I. HIGHLIGHTS OF THE YEARRIKEN Accel. Prog. Rep. 54 (2021)

RIKEN Accel. Prog. Rep. 54 (2021)

Reaction cross sections on a deuteron as a probe of nuclear radii†
W. Horiuchi,∗1 Y. Suzuki,∗2,∗3 T. Uesaka,∗3,∗4 and M. Miwa∗3,∗4

where r = (s, z) with z being the beam direction.
In most measurements, the interaction cross section
σd:I is observed but not σd . Since σd includes all inelastic cross sections, σd > σd:I always holds. However, acalculation of σd:I demands all bound-state wave functions of the projectile, which is diﬃcult in general. For
the deuteron target, provided the projectile has only
one bound state, i.e., its ground state, one can evaluthe same inputs required to evaluate
ate σd − σd:I with
σd as ∆0 σ = db (P0 (b) − Pd (b)) with
 P
2P
11

P0 (b) = dr |ϕd (r)|2 eiχp (b+ 2 s)+iχn (b− 2 s)  . (2)If the projectile has more than one bound state, ∆0 σgives the lower bound of σd − σd:I .
Figure 1 displays σd , σp , and σn for 30 Ne as a†∗1∗2∗3∗4Condensed from the article in Phys. Rev. C 102, 054601(2020)
Department of Physics, Hokkaido University
Department of Physics, Niigata University
RIKEN Nishina Center
Department of Physics, Saitama University

1400

Cross sections (mb)

Total reaction or interaction cross section measurement has been used as a standard tool to determine
the nuclear radii of unstable nuclei. The total reaction
cross section of a proton target is known to exhibit
strong incident energy dependence that can be used todeduce both the neutron and proton radii.1,2) A neutron target may also be useful for the structual studyof the unstable nuclei as it has a diﬀerent sensitivity
compared to that of the proton target but no neutron
target exists. Since the deuteron is composed of neutrons and protons, the total reaction cross section ona deuteron target must include both information on
the nucleus-neutron and the nucleus-proton scattering
proﬁles.
To describe high-energy nucleus-deuteron reactions,
we employ the Glauber model,3) wherein the nucleusnucleon total reaction cross section σN (N = n, p) can P 2
be obtained by σN = db (1 − eiχN (b)  ). Under
the optical-limit approximation, the optical phase-shift
Pfunction eiχN (b) at the impact parameter vector b canbe evaluated using the projectile’s density and nucleonnucleon (NN ) scattering proﬁles. A unique advantageof the deuteron target is that one can calculate thephase-shift function accurately using its ground-state
wave function ϕd (r). The nucleus-deuteron total reaction cross section σd = db (1 − Pd (b)), and it can beobtained with2PP11Pd (b) =  dr |ϕd (r)|2 eiχp (b+ 2 s)+iχn (b− 2 s)  , (1)1200

30Ne1000

σdσpσnσd:IΔ0σ (x5)800600400(x5)2000100

1000

Incident energy (MeV/nucleon)

Fig. 1. Various cross sections for 30 Ne adopted from theoriginal paper. See text for details.

function of incident energy. σd is always signiﬁcantly
smaller than σp + σn by about 70–90% of σp + σn due
to the “eclipse” of the constituent neutron and proton.3) The energy dependence of these cross sections
follows that of the NN total cross section. The information of the nucleus-neutron scattering proﬁle isincluded in σd . As already mentioned, the deuterontarget has the advantages that the upper bound of theinteraction cross section can be evaluated reliably using the deuteron wave function. Further, Fig. 1 displays the upper bound of the interaction cross section
σd:I and ∆0 σ for 30 Ne. ∆0 σ has at maximum 60–
70 mb at around 80 MeV/nucleon, which is about 6%of σd . In addition, ∆0 σ decreases with increasing incident energy. The ratio ∆0 σ/σd becomes at most few
percent beyond 300 MeV/nucleon. For the unstable
nuclei near the dripline that has only one bound state,a reliable interaction cross section can be obtained,
and this greatly improves the accuracy of the radius
extraction. We conclude that measuring the total reaction cross sections on both deuteron and proton targets is the most unambiguous and promising approachto determine the neutron and proton radii of unstable
nuclei.
References
1) W. Horiuchi, Y. Suzuki, T. Inakura, Phys. Rev. C 89,011601(R) (2014).
2) W. Horiuchi, S. Hatakeyama, S. Ebata, Y. Suzuki,Phys. Rev. C 93, 044611 (2016).
3) R. J. Glauber, Lectures in Theoretical Physics, edited byW. E. Brittin, L. G. Dunham (Interscience, New York,1959), Vol. 1, p. 315.

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