Neutron study of magnetic correlations in rare-earth-free Mn-Bi magnets

We report the results of an unpolarized small-angle neutron scattering (SANS) study on Mn-Bi-based rare-earth-free permanent magnets. The magnetic SANS cross section is dominated by long-wavelength transversal magnetization fluctuations, and has been analyzed in terms of the Guinier-Porod model and the distance distribution function. This provides the radius of gyration which, in the remanent state, ranges between about $220-240\mathrm{nm}$ for the three different alloy compositions investigated. Moreover, computation of the distance distribution function, in conjunction with results for the so-called $s$ parameter obtained from the Guinier-Porod model, indicates that the magnetic scattering of a ${\mathrm{Mn}}_{45}{\mathrm{Bi}}_{55}$ sample has its origin in slightly shape-anisotropic structures.

Figure 1

Sketch of the SANS setup. The external magnetic field to the sample, ${\mathbf{H}}_0$, is applied perpendicular to the incident neutron beam. The scattering vector $\mathbf{q}$ is defined as the difference between the wave vectors of the incident (${\mathbf{k}}_0$) and the scattered (${\mathbf{k}}_1$) neutrons, i.e., $\mathbf{q}={\mathbf{k}}_0-{\mathbf{k}}_1$. Its magnitude for elastic scattering, $q=(4\pi /\lambda )sin(\psi /2)$, depends on the mean wavelength $\lambda$ of the neutrons (selected by the velocity selector) and on the scattering angle $\psi$. The angle $\theta$ specifies the orientation of $\mathbf{q}$ on the two-dimensional position-sensitive detector. In the small-angle approximation the component of $\mathbf{q}$ along ${\mathbf{k}}_0$ is neglected, so that $\mathbf{q}\cong \{0,q_y,q_z\}=q\{0,sin\theta ,cos\theta \}$ for ${\mathbf{H}}_0\perp {\mathbf{k}}_0$.

Figure 2

Room-temperature magnetization curves $M\left(H_0\right)$ of various Mn-Bi samples (see inset).

Figure 3

Illustration of the neutron data analysis procedure. Shown is the total two-dimensional SANS cross section of a ${\mathrm{Mn}}_{55}{\mathrm{Bi}}_{45}$ rare-earth-free permanent magnet (${\mathbf{H}}_0\perp {\mathbf{k}}_0$; logarithmic color scale). (a) Total (nuclear and magnetic) SANS cross section $d\mathrm{\Sigma }/d\mathrm{\Omega }$ at ${\mu }_0{H}_0=2.2\mathrm{T}$ (${\mathbf{H}}_0$ is horizontal in the plane, see inset). (b)  $d\mathrm{\Sigma }/d\mathrm{\Omega }$ at remanence ($0\mathrm{T}$). (c) Magnetic (difference) SANS cross section $d{\mathrm{\Sigma }}_M/d\mathrm{\Omega }$ at remanence. The dashed white line emphasizes the slight elongation of $d{\mathrm{\Sigma }}_M/d\mathrm{\Omega }$ along ${\mathbf{H}}_0$ [54].

Figure 4

Two-dimensional magnetic (difference) SANS cross section $d{\mathrm{\Sigma }}_M/d\mathrm{\Omega }$ of Mn–Bi rare-earth-free permanent magnets at the remanent state (${\mathbf{H}}_0\perp {\mathbf{k}}_0$; logarithmic color scale).

Figure 5

$2\pi$-azimuthally-averaged $d{\mathrm{\Sigma }}_M/d\mathrm{\Omega }$ of Mn–Bi rare-earth-free permanent magnets in the remanent state (${\mathbf{H}}_0\perp {\mathbf{k}}_0$; log-log scale). Solid lines: Fit to the generalized Guinier-Porod model [Eqs. (2, 3, 4, 5)]. Error bars are selectively shown only for the ${\mathrm{Mn}}_{55}{\mathrm{Bi}}_{45}$ sample.

Figure 6

Magnetic field dependence of the Guinier radii $R_G$ resulting from the generalized Guinier-Porod model. Lines are a guide to the eyes.

Figure 7

Distance distribution functions $p\left(r\right)$ [Eq. (6)] of the remanent-state Mn-Bi data shown in Fig. 5. Dashed line: Analytical $p\left(r\right)\propto r^2(1-\frac{3r}{4R}+\frac{r^3}{16R^3})$ of a sphere of radius $R=290\mathrm{nm}$, corresponding to a Guinier radius of $R_G=\sqrt{\frac{3}{5}}R=225\mathrm{nm}$.