# Neutron skin of 27Al with Skyrme and Korea-IBS-Daegu-SKKU density
functionals

Hana Gil  Center for Extreme Nuclear Matters, Korea University, Seoul 02841,
Korea  Chang Ho Hyun<EMAIL_ADDRESS>Department of Physics Education, Daegu
University, Gyeongsan 38453, Korea  K. S. Kim<EMAIL_ADDRESS>School of
Liberal Arts and Science, Korea Aerospace University, Goyang 10540, Korea

###### Abstract

Recent measurement of the parity-violating (PV) asymmetry in the elastic
electron scattering on 27Al target evokes the interest in the distribution of
the neutron in the nucleus. In this work, we calculate the neutron skin
thickness ($R_{np}$) of 27Al with nonrelativistic nuclear structure models. We
focus on the role of the effective mass, symmetry energy and pairing force.
Models are selected to have effective masses in the range $(0.58-1.05)M$ where
$M$ is the nucleon mass in free space, and stiffness of the symmetry energy is
varied by choosing the slope of the symmetry energy in the range 9.4 – 100.5
MeV. Effect of pairing force is investigated by calculating $R_{np}$ with and
without pairing, and using two different forms of the pairing force. With nine
models, we obtain $R_{np}=0.001-0.014$ fm. The result is independent of the
effective mass, symmetry energy, and the form of pairing force. However,
$R_{np}$ is negative when the pairing force is switched off, so the pairing
force plays an essential role to make $R_{np}$ positive and constrained in a
narrow range. We also calculate the PV asymmetry ($A_{\rm pv}$) in the elastic
electron-27Al scattering in the Born approximation at the kinematics of the
Qweak experiment. We obtain a very narrow-ranged result $A_{\rm pv}=$ (2.07 –
2.09) $\times 10^{-6}$. The result is consistent with the experiment and
insensitive to the effective mass, symmetry energy and pairing force.

Introduction. Measurement of parity violation in the elastic electron
scattering on nuclei is expected to shed light on the distribution of neutrons
within an atomic nucleus. For the neutron-rich nuclei neutron skin thickness
($R_{np}$) is known to be strongly correlated to the density dependence of the
symmetry energy. With precise measurement of the parity-violating (PV)
asymmetry ($A_{\rm pv}$), the neutron skin thickness of the target nuclei can
be determined, and it can constrain the density dependence of the symmetry
energy. It is shown that the slope parameter $L$ of the symmetry energy is
most closely correlated to the neutron skin thickness of 208Pb ksym .

Recently, measurement of $A_{\rm pv}$ with 27Al has been performed by the
Qweak Collaboration prl2022 . Major goal of the Collaboration is to determine
proton’s weak charge from the elastic $\vec{e}p$ scattering. Largest
background of the experiment comes from the aluminum alloy cell that contains
the hydrogen target. To extract an accurate value of proton’s weak charge,
background should be understood precisely. Measurement of $A_{\rm
pv}(^{27}{\rm Al})$ was guided by a theoretical work prc2014 , in which
$A_{\rm pv}(^{27}{\rm Al})$ is calculated with the FSUGold relativistic mean
field (RMF) model. The experimental result is obtained in good agreement with
the predictions in prc2014 . From the measured value $A_{\rm pv}=(2.16\pm
0.11\pm 0.16)\times 10^{-6}$, the neutron skin thickness of 27Al is obtained
as $R_{np}=-0.04\pm 0.12$ fm. Since there is only one more neutron than the
proton in 27Al, neutron skin will not be thick, and the result turns out to be
consistent with the guess. However, the range of $R_{np}$ determined from
$A_{\rm pv}(^{27}{\rm Al})$ is significantly uncertain containing both
negative and positive values. Moreover, in the theoretical work prc2014 , only
one RMF model is used in the calculation of $A_{\rm pv}$. For an improved
constraint on the neutron skin thickness of 27Al, a systematic investigation
is required.

In recent works, we applied the KIDS (Korea-IBS-Daegu-SKKU) density functional
kidsnm  to the quasielastic electron and neutrino scattering off nuclei, and
showed that the model reproduces the experimental data at high accuracy kids-
scatt1 ; kids-scatt2 ; kids-scatt3 . In the present work, we calculate
$R_{np}$ of 27Al and $A_{\rm pv}$ in the elastic electron-27Al scattering with
nonrelativistic energy density functionals (EDFs). Nuclear models consist of
seven Skyrme forces and two KIDS models. Nuclear properties in the vicinity of
surface could be sensitive to the momentum-dependent terms in the Skyrme
force. Contribution of the momentum-dependent terms appears directly in the
effective mass of the nucleon, so the role of these terms to $R_{np}$ and
$A_{\rm pv}$ is examined by choosing models with effective masses in a wide
range at the saturation density. Since 27Al is an open-shell and odd-number
nucleus, the effect of pairing can play a critical role in the properties of
the nucleus. Dependence on the pairing force is examined by adopting two
different forms of the pairing force: one is the constant gap form, and the
other is the constant force form. In addition, by switching the contribution
of the pairing on and off, the effect of pairing force is thoroughly explored.

