# Uniaxial Polarization Analysis of Bulk Ferromagnets: Theory and First
Experimental Results

Artem Malyeyev<EMAIL_ADDRESS>Ivan Titov  Department of Physics and
Materials Science, University of Luxembourg, 162A Avenue de la Faïencerie,
L-1511 Luxembourg, Grand Duchy of Luxembourg  Charles D. Dewhurst  Institut
Laue-Langevin, 6 Rue Jules Horowitz, B.P. 156, F-38042 Grenoble Cedex 9,
France  Kiyonori Suzuki  Department of Materials Science and Engineering,
Monash University, Clayton, Victoria 3800, Australia  Dirk Honecker
<EMAIL_ADDRESS>ISIS Neutron and Muon Source, Rutherford Appleton
Laboratory, Didcot, United Kingdom  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

Based on Brown’s static equations of micromagnetics, we compute the uniaxial
polarization of the scattered neutron beam of a bulk magnetic material. The
theoretical expressions are compared to experimental data on a soft magnetic
nanocrystalline alloy. The micromagnetic SANS theory provides a general
framework for polarized real-space neutron methods, and it opens up a new
avenue for magnetic neutron data analysis on magnetic microstructures.

##  I Introduction

Polarized neutron scattering is one of the most powerful techniques for
investigating the structure and dynamics of condensed matter, in particular
magnetic materials and superconductors [1]. Based on the seminal papers by
Bloch, Schwinger, and Halpern and Johnson [2, 3, 4, 5], the theory of
polarized neutron scattering has been worked out in the early 1960’s by Maleev
and Blume [6, 7]. Several classic experimental studies [8, 9, 10, 11, 12, 13,
14, 15] have demonstrated the basic principles and paved the way for todays
three-dimensional cryogenic polarization-analysis device (CRYOPAD) [16, 17,
18, 19]. With this technique it becomes possible to measure 16 correlation
functions, which provide important information on the nuclear and magnetic
structure of materials (see Refs. 20, 21 for textbook expositions of polarized
neutron scattering).

However, for the scattering of cold (long-wavelength) neutrons along the
forward direction—as implemented on a small-angle neutron scattering (SANS)
instrument—it has only in recent years become possible to perform ‘routinely’
neutron-polarization analysis (retaining the full two-dimensional scattering
information); more specifically, uniaxial (also called longitudinal or one
dimensional) polarization analysis, where the polarization of the scattered
neutrons is analyzed along the direction of the initial polarization [9].
Clearly, this progress is due to the development of efficient 3He spin filters
(e.g. [22, 23, 24]), which, in contrast to e.g. single-crystal analyzers, can
be used over a rather broad wavelength range and cover a large detector
acceptance angle. Note also that Refs. 25, 26, 27 report on the development of
a novel neutron spin filter, based on the strong spin dependence of the
neutron scattering on protons. For the combination of uniaxial polarization
analysis with SANS, the term POLARIS has been coined [28]. In contrast to
CRYOPAD, which generally demands the sample to be in a zero magnetic field
environment, POLARIS allows for the application of large magnetic fields.

The POLARIS method has been successfully employed for studying the
superparamagnetic response of concentrated ferrofluids [28], proton domains in
deuterated solutions [29, 30, 31], the multiferroic properties of
$\mathrm{HoMn}_{3}$ single crystals [32], the role of nanoscale
heterogeneities for the magnetostriction of Fe-Ga alloys [33, 34], local weak
ferromagnetism in $\mathrm{BiFeO}_{3}$ [35], nanometer-sized magnetic domains
and coherent magnetization reversal in an exchange-bias system [36],
precipitates in Heusler-based alloys [37], the magnetic microstructure of
nanoscaled bulk magnets [38, 39], the internal spin structure of nanoparticles
[40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50], or Invar alloys [51].
Polarization analysis allows further to reveal the direction of the magnetic
anisotropy in single-crystalline spin systems, e.g. an easy plane versus an
easy axis anisotropy or the confinement of the propagation vector along
certain crystallographic directions in chiral and other exotic magnets [52,
53]. In all of the above mentioned studies, the spin-resolved SANS cross
sections were obtained and analyzed, but the polarization of the scattered
neutrons was not further investigated.

Historically, this is the domain of the neutron depolarization technique (see
e.g. [54, 55, 56, 57, 58, 10, 11, 59, 60, 61] and references therein), where
one measures the change in the polarization of a polarized neutron beam after
transmission through a partially-magnetized magnetic material. Analysis of the
$3\times 3$ depolarization matrix yields information on e.g. the average
domain size and the domain magnetization. This type of polarization analysis
on a SANS instrument has been termed ‘vector analysis of polarization’ by
Okorokov and Runov [19]. Alternatively, it has been demonstrated that the
neutron spin-echo technique can resolve magnetic small-angle scattering [62,
63]. Spin-echo small-angle neutron scattering (SESANS) provides information on
correlations on a length scale from about $10\,\mathrm{nm}$ to
$10\,\mu\mathrm{m}$. In SESANS, the neutron spin precesses in a constant
magnetic field and the neutron runtime difference due to sample scattering
results in the dephasing of the neutron spins and in a loss of the measured
beam polarization. Magnetic scattering can result in neutron spin-flip events
that act as an additional optical element reversing the sense of the Larmor
precession. The change of the neutron spin due to magnetic scattering can be
exploited to study magnetic systems.

More recently, the method of polarized neutron dark-field contrast imaging
(DFI) has been introduced for spatially-resolved small-angle scattering
studies of magnetic microstructures [64]. First experimental results on a
sintered Nd-Fe-B magnet demonstrated that not only dark-field contrast from
half-polarized SANS measurements can be observed, but that it also becomes
possible to separate and retrieve dark-field contrast for all spin-flip and
non-spin flip channels separately. The polarized DFI method bears great
potential for analyzing real-space magnetic correlations on a macroscopic
length scale, well beyond of what can be probed with a conventional SANS
instrument. Similarly, first measurements of micron-sized magnetic
correlations have been performed with an alternative neutron precession
technique called spin-echo modulated small-angle neutron scattering (SEMSANS)
[65]. With this setup, the spin manipulations are performed before the sample
so that the measurement is not sensitive to large stray fields (related e.g.
to the sample environment), and even allows studies under beam depolarizing
conditions. The polarization analyzer discriminates the polarization parallel
to the analyzing direction, such that the scattering cross sections for the
opposite neutron spin state are probed at the sample. The two-dimensional
neutron polarization modulation observed on the detector is then integrated to
yield the one-dimensional correlation function of the system.

In this work, we present a micromagnetic SANS theory for the uniaxial
polarization of the scattered neutrons of bulk magnetic materials. The
continuum theory of micromagnetics allows one to characterize the large-scale
magnetization distribution of polycrystalline magnets, which is determined
e.g. by magnetic-anisotropy and saturation-magnetization fluctuations,
antisymmetric exchange, and dipolar stray fields. Since the validity of
micromagnetic theory extends to the micrometer regime, the here-developed
theoretical framework may as well serve as a basis for the above-mentioned
polarized neutron techniques (SESANS, DFI, SEMSANS). The derived theoretical
expressions are tested against experimental SANS data on a soft magnetic
nanocrystalline alloy.

The paper is organized as follows: Section II summarizes the elementary
equations of polarized neutron scattering. Section III sketches the basic
steps and ideas of the micromagnetic SANS theory. Section IV displays the
final expressions for the polarization of the scattered neutrons, discusses
special sector averages, and shows the results for the $2\pi$-azimuthally-
averaged saturated state. Section V furnishes the details of the polarized
SANS experiment and on the investigated sample, while section VI presents and
discusses the analysis of the experimental results. Section VII summarizes the
main findings of this study. Appendix A features the expressions for the spin-
resolved SANS cross sections in terms of the Fourier components of the
magnetization, which enter the final expressions for the polarization, while
Appendix B showcases some computed examples for the polarization.

##  II Uniaxial SANS Polarization Analysis

Fig. 1 depicts a sketch of a typical uniaxial neutron polarization analysis
setup. We consider the most relevant cases where the externally applied
magnetic field $\mathbf{H}_{0}$, which defines the polarization axis for both
the incident and scattered neutrons, is either perpendicular or parallel to
the wave vector $\mathbf{k}_{0}$ of the incident neutron beam. Note that in
both scattering geometries $\mathbf{H}_{0}$ is assumed to be parallel to the
$\mathbf{e}_{z}$-direction of a Cartesian laboratory coordinate system.

Figure 1: Sketch of the SANS setup and of the two most often employed
scattering geometries in magnetic SANS experiments. (a) Applied magnetic field
$\mathbf{H}_{0}$ perpendicular to the incident neutron beam
($\mathbf{k}_{0}\perp\mathbf{H}_{0}$); (b)
$\mathbf{k}_{0}\parallel\mathbf{H}_{0}$. The momentum-transfer or scattering
vector $\mathbf{q}$ corresponds to 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, $q=|\mathbf{q}|=(4\pi/\lambda)\sin(\psi)$, depends on the mean
wavelength $\lambda$ of the neutrons and on the scattering angle $2\psi$. For
a given $\lambda$, sample-to-detector distance $L_{\mathrm{SD}}$, and distance
$r_{\mathrm{D}}$ from the center of the direct beam to a certain pixel element
on the detector, the $q$-value can be obtained using $q\cong
k_{0}\frac{r_{\mathrm{D}}}{L_{\mathrm{SD}}}$. The symbols ‘P’, ‘F’, and ‘A’
denote, respectively, the polarizer, spin flipper, and analyzer, which are
optional neutron optical devices. Note that a second flipper after the sample
has been omitted here. In spin-resolved SANS (POLARIS) using a 3He spin
filter, the transmission (polarization) direction of the analyzer can be
switched by $180^{\circ}$ by means of an rf pulse. SANS is usually implemented
as elastic scattering ($k_{0}=k_{1}=2\pi/\lambda$), and the component of
$\mathbf{q}$ along the incident neutron beam [i.e. $q_{x}$ in (a) and $q_{z}$
in (b)] is neglected. The angle $\theta$ may be conveniently used in order to
describe the angular anisotropy of the recorded scattering pattern on a two-
dimensional position-sensitive detector. Image taken from Ref. 66.

In a classical picture, the polarization $\mathbf{P}$ of a neutron beam
containing $N$ spins can be defined as the average over the individual
polarizations $\mathbf{P}_{j}$ of the neutrons as [67]:

$\displaystyle\mathbf{P}=\frac{1}{N}\sum^{N}_{j=1}\mathbf{P}_{j},$ (1)

where $0\leq|\mathbf{P}|\leq 1$. In experimental SANS studies the beam is
usually partially polarized along a certain guide-field direction
(quantization axis), which we take here as the $z$-direction. Assuming that
the expectation values of the perpendicular polarization components vanish,
i.e. $P_{x}=P_{y}=0$, and that $P_{z}=P$, one can then introduce the fractions

$\displaystyle p^{+}=\frac{1}{2}\left(1+P\right)\hskip
14.22636pt\mathrm{and}\hskip 14.22636ptp^{-}=\frac{1}{2}\left(1-P\right)$ (2)

of neutrons in the spin-up ($+$) and spin-down ($-$) state, with

$\displaystyle p^{+}+p^{-}=1\hskip 14.22636pt\mathrm{and}\hskip
14.22636ptp^{+}-p^{-}=P.$ (3)

Obviously, for an unpolarized beam $p^{+}=p^{-}=0.5$ and $P=0$, while $P=+1$
($p^{+}=1$) or $P=-1$ ($p^{-}=1$) for a fully polarized beam.

