###### Abstract

A nonrelativistic quantum theory of dark matter particles in a spherical halo
is developed by using a new asymmetric equation, which is complementary to the
Schrödinger equation. The theory predicts that each dark matter halo has its
core and envelope with very distinct physical properties. The core is free of
any quantum structure and its dark matter particles are in random motion and
frequently collide with each other. However, the envelope has a global quantum
structure that contains quantized orbits populated by the particles.
Applications of the theory to dark matter halos with given density profiles
are described, and physical implications and predictions of the theory are
discussed.

Quantum Theory of Cold Dark Matter Halos

Z. E. Musielak

##  1 Introduction

Many theories of dark matter (DM) have been proposed [1-20]. One popular
theory predicts the existence of weakly interacting massive particles (WIMPs)
of dark matter (DM). However, so far, all attempts to discover WIMPs
experimentally have failed (e.g., [21-24]); thus, the nature and origin of DM
still remains unknown. According to the Planck 2018 mission [25], DM
constitutes 26.8% of the total mass-energy density of the Universe, which is
almost 5.5 times more than the amount of ordinary matter (OM).

Recent results [19] show that the Schrödinger equation (SE) of nonrelativistic
quantum mechanics [26] is consistent with the irreducible representations
(irreps) of the extended Galilean group [27-31]. The validity of the SE and
its theoretical predictions have been verified by numerous experiments
performed since Schrödinger published it [32]. The obtained theoretical and
experimental results have demonstrated that the SE describes OM and its
quantum properties. Thus, it seems natural to apply the SE to construct models
of DM, which require solving the SE on large galactic scales [33,34]. However,
more detailed calculations [35,36] have revealed that such models constructed
for different galactic halos require DM particles to have different masses,
which is difficult to justify physically.

The irreps used to obtain the SE also allow for the existence of a new
asymmetric equation (NAE) that is complementary to the SE [19]; note that both
the SE and NAE can be used to describe propagation of classical waves [37].
Moreover, since all models of DM based on the SE have failed [35,36], an
attempt was made to construct a model of two DM particles interacting only
gravitationally by using the NAE [20]. The model has two main shortcomings,
namely, it does not consider any halo’s effects on the particles, and it is
based on the assumption that the density is constant. In this paper, a quantum
theory of a DM halo is developed based on the NAE; the main advantage of the
theory is that it removes all the shortcomings of the previous model.

The developed theory predicts a quantum structure of the halo that resembles
an atom; means that each halo has a core and an envelope that surrounds it.
The core occupies a small fraction ($<10\%$) of the volume of the halo, and
its density is constant, which makes the core free of any quantum structure.
As a result, the DM particles move randomly inside the core and frequently
collide with each other. On the other hand, the envelope that surrounds the
core contains quantized orbits with DM particles confined to them. A narrow
region of the envelope in the vicinity of the core is of special interest as
it allows for exchanging DM particles and quanta of energy (dark gravitons)
between the core and the envelope. Applications of the theory to DM halos with
constant density and with density varying linearly with halo’s radius are
presented, and the obtained results are discussed.

This paper is organized as follows: nonrelativistic asymmetric equations are
derived in Section 2; previous theories of DM and their validity are discussed
in Section 3; a quantum theory of DM particles confined to a spherical DM halo
is formulated in Section 4; physical implications of the theory and its
predictions are discussed in Section 5; and conclusions are given in Section
6.

##  2 Nonrelativistic asymmetric equations

###  2.1 Group theory derivation of asymmetric equations

The structure of the extended Galilean group is
$\mathcal{G}_{e}=[O(3)\otimes_{s}B(3)]\otimes_{s}[T(3+1)\otimes U(1)]$, where
$O(3)$ and $B(3)$ are subgroups of rotations and boosts, $T(3+1)$ is an
invariant subgroup of combined translations in space and time, and $U(1)$ is a
one-parameter unitary subgroup; the irreps of $\mathcal{G}_{e}$ are well-known
[27-31].

A scalar function $\phi(t,\mathbf{x})$ transforms as one of the irreps of
$\mathcal{G}_{e}$ if, and only if, the following eigenvalue equations
$i\partial_{t}\phi=\omega\phi$ and $-i\nabla\phi=\mathbf{k}\phi$ are
satisfied, where $\partial_{t}=\partial/\partial t$, and $\omega$ and
$\mathbf{k}$ are labels of the irreps. The eigenvalue equations can be used to
derive the following two second-order asymmetric equations [19,20]

$\left[i{{\partial}\over{\partial
t}}+\left(\frac{\omega}{k^{2}}\right)\nabla^{2}\right]\phi(t,\mathbf{x})=0\ ,$
(1)

and

$\left[{{\partial^{2}}\over{\partial
t^{2}}}-i\left(\frac{\omega}{k}\right)^{2}{\mathbf{k}}\cdot\nabla\right]\phi(t,\mathbf{x})=0\
.$ (2)

where the ratios of the labels of the irreps in both equations are constant as
required by the eigenvalue equations. Comparison of these equations shows that
their mathematical structure is complementary, but their wavefunctions must be
different.

The ratios of eigenvalues in the above equations can be expressed in terms of
the universal constants, so the equations can be used to formulate quantum
theories of OM and DM [19,20]. However, if the ratios are expressed in terms
of the wave characteristic speed and frequency, then the equations become the
wave equations that describe propagation of classical waves in uniform or
nonuniform media [36]; classical waves are not discussed in this paper.

###  2.2 The Schrödinger equation

By using the de Broglie relationship [26], the ratio of eigenvalues in Eq. (1)
can be evaluated as $\omega/k^{2}=\hbar/2m$, which turns the equation into the
Schrödinger equation [32]. Denoting the wavefunction by $\phi_{S}$, the SE can
be written in its standard form [26] as

$\left[i{{\partial}\over{\partial
t}}+\frac{\hbar}{2m}\nabla^{2}\right]\phi_{S}(t,\mathbf{x})=0\ ,$ (3)

where $m$ is the mass of a particle.

The eigenvalue equations can also be used to define operators of energy,
$\hat{E}$, and momentum, $\mathbf{\hat{P}}$. This is achieved by multiplying
the eigenvalue equations by $\hbar$, and defining $\hat{E}=i\hbar\partial_{t}$
and $\mathbf{\hat{P}}=-i\hbar\nabla$. Then, the eigenvalue equations for these
operators are: $\hat{E}\phi_{S}=E\phi_{S}$ and
$\mathbf{\hat{P}}\phi_{S}={\mathbf{p}}\phi_{S}$, with the eigenvalues
$E=\hbar\omega$ and $\mathbf{p}=\hbar\mathbf{k}$. Using these results, the SE
can be written in its operator’s form as

$\hat{E}\phi_{S}=\frac{1}{2m}\left(\mathbf{\hat{P}}\cdot\mathbf{\hat{P}}\right)\phi_{S}\
,$ (4)

which gives the following relationship between the eigenvalues

$E=\frac{\mathbf{p}\cdot\mathbf{p}}{2m}=\frac{p^{2}}{2m}=E_{k}\equiv E_{SE}\
.$ (5)

This shows that the SE is based on the nonrelativistic kinetic energy,
$E_{k}=E_{SE}$, which is a well-known result (e.g., [26]).

###  2.3 A new asymmetric equation

The de Broglie relationship [26] may also be used to determine the ratio of
eigenfunctions $\omega^{2}/k^{2}=\varepsilon_{o}/2m$, where
$\varepsilon_{o}=\hbar\omega_{o}$ is a fixed quanta of energy, and
$\omega_{o}$ is a fixed frequency [19,20]. Then, Eq. (2) becomes

$\left[{{\partial^{2}}\over{\partial
t^{2}}}-i\frac{\varepsilon_{o}}{2m}{\mathbf{k}}\cdot\nabla\right]\phi_{A}(t,\mathbf{x})=0\
,$ (6)

where $\phi_{A}(t,\mathbf{x})$ is the wavefunction of this new asymmetric
equation (NAE).

