Quasiparticle nature of excited states in random-phase approximation

Quantum models of nuclear reactions require an effective way of taking into account the complex coupling of the components of different excited states as well as the nuclear residual interactions of the composite system. This is usually achieved with the use of distributions giving nonuniform weights for each of the modes involved. The response function of the quantum model permits the inclusion of the residual interaction and thus a unified description of collective and single-particle excitations. We analyze the particle-hole nature of the RPA modes of the response function in the context of multistep direct (MSD) nuclear reactions. The energy distribution of the particle-hole states that contribute to the response function is studied. Although many states make small contributions to the low-energy collective states, we show that the energy distribution of the higher-energy states is concentrated around the energies of the noninteracting components. The spreading energy of the strength function is determined for different nuclei, and a general fit accounting for both the collective and noncollective parts of the spectra is proposed. In addition we also test the randomness assumption commonly applied in models of MSD reactions.

Figure 1

(a) Energy deviation $\delta E$ and (b) the state strength ${\left|a_{\mu }^x\right|}^2$ as functions of the excitation energy for different values of the angular momentum and parity for ${}^{56}\mathrm{Ni}$. Panel (c) shows the excitation energy dependence of the width $\sigma$ of the states.

Figure 2

The percentage contribution of states near the noninteracting energy components for $3^{-}$ excited states of ${}^{56}\mathrm{Ni}$. The low energy components contribute more to the collective states, while the higher energy part of the spectra is composed mainly of single particle-hole contributions.

Figure 3

(a) Relative contribution of RPA components to the transition matrix for a collective (a) and two noncollective (b), (c) $3^{-}$ excited states of ${}^{56}\mathrm{Ni}$. The dark dotted line represents the case where only one state is taken into account. Vertical lines represent the ${\left|a_{\mu }\right|}^2$ component for each case. All curves are normalized to their relative maximum value.

Figure 4

(a) The total averaged contribution of the RPA components of the diagonal (solid dark lines) and off-diagonal (dashed gray lines) terms of the RPA components for $3^{-}$ states of ${}^{56}\mathrm{Ni}$: Off diagonals contributions vanish rapidly for noncollective states independently of the angular momentum and parity. (b) The number of states within the excitation energy interval $\mathrm{\Delta }E$ in the statistics: The discrete nature of the low-energy collective states does not permit enough states for a randomness assumption to be valid.

Figure 5

Low energy response function for $3^{-}$ exited states of ${}^{56}\mathrm{Ni}$. The histogram is constructed using the RPA eigenvector components. The $E_{\mu }=11.55$ MeV particle-hole state is represented by the vertical solid line. The dashed curve is the Breit-Wigner fit and its mean value $\overline{E}=12.50$ MeV is represented by the vertical dashed line. The displacement of the mean value of the distribution in comparison to the particle-hole mode is caused by configuration mixing present in the collectivity of the low lying excited states.

Figure 6

Response function for three high energy particle-hole configurations of the $3^{-}$ exited states of ${}^{56}\mathrm{Ni}$. The $E_{\mu }$ particle-hole states and the mean distribution value (vertical lines) coincide: dark denotes $E_{\mu }=33.13$ MeV, gray $E_{\mu }=50.65$ MeV, and brown $E_{\mu }=68.51$ MeV.

Figure 7

Single-particle contribution to the response function in the high energy region of the spectrum of exited states of Fig. 5. The vertical solid dark and gray lines correspond to $E_m=14.33$ MeV proton and $E_m=61.46$ MeV neutron single-particle states, respectively.

Figure 8

The width $\gamma$ of the Breit-Wigner distribution (upper panels) and the energy shift $\delta$ between the mean distribution value and the noninteracting p-h components (lower panels) for the (a), (c) $3^{-}$ and (b), (d) $4^{+}$ excited states of ${}^{56}\mathrm{Ni}$. The squared dotes represent averages around bins of 4.5 MeV, defined by the dotted lines. The standard deviation of the mean value within each bin varies over the spectra: larger values are found for the lower energy part of the spectra while smaller numbers are obtained for intermediate/high energies due to the absence of collective states.

Figure 9

Spreading width of the BW distribution for four different nuclei. Starting from the top: (a) ${}^{16}\mathrm{O}$, (b) ${}^{56}\mathrm{Ni}$, (c) ${}^{90}\mathrm{Zr}$ and (d) ${}^{120}\mathrm{Sn}$. The function in Eq. (18) is adjusted to the data with the coefficients given in Table 1. Dashed and solid lines represent fits with $a_2=0$ and $a_1=0$, respectively.

Figure 10

Energy level spacing distribution $P\left(s\right)$ (upper left panel) of the particle-hole configuration energies $E_{\mu }$, where $s=E_{\mu }-E_{\mu +1}$, and the distribution of the squared interaction amplitude (upper right panel). Both upper plots were obtained using the entire spectrum of particle-hole configurations ($E_{\mu }$). The middle panel shows the $\overline{s}$ (solid squares) and $\overline{v^2}$ (solid circles) computed taking into account states that contribute within a range of $\mathrm{\Delta }{E}_{\mu }=10\mathrm{MeV}$ around each point on the horizontal axis. In the bottom panel is the local spreading width for different regions of the spectrum. The horizontal dashed line was obtained using the mean values of the histograms on the upper panels. The $3^{-}$ excited states of ${}^{56}\mathrm{Ni}$ were used.