# Interaction of light with subwavelength particles: Revealing the physics of
electric dipole moment in the classical scattering problem

Yuriy A. Akimov<EMAIL_ADDRESS>Institute of High Performance
Computing (IHPC), Agency for Science, Technology and Research (A*STAR),
1 Fusionopolis Way, #16-16 Connexis, Singapore 138632

###### Abstract

Scattering problems are the classical tools for modeling of light-matter
interaction. In this paper, we investigate the solution of dipole scattering
problem under different incident radiation. In particular, we compare the two
cases of incident plane and spherically incoming fields. With this comparison,
we disclose the two distinct groups of current-sourced and current-free
scattered fields which exhibit independent dynamics and dissimilar effects of
the scatterer. We demonstrate how these fields by interfering each other make
the resultant electric dipole moment of the scattered fields resonant and,
thus, give rise to all the spectral features observed in the classical
solution for dipole scattering of light.

##  I Introduction

Classical scattering problems play a key part in our understanding of light-
matter interaction. Scattering by subwavelength particles explains the blue
color of the daytime sky and the reddish color of the low Sun, as well as
enables the qualitative description of matter polarization at the atomic scale
[1, 2, 3]. Despite the wide use of scattering problems, their solutions still
keep a number of unanswered fundamental questions. For instance, subwavelength
particles are known to exhibit frequency-selective effects such as strong
extinction [3, 4] or nonradiating anapole states [6, 5]. These effects are
commonly explained by the interference of the particle’s electric dipole and
higher-order multipole currents induced by the incident fields [2, 7]. Such
explanation, however, does not hold for deeply subwavelength metal particles,
which exhibit the same resonant effects [8, 9], but with negligible higher-
order multipole currents. This suggests that the resonances observed in
classical scattering problems are caused by other mechanisms which do not
require higher-order currents and can work even on the level of a single
electric dipole moment. To address this issue, we reconsider the classical
quasi-static solution of dipole scattering by including the radiative decay to
the scattered fields that allows us to look at the solution from a new
physically wider perspective.

##  II Dipole scattering

Let us consider the quasi-static regime of scattering [1, 3, 2] given by
Maxwell’s equations for a spherical particle of radius $R$ and dielectric
permittivity $\varepsilon_{i}$ embedded in a material of dielectric
permittivity $\varepsilon_{e}$. If the particle is exposed to a harmonic
incident electric field ${\bf E}_{\rm inc}$, then the total electric field
outside the particle, ${\bf E}(r>R)={\bf E}_{\rm ext}$, is given by the
superposition of the incident ${\bf E}_{\rm inc}$ and scattered ${\bf E}_{\rm
sca}$ fields:

${\bf E}_{\rm ext}={\bf E}_{\rm inc}+{\bf E}_{\rm sca}.$ (1)

For deeply subwavelength particles with $|k_{e}|R\ll 1$, where
$k_{e}=2\pi\sqrt{\varepsilon_{e}}/\lambda$ is the external medium wavenumber
at the vacuum wavelength $\lambda$, the scattered field is dominated by the
quasi-static dipolar fields [1, 3, 2]. At small distances $r$ compared to the
wavelength $\lambda$, the scattered field is given by

$\displaystyle{\bf E}_{\rm sca}=AE_{0}\frac{R^{3}}{r^{3}}[{\bf e}_{r}(2+{\rm
i}f_{\rm rad}(r))\cos\theta+\hskip 28.45274pt$ $\displaystyle\vspace{3cm}{\bf
e}_{\theta}(1-{\rm i}f_{\rm rad}(r))\sin\theta]\mathop{\rm e}\nolimits^{-{\rm
i}\omega t},$ (2)

where $A$ is the scattered field amplitude (see Appendices A and B for
derivation) yet to be determined, $E_{0}$ is the amplitude of the incident
field assumed to be $z$-polarized, $\omega=2\pi c/\lambda$ is the angular
frequency of the incident field with $c$ being the speed of light,
$(r,\theta,\phi)$ are the spherical coordinates, and ${\bf e}_{r},{\bf
e}_{\theta},{\bf e}_{\phi}$ are the respective spherical unit vectors. In
addition to the dipolar decay, ${\bf E}_{\rm sca}\propto r^{-3}$, prevailing
in the scattered field, we also account for the radiative decay given by small
imaginary contributions with $f_{\rm rad}(r)=(2/3)(k_{e}r)^{3}\ll 1$ being the
leading radiative terms in the full-wave solution of dipolar fields, as shown
in Appendix B. The dominance of the dipolar fields for small spheres holds for
the electric field excited inside the particle, ${\bf E}(r<R)={\bf E}_{\rm
int}$, if $|k_{i}|R\ll 1$ with $k_{i}=2\pi\sqrt{\varepsilon_{i}}/\lambda$. In
this case, ${\bf E}_{\rm int}$ is given by the uniform $z$-polarized field:

${\bf E}_{\rm int}=BE_{0}({\bf e}_{r}\cos\theta-{\bf
e}_{\theta}\sin\theta)\mathop{\rm e}\nolimits^{-{\rm i}\omega t},$ (3)

where $B$ is the internal field amplitude (see Appendix B for details). The
unknown amplitudes $A$ and $B$ in Eqs. (2) and (3) can be determined by
applying the boundary conditions consisting of the continuity of (i) the
normal component ${\bf e}_{r}\cdot\varepsilon_{0}\varepsilon{\bf E}$ of the
electric displacement field, where the space-dependent permittivity
$\varepsilon$ is given by $\varepsilon_{i}$ at $r<R$ and $\varepsilon_{e}$ at
$r>R$, and (ii) the tangential components ${\bf e}_{\theta}\cdot{\bf E}$ and
${\bf e}_{\phi}\cdot{\bf E}$ of the electric field at the sphere interface
$r=R$.

As the fields scattered by small particles are dipolar by their composition,
they can be considered as the fields emitted by a point electric dipole with
the effective moment of

${\bf p}_{\rm eff}={\bf
e}_{z}4\pi\varepsilon_{0}\varepsilon_{e}AR^{3}E_{0}\mathop{\rm
e}\nolimits^{-{\rm i}\omega t}.$ (4)

Then, the processes underlying light scattering by small spheres, such as
field energy or momentum transfer, can be described in terms of the effective
electric dipole moment ${\bf p}_{\rm eff}$ [3, 2].

##  III Scattering current approach

An alternative to the effective dipole description is the scattering current
approach [2, 7]. If the scattered field ${\bf E}_{\rm sca}$ is defined in the
entire space, i.e. when ${\bf E}={\bf E}_{\rm inc}+{\bf E}_{\rm sca}$ for both
the internal and external domains, then Maxwell’s equations can be reduced to

$\displaystyle\nabla\times\nabla\times{\bf E}_{\rm
sca}-\omega^{2}\mu_{0}\varepsilon_{0}\varepsilon_{e}{\bf E}_{\rm sca}={\rm
i}\omega\mu_{0}{\bf J}_{\rm sca},$ (5)

with the current density

$\displaystyle{\bf J}_{\rm sca}=-{\rm
i}\omega\varepsilon_{0}(\varepsilon-\varepsilon_{e}){\bf E},$ (6)

serving as the source of the scattered fields excited at angular frequency
$\omega$, where $\varepsilon_{0}$ and $\mu_{0}$ are the electric and magnetic
constants. Noteworthy that the scattering current density ${\bf J}_{\rm sca}$
appears nonzero inside the sphere only, where the total field ${\bf E}$ is
given by the uniform ${\bf E}_{\rm int}$. If we decompose the current density
${\bf J}_{\rm sca}$ over different-order moments, the latter can be used for
description of the scattered fields and optical effects associated with them.
For deeply subwavelength spheres with $|k_{e}|R,|k_{i}|R\ll 1$, the scattering
current is dominated by the electric dipole moment

$\displaystyle{\bf p}_{\rm sca}$ $\displaystyle=$
$\displaystyle\displaystyle\frac{{\rm i}}{\omega}\int{\bf J}_{\rm sca}\;{\rm
d}^{3}r=$ (7) $\displaystyle\displaystyle{\bf
e}_{z}\frac{4\pi}{3}\varepsilon_{0}(\varepsilon_{i}-\varepsilon_{e})BR^{3}E_{0}\mathop{\rm
e}\nolimits^{-{\rm i}\omega t}.$

The moment ${\bf p}_{\rm sca}$ is widely used along with ${\bf p}_{\rm eff}$
as a measure of the dipolar fields scattered by small particles, assuming that
the two moments coincide.

##  IV Scattering of plane fields

Although the scattering current approach has gotten wide acceptance in the
optics community, its hypothesis of equal scattering and effective dipole
moments remains unproven. To test it, we can consider dipole scattering of
plane incident fields (see Appendix B for details):

${\bf E}^{\rm pl}_{\rm inc}=E_{0}({\bf e}_{r}\cos\theta-{\bf
e}_{\theta}\sin\theta)\mathop{\rm e}\nolimits^{-{\rm i}\omega t},$ (8)

by deeply subwavelength spheres. Application of the boundary conditions for
electric fields at the sphere interface gives us the following amplitudes of
the scattered and internal fields:

$\displaystyle A^{\rm pl}=A({\bf E}_{\rm inc}={\bf E}^{\rm pl}_{\rm
inc})=\frac{A_{\rm qs}}{1-{\rm i}A_{\rm qs}f_{\rm rad}(R)},$ (9)
$\displaystyle B^{\rm pl}=B({\bf E}_{\rm inc}={\bf E}^{\rm pl}_{\rm
inc})=\frac{B_{\rm qs}}{1-{\rm i}A_{\rm qs}f_{\rm rad}(R)},$ (10)

where

$\displaystyle A_{\rm
qs}=\frac{\varepsilon_{i}-\varepsilon_{e}}{\varepsilon_{i}+2\varepsilon_{e}},\quad
B_{\rm qs}=\frac{3\varepsilon_{e}}{\varepsilon_{i}+2\varepsilon_{e}},$ (11)

are the quasi-static amplitudes commonly obtained in the limiting case of
$f_{\rm rad}(R)\rightarrow 0$ [1, 3, 2].

The scattered fields and scattering currents derived result in the equal
effective and scattering electric dipole moments:

${\bf p}^{\rm pl}_{\rm eff}={\bf p}^{\rm pl}_{\rm sca}={\bf
e}_{z}\frac{4\pi\varepsilon_{0}\varepsilon_{e}R^{3}E_{0}A_{\rm qs}}{1-{\rm
i}A_{\rm qs}f_{\rm rad}(R)}\mathop{\rm e}\nolimits^{-{\rm i}\omega t}.$ (12)

The coincidence of ${\bf p}^{\rm pl}_{\rm eff}$ and ${\bf p}^{\rm pl}_{\rm
sca}$ is commonly interpreted as the validation of the scattering current
approach [3]. Eventually, this approach is widely used, e.g., in discrete
dipole approximation for simulation of scattering by large objects of
arbitrary shape [10, 11, 12, 13], in multipole current decomposition of
scattering peculiarities, such as nonradiating anapole configurations, bright
and dark mode resonances, etc. [14, 15, 16, 18, 19, 20, 17, 21, 22].

