# Oscillation conditions of half-linear delay difference equations of second-
order

C. Jayakumar  C. Jayakumari
Department of Mathematics, Mahendra Arts & Science College (Autonomous),
Kalipatti 637501, Tamil Nadu, India<EMAIL_ADDRESS>,  A. Murugesan
A. Murugesan
Department of Mathematics, Government Arts College (Autonomous), Salem 636007,
Tamil Nadu, India<EMAIL_ADDRESS>,  Vediyappan Govindan  Vediyappan
Govindan
Department of Mathematics, Hindustan Institute of Technology and Science,
Padur 603103, Tamil Nadu, India<EMAIL_ADDRESS>and  Choonkil Park∗
Choonkil Park
Department of Mathematics, Research Institute for Convergence of Basic
Science, Hanyang University, Seoul 04763, Korea<EMAIL_ADDRESS>
###### Abstract.

The goal of our work is to investigate the oscillation and asymptotic
properties of a class of difference equations

$\Delta(r(\zeta)(\Delta
x(\zeta))^{\alpha})+q(\zeta)x^{\alpha}(\zeta-\sigma)=0;\quad\zeta\geq\zeta_{o}$
($*$)

and

$\Delta(r(\zeta)(\Delta
x(\zeta))^{\alpha})+q(\zeta)x^{\alpha}(\zeta-\sigma+1)=0;\quad\zeta\geq\zeta_{o}$
($**$)

with the condition
$\sum^{\infty}\frac{1}{r^{\frac{1}{\alpha}}(\zeta)}<\infty$. In contrast to
most previous studies, the oscillation of the investigated equation is
obtained with only one condition.

###### Key words and phrases:

oscillation; non-oscillation; delay; half-linear; advanced type; second-order
difference equation
∗Corresponding authors: C. Park (email<EMAIL_ADDRESS>fax:
+82-2-2281-0019, orcid: 0000-0001-6329-8228).

###### 2010 Mathematics Subject Classification:

Primary 39A10, 39A12.

##  1\. Introduction

We investigate the oscillation properties of a class of second-order half-
linear difference equations

$\Delta(r(\zeta)(\Delta
x(\zeta))^{\alpha})+q(\zeta)x^{\alpha}(\zeta-\sigma)=0,\quad\zeta\geq\zeta_{o},$
(1.1)

and

$\Delta(r(\zeta)(\Delta
x(\zeta))^{\alpha})+q(\zeta)x^{\alpha}(\zeta-\sigma+1)=0,\quad\zeta\geq\zeta_{o},$
($1.1^{\prime}$)

where the difference operator (forward) $\Delta$ is given by $\Delta
x(\zeta)=x(\zeta+1)-x(\zeta)$, $\sigma\geq 0$ is an integer for (1.1) and
$\sigma\geq 1$ for ($1.1^{\prime}$), and $\alpha=\frac{m}{n}$ for relatively
prime odd positive integers $m$ and $n$. Assume that the real sequences
$\\{r(\zeta)\\}^{\infty}_{\zeta=\zeta_{0}}$ and
$\\{q(\zeta)\\}^{\infty}_{\zeta=\zeta_{0}}$ meet the following requirements:

  * $(H_{1})$

$\\{r(\zeta)\\}^{\infty}_{\zeta=\zeta_{0}}$ is a positive sequence;

  * $(H_{2})$

$q(\zeta)\geq 0$ and $q(\zeta)\not\equiv 0$ for infinitely many values of
$\zeta\in\mathbb{N}(\zeta_{0})=\\{\zeta_{0},\zeta_{0}+1,\zeta_{0}+2,\cdots\\}$.

“A solution of (1.1) is a non-trivial real sequence $\\{x(\zeta)\\}$ that is
defined for $\zeta\geq\zeta_{0}-\sigma$ and satisfies (1.1) for all
$\zeta\geq\zeta_{0}$. Obviously, if $x(\zeta)=A_{\zeta}$ for
$\zeta=\zeta_{0}-\sigma$, $\zeta_{0}-\sigma+1,\ldots,\zeta_{0}-1$ are given,
then (1.1) has a unique solution satisfying the above initial conditions.
(1.1) is said to be oscillatory if a solution $\\{x(\zeta)\\}$ of (1.1) is
neither eventually positive nor eventually negative, and if it is not, it is
said to be non-oscillatory. If all the solutions of (1.1) are non-oscillatory,
then it is called non-oscillatory”[20].

The equation (1.1) is said to be in canonical form if

$R(\zeta):=\sum_{s=\zeta_{0}}^{\zeta-1}\frac{1}{r^{\frac{1}{\alpha}}(s)}\rightarrow\infty\quad
as\quad\zeta\rightarrow\infty.$ (1.2)

The equation (1.1) is in non-canonical form if

$\theta(\zeta):=\sum_{s=\zeta}^{\infty}\frac{1}{r^{\frac{1}{\alpha}}(s)}<\infty.$
(1.3)

In recent years, the oscillation conditions for difference equations have been
examined for first and higher orders (see [4, 5, 8, 9, 13, 14, 18, 17, 20]).
We recommend [1, 2, 3, 15] and the nearly 500 sources listed therein for a
general theory of oscillation of difference equations. Second-order difference
equations have considered far less attention in the study than first-order
difference equations, despite the fact that they occur in mathematical
biology, economics, and other fields (see [5]). Many papers [9, 11, 13, 14,
18] have some recent results on second-order difference equations. On the
other hand, it looks that much less is known about the oscillation properties
of (1.1). Most of the papers [5, 6, 10, 11, 12, 16, 21, 24] deal with the
qualitative properties of solutions to differential and difference equations.

The structure of non-oscillatory solutions differ tremendously between non-
canonical and canonical difference equations. Generally, the first order
difference of a positive solution $\\{x(\zeta)\\}$ of the equation (1.1) is of
constant sign eventually, but Eq. (1.2) assures that the solution is
eventually increasing. The equation (1.1) has been analyzed in canonical form
a number of times.

The goal of our study is to analyze the oscillation condition and asymptotic
properties of the equation (1.1) in the form of non-canonical. As a result,
we’ll assume that (1.3) holds in the next sections. We review at a few key
oscillation outcomes for second-order non-canonical equations that inspired us
to do this research.

In [25], Zhang et al. obtained oscillation conditions for the following
equation

$\Delta(p(\zeta)(\Delta
x(\zeta))^{\alpha})+q(\zeta+1)f(x(\zeta+1))=0,\quad\zeta=0,1,2,\ldots.$

In [19], Li examined and established oscillatory criteria for the solutions of
the following equation

$\Delta(p(\zeta)g(\Delta
x(\zeta)))+q(\zeta+1)f(x(\zeta+1))=0,\quad\zeta=0,1,2,\ldots.$

Saker [23] investigated the following equation

$\Delta(p(\zeta)\Delta
x(\zeta))+q(\zeta)f(x(\zeta-\sigma))=0,\quad\zeta=0,1,2,3,\ldots.$ (1.4)

and established conditions for oscillation of (1.4).

Dinakar et al. [7] established sufficient conditions for the oscillation of
the following equation

$\Delta(a(\zeta)(\Delta
x(\zeta)^{\alpha}))+q(\zeta)x^{\alpha}(\sigma(\zeta))=0,\quad\zeta\geq\zeta_{0},$
(1.5)

and proved that every solutions of (1.5) is oscillatory if

$\sum_{\zeta=\zeta_{0}}^{\infty}\left(\frac{1}{a(\zeta)}\sum_{s=\zeta_{0}}^{\zeta-1}A^{\alpha}(\sigma(s))q(s)\right)^{\frac{1}{\alpha}}=\infty,$

where $A(\zeta)=\sum_{s=\zeta}^{\infty}\frac{1}{a^{\frac{1}{\alpha}}(s)}$.

In [9], Grace and Alzabut established new oscillations criteria for non-linear
second-order equation

$\Delta(a(\zeta)(\Delta
y(\zeta))^{\alpha})+q(\zeta)x^{\gamma}(\zeta-m+1)+c(\zeta)x^{\mu}(\zeta+m^{*}+1)=0,$

by comparison with the oscillation results of first order difference equation.

In [20], Murugesan and Jayakuma derived oscillatory criteria for the
difference equation in non-canonical form of advanced type

$\Delta(r(\zeta)(\Delta
x(\zeta))^{\alpha})+q(\zeta)x^{\alpha}(\zeta+\sigma)=0,\quad\zeta\geq\zeta_{o}$

For the purpose of simplicity and generality, we can only deal with eventually
positive solutions of (1.1). Any functional inequality is assumed to hold
eventually.

##  2\. Main results

######  Theorem 2.1.

If

$\sum_{\zeta=\zeta_{0}}^{\infty}\left(\frac{1}{r(\zeta)}\sum_{s=\zeta_{0}}^{\zeta-1}q(s)\right)^{\frac{1}{\alpha}}=\infty,$
(2.1)

then every solution $\\{x(\zeta)\\}$ of (1.1) oscillates or
$\lim_{\zeta\rightarrow\infty}x(\zeta)=0$.

###### Proof.

Assume to the contrary that (1.1) has an eventually positive solution
$\\{x(\zeta)\\}$. Then either $\Delta x(\zeta)>0$ or $\Delta x(\zeta)<0$ holds
eventually, say, $\zeta\geq\zeta_{1}$ and $\zeta_{1}\geq\zeta_{0}$.

If $\Delta x(\zeta)<0$, then we have
$\lim_{\zeta\rightarrow\infty}x(\zeta)=c\geq 0$ exists. Let $c>0$. Summing
(1.1) from $\zeta_{1}$ to $\zeta-1$, we have

$r(\zeta)(\Delta x(\zeta))^{\alpha}-r(\zeta_{1})(\Delta
x(\zeta_{1}))^{\alpha}+\sum_{s=\zeta_{1}}^{\zeta-1}q(s)x^{\alpha}(s-\sigma)=0,$

which gives that

$r(\zeta)(\Delta
x(\zeta))^{\alpha}\leq-\sum_{s=\zeta_{1}}^{\zeta-1}q(s)x^{\alpha}(s-\sigma)$

and hence

$\Delta
x(\zeta)\leq-\frac{c}{r^{\frac{1}{\alpha}}(\zeta)}\left(\sum_{s=\zeta_{1}}^{\zeta-1}q(s)\right)^{\frac{1}{\alpha}}.$

Summing the above from $\zeta_{1}$ to $\zeta-1$, we have

$x(\zeta)-x(\zeta_{1})\leq-c\sum_{u=\zeta_{1}}^{\zeta-1}\frac{1}{r^{\frac{1}{\alpha}}(u)}\left(\sum_{s=\zeta_{1}}^{u-1}q(s)\right)^{\frac{1}{\alpha}}.$

Taking $\zeta\rightarrow\infty$ and using (2.1), we see that
$x(\zeta)\rightarrow-\infty$ as $\zeta\rightarrow\infty$. This is a
contradiction and hence $c=0$.