When there is an additional analyzer behind the sample, configured such that
it selects only neutrons with spins either parallel or antiparallel to the
initial polarization, then one can distinguish four scattering cross sections
(scattering processes) [7, 9, 67]: two of which conserve the neutron-spin
direction ($++$ and $--$), the so-called non-spin-flip cross sections

$\displaystyle\frac{d\Sigma^{++}}{d\Omega}$ $\displaystyle=$ $\displaystyle
K\left[b^{-2}_{\mathrm{H}}|\widetilde{N}|^{2}+b^{-1}_{\mathrm{H}}(\widetilde{N}\widetilde{Q}_{z}^{\ast}+\widetilde{N}^{\ast}\widetilde{Q}_{z})+|\widetilde{Q}_{z}|^{2}\right],$
$\displaystyle\frac{d\Sigma^{--}}{d\Omega}$ $\displaystyle=$ $\displaystyle
K\left[b^{-2}_{\mathrm{H}}|\widetilde{N}|^{2}-b^{-1}_{\mathrm{H}}(\widetilde{N}\widetilde{Q}_{z}^{\ast}+\widetilde{N}^{\ast}\widetilde{Q}_{z})+|\widetilde{Q}_{z}|^{2}\right],$

and two cross sections which reverse the neutron spin ($+-$ and $-+$), the so-
called spin-flip cross sections

$\displaystyle\frac{d\Sigma^{+-}}{d\Omega}$ $\displaystyle=$ $\displaystyle
K\left[|\widetilde{Q}_{x}|^{2}+|\widetilde{Q}_{y}|^{2}-i(\widetilde{Q}_{x}\widetilde{Q}_{y}^{\ast}-\widetilde{Q}_{x}^{\ast}\widetilde{Q}_{y})\right],$
$\displaystyle\frac{d\Sigma^{-+}}{d\Omega}$ $\displaystyle=$ $\displaystyle
K\left[|\widetilde{Q}_{x}|^{2}+|\widetilde{Q}_{y}|^{2}+i(\widetilde{Q}_{x}\widetilde{Q}_{y}^{\ast}-\widetilde{Q}_{x}^{\ast}\widetilde{Q}_{y})\right].$

In these expressions, $K=8\pi^{3}V^{-1}b_{\mathrm{H}}^{2}$, where $V$ denotes
the scattering volume, and $b_{\mathrm{H}}=2.70\times
10^{-15}\,\mathrm{m}\,\mu^{-1}_{\mathrm{B}}=2.91\times
10^{8}\,\mathrm{A^{-1}m^{-1}}$ is a constant (with $\mu_{\mathrm{B}}$ the Bohr
magneton), which relates the atomic magnetic moment $\mu_{\mathrm{a}}$ to the
atomic magnetic scattering length $b_{\mathrm{m}}$, given by [9]:

$\displaystyle
b_{\mathrm{m}}=\frac{\gamma_{\mathrm{n}}r_{0}}{2}\frac{\mu_{\mathrm{a}}}{\mu_{\mathrm{B}}}f(\mathbf{q})\cong
2.70\times
10^{-15}\,\mathrm{m}\frac{\mu_{\mathrm{a}}}{\mu_{\mathrm{B}}}f(\mathbf{q})\cong
b_{\mathrm{H}}\mu_{\mathrm{a}},$ (6)

where $\gamma_{\mathrm{n}}=1.913$ denotes the neutron magnetic moment
expressed in units of the nuclear magneton, $r_{0}=2.818\times
10^{-15}\,\mathrm{m}$ is the classical radius of the electron, and
$f(\mathbf{q})$ is the normalized atomic magnetic form factor, which we set to
unity, $f\cong 1$, along the forward direction. The function
$\widetilde{N}(\mathbf{q})$ denotes the Fourier transform of the nuclear
scattering-length density $N(\mathbf{r})$. The partial SANS cross sections,
equations (II) and (II), are written here in terms of the Cartesian components
of the Halpern–Johnson vector $\widetilde{\mathbf{Q}}$ (sometimes also denoted
as the magnetic interaction or magnetic scattering vector) [5]:

$\displaystyle\widetilde{\mathbf{Q}}=\hat{\mathbf{q}}\times\left(\hat{\mathbf{q}}\times\widetilde{\mathbf{M}}\right)=\hat{\mathbf{q}}\left(\hat{\mathbf{q}}\cdot\widetilde{\mathbf{M}}\right)-\widetilde{\mathbf{M}},$
(7)

where $\hat{\mathbf{q}}$ is the unit scattering vector, and
$\widetilde{\mathbf{M}}(\mathbf{q})=\\{\widetilde{M}_{x}(\mathbf{q}),\widetilde{M}_{y}(\mathbf{q}),\widetilde{M}_{z}(\mathbf{q})\\}$
represents the Fourier transform of the magnetization vector field
$\mathbf{M}(\mathbf{r})=\\{M_{x}(\mathbf{r}),M_{y}(\mathbf{r}),M_{z}(\mathbf{r})\\}$
of the sample under study. The three-dimensional Fourier-transform pair of the
magnetization is defined as follows:

$\displaystyle\widetilde{\mathbf{M}}(\mathbf{q})=\frac{1}{(2\pi)^{3/2}}\int\limits_{-\infty}^{+\infty}\int\limits_{-\infty}^{+\infty}\int\limits_{-\infty}^{+\infty}\mathbf{M}(\mathbf{r})\exp\left(-i\mathbf{q}\cdot\mathbf{r}\right)d^{3}r,$
(8)
$\displaystyle\mathbf{M}(\mathbf{r})=\frac{1}{(2\pi)^{3/2}}\int\limits_{-\infty}^{+\infty}\int\limits_{-\infty}^{+\infty}\int\limits_{-\infty}^{+\infty}\widetilde{\mathbf{M}}(\mathbf{q})\exp\left(i\mathbf{q}\cdot\mathbf{r}\right)d^{3}q.$
(9)

The Halpern–Johnson vector is a manifestation of the dipolar origin of
magnetic neutron scattering and it emphasizes the fact that only the
components of $\mathbf{M}$ which are perpendicular to $\mathbf{q}$ are
relevant for magnetic neutron scattering. We note that different symbols for
the Halpern–Johnson vector such as $\mathbf{M}_{\perp}$, $\mathbf{Q}_{\perp}$,
$\mathbf{S}_{\perp}$, or $\mathbf{q}$, as in the original paper by Halpern and
Johnson [5], can be found in the literature. Likewise, in many textbooks (e.g.
[21, 68]) $\widetilde{\mathbf{Q}}$ is defined with a minus sign and normalized
by the factor $2\mu_{\mathrm{B}}$, which makes it dimensionless.
$\widetilde{\mathbf{Q}}$ is a linear vector function of the components of
$\widetilde{\mathbf{M}}$. Both $\widetilde{\mathbf{Q}}(\mathbf{q})$ and
$\widetilde{\mathbf{M}}(\mathbf{q})$ are in general complex vectors. For
$\mathbf{k}_{0}\perp\mathbf{H}_{0}$ and
$\mathbf{k}_{0}\parallel\mathbf{H}_{0}$ one finds, respectively (subscripts
$\perp$ and $\parallel$ refer to the respective scattering geometry, compare
Fig. 1):

$\displaystyle\hat{\mathbf{q}}_{\perp}$ $\displaystyle=$
$\displaystyle\\{0,\sin\theta,\cos\theta\\},$ (10)
$\displaystyle\hat{\mathbf{q}}_{\parallel}$ $\displaystyle=$
$\displaystyle\\{\cos\theta,\sin\theta,0\\}.$ (11)

Inserting these expressions into equation (7) yields:

$\displaystyle\widetilde{\mathbf{Q}}_{\perp}=\left\\{\begin{array}[]{c}-\widetilde{M}_{x}\\\
-\widetilde{M}_{y}\cos^{2}\theta+\widetilde{M}_{z}\sin\theta\cos\theta\\\
\widetilde{M}_{y}\sin\theta\cos\theta-\widetilde{M}_{z}\sin^{2}\theta\end{array}\right\\},$
(15)
$\displaystyle\widetilde{\mathbf{Q}}_{\parallel}=\left\\{\begin{array}[]{c}-\widetilde{M}_{x}\sin^{2}\theta+\widetilde{M}_{y}\sin\theta\cos\theta\\\
\widetilde{M}_{x}\sin\theta\cos\theta-\widetilde{M}_{y}\cos^{2}\theta\\\
-\widetilde{M}_{z}\end{array}\right\\}.$ (19)

Inspection of equations (II) and (II) shows that the transversal components
$\widetilde{Q}_{x}$ and $\widetilde{Q}_{y}$ give rise to spin-flip scattering,
while the longitudinal component $\widetilde{Q}_{z}$ results in non-spin-flip
scattering. Furthermore, if we set $\theta=0^{\circ}$ in equation (15), which
corresponds to the case that the scattering vector is along the neutron
polarization, we see that

$\displaystyle\widetilde{\mathbf{Q}}_{\perp}^{\theta=0^{\circ}}=\left\\{\begin{array}[]{c}-\widetilde{M}_{x}\\\
-\widetilde{M}_{y}\\\ 0\end{array}\right\\},$ (23)

so that nuclear coherent and magnetic scattering are fully separated in the
perpendicular scattering geometry. In the case
$\mathbf{k}_{0}\parallel\mathbf{H}_{0}$ [equation (19)], spin-flip scattering
probes only the transversal magnetization Fourier components
$\widetilde{M}_{x,y}$, whereas the longitudinal scattering is entirely
contained in the non-spin-flip channel, in contrast to the
$\mathbf{k}_{0}\perp\mathbf{H}_{0}$ geometry. We emphasize that nuclear-spin-
dependent SANS is not taken into account in this paper, so that the
corresponding scattering contributions do not show up in equations (II) and
(II).

The total SANS cross section $d\Sigma/d\Omega$ can be expressed in terms of
the initial spin populations $p^{\pm}$ as [7, 9, 67]:

$\displaystyle\frac{d\Sigma}{d\Omega}=p^{+}\frac{d\Sigma^{++}}{d\Omega}+p^{+}\frac{d\Sigma^{+-}}{d\Omega}+p^{-}\frac{d\Sigma^{--}}{d\Omega}+p^{-}\frac{d\Sigma^{-+}}{d\Omega}.$

Inserting the above expressions for $p^{+}$ and $p^{-}$ [equations (2) and
(3)] and for the partial SANS cross sections $d\Sigma^{\pm\pm}/d\Omega$ and
$d\Sigma^{\pm\mp}/d\Omega$ [equations (II) and (II)], equation (II) evaluates
to:

$\displaystyle\frac{d\Sigma}{d\Omega}=K\left[b^{-2}_{\mathrm{H}}|\widetilde{N}|^{2}+|\widetilde{\mathbf{Q}}|^{2}+\mathbf{P}\cdot
b^{-1}_{\mathrm{H}}(\widetilde{N}\widetilde{\mathbf{Q}}^{\ast}+\widetilde{N}^{\ast}\widetilde{\mathbf{Q}})-i\mathbf{P}\cdot(\widetilde{\mathbf{Q}}\times\widetilde{\mathbf{Q}}^{\ast})\right],$
(25)

which, using $\mathbf{P}=\\{0,0,P_{z}=\pm P\\}$, can be rewritten as:

$\displaystyle\frac{d\Sigma^{\pm}}{d\Omega}=K\left[b^{-2}_{\mathrm{H}}|\widetilde{N}|^{2}+|\widetilde{\mathbf{Q}}|^{2}\pm
Pb^{-1}_{\mathrm{H}}(\widetilde{N}\widetilde{Q}_{z}^{\ast}+\widetilde{N}^{\ast}\widetilde{Q}_{z})\mp
iP(\widetilde{Q}_{x}\widetilde{Q}_{y}^{\ast}-\widetilde{Q}_{x}^{\ast}\widetilde{Q}_{y})\right].\,\,\,\,\,\,$
(26)

Since the cross section is a scalar quantity and the polarization is an axial
vector (or pseudovector), equation (25) shows that the system under study must
itself contain an axial vector. As emphasized by Maleev [69], examples for
such built-in pseudovectors are related to the interaction of a
polycrystalline sample with an external magnetic field (inducing an average
magnetization directed along the applied field), the existence of a
spontaneous magnetization in a ferromagnetic single crystal, the antisymmetric
Dzyaloshinskii–Moriya interaction (DMI), mechanical (torsional) deformation,
or the presence of spin spirals. If on the other hand there is no preferred
axis in the system, then $d\Sigma/d\Omega$ is independent of $\mathbf{P}$.
Examples include a collection of randomly oriented noninteracting nuclear
(electronic) spins, which describe the general case of nuclear (paramagnetic)
scattering at not-too-low temperatures and large applied fields, or a multi-
domain ferromagnet with a random distribution of the domains. The same
condition—existence of an axial system vector—applies for neutrons to be
polarized in the scattering process [compare the last two terms in equation
(30) below].

