# Magnetic neutron scattering from spherical nanoparticles with Néel surface
anisotropy: Analytical treatment

Michael P. Adams<EMAIL_ADDRESS>Department of Physics and Materials
Science, University of Luxembourg, 162A avenue de la Faiencerie, L-1511
Luxembourg, Grand Duchy of Luxembourg  Andreas Michels<EMAIL_ADDRESS>Department of Physics and Materials Science, University of Luxembourg, 162A
avenue de la Faiencerie, L-1511 Luxembourg, Grand Duchy of Luxembourg  Hamid
Kachkachi<EMAIL_ADDRESS>Université de Perpignan via Domitia,
Laboratoire PROMES CNRS UPR8521, Rambla de la Thermodynamique, Tecnosud,
F-66100 Perpignan, France

###### Abstract

The magnetization profile and the related magnetic small-angle neutron
scattering cross section of a single spherical nanoparticle with Néel surface
anisotropy is analytically investigated. We employ a Hamiltonian that
comprises the isotropic exchange interaction, an external magnetic field, a
uniaxial magnetocrystalline anisotropy in the core of the particle, and the
Néel anisotropy at the surface. Using a perturbation approach, the
determination of the magnetization profile can be reduced to a Helmholtz
equation with Neumann boundary condition, whose solution is represented by an
infinite series in terms of spherical harmonics and spherical Bessel
functions. From the resulting infinite series expansion, we analytically
calculate the Fourier transform, which is algebraically related to the
magnetic small-angle neutron scattering cross section. The approximate
analytical solution is compared to the numerical solution using the Landau-
Lifshitz equation, which accounts for the full nonlinearity of the problem.

##  I Introduction

Magnetic small-angle neutron scattering (SANS) is a powerful technique for
investigating spin structures on the mesoscopic length scale ($\sim
1$$-$$100\,\mathrm{nm}$) and inside the volume of magnetic materials [1, 2].
Recent SANS studies on magnetic nanoparticles, in particular employing spin-
polarized neutrons, unanimously demonstrate that their spin textures are
highly complex and exhibit a variety of nonuniform, canted, or core-shell-type
configurations (see, e.g. Refs. [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]
and references therein). The magnetic SANS data analysis largely relies on
structural form-factor-models for the cross section, borrowed from nuclear
SANS, which do not properly account for the existing spin inhomogeneity inside
magnetic nanoparticles or nanomagnets (NM).

Progress in magnetic SANS theory [16, 17, 18, 19, 20, 21, 22, 23, 24, 25]
strongly suggests that for the analysis of experimental magnetic SANS data the
spatial nanometer scale variation of the orientation and magnitude of the
magnetization vector field must be taken into account, and that macrospin-
based models—assuming a uniform magnetization—are not adequate. The starting
point for a proper analysis of the scattering problem is a micromagnetic
continuum expression for the magnetic energy of the system. In the static
case, this then leads to the so-called Brown’s equations, a set of nonlinear
partial differential equations for the magnetization along with complex
boundary conditions on the surface of the magnet. From these equations the
Fourier image and the magnetic SANS cross section may be obtained.

In this paper, we present an analytical treatment of the magnetic SANS cross
section of a spherical NM with Néel’s surface anisotropy [26]. The manuscript
is organized as follows: In Section II, we calculate the real-space spin
structure of the spherical NM using classical micromagnetic theory within the
second-order perturbation approach. In Section III, we compute the three-
dimensional Fourier transform of the real-space spin structure, which directly
yields the magnetic neutron scattering cross section and the pair-distance
distribution function. The analytical results are benchmarked by comparing
them to numerical finite-difference simulations using the Landau-Lifshitz
equation of motion. Finally, Section IV summarizes the main findings of this
study.

We also make reference to our numerical study [27], where in contrast to the
present analytical work the full nonlinearity of the problem is considered.

##  II Micromagnetic Theory

In the static micromagnetic approach [28], the magnetic configuration of a
system is described by the continuous magnetization vector field
$\mathbf{M}(\mathbf{r})$, which is subject to a constant magnitude
$\|\mathbf{M}(\mathbf{r})\|=M_{0}$. The saturation magnetization $M_{0}$ is
only a function of temperature. The normalized magnetization vector field is
then defined as

$\displaystyle\mathbf{m}(\mathbf{r})=\mathbf{M}(\mathbf{r})/M_{0}=[m_{x}(\mathbf{r}),m_{y}(\mathbf{r}),m_{z}(\mathbf{r})].$
(1)

Our Hamiltonian for the NM includes the isotropic exchange interaction, the
Zeeman energy, a uniaxial magnetic anisotropy for spins in the core and Néel’s
surface anisotropy for those on the surface. In the continuum approach, it
reads:

$\displaystyle\mathcal{H}$
$\displaystyle=-A\sum_{\alpha\in\\{x,y,z\\}}\int_{V}m_{\alpha}\Delta
m_{\alpha}\;d^{3}r$ $\displaystyle-
M_{0}\mathbf{B}_{0}\int_{V}\mathbf{m}\;d^{3}r-K_{c}\int_{V}\left(\mathbf{m}\cdot\mathbf{e}_{\mathrm{A}}\right)^{2}\;d^{3}r$
$\displaystyle+A\sum_{\alpha\in\\{x,y,z\\}}\oint_{\partial V}m_{\alpha}\nabla
m_{\alpha}\cdot\mathbf{n}\;d^{2}r$
$\displaystyle-\frac{K_{s}}{2}\sum_{\alpha\in\\{x,y,z\\}}\oint_{\partial
V}\left|n_{\alpha}\right|m_{\alpha}^{2}\;d^{2}r$ (2)

where $A$ is the exchange-stiffness constant, $\nabla$ is the Del operator,
$\Delta$ is the Laplace operator, $\mathbf{B}_{0}=\mu_{0}\mathbf{H}_{0}$ is a
constant applied magnetic field, $K_{c}>0$ denotes the uniaxial core
anisotropy constant, $\mathbf{e}_{\mathrm{A}}$ is a unit vector specifying the
arbitrary core anisotropy axis, $K_{s}>0$ is the Néel surface anisotropy
constant [26] with

$\displaystyle\mathbf{n}=[\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta]$
(3)

being the surface normal to the boundary of the NM [29, 30]. In (2), the two
surface integrals take into account the boundary conditions for the
magnetization on the surface ($\partial V$) of the NM of volume $V$, which
result from the exchange interaction and the Néel term.

For small deviations from the homogeneous magnetization state, a perturbation
approach is applicable. Let $\mathbf{m}_{0}$ be the principal unit vector
(average direction) associated with $\mathbf{m}(\mathbf{r})$ and let the
vector function $\bm{\psi}(\mathbf{r})\perp\mathbf{m}_{0}$ describe the spin
misalignment. One can then write:

$\mathbf{m}(\mathbf{r})=\mathbf{m}_{0}\sqrt{1-\|\bm{\psi}(\mathbf{r})\|^{2}}+\bm{\psi}(\mathbf{r}),\quad\|\mathbf{m}(\mathbf{r})\|=1.$
(4)

Assuming that $\psi_{x},\psi_{y},\psi_{z}\ll 1$, the following second-order
Maclaurin expansion in $\bm{\psi}$ is used to find an approximate closed-form
solution for $\mathbf{m}(\mathbf{r})$:

$\mathbf{m}\left(\mathbf{r}\right)\cong\mathbf{m}_{0}+\bm{\psi}\left(\mathbf{r}\right)-\frac{1}{2}\|\bm{\psi}(\mathbf{r})\|^{2}\mathbf{m}_{0},$
(5)

where $\mathbf{m}_{0}$ is taken as a known constant vector in subsequent
calculations. By choosing the orthonormal vector base [31]

$\displaystyle\begin{cases}\displaystyle\mathbf{g}_{0}=\mathbf{m}_{0},\\\
\displaystyle\mathbf{g}_{1}=\frac{\displaystyle\mathbf{m}_{0}\times\mathbf{e}_{\mathrm{A}}}{\displaystyle\|\mathbf{m}_{0}\times\mathbf{e}_{\mathrm{A}}\|},\\\
\displaystyle\mathbf{g}_{2}=\frac{\displaystyle(\mathbf{m}_{0}\cdot\mathbf{e}_{\mathrm{A}})\cdot\mathbf{m}_{0}-\mathbf{e}_{\mathrm{A}}}{\displaystyle\|(\mathbf{m}_{0}\cdot\mathbf{e}_{\mathrm{A}})\cdot\mathbf{m}_{0}-\mathbf{e}_{\mathrm{A}}\|},\end{cases}$
(6)

the parametrization

$\displaystyle\bm{\psi}(\mathbf{r})=\psi_{1}(\mathbf{r})\mathbf{g}_{1}+\psi_{2}(\mathbf{r})\mathbf{g}_{2},$
(7)

and by introducing the dimensionless coordinates $\bm{\xi}=\mathbf{r}/R$ (with
$\xi=\|\bm{\xi}\|=r/R$), where $\mathbf{r}$ is the position vector,

$\displaystyle\bm{r}=[r\sin\theta\cos\phi,r\sin\theta\sin\phi,r\cos\theta],$
(8)

and $R$ denotes the radius of the NM, the minimization of the Hamiltonian (2)
leads to the well-known Helmholtz equation with Neumann boundary conditions on
the unit sphere [30, 29]:

$\displaystyle[\Delta_{\bm{\xi}}-\kappa_{\beta}^{2}]\psi_{\beta}$
$\displaystyle=0\;,\quad\beta\in\\{1,2\\}$ (9)
$\displaystyle\left.\frac{d\psi_{\beta}}{d\xi}\right|_{\xi=1}$
$\displaystyle=\sum_{\alpha\in\\{x,y,z\\}}\chi_{\alpha}^{\beta}|n_{\alpha}|,$
(10)

where the constants are defined as:

$\displaystyle\kappa_{1}^{2}$
$\displaystyle=\mathbf{m}_{0}\cdot\mathbf{b}_{0}+2k_{c}(\mathbf{m}_{0}\cdot\mathbf{e}_{\mathrm{A}})^{2},$
(11) $\displaystyle\kappa_{2}^{2}$
$\displaystyle=\mathbf{m}_{0}\cdot\mathbf{b}_{0}+2k_{c}\left[2(\mathbf{m}_{0}\cdot\mathbf{e}_{\mathrm{A}})^{2}-1\right],$
(12) $\displaystyle\chi_{\alpha}^{\beta}$
$\displaystyle=k_{s}(\mathbf{m}_{0}\cdot\mathbf{e}_{\alpha})(\mathbf{g}_{\beta}\cdot\mathbf{e}_{\alpha}),$
(13)

with the dimensionless quantities

$\displaystyle k_{c}$ $\displaystyle=\frac{R^{2}K_{c}}{A},$ $\displaystyle
k_{s}$ $\displaystyle=\frac{RK_{s}}{A},$ $\displaystyle\mathbf{b}_{0}$
$\displaystyle=\frac{R^{2}M_{0}}{A}\mathbf{B}_{0}.$ (14)

The $\mathbf{e}_{\alpha}$ (with $\alpha=x,y,z$) in (13) denote the unit
vectors of the Cartesian laboratory coordinate frame (in which $\mathbf{n}$
and $\mathbf{r}$ are defined). We emphasize that there are only two
independent differential equations for $\bm{\psi}$, which is a consequence of
the constraint $\|\mathbf{m}(\mathbf{r})\|=1$.

