#  Uniqueness of Low Rank Matrix Completion and Schur Complement

Fei Wang<EMAIL_ADDRESS>Fields Institute, Toronto and Department of
Combinatorics and Optimization, University of Waterloo, Waterloo, Canada

(June, 2022)

###### Abstract

In this paper we study the low rank matrix completion problem using tools from
Schur complement. We give a sufficient and necessary condition such that the
completed matrix is globally unique with given data. We assume the observed
entries of the matrix follow a special ”staircase” structure. Under this
assumption, the matrix completion problem is either globally unique or has
infinitely many solutions (thus excluding local uniqueness). In fact, the
uniqueness of the matrix completion problem totally depends on the rank of the
submatrices at the corners of the ”staircase”. The proof of the theorems make
extensive use of the Schur complement.

Keywords: Low-rank matrix completion, matrix recovery, chordal graph

AMS subject classifications: 65J22, 90C22, 65K10, 52A41, 90C46

##  1 Introduction

Given a matrix $Z$ with partially sampled data $z$, the matrix completion
problem asks whether the missing entries from $z$ can be recovered. Clearly,
there are infinitely many ways to fill the missing entries, so in order for
the matrix completion problem to be meaningful, we require the rank of
recovered matrix to be not greater than a given target rank $r$ with the
assumption that the rank of the original matrix $Z$ is also not greater than
$r$.

The low rank matrix completion problem has come up naturally in many different
applications, for example, in compressed sensing [5], image analysis [19],
model reduction [13], machine learning [1], control [15] and so on. Numerous
algorithms has been developed in recent years. Most of the algorithms relax
the non-convex rank constraint to a convex constraint and solve a convex
optimization problem.

###  1.1 Model on matrix completion

We now introduce the framework for the matrix completion problem. Suppose that
we are given a low rank $m\times n$ real matrix $Z\in\mathbb{R}^{m\times n}$
with $r=\operatorname{{rank}}(Z)$ where a subset of entries are _sampled_.

The hard nonconvex _low rank matrix completion_ problem, LRMC , for recovering
the low rank matrix $Z$ can be reformulated as follows:

$(\mbox{\bf LRMC}\,\,)\qquad\qquad\begin{array}[]{cl}\mbox{find}&M\\\
\text{s.t.}&{\mathcal{P}}_{\hat{E}}(M)=z\\\ &\operatorname{{rank}}(M)\leq
r\end{array}$ (1.1)

where _$\hat{E}$_ is the set of indices containing the known (_sampled_)
entries of $Z$, ${\mathcal{P}}_{\hat{E}}(\cdot):\mathbb{R}^{m\times
n}\rightarrow\mathbb{R}^{|\hat{E}|}$ is the projection onto the corresponding
entries in $\hat{E}$, and $z={\mathcal{P}}_{\hat{E}}(Z)$ is the vector of
known entries formed from $Z$.

###  1.2 Unique recovery and related work

Suppose we know the target rank and sampled entries $z$, a natural question to
ask is which elements can be uniquely determined and which elements can not?
Also under which situation the whole low rank matrix $L$ admits a unique
completion? It is known that if the linear mapping ${\mathcal{P}}_{\hat{E}}$
satisfies some restricted isometry properties (RIP) [5, 6], then $Z$ is the
unique matrices satisfying $\operatorname{{rank}}(L)\leq
r,{\mathcal{P}}_{\hat{E}}(L)=z$. The RIP provides a sufficient condition for
exact recovery and checking if a linear map $\mathcal{A}$ satisfies RIP
property is NP-hard [2]. However the necessary condition for a matrix to be
uniquely completed is less well-studied. In [17], using tools from rigidity
theory, a randomized algorithm for testing sufficient and necessary conditions
for local completion and for testing sufficient conditions for global
completion is proposed. In [11, 12, 10], the authors use an algebraic
combinatorial approach to study the matrix completion problem and a sufficient
and necessary criterion is given in [11]. However, the matrix to be completed
is assumed to be “generic” in [11, 12, 10].

We are mostly interested in exploiting the clique structures in the sampled
data. The cliques in the sampled data can be used to do facial reductions for
solving the corresponding optimization problems more efficiently [14, 9]. In
the following sections, we will provide a sufficient and necessary condition
for the uniqueness of completion without using the generic assumption by
exploiting the properties of the intersections of bicliques.

##  2 Matrix completion by clique intersection

We first recall the basic property of Schur complement:

######  Lemma 2.1.

[8] Consider the partitioned matrix

$M=\begin{bmatrix}A&B\\\ C&D\end{bmatrix}\in\mathbb{R}^{m\times n},$

if we assume that $\operatorname{Range}(B)\subseteq\operatorname{Range}(A)$
and $\operatorname{Range}(C^{T})\subseteq\operatorname{Range}(A^{T})$, then
$M/A=D-CA^{\dagger}B$ is well-defined and

$\begin{bmatrix}I&0\\\ -CA^{\dagger}&I\end{bmatrix}\begin{bmatrix}A&B\\\
C&D\end{bmatrix}\begin{bmatrix}I&-A^{\dagger}B\\\
0&I\end{bmatrix}=\begin{bmatrix}A&0\\\ 0&M/A\end{bmatrix}$

and hence

$\operatorname{{rank}}(M)=\operatorname{{rank}}(A)+\operatorname{{rank}}(M/A).$

###  2.1 Two-biclique intersection

As a consequence of Lemma 2.1, we have the following theorem about the
completion of $2\times 2$ block matrix with one missing block.

######  Theorem 2.2.

Consider the partitioned matrix

$M=\begin{bmatrix}A&B\\\ C&D\end{bmatrix}\in\mathbb{R}^{m\times n},$

with $\operatorname{{rank}}(M)=r$, and the block submatrices $A,B,C$ are fixed
(sampled). Then $M$ is unique in LRMC if and only if
$\operatorname{{rank}}(A)=r$.

###### Proof.

First assume $\operatorname{{rank}}(A)=r$. Let $M/A=D-CA^{\dagger}B$ be the
generalized Schur complement, e.g., [16], where $A^{\dagger}$ denotes the
Moore-Penrose generalized inverse. Then we have
$\operatorname{{rank}}(M/A)+\operatorname{{rank}}(A)=\operatorname{{rank}}(M)$.
Therefore $\operatorname{{rank}}(M/A)=0$ and $D=CA^{\dagger}B$ is unique.

For necessity, assume $\operatorname{{rank}}(A)<r$, we first assume
$\operatorname{Range}(B)\subseteq\operatorname{Range}(A)$ and
$\operatorname{Range}(C^{T})\subseteq\operatorname{Range}(A^{T})$. Then we by
Lemma 2.1 we have the following equality

$\operatorname{{rank}}(D-CA^{\dagger}B)+\operatorname{{rank}}(A)=\operatorname{{rank}}(M)$

Since $\operatorname{{rank}}(A)<r$ and $\operatorname{{rank}}(M)=r$, we have
$\operatorname{{rank}}(D-CA^{\dagger}B)=\operatorname{{rank}}(M)-\operatorname{{rank}}(A)=\bar{r}>0$.
We can then let $D=CA^{\dagger}B+E$ where $E$ is an arbitrary matrix of rank
$\bar{r}$. Therefore $D$ is not unique.

Now suppose either $\operatorname{Range}(B)\subsetneq\operatorname{Range}(A)$
or $\operatorname{Range}(C^{T})\subsetneq\operatorname{Range}(A^{T})$. Without
loss we can assume
$\operatorname{Range}(C^{T})\subsetneq\operatorname{Range}(A^{T})$. Then we
have $\operatorname{Null}(A)\subsetneq\operatorname{Null}(C)$ which means
there exists a column vector $x$ such that $Ax=0,Cx\neq 0$. Now we can add the
column vector $\begin{bmatrix}Ax\cr Cx\end{bmatrix}=\begin{bmatrix}0\cr
Cx\end{bmatrix}$ to any column of $\begin{pmatrix}B\cr D\end{pmatrix}$ without
changing the rank of $M$ and we get a different $D$. So $D$ is not unique. ∎

We now study the simple case such that the observed data pattern $\hat{E}$
consists of two bicliques .

######  Corollary 2.3.

Let $Z$ be the following matrix with two intersecting bicliques such that
$\operatorname{{rank}}(Z)=r$ and submatrices $X$ and $Y$ are fixed,

$Z=\left[\begin{array}[]{ccc}\lx@intercol\hfil
Z_{1}\hfil\lx@intercol\vrule\lx@intercol&X_{1}&X_{2}\\\
\cline{1-2}\cr\lx@intercol\hfil
Y_{1}\hfil\lx@intercol\vrule\lx@intercol&\lx@intercol\hfil
Q\hfil\lx@intercol\vrule\lx@intercol&X_{3}\\\ \cline{2-3}\cr
Y_{2}&\lx@intercol\hfil
Y_{3}\hfil\lx@intercol\vrule\lx@intercol&Z_{2}\end{array}\right],\quad
X=\left[\begin{array}[]{cc}X_{1}&X_{2}\\\ \cline{1-1}\cr\lx@intercol\hfil
Q\hfil\lx@intercol\vrule\lx@intercol&X_{3}\end{array}\right],\quad
Y=\left[\begin{array}[]{cc}\lx@intercol\hfil
Y_{1}\hfil\lx@intercol\vrule\lx@intercol&Q\\\ \cline{2-2}\cr
Y_{2}&Y_{3}\end{array}\right].$ (2.1)

Let $Q$ be the submatrix that lies in both $X$ and $Y$. If
$\operatorname{{rank}}(Q)=r$, then $Z$ is unique in LRMC.

###### Proof.

If $Q$ has rank $r$, then the top left union of four blocks must also be rank
$r$. Therefore, Theorem 2.2 implies $Z_{1}$ is unique. Similarly, $Z_{2}$ is
unique by looking at the four blocks at the bottom right. ∎

However, the converse may not be true. Consider the following example

$Z=\left[\begin{array}[]{cccc}\lx@intercol\hfil
Z_{1}\hfil\lx@intercol\vrule\lx@intercol&6&5&3\\\
\cline{1-3}\cr\lx@intercol\hfil
1\hfil\lx@intercol\vrule\lx@intercol&2&\lx@intercol\hfil
3\hfil\lx@intercol\vrule\lx@intercol&2\\\ \cline{2-4}\cr 3&4&\lx@intercol\hfil
2\hfil\lx@intercol\vrule\lx@intercol&Z_{2}\end{array}\right],\quad
Q=\left[\begin{array}[]{cc}2&3\end{array}\right],$ (2.2)

Assume $\operatorname{{rank}}(Z)=2$ and
$\operatorname{{rank}}(Q)=1<\operatorname{{rank}}(Z)$. However, $Z_{1}$ and
$Z_{2}$ are still unique and by basic linear algbera we have $Z_{1}=4$ and
$Z_{2}=1$.

From Theorem 2.2, if we have two bicliques such that their intersection has
the target rank, we can now merge these two bicliques into one bigger biclique
and recover the corresponding missing entries of $Z$. We can then use this
bigger biclique to merge with other bicliques. This process can carry on until
all the missing entries are recovered.

###  2.2 More bicliques and the staircase

We can generate Theorem 2.2 to include three bicliques.

######  Theorem 2.4.

Consider the partitioned matrix

$M=\begin{bmatrix}E&F\\\ A&B\\\ C&D\end{bmatrix}\in\mathbb{R}^{m\times n},$

with $\operatorname{{rank}}(M)=r$, where $A,B,C,F$ are fixed (sampled). Then
$M$ is unique in LRMC if and only if $\operatorname{{rank}}(A)=r$ and
$\operatorname{{rank}}(B)=r$.

###### Proof.

Suppose $\operatorname{{rank}}(A)=\operatorname{{rank}}(B)=r$, then it is
obvious that $D,E$ are unique by Theorem 2.2.

For necessity, without loss we assume $\operatorname{{rank}}(A)=\bar{r}<r$. By
a permutation, let

$M=\begin{bmatrix}A&B\\\ E&F\\\ C&D\end{bmatrix}.$

If $\operatorname{{rank}}(\begin{bmatrix}A\\\ E\end{bmatrix})<r$, then by
Therem 2.2, $D$ is not unique so $M$ is not unique.

If $\operatorname{{rank}}(\begin{bmatrix}A\\\ E\end{bmatrix})=r$, then we have
$\operatorname{Range}(B)\subseteq\operatorname{Range}(A)$. Now we partition
$A,E,C$ such that

$M=\begin{bmatrix}A_{1}&A_{2}&B\\\ E_{1}&E_{2}&F\\\
C_{1}&C_{2}&D\end{bmatrix}$

where $A_{1}$ has full column rank $\bar{r}$. So we have
$\operatorname{Range}(A_{1})=\operatorname{Range}(A)$.

