Multi-step Direct Reaction Models and Collectivity
69
The Blomqvist–Molinari formula [20] is used for the harmonic oscillator parameter
b =
197.33
√
940 × ¯
hω
fm,
¯
hω =
45A
−1/3
− 25A
−2/3
MeV.
3 Results
We take the 3 − excited states of the double-magic target Ni 56 (A= 56, and
Z= 28) for the particle–hole response function analysis. We show in Fig. 1 the
contribution of a low-energy p-h component along the excitation energy spectrum.
We immediately notice that this mode dominates the lower (left side) energy part
and has a long tail toward more energetic states. A Gaussian function fit to it
gives g 0 = 0.67, σ = 2.5 MeV, and E 0 = 12.12 MeV, while the Breit–Wigner
provides w 0 = 0.44, γ = 0.82 MeV, and E 0 = 13.12 MeV. The former furnishes a
better description of the large contributions but ignores the important contributions
from the high energy part of the spectrum. One may notice that the peaked BW
distribution has its mean value shift to the right compared to the non-interacting
energy component. In addition, we also attempt to adjust the exponential function
of Eq. (4) to the tail of the distribution with e 0 = 5.0 × 10 −5 , β = 20.0 MeV,
and E 0 = 25.0 MeV. This part of the histogram seems to be well described by
both the BW and the exponential distributions. As a more general representation for
the strength function, we propose a linear combination of the Gaussian function,
accounting for the lower energy part of the spectra, plus either the BW or the
exponential distributions for a better approximation of the higher energy tail of the
histogram. With the BW tail we have
F (E) = G(E) + BW(E),
(6)
where the best fit is given by g 0 = 0.92, w 0 = 0.04, σ = γ = 2.5 MeV with a
mean value for both functions E 0 = 12.12 MeV. If an exponential approximation is
used with
K(E) = G(E) + Exp(E),
(7)
we obtain g 0 = 1 and e 0 = 5.5 × 10 −5 with the other parameters held constant.
Figure 2 presents a comparison of the two cases and the histogram data. The very
final part of the tail of the distribution is better reproduced by the BW curve (K)
while the exponential (F) fits nicely in the middle part of the spectra.
Before closing, we present in Fig. 3 the particle–hole spectrum of proton states
obtained with the simplified model described in Sect. 2.2. The components are
formed by all quasi-bound p-h pairs with particle energies lying below the sum of
the Coulomb plus centrifugal barrier. This model permits a very large basis, which
69
The Blomqvist–Molinari formula [20] is used for the harmonic oscillator parameter
b =
197.33
√
940 × ¯
hω
fm,
¯
hω =
45A
−1/3
− 25A
−2/3
MeV.
3 Results
We take the 3 − excited states of the double-magic target Ni 56 (A= 56, and
Z= 28) for the particle–hole response function analysis. We show in Fig. 1 the
contribution of a low-energy p-h component along the excitation energy spectrum.
We immediately notice that this mode dominates the lower (left side) energy part
and has a long tail toward more energetic states. A Gaussian function fit to it
gives g 0 = 0.67, σ = 2.5 MeV, and E 0 = 12.12 MeV, while the Breit–Wigner
provides w 0 = 0.44, γ = 0.82 MeV, and E 0 = 13.12 MeV. The former furnishes a
better description of the large contributions but ignores the important contributions
from the high energy part of the spectrum. One may notice that the peaked BW
distribution has its mean value shift to the right compared to the non-interacting
energy component. In addition, we also attempt to adjust the exponential function
of Eq. (4) to the tail of the distribution with e 0 = 5.0 × 10 −5 , β = 20.0 MeV,
and E 0 = 25.0 MeV. This part of the histogram seems to be well described by
both the BW and the exponential distributions. As a more general representation for
the strength function, we propose a linear combination of the Gaussian function,
accounting for the lower energy part of the spectra, plus either the BW or the
exponential distributions for a better approximation of the higher energy tail of the
histogram. With the BW tail we have
F (E) = G(E) + BW(E),
(6)
where the best fit is given by g 0 = 0.92, w 0 = 0.04, σ = γ = 2.5 MeV with a
mean value for both functions E 0 = 12.12 MeV. If an exponential approximation is
used with
K(E) = G(E) + Exp(E),
(7)
we obtain g 0 = 1 and e 0 = 5.5 × 10 −5 with the other parameters held constant.
Figure 2 presents a comparison of the two cases and the histogram data. The very
final part of the tail of the distribution is better reproduced by the BW curve (K)
while the exponential (F) fits nicely in the middle part of the spectra.
Before closing, we present in Fig. 3 the particle–hole spectrum of proton states
obtained with the simplified model described in Sect. 2.2. The components are
formed by all quasi-bound p-h pairs with particle energies lying below the sum of
the Coulomb plus centrifugal barrier. This model permits a very large basis, which