Saturation magnetization, Bs, and coercive force, Hc, depending on silicon content, X, in the nanocrystalline alloy Fe87-XSiXB9Nb3Cu1.
Nanocrystalline Fe-Si-B-Nb-Cu alloys (Finemets) obtained by crystallization of amorphous ribbons quenched from the melt exhibit high soft magnetic properties: the permeability greater than 105, coercive force of about 0.002 Oe and the saturation magnetization exceeding 10 kGs (Yoshizawa et al., 1988). In addition, their permeability can be purposefully controlled by inducing a magnetic anisotropy during annealing in a magnetic field (Herzer, 1992; Yoshizawa & Yamauchi, 1989) or in the field of tensile stress (Glazer et al., 1991; Herzer, 1994; Hofmann & Kronmüller, 1996). Magnetic anisotropy energy exceeding 5000 J/m3 is attained by annealing under a tensile stress of 400-600 MPa. For the first time the effect of tensile load applied along the ribbon at nanocrystallization annealing was investigated in (Glazer et al., 1991) and it was shown that after such treatment in a sample of the alloy Fe73.5Si13.5B9Nb3Cu1 a state with the magnetic anisotropy of the “easy-plane” type is formed. The plane of the magnetic anisotropy is oriented perpendicularly to the direction of stretching. For the first time it was assumed that the transverse magnetic anisotropy is due to residual elastic strain in the lattice of nanocrystals having a negative constant of the magnetostriction.
It is known that the type of the magnetic anisotropy induced in the Fe87-XSiXB9Nb3Cu1 alloy during tensile stress annealing (TSA) depends on the silicon content X (Serikov et al., 2006). If X ≤ 8, in specimens after TSA the magnetic anisotropy of easy axis type is induced, whose axis is oriented along the tape (i.e. in the direction of load application during TSA). In the process of reversal magnetization along the tape axis, the permeability and the residual magnetization increase, coercive force decreases, the hysteresis loop becomes rectangular. If the concentration of silicon is equal to or greater than 11 at.%, then a transverse magnetic anisotropy of easy-plane type is formed after TSA. In this case, the permeability is constant over a wide range of magnetic fields (up to 125 Oe), the magnetic hysteresis loop becomes inclined, and the magnetization is oriented predominantly in a plane perpendicular to the direction of load application during the annealing.
If the formation of a state with magnetic anisotropy after annealing and cooling under a tensile load is associated with residual elastic stresses (Glazer et al. 1991; Herzer, 1994), − a result of the so-called Villari effect, then the type of magnetic anisotropy can be explained by the nature of magnetoelastic interactions in the nanoparticles (Serikov et al., 2006; Filippov, 2006) mainly consisting of Fe-Si crystals having a bcc lattice. This argument is supported by the fact that, in bulk crystalline samples of the Fe1-xSix alloy, with increasing concentration of silicon the magnetostriction constant changes its sign from positive to negative at x ~ 0.12 (Bozorth, 1993; Bertotti & Fiorillo, 1994c), and that the silicon concentration in nanoparticles is a few percent higher then the average Si concentration X in the alloy (Serikov et al., 2006). For example, if X is equal to 8 at.% then silicon in nanocrystals is contained in an amount of about 12 at.%.
In the Fe73.5Si15.5B7Nb3Cu1 alloy after TSA, residual deformations of a bcc nanocrystal lattice were detected by X-ray diffraction analysis judging from the shifts of one Bragg reflection (Ohnuma et al., 2003, 2005). In these papers it has been shown that the crystal lattice is stretched in the direction of load application and is compressed in the transverse direction, the strain and the magnetic anisotropy energy being proportional to the load.
Detailed studies of the residual strains have shown that deformations of the bcc lattice of Fe-Si nanocrystals are not isotropic: extension and compression along the <100> directions are maximum, and in the <111> − are minimum (Chernenkov et al., 2010). Therefore, the formation of a state with transverse magnetic anisotropy in the nanocrystals of the Fe73.5Si13.5B9Nb3Cu1 alloy, which is characterized by the orientation of magnetic moments transverse to the direction of stretching, is due to the negative magnetoelastic coupling that is caused by residual anisotropic lattice deformations of the Fe-Si nanocrystals with a large fraction of the Fe3Si phase. The research presented was undertaken to ascertain in detail the atomic structure of nanocrystals of Fe-Si-B-Nb-Cu alloys, depending on the silicon content and the thermal treatment of the samples.