Suppose that $\Delta x(\zeta)>0$. Let us define a sequence $\\{w(\zeta)\\}$
given by

$w(\zeta):=\frac{r(\zeta)(\Delta
x(\zeta))^{\alpha}}{x^{\alpha}(\zeta-\sigma)}>0.$

Clearly, $\\{w(\zeta)\\}$ satisfies

$\Delta w(\zeta)=-q(\zeta)-\frac{\alpha w(\zeta+1)\Delta
x(\zeta-\sigma)}{x(\zeta-\sigma)}\leq-q(\zeta).$ (2.2)

Summing (2.2) from $\zeta_{1}$ to $\zeta-1$, we obtain

$w(\zeta)\leq w(\zeta_{1})-\sum_{s=\zeta_{1}}^{\zeta-1}q(s).$

In view of (2.1) and (1.3), we get that
$\\{\sum_{s=\zeta_{1}}^{\zeta-1}q(s)\\}$ is an unbounded sequence, which is a
contradiction. Hence the case $\Delta x(\zeta)>0$ is not possible and hence we
complete the proof. ∎

######  Theorem 2.2.

If one of the following conditions

$\sum_{s=\zeta_{0}}^{\infty}\left(\frac{1}{r(\zeta)}\sum_{s=\zeta_{1}}^{\zeta-1}q(s)\theta^{\alpha}(s-\sigma)\right)^{\frac{1}{\alpha}}=\infty$
(2.3)

and

$\sum_{s=\zeta_{0}}^{\infty}q(s)\theta^{\alpha+1}(s+1)=\infty$ (2.4)

holds, then (1.1) oscillates.

###### Proof.

On the contrary, let us suppose that (1.1) has an eventually positive solution
$\\{x(\zeta)\\}$. Then, either $\Delta x(\zeta)>0$ or $\Delta x(\zeta)<0$
eventually, $\zeta\geq\zeta_{1}$, $\zeta_{1}\geq\zeta_{0}$.

Case I:

Assume that $\Delta x(\zeta)<0$ and (2.3) holds. Now, for $l>\zeta$, we have

$\displaystyle x(l)-x(\zeta)$ $\displaystyle=\sum_{s=\zeta}^{l-1}\Delta x(s)$
$\displaystyle\leq\left(r(\zeta)(\Delta
x(\zeta))^{\alpha}\right)^{\frac{1}{\alpha}}\sum_{s=\zeta}^{l-1}\frac{1}{r^{\frac{1}{\alpha}}(s)}.$

Thus, as $\zeta\rightarrow\infty$, we obtain

$-x(\zeta)\leq\left(r(\zeta_{1})(\Delta
x(\zeta_{1}))^{\alpha}\right)^{\frac{1}{\alpha}}\theta(\zeta).$ (2.5)

Using (2.5) in (1.1), we have

$\Delta(r(\zeta)(\Delta x(\zeta))^{\alpha})\leq
L^{\alpha}q(s)\theta^{\alpha}(\zeta-\sigma),$

where $L=\left(r(\zeta_{1})(\Delta
x(\zeta_{1}))^{\alpha}\right)^{\frac{1}{\alpha}}<0.$ Summing the above from
$\zeta_{1}$ to $\zeta-1$, we obtain

$\displaystyle r(\zeta)(\Delta x(\zeta_{1}))^{\alpha}$ $\displaystyle\leq
r(\zeta_{1})\Delta
x(\zeta_{1})+L^{\alpha}\sum_{s=\zeta_{1}}^{\zeta-1}q(s)\theta^{\alpha}(s-\sigma)$
$\displaystyle\leq
L^{\alpha}\sum_{s=\zeta_{1}}^{\zeta-1}q(s)\theta^{\alpha}(s-\sigma).$

It follows from the last inequality that

$x(\zeta)-x(\zeta_{1})\leq
L\left(\sum_{u=\zeta_{1}}^{\zeta-1}\frac{1}{r(u)}\sum_{s=\zeta_{1}}^{u-1}q(s)\theta^{\alpha}(s-\sigma)\right)^{\frac{1}{\alpha}}.$

By using (2.3), we see that $\lim_{\zeta\rightarrow}x(\zeta)=-\infty$ which is
a contradiction on the assumption of $\\{x(\zeta)\\}$.

Case II: Assume that $\Delta x(\zeta)<0$ and (2.4) holds. Set

$u(\zeta)=\frac{r(\zeta)(\Delta
x(\zeta))^{\alpha}}{x^{\alpha}(\zeta-\sigma)}.$ (2.6)

Obviously, $u(\zeta)<0$, $\zeta\geq\zeta_{1}$. Since $\\{r(\zeta)(\Delta
x(\zeta))^{\alpha}\\}$ is nonincreasing, we obtain

$r(s)(\Delta x(s))^{\alpha}\leq r(\zeta)(\Delta x(\zeta))^{\alpha},\quad
s\geq\zeta\geq\zeta_{1}.$

Summing the above from $\zeta$ to $l-1$, we have

$x(l)\leq x(\zeta)+r^{\frac{1}{\alpha}}(\zeta)\Delta
x(\zeta)\sum_{s=\zeta}^{l-1}\frac{1}{r^{\frac{1}{\alpha}}(s)}.$

Letting $l\rightarrow\infty$ in the above inequality, we have

$0\leq x(\zeta)+r^{\frac{1}{\alpha}}(\zeta)\Delta
x(\zeta)\theta(\zeta),\quad\zeta\geq\zeta_{1}.$

So we get

$\frac{r^{\frac{1}{\alpha}}(\zeta)\Delta
x(\zeta)}{x(\zeta)}\theta(\zeta)\geq-1,\quad\zeta\geq\zeta_{1}$

or

$-1\leq\theta(\zeta)u^{\frac{1}{\alpha}}(\zeta)<0,\quad\zeta\geq\zeta-1.$

From (2.6), we have

$\Delta u(\zeta)+q(\zeta)+\frac{\alpha
u^{\frac{1+\alpha}{\alpha}}(\zeta)}{r^{\frac{1}{\alpha}}(\zeta-\sigma)}\leq
0,\quad\zeta\geq\zeta_{1}.$ (2.7)

Multiplying (2.7) by $\theta^{\alpha+1}(\zeta)$ and summing from $\zeta_{1}$
to $l-1$, we have

$\theta^{\alpha+1}(l)u(l)-\theta^{\alpha+1}(\zeta_{1})u(\zeta_{1})+\sum_{s=\zeta_{1}}^{l-1}q(s)\theta^{\alpha+1}(s+1)\\\
-(\alpha+1)\sum_{s=\zeta_{1}}^{l-1}\frac{u(s)\theta^{\alpha}(s)}{r^{\frac{1}{\alpha}}(s)}+\alpha\sum_{s=\zeta_{1}}^{l-1}\frac{\theta^{\alpha+1}(s+1)u^{\frac{\alpha+1}{\alpha}}(s)}{r^{\frac{1}{\alpha}}(s-\sigma)}\leq
0.$ (2.8)

Now

$\left|\sum_{s=\zeta_{1}}^{\infty}\frac{\theta^{\alpha}(s)u(s)}{r^{\frac{1}{\alpha}}(s)}\right|\leq\sum_{s=\zeta_{1}}^{\infty}\frac{1}{r^{\frac{1}{\alpha}}(s)}|\theta^{\alpha}(s)u(s)|\leq\sum_{s=\zeta_{1}}^{\infty}\frac{1}{r^{\frac{1}{\alpha}}(s)}<\infty$

and

$\left|\sum_{s=\zeta_{1}}^{\infty}\frac{\theta^{\alpha+1}(s+1)u^{\frac{\alpha+1}{\alpha}}(s)}{r^{\frac{1}{\alpha}}(s-\sigma)}\right|\leq\sum_{s=\zeta_{1}}^{\infty}\frac{1}{r^{\frac{1}{\alpha}}(s-\sigma)}\left|\theta^{\alpha}(s)u(s)\right|^{\frac{\alpha+1}{\alpha}}<\sum_{s=\zeta_{1}}^{\infty}\frac{1}{r^{\frac{1}{\alpha}}(s-\sigma)}<\infty.$

By applying the above inequalities, letting $l\rightarrow\infty$ in (2.8), we
see that

$\sum_{s=\zeta_{1}}^{\infty}q(s)\theta^{\alpha+1}(s+1)<\infty,$

which contradicts (2.4).

So in either case $\Delta x(\zeta)<0$ is impossible.

If $\Delta x(\zeta)>0$, then we repeat the same procedure as in the proof of
Theorem 2.1 and so we omitt the proof. ∎

######  Lemma 2.3.

[7] Suppose that

$\sum_{\zeta=\zeta_{0}}^{\infty}q(\zeta)=\infty.$ (2.9)

If $\\{x(\zeta)\\}^{\infty}_{\zeta=\zeta_{0}}$ is an eventually positive
solution of (1.1), then $\Delta x(\zeta)<0$.

######  Theorem 2.4.

If

$\limsup_{\zeta\rightarrow\infty}\left(\theta^{\alpha}(\zeta)\sum_{s=\zeta_{1}}^{\zeta-1}q(s)\right)>1,$
(2.10)

then (1.1) oscillates.

