We established that the association $x \mapsto \bot_x$ for an object $x \in \mathbb{E}_n$ extends to a functor $\textbf{Gro}^\prime(\mathbb{E}) \xrightarrow{\bot} \mathbb{E}_0$. We may also observe that the structure map $\eta : x \rightarrow s_0^{(0)}\bot_x$ gives us a costar in $E_0$ via the vertex maps $x_i \rightarrow \bot_x$. This should define a map

$$\bot : E \rightarrow \text{\textbf{coSpan}}(\mathbb{E}_0)$$

This map seems to describe a cospan representation in the examples. Cospan representation for double categories is discussed in [4] and a similar trend seems to appear for simplicial categories. We leave these considerations for future work as well.

§.§ Simplicially enriched categories

Categories enriched over simplicial sets (with its cartesian monoidal structure) are a fundamental structure in categorical homotopy theory. Unfortunately they are named simplicial categories in a lot of the literature on the subject. Our references for this subject are [20] and [22].

A simplicially enriched category $\mathcal{C}$, or sSet-category for short, consists of:

($\bullet$) A collection of objects $x,y,z \dots$

($\triangle$) A simplicial set $\mathcal{C}(x,y)$ for any two objects $x,y$

($=$) A $0$-simplex
$$1_x : \Delta^0 \rightarrow \mathcal{C}(x,x)$$

called the identity, for each object $x$

($\circ$) A composition operation

$$\circ: \mathcal{C}(y,z) \times \mathcal{C}(x,y) \rightarrow \mathcal{C}(x,z)$$

which is a morphism of simplicial sets, for any three objects $x,y,z$

satisfying associativity and unital laws depicted as commutative diagrams

(v1) at (-3.5,3) $\Delta^0 \times \mathcal{C}(x,y)$;
(v2) at (0.5,3) $\mathcal{C}(x,x) \times \mathcal{C}(x,y)$;
(v3) at (-1.5,1.5) $\mathcal{C}(x,y)$;
[->]  (v1) edge node[above, font= ] $1_x \times 1_{\mathcal{C}(x,y)}$ (v2);
[->]  (v2) edge node[right, font= ]$\circ$ (v3);
[->]  (v1) edge node[left, font= ]$\cong$(v3);

(v1) at (-3.5,3) $\mathcal{C}(x,y) \times \Delta^0$;
(v2) at (0.5,3) $\mathcal{C}(x,y) \times \mathcal{C}(y,y)$;
(v3) at (-1.5,1.5) $\mathcal{C}(x,y)$;
[->]  (v1) edge node[above, font= ] $ 1_{\mathcal{C}(x,y)} \times 1_y$ (v2);
[->]  (v2) edge node[right, font= ]$\circ$ (v3);
[->]  (v1) edge node[left, font= ]$\cong$ (v3);

(v1) at (-3,3.5) $(\mathcal{C}(z,w) \times \mathcal{C}(y,z)) \times \mathcal{C}(x,y)$;
(v2) at (3.5,3.5) $\mathcal{C}(z,w) \times (\mathcal{C}(y,z) \times \mathcal{C}(x,y))$;
(v3) at (-3,1.5) $\mathcal{C}(y,w) \times \mathcal{C}(x,y)$;
(v4) at (3.5,1.5) $\mathcal{C}(z,w) \times \mathcal{C}(x,z)$;
(v5) at (0.3,0) $\mathcal{C}(x,w)$;
[->]  (v1) edge node[above, font= ] $\cong$  (v2);
[->]  (v1) edge node[left, font= ]$\circ \times 1_{\mathcal{C}(x,y)}$ (v3);
[->]  (v2) edge node[right, font= ]$ 1_{\mathcal{C}(z,w)} \times \circ$ (v4);
[->]  (v3) edge node[below, font= ]$\circ$ (v5);
[->]  (v4) edge node[below, font= ]$\circ$ (v5);

for all objects $x,y,z,w$.

Examples of simlicially enriched categories are numerous and occur naturally. Most prominently the category of simplicial sets is itself simplicially enriched. For two simplicial sets $X,Y$ the $n$-simplicies of the mapping space are given by

$$\text{Map}(X,Y)_n = \text{\textbf{sSet}}(X \times \Delta^n, Y)$$

with the obvious face and degeneracy relations. The category of topological spaces can be seen to be simplicially enriched in a similar fashion. For two spaces $X,Y$ we can declare the $n$-simplicies of the mapping space to be continuous maps

$$X \times |\Delta^n| \rightarrow Y$$

where $|\Delta^n|$ is the standard topological $n$-simplex.

This way in both examples $0$-simplicies in $\text{Map}(X,Y)$ are ordinary maps of simplicial sets, $1$-simplicies are homotopies and higher simplicies are higher homotopies.

We may (and will) use this interpretation of simplicial enrichment even for abstract sSet-categories $\mathcal{C}$. Let $x,y \in \mathcal{C}$ be two objects. A $0$-simplex in $f \in \mathcal{C}(x,y)_0$ may be depicted as a usual morphism $x \xrightarrow{f} y$. A $1$-simplex $\alpha \in \mathcal{C}(x,y)_1$ will have two vertices in the mapping space, say $d_0\alpha = f$ and $d_1\alpha = g$. We may depict $\alpha$ as

(v1) at (-4,3) $x$;
(v2) at (-2,3) $y$;
[->, bend left = 50]  (v1) edge node [above,font = ]$f$ (v2);
[->, bend right = 50]  (v1) edge node [below,font = ]$g$ (v2);
(v3) at (-3,3.5) ;
(v4) at (-3,2.5) ;
[->]  (v3) edge[double] node[right,font = ]$\alpha$ (v4);

The above picture may be thought of as being produced by suspending the $1$-simplex $f \xrightarrow{\alpha} g$ between the objects $x$ and $y$. It is helpful in visualising sSet-categories to think of the whole mapping space $\mathcal{C}(x,y)$ as being suspended between $x$ and $y$.

While in an ordinary category we compose arrows in a sSet-category $\mathcal{C}$ we compose higher simplicies as well. Nonetheless we have an underlying category $\mathcal{C}_0$ associated to $\mathcal{C}$ with the same objects and morphism sets given as

$$\mathcal{C}_0(x,y) = \mathcal{C}(x,y)_0$$

Before returning to homotopy colimits we define another important notion: tensoring and cotensoring in a sSet-category. A sSet-category is said to be tensored over simplicial sets if for any object $x \in \mathcal{C}$ and simplicial set $K$ there is an object $K \odot x \in \mathcal{C}$ such that for any other object $y$ we have a natural isomorphism

$$\mathcal{C}_0(K \odot x, y) \cong \textbf{sSet}(K, \mathcal{C}(x,y))$$

Similarly $\mathcal{C}$ is cotensored if there is an object $\{K,y \}$ for all $y \in \mathcal{C}$ and $K \in \text{\textbf{sSet}}$ such that for all objects $x$ there is a natural isomorphism

$$\textbf{sSet}(K, \mathcal{C}(x,y)) \cong \mathcal{C}_0(x, \{K,y \})$$

The category of simplicial sets is tensored and cotensored over itself via
$$K \odot X = K \times X$$
$$\{K,X\} = \text{Map}(K,X)$$

for all $K,X \in \text{\textbf{sSet}}$. The category of topological spaces is tensored and cotensored as well. If $X$ is a space then define
$$K \odot X = |K| \times X$$
$$\{K,X\} = \text{C}(|K|,X)$$

where $|K|$ denotes the geometric realization of $K$ and $\text{C}(|K|,X)$ denotes the set of continuous maps from $|K|$ to $X$ endowed with the compact-open topology. In this case we have to restrict ourselves to a convenient category of spaces containing all CW-complexes, like for example the category of compactly generated Hausdorff spaces.

Tensoring and cotensoring bring a sSet-category $\mathcal{C}$ closer to intition. For example a homotopy $\Delta^1 \rightarrow \mathcal{C}(x,y)$ can be realized as a map

$$\Delta^1 \odot x \rightarrow y$$

Also for a map $f: x \rightarrow y$ we can define its mapping cylinder to be given as a pushout

(v1) at (-3,3.5) $x$;
(v2) at (-1,3.5) $y$;
(v3) at (-3,1.5) $\Delta^1 \odot x$;
(v4) at (-1,1.5) $M_f$;
[->]  (v1) edge node [above,font = ]$f$ (v2);
[->]  (v1) edge node [left,font = ]$d^0 \odot 1_x$ (v3);
[->, dashed]  (v3) edge (v4);
[->, dashed]  (v2) edge (v4);

We also have enough structure to talk about homotopy colimits in $\mathcal{C}$. Let $\mathcal{J}$ be an indexing category and
$$F : \mathcal{J} \rightarrow \mathcal{C}$$

be a diagram. Then the homotopy colimit of $F$ is defined via the coend formula

$$\text{hocolim}F = \int^{i \in \mathcal{J}} i/\mathcal{J} \odot F(i)$$

Here $i/J$ for an object $i\in \mathcal{J}$ is the usual category over $i$ with

($\bullet$) objects pairs $(j,f)$ where $j$ is an object in $\mathcal{J}$ and $f: i \rightarrow j$ is a morphism

($\rightarrow$) morphisms between two objects $(j,f)$ and $(j^\prime, f^\prime)$ are commutative triangles

(v1) at (-2.5,2) $i$;
(v2) at (-3.5,3) $j$;
(v3) at (-1.5,3) $j^\prime$;
[->]  (v1) edge node [left,font = ]$f$ (v2);
[->]  (v1) edge node [right,font = ]$f^\prime$ (v3);
[->] (v2) edge (v3);

Coends are a special form of colimit (see for example [16] or the dedicated manuscript [14]) which we find mysterious, so we will elaborate the homotopy colimit construction in more elementary terms.

We first consider the case where $\mathcal{J} = \Delta^n$ so that the functor $F$ will give us an $n$-simplex, say

$$\sigma : x_0 \xrightarrow{f_1} x_1 \xrightarrow{f_2} \dots \xrightarrow{f_n} x_n$$

in the category $\mathcal{C}_0$. Then we may think of the homotopy colimit of $\sigma$ as being given by the colimit of the following staircase diagram:

(v2) at (-5,-3) $x_0 \odot \Delta^n$;
(v1) at (-5,-1.5) $x_0 \odot \Delta^{n-1}$;
(v3) at (-2,-1.5) $x_1 \odot \Delta^{n-1}$;
(v4) at (-2,0) $x_1 \odot \Delta^{n-2}$;
(v5) at (1,0) $x_2 \odot \Delta^{n-2}$;
(v6) at (1,1.5) $\dots$;
at (1.5,2) $\dots$;
(v7) at (1.5,2.5) $\dots$;
(v8) at (3.5,2.5) $x_{n-1} \odot \Delta^1$;
(v9) at (3.5,4) $x_{n-1}$;
(v10) at (5.5,4) $x_n$;
[->]  (v1) edge node [left,font = ]$1_{x_0} \odot d^0$ (v2);
[->]  (v1) edge node [above,font = ]$f_1 \odot \Delta^{n-1}$(v3);
[->]  (v4) edge node [left,font = ]$1_{x_1} \odot d^0$ (v3);
[->]  (v4) edge node [above,font = ]$f_2 \odot \Delta^{n-2}$ (v5);
[->]  (v6) edge (v5);
[->]  (v7) edge (v8);
[->]  (v9) edge node [left,font = ]$1_{x_{n-1}} \odot d^0$ (v8);
[->]  (v9) edge node [above,font = ]$f_n$ (v10);

We will denote this colimit by $M_\sigma$ and refer to it as the (higher) mapping cylinder of $\sigma$.

Next we may wonder about the association
$$\sigma \mapsto M_\sigma$$

of the mapping cylinder to an $n$-simplex in $\mathcal{C}_0$. As expected this extends to define a functor

$$M_* : \text{\textbf{Gro}}(\mathcal{C}_0)^{op} \rightarrow \mathcal{C}_0$$

To construct this functor we consider a map $\theta: [m] \rightarrow [n]$ in $\Delta$ and then tries to see that there is an induced map $M_{\theta(\sigma)} \rightarrow M\sigma$ by constructing a map between the diagrams that form these mapping cylinders. This is not a difficult exercice and we leave it to the reader. We also invite the reader to contemplate the induced map geometrically in light of the picture we drew in the first section.

