2022

[1,2]Igor Pshenichnov

1]Moscow Institute of Physics and Technology, Institutskiy per. 9,
Dolgoprudny, Moscow Region, 141701, Russia

2]Institute for Nuclear Research of the Russian Academy of Sciences, Prospekt
60-letiya Oktyabrya 7a, Moscow, 117312, Russia

# Peeling away neutron skin in ultracentral collisions of relativistic nuclei

Nikita Kozyrev<EMAIL_ADDRESS>Aleksandr Svetlichnyi
<EMAIL_ADDRESS>Roman Nepeivoda<EMAIL_ADDRESS><EMAIL_ADDRESS>[  [

###### Abstract

In heavy nuclei the ratio between local densities of neutrons and protons
increases towards the nuclear periphery. The excess of neutrons is known as
the neutron skin (NS) with a subtle difference ($<0.5$ fm) between the r.m.s
radii of the distributions of neutrons and protons. We show that the presence
of NS in 208Pb leads to extra spectator neutrons in ultracentral 208Pb–208Pb
collisions at the CERN SPS and LHC. The yields of spectator neutrons and
protons were calculated within a new version of Abrasion-Ablation Monte Carlo
for Colliders model (AAMCC-MST) taking into account NS and pre-equilibrium
clustering of spectator matter. While the average numbers of spectator
neutrons and protons in ultracentral collisions vary insignificantly, the
cross sections of emission of certain numbers of spectator neutrons in events
with 0,1,…5 spectator protons are changed by 50–250%, depending on the
thickness of NS. These cross sections are less sensitive to other parameters
in calculations, and their measurements in the ALICE experiment at the LHC
will make it possible to restrict the existing variety of neutron density
parameterizations in 208Pb.

###### keywords:

Heavy-Ion Collisions, Nuclear Structure and Reactions

##  1 Introduction

In experiments on nucleus-nucleus collisions at the Large Hadron Collider
(LHC) at CERN the main attention is paid to hot and dense matter created in
the domain of overlap of colliding nuclei. The largest overlap volume is
achieved in very central (ultracentral) collision events, and it is smaller in
peripheral collisions. Non-interacting nucleons outside the overlap domain
retain their momenta, move forward and form excited spectator matter
(prefragment). Multiple production of nuclear spectator fragments in explosive
prefragment decays is a well-known phenomenon of multifragmentation
Botvina1995  mostly investigated at lower collision energies. The study of the
properties of spectator matter, in particular, its neutron and proton content
is an interesting topic by itself. The n/p-ratio of free spectator nucleons
has been measured in the NA49 experiment at the CERN SPS by means of forward
calorimeters Appelshauser1998 . The ALICE experiment at the LHC is equipped
with two pairs of neutron (NZDC) and proton (PZDC) Zero Degree Calorimeters
(ZDC) Puddu2007  used, in particular, for the determination of event
centrality Abelev2013  by detecting forward spectator neutrons.

In this work, we show that the detection of spectator neutrons and protons can
also be used to study the neutron skin (NS) in colliding nuclei. NS is a well-
known phenomenon of an increased ratio between local neutron and proton
densities at the periphery of a heavy or a neutron-rich nucleus. The dominance
of neutrons over protons at the nuclear periphery was discovered in early
experiments to study nuclear structure Barrett1977 ; Tanihata1985 . The radial
density distributions of neutrons and protons in medium-weight and heavy
nuclei are commonly characterized by the two-parameter Fermi functions:

$\rho_{n,p}(r)=\frac{\rho_{0n,p}}{1+\exp\left\lfloor(r-R_{n,p})/a_{n,p}\right\rfloor}\
,$ (1)

with the following normalization to the nuclear mass number $A$:

$\int d^{3}r(\rho_{n}(r)+\rho_{p}(r))=A\ .$ (2)

The respective half-density radii, $R_{n}$ and $R_{p}$, and diffuseness
parameters, $a_{n}$ and $a_{p}$ are taken individually for neutrons and
protons, as well as the densities, $\rho_{0n}$ and $\rho_{0p}$, in the nuclear
center. As known, in light neutron-rich nuclei the effect of peripheral
neutron excess is caused by a weak connection between a nuclear core and a few
neutrons that form a neutron halo (NH) Hansen1995 . The neutron halo is
considered also in medium-weight and heavy nuclei when $R_{n}=R_{p}$ and
$a_{n}>a_{p}$ Trzcinska2001 . In the case of $R_{n}>R_{p}$ and $a_{n}=a_{p}$
the surface excess of neutrons is usually considered as a neutron skin
Trzcinska2001 . Due to the relations $R_{n}>R_{p}$ and $a_{n}>a_{p}$ validated
for many nuclei, including 208Pb, the classification into NH and NS cases is
not quite straightforward. Therefore, we call hereafter the surface nuclear
layer with the predominance of neutrons over protons simply as NS. The
thickness of NS is commonly characterized by the difference between the root-
mean-square (r.m.s.) radii calculated for the point distributions of neutrons
and protons:

$\Delta r_{np}=\left\langle r_{n}^{2}\right\rangle^{1/2}-\left\langle
r_{p}^{2}\right\rangle^{1/2}\ .$ (3)

Here $r_{n}$ and $r_{p}$ represent the distance between the center of each
neutron or proton, respectively, and the center of the nucleus. The key point
in accurate determination of $\Delta r_{np}$ in 208Pb and other nuclei
consists in reducing the uncertainty of $\left\langle
r_{n}^{2}\right\rangle^{1/2}$, because proton r.m.s. radii of nuclei were
accurately measured Angeli2013 . The values of $\Delta r_{np}$ were calculated
in several papers, in particular, in Refs. Brown2000 ; Centelles2010 ;
Warda2010 . Various experimental methods were used Trzcinska2001 ; Tarbert2014
; Adhikari2021  to measure $\left\langle r_{n}^{2}\right\rangle^{1/2}$ and
define $\Delta r_{np}$ for 208Pb. However, the existing theoretical and
experimental results on the NS thickness in 208Pb and other nuclei are
characterized by large uncertainties and often contradict each other.

The NS thickness is one of the most fundamental properties of nuclei
associated with the nuclear symmetry energy and, therefore, requiring its
reliable determination. As follows from calculations Dobaczewski1996 , the
excess of neutrons at the periphery of nuclei close to the border of stability
is well above the regular increase caused by a simple excess of neutrons over
protons. This is in contrast to the absence of a significant increase of the
proton radius when approaching the proton-drip line reported in the same paper
Dobaczewski1996 . The relationships between the neutron-rich skin of 208Pb and
the properties of neutron-star crusts have been demonstrated Horowitz2001 .
The correlations between the NS thickness in heavy nuclei, the derivative of
the nuclear symmetry energy at the reduced nuclear density and the radii of
moderate mass neutron stars were investigated in Ref. Steiner2005 . In view of
the existing uncertainties of NS properties, these and other studies strongly
motivate the search for new methods to restrict $\Delta r_{np}$ in heavy
nuclei, in particular, in 208Pb.

It was shown Fang2011 ; Fang2010 ; Yan2019 ; Aumann2017 ; Bertulani2019  that
certain characteristics calculated for nucleus-nucleus collisions at low and
relativistic energies are sensitive to the parameters of NS. The total
reaction cross section, the neutron (proton) removal cross sections as well as
their ratios calculated for collisions of neutron-rich 48Ca of $100A$ MeV with
12C target by means of the statistical abrasion ablation (SAA) model linearly
depend on the NS thickness in 48Ca projectile Fang2011 . Linear correlations
between the neutron removal cross sections and the NS thickness were predicted
by SAA for peripheral collisions of neutron-rich Na, P, Ca and Ni nuclei of
$1A$ GeV energy with 12C Fang2010 . As shown Yan2019 , the ratio between
neutron and proton yields, as well as the tritium to 3He ratio calculated for
collisions of neutron-rich 42-56Ca isotopes of $50A$ MeV with 40Ca demonstrate
strong linear correlations with $\Delta r_{np}$ in projectile nuclei. In Refs.
Aumann2017 ; Bertulani2019  the potential of constraining the density
dependence of the symmetry energy close to saturation density through
measurements of neutron-removal cross sections in high-energy nuclear
collisions of 0.4 to 1 GeV/nucleon is investigated. In order to confirm all
these theoretical results Fang2011 ; Fang2010 ; Yan2019 ; Aumann2017 ;
Bertulani2019  in experiments, beams of various neutron rich Na, P, Ca and Ni
isotopes are necessary, while many of these isotopes are very short-lived.
This creates difficulties in conducting such experiments.

In contrast to Refs. Fang2011 ; Fang2010 ; Yan2019 ; Aumann2017 ;
Bertulani2019 , the production of secondary particles other than nucleons in
208Pb–208Pb and p–208Pb interactions at much higher collision energies
available at the LHC were considered in Refs. De2017 ; Paukkunen2015 ;
Alvioli2019 . An impact of the neutron skin of 208Pb on the inclusive prompt
photon production in 208Pb–208Pb collisions at the LHC has been predicted
De2017 . As shown by calculations Paukkunen2015 ; Alvioli2019 , the
differences in spatial distributions of neutrons and protons in colliding
nuclei induce small differences between the production of $W^{+}$ and $W^{-}$
bosons in peripheral p–208Pb collisions. The authors of De2017 ; Paukkunen2015
; Alvioli2019  have proposed relativistic proton-nucleus and nucleus-nucleus
collisions as tools to study NS, but also pointed out the difficulties that
can arise in detecting such subtle NS effects in experiments at the LHC.

