# On the concept of _Algebraic Crystallography_

Dominique Bourn

###### Abstract

Category Theory provides us with a clear notion of what is an internal
structure. This will allow us to focus our attention on a certain type of
relationship between _context_ and _structure_.

##  1 Introduction

It is known that, in the unital and strongly unital categories [4] and in the
subtractive varieties [21] or categories [15] as well, on any object $X$ there
is at most one structure of abelian group object. But, in these contexts, this
did not seem so surprising because the four cases were closely related
(strongly unital=unital+subtractive [15]) and because of the kind of their
varietal origins: using an Eckmann-Hilton argument [13], this uniqueness
property arised naturally because, and when, some term in the definition of
the varietal examples in question became a homomorphism in this variety.

Similar situations for other algebraic structures were even known from a long
time; for instance, it was clear that in a pointed Jónnson-Tarski variety, on
any algebra $X$ there is at most one internal commutative monoid structure;
the same property holds for the commutative and associative (=autonomous)
Mal’tsev operations in the Mal’tsev varieties [20]. And again the limpid
varietal contexts supplied the same simple explanation for this phenomenon.

But recently we were led to observe that the uniqueness structure for abelian
group objects still holds in the new context of Congruence Hyperextensible
categories [8]. This, in restrospect, emphasized that the uniqueness of the
autonomous Mal’tsev operations was already noticed in the Congruence Modular
Varieties [14] as well.

This phenomenon of uniqueness of some kinds of algebraic structure being now
clearly extended to much larger contexts than the ones of the first paragraph,
and the explanation by the existence of some kinds of terms in the definition
of the varieties being no longer valid, it cannot remain possible to accept
this uniqueness so easily and to keep it as an unquestioned property. So, we
propose the following:

######  Definition 1.1.

A finitely complete category $\mathbb{E}$ is called crystallographic with
respect to a given algebraic structure $S$ when, on any object $X$ in
$\mathbb{E}$, there is at most one internal algebraic structure of this kind.

This terminology is chosen because, in such a category $\mathbb{E}$, the
algebraic structure $S$ in question is growing so punctually scarce. There are
two extremal cases:
i) when on any object $X$ in $\mathbb{E}$ there is one and only one
$S$-structure, we say that $\mathbb{E}$ is _intensively crystallographic_ with
respect to the structure $S$;
ii) when the only $S$-structures are the terminal object $1$ and its
subobjects, we say that $\mathbb{E}$ _trivializes_ the structure $S$.
Concerning the structure of group, any additive category is intensively
chrystallographic with respect to it, and any Mal’tsev variety whose Mal’tsev
term $p$ satisfies the Pixley axiom $p(x,y,x)=x$ [18] trivializes it.

Finally, when a structure $\Sigma$ is such that any category
$\Sigma(\mathbb{E})$ of internal $\Sigma$-structures in $\mathbb{E}$
trivializes the structure $S$, we say that the structure $\Sigma$ trivializes
the structure $S$.

The aim of this work is to produce examples and to establish the very first
properties and general questionings about this notion of _Algebraic
Crystallography_. It will lead to an unexpected and rather spectacular outcome
with an example of a variety $\mathbb{H}$ which is crystallographic for the
structure of abelian group and in which the punctual sarcity of abelian group
objects seems to be offset by a kind of multiplication of this structure
inside it. Indeed, the category $\mathsf{Ab}\mathbb{H}$ of abelian objects in
this variety $\mathbb{H}$ will appear:
i) to fully faithfully embed the category $\mathsf{Ab}$ of abelian groups by a
functor $h:\mathsf{Ab}\rightarrow\mathsf{Ab}\mathbb{H}$
and in an independant way
ii) to faithfully contain any category $K$-$\mathsf{Vect}$ of $K$-vector
spaces, provided that the field $K$ is not of characteristic $2$, by a functor
$w_{K}:K$-$\mathsf{Vect}\rightarrow\mathsf{Ab}\mathbb{H}$.
So, the abelian group $(V,+)$ underlying a $K$-vector space $V$ will be
represented by two distinct objects $h(V,+)$ and $w_{K}(V,+)$ in the abelian
category $\mathsf{Ab}\mathbb{H}$.
We give an example of the same kind of multiplication process in a non-pointed
context.

##  2 First observations from the unital setting

In this article any category will be suppose finitely complete. The kernel
equivalence relation of a map $f:X\to Y$ is denoted by $R[f]$. Given any
algebraic structure $S$ and any category $\mathbb{E}$, we denote by
$S(\mathbb{E})$ the category of internal $S$-structures in $\mathbb{E}$ and by
$U^{S}_{\mathbb{E}}:S(\mathbb{E})\to\mathbb{E}$ the canonical forgetful
functor. The category $S(\mathbb{E})$ is finitely complete as soon as so is
$\mathbb{E}$, and its terminal object is the unique possible $S$-structure on
the terminal object $1$ of $\mathbb{E}$. The functor $U^{S}_{\mathbb{E}}$ is
clearly left exact and it reflects isomorphims.

Suppose now that the $S$-structure in question has no constant. Since, given
any subobject $J\rightarrowtail 1$ of $1$ in $\mathbb{E}$, we get $J^{n}\simeq
J$ for any $1\leq n$ in a unique possible way, there is one and only one
$S$-structure on this object $J$. Whence a canonical fully faithful embedding
$J_{\mathbb{E}}:Sub_{1}\mathbb{E}\rightarrowtail S(\mathbb{E})$ from the fully
faithful subcategory $Sub_{1}\mathbb{E}$ of the subobjects of $1$ in
$\mathbb{E}$ such that
$U^{S}_{\mathbb{E}}.J_{\mathbb{E}}:Sub_{1}\mathbb{E}\to\mathbb{E}$ is nothing
but the inclusion $Sub_{1}\mathbb{E}\rightarrowtail\mathbb{E}$.

###  2.1 General unital setting

A _unital_ category [4] is a pointed category $\mathbb{E}$ such that the
canonical pair of injections:

$X\stackrel{{\scriptstyle l_{X}}}{{\rightarrowtail}}X\times
Y\stackrel{{\scriptstyle
r_{Y}}}{{\leftarrowtail}}Y\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(1)$

is jointly strongly epic, i.e. such that the only (up to isomorphism)
subobject of $X\times Y$ containing $l_{X}$ and $r_{Y}$ is $1_{X\times Y}$.

A pointed variety $\mathbb{V}$ is unital if and only if it is a Jónnson-Tarski
variety [2], i.e. if it has a unique constant $0$ and a binary term $+$
satisfying $x+0=x=0+x$. So, the main examples of unital category are the
categories $\mathsf{Mag}$ and $\mathsf{Mon}$ of unitary magmas and monoids.
The unital setting appears to be the right one in which there is an _intrinsic
notion of commutative pairs of subobjects_ : given any pair
$u:U\rightarrowtail X$, $v:V\rightarrowtail X$ of subobjects, there is at most
one factorization $\varphi$:

---
$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{l_{U}}$$\scriptstyle{u}$$\textstyle{U\times
V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\textstyle{X}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r_{V}}$$\scriptstyle{v}$

making the previous diagram commute. So, when such a map does exist, _the
subobjects $u$ and $v$ are said to commute_ and the map $\varphi$ is called
the cooperator of the pair. This situation is denoted by $[u,v]=0$. We can
then follow the usual variations: a subobject $u:U\rightarrowtail X$ is
central when $[u,1_{X}]=0$; an object $X$ is commutative when
$[1_{X},1_{X}]=0$.

By its cooperator $\varphi:X\times X\to X$, any commutative object $X$ is
endowed with a structure of internal unitary magma _which turns out to be an
internal commutative monoid_. Moreover, any morphism $f:X\to Y$ between two
commutative objects preserves these monoid structures. When this commutative
monoid is an internal abelian group, the object $X$ is said to be _abelian_.
Whence, in this context, the two natural intrinsic fully faithful
subcategories:
$\mathsf{Ab}(\mathbb{E})\hookrightarrow\mathsf{CoM}(\mathbb{E})\hookrightarrow\mathbb{E}$
of the commutative and abelian objects.

A collateral aspect of these observations was noticed: _in a unital setting,
on an object $X$, there is atmost one structure of internal unitary magma_,
but it was not thorough. We can now assert:

######  Proposition 2.1.

A unital category $\mathbb{E}$ is crystallographic with respect to the
structure of unitary magma and, a fortiori, to the structure of commutative
monoid and abelian group.

###  2.2 Special unital settings

There are two extremal cases of unital categories: [4]: i) the only
commutative object is the terminal object $1$; ii) any object $X$ is
commutative.

The second case holds when the diagram $(1)$ is actually the diagram of the
sum of $X$ and $Y$; the pointed category $\mathbb{E}$ is then called _linear_.
The first case holds when we have: $[u,v]=0\implies u\cap
v=0_{X}:1\rightarrowtail X$; the pointed category $\mathbb{E}$ is then said
_stiffly unital_. These two extremal situations give rise to two extremal
cases of crystallography:

######  Definition 2.1.

A category $\mathbb{E}$ trivializes (resp. strongly trivializes) a structure
$S$ when the subobjects $J$ of $1$ are the only ones to have a $S$-structure
(resp. the terminal object $1$ is the only one to have a $S$-structure).

A category $\mathbb{E}$ is intensively chrystallographic with respect to a
structure $S$, when, on any object $X$, there is one and only one internal
structure $S$.

Accordingly, the category $\mathbb{E}$ (resp. strongly) trivializes the
structure $S$ if and only if the canonical fully faithful embedding
$J_{\mathbb{E}}:Sub_{1}\mathbb{E}\rightarrowtail S(\mathbb{E})$ is an
equivalence of categories, where $Sub_{1}\mathbb{E}$ is the fully faithful
subcategory of the subobjects of $1$ in $\mathbb{E}$ (resp. if and only if the
canonical embedding $1\rightarrowtail S(\mathbb{E})$ is an equivalence of
categories where $1$ is the discrete category with only one object). And the
category $\mathbb{E}$ is intensively chrystallographic with respect to the
structure $S$ if and only if the forgetful functor
$U^{S}_{\mathbb{E}}:S(\mathbb{E})\to\mathbb{E}$ is an equivalence of
categories.

