INTRODUCTION

The Mössbauer spectra of samples, in which iron atoms are located in many structurally and magnetically nonequivalent positions, have a complex hyperfine structure consisting of a large number of partial spectra. Some of the parameters of these partial spectra may have the same values or be related in some way due to the features of the crystalline and magnetic structures of the sample. The interpretation of such spectra involves the creation of models consisting of a set of models of individual partial spectra with physically determined relations between their variable parameters.

SpectrRelax [1, 2] is a program for processing and analyzing Mössbauer spectra. It makes it possible to look up and process experimental spectra, simulate spectra, and find the optimal values of the parameters of the interpretation model when fitting the envelope to the experimental spectrum. The models can use an unlimited number of spectral lines with independent or related parameters. SpectrRelax consists of several large and sufficiently independent blocks: a user interface, functions for minimizing the chi-square functional and searching for optimal values of variable parameters, a module for calculating mathematical expressions specified by the user, and a library of partial spectral models. The SpectrRelax program implements the ability to set arbitrary relations between parameters created by introducing additional variable parameters and creating analytical expressions [1, 2]. However, with a large number of partial spectra and relations between their numerous parameters, whose number can reach several tens, it becomes difficult to work with these models for deciphering spectra, and the likelihood of making mistakes in setting conditions and relations increases.

Here, we describe the method provided in the SpectrRelax program for creating new user models of partial spectra based on existing models built into the program without changing the program itself by writing code fragments in the built-in Lua programming language [3]. The extension of the capabilities of the SpectrRelax program is demonstrated by the example of studying hyperfine interactions of 57Fe nuclei in Tb(Fe0.8Al0.2)2 and Ho(Fe0.8Mn0.2)2 quasi-binary alloys with the C15 Laves phase structure.

CREATING CUSTOM MODELS IN SpectrRelax

To simplify the interpretation of Mössbauer spectra, which requires a model with a large number of mathematical expressions and relations, we built the Lua language into SpectrRelax to add new models to the library without changing the program itself. When creating new models, existing models can be also used. Lua (translated from Portuguese as “moon”) is a simple, interpreted programming language created at the Pontifical Catholic University of Rio de Janeiro, Brazil for configuring complex software [3]. Lua was designed to be embedded in other applications and is used for this purpose in hundreds of programs around the world.

To add a custom model, it is necessary to create a .lua file with any name that is not used by SpectrRelax. The code in the file is called from SpectrRelax when the program is started and returns a Lua table with a description of the model, the fields of which are presented in Table 1.

Table 1.   The fields of the model description table
Full size table

The “parameters” field in Table 1 is a list of parameters, whose elements define descriptions of the parameters of the model. The fields of each description are presented in Table 2. The “initialize” function takes a single isotope parameter, which contains the description of the Mössbauer isotope in SpectrRelax, and returns an envelope calculation function, which may be different for different isotopes.

Table 2.   The parameter description fields in the parameters table
Full size table

The envelope calculation function is called with the parameters presented in Table 3.

Table 3.   The parameters of the envelope calculation function
Full size table

When calculating the values of the model function, the method of automatic differentiation using double numbers [4] was applied; thus, there is no need to write a separate code for calculating partial derivatives.

We demonstrate the creation of a complex multicomponent model of Mössbauer spectra in the Lua language in the SpectrRelax program using the “Laves” user’s model as an example, which takes into account the local magnetic inhomogeneity of the positions of Fe atoms in the R(Fe1 – xMx)2 Laves phases.

HYPERFINE INTERACTIONS OF 57Fe NUCLEI IN QUASIBINARY R(Fe1 – xMx)2 ALLOYS AND THE LAVES MODEL

Intermetallic compounds of rare-earth metals (R) and 3d transition metals (M) of the RM2 type with the structure of Laves phases are still of great interest to researchers primarily because of the giant magnetostriction [5] and large magnetocaloric effect found in them [6]. At the same time, due to their relatively simple, highly symmetrical crystal structure, they are excellent model objects for theoretical and experimental research in the condensed matter physics.

