# Size limit of superparamagnetic inclusions in dust grains and difficulty of
magnetic grain alignment in protoplanetary disks

Haifeng Yang C.N. Yang Junior Fellow  Institute for Advanced Study, Tsinghua
University, Beijing, 100084, China

###### Abstract

Alignment of non-spherical grains with magnetic fields is an important problem
as it lays the foundation of probing magnetic fields with polarized dust
thermal emissions. In this paper, we investigate the feasibility of magnetic
alignment in protoplanetary disks (PPDs). We use an alignment condition that
Larmor precession should be fast compared with the damping timescale. We first
show that the Larmor precession timescale is some three orders of magnitude
longer than the damping time for millimeter-sized grains under conditions
typical of PPDs, making the magnetic alignment unlikely. The precession time
can be shortened by superparamagnetic inclusions (SPIs), but the reduction
factor strongly depends on the size of the SPI clusters, which we find is
limited by the so-called “Néel’s relaxation process.” In particular, the size
limit of SPIs is set by the so-called “anisotropic energy constant” of the SPI
material, which describes the energy barrier needed to change the direction of
the magnetic moment of an SPI. For the most common iron-bearing materials, we
find maximum SPI sizes corresponding to a reduction factor of the Larmor
precession timescale of order $10^{3}$. We also find that reaching this
maximum reduction factor requires fine-tuning on the SPI sizes. Lastly, we
illustrate the effects of the SPI size limits on magnetic alignment of dust
grains with a simple disk model, and we conclude that it is unlikely for
relatively large grains of order 100 $\mu$m or more to be aligned with
magnetic fields even with SPIs.

Protoplanetary disks — ISM: magnetic fields — Polarization

##  1 Introduction

The polarization of starlight was first observed in 1949 (Hiltner, 1949). It
was soon attributed to the alignment of dust grains in the foreground
interstellar medium. Since then, the alignment of dust grains, especially with
respect to magnetic fields, has many developments. Many theories were
developed to explain how dust grains are aligned with magnetic fields, such as
Davis-Greenstein mechanism (Davis & Greenstein, 1951), hydrogen formation
torque (Purcell, 1979), radiative alignment torque (B-RAT; Dolginov &
Mytrophanov 1976; Draine & Weingartner 1997; Lazarian & Hoang 2007), and
recently mechanical alignment torque (B-MAT; Hoang et al. 2018). We refer
interested readers to Andersson et al. (2015) and references therein.

Superparamagnetism (SPM) was first introduced to the astronomical literatures
of grain alignment by Jones & Spitzer (1967). They pointed out that SPM can
enhance magnetic relaxation, the process invoked by Davis-Greenstein mechanism
to dissipate oscillating magnetic moments and to align grains, which was found
insufficient to align regular paramagnetic dust grains with magnetic fields.
Mathis (1986) adopted this theory with the assumption that grains containing
any small superparamagnetic particle, the so-called superparamagnetic
inclusions (SPIs), can be aligned with magnetic fields. Under this theory, the
fact that bigger grains are better aligned is well explained since bigger
grains are more likely to contain SPIs. Fe-Ni inclusions appear to present in
interplanetary dust particles, and their spatial frequency supports Mathis’s
theory in explaining the wavelength dependence of polarization (Goodman &
Whittet, 1995). Magnetic nanoparticles and inclusions were also discussed
recently by Draine & Hensley (2013) focusing on the impacts of such inclusions
on the dust thermal emission and polarization.

Observationally, tracing magnetic fields with polarized thermal emission from
grains aligned with magnetic fields is a classical and successful method. It
is clear that grains in diffuse interstellar medium are aligned with magnetic
field from starlight polarization (Mathewson & Ford, 1970). Recently, this
picture receives firm supports from the Planck all-sky survey data (Planck
Collaboration et al., 2015), and from the interferomatric polarimetry data
(see Hull & Zhang 2019 and reference therein). It is in no doubt that grains
on scales larger than disks are aligned with magnetic fields.

In the past six years, thanks to the improvement of interferomatric
polarimetry, especially with the Atacama Large Millimeter/submillimeter Array
(ALMA), we have become able to resolve polarization maps down to the disk
scale. The results have been surprising. Most systems show uniform
polarization patterns, especially at shorter wavelengths, e.g. HL Tau
(Stephens et al., 2014, 2017), IM Lup (Hull et al., 2018), DG Tau (Bacciotti
et al., 2018), and HD163296 (Dent et al., 2019), which are better explained
with the self-scattering of dust grains than with magnetically aligned grains
(Kataoka et al., 2015; Yang et al., 2016a; Ohashi & Kataoka, 2019; Lin et al.,
2019). Cox et al. (2018)’s survey found a trend that the polarization is
uniform on scales smaller than 100 AU at percent level, whereas the
polarization is less organized on larger scales at a higher level ($\sim
5\%$). This is consistent with the picture that dust grains at disk scales are
not aligned. There are also some systems showing complicated polarization
features, such as the binary system BHB07-11 (Alves et al., 2018), the
southern part of HD142527 (Kataoka et al., 2016; Ohashi et al., 2018), and HL
Tau (Kataoka et al., 2017). However, the origin of these additional
complexities is not certain at this time.

Theoretically, the transition from magnetic alignment to no-magnetic-alignment
going from the more diffuse surrounding region to the disk is not surprising.
The protoplanetary disk (PPD) environment is very different from the diffuse
ISM in many ways: higher density, bigger grain sizes, different temperature
and radiation energy density, etc. A rough comparison given in Sec. 2 shows
that the Larmor precession timescale can be some three orders of magnitude
larger than the gaseous damping timescale, making magnetic alignment
impossible (see also Tazaki et al. 2017). Hoang (2017) pointed out that if the
dust grains contain large SPIs (on the order of $10^{5}$ iron atoms each),
magnetic alignment of millimeter dust grains can still be possible in PPDs.
The enhancement from SPIs strongly depends on the size of each SPI. In this
paper we will focus on estimating the maximum size of SPIs and discuss how the
size limit affects the magnetic alignment of dust grains in PPDs.

The structure of this paper is as follows. In Sec. 2, we introduce the basic
timescales involved in the magnetic alignment process, and give a simple
comparison for regular paramagnetic materials. In Sec. 3, we introduce how
SPIs can enhance the magnetic susceptibility and how the cluster size is
limited by the relaxation process. In Sec. 4, we estimate the size limit of
SPIs assuming a few commonly adopted forms of iron. In Sec. 5, we adopt a
simple disk model and discuss the feasibility of magnetic alignment in PPDs In
Sec. 6, we discuss caveats and uncertainties in this work, as well as its
implications for several observed disks. We summarize our results in Sec. 7.

##  2 Basic magnetic alignment theory

###  2.1 Criteria for magnetic alignment

Dust grain alignment with magnetic fields is a complicated process with many
timescales involved. An outline of the process (regardless of alignment
mechanism) is as follows (see e.g. Lazarian 2007). Initially, a randomly-
spinning dust grain nutates about its angular momentum axis. Over a timescale
$t_{\mathrm{int}}$, the principal axis of the dust grain becomes aligned with
the its angular momentum through some relaxation processes. The dust grain’s
angular momentum also precesses around the external magnetic field on the
Larmor precession timescale ($t_{L}$). At the end, some torques gradually
force the angular momentum of the dust grain to be aligned with the magnetic
field, over a timescale of $t_{\mathrm{al}}$. At the same time, the random
bombardment of gas particles tries to disturb the angular momentum, over the
gaseous damping timescale ($t_{d}$).

One typically considers that these three timescales must follow a hierarchical
in equality – i.e., $t_{L}<t_{\mathrm{al}}<t_{d}$ – in order for grain
alignment to proceed successfully. However, the alignment timescale
$t_{\mathrm{al}}$ is considerably more complicated than the other two
timescales to compute in general, because it depends on the specific torquing
mechanism at play. For the purpose of this paper, we work with the
insufficient but still necessary condition $t_{L}<t_{d}$ to determine
conditions under which dust grains can align with ambient magnetic
fields111Note that this criterion effectively ignored the suprathermal
rotation, which will be discussed in Sec. 6.. Even though $t_{L}>t_{d}$ is
unrealistic in most interstellar medium, it becomes a strong possibility in
PPDs as grains grow to millimeter in size and gas density increases by many
orders of magnitude compared with the ISM values, as we will see more
quantitatively in the following section.