Any linear category is intensively chrystallogaphic with respect to the
structure of monoid. The category $\mathsf{BooRg}$ of boolean rings and the
category $\mathsf{HsLat}$ of Heyting semi-lattices strongly trivialize the
structure of monoid, see [2]; this is the case as well for the dual
$(\mathsf{Set}_{*})^{op}$ of the category of pointed sets and more generally
for the dual $(\mathbb{E}_{*})^{op}$ of the category of pointed objects in
$\mathbb{E}$ when it is topos, see [4]. Finally we come to the following:

######  Definition 2.2.

An algebraic structure $\Sigma$ (resp. strongly) trivializes another algebraic
structure $S$ when, for any category $\mathbb{E}$, the category
$\Sigma(\mathbb{E})$ of internal $\Sigma$-structures in $\mathbb{E}$ (resp.
strongly) trivializes the structure $S$.

An algebraic structure $S$ is self-chrystallographic when, for any category
$\mathbb{E}$, the category $S(\mathbb{E})$ is intensively crystallographic
with respect to the structure $S$ itself.

So, the algebraic structure $\Sigma$ trivializes the structure $S$ if and only
if the canonical fully faithful embedding
$Sub_{1}\mathbb{E}\stackrel{{\scriptstyle\simeq}}{{\to}}Sub_{1}\Sigma(\mathbb{E})\to
S(\Sigma(\mathbb{E}))$ is an equivalence of categories. It strongly
trivializes it if and only if the canonical embedding $1\rightarrowtail
S(\Sigma(\mathbb{E}))$ is an equivalence of categories. Since
$S(\Sigma(\mathbb{E}))=\Sigma(S(\mathbb{E}))$, the _trivialization_ is clearly
a symmetric relation between algebraic structures. On the other hand, an
algebraic structure $S$ is self-chrystallographic if and only if the forgetful
functor $U^{S}_{S(\mathbb{E})}:S(S(\mathbb{E}))\to S(E)$ is an equivalence of
categories. The most basic examples are the following ones:
i) the structure of commutative monoid is self-chrystallographic;
ii) and the structure of abelian group as well;
iii) the structure of group and idempotent unitary magma strongly trivialize
each other.

###### Proof.

The two first points are straightforward. Let $(G,\circ,1)$ be a group and $*$
the binary law underlying the internal unitary magma structure in
$\mathsf{Gp}$. By Eckmann-Hilton, we get $\circ=*$. From $x^{2}=x$, we get
$x=1$, since this equality now holds inside a group. ∎

######  Corollary 2.1.

The fundamental group of a topological idempotent unitary magma is necessarily
trivial. More generally, when a structure $S$ is trivialized by the group
structure, any topological $S$-structure produces a trivial fundamental group.

###  2.3 Subtractive categories

The concept of subtractive category, introduced in [15], is the categorical
characterization of the pointed subtractive varieties in the sense of [21]. A
pointed category is subtractive when, given any reflexive relation $R$ on an
object $X$, if $(0_{X},1_{X}):X\rightarrowtail X\times X$ factorizes through
$R$, then so does $(1_{X},0_{X}):X\rightarrowtail X\times X$. It was shown in
[10] that on any object $X$ there is at most one structure of subtraction
[$s(x,x)=0+s(x,0)=0$] and abelian group; so, the induced inclusion functors
$\mathsf{Ab}\mathbb{E}\rightarrowtail\mathsf{Subt}\mathbb{E}\rightarrowtail\mathbb{E}$
are fully faithfull, and we get:

######  Corollary 2.2.

Any subtractive category is crystallographic with respect to the structure of
subtraction and abelian group.

##  3 Further examples from the Mal’tsev setting

###  3.1 General Mal’sev setting

A Mal’tsev structure is a set $X$ endowed with a ternary operation $p:X\times
X\times X\to X$ satisfying the Mal’tsev identities $p(x,y,y)=x=p(y,y,x)$. We
denote by $\mathsf{Mal}$ the category of Mal’tsev structures. An affine
structure is a Mal’tsev structure such that $p$ is associative
($p(x,y,p(z,u,v))=p(p(x,y,z),u,v)$) and commutative ($p(x,y,z)=p(z,y,x)$). We
denote by $\mathsf{Aff}$ the category of affine structures.

A category $\mathbb{E}$ is a Mal’tsev one when any internal reflexive relation
is actually an equivalence relation, [11] and [12]. Clearly any category
$\mathsf{Mal}\mathbb{E}$ of internal Mal’tsev structures in $\mathbb{E}$ is a
Mal’tsev one. Let us recall the following characterizations:

######  Theorem 3.1.

The following conditions are equivalent:
1) $\mathbb{E}$ is a Mal’tsev category;
2) any sub-reflexif graph of an internal groupoid in $\mathbb{E}$ is an
internal groupoid;
3) any fiber $Pt_{Y}{\mathbb{E}}$ of the fibration of points is unital;
4) any fiber $Pt_{Y}{\mathbb{E}}$ of the fibration of points is subtractive.

The point 3) is grounded on the following definition: we denote by
$Pt\mathbb{E}$ the category whose objects are the split epimorphisms in
$\mathbb{E}$ with a given splitting and morphisms the commutative squares
between these data; by $\P_{\mathbb{E}}:Pt\mathbb{E}\rightarrow\mathbb{E}$ we
denote the functor associating its codomain with any split epimorphism. It is
left exact and a fibration whose cartesian maps are the pullbacks of split
epimorphisms; it is called the _fibration of points_. The fiber above $Y$ is
denoted by $Pt_{Y}\mathbb{E}$. The points 2) and 3) come from [3] and the
point 4) from [15]. The following observations are important for our purpose:

######  Theorem 3.2.

Let $\mathbb{E}$ be any Mal’tsev category. Then:
1) on any object $X$ there is at most one internal Mal’tsev structure which is
necessarily an affine one;
2) on any internal reflexive graph in $\mathbb{E}$, there is at most one
internal category structure, which is necessarily a groupoid one;
3) any internal groupoid is necessarily an affine one.

The two first points come from [12]. A groupoid is affine when the associative
Mal’tsev operation defined on the parallel arrows by
$p(\phi,\chi,\psi)=\phi.\chi^{-1}.\psi$ is moreover commutative, and
consequently produces an affine structure. The third point comes from [5]
where the _affine groupoids_ were introduced under the name of _abelian
groupoids_. From the point 1), we get immediately:

######  Proposition 3.1.

Given any Mal’tsev category $\mathbb{E}$,
1) it is crystallographic with respect to the Mal’tsev structure and, a
fortiori, with respect to the affine structure
2) any fiber $Pt_{Y}\mathbb{E}$ is crystallographic with respect to the
(abelian) group structure.

###  3.2 Special Mal’sev settings

Similarly to what happens in the unital setting and from the characterization
Theorem 3.1, there are two extremal cases of Mat’sev categories: when i) any
fiber $Pt_{Y}\mathbb{E}$ is stiffly unital; and when ii) any fiber
$Pt_{Y}\mathbb{E}$ is linear.

In the first case, the category $\mathbb{E}$ is said to be a _stiffly
Mal’tsev_ one, see [7] from which we shall recall the three first points of
the following characterization:

######  Theorem 3.3.

Let $\mathbb{E}$ be any Mal’tsev category. The following conditions are
equivalent:
1) the category $\mathbb{E}$ is a stiffly Mal’tsev one;
2) the only abelian equivalence relations are the discrete ones;
3) any fiber $Pt_{X}\mathbb{E}$ trivializes the group structure;
4) the only internal groupoids are the equivalence relations.

The equivalence between 3) and 4) is shown below in Proposition 5.6.
Accordingly:

######  Corollary 3.1.

A stiffly Mal’tsev category $\mathbb{E}$ trivializes the Mal’tsev structure. A
stiffly Mal’tsev category $\mathbb{E}$ strongly trivializes the group
structure.

###### Proof.

In a Mal’tsev category, an object $X$ is affine if and only if the terminal
map $\tau_{X}:X\to 1$ has an abelian kernel equivalence relation. So, our
first assertion is a consequence of the point 2) of the previous
characterization theorem which makes $\tau_{X}$ a monomorphism. An internal
group object is always abelian in a Mal’tsev context; so, it is nothing but an
affine object $X$ with a global element $e:1\rightarrowtail X$. The second
assertion is then straightforward. ∎

As for the second extremal case, let us recall the following:

######  Theorem 3.4.

The following conditions are equivalent:
1) the category $\mathbb{E}$ is a naturally Mal’tsev category;
2) on any reflexif graph in $\mathbb{E}$ there is a unique structure of
internal groupoid;
3) any fiber $Pt_{Y}{\mathbb{E}}$ is linear;
4) any fiber $Pt_{Y}{\mathbb{E}}$ is additive.

The equivalence between 1) and 2) is the ground result of [16] where a
naturally Mal’tsev category is defined as a category in which any object $X$
is endowed with a natural Mal’tsev structure. So, a naturally Mal’tsev
category is a Mal’tsev one, and the natural Mal’tsev structure in question is
then necessarily an affine one. Clearly the category $\mathsf{Aff}$ of affine
structures and any category $\mathsf{Aff}\mathbb{E}$ are naturally Mal’tsev
ones. The equivalence between 1), 3) and 4) is in [4]. From that, we get
immediately:

######  Proposition 3.2.

1) Any naturally Mal’tsev category $\mathbb{E}$ is intensively
crystallographic with respect to the Mal’tsev structure.
2) The affine structure is self-crystallographic.

##  4 Fiberwise extension of the notion of algebraic crystallography

From 2) in Theorem 3.2 and in Theorem 3.4, we shall enlarge the set of
examples of the crystallographic settings. For that, we shall think to the
composition of arrows in a category or a groupoid as an algebraic structure on
its underlying reflexive graph.