##  V Scattering of spherically incoming fields

Despite the demonstrated ${\bf p}^{\rm pl}_{\rm eff}={\bf p}^{\rm pl}_{\rm
sca}$, the general assumption of the scattering current approach, ${\bf
p}_{\rm eff}={\bf p}_{\rm sca}$, for an arbitrary incident field remains an
unproven hypothesis. To prove it, Eq. (5) must be solved in a general form for
an arbitrary incident field. Since such a solution is not as straightforward,
the correlation between ${\bf p}_{\rm eff}$ and ${\bf p}_{\rm sca}$ remains
under question and may potentially not hold for nonplane incident fields. To
further test it, we change the incident plane fields to spherically incoming
dipolar fields (see Appendices B for details):

$\displaystyle{\bf E}^{\rm sph,in}_{\rm inc}=\frac{{\rm i}E_{0}}{2f_{\rm
rad}(r)}[{\bf e}_{r}(2-{\rm i}f_{\rm rad}(r))\cos\theta+\hskip 28.45274pt$
$\displaystyle{\bf e}_{\theta}(1+{\rm i}f_{\rm rad}(r))\sin\theta]\mathop{\rm
e}\nolimits^{-{\rm i}\omega t}.$ (13)

In this case, we obtain the following amplitudes of the scattered and internal
fields:

$\displaystyle A^{\rm sph,in}$ $\displaystyle=$ $\displaystyle A({\bf E}_{\rm
inc}={\bf E}^{\rm sph,in}_{\rm inc})=$ (14)
$\displaystyle\displaystyle\frac{1+{\rm i}A_{\rm qs}f_{\rm rad}(R)}{2{\rm
i}f_{\rm rad}(R)[1-{\rm i}A_{\rm qs}f_{\rm rad}(R)]},$ $\displaystyle B^{\rm
sph,in}$ $\displaystyle=$ $\displaystyle\displaystyle B({\bf E}_{\rm inc}={\bf
E}^{\rm sph,in}_{\rm inc})=\frac{B_{\rm qs}}{1-{\rm i}A_{\rm qs}f_{\rm
rad}(R)}.$ (15)

The scattered fields result in the effective electric dipole moment

$\displaystyle{\bf p}^{\rm sph,in}_{\rm eff}={\bf
e}_{z}\frac{2\pi\varepsilon_{0}\varepsilon_{e}R^{3}E_{0}[1+{\rm i}A_{\rm
qs}f_{\rm rad}(R)]}{{\rm i}f_{\rm rad}(R)[1-{\rm i}A_{\rm qs}f_{\rm
rad}(R)]}\mathop{\rm e}\nolimits^{-{\rm i}\omega t},$ (16)

while the scattering current gives us

$\displaystyle{\bf p}^{\rm sph,in}_{\rm sca}={\bf
e}_{z}\frac{4\pi\varepsilon_{0}\varepsilon_{e}R^{3}E_{0}A_{\rm qs}}{1-{\rm
i}A_{\rm qs}f_{\rm rad}(R)}\mathop{\rm e}\nolimits^{-{\rm i}\omega t}.$ (17)

The observed mismatch, ${\bf p}^{\rm sph,in}_{\rm eff}\neq{\bf p}^{\rm
sph,in}_{\rm sca}$, demonstrates that the scattering current approach
generally fails for nonplane incident fields. This failure can be understood
from Eq. (6) considered from the point of view of the relation between the
scattering current density ${\bf J}_{\rm sca}$ and the scattered field ${\bf
E}_{\rm sca}$. Following Eq. (6), the zero scattering current ${\bf J}_{\rm
sca}$ is associated with the null total field ${\bf E}$, but not the null
scattered field ${\bf E}_{\rm sca}$. This means that Maxwell’s equation
generally support existence of the scattered fields which are not current-
sourced.

Figure 1: Distribution of dipolar electric fields at $\lambda=400$ nm given by
the quasi-static solution for (a) planar and (b) spherically incoming fields
incident on a silver particle of $R=10$ nm embedded in SiO2. (c) Difference of
the fields plotted in (a) and (b).

The paradox of source-free scattered fields comes from the formulation of
scattering problems. In addition to the conventional electrodynamic
formulation, following which all electromagnetic fields are generated by
electric currents, scattering problems introduce the so-called free fields
which are sourceless. Namely, the incident fields used in scattering problems
are given in the form of such source-free fields [1, 3, 2]. Eventually,
scattered fields appear composed of (i) current-sourced fields that can be
treated in terms of their excitation currents and (ii) source-free fields
accompanied by the zero induced current. The latter appear in scattering
problem as a part of the trivial solutions of Maxwell’s equations given by the
null total electric field ${\bf E}$ and the null scattering current ${\bf
J}_{\rm sca}$ (see Appendices A for details). In other words, the source-free
scattered fields originate from the trivial response of Maxwell’s equations to
the introduced source-free incident fields.

Appearance of ${\bf p}^{\rm sph,in}_{\rm sca}={\bf p}^{\rm pl}_{\rm sca}$
demonstrates that scattered fields for the plane and spherical dipolar
incident waves differ from each other by the source-free fields only. Indeed,
the total field distributions for planar and spherical dipole incident waves
are exactly the same, while their scattered fields differ by the amplitude
only, as shown in Fig. 1 (a) and (b) for a silver particle of $R=10$ nm
embedded in SiO2. As a result, the field difference plotted in Fig. 1 (c)
demonstrates the source-free scattered fields accompanied by the trivial
solution for the total field ${\bf E}$ and, hence, for the scattering current
density ${\bf J}_{\rm sca}$.

##  VI Scattering of spherically outgoing fields

Now, let us separate the effects of source-free and current-sourced fields. As
we discussed above, the source-free scattered fields mathematically appear as
a part of the trivial solution of Maxwell’s equations, where they fully
compensate a part of the incident fields. As the scattered fields are
spherically outgoing, the compensated part of the incident field must be
spherically outgoing as well. To demonstrate the purely source-free scattered
fields, we consider scattering of the outgoing dipolar spherical field (see
Appendices B for details):

$\displaystyle{\bf E}^{\rm sph,out}_{\rm inc}=\frac{-{\rm i}E_{0}}{2f_{\rm
rad}(r)}[{\bf e}_{r}(2+{\rm i}f_{\rm rad}(r))\cos\theta+\hskip 28.45274pt$
$\displaystyle{\bf e}_{\theta}(1-{\rm i}f_{\rm rad}(r))\sin\theta]\mathop{\rm
e}\nolimits^{-{\rm i}\omega t}~{}~{}$ (18)

that does not contain any spherically incoming component. After application of
the boundary conditions, we obtain the following amplitudes of the scattered
and internal fields:

$\displaystyle\displaystyle A^{\rm sph,out}=A({\bf E}_{\rm inc}={\bf E}^{\rm
sph,out}_{\rm inc})=\frac{{\rm i}}{2f_{\rm rad}(R)},$ (19)
$\displaystyle\displaystyle B^{\rm sph,out}=B({\bf E}_{\rm inc}={\bf E}^{\rm
sph,out}_{\rm inc})=0,$ (20)

revealing the nonzero source-free scattered field with the effective electric
dipole moment

$\displaystyle{\bf p}^{\rm sph,out}_{\rm eff}=-{\bf
e}_{z}\frac{2\pi\varepsilon_{0}\varepsilon_{e}R^{3}E_{0}}{{\rm i}f_{\rm
rad}(R)}\mathop{\rm e}\nolimits^{-{\rm i}\omega t}$ (21)

and the null scattering current with the electric dipole moment

$\displaystyle{\bf p}^{\rm sph,out}_{\rm sca}=0.$ (22)

The solution obtained is trivial for electric field with the net zero total
field ${\bf E}$ everywhere in space, as shown in Fig. 1 (c). This solution
demonstrates the evident failure of the scattering current approach for
description of such type of source-free scattered fields.

The failure of the scattering current approach for description of scattered
fields comes from its limitation to electrodynamic formulation where any
electromagnetic fields must have sources in terms of their currents. By
rejecting source-free fields, the scattering current approach naturally fails
for complete description of scattering problems where free incident fields are
used. To be fully compliant with the electrodynamic formulation, the
scattering current approach must eliminate the use of free fields, including
the incident one. This requires Eq. (5) to be rewritten as follows:

$\displaystyle\nabla\times\nabla\times{\bf
E}-\omega^{2}\mu_{0}\varepsilon_{0}\varepsilon_{e}{\bf E}={\rm
i}\omega\mu_{0}{\bf J}_{\rm sca},$ (23)

where ${\bf J}_{\rm sca}$ is now appears the source of ${\bf E}$, not ${\bf
E}_{\rm sca}$. According to this equation, ${\bf p}_{\rm sca}$ should be
considered as the electric dipole moment of the total fields, not the
scattered ones. Eventually, this clarifies the equality ${\bf p}^{\rm
sph,in}_{\rm sca}={\bf p}^{\rm pl}_{\rm sca}$ observed for the plane and
spherically incoming ${\bf E}_{\rm inc}$, whose total fields ${\bf E}$ are
identical, as shown in Fig. 1.

Figure 2: Scattering and absorption cross-sections of silver nanoparticles of
$R=10$ nm embedded in silicon dioxide as functions of the wavelength of
incident plane waves. Total cross-sections are shown for comparison and
calculated based on the Mie theory that includes all types of multipole
fields. The dipolar contributions are calculated based on the electric dipole
polarizabilities $\alpha^{\rm pl}_{\rm j=0}$ and $\alpha^{\rm pl}_{\rm j\neq
0}$.

##  VII Two types of scattered fields

With the revealed structure of scattered fields, we can reconsider the
solution obtained for plane incident fields. The plane incident field given by
Eq. (8) can be decomposed over dipolar spherically incoming (13) and outgoing
(18) fields as follows:

${\bf E}^{\rm pl}_{\rm inc}={\bf E}^{\rm sph,in}_{\rm inc}+{\bf E}^{\rm
sph,out}_{\rm inc}.$ (24)

As the dipolar spherically incoming incident field does not contain any
spherically outgoing component that could be compensated with the scattered
fields, the part ${\bf E}^{\rm sph,in}_{\rm inc}$ of the plane incident field
${\bf E}^{\rm pl}_{\rm inc}$ experiences purely current-sourced scattering. At
the same time, the spherically outgoing part ${\bf E}^{\rm sph,out}_{\rm inc}$
of the plane incident field is fully compensated by the scattered fields and,
thus, experiences purely current-free scattering. In other words,
decomposition of the plane fields into spherically incoming and outgoing
fields helps us split the effective electric dipole moment of the plane
incident field into two fundamentally different moments:

${\bf p}^{\rm pl}_{\rm eff}={\bf p}^{\rm pl}_{\rm j=0}+{\bf p}^{\rm pl}_{\rm
j\neq 0}.$ (25)

where

$\displaystyle\displaystyle{\bf p}^{\rm pl}_{\rm j=0}={{\bf p}^{\rm
sph,out}_{\rm eff}}={\bf e}_{z}\varepsilon_{0}\varepsilon_{e}\alpha^{\rm
pl}_{\rm j=0}E_{0}\mathop{\rm e}\nolimits^{-{\rm i}\omega t},$ (26)
$\displaystyle\displaystyle{\bf p}^{\rm pl}_{\rm j\neq 0}={{\bf p}^{\rm
sph,in}_{\rm eff}}={\bf e}_{z}\varepsilon_{0}\varepsilon_{e}\alpha^{\rm
pl}_{\rm j\neq 0}E_{0}\mathop{\rm e}\nolimits^{-{\rm i}\omega t}$ (27)

are the electric dipole moments of the current-free and current-sourced
scattering given in terms of their polarizabilities

$\displaystyle\displaystyle\alpha^{\rm pl}_{\rm j=0}=-\frac{2\pi R^{3}}{{\rm
i}f_{\rm rad}(R)}=\frac{3{\rm i}\pi}{k_{e}^{3}},$ (28)
$\displaystyle\displaystyle\alpha^{\rm pl}_{\rm j\neq 0}=-\alpha^{\rm pl}_{\rm
j=0}\frac{1+{\rm i}A_{\rm qs}f_{\rm rad}(R)}{1-{\rm i}A_{\rm qs}f_{\rm
rad}(R)}.$ (29)

This separation is further confirmed by Mie theory [23] and generalized
Lorenz-Mie theory (see Appendix A) that provide the exact multipole solution
for scattering of electromagnetic waves on a spherical particle, following
which spherically outgoing parts of the incident fields experience current-
free scattering for any polarization, orbital and azimuthal structure.

By interfering, ${\bf p}^{\rm pl}_{\rm j=0}$ and ${\bf p}^{\rm pl}_{\rm j\neq
0}$ make the resultant electric dipole moment ${\bf p}^{\rm pl}_{\rm eff}$
resonant and, hence, define all spectral peculiarities in the scattering and
absorption cross-sections for deeply subwavelength spheres [1, 3, 2]:

$\displaystyle\displaystyle\sigma_{\rm sca}^{\rm
pl}=\frac{3\pi}{2k_{e}^{2}}\left|\frac{{\bf p}^{\rm pl}_{\rm eff}\cdot{\bf
E}_{0}^{*}}{\varepsilon_{0}\varepsilon_{e}\alpha^{\rm pl}_{\rm
j=0}|E_{0}|^{2}}\right|^{2},$ (30) $\displaystyle\displaystyle\sigma_{\rm
abs}^{\rm pl}=\frac{3\pi}{k_{e}^{2}}{\rm Re}\left(\frac{{\bf p}^{\rm pl}_{\rm
eff}\cdot{\bf E}_{0}^{*}}{\varepsilon_{0}\varepsilon_{e}\alpha^{\rm pl}_{\rm
j=0}|E_{0}|^{2}}\right)-\sigma_{\rm sca}^{\rm pl}.$ (31)

For instance, the destructive interference of the two moments, ${\bf p}^{\rm
pl}_{\rm j\neq 0}=-{\bf p}^{\rm pl}_{\rm j=0}$, results in the nonradiating
states of purely electric dipole structure that do not require toroidal or any
other higher-order moments. These states exist at either $k_{e}=0$ or
$\varepsilon_{i}=\varepsilon_{e}$. The constructive interference of the two
moments, ${\bf p}^{\rm pl}_{\rm j\neq 0}={\bf p}^{\rm pl}_{\rm j=0}$, results
in the super-radiating state that require $\varepsilon_{i}=-2\varepsilon_{e}$.

