# Negative energy states in pionic hydrogen††thanks: Preprint Alberta Thy
1-22. Results of this paper were partially presented at Matter to the Deepest,
September 15-17, 2021, Institute of Physics, University of Silesia, Poland.

Andrzej Czarnecki  Department of Physics, University of Alberta, Edmonton,
Alberta T6G 2E1, Canada

###### Abstract

Probabilities of finding an antiparticle in an atom or ion containing a
particle of spin 1/2 or spin 0 are determined. The spin 1/2 case was
previously solved by Hans Bethe and his work is summarized. The spin 0 case is
treated numerically for an arbitrary atomic number and analytically for small
atomic numbers. The main tool for the spin 0 case is the Feshbach-Villars
representation of the Klein-Gordon equation.

##  1 Introduction

Solution of a Dirac equation with a Coulomb potential has a well-defined
energy, equal to the electron rest energy decreased by the binding, which
amounts to about $13.6$ eV in case of hydrogen. However, a decomposition of
the full solution of the Dirac equation into plane waves contains both
positive and negative energy solutions of the free Dirac equation. Positive
energy solutions alone do not form a complete basis. Negative energy
components, describing antiparticles resulting from the virtual pair
production, contribute very little to the norm of the wave function, only a
fraction of a percent even for heavy ions like the hydrogen-like lead with the
atomic number $Z=82$. For smaller $Z$, this contribution decreases further and
for small $\alpha_{Z}=Z\alpha$, where $\alpha\simeq 1/137$ denotes the fine
structure constant, becomes approximately $8\alpha_{Z}^{5}/(15\pi)$ [1] in the
ground state. Throughout this paper we focus on the ground state only.

Despite the smallness of their contribution to the norm, the negative energy
states have been found to contribute significantly to some processes, on par
with positive energies. For example, when a muon bound in an atom decays,
there is some probability that the resulting electron remains bound. In this
process, negative energy components of the muon and of the electron wave
functions play an important role [2, 3]. This is counterintuitive. For
example, in an earlier study of the bound muon decay, these negative-energy
contributions were neglected, which led to a significant error [4].

How large are negative energy contributions in the case of a spinless particle
like a pion, bound in a hydrogen-like atom? In the present paper, this
question is answered. This study is motivated by experiments with pionic atoms
carried out at the Paul Scherrer Institute [5, 6]. We find that the
probability of finding an antipion in a pionic atom is
$2\alpha_{Z}^{5}/(15\pi)$, a factor of 4 smaller than in the fermionic case.

Section 2 reviews Bethe’s work on antiparticles in the Dirac equation. In
Section 3 we summarize a two-component wave function formalism for the Klein-
Gordon equation which makes the negative energy contributions explicit.
Probability of finding antiparticles is computed numerically using the
momentum-space wave function (Subsection 3.1). An analytic result is found for
small $\alpha_{Z}$ using an integral equation for the wave function
(Subsection 3.2). Section 4 contains conclusions. An appendix reviews
solutions of the Schrödinger and the Klein-Gordon equations with a Coulomb
potential and summarizes our convention for Laguerre polynomials.

##  2 Negative energy content: the case of spin 1/2

###  2.1 Integral Dirac equation

Let $\psi\left(\bm{r}\right)$ be the spinor wave function of an electron in
the ground state of a hydrogen-like ion, with spin up. Define its Fourier
component $\phi\left(\bm{k},\tau\right)$ with spin projection $\tau$ (with
units such that $\hbar=c=1$),

$\psi\left(\bm{r}\right)=\sum_{\tau}\int\frac{\mbox{d}^{3}k}{\left(2\pi\right)^{3}}\phi\left(\bm{k},\tau\right)u_{\tau}\left(\bm{k}\right)e^{i\bm{k}\cdot\bm{r}}.$
(1)

$u_{\tau}$ are spatially-constant Dirac amplitudes for a free electron
normalized by $u_{\sigma}^{\dagger}u_{\tau}=\delta_{\sigma\tau}$,

$\displaystyle u_{1,2}\left(\bm{k}\right)$
$\displaystyle=\sqrt{\frac{E+m}{2E}}\left(\begin{array}[]{c}\varphi_{\pm}\\\
\frac{\bm{\sigma}\cdot\bm{k}}{E+m}\varphi_{\pm}\end{array}\right),\qquad\varphi_{+}=\left(\begin{array}[]{c}1\\\
0\end{array}\right),\quad\varphi_{-}=\left(\begin{array}[]{c}0\\\
1\end{array}\right),$ (8) $\displaystyle u_{3,4}\left(\bm{k}\right)$
$\displaystyle=\sqrt{\frac{E+m}{2E}}\left(\begin{array}[]{c}\pm\frac{\bm{\sigma}\cdot\bm{k}}{E+m}\varphi_{\mp}\\\
\mp\varphi_{\mp}\end{array}\right),\quad E=\sqrt{m^{2}+k^{2}},\quad
k=|\bm{k}|.$ (11)

They satisfy the Dirac equation in the following form,

$\displaystyle\left(\bm{\alpha}\cdot\bm{k}+\beta
m\right)u_{\sigma}\left(\bm{k}\right)$
$\displaystyle=E_{\sigma}u_{\sigma}\left(\bm{k}\right),$ (12) $\displaystyle
E_{1,2}$ $\displaystyle=E,\quad E_{3,4}=-E.$ (13)

We are interested in small atomic numbers $Z$ such that $\alpha_{Z}\ll 1$. The
dominant Fourier component is $\phi\left(\bm{k},1\right)$. The other positive
energy component vanishes, $\phi\left(\bm{k},2\right)=0$, and components with
$\tau=3,4$ describe the tiny negative energy content. All components are
obtained by projection,

$\phi\left(\bm{k},\sigma\right)=\int\mbox{d}^{3}re^{-i\bm{k}\cdot\bm{r}}\left[u_{\sigma}^{\dagger}\psi\left(\bm{r}\right)\right].$
(14)

The integral form of the Dirac equation,

$\left[V\left(r\right)+\bm{\alpha}\cdot\bm{k}+\beta m\right]\psi=W\psi,$ (15)

is also derived with this projection. Here $V$ denotes the Coulomb potential
energy, $V\left(r\right)=-\frac{\alpha_{Z}}{r}$, and $W$ is the total energy,
$W\simeq m-\frac{\alpha_{Z}^{2}m}{2}$. If Eq. (15) is multiplied with
$\int\mbox{d}^{3}re^{-i\bm{k}\cdot\bm{r}}u_{\sigma}^{\dagger}\left(\bm{k}\right)$,
the first term becomes

$\displaystyle V_{\sigma}^{\prime}(\bm{k})$
$\displaystyle=\int\mbox{d}^{3}re^{-i\bm{k}\cdot\bm{r}}\left[u_{\sigma}^{\dagger}\left(\bm{k}\right)\psi\left(\bm{r}\right)\right]V\left(\bm{r}\right)$
(16)
$\displaystyle=\int\frac{\mbox{d}^{3}q}{\left(2\pi\right)^{3}}\int\mbox{d}^{3}re^{-i\left(\bm{k}-\bm{q}\right)\cdot\bm{r}}\left[u_{\sigma}^{\dagger}\left(\bm{k}\right)\psi\left(\bm{r}\right)\right]\overbrace{\int\mbox{d}^{3}r^{\prime}e^{-i\bm{q}\cdot\bm{r}^{\prime}}V\left(\bm{r}^{\prime}\right)}^{V\left(\bm{q}\right)}$
(17)
$\displaystyle=\int\frac{\mbox{d}^{3}q}{\left(2\pi\right)^{3}}V\left(\bm{q}\right)\int\mbox{d}^{3}re^{-i\left(\bm{k}-\bm{q}\right)\cdot\bm{r}}\left[u_{\sigma}^{\dagger}\left(\bm{k}\right)\psi\left(\bm{r}\right)\right].$
(18)

Substitute $\psi$ from Eq. (1),

$\displaystyle V_{\sigma}^{\prime}(\bm{k})$
$\displaystyle=\int\frac{\mbox{d}^{3}q}{\left(2\pi\right)^{3}}V\left(\bm{q}\right)\sum_{\tau}\phi\left(\bm{k}-\bm{q},\tau\right)\underbrace{\left[u_{\sigma}^{\dagger}\left(\bm{k}\right)u_{\tau}\left(\bm{k}-\bm{q}\right)\right]}_{\left\langle\bm{k},\sigma|\bm{k}-\bm{q},\tau\right\rangle}.$
(19)

The conjugate of (12) is
$u_{\sigma}^{\dagger}\left(\bm{k}\right)\left(\bm{\alpha}\cdot\bm{k}+\beta
m\right)=E_{\sigma}u_{\sigma}^{\dagger}\left(\bm{k}\right)$, so the last three
terms of (15) give

$\displaystyle\int\mbox{d}^{3}re^{-i\bm{k}\cdot\bm{r}}u_{\sigma}^{\dagger}\left(\bm{k}\right)\left(\bm{\alpha}\cdot\bm{k}+\beta
m-W\right)\psi$
$\displaystyle=\left[E_{\sigma}-W\right]\phi\left(\bm{k},\sigma\right).$ (20)

Fourier-transforming the Coulomb potential,
$\int\mbox{d}^{3}re^{-i\bm{q}\cdot\bm{r}}V\left(\bm{r}\right)$
$=-\frac{4\pi\alpha_{Z}}{q^{2}}$,

$\left(W-E_{\sigma}\right)\phi\left(\bm{k},\sigma\right)=-4\pi\alpha_{Z}\int\frac{\mbox{d}^{3}q}{\left(2\pi\right)^{3}}\frac{1}{q^{2}}\sum_{\tau}\phi\left(\bm{k}-\bm{q},\tau\right)\left\langle\bm{k},\sigma|\bm{k}-\bm{q},\tau\right\rangle.$
(21)

###  2.2 Solution of the Dirac equation for spin 1/2

In Eq. (21) set $W\simeq m$,
$E_{\sigma}\left(\bm{k}\right)=-\sqrt{m^{2}+k^{2}}=-E$, and change the
integration momentum $\bm{q}\to\bm{p}=\bm{k}-\bm{q}$

$\left(m+E\right)\phi\left(\bm{k},\sigma\right)=-4\pi\alpha_{Z}\int\frac{\mbox{d}^{3}p}{\left(2\pi\right)^{3}}\frac{1}{\left(\bm{p}-\bm{k}\right)^{2}}\sum_{\tau}\phi\left(\bm{p},\tau\right)\left\langle\bm{k},\sigma|\bm{p},\tau\right\rangle.$
(22)