###### Proof.

Let us assume on the contrary, without loss of generality, that (1.1) has an
eventually positive solution $\\{x(\zeta)\\}$. Then, either $\Delta
x(\zeta)<0$ or $\Delta x(\zeta)>0$ eventually, for $\zeta\geq\zeta_{1}$,
$\zeta_{1}\geq\zeta_{0}$. First now, we suppose that $\Delta x(\zeta)<0$.
Since $\\{r^{\frac{1}{\alpha}}(\zeta)\Delta x(\zeta)\\}$ is decreasing, we get

$\displaystyle x(\zeta-\sigma)\geq x(\zeta)\geq$
$\displaystyle\sum_{s=\zeta}^{\infty}\frac{1}{r^{\frac{1}{\alpha}}(s)}\left(-r^{\frac{1}{\alpha}}(s)\Delta
x(s)\right)$ $\displaystyle\geq-\theta(\zeta)r^{\frac{1}{\alpha}}(\zeta)\Delta
x(\zeta)\geq 0.$ (2.11)

Summing (1.1) from $\zeta_{1}$ to $\zeta-1$ and using (2), we obtain

$\displaystyle-r(\zeta)(\Delta x(\zeta))^{\alpha}\geq$
$\displaystyle\sum_{s=\zeta_{1}}^{\zeta-1}q(s)x^{\alpha}(s-\sigma)$
$\displaystyle\geq x^{\alpha}(\zeta-\sigma)\sum_{s=\zeta_{1}}^{\zeta-1}q(s)$
$\displaystyle\geq-r(\zeta)(\Delta
x(\zeta))^{\alpha}\theta^{\alpha}(\zeta)\sum_{s=\zeta_{1}}^{\zeta-1}q(s).$

Taking the limsup, we have a contradiction to (2.10).

Now suppose that $\Delta x(\zeta)>0$. Then by (1.3) and (2.10), we have (2.9)
holds. Then by Lemma 2.5, we have a contradiction and the proof is now
complete. ∎

The comparison with the second-order linear delay difference equation in
canonical form

$\Delta(\widetilde{r}(\zeta)\Delta
x(\zeta-1))+\widetilde{q}(\zeta)x(\zeta-\sigma)=0$ (2.12)

yields the following oscillation criterion for ($1.1^{\prime}$).

######  Lemma 2.5.

[8] If the difference inequality

$\Delta(\widetilde{r}(\zeta)\Delta
x(\zeta-1))+\widetilde{q}(\zeta)x(\zeta-\sigma)\leq 0$

has an eventually positive solution, then Eq. (2.12) has an eventually
positive solution.

######  Lemma 2.6.

If $\alpha\geq 1$ and $\\{x(\zeta)\\}$ is an eventually positive solution of
($1.1^{\prime}$), then

$\left(\frac{r^{\frac{1}{\alpha}}(\zeta)\Delta
x(\zeta)}{x(\zeta-\sigma+1)}\right)^{\alpha-1}\leq\theta^{1-\alpha}(\zeta).$
(2.13)

###### Proof.

Suppose that ($1.1^{\prime}$) has an eventually positive solution
$\\{x(\zeta)\\}$. Then we get $\Delta x(\zeta)>0$ or $\Delta x(\zeta)<0$
eventually for $\zeta\geq\zeta_{1}$, $\zeta_{1}\geq\zeta_{0}$.

Let us assume first that $\Delta x(\zeta)<0$. Since $\alpha\geq 1$, repeatting
the procedure as in the proof of Theorem 2.4, we obtain

$x(\zeta-\sigma+1)\geq-\theta(\zeta)r^{\frac{1}{\alpha}}(\zeta)\Delta
x(\zeta)\geq 0,$

which implies (2.13).

Assume now that $\Delta x(\zeta)>0$. Note that
$R(\zeta-\sigma)+\theta(\zeta-\sigma)=\theta(\zeta_{0})>0$ together with (1.3)
implies $R(\zeta-\sigma)\geq\theta(\zeta)$. Now

$\displaystyle x(\zeta-\sigma+1)\geq x(\zeta-\sigma)$
$\displaystyle\geq\sum_{s=\zeta_{1}}^{\zeta-\sigma-1}\Delta x(s)$
$\displaystyle=\sum_{s=\zeta_{1}}^{\zeta-\sigma-1}\frac{1}{r^{\frac{1}{\alpha}}(s)}r^{\frac{1}{\alpha}}(s)\Delta
x(s)$ $\displaystyle\geq R(\zeta-\sigma)r^{\frac{1}{\alpha}}(\zeta)\Delta
x(\zeta)$ $\displaystyle\geq\theta(\zeta)r^{\frac{1}{\alpha}}(\zeta)\Delta
x(\zeta),$

which is obviously the same as (2.13) and this completes the proof. ∎

######  Theorem 2.7.

Let $\alpha\geq 1$. Suppose that Eq. (2.12) with

$\widetilde{r}(\zeta)=\theta(\zeta)\theta(\zeta+1)r^{\frac{1}{\alpha}}(\zeta),$
$\widetilde{q}(\zeta)=\frac{1}{\alpha}\theta(\zeta+1)\theta^{\alpha-1}(\zeta)\theta(\zeta-\sigma+1)q(\zeta)$

oscillates. Then ($1.1^{\prime}$) oscillates.

###### Proof.

Let us assume on the contrary that $\\{x(\zeta)\\}$ is an eventually positive
solution of ($1.1^{\prime}$). Now the equation ($1.1^{\prime}$) can be
rewritten as

$\Delta\left(r^{\frac{1}{\alpha}}(\zeta)\Delta
x(\zeta)\right)+\frac{1}{\alpha}\left(r^{\frac{1}{\alpha}}(\zeta)\Delta
x(\zeta)\right)^{1-\alpha}q(\zeta)x^{\alpha}(\zeta-\sigma+1)\leq 0.$

By Lemma 2.6, we conclude that $\\{x(\zeta)\\}$ is a solution of the second-
order inequality

$\Delta\left(r^{\frac{1}{\alpha}}(\zeta)\Delta
x(\zeta)\right)+\frac{1}{\alpha}\theta^{\alpha-1}(\zeta)q(\zeta)x(\zeta-\sigma+1)\leq
0,$

for $\zeta\geq\zeta_{1}$, $\zeta\geq\zeta_{0}$. Now the inequality can be
modified into

$\frac{1}{\theta(\zeta+1)}\Delta\left(\theta(\zeta)\theta(\zeta+1)r^{\frac{1}{\alpha}}(\zeta)\Delta\left(\frac{x(\zeta)}{\theta(\zeta)}\right)\right)+\frac{1}{\alpha}\theta^{\alpha-1}(\zeta)q(\zeta)x(\zeta-\sigma+1)\leq
0.$ (2.14)

Put

$u(\zeta-1)=\frac{x(\zeta)}{\theta(\zeta)}.$

Then the inequality (2.14) reduces to

$\Delta(\widetilde{r}(\zeta)\Delta
u(\zeta-1))+\widetilde{q}(\zeta)u(\zeta-\sigma)\leq 0.$ (2.15)

This shows that $\\{u(\zeta)\\}$ is an eventually positive solution of (2.15).
Then by Lemma 2.5, (2.12) has an eventually positive solution which is a
contradiction and hence the proof is completed. ∎

######  Example 2.8.

Consider the difference equation

$\Delta\left(2^{\frac{\zeta}{3}}(\Delta
x(\zeta))^{\frac{1}{3}}\right)+\lambda_{0}2^{\zeta}x^{\frac{1}{3}}(\zeta-1)=0;\quad\zeta\geq
1,$ (2.16)

where $r(\zeta)=2^{\frac{\zeta}{3}}$, $q(\zeta)=\lambda_{0}2^{\zeta}$,
$\alpha=\frac{1}{3}$ and $\sigma=1$. We can easily verify that (2.1) holds and
by Theorem 2.1, every eventually non-oscillatory solution of (2.16) converges
to zero as $\zeta\rightarrow\infty$. Also, we can easily verify that (2.10)
holds and hence by Theorem 2.4, (2.16) oscillates if $\lambda_{0}>1$.

######  Example 2.9.

Consider the difference equation

$\Delta\left((\zeta(\zeta-1))^{\frac{1}{3}}(\Delta
x(\zeta))^{\frac{1}{3}}\right)+\zeta^{\frac{4}{3}}x^{\frac{1}{3}}(\zeta-1)=0;\quad\zeta\geq
1,$ (2.17)

where $r(\zeta)=(\zeta(\zeta-1))^{\frac{1}{3}}$,
$q(\zeta)=\zeta^{\frac{4}{3}}$, $\alpha={\frac{1}{3}}$ and $\sigma=1$. We can
easily show that $\theta(\zeta)=\frac{1}{\zeta-1}$ and also

$\sum_{s=1}^{\infty}q(s)\theta^{\alpha+1}(s+1)=\infty.$

Hence (2.17) oscillates by Theorem 2.2.

######  Example 2.10.

Consider the difference equation

$\Delta\left((\zeta(\zeta+1))^{\frac{5}{3}}(\Delta
x(\zeta))^{\frac{5}{3}}\right)+\frac{4(\zeta^{2}-1)\zeta^{\frac{2}{3}}}{3}x(\zeta-1)=0;\quad\zeta=1,2,3,\ldots,$
(2.18)

where $r(\zeta)=(\zeta(\zeta+1))^{\frac{5}{3}}$,
$q(\zeta)=\frac{4(\zeta^{2}-1)\zeta^{\frac{2}{3}}}{3}$, $\alpha=\frac{5}{3}$
and $\sigma=2$. We can easily calculate that $\theta(\zeta)=\frac{1}{\zeta}$,

$\widetilde{r}(\zeta)=\theta(\zeta)\theta(\zeta+1)r^{\frac{1}{\alpha}}(\zeta)=1,$
(2.19)

and

$\widetilde{q}(\zeta)=4.$ (2.20)

Clearly, $\sum_{\zeta=1}^{\infty}\widetilde{q}(\zeta)=\infty$. Then by [22,
Theorem 3.1], every solution of (2.12) is oscillatory with $\sigma=2$, (2.19)
and (2.20). One such a solution of (2.12) is
$\\{(-1)^{\zeta}\\}^{\infty}_{\zeta=1}$. By Theorem 2.7, (2.18) oscillates.