Let $M_{1},M_{2},M_{3},M_{4}$ be the four Schur complements corresponding to
$E_{2},F,C_{2},D$ such that $M_{1}=E_{2}-E_{1}A_{1}^{\dagger}A_{2}$,
$M_{2}=F-E_{1}A_{1}^{\dagger}B$, $M_{3}=C_{2}-C_{1}A_{1}^{\dagger}A_{2}$,
$M_{4}=D-C_{1}A_{1}^{\dagger}B$. Since
$\operatorname{Range}(A_{2})\subseteq\operatorname{Range}(A_{1})$,
$\operatorname{Range}(B)\subseteq\operatorname{Range}(A_{1})$ and
$\operatorname{Range}(E_{1}^{T})\subseteq\operatorname{Range}(A_{1}^{T})$,
$\operatorname{Range}(C_{1}^{T})\subseteq\operatorname{Range}(A_{1}^{T})$, we
have

$\operatorname{{rank}}(M)=\operatorname{{rank}}(A_{1})+\operatorname{{rank}}(\begin{bmatrix}M_{1}&M_{2}\cr
M_{3}&M_{4}\end{bmatrix})=r.$

Also

$\operatorname{{rank}}(\begin{bmatrix}A\cr
E\end{bmatrix})=\operatorname{{rank}}(A_{1})+\operatorname{{rank}}(M_{1})=r.$

Therefore
$\operatorname{{rank}}(M_{1})=\operatorname{{rank}}(\begin{bmatrix}M_{1}&M_{2}\cr
M_{3}&M_{4}\end{bmatrix})=r-\bar{r}$ and we have

$M_{4}=M_{3}M_{1}^{\dagger}M_{2}.$ (2.3)

Now $M_{1}\neq 0$, since $\operatorname{{rank}}(M_{1})=r-\bar{r}>0$, we can
perturb $E_{2}$ such that $\bar{E}_{2}=E_{2}+M_{1}$ and perturb $D$ such that
$\bar{D}=D-\frac{1}{2}M_{4}$ and the corresponding full perturbed matrix is
$\bar{M}$. After similar arguments we can get

$\displaystyle\operatorname{{rank}}(\bar{M})$ $\displaystyle=$
$\displaystyle\operatorname{{rank}}(A_{1})+\operatorname{{rank}}(\begin{bmatrix}2M_{1}&M_{2}\cr
M_{3}&\frac{1}{2}M_{4}\end{bmatrix})$ $\displaystyle=$
$\displaystyle\operatorname{{rank}}(A_{1})+\operatorname{{rank}}(2M_{1})+\operatorname{{rank}}(\frac{1}{2}M_{4}-M_{3}(2M_{1})^{\dagger}M_{2})$
$\displaystyle=$
$\displaystyle\operatorname{{rank}}(A_{1})+\operatorname{{rank}}(2M_{1})\quad\text{
(due to {(\ref{eq:m14})})}$ $\displaystyle=$
$\displaystyle\bar{r}+r-\bar{r}=r.$

Therefore $M$ is not unique. ∎

The following proposition regarding the general case of 3 bicliques is a
direct consequence of Theorem 2.4.

######  Proposition 2.5.

Consider the partitioned matrix

$M=\begin{bmatrix}F&H&E\\\ A&G&B\\\ C&K&D\end{bmatrix}\in\mathbb{R}^{m\times
n},$

with $\operatorname{{rank}}(M)=r$, where $A,B,C,E,G$ are fixed (sampled). Then
$M$ is unique in LRMC if and only if $\operatorname{{rank}}(A)=r$ and
$\operatorname{{rank}}(B)=r$.

###### Proof.

Without loss we assume $\operatorname{{rank}}(A)<r$. Now let $B=[G,\;B]$,
$E=[H,\;E]$, the result follows directly from Theorem 2.4. ∎

We can then generate Theorem 2.4 to the cases with 4 and 5 bicliques in the
following Theorem and propositions.

######  Theorem 2.6.

Consider the partitioned matrix

$M=\begin{bmatrix}F&H&E\\\ A&G&B\\\ C&K&D\end{bmatrix}\in\mathbb{R}^{m\times
n},$

with $\operatorname{{rank}}(M)=r$, where $F,A,G,K,D$ are fixed. Then $M$ is
unique in LRMC if and only if
$\operatorname{{rank}}(A)=\operatorname{{rank}}(G)=\operatorname{{rank}}(K)=r$.

###### Proof.

If $\operatorname{{rank}}(A)<r$ or $\operatorname{{rank}}(K)<r$, then
according to Theorem 2.4, $M$ is not unique. Therefore we only need to
consider the case when $\operatorname{{rank}}(G)<r$.

If $\operatorname{{rank}}(\begin{bmatrix}H\\\ G\end{bmatrix})<r$ or
$\operatorname{{rank}}(\begin{bmatrix}G&B\end{bmatrix})<r$, then again
according to Theorem 2.4, $M$ is not unique. Therefore we consider the case
where $\operatorname{{rank}}(\begin{bmatrix}H\\\ G\end{bmatrix})=r$ and
$\operatorname{{rank}}(\begin{bmatrix}G&B\end{bmatrix})=r$.

By a permutation, let

$M=\begin{bmatrix}G&B&A\\\ H&E&F\\\ K&D&C\end{bmatrix}\in\mathbb{R}^{m\times
n}.$

Let $P=\begin{bmatrix}G&B\\\ H&E\end{bmatrix}$, since
$\operatorname{{rank}}(G)<r$, by Theorem 2.2, there exists a different
$\bar{E}$ and $\bar{P}$ such that $\operatorname{{rank}}(\bar{P})=r$, we let
$\bar{C}=\begin{bmatrix}K&D\end{bmatrix}\bar{P}^{\dagger}\begin{bmatrix}A\\\
F\end{bmatrix})$, since $\operatorname{Range}(\begin{bmatrix}A\\\
F\end{bmatrix})\subseteq\operatorname{Range}(\begin{bmatrix}G\\\
H\end{bmatrix})$ and $\operatorname{Range}(\begin{bmatrix}K^{T}\\\
D^{T}\end{bmatrix})\subseteq\operatorname{Range}(\begin{bmatrix}G^{T}\\\
B^{T}\end{bmatrix})$, we have
$\operatorname{{rank}}(\bar{M})=\operatorname{{rank}}(\bar{P})+\operatorname{{rank}}(\bar{C}-\begin{bmatrix}K&D\end{bmatrix}\bar{P}^{\dagger}\begin{bmatrix}A\\\
F\end{bmatrix})=\operatorname{{rank}}(\bar{P})=r$. The corresponding $\bar{M}$
is different from $M$ and the proof is finished. ∎

######  Proposition 2.7.

Consider the partitioned matrix

$M=\begin{bmatrix}F&H&E\\\ A&G&B\\\ C&K&D\\\
J&I&L\end{bmatrix}\in\mathbb{R}^{m\times n},$

with $\operatorname{{rank}}(M)=r$, and $F,A,G,K,D,L$ are fixed. Then the
matrix $M$ is unique in LRMC if and only if
$\operatorname{{rank}}(A)=\operatorname{{rank}}(G)=\operatorname{{rank}}(K)=\operatorname{{rank}}(D)=r$.

###### Proof.

Direct consequences from Theorem 2.4 and Theorem 2.6. ∎

With all the prerequisite theorems and propositions that have been proved, we
are now able to extend to the most general case where we have a stair case of
known block matrices. We conclude that the whole matrix is unique if and only
if every “corner” matrix has rank $r$. This can be proved by repeatedly using
the previous theorems and propositions.

######  Theorem 2.8.

Given a low rank matrix $Z\in\mathbb{R}^{m\times n}$ and a partial sampling
${\mathcal{P}}_{\hat{E}}(Z)=z$. If by a permutation there exists a chain of
bicliques (submatrices of $Z$) $\alpha_{1},...,\alpha_{l}$ with the
corresponding edge sets $E_{1},\cdots,E_{l}$ and vertex sets in pairs
$(U_{1},V_{1}),(U_{2},V_{2}),\cdots,(U_{l},V_{l})$ . Assume
$\cup_{i=1}^{l}E_{i}=\hat{E}$ and $E_{i}\cap E_{j}=\emptyset,\,\forall j\geq
i+2$ and the union of all the vertices of the bicliques satisfy
$\cup_{i=1}^{l}\alpha_{i}=\\{1,...,m\\}\times\\{1,...,n\\}$. In addition

  * •

$U_{i}\cap U_{i+2}=U_{i+1}$, $V_{i}\cap V_{i+1}=V_{i},V_{i+2}\cap
V_{i+1}=V_{i+2}$ for $i$ being even number between $1$ and $l-2$

  * •

$V_{i}\cap V_{i+2}=V_{i+1}$, $U_{i}\cap U_{i+1}=U_{i},U_{i+2}\cap
U_{i+1}=U_{i+2}$ for $i$ being odd number between $1$ and $l-2$

Then the matrix $Z$ can be uniquely recovered if and only if
$\operatorname{{rank}}(X_{\alpha_{i}\cap\alpha_{i+1}})=r,i=1,...,l-1$.

###### Proof.

Suppose the bicliques satisfy
$\operatorname{{rank}}(X_{\alpha_{i}\cap\alpha_{i+1}})=r,i=1,...,l-1$, then it
is abvious to see the uniqueness of the completion by repeatedly using Theorem
2.2.

For the other direction, suppose one of the bicluqes $X_{\alpha_{i}}$
satisfies $\operatorname{{rank}}(X_{\alpha_{i}})\neq r$, then one can recover
a different matrix $\bar{Z}$ with the same rank and sampled data by reducing
the problem to the sample cases as shown in Theorem 2.2, 2.4, 2.6 and
Proposition 2.5, 2.7. ∎

An illustration of the “staircase” structure in the assumption of Theorem 2.8
is shown in equation (2.4).

$Z=\left(\begin{array}[]{cccccccc}\vrule\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol&&&&&&&\\\
\cline{2-3}\cr\vrule\lx@intercol\hfil
Z_{1}\hfil\lx@intercol&&\lx@intercol\hfil
Z_{2}\hfil\lx@intercol\vrule\lx@intercol&&&&&\\\
\cline{1-2}\cr&&\vrule\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol&&&&&\\\
\cline{4-4}\cr&&\vrule\lx@intercol\hfil Z_{3}\hfil\lx@intercol&&\ddots&&&\\\
\cline{3-4}\cr&&&&&\ddots&&\\\
&&&&&\vrule\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol&&\\\
\cline{7-8}\cr&&&&&\vrule\lx@intercol\hfil Z_{l-1}\hfil\lx@intercol&&\\\
\cline{6-8}\cr\end{array}\right),$ (2.4)

##  3 Graph Representation of the Problem

Our sampling yields elements $b={\mathcal{P}}_{\hat{E}}(Z)$. With the matrix
$Z$ and the sampled elements we can associate a bipartite graph
$G_{Z}=(U_{m},V_{n},\hat{E})$, where

$U_{m}=\\{1,\ldots,m\\},\quad V_{n}=\\{1,\ldots,n\\}.$

For our needs we associate $Z$ with the undirected graph, $G=(V,E)$, with node
set $V=\\{1,\ldots,m,m+1,\ldots,m+n\\}$ and edge set $E$ that satisfies

$\big{\\{}\\{ij\in V\times V:i<j\leq m\\}\cup\\{ij\in V\times V:m+1\leq
i<j\leq m+n\\}\big{\\}}\subseteq E\subseteq\\{ij\in V\times V:i<j\\}.$

Note that as above, $\bar{E}$ is the set of edges excluding the trivial ones,
that is,

$\bar{E}=E\backslash\bigg{\\{}\\{ij\in V\times V:i\leq j\leq m\\}\cup\\{ij\in
V\times V:m+1\leq i\leq j\leq m+n\\}\bigg{\\}}.$

Recall that a biclique $\alpha$ in the graph $G_{Z}$ is a complete bipartite
subgraph in $G_{Z}$ with corresponding complete submatrix $z[\alpha]$. This
corresponds to a nontrivial111For $G$ we have the additional trivial cliques
of size $k$, $C=\\{i_{1},\ldots,i_{k}\\}\subset\\{1,\ldots,m\\}$ and
$C=\\{j_{1},\ldots,j_{k}\\}\subset\\{m+1,\ldots,m+n\\}$, that are not of
interest to our algorithm. clique in the graph $G$, a complete subgraph in
$G$. The cliques of interest are $C=\\{i_{1},\ldots,i_{k}\\}$ with
cardinalities

$|C\cap\\{1,\ldots,m\\}|=p\neq 0,\quad|C\cap\\{m+1,\ldots,m+n\\}|=q\neq 0.$
(3.1)

The submatrix $z[\alpha]$ of $Z$ for the corresponding biclique from the
clique $C$ is

$z[\alpha]\equiv X\equiv\\{Z_{i(j-m)}:ij\in C\\},\quad\text{sampled $p\times
q$ rectangular submatrix}.$ (3.2)

These non-trivial cliques in $G$ that correspond to bicliques of $G_{Z}$ are
at the center of our interest.

###  3.1 Chordal graphs and clique trees

Chordal graph is a special graph with a chordless structure. An undirected
graph is chordal if every cycle of length greater than three has a chord.

A clique tree of a graph $G=(V,E)$ is a tree which has the cliques of $G$ as
its vertices. A clique tree $T$ has the induced subtree property if for every
$v\in V$, the cliques that contain $v$ form a subtree (connected subgraph) of
$T$. It has been shown that chordal graphs are exactly the graphs for which a
clique tree with the induced subtree property exists. ([4, Theorem 2.7] and
[7, Theorem 3]).

We show the graph $\hat{E}$ in Theorem 2.8 has a unique clique tree with the
induced subtree property. Therefore it is chordal.

######  Theorem 3.1.

Given a graph $G$ with edge set $\hat{E}$, if by a permutation there exists a
chain of bicliques $\alpha_{1},...,\alpha_{l}$ with the corresponding edge
sets $E_{1},\cdots,E_{l}$. Assume $E_{1},\cdots,E_{l}$ and
$\alpha_{1},...,\alpha_{l}$ satisfy the assumptions in Theorem 2.8. Then the
graph $\hat{E}$ is chordal and it has a unique clique tree with the induced
subtree property.

