# Differential charge radii: self-consistency and proton-neutron interaction
effects

U. C. Perera  Department of Physics and Astronomy, Mississippi State
University, MS 39762  A. V. Afanasjev  Department of Physics and Astronomy,
Mississippi State University, MS 39762

###### Abstract

The analysis of self-consistency and proton-neutron interaction effects in the
buildup of differential charge radii has been carried out in covariant density
functional theoretical calculations without pairing interaction. Two
configurations of the 218Pb nucleus, generated by the occupation of the
neutron $1i_{11/2}$ and $2g_{9/2}$ subshells, are compared with the ground
state configuration in 208Pb. The interaction of added neutron(s) and the
protons forming the $Z=82$ proton core is responsible for a major contribution
to the buildup of differential charge radii. It depends on the overlaps of
proton and neutron wave functions and leads to a redistribution of single-
particle density of occupied proton states which in turn modifies the charge
radii. Self-consistency effects affecting the shape of proton potential, total
proton densities and the energies of the single-particle proton states provide
only secondary contribution to differential charge radii. The buildup of
differential charge radii is a combination of single-particle and collective
phenomena. The former is due to proton-neutron interaction, the impact of
which is state dependent, and the latter reflects the fact that all occupied
proton single-particle states contribute to this process. The neglect of
either one of these aspects of the process by ignoring proton-neutron
interaction and self-consistency effects as it is done in
macroscopic+microscopic approach or by introducing the core as in spherical
shell model introduces uncontrollable errors and restricts the applicability
of such approaches to the description of differential charge radii. The
analysis also indicates that both the Coriolis interaction in odd and odd-odd
deformed nuclei and the residual interaction between unpaired proton and
neutron in odd-odd nuclei could affect the odd-even staggering in charge radii
if their impact on the wave function of the ground state of these nuclei is
appreciable.

##  I Introduction

Charge radii are among the most fundamental properties of atomic nuclei and
during the last decade there was a significant increase in experimental and
theoretical studies of this physical observable. The experimental results were
reviewed in Refs. [1, 2] and recent experimental investigations were
summarized in the introduction of Ref. [3]. The introduction to the latter
publication provides also the overview of theoretical efforts. Theoretical
calculations within different density functional theories (DFTs) provide a
quite accurate global description of experimental charge radii presented in
the compilation of Ref. [1]: the rms deviations of calculated charge radii
$r_{ch}$ from experimental ones are at the level of $\approx 0.03$ fm [4]
which corresponds to high average precision of 0.625% in the prediction of
charge radii (see Ref. [3]).

The changes of the charge radii within the isotopic chain are measured with
high precision using laser spectroscopy (see Refs. [1, 2]). Thus, the
differential mean-square (ms) charge radii (see Eq. (3) below for definition),
measured with high precision within the isotopic chains, become an important
quantity. They have been studied within ab initio approaches (see, for
example, Refs. [5, 6]), non-relativistic DFTs based on zero range Skyrme
forces (see Refs. [7, 8, 9, 10]), Fayans functional (see Refs. [11, 12, 13])
and finite range Gogny forces (see Refs. [14, 5]), non-relativistic Hartree-
Fock-Bogoluibov (HFB) approach with a finite-range Yukawa interaction [15, 16,
17] and covariant density functional theory (CDFT) (see Refs. [18, 19, 20, 3,
10]).

Different aspects of the buildup of differential charge radii within the
isotopic chain with increasing neutron number have been investigated in these
papers. In particular, it was found that the evolution of the charge radii
within the isotopic chain with increasing neutron number is defined by the
pull on the proton states generated by neutrons gradually added to the nuclear
system [8, 21, 22, 3]. The most investigated case here is the kink in charge
radii at $N=126$ and the evolution of charge radii above $N=126$ in the Pb
isotopic chain. The pattern of these effects critically depends on the
occupation of the $2g_{9/2}$ and $1i_{11/2}$ orbitals, on their relative
energies, and on how close they are in energy [8, 21, 3]: the kink is
generated only when neutron $1i_{11/2}$ orbitals are substantially occupied in
the nuclei with $N>126$.

Ref. [21] provided a hint on microscopic origin of this difference by
revealing that it is traced back to the nodal structure of these two orbitals
($n=1$ for $1i_{11/2}$ and $n=2$ for $2g_{9/2}$, where $n$ stands for
principal quantum number) and the overlap of their wavefunctions with those of
the proton states. However, as follows from the present study the
interpretation of the pull of these neutron states on proton orbitals via the
symmetry energy does not corresponds to real physical situation since it is
related to proton-neutron interaction. The detailed global analysis of the
impact of the occupation of neutron single-particle orbitals in the vicinity
of spherical neutron shell closures generalized the results of Ref. [21] to
whole nuclear chart (see discussion of Fig. 32 in Ref. [3]). It revealed
strong correlations between the principal quantum number $n$ of the single
neutron orbital occupied above the neutron shell closure and the impact of the
occupation of this orbital on differential charge radii: in a given isotopic
chain the largest impact on differential charge radii is provided by the
occupation of the neutron orbital with the lowest $n$. As a consequence, a
significant occupation of the $n=1$ neutron subshell above the neutron shell
closure is required for a creation of the kink in differential charge radii at
this closure (see Ref. [3]).

Despite all these studies, there are the aspects of the process of the buildup
of differential charge radii which are not completely understood on
microscopic level and which have not been discussed in the literature. Thus,
the goal of the present paper is to fill these gaps in our knowledge and to
perform detailed studies of the impact of self-consistency effects and the
interaction between neutron(s) added to a reference nucleus and the protons
forming the proton subsystem on the buildup of differential charge radii. Of
particular interest is the balance of these two types of contributions and the
microscopic mechanisms affecting the changes of charge radii in the isotopic
chain with increasing neutron number. The results of these studies will also
allow to answer the question of the applicability of different theoretical
frameworks to the description of differential charge radii and potential
contributing factors to odd-even staggering in charge radii.

The paper is organized as follows. Sec. II provides a brief outline of
theoretical formalism and the discussion of physical observables under study.
The impact of self-consistency and proton-neutron interaction effects on
differential charge radii is discussed in Sec. III. Sec. IV is dedicated to
general observations following from this study. Finally, Sec. V summarizes the
results of our paper.

##  II Theoretical formalism and physical observables

Theoretical calculations have been performed within the framework of covariant
density functional theory (CDFT) [23] employing the modified version of the
computer code restricted to spherical symmetry used in Ref. [24]. Since the
details of the CDFT framework are widely available (see, for example, Ref.
[23]), we focus on the physical quantities of the interest. The pairing
correlations are neglected in the calculations in order to better understand
the underlying physical mechanisms. The calculations are performed with the
NL3* covariant energy density functional (CEDF) [25]. Its global performance
in the description of the masses and charge radii is well documented (see
Refs. [4, 3]). It was also recently used in the study of bubble nuclei (see
Ref. [26]) the results of which have substantial overlap with some aspects of
the present study; this is one of main reasons for the selection of this
functional. Note that it is was verified that main conclusions obtained in the
present paper do not depend on the selection of the functional.

In order to better apprehend the role of self-consistency effects on
differential charge radii we consider ground state configuration in 208Pb and
two configurations of the 218Pb nucleus labeled below as ”Conf-$1i_{11/2}$”
and ”Conf-$2g_{9/2}$”. In these configurations of 218Pb, ten neutrons outside
the 208Pb core are located in the $1i_{11/2}$ and $2g_{9/2}$ spherical
subshells, respectively. We selected 218Pb in order to maximize the effect of
the addition of neutrons in a given spherical subshell on proton charge
radii111One can definitely consider the $1i_{11/2}$ and $2g_{9/2}$
configurations in odd-$A$ 209Pb nucleus and this will completely justify the
neglect of pairing which collapses because of the blocking of odd neutron.
However, this will not change the results and conclusions of the paper.. Note
that the maximum number of neutrons which can be put into the $2g_{9/2}$ and
$1i_{11/2}$ spherical subshells is 10 and 12, respectively. Thus, the
selection of 218Pb nucleus corresponds to full filling of the $2g_{9/2}$
neutron subshell and almost (two neutrons short) full filling of the
$1i_{11/2}$ neutron subshell.