##  3\. Conclusion

The results of this study are unique and also have a high degree of
generality. In contrast to most previous studies, our oscillation result of
the investigated equation (1.1) is obtained with only one condition. Also, we
have derived oscillation criteria for ($1.1^{\prime}$) by comparison with the
second-order linear delay difference equation in canonical form. To illustrate
our results, three examples are provided.

## Declarations

Availablity of data and materials
Not applicable.

Human and animal rights
We would like to mention that this article does not contain any studies with
animals and does not involve any studies over human being.

Conflict of interest
The authors declare that they have no competing interests.

Fundings
The authors declare that there is no funding available for this paper.

Authors’ contributions
The authors equally conceived of the study, participated in its design and
coordination, drafted the manuscript, participated in the sequence alignment,
and read and approved the final manuscript.

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# Near K-Edge Photoionization and Photoabsorption of Singly, Doubly, and
Triply Charged Silicon Ions

Stefan Schippers  I. Physikalisches Institut, Justus-Liebig-Universität
Gießen, Heinrich-Buff-Ring 16, 35392 Giessen, Germany Stefan Schippers
<EMAIL_ADDRESS>Sebastian Stock  Helmholtz-Institut Jena, Fröbelstieg 3,
07743 Jena, Germany  Theoretisch-Physikalisches Institut, Friedrich-Schiller-
Universität Jena, 07743 Jena, Germany  Ticia Buhr  I. Physikalisches Institut,
Justus-Liebig-Universität Gießen, Heinrich-Buff-Ring 16, 35392 Giessen,
Germany  Alexander Perry-Sassmannshausen  I. Physikalisches Institut, Justus-
Liebig-Universität Gießen, Heinrich-Buff-Ring 16, 35392 Giessen, Germany
Simon Reinwardt  Institut für Experimentalphysik, Universität Hamburg, Luruper
Chaussee 149, 22761 Hamburg, Germany  Michael Martins  Institut für
Experimentalphysik, Universität Hamburg, Luruper Chaussee 149, 22761 Hamburg,
Germany  Alfred Müller  Institut für Atom- und Molekülphysik, Justus-Liebig-
Universität Gießen, Leihgesterner Weg 217, 35392 Giessen, Germany  Stephan
Fritzsche  Helmholtz-Institut Jena, Fröbelstieg 3, 07743 Jena, Germany
Theoretisch-Physikalisches Institut, Friedrich-Schiller-Universität Jena,
07743 Jena, Germany

###### Abstract

Experimental and theoretical results are presented for double, triple, and
quadruple photoionization of Si+ and Si2+ ions and for double photoionization
of Si3+ ions by a single photon. The experiments employed the photon-ion
merged-beams technique at a synchrotron light source. The experimental photon-
energy range 1835–1900 eV comprises resonances associated with the excitation
of a $1s$ electron to higher subshells and subsequent autoionization.
Energies, widths, and strengths of these resonances are extracted from high-
resolution photoionization measurements, and the core-hole lifetime of K-shell
ionized neutral silicon is inferred. In addition, theoretical cross sections
for photoabsorption and multiple photoionization were obtained from large-
scale Multi-Configuration Dirac-Hartree-Fock (MCDHF) calculations. The present
calculations agree with the experiment much better than previously published
theoretical results. The importance of an accurate energy calibration of
laboratory data is pointed out. The present benchmark results are particularly
useful for discriminating between silicon absorption in the gaseous and in the
solid component (dust grains) of the interstellar medium.

atomic data — atomic processes — line: identification — opacity

##  1 Introduction

Silicon is a relatively abundant element in the Universe and, in particular, a
major component of interstellar dust. The binding energy of Si $K$-shell
electrons amounts to about 1840 eV (Deslattes et al., 2003). Correspondingly,
Si $K$-shell absorption by the interstellar medium (ISM) can be observed by
x-ray satellites when directed towards distant x-ray emitting cosmic objects
such as x-ray binaries (Rogantini et al., 2020). For the correct determination
of the silicon abundance in the ISM it is important to know how much silicon
is in the gaseous state and how much is (chemically) bound to dust grains
(see, e.g., Jenkins, 2009). Since high-resolution x-ray absorption
spectroscopy is sensitive to chemical shifts of characteristic absorption
lines, astrophysical models of the ISM have recently been augmented by
absorption coefficients of silicon-containing minerals that were obtained from
laboratory measurements at a synchrotron light source (Zeegers et al., 2017,
2019). Here we provide complementary laboratory data for $K$-shell x-ray
absorption by atomic Si+, Si2+, and Si3+ ions that address the gaseous
component of the ISM.

Previous experimental work on the photoabsorption of silicon ions considered
only absorption by outer electronic shells (Mosnier et al., 2003; Bizau et
al., 2009; Kennedy et al., 2014). With the very recent exception of negatively
charged Si- (Perry-Sassmannshausen et al., 2021), $K$-shell x-ray absorption
data for silicon ions were hitherto exclusively obtained from theoretical
calculations (Verner et al., 1993; Palmeri et al., 2008; Witthoeft et al.,
2011; Kučas et al., 2012, 2015; Hasoglu et al., 2021). Because of the many-
body nature of the theoretical problem the theoretical calculations have to
resort to approximations yielding results that bear (usually unknown)
uncertainties. In this situation, benchmarking by laboratory experiments
(Schippers & Müller, 2020) is vital for arriving at sufficiently accurate
results that allow one, in the present context, to reliably discriminate
between the gaseous and solid components of the ISM.

Similar to our previous work on $L$-shell ionization of Fe+, Fe2+, Fe3+ and
Ar+ ions (Schippers et al., 2017; Beerwerth et al., 2019; Schippers et al.,
2021; Müller et al., 2021), we have investigated ionization of silicon ions in
different charge states. Here, we present experimental and theoretical data
for $m$-fold photoionization of Siq+ ions with primary charge state $q$
leading to the production of Sir+ ions with product charge-state $r=q+m$

$h\nu+\mathrm{Si}^{q+}\to\mathrm{Si}^{r+}+m\mathrm{e}^{-}.$ (1)

The paper is organized as follows. Section 2 discusses experimental issues, in
particular the role of metastable ions in the silicon ion beams and the energy
calibration of the photon energy scale. Section 3 describes calculations of
the absorption cross-sections and of the complex deexcitation cascades that
set in after the initial creation of a $K$-shell hole. The results are
presented and discussed in section 4 emphasizing their relevance for the
identification of gaseous silicon in the ISM. The concluding section 5 briefly
summarizes the findings and provides an outlook on future directions of
laboratory astrophysics in the area of atomic inner-shell absorption.

##  2 Experiment

The measurements were carried out at the synchrotron light source PETRA III
operated by DESY in Hamburg, Germany. More specifically, the photon-ion
merged-beam technique (recently reviewed by Schippers et al., 2016a) was
employed using the PIPE end-station (Schippers et al., 2014; Müller et al.,
2017; Schippers et al., 2020) at the photon-beamline P04 (Viefhaus et al.,
2013). The beamline’s monochromator is equipped with two diffraction gratings
with different rulings of 400 and 1200 lines/mm. The latter was used for the
present experiment because it offers the highest photon flux in the photon-
energy region of interest. The experimental procedures have been described
previously (Schippers et al., 2014; Müller et al., 2017). Therefore, we only
provide the details here that are relevant for the presently-studied ion
species.

Beams of Siq+ ions ($q$=1–3) were produced by evaporating SiO powder from an
electrically heated oven into an electron-cyclotron-resonance (ECR) ion
source. The ion source was operated on a positive potential of 6 kV such that
positively charged ions were accelerated from the ion source towards the
electrically grounded ion beam-line. Downbeam, a dipole electromagnet served
for selecting the desired ion species according to their mass-to-charge ratio.
Subsequently, the ion beam was collimated and centered onto the counter-
propagating photon beam by tuning electrostatic deflectors and lenses
appropriately. Typical ion currents in the interaction region were 4.5 nA of
28Si+ and 2.5 nA of 28Si3+. The 28Si2+ ion current was significantly higher
(20 nA) since it was heavily contaminated with, e.g., N+ or CO2+ ions, which
have the same mass-to-charge ratio as 28Si2+. Similarly, it cannot be excluded
that the Si+ beam was contaminated by N${}_{2}^{+}$ or CO+ ions. It has to be
pointed out that the product ions with charge states increased by 1, 2, 3,
$\ldots$ units can all be individually separated by the dipole magnet (the
demerger) in front of the ion detector so that each final channel could be
individually investigated.

Relative cross sections for multiple photoionization (cf. Equation 1) were
determined over a preselected range of photon energies by normalizing the
photon-energy dependent product-ion count rate on the photon flux and on the
ion-current, which were measured with a calibrated photodiode and a Faraday
cup, respectively. For the Si3+ primary ions, the relative cross sections were
put on an absolute scale by additionally accounting for the separately
measured geometrical beam overlap (for details see Schippers et al., 2014;
Müller et al., 2017). The uncertainty of the experimental absolute cross-
section scale amounts to $\pm$15% (Schippers et al., 2014).

The measurement of absolute cross sections requires a careful adjustment of
the mutual overlap of the photon and ion beams. This procedure was only
applied to the Si3+ ion beam. An experimental determination of absolute cross
sections for the multiple ionization of Si+ and Si2+ primary ions was not
possible because of the unknown fractions of contaminating ions in these ion
beams. Therefore, the Si+ and Si2+ cross sections were normalized to
theoretical absorption cross sections as explained below.

###  2.1 Metastable primary ions

The Siq+ ion beams contained ground-level ions and possibly also unknown
fractions of long-lived metastable ions with lifetimes longer than the ions’
flight time (a few microseconds) through the apparatus. For aluminium-like
Si+, where we assume a statistical population of the experimentally unresolved
two fine-structure components of the $3s^{2}\,3p\;^{2}P$ ground term, the
long-lived metastable levels are the $3s\,3p^{2}\;{}^{4}P_{J}$ levels with
excitation energies of $\sim$5.3 eV and lifetimes in the range 0.13–1.2 ms
(Froese Fischer et al., 2006). Magnesium-like Si2+ has long-lived
$3s\,3p\;^{3}P_{J}$ excited levels with excitation energies of $\sim$6.6 eV
and lifetimes of 58 $\mu$s ($J=1$), 7.8 s ($J=2$), and $\infty$ ($J=0$) with
regard to single-photon emission (Froese Fischer et al., 2006). Sodium-like
Si3+ does not have long-lived singly excited metastable levels. However, the
core-excited $2p^{5}\,3s\,3p\;^{4}D_{7/2}$ level is metastable against
autoionization with a lifetime in the 1–10 $\mu$s range (Howald et al., 1986).
Its excitation energy is 106 eV (Schmidt et al., 2007).

