Analysis of magnetic neutron-scattering data of two-phase ferromagnets

We have analyzed magnetic-field-dependent small-angle neutron scattering (SANS) data of soft magnetic two-phase nanocomposite ferromagnets in terms of a recent micromagnetic theory for the magnetic SANS cross section [Honecker and Michels, Phys. Rev. B 87, 224426 (2013)]. The approach yields a value for the average exchange-stiffness constant and provides the Fourier coefficients of the magnetic anisotropy field and the magnetostatic field, which is related to jumps of the magnetization at internal interfaces.

Figure 1

Sketch of the scattering geometry.

Figure 2

Illustration of the neutron-data analysis procedure according to Eq. (11). The total $d\Sigma /d\Omega$ ($•$) at $q^{☆}=0.114{\mathrm{nm}}^{-1}$ is plotted versus the response functions $R_H$ and $R_M$ at $A=4.7\mathrm{pJ}/\mathrm{m}$ and experimental field values (in mT) of 1270, 312, 103, 61, 42, 33. The plane represents a fit to Eq. (11). The intercept of the plane with the $d\Sigma /d\Omega$ axis provides the residual SANS cross section $d{\Sigma }_{\mathrm{res}}/d\Omega$, while $S_H$ and $S_M$ are obtained from the slopes of the plane (slopes of the thick black and red lines).

Figure 3

Qualitative comparison between experiment, analytical theory, and numerical micromagnetic simulations. Upper row: Experimental spin-misalignment SANS cross sections $d{\Sigma }_M/d\Omega$ of ${\mathrm{Fe}}_{89}{\mathrm{Zr}}_7{\mathrm{B}}_3\mathrm{Cu}$ (Ref. 21) in the plane of the two-dimensional detector at selected applied magnetic fields (see insets). The $d{\Sigma }_M/d\Omega$ were obtained by subtracting the scattering at a saturating field of $1994\mathrm{mT}$. ${\mathbf{H}}_0$ is horizontal. Middle row: Prediction by the micromagnetic theory for $d{\Sigma }_M/d\Omega$, Eq. (4), at the same field values as above. For both ${\tilde{M}}_z\left(qR\right)$ and $h\left(qR\right)$ we have used the form factor of the sphere with a radius of $R=6\mathrm{nm}$. Furthermore, the following materials parameters were used: $A=3.1\mathrm{pJ}/\mathrm{m}$; ${\mu }_0{M}_s=1.26\mathrm{T}$; ${\mu }_0{H}_p=0.01\mathrm{T}$; ${\mu }_0\Delta M=0.05\mathrm{T}$; this particular choice of parameters is motivated by the outcome of our data analysis (see below). Lower row: Results of full-scale three-dimensional micromagnetic simulations for $d{\Sigma }_M/d\Omega$ (Refs. 11, 12, 13). Linear color scale is used in all subfigures. For each data set, we have, respectively, normalized $d{\Sigma }_M/d\Omega$ to its highest value at the lowest field of $45\mathrm{mT}$. Yellow color corresponds to “high” $d{\Sigma }_M/d\Omega$ and blue color to “low” values of $d{\Sigma }_M/d\Omega$.

Figure 4

Azimuthally-averaged $d\Sigma /d\Omega$ of (a) ${\mathrm{Fe}}_{89}{\mathrm{Zr}}_7{\mathrm{B}}_3\mathrm{Cu}$ (Ref. 21) and (b) ${\left({\mathrm{Fe}}_{0.985}{\mathrm{Co}}_{0.015}\right)}_{90}{\mathrm{Zr}}_7{\mathrm{B}}_3$ (Ref. 24) at selected applied magnetic fields (log-log scale). Field values (in mT) from bottom to top: (a) 1994, 321, 163, 85, 45; (b) 1270, 312, 103, 61, 33. Solid lines in (a) and (b): Fit to the micromagnetic theory, Eq. (11); the solid lines connect the computed $d\Sigma /d\Omega$ at each value of $q$ and $H_i$. ($\circ$) Residual scattering cross sections $d{\Sigma }_{\mathrm{res}}/d\Omega$. The insets depict the respective (reduced) weighted mean-square deviation between experiment and fit, ${\chi }^2/\nu$, as a function of the exchange-stiffness constant $A$.

Figure 5

Best-fit results for the scattering function of the anisotropy field $S_H=8{\pi }^3{V}^{-1}{b}_H^2{\left|h\left(q\right)\right|}^2$ and for the scattering function of the longitudinal magnetization $S_M=8{\pi }^3{V}^{-1}{b}_H^2{\left|{\tilde{M}}_z\left(q\right)\right|}^2$ of (a) ${\mathrm{Fe}}_{89}{\mathrm{Zr}}_7{\mathrm{B}}_3\mathrm{Cu}$ and (b) ${\left({\mathrm{Fe}}_{0.985}{\mathrm{Co}}_{0.015}\right)}_{90}{\mathrm{Zr}}_7{\mathrm{B}}_3$ (log-log scale). $d{\Sigma }_{\mathrm{nuc}}/d\Omega =8{\pi }^3{V}^{-1}{\left|\tilde{N}\right|}^2$ denotes the nuclear SANS, which was, respectively, obtained by subtracting the $|{\tilde{M}}_z{|}^2$ scattering from the residual SANS cross section $d{\Sigma }_{\mathrm{res}}/d\Omega$ [compare Eq. (3)].