Finally we may define (or conclude depending on where we start)
$$\text{hocolim F} = \text{colim} \{ M_* \circ \text{\textbf{Gro}}(F)\}$$

where the functor on the right is the composite

$$\text{\textbf{Gro}}(\mathcal{J})^{op} \xrightarrow{\text{\textbf{Gro}}(F)} \text{\textbf{Gro}}(\mathcal{C}_0)^{op} \xrightarrow{M_*} \mathcal{C}_0$$

All we have done is form the mapping cylinder corresponding to each simplex in $\mathcal{J}$ and then we glued these higher cylinders along their faces faces. Homotopy limits are defined by dualizing everything.

The above are labelled "local homotopy colimits" in [22]. The reason being they do not necessarily satisfy the "global" properties to define derived functors. In the setting of simplicial enrichment one defines homotopy colimits without ever mentioning weak equivalences at all.

§.§ Double colimits and homotopy colimits

Now we reveal the promised connection between double colimits in higher equipments and homotopy colimits. The fist were defined for a simplicial category and the latter can be defined in various contexts (sSet-categories, model categories etc, see [2]). So we start by building a bridge between the two worlds.

Let $\mathbb{E}$ be a simplicial category.There is a sSet-category $\mathbb{E}_v$ with:

($\bullet$) objects those of $\mathbb{E}_0$

($\triangle$) $n$-simplicies of the mapping space for two objects $x,y \in \mathbb{E}_0$ given by

$$\mathbb{E}_v(x,y)_n = \mathbb{E}_n(s_0^{(n)}x, s_0^{(n)}y)$$

($\rightsquigarrow$) for a map $\theta: [m] \rightarrow [n]$ in $\Delta$ the induced map given as

$$\mathbb{E}(\theta) : \mathbb{E}_n(s_0^{(n)}x, s_0^{(n)}y) \rightarrow \mathbb{E}_m(s_0^{(m)}x, s_0^{(m)}y)$$

($\circ$) composition given in the obvious way by composition in the categories $\mathbb{E}_n$

The above construction is derived from the analogy between the following figures

(v1) at (-4,2.5) $x$;
(v2) at (-2.5,3.5) $x$;
(v3) at (-1,2.5) $x$;
(v4) at (-4,0.5) $y$;
(v5) at (-2.5,1.5) $y$;
(v6) at (-1,0.5) $y$;
[-]  (v1) edge[double] (v2);
[-]  (v2) edge[double] (v3);
[-]  (v1) edge[double] (v3);
[->]  (v1) edge node[left, font= ]$f_0$ (v4);
[->]  (v2) edge node[left, font= ]$f_1$ (v5);
[->]  (v3) edge node[right, font= ]$f_2$ (v6);
[-]  (v4) edge[double] (v5);
[-]  (v5) edge[double] (v6);
[-]  (v4) edge[double] (v6);

(v1) at (-2,3.5) $x$;
(v2) at (-2,0.5) $y$;
[->, bend right = 50]  (v1) edge node[left, font= ]$f_0$ (v2);
[->, bend left  = 50]  (v1) edge node[right, font= ]$f_2$ (v2);
[->]  (v1) edge node[ font= ]$f_1$ (v2);

(v3) at (-2.8,2.2) ;
(v4) at (-2,1.6) ;
[->, bend right = 15]  (v3) edge[double] (v4);

(v5) at (-1.2,2.2) ;
[->, bend right = 15]  (v4) edge[double] (v5);

[->, bend left = 10, dashed]  (v3) edge[double] (v5);

On the left we have depicted a map in $\mathcal{\mathbb{E}}_2$ between $s_0^{(2)}x$ and $s_0^{(2)}y$ and on the right we have depicted a $2$-simplex in the mapping space between $x$ and $y$ in the sSet-category $\mathbb{E}_v$.

This construction is also analogous to the construction of the vertical $2$-category associated to a double category. So we have discovered something important:

The vertical direction of a simplicial category (when looked at as a two-fold structure) consists of a simplicially enriched category!

We dare rephrase the above and say that double categories are to $2$-categories what simplicial categories are to simplicially enriched categories, and write

$$\frac{\text{double categories}}{2\text{-categories}} = \frac{\text{simplicial categories}}{\text{\textbf{sSet}-categories}}$$

The connection between simplicial categories and simplicially enriched categories is not unknown, although our construction and interpretation seems to be novel (to the best of our knowledge). If $\mathcal{C}$ is a sSet-category, besides $\mathcal{C}_0$ we may construct categories $\mathcal{C}_n$ with

($\bullet$) objects those of $\mathcal{C}$

($\rightarrow$) morphism set between two objects $x$ and $y$ being the $n$-simplicies of their mapping space

$$\mathcal{C}_n(x,y) = \mathcal{C}(x,y)_n$$

($\circ$) composition as prescribed by the composition map in $\mathcal{C}$

The reader can utilize the above pictures again to grasp the categories $\mathcal{C}_n$, and also the fact that the mapping $[n] \mapsto \mathcal{C}_n$ gives us a simplicial category, which we denote $|\mathcal{C}|$, with faces and degeneracies defined in the obvious way (this is mentioned in [20] when discussing the coherent nerve construction).

Both constructions extend to give us functors and as a matter of fact we have an adjunction

$$|\cdot| : \textbf{sSet-Cat} \rightleftarrows \textbf{sCat} : (\cdot)_v$$

with $|\cdot|$ the left adjoint and $(\cdot_v)$ the right adjoint. Moreover we have $|\mathcal{C}|_v = \mathcal{C}$ for all $\mathcal{C}$ and $|\cdot|$ fully faithful. Next we state a theorem whose proof is easy but meaning essential in light of this work.

Let $\mathbb{E}$ be a simplicial category. If $\mathbb{E}$ has double colimits then $\mathbb{E}_v$ is cotensored.

Let $x$ be an object in $\mathbb{E}$ and $K$ be a simplicial set. Consider the constant diagram

$$K_x : K \rightarrow \mathbb{E}$$

which sends everything to $x$. Define

$$K \odot x = \text{dcolim}{K_x}$$

In particular

$$\Delta^n \odot x = \bot_{s^{(n)}x}$$

Let $y$ be another object. Then for all $[n] \in \Delta$ we have

\begin{align*}
\textbf{sSet}(\Delta^n, \mathbb{E}_v(x,y)) &= \mathbb{E}_v(x,y)_n \\ &= E_n(s_0^{(n)}x, s_0^{(n)}y) \\
&\cong \mathbb{E}_0(\text{dcolim}{\Delta^n_x}, y) \\
&= \mathbb{E}_0(\Delta^n \odot x, y) \\
&= \mathbb{E}_{v0}(\Delta^n \odot x, y)
\end{align*}

Since every simplicial set $K$ is a colimit of its simplicies we conclude

$$\textbf{sSet}(K, \mathbb{E}_v(x,y)) \cong \mathbb{E}_{v0}(K\odot x, y)$$

for all $x$ and $y$.

Dually we have

$$\{K,x \} = \text{dlim}{K_x}$$

and in particular

$$\{ \Delta^n, x \} = \top_{s^{(n)}x}$$

Now we state and prove the main theorem of this work:

In light of the construction theorem for double colimits (Theorem <ref>)  it is enough to consider $\mathcal{J} = \Delta^n$. In this case, for an $n$-simplex $\sigma$ in $\mathbb{E}_0$, the tower representation (Proposition <ref>) of $\sigma^*$ together with Proposition  <ref> imply that morphism in $E_n$
$$\sigma^* \rightarrow s^ny$$
for some $y \in \mathbb{E}_0$ corresponds precisely to morphisms from the step diagram corresponding to $M_\sigma$ to $y$.

We would like to conclude by duality that the theorem holds for homotopy limits as well by our proof relies on the fact that colimits interact well with cotabulators of universal extensions. We address this issue in the next section.

§.§ Duality

The equipment property as postulated previously states "every morphism from boundaries extends universally". We can also consider the dual which states "every morphism to a boundary lifts universally". Perhaps we can refer to the first as the "left equipment property" and the latter as "right equipment property".

More precisely, let $\mathbb{E}$ be a simplicial category, $y \in \mathbb{E}_n$, $x^\bullet : \partial \Delta^n \rightarrow \mathbb{E}$ and $f^\bullet: x^\bullet \rightarrow \partial y$ be a morphism. Then we
say $\mathbb{E}$ is a right equipment if given the above there if $x \in \mathbb{E}_n$ and $f: x\rightarrow y$ satisfying:

i) $\partial f = f^\bullet$ and as a consequence $\partial x = x^\bullet$

ii) Any morphism $g : z \rightarrow y$ such that $\partial g$ factors through $f^\bullet$ factors uniquely through $f$.

Denote such a universal lift $(f^\bullet)y$

In a right equipment we can perform all the constructions we did on the left side and obtain a theorem about homotopy limits. We will present just a sketch. First we see that $\textbf{sSet}^\sharp$ is a right equipment. Given $X^\bullet$, $Y$ and $f^\bullet$ as above we can obtain the universal lift as a pullback

(v4) at (-3.5,3) $(f^\bullet)Y$;
(v1) at (-1.5,3) $\partial Y$;
(v3) at (-3.5,1) $X^\bullet$;
(v2) at (-1.5,1) $Y$;
[left hook->]  (v1) edge node[right]$\partial$ (v2);
[->]  (v3) edge node[above]$f^\bullet$ (v2);
[->, dashed]  (v4) edge (v3);
[->, dashed]  (v4) edge (v1);

Let $\mathbb{E}$ be a right equipment. Then we may define the (right) companion ${}^*\sigma$ of an $n$-simplex

$$\sigma : x_0 \xrightarrow{f_1} x_1 \xrightarrow{f_2} \dots \xrightarrow{f_n} x_n$$

in $\mathbb{E}_0$ together with a structure map $\zeta_\sigma : {}^*\sigma \rightarrow s^nx_n$ recursively as

$${}^*\sigma= \begin{cases}
\sigma &\text{if} \ \  n=0 \\
(\zeta_{d_n\sigma}, \dots, \zeta_{d_0\sigma})s^nx_n &\text{if} \ \ n>0
\end{cases}$$

We define $\zeta_\sigma = 1_\sigma $ for $\sigma \in \mathbb{E}_0$ and we define $\zeta_\sigma$ to be the induced map $(\zeta_{d_n\sigma}, \dots, \zeta_{d_0\sigma})s^nx_n \rightarrow s^nx_n$.

This construction preserves faces but not degeneracies. As previously the axioms guarantee a comparison map

$$s_i{}^*\sigma \rightarrow {}^*(s_i\sigma)$$

so that we obtain a lax morphism

$${}^*(\cdot) : \mathbb{E}_0 \rightarrow \mathbb{E}$$

We may define the limit of a lax horizontal diagram in $\mathbb{E}$ in the obvious way. Then $\mathbb{E}_v$ is tensored if $\mathbb{E}$ has double limits. The tensoring is given by

$$\{\Delta^n, x\} = \top_{s^{(n)}x}$$

and in general $\{K, x\} = \text{dlim}K_x$ for $K\in \textbf{sSet}$.

Let $\mathcal{J}$ be an indexing category and $F: \mathcal{J} \rightarrow \mathbb{E}_0$ be a functor. Then we have

$$\text{dlim} {}^*F \cong \text{holim} F$$

where ${}^*F$ is the composite

$$\mathcal{J} \xrightarrow{F} E_0 \xrightarrow{{}^*(\cdot)} \mathbb{E}$$

and the homotopy colimit on the left side is interpreted in the simplicially enriched category $\mathbb{E}_v$.

The example of simplicial sets seems to indicate that we consider the case in which both the left and right equipment property are satisfied and left companions agree with right companions. We leave these considerations for future work.

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11institutetext:  Budker Institute of Nuclear Physics SB RAS, Novosibirsk,
630090, Russia
Novosibirsk State University, Novosibirsk, 630090, Russia

Neutrino, muon,pion, and other elementary particle detectors; cosmic ray
detectors Liquid noble gases Dark matter

# Neutral bremsstrahlung electroluminescence in noble liquids

E. Borisova 1122 A. Buzulutskov111Corresponding author. Email:
<EMAIL_ADDRESS>11221122

###### Abstract

Proportional electroluminescence (EL) is the physical effect used in two-phase
dark matter detectors, to optically record in the gas phase the ionization
signal produced by particle scattering in the liquid phase. In our previous
works the presence of a new EL mechanism in noble gases, namely that of
neutral bremsstrahlung (NBrS), was demonstrated both theoretically and
experimentally, in addition to the ordinary EL mechanism due to excimer
emission. In this work we show that the similar theoretical approach can apply
to noble liquids, namely to liquid helium, neon, argon, krypton and xenon. In
particular, the photon yields and spectra for NBrS EL in noble liquids have
for the first time been calculated, using the electron energy and transport
parameters obtained in the framework of Cohen-Lekner and Atrazhev theory. The
relevance of the results obtained to the development of noble liquid detectors
for dark matter searches and neutrino experiments is discussed.