In the first approximation, neglect $\bm{p}$ where possible, arguing that it
introduces higher order corrections in $\alpha_{Z}$. Since the second spinor
component of the spin-up wave function $\psi$ vanishes, form such a linear
combination of $u_{3,4}$ in Eq. (11) that its second component is also zero,
$u_{n}=\frac{1}{\sqrt{2E\left(E-m\right)}}\left(\begin{array}[]{c}E-m\\\ 0\\\
-k_{z}\\\ -k_{+}\end{array}\right)$. Since the wave function in the momentum
space is peaked at zero momentum, $\left(\bm{p}-\bm{k}\right)^{2}$ in the
denominator can be approximated by $k^{2}$ and taken out of the integral.
Also, neglecting corrections $\mathcal{O}\left(\bm{p}/m\right)$, only
$\sigma=n$ and $\tau=1$ contribute,
$\left\langle\bm{k},n|0,1\right\rangle=\sqrt{\left(E-m\right)/2E}$,

$\displaystyle\phi_{-}\left(\bm{k}\right)$
$\displaystyle\simeq-\frac{4\pi\alpha_{Z}}{\left(m+E\right)k^{2}}\sqrt{\frac{E-m}{2E}}\int\frac{\mbox{d}^{3}p}{\left(2\pi\right)^{3}}\phi\left(\bm{p},1\right)$
(23)
$\displaystyle=-\frac{4\pi\alpha_{Z}}{\left(m+E\right)k^{2}}\sqrt{\frac{E-m}{2E}}\psi\left(0\right),$
(24)

where for the spatial wave function at the origin the non-relativistic result
can be used,
$\psi\left(\bm{r}=0\right)\simeq\sqrt{\frac{\alpha_{Z}^{3}m^{3}}{\pi}}$. This
is the only characteristic of the wave function we need to determine the
negative energy amplitude to the leading order in $\alpha_{Z}$. This reflects
creation of particle-antiparticle pairs only in the vicinity of the origin,
where the potential is strong. The resulting probability of finding negative
energy states is

$\displaystyle P_{-}(Z)$
$\displaystyle=\int\frac{\mbox{d}^{3}k}{\left(2\pi\right)^{3}}\left|\phi_{-}(\bm{k})\right|^{2}$
(25)
$\displaystyle=4\frac{m^{3}\alpha_{Z}^{5}}{\pi}\int_{0}^{\infty}\mbox{d}kk^{2}\left[\frac{1}{\left(m+E\right)\left(E^{2}-m^{2}\right)}\sqrt{\frac{E-m}{E}}\right]^{2}.$
(26)

Use $k\mbox{d}k=E\mbox{d}E$ and change variables to $E=m\epsilon$,

$\displaystyle P_{-}(Z)$
$\displaystyle=4\frac{\alpha_{Z}^{5}}{\pi}\int_{0}^{\infty}\frac{\mbox{d}\epsilon}{\left(\epsilon+1\right)^{7/2}\left(\epsilon-1\right)^{1/2}}=\frac{8\alpha_{Z}^{5}}{15\pi}.$
(27)

This agrees with the numerical evaluation of $P_{-}(Z)$ presented in Fig. 1.
Dots in that figure show $P_{-}/\alpha_{Z}^{4}$ from a numerical integration
of the negative energy components of the exact solution of the Dirac equation
with the Coulomb potential, obtained in [3]. When $Z$ is small, these dots
come close to the straight line predicted by Eq. (27). However, already for
$Z=8$, the straight line exceeds the numerical value by 59 per cent, even
though $\left(Z=8\right)\cdot\alpha$ is less than $0.06$. Very likely higher-
order effects in $\alpha_{Z}$, not included in Eq. (27), are logarithmically
enhanced.

Figure 1: Probability of finding spin 1/2 electrons with negative energies,
$P_{-}=\int\left(\left|\phi\left(\bm{p},3\right)\right|^{2}+\left|\phi\left(\bm{p},4\right)\right|^{2}\right)\frac{d^{3}p}{\left(2\pi\right)^{3}}$,
divided by $\alpha_{Z}^{4}$, as a function of the atomic number $Z$, evaluated
with $\phi\left(\bm{p},\sigma\right)$ computed in Ref. [3] (dots for every
third integer $Z$). For $Z=5$ the value is about 0.004, in agreement with
$\frac{8}{15\pi}\cdot 5\alpha=0.006$ predicted by Eq. (27). The first dot is
for $Z=2$, just above 0.002, in even better agreement with
$\frac{8}{15\pi}\cdot 2\alpha=0.0025$. When $Z\alpha\to 1$, $P_{-}$ seems to
tend to 0.0329. The solid line shows the small $Z$ behavior predicted by Eq.
(27).

##  3 Pionic atoms and the Klein-Gordon equation

We now proceed to an idealized description of a hydrogen-like ion with the
electron replaced by a negative pion $\pi^{-}$, assumed to be point-like,
stable, and not strongly interacting. The probability of negative energy
components in its wave function is the spin 0 analogue of Eq. (27), which was
derived for spin 1/2. We set out to derive it.

The spin 0 wave function is described by the Klein-Gordon (KG) equation.
Decomposition of KG wave functions into plane waves with positive and negative
energies was studied by Feshbach and Villars (FV) [7]. We shall first
summarize the integral equation they derived and then solve it with the
approximation method described in Section 2.2.

###  3.1 Feshbach-Villars representation of the KG wave function

Focus on the Coulomb problem with $V\left(r\right)=-\frac{\alpha_{Z}}{r}$ and
no vector potential. The KG equation is

$\left[\left(i\partial_{t}-V\right)^{2}+\nabla^{2}-m^{2}\right]\psi\left(\bm{r},t\right)=0.$
(28)

The two component wave function, which we denote with a capital letter $\Psi$,
satisfying a first-order equation in time, is

$\Psi\left(\bm{r},t\right)=\left(\begin{array}[]{c}\phi\\\
\chi\end{array}\right)=\frac{1}{\sqrt{2}m}\left(\begin{array}[]{c}m+i\partial_{t}-V\\\
m-i\partial_{t}+V\end{array}\right)\psi\left(\bm{r},t\right).$ (29)

The solution has the form
$\Psi\left(\bm{r},t\right)=\Psi\left(\bm{r}\right)e^{-iWt}$ where $W$ is the
energy eigenvalue. For the Coulomb problem
$W=m-\frac{m\alpha_{Z}^{2}}{2}+\mathcal{O}\left(\alpha_{Z}^{4}\right)$. Assume
that $W$ has been determined and focus on the time-independent part of the
wave function. Use such units of energy that $m=1$. In momentum space,

$\Psi\left(\bm{p}\right)=\int\mbox{d}^{3}re^{-i\bm{p}\cdot\bm{r}}\Psi\left(\bm{r}\right),$
(30)

the wave function can be decomposed into plane waves with positive and
negative energies,

$\Psi\left(\bm{p}\right)=u\left(p\right)\Psi_{0}^{\left(+\right)}\left(\bm{p}\right)+v\left(p\right)\Psi_{0}^{\left(-\right)}\left(\bm{p}\right),\quad
p=\left|\bm{p}\right|,$ (31)

with $\Psi_{0}^{\left(\pm\right)}\left(\bm{p}\right)$ being an orthonormal
basis, analogous to $u_{1,\dots,4}$ in Eqs. (8) and (11). This basis
diagonalizes the free-particle Hamiltonian, explicitly decoupling positive and
negative energy solutions (see Eq. (42)). Coefficients $u,v$ are related to
$\phi,\chi$ by a unitary transformation; using $E_{p}=\sqrt{1+p^{2}}$,

$\Psi^{\\#}=\left(\begin{array}[]{c}u\left(p\right)\\\
v\left(p\right)\end{array}\right)=U^{-1}\left(\begin{array}[]{c}\phi\left(p\right)\\\
\chi\left(p\right)\end{array}\right),\,U^{-1}=\frac{1}{2\sqrt{E_{p}}}\left(\begin{array}[]{cc}E_{p}+1&E_{p}-1\\\
E_{p}-1&E_{p}+1\end{array}\right).$ (32)

Fourier components $\phi,\chi$ can be expressed in closed form, obtained from
the configuration space wave function (see Appendix A),

$\displaystyle\phi\left(p\right)$
$\displaystyle=\phantom{-}a\left(p\right)\left(\sqrt{p^{2}+\nu}s_{1}+\frac{1-\nu+\sqrt{1-\nu}}{\sqrt{\nu}}s_{2}\right)$
(33) $\displaystyle\chi\left(p\right)$
$\displaystyle=-a\left(p\right)\left(\sqrt{p^{2}+\nu}s_{1}+\frac{1-\nu-\sqrt{1-\nu}}{\sqrt{\nu}}s_{2}\right)$
(34) $\displaystyle a\left(p\right)$
$\displaystyle=\frac{2^{1-\nu}\sqrt{\pi}\sqrt[4]{\nu\left(1-\nu\right)}\Gamma\left(1-\nu\right)}{p\sqrt{\Gamma\left(2-2\nu\right)}}\left(\frac{p^{2}+\nu}{\nu}\right)^{\frac{\nu}{2}-1},$
(35) $\displaystyle s_{n=1,2}$
$\displaystyle=\sin\left(\left(n-\nu\right)\arctan\frac{p}{\sqrt{\nu}}\right),\quad\nu=\frac{1}{2}-\sqrt{\frac{1}{4}-\alpha_{Z}^{2}}.$
(36)

In the weak field limit when $\alpha_{Z}\ll 1$, $\nu$ is of order
$\alpha_{Z}^{2}$ and so is the typical $p^{2}$. Then $\chi\ll\phi$ and $\chi$
is analogous to the small component of the Dirac wave function. Similarly,
$v\ll u$. It is $v\left(p\right)$ that determines the probability $P_{-}$ of
finding antiparticles,

$P_{-}\left(Z\right)=\int\frac{\mbox{d}^{3}p}{\left(2\pi\right)^{3}}\left|v\left(p\right)\right|^{2}.$
(37)

The result is plotted in Fig. 2 for $Z$ up to 68.

Figure 2: Probability of finding spin 0 particles with negative energies (the
Klein-Gordon case), divided by $\alpha_{Z}^{4}$, as a function of the atomic
number $Z$ (dots). Note that in the KG case, the field becomes supercritical
at $\alpha_{Z}=1/2$ rather than 1 as in the Dirac case [8, 9]. The solid line
shows the small $Z$ behavior in Eq. (38).

Note that for larger $Z$, when $\alpha_{Z}>1/2$, the field becomes
supercritical [8, 9], unlike in the Dirac equation case which requires
$\alpha_{Z}>1$ for super-criticality. For small $\alpha_{Z}$, numerical
results plotted in Fig. 2 indicate the behavior

$P_{-}\left(Z\to 0\right)=\frac{2\alpha_{Z}^{5}}{15\pi},$ (38)

a four times smaller slope that in the Dirac equation case, Eq. (27). Eq. (38)
can be confirmed analytically with the help of an integral equation, as we now
proceed to demonstrate.