In view of the above-mentioned difficulties of the experiments revealing the
presence of NS in nuclei, it is worthwhile to propose measurements of other
characteristics of nucleus-nucleus collisions sensitive to the parameters of
NS. As shown in Ref. Li2020 , the charged hadron multiplicity difference
between central collisions of isobar nuclei 96Ru–96Ru and 96Zr–96Zr at RHIC is
sensitive to the neutron skin thickness. In this work we demonstrate that the
calculated cross sections of emission of certain numbers of forward spectator
neutrons and protons produced specifically in ultracentral collisions of
relativistic 208Pb nuclei are very sensitive to the parameters of NS in 208Pb
and less sensitive to other parameters in calculations. Indeed, spectator
matter in ultracentral collisions is represented mostly by nucleons which are
peeled away from the nuclear periphery representing NS. It is expected that in
collisions of relativistic nuclei spectator nucleons are kinematically
separated from participant nucleons, and the excess of neutrons with respect
to protons can be detected with neutron and proton ZDCs in the ALICE
experiment at the LHC Puddu2007 ; Abelev2013 .

Our suggestions Pshenichnov2021properties ; Dmitrieva2021 ; Kozyrev2021 ;
Pshenichnov2022  to study ultracentral collisions of 208Pb–208Pb at the CERN
SPS and LHC complemented proposals De2017 ; Paukkunen2015 ; Alvioli2019  to
probe NS in peripheral collisions of these nuclei at the LHC. Previous studies
of 208Pb–208Pb collisions at both facilities Appelshauser1998 ; Abelev2013
included detection of spectator neutrons and protons. Therefore, the effects
of NS in ultracentral 208Pb–208Pb collisions can be investigated and compared
at significantly different collision energies $\sqrt{s_{\mathrm{NN}}}=17.21$
GeV of the CERN SPS and $\sqrt{s_{\mathrm{NN}}}=5.02$ TeV of the LHC. The
motivation to study spectator neutrons and protons specifically in most
central collisions of heavy relativistic nuclei was first formulated in our
papers Pshenichnov2021properties ; Dmitrieva2021 ; Kozyrev2021 ;
Pshenichnov2022  based on an early version of Abrasion-Ablation Monte Carlo
for Colliders (AAMCC) model. Later, the probing of neutron-skin thickness with
spectator neutrons in ultracentral high-energy isobaric 96Zr–96Zr and
96Ru–96Ru collisions was considered independently by Liu and co-authors
Liu2022 .

##  2 Modeling spectator matter with AAMCC-MST

A new version of the AAMCC model is used in the present work to calculate the
properties of spectator matter in relativistic collisions of heavy nuclei on
the event-by-event basis. The previous AAMCC version svetlichnyi2021using ;
svetlichnyi2020formation ; nepeivoda2021dependence ; Pshenichnov2021properties
; Dmitrieva2021  has been supplemented with a pre-equilibrium decay model
based on the MST-clusterisation to build the new version of the model called
AAMCC-MST Nepeivoda2022 . Each collision event is modelled with AAMCC-MST in
several stages. Firstly, the sizes and shapes of spectator prefragments from
both colliding nuclei are defined at the abrasion stage using the Glauber
Monte Carlo (Glauber MC) model Loizides2018  as described in Section 2.1.
Secondly, the excitation energy of the prefragments is calculated with a
hybrid approach combining the Ericson formula Ericson1960  and ALADIN
parameterization Botvina1995 , see Section 2.2. Thirdly, at the ablation stage
the minimum spanning tree (MST) clustering algorithm is applied to
prefragments to define secondary clusters and their excitation energies, see
Section 2.3. Finally, cluster decays are simulated via the evaporation model
Weisskopf1940 , Fermi Break-up model Fermi1950  and Statistical
Multifragmentation Model (SMM) Bondorf1995  from Geant4 toolkit
Agostinelli2003 , see Section 2.4.

###  2.1 Modeling collisions with Glauber Monte Carlo

The Glauber MC model Loizides2018  is used to simulate the initial geometry of
each collision event. At this first abrasion step, the positions of nucleons
in colliding nuclei are sampled according to the nucleon density distributions
in these nuclei, taking into account a minimum possible distance between the
centers of nucleons representing the repulsive core of the nucleon-nucleon
potential. Examples of distributions of neutron and proton densities in 208Pb
simulated on the basis of the two-parameter Fermi functions given by Eq. (1)
are shown in Fig. 1. The ratio of these distributions
$\rho_{p}(r)/\rho_{n}(r)$ is also shown in Fig. 1. It becomes small at the
nuclear periphery revealing the NS effect in 208Pb. The parameters of the
Pbpnrw nuclear density profile used by default in Glauber MC for 208Pb are
$R_{n}=6.69$ fm, $a_{n}=0.56$ fm and $R_{p}=6.68$ fm, $a_{p}=0.447$ fm
Loizides2018 .

Figure 1: Radial distributions of neutron and proton densities $\rho_{n,p}(r)$
and their ratio $\rho_{p}(r)/\rho_{n}(r)$ in 208Pb calculated with GlauberMC
profile Pbpnrw Loizides2018  with $R_{n,p}$ and $a_{n,p}$ given in Table 1.

In the present work six different combinations of neutron and proton density
distributions in 208Pb were implemented in AAMCC-MST. This allows us to study
the sensitivity of neutron and proton content of spectator matter from
ultracentral 208Pb–208Pb collisions to the parameters of NS in 208Pb. All
profiles were taken following Eqs. (1) and (2) with their parameters listed in
Table 1. These profiles were grouped into three pairs to compare AAMCC-MST
results pairwise. The density profiles in the first pair, denoted as NFB-8 and
NL1, were taken from Ref. Centelles2010 . They represent calculated density
distributions which characterize, respectively, the most thin ($\Delta
r_{np}=0.115$ fm) and thick ($\Delta r_{np}=0.321$ fm) NS among all the
nuclear structure models of 208Pb mentioned in Ref. Centelles2010 . The
parameterizations Pbpnrw and PREX in the second pair are based on the results
of Crystal Ball at MAMI and A2 collaboration Tarbert2014  and PREX
Adhikari2021  collaboration, respectively. These collaborations conducted
measurements for 208Pb by two different techniques and used different
theoretical models to fit experimental results. In Ref. Tarbert2014  an
interpolated fit of a theoretical model to the measured cross sections
provided $R_{n}=6.70\pm 0.03$ fm and $a_{n}=0.55\pm
0.01\mathrm{(stat.)}-0.03+0.02\mathrm{(sys.)}$ fm. Following this result, the
parameterization Pbpnrw with $R_{n}=6.69$ fm and $a_{n}=0.56$ fm was adopted
in Glauber MC as a default option for 208Pb Loizides2018 . In Ref.
Adhikari2021  $\Delta r_{np}=0.283\pm 0.071$ fm was extracted from
measurements of parity violation in elastic scattering of polarised electrons
from 208Pb, but the corresponding $R_{n}$ and $a_{n}$ were not reported. As
known Warda2010 , even with a well-established parametrization of proton
density in 208Pb with $R_{p}=6.68$ fm and $a_{p}=0.447$ fm Tarbert2014 ;
Loizides2018 , many combinations of $R_{n}$ and $a_{n}$ lead to the same
$\Delta r_{np}$.

In order to find $R_{n}$ and $a_{n}$ which reasonably reproduce $\Delta
r_{np}=0.283$ fm reported by the PREX collaboration Adhikari2021 , the average
value of $R_{n}=6.810$ fm from several versions of the models considered in
Ref. Centelles2010  was taken. Then, the diffuseness parameter $a_{n}=0.60$ fm
for neutrons was calculated from the following relations Warda2010 :

$\left\langle
r_{p}^{2}\right\rangle=\frac{3}{5}R_{p}^{2}\left(1+\frac{7}{3}\frac{\pi^{2}a_{p}^{2}}{R_{p}^{2}}\right),$
(4) $\left\langle a_{n}^{2}\right\rangle=\frac{5}{7\pi^{2}}\left(\Delta
r_{np}+\sqrt{\left\langle
r_{p}^{2}\right\rangle}\right)^{2}-\frac{3}{7}\frac{R_{n}^{2}}{\pi^{2}}.$ (5)

The values of $R_{n}=6.810$ fm, $a_{n}=0.60$ fm, $R_{p}=6.68$ fm and
$a_{p}=0.447$ fm thus represent the nuclear density profile of 208Pb denoted
as PREX, see Table 1. The Pbpnrw and PREX profiles also represent thin
($\Delta r_{np}=0.150$ fm) and thick ($\Delta r_{np}=0.283$ fm) NS,
respectively.

With the aim to evaluate the uncertainty of selecting $R_{n}$ and $a_{n}$ to
reproduce $\Delta r_{np}=0.283$ fm reported by the PREX collaboration, two
additional parameterizations denoted as PREX1 and PREX2 were introduced as a
third pair in Table 1. These parameterizations provide the same $\Delta
r_{np}=0.283$ fm, but with the minimum and maximum values of $R_{n}$
corresponding, respectively, to NFB-8 and NL1 profiles Centelles2010 .
Following Ref. Trzcinska2001 , PREX1 profile with its parameters $R_{n}=R_{p}$
and $a_{n}>a_{p}$ can be considered as a pure case of a neutron halo (NH) in
208Pb. Other profiles, excluding NFB-8, have $R_{n}>R_{p}$ and $a_{n}>a_{p}$
and thus should be considered as nuclear density profiles with NS.

