present, implies in the use of superoperators like in Ref. 9. Jones and Srivastava8 assume that random fields give rise to transitions between adjacent states, i.e., only matrix elements 具 S z ⫹1 兩 S ⫹ 兩 S z 典 and 具 S z 兩 S ⫺ 兩 S z ⫹1典 are different from zero and hence the matrix is tridiagonal. It is shown that in the continuum limit the Mo¨ssbauer line shape can be obtained from the numerical solution of a differential equation. In the low-temperature regime and in the presence of an ⫺1 8 , external field they derive a relaxation rate ␶ ⫾JS ⫺1 ␶ ⫾JS ⫽A ␲ 1/2␣ 3/2共 1⫺h 2 兲共 1⫾h 兲 exp关 ⫺ ␣ 共 1⫾h 兲 2 兴 , 共2.11兲

⫺1 ␣ Ⰷ1: ␶ ⫾Br ⬇ 共 ␥ 0 K/ ␲ 1/2M S 兲 ␣ 1/2共 1⫺h 2 兲共 1⫾h 兲

⫻exp关 ⫺ ␣ 共 1⫾h 兲 2 兴 .

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共2.6兲

Bessais, Ben Jaffel, and Dormann7 solved the FokkerPlanck equation by introducing Fourier and Chebyshev se-

where A is a parameter proportional to the square of the random field and could depend on K,M S ,T, etc. In a later publication6 Jones and Srivastava questioned whether classical fluctuations could be responsible for superparamagnetic

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relaxation since their correlation times, which are supposed to be of the order of 10⫺13 s are shorter than the precession time of ⭓10⫺12 s that is needed for a magnetization oriented at ⌰⬇85 degrees to turn to the other side of the barrier. In our model presented below the superparamagnetic relaxation is supposed to be due to interaction of the spins with longwavelength 共low-frequency兲 phonons and therefore no correlation problem arises. For the case of magnetic molecules with a net spin of S ⫽10 Villain et al.10 calculated a spin-relaxation rate 共without external field兲 based on normal spin-phonon paramagnetic relaxation theory and a Debye model. In the low-temperature limit the rate ␶ V⫺1 关approximated from Eq. 共2.11兲 of Ref. 10兴 is

␶ V⫺1 ⬇3 兩 B 01 兩 2 共 KV 兲 3 共 2 ␲ ប 4 ␳ v 5 ␣ S 4 兲 ⫺1 exp共 ⫺ ␣ 兲 ,

共2.12兲

where ␳ and v is the density and sound velocity, respectively, of the particle and B 01 is a spin-phonon interaction constant. It is our main intention in the following to work out expressions comparable to Eqs. 共2.3兲–共2.7兲, 共2.11兲, and 共2.12兲 based on a spin-phonon-interaction-like relaxation model. In contrast to Ref. 10 we include an external field in the calculations, take electronic matrix elements into account, and allow for higher-order terms in the spin-phonon coupling Hamiltonian. Finally we discuss the calculation of Mo¨ssbauer spectra of superparamagnetic samples. III. MODEL

With the simplified assumptions already made above 共uniaxial anisotropy, coherent rotation, no interparticle interaction兲 we adopt now the picture for superparamagnetic relaxation described in the following 共see also Ref. 3兲. The magnetization of the equally sized 共monodispersed兲 particles is considered as arising from a large spin S whose direction relative to the anisotropy axis fluctuates due to interaction with phonons. In order to reach the opposite easy direction ⫺z the spin S must pass through intermediate levels. By analogy with the classical anisotropy energy 共1.1兲, E⫽KV⫺KV cos2 ␪ ⫺M S VB cos ␪ ⫽KV⫺A 共 S cos ␪ 兲 2 ⫺g ␮ B B 共 S cos ␪ 兲 ,

共3.1兲

共3.2兲

where KV⫽AS 2 共height of energy barrier without external field兲, M S V⫽g ␮ B S, and 2h ⬘ ⫽g ␮ B B (B along the ⫹z direction兲. Energy eigenvectors of H sp are 兩 S z 典 ⫽ 兩 m 典 and eigenvalues are E m ⫽⫺Am 2 ⫺2h ⬘ m

⬘ ⫽AS whereas are two energy minima up to a limit field h lim for higher fields only one minimum exists. Without external field (h ⬘ ⫽0) the barrier height is E B ⫽AS 2 and for h ⬘ ⬎0 it can be written as E B⫾ ⫽AS 2 共 1⫾h 兲 2 ,

共3.3兲

with m⫽⫺S,⫺S⫹1, . . . ,⫺h ⬘ A 共top of barrier兲, . . . S ⫺1,S, where we have assumed that h ⬘ /A is a positive integer. The function E m vs m is shown in Fig. 1. It exhibits a maximum at m⫽⫺h ⬘ /A with E m (⫺h ⬘ /A)⫽h ⬘ 2 /A. There

共3.4兲

⬘ ⭐1 and the positive 共negative兲 sign rewhere 0⭐h⬅h ⬘ /h lim fers to whether m is on the right 共left兲-hand side of the maximum in Fig. 1 (m⭓⫺hS or m⭐⫺hS, respectively兲. We calculate successive energy differences E m ⫺E m⫹k or E m ⫺E m⫺n that differ by ⌬m⫽k or n on the right- or lefthand side of the maximum in Fig. 1, respectively, E m ⫺E m⫹k ⫽Ak 2 共 1⫹2q 兲

we write, up to a constant, a spin-Hamiltonian H sp for the 共anisotropy兲 energy of S in a field B, H sp ⫽⫺AS z2 ⫺2h ⬘ S z ,

FIG. 1. Anisotropy energy E m vs eigenvalues m under an external field with B⫽2hAS/g ␮ B 共see text兲. The arrows indicate an example of transition 兩 ⫺S 典 → 兩 S 典 via phonon anihilation 共left兲 and creation 共right兲 with ⌬m⫽n⫽k⫽2.