In the domain of magnetic SANS it is customary to denote experiments with a
polarized incident beam only, and no spin analysis of the scattered neutrons,
with the acronym SANSPOL. The two SANSPOL cross sections $d\Sigma^{+}/d\Omega$
and $d\Sigma^{-}/d\Omega$ (also sometimes denoted as the half-polarized SANS
cross sections) combine non-spin-flip and spin-flip scattering contributions,
according to ($p^{\pm}=1$):

$\displaystyle\frac{d\Sigma^{+}}{d\Omega}$ $\displaystyle=$
$\displaystyle\frac{d\Sigma^{++}}{d\Omega}+\frac{d\Sigma^{+-}}{d\Omega},$ (27)
$\displaystyle\frac{d\Sigma^{-}}{d\Omega}$ $\displaystyle=$
$\displaystyle\frac{d\Sigma^{--}}{d\Omega}+\frac{d\Sigma^{-+}}{d\Omega}.$ (28)

Finally, noting that an unpolarized beam can be viewed as consisting of $50\%$
spin-up and $50\%$ spin-down neutrons [compare equations (2) and (3)], the
unpolarized SANS cross section is obtained as [compare equation (II)]:

$\displaystyle\frac{d\Sigma}{d\Omega}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\frac{d\Sigma^{++}}{d\Omega}+\frac{d\Sigma^{+-}}{d\Omega}+\frac{d\Sigma^{--}}{d\Omega}+\frac{d\Sigma^{-+}}{d\Omega}\right)$
(29) $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\frac{d\Sigma^{+}}{d\Omega}+\frac{d\Sigma^{-}}{d\Omega}\right).$

The polarization $P_{\mathrm{f}}$ of the scattered beam along the direction of
the incident neutron polarization $P$ is obtained from the following relation
[7, 9, 67]:

$\displaystyle P_{\mathrm{f}}\frac{d\Sigma}{d\Omega}$ $\displaystyle=$
$\displaystyle
p^{+}\frac{d\Sigma^{++}}{d\Omega}+p^{-}\frac{d\Sigma^{-+}}{d\Omega}-p^{-}\frac{d\Sigma^{--}}{d\Omega}-p^{+}\frac{d\Sigma^{+-}}{d\Omega}$
(30) $\displaystyle=$ $\displaystyle
KP\left[b^{-2}_{\mathrm{H}}|\widetilde{N}|^{2}+|\widetilde{Q}_{z}|^{2}-|\widetilde{Q}_{x}|^{2}-|\widetilde{Q}_{y}|^{2}\right]$
$\displaystyle+$ $\displaystyle
K\left[b^{-1}_{\mathrm{H}}(\widetilde{N}\widetilde{Q}_{z}^{\ast}+\widetilde{N}^{\ast}\widetilde{Q}_{z})+i(\widetilde{Q}_{x}\widetilde{Q}_{y}^{\ast}-\widetilde{Q}_{x}^{\ast}\widetilde{Q}_{y})\right].$

The first four terms in the second line on the right-hand side of equation
(30) demonstrate that nuclear scattering (to be more precise, the nuclear
coherent scattering, the isotopic disorder scattering, and $1/3$ of the
nuclear-spin-dependent scattering) and scattering due to the longitudinal
component $\widetilde{Q}_{z}$ of the magnetic scattering vector
$\widetilde{\mathbf{Q}}$ do not reverse the initial polarization, while the
two transversal components $\widetilde{Q}_{x}$ and $\widetilde{Q}_{y}$ give
rise to spin-flip scattering. The last two terms in equation (30) do create
polarization: these are the familiar nuclear-magnetic interference terms
($\widetilde{N}\widetilde{Q}_{z}^{\ast}+\widetilde{N}^{\ast}\widetilde{Q}_{z}$),
which are commonly used to polarize beams, and the chiral term
$i(\widetilde{Q}_{x}\widetilde{Q}_{y}^{\ast}-\widetilde{Q}_{x}^{\ast}\widetilde{Q}_{y})$,
which is of relevance in inelastic scattering (dynamic chirality) [70, 69,
71], in elastic scattering on spiral structures and weak ferromagnets (canted
antiferromagnets) [72], or in the presence of the DMI in microstructural-
defect-rich magnets [73, 74]. We remind the reader that nuclear-spin-dependent
scattering is not taken into account in the expressions for the magnetic SANS
cross sections. In the general expression for the polarization of the
scattered neutrons, a term
$i\mathbf{P}\times(\widetilde{N}\widetilde{\mathbf{Q}}^{\ast}-\widetilde{N}^{\ast}\widetilde{\mathbf{Q}})$
appears [67], which is ignored in equation (30). This term rotates the
polarization perpendicular to the initial polarization and cannot be observed
in the uniaxial setup. We emphasize that in linear neutron polarimetry it is
not possible to distinguish between a rotation of the polarization vector and
a change of its length [9, 69].

From equation (30) it follows that the polarization
$P_{\mathrm{f}}(\mathbf{q})$ of the scattered neutron beam at momentum-
transfer vector $\mathbf{q}$ can be expressed as [6, 7, 75]:

$\displaystyle P_{\mathrm{f}}$ $\displaystyle=$
$\displaystyle\frac{p^{+}\frac{d\Sigma^{++}}{d\Omega}+p^{-}\frac{d\Sigma^{-+}}{d\Omega}-p^{-}\frac{d\Sigma^{--}}{d\Omega}-p^{+}\frac{d\Sigma^{+-}}{d\Omega}}{p^{+}\frac{d\Sigma^{++}}{d\Omega}+p^{+}\frac{d\Sigma^{+-}}{d\Omega}+p^{-}\frac{d\Sigma^{--}}{d\Omega}+p^{-}\frac{d\Sigma^{-+}}{d\Omega}},$
(31)

which for $p^{+}=1$ ($p^{-}=0$) and $p^{-}=1$ ($p^{+}=0$) reduce to,
respectively:

$\displaystyle
P_{\mathrm{f}}^{+}=\frac{\frac{d\Sigma^{++}}{d\Omega}-\frac{d\Sigma^{+-}}{d\Omega}}{\frac{d\Sigma^{++}}{d\Omega}+\frac{d\Sigma^{+-}}{d\Omega}}=1-2\frac{\frac{d\Sigma^{+-}}{d\Omega}}{\frac{d\Sigma^{+}}{d\Omega}},$
(32a) $\displaystyle
P_{\mathrm{f}}^{-}=\frac{\frac{d\Sigma^{-+}}{d\Omega}-\frac{d\Sigma^{--}}{d\Omega}}{\frac{d\Sigma^{--}}{d\Omega}+\frac{d\Sigma^{-+}}{d\Omega}}=-\left(1-2\frac{\frac{d\Sigma^{-+}}{d\Omega}}{\frac{d\Sigma^{-}}{d\Omega}}\right).$
(32b)

Note that for the following analysis, we drop the minus sign in front of the
round brackets in equation (32b). For a quantitative analysis of
$P_{\mathrm{f}}^{\pm}$, a theoretical model for the magnetization Fourier
components $\widetilde{M}_{x,y,z}(\mathbf{q})$ and for
$\widetilde{N}(\mathbf{q})$ is required. This will be discussed in the next
section.

##  III Sketch of Micromagnetic SANS Theory

In Ref. 76 we have presented a theory for the magnetic SANS cross section of
bulk ferromagnets. The approach, which considers the response of the
magnetization to spatially inhomogeneous magnetic anisotropy fields and
magnetostatic fields, is based on the continuum theory of micromagnetics,
valid close to magnetic saturation, and takes the antisymmetric DMI into
account. Here, we recall the basic steps and ideas. The starting point are the
static equations of micromagnetics for the bulk, which can be conveniently
written as [77, 78, 79]:

$\mathbf{M}(\mathbf{r})\times\mathbf{H}_{\mathrm{eff}}(\mathbf{r})=0.$ (33)

Equations (33) express the fact that at static equilibrium the torque on the
magnetization $\mathbf{M}(\mathbf{r})$ due to an effective magnetic field
$\mathbf{H}_{\mathrm{eff}}(\mathbf{r})$ vanishes everywhere inside the
material. The effective field is defined as the functional derivative of the
ferromagnet’s total energy-density functional $\epsilon$ with respect to the
magnetization:

$\displaystyle\mathbf{H}_{\mathrm{eff}}$ $\displaystyle=$
$\displaystyle-\frac{1}{\mu_{0}}\frac{\delta\epsilon}{\delta\mathbf{M}}$ (34)
$\displaystyle=$
$\displaystyle\mathbf{H}_{0}+\mathbf{H}_{\mathrm{d}}+\mathbf{H}_{\mathrm{p}}+\mathbf{H}_{\mathrm{ex}}+\mathbf{H}_{\mathrm{DMI}},$

where $\mathbf{H}_{0}$ is a uniform applied magnetic field,
$\mathbf{H}_{\mathrm{d}}(\mathbf{r})$ is the magnetostatic field,
$\mathbf{H}_{\mathrm{p}}(\mathbf{r})$ is the magnetic anisotropy field,
$\mathbf{H}_{\mathrm{ex}}=l_{\mathrm{M}}^{2}\nabla^{2}\mathbf{M}$ is the
exchange field, and
$\mathbf{H}_{\mathrm{DMI}}=-l_{\mathrm{D}}\nabla\times\mathbf{M}$ is due to
the DMI (assuming for simplicity a cubic symmetry); $\nabla^{2}$ is the
Laplace operator and $\nabla=\partial/\partial
x\,\mathbf{e}_{x}+\partial/\partial y\,\mathbf{e}_{y}+\partial/\partial
z\,\mathbf{e}_{z}$ is the gradient operator, where the unit vectors along the
Cartesian laboratory axes are, respectively, denoted with $\mathbf{e}_{x}$,
$\mathbf{e}_{y}$, and $\mathbf{e}_{z}$ ($\mu_{0}$: vacuum permeability). The
micromagnetic length scales

$l_{\mathrm{M}}=\sqrt{\frac{2A}{\mu_{0}M_{0}^{2}}}$ (35)

and

$l_{\mathrm{D}}=\frac{2D}{\mu_{0}M_{0}^{2}}$ (36)

are, respectively, related to the magnetostatic interaction and to the DMI.
The values for the DMI constant $D$ and for the exchange-stiffness constant
$A$ are assumed to be uniform throughout the material, in contrast to the
local saturation magnetization $M_{\mathrm{s}}(\mathbf{r})$, which is assumed
to depend explicitly on the position $\mathbf{r}$ (see also Ref. 80);
$M_{0}=V^{-1}\int_{V}M_{\mathrm{s}}(\mathbf{r})dV$ denotes the macroscopic
saturation magnetization of the sample, which can be measured with a
magnetometer. Typical values for the above length scales are
$l_{\mathrm{M}}\cong 5-10\,\mathrm{nm}$ [79] and $l_{\mathrm{D}}\cong
1-2\,\mathrm{nm}$. However, due to the lack of an established database for
$D$-values, the latter estimate should be considered with some care.

