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

Artem Malyeyev<EMAIL_ADDRESS>Department of Physics and Materials
Science, University of Luxembourg, 162A Avenue de la Faïencerie, L-1511
Luxembourg, Grand Duchy of Luxembourg  Ivan Titov  Department of Physics and
Materials Science, University of Luxembourg, 162A Avenue de la Faïencerie,
L-1511 Luxembourg, Grand Duchy of Luxembourg  Philipp Bender  Now at: Heinz
Maier-Leibnitz Zentrum (MLZ), Technische Universität München, D-85748
Garching, Germany  Department of Physics and Materials Science, University of
Luxembourg, 162A Avenue de la Faïencerie, L-1511 Luxembourg, Grand Duchy of
Luxembourg  Mathias Bersweiler  Department of Physics and Materials Science,
University of Luxembourg, 162A Avenue de la Faïencerie, L-1511 Luxembourg,
Grand Duchy of Luxembourg  Vitaliy Pipich  Forschungszentrum Jülich GmbH,
Jülich Centre for Neutron Science (JCNS) at Heinz Maier-Leibnitz Zentrum
(MLZ), Lichtenbergstraße 1, D-85748 Garching, Germany  Sebastian Mühlbauer
Heinz Maier-Leibnitz Zentrum (MLZ), Technische Universität München, D-85748
Garching, Germany  Semih Ener  Institute of Materials Science, Technical
University of Darmstadt, D-64287 Darmstadt, Germany  Oliver Gutfleisch
Institute of Materials Science, Technical University of Darmstadt, D-64287
Darmstadt, Germany  Andreas Michels<EMAIL_ADDRESS>Department of
Physics and Materials Science, University of Luxembourg, 162A Avenue de la
Faïencerie, L-1511 Luxembourg, Grand Duchy of Luxembourg

###### Abstract

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 indicate that the magnetic
scattering of a Mn45Bi55 sample has its origin in slightly shape-anisotropic
structures.

##  I Introduction

Permanent magnets are the subject of an intense worldwide research effort,
which is due to their technological relevance as integral components in many
electronics devices or motors [1, 2]. Currently, the worldwide permanent
magnet market is dominated by two classes of magnets: (i) High-performance Nd-
Fe-B with a maximum energy product of $(BH)_{\mathrm{max}}\cong
400\,\mathrm{kJm}^{-3}$ at $300\,\mathrm{K}$, and (ii) low-performance ferrite
magnets with a $(BH)_{\mathrm{max}}\lesssim 40\,\mathrm{kJm}^{-3}$. There is a
need for a medium-performance and cost-effective material working at
temperatures as high as $500\,\mathrm{K}$ (typical operating temperature of
motors), where the $(BH)_{\mathrm{max}}$ of ternary Nd-Fe-B is unacceptably
low; in other words, a low-cost permanent magnet material is required which
may replace Nd-Fe-B in such applications where the full potential of the
latter is not exploited. Rare-earth-free Mn-based permanent magnets are a
prime candidate for filling the gap between Nd-Fe-B and the ferrites [3]. Mn-
based magnets in general [4, 5, 6, 7] and the low-temperature phase of Mn–Bi
binary alloy in particular [8, 9, 10, 11, 12, 13] have received a lot of
attention lately, not the least because of a positive temperature coefficient
of the magnetic anisotropy rendering high-temperature applications attractive
[11].

Most of the published studies on Mn–Bi-based magnets have focused on integral
measurement techniques and on engineering aspects [14, 15, 16, 17, 18, 19, 20,
21]. Yet, the macroscopic magnetic properties arise, at least partly, from
spatial variations in the magnitude and orientation of the magnetization
vector field $\mathbf{M}(\mathbf{r})$ on a mesoscopic length scale (a few nm
up to the micron scale). Therefore, a deeper understanding of the correlations
and long-wavelength magnetization fluctuations is of paramount importance both
from the basic science point of view as well as from a materials science
perspective aiming to optimize the properties of the material.