In theoretical works studying the dynamics of magnetic grain alignment, fast
Larmor precession has been mostly assumed, so that the calculated torques are
averaged over one precession before the dynamics of the dust grain is studied
(e.g., Lazarian & Hoang 2007; Hoang et al. 2018). In this work, we use
$t_{L}<0.1t_{d}$ as the criterion for fast Larmor precession assumption. In
this regime, the Larmor precession timescale will not be the limiting factor,
and magnetic grain alignment becomes possible. We will come back to discuss
this criterion and its caveats in more detail in Sec. 6.

###  2.2 Gaseous damping timescale

The random bombardment of gas particles on dust grains tends to misalign any
ordered orientation of the dust grains. This happens roughly on a timescale of
(Roberge et al., 1993):

$\begin{split}t_{\mathrm{d}}=&\frac{2\sqrt{\pi}}{5}\frac{\rho_{s}a}{n_{\rm
g}m_{\rm g}v_{\rm g,th}}\\\ =&3.54\times
10^{12}~{}\mathrm{s}\times\left(\frac{\rho_{s}}{3~{}\mathrm{g/cm^{3}}}\right)\\\
&\times\left(\frac{a}{0.1~{}\mathrm{\mu m}}\right)\left(\frac{n_{\rm
g}}{20~{}\mathrm{cm^{-3}}}\right)^{-1}\left(\frac{T_{g}}{85~{}\mathrm{K}}\right)^{-1/2},\end{split}$
(1)

where $\rho_{s}$ is the mass density of the (solid) dust grain, $a$ is the
grain size, $n_{g}$ is the number density of gas particles, and $T_{g}$ is the
gas temperature. This is a rather long time scale for the diffuse ISM
conditions we used above. If we take a PPD with total mass of
$0.01\rm\,M_{\sun}$ as normalization, uniformly distributed in a cylinder with
$100$ AU as radius and $10$ AU as scale height, we have:

$\begin{split}t_{\mathrm{d}}=&2.6\times
10^{8}~{}\mathrm{s}\times\left(\frac{\rho_{s}}{3~{}\mathrm{g/cm^{3}}}\right)\left(\frac{a}{1~{}\mathrm{mm}}\right)\\\
&\times\left(\frac{n_{\rm g}}{5\times
10^{9}~{}\mathrm{cm^{-3}}}\right)^{-1}\left(\frac{T_{g}}{25~{}\mathrm{K}}\right)^{-1/2},\end{split}$
(2)

which is shorter than typical dynamical timescales in PPDs.

###  2.3 Larmor precession timescale

The Larmor precession process is the precession of a magnetic moment around a
magnetic field. A spinning dust grain possess a magnetic moment due to the
Barnett effect(Barnett, 1915). Its magnetization is $M=\chi\Omega/\gamma$
(Draine, 2004), where $\chi$ is the magnetic susceptibility, $\Omega$ is the
angular velocity, $\gamma=g\mu_{B}/\hbar$ is the gyromagnetic ratio, and
$\mu_{B}$ is the Bohr magneton. The $g$-factor is about 2 for electrons.

For regular paramagnetic material, the magnetic susceptibility follows Curie’s
Law (Morrish, 2001):

$\chi=\frac{n\mu^{2}}{3kT}=10^{-3}\hat{\chi}\left(\frac{T}{25\rm\,K}\right)^{-1},$
(3)

where $n$ is the number density of magnetic units, $\mu$ is the magnetic
moment of each unit, and $\hat{\chi}$ is a dimensionless parameter that is on
the order of unity for regular paramagnetic materials (Draine, 1996; Lazarian,
2007).

The Larmor precession timescale is then:

$\begin{split}t_{L}=\frac{2\pi I\gamma}{\chi VB}=&4.3\times
10^{6}\mathrm{s}\times\hat{\chi}^{-1}\left(\frac{\rho_{s}}{3~{}\mathrm{g/cm^{3}}}\right)\\\
&\times\left(\frac{T_{\mathrm{d}}}{25~{}\mathrm{K}}\right)\left(\frac{B}{5~{}\mathrm{\mu
G}}\right)^{-1}\left(\frac{a}{0.1~{}\mathrm{\mu m}}\right)^{2}.\end{split}$
(4)

We can see that this timescale is some six orders smaller than the gaseous
damping timescale for parameters appropriate for the diffuse ISM. But if we
normalize the field strength to $5$ mG, which is typical for a PPD with
$10^{-8}\rm\,M_{\sun}/yr$ accretion rate at tens of AU scale (Bai, 2011), and
the grain size to $1$ mm, we get:

$\begin{split}t_{L}=&4.3\times
10^{11}\mathrm{s}\times\hat{\chi}^{-1}\left(\frac{\rho_{s}}{3~{}\mathrm{g/cm^{3}}}\right)\\\
&\times\left(\frac{T_{\mathrm{d}}}{25~{}\mathrm{K}}\right)\left(\frac{B}{5~{}\mathrm{mG}}\right)^{-1}\left(\frac{a}{1~{}\mathrm{mm}}\right)^{2},\end{split}$
(5)

which is about $10^{3}t_{d}$.

##  3 Superparamagnetic inclusions

###  3.1 Basic picture

Superparamagnetic inclusions (SPIs) are small (nano-sized) particles of
ferromagnetic material222In this work, we don’t distinguish ferromagnetic
material with ferrimagnetic material, or even speromagnetic material. They all
behave as macro-spins, as discussed in the text, but maybe with different
number of effective Bohr magneton per atom ($p$ in Eq.(7)).. Within one such
particle, all the atoms are spontaneously magnetized and behave like a single
large magnetic moment, the so-called “macro-spin” (Bean & Livingston, 1959).
They are not big enough to create domain walls yet, and are usually referred
to as “single-domain particles”. In the absence of external magnetic fields,
these macro-spins are randomly oriented and behave like paramagnetic materials
as an ensemble.

Let’s first consider the simplest case, where an SPI has no preferred
direction for magnetization. This isotropic case is mathematically identical
to the regular paramagnetic case, and we have the bulk magnetic susceptibility
of an ensemble of identical SPIs as (Jones & Spitzer, 1967):

$\chi_{sp}=\frac{\mathbb{N}\mu^{2}}{3kT},$ (6)

where $\mathbb{N}$ is the number density of SPIs inside the dust grain. Let
$n_{tot}$ be the total number density of atoms and assume a fraction of
$f_{sp}$ atoms are magnetic atoms embedded in SPIs. Let $\mu$ be the magnetic
moment of the “macro-spins”. If each SPI contains $N_{cl}$ magnetic atoms, we
have $\mathbb{N}=f_{sp}n_{tot}/N_{cl}$, and $\mu=N_{cl}p\mu_{B}$, where
$p\mu_{B}$ is the averaged Bohr magneton for each magnetic atom. With these we
have (Draine 1996):

$\begin{split}\chi_{sp}=&\frac{f_{sp}n_{tot}N_{cl}(p\mu_{B})^{2}}{3kT}\\\
=&0.72\times
10^{-2}N_{cl}f_{sp}\left(\frac{n_{tot}}{10^{23}\rm\,cm^{-3}}\right)\left(\frac{p}{3}\right)^{2}\left(\frac{T}{25\rm\,K}\right)^{-1}.\end{split}$
(7)

Compared with Eq. (3), we have:

$\hat{\chi}=7.2N_{cl}f_{sp}(p/3)^{2}.$ (8)

In reality, SPIs will have a preferred direction for magnetization. For
example, a prolate particle prefers to be magnetized along its long axis (Bean
& Livingston, 1959). This is called shape anisotropy. Another example is
metallic iron, which has a cubic crystalline structure. Metallic iron has less
energy when magnetized along one of the principal axes of its crystalline
structure (so-called “easy axis”, Dai & Qian 2017). This is called
magnetocrystalline anisotropy.

Even though the energy would be different in the presence of anisotropy, the
susceptibility remains the same in thermal equilibrium states. Let $K$ be the
so-called “anisotropy constant”, such that $KV$ is the energy needed to change
the direction of the magnetic moment. With a simple prescription, Bean &
Livingston (1959) showed that the ensemble-averaged magnetic susceptibility
remains the same for both of the two limiting cases, when $KV\gg kT$ and when
$KV\ll kT$. Thus Eq. (7) works even for anisotropic SPIs.