The solution of the linearized version of equation (33) (see Ref. 76 for
details) provides closed-form expressions for the transversal Fourier
components $\widetilde{M}_{x}(\mathbf{q})$ and
$\widetilde{M}_{y}(\mathbf{q})$. Together with theoretical models (or even
experimental data) for the longitudinal magnetization Fourier component
$\widetilde{M}_{z}(\mathbf{q})$ and for the nuclear scattering amplitude
$\widetilde{N}(\mathbf{q})$, these analytical solutions can be employed to
compute the corresponding terms in the SANS cross sections (see Appendix A)
and, hence, to evaluate the final polarizations. Averaging over the directions
of the magnetic anisotropy field in the plane perpendicular to the applied
field, the magnetic terms for the transversal magnetic field geometry
($\mathbf{k}_{0}\perp\mathbf{H}_{0}$, $q_{x}=0$) are:

$|\widetilde{M}_{x}|^{2}=\frac{p^{2}}{2}\frac{\widetilde{H}^{2}_{\mathrm{p}}\left(\left[1+p\sin^{2}\theta\right]^{2}+p^{2}l_{\mathrm{D}}^{2}q^{2}\cos^{2}\theta\right)+2\widetilde{M}_{z}^{2}(1+p)^{2}l_{\mathrm{D}}^{2}q^{2}\sin^{2}\theta}{\left(1+p\sin^{2}\theta-p^{2}l_{\mathrm{D}}^{2}q^{2}\cos^{2}\theta\right)^{2}},$
(37)
$|\widetilde{M}_{y}|^{2}=\frac{p^{2}}{2}\frac{\widetilde{H}_{\mathrm{p}}^{2}\left(1+p^{2}l_{\mathrm{D}}^{2}q^{2}\cos^{2}\theta\right)+2\widetilde{M}_{z}^{2}\left(1+pl_{\mathrm{D}}^{2}q^{2}\right)^{2}\sin^{2}\theta\cos^{2}\theta}{\left(1+p\sin^{2}\theta-p^{2}l_{\mathrm{D}}^{2}q^{2}\cos^{2}\theta\right)^{2}},$
(38)
$CT_{yz}=\widetilde{M}_{y}\widetilde{M}_{z}^{\ast}+\widetilde{M}_{y}^{\ast}\widetilde{M}_{z}=-\frac{2\widetilde{M}_{z}^{2}p\left(1+pl_{\mathrm{D}}^{2}q^{2}\right)\sin\theta\cos\theta}{1+p\sin^{2}\theta-p^{2}l_{\mathrm{D}}^{2}q^{2}\cos^{2}\theta}.$
(39)
$-2i\chi=\frac{2\widetilde{H}^{2}_{\mathrm{p}}p^{3}\left(2+p\sin^{2}\theta\right)l_{\mathrm{D}}q\cos^{3}\theta+4\widetilde{M}_{z}^{2}p(1+p)^{2}l_{\mathrm{D}}q\sin^{2}\theta\cos\theta}{\left(1+p\sin^{2}\theta-p^{2}l_{\mathrm{D}}^{2}q^{2}\cos^{2}\theta\right)^{2}},$
(40)

The results for the parallel field geometry
($\mathbf{k}_{0}\parallel\mathbf{H}_{0}$, $q_{z}=0$) are:

$|\widetilde{M}_{x}|^{2}=\frac{p^{2}}{2}\frac{\widetilde{H}_{\mathrm{p}}^{2}\left(1+p(2+p)\sin^{2}\theta\right)+2\widetilde{M}_{z}^{2}(1+p)^{2}l_{\mathrm{D}}^{2}q^{2}\sin^{2}\theta}{\left(1+p\right)^{2}},$
(41)
$|\widetilde{M}_{y}|^{2}=\frac{p^{2}}{2}\frac{\widetilde{H}_{\mathrm{p}}^{2}\left(1+p(2+p)\cos^{2}\theta\right)+2\widetilde{M}_{z}^{2}(1+p)^{2}l_{\mathrm{D}}^{2}q^{2}\cos^{2}\theta}{\left(1+p\right)^{2}},$
(42)
$CT_{xy}=\widetilde{M}_{x}\widetilde{M}_{y}^{\ast}+\widetilde{M}_{x}^{\ast}\widetilde{M}_{y}=-p^{2}\frac{\left(\widetilde{H}_{\mathrm{p}}^{2}p(2+p)+2\widetilde{M}_{z}^{2}(1+p)^{2}l_{\mathrm{D}}^{2}q^{2}\right)\sin\theta\cos\theta}{\left(1+p\right)^{2}},$
(43) $\chi=0.$ (44)

In equations (37)$-$(44), $\widetilde{H}^{2}_{\mathrm{p}}(q\xi_{\mathrm{H}})$
denotes the magnitude-square of the Fourier transform of the magnetic
anisotropy field, and $\widetilde{M}^{2}_{z}(q\xi_{\mathrm{M}})$ is the
Fourier component of the longitudinal magnetization. We also emphasize that

$\widetilde{M}_{z}\cong\widetilde{M}_{\mathrm{s}}$ (45)

is assumed in the approach-to-saturation regime. These functions characterize
the strength and spatial structure of, respectively, the magnetic anisotropy
field $\mathbf{H}_{\mathrm{p}}(\mathbf{r})$, with correlation length
$\xi_{\mathrm{H}}$, and of the local saturation magnetization
$M_{\mathrm{s}}(\mathbf{r})$, with correlation length $\xi_{\mathrm{M}}$.

$p(q,H_{\mathrm{i}})=\frac{M_{0}}{H_{\mathrm{eff}}(q,H_{\mathrm{i}})}=\frac{M_{0}}{H_{\mathrm{i}}\left(1+l_{\mathrm{H}}^{2}q^{2}\right)}$
(46)

is a dimensionless function, where

$H_{\mathrm{eff}}(q,H_{\mathrm{i}})=H_{\mathrm{i}}\left(1+l_{\mathrm{H}}^{2}q^{2}\right)$
(47)

is the effective magnetic field [not to be confused with
$\mathbf{H}_{\mathrm{eff}}(\mathbf{r})$ in equation (33)], which depends on
the internal magnetic field $H_{\mathrm{i}}=H_{0}-NM_{0}$ ($N$: demagnetizing
factor), on $q=|\mathbf{q}|$, and on the exchange length of the field

$l_{\mathrm{H}}=\sqrt{\frac{2A}{\mu_{0}M_{0}H_{\mathrm{i}}}}.$ (48)

The latter quantity is a measure for the size of inhomogeneously magnetized
regions around microstructural lattice defects [81]. The Fourier coefficient
of the longitudinal magnetization $\widetilde{M}_{z}(\mathbf{q})$ provides
information on the spatial variation of the saturation magnetization
$M_{\mathrm{s}}(\mathbf{r})$; for instance,
$|\widetilde{M}_{z}|^{2}\propto(\Delta M)^{2}$ in a multiphase magnetic
nanocomposite, where $\Delta M$ denotes the jump of the magnetization
magnitude at internal (particle-matrix) interfaces [82]. Moreover, while the
squared Fourier components and the cross terms are even functions of
$\mathbf{q}$, it is easily seen that the chiral term $-2i\chi(\mathbf{q})$
[equation (40)] is asymmetric in $\mathbf{q}$, which is due to the DMI term:
at small fields, when the term $\propto\widetilde{H}^{2}_{\mathrm{p}}$ in the
numerator of equation (40) dominates, two extrema parallel and antiparallel to
the field axis are observed, whereas at larger fields, when the term
$\propto\widetilde{M}_{z}^{2}$ dominates, additional maxima and minima appear
approximately along the detector diagonals [76].

By inserting equations (37)$-$(44) into the SANS cross sections (see Appendix
A) and by specifying particular models for the nuclear scattering function
$\widetilde{N}^{2}(q\xi_{\mathrm{nuc}})$, the longitudinal magnetic Fourier
component $\widetilde{M}^{2}_{z}(q\xi_{\mathrm{M}})$, and for the Fourier
coefficient of the magnetic anisotropy field
$\widetilde{H}^{2}_{\mathrm{p}}(q\xi_{\mathrm{H}})$, one obtains $P^{\pm}$ as
a function of the magnitude and orientation of the scattering vector
$\mathbf{q}$, the applied magnetic field $\mathbf{H}_{0}$, the magnetic-
interaction parameters ($A$, $D$, $M_{0}$, $\Delta M$, $H_{\mathrm{p}}$,
$\xi_{\mathrm{M}}$, $\xi_{\mathrm{H}}$), and microstructural quantities
(particle-size distribution, crystallograpic texture, etc.). Later on in the
paper, for graphically displaying the $P^{\pm}$, we have assumed that
$\widetilde{N}^{2}$, $\widetilde{M}_{z}^{2}$, and
$\widetilde{H}_{\mathrm{p}}^{2}$ are all isotropic (i.e.
$\theta$-independent), as is appropriate for polycrystalline texture-free bulk
ferromagnets, and that they can be represented by Lorentzian-squared
functions, i.e.

$\widetilde{M}^{2}_{z}(q\xi_{\mathrm{M}})=\frac{A_{\mathrm{M}}^{2}\xi_{\mathrm{M}}^{6}}{\left(1+q^{2}\xi_{\mathrm{M}}^{2}\right)^{2}},$
(49)
$\widetilde{H}_{\mathrm{p}}^{2}(q\xi_{\mathrm{H}})=\frac{A_{\mathrm{H}}^{2}\xi_{\mathrm{H}}^{6}}{\left(1+q^{2}\xi_{\mathrm{H}}^{2}\right)^{2}},$
(50)
$\widetilde{N}^{2}(q\xi_{\mathrm{nuc}})=\alpha(q)b_{\mathrm{H}}^{2}\widetilde{M}^{2}_{z}(q\xi_{\mathrm{M}}),$
(51)

where the amplitudes $A_{\mathrm{M}}^{2}$ and $A_{\mathrm{H}}^{2}$ (both in
units of $\mathrm{A^{2}}/\mathrm{nm^{2}}$) are, respectively, related to the
mean-square magnetization fluctuation and anisotropy-field variation. Of
course, other scattering functions such as the form factor of a sphere and
various structure-factor models (e.g. a Percus-Yevick hard-sphere structure
factor) can be straightforwardly implemented [81]. The characteristic
structure sizes of $\widetilde{M}_{z}^{2}$ and
$\widetilde{H}_{\mathrm{p}}^{2}$ are generally different. We remind the reader
that these are related, respectively, to the spatial extent of regions with
uniform saturation magnetization ($\xi_{\mathrm{M}}$) and magnetic anisotropy
field ($\xi_{\mathrm{H}}$). Measurement of the magnetic field dependent
Guinier radius provides a means to determine these correlation lengths as well
as the exchange-stiffness constant $A$ [83]. A simple example where
$\xi_{\mathrm{M}}=\xi_{\mathrm{H}}$ is a collection of homogeneous and defect-
free magnetic nanoparticles in a magnetic and homogeneous matrix. If, on the
other hand, atomically-sharp grain boundaries are introduced into such
particles, then the direction of the magnetic anisotropy field changes due to
the changing set of crystallographic directions at the intraparticle
interfaces, but the value of $M_{\mathrm{s}}$ may remain the same, so that
$\xi_{\mathrm{H}}<\xi_{\mathrm{M}}$. In Ref. 82 it was shown, assuming
$\xi_{\mathrm{H}}=\xi_{\mathrm{M}}$ and using the sphere form factor for both
$\widetilde{M}^{2}_{z}$ and $\widetilde{H}_{\mathrm{p}}^{2}$, that it is the
ratio of $H_{\mathrm{p}}/\Delta M$ (related to the amplitudes $A_{\mathrm{H}}$
and $A_{\mathrm{M}}$) which determines the angular anisotropy and the
asymptotic power-law dependence of $d\Sigma/d\Omega$ as well as the
characteristic decay length of spin-misalignment fluctuations. The ratio of
nuclear to longitudinal magnetic scattering is denoted with $\alpha$, which
for the general case (that the nuclear correlation length $\xi_{\mathrm{nuc}}$
is different from $\xi_{\mathrm{M}}$) is a function of $q$. Here, we do not
specify a particular $\xi_{\mathrm{nuc}}$ and assume this characteristic size
to be contained in $\alpha(q)$.

