Let us consider an ensemble of single-domain particles isotropically distributed in space. If this ensemble of particles is magnetized to saturation and then magnetizing field *H* is switched off, maximum angle θ*m* of deviation of the particle magnetization vectors from the field-application direction is equal to angle α between the hard and easy magnetization axes in this magnet. For magnets with cubic and uniaxial anisotropies, α = 55° and 90°, respectively. Thus, in the state of residual magnetization, the magnetization vectors of single-domain particles are isotropically distributed in solid angle Ω = 2θ*m* = 2α.

If the particles have a domain structure, θ*m* > α because closure domains and domain walls, in which the magnetic moments make an angle with the easy magnetization axes, exist in the domain structure along with domains, the magnetization vectors of which are directed along the easy magnetization axes.

It is known that the relative areas of the Zeeman splitting lines in a Mössbauer spectrum of 57Fe nuclei in a homogeneously magnetized sample depend on angle θ between the γ-ray propagation direction and magnetization direction in the sample [9]:

$${S}_{\text{1,6}}=3\left(1+{\text{cos}}^{2}\theta \right)$$

$${S}_{\text{2,5}}=4{\text{sin}}^{2}\theta$$

2

$${S}_{\text{3,4}}=1+{\text{cos}}^{2}\theta$$

Let the geometry of the experiment be as follows the γ-ray propagation direction coincides with the *H* direction. Then, the following expression can be written for parameter *k* equal to the ratio of the second-to-first (or fifth-to-sixth) absorption line areas in the Mössbauer spectrum of 57Fe nuclei in the sample:

$$k=\frac{{S}_{\text{2,5}}}{{S}_{\text{1,6}}}=\frac{4(1-\stackrel{-}{{cos}^{2}}{\theta }_{i})}{3(1+\stackrel{-}{{cos}^{2}}{\theta }_{i})}$$

3

,

where θ*i* is the angle between the γ-ray propagation direction and the magnetization direction of the *i* - particle (0 ≤ θ*i* ≤ θ*m*):

$$\stackrel{-}{{cos}^{2}}{\theta }_{i}=\frac{\underset{0}{\overset{{\theta }_{m}}{\int }}\underset{0}{\overset{2\pi }{\int }}{cos}^{2}\theta \bullet sin\theta d\theta d\phi }{\underset{0}{\overset{{\theta }_{m}}{\int }}\underset{0}{\overset{2\pi }{\int }}sin\theta d\theta d\phi }=\frac{{cos}^{3}{\theta }_{m}-1}{3(cos{\theta }_{m}-1)}$$

4

,

For single-domain particles, we have \(\stackrel{-}{{cos}^{2}}{\theta }_{i}=\text{0,63}\) (cubic anisotropy) and \(\stackrel{-}{{cos}^{2}}{\theta }_{i}=\text{0,33}\) (uniaxial anisotropy).

Substituting these values into (3), we arrive at the following criteria: if *k* + Δ*k*≤0.30 (cubic anisotropy) and *k* + Δ*k*≤0.67 (uniaxial anisotropy), the powder particles are single-domain (Δ*k* is the experimental error in determining parameter *k*). If *k* –Δ*k* > 0.30 (cubic anisotropy) and *k* – Δ*k* > 0.67 (uniaxial anisotropy), the powder particles have a domain structure; the relative number of domains in the particles can be estimated from the *k* value.

Rare-earth ferrite garnets (REFG) having the magnetic compensation point *T**cm* are the most favorable magnets for the organization of the experiment. Indeed, fairly large REFG particles may become single-domain near *T**cm* due to the small value of spontaneous magnetization [10]. In addition, the particles that are single-domain near *T**cm* pass to the multidomain state while moving away from *T**cm*. Therefore, one can experimentally validate the found criteria gradually increasing the difference in temperature from *T**cm*. Note that the interest in REFG is related to the prospects of formation of materials based on the domain structure of these ferrimagnets for the element base of magnetic microelectronic devices [11–13].