共3.5a兲

with m⫽⫺hS⫹kq, q⫽0,1, . . . ,S(1⫹h)/k⫺1, and E m ⫺E m⫺n ⫽An 2 共 1⫹2q 兲

共3.5b兲

with m⫽⫺hS⫺nq, q⫽0,1, . . . ,S(1⫺h)/n⫺1, where both m and q are counted from top to bottom of the barrier potential. In order to obtain an expression for the transition probabilities between different spin levels of H sp we write the Hamiltonian of the spin-phonon system as H⫽H sp ⫹H ph ⫹H sp⫺ph ,

共3.6兲

where the eigenstates of the phonon operator H ph are the usual phonon states, characterized by phonon occupation numbers n j . The spin-phonon interaction H sp⫺ph induces transitions in the spin and phonon subsystems and may be described by a dynamic spin-Hamiltonian9,11,12

SPIN-LATTICE-RELAXATION-LIKE MODEL FOR . . .

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TABLE I. ‘‘Standard’’ values used in numerical calculations. B lp is measured in energy units cm⫺1 . Diameter d⫽9⫻10⫺9 m⫽90 Å Anisotropy constant K⫽7⫻104 Jm⫺3 Density ␳ ⫽5000 kgm⫺3 Lattice constant a⫽8.4⫻10⫺10 m⫽8.4 Å C⬅3(B lp ) 2 / ␲ ប 4 ␳ v 5 ⫽1.26⫻1069 (B lp /cm⫺1 ) 2 J⫺3 s⫺1 ␭ min ⫽2a⫽16.8 Å E phmax ⫽2 ␲ ប v /␭ min ⫽1.18⫻10⫺21 J

H sp⫺ph ⫽

B lp O lp 共 ␧ ␣ ␯ ⫹␧ ␣2 ␯ ⫹••• 兲 , 兺 l,p

␻

共3.7兲

where ␧ a v is an angle-averaged lattice strain operator given by (M ⫽crystal mass, ␻ j and kជ j ⫽frequency and wave vector of phonon mode j) ␧ ␣␯⫽

冊 兺冉 j

ប 2M ␻ j

兰 0 D g( ␻ )d ␻ , where g( ␻ ) is the Debye distribution function g( ␻ )⫽(3V/2␲ 2 v 3 ) ␻ 2 and ␻ D ⫽(6 ␲ 2 /V) 1/3v , and obtain 共density ␳ ⫽M /V) m⬘ ⫽ wm

3 共 B lp 兲 2 兩 具 m ⬘ 兩 O lp 兩 m 典 兩 2 2 ␲ ប 4␳ v 5

1/2

ជជ ជជ k j关 a ⫹ j exp共 ⫺ik r j 兲 ⫹a j exp共 ik r j 兲兴 . 共3.8兲

O lp are dynamical crystal-field operators of order l in the spin variables and B lp the corresponding coupling constants to the phonons.11,12 For the present we do not restrict the order of the O lp since the superparamagnetic spin-strain interaction may be described by higher order than l⫽1 8 or l ⫽2 10. The phonon creation and annihilation operators a ⫹ j and a j act on the combined spin-phonon states 兩 m, j 典 ⫽ 兩 m 典 兩 •••n j ••• 典 leading to phonon matrix elements and 具 •••n j ⫹1••• 兩 a ⫹j 兩 •••n j ••• 典 ⫽(n j ⫹1) 1/2 具 •••n j 1/2 ⫺1••• 兩 a j 兩 •••n j ••• 典 ⫽n j with the thermal average n j ⫽ 具 n j 典 ⫽ 兵 exp关h␻ j /(kBT)兴⫺1其⫺1 共Bose-Einstein factor兲. The m⬘ ⬅w( 兩 m, j 典 → 兩 m ⬘ , j ⬘ 典 ) of a transition transition rate w m 兩 m, j 典 → 兩 m ⬘ , j ⬘ 典 induced by H sp⫺ph can then be calculated by the golden rule m⬘ wm ⫽ 共 2 ␲ /ប 兲 兩 具 m ⬘ , j ⬘ 兩 H sp⫺ph 兩 m, j 典 兩 2 ␦ 共 E m ⬘ ⫺E m ⫾ប ␻ j 兲 , 共3.9兲

where the ␦ function accounts for energy conservation, the ‘‘⫺’’ sign is valid for E m ⬘ ⬎E m 共phonon annihilation兲 and the ‘‘⫹’’ sign for E m ⬎E m ⬘ 共phonon creation兲. In calculating m the matrix elements w m ⬘ , in principle, one has to invert the sign of the magnetic field,13 but also the sign of the spin numbers has to be inverted, so that H sp⫺ph remains unaltered. Introducing Eqs. 共3.8兲 in 共3.9兲, restricting ␧ a v to first order 共one-phonon process兲 and taking for simplicity only one term (l, p) in the electronic matrix element into account, results for phonon annihilation in m⬘ ⫽ wm