The charge radii are defined as

$\displaystyle r_{ch}=\sqrt{\left<r^{2}\right>_{p}+0.64}\,\,\,\,{\rm fm}$ (1)

where the mean square proton point radius is given by

$\displaystyle\left<r^{2}\right>_{p}=\frac{\int
r^{2}\rho_{tot}^{p}(\vec{r}\,)d^{3}r}{\int\rho_{tot}^{p}(\vec{r}\,)d^{3}r}$
(2)

and the factor 0.64 accounts for the finite-size effects of the proton222
Small contributions to the charge radii originating from the electric neutron
form factor and electromagnetic spin-orbit coupling [27, 28] are neglected in
the NL3* functional (as well as in the fitting protocols of all existing
CEDFs). More precise expressions for charge radii in CDFT are available but
their use would require the refit of the CEDFs (see Refs. [29, 30] and
discussion in Sec. VIII of Ref. [3]). However, this neglect is not critical
since spin-orbit contribution to charge radii decreases with increasing the
mass of nuclei [29, 31] and its contribution to differential charge radii of
the Pb isotopes is expected to be negligible [31].. Then differential mean-
square charge radius is given by333This quantity is frequently written as a
function of mass number $A$. However, we prefer to define it as a function of
neutron number $N$ since this allows to see the behavior of the
$\delta\left<r^{2}\right>_{p}^{N,N^{\prime}}$ curves at neutron shell
closures.

$\displaystyle\delta\left<r^{2}\right>_{p}^{N,N^{\prime}}$ $\displaystyle=$
$\displaystyle\left<r^{2}\right>_{p}(N)-\left<r^{2}\right>_{p}(N^{\prime})=$
(3) $\displaystyle=r^{2}_{ch}(N)-r^{2}_{ch}(N^{\prime}).$

Note that $N^{\prime}$ is the neutron number of the reference nucleus (208Pb
in this paper).

The total nucleonic density $\rho_{tot}(r)$ in a given subsystem (proton or
neutron) is built from the contributions of individual particles as follows:

$\displaystyle\rho_{tot}(r)=\sum_{i}m_{i}\rho^{sp}_{i}(r),$ (4)

where $m_{i}$ is the multiplicity of the occupation of the $i-$th subshell
[$m_{i}=(2j_{i}+1)$ for a fully occupied subshell with angular momentum
$j_{i}$] and $\rho^{sp}_{i}(r)$ is the density of the single-particle state
belonging to the $i-$th subshell with the normalization

$\displaystyle\int\rho^{sp}_{i}(\vec{r}\,)d^{3}r=4\pi\int
r^{2}\rho^{sp}_{i}(r)dr=1.0.$ (5)

Taking into account that $\int\rho_{tot}^{p}(\vec{r}\,)d^{3}r=Z$ and that all
proton subshells below the $Z=82$ shell gap are fully occupied in the proton
subsystem of the Pb isotopes, Eq. (2) can be rewritten as

$\displaystyle\left<r^{2}\right>_{p}=\frac{1}{Z}\sum_{i}(2j_{i}+1)\left<r^{2}\right>_{i}^{p}$
(6)

where

$\displaystyle\left<r^{2}\right>_{i}^{p}=\int
r^{2}\rho_{i}^{p}(\vec{r}\,)d^{3}r$ (7)

is the proton mean square radius of the single-particle state belonging to the
$i$-th subshell. As a consequence, the differential charge radius of two
isotopes can be redefined as

$\displaystyle\delta\left<r^{2}\right>_{p}^{N,N^{\prime}}=\frac{1}{Z}\sum_{i}(2j_{i}+1)[\left<r^{2}\right>_{i}^{p}(N)-\left<r^{2}\right>_{i}^{p}(N^{\prime})]$

and its magnitude could be traced back to the modifications in proton mean
square radius of the single-particle states generated by the transition from
the nucleus with neutron number $N^{\prime}$ to the nucleus with $N$. The
quantity

$\displaystyle\Delta\left<r^{2}\right>_{i}^{N,N^{\prime}}=\left<r^{2}\right>_{i}^{p}(N)-\left<r^{2}\right>_{i}^{p}(N^{\prime})$
(9)

is denoted here as differential single-particle proton radius of the single-
particle state belonging to the $i$-th proton subshell.

Note that for simplicity of the discussion of the role of the single-particle
states we consider also single-particle rms radii $r^{p}_{i}$ of the proton
states defined as

$\displaystyle r^{p}_{i}=\sqrt{\left<r^{2}\right>_{i}^{p}}.$ (10)

The proton ($i=\pi$) and neutron ($i=\nu$) nucleonic potentials are defined in
the CDFT as follows:

$\displaystyle V_{\pi}$ $\displaystyle=$ $\displaystyle V+S+V_{Coul},$ (11)
$\displaystyle V_{\nu}$ $\displaystyle=$ $\displaystyle V+S,$ (12)

where scalar potential is given by

$\displaystyle S(r)=g_{\sigma}\sigma(r),$ (13)

meson defined part of the vector potential is written as

$\displaystyle V(r)=g_{\omega}\omega_{0}(r)+g_{\rho}\tau_{3}\rho_{0}(r),$ (14)

and

$\displaystyle V_{Coul}(r)=eA_{0}(r)$ (15)

is the Coulomb potential.

##  III The role of self-consistency and proton-neutron interaction effects

###  III.1 Nucleonic potentials and densities

Figure 1: Proton and neutron nucleonic potentials in the configurations under
study. Dotted horizontal line shows zero energy threshold. Note that
$V_{Coul}=0$ in the panel (b).

To better understand the impact of the occupation of different neutron
spherical subshells on charge radii of the $Z=82$ core we first consider how
nucleonic potentials and densities change on transition from the 208Pb nucleus
to the ”Conf-$1i_{11/2}$” and ”Conf-$2g_{9/2}$” configurations in the 218Pb
nucleus.

Proton and neutron nucleonic potentials of the configurations under study are
shown in Fig. 1. The addition of ten neutrons to the 208Pb core only slightly
increases the radius of neutron potential and the shapes of neutron potentials
of the ”Conf-$1i_{11/2}$” and ”Conf-$2g_{9/2}$” configurations in 218Pb are
almost identical in the energy range between -60 MeV and 0 MeV [see Fig.
1(b)]. The largest changes are seen at the bottom of the neutron potentials
which affects mostly deep lying neutron states. However, these potentials
remain nearly flat bottom for all configurations of interest.

It is interesting that the addition of ten neutrons to 208Pb triggers larger
changes in the radial profile of proton potential than that for neutron one
(compare panels (a) and (b) in Fig. 1). The proton potentials of the
”Conf-$1i_{11/2}$” and ”Conf-$2g_{9/2}$” configurations in 218Pb are almost
identical in the energy range between -20 MeV and 0.0 MeV [see Fig. 1(a)]. At
lower energies, the radius of the proton potential of the ”Conf-$1i_{11/2}$”
configuration is smaller than that of the ”Conf-$2g_{9/2}$” one and this
difference increases with decreasing energy.

Figure 2: Proton and neutron densities as a function of radial coordinate $r$
for indicated configurations.  Figure 3: The energies of proton single-
particle states in the proton potentials of indicated nucleonic
configurations. Note that only the states below the magic $Z=82$ shell closure
are shown. The figure is split into two panels in order to make it more
understandable and to avoid overlapping single-particle states.

Fig. 2 shows total proton and neutron densities of the configurations under
study. Neutron and proton densities of 208Pb serve as the references with
respect of which the effect of the occupation of the neutron $1i_{11/2}$ and
$2g_{9/2}$ subshells on total neutron and proton densities is discussed. The
occupation of the neutron $1i_{11/2}$ subshell in 218Pb (”Conf-1$i_{11/2}$”)
builds the neutron density mostly in the near-surface and surface regions [see
Fig. 2(b)] since the single peak of its single-particle density is located at
$r\approx 6.3$ fm (see Fig. 6(z) in Ref. [26]). In contrast, the occupation of
the neutron $2g_{9/2}$ subshell in 218Pb (”Conf-$2g_{9/2}$”) builds the
neutron density both in the subsurface and surface regions [see Fig. 2(d)].
This is due to the nodal structure of the wave function of the $2g_{9/2}$
subshell which has two peaks in the single-particle density distribution: the
large peak at $r\approx 4.4$ fm and the smaller one at $r\approx 7.8$ fm (see
Fig. 6(x) in Ref. [26]).

