The fundamentals of precessing ball solitons (PBS), arising as a result of the energy fluctuations during spin-flop phase transition induced by a magnetic field in antiferromagnets with uniaxial anisotropy, are presented. The PBS conditions exist within a wide range of amplitudes and energies, including negative energies relative to an initial condition. For each value of the magnetic field, there exists a precession frequency for which a curve of PBS energy passes through a zero value (in bifurcation point), and hence, in the vicinity of this point the PBS originate with the highest probability. The characteristics of PBS, including the time dependences of configuration, energy, and precession frequency, are considered. As a result of dissipation, the PBS transform into the macroscopic domains of a new phase.

Magnetic solitons with spherical symmetry, which can arise in crystals with magnetic ordering during the phase transitions induced by a magnetic field, were considered in some papers [

In the present article, it is shown that dissipation of energy is accompanied not only by the change of configuration of solitons but also by the change of the precession frequency. Taking into account this time dependency, we can carry out a more comprehensive analysis of the quantities of PBS and to consider the whole process of transformation of PBS into macroscopic domains of a new phase.

In the first chapter, we represent a more correct analysis of equations and expressions for PBS than in [

To analyze magnetic solitons in an antiferromagnet with uniaxial anisotropy, we used the following expression for the thermodynamic potential:

The equations of motion for the

The solutions of (

From (

First of all, take notice that inserting (

To analyze the approximate soliton solutions of (

Taking into consideration that

Using the (

Let’s confine ourselves to low temperature approximation, that is, suppose that

This equation should be supplemented by the following expression obtained from (

Only in the case of immovable PBS, that is, at

Transforming (

Using (

As in [

PBS configurations, which are particular solutions of (_{2}O_{3} have been used:

Frequency dependencies of the energy, amplitude, and effective radius if

The same frequency dependencies as in Figure

Probability of the spontaneous creation of PBS related to the fluctuations of energy at nonzero temperatures is proportional to probability of such fluctuations. For the metastable state of our system, we use the following expression for the probability of PBS creation with the energy

Here,

Energy dependence for probability of the spontaneous PBS creation near the point of bifurcation.

Further evolution of PBS is described by (

In this part, we consider the change of PBS related to the dissipation only, that is, at

In conformity with (

In Figures 4 to 7, several examples are adduced at

If

The frequency dependencies of energy, amplitude, and

The same frequency dependencies as in Figure

The same dependencies as in Figure

The same dependencies as in Figure

If

Equations (

The time dependency of the frequency if

Figures

Time dependency of the frequency if

In Figure

The time dependency of the frequency if

Time dependency of the frequency if

Sequence of change of the form and sizes for PBS at

A sequence of configurations of PBS if

Asymptotical disappearance of PBS at

Fading of PBS if

Now, we consider the influence of movement on soliton form. Using (

Let us present the

If we designate the value

In turn, as a result of the velocity reduction, PBS asymptotically becomes more spherical in the form (more precisely, owing to anisotropy, ellipsoid of rotation).

To estimate approximately the influence of demagnetizing field in the case of PBS, let’s use the formula that concerns homogeneity magnetize sphere only:

Corresponding to (

If to us the expression (

We can use this expression to estimate the value of demagnetizing field. With constants that were used in our examples, we receive

It is possible to write down a more correct equation for PBS, in comparison to (_{2}O_{3} [

(1) It is shown that dissipation of energy for precessing ball solitons is accompanied by the change of the precession frequency and by the deceleration of spatial movement.

(2) Kinetics of the PBS is defined by the sign of initial precession frequency and the amplitude of the originated PBS. PBS transforms into the domains of the new phase if the initial frequency of precession is negative (

(3) If

(4) At spatial movement of PBS, its form is deformed, but it does not change the size and the amplitude. The dissipation results not only in change of precession frequency, but also in reduction of the velocity of movement. As a result of the velocity reduction, the shape of PBS is approaching to spherical.

No. | ||||
---|---|---|---|---|

1 | 0 | −0.0025 | 2.7892 | 635 |

2 | 7.6896 | −0.01 | 0.8513 | 765 |

3 | 8.0899 | −0.0145 | −3.6714 | 885 |

4 | 8.2097 | −0.02 | −20.08 | 1080 |

5 | 8.2236 | −0.0233 | −48.8 | 1290 |

6 | 8.2243 | −0.027 | −130.7 | 1610 |

7 | 8.2263 | −0.03 | −321.23 | 2000 |

8 | −0.032 | −791.8 | 2580 |

The author is grateful to Professor A. Zvezdin for very useful discussions about the solitonic problem.

_{2}O

_{3}