To determine the expression for energy that underlies the NAE, the eigenvalue
equations for the operators of energy, $\hat{E}$, and momentum,
$\mathbf{\hat{P}}$ must be determined. Since the wavefunctions of the NAE and
SE are different, the eigenvalue equations for $\phi_{A}$ will be different
than those found for $\phi_{S}$ in Section II.B. The main reason is that the
operators $\hat{E}$ and $\mathbf{\hat{P}}$ acting on different eigenfunctions
give different eigenvalues.

Thus, for the energy, $\hat{E}=i\hbar\partial_{t}$, and momentum,
$\mathbf{\hat{P}}=-i\hbar\nabla$, operators, the eigenvalue equations are:
$\hat{E}\phi_{A}=\sqrt{\varepsilon_{o}E}\ \phi_{A}$ and
$(\hbar\mathbf{k}\cdot\mathbf{\hat{P}})\phi_{A}=(\mathbf{p}\cdot\mathbf{\hat{P}})\phi_{A}=p^{2}\phi_{A}$,
with the eigenvalues being $\varepsilon_{o}=\hbar\omega_{o}$, $E=\hbar\omega$
and $\mathbf{p}=\hbar\mathbf{k}$. Then, Eq. (6) can be written in the
following form

$\left(\frac{1}{\varepsilon_{o}}\right)\hat{E}^{2}\phi_{A}=\frac{1}{2m}\left(\mathbf{p}\cdot\mathbf{\hat{P}}\right)\phi_{A}\
,$ (7)

and the relationship between the eigenvalues becomes

$E=\frac{p^{2}}{2m}=E_{k}\equiv E_{NAE}\ ,$ (8)

which shows that $E_{NAE}=E_{SE}=E_{k}$ and, as expected, both the NAE and SE
are based on the nonrelativistic kinetic energy.

The presented results demonstrate that the main difference between the SE and
NAE is that the former allows for the quanta of energy $\hbar\omega$ to be of
any frequency; however, the latter is valid only when the quanta of energy
$\varepsilon_{o}$ is fixed at one frequency $\omega_{o}$. Moreover, the
evolution of the wavefunction $\phi_{A}(t,\mathbf{x})$ described by the NAE
depends on direction as shown by the presence of the term
${\mathbf{k}}\cdot\nabla\phi$, whose values are different in different
directions with respect to ${\mathbf{k}}$; the SE does not show such a
directional dependence.

##  3 Asymmetric equations and quantum theories of dark matter

The fact that the SE describes the quantum structure of OM has been known for
almost 100 years (e.g., [26]). There were also attempts to use the SE to
formulate quantum theories of DM. The basic idea was proposed by Sin [33], who
postulated the existence of extremely light bosonic DM particles with masses
of the order of $10^{-24}$ eV, which allows for solving the SE on the galactic
scale because of the very long Compton wavelength of such particles. The
gravitational potential added to the SE was calculated by solving the Poisson
equation with the DM density as the forcing term. The work was followed by Hu
et al. [34], whose DM particles had masses of the order of $10^{-22}$ eV.
However, more detailed studies [35,36] revealed that these theories require
different masses of DM particles in different galactic halos, which is
difficult to justify from a physical point of view.

Since the SE equation describes the quantum structure of OM, and since the
developed quantum theories of DM based on the SE failed, it was suggested that
the NAE, being complementary to SE, may represent the quantum structure of DM
[19,20]. If this is correct, then DM particles may only exchange the quanta of
energy $\varepsilon_{o}=\hbar\omega_{o}$, whose frequency $\omega_{o}$ is
fixed for DM particles. This means that while OM emits or absorbs radiation at
a broad range of frequencies, DM’s emission or absorption is restricted to
only one specific frequency $\omega_{o}$ that is characteristic for DM
particles.

An attempt to formulate a nonrelativistic quantum theory of cold DM based on
the NAE was done in [20]. In this theory, a pair of DM particles interact only
gravitationally and the particles are represented by a scalar wavefunction,
whose evolution in time and space is described by the NAE. The obtained
solutions for the wavefunction show that the particles may exchange the quanta
of energy $\varepsilon_{o}$ between themselves or with other particles in the
halo. Moreover, the velocities of the particles in the pair may change due
collissions with other DM particles.

The main disadvantage of the theory developed in [20] is that all
gravitational effects of the halo on the pair of DM particles were neglected,
and that the density in the halo was assumed to be constant, which is not
supported by the known halo models (e.g., [38-41]). In the following, the
theory is generalized to account for the effects of the halo including non-
constant density.

##  4 Dark matter halos and their quantum structure

###  4.1 Quantum theory of halos

Consider a spherical halo of DM particles with radius $R_{h}$ and total mass
$M_{h}$. Let a DM particle of mass $m$ be located at point $P$, whose distance
from the center of the halo is $R$ (spherical coordinate). The force acting on
the particle is $F_{g}(R)=GM(R)\ m/R^{2}$, and this force is balanced by the
centrifugal force $F_{c}(R)=mR\ \Omega_{c}^{2}(R)$, which gives the circular
orbital frequency of the particle

$\Omega_{c}^{2}(R)=\frac{GM(R)}{R^{3}}\ .$ (9)

where $M(R)=4\pi\int_{0}^{R}\rho(\tilde{R})\tilde{R}^{2}d\tilde{R}$ with
$\rho(R)$ being the density profile of DM inside the halo, which must be
specified.

The new asymmetric equation given by Eq. (6) describes the temporal and
spatial evolution of the wavefunction $\phi_{A}(t,R)$ for free DM particles.
In a galactic DM halo, the particles are affected by the halo’s gravitational
field and undergo gravitational interactions between themselves; in addition,
the DM particles may also collide. In the theory developed in this paper, the
effects of halo’s gravitational field on the particles are fully accounted
for. However, gravitational interactions between two DM particles are
neglected because they are very weak [20]. Moreover, as the obtained results
show, collisions are only important in the cores and their roles in halo’s
envelopes are negligible. This shows that the NAE for free particles must be
modified to take into account interactions of DM particles with the halo’s
gravitational field. To achieve it, $\Omega_{c}^{2}(R)$ is added to Eq. (6) as
it plays the role of gravitational potential.

The resulting equation

$\left[{{\partial^{2}}\over{\partial
t^{2}}}-i\frac{\varepsilon_{o}}{2m}|\mathbf{k}\cdot\mathbf{\hat{R}}|\frac{\partial}{\partial
R}+\Omega^{2}_{c}(R)\right]\phi_{A}(t,R)=0\ ,$ (10)

becomes the governing equation for the theory developed in this paper. After
separating the variables, $\phi_{A}(t,R)=\chi(t)\eta(R)$, the equation for
$\eta(R)$ is

$\frac{d\eta}{\eta}=i\frac{2m}{\varepsilon_{o}|\mathbf{k}\cdot\mathbf{\hat{R}}|}\left[\mu^{2}-\Omega^{2}_{c}(R)\right]dR\
,$ (11)

where $\mathbf{\hat{R}}=\mathbf{R}/R$, and $\mu^{2}$ is the separation
constant to be determined.

Then, the real part of the solution of Eq. (11) is

$\eta(R)=\eta_{o}\cos\left(\frac{2Gm^{2}[\mu^{2}R-I_{c}(R)]}{\varepsilon_{o}^{2}|\mathbf{\hat{k}}\cdot\mathbf{\hat{R}}|}\right)\
,$ (12)

where $I_{c}(R)=\int_{0}^{R}\Omega_{c}^{2}(\tilde{R})d\tilde{R}$, and
$\mathbf{k}=k\mathbf{\hat{k}}$, with $k=1/\lambda_{o}=Gm^{2}/\varepsilon_{o}$
= const [20].