Volume V⫽ ␲ d 3 /6⫽3.82⫻10⫺25 m3 Anisotropy energy KV⫽AS 2 ⫽2.67⫻10⫺20 J Sound velocity v ⫽3000 ms⫺1 Spin S⫽3222 共five spins per a 3 ) C ⬘ ⫽C(AS 2 ) 3 ⫽2.4⫻1010(B lp /cm⫺1 ) 2 s⫺1 ␭ max ⫽d⫽90 Å E phmin ⫽2 ␲ ប v /␭ max ⫽2.21⫻10⫺22 J

␲ p 2 共 B 兲 兩 具 m ⬘ 兩 O lp 兩 m 典 兩 2 M l

兺j

k 2j ␦ 共 E m ⬘ ⫺E m ⫺ប ␻ j 兲 . ␻ j exp共 ប ␻ j /k B T 兲 ⫺1 共3.10兲

We introduce the Debye model in the long-wavelength limit 共acoustic branch with dispersion relation ␻ ⫽ v q, v ⫽weighted average of transverse and longitudinal sound velocities兲 by replacing the sum by an integration

共 E m ⬘ ⫺E m 兲 3

, exp关共 E m ⬘ ⫺E m 兲 / 共 k B T 兲兴 ⫺1 共3.11兲

where E m ⬘ ⬎E m for phonon annihilation and E m ⬎E m ⬘ for phonon creation. For the ratio of up and down transitions follows m m⬘ /w m ⬘ ⫽exp关⫺(Em⬘⫺Em)/(kBT)兴 in accordance with a wm Boltzmann population of the levels. It may be interesting to observe that in the framework of the Debye model the transition probability for small spinenergy differences tends to zero and not to infinity as one would take for granted intuitively. The reason of this is the cubic dependence on energy in the transition probabilities which stems from the depletion of low-energy phonons 关 g( ␻ )⬀ ␻ 2 兴 in the Debye model. For small isolated particles however the minimum one-phonon energy is of the order of 2 ␲ ប v /␭. This could be greater than the energy differences 共3.5兲 for low k or n values, so that no energy conserving spin-transition could occur. Nevertheless, for the following we presume the presence of sufficient low-energy phonons which stem, e.g., from the matrix in what the superparamagnetic particles are embedded. In order to extract later on concrete values for relaxation times from the theoretical expressions developed below we fix the numerical values related to the superparamagnetic particle listed in Table I. These values correspond to the values used in Ref. 14 for a ferrofluid 共powder兲, but are also typical for other superparamagnetic particles. IV. THREE-LEVEL SYSTEM

With the intention of delineating the basic ideas for calculating relaxation times we start with a simple three-level system where p S , p ⫺hS , and p ⫺S are the fractional populations of the corresponding levels. We have then, by detailed balance, the rate equations S p ⫺hS ⫺w ⫺hS pS , p˙ S ⫽w ⫺hS S

共4.1a兲

⫺S ⫺hS p ⫺hS ⫺w ⫺S p ⫺S , p˙ ⫺S ⫽w ⫺hS

共4.1b兲

⫺hS S ⫺S p˙ ⫺hS ⫽w ⫺S p ⫺S ⫹w ⫺hS p S ⫺ 共 w ⫺hS ⫹w ⫺hS 兲 p ⫺hS S 共4.1c兲

and the condition

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p S ⫹p ⫺hS ⫹p ⫺S ⫽1.

共4.2兲

With Eqs. 共4.2兲 and 共4.1b兲 one can eliminate p ⫺hS from the first equation and obtains a second-order linear differential equation for p S S ⫺S p¨ S ⫹p˙ S 兵 关 1⫹exp共 ⫺ ␣ ⫹ 兲兴 w ⫺hS ⫹ 关 1⫹exp共 ⫺ ␣ ⫺ 兲兴 w ⫺hS 其 S ⫺S ⫹p S w ⫺hS w ⫺hS 兵 关 1⫹exp共 ⫺ ␣ ⫹ 兲兴

⫻ 关 1⫹exp共 ⫺ ␣ ⫺ 兲兴 ⫺1 其 S ⫺S ⫽w ⫺hS w ⫺hS exp共 ␣ ⫺ 兲 ,

p˙ S ⫺ p˙ ⫺S ⫽⫺2J⫽⫺ 共 2/⌳ 兲共 p S ⫺⌫p ⫺S 兲

共4.10a兲

and p˙ S ⫺⌫p˙ ⫺S ⫽⫺J⫺⌫.J⫽ 关共 1⫹⌫ 兲 /2兴共 p˙ S ⫺p˙ ⫺S 兲 共4.10b兲 which results, up to a constant, in p S ⫺⌫p ⫺S ⫽ 关 (1⫹⌫)/ 2兴 (p S ⫺ p ⫺S ) and p˙ S ⫺ p˙ ⫺S ⫽(⫺2/⌳) 关 (1⫹⌫)/2兴 (p S ⫺p ⫺S ). Thus we obtain a relaxation rate

␶ ⫺1 ⫽ 共 1⫹⌫ 兲 /⌳.