###  3.2 Size limit determined by the relaxation process

Billas et al. (1994) showed that single-domain particles as small as
$N_{cl}=30$ can show superparamagnetism. In this subsection, we will discuss
what determines the maximum size for SPI, which is more important than lower
limit.

Néel (1949) first proposed that single-domain particles experience random
Brownian-like motions that can change the orientation of its magnetic moment
(see also Bean & Livingston 1959). This happens on a timescale (the so-called
“Néel’s relaxation timescale”) of:

$t_{N}\equiv t_{0}\exp\left(\frac{KV}{kT}\right),$ (9)

where $t_{0}$ is called “attempt timescale” typically on the order of
$10^{-9}\rm\,s$. Setting the $t_{N}$ equal to the timescale of interest
($\tau$), we can define a critical blocking volume as:

$V_{\mathrm{cr}}=\frac{kT}{K}\ln\left(\frac{\tau}{t_{0}}\right).$ (10)

The typical dynamical timescale of a $100$ AU sized PPD is about $1000$ yr,
which yields $\ln(\tau/t_{0})\approx 45$. We can see that the critical volume
strongly depends on the temperature, and the cluster size measured at room
temperature doesn’t apply directly to an astronomical environment, which
hasn’t been considered before. The critical number of magnetic atoms can then
be calculated, for iron-based ferromagnetic material, through:

$N_{\mathrm{cr}}=\frac{\rho
V_{\mathrm{cr}}f_{\mathrm{Fe}}}{56m_{p}}=\frac{\rho
f_{\mathrm{Fe}}kT}{56m_{p}K}\ln\left(\frac{\tau}{t_{0}}\right),$ (11)

where $f_{\mathrm{Fe}}$ is the mass fraction of iron atoms in the
ferromagnetic material constituting the SPIs, and $56m_{p}$ is the mass of an
iron atom.

The magnetization of a dust grain with SPIs is illustrated with a simplified
model in Fig. 1. In this model, we consider two dust grains with uniformly
sized SPIs. The left one has smaller SPIs, whereas the right one contains
bigger SPIs. At $t<0$, there is no external magnetic field. The magnetic
moment of each SPIs are randomly oriented such that both dust grains have no
bulk magnetization. At $t=0$, we turn on external magnetic fields, and observe
the magnetization of dust grains at $\tau$, the dynamical timescale of our
interest. We will find that all the small SPIs in the left dust grain turn
into the external magnetic field direction (with thermal fluctuation). The
resulting magnetic susceptibility of the dust grain is the superparamagnetic
susceptibility $\chi_{sp}(N_{cl})$ in Eq. (7). In contrast, the large SPIs in
the right dust grain do not have enough time to overcome the anisotropic
energy barrier. Their magnetic moments are effectively “blocked” and do not
contribute to the magnetization of the dust grain. The dependence of the
magnetic susceptibility on the size of SPI clusters can be approximated as
(see Appendix A for a more quantitative discussion):

$\chi=\left\\{\begin{array}[]{ll}0,&N_{cl}>N_{\mathrm{cr}}\\\
\chi_{sp}(N_{cl}),&N_{cl}<N_{\mathrm{cr}}\end{array}\right..$ (12)

We can see that the maximum enhancement of the magnetic susceptibility for an
ensemble of SPIs of equal sizes is:

$\chi_{\mathrm{max}}=\chi_{sp}(N_{\mathrm{cr}}),$ (13)

which is achieved when all SPIs in a dust grain are of the same size with
$N_{\mathrm{cr}}$ magnetic atoms.

Figure 1: A schematic illustration of the magnetization of dust grains with
SPIs of different sizes. The left column represents a dust grain with small
SPIs. The right column represents a dust grain with big SPIs. The top row is
their initial states before applying external magnetic fields. Starting from
$t=0$, an external magnetic field going up is applied. The bottom row is the
magnetization of these two dust grains at time $\tau$, the dynamical timescale
of interest.

###  3.3 Distribution of SPIs

So far we have only considered ensembles of SPIs with the same size. In order
to understand the effects of a distribution of SPI sizes, we adopt a very
simple power-law distribution with index $-q$:
$dn(N_{cl})/dN_{cl}=CN_{cl}^{-q}$, with $N_{1}<N_{cl}<N_{2}$ is the cluster
size and $C$ is an arbitrary constant to be determined from the iron
abundance.

We are particularly interested in the scenario when
$N_{1}<N_{\mathrm{cr}}<N_{2}$. With Eq. (12), the magnetic susceptibility of
this ensemble of SPIs can be calculated as:

$\begin{split}\chi&=\frac{1}{B}\int_{N_{1}}^{N_{2}}\frac{dn(N_{cl})}{dN_{cl}}\left<\mu_{z}\right>dN_{cl}\\\
&=\int_{N_{1}}^{N_{\mathrm{cr}}}CN_{cl}^{-q}\frac{N_{cl}^{2}p^{2}\mu_{B}^{2}}{3kT}dN_{cl}\\\
&=\frac{Cp^{2}\mu_{B}^{2}}{3(3-q)kT}\left.N_{cl}^{3-q}\right|_{N_{1}}^{N_{\mathrm{cr}}},\end{split}$
(14)

where $\left<\mu_{z}\right>$ is the averaged magnetic moment along magnetic
field direction (see Appendix for more detail). With some arithmetic of
finding the constant $C$, we can express our results as:

$\chi=\chi_{\mathrm{max}}\times\left\\{\begin{aligned}
&\frac{2-q}{3-q}\left(\frac{N_{\mathrm{cr}}}{N_{2}}\right)^{2-q},&q<2;\\\
&\frac{q-2}{3-q}\left(\frac{N_{1}}{N_{\mathrm{cr}}}\right)^{q-2},&2<q<3.\\\
\end{aligned}\right.$ (15)

We can see that the end magnetic susceptibility of this ensemble is always
smaller than $\chi_{\mathrm{max}}$. The reduction factor is roughly the ratio
of the number density of iron atoms within SPIs with sizes close to
$N_{\mathrm{cr}}$ to the total number density of iron atoms within SPIs.
Because of this reduction factor, reaching the maximum value of
$\chi_{sp}=\chi_{\mathrm{max}})$ implicitly assumes that all SPIs have sizes
close to the critical size $N_{\mathrm{cr}}$.

##  4 Estimate the critical size of SPIs

The critical size of an SPI is an important quantity for determining how big
of an enhancement it will have on the magnetic susceptibility of the host dust
grain. In this section we perform estimates of SPI critical sizes, in terms of
the number of magnetic atoms $N_{\mathrm{cr}}$, for various materials that
might plausibly be contained in astrophysical dust grains.

As seen in Eq. (10), the anisotropy constant $K$ determines the critical
volume, when the temperature is fixed. In nature, there are two most important
contributions to $K$: the shape anisotropy and the magnetocrystalline
anisotropy. In this work, we will ignore the first one and assume spherical
SPIs. Including the shape anisotropy will increase the anisotropy constant and
decrease the size estimates given below. In other words, our estimates are
conservative upper limits.

The critical volume is defined in Eq. (10) and values at $25$ K are reported
here. We assume room temperature densities of stoichiometric materials.
Besides the critical number of magnetic atoms in one SPI ($N_{\mathrm{cr}}$),
we also report the magnetic susceptibility $\hat{\chi}_{\mathrm{max}}$, taking
$f_{sp}=0.1$, and $n_{tot}=10^{23}\rm\,cm^{-3}$.

###  4.1 $\rm Fe_{3}O_{4}$ (magnetite)

At low temperature ($T<120$ K), magnetite has a monoclinic structure (Iizumi
et al., 1982). The magnetic anisotropy energy was determined by Abe et al.
(1976) as:

$E_{a}=K_{a}\alpha_{a}^{2}+K_{b}\alpha_{b}^{2}+K_{aa}\alpha_{a}^{4}+K_{bb}\alpha_{b}^{4}+K_{ab}\alpha_{a}^{2}\alpha_{b}^{2}-K_{u}\alpha_{111}^{2},$
(16)

with $K_{a}=25.2,\,K_{b}=3.7,\,K_{u}=2.1,\,K_{aa}=1.8,\,K_{bb}=2.4,\,$ and
$K_{ab}=7.0$ in $10^{5}\rm\,erg/cm^{3}$, and all $\alpha$’s are directional
cosines (see Abe et al. 1976 for their definitions). The easy axes are along
$(001)$ and $(00\bar{1})$ directions. The magnetization can change from one
easy axis to another through saddle points along $(010)$ or $(0\bar{1}0)$
directions, and the energy barrier is $K\approx 6.1\times
10^{5}\rm\,erg/cm^{3}$. This translates to a critical volume as
$V_{\mathrm{cr}}=2.5\times 10^{-19}\rm\,cm^{3}$. Li et al. (2007) found a spin
density as $3.54\mu_{B}$ per formula at $10$ K along [100] direction, which we
will use in this paper. For every iron atom, we have $p=1.18$ Bohr magnetons.
Taking $\rho=5.17\rm\,g/cm^{3}$, we get $N_{\mathrm{cr}}=1.0\times 10^{4}$ and
$\hat{\chi}_{\mathrm{max}}=1.1\times 10^{3}$.