2. Nanocrystalline Fe-Si-B-Nb-Cu alloys and their magnetic properties
The investigations were carried out using Fe87-XSiXB9Nb3Cu1 alloys (X = 0, 4, 6, 8, 9.5, 11, 13.5) on samples in the form of ribbons 20 mcm thick and 1 mm wide obtained in the initial amorphous state by melt-spinning technique, where a molten metal alloy is ejected through an orifice onto a rotating copper wheel. The nanocrystallization annealing (NCA) and the stress annealing (TSA) under a tensile stress of 440 MPa were carried out in air at a temperature of 520 С for 120 minutes (under these conditions, no secondary recrystallization still occurs and coercive force shows up no sharp changes in the annealing-temperature range of 510-570°С).
As a result of the heat treatments, samples from the alloys in three different states: initial (or immediately after quenching on the wheel), nanocrystalline (after NCA) and subjected to TSA were obtained. The magnetic state of the samples after NCA and TSA was controlled by the shape of hysteresis loops measured by the ballistic method using an F-190 microwebermeter (Serikov et al., 2006). The hysteresis loops of the samples after NCA and TSA treatments measured upon reversal magnetization along the ribbon are compared in Fig. 1. It is seen that with increasing silicon content X from 0 to 13.5 the samples subjected to NCA show a decrease in the saturation magnetization Bs from 14.5 to 12.5 kGs; the coercive force Hc decrease from 1.2 Oe at X = 0 to 0.006 Oe at X = 13.5 (the data for all compositions are given in Table 1). After TSA, the coercive force decreases only for the alloy without silicon (X = 0). The greatest changes after TSA are observed in the shape of the hysteresis loops, and the character of changes strongly depends on the silicon content.
For samples after TSA with X < 8, the hysteresis loop shape transforms from inclined into rectangular, with a high magnitude of the remanent magnetization. This indicates the inducing the longitudinal magnetic anisotropy (Fig. 1, X = 0 and 6). For samples with X = 8, the loop after the TSA is still steeper than after NCA, and the difference in the slope of the loops is much less than at a lower silicon content. When X = 9.5, the loops of samples after NCA and TSA are also close to each other, but the loop of sample after TSA is lower than for sample after the NCA, i.e., remanent magnetization decreases as a result of TSA. For samples with X > 9.5 after TSA treatment, the loops become inclined. This indicates the appearance of a transverse magnetic anisotropy. The sample with X = 13.5 after TSA has a constant magnetic permeability in the range of magnetic fields from -85 to 85 Oe, and its hysteresis loop has an inclined shape.
3. Nanocrystal structure
3.1. X-ray diffraction analysis
The X-ray diffraction patterns of Fe87-XSiXB9Nb3Cu1 alloys were measured in transmission geometry using a four-circle diffractometer and monochromated Mo Kα radiation (λ = 0.71 Å). The experimental scheme is shown in Fig. 2.
The samples in the form of a rectangular plate were prepared from ribbon fragments, which were glued to the thin narrow ring-like holder parallel to each other in several overlapping layers of thickness about 40 mcm. During the θ-2θ scanning, when ω angle is equal to θ, the scattering vector q was in the sample plane. The ω angle is equal to zero, when a sample plate is perpendicular to the direction of the incident X-ray beam. By rotating the sample around the horizontal axis by 90 , the transition from the longitudinal scan, when the scattering vector is directed along the axis of the ribbons, to the transverse scan was carried out. In the case of the samples annealed under stress at the longitudinal scan the scattering vector was parallel to the direction of the tensile load application and perpendicular to it in the transverse scan. For each sample, the diffraction patterns were measured for two orientations, i.e. in the form of longitudinal and transverse scans.
The X-ray diffraction patterns of Fe87-XCu1Nb3SiXB9 (X = 0, 6, 8, 9.5, 11, 13.5) alloy samples in the initial state are shown in Fig. 3. Within the accuracy of measurements the longitudinal and transverse scans for the same sample are identical. All these diffraction patterns have a shape typical of the quasi-amorphous material (Yoshizawa et al., 1988; Noskova et al., 1992). However, the position of the first peak (2θ ≈ 20 ) coinciding with the Bragg reflection (110) for the bcc structure of iron gives reason to suppose that in the initial state in the alloys there are the regions with bcc short-range order in the arrangement of atoms. Since the iron atoms are a major component of the alloy and the shortest distance between two atoms in the α-Fe is 2.48 Å, then the position of the main peak in the diffraction pattern of disordered (amorphous) state can be estimated as 2θamorphous = 16.46 (Fischer et al., 2006).