Noteworthy that amongst the two electric dipole moments only the current-
sourced one, ${\bf p}^{\rm pl}_{\rm j\neq 0}$, depends on the sphere
permittivity and, thus, solely defines its effect on the scattering and
absorption cross-sections. As $A_{\rm qs}$ is resonant to $\varepsilon_{i}$
given by the pole $(\varepsilon_{i}+2\varepsilon_{e})^{-1}$ in Eq. (11), the
effect of $\varepsilon_{i}$ on the cross-sections differs inside the resonance
and out of it.

For large wavelength out of the resonance, when $|A_{\rm qs}|\sim 1$, the
current-sourced electric dipole moment ${\bf p}^{\rm pl}_{\rm j\neq
0}\approx-{\bf p}^{\rm pl}_{\rm j=0}$. This regime, known as the Rayleigh
scattering, is influenced by the nonradiating state with $k_{e}=0$. The above
state is material-independent and common to all subwavelength particles. This
regime features $\sigma_{\rm abs}^{\rm pl},\sigma_{\rm abs}^{\rm pl}\ll\pi
R^{2}$ and can be seen in Fig. 2 for $R=10$ nm Ag nanoparticles embedded in
SiO2 at $\lambda>400$ nm when $|\varepsilon_{i}|>2\varepsilon_{e}$.

In the resonant wavelength range, where $|A_{\rm qs}|>1$, the current-sourced
electric dipole moment deviates more from its destructive value $-{\bf p}^{\rm
pl}_{\rm j=0}$, giving rise to stronger ${\bf p}^{\rm pl}_{\rm eff}$ and,
hence, scattering and absorption. This regime is influenced by the super-
radiating state and seen in Fig. 2 around the surface-plasmon-polariton
resonance at $\lambda=$395 nm with the maximum scattering and absorption
cross-sections of $0.3\pi R^{2}$ and $5.6\pi R^{2}$ when
$\varepsilon_{i}\approx-2\varepsilon_{e}$.

For smaller wavelength out of the resonance, at which $|A_{\rm qs}|$ drops
below 1, the destructive interference restores influenced by the nonradiating
state with $\varepsilon_{i}=\varepsilon_{e}$. The corresponding anapole
scattering features strong suppression of $\sigma_{\rm sca}^{\rm pl}$ and
$\sigma_{\rm abs}^{\rm pl}$, as seen in Fig. 2 at $\lambda\approx$ 318 nm when
$\varepsilon_{i}\approx\varepsilon_{e}$. Noteworthy that the quadratic
dependence of $\sigma_{\rm sca}^{\rm pl}$ on ${\bf p}_{\rm eff}^{\rm pl}$
makes its drop dipper compared to $\sigma_{\rm abs}^{\rm pl}$ that features
dominant linear dependence on ${\bf p}_{\rm eff}^{\rm pl}$. For instance, the
anapole scattering shown in Fig. 2 experiences $\sim 500$ times reduction for
$\sigma_{\rm sca}^{\rm pl}$ and only $\sim 40$ times for $\sigma_{\rm
abs}^{\rm pl}$ with respect to their resonant values.

##  VIII Conclusion

In conclusion, classical scattering problems being the model tasks that use
free electromagnetic fields for incident radiation inevitably face the issue
of free scattered fields. This issue makes the conventional current moment
analysis generally inapplicable to the scattered fields. As a result, the
independent current-sourced and current-free field moments should be used
instead. Together, the two moments completely define all the spectral
features, as has been demonstrated in the paper for the classical case of
dipole scattering by deeply subwavelength spherical particles.

##  Appendix A Generalized Lorenz-Mie theory

Generalized Lorenz-Mie theory describes scattering and absorption of free
electromagnetic fields of arbitrary structure incident on a spherical particle
[24, 25]. Following it, Maxwell’s equations

$\displaystyle\displaystyle\nabla\times{\bf
H}=\varepsilon_{0}\varepsilon\frac{{\rm\partial}{\bf E}}{{\rm\partial}t}+{\bf
J},\quad\nabla\times{\bf E}=-\mu_{0}\frac{{\rm\partial}{\bf
H}}{{\rm\partial}t},$ (32)

can be solved for electromagnetic fields excited in domains with uniform
dielectric permittivity $\varepsilon$ in terms of the transverse magnetic (TM)
and transverse electric (TE) fields as follows [23, 26]:

$\displaystyle\displaystyle{\bf H}=\left({\bf H}^{\rm TM}-{\rm
i}\frac{\lambda}{2\pi}\sqrt{\frac{\varepsilon_{0}}{\mu_{0}}}\nabla\times{\bf
E}^{\rm TE}\right)\mathop{\rm e}\nolimits^{-{\rm i}\omega t},$ (33)
$\displaystyle\displaystyle{\bf E}=\left({\bf E}^{\rm TE}+{\rm
i}\frac{\lambda}{2\pi\varepsilon}\sqrt{\frac{\mu_{0}}{\varepsilon_{0}}}\nabla\times{\bf
H}^{\rm TM}\right)\mathop{\rm e}\nolimits^{-{\rm i}\omega t},$ (34)

where ${\bf H}^{\rm TM}$ and ${\bf E}^{\rm TE}$ are the amplitudes of the
governing fields for the TM and TE polarizations. In the spherical geometry,
the governing TM and TE fields can be decomposed over the vector spherical
harmonics of different orbital and azimuthal indices $l,m$ [23, 26]:

$\displaystyle\displaystyle{\bf H}^{\rm
TM}=\sum_{l=0}^{\infty}\sum_{m=-l}^{l}\sum_{j=1}^{2}{H}_{lm}^{(j)}h_{l}^{(j)}(kr){\bf
Y}_{lm}^{(3)}(\theta,\phi),$ (35) $\displaystyle\displaystyle{\bf E}^{\rm
TE}=\sum_{l=0}^{\infty}\sum_{m=-l}^{l}\sum_{j=1}^{2}{E}_{lm}^{(j)}h_{l}^{(j)}(kr){\bf
Y}_{lm}^{(3)}(\theta,\phi).$ (36)

In this decomposition, the angular dependence of the excited fields is fully
given by the vector spherical harmonics [27]

${\bf Y}_{lm}^{(3)}(\theta,\phi)={\bf
e}_{\phi}\frac{{\rm\partial}Y_{lm}(\theta,\phi)}{{\rm\partial}\theta}-\frac{{\bf
e}_{\theta}}{\sin\theta}\frac{{\rm\partial}Y_{lm}(\theta,\phi)}{{\rm\partial}\phi},$
(37)

defined with

$Y_{lm}(\theta,\phi)=\sqrt{\frac{2l+1}{4\pi}\frac{(l-m)!}{(l+m)!}}\,P_{l}^{m}(\cos\theta)\mathop{\rm
e}\nolimits^{{\rm i}m\phi},$ (38)

where $P_{l}^{m}(\cos\theta)$ are the associated Legendre polynomials, while
the radial dependence is fully given by the spherical Hankel functions
$h_{l}^{(1,2)}(kr)$ defined with $k=2\pi\sqrt{\varepsilon}/\lambda$. The
amplitudes ${H}_{lm}^{(1,2)}$ and ${E}_{lm}^{(1,2)}$ are generally independent
of each other and give all possible distributions of electromagnetic fields
supported by Maxwell’s equations for uniform domains.

Depending on the spatial distribution, incident fields can possess any values
of $[{H}_{lm}^{(1,2)}]_{\rm inc}$ and $[{E}_{lm}^{(1,2)}]_{\rm inc}$.
Scattered fields, being spherically diverging, always feature
$[{H}_{lm}^{(2)}]_{\rm sca}=0$ and $[{E}_{lm}^{(2)}]_{\rm sca}=0$, while
$[{H}_{lm}^{(1)}]_{\rm sca}$ and $[{E}_{lm}^{(1)}]_{\rm sca}$ appear linear to
the incident field amplitudes:

$\displaystyle\displaystyle[{H}_{lm}^{(1)}]_{\rm
sca}=-\sum_{j=1}^{2}a^{(j)}_{lm}[{H}_{lm}^{(j)}]_{\rm inc},$ (39)
$\displaystyle\displaystyle[{E}_{lm}^{(1)}]_{\rm
sca}=-\sum_{j=1}^{2}b^{(j)}_{lm}[{E}_{lm}^{(j)}]_{\rm inc},$ (40)

where the coefficients $a^{(1,2)}_{lm}$ and $b^{(1,2)}_{lm}$ give
contributions of the TM and TE incident fields with different orbital and
azimuthal structure to the scattered fields. As for internal fields, they
always feature $[{H}_{lm}^{(2)}]_{\rm inc}=[{H}_{lm}^{(1)}]_{\rm inc}$ and
$[{E}_{lm}^{(2)}]_{\rm inc}=[{E}_{lm}^{(1)}]_{\rm inc}$ for finiteness in the
sphere’s center, $r=0$. Similarly to scattered fields, internal fields appear
linear to $[{H}_{lm}^{(1,2)}]_{\rm inc}$ and $[{E}_{lm}^{(1,2)}]_{\rm inc}$:

$\displaystyle\displaystyle[{H}_{lm}^{(1)}]_{\rm
int}=\sum_{j=1}^{2}d^{(j)}_{lm}[{H}_{lm}^{(j)}]_{\rm inc},$ (41)
$\displaystyle\displaystyle[{E}_{lm}^{(1)}]_{\rm
int}=\sum_{j=1}^{2}c^{(j)}_{lm}[{E}_{lm}^{(j)}]_{\rm inc},$ (42)

where the coefficients $d^{(1,2)}_{lm}$ and $c^{(1,2)}_{lm}$ give the TM and
TE contributions of the incident fields with different orbital and azimuthal
structure to the internal fields.

Following the continuity of the tangential electric and magnetic fields on the
particle surface $r=R$, the field coefficients given for spherically outgoing
incident fields are

$\displaystyle\displaystyle a^{(1)}_{lm}=b^{(1)}_{lm}=1,\quad
c^{(1)}_{lm}=d^{(1)}_{lm}=0.$ (43)

These coefficients correspond to the current-free scattered fields that appear
as a part of the trivial solution of Maxwell’s equations for all
polarizations, orbital and azimuthal compositions. As for spherically incoming
incident fields, their contributions are size- and material-dependent
following nontrivial solutions of Maxwell’s equations with nonzero
polarization currents induced in the spherical particle:

$\displaystyle\displaystyle
a_{lm}^{(2)}=\frac{q_{i}\psi_{l}(q_{i})\zeta_{l}^{\prime}(q_{e})-q_{e}\zeta_{l}(q_{e})\psi_{l}^{\prime}(q_{i})}{q_{i}\psi_{l}(q_{i})\xi_{l}^{\prime}(q_{e})-q_{e}\xi_{l}(q_{e})\psi_{l}^{\prime}(q_{i})},$
(44) $\displaystyle\displaystyle
b_{lm}^{(2)}=\frac{q_{e}\psi_{l}(q_{i})\zeta_{l}^{\prime}(q_{e})-q_{i}\zeta_{l}(q_{e})\psi_{l}^{\prime}(q_{i})}{q_{e}\psi_{l}(q_{i})\xi_{l}^{\prime}(q_{e})-q_{i}\xi_{l}(q_{e})\psi_{l}^{\prime}(q_{i})},$
(45) $\displaystyle\displaystyle
c_{lm}^{(2)}=\frac{1}{2}\frac{q_{i}\psi_{l}(q_{e})\zeta_{l}^{\prime}(q_{e})-q_{i}\zeta_{l}(q_{e})\psi_{l}^{\prime}(q_{e})}{q_{e}\psi_{l}(q_{i})\xi_{l}^{\prime}(q_{e})-q_{i}\xi_{l}(q_{e})\psi_{l}^{\prime}(q_{i})},$
(46) $\displaystyle\displaystyle
d_{lm}^{(2)}=\frac{1}{2}\frac{q_{i}\zeta_{l}(q_{e})\xi_{l}^{\prime}(q_{e})-q_{i}\xi_{l}(q_{e})\zeta_{l}^{\prime}(q_{e})}{q_{i}\psi_{l}(q_{i})\xi_{l}^{\prime}(q_{e})-q_{e}\xi_{l}(q_{e})\psi_{l}^{\prime}(q_{i})},$
(47)

where $\xi_{l}(q)=qh_{l}^{(1)}(q)$, $\zeta_{l}(q)=qh_{l}^{(2)}(q)$,
$\psi_{l}(q)=[\xi_{l}(q)+\zeta_{l}(q)]/2$, are the Riccati–Bessel functions,
with $q_{i}=k_{i}R$ and $q_{e}=k_{e}R$.