###  4.2 $\gamma-\rm Fe_{2}O_{3}$ (maghemite)

$\gamma-\rm Fe_{2}O_{3}$ (maghemite) is ferrimagnetic iron oxide with similar
structures to magnetite at room temperature. Its formula is often supposed to
be $\rm(Fe^{3+})_{A}(Fe^{3+}_{5/3}\square_{1/3})_{B}O_{4}$, where $\square$
represents a vacancy. A perfect crystal has $3.33\mu_{B}$ per formula. Hence
we have $p=1.25$. In bulk samples, the moments were usually found to be about
$87\sim 94\%$ of this value (Coey & Khalafalla, 1972). Pisane et al. (2017)
fitted the effective magnetic anisotropy as a function of particle size with
three terms. The leading term (and the dominating term for big particles)
corresponds to $K=1.9\times 10^{5}\rm\,erg/cm^{3}$. This translates into a
critical volume of $V_{\mathrm{cr}}=8.2\times 10^{-19}\rm\,cm^{3}$. Taking
$\rho=4.9\rm\,g/cm^{3}$, we have $N_{\mathrm{cr}}=3.0\times 10^{4}$, and
$\hat{\chi}_{\mathrm{max}}=3.7\times 10^{3}$.

###  4.3 Metallic iron

The leading term in the magnetic anisotropy energy of metallic iron has the
form of
$K_{1}(\alpha_{1}^{2}\alpha_{2}^{2}+\alpha_{2}^{2}\alpha_{3}^{2}+\alpha_{1}^{2}\alpha_{3}^{2})$,
where $\alpha_{i}$, $i=1,2,3$ are directional cosines. At temperatures below
$100$ K, $K_{1}\approx 5.4\times 10^{5}\rm\,erg/cm^{3}$ (Dai & Qian, 2017).
The energy barrier to change the magnetization from one easy axis to another
is $K=(1/4)K_{1}=1.35\times 10^{5}\rm\,erg/cm^{3}$. Hence
$V_{\mathrm{cr}}=1.15\times 10^{-18}\rm\,cm^{3}$. We will follow Draine (1996)
and take $p=3$, which was inspired by Billas et al. (1994)’s work showing that
small clusters of iron have $3\mu_{B}$ per atom. Taking
$\rho=7.87\rm\,g/cm^{3}$, we have $N_{\mathrm{cr}}=9.7\times 10^{4}$ and
$\hat{\chi}_{\rm max}=7.0\times 10^{4}$.

###  4.4 Other forms of iron and summary

Hematite ($\alpha-\rm Fe_{2}O_{3}$) is another possible form of iron, which is
more stable than maghemite ($\gamma-\rm Fe_{2}O_{3}$) discussed above. It is,
however, antiferromagnetic, and perfect crystal can be essentially considered
as non-magnetic. In reality, some defects may exist to contribute to the
magnetization but it should be negligible comparing with other ferromagnetic
materials discussed above. The same goes for FeO. For a summary of magnetic
properties of the iron oxides, we refer interested readers to Cornell &
Schwertmann (2003), especially their Table 6.2. We do not consider sulfuric
iron in this work.

All results discussed above are summarized in Table 1. We can see the maximum
cluster size and the enhancement of magnetic susceptibility strongly depends
on the form of iron. Even though $\hat{\chi}_{\mathrm{max}}$ on the order of
$10^{4}$ is still possible with metallic iron, it is less likely to exist in
real dust grains as it can easily be oxidated. We suggest that $10^{3}$ is a
more realistic enhancement factor $\hat{\chi}_{\mathrm{max}}$ on the magnetic
susceptibility through SPIs.

Material |  $K$ ($\rm erg/cm^{3}$) | $p$ |  $V_{\mathrm{cr}}$ ($\rm cm^{3}$) | $N_{\mathrm{cr}}$ | $\hat{\chi}_{\rm max}$
---|---|---|---|---|---
$\rm Fe_{3}O_{4}$ | $6.1\times 10^{5}$ | $1.18$ | $2.5\times 10^{-19}$ | $1.0\times 10^{3}$ | $1.1\times 10^{3}$
$\rm\gamma-Fe_{2}O_{3}$ | $1.9\times 10^{5}$ | $1.25$ | $8.2\times 10^{-19}$ | $3.0\times 10^{4}$ | $3.7\times 10^{3}$
Fe | $1.35\times 10^{5}$ | $3$ | $1.15\times 10^{-18}$ | $9.7\times 10^{4}$ | $7.0\times 10^{4}$
Table 1: Anisotropy constant $K$, effective number of Bohr magnetons per iron
atom $p$, critical volume $V_{\mathrm{cr}}$ (Eq. (10)), critical cluster size
$N_{\mathrm{cr}}$, and reduced magnetic susceptibility $\hat{\chi}_{\rm max}$
for various iron bearing ferromagnetic materials that may exist in dust
grains. See text for discussion and references.

##  5 Magnetic alignment in PPDs

In Sec. 2, we gave rough estimates of timescales relevant for magnetic
alignment and compared them to motivate this study. In this section, we
perform a more detailed study on whether magnetic alignment is feasible in a
fiducial PPD model.

###  5.1 Disk model

As an illustration, we adopt the well-known Chiang & Goldreich (1997) model.
It is a passive disk with Minimum Mass Solar Nebula density profile
$\Sigma=(r/\mathrm{AU})^{-3/2}\Sigma_{0}$, with
$\Sigma_{0}=10^{3}\rm\,g/cm^{2}$ (Weidenschilling, 1977).It has a superheated
surface layer and a cooler interior region, where most grown mm-sized dust
grains reside. The temperature in the interior region and the scale height in
the model under both hydrostatic equilibrium and radiative equilibrium were
fitted as

$T=\begin{dcases}150\,\mathrm{K}\times\left(\frac{r}{1\rm\,AU}\right)^{-3/7}&0.4\,\mathrm{AU}<r<84\,\mathrm{AU}\\\
21\,\mathrm{K}&84<r<100\,\mathrm{AU}\\\ \end{dcases},$ (17)

and

$\frac{H}{r}=\begin{dcases}0.17\left(\frac{r}{1\rm\,AU}\right)^{2/7}&0.4\,\mathrm{AU}<r<84\,\mathrm{AU}\\\
0.59\left(\frac{r}{84\rm\,AU}\right)^{1/2}&84<r<100\,\mathrm{AU}\\\
\end{dcases}.$ (18)

At a given radius, the midplane density is used to calculate timescales, which
is $\Sigma/\sqrt{2\pi H}$. We adopt a mean molecular weight of $2.3$ and
assume the temperature to be the same for gas and dust.

For the magnetic field structure, we adopt the estimate from Bai (2011):

$B=1.0\,\mathrm{G}\times\left(\frac{\dot{M}}{10^{-8}\rm\,M_{\sun}/yr}\right)^{1/2}\left(\frac{r}{1\rm\,AU}\right)^{-11/8},$
(19)

where $\dot{M}$ is the mass accretion rate, which is assumed to be
$10^{-8}\rm\,M_{\sun}/yr$, typical for classical T Tauri stars (see Dutrey et
al. 2014 and references therein).

###  5.2 Timescale comparison

As discussed in Sec. 2.1, we are specially interested in two conditions:
$t_{L}=t_{d}$, and $t_{L}=0.1t_{d}$. These two conditions define three
different regimes:

  * •

$t_{L}>t_{d}$: No magnetic alignment.

  * •

$0.1t_{d}<t_{L}<t_{d}$: More complicated dynamical study needed.

  * •

$t_{L}<0.1t_{d}$: Magnetic alignment is possible with fast Larmor precession.