In this paper we report the results of unpolarized magnetic small-angle
neutron scattering (SANS) experiments on cold-compacted isotropic Mn–Bi
magnets. The magnetic SANS method is ideally suited to characterize the
magnetic structure and interactions on the mesoscopic length scale, since it
provides information on both variations of the magnitude and orientation of
the magnetization $\mathbf{M}(\mathbf{r})$ in the bulk of the material. This
technique has previously been applied to study e.g. the structures of magnetic
nanoparticles [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], soft magnetic
nanocomposites and complex alloys [33, 34, 35, 36, 37, 38, 39], proton domains
[40, 41, 42], magnetic steels [43, 44, 45, 46, 47], or Heusler-type alloys
[48, 49, 50, 51, 52] (see Ref. 53 for a recent review on magnetic SANS). When
conventional SANS is extended by very small-angle neutron scattering, as in
the present case, then the accessible real-space length scale may range from a
few nanometers up to the micron regime. Here, we aim to estimate the
characteristic size of microstructural-defect-induced spin perturbations in
the polycrystalline microstructure of Mn–Bi magnets.

The paper is organized as follows: Section II furnishes the details on the
sample synthesis and on the neutron experiments. Section III displays the
expressions for the SANS cross section, the generalized Guinier-Porod model,
and the distance distribution function. Section IV then presents and discusses
the neutron results. Finally, Sec. V summarizes the main findings of this
study. We refer to the Supplemental Material for additional supporting
information [54].

##  II Experimental

All Mn–Bi samples were synthesized by using conventional melting and milling,
similar to Refs. 55, 56, 11. Initial ingots were prepared by arc melting high-
purity elements ($99.8\,\%$ for Mn and $99.99\,\%$ for Bi) and were annealed
under Ar atmosphere for $24\,\mathrm{h}$ at $300\,^{\circ}$C followed by
quenching in water at room temperature. Subsequently, the resulting ingots
were hand crushed under N2 atmosphere into powder (with a particle size
$<60\,\mu$m) and ball milled for $2\,\mathrm{h}$ in hexane with a ball-to-
powder weight ratio of $1$$:$$10$ at $150\,\mathrm{RPM}$ [21]. The ball milled
powder was washed in ethanol, magnetically separated, dried, and cold
compacted at a pressure of $\sim 1.0\,\mathrm{GPa}$ into $10\times 5\times
1\,\mathrm{mm}$ pellets. Magnetization isotherms were recorded using a
vibrating sample magnetometer (Cryogenic,
$\mu_{0}H_{\mathrm{max}}=14\,\mathrm{T}$). For more details on sample
preparation and characterization using x-ray diffraction and scanning electron
microscopy see Ref. 55.

The unpolarized SANS experiments were conducted at room temperature at the
very small-angle neutron scattering instrument KWS-3 [57] at the Heinz Maier-
Leibnitz Zentrum (MLZ), Garching, Germany. We employed samples grinded down to
a thickness of $0.1\,\mathrm{mm}$. The external magnetic field
$\mathbf{H}_{0}$ was applied perpendicular to the incident neutron beam
($\mathbf{H}_{0}\perp\mathbf{k}_{0}$), and a mean wavelength of
$\lambda=12.8\,\text{\AA}$ with a bandwidth of $\Delta\lambda/\lambda\cong
10\,\%$ (FWHM) was chosen; see Fig. 1 for a sketch of the neutron setup. The
covered momentum transfer ranges between about
$0.002\,\mathrm{nm}^{-1}\lesssim q\lesssim 0.2\,\mathrm{nm}^{-1}$. The neutron
experiments were performed by first applying a field of $2.2\,\mathrm{T}$ and
then reducing the field following the magnetization curve (compare Fig. 2).
SANS data reduction (correction for background scattering, transmission,
detector efficiency) was carried out using the QTI-SAS software package [58].
Additional neutron measurements on the instrument SANS-1 at MLZ [59] were
performed (see Ref. 54 for details).

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
$\mathbf{H}_{0}\perp\mathbf{k}_{0}$.