The maxima of the cos function are when its argument is $\pm 2n\pi$, with
$n=0$, 1, 2, 3, … . After using $\varepsilon_{o}=\hbar\omega_{o}$, the
condition for the maxima can be written as

$\mu^{2}-\frac{1}{R}I_{c}(R)=\pm
n\pi\omega_{o}^{2}|\mathbf{\hat{k}}\cdot\mathbf{\hat{R}}|\frac{\hbar^{2}}{Gm^{3}R}\
.$ (13)

To determine the separation constant $\mu^{2}$, the integral $I_{c}(R)$ must
be evaluated in the entire halo that is from $R=0$ to $R=R_{h}$. It can be
shown that the result of this integration is

$I_{c}(R_{h})=\int_{0}^{R_{h}}\Omega_{c}^{2}(\tilde{R})d\tilde{R}=C_{\rho}R_{h}\Omega_{h}^{2}\
,$ (14)

where $C_{\rho}$ is a dimensionless constant whose value is different for
different density profiles (see specific examples in Sections 4.2 and 4.3).
Then, the separation constant $\mu^{2}$ is given by

$\mu^{2}=C_{\rho}\Omega_{h}^{2}\pm
n\pi\omega_{o}^{2}|\mathbf{\hat{k}}\cdot\mathbf{\hat{R}_{h}}|\frac{\hbar^{2}}{Gm^{3}R_{h}}\
.$ (15)

Taking $\mu^{2}=C_{\rho}\Omega_{n}^{2}$, with $\Omega_{n}$ being the quantized
orbital frequencies, Eq. (15) becomes

$\Omega_{n}^{2}=\Omega_{h}^{2}\pm n\pi\kappa_{h}\omega_{o}^{2}\ ,$ (16)

where $+$ and $-$ is for the inside and outside of the halo, respectively, and
$\kappa_{h}=\hbar^{2}/(Gm^{3}C_{\rho}R_{h})$ is the dimensional constant that
connects the universal constants $G$ and $\hbar$ and the mass $m$ of DM
particles to the halo radius $R_{h}$ and the density gradient constant
$C_{\rho}$. Moreover, the term $|\mathbf{\hat{k}}\cdot\mathbf{\hat{R}_{h}}|=1$
because the unit vector $\mathbf{\hat{R}_{h}}$ is not restricted and thus it
can always be aligned with the unit vector $\mathbf{\hat{k}}$. Multiplying Eq.
(16) by $\hbar^{2}$, the spectrum of quantum energies corresponding to the
orbits $\Omega_{n}^{2}$ inside (’+’) and outside (’-’) the halo is obtained

$E_{n}^{2}=E_{h}^{2}\pm n\pi\kappa_{h}\varepsilon_{o}^{2}\ .$ (17)

To calculate the radius of the quantized orbits of the halo, the conservation
of angular momentum, $L_{n}=mR_{n}^{2}\Omega_{n}=$ const, must be used. Since
the value of $L_{0}$ is fixed and known,
$L_{0}=mR_{0}^{2}\Omega_{0}=mR_{h}^{2}\Omega_{h}=L_{h}=$ const, the radius
$R_{n}$ is obtained from $L_{n}=L_{h}$ with $n\geq 0$, which gives

$R_{n}=R_{h}\sqrt{\frac{\Omega_{h}}{\Omega_{n}}}\ ,$ (18)

where $\Omega_{n}=\sqrt{\Omega_{h}^{2}+n\pi\kappa_{h}\omega_{o}^{2}}$ for the
quantized orbits inside the halo ($R_{n}<R_{h}$), and
$\Omega_{n}=\sqrt{\Omega_{h}^{2}-n\pi\kappa_{h}\omega_{o}^{2}}$ for the
quantized orbits outside the halo ($R_{n}>R_{h}$), with
$\Omega_{h}^{2}>n\pi\kappa_{h}\omega_{o}^{2}$.

For the orbits inside (’+’) the halo, there is no restriction on $n$ as when
$n\rightarrow\infty$, then $R_{n}\rightarrow 0$. However, for large values of
$n$, the orbits become so dense that they form a continuum, which is
identified with a central core of the halo. In other words, the halo has its
central core, in which DM particles can move on any orbit. Since these orbits
form the continuum, DM particles may easily collide with each other. From a
physical point of view, the situation is different in the envelope that
surrounds the core as all orbits in the envelope are quantized and there is a
finite number of them between the core and the edge of the halo; more details
can be found in Section 5.

Now, for the orbits outside (’-’) the halo, the value of $n$ is restricted by
the condition $\Omega_{h}^{2}>n\pi\kappa_{h}\omega_{o}^{2}$, which means that
there is a value of $n$ for which $\Omega_{h}^{2}\approx
n\pi\kappa_{h}\omega_{o}^{2}$; in this case, when $\Omega_{n}\rightarrow 0$,
then $R_{n}\rightarrow\infty$. Since this happens for a finite value of $n$,
the number of quantized orbits outside the halo can be estimated by using
$n\approx\Omega_{h}^{2}/(\pi\kappa_{h}\omega_{o}^{2})$, with understanding
that $n$ is the largest integer resulting from this relationship.

The described quantum structure of the halo and its energy spectrum resemble
an atom with its available energy levels. In the following, the theory is now
applied to a halo with constant density, and then to another halo with a given
density profile $\rho(R)$; the physical pictures of the DM halos emerging from
these models are discussed.

###  4.2 Halos with constant density

The simplest density profile is $\rho(R)=\rho_{o}=$ const, with $\rho_{o}$
being specified. Then, the mass of halo inside the radius $R$ is given by
$M(R)=4\pi\rho_{o}R^{3}/3$, and the mass of the entire halo is
$M_{h}=M(R_{h})=4\pi\rho_{o}R_{h}^{3}/3$. Using these masses, the
corresponding orbital frequencies are: $\Omega_{c}=4\pi G\rho_{o}/3$, and
$\Omega_{h}=\Omega_{c}(R_{h})=GM_{h}/R_{h}^{3}=4\pi G\rho_{o}/3$, which shows
that the orbital frequency remains the same at each point inside the halo
[20]; however, it must be noted that the orbital velocity
$v_{h}=R_{h}\Omega_{h}$ at the edge of the halo is smaller by the factor
$\sqrt{2}$ than the escape velocity $v_{esc}$ from the halo.

For this model, the evaluation of the integral $I_{c}(R_{h})$ gives the
dimensionless constant $C_{\rho}=1$, which means that
$\kappa_{h}=\hbar^{2}/(Gm^{3}R_{h})$. By applying the theory developed in
Section 4.1, it is seen that there no quantized orbits inside the halo because
all orbital frequencies in this model are the same; thus, the quantization
rules given by Eqs (16) and (17) with the ’+’ sign cannot be applied inside
the halo. However, the quantization rules with the ’-’ sign can be applied to
space that surrounds the halo, and they show that if any DM or OM particles
are present there, then they must be confined to the orbits that are
quantized; the developed theory allows finding such orbits and their location
around the halo after $m$ and $\omega_{o}$ are either specified (see Section
4.3) or determined by future DM experiments.

###  4.3 Halos with linear density profile

Let a halo of radius $R_{h}$ and total mass $M_{h}$ have its density profile
given by

$\rho(R)=\rho_{c}\left(1-\frac{R}{R_{h}}\right)\ ,$ (19)

where $\rho_{c}=3M_{h}/\pi R_{h}^{3}$ is the central density of the halo; this
density profile is much simpler than the so-called Einasto or NFW profiles
(e.g., [38-41]); nevertheless, the model illustrates well all important
aspects of the developed quantum theory of DM halo.

The mass distribution $M(R)$ resulting from this density profile gives

$\Omega_{c}^{2}(R)=4\Omega_{h}^{2}\left[1-\frac{3}{4}\left(\frac{R}{R_{h}}\right)\right]\
.$ (20)

In the limits of $R=R_{h}$ and $R=0$, one obtains
$\Omega_{c,min}^{2}(R_{h})=\Omega_{h}^{2}$ and
$\Omega_{c,max}^{2}(0)=4\Omega_{h}^{2}$, respectively, which represents the
range of orbital frequencies allowed in this model.