From Eq.(5) and (2):

$\begin{split}\frac{t_{L}}{t_{d}}=&1.65\times
10^{3}\left(\frac{\hat{\chi}}{a_{\mathrm{mm}}}\right)^{-1}\left(\frac{T}{25\rm\,K}\right)^{3/2}\\\
&\times\left(\frac{n_{g}}{5\times
10^{9}\rm\,cm^{-3}}\right)\left(\frac{B}{5\rm\,mG}\right)^{-1},\end{split}$
(20)

where $a_{\mathrm{mm}}\equiv(a/1\,\mathrm{mm})$. We can see that $\hat{\chi}$
and grain size $a$ are degenerate. Due to this degeneracy, the timescale ratio
is calculated as a function of radius in the PPD, and the dimensionless factor
$\hat{\chi}/a_{\mathrm{mm}}$. The results are plotted in Fig. 2.

Figure 2: The ratio $t_{L}/t_{d}$ as a function of the radius and
$\hat{\chi}/a_{\mathrm{mm}}$, with $a_{\mathrm{mm}}\equiv a/(1\rm\,mm)$. The
$t_{L}/t_{d}=0.1$ and $t_{L}/t_{d}=1$ contours are plotted to aid the
interpretation.

First of all, we can see that grains in the inner disk are harder to align
compared with those in the outer disk. This is because the gas density power
low index ($-2.8$ in our model within $84$ AU) is usually more negative than
the magnetic fields power low index ($-1.4$ in our model). As we decrease the
radius, the gaseous damping timescale increases faster than the Larmor
precession timescale. In order to achieve fast Larmor precession
($t_{L}<0.1t_{d}$) at a radius of 30 AU, which is the typical resolution of
ALMA polarization observations of the nearest star forming regions, we need
roughly $\hat{\chi}/a_{\mathrm{mm}}>\sim 10^{4}$.

This result can be easily translated into more meaningful statements after
fixing $\hat{\chi}$. Fig. 3 shows the largest grain sizes with fast Larmor
precession ($t_{L}/t_{d}<0.1$) at each radius in the disk. If one take
$\hat{\chi}=10^{3}$, the upper limit we suggested in Sec. 4, we get
$a=100\rm\,\mu m$. This means that grains with sizes of $100\rm\,\mu m$ or
bigger are unlikely to be aligned with magnetic field within $30$ AU of the
central star, even with the aid of SPI. The situation is even worse for
regular paramagnetic material ($\hat{\chi}=1$), where we need $a>0.1\rm\,\mu
m$ to have fast Larmor precession.

The situation is a lot better in the outer disk at scales larger than $100$
AU. We will only need $\hat{\chi}/a_{\mathrm{mm}}\sim 10^{3}$. Alignment of
$1$ mm dust grains becomes marginally possible if they contain large SPI
clusters ($\hat{\chi}>\sim 10^{3}$).

Figure 3: The largest grain sizes with fast Larmor precession
($t_{L}/t_{d}<0.1$) at each radius in the disk. Both regular paramagnetic dust
grains ($\hat{\chi}=1$) and grains with substantial SPIs ($\hat{\chi}=10^{3}$)
are plotted.

##  6 Discussion

###  6.1 Caveats and uncertainties in this work

The above discussions have relied on a number of simplifying assumptions. In
this section we enumerate these assumptions and discuss the implications that
relaxing them could have for our results.

####  6.1.1 Suprathermal rotation

Our first assumption is that the dust grain rotational motions are distributed
thermally, meaning that we have effectively ignored the possibility of
suprathermal rotation. Purcell (1979) first pointed out that the dust grains
can be spun up to suprathermal motion (rotation energy much larger than
thermal energy) by torques arising from the formation of molecular Hydrogen on
the surface of the dust grains. A suprathermally rotating grain would have a
larger gaseous damping timescale, which would make grain alignment easier.
However, this Hydrogen formation torque is unlikely to work for large
millimeter-sized dust grains, since the rotation energy decreases with
increasing number of Hydrogen formation sites. In PPDs, more plausible
mechanisms for producing suprathermal rotation in millimeter-sized dust grains
are radiative alignment torques (RAT; Lazarian & Hoang 2007) and mechanical
torques from differential motion with the surrounding gas (MAT; Hoang et al.
2018). Indeed, both Lazarian & Hoang (2007) and Hoang et al. (2018) showed
that grains can be aligned towards the so-called “high-J” attractors with
angular momenta that are factors of several tens to hundreds of times larger
than the thermal angular momentum. However, Hoang et al. (2018) also showed
that the mechanical alignment torque goes as $1/\sqrt{N_{\mathrm{facet}}}$,
with $N_{\mathrm{facet}}$ being the number of facets on the grain surface.
This cancellation effect may prevent the MAT from working for large
millimeter-sized grains. Such grains may also have limited helicity, which is
crucial for both the RAT and the MAT to operate, so it is possible that this
cancellation effect applies to RAT as well. More detailed study on
suprathermal rotation of large millimeter-sized grains in dense environment is
needed to better understand how they might modify our results.

####  6.1.2 Alignment Condition

Our second assumption is that of fast Larmor precession, coupled with the lack
of a precise understanding regarding what constitutes “fast enough”
precession. Previous work on magnetic grain alignment has focused on two
limiting cases. In the first limit, that of fast Larmor precession ($t_{L}\ll
t_{d}$), torques are taken to have values that are averaged over the
precession cycle prior to investigating grain dynamics (e.g., B-RAT work in
Lazarian & Hoang 2007 and B-MAT work in Hoang et al. 2018). In the other
limit, that of slow Larmor precession ($t_{L}\gg t_{d}$), the effects of
magnetic fields are completely ignored (e.g. k-RAT work in Lazarian & Hoang
2007 and k-MAT work in Hoang et al. 2018). However, as we can see from Fig. 2,
it is not unreasonable to expect that the ratio $t_{L}/t_{d}$ may fall in the
range $0.1-1$ in PPD environments, in which case we can neither assume
precession-averaged torques nor ignore the magnetic field completely. Whether
dust grains can be aligned magnetically in this regime is thus not currently
clear, and this uncertainty limits the predictive power of magnetic alignment
theories. Solving this problem will require studying the grain dynamics in
three dimensions (i.e., without averaging over Larmor precession) for each
potential alignment mechanism.

####  6.1.3 Form of iron

Our third assumption is the iron abundance. In deriving
$\hat{\chi}_{\mathrm{max}}$, we took $f_{sp}=0.1$. Draine (1996) suggested
that about $10\%$ of atoms in a grain are iron atoms. $f_{sp}=0.1$ means all
iron atoms are in the form of SPIs, and no iron atoms are in other forms, such
as silicate. This optimistic assumption can easily be wrong and reduce
$\hat{\chi}_{\mathrm{max}}$ by one order of magnitude or more, making magnetic
alignment harder. We also see that the maximum cluster size depends on the
detailed form of iron. Even for ferromagnetic materials, the critical cluster
size can vary by orders of magnitude. A chemical study of dust compositions
may help resolve this uncertainty.

####  6.1.4 Disk model

Our fourth assumption is the specific form of the PPD disk model. The MMSN
model, by construction, contains only the minimum amount of mass required to
form our solar system. It is thus likely to have less mass than a real PPD.
Increasing the mass and density of a PPD would increase the gaseous damping
timescale, which would make magnetic grain alignment more difficult.

###  6.2 Effects of temperature dependence on cluster sizes

In Sec. 5, we used $\hat{\chi}$ as a free parameter to discuss the possibility
of magnetic alignment in PPDs, even though the $\hat{\chi}_{\mathrm{max}}$ has
a temperature dependence. In this section, we discuss the implications of this
temperature dependence.

One may be tempted to set $\hat{\chi}$ at each radius in the disk to the local
$\hat{\chi}_{\mathrm{max}}$ defined by its local temperature. However, doing
so would mean that the SPI sizes in dust grains are changing at each location
in the disk, in such a way as to maximize the effects of SPM. This behavior is
not physical, both because dust grains at different radii should have similar
origins and because the size of SPIs in any given dust grain are unlikely to
change as it migrates to other locations of the disk.

However, the temperature dependence of maximum cluster sizes may still have an
impact on the magnetic susceptibility of dust grains as a function of disk
radius. If the dust grains contain SPIs with sizes larger than the critical
$N_{cr}$ (see Eq. 11), then it is possible that these SPIs will be blocked in
the low-temperature regions of the disk, while contributing heavily to the
magnetic susceptibility in the high-temperature regions of the disk. Such a
model would predict a higher degree of magnetic alignment near the center of
the disk. It is interesting to note that this behavior is opposite that
expected from the usual Curie’s Law (Eq. 3).