In the calculation of $A_{\rm pv}$, following the prescription in prc2014 , we
assume the distribution of proton and neutron spherically symmetric, and
calculate the multipole form factors of the proton by using the hole state in
the $d_{5/2}$ level. With the nine EDF models, $A_{\rm pv}$ is calculated at
the laboratory angle $\left<\theta_{\rm lab}\right>=7.61^{\circ}\pm
0.02^{\circ}$ and incident electron energy $\left<E_{\rm lab}\right>=1.157$
GeV.

| SkI3 | SkMP | SLy4 | KDE0v1 | SkM* | LNS | KIDS0-m*97 | KIDS0 | MSk7
---|---|---|---|---|---|---|---|---|---
$m^{*}_{s}/M$ | 0.58 | 0.65 | 0.69 | 0.74 | 0.79 | 0.83 | 0.90 | 0.99 | 1.05
$m^{*}_{v}/M$ | 0.80 | 0.59 | 0.80 | 0.81 | 0.65 | 0.73 | 0.70 | 0.81 | 1.05
$K_{0}$ | 258.2 | 230.9 | 229.9 | 227.5 | 216.6 | 210.8 | 240.0 | 240.0 | 231.2
$J$ | 34.8 | 29.9 | 32.0 | 34.6 | 30.0 | 33.4 | 32.8 | 32.8 | 28.0
$L$ | 100.5 | 70.3 | 45.9 | 54.7 | 45.8 | 61.5 | 49.1 | 49.1 | 9.4
Ref. | ski3  | skmp  | sly4  | kde  | skmstar  | lns  | kids0a ; kids0b ; kids0c  | kids0a ; kids0b ; kids0c  | msk7
Table 1: Skyrme force models considered in the work. $m^{*}_{s}$ and
$m^{*}_{v}$ denote the isoscalar and isovector effective masses at the
saturation density, respectively. Incompressibility in symmetric nuclear
matter $K_{0}$, and symmetry energy parameters $J$, $L$ are in units of MeV.

Model and Formalism. Since the role of symmetry energy and momentum-dependent
interaction is a focus of the work, models having wide ranges for these
quantities are selected. In Tab. 1 we summarize the effective masses at the
saturation density and parameters that determine the density dependence of the
nuclear matter equation of state for the nine Skyrme and KIDS models. Models
are chosen to have the isoscalar ($m^{*}_{s}$) and isovector ($m^{*}_{v}$)
effective masses distributed uniformly in a wide range $(0.58-1.05)M$.
Incompressibility of the symmetric matter $K_{0}$ is within the range of
experimental uncertainty k0 . Density dependence of the symmetry energy is
characterized by the slope $L$ in the vicinity of the saturation density. SkI3
and MSk7 models have anomalously large and small $L$, but the other models
give values of $L$ within a range consistent with neutron star observation
ksym ; symene .

Pairing force is essential in explaining the odd-even staggering of the mass
and charge radius. To see the dependence on the pairing, we employ two
different forms of the pairing correlation. One is the constant gap formula

$\Delta_{q}=\frac{11.2}{\sqrt{A}},$ (1)

where $A$ denotes the mass number, and the other is the constant force

$G_{q}=\frac{29}{A}$ (2)

in the gap equation $\Delta_{q}=G_{q}\sum_{\beta\in
q}\sqrt{w_{\beta}(1-w_{\beta})}$ where $w_{\beta}$ denotes the occupation
probability in a given orbit. In addition to using two different forms for the
pairing correlation, role of the pairing is investigated by calculating
$R_{np}$ with and without the pairing.