Here and below, to describe the shape of reflections, the pseudo-Voigt function defined as a linear combination of Lorentzian and Gaussian with the same halfwidth (FWHM) was selected. After correction for instrumental resolution, integral width of the reflexion was computed, and the average diameter of the bcc grains was calculated by Scherrer formula (Warren, 1969). The dependence of the average sizes of the regions with the bcc order in the atomic arrangement on the silicon concentration is shown in the inset in Fig. 3. Therefore, the structure of the alloy in the initial state can be defined as a fine grained, highly defective bcc structure with the grain sizes about 2 nm. This statement is supported by the fact that the broad diffuse maxima are located close to the calculated positions for the bcc reflections, which are shown in Fig. 3 by triangles. The average size of the ordered regions that contribute to the main peak in the diffraction patterns slightly depends on the concentration of silicon X. One point at X = 8 drops out of a linear dependence, which is apparently due to a significant number of relatively large, micron-sized grains of iron that give the narrow Bragg peaks (200) and (211) and narrow and intense contribution to the main peak (110) in the diffraction pattern. Previously, a minor amount of such iron grains was observed in the nanocrystalline samples with X = 0 and 8 by TEM (Lukshina et al., 2002).
In the diffraction patterns of the Fe87-XSiXB9Nb3Cu1 alloy samples subjected to NCA shown in Fig. 4, there are peaks with the Miller indices (110), (200), (211), (220), (310), (222), (321), and (330) allowed for the bcc lattice of α-Fe(Si). Whereas the intensity of these reflections is sufficient for quantitative analysis, the intensity of other reflections is too small for it. The unit cell parameter of the ordered Fe3Si phase (structure of D03 type) is about twice as high as that of the bcc Fe(Si) phase. Hereafter we will use the peak indexing for the bcc lattice. Therefore, when the Fe3Si phase appears in a sufficient quantity, the superstructure peaks with half-integer indices, e.g. (½ ½ ½), (1½ ½ ½), (1½ 1½ ½), etc., as well as the peaks with an odd sum of integer indices (h + k + l), for example (100), (111), (210), etc. should appear. With the available resolution of the diffractometer, the bcc peaks (hkl) of α-Fe (Si) phase in the diffraction patterns coincide with the D03-peaks (2h 2k 2l) of the Fe3Si phase.
After annealing at temperature 520°C (NCA treatment), the nanocrystals of α-Fe(FeSi) have appeared in all the alloys, independently of the silicon concentration. The average size of the nanocrystals is 10-12 nm.
The intensity of Bragg reflections at higher angles is rather weak. With increasing silicon content (X) in the samples, and, as a consequence, in the nanocrystals, the Bragg peaks are slightly shifted to the higher angles. It means that the bcc unit cell parameter decreases as in the bulk Fe-Si single crystals (Bozorth, 1993; Bertotti & Fiorillo, 1994a). The dependencies of the unit cell parameter on X in nancrystals and in bulk samples shown in the inset in Fig. 4 are very similar. A sharper decline in the curve a(X) for nancrystals is probably due to the higher concentration of silicon in nanocrystals than that (CSi = X/100) in bulk Fe-Si crystal.
A large value of a(X) for alloy with X = 0 can be explained by the presence in nancrystals of Nb atoms (atomic radii 0.146 nm) substituting for the Fe atoms (atomic radii 0.126 nm) in the bcc lattice of iron. With increasing X, the Nb atoms are replaced by Si atoms (atomic radii 0.118 nm), and the unit cell parameter rapidly decreases. A kink in the curve a(X) for Fe-Si crystals at CSi = 0.10 is caused by the formation of the Fe3Si phase with the D03 structure. The D03 unit cell is composed of eight bcc-unit cells, and its parameter (0.5652 nm) is less then the doubled unit cell parameter of bcc-Fe (0.5732 nm). With increasing X, the volume fraction of D03 increases, and the unit cell parameter of the Fe-Si alloy decreases faster (CSi > 0.10) then in the α-FeSi phase (CSi < 0.10). The similar behaviour of a(X) is observed in the nanocrystals of the Fe87-XSiXB9Nb3Cu1 alloys.
The appearance of the Fe3Si phase in the Fe87-XSiXB9Nb3Cu1 alloys with X = 11 and 13.5 is evidenced by two superstructure peaks (½ ½ ½) and (100) at angles 2θ ≈ 12 and 14 in the diffraction patterns. Other peaks of the Fe3Si phase are not distinguishable because they overlap with peaks of α-FeSi phase or their intensity is rather low.