The method was verified on particles of a Gd3Fe5O12 ferrite single crystal with cubic anisotropy prepared according to the standard technology from pure initial oxides Gd2O3 and Fe2O3 [14]. The magnetization was measured in the temperature range of 150–400 K on a VM2-A vibrational magnetometer. It was established that the ferrite compensation point is *T**cm* = 286 K (Fig. 1). The Mössbauer spectra of 57Fe nuclei were recorded on a YGRS-4M spectrometer with a 57Co(Cr) γ-radiation source. A ferrite single crystal powdered in an agate mortar was filtered through a set of hair sieves to obtain spherical particles with a diameter *d* = 40 ± 5 µm. These particles were used to make a sample (absorbent for Mössbauer measurements) by depositing the powder in a glue mixture on a thin mica disk. The absorbent thickness with respect to natural iron was 20 mg/cm2. For temperature measurements, the sample was placed in a temperature chamber combined with a cryostat with a fine temperature control in the range of 120–500 K. Automatic temperature control system maintained a specified temperature with an error of ± 0.5 K. Before measuring the Mössbauer sample, the spectrum was brought into the residual-magnetization state at a specified temperature. The sample temperature was brought to the desired value, and then the magnetic field applied perpendicular to the sample plane was increased from zero to the value *H**S* = 2 kOe, which is sufficient for magnetic saturation of the sample, after which it was reduced to zero.

Before each measurement, the sample was preliminarily demagnetized in an alternating magnetic field with amplitude decreasing to zero. Typical spectra measured in the vicinity of *T**cm* are shown in Fig. 2. They are a superposition of two Zeeman sextets caused by iron ions in the *a* and *d* ferrite \(\left\{{Gd}_{3}^{3+}{\}}_{c}\right[{Fe}_{2}^{3+}{]}_{a}({Fe}_{3}^{3+}{)}_{d}{O}_{12}^{2-}\) sublattices. Note that the Mössbauer spectrum does not distinguish symmetric and antisymmetric magnetic-moment orientations of ions in the ferrite microparticle sublattices with respect to the γ-ray propagation direction (Fig. 3), because (sin(180° + α) = –sinα) the relative areas of the absorption lines of the Mössbauer spectrum of 57Fe nuclei are determined by squared trigonometric functions (2).

Taking into account this fact, parameter *k* can be determined as a ratio of the integral areas of the Mössbauer spectrum absorption lines corresponding to the *a* and *d* ferrite sublattices. In view of the aforesaid, particles may have either ferromagnetic or ferrimagnetic ordering in the method of analyzing the domain structure in magnetic powder microparticles based on Mössbauer spectroscopy. In both cases, the Mössbauer spectroscopy data are identical. The areas of the absorption lines of the Mössbauer spectra were determined using the UnivemMS program. The following values were obtained for the parameter *k*: *k* = 0.26 ± 0.04 at *T* = *T**cm* + 2 K, k = 0.45 ± 0.04 at *T* = *T**cm* +33 K, and k = 0.64 ± 0.04 at *T* = *T**cm* + 45 K. It can be seen that ferrite particles are single-domain near *T**cm*. While moving away from *T**cm*, particles pass to the multidomain state and the number of domains in the particles increases. A similar situation was observed at temperatures below *T**cm*. In the domain structure, domains occupy much larger volume in comparison with domain walls; therefore, the increase in parameter *k* while moving away from *T**cm* is mainly due to the increase in the number of closure domains, which are magnetized at an angle with respect to the easy magnetization axes.

Note that ferrite particles near *T**cm* are weak magnetic; therefore, particles were not attached to each other during the preparation of the sample at room temperature and sample isotropy was provided automatically. In practice, strong magnetic materials are more often applied. Their single-domain particles may stick to each other as elementary magnets forming chains and lumps, which may violate the sample isotropy underlying the method proposed. Therefore, samples should be fabricated at temperatures above Curie temperature *T**c* with a specially chosen adhesive material hardening near *T**c*.