The fractional populations of metastable excited levels depend on their
excitation energies and on the plasma conditions in the ion source. In outer-
shell photoionization experiments, their presence is often revealed by
characteristic threshold and resonance features in the measured
photoionization cross-sections. By comparing experimental and theoretical
photoionization cross-sections Kennedy et al. (2014) inferred a
$3s\,3p^{2}\;{}^{4}P$ fraction of about 10% for a Si+ ion beam that was also
produced with an ECR ion source. Similarly, Mosnier et al. (2003) arrived at a
2–3% $3s\,3p\;^{3}P$ fraction of their Si2+ ion beam. In the present inner-
shell photoionization measurements we did not recognise any specific cross-
section features that can be attributed to the presence of metastable levels.
Nevertheless, such metastable levels cannot be excluded for the present Si+
and Si2+ beams similar as in the previous experiments by Kennedy et al. (2014)
and Mosnier et al. (2003).

In view of its high excitation energy, one does not expect a significant
population of the autoionizing Si3+($2p^{5}\,3s\,3p\;^{4}D_{7/2}$) metastable
level. However, even a tiny contamination of the Si3+ beam can lead to a
noticeable production of Si4+ ions via autoionization. In the present
experiment, this created a large background in the Si3+ single-ionization
channel, such that no meaningful experimental result could be obtained for the
production of Si4+ by photoionization of Si3+. We conclude that, apart from
this limitation, metastable primary ions do not play a significant role in the
present study.

###  2.2 Photon-energy calibration

The photon-energy scale was calibrated by a separate measurement of the
krypton $L_{3}$ and $L_{2}$ absorption edges at about 1677 eV and 1730 eV
(Figure 1) using a combination of a gas jet and a photoelectron spectrometer
(see Müller et al., 2017, 2018, for details). The recommended values for the
Kr $L_{3}$ and $L_{2}$ threshold energies are 1679.07(39) and 1730.90(50) eV
(Deslattes et al., 2003). Unfortunately, it is not clear where to read these
values off the measured absorption curves. Therefore, we have applied energy
shifts to our measured data such that the presently measured krypton
absorption edges line up with the corresponding absorption curves of
Wuilleumier (1971) as displayed in Figure 1.

Wuilleumier (1971) provided wavelengths in Cu x units referring to the Cu
$K\alpha_{1}$ line. For the conversion of these units to electronvolts, the
CODATA 2018 set of fundamental physical constants was used (Tiesinga et al.,
2021). The resulting conversion factor agrees with the one provided earlier by
Deslattes et al. (2003) within its negligible (in the present context)
uncertainty.

Figure 1: Calibration of the photon-energy scale by shifting the present
krypton absorption data (symbols) to the experimental results of Wuilleumier
(1971) (full lines, digitized) for the $L_{3}$ and $L_{2}$ absorption edges
displayed in panels a) and b), respectively. The full line in panel c) is the
extrapolated difference between calibrated photon energies Ecal and nominal
photon energies $E_{\mathrm{nom}}$ for the entire energy-range of interest.
The gray shaded area visualizes the associated uncertainties as determined
from error propagation (see text). In all panels, the abscissae are the
calibrated photon energy.

Absorption at the Kr $L_{3}$ and $L_{2}$ edges was also measured by others.
The results of Kato et al. (2007) differ from those of Wuilleumier (1971) by
$\sim$1.25 and $\sim$0.55 eV, respectively. Kato et al. (2007) calibrated
their energy scale to the Kr $2p_{3/2}^{-1}5p$ excitation energy in the
vicinity of the $L_{3}$ threshold as determined by Ibuki et al. (2002). These
latter authors did not provide an estimate for the uncertainty of their energy
scale. The same holds for the Kr $L_{3}$ absorption measurements of Schmelz et
al. (1995). Nagaoka et al. (2000) and Okada et al. (2005) calibrated their
energy scales by referring to the work of Wuilleumier (1971) as is also done
here.

The energy shifts applied to our absorption curves were 4.65 and 4.7 eV for
the $L_{3}$ edge and the $L_{2}$ edge, respectively. We attribute a read-off
uncertainty of 0.05 eV to this calibration method. For the extrapolation to
higher energies we used a linear fit through the two calibration points. By
this extrapolation, the read-off error propagates to energy uncertainties of
$\pm$0.2 eV at 1840 eV and of $\pm$0.3 eV at 1920 eV (Figure 1c). Taking the
0.5 eV uncertainty of the recommended $L_{2}$-threshold energy (Deslattes et
al., 2003) and a 0.2 eV uncertainty associated with imperfections of the
photon beamline (see below) additionally into account, we arrive at an
uncertainty of $\pm$1 eV of the present calibrated photon-energy scale for the
photon energies in the current experimental range of 1835–1900 eV.

In calibrating the photon-energy scale of the silicon ions, we have
additionally considered the Doppler shift that is caused by the unidirectional
movement of the ions in the Siq+ ion beams (see, e.g., Schippers et al., 2014,
for details). This does not introduce any significant additional uncertainty
of the photon-energy calibration.

##  3 Theory

Table 1: Configurations that are taken into account in the present calculations of the deexcitation cascades following the direct $1s$ ionization and the $1s\to 3p$ excitation of Si+. All in all, nearly a quarter million of level-to-level transition rates had to be calculated to describe the cascade processes. Ion |  $1s$ ionization |  $1s\to 3p$ excitation
---|---|---
Si+ |  | $1s^{1}\,2s^{2}\,2p^{6}\,3s^{2}\,3p^{2}$
Si2+ | $1s^{2}\,2s^{2}\,2p^{6}\,3s^{2}$ | $1s^{2}\,2s^{2}\,2p^{6}\,3s^{2}$
| $1s^{2}\,2s^{2}\,2p^{6}\,3s^{1}\,3p^{1}$ | $1s^{2}\,2s^{2}\,2p^{6}\,3s^{1}\,3p^{1}$
| $1s^{2}\,2s^{2}\,2p^{5}\,3s^{2}\,3p^{1}$ | $1s^{2}\,2s^{2}\,2p^{6}\,3p^{2}$
| $1s^{2}\,2s^{2}\,2p^{5}\,3s^{1}\,3p^{2}$ | $1s^{2}\,2s^{2}\,2p^{5}\,3s^{2}\,3p^{1}$
| $1s^{1}\,2s^{2}\,2p^{6}\,3s^{2}\,3p^{1}$ | $1s^{2}\,2s^{2}\,2p^{5}\,3s^{1}\,3p^{2}$
|  | $1s^{2}\,2s^{1}\,2p^{6}\,3s^{2}\,3p^{1}$
|  | $1s^{2}\,2s^{1}\,2p^{6}\,3s^{1}\,3p^{2}$
|  | $1s^{2}\,2s^{2}\,2p^{4}\,3s^{2}\,3p^{2}$
|  | $1s^{2}\,2s^{1}\,2p^{5}\,3s^{2}\,3p^{2}$
|  | $1s^{2}\,2p^{6}\,3s^{2}\,3p^{2}$
Si3+ | $1s^{2}\,2s^{2}\,2p^{6}\,3s^{1}$ | $1s^{2}\,2s^{2}\,2p^{6}\,3s^{1}$
| $1s^{2}\,2s^{2}\,2p^{6}\,3p^{1}$ | $1s^{2}\,2s^{2}\,2p^{6}\,3p^{1}$
| $1s^{2}\,2s^{2}\,2p^{5}\,3s^{2}$ | $1s^{2}\,2s^{2}\,2p^{5}\,3s^{2}$
| $1s^{2}\,2s^{2}\,2p^{5}\,3s^{1}\,3p^{1}$ | $1s^{2}\,2s^{2}\,2p^{5}\,3s^{1}\,3p^{1}$
| $1s^{2}\,2s^{1}\,2p^{6}\,3s^{2}$ | $1s^{2}\,2s^{2}\,2p^{5}\,3p^{2}$
| $1s^{2}\,2s^{1}\,2p^{6}\,3s^{1}\,3p^{1}$ | $1s^{2}\,2s^{1}\,2p^{6}\,3s^{2}$
| $1s^{2}\,2s^{2}\,2p^{4}\,3s^{2}\,3p^{1}$ | $1s^{2}\,2s^{1}\,2p^{6}\,3s^{1}\,3p^{1}$
| $1s^{2}\,2s^{1}\,2p^{5}\,3s^{2}\,3p^{1}$ | $1s^{2}\,2s^{1}\,2p^{6}\,3p^{2}$
| $1s^{2}\,2p^{6}\,3s^{2}\,3p^{1}$ | $1s^{2}\,2s^{2}\,2p^{4}\,3s^{2}\,3p^{1}$
|  | $1s^{2}\,2s^{2}\,2p^{4}\,3s^{1}\,3p^{2}$
|  | $1s^{2}\,2s^{1}\,2p^{5}\,3s^{2}\,3p^{1}$
|  | $1s^{2}\,2s^{1}\,2p^{5}\,3s^{1}\,3p^{2}$
Si4+ | $1s^{2}\,2s^{2}\,2p^{6}$ | $1s^{2}\,2s^{2}\,2p^{6}$
| $1s^{2}\,2s^{2}\,2p^{5}\,3s^{1}$ | $1s^{2}\,2s^{2}\,2p^{5}\,3s^{1}$
| $1s^{2}\,2s^{2}\,2p^{5}\,3p^{1}$ | $1s^{2}\,2s^{2}\,2p^{5}\,3p^{1}$
| $1s^{2}\,2s^{1}\,2p^{6}\,3s^{1}$ | $1s^{2}\,2s^{1}\,2p^{6}\,3s^{1}$
| $1s^{2}\,2s^{1}\,2p^{6}\,3p^{1}$ | $1s^{2}\,2s^{1}\,2p^{6}\,3p^{1}$
| $1s^{2}\,2s^{2}\,2p^{4}\,3s^{2}$ | $1s^{2}\,2s^{2}\,2p^{4}\,3s^{2}$
| $1s^{2}\,2s^{2}\,2p^{4}\,3s^{1}\,3p$ | $1s^{2}\,2s^{2}\,2p^{4}\,3s^{1}\,3p^{1}$
| $1s^{2}\,2s^{1}\,2p^{5}\,3s^{2}$ | $1s^{2}\,2s^{2}\,2p^{4}\,3p^{2}$
| $1s^{2}\,2s^{1}\,2p^{5}\,3s^{1}\,3p^{1}$ |
Si5+ | $1s^{2}\,2s^{2}\,2p^{5}$ | $1s^{2}\,2s^{2}\,2p^{5}$
| $1s^{2}\,2s^{1}\,2p^{6}$ |
| $1s^{2}\,2s^{2}\,2p^{4}\,3s^{1}$ |

