II-2. Nuclear Physics (Theory)

RIKEN Accel. Prog. Rep. 55 (2022)

RIKEN Accel. Prog. Rep. 55 (2022)

Second and fourth moments of the charge density and neutron-skinthickness of atomic nuclei†
T. Naito,∗1,∗2 G. Colò,∗3,∗4 H. Z. Liang,∗1,∗2 and X. Roca-Maza∗3,∗4
Recently, the fourth moment of the charge distribution has been highlighted as a possible proxy to access information of the neutron root-mean-square radius.1–3) This is because the neutron distributions ρn ofatomic nuclei do contribute to their charge density distributions ρch . In other words, precise measurementsof ρch may provide information on ρn as well as ρp .
For 2 instance,Ref.
  the feasibility to extract 3) showedr n using r2 ch and r4 ch . This, however, relies
on a correlation for speciﬁc nuclei based on a speciﬁctype of models, and hence, it is questionable whether
that method can be applied, in general.
To answer this question, in this paper, we discuss
the feasibility of extracting r2 n from the second andfourth moments of ρch , avoiding as much as possible
the use of model-induced correlations. We also explore
a method to extract the neutron-skin thickness by employing the information of ρch of two neighboring eveneven isotopes to cancel a large part of the spin-orbit
contributions and reduce the uncertainty due to thenucleon form factors and the pairing correlation.
We will show that the key issue is to extract r2 n from
 
 
 2
r ch and r4 ch to accurately determine r4 p .
The second and fourth moments of ρch readwhere Nτ = Z for the proton (τ = p) distribution
and Nτ = N for the neutron (τ = n) distribution, κτis the anomalous magnetic moment of the nucleon τ ,2and rMτ
is the second moment of magnetic distributionof the nucleon τ . The index a = (n, κ, m) is the setof quantum numbers of a single-particle orbital, theoccupation number of which is Naτ . It is also assumed
that the second moment of a single-particle orbital isapproximately equal to r2 τ , which is probably a good
approximation except in weaklysystems.
 bound
We then consider whether r2 n can be extracted  
from experimental data of r2 ch and r4 ch by using
Eqs. (1a) and (1b). Our goal was to reduce model
assumptions to a minimum. To this end, we have discussed in detail two contributions
to the neutron mo
ment: rn SOτ and r4 p . As for the latter, we have 
seen that it can be related to r2 p in a quite robust
manner. Therefore, we deem that we have been able
tothe mildest assumptions under which the 2determiner n of a single isotope can be extracted.
Our main result has been the introduction of a novel
method to extract r2 n from ρch using the information of two neighboring even-even nuclei. The uncertainties due to the nucleon form factors and the approximation for rn SOτ are suppressed, whereas theuncertainties introduced by the pairing are negligible.
We advocate that this method, namely the consideration of two neighboring isotopes, is more reliable.
Despite our eﬀorts, we conclude that the main obstacle to an accurate determination  of the neutron radiusis the contribution from r4 p . Even if it is strongly 
correlated to r2 p , the resulting uncertainty cannot beneglected. This uncertainty is strongly enhanced when
propagated from the fourth moment to the neutron
radius. Eventually, extracting theneutron radiusor
the neutron-skin thickness from r2 ch and r4 ch does
not seem to be feasible based on the present discussion. Despite these pessimistic conclusions, the equations derived in this paper may be useful for further
understanding and investigation and even more usefulif, in the future, a clever way to better determine theproton fourth moment can be envisaged.

 
 2
N 22+ rEn
r ch = r2 p + rEp
Z 2

N 2
+ r SOp +(1a)r SOn ,Z  2 4
10N 2  2
2r nrEpr p + rEn
r ch = r4 p +3Z
 N 4N  44+ r4 SOp ++ rEn+ rEp
r SOn ,
ZZ
(1b)
24
where rEτ
and rEτ
are the second and fourth momentsof the charge distribution of the nucleon τ , respectively.
The spin-orbit contributions rn SOτ read 2κτNaτ l · σ ,(2a)r SOτ  2
Mτ Nτ a∈occ

 4
r SOτ 2κτ  2 10r2 − rEτκτr τ + Mτ
+2Mτ N τ 5
32Mτ2Naτ l · σ ,(2b)×References
1) H. Kurasawa, T. Suzuki,Prog. Theor. Exp.
Phys. 2019, 113D01 (2019).
2) P. -G. Reinhard et al., Phys. Rev. C 101, 021301 (2020).
3) H. Kurasawa et al., Prog. Theor. Exp. Phys. 2021,013D02 (2021).

a∈occ
†∗1
∗2
∗3
∗4Condensed from the article in Phys. Rev. C 104, 024316(2021)
Department of Physics, University of Tokyo
RIKEN Nishina Center
Dipartimento di Fisica, Università degli Studi di Milano
INFN, Sezione di Milano- 41 -1