Table 1: Half-density radii $R_{n,p}$, diffuseness parameters $a_{n,p}$ and the corresponding neutron skin thickness $\Delta r_{np}$ for six different parameterizations of neutron and proton densities in 208Pb used in this work in AAMCC-MST calculations. The values of $b_{max}$ for the impact parameter interval $[0,b_{max}]$ corresponding to 0–5% centrality at $\sqrt{s_{\mathrm{NN}}}=17.21$ GeV and $5.02$ TeV, respectively, are also given. |  $R_{n}$, |  $a_{n}$, |  $R_{p}$, |  $a_{p}$, |  $\Delta r_{np}$, |  $b_{max}$,
---|---|---|---|---|---|---
| fm | fm | fm | fm | fm | fm
|  |  |  |  |  | 17.21 GeV | 5.02 TeV
NFB-8 | $6.679$ | $0.546$ | $6.683$ | $0.451$ | $0.115$ | $3.32$ | $3.45$
NL1 | $6.940$ | $0.587$ | $6.718$ | $0.463$ | $0.321$ | $3.45$ | $3.57$
Pbpnrw | $6.69$ | $0.56$ | $6.68$ | $0.447$ | $0.15$ | $3.37$ | $3.49$
PREX | $6.81$ | $0.60$ | $6.68$ | $0.447$ | $0.283$ | $3.42$ | $3.54$
PREX1 | $6.68$ | $0.66$ | $6.68$ | $0.447$ | $0.283$ | $3.46$ | $3.59$
PREX2 | $6.94$ | $0.53$ | $6.68$ | $0.447$ | $0.283$ | $3.38$ | $3.50$

In contrast to the uncertainties of the parameters of neutron distributions in
208Pb, the profiles Pbpnrw and PREX, as well as PREX1 and PREX2, are based on
the well-established parameters for proton density distributions: $R_{p}=6.68$
fm and $a_{p}=0.447$ fm, see Ref. Loizides2018  for details. Therefore, these
values of $R_{p}$ and $a_{p}$ were used in the calculations with these four
density profiles.

In modelling of each event the centers of the colliding nuclei are placed
initially at ($\pm\frac{b}{2}$, 0, 0), where $b$ is the impact parameter of
the collision. The impact parameter $b$ is defined as the distance of the
closest approach between the centers of colliding nuclei. It is sampled
following the distribution of the number of events per unit interval of $b$
taken as $\frac{dN}{db}\propto b$. Ultracentral 208Pb–208Pb collisions
modelled in this work were defined as events of 0–5% centrality corresponding
to the impact parameter interval of $[0,b_{max}]$. Because of slightly
different radial extensions of nuclear density in each of the six
parametrerizations, $b_{max}$ values corresponding to 0–5% centrality are also
different, and they are listed in Table 1 separately for 208Pb–208Pb
collisions at $\sqrt{s_{\mathrm{NN}}}=17.21$ GeV and $5.02$ TeV. As in other
implementations of the Glauber model, it is assumed that in the course of each
collision all nucleons move strictly parallel to the beam axis $z$ without
changing their positions in the transverse ($x,y$) plane.

In Glauber MC model there are two options to calculate the probability of a
nucleon-nucleon collision $P(b_{\mathrm{NN}})$ as a function of NN impact
parameter $b_{\mathrm{NN}}$. In the first option nucleons are considered as
hard spheres with the diameter $D=\sqrt{\sigma^{\mathrm{NN}}_{inel}/\pi}$. Two
nucleons from colliding nuclei are considered interacting, and thus termed
participants, if the distance between their centers in the transverse ($x,y$)
plane is less than $D$. With this default option:

$P(b_{\mathrm{NN}})=\Theta(D-b_{\mathrm{NN}})\ .$ (6)

The fluctuations of nucleon shape or parton degrees of freedom are taken into
account with another option of GlauberMC Loizides2016  to calculate the NN
collision probability as

$P(b_{\mathrm{NN}})=\Gamma\Big{(}1/w,\frac{b_{\mathrm{NN}}^{2}}{D^{2}w}\Big{)}\Big{/}\Gamma(1/w)\
.$ (7)

Here $\Gamma$ is the gamma function and $w$ is a parameter which controls the
transition from the hard-sphere ($w\rightarrow 0$) to the Gaussian
($w\rightarrow 1$) representation of $P(b_{\mathrm{NN}})$. As pointed out in
Ref. Loizides2018 , an intermediate value of $w$ = 0.4 reproduces well the
values of the total and elastic $pp$ cross sections measured at the LHC. As
also shown Loizides2018 , by using $w$ = 0.4 in Eq. (7) the number of binary
nucleon-nucleon collisions $N_{coll}$ is reduced up to 10% with respect to
calculations with hard-sphere option Eq. (6) in modelling peripheral
208Pb–208Pb collisions at $\sqrt{s_{\mathrm{NN}}}=5.02$ TeV. However, a
smaller reduction of $N_{coll}$ by $\sim 1$% is estimated for central (0–5%)
events.

The dependence of $\sigma^{\mathrm{NN}}_{inel}$ on the square of center-of-
mass energy $s$ in nucleon-nucleon collisions is parameterized by
$\sigma^{\mathrm{NN}}_{inel}(s)={\cal A}+{\cal B}\times\ln^{2}(s)$, where the
parameters ${\cal A}$ and ${\cal B}$ are extracted from experimental data, see
Ref. Loizides2018  for details. While each nucleon can interact several times
with nucleons of the collision partner, same inelastic nucleon-nucleon cross
section is assumed for the first and for all subsequent collisions despite of
nucleon energy loss. This assumption is based on the weak dependence of
$\sigma^{\mathrm{NN}}_{inel}$ on $\mathrm{NN}$ collision energy typical for
high energies. It is assumed that all other nucleons that are not participants
form two excited spectator prefragments representing the remnants of colliding
nuclei.

###  2.2 Calculation of prefragment excitation energy

Each of the two prefragments is produced by removing $a$ participating
nucleons from the respective initial nucleus with the mass number $A$ by the
end of the abrasion stage. Since an individual nucleon can interact with one
or more nucleons from another nucleus, the number of participants in each of
the two nuclei usually differs. Such fluctuations are properly reflected by
Monte Carlo modeling, and the prefragment mass number is calculated
individually on each side as $A_{pf}=A-a$. Two different methods to calculate
the prefragment excitation energy $E^{\star}$ are employed in AAMCC-MST
depending on the relative mass of the prefragment $\alpha_{\rm pf}=A_{\rm
pf}/A$.

In the case of a peripheral collision event when very few nucleons are
removed, $a\ll A$, it is assumed that the properties of a nuclear residue
remain similar to those of the initial nucleus. As discussed Gaimard1991 ,
this justifies the calculation of the total prefragment excitation energy by
considering a set of nucleon holes created in the initial nucleus. With this
method the distribution of the excitation energy of a prefragment is computed
from the density of states of the nucleus $A$ with $a$ nucleon holes.
Following the calculations on the basis of Ericson formula Ericson1960 , this
results in the mean excitation energy per removed nucleon in events with $a$
removed nucleons Scheidenberger2004 :

$\frac{\langle E^{\star}\rangle}{a}=E_{\rm max}\frac{a}{a+1}\ .$ (8)

Here $E_{\rm max}=40$ MeV sets the maximum energy of a single hole state in
the initial heavy nucleus Scheidenberger2004 . From Eq. (8) the average
excitation energy per prefragment nucleon $\langle\epsilon^{*}\rangle=\langle
E^{*}/A_{\rm pf}\rangle$ can be obtained:

$\langle\epsilon^{\star}\rangle=E_{\rm max}\frac{(1-\alpha_{\rm
pf})^{2}}{\alpha_{\rm pf}\big{(}1-\alpha_{\rm pf}+A^{-1}\big{)}}\ \ ,$ (9)

with $\alpha_{\rm pf}=A_{pf}/A$ defined as a relative mass of the prefragment.
In central collisions the numbers of removed nucleons are comparable to their
total numbers in the initial nuclei, $a\sim A$, and essential changes of the
both nuclear cores are expected. In this case the validity of Eq. (9) becomes
questionable, and it is replaced by a phenomenological dependence:

$\langle\epsilon^{*}\rangle=\epsilon_{\rm max}\sqrt{1-\alpha_{\rm pf}}\ \ .$
(10)

This relation between $\langle\epsilon^{*}\rangle$ and $\alpha_{\rm pf}$ is
similar to the one used in Ref. Botvina1995 , but without its linear term. The
value of $\epsilon_{\rm max}=11.5$ MeV is used in the present work as a model
parameter. A dispersion given by

$\sigma=\sigma_{0}(1+b_{0}\cdot(1-\alpha_{\rm pf}))$ (11)

around the average values of $\alpha_{\rm pf}$ was assumed with
$\sigma_{0}=0.005$ and $b_{0}=2$ taken as free parameters.

It was found that the calculated average multiplicities of spectator neutrons
$\langle N_{n}\rangle$ and protons $\langle N_{p}\rangle$ as functions of the
collision impact parameter $b$ are mostly defined by the dependence of
$\langle\epsilon^{*}\rangle$ on $\alpha_{\rm pf}$, rather than by the
dispersion $\sigma$. Due to the overestimation of $E^{\star}$ in peripheral
collisions, $\langle N_{n}\rangle$ and $\langle N_{p}\rangle$ are also
overestimated in calculations based on Eq. (10) with respect to the neutron
multiplicities measured in peripheral 208Pb–208Pb collisions
CollaborationALICE2020 . In contrast, calculations based on the Ericson
formula, Eq. (8), overestimate the multiplicities of spectator neutrons and
protons in semi-central and central 208Pb–208Pb collisions when compared with
the data CollaborationALICE2020 . In order to resolve this issue, a hybrid
parameterization of the prefragment excitation energy was introduced in AAMCC-
MST in Ref. Nepeivoda2022  to improve the description of ALICE data
CollaborationALICE2020 . In peripheral events for prefragments with
$\alpha_{\rm pf}>\alpha_{\mathrm{s}w}$ the excitation energy is calculated on
the basis of Ericson formula, while for events with low relative mass of
prefragments, $\alpha_{\rm pf}<\alpha_{\mathrm{s}w}$, their excitation energy
is calculated according to the ALADIN parameterization. Then, the dependence
(9) is switched to (10) at the value of $\alpha_{\rm sw}$ determined from the
condition of continuity of the expressions (9) and (10) used at low and high
$\epsilon^{\star}$, respectively. Neglecting $1/A$ for heavy nuclei and
equating the expressions (9) and (10) gives

$\alpha_{\rm sw}=\frac{\sqrt{1+4\kappa}-1}{2\kappa},\ \
\kappa=\frac{\epsilon_{\rm max}^{2}}{E_{\rm max}^{2}}\ .$ (12)

With the parameters used in the present work for 208Pb projectiles Eq. (12)
provides $\alpha_{\rm sw}=0.9287$. This means that for central 208Pb–208Pb
collisions, which are of the main interest in the present work, the method
based on the ALADIN parameterization, Eq. (10), is applied.