共4.3兲

共4.11兲

Since p˙ ⫺hS ⫽0 we can regard p S and p ⫺S as belonging to a two-level system and obtain, in analogy with Eq. 共2.1兲 from Eqs. 共4.6兲 and 共4.9兲 p˙ S ⫽ p ⫺S / ␶ ⫺ ⫺ p S / ␶ ⫹ ⫽⫺J⫽⫺p S /⌳ ⫹⌫p ⫺S /⌳ and thus

where ␣ ⫾ ⫽E B⫾ /k B T. The solution is p S ⫽c 1S exp共 ⫺␭ 1 t 兲 ⫹c 2S exp共 ⫺␭2t 兲 ⫹exp共 ⫺ ␣ ⫺ 兲 ⫻ 兵 关 1⫹exp共 ⫺ ␣ ⫹ 兲兴关 1⫹exp共 ⫺ ␣ ⫺ 兲兴 ⫺1 其 ⫺1 , where the constants c 1S and c 2S depend on the initial conditions, and the two characterisitic frequencies are ⫺1 S ⫺S ␶ 1,2 ⫽␭ 1,2⫽ 共 A ⫹ ⫹A ⫺ 兲 /2⫾ 关 w ⫺hS w ⫺hS ⫹ 共 A ⫹ ⫺A ⫺ 兲 2 /4兴 1/2 共4.5兲 ⫾S . For p ⫺S one deduces an with A ⫾ ⫽ 关 1⫹exp(⫺␣⫾)兴w⫺hS identic homogeneous equation and thus the same character⫺1 . The rate Eqs. 共4.1兲 and consequently istic frequencies ␶ 1,2 ⫺1 ␶ 1,2 are invariant under a simultaneous change of (S,h) →(⫺S,⫺h). It is not possible from Eq. 共4.1兲 to define relaxation times like ␶ ⫹ and ␶ ⫺ and also not a single relaxation time unless one characteristic time ␶ 1,2 in Eqs. 共4.5兲 and 共4.4兲 is predominant. In general, the return of p S and p ⫺S to equilibrium after release of a transient is determined by both times ␶ 1 and ␶ 2 . Another possibility to define relaxation times, in the case of small perturbations, is to focus on a pair of levels, in our case especially 兩 S 典 and 兩 ⫺S 典 , under the assumption that the occupation of all other levels remains constant 共stationarity condition兲. In this case, in fact, we deal with a pseudo-twostate relaxation system. Then p˙ ⫺hS ⫽0 in Eq. 共4.1c兲 and from this we obtain10

J⬅⫺p˙ S S ⫽w ⫺hS p S ⫺w ⫺hS p ⫺hS S ⫺S ⫺hS p ⫺S ⫽w ⫺hS p ⫺hS ⫺w ⫺S

⫺1 ⫺1 ␶ ⫺ ⫽⌳/⌫ and ␶ ⫺1 ⫽ ␶ ⫹ ⫹␶⫺

␶ ⫹ ⫽⌳,

共4.4兲

⫽p˙ ⫺S

PRB 62

共4.12兲

⫺1

with ␶ in accordance with Eq. 共4.11兲. For the three-level system, suppressing the electronic matrix elements, the result is thus ⫺1 3 3 ␶⫾ ⫽CE B⫹ E B⫺

⫻

exp共 ⫺ ␣ ⫾ 兲 3 3 E B⫺ 关 1⫺exp共 ⫺ ␣ ⫹ 兲兴 ⫹E B⫹ 关 1⫺exp共 ⫺ ␣ ⫺ 兲兴

.

共4.13兲 For h⫽0

␶ ⫺1 ⫽

C ⬘ exp共 ⫺ ␣ 兲 , 2 1⫺exp共 ⫺ ␣ 兲

共4.14兲

where ␣ ⫽E B /(k B T)⫽AS 2 /(k B T) and C ⬘ ⫽C(AS 2 ) 3 共cf. Table I兲. An equivalent formula has been given in Ref. 14. In the limit of high temperature ( ␣ →0), ␶ ⫺1 from Eq. 共4.14兲 tends to infinity, which intuitively is physically suggestive. Assuming B lp ⫽10 cm⫺1 the pre-exponential factor C ⬘ /2 in Eq. 共4.14兲 is approximately 1012 s⫺1 which is within the correct order of magnitude. However, as we will see below, respecting the suppressed electronic matrix elements leads to completely unrealistically fast relaxation rates. We must therefore consider the multilevel system according to the ‘‘real spin’’ S. V. MULTILEVEL SYSTEM

共4.6兲

and can therefore express p S by p ⫺S , p S ⫽J⌳⫹⌫ p ⫺S ,

共4.7兲

S ⫺S ⫺1 /w ⫺hS ⌳⫽ 共 w ⫺hS 兲 ⫺1 ⫹ 共 w ⫺hS 兲共 w ⫺hS 兲 S S

共4.8兲

with

and S ⫺hS ⫺S /w ⫺hS /w ⫺hS ⌫⫽ 共 w ⫺hS 兲共 w ⫺S 兲 ⫽exp共 ␣ ⫹ 兲 exp共 ␣ ⫺ 兲 . S 共4.9兲

We have therefore J⫽( p S ⫺⌫ p ⫺S )/⌳ and the equations

In general, instead of three levels a multilevel system with 2S⫹1 levels is present 共6445 levels in our example兲. We calculate now the relaxation times ␶ ⫾ under the stationarity conditions p˙ i ⫽0,i⫽S,⫺S. For simplicity we only consider spin transitions with constant quantum number differences ⌬m 共variable energy differences兲, where ⌬m⫽k on the right-hand side in Fig. 1 and ⌬m⫽n on the left-hand side. The corresponding energy differences are given by Eq. 共3.5兲. In analogy to Eq. 共4.1兲 the master equations in the different regions of m in Fig. 1 read: at the right-hand side in Fig. 1, ⫺S p S⫺k ⫺w S⫺k pS p˙ S ⫽w S⫺k S

with stationarity condition p˙ m ⫽0⇒

共5.1a兲

PRB 62

SPIN-LATTICE-RELAXATION-LIKE MODEL FOR . . .

