Ⅱ-2. Nuclear Physics (Theory)
RIKEN
Accel. Prog. Rep. 56 (2023)

RIKEN Accel. Prog. Rep. 56 (2023)

Microscopic collective inertial masses for nuclear reactions in thepresence of the nucleonic eﬀective mass†
K. Wen∗1 and T. Nakatsukasa∗1,∗2
A theoretical description of low-energy nuclear reactions with a solid microscopic foundation is still achallenging subject in nuclear physics. It may provide
us with a deep insight into the reaction mechanisms
and quantum dynamics of many-nucleon systems. Ourstrategy is as follows. First, we ﬁnd a collective subspace spanned by a few selected collective canonicalvariables, which is decoupled from the intrinsic excitations. The collective subspace and collective variables
can be extracted using the adiabatic self-consistent collective coordinate (ASCC) method.1) The collective
subspace in the adiabatic regime is given by a seriesof time-even Slater determinants and generators of thecollective coordinates and momenta locally deﬁned foreach state. We apply the method to low-energy nuclear
reactions, to identify the optimal reaction path andcanonical variables. The self-consistently determinedcollective reaction path is requantized. The procedure
results in the reaction model= Euℓ (R),(1)Jx [MeV-1h2]{}ℓ(ℓ + 1)d1d−++ V (R) uℓ (R)dR 2M (R) dR
2J (R)

nuclei for the velocity-independent mean-ﬁeld potential. We also examined those of the cranking formula,which turned out to be almost identical to the ASCC
mass at R → ∞. However, this is not true in general. In particular, the nonlocal mean-ﬁeld potential
produces the eﬀective mass m∗ /m < 1. In this case,in order to guarantee the Galilean invariance of theenergy density functional, we need densities that aretime-odd with respect to the time-reversal transformation. The time-odd mean ﬁelds play important rolesin the time-dependent dynamics because the time-odd
densities are nonzero in general for the time-dependent
states. Since the ASCC equations are derived fromthe time-dependent mean-ﬁeld theory, they are able
to take into account the eﬀects of the time-odd meanﬁelds in calculations of the inertial masses. We havefound that the ASCC inertial masses reproduce thecorrect asymptotic values, whereas the cranking formula fails to do so.
Figure 1 shows the results for the moments of inertia J (R).
The velocity dependence
in the mean-ﬁeld
6ASCC
Cranking
potential violates the local
Galilean invariance. The
Rigid body5
Point masscalculated
cranking momentsof inertia (dashed line)
are signiﬁcantly
smaller than the rigid-body value Jrig4(dotted line). They are also smaller than the point3
particle value µred R2 (dash-dot line) at large R. In
contrast,2 the ASCC calculation (solid)
includes theα + 16O
residual eﬀects of the time-odd mean ﬁelds, which re1
store the local Galilean invariance, nicely reproducingµred R2 0at R > 7.5 fm. Near the equilibrium state
4.5
5
5.5
6
6.5
77.5
(R ∼ 5.5 fm),
it also
reproduces
therigid-bodyvalue.

where R is the relative distance between the projectile and the target nuclei. The wave function for the
relative motion is given by ψ(R) = ul (R)Ylm (R̂)/R.
The key ingredients of the requantization procedureare the potential V (R), the inertial mass parameterswith respect to the collective coordinates M (R), andthe rotational moment of inertia J (R) in Eq. (1). For
the nuclear reaction, the relative distance R between
two colliding nuclei should be properly chosen collective coordinates in the asymptotic region (R → ∞).
The corresponding inertial mass should be the reduced
mass µred = AP AT m/(AP +AT ), where AP (AT ) is themass number of the projectile (target) nucleus and m isthe bare nucleon mass. The moment of inertia J (R) inEq. (1) also has an asymptotic value, J (R) → µred R2
at R → ∞. Therefore, the theory can be tested byexamining its asymptotic limit. However, the inertial
masses in the interior region where two nuclei toucheach other are highly nontrivial. Thus, a microscopic
theory for calculating the mass over the entire reaction
path is necessary.
In our previous studies,2) we calculated the ASCC
inertial masses for the relative motion between two†∗1∗2

R [fm]

18Jx [MeV-1h2]

161412108
616O + 16O

42
055.566.577.588.5

R [fm]

Fig. 1. Rotational moments of inertia as a function of therelative distance R for 16 O+16 O. See text for details.

References
1) T. Nakatsukasa et al., Rev. Mod. Phys. 88, 045004(2016).
2) K. Wen et al., Phys. Rev. C 94, 054618 (2016); ibid. 96,014610 (2017).

Condensed from the article in Phys. Rev. C 105, 034603(2022)
Center for Computational Sciences, University of Tsukuba
RIKEN Nishina Center- 37 -