###  3.2 Integral KG equation

Following Feshbach and Villars, write down the first order equation for the
wave function $\Psi^{\\#}$ decomposed into free particle solutions, Eq. (32).
In momentum space, position operator is represented by $i\bm{\nabla}_{p}$.
Using $\tau_{2}=\begin{pmatrix}0&-i\\\ i&0\end{pmatrix}$ and
$\tau_{3}=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}$,

$\displaystyle i\partial_{t}\Psi^{\\#}=U^{-1}i\partial_{t}\Psi$
$\displaystyle=U^{-1}\left\\{\left(\tau_{3}+i\tau_{2}\right)\frac{\bm{p}^{2}}{2m}+\tau_{3}m+V\left(i\bm{\nabla}_{p}\right)\right\\}U\Psi^{\\#}.$
(39)

With identities

$\displaystyle U^{-1}\tau_{3}U$
$\displaystyle=\frac{\left(E_{p}^{2}+1\right)\tau_{3}+\left(1-E_{p}^{2}\right)i\tau_{2}}{2E_{p}},$
(40) $\displaystyle U^{-1}\tau_{2}U$
$\displaystyle=\frac{\left(E_{p}^{2}+1\right)\tau_{2}+\left(E_{p}^{2}-1\right)i\tau_{3}}{2E_{p}},$
(41)

the free-particle part of the Hamiltonian is diagonal,

$U^{-1}\left[\left(\tau_{3}+i\tau_{2}\right)\frac{\bm{p}^{2}}{2}+\tau_{3}\right]U=E_{p}\tau_{3},$
(42)

and the wave equation becomes

$\displaystyle i\partial_{t}\Psi^{\\#}$
$\displaystyle=E_{p}\tau_{3}\Psi^{\\#}+U^{-1}\left(\bm{p}\right)V\left(i\bm{\nabla}_{p}\right)U\left(\bm{p}\right)\Psi^{\\#}\left(\bm{p}\right).$
(43)

For the Coulomb potential,

$i\partial_{t}\Psi^{\\#}=E_{p}\tau_{3}\Psi^{\\#}-4\pi\alpha_{Z}U^{-1}\left(\bm{p}\right)\int\frac{\mbox{d}^{3}q}{\left(2\pi\right)^{3}}\frac{U\left(\bm{q}\right)\Psi^{\\#}\left(\bm{q}\right)}{\left(\bm{p}-\bm{q}\right)^{2}}.$
(44)

We are interested in the equation for the lower component $v$. With
$i\partial_{t}\to W$ and neglecting $v$ in the right hand side since $v\ll u$,

$\displaystyle\left(W+E_{p}\right)v$
$\displaystyle=-4\pi\alpha_{Z}\int\frac{\mbox{d}^{3}q}{\left(2\pi\right)^{3}}\frac{\left(E_{p}-E_{q}\right)u}{2\sqrt{E_{p}E_{q}}\left(\bm{p}-\bm{q}\right)^{2}}.$
(45)

Following the approximation discussed below Eq. (22), we neglect $q$ where
possible under the integral and find

$\displaystyle v\left(p\right)$
$\displaystyle\simeq-\frac{2\pi\alpha_{Z}\left(E_{p}-1\right)}{\sqrt{E_{p}}p^{2}\left(1+E_{p}\right)}\int\frac{\mbox{d}^{3}q}{\left(2\pi\right)^{3}}u\left(q\right)$
(46)
$\displaystyle\simeq-\frac{2\pi\alpha_{Z}\psi(0)}{\sqrt{E_{p}}\left(1+E_{p}\right)^{2}},$
(47)

as obtained in Ref. [10]. To check this approximation, we plot in Fig. 3 the
numerical solution of the integral equation (45) (solid line), and the
analytical result in Eq. (47) (dashed). As $Z$ tends to zero, the two curves
become closer. This illustrates that the momentum wave function strongly
decreases with increasing momentum; the typical momentum is $\alpha_{Z}$.

Figure 3: Numerical (solid line) and approximate analytical (dashed) solutions
of the integral equation, Eq. (45). The error decreases with decreasing $Z$:
neglecting $q$ under the integral is sound. For each $Z$, curves were rescaled
to make the area under the solid curve equal 1.

The integration based on the analytical formula in Eq. (47) is elementary,

$\displaystyle P_{-}\left(Z\to 0\right)$
$\displaystyle=\int\frac{\mbox{d}^{3}p}{\left(2\pi\right)^{3}}\left|v\left(p\right)\right|^{2}$
$\displaystyle=2\frac{\alpha_{Z}^{5}}{\pi}\int_{1}^{\infty}\frac{\sqrt{E_{p}-1}\mbox{d}E_{p}}{\left(1+E_{p}\right)^{7/2}}=\frac{2\alpha_{Z}^{5}}{15\pi},$
(48)

in agreement with Eq. (38).

##  4 Conclusions

We have determined the probability of finding an antiparticle in two systems:
the previously studied spin 1/2 particle in the Coulomb potential of a point-
like, static nucleus with atomic number $Z$, and an analogous system with a
spin 0 particle (an idealized pionic atom or ion). In both cases the
probability is suppressed by five powers of $Z\alpha$, and, for small $Z$, is
smaller by a factor 4 in the spin 0 case. We found that both cases, described
by the Dirac and the Klein-Gordon equations, can be treated in an analogous
manner. In the future, it would be interesting to interpret these results in
terms of Feynman diagrams.

## Acknowledgements

I thank Alexander Penin and Vladimir Shabaev for useful discussions and M.
Jamil Aslam and M. Mubasher for reading the manuscript and suggesting
improvements. This research was supported by the Natural Sciences and
Engineering Research Council of Canada (NSERC), by WestGrid (www.westgrid.ca),
and by Compute Canada (www.computecanada.ca).

##  Appendix A Klein-Gordon equation with a Coulomb potential

We consider a pion in the Coulomb field of an infinitely heavy point-like
nucleus with charge $Ze$. We first summarize the solution of the radial
Schrödinger equation with a Coulomb potential, to emphasize its similarity
with the KG case, treated in detail. The Schrödinger equation reads (as in the
main text, we use such units that $\hbar=c=1$, but we keep $m$ explicit)

$\left[\left(\partial_{r}+\frac{1}{r}\right)^{2}-\frac{l\left(l+1\right)}{r^{2}}+\frac{2\alpha_{Z}m}{r}+2m\left(E-m\right)\right]R^{\text{Sch}}\left(r\right)=0,$
(49)

where $E-m=-\frac{\alpha_{Z}^{2}m}{2\left(1+n_{r}+l\right)^{2}}$ is the
binding energy, with $n_{r}=0,1,\dots,$ denoting the radial excitation, and
$l=0,1,\dots,$ denoting the angular momentum. With the distance given by
$x=r/a$ in units of the Bohr radius $a=1/\left(\alpha_{Z}m\right)$, the
resulting radial wave functions [11] are

$R_{nl}^{\text{Sch}}\left(x\right)=-\frac{2}{n^{2}}\sqrt{\frac{\left(n+l\right)!}{\left(n-l-1\right)!}}\exp\left(-\frac{x}{n}\right)\left(\frac{2x}{n}\right)^{-l-1}L_{n+l}^{\left(-2l-1\right)}\left(\frac{2x}{n}\right).$
(50)

Laguerre polynomials $L_{n}^{\left(\alpha\right)}$ are defined in Appendix B.
In the ground state, $n=1$, $l=0$, the radial wave function becomes
$2\exp\left(-x\right)$.

For the KG equation we have, from
$\left[\nabla^{2}-m^{2}+\left(W+\frac{\alpha_{Z}}{r}\right)^{2}\right]\psi=0$,

$\left[\left(\partial_{r}+\frac{1}{r}\right)^{2}-\frac{l\left(l+1\right)}{r^{2}}+\frac{\alpha_{Z}^{2}}{r^{2}}+\frac{2\alpha_{Z}W}{r}+W^{2}-m^{2}\right]R^{\text{KG}}\left(r\right)=0.$
(51)

We rescale the distance variable, $\rho=\sqrt{m^{2}-W^{2}}r$ and replace

$l\to\lambda=\sqrt{\left(l+\frac{1}{2}\right)^{2}-\alpha_{Z}^{2}}-\frac{1}{2},$
(52)

to derive the radial equation in a dimensionless form,

$\left[\left(\partial_{\rho}+\frac{1}{\rho}\right)^{2}-\frac{\lambda\left(\lambda+1\right)}{\rho^{2}}+\frac{\varepsilon}{\rho}-1\right]R=0,\quad\varepsilon\equiv\frac{2\alpha_{Z}W}{\sqrt{m^{2}-W^{2}}}.$
(53)

For large $\rho$,

$\left(\rho R\right)^{\prime\prime}=\rho R\Rightarrow
R\sim\frac{e^{-\rho}}{\rho},$ (54)

while for small $\rho$, $\partial_{\rho}^{2}\left(\rho
R\right)=\frac{\lambda\left(\lambda+1\right)}{\rho^{2}}\rho R$, so
$R\sim\rho^{\lambda}$. With the substitution
$R=\rho^{\lambda}e^{-\rho}L\left(\rho\right)$, the equation for $L$ becomes

$\rho
L^{\prime\prime}+2\left(\lambda+1-\rho\right)L^{\prime}+\left[\varepsilon-2\left(\lambda+1\right)\right]L=0$
(55)

Substituting a power series for $L$, $L=\sum_{k=0}^{\infty}a_{k}\rho^{k},$
gives a recurrence relation,

$\displaystyle\left(2\left(\lambda+1\right)+k\right)\left(k+1\right)a_{k+1}+\left[\varepsilon-2\left(\lambda+1\right)-2k\right]a_{k}$
$\displaystyle=0,$ (56)

The series terminates if for some $k$ the coefficient of $a_{k}$ vanishes,
that is when $\frac{e-2\left(\lambda+1\right)}{2}=n_{r}=0,1,\dots$. This gives
the quantization condition for the energy,

$\displaystyle\frac{\alpha_{Z}W}{\sqrt{m^{2}-W^{2}}}$
$\displaystyle=1+\lambda+n_{r},$ (57)

so that finally

$W=\frac{m}{\sqrt{1+\frac{\alpha_{Z}^{2}}{\left(1+\lambda+n_{r}\right)^{2}}}}.$
(58)

When the condition $\varepsilon-2\left(\lambda+1\right)=2n_{r}$ is fulfilled,
Eq. (55) becomes

$\rho L^{\prime\prime}+2\left(\lambda+1-\rho\right)L^{\prime}+2n_{r}L=0.$ (59)

Change the variable to $x=2\rho$ and recognize the generalized Laguerre
equation,

