# The Jellium Edge and the Size Effect of the Chemical Potential and Surface
Energy in Metal Slabs

Kazuhiko Seki<EMAIL_ADDRESS>Nanomaterials Research Institute(NMRI),
National Institute of Advanced Industrial Science and Technology (AIST)
AIST Tsukuba Central 5, Higashi 1-1-1, Tsukuba, Ibaraki, Japan, 305-8565,
Japan

###### Abstract

Although free electron models have been established in order to capture the
essential physics of interfacial and bulk properties in metals, some issues
still remain regarding the application of free electron models to thin metal
films. One of the issues relates to whether the geometric edge coincides with
the potential edge in order to satisfy the charge neutrality condition when
the potential profile is modeled as a rectangular potential well. We show that
they coincide by rigorously taking into account the quantization effect
arising from electron confinement in a thin metal slab. As a result, the
overall behaviors of the chemical potential and surface energy show an
increasing trend by decreasing the thickness of the slab. The chemical
potential and surface energy show an oscillatory thickness dependence by
further taking into account the discreteness of the total number of free
electrons.

##  I Introduction

Understanding quantum effects in nano-structured metals is important for
developing nano-devices. Although free electron models have been used to
capture the essential physics of interfacial and bulk properties in metals,
Kittel and Kroemer (1980); Lang and Kohn (1970) free electron models for thin
metal films have not yet been fully investigated. In thin metal films, the
separation of quantum states and maximum states occupied by electrons depend
on the film thickness and should be evaluated with care. One of the issues in
free electron models for thin metal films relates to whether the geometric
edge coincides with the potential edge in order to satisfy the charge
neutrality condition when the potential profile is modeled as a rectangular
potential well. Such details in this model affect the quantum effect in thin
metal films. Stratton (1965); Kiejna and Wojciechowski (1996); Kostrobij and
Markovych (2018); Schulte (1976)

It is necessary to consider the background positive charge and electron
density to correctly determine the electrostatic potential in metals. Bardeen
(1936); Lang and Kohn (1970); Kiejna and Wojciechowski (1996) The simplest
model is the so-called jellium model, wherein the background positive charge
is expressed as a uniform profile in the bulk of the metal sample and is
sharply terminated at the surface. The cut-off edge is called the jellium edge
or the geometric edge. The jellium edge is determined by imposing the charge
neutrality condition, where the electron density is calculated quantum
mechanically. It has been frequently stated that the jellium edge differs from
the potential edge of the rectangular potential of free electrons, and others
have concluded that the jellium edge and potential edge are shifted relative
to each other. Bardeen (1936); van Himbergen and Silbey (1978); Stratton
(1965); Kiejna and Wojciechowski (1996); Huntington (1951); Kostrobij and
Markovych (2015, 2018); Han and Liu (2009); Schulte (1976) On the other hand,
the shift in the jellium edge away from the potential edge is not considered
when calculating the chemical potential of metal films in some cases. Dymnikov
(2011); Paskin and Singh (1965); Smith (1965); Thompson and Blatt (1963); Wu
and Zhang (2008) It has been argued whether or not the jellium edge is
different from the potential surface. Stratton (1965); Kiejna and
Wojciechowski (1996); Kostrobij and Markovych (2018) In this manuscript, we
scrutinize whether the jellium edge coincides with or differs from the
potential edge of the rectangular potential for free electrons.

If the Fermi sphere is considered in the ground state (at temperature $T=0$
K), the number of free electrons in the volume $L^{3}$ is expressed by Kittel
and Kroemer (1980)

$\displaystyle N=2\frac{(4\pi/3)K_{F}^{3}}{(2\pi/L)^{3}}.$ (1)

When the number of positive charges from ions is equal to the number of free
electrons, the average density of positive charge is

$\displaystyle\rho=\frac{N}{L^{3}}=\frac{K_{F}^{3}}{3\pi^{2}},$ (2)

where $K_{F}$ denotes the bulk Fermi wave vector. The average positive charge
density given by Eq. (2) is a consequence of the Sommerfeld model for free
electrons in an infinite box potential. For the infinite barrier model, the
number of positive charges increases as $L^{3}$ in such a way that $N/L^{3}$
approaches a constant value given by Eq. (2).

A free electron model of a finite width was introduced by imposing the Born-
von Karman periodic boundary condition in the x and y directions, as well as a
fixed boundary condition at $z=L_{z}$ without taking the thermodynamic limit.
Bardeen (1936) The electron density profile along the z-direction shows
oscillation with a wavelength given by $\pi/K_{F}$, called Friedel
oscillations. Kiejna and Wojciechowski (1996) Friedel oscillations will be
discussed later for some particular cases.

On the basis of the electron density, the jellium edge is obtained by applying
the charge neutrality condition, where the average positive charge density
should be correctly evaluated. In this paper, we rigorously evaluate the
average positive charge density by taking into account the quantum effect
arising from the finite width of the metal slab. By using the charge
neutrality condition together with the rigorous results regarding the average
positive charge density and the electron density profile, we show that the
jellium edge coincides with the potential edge. On this basis, we study the
size dependence of the chemical potential and the surface energy due to
quantum confinement of free electrons in a slab.

##  II Theory

We consider a slab of finite thickness $L_{z}$ in the z direction. The slab is
extended infinitely in the other directions and, the Born-von Karman periodic
boundary condition is imposed along the x and y directions with characteristic
lengths $L_{x}$ and $L_{y}$, respectively. For simplicity, we first consider
the case where the potential is zero at $0\leq z\leq L_{z}$ and is infinitely
large at $z=0$ and $z=L_{z}$. The wave function inside the slab is expressed
as van Himbergen and Silbey (1978); Kiejna and Wojciechowski (1996)

$\displaystyle\psi_{k}(x,y,z)=\sqrt{\frac{2}{L_{x}L_{y}L_{z}}}\exp\left[i(k_{x}x+k_{y}y)\right]\sin(k_{z}z),$
(3)

where the components of the wave vectors are given by

$\displaystyle
k_{x}=\pm\frac{2\pi}{L_{x}}n_{x},k_{y}=\pm\frac{2\pi}{L_{y}}n_{y}\mbox{
}n_{x,y}=0,1,2,\cdots$ (4) $\displaystyle k_{z}=\frac{\pi}{L_{z}}n_{z},\mbox{
}n_{z}=1,2,3,\cdots.$ (5)

In the ground state (at temperature $T=0$ K), $n_{x}$, $n_{y}$, and $n_{z}$
satisfy
$\hbar^{2}(k_{x}^{2}+k_{y}^{2}+k_{z}^{2})/(2m)\leq\hbar^{2}k_{F}^{2}/(2m)$,
where $m$ and $\hbar$ are the electron mass and the Planck constant divided by
$2\pi$, respectively. $k_{F}$ indicates the Fermi wave vector, i.e., the
largest wave vector occupied by electrons in the ground state. As shown later,
$k_{F}$ depends on the thickness $L_{z}$ and should be distinguished from the
bulk Fermi wave vector denoted by $K_{F}$ except when the limit of
$L_{z}\rightarrow\infty$ is taken.