In our graphical representations, we will frequently use the following values:
$k_{c}=0.1$ and $k_{s}=3.0$, which (using $R=5\,\mathrm{nm}$ and
$A=10^{-11}\,\mathrm{J/m}$) correspond to $K_{c}=40\,\mathrm{kJ/m^{3}}$ and
$K_{s}=6\,\mathrm{mJ/m^{2}}$ [32, 33]. For $M_{0}=1.7\times
10^{6}\,\mathrm{A/m}$, the relation between $b_{0}$ (dimensionless) and the
external field is $B_{0}=4/17b_{0}\times 1\mathrm{T}$.

The fundamental solution of the homogeneous Helmholtz equation (9) is well
known [34, 35]. Its nonsingular part can be expressed in spherical coordinates
as an infinite series in terms of spherical harmonics $Y_{\ell
m}(\theta,\phi)$ and spherical Bessel functions of the first kind
$j_{n}(\mathrm{i}\kappa_{\beta}\xi)$,

$\displaystyle\psi_{\beta}=\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}c_{\ell
m}^{\beta}j_{\ell}(\mathrm{i}\kappa_{\beta}\xi)Y_{\ell m}(\theta,\phi).$ (15)

The imaginary number ‘$\mathrm{i}$’ in the argument of the spherical Bessel
function is due to the negative sign in the Helmholtz equation (9). The
expansion coefficients $c_{\ell m}^{\beta}$ are obtained from the Neumann
boundary condition (10) using the method of least squares (see Appendix A).
From there it is seen that the zero-order term with $\ell=0$ vanishes, which
physically makes sense, since the spin misalignment in our model is caused by
the Néel surface anisotropy and, thus, due to symmetry reasons there is no
misalignment at the center of the NM, i.e.
$\psi_{\beta}(\xi=0,\theta,\phi)\equiv 0$. By contrast, the largest spin
misalignment is found at the boundary of the NM, i.e. $\xi=1$. Further, we
find that the coefficients $c_{\ell m}^{\beta}$ vanish in the case of odd
$\ell$ and $m$, they are real-valued, and even with respect to the index $m$,
i.e. $c_{\ell m}^{\beta}=c_{\ell,-m}^{\beta}$. Taking these properties into
account, one can conveniently express the solution in terms of the associated
Legendre polynomials $P_{\ell}^{m}(\cos\theta)$ with $\ell=2\nu$ and $m=2\mu$
[note that we use the convention that $Y_{\ell m}(\theta,\phi)=N_{\ell
m}P_{\ell}^{m}(\cos\theta)\mathrm{e}^{\mathrm{i}m\phi}$ (p. 378 (14.30.1) in
[36])]:

$\displaystyle\psi_{\beta}$
$\displaystyle=\sum_{\nu=1}^{\infty}\sum_{\mu=0}^{\nu}a_{\nu\mu}^{\beta}\Upsilon_{\nu}(\kappa_{\beta}\xi)P_{2\nu}^{2\mu}(\cos\theta)\cos(2\mu\phi),$
(16)

where we define (compare p. 624–626 in [34])

$\displaystyle\Upsilon_{\nu}(\tau)$
$\displaystyle=j_{2\nu}(\mathrm{i}\tau)=\frac{\sqrt{\pi}}{2}\sum_{s=0}^{\infty}\frac{(-1)^{\nu}(\tau/2)^{2(s+\nu)}}{s!\Gamma(2\nu+s+3/2)},$
(17)

and the expansion coefficients are given by

$\displaystyle a_{\nu\mu}^{\beta}$
$\displaystyle=\frac{2k_{s}N_{2\nu,2\mu}}{1+\delta_{\mu,0}}\frac{\mathbf{g}_{\beta}\cdot\operatorname{diag}\left[I_{2\nu,2\mu}^{x},I_{2\nu,2\mu}^{y},I_{2\nu,2\mu}^{z}\right]\cdot\mathbf{m}_{0}}{\kappa_{\beta}\Upsilon_{\nu}^{\prime}(\kappa_{\beta})}$
(18)

with

$\displaystyle I_{\ell m}^{\alpha}$
$\displaystyle=\int_{0}^{2\pi}\int_{0}^{\pi}Y_{\ell,m}^{*}(\theta,\phi)|n_{\alpha}|\sin\theta\;d\theta
d\phi$ (19)

and

$\displaystyle N_{\ell m}$
$\displaystyle=\sqrt{\frac{2\ell+1}{4\pi}\frac{(\ell-m)!}{(\ell+m)!}}.$ (20)

In (18), $\delta_{\mu,0}$ is the Kronecker delta function,
$\operatorname{diag}[...]$ denotes a $3\times 3$ diagonal matrix, and
$\Upsilon_{\nu}^{\prime}(\tau)$ is the first-order derivative of (17) with
respect to $\tau$. For some small values of $\ell$ and $m$, the exact
solutions of the integrals $I_{\ell m}^{\alpha}$ are listed in Appendix B.

From (18) it is seen that the functions $\psi_{\beta}$ depend linearly on
$k_{s}$, such that for $k_{s}=0$ the magnetization of the NM is homogeneous
(as expected). Since we assume that $\psi_{x},\psi_{y},\psi_{z}\ll 1$, it is
clear that the validity of our solution is restricted to a finite range $0\leq
k_{s}\leq k_{s,\mathrm{max}}$. Taking only the terms with $\nu=1$ into account
(corresponding to $\ell=2$), the remaining (second-order) expression reads:

$\displaystyle\psi_{\beta}(\xi,\theta,\phi)$
$\displaystyle\approx-\frac{15k_{s}}{32}\frac{\Upsilon_{1}(\kappa_{\beta}\xi)}{\kappa_{\beta}\Upsilon_{1}^{\prime}(\kappa_{\beta})}\mathbf{g}_{\beta}\cdot\bm{\mathcal{V}}(\theta,\phi)\cdot\mathbf{m}_{0},$
(21)

where

$\displaystyle\bm{\mathcal{V}}(\theta,\phi)$
$\displaystyle=\operatorname{diag}\begin{bmatrix}\cos^{2}\theta-1/3-\sin^{2}\theta\cos(2\phi)\\\
\cos^{2}\theta-1/3+\sin^{2}\theta\cos(2\phi)\\\
-2(\cos^{2}\theta-1/3)\end{bmatrix}.$ (22)

A reasonable approximation for small $\kappa_{\beta}$ in (21) is obtained by
taking into account the first two terms in the infinite series (17) for
$\Upsilon_{\nu}(\tau)$. This results in the following expression [compare
(21)]:

$\displaystyle\frac{\Upsilon_{1}(\kappa_{\beta}\xi)}{\kappa_{\beta}\Upsilon_{1}^{\prime}(\kappa_{\beta})}\approx\frac{1}{4}\frac{\kappa_{\beta}^{2}\xi^{4}+14\xi^{2}}{\kappa_{\beta}^{2}+7}.$
(23)

In the limit $\kappa_{\beta}\rightarrow 0$, this expression reduces to a
quadratic function in $\xi$

$\displaystyle\lim_{\kappa_{\beta}\rightarrow
0}\left\\{\frac{\Upsilon_{1}(\kappa_{\beta}\xi)}{\kappa_{\beta}\Upsilon_{1}^{\prime}(\kappa_{\beta})}\right\\}=\frac{\xi^{2}}{2}.$
(24)

The case of an infinite applied magnetic field $\mathbf{B}_{0}$, or of a
strong uniaxial core anisotropy [compare (11) and (12)], corresponds to the
limit

$\displaystyle\lim_{\kappa_{\beta}\rightarrow\infty}\left\\{\frac{\Upsilon_{1}(\kappa_{\beta}\xi)}{\kappa_{\beta}\Upsilon_{1}^{\prime}(\kappa_{\beta})}\right\\}=0,$
(25)

which recovers the expected result of zero spin misalignment. Note that the
limit $\kappa_{\beta}\rightarrow\infty$ is only obtained using all terms of
the infinite series (17).

Of particular interest is the behavior of $\psi_{\beta}$ as a function of the
radius $R$ of the NM. Inspecting the Hamiltonian (2), it becomes clear that
the surface anisotropy energy scales as $R^{2}$, while the uniaxial core
anisotropy energy scales as $R^{3}$. Since the core and the surface anisotropy
act in opposite ways (trying to make the spin structure more homogeneous,
respectively, more inhomogeneous), we see that an increasing radius $R$
corresponds to a decreasing $\psi_{\beta}$. This behavior reflects the NM’s
surface-area-to-volume ratio. With (21) it is not possible to make any
prediction in this regard, because until this point we did not include the
principal unit vector $\mathbf{m}_{0}$ into the minimization of the
Hamiltonian. Generally, $\mathbf{m}_{0}$ is a function of $k_{s}$, $k_{c}$,
$\mathbf{b}_{0}$, and $\mathbf{e}_{\mathrm{A}}$.

Figure 1: Normalized effective energy potential of the Néel surface anisotropy
as a function of the Cartesian components of the average magnetization vector
$\mathbf{m}_{0}=[\sin\beta\cos\alpha,\sin\beta\sin\alpha,\cos\beta]$, computed
via numerical integration of the surface contribution in (2) and by using the
second-order approximation (21). Parameters are:
$\mathbf{e}_{\mathrm{A}}=[0,0,1]$, $\mathbf{b}_{0}=\mathbf{0}$, $k_{c}=0.1$,
and $k_{s}=3.0$. The minima of the Néel surface contribution are in this case
along the cubic space diagonals $\mathbf{m}_{0}=[\pm 1,\pm 1,\pm 1]/\sqrt{3}$,
while the maxima correspond to the Cartesian axes
$\pm\mathbf{e}_{x},\pm\mathbf{e}_{y},\pm\mathbf{e}_{z}$. The effective energy
potential has cubic symmetry and is approximately proportional to a function
of the type $\simeq m_{0,x}^{4}+m_{0,y}^{4}+m_{0,z}^{4}$ (see also [29]).