$x\frac{\mbox{d}^{2}L}{\mbox{d}x^{2}}+\left(2\lambda+2-x\right)\frac{\mbox{d}L}{\mbox{d}x}+n_{r}L=0,$
(60)

whose solutions are
$L\left(x\right)=L_{n_{r}}^{\left(2\lambda+1\right)}\left(x\right)$.
Remembering $\rho=\sqrt{m^{2}-W^{2}}r$ we get

$\displaystyle x$
$\displaystyle=2\rho=\frac{2mr\alpha_{Z}}{\sqrt{\left(1+\lambda+n_{r}\right)^{2}+\alpha_{Z}^{2}}},$
(61) $\displaystyle R$
$\displaystyle=Nx^{\lambda}e^{-x/2}L_{n_{r}}^{\left(2\lambda+1\right)}\left(x\right).$
(62)

The normalization $N$ is often defined by the condition (but see the
discussion below Eq. (68))

$\displaystyle 1$
$\displaystyle=N^{2}\int_{0}^{\infty}x^{2\lambda}e^{-x}\left[L_{n_{r}}^{\left(2\lambda+1\right)}\left(x\right)\right]^{2}r^{2}\mbox{d}r$
(63) $\displaystyle=N^{2}s^{3}\cdot
2\left(n_{r}+\lambda+1\right){n_{r}+2\lambda+1\choose
n_{r}}\Gamma\left(2\lambda+2\right).$ (64)

In summary, the solution of the Klein-Gordon equation with the Coulomb
potential is (see also [12])

$\displaystyle R_{n_{r}l}$
$\displaystyle=\left(\frac{\alpha_{Z}m}{\sqrt{\left(1+\lambda+n_{r}\right)^{2}+\alpha_{Z}^{2}}}\right)^{3/2}\frac{2\sqrt{n_{r}!}x^{\lambda}e^{-x/2}L_{n_{r}}^{\left(2\lambda+1\right)}\left(x\right)}{\sqrt{\left(1+\lambda+n_{r}\right)\Gamma\left(2+2\lambda+n_{r}\right)}},$
(65) $\displaystyle x$
$\displaystyle=\frac{2\alpha_{Z}mr}{\sqrt{\left(1+\lambda+n_{r}\right)^{2}+\alpha_{Z}^{2}}},\qquad\lambda=\sqrt{\left(l+\frac{1}{2}\right)^{2}-\alpha_{Z}^{2}}-\frac{1}{2},$
(66) $\displaystyle W$
$\displaystyle=\frac{m}{\sqrt{1+\frac{\alpha_{Z}^{2}}{\left(1+\lambda+n_{r}\right)^{2}}}}.$
(67)

Here $n_{r}$ is the degree of the radial excitation and $l$ is the orbital
quantum number. The ground state corresponds to $n_{r}=l=0$, thus
$\lambda\to\sqrt{\frac{1}{4}-\alpha_{Z}^{2}}-\frac{1}{2}<0$. It is convenient
to introduce a positive parameter
$\nu=\frac{1}{2}-\sqrt{\frac{1}{4}-\alpha_{Z}^{2}}$ and use
$\sqrt{\left(1-\nu\right)^{2}+\alpha_{Z}^{2}}=\sqrt{1-\nu}$

$\displaystyle R_{00}\left(r\right)$
$\displaystyle=\left(2\sqrt{\nu}m\right)^{3/2-\nu}\frac{r^{-\nu}e^{-\sqrt{\nu}mr}}{\sqrt{\Gamma\left(3-2\nu\right)}}.$
(68)

Return now to the issue of normalization. It is convenient to define such
$\psi\left(\bm{r}\right)$ that
$2\left[W-V\left(r\right)\right]\left|\psi\left(\bm{r}\right)\right|^{2}$ is
interpreted as charge density (with the charge of the negative pion taken as
the unit,
$2\int\mbox{d}^{3}r\left[W-V\left(r\right)\right]\left|\psi\left(\bm{r}\right)\right|^{2}=1$).
To this end, in case of the ground state, include the spherical harmonic
$Y_{00}\left(\theta,\phi\right)=1/\sqrt{4\pi}$ and define [13]

$\psi\left(\bm{r}\right)=\frac{\left(1-\nu\right)^{1/4}}{\sqrt{8\pi}}R_{00}\left(r\right).$
(69)

In Eq. (29), $\psi\left(\bm{r},t\right)$ equals
$\psi\left(\bm{r}\right)e^{-iWt}$ with $W=\sqrt{1-\nu}m$.

##  Appendix B Generalized Laguerre functions: conventions

We use Laguerre functions $L_{n}^{\left(\alpha\right)}$ according to the
convention of Ref. [14, 15], which differs from Landau and Lifshitz [11],
whose functions we denote by $\mathcal{L}_{n}^{m}$. Here we explain the
connection between them. We use Rodrigues formula in the form

$L_{n}^{\left(\alpha\right)}\left(z\right)=\frac{e^{z}z^{-\alpha}}{n!}\frac{\mbox{d}^{n}}{\mbox{d}z^{n}}\left(e^{-z}z^{n+\alpha}\right),$
(70)

while Landau and Lifshitz use

$\mathcal{L}_{n}^{m}\left(z\right)=\frac{n!e^{z}}{\left(n-m\right)!}\frac{\mbox{d}^{n}}{\mbox{d}z^{n}}\left(e^{-z}z^{n-m}\right).$
(71)

Therefore,

$\mathcal{L}_{n}^{m}\left(z\right)=\frac{\left(n!\right)^{2}z^{-m}}{\left(n-m\right)!}L_{n}^{\left(-m\right)}\left(z\right).$
(72)

## References

  * [1] H. A. Bethe, _Bemerkungen über die Wasserstoff-Eigenfunktionen in der Diracschen Theorie_ , Zeitschrift für Naturforschung A 3, 470 – 477 (1948).
  * [2] M. J. Aslam, A. Czarnecki, A. Morozova, and G. P. Zhang, _Decay of a bound muon into a bound electron_ , Proceedings of Science 367, 148 (2019), https://pos.sissa.it/367/148/.
  * [3] M. J. Aslam, A. Czarnecki, G. Zhang, and A. Morozova, _Decay of a bound muon into a bound electron_ , Phys. Rev. D 102, 073001 (2020), 2005.07276.
  * [4] C. Greub, D. Wyler, S. Brodsky, and C. Munger, _Atomic alchemy: Weak decays of muonic and pionic atoms into other atoms_ , Phys. Rev. D52, 4028–4037 (1995), hep-ph/9405230.
  * [5] M. Hori, H. Aghai-Khozani, A. Sótér, A. Dax, and D. Barna, _Laser spectroscopy of pionic helium atoms_ , Nature 581, 37–41 (2020).
  * [6] A. Sótér, _Laser spectroscopy of antiprotonic and pionic helium atoms_ , Ph.D. thesis, Ludwig-Maximilians-Universität München (2016).
  * [7] H. Feshbach and F. Villars, _Elementary relativistic wave mechanics of spin 0 and spin 1/2 particles_ , Rev. Mod. Phys. 30, 24–45 (1958).
  * [8] W. Greiner, B. Muller, and J. Rafelski, _Quantum Electrodynamics Of Strong Fields_ , Springer, Berlin (1985).
  * [9] A. Klein and J. Rafelski, _Bose Condensation in Supercritical External Fields_ , Phys. Rev. D 11, 300 (1976).
  * [10] M. Horbatsch and D. V. Shapoval, _Analysis of the Klein-Gordon Coulomb problem in the Feshbach-Villars representation_ , Phys. Rev. A 51, 1804–1807 (1995).
  * [11] L. D. Landau and E. M. Lifshits, _Quantum Mechanics: Non-Relativistic Theory_ , Butterworth-Heinemann, Oxford (1991).
  * [12] V. Bagrov and D. Gitman, _Exact Solutions of Relativistic Wave Equations_ , Springer (1990).
  * [13] W. Greiner, _Relativistic Quantum Mechanics. Wave Equations_ , Springer, Berlin, 3 edition (2000).
  * [14] F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, editors, _NIST Handbook of Mathematical Functions_ , Cambridge University Press, Cambridge (2011).
  * [15] J. Schwinger, _Quantum Mechanics: symbolism of atomic measurements_ , Springer, Berlin (2001), edited by B.-G. Englert.


# Study of the thermochromic phase transition in CuMo1-xWxO4 solid solutions
at the W L3-edge by resonant X-ray emission spectroscopy

Inga Pudza<EMAIL_ADDRESS>Aleksandr Kalinko<EMAIL_ADDRESS>Arturs Cintins<EMAIL_ADDRESS>Alexei Kuzmin<EMAIL_ADDRESS>Institute of Solid State Physics, University of Latvia, Kengaraga street 8,
LV-1063 Riga, Latvia  Department of Chemistry and Center for Sustainable
Systems Design, Paderborn University, 33098 Paderborn, Germany

###### Abstract

Polycrystalline CuMo1-xWxO4 solid solutions were studied by resonant X-ray
emission spectroscopy (RXES) at the W L3-edge to follow a variation of the
tungsten local atomic and electronic structures across thermochromic phase
transition as a function of sample composition and temperature. The
experimental results were interpreted using ab initio calculations. The
crystal-field splitting parameter $\Delta$ for the 5d(W)-states was obtained
from the analysis of the RXES plane and was used to evaluate the coordination
of tungsten atoms. Temperature-dependent RXES measurements were successfully
employed to determine the hysteretic behaviour of the structural phase
transition between the $\alpha$ and $\gamma$ phases in CuMo1-xWxO4 solid
solutions on cooling and heating, even at low ($x<0.10$) tungsten content. It
was found that tungsten ions have octahedral coordination for $x>0.15$ in the
whole studied temperature range (90-420 K), whereas their coordination changes
from tetrahedral to octahedral upon cooling for smaller ($x\leq 0.15$)
tungsten content. Nevertheless, some amount of tungsten ions was found to co-
exists in the octahedral environment at room temperature for $x<0.15$. The
obtained results correlate well with the color change in these solid
solutions.

###### keywords:

CuMo1-xWxO4 , crystal-field splitting , resonant X-ray emission spectroscopy
(RXES) , high-energy resolution fluorescence detected X-ray absorption near-
edge structure (HERFD-XANES)

††journal: Acta Materialia

##  1 Introduction

CuMo1-xWxO4 solid solutions represent a class of functional materials
demonstrating a wide range of physical and chemical properties such as
thermochromic [1, 2, 3, 4, 5, 6], piezochromic [2, 5, 7], halochromic [8],
photoelectrochemical [9], thermosalient [5] and catalytic [10].

