Hall effect for neutrons scattered by an $A$-phase MnSi crystal

We study a neutron diffraction by A phase of MnSi using a dynamical theory of diffraction and three wave approximation. We show that the neutron diffraction is asymmetrical with respect to an incident plane. The asymmetry depends on a sign of an external magnetic field. This phenomenon can be considered as the Hall effect for neutrons.

Figure 1

The neutron diffraction geometry. The subscripts $i,r,d_1$, and $d_2$ denote incident, reflected, and two diffracted beams. Unpolarized neutrons fall on a MnSi surface with a glancing angle $\alpha$. The angle $\beta$ determines the orientation of the incident plane with respect to the $\left(x,z\right)$ plane. Rings with arrows symbolize a Skyrmion lattice with lattice vectors ${\vec{a}}_1$ and ${\vec{a}}_2$. Corresponding reciprocal vectors are ${\vec{g}}_1,{\vec{g}}_2$, and ${\vec{g}}_3$. These vectors lie in the plane $\left(y,x\right)$. (a) The general view of the system. (b) The $z$-direction view. (b) The most symmetric case when $\beta =0$.

Figure 2

Diffracted intensity as a function of the angles $\alpha$ and $\beta$. ${\alpha }_0=1.842$ grad. (a) and (b) Correspond to diffraction directions ${\vec{k}}_{d_1}$ and ${\vec{k}}_{d_2}$. Ovals show the regions where only one reciprocal vector satisfies the Bragg condition. In these regions two wave approximation is valid. Diffracted intensity in these regions obey the relation $I_{d_1}(\alpha ,\beta )=I_{d_2}(\alpha ,-\beta )$. Circles show the regions in which two reciprocal vectors satisfy the Bragg condition and three wave approximation has to be used. In these regions skew scattering appears leading to difference between diffraction peaks ${\vec{k}}_{d_1}$ and ${\vec{k}}_{d_2}$ and violation of the relation $I_{d_1}(\alpha ,\beta )=I_{d_2}(\alpha ,-\beta )$.

Figure 3

(a) The quantity $2\Delta I(\alpha ,\beta )/[I_{d_1}({\alpha }_0,0)+I_{d_2}({\alpha }_0,0)]$ as a function of the angles $\alpha$ and $\beta$. (b) The dependence of the averaged quantity $2\Delta I(\alpha ,\beta )/[I_{d_1}({\alpha }_0,0)+I_{d_2}({\alpha }_0,0)]$ on the angles $\alpha$ and $\beta$.

Figure 4

Intensity of the waves diffracted in directions ${\vec{k}}_{d_{1,2}}$ (see Fig. 1) as a function of the glancing angle $\alpha$. The solid line corresponds to neutrons traveling to the right from the incident plane, and the dashed line corresponds to neutrons traveling to the left. $I_0$ is the intensity of an incident beam.

Figure 5

The dependence of the averaged diffracted intensity on angles $\alpha$ and $\beta$. ${\alpha }_0=1.842$ grad. (a) and (b) Correspond to diffraction directions ${\vec{k}}_{d_1}$ and ${\vec{k}}_{d_2}$. Averaging was done over both angles $\alpha$ and $\beta$. Averaging area is $0.02{}^{\circ }$. Circles show the regions in which two reciprocal vectors satisfy the Bragg condition and three wave approximation has to be used. In these regions skew scattering appears leading to a difference between diffraction peaks ${\vec{k}}_{d_1}$ and ${\vec{k}}_{d_2}$ and violation of the relation $I_{d_1}(\alpha ,\beta )=I_{d_2}(\alpha ,-\beta )$.