m⫺k m m m⫹k wm p m ⫺w m⫺k p m⫺k ⫽w m⫹k p m⫹k ⫺w m p m ⬅J k , 共5.1b兲

m⫽S⫺k,S⫺2k, . . . ,⫺hS⫹k;

共5.1c兲

at the top of barrier 0⫽p˙ ⫺hS ⇒

⫻

1⫺exp关 ⫺An 2 共 1⫹2i 兲 / 共 k B T 兲兴

. n 兩 ⫺hS⫺ni 典 兩 2 共 1⫹2i 兲 3 具 ⫺hS⫺n 共 i⫹1 兲 兩 S ⫺ 共5.7兲

With C ⬘ ⬅C(AS 2 ) 3 共cf. Table I兲 this leads to a general expression for ␶ ⫾ ,

⫺hS ⫺hS⫹k p ⫺hS⫹k ⫺w ⫺hS p ⫺hS w ⫺hS⫹k

␶ ⫾⫽

⫺hS⫺n ⫺hS ⫽w ⫺hS p ⫺hS ⫺w ⫺hS⫺n p ⫺hS⫺n ⬅J ⫺hS ;

exp共 ⫺ ␣ ⫾ 兲 C⬘

共5.2兲 at the left-hand side in Fig. 1, ⫻

共5.3a兲

with stationarity condition p˙ m ⫽0⇒

⫹

m⫺n m m m⫹n p m ⫺w m⫺n p m⫺n ⫽w m⫹n p m⫹n ⫺w m p m ⬅J n m⫽⫺hS wm

⫺n, . . . ,⫺S⫹n.

p S ⫽J k ⌳ k ⫹J n ⌳ n ⫹⌫ p ⫺S

共5.4兲

with S S⫺2k ⫺1 S /w S⫺k /w S⫺k ⌳ k ⫽ 共 w S⫺k 兲 ⫺1 ⫹ 共 w S⫺k 兲共 w S⫺k 兲 ⫹ 共 w S⫺k 兲 S S S S⫺k S⫺2k S⫺3k ⫺1 S ⫻ 共 w S⫺2k /w S⫺k /w S⫺k 兲共 w S⫺2k 兲 ⫹•••⫹ 共 w S⫺k 兲 S

冋 冉冊

冉冊 S n

2

共 1⫹2i 兲

2

i2

册

冋 冉冊 册 冋 冉冊 册

6 S(1⫺h)/n⫺1

兺

i⫽0

exp ⫺ ␣ n S

n S

2

i2

2

共 1⫹2i 兲

n 兩 ⫺hS⫺ni 典 兩 2 共 1⫹2i 兲 3 具 ⫺hS⫺n 共 i⫹1 兲 兩 S ⫺

冎

.

k 兩 具 ⫺hS⫹ki 兩 S ⫺ 兩 ⫺hS⫹k 共 i⫹1 兲 典 兩 2

S ⫺hS⫹k ⫺hS ⌳ n ⫽ 共 w S⫺k /w S⫺k /w ⫺hS⫹k 兲 ••• 共 w ⫺hS 兲 S

⫽

⫺hS⫺n ⫺1 ⫺hS ⫺hS⫺n ⫺hS⫺2n ⫺1 ⫻ 关共 w ⫺hS /w ⫺hS 兲 ⫹ 共 w ⫺hS⫺n 兲共 w ⫺hS⫺n 兲

关 S 共 1⫹h 兲 ⫺ki 兴 ! 关 S 共 1⫺h 兲 ⫹k 共 i⫹1 兲兴 ! 关 S 共 1⫹h 兲 ⫺k 共 i⫹1 兲兴 ! 关 S 共 1⫺h 兲 ⫹ki 兴 !

共5.9a兲

⫺hS ⫺hS⫺n ⫺S⫹2n ⫺S⫹n ⫹•••⫹ 共 w ⫺hS⫺n /w ⫺hS /w ⫺S⫹2n 兲 ••• 共 w ⫺S⫹n 兲 ⫺S ⫻共 w ⫺S⫹n 兲 ⫺1 兴

k S

k S

共5.8兲 The electronic matrix elements contained in Eq. 共5.8兲 depend strongly on the involved quantum numbers. Because of S ⫺ 兩 S,m 典 ⫽ 关 S(S⫹1)⫺m(m⫺1) 兴 1/2兩 S,m⫺1 典 and S(S⫹1) ⫺m(m⫺1)⫽(S⫹m)(S⫺m⫹1) we can write for the matrix elements

S⫺k S⫺2k ⫺hS⫹2k ⫺hS⫹k hS ⫻ 共 w S⫺2k /w S⫺k /w ⫺hS⫹2k 兲 ••• 共 w ⫺hS⫹k 兲共 w ⫺hS⫹k 兲 ⫺1 ,

and n 兩 具 ⫺hS⫺n 共 i⫹1 兲 兩 S ⫺ 兩 ⫺hS⫺ni 典 兩 2

共5.5兲

and

⫽

S ⫺hS⫹k ⫺hS ⫺hS ⫺hS⫺n /w S⫺k /w ⫺hS⫹k /w ⫺hS ⌫⫽ 共 w S⫺k 兲 ••• 共 w ⫺hS 兲共 w ⫺hS⫺n 兲 S

关 S 共 1⫺h 兲 ⫺ni 兴 ! 关 S 共 1⫹h 兲 ⫹n 共 i⫹1 兲兴 ! . 关 S 共 1⫺h 兲 ⫺n 共 i⫹1 兲兴 ! 关 S 共 1⫹h 兲 ⫹ni 兴 !