##  Appendix B Dipolar fields

In generalized Lorenz-Mie theory, the fields possesing a $z$-polarized
electric dipole moment are given by the TM contributions with the orbital
index $l=1$ and azimuthal index $m=0$:

$\displaystyle\displaystyle{\bf H}^{\rm TM}_{\rm
ED}=\sum_{j=1}^{2}{H}_{10}^{(j)}h_{1}^{(j)}(kr){\bf
Y}_{10}^{(3)}(\theta,\phi).$ (48)

Eventually, spatial distribution of electric field is given by

$\displaystyle{\bf E}_{\rm ED}={\rm
i}\frac{\lambda}{2\pi\varepsilon}\sqrt{\frac{\mu_{0}}{\varepsilon_{0}}}\nabla\times{\bf
H}^{\rm TM}_{\rm ED}\mathop{\rm e}\nolimits^{-{\rm i}\omega t}.$ (49)

Applying Eq. (49) to the internal area of the sphere and assuming $|k_{i}|r\ll
1$, we get the field distribution used in Eq. (3) for the internal field
excited inside the deeply subwavelegth particles:

$\displaystyle\displaystyle{\bf E}_{\rm int}=C_{\rm int}({\bf
e}_{r}\cos\theta-{\bf e}_{\theta}\sin\theta)\mathop{\rm e}\nolimits^{-{\rm
i}\omega t}+O(|k_{i}r|^{2})~{}~{}$ (50)

with the field amplitude defined as

$C_{\rm int}=-{\rm
i}\left(\frac{4}{3\pi}\frac{\mu_{0}}{\varepsilon_{0}\varepsilon_{i}}\right)^{1/2}{H}_{10}^{(1)}.$
(51)

For the external area of the sphere, we separately consider spherically
outgoing and incoming fields. Under $|k_{e}|r\ll 1$, Eq. (49) gives us the
following distribution for the outgoing dipolar field

$\displaystyle\displaystyle{\bf E}_{\rm ext}^{\rm sph,out}=-{\rm
i}\frac{C_{\rm ext}^{\rm sph,out}}{2f_{\rm rad}(r)}[{\bf
e}_{r}(2+k_{e}^{2}r^{2}+{\rm i}f_{\rm rad}(r))\cos\theta+~{}~{}$
$\displaystyle\displaystyle{\bf
e}_{\theta}\left(1-\frac{k_{e}^{2}r^{2}}{2}-{\rm i}f_{\rm
rad}(r)\right)\sin\theta]\mathop{\rm e}\nolimits^{-{\rm i}\omega
t}+O(|k_{e}r|^{\frac{1}{2}})~{}~{}~{}$ (52)

with the amplitude of

$C_{\rm ext}^{\rm sph,out}=-{\rm
i}\left(\frac{4}{3\pi}\frac{\mu_{0}}{\varepsilon_{0}\varepsilon_{e}}\right)^{1/2}{{H}_{10}^{(1)}}.$
(53)

This field was used in Eq. (2) for the scattered field and in Eq. (18) for the
spherically outgoing incident field, where we left only the leading real and
imaginary terms. As for spherically incoming external fields under
$|k_{e}|r\ll 1$, Eq. (49) results in

$\displaystyle\displaystyle{\bf E}_{\rm ext}^{\rm sph,in}={\rm i}\frac{C_{\rm
ext}^{\rm sph,in}}{2f_{\rm rad}(r)}[{\bf e}_{r}(2+k_{e}^{2}r^{2}-{\rm i}f_{\rm
rad}(r))\cos\theta+~{}~{}~{}$ $\displaystyle\displaystyle{\bf
e}_{\theta}\left(1-\frac{k_{e}^{2}r^{2}}{2}+{\rm i}f_{\rm
rad}(r)\right)\sin\theta]\mathop{\rm e}\nolimits^{-{\rm i}\omega
t}+O(|k_{e}r|^{\frac{1}{2}})~{}~{}~{}~{}$ (54)

where the field amplitude $C_{\rm ext}^{\rm sph,in}$ is given by

$C_{\rm ext}^{\rm sph,in}=-{\rm
i}\left(\frac{4}{3\pi}\frac{\mu_{0}}{\varepsilon_{0}\varepsilon_{e}}\right)^{1/2}{{H}_{10}^{(2)}}.$
(55)

This distirbution was used in Eq. (13) for the spherically incoming incident
field, where we left the dominant real and imaginary terms. Regarding the
plane field used in Eq. (8) for ${\bf E}^{\rm pl}_{\rm inc}$, it is given by
the superposition of spherically incoming ${\bf E}_{\rm ext}^{\rm sph,in}$ and
outgoing ${\bf E}_{\rm ext}^{\rm sph,out}$ fields taken with equal amplitudes
${H}_{10}^{(1)}$ and ${H}_{10}^{(2)}$:

$\displaystyle\displaystyle{\bf E}_{\rm ext}^{\rm pl}=C_{\rm ext}^{\rm
pl}({\bf e}_{r}\cos\theta-{\bf e}_{\theta}\sin\theta)\mathop{\rm
e}\nolimits^{-{\rm i}\omega t}+O(|k_{e}r|^{2}),~{}~{}$ (56)

where the resultant field amplitude is

$C_{\rm ext}^{\rm pl}=-{\rm
i}\left(\frac{4}{3\pi}\frac{\mu_{0}}{\varepsilon_{0}\varepsilon_{e}}\right)^{1/2}{H}_{10}^{(1)}.$
(57)

## References

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# Trimming and Building Freezing Sets

Laurence Boxer Department of Computer and Information Sciences, Niagara
University, NY 14109, USA and
Department of Computer Science and Engineering, State University of New York
at Buffalo
email<EMAIL_ADDRESS>
###### Abstract

We develop new tools for the construction of fixed point sets in digital
topology. We define excludable points and show that these may be excluded from
all freezing sets. We show that articulation points are excludable.

We also present results concerning points that must belong to a freezing set
and often are easily recognized. These include points of degree 1 and some
local extrema.

Key words and phrases: digital topology, digital image, freezing set,
1-coordinate local extreme point

MSC: 54B20, 54C35

##  1 Introduction

Freezing sets are part of the fixed point theory of digital topology. They
were introduced in [4] and studied subsequently in [5, 6, 7, 8, 9, 10, 11].
Given a digital image $(X,\kappa)$, it is desirable to know of a
$\kappa$-freezing set for $X$ that is as small as possible.

In this paper, we study conditions that can be used to exclude or force
membership in a freezing set. Specifically,

  * •

We show all articulation points can be excluded from freezing sets.

  * •

We show all points of degree 1 must be included in freezing sets.

  * •

We develop the notion of a 1-coordinate local extreme point and show that such
a point that satisfies an additional condition must be a member of every
freezing set of the image.

##  2 Preliminaries

We use ${\mathbb{N}}$ for the set of natural numbers, ${\mathbb{Z}}$ for the
set of integers, and $\\#X$ for the number of distinct members of $X$.

We typically denote a (binary) digital image as $(X,\kappa)$, where
$X\subset{\mathbb{Z}}^{n}$ for some $n\in{\mathbb{N}}$ and $\kappa$ represents
an adjacency relation of pairs of points in $X$. Thus, $(X,\kappa)$ is a
graph, in which members of $X$ may be thought of as black points, and members
of ${\mathbb{Z}}^{n}\setminus X$ as white points, of a picture of some “real
world” object or scene.

###  2.1 Adjacencies

This section is largely quoted or paraphrased from [7].

Let $u,n\in{\mathbb{N}}$, $1\leq u\leq n$. A digital image $X$ that satisfies
$X\subset{\mathbb{Z}}^{n}$ and
$x=(x_{1},\ldots,x_{n}),~{}y=(y_{1},\ldots,y_{n})\in X$ are $c_{u}$-adjacent
if and only if

  * •

$x\neq y$, and

  * •

for at most $u$ indices $i$, $\mid x_{i}-y_{i}\mid=1$, and

  * •

for all indices $j$ such that $\mid x_{j}-y_{j}\mid\neq 1$, we have
$x_{j}=y_{j}$.

The $c_{u}$ adjacencies are the adjacencies most used in digital topology,
especially $c_{1}$ and $c_{n}$.

In low dimensions, it is also common to denote a $c_{u}$ adjacency by the
number of points that can have this adjacency with a given point in
${\mathbb{Z}}^{n}$. E.g.,

  * •

For subsets of ${\mathbb{Z}}^{1}$, $c_{1}$-adjacency is 2-adjacency.

  * •

For subsets of ${\mathbb{Z}}^{2}$, $c_{1}$-adjacency is 4-adjacency and
$c_{2}$-adjacency is 8-adjacency.

  * •

For subsets of ${\mathbb{Z}}^{3}$, $c_{1}$-adjacency is 6-adjacency,
$c_{2}$-adjacency is 18-adjacency, and $c_{3}$-adjacency is 26-adjacency.

We use the notations $y\leftrightarrow_{\kappa}x$, or, when the adjacency
$\kappa$ can be assumed, $y\leftrightarrow x$, to mean $x$ and $y$ are
$\kappa$-adjacent. The notations $y\leftrightarroweq_{\kappa}x$, or, when
$\kappa$ can be assumed, $y\leftrightarroweq x$, mean either $y=x$ or
$y\leftrightarrow_{\kappa}x$.

A sequence $P=\\{y_{i}\\}_{i=0}^{m}$ in a digital image $(X,\kappa)$ is a
$\kappa$-path from $a\in X$ to $b\in X$ if $a=y_{0}$, $b=y_{m}$, and
$y_{i}\leftrightarroweq_{\kappa}y_{i+1}$ for $0\leq i<m$.

$X$ is $\kappa$-connected [17], or connected when $\kappa$ is understood, if
for every pair of points $a,b\in X$ there exists a $\kappa$-path in $X$ from
$a$ to $b$.

A (digital) $\kappa$-closed curve is a path $S=\\{s_{i}\\}_{i=0}^{m-1}$ such
that $s_{0}\leftrightarrow_{\kappa}s_{m-1}$, and $i\neq j$ implies $s_{i}\neq
s_{j}$. If also $0\leq i<m$ implies

$N(S,x_{i},\kappa)=\\{x_{(i-1)\mod m},~{}x_{(i+1)\mod m}\\}$

then $S$ is a (digital) $\kappa$-simple closed curve. We say the members of
$S$ are circularly labeled if they are indexed as described above.

Let $X\subset{\mathbb{Z}}^{n}$. The boundary of $X$ [16] is

$Bd(X)=\\{x\in X\,|\mbox{ there exists }y\in{\mathbb{Z}}^{n}\setminus X\mbox{
such that }y\leftrightarrow_{c_{1}}x\\}.$

###  2.2 Digitally continuous functions

This section is largely quoted or paraphrased from [7].

Digital continuity is defined to preserve connectedness, as at Definition 2.1
below. By using adjacency as our standard of “closeness,” we get Theorem 2.2
below.

######  Definition 2.1.

[2] (generalizing a definition of [17]) Let $(X,\kappa)$ and $(Y,\lambda)$ be
digital images. A function $f:X\rightarrow Y$ is $(\kappa,\lambda)$-continuous
if for every $\kappa$-connected $A\subset X$ we have that $f(A)$ is a
$\lambda$-connected subset of $Y$.

