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Alben et al. [42] strongly depends on the degree of magnetic coupling. This stress anisotropy is induced, as has been noted previously, inside the grains. The maximum value (around 1000 J/m3) is clearly lower than 8000 J/m3, corresponding to the magnetocrystalline anisotropy of the a-Fe(Si) grains; therefore, the origin of the stress anisotropy should be strongly connected to the internal stresses in the FeSi nanocrystals. An interesting question should be one related to the coupling between these two phases with a large interface area such as is the case of Fe-rich nanocrystalline alloys. For this, a deep knowledge about the nature of the interface results are to be determinant. Unavoidable mixing of atoms of the interface gives rise to the formation of thin layers of alloys of unknown composition, which makes this study complicated.

### 2.5. Saturation Magnetostriction Behavior

The effective magnetostriction, kf, observed in nanocrystalline alloys at different stages of crystallization, has been interpreted as a volumetrically weighted balance among two contributions with opposite signs originating from the bcc-FeSi grains (AJr) and residual amorphous matrix (A^m) according to Herzer [42]:

where p is the volumetric fraction of the crystalline phase. Therefore, assuming negative and positive sign as for the nanocrystalline and amorphous phase, respectively, the variations of \f (including the change of sign observed in some nanocrystalline composition) can be interpreted as a consequence of the variations of the p parameter. Although this simple approximation gives the qualitative explanation of the effective magnetostriction in Fe-based nanocrystalline alloys [42], it does not consider many effects occurring in the real materials. More exact calculations take into account that the magnetostriction of the residual amorphous phase is not constant but depends on the crystalline fraction due to the enrichment with B and Nb [61, 78]. Consequently, Eq. (6) can be modified in the form [78]:

where k is a parameter that expresses changes of the magnetostriction in the residual amorphous phase with evolution of the crystallization. In many cases, even this model does not fit the experimental results, demonstrating that the effective magnetostriction in nanocrystalline material cannot be described by a simple superposition of the crystalline and amorphous components [61]. In the case of the FeZrBCu nanocrystalline system in which the bcc-Fe phase is formed, the model described does not fit the experimental data, even through the k^m(p) dependence as was shown by Slawska-Waniewska and Zuberek in [79, 80]. They considered this to be an additional contribution to the effective magnetostriction, which arises from the enhanced surface to volume ratio describing interfacial effects [79-83]. Therefore, the Eq. (7) of the effective magnetostriction could be approximated by:

where the last term describes the effects at the interfaces and depends on the surface-to-volume ratio for the individual grain, as well as on the magnetostriction constant AS, which characterizes the crystal-amorphous interface.

Equation (8) is the basic dependence, which can be used to interpret the experimental data on the effective magnetostriction in Fe-based nanocrystalline alloys at different stages of crystallization. The composition of the Fe(Si) grains (depending on the annealing temperature) should be considered, giving rise to different values of the magnetostriction constant for the crystalline phase. The appropriate values of Acsr can be obtained from the compositional dependence of the saturation magnetostriction in the polycrystalline a-Fe100—xSix, shown in Figure 8 [42, 84]. Thus, the first term in the Eq. (8) can be treated as the well-defined one.

Figure 9 presents the crystallization behavior and accompanying changes in the magnetostriction of the classical Finemet (Fe73-5Cu1Nb3Si13i5B9) alloy published by Gutierrez et al. [85]. The analysis of these data, according to Eq. (8), allows (i) estimation of the contribution from the crystalline phase (see Fig. 8a, where the values of A°sr were found from the combined Figs. 8 and 9a), and then (ii) fitting of the experimental (Af — pAcsr) on p dependence in Eq. (8). The results, both experimental points and the fitted curve (solid line) are shown in Figure 9b.

Assuming spherical a-Fe(Si) grains, with radius R, the last term of Eq. (8) can be expressed as 3ASS/R, and the magnetostriction constant, which describes the interface Ass, can be estimated. For the soft nanocrystalline alloys (Finemet and FeZrBCu alloys) [42, 61, 79], R = 5 nm, and, thus, Ass has been found to vary in the range 4.5-7.1 x 10—6 nm. These results are one order of magnitude smaller than values of the surface magnetostriction obtained in multilayer systems. However, investigations of Fe/C multilayers have shown that not only the value but also the sign of the surface magnetostriction constant depends on the structure of the iron layer, and it has been found that for crystalline iron, Ass (bcc-Fe) = 45.7 x 10—6 nm, whereas, for the amorphous iron, AS (am-Fe) = —31 x 10—6 nm [86]. Thus, the value of the interface magnetostriction obtained in the nanocrystalline systems is within the range of the surface magnetostriction constant estimated for thin iron layers. It should be noted that, contrary to Fe/c multilayers, in the nanocrystalline materials, both the crystalline and amorphous phases are

Figure 8. Saturation magnetostriction of the polycrystalline a-Fe100—x • Six. Reprinted with permission from [80], A. Slawska-Waniewska et al., Mater. Sci. Eng. A (Supplement), 220 (1997). © 1997, Elsevier Science.

Figure 8. Saturation magnetostriction of the polycrystalline a-Fe100—x • Six. Reprinted with permission from [80], A. Slawska-Waniewska et al., Mater. Sci. Eng. A (Supplement), 220 (1997). © 1997, Elsevier Science.

crystalline fraction

Figure 9. Si content in a-Fe(Si) grains (a) and magnetostriction of the Fe73_5Cu1Nb3Si13_5B9 alloy (b) versus crystalline fraction. Reprinted with permission from [80], A. Slawska-Waniewska et al., Mater. Sci. Eng. A (Supplement), 220 (1997). © 1997, Elsevier Science.

crystalline fraction

Figure 9. Si content in a-Fe(Si) grains (a) and magnetostriction of the Fe73_5Cu1Nb3Si13_5B9 alloy (b) versus crystalline fraction. Reprinted with permission from [80], A. Slawska-Waniewska et al., Mater. Sci. Eng. A (Supplement), 220 (1997). © 1997, Elsevier Science.

magnetic, and they are coupled through exchange and dipolar interactions. It must thus be expected that the magnetic interactions in the system can affect the magnetoelastic coupling constant at the grain-matrix interfaces. In addition, the structure and properties of the particular surfaces that are in contact, as well as local strains at the grain boundaries also should be considered.

The problem of the surface/interface magnetostriction, however, requires further studies, which, in particular, should include measurements at temperatures above the Curie point of the amorphous matrix, where only ferromagnetic grains should contribute to the effective magnetostriction, simplifying a separation between bulk and surface contributions.

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