Because of the isovector force, which tries to keep the neutron and proton
density profiles alike, the neutron density changes discussed above feed back
into proton densities [see Figs. 2(a) and (c)]. As compared with proton
densities in 208Pb, the occupation of either neutron $1i_{11/2}$ or $2g_{9/2}$
subshells leads to the reduction of the proton densities in the central and
subsurface regions and increases the density in the surface region. These
changes are more pronounced in the case of the occupation of the neutron
$1i_{11/2}$ subshell (the ”Conf-1$i_{11/2}$” configuration in 218Pb).

Table 1: The contribution of different spherical subshells (column 7) into the build-up of differential charge radii of the ”Conf-1$i_{11/2}$” configuration in 218Pb. The ground state configuration in 208Pb is used here as a reference. Proton subshells occupied below the $Z=82$ shell closure are shown in column 1. Their single-particle energies $e_{i}$ [in MeV] are shown in columns 2 and 3 for two configurations under study. Their proton single-particle rms radii $r_{i}^{p}$ are displayed in columns 4 and 5. The column 6 shows the change of proton single-particle radii $\delta r^{p}_{i}=r^{p}_{i}(^{218}$Pb[Conf-$1i_{11/2}])-r^{p}_{i}(^{208}$Pb). Column 7 shows differential single-particle proton radii $\Delta\left<r^{2}\right>_{i}^{N,N^{\prime}}$ of the single-particle state belonging to the $i$-th proton subshell. The overlaps of proton and neutron wave functions $<\Psi_{n}^{k}|\Psi_{p}^{i}>$ are shown in column 8. The results for spherical subshells with principal quantum number $n=1$ are shown in bold. The total quantities given in the last line are calculated using Eq. (II) (column 7) and equations similar to it (columns 4, 5 and 6).  subshell |  $e_{i}$ (208Pb) |  $e_{i}$ (218Pb) |  $r^{p}_{i}$ (208Pb) |  $r^{p}_{i}$ (218Pb) | $\delta r^{p}_{i}$ | $\Delta\left<r^{2}\right>_{i}^{N,N^{\prime}}$ | $<\Psi_{n}^{k}|\Psi_{p}^{i}>$
---|---|---|---|---|---|---|---
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8
$1s_{1/2}$ | -48.905 | -48.271 | 4.064254 | 4.235252 | 0.170998 | 1.419199 | 0.570778
$1p_{3/2}$ | -43.211 | -43.156 | 4.663197 | 4.846186 | 0.182989 | 1.740112 | 0.729091
$1p_{1/2}$ | -42.529 | -42.598 | 4.582763 | 4.771297 | 0.188534 | 1.763556 | 0.694871
$1d_{5/2}$ | -36.118 | -36.727 | 5.105879 | 5.283008 | 0.177129 | 1.840178 | 0.795503
$1d_{3/2}$ | -34.559 | -35.462 | 4.981504 | 5.159900 | 0.178396 | 1.809189 | 0.795142
$2s_{1/2}$ | -30.886 | -32.032 | 4.450498 | 4.528233 | 0.077735 | 0.697962 | -0.638347
$1f_{7/2}$ | -28.068 | -29.330 | 5.479627 | 5.643146 | 0.163519 | 1.818783 | 0.899794
$1f_{5/2}$ | -25.298 | -27.044 | 5.334254 | 5.490660 | 0.156406 | 1.693085 | 0.837178
$2p_{3/2}$ | -20.924 | -22.559 | 4.985584 | 5.017296 | 0.031712 | 0.317213 | -0.571543
$2p_{1/2}$ | -19.865 | -21.566 | 5.004156 | 5.031929 | 0.027773 | 0.278730 | -0.583835
$1g_{9/2}$ | -19.396 | -21.242 | 5.816346 | 5.962968 | 0.146622 | 1.727100 | 0.910026
$1g_{7/2}$ | -15.205 | -17.693 | 5.682103 | 5.807857 | 0.125754 | 1.444908 | 0.932110
$2d_{5/2}$ | -11.163 | -13.089 | 5.522224 | 5.527597 | 0.005373 | 0.059372 | -0.521033
$1h_{11/2}$ | -10.360 | -12.684 | 6.129798 | 6.257349 | 0.127551 | 1.579994 | 0.975584
$2d_{3/2}$ | -9.513 | -11.537 | 5.580005 | 5.584444 | 0.004439 | 0.049559 | -0.554369
$3s_{1/2}$ | -8.405 | -10.360 | 5.489444 | 5.476911 | -0.012533 | -0.137438 | 0.448467
Total |  |  | 5.450221 | 5.569299 | 0.119078 | 1.312261 |

Note that there are intricate and sometimes counterintuitive interplays
between the changes in the densities and nucleonic potentials the origin of
which was discussed in detail in Sec. V of Ref. [26]. For example, the density
changes on going from the ground state configuration in 208Pb to the
configurations in 218Pb show larger increase in the radial profile of neutron
densities as compared with proton ones (see Fig. 2). In contrast, the
respective changes in the radial profiles of proton potentials of the
configurations in 218Pb with respect of that in 208Pb are larger than those
for neutron potentials (see Fig. 1).

###  III.2 Proton single-particle states

Another consequence of the increase of neutron number on going from 208Pb to
218Pb is the lowering of the energies of the single-particle states in proton
potential (see Fig. 3). The energies of spherical subshells located at
energies higher than $-40$ MeV in the ”Conf-1$i_{11/2}$” and ”Conf-$2g_{9/2}$”
configurations of 218Pb are lower than those in the ground state configuration
of 208Pb. Such lowering implies some reduction of the rms proton radius as
compared with the one at the energy of the subshell corresponding to the
ground state configuration of 208Pb. However, as follows from further
discussion the effect is rather marginal.

Note that there is either small or no energy splitting between the energies of
a given proton subshell calculated in these two configurations of 218Pb if the
principal quantum number $n$ of the subshell is either $n=2$ or $n=3$ (see
Fig. 3). In contrast such energy splitting is typically large for the
subshells with $n=1$. These features are not very important for charge radii
in the calculations without pairing. However, they are more important in the
calculations with pairing since the shifts of the energies of spherical
subshells can affect their occupation probabilities in the vicinity of the
Fermi level (see discussion in Refs. [22, 3]).

Deeply lying $1s_{1/2}$, $1p_{3/2}$ and $1p_{1/2}$ spherical subshells are
affected by the properties and modifications (as compared with 208Pb one) of
the bottom of the proton potential which shows the development of wine bottle
potential features (see Ref. [26]). These features are most/least pronounced
in the ”Conf-$2g_{9/2}$”/”Conf-1$i_{11/2}$” configurations of 218Pb (see Fig.
3). As a consequence, for a given spherical subshell the calculated energies
are almost the same in the ground state configuration of 208Pb and in the
”Conf-1$i_{11/2}$” configurations of 218Pb but significantly lower for the
”Conf-$2g_{9/2}$” configuration of 218Pb. Note that the presence of
classically forbidden region in proton potential at small radial coordinate
$r$ can significantly modify the density distribution of the $1s_{1/2}$
subshell in the ”Conf-$2g_{9/2}$” configuration of 218Pb and can have some
impact on density distribution of this subshell in other two configurations
under study (see detailed discussion in Sec. IV of Ref. [26]).