So, given any category $\mathbb{E}$, let us denote by $RGh\mathbb{E}$ the
category of internal reflexif graphs in $\mathbb{E}$:

$\textstyle{X_{\bullet}:\;\;\;\;\;X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{1}^{X_{\bullet}}}$$\scriptstyle{d_{0}^{X_{\bullet}}}$$\textstyle{X_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s_{0}^{X_{\bullet}}}$

which is finitely complete and by $U_{0}:RGh\mathbb{E}\to\mathbb{E}$ the left
exact forgetful functor; it is a fibration whose cartesian map above
$f_{0}:X_{0}\to Y_{0}$ is given by the inverse image along $f_{0}$. We denote
by $RGh_{X}\mathbb{E}$ the fibre above $X$; it has an initial object given by
the discrete equivalence relation $\Delta_{X}$ on $X$ and a terminal one given
by the indiscrete equivalence relation $\nabla_{X}$. Similarly we shall denote
by $Cat\mathbb{E}$ and $Grd\mathbb{E}$ the categories of internal categories
and groupoids whose objects are the 3-truncated simplicial objects in
$\mathbb{E}$ (including all the natural degeneracy maps which do not appear in
the following diagram):

$\textstyle{X_{\bullet}:\;\;\;\;\;X_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{4}^{X_{\bullet}}}$$\scriptstyle{d_{3}^{X_{\bullet}}}$$\scriptstyle{d_{1}^{X_{\bullet}}}$$\scriptstyle{d_{0}^{X_{\bullet}}}$$\textstyle{X_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{2}^{X_{\bullet}}}$$\scriptstyle{d_{1}^{X_{\bullet}}}$$\scriptstyle{d_{0}^{X_{\bullet}}}$$\textstyle{X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{1}^{X_{\bullet}}}$$\scriptstyle{d_{0}^{X_{\bullet}}}$$\textstyle{X_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s_{0}^{X_{\bullet}}}$

where, in the first case, $X_{2}$ and $X_{3}$ are respectively obtained by the
pullback of $d_{0}$ along $d_{1}$ and the pullback of $d_{0}$ along $d_{2}$
and where, in the second one, any commutative square is a pullback. The same
inverse image process as above makes fibrations the following forgetful
functors:

|
---|---
$\textstyle{Grd\mathbb{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{U_{0}}$$\textstyle{Cat\mathbb{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{U_{0}}$$\textstyle{RGh\mathbb{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{U_{0}}$$\textstyle{\mathbb{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{E}}$

whose fibers, denoted respectively $Grd_{X}\mathbb{E}$ and
$Cat_{X}\mathbb{E}$, have the same $\Delta_{X}$ and $\nabla_{X}$ as initial
and terminal objects. Now the points 2) of Theorems 3.2, 3.3 and 3.4 give rise
to the following:

######  Corollary 4.1.

Let $\mathbb{E}$ be a Mal’tsev category. Then any fiber $RGh_{X}\mathbb{E}$:
1) is crystallographic with respect to the structure of affine groupoid;
2) trivializes the structure of groupoid if and only if $\mathbb{E}$ is a
stiffly Mal’tsev category;
3) is intensively crystallographic with respect to the structure of affine
groupoid if and only if $\mathbb{E}$ is a naturally Mal’tsev category.

###### Proof.

Only the point 2) demands some precisions: in a Mal’tsev category the
subobjects of the terminal object $\nabla_{X}$ in the fiber
$RGh_{X}\mathbb{E}$ (namely the reflexive relations) are the equivalence
relations. ∎

##  5 Congruence hyperextensible and Gumm categories

###  5.1 Congruence modular varieties and Gumm categories

A congruence modular variety is a variety in which the modular formula holds
for congruences:

$(T\vee S)\wedge R=T\vee(S\wedge
R),\rm{\;for\;any\;triple\;}(T,S,R)\;\rm{\;such\;that:\;}T\subset R$

In [14], they were characterized in geometric terms by the validity of the
_Shifting Lemma_ : given any triple of equivalence relations $(T,S,R)$ such
that $R\cap S\subset T$, the following left hand side situation implies the
right hand side one:

$\textstyle{x\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{S}$$\scriptstyle{T}$$\scriptstyle{R}$$\textstyle{y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{R}$$\textstyle{y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{T}$$\textstyle{x^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{S}$$\textstyle{y^{\prime}}$$\textstyle{y^{\prime}}$

One of the main interest of the Shifting lemma is that it is freed of any
condition involving finite colimits. So, thanks to the Yoneda embedding, it
keeps a meaning in any finitely complete category $\mathbb{E}$. This led to
the notion of Gumm category [9]. Any regular Mal’tsev category is a Gumm one.
Let us recall, from the same article, the following:

######  Proposition 5.1.

Let $\mathbb{E}$ be any Gumm category. Then:
1) on any object $X$ there is at most one affine structure;
2) on any reflexive graph, there is at most one structure of category.

The first point was already noticed by Gumm in the congruence modular
varieties. On the contrary to what happens in Mal’tsev ones, in Gumm
categories there are internal categories which are not groupoids. Now we get:

######  Corollary 5.1.

Let $\mathbb{E}$ be any Gumm category:
1) it is chrytallographic with respect to the affine structure;
2) any fiber $RGh_{X}\mathbb{E}$ is chrytallographic with respect to the
structure of category.

###  5.2 Conruence hyperextensible categories

On the model of Mal’tsev ones, Gumm categories have a characterization through
the fibration of points [6]:

######  Theorem 5.1.

Given a category $\mathbb{E}$, the following conditions are equivalent:
1) $\mathbb{E}$ is Gumm category;
2) any fiber $Pt_{Y}\mathbb{E}$ is congruence hyperextensible.

######  Definition 5.1.

A pointed category $\mathbb{E}$ is said to be congruence hyperextensible when
given any punctual span in $\mathbb{E}$ (= any commutative diagram of split
epimorphisms):

$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{g}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau_{Y}}$$\scriptstyle{t}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s}$$\scriptstyle{\tau_{X}}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{0_{Y}}$$\scriptstyle{0_{X}}$

and any equivalence relation $T$ on $W$ with $R[f]\cap R[g]\subset T$, we get:
$R[f]\cap g^{-1}(t^{-1}(T))\subset T$.

This means that the following situation holds in $W$:

$\textstyle{tg(x)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{R[g]}$$\scriptstyle{T}$$\scriptstyle{R[f]}$$\textstyle{x\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{R[f]}$$\textstyle{x\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{T}$$\textstyle{tg(x^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{R[g]}$$\textstyle{x^{\prime}}$$\textstyle{x^{\prime}}$

Any Jónsson-Tarski variety $\mathbb{V}$ or regular unital category
$\mathbb{E}$ is congruence hyperextensible. It is a fortiori the case for any
regular pointed protomodular category [2], and thus for any Slomińsky variety
[19]. Here are some examples of varieties which are outside of these classes:

######  Proposition 5.2.

Let $\mathbb{V}$ be a pointed variety with $2k+1$ ternary terms $p_{i}$ such
that:

$\displaystyle p_{1}(a,0,0)=a\;\;,$
$\displaystyle\;\;\;p_{i}(a,0,a)=a\;\;\;2\leq i\leq 2k\;\;,$ $\displaystyle
p_{2k+1}(0,0,a)=a$ $\displaystyle p_{2i-1}(a,a,b)=p_{2i}(a,a,b)$
$\displaystyle\;\;\;\;1\leq i\leq k$ $\displaystyle
p_{2i}(a,b,b)=p_{2i+1}(a,b,b)$ $\displaystyle\;\;\;\;1\leq i\leq k$
$\displaystyle\;\;\;\;\;\;\;p_{i}(b,b,b)=b,\;$ $\displaystyle\;1\leq i\leq
2k+1$

Then the variety $\mathbb{V}$ is congruence hyperextensible but not a Jónsson-
Tarski one.

###### Proof.

Starting with a punctual relation in $\mathbb{V}$ as above and with an
equivalence relation $T$ on $W$ such that $(0Wb)T(0Wc)$. First let us show
that:
$p_{2}(aWb,0Wc,aWc)Tp_{2k}(aWb,0Wc,aWc)$. For all $i$ with $1\leq i\leq k$, we
get:
$p_{2i}(aWb,0Wc,aWc)=p_{2i}(a,0,a)Wp_{2i}(b,c,c)=p_{2i+1}(a,0,a)Wp_{2i+1}(b,c,c)=$
$p_{2i+1}(aWb,0Wc,aWc)$. So: $p_{2i}(aWb,0Wc,aWc)Tp_{2i+1}(aWb,0Wb,aWc)$.
Then: $p_{2i+1}(aWb,0Wb,aWc)=p_{2i+1}(a,0,a)Wp_{2i+1}(b,b,c)$
$=p_{2i+2}(a,0,a)Wp_{2i+2}(b,b,c)=p_{2i+2}(aWb,0Wb,aWc)$, and finally:
$p_{2i+2}(aWb,0Wb,aWc)Tp_{2i+2}(aWb,0Wc,aWc)$.

We get also: $aWb=p_{1}(a,0,0)Wp_{1}(b,b,b)=p_{1}(aWb,0Wb,0Wb)$. Whence:
$(aWb)Tp_{1}(aWb,0Wb,0Wc)$ $=(aWb)T(aWp_{1}(b,b,c))=(aWb)T(aWp_{2}(b,c,c))$
$=(aWb)T(p_{2}(a,0,a)Wp_{2}(b,c,c))=(aWb)Tp_{2}(aWb,0Wc,aWc)$.

Finally: $p_{2k}(aWb,0Wc,aWc)=p_{2k}(a,0,a)Wp_{2k}(b,c,c)$
$=p_{2k+1}(0,0,a)Wp_{2k+1}(b,c,c)=p_{2k+1}(0Wb,0Wc,aWc)$. Whence finally:
$p_{2k}(aWb,0Wc,aWc)Tp_{2k+1}(0Wc,0Wc,aWc)=p_{2k}(aWb,0Wc,aWc)T(aWc)$.

The last assertion is proved when $n=3$ in [8]. ∎

We shall say that a congruence hyperextensible variety of the previous kind is
a CHyper variety of type $2k+1$. _Any CHyper variety of type $2k+1$ is of type
$2k+3$_:

###### Proof.

Let $\mathbb{V}$ be a CHyper variety of type $2k+1$. Let us set:

$\bar{p}_{i}=p_{i},\;\forall i\;1\leq i\leq
2j;\;\bar{p}_{2j+1}=p_{2j}=\bar{p}_{2j+2};\;\bar{p}_{i}=p_{i-2},\;\;\forall
i\;\;2j+3\leq i\leq 2k+1$

These $\bar{p}_{i}$ make the variety $\mathbb{V}$ a CHyper variety of type
$2k+3$. ∎

Recall now the main observation from [8]:

######  Proposition 5.3.

Let $\mathbb{E}$ be a congruence hyperextensible category. Any subtraction $s$
on an object $X$ is the difference mapping associated with an internal group
structure which is necessarily abelian. On any object $X$, there is at most
one subtraction $s$; any morphism $f:X\to Y$ between objects with subtraction
is a subtraction homomorphism.

Whence the following result which, as explained in the introduction, led to
the notion:

######  Corollary 5.2.

Any congruence hyperextensible category is crystallographic with respect to
structure of subtraction and (abelian) group. Given a Gumm category
$\mathbb{E}$, any fiber $Pt_{Y}\mathbb{E}$ is crystallographic with respect to
the (abelian) group structure.