##  IV Polarization of Scattered Beam

When the expressions for the elastic differential spin-flip and SANSPOL cross
sections $\frac{d\Sigma^{\pm\mp}}{d\Omega}$ and
$\frac{d\Sigma^{\pm}}{d\Omega}$ [equations (A)$-$(A)] are inserted into
equations (32a) and (32b), and use is made of the expressions for the
magnetization Fourier components [equations (37)$-$(44)], one obtains,
respectively, for the transversal and longitudinal scattering geometry:

$\displaystyle\centering P_{\mathrm{f}\perp}^{+}(\mathbf{q})\@add@centering$
$\displaystyle=$ $\displaystyle
1-2\frac{h_{1}(\mathbf{q})-i\chi(\mathbf{q})}{h_{2}(\mathbf{q})-b_{\mathrm{H}}^{-1}CT_{\widetilde{N}\widetilde{M}_{z}}\sin^{2}\theta+b_{\mathrm{H}}^{-1}CT_{\widetilde{N}\widetilde{M}_{y}}\sin\theta\cos\theta-i\,\chi(\mathbf{q})},$
(52a) $\displaystyle P_{\mathrm{f}\perp}^{-}(\mathbf{q})$ $\displaystyle=$
$\displaystyle
1-2\frac{h_{1}(\mathbf{q})+i\chi(\mathbf{q})}{h_{2}(\mathbf{q})+b_{\mathrm{H}}^{-1}CT_{\widetilde{N}\widetilde{M}_{z}}\sin^{2}\theta-
b_{\mathrm{H}}^{-1}CT_{\widetilde{N}\widetilde{M}_{y}}\sin\theta\cos\theta+i\,\chi(\mathbf{q})},$
(52b) $\displaystyle\centering
P_{\mathrm{f}\parallel}^{+}(\mathbf{q})\@add@centering$ $\displaystyle=$
$\displaystyle
1-2\frac{g_{1}(\mathbf{q})}{g_{2}(\mathbf{q})-b_{\mathrm{H}}^{-1}CT_{\widetilde{N}\widetilde{M}_{z}}},$
(52c) $\displaystyle P_{\mathrm{f}\parallel}^{-}(\mathbf{q})$ $\displaystyle=$
$\displaystyle
1-2\frac{g_{1}(\mathbf{q})}{g_{2}(\mathbf{q})+b_{\mathrm{H}}^{-1}CT_{\widetilde{N}\widetilde{M}_{z}}}.$
(52d)

The functions $h_{1}(\mathbf{q})$, $h_{2}(\mathbf{q})$, $g_{1}(\mathbf{q})$,
and $g_{2}(\mathbf{q})$ are independent of the incident neutron beam
polarization and are defined as:

$\displaystyle
h_{1}(\mathbf{q})=|\widetilde{M}_{x}|^{2}+|\widetilde{M}_{y}|^{2}\cos^{4}\theta$
$\displaystyle+|\widetilde{M}_{z}|^{2}\sin^{2}\theta\cos^{2}\theta-
CT_{yz}\sin\theta\cos^{3}\theta,$ (53) $\displaystyle
h_{2}(\mathbf{q})=b_{\mathrm{H}}^{-2}|\widetilde{N}|^{2}+|\widetilde{M}_{x}|^{2}+|\widetilde{M}_{y}|^{2}\cos^{2}\theta$
$\displaystyle+|\widetilde{M}_{z}|^{2}\sin^{2}\theta-
CT_{yz}\sin\theta\cos\theta,$ (54) $\displaystyle
g_{1}(\mathbf{q})=|\widetilde{M}_{x}|^{2}\sin^{2}\theta+|\widetilde{M}_{y}|^{2}\cos^{2}\theta$
$\displaystyle-CT_{xy}\sin\theta\cos\theta,$ (55) $\displaystyle
g_{2}(\mathbf{q})=b_{\mathrm{H}}^{-2}|\widetilde{N}|^{2}+|\widetilde{M}_{x}|^{2}\sin^{2}\theta+|\widetilde{M}_{y}|^{2}\cos^{2}\theta$
$\displaystyle+|\widetilde{M}_{z}|^{2}-CT_{xy}\sin\theta\cos\theta.$ (56)

At complete magnetic saturation, when
$\mathbf{M}(\mathbf{r})=\\{0,0,M_{\mathrm{s}}(\mathbf{r})\\}$, these
expressions reduce to:

$\displaystyle
h^{\mathrm{sat}}_{1}(\mathbf{q})=|\widetilde{M}_{\mathrm{s}}|^{2}\sin^{2}\theta\cos^{2}\theta,$
(57) $\displaystyle
h^{\mathrm{sat}}_{2}(\mathbf{q})=b_{\mathrm{H}}^{-2}|\widetilde{N}|^{2}+|\widetilde{M}_{\mathrm{s}}|^{2}\sin^{2}\theta,$
(58) $\displaystyle g^{\mathrm{sat}}_{1}(\mathbf{q})=0,$ (59) $\displaystyle
g^{\mathrm{sat}}_{2}(\mathbf{q})=b_{\mathrm{H}}^{-2}|\widetilde{N}|^{2}+|\widetilde{M}_{\mathrm{s}}|^{2},$
(60)

where $\widetilde{M}_{\mathrm{s}}(\mathbf{q})$ is the Fourier transform of
$M_{\mathrm{s}}(\mathbf{r})$. As can be seen from equations (52a) and (52b),
the difference between $P_{\mathrm{f}}^{+}$ and $P_{\mathrm{f}}^{-}$ resides,
for $\mathbf{k}_{0}\perp\mathbf{H}_{0}$, in the nuclear-magnetic interference
terms $\propto\widetilde{N}\widetilde{M}_{z}$ and
$\propto\widetilde{N}\widetilde{M}_{y}$, and in $\chi(\mathbf{q})$, while for
$\mathbf{k}_{0}\parallel\mathbf{H}_{0}$ both polarizations differ only by the
term $\propto\widetilde{N}\widetilde{M}_{z}$ [equations (52c) and (52d)]. We
would also like to remind the reader that the Fourier coefficients in the
above expressions are evaluated in the plane of the detector, which for the
perpendicular scattering geometry corresponds to the plane $q_{x}=0$, and to
the plane $q_{z}=0$ for the parallel geometry (compare Fig. 1).

The $\widetilde{N}\widetilde{M}_{y}$ contribution to equations (52a) and (52b)
requires a special consideration. This term is expected to be negligible for a
polycrystalline statistically-isotropic ferromagnet with vanishing
fluctuations of the saturation magnetization. This can be seen by scrutinizing
the following expression for the $\widetilde{M}_{y}$ magnetization Fourier
component in the perpendicular scattering geometry, corresponding to the plane
$q_{x}=0$ [66]:

$\widetilde{M}_{y}=\frac{p\left(\widetilde{H}_{\mathrm{p}y}-\widetilde{M}_{z}\frac{q_{y}q_{z}}{q^{2}}\left(1+pl_{\mathrm{D}}^{2}q^{2}\right)-i\widetilde{H}_{\mathrm{p}x}pl_{\mathrm{D}}q_{z}\right)}{1+p\frac{q_{y}^{2}}{q^{2}}-p^{2}l_{\mathrm{D}}^{2}q_{z}^{2}},$
(61)

where $\widetilde{H}_{\mathrm{p}x}$ and $\widetilde{H}_{\mathrm{p}y}$ denote
the Cartesian components of the Fourier transform of the magnetic anisotroy
field. If we assume that the nuclear scattering is isotropic and that
$\widetilde{H}_{\mathrm{p}x}$ and $\widetilde{H}_{\mathrm{p}y}$ vary randomly
in the plane perpendicular to the field, then the corresponding averages over
the direction of the anisotropy field vanish. The only remaining term in the
$\widetilde{N}\widetilde{M}_{y}$ contribution is then ($q_{y}/q=\sin\theta$;
$q_{z}/q=\cos\theta$):

$\displaystyle CT_{\widetilde{N}\widetilde{M}_{y}}$ $\displaystyle=$
$\displaystyle\widetilde{N}\widetilde{M}_{y}^{\ast}+\widetilde{N}^{\ast}\widetilde{M}_{y}$
(62) $\displaystyle=$
$\displaystyle-\frac{2\widetilde{N}\widetilde{M}_{z}p\left(1+pl_{\mathrm{D}}^{2}q^{2}\right)\sin\theta\cos\theta}{1+p\sin^{2}\theta-p^{2}l_{\mathrm{D}}^{2}q^{2}\cos^{2}\theta},$

where we have furthermore assumed that both $\widetilde{N}(q)$ and
$\widetilde{M}_{z}(q)$ are real-valued, as is done throughout this paper. Note
that equation (62) still needs to be multiplied with the term
$\sin\theta\cos\theta$ in order to obtain the corresponding contribution to
$P_{\mathrm{f}\perp}^{\pm}$ [compare equations (52a) and (52b)]. For
homogeneous single-phase materials with $M_{\mathrm{s}}=$ constant, the
$\widetilde{N}\widetilde{M}_{y}$ contribution is expected to be negligible,
and we are not aware that is has been reported experimentally. However, for
materials exhibiting strong spatial nanoscale variations in the saturation
magnetization, i.e. $M_{\mathrm{s}}=M_{\mathrm{s}}(\mathbf{r})$, such as
magnetic nanocomposites or porous ferromagnets, it should be possible to
observe this scattering contribution in polarized SANS experiments.

###  IV.1 Sector Averages

Carrying out a $2\pi$ azimuthal average of the
$P_{\mathrm{f}}^{\pm}(\mathbf{q})$, which are maps with numbers varying
between $\pm 1$, may result in a significant loss of information (compare e.g.
Figs. 10 and 11 below). It is therefore often advantageous to consider cuts of
$P_{\mathrm{f}}^{\pm}$ along certain directions in $\mathbf{q}$-space. This
might also be of relevance for other spin-manipulating techniques such as
SEMSANS, which is a one dimensional technique that only measures correlations
in the encoding direction [65]. Sector averages are straightforwardly obtained
by evaluating equations (52a)$-$(52d) [using equations (IV)$-$(60)] for
certain angles $\theta$. For instance, for the perpendicular scattering
geometry and for $\mathbf{q}$ along the vertical direction on the detector
($\theta=90^{\circ}$), we obtain [$\chi(q,\theta=90^{\circ})=0$, compare
equation (40)]:

$\displaystyle\centering
P_{\mathrm{f}\perp}^{+}(q,\theta=90^{\circ})\@add@centering$ $\displaystyle=$
$\displaystyle
1-2\frac{|\widetilde{M}_{x}|^{2}}{b_{\mathrm{H}}^{-2}|\widetilde{N}|^{2}+|\widetilde{M}_{x}|^{2}+|\widetilde{M}_{z}|^{2}-b_{\mathrm{H}}^{-1}CT_{\widetilde{N}\widetilde{M}_{z}}},$
(63a) $\displaystyle P_{\mathrm{f}\perp}^{-}(q,\theta=90^{\circ})$
$\displaystyle=$ $\displaystyle
1-2\frac{|\widetilde{M}_{x}|^{2}}{b_{\mathrm{H}}^{-2}|\widetilde{N}|^{2}+|\widetilde{M}_{x}|^{2}+|\widetilde{M}_{z}|^{2}+b_{\mathrm{H}}^{-1}CT_{\widetilde{N}\widetilde{M}_{z}}},$
(63b)

where [compare equation (37)]

$|\widetilde{M}_{x}|^{2}(q,\theta=90^{\circ})=\frac{p^{2}}{2}\left(\widetilde{H}^{2}_{\mathrm{p}}+2\widetilde{M}_{z}^{2}l_{\mathrm{D}}^{2}q^{2}\right).$
(64)

At saturation ($M_{x}=0$), $P_{\mathrm{f}\perp}^{\pm}(q,\theta=90^{\circ})=1$,
except for the case $\alpha=1$, where
$P_{\mathrm{f}\perp}^{+}(q,\theta=90^{\circ})=-1$. We also see that
information on the DMI is contained in $|\widetilde{M}_{x}|^{2}$ via the
length scale $l_{\mathrm{D}}$. For $l_{\mathrm{D}}=0$,
$|\widetilde{M}_{x}|^{2}=\frac{p^{2}}{2}\widetilde{H}^{2}_{\mathrm{p}}$.