Table 2: The same as in Table 1 but for the ”Conf-$2g_{9/2}$” configuration in 218Pb. The results for spherical subshells with principal quantum number $n=2$ are shown in bold.  subshell |  $e_{i}$ (208Pb) |  $e_{i}$ (218Pb) |  $r^{p}_{i}$ (208Pb) |  $r^{p}_{i}$ (218Pb) | $\delta r^{p}_{i}$ | $\Delta\left<r^{2}\right>_{i}^{N,N^{\prime}}$ | $<\Psi_{n}^{k}|\Psi_{p}^{i}>$
---|---|---|---|---|---|---|---
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8
$1s_{1/2}$ | -48.905 | -50.239 | 4.064254 | 4.062288 | -0.001966 | -0.015978 | 0.399380
$1p_{3/2}$ | -43.211 | -44.539 | 4.663197 | 4.631773 | -0.031424 | -0.292086 | 0.345824
$1p_{1/2}$ | -42.529 | -43.915 | 4.582763 | 4.535377 | -0.047386 | -0.432074 | 0.373550
$1d_{5/2}$ | -36.118 | -37.305 | 5.105879 | 5.092180 | -0.013699 | -0.139704 | 0.273309
$1d_{3/2}$ | -34.559 | -35.720 | 4.981504 | 4.962466 | -0.019038 | -0.189316 | 0.282916
$2s_{1/2}$ | -30.886 | -31.484 | 4.450498 | 4.565793 | 0.115295 | 1.039534 | 0.182596
$1f_{7/2}$ | -28.068 | -29.174 | 5.479627 | 5.496088 | 0.016461 | 0.180670 | 0.119881
$1f_{5/2}$ | -25.298 | -26.329 | 5.334254 | 5.363151 | 0.028897 | 0.309126 | 0.186386
$2p_{3/2}$ | -20.924 | -22.315 | 4.985584 | 5.158503 | 0.172919 | 1.754108 | 0.475303
$2p_{1/2}$ | -19.865 | -21.544 | 5.004156 | 5.170061 | 0.165905 | 1.687954 | 0.447216
$1g_{9/2}$ | -19.396 | -20.544 | 5.816346 | 5.862341 | 0.045995 | 0.537159 | -0.000955
$1g_{7/2}$ | -15.205 | -16.330 | 5.682103 | 5.757601 | 0.075498 | 0.863670 | -0.000587
$2d_{5/2}$ | -11.163 | -13.214 | 5.522224 | 5.663334 | 0.14111 | 1.578392 | 0.754258
$1h_{11/2}$ | -10.360 | -11.659 | 6.129798 | 6.199679 | 0.069881 | 0.861599 | -0.182783
$2d_{3/2}$ | -9.513 | -11.887 | 5.580005 | 5.686162 | 0.106157 | 1.195982 | 0.719801
$3s_{1/2}$ | -8.405 | -10.335 | 5.489444 | 5.540033 | 0.050589 | 0.557970 | -0.7388235
Total |  |  | 5.450221 | 5.505978 | 0.055757 | 0.6110 |
Figure 4: Single-particle wave functions of proton (black curves) spherical
subshells in 208Pb nucleus compared with those of the neutron $2g_{9/2}$
(green lines) and $1i_{11/2}$ (red lines) subshells. Note that the latter two
states are shown in each panel. Proton subshell labels are shown on each
panel.

###  III.3 Differential charge radii: microscopic origin of the kinks

It is well established in different model calculations that dominant or
significant occupation of the neutron $1i_{11/2}$ subshell above the $N=126$
shell closure is critical for explaining the kink in differential charge radii
at neutron $N=126$ shell closure [18, 8, 21, 16, 20, 22, 3]. The functionals
(typically CEDFs) in which the $\nu 1i_{11/2}$ subshell is located below the
$\nu 2g_{9/2}$ subshell reasonably well describe this kink and the slopes of
differential charge radii below and above $N=126$. In contrast, the
functionals (typically non-relativistic ones) with opposite location of two
subshells either fail to describe the kink or significantly underestimate its
magnitude.

This difference in the slope of differential charge radii above the $N=126$
shell closure as emerging due to the occupation of above mentioned neutron
spherical subshells has been discussed earlier within unpaired RMF
calculations with the NL3* CEDF (see discussion of Fig. 4 in Ref. [3]). The
differential charge radii $\delta\left<r^{2}\right>_{p}^{136,126}$ are 1.31
fm2 and 0.61 fm2 for the ”Conf-1$i_{11/2}$” and ”Conf-$2g_{9/2}$”
configurations of 218Pb, respectively. However, the microscopic origin of this
significant difference has not been investigated. To fill the gap in our
knowledge, we show in Tables 1 and 2 the contributions of different spherical
subshells into build-up of differential charge radii of the ”Conf-1$i_{11/2}$”
and ”Conf-$2g_{9/2}$” configurations of 218Pb. Note that for the sake of
comparison with single-particle wave functions and densities we also consider
proton single-particle rms radii $r^{p}_{i}$ in these tables.

Table 1 clearly shows that when the neutron $1i_{11/2}$ subshell is occupied
in 218Pb the largest changes in proton single-particle rms radii $\delta
r^{p}_{i}$ take place for the proton subshells with principal quantum number
$n=1$. The $\delta r^{p}_{i}$ values for the $s$, $p$ and $d$ $n=2$ subshells
are smaller than the average $\delta r^{p}_{i}$ value over the $n=1$ subshells
by a factor of approximately 2, 5 and 25, respectively. Proton single-particle
rms radius of the $3s_{1/2}$ subshell even decreases on transition from the
ground state configuration of 208Pb to the ”Conf-1$i_{11/2}$” configuration of
218Pb. All these changes are reflected in differential single-particle proton
radii $\Delta\left<r^{2}\right>_{i}^{N,N^{\prime}}$ (see column 7 of Table 1)
so that 96.7% of differential charge radius
$\delta\left<r^{2}\right>_{p}^{136,126}=1.31$ fm2 [see Eq. (3)] of the
”Conf-1$i_{11/2}$” configuration in 218Pb with respect of the ground state
configuration in 208Pb are built by the $n=1$ proton subshells.

The situation drastically changes when the neutron $2g_{9/2}$ subshell is
occupied in 218Pb [configuration ”Conf-$2g_{9/2}$”] (see Table 2). In this
case, the largest $\delta r^{p}_{i}$ values are seen for the $2s_{1/2}$,
$2p_{3/2}$, $2p_{1/2}$, $2d_{5/2}$ and $2d_{3/2}$ proton subshells. The proton
rms radii of low lying $1s_{1/2}$, $1p_{3/2}$, $1p_{1/2}$, $1d_{5/2}$ and
$1d_{3/2}$ subshells even decrease on transition from the ground state
configuration of 208Pb to the ”Conf-2$g_{9/2}$” configuration of 218Pb. This
is the consequence of self-consistency effects discussed in Secs. III.1 and
III.2 and the effects discussed in Sec. III.4. In addition, the increase of
proton rms radii is rather modest for remaining $n=1$ subshells and for the
$3s_{1/2}$ subshell. All these changes are reflected in differential single-
particle proton radii $\Delta\left<r^{2}\right>_{i}^{N,N^{\prime}}$ (see
column 7 of Table 2) so that 53.3% of differential charge radius
$\delta\left<r^{2}\right>_{p}^{136,126}=0.61$ fm2 of two configurations under
study are built by the $n=2$ proton subshells. This is despite the low
multiplicity of the occupied $n=2$ subshells which represent only
approximately 22% of the occupied single-particle states of the $Z=82$ core.

Figure 5: The same as Fig. 4 but only for the proton $s$ states and with
enlarged range on vertical axis.

The impact of the occupation of neutron $2g_{9/2}$ subshell on proton single-
particle rms radii of the $n=2$ proton subshells (see Table 2) is on average
comparable to the one of neutron $1i_{11/2}$ subshell on proton single-
particle rms radii of the $n=1$ proton subshells (see Table 1). Thus, other
factors have to be involved to explain large difference in differential charge
radii of the ”Conf-1$i_{11/2}$” and ”Conf-2$g_{9/2}$” configurations in 218Pb.
Indeed, the analysis of Eq. (II) clearly indicates that differential charge
radii between two isotopes are defined not only by the differential single-
particle radii $\Delta\left<r^{2}\right>_{i}^{N,N^{\prime}}$ of occupied
single-particle states

but also by the abundance of the subshells with a given $n$ among occupied
subshells and their multiplicity $m_{i}$. Low $n$ subshells are most abundant
in any nucleonic potential (see Refs. [32, 26]). Indeed, there are 10 $n=1$, 5
$n=2$ and 1 $n=3$ occupied subshells in the $Z=82$ core of the Pb isotopes
(see Fig. 3). In addition the $n=1$ subshells have the highest multiplicity
$m_{max}$ among the occupied subshells: $m_{max}=12$, 6 and 2 for the $n=1$, 2
and 3 subshells, respectively. As a result, 62, 18 and 2 protons of the $Z=82$
core are located in the $n=1$, 2 and 3 subshells, respectively. The
combination of all above mentioned in this subsection factors allows to
explain large difference in differential charge radii of the
”Conf-1$i_{11/2}$” and ”Conf-2$g_{9/2}$” configurations in 218Pb.