##  III SANS cross section, generalized Guinier-Porod model, and distance
distribution function

In this section the expressions for the unpolarized SANS cross section, for
the Guinier-Porod model, as well as for the distance distribution function are
displayed. For more background details on magnetic SANS the reader is referred
to Refs. 60, 53.

###  III.1 Unpolarized SANS cross section

For the perpendicular scattering geometry
($\mathbf{H}_{0}\perp\mathbf{k}_{0}$) the elastic unpolarized SANS cross
section $d\Sigma/d\Omega$ at momentum-transfer vector $\mathbf{q}$ can be
written as [60, 53]:

$\begin{split}\frac{d\Sigma}{d\Omega}(\mathbf{q})=&\frac{8\pi^{3}}{V}b_{H}^{2}\left(b_{H}^{-2}\right.|\widetilde{N}|^{2}+|\widetilde{M}_{x}|^{2}+|\widetilde{M}_{y}|^{2}\cos^{2}\theta+\\\
&+|\widetilde{M}_{z}|^{2}\sin^{2}\theta-\Big{.}(\widetilde{M}_{y}\widetilde{M}_{z}^{\ast}+\widetilde{M}_{y}^{\ast}\widetilde{M}_{z})\sin\theta\cos\theta\Big{)},\end{split}$
(1)

where $V$ is the scattering volume, $b_{H}=2.91\times
10^{8}\,\mathrm{A^{-1}m^{-1}}$ is the magnetic scattering length in the small-
angle regime, $\widetilde{N}(\mathbf{q})$ and
$\widetilde{\mathbf{M}}(\mathbf{q})$ denote, respectively, the Fourier
transforms of the nuclear scattering length density $N(\mathbf{r})$ and of the
magnetization vector field $\mathbf{M}(\mathbf{r})$, the angle $\theta$ is
measured between $\mathbf{q}$ and $\mathbf{H}_{0}$, and the asterisk
“$\,{}^{\ast}\,$” marks the complex-conjugated quantity. We would like to
emphasize that the magnetization of a bulk ferromagnet is a function of the
position $\mathbf{r}=\\{x,y,z\\}$ inside the material, i.e.,
$\mathbf{M}=\mathbf{M}(x,y,z)$, and that, consequently,
$\widetilde{\mathbf{M}}=\widetilde{\mathbf{M}}(q_{x},q_{y},q_{z})$. However,
the above Fourier components represent projections into the plane of the two-
dimensional detector, i.e., the $q_{y}$-$q_{z}$-plane for
$\mathbf{H}_{0}\perp\mathbf{k}_{0}$ ($q_{x}\cong 0$) (compare Fig. 1). This
shows that SANS predominantly measures correlations in the plane perpendicular
to the incident neutron beam.

In our data analysis we subtract the total nuclear and magnetic SANS cross
section at the highest available field from the data at lower fields. This
eliminates the nuclear SANS contribution in Eq. (1) and yields the purely
magnetic SANS cross section $d\Sigma_{M}/d\Omega$; to be more precise, the
subtraction procedure results in a magnetic SANS cross section which depends
on the differences of the magnetization Fourier components at the two fields
considered, e.g.,
$\Delta|\widetilde{M}_{x}|^{2}=|\widetilde{M}_{x}|^{2}(H_{0})-|\widetilde{M}_{x}|^{2}(H_{\mathrm{max}})$
(and similarly for the other Fourier components). The field dependence of the
transversal magnetization Fourier components $\widetilde{M}_{x}$ and
$\widetilde{M}_{y}$ is different from, and usually much larger than, the
longitudinal component $\widetilde{M}_{z}$ (see Fig. 8 in [61]); more
specifically, $\widetilde{M}_{x,y}$ are usually larger at lower field than at
higher field, whereas $\widetilde{M}_{z}$ may weakly increase with increasing
field. Effectively, for Mn–Bi, this entails that the difference SANS cross
section is non-negative at all $\mathbf{q}$ and $H_{0}$ investigated.