###### Proof.

Consider the set of bicliques $\alpha_{1},\cdots,\alpha_{l}$, each clique
$\alpha_{i}$ is a node and connect $\alpha_{i}$ and $\alpha_{i+1}$ by an edge,
then it forms a clique tree. For any vertex $v$ in $\hat{E}$, it either
belongs to $\alpha_{i}$ or belongs to $\alpha_{i}\cap\alpha_{i}+1$ for some
$i$, therefore the set of cliques containing $v$ is connected. Hence it has
the induced subtree property. Therefore by Theorem 3 in [7] it is a chordal
graph. Now consider a different clique tree, then there exists $\alpha_{i}$
which is not connected $\alpha_{i+1}$, but there exists
$v\in\alpha_{i}\cap\alpha_{i}+1$, therefore the induced subtree property is
not satisfied. ∎

######  Proposition 3.2.

Given a graph $G$ with $\hat{E}$ which satisfies the chain clique properties
defined in Theorem 3.1, then there exists a polynomial algorithm to determine
if the correponding low rank matrix completion problem 1.1 has a unique
solution.

###### Proof.

Given a chordal graph, finding the clique tree with the induced subtree
property can be solved in polynomial time by the maximum cardinality search
algorithm (MCS) [18, 3].

After we the clique tree is found, checking if the intersection of two
neighbouring clique has rank $r$ can also be done in polynomial time. ∎

##  4 Completion of Positive Semidefinite Matrices is not true

We recall the following theorem about symmetric matrix:

######  Theorem 4.1.

Suppose M is symmetric and partitioned as

$M=\begin{bmatrix}A&B\\\ B^{*}&C\end{bmatrix},$

in which $A$ and $C$ are square. Then $M\succeq 0$ if and only if $A\succeq
0,\operatorname{Range}(B)\subseteq\operatorname{Range}(A),$ and $M/A\succeq
0$.

For positive semidefinite matrix completion, we have the following
proposition:

######  Proposition 4.2.

Consider the partitioned matrix

$M=\begin{bmatrix}A&B\\\ B^{T}&C\end{bmatrix}\in\mathbb{R}^{m\times n},$

where $\operatorname{{rank}}(M)=r$ and $M\succeq 0$ in which $A$, $C$ are
square, and the blocks $A,B$ are fixed. Then there exists a unique positive
semidefinite matrix $M$ if and only if $\operatorname{{rank}}(A)=r$.

###### Proof.

First assume $\operatorname{{rank}}(A)=r$. Let $M/A=C-B^{T}A^{\dagger}B$ be
the generalized Schur complement, where $A^{\dagger}$ denotes the Moore-
Penrose generalized inverse. The existence of $M\succeq 0$ ensures
$\operatorname{Range}(B)\subseteq\operatorname{Range}(A)$ and $A\succeq 0$,
therefore $M/A$ is well-defined. We then have
$\operatorname{{rank}}(M/A)+\operatorname{{rank}}(A)=\operatorname{{rank}}(M)$.
Hence $\operatorname{{rank}}(M/A)=0$ and $C=B^{T}A^{\dagger}B\succeq 0$ is
unique.

For necessity, assume $\operatorname{{rank}}(A)<r$, the existence of $M\succeq
0$ ensures $\operatorname{Range}(B)\subseteq\operatorname{Range}(A)$ and
$A\succeq 0$, therefore $M/A$ is well-defined and

$\operatorname{{rank}}(M/A)+\operatorname{{rank}}(A)=\operatorname{{rank}}(M)$

Since $\operatorname{{rank}}(A)<r$ and $\operatorname{{rank}}(M)=r$, we have
$\operatorname{{rank}}(C-B^{T}A^{\dagger}B)=\operatorname{{rank}}(M)-\operatorname{{rank}}(A)=\bar{r}>0$.
We can then let $C=B^{T}A^{\dagger}B+E$ where $E$ is an arbitrary positive
semidefinite matrix of rank $\bar{r}$. Hence by Theorem 4.1

$\bar{M}=\begin{bmatrix}A&B\\\ B^{T}&B^{T}A^{\dagger}B+E\end{bmatrix}\succeq
0.$

Therefore $M$ is not unique. ∎

Now ee consider the following question: let $M$ be the partitioned matrix

$M=\begin{bmatrix}A&B&D\\\ B^{T}&C&E\\\
D^{T}&E^{T}&F\end{bmatrix}\in\mathbb{R}^{m\times n},$

$M$ is symmetric and positive semidefinite and the diagonal elements of $M$
are all nonzeros. Suppose $A,B,C,E,F$ are fixed and
$\operatorname{{rank}}(M)=r$. Then it is easy to see that $M$ can be uniquely
completed if $\operatorname{{rank}}(C)=r$. We now ask the question: is the
converse also true? The answer is NO.

We consider the following example

$H=\left[\begin{array}[]{ccc}5&4&-2\\\ 4&16&-8\\\ -2&-8&4\end{array}\right]$

Suppose the top right element and bottom left element $-2$ are missing, the
partially observed entries are the union of two 2 by 2 cliques, i.e,

$\left[\begin{array}[]{ccc}5&4&?\\\ 4&16&-8\\\ ?&-8&4\end{array}\right]$

We partition $H$ as matrix $M$. The intersection of the two cliques is 16
which is rank 1, and $\operatorname{{rank}}(H)=2$. Let the missing element $D$
be $x$.

We compute $\operatorname{{rank}}(\begin{bmatrix}5&4\\\ 4&16\end{bmatrix})=2$.
In order for the rank of $H$ to be 2, the Schur complement

$4-(x,-8)\begin{bmatrix}5&4\\\ 4&16\end{bmatrix}^{-1}\begin{bmatrix}x\\\
-8\end{bmatrix}=-(\frac{x^{2}}{4}+x+1)=-(\frac{1}{2}x+1)^{2}$

must be zero. (otherwise the rank of $H$ is 3.)

Therefore $x=-2$ is the only solution, the missing entry can be uniquely
completed.

##  5 Conclusion

In this paper, we show that if the missing entries of a matrix of rank $r$ has
a ”staircase” pattern, then the uniqueness of the recovery problem given the
target rank depends totally on the submatrices at the corner of the
”staircase”. The proof of the main theorem makes extensive use of Schur
complement.

## Acknowledgment

The author thanks Henry Wolkowicz and Anders Forsgren for the very helpful
discussion during this work.

## References

  * [1] Yonatan Amit, Michael Fink, Nathan Srebro, and Shimon Ullman.  Uncovering shared structures in multiclass classification.  In Proceedings of the 24th international conference on Machine learning, pages 17–24, 2007.
  * [2] Afonso S Bandeira, Edgar Dobriban, Dustin G Mixon, and William F Sawin.  Certifying the restricted isometry property is hard.  IEEE transactions on information theory, 59(6):3448–3450, 2013\.
  * [3] Jean RS Blair and Barry Peyton.  An introduction to chordal graphs and clique trees.  In Graph theory and sparse matrix computation, pages 1–29. Springer, 1993.
  * [4] Peter Buneman.  A characterisation of rigid circuit graphs.  Discret. Math., 9(3):205–212, 1974.
  * [5] Emmanuel J Candes, Justin K Romberg, and Terence Tao.  Stable signal recovery from incomplete and inaccurate measurements.  Communications on Pure and Applied Mathematics: A Journal Issued by the Courant Institute of Mathematical Sciences, 59(8):1207–1223, 2006.
  * [6] Maryam Fazel, E Candes, Benjamin Recht, and P Parrilo.  Compressed sensing and robust recovery of low rank matrices.  In 2008 42nd Asilomar Conference on Signals, Systems and Computers, pages 1043–1047. IEEE, 2008.
  * [7] Fǎnicǎ Gavril.  The intersection graphs of subtrees in trees are exactly the chordal graphs.  Journal of Combinatorial Theory, Series B, 16(1):47–56, 1974.
  * [8] Roger A Horn and Fuzhen Zhang.  Basic properties of the schur complement.  In The Schur Complement and Its Applications, pages 17–46. Springer, 2005.
  * [9] Shimeng Huang and Henry Wolkowicz.  Low-rank matrix completion using nuclear norm minimization and facial reduction.  Journal of Global Optimization, 72(1):5–26, 2018.
  * [10] Bill Jackson, Tibor Jordán, and Shin-ichi Tanigawa.  Unique low rank completability of partially filled matrices.  Journal of Combinatorial Theory, Series B, 121:432–462, 2016.
  * [11] Franz Király and Ryota Tomioka.  A combinatorial algebraic approach for the identifiability of low-rank matrix completion.  arXiv preprint arXiv:1206.6470, 2012.
  * [12] Franz J Király, Louis Theran, and Ryota Tomioka.  The algebraic combinatorial approach for low-rank matrix completion.  J. Mach. Learn. Res., 16(1):1391–1436, 2015.
  * [13] Zhang Liu and Lieven Vandenberghe.  Interior-point method for nuclear norm approximation with application to system identification.  SIAM Journal on Matrix Analysis and Applications, 31(3):1235–1256, 2010.
  * [14] Shiqian Ma, Fei Wang, Linchuan Wei, and Henry Wolkowicz.  Robust principal component analysis using facial reduction.  Optimization and Engineering, 21(3):1195–1219, 2020.
  * [15] Mehran Mesbahi and George P Papavassilopoulos.  On the rank minimization problem over a positive semidefinite linear matrix inequality.  IEEE Transactions on Automatic Control, 42(2):239–243, 1997.
  * [16] Diane Valerie Ouellette.  Schur complements and statistics.  Linear Algebra and its Applications, 36:187–295, 1981.
  * [17] Amit Singer and Mihai Cucuringu.  Uniqueness of low-rank matrix completion by rigidity theory.  SIAM Journal on Matrix Analysis and Applications, 31(4):1621–1641, 2010.
  * [18] Robert Endre Tarjan.  Data structures and network algorithms.  SIAM, 1983.
  * [19] Xiaowei Zhou, Can Yang, Hongyu Zhao, and Weichuan Yu.  Low-rank modeling and its applications in image analysis.  ACM Computing Surveys (CSUR), 47(2):1–33, 2014.


# Photoexcitation of 76Ge

R. Schwengner  Helmholtz-Zentrum Dresden-Rossendorf, 01328 Dresden, Germany
R. Massarczyk  Los Alamos National Laboratory, Los Alamos, New Mexico 87545,
USA  K. Schmidt  Helmholtz-Zentrum Dresden-Rossendorf, 01328 Dresden, Germany
K. Zuber  Institut für Kern- und Teilchenphysik, Technische Universität
Dresden, 01069 Dresden, Germany  R. Beyer  Helmholtz-Zentrum Dresden-
Rossendorf, 01328 Dresden, Germany  D. Bemmerer  Helmholtz-Zentrum Dresden-
Rossendorf, 01328 Dresden, Germany  S. Hammer  Helmholtz-Zentrum Dresden-
Rossendorf, 01328 Dresden, Germany  A. Hartmann  Helmholtz-Zentrum Dresden-
Rossendorf, 01328 Dresden, Germany  T. Hensel  Helmholtz-Zentrum Dresden-
Rossendorf, 01328 Dresden, Germany  Institut für Kern- und Teilchenphysik,
Technische Universität Dresden, 01069 Dresden, Germany  H. F. Hoffmann
Helmholtz-Zentrum Dresden-Rossendorf, 01328 Dresden, Germany  Institut für
Kern- und Teilchenphysik, Technische Universität Dresden, 01069 Dresden,
Germany  A. R. Junghans  Helmholtz-Zentrum Dresden-Rossendorf, 01328 Dresden,
Germany  T. Kögler  Helmholtz-Zentrum Dresden-Rossendorf, 01328 Dresden,
Germany  OncoRay – National Center for Radiation Research in Oncology, Faculty
of Medicine and University Hospital Carl Gustav Carus, Technische Universität
Dresden, 01309 Dresden, Helmholtz-Zentrum Dresden-Rossendorf, 01328 Dresden,
Germany  S. E. Müller  Helmholtz-Zentrum Dresden-Rossendorf, 01328 Dresden,
Germany  M. Pichotta  Institut für Kern- und Teilchenphysik, Technische
Universität Dresden, 01069 Dresden, Germany  S. Turkat  Institut für Kern- und
Teilchenphysik, Technische Universität Dresden, 01069 Dresden, Germany  J. A.
B. Turko  Helmholtz-Zentrum Dresden-Rossendorf, 01328 Dresden, Germany
OncoRay – National Center for Radiation Research in Oncology, Faculty of
Medicine and University Hospital Carl Gustav Carus, Technische Universität
Dresden, 01309 Dresden, Helmholtz-Zentrum Dresden-Rossendorf, 01328 Dresden,
Germany  S. Urlaß  Helmholtz-Zentrum Dresden-Rossendorf, 01328 Dresden,
Germany  Conseil Européen pour la Recherche Nucléaire (CERN), 1211 Meyrin,
Geneva, Switzerland  A. Wagner  Helmholtz-Zentrum Dresden-Rossendorf, 01328
Dresden, Germany

###### Abstract

The dipole strength of the nuclide 76Ge was studied in photon-scattering
experiments using bremsstrahlung produced with electron beams of energies of
7.8 and 12.3 MeV at the $\gamma$ELBE facility. We identified 210 levels up to
an excitation energy of 9.4 MeV and assigned spin $J$ = 1 to most of them. The
quasicontinuum of unresolved transitions was included in the analysis of the
spectra and the intensities of branching transitions were estimated on the
basis of simulations of statistical $\gamma$-ray cascades. The photoabsorption
cross section up to the neutron-separation energy was determined and is
compared with predictions of the statistical reaction model. The derived
photon strength function is compared with results of experiments using other
reactions.