Using Eq. (20), the integral given by Eq. (14) can be evaluated, and the
result is $I_{c}(R_{h})=(5/2)R_{h}\Omega_{h}^{2}$, which shows that
$C_{\rho}=5/2.$ Then, the resulting spectrum of quantized orbits and energies
is given by Eqs (16) and (17), respectively, with the dimensionless constant
being $\kappa_{h}=\hbar^{2}/(5Gm^{3}R_{h}/2)$. Using Eq. (18) and the obtained
range of orbital frequencies, one finds $R_{max}=R_{h}$, which corresponds to
$\Omega_{c,min}=\Omega_{h}$, and $R_{min}=0.7R_{h}$, which corresponds to
$\Omega_{c,max}=2\Omega_{h}$. Identifying $R_{min}$ as the radius of a core,
in which the density is $\rho_{c}=$ const, it is seen that 70% of of the size
of the halo is its core, but the envelope is only 30%.

As already shown in Section 4.2, the quantization rules given by Eqs (16) and
(17) apply only to the envelope (the ’+’ sign) and to space that surrounds the
halo (the ’-’ sign). The reason is that the core’s density is constant, which
means that all its orbital frequencies are the same, hence, they cannot be
quantized. The highest orbital frequency is at $R_{min}$, and then its value
decreases towards the edge of the halo, where it becomes $\Omega_{h}$. In the
vicinity of the halo, the orbital frequency continues to decrease with
distance and approaches zero when $R\rightarrow\infty$.

To perform some estimates, let the total mass of the halo be $M_{h}\sim
10^{10}M_{sun}\approx 10^{40}$ $kg$, and its radius be $R_{h}\sim 10^{20}$
$m$. Then, $\Omega_{h}^{2}\sim 10^{-30}$ $s^{-1}$, which corresponds to the
orbital velocity at $R=R_{h}$ to be about $100$ $km$ $s^{-1}$ [42,43]. In
addition, the dimensionless constant $\kappa_{h}$ can also be calculated by
specifying the mass $m$ of the DM particles. Taking $m\approx 10m_{p}\approx
10^{-26}$ $kg$, where $m_{p}$ is the proton mass, the resulting
$\kappa_{h}\sim 1$; however, if $m\approx 100m_{p}$, which is the Higgs boson
mass, then, $\kappa_{h}\sim 10^{-3}$.

Since $\omega_{o}$ and $m$ are currently unknown, one may take
$\Omega_{h}/\omega_{o}\sim 10$ and $\kappa_{h}\sim 1$, which allows obtaining
$\Omega_{n}$ and using Eq. (18) to find the values of $R_{n}$ in the envelope
as well as in the vicinity of the halo. It is seen that $R_{min}$ in the
envelope is reached by a finite number of $n$, and that this number increases
by three orders of magnitude when $\kappa_{h}\sim 10^{-3}$ is used; in both
cases, the density of orbits near the core increases, and this increase is
more prominent in the second case. On the other hand, the number of quantized
orbits outside the halo can be calculated by using
$n\approx\Omega_{h}^{2}/(\pi\kappa_{h}\omega_{o}^{2})$ and the above values of
$\Omega_{h}$, $\kappa_{h}$ and $\omega_{o}$. For $\kappa_{h}\sim 1$, one finds
$n\sim 30$, and for $\kappa_{h}\sim 10^{-3}$, it is $n\sim 3\cdot 10^{4}$,
which shows a strong dependence of the number of quantized orbits on the value
of $\kappa_{h}$, or more precisely, on the value of the currently unknown mass
of the DM particles.

Finally, it must be pointed out that the size of the core and envelope, as
well as the number of quantized orbits may significantly change when more
realistic density profiles for DM halos (e.g., [38-41]) are considered, and
when the values of $m$ and $\omega_{o}$ for DM are experimentally established.

##  5 Physical implications and predictions

The developed quantum theory based on a new asymmetric equation is
significantly different than the QM based on the Schrödinger equation. The
main difference is that the theory can be applied to a spherical halo of DM
particles, and it predicts its quantum structure that contains a core, which
is free of quantum orbits, and a surrounding envelope that may contain many
quantized orbits. This shows that the halo’s quantum structure resembles that
known in atoms.

There are two physical constants $\varepsilon_{o}$ and $\kappa_{h}$ in the
theory that play essential roles in establishing the quantum structure of the
DM halo on its global scale. One of these constants is the quanta of energy
$\varepsilon_{o}=\hbar\omega_{o}$, which is fixed for DM; however, the value
of its characteristic frequency $\omega_{o}$ is unknown. DM particles may
absorb and emit $\varepsilon_{o}$, and they may also interact gravitationally
with each other by exchanging virtual quanta of energy; for this reason,
$\varepsilon_{o}$ is called a dark graviton [20]. Since the theory requires
that $\omega_{o}$ = const, DM particles can only emit or absorb radiation with
this one frequency.

The other constant $\kappa_{h}=\hbar^{2}/(Gm^{3}C_{\rho}R_{h})$ is
dimensional, and an interesting combination of the universal constants $\hbar$
and $G$, the mass $m$ of a DM particle, the density profile constant
$C_{\rho}$, and the radius of the DM halo $R_{h}$. The value of $\kappa_{h}$
directly affects the orbit and energy quantization rules given by Eqs (16) and
(17), which can also be used to determine the radius $R_{n}$ of quantized
orbits (see Eq. 18). The results presented in Section 4.1 show that there is
no restriction on $n$ for the orbits inside the halo because when
$n\rightarrow\infty$, then $R_{n}\rightarrow 0$. However, for large values of
$n$, the orbits become so dense that they practically form a continuum, which
is identified with a core of the halo. The quantized orbits exist in the
envelope that surrounds the core, and they also exist in space outside of the
halo. These different parts of halo are now discussed.

The fact that the orbits form a continuum inside the core means that the
density inside the core is constant, which implies that there are no quantized
orbits in the core (see Sections 4.2 and 4.3). As a result, the DM particles
in the core move randomly and undergo frequent collisions. There is a narrow
region in the immediate vicinity of the core in which many orbits may exist
close to each other, and form a transition between the core and envelope. In
this narrow region, the density of orbits maybe so high that the DM particles
on such orbits may directly interact with the randomly moving and colliding
particles of the core causing some particle’s exchange. Moreover, the core may
be filled with free dark gravitons (the fixed quanta of energy $\varepsilon$
[20]), which may contribute to the gravitational wave background [44] and make
it different than such wave background of the envelope that surrounds this
core; the difference may be detected observationally by the NANOGrav detector
[45] other similar future experiments.

The envelope that surrounds the core in the halo contains the quantized
orbits, and the DM particles are confined to these orbits. The main reason for
this confinment is the lack of transitions between the orbits, which is caused
by the fact that the orbital frequency differences
$\Delta\Omega_{n}=\Omega_{n+1}-\Omega_{n}$ are not exact multiples of
$\varepsilon_{o}$, because they also depend on $\kappa_{h}$. As a result, DM
particles cannot move from one orbit to another by emitting or absorbing
$\varepsilon_{o}$. Instead, the particles are confined to their orbits with no
other quantum restrictions, since they are spinless and have no charge. An
exception could be a narrow region very close to the core, where many orbits
may exit, and the quantum jumps (by absorbing or emitting $\varepsilon_{o}$)
between these orbits and the core may take place.

In the considered model of halos, all DM particles are inside the halo with
space surrounding the halo being matter free. Nevertheless, the presence of
the halo itself is resposnible for the existence of quantized orbits outside
the halo. The developed theory demonstrates that the number of such orbits is
a sensitive function of the two basic parameters of the theory, namely,
$\omega_{o}$ and $\kappa_{h}$, which are currently unknown; the smallest are
these parameters, the largest is the number of quantized orbits outside the
halo.