The diffraction patterns measured in the longitudinal and transverse scans coincide (Fig. 4). The non-linear background is most probably due to the amorphous matrix surrounding the nanocrystallites in the Fe87-XSiXB9Nb3Cu1 alloy. The phase analysis of the alloy with X = 13.5 has shown that contributions of α-FeSi (bcc), Fe3Si (D03), and amorphous phase (matrix) are comparable (Chernenkov et al., 2010). The number of separate peaks of the Fe3Si phase is not enough to obtain the reliable value of the Fe3Si fraction, but it is predominant. With decreasing X, the Fe3Si fraction is reduced, and only α-Fe(Si) nanocrystals and amorphous matrix are retained.
In contrast to the alloys in the initial state and alloys subjected to the nanocrystallization annealing, a shift of peaks is observed in the diffraction patterns from alloys subjected to the tensile stress annealing (Fig. 5).
The diffraction peak profiles measured in the longitudinal and transverse scans for the Fe87-XSiXB9Nb3Cu1 alloys with X = 6 and 13.5 are shown in Fig. 6 and 7. The peaks in the longitudinal scans are shifted to the lower angles. This means that the inerplanar distances in the nanocrystals increase along the direction of the stress application during TSA. The peaks in the transverse scans are shifted in opposite direction, and consequently the inerplanar distances decrease.
The relative shifts of the peaks (Fig. 6 and 7) are different, and correspond to the anisotropic residual deformation of the nanocrystals. The largest shifts are observed for reflections (200) and (310), whereas no shifts are seen for the reflections (222) within the limits of the experimental resolution. The character of shifting the reflections in the diffraction patterns of the Fe87-XSiXB9Nb3Cu1 alloys with other values of X is similar to that displayed for X = 6 and 13.5 in Fig. 6 and 7.
The anisotropy of the distortion of the nanocrystals can be characterized by the relative change (Δd/d) of the interplanar spacing d in their lattice. The positions of all the peaks (hkl) in the diffraction pattern have been determined by the least-square refinement. The peak profile was described by a pseudo-Voigt function, a linear combination of a Lorentzian and a Gaussian of the same full width at half maximum. Each peak was considered to be a sum of two pseudo-Voigt functions corresponding to the contribution of two lines Kα1 and Kα2 in the spectrum of the MoKα. The value of Δd was determined as a difference between the d values calculated from the refined positions of the peaks (hkl) in the longitudinal (transverse) scan for the sample subjected to TSA and in the scan for the sample subjected to NCA. As a result of this procedure, relative values of the residual strains of the nanocrystals, Δd/d, were obtained for different directions [hkl] in the lattice, parallel and perpendicular to the direction of the load application during the TSA treatment, independently accounting for the residual lattice extensions and compressions, respectively. Their dependences on the angle Φ between the direction of the [hkl] vector and the nearest  axis are shown in Fig. 8. The residual extensions and compressions change with the silicon content (Fig. 9). The general trend in the Δd/d dependence on X, i.e. the decrease of Δd/d, is in accordance with the enhancement of hardness of the Fe-Si alloys with increasing CSi (Bertotti & Fiorillo, 1994b). Some softening, i.e. an increase of Δd/d, is observed at X = 11, where a sharp decrease of the a(X) takes place (Fig. 4). At lower X, the values of the relative residual strains, Δd/d, in the longitudinal and transverse directions, in the form of extension and compression of the lattice, respectively, are approximately equal, but at X = 11 and 13.5, the residual distortions of nanocrystals in the direction of the tensile stress application are about twice as high as in the transverse one (Fig. 9).
The distortions of the nanocrystal structure in the Fe87-XSiXB9Nb3Cu1 alloy sample subjected to the TSA treatment observed experimentally can be interpreted as follows. In the case of measurements in the transmission geometry, the sample is adjusted so that, during θ–2θ scanning, the scattering vector q is always in the sample plane. In this case, all the nanocrystals, for which the normals to the (hkl) planes are parallel to the scattering vector qhkl within the limits of the instrumental resolution, will contribute to the (hkl) reflection in the X-ray diffraction pattern and the position of the maximum in θ angles θhkl will indicate the mean distance between the (hkl) planes in the direction of scanning.