###### pacs:

21.10.Tg, 21.60.Cs, 23.20.-g, 25.20.Dc, 27.40.+z

##  I Introduction

The search for signals of the neutrinoless double-$\beta$ ($0\nu\beta\beta$)
decay is currently one of the most challenging experimental efforts expected
to gain information about the validity of the standard model of particle
physics. The existence of this decay mode would imply that neutrinos are
identical with antineutrinos, their antiparticles, and that special conditions
for their vertices exist, realized for example through a non-zero neutrino
mass zub20 . The discovery of this process would prove that the lepton number
is violated by two units and thus, physics goes beyond the standard model.
Experiments searching for this very rare decay mode need large amounts of
target material under very low background conditions. Hence, the experiments
are performed deep underground. Among the nuclides, where $\beta\beta$ decay
is possible in contrast to $\beta$ decay, there is the nuclide 76Ge, which can
be used at the same time as target and detector material. Collaborations using
this target and detector material are, for example, MAJORANA alv19  and GERDA
ago19 . Up to now, GERDA is the experiment that sets the strongest limit on
the half-life of the $0\nu\beta\beta$ decay of 76Ge to about $10^{26}$ years
ago18 . The next generation of experiments, such as LEGEND abg17 , tries to
reach $10^{28}$ years. Thus, lower background levels have to be reached and
every possible background reaction channel has to be understood as well as
possible. Another current project is CDEX-1 kan13 . If the $0\nu\beta\beta$
decay is realized in nature, the $\beta$ spectrum is discrete rather than
continuous. The signal for this process is a peak at the $Q$ value of
(2039.006 $\pm$ 0.050) keV in the sum-energy spectrum.

An important issue for this search is the exclusion of signals from other
reactions on 76Ge that involve the emission of $\gamma$ rays of about this
energy. Earlier experiments using the 76Ge($n,\gamma$) reaction revealed a
$\gamma$ ray at 2035.5 keV mei12 . A 2040.7 keV $\gamma$ ray was identified in
the $\beta$ decay of 76Ga cam71 . In 76Ge($n,n^{\prime}\gamma$) experiments,
$\gamma$ rays at 2034.8 keV neg13  and at 2038.9 keV muk17  were observed.
Various experiments using neutron-induced reactions or activations on Ge
isotopes are currently performed at several laboratories. In addition to
these, one may think of other reactions that can serve to identify so far
unknown $\gamma$ transitions in 76Ge. One of these reactions is photon
scattering $(\gamma,\gamma^{\prime})$, also called nuclear resonance
fluorescence, in which the incident photons transfer preferentially angular
momentum $L$ = 1 and hence excite states of spin $J$ = 1 from the ground state
in even-even nuclei. In the present work, we performed photon-scattering
experiments at the bremsstrahlung facility $\gamma$ELBE sch05  of Helmholtz-
Zentrum Dresden-Rossendorf (HZDR) to study states up to the neutron-separation
energy $S_{n}$ = 9.4 MeV and their deexcitation to low-lying states, in
particular the possible occurrence of $\gamma$ rays with energies in the
interesting region around 2039 keV.

In addition to the just described interest from the search for the
$0\nu\beta\beta$ decay of 76Ge, photon scattering from 76Ge is also of
interest for nuclear structure and reaction physics. Photoabsorption cross
sections $\sigma_{\gamma}$ and the related photon strength functions
$f_{1}(E_{\gamma})$ for $L$ = 1 have attracted growing interest gor19 ; kaw20
because of their importance as inputs to calculations of reaction cross
sections within the statistical reaction model hau52 . In photoabsorption, the
two quantities are connected via the relation $\sigma_{\gamma}=g(\pi\hbar
c)^{2}$ $E_{\gamma}$ $f_{1}(E_{\gamma})$ with $g=(2J_{x}+1)/(2J_{0}+1)$, where
$J_{x}$ and $J_{0}$ are the spins of the excited and ground states,
respectively. It was shown, for example, that the so-called pygmy dipole
resonance (PDR) bar73 ; sav13 ; bra19 , an extra strength found on top of the
tail of the isovector giant dipole resonance (GDR), influences neutron-capture
reaction rates bea12 ; tso15 , which are important for the synthesis of heavy
elements in astrophysical processes arn07 ; kap11 .

In an earlier photon-scattering experiment on 76Ge using unpolarized
bremsstrahlung at the former Stuttgart Dynamitron and polarized bremsstrahlung
at the former Gießen linear accelerator, 30 states with $J$ = 1 were
identified between 2.9 and 9.1 MeV and parities were assigned to 17 of them
jun95 . Further experiments were performed using bremsstrahlung at the
S-Dalinac electron accelerator of Technische Universität Darmstadt, Germany,
son11  and quasimonoenergetic, polarized photons at the high-intensity
$\gamma$-ray source (HI$\gamma$S) wel09  of the Triangle Universities Nuclear
Laboratory (TUNL) in Durham, North Carolina. The experiments are briefly
described in Ref. wer15  while 128 states with $1^{-}$ assignments and 2 with
$1^{+}$ assignments between 4.4 and 8.9 MeV are compiled in a Ph.D. thesis
coo15 . In the present work, we found 210 states and assigned $J$ = 1 to most
of them. In addition, we determined the photon-scattering cross section for 10
keV bins of excitation energy up to $S_{n}$. In this analysis, the intensity
in the quasi-continuum part of the spectrum was taken into account. Moreover,
we estimated average intensities of inelastic transitions to low-lying excited
states and branching ratios of the ground-state transitions by means of
simulations of statistical $\gamma$-ray cascades. Using these quantities, we
determined the photoabsorption cross section.

Photon strength functions in 76Ge have previously been studied using $\beta$
decay of 76Ga in connection with the so-called Oslo method spy14 . Preliminary
results for a photon strength function deduced from photon-scattering
experiments with quasimonoenergetic photons at HI$\gamma$S were presented in
Ref. ton17 .

##  II Experimental Methods and results

###  II.1 The photon-scattering method

In photon-scattering experiments, the energy- and solid-angle-integrated
scattering cross section $I_{s}$ of an excited state at the energy $E_{x}$ is
deduced from the measured intensity of the respective transition to the ground
state. It can be determined relative to known integrated scattering cross
sections. In the present experiments, we used the integrated scattering cross
sections $I_{s}(E_{x}^{\rm B})$ of states in 11B ajz90  and their angular
correlations including mixing ratios rus09B  as a reference:

$\displaystyle\frac{I_{s}(E_{x})}{I_{s}(E_{x}^{\rm
B})}=\left(\frac{I_{\gamma}(E_{\gamma},\theta)}{W(E_{\gamma},\theta)\Phi_{\gamma}(E_{x})N_{N}}\right)$
$\displaystyle\times\left(\frac{I_{\gamma}(E_{\gamma}^{\rm
B},\theta)}{W(E_{\gamma}^{\rm B},\theta)\Phi_{\gamma}(E_{x}^{\rm B})N_{N}^{\rm
B}}\right)^{-1}.$ (1)

Here, $I_{\gamma}(E_{\gamma},\theta)$ and $I_{\gamma}(E_{\gamma}^{\rm
B},\theta)$ denote the efficiency-corrected measured intensities of a
considered ground-state transition at energy $E_{\gamma}$ and of a ground-
state transition in 11B at $E_{\gamma}^{\rm B}$, respectively, observed at an
angle $\theta$ to the beam. $W(E_{\gamma},\theta)$ and $W(E_{\gamma}^{\rm
B},\theta)$ describe the angular correlations of these transitions. The
quantities $N_{N}$ and $N_{N}^{\rm B}$ are the areal densities of nuclei in
the 76Ge and 11B targets, respectively. The quantities $\Phi_{\gamma}(E_{x})$
and $\Phi_{\gamma}(E_{x}^{\rm B})$ stand for the photon fluxes at the energy
of the considered level and at the energy of a level in 11B, respectively.

The integrated scattering cross section is related to the partial width of the
ground-state transition $\Gamma_{0}$ according to

$I_{s}=\int\limits^{+\infty}_{0}\sigma_{\gamma\gamma}~{}dE=\left(\frac{\pi\hbar
c}{E_{x}}\right)^{2}\frac{2J_{x}+1}{2J_{0}+1}\frac{\Gamma_{0}^{2}}{\Gamma},$
(2)

where $\sigma_{\gamma\gamma}$ is the elastic scattering cross section,
$E_{x}$, $J_{x}$ and $\Gamma$ denote energy, spin and total width of the
excited level, respectively, and $J_{0}$ is the spin of the ground state. If a
given level deexcites to low-lying excited states (inelastic scattering) in
addition to the deexcitation to the ground state (elastic scattering), then
the branching ratio $b_{0}=\Gamma_{0}/\Gamma$ of the ground-state transition
has to be known to deduce $\Gamma_{0}$. The $\gamma$-ray intensities and,
hence, the deduced quantities $I_{s}$ and $\Gamma_{0}$ are also distorted if a
level is populated from higher-lying levels. This feeding can be reduced by
choosing beam energies not far above the considered levels.

Spins of excited states are deduced by comparing experimental ratios of
$\gamma$-ray intensities, measured at two angles, with theoretical
predictions. The optimum combination includes angles of 90∘ and 127∘ to the
beam direction, because the respective ratios for the spin sequences $0-1-0$
and $0-2-0$ differ most at these angles. The expected values are
$W(90^{\circ})/W(127^{\circ})_{0-1-0}=0.74$ and
$W(90^{\circ})/W(127^{\circ})_{0-2-0}=2.15$ by taking into account opening
angles of $16^{\circ}$ and $14^{\circ}$ of the collimators in front of the
detectors placed at $90^{\circ}$ and $127^{\circ}$, respectively, in the setup
at $\gamma$ELBE sch05 .

###  II.2 The target

The target consisted of 1.8760 g of germanium, enriched to 93.4% in 76Ge, in a
square shape of 15 mm $\times$ 11 mm. The germanium target was combined with
0.300 g of boron, enriched to 99.5% in 11B and formed to a disk of 20 mm in
diameter. The known integrated scattering cross sections of levels in 11B were
used to determine the photon flux (see Sec. II.4). The photon-flux density was
proven to be nearly constant in a spot of about 25 mm in diameter rus07 . For
the calculation of cross sections for 76Ge, the ratio of the 76Ge and 11B
target areas was taken into account.

###  II.3 Detector response

For the determination of the integrated scattering cross sections according to
Eq. (II.1) the relative efficiencies of the detectors and the photon flux are
needed. The detector response was simulated using the program package GEANT4
ago03 ; all06 ; all16 . The reliability of the simulations was tested by
comparing simulated spectra with measured ones as described, for example, in
Refs. rus07 ; sch07 ; rus08 ; mas12 ; mar10 . The determination of the
absorption cross section requires in addition a correction of the experimental
spectra for photons scattered by atomic processes induced by the impinging
photons in the target material, and for ambient background radiation, which is
described in Sec. III.

The absolute efficiencies of the high-purity germanium (HPGe) detectors in the
setup at $\gamma$ELBE were determined experimentally up to 2.4 MeV from
measurements with a 226Ra calibration source. For interpolation, an efficiency
curve calculated with GEANT4 and scaled to the absolute experimental values
was used. A check of the simulated efficiency curve up to about 9 MeV was
performed via various $(p,\gamma)$ reactions at the HZDR Tandetron
accelerator. The efficiency values deduced from these measurements agree with
the simulated values within their uncertainties tro09 . Similar results were
obtained for the resonances at 4.44 and 11.66 MeV in 12C populated in the
11B$(p,\gamma)$ reaction at the van-de-Graaff accelerator of the Triangle
Universities Nuclear Laboratory (TUNL) in Durham, North Carolina car10 .

###  II.4 Experiments and results

The nuclide 76Ge was studied in two experiments at $\gamma$ELBE sch05 .
Bremsstrahlung was produced using electron beams of 7.8 and 12.3 MeV kinetic
energy, respectively. In the measurement at 7.8 MeV, the electron beam hit a
niobium foil of 7 $\mu$m thickness acting as a radiator at an average current
of about 700 $\mu$A. In the measurement at 12.3 MeV, the niobium foil had a
thickness of 12.5 $\mu$m and the average current was also about 700 $\mu$A. A
10 cm thick aluminum absorber (beam hardener) was placed behind the radiator
to reduce the low-energy intensity of the bremsstrahlung spectrum in the
measurement at 12.3 MeV. The photon beam was collimated by a 260-cm-long pure-
aluminum collimator with a conical borehole of 8 mm in diameter at the
entrance, 90 cm behind the radiator, and 24 mm in diameter at the exit. The
target, placed 200 cm behind the collimator exit, was irradiated with a
typical flux of about $10^{9}$ s-1 in a spot of 38 mm in diameter. Scattered
photons were measured with four HPGe detectors with a full-energy efficiency
of 100% relative to a NaI detector of 7.6 cm in diameter and 7.6 cm in length.
All HPGe detectors were surrounded by escape-suppression shields made of
bismuth germanate (BGO) scintillation detectors of 3 cm in thickness. Two HPGe
detectors were placed vertically at 127∘ relative to the photon-beam direction
and a distance of 32 cm from the target. The other two HPGe detectors were
positioned in a horizontal plane at 90∘ to the beam and a distance of 28 cm
from the target. Absorbers of 8 mm Pb plus 3 mm Cu were placed in front of the
detectors at 90∘ and of 3 mm Pb plus 3 mm Cu in front of the detectors at
127∘. Spectra of scattered photons were measured for 132 h each in the
experiments at 7.8 and 12.3 MeV electron energy. Part of a spectrum including
events measured with the two detectors placed at 127∘ relative to the beam at
an electron energy of 12.3 MeV is shown in Fig. 1.