The strongest thermochromic effect is observed on cooling for small tungsten
content ($x<0.15$), when the change of the material color from green to brown
is due to the first-order phase transition caused by the Mo(W) displacement
from the tetrahedral site (as in $\alpha$-CuMoO4) to the octahedral site (as
in $\gamma$-CuMoO4) (Fig. 1) [1, 5, 11]. The similar color change occurs also
in $\alpha$-CuMoO4 on heating [12], however, in this case, no phase transition
takes place, the coordination of metal ions remains unchanged, and the
variation of color is due to the lattice expansion and strong enhancement of
specific thermal disorder [13]. For tungsten content above $x\sim 0.15$,
CuMo1-xWxO4 solid solutions crystallize in the wolframite-type structure with
the octahedral coordination of metal ions and do not manifest pronounced
thermochromic behaviour. Therefore, the knowledge of the local environment of
metal ions in these mixed compounds is crucial for understanding and
controlling their properties.

Conventionally, X-ray absorption spectroscopy (XAS) is used to probe the local
environment of atoms in complex materials [14, 15, 16]. However, in the case
of tungstates and molybdates, often composed from strongly distorted
polyhedra, the recovery of structural information from XAS requires the use of
a computationally heavy and time-consuming approach based on reverse Monte
Carlo (RMC) simulations [17, 18, 19, 20]. Moreover, we have previously shown
that a reliable structural analysis requires the knowledge of X-ray absorption
spectra at least for all metal edges [17, 21]. In the case of solid solutions,
the low content of one of the component limits the quality of the experimental
data, which additionally complicates the analysis. Finally, the close values
of the metal–oxygen interatomic distances in the radial distribution functions
(RDFs) obtained using the RMC simulation make it often difficult to
unambiguously identify the atoms belonging to the first coordination shell of
metal and, as a result, the type of its coordination polyhedron.

In tungsten oxides and related materials, the splitting of the tungsten 5d
band depends on the crystal field of ligands and can be experimentally probed
by the W L3-edge XAS [22, 23], that can be used to distinguish between
different local symmetries of the tungsten coordination polyhedron. In
particular, the tungsten 5d-band splits into 5d(t2g) and 5d(eg) sub-bands in
octahedral and 5d(t2) and 5d(e) sub-bands in tetrahedral coordination.
However, the small value of the crystal field splitting (about several
electron volts) and the large value of the natural width of the excited
2p3/2(W) level ($\sim$4.5 eV [24]) often mask the difference between the two
coordinations [25, 26, 27]. Therefore, a more sophisticated experimental
approach should be used to more reliably address this issue.

In this study, we used the resonant X-ray emission spectroscopy (RXES) at the
W L3-edge to determine a variation of the tungsten local atomic and electronic
structure in CuMo1-xWxO4 solid solutions across thermochromic phase transition
as a function of sample composition and temperature. We demonstrate that the
analysis of the W L3-edge white line splitting provides a robust tool for
distinguishing tetrahedral and octahedral coordinations of tungsten ions in
these solid solutions.

##  2 Experimental details

Polycrystalline CuMo1-xWxO4 powders were synthesized using a solid-state
reaction method by heating a mixture of CuO and MoO3 powders with a
stoichiometric amount of WO3 at 650∘C in air for 8 hours followed by cooling
down to the room temperature. One group of samples with $x\leq 0.15$ was
greenish, and the second group with $x\geq 0.20$ was brownish. All samples
were characterized by X-ray powder diffraction and micro-Raman spectroscopy
(see in the Supplementary Information).

The temperature-dependent RXES experiments were performed at the P64 Advanced
X-Ray Absorption Spectroscopy beamline of the PETRA III (HASYLAB/DESY) storage
ring using recently constructed RXES endstation [28, 29]. The X-ray beam from
an undulator was monochromatized by a liquid-nitrogen-cooled double-crystal
Si(311) monochromator and focused in vertical and horizontal planes (100
$\mu$m $\times$ 240 $\mu$m) by two mirrors with Rh coating positioned
downstream of the monochromator. The high stability and high intensity
(5$\cdot$1011 photons/s) of the P64 beamline provided the possibility to
accumulate high quality/resolution RXES data. The ionization chamber was used
for incident X-ray intensity monitoring. The XES signal was dispersed by von
Hamos-type spectrometer equipped with cylindrical bent Si(444) analyser
crystals [29] and collected by Dectris 2D Pilatus 300K detector. The
monochromator was calibrated at the W L3-edge using tungsten foil and the
energy scale of the spectrometer was set according to the position of the
elastic line on the detector. The inelastically scattered X-ray signal and W
L$\alpha_{1}$ and L$\alpha_{2}$ fluorescence were acquired in the RXES plane.
The high-resolution W L3-edge XANES (HERFD-XANES) was extracted from the RXES
plane at fixed emission energy 8398.5$\pm$0.2 eV. The temperature of the
sample was controlled using liquid nitrogen (LN2) cryostat Linkam THMS600 in
the range of 90-450 K. All samples were measured in the form of sintered
powders.

The W L3-edge XANES spectra of the same powder samples recorded at room-
temperature in transmission mode at the Elettra XAFS bending-magnet beamline
[30] were used for comparison. The experimental details can be found in [13].

##  3 FDMNES calculations

The simulation of the experimental W L3-edge XANES spectra was performed based
on the finite difference method (FDM) by the ab initio real-space FDMNES code
[31, 32], thus avoiding the muffin-tin potential approximation. The XANES
spectra were calculated using real energy-dependent Hedin-Lundqvist exchange-
correlation potential [33] taking into account spin-orbit interactions and the
self-consistent cluster potential. The calculations were performed for 4.0 Å
radius clusters constructed around the absorbing tungsten atom from the
crystallographic structures of interest, since the effect of the crystal field
is related to the interaction between tungsten and nearest oxygen atoms.

All calculated XANES spectra were convoluted with a Lorentzian broadening
function with a FWHM given by $\Gamma_{hole}$=1 eV to mimic the broadening due
to the core-hole lifetime and experimental resolution. The effect of the core-
hole was also examined by performing calculations for the excited (with the
core hole) and non-excited (without the core hole) absorbing tungsten atom.

The structures of $\alpha$-CuMoO4 and $\gamma$-CuMoO4 contain three non-
equivalent sites for copper (Cu1, Cu2, Cu3) and molybdenum (Mo1, Mo2, Mo3)
atoms [34]. However, all molybdenum atoms occupy either distorted tetrahedral
(in $\alpha$-CuMoO4) or distorted octahedral (in $\gamma$-CuMoO4) positions.
In CuWO4, there is only one tungsten position with distorted octahedral
coordination [35].

The calculated W L3-edge XANES spectra are shown in Fig. 2 for CuWO4 and
tungsten atom substituting molybdenum atom at the Mo2 site in $\alpha$-CuMoO4
(distorted tetrahedral environment) and $\gamma$-CuMoO4 (distorted octahedral
environment). The XANES spectra calculated for tungsten atoms placed at the
Mo1 or Mo3 sites are close to that at the Mo2 site and are not shown. Note
that all calculations were performed based on the crystallographic structures
obtained from diffraction [34, 35], without any structure relaxation due to
the substitution.

Figure 1: Crystal structures and unit cells of high-temperature green
($\alpha$) and low-temperature brownish-red ($\gamma$) CuMo1-xWxO4 phases [5,
34]. MoO4 and MoO6 polyhedra are indicated in the $\alpha$\- and
$\gamma$-phases, respectively. Small red balls are oxygen atoms, medium-sized
blue balls are copper atoms, and large green balls are molybdenum/tungsten
atoms. Photographs of the two phases are also shown. Figure 2: (a) Calculated
W L3-edge XANES spectra for tungsten atoms in distorted octahedral (CuWO4 and
W substituting molybdenum atoms at the Mo2 site in $\gamma$-CuMoO4) and
tetrahedral (W substituting molybdenum atoms at the Mo2 site in
$\alpha$-CuMoO4) environments. The results for the excited state with the core
hole (W*, solid curves) and non-excited state without the core hole (W, dashed
curves) are given. Tungsten–oxygen polyhedra with indicated interatomic
distances in (b) CuWO4, (c) W at the Mo2 site in $\gamma$-CuMoO4 and (d) W at
the Mo2 site in $\alpha$-CuMoO4 are also shown. Inset shows the experimental W
L3-edge XANES spectra in CuMo1-xWxO4 solid solutions measured in the
transmission mode.

##  4 Results and discussion

###  4.1 Crystal field splitting in the W L3-edge XANES & RXES

At ambient conditions, CuMo1-xWxO4 solid solutions exist in one of three
crystal structures with triclinic $P\bar{1}$ symmetry, which are isostructural
to the high-pressure phases of CuMoO4 [34]. In the $\alpha$-phase, $2/3$ and
$1/3$ of Cu2+ cations form distorted CuO6 octahedra and CuO5 square-pyramids,
respectively, whereas Mo6+ cations form MoO4 tetrahedra. The crystal structure
of $\gamma$-CuMoO4 is built up of distorted CuO6 and MoO6 octahedra. Finally,
CuMoO4-III phase has a wolframite-type structure, in which both Cu2+ and Mo6+
ions occupy octahedral sites. The sequence
($\alpha\rightarrow\gamma\rightarrow$ III) of structural transitions upon
increasing W content is similar to that under hydrostatic pressure [34].
Therefore, the ability to tune the functional properties of CuMo1-xWxO4 solid
solutions by the introduction of tungsten ions is often explained by the
effect of locally-induced chemical pressure that appears due to a small
difference between the ionic radii of Mo6+ (0.59 Å) and W6+ (0.60 Å) ions
[36]. Besides, the increase of the $\alpha$ $\leftrightarrow$ $\gamma$ phase
transition temperature at high tungsten content was attributed to the stronger
preference of tungsten ions to the octahedral environment [1, 2].

According to the crystal-field theory [37], the d-orbitals of the metal ion
split in the octahedral field of six oxygen atoms into two groups (triply
degenerate t2g orbitals and doubly degenerate eg orbitals) with the energy
difference $\Delta_{oct}$, where the t2g orbitals have lower energy than the
eg ones. In tetrahedral coordination, the splitting of the d orbitals occurs
into two groups with the energy difference $\Delta_{tet}$ between the lower
energy, doubly degenerate e orbitals and the higher energy, triply degenerate
t2 orbitals. The splitting magnitudes for regular tetrahedral $\Delta_{tet}$
and octahedral $\Delta_{oct}$ coordinations are related as
$\Delta_{tet}=4/9\Delta_{oct}$ for similar W–O distances [37]. For oxidation
state W6+, all five d-orbitals are vacant, therefore, two electron transitions
from 2p3/2 to split 5d states are possible and will have different X-ray
absorption intensities. For tetrahedral WO4 coordination, the intensity ratio
will be e:t2=2:3, whereas for octahedral WO6 coordination, it will be
t2g:eg=3:2. Thus, the knowledge of splitting value or intensity ratio for the
two transitions in CuMo1-xWxO4 solid solutions allows one to conclude on the
type of tungsten ion coordination. Further, we will demonstrate how the
analysis of the W L3-edge RXES data provides an opportunity to gain this
information.