The number of allowed values for $k_{x}$ and $k_{y}$ for a certain value of
$k_{z}$ is given by

$\displaystyle\frac{L_{x}L_{y}}{(2\pi)^{2}}\pi(k_{F}^{2}-k_{z}^{2})=\frac{1}{4\pi}L_{x}L_{y}\left(k_{F}^{2}-k_{z}^{2}\right).$
(6)

The electron density profile in the ground state is obtained as van Himbergen
and Silbey (1978); Sugiyama (1960)

$\displaystyle n_{e}(z)$
$\displaystyle=2\frac{1}{4\pi}L_{x}L_{y}\sum_{n_{z}=1}^{n_{\rm
Fz}}\left(k_{F}^{2}-k_{z}^{2}\right)\left|\psi_{k}(x,y,z)\right|^{2}$ (7)
$\displaystyle=\frac{1}{2\pi L_{z}}\sum_{n_{z}=1}^{n_{\rm
Fz}}\left(k_{F}^{2}-k_{z}^{2}\right)\left[1-\cos\left(2k_{z}z\right)\right],$
(8)

where spin multiplicity is included. In Eq. (8), $n_{\rm Fz}$ is the quantum
number for the highest occupied $n_{z}$ and is given by $n_{\rm
Fz}=[k_{F}L_{z}/\pi]\approx k_{F}L_{z}/\pi$, where $[x]$ indicates the integer
part of $x$. The summation in Eq. (8) with $k_{z}=(\pi/L_{z})n_{z}$ is
evaluated analytically and is simplified by using Mathematica Wolfram
Research, Inc. (2018); the electron density can be expressed as

$n_{e}(z)=\frac{k_{F}^{2}}{2\pi L_{z}}\left(n_{\rm
Fz}+\frac{1}{2}\right)-\frac{\pi}{12L_{z}^{3}}n_{\rm Fz}(n_{\rm Fz}+1)(2n_{\rm
Fz}+1)\\\ +\frac{1}{8\pi L_{z}^{3}}\left[(2n_{\rm
Fz}+3)\pi^{2}\cos\left(\frac{2(n_{\rm Fz}+1)\pi
z}{L_{z}}\right)\csc^{2}\left(\frac{\pi z}{L_{z}}\right)\right.\\\
+2\left(\pi^{2}(n_{\rm
Fz}+1)^{2}-k_{F}^{2}L_{z}^{2}\right)\sin\left(\frac{(2n_{\rm Fz}+1)\pi
z}{L_{z}}\right)\csc\left(\frac{\pi
z}{L_{z}}\right)\left.-\pi^{2}\sin\left(\frac{(2n_{\rm Fz}+3)\pi
z}{L_{z}}\right)\csc^{3}\left(\frac{\pi z}{L_{z}}\right)\right].$ (9)

The total number of electrons is calculated using

$\displaystyle\int_{0}^{L_{z}}n_{e}(z)dz$ $\displaystyle=\frac{1}{2\pi
L_{z}}\sum_{n_{z}=1}^{n_{\rm
Fz}}\left(k_{F}^{2}-k_{z}^{2}\right)\int_{0}^{L_{x}}dz\left[1-\cos\left(2\frac{\pi}{L_{z}}n_{z}z\right)\right]$
$\displaystyle=\frac{1}{2\pi L_{z}^{2}}\sum_{n_{z}=1}^{n_{\rm
Fz}}\left(k_{F}^{2}L_{z}^{2}-n_{z}^{2}\pi^{2}\right)$ (10)
$\displaystyle=\frac{k_{F}^{2}n_{\rm Fz}}{2\pi}-\frac{\pi}{12L_{z}^{2}}n_{\rm
Fz}(n_{\rm Fz}+1)(2n_{\rm Fz}+1).$ (11)

Equation (11) is known. Thompson and Blatt (1963)

The total number of positive charges from ions is equal to the total number of
free electrons denoted by $N$. The total number of positive charges $N$ can be
calculated by calculating a relation for the total number of electrons, given
by

$\displaystyle 2\frac{1}{4\pi}L_{x}L_{y}\sum_{n_{z}=1}^{n_{\rm
Fz}}\left(k_{F}^{2}-k_{z}^{2}\right)=N,$ (12)

which can be rewritten as Czoschke  _et al._ (2005)

$\displaystyle\frac{L_{x}L_{y}n_{\rm
Fz}}{2\pi}\left[k_{F}^{2}-\frac{1}{6L_{z}^{2}}(n_{\rm Fz}+1)(2n_{\rm
Fz}+1)\pi^{2}\right]=N.$ (13)

We find from Eq. (13) that

$\displaystyle k_{F}^{2}=\frac{\pi^{2}}{6L_{z}^{2}}(n_{\rm Fz}+1)(2n_{\rm
Fz}+1)+\frac{2\pi N}{L_{x}L_{y}n_{\rm Fz}}.$ (14)

By substituting Eq. (14) into Eq. (11), we obtain

$\displaystyle\frac{1}{L_{z}}\int_{0}^{L_{z}}dzn_{e}(z)=\frac{N}{L_{x}L_{y}L_{z}}.$
(15)

The average positive charge density (denoted by $\rho$) should be given by the
number of positive charges $N$ averaged over the volume (given by
$L_{x}L_{y}L_{z}$), i.e., $\rho=N/(L_{x}L_{y}L_{z})$. Equation (15) indicates
that charge neutrality is satisfied; it is not necessary to introduce a new
width for the ions instead of $L_{z}$ to describe the shift of the jellium
edge from the potential edge. Therefore, the jellium edge coincides with the
potential edge.