In addition to its explicit linear dependence on temperature, the critical
volume has another implicit temperature dependence through the anisotropic
constant $K$. For metallic iron, this dependence can be safely ignored as it
changes very slowly and smoothly within the temperature range $0$–$300$ K
(decreasing by roughly only $15\%$ across the range; Dai & Qian 2017). For
magnetite ($\rm Fe_{3}O_{4}$), however, the situation is considerably more
complicated. The first complication is that magnetite undergoes a Verwey
transition (Verwey, 1939; Walz, 2002) and thus changes crystal structure at a
temperature of around $120$ K (Iizumi et al., 1982); the structure changes
from cubic spinel above this temperature to monoclinic below it. Our results
for magnetite above (Sec. 4) would thus need to be revisited for environments
with temperature higher than $120$ K. A second complication is that magnetite
has so-called “isotropic point” near $130$ K. Around this temperature, the
cubic anisotropy constant changes signs, such that easy axis changes from the
cubic diagonals ($T>130$ K) to cubic edges ($T<130$ K); in the immediate
vicinity of $T\sim 130$ K, the anisotropy constant is close to zero. A near-
zero anisotropy constant permits an arbitrarily large SPI cluster size333The
cluster size may still be limited by the domain size – which is usually larger
than the blocking size and which is ignored in this work – as well as other
sources of anisotropy such as the shape anisotropy.. If dust grains contain
large clusters of magnetite, they may be blocked at temperatures other than
$\sim 130$ K, in which case our model would predict a ring of magnetically
aligned dust grains only in the region of the disk where $T\approx 130$ K.
However, we note that for typical PPDs such a high temperature is reached only
on the AU scale (see Eq. 18), which is unlikely to be resolved by ALMA in
polarized emission.

###  6.3 Applications to observed disks

####  6.3.1 Small grains near the surface of AB Aur disk

Li et al. (2016) used mid-infrared polarimetry to observe the disk around AB
Aur at a wavelength of about $10.3\rm\,\mu m$. The polarization was
interpreted as arising from dust grains aligned with poloidal magnetic fields.

At this wavelength, dust grains with sizes on the order of $\sim\rm\mu m$ are
most important for emission. Such small dust grains are lifted more easily by
turbulence and thus reach higher above the disk midplane than (sub)millimeter
grains. Both the dust grains and the their environment are thus different from
what we have discussed in Sec. 5. For the sake of example, suppose the grains
are lifted to twice the scale height, where the gas density is an order of
magnitude smaller than in the disk midplane. The magnetic field is also likely
to be different in this region, but we will assume the same value as in Eq.
(19). Given these conditions, we derive the criterion for fast Larmor
precession to be $\hat{\chi}/a_{\mathrm{mm}}=10^{3}$ at $30$ AU (Li et al.
2016 has a 50 AU resolution). For $\mu$m-sized grains, SPIs are thus
unnecessary to achieve magnetic alignment because $\hat{\chi}=1$ is sufficient
to guarantee fast Larmor precession. Mid-IR polarimetry probing the small dust
grains at the disk surface should therefore be capable of studying magnetic
field structures in PPD.

####  6.3.2 Class 0 systems

Class 0 disks represent the earliest stages of planet formation. They are
likely to have stronger magnetic fields (Yen et al., 2017) and may be more
massive than their later-stage counterparts (Tobin et al., 2020). If we adopt
a representative accretion rate, $10^{-6}\rm\,M_{\sun}/yr$, we would increase
the strength of magnetic field by one order of magnitude, compared with the
value adopted in Eq. (19). It is possible that Class 0 disks are also more
massive than Class I/II disks. For example, Tobin et al. (2020) reported the
mass of Class 0 and Class I disks has mean dust mass of $25.9M_{\oplus}$ and
$14.9M_{\oplus}$, respectively. If we assume the mass of the Class 0 disk is
the same as the one adopted in Sec. 5 – hence the density is also the same –
then applying the fast Larmor precession criterion yields
$\hat{\chi}/a_{\mathrm{mm}}=10^{3}$. Valdivia et al. (2019) infer that the
dust grains in Class 0 sources have grown to at least $10\rm\,\mu$m. Such dust
grains require at least $\hat{\chi}=10$ to have fast Larmor precession, which
requires minimal SPI enhancement to achieve. At the same time, $10\rm\,\mu$m
dust grains are less efficient at producing polarization through self-
scattering at (sub)millimeter wavelengths, avoiding a known confounding effect
for studying magnetic fields (Kataoka et al., 2015). Class 0 disks are thus
reasonable targets for detecting magnetic fields through spatially resolved
(sub)millimeter polarimetry.

##  7 Summary

In this paper, we discussed the feasibility of magnetic alignment of dust
grains with superparamagnetic inclusions in protoplanetary disks. The major
results are summarized as follow:

  1. 1.

Under the Néel’s relaxation theory, we show that there exists a critical size
of SPIs within dust grains. SPIs larger than this critical size cannot respond
to external magnetic fields and do not contribute to the magnetic
susceptibility of the grain. There is thus a maximum enhancement that SPIs can
provide to a grain’s magnetic susceptibility, and therefore a corresponding
maximum reduction factor for the Larmor precession timescale of the grain.

  2. 2.

We explore the effect on magnetic susceptibility for a dust grain containing
an ensemble of SPIs having a power-law size distribution. We find that if the
ensemble contains SPIs bigger than the threshold $N_{\mathrm{cr}}$, the grain
as a whole will be unlikely to achieve the maximum magnetic susceptibility
$\chi_{\mathrm{max}}$ (Eq. 13) without fine-tuning (i.e., all SPIs in the dust
grain would need to have sizes close to the maximum cluster size). This is
because SPIs larger than the critical size do not respond to external magnetic
fields and do not contribute to the ensemble magnetic susceptibility (Eq. 12).

  3. 3.

We estimate the maximum sizes for SPIs composed of several plausible
ferromagnetic materials, given their magnetic anisotropy constant at the
temperature of astronomical interest ($25$ K). Our results are tabulated in
Table 1. We suggest that $10^{3}$ is a more realistic upper limit for the SPI
enhancement factor $\hat{\chi}$. This value is two orders of magnitude smaller
than that obtained from previous work, implying that magnetic alignment is
more difficult than previously thought.

  4. 4.

We explore the feasibility of magnetic grain alignment in the disk midplane of
a MMSN model for a PPD. We find that (1) magnetic alignment is impossible
unless SPIs are ubiquitous throughout the disk, even if the dust grains are as
small as $10\rm\,\mu m$; (2) it is difficult to align grains larger than 100
$\rm\mu m$ even with SPIs, particularly at small orbital radii where the high
gas density leads to short damping timescales.

We conclude that large millimeter-sized dust grains in the midplane of PPDs
are unlikely to be aligned with the ambient magnetic fields (c.f. Fig. 3). An
important implication of this finding is that observations that are primarily
sensitive to the emission from this population of grains – such as
(sub)millimeter-wavelength polarimetric observations of Class I/II PPDs – are
unlikely to be tracing the disk magnetic field structure. We suggest instead
that observations of disk surfaces – probed by mid-infrared polarimetry – and
the early Class 0 PPDs are better-suited to studying the magnetic fields
structures in PPDs.

## Acknowledgements

We thank the anonymous referee for detailed and constructive comments that
greatly improved the manuscript. We thank Xuening Bai, Zhi-Yun Li, Daniel
Harsono, and Vincent Guillet for fruitful discussions. We thank Zheng Liu,
Zhiyuan Yao and Shuai Yin for discussions over the physics behind the
superparamagnetism. We thank Dominic Pesce, Zhi-Yun Li, and Xuening Bai for
comments and suggestions that helped to improve the manuscript.

##  Appendix A A simple model for SPI dynamics

In order to see how the magnetic susceptibility of an ensemble of identical
SPIs changes as a function of their volume, we use a simplified model with
three-axis assumption: all the SPIs have their easy axis along one of an
arbitrary set of Cartesian axes and they are split evenly among three axes.
Also, we are interested in the regime where $V>kT/K$, so the equilibrium state
have averaged magnetic moment as $\mu^{2}B/kT$ along easy axis and $0$ along
perpendicular directions. Initially, all the SPIs have their magnetic moment
randomly distributed, so that there are equal numbers of $\mu$’s along the six
directions ($\pm x,\pm y,$ and $\pm z$).