###  III.2 Generalized Guinier-Porod model

The magnetic SANS cross section $d\Sigma_{M}/d\Omega$ was analyzed in terms of
the generalized Guinier-Porod model, developed by Hammouda [62] in order to
describe the $2\pi$-azimuthally-averaged scattering from both spherical and
nonspherical objects. The model is purely empirical and essentially decomposes
the $I(q)=\frac{d\Sigma_{M}}{d\Omega}(q)$ curve into a Guinier region for
$q\leq q_{1}$ and into a Porod region for $q\geq q_{1}$. Both parts of the
scattering curve are then joined by demanding the continuity of the Guinier
and Porod laws (and of their derivatives) at $q_{1}$; more specifically [62]:

$\displaystyle I(q)$ $\displaystyle=$
$\displaystyle\frac{G}{q^{s}}\exp\left(-\frac{q^{2}R_{G}^{2}}{3-s}\right)\hskip
7.11317pt\mathrm{for}\hskip 7.11317ptq\leq q_{1},$ (2) $\displaystyle I(q)$
$\displaystyle=$ $\displaystyle\frac{D}{q^{n}}\hskip
7.11317pt\mathrm{for}\hskip 7.11317ptq\geq q_{1},$ (3)

where the scaling factors $G$ and $D$, the Guinier radius $R_{G}$, the
dimensionality factor $s$, and the Porod power-law exponent $n$ are taken as
independent parameters. From the continuity of the Guinier and Porod functions
and their derivatives it follows that:

$\displaystyle q_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{R_{G}}\left[\frac{(n-s)(3-s)}{2}\right]^{1/2},$ (4)
$\displaystyle D$ $\displaystyle=$ $\displaystyle
Gq_{1}^{n-s}\exp\left(-\frac{q_{1}^{2}R_{G}^{2}}{3-s}\right),$ (5)

where $n>s$ and $s<3$ must be satisfied. Note that $q_{1}$ is not a fitting
parameter, but an internally computed value [via Eq. (4)]. For a dilute set of
homogeneous spherical particles with sharp interfaces one expects $s=0$,
$n=4$, and $R_{G}^{2}=\frac{3}{5}R^{2}$, where $R$ is the particle radius.

###  III.3 Distance distribution function

In addition to the above analysis using the generalized Guinier-Porod model,
we have model-independently calculated the distance distribution function
[63]:

$p(r)=r^{2}\int_{0}^{\infty}\frac{d\Sigma_{M}}{d\Omega}(q)j_{0}(qr)q^{2}dq,$
(6)

where $j_{0}(qr)=\sin(qr)/(qr)$ denotes the zeroth-order spherical Bessel
function. This provides information on the characteristics (e.g., size and
shape) of the scattering objects [64, 65], and on the presence of
interparticle correlations [66, 67].

##  IV Results and Discussion

Figure 2: Room-temperature magnetization curves $M(H_{0})$ of various Mn–Bi
samples (see inset).

The room-temperature magnetization curves of the Mn–Bi samples are shown in
Fig. 2. The coercivity $H_{c}$ of the samples is found to be between
$0.47$$-$$0.56\,\mathrm{T}$ for all compositions, while the saturation
magnetization $M_{s}$ varies from about $33\,\mathrm{Am^{2}kg^{-1}}$
(Mn55Bi45) to $36\,\mathrm{Am^{2}kg^{-1}}$ (Mn50Bi50) to
$34\,\mathrm{Am^{2}kg^{-1}}$ (Mn45Bi55). These values fall short of the
theoretical saturation magnetization of the low-temperature Mn–Bi phase
($80\,\mathrm{Am^{2}kg^{-1}}$) and indicate a magnetic content of $\sim
40$$-$$45\,\%$. A field larger than $1.8\,\mathrm{T}$ is sufficient to close
the hysteresis loop and to reach the reversible part of the $M(H_{0})$ curve.
This is important because in the neutron-data analysis the measurement at
$2.2\,\mathrm{T}$ is used for subtraction to eliminate the nuclear scattering.