###### pacs:

95.55.Vj

###### pacs:

61.25.Bi

###### pacs:

95.35.+d

##  1 Introduction

In noble liquid detectors for dark matter searches [1] and low-energy neutrino
experiments [2], the scattered particle produces two types of signals: that of
primary scintillation, produced in the liquid and recorded promptly (”S1”),
and that of primary ionization, produced in the liquid and recorded with a
delay (”S2”). In two-phase (liquid-gas) detectors [3], to record the S2 signal
proportional electroluminescence (EL) is used produced by drifting electrons
in the gas phase under high enough electric fields.

According to modern concepts [4], there are three mechanisms responsible for
proportional EL in noble gases: that of excimer (e.g. Ar${}^{*}_{2}$) emission
in the vacuum ultraviolet (VUV) [5], that of emission due to atomic
transitions in the near infrared (NIR) [6, 7], and that of neutral
bremsstrahlung (NBrS) emission in the UV, visible and NIR range [8]. These
three mechanisms are referred to as excimer (ordinary) EL, atomic EL and NBrS
EL, respectively.

NBrS EL is due to bremsstrahlung of drifting electrons scattered on neutral
atoms:

$\displaystyle e^{-}+\mathrm{A}\rightarrow e^{-}+\mathrm{A}+h\nu\;.$ (1)

The presence of NBrS EL in two-phase Ar detectors has for the first time been
demonstrated in our previous work [8], both theoretically and experimentally.
Recently, the similar theoretical approach has been applied to all noble
gases, i.e. overall to He, Ne, Ar, Kr and Xe, to calculate the photon yields
and spectra for NBrS EL [9]. NBrS EL in noble gases was further studied
experimentally in [10, 11, 12, 13, 14, 15, 16, 17] and theoretically in [18].

On the other hand, much less is known about proportional EL in noble liquids
[4, 19, 20, 21, 22, 23, 24]. In a sense, the experimental data are even
confusing. Indeed, in liquid Ar the observed threshold in the electric field
for proportional EL, of about 60 kV/cm [4, 23], was 2 orders of magnitude less
than expected for excimer EL [24]. In liquid Xe, the EL threshold was more
reasonable, around 400 kV/cm, but some puzzling EL events were observed below
this threshold [21].

In our previous works [8, 4] it was suggested that these puzzling events at
unexpectedly low fields might be induced by proportional EL produced by
drifting electrons in a noble liquid due to NBrS effect, the latter having no
threshold in the electric field. In this work we verify this hypothesis,
namely we extend the theoretical approach developed for noble gases to noble
liquids in order to develop a quantitative theory that can predict the photon
yields and spectra for NBrS EL in all noble liquids.

What is new in this work is that the electron energy and transport parameters
in noble liquids are calculated in the framework of rigorous Cohen-Lekner [25]
and Atrazhev [26] theory. In this theory, the electron transport through the
liquid is considered as a sequence of single scatterings on the effective
potential. Therefore, such a parameter as electron scattering cross section
can be used in the liquid in a way similar to that of the gas [3]. An
important concept of the theory is the distinction between the energy transfer
scattering, which changes the electron energy, and that of momentum transfer,
which only changes the direction of the electron velocity. Both processes have
been assigned separate cross sections [25, 26, 24]: that of energy transfer
(or else effective) and that of momentum transfer. These are obvious analogs
of those in the gas, namely of the total elastic and momentum transfer
(transport) cross sections, respectively. The latest modifications of the
theory can be found elsewhere [27, 28].

Accordingly, in this work the photon yields and spectra are calculated for
NBrS EL in all noble liquids: in liquid He, Ne, Ar, Kr and Xe. The relevance
of the results obtained to the development of noble liquid detectors for dark
matter searches and neutrino detection is also discussed.

##  2 Theoretical formulas

To calculate the photon yields and spectra for NBrS EL in noble liquids we
used the approach developed for noble gases in [8]. Let us briefly recall the
main points of this approach.

The differential cross section for NBrS photon emission is expressed via
electron-atom total elastic cross section ($\sigma_{el}(E)$) [8, 29, 30, 31,
32, 33]:

$\displaystyle\frac{d\sigma}{d\nu}=\frac{8}{3}\frac{r_{e}}{c}\frac{1}{h\nu}\left(\frac{E-h\nu}{E}\right)^{1/2}\times\hskip
40.0pt$ $\displaystyle\times\ [(E-h\nu)\ \sigma_{el}(E)\ +\ E\
\sigma_{el}(E-h\nu)]\;,$ (2)

where $r_{e}=e^{2}/mc^{2}$ is the classical electron radius, $c=\nu\lambda$ is
the speed of light, $E$ is the initial electron energy and $h\nu$ is the
photon energy.

To be able to compare results at different medium densities and temperatures,
we need to calculate the reduced EL yield ($Y_{EL}/N$) as a function of the
reduced electric field ($\mathcal{E}/N$), where $\mathcal{E}$ is the electric
field and $N$ is the atomic density. The reduced EL yield is defined as the
number of photons produced per unit drift path and per drifting electron,
normalized to the atomic density; for NBrS EL it can be described by the
following equation [8]:

$\displaystyle\left(\frac{Y_{EL}}{N}\right)_{NBrS}=\int\limits_{\lambda_{1}}^{\lambda_{2}}\int\limits_{h\nu}^{\infty}\frac{\upsilon_{e}}{\upsilon_{d}}\frac{d\sigma}{d\nu}\frac{d\nu}{d\lambda}f(E)\
dE\ d\lambda\;,$ (3)

where $\upsilon_{e}=\sqrt{2E/m_{e}}$ is the electron velocity of chaotic
motion, $\upsilon_{d}$ is the electron drift velocity,
$\lambda_{1}-\lambda_{2}$ is the sensitivity region of the photon detector,
$d\nu/d\lambda=-c/\lambda^{2}$, $f(E)$ is the electron energy distribution
function normalized as

$\displaystyle\int\limits_{0}^{\infty}f(E)\ dE=1\;.$ (4)

The distribution functions with a prime, $f^{\prime}=f/E^{1/2}$, is often used
instead of $f$, normalized as

$\displaystyle\int\limits_{0}^{\infty}E^{1/2}f^{\prime}(E)\ dE=1\;.$ (5)

$f^{\prime}$ is considered to be more enlightening than $f$, since in the
limit of zero electric field it tends to Maxwellian distribution.

Consequently, the spectrum of the reduced EL yield is

$\displaystyle\frac{d(Y_{EL}/N)_{NBrS}}{d\lambda}=\int\limits_{h\nu}^{\infty}\frac{\upsilon_{e}}{\upsilon_{d}}\frac{d\sigma}{d\nu}\frac{d\nu}{d\lambda}f(E)\
dE\ \;.$ (6)

In our previous works [8, 9], the electron energy distribution function and
drift velocity in noble gases, at a given reduced electric field, were
calculated using Boltzmann equation solver [34].

In this work, we follow exactly the Atrazhev paper [26] to calculate the
electron energy distribution function and drift velocity in noble liquids.
Another modification is that the total elastic cross section in Eq. 2 is
replaced with the energy transfer cross section for electron transport through
the liquid. With these two modifications all the Eqs. 2,3,4,6 can directly
apply to noble liquids.

##  3 Cross sections, electron energy distribution functions and drift
velocities in noble liquids

According to Cohen-Lekner and Atrazhev theory the drift and heating of excess
electrons by an external electric field in the liquid are determined by two
parameters, the collision frequency of energy transfer ($\nu_{e}$) and that of
momentum transfer ($\nu_{m}$) [26]:

$\displaystyle\nu_{e}=\delta N\sigma_{e}(E)(2E/m)^{1/2}\>,$ (7)
$\displaystyle\nu_{m}=N\sigma_{m}(E)(2E/m)^{1/2}\>,$ (8)
$\displaystyle\sigma_{m}(E)=\sigma_{e}(E)\widetilde{S}(E)\,.$ (9)

Here $N$ is the atomic density of the medium; $E$ is the electron energy;
$\delta=2m/M$ is twice the electron-atom mass ratio; $\sigma_{e}(E)$ and
$\sigma_{m}(E)$ is the energy transfer (effective) and momentum transfer
electron scattering cross section in the liquid, respectively;
$\widetilde{S}(E)$ is the function, which takes into account liquid structure.

To calculate collision frequencies one need to know $\sigma_{e}(E)$ and
$\sigma_{m}(E)$; for liquid Ar, Kr and Xe these were given in [26]: see Fig. 1
(top). For comparison, Fig. 1 (bottom) presents the total elastic cross
sections for gaseous Ne, Ar, Kr, and Xe taken from the BSR database [35];
since for He it is not available, the momentum transfer cross section is shown
instead taken from the Biagi database [36].

Figure 1: Top: Electron scattering cross sections in liquid Ar, Kr and Xe as a
function of electron energy, namely that of energy transfer (or effective),
$\sigma_{e}$, and that of momentum transfer, $\sigma_{m}$, both taken from
[26]. Bottom: Electron scattering cross section in noble gases as a function
of electron energy: that of total elastic for Ne, Ar, Kr, and Xe, taken from
the BSR database [35], and that of momentum transfer for He, taken from the
Biagi database [36].

The electron distribution function $f^{\prime}(E)$ in a strong electric field
is expressed via both collision frequencies [26]:

$\displaystyle
f^{\prime}(E)=f(0)exp\left(-\int\limits_{0}^{E}\frac{3m\nu_{e}(E)\nu_{m}(E)}{2e^{2}\mathcal{E}^{2}}dE\right).$
(10)

The constant $f(0)$ is determined from the normalization condition of Eq. 5.

Using the electron energy distribution functions, one can calculate the
electron drift velocity in the liquid [26]:

$\displaystyle\upsilon_{d}=-\frac{2}{3}\frac{e\mathcal{E}}{m}\int\limits_{0}^{\infty}\frac{E^{3/2}}{\nu_{m}(E)}\frac{df^{\prime}}{dE}dE.$
(11)

It is shown in Fig. 2 as a function of the reduced electric field, the latter
being expressed in Td units: 1 Td = $10^{-17}$ V cm2. It is possible to check
the correctness of the distribution functions by comparing the calculated and
measured electron drift velocities: this is done in Fig. 2 using the
experimental data compiled in [37]. It can be seen that the theoretical and
experimental drift velocities are in a reasonable agreement, within a factor
of 2, thus confirming the correctness of the calculated distribution functions
for liquid Ar, Kr and Xe.

Figure 2: Comparison of electron drift velocity ($\upsilon_{d}$) in liquid Ar,
Kr and Xe theoretically calculated in this work (curves) with that measured in
experiment [37] (data points). The color of the curve and the data points is
the same for a given noble liquid.

It should be remarked that in light noble liquids, He and Ne, the Cohen-Lekner
and Atrazhev theory cannot apply to calculate the electron energy distribution
functions, since the appropriate cross sections for electron transport in the
liquid, $\sigma_{e}(E)$ and $\sigma_{m}(E)$, are not available in the
literature. Thereby in the following for these liquids a ”compressed gas”
approximation will be used, similarly to that developed in [9]. In this
approximation, the Eqs. 2,3,4,6 apply directly as for the gas, i.e. with
electron energy distribution function and drift velocity obtained using
Boltzmann equation solver, with the input elastic cross sections taken for the
gas from Fig. 1 (bottom), and with the atomic density $N$ equal to that of the
liquid.