Figure 1:  Part of a spectrum of photons scattered from 76Ge combined with
11B, measured during the irradiation with bremsstrahlung produced by electrons
of an energy of $E_{e}^{\rm kin}$ = 12.3 MeV. This spectrum is the sum of the
spectra measured with the two detectors placed at 127∘ relative to the beam at
$\gamma$ELBE. The arrow labeled $S_{n}$ indicates the neutron-separation
energy.

The absolute photon fluxes in the two measurements at $\gamma$ELBE were
determined from intensities and known integrated scattering cross sections of
transitions in 11B. The 7283 keV transition in 11B was found to form an
unresolved doublet with another transition, which is negligible at $E_{e}$ =
7.8 MeV, but comparable in intensity with the transition in 11B at $E_{e}$ =
12.3 MeV. The latter has therefore not been considered for the flux
determination. For interpolation, the photon flux was calculated using a
bremsstrahlung computer code hau05  based on the Born approximation with
Coulomb correction roc72  and including an atomic screening correction sal87 .
In addition, the flux was corrected for the attenuation by the beam hardener
by applying a parametrization of the results of a corresponding GEANT4
simulation. The calculated flux curves were adjusted to the experimental
values obtained at the energies of levels in 11B. The experimental flux values
and the calculated curves are presented in Fig. 2.

Figure 2:  Average absolute photon flux on the 11B target deduced from
intensities of known transitions in 11B for the measurements at $E_{e}$ = 7.8
MeV (triangles) and $E_{e}$ = 12.3 MeV (circles) at $\gamma$ELBE. The curves
represent the calculated flux described in the text.

The measurements at two electron energies allowed us to identify inelastic
transitions that feed low-lying from high-lying levels. Transitions found in
the measurement at $E_{e}^{\rm kin}$ = 7.8 MeV are assumed to be ground-state
transitions. Additional transitions observed up to 7.8 MeV in the measurement
at $E_{e}^{\rm kin}$ = 12.3 MeV are considered to be inelastic transitions
from high-lying to low-lying excited states. By comparing the respective
spectra, these inelastic transitions were sorted out. Besides, there is a
number of transitions with energies that fit the difference between the energy
of a higher-lying level and the first or second excited $2^{+}$ states. These
transitions are also assumed to be inelastic transitions, if their intensity
is smaller than that of the ground-state transition from the considered
higher-lying level. The remaining ground-state transitions were used to derive
the corresponding level energies, the integrated scattering cross sections of
the states, and spin assignments deduced from angular distributions of the
ground-state transitions. These quantities are compiled in Table
LABEL:tab:gam. The integrated scattering cross sections of levels up to
$E_{x}$ = 7.0 MeV were taken from the measurement at 7.8 MeV electron energy,
because they are affected by feeding intensities in the 12.3 MeV measurement.
We note that in principle low-lying states can also be fed from other states
below 7.8 MeV. However, previous investigations have shown that the states
below about 6 MeV are fed by states mainly above about 9 MeV sch08 ; ben09 .

Table 1: Levels assigned to 76Ge. $E_{x}$(keV)111 Excitation energy. The uncertainty of this and the other quantities in the table is given in parentheses in units of the last digit. This energy value was deduced from the $\gamma$-ray energy measured at 127∘ including a recoil and Doppler-shift correction. |  $I_{\gamma}(90^{\circ})/I_{\gamma}(127^{\circ})$222 Ratio of the intensities measured at angles of 90∘ and 127∘. The expected values for an elastic dipole transition (spin sequence $0-1-0$) and for an elastic quadrupole transition (spin sequence $0-2-0$) are 0.74 and 2.15, respectively. |  $J_{x}^{\pi}$333 Spin deduced from the angular distribution of the ground-state transition. A tentative assignment of (1) is given, if the angular distribution is compatible with dipole as well as isotropic behavior. |  $I_{s}$(eV b)444 Energy-integrated scattering cross section. Below an excitation energy of 7.0 MeV the value was deduced from the measurement at 7.8 MeV electron energy, otherwise the value was deduced from the measurement at 12.3 MeV.
---|---|---|---
564.5(1) | 1.05(14) |  $2^{+}$555 Spin and parity taken from Ref. muk17 . |
1109.2(1) | 1.11(15) |  $2^{+}$555 Spin and parity taken from Ref. muk17 . |
2504.2(4) |  |  $2^{+}$555 Spin and parity taken from Ref. muk17 . |
2655.1(3) | 0.79(27) | (1) | 5.6(10)
2919.3(2) | 1.33(24) |  $1^{+}$555 Spin and parity taken from Ref. muk17 . | 12.1(12)
3006.7(2) | 0.90(16) |  $1^{+}$555 Spin and parity taken from Ref. muk17 . | 9.6(10)
3140.9(2) | 1.05(15) |  $1^{+}$555 Spin and parity taken from Ref. muk17 . | 12.0(12)
3200.0(2) | 1.23(23) |  | 9.7(11)
3418.9(1) | 0.86(4) |  $1^{+}$666 Spin and parity taken from Ref. jun95 . | 46(4)
3680.4(1) | 0.84(3) |  $1^{-}$666 Spin and parity taken from Ref. jun95 . | 52(4)
3763.2(1) | 0.83(6) |  $1^{+}$666 Spin and parity taken from Ref. jun95 . | 25.8(24)
3951.0(4) | 1.10(28) |  | 4.5(7)
4024.0(2) | 0.70(20) |  $1^{(-)}$666 Spin and parity taken from Ref. jun95 . | 6.2(8)
4035.0(2) | 0.76(15) | 1 | 9.1(10)
4115.9(2) | 0.81(12) | 1 | 14.7(16)
4250.8(3) | 0.68(20) | 1 | 6.7(11)
4624.0(2) | 0.84(8) |  $1^{+}$666 Spin and parity taken from Ref. jun95 . | 26.9(24)
4661.0(4) | 0.53(16) | 1 | 6.4(10)
4678.1(1) | 0.75(5) | 1 | 33.0(28)
4722.2(2) | 0.88(15) | (1) | 9.6(12)
4741.0(2) | 0.94(15) |  | 10.3(12)
4788.9(3) | 1.17(26) |  | 7.1(11)
4837.0(4) | 0.88(20) | (1) | 7.3(12)
4845.9(3) | 0.76(13) | 1 | 10.9(13)
4874.5(2) | 1.5(6) |  | 8.7(13)
4916.5(1) | 0.75(4) | 1 | 50(4)
4935.9(2) | 0.91(7) | 1 | 19.5(18)
5116.4(2) | 0.85(9) | 1 | 17.3(18)
5166.7(2) | 0.91(9) | (1) | 17.3(16)
5185.8(1) | 0.98(6) | (1) | 38(3)
5202.3(2) | 0.80(8) | 1 | 22.3(20)
5222.0(3) | 1.04(15) |  | 13.3(16)
5266.8(3) | 0.65(13) | 1 | 11.6(16)
5273.6(6) | 0.80(24) | (1) | 5.0(10)
5284.9(2) | 0.60(10) | 1 | 13.1(15)
5304.1(3) | 0.56(13) | 1 | 10.2(13)
5365.6(3) | 0.68(12) | 1 | 10.1(13)
5379.5(4) | 0.59(16) | 1 | 6.2(10)
5390.6(5) | 0.81(22) | (1) | 5.4(10)
5418.6(4) | 0.76(27) | (1) | 5.4(10)
5434.3(3) | 0.74(22) | 1 | 7.9(11)
5492.7(2) | 0.62(13) | 1 | 21.3(25)
5540.1(2) | 0.76(5) | 1 | 31.6(27)
5567.4(2) | 0.92(8) | (1) | 20.6(19)
5581.0(2) | 0.66(8) | 1 | 16.0(16)
5665.2(3) | 0.72(10) | 1 | 20.8(24)
5677.6(3) | 0.75(11) | 1 | 23.5(26)
5698.8(2) | 0.74(6) |  $1^{-}$666 Spin and parity taken from Ref. jun95 . | 52(5)
5708.4(6) | 0.98(21) | (1) | 8.3(14)
5748.3(1) | 0.81(5) |  $1^{-}$666 Spin and parity taken from Ref. jun95 . | 67(5)
5785.0(2) | 0.72(5) | 1 | 52(4)
5794.1(2) | 0.66(7) | 1 | 26.5(25)
5820.8(6) |  |  | 22(5)
5825.3(8) | 0.76(10) | 1 | 12(3)
5846.5(7) | 0.81(28) |  | 8.8(21)
5864.8(6) | 0.83(22) |  | 10.8(22)
5908.8(3) |  |  | 8.1(13)
5954.8(2) | 0.71(5) | 1 | 41(3)
5983.0(2) | 0.69(4) |  $1^{-}$666 Spin and parity taken from Ref. jun95 . | 49(4)
6048.4(4) | 0.81(20) | 1 | 7.8(13)
6081.4(4) | 0.90(16) | (1) | 12.4(17)
6102.0(9) |  |  | 5.3(12)
6113.6(3) | 0.77(9) | 1 | 21.1(22)
6130.3(2) | 0.76(7) | 1 | 30.9(29)
6145.6(2) | 0.77(9) | 1 | 24.2(25)
6162.4(9) | 0.9(4) |  | 4.5(14)
6191.3(2) | 0.72(6) | 1 | 41(4)
6223.4(7) |  |  | 6.0(14)
6228.2(4) | 0.79(11) | 1 | 14.5(20)
6234.8(9) |  |  | 4.6(12)
6240.7(3) | 0.70(14) | 1 | 13.3(18)
6272.7(3) | 0.79(9) | 1 | 24.0(25)
6285.3(2) | 0.68(6) | 1 | 46(4)
6315.4(4) | 0.66(11) | 1 | 21.5(26)
6330.2(2) | 0.77(5) | 1 | 63(5)
6366.2(11) | 1.0(3) |  | 8.7(21)
6393.2(5) | 0.81(19) | 1 | 15.5(26)
6408.1(5) | 0.63(29) | 1 | 17(3)
6436.1(9) | 1.2(4) |  | 12.0(28)
6448.3(11) | 0.9(4) |  | 8.2(23)
6472.2(3) | 0.79(9) | 1 | 41(4)
6497.9(3) | 0.72(13) | 1 | 18.7(27)
6513.3(4) | 0.75(11) | 1 | 31(4)
6572.0(6) | 1.08(21) |  | 11.8(18)
6601.2(2) | 0.77(6) | 1 | 53(5)
6611.1(6) |  |  | 13.0(19)
6629.0(3) | 0.79(9) | 1 | 27.0(28)
6641.9(5) | 1.00(15) |  | 16.5(21)
6661.4(9) |  |  | 9.6(19)
6670.6(3) | 0.85(9) | 1 | 33(3)
6741.6(6) | 0.86(20) | (1) | 11.4(19)
6764.8(4) | 0.74(11) | 1 | 20.4(24)
6786.7(2) | 0.83(6) | 1 | 59(5)
6816.5(3) | 0.75(7) | 1 | 44(4)
6835.5(2) | 0.70(5) | 1 | 82(7)
6846.2(3) | 0.66(7) | 1 | 44(4)
6880.3(4) | 0.70(6) | 1 | 46(5)
6884.2(10) |  |  | 18(2)
6898.9(5) | 0.55(8) | 1 | 41(6)
6908.0(18) | 1.2(3) |  | 19(5)
6938.6(7) | 0.61(16) | 1 | 15.1(27)
6959.9(3) | 0.71(8) | 1 | 45(5)
6985.1(5) | 0.53(9) | 1 | 26(3)
6998.7(3) | 0.68(5) |  $1^{-}$666 Spin and parity taken from Ref. jun95 . | 79(7)
7011.0(9) | 0.46(16) | 1 | 12.3(25)
7025.8(3) | 0.73(8) |  $1^{(-)}$666 Spin and parity taken from Ref. jun95 . | 51(5)
7047.9(9) | 0.58(22) | 1 | 10.4(26)
7081.2(9) | 0.53(29) | 1 | 9.0(22)
7091.4(4) | 0.74(11) | 1 | 28(3)
7102.4(6) | 0.75(26) | 1 | 13.8(24)
7121.3(3) | 0.80(8) | 1 | 42(4)
7130.1(3) | 0.70(9) | 1 | 36(4)
7147.3(4) | 0.76(13) | 1 | 23(3)
7171.6(9) |  |  | 10.8(26)
7250.5(2) | 0.79(8) |  $1^{-}$666 Spin and parity taken from Ref. jun95 . | 76(7)
7289.7(4) | 0.9(5) |  | 51(5)
7300.7(3) | 0.75(9) |  $1^{-}$666 Spin and parity taken from Ref. jun95 . | 56(5)
7406.7(3) | 0.76(11) | 1 | 60(6)
7415.6(4) | 0.98(18) |  | 37(5)
7452.2(5) |  |  | 24(4)
7478.6(5) | 0.8(3) |  | 21(3)
7485.0(3) | 0.65(26) | 1 | 33(4)
7521.2(5) | 0.75(14) | 1 | 26(4)
7536.6(4) | 0.89(15) | (1) | 30(4)
7548.8(7) | 0.95(26) | (1) | 19(4)
7584.6(4) | 0.67(9) | 1 | 46(5)
7642.6(4) | 0.76(9) | 1 | 69(7)
7650.8(4) | 0.79(10) | 1 | 58(6)
7677.7(4) | 0.78(12) | 1 | 44(5)
7694.0(3) | 0.75(10) | 1 | 59(7)
7722.7(4) | 0.93(19) | (1) | 36(6)
7776.9(7) | 0.83(17) | (1) | 28(5)
7783.8(9) |  |  | 24(5)
7796.6(4) | 0.70(10) | 1 | 66(7)
7803.7(6) | 0.76(13) | 1 | 42(5)
7814.3(7) | 0.79(16) | 1 | 26(4)
7817.2(2) |  |  | 20.7(17)
7836.3(6) |  |  | 17(4)
7849.3(5) | 0.77(24) | (1) | 22(4)
7861.2(4) | 0.51(20) | 1 | 31(6)
7883.3(10) | 0.69(26) | 1 | 16(4)
7893.6(12) | 1.0(4) |  | 11(3)
7913.4(2) | 0.80(7) | 1 | 85(8)
7949.9(2) | 0.86(12) | 1 | 42(5)
7975.6(7) | 0.93(16) | (1) | 29(5)
7995.8(4) | 0.85(18) | (1) | 25(4)
8017.5(14) | 0.7(3) | (1) | 19(5)
8026.5(8) | 0.86(25) | (1) | 34(7)
8049.3(6) | 0.82(21) | (1) | 34(5)
8063.4(8) | 0.71(27) | 1 | 20(4)
8094.2(8) |  |  | 12(4)
8102.8(5) | 1.23(28) |  | 24(5)
8109.5(8) |  |  | 13(3)
8134.5(11) | 1.4(5) |  | 13(4)
8151.6(5) | 0.85(12) |  $1^{(-)}$666 Spin and parity taken from Ref. jun95 . | 52(7)
8160.2(9) |  |  | 25(5)
8177.8(4) | 0.70(8) | 1 | 56(6)
8187.8(5) | 0.86(11) | 1 | 49(6)
8236.4(4) | 0.90(14) | (1) | 44(6)
8252.9(9) |  |  | 25(6)
8259.6(6) | 0.86(16) | (1) | 43(7)
8284.5(3) | 0.88(13) | (1) | 72(8)
8294.3(12) | 0.86(25) |  | 21(4)
8303.5(5) | 0.77(13) | 1 | 49(6)
8317.8(3) | 0.74(9) | 1 | 77(7)
8328.9(7) | 0.77(15) | 1 | 27(4)
8347.7(9) | 0.84(25) |  | 19(4)
8357.4(7) | 0.91(29) | (1) | 28(5)
8397.3(5) |  |  | 38(6)
8418.0(15) |  |  | 9(3)
8425.2(3) | 1.18(26) |  | 53(7)
8446.1(7) | 0.5(4) | 1 | 16(3)
8461.9(9) |  |  | 13(3)
8500.0(3) | 0.84(11) | 1 | 62(7)
8520.7(6) |  |  | 23(6)
8535.1(5) | 0.61(20) | 1 | 33(5)
8546.1(5) | 0.68(11) |  $1^{-}$666 Spin and parity taken from Ref. jun95 . | 73(10)
8552.3(8) | 0.42(20) | 1 | 34(7)
8566.9(3) | 0.78(12) | 1 | 49(6)
8602.3(5) |  |  | 19(4)
8625.7(7) | 0.49(25) | 1 | 28(11)
8649.1(8) | 1.20(28) |  | 22(4)
8662.0(4) | 0.89(12) | (1) | 52(6)
8696.2(7) |  |  | 20(10)
8740.7(4) | 0.92(13) | (1) | 41(5)
8752.8(4) | 1.05(14) |  $1^{-}$666 Spin and parity taken from Ref. jun95 . | 43(5)
8768.4(9) | 0.62(25) | 1 | 15(4)
8806.2(5) | 1.2(5) |  | 27(7)
8843.7(4) | 0.67(11) | 1 | 39(6)
8888.5(9) |  |  | 19(5)
9014.2(14) |  |  $1^{-}$666 Spin and parity taken from Ref. jun95 . | 24(9)
9019.5(10) | 0.98(15) | (1) | 37(12)
9033.1(9) | 1.0(3) |  | 15(4)
9051.7(12) | 0.89(17) | (1) | 17(5)
9058.5(11) | 0.92(29) |  | 18(5)
9163.3(9) | 0.69(23) | 1 | 13.9(28)
9175.5(8) | 0.59(16) | 1 | 19.1(29)
9187.4(4) | 0.80(12) | 1 | 35(4)
9254.6(7) | 1.08(29) |  | 21(4)
9264.1(6) | 1.1(3) |  | 22(3)
9305.0(4) |  |  | 18.0(26)
9315.8(4) | 1.0(5) |  | 23(3)
9337.8(6) | 1.0(3) |  | 17(3)
9354.5(8) | 0.92(29) | (1) | 15.4(27)
9365.9(5) | 0.81(18) | 1 | 29(4)
9377.9(4) | 0.95(18) | (1) | 38(5)
9399.4(6) | 0.60(17) | 1 | 19(3)
9409.9(4) | 0.69(11) | 1 | 49(5)
9417.6(5) | 0.52(13) | 1 | 29(4)
9556.6(5) | 0.71(22) | 1 | 11.5(22)
|  |  |