###  5.3 Congruence distributive varieties and categories

It is well known that any congruence distributive variety in which the
following congruence formula holds:

$(T\wedge R)\vee(T\wedge S)=T\wedge(R\vee S)$

is congruence modular. Let us recall that a Mal’tsev variety $\mathbb{V}$ is
congruence distributive if and only if its Mal’tsev term $p$ satisfies the
Pixley axiom $p(x,y,x)=x$ [18]. This is the case in particular for the
category $\mathsf{Heyt}$ of Heyting algebras. In this section, we shall show
that congruence distributive varieties and categories trivialize the group
structure.

######  Definition 5.2.

A category $\mathbb{E}$ is said to be weakly congruence distributive when the
following implication holds:

$(T\wedge R=\Delta_{X}\;{\;\rm and\;}\;T\wedge S=\Delta_{X})\Rightarrow
T\wedge(R\vee S)=\Delta_{X}$

whenever the supremum $R\vee S$ of the pair $(R,S)$ of equivalence relations
does exist.

######  Proposition 5.4.

Given any weakly congruence distributive category $\mathbb{E}$, any fiber
$Pt_{X}\mathbb{E}$ trivializes the group structure; the category $\mathbb{E}$
trivializes the associative Mal’tsev structure.

###### Proof.

In any category $\mathbb{E}$, given any pair $(X,Y)$ of objects, we get:
$R[p_{X}]\vee R[p_{Y}]=\nabla_{X\times Y}$, since we get it in $Set$.
1) Let us begin by showing that $\mathbb{E}$ trivialize the group structure.
So, let us denote by $m:X\times X\to X$ the binary operation associated with
an internal monoid structure. It is a group structure if and only if the
following square is a pullback:

$\textstyle{X\times
X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{0}}$$\scriptstyle{m}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(1_{X},()^{-1})}$$\scriptstyle{\tau_{X}}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e}$

Then the unit $e$ produces the left hand side section. Now, observe that both
$(m,p_{0})$ and $(m,p_{1})$ are monomorphisms; this means that

$R[m]\wedge R[p_{0}]=\Delta_{X\times X}\;{\;\rm and\;}\;R[m]\wedge
R[p_{1}]=\Delta_{X\times X}$

Whence: $\Delta_{X\times X}=R[m]\wedge(R[p_{0}]\vee
R[p_{1}])=R[m]\vee\nabla_{X\times X}=R[m]$. Accordingly the morphism $m$ is a
monomorphism; being split (by $e\times 1_{X}$), it is an isomorphism. Coming
back to the pullback above, the upward square is a pullback, and $\tau_{X}$ is
an isomorphism as well. Accordingly, $X$ is isomorphic to the terminal object.
2) We can then reproduce this proof in any slice category $\mathbb{E}/X$ or
equivalently in any fiber $Pt_{X}\mathbb{E}$, since, similarly to 1), given
any pullback in $\mathbb{E}$:

$\textstyle{X\times_{Z}Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{X}}$$\scriptstyle{\gamma}$$\scriptstyle{p_{Y}}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{Z}$

we get $R[p_{X}]\vee R[p_{Y}]=R[\gamma]$ in $\mathbb{E}$: if $S$ is any
equivalence relation on $X\times_{Z}Y$ in $\mathbb{E}$ such that
$R[p_{X}]\subset S$ and $R[p_{Y}]\subset S$, then the same inclusions hold for
$S\wedge R[\gamma]$ which now lies in $\mathbb{E}/Y$. So, we get
$R[\gamma]=S\wedge R[\gamma]$ in the category $\mathbb{E}/Y$, whence
$R[\gamma]\subset S$ in $\mathbb{E}$.
3) Now let $p:X\times X\times X\to X$ be any internal associative Mal’tsev
structure. We then get an internal group structure on the object
$(p_{0}^{X},s_{0}^{X}):X\times X\rightleftarrows X$ of the fiber
$Pt_{X}\mathbb{E}$, by setting $(x,y)*(x,z)=(x,p(y,x,z))$. So, $p_{0}^{X}$ is
an isomorphism; accordingly $\tau_{X}:X\to 1$ is a monomorphism and $X$ a
subobject of $1$. ∎

It is a fortiori the case for any congruence distributive variety or category.
In the Mal’tsev context, we have the converse with the following precisions:

######  Proposition 5.5.

Given any regular Mal’tsev category $\mathbb{E}$, the following conditions are
equivalent:
1) $\mathbb{E}$ is weakly congruence distributive;
2) any fiber $Pt_{X}\mathbb{E}$ trivializes the group structure;
3) the category $\mathbb{E}$ is a stiffly Mal’tsev one.
Given any exact Mal’tsev category $\mathbb{E}$, the following conditions are
equivalent:
1) $\mathbb{E}$ is congruence distributive;
2) any fiber $Pt_{X}\mathbb{E}$ trivializes the group structure;
3) the category $\mathbb{E}$ is a stiffly Mal’tsev one.

###### Proof.

The first equivalence comes from Lemma 2.9.10 and Proposition 1.11.34 in [2],
while the second one is Theorem 2.9.11. ∎

Accordingly, a Mal’tsev variety is a stiffly Mal’tsev category if and only if
it satisfies the Pixley axiom. Finally let us observe that more generally we
get the following:

######  Proposition 5.6.

Given any category $\mathbb{E}$, the following conditions are equivalent:
1) any fiber $Pt_{X}\mathbb{E}$ trivializes the group structure;
2) any internal groupoid is an equivalence relation.
Such a category $\mathbb{E}$ trivializes the associative Mal’tsev structure.

###### Proof.

We shall work in the fiber $Grd_{X}\mathbb{E}$ which is a protomodular
category [2]; accordingly, in this fiber, pullbacks reflect monomorphisms. Let
$X_{\bullet}$ be any groupoid in this fiber and consider the following
pullback in $Grd_{X}\mathbb{E}$:

$\textstyle{EndX_{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Delta_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\nabla_{X}}$

where $EndX_{\bullet}$ is the sub-groupoid of the endomorphisms of
$X_{\bullet}$. It coincides with a group structure in the fiber
$Pt_{X}\mathbb{E}$. So, when 1) holds, the left hand side vertical map is an
isomorphism, and consequently the right hand side one is a monomorphism, and
thus $X_{\bullet}$ is an equivalence relation. Conversely any group structure
in the fiber $Pt_{X}\mathbb{E}$ produces a groupoid $X_{\bullet}$ which
coincides with its endsome $EndX_{\bullet}$; in other words the upper
horizontal arrow in the previous diagram is an isomorphism. So, if we suppose
2), namely that the right hand side vertical map is a monomorphism, so is the
left hand side one. Being split, this same map is an isomorphism, and the
group in question is trivial in $Pt_{X}\mathbb{E}$. The last assertion is
checked as in the previous proposition. ∎

In this context, we then get another possible level of trivialization:

######  Definition 5.3.

A category $\mathbb{E}$ weakly trivializes a structure $S$ when the only
$S$-structures in $\mathbb{E}$ are given by _some class_ of subobjects of $1$.

######  Proposition 5.7.

When $\mathbb{E}$ is a category such that any fiber $Pt_{X}\mathbb{E}$
trivializes the group structure, then any fiber $RGh_{X}\mathbb{E}$ weakly
trivializes the structure of groupoid.

##  6 First principles and questionings about the algebraic crystallography

###  6.1 Commutativity

The three first examples of crystallographic context (unital, additive,
Mal’tsev) are actually dealing with commutative structures. The reason is
simple. Any of the structures $S$ in question (monoid, group, Mal’tsev
structure) is endowed with a duality operator making the following diagram
commute:

|  |  |
---|---|---|---
$\textstyle{S(\mathsf{Set})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\;)^{op}}$$\scriptstyle{U^{S}}$$\textstyle{S(\mathsf{Set})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{U^{S}}$$\textstyle{\mathsf{Set}}$

The uniqueness involved in the definition of a crystallographic context
implies that on an object $X$ any structure coincides with its dual.

###  6.2 Questions

Of course, a first natural question is: is it possible to characterize in some
way the structures which admit a crystallographic context or an intensive
crystallographic context? Concerning the classical (=non-fiberwise) algebraic
structures, we can ask when the first situation implies the second one.

Another way to think to this question would be: are the commutative monoid,
abelian group and affine structure the only ones to be self-crystallogra- phic
structures?

###  6.3 Trivializing chrystallographic context

Certainly, as we already noticed, trivializing contexts are not so scarce. Let
us add the following very special one, recalling from [1] the following:

######  Definition 6.1.

An implication algebra is a triple $(X,\triangleright,1)$ of a set, a binary
operation and a constant such that:

$\displaystyle x\triangleright x=1\;\;(1)$ $\displaystyle(x\triangleright
y)\triangleright x=x\;\;(2)$ $\displaystyle(x\triangleright y)\triangleright
y=(y\triangleright x)\triangleright x\;\;(3)$ $\displaystyle
x\triangleright(y\triangleright z)=y\triangleright(x\triangleright z)\;\;(4)$

######  Lemma 6.1.

In any impication algebra we get:

$1\triangleright x=x\;\;\;{\rm and}\;\;\;x\triangleright 1=1$

Accordingly any impication algebra has an underlying opsubtraction.

###### Proof.

First set $y=x$ in (2). Then set $x=y$ in the same (2). ∎

Let us denote by $\mathsf{Imp}$ the category of implication algebras.

######  Proposition 6.1.

The category $\mathsf{Imp}$ trivializes the structure of implication algebra.
In other words, the structure of implication algebra is self-trivializing.

###### Proof.

Let $*$ give an internal implication algebra structure on
$(X,\triangleright,1)$ in $\mathsf{Imp}$. It must be a
$\mathsf{Imp}$-homomorphism, that is to satisfy:

$(x\triangleright x^{\prime})*(y\triangleright
y^{\prime})=(x*y)\triangleright(x^{\prime}*y^{\prime})$

Setting $x=x^{\prime}=y$, we get: $y\triangleright y^{\prime}=y*y^{\prime}$.
So we must have:

$(x\triangleright x^{\prime})\triangleright(y\triangleright
y^{\prime})=(x\triangleright y)\triangleright(x^{\prime}\triangleright
y^{\prime})$

Setting $x=x^{\prime}=y^{\prime}$ and $y=1$ , we get: $y^{\prime}=1$; so,
$(X,\triangleright,1)$ is the terminal object in $\mathsf{Imp}$. ∎

According to the proofs of the previous lemma and proposition, the same result
holds for the structure of _implicative opsubtraction_ defined by
$s(x,x)=0,s(0,x)=x$ and $s(x,0)=0$ or, by duality, for the structure of
_opimplicative subtraction_. Let $(X,p)$ be any Mal’tsev structure such that
$p$ satisfies the Pixley axiom; then, given any element $x_{0}\in X$, the
binary operation defined by $s_{x_{0}}(y,z)=p(y,z,x_{0})$ produces an
opimplicative subtraction on $X$ with $x_{0}$ as constant. Incidentally, we
get the following:

######  Proposition 6.2.

The structure of opimplicative subtraction (resp. implicative opsubtraction)
trivializes the structure of unitary magma. The fundamental group of a
topological opimplicative subtraction (resp. implicative opsubtraction) is
trivial. Accordingly, the fundamental group, at any point, of a topological
Mal’tsev algebra satisfying the Pixley axiom is trivial.