Table 1: Properties of noble gases and liquids, and parameters of neutral bremsstrahlung (NBrS) electroluminescence (EL) theoretically calculated in this work. No |  Parameter |  He |  Ne |  Ar |  Kr |  Xe
---|---|---|---|---|---|---
(1) |  Boiling temperature at 1.0 atm, $T_{b}$ [38] (K) |  $4.215$ |  $27.07$ |  $87.29$ |  $119.80$ |  $165.05$
(2) |  Gas atomic density at $T_{b}$ and 1.0 atm, derived from [38] (cm-3) |  $2.37\cdot 10^{21}$ |  $3.41\cdot 10^{20}$ |  $8.62\cdot 10^{19}$ |  $6.18\cdot 10^{19}$ |  $5.75\cdot 10^{19}$
(3) |  Liquid atomic density at $T_{b}$ and 1.0 atm, derived from [38] and from [39] for Xe (cm-3) |  $1.89\cdot 10^{22}$ |  $3.59\cdot 10^{22}$ |  $2.10\cdot 10^{22}$ |  $1.73\cdot 10^{22}$ |  $1.35\cdot 10^{22}$
(4) |  Threshold in electric field for excimer EL in noble liquid deduced from the corresponding threshold in noble gas by reduction to the atomic density of the liquid, obtained using data of [9] (kV/cm) |  $1134$ |  $538$ |  $840$ |  $519$ |  $472$
(5) |  Number of photons for NBrS EL in noble liquid produced by drifting electron in 1 mm thick EL gap at $T_{b}$ and 1.0 atm, at electric field of 100 kV/cm |  $0.13$ |  $2.5$ |  $0.93$ |  $1.6$ |  $1.1$
(6) |  The same at 500 kV/cm |  $4.3$ |  $40$ |  $12$ |  $24$ |  $30$

The values of the atomic densities for the gas and liquid phases at boiling
temperatures at 1 atm are presented in Table 1. We will see in the following
on the example of heavy noble liquids that ”compressed-gas” approximation
works well: the difference in photon yields for NBrS EL between the ”liquid”
and ”compressed-gas” approximation is not that large, remaining within a
factor of 1.5.

It should be also remarked, that all the calculations in this work were
provided for atomic densities of the medium, liquid or gas, corresponding to
the boiling temperature of a given noble element at 1 atm.

##  4 Operational range of reduced electric fields in noble liquids for NBrS
EL

It is obvious that NBrS EL in noble liquids is much weaker than excimer EL and
thus becomes insignificant above the electric field threshold for excimer EL.
Table 1 gives an idea of these thresholds in noble liquid deduced from the
corresponding threshold in noble gas by reduction to the atomic density of the
liquid, obtained using the data of [9].

To compare with the results for noble gases, one also need to determine the
operational range of reduced electric fields for NBrS EL in noble liquids from
the experimental works, where it was presumably observed and where the
operation electric field can be reliably estimated. Basically 3 works do fit
to these conditions: that of [40], operating the gas electron multiplier
(GEM,[41]) in liquid Ar, that of [23], operating the thick GEM (THGEM, [42])
in liquid Ar, and that of [21], operating the thin anode wire in liquid Xe.
Deduced from the absolute electric field values given in [40] and [21], the
required range of reduced electric fields within which NBrS EL was presumably
observed amounts to 0.1-5 Td. In particular, for liquid Ar this range
corresponds to electric fields ranging from 21 to 1040 kV/cm. We will restrict
our calculations to this range of fields.

##  5 NBrS EL spectra and yields in noble liquids

Figs. 3 show the NBrS spectra of the reduced EL yield for liquid Ar, Kr and Xe
at different reduced electric fields. The spectra were calculated by numerical
integration of Eq. 6.

One can see that NBrS EL spectra are similar in all noble liquids; moreover
they look almost identical to those obtained in noble gases at the same
reduced electric field: compare Fig. 3 to Fig. 10 of [9] at 5 Td. The spectra
are rather flat, extending from the UV to visible and NIR range at higher
reduced electric field, e.g. at 5 Td. In each noble liquid, the NBrS EL
spectrum has a broad maximum that gradually moves to longer wavelength with
decreasing electric field. At lower reduced electric field, in particular at
0.3 Td corresponding to 60 kV/cm in liquid Ar, the spectra have moved
completely to the visible and NIR ranges. In all noble liquids, the spectra
are mostly above 200 nm (in the UV, visible and NIR range), i.e. just in the
sensitivity region of commonly used photomultiplier tubes (PMTs) and silicon
photomultipliers (SiPMs).

Figure 3: Spectra of the reduced EL yield for NBrS EL in liquid Ar, Kr and Xe
at different reduced electric fields (0.3, 1 and 5 Td), calculated using Eq.
6. Figure 4: Reduced EL yield for NBrS EL at 0-1000 nm in liquid Ar, Kr and Xe
as a function of the reduced electric field, calculated in this work in the
framework of Cohen-Lekner and Atrazhev theory using Eq. 3 (solid lines). For
comparison, the reduced yield for NBrS EL at 0-1000 nm in noble gases is shown
calculated in [9] using Boltzmann equation solver (dashed lines). The color of
the curves are the same for a given noble element. The top scale shows the
corresponding absolute electric field in liquid Ar. Figure 5: Absolute EL
yield (number of photons per drifting electron per 1 cm) for NBrS EL at 0-1000
nm in noble liquids as a function of the absolute electric field, calculated
in this work. For heavy noble liquids (Ar, Kr and Xe) the rigorous Cohen-
Lekner and Atrazhev theory was used to calculate the electron energy and
transport parameters in the liquid, while for light noble liquids (He and Ne)
the ”compressed gas” approximation was used.

The EL yield for NBrS EL in noble liquids is presented in Figs. 4 obtained by
numerical integration of Eq. 3: the reduced EL yield is shown as a function of
the reduced electric field. For comparison, the reduced yield for NBrS EL in
noble gases is shown calculated in [9] using Boltzmann equation solver.

Surprisingly, this ”compressed-gas” approximation, successfully applied before
to describe NBrS EL in noble gases, has led to almost the same results as that
of the rigorous ”liquid” theory in terms of the reduced EL yields and spectra
when formally extrapolated to the atomic density of the noble liquid: for a
given noble element and given reduced electric field the difference between
them remains within a factor of 1.5 up to reduced electric field of 5 Td.

This fact indicates that the scaling law, stating that the reduced EL yield
($Y/N$) is a function of the reduced electric field ($\mathcal{E}/N$), is
valid not only for noble gases, but also for noble liquids to some extent, at
least as concerned the NBrS EL effect. It also indicates on the applicability
of the ”compressed gas” approximation to noble liquids at moderate reduced
electric fields, below 5 Td, thus justifying its use for light noble liquids,
He and Ne, where the Cohen-Lekner and Atrazhev theory cannot be used due to
the lack of the data.

Furthermore, Fig. 5 shows the practical photon yield suitable for verifying in
experimental conditions, namely the number of photons produced by drifting
electron per 1 cm in all noble liquids, as a function of the absolute electric
field. In this figure, for heavy noble liquids (Ar, Kr and Xe) the rigorous
Cohen-Lekner and Atrazhev theory was used to calculate the electron energy and
transport parameters in the liquid, while for light noble liquids (He and Ne)
the ”compressed gas” approximation was used, with the calculations identical
to those of [9]. The appropriate NBrS EL spectra and yields for He and Ne can
be found in [9].

Table 1 (items 5 and 6) gives an idea of the magnitude of the NBrS EL effect
in a practical parallel-plate EL gap, of a thickness of 1 mm: at a field of
500 kV/cm the photon yield varies as 4, 40, 12, 24 and 30 photons for He, Ne,
Ar, Kr and Xe, respectively. On the other hand, at 100 kV/cm the photon yield
is reduced by about an order of magnitude down to about 1 photon per drifting
electron in almost all noble liquids. It is remarkable that up to 600 kV/cm,
liquid Ne has the highest EL yield for NBrS EL, obviously due to much lower
elastic cross section between 1 and 10 eV of the electron energy compared to
other noble elements (see Fig. 1 (bottom)), resulting in stronger electron
heating by the electric field and thus in more intense NBrS photon emission.

##  6 Possible applications and discussion

In order to produce noticeable NBrS EL in noble liquids, one should provide
high enough electric fields, ranging from 50 to 500 kV/cm, in practical
devices. Based on previous experience, such devices might be GEMs [40], THGEMs
[23] and thin anode wires [21]. The parallel-plate EL gap of a thickness of 1
mm can also be considered, albeit being not tested in real experiment in noble
liquids at such high fields. It should be remarked that the larger EL gap
thickness, e.g. 1 cm, can hardly be used in practice due to the existing limit
on high voltage breakdowns in noble liquids: the absolute voltage before
breakdown cannot exceed values of about 100 kV in liquid He [43] and several
hundreds kV in other noble liquids [4, 44, 45].

It looks natural to use GEMs or THGEMs as EL plates instead of parallel-plate
EL gaps in noble liquids, since the former are more resistant to breakdowns
than the latter. Note that the NBrS EL spectrum is mostly in the visible and
NIR range: see Fig. 3. This implies a possible practical application of NBrS
EL in noble liquid detectors, namely the method of direct optical readout of
S2 signal in the visible range, i.e. without using a wavelength shifter (WLS).
A similar technique has been recently demonstrated in two-phase Ar detector
with direct SiPM-matrix readout using NBrS EL in the gas phase [16]. These
results have lead us to the idea of using THGEM plates in combination with
SiPM-matrices that have high sensitivity in the visible and NIR range, to
optically record the S2 signal in single-phase noble liquid detectors for dark
matter search and neutrino experiments. In addition, the recently proposed
transparent very-thick GEM [46] can be used as EL plate, with enhanced light
collection efficiency.

We can verify the theory of NBrS EL in noble liquids by experiments where it
was presumably observed, where the electric field is explicitly known and
where it is known how to convert the emitted photons into recorded
photoelectrons. At first glance, only two works qualify for these criteria:
[23] and [21].

In particular, in [23] the operation electric field in the center of THGEM
hole (1.5 mm height), of 60 kV/cm [40], corresponds to $\mathcal{E}/N$=0.3 Td
in liquid Ar, resulting in about 0.6 photons per drifting electron predicted
by the NBrS EL theory according to Figs. 4 and 5. However, this is more than 2
orders of magnitude smaller than the light gain reported in [23]. We therefore
suggest to interpret the results of [23] as caused by the presence of gas
bubbles associated to THGEM holes, inside which proportional EL in the gas
phase took place, similarly to what happens in Liquid Hole Multipliers [47].

Figure 6: Number of photoelectrons recorded in liquid Xe by a PMT as a
function of the voltage on $10~{}\mu m$ thick anode wire [21]: the
experimental data (data points) and linear fit of proportional EL to the data
(solid line) are shown, the latter defining the threshold of excimer EL. Top
scale shows the corresponding reduced electric field on anode wire surface.
For comparison, the theoretical assessment of the number of photoelectrons due
to NBrS EL obtained in this work is shown (area between dashed lines).

In [21], where puzzling EL events were observed in liquid Xe under the
threshold of excimer EL, the operation fields near the anode wire were much
higher, around 400 kV/cm. Fig. 6 shows the experimental data and linear fit of
proportional EL to the data, the latter defining the threshold of excimer EL.
In addition, the experimental conditions were explicitly described. This
allowed us to predict the number of photoelectrons recorded by the PMT due to
NBrS EL, although with some difficulties associated with highly inhomogeneous
field near the wire. Due to the latter, Eq. 3 if applied directly gives only
the lower limit of the event amplitude, since it does not take into account
the electron diffusion, which significantly increases the travel time of the
electron to the wire and thus the overall photon yield. We tried to take into
account the diffusion effect: as a result, the theoretical prediction in Fig.
6 is shown in the form of an area between two dashed curves, thus setting the
theoretical uncertainty. Within this uncertainty, the NBrS EL theory well
describes the puzzling underthreshold events, namely their absolute amplitudes
and the dependence on the anode voltage, which might be treated as the first
experimental evidence for NBrS EL in noble liquids.

##  7 Conclusion

In this work we systematically studied the effect of neutral bremsstrahlung
(NBrS) electroluminescence (EL) in all noble liquids: the photon yields and
spectra for NBrS EL have for the first time been theoretically calculated in
liquid He, Ne, Ar, Kr and Xe. For heavy noble liquids, the calculations were
done in the framework of Cohen-Lekner and Atrazhev theory describing the
electron energy and transport parameters in the liquid medium.

Surprisingly, the ”compressed-gas” approximation, successfully applied before
to describe NBrS EL in noble gases, has led to almost the same results as that
of the rigorous ”liquid” theory in terms of the reduced EL yields and spectra
when formally extrapolated to the atomic density of the noble liquid.

The predicted magnitude of the NBrS EL effect in a practical parallel-plate EL
gap, of a thickness of 1 mm, is noticeable: at a field of 500 kV/cm the photon
yield varies from 12 to 30 and 40 photons per drifting electron in liquid Ar,
Xe and Ne respectively. The NBrS EL spectra in noble liquids are in the
visible and NIR range.

The practical applications of the results obtained might be the use of THGEMs
as EL plates in combination with SiPM-matrices, to optically record the S2
signal in single-phase noble liquid detectors for dark matter search and
neutrino experiments.

###### Acknowledgements.

This work was supported by Russian Science Foundation (project no.
19-12-00008). It was done within the R&D program of the DarkSide-20k
experiment.