###  2.3 Preequilibrium clustering of spectator matter

To the best of our knowledge, in all versions of the abrasion-ablation model
spectator matter (a prefragment) is considered as an entire nuclear system
with intense interactions between constituent nucleons. Therefore, it is
assumed that the thermodynamic equilibrium in the entire prefragment is
achieved by the time of its decay. However, the validity of this assumption
can be questioned in the case of central nucleus-nucleus collisions, in which
the prefragment has a shape of a narrow crescent, see Fig. 2. It can be
expected that in such collisions the hot prefragment disintegrates into
several clusters already at the preequilibrium stage and later the
thermodynamic equilibrium is established locally in each cluster. Similarly to
the assumptions adopted in Ref. Botvina2022 , an excited nuclear system
undergo a subdivision into primary equilibrated nucleon clusters, and final
state nuclei are produced after the decay of these excited clusters.

In order to account for preequilibrium disintegration of prefragments we have
developed an algorithm of clusterization based on the construction of a
minimum spanning tree (MST) Prim1957 . In this algorithm, groups of nucleons
of spectator matter are identified as clusters on the basis of their proximity
in three-dimensional coordinate space.

Figure 2: Transverse projection of a central 208Pb$-^{208}$Pb collision at
$\sqrt{s_{NN}}=17.2$ GeV (top). Spectator nucleons from each of the colliding
nuclei in the same event are shown separately in the bottom panel.

First, it is assumed that all nucleons represent the vertices of a complete
weighted undirected graph. The edge weights of the graph are equal to the
modulus of the distance between the corresponding vertex nucleons. Second, a
tree without breaking connectivity with the minimum possible sum of all the
edge weights is constructed by means of Kruskal’s algorithm. Third, the
critical distance cut is applied to search for inconsistent edges and
subdivide the MST tree. This is made by removing edges with weights greater
than some critical distance $d$ considered as a free parameter of the model.
Finally, the depth-first search is used to define clusters.

The application of the MST algorithm to large prefragments resulting from
peripheral collisions should not lead to their separation into many clusters
at the preequilibrium stage. Therefore, it is reasonable to set $d$
significantly longer than the average distance between the centers of nucleons
in nuclear matter at the normal nuclear density. In particular, in modelling
peripheral collisions assuming $d_{0}=2.7$ fm, prefragments remained
sufficiently stable. At the same time, prefragements with specific shapes
produced in central collisions were dissociated into clusters in calculations
with $d_{0}=2.7$ fm.

One can expect a higher average multiplicity of nucleons and nucleon clusters
estimated by the MST algorithm for dilute nuclear matter with reduced density
$\rho<\rho_{0}$ with respect to prefragments of the normal density $\rho_{0}$.
The AAMCC-MST model is based on the Glauber theory, and estimating the
prefragment density at the preequilibrium stage is not a simple task. During a
short collision time any changes in the spatial coordinates of nucleons due to
the radial expansion of the spectator matter in the plane transverse to the
beam axis are neglected in the Glauber theory. Alternatively, the lower limit
for the prefragment density at the preequilibrium stage can be estimated from
the average prefragment density $\langle\rho\rangle$ at a later stage, after
reaching the thermodynamic equilibrium.

As estimated from experimental data Viola2004 ; De2006 , at high
$\epsilon^{\star}$, $\langle\rho\rangle$ drops significantly, and at
$\epsilon^{\star}\geq 5$ MeV it reaches a plateau value of
$\langle\rho\rangle\sim(1/3-1/2)\rho_{0}$. Similarly, in the Statistical
Multifragmentation Model (SMM) Bondorf1995  it is assumed that at the moment
of an explosive decay of a hot termalized nuclear system with
$\epsilon^{\star}\geq 3-4$ MeV into fragments, its average density is also
lower than $\rho_{0}$: $\langle\rho\rangle\sim(1/4-1/3)\rho_{0}$.

The dependence of $\langle\rho(\epsilon^{\star})/\rho_{0}\rangle$ on the
excitation energy $\epsilon^{\star}$ at the time of preequilibrium
clusterization is postulated in the AAMCC-MST model taking into account the
above mentioned results Viola2004 ; De2006 . It is assumed that
$\langle\rho(\epsilon^{\star})/\rho_{0}\rangle=1$ at
$\epsilon^{\star}\leq\epsilon_{0}$. At higher excitation energies
$\epsilon^{\star}>\epsilon_{0}$ a phenomenological approximation of
$\langle\rho(\epsilon^{\star})/\rho_{0}\rangle$ by a power function is
introduced to account for the expansion:

$\langle\rho(\epsilon^{\star})/\rho_{0}\rangle=(\epsilon^{\star}/\epsilon_{0})^{\gamma}\
.$ (13)

Here $\gamma=-1.02\pm 0.10$ and $\epsilon_{0}=2.17\pm 0.23$ MeV are the
parameters which ensure the continuity of
$\langle\rho(\epsilon^{\star})/\rho_{0}\rangle$ below and above
$\epsilon_{0}$.

Following Eq. (13) a phenomenological dependence of the MST clustering
parameter $d$ on the excitation energy at the preequilibrium stage is
postulated in the AAMCC-MST model. It is expected that due to more intense
motion of intracluster nucleons and outward pressure associated with the
increase of $\epsilon^{\star}$, the internal density of clusters decreases on
the way to the thermal equilibrium. However, due to a short duration of the
preequilibrium stage, there is no significant global expansion of the cluster
system. Therefore, with an increase in the excitation energy of the system,
the distances between the cluster boundaries also decrease because they
correlate on average with $d$. Since the exact dependence of $d$ on the
internal density of clusters is unknown, a phenomenological dependence of
$d\propto\rho^{1/3}(\epsilon^{\star})$ is assumed. Together with Eq. (13) it
finally gives:

$d=\begin{cases}d_{0},\ \epsilon^{\star}<\epsilon_{0}\\\ d_{0}\
(\epsilon^{\star}/\epsilon_{0})^{\gamma/3},\
\epsilon^{\star}>\epsilon_{0}\end{cases}\text{, }$ (14)

where $d_{0}=2.7$ fm is the clustering parameter at the normal nuclear density
$\rho_{0}$. The decrease of $d$ with the increase of $\epsilon^{\star}$
reflects a decrease of the connectivity of the prefragment system with the
growth of $\epsilon^{\star}$. It is assumed that the excitation energy
$E^{\star}$ of the spectator prefragment is distributed between the resulting
clusters proportionally to the number of nucleons they contain.

###  2.4 Modeling decays of clusters

Statistical models of the Geant4 Allison2016  toolkit are used in AAMCC-MST to
simulate decays of clusters formed at the preequilibrium stage. For clusters
with $\epsilon^{\star}\leq 4$ MeV, the Weisskopf-Ewing model Weisskopf1937  is
used to simulate sequential evaporation of nucleons and light nuclei (2H, 3H,
3He, 4He). The Statistical Multifragmentation Model (SMM) Bondorf1995  of
Geant4 Allison2016  is employed at $\epsilon^{\star}>4$ MeV. The Fermi Break-
Up model Fermi1950  from the same toolkit is used to simulate decays of
excited light nuclei up to 18O.

Specific versions of Geant4 decay models were employed on the basis of the
correspondence of their results to Fortran versions of the respective models
Bondorf1995 . Namely, the SMM and evaporation models were taken from the
Geant4 of version 10.4, while the Fermi Break-up model was adopted from the
Geant4 of version 9.6.

##  3 Results and discussion

###  3.1 Average numbers of spectator neutrons and protons

Six different parameterizations of neutron and proton densities in 208Pb
listed in Table 1 were used in AAMCC-MST to calculate the average numbers of
spectator neutrons $\left\langle N_{n}\right\rangle$ and protons $\left\langle
N_{p}\right\rangle$ in central 208Pb–208Pb collisions at the CERN SPS. The
data of the NA49 collaboration Appelshauser1998  obtained for 208Pb–208Pb
collisions with $\langle b\rangle=2$ fm at the CERN SPS at
$\sqrt{s_{\mathrm{NN}}}=17.21$ GeV make it possible to validate the AAMCC-MST
results. As reported Appelshauser1998 , forward-going spectator matter in
central 208Pb–208Pb collisions consists on average of 9 neutrons, 7 protons,
and 0.5 deuterons. The values of $\left\langle N_{n}\right\rangle$,
$\left\langle N_{p}\right\rangle$ and $\left\langle N_{d}\right\rangle$
calculated with AAMCC-MST for 208Pb–208Pb collisions at $b$ = 2 fm are
compared with the data Appelshauser1998  in Table 2. The values of
$\left\langle N_{p}\right\rangle$ calculated with all nuclear density
parameterizations agree with the measured $\left\langle N_{p}\right\rangle$
within the experimental uncertainties. This indicates that the parameters of
the proton density distributions used in AAMCC-MST calculations are realistic.
However, the average numbers of spectator neutrons $\left\langle
N_{n}\right\rangle$ are overestimated with all NS options. One can note that
the average $\langle b\rangle=2$ fm for central events was reported in Ref.
Appelshauser1998  rather than a specific interval of $b$. The observed
disagreement between the measurements and calculations can be potentially
caused by assuming the specific value of $b=2$ fm in calculations with AAMCC-
MST. Nevertheless, an obvious correlation between $\left\langle
N_{n}\right\rangle$ and $\Delta r_{np}$ is seen in Table 2. As expected,
slightly more neutrons are emitted in modelling with a thicker NS represented
by NL1, PREX, PREX1 and PREX2 options. The calculated average numbers of
deuterons are overestimated in comparison to the measured $\left\langle
N_{d}\right\rangle$, but some variations of deuteron yields are also seen with
respect to the NS thickness. Namely, the highest $\left\langle
N_{d}\right\rangle$ of 0.87 is obtained with the thinnest NS in NFB-8, while
the lowest $\left\langle N_{d}\right\rangle$ of 0.77 is associated with the
most thick NS in NL1. Due to the equal presence of neutrons and protons in
deuterons, neutron-rich spectator matter is less favorable for the formation
of deuterons.

