92
T. R. Whitehead et al.
potential is then folded with the relevant nuclear density distribution for the isotope
under investigation, calculated using a Skyrme effective interaction fitted to the
nuclear equation of state derived from the same chiral potential. The result is an
energy-dependent nucleon–nucleus optical potential in position space. This local
density approximation (LDA) [6] is known to give a poor description of the optical
potential surface diffuseness, and therefore in the present work we employ an
improved local density approximation (ILDA) that accounts for the non-zero range
of the nuclear force:
U(E, r) I LDA =
1
(t
√
π) 3
U(E, r
)e
−|r−r | 2
t 2
d
3 r
,
(1)
where t is a distance scale associated with the average range of the nucleon–nucleon
interaction. In the present study we vary t within the range 1.15 fm < t < 1.25 fm.
2 Results
We have implemented the nuclear optical potentials described above in the TALYS
reaction code [7]. In the top two rows of Fig. 1 we plot the proton–nucleus differential elastic scattering cross sections at the two energies E = 25, 45 MeV for the
isotopes 40 Ca, 44 Ca, and 48 Ca. Experimental data are shown as the red points, while
the results from the microscopic optical potentials are shown with the blue band.
The uncertainties giving rise to the theoretical error band are obtained by varying
the ILDA range parameters for both the central and spin–orbit components. In the
future we plan to estimate also the uncertainties arising from the choice of chiral
potential by varying the momentum-space cutoff, the order in the chiral expansion,
and the regulating function. We also show in the top two rows of Fig. 1 the results
(green curves) from the Koning–Delaroche phenomenological optical potential as
it is implemented in the TALYS reaction code. We see that the microscopic optical
potentials from chiral effective field theory give an overall reasonable description
of the elastic scattering cross section within the chosen energy regime. However, at
higher energies and larger scattering angles, the description starts to deteriorate.
Comparing the real and imaginary components of the microscopic optical
potential to those from phenomenology, we find excellent agreement in the real
part but the microscopic imaginary optical potential is too strongly absorptive.
This feature is ubiquitous in nuclear matter optical potential calculations and is
the reason why semi-microscopic optical potentials used today implement energydependent strength factors [8]. To test this, we show in the bottom two rows of
Fig. 1, the results from our microscopic optical potentials for which the imaginary
part has been replaced by that from the KD optical potential. We see that this
replacement dramatically improves the differential elastic scattering cross sections
across all energies and target isotopes. In the future we plan to investigate higherorder perturbative contributions to the self-energy and their effect on the imaginary
part of the optical potential.
T. R. Whitehead et al.
potential is then folded with the relevant nuclear density distribution for the isotope
under investigation, calculated using a Skyrme effective interaction fitted to the
nuclear equation of state derived from the same chiral potential. The result is an
energy-dependent nucleon–nucleus optical potential in position space. This local
density approximation (LDA) [6] is known to give a poor description of the optical
potential surface diffuseness, and therefore in the present work we employ an
improved local density approximation (ILDA) that accounts for the non-zero range
of the nuclear force:
U(E, r) I LDA =
1
(t
√
π) 3
U(E, r
)e
−|r−r | 2
t 2
d
3 r
,
(1)
where t is a distance scale associated with the average range of the nucleon–nucleon
interaction. In the present study we vary t within the range 1.15 fm < t < 1.25 fm.
2 Results
We have implemented the nuclear optical potentials described above in the TALYS
reaction code [7]. In the top two rows of Fig. 1 we plot the proton–nucleus differential elastic scattering cross sections at the two energies E = 25, 45 MeV for the
isotopes 40 Ca, 44 Ca, and 48 Ca. Experimental data are shown as the red points, while
the results from the microscopic optical potentials are shown with the blue band.
The uncertainties giving rise to the theoretical error band are obtained by varying
the ILDA range parameters for both the central and spin–orbit components. In the
future we plan to estimate also the uncertainties arising from the choice of chiral
potential by varying the momentum-space cutoff, the order in the chiral expansion,
and the regulating function. We also show in the top two rows of Fig. 1 the results
(green curves) from the Koning–Delaroche phenomenological optical potential as
it is implemented in the TALYS reaction code. We see that the microscopic optical
potentials from chiral effective field theory give an overall reasonable description
of the elastic scattering cross section within the chosen energy regime. However, at
higher energies and larger scattering angles, the description starts to deteriorate.
Comparing the real and imaginary components of the microscopic optical
potential to those from phenomenology, we find excellent agreement in the real
part but the microscopic imaginary optical potential is too strongly absorptive.
This feature is ubiquitous in nuclear matter optical potential calculations and is
the reason why semi-microscopic optical potentials used today implement energydependent strength factors [8]. To test this, we show in the bottom two rows of
Fig. 1, the results from our microscopic optical potentials for which the imaginary
part has been replaced by that from the KD optical potential. We see that this
replacement dramatically improves the differential elastic scattering cross sections
across all energies and target isotopes. In the future we plan to investigate higherorder perturbative contributions to the self-energy and their effect on the imaginary
part of the optical potential.