^{1}

The excitation spectrum of the Néel ensemble of antiferromagnetic nanoparticles with uncompensated magnetic moment is deduced in the two-sublattice approximation following the exact solution of equations of motion for magnetizations of sublattices. This excitation spectrum represents four excitation branches corresponding to the normal modes of self-consistent regular precession of magnetizations of sublattices and the continuous spectrum of nutations of magnetizations accompanying these normal modes. Nontrivial shape of the excitation spectrum as a function of the value of uncompensated magnetic moment corresponds completely to the quantum-mechanical calculations earlier performed. This approach allows one to describe also Mössbauer absorption spectra of slowly relaxing antiferromagnetic and ferrimagnetic nanoparticles and, in particular, to give a phenomenological interpretation of macroscopic quantum effects observed earlier in experimental absorption spectra and described within the quantum-mechanical representation.

A wide application of materials containing nanosized antiferromagnetic particles in different branches of nanotechnology is primarily due to a number of specific structural, magnetic, and thermodynamic properties of these materials found within long-term fundamental studies. However, these real materials are still characterized by different experimental techniques on the basis of phenomenological Néel approach describing a superposition of antiferromagnetism and superparamagnetism of uncompensated magnetic moments on two magnetic sublattices [

Meanwhile, a quantum-mechanical model for describing thermodynamic properties of an ensemble of ideal (compensated) antiferromagnetic nanoparticles is recently developed [

The quantum-mechanical model [

However, the normal modes of the uniform precession are only partial solutions of equations of motion of uniform magnetizations of sublattices, while the general solution of these equations should contain nutations at the background of the uniform precession in analogy with problems of a sphere pendulum and a heavy gyroscope [

The continuous models described in [

In analogy with [

In accordance with the classical theory of the antiferromagnetic resonance [

It is convenient to look for axially symmetric solutions of equations of motion (

The solutions of (

Normal modes of the self-consistent precession of the sublattice magnetization vectors

For ideal antiferromagnetic particles (

(a) Correlation between projections of the sublattice magnetization vectors

The normal modes correspond to four excitation branches in the energy spectrum of ideal antiferromagnetic particles (Figure

Remaining within the generally accepted assumption on the smallness of the anisotropy energy with respect to the exchange energy

Since the middle of the 20th century, modes 3 and 4 have fallen beyond the focus of interest of researchers primarily because of the approximation of small deviations of the vectors

In the presence of uncompensated magnetic moment in antiferromagnetic particles (

At the same time, the excitation branches in the energy spectrum that correspond to normal precession modes change essentially their form already at very low values of the uncompensated moment (Figure _{2} while the shape of branch 1 acquires that of a three-well potential with absolute energy minima for the mutually opposite orientation of vectors_{1} and_{2} along each of two directions along the easy axis and with the local energy minimum that corresponds to the precession of these vectors in the equatorial plane. Branches 2 and 3 for _{1} with a smaller value of magnetic moment varies for these branches over the entire range of possible values of polar angle _{2} is reoriented in the limited range of _{1} and_{2} along each of two directions along the easy axis and one maximum for the opposite orientation of both the vectors_{1} and_{2} (Figure

Note that for

Modes of the uniform precession of magnetizations of sublattices described in the previous sections are partial solutions of the combined equations of motion (

Equation (

Dissipationless trajectories of motion of sublattice magnetizations

If condition (

Ranges of the allowed projections of the total magnetic moment

Let us consider now the next energy range, where three normal precession modes of the sublattice magnetizations coexist (see the uppermost panels of Figures

Finally, in the third energy range

In the presence of uncompensated magnetic moment in antiferromagnetic particles (

First of all, at

At

Let us consider the next energy range, where also three normal precession modes of the sublattice magnetizations coexist (see Figures

Finally, in the third energy range _{2} extend in both the potential energy wells for all possible

A detailed analysis of the functional dependence of the character of nutations on

Now let us consider the effect of the nontrivial excitation spectrum of the Néel ensemble of antiferromagnetic nanoparticles with uncompensated magnetic moment (Figures

In this case, the equilibrium state of the ensemble of particles at a given temperature

In our case, calculations of the Mössbauer spectra of the ensemble of chaotically oriented antiferromagnetic particles can be performed in accordance with the results obtained in [

To calculate the absorption spectrum of the Néel ensemble of antiferromagnetic particles, one must also perform averaging over a random quantity of the uncompensated magnetic moment, for example, according to the Gaussian distribution of parameter

Mössbauer absorption spectra of ^{57}Fe nuclei in the ensemble of slowly relaxing ideal (

These calculations confirm also the fundamental conclusion following the quantum model that the uncompensated spin (magnetic moment) in the limit of the slow magnetization relaxation does not alter the qualitative character of the shape evolution for the spectra of ideal antiferromagnetic particles with temperature [

Mössbauer absorption spectra (solid lines) of

Equations (

Then, the general equations for calculations of partial absorption spectra

As seen from (

The normal precession modes of vectors

Using the mathematical formalism described in the previous sections, one can calculate also Mössbauer absorption spectra of an ensemble of antiferromagnetic particles with uncompensated magnetic moment for arbitrary values of parameter

Mössbauer absorption spectra (solid lines) of

The normal precession modes and nutations of the magnetization vectors of sublattices as well as the excitation energy spectrum and absorption spectra of particles with

Thus, a theory for describing excitation spectrum and thermodynamic properties of the Néel ensemble of antiferromagnetic nanoparticles with uniaxial magnetic anisotropy and uncompensated magnetic moment is developed by means of generalization of the continual model of magnetic dynamics of ideal antiferromagnetic nanoparticles in the two-sublattice approximation. This approach clarifies the principal difference in thermodynamic behavior of ferromagnetic and antiferromagnetic particles revealed in spectroscopic measurements. In the framework of the theory, a formalism for calculating Mössbauer absorption spectra of an ensemble of antiferromagnetic particles is implemented, which can be efficiently used for analyzing a large array of experimental spectra taken on these materials so far. The resulting model can be easily realized on PC, which makes it possible to take into account the physical mechanisms of the formation of the hyperfine structure of the absorption spectra in the real situation and to numerically describe the qualitative features of the temperature evolution of the spectral shape, which is observed in most experimental spectra of ^{57}Fe nuclei in antiferromagnetic nanoparticles.

Calculations performed within the macroscopic theory confirm completely the main conclusions of the earlier developed quantum-mechanical model [

It is worth mentioning that the above model of magnetic dynamics and the formalism for calculations of the Mössbauer spectra of an ensemble of antiferromagnetic particles can be easily generalized to the case of antiferromagnetic and ferrimagnetic nanoparticles in an external magnetic field. Solution of the problem in the last case will allow a more accurate description of not only the Mössbauer spectra in a magnetic field [

The author declares that there are no conflicts of interest regarding the publication of this paper.

The author is grateful to the Russian Foundation for Basic Research for a financial support.

^{1}H-NMR study of the spin dynamics of fine superparamagnetic nanoparticles