Table 2: Average numbers of spectator neutrons $\left\langle N_{n}\right\rangle$, protons $\left\langle N_{p}\right\rangle$ and deuterons $\left\langle N_{d}\right\rangle$ in ultracentral ($b=2$ fm) 208Pb–208Pb collisions at $\sqrt{s_{\mathrm{NN}}}=17.21$ GeV measured by NA49 collaboration Appelshauser1998  and calculated with AAMCC-MST with different nuclear density profiles. |  $\Delta r_{np}$, fm | $\left\langle N_{n}\right\rangle$ | $\left\langle N_{p}\right\rangle$ | $\left\langle N_{d}\right\rangle$
---|---|---|---|---
NFB-8 | 0.115 | 12.45 | 6.82 | 0.87
NL1 | 0.321 | 13.88 | 6.44 | 0.77
Pbpnrw | 0.15 | 12.73 | 6.74 | 0.85
PREX | 0.283 | 13.59 | 6.42 | 0.80
PREX1 | 0.283 | 13.79 | 6.49 | 0.78
PREX2 | 0.283 | 13.41 | 6.32 | 0.82
NA49 |  | $9.0\pm 1.8$ | $7.0\pm 1.4$ | 0.5

The values of $\left\langle N_{n}\right\rangle$, $\left\langle
N_{p}\right\rangle$ were calculated with AAMCC-MST in Ref. Nepeivoda2022  as
functions of $b$ and were compared with preliminary ALICE data
CollaborationALICE2020  for 208Pb–208Pb collisions at
$\sqrt{s_{\mathrm{NN}}}=5.02$ TeV. The value of $\left\langle
N_{p}\right\rangle$ calculated at $b=2$ fm was found in good agreement with
measured $\left\langle N_{p}\right\rangle\sim 3$. At the same time, calculated
$\left\langle N_{n}\right\rangle$ underestimated the data for central
collisions at $\sqrt{s_{\mathrm{NN}}}=5.02$ TeV, in contrast to $\left\langle
N_{n}\right\rangle$ calculated in the present work at
$\sqrt{s_{\mathrm{NN}}}=17.21$ GeV.

###  3.2 NFB-8 versus NL1 profile and dependence on collision energy

The dependence of the yields of spectator nucleons in ultracentral 208Pb–208Pb
events on collision energy can be understood from the comparison of AAMCC-MST
results obtained at $\sqrt{s_{\mathrm{NN}}}=17.21$ GeV and at 5.02 TeV for the
same centrality of 0–5%.

Table 3: Average numbers of spectator neutrons $\left\langle N_{n}\right\rangle$ and protons $\left\langle N_{p}\right\rangle$ in ultracentral 208Pb–208Pb collisions for 0–5% centrality at $\sqrt{s_{\mathrm{NN}}}=17.21$ GeV (SPS) and $\sqrt{s_{\mathrm{NN}}}=5.02$ TeV (LHC) calculated with NFB-8 and NL1 density profiles. | NFB-8 | NL1
---|---|---
|  $\left\langle N_{n}\right\rangle$ |  $\left\langle N_{p}\right\rangle$ |  $\left\langle N_{n}\right\rangle$ |  $\left\langle N_{p}\right\rangle$
SPS |  14.38 |  7.81 |  16.24 |  7.72
LHC |  8.67 |  4.60 |  9.92 |  4.38

As seen from Table 3, $\left\langle N_{n}\right\rangle$ and $\left\langle
N_{p}\right\rangle$ calculated with NFB-8, as well as with NL1 profiles,
noticeably decrease with the increase of collision energy. Such a depletion of
spectator matter in collisions at the LHC is explained by the increase of the
nucleon-nucleon cross section $\sigma^{\mathrm{NN}}_{inel}$ as discussed in
Section 2. Because of this increase, the number of NN-collisions is larger and
results in the increase of the number of participant nucleons at high energy.
Consequently, the number of remaining spectator nucleons is reduced at the
LHC.

Figure 3: Cross sections $\sigma(N_{n},N_{p})$ of emission of given numbers of
spectator neutrons $N_{n}$ and protons $N_{p}$ in 208Pb–208Pb collisions for
0–5% centrality at $\sqrt{s_{\mathrm{NN}}}=17.21$ GeV (top) and
$\sqrt{s_{\mathrm{NN}}}=5.02$ TeV (bottom) calculated with NFB-8 nuclear
density profile in 208Pb.

The reduction in the numbers of $N_{n}$ and $N_{p}$ at the LHC energy is also
seen in the partial cross sections $\sigma(N_{n},N_{p})$ of emission of given
numbers of neutrons and protons calculated with AAMCC-MST with NFB-8 profile
for collisions at 0–5% centrality. These cross sections are presented in Fig.
3 for two collision energies. As follows from calculations, the maximum of
$\sigma(N_{n},N_{p})\sim 3$ mb at the SPS is predicted for the emission of
11–17 neutrons accompanied by 6–9 protons. However, the maximum of
$\sigma(N_{n},N_{p})\sim 6$ mb at the LHC is obtained for the emission of only
2–4 neutrons along with 1–2 protons. While in the simulated ultracentral
208Pb–208Pb collisions about 25 spectator neutrons and 15 spectator protons
are emitted with noticeable cross sections at the SPS, such high
multiplicities of spectator nucleons are very improbable in collisions at the
LHC, see Fig. 3.

Figure 4: Difference between the cross sections $\sigma(N_{n},N_{p})$ of
emission of given numbers of spectator neutrons $N_{n}$ and protons $N_{p}$
calculated with NL1 and NFB-8 nuclear density profiles for 208Pb–208Pb
collisions for 0–5% centrality at $\sqrt{s_{\mathrm{NN}}}=17.21$ GeV (top) and
$\sqrt{s_{\mathrm{NN}}}=5.02$ TeV (bottom).

The main goal of the present work consists in evaluating the sensitivity of
the parameters of NS to the yields of spectrator neutrons and protons in
ultracentral 208Pb–208Pb collisions. With this in mind the cross sections
$\sigma(N_{n},N_{p})$ were calculated not only with NFB-8 nuclear density
profiles representing a thin NS as in Fig. 3, but also with NL1 corresponding
to thick NS. The changes in $\sigma(N_{n},N_{p})$ caused by the replacement of
NFB-8 by NL1 in calculations can be seen from Fig. 4 where the absolute
difference $\sigma_{\rm NL1}-\sigma_{\rm NFB-8}$ between two calculations is
presented as function of $N_{n}$ and $N_{p}$. Also in this case, the results
are given at $\sqrt{s_{\mathrm{NN}}}=17.21$ GeV and 5.02 TeV to understand the
dependence on collision energy.

As seen from Fig. 4, additional surface neutrons in NL1 lead to larger
$\sigma(N_{n},N_{p})$ for $N_{n}$ between 10 and 27 and $N_{p}$ between 2 and
14 in 208Pb–208Pb collisions at the SPS. Qualitatively, a similar increase of
$\sigma(N_{n},N_{p})$ is seen also for collisions at the LHC, but for smaller
numbers of spectator nucleons: $N_{n}=4-18$ and $N_{p}=0-7$. More neutrons and
less protons are obtained in general in calculations with NL1 representing a
thick NS ($\Delta r_{np}=0.321$ fm) in comparison to NFB-8 option ($\Delta
r_{np}=0.115$ fm).

Figure 5: Cross sections $\sigma(N_{n},N_{p})$ of emission of given numbers of
spectator neutrons $N_{n}$ in events with specific number of spectator protons
$N_{p}=$2, 3, 4 or 5 in 208Pb–208Pb collisions of 0–5% centrality at
$\sqrt{s_{\mathrm{NN}}}=17.21$ GeV calculated with NFB-8 and NL1 density
profiles in 208Pb.

Figure 6: Relative difference between the cross sections $\sigma(N_{n},N_{p})$
of emission of a given number of spectator neutrons $N_{n}$ accompanied by a
specific number of spectator protons $N_{p}$ calculated with NL1 and NFB-8
profiles in 208Pb–208Pb collisions for 0–5% centrality. Top panel: at
$\sqrt{s_{\mathrm{NN}}}=17.21$ GeV for $N_{p}=2,3,4,5$. Bottom panel: at
$\sqrt{s_{\mathrm{NN}}}=5.02$ TeV for $N_{p}=0,1,2,3$. Shadowed bands
represent statistical uncertainties of calculations.