A transition with an energy around 2039 keV, close to the one of the expected
signal for the $0\nu\beta\beta$ decay, has not been clearly identified in the
present experiments. In particular, the ground-state transition from the 3951
keV level, known from 76Ga $\beta$ decay cam71 , has been detected in the
present study (see Table LABEL:tab:gam), but there is no indication of the
2040 keV transition depopulating the 3951 to the 1911 keV level in the
measurement at $E_{e}$ = 7.8 MeV. There may be a tiny bump in the measurement
at $E_{e}$ = 12.3 MeV, which occurs due to feeding of the 3951 keV level from
higher-lying levels. This situation is illustrated in Fig. 3. The tentative
spin and parity assignment of $(1,2^{+})$ given in Ref. cam71  for the 3951
keV level could not be made more precise because of the uncertain angular
distribution of the weak ground-state transition. The spectrum at 12.3 MeV
contains, for example, transitions from the $0^{+}_{3}$ state at 2697 and from
the $2^{+}_{7}$ state at 3130 keV, which are not seen in the spectrum at 7.8
MeV. This proves that these low-lying states are mainly fed by states above
about 8 MeV.

Figure 3:  Parts of spectra measured at $E_{e}$ = 7.8 MeV (bottom) and $E_{e}$
= 12.3 MeV (top) showing the energy section around the expected 2039 keV
signal. Note that the 7.8 MeV spectrum was scaled up by a factor of 1.8 to
make the two spectra better comparable. The labels $2^{+}_{7}\rightarrow
2^{+}_{2}$ and $0^{+}_{3}\rightarrow 2^{+}_{1}$ mark the corresponding
transitions in 76Ge, 2039 is the energy of the $0\nu\beta\beta$ signal and 11B
denotes a transition in this nuclide.

##  III Determination of the dipole-strength distribution

The determination of the dipole-strength distribution and the related
photoabsorption cross section requires the knowledge of the intensity
distribution of the ground-state transitions and their branching ratios. As
these cannot be derived directly from the measured spectra, we applied
statistical methods to discriminate between $\gamma$ rays from nuclear
excitations and photons scattered by atomic processes, and to disentangle the
intensity distributions of elastic and inelastic transitions in the
quasicontinuum of nuclear levels.

First, a spectrum of the ambient background adjusted to the intensities of the
transitions from 40K and 208Tl decay in the in-beam spectrum was subtracted
from the measured spectrum. To correct the measured spectrum for the detector
response, spectra of monoenergetic $\gamma$ rays were calculated in steps of
10 keV by using the simulation code GEANT4. Starting from the high-energy end
of the experimental spectrum, the simulated spectra were subtracted
sequentially (spectrum-stripping method fur68 ).

The background radiation produced by atomic processes in the 76Ge target was
obtained from a GEANT4 simulation. The simulated atomic background is compared
with the response-corrected spectrum at $E_{e}$ = 7.8 MeV in Fig. 4 and at
$E_{e}$ = 12.3 MeV in Fig. 5. The atomic background amounts in average to only
a few percent of the intensity in the spectrum, but coincides with that above
the neutron threshold, which proves the right magnitude. This behavior is
similar to that found in previous studies rus09M ; rus08 ; schw13 ; mas14 ;
mak14 ; mas15 ; mak16 ; shi18  and shows that the experimental spectrum
contains a considerable amount of nuclear strength in a quasicontinuum. This
is formed by a large number of unresolved transitions with small intensities
that are the result of the increasing nuclear level density at high energy in
combination with the finite detector resolution. Because of the different
orders of magnitude, the nuclear intensity distribution resulting from the
subtraction of the simulated atomic background is not very sensitive to
uncertainties of the latter, for which we assume 5%. The nuclear intensity
distribution contains ground-state (elastic) transitions and, in addition,
branching (inelastic) transitions to lower-lying excited states as well as
transitions from those states to the ground state (cascade transitions). The
different types of transitions cannot be clearly distinguished. However, for
the determination of the photoabsorption cross section and the partial widths
$\Gamma_{0}$, the intensities of the ground-state transitions are needed.
Therefore, contributions of inelastic and cascade transitions have to be
subtracted from the spectra. We corrected the intensity distributions by
simulating $\gamma$-ray cascades from the levels in the entire energy region
using the code $\gamma$DEX sch12 ; mas12 . This code works analogously to the
strategy of the code DICEBOX bec98  developed for $(n,\gamma)$ reactions, but
in addition it includes also the excitation from the ground state. In the
present simulations, level schemes (nuclear realizations) including states
with $J$ = 0, …, 5 were created. Known low-lying levels were taken into
account up to about 3 MeV. Partial widths were varied in the individual
nuclear realizations applying the Porter-Thomas distribution por56 . Level
densities were calculated by using the constant-temperature model gil65  with
the parameters $T$ = 0.92(1) MeV and $E_{0}$ = 0.13(5) MeV adjusted to
experimental level densities egi09 . In the individual nuclear realizations,
the values of $T$ and $E_{0}$ were varied randomly within a Gaussian
distribution with a standard deviation corresponding to the uncertainties
given in Ref. egi09 . The parity distribution of the level densities was
modeled according to the information given in Ref. alq03 .

The first inputs for the photon strength function simulations were assumed to
be Lorentz-shaped. For the $E1$ strength, a sum of three Lorentz functions
(TLO) that account for a triaxial deformation of the nucleus was used with
parameters described in Refs. jun08 ; gro17 . In the present case, deformation
parameters of $\beta_{2}$ = 0.26 ram01  and $\gamma=26^{\circ}$ del10  were
applied. The parameters for the $M1$ and $E2$ strengths were taken from global
parametrizations of $M1$ spin-flip resonances and $E2$ isoscalar resonances,
respectively cap10 . Low-lying levels were also taken into account. Spectra of
$\gamma$-ray cascades were generated for groups of levels in energy bins of
$\Delta E$ = 100 keV. Starting from the high-energy end of the intensity
distribution, that contains ground-state transitions only, the simulated
intensities of the ground-state transitions were normalized to the
experimental ones in the considered bin. The intensity distribution of the
branching transitions was subtracted from the total intensity distribution.
Applying this procedure step-by-step for each energy bin moving toward the
low-energy end of the spectrum, one obtains the intensity distribution of the
ground-state transitions. Simultaneously, the branching ratios $b_{0}(E)$ of
the ground-state transitions are determined for each energy bin. In an
individual nuclear realization, the branching ratio $b_{0}(E)$ is calculated
as the ratio of the sum of the intensities of the ground-state transitions
from all levels in $\Delta E$ to the total intensity of all transitions
depopulating those levels to either any low-lying levels or the ground state
rus09M ; mas12 ; sch12 ; mas14L ; schw13 ; mak14 . Branching ratios $\langle
b_{0}(E)\rangle$, averaged over the many nuclear realizations in the present
cascade simulations, are illustrated in Fig. 6.

Figure 4: Spectrum of the two detectors at 127∘ and $E_{e}$ = 7.8 MeV,
corrected for detector response (blue), and simulated spectrum of photons
scattered from the target to the detectors by atomic processes (black). Figure
5: Spectrum of the two detectors at 127∘ and $E_{e}$ = 12.3 MeV corrected for
detector response (blue), and simulated spectrum of photons scattered from the
target to the detectors by atomic processes (black). Figure 6: Average
branching ratios of ground-state transitions resulting from the simulations of
statistical $\gamma$-ray cascades up to $S_{n}$ as described in the text.

The uncertainty of the number of counts $N(E)$ in an energy bin of the
experimental intensity distribution was deduced as

$\delta
N(E)=\sqrt{N(E)}+\sum_{E^{\prime}}{\left[\sqrt{N(E^{\prime}>E)}~{}b(E^{\prime}\rightarrow
E)\right]},$ (3)

where $b(E^{\prime}\rightarrow E)$ is the branching intensity from bin
$E^{\prime}$ to bin $E$. We transform $N(E)$ to the scattering cross section
according to

$\sigma_{\gamma\gamma}(E)=N(E)/\left[\epsilon(E)~{}\Phi_{\gamma}(E)~{}W(E)~{}N_{N}~{}\Delta
t~{}\Delta E~{}\right]$ (4)

with the quantities defined in Eq. (II.1), the absolute detector efficiency
$\epsilon(E)$, the measuring time $\Delta t$, and the bin width $\Delta E$.
The absorption cross section in each bin is obtained as
$\sigma_{\gamma}(E)=\sigma_{\gamma\gamma}(E)/b_{0}(E)$ for each nuclear
realization. Finally, the absorption cross sections of each bin were obtained
by averaging over the values of the individual nuclear realizations.