For the perpendicular scattering geometry and $\mathbf{q}$ along the
horizontal direction ($\theta=0^{\circ}$), we obtain:

$\displaystyle\centering
P_{\mathrm{f}\perp}^{+}(q,\theta=0^{\circ})\@add@centering$ $\displaystyle=$
$\displaystyle
1-2\frac{|\widetilde{M}_{x}|^{2}+|\widetilde{M}_{y}|^{2}-i\chi}{b_{\mathrm{H}}^{-2}|\widetilde{N}|^{2}+|\widetilde{M}_{x}|^{2}+|\widetilde{M}_{y}|^{2}-i\chi},$
$\displaystyle P_{\mathrm{f}\perp}^{-}(q,\theta=0^{\circ})$ $\displaystyle=$
$\displaystyle
1-2\frac{|\widetilde{M}_{x}|^{2}+|\widetilde{M}_{y}|^{2}+i\chi}{b_{\mathrm{H}}^{-2}|\widetilde{N}|^{2}+|\widetilde{M}_{x}|^{2}+|\widetilde{M}_{y}|^{2}+i\chi},$

where [compare equations (37)$-$(38)]

$|\widetilde{M}_{x}|^{2}(q,\theta=0^{\circ})=|\widetilde{M}_{y}|^{2}=\frac{p^{2}}{2}\frac{\widetilde{H}^{2}_{\mathrm{p}}\left(1+p^{2}l_{\mathrm{D}}^{2}q^{2}\right)}{\left(1-p^{2}l_{\mathrm{D}}^{2}q^{2}\right)^{2}}$
(66)

and [compare equation (40)]

$i\chi(q,\theta=0^{\circ})=-\frac{2\widetilde{H}^{2}_{\mathrm{p}}p^{3}l_{\mathrm{D}}q}{\left(1-p^{2}l_{\mathrm{D}}^{2}q^{2}\right)^{2}}.$
(67)

At saturation ($M_{x}=M_{y}=\chi=0$) and for nonzero nuclear scattering,
$P_{\mathrm{f}\perp}^{\pm}(q,\theta=0^{\circ})=1$. For $l_{\mathrm{D}}=0$,
$|\widetilde{M}_{x}|^{2}+|\widetilde{M}_{y}|^{2}=p^{2}\widetilde{H}^{2}_{\mathrm{p}}$
and $P_{\mathrm{f}\perp}^{\pm}(q,\theta=0^{\circ})$ contains information on
the transversal spin components.

###  IV.2 Saturated State

At saturation and for $\mathbf{k}_{0}\perp\mathbf{H}_{0}$, it is readily
verified from equations (52a) and (52b) using equations (57)$-$(60),
$\chi(\mathbf{q})=0$, and $CT_{\widetilde{N}\widetilde{M}_{y}}=0$ that

$\displaystyle
P_{\mathrm{f}\perp}^{+}(q,\theta)=1-2\frac{\sin^{2}\theta\cos^{2}\theta}{\alpha-2\sqrt{\alpha}\sin^{2}\theta+\sin^{2}\theta}$
(68a) $\displaystyle
P_{\mathrm{f}\perp}^{-}(q,\theta)=1-2\frac{\sin^{2}\theta\cos^{2}\theta}{\alpha+2\sqrt{\alpha}\sin^{2}\theta+\sin^{2}\theta}$
(68b)

depend exclusively on the ratio

$\alpha(q)=\frac{\widetilde{N}^{2}(q)}{b_{\mathrm{H}}^{2}\widetilde{M}^{2}_{s}(q)}$
(69)

of nuclear to longitudinal magnetic scattering [compare equation (51)]. The
possible angular anisotropy of $\alpha$ is not considered in this paper. Since
for $\mathbf{k}_{0}\parallel\mathbf{H}_{0}$ the spin-flip SANS cross section
vanishes at saturation ($g_{1}=0$), we see that
$P_{\mathrm{f}\parallel}^{\pm}=+1$. The azimuthally-averaged
($\frac{1}{2\pi}\int(\cdots)d\theta$) versions of equations (68a) and (68b)
read:

$\displaystyle P_{\mathrm{f}\perp}^{+}(q)$ $\displaystyle=$
$\displaystyle\frac{2\sqrt{\alpha}\left(-1+\sqrt{\alpha}+\left|-1+\sqrt{\alpha}\right|\right)}{\left(1-2\sqrt{\alpha}\right)^{2}},$
(70a) $\displaystyle P_{\mathrm{f}\perp}^{-}(q)$ $\displaystyle=$
$\displaystyle 1-\frac{1}{\left(1+2\sqrt{\alpha}\right)^{2}}.$ (70b)

The function $\alpha(q)$ can be a monotonically increasing or decreasing
function of $q$, and it can even exhibit local extrema. In the following, we
will consider the cases of $\alpha=$ constant and $\alpha=\alpha(q)$ using the
experimental data of the soft magnetic Fe-based alloy NANOPERM [39].

####  IV.2.1 $\alpha=$ constant

Fig. 2 displays the two-dimensional polarization
$P_{\mathrm{f}\perp}^{\pm}(\mathbf{q})$ of the scattered neutrons in the
saturated state as a function of $\alpha=$ constant. The case of constant
$\alpha$ is very rarely realized in experimental situations, and we consider
it here only as a starting point for our discussion and for the comparison to
the experimentally more relevant situation of $\alpha=\alpha(q)$. For
$\alpha\rightarrow 0$ it follows that
$P_{\mathrm{f}\perp}^{\pm}=1-2\cos^{2}\theta$ [Fig. 2(a) and (e)], while
$P_{\mathrm{f}\perp}^{+}=1-2\sin^{2}\theta$ [Fig. 2(c)] and
$P_{\mathrm{f}\perp}^{-}=1-2\sin^{2}\theta\cos^{2}\theta/(1+3\sin^{2}\theta)$
[Fig. 2(g)] for $\alpha=1$. When nuclear coherent scattering is dominating
($\alpha\rightarrow\infty$), we see that the $P_{\mathrm{f}\perp}^{\pm}$ both
tend to unity, as expected. The corresponding $2\pi$-azimuthally-averaged
functions [equations (70a) and (70b)] are plotted in Fig. 3. One readily
verifies that $P_{\mathrm{f}\perp}^{+}=0$ for $\sqrt{\alpha}\leq 1$, which
further underlines the loss of information when a $2\pi$ azimuthal average is
carried out [compare e.g. Fig. 2(b)].

Figure 2: Plot of $P_{\mathrm{f}\perp}^{+}(q_{y},q_{z})$ (upper row) and
$P_{\mathrm{f}\perp}^{-}(q_{y},q_{z})$ (lower row) in the saturate state for
different values of $\alpha$ (see insets) [equations (68a) and (68b)].

Figure 3: Plot of $P_{\mathrm{f}\perp}^{+}$ and $P_{\mathrm{f}\perp}^{-}$ (see
inset) in the saturate state as a function of $\alpha$ [equations (70a) and
(70b)].

####  IV.2.2 $\alpha=\alpha(q)$

Fig. 4 shows the experimentally determined ratio $\alpha_{\mathrm{exp}}(q)$
[39] of nuclear to magnetic scattering of the two-phase alloy NANOPERM. Within
the experimental $q$-range of
$0.03\,\mathrm{nm}^{-1}<q<0.3\,\mathrm{nm}^{-1}$, these data for
$\alpha_{\mathrm{exp}}(q)$ have been fitted by a power-law in $1/q$ to obtain
the functions $P_{\mathrm{f}\perp}^{\pm}(q)$, which are depicted in Fig. 5.
The used fit function for $\alpha_{\mathrm{exp}}(q)$ is:

$\displaystyle\alpha_{\mathrm{exp}}(q)=\frac{0.14853}{q}-\frac{0.0264491}{q^{2}}+\frac{0.00176887}{q^{3}}$
$\displaystyle-\frac{4.95094\times 10^{-5}}{q^{4}}+\frac{5.01767\times
10^{-7}}{q^{5}}.$ (71)

This expression will be used in the analysis of the experimental data (see
section VI).

Figure 4: ($\bullet$) Experimental ratio $\alpha_{\mathrm{exp}}(q)$ of nuclear
to magnetic scattering of the two-phase alloy NANOPERM [39]
($\mathbf{k}_{0}\perp\mathbf{H}_{0}$; $\mu_{0}H_{0}=1.27\,\mathrm{T}$; log-log
plot). Solid line: power-law fit to parametrize the experimental data
[equation (IV.2.2)]. The fit has been restricted to the interval
$0.03\,\mathrm{nm}^{-1}<q<0.3\,\mathrm{nm}^{-1}$, but the fit function is
displayed for $0.01\,\mathrm{nm}^{-1}<q<1.0\,\mathrm{nm}^{-1}$.

Figure 5: Plot of $P_{\mathrm{f}\perp}^{+}(q)$ (solid line) and
$P_{\mathrm{f}\perp}^{-}(q)$ (dashed line) of NANOPERM using equations (70a)
and (70b) with $\alpha_{\mathrm{exp}}(q)$ given by equation (IV.2.2) and
$0.03\,\mathrm{nm}^{-1}<q<0.3\,\mathrm{nm}^{-1}$.

###  IV.3 Nonsaturated State

Appendix B features some theoretical results for
$P_{\mathrm{f}\perp}^{\pm}(\mathbf{q})$ for various combinations of the
magnetic interaction parameters (applied magnetic field, ratio of
$\widetilde{H}_{\mathrm{p}}^{2}$ to $\widetilde{M}^{2}_{z}$,
$\chi(\mathbf{q})\neq 0$). For a statistically isotropic ferromagnet, the two-
dimensional distribution of the polarization of the scattered neutrons is
isotropic ($\theta$-independent) for the longitudinal scattering geometry
($\mathbf{k}_{0}\parallel\mathbf{H}_{0}$), as are the corresponding SANS cross
sections. This is in contrast to the $P_{\mathrm{f}\perp}^{\pm}(\mathbf{q})$
for the transversal geometry ($\mathbf{k}_{0}\perp\mathbf{H}_{0}$), which are
highly anisotropic. In the following, we will use the theoretical expressions
for $P_{\mathrm{f}\perp}^{\pm}(\mathbf{q})$ to analyze experimental data on
the soft magnetic two-phase nanocrystalline alloy NANOPERM.

##  V Experimental Details

The polarized neutron experiment was carried out at room temperature at the
instrument D22 at the Institut Laue-Langevin, Grenoble, France. Incident
neutrons with a mean wavelength of $\lambda=8\,\mathrm{{\AA}}$ and a
wavelength broadening of $\Delta\lambda/\lambda=10\,\%$ (FWHM) were selected
by means of a velocity selector. The beam was polarized using a
$1.2\,\mathrm{m}$-long remanent Fe-Si supermirror transmission polarizer
($m=3.6$), which was installed immediately after the velocity selector. A
radio-frequency (rf) spin flipper, installed close to the sample position,
allowed us to reverse the initial neutron polarization. The external magnetic
field (provided by an electromagnet) was applied perpendicular to the wave
vector $\mathbf{k}_{0}$ of the incident neutrons (compare Fig. 1). Measurement
of the four partial POLARIS cross sections $d\Sigma^{++}/d\Omega$,
$d\Sigma^{--}/d\Omega$, $d\Sigma^{+-}/d\Omega$, and $d\Sigma^{-+}/d\Omega$ was
accomplished through a polarized 3He spin-filter cell, which was installed
inside the detector housing, about $1\,\mathrm{m}$ away from the sample
position. The polarization between polarizer, rf flipper, and 3He filter was
maintained by means of magnetic guide fields of the order of $1\,\mathrm{mT}$.
The efficiencies of the polarizer, spin flipper, and 3He analyzer were,
respectively, $90\,\%$, $99\,\%$, and $87.5\,\%$. The scattered neutrons were
detected by a multitube detector which consists of $128\times 128$ pixels with
a resolution of $8\times 8\,\mathrm{mm}$. Neutron-data reduction, including
corrections for background scattering and spin leakage [84], was performed
using the GRASP [85] and BerSANS [86, 87] software packages.

The sample under study was a two-phase magnetic nanocomposite from the
NANOPERM family of alloys [88] with a nominal composition of
$(\mathrm{Fe}_{0.985}\mathrm{Co}_{0.015})_{90}\mathrm{Zr}_{7}\mathrm{B}_{3}$
[89, 90]. The alloy was prepared by melt spinning, followed by a subsequent
annealing treatment for $1\,\mathrm{h}$ at $883\,\mathrm{K}$, which resulted
in the precipitation of bcc iron nanoparticles in an amorphous magnetic
matrix. The average iron particle size of $D=15\pm 2\,\mathrm{nm}$ was
determined by the analysis of wide-angle x-ray diffraction data. The
crystalline particle volume fraction is about $65\,\%$ and the saturation
magnetization of the alloy amounts to $\mu_{0}M_{0}=1.64\,\mathrm{T}$. The
exchange-stiffness constant $A=(4.7\pm 0.9)\times 10^{-12}\,\mathrm{J/m}$ has
previously been determined by the analysis of the field-dependent unpolarized
SANS cross section [91]. For the SANS experiments, several circular discs with
a diameter of $10\,\mathrm{mm}$ and a thickness of about $20\,\mu\mathrm{m}$
were stacked and mounted on a Cd aperture (for further details see Refs. 39,
91).