###  III.4 Microscopic origin of the pull of neutron subshells on proton ones

In order to better understand the state dependence of the pull provided by a
neutron in a given state on the proton in the $nlj$ subshell, Figs. 4 and 5
compare the proton wave functions of all occupied proton subshells in the
$Z=82$ core of the 208Pb nucleus with the neutron wave functions of the
neutron $1i_{11/2}$ and $2g_{9/2}$ subshells calculated in the
”Conf-1$i_{11/2}$” and ”Conf-2$g_{9/2}$” configurations of 218Pb,
respectively. In addition, the overlap of respective proton and neutron wave
functions defined as

$\displaystyle<\Psi_{n}^{k}|\Psi_{p}^{i}>=\int\Psi_{n}^{k}({\vec{r}}\,)\Psi_{p}^{i}({\vec{r}}\,)d^{3}r$
(16)

is presented in the last columns of Tables 1 and 2. Here proton state index
$i$ runs over all occupied proton subshells while neutron index $k$ is equal
either to $k=1i_{11/2}$ or $k=2g_{9/2}$. Positive (negative) values of these
overlaps indicate that the wave functions $\Psi_{n}^{k}$ and $\Psi_{p}^{i}$
are spatially mostly in phase (out of phase).

Let us first consider the overlaps of proton wave functions with the neutron
$\nu 1i_{11/2}$ one [see Table 1]. The largest overlap exists for the $\pi
1h_{11/2}$ state ($<\Psi_{n}^{k}|\Psi_{p}^{i}>=0.98$). Indeed, these two
states have the wave functions which are most similar among considered cases
[see Fig. 4(p) and compare it with other panels of this figure]. The degree of
the similarity ($<\Psi_{n}^{k}|\Psi_{p}^{i}>\approx 0.92$) of the wave
function of the neutron $\nu 1i_{11/2}$ subshell is somewhat smaller with the
wave functions of the proton $\pi 1g_{7/2}$ and $\pi 1g_{9/2}$ subshells [see
Fig. 4(m) and (o)]. With decreasing the single-particle energy of spherical
$n=1$ proton subshell the degree of the similarity between neutron and proton
wave functions given by $<\Psi_{n}^{k}|\Psi_{p}^{i}>$ decreases but still
remains high (see the last column of Table 1 and Figs. 4(l), (j), (h), (f),
(d) and (a)]. Among the proton $n=1$ subshells the lowest overlap
$<\Psi_{n}^{k}|\Psi_{p}^{i}>=0.57$ exists for proton $1s_{1/2}$ subshell which
is the only $n=1$ subshell with the maximum of the wave function at the center
of nucleus [see Fig. 4(a)].

The situation is completely different for the proton subshells with $n=2$: the
evolution of their wave functions as a function of radial coordinate $r$ is
mostly out of phase with that of the wave function of the proton $1i_{11/2}$
subshell [see Figs. 4(b), (e), (g), (i) and (k)]. This is due to the
differences in the nodal structure of these wave functions. As a consequence,
large negative overlaps $<\Psi_{n}^{k}|\Psi_{p}^{i}>$ exist for these pairs of
the states (see last column of Table 1). There are large differences due to
underlying nodal structure between the wave functions of the proton $3s_{1/2}$
and neutron $1i_{11/2}$ subshells (see Fig. 4(c)). However, the overlap
$<\Psi_{n}^{k}|\Psi_{p}^{i}>$ for this pair of the subshells is positive due
to the fact that the overlap is dominated by the behavior of the wave
functions at large radial coordinates.

Figure 6: The correlations between the overlaps $<\Psi_{n}^{k}|\Psi_{p}^{i}>$
and the changes of proton single-particle radii $\delta r^{p}_{i}$ for proton
subshells of the $Z=82$ core. Solid red circles, open black squares and green
triangles are used for the $n=1$, $n=2$ and $n=3$ proton subshells. Panels (a)
and (b) show the results when neutron $1i_{11/2}$ and $2g_{9/2}$ subshells are
occupied in the configurations of the 218Pb nucleus, respectively.  Figure 7:
The same as in Fig. 6 but for the correlations between the overlaps
$<\Psi_{n}^{k}|\Psi_{p}^{i}>$ and differential single-particle proton radii
$\Delta\left<r^{2}\right>_{i}^{N,N^{\prime}}$.

The occupation of the neutron $2g_{9/2}$ subshell leads to completely
different pattern of behavior (see Figs. 4 and 5 and Table 2). The largest
overlaps exist for the $n=2$ proton subshells: the only exception is the
overlap which includes proton $2s_{1/2}$ subshell which has a maximum of its
wave function at $r=0$. These overlaps become smaller or even negative for the
cases which include $n=1$ and $n=3$ proton subshells (see the last column in
Table 2).

Figure 8: Proton single-particle density redistributions
$\delta\rho_{i}^{p}(r)$ caused by the occupation of indicated neutron
subshells in 218Pb (see text for details).

Fig. 6 shows the correlations between the overlaps
$<\Psi_{n}^{k}|\Psi_{p}^{i}>$ for the neutrons in the $1i_{11/2}$ and
$2g_{9/2}$ subshells and the proton subshells occupied in the $Z=82$ core and
the changes in single-particle proton rms radii $\delta r_{i}^{p}$ of these
subshells triggered by the occupation of respective neutron subshells. In
general, the largest $\delta r_{i}^{p}$ values appear for the proton subshells
which have the same principal quantum number $n$ as occupied neutron subshell.
This also corresponds to the largest positive overlaps
$<\Psi_{n}^{k}|\Psi_{p}^{i}>$. Small or negative overlaps, which correspond to
the case of different principal quantum numbers $n$ of proton and neutron
subshells, typically lead to relatively small $\delta r_{i}^{p}$ values.

These correlations are very pronounced in the case of the occupation of the
neutron $1i_{11/2}$ subshell since its wave function has a simple structure
with a single maximum at $r\approx 6$ fm (see Fig. 4). Significant changes in
the single-particle rms radius are seen for proton $n=1$ subshells which have
large overlaps $<\Psi_{n}^{k}|\Psi_{p}^{i}>$ but rather small $\delta
r_{i}^{p}$ values exist for the proton $n=2$ subshells which have negative
overlaps (see Fig. 6 and Table 1).

Such correlations are somewhat less pronounced in the case of the occupation
of the $2g_{9/2}$ neutron subshell the wave function of which has maximum at
$r\approx 4$ fm and minimum at $r=7.5$ fm (see Fig. 4). The largest overlaps
$<\Psi_{n}^{k}|\Psi_{p}^{i}>$ are seen with proton $n=2$ $2p_{3/2}$,
$2p_{1/2}$, $2d_{5/2}$ and $2d_{3/2}$ subshells which produce the largest
changes in the proton single-particle rms radii (see Fig. 6 and Table 2).
Smaller and sometimes negative changes in the proton single-particle rms radii
are produced for the $n=1$ and $n=3$ proton subshells when neutron $2g_{9/2}$
subshell becomes occupied.

Note that similar to above discussed correlations are also seen between
$<\Psi_{n}^{k}|\Psi_{p}^{i}>$ and differential single-particle proton radii
$\Delta\left<r^{2}\right>_{i}^{N,N^{\prime}}$ (see Fig. 7) with the latter
quantities defining the differential charge radius between two isotopes [see
Eq. (II)].