The simulations were performed iteratively, where the strength function
obtained from an iteration step was used as the input for the next step. We
note that the simulations are little sensitive to the shape of the first input
strength function, which was tested, for example, in Refs. sch08 ; sch12 . The
iteration was stopped when the input strength function and the output strength
function were in agreement within their respective uncertainties. The cross
section obtained in the last iteration step is adopted as the final absorption
cross section. Toward low energy, the uncertainties increase due to the use of
the spectrum-stripping method and the cross sections do not converge. Besides,
the assumption of a statistical quasicontinuum becomes invalid and individual
states become important. Therefore, cross sections cannot be determined
reliably below about 6 MeV. The procedure just decsribed was also performed
for the measurement at $E_{e}$ = 7.8 MeV to extend the cross section data to
low excitation energy. The absorption cross sections obtained in this way are
listed in Tables 2, 3, and graphed in Fig. 7. The uncertainties of the cross-
section values include statistical uncertainties of the spectrum, the given
uncertainties of the efficiency and the subtracted simulated background
spectrum, uncertainties of the flux resulting from the integrated cross
sections of the 11B levels and the uncertainties of the level-density
parameters given in the text above. Systematic uncertainties of level-density
models can result in additional uncertainties of up to about 20%, which are
not included here. These deviations of modeled from experimentally determined
level densities and between the various level-density models are, for example,
discussed for the case of 76Ge in Ref. voi19 .

Table 2: Photoabsorption cross section of 76Ge deduced from the present $(\gamma,\gamma^{\prime})$ experiment at $E_{e}$ = 7.8 MeV. $E_{\gamma}$ (MeV) |  $\sigma$ (mb) 111Absorption cross section resulting from the experimental intensity distribution including the quasicontinuum, corrected for branching intensities and branching ratios obtained from $\gamma$-ray cascade simulations. The uncertainties include statistical uncertainties of the spectra (see Sec. III), the given uncertainties of the efficiencies and the subtracted simulated background spectra, uncertainties of the flux resulting from the integrated cross sections of the 11B levels and the given uncertainties of the level-density parameters.
---|---
5.0 | 0.18(3)
5.1 | 0.18(5)
5.2 | 0.28(7)
5.3 | 0.33(7)
5.4 | 0.28(7)
5.5 | 0.25(5)
5.6 | 0.30(5)
5.7 | 0.66(12)
5.8 | 0.87(15)
5.9 | 0.62(10)
6.0 | 0.70(13)
6.1 | 0.79(12)
6.2 | 0.70(10)
6.3 | 1.36(22)
6.4 | 1.48(23)
6.5 | 1.00(13)
6.6 | 1.62(21)
6.7 | 1.93(27)
6.8 | 2.9(4)
6.9 | 3.3(4)
7.0 | 2.9(3)
7.1 | 3.8(4)
7.2 | 3.8(4)
7.3 | 3.9(4)
7.4 | 4.4(4)
7.5 | 2.51(26)
Table 3: Photoabsorption cross section of 76Ge deduced from the present $(\gamma,\gamma^{\prime})$ experiment at $E_{e}$ = 12.3 MeV. $E_{\gamma}$ (MeV) |  $\sigma$ (mb) 111Absorption cross section resulting from the experimental intensity distribution including the quasicontinuum, corrected for branching intensities and branching ratios obtained from $\gamma$-ray cascade simulations. The uncertainties include statistical uncertainties of the spectra (see Sec. III), the given uncertainties of the efficiencies and the subtracted simulated background spectra, uncertainties of the flux resulting from the integrated cross sections of the 11B levels and the given uncertainties of the level-density parameters.
---|---
6.0 | 2.2(12)
6.1 | 1.8(8)
6.2 | 1.9(8)
6.3 | 2.7(9)
6.4 | 2.2(5)
6.5 | 2.7(6)
6.6 | 3.0(4)
6.7 | 2.8(3)
6.8 | 4.5(3)
6.9 | 3.7(3)
7.0 | 4.7(3)
7.1 | 4.8(3)
7.2 | 4.1(2)
7.3 | 5.4(3)
7.4 | 5.1(3)
7.5 | 4.9(3)
7.6 | 5.8(3)
7.7 | 6.5(3)
7.8 | 6.5(3)
7.9 | 6.9(3)
8.0 | 7.6(4)
8.1 | 7.1(3)
8.2 | 8.2(4)
8.3 | 9.2(4)
8.4 | 8.0(4)
8.5 | 9.2(4)
8.6 | 8.0(4)
8.7 | 8.6(4)
8.8 | 8.7(4)
8.9 | 9.9(5)
9.0 | 8.3(4)
9.1 | 8.1(4)
9.2 | 8.2(4)
9.3 | 9.2(5)
9.4 | 9.1(5)
9.5 | 4.7(3)
9.6 | 1.93(9)
9.7 | 0.92(4)
9.8 | 0.88(4)
9.9 | 0.72(3)
10.0 | 0.74(3)
Figure 7: Photoabsorption cross sections of 76Ge resulting from the present
$(\gamma,\gamma^{\prime})$ experiments at $E_{e}$ = 7.8 MeV (blue triangles)
and at $E_{e}$ = 12.3 MeV (red circles), from $(\gamma,n)$ data (green
squares) taken from Ref. car76 , from calculations using the TALYS code as
given in the TENDL-2019 library (black solid curve), and from the TLO with
deformation parameters given in the text (black dashed curve).

##  IV Discussion

The photoabsorption cross section resulting from the present experiments is
compared with the cross section of the $(\gamma,n)$ reaction car76  in Fig. 7.
In addition, the TLO with the deformation parameters given in Sec. III and the
photoabsorption cross section given in the latest TALYS-based evaluated
nuclear data library (TENDL-2019) ten19  are displayed. In the latter, the
standard Lorentzian (Brink-Axel model) bri55 ; axe62 ; kaw20  was used as a
strength function in the $(\gamma,\gamma^{\prime})$ reaction kon20 . The
present $(\gamma,\gamma^{\prime})$ cross section shows extra strength above
the TLO in the energy region from about 6 MeV to $S_{n}$, which is attributed
to the PDR. The shape of the experimental cross section is fairly well
approximated by the TENDL cross section because of its relatively smooth
behavior. The cross section of 76Ge is compared with those of the neighboring
isotope 74Ge mas15  and of the isotone 78Se sch12 ; mak16  resulting from
analogous experiments and methods in Fig. 8. In the PDR region from about 6 to
9 MeV, the cross section of 76Ge appears to be in average by a factor of about
two higher than that of 74Ge, but by a factor of about two smaller than that
of 78Se. Toward low energy, the cross section obtained from the low-energy
measurement on 76Ge drops more rapidly than the ones in 74Ge and 78Se. These
relatively large differences to nuclides with two neutrons less or two protons
more, respectively, are remarkable, and interesting to be tested by nuclear
models.

Figure 8: Photoabsorption cross sections of 74Ge (green squares), 76Ge (blue
triangles and red circles), and 78Se (black triangles), resulting from
$(\gamma,\gamma^{\prime})$ experiments at $\gamma$ELBE. The data for 74Ge were
taken from Ref. mas15  and the data for 78Se were taken from Refs. sch12 ;
mak16 .

The photon strength function deduced from the present photoabsorption cross
section of 76Ge is compared with preliminary results from a
$(\gamma,\gamma^{\prime})$ experiment with quasimonoenergetic photons at the
HI$\gamma$S facility ton17  and with data obtained from an experiment applying
the so-called Oslo method in connection with the $\beta$ decay of 76Ga spy14
in Fig. 9. The data from the HI$\gamma$S and the $\beta$ decay experiments
exceed the present data below 6 MeV. Between 7 and 9 MeV, the HI$\gamma$S data
amount in average to about 70% of the present data. They were obtained from an
analysis of mainly resolved elastic transitions ton21 . This means that
average branching ratios $\langle b_{0}(E)\rangle$ in energy bins are
overestimated at high excitation energy and, hence, the photoabsorption cross
section is underestimated. Besides, strength in the quasicontinuum was not
taken into account in that analysis. Similar discrepancies resulting from
missing strength in the quasicontinuum were also reported in other recent
studies mak16 ; mue20 ; wil20 . On the other hand, the present $\langle
b_{0}(E)\rangle$ include uncertainties of the inputs for the statistical
cascade simulations, for example, of the used level-density model.

Whereas the strength functions deduced from photoabsorption cross sections
contain exclusively ground-state transitions from $J$ = 1 states, the strength
functions obtained from light-ion induced reactions or from $\beta$ decay
comprise a large number of transitions linking many states of various spins up
to about $J$ = 10, which may cause different characteristics of the strength
functions. The strength function of 76Ge from $\beta$ decay continues to low
transition energies that belong to cascade transitions between close-lying
levels at high excitation energy. It shows the characteristic upbend below
about 2.5 MeV that was also observed in the isotopic neighbors 73Ge and 74Ge
ren16 . This low-energy enhancement of strength has been described in shell-
model calculations as resulting from the large strengths of many $M1$
transitions between states generated by recoupling the spins of protons and
neutrons in high-$j$ orbitals sch13L ; bro14Fe ; sie17 . Interestingly, a bump
appears between about 3 and 4.5 MeV in 76Ge in addition to the low-energy
upbend. In the shell-model calculations, such a bump appears in open-shell
nuclei and has been related to the scissors resonance, which develops in
deformed nuclides sch17 . A pronounced bump around 3 MeV in addition to the
low-energy upbend was also observed in Sm isotopes sim16 ; naq19 . The
$(\gamma,\gamma^{\prime})$ data principally contain also inelastic and cascade
transitions between close-lying states at high excitation energy that
contribute to the low-energy enhancement sch19 . Those transitions are hidden
in the huge background in the low-energy parts of the $\gamma$-ray spectra. An
identification of the low-energy enhancement in $(\gamma,\gamma^{\prime})$
experiments may be feasible by applying coincidence techniques.

Figure 9:  Photon strength functions for 76Ge from the present
$(\gamma,\gamma^{\prime})$ experiment (blue triangles and red circles), from
$\beta$ decay of 76Ga (black squares), taken from Ref. spy14 , and from a
$(\gamma,\gamma^{\prime})$ experiment at HI$\gamma$S (green triangles), taken
from Ref. ton17 .

##  V Summary

The dipole-strength distribution in 76Ge was studied up to the neutron-
separation energy in photon-scattering experiments at the $\gamma$ELBE
bremsstrahlung facility using two electron energies. A total of 210 levels was
identified. Spins $J$ = 1 were deduced from angular correlations of ground-
state transitions. A $\gamma$ transition in the region of interest for the
$0\nu\beta\beta$ decay has not been observed.

The intensity distribution obtained from the measured spectra was corrected
for the detector response and a simulated spectrum of photons scattered from
the target by atomic interactions was subtracted. The remaining spectrum
contains a continuum part in addition to the resolved peaks, which was
included in the determination of the photoabsorption cross section. An
assignment of inelastic transitions to particular levels and, thus, the
determination of branching ratios was, in general, not possible. Therefore, we
performed simulations of statistical $\gamma$-ray cascades to estimate
intensities of branching transitions. These were subtracted from the
experimental intensity distribution and the remaining intensities of ground-
state transitions were corrected on average for their branching ratios. In
this way, a continuous photoabsorption cross section was derived for the
energy range from about 5 MeV up to the neutron threshold at 9.4 MeV.

The absorption cross section of 76Ge displays an extra strength on top of the
tail of a Lorentz function for the GDR in the range between 6 and 9 MeV that
can be considered as the PDR. The shape of the PDR is relatively smooth and
approximated by cross sections calculated in the statistical model as given in
the TENDL library. The PDR is more pronounced and by a factor of about two
higher in magnitude than in the isotope 74Ge, but on the other hand by a
factor of about two smaller than in the isotone 78Se. The strength function of
76Ge resulting from the present work is comparable with the ones from other
experiments in the PDR region, but drops rapidly toward small energy.

##  VI Acknowledgments

We thank the operating crew of the ELBE accelerator for the cooperation. R. M.
acknowledges support by the U.S. Department of Energy, Office of Science,
Office of Nuclear Physics under grant No. LANLEM78.