Figure 2: Comparison between the numerical solution using the Landau-Lifshitz
equation (upper row) and the second-order analytical solution (21) for
$\|\bm{\psi}(\bm{\xi})\|=\sqrt{\psi_{1}^{2}(\bm{\xi})+\psi_{2}^{2}(\bm{\xi})}$
(lower row). (a) and (e) show $\|\bm{\psi}\|$ on the boundary surface
($\xi=1$), while (b)$-$(d) and (f)$-$(h) display selected planar cuts in
$(\xi_{x},\xi_{y},\xi_{z})$ space. The following parameters are used:
$\mathbf{e}_{\mathrm{A}}=\mathbf{e}_{z}$, $\mathbf{b}_{0}=[0.4,0,0.4]$
($B_{0}\cong 133\,\mathrm{mT}$), $k_{c}=0.1$, $k_{s}=3.0$, and
$\mathbf{m}_{0}=[\sin\beta\cos\alpha,\sin\beta\sin\alpha,\cos\beta]$ where
$\alpha=0^{\circ}$ and $\beta=40^{\circ}$.

Figure 3: Real-space spin structure in the $\xi_{x}$-$\xi_{z}$ plane computed
using (4) and (21). Parameters are the same as in Fig. 2. The external field
$B_{0}\cong 133\,\mathrm{mT}$ is applied in the $\xi_{x}$-$\xi_{z}$ plane and
inclined by an angle of $\beta=40^{\circ}$ relative to the $\xi_{z}$ axis
(compare to [29]).

In the special case when the uniaxial anisotropy axis and the applied magnetic
field are both directed parallel to the $z$ axis
($\mathbf{e}_{\mathrm{A}}=[0,0,1]$ and $\mathbf{b}_{0}=[0,0,b_{0}]$), the
principal unit magnetization vector may be written as:

$\displaystyle\mathbf{m}_{0}=[1/\sqrt{2}\sin\beta,1/\sqrt{2}\sin\beta,\cos\beta],$
(26)

where $\beta\in[0,\arccos(1/\sqrt{3})]$. This choice is justified by the
effective cubic symmetry of the Néel anisotropy as shown in Fig. 1. This
result was already predicted by Garanin and Kachkachi [29]. The solutions for
$\psi_{1,2}(\xi,\theta,\phi)$ [using the particular $\mathbf{m}_{0}$ (26)]
then read:

$\displaystyle\psi_{1}$
$\displaystyle\approx\frac{15k_{s}}{32}\frac{\Upsilon_{1}(\kappa_{1}\xi)}{\kappa_{1}\Upsilon_{1}^{\prime}(\kappa_{1})}\sin^{2}\theta\cos(2\phi)\sin\beta,$
(27) $\displaystyle\psi_{2}$
$\displaystyle\approx\frac{15k_{s}}{32}\frac{\Upsilon_{1}(\kappa_{2}\xi)}{\kappa_{2}\Upsilon_{1}^{\prime}(\kappa_{2})}(1-3\cos^{2}\theta)\sin\beta\cos\beta.$
(28)

In Fig. 2, the analytical solution (21) (lower row) is compared to the
numerical solution based on the Landau-Lifshitz equation
$\dot{\mathbf{m}}=-\gamma\mathbf{m}\times\mathbf{B}_{\mathrm{eff}}-\alpha\mathbf{m}\times(\mathbf{m}\times\mathbf{B}_{\mathrm{eff}})$
(upper row) [37], where $\gamma$ is the gyromagnetic ratio, $\alpha$ is the
damping constant, and the dot denotes the first-order time derivative (see our
numerical study [27] for further details). Shown is the vector norm of the
$\bm{\psi}(\bm{\xi})$ function scaled to its maximum value. From Fig. 2 it is
seen that our analytical approximation is in qualitative agreement with the
results from the numerical simulation. The corresponding real-space spin
structure $\mathbf{m}(\bm{\xi})$ is displayed in Fig. 3, where the surface
spin disorder becomes clearly visible.

It is also instructive to compare our solution (21) with that obtained using
the Green function approach [29, 30]. In particular, for $\xi$ located close
to the surface, where the maximum spin misalignment with respect to
$\mathbf{m}_{0}$ occurs, the Green function method yields the following
approximate expression:

$\displaystyle\psi_{\beta}(\bm{\xi})$
$\displaystyle\approx-\frac{15k_{s}}{32}\left[1-\frac{\kappa_{\beta}^{2}}{14}\right]\xi^{2}\mathbf{g}_{\beta}\cdot\bm{\mathcal{V}}(\bm{\xi})\cdot\mathbf{m}_{0},$
(29) $\displaystyle\bm{\mathcal{V}}(\bm{\xi})$
$\displaystyle=-\operatorname{diag}\begin{bmatrix}\xi_{x}^{2}/\xi^{2}-1/3\\\
\xi_{y}^{2}/\xi^{2}-1/3\\\ \xi_{z}^{2}/\xi^{2}-1/3\end{bmatrix}.$ (30)

This expression is also found when (21) is expanded in $\kappa_{\beta}$ at the
surface of the NM ($\xi=1$). While the infinite series approach using
spherical harmonics and spherical Bessel functions yields an exact solution of
the Helmholtz equation, the Green’s function approach provides an approximate
explicit expression of $\psi_{\beta}$ in terms of the coefficients
$\kappa_{\beta}$. Indeed, as was shown in [30], in the presence of core
anisotropy, the Green function as the kernel of the Helmholtz equation is only
obtained as a perturbative series in $\kappa_{\beta}$. As such, (29) is
restricted to small values of $\kappa_{\beta}$, i.e. assuming that the core
anisotropy and applied magnetic field are much smaller than the exchange
coupling. This is manifest in (29) by the presence of the factor
$1-\kappa_{\beta}^{2}/14$ which implies that the contribution of spin
misalignment may diverge for too large $\kappa_{\beta}$ (i.e. for a strong
field and/or large core anisotropy).

##  III Magnetic SANS cross section

The quantity of interest in experimental SANS studies is the elastic magnetic
differential scattering cross section $d\Sigma_{M}/d\Omega$, which is usually
recorded on a two-dimensional position-sensitive detector. For the most
commonly used scattering geometry in magnetic SANS experiments, where the
applied magnetic field $\mathbf{B}_{0}\parallel\mathbf{e}_{z}$ is
perpendicular to the wave vector $\mathbf{k}_{0}\parallel\mathbf{e}_{x}$ of
the incident neutrons (see Fig. 4), $d\Sigma_{M}/d\Omega$ (for unpolarized
neutrons) can be written as [1]:

$\displaystyle\frac{d\Sigma_{M}}{d\Omega}(\mathbf{q})=\frac{8\pi^{3}}{V}b_{H}^{2}\left(|\widetilde{M}_{x}|^{2}+|\widetilde{M}_{y}|^{2}\cos^{2}\theta_{q}\right.$
$\displaystyle\left.+|\widetilde{M}_{z}|^{2}\sin^{2}\theta_{q}-(\widetilde{M}_{y}\widetilde{M}_{z}^{\ast}+\widetilde{M}_{y}^{\ast}\widetilde{M}_{z})\sin\theta_{q}\cos\theta_{q}\right),$
(31)

where $V$ is the scattering volume and $b_{H}=2.91\times
10^{8}\,\mathrm{A}^{-1}\mathrm{m}^{-1}$ the magnetic scattering length in the
small-angle regime (the atomic magnetic form factor is approximated by $1$,
since we are dealing with forward scattering);
$\widetilde{\mathbf{M}}(\mathbf{q})=[\widetilde{M}_{x}(\mathbf{q}),\widetilde{M}_{y}(\mathbf{q}),\widetilde{M}_{z}(\mathbf{q})]$
represents the magnetization vector field $\mathbf{M}(\mathbf{r})$ in Fourier
space, $\theta_{q}$ denotes the angle between $\mathbf{q}$ and
$\mathbf{B}_{0}$, and the asterisk ‘$*$’ stands for the complex conjugate.
Note that in the perpendicular scattering geometry, the Fourier components are
evaluated in the plane $q_{x}=0$.

Figure 4: Sketch of the perpendicular scattering geometry
($\mathbf{B}_{0}\perp\mathbf{k}_{0}$). The scattering vector $\mathbf{q}$
corresponds to the difference between the wave vectors of the incident
($\mathbf{k}_{0}$) and scattered ($\mathbf{k}_{1}$) neutrons. The angle
$\theta_{q}$ specifies the orientation of $\mathbf{q}$ on the detector. In the
small-angle approximation, the component of $\mathbf{q}$ along
$\mathbf{k}_{0}$ is neglected.

The Fourier transform of the three-dimensional magnetization vector field
(with a tilde above the symbol) is defined as follows

$\displaystyle\widetilde{\mathbf{M}}(\mathbf{q})$
$\displaystyle=\frac{1}{(2\pi)^{3/2}}\int_{V}\mathbf{M}(\mathbf{r})\exp\left(-\mathrm{i}\mathbf{q}\cdot\mathbf{r}\right)d^{3}r,$
(32) $\displaystyle\mathbf{M}(\mathbf{r})$
$\displaystyle=\frac{1}{(2\pi)^{3/2}}\int_{V}\widetilde{\mathbf{M}}(\mathbf{q})\exp\left(\mathrm{i}\mathbf{q}\cdot\mathbf{r}\right)d^{3}q.$
(33)

For subsequent calculations, we introduce the following dimensionless
quantities

$\displaystyle\bm{\upsilon}$ $\displaystyle=\mathbf{q}R,$
$\displaystyle\widetilde{\bm{\mathcal{M}}}$
$\displaystyle=\frac{(2\pi)^{3/2}}{4\pi R^{3}M_{0}}\widetilde{\mathbf{M}},$
(34)

and we express the dimensionless scattering vector in spherical coordinates as

$\displaystyle\bm{\upsilon}=[\upsilon\sin\theta_{q}\cos\phi_{q},\upsilon\sin\theta_{q}\sin\phi_{q},\upsilon\cos\theta_{q}].$
(35)

Next, in (32) we use the following first-order approximation for the real-
space magnetization vector $\mathbf{m}(\bm{\xi})$ [see (5) and (7)]

$\displaystyle\mathbf{m}(\bm{\xi})=\mathbf{m}_{0}+\sum_{\beta=1}^{2}\mathbf{g}_{\beta}\psi_{\beta}(\bm{\xi}).$
(36)

As shown in Appendix C, the final expression for the Fourier transform of the
magnetization is then given by:

$\displaystyle\widetilde{\bm{\mathcal{M}}}(\bm{\upsilon})=\frac{j_{1}(\upsilon)}{\upsilon}\mathbf{m}_{0}$
$\displaystyle+\sum_{\beta=1}^{2}\mathbf{g}_{\beta}\sum_{\nu=1}^{\infty}\sum_{\mu=0}^{\nu}(-1)^{\nu}a_{\nu\mu}^{\beta}\rho_{\nu}^{\beta}(\upsilon)P_{2\nu}^{2\mu}(\cos\theta_{q})\cos(2\mu\phi_{q}),$
(37)

where

$\displaystyle\rho_{\nu}^{\beta}(\upsilon)=-\frac{\upsilon
j_{2\nu-1}(\upsilon)\Upsilon_{\nu}(\kappa_{\beta})-\kappa_{\beta}\mathcal{\aleph}_{\nu}(\kappa_{\beta})j_{2\nu}(\upsilon)}{\upsilon^{2}+\kappa_{\beta}^{2}},$
(38) $\displaystyle\mathcal{\aleph}_{\nu}(\tau)$
$\displaystyle=\frac{\sqrt{\pi}}{2}\sum_{s=0}^{\infty}\frac{(-1)^{\nu}(\tau/2)^{2(s+\nu)-1}}{s!\Gamma(2\nu+s+1/2)},$
(39)

and $\Upsilon_{\nu}(\kappa_{\beta})$ is given by (17). The zero-order term
$\propto j_{1}(\upsilon)/\upsilon$ in (37) represents the form factor of a
homogeneously magnetized sphere [2]. In the limiting case of an infinite
applied magnetic field, which is equivalent to the limit
$\kappa_{\beta}\rightarrow\infty$, the additional terms [second line in (37)]
vanish [compare (25)] and the spherical form factor remains. On the other
hand, if $k_{s}=0$, the additional terms also vanish because from the physical
point of view, the Néel surface anisotropy cancels and from (18) we know that
the coefficients $a_{\nu\mu}^{\beta}$ are linear in $k_{s}$. Taking only the
terms with $\nu=1$ into account and setting $\phi_{q}=\pi/2$
($\upsilon_{x}=0$), corresponding to the scattering geometry where the applied
magnetic field $\mathbf{B}_{0}\parallel\mathbf{e}_{z}$ is perpendicular to the
wave vector $\mathbf{k}_{0}\parallel\mathbf{e}_{x}$ of the incident neutrons
(Fig. 4), the expression for $\widetilde{\bm{\mathcal{M}}}(\bm{\upsilon})$ can
be written as [compare with (21)]:

$\displaystyle\widetilde{\bm{\mathcal{M}}}(\bm{\upsilon})$
$\displaystyle\approx\frac{j_{1}(\upsilon)}{\upsilon}\mathbf{m}_{0}$
$\displaystyle-\frac{15k_{s}}{32}\sum_{\beta=1}^{2}\mathcal{R}_{\beta}(\upsilon)\left(\mathbf{g}_{\beta}\cdot\bm{\mathcal{V}}(\theta_{q},\pi/2)\cdot\mathbf{m}_{0}\right)\mathbf{g}_{\beta},$
(40)

where the radial function is

$\displaystyle\mathcal{R}_{\beta}(\upsilon)$ $\displaystyle=\frac{\upsilon
j_{1}(\upsilon)}{\upsilon^{2}+\kappa_{\beta}^{2}}\frac{\Upsilon_{1}(\kappa_{\beta})}{\kappa_{\beta}\Upsilon_{1}^{\prime}(\kappa_{\beta})}-\frac{j_{2}(\upsilon)}{\upsilon^{2}+\kappa_{\beta}^{2}}\frac{\aleph_{1}(\kappa_{\beta})}{\Upsilon_{1}^{\prime}(\kappa_{\beta})}.$
(41)

$\mathcal{R}_{\beta}(\upsilon)$ can be approximated for small $\kappa_{\beta}$
and when only terms up to $s=1$ in the infinite series (17) and (39) are kept:

$\displaystyle\mathcal{R}_{\beta}(\upsilon)$
$\displaystyle=\frac{1}{4}\frac{\upsilon
j_{1}(\upsilon)}{\upsilon^{2}+\kappa_{\beta}^{2}}\frac{\kappa_{\beta}^{2}+14}{\kappa_{\beta}^{2}+7}-\frac{7}{4}\frac{j_{2}(\upsilon)}{\upsilon^{2}+\kappa_{\beta}^{2}}\frac{\kappa_{\beta}^{2}+10}{\kappa_{\beta}^{2}+7}.$
(42)

For small $\upsilon$-values, one finds the following limit:

$\displaystyle\lim_{\upsilon\rightarrow 0}\mathcal{R}_{\beta}(\upsilon)=0,$
(43)

which is consistent with

$\displaystyle\int_{V}\bm{\psi}(\mathbf{r})\;d^{3}r=\bm{0}.$ (44)

This can be seen by inspecting the definition of the Fourier transform in
(32). Note that for $q\rightarrow 0$ the Fourier transform is proportional to
the average of the magnetization vector field $\mathbf{M}$ and the maximum of
this average is given by the homogeneous magnetization state. Using this
result, the $\upsilon\rightarrow 0$ limit for the first-order approximation in
$\bm{\psi}$ of the Fourier transform of the magnetization yields:

$\displaystyle\lim_{\upsilon\rightarrow
0}\widetilde{\bm{\mathcal{M}}}(\upsilon,\theta_{q},\phi_{q})=\frac{1}{3}\mathbf{m}_{0}.$
(45)

Beyond the linear approximation in $\bm{\psi}$, a nonvanishing term appears in
$\widetilde{\bm{\mathcal{M}}}$ in the limit $\upsilon\rightarrow 0$, which
reduces the Fourier components relative to the homogeneous magnetization
state. In the second order in $\bm{\psi}$, the result is [compare (5)]:

$\displaystyle\lim_{\upsilon\rightarrow
0}\widetilde{\bm{\mathcal{M}}}(\upsilon,\theta_{q},\phi_{q})=\left[\frac{1}{3}-\frac{1}{2}\int_{V}\|\bm{\psi}(\bm{\xi})\|^{2}\;d^{3}\xi\right]\cdot\mathbf{m}_{0}$
(46)

Using (34) and

$\displaystyle\frac{d\Sigma_{M}}{d\Omega}=\frac{16\pi^{2}R^{6}M_{0}^{2}b_{H}^{2}}{V}\mathcal{S}_{M},$
(47)

the dimensionless two-dimensional magnetic SANS cross section
$\mathcal{S}_{M}(\upsilon,\theta_{q})$ can be straightforwardly obtained as
[compare (31)]

$\displaystyle\mathcal{S}_{M}(\upsilon,\theta_{q})$
$\displaystyle=|\widetilde{\mathcal{M}}_{x}|^{2}+|\widetilde{\mathcal{M}}_{y}|^{2}\cos^{2}\theta_{q}+|\widetilde{\mathcal{M}}_{z}|^{2}\sin^{2}\theta_{q}$
$\displaystyle-(\widetilde{\mathcal{M}}_{y}\widetilde{\mathcal{M}}_{z}^{\ast}+\widetilde{\mathcal{M}}_{y}^{\ast}\widetilde{\mathcal{M}}_{z})\sin\theta_{q}\cos\theta_{q}.$
(48)

In limit $k_{s}\rightarrow 0$, the resulting cross section from (37) is

$\displaystyle\lim_{k_{s}\rightarrow 0}\mathcal{S}_{M}(\upsilon,\theta_{q})$
$\displaystyle=\left(\frac{j_{1}(\upsilon)}{\upsilon}\right)^{2}\left(m_{0,x}^{2}+m_{0,y}^{2}\cos^{2}\theta_{q}\right.$
$\displaystyle+m_{0,z}^{2}\sin^{2}\theta_{q}-\left.2m_{0,y}m_{0,z}\sin\theta_{q}\cos\theta_{q}\right).$
(49)

The relation (49) nicely demonstrates that, depending on the orientation of
the uniformly magnetized particle, different angular anisotropies become
visible on the detector: For $\mathbf{m}_{0}\parallel\mathbf{e}_{x}$ (i.e.
$m_{0y}=m_{0z}=0$), the scattering pattern is isotropic, while it exhibits a
$\cos^{2}\theta_{q}$ ($\sin^{2}\theta_{q}$) type shape when
$\mathbf{m}_{0}\parallel\mathbf{e}_{y}$
($\mathbf{m}_{0}\parallel\mathbf{e}_{z}$).

Figure 5: Results for the two-dimensional Fourier components
$|\widetilde{\mathcal{M}}_{x}|^{2}$, $|\widetilde{\mathcal{M}}_{y}|^{2}$,
$|\widetilde{\mathcal{M}}_{z}|^{2}$,
$CT=-(\widetilde{\mathcal{M}}_{y}\widetilde{\mathcal{M}}_{z}^{\ast}+\widetilde{\mathcal{M}}_{y}^{\ast}\widetilde{\mathcal{M}}_{z})$,
and for the total magnetic SANS cross section
$\mathcal{S}_{M}(\upsilon,\theta_{q})$ [(48)] using expression (40). The upper
row shows the results taking into account only the zero-order term in (40),
which corresponds to the case of a homogeneously magnetized particle. The
lower row displays the results when the second-order term ($\nu=1$) in (40) is
taken into account. The parameters are:
$\mathbf{e}_{\mathrm{A}}=\mathbf{e}_{z}$, $\mathbf{b}_{0}=0.1\mathbf{e}_{z}$
[$B_{0}\cong 24\,\mathrm{mT}$], $k_{c}=0.1$, $k_{s}=3$, and
$\mathbf{m}_{0}=[\sin\beta\cos\alpha,\sin\beta\sin\alpha,\cos\beta]$. Since
the Néel surface anisotropy has effectively a cubic symmetry (see Fig. 1), we
average $\mathcal{S}_{M}$ over the angles
$\alpha=(45^{\circ},135^{\circ},225^{\circ},315^{\circ})$ and
$\beta=20^{\circ}$. Logarithmic color scale is used.

Figure 6: Results for (a) the azimuthally-averaged SANS cross section
$\mathcal{I}(\upsilon)$, (b) the pair-distance distribution function
$\mathcal{P}(\xi)$, (c) the correlation function $\mathcal{C}(\xi)$, and (d)
the quantities $\epsilon\mathcal{I}(k_{s})$ (56) and
$\epsilon\mathcal{P}(k_{s})$ (57) ($B_{0}\cong 24\,\mathrm{mT}$). For the
homogeneous case (blue curves in (a) - (c)), the surface anisotropy is set to
$k_{s}=0$, and for the inhomogeneous case (red curves (a) - (c)), we use the
same parameters as in Fig. 5. The functional dependence of
$\mathcal{I}(\upsilon)$, $\mathcal{P}(\xi)$, and $\mathcal{C}(\xi)$ for the
uniformly magnetized particle all correspond to the analytically well-known
cases, i.e. (51), (54), and (55).