Figure 3: Illustration of the neutron data analysis procedure. Shown is the
total two-dimensional SANS cross section of a Mn55Bi45 rare-earth-free
permanent magnet ($\mathbf{H}_{0}\perp\mathbf{k}_{0}$; logarithmic color
scale). (a) Total (nuclear and magnetic) SANS cross section $d\Sigma/d\Omega$
at $\mu_{0}H_{0}=2.2\,\mathrm{T}$ ($\mathbf{H}_{0}$ is horizontal in the
plane, see inset). (b) $d\Sigma/d\Omega$ at remanence ($0\,\mathrm{T}$). (c)
Magnetic (difference) SANS cross section $d\Sigma_{M}/d\Omega$ at remanence.
The dashed white line emphasizes the slight elongation of
$d\Sigma_{M}/d\Omega$ along $\mathbf{H}_{0}$ [54]. Figure 4: Two-dimensional
magnetic (difference) SANS cross section $d\Sigma_{M}/d\Omega$ of Mn–Bi rare-
earth-free permanent magnets at the remanent state
($\mathbf{H}_{0}\perp\mathbf{k}_{0}$; logarithmic color scale).

Figure 3 illustrates part of our neutron data analysis procedure, which is
based on the subtraction of the total $d\Sigma/d\Omega$ at the highest field
[Fig. 3(a)] from data at lower fields [Fig. 3(b)]. This eliminates the strong
and presumably isotropic nuclear SANS contribution [compare Eq. (1)] and
provides access to the purely magnetic SANS cross section
$d\Sigma_{M}/d\Omega$ [Fig. 3(c)] [60, 53]. As can be seen in Fig. 4, the in
this way obtained $d\Sigma_{M}/d\Omega$ are anisotropic, elongated along the
direction parallel to the applied magnetic field $\mathbf{H}_{0}$; compare the
sector-averaged data in [54]. By comparison to the expression for
$d\Sigma/d\Omega$ in the $\mathbf{H}_{0}\perp\mathbf{k}_{0}$ geometry [Eq.
(1)] this angular anisotropy can be related to the transversal Fourier
component $|\widetilde{M}_{y}|^{2}\cos^{2}\theta$ in $d\Sigma_{M}/d\Omega$.
The feature is observable for all Mn–Bi samples in the remanent state (Fig.
4), and it suggests the presence of long-range spin-misalignment correlations
on a real-space length scale of at least a few ten to a few hundreds of
nanometers.

In order to quantify the range of the magnetic correlations we have
azimuthally-averaged the two-dimensional magnetic SANS cross sections and
fitted the resulting data to the generalized Guinier-Porod (GP) model [Eqs.
(2)$-$(5)]. The results of the weighted nonlinear least-squares fitting
procedure for the remanent-state data are displayed in Fig. 5 (solid lines)
and demonstrate that the GP model can very well describe the $q$-dependence of
$d\Sigma_{M}/d\Omega$ [54]. The obtained Guinier radii $R_{G}$ are shown in
Fig. 6, while Table 1 lists (for the remanent state) the results for the
remaining fit parameters, the dimensionality parameter $s$ and the asymptotic
power-law exponent $n$.

The origin of magnetic SANS is due to spatial mesoscale variations in the
magnitude and orientation of the magnetization. Such magnetization
fluctuations may be caused by microstructural defects (e.g., dislocations,
interfaces, pores) via the magnetoelastic coupling of the magnetization to the
strain field of the defect [68]. The range and the amplitude of defect-induced
spin disorder can be suppressed by an applied field. In the following we
associate the value of $R_{G}$ with the size of such perturbed, nonuniformly
magnetized regions around defects.