## References

  * (1) K. Zuber, Neutrino Physics (CRC Press, Boca Raton, Florida, 2020).
  * (2) S. I. Alvis et al., Phys. Rev. C 100, 025501 (2019).
  * (3) M. Agostini et al., Science 365, 1445 (2019).
  * (4) M. Agostini et al., Phys. Rev. Lett. 120, 132503 (2018).
  * (5) N. Abgrall et al., AIP Conf. Proc. 1894, 020027 (2017).
  * (6) K. J. Kang et al., Front. Phys. 8, 412 (2013).
  * (7) G. Meierhofer et al., Eur. Phys. J. A 48, 20 (2012).
  * (8) D. C. Camp and B. P. Foster, Nucl. Phys. A 177, 401 (1971).
  * (9) A. Negret, C. Borcea, and A. J. M. Plompen, Phys. Rev. C 88, 027601 (2013).
  * (10) S. Mukhopadhyay, B. P. Crider, B. A. Brown, S. F. Ashley, A. Chakraborty, A. Kumar, M. T. McEllistrem, E. E. Peters, F. M. Prados-Estévez, and S. W. Yates, Phys. Rev. C 95, 014327 (2017).
  * (11) R. Schwengner et al., Nucl. Instrum. Methods Phys. Res., Sect. A 555, 211 (2005).
  * (12) S. Goriely et al., Eur. Phys. J. A 55, 172 (2019).
  * (13) T. Kawano et al., Nucl. Data Sheets 163, 109 (2020).
  * (14) W. Hauser and H. Feshbach, Phys. Rev. 87, 366 (1952).
  * (15) G. A. Bartholomew, E. D. Earle, A. J. Ferguson, J. W. Knowles, and M. A. Lone, in Advances in Nuclear Physics, edited by M. Baranger and E. Vogt, Vol. 7 (Springer, Boston, MA, 1973), pp. 229 - 324.
  * (16) D. Savran, T. Aumann, and A. Zilges, Prog. Part. Nucl. Phys. 70, 210 (2013).
  * (17) A. Bracco, E. G. Lanza, and A. Tamii, Prog. Part. Nucl. Phys. 106, 360 (2019).
  * (18) M. Beard, S. Frauendorf, B. Kämpfer, R. Schwengner, and M. Wiescher, Phys. Rev. C 85, 065808 (2012).
  * (19) N. Tsoneva, S. Goriely, H. Lenske, and R. Schwengner, Phys. Rev. C 91, 044318 (2015).
  * (20) M. Arnould, S. Goriely, and K. Takahashi, Phys. Rep. 450, 97 (2007).
  * (21) F. Käppeler, R. Gallino, S. Bisterzo, and W. Aoki, Rev. Mod. Phys. 83, 157 (2011).
  * (22) A. Jung et al., Nucl. Phys. A 584, 103 (1995).
  * (23) K. Sonnabend et al., Nucl. Instrum. Methods Phys. Res., Sect. A 640, 6 (2011).
  * (24) H. R. Weller, M. W. Ahmed, H. Gao, W. Tornow, Y. K. Wu, M. Gai, and R. Miskimen, Prog. Part. Nucl. Phys. 62, 257 (2009).
  * (25) V. Werner et al., EPJ Web of Conf. 93, 01031 (2015).
  * (26) N. M. Cooper, dissertation, Yale University 2015, unpublished.
  * (27) A. Spyrou et al., Phys. Rev. Lett. 113, 232502 (2014).
  * (28) A. P. Tonchev et al., EPJ Web of Conf. 146, 01013 (2017).
  * (29) F. Ajzenberg-Selove, Nucl. Phys. A506, 1 (1990).
  * (30) G. Rusev, A. P. Tonchev, R. Schwengner, C. Sun, W. Tornow, and Y. K. Wu, Phys. Rev. C 79, 047601 (2009).
  * (31) S. Agostinelli et al., Nucl. Instrum. Methods A 506, 250 (2003).
  * (32) J. Allison et al., IEEE Transactions on Nuclear Science 53, 270 (2006).
  * (33) J. Allison et al., Nucl. Instrum. Methods Phys. Research A 835, 186 (2016).
  * (34) G. Rusev, Dissertation, Technische Universität Dresden 2007, Report FZD-478 (ISSN 1437-322X), http://www.hzdr.de/publications/010008/10008.pdf.
  * (35) R. Schwengner et al., Phys. Rev. C 76, 034321 (2007).
  * (36) G. Rusev et al., Phys. Rev. C 77, 064321 (2008).
  * (37) R. Massarczyk et al., Phys. Rev. C 86, 014319 (2012).
  * (38) M. Marta et al., Phys. Rev. C 81, 055807 (2010).
  * (39) E. Trompler, Diploma thesis, Technische Universität Dresden 2009, Report FZD-523 (ISSN 1437-322X), http://www.hzdr.de/publications/013364/13364.pdf
  * (40) S. Carson, C. Iliadis, J. Cesaratto, A. Champagne, L. Downen, M. Ivanovic, J. Kelley, R. Longland, J. R. Newton, G. Rusev, and A. P. Tonchev, Nucl. Instrum. Method A 618, 190 (2010).
  * (41) E. Haug, Rad. Phys. Chem. 77, 207 (2008).
  * (42) G. Roche, C. Ducos, and J. Proriol, Phys. Rev. A 5, 2403 (1972).
  * (43) F. Salvat, J. D. Martinez, R. Mayol, and J. Parellada, Phys. Rev. A 36, 467 (1987).
  * (44) R. Schwengner et al., Phys. Rev. C 78, 064314 (2008).
  * (45) N. Benouaret et al., Phys. Rev. C 79, 014303 (2009).
  * (46) A. K. Furr, E. L. Robinson, and C. H. Robins, Nucl. Instrum. Methods 63, 205 (1968).
  * (47) G. Rusev et al., Phys. Rev. C 79, 061302(R) (2009).
  * (48) R. Schwengner et al., Phys. Rev. C 87, 024306 (2013).
  * (49) R. Massarczyk, G. Rusev, R. Schwengner, F. Dönau, C. Bhatia, M. E. Gooden, J. H. Kelley, A. P. Tonchev, and W. Tornow, Phys. Rev. C 90, 054310 (2014).
  * (50) A. Makinaga et al., Phys. Rev. C 90, 044301 (2014).
  * (51) R. Massarczyk et al., Phys. Rev. C 92, 044309 (2015).
  * (52) A. Makinaga et al., Phys. Rev. C 94, 044304 (2016).
  * (53) T. Shizuma et al., Phys. Rev. C 98, 064317 (2018).
  * (54) G. Schramm et al., Phys. Rev. C 85, 014311 (2012).
  * (55) F. Bečvář, Nucl. Instrum. Methods, Sect. A 417, 434 (1998).
  * (56) C. E. Porter and R. G. Thomas, Phys. Rev. 104, 483 (1956).
  * (57) A. Gilbert and A. G. W. Cameron, Can. J. Phys. 43, 1446(1965).
  * (58) T. von Egidy and D. Bucurescu, Phys. Rev. C 80, 054310 (2009).
  * (59) S. I. Al-Quraishi, S. M. Grimes, T. N. Massey, and D. A. Resler, Phys. Rev. C 67, 015803 (2003).
  * (60) A. R. Junghans, G. Rusev, R. Schwengner, A. Wagner, and E. Grosse, Phys. Lett. B 670, 200 (2008).
  * (61) E. Grosse, A. R. Junghans, and R. Massarczyk, Eur. Phys. J. A 53, 225 (2017).
  * (62) S. Raman, C. W. Nestor Jr., and P. Tikkanen, At. Data Nucl. Data Tables 78, 128 (2001).
  * (63) J.-P. Delaroche, M. Girod, J. Libert, H. Goutte, S. Hilaire, S. Péru, N. Pillet, and G. F. Bertsch, Phys. Rev. C 81, 014303 (2010).
  * (64) R. Capote et al., Nucl. Data Sheets 110, 3107 (2009).
  * (65) R. Massarczyk et al., Phys. Rev. Lett. 112, 072501 (2014).
  * (66) A. V. Voinov et al., Phys. Rev. C 99, 054609 (2019).
  * (67) P. Carlos, H. Beil, R. Bergere, J. Fagot, A. Lepretre, A. Veyssiere, and G. V. Solodukhov, Nucl. Phys. A 258, 365 (1976).
  * (68) A. J. Koning, D. Rochman, J. Sublet, N. Dzysiuk, M. Fleming, and S. van der Marck, Nucl. Data Sheets 155, 1 (2019).
  * (69) D. M. Brink, Ph. D. thesis, Oxford University, 1955, unpublished.
  * (70) P. Axel, Phys. Rev. 126, 671 (1962).
  * (71) A. Koning, private communication.
  * (72) A. P. Tonchev, private communication.
  * (73) M. Müscher et al., Phys. Rev. C 102, 014317 (2020).
  * (74) J. Wilhelmy et al., Phys. Rev. C 102, 044327 (2020).
  * (75) T. Renstrøm et al., Phys. Rev. C 93, 064302 (2016).
  * (76) R. Schwengner, S. Frauendorf, and A. C. Larsen, Phys. Rev. Lett. 111, 232504 (2013).
  * (77) B. Alex Brown and A. C. Larsen, Phys. Rev. Lett. 113, 252502 (2014).
  * (78) K. Sieja, Phys. Rev. Lett. 119, 052502 (2017).
  * (79) R. Schwengner, S. Frauendorf, and B. A. Brown, Phys. Rev. Lett. 118, 092502 (2017).
  * (80) A. Simon et al., Phys. Rev. C 93, 034303 (2016).
  * (81) F. Naqvi, A. Simon, M. Guttormsen, R. Schwengner, S. Frauendorf, C. S. Reingold, J. T. Burke, N. Cooper, R. O. Hughes, S. Ota, and A. Saastamoinen, Phys. Rev. C 99, 054331 (2019).
  * (82) R. Schwengner and G. Rusev, Phys. Rev. C 100, 054320 (2019).


# The Number of Singularities in the Intersections of Convex Planar Translates

Cameron (Braden) Strachan

(Math 518 Undergraduate Thesis Winter 2020)

###### Abstract

This purpose of this paper is to prove the following result: let $\phi$ be a
strictly convex, smooth, convex body in $\mathbb{E}^{2}$, if the intersection
of $n$ translates of $\phi$ has a non-empty interior, and all of the
translates contribute to the intersection, then the intersection of these $n$
translates will have exactly $n$ points of singularity along its boundary.
Furthermore this result is sharp, in the sense that, removing any one of the
assumptions from our statement will render the result unable to hold in
general.

###### Contents

  1. 1 Introduction
    1. 1.1 A Note on Notations
  2. 2 Basic Convexity
    1. 2.1 The Basics, Developing an Intuition

## Acknowledgments

I would like to acknowledge two people in particular, Professor Karoly Bezdek
and Ilya Ivanov, for giving me guidance throughout the research and production
of this paper. The ability to discuss ideas for a proof with more experienced
individuals is invaluable, not only for the proof or the paper’s sake, but
also in stimulating my own mathematical thinking and learning. I am grateful
to them for supporting me throughout this semester.

##  1 Introduction

In this paper I strive to guide the reader first through the basics of
convexity, and then incrementally advancing to the more difficult/complex
aspects of convexity. If the reader is already well adept in the field of
convex geometry, then skipping right to section 3 would be of no concern.
Through section 2 I try to not assume any prior knowledge of convexity, and
attempt to develop the theory naturally with an emphasis on examples. The most
important part of section 2 is by far subsection 2.4, which is about convex
bodies. As mentioned in the abstract, the main result of this paper focuses on
this class of sets. Section 2.4 also contains some basics regarding the Gauss
mapping of a convex body, something that is heavily used in proving the new
results.

Section 3 outlines in more detail the main result of this paper. Also
contained in this section is the justification of why the main result is
sharp. I give explicit counterexamples to each assumption we take away, and
then finish the section with an exposition on the limitations of this theorem
in higher dimensions, specifically in $\mathbb{E}^{3}$.

Section 4 is where I prove the main result of the theorem by induction on the
number of translates we are intersecting. Also in this section I develop what
I am calling “Chord theory” which focuses on chords, or line segments whose
end points belong to the boundary of a convex body. Various lemmas are proven
in this section to prove the main result; in particular, I consider Theorem
4.8 and Lemma 4.9 to be the most notable proven results besides the main
result.

###  1.1 A Note on Notations

Before concluding the introduction I want to introduce some of the notation I
will be using throughout this paper: $A\subseteq B$ will denote being a subset
or equal to, where as $A\subsetneq B$ will denote a proper subset, $A\setminus
B$ will denote set minus, and $int(A)$, $bd(A)$ will denote the interior and
boundary respectively, of the set $A$. One thing I will assume of the reader
is that they know the basics of analysis, such as knowing what an open set
compared to a closed set is, along with other basics as converging sequences,
compactness, and disjointedness. If the reader is unfamiliar with these terms,
any introductory analysis text will suffice in familiarizing the reader.

##  2 Basic Convexity

###  2.1 The Basics, Developing an Intuition

Convexity is a relatively new field in mathematics, while some contributions
to the field date back to Archimedes and Euclid. The subject really didn’t
take on its own identity until the turn of the 20th century, thanks to the
help of great mathematicians such as Hermann Brunn and Hermann Minkowski. They
studied convexity in small dimensions which guided the field to generalize
convexity to higher dimensions with the efforts of countless other
mathematicians.

I want to emphasize that we stand on top of the hill of convex geometry, built
up by thousands of hours of work done by our predecessors. Looking down,
hoping to give future mathematicians a higher view.

With this perspective, we start climbing the hill from the bottom, which leads
us to start with the definition of the space where convexity was first
developed.

######  Definition 2.1.

$\mathbb{E}^{n}$ will denote the $n$-dimensional Euclidean space, that is

$\mathbb{E}^{n}=\\{(x_{1},x_{2},\dots
x_{n})|~{}x_{1},x_{2},\dots,x_{n}\in\mathbb{R}\\}$

and we equip this space with the 2-norm

$\|(x_{1},x_{2},\dots,x_{n})\|=\sqrt{x_{1}^{2}+x_{2}^{2}+\dots+x_{n}^{2}}$

The Euclidean space is full of great mathematics to be learned, some of which
I’m sure the reader has encountered before. However, for the purposes of this
paper we restrict our view of this space to a special family of subsets of
$\mathbb{E}^{n}$ called convex sets.