The absolute values of the $<\Psi_{n}^{k}|\Psi_{p}^{i}>$ overlaps presented in
column 8 of Tables 1 and 2 are similar to those obtained in Skyrme DFT
calculations with NRAPRii energy density functional (see Fig. 5 of Ref. [21]).
This clearly indicates a similar mechanism of the buildup of differential
charge radii in non-relativistic and covariant DFTs.

Because of the dependence of charge radii and their changes on proton single-
particle densities (see Sec. II), a deeper microscopic insight is provided by
the analysis of the redistributions of the proton single-particle densities
defined as

$\displaystyle\delta\rho_{i}^{p}(r)=\rho_{i}^{p}(r)[^{218}{\rm Pb}-{\rm
conf}]-\rho_{i}^{p}(r)[^{208}{\rm Pb}]$ (17)

when different neutron subshells are occupied in 218Pb. Since the single-
particle density is normalized to unity [see Eq. (5)], the addition of
neutron(s) to the 208Pb nucleus will only lead to redistribution of the proton
single-particle density under the condition that

$\displaystyle\int r^{2}\delta\rho_{i}^{p}(r)=0$ (18)

These redistributions are clearly seen in Fig. 8. Let us consider as an
example the changes in the single-particle proton densities of the $1p_{3/2}$
subshell [see Fig. 8(h)]. The occupation of the neutrons in the $2g_{9/2}$
subshell (the ”Conf-2$g_{9/2}$” configuration in 218Pb) leads to an increase
of proton densities of the $1p_{3/2}$ subshell at $r\approx 3.2$ fm and their
decrease at $r\approx 5.5$ fm as compared with that in the ground state
configuration of 208Pb [green line in Fig. 8(h)]. In contrast, the occupation
of the neutrons in the $1i_{11/2}$ subshell (the ”Conf-1$i_{11/2}$”
configuration in 218Pb) has an opposite effect: it leads to the decrease of
proton densities of the $1p_{3/2}$ subshell at $r\approx 3$ fm and their
increase444The reader should not be confused by larger density changes at low
radial coordinates as compared with those at larger values of $r$. This is
because the density plots as a function of radial coordinate tend to
overemphasize the importance of the central region since they ignore the fact
that the number of particles $dn$ in a spherical shell of thickness $dr$ is
given by $4\pi r^{2}\rho(r)dr$ (see example in Sec. III of Ref. [26]). at
$r\approx 6$ fm [red line in Fig. 8(h)]. Note that the number of the
oscillations of density redistributions increases with the increase of
principal quantum number $n$ of the proton and neutron subshells involved
(compare, for example, the panels (c) and (h) of Fig. 8).

For almost all proton subshells of the $Z=82$ core the occupation of the
$1i_{11/2}$ or $2g_{9/2}$ neutron subshells leads to drastically different
redistributions of proton single-particle densities which are frequently out
of phase of each others as a function of radial coordinate $r$ (see Fig. 8).
Thus, for a given proton subshell this leads to different changes in the
proton single-particle radii $\delta r_{i}^{p}$ and substantial differences in
differential single-particle radii
$\Delta\left<r^{2}\right>_{i}^{N,N^{\prime}}$ (compare Tables 1 and 2).

##  IV General observations

###  IV.1 The origin of differential charge radii

The detailed analysis presented in Sec. III clearly indicates that the change
of differential charge radius with increasing neutron number has a single-
particle origin. Added neutron interacts with proton in a given subshell and
this leads to a redistribution of single-particle densities of occupied proton
states which in turn modifies the charge radii. The outcome of this
interaction depends on the relative properties of the wave functions of
interacting proton and neutron. Large positive overlap
$<\Psi_{n}^{k}|\Psi_{p}^{i}>$ between their wave functions leads to a
substantial increase of differential single-particle charge radius of proton
subshell in which interacting proton is located. In contrast, small or
negative overlap of their wave functions typically leads to either small
increase or even decrease in differential single-particle charge radius.

Ref. [21] has discussed the pull on the proton states provided by the
occupation of neutron states and resulting change in charge radii as emerging
from strong nuclear symmetry energy. However, our detailed analysis clearly
indicates that the changes in charge radii with increasing neutron number are
governed by the proton-neutron interaction.

The buildup of differential charge radii
$\delta\left<r^{2}\right>_{p}^{N,N^{\prime}}$ between two isotopes is also a
collective phenomenon since all occupied proton states contribute to it [see
Eq. (II)] and none of these states provides a dominant contribution to
$\delta\left<r^{2}\right>_{p}^{N,N^{\prime}}$ (see Tables 1 and 2).

###  IV.2 Applicability of alternative models to the description of
differential charge radii in deformed nuclei

As reviewed in the introduction of Ref. [3], the absolute majority of the
studies of differential charge radii has been performed either in DFT or in ab
initio approaches. However, it is well known that DFT models have some
deficiencies in the description of spectroscopic properties related to the
energies of the single-particle states and their wave functions [33, 34, 35,
36, 37]. Moreover, the performance of ab initio models in the description of
single-particle spectra in odd-$A$ nuclei is comparable with that for the DFT
models but such calculations are available only for light nuclei (see Refs.
[38, 39, 40, 41]). In contrast, spherical shell models with empirical
interactions provide a better description of experimental spectroscopic data
in spherical nuclei located in the vicinity of doubly magic nuclei and
microscopic+macroscopic (mic+mac) models based on phenomenological potentials
such as the Woods-Saxon one does the same in the region of deformed nuclei.
However, these models are not expected to be adequate for the description of
differential charge radii due to the reasons mentioned below.

The lack of self-consistency effects and the interaction between protons and
neutrons will affect the description of differential charge radii in the
mic+mac model. This is because the addition of neutron does not affect the
proton subsystem in a self-consistent manner on the level of single-particle
subshells via the mechanisms discussed in Sec. III. For example, in the Woods-
Saxon potential it affects the total radius $R$ of nucleus only via mass
dependence $R=1.2A^{1/3}$. This means that the occupation of the neutron
$1i_{11/2}$ and $2g_{9/2}$ subshells in the $N>126$ Pb isotopes will lead to
the same differential charge radii contrary to the results of self-consistent
calculations (see Refs. [18, 8, 3]). Moreover, there is a lack of self-
consistency in the definition of the radial properties of the density
distributions in the macroscopic (liquid drop) and microscopic (single-
particle potential) parts of the mic+mac model. To our knowledge, this aspect
of the problem has not been studied in detail. However, the physical
observables similar to $\delta\left<r^{2}\right>_{p}^{N,N^{\prime}}$, namely,
relative charge quadrupole moments of superdeformed bands are affected by the
lack of self-consistency between microscopic and macroscopic parts (see Ref.
[42]).

Although the spherical shell model takes into account the proton-neutron
interaction it suffers from the introduction of the core. As a consequence,
the pull provided by extra neutron(s) on the proton single-particle states
forming the core is ignored and this affects drastically the calculated charge
radii of the nuclei with valence nucleons outside the core. This introduces
uncontrollable errors in the calculations of differential charge radii and
thus severely limits the applicability of spherical shell model to the
description of this observable. Few existing calculations of differential
charge radii in spherical shell model (see Refs. [43, 44]) suffer from this
problem. For example, they cannot reproduce the kink in the differential
charge radii of the Sn isotopes at $N=82$ and Pb isotopes at $N=126$ [44].
This problem can be rectified by employing no-core shell model but because of
numerical reasons such models are applicable only to light nuclei [45, 46].

###  IV.3 Potential mechanisms affecting odd-even staggering (OES) in charge
radii

Several mechanisms of regular and inverted OES in charge radii have been
discussed and reviewed in Sec. IX of Ref. [3]. These include shape coexistence
leading to deformation staggering in even and odd-$A$ nuclei (see Sec. IXA of
Ref. [3]), pairing correlations (see Sec. IXB of Ref. [3]), particle-vibration
coupling (PVC) in odd-$A$ nuclei (see Sec. IXC of Ref. [3]) and some other
mechanisms (see Sec. IXD of Ref. [3]). The PVC mechanism is responsible for a
substantial fragmentation of the wave function of the ground states in
spherical odd-$A$ nuclei (see Sec. IXC of Ref. [3]) and rearrangement of the
energies of the predominantly single-particle states (see Ref. [20]).

