[16] Below, as an example, we address a uniaxial homogeneously magnetized grain having the shape of an ellipsoid of revolution and neglect all types of anisotropy except the natural crystalline and form anisotropies.
[17] In this case, as was shown in [Afremov and Belokon, 1976, 1977], if the dimensionless constants of crystalline and form anisotropy are positive ( k_{A} > 0 and k_{N} > 0, respectively), the magnetic moment of a grain is oriented along the effective axis of anisotropy, making the angle
(7) |
with the axis of the crystalline anisotropy, and its orientation in the equilibrium state can be either parallel or antiparallel to the chosen direction. The critical field H_{0} of the transition of one equilibrium state into another is determined by the effective constant of anisotropy K:
(8) |
[18] This result is remarkable because it indicates the presence of a spectrum of critical fields in a system of particles differing in k_{N} and a. Moreover, of the entire set of particles with fixed angle a, particles meeting the condition k_{N} = -k_{A} cos 2a possess the weakest critical field H_{0} = k_{A} I_{s} sin 2a.
[19] Analysis of equilibrium magnetic moment states in grains with k_{A} 0, k_{N} 0 and k_{A} 0, k_{N} 0 [Afremov and Belokon, 1979] shows that equilibrium orientations of the magnetic moment are determined (??) in these cases in the same way as at k_{A} 0, k_{N} 0 whereas, in a magnetic field directed along such an effective easy axis, an irreversible change in the magnetic moment takes place upon reaching the critical field
(9) |
[20] With k_{A} 0, k_{N} 0, the effective easy axis coincides with the intersection of "easy" planes specified by the anisotropy of form and the natural crystalline anisotropy. The orientation of the grain magnetic moment changes irreversibly upon reaching the critical field [Afremov and Belokon, 1979]
(10) |
The study of nonspherical particles under uniaxial compression [Afremov and Belokon, 1980a, 1980b] showed that the magnetic moment of a grain is oriented along an effective axis whose position and the effective constant of anisotropy are determined by anisotropy of the crystalline, form and stress types. If the external field is applied along the effective axis and makes an angle b with the axis of the crystalline anisotropy, the position of the effective axis and the critical field of the grain magnetic moment remagnetization depend on the above types of anisotropy as follows:
(11) |
(12) |
where l_{i} are magnetostriction constants of a uniaxial crystal and s denotes uniaxial stresses. If the stresses are perpendicular to the plane ( k_{A}, k_{N} ), we have
(13) |
(14) |
As in the case of two types of anisotropy, the critical field can behave nonmonotonically with variations in the applied stresses. With the stresses oriented parallel to the plane ( k_{A}, k_{N} ), the critical field reaches the minimum value H_{0 min} = k_{A}I_{s} sin(2a - y_{0}) at
[21] Summarizing, we can note the following.
[22] (1) The grain elongation and uniaxial stresses can change significantly the magnetic state of a particle.
[23] (2) The orientation of the effective easy axis and the critical field of an irreversible change in the magnetic moment of a grain are basically dependent on the mutual orientations of the uniaxial stresses and the axes of the crystalline and form anisotropies, as well as on the signs of s, k_{A} and k_{N}.
[24] (3) The application of any external force in the absence of a magnetic field can lead to the destruction of remanent magnetization.
Figure 1 |
[26] In our analysis, we neglect the magnetoelastic interphase interaction, which can be more or less valid only in the case of a strongly disordered distribution of magnetic atoms in the boundary layer.
[27] first ( ) state: the magnetic moments of both phases are parallel and directed along the 0z axis;
[28] second ( ) state: the magnetizations of the phases are antiparallel, with the magnetic moment of the first phase m_{1} being directed along the 0z axis;
[29] third ( ) state: the magnetic moments of the phases are parallel and opposite in direction to the 0z axis; and
[30] fourth ( ) state: the magnetic moments of the phases are antiparallel, with the first phase being magnetized in the direction opposite to the 0z axis.
[31] If the magnetostatic interaction between the phases prevails over the exchange interaction, N^{}_{21} = N_{21}-A^{} > 0 ( N_{ik} are the demagnetizing coefficients, A^{} = 2A_{in}/da I_{s1}I_{s2}, A_{in} is the exchange constant, and d is the width of the interphase transition zone having the same order of magnitude as the lattice parameter), then the first and third states are metastable because the free energy of the grain F = N_{21}I_{s1}I_{s2} in these states is higher than the free energy in the second and fourth states F = -N_{21}I_{s1}I_{s2}. The second and fourth states are metastable if N^{}_{21} < 0.
[32] Analysis of the grain free energy shows that the third-to-second state transition occurs in the field [Afremov and Panov, 1996a, 1996b]
(15) |
Similar calculations yield the following critical fields:
(16) |
for the third-to-fourth state transition,
(17) |
for the fourth-to-first state transition,
(18) |
for the second-to-first state transition,
(19) |
for the third-to-first state transition, and
(20) |
for the second-to-fourth state transition.
[34] Because of the small volume of particles, one may expect that, due to thermal
fluctuations of magnetic moments of phases, transitions from one equilibrium
state into another can be realized in a field
[35] The reorientation of the magnetic moment of a phase is determined by the height of the potential barrier E_{ik} separating the i th and k th states. The frequency of the i-k state transition is expressed as
(21) |
Here f_{0} 10^{7}-10^{10} s ^{-1} is the characteristic frequency of "attempts" to overcome the potential barrier, k_{B} is the Boltzmann constant, and E_{ik} = F_{ik max} - F_{ik min} where F_{ik min} is the free energy of the i th equilibrium state characterizing the particle before the transition and F_{ik max} is the maximum free energy separating the i th and k th states.
[36] If the transition probability from one state to another is known, four equations of "motion" can be constructed for the vector of states:
(22) |
Considering the normalization condition N_{1}+N_{2}+N_{3}+N_{4} = 1, these equations can be reduced to the form
(23) |
where
(24) |
The solution of (23) can be conveniently represented with the use of a matrix exponential:
(25) |
With the known vector of the initial state
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