共5.9b兲

⫺S⫹n ⫺S ⫻共 w ⫺S /w ⫺S⫹n 兲

VI. RESULTS AND DISCUSSION

⫽exp共 ␣ ⫹ 兲 exp共 ⫺ ␣ ⫺ 兲 .

共5.6兲

In the calculation of ⌳ k and ⌳ n we have now to include the electronic matrix elements. Because of Eqs. 共5.1兲–共5.3兲, J k ⫽J n and therefore p S - ⌫ p ⫺S ⫽J k (⌳ k ⫹⌳ n ). With Eq. 共5.5兲 and in complete analogy to Eqs. 共4.6兲–共4.12兲 we obtain exp关 ␣ ⫹ 兴

S(1⫹h)/k⫺1

C 共 Ak 2 兲 3

i⫽0

⫹

兺

exp ⫺ ␣

i⫽0

1⫺exp ⫺ ␣ ⫻

冋 冉冊 册

6 S(1⫹h)/k⫺1

k 兩 ⫺hS⫹ 共 k 共 i⫹1 兲 典 兩 2 共 1⫹2i 兲 3 兩 具 ⫺hS⫹ki 兩 S ⫺

共5.3b兲

From these recursion formulas one obtains

⫻

再冉 冊 S k

1⫺exp ⫺ ␣

⫺S ⫺S⫹n p ⫺S⫹n ⫺w ⫺S p ⫺S p˙ ⫺S ⫽w ⫺S⫹n

⌳ k ⫹⌳ n ⫽

3377

兺

exp关 ⫺Ak 2 共 1⫹i 兲 2 / 共 k B T 兲兴

exp关 Ak 2 共 1⫹2i 兲 / 共 k B T 兲兴 ⫺1 k 兩 ⫺hS⫹k 共 i⫹1 兲 典 兩 2 共 1⫹2i 兲 3 兩 具 ⫺hS⫹ki 兩 S ⫺

exp共 ␣ ⫹ 兲

S(1⫺h)/n⫺1

C 共 An 2 兲 3

i⫽0

兺

exp关 ⫺An 2 i 2 / 共 k B T 兲兴

In order to analize the general expression 共5.8兲 we treat first the case of zero external field. For h⫽0, assuming k ⫽n and introducing Eq. 共5.9兲 in Eq. 共5.8兲 we find

␶⫽

冉冊

2 ex p 共 ␣ 兲 S k C⬘ S/k⫺1

⫻

兺 i⫽0

6

冋 冉冊 册

k 2 2 i S 兵 共 S⫺ki 兲共 S⫺ki⫺1 兲 ••• 关 S⫺ki⫺ 共 k⫺1 兲兴 其 共 1⫹2i 兲 ⫺3 exp ⫺ ␣

冉 冉冊

冊

k 2 共 1⫹2i 兲 S ⫻ . 共6.1兲 关共 S⫹ki⫹k 兲共 S⫹ki⫹k⫺1 兲 ••• 共 S⫹ki⫹1 兲兴 1⫺exp ⫺ ␣

We estimate this for several k values. For high k or n values, e.g., for n⫽k⫽S 共three-level model above兲 the denomina-

3378

H.-D. PFANNES et al.

tors are of the order (2S)! and thus ␶ tends to zero. The case k⫽n is not fundamentally different from k⫽n. We start therefore considering k⫽n⫽1 and estimate ␶ at

␶⬎

2 exp共 ␣ 兲 C⬘

S6

␶⫽

1⫺exp共 ⫺ ␣ /S 2 兲 2 ␣ 2 ⬇ S exp共 ␣ 兲 . S 共 S⫹1 兲 C⬘

冉冊

2 exp共 ␣ 兲 S 2 C⬘

In the usual temperature range T⫽1⫺1000 K, and with KV from Table I, ␣ varies from approximately 2000 to 2 and ␣ /S 2 from 2⫻10⫺4 to 2⫻10⫺7 so that we can estimate a lower limit for ␶ , using the values of Table I, as ␶ ⭓8.66 ⫻10⫺4 s•(B 11 /cm⫺1 ) ⫺2 ␣ •exp(␣). In a wide range of B 11 this is far too long compared to the experimental values. For k⫽2 we obtain from Eq. 共6.1兲

共6.2兲

6 S/2⫺1

兺

i⫽0

exp共 ⫺4 ␣ i 2 /S 2 兲

1⫺exp关 ⫺4 ␣ 共 1⫹2i 兲 /S 2 兴 共 1⫹2i 兲 3 关共 S⫺2i 兲共 S⫺2i⫺1 兲共 S⫹2i⫹2 兲共 S⫹2i⫹1 兲兴

As an upper limit of ␶ we estimate

␶ ⬍ 共 1.6/8C ⬘ 兲 ␣ exp共 ␣ 兲

共6.4a兲

␶ ⬇ 关 8.33⫻10⫺12 s/ 共 B 22 /cm ⫺1 兲 2 兴 • ␣ exp共 ␣ 兲 . 共6.4b兲 If we concisely write ␶ ⫺1 in the form 共6.5兲 as it is often done we obtain in the assuming a weakly varying prefactor ␶ ⫺1 0 temperature range 1⫺1000 K

␶

⫽ 共 6⫻10

兲 s

⫺1

⬇1.5⫻10⫺19 s共 B 33 /cm⫺1 兲 ⫺2 ␣ exp共 ␣ 兲 .