If either of $X$ or $Y$ is a subset of the other, we use the abbreviation
$\kappa$-continuous for $(\kappa,\kappa)$-continuous.

When the adjacency relations are understood, we will simply say that $f$ is
_continuous_. Continuity can be expressed in terms of adjacency of points:

######  Theorem 2.2.

[17, 2] A function $f:X\to Y$ is continuous if and only if $x\leftrightarrow
x^{\prime}$ in $X$ implies $f(x)\leftrightarroweq f(x^{\prime})$.

See also [14, 15], where similar notions are referred to as immersions,
gradually varied operators, and gradually varied mappings.

A digital isomorphism (called homeomorphism in [1]) is a
$(\kappa,\lambda)$-continuous surjection $f:X\to Y$ such that $f^{-1}:Y\to X$
is $(\lambda,\kappa)$-continuous.

The literature uses path polymorphically: a $(c_{1},\kappa)$-continuous
function $f:[0,m]_{{\mathbb{Z}}}\to X$ is a $\kappa$-path if
$f([0,m]_{{\mathbb{Z}}})$ is a $\kappa$-path from $f(0)$ to $f(m)$ as
described above.

We use $\operatorname{id}_{X}$ to denote the identity function,
$\operatorname{id}_{X}(x)=x$ for all $x\in X$.

Given a digital image $(X,\kappa)$, we denote by $C(X,\kappa)$ the set of
$\kappa$-continuous functions $f:X\to X$.

Given $f\in C(X,\kappa)$, a fixed point of $f$ is a point $x\in X$ such that
$f(x)=x$. $Fix(f)$ will denote the set of fixed points of $f$. We say $f$ is a
retraction, and the set $Y=f(X)$ is a retract of $X$, if
$f|_{Y}=\operatorname{id}_{Y}$; thus, $Y=Fix(f)$.

######  Definition 2.3.

[4] Let $(X,\kappa)$ be a digital image. We say $A\subset X$ is a freezing set
for $X$ if given $g\in C(X,\kappa)$, $A\subset Fix(g)$ implies
$g=\operatorname{id}_{X}$. A freezing set $A$ is minimal if no proper subset
of $A$ is a freezing set for $(X,\kappa)$.

The following elementary assertion was noted in [4].

######  Lemma 2.4.

Let $(X,\kappa)$ be a connected digital image for which $A$ is a freezing set.
If $A\subset A^{\prime}\subset X$, then $A^{\prime}$ is a freezing set for
$(X,\kappa)$.

Let $X\subset{\mathbb{Z}}^{n}$, $x=(x_{1},\ldots,x_{n})\in Z^{n}$, where each
$x_{i}\in{\mathbb{Z}}$. For each index $i$, the projection map (onto the
$i^{th}$ coordinate) $p_{i}:X\to{\mathbb{Z}}$ is given by $p_{i}(x)=x_{i}$.

###  2.3 Tools for determining fixed point sets

######  Theorem 2.5.

[4] Let $A$ be a freezing set for the digital image $(X,\kappa)$ and let
$F:(X,\kappa)\to(Y,\lambda)$ be an isomorphism. Then $F(A)$ is a freezing set
for $(Y,\lambda)$.

######  Proposition 2.6.

[13] Let $(X,\kappa)$ be a digital image and $f\in C(X,\kappa)$. Suppose
$x,x^{\prime}\in Fix(f)$ are such that there is a unique shortest
$\kappa$-path $P$ in $X$ from $x$ to $x^{\prime}$. Then $P\subset Fix(f)$.

The following lemma may be understood as saying that if $q$ and $q^{\prime}$
are adjacent with $q$ in a given direction from $q^{\prime}$, and if $f$ pulls
$q$ further in that direction, then $f$ also pulls $q^{\prime}$ in that
direction.

######  Lemma 2.7.

[4] Let $(X,c_{u})\subset{\mathbb{Z}}^{n}$ be a digital image, $1\leq u\leq
n$. Let $q,q^{\prime}\in X$ be such that $q\leftrightarrow_{c_{u}}q^{\prime}$.
Let $f\in C(X,c_{u})$.

  1. 1.

If $p_{i}(f(q))<p_{i}(q)<p_{i}(q^{\prime})$ then
$p_{i}(f(q^{\prime}))<p_{i}(q^{\prime})$.

  2. 2.

If $p_{i}(f(q))>p_{i}(q)>p_{i}(q^{\prime})$ then
$p_{i}(f(q^{\prime}))>p_{i}(q^{\prime})$.

The following has been relied on implicitly in several previous papers. It is
an extension of Lemma 2.7 showing that a continuous function can pull a
digital arc that is monotone with respect to a given coordinate $i$ in the
direction of monotonicity.

######  Proposition 2.8.

Let $(X,c_{u})\subset{\mathbb{Z}}^{n}$ be a digital image, $1\leq u\leq n$.
Let $q,q^{\prime}\in X$ be such that $q\leftrightarrow_{c_{u}}q^{\prime}$. Let
$f\in C(X,c_{u})$.

  1. 1.

Suppose $\\{q_{j}\\}_{j=0}^{m}$ is a $c_{u}$-path in $X$ such that $q_{0}=q$,
$q_{1}=q^{\prime}$, $p_{i}(f(q))<p_{i}(q)$, and for $0\leq j<m$ we have
$p_{i}(q_{j})<p_{i}(q_{j+1})$. Then $p_{i}(f(q_{j}))<p_{i}(q_{j})$ for $0\leq
j\leq m$.

  2. 2.

Suppose $\\{q_{j}\\}_{j=0}^{m}$ is a $c_{u}$-path in $X$ such that $q_{m}=q$,
$q_{m-1}=q^{\prime}$, $p_{i}(f(q))>p_{i}(q)$, and for $0\leq j<m$ we have
$p_{i}(q_{j})>p_{i}(q_{j-1})$. Then $p_{i}(f(q_{j}))>p_{i}(q_{j})$ for $0\leq
j\leq m$.

###### Proof.

We prove the first assertion; the second is proven similarly. We argue by
induction. We know

$p_{i}(f(q_{0}))=p_{i}(f(q))<p_{i}(q)=p_{i}(q_{0}).$

Suppose we have $p_{i}(f(q_{k}))<p_{i}(q_{k})$ for some $k<m$. Then by Lemma
2.8, $p_{i}(f(q_{k+1}))<p_{i}(q_{k+1})$. This completes our induction. ∎

######  Definition 2.9.

[5] Let $\kappa\in\\{c_{1},c_{2}\\}$. We say a $\kappa$-connected set
$S=\\{x_{i}\\}_{i=1}^{n}\subset{\mathbb{Z}}^{2}$ for $n>1$ is a (digital) line
segment if the members of $S$ are collinear.

######  Remark 2.10.

[5] A digital line segment must be vertical, horizontal, or have slope of $\pm
1$. We say a segment with slope of $\pm 1$ is slanted.

######  Lemma 2.11.

Let $(X,c_{1})\subset{\mathbb{Z}}^{2}$ be a connected digital image. Let $L$
be a horizontal or vertical digital line segment of at least 2 points
contained in $X$. Let $f\in C(X,c_{1})$. Suppose

the endpoints of $L$ are in $Fix(f)$. (1)

Then $L\subset Fix(f)$.

###### Proof.

This follows from Proposition 2.6, since $L$ is the unique shortest
$c_{1}$-path in $X$ between the endpoints of $L$. ∎

We do not have an analog of Lemma 2.11 for slanted segments, as shown by the
following.

######  Example 2.12.

Let

$X=\\{(0,0),(0,1),(0,2),(1,0),(1,1),(2,0)\\}.$

Let $S=\\{(0,2),(1,1),(2,0)\\}$. Let $f:X\to X$ be given by

$f(x)=\left\\{\begin{array}[]{ll}x&\mbox{if }x\neq(1,1);\\\ (0,0)&\mbox{if
}x=(1,1).\end{array}\right.$

It is easily seen that $f\in C(X,c_{1})$, that $S$ is a slanted segment with
endpoints $(0,2)$ and $(2,0)$ in $Fix(f)$, but $(1,1)\in S\setminus Fix(f)$.

We have the following analog of Lemma 2.11 for the $c_{2}$ adjacency.

######  Lemma 2.13.

Let $(X,c_{2})\subset{\mathbb{Z}}^{2}$ be a connected digital image. Let $L$
be a slanted digital line segment contained in $X$. Let $f\in C(X,c_{2})$.
Suppose (1). Then $L\subset Fix(f)$.

###### Proof.

This follows from Proposition 2.6, since $L$ is the unique shortest
$c_{2}$-path in $X$ between the endpoints of $L$. ∎

We do not have an analog of Lemma 2.13 for horizontal or vertical digital line
segments, as shown in the following.

######  Example 2.14.

Let $X=[0,2]_{{\mathbb{Z}}}^{2}\subset{\mathbb{Z}}^{2}$. The function $f:X\to
X$ given by

$f(x)=\left\\{\begin{array}[]{ll}(1,1)&\mbox{if }x\in\\{(0,1),(1,0)\\};\\\
x&\mbox{otherwise,}\end{array}\right.$

belongs to $C(X,c_{2})$. The digital line segments

$S_{1}=\\{(0,0),(1,0),(2,0)\\},~{}~{}~{}~{}S_{2}=\\{(0,0),(0,1),(0,2)\\}$

have endpoints in $Fix(f)$ and are respectively horizontal and vertical, but
neither of $S_{1},S_{2}$ is a subset of $Fix(f)$.

######  Theorem 2.15.

[4] Let $X\subset{\mathbb{Z}}^{n}$ be finite. Then for $1\leq u\leq n$,
$Bd(X)$ is a freezing set for $(X,c_{u})$.

######  Theorem 2.16.

Let $(X,\kappa)$ be a connected digital image. Let $A$ be a freezing set for
$(X,\kappa)$. Let $x_{0}\in X$ be a point that has degree 1. Then $x_{0}\in
A$.

###### Proof.

Let $x_{1}$ be the unique member of $X$ such that
$x_{0}\leftrightarroweq_{\kappa}x_{1}$. If $x_{0}\not\in A$ then the function
$f:X\to X$ given by

$f(x)=\left\\{\begin{array}[]{ll}x&\mbox{if }x\neq x_{0};\\\ x_{1}&\mbox{if
}x=x_{0},\end{array}\right.$

is easily seen to belong to $C(X,\kappa)$. Also,
$f|_{A}=\operatorname{id}_{A}$, and $f\neq\operatorname{id}_{X}$. The
assertion follows. ∎

##  3 Excludable sets

######  Definition 3.1.

Let $(X,\kappa)$ be a connected digital image. Let $W\subset X$. We say $W$ is
excludable from freezing sets for $(X,\kappa)$ (excludable for short) if for
every freezing set $A$ of $(X,\kappa)$, if $A\setminus W\neq\emptyset$ then
$A\setminus W$ is also a freezing set. If $p\in W$ then $p$ is an excludable
point.

######  Remark 3.2.

By Definition 3.1, if $A$ is a minimal freezing set and $W$ is excludable for
$(X,\kappa)$, then $A\cap W=\emptyset$.

######  Remark 3.3.

We require $A\setminus W\neq\emptyset$ in Definition 3.1 because, without such
a requirement, we could have

$f|_{A\setminus W}=f|_{\emptyset}=\operatorname{id}_{\emptyset}$

for all $f\in C(X,\kappa)$, including those $f$ for which
$f\neq\operatorname{id}_{X}$.

######  Proposition 3.4.

Let $(X,\kappa)$ be a finite connected digital image. Let $W$ be an excludable
set for $X$. Then every subset of $W$ is excludable.

###### Proof.

This follows from Lemma 2.4. ∎

######  Proposition 3.5.

Let $F:(X,\kappa)\to(Y,\lambda)$ be an isomorphism of digital images. If $W$
is an excludable set for $X$ then $F(W)$ is an excludable set for $Y$.

###### Proof.

Let $A$ be a freezing set for $X$. Let $f\in C(Y,\lambda)$ such that

$f|_{F(A)\setminus F(W)}=\operatorname{id}_{F(A)\setminus F(W)}$ (2)

Then $g=F^{-1}\circ f\circ F\in C(X,\kappa)$.