Fig. 5 shows $\mathcal{S}_{M}(\upsilon,\theta_{q})$ along with the
contribution of the individual Fourier components to (48). The upper row in
Fig. 5 presents the result taking into account only the zero-order term
($j_{1}(\upsilon)/\upsilon\mathbf{m}_{0}$) from (40), while in the lower row
the second-order term ($\nu=1$) is additionally included. Since the zero-order
term represents the case of a homogeneously magnetized NM, this comparison
provides useful insights about the impact of the Néel surface anisotropy on
the magnetic SANS cross section. In the case of a uniformly magnetized NM
(upper row) the Fourier components $|\widetilde{\mathcal{M}}_{x}|^{2}$,
$|\widetilde{\mathcal{M}}_{y}|^{2}$, and $|\widetilde{\mathcal{M}}_{z}|^{2}$
are isotropic (rotational symmetry), while including the second-order terms
(lower row) leads to an anisotropic behavior of the transverse components
$|\widetilde{\mathcal{M}}_{x}|^{2}$ and $|\widetilde{\mathcal{M}}_{y}|^{2}$.
The cross term ($CT$) averages to zero for both situations, and the dominating
contribution to the magnetic SANS cross section is (for the parameters chosen
in Fig. 5) given by the $|\widetilde{\mathcal{M}}_{z}|^{2}$ component.
Therefore, it may be concluded that the impact of the Néel surface anisotropy
on $\mathcal{S}_{M}(\upsilon,\theta_{q})$ is relatively small. By comparing
the $\mathcal{S}_{M}(\upsilon,\theta_{q})$ from the upper and the lower row,
it is seen that by including the Néel surface anisotropy the circular symmetry
of the zeros of $\mathcal{S}_{M}$ (deep blue colors) is broken. This feature
becomes more clearly visible by analyzing the azimuthal average of
$\mathcal{S}_{M}(\upsilon,\theta_{q})$, which is readily computed as:

$\displaystyle\mathcal{I}(\upsilon)=\frac{1}{2\pi}\int_{0}^{2\pi}\mathcal{S}_{M}(\upsilon,\theta_{q})d\theta_{q}.$
(50)

In the limit $k_{s}\rightarrow 0$, the azimuthal average corresponding to (49)
is:

$\displaystyle\lim_{k_{s}\rightarrow
0}\mathcal{I}(\upsilon)=\left(\frac{j_{1}(\upsilon)}{\upsilon}\right)^{2}\frac{\|\mathbf{m}_{0}\|^{2}+(\mathbf{e}_{x}\cdot\mathbf{m}_{0})^{2}}{2}.$
(51)

Moreover, we have also calculated the pair-distance distribution function,

$\displaystyle\mathcal{P}(\xi)=\xi^{2}\int_{0}^{\infty}\mathcal{I}(\upsilon)j_{0}(\upsilon\xi)\upsilon^{2}d\upsilon,$
(52)

and the correlation function

$\displaystyle\mathcal{C}(\xi)=\mathcal{P}(\xi)/\xi^{2}.$ (53)

In the limit $k_{s}\rightarrow 0$, the pair-distance distribution and the
correlation function corresponding to (51) are:

$\displaystyle\lim_{k_{s}\rightarrow 0}\mathcal{P}(\xi)$
$\displaystyle=\frac{\pi\xi^{2}}{6}\left(1-\frac{3\xi}{4}+\frac{\xi^{3}}{16}\right)\frac{\|\mathbf{m}_{0}\|^{2}+(\mathbf{e}_{x}\cdot\mathbf{m}_{0})^{2}}{2},$
(54) $\displaystyle\lim_{k_{s}\rightarrow 0}\mathcal{C}(\xi)$
$\displaystyle=\frac{\pi}{6}\left(1-\frac{3\xi}{4}+\frac{\xi^{3}}{16}\right)\frac{\|\mathbf{m}_{0}\|^{2}+(\mathbf{e}_{x}\cdot\mathbf{m}_{0})^{2}}{2}.$
(55)

These functions are graphically displayed in Fig. 6. Due to the surface-
anisotropy induced spin disorder, the form-factor maxima of
$\mathcal{I}(\upsilon)$ [Fig. 6(a)] are shifted to larger $q$ values (i.e.
smaller structures). Moreover, as already observed in numerical micromagnetic
continuum simulations [38, 39], the oscillations are damped for the case of
surface spin disorder, which mimics the effect of a particle-size distribution
or of instrumental resolution. In agreement with this observation is the
finding that the maximum of the $\mathcal{P}(\xi)$ function [Fig. 6(b)]
appears at smaller distances $\xi$ as compared to the homogeneous case.
Likewise, due to spin disorder, the $\mathcal{C}(\xi)$ function [Fig. 6(c)]
exhibits a larger amplitude [18].

To analyze the role of the surface anisotropy more quantitatively, we have
computed the following quantities, which describe the deviation of the one-
dimensional SANS cross section and of the pair-distance distribution function
from the homogeneous particle case:

$\displaystyle\epsilon\mathcal{I}(k_{s})=\frac{\displaystyle\int_{0}^{\infty}|\mathcal{I}(k_{s}=0,\upsilon)-\mathcal{I}(k_{s},\upsilon)|d\upsilon}{\displaystyle\int_{0}^{\infty}|\mathcal{I}(k_{s}=0,\upsilon)|d\upsilon},$
(56)
$\displaystyle\epsilon\mathcal{P}(k_{s})=\frac{\displaystyle\int_{0}^{2}|\mathcal{P}(k_{s}=0,\xi)-\mathcal{P}(k_{s},\xi)|d\xi}{\displaystyle\int_{0}^{2}|\mathcal{P}(k_{s}=0,\xi)|d\xi}.$
(57)

Fig. 6(d) depicts both $\epsilon\mathcal{I}(k_{s})$ and
$\epsilon\mathcal{P}(k_{s})$ as a function of $k_{s}$. The difference is only
of the order of a few percent, which suggests that the effect of surface
anisotropy on the SANS observables is relatively weak within the present
analytical approximation; see the numerical work [27], which takes into
account the full nonlinearity of the micromagnetic equations. However, this is
only true for the magnetic interactions considered here. Taking into account
the anisotropic and long-range dipole-dipole interaction and the asymmetric
Dzyaloshinskii-Moriya interaction will very likely result in more
inhomogeneous spin structures and in larger deviations from the macrospin
model [38, 39, 40]. Likewise, for NM of elongated shapes, the surface
anisotropy renders an additional first-order contribution to the effective
energy [29], in addition to the second-order cubic contribution discussed
above. This new shape-induced contribution could also lead to an enhancement
of the spin misalignment. The analytical calculations presented here provide a
general framework for future studies of more complicated (anisotropic)
magnetic interactions.

##  IV Conclusion

We have analytically computed the magnetization distribution and the ensuing
magnetic small-angle neutron scattering (SANS) cross section of a spherical
nanoparticle. Our micromagnetic Hamiltonian takes into account the isotropic
exchange interaction, an external magnetic field, a uniaxial anisotropy for
the particle’s core, and Néel anisotropy on its boundary. The resulting
Helmholtz equation has been solved by expanding the real-space magnetization
in terms of spherical Bessel functions and spherical harmonics. The central
results are the infinite series (16) and its second-order expansion (21) for
the real-space magnetization, and the corresponding Fourier transforms (37)
and (40). Using these expressions, the two-dimensional magnetic SANS cross
section $\mathcal{S}_{M}(\upsilon,\theta_{q})$, the azimuthally-averaged SANS
signal $\mathcal{I}(\upsilon)$, and the correlation functions
$\mathcal{P}(\xi)$ and $\mathcal{C}(\xi)$ were obtained and compared to the
case of a homogeneous spin configuration (uniform magnetization vector field).
The signature of Néel’s surface anisotropy (of constant $k_{s}$) has been
identified in all of these functions. However, its effect is relatively small,
even for large values of $k_{s}$. Taking into account the magnetodipolar
and/or the Dzyaloshinskii-Moriya interaction, or shape asymmetry, will likely
result in configurations with stronger spin misalignment (e.g. in vortex-type
textures or skyrmions) and thereby in more prominent signatures in the SANS
cross section and correlation function. These interactions are beyond the
scope of the current analytical approach and will be considered in our future
(numerical) works [27].

###### Acknowledgements.

Michael Adams and Andreas Michels thank the National Research Fund of
Luxembourg for financial support (AFR Grant No. 15639149).

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##  Appendix A Solution of the Boundary Value Problem of the Helmholtz
Equation

The coefficients $c_{\ell m}^{\beta}$ in the fundamental solution (15) of the
Helmholtz equation (9) must be determined such that the Neumann boundary
condition (10) is satisfied. For this purpose, we use the method of least
squares, where we make use of the orthogonality properties of the spherical
harmonics $Y_{\ell m}(\theta,\phi)$. The normal derivative of (15) at the
surface of the NM ($\xi=1$) is:

$\displaystyle\left.\frac{d\psi_{\beta}}{d\xi}\right|_{\xi=1}=\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}c_{\ell
m}^{\beta}[j_{\ell}(\mathrm{i}\kappa_{\beta}\xi)]_{\xi=1}^{\prime}Y_{\ell
m}(\theta,\phi),$ (58)

where

$\displaystyle[j_{\ell}(\mathrm{i}\kappa_{\beta}\xi)]_{\xi=1}^{\prime}=\left.\frac{d}{d\xi}j_{\ell}(\mathrm{i}\kappa_{\beta}\xi)\right|_{\xi=1}.$
(59)

Our goal is now to minimize the following error functional with respect to the
coefficients $c_{\ell m}^{\beta}$:

$\displaystyle\varepsilon[c_{\ell
m}^{\beta}]=\int_{0}^{2\pi}\int_{0}^{\pi}\left|\sum_{\alpha\in\\{x,y,z\\}}\chi_{\alpha}^{\beta}|n_{\alpha}|\right.$
$\displaystyle-\left.\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}c_{\ell
m}^{\beta}[j_{\ell}(\mathrm{i}\kappa_{\beta}\xi)]_{\xi=1}^{\prime}Y_{\ell
m}(\theta,\phi)\right|^{2}\sin\theta\;d\theta d\phi.$ (60)

The minimum of this error functional is found from the condition that the
partial derivatives of $\varepsilon[c_{\ell m}^{\beta}]$ with respect to the
$(c_{ij}^{\beta})^{\ast}$ vanish:

$\displaystyle\frac{\partial\varepsilon[c_{\ell
m}^{\beta}]}{\partial(c_{ij}^{\beta})^{\ast}}=0,$ (61)

where ‘$*$’ denotes the complex conjugate. Using the orthogonality relation
(62) (p. 378 (14.30.8) in [36]) and by defining the integral $I_{ij}^{\alpha}$
by (63),

$\displaystyle\int_{0}^{2\pi}\int_{0}^{\pi}Y_{\ell
m}(\theta,\phi)Y_{ij}^{*}(\theta,\phi)\sin\theta\;d\theta d\phi$
$\displaystyle=\delta_{\ell i}\delta_{mj},$ (62)
$\displaystyle\int_{0}^{2\pi}\int_{0}^{\pi}Y_{ij}^{*}(\theta,\phi)|n_{\alpha}|\sin\theta\;d\theta
d\phi$ $\displaystyle=I_{ij}^{\alpha},$ (63)

the solution of (61) can be written as:

$\displaystyle
c_{ij}^{\beta}=\frac{1}{[j_{j}(\mathrm{i}\kappa_{\beta}\xi)]_{\xi=1}^{\prime}}\sum_{\alpha\in\\{x,y,z\\}}\chi_{\alpha}^{\beta}I_{ij}^{\alpha}.$
(64)

Alternatively, using again the indices $\ell$ and $m$, expressing the
coefficient $\chi_{\alpha}^{\beta}$ as in (13), and by using the matrix-vector
product, the coefficients $c_{\ell m}^{\beta}$ can be more explicitly written
as:

$\displaystyle c_{\ell
m}^{\beta}=\frac{k_{s}\mathbf{g}_{\beta}\cdot\operatorname{diag}\left[I_{\ell
m}^{x},I_{\ell m}^{y},I_{\ell
m}^{z}\right]\cdot\mathbf{m}_{0}}{[j_{\ell}(\mathrm{i}\kappa_{\beta}\xi)]_{\xi=1}^{\prime}}.$
(65)

For some low-orders of $\ell$ ans $m$, the exact solutions of the integrals
$I_{\ell m}^{\alpha}$ are presented in Appendix B.