The present theoretical calculations were performed in the framework of the
Multi-Configuration Dirac-Hartree-Fock (MCDHF) method by applying the Grasp
(Jönsson et al., 2007) and Ratip (Fritzsche, 2012) atomic structure codes for
the calculation of atomic energy levels and transition rates. In particular,
the Auger cascades were analyzed and calculated that follow the creation of a
$K$-shell hole in initially Siq+ ions by direct $1s$ ionisation or by $1s$
$\to$ $np$ excitation. These cascade computations (Fritzsche et al., 2021)
support quantitative estimates for the distribution of Sir+ product ions (cf.
Equation 1), similar to earlier studies for other atoms and ions (Stock et
al., 2017; Beerwerth & Fritzsche, 2017; Buth et al., 2018; Beerwerth et al.,
2019; Schippers et al., 2017, 2021).

In these cascade computations, the Siq+ ions were assumed to be initially in
their respective ground configuration with statistically populated fine-
structure levels. Cross sections for the direct photoionization of these ions
were obtained for each atomic subshell, while the resonant $1s\to np$
photoexcitation was considered for all shells with $3\leq n\leq 7$ for Si+ and
$3\leq n\leq 8$ for Si2+ and Si3+. From the incoherent summation of all these
direct-photoionization and resonant-excitation contributions, the total
photoabsorption cross-sections were determined for Si+, Si2+, and Si3+ ions.

Figure 2:  Experimental cross sections for double (blue triangles up, $m$=2),
triple (orange triangles down, $m$=3), and quadruple (magenta diamonds, $m$=4)
photoionization of a) Si+, b) Si2+, and c) Si3+. The grey circles in panels a)
and b) represent the sum (Equation 2) of the cross sections
$\sigma_{1,\Sigma}=\sigma_{1,3}+\sigma_{1,4}+\sigma_{1,5}$ and
$\sigma_{2,\Sigma}=\sigma_{2,4}+\sigma_{2,5}+\sigma_{2,6}$, respectively. For
Si3+, only the double ionization cross-section $\sigma_{3,5}$ was measured.
The open circle at $\sim$1887 eV in panel c) represents the measured absolute
data point. Its error bar comprises the statistical uncertainty and a 15%
systematic uncertainty. The dashed curves are the present theoretical results
for single ionization (olive), double ionization (blue), triple ionization
(orange), and quadruple ionization (magenta). The black dashed curves are the
corresponding cross-section sums. The theoretical cross sections were
convolved with a Gaussian with a FWHM of $\Delta E=3$ eV corresponding to the
experimental photon energy spread and shifted by a) $-1.7$ eV, b) $-1.8$ eV,
and c) $-1.2$ eV in order to line up the theoretical resonance features with
the experimental ones. Single ionization could not be measured due to the low
cross sections and comparatively high background levels, particulary for Si3+
where a strong background was caused by autoionizing metastable levels (see
Section 2.1).

While different autoionization processes might contribute to the electron
emission from inner-shell excited ions, only those single-step Auger processes
were taken into account into the cascade, that are energetically allowed in a
configuration-averaged representation of the atomic fine-structure. This
simplifies the cascade computations and makes them tractable, although further
contributions from the simultaneous excitation (shake-up) or ionization
(shake-off) of another electron were found essential as well for accurately
predicting the final charge-state distributions of light and especially
negatively-charged ions (Müller et al., 2015; Schippers et al., 2016b; Perry-
Sassmannshausen et al., 2020). For multiply-charged ions, however, these shake
processes are typically suppressed, though not always negligible. Table 1
lists all those electron configurations that were included into the present
cascade computations for Si+ ions. The number of cascade steps, i.e., the so-
called depth of the cascade, were taken to be 4 for Si+, 3 for Si2+ , and 2
for Si3+. For the analysis of the cascades, therefore, the highest product-ion
charge state was $r=5$ (cf. Equation 1), while higher charge states were not
accessible within the given approach and by excluding autoionization processes
with an additional excitation or ionization of electrons.

Theoretical cross sections for multiple photoionization were eventually
obtained by multiplying the calculated absorption cross sections with the
branching ratios from the cascade computations. These computations were
carried out separately for the different $1s$ core ionized and $1s\to np$ core
excited configurations. For the comparison with the experimental results, the
theoretical cross sections were convolved with a Gaussian with the full width
at half maximum (FWHM) corresponding to the experimental photon energy spread
$\Delta E$.

##  4 Results and discussion

###  4.1 Cross sections for multiple ionization

Figure 2 provides an overview over the measured and calculated cross sections
$\sigma_{q,r}$ for multiple photoionization of Siq+ ions (cf. Equation 1) with
primary charge states $q=1,2,3$ together with the cross-section sums

$\sigma_{q,\Sigma}=\sum_{r}\sigma_{q,r}.$ (2)

The experimental energy ranges comprise the thresholds for direct ionization
of one $K$-shell electron. The experimental photon-energy spread was $\Delta
E\approx 3$ eV. This is sufficient for resolving the individual $1s\to np$
resonance groups for $n=3$ and $n=4$. For Si3+, even the $1s\to 5p$ resonance
group can be discerned.

The logarithmic cross-section scales cover up to four orders of magnitude. For
Si+, the main ionization channel is triple ionization. For Si2+, triple
ionization dominates only for energies above $\sim$1875 eV where direct
$K$-shell ionization becomes energetically possible. At lower energies, double
ionization is stronger. For Si3+, double ionization is the dominating
ionization channel in the entire experimental photon-energy range. The single-
ionization channel could not be measured as explained above (Section 2.1).
Other Si3+ ionization channels were scrutinized, but no detectable signal was
found. This indicates that the associated cross sections are small as is also
corroborated by our theoretical calculations.

Our theoretical ab-initio cross sections are in remarkable agreement with the
measured ones considering the simplifications that had to be applied in order
to make the calculations of the complex deexcitation cascades tractable. We
like to point out that the calculations were performed independently of the
measurements.

The experimental and theoretical resonances line up if energy shifts of
$-1.7$, $-1.8$, and $-1.2$ eV are applied to the Si+, Si2+, and Si3+ cross
sections, respectively. These differences between our theoretical and
experimental resonance positions are less than a factor of 2 larger than the 1
eV uncertainty of the experimental energy scale. Our theoretical calculations
do not account for any of the metastable primary ions discussed in Section
2.1. The comparison between the theoretical and the experimental resonance
structures suggests that metastable ions do indeed not play a significant role
in the present study as already concluded above.

Only the theoretical and experimental cross-sections $\sigma_{3,5}$ can be
directly compared on an absolute scale. The mutual agreement is within the
$\pm$15% experimental uncertainty over nearly the entire experimental energy
range. The largest discrepancy between the experimental and present
theoretical cross sections concerns the (small) Si+ double-ionization cross-
section which is underestimated by a factor of $\sim$2\. At energies between
1845 and 1858 eV, slight discrepancies occur which are due to a limited
ability of the theory to correctly describe the experimentally observed
resonance structure. There are no such obvious discrepancies for Si+ and Si2+.

We conclude that the present theoretical approach is capable of reliably
predicting the multiple photoionization cross-sections of low-charged silicon
ions. Similar accuracy can be expected for neighboring elements from the
periodic table. The benchmarking by our experimental results leads to rather
small corrections of the theoretical resonance energies which are only
slightly larger than the uncertainty of the present experimental photon-energy
scale.

###  4.2 Absorption cross-sections

Figure 3:  Panel a): Experimental cross-section sum (symbols) for negatively
charged silicon ions (Perry-Sassmannshausen et al., 2021). The dash-dotted
curve is the photoabsorption cross section for neutral silicon atoms as
recommended by Henke et al. (1993). Panels b), c), and d): Experimental cross-
section sums (symbols) from Figure 2 for Si+, Si2+, and Si3+, respectively, in
comparison with corresponding present theoretical photoabsorption cross-
sections (shaded curves) and those of Verner et al. (1993, dashed curves) and
Witthoeft et al. (2011, full blue curves). For the comparison, the theoretical
cross sections were convolved with a Gaussian with a FWHM of 3 eV in order to
account for the experimental photon energy spread. In addition, the present
theoretical cross sections were shifted by $-1.7$ eV, $-1.8$ eV, and $-1.2$
eV, respectively, in order to line up the theoretical resonance features with
the experimental ones. The vertical arrows mark the present theoretical
thresholds for direct $K$-shell ionization with the above energy shifts
applied.

For Si+ and Si2+, the experimental cross-section sums $\sigma_{q,\Sigma}$ were
put on absolute scales by multiplying all individual cross sections
$\sigma_{q,r}$ for a given primary charge state $q$ by the same factor such
that the sums line up with the respective theoretical absorption cross-
sections of Verner et al. (1993, Figure 3b–3d). We have applied the same
approach already earlier to the photoabsorption of negatively charged Si- ions
(Perry-Sassmannshausen et al., 2021) where we used the theoretical absorption
cross section for neutral silicon by Henke et al. (1993) as a reference. These
data are shown in Figure 3a for comparison. Apparently, the cross sections for
non-resonant absorption are nearly independent of the primary charge state. At
1890 eV the cross sections for Si0 (Henke et al., 1993) and for Si+, Si2+, and
Si3+ (Verner et al., 1993) all amount to about 0.15 Mb.