The presence of the 57Fe Mössbauer atom in the RM2 Laves phases opens up the possibility of studying hyperfine interactions (HFIs) in them. Substitution of both elements by other atoms in these compounds makes it possible to study the variety of HFI mechanisms and features, as well as the relationship between HFI parameters and local characteristics and properties of the compound. Despite a sufficiently large number of works devoted to the study of HFI in Laves phases (see, for example, [7–15]), the HFI mechanisms in them have not been sufficiently studied to date. Note, in particular, such incompletely resolved issues as the nature of the anisotropy of magnetic HFIs, as well as the nature of the behavior of the HFI parameters upon substitution of one element by another.

One essential step in the HFI study was the use of a tensor description of magnetic HFI [8, 10], which makes it possible to obtain all the necessary HFI parameters directly from the Mössbauer spectra and to determine the orientation of the easy magnetization axis (EMA) with a high accuracy. The use of the tensor description was found to be quite productive in the study of Laves phases with several atoms of different types in the crystallographic position of rare-earth atoms [8, 10, 14], which weakly affect HFI of 57Fe nuclei.

Difficulties arise in the study of iron-containing Laves phases with the substitution of Fe atoms in the crystallographic position. In this case, both the parameters of the exchange interactions of the Fe atom with surrounding atoms and the HFI parameters change significantly and positions of 57Fe Mössbauer atoms occur that are nonequivalent both magnetically and compositionally [10]. The Mössbauer spectra of 57Fe nuclei reperesent complex hyperfine structure generally consisting of a large number of partial spectra with interrelated hyperfine parameters. With no allowance for the relations, it seems imssible to find the values of these parameters because of their large number and, as a rule, a poorly resolved experimental spectrum. In such a situation, it is necessary to create and implement an interpretation model that takes into account, if possible, all the features of the crystal and magnetic structure of a multicomponent compound within the framework of the tensor description of magnetic HFIs.

Our model for processing the spectra of quasibinary R(Fe1 – xMx)2 alloys with the C15 Laves phase structure, as proposed and implemented in the SpectrRelax program, is based on the following assumptions:

(1) The compounds have a cubic structure of the C15 Laves phase (space group \(Fd\bar {3}m\)).

(2) The magnetic structure of the compound is collinear: regardless of their local atomic configuration, the magnetic moments of all atoms are directed along common EMA.

(3) Hyperfine magnetic fields at 57Fe nuclei are specified within the framework of the tensor description of magnetic hyperfine interactions and depend on hyperfine isotropic His and anisotropic Han fields and the orientation of the EMA, which is specified by the polar ϑ and azimuthal φ angles relative to the crystallographic axes of the compound [10, 14].

(4) Hyperfine isotropic fields His and shifts δ of the Mössbauer spectra of Fe atoms with different nearest atomic neighborhood are different while the hyperfine anisotropic fields and quadrupole interaction constants e2qQ are the same for all Fe atoms and do not depend on the nearest atomic neighborhood.

(5) The energy of the electric quadrupole interaction is accounted for in the second order of smallness [16].

(6) The local inhomogeneity of Fe atoms caused by the variation in the number of atoms of different types in the second and further coordination shells of the Fe atom is taken into account via describing the shape of the resonance lines by the pseudo Voigt function [1, 2].

(7) Since there are from 0 to 6 M atoms in the substituted rare-earth Laves phases R(Fe1 – xMx)2 in the nearest atomic neighborhood of Fe atoms in a general case, it is almost impossible to implement the spectrum interpretation model using only user-defined mathematical expressions, as was done for \({\text{Z}}{{{\text{r}}}_{{1 - x}}}{\text{S}}{{{\text{c}}}_{x}}{\text{F}}{{{\text{e}}}_{2}}\) in [14]. For this purpose, it is necessary to create a complex partial spectrum model in the Lua programming language. In the considered case of Laves phases R(Fe1 – xMx)2, for each partial spectrum corresponding to a certain number of atoms M in the nearest atomic neighborhood of Fe atoms, the Laves user model consisting of four interconnected Zeeman sextets was created based on relations (19 )–(30) given in [14], which takes into account the local magnetic inhomogeneity of the positions of Fe atoms (see Appendix).