The shift of the jellium edge from the potential edge follows from the
following argument. If we loosely evaluate Eq. (8) by integration, we have
Kiejna and Wojciechowski (1996); Sugiyama (1960)

$\displaystyle n_{e}^{(a)}(z)$ $\displaystyle=\frac{L_{z}}{\pi}\frac{1}{2\pi
L_{z}}\sum_{n_{z}=1}^{n_{\rm
Fz}}\left(k_{F}^{2}-k_{z}^{2}\right)\left[1-\cos\left(2k_{z}z\right)\right]\frac{\pi}{L_{z}}$
(16)
$\displaystyle\approx\frac{1}{2\pi^{2}}\int_{0}^{K_{F}}dk_{z}\left(K_{F}^{2}-k_{z}^{2}\right)\left[1-\cos\left(2k_{z}z\right)\right]$
(17)
$\displaystyle=\rho\left(1+\frac{3\cos(2K_{F}z)}{(2K_{F}z)^{2}}-\frac{3\sin(2K_{F}z)}{(2K_{F}z)^{3}}\right),$
(18)

where $K_{F}$ appeared by evaluating $k_{F}$ in the limit of
$L_{z}\rightarrow\infty$, and $\rho=K_{F}^{3}/(3\pi^{2})$ is given by Eq. (2).
Similarly, we can loosely evaluate the average positive charge density by
integration using Eq. (12): Kiejna and Wojciechowski (1996); Sugiyama (1960)

$\displaystyle\frac{1}{2\pi^{2}}L_{x}L_{y}L_{z}\int_{0}^{K_{F}}dk_{z}\left(K_{F}^{2}-k_{z}^{2}\right)=\frac{K_{F}^{3}}{3\pi^{2}}L_{x}L_{y}L_{z}=N.$
(19)

Therefore, the average positive charge density is given by
$N/(L_{x}L_{y}L_{z})$ and is obtained as $\rho$. We also find from Eq. (18)

$\displaystyle\int_{0}^{\infty}dz\left[n_{e}^{(a)}(z)-\rho\right]=-\rho\frac{3\pi}{8K_{F}}.$
(20)

If one notices that a single edge is present in the limit of
$\pi/L_{z}\rightarrow 0$, the result suggests that the jellium edge is
$3\pi/(8K_{F})$ and is shifted inside from the potential edge to satisfy the
charge neutrality condition; Bardeen (1936); van Himbergen and Silbey (1978);
Kiejna and Wojciechowski (1996); Huntington (1951); Kostrobij and Markovych
(2015); Han and Liu (2009); Schulte (1976) Equation (20) indicates
$\int_{0}^{L_{z}}dzn_{e}^{(a)}(z)=\rho[L_{z}-(3\pi)/(8K_{F})]$ when $L_{z}$ is
sufficiently larger than the Wigner-Seitz radius. The difference between the
density calculated from the wave function and that calculated from the density
of states is a mathematical consequence of taking the limit
$\pi/L_{z}\rightarrow 0$. The difference disappears as long as we rigorously
evaluate the quantized effect represented by summation, thus the difference is
unphysical in thin slabs. We conclude that the jellium edge coincides with the
potential edge for free electrons at least when the potential barrier along
the z-direction is symmetric and infinite. Below, we show that the same
conclusion can be reached for more general potentials.

We consider a slab of finite thickness $L_{z}$ in the z-direction and extended
infinitely in the x and y directions, where the Born-von Karman periodic
boundary condition is imposed in x and y directions as before. The potential
is zero at $0\leq z\leq L_{z}$. A constant potential denoted by $w_{R}$ is
assumed for $L_{z}\leq z$, and a constant potential denoted by $w_{L}$ is
assumed for $z\leq 0$. The values of both $w_{R}$ and $w_{L}$ are assumed to
be positive. The wave function is denoted by
$\psi_{n}(z)\exp\left[i(k_{x}x+k_{y}y)\right]/\sqrt{L_{x}L_{y}}$, and the
eigen-states are denoted by $n=1,2,\cdots$. Because the eigen-states are
orthonormal, the electron density can be written as

$\displaystyle\int_{-\infty}^{\infty}dzn_{e}(z)$
$\displaystyle=\frac{1}{2\pi}\sum_{n_{z}=1}^{n_{\rm
Fz}}\left(k_{F}^{2}-k_{z}^{2}\right)\int_{-\infty}^{\infty}dz\left|\psi_{n}(z)\right|^{2}$
$\displaystyle=\frac{1}{2\pi}\sum_{n_{z}=1}^{n_{\rm
Fz}}\left(k_{F}^{2}-k_{z}^{2}\right).$ (21)

The total number of positive charges obeys Eq. (12) in the ground state. By
combining Eq. (21) and Eq. (12), we obtain,

$\displaystyle\frac{1}{L_{z}}\int_{-\infty}^{\infty}dzn_{e}(z)=\frac{N}{L_{x}L_{y}L_{z}}.$
(22)

Equation (22) expresses the charge neutrality condition, where the positive
charge density is obtained by averaging the total number of positive charges
over the potential width denoted by $L_{z}$, yielding
$\rho=N/(L_{x}L_{y}L_{z})$. If the jellium edge differs from the potential
edge, then $L_{z}$ on the left-hand side of Eq. (22) must be changed.
Therefore, we conclude that the jellium edge coincides with the potential
edge.

##  III Chemical potential results

By substituting the positive charge density given by
$\rho=N/(L_{x}L_{y}L_{z})$ into Eq. (14) and using the Wigner-Seitz radius
given by $r_{s}=[3/(4\pi\rho)]^{1/3}$, we obtain a closed equation giving the
thickness dependence of the Fermi wave vector as

$\displaystyle k_{F}^{2}=\frac{\pi^{2}}{6L_{z}^{2}}(n_{\rm Fz}+1)(2n_{\rm
Fz}+1)+\frac{3}{2n_{\rm Fz}}\frac{L_{z}}{r_{s}^{3}}.\,$ (23)

where $n_{\rm Fz}=[k_{F}L_{z}/\pi]\approx k_{F}L_{z}/\pi$. By introducing the
approximation given by $n_{\rm Fz}\approx k_{F}L_{z}/\pi$, we find

