Third-order effect in magnetic small-angle neutron scattering by a spatially inhomogeneous medium

Magnetic small-angle neutron scattering (SANS) is a powerful tool for investigating nonuniform magnetization structures inside magnetic materials. Here, we consider a ferromagnetic medium with weakly inhomogeneous uniaxial magnetic anisotropy, saturation magnetization, and exchange stiffness, and derive, to second order in the amplitudes of the inhomogeneities, the micromagnetic solutions for the equilibrium magnetization textures. Further, we compute the corresponding magnetic SANS cross section up to the third order. For the special case of scattering geometry where the incident neutron beam is perpendicular to the applied magnetic field, twice the cross section along the direction orthogonal to both the field and the neutron beam cancels the cross section along the field direction in the second order. This cancellation does not depend on the defect shape and amplitudes of the exchange inhomogeneities. Hence, such a cross-section difference has only a third-order contribution in the amplitudes of the inhomogeneities. It provides a separate gateway for a deeper analysis of the sample's magnetic structure. We derive and analyze analytical expressions for the dependence of this difference on the scattering-vector magnitude for the case of spherical Gaussian inhomogeneities.

Figure 1

Dependence of the mean-squared inhomogeneity function $\langle {\tilde{I}}^2\rangle =N\langle a_n^2\rangle {(v/V)}^2{\rm Y}(qs,\tau )$ on the inclusion shape $\tau$ for different values of $\mu =qs$ in the range from 0 to 2 in equal steps of $0.1$. $\langle {\tilde{I}}^2\rangle$ has an extremum at $\tau =1$, corresponding to spherical defects. The left side of the plot corresponds to planar defects, while the right side to needlelike ones.

Figure 2

Angular dependence of the anisotropic terms in the perpendicular magnetic SANS cross section at different values of $h_q$. (a) displays the first and (b) the third term in Eq. (49).

Figure 3

UFO-like magnetic SANS cross-section shape (omitting the constant prefactor of $8{\pi }^3{b}_{\mathrm{H}}^2{M}_0^2\langle a_n^2\rangle v^{2}N/V$) at small $h=0.01$ and $q$ in a sample with spherical ($\tau =1$) Gaussian inclusions. The other parameters for this plot are $\kappa =1$, $s=1$, $L_0=1$. The outer contour corresponds to a cross-section value of 15, which increases for inner contours in equal steps of 100. There is a very sharp maximum at the center. The inset shows experimental data displaying the UFO-like SANS cross-section shape from Fig. 2 in Ref. [12].

Figure 4

The functions $g_{\mathrm{A}}(\mu ,h,\lambda )$ (solid lines) and their approximation by decaying exponentials (dotted lines) for different values of $h$ at fixed $\lambda$ (upper plot) and for different values of $\lambda$ at fixed $h$ (lower plot).

Figure 5

The functions $g_{\mathrm{MS}}(\mu ,h,\lambda )$. The curves correspond to the same values of parameters as those in Fig. 4.