Here we extend the discussion presented in Sec. IX of Ref. [3] and outline
other potential mechanisms which could affect OES in charge radii. This
discussion is based on the fact that it is the single-particle content of
unpaired neutron states in odd-$N$ nuclei which defines the pull on proton
densities (see present paper and Refs. [21, 3]). If the deformation changes
between neighbouring even-even and odd-$A$ nuclei are small, this is dominant
mechanism defining OES in charge radii.

If the structure of extra neutron states does not change with their sequential
addition, the differential charge radii show a linear evolution as a function
of neutron number (see discussion of Fig. 4 in Ref. [3]) and no OES in charge
radii is present. However, if the structure of the neutron state in odd-$N$
nuclei is affected by the interaction(s) or effect(s) not present in its even-
even neighbors (such as blocking effect for pairing [11, 12, 13]) and/or
particle-vibration coupling [20, 3]), this leads to OES in charge radii.

The transition to deformed nuclei opens additional channels which are not
present in spherical ones but which can affect OES in charge radii. These are
the Coriolis interaction in odd and odd-odd nuclei and residual interaction of
unpaired proton and neutron in odd-odd nuclei: each of them affects the
structure of the wave function in odd-$N$ nuclei as compared with the ones of
their even $N\pm 1$ neighbors and this can contribute to OES in charge radii.
To our knowledge these channels have not been indicated in literature as
possible sources of OES in charge radii because they are neglected in the DFT
calculations.

The admixtures of the vibrational phonons to the structure of the ground
states in deformed rare-earth and actinide odd-mass nuclei is relatively small
(see Ref. [47, 48, 49]) especially when compared with spherical nuclei (see
Refs. [50, 51, 37]). Thus, their role in building OES of charge radii is
expected to be reduced as compared with the one in spherical nuclei (see Refs.
[20, 3]). On the other hand, the Coriolis interaction can be active in the
ground states of deformed odd-$N$ nuclei but it is absent in the ground states
of even-even nuclei (see Refs. [52, 32]). This interaction leads to a $\Delta
K=1$ mixing555Here $K$ is the projection of single-particle angular momentum
on the axis of symmetry of deformed nucleus. of different states both in odd
(see Refs. [52]) and odd-odd (see Refs. [53, 54]) nuclei. If the wavefunction
of the ground state of the nucleus with odd-$N$ is strongly affected by
Coriolis mixing, this can affect OES of charge radii.

The situation in odd-$Z$ isotopic chains (see Fig. 27 in Ref. [3]) which
includes odd-odd nuclei becomes even more complicated for theoretical
interpretation since in addition to Coriolis interaction there is the
possibility for the $\Delta K=0$ mixing due to residual interaction of
unpaired proton and neutron in odd-odd nuclei [53, 54]. Note that such mixing
is typically calculated in the rotor+particle models (PRM) and it is neglected
in the state-of-the-art DFT applications. In some cases the $\Delta K=0$
mixing can be strong so it can potentially have significant impact on the
structure of the wave function of odd neutron which generates the pull on
proton densities.

Let us consider the isotopic chain of the Eu isotopes which is one of the best
examples of inverted OES in charge radii (see Fig. 27 in Ref. [3]). As
discussed in Sec. IX of Ref. [3] the inversion of OES at $N\approx 88$ is most
likely triggered by the transition from spherical or quasispherical nuclei to
deformed ones with increasing neutron number. However, such inversion of OES
exists also for $N=90,91$ and 92 which covers deformed nuclei (see Fig. 27(f)
in Ref. [3]). The experimental observations (Ref. [55]) and PRM calculations
(Ref. [56]) indicate strong $\Delta K=0$ mixing for the
$K^{\pi}=1^{\pm},3^{\pm}$ and $4^{\pm}$ bandheads in the 154Eu nucleus
including $K^{\pi}=3^{-}$ ground state with the $\pi 5/2[413]-\nu 11/2[505]$
structure. Such mixing is observed and calculated in two-quasiparticle
configurations of 156Eu [57] but ground state of this nucleus seems not to be
affected by it. These features may lead to the inversion of OES around $N=91$
and its absence above $N=92$. In reality, at the neutron numbers of interest
the neighboring $Z=62$ Sm isotopes show regular OES in charge radii (see Fig.
26(l) in Ref. [3]) which suggests that the inversion of OES in charge radii of
the Eu ($Z=63$) isotopes are most likely due to the properties of odd-odd
nuclei. The detailed DFT calculations with residual interaction of unpaired
proton and neutron in odd-odd nuclei are needed to confirm or reject such an
interpretation. However, such investigation goes beyond the scope of the
present study.

##  V Conclusions

The self-consistency and proton-neutron interaction effects in the buildup of
differential charge radii have been considered by comparing two configurations
of the 218Pb nucleus, generated by the occupation of the neutron $1i_{11/2}$
and $2g_{9/2}$ subshells, with the ground state configuration in 208Pb. The
main contribution to differential charge radii has a single-particle origin
and comes from the interaction of added neutron(s) and the protons forming the
$Z=82$ proton core. This interaction depends on the overlaps of the proton and
neutron wave functions and leads to a redistribution of single-particle
density of occupied proton states which in turn modifies the charge radii. In
contrast, self-consistency effects affecting the shape of proton potential,
proton densities and the energies of the single-particle states in the proton
potential provide only a small contribution. Note that the buildup of
differential charge radii between two isotopes is also a collective phenomenon
since all occupied proton single-particle states contribute to it. Although
these results were obtained for the Pb isotopes, they are general and
applicable to any isotopic chain.

This combination of single-particle and collective aspects in the building of
differential charge radii limits the applicability of different classes of the
models to the description of this physical observable. The models which ignore
self-consistency effects and proton-neutron interaction (such as
microscopic+macroscopic model) cannot describe this physical observable. The
models which ignore collective aspect of the problem by introducing the core
(such as spherical shell model) introduce uncontrollable errors in the
description of differential charge radii.

The pull provided by an extra neutron on proton states depends on the
structure of the state occupied by it. For deformed even-$Z$ nuclei, the wave
function of the ground states in odd-$N$ nuclei is more complicated than that
of the ground states in neighboring even-even nuclei because of the presence
of the Coriolis interaction. The addition of proton (odd-$Z$ nuclei) leads to
a further complication of the wave function since Coriolis interaction is
present in all nuclei of isotopic chain under study and, in addition, the
structure of odd-odd nuclei is affected by the residual interaction between
unpaired proton and neutron. The analysis indicates than both the Coriolis
interaction in odd and odd-odd nuclei and the residual interaction between
unpaired proton and neutron in odd-odd nuclei could affect the odd-even
staggering in charge radii if their impact on the wave function of the ground
state of these nuclei is appreciable. Note that these interactions are usually
neglected in the DFT calculations.

##  VI ACKNOWLEDGMENTS

This material is based upon work supported by the U.S. Department of Energy,
Office of Science, Office of Nuclear Physics under Award No. DE-SC0013037.

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Remembering Steven Weinberg

Gian Francesco Giudice

CERN, Theoretical Physics Department, Geneva, Switzerland

Contribution to the volume “In Memory of Steven Weinberg” to appear in Nuclear
Physics B.

Meeting Steven Weinberg has been one of the highest privileges of my
professional life. In 1990, I was hired as a postdoc in his research group and
spent two years in Austin, Texas. Those were extraordinarily stimulating
years. Weinberg’s group was relatively small, but comprised some of the
brightest minds in physics of the time: Joe Polchinski, Willy Fischler, Vadim
Kaplunovsky, Philip Candelas, among others. Even without Weinberg, this would
have been one of the strongest groups in theoretical particle physics in the
world. With Weinberg, it was nirvana for a young researcher like me.

Steve was the greatest hero in physics I could ever imagine. True, I have
never had the opportunity of meeting Feynman, who may have been more
impressive to a young researcher with his captivating enthusiasm and
overwhelming brilliance. Steve was not the kind who overpowers you with words.
During conversations he remained reflective and pensive, but anything that
came out of his mouth was pure wisdom, expressed with a lucidity and logical
rigor to leave you breathless.