It is challenging to measure $\left\langle N_{n}\right\rangle$ and
$\left\langle N_{p}\right\rangle$ in experiments because of expected wide
ranges of $N_{n}$ and $N_{p}$ presented in Fig. 3. Events of low and high
spectator nucleon multiplicity have to be detected. With a limited acceptance
and efficiency of forward detectors sophisticated corrections for spectator
nucleons lost in multinucleon events are necessary, as explained, in
particular, in Ref. Dmitrieva2018 . According to Table 3 only $\sim 15$%
differences in $\left\langle N_{n}\right\rangle$ are expected for thin and
thick NS, and such a difference may be hidden in the uncertainties of
detection of multinucleon events caused by efficiency limitations. Therefore,
the cross sections $\sigma(N_{n},N_{p})$ for relatively small $N_{n}$ and
$N_{p}$ can be considered as more sensitive probes of the parameters of NS. In
selecting suitable intervals of $N_{n}$ and $N_{p}$ one can turn to Fig. 4 to
find neutron and proton multiplicities with the most prominent changes of
$\sigma(N_{n},N_{p})$ caused by variations of the NS thickness in
calculations. In particular, as seen from Fig. 4, it is not reasonable to plan
measurements of the sum of $\sigma(N_{n},N_{p})$ over a wide range of
$N_{n}=1-15$, but for specific numbers of $N_{p}=0,1,2,...5$. Indeed, the
differences between $\sigma(N_{n},N_{p})$ calculated with NL1 and NFB-8
profiles have different signs, and thus their contributions to this sum
compensate each other. This is seen from Fig. 5 presenting
$\sigma(N_{n},N_{p})$ calculated for 208Pb–208Pb collisions at 0–5% centrality
and $\sqrt{s_{\mathrm{NN}}}=17.21$ GeV in a wide range of $N_{n}$, but for
specific $N_{p}=2,3,4$ or 5. The selection of $2\leq N_{p}\leq 5$ is based on
the difference between $\sigma(N_{n},N_{p})$ calculated with NL1 and NFB-8
shown in Fig. 4. Indeed, the largest absolute difference up to 0.6 mb is found
for $2\leq N_{p}\leq 5$ ultracentral 208Pb–208Pb collisions at
$\sqrt{s_{\mathrm{NN}}}=17.21$ GeV.

As seen from Fig. 5, the maximum of $\sigma(N_{n},N_{p})$ calculated with a
thicker NS corresponding to NL1 profile is shifted to larger $N_{n}$ compared
to the calculations with NFB-8, but the sum of $\sigma(N_{n},N_{p})$ within
the considered range of $N_{n}$ remains almost the same for calculations with
NL1 and NFB-8. Therefore, the determination of $\sigma(N_{n},N_{p})$ for
specific $N_{n}$ and $N_{p}$ can be proposed in future experiments, rather
than measuring the sum of $\sigma(N_{n},N_{p})$ over $N_{n}$.

The results for $\sigma(N_{n},N_{p})$ calculated with NL1 and NFB-8 for
ultracentral 208Pb–208Pb collisions at $\sqrt{s_{\mathrm{NN}}}=5.02$ TeV were
found qualitatively similar to those shown in Fig. 5 for
$\sqrt{s_{\mathrm{NN}}}=17.21$ GeV, and they are not presented here. In
comparison to lower collision energy, $\sigma(N_{n},N_{p})$ calculated at
$\sqrt{s_{\mathrm{NN}}}=5.02$ TeV demonstrate a larger difference between the
results obtained with NL1 and NFB-8 (up to 1.6 mb). However, this maximum
value is found at lower $N_{n}$ and $N_{p}=0,1,2,3$ because of the above-
described depletion of the spectator matter in 208Pb–208Pb collisions caused
by the increase of NN-cross section with collision energy.

The degree of sensitivity of $\sigma(N_{n},N_{p})$ to variations of the
parameters of NS can be demonstrated by calculating the relative difference
between $\sigma(N_{n},N_{p})$ obtained with two different nuclear density
profiles. The relative variation
$(\sigma_{\mathrm{NL1}}-\sigma_{\mathrm{NFB-8}})/\sigma_{\mathrm{NL1}}$
calculated for NL1 and NFB-8 profiles as function of $N_{n}$ is presented in
Fig. 6 for 208Pb–208Pb collisions $\sqrt{s_{\mathrm{NN}}}=17.21$ GeV and 5.02
TeV. For the considered two collision energies specific numbers of spectator
protons were taken as $N_{p}=2,3,4,5$ and $N_{p}=0,1,2,3$, respectively, as
explained above.

As seen from Fig. 6, the value of
$(\sigma_{\mathrm{NL1}}-\sigma_{\mathrm{NFB-8}})/\sigma_{\mathrm{NL1}}$ drops
to -250% and -150% for low $N_{n}$, and reaches 50% and 60% for high $N_{n}$
at $\sqrt{s_{\mathrm{NN}}}=17.21$ GeV and 5.02 TeV, respectively. This
relative difference is larger for $N_{p}=5$ and $N_{p}=3$ in comparison to
$N_{p}=2$ and $N_{p}=0$, respectively, for the considered collision energies.
One can conclude that while $\sigma(N_{n},N_{p})$ are small for $N_{n}\leq 7$
and $N_{p}\leq 5$, precise measurements of these cross sections sensitive to
the presence of NS in 208Pb in ultracentral 208Pb–208Pb collisions can help to
validate the predictions of nuclear structure models Centelles2010 .
Alternatively, larger $\sigma(N_{n},N_{p})$ for $N_{n}>10$ and $N_{p}\leq 5$
can be proposed for measurements at both collision energies for the same
purpose.

###  3.3 Calculations with Pbpnrw and PREX profiles

Calculations with AAMCC-MST also demonstrate that precise measurements of
$\sigma(N_{n},N_{p})$ can help to confirm or refute the results of Crystal
Ball at MAMI/A2 collaboration Tarbert2014  and PREX Adhikari2021
collaboration. For a quantitative assessment of the sensitivity of
$\sigma(N_{n},N_{p})$ to the choice of nuclear density profiles, the ratio
$(\sigma_{\mathrm{PREX}}-\sigma_{\mathrm{Pbpnrw}})/\sigma_{\mathrm{PREX}}$ was
calculated with results presented in Fig. 7.

Figure 7:  Same as in Fig. 6, but calculated with PREX and Pbpnrw nuclear
density profiles.

With respect to NS thickness, there is a similarity between AAMCC-MST results
obtained with NFB-8 and Pbpnrw as profiles with thin NS ($\Delta r_{np}<0.15$
fm) on one side, and with thick NS ($\Delta r_{np}>0.28$ fm) with NL1 and PREX
profiles on the other side. This explains the similarity of results presented
in Figs. 6 and 7. However, the magnitude of variations caused by the
replacement of Pbpnrw by PREX and reflected by
$(\sigma_{\mathrm{PREX}}-\sigma_{\mathrm{Pbpnrw}})/\sigma_{\mathrm{PREX}}$ is
found somehow smaller in comparison to
$(\sigma_{\mathrm{NL1}}-\sigma_{\mathrm{NFB-8}})/\sigma_{\mathrm{NL1}}$. It is
typically at the level of -150% and -80% for small $N_{n}$ and up to 30% and
50% for high $N_{n}$ at $\sqrt{s_{\mathrm{NN}}}=17.21$ GeV and 5.02 TeV,
respectively. Such a reduction of variations of $\sigma(N_{n},N_{p})$ is
expected because of a smaller difference in $\Delta r_{np}$ between PREX and
Pbpnrw in comparison to the difference in $\Delta r_{np}$ between NL1 and
NFB-8. Nevertheless, one can conclude that also in the former case precise
measurements of $\sigma(N_{n},N_{p})$ for $N_{n}\leq 7$ and $N_{p}\leq 5$
sensitive to the presence of NS in 208Pb in ultracentral 208Pb–208Pb
collisions can help to validate the results of the experiments Tarbert2014 ;
Adhikari2021 . Larger $\sigma(N_{n},N_{p})$ calculated for $N_{n}>10$ and
$N_{p}\leq 5$ can be also proposed for measurements at both collision energies
for the same purpose.

###  3.4 Calculations with PREX1 and PREX2 profiles

As shown in Sections 3.2 and 3.3, the changes in the thickness $\Delta r_{np}$
of NS in calculations with AAMCC-MST lead to a significant change in
$\sigma(N_{n},N_{p})$ for certain $N_{n}$ and $N_{p}$. In calculations with a
thicker NS, the periphery of 208Pb contains more neutrons. As a result,
$\sigma(N_{n},N_{p})$ in ultracentral collisions becomes larger for higher
numbers $N_{n}$ of spectator neutrons.

Figure 8: Difference between the cross sections $\sigma(N_{n},N_{p})$ of
emission of given numbers of spectator neutrons $N_{n}$ and protons $N_{p}$
calculated with PREX2 and PREX1 nuclear density profiles for 208Pb–208Pb
collisions for 0–5% centrality at $\sqrt{s_{\mathrm{NN}}}=17.21$ GeV (top) and
$\sqrt{s_{\mathrm{NN}}}=5.02$ TeV (bottom).

It is interesting to investigate whether the proposed method to constrain
$\Delta r_{np}$ by measuring $\sigma(N_{n},N_{p})$ in ultracentral nucleus-
nucleus collisions can also help in disentangling other fine details of
nuclear periphery, like the relations between $R_{n}$, $R_{p}$, $a_{n}$ and
$a_{p}$. With this purpose the difference in $\sigma(N_{n},N_{p})$ calculated
with PREX2 ($R_{n}>R_{p}$, $a_{n}>a_{p}$) and PREX1 ($R_{n}=R_{p}$,
$a_{n}>a_{p}$) profiles was calculated and presented in Fig. 8. Both profiles
are characterized by $\Delta r_{np}=0.283$ fm. However, as described above in
Section 2.1, PREX1 profile represents the case of a neutron halo in 208Pb with
a sufficiently large diffuseness parameter $a_{n}$ to compensate the equality
$R_{n}=R_{p}$ and still provide the same $\Delta r_{np}=0.283$ fm as PREX2
profile with its $R_{n}>R_{p}$, see Table 1.

Figure 9: Same as in Fig.6, but calculated with PREX2 and PREX1 nuclear
density profiles.