Figure 5: $2\pi$-azimuthally-averaged $d\Sigma_{M}/d\Omega$ 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)$-$(5)]. Error bars are selectively shown only for the Mn55Bi45 sample. Table 1:  Results of the fit analysis on Mn–Bi rare-earth-free permanent magnets using the generalized Guinier-Porod model [62] (remanent state). | Mn55Bi45 | Mn50Bi50 | Mn45Bi55
---|---|---|---
$R_{G}$ ($\mathrm{nm}$) | 224 $\pm$ 8 | 242 $\pm$ 12 | 218 $\pm$ 13
$s$ | 0.30 $\pm$ 0.07 | 0.35 $\pm$ 0.10 | 1.07 $\pm$ 0.09
$n$ | 3.38 $\pm$ 0.04 | 3.42 $\pm$ 0.04 | 3.58 $\pm$ 0.03
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.

The Guinier radii in Fig. 6 do not exhibit a systematic variation with the
composition of the Mn–Bi samples. At remanence, their values range between
$R_{G}\sim 220$$-$$240\,\mathrm{nm}$. While the $R_{G}$ for the Mn55Bi45
sample are field independent within error bars, the Mn45Bi55 specimen exhibits
a decrease of $R_{G}$ with increasing field, from about $220\,\mathrm{nm}$ at
remanence to $\sim 100\,\mathrm{nm}$ at $1.5\,\mathrm{T}$. Such a behavior is
in qualitative agreement with the suppression of spin-misalignment
fluctuations around defects with increasing applied field [69]. On the other
hand, the Mn50Bi50 specimen seems to exhibit an increase of $R_{G}$ with
increasing field, from about $240\,\mathrm{nm}$ at remanence to $\sim
285\,\mathrm{nm}$ at $1.0\,\mathrm{T}$. However, in view of the large
uncertainties in the $R_{G}$-values of this sample, an unambiguous
determination of the field behavior of the $R_{G}(H_{0})$ data set is
difficult.

The Porod exponents of $n\sim 3.4$$-$$3.6$ are systematically reduced below
the sharp-interface value of $n=4$. In the context of particle scattering this
observation could be interpreted as a smoothing of the surfaces of the
scattering objects [62]. However, for magnetic SANS, where continuous rather
than sharp scattering-length density variations are at the origin of the
scattering, asymptotic power-law exponents smaller than 4 have only been
reported for amorphous magnets [35]. Similarly, exponentially correlated
magnetization fluctuations would give rise to $n=4$, corresponding to a
Lorentzian-squared cross section. Therefore, the unusually low $n$-values
observed in Mn–Bi remain to be explored by future experimental and theoretical
neutron studies.

Within the generalized Guinier-Porod model the $s$-parameter models
nonspherical objects [62]. For three-dimensional globular particles (or
domains), $s$ is expected to take on a value of $s=0$. The Mn55Bi45 and
Mn50Bi50 samples are close to this value, whereas Mn45Bi55 exhibits $s=1.07$,
which would indicate scattering due to elongated rod-like objects. The latter
observation is surprising in view of the fact that extended electron-
microscopy investigations on similar samples, albeit on a different length
scale, did not reveal the presence of shape-anisotropic particles [55].

In order to further understand the differences between the samples (regarding
the $s$-parameter), we have model-independently calculated the distance
distribution function $p(r)$ [Eq. (6)]. The results for $p(r)$ in Fig. 7 are
qualitatively consistent with the numerical fit analysis using the generalized
Guinier-Porod model. The Mn55Bi45 and Mn50Bi50 samples both exhibit a $p(r)$
which is typical for globular scatterers [70]. Yet, a small shoulder at the
larger distances points towards the presence of slightly anisotropic
structures. By contrast, the $p(r)$ of the Mn45Bi55 sample clearly shows a
broad maximum at $r\cong 470\,\mathrm{nm}$ followed by a long tail at the
larger $r$, suggesting that the scattering originates from shape-anisotropic
elongated objects (compare Fig. 5 in the review by Svergun and Koch [64]). The
broad maximum of $p(r)$ at the smaller distances of the Mn45Bi55 specimen
corresponds to the shorter dimension of the structure. This finding is in line
with the behavior of the $s$-parameter obtained from the Guinier-Porod model.