Let $b\in F(A)\setminus F(W)$. Then $a=F^{-1}(b)\in A\setminus W$. We have

$g(a)=F^{-1}(f(b))=F^{-1}(b)=a.$

Since every $a\in A\setminus W$ satisfies $a=F^{-1}(b)$ for some $b\in
F(A)\setminus F(W)$, we have $g|_{A\setminus W}=\operatorname{id}_{A\setminus
W}$. Since $W$ is excludable, $g=\operatorname{id}_{X}$. Therefore,

$f=F\circ g\circ F^{-1}=F\circ\operatorname{id}_{X}\circ
F^{-1}=\operatorname{id}_{Y}.$

Thus, $F(W)$ is excludable. ∎

##  4 Articulation points and freezing sets

An articulation point or cut point of a connected graph $(X,\kappa)$ is a
point $x\in X$ such that $(X\setminus\\{x\\},\kappa)$ is not connected (see
Figure 1). In this section, we show that if the set of articulation points is
removed from a freezing set, what is left is still a freezing set.

Figure 1: A digital image shown with the $c_{2}$ adjacency. The set of
articulation points is $\\{(2,3),\,(3,0),\,(4,0)\\}$.

######  Lemma 4.1.

Let $M$ be the set of articulation points for the connected digital image
$(X,\kappa)$. Let $K$ be a $\kappa$-component of $X\setminus M$. Then there is
a $\kappa$-retraction of $X$ to $X\setminus K$.

###### Proof.

Without loss of generality, $M\neq\emptyset$.

Since $X$ is connected, there exists $x_{0}\in X\setminus(K\cup M)$ such that
$x_{0}$ is $\kappa$-adjacent to a point of $M$. By choice of $M$, no point of
$K$ is adjacent to $x_{0}$. Let $r:X\to X$ be the function

$r(x)=\left\\{\begin{array}[]{ll}x&\mbox{if }x\in X\setminus K;\\\
x_{0}&\mbox{if }x\in K.\end{array}\right.$

It is easily seen that $r$ is a $\kappa$-retraction of $X$ to $X\setminus K$.
∎

######  Lemma 4.2.

Let $x_{0}$ be an articulation point for the connected digital image
$(X,\kappa)$. Let $K_{1}$ and $K_{2}$ be distinct $\kappa$-components of
$X\setminus\\{x_{0}\\}$. Let $f\in C(X,\kappa)$ such that for some $x_{1}\in
K_{1}$ and $x_{2}\in K_{2}$, $\\{x_{1},x_{2}\\}\subset Fix(f)$. Then $x_{0}\in
Fix(f)$.

###### Proof.

Let $P_{i}$ be a shortest $\kappa$-path in $X$ from $x_{i}$ to $x_{0}$,
$i\in\\{1,2\\}$. Then $P_{1}\cup P_{2}$ is a path from $x_{1}$ to $x_{2}$.

By choice of $x_{0}$ we must have $x_{0}\in f(P_{1})$. If $f(x_{0})\neq
x_{0}$, $f(P_{1})$ is a path from $x_{1}=f(x_{1})$ to $x_{0}$ to $f(x_{0})$
that has length greater than that of $P_{1}$, which is impossible. The
assertion follows. ∎

######  Theorem 4.3.

Let $W$ be the set of articulation points for the finite connected digital
image $(X,\kappa)$, with $W\neq\emptyset$. If $W$ is a proper subset of $X$,
then $W$ is excludable.

###### Proof.

Let $A$ be a freezing set for $X$. Let $f\in C(X,\kappa)$ such that
$f|_{A\setminus W}=\operatorname{id}_{A\setminus W}$. Let $x_{0}\in W$. Then
there exist distinct components $K_{1},K_{2}$ of $X\setminus\\{x_{0}\\}$. By
Lemma 4.1, there exists a retraction $r$ of $X$ to $X\setminus K_{1}$. It
follows that $A\cap K_{1}\neq\emptyset$, for otherwise
$r|_{A}=\operatorname{id}_{A}$ yet $r\neq\operatorname{id}_{X}$, contrary to
$A$ being a freezing set. Similarly, $A\cap K_{2}\neq\emptyset$.

By Lemma 4.2, $f(x_{0})=x_{0}$. Since $x_{0}$ was taken as an arbitrary member
of $W$, we have

$f|_{A\setminus W}=\operatorname{id}_{A\setminus
W}~{}~{}~{}\Rightarrow~{}~{}~{}f|_{A\cup W}=\operatorname{id}|_{A\cup
W}~{}~{}~{}\Rightarrow~{}~{}~{}f|_{A}=\operatorname{id}_{A}~{}~{}~{}\Rightarrow~{}~{}~{}f=\operatorname{id}_{X}.$

Thus $W$ is excludable. ∎

######  Remark 4.4.

Theorem 4.3 implies that if $(X,\kappa)$ is a wedge of two finite digital
images, $(X,\kappa)=(X_{1},\kappa)\vee(X_{2},\kappa)$, then the “wedge point”
of $X$ is excludable, hence does not belong to any minimal freezing set for
$(X,\kappa)$.

##  5 1-D local extrema

Let $(X,\kappa)$ be a digital image, where $X\subset{\mathbb{Z}}^{n}$. Let
$x=(x_{1},\ldots,x_{n})\in X$, where each $x_{i}\in{\mathbb{Z}}$. We say $x$
is a 1-coordinate local maximum at index $i$ for $(X,\kappa)$ if for some
index $i$ and all $y=(y_{1},\ldots,y_{n})\in N(X,x,\kappa)$, $x_{i}>y_{i}$;
and $x^{\prime}=(x_{1},\ldots,x_{i-1},x_{i}-1,x_{i+1},\ldots,x_{n})$ is a
justifying neighbor at index $i$ of $x$. We say $x$ is a 1-coordinate local
minimum at index $i$ for $(X,\kappa)$ if for some index $i$ and all
$y=(y_{1},\ldots,y_{n})\in N(X,x,\kappa)$, $x_{i}<y_{i}$; and
$x^{\prime}=(x_{1},\ldots,x_{i-1},x_{i}+1,x_{i+1},\ldots,x_{n})$ is a
justifying neighbor of $x$. We say $x$ is a 1-coordinate local extremum for
$(X,\kappa)$ if $x$ is either a 1-coordinate local maximum or a 1-coordinate
local minimum for $(X,\kappa)$. Such points are often easily recognized, and,
we will show, often are members of minimal freezing sets.

Note if $x$ and $x^{\prime}$ are, respectively, a 1-coordinate local extremum
and its justifying neighbor, then $x$ and $x^{\prime}$ differ in exactly one
index. Thus, $x\leftrightarrow_{c_{1}}x^{\prime}$.

Neither being nor not being a 1-coordinate local extremum is necessarily
preserved by isomorphism, as the following shows.

######  Example 5.1.

Let

$X=\\{(x,y)\in{\mathbb{Z}}^{2}\mid|x|+|y|=2\\}~{}~{}~{}~{}\mbox{(see
Figure~{}\ref{fig:diamondRadius2})}.$

Figure 2: The digital image $(X,c_{2})$ of Example 5.1

Assume the points of $X$ are indexed circularly with
$X=\\{x_{i}\\}_{i=0}^{7}$, with $x_{0}=(2,0)$, $x_{1}=(1,1)$, etc. Then
$(X,c_{2})$ is a digital simple closed curve, for which the “corner points”
$x_{0},x_{2},x_{4},x_{6}$ are the 1-coordinate local extrema. Let $f:X\to X$
be the function

$f(x_{i})=x_{i+1\mod 8}.$

Then $f$ is a $(c_{2},c_{2})$-isomorphism. For each index $i$,

  * •

if $x_{i}$ is a 1-coordinate local extremum, then $f(x_{i})$ is not a
1-coordinate local extremum; and

  * •

if $x_{i}$ is not a 1-coordinate local extremum, then $f(x_{i})$ is a
1-coordinate local extremum.

We have the following.

######  Theorem 5.2.

Let $(X,c_{u})$ be a connected digital image, where $X\subset{\mathbb{Z}}^{n}$
and $1\leq u\leq n$. Let $x_{0}=(x_{1},\ldots,x_{n})\in X$, where each
$x_{i}\in{\mathbb{Z}}$. Suppose, for some index $i$, $x_{0}$ is a 1-coordinate
local extremum for $(X,c_{u})$ with a justifying neighbor $x^{\prime}\in X$ at
index $i$. Let $A$ be a freezing set for $(X,c_{u})$. Then $x_{0}\in A$.

###### Proof.

There are at most $u$ indices $j$, one of which is $j=i$, at which
$|x_{j}-p_{j}(x^{\prime})|=1$ and for all other indices $k$,
$x_{k}=p_{k}(x^{\prime})$.

Therefore, if $A\subset X$ and $x_{0}\not\in A$, consider the function $f:X\to
X$ given by

$f(x)=\left\\{\begin{array}[]{ll}x&\mbox{if }x\neq x_{0};\\\
x^{\prime}&\mbox{if }x=x_{0},\end{array}\right.$

Then for $x\leftrightarrow_{c_{u}}y$, $f(x)$ and $f(y)$ differ in at most
$u-1$ indices. It follows easily that $f\in C(X,c_{u})$. Further,
$f|_{A}=\operatorname{id}_{A}$ and $f\neq\operatorname{id}_{X}$. Hence
$x_{0}\not\in A$ implies $A$ is not a freezing set. The assertion follows. ∎

##  6 Example

######  Example 6.1.

Let

$X=\\{(0,1),(1,2),(2,1),(3,0),(3,1),(3,2),(4,1),(4,2),(4,3),(5,2)\\}.$

See Figure 3.

Figure 3: The image $(X,c_{2})$ of Example 6.1

Let

$A=\\{(0,1),(3,0),(4,3),(5,2)\\}.$

We claim $A$ is a minimal freezing set for $(X,c_{2})$. We show this as
follows.

Theorem 2.15 tells us there is a minimal freezing set $B$ for $(X,c_{2})$ that
is a subset of $Bd(X)$. Therefore $(3,1)$ and $(4,2)$ can be excluded from
$B$. We proceed to show $B=A$.

We must have $(0,1)\in B$, by Theorem 2.16.

Since each of $(1,2)$ and $(2,1)$ is an articulation point of $(X,c_{2})$, by
Theorem 4.3, they are excluded from $B$. Note also that $(1,2)$ is a
1-coordinate local maximum in $X$, but lacks a justifying neighbor in $X$, so
Theorem 5.2 does not apply to $(1,2)$.

Each of $(3,0)$, $(4,3)$, and $(5,2)$ is a 1-coordinate local extreme point of
$X$ with justifying neighbors $(3,1)\in X$, $(4,2)~{}\in X$, and $(4,2)~{}\in
X$, respectively, so by Theorem 5.2, $\\{(3,0),(4,3),(5,2)\\}\subset B$.

Thus $A\subset B$.

Let $f\in C(X,c_{2})$ such that $f|_{B}=\operatorname{id}_{B}$.

Since $(2,1)$ belongs to the unique shortest-length $c_{2}$-path between the
fixed points $(0,1)$ and $(3,0)$, Proposition 2.6 lets us conclude $(2,1)\in
Fix(f)$. Since $(4,1)$ belongs to the unique shortest-length $c_{2}$-path
between the fixed points $(3,0)$ and $(5,2)$, Proposition 2.6 or Lemma 2.13
lets us conclude $(4,1)\in Fix(f)$. Similarly, $(3,2)$ belongs to the unique
shortest-length $c_{2}$-path between the fixed points $(2,1)$ and $(4,3)$, so
Theorem 2.6 or Lemma 2.13 lets us conclude $(3,2)\in Fix(f)$. Since $f$ is an
arbitrary member of $C(X,c_{2})$, we conclude that

$\\{(2,1),(4,1),(3,2)\\}\subset X\setminus B.$

Thus $A=B$. Hence by choice of $B$, $A$ is minimal.