Parity-violating asymmetry $A_{\rm pv}$ is defined by

$A_{\rm
pv}=\frac{d\sigma_{+}/d\Omega-d\sigma_{-}/d\Omega}{d\sigma_{+}/d\Omega+d\sigma_{-}/d\Omega},$
(3)

where the subscripts $+$ and $-$ denote the positive and negative helicity of
the incoming electrons, respectively. If both neutrons and protons are
distributed equally and symmetrically within a nucleus, $A_{\rm pv}$ is
simplified to

$A^{0}_{\rm pv}=-\frac{G_{F}\,Q_{W}}{4\sqrt{2}\pi\alpha Z}q^{2},$ (4)

where $G_{F}$ is the Fermi constant, $\alpha$ is the fine structure constant,
and $Z$ is the proton number of a nucleus. Weak charge of a nucleus $Q_{W}$ is
given by

$Q_{W}=Q_{n}N+Q_{p}Z,$ (5)

where $Q_{n}=-1$ and $Q_{p}=1-4\sin^{2}\theta_{W}$ at the tree level, and $N$
is the neutron number of a nucleus. $q^{2}$ is the three momentum transfer
$q^{2}=4E_{i}E_{f}\sin^{2}\frac{\theta}{2}+(E_{i}-E_{f})^{2}$, which reduces
to

$q^{2}=4E^{2}_{i}\sin^{2}\frac{\theta}{2}$ (6)

in the elastic scattering.

In reality, however, distribution of the proton is different from that of the
neutron for various reasons. Taking into account the difference of the neutron
and proton distribution, $A_{\rm pv}$ is given in the Born approximation as
prc2014

$A_{\rm pv}=A^{0}_{\rm
pv}\frac{C_{0}C^{W}_{0}+C_{2}C^{W}_{2}+C_{4}C^{W}_{4}}{C^{2}_{0}+C^{2}_{2}+C^{2}_{4}}.$
(7)

$C_{0}$, $C_{2}$ and $C_{4}$ are the form factors of the proton defined by

$\displaystyle C_{0}(q)$ $\displaystyle=$ $\displaystyle\frac{1}{Z}\int
d^{3}r\rho^{p}_{0}(r)j_{0}(qr),$ (8) $\displaystyle C_{2}(q)$ $\displaystyle=$
$\displaystyle\frac{1}{Z}\sqrt{\frac{10}{7}}\int
d^{3}r\frac{1}{4\pi}|R^{p}_{d5/2}(r)|^{2}j_{2}(qr),$ (9) $\displaystyle
C_{4}(q)$ $\displaystyle=$ $\displaystyle\frac{1}{Z}\sqrt{\frac{2}{7}}\int
d^{3}r\frac{1}{4\pi}|R^{p}_{d5/2}(r)|^{2}j_{4}(qr),$ (10)

where $\rho^{p}_{0}(r)$ is the proton distribution in the spherical
approximation, and $R^{p}_{d5/2}(r)$ is the radial function of the proton wave
function in the $d_{5/2}$ state. Weak counterparts of the form factors are

$\displaystyle C^{W}_{0}(q)$ $\displaystyle=$
$\displaystyle\frac{1}{Q_{W}}[Q_{n}NF_{n}(q)+Q_{p}ZC_{0}(q)],$ (11)
$\displaystyle C^{W}_{2}(q)$ $\displaystyle=$
$\displaystyle\frac{Q_{p}}{Q_{W}}ZC_{2}(q),$ (12) $\displaystyle C^{W}_{4}(q)$
$\displaystyle=$ $\displaystyle\frac{Q_{p}}{Q_{W}}C_{4}(q),$ (13)

where $F_{n}(q)=\frac{1}{N}\int d^{3}r\rho^{n}_{0}(r)e^{i\vec{q}\cdot\vec{r}}$
and $\rho^{n}_{0}(r)$ is the spherical neutron distribution.

Model dependence and validity of approximations can be checked by comparing
the elastic cross section with experimental data. Cross section of the
unpolarized electrons can be approximated as

$\frac{d\sigma}{d\Omega}\simeq\sigma_{m}F_{L}(q)^{2},$ (14)

where $\sigma_{m}$ is the Mott cross section

$\sigma_{m}=\frac{Z^{2}\alpha^{2}}{4E^{2}_{i}}\frac{\cos^{2}\frac{\theta}{2}}{\sin^{4}\frac{\theta}{2}},$
(15)

and the longitudinal form factor is

$F_{L}(q)^{2}=G_{E}(q)^{2}F_{p}(q)^{2}.$ (16)

$G_{E}(q)$ is the electric form factor of the proton, and $F^{2}_{p}$ is
obtained from

$F_{p}(q)^{2}=C_{0}(q)^{2}+C_{2}(q)^{2}+C_{4}(q)^{2}.$ (17)

Exact form of $G_{E}(q)$ is adopted from the quasielastic electron scattering
prc2003 , where $G_{E}(q)=1/(1+q^{2}/\Lambda^{2})^{2}$ and $\Lambda^{2}=0.71$
(GeV/$c$)2.