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# On channeling of charged particles in a single dielectric capillary

S.B. Dabagov  INFN Laboratori Nazionali di Frascati, Via E. Fermi 54, I-00044
Frascati (RM), Italy  RAS P.N. Lebedev Physical Institute, Leninsky Pr. 53,
119991 Moscow, Russia  NR Nuclear University MEPhI, Kashirskoe Sh. 31, 115409
Moscow, Russia  A.V. Dik  RAS P.N. Lebedev Physical Institute, Leninsky Pr.
53, 119991 Moscow, Russia

###### Abstract

We analytically analyse the motion of a nonrelativistic charged particle in a
cylindrical single capillary. The effective potential for interaction of a
charged particle with the inner surface of a capillary is derived as a sum of
the averaged atomic potential of the capillary wall surface and the induced
potential defined by collective surface excitations. We have shown that under
certain conditions this potential becomes attractive and may hold a particle
in a bound state due to the surface excitations that defines a so-called
”surface channeling” regime of motion. For the first time we have evaluated
the induced potential revealing two limits to form either a well or a barrier
that are delineated by the ratio of the insulator plasmon frequency to the
frequency defined by the particle motion in a capillary.

Capillary guiding Quantum particle Surface interaction Surface channeling

###### pacs:

34.50.Fa, 41.85.-p, 41.85.Ct, 61.85.+p

##  I Introduction

Controlling beams of charged particles, which determines the development in
accelerator, storage-ring and collider techniques, had occupied the minds of
physicists and engineers throughout the last century. Research in this area
remains relevant to this day directing us to fundamental studies on basic
knowledge of the origin and structure of the world (see the recent
comprehensive review [1]). The main principles of this branch of research are
formulated through the interaction of charged particle beams in strong
external electromagnetic fields, which can be created using solenoids,
dipoles, quadrupoles, etc., magnets of various types, laser-plasma sources [2,
3].

As known a strong electromagnetic interaction of charged particles takes also
place in solids (dense media), as well as in proximity to their surfaces,
sometimes offering the field gradients far beyond our technical capabilities.
The formation of strong fields of a certain geometry of small dimensions
allows actually handling the particle beams, endowing them with requested
characteristics for use in various experiments in both fundamental and applied
research [4].

One of the promising methods for shaping the beams is the use of capillaries
or capillary structures of various geometries. Despite the difference in the
physics of the processes, a clear example is capillary x-ray optics (for the
physics see in [5]), which has grown from a beautiful idea into a widely used
x-ray radiation control tool. In fact, even before the development of
capillary/polycapillary x-ray optics, in the early 1980s, the possibility of
using curved reflecting surfaces as an alternative method for the formation of
ion beams was discussed in a number of experimental works with theoretical
estimates based on the physics of charged particle channeling in crystals [6].
The interaction of an incident ion with a reflecting surface was described in
the approximation of a continuous surface atomic potential, confirming very
small angles of ion reflection from a smooth surface regardless of the change
in the beam charge characteristics. In order to increase the angles of
deflection of the beams, to use curved surfaces, in particular, dielectric
capillaries, was proposed ([7] and Refs. therein). However, these studies were
not properly continued, while the further use of capillaries (capillary
bundles) is presently known in relation to the beams of x rays and thermal
neutrons [8, 9].

On the contrary, since the first observations of the ions passage through
dielectric capillaries [10], this topic has attracted a growing attention of
physicists. The phenomenon is explained by the fact that when the ion beam is
transmitted by the capillary, at the beginning, the part of the beam settles
on the inner capillary surface and, having accumulated there, creates a
repulsive potential, which becomes a channel-guiding potential for a charged
particle [10, 11].

The undoubted interest in the development of new methods for handling particle
beams based on capillary systems has resulted in a wide spectrum of
experimental activities dedicated to this phenomenon, while theoretical works
counts disproportionally much less published articles (see in the detailed
review [12] and Refs. therein). Theoretical studies have mainly aimed at
simulating the motion of charged particle beams inside capillary insulators
using the Monte Carlo method [11, 13].

In this work, the interaction of a charged particle with the inner wall of a
dielectric capillary has been analytically evaluated, assuming the wall
thickness to be large for creating collective features in the interaction
potential. In addition, our approach shortens and slightly simplifies general
known calculations without losing the subtle features of the particle
interaction with the inner capillary wall.

##  II Surface excitation

Let us consider the electromagnetic field in a nonmagnetic substance of the
permittivity $\epsilon(\omega)$ that occupies the infinite space with a
cylindrical cavity of radius $R_{0}$ inside (Fig. 1). Then, in absence of free
currents and charges, based on Maxwell’s equations, the Fourier components of
the field scalar and vector potentials are defined by the following equation

Figure 1: Scheme of a cylindric cavity of radius $R_{0}$ in the infinite
dielectric, characterised by the permittivity $\epsilon(\omega)$, with a
particle of the charge $e$ moving inside the cavity. For the cavity we choose
$\epsilon=1$.

$\left(\nabla^{2}_{\bot}+\frac{\partial^{2}}{\partial
z^{2}}+\frac{\omega^{2}}{c^{2}}\epsilon(\omega)\right)\left(\begin{array}[]{ll}\varphi_{\omega}\\\
\mathbf{A}_{\omega}\end{array}\right)=0\>,$ (1)

where $\nabla^{2}_{\bot}$ is the part of the Laplace operator of the
coordinates transverse to the $Oz$ axis, which coincides with the cavity axis.
The field potentials satisfy the following gauge relation

$\nabla\mathbf{A}_{\omega}-i\frac{\omega}{c}\epsilon(\omega)\varphi_{\omega}=0$
(2)

As we are interested in characterising the damping surface waves excited at
the boundary cavity interface, the solution of Eq.(1) under the condition (2)
in a cylindrical coordinate system can be presented in the form (for ease of
perception the index $\omega$ is omitted)

$\varphi=\sum\limits_{q,n}\varphi^{0}_{n}f_{n}(\nu r)e^{i(qz+n\phi)}\>,$ (3)
$A_{z}=\sum\limits_{q,n}A^{0}_{n}f_{n}(\nu r)e^{i(qz+n\phi)}\>,$ (4)
$A_{r}=\sum\limits_{q,n}\left[-i\frac{q}{\nu}\left(qA_{n}^{0}+\frac{\omega}{c}\epsilon(\omega)\varphi_{n}^{0}\right)f_{n}^{\prime}(\nu
r)+\right.$ $\left.+\frac{1}{r}B_{n}^{0}f_{n}(\nu r)\right]e^{i(qz+n\phi)}\>,$
(5)
$A_{\phi}=\sum\limits_{q,n}\left[\frac{n}{\nu^{2}r}\left(qA_{n}^{0}+\frac{\omega}{c}\epsilon(\omega)\varphi_{n}^{0}\right)f_{n}(\nu
r)+\right.$ $\left.+i\frac{\nu}{n}B_{n}^{0}f^{\prime}_{n}(\nu
r)\right]e^{i(qz+n\phi)}\>,$ (6) $f_{n}(\nu
r)=\left\\{\begin{array}[]{ll}K_{n}(\nu
r),\nu=\sqrt{q^{2}-\epsilon\omega^{2}/c^{2}},r>R_{0}\\\ I_{n}(\nu
r),\nu=\sqrt{q^{2}-\omega^{2}/c^{2}},r\leq R_{0}\end{array}\right.$ (7)

Here $\epsilon(\omega)=1$ inside the hollow cavity $r<R_{0}$,
$\\{\varphi^{0}_{n},A_{n}^{0},B_{n}^{0}\\}$ are the constants, $n$ is the
integer, $K_{n}(\xi),I_{n}(\xi)$ are the modified Bessel functions. From the
condition of continuity of the components of the electric and magnetic fields
tangential to the interface, we derive the dispersion law $q(\omega)$ in the
new notation $x=qc/\omega$ and $\alpha=R_{0}\omega/c$

$\frac{x^{2}n^{2}(\epsilon-1)^{2}}{\alpha^{2}(x^{2}-1)(x^{2}-\epsilon)}=\left[\sqrt{x^{2}-1}\frac{K^{\prime}_{n}(\alpha\sqrt{x^{2}-\epsilon})}{K_{n}(\alpha\sqrt{x^{2}-\epsilon})}-\right.$
$\left.-\sqrt{x^{2}-\epsilon}\frac{I^{\prime}_{n}(\alpha\sqrt{x^{2}-1})}{I_{n}(\alpha\sqrt{x^{2}-1})}\right]\left[\epsilon\sqrt{x^{2}-1}\frac{K^{\prime}_{n}(\alpha\sqrt{x^{2}-\epsilon})}{K_{n}(\alpha\sqrt{x^{2}-\epsilon})}-\right.$
$\left.-\sqrt{x^{2}-\epsilon}\frac{I^{\prime}_{n}(\alpha\sqrt{x^{2}-1})}{I_{n}(\alpha\sqrt{x^{2}-1})}\right]$
(8)

Solving analytically Eq.(8) is a rather routine and complex procedure.
However, we can solve it numerically (Fig. 2). The curves of the figure
present two various solutions for different values of the parameter $\alpha$
and the first orders of Bessel function. As seen, the equation (8) can be
solved only at negative $\epsilon(\omega)$ in very narrow interval near the
point $\epsilon(\omega)=-1$. At higher values of the parameter $\alpha$ and/or
the Bessel function order we get the dependences presented in the bottom plot
of Fig. 2 but with the difference that the range of valid values
$\epsilon(\omega)$ rapidly decreases and tends to the value
$\epsilon(\omega)=-1$.

Figure 2: The solutions of the disperse equation (8) for two values of the
dimensionless parameter $\alpha$, which characterises the ratio of the cavity
radius to the wavelength of surface excitations, and the first three orders of
Bessel function. Both dependences reveal identical evolution with an increase
in the values $\alpha$ and $n$, while the valid values of $\epsilon(\omega)$
are essentially different.

Thus, neglecting the wavelengths of surface excitations of the order of the
cavity radius $\lambda\sim R_{0}$, we can conclude that the wave-vector of
surface excitations should be within the range $0\leq q\leq q_{0}$ ($q_{0}$ is
the wave-vector corresponding to cut-off frequency), while the corresponding
frequency $\omega_{s}$ can be reduced from the solution of the equation
$\epsilon(\omega_{s})=-1$.

Let examine now two physically significant limiting cases determined by the
parameter $\alpha=R_{0}/\lambda_{s}$, the ratio of the resonator radius
$R_{0}$ to the wavelength of surface excitations $\lambda_{s}$. For $\alpha\ll
1$, Eq.(8) can be solved only at $\epsilon(\omega)=-1$, while for $\alpha\gg
1$ the solution can be presented in the form of a series expansion in powers
of a small parameter $(1/\alpha)$

$x=\sqrt{\frac{\epsilon}{1+\epsilon}}+\frac{1}{2\alpha}\frac{\epsilon-1}{\sqrt{\epsilon(1+\epsilon)}}+...$
(9)

For large $\alpha$, i.e. for the wavelengths $\lambda_{s}\ll R_{0}$, the
dispersion relation $q(\omega)$ slightly differs from the case of a flat
reflecting plane [14]. Thus, if we neglect the wavelengths
$\lambda_{s}=c/\omega_{s}\sim R_{0}$, the frequency of surface excitations
$\omega_{s}$ is assumed to be constant, while the wave vector varies within
the range $0\leq q\leq q_{0}$.

At these assumptions, the charged particle interaction with the cavity surface
is described by the scalar potential operator of surface excitations

$\hat{\Phi}(\mathbf{r})=\hbar\sum\limits_{q,s}f_{s}(qr)\times$
$\times\left(g_{qs}\hat{c}_{qs}e^{i(qz+s\phi)}+g^{*}_{qs}\hat{c}^{+}_{qs}e^{-i(qz+s\phi)}\right)\>,$
(10)

where the excitation amplitudes $g_{qs}$ are calculated in Appendix 44, while
the function $f_{s}(qr)$ is defined by Eq.(7) at $\nu\approx q$.