##  7 Bounds on size of freezing set

######  Theorem 7.1.

Let $(X,c_{u})$ be a connected finite digital image, where
$X\subset{\mathbb{Z}}^{n}$ and $2\leq u\leq n$. Let $D_{1}$ be the set of
points that have degree 1 in $(X,c_{u})$. Let $T$ be the set of 1-coordinate
local extrema of $(X,c_{u})$ that have justifying points in $X$. Let $W$ be
the set of articulation points of $(X,c_{u})$. Then there is a minimal
freezing set $A\subset Bd(X)$ for $(X,c_{u})$ such that

$\\#(D_{1}\cup T)\leq\\#A.$

For $u>1$, we have

$\\#(D_{1}\cup T)\leq\\#A\leq\\#Bd(X)-\\#W.$

###### Proof.

By Theorem 2.15, there is a minimal freezing set $A\subset Bd(X)$ for
$(X,c_{u})$. We have $T\subset Bd(X)$, $D_{1}\subset Bd(X)$, and for $u\geq
2$, $W\subset Bd(X)$. Thus the conclusion follows from Theorems 4.3, 2.16, and
5.2. ∎

######  Remark 7.2.

We need $u>1$ in the second assertion of Theorem 7.1, since if
$X=[0,2]_{{\mathbb{Z}}}\subset{\mathbb{Z}}$, we have that $1$ is an
articulation point for $(X,c_{1})$ but is not a member of $Bd(X)$.

##  8 Further remarks

We have presented the notion of excludable points in digital topology, and
have shown that these may be excluded from all freezing sets. We have shown
that articulation points are excludable. We have shown that points of degree 1
and certain local extrema are points that must be included in freezing sets.

## References

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  * [5] L. Boxer, Convexity and freezing sets in digital topology, Applied General Topology 22 (1) (2021), 121 - 137
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https://arxiv.org/abs/2307.04969

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# A Transparent Window into Early-Type Stellar Variability

Adam S. Jermyn Center for Computational Astrophysics, Flatiron Institute, New
York, NY 10010, USA  Evan H. Anders CIERA, Northwestern University, Evanston
IL 60201, USA  Matteo Cantiello Center for Computational Astrophysics,
Flatiron Institute, New York, NY 10010, USA  Department of Astrophysical
Sciences, Princeton University, Princeton, NJ 08544, USA

###### Abstract

Subsurface convection zones are ubiquitous in early-type stars. Driven by
narrow opacity peaks, these thin convective regions transport little heat but
play an important role in setting the magnetic properties and surface
variability of stars. Here we demonstrate that these convection zones are
_not_ present in as wide a range of stars as previously believed. In
particular, there are regions which 1D stellar evolution models report to be
convectively unstable but which fall below the critical Rayleigh number for
onset of convection. For sub-solar metallicity this opens up a _stability
window_ in which there are no subsurface convection zones. For LMC metallicity
this surface stability region extends roughly between $8M_{\odot}$ and
$16M_{\odot}$, increasing to $8M_{\odot}$ – $35M_{\odot}$ for SMC metallicity.
Such windows are then an excellent target for probing the relative influence
of subsurface convection and other sources of photometric variability in
massive stars.

Stellar physics (1621); Stellar evolutionary models (2046); Stellar convection
zones (301)

††journal: ApJ††software:  MESA (Paxton et al., 2011, 2013, 2015, 2018, 2019,
http://mesa.sourceforge.net), MESASDK 20190830 (Townsend, 2019a, b),
matplotlib (Hunter, 2007), NumPy (van der Walt et al., 2011)

##  1 Introduction

Subsurface convection zones are ubiquitous in early-type stars (Cantiello et
al., 2009; Cantiello & Braithwaite, 2019). Driven by narrow opacity peaks
(Stothers & Chin, 1993), these thin convective regions transport little heat
but play an important role in setting the magnetic fields (Cantiello &
Braithwaite, 2011; Jermyn & Cantiello, 2020) and chemical mixing (Michaud,
1970).

Subsurface convection zones may also play a role in setting the photometric
variability of early-type stars, though the precise connection between these
is not well-understood. In particular, one of the most exciting potential
observational probes of these subsurface convection zones is stochastic low-
frequency variability (SLF, Blomme et al., 2011; Bowman et al., 2019), but
there is active debate into the origin of the SLF phenomenon (Lecoanet et al.,
2019). Subsurface convective motions form one possible explanation (Cantiello
et al., 2021), as do internal gravity waves excited by core convection (Bowman
et al., 2019) and wind mass loss (Krtička & Feldmeier, 2021).

Here we demonstrate that these convection zones are _not_ present in as wide a
range of stars as previously believed. In particular, there are regions which
1D stellar evolution models report to be convectively unstable but which fall
below the critical Rayleigh number for onset of convection. This means that
the radiative conductivity is large enough to turn off the convective
instability in these regions.

We begin in Section 2 with an overview of the theory of stellar convection and
the critical Rayleigh number. In Section 3 we compute stellar evolutionary
tracks and use these to demonstrate that over certain mass and age windows
some subsurface convection zones are subcritical. Importantly, this leaves a
metallicity-dependent _stability window_ where there are no subsurface
convection zones. This region opens at around $8M_{\odot}$ and, for the
metallicity of the Small Magellanic Cloud (SMC, $Z=0.002$), extends up to
$35M_{\odot}$. The stability window is then an excellent target for
understanding how much of the SLF phenomenon is due to subsurface convection,
as it should be possible to compare the SLF properties of stars inside and
outside the window to see if there is any abrupt change, or if the properties
better track the more gradual evolution in core properties. We conclude with a
discussion of this and other observational implications in Section 4.

##  2 Theory

In 1D stellar evolution calculations, convection is assumed to be active
whenever the Ledoux condition for instability holds, namely

$\displaystyle\nabla_{\rm rad}>\nabla_{\rm
ad}+\nabla_{\mu}\left.\frac{\partial\ln T}{\partial\ln\mu}\right|_{P,\rho}.$
(1)

Here $\mu$ is the mean molecular weight,

$\displaystyle\nabla_{\mu}\equiv\frac{d\ln\mu}{d\ln P}$ (2)

is the composition gradient,

$\displaystyle\nabla_{\rm ad}\equiv\left.\frac{\partial\ln T}{\partial\ln
P}\right|_{s}$ (3)

is the adiabatic temperature gradient,

$\displaystyle\nabla_{\rm rad}\equiv\frac{3\kappa LP}{64\pi GM\sigma T^{4}}$
(4)

is the radiative temperature gradient, $\kappa$ is the opacity, $\rho$ is the
density, $L$ is the luminosity, $P$ is the pressure, $T$ is the temperature,
$G$ is the gravitational constant, $M$ is the mass below the point of
interest, and $\sigma$ is the Stefan-Boltzmann constant.

However there is an additional requirement for convection, namely that the
Rayleigh number

$\displaystyle\mathrm{Ra}\equiv\frac{g(\nabla_{\rm rad}-\nabla_{\rm ad})\delta
r^{3}}{\nu\alpha}\left(\frac{\delta
r}{h}\right)\left(\frac{4-3\beta}{\beta}\right)$ (5)

exceed the critical value $\mathrm{Ra}_{\rm crit}$ (e.g. Chandrasekhar, 1961;
Shore, 1992). The Rayleigh number expresses the importance of buoyancy
compared to diffusive processes; when $\rm{Ra}<\rm{Ra}_{\rm{crit}}$, buoyant
forces are not strong enough to overcome diffusive processes, and there is no
convection. Here

$\displaystyle\alpha\equiv\frac{16\sigma T^{3}}{3\kappa c_{p}\rho^{2}}$ (6)

is the thermal diffusivity, $c_{p}$ is the specific heat at constant pressure,
$g$ is the acceleration of gravity, $\delta r$ is the thickness of the Ledoux-
unstable layer, $\nu$ is the kinematic viscosity, $h\equiv-dr/d\ln P$ is the
pressure scale height, and $\beta=P_{\rm gas}/P$ is the radiation parameter.
The factor depending on $\beta$ arises from the thermal expansion coefficient,
and is equal to the ratio of the density susceptibility to the temperature
susceptibility ($\chi_{\rho}/\chi_{T}$). See Appendix B for details on how we
compute $\nu$.

The critical Rayleigh number depends on the precise setup of the convection
zone, including geometry, the shape of the opacity profile, rotation, and the
diffusivities (Net et al., 2008; Garcia et al., 2018; Calkins et al., 2015).
For simplicity, we adopt Chandrasekhar’s theory (Chandrasekhar, 1961, Chapter
29), and assume that convection occurs by either direct onset or
overstability, whichever has a lower $\mathrm{Ra}_{\rm crit}$. For non-
rotating systems this produces $\mathrm{Ra}_{\rm crit}\sim 10^{3}$, while for
rotating systems it is somewhat greater because rotation stabilizes against
convection. See Appendix D for details of how we compute $\mathrm{Ra}_{\rm
crit}$.

##  3 Results

Figure 1: The Rayleigh number normalized to the critical $\mathrm{Ra}_{\rm
crit}$ is shown on a Hertzsprung-Russel diagram in terms of $\log T_{\rm eff}$
and $\log L$ for stellar models ranging from $1.1-60M_{\odot}$ with Milky Way
metallicity $Z=0.014$. The four panels correspond to four different subsurface
convection zones, namely HI, HeI, HeII, and the FeCZ. White space indicates
regions which lack a given convection zone in our models (i.e. equation (1) is
not satisfied). To calculate $\mathrm{Ra}_{\rm crit}$ we assumed a rotation
period of $10^{3}\,\mathrm{d}$.

We calculated stellar evolutionary tracks for stars ranging from
$1.1-60M_{\odot}$ using revision 15140 of the Modules for Experiments in
Stellar Astrophysics (MESA Paxton et al., 2011, 2013, 2015, 2018, 2019)
software instrument. Details on the MESA microphysics inputs are provided in
Appendix A. Our models use convective premixing (Paxton et al., 2019) and
determine the convective boundary using the Ledoux criterion.

We used equation 5 to calculate the Rayleigh numbers in subsurface convective
layers caused by hydrogen, helium, and iron recombination (HI, HeI, HeII, and
FeCZ). As they trace different ionization stages of H, He and Fe, these
convective regions occur in specific temperature ranges (Cantiello &
Braithwaite, 2019). Moreover the FeCZ is only found in stars above a certain
luminosity threshold, which in turns depends on the stellar metallicity
(Cantiello et al., 2009).

Figure 1 shows the Rayleigh number normalized to the critical
$\mathrm{Ra}_{\rm crit}$ ($\sim 10^{3}$, see Appendix D) on a Hertzsprung-
Russel diagram (HRD) for the four different subsurface convection zones with
Milky Way metallicity $Z=0.014$. A few features are striking. First, the HeI
convection zone is _almost always_ subcritical. This is a missing convection
zone: it is present in stellar models, and is still a superadiabatic layer in
stars, but it likely does not undergo convective motions in most stars and has
$\nabla=\nabla_{\rm rad}$ because convection transports no heat there.

Similarly the HI zone is subcritical for masses $2-3M_{\odot}$ and the HeII
zone is subcritical above $5M_{\odot}$. The FeCZ by contrast is always
supercritical where it exists.

Figure 2: Profiles of various quantities are shown for mid-main-sequence
models of $2.4M_{\odot}$ (left) and $9M_{\odot}$ (right) stars at the Milky
Way metallicity ($Z=0.014$) as functions of $\log T/\mathrm{K}$. The upper
panels show the radiative ($\nabla_{\rm rad}$) and adiabatic ($\nabla_{\rm
ad}$) temperature gradients. The middle panels show the pressure scale height
$h\equiv P/\rho g$ and the convection speed computed by Mixing Length Theory.
The lower panels show the viscosity $\nu$ and the thermal diffusivity
$\alpha$.

The differences in Ra between these different layers can be explained by a few
factors. First, the four CZs exist at very different densities. At lower
density photons have a longer mean free path, and since photons carry both
heat and momentum this results in a higher thermal and kinetic diffusivity.
Examples of these trends are shown in the lower two panels of Figure 2.
Moreover, the size of the convective region $\delta r$ is comparable to the
local scale height, and this gets smaller for regions closer to the surface
(Cantiello & Braithwaite, 2019, see also the middle panels of Figure 2).

A careful examination of Figure 1 reveals that there is _always_ a subsurface
convection zone. At Milky Way metallicity ($Z=0.014$) at least one of the four
CZs is active for every mass we have examined and across the HRD. This is not
the case for all $Z$, however.