The ends of the <100> crystallographic axes of the set of such nanocrystals on the sphere of a unit radius will circumscribe circles, belonging to the planes normal to the scattering vector q (Fig. 10a). Thus, when scanning along the ribbon, we can trace, from the positions of the (hkl) maxima, changes in the interplanar distances of the nanocrystals, in which corresponding (hkl) planes are normal to the direction of applying the tensile load P during the TSA treatment (in this case, qhkl P). In the case of the transverse scan, the positions of the (hkl) peaks also characterize the distances between the (hkl) planes with the normals oriented crosswise the ribbon or crosswise P (qhkl P). In the case of the (200) reflection, for which the orientation of the axes is shown in Fig. 10b, the angle Φ between the normal to the (200) planes parallel to the scattering vector q200 and the  direction in the nanocrystals is ~55 . The largest distortion of the nanocrystalline lattice is exactly observed along the <100> axis (Figs. 8 and 9; Table 2). Thus, after the TSA treatment, the deformation of the initial cubic unit cell has a tetragonal character: an extension along  and contraction along  and ; the opposite pattern is observed for q P. The contraction can be explained by displacements of atoms in the bcc lattice during the TSA treatment. If the atoms can be considered as hard spheres, they touch each other in the bcc cell in the directions of the cube diagonals. Their close contact is retained with the extension of the unit cell along  if eight atoms in the cube vertices are shifted to the central atom along the diagonals of the <110> cube faces.
|along the ribbon - stretching|
|across the ribbon – compressing|
The specific case of the orientation of the nanocrystalline axes for the (222) reflection is shown in Fig. 10c. The <100> axes of the nanocrystals that contribute to the (222) reflection are on the cone surface with the apex angle ~55 × 2 = 110 , and the angle Φ is zero. It can be assumed that the absence of the deformation or its minimum value in the  direction is provided by a strong interaction of the nearest neighboring atoms arranged along the cube diagonals in the bcc lattice. The interaction hinders an increase in the distances between nearest neighboring atoms during extension. In the case of the tetragonal distortions, the distance between the nearest neighboring atoms is unchanged and only the directions of the bonds between the nearest neighboring atoms are slightly changed.
The residual structure distortions should be discussed using the elastic deformation tensor. However, it is hardly possible to do this, since the available structural information is very limited and the chemical composition of the nanocrystals is likely inhomogeneous: along with Fe and Si, the composition can also include other atoms. Moreover, the deformed nanocrystals are in the rigid amorphous matrix, in which, after the TSA treatment, residual stresses are likely exist as well.
The local atomic structure, phase composition and orientation of magnetization as a function of silicon content − X were investigated in the alloys after the NCA and TSA by nuclear gamma resonance (NGR, Mössbauer spectroscopy). The Mössbauer spectra were measured on a YaGRS-4M spectrometer in the regime of constant velocity at 20°C with a 57Co in a chromium matrix used as the source. From the Mössbauer spectra, the distribution functions of hyperfine fields (HFFs) - P (H) were constructed.
Mössbauer spectra of the alloys with different silicon content after the nanocrystallizing annealing and after the annealing under tensile load are shown in Fig. 11. Whereas, the distribution function P (H), in Fig. 12. The intensity of contributions at fields below 100 kOe is quite small; therefore, this range is not shown here. When analyzing distributions of HFFs, the Mössbauer data on the structure of FeSi alloys (Litvinov et al., 1982; Randrianantoandro et al., 1999; Stearns, 1963) were used.
Based on the Mössbauer spectra, the coefficients that characterize the deviation of the magnetic moments from the sample plane (Wertheim, 1964) were determined as a ratio of intensities of the second (fifth) and first (sixth) lines of the Mössbauer sextet. If the magnetic moment is perpendicular to the sample plane, i.e., lies along the direction of incidence of γ quanta, then А2/А1 = 0; in the case of an isotropic distribution of the magnetic moment, А2/А1 = 0.67; if the magnetic moments are located in the sample plane, А2/А1 = 1.33.
Fig. 13 displays the concentration dependence of the coefficient А2/А1 for the alloys studied, which is compared with the magnitudes of the constant of induced anisotropy, KTSA, found from the hysteresis loops measured on the samples subjected to TSA (Serikov et al., 2006). It is seen that in all the curves there is a feature near the Si content X ~ 8. In the case of heat treatment without a tensile stress, the magnetic moments are off the sample plane at small silicon concentrations, which appears to indicate a strong influence of borides on the iron grains (Kleinerman et al., 2004); a further growth of silicon content leads to the orientation of magnetic moments in the sample plane. After TSA the magnetic moments lie in the sample plane already at small silicon concentrations, but are oriented along the ribbon axis; later, they change the orientation to the transverse and then become somewhat off the sample plane.