It must be pointed out that the predicted quantum structure of the halo is
only valid for the elementary DM particles, for which the wave-particle
duality is important, and whose masses are within the range of the proton and
Higgs masses. For classical particles, whose masses are many orders of
magnitude higher, the parameter $\kappa_{h}$ becomes very small and the
quantum effects are negligible; this seems to be consistent with the fact that
DM cannot form gravitationally bounded objects larger than pairs of DM
particles [20]. Moreover, the theory cannot be applied to OM particles in
gaseous nebulae because their electromagnetic and plasma effects would
supersede all gravitational effects described in this paper for DM halos.

The main aim of this paper is to present a quantum theory of DM halos and
discuss physical implications and predictions of the theory. An important
topic of the origin of DM is outside the scope of the presented approach.
Nevertheless, the topic has been considered in the literature and different
explanations have been proposed (e.g., [1-8]). For example, a relationship
between dissipative DM and intermediate black holes was suggested [46], and it
was also proposed that primordial black holes can be responsible for the
origin of either DM or large-scale cosmic structures, depending on their
initial masses [47]. Another interesting idea is that microscopic warmholes
and extra-dimensions can generate both DM and dark energy [48]. However, as
pointed out by these authors, the problem of the origin and physical nature of
DM remains one of the most challenging and unsolved problems in modern
cosmology.

##  6 Conclusions

A quantum theory of DM particles in a spherical halo is developed based on a
new asymmetric equation [19,20], which is complementary to the Schrödinger
equation. The two physical parameters that determine the theory and its
predictions are the quanta of energy $\varepsilon_{o}=\hbar\omega_{o}$, with
$\omega_{o}$ being a fixed frequency, and a new dimensional parameter
$\kappa_{h}=\hbar^{2}/(Gm^{3}C_{\rho}R_{h})$, which combines the universal
constants $\hbar$ and $G$, the mass $m$ of DM particles, the density profile
constant $C_{\rho}$, and the radius of the DM halo $R_{h}$.

The theory predicts that the halo contains a core that is surrounded by an
envelope. The core is filled with free and randomly moving DM particles that
collide with each other. The envelope contains the quanitized orbits on the
halo’s global scale with the lowest frequency (energy) orbit located at the
edge of the halo. The orbital frequency (energy) inreases towards the core and
the quantized orbits become more densely populated to form a continuum inside
the core. The described quantum structure of the halo resembles an atom.

To determine the distribution of quantized orbits in the envelope, the density
profile in the halo as well as its radius and total mass must be specified.
This means that the distribution of quantized orbits is consistent with the
global physical parameters of the halo, and that the orbits are populated by
DM particles, so that the density profile is accounted for. Since DM particles
that populate the orbits are spinless and have no charge, there no are other
quantum limits on the number of particles on each orbit.

DM particles are allowed to emit or absorb the quanta of energy,
$\varepsilon_{o}$, called dark graviton. However, since differences between
the quantized orbits are not exact multiples of $\varepsilon_{o}$, as they
also depend on $\kappa_{h}$, DM particles are permanently confined to their
orbits. An exception could be a narrow region in the immediate vicinity of the
core, where many orbits may exit. DM particles may undergo quantum jumps
between these orbits and the core by absorbing or emitting $\varepsilon_{o}$,
which requires that the differences between some orbits and the core are
exactly equal to the quanta of energy $\varepsilon_{o}$. The existence of the
dark gravitons, or a sea of these gravitons in the core, may cause its
gravitational wave background to be different than that generated by the
envelope of the halo.

Acknowledgment: The author is indebted to an anonymous referee for providing
comments and suggestions that allowed significantly improved this paper. The
author also thanks Dora Musielak for valuable comments on the earlier version
of this manuscript. This work was partially supported by Alexander von
Humboldt Foundation.

References

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Current affiliation: Maastricht University, P.O. Box 616, 6200 MD Maastricht,
The Netherlands Current affiliation: Nikhef, Science Park 105, 1098 XG
Amsterdam, The Netherlands These authors contributed equally to this work.
Istituto Nazionale di Fisica Nucleare, Sezione di Genova, via Dodecaneso 33,
16146 Genova, Italy These authors contributed equally to this work.

# Effects of mixing and annealing on the optical and mechanical properties of
TiO2:Ta2O5 amorphous coatings

Alex Amato  Université de Lyon, Université Claude Bernard Lyon 1, CNRS,
Institut Lumière Matière, F-69622 Villeurbanne, France
<EMAIL_ADDRESS>Michele Magnozzi  OptMatLab, Dipartimento di
Fisica, Università di Genova, via Dodecaneso 33, 16146 Genova, Italy
<EMAIL_ADDRESS>Nikita Shcheblanov  Navier, Ecole des Ponts, Univ
Gustave Eiffel, CNRS, Marne-la-Vallée, France  Anaël Lemaître  Navier, Ecole
des Ponts, Univ Gustave Eiffel, CNRS, Marne-la-Vallée, France  Gianpietro
Cagnoli  Université de Lyon, Université Claude Bernard Lyon 1, CNRS, Institut
Lumière Matière, F-69622 Villeurbanne, France  Massimo Granata  Christophe
Michel  Laboratoire des Matériaux Avancés - IP2I, CNRS, Université de Lyon,
Université Claude Bernard Lyon 1, F-69622 Villeurbanne, France  Gianluca Gemme
Istituto Nazionale di Fisica Nucleare, Sezione di Genova, via Dodecaneso 33,
16146 Genova, Italy  Laurent Pinard  Laboratoire des Matériaux Avancés - IP2I,
CNRS, Université de Lyon, Université Claude Bernard Lyon 1, F-69622
Villeurbanne, France  Maurizio Canepa  OptMatLab, Dipartimento di Fisica,
Università di Genova, via Dodecaneso 33, 16146 Genova, Italy

###### Abstract

Amorphous mixed titania-tantala coatings are key components of Bragg
reflectors in the gravitational wave detectors (GWDs). Attaining the lowest
possible values of optical absorption and mechanical losses in the coatings is
of paramount importance for GWDs, and this requires a complex optimization of
the coating deposition and post-deposition annealing. We present here a
systematic investigation of the optical properties and internal friction of
amorphous mixed titania-tantala coatings grown by ion beam sputtering. We
consider coatings with six different cation mixing ratios - defined as
Ti/(Ti+Ta) - and we study them both in the as-deposited and annealed state.
All coatings have been subject to the same annealing of 500 ∘C for 10 hours in
air, which is the post-deposition treatment adopted so far for Bragg
reflectors in GWDs applications. By exploiting spectroscopic ellipsometry data
and modelling, along with ancillary techniques, we retrieved the dielectric
function of the coatings in a wide spectral range. When varying the mixing
ratio and performing the annealing we find monotonic - in some cases, almost
linear - trends for most of the aforementioned properties. Remakably, the
post-annealing Urbach energy displays a definite minimum for a mixing ratio
around 20$\%$, very close to the composition of the coatings showing the
lowest optical absorption for GWDs applications. We suggest that the observed
minimum in the Urbach energy depends not only on the mixing ratio, but also on
the annealing parameters. On the other hand, the minimum coating loss angle
was found to be weakly dependent on the considered measurement frequency and
to lie within a rather broad range of Ti content (cation ratios of 21% and
44%), suggesting that the search for an absolute minimum following post-
deposition annealing should be rather sought in the study of the best
annealing parameters for each specific cation ratio considered. This work
constitutes a reference for the optical properties of the amorphous mixed
titania-tantala coatings, and highlights the relevance of the Urbach energy in
the optimization process of materials for high-performing Bragg reflectors.