###### Proof.

Let $(X,*)$ be a unitary magma, and $s$ the internal opimplicative subtraction
on it. We have:
$s(x*x^{\prime},y*y^{\prime})=s(x,y)*s(x^{\prime},y^{\prime})$. Taking
$x^{\prime}=y=1$, we get: $s(x,y^{\prime})=x*1=x$. Then take $x=y^{\prime}$;
so, $x=1$. The second assertion comes from Corollary 2.1. A topological
Mal’tsev algebra is any object of a category $\mathbb{T}$-$Top$ of internal
$\mathbb{T}$-algebras in the category $Top$ of topological spaces, where
$\mathbb{T}$ is a Mal’tsev variety, see [17] for instance. The last assertion
is then straightforward from the opimplicative subtraction $s_{x_{0}}$. ∎

###  6.4 Paradoxical aspect of the notion

On the one hand, this kind of relationship _”context vs structure”_ has a
classical positive aspect on the side of the context:
_in this given context, any object $X$ has at most one structure_.

On the other hand, on the side of the structure, it could be thought as a kind
of photographic negative:
_on any object in the context, there is no more than one structure of this
type_.

So, emerge a paradoxical question: could it be possible to extract some
interesting (positive) information about a structure from the contexts in
which this structure becomes so punctually scarce? Or, for example and more
precisely, what are we exactly learning about the notion of (abelian) group by
knowing that congruence hyperextensive categories produce a crystallographic
context with respect to it?

##  7 Some very large abelian and naturally Mal’tsev categories

As said in the introduction, in this last section, we shall produce some
constructions which seem to offset the punctual scarcity of a structure in a
chrystallographic context by a kind of multiplication of this structure inside
this context; first in the pointed case of abelian groups, then in the non-
pointed case of affine structures.

###  7.1 A very large abelian category

Consider the congruence hyperextensible variety $Hex_{3}$ defined by the only
following three ternary terms and the following equations:

$\displaystyle p_{1}(a,0,0)=a,$ $\displaystyle p_{2}(a,0,a)=a,$ $\displaystyle
p_{3}(0,0,a)=a,$ $\displaystyle p_{1}(a,a,b)=p_{2}(a,a,b),$ $\displaystyle
p_{2}(a,b,b)=p_{3}(a,b,b),$ $\displaystyle p_{i}(b,b,b)=b,\;\;1\leq i\leq 3.$

We get a fully faithful functor $h:\mathsf{Gp}\to Hex_{3}$ defined in the
following way: starting from a group $(G,.)$, construct a $Hex_{3}$-algebra on
the set $G$ by setting:

$p_{1}(x,y,z)=x.y^{-1}.z,\;\;\ {\rm and}\;\;p_{2}(x,y,z)=z=p_{3}(x,y,z)$

By restriction we get a fully faithful embedding
$h:\mathsf{Ab}\to\mathsf{Ab}(Hex_{3})$.

Now consider any field $K$ whose characteristic is not $2$. We get a faithful
functor $w_{K}:K$-$Vect\to\mathsf{Ab}(Hex_{3})$: starting from a $K$-vector
space $V$, construct a $Hex_{3}$-algebra on the set $V$ by setting:

$\bar{p}_{1}(x,y,z)=x+\frac{-y+z}{2},\;\;\bar{p}_{2}(x,y,z)=\frac{x+z}{2},\;\;{\rm
and}\;\;\bar{p}_{3}(x,y,z)=\frac{x-y}{2}+z$

This algebra is made abelian by the $Hex_{3}$-homomorphism $+:V\times V\to V$.

So, the category $\mathsf{Ab}Hex_{3}$ of abelian objects in $Hex_{3}$ appears:
i) to fully faithfully embed the category $\mathsf{Ab}$ of abelian groups by
the functor $h:\mathsf{Ab}\rightarrow\mathsf{Ab}Hex_{3}$
and in an independant way
ii) to faithfully contain any category $K$-$\mathsf{Vect}$ of $K$-vector
spaces, provided that the field $K$ is not of characteristic $2$, by the
functor $w_{K}:K$-$\mathsf{Vect}\rightarrow\mathsf{Ab}Hex_{3}$.
So, the underlying abelian group $(V,+)$ of a $K$-vector space is represented
by two distinct objects, $h(V,+)$ and $w_{K}(V,+)$, in the abelian category
$\mathsf{Ab}Hex_{3}$.

###  7.2 A very large exact naturally Mal’tsev category

We are now going to extend the previous kind of situation to a non-pointed
context.

Consider the congruence modular variety $CM_{3}$ defined by the only following
three ternary terms and the following equations:

$\displaystyle p_{1}(a,b,b)=a,$ $\displaystyle p_{2}(a,b,a)=a,$ $\displaystyle
p_{3}(b,b,a)=a,$ $\displaystyle p_{1}(a,a,b)=p_{2}(a,a,b),$ $\displaystyle
p_{2}(a,b,b)=p_{3}(a,b,b).$

We get a fully faithful functor $m:\mathsf{Mal}\to CM_{3}$ defined in the
following way: starting from an algebra $(X,p)$ in $Mal$, construct a
$CM_{3}$-algebra on the set $X$ by setting:

$p_{1}=p\;\;\;\;\;\;{\rm and}\;\;\;\;\;\;p_{2}(x,y,z)=z=p_{3}(x,y,z)$

By restriction, we get a fully faithful functor
$m:\mathsf{Aff}\to\mathsf{Aff}(CM_{3})$.

Now consider any field $K$ whose characteristic is not $2$. We get a faithful
functor $a_{K}:K$-$\mathsf{Aff}\to\mathsf{Aff}(CM_{3})$: starting from a
$K$-affine space $X$, construct an affine $CM_{3}$-algebra on the set $X$ by
setting:

$\bar{p}_{1}(x,y,z)=\beta(\dot{x}+\frac{-\dot{y}+\dot{z}}{2}),\;\;\bar{p}_{2}(x,y,z)=\beta(\frac{\dot{x}+\dot{z}}{2}),\;\;\bar{p}_{3}(x,y,z)=\beta(\frac{\dot{x}-\dot{y}}{2}+z)$

where $\beta$ is the barycentric mapping which is actually a
$K$-$\mathsf{Aff}$-homomorphism. The affine structure $X\times X\times X\to X$
on the algebra $a_{K}(X)$ in the variety $CM_{3}$ is then produced by the
$CM_{3}$-homomorphism $p(x,y,z)=\beta(\dot{x}-\dot{y}+\dot{z})$.

So, the category $\mathsf{Aff}CM_{3}$ of affine structures in the variety
$CM_{3}$ appears:
i) to fully faithfully embed the category $\mathsf{Aff}$ of affine structures
by the functor $m:\mathsf{Aff}\rightarrow\mathsf{Aff}CM_{3}$
and in an independant way
ii) to faithfully contain any category $K$-$\mathsf{Aff}$ of $K$-affine
spaces, provided that the field $K$ is not of characteristic $2$, by the
functor $a_{K}:K$-$\mathsf{Aff}\rightarrow\mathsf{Aff}CM_{3}$.
So, the affine structure $(X,\beta)$ underlying a $K$-affine space $(X,\beta)$
is represented by two distinct objects, $m(X,\beta)$ and $a_{K}(X,\beta)$, in
the naturally Mal’tsev category $\mathsf{Aff}CM_{3}$.

Keywords: Internal structures, Mal’tsev, congruence modular and congruence
distributive varieties, unital, Mal’tsev and Gumm categories, context vs
structure.
Amsclass: 08A05,08B05,08B10,18C10,18C40,18E13.

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  * [19] J. Slomiński, _On the determining of the form of congruences in abstract algebras with equationally definable constant elements_ , Fundamenta Mathematicae 48 (1960), 325-341.
  * [20] J.D.H. Smith, _Malcev varieties_ , Springer L.N. in Math. 554 (1976).
  * [21] A. Ursini, _On subtractive varieties_ , Algebra universalis 31 (1994), 204-222.

Univ. Littoral Côte d’Opale, UR 2597, LMPA,
Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville,
F-62100 Calais, France<EMAIL_ADDRESS>


Present address: ]Westinghouse Electric Germany GmbH, Dudenstrasse 44, D-68167
Mannheim.

Present address: ]Physik Department E12, Technische Universtät München,
D-85748 Garching

# Cross section measurements of $\alpha$-induced reactions on 92,94Mo and
112Sn for $p$-process studies

W. Rapp  [  I. Dillmann<EMAIL_ADDRESS>[  F. Käppeler  Institut für
Kernphysik, Forschungszentrum Karlsruhe, Postfach 3640, D-76021 Karlsruhe,
Germany  U. Giesen  H. Klein  Physikalisch-Technische Bundesanstalt,
Bundesallee 100, D-38116 Braunschweig, Germany  T. Rauscher  Departement für
Physik, Universität Basel, Klingelbergstrasse 82, CH-4056 Basel  D. Hentschel
S. Hilpp  Institut für Nukleare Entsorgung, Forschungszentrum Karlsruhe,
Postfach 3640, D-76021 Karlsruhe, Germany

###### Abstract

The 92Mo($\alpha,n$)95Ru, 94Mo($\alpha,n$)97Ru, and
112Sn($\alpha,\gamma$)116Te cross sections were measured at the upper end of
the $p$-process Gamow window between 8.2 MeV and 11.1 MeV. Our results are
slightly lower than global Hauser-Feshbach calculations from the code NON-
SMOKER, but still within the uncertainty of the prediction. The
112Sn($\alpha,\gamma$)116Te cross section agrees well with a recently measured
thick-target cross section in the same energy range. For the
92,94Mo($\alpha,n$) reactions the present data close to the reaction
thresholds could eliminate previous uncertainties within a factor of 20, and
we can present now useful comparisons to statistical model calculations with
different optical potentials.

nucleosynthesis, $p$ process, $\alpha$-induced cross sections

###### pacs:

12.40.Ee, 25.55.-e, 26.30.-k, 27.60.+j

††preprint: APS/123-QED

##  I Introduction

A “$p$ process” was postulated to produce 35 stable but rare isotopes between
74Se and 196Hg on the proton-rich side of the valley of stability. Unlike the
remaining 99% of the heavy elements beyond iron these isotopes cannot be
created by (slow or rapid) neutron captures Burbidge et al. (1957), and their
solar and isotopic abundances are 1-2 orders of magnitude lower than the
respective $s$\- and $r$-process nuclei Anders and Grevesse (1989); Rosman and
Taylor (1998). However, so far it seems to be impossible to reproduce the
solar abundances of all $p$ isotopes by one single process. In current
understanding, several (independently operating) processes seem to contribute.