From (65) several conclusions can be drawn. First, the zero-order term
($\ell=m=0$) in (15) vanishes, which can easily be shown by rewriting the
coefficient $c_{00}^{\beta}$ as:

$\displaystyle
c_{00}^{\beta}=\frac{k_{s}\sqrt{\pi}(\mathbf{g}_{\beta}\cdot\mathbf{m}_{0})}{[j_{0}(\mathrm{i}\kappa_{\beta}\xi)]_{\xi=1}^{\prime}}=0,$
(66)

which is due to the orthogonality of $\mathbf{m}_{0}$ and $\mathbf{g}_{\beta}$
with $\beta\in\\{1,2\\}$ [see (6)]. Moreover, we see that

$\displaystyle\psi_{\beta}(\xi=0,\theta,\phi)=0,$ (67)

which is a consequence of the behavior of the spherical Bessel functions of
the first kind at the origin. Since the $j_{\ell}$ with $\ell\geq 1$ are all
vanishing at the origin $\xi=0$, this implies that the total $\psi_{\beta}$
also vanishes at the origin. Note that $j_{0}(0)=1$, but does not contribute
to $\psi_{\beta}$ due to $c_{00}^{\beta}=0$. From the physical point of view
this make sense, since the spin misalignment is caused by the Néel surface
anisotropy and thus, due to symmetry reasons, there is no spin disorder at the
center of the spherical NM; the highest misalignment is found at its surface.
Second, in the table of Appendix B it is seen that the coefficients
$I_{\ell,m}^{\alpha}$ vanish for odd $\ell$ or $m$, and that
$I_{\ell,m}^{\alpha}=I_{\ell,-m}^{\alpha}$, so that the expansion coefficients
also exhibit this symmetry

$\displaystyle c_{\ell,m}^{\beta}=c_{\ell,-m}^{\beta}.$ (68)

Taking these properties into account, one can express the solution (15) more
conveniently in terms of the associated Legendre polynomials
$P_{\ell}^{m}(\cos\theta)$ with $\ell=2\nu$ and $m=2\mu$ [note that we use the
convention that $Y_{\ell m}(\theta,\phi)=N_{\ell
m}P_{\ell}^{m}(\cos\theta)\mathrm{e}^{\mathrm{i}m\phi}$ (p. 378 (14.30.1) in
[36])]:

$\displaystyle\psi_{\beta}$
$\displaystyle=\sum_{\nu=1}^{\infty}\sum_{\mu=0}^{\nu}a_{\nu\mu}^{\beta}\Upsilon_{\nu}(\kappa_{\beta}\xi)P_{2\nu}^{2\mu}(\cos\theta)\cos(2\mu\phi),$
(69)

where $\Upsilon_{\nu}(\tau)$ was defined in (17) and $a_{\nu\mu}^{\beta}$ in
(18).

The infinite series in (18) is a consequence of the relation between the
spherical Bessel functions of the first kind and the ordinary Bessel functions
of the first kind (p. 262 (10.47.3) in [36])

$\displaystyle j_{l}(\tau)=\sqrt{\frac{\pi}{2\tau}}J_{l+\frac{1}{2}}(\tau),$
(70)

and the well known series representation (p. 262 (10.47.3) in [36])

$\displaystyle
J_{l}(\tau)=\sum_{s=0}^{\infty}\frac{(-1)^{s}(\tau/2)^{2s+l}}{s!\Gamma(l+s+1)}.$
(71)

##  Appendix B Integral Coefficients

The integrals $I_{\ell m}^{\alpha}$ can be simplified, since both $n_{\alpha}$
and $Y_{\ell m}^{\ast}(\theta,\phi)$ are separable functions in $\theta$ and
$\phi$. By rewriting the spherical harmonics in terms of complex exponentials
and associated Legendre polynomials and by using the definition of the
Cartesian components of the surface normal vector $\mathbf{n}$ from (3) [note
that we use the convention that $Y_{\ell m}(\theta,\phi)=N_{\ell
m}P_{\ell}^{m}(\cos\theta)\mathrm{e}^{\mathrm{i}m\phi}$ (p. 378 (14.30.1) in
[36]), the integrals are expressed as follows:

$\displaystyle I_{\ell m}^{x}$ $\displaystyle=N_{\ell m}K_{\ell
m}^{(1)}U_{m}^{(1)},$ (72) $\displaystyle I_{\ell m}^{y}$
$\displaystyle=N_{\ell m}K_{\ell m}^{(1)}U_{m}^{(2)},$ (73) $\displaystyle
I_{\ell m}^{z}$ $\displaystyle=N_{\ell m}K_{\ell m}^{(2)}U_{m}^{(3)},$ (74)

where

$\displaystyle K_{\ell m}^{(1)}$
$\displaystyle=\int_{0}^{\pi}P_{\ell}^{m}(\cos\theta)|\sin\theta|\sin\theta
d\theta,$ (75) $\displaystyle K_{\ell m}^{(2)}$
$\displaystyle=\int_{0}^{\pi}P_{\ell}^{m}(\cos\theta)|\cos\theta|\sin\theta
d\theta,$ (76) $\displaystyle U_{m}^{(1)}$
$\displaystyle=\int_{0}^{2\pi}\mathrm{e}^{-\mathrm{i}m\phi}|\cos\phi|\;d\phi=(-1)^{|m|/2}\frac{2(1+(-1)^{|m|})}{1-m^{2}+\delta_{|m|,1}},$
(77) $\displaystyle U_{m}^{(2)}$
$\displaystyle=\int_{0}^{2\pi}\mathrm{e}^{-\mathrm{i}m\phi}|\sin\phi|\;d\phi=\frac{2(1+(-1)^{|m|})}{1-m^{2}+\delta_{|m|,1}},$
(78) $\displaystyle U_{m}^{(3)}$
$\displaystyle=\int_{0}^{2\pi}\mathrm{e}^{-\mathrm{i}m\phi}\;d\phi=2\pi\delta_{0,m},$
(79) $\displaystyle N_{\ell m}$
$\displaystyle=\sqrt{\frac{2\ell+1}{4\pi}\frac{(\ell-m)!}{(\ell+m)!}}.$ (80)

The integrals $U_{m}^{(1)}$, $U_{m}^{(2)}$, and $U_{m}^{(3)}$ are solvable
straightforwardly by using Euler’s formula for the complex exponential and by
splitting the region of integration according to the absolute values of the
trigonometric functions. In the denominator of $U_{m}^{(1)}$ and $U_{m}^{(2)}$
we included the Kronecker delta $\delta_{|m|,1}$ to take account of the cases
$m=\pm 1$. It is common to express the integrals $K_{\ell m}^{(1)}$ and
$K_{\ell m}^{(2)}$ by the substitution $x=\cos\theta$ ($dx=-\sin\theta
d\theta$):

$\displaystyle K_{\ell m}^{(1)}$
$\displaystyle=\int_{-1}^{1}P_{\ell}^{m}(x)\sqrt{1-x^{2}}\;dx,$ (81)
$\displaystyle K_{\ell m}^{(2)}$
$\displaystyle=\int_{-1}^{1}P_{\ell}^{m}(x)|x|\;dx.$ (82)

Using $U_{m}^{(3)}=2\pi\delta_{0,m}$, we need only to compute
$K_{\ell,0}^{(2)}$ such that the associated Legendre polynomials in
$K_{\ell,m}^{(2)}$ are reduced to the Legendre polynomials [with one index
only (p. 352 in [36])]. By considering the symmetry properties of the Legendre
polynomials it becomes clear that the integral must vanish for odd $\ell$ and
can be simplified for even $\ell$ in the following way:

$\displaystyle K_{\ell,0}^{(2)}$
$\displaystyle=(1+(-1)^{\ell})\int_{0}^{1}P_{\ell}(x)x\;dx.$ (83)

The closed-form of $K_{\ell,0}^{(2)}$ is then found in terms of the Gamma
function (p. 771 (7.126.1) in [41]):

$\displaystyle K_{\ell,0}^{(2)}$
$\displaystyle=\frac{\sqrt{\pi}(1+(-1)^{\ell})}{4\Gamma(3/2-\ell/2)\Gamma(\ell/2+2)}.$
(84)

The overall solution for $I_{\ell m}^{z}$ is then written as:

$\displaystyle I_{\ell
m}^{z}=\frac{\pi(1+(-1)^{\ell})\sqrt{2\ell+1}}{4\Gamma(3/2-\ell/2)\Gamma(\ell/2+2)}\delta_{0,m}.$
(85)

Since we only have to calculate $K_{\ell m}^{(1)}$ for even $m$ (since
$U_{m}^{(1)}$ and $U_{m}^{(2)}$ include the term $1+(-1)^{|m|}$), we may use
the index $m=2\mu$, and by exploiting the parity of the associated Legendre
polynomials $P_{\ell}^{m}(-x)=(-1)^{\ell+m}P_{\ell}^{m}(x)$, we find:

$\displaystyle K_{\ell,2\mu}^{(1)}$
$\displaystyle=(1+(-1)^{\ell})\int_{0}^{1}P_{\ell}^{2\mu}(x)\sqrt{1-x^{2}}\;dx.$
(86)

From these results it is seen that the integrals $I_{\ell m}^{\alpha}$ with
$\alpha\in\\{x,y,z\\}$ vanish for odd $\ell$ and $m$ (note that in (86) this
is only the case for $m=2\mu$). For the remaining integrals
$K_{2\nu,2\mu}^{(1)}$ in (86), where $\ell=2\nu$, we do not give an expression
in closed form, but there must exist one in terms of the Gamma function or the
Beta function, since (p. 324 (3.251.2) in [41])

$\displaystyle\int_{0}^{1}x^{s}\sqrt{1-x^{2}}\;dx=\frac{1}{2}\mathrm{B}\left(\frac{s+1}{2},\frac{3}{2}\right),$
(87)

where $\mathrm{B}(\cdot,\cdot)$ is the Beta function (Euler integral), and the
associated Legendre functions $P_{2\nu}^{2\mu}$ of even order and degree are
true polynomials, as seen for example from the related Rodrigues formula (p.
360 (14.7.14) in [36]). Since the remaining integrals are numerically easy to
compute, we used Mathematica for the reconstruction of the analytically exact
results, which we present in the Table below for some low orders of $\ell$ and
$m$.