As a result, a partial spectrum model appears, which can be used along with other models provided in the SpectrRelax program (including the restoration of distributions of hyperfine spectrum parameters):

$${{N}_{{{\text{Laves}}}}}\left( {\begin{array}{*{20}{c}} {v;I,\frac{{2I}}{{1I}},\frac{{3I}}{{1I}},\frac{{4I}}{{1I}},\frac{{{{I}_{2}}}}{{{{I}_{1}}}},\frac{{{{I}_{3}}}}{{{{I}_{1}}}},\delta ,{{e}^{2}}qQ,} \\ {\varphi ,\theta ,~{{H}_{{{\text{is}}}}},{{H}_{{{\text{an}}}}},{{\Gamma }_{1}},\frac{{{{\Gamma }_{2}}}}{{{{\Gamma }_{1}}}},\frac{{{{\Gamma }_{3}}}}{{{{\Gamma }_{1}}}},\alpha } \end{array}} \right).$$
(1)

Here, \(I\) is the intensity (area) of the partial spectrum in units of the total area of the spectrum, \(\frac{{2I}}{{1I}},\frac{{3I}}{{1I}},\frac{{4I}}{{1I}}\) are the ratios of the intensities of the second, third, and fourth to the intensity of the first partial spectrum, respectively; δ is the shift of all four partial spectra; \({{e}^{2}}qQ\) is the quadrupole interaction constant for all four partial spectra; \(\varphi ,\theta \) are the azimuth and polar angles that specify the direction of the easy magnetization axis relative to the principal axes of the electric field gradient tensor; \({{H}_{{{\text{is}}}}}\) and \({{H}_{{{\text{an}}}}}\) are the isotropic and anisotropic hyperfine magnetic fields at 57Fe nuclei; \({{\Gamma }_{1}}\) is the width of the first resonance line of the Zeeman sextet; \(\frac{{{{\Gamma }_{2}}}}{{{{\Gamma }_{1}}}},\frac{{{{\Gamma }_{3}}}}{{{{\Gamma }_{1}}}}\) are the ratios of the widths of the second and third to the intensity of the first resonance line in the Zeeman sextet, respectively; and \(\alpha \), the fraction of the Gaussian contribution to the intensity of the resonance line, varies from 0 (Lorentzian form) to 1 (Gaussian form).

As can be seen, taking into account a priori information on the crystal and magnetic structures of compounds of the RFe2 type in the framework of the tensor description of the anisotropy of hyperfine magnetic interactions into account makes it possible to take into account the magnetic nonequivalence of Fe atoms for each position with a various number of atoms M (mM) in the nearest neighborhood, and significantly reduce the number of variable parameters that specify positions of the spectrum components. In this case, all these parameters {\(\delta ,{{e}^{2}}qQ,~\varphi ,\theta ,~{{H}_{{{\text{is}}}}},{{H}_{{{\text{an}}}}}\)} have a physical meaning and are, as a rule, the subject of a search.

SUPERFINE INTERACTIONS OF 57Fe NUCLEI IN QUASIBINARY Tb(Fe0.8Al0.2)2 AND Ho(Fe0.8Mn0.2)2 ALLOYS

As an example of using the Laves user’s model, we present the result of processing the Mössbauer spectra of 57Fe nuclei in the quasibinary Tb(Fe0.8Al0.2)2 and Ho(Fe0.8Mn0.2)2 alloys. The spectrum interpretation model consisted of five most probable for a given concentration of Al and Mn Laves models of partial spectra, the parameters of which are related to each other, as well as the user-defined mathematical expressions. The results of the model interpretation of the Mössbauer spectra are shown in Figs. 1 and 2 and Table 4. As can be seen, the interpretation model describes the experimental spectra well: there are no obvious systematic deviations in the difference spectra (Fig. 1) at values of the chi-square functionals close to unity.

Fig. 1.
figure 1

The results of model interpretation of the spectra of 57Fe nuclei in Tb(Fe0.8Al0.2)2 and Ho(Fe0.8Mn0.2)2 alloys.

Full size image
Fig. 2.
figure 2

The intensities Im of the partial spectra of the 57Fe nuclei, the values of the δ spectrum shift and isotropic field His depending on the m number of Fe atoms replaced by Al and Mn atoms in the nearest neighborhood of the Fe atom.