The qualitative analysis of the concentration dependence of the distributions P(H) shown in Fig. 12 gives the following results. For X = 0 the distribution function over the local surroundings of iron atoms has a peak at H = 330 kOe slightly broadened, but coinciding with the line of 8:0 (8 Fe atoms and 0 Si atoms in the first coordination shell of the Fe atom) for pure iron. The addition of silicon in the alloy (X = 4 and 6) leads to a slight growth of the field at the 8:0 peak maximum of the function P(H). Apparently, the effect of Si atoms in the remote (third or fourth) coordination shells of Fe atoms is thus manifested in the change of the hyperfine field at the iron nuclei. In the alloy with X = 9.5 there is a significant fraction of the structural component of 8:0 type with a field close in value to 330 kOe, which almost completely disappears from the P(H) at X = 11. When X is in the range between 6 and 9.5, there appears a first sign of the local configuration 4:4, i.e., of four Si atoms in the first coordination shell of iron, typical of the D03 ordering. Because of the six silicon atoms in the second coordination shell of an iron atom that has only iron in the first shell – a configuration characteristic of the D03 phase, there takes place the decrease of the hyperfine field of the maximum of P(H) that corresponds to coordination 8:0. When X = 13.5, more than 80% of the nanocrystal volume is occupied by clusters of the ordered D03 phase. The concentration dependence of the relative volume fraction of the D03 phase Vrel shown in Fig. 14 was constructed out using the structural and magnetic parameters from (Yelsukov et al., 1989).
An observed fact that after TSA the peaks of 8:0 and 7:1 coordinations are broader than those after the heat treatment without stresses can be explained by the effect of residual strains (X = 6). For alloys with X ≥ 8, the distribution functions P(H) are identical in the main, but the presence of surroundings with two (6:2) and three (5:3) silicon atoms in the first coordination shell of iron atom are indicative of the silicon-concentration heterogeneity of nanocrystals and the low degree of the D03 type order in them.
4. Residual distortions and magnetic anisotropy
Thus, it is shown that in the samples subjected to TSA, there is a significant residual distortion of the nanocrystal lattice, the anisotropic character of which does not change when the concentration of silicon X varies in the range from 0 to 13.5. The relative lattice distortion along the <100> axis reaches almost 0.01, which is more than two orders of magnitude greater than the saturation magnetostriction in crystalline α-FeSi alloys (Bozorth, 1993). Most likely, the residual strain in the α-Fe(Si) nanocrystals is retained owing to rigidity of the surrounding amorphous matrix (Ohnuma et al., 2005).
The formation of the magnetic anisotropy after annealing and cooling under tensile load (TSA treatment) can be attributed to the residual elastic stresses in the alloy nanoparticles (Glaser et al., 1991; Herzer, 1994). Then, the change of the magnetic-anisotropy type from longitudinal to transverse is explained, respectively, by changing the sign of the magnetoelastic coupling in the nanocrystals from positive to negative with increasing the silicon content. The magnetoelastic Villari effect consisting in the change of magnetization in magnetic materials under the influence of mechanical stretching is the inverse phenomenon of the magnetostriction. If at the positive Villari effect, the magnetization along the direction of elongation increases, then at the negative magnetoelastic effects, on the contrary, an elastic elongation leads to a decrease of magnetization. In the bulk α-Fe(Si) crystals with the positive magnetostriction (and the positive Villari effect), the magnetization is predominantly oriented along one of the easy axes <100> forming the smallest angle with the direction of elongation. Upon saturation of the effect, which corresponds to the strain ~ 20-25 × 10-6, almost 100% of the magnetic moments are directed along this axis. If the magnetoelastic coupling takes up a negative value, as in the ordered Fe3Si alloy, the elongation would result in a deviation of the magnetic moments of individual Fe atoms from the longitudinal direction toward the transverse direction. They will be mainly oriented along one of the easy axes <100> that is perpendicular to the direction of the elongation or has the smallest angle with the plane transverse to it. The critical point of silicon concentration CSi, in which the magnetostriction constant changes its sign (see, for example, λ100 in Fig. 15), is the value of CSi ≈ 0.12 (Bertotti & Fiorillo, 1994c).