Figure 7: Distance distribution functions $p(r)$ [Eq. (6)] of the remanent-
state Mn–Bi data shown in Fig. 5. Dashed line: Analytical $p(r)\propto
r^{2}(1-\frac{3r}{4R}+\frac{r^{3}}{16R^{3}})$ of a sphere of radius
$R=290\,\mathrm{nm}$, corresponding to a Guinier radius of
$R_{G}=\sqrt{\frac{3}{5}}R=225\,\mathrm{nm}$.

The Guinier radius $R_{G}$, which is one of the central outcomes of our
neutron analysis (Fig. 6), represents the characteristic size over which
microstructural-defect-induced perturbations in the spin structure are
transmitted by the exchange interaction into the surrounding crystal lattice;
in other words, $R_{G}$ is considered to be a measure for the size of
inhomogeneously magnetized regions around lattice imperfections. This length
scale is of relevance for the understanding of the coercivity mechanism in
Mn–Bi magnets—domain nucleation versus pinning—which is currently discussed in
the literature [71, 56, 72]. For instance, the nucleation of a reverse domain
in a grain usually starts at a defect site, where the magnetic anisotropy may
be reduced relative to the bulk phase. Therefore, the presented neutron
methodology (analysis of difference data using the generalized Guinier-Porod
model and calculation of the distance ditribution function) provides a means
to systematically correlate the spin-misalignment length, which is a property
of the defect, to the macroscopic parameters (e.g., coercivity, maximum energy
product) of a permanent magnet. Moreover, previous studies (e.g., [56])
demonstrated enhanced coercivity over a wide temperature range with shifting
alloy composition towards Bi, which was explained by differences in the grain-
size distribution. Our SANS analysis indicates that an increase of the Bi
content results in increasingly elongated magnetic structures (Fig. 7). Thus,
a further increase of Bi might be a valid approach to enhance the magnetic
hardness of the compound via shape anisotropy. In this respect, magnetic SANS
permits the determination of the relevant figures of merit ($R_{G}$, $s$,
$n$), which are otherwise not accessible by integral measurement techniques.

##  V Conclusion and outlook

We have investigated the magnetic microstructure of rare-earth-free Mn–Bi
magnets by means of unpolarized very small-angle neutron scattering (SANS).
The magnetic scattering cross section, which has been obtained by subtracting
the total nuclear and magnetic scattering signal at $2.2.\,\mathrm{T}$ from
data at lower fields, has been described in terms of the generalized Guinier-
Porod model. The value of the Guinier radius is interpreted as the size of
inhomogeneously magnetized regions around microstructural defects. We find
that the spin-misalignment correlations are in the range of $\sim
100$$-$$300\,\mathrm{nm}$ for the compositions studied. Moreover, in
particular using the distance distribution function, our analysis indicates
that the magnetic scattering of the Mn45Bi55 sample is related to shape-
anisotropic structures, while the scattering of Mn55Bi45 and Mn50Bi50 has its
origin in more globular-like objects. The neutron-data subtraction procedure
(low field minus high field) eliminates the nuclear scattering contribution,
which is not further analyzed. In this respect, neutron imaging techniques
could be employed for the characterization of the nuclear grain microstructure
and morphology inside the bulk of the magnet [73]. In future investigations
the usage of polarized neutrons will be beneficial (e.g., [74, 26, 28, 50]),
since it then becomes possible to directly measure the purely magnetic SANS
cross section without the coherent nuclear contribution. Likewise, extending
the range of momentum transfers to the so-called ultra SANS regime
($q_{\mathrm{min}}\cong 10^{-4}\,\mathrm{nm}^{-1}$) permits following the
correlations up to the $10$ micron range. Temperature-dependent neutron
measurements will allow one to obtain mesoscale information on the relation
between the magnetic microstructure and the positive temperature coefficient
of the magnetic anisotropy found for this material.

## Acknowledgements

Artem Malyeyev, Philipp Bender, and Andreas Michels acknowledge financial
support from the National Research Fund of Luxembourg (AFR and CORE SANS4NCC
grants). We thank the Heinz Maier-Leibnitz Zentrum for the provision of
neutron beamtime.

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