$\displaystyle I_{\ell m}^{x}=\int_{0}^{2\pi}\int_{0}^{\pi}Y_{\ell
m}^{*}(\theta,\phi)|\sin\theta\cos\phi|\sin\theta\;d\theta
d\phi=(-1)^{|m|/2}\frac{2(1+(-1)^{|m|})(1+(-1)^{\ell})}{1-m^{2}+\delta_{|m|,1}}\sqrt{\frac{2\ell+1}{4\pi}\frac{(\ell-m)!}{(\ell+m)!}}\int_{0}^{1}P_{\ell}^{m}(x)\sqrt{1-x^{2}}\;dx$
---
$\ell$ $m$ |  $\displaystyle 0$ |  $\displaystyle\pm 1$ |  $\displaystyle\pm 2$ |  $\displaystyle\pm 3$ |  $\displaystyle\pm 4$ |  $\displaystyle\pm 5$ |  $\displaystyle\pm 6$
$\displaystyle 0$ |  $\displaystyle\sqrt{\pi}$ |  |  |  |  |  |
$\displaystyle 1$ |  $\displaystyle 0$ |  $\displaystyle 0$ |  |  |  |  |
$\displaystyle 2$ |  $\displaystyle-\frac{\sqrt{5\pi}}{8}$ |  $\displaystyle 0$ |  $\displaystyle\frac{1}{8}\sqrt{\frac{15\pi}{2}}$ |  |  |  |
$\displaystyle 3$ |  $\displaystyle 0$ |  $\displaystyle 0$ |  $\displaystyle 0$ |  $\displaystyle 0$ |  |  |
$\displaystyle 4$ |  $\displaystyle-\frac{3\sqrt{\pi}}{64}$ |  $\displaystyle 0$ |  $\displaystyle\frac{1}{32}\sqrt{\frac{5\pi}{2}}$ |  $\displaystyle 0$ |  $\displaystyle-\frac{1}{64}\sqrt{\frac{35\pi}{2}}$ |  |
$\displaystyle 5$ |  $\displaystyle 0$ |  $\displaystyle 0$ |  $\displaystyle 0$ |  $\displaystyle 0$ |  $\displaystyle 0$ |  $\displaystyle 0$ |
$\displaystyle 6$ |  $\displaystyle-\frac{5\sqrt{13\pi}}{1024}$ |  $\displaystyle 0$ |  $\displaystyle\frac{\sqrt{1365\pi}}{2048}$ |  $\displaystyle 0$ |  $\displaystyle-\frac{3}{1024}\sqrt{\frac{91\pi}{2}}$ |  $\displaystyle 0$ |  $\displaystyle\frac{\sqrt{3003\pi}}{2048}$
$\displaystyle I_{\ell m}^{y}=\int_{0}^{2\pi}\int_{0}^{\pi}Y_{\ell
m}^{*}(\theta,\phi)|\sin\theta\sin\phi|\sin\theta\;d\theta
d\phi=\frac{2(1+(-1)^{|m|})(1+(-1)^{\ell})}{1-m^{2}+\delta_{|m|,1}}\sqrt{\frac{2\ell+1}{4\pi}\frac{(\ell-m)!}{(\ell+m)!}}\int_{0}^{1}P_{\ell}^{m}(x)\sqrt{1-x^{2}}\;dx$
$\ell$ $m$ |  $\displaystyle 0$ |  $\displaystyle\pm 1$ |  $\displaystyle\pm 2$ |  $\displaystyle\pm 3$ |  $\displaystyle\pm 4$ |  $\displaystyle\pm 5$ |  $\displaystyle\pm 6$
$\displaystyle 0$ |  $\displaystyle\sqrt{\pi}$ |  |  |  |  |  |
$\displaystyle 1$ |  $\displaystyle 0$ |  $\displaystyle 0$ |  |  |  |  |
$\displaystyle 2$ |  $\displaystyle-\frac{\sqrt{5\pi}}{8}$ |  $\displaystyle 0$ |  $\displaystyle-\frac{1}{8}\sqrt{\frac{15\pi}{2}}$ |  |  |  |
$\displaystyle 3$ |  $\displaystyle 0$ |  $\displaystyle 0$ |  $\displaystyle 0$ |  $\displaystyle 0$ |  |  |
$\displaystyle 4$ |  $\displaystyle-\frac{3\sqrt{\pi}}{64}$ |  $\displaystyle 0$ |  $\displaystyle-\frac{1}{32}\sqrt{\frac{5\pi}{2}}$ |  $\displaystyle 0$ |  $\displaystyle-\frac{1}{64}\sqrt{\frac{35\pi}{2}}$ |  |
$\displaystyle 5$ |  $\displaystyle 0$ |  $\displaystyle 0$ |  $\displaystyle 0$ |  $\displaystyle 0$ |  $\displaystyle 0$ |  $\displaystyle 0$ |
$\displaystyle 6$ |  $\displaystyle-\frac{5\sqrt{13\pi}}{1024}$ |  $\displaystyle 0$ |  $\displaystyle-\frac{\sqrt{1365\pi}}{2048}$ |  $\displaystyle 0$ |  $\displaystyle-\frac{3}{1024}\sqrt{\frac{91\pi}{2}}$ |  $\displaystyle 0$ |  $\displaystyle-\frac{\sqrt{3003\pi}}{2048}$
$\displaystyle I_{\ell m}^{z}=\int_{0}^{2\pi}\int_{0}^{\pi}Y_{\ell
m}^{*}(\theta,\phi)|\cos\theta|\sin\theta\;d\theta
d\phi=\frac{\pi(1+(-1)^{\ell})\sqrt{2\ell+1}}{4\operatorname{\Gamma}\left(3/2-\ell/2\right)\operatorname{\Gamma}\left(\ell/2+2\right)}\delta_{m,0}$
$\ell$ $m$ |  $\displaystyle 0$ |  $\displaystyle\pm 1$ |  $\displaystyle\pm 2$ |  $\displaystyle\pm 3$ |  $\displaystyle\pm 4$ |  $\displaystyle\pm 5$ |  $\displaystyle\pm 6$
$\displaystyle 0$ |  $\displaystyle\sqrt{\pi}$ |  |  |  |  |  |
$\displaystyle 1$ |  $\displaystyle 0$ |  $\displaystyle 0$ |  |  |  |  |
$\displaystyle 2$ |  $\displaystyle\frac{\sqrt{5\pi}}{4}$ |  $\displaystyle 0$ |  $\displaystyle 0$ |  |  |  |
$\displaystyle 3$ |  $\displaystyle 0$ |  $\displaystyle 0$ |  $\displaystyle 0$ |  $\displaystyle 0$ |  |  |
$\displaystyle 4$ |  $\displaystyle-\frac{\sqrt{\pi}}{8}$ |  $\displaystyle 0$ |  $\displaystyle 0$ |  $\displaystyle 0$ |  $\displaystyle 0$ |  |
$\displaystyle 5$ |  $\displaystyle 0$ |  $\displaystyle 0$ |  $\displaystyle 0$ |  $\displaystyle 0$ |  $\displaystyle 0$ |  $\displaystyle 0$ |
$\displaystyle 6$ |  $\displaystyle\frac{\sqrt{13\pi}}{64}$ |  $\displaystyle 0$ |  $\displaystyle 0$ |  $\displaystyle 0$ |  $\displaystyle 0$ |  $\displaystyle 0$ |  $\displaystyle 0$

Table 1: Values of the integrals (19) for some small values of $\ell$ and $m$.

##  Appendix C Derivation of the Fourier Transform of the Magnetization

The Fourier transform of the magnetization vector field
$\mathbf{M}(\mathbf{r})$ is written as:

$\displaystyle\widetilde{\mathbf{M}}(\mathbf{q})=\frac{1}{(2\pi)^{3/2}}\int_{V}\mathbf{M}(\mathbf{r})\exp\left(-\mathrm{i}\mathbf{q}\cdot\mathbf{r}\right)d^{3}r.$
(88)

In the sequel, we will use dimensionless quantities. For this purpose, we
define the dimensionless scattering vector $\bm{\upsilon}=\mathbf{q}R$, where
$R$ is the radius of the nanomagnet, the dimensionless position vector
$\bm{\xi}=\mathbf{r}/R$, and the dimensionless magnetization vector
$\mathbf{m}=\mathbf{M}/M_{0}$, where $M_{0}$ is the saturation magnetization.
Substituting in (88) results in

$\displaystyle\widetilde{\mathbf{M}}(\bm{\upsilon})=\frac{R^{3}M_{0}}{(2\pi)^{3/2}}\int_{V}\mathbf{m}(\bm{\xi})\exp\left(-\mathrm{i}\bm{\upsilon}\cdot\bm{\xi}\right)\;d^{3}\xi.$
(89)

The dimensionless Fourier transform $\widetilde{\bm{\mathcal{M}}}$ is then
defined as

$\displaystyle\widetilde{\bm{\mathcal{M}}}(\bm{\upsilon})=\frac{1}{4\pi}\int_{V}\mathbf{m}(\bm{\xi})\exp\left(-\mathrm{i}\bm{\upsilon}\cdot\bm{\xi}\right)\;d^{3}\xi,$
(90)

with

$\displaystyle\widetilde{\mathbf{M}}=\frac{4\pi
R^{3}M_{0}}{(2\pi)^{3/2}}\widetilde{\bm{\mathcal{M}}}.$ (91)

The next step consists in calculating the Fourier integral of the first-order
approximation (36) of the magnetization vector $\mathbf{m}$. Since (36) is
formulated in dimensionless spherical coordinates $\xi,\theta,\phi$, it is
convenient to express the scattering vector $\bm{\upsilon}$ in spherical
coordinates as well:

$\displaystyle\bm{\xi}$
$\displaystyle=[\xi\sin\theta\cos\phi,\xi\sin\theta\sin\phi,\xi\cos\theta],$
(92) $\displaystyle\bm{\upsilon}$
$\displaystyle=[\upsilon\sin\theta_{q}\cos\phi_{q},\upsilon\sin\theta_{q}\sin\phi_{q},\upsilon\cos\theta_{q}],$
(93)

so that the plane-wave expansion of the complex exponential can be used [42]