The largest fraction of $p$ isotopes is created in the “$\gamma$ process” by
sequences of photodissociations and $\beta^{+}$ decays Woosley and Howard
(1978, 1990); Rayet et al. (1990); Arnould and Goriely (2003). This occurs in
explosive O/Ne burning during SNII explosions and reproduces the solar
abundances for the bulk of $p$ isotopes within a factor of $\approx$3 Rayet et
al. (1990, 1995). The SN shock wave induces temperatures of 2-3 GK in the
outer (C, Ne, O) layers, sufficient for triggering the required
photodisintegrations. More massive stellar models (M$\geq$20 M⊙) seem to reach
the required temperatures for efficient photodisintegration already at the end
of hydrostatic O/Ne burning Rauscher et al. (2002). Historically, the $p$
process was thought to proceed via proton captures but today they are found to
play no role since the required amount of free protons is not available in the
Ne and O layers within the $p$-process timescale of a few seconds and proton
captures are too slow for elements with large $Z$. Therefore, the term
”$\gamma$ process” is also often used synonymously because seed nuclei pre-
existing in the stellar plasma are photodissociated and more proton-rich
isotopes are produced, initially by $(\gamma,~{}n)$ reactions. When
($\gamma,~{}p$) and ($\gamma,~{}\alpha$) reactions become comparable or faster
than neutron emission within an isotopic chain, the reaction path branches out
and feeds nuclei with lower charge number $Z$. The decrease in temperature
after passage of the shock leads to a freeze-out via neutron captures and
mainly $\beta^{+}$ decays, resulting in the typical $p$-process abundance
pattern with maxima at 92Mo ($N$=50) and 144Sm ($N$=82).

However, the $\gamma$-process scenario suffers from a strong underproduction
of the most abundant $p$ isotopes, 92,94Mo and 96,98Ru, due to lack of seed
nuclei with $A$$>$90\. For these missing abundances, alternative processes and
sites have been proposed, either related to strong neutrino fluxes in the
deepest ejected layers of a SNII ($\nu p$ process Fröhlich et al. (2006)), or
rapid proton-captures in proton-rich, hot matter accreted on the surface of a
neutron star ($rp$ process Schatz et al. (1998, 2001)).

Modern, self-consistent studies of the $\gamma$ process have problems to
synthesize $p$ nuclei in the regions $A<124$ and $150\leq A\leq 165$ Rauscher
et al. (2002). It is not yet clear whether the observed underproductions are
only due to a problem with astrophysical models or also with the nuclear
physics input, i.e. the reaction rates used. Thus, the reduction of
uncertainties in nuclear data is strictly necessary for a consistent
understanding of the $p$ process. Experimental data can improve the situation
in two ways, either by directly replacing predictions with measured cross
sections in the relevant energy range or by testing the reliability of
predictions at other energies when the relevant energy range is not
experimentally accessible.

Accordingly, model calculations of the $p$ process require large nuclear
reaction networks involving about 1800 isotopes and more than ten thousand
reaction rates. Since the largest fraction of such a network contains proton-
rich, unstable isotopes, which are not accessible for cross section
measurements with present experimental techniques, almost all of these rates
have to be determined theoretically by means of the statistical Hauser-
Feshbach (HF) formalism Hauser and Feshbach (1952). Hence, there is no
alternative to employing statistical model calculations, e.g. those performed
with the codes NON-SMOKER Rauscher and Thielemann (2000, 2001) or MOST Goriely
(2005). Comparison with experimental results has shown that theoretical HF
predictions with global parameter sets reproduce the neutron- and proton-
induced cross sections within a factor of two or even better. This deviation
increases for $\alpha$-induced reactions. It has been noted that the
difficulties in calculating these rates are due to the poor knowledge of the
$\alpha$-optical potential at astrophysically relevant energies Mohr et al.
(1997); Demetriou et al. (2002). The optical potential governs the
transmission coefficient in the HF model. Additional input to this model are
nuclear level densities and particle separation energies. The latter data are
well defined since nuclear masses in the $p$-process network are
experimentally known.

It is only one decade ago that charged particle-induced reaction rates at
$p$-process energies ($E_{p}$$\approx$1-10 MeV, $E_{\alpha}$$\approx$5-15 MeV)
became subject of experimental efforts (for a list of reactions, see KADoNiS
(2005)). For example, a series of $\alpha$-scattering experiments at energies
below the Coulomb barrier helped to improve the optical $\alpha$-nucleus
potential Demetriou et al. (2002); Avrigeanu et al. (2003); Fülöp et al.
(2001); Rauscher (2003a, b). However, firm conclusions are still difficult to
draw due to the lack of experimental cross sections.

Nevertheless, extrapolations to the low astrophysical energies remain
necessary since the Coulomb barrier hampers scattering experiments in the
astrophysically relevant energy range. Scattering data within a confined
energy range can be described equally well by different optical potentials
with different extrapolation behavior. This ambiguity, which is called the
family problem, makes extrapolations very uncertain McFadden and Satchler
(1966); Fülöp et al. (1996); Mohr et al. (1997); Galaviz et al. (2005). These
uncertainties can be avoided if experimental cross sections are available for
comparison with HF predictions in or close to the astrophysical Gamow window.

Up to now only five $(\alpha,\gamma)$ measurements with relevance for the $p$
process were performed within the Gamow window between 5 and 15 MeV: 70Ge
Fülöp et al. (1996), 96Ru Rapp et al. (2002), 106Cd Gyürky et al. (2006),
112Sn Özkan et al. (2007), and 144Sm Somorjai et al. (1998). With exception of
the latter, the first four experiments agree within a factor of three with the
NON-SMOKER prediction using a global parameter set with the McFadden-Satchler
$\alpha$ potential McFadden and Satchler (1966). For the $(\alpha,n)$
particle-exchange reactions a much larger amount of data exists, but almost
all of these measurements are restricted to the upper end of the Gamow window.
It was noted in Rapp et al. (2006); Rauscher (2006) that especially in the
heavy mass region (beyond $N$=82) the knowledge of $(\alpha,x)$ cross sections
is crucial, whereas at lower masses the $p$-induced reactions have more
importance. However, measurements of $\alpha$-induced reactions are mainly
limited – apart from some exceptions for $(\alpha,n)$ cross sections – to the
lower mass region below 144Sm. In an attempt to constrain the $\alpha$+nucleus
optical potential at low energy, it has been suggested to use ($n,\alpha$)
reactions and some reactions around $A\approx 147-152$ have been studied in
this way Gledenov et al. (2000); Koehler et al. (2000, 2004).

Finally, it has to be emphasized that with a few exceptions
($\alpha,~{}\gamma$) and ($p,~{}\gamma$) data in the relevant energy range are
more useful for a direct application in reaction networks than
($\gamma,~{}\alpha$) or ($\gamma,~{}p$) data despite of the fact that the $p$
process proceeds via photodisintegration reactions. This may seem puzzling at
first sight but has two reasons. First, reactions in the stellar plasma
involve thermally excited targets and not only targets in the ground state as
in the laboratory. Measured rates have to be corrected for this effect. It can
be shown that the thermal effects are much more pronounced in the
photodisintegration than in the capture channel. Thus the measured rate is
closer to the actual rate when considering capture reactions. The reverse rate
can then be derived applying detailed balance. Second, in order to avoid
inconsistencies, forward and reverse rates for a given reaction in a reaction
network have to stem from the same source, related to each other by detailed
balance. Because of the exponential factor $\exp(-Q_{\mathrm{forward}})$ in
the expression for the relation of reverse to forward rate (see, e.g.,
Rauscher and Thielemann (2000)), it is preferrable to have data for the
direction of positive $Q$ value to limit numerical errors in the simulation.
In most cases this is the capture direction, exclusively in the case of proton
captures close to stability. However, in the intermediate to heavy mass region
the $p$ process also includes $\alpha$ emitters with negative $\alpha$ capture
$Q$ values.

In this work we study ($\alpha,n$) reactions on light Mo isotopes and $\alpha$
capture on 112Sn, motivated by the need to better understand the above
described deficiencies in the $p$ process models. Previous investigations of
astrophysical relevance were restricted to $\alpha$ scattering on 92Mo Fülöp
et al. (2001), 106Cd Kiss et al. (2006), and on 112Sn Galaviz et al. (2005)
attempting to derive an improved optical potential for these cases. The
experimental setup for the measurement of $\alpha$-induced cross sections is
described in Sec. II. The resulting cross sections and a comparison with
previous experimental results for 92Mo($\alpha,n$)95Ru, 94Mo($\alpha,n$)97Ru,
and 112Sn($\alpha,\gamma$)116Te are given in Sec. III. The impact of using
different optical $\alpha$+nucleus potentials in theoretical calculations for
these reactions is investigated in Sec. IV.

##  II Cross section measurements

All cross section measurements have been carried out at the cyclotron of the
Physikalisch-Technische Bundesanstalt (PTB) in Braunschweig/ Germany Brede et
al. (1980) with the activation technique by irradiation of thin sample layers
and subsequent counting of the induced activity. A detailed description of the
measurement technique and the data analysis can be found in Ref. Rapp et al.
(2002).

###  II.1 Preparation and characterization of samples

The sample material was deposited on 0.4 mm thick Ta backings, which were
thick enough to stop the $\alpha$-beam completely to ensure a reliable charge
collection. Background reactions are efficiently suppressed by the high
Coulomb barrier of Ta because $\alpha$ capture on 181Ta leads to the stable
product nucleus 185Re. The 181Ta$(\alpha,n)$184Re channel opens at 9.86 MeV
but does not interfere with our measurements. The contribution from the
extremely rare isotope 180Tam (natural elemental abundance 0.012% Rosman and
Taylor (1998)) could be neglected.