For energies where there are no signatures from resonant processes, our
theoretical cross-section sums for Si+, Si2+, and Si3+ agree with the results
of Verner et al. (1993), which do not account for resonant photoabsorption.
The same is true for the more recent theoretical results of Witthoeft et al.
(2011), who also did include resonant photoabsorption in their calculations.
However, their resonance positions deviate significantly by up to $\sim$9 eV
from the measured ones. Our present energy-shifted (by less than 2 eV, see
above) theoretical results fit much better to the experimentally observed
resonance positions than the results of Witthoeft et al. (2011).

Another significant discrepancy between the present and the previous results
concerns the thresholds for direct $K$-shell photoionization. Our theoretical
results are 1860, 1874, and 1891 eV for Si+, Si2+, and Si3+, respectively,
with the above mentioned energy shifts applied. As can be seen from Figure 3,
the threshold energies as predicted by Verner et al. (1993) are lower by more
than 20 eV. This shows that our experimental benchmarks are vital for arriving
at an accurate positioning of the various absorption features. From the
comparison with the theoretical absorption cross sections we conclude that,
within the present experimental uncertainties, the experimental cross-section
sums $\sigma_{q,\Sigma}$ represent the Siq+ photoabsorption cross-sections.

###  4.3 High-resolution measurements of resonances

Figure 4:  Measured cross sections $\sigma_{1,4}$ for triple photoionization
of Si+ ions using different photon-energy spreads $\Delta E$ of a) 1.45 eV
(500 $\mu$m), b) 0.80 eV (250 $\mu$m), c) 0.45 eV (100 $\mu$m) and d) 0.30 eV
(20 $\mu$m) with the numbers in parentheses denoting the nominal monochromator
exit-slit widths. The thick full lines represent the results from fits of the
sum of four Voigt line profiles to the measured data. The four data sets were
fitted simultaneously using a common set of resonance parameters. The
resulting fit values are listed in Table 2. The individual line profiles are
visualized by the thin full lines in panel d). Additional fit parameters were
$\Delta E$ (the Gaussian FWHM, $\pm$0.02 eV fit uncertainty in all cases), a
constant background (practically negligible) and an overall energy shift
amounting to a) $-0.19\pm$0.01 eV, b) $-0.150\pm 0.007$ eV, c) $-0.089\pm
0.005$ eV, and d) 0 eV.The shaded curves in panel d) is the present
theoretical cross section $\sigma_{1,4}$ shifted in energy by $-1.7$ eV and
convolved with a Gaussian with a FWHM of 0.3 eV.

The most prominent resonance feature in each absorption cross-section is the
lowest-energy resonance group, which is associated with $1s\to 3p$ excitation
(Figure 3). Its relative strength increases with increasing primary charge
state. This also holds for the resonances associated with $1s$ excitation to
higher $np$ subshells. In order to provide accurate resonance data we have
measured the most prominent resonance features at lower photon-energy spreads.
This was achieved at the expense of photon flux by narrowing the monochromator
exit-slit.

Table 2:  Results of the resonance fits to the Si+ high-resolution data
($\sigma_{1,4}$) for the $1s^{-1}\,3s^{2}\,3p^{2}$ resonance group (Figure 4).
The resonance energies $E_{r}$, Lorentzian line widths $\Gamma$, and resonance
strength $S_{1,4}$ were obtained from fitting Voigt line profiles to the
experimental data. The column labelled by $S_{\Sigma}$ contains the derived
absorption line strengths (see text). Numbers in parentheses denote the
uncertainties that were obtained from the fit.

Designation | $E_{\mathrm{r}}$ | $\Gamma$ | $S_{1,4}$ | $S_{\Sigma}$
---|---|---|---|---
| (eV) | (eV) | (Mb eV)††footnotemark:  | (Mb eV)
$1s^{-1}\,3s^{2}\,3p^{2}\;{}^{2\\!}P$ | 1841.26(1) | 0.38(22) | 0.91(07) | 1.42(11)
$1s^{-1}\,3s^{2}\,3p^{2}\;{}^{2\\!}D$ | 1841.82(1) | 0.39(09) | 0.70(14) | 1.09(22)
$1s^{-1}\,3s^{2}\,3p^{2}\;{}^{4\\!}P$ | 1842.35(5) | 0.34(13) | 0.15(07) | 0.23(11)
$1s^{-1}\,3s^{2}\,3p^{2}\;{}^{2\\!}S$ | 1843.43(2) | 0.94(04) | 0.26(01) | 0.41(02)

††footnotetext: 1 Mb = $10^{-18}$ cm2

Figure 5: Measured cross sections $\sigma_{2,4}$ for double photoionization of
Si2+ ions using different photon-energy spreads $\Delta E=0.45$ eV and $\Delta
E=0.30$ eV. The thick full lines represent the results from fits of the sum of
two (for a and c) and four (for b and d) Voigt line profiles to the measured
data. The data sets in panels a) and c) were fitted simultaneously using a
common set of resonance parameters. The same holds for the data sets from
panels b) and d). The resulting fit values are listed in Table 3. In panel c),
the individual line profiles are visualized by the thin full lines. Additional
fit parameters were $\Delta E$ (kept fixed), a constant background
(subtracted) and an overall energy shift amounting to a) $-0.26\pm$0.02 eV, b)
$-0.05\pm 0.02$ eV, c) $0$ eV, and d) $0$ eV. The shaded curve in panels c)
and d) is the present theoretical cross section $\sigma_{2,4}$ shifted in
energy by $-1.8$ eV and convolved with a Gaussian with a FWHM of 0.3 eV.

Table 3: Results of the resonance fits to the Si2+ high-resolution data ($\sigma_{2,4}$) for the $1s^{-1}\,3s^{2}\,np$ resonance groups with $n=3,4,5$ (Figure 5). For the explanation of the notation see Table 2.  Designation | $E_{\mathrm{r}}$ | $\Gamma$ | $S_{2,4}$ | $S_{\Sigma}$
---|---|---|---|---
| (eV) | (eV) | (Mb eV) | (Mb eV)
$1s^{-1}\,3s^{2}\,3p\;^{1\\!}P$ | 1845.10(01) | 0.28(03) | 2.48(16) | 3.33(22)
$1s^{-1}\,3s^{2}\,3p\;^{3\\!}P$ | 1845.84(15) | 0.76(14) | 0.72(16) | 0.97(22)
$1s^{-1}\,3s^{2}\,4p\;^{1\\!}P$ | 1862.33(24) | 0.37(08) | 0.55(10) | 0.60(11)
$1s^{-1}\,3s^{2}\,4p\;^{3\\!}P$ | 1863.10(20) | 1.77(37) | 0.48(18) | 0.52(20)
$1s^{-1}\,3s^{2}\,5p\;^{1\\!}P$ | 1867.58(08) | 0.00(20) | 0.07(05) |
$1s^{-1}\,3s^{2}\,5p\;^{3\\!}P$ | 1867.85(08) | 0.84(15) | 0.32(08) |

Figure 4 displays high-resolution measurements of the
Si+($1s^{-1}\,3s^{2}\,3p^{2}$) resonance group, which consists of the four LS
terms ${}^{2}S$, ${}^{2}P$, ${}^{4}P$, and ${}^{2}D$ resulting from the
coupling of the single $1s$ electron to the three $3p^{2}$ parent terms
${}^{1}S$, ${}^{3}P$, and ${}^{1}D$. For these measurements the dominant
triple-ionization channel ($\sigma_{1,4}$) was chosen. The experimental
resolving power increases when going from panel a) to panel d). The numerical
values of the respective photon energy spreads $\Delta E$ were obtained by a
simultaneous fit of four Voigt line profiles to the four measured data sets.
The resonance parameters that resulted from the fit, i.e., the resonance
energies $E_{r}$, the Lorentzian line widths $\Gamma$, and the resonance
strengths $S_{1,4}$ are listed in Table 2.

Figure 6: Measured (symbols) and presently calculated (shaded curve) cross section $\sigma_{3,5}$ for double photoionization of Si3+ ions using a photon-energy spread $\Delta E=0.45$ eV. The resonances are associated with the excitation of an $1s$ electron to the $3p$, $4p$, $5p$, and $6p$ subshells. The thick full line represents the result from a fit of the sum of six Voigt line profiles to the measured data. The resulting fit values are listed in Table 4. Additional fit parameters were $\Delta E$ (kept fixed) and a constant background (subtracted from the data). The shaded curve is the present theoretical cross section $\sigma_{3,5}$ shifted in energy by $-1.2$ eV and convoluted with a Gaussian with a FWHM of 0.45 eV. Table 4:  Results of the resonance fits to the Si3+ high-resolution data ($\sigma_{3,5}$) for the $1s^{-1}\,3s\,np$ resonance groups with $n=3,4,5$ (Figure 6). For the explanation of the notation see Table 2.  Designation | $E_{\mathrm{r}}$ | $\Gamma$ | $S_{3,5}$ | $S_{\Sigma}$
---|---|---|---|---
| (eV) | (eV) | (Mb eV) | (Mb eV)
$1s^{-1}\,3s\,3p\,(^{3\\!}P)\;^{2\\!}P$ | 1849.64(01) | 0.45(02) | 5.20(09) | 5.20(09)
$1s^{-1}\,3s\,3p\,(^{1\\!}P)\;^{2\\!}P$ | 1853.24(01) | 0.35(04) | 1.23(06) | 1.23(06)
$1s^{-1}\,3s\,4p\,(^{3\\!}P)\;^{2\\!}P$ | 1871.91(02) | 0.34(05) | 1.19(08) | 1.19(08)
$1s^{-1}\,3s\,4p\,(^{1\\!}P)\;^{2\\!}P$ | 1872.96(04) | 0.45(12) | 0.56(08) | 0.56(08)
$1s^{-1}\,3s\,5p\;^{2\\!}P$ | 1880.20(05) | 1.00(44) | 0.79(06) | 0.79(06)
$1s^{-1}\,3s\,6p\;^{2\\!}P$ | 1883.86(04) | 0.23(12) | 0.25(04) | 0.25(04)

In the fit, an overall energy shift was used as an additional fit parameter
individually for each data set (see caption of Figure 4). The maximum shift
amounts to $-0.19$ eV which is much less than the 1 eV uncertainty of the
experimental photon-energy scale. Ideally, one would expect that the resonance
positions were independent of the width of the monochromator’s exit-slit. We
attribute the observed shifts to a slight asymmetry of the photon-energy
distribution and to mechanical imperfections of the photon-beamline’s
mechanics. A pertaining 0.2 eV uncertainty has already been considered in the
error budget of our photon-energy calibration (see Section 2.2).