Figure 3: The region where all subsurface superadiabatic layers have sub-
critical Rayleigh numbers (the stability window) is shown on a Hertzsprung-
Russel diagram in terms of $\log T_{\rm eff}$ and $\log L$ for stellar models
ranging from $7-40M_{\odot}$. The four panels correspond to four different
metallicities, ranging from that of the Small Magellanic Cloud ($Z=0.002$) to
the Milky Way ($Z=0.014$). At Milky Way metallicity and above the window with
no subsurface convection disappears because the FeCZ moves down in $\log
T_{\rm eff}$ and $L$ and meets the HeII CZ. To calculate $\mathrm{Ra}_{\rm
crit}$ we assumed a rotation period of $10^{3}\,\mathrm{d}$.

Towards smaller $Z$ the FeCZ moves up in $\log T_{\rm eff}$ and $\log L$ and a
window opens where there are no subsurface convection zones. This is shown in
Figure 3, where we have plotted the window with no subsurface CZs for four
different metallicities ranging from the Small Magellanic Cloud (SMC,
$Z=0.002$) up to the Milky Way (where the window vanishes). We call this the
_stability window_ , because it is a region where subsurface convection is
stabilized by microscopic diffusivities.

At SMC metallicity the stability window is wide, from $8-35M_{\odot}$ and
covering the whole main-sequence for most of those masses. As $Z$ rises the
upper end of the window comes down, tracing the FeCZ, first to $16M_{\odot}$
at Large Magellanic Cloud (LMC) metallicity ($Z=0.006$) then $10M_{\odot}$ at
$Z=0.01$. Finally by Milky Way metallicity of $Z=0.014$ the start of the FeCZ
pulls down to $8M_{\odot}$, matching the upper mass at which HeII convection
happens, and the window disappears. For convenience the same quantities are
shown in Figure 4 versus spectroscopic luminosity $\mathscr{L}\equiv T_{\rm
eff}^{4}/g$, where $g$ is the surface gravity.

Figure 4: The region where all subsurface superadiabatic layers have sub-
critical Rayleigh numbers (the stability window) is shown on a Hertzsprung-
Russel diagram in terms of $\log T_{\rm eff}$ and $\log\mathscr{L}$ for
stellar models ranging from $7-40M_{\odot}$. Here $\mathscr{L}\equiv
L/M\propto T_{\rm eff}^{4}/g$ is the spectroscopic luminosity. The four panels
correspond to four different metallicities, ranging from that of the Small
Magellanic Cloud ($Z=0.002$) to the Milky Way ($Z=0.014$). At Milky Way
metallicity and above the window with no subsurface convection disappears
because the FeCZ moves down in $\log T_{\rm eff}$ and $L$ and meets the HeII
CZ. To calculate $\mathrm{Ra}_{\rm crit}$ we assumed a rotation period of
$10^{3}\,\mathrm{d}$.

Rotation modifies our scenario in two ways. First, it slightly changes the
Rayleigh number in these subsurface superadiabatic zones by changing the
hydrostatic structure of the star, but this is quite a small effect. The
reason for this is that these zones are generated by opacity bumps which occur
at nearly fixed $T$. Taking the density to be unchanged, and treating these
zones as of order the pressure scale height $h$ in radial extent, we find

$\displaystyle\mathrm{Ra}_{\rm rotating}\sim\frac{h^{3}g_{\rm
eff}}{\alpha\nu}\propto\frac{1}{g_{\rm
eff}^{2}}\propto\frac{1}{(1-\Omega^{2}/\Omega_{\rm c}^{2})^{2}},$ (7)

where $g_{\rm eff}$ is the effective acceleration of gravity accounting for
rotation, $\Omega$ is the angular velocity, and $\Omega_{\rm c}$ is the
critical rotation rate. Except very near critical rotation this is a small
effect that does not compete with the many orders of magnitude variation in
$\mathrm{Ra}$ across the HRD.

Secondly, and more importantly, rotation increases the critical Rayleigh
number, stabilizing more stars against convection and growing the stability
window. In the calculations above we assumed a very slow rotation rate of
$10^{3}\,\mathrm{d}$. We do this because $\mathrm{Ra}_{\rm crit}$ has a
different value when no rotation is assumed than when an extremely slow
rotation is assumed. Using slow but nonzero rotation rates slightly expands
the stability windows. The effects of increasing rotation further are small,
however, and we see no difference in the stability windows calculated with
rotation periods varying from $P=100\,\mathrm{d}$ down to $P=2\,\mathrm{d}$.

##  4 Discussion

How much do these missing convection zones matter? For stellar structure, not
much: subsurface convection zones carry very little heat, the temperature
gradient $\nabla\approx\nabla_{\rm rad}$, so in the absence of convective heat
transport the near-surface temperature profiles of these stars are likely not
that different. However, for more subtle measurements such as magnetism and
photometric variability the properties of subsurface convection zones are
critically important.

For instance in Jermyn & Cantiello (2020) and Jermyn & Cantiello (2021) we
made predictions of the interaction between surface magnetism and subsurface
convection zones. At Milky Way metallicities and above these are unaffected,
but at lower metallicities (e.g. SMC and LMC) those predictions are
significantly altered because there is a window in mass where no subsurface
convection occurs (Figure 3), and we now expect:

  1. 1.

There should be no magnetic desert for stars in the stability window because
there is no subsurface convection that can erase a magnetic field.

  2. 2.

A corollary is that stars should typically have stronger surface magnetic
fields in the window than those just below the window in mass, because there
can be no surface convective dynamo fields in these stars, only fossil fields,
which tend to be stronger.

  3. 3.

If Discrete Absorption Components (DACs, Kaper et al., 1996) are generated by
magnetic spots then DACs should be steady rather than time-varying in the
stability window, because there is no subsurface convection present to
generate a disordered magnetic field.

  4. 4.

If wind clumping is seeded close to the stellar surface via convectively-
driven fluctuations in density and velocity (e.g. Puls et al., 2006, 2008;
Cantiello et al., 2009) then the clumping properties of stars in the window
could be very different.

Similarly, the stability window provides an excellent opportunity to test
theories of photometric variability in early-type stars. In particular there
is active debate over the origin of surface low-frequency variability
(Lecoanet et al., 2019). We are aware of three proposed explanations:
subsurface convective motions (Cantiello et al., 2021), internal gravity waves
excited by core convection (Bowman et al., 2019) and wind mass loss (Krtička &
Feldmeier, 2021). Stars inside the stability window, however, have no
subsurface convection. Any photometric variability observed in the window,
then, cannot come from convection and must come from either one of the two
other proposals or else some process not yet considered. If the SLF phenomenon
has multiple causes, we should expect a discontinuity on the edge of the
window, and if it is monocausal we should either expect it to be indifferent
to the window (ruling out a convective origin) or to be excluded from the
window (requiring a convective origin).

The Flatiron Institute is supported by the Simons Foundation. This research
was supported in part by the National Science Foundation under Grant No.
PHY-1748958. We thank Dominic Bowman for suggesting the addition of Figure 4
and Daniel Lecoanet for advice on computing the critical Rayleigh number in
rotating convection.

##  Appendix A MESA

The MESA EOS is a blend of the OPAL (Rogers & Nayfonov, 2002), SCVH (Saumon et
al., 1995), FreeEOS (Irwin, 2004), HELM (Timmes & Swesty, 2000), and PC
(Potekhin & Chabrier, 2010) EOSes.

Radiative opacities are primarily from OPAL (Iglesias & Rogers, 1993, 1996),
with low-temperature data from Ferguson et al. (2005) and the high-
temperature, Compton-scattering dominated regime by Buchler & Yueh (1976).
Electron conduction opacities are from Cassisi et al. (2007).

Nuclear reaction rates are from JINA REACLIB (Cyburt et al., 2010) plus
additional tabulated weak reaction rates Fuller et al. (1985); Oda et al.
(1994); Langanke & Martínez-Pinedo (2000). Screening is included via the
prescription of Chugunov et al. (2007). Thermal neutrino loss rates are from
Itoh et al. (1996).

Models were constructed on the pre-main sequence with $Y=0.24+2Z$ and
$X=1-Y-Z$ and evolved from there. We neglect rotation and associated chemical
mixing.

##  Appendix B Viscosity

Computing the viscosity of a plasma is complicated. For simplicity we use the
viscosity of pure hydrogen plus radiation, so that

$\displaystyle\nu\approx\nu_{\rm H}+\nu_{\rm rad}.$ (B1)

The radiation component is (Spitzer, 1962)

$\displaystyle\nu_{\rm rad}=\frac{4aT^{4}}{15c\kappa\rho^{2}},$ (B2)

where $a$ is the radiation gas constant. We obtain the hydrogen viscosity
using the Braginskii-Spitzer formula (Braginskii, 1957; Spitzer, 1962)

$\displaystyle\nu_{\rm ie}\approx 2.21\times
10^{-15}\frac{(T/\mathrm{K})^{5/2}}{(\rho/\mathrm{g\,cm^{-3}})\ln\Lambda}\mathrm{cm^{2}\,s^{-1}},$
(B3)

where $\ln\Lambda$ is the Coulomb logarithm, given for hydrogen by (Wendell et
al., 1987)

$\displaystyle\ln\Lambda=-17.9+1.5\ln\frac{T}{\mathrm{K}}-0.5\ln\frac{\rho}{\mathrm{g\,cm^{-3}}}$
(B4)

for temperatures $T<4.2\times 10^{5}\,\mathrm{K}$ and

$\displaystyle\ln\Lambda=-11.5+\ln\frac{T}{\mathrm{K}}-0.5\ln\frac{\rho}{\mathrm{g\,cm^{-3}}}$
(B5)

for $T>4.2\times 10^{5}\,\mathrm{K}$. Corrections owing to different
compositions are generally small relative to the many orders of magnitude we
are interested in here. For instance the difference between pure hydrogen and
a cosmic mixture of hydrogen and helium is under $30\%$ (Balbus & Henri,
2008).

##  Appendix C Data Availability

The plotting scripts used in this work, as well as the MESA run directories
used in producing Figure 2, may be found in this GitHub repository in the
commit with short-sha d7cdf2f. The inlists and run scripts used in producing
the HR diagrams in this work are available in this other GitHub repository on
the main branch in the commit with short-sha 964e07d. The data those scripts
produced are available in Jermyn et al. (2022).

##  Appendix D Critical Rayleigh Number

The critical Rayleigh number depends on both the value of the Taylor number
(rotation) and the Prandtl number (diffusivity ratio),

$\displaystyle\mathrm{Ta}\equiv\frac{4\delta r^{4}\Omega^{2}}{\nu^{2}}$ (D1)

and

$\displaystyle\mathrm{Pr}\equiv\frac{\nu}{\alpha}.$ (D2)

Recent work has explored the intricacies of convective onset in rotating,
spherical domains (Net et al., 2008; Garcia et al., 2018). For simplicity, we
use the derivation of Chandrasekhar (1961) to determine convective onset. For
a given value of Ta and Pr, we use his Ch. 3, eqns. 222 & 227 to respectively
calculate the critical Rayleigh number of an overstable onset and direct
onset,

$\displaystyle\mathrm{Ra}_{\mathrm{crit,over}}=2\pi^{4}\frac{(1+\mathrm{Pr})}{x}\left((1+x)^{3}+\frac{\mathrm{Pr}^{2}\mathrm{Ta}/\pi^{4}}{(1+\mathrm{Pr})^{2}}\right),$
(D3)
$\displaystyle\mathrm{Ra}_{\mathrm{crit,direct}}=\pi^{4}\left(\frac{(1+x)^{3}+\mathrm{Ta}/\pi^{4}}{x}\right),$
(D4)

where $x$ is a nondimensional wavenumber of convective onset, which must be
compared to a reference value (eqn. 225),

$x^{*}=\left(\frac{\mathrm{Ta}(1-\mathrm{Pr})}{\pi^{4}(1+\mathrm{Pr})}\right)^{1/3}-1.$
(D5)

If $x<x^{*}$, $\mathrm{Ra}_{\mathrm{crit}}=\mathrm{Ra}_{\mathrm{crit,over}}$,
otherwise $\mathrm{Ra}_{\mathrm{crit}}=\mathrm{Ra}_{\mathrm{crit,direct}}$.
For a given value of Ta and Pr, we evaluate $\mathrm{Ra}_{\mathrm{crit}}$ for
$x\in[10^{-3},10]$, and take the minimum value to be the true value of
$\mathrm{Ra}_{\mathrm{crit}}$.

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