The XANES spectra of CuMo1-xWxO4 solid solutions at the W L3-edge contain a
strong resonance (see the inset in Fig. 2(a)), the so-called ”white line“
[38], located just below the continuum threshold and corresponding to the
dipole-allowed electron transition from the 2p3/2(W) level to a quasi-bound
5d(W)+2p(O) mixed-state in the presence of the core hole [25].

Although the W L3-edge XANES is sensitive to the coordination of the absorbing
atom, a very short natural lifetime of the excited state with the core hole at
the 2p3/2(W) level leads to significant intrinsic broadening ($\sim$4.5 eV
[24]) of the absorption spectrum. As a result of the limited resolution, the
crystal field splitting of the 5d(W) states is smeared out and cannot be
usually resolved in the conventional transmission experiment (see the inset in
Fig. 2(a)). However, some splitting of the white line maximum is nevertheless
observed in the XANES spectrum measured in the transmission mode for a high
tungsten content ($x=0.75$).

Unlike the conventional XANES, RXES is a second-order process involving X-ray
absorption and then X-ray emission described by the Kramers-Heisenberg
formalism [39, 40, 41]. Experimental resolution is governed by the lifetime of
intermediate and final states involved in the RXES process and Heisenberg’s
uncertainty principle. Note that the intermediate state of RXES is the same as
the final state of the first-order optical process in XAS [42, 43, 44].

To interpret the experimental data, we performed the W L3-edge XANES
calculations using the FDMNES code [31, 32]. As a result, information was
obtained on the splitting of the 5d(W) orbitals for three local geometries of
tungsten ions similar to those in CuWO4, $\alpha$-CuMoO4 and $\gamma$-CuMoO4
(Fig. 2). The white line in the calculated XANES spectra is split by about 4.0
eV and 3.0 eV, respectively, in CuWO4 and $\gamma$-CuMoO4 (W substitutes Mo at
the Mo2 site) due to the distorted octahedral crystal-field, whereas a smaller
splitting of about 2.0 eV is characteristic of a tungsten ion in the
tetrahedral environment, as when it is located at the Mo2 site in
$\alpha$-CuMoO4.

The splitting of the 5d(W) orbitals can be observed experimentally by
detecting the RXES due to 3d5/2(W)$\rightarrow$2p3/2(W) electron transition,
since the lifetime of the final state with a hole at the 3d5/2(W) orbital is
longer than that with a hole at the 2p3/2(W) orbital (the W L3-edge) [45].
Therefore, the lifetime broadening present in conventional XANES spectra is
significantly reduced in the RXES experiment. As a result, additional features
become visible in the HERFD-XANES spectra, which are blurred or poorly
resolved with the traditional approach. A high-resolution spectrometer with
the high brightness X-ray source and highly effective detectors are required
to acquire them. At the same time, high penetration of hard X-ray radiation
allows one to probe the sample bulk.

Figure 3:  (a) RXES intensity map as a function of incident and emitted
energies for CuMo0.50W0.50O4 solid solution at 300 K. (b) Vertical cuts of the
RXES plane at the incident energies Ei=10190 eV and 10242 eV, shown in (a) by
two vertical red dashed lines. Principal observed bands
(L$\alpha_{1},\alpha_{2}$ and B1-B4) are labeled. (c) A schematic diagram of
2p3d RXES process for octahedral tungsten coordination showing initial,
intermediate and final states. (d) The W L3-edge HERFD-XANES spectra for
different CuMo1-xWxO4 solid solutions measured at the emission energy
Ee=8398.5$\pm$0.2 eV, indicated by the horizontal orange solid line in (a).
(e,f) High energy resolution off-resonant X-ray spectra for different
CuMo1-xWxO4 solid solutions obtained at Ei=10190 eV below the W L3-edge. (g,h)
High energy resolution off-resonant X-ray emission spectra for different
CuMo1-xWxO4 solid solutions obtained at Ei=10242 eV above the W L3-edge.

In the RXES experiment, the intensities and energies of the incoming and
emitted X-rays are monitored. One way of presenting the RXES intensity plane
is to plot the energy of emitted photons against the incident photon energy as
shown in Fig. 3(a). Different plane cuts allow one to obtain valuable
information on the structural and electronic properties of the material. For
example, the W L3-edge HERFD-XANES spectrum can be obtained from Fig. 3(a) by
integrating the emission intensity in the narrow emission energy region of
$\pm$0.2 eV around 8398.5 eV. Thus extracted HERFD-XANES spectra of
CuMo1-xWxO4 solid solutions at 300 K are shown in Fig. 3(d).

Besides, the HERFD-XANES spectra, one can additionally obtain (Figs. 3(a,b))
high energy resolution off-resonant X-ray emission spectra by extracting them
slightly below (bands B1 and B2 in Figs. 3(e,f)) and above (bands B3 and B4 in
Figs. 3(g,h)) the W L3 absorption edge, respectively. As one can see, both
types of emission spectra contain information on the crystal field splitting,
while with a smaller signal-to-noise ratio compared to HERFD-XANES (Fig. 3(d))
due to weaker intensity. Besides, in the non-resonant case (Figs. 3(g,h)), the
tails of the main X-ray emission line (W L${}_{\alpha_{1}}$ in Fig. 3(b))
produce some background under the B3 and B4 bands, which also limits the
accuracy of the analysis and should be removed in advance.

Further, we will discuss the effect of concentration and temperature on the
local structure of tungsten ions in CuMo1-xWxO4 solid solutions observed in
the RXES spectra.

###  4.2 Effect of composition

The concentration dependences of the RXES spectra of CuMo1-xWxO4 solid
solutions at 300 K are shown in Figs. 3(d-h). The origin of the observed bands
is due to three different mechanisms, already discussed above.

The emission spectra for the incident X-ray energy Ei=10190 eV below the W L3
absorption edge are shown in Figs. 3(e,f) and correspond to off-resonant
condition [46]. Two emission bands B1 at $\sim$8312 eV and B2 at $\sim$8376 eV
can be distinguished in the spectra and correspond to the transitions from
3d3/2 and 3d5/2 levels, respectively (Figs. 3(b,c)). The intensity of the
lower energy band B1 is smaller than that of the band B2, correlating with the
intensity of the W L$\alpha_{1}$ and L$\alpha_{2}$ bands.

Both bands B1 and B2 are split into two peaks due to the crystal field of
oxygen ligands. The distance between these peaks and the ratio $I_{1}/I_{2}$
of their intensities were evaluated by fitting the spectra with two Lorentzian
functions and were used to estimate the symmetry of the crystal field. The
difference between the position of the two Lorentzian functions is related to
the crystal-field splitting parameter $\Delta$ or 10Dq used in the crystal-
field theory [37].

One can see in Figs. 3(e,f), that the change of the spectral shape upon
tungsten content increase indicates that the local coordination of tungsten
ions changes from tetrahedral ($x=0.04$) to octahedral ($x>0.15$).

The emission spectra for the incident X-ray energy Ei=10242 eV well above the
W L3 absorption edge are reported in Figs. 3(g,h). In this case, four emission
bands W L$\alpha_{2}$, B3, W L$\alpha_{1}$ and B4 are observed in Fig. 3(b).
The origin of the bands B3 and B4 is similar to that of B1 and B2, i.e., is
due to the transitions from 3d3/2 and 3d5/2 levels, respectively (Fig. 3(c)).
Again the lower energy band B3 has a much weaker intensity than that of the
band B4, and both bands are split by the crystal field to two peaks (Figs.
3(g,h)). Opposite to the case of the bands B1 and B2, the bands B3 and B4 are
located on top of the background due to the W L$\alpha_{1}$ and partially W
L$\alpha_{2}$ fluorescence, which makes it more difficult to determine the
magnitude of the peaks splitting and the ratio of their intensities.
Nevertheless, the obtained parameters agree well with that obtained from the
analysis of the bands B1 and B2 suggesting the octahedral coordination of
tungsten ions for $x>0.15$.

Finally, the analysis was also performed for the W L3-edge HERFD-XANES spectra
(Fig. 3(d)). Here the pronounced splitting of the white line due to the
octahedral crystal field is clearly visible for $x>0.15$ and agrees with the
results obtained from X-ray fluorescence spectra in Figs. 3(e-h). The white
line was fitted with two Lorentzian functions to estimate its splitting
magnitude. The results for the crystal-field splitting parameter $\Delta$
obtained from the off-resonant XES and HERFD-XANES spectra of CuMo1-xWxO4
solid solutions at 300 K and 90 K are combined in Fig. 4.

At 300 K, CuWO4 ($x=1$) and CuMo0.96W0.04O4 ($x=0.04$) have wolframite [35]
and $\alpha$-CuMoO4 [34] phases, respectively, which correspond to the
octahedral and tetrahedral coordination of tungsten ions. Upon cooling down to
90 K, CuWO4 remains always in the triclinic phase (space group $P\bar{1}$)
[47]. At the same time, CuMo0.96W0.04O4 transforms on cooling to
$\gamma$-CuMoO4 phase [1, 2] with distorted octahedral coordination of Mo6+
and W6+ ions.

For intermediate compositions of CuMo1-xWxO4 solid solutions, a mixture of
$\alpha$-CuMoO4 and $\gamma$-CuMoO4 phases co-exists in the samples with the
relative amounts depending on the tungsten content and temperature range [1,
5, 48]. In particular, when samples are cooled from 300 K down to 90 K, a
transition from tetrahedral-to-octahedral tungsten coordination is observed
for $x<0.20$ but not for $x\geq 0.20$ (Fig. 4).

Our theoretical XANES calculations for distorted WO6 and WO4 environments
(Fig. 2) suggest a value of the parameter $\Delta\sim 3-4$ eV for octahedral
coordination and about 2 eV for tetrahedral one. Therefore, taking into
account the experimentally observed $\alpha$-to-$\gamma$ transition and
theoretical predictions, two ranges of the crystal-field splitting parameter
$\Delta$ can be identified (see green and brown regions in Fig. 4): the
tetrahedral coordination of tungsten ions is expected for $\Delta<3$, whereas
the octahedral coordination for $\Delta>3$. Thus, the simple analysis of the
experimental off-resonant XES and HERFD-XANES spectra can provide useful
information on the local coordination of tungsten ions. In the next section,
we will demonstrate the sensitivity of the parameter $\Delta$ to the
hysteretic phase transitions in CuMo1-xWxO4.

Furthermore, we found that the parameter $\Delta$ correlates well with the
solid solution optical properties (their color). Namely, at 300 K, the samples
with $x\leq 0.15$ have the crystal-field splitting parameter $\Delta<3$ eV and
greenish colour, whereas the samples with $x\geq 0.20$ have $\Delta>3$ eV and
brown colour. Note, that upon cooling of the samples with $x\leq 0.15$ down to
90 K, their color changes to brown, in agreement with previous studies [34],
and the parameter $\Delta$ increases above 3 eV indicating the transition to
the phase with the octahedral tungsten coordination.