关 B 22 /cm ⫺1 兴 2 )•exp共 ⫺ ␣ 兲

or, e.g., for T⫽300 K ( ␣ ⫽6.45), where the transition of slow to fast relaxation in the Mo¨ssbauer spectra of Ref. 14 occurs, ␶ ⫺1 ⬇2.94⫻107 s⫺1 (B 22 /cm⫺1 ) 2 , which for B 22 ⭐1 cm⫺1 would correspond to a completely static Mo¨ssbauer pattern. We have no detailed information on the value of B 22 . The value k⫽2 , i.e., a second-order dynamical spin Hamiltonian, describes normal spin-orbit coupling treated in second-order perturbation theory for which a value of the coupling parameter B 22 in the range of 0.2 cm⫺1 for S-state ions, e.g., Fe3⫹ , may be adequate.12 This results in ␶ ⫺1 0 ⬇2.4⫻106•••9 s⫺1 in contrast to the experimental range of

␴ 共 ⫾h 兲 ⫽

兺

i⫽0

␶ ⫾ ⫽C ⬘ ⫺1 exp共 ␣ ⫾ 兲共 S/2兲 6 关 ␴ 共 ⫹h 兲 ⫹ ␴ 共 ⫺h 兲兴

2

冋 冉冊

册

2 2 共 1⫹2i 兲 S . 共 1⫹2i 兲 3 关 S 共 1⫾h 兲 ⫺2i 兴关 S 共 1⫾h 兲 ⫺2i⫺1 兴 关 S 共 1⫿h 兲 ⫹2i⫹2 兴关 S 共 1⫿h 兲 ⫹2i⫹1 兴 2 S

In complete analogy to the case h⫽0 and in good approximation up to h⬇0.95 we estimate ␶ ⫾ as

␶ ⫾ ⬇ 共 0.2C ⬘ ⫺1 兲 ␣ 共 1⫺h 2 兲 ⫺2 exp共 ␣ ⫾ 兲 .

共6.9兲

In terms of a relaxation rate, with C ⬘ from Table I and B 22 ⫽13 cm⫺1 this results in our final estimation

共6.7兲

with

冋 冉冊 册

exp ⫺ ␣

共6.6兲

It seems to be reasonable that the value of the coupling constants decreases strongly with the order of the dynamical spin Hamiltonian involved. If we take, e.g., B 33 ⫽10⫺4 •B 22 with B 22 ⫽13 cm⫺1 we obtain for the relaxation rate ␶ ⫺1 0 ⬇1.8⫻1010•••13 s⫺1 , i.e., the correct order of magnitude. However, for k⬎3 the values of B lp needed for obtaining reasonable ␶ 0 values are unrealistically small. Considering the different B lp values above we adopt k⫽2 as a probable value for the order of the dynamical spin Hamiltonian. An external field is easily included in the calculations. For k⫽n⫽2 we obtain from Eqs. 共5.8兲 and 共5.9兲 the final result,

共6.5兲

(S/2)(1⫾h)⫺1

共6.3兲

␶ ⬇ 共 1/27C ⬘ S 2 兲 ␣ exp共 ␣ 兲

⬅ ␶ ⫺1 0 exp共 ⫺ ␣ 兲 7•••10

.

(1010•••13) s⫺1 . However, for Fe 2⫹ the coupling constant B 22 could be considerably higher, e.g., for B 22 ⬇13 cm⫺1 , which seems to be in a reasonable range, we would obtain the cor⫺1 ( ␣ ⫽6.45) rect order of magnitude of ␶ ⫺1 0 . In this case ␶ 9 ⫺1 ⫽5⫻10 s which corresponds to an intermediate relaxation Mo¨ssbauer pattern as experimentally found in Ref. 14. For comparison we also consider the case of k⫽3 and obtain from Eq. 共6.3兲 the approximation

and thus, with the values of Table I,

⫺1

PRB 62

i2

1⫺exp ⫺ ␣

共6.8兲

⫺1 ␶⫾ ⬇2⫻1013 s⫺1 ␣ ⫺1 共 1⫺h 2 兲 2 exp共 ⫺ ␣ ⫾ 兲 . 共6.10兲

Comparing Eqs. 共6.4兲–共6.10兲 with the formulas 共2.3兲, 共2.6兲, 共2.11兲, and 共2.12兲 given by Ne´el, Brown, and others, we state that the most prominent feature in ␶ ⫺1 , namely the

PRB 62

SPIN-LATTICE-RELAXATION-LIKE MODEL FOR . . .