In analogy to the work of Schlachter et al. (2004) for carbon, we note that
the $1s^{-1}\,3s^{2}\,3p^{2}\;{}^{2\\!}P$ and
$1s^{-1}\,3s^{2}\,3p^{2}\;{}^{4\\!}P$ levels, which are populated by $1s\to
3p$ excitation of Si+, can also be prepared by the removal of a $1s$ electron
from neutral ground-term neutral Si($1s^{2}\,3s^{2}\,3p^{2}\;{}^{3\\!}P$).
Therefore, the widths of the $1s^{-1}\,3s^{2}\,3p^{2}\;{}^{2\\!}P$ and
$1s^{-1}\,3s^{2}\,3p^{2}\;{}^{4\\!}P$ levels from Table 2 correspond to the
core-hole lifetime $\tau$ of initially neutral silicon. Both widths agree
within their experimental uncertainties. From the less uncertain value we
obtain $\tau=h/\Gamma=1.4\pm 0.5$ fs in agreement with the present
calculations and with the value of 1.47 fs calculated by Palmeri et al.
(2008).

In Figure 4d, the present theoretical results for $\sigma_{1,4}$ are compared
with the experimental high-resolution results. For the comparison, the
theoretical resonances were shifted in energy by $-1.7$ eV and convoluted with
a Gaussian with a FWHM of 0.3 eV. This resolution is a factor of 10 higher as
compared to the one in Figure 3 and reveals some discrepancies between
experiment and theory which are missed when looked at with larger photon-
energy spread. The calculated splittings between the various
$1s^{-1}\,3s^{2}\,3p^{2}$ terms are larger than what is found experimentally.

The computed resonance strengths of individual terms contributing to
$\sigma_{1,4}$ (see Figure 4d) are larger than the experimental ones by about
50%. Since the present theoretical and experimental absorption cross-sections
$\sigma_{1,\Sigma}$ agree with one another within the experimental uncertainty
(Figure 3b) it must be concluded that the mismatch for $\sigma_{1,4}$ is due
to the overestimation of the corresponding branching ratios for the
$1s^{-1}\,3s^{2}\,3p^{2}$ resonances.

The theoretical branching ratios for the triple-ionization channel are almost
the same for all of the $1s^{-1}\,3s^{2}\,3p^{2}$ resonances, i.e., their
differences are smaller than the uncertainties of the individual $S_{1,4}$
values. This allows us to scale the fitted individual resonance strengths to
the absorption cross-section. From a fit to the $1s^{-1}\,3s^{2}\,3p^{2}$ peak
in Figure 3b we obtain a strength of 3.15(17) Mb eV for the sum of the four
tabulated resonances. The sum of the $S_{1,4}$ values in Table 2 amounts to
2.02(17) Mb eV. Using these values we have calculated the tabulated absorption
resonance-strengths as $S_{\Sigma}=S_{1,4}\times 3.15/2.02$. In the same way
we have calculated $S_{\Sigma}$ separately for the $1s^{-1}\,3s^{2}\,3p$
[factor 4.30(3)/3.20(22)] and $1s^{-1}\,3s^{2}\,4p$ [factor (1.12(4)/1.03(21)]
resonances of the Si2+ absorption spectrum (Table 3). For Si3+, the absorption
spectrum is essentially identical with $\sigma_{3,5}$ and, thus,
$S_{\Sigma}=\sigma_{3,5}$ for all resonances in Table 4.

The fits to the Si2+ and Si3+ high-resolution data are shown in Figures 5 and
6, respectively. The agreement between experiment and the present theory
(shifted by $-1.8$ eV for Si2+ and by $-1.2$ eV for Si3+, see Figure 3) is
remarkable, particularly, for $\sigma_{2,4}$ (Figure 5). There is less
agreement for Si3+, as already noted in the low-resolution data in Figures 2
and 3. The too large separation of the first two peaks is probably caused by
subtleties concerning the atomic fine structure associated with the angular
momentum coupling of three open atomic subshells.

###  4.4 Comparison with other forms of silicon

Figure 7:  a) Present results for the Si+, Si2+, and Si3+ absorption features
that are associated with $1s\to 3p$ excitations. Each individual resonance is
represented as a Lorentzian. The corresponding resonance parameters $E_{r}$,
$\Gamma$ and $S_{\Sigma}$ are provided in Tables 2, 3, and 4. b) Absorption
cross-sections of some silicon containing crystalline minerals (Zeegers et
al., 2019). The horizontal arrows denote the experimental uncertainties of the
photon energy scales amounting to a) $\pm 1$ eV or $\pm 0.0036$ Å and b) $\pm
1.5$ eV or $\pm 0.0055$ Å (see text).

Figure 7 shows a comparison between the strongest Si $K$-shell absorption
features for Si+, Si2+, and Si3+ (panel a) with the absorption by some silicon
containing minerals that potentially occur in interstellar dust (panel b,
Zeegers et al., 2019). In order to be able to discriminate between the
different forms of silicon in x-ray absorption spectra, the energies of the
absorption features must be known with sufficient precision. The systematic
uncertainty of the present experimental energy scale is $\pm 1$ eV (Section
2.2). Zeegers et al. (2019) did not quantify the uncertainty of their energy
scale. Instead, they referred for their energy calibration to previous work
(Li et al., 1995; Nakanishi & Ohta, 2009) which, in turn, is based on earlier
investigations. The energy scale of Nakanishi & Ohta (2009) can be traced back
to the work of Wong et al. (1999) who mention that their energy calibration
varies by 2–3 eV depending on the operating conditions of their synchrotron
light source. This suggests that the uncertainty of the energy scale of
Zeegers et al. (2019, Figure 7b) is at least $\pm 1.5$ eV.

Different minerals exhibit different chemical shifts of the Si $K$-shell
absorption features which vary by up to 1.5 eV (Li et al., 1995). This is
still less than the combined energy uncertainties of the present energy scale
and the one of Zeegers et al. (2019). Therefore, we conclude that, in general,
x-ray absorption spectroscopy can discriminate between absorption by gaseous
silicon and absorption by silicon containing minerals. In particular, the Si+
absorption feature in Figure 7a appears at an unambiguous location ($\sim
1841.5$ eV). This is also true for the corresponding absorption feature of Si-
which occurs at an even lower energy of 1838.4 eV (Figure 3a, Perry-
Sassmannshausen et al., 2021). To the best of our knowledge, there are no such
experimental data for gaseous neutral silicon. In view of the findings for Si-
and Si+ its dominating absorption feature can be expected close to 1840 eV.

Gatuzz et al. (2020) investigated the gaseous component of the ISM using the
theoretical absorption data of Witthoeft et al. (2011). In their data, the
$1s\to 3p$ resonance group for neutral silicon is located at $\sim$1839.5 eV
in accord with the above considerations. For the other charge states there are
significant discrepancies as already noted above. The deviations between the
theoretical resonance positions of Witthoeft et al. (2011) and the present
experimental ones are 6.7, 4.4, and 9.1 eV for Si+, Si2+, and Si3+,
respectively (Figure 3). All these differences are substantially larger than
the uncertainty of the experimental energy scale and they are as large or
larger than the energy differences between adjacent charge states (Figure 7a).
This will lead to a wrong assignment of astronomically observed resonance
features if their analysis is based on these theoretical absorption cross
sections.

##  5 Conclusions and outlook

Using the photon-ion merged-beams technique at a synchrotron light source we
have measured cross sections for multiple photoionization of low-charged Si+,
Si2+, and Si3+ ions and derived precise absorption data (resonance positions,
widths, and strengths) which can be directly used for the astrophysical
modelling of the silicon $K$-shell absorption by interstellar gas clouds and
other cosmic objects as well as for the benchmarking the theoretical
calculations. From the widths of the core excited resonances in Si+ we
inferred a value for the core-hole lifetime of $1s$-ionized neutral silicon of
$1.4\pm 0.5$ fs.

Previously theoretically predicted absorption features deviate significantly
in energy from the present experimental findings, whereas the present large-
scale MCDHF calculations agree much better with the experimental results. In
addition, the present computations also capture the hole-deexcitation cascades
following the initial creation of a $K$-shell hole. The obtained product
charge-state distributions, which are required for an accurate modelling of
the charge balance in astrophysical plasmas, agree remarkably well with the
experimental results, despite of the simplifications that had to be applied to
keep the calculations tractable. Current code development aims at
systematically expanding such cascade calculations to improve the treatment of
the deexcitation processes (Fritzsche, 2019; Fritzsche et al., 2021).

When comparing absorption data from different sources the calibration of the
photon energy scales is an issue of concern. The present experimental
uncertainty of $\pm 1$ eV is sufficient for discriminating between absorption
by gaseous and solid silicon-containing matter in the x-ray absorption spectra
from the currently operated x-ray telescopes. The accuracy demands will
increase with increasing resolving power of future missions such as Athena
(Barret et al., 2020). Meeting these demands will require the development of
more accurate calibration standards at synchrotron light sources. Promising
candidates are few-electron atomic ions which promise calibration
uncertainties of less than 10 meV (see, e.g., Müller et al., 2018; Stierhof et
al., 2022). Corresponding activities are under way at the PIPE setup.

We acknowledge DESY (Hamburg, Germany), a member of the Helmholtz Association
HGF, for the provision of experimental facilities. Parts of this research were
carried out at PETRA III and we would like to thank Jens Buck, Moritz Hoesch,
Frank Scholz, and Jörn Seltmann for assistance in using beamline P04. This
work was supported by the German Federal Ministry for Education and Research
(BMBF, grant numbers 05K16RG1, 05K16GUC, and 05K16SJA) and by the Deutsche
Forschungsgemeinschaft (DFG, grant number 389115454).

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