_keywords: amorphous semiconductors, thin films, Urbach energy, internal
friction, loss angle, ellipsometry, gravitational waves._

##  1 Introduction

Amorphous oxides are key constituents of high-performance Bragg reflectors in
technologies as diverse as optical atomic clocks1, passive optical
gyroscopes2, ultrastable frequency laser cavities3 and Fabry-Perot cavities of
present and future ground-based gravitational-wave detectors4, 5, 6, 7 (GWDs).
In such advanced applications, optical multilayer coatings must display
exceedingly low levels of optical absorption and thermal noise8, as these two
properties often set the limits of technological possibilities9. While the
precise microstructural features responsible for either optical absorption 10
or thermal noise 11 are still subject to discussion, it is widely expected
that they originate from different types of defects of the amorphous
structure. The best amorphous materials for high-performance Bragg reflectors,
hence, should be in the most relaxed and lowest potential energy state.

Optical multilayer coatings for GWDs are composed of alternating high- and
low-refractive index amorphous oxides deposited at room temperature using ion
beam sputtering (IBS) 12. Deposition is routinely followed by thermal
annealing to decrease the internal stress and the optical absorption of the
coatings 13. Crucially, annealing must avoid crystallization in order to
preserve material homogeneity and isotropy, and to prevent optical scattering.
The limits of the annealing temperature and duration are thus set by the
material in the stack which crystallizes first 14.

Silica is the unchallenged low-index material for room-temperature
applications as IBS deposition and annealing at moderate temperatures suffice
to confer it excellent properties15. On the other hand, there is no evident
solution of comparable quality for high-index layers. Good high-index
materials, such as Ta2O5, TiO2 or HfO2 present too high a thermal noise, as
deposited, and meanwhile crystallize easily upon annealing: for these reasons,
they cannot be used in pure form.

A strategy of primary importance towards the improvement of mirror coatings
therefore consists in identifying adequate high-index _mixed oxides_16, 17.
Indeed, it is known in many different contexts, ranging from metallic glasses
to suspensions, that mixing strongly improves the stability of glasses against
crystallization 18, 19, 20. The goal is then to optimize a complex trade-off
between refractive index, stability against crystallization, optical
absorption, and thermal noise. This must be accomplished in consideration of
the entire stack. In fact, thermal noise depends on the amount of materials
used for the coatings and improving the index contrast between layers reduces
the number of doublets while retaining the same reflectivity, thus
contributing to reduce the overall losses and thermal noise.

The simplest mixed oxides are typically envisioned as a combination of two
pure oxides, one which is stable at the relevant annealing temperatures, and
another one which has a higher index. Important examples include TiO2-Ta2O521,
22, 14, 23, SiO2-TiO224, TiO2-Nb2O525, GeO2-TiO226, and ZrO2-Ta2O520.

Today, the physical mechanisms at work in mixing remain to be clearly
understood, with the consequence that the identification of optimal binary
mixtures largely rests on educated guesses and a trial-and-error process 23.
In particular, the oxide mixture currently used in GWDs mirrors is TiO2-Ta2O5
with a Ti/(Ti+Ta) ratio around 20$\%$ because this was found to yield the
lowest mechanical and optical losses after a thermal annealing of 500 ∘C in
air 14, 21. However, the relation between the mixing ratios, the annealing
parameters, and the resulting optical/mechanical properties of these mixtures,
remain insufficiently characterized.

In this work, we study TiO2-Ta2O5 coatings by systematically varying the
cation mixing ratio (defined in percentage as Ti/(Ti+Ta)), before and after
the standard annealing procedure for GWDs applications (10 hours at 500 ∘C in
air). We measure the dielectric function in a wide spectral range, accurately
characterizing the fundamental absorption edge of the coatings. This edge is
characterized not only by the energy gap, but also by a measure of the tail
states within the gap - that is, the Urbach energy. This parameter is of
particular relevance because it has been associated with topological or
compositional disorder27, which must be kept at a minimum in the materials
composing high-performance Bragg reflectors. All the optical properties
considered in this work scale monotonously - with some tendence to linearity -
with the cation ratio, save for the Urbach energy of the annealed coatings,
which shows a very clear minimum for cation ratio around 20$\%$ \- that is,
the composition at which the lowest optical absorption has been measured in
this kind of coatings8. On the other hand, when measuring the mechanical loss,
we find that the post-annealing values do not show a clear minimum, but are
relatively low over a broad range of Ti/(Ta+Ti) ratios.

##  2 Materials and Methods

###  2.1 Coating samples

Our samples consist in six sets of amorphous oxide coatings of varying cation
ratio Ti/(Ta+Ti)$=0\%$, 21$\%$, 44$\%$, 71$\%$, 85$\%$ and 96$\%$, as measured
through Rutherford backscattering spectrometry. They were deposited by IBS at
the Laboratoire des Matériaux Avancés111http://lma.in2p3.fr/ using a custom
made coater machine, the so-called Grand Coater 28, developed to deposit the
large high-reflective mirrors for GWDs.

For optical investigation, in order to avoid backside reflections, the
coatings were deposited on single-side polished silicon wafers. For mechanical
measurements, which require low-loss substrates, they where deposited on
silica disk-shaped resonators (2" or 3" of diameter and 1 mm thick). The Grand
Coater enabled us to deposit these coatings at the same time on all the
required substrates, thus guaranteeing that the optical and mechanical
investigation are carried out on identical samples.

All samples have been analysed as-deposited and after annealing at 500 ℃ for
10 hours (ramp rate of 100 ℃/hour), which is the post-fabrication annealing
procedure for GWDs coatings so far adopted.

Annealing induced the partial crystallization of the two coatings with the
highest cation ratio (85$\%$ and 96$\%$). In this work we focus on the
properties of amorphous coatings, therefore we analyzed these two samples only
in the not-annealed state.

###  2.2 Optical Characterization

The optical characterization relied on spectroscopic ellipsometry (SE), which
is especially well suited to study thin films 29, 30, 31, and which gives
access to both electronic and optical properties via a model-based data
analysis.

Ellipsometry measurements were performed with a J.A. Woollam Co. Variable
Angle Spectroscopic Ellipsometer. The considered spectral range was 190-2500
nm (0.5-6.5 eV), thus encompassing a large portion of the spectrum both below
and above the energy gap. For each dataset the ellipsometry angles $\Psi$ and
$\Delta$ plus the percentage of depolarization were acquired at three angles
of incidence (AOI) (55∘, 60∘, 65∘); the spectral density was set to one
datapoint each nm, totalling 2310 datapoints per spectrum. SE data were
analyzed with the software WVASE 3.718 by J.A. Wollam Co.

In order to model the SE data, we adapted a well-proven strategy based on the
Cody Lorentz (C-L) model32, which was designed to describe the optical
response of amorphous semiconductors in a wide spectral range. In the C-L
model, the imaginary part of the dielectric function reads:33

$\varepsilon_{2}=\begin{cases}L(E)G(E),&E>E_{t}\medspace,\\\ U(E),&0<E\leq
E_{t}\medspace,\end{cases}$ (1)

where $L(E)$ is the Lorentz oscillator and $G(E)$ describes the Cody behaviour
for a parabolic electronic density of states

$G(E)=\frac{(E-E_{g})^{2}}{(E-E_{g})^{2}+E_{p}^{2}}\medspace,$ (2)

where $E_{g}$ is the energy gap and $E_{p}$ is the limit energy separating the
Cody behaviour from the Lorentz oscillator. $U(E)$ is related to the Urbach
tails via the following equation,

$U(E)=\frac{E_{t}L(E_{t})G(E_{t})}{E}e^{\frac{E-E_{t}}{E_{U}}}\medspace,$ (3)

where $E_{t}$ is the limit between the localized transitions of exponential
tails and transitions involving only extended states and $E_{U}$ is the Urbach
energy. In addition to the C-L model we also fitted the experimental SE data
point by point, that is, without a parametric dispersion model. In a point-by-
point fit, the calculated optical constants at different wavelengths are
independent of one another34; this guarantees the best agreement with the
experimental data, however the resulting optical constants are more noisy with
respect to the ones obtained with a parametric model. We use the point-by-
point fit as a further validation of the C-L model and results.35.