$\displaystyle k_{F}$
$\displaystyle=\frac{\pi}{4L_{z}}\left[1+\frac{7\pi^{2/3}}{f(\ell_{z})^{1/3}}+\frac{f(\ell_{z})^{1/3}}{3\pi^{2/3}}\right]$
(24)
$\displaystyle\approx\left(\frac{9}{4}\pi\right)^{1/3}\frac{1}{r_{s}}+\frac{\pi}{4L_{z}}+\frac{7\pi^{5/3}r_{s}}{24\times
18^{1/3}L_{z}^{2}},$ (25)

where $f(\ell_{z})=2^{3}\times 3^{5}\ell_{z}^{3}+81\pi^{2}+6\sqrt{16\times
3^{8}\ell_{z}^{6}+4\times 3^{7}\ell_{z}^{3}\pi^{2}-75\pi^{4}}$ and
$\ell_{z}=L_{z}/r_{s}$. Apart from a constant in terms of $L_{z}$, the
chemical potential of free electrons can be expressed as
$\mu=\hbar^{2}k_{F}^{2}/(2m)$. Consequently, the size dependence of the
chemical potential can be rewritten as

$\displaystyle\mu\approx\frac{\hbar^{2}}{2m}\left[\left(\frac{9}{4}\pi\right)^{1/3}\frac{1}{r_{s}}+\frac{\pi}{4L_{z}}+\frac{7\pi^{5/3}r_{s}}{24\times
18^{1/3}L_{z}^{2}}\right]^{2}.$ (26)

In the limit $L_{z}\rightarrow\infty$, we recover the bulk chemical potential
expressed as Kittel and Kroemer (1980)

$\displaystyle\mu_{b}=\frac{\hbar^{2}}{2m}\left(3\pi^{2}\rho\right)^{2/3}=\frac{\hbar^{2}}{2mr_{s}^{2}}\left(\frac{9\pi}{4}\right)^{2/3}.$
(27)

In Fig 1, we show the chemical potential $\hbar^{2}k_{F}^{2}/(2m)$ of free
electrons as a function of $L_{z}$ calculated from Eq. (26). We also show the
exact chemical potential obtained using Eq. (14) and $n_{\rm
Fz}=[k_{F}L_{z}/\pi]$. The figure is invariant under changes in the electron
density, which indicates that the free electron model is characterized by a
single length scale given by the Wigner-Seitz radius. Equation (26) indicates
that the chemical potential is a monotonic decreasing function of $L_{z}$. The
separation between quantum states increases by decreasing the thickness of the
slab; the largest energy value of electrons in the ground state ($\mu$)
increases by increasing the separation between quantum states. This
qualitative feature was already pointed out previously. Czoschke  _et al._
(2005); Korotun (2015) The exact numerical result in Fig. 1 shows the
oscillatory dependence on $L_{z}$. The oscillatory $L_{z}$ dependence
originates from the discreteness of the total number of free electrons. This
result agrees with the oscillatory $L_{z}$ dependence of the chemical
potential reported previously. Czoschke  _et al._ (2005); Korotun (2015);
Kostrobij and Markovych (2015, 2018)

In Fig. 2, we show the electron density profile given by Eq. (9) and a
comparison with the electron density obtained from Eq. (18), where the limit
$\pi/L_{z}\rightarrow 0$ is employed. The electron density is made
dimensionless by multiplying it with $r_{s}^{3}$. The figure is invariant
under changes in the electron density. The result in Eq. (9) indicates that
the electron density satisfies the boundary condition at $z=0$ and $z=L_{z}$.
The electron density in the middle of the thickness is higher than the average
density because of the depletion of the electron density near the boundaries
due to quantum confinement effect. By increasing the slab thickness, the
difference between the average density and the density in the middle of the
thickness decreases. When the positive charge density with the average
electron density is uniformly distributed over the thickness determined by the
potential edges, the electron density in the middle of the slab is higher than
the positive charge density. If the positive charge density is assumed to be
equal to the electron density in the middle of the thickness, the positive
charge density should be distributed over the thickness narrower than that
determined from the potential edges to maintain the overall charge neutrality
in the slab. In this case, the positive charge density differs from the
average electron density; as a result, the geometric edges differ from the
potential edges. Both types of edges coincide when the positive charge density
with the average electron density is uniformly distributed.

The electron density profile obtained from Eq. (9) shows oscillation with
wavelength $L_{z}/n_{\rm Fz}$. Using $n_{\rm Fz}=[k_{F}L_{z}/\pi]\approx
k_{F}L_{z}/\pi$, and Eq. (24) for $k_{F}$, the wavelength can be written as

$\displaystyle\frac{L_{z}}{n_{\rm Fz}}\approx
k_{F}/\pi\approx\left(\frac{2\pi}{3}\right)^{2/3}r_{s}\approx 1.64r_{s}.$ (28)

The oscillation shown in Fig. 2 is well characterized by the wavelength. If
the bulk value of the Fermi wave vector obtained from Eq. (2) is introduced
into Eq. (28), we find $K_{F}/\pi$. The result in Eq. (18) also shows
oscillation with a wavelength given by $K_{F}/\pi$. However, it differs
significantly from the exact result. The density of electrons obtained in the
limit $\pi/L_{z}\rightarrow 0$ is lower than the exact result in the middle of
the slab thickness. Moreover, we see that the approximate electron density
does not fulfill the boundary condition at $z=L_{z}$. The electron density
approaches zero only at $z=0$.

Before closing this section, we comment on the energy required to excite an
electron from the ground state. If we denote the right-hand side of Eq. (14)
by $k_{F}^{2}(n_{\rm Fz})$, the excitation energy can be estimated from
$\Delta\epsilon_{F}=\hbar^{2}/(2m)[k_{F}^{2}(n_{\rm Fz}+1)-k_{F}^{2}(n_{\rm
Fz})]$ as

$\displaystyle\Delta\epsilon_{F}$
$\displaystyle=\frac{\hbar^{2}}{12mL_{z}^{2}}\left(\pi^{2}\left(5+4n_{\rm
Fz}\right)-\frac{9(L_{z}/r_{s})^{3}}{n_{\rm Fz}(n_{\rm Fz}+1)}\right)$ (29)
$\displaystyle=\frac{\pi^{2}\hbar^{2}}{mL_{z}^{2}}\left(1-\frac{1}{3}\frac{K_{F}^{2}r_{s}^{3}}{L_{z}}\right),$
(30)

where Eq. (25) is used. For sodium, $\Delta\epsilon_{F}$ is estimated to be
about $0.026$ (eV) when the slab thickness is $5.3$ (nm) using Eq. (30). If
the Wigner-Seitz radius is given by $r_{s}=3$ (Bohr), $\Delta\epsilon_{F}$ is
also about $0.026$ (eV) when the slab thickness is $5.3$ (nm). The results
suggest the length scale of the Kubo effect at room temperature; the
quantization of one electronic level at the Fermi level results in remarkable
effects in thermodynamic properties of fine metals. Kubo (1962) Equation (30)
indicates that the Kubo effect can be observed for the thicker metal slabs if
the temperature is lowered from room temperature.