Figure 4: Crystal-field splitting parameter $\Delta$ as a function of tungsten
content $x$ in CuMo1-xWxO4 solid solutions at 90 K and 300 K calculated from
off-resonant XES (Fig. 3(f)) and HERFD-XANES (Fig. 3(d)) spectra. Schematic
energy level diagram showing crystal-field splitting of tungsten 5d-states in
the tetrahedral and octahedral environment is also shown. Note that
$\Delta_{tet}$ is smaller than $\Delta_{oct}$.

###  4.3 Effect of temperature

It is known that the first-order structural phase transition between $\alpha$
and $\gamma$ phases in CuMo1-xWxO4 solid solutions for $x<0.15$ has strong
hysteretic behaviour, which can be controlled by the tungsten content [1, 5].
The hysteresis loop was first experimentally detected by optical, calorimetry
and magnetic measurement [1].

In our previous studies [21, 49], we have shown for CuMoO4 and CuMo0.90W0.10O4
that the Mo K-edge XANES is sensitive to the $\alpha$-to-$\gamma$ phase
transition due to the significant change in the local coordination of
molybdenum ions from tetrahedral to octahedral one. Moreover, the analysis of
the Mo K-edge XANES allowed us to determine the transition hysteresis loop and
to estimate the fraction of $\alpha$-phase upon cooling and heating. At the
same time, the sensitivity of the Mo K-edge XANES to the $\alpha$-to-$\gamma$
transition is based on a variation of the pre-edge peak amplitude which is
hidden by the natural broadening of the core level (5.8 eV at the Mo K-edge
[24]). Therefore, the high resolution of the off-resonant XES and HERFD-XANES
spectra (Fig. 3) should be more beneficial for tracking the
$\alpha$-to-$\gamma$ phase transition.

Figure 5: Parts of the RXES planes at 90 K, 300 K and 420 K on heating (left
panels) and temperature dependent W L3-edge HERFD-XANES spectra measured on
cooling and heating (right panel) for CuMo0.96W0.04O4. Figure 6: Temperature
dependencies of the magnitude of the peak splitting ($\Delta$) and the ratio
of the white line peak intensities ($I_{1}/I_{2}$) for CuMo0.96W0.04O4 (a) and
CuMo0.90W0.10O4 (b). A fraction of the $\alpha$-phase extracted from the
analysis of the Mo K-edge XANES spectra of CuMo0.90W0.10O4 [21] is also
displayed for comparison (solid triangles). Inset shows a photo of two
CuMo0.96W0.04O4 samples at room temperature: the brown sample is in the
$\gamma$-phase (after treatment at 77 K in liquid nitrogen), and the green
sample is in the $\alpha$-phase.

An example of the temperature-dependent W L3-edge HERFD-XANES spectra of
CuMo0.96W0.04O4 is shown in Fig. 5. The change of the spectral shape reflects
a transition from tetrahedral ($\alpha$-phase) to octahedral ($\gamma$-phase)
tungsten coordination on cooling from 300 K to 120 K, whereas from octahedral
to tetrahedral tungsten coordination on heating from 90 K to 420 K. Note that
the effect is well observed in RXES planes as the temperature dependence of
the relative position of two bright spots, which are well resolved in the
octahedral coordination but change relative intensity and move closer in the
tetrahedral one.

The phase transition hysteresis loops for CuMo0.96W0.04O4 and CuMo0.90W0.10O4
solid solutions are shown in Fig. 6. They were determined from the temperature
dependencies of the magnitude of the peak splitting and the ratio
$I_{1}/I_{2}$ of the peak intensities, which were obtained from the W L3-edge
HERFD-XANES and off-resonant XES spectra normalized on a scale from 0 to 1 to
display them on the same graph. Note that the two approaches have slightly
different sensitivity to the change of tungsten coordination leading to some
scatter of points. The results for CuMo0.90W0.10O4 (Fig. 6(b)) are also
compared with a fraction of the $\alpha$-phase extracted from the analysis of
the Mo K-edge XANES spectra reported in our previous work [21]. One can see a
small difference between the starting temperatures of the $\alpha$-to-$\gamma$
phase transition obtained from the W L3 and Mo K edges for CuMo0.90W0.10O4
sample. When probed at the Mo K-edge, the transition starts at a slightly
lower temperature than when measured at the W L3 edge. This seeming
contradiction has a simple explanation by the fact, that some part of the
tungsten atoms is located in the octahedral environment already at room
temperature, as was concluded above from Fig. 4.

##  5 Conclusions

Resonant X-ray emission spectroscopy at the W L3-edge was used to study the
changes in the local atomic structure of tungsten ions across the
thermochromic phase transition in CuMo1-xWxO4 solid solutions as a function of
sample composition and temperature.

We demonstrated that the analysis of the RXES plane provides useful
information on the coordination of tungsten atoms in the sample bulk and
allows one to determine the crystal-field splitting parameter $\Delta$ for the
5d(W)-states. Moreover, this information can be extracted from the RXES plane
using two different approaches by analysing the high-energy resolution
fluorescence detected X-ray absorption near-edge structure (HERFD-XANES) and
the high energy resolution off-resonant X-ray emission spectra excited below
and above resonance conditions. The analysis of the RXES planes shows a clear
advantage over conventional XANES due to revealing spectral features with much
higher resolution.

The method is well suited for in-situ measurements and was used here to
determine the hysteretic behaviour of the first-order structural phase
transition between $\alpha$ and $\gamma$ phases in CuMo1-xWxO4 solid solutions
on cooling and heating, even at low ($x<0.10$) tungsten content.

We found that tungsten ions in CuMo1-xWxO4 solid solutions have octahedral
coordination for $x>0.15$ at all temperatures, whereas their coordination
changes from tetrahedral to octahedral upon cooling for smaller tungsten
content. Nevertheless, some amount of tungsten ions co-exists in the
octahedral environment at room temperature for $x<0.15$. The obtained results
correlate well with the optical properties of these materials, in particular,
color change from green to brown upon cooling or increasing tungsten content.

The electronic structure of CuMo1-xWxO4 solid solutions controls their
thermochromic properties, which are related to a variation of the transmission
window in optical spectra [1, 6]. A change in the local coordination of
tungsten atoms from tetrahedral to octahedral affects the band gap, which is
determined by the oxygen-to-metal charge transfer [1]. The band gap is smaller
in the case of octahedral coordination of tungsten [6], i.e., at higher
tungsten content or lower temperature.

To conclude, this study demonstrates the possibilities of the RXES technique
to probe the crystal field effect in functional thermochromic materials with
controllable properties on the example of CuMo1-xWxO4 solid solutions.

## Declaration of Competing Interest

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

## Acknowledgements

Financial support provided by Scientific Research Project for Students and
Young Researchers Nr. SJZ/2019/1 realized at the Institute of Solid State
Physics, University of Latvia is greatly acknowledged. The used infrastructure
of the von Hamos spectrometer was realized in the frame of projects FKZ
05K13UK1 and FKZ 05K14PP1. The experiment at the PETRA III synchrotron was
performed within the project No. I-20180615 EC. The synchrotron experiments
have been supported by the project CALIPSOplus under the Grant Agreement
730872 from the EU Framework Programme for Research and Innovation HORIZON
2020. The experiment at the Elettra synchrotron was performed within the
project No. 20150303. Institute of Solid State Physics, University of Latvia
as the Center of Excellence has received funding from the European Union’s
Horizon 2020 Framework Programme H2020-WIDESPREAD-01-2016-2017-TeamingPhase2
under grant agreement No. 739508, project CAMART2.

## Supplementary materials

Supplementary material associated with this article can be found, in the
online version, at doi: XXX.j.actamat.2020.YYY