Figure 1: Square of the form factor (a) $C^{2}_{0}$ and (b) $C^{2}_{2}$ for
the nine EDF models.

Result. Figure 1 shows the squares of $C_{0}$ and $C_{2}$ as functions of
three momentum transfer $q$ for the nine EDF models. Order of magnitude of
each term, and the dependence on $q$ are consistent with the results in
prc2014 ; prc2017 . At $q\lesssim 1$ fm-1, models give very similar result,
but as $q$ increases, dependence on the model becomes evident. In our
calculation of $A_{\rm pv}$, $q\simeq 0.78$ fm-1, so the result is expected to
be weakly dependent on the model.

Figure 2: Cross sections for the elastic scattering of electrons off 27Al at
$E=250$ MeV. Experimental data are taken from prc1974 .

Figure 2 presents the cross section in the elastic electron scattering on 27Al
at $E=250$ MeV as a function of the laboratory angle $\theta$. Experimental
data are available in prc1974 . Up to 50 degrees, eight models give very
similar result and agree well with data, but the LNS model overshoots the
experiment slightly. From about middle of 50 degrees, model dependence becomes
transparent. Models are divided into four groups, and the grouping persists to
about 64 degrees. SkI3 and SkM* are located in the lowest place, and KIDS0,
KIDS0-m*97, MSk7 and SLy4 models are overlapping very closely above the SkI3
and SkM* group. KDE0v1 and SkMP models show the best agreement to data until
58 degrees, and LNS model is located at the highest position. Angle that
corresponds to $q=0.78$ fm-1 at which $A_{\rm pv}$ is calculated is about 40
degrees. At this angle, all the models agree well with data and model
dependence is negligible.

| SkI3 | SkMP | SLy4 | KDE0v1 | SkM* | LNS | KIDS0-m*97 | KIDS0 | MSk7 | Exp.
---|---|---|---|---|---|---|---|---|---|---
| constant gap pairing Eq. (1) |
$E/A$ | 7.782 | 8.651 | 7.997 | 8.564 | 8.461 | 8.562 | 8.200 | 8.273 | 8.288 | 8.332
$R_{n}$ | 3.028 | 2.939 | 3.025 | 2.941 | 2.967 | 2.831 | 2.994 | 2.985 | 2.989 |
$R_{p}$ | 3.014 | 2.936 | 3.020 | 2.938 | 2.966 | 2.823 | 2.981 | 2.974 | 2.985 |
$R_{c}$ | 3.115 | 3.035 | 3.121 | 3.038 | 3.066 | 2.937 | 3.080 | 3.072 | 3.083 | 3.061
$R_{np}/10^{-2}$ | 1.32 | 0.31 | 0.43 | 0.28 | 0.10 | 0.82 | 1.30 | 1.07 | 0.37 | $-4\pm 12$
$A_{\rm pv}/10^{-6}$ | 2.068 | 2.089 | 2.086 | 2.090 | 2.092 | 2.093 | 2.076 | 2.079 | 2.091 | 1.89 – 2.43
| no pairing |
$R_{np}/10^{-2}$ | $-3.94$ | $-3.31$ | $-3.57$ | $-3.82$ | $-3.59$ | $-3.81$ | $-3.17$ | $-3.38$ | $-3.54$ |
| constant force pairing Eq. (2) |
$R_{np}/10^{-2}$ | 1.38 | 0.38 | 0.51 | 0.28 | 0.14 | 0.92 | 1.33 | 1.13 | 0.42 |
$A_{\rm pv}/10^{-6}$ | 2.067 | 2.087 | 2.084 | 2.089 | 2.091 | 2.092 | 2.073 | 2.076 | 2.090 |
Table 2: Model results for the properties of 27Al and parity-violating
asymmetry. Binding energy per nucleon $E/A$ is in MeV, and radii are in fm.
Experimental values of $E/A$ and $R_{c}$ are taken from AME2020 ame2020  and
those of $R_{np}$ and $A_{\rm pv}$ are from prl2022 . Two rows in the bottom
show the result with constant force form of the pairing.