##  VI Experimental Results and Discussion

The two-dimensional experimental distribution of the polarization of NANOPERM
is depicted in Figs. 6 ($P_{\mathrm{f}\perp}^{+}$) and 7
($P_{\mathrm{f}\perp}^{-}$) at selected field values together with a
qualitative comparison to the simulated polarization based on the
micromagnetic SANS theory (see Refs. 39 for some selected spin-resolved SANS
cross sections). The theory uses as input values the experimental ratio
$\alpha_{\mathrm{exp}}(q)$ [equation (IV.2.2)] and the structural
($\xi_{\mathrm{M}}=\xi_{\mathrm{H}}=D/2=7.5\,\mathrm{nm}$) and magnetic
($A,M_{0}$) interaction parameters. In agreement with the previous
micromagnetic SANS data analysis of this sample [39, 91], we have set the
ratio $A_{\mathrm{H}}/A_{\mathrm{M}}=0.2$. We also assumed that both spin-flip
channels are equal, i.e. $d\Sigma^{+-}/d\Omega=d\Sigma^{-+}/d\Omega$, a
constraint that was already imposed during the spin-leakage correction. The
overall qualitative agreement between experiment and theory (no free
parameters) is evident, although the angular anisotropy of the data does not
exhibit a large variation with field. Only at the smallest momentum-transfers
can one notice a change in the anisotropy with decreasing field (in particular
in $P_{\mathrm{f}\perp}^{-}$), which is related to the emerging spin-
misalignment scattering; compare e.g. scattering terms
$\propto|\widetilde{M}_{y}|^{2}\cos^{4}\theta$ and $\propto
CT_{yz}\sin\theta\cos\theta$ in equations (A) and (A). We also note the
existence of (seemingly isotropic) scattering contributions at small
$q\lesssim 0.1\,\mathrm{nm}^{-1}$ (especially at $1.27\,\mathrm{T}$), which
are likely due to large-scale structures that are not contained in the
micromagnetic theory [compare Fig. 6(a) and (e) and Fig. 7(a) and (e)].

Figure 6: Qualitative comparison between experiment and theory. (a)$-$(d) Two-
dimensional experimental polarization $P_{\mathrm{f}\perp}^{+}(\mathbf{q})$ of
the scattered neutrons of NANOPERM
[$(\mathrm{Fe}_{0.985}\mathrm{Co}_{0.015})_{90}\mathrm{Zr}_{7}\mathrm{B}_{3}$]
at a series of applied magnetic fields (see insets). $\mathbf{H}_{0}$ is
horizontal in the plane. The range of momentum transfers is restricted to
$q\lesssim 0.33\,\mathrm{nm}^{-1}$. (e)$-$(h) Prediction by the analytical
micromagnetic theory (no free parameters) using the experimental ratio
$\alpha_{\mathrm{exp}}(q)$ [equation (IV.2.2)] and the structural
($\xi_{\mathrm{M}}=\xi_{\mathrm{H}}=D/2=7.5\,\mathrm{nm}$) and magnetic
($A,M_{0}$) interaction parameters of NANOPERM (see text and Refs. 39, 91).
The central white octagons mark the position of the beam stop.

Figure 7: Similar to Fig. 6, but for $P_{\mathrm{f}\perp}^{-}(\mathbf{q})$.

Due to the relatively large statistical noise in the two-dimensional
$P_{\mathrm{f}\perp}^{\pm}$ maps we did not fit the experimental data directly
to the theoretical expressions. Therefore, in the following, we consider one-
dimensional experimental polarization data, which were obtained by averaging
the two-dimensional polarized SANS cross sections over $\pm 8^{\circ}$ along
the vertical direction ($\theta=90^{\circ}$). These averages were used in
equations (32a) and (32b) to obtain the $P_{\mathrm{f}\perp}^{\pm}(q)$. The
resulting data in Fig. 8 were then fitted using the general equations (52a)
and (52b) (also averaged over $\pm 8^{\circ}$ along $\theta=90^{\circ}$).
Adjustable parameters are the amplitudes (scaling parameters) $A_{\mathrm{M}}$
and $A_{\mathrm{H}}$ of, respectively, $\widetilde{M}^{2}_{z}$ and
$\widetilde{H}^{2}_{\mathrm{p}}$ as well as the corresponding correlation
lengths $\xi_{\mathrm{M}}$ and $\xi_{\mathrm{H}}$ [compare equations (49) and
(50)]. The field-dependent micromagnetic exchange length $l_{\mathrm{H}}$,
which is contained in the dimensionless function $p(q,H_{\mathrm{i}})$
[compare equations (46)$-$(48)], is computed at each field using the materials
parameters $A$ and $M_{0}$; $A$ is treated here as an additional adjustable
parameter. For $\alpha(q)$ we used equation (IV.2.2), and the DMI has been
ignored in the data analysis ($l_{\mathrm{D}}=0$). Since the
$P_{\mathrm{f}\perp}^{\pm}$ differ only by the
$\widetilde{N}\widetilde{M}_{y}$ and $\widetilde{N}\widetilde{M}_{z}$
interference terms, we have fitted the $P_{\mathrm{f}\perp}^{\pm}(q)$ data
corresponding to the same field simultaneously. The applied field $H_{0}$ has
been corrected for demagnetizing effects.

The fits in Fig. 8 (solid lines) provide a reasonable description of the
experimental data. The obtained values for $\xi_{\mathrm{M}}$ and
$\xi_{\mathrm{H}}$ are shown in Fig. 9; $\xi_{\mathrm{H}}\cong
6$$-$$15\,\mathrm{nm}$ is at all fields consistently of the order of the
particle size, while $\xi_{\mathrm{M}}$ takes on larger values between about
$22$$-$$65\,\mathrm{nm}$. For the exchange-stiffness constant, we obtain (from
the four local fits) best-fit values that range between $A=(4.8$$-$$9.7)\times
10^{-12}\,\mathrm{J/m}$. These values agree very well with data in the
literature [91, 92].

Clearly, more experiments are needed in order to further scrutinize the
predictions of the present micromagnetic theory for the uniaxial polarization
analysis of bulk ferromagnets. In this respect, the development of
computational tools to directly analyze the two-dimensional polarization maps
using different form-factor and structure-factor expressions for
$\widetilde{M}^{2}_{z}$ and $\widetilde{H}^{2}_{\mathrm{p}}$, and possibly the
inclusion of a particle-size distribution function, would be desirable.
Likewise, SANS measurements at a preferably saturating magnetic field are
necessary to determine the nuclear SANS cross section, e.g. via a horizontal
average of the non-spin-flip SANS cross section.

Figure 8: (Data points) Experimental polarizations
$P_{\mathrm{f}\perp}^{+}(q,\theta=90^{\circ})$ (a) and
$P_{\mathrm{f}\perp}^{-}(q,\theta=90^{\circ})$ (b) of the scattered neutrons
of NANOPERM
[$(\mathrm{Fe}_{0.985}\mathrm{Co}_{0.015})_{90}\mathrm{Zr}_{7}\mathrm{B}_{3}$]
at a series of internal magnetic fields (see inset). For the clarity of
presentation, error bars are only shown for one field. (Solid lines)
Prediction by the analytical micromagnetic theory [equations (52a) and (52b)]
using the ratio $\alpha_{\mathrm{exp}}(q)$ [equation (IV.2.2)]. Note the
different scales on the ordinates in (a) and (b).

Figure 9: Resulting best-fit values for the correlation lengths
$\xi_{\mathrm{M}}$ and $\xi_{\mathrm{H}}$ (see inset). Lines are a guide to
the eyes.

##  VII Summary and Conclusions

We have provided a micromagnetic theory for the uniaxial polarization of the
scattered neutrons of bulk ferromagnets, as it can be measured by means of the
small-angle neutron scattering (SANS) method. The theoretical expressions
contain the effects of an isotropic exchange interaction, the
Dzyaloshinskii–Moriya interaction, magnetic anisotropy, magnetodipolar
interaction, and an external magnetic field. The theory has been employed to
analyze experimental data on a soft magnetic nanocrystalline alloy; it may
provide information on the magnetic interactions (exchange and DMI constants)
and on the spatial structures of the magnetic anisotropy and magnetostatic
fields. Given that uniaxial polarization analysis is becoming more and more
available on SANS instruments worldwide and in view of the recent seminal
progress made regarding several techniques which exploit the neutron
polarization degree of freedom to characterize large-scale magnetic structures
(SESANS, DFI, SEMSANS), we believe that the results of this paper open up a
new avenue for magnetic neutron data analysis on mesoscopic magnetic systems.
The presented micromagnetic SANS framework forms the basis for all of these
new and promising neutron techniques.

###### Acknowledgements.

Andreas Michels and Artem Malyeyev thank the National Research Fund of
Luxembourg for financial support (AFR Grant No. 12417141). This work is
partially based on experiments performed at the Institut Laue-Langevin,
Grenoble, France.

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##  Appendix A Non-spin-flip, spin-flip, and SANSPOL cross sections

In this Appendix, we display the expressions for the polarized SANS cross
sections in terms of the Fourier components
$\widetilde{M}_{x,y,z}(\mathbf{q})$ of the magnetization. The two non-spin-
flip and the two spin-flip POLARIS cross sections carry, respectively, the
superscripts $++$ and $--$ and $+-$ and $-+$, and the subscripts $\perp$ and
$\parallel$ refer to the respective scattering geometry (compare Fig. 1) [93]:

$\displaystyle\frac{d\Sigma^{\pm\pm}_{\perp}}{d\Omega}=K\left(b_{\mathrm{H}}^{-2}|\widetilde{N}|^{2}+|\widetilde{M}_{y}|^{2}\sin^{2}\theta\cos^{2}\theta+|\widetilde{M}_{z}|^{2}\sin^{4}\theta\right.$
$\displaystyle\left.-CT_{yz}\sin^{3}\theta\cos\theta\mp
b_{\mathrm{H}}^{-1}CT_{\widetilde{N}\widetilde{M}_{z}}\sin^{2}\theta\right.$
$\displaystyle\left.\pm
b_{\mathrm{H}}^{-1}CT_{\widetilde{N}\widetilde{M}_{y}}\sin\theta\cos\theta\right),$
$\displaystyle\frac{d\Sigma^{\pm\pm}_{\parallel}}{d\Omega}=K\left(b_{\mathrm{H}}^{-2}|\widetilde{N}|^{2}+|\widetilde{M}_{z}|^{2}\mp
b_{\mathrm{H}}^{-1}CT_{\widetilde{N}\widetilde{M}_{z}}\right),$
$\displaystyle\frac{d\Sigma^{\pm\mp}_{\perp}}{d\Omega}=K\left(|\widetilde{M}_{x}|^{2}+|\widetilde{M}_{y}|^{2}\cos^{4}\theta+|\widetilde{M}_{z}|^{2}\sin^{2}\theta\cos^{2}\theta\right.$
$\displaystyle\left.-CT_{yz}\sin\theta\cos^{3}\theta\mp i\chi\right),$
$\displaystyle\frac{d\Sigma^{\pm\mp}_{\parallel}}{d\Omega}=K\left(|\widetilde{M}_{x}|^{2}\sin^{2}\theta+|\widetilde{M}_{y}|^{2}\cos^{2}\theta\right.$
$\displaystyle\left.-CT_{xy}\sin\theta\cos\theta\right),$ (75)

where $K=8\pi^{3}V^{-1}b_{\mathrm{H}}^{2}$, and the chiral function
$\chi(\mathbf{q})$ is given by:

$\displaystyle\chi=\left(\widetilde{M}_{x}\widetilde{M}_{y}^{\ast}-\widetilde{M}_{x}^{\ast}\widetilde{M}_{y}\right)\cos^{2}\theta$
$\displaystyle-\left(\widetilde{M}_{x}\widetilde{M}_{z}^{\ast}-\widetilde{M}_{x}^{\ast}\widetilde{M}_{z}\right)\sin\theta\cos\theta.$
(76)

Note that $\chi(\mathbf{q})=0$ for $\mathbf{k}_{0}\parallel\mathbf{H}_{0}$.
The two SANSPOL cross sections
$\frac{d\Sigma^{+}}{d\Omega}=\frac{d\Sigma^{++}}{d\Omega}+\frac{d\Sigma^{+-}}{d\Omega}$
and
$\frac{d\Sigma^{-}}{d\Omega}=\frac{d\Sigma^{--}}{d\Omega}+\frac{d\Sigma^{-+}}{d\Omega}$
read:

$\displaystyle\frac{d\Sigma^{\pm}_{\perp}}{d\Omega}=K\left(b_{\mathrm{H}}^{-2}|\widetilde{N}|^{2}+|\widetilde{M}_{x}|^{2}+|\widetilde{M}_{y}|^{2}\cos^{2}\theta\right.$
$\displaystyle\left.+|\widetilde{M}_{z}|^{2}\sin^{2}\theta-
CT_{yz}\sin\theta\cos\theta\right.$ $\displaystyle\left.\mp
b_{\mathrm{H}}^{-1}CT_{\widetilde{N}\widetilde{M}_{z}}\sin^{2}\theta\pm
b_{\mathrm{H}}^{-1}CT_{\widetilde{N}\widetilde{M}_{y}}\sin\theta\cos\theta\right.$
$\displaystyle\left.\mp i\chi\right),$ (77)
$\displaystyle\frac{d\Sigma^{\pm}_{\parallel}}{d\Omega}=K\left(b_{\mathrm{H}}^{-2}|\widetilde{N}|^{2}+|\widetilde{M}_{x}|^{2}\sin^{2}\theta+|\widetilde{M}_{y}|^{2}\cos^{2}\theta\right.$
$\displaystyle\left.+|\widetilde{M}_{z}|^{2}-CT_{xy}\sin\theta\cos\theta\mp
b_{\mathrm{H}}^{-1}CT_{\widetilde{N}\widetilde{M}_{z}}\right).$

The magnetic-magnetic and nuclear-magnetic cross terms have been abbreviated
as follows:

$\displaystyle CT_{yz}$ $\displaystyle=$
$\displaystyle\widetilde{M}_{y}\widetilde{M}_{z}^{\ast}+\widetilde{M}_{y}^{\ast}\widetilde{M}_{z},$
(79a) $\displaystyle CT_{xy}$ $\displaystyle=$
$\displaystyle\widetilde{M}_{x}\widetilde{M}_{y}^{\ast}+\widetilde{M}_{x}^{\ast}\widetilde{M}_{y},$
(79b) $\displaystyle CT_{\widetilde{N}\widetilde{M}_{z}}$ $\displaystyle=$
$\displaystyle\widetilde{N}\widetilde{M}_{z}^{\ast}+\widetilde{N}^{\ast}\widetilde{M}_{z},$
(79c) $\displaystyle CT_{\widetilde{N}\widetilde{M}_{y}}$ $\displaystyle=$
$\displaystyle\widetilde{N}\widetilde{M}_{y}^{\ast}+\widetilde{N}^{\ast}\widetilde{M}_{y}.$
(79d)

In actual SANSPOL and POLARIS experiments the neutron optics (polarizer, spin
flipper, analyzer) do not work perfectly and polarization corrections become
necessary. The incident beam polarization efficiency may be denoted by
$p^{+}=I^{+}/(I^{+}+I^{-})$, where $I^{\pm}$ are, respectively, the number of
neutrons with spins aligned antiparallel and parallel with respect to
$\mathbf{H}_{0}$; note that $p^{+}=1/2$ for an unpolarized beam. The
efficiency of the spin flipper is $\epsilon^{\pm}$ with $\epsilon^{+}=0$ for
flipper off and $\epsilon^{-}=\epsilon\cong 1$ for flipper on. It is important
to emphasize that the half-polarized SANSPOL cross sections
$d\Sigma^{+}/d\Omega$ and $d\Sigma^{-}/d\Omega$ can be obtained directly and
corrected for nonideal neutron polarization provided that the parameters
$p^{+}$ and $\epsilon$ are known from reference measurements. For POLARIS, it
is necessary to measure all four partial cross sections
$d\Sigma^{++}/d\Omega$, $d\Sigma^{--}/d\Omega$, $d\Sigma^{+-}/d\Omega$, and
$d\Sigma^{-+}/d\Omega$ in order to correct for spin leakage between the
different channels [84]. Such corrections can e.g. be accomplished by means of
the BerSANS [86, 87], Pol-Corr [94], and GRASP [85] software tools.

Moreover, we note that $d\Sigma^{+-}/d\Omega=d\Sigma^{-+}/d\Omega$ for many
polycrystalline bulk ferromagnets [38]. However, in our theoretical treatment
we explicitly take into account the polarization dependence of the SANSPOL and
spin-flip cross sections via the chiral function $\chi(\mathbf{q})$. This is
relevant e.g. for systems where inversion symmetry is broken and the DMI is
operative [73, 74].

##  Appendix B Selected results for the polarization of the scattered neutrons
of bulk ferromagnets

In this Appendix we provide some selected graphical representations for the
dependency of the polarization of the scattered neutrons on the magnitude and
orientation of the scattering vector, the applied magnetic field, the ratio of
$A_{\mathrm{M}}$ to $A_{\mathrm{H}}$, the ratio of nuclear to longitudinal
magnetic scattering, and the DMI (via the exchange length $l_{\mathrm{D}}$).
Only results for $\mathbf{k}_{0}\perp\mathbf{H}_{0}$ are shown. The following
materials parameters are used: $A=4.7\,\mathrm{pJ/m}$;
$\mu_{0}M_{0}=1.64\,\mathrm{T}$ ($l_{\mathrm{M}}\cong 2.1\,\mathrm{nm}$);
$D=2\,\mathrm{mJ/m^{2}}$ ($l_{\mathrm{D}}\cong 1.9\,\mathrm{nm}$) [91].

Fig. 10 shows the two-dimensional final polarization
$P_{\mathrm{f}\perp}^{\pm}(q_{y},q_{z})$ and Fig. 11 depicts the corresponding
$2\pi$-azimuthally-averaged data $P_{\mathrm{f}\perp}^{\pm}(q)$ for different
values of the applied magnetic field $H_{\mathrm{i}}$, $\alpha=\alpha(q)$
[equation (IV.2.2)], $A_{\mathrm{H}}/A_{\mathrm{M}}=0.5$, and
$l_{\mathrm{D}}=0$. The local extrema in $P_{\mathrm{f}\perp}^{\pm}(q)$ at
small $q$ are due to $\alpha_{\mathrm{exp}}(q)$; setting $\alpha=$ constant
results in smooth and continuously decaying functions. Figs. 12 and 13 display
the polarization $P^{\pm}(\mathbf{q})$ for different ratios of
$A_{\mathrm{H}}/A_{\mathrm{M}}$ (Fig. 12) and for different (constant)
$\alpha$-values (Fig. 13) at a constant field of
$\mu_{0}H_{\mathrm{i}}=0.3\,\mathrm{T}$ and for $l_{\mathrm{D}}=0$. Including
the DMI results in asymmetric $P^{\pm}$ patterns at nonsaturating fields (see
Fig. 14). For the calculation of the $P^{\pm}$ according to equation (52a), we
have up to now assumed Lorentzian-squared functions for
$\widetilde{M}_{z}^{2}(q\xi_{\mathrm{M}})$ and
$\widetilde{H}_{\mathrm{p}}^{2}(q\xi_{\mathrm{H}})$ [compare equations
(49)$-$(50)] with $\xi_{\mathrm{M}}=\xi_{\mathrm{H}}=D/2=7.5\,\mathrm{nm}$.
The effect of a hard-sphere form factor for $\widetilde{M}_{z}^{2}$ and
$\widetilde{H}_{\mathrm{p}}^{2}$ with different values for $\xi_{\mathrm{M}}$
and $\xi_{\mathrm{H}}$ is depicted in Fig. 15. Here, peak-type features may
appear in $P^{\pm}$, which might be detected in highly monodisperse
particulate systems.

Figure 10: Plot of $P_{\mathrm{f}\perp}^{+}(q_{y},q_{z})$ (upper row) and
$P_{\mathrm{f}\perp}^{-}(q_{y},q_{z})$ (lower row) for different applied
magnetic fields $H_{\mathrm{i}}$ (see insets). $\alpha=\alpha(q)$ [equation
(IV.2.2)], $A_{\mathrm{H}}/A_{\mathrm{M}}=0.5$, and $l_{\mathrm{D}}=0$.

Figure 11: $2\pi$-azimuthally-averaged $P_{\mathrm{f}\perp}^{+}(q)$ (a) and
$P_{\mathrm{f}\perp}^{-}(q)$ (b) of the data shown in Fig. 10.

Figure 12: Plot of $P_{\mathrm{f}\perp}^{+}(q_{y},q_{z})$ (upper row) and
$P_{\mathrm{f}\perp}^{-}(q_{y},q_{z})$ (lower row) for different ratios of
$A_{\mathrm{H}}/A_{\mathrm{M}}$ (see insets). $\alpha=\alpha(q)$ [equation
(IV.2.2)], $\mu_{0}H_{\mathrm{i}}=0.3\,\mathrm{T}$, and $l_{\mathrm{D}}=0$.

Figure 13: Plot of $P_{\mathrm{f}\perp}^{+}(q_{y},q_{z})$ (upper row) and
$P_{\mathrm{f}\perp}^{-}(q_{y},q_{z})$ (lower row) for different values of
$\alpha=\mathrm{constant}$ (see insets). $A_{\mathrm{H}}/A_{\mathrm{M}}=0.2$,
$\mu_{0}H_{\mathrm{i}}=0.3\,\mathrm{T}$, and $l_{\mathrm{D}}=0$.

Figure 14: Effect of the DMI. Plot of $P_{\mathrm{f}\perp}^{+}(q_{y},q_{z})$
(upper row) and $P_{\mathrm{f}\perp}^{-}(q_{y},q_{z})$ (lower row) as a
function of $H_{\mathrm{i}}$ (see insets). $\alpha=\alpha(q)$ [equation
(IV.2.2)], $A_{\mathrm{H}}/A_{\mathrm{M}}=1$, and
$l_{\mathrm{D}}=1.9\,\mathrm{nm}$.

Figure 15: Results for the azimuthally-averaged $P_{\mathrm{f}\perp}^{+}(q)$
using the sphere form factor (instead of Lorentzian squared functions) for
both $\widetilde{M}_{z}^{2}(q\xi_{\mathrm{M}})$ and
$\widetilde{H}_{\mathrm{p}}^{2}(q\xi_{\mathrm{H}})$. (a) Field dependence (see
inset) of $P_{\mathrm{f}\perp}^{+}(q)$ for $\xi_{\mathrm{M}}=7.5\,\mathrm{nm}$
and $\xi_{\mathrm{H}}=8.5\,\mathrm{nm}$. (b) $P_{\mathrm{f}\perp}^{+}(q)$ at
$\mu_{0}H_{\mathrm{i}}=0.3\,\mathrm{T}$, $\xi_{\mathrm{M}}=7.5\,\mathrm{nm}$,
but for increasing $\xi_{\mathrm{H}}$ (see inset). (c)
$P_{\mathrm{f}\perp}^{+}(q)$ at $\mu_{0}H_{\mathrm{i}}=0.3\,\mathrm{T}$,
$\xi_{\mathrm{H}}=7.5\,\mathrm{nm}$, but for increasing $\xi_{\mathrm{M}}$
(see inset). $\alpha=\alpha(q)$ [equation (IV.2.2)],
$A_{\mathrm{H}}/A_{\mathrm{M}}=0.2$, and $l_{\mathrm{D}}=0$.


(NSF) through DMS-2011444.

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