###  2.3 Mechanical Loss Characterization

In amorphous materials, thermally driven random transitions cause a local
variation of the elastic moduli of the film giving rise to dissipation. This
mechanical dissipation is measured through the resonant method36 using a
Gentle Nodal Suspension (GeNS)37, 38, 13. In particular, at each cycle of
vibration the dissipated energy $E_{\text{diss}}$ is related to the total
energy stored in system $E_{\text{st}}$ through the relation

$E_{\text{diss}}=2\pi\phi E_{\text{st}}\medspace,$ (4)

where $\phi$ is the so-called loss angle that quantifies the internal
friction. The loss angle is obtained from the characteristic decay time
$\tau_{k}$ of several resonance modes $k$ with frequency $\nu_{k}$ in the
range $0.1-40$ kHz through the relation $\phi_{k}=(\pi\nu_{k}\tau_{k})^{-1}$.
In composite resonators, made of substrate and coating, the coating loss angle
is worked out by subtraction of the measurements of the bare substrate from
that of the coated sample39, 14. Once the coating loss was obtained, we used a
power law to describe the frequency dependence across the measured frequency
range. The power-law dependence of the loss angle is derived from an
exponential behaviour of the barrier height in a two-level system (TLS)40 as
it has been confirmed in silica samples and over a wide temperature range41
and in Ti-doped tantala molecular dynamics simulations11.

In our study, for each Ti concentration, two nominally identical substrates
have been coated at the same time in order to increase the statistics of the
results and have a double check of the measured loss angles.

##  3 Results

Figures 1a) and 1b) show the ellipsometry data ($\Psi$ and $\Delta$) measured
on the as-deposited (_i.e._ , not annealed) samples. For the sake of clarity,
only the data at 60∘ AOI are reported here. Generally speaking, each spectrum
in Fig. 1 exhibits three regimes depending on the spectral region. In the
region of transparency (low photon energies) the spectra are characterized by
periodic structures such as peaks or troughs due to the interference of
multiple reflections inside the coating. In the region of strong absorption
(high photon energies) the spectra are featureless. Between the two mentioned
regions lies the onset of absorption. The 3D plots in Fig. 1a) and 1b) allow
to compare the features of the samples as the mixing ratio is varied. The most
notable trend, which is visible both in the $\Psi$ and $\Delta$ spectra, is
the redshift of the onset of absorption as the mixing ratio is increased.

Figure 1: a), b): $\Psi$, $\Delta$ spectra measured on the as-deposited
(_i.e._ , not annealed) samples. The spectra are stacked according to the
mixing ratio in each sample, expressed as percent Ti/(Ti+Ta). c), d): $\Psi$
spectra of not annealed (blue curves) and annealed (red curves) coatings with
cation ratio of 21$\%$ and 44$\%$, respectively.

Figures 1c) and 1d) report the $\Psi$ spectra of the sample with Ti/(Ti+Ta)
corresponding to 21$\%$ and 44$\%$, respectively, in the pristine (blue
curves) and annealed (red curves) state. In both cases, the variations induced
by annealing are mainly located in the spectral region corresponding to the
onset of absorption; the largest of these variations (up to 9∘ in $\Psi$) are
observed in the sample with cation ratio of 21$\%$.

Figure 2: Analysis of the SE data measured on the sample with Ti/(Ti+Ta):
21$\%$, annealed. a), b): comparison between experimental data and the results
of Cody-Lorentz (C-L) model fit. Below the graphs are reported the residual
from the C-L fit (gray lines) and from the point-by-point fit (violet lines).
c): $\alpha$ obtained by C-L model (gray line) and point-by-point fit (violet
circles). d): Refractive index (light green line) and extinction coefficient
(dark green line) obtained from the C-L model.

As an example of SE data modelling and fitting, we report in Fig. 2
representative data from the sample with cation ratio of 21$\%$, annealed.
Panels 2a) and 2b) report the $\Psi$ and $\Delta$ at 60∘ AOI, both
experimental and calculated with the C-L model. Below those graphs are
reported the fit residual obtained with the C-L fit and with the point-by
point fit. The good agreement between these two data analysis methods can be
appreciated in 2c), where the absorption coefficient $\alpha$ is reported on a
logarithmic scale. The refractive index and extinction coefficient of the
coating, obtained with the C-L model, are reported in 2d).

In Table 1 and Fig. 3 several properties of the films, resulting from the
analysis of the SE data, are reported for varying the mixing ratio. In the
table, the thickness of coatings before and after the annealing are reported,
along with the percent variation. The data show that the thickness of the
films systematically increases as a result of the annealing; the percent
variation decreases monotonically as the Ti content increases. In Fig. 3a) the
refractive index at $\lambda=1064$ nm is seen to increase as the content of Ti
atoms increases. Furthermore, the annealing reduces the refractive index by
$\sim$1-2%, a finding compatible with the results previously reported on pure
tantala coatings42. By combining the thickness data in Table 1 with the
measurements of the mass and diameter of the films, we calculated their
density which is reported in Fig. 3b); the mass of the films remained constant
upon annealing. The density of the coating decreases as the content of Ti
atoms increases, as expected from the different atomic mass of the two
transition metals. No direct measurement data were available on the samples
with the highest Ti concentration, however an approximate value can be
obtained by exploiting the linear relation between refractive index and
density, which has been demonstrated for TiO2 filmsBundesmann_2017. Based on
such relation, having calculated $n$@550 nm = 2.39 for the not-annealed
coating with 96$\%$ Ti, the density of that coating can be estimated as
3.6$\pm$0.2 g/cm3 (point marked with an asterisk in Fig. 3b)).

In Fig. 3c) the energy gap is reported, which shows a monotonic decrease as
the Ti concentration increases and no clear variation trend induced by the
annealing. Finally, in Fig. 2d) the Urbach energy values are plotted.
Interestingly, the Urbach energy trends for the pristine and annealed samples
are qualitatively different. Pristine samples exhibit a monotonous decrease of
the Urbach energy up to the highest values of Ti concentration; on the other
hand, the datapoints for annealed samples indicate different Urbach energies,
with a clear minimum at the mixing ratio of 21$\%$. At the same mixing ratio
the largest difference of Urbach energy is observed between pristine and
annealed samples.

$\%$ Ti/(Ti+Ta) |  0 |  21 |  44 |  71 |  85 |  96
---|---|---|---|---|---|---
Thickness, not annealed (nm) |  579 |  504 |  480 |  580* |  817 |  532
Thickness, annealed (nm) |  592 |  514 |  489 |  520* |  827** |  538**
$\%$ variation upon annealing |  2.2 |  2.0 |  1.9 |  - |  1.2 |  1.1

*different samples; **crystallized

Table 1:  Thickness of coatings determined by SE for pristine and annealed
samples. The last row reports the percent variation in thickness as a
consequence of the annealing. Figure 3: Properties of coatings as a function
of the mixing ratio, determined from the analysis of ellipsometry data. Not
annealed and annealed coatings are indicated with blue and red markers,
respectively. a) Refractive index at $\lambda=1064$ nm. b) Density. The point
marked with an asterisk is calculated exploiting the linear relation between
refractive index and density in titania coatings, as reported in Ref. [43]. c)
Energy gap. d) Urbach energy. Following the annealing protocol described in
the text, the mixing ratio of 21$\%$ yields both the lowest Urbach energy and
the largest annealing-induced decrease. Figure 4: a) Coating loss angle as
function of frequency of TiO2:Ta2O5 with different cation mixing ratio, before
and after the annealing. For each ratio, the squares and circles refer to
measurements from two different disk-shaped resonators, coated at the same
time and therefore nominally identical.

In Fig. 4 the coating loss angle data are plotted as a function of frequency.
The loss angle is seen to increase with frequency for all samples. Although
the slope varies, the loss angle is systematically smaller in the annealed
samples; the lowest values are those corresponding to mixing ratios of 21$\%$
and 44$\%$, with no clear minimum associated to either of the two cases in the
investigated frequency range.