Table 2 displays the results for the properties of 27Al and $A_{\rm pv}$.
Table is divided into three sections: upper section corresponds to the result
with constant gap formula Eq. (1), middle section is without pairing, and the
lowest section is with constant force formula Eq. (2). Data of the binding
energy per nucleon ($E/A$) and the charge radius ($R_{c}$) are taken from
AME2020 ame2020 , and those of $R_{np}$ and $A_{\rm pv}$ are from prl2022 .
$R_{c}$ are also furnished with good agreement to the experimental data.
Deviations from the data defined by

$D_{\cal O}=\left|\frac{{\cal O}_{\rm cal}-{\cal O}_{\rm exp}}{{\cal O}_{\rm
exp}}\right|$ (18)

are less than 1 % in six models for $R_{c}$, but for $E/A$ only two models
KIDS0 and MSk7 satisfy the accuracy. Neutron radius $R_{n}$ takes a range
2.84–3.03 fm, which is comparable to the values obtained from the PV
experiment 2.77–3.01 fm prl2022 . Model dependence of $R_{n}$ is non-
negligible, but with the models satisfying $D_{R_{c}}<1$ %, $R_{n}$ is
constrained in the range 2.94–2.99 fm, so the uncertainty due to the model
reduces significantly.

For the skin thickness, we obtain the result both with and without pairing.
Without pairing the skin thickness is negative in all the models, and the
magnitudes are similar. Including the pairing force, $R_{np}$ becomes positive
and determined in a range $0.001\lesssim R_{np}\lesssim 0.014$ fm. Our result
is consistent with the result of PV experiment, and constrains $R_{np}$ in a
narrower range. Dependence on the model is, contrary to $R_{n}$, not affected
by the accuracy of $R_{c}$, so the six models that satisfy $D_{R_{c}}<1$ %
have the same range of $R_{np}$ as the nine models.

Parity-violating asymmetry is obtained within a range $2.07\times
10^{-6}\lesssim A_{\rm pv}\lesssim 2.09\times 10^{-6}$. There is no particular
correlation with the effective mass or the symmetry energy. The result is
almost a constant, so we could not extract the range of $R_{n}$ from the
correlation between $A_{\rm pv}$ and $R_{n}$ as is done in prl2022 . Survey
with nine EDF models shows that $R_{np}$ and $A_{\rm pv}$ of 27Al are
determined precisely and model-independently. They are weakly correlated to
the effective mass and the symmetry energy.

With constant force pairing formula in Eq. (2), $R_{np}$ and $A_{\rm pv}$ show
slight increase and decrease, respectively, compared to the result of the
constant gap. However, the ranges of these quantities are identical in both
constant gap and constant force. To the extent we consider, $R_{np}$ and
$A_{\rm pv}$ turn out to be insensitive to the form of pairing force.

Summary. Measurement of the PV asymmetry in the elastic electron scattering on
27Al was performed by the Qweak Collaboration, and the result has been
reported recently. The experimental result agrees well with the theoretical
prediction calculated with the FSUGold RMF model. Neutron skin thickness
estimated from the measured $A_{\rm pv}$ is small enough to be consistent with
$N\simeq Z$ of 27Al. However, the value is distributed in both negative and
positive regions, and it is not clear what is the dominant source that
constrains $R_{np}$ in a narrow range. Attempting to find answers to the
questions, we performed a survey in the nonrelativistic model space. We
focused on the correlation of $R_{np}$ and $A_{\rm pv}$ with the effective
mass, symmetry energy and pairing force.

With the nine EDF models, we obtain $0.001\lesssim R_{np}\lesssim 0.014$ fm.
It is remarkably independent of the effective mass, symmetry energy and the
form of pairing force. The result is consistent with the range of $R_{np}$
extracted from the PV experiment, but in our result, negative values are
excluded and upper limit is much lower than the empirical value. We note that
the radii of the neutron and the proton are non-negligibly model-dependent,
but their difference $R_{np}$ falls into a narrow range. We found that the
pairing force plays a crucial role in making $R_{np}$ positive and reside in a
narrow range. $R_{np}$ calculated without pairing force is consistently
negative and the magnitude is marginally dependent on the model. Including the
pairing force, neutrons are expelled outward and $R_{np}$ becomes positive. It
is remarked in prc2014  that “Very likely theory makes a sharp prediction for
$R_{n}-R_{p}$ for 27Al that is essentially independent of the density
dependence of the symmetry energy”. Our results confirm the guess.

More interesting is the result of $A_{\rm pv}$, which takes values in
$2.068$–$2.093\times 10^{-6}$. There is no correlation with effective mass,
symmetry energy and pairing force. This behavior is different from the result
of RMF models in which $A_{\rm pv}$ takes a range $2.10$–$2.22\times 10^{-6}$
prl2022 . Accurate determination of $A_{\rm pv}$ from either theory or
experiment could reduce the uncertainty arising from the nuclear background,
and make the measurement of weak charge more precise.

Acknowledgments. This work was supported by the National Research Foundation
of Korea (NRF) grant funded by the Korea government (No. 2018R1A5A1025563 and
No. 2020R1F1A1052495).

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