The Mo samples were produced by sputtering natural metallic Mo in a pure Ar
atmosphere. These samples were 10 mm in diameter and exhibited an area density
between 5$\times$1017 and 3$\times$1018 Mo atoms/cm2, corresponding to layer
thicknesses between 100 nm and 600 nm. The 112Sn samples were prepared from 10
mg of enriched metal powder containing 98.9% of 112Sn. This batch was
dissolved in 30% hydrochloric acid and diluted with 90% acetone. The solution
was then deposited on the Ta backings, where uniform SnCl2 layers with an area
density between 6.2 and 9.8$\times$1017 Sn atoms/cm2 (layer thicknesses
$\approx$300-450 nm) were obtained after evaporation of the acetone. The
adhesion of the layers was improved by soft sand-blasting of the Ta backings
before the deposition.

The masses of the Mo and Sn samples were determined by X-ray fluorescence
analysis using a crystal spectrometer (Siemens SRS 3000), which was operated
with a LiF (100) crystal and a rhodium anode. The fluorescence yield was
measured with a gas counter and a NaI detector. The efficiency of the
spectrometer was calibrated by the radiation emitted from an empty backing and
by nine well defined Mo and Sn reference samples. The latter were prepared
from standard solutions of (NH4)2MoO4 and SnCl2, respectively. The sample mass
was derived from the measured Kα1 and Kβ1 X-ray lines. In case of the Sn
sample an additional Ta diaphragm 10 mm in diameter was placed on top of the
sample to restrict the measurement exactly to the area covered in the
irradiations with the $\alpha$ beam. While the statistical uncertainties were
smaller than 0.3%, the definition of the sample mass was affected by
systematic uncertainties of 4.0% and 4.6% for the Mo and the Sn samples,
respectively.

###  II.2 Irradiations and $\gamma$ counting

A sketch of the beamline setup at the PTB is shown in Fig. 1. The activation
chamber is designed as a Faraday cup, and the charge deposited on the sample
was recorded in small time steps by a current integrator for later correction
of beam fluctuations. Secondary electrons were suppressed by a negatively
charged (US=-150 V) diaphragm at the entrance of the activation chamber.

Figure 1: Experimental setup at the PTB Braunschweig.

For each energy step the $\alpha$-beam spot was adjusted by means of a quartz
window so that the sample was illuminated as complete and homogeneously as
possible by wobbling of the $\alpha$-beam. The energy of the $\alpha$ beam was
defined within $\pm$25 keV uncertainty by means of the field calibration of
two analyzing magnets as well as by a time-of-flight measurement of the
particle velocity Böttger (2002). The effective $\alpha$-energy $E^{\rm{eff}}$
was calculated from the expression 4.99 in Iliadis (2007)

$E^{\rm{eff}}=E_{\rm{c.m.}}-\Delta+\Delta\cdot\left(-\frac{\sigma_{2}}{\sigma_{1}-\sigma_{2}}+\left[\frac{\sigma_{1}^{2}+\sigma_{2}^{2}}{2(\sigma_{1}-\sigma_{2})^{2}}\right]^{1/2}\right).$
(1)

where $\Delta$ is the target thickness calculated with the Monte Carlo program
SRIM 2003 Ziegler and Biersack (2003), $E_{\rm{c.m.}}$ the respective center-
of-mass energy, and $\sigma_{1}$, $\sigma_{2}$ the measured cross sections of
two neighboring points. As can be seen from Eq. 1 the error bars of
$E^{\rm{eff}}$ become asymmetric, with the smaller component in positive
direction due to the correction factor in brackets (see Tables 3, 4, and 5).

Table 1:  Decay properties of product nuclei and corrections for $\gamma$-ray counting Product | $t_{1/2}$ | $E_{\gamma}$ | $I_{\gamma}$ | Reference | Summing corrections [%]
---|---|---|---|---|---
nucleus |  | [keV] | per decay |  |  $\gamma$ Cascades | X Rays
95Ru | (1.643 $\pm$ 0.014) h | 336.4 | 0.702 $\pm$ 0.005 | Burrows (1993) | 7 $\pm$ 1 | 11.2 $\pm$ 0.4
|  | 626.8 | 0.178 $\pm$ 0.005 |  | 7 $\pm$ 1 | “
97Ru | (2.9 $\pm$ 0.1) d | 215.7 | 0.856 $\pm$ 0.013 111No uncertainty given in literature. Assumed uncertainty 1.6%, as for the 324.4 keV line. | Artna-Cohen (1993) |  $<$0.2 | 12.4 $\pm$ 1.5
|  | 324.4 | 0.1079 $\pm$ 0.0017 |  | “ | “
116Te | (2.49 $\pm$ 0.04) h | 93.7 | 0.331 $\pm$ 0.021 | Blachot (2001) |  $<$0.2 | 15.6 $\pm$ 1.7

During the activations, the samples were irradiated with average $\alpha$-beam
currents of 5 $\mu$A for periods between 20 minutes and 14 hours, depending on
isotope and cross section. For the Sn samples the waiting time between the end
of activation and the start of activity counting was extended to one hour to
eliminate the 38K decay from the concomitant 35Cl($\alpha,n$)38K reaction
($t_{1/2}$= 7.5 min). The induced $\gamma$ activities were measured with a
calibrated HPGe detector. Natural backgrounds were suppressed by a shielding
consisting of an inner layer of 5 mm copper surrounded by 10 cm of lead. The
samples were analyzed as described in Ref. Rapp et al. (2002) using the
$\gamma$-ray energies and intensities of the characteristic lines listed in
Table 1. In cases, where two $\gamma$-ray lines could be used in data
analysis, the results were averaged by weighting with the respective
transition probabilities.

Summing corrections due to the coincident detection of $\gamma$-cascades were
evaluated by means of simulations with the Monte Carlo code CASC Jaag (1993)
using the decay schemes of the corresponding product nuclei 95Ru, 97Ru, and
116Te Burrows (1993); Artna-Cohen (1993); Blachot (2001). The resulting
corrections for $\gamma$ summing are given in Table 1 together with the
summing corrections due to the coincident detection of X rays. The analysis of
the induced activity in the 112Sn samples via the 93.7 keV transition in the
decay of 116Te was complicated by a constant background activity of
(0.031$\pm$0.003) cts/s, corresponding presumably to the Po-Kβ-line, as well
as by the characteristic 93.3 keV $\gamma$ line from the decay of 67Ga. The
latter background was produced by 63Cu($\alpha,\gamma$) reactions due to a
copper contamination of the samples. The background component from the 67Ga
decay could be corrected by means of the associated 67Ga line at 184.6 keV.

###  II.3 Uncertainties

The systematic uncertainties (Table 2) are governed by the target thickness
and the detector efficiency. For 97Ru, the uncertainty in the half-life gives
rise to an additional contribution of 3.4%. The same holds for the $\gamma$
intensity of the 93.7 keV line from the decay of 116Te with an uncertainty of
6.3% Blachot (2001). The overall uncertainties are 5.3% and 6.3% for the
92Mo$(\alpha,n)$ and 94Mo$(\alpha,n)$, and 8.7% for the
112Sn$(\alpha,\gamma)$116Te cross sections.

Table 2: Systematic uncertainties (in %).

Source of uncertainty |  92Mo($\alpha,n$) |  94Mo($\alpha,n$) |  112Sn($\alpha,\gamma$)
---|---|---|---
Target thickness | 4.0 | 4.0 | 4.6
Efficiency | 3.0 | 3.0 | 3.0
Beam current | 1.0 | 1.0 | 1.0
$\gamma$ intensity | 0.7 | 0.5 | 6.3
Decay constants | 0.9 | 3.4 | 1.6
Summing corrections | 1.1 | 1.5 | 1.7
Overall uncertainty | 5.3 | 6.3 | 8.722footnotemark: 2

11footnotetext: Does not include effect of background line (see text).

##  III Results and comparison with previous data

The results are summarized in Tables 3, 4, and 5, and in Fig. 2 in form of
cross sections and astrophysical $S$ factors. The cross sections of all three
reactions (Fig. 2) are systematically lower by factors of two to three
compared to the global Hauser-Feshbach predictions from NON-SMOKER (dashed
line).

For 112Sn($\alpha,\gamma$)116Te our data points follow the Hauser-Feshbach
dependence within the uncertainties, whereas the results from Özkan et al.
Özkan et al. (2007), which were obtained with rather thick targets and
represent averaged cross sections over energy bins of 440 to 565 keV, seem to
exhibit a slightly different energy trend. Unfortunately, the present
measurements at lower energies were severely hampered by an interfering
background line from 63Cu$(\alpha,\gamma)$67Ga reactions on the Cu impurity of
the samples. This background could be only normalized by a much weaker 67Ga
line at 184.6 keV, resulting in additional systematic and statistical
uncertainties.

For the $(\alpha,n)$ cross sections some older measurements were available for
comparison (Fig. 3). The data given by Levkovskij Levkovskij (1991) is
reproducing the NON-SMOKER cross sections in both cases perfectly for energies
above 13 MeV, but there is a systematic deviation (compared to all other data)
below this energy, presumably due to an undetected background component. This
problem is particularly apparent at the lowest data point at 8.4 MeV, whereas
the 92Mo$(\alpha,n)$95Ru threshold is at 9.39 MeV. A similar problem exists
for 94Mo$(\alpha,n)$.

The values from Esterlund et al. Esterlund and Pate (1965) for
92Mo$(\alpha,n)$ and Graf et al. Graf and Muenzel (1974) for
92,94Mo$(\alpha,n)$ seem to be low compared to the energy trend below 15 MeV,
but agree like the Levkovskij data rather well with the predictions at higher
energies. The only data which follow the Hauser-Feshbach trend also at lower
energies down to 11.8 MeV and at the same time fits well with our highest data
point are the 92Mo$(\alpha,n)$ cross sections of Denzler et al. Denzler et al.
(1995). This comparison shows the necessity to measure $(\alpha,n)$ cross
sections – especially for $p$-process studies – down to the threshold.