##  IV Surface energy results

Similarly, we can calculate the surface energy. First, we express the total
energy as Czoschke  _et al._ (2005)

$\displaystyle E_{t}$
$\displaystyle=\frac{\hbar^{2}}{2m}\frac{2L_{x}L_{y}}{(2\pi)^{2}}\sum_{n_{z}=1}^{n_{\rm
Fz}}\int_{0}^{\sqrt{k_{F}^{2}-k_{z}^{2}}}dk_{\parallel}\,2\pi
k_{\parallel}\left(k_{\parallel}^{2}+k_{z}^{2}\right)$ (31)
$\displaystyle=\frac{\hbar^{2}}{2m}\frac{L_{x}L_{y}}{4\pi}\sum_{n_{z}=1}^{n_{\rm
Fz}}\left(k_{F}^{4}-k_{z}^{4}\right)$ (32)
$\displaystyle=\frac{\hbar^{2}}{2m}\frac{L_{x}L_{y}}{4\pi}\left[n_{\rm
Fz}k_{F}^{4}-\frac{\pi^{4}}{30L_{z}^{4}}n_{\rm Fz}(n_{\rm Fz}+1)(2n_{\rm
Fz}+1)(3n_{\rm Fz}^{2}+3n_{\rm Fz}-1)\right],$ (33)

where $k_{z}$ is given by Eq. (5). Then, we decompose the total energy into
the bulk energy per unit volume and the two parts of the surface energy per
unit area of the slab as Czoschke  _et al._ (2005)

$\displaystyle E_{t}$
$\displaystyle=\epsilon_{b}L_{x}L_{y}L_{z}+2\epsilon_{s}L_{x}L_{y},$ (34)

where the bulk energy is obtained using $n_{\rm Fz}\approx k_{F}L_{z}/\pi$ as

$\displaystyle\epsilon_{b}$
$\displaystyle=\lim_{L_{z}\rightarrow\infty}\frac{E_{t}}{L_{x}L_{y}L_{z}}$
(35) $\displaystyle=\frac{\hbar^{2}K_{F}^{5}}{10m\pi^{2}},$ (36)

where $k_{F}$ is replaced with $K_{F}$ since the bulk energy can be obtained
in the limit of $L_{z}\rightarrow\infty$. The obtained bulk energy is equal to
the total energy given by $(3N/5)\hbar^{2}K_{F}^{2}/(2m)$ divided by
$L_{x}L_{y}L_{z}$ as it should be. Kittel and Kroemer (1980) Now, we calculate
the surface energy. The surface energy is obtained from Eq. (33) using the
decomposition given by Eq. (34) as

$\displaystyle\frac{\epsilon_{s}}{\epsilon_{s0}}$
$\displaystyle=2\left(\frac{4}{9\pi}\right)^{4/3}\left[n_{\rm
Fz}(k_{F}r_{s})^{4}-\frac{\pi^{4}r_{s}^{4}}{30L_{z}^{4}}n_{\rm Fz}(n_{\rm
Fz}+1)(2n_{\rm Fz}+1)(3n_{\rm Fz}^{2}+3n_{\rm
Fz}-1)\right]-\frac{8}{5\pi}K_{F}L_{z}$ (37) $\displaystyle\approx
1+\frac{\pi}{2K_{F}L_{z}}-\frac{\pi^{2}}{24(K_{F}L_{z})^{2}},$ (38)

where the surface energy in the limit of $L_{z}\rightarrow\infty$ is given by

$\displaystyle\epsilon_{s0}$
$\displaystyle=\frac{1}{32\pi}\frac{\hbar^{2}K_{F}^{4}}{m}.$ (39)

Equation (39) was obtained earlier. Waber  _et al._ (1972)

In Fig. 3, we compare the surface energy given by Eq. (39) with the
experimental values. The metal species are chosen so that the Sommerfeld
parameter of the heat capacity is close to that estimated from the free
electron model. Specifically, the criterion is that the ratio of the measured
to the free electron values of the Sommerfeld parameter is between $0.7$ and
$1.3$. The solid line is the surface energy calculated from the free electron
model of the slab obtained from Eq. (39). Compared to the Sommerfeld
parameter, the larger deviation of the experimental values from the
theoretical results of the free electron model is found for the surface energy
of the slab. The deviation could originate from the oversimplification in the
free electron model of the slab, where the inhomogeneity in the background
positive charges in particular near the surface is ignored. Though we did not
consider finite barriers, the effect of finite barrier hight can be taken into
account in a straightforward manner as sketched in Theory section and can be
ignored as long as the lowest barrier hight sufficiently exceeds the chemical
potential. In the same figure, we show the known result given by
$\epsilon_{s0}\approx\hbar^{2}K_{F}^{4}/(160\pi m)$ which is derived by
applying the charge neutrality condition using an average positive charge
density that is different from the expression presented here. Huntington
(1951); Sugiyama (1960) We consider the free electron model of the slab; the
surface energy is calculated from the natural decomposition of the total
energy into the bulk part and the surface parts. Then, the summation appeared
due to quantization in the direction of the slab thickness is rigorously
evaluated. The deviation introduced by approximating the summation by
integration is well captured in the electron density shown in Fig. 2. As a
result, the dashed line significantly differs from the solid line in Fig. 3.

In Fig. 4, we show the surface energy as a function of $L_{z}$ calculated from
Eq. (37). The surface energy is normalized by that taking the limit
$L_{z}\rightarrow\infty$. We also show the surface energy by taking into
account the discreteness of the total number of free electrons. By
substituting $k_{F}$ obtained from Eq. (23) into Eq. (37) using $n_{\rm
Fz}=[k_{F}L_{z}/\pi]$, we obtained the exact result of the surface energy. The
figure is invariant under changes in the electron density. Equation (38)
indicates that the surface energy is a monotonic decreasing function of
$L_{z}$ like the chemical potential. The exact numerical result in Fig. 4
shows the oscillatory dependence on $L_{z}$. The oscillatory $L_{z}$
dependence originates from the discreteness of the total number of free
electrons as in the case of the chemical potential.