Now let’s apply magnetic field $B$ along $z$ axis at $t=0$. Since the Barnett
equivalent field (for thermal angular velocity of a $10\rm\,\mu m$ dust grain)
$H_{B}=\Omega/\gamma\sim~{}10^{-7}\rm\,Oersted$ and the corresponding magnetic
energy is at most on the order of $10^{5}\mu_{B}H_{B}/k\sim 10^{-7}\rm\,K$,
which is very small compared with the thermal energy (on the order of $10$ K).
We conclude that the magnetic field cannot overcome the anisotropy energy on
its own, which is even bigger than the thermal energy. We need the Brownian-
like process Néel proposed that each magnetic moment tries to change its
orientation every $t_{0}$. Since the SPIs with easy axis along $x$ or $y$
direction cannot response to the magnetic field along $z$ direction, we focus
on those with easy axis along $z$ axis, and define $f_{+}$ and $f_{-}$ as
fraction of particles along $z+$ and $z-$, respectively. The energies for
these two states are $-\mu B$ and $\mu B$, but there is an energy barrier of
$KV$ if the magnetic moment tries to switch to the opposite state. As such,
the probability of one successful transition is:

$\left\\{\begin{split}P(z+\to z-)=\exp\left(-\frac{KV+\mu B}{kT}\right)\\\
P(z-\to z+)=\exp\left(-\frac{KV-\mu B}{kT}\right)\end{split}\right..$ (A1)

The dynamical equation for $(f_{+},f_{-})$ is then:

$\frac{d}{dt}\left(\begin{array}[]{c}f_{+}\\\
f_{-}\end{array}\right)=\frac{1}{t_{0}}\left(\begin{array}[]{cc}-\exp\left(-\frac{KV+\mu
B}{kT}\right)&\exp\left(-\frac{KV-\mu B}{kT}\right)\\\ \exp\left(-\frac{KV+\mu
B}{kT}\right)&-\exp\left(-\frac{KV-\mu B}{kT}\right)\\\
\end{array}\right)\left(\begin{array}[]{c}f_{+}\\\ f_{-}\end{array}\right).$
(A2)

The solution to this equation is:

$\left(\begin{array}[]{c}f_{+}\\\
f_{-}\end{array}\right)=\exp\left(\frac{t}{t_{N}}\mathcal{M}\right)\left(\begin{array}[]{c}f_{+,0}\\\
f_{-,0}\end{array}\right),$ (A3)

where $t_{N}$ is the Néel’s relaxation timescale, and $\mathcal{M}$ is a
matrix defined as
$\mathcal{M}\equiv[[-e^{-\beta},e^{\beta}],[e^{-\beta},-e^{\beta}]]$, with
$\beta\equiv\mu B/kT$.

It is possible to calculate the above matrix exponential explicitly with the
aid of the following matrix transformation:

$\mathcal{M}=\mathcal{P}\left(\begin{array}[]{cc}0&0\\\ 0&-2\cosh{\beta}\\\
\end{array}\right)\mathcal{P}^{-1},$ (A4)

with $\mathcal{P}\equiv[[e^{\beta},1],[e^{-\beta},-1]]$. Eq. (A3) can then be
rewritten as:

$\left(\begin{array}[]{c}f_{+}\\\
f_{-}\end{array}\right)=\mathcal{P}\left(\begin{array}[]{cc}1&0\\\
0&\exp\left(-\frac{2t}{t_{N}}\cosh{\beta}\right)\\\
\end{array}\right)\mathcal{P}^{-1}\left(\begin{array}[]{c}f_{+,0}\\\
f_{-,0}\end{array}\right).$ (A5)

Now let’s consider the two limiting cases. For $t\gg t_{N}$, the exponential
in Eq. (A5) is basically $0$. We have:

$\left(\begin{array}[]{c}f_{+}\\\
f_{-}\end{array}\right)=\frac{f_{+,0}+f_{-,0}}{2\cosh\beta}\left(\begin{array}[]{c}e^{\beta}\\\
e^{-\beta}\end{array}\right).$ (A6)

This means the end state is independent of initial state and the system has
reached equilibrium (we always have $f_{+,0}+f_{-,0}=1$). In this case, the
averaged magnetic moment along magnetic field direction is
$\left<\mu_{z}\right>=f_{+}\mu-f_{-}\mu=\mu\tanh\beta\approx\mu^{2}B/kT$. The
magnetic susceptibility of this ensemble is thus the same as the
superparamagnetic magnetic susceptibility $\chi_{sp}$ in Eq. (7) (remember
that there are two third of SPIs with perpendicular easy axes and do not
contribute to the magnetic susceptibility).

For $t\ll t_{N}$, we have (this is easier to derive directly from Eq. A3):

$\left(\begin{array}[]{c}f_{+}\\\
f_{-}\end{array}\right)\approx\left(\mathcal{I}+\frac{t}{t_{N}}\mathcal{M}\right)\left(\begin{array}[]{c}f_{+,0}\\\
f_{-,0}\end{array}\right)=\left(\begin{array}[]{c}f_{+,0}\\\
f_{-,0}\end{array}\right)+\frac{t}{t_{N}}\left(\begin{array}[]{c}-f_{+,0}e^{-\beta}+f_{-,0}e^{\beta}\\\
f_{+,0}e^{-\beta}-f_{-,0}e^{\beta}\end{array}\right).$ (A7)

From the end result, we can see that a fraction of $(t/t_{N})e^{-\beta}$ SPIs
have changed from $+$ state to $-$ state, whereas a fraction of
$(t/t_{N})e^{\beta}$ SPIs have changed from $-$ state to $+$ state. In this
case, for an initial condition $f_{+,0}=f_{-,0}=0.5$, we get
$\left<\mu_{z}\right>=f_{+}\mu-
f_{-}\mu=(2t\mu/t_{N})\sinh\beta\approx(2t/t_{N})\mu^{2}B/kT$.

Figure 4: The averaged magnetic moment along magnetic field direction
$\left<\mu_{z}\right>$ of SPIs with magnetic moment $\mu$ and Néel’s
relaxation timescale $t_{N}$, as a function of time $t$ exposed in an external
magnetic field $B$. The SPIs considered here all have their easy axis along
the magnetic field direction $\pm z$.

The accurate solution, together with the above two asymptotic solutions, is
plotted in Fig. 4. $\mu_{z}$ linearly increases to its equilibrium value.
However, since the Néel’s relaxation timescale increases exponentially with
the volume $V$, the magnetic susceptibility of the ensemble of SPIs changes
with volume $V$ very sharply across the critical volume defined in Eq. (10)
which, to a good approximation, can be summarized as follows:

$\chi=\left\\{\begin{array}[]{ll}0,&V>V_{\mathrm{cr}}\\\
\chi_{sp},&V<V_{\mathrm{cr}}\end{array}\right..$ (A8)