## References

  * [1] M. Gaudon, C. Carbonera, A. E. Thiry, A. Demourgues, P. Deniard, C. Payen, J. F. Létard, S. Jobic, Adaptable thermochromism in the CuMo1-xWxO4 series ($0\leq x<0.1$): a behavior related to a first-order phase transition with a transition temperature depending on x, Inorg. Chem. 46 (2007) 10200–10207.  doi:10.1021/ic701263c.
  * [2] M. Gaudon, P. Deniard, A. Demourgues, A. E. Thiry, C. Carbonera, A. Le Nestour, A. Largeteau, J. F. Létard, S. Jobic, Unprecedented “one-finger-push”-induced phase transition with a drastic color change in an inorganic material, Adv. Mater. 19 (2007) 3517–3519.  doi:10.1002/adma.200700905.
  * [3] M. Gaudon, B. Basly, Y. Fauque, J. Majimel, M. H. Delville, Thermochromic phase transition on CuMo0.9W0.1O4@SiO2 core-shell particles, Inorg. Chem. 48 (2009) 2136–2139.  doi:10.1021/ic802057c.
  * [4] I. Yanase, T. Mizuno, H. Kobayashi, Structural phase transition and thermochromic behavior of synthesized W-substituted CuMoO4, Ceram. Int. 39 (2013) 2059–2064.  doi:10.1016/j.ceramint.2012.08.059.
  * [5] L. Robertson, N. Penin, V. Blanco-Gutierrez, D. Sheptyakov, A. Demourgues, M. Gaudon, CuMo0.9W0.1O4 phase transition with thermochromic, piezochromic, and thermosalient effects, J. Mater. Chem. C 3 (2015) 2918–2924.  doi:10.1039/C4TC02463J.
  * [6] X. Wu, C. Fu, J. Cao, C. Gu, W. Liu, Effect of W doping on phase transition behavior and dielectric relaxation of CuMoO4 obtained by a modified sol-gel method, Mater. Res. Express 7 (2020) 016309.  doi:10.1088/2053-1591/ab6546.
  * [7] V. Blanco-Gutierrez, L. Cornu, A. Demourgues, M. Gaudon, CoMoO4/CuMo0.9W0.1O4 mixture as an efficient piezochromic sensor to detect temperature/pressure shock parameters, ACS Appl. Mater. Interfaces 7 (2015) 7112–7117.  doi:10.1021/am508652h.
  * [8] M. Gaudon, C. Riml, A. Turpain, C. Labrugere, M. H. Delville, Investigation of the chromic phase transition of CuMo0.9W0.1O4 induced by surface protonation, Chem. Mater. 22 (2010) 5905–5911.  doi:10.1021/cm101824d.
  * [9] J. C. Hill, Y. Ping, G. A. Galli, K.-S. Choi, Synthesis, photoelectrochemical properties, and first principles study of n-type CuW1-xMoxO4 electrodes showing enhanced visible light absorption, Energy Environ. Sci. 6 (2013) 2440–2446.  doi:10.1039/C3EE40827B.
  * [10] Q. Liang, Y. Guo, N. Zhang, Q. Qian, Y. Hu, J. Hu, Z. Li, Z. Zou, Improved water-splitting performances of CuW1-xMoxO4 photoanodes synthesized by spray pyrolysis, Sci. China Mater. 61 (2018) 1297–1304.  doi:10.1007/s40843-018-9287-5.
  * [11] N. Joseph, J. Varghese, M. Teirikangas, H. Jantunen, A temperature-responsive copper molybdate polymorph mixture near to water boiling point by a simple cryogenic quenching route, ACS Appl. Mater. Interfaces 12 (2020) 1046–1053.  doi:10.1021/acsami.9b17300.
  * [12] G. Steiner, R. Salzer, W. Reichelt, Temperature dependence of the optical properties of CuMoO4, Fresenius J. Anal. Chem. 370 (2001) 731–734.  doi:10.1007/s002160000630.
  * [13] I. Jonane, A. Anspoks, G. Aquilanti, A. Kuzmin, High-temperature X-ray absorption spectroscopy study of thermochromic copper molybdate, Acta Mater. 179 (2019) 26–35.  doi:10.1016/j.actamat.2019.06.034.
  * [14] S. Bordiga, E. Groppo, G. Agostini, J. A. van Bokhoven, C. Lamberti, Reactivity of surface species in heterogeneous catalysts probed by In situ X-ray absorption techniques, Chem. Rev. 113 (2013) 1736–1850.  doi:10.1021/cr2000898.
  * [15] N. A. Young, The application of synchrotron radiation and in particular X-ray absorption spectroscopy to matrix isolated species, Coord. Chem. Rev. 277-278 (2014) 224–274.  doi:10.1016/j.ccr.2014.05.010.
  * [16] V. R. Mastelaro, E. D. Zanotto, X-ray absorption fine structure (XAFS) studies of oxide glasses-A 45-year overview, Materials 11 (2018) 204.  doi:10.3390/ma11020204.
  * [17] J. Timoshenko, A. Anspoks, A. Kalinko, I. Jonane, A. Kuzmin, Local structure of multiferroic MnWO4 and Mn0.7Co0.3WO4 revealed by the evolutionary algorithm, Ferroelectrics 483 (2015) 68–74.  doi:10.1080/00150193.2015.1058687.
  * [18] A. Kalinko, M. Bauer, J. Timoshenko, A. Kuzmin, Molecular dynamics and reverse Monte Carlo modeling of scheelite-type AWO4 (A= Ca, Sr, Ba) W L3-edge EXAFS spectra, Phys. Scr. 91 (2016) 114001.  doi:10.1088/0031-8949/91/11/114001.
  * [19] J. Timoshenko, A. Anspoks, A. Kalinko, A. Kuzmin, Local structure of cobalt tungstate revealed by EXAFS spectroscopy and reverse Monte Carlo/evolutionary algorithm simulations, Z. Phys. Chem. 230 (2016) 551–568.  doi:10.1515/zpch-2015-0646.
  * [20] A. Kuzmin, J. Timoshenko, A. Kalinko, I. Jonane, A. Anspoks, Treatment of disorder effects in X-ray absorption spectra beyond the conventional approach, Rad. Phys. Chem. 175 (2020) 108112.  doi:10.1016/j.radphyschem.2018.12.032.
  * [21] I. Jonane, A. Cintins, A. Kalinko, R. Chernikov, A. Kuzmin, Low temperature X-ray absorption spectroscopy study of CuMoO4 and CuMo0.90W0.10O4 using reverse Monte-Carlo method, Rad. Phys. Chem. 175 (2020) 108411.
  * [22] S. Yamazoe, Y. Hitomi, T. Shishido, T. Tanaka, XAFS study of tungsten L1\- and L3-edges: structural analysis of WO3 species loaded on TiO2 as a catalyst for photo-oxidation of NH3, J. Phys. Chem. C 112 (2008) 6869–6879.  doi:10.1021/jp711250f.
  * [23] U. Jayarathne, P. Chandrasekaran, A. F. Greene, J. T. Mague, S. DeBeer, K. M. Lancaster, S. Sproules, J. P. Donahue, X-ray absorption spectroscopy systematics at the tungsten L-edge, Inorg. Chem. 53 (2014) 8230–8241.  doi:10.1021/ic500256a.
  * [24] O. Keski-Rahkonen, M. O. Krause, Total and partial atomic-level widths, At. Data Nucl. Data Tables 14 (1974) 139–146.  doi:10.1016/S0092-640X(74)80020-3.
  * [25] A. Balerna, E. Bernieri, E. Burattini, A. Kuzmin, A. Lusis, J. Purans, P. Cikmach, XANES studies of MeO3-x (Me = W, Re, Ir) crystalline and amorphous oxides, Nucl. Instrum. Methods Phys. Res. A 308 (1991) 240–242.  doi:10.1016/0168-9002(91)90637-6.
  * [26] A. Kuzmin, J. Purans, X-ray absorption spectroscopy study of the local environment around tungsten and molybdenum ions in tungsten-phosphate and molybdenum-phosphate glasses, Proc. SPIE 2968 (1997) 180–185.  doi:10.1117/12.266831.
  * [27] P. Charton, L. Gengembre, P. Armand, TeO2–WO3 glasses: infrared, XPS and XANES structural characterizations, J. Solid State Chem. 168 (2002) 175–183.  doi:10.1006/jssc.2002.9707.
  * [28] W. A. Caliebe, V. Murzin, A. Kalinko, M. Görlitz, High-flux XAFS-beamline P64 at PETRA III, AIP Conf. Proc. 2054 (2019) 060031.  doi:10.1063/1.5084662.
  * [29] A. Kalinko, W. A. Caliebe, R. Schoch, M. Bauer, A von Hamos-type hard X-ray spectrometer at the PETRA III beamline P64, J. Synchrotron Rad. 27 (2020) 31–36.  doi:10.1107/S1600577519013638.
  * [30] A. Di Cicco, G. Aquilanti, M. Minicucci, E. Principi, N. Novello, A. Cognigni, L. Olivi, Novel XAFS capabilities at ELETTRA synchrotron light source, J. Phys.: Conf. Ser. 190 (2009) 012043.  doi:10.1088/1742-6596/190/1/012043.
  * [31] Y. Joly, X-ray absorption near-edge structure calculations beyond the muffin-tin approximation, Phys. Rev. B 63 (2001) 125120.  doi:10.1103/PhysRevB.63.125120.
  * [32] O. Bunău, Y. Joly, Self-consistent aspects of x-ray absorption calculations, J. Phys.: Condens. Matter 21 (2009) 345501.  doi:10.1088/0953-8984/21/34/345501.
  * [33] L. Hedin, B. I. Lundqvist, Explicit local exchange-correlation potentials, J. Phys. C: Solid State Phys. 4 (1971) 2064–2083.  doi:10.1088/0022-3719/4/14/022.
  * [34] M. Wiesmann, H. Ehrenberg, G. Miehe, T. Peun, H. Weitzel, H. Fuess, $p$-$T$ phase diagram of CuMoO4, J. Solid State Chem. 132 (1997) 88–97.  doi:10.1006/jssc.1997.7413.
  * [35] J. Forsyth, C. Wilkinson, A. Zvyagin, The antiferromagnetic structure of copper tungstate, CuWO4, J. Phys.: Condens. Matter 3 (1991) 8433.  doi:10.1088/0953-8984/3/43/010.
  * [36] R. D. Shannon, Revised effective ionic radii and systematic studies of interatomic distances in halides and chalcogenides, Acta Cryst. A 32 (1976) 751–767.  doi:10.1107/S0567739476001551.
  * [37] I. B. Bersuker, Electronic Structure and Properties of Transition Metal Compounds: Introduction to the Theory, 2nd Edition, John Wiley & Sons, Hoboken, New Jersey, 2010.  doi:10.1002/9780470573051.
  * [38] M. Brown, R. E. Peierls, E. A. Stern, White lines in x-ray absorption, Phys. Rev. B 15 (1977) 738–744.  doi:10.1103/PhysRevB.15.738.
  * [39] H. A. Kramers, W. Heisenberg, Über die streuung von strahlung durch atome, Z. Physik 31 (1925) 681–708.  doi:10.1007/BF02980624.
  * [40] P. Glatzel, U. Bergmann, High resolution 1s core hole X-ray spectroscopy in 3d transition metal complexes – electronic and structural information, Coord. Chem. Rev. 249 (2005) 65–95.  doi:10.1016/j.ccr.2004.04.011.
  * [41] J.-P. Rueff, A. Shukla, Inelastic x-ray scattering by electronic excitations under high pressure, Rev. Mod. Phys. 82 (2010) 847–896.  doi:10.1103/RevModPhys.82.847.
  * [42] P. Glatzel, M. Sikora, G. Smolentsev, M. Fernández-García, Hard X-ray photon-in photon-out spectroscopy, Catal. Today 145 (2009) 294–299.  doi:10.1016/j.cattod.2008.10.049.
  * [43] J.-P. Rueff, A. Shukla, A RIXS cookbook: Five recipes for successful RIXS applications, J. Electron Spectrosc. Relat. Phenom. 188 (2013) 10–16.  doi:10.1016/j.elspec.2013.04.014.
  * [44] M. Rovezzi, P. Glatzel, Hard x-ray emission spectroscopy: a powerful tool for the characterization of magnetic semiconductors, Semicond. Sci. Technol. 29 (2) (2014) 023002.  doi:10.1088/0268-1242/29/2/023002.
  * [45] A. Wach, J. Sá, J. Szlachetko, Comparative study of the around-Fermi electronic structure of 5$d$ metals and metal-oxides by means of high-resolution X-ray emission and absorption spectroscopies, J. Synchrotron Rad. 27 (2020) 689–694.  doi:10.1107/S1600577520003690.
  * [46] W. Błachucki, J. Hoszowska, J.-C. Dousse, Y. Kayser, R. Stachura, K. Tyrała, K. Wojtaszek, J. Sá, J. Szlachetko, High energy resolution off-resonant spectroscopy: A review, Spectrochimica Acta B 136 (2017) 23–33.  doi:10.1016/j.sab.2017.08.002.
  * [47] von Hans Weitzel, Magnetische struktur von CoWO4, NiWO4 und CuWO4, Solid State Commun. 8 (1970) 2071–2072.  doi:10.1016/0038-1098(70)90221-8.
  * [48] M. Benchikhi, R. El Ouatib, S. Guillemet-Fritsch, L. Er-Rakho, B. Durand, Investigation of structural transition in molybdates CuMo1-xWxO4 prepared by polymeric precursor method, Process. Appl. Ceram. 11 (2017) 21–26.  doi:10.2298/PAC1701021B.
  * [49] I. Jonane, A. Cintins, A. Kalinko, R. Chernikov, A. Kuzmin, X-ray absorption near edge spectroscopy of thermochromic phase transition in CuMoO4, Low Temp. Phys. 44 (2018) 434–437.  doi:10.1063/1.5034155.