##  V Conclusion

We have considered the electron density and the number of positive charges by
rigorously taking into account the quantum effect while keeping the discrete
sum in the free electron model for a thin metal slab. We showed that the
jellium edge coincides with the rectangular potential edge. The effect of
quantum confinement on the chemical potential and surface energy in a thin
slab was subsequently studied. The thickness dependence of the chemical
potential was derived as Eq. (26); the chemical potential (Fermi energy)
increases as the thickness of the slab decreases because the separation
between quantized states becomes wider. The thickness dependence of the
surface energy was derived as Eq. (38) and showed the similar dependence on
the thickness of the slab. More accurate results for the chemical potential
and surface energy that reflect the discrete nature of the number of electrons
in the Wigner-Seitz cell showed the oscillatory thickness dependence
superimposed on top of the continuous thickness dependence mentioned above.

In our analysis, quantized effects due to confinement of electrons in a thin
slab were considered according to the free electron model, where the length
scale is characterized by the Wigner-Seitz radius. Lattice structures and
lattice constants could affect the physical quantities as the slab thickness
decreases. A full quantitative characterization of a particular metal film
based on the aforementioned structure is beyond the scope of the present
study. The electrostatic interaction, the exchange interaction and the
correlation interaction were also ignored. Nevertheless, we qualitatively
discussed the quantum size effect on the chemical potential and the surface
energy in a thin metal slab. In some theories, van Himbergen and Silbey
(1978); Kiejna and Wojciechowski (1996); Kostrobij and Markovych (2015, 2018);
Han and Liu (2009); Schulte (1976) the shift of the jellium edge from the
potential edge was calculated, and the chemical potential was affected by the
shift. We showed that such a shift is unnecessary if both the electron density
and the total number of positive charges are evaluated by taking into account
the finite width of the metal slab. By using the charge neutrality condition
together with the rigorous results for the average positive charge density and
the electron density profile, we showed that the jellium edge indeed coincides
with the potential edge.

## References

  * Kittel and Kroemer (1980) C. Kittel and H. Kroemer,  _Thermal Physics_ (W. H. Freeman, 1980).
  * Lang and Kohn (1970) N. D. Lang and W. Kohn, Phys. Rev. B 1, 4555 (1970).
  * Stratton (1965) R. Stratton, Phys. Lett. 19, 556 (1965).
  * Kiejna and Wojciechowski (1996) A. Kiejna and K. Wojciechowski,  _Metal Surface Electron Physics_ (Pergamon, Oxford, 1996).
  * Kostrobij and Markovych (2018) P. P. Kostrobij and B. M. Markovych, ArXiv e-prints (2018), arXiv:1804.08884 [cond-mat.stat-mech] .
  * Schulte (1976) F. Schulte, Surf. Sci. 55, 427 (1976).
  * Bardeen (1936) J. Bardeen, Phys. Rev. 49, 653 (1936).
  * van Himbergen and Silbey (1978) J. E. van Himbergen and R. Silbey, Phys. Rev. B 18, 2674 (1978).
  * Huntington (1951) H. B. Huntington, Phys. Rev. 81, 1035 (1951).
  * Kostrobij and Markovych (2015) P. P. Kostrobij and B. M. Markovych, Phys. Rev. B 92, 075441 (2015).
  * Han and Liu (2009) Y. Han and D.-J. Liu, Phys. Rev. B 80, 155404 (2009).
  * Dymnikov (2011) V. D. Dymnikov, Phys. Solid State 53, 901 (2011).
  * Paskin and Singh (1965) A. Paskin and A. D. Singh, Phys. Rev. 140, A1965 (1965).
  * Smith (1965) B. Smith, Phys. Lett. 18, 210 (1965).
  * Thompson and Blatt (1963) C. Thompson and J. Blatt, Phys. Lett. 5, 6 (1963).
  * Wu and Zhang (2008) B. Wu and Z. Zhang, Phys. Rev. B 77, 035410 (2008).
  * Sugiyama (1960) A. Sugiyama, J. Phys. Soc. Jpn. 15, 965 (1960).
  * Wolfram Research, Inc. (2018) Wolfram Research, Inc.,  _Mathematica, Version 11.3_ (Champaign, IL, 2018).
  * Czoschke  _et al._ (2005) P. Czoschke, H. Hong, L. Basile, and T.-C. Chiang, Phys. Rev. B 72, 075402 (2005).
  * Korotun (2015) A. V. Korotun, Phys. Solid State 57, 391 (2015).
  * Kubo (1962) R. Kubo, J. Phys. Soc. Jpn. 17, 975 (1962).
  * Waber  _et al._ (1972) J. Waber, E. Kennard, and Y.-P. Tsui, J. Phys. Colloques 33, C3 (1972).

Figure 1: (Color online) Normalized chemical potential is shown against the
thickness of the slab. The thickness of the slab is normalized by the Wigner-
Seitz radius given by $r_{s}=[3/(4\pi\rho)]^{1/3}$. The dashed line is
calculated using Eq. (24) and normalized by the bulk value given by $\mu_{b}$
[Eq. (27)]. The (red) dots are calculated from Eq. (26). The solid line
indicates the exact result shown in the main text.  Figure 2: (Color online)
The electron density is shown as a function of distance from the surface when
$L_{z}=10r_{s}$. The electron density is made dimensionless by multiplying it
with $r_{s}^{3}$. Both the electron density and the distance from the surface
of the slab are normalized by the Wigner-Seitz radius
$r_{s}=[3/(4\pi\rho)]^{1/3}$. The solid line shows the exact result obtained
from Eq. (9). The normalized average density is shown by the (red) dashed-
dotted line, which is given by $3/(4\pi)$ because $\rho=3/(4\pi r_{s}^{3})$.
The dashed line shows the result from Eq. (18) obtained in the limit
$\pi/L_{z}\rightarrow 0$.  Figure 3: (Color online) The surface energy is
shown against the Wigner-Seitz radius. The solid line is obtained from Eq.
(39) and the dashed line indicates the classical result of
$\hbar^{2}k_{F}^{4}/(160\pi m)$. The (red) dots indicate the experimental
values of the surface energies of metals.  Figure 4: (Color online) Normalized
surface energy is shown against the thickness of the slab. The thickness of
the slab is normalized by the Wigner-Seitz radius given by
$r_{s}=[3/(4\pi\rho)]^{1/3}$. The dashed line is calculated from Eq. (37) with
$n_{\rm Fz}\approx k_{F}L_{z}/\pi$ and normalized by the bulk value of the
surface energy [Eq. (39)]. The (red) dots are calculated from Eq. (38). The
solid line indicates the exact result obtained from Eq. (37) and Eq. (23)
using $n_{\rm Fz}=[k_{F}L_{z}/\pi]$.