##  III Interaction potential

Since we deal with a nonrelativistic charge moving in a cylindrical cavity,
the potential energy of interaction with the atomic system of a dielectric can
be calculated as the sum of Coulomb potentials, represented as a Fourier
series, split at the cut-off frequency $q_{0}$ [15]

$U_{int}(\mathbf{r})=\frac{4\pi}{V_{g}}\sum\limits_{\mathbf{r}_{j}}\left(\sum\limits_{\mathbf{q}>\mathbf{q}_{0}}\frac{e_{p}e_{j}}{q^{2}}e^{i\mathbf{q}(\mathbf{r}_{p}-\mathbf{r}_{j})}+\right.$
$\left.+\sum\limits_{\mathbf{q}<\mathbf{q}_{0}}\frac{e_{p}e_{j}}{q^{2}}e^{i\mathbf{q}(\mathbf{r}_{p}-\mathbf{r}_{j})}\right)\>,$
(11)

where $e_{p(j)},\mathbf{r}_{p(j)}$ are, respectively, the charges and radius-
vectors of the particle $p$ and the surface nuclei and electrons $j$, $V_{g}$
is the normalisation volume. The 1st term describes the interaction at small
distances $l<q_{0}^{-1}$ that does not allow the averaging at calculations
111For instance, for metals, $q_{0}^{-1}$ is of the order of the atom size
[16]. , while the 2nd - at $l>q_{0}^{-1}$, i.e. the interaction with
collective surface excitations, can be evaluated by averaging single
interaction features over all the system. As aforementioned, the latter is
described by the potential operator (10). The 1st term of Eq.(11), in turn, is
described as the sum of potential energies $V(\mathbf{r}_{p})$ for the
’particle - screened atomic’ interaction over all surface atoms [17] that in
view of the equation of motion for a particle transforms into the averaged
atomic potential of the surface.

Generally, for a particle of charge $e$ and mass $m$ moving inside a
cylindrical hollow cavity formed by an infinite insulator we can write the
Hamiltonian in the form

$\hat{H}=\int\hat{\psi}^{+}(\mathbf{r},z)\left(-\frac{\hbar^{2}}{2m}\nabla^{2}+V(\mathbf{r},z)\right)\hat{\psi}(\mathbf{r},z)d^{2}rdz+$
$+\hat{H}_{s}+\int\hat{\psi}^{+}(\mathbf{r},z)\hat{\Phi}(\mathbf{r},z)\hat{\psi}(\mathbf{r},z)d^{2}rdz\>,$
(12)

where $\mathbf{r}=(r,\phi)$ is the radius-vector in the transverse plane
$xOy$, while the $Oz$-axis coincides with the cavity longitudinal axis,
$\hat{\psi}$ is the particle field operator (fermion-operator), $\hat{\Phi}$
is the operator describing the particle interaction with the surface
excitations (boson-operator). The potential energy of the atom-particle
interaction $V(\mathbf{r})$ corresponds to the 1st term of Eq.(11), while the
operator $\hat{H}_{s}$ \- to the energy of non-interacting surface
excitations. Here, the particle field operator $\hat{\psi}$ is defined by the
sum over the quantum states of the motion

$\hat{\psi}(\mathbf{r},z)=\sum\limits_{k,n,l}\varphi_{knl}(\mathbf{r},z)\hat{a}_{knl}\>,$
(13)

where $\hat{a}_{knl}$ is the annihilation operator, $k$ is the z-projection of
the charged particle wave vector, $n,\>l$ are the radial and azimuthal quantum
numbers ($\mathbf{n}=(n,l)$). The wave function $\varphi_{knl}$ can be divided
into longitudinal and transverse components

$\varphi_{knl}(\mathbf{r},z)=\frac{1}{\sqrt{L}}u_{nl}(\mathbf{r})e^{ikz}$ (14)

with the cavity longitudinal size $L$. At these definitions the interaction
operator can be reduced to a simpler operator equation

$\hat{H}_{i}=\hbar\sum\limits_{q,s,k,\mathbf{n},\mathbf{n}^{\prime}}\left(\Gamma_{\mathbf{n}^{\prime}\mathbf{n}}(q,s)\hat{c}_{qs}\hat{a}^{+}_{k+q,\mathbf{n}^{\prime}}\hat{a}_{k\mathbf{n}}+\right.$
$\left.+\overline{\Gamma}_{\mathbf{n}^{\prime}\mathbf{n}}(q,s)\hat{c}^{+}_{qs}\hat{a}^{+}_{k\mathbf{n}}\hat{a}_{k+q,\mathbf{n}^{\prime}}\right)\>,$
(15)

where

$\Gamma_{\mathbf{n}^{\prime}\mathbf{n}}(q,s)=g_{qs}\int
f_{qs}(r)e^{is\phi}u^{*}_{\mathbf{n}^{\prime}}(\mathbf{r})u_{\mathbf{n}}(\mathbf{r})d^{2}r$
(16)

The time evolution of the boson operators $\hat{c}_{qs}$ in Eq.(15) can be
estimated introducing new operators
$\hat{C}_{qs}=\hat{c}_{qs}e^{i\omega_{qs}t}$,
$\hat{C}^{+}_{qs}=\hat{c}^{+}_{qs}e^{-i\omega_{qs}t}$ and
$\hat{A}_{k\mathbf{n}}=\hat{a}_{k\mathbf{n}}e^{i\Omega_{\mathbf{n}}(k)t}$,
$\hat{A}^{+}_{k\mathbf{n}}=\hat{a}^{+}_{k\mathbf{n}}e^{-i\Omega_{\mathbf{n}}(k)t}$
for the quantum energies of both surface excitations $\hbar\omega_{qs}$ and
particle $\hbar\Omega_{\mathbf{n}}(k)$. For getting the expression of
effective Hamiltonian $\hat{H}_{eff}$ we have followed the discussion of known
work [18], where the calculation procedure was in detail described. The only
difference is in the particle freedom, i.e. in our case the particle is free
in one direction, while in [18] \- in two ones. Hence, in further we can
briefly point out just the main steps.

Assuming the weak interaction we deal with, the new boson operators should
vary in time much faster than those of fermion. That allows easy integrating
the Heisenberg equation that results in the following expressions for the
boson operators

$\hat{C}_{qs}(t)=\sum\limits_{k,\mathbf{n},\mathbf{n}}\overline{\Gamma}_{\mathbf{n}^{\prime}\mathbf{n}}(q,s)\hat{A}^{+}_{k\mathbf{n}}\hat{A}_{k+q,\mathbf{n}^{\prime}}\times$
$\times\frac{e^{-i(\Omega_{\mathbf{n}^{\prime}}(k+q)-\Omega_{\mathbf{n}}(k)-\omega_{q,s})t}}{\Omega_{\mathbf{n}^{\prime}}(k+q)-\Omega_{\mathbf{n}}(k)-\omega_{q,s}-i\delta}\>,$
(17)
$\hat{C}^{+}_{qs}(t)=\sum\limits_{k,\mathbf{n},\mathbf{n}}\Gamma_{\mathbf{n}^{\prime}\mathbf{n}}(q,s)\hat{A}^{+}_{k+q,\mathbf{n}^{\prime}}\hat{A}_{k,\mathbf{n}}\times$
$\times\frac{e^{i(\Omega_{\mathbf{n}^{\prime}}(k+q)-\Omega_{\mathbf{n}}(k)-\omega_{q,s})t}}{\Omega_{\mathbf{n}^{\prime}}(k+q)-\Omega_{\mathbf{n}}(k)-\omega_{q,s}+i\delta}$
(18)

Using these expressions the effective Hamiltonian for the interaction of a
charged particle with the field of surface excitations can be presented as
follows

$\hat{H}_{eff}=\sum\limits_{k,q,\mathbf{n},\mathbf{n}^{\prime},s,\mathbf{j}}\frac{\hbar\Gamma_{\mathbf{n}^{\prime},\mathbf{n}}(q,s)\overline{\Gamma}_{\mathbf{j},\mathbf{n}}(q,s)\hat{a}^{+}_{k\mathbf{n}^{\prime}}\hat{a}_{k\mathbf{j}}}{\Omega_{\mathbf{n}^{\prime}}(k)-\Omega_{\mathbf{n}}(k-q)-\omega_{qs}+i\delta}$
(19)

In our case the particle longitudinal energy is much greater than the
transverse one $E_{\parallel}=\hbar^{2}k^{2}/(2m)\gg\hbar\omega_{\bot}^{n,l}$.
Hence, the denominator in Eq.(19) becomes equal to
$\triangle(k,q,s)=(\hbar/m)kq-(\hbar/2m)q^{2}-\omega_{q,s}+i\delta$, and,
since
$\sum_{\mathbf{n}}\left<u_{\mathbf{n}}(\mathbf{r}^{\prime})\left|\right.u_{\mathbf{n}}(\mathbf{r})\right>=\delta(\mathbf{r}-\mathbf{r}^{\prime})$,
the effective Hamiltonian is defined by the matrix element

$\hat{H}_{eff}=\int\hat{\psi}^{+}(\mathbf{r})\Phi(k,r)\hat{\psi}(\mathbf{r})d^{2}rdz\>,$
(20)

where

$\Phi(k,r)=\sum\limits_{q,s}\frac{\hbar\left|g_{qs}f_{qs}(r)\right|^{2}}{\triangle(k,q,s)}$
(21)

The real part of Eq.(20) is responsible for elastic processes in a complex
potential energy, while its imaginary part - for inelastic ones.

The change in the energy of surface excitations due to the interaction with
the particle can be estimated calculating the time derivative of the operator
of the energy of surface excitations

$\frac{\partial\hat{E}_{s}}{\partial
t}=\sum\limits_{q,s}\hbar\omega_{qs}\frac{\partial}{\partial
t}\hat{c}^{+}_{qs}(t)\hat{c}_{qs}(t)$ (22)

For that we use the expressions (17), (18), which describe the time evolution
for the boson operators. Substituting them into (22) and neglecting the recoil
term $\frac{\hbar}{2m}q^{2}$, we obtain

$\frac{\partial\hat{E}_{s}}{\partial
t}=\sum\limits_{q,s,\mathbf{n},\mathbf{n}^{\prime},\mathbf{n}_{1},\mathbf{n}_{1}^{\prime},k,\kappa}i\hbar\omega_{qs}\overline{\Gamma}_{\mathbf{n}^{\prime}\mathbf{n}}\Gamma_{\mathbf{n}_{1}^{\prime}\mathbf{n}_{1}}\hat{A}^{+}_{k,\mathbf{n}}\hat{A}_{k+q,\mathbf{n}^{\prime}}\times$
$\times\hat{A}^{+}_{\kappa+q,\mathbf{n}_{1}^{\prime}}\hat{A}_{\kappa,\mathbf{n}_{1}}\left(\frac{1}{\frac{\hbar}{m}kq-\omega_{qs}-i\delta}-\right.$
$\left.-\frac{1}{\frac{\hbar}{m}\kappa
q-\omega_{qs}+i\delta}\right)e^{i\frac{\hbar}{m}(\kappa-k)qt}$ (23)

In this expression we assumed the time dependence for the fermion operators to
be mostly defined by the exponent term. If now we apply the following rule

$\frac{1}{x-a\pm i\delta}=\mathcal{P}\frac{1}{x-a}\mp i\pi\delta(x-a)$ (24)

and take into account that at large times only the terms with $\kappa=k$ make
a notable contribution, then the expression (23) can be reduced to the
following

$\frac{\partial\hat{E}_{s}}{\partial
t}=-2\pi\sum\limits_{q,s,\mathbf{n},\mathbf{n}^{\prime},\mathbf{n}_{1},\mathbf{n}_{1}^{\prime},k}\hbar\omega_{qs}\overline{\Gamma}_{\mathbf{n}^{\prime}\mathbf{n}}\Gamma_{\mathbf{n}_{1}^{\prime}\mathbf{n}_{1}}\times$
$\times\hat{A}^{+}_{k,\mathbf{n}}\hat{A}_{k+q,\mathbf{n}^{\prime}}\hat{A}^{+}_{k+q,\mathbf{n}_{1}^{\prime}}\hat{A}_{k,\mathbf{n}_{1}}\delta\left(\frac{\hbar}{m}kq-\omega_{qs}\right)$
(25)

Assuming that at the initial moment the particle was in a state with quantum
numbers $(k,\mathbf{n})$, the particle energy loss can be defined considering
the time derivative of the particle energy to be equal to that of surface
excitations with the opposite sign (hereinafter, the longitudinal velocity
$v=\frac{\hbar}{m}k$ is introduced)

$\frac{\partial E_{p}}{\partial
t}=-2\pi\sum\limits_{q,s}\hbar\omega_{qs}\left|\Gamma_{\mathbf{n}\mathbf{n}}(q,s)\right|^{2}\delta\left(vq-\omega_{qs}\right)$
(26)

To obtain an expression for the particle energy loss as a function of its
coordinate, let assume that
$\left|u_{\mathbf{n}}^{*}u_{\mathbf{n}}\right|^{2}=\delta(r-r_{0})\delta(\phi-\phi_{0})/r$,
i.e. the particle is at the point $(r_{0},\phi_{0})$. Moreover, taking into
account the dispersion expression for surface excitations, namely, the
constancy of the frequency $\omega_{s}$ for any $q<q_{0}$ and passing from the
sum over $q$ to the integral, we obtain the following expression for the
energy loss of a particle in dependence on its position (to avoid confusion,
the index $s$ in (26) is substituted by $n$)

$\frac{\partial E_{p}}{\partial
t}=-\frac{L}{v}\hbar\omega_{s}\sum\limits_{n}\left|g_{qn}f_{qn}(r_{0})\right|^{2}_{q=\omega_{s}/v}$
(27)

For instance, the particle energy loss due to the interaction with surface
plasmons equals to

$-\frac{dE_{p}}{dz}=\left(\frac{e\omega_{s}}{v}\right)^{2}\sum\limits_{n=-\infty}^{\infty}\frac{K_{n}\left(\frac{\omega_{s}}{v}R_{0}\right)}{I_{n}\left(\frac{\omega_{s}}{v}R_{0}\right)}I_{n}^{2}\left(\frac{\omega_{s}}{v}r_{0}\right)\,,$
(28)

In the limiting case $R_{0}\to\infty$, Eq.(28) defines the energy loss of a
charged particle moving along the plane [19] (see Appendix 46)

$-\frac{dE}{dz}=e^{2}\left(\frac{\omega_{s}}{v}\right)^{2}K_{0}\left(2\frac{\omega_{s}}{v}x\right)\>,$
(29)

where $x=R_{0}-r_{0}$ is the particle-to-plane distance.