Full size image
Table 4.   The hyperfine parameters of the Mössbauer spectra of Tb(Fe0.8Al0.2)2 and Ho(Fe0.8Mn0.2)2
Full size table

The relative intensities of the partial spectra corresponding to the positions of Fe atoms with a various number of Al and Mn atoms in their nearest neighborhood, with some deviations, are in good agreement with the binomial distribution (Fig. 2). The results of model interpretation made it possible to obtain the values of the Cowley short-range order parameter [17], which were calculated by specifying a mathematical expression directly in the SpectrRelax program:

$${{\alpha }_{{{\text{Cowley}}}}} = 1 - \frac{1}{{nx}}\frac{{\sum\limits_{m = 0}^n {m{{I}_{m}}} }}{{\sum\limits_{m = 0}^n {{{I}_{m}}} }},$$
(2)

where n = 6 is the number of atoms in the first cationic coordination shell of the Fe atom, x = 0.2 is the concentration of Al or Mn atoms in the quasibinary Tb(Fe0.8Al0.2)2 and Ho(Fe0.8Mn0.2)2 alloys, respectively. As a result, the values of \(\alpha _{{{\text{Cowely}}}}^{{{\text{Al}}}} = - 0.22 \pm 0.05\) and \(\alpha _{{{\text{Cowely}}}}^{{{\text{Mn}}}} = - 0.05 \pm 0.04\) were obtained, which mean that the distribution of Fe and Mn atoms over the positions of transition metal atoms in Ho(Fe0.8Mn0.2)2 is close to binomial, while in the distribution of Fe and Al atoms, Fe–Al bonds have some energy preference compared to Al–Al and Fe–Fe bonds.

As a result of the model interpretation of the spectra, it was found that the shifts of the spectra and the isotropic His fields change almost linearly with a change in the number of Al and Mn atoms in the nearest neighborhood of the Fe atom (Fig. 2). In this case, the average values of changes in the shift of the spectrum \(\Delta \delta \) and the isotropic field \(\Delta {{H}_{{{\text{is}}}}}\) upon substitution of an Fe atom by an Al and Mn atom in the nearest neighborhood of the Fe atom (Table 4) show that the changes in the hyperfine parameters are most sensitive to the substitution of Fe atoms by Al atoms. The explanation for this should be sought, first of all, in the significant difference in the electronic configurations of the Fe and Al atoms compared to the small difference in the electronic configurations of the Fe and Mn atoms, which are adjacent in Mendeleev’s table.

The relatively large value of the anisotropic field \(\sim - 12\,\,{\text{kOe}}\) for the Tb(Fe0.8Al0.2)2 alloy compared to that \(\sim {\kern 1pt} - 6\,\,{\text{kOe}}\) for the Ho(Fe0.8Mn0.2)2 alloy (Table 4) is primarily due to the large contribution of the dipole field from the effective spin of the Tb3+ atom compared with the Ho3+ atom (see, for example, [10]). When searching for the EMA direction, it was found that the experimental spectra are weakly sensitive to variations in the \({{\varphi }}\) azimuth angle in the vicinity of 45°, so it was fixed. At room temperature, EMA deviates by ~13° from the \(\left[ {100} \right]\) crystallographic axis for Ho(Fe0.8Mn0.2)2 and by ~17° from the \(\left[ {111} \right]\) axis in the \(\left( {\bar {1}10} \right)\) plane for Tb(Fe0.8Al0.2)2 (Table 4).

CONCLUSIONS

The capabilities of a program for processing and analyzing Mössbauer spectra SpectrRelax were expanded, which make it possible to create new user models of partial spectra based on those already existing in the program without changing the program itself. For this purpose, a file with structured code fragments in the built-in Lua programming language that describe the model has been created.

These new capabilities have been demonstrated based on the example of creating the Laves user model which, within the framework of the tensor description of the anisotropy of hyperfine magnetic interactions, takes into account the features of the crystal and magnetic structures of Laves cubic phases of the RM2 type.

The use of the five most probable Laves models for processing the spectra of 57Fe nuclei quasibinary in Tb(Fe0.8Al0.2)2 and Ho(Fe0.8Mn0.2)2 alloys made it possible to take into account simultaneously the compositional and magnetic nonequivalence of positions of Fe atoms in these alloys. As a result, new data on the orientation of the easiest magnetization axis, the anisotropy of the hyperfine magnetic field, and the effect of substitution of Al and Mn atoms for Fe atoms on the hyperfine parameters of Mössbauer spectra was obtained.