11institutetext: CERN, Geneva, Switzerland 22institutetext: University of
Wisconsin–Madison, Madison, United States 33institutetext: Max-Planck-Institut
für Physik, München, Germany 44institutetext: Sun Yat-Sen University, China
55institutetext: SCIPP, UC Santa Cruz, United States

# Scalable ATLAS pMSSM computational workflows using containerised REANA
reusable analysis platform

Marco Donadoni 11 Matthew Feickert 22 Lukas Heinrich 33 Yang Liu 44 Audrius
Mečionis 11 Vladyslav Moisieienkov 11 Tibor Šimko Corresponding author 11
<EMAIL_ADDRESS>Giordon Stark 55 Marco Vidal García 11

###### Abstract

In this paper we describe the development of a streamlined framework for
large-scale ATLAS pMSSM reinterpretations of LHC Run-2 analyses using
containerised computational workflows. The project is looking to assess the
global coverage of BSM physics and requires running O(5k) computational
workflows representing pMSSM model points. Following ATLAS Analysis
Preservation policies, many analyses have been preserved as containerised
Yadage workflows, and after validation were added to a curated selection for
the pMSSM study. To run the workflows at scale, we utilised the REANA reusable
analysis platform. We describe how the REANA platform was enhanced to ensure
the best concurrent throughput by internal service scheduling changes. We
discuss the scalability of the approach on Kubernetes clusters from 500 to
5000 cores. Finally, we demonstrate a possibility of using additional ad-hoc
public cloud infrastructure resources by running the same workflows on the
Google Cloud Platform.

##  1 Introduction

We have developed a streamlined framework for large-scale pMSSM
reinterpretations of ATLAS analyses of LHC Run-2 using containerised
computational workflows. The project is looking to assess the global coverage
of BSM physics and requires running numerous computational workflows
representing pMSSM model points. The framework builds upon the idea of RECAST-
ing analyses Cranmer:2010hk  and takes into account the experiences with the
previous ATLAS pMSSM reinterpretations from LHC Run-1 period ATLAS:2015wrn .

Following the ATLAS analysis preservation policies, many ATLAS analyses have
been preserved as containerised Yadage workflows. After validation they are
added to a curated selection of analyses suitable for the pMSSM study. Figure
1 shows one such repository for the supersymmetry searches.

Figure 1: A screenshot of the ATLAS SUSY group analyses preserved on GitLab.
Each repository is labeled with the internal ATLAS analysis identifier and
contains both workflow files and additional data files needed for the
computational processing.

One typical pMSSM computational workflow is presented in Figure 2. The
workflow consists of three time-consuming ntupling steps that process data
files and run in parallel. The workflow ends with a latter fitting steps that
run afterwards. The dependency of steps in the computational graph is rather
simple. The complexity of the problem lies in having to run several thousands
of these workflows in order to cover a sufficient number of pMSSM model
points.

Figure 2: A typical pMSSM workflow. The computational runtime is about 10
minutes without systematics (test payload) and about 10 hours with all
systematics (real payload).

It was the goal of the present work to study the feasibility of running
several thousands of these containerised workflows in parallel in an automated
way in order to facilitate typical pMSSM studies.

##  2 Method

The computational workflows were run at scale using the REANA reusable
analysis platform Simko:2018zzz . The computational backend was the Kubernetes
cluster of various sizes (from 500 cores up to 5000 cores). We have been
varying several parameters of the cluster such as the number of nodes and the
required memory and studied the maximum number of pMSSM workflows that the
platform can handle concurrently. After performing several such computational
experiments, we have improved the scheduling efficiency of REANA to increase
the running bandwidth for the pMSSM style of workflows.

Figure 3: The sequence diagram showing how REANA schedules incoming workflows
after submission. The submitted workflows are announced via message queue that
is later processed by the workflow scheduler in Figure 4.

Figure 3 shows the sequence diagram of the workflow submission stage. The
incoming workflows are stored in a queue that is later processed by the
scheduler. The first task was to improve the performance of the REANA
platform’s server submission end points to allow many concurrent workflow
starting requests.

Figure 4: The sequence diagram showing how REANA schedules queued workflows.
The system checks for available resources before allowing workflow runs for
execution. The checking and rescheduling workflow offers several possibilities
for optimisations. The workflows accepted for execution are further processed
in Figure 5.

Figure 4 shows the next stage of the process, namely how the submitted
workflows are being consumed from the incoming queue. The scheduler first
checks whether the incoming workflow does not exceed the limits on the total
number of workflow the system could handle as well as currently available free
memory on the Kubernetes cluster. If the checks succeed, the workflow is
accepted for execution. In the opposite case the incoming workflow is being
rescheduled and attempted to be accepted for execution several times whilst
waiting for the Kubernetes cluster resources to liberate. If the workflow
cannot be scheduled for a substantial amount of time, a failure is declared.

Figure 5: The sequence diagram showing how the REANA executes scheduled
workflows. Note the interplay between the scheduler and the Kubernetes
cluster. The pod creation offers another space for optimisations. The workflow
execution status monitoring is carried out by a watching loop. The workflow
jobs are started for each workflow step. The termination procedures are
further illustrated in Figure 6.

Figure 5 shows the stage of the running of the workflow after it has been
accepted for execution. Note the interplay of the REANA platform with the
underlying Kubernetes cluster: the job is scheduled using the Kubernetes
native job scheduler mechanism which include additional scheduling delays that
needed to be taken into account for optimisation. The progress of the workflow
is monitored until the workflow execution terminates. The workflow steps are
launched when the worker nodes are free to run the workload. The status of
jobs is published in the message queue.

Figure 6: The sequence diagram showing how REANA updates workflow statuses and
terminates finished workflows. The procedure involves consuming the message
queue, closing the Kubernetes pods, and updating the database about the status
of the workflow run. In case of launching several thousands of concurrent
workflows, these processes also have to be optimised.

