Level density of 2${}^{+}$ states in ${}^{40}$Ca from high-energy-resolution ($p$,$p^{\prime }$) experiments

The level density of 2${}^{+}$ states in ${}^{40}$Ca has been extracted in the energy region of the isoscalar giant quadrupole resonance from a fluctuation analysis of high-energy-resolution ($p,p^{\prime }$) data taken at incident energies of 200 MeV at the K600 magnetic spectrometer of iThemba LABS, South Africa. Quasifree-scattering cross sections were calculated to estimate their role as a background contribution to the spectra and found to be small. The shape of the background was determined from the discrete wavelet transform of the spectra using a biorthogonal wavelet function normalized at the lowest particle separation threshold. The experimental results are compared to widely used phenomenological and microscopic models.

Figure 1

Excitation energy spectra for ${}^{40}$Ca for $E_x\approx 6$–30 MeV at scattering angles ${\theta }_{\mathrm{Lab}}=7^{\circ }$, 11${}^{\circ }$, and 15${}^{\circ }$. Note that the ISGQR is expected to be most strongly excited at ${\theta }_{\mathrm{Lab}}$ $=$ 11${}^{\circ }$. See also Ref. [20].

Figure 2

Results of the DWIA calculations for quasifree reaction cross sections (SUM QF) of the ${}^{40}$Ca($p,p^{\prime }$)reaction at $E_p=200$ MeV and ${\theta }_{\mathrm{Lab}}={11}^{\circ }$ (solid line) and their decomposition into contributions from $(p,2p)$ (dashed line) and $(p,pn)$ (dotted line) reactions.

Figure 3

Quasifree background contribution (SUM QF) to the spectrum of the ${}^{40}$Ca($p,p^{\prime }$) reaction at $E_p=200$ MeV and ${\theta }_{\mathrm{Lab}}={11}^{\circ }$.

Figure 4

Examples of wavelet functions used in the present analysis. (a) BIOR3.9 wavelet. (b) BIOR6.8 wavelet.

Figure 5

DWT decomposition of the excitation energy spectrum of the ${}^{40}$Ca($p,p^{\prime }$) reaction measured at $E_p=200$ MeV for ${\theta }_{\mathrm{Lab}}={11}^{\circ }$ into approximations $Ai$ and details $Di$.

Figure 6

Comparison of the spectrum of the ${}^{40}$Ca($p,p^{\prime }$) reaction measured at $E_p=200$ MeV for (a) ${\theta }_{\mathrm{Lab}}={11}^{\circ }$ and (b) ${\theta }_{\mathrm{Lab}}={15}^{\circ }$ to approximation $A9$ (dashed-dotted lines) determined from the DWT (cf. Fig. 5). The shape of the ISGQR is well reproduced. The resulting background (approximation $A10$) is shown for different biorthogonal wavelet functions: BIOR6.8 (solid lines), BIOR5.5 (dotted lines), BIOR4.4 (dashed lines), and BIOR3.9 (dashed-doubly dotted lines).

Figure 7

Illustration of the autocorrelation analysis: (a) experimental ${}^{40}$Ca($p,p^{\prime }$) spectrum including background obtained with the use of the discrete wavelet transform; (b) background-subtracted smoothed spectra $g$($E_{\mathrm{x}}$) and $g_{>}$($E_{\mathrm{x}}$), respectively; (c) stationary fluctuating spectrum $d$$\left(E_{\mathrm{x}}\right)$ obtained by dividing the two smoothed spectra; and (d) autocorrelation functions from experiment and Eq. (