##  4 Discussion

So far we have determined by measurements and data analysis how several
properties in mixed titania-tantala coatings vary as a function of the mixing
ratio and annealing. Here, we will attempt to link the observed variations and
trends to their underlying physical causes, to draw comparisons with the
existing reports in the literature where possible, and finally to extract
useful information for optimizing the performance of this kind of coatings.

The interpretation of the dielectric function of mixed oxides is usually based
on generalisations of the Clausius-Mossotti (CM) formula, resulting in the so-
called "oxide additivity rule", which allows to predict the dielectric
function (and therefore the square of the refractive index) of a very large
number of binary and ternary metal oxide alloys, in general with an accuracy
better than a few percent 44. Therefore, the CM formula expresses the square
of the refractive index of the mixed oxide TiO2:Ta2O5 as a linear combination
of the polarizabilities of the single oxides Ta2O5 and TiO2. Our data in Fig.
3a) show a monotonic relation between refractive index and mixing ratio, and
are thus consistent with the aforementioned rule.

The relation between density and mixing ratio (Fig. 3.b)) is clearly
monotonic, which results from the atomic mass contrast between the two
transition metals ($m_{a,(Ti)}=47.9$u and $m_{a,(Ta)}=180.9$u). The relation,
however, is only approximately linear for non-annealed samples. Annealing
induces a slight ($\sim$1-2%), but measurable reduction of the film density.
This effect is better seen from the associated changes in film thickness in
Table 1, from which we observe that the relative change is smaller at higher
Ti content. As for pure tantala coatings, our findings are fully compatible
with previous reports both from the standpoint of the absolute density,
7.4$\pm$0.1 g/cm3, which compares with Ref. [45], and with regard to the
density change upon annealing for films of similar thickness ($\sim$1-2% in
Refs. [46, 14, 47, 42]).

We shall now consider the energy gap, displayed in Fig. 2c). The overall trend
for varying cation ratio is approximately linear, with the exception of the
pure tantala data points; this observation is consistent with that reported by
Fazio et al.23. Unlike refractive index and density, the energy gap is not
measurably affected by annealing: in Fig. 2c), the annealed and non-annealed
data points remain consistent within error bars, regardless of the mixing
ratio. This, again, is consistent with, and extends, previous results on pure
tantala coatings with similar thicknesses, the energy gap of which was found
not to be affected by annealing up to 600 ∘C42. Moreover, this insensitivity
with annealing is in line with the idea that the gap is controlled by the
nearest neighborhood of metallic atoms, but not by broader structural features
(medium range order) that are the most affected by structural relaxation48.

The data discussed so far show a monotonic dependence on the cation ratio,
which in some cases is even compatible with a linear relation. For this
reason, the trends of refractive index, density, and energy gap in this kind
of amorphous mixed oxides coatings can generally be understood in terms of the
aforementioned additivity rule. The Urbach energy, reported in Fig. 2 d),
shows instead a quite different picture. Its trend is still monotonic in the
as-deposited coatings, but shows a clear minimum in the annealed ones,
corresponding to the cation ratio of 21$\%$. Moreover, for that mixing value,
the Urbach energy is decreased by $\sim$40 meV upon annealing, which is the
largest reduction among the investigated samples. Therefore, the Urbach energy
manifests two very different trends before and after the annealing. In the
first case, it seems to be governed by the mixing ratio, in other words: the
higher the Ti content, the lower the Urbach energy. In the second case, we
register a significant decrease in the Urbach energy for mixing ratio 0$\%$
and 21$\%$, but very small and no variation at all for mixing ratio 44$\%$ and
71$\%$, respectively. This means that the annealing parameters employed so far
(10 hours at 500 ℃) can have large effects on the Urbach energy, but only for
certain mixing ratios.

In order to explain for this peculiar behavior, it is useful to recall that
from an atomistic point of view, Urbach tails have been related to _extended_
structural patterns, involving a correlation of interatomic bond lengths or
angles overal many interatomic distances49, 50, 51. According to this model,
Urbach tails are mainly related not only to the short range but also to the
intermediate range. Using a combination of experimental and modelling tools,
Prasai et al. demonstrated that a thermal annealing in amorphous zirconia-
tantala coatings leaves the short-range order almost unchanged, but has a
significant effect on the intermediate-range order52. This explains that
annealing can have large effects on the Urbach energy of amorphous mixed oxide
coatings as already shown elsewhere47. Crucially, however, our data show that
these effects depend on the mixing ratio, as they are sizable for low Ti
content, but very small or negligible for medium Ti content. We hypothesize
that the observed behavior is determined by the fact that for medium Ti
content, the maximum temperature (and/or time duration) of the annealing are
not sufficient to remove the extended structural patterns mentioned at the
beginning of this paragraph. Indeed, the same annealing parameters can have
profoundly different effects on the microscopic structure of the coatings with
different mixing ratios - in the extreme case, the same annealing can lead the
coatings with high Ti content to crystallization, while leaving those with low
Ti content perfectly amorphous. Following our hypothesis, we suggest that the
minimum observed in the Urbach energy for cation ratio of 21$\%$ could not be
the one and only for this kind of materials - other minima could be found at
different cation ratios _and_ with a different annealing protocol (_i.e._ ,
with a different combination of maximum temperature and time duration).
Interestingly, the minimum of the Urbach energy reported in this work is
mirrored by the minimum in the optical absorption measured in the region of
transparency (1064 nm) in similar coatings23; this implies that both the
extended disorder (which gives rise to the Urbach tails) and the localized,
individual defects (which give rise to optical absorption deep within the
energy gap) are minimized in annealed titania-tantala coatings with mixing
ratio around 20$\%$.

The available data show that annealing induces a significant reduction of the
loss angle, a finding which confirms the previous results from the
literature35, 23. However, unlike the Urbach energy, the measured loss angles
in the annealed samples do not have a well-defined minimum. In fact, the loss
angle data-points relative to the annealed 21$\%$ and 44$\%$ samples lie very
close to each other, one being slightly lower than the other depending on the
considered frequency. This further strengthens our hypothesis that the search
for an absolute minimum following post-deposition heat treatment must be
sought in the individual study of the best annealing for each different type
of titania-tantala mixing concentration - a study in which temperature-
depentent, _in situ_ SE is a valid research tool53, 54. Furthermore,
considering the frequency trend of the loss angle, the search for a minimum
after annealing may depend on the frequency range of interest.

##  5 Conclusions

We have systematically studied the dielectric function and loss angles in
amorphous titania-tantala coatings, determining how they vary as a function of
the cation mixing ratio and after a thermal annealing. We found that the
aforementioned properties scale monotonously with the mixing ratio, with the
notable exception of the Urbach energy of the annealed samples which has a
clear minimum at the cation ratio of 21$\%$. A minimum in the Urbach energy
can be correlated to the least amount of extended defects in the amorphous
structure, which is a potentially good configuration for a coating material in
GWDs applications. Indeed, a very similar mixing ratio was found to yield the
lowest optical absorption in the transparency region. On the other hand, our
data on mechanical losses show no clear minimum for intermediate Ti content, a
feature that can be ascribed to the choice of the annealing parameters which
might be different for obtaining the lowest loss at each Ti content.
Therefore, we suggest that the minimum we found in the Urbach energy is not
the one and only possibile in amorphous titania-tantala coatings - other
minima could be found at different mixing ratios, depending on the annealing
protocol. This work constitutes a reference for the optical properties of
amorphous titania-tantala coatings and proposes the Urbach energy a parameter
to guide the optimization process of amorphous mixed oxide coatings for high-
performance Bragg reflectors.

## Conflicts of interest

There are no conflicts of interest to declare.

## Acknowledgments

A subset of the optical data contained in this manuscript was included in the
Technical Summary of the Optical Interference Coatings 2022 conference. This
work was supported by the Virgo Coating R$\&$D (VCR$\&$D) Collaboration. We
acknowledge the support of Italian Ministry for University and Research (MUR)
through the project ’Dipartimenti di Eccellenza 2017-2022’ (DIFILAB).

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