Ones we use Eq.(20), the equation of transverse motion inside a cavity can be
written in the following way

$\left(-\frac{\hbar^{2}}{2m}\nabla^{2}_{\bot}+\frac{1}{L}\int
V(\mathbf{r}_{\bot},z)dz+\Phi(k,r)\right)u_{\mathbf{n}}(\mathbf{r}_{\bot})=$
$=\hbar\omega_{\bot}^{\mathbf{n}}u_{\mathbf{n}}(\mathbf{r}_{\bot})\,,$ (30)

which is actually describes the channeled motion. Indeed, neglecting the
change in the particle longitudinal momentum results in averaging the atomic
potential over the surface plane. The surface potential is calculated by
summing the potentials of individual atoms, where the number of atoms per unit
length along the cavity axis is much greater than unity $N=n2\pi R_{0}\gg 1$.
After integration over the surface, the 2nd term in Eq.(30) becomes dependent
only on the distance from the cavity axis $(1/L)\int
V(\mathbf{r}_{\bot},z)dz\equiv\bar{V}(r)$.

Solving the equation for elastic processes, we consider only the real part of
induced potential, while the potential imaginary part leads to a finite
lifetime for each quantum state. Thus, the induced potential with the
amplitude of surface excitations $g_{qn}=g_{n}(qR_{0})$ is defined by

$U_{ind}(r)\equiv Re\left[\Phi(v,r)\right]=$
$=\frac{L}{2\pi}\sum\limits_{n=-\infty}^{\infty}\mathcal{P}\int\limits_{0}^{q_{0}}\frac{\hbar\left|g_{n}\left(qR_{0}\right)\right|^{2}}{vq-\omega_{s}}I_{n}^{2}\left(qr\right)dq\>,$
(31)

where $\mathcal{P}$ means the principal value of the integral.

Hence, we obtain the effective potential for a channeled particle inside a
cylindrical cavity as a sum of averaged and induced potentials
$U_{eff}(r)=\bar{V}(r)+U_{ind}(r)$. The 1st term is evaluated as a continuous
potential via known technique in channeling physics [20, 21] and does not
result in any unexpected behaviours, while the 2nd one brings us to new
features.

An approximate analysis of general induced potential for two extreme cases,
i.e. $\omega_{s}R_{0}/v\gg 1$ and $\omega_{s}R_{0}/v\ll 1$, is shown in Fig.
3. At $\omega_{s}R_{0}/v\ll 1$, the induced potential is defined as follows
(see Appendix 56) 222All the results stated below are valid under the
condition $q_{0}v>\omega_{s}$, which corresponds to the presence at least one
singularity within the integration interval in Eq.(31).

Figure 3: Dimensionless induced potential versus the distance from the cavity
axis. a). The graph at the condition $\omega_{s}R_{0}/v\gg 1$ shows two curves
corresponding different values of longitudinal velocity. As seen, at the
particle velocity increase the potential shape does not change, and the curve
minimum tends to the cavity center. b). The general induced potential at
$\omega_{s}R_{0}/v\ll 1$, while a concrete plotting is for $q_{0}R_{0}=10^{3}$
and $\omega_{s}=10^{15}$ c-1 with the characteristic value $q_{0}\sim 10^{7}$
c-1 corresponding the cavity radius $R_{0}\sim 10^{-4}$ cm.

$U_{ind}(r)=-\frac{e^{2}\omega_{s}}{2\pi
v}\frac{\epsilon_{0}-1}{(\epsilon_{0}+1)}\times$
$\times\ln{\left(\frac{q_{0}v}{\omega_{s}}-1\right)}\ln{\left(1-\frac{r^{2}}{R_{0}^{2}}\right)}+C$
(32)

with

$C=\frac{\epsilon_{0}-1}{\epsilon_{0}}\left[\ln{\left(\frac{q_{0}v}{\omega_{s}}-1\right)}\left(\ln{\left(\frac{\omega_{s}R_{0}}{2v}\right)}+\gamma_{0}\right)-\right.$
$\left.-\left(\mathrm{Li}_{2}\left(1-\frac{q_{0}v}{\omega_{s}}\right)-\mathrm{Li}_{2}\left(1\right)\right)\right]\>,$
(33)

where $\gamma_{0}$ is the Euler’s constant, $\mathrm{Li}_{2}(\xi)$ is the
polylogarithm function. On the contrary, at $\omega_{s}R_{0}/v\gg 1$ the
induced potential is evaluated by the following expression (see Appendix 56)
333Here, for clarity of the result, we set $q_{0}v\gg\omega_{s}$.

$U_{ind}(r)=-\frac{e^{2}\omega_{s}}{4\pi
v}\frac{\epsilon_{0}-1}{(\epsilon_{0}+1)}G\left(\frac{3\omega_{s}R_{0}}{2v}\left(1-\frac{r}{R_{0}}\right)\right)$
(34)

where the function

$G\left(\xi\right)=\int\limits_{0}^{\infty}\mathrm{Ei}\left(\xi\left(1+\frac{\mathrm{ch}{(t)}}{3}\right)\right)e^{-\xi(1+\mathrm{ch}{(t)}/3)}dt\>,$
(35)

reaches its maximum $G(\xi_{m})=2.59815$ at $\xi_{m}=0.545901$
($\mathrm{Ei}(\xi)$ is the exponential integral function). That allows
obtaining the potential minimum position

$\left.\frac{r}{R_{0}}\right|_{max}=1-0,3639\cdot\frac{v}{\omega_{s}R_{0}}$
(36)

with the minimum value of the induced potential

$U^{min}_{ind}=-0.2\cdot\frac{e^{2}\omega_{s}}{v}\frac{\epsilon_{0}-1}{(\epsilon_{0}+1)}$
(37)

If the particle velocity increases, the potential minimum position very slowly
shifts closer to the cavity center, keeping the potential shape itself
practically unchanged. At the cavity surface the induced potential
logarithmically diverges $U_{ind}(r\to R_{0})\sim\ln{(R_{0}-r)}$, while at the
cavity center it equals to

$U_{ind}(0)=-\frac{e^{2}\omega_{s}}{2v}\frac{\epsilon_{0}-1}{(\epsilon_{0}+1)}e^{-2\omega_{s}R_{0}/v}\mathrm{Ei}\left(2\frac{\omega_{s}R_{0}}{v}\right)$
(38)

In both cases considered, approaching the cavity surface, the induced
potential increases. However, the induced potential must be cut off for the
distances $R_{0}-r\sim q_{0}^{-1}$, therefore, the maximum value of the
induced potential in both limits can be estimated as 444In the following
estimates, we set $\dfrac{\epsilon_{0}-1}{\epsilon_{0}+1}\sim 1$.

$U^{min}_{ind}\simeq\frac{e^{2}\omega_{s}}{v}\ln{\left(\frac{q_{0}v}{\omega_{s}}-1\right)}\ln{\left(\frac{\omega_{s}}{q_{0}v}\right)}\,,$
(39)

in which the logarithms do not essentially contribute to the value of induced
potential. That allows us in further estimates to omit the logarithmic
factors, i.e. $U^{min}_{ind}\sim e^{2}\omega_{s}/v$.

Eq.(31) is obtained in view of smallness of the transverse energy in
comparison with the longitudinal one, $E_{\parallel}\gg E_{\bot}$, which also
imposes definite restrictions on the particle longitudinal velocity $v$.
Supposing the maximum of transverse energy $E_{\bot}$ to be equal to the
maximum of induced potential, we can write the limitation for longitudinal
energy $E_{\parallel}\gg\left(e^{2}\omega_{s}\sqrt{m}\right)^{2/3}$ as well as
for normalised longitudinal speed of a particle $\beta\equiv
v/c\gg\left(e^{2}\omega_{s}/(mc^{3})\right)^{1/3}$.

For example, for a channeled electron moving inside a cavity characterised by
surface oscillations of a plasma frequency $\omega_{s}\sim 10^{15}$ c-1, the
reduced potential is valid under the condition $E_{\parallel}\gg 1$ eV. In
turn, it corresponds to the velocity constraint $10^{-3}\ll\beta<1$, where the
upper limit should be correlated with the applicability of nonrelativistic
description.

As follows from Eq.(39), the maximum value of induced potential depends mainly
on the particle charge and its longitudinal velocity, resulting for electron
in the estimate $U_{ind}^{max}\sim\beta^{-1}$ meV. Hence, the magnitude of the
induced potential for one particle is not large in comparison with the
averaged atomic potential $\bar{V}_{max}\sim 10$ eV. However, it might be
essentially different for a beam (multi-particle) passing through the cavity.

##  IV Conclusion

In this paper, the problem of interaction of a charged particle with the inner
surface of a cylindrical cavity in an infinite insulator has been analysed
within definite analytical approach. Based on the Hamiltonian formalism, we
have succeeded in reducing the interaction potential for a charged particle in
the field of surface excitations.

Neglecting excitations of the wavelengths comparable to the cavity radius, the
interaction potential has been explicitly written revealing its complex
nature. An imaginary part of the potential leads to a finite width of the
energy levels and is not examined in general form. However, as an example, the
energy losses of a particle per unit of traveled distance is obtained at its
interaction with surface plasmons. The analysis of a real part of the
potential, instead, has been carried out for two limiting cases.

We have shown for the first time that at the limit $\omega_{s}R_{0}/v\ll 1$
the potential of interaction of a charged particle with the cavity surface
acts as a scattering potential (forming a reflecting barrier), while at
$\omega_{s}R_{0}/v\gg 1$ it reveals a potential well near the surface. The
width of the potential well depends on the speed of the particle, i.e. the
higher the speed of the particle the wider the well. In both cases considered,
the real potential logarithmically tends to plus infinity. The maximum value
of the induced potential mainly depends on the particle charge and its
longitudinal velocity. The estimates performed show that the averaged atomic
potential is much higher than that induced for one particle, while for a beam
of many particles channeled in a capillary the maximum value of the induced
potential is expected to be essentially different.