The dependence of the magnetostriction constant λ100 on the CSi largely reflects an increase in the volume fraction of the ordered Fe3Si phase with increasing concentration of silicon (Hilfrich et al., 1994), which, as known (Bertotti & Fiorillo, 1994c), has a negative value of the magnetostriction constant (λ100 ≈ -20 × 10-6, λ111 = -5 × 10-6). With the negative magnetoelastic coupling, which is typical of the ordered Fe3Si phase, the relative compression of the lattice should cause an increase in the magnetization along the direction of reduction.
The microstructure of the nanocrystalline FeSiBNbCu alloy can be represented as a huge number of the isotropically oriented α-Fe (Si) nanocrystals of about 10 nm in average grain diameter with the bcc lattice, and the clusters of nonmagnetic fcc Cu(Fe) grains about 5 nm placed in the residual amorphous matrix phase Fe(Nb)-B (Yoshizawa, 2006). The concentration of silicon in the nanoparticles is a few percent higher than its average concentration in the alloy (Serikov et al., 2006). For example, at an average value of X = 13.5, the silicon content in the nanocrystals reaches 18 at.%. Therefore, if we assume that the magnetoelastic properties of the crystalline Fe-Si alloys can be transferred to the nanocrystalline ones, then the critical point at which the sign of the magnetoelastic coupling changes will be the average silicon concentration X ≈ 9 in the Fe87-XSiXB9Nb3Cu1 alloy.
At this point it should be noted the role of Fe3Si phase in the formation of the transverse magnetic anisotropy. The iron-silicon alloy in the ordered Fe3Si state (D03 structure) is characterized by a negative constant of the magnetoelastic coupling. The transverse orientation of the magnetization with respect to the extension direction is characteristic of it, the so-called the transverse Villari effect; and negative magnetostriction − in the magnetic field, the length of the sample in the direction of the field application decreases. Therefore, the transverse magnetic anisotropy induced in the nanocrystalline alloy ribbons after the SA treatment should be attributed to the presence of a substantial fraction of Fe3Si phase in the nanocrystals. The presence of Fe3Si phase in the nanocrystals grown during NCA or TSA was supported by the following experimental observations. Firstly, as well as in the crystalline α-FeSi alloys the line a(X) (inset in Fig. 4) describing the dependence of the nanocrystal lattice parameter on the silicon concentration has an inflection point at X ≈ 9.5 associated with the D03 ordering. Secondly, in the low-angle part of the diffraction patterns of the Fe87-XSiXB9Nb3Cu1 alloy samples subjected to nanocrystallization (by NCA or TSA) with X = 11 and 13.5, there are the superstructure peaks (½ ½ ½) and (100). And finally, the distribution functions P(H) obtained from the Mössbauer spectra has a weak manifestation of the contribution from the coordination of 4:4 (four Si and four Fe atoms in the first shell of an iron atom) at X = 8, and a further significant increase of it with X ≥ 9. The relative volume fraction of Fe3Si phase obtained from these data reaches 80% at X = 13.5. The observed changes in the phase composition of the nanocrystals depending on the silicon concentration correlate with the magnetic properties (a form of the magnetic hysteresis loops and the magnetic anisotropy constant). When X ≥ 9.5 the constant KTSA is negative and grows in magnitude. Most clearly, the transverse magnetic anisotropy is pronounced in the form of the loops for samples with X = 11 and 13.5. For the same values of X, the magnetization comes from the sample plane as observed after the TSA treatment.
Fig. 16a represents schematically one of the 48 equivalent spherical triangles. The direction of tensile stress application relative to the nanocrystalline axes is shown by the radius–vector for three most interesting cases, namely, along , , and . We remind that the nanocrystals have random orientation in the Fe–Si–Nb–Cu–B alloy and, hence, a random one relative to the ribbon axis. The distortions of the nanocrystal lattice are different for the radius–vectors within this spherical triangle. The nearly tetragonal distortions correspond to the case where the radius–vector is directed along an easy-magnetization <100> axis of a nanocrystal, for example,  in Fig. 16a. After TSA in the nanocrystals of the Fe87-XSiXB9Nb3Cu1 alloys with the positive magnetoelastic coupling constant (longitudinal Villari effect) at X < 9, the magnetization will be oriented along one of the easy axes <100> that makes the smallest angle with the axis of residual strain or with the direction of load application P. In this case the maximum deviation from the longitudinal direction can be ~ 55 for those nanocrystals that have  axis parallel to the direction of tensile load application P. But for this direction the residual distortion is minimum, if any is. If the nanocrystal axis  P, then the deviation of magnetization from the longitudinal axis of the ribbon will be 45 . The value of the residual strain along the  - Δd110/d110 is not more then ½ of Δd100/d100. Since the greatest value of the relative lattice distortion was observed in nanocrystals oriented with one of the axes <100> along P, then the magnetization in these nanocrystals is oriented parallel to P, making the largest contribution to the formation of the longitudinal magnetic anisotropy. Since the density of the magnetic anisotropy energy is proportional to the interplanar spacing strain Δd/d, then the nanocrystals with tetragonal deformation of lattice will give the largest contribution to the energy of the longitudinal magnetic anisotropy, when one of the easy magnetization axes <100> is parallel to the ribbon axis and P.