As seen from Fig. 8, the difference between $\sigma(N_{n},N_{p})$ calculated
with PREX2 and PREX1 profiles is smaller in comparison to the difference in
cross sections calculated with NL1 and NFB-8 profiles presented in Fig. 4.
This means that very precise measurements and calculations are necessary to
understand whether theory and experiment are consistent with $R_{n}=R_{p}$ or
rather with $R_{n}>R_{p}$. Nevertheless, one can conclude that the most
sensitive to the presence or absence of NH cross sections
$\sigma(N_{n},N_{p})$ are predicted basically in the intervals $N_{n}=1-10$
and $N_{p}=1-5$. The evolution of $\sigma(N_{n},N_{p})$ with collision energy,
which is explained in Section 2 by the depletion of spectator matter in
collisions at the LHC, is seen also for the calculations with PREX1 and PREX2
profiles, see Fig. 8. While noticeable differences in $\sigma(N_{n},N_{p})$
are confined within $3<N_{n}<10$ and $N_{p}=2,3,4,5$ at
$\sqrt{s_{\mathrm{NN}}}=17.21$ GeV, at high collision energy
$\sqrt{s_{\mathrm{NN}}}=5.02$ TeV the differences are confined within
$0<N_{n}<4$ and $N_{p}=0,1,2,3$.

Finally, the ratio
$(\sigma_{\mathrm{PREX2}}-\sigma_{\mathrm{PREX1}})/\sigma_{\mathrm{PREX2}}$
presented in Fig. 9 can be considered at two collision energies. As seen, this
ratio is up to 60% at $\sqrt{s_{\mathrm{NN}}}=17.21$ GeV and up to 50% at
$\sqrt{s_{\mathrm{NN}}}=5.02$ TeV. On one hand, large statistical
uncertainties of calculations make it difficult to draw definite conclusions
on the difference in the cross sections calculated with PREX2 and PREX1
profiles for $N_{n}>10$ at the LHC. On the other hand, in comparison to events
with high multiplicity of spectator nucleons, events with $N_{n}\leq 3$ and
$N_{p}\leq 3$ are preferable for detecting by ALICE Zero Degree Calorimeters
because the corrections for ZDC efficiency are more simple for low nucleon
multiplicities Dmitrieva2018 ; Dmitrieva2021

###  3.5 Sensitivity to other parameters in calculations

As shown above, the relative differences between the cross sections
$\sigma(N_{n},N_{p})$ of emission of certain numbers of spectator nucleons in
ultracentral 208Pb–208Pb collisions calculated with different NS parameters
are up to 250% for certain $N_{n}$ and $N_{p}$. This demonstrates the
sensitivity of calculated $\sigma(N_{n},N_{p})$ to the parameters of NS in
208Pb. However, the NS parameters can be reliably constrained by the
comparison of calculated and measured $\sigma(N_{n},N_{p})$ only if the
changes of $\sigma(N_{n},N_{p})$ due to variations of other parameters of the
AAMCC-MST model are essentially smaller.

The systematic uncertainties of the numbers of binary collisions $N_{coll}$
and participants $N_{part}$ calculated with Glauber MC model were estimated
below 5%, see Ref. Loizides2018  for details. The combined uncertainties of
$N_{coll}$ and $N_{part}$ due to the choice of the minimal internucleon
separation, the algorithm of placing nucleon centers in nuclei (recentering)
and $\sigma^{\mathrm{NN}}_{inel}$ were thoroughly evaluated. Therefore, in the
present work only the sensitivity of $\sigma(N_{n},N_{p})$ to the main AAMCC-
MST parameters and assumptions with a direct influence on the emission of
forward spectator nucleons in ultracentral collisions is investigated.

The results presented above in Sec. 3.1–Sec. 3.4 were obtained by simulating
the nucleon-nucleon collisions assuming the hard-sphere shape for nucleons,
Eq. (6). In Glauber MC model nucleon-nucleon collisions can be also simulated
taking into account the fluctuations of the nucleon shape or partonic degrees
of freedom Loizides2016 , Eq. (7). In particular, in Ref. Loizides2018
$w=0.4$ was suggested leading to a Gaussian shape of this collision
probability distribution. Therefore, it is necessary to evaluate the
sensitivity of $\sigma(N_{n},N_{p})$ to the replacement of the hard-sphere
option ($w=0$) by the fluctuating shape option ($w=0.4$) for NN collisions.
For this purpose 208Pb–208Pb collisions of 0–5% centrality at
$\sqrt{s_{\mathrm{NN}}}=5.02$ TeV were modelled with PREX nuclear density
profile employing the both options.

Figure 10: Relative difference between the cross sections
$\sigma(N_{n},N_{p})$ of emission of given numbers of spectator neutrons
$N_{n}$ and protons $N_{p}$ calculated with $w=0$ and $w=0.4$ for 208Pb–208Pb
collisions of 0–5% centrality at $\sqrt{s_{\mathrm{NN}}}=5.02$ TeV. PREX
nuclear density profile was used.

Figure 11: Cross sections $\sigma(N_{n},N_{p})$ of emission of given numbers
of spectator neutrons $N_{n}$ in events with specific number of spectator
protons $N_{p}=$ 0, 1, 2 or 3 in ultracentral 208Pb–208Pb collisions for 0–5%
centrality at $\sqrt{s_{\mathrm{NN}}}=5.02$ TeV calculated with Pbpnrw and
PREX density profiles in 208Pb with nominal hard-sphere $w=0$ and Gaussian
$w=0.4$ options in calculating NN-collision probability.

The relative difference between the cross sections obtained with $w=0$ and
$w=0.4$ is presented in Fig. 10. As seen, the transition from $w=0$ to $w=0.4$
in Eq. (7) leads to a noticeable increase in $\sigma(N_{n},N_{p})$ for
$N_{n}=$ 0, 1 and $N_{p}=$ 0, 1 up to 80%, and it is comparable to the
difference between the respective cross sections calculated with PREX and
Pbpnrw nuclear density profiles presented in Fig. 7. Nevertheless, much
smaller relative changes of $\sim 10$% are seen for $\sigma(N_{n},N_{p})$ for
$N_{p}=$ 2, 3 and intermediate neutron multiplicities $2<N_{n}<15$. Such
changes are smaller than the relative difference ($\pm 50$%) between the cross
sections obtained with PREX and Pbpnrw nuclear density profiles for the same
multiplicities of spectator neutrons and protons, see Fig. 7. Despite of the
variations in modelling nucleon-nucleon collisions caused by replacing $w=0$
with $w=0.4$, the absolute values of $\sigma(N_{n},N_{p})$ calculated for
$N_{p}=$0, 1, 2, 3 and $7<N_{n}<15$ with PREX and Pbpnrw profiles in
ultracentral collisions still remain distinguishable, as shown in Fig. 11. The
conclusion that the details of the NN interaction model play a minor role in
central collisions is in line with findings of Ref. Alvioli2012 .

Table 4: Average numbers of spectator neutrons $\left\langle N_{n}\right\rangle$, protons $\left\langle N_{p}\right\rangle$ and deuterons $\left\langle N_{d}\right\rangle$ in ultracentral ($b=2$ fm) 208Pb–208Pb collisions at $\sqrt{s_{\mathrm{NN}}}=17.21$ GeV calculated with AAMCC-MST with nominal hard-sphere $w=0$ and Gaussian $w=0.4$ options in calculating NN collision probability and without and with accounting for the nucleon-nucleon correlations. No neutron skin was assumed in all calculations, $\Delta r_{np}=0$. NA49 data Appelshauser1998  are given for comparison. | $\left\langle N_{n}\right\rangle$ | $\left\langle N_{p}\right\rangle$ | $\left\langle N_{d}\right\rangle$
---|---|---|---
hard-sphere, uncorrelated | $12.13$ | $7.48$ | $0.82$
hard-sphere, correlated | $11.87$ | $7.41$ | $0.67$
Gaussian, uncorrelated | $12.99$ | $8.03$ | $0.91$
Gaussian, correlated | $12.72$ | $7.91$ | $0.75$
NA49 | $9.0\pm 1.8$ | $7.0\pm 1.4$ | 0.5

In Ref. Alvioli2011  the production of spectator nucleons in collisions of
ultrarelativistic nuclei was modelled taking into account short-range nucleon-
nucleon correlations in nuclei. The authors of Refs. Alvioli2009 ; Alvioli2011
made publicly available the sets of configurations of nucleons in several
nuclei, including 208Pb, generated with accounting for the nucleon-nucleon
correlations. In dedicated AAMCC-MST runs these configurations were introduced
as initial positions of nucleons in colliding nuclei to model 208Pb–208Pb
collisions for 0–5% centrality at $\sqrt{s_{\mathrm{NN}}}=5.02$ TeV. Then, the
results of calculations with accounting for the NN correlations can be
compared with results obtained with uncorrelated nucleons. In the available
configurations of nucleons Alvioli2009 ; Alvioli2011  the effects of NS were
apparently neglected. For consistency, in respective AAMCC-MST calculations
disregarding the NN correlation the values of the half-density radii and
diffuseness parameters were set equal for neutrons and protons:
$R_{n}=R_{p}=6.624$ fm and $a_{n}=a_{p}=0.549$ fm. This resulted in $\Delta
r_{np}=0$ in calculations with and without NN correlations. In the latter case
also neglecting NS, nucleons were placed according to the standard procedure
adopted in the Glauber MC model Loizides2018 .

Figure 12: Relative difference between the cross sections
$\sigma(N_{n},N_{p})$ of emission of given numbers of spectator neutrons
$N_{n}$ and protons $N_{p}$ calculated without and with accounting for
nucleon-nucleon correlations for 208Pb–208Pb collisions of 0–5% centrality at
$\sqrt{s_{\mathrm{NN}}}=5.02$ TeV. No neutron skin was assumed in all
calculations, $\Delta r_{np}=0$.

With the results presented in Table 4 and Fig. 12 the impact of accounting for
the NN correlations in nuclei on the yields of spectator neutrons, protons and
deuterons can be estimated for ultracentral 208Pb–208Pb collisions at
$\sqrt{s_{\mathrm{NN}}}=17.2$ GeV and also at $5.02$ TeV. As seen from Table
4, the average numbers of spectator neutrons $\left\langle N_{n}\right\rangle$
and protons $\left\langle N_{p}\right\rangle$ are changed by less than 1–2%
when these numbers are calculated with accounting for the NN correlations.
Such small variations are also estimated in calculations with accounting for
the fluctuating shape of nucleons, with $w=0.4$ in Eq. (7). The changes in the
smaller average numbers of deuterons $\left\langle N_{d}\right\rangle$ are
found larger, up to 18%. In general, with the exception of the production of
deuterons, the agreement with NA49 data Appelshauser1998  is not improved by
introducing the NN correlations.

The relative difference between $\sigma(N_{n},N_{p})$ calculated without and
with the nucleon-nucleon correlations for 208Pb–208Pb collisions of 0–5%
centrality at $\sqrt{s_{\mathrm{NN}}}=5.02$ TeV is presented in Fig. 12.

As seen from Fig. 12, the changes in $\sigma(N_{n},N_{p})$ calculated with
accounting for nucleon-nucleon correlation are within $\pm 10$% for all
considered $N_{p}$ and for $N_{n}<10$. While in these calculations the
presence of NS in 208Pb was neglected, a similar sensitivity of
$\sigma(N_{n},N_{p})$ to the NN correlations ($\sim 10$%) can be expected in
calculations with more realistic nuclear density profiles with NS, Table 1.

In AAMCC-MST, the yields of spectator neutrons and protons depend also on the
critical distance $d$ used in the MST-clustering algorithm presented in Sec.
2.3. The parameter $d_{0}=2.7$ fm is used in Eq. (14) to calculate $d$, and it
is typically larger than the average internucleon distance ($\sim 2$ fm) in
nuclei in the ground state. While the NA49 data Appelshauser1998  are
described with $d_{0}=2.7$ fm in general, see Table 2, a smaller value
$d_{0}=2.4$ fm can be also considered to evaluate the sensitivity of
$\sigma(N_{n},N_{p})$ to $d_{0}$.

In Table 5 the average numbers of spectator neutrons $\left\langle
N_{n}\right\rangle$ and protons $\left\langle N_{p}\right\rangle$ calculated
for 208Pb–208Pb collisions of 0–5% centrality at $\sqrt{s_{\mathrm{NN}}}=5.02$
TeV are presented. In this table the results are divided into two groups
representing, respectively, calculations with Pbpnrw and PREX nuclear density
profiles. Some variations of $\left\langle N_{n}\right\rangle$ up to 5% are
seen for each nuclear density profile depending on $w$ and $d_{0}$ used in
calculations. However, the difference in $\left\langle N_{n}\right\rangle$,
and partially in $\left\langle N_{p}\right\rangle$, obtained with Pbpnrw and
PREX is preserved. This means that in ultracentral collisions $\left\langle
N_{n}\right\rangle$ and $\left\langle N_{p}\right\rangle$ are defined mainly
by the specific nuclear density profile used in modelling with AAMCC-MST
rather than by other calculational parameters.

Table 5: Average numbers of spectator neutrons $\left\langle N_{n}\right\rangle$ and protons $\left\langle N_{p}\right\rangle$ in 208Pb–208Pb collisions of 0–5% centrality at $\sqrt{s_{\mathrm{NN}}}=5.02$ TeV calculated with AAMCC-MST for Pbpnrw and PREX nuclear density profiles with various parameters in modelling NN collisions (hard-sphere, $w=0$ and Gaussian, $w=0.4$) and in estimating critical distance in MST-clusering ($d_{0}=2.7$ fm and $d_{0}=2.4$ fm). | $\left\langle N_{n}\right\rangle$ | $\left\langle N_{p}\right\rangle$
---|---|---
Pbpnrw, hard-sphere, $d_{0}=2.7$ fm | 8.96 | 4.59
Pbpnrw, Gaussian, $d_{0}=2.7$ fm | 9.34 | 4.8
Pbpnrw, hard-sphere, $d_{0}=2.4$ fm | 9.39 | 4.98
PREX, hard-sphere, $d_{0}=2.7$ fm | 9.7 | 4.34
PREX, Gaussian, $d_{0}=2.7$ fm | 10.19 | 4.61
PREX, hard-sphere, $d_{0}=2.4$ fm | 10.11 | 4.71

Figure 13: Relative difference between the cross sections
$\sigma(N_{n},N_{p})$ of emission of given numbers of spectator neutrons
$N_{n}$ and protons $N_{p}$ calculated with $d_{0}=2.7$ fm and $d_{0}=2.4$ fm
for 208Pb–208Pb collisions of 0–5% centrality at $\sqrt{s_{\mathrm{NN}}}=5.02$
TeV. PREX density profile was used.

In Fig. 13 the relative difference between $\sigma(N_{n},N_{p})$ calculated
for 208Pb–208Pb collisions of 0–5% centrality at $\sqrt{s_{\mathrm{NN}}}=5.02$
TeV with the nominal $d_{0}=2.7$ fm and reduced $d_{0}=2.4$ fm is presented.
PREX density profile was used in this modelling. As seen from Fig. 13, the
variation of $\sigma(N_{n},N_{p})$ for $N_{n}<10$ and $N_{p}<4$ is less than
10%. This means that the cross sections calculated for the specified
multiplicities of spectator nucleons are relatively stable with respect to
variations of $d_{0}$.

Figure 14: Cross sections $\sigma(N_{n},N_{p})$ of emission of given numbers
of spectator neutrons $N_{n}$ in events with specific number of spectator
protons, $N_{p}=$ 0, 1, 2 or 3, in 208Pb–208Pb collisions of 0–5% centrality
at $\sqrt{s_{\mathrm{NN}}}=5.02$ TeV calculated with Pbpnrw and PREX density
profiles in 208Pb with the nominal $d_{0}=2.7$ fm and reduced $d_{0}=2.4$ fm.

In order to prove that the uncertainties of AAMCC-MST results due to the
choice of $d_{0}$ still make possible to distinguish different NS
parameterizations, calculations with the Pbpnrw profile were also performed
with $d_{0}=2.7$ fm and $d_{0}=2.4$ fm. The results presented in Fig. 14
demonstrate that the values of $\sigma(N_{n},N_{p})$ remain well separated for
$N_{p}=0$ and 1 for all $N_{n}$ despite of variations of $d_{0}$. The cross
sections with $N_{p}=2$ and 3 remain distinguishable for $N_{n}<5$. Therefore,
the cross sections measured for the above-mentioned $N_{n}$ and $N_{p}$ can be
safely used to constrain the parameters of NS by comparison with calculations.

##  4 Conclusions

As demonstrated by calculations with the AAMCC-MST model, the yields of
spectator neutrons and protons in ultracentral (0–5% centrality) 208Pb–208Pb
collisions at $\sqrt{s_{\mathrm{NN}}}=17.21$ GeV and $5.02$ TeV are sensitive
to the parameters of NS in 208Pb. Spectator matter in ultracentral 208Pb–208Pb
collisions is represented mostly by nucleons which are peeled away from the
nuclear periphery representing NS. Depending on the thickness $\Delta r_{np}$
of NS, and on the relations between the half-density radii of the
distributions of neutrons and protons ($R_{n}=R_{p}$ vs $R_{n}>R_{p}$), very
modest changes of the average numbers of spectator neutrons were calculated.

In contrast, the cross sections $\sigma(N_{n},N_{p})$ of emission of given
numbers of spectator neutrons $N_{n}$ accompanied by specific number of
spectator protons $N_{p}$ calculated with different parameterizations of NS
diverge up to 250% for certain numbers of $N_{n}$ and $N_{p}$. In particular,
$\sigma(N_{n},N_{p})$ calculated for $N_{n}=1,2...7$ and $N_{p}=0,1,2,3$ in
ultracentral 208Pb–208Pb collisions at the LHC were found to be very sensitive
to $\Delta r_{np}$ in 208Pb. Accurate measurements of $\sigma(N_{n},N_{p})$
may also help to understand whether theory and experiment are consistent with
$R_{n}=R_{p}$ or rather with $R_{n}>R_{p}$. The cross sections
$\sigma(N_{n},N_{p})$ of emission of low numbers of spectator neutrons and
protons ($N_{n}\leq 7$ and $N_{p}\leq 3$) can be proposed for measurements in
the ALICE experiment at the LHC providing that the yields of spectator
nucleons will be thoroughly corrected for the acceptance and efficiency of the
ALICE Zero Degree Calorimeters Puddu2007 ; Dmitrieva2018 . As reported by
ALICE, the multiplicities of forward spectator neutrons and protons can be
measured individually in events of different centralities
CollaborationALICE2020 , while the collisions of 0–5% centrality can be
identified by different well-established methods Abelev2013 . The cross
sections of emission of 1–3 forward neutrons resulting from electromagnetic
dissociation of 208Pb at the LHC were already measured Abelev2012n .

In order to summarize the studies of sensitivities of $\sigma(N_{n},N_{p})$ to
the parameters of calculations considered in this work, one can conclude that
$\sigma(N_{n},N_{p})$ for $N_{p}=$0, 1 are sufficiently stable ($\pm 10$%) for
intermediate $2<N_{n}<10$. The cross sections $\sigma(N_{n},N_{p})$ for
$N_{p}=$2, 3 can be also proposed to probe NS effects, but for $N_{n}<5$.
Their variations are also sufficiently small ($\sim 10$%) in comparison to the
difference in $\sigma(N_{n},N_{p})$ ($\sim$50–250%) calculated with different
NS parameters.

In summary, one can expect that $\sigma(N_{n},N_{p})$ measured in ultracentral
208Pb–208Pb collisions will help to constrain the existing variety of the
parameters of NS predicted by various nuclear structure models and estimated
from experimental data for 208Pb.

##  5 Acknowledgments

One of the authors (I.P.) is grateful to Dariusz Miskowiec and Chiara
Oppedisano for the discussions which stimulated the investigation of the
effects of neutron skin in ultrarelaticistic 208Pb–208Pb collisions.

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