presence of an Arrhenius-like factor exp(⫺␣) is also found in our formula. Moreover, since in Eqs. 共6.4兲 and 共6.9兲 ␶ is proportional to ␣ , the asymptotic behavior at high temperature ( ␶ →0 for ␣ →0) is the same as in Eqs. 共2.5兲, 共2.7兲, and 共2.12兲, but different from Eqs. 共2.3兲 and 共2.6兲. Though in principle we can define a single ␶ only in the lowtemperature limit, the formulas that we deduced are meaningful also at higher temperatures. The field dependence in our formulas is similar to that in ⫺1 is proportional to (1⫺h 2 )(1 Brown’s expression. Our ␶ ⫾ ⫺1 ⫹h)(1⫺h) whereas ␶ ⫾Br from Eq. 共2.6兲 is proportional to ⫺1 (1⫺h 2 )(1⫾h). For ␶ ⫺ the field dependent prefactor terms ⫺1 differ by a factor 2, at most 共for h→1), and for ␶ ⫹ our expression tends slightly faster to zero as Brown’s. However ⫺1 ␶⫾ (h⫽1) calculated without approximations directly from Eq. 共5.8兲 remains finite, which seems to be more correct physically. The phonon energies (h⫽0) involved in the spintransitions range from E ⫾S⫿2 ⫺E ⫾S ⫽3.3⫻10⫺23J 共step at minimum of barrier potential兲 to E 0 ⫺E ⫾2 ⫽1.03⫻10⫺26J 共step near to the top of the barrier兲. The minimum phonon energy is E phmin ⫽2.21⫻10⫺22J 共cf. Table I兲. Thus for 共direct兲 transitions near to the top of the barrier phonons with an energy of approximately 5⫻10⫺5 •E phmin are necessary. We postulated the presence of such low energy phonons. The frequency of these phonons would be of the order of 107 ⫺5⫻1010 s⫺1 . We also believe that there is no correlation problem present as it was raised in Ref. 6. However, within the scope of the above developed model no superparamagnetic relaxation would occur in nanosized magnetic particles that are ideally isolated from each other or from a heat bath. ⫺1 from Eqs. 共6.8兲 or 共6.10兲, one is Once disposing on ␶ ⫾ able, at least in the low-temperature regime, to calculate the corresponding superparamagnetic Mo¨ssbauer relaxation pattern. The only ‘‘free’’ parameter is the coupling constant B 22 . This can be done in the simplest case by using the formalisms described in Refs. 9 and 15–17 with an effective spin S⫽ 12 . Depending on the absolute values of ␶ ⫹ and ␶ ⫺ , both slow and fast relaxation can be present in the spectrum at the same temperature.18 When the characteristic time of the Mo¨ssbauer method is longer than ␶ ⫾ , a static 共six-line兲 spectrum is observed whose splitting is reduced by the factor ( ␶ ⫹ ⫺ ␶ ⫺ )/( ␶ ⫹ ⫹ ␶ ⫺ ). Since our ␶ ⫾ from Eq. 共6.10兲 depends stronger on h than Brown’s expression 共2.6兲 and is proportional to ␣ this effect should manifest itself more distinctly than hitherto supposed. For somewhat higher temperatures the fluctuations of S around the anisotropy directions can be approximately taken into account by diminishing the magnetic splitting as a function of the temperature according to the thermal average value 共collective excitations, Ref. 1兲. The correct way, however, to simulate a Mo¨ssbauer spectrum for arbitrary temperature and hyperfine interaction would consist in using the ‘‘real’’ spin S in a superoperator formalism like in Refs. 9 or 17, where the relaxation supermatrix would contain the transition probabilities 共3.11兲. Unfortunately, this straightforward procedure is not feasible since the method involves the inversion of a non-Hermitian complex matrix of huge order, vis. 3.3⫻108 in our case (S⫽3222, isotope 57Fe). However, for most superparamagnetic materials the magnetic hyperfine

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interaction is larger than the electric quadrupole interaction and then in calculating a spectrum the simplified expression 共2.9兲 can be used. With this the inversion of a matrix whose order is only 2S⫹1 is required and this is feasible on a workstation. In real samples most of the assumptions made in the beginning are not fulfilled, but in some cases the simulation of spectra can be easily adapted to the real conditions by summing up discrete spectra, e.g., to obtain powder averages or to introduce size dispersion. However, the presence of various easy directions is more difficult to take into account. Other difficulties remain also, like the influence of the surface and surface-near regions in the particles which may result in different anisotropy constants, hyperfine fields, etc. Particle interaction has been discussed by Mo” rup and Tronc19 in terms of magnetic dipole interaction between the particles. At very low temperature, in principle, there exists the possibility that the spin S fluctuates between states on opposite sides of the barrier, without surmounting it, by ‘‘spin tunneling,’’ e.g., by phonon assisted ‘‘macroscopic quantum tunneling’’ 共MQT兲.20,21 It seems, however, that MQT is only present in very small (S⬇10⫺100) ideal clusters, such as crystals consisting of magnetic molecules.22,23

VII. CONCLUSION

The thermally activated superparamagnetic relaxation of the magnetization of small noninteracting identical particles under the influence of an external magnetic field can be described by a spin-phonon-interaction-like model in which the total spin S of the monodomain particle interacts with strain fields of the crystal. Transition probabilities between the S z levels are calculated on the basis of a dynamical spin Hamiltonian and the Debye model. For low temperatures a twostate relaxation system is considered, where the relaxation occurs between the states 兩 S 典 and 兩 ⫺S 典 and relaxation rates ⫺1 ␶⫾ are introduced and calculated for transitions with constant ⌬S z . It is found that ⌬S z ⫽2 and a spin-phonon coupling constant of 13 cm⫺1 reproduces the experimental val⫺1 . The so deduced expression exhibits the same ues of ␶ ⫾ Arrhenius-like factor and a similar, but somewhat steeper dependence on the external field, as the classical formulas by Ne´el and Brown, derived on the ground of completely different models. The temperature dependence of the preexponential factor of our expression is meaningful also for high temperatures. It is possible, on the basis of the spintransition probabilities introduced, to calculate Mo¨ssbauer spectra valid at arbitrary temperatures.

ACKNOWLEDGMENTS

The support of FAPEMIG, CAPES, and CNPq is gratefully acknowledged. One of us, H.D.P., is grateful to Professor G. Langouche and Dr. J. Dekoster from the IKS, Katholieke Universiteit Leuven, and Professor W. Keune from the Gerhard Mercartor Universita¨t Duisburg for the hospitality during a research stay at these institutions.

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H.-D. PFANNES et al.

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