$\displaystyle\exp\left(-\text{i}\bm{\upsilon}\cdot\bm{\xi}\right)=4\pi\sum_{k=0}^{\infty}\sum_{n=-k}^{k}(-\text{i})^{k}j_{k}(\upsilon\xi)Y_{kn}^{*}(\theta,\phi)Y_{kn}(\theta_{q},\phi_{q}).$
(94)

The Fourier integral (90) is then expanded into the following infinite series:

$\displaystyle\widetilde{\bm{\mathcal{M}}}(\bm{\upsilon})=\sum_{k=0}^{\infty}\sum_{n=-k}^{k}(-\text{i})^{k}\hat{\bm{\mathcal{M}}}_{kn}(\upsilon)Y_{kn}(\theta_{q},\phi_{q}),$
(95)

where

$\displaystyle\hat{\bm{\mathcal{M}}}_{kn}(\upsilon)=\int_{0}^{2\pi}\int_{0}^{\pi}\int_{0}^{1}\mathbf{m}(\bm{\xi})j_{k}(\upsilon\xi)Y_{kn}^{*}(\theta,\phi)\xi^{2}\sin\theta\;d\xi
d\theta d\phi.$ (96)

We now use the infinite series (15) for $\psi_{\beta}$ to express the first-
order approximation of the magnetization, which leads to

$\displaystyle\mathbf{m}(\bm{\xi})=\mathbf{m}_{0}+\sum_{\beta=1}^{2}\mathbf{g}_{\beta}\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}c_{\ell
m}^{\beta}j_{\ell}(\mathrm{i}\kappa_{\beta}\xi)Y_{\ell m}(\theta,\phi).$ (97)

Since the integral transform (96) is linear, each term in (97) can be
separately transformed. For the zero-order term, we obtain

$\displaystyle\int_{0}^{2\pi}\int_{0}^{\pi}\int_{0}^{1}\mathbf{m}_{0}j_{k}(\upsilon\xi)Y_{kn}^{*}(\theta,\phi)\xi^{2}\sin\theta\;d\xi
d\theta
d\phi=\sqrt{4\pi}\frac{j_{1}(\upsilon)}{\upsilon}\mathbf{m}_{0}\delta_{k,0}\delta_{n,0}.$
(98)

This result is well known to the neutron-scattering community as the spherical
form factor, corresponding to a uniformly-magnetized spherical particle [2].
In the second step, we carry out the integration of the higher-order terms
from (97). The radial and the angular parts in the higher-order terms of (97)
are multiplicative, such that the volume integral (96) is separable into:

$\displaystyle A_{\ell k}^{\beta}(\upsilon)$
$\displaystyle=\int_{0}^{1}j_{\ell}(\mathrm{i}\kappa_{\beta}\xi)j_{k}(\upsilon\xi)\xi^{2}d\xi,$
(99) $\displaystyle B_{\ell m}^{kn}$
$\displaystyle=\int_{0}^{2\pi}\int_{0}^{\pi}Y_{\ell
m}(\theta,\phi)Y_{kn}^{*}(\theta,\phi)\sin\theta d\theta d\phi.$ (100)

As an intermediate result, the integral (96) is then rewritten as

$\displaystyle\hat{\bm{\mathcal{M}}}_{kn}(\upsilon)$
$\displaystyle=\sqrt{4\pi}\frac{j_{1}(\upsilon)}{\upsilon}\mathbf{m}_{0}\delta_{k,0}\delta_{n,0}+\sum_{\beta=1}^{2}\mathbf{g}_{\beta}\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}c_{\ell
m}^{\beta}A_{\ell k}^{\beta}(\upsilon)B_{\ell m}^{kn}.$ (101)

The integral (100) directly corresponds to the orthogonality relation of the
spherical harmonics, and thereby we have $B_{\ell m}^{kn}=\delta_{\ell
k}\delta_{mn}$ (p. 378 (14.30.8) in [36]). Due to the term $\delta_{\ell k}$
in $B_{\ell m}^{kn}$, and since $B_{\ell m}^{kn}$ and $A_{\ell k}^{\beta}$ are
multiplicative in (101), the following spherical Hankel transform results:

$\displaystyle A_{kk}^{\beta}(\upsilon)$
$\displaystyle=\int_{0}^{1}j_{k}(\mathrm{i}\kappa_{\beta}\xi)j_{k}(\upsilon\xi)\xi^{2}d\xi.$
(102)

In order to calculate these integrals, we replace [using (70)] the spherical
Bessel functions of the first kind $j_{k}(\cdot)$ with the ordinary Bessel
functions of the first kind $J_{\iota}(\cdot)$. This yields:

$\displaystyle
A_{kk}^{\beta}(\upsilon)=\sqrt{\frac{\pi}{2\upsilon}}\sqrt{\frac{\pi}{2\mathrm{i}\kappa_{\beta}}}W_{k}^{\beta}(\upsilon),$
(103)

where

$\displaystyle
W_{k}^{\beta}(\upsilon)=\int_{0}^{1}J_{k+\frac{1}{2}}(\mathrm{i}\kappa_{\beta}\xi)J_{k+\frac{1}{2}}(\upsilon\xi)\xi
d\xi.$ (104)

is the Hommel integral. The result for the integral $W_{k}^{\beta}$ can be
found in the textbook by Gradshteyn and Ryzhik (p. 664 (6.521.1) in [41]) and
is written as

$\displaystyle W_{k}^{\beta}(\upsilon)=-\frac{\upsilon
J_{k-\frac{1}{2}}(\upsilon)J_{k+\frac{1}{2}}(\mathrm{i}\kappa_{\beta})-\mathrm{i}\kappa_{\beta}J_{k-\frac{1}{2}}(\mathrm{i}\kappa_{\beta})J_{k+\frac{1}{2}}(\upsilon)}{\upsilon^{2}+\kappa_{\beta}^{2}}.$
(105)

Again, upon applying the relation (70), we write the solution for
$A_{kk}^{\beta}(\upsilon)$ more convenient in terms of spherical Bessel
functions of the first kind as

$\displaystyle A_{kk}^{\beta}(\upsilon)$ $\displaystyle=-\frac{\upsilon
j_{k-1}(\upsilon)j_{k}(\mathrm{i}\kappa_{\beta})-\mathrm{i}\kappa_{\beta}j_{k-1}(\mathrm{i}\kappa_{\beta})j_{k}(\upsilon)}{\upsilon^{2}+\kappa_{\beta}^{2}}.$
(106)

Now that the integrals are obtained, we have to substitute (101) in (95). The
term on the first line of (101) only accounts for $k=0$ and $n=0$ in (95).
Therefore, the contribution of this term to $\widetilde{\bm{\mathcal{M}}}$ is
$j_{1}(\upsilon)/\upsilon\mathbf{m}_{0}$, since
$Y_{00}(\theta,\phi)=1/\sqrt{4\pi}$. Since $B_{\ell m}^{kn}=\delta_{\ell
k}\delta_{mn}$, the final result is

$\displaystyle\widetilde{\bm{\mathcal{M}}}(\bm{\upsilon})$
$\displaystyle=\frac{j_{1}(\upsilon)}{\upsilon}\mathbf{m}_{0}+\sum_{\beta=1}^{2}\mathbf{g}_{\beta}\sum_{k=0}^{\infty}\sum_{n=-k}^{k}(-\text{i})^{k}c_{kn}^{\beta}A_{kk}^{\beta}(\upsilon)Y_{kn}(\theta_{q},\phi_{q}).$
(107)

Using the properties of the coefficients $c_{kn}^{\beta}$ studied in Appendix
A, we reformulate (107) in terms of the associated Legendre polynomials, the
indices $k=2\nu$ and $n=2\mu$, and the coefficients $a_{\nu\mu}^{\beta}$:

$\displaystyle\widetilde{\bm{\mathcal{M}}}(\bm{\upsilon})=\frac{j_{1}(\upsilon)}{\upsilon}\mathbf{m}_{0}+\sum_{\beta=1}^{2}\mathbf{g}_{\beta}\sum_{\nu=1}^{\infty}\sum_{\mu=0}^{\nu}(-1)^{\nu}a_{\nu\mu}^{\beta}\rho_{\nu}^{\beta}(\upsilon)P_{2\nu}^{2\mu}(\cos\theta_{q})\cos(2\mu\phi_{q}),$
(108)

where the radial function is given by

$\displaystyle\rho_{\nu}^{\beta}(\upsilon)=-\frac{\upsilon
j_{2\nu-1}(\upsilon)j_{2\nu}(\mathrm{i}\kappa_{\beta})-\mathrm{i}\kappa_{\beta}j_{2\nu-1}(\mathrm{i}\kappa_{\beta})j_{2\nu}(\upsilon)}{\upsilon^{2}+\kappa_{\beta}^{2}}.$
(109)

Using once again (70) and the well-known series (71) for the Bessel functions
of the first kind, we redefine the spherical Bessel functions of imaginary
arguments [as they appear in (109)] as:

$\displaystyle\Upsilon_{\nu}(\tau)$
$\displaystyle=j_{2\nu}(\mathrm{i}\tau)=\frac{\sqrt{\pi}}{2}\sum_{s=0}^{\infty}\frac{(-1)^{\nu}(\tau/2)^{2(s+\nu)}}{s!\Gamma(2\nu+s+3/2)},$
(110) $\displaystyle\mathcal{\aleph}_{\nu}(\tau)$
$\displaystyle=\mathrm{i}j_{2\nu-1}(\mathrm{i}\tau)=\frac{\sqrt{\pi}}{2}\sum_{s=0}^{\infty}\frac{(-1)^{\nu}(\tau/2)^{2(s+\nu)-1}}{s!\Gamma(2\nu+s+1/2)},$
(111)

such that it becomes clear that $\widetilde{\bm{\mathcal{M}}}(\bm{\upsilon})$
is a purely real-valued function. The radial function is then rewritten as

$\displaystyle\rho_{\nu}^{\beta}(\upsilon)=-\frac{\upsilon
j_{2\nu-1}(\upsilon)\Upsilon_{\nu}(\kappa_{\beta})-\kappa_{\beta}\mathcal{\aleph}_{\nu}(\kappa_{\beta})j_{2\nu}(\upsilon)}{\upsilon^{2}+\kappa_{\beta}^{2}}.$
(112)

By comparing this result to (102), we find the following pair of spherical
Hankel transforms

$\displaystyle\rho_{\nu}^{\beta}(\upsilon)=\int_{0}^{1}\Upsilon_{\nu}(\kappa_{\beta}\xi)j_{\nu}(\upsilon\xi)\xi^{2}d\xi.$
(113)

By comparing the result (108) to (16) it becomes clear that the angular part
is (due to the orthogonality relation of the spherical harmonics) shape
invariant under Fourier transformation, while the spherical Hankel transform
(113) of the radial function $\Upsilon_{\nu}(\kappa_{\beta}\xi)$ remains.