For X > 9, the magnetoelastic coupling in nanocrystals becomes negative (transverse Villari effect). The tetragonal distortions correspond to the case when the radius vector is directed along the easy axis <100> of the nanocrystal in Fig. 16b - . Owing to the transverse Villari effect, the magnetization of these nanocrystals is to be directed with equal probability along either  or . The residual distortion along  results in the nanocrystal magnetization along  as shown in Fig. 16b. The magnetization of the nanocrystals with the axis of <111> type parallel to the ribbon axis (i.e. the tensile stress direction) is to be directed along one of the easy axes <100>. Their deviation from the ribbon axis is about 55 . Thus, these nanocrystals also give contribution to the transverse magnetization (Fig. 16b). The radius–vectors <111> and <100> correspond to two limiting cases: the magnetization direction of a nanocrystal makes an angle of 55 and 90 with the ribbon axis, and of 35 and 0 with the transverse plane, respectively. The magnetization directions of all the nanocrystals must lie within ±35 to the plane normal to P. Taking into account the random orientation of nanocrystals and the anisotropy of lattice strains in the nanocrystals, we can estimate the ratio of the magnetization components that are perpendicular and parallel to the ribbon axis as at least 3 to 1. The anisotropy of the residual strains facilitates an increase in the transversal component nature of the magnetic anisotropy induced during TSA as the greatest contribution to the energy of the transverse magnetic anisotropy is provided by the nanocrystals in which one of the easy axes is oriented along the ribbon and stretched along the same axis. At the same time, the anisotropy of the ribbon shape is an important factor. Therefore, the magnetization vector in the nanocrystals predominantly oriented along the easy magnetization axis that is closer to the ribbon plane.
Summing up the results of X-ray diffraction and NGR-spectroscopy studies on the structure of the nanocrystals in soft magnetic alloys Fe87-XSiXB9Nb3Cu1 (X = 0 - 13.5) in the initial state, i.e. after quenching from the melt, after the nanocrystallization annealing and the tensile stress annealing, we can conclude that all the alloys in the initial state independently of the silicon concentration (X) are in a state with an ultra fine grain (average grain size of about 2 nm). Their structure is isotropic, it does not depend on the direction of observation (i.e., along or across the ribbon, or direction of the ribbon output upon quenching) and silicon content. After the nanocrystallization annealing, the intensity distribution in the diffraction patterns remains isotropic, but with increasing crystallite size up to 10-12 nm. A complete set of reflections, corresponding the bcc lattice of the alloy α-Fe (Si) becomes allowed. When the content of silicon X = 11 and 13.5, in the diffraction patterns there appear superstructure peaks of the D03 phase - (½ ½ ½) and (100). After TSA along the direction of the tensile load application, there is an increase in the interplanar distances and the decrease in the transverse direction. The strain is anisotropic: within the accuracy of the experiment, no distortions in the <111> directions are observed, and the distortions in the <100> directions are maximum and reach 1%. It is shown that the type (longitudinal or transverse) of the magnetic anisotropy depends on the sign (positive or negative) of the magnetoelastic coupling in nanocrystals, which, in turn, is determined by the contribution of the ordered phase Fe3Si, characterized by a negative constant of magnetoelastic coupling.
For nanocrystalline alloy Fe87-XSiXB9Nb3Cu1, a phenomenological model of the mechanism of inducing the longitudinal magnetic anisotropy at X < 9 and the transverse magnetic anisotropy at X > 9 due to the anisotropic distortions of nanocrystals was proposed.
This study was supported by the Russian Foundation for Basic Research (project no. 10-02-00435) and by the Presidium of Russian Academy of Sciences (project no. 09-Π-2-1035).