##  Appendix A The $g_{q}$-amplitudes

To determine the amplitudes $g_{q}$ we solve the electrostatic task for a
scalar potential of a charge $e$ inside an infinite cylindrical cavity

$\nabla^{2}\varphi=\left\\{\begin{array}[]{ll}-4\pi
e\delta(\mathbf{r}-\mathbf{r}_{0}),0\leq r\leq R_{0}\\\
0,R_{0}<r\end{array}\right.$ (40)

valid at the following interface matching condition

$\varphi_{R_{0}+0}=\varphi_{R_{0}-0},\quad\left.\frac{\partial\varphi}{\partial
r}\right|_{R_{0}-0}=\epsilon_{0}\left.\frac{\partial\varphi}{\partial
r}\right|_{R_{0}+0},$

where $\epsilon_{0}$ is the static dielectric constant. Presenting the
potential as a sum
$\varphi=\sum\limits_{q,n}\tilde{\varphi}_{n}(q,r)e^{iqz+in\phi}$ we get at
$\mathbf{r}_{0}=(0,r_{0},0)$ the expression

$\tilde{\varphi}_{n}(q,r)=\left\\{\begin{array}[]{lll}A_{1}K_{n}{qr},r\geq
R_{0}\\\ A_{2}K_{n}(qr)+B_{2}I_{n}(qr),R_{0}\geq r\geq r_{0}\\\
A_{3}I_{n}(qr),r_{0}\geq r\geq 0\end{array}\right.$ (41)

with the following constants

$A_{1}=\frac{e}{L}I_{n}(qr_{0})\left(1+\phantom{\frac{K_{n}}{K_{n}}}\right.$
$+\left.\frac{(\epsilon_{0}-1)K^{\prime}_{n}(qR_{0})I_{n}(qR_{0})}{I_{n}^{\prime}(qR_{0})K_{n}(qR_{0})-\epsilon_{0}I_{n}(qR_{0})K^{\prime}_{n}(qR_{0})}\right),$
$A_{2}=\frac{e}{L}I_{n}(qr_{0}),$
$A_{3}=\frac{e}{L}\left[K_{n}(qr_{0})+\phantom{\frac{K_{n}}{K_{n}}}\right.$
$+\left.\frac{(\epsilon_{0}-1)K^{\prime}_{n}(qR_{0})K_{n}(qR_{0})I_{n}(qr_{0})}{I_{n}^{\prime}(qR_{0})K_{n}(qR_{0})-\epsilon_{0}I_{n}(qR_{0})K^{\prime}_{n}(qR_{0})}\right],$
$B_{2}=\frac{e}{L}\frac{(\epsilon_{0}-1)K^{\prime}_{n}(qR_{0})K_{n}(qR_{0})I_{n}(qr_{0})}{I_{n}^{\prime}(qR_{0})K_{n}(qR_{0})-\epsilon_{0}I_{n}(qR_{0})K^{\prime}_{n}(qR_{0})}$

Correspondingly, the interaction energy of the induced charges with the
particle can be presented as follows

$U_{ind}=\frac{e^{2}}{L}\times$
$\times\sum\limits_{q,n}\frac{(\epsilon_{0}-1)K^{\prime}_{n}(qR_{0})K_{n}(qR_{0})I^{2}_{n}(qr_{0})}{I_{n}^{\prime}(qR_{0})K_{n}(qR_{0})-\epsilon_{0}I_{n}(qR_{0})K^{\prime}_{n}(qR_{0})}$
(42)

Comparing the this expression with Eq.(21) at $k=0$, we can get the following
definition for the amplitudes $g_{q}$

$\left|g_{qn}\right|^{2}=\left|g_{n}(qR_{0})\right|^{2}=-\frac{e^{2}\omega_{q}}{\hbar
L}\times$
$\times\frac{(\epsilon_{0}-1)K^{\prime}_{n}(qR_{0})K_{n}(qR_{0})}{I_{n}^{\prime}(qR_{0})K_{n}(qR_{0})-\epsilon_{0}I_{n}(qR_{0})K^{\prime}_{n}(qR_{0})}\,,$
(43)

which in a particular case of interactions with surface plasmons becomes equal
to

$\left|g_{n}(qR_{0})\right|^{2}_{\epsilon_{0}\to\infty}=\frac{e^{2}\omega_{q}}{\hbar
L}\frac{K_{n}(qR_{0})}{I_{n}(qR_{0})}$ (44)

##  Appendix B The sum calculation

Analysing $e^{i(k_{z}z+n\phi)}$ for a plane wave propagating along the
$zOy$-plane at $R_{0}\to\infty$, we can define $\phi\approx y/R_{0}$ near
curved surface of very large radius that corresponds to the wave at
$n=k_{y}R_{0}$. Thus, we can rewrite the sum in Eq.(28) in the following
(below for brevity we omit the index $y$ of the wave vector $k_{y}$)

$\lim_{R_{0}\to\infty}\sum\limits_{n=-\infty}^{\infty}\frac{K_{n}\left(\frac{\omega_{s}}{v}R_{0}\right)}{I_{n}\left(\frac{\omega_{s}}{v}R_{0}\right)}I_{n}^{2}\left(\frac{\omega_{s}}{v}(R_{0}-x)\right)\approx$
$\approx\lim_{R_{0}\to\infty}2\int\limits_{0}^{\infty}\frac{K_{kR_{0}}\left(z_{1}kR_{0}\right)}{I_{kR_{0}}\left(z_{1}kR_{0}\right)}I_{kR_{0}}^{2}\left(z_{2}kR_{0}\right)R_{0}dk$
(45) $z_{1}=\frac{\omega_{s}}{vk};\quad
z_{2}=z_{1}\left(1-\frac{x}{R_{0}}\right)$

Neglecting small $k\ll R_{0}^{-1}$ we present the Bessel functions in the form
of asymptotic expansion [22] ($x/R_{0}\ll 1$)

$K_{kR_{0}}\left(z_{1}kR_{0}\right)\approx\sqrt{\frac{\pi}{2kR_{0}}}\frac{e^{-kR_{0}\eta_{1}}}{(1+z_{1}^{2})^{1/4}}$
$I_{kR_{0}}\left(z_{1}kR_{0}\right)\approx\frac{1}{\sqrt{2\pi
kR_{0}}}\frac{e^{kR_{0}\eta_{1}}}{(1+z_{1}^{2})^{1/4}}$
$I_{kR_{0}}\left(z_{2}kR_{0}\right)\approx\frac{1}{\sqrt{2\pi
kR_{0}}}\frac{e^{kR_{0}\eta_{2}}}{(1+z_{2}^{2})^{1/4}}$
$\eta_{1}=\sqrt{1+z_{1}^{2}}+\ln{\frac{z_{1}}{1+\sqrt{1+z_{1}^{2}}}}$
$\eta_{2}\approx\sqrt{1+z_{1}^{2}}+\ln{\frac{z_{1}}{1+\sqrt{1+z_{1}^{2}}}}-\frac{x}{R_{0}}\sqrt{1+z_{1}^{2}}$

Substituting these expressions into Eq.(45), we obtain

$\lim_{R_{0}\to\infty}\sum\limits_{n=-\infty}^{\infty}\frac{K_{n}\left(\frac{\omega_{s}}{v}R_{0}\right)}{I_{n}\left(\frac{\omega_{s}}{v}R_{0}\right)}I_{n}^{2}\left(\frac{\omega_{s}}{v}(R_{0}-x)\right)=$
$=\int\limits_{0}^{\infty}\frac{e^{-2x\sqrt{k^{2}+\omega_{s}^{2}/v^{2}}}}{\sqrt{k^{2}+\omega_{s}^{2}/v^{2}}}dk=K_{0}\left(2\frac{\omega_{s}}{v}x\right)$
(46)

Having assumed an infinite upper integration limit, while a finite - equal to
$\sqrt{q_{0}^{2}-\omega_{s}^{2}/v^{2}}$, the result is conformed with that
presented in [18].

##  Appendix C The sum and integral calculations

In new notations $x=r/R_{0}$, $a=q_{0}R_{0}$ and $z_{0}=\omega_{s}R_{0}/v$ the
induced potential is written in the form

$U_{ind}(r)=-\frac{e^{2}\omega_{s}}{2\pi v}F(x,a,z_{0})\,,$ (47)

where

$F(x,a,z_{0})=\ln{\left(\frac{a}{z_{0}}-1\right)}g(x,z_{0})+$
$+\sum\limits_{k=1}^{\infty}\frac{(a-z_{0})^{k}-(-z_{0})^{k}}{kk!}\frac{\partial^{k}}{\partial
z_{0}^{k}}g(x,z_{0})$ (48)
$g(x,z_{0})=\sum\limits_{n=-\infty}^{\infty}\frac{(\epsilon_{0}-1)K_{n}^{\prime}(z_{0})K_{n}(z_{0})I_{n}^{2}(xz_{0})}{I_{n}^{\prime}(z_{0})K_{n}(z_{0})-\epsilon_{0}K_{n}^{\prime}(z_{0})I_{n}(z_{0})}$
(49)

###  C.1 $z_{0}\gg 1$:

For this limit, we can use the asymptotic expressions for the Bessel functions
[22]. Then at $0\ll x\leq 1$ we can calculate the sum by replacing it with the
integral, which successfully can be simplified within the saddle-point method

$g(x,z_{0})\approx-\frac{\epsilon_{0}-1}{2(\epsilon_{0}+1)}\times$ $\times
e^{-2(1-x)(1-x/4)z_{0}}K_{0}\left(\frac{(1-x)x}{2}z_{0}\right)$ (50)

Substituting (50) into (48), we obtain

$F(x,a,z_{0})=-\frac{\epsilon_{0}-1}{2(\epsilon_{0}+1)}e^{-z_{0}\alpha(x)}\times$
$\times\int\limits_{0}^{\infty}\left(\mathrm{Ei}\left[(z_{0}-a)(\alpha(x)+\beta(x)\mathrm{ch}{(t)})\right]\right.-$
$-\left.\mathrm{Ei}\left[z_{0}(\alpha(x)+\beta(x)\mathrm{ch}{(t)})\right]\right)e^{-z_{0}\beta(x)\mathrm{ch}{(t)}}dt\,,$
(51)

where $\alpha(x)=2(1-x)(1-x/4),\beta(x)=(1-x)x/2$. For $x\to 1$,
$(\alpha(x),\beta(x))\to 0$, that corresponds to the postion at the surface,
we get the function $F(x,a,z_{0})$ as follows

$F(x,a,z_{0})\approx-\frac{\epsilon_{0}-1}{2(\epsilon_{0}+1)}e^{-z_{0}\alpha(x)}\times$
$\times\ln{\left(\frac{a}{z_{0}}-1\right)}K_{0}\left(\beta(x)z_{0}\right)$
(52)

This result proves, that approaching the surface, the potential tends
logarithmically to minus infinity. If $a\gg z_{0}$, then the 1st term in (51)
can be neglected, and the integral itself very weakly depends on the functions
$\alpha(x)$ and $\beta(x)$. Thus, rather precisely the expression (51) can be
replaced with

$F(x,a,z_{0})=\frac{\epsilon_{0}-1}{2(\epsilon_{0}+1)}G\left(\frac{3}{2}z_{0}(1-x)\right)\,,$
(53)

where

$G(\xi)=\int\limits_{0}^{\infty}\mathrm{Ei}\left(\xi\left(1+\frac{\mathrm{ch}{(t)}}{3}\right)\right)e^{-\xi(1+\mathrm{ch}{(t)}/3)}dt$

The reduced function has a maximum $G(\xi_{m})=2.59815$ at $\xi_{m}=0.545901$.
It is notable that in the limiting case $R_{0}\to\infty$ the form of the
potential remains unchanged and is determined by the function
$G(3\omega_{s}x/2v)$ ($x$ is the distance from the particle to the plane).

###  C.2 $z_{0}\ll 1$:

At $z_{0}\ll 1$, using expressions for Bessel functions of a small argument,
we obtain the following expression for the function $g(x,z_{0})$

$g(x,z_{0})\approx\frac{\epsilon_{0}-1}{\epsilon_{0}+1}\ln{\left(1-x^{2}\right)}+\frac{\epsilon_{0}-1}{\epsilon_{0}}\left[\ln{\left(\frac{z_{0}}{2}\right)}+\gamma_{0}\right]\,,$
(54)

where $\gamma_{0}$ is the Euler’s constant. Substitution of (54) into (48)
followed by summing results in

$F(x,a,z_{0})=\frac{\epsilon_{0}-1}{\epsilon_{0}+1}\ln{\left(\frac{a}{z_{0}}-1\right)}\ln{\left(1-x^{2}\right)}$
$+C(\epsilon_{0},a,z_{0})$ (55)

with

$C(\epsilon_{0},a,z_{0})=\frac{\epsilon_{0}-1}{\epsilon_{0}}\left[\ln{\left(\frac{a}{z_{0}}-1\right)}\left(\ln{\left(\frac{z_{0}}{2}\right)}+\gamma_{0}\right)\right.$
$\left.-\left(\mathrm{Li}_{2}\left(1-\frac{a}{z_{0}}\right)-\mathrm{Li}_{2}\left(1\right)\right)\right]$
(56)

Here $\mathrm{Li}_{2}(\xi)$ is the polylogarithm function. As seen from (55),
the interaction potential at $z_{0}\ll 1$ is attractive with the tendency of
logarithmic dropp-off when the projectile approaches the cavity wall.

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