Figure 6 shows the termination stage of the workflow. When all the steps are
finished and the results are produced, the system has to delete the Kubernetes
pod and update the status of the workflow in both the message queue and the
database. This constituted another layer of optimisations in order to handle
any status handling processes in an asynchronous manner whilst the platform is
starting the new incoming workflows.

##  3 Results

We have improved the REANA platform scheduling performance in order to
maximise the scheduling throughput of incoming workflows at the various stages
of the workflow life cycle as described in Section 2. A special attention was
paid to measure the CPU and Memory usage of the cluster nodes.

Figure 7 shows a typical snapshot of the status of cluster nodes running the
pMSSM workloads. We have used nodes of the m2.xlarge flavour which consist of
16 GiB of available memory and 8 virtual cores. One can see the efficient use
of cores of the cluster resulting from tuning REANA parameters such as the
number of nodes running workflow orchestration tasks, the number of nodes
running the pMSSM workflow step jobs themselves, as well as the memory request
limits for each ntupling job of the first pMSSM workflow stages.

        $ kubectl top nodes
        NAME                                CPU(cores)   CPU%   MEMORY(bytes)   MEMORY%
        reanaatlas1-3slyowp42qex-node-15    7858m        98%    12033Mi         82%
        reanaatlas1-3slyowp42qex-node-16    7848m        98%    12083Mi         83%
        reanaatlas1-3slyowp42qex-node-17    7846m        98%    12210Mi         83%
        reanaatlas1-3slyowp42qex-node-18    7773m        97%    8995Mi          61%
        reanaatlas1-3slyowp42qex-node-19    7864m        98%    11516Mi         79%
        reanaatlas1-3slyowp42qex-node-20    7843m        98%    12177Mi         83%
        reanaatlas1-3slyowp42qex-node-21    7376m        92%    8698Mi          59%
        reanaatlas1-3slyowp42qex-node-22    7817m        97%    11201Mi         77%
        reanaatlas1-3slyowp42qex-node-23    7748m        96%    9978Mi          68%
        reanaatlas1-3slyowp42qex-node-24    7854m        98%    12161Mi         83%
        reanaatlas1-3slyowp42qex-node-25    7868m        98%    12293Mi         84%
        reanaatlas1-3slyowp42qex-node-26    7787m        97%    10991Mi         75%

Figure 7: An example of the benchmark tests running in the CERN Computer
Centre. The REANA scheduling parameters were optimised to maximise the CPU
utilisation and the Memory consumption on the cluster for the typical pMSSM
ntupling job parallelism (see Figure 2). Note the very good efficiency of CPU
cores in the above screenshot.

Figure 8 shows the results of one of our scalability experiment that consisted
of submitting 200 new pMSSM workflows every 10 minutes. A cluster with 448
cores presented on the left cannot keep up with such a workload: note the
increasing scheduling waiting times (plotted in the orange colour) as well as
increasing workflow run times (plotted in blue). The overflow happens because
the cluster is allowing more workflows than it can hold. However, note how the
same cluster with 1072 cores presented on the right of the Figure holds the
same workload very comfortably.

Figure 8: A scalability test submitting 200 workflows every 10 minutes. A
cluster with 448 cores (left) cannot keep up with the load. A cluster with
1072 cores (right) can comfortably hold the incoming workload.

Figure 9 shows the same kind of experiment executed over a longer period of
time. This helped to ensure that the platform can sustain the constantly
increasing stream of incoming workloads.

Figure 9: The workload burndown throughput rate is sustainable over a long
period of time.

We have run several benchmarking experiments in the CERN Computer Centre and,
to test the portability, performed a few runs also on the Google Cloud
Platform. This allowed to prove the applicability of the approach on various
compute backends, facilitating future reproducibility of containerised
workflows irrespective of their original computing environments.

##  4 Conclusions

ATLAS searches for new physics are being effectively preserved together with
containerised computational workflow recipes as part of the ATLAS RECAST
project. This enables their future reuse and reinterpretation and greatly
facilitates the running of efficient pMSSM studies over a large collection of
individual analyses.

We have launched several ATLAS pMSSM workflows on the REANA reproducible
analysis platform and studied the performance from workflow scheduling up to
workflow execution and termination procedures with the aim of allowing running
several thousands of these workflows to cover a sufficient number of pMSSM
model points.

The REANA platform has been internally optimised to allow faster workflow
scheduling, processing and terminating procedures on an individual workflow
level as well as under the stressing conditions of processing many incoming
concurrent workloads. A set of benchmarking experiments allowed to optimise
and tune the REANA system for the pMSSM workloads on the Kuberentes clusters
ranging from medium to large sizes (from 500 to 5000 cores). It was essential
to adjust REANA scheduling parameters to the type of the pMSSM workload in
order to ensure the best throughput and the efficient cluster CPU and memory
resource utilisation.

The developed system was tested on the CERN Computer Centre as well as on the
Google Cloud Platform in order to ensure the reproducibility of the approach
and is fully ready to run large-scale ATLAS pMSSM reinterpretations of LHC
Run-2 analyses. The first results by the ATLAS collaborations are being
published ATLAS:2023oun .

## Acknowledgements

L.H. is supported by the Excellence Cluster ORIGINS, which is funded by the
Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under
Germany’s Excellence Strategy - EXC-2094-390783311.

M.F. is supported by the U.S. National Science Foundation (NSF) under
Cooperative Agreement OAC-1836650 (IRIS-HEP).

## References

  * (1) K. Cranmer, I. Yavin, “RECAST: Extending the Impact of Existing Analyses”, JHEP 1104:028 (2011), https://doi.org/10.48550/arXiv.1010.2506.
  * (2) ATLAS Collaboration, “Summary of the ATLAS experiment’s sensitivity to supersymmetry after LHC Run 1 - interpreted in the phenomenological MSSM”, JHEP 10 (2015), 134. https://doi.org/10.48550/arXiv.1508.06608.
  * (3) T. Šimko, L. Heinrich, H. Hirvonsalo, D. Kousidis, D. Rodríguez, “REANA: A system for reusable research data analyses”, EPJ Web of Conferences 214, 06034 (2019), https://doi.org/10.1051/epjconf/201921406034.
  * (4) ATLAS Collaboration, “ATLAS Run 2 searches for electroweak production of supersymmetric particles interpreted within the pMSSM”, ATLAS-CONF-2023-055 (13 Sep 2023). https://cds.cern.ch/record/2870222.