Table 3:  Cross sections and $S$ factors for 92Mo$(\alpha,n)$95Ru. $E{\rm{}^{eff}_{c.m.}}$ | Cross section |  $S$ factor
---|---|---
[MeV] | [mbarn] |  $\times$1020[MeV barn]
10.989 ${}^{+0.038}_{-0.039}$ | 9.26 $\pm$ 0.65 | 2.13 $\pm$ 0.15
10.981 ${}^{+0.046}_{-0.047}$ | 10.6 $\pm$ 0.9 | 2.47 $\pm$ 0.21
10.543 ${}^{+0.033}_{-0.037}$ | 3.88 $\pm$ 0.33 | 2.38 $\pm$ 0.20
10.019 ${}^{+0.039}_{-0.043}$ | 2.53 $\pm$ 0.26 | 5.39 $\pm$ 0.55
9.638 ${}^{+0.066}_{-0.087}$ | 1.02 $\pm$ 0.09 | 5.71 $\pm$ 0.50
9.112 ${}^{+0.036}_{-0.047}$ | 0.194 $\pm$ 0.015 | 4.57 $\pm$ 0.35
Table 4:  Cross sections and $S$ factors for 94Mo$(\alpha,n)$97Ru. $E{\rm{}^{eff}_{c.m.}}$ | Cross section |  $S$ factor
---|---|---
[MeV] | [mbarn] |  $\times$1020[MeV barn]
10.997 ${}^{+0.039}_{-0.040}$ | 20.90 $\pm$ 2.00 | 4.81 $\pm$ 0.46
10.966 ${}^{+0.070}_{-0.073}$ | 18.50 $\pm$ 1.50 | 4.56 $\pm$ 0.37
10.548 ${}^{+0.038}_{-0.044}$ | 8.14 $\pm$ 0.77 | 5.06 $\pm$ 0.48
10.034 ${}^{+0.034}_{-0.036}$ | 5.47 $\pm$ 0.62 | 11.50 $\pm$ 1.30
9.676 ${}^{+0.037}_{-0.043}$ | 2.49 $\pm$ 0.25 | 12.96 $\pm$ 1.30
9.114 ${}^{+0.042}_{-0.056}$ | 0.58 $\pm$ 0.06 | 13.92 $\pm$ 1.44
8.572 ${}^{+0.067}_{-0.102}$ | 0.12 $\pm$ 0.01 | 14.52 $\pm$ 1.21
8.136 ${}^{+0.061}_{-0.095}$ | 0.021 $\pm$ 0.002 | 10.49 $\pm$ 1.00
Table 5:  Cross sections and $S$ factors for 112Sn$(\alpha,\gamma)$116Te. $E{\rm{}^{eff}_{c.m.}}$ | Cross section |  $S$ factor
---|---|---
[MeV] | [$\mu$barn] |  $\times$1023[MeV barn]
11.067 ${}^{+0.042}_{-0.038}$ | 408 $\pm$ 57 | 1.10 $\pm$ 0.15
10.615 ${}^{+0.040}_{-0.043}$ | 295 $\pm$ 43 | 2.61 $\pm$ 0.38
10.097 ${}^{+0.037}_{-0.045}$ | 77 $\pm$ 15 | 2.93 $\pm$ 0.57
9.180 ${}^{+0.036}_{-0.048}$ | 9.52 $\pm$ 3.68 | 6.51 $\pm$ 2.52
8.209 ${}^{+0.043}_{-0.064}$ | 0.48 $\pm$ 0.38 | 11.78 $\pm$ 9.33
Figure 2: Cross section and $S$ factor for 92Mo$(\alpha,n)$95Ru,
94Mo$(\alpha,n)$97Ru, and 112Sn$(\alpha,\gamma)$116Te in comparison with
standard NON-SMOKER predictions (dashed line). The grey area shows the
estimated uncertainty of the prediction (factor 10). Figure 3: (Color online)
Comparison of previous experiments at higher energies (up to 20 MeV) with our
results and with the standard NON-SMOKER predictions (dashed line). The grey
area shows the uncertainty of the prediction (factor 10).

##  IV Test of the optical $\alpha$+nucleus potential

The present experimental $S$ factors were compared to NON-SMOKER Hauser-
Feshbach calculations employing different optical $\alpha$+nucleus potentials
while keeping the standard inputs for the other parameters. The standard
optical potential used in NON-SMOKER Rauscher and Thielemann (2000) is the one
by McFadden and Satchler McFadden and Satchler (1966), derived from scattering
data above 60 MeV. This is also the potential used in the standard NON-SMOKER
predictions shown in Figs. 2 and 3. Another global potential is the one by
England et al. England et al. (1982), also fitted to data at higher energies.
A more recent global potential, which also included reaction data, was given
by Avrigeanu et al. Avrigeanu et al. (2003). Finally, Fröhlich et al. (see
Rauscher (2003c), erratum in Rauscher (2003d)) used reaction data at $A\approx
150$ to derive a low-energy potential which also reproduces data at lower
masses (e.g., Rauscher (2003c); Gyürky et al. (2006)).

The reaction 92Mo$(\alpha$,n)95Ru is mainly sensitive to the optical $\alpha$
potential in the measured energy range. Only up to a few keV above the
threshold, the averaged neutron widths are smaller than the $\alpha$ widths
and therefore the sensitivity is higher to the neutron widths close to the
threshold. As can be seen in Fig. 4, the theoretical results obtained with the
different optical potentials display the same energy behavior but with
different absolute values. This energy dependence describes that of the data
reasonably well. The $S$ factors obtained with the McFadden and Satchler
potential have to be divided by 1.8, whereas the ones obtained with the
Fröhlich potential are a factor of 1.8 too low. All other potentials used lead
to larger deviations. An optical potential for 92Mo+$\alpha$ was derived from
scattering data and carefully extrapolated to low energy in Ref. Fülöp et al.
(2001). We do not show the results with this potential because they coincide
with the Avrigeanu et al. results at the low energy range shown here, whereas
they coincide with the McFadden and Satchler $S$ factors at the high end of
the energy range. Thus, its energy dependence is slightly different from that
of the other potentials and the data.

Also the reaction 94Mo$(\alpha,n)$97Ru is almost exclusively sensitive to the
$\alpha$ widths and shows only a small dependence on the neutron widths,
except close to the threshold. Inspection of Fig. 4 evidently shows that the
energy dependence of the data cannot be reproduced by any of the potentials
tried here. The peak $S$ factor is shifted to higher energy in the data as
compared to the one found in the calculated $S$ factor. The cause of this
remains unclear. The importance of the neutron widths does not extend beyond a
few tens of keV above the ($\alpha$,n) threshold. The impact of width
fluctuation corrections is also confined to a small window above the reaction
threshold. Similar to the case above, multiplication of the results obtained
with the Fröhlich potential with a factor 1.8 and division of the ones
obtained with the McFadden and Satchler potential by 1.7 lead to an improved
reproduction of the data. However, because of the distinctly different energy
dependence none of these solutions are satisfactory.

Figure 4: (Color online) Comparison of $S$ factors obtained with different
optical potentials in the NON-SMOKER code for the reactions
92Mo$(\alpha,n)$95Ru and 94Mo$(\alpha,n)$97Ru (see text for details). Figure
5: (Color online) Comparison of $S$ factors obtained with different optical
potentials in the NON-SMOKER code for the reaction 112Sn$(\alpha,\gamma)$116Te
(see text for details).

It may be surprising at first glance that the $S$ factors of the reaction
112Sn$(\alpha,\gamma)$116Te are much more sensitive to the $\alpha$ widths
than to the $\gamma$ widths. However, due to the Coulomb barrier, the $\alpha$
widths are smaller by one to several orders of magnitude than the $\gamma$
widths in the energy range covered in the measurement. There are two data sets
available, the one obtained in this work and the one by Özkan et al. Özkan et
al. (2007). Both of them are compared to theoretical results in Fig. 5.
Concerning the theoretical predictions using different optical $\alpha$
potentials it is not surprising that the Özkan et al. data at the high energy
range are best described employing the potential I of Demetriou et al.
Demetriou et al. (2002). A preliminary version of these data was part of the
procedure to derive the Demetriou et al. potential, a global potential for
heavy targets. Similar agreement was already found in a study of $\alpha$
scattering on 112Sn Galaviz et al. (2005). In that study, an optical potential
was derived from scattering data and then used to compute the
112Sn$(\alpha,\gamma)$116Te $S$ factors. Peculiarly, however, it was found
that the derived potential did not reproduce the Özkan et al. reaction data
well. Much better agreement was achieved with the Demetriou et al. potential
and best agreement was obtained when using the Fröhlich potential, although
the latter did not describe the scattering data. Here, we confirm these
findings Galaviz et al. (2005). We do not show the result with the optical
potential derived in Galaviz et al. (2005) separately because it basically
coincides with the one obtained with the McFadden and Satchler potential.

Interestingly, our new experimental data seem to indicate a slightly steeper
energy dependence than the Özkan data. This yields good agreement (within the
error bars) with the prediction making use of the Fröhlich-Rauscher potential
(with exception of the data point at 10.615 MeV). The calculation using the
Demetriou potential Demetriou et al. (2002) exceeds our data (including error
bar) at the three highest energies and is compatible to our data only at the
two lowest energies. Comparing to the Özkan et al. data, the Fröhlich-Rauscher
potential gives acceptable agreement only below 9.5 MeV and underestimates the
$S$ factors above that energy. On the other hand, the Demetriou et al.
potential excellently reproduces the Özkan et al. data above 8.5 MeV but
strongly overestimates them below that energy.

##  V Summary

We have measured the $(\alpha,n)$ cross sections of the $p$ nuclei 92,94Mo and
the $(\alpha,\gamma)$ cross section of the lightest tin isotope 112Sn. For
94Mo$(\alpha,n)$ no data for comparison at lower energies was available, but
for 92Mo our energy trend seems to fit well with the results of Denzler et al.
Denzler et al. (1995) starting at 12 MeV. For 112Sn$(\alpha,\gamma)$116Te we
can essentially reproduce the thick-target cross sections from Özkan et al.
Özkan et al. (2007) although we find slightly lower values above 10 MeV.
Additionally, we studied the sensitivity of Hauser-Feshbach predictions
performed with the code NON-SMOKER for these reactions. We find that the cross
sections and $S$ factors are mostly sensitive to the $\alpha$ widths and thus
to the optical $\alpha$+nucleus potential employed in the calculations. The
standard, global predictions – utilizing the global potential by McFadden-
Satchler McFadden and Satchler (1966) – systematically overestimate the data
by a factor of two. The potential of Fröhlich and Rauscher Rauscher (2003c, d)
describes well the reaction data of 112Sn$(\alpha,\gamma)$116Te but
underestimates the 92,94Mo data by a factor of about two. The energy
dependence of 94Mo($\alpha$,n)97Ru cannot be reproduced by the statistical
model calculations.

###### Acknowledgements.

We thank the operating team O. Döhr, H. Eggestein, T. Heldt, and M. Hoffmann
for providing the $\alpha$-beam at the PTB cyclotron. This work was partially
supported by the Swiss NSF (grant 2000-113984/1).

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