## References

  * Abe et al. (1976) Abe, K., Miyamoto, Y., & Chikazumi, S. 1976, Journal of the Physical Society of Japan, 41, 1894, doi: 10.1143/JPSJ.41.1894
  * Alves et al. (2018) Alves, F. O., Girart, J. M., Padovani, M., et al. 2018, A&A, 616, A56, doi: 10.1051/0004-6361/201832935
  * Andersson et al. (2015) Andersson, B. G., Lazarian, A., & Vaillancourt, J. E. 2015, ARA&A, 53, 501, doi: 10.1146/annurev-astro-082214-122414
  * Bacciotti et al. (2018) Bacciotti, F., Girart, J. M., Padovani, M., et al. 2018, ApJ, 865, L12, doi: 10.3847/2041-8213/aadf87
  * Bai (2011) Bai, X.-N. 2011, ApJ, 739, 50, doi: 10.1088/0004-637X/739/1/50
  * Barnett (1915) Barnett, S. J. 1915, Physical Review, 6, 239, doi: 10.1103/PhysRev.6.239
  * Bean & Livingston (1959) Bean, C. P., & Livingston, J. D. 1959, Journal of Applied Physics, 30, S120, doi: 10.1063/1.2185850
  * Billas et al. (1994) Billas, I. M., Châtelain, A., & de Heer, W. A. 1994, Science, 265, 1682, doi: 10.1126/science.265.5179.1682
  * Chiang & Goldreich (1997) Chiang, E. I., & Goldreich, P. 1997, ApJ, 490, 368, doi: 10.1086/304869
  * Coey & Khalafalla (1972) Coey, J. M. D., & Khalafalla, D. 1972, Phys. Stat. Sol. (a), 11, 229
  * Cornell & Schwertmann (2003) Cornell, R. M., & Schwertmann, U. 2003, The Iron Oxides: Structure, Properties, Reactions, Occurences and Uses (Wiley‐VCH Verlag GmbH & Co. KGaA)
  * Cox et al. (2018) Cox, E. G., Harris, R. J., Looney, L. W., et al. 2018, ApJ, 855, 92, doi: 10.3847/1538-4357/aaacd2
  * Dai & Qian (2017) Dai, D., & Qian, K. 2017, Ferromagnetism (in Chinese) (Beijing: Science Press)
  * Davis & Greenstein (1951) Davis, Jr., L., & Greenstein, J. L. 1951, ApJ, 114, 206, doi: 10.1086/145464
  * Dent et al. (2019) Dent, W. R. F., Pinte, C., Cortes, P. C., et al. 2019, MNRAS, 482, L29, doi: 10.1093/mnrasl/sly181
  * Dolginov & Mytrophanov (1976) Dolginov, A. Z., & Mytrophanov, I. G. 1976, Ap&SS, 43, 257, doi: 10.1007/BF00640009
  * Draine (1996) Draine, B. T. 1996, Astronomical Society of the Pacific Conference Series, Vol. 97, Optical and Magnetic Properties of Dust Grains, ed. W. G. Roberge & D. C. B. Whittet, 16
  * Draine (2004) Draine, B. T. 2004, Astrophysics of Dust in Cold Clouds, ed. D. Pfenniger & Y. Revaz (Berlin, Heidelberg: Springer Berlin Heidelberg), 213–304.  https://doi.org/10.1007/3-540-31636-1_3
  * Draine & Hensley (2013) Draine, B. T., & Hensley, B. 2013, ApJ, 765, 159, doi: 10.1088/0004-637X/765/2/159
  * Draine & Weingartner (1997) Draine, B. T., & Weingartner, J. C. 1997, ApJ, 480, 633, doi: 10.1086/304008
  * Dutrey et al. (2014) Dutrey, A., Semenov, D., Chapillon, E., et al. 2014, in Protostars and Planets VI, ed. H. Beuther, R. S. Klessen, C. P. Dullemond, & T. Henning, 317
  * Goodman & Whittet (1995) Goodman, A. A., & Whittet, D. C. B. 1995, ApJ, 455, L181, doi: 10.1086/309840
  * Hiltner (1949) Hiltner, W. A. 1949, Nature, 163, 283, doi: 10.1038/163283a0
  * Hoang (2017) Hoang, T. 2017, arXiv e-prints, arXiv:1704.01721.  https://arxiv.org/abs/1704.01721
  * Hoang et al. (2018) Hoang, T., Cho, J., & Lazarian, A. 2018, ApJ, 852, 129, doi: 10.3847/1538-4357/aa9edc
  * Hull & Zhang (2019) Hull, C. L. H., & Zhang, Q. 2019, Frontiers in Astronomy and Space Sciences, 6, 3, doi: 10.3389/fspas.2019.00003
  * Hull et al. (2018) Hull, C. L. H., Yang, H., Li, Z.-Y., et al. 2018, ApJ, 860, 82, doi: 10.3847/1538-4357/aabfeb
  * Iizumi et al. (1982) Iizumi, M., Koetzle, T. F., Shirane, G., et al. 1982, Acta Cryst., B38, 2121
  * Jones & Spitzer (1967) Jones, R. V., & Spitzer, Lyman, J. 1967, ApJ, 147, 943, doi: 10.1086/149086
  * Kataoka et al. (2017) Kataoka, A., Tsukagoshi, T., Pohl, A., et al. 2017, ApJ, 844, L5, doi: 10.3847/2041-8213/aa7e33
  * Kataoka et al. (2015) Kataoka, A., Muto, T., Momose, M., et al. 2015, ApJ, 809, 78
  * Kataoka et al. (2016) Kataoka, A., Tsukagoshi, T., Momose, M., et al. 2016, ApJ, 831, L12
  * Lazarian (2007) Lazarian, A. 2007, J. Quant. Spec. Radiat. Transf., 106, 225, doi: 10.1016/j.jqsrt.2007.01.038
  * Lazarian & Hoang (2007) Lazarian, A., & Hoang, T. 2007, MNRAS, 378, 910, doi: 10.1111/j.1365-2966.2007.11817.x
  * Li et al. (2016) Li, D., Pantin, E., Telesco, C. M., et al. 2016, ApJ, 832, 18, doi: 10.3847/0004-637X/832/1/18
  * Li et al. (2007) Li, Y., Montano, P., Barbiellini, B., et al. 2007, Journal of Physics and Chemistry of Solids, 68, 1556 , doi: https://doi.org/10.1016/j.jpcs.2007.03.037
  * Lin et al. (2019) Lin, Z.-Y. D., Li, Z.-Y., Yang, H., et al. 2019, arXiv e-prints, arXiv:1912.10012.  https://arxiv.org/abs/1912.10012
  * Mathewson & Ford (1970) Mathewson, D. S., & Ford, V. L. 1970, MmRAS, 74, 139
  * Mathis (1986) Mathis, J. S. 1986, ApJ, 308, 281, doi: 10.1086/164499
  * Morrish (2001) Morrish, A. H. 2001, The Physical Principles of Magnetism (Wiley-IEEE Press)
  * Néel (1949) Néel, L. 1949, Ann. Géophys., 5, 99
  * Ohashi & Kataoka (2019) Ohashi, S., & Kataoka, A. 2019, ApJ, 886, 103, doi: 10.3847/1538-4357/ab5107
  * Ohashi et al. (2018) Ohashi, S., Kataoka, A., Nagai, H., et al. 2018, ApJ, 864, 81, doi: 10.3847/1538-4357/aad632
  * Pisane et al. (2017) Pisane, K. L., Singh, S., & Seehra, M. S. 2017, Applied Physics Letters, 110, 222409, doi: 10.1063/1.4984903
  * Planck Collaboration et al. (2015) Planck Collaboration, Ade, P. A. R., Aghanim, N., et al. 2015, A&A, 576, A104, doi: 10.1051/0004-6361/201424082
  * Purcell (1979) Purcell, E. M. 1979, ApJ, 231, 404, doi: 10.1086/157204
  * Roberge et al. (1993) Roberge, W. G., Degraff, T. A., & Flaherty, J. E. 1993, ApJ, 418, 287, doi: 10.1086/173390
  * Stephens et al. (2014) Stephens, I. W., Looney, L. W., Kwon, W., et al. 2014, Nature, 514, 597
  * Stephens et al. (2017) Stephens, I. W., Yang, H., Li, Z.-Y., et al. 2017, ApJ, 851, 55, doi: 10.3847/1538-4357/aa998b
  * Tazaki et al. (2017) Tazaki, R., Lazarian, A., & Nomura, H. 2017, ApJ, 839, 56, doi: 10.3847/1538-4357/839/1/56
  * Tobin et al. (2020) Tobin, J. J., Sheehan, P. D., Megeath, S. T., et al. 2020, ApJ, 890, 130, doi: 10.3847/1538-4357/ab6f64
  * Valdivia et al. (2019) Valdivia, V., Maury, A., Brauer, R., et al. 2019, MNRAS, 488, 4897, doi: 10.1093/mnras/stz2056
  * Verwey (1939) Verwey, E. J. W. 1939, Nature, 144, 327
  * Walz (2002) Walz, F. 2002, Journal of Physics: Condensed Matter, 14, R285, doi: 10.1088/0953-8984/14/12/203
  * Weidenschilling (1977) Weidenschilling, S. J. 1977, Ap&SS, 51, 153, doi: 10.1007/BF00642464
  * Yang et al. (2016a) Yang, H., Li, Z.-Y., Looney, L., & Stephens, I. 2016a, MNRAS, 456, 2794
  * Yen et al. (2017) Yen, H.-W., Koch, P. M., Takakuwa, S., et al. 2017, The Astrophysical Journal, 834, 178, doi: 10.3847/1538-4357/834/2/178