He liked to work alone and not to be constantly disturbed. To enter his office
one had to go through the office of his loyal secretary Adele, who was
protecting his privacy as a mastiff guards over its master’s property. Steve
was not spending everyday’s life with common mortals, such as postdocs, but he
wasn’t ignoring them either. The encounters followed a recurrent pattern and,
although relatively rare, were always meaningful and memorable.

Once a week, there was the brown bag seminar, when Weinberg’s group got
together for lunch. Every week, a different member of the group was presenting
some work in progress or an interesting paper they read, while the others were
chewing their sandwiches. Speaking in front of so many eminent theorists and
addressing their sharp questions was an experience one couldn’t forget.
Whenever Steve started to utter a question, I was holding my breath. He was
often interrupting the speaker, not to prove them wrong but to understand the
subject in his own way.

Another regular occasion for encounters were the lunches at the UT Faculty
Club, which happened whenever an external speaker came to Austin to give the
main seminar. All professors and postdocs of Weinberg’s group were invited,
together with the guest speaker. The Faculty Club was a few blocks away from
the Physics Department, an easy and pleasant walk across campus. Nevertheless
Steve was preferring to take his car, kindly inviting people for the ride.
Almost everyone was declining, with the excuse that the weather was favorable
for a short stroll (even when the outside temperature was above 100
Fahrenheit). I later understood that the true reason was Weinberg’s poor
driving skills (nobody is perfect), as his attention was easily diverted away
from the traffic. I was one of the few who always accepted his invitations.
Maybe it was just because, as an Italian, I have low standards for driving,
but for me listening to Steve’s conversation for five minutes was well worth
the risk of a car accident.

At my first visit to the Faculty Club, I was instructed about the rigid lunch
etiquette by my friend and office mate Ramy Brunstein, a fellow postdoc more
experienced than myself. Rule number one: there should be a single
conversation at the table and you should never start small talk with the
person sitting next to you. Rule number two: nobody (other than Weinberg) was
allowed to change the topic of conversation. It was delightful to listen to
Steve speaking about physics in an informal setting. I will never forget the
lunch in which our guest, a well-known phenomenologist, was explaining how
subtle is the art of building models for new physics. Referring to Steve and
himself collectively, he said: “Because we model-builders…” Steve promptly
interrupted him. “I don’t consider myself a model-builder,” he said with an
ironic smile on his face. “In my life, I have built only one model.”

Physics was not the only topic of conversation at Faculty Club lunches. Steve
was a history connoisseur and enjoyed recounting episodes about the Civil War
or making remarks about military strategy in famous battles of the Middle
Ages. He also had a series of funny anecdotes about history. Once, after
smiling at me as a way of asking for permission, he said that he wanted to
tell a true story that happened in Italy, soon after the Racial Laws were
introduced in 1938. The podestà (mayor) of a remote village in a forlorn
valley in central Sicily was a scrupulous fascist, eager to show his loyalty
to Mussolini. After receiving the dispatch listing the racial restrictions to
be implemented in every town in Italy, he quickly telegraphed the Duce:
“Racial campaign ready. Please send Jews”.

Alas, after two years of postdoc, I started noticing that his repertoire was
rather limited and that he kept repeating the same jokes to different guests.
Experience with Faculty Club lunches helped me to realize that explaining
complex concepts in physics came more natural to him than telling jokes.

Seminars too followed a well-established ritual. Weinberg would enter the
seminar room last and sit in the front row, with a pile of mail that was just
handed to him by the devoted Adele. Soon after the beginning of the seminar,
Steve would start asking questions. He was never overtly aggressive with
speakers but he always wanted to get to the bottom of every issue. If
something wasn’t clear to him, he would stop the speaker and ask for
explanations until he had fully understood the point. Eventually, if the
answers didn’t satisfy him, he would simply ignore the seminar and start
opening noisily his mail (I mean tearing envelopes, unfolding letters and
reading them; a concept of mail that young people today may not fully
comprehend). Disinterest was his way of expressing dissent and humiliating the
speakers.

Those were the years of the SSC catastrophe. Steve had become a vocal advocate
of the project. He was regularly interviewed by journalists and often went to
Washington to lobby for the SSC or participate in congressional hearings.
Together with other postdocs, we were watching the interviews on TV. Mostly we
were having fun laughing at Steve’s difficulties with the ear-buds, which kept
on falling from his ears during the interview. When back from trips to
Washington, Steve liked to share with us his impressions of various senators
and offer his opinions about the process. For me it was a thrilling
experience: it felt like learning about the unfolding events directly from the
man who was shaping the History of Physics. At the time, Steve seemed
generally optimistic about the outcome of the SSC decision process in
Washington.

I left Austin in the Fall of 1992. About one year later the SSC was cancelled.
Weinberg mourned the decision and later said: “A society that decides it will
only support applied science and not waste money on pure science is likely to
wind up with neither.” In the meanwhile, I had arrived at CERN, where LEP was
in full swing, the LHC construction would be approved in December 1994, and
LEP2 would start its gradual energy increase in 1996. I left the US to go to
Europe just in time to participate in a new era of high-energy physics at
CERN.

I remained in touch with Steve. I wouldn’t call it a close friendship, but we
exchanged email messages from time to time. Sometimes it was about physics: he
liked the idea of Split Supersymmetry, and was intrigued by the method of
calculating supersymmetry-breaking effects from wave-function renormalization
and by anomaly mediation (work I did in collaboration with Nima Arkani-Hamed,
Savas Dimopoulos, Markus Luty, Hitoshi Murayama, Riccardo Rattazzi, Andrea
Romanino). Sometimes he was writing just to ask for news from CERN. We met in
several occasions when he travelled to Europe. He once came to Padua, my
hometown, and we spent some time together, also with his wife Louise. Faithful
to his principles, he refused to enter religious buildings, let alone the
synagogue, so missing a visit, which I had organized for him through some
friends, to see a small Paduan cemetery hidden in a charming garden among old
houses, where the 16th century Talmudist Meir Katzenellenbogen is buried.

Weinberg had a profound influence on me as a physicist. Because of the limited
amount of time I spent with him, the influence came mostly from his writings.
What I admire in those papers are his lucid thinking, logical clarity and
brilliant insight. Few theorists match his vision of physics and foresight to
anticipate new research directions. Take the example of effective field
theories, which today are part of the toolkit of every particle physicist.
Many of the founding ideas on effective theories came from his 1979 paper on
Phenomenological Lagrangians.

Just as happened with his 1967 article (namely the paper that contains the
only model Steve has ever proposed and that was later rewarded with a Nobel
prize), also the message about effective field theories took a while to
percolate through the physics community. I remember that, when I was a PhD
student in the late 80s, some well-known authors were still claiming that
renormalizability is a necessary requirement for any realistic quantum field
theory. Luckily my PhD advisor Riccardo Barbieri, one of the greatest
theorists of his time, was well aware of Wilson’s and Weinberg’s lessons and
could pass them to me.

Today the database Inspire lists 46 papers by Weinberg with more than 500
citations, and 78% of them are single-authored. Each of them is a masterpiece
that broke new ground in a different subject, testifying to the amazing breath
and importance of Weinberg’s contribution to physics. Any theorist would
consider a supreme accomplishment of their career to have written only one of
them. Weinberg wrote 46 such papers. Every student in theoretical particle
physics should read all of them at least once, not so much to learn about
their content, which by now is standard material that can be found in any
textbook on the subject, but to learn what it means to do research in
theoretical physics.

Not only his technical papers, but his books too shaped my views on physics.
His monumental textbooks, from gravitation and cosmology to quantum field
theory, are still an essential part of any theoretical physicist’s education.
I am particularly fond of his books addressed to the general public, which
contain powerful lessons on the meaning of science and its role in society.
Reading his essays was crucial for me to form my own perspective on these
issues. Before giving public talks, I often refresh my memory by reading once
again some of the chapters of his popular books. Besides always being a
pleasure, it is also a way for me to return to the old master and listen to
familiar advice.

Weinberg is a hero for all of us who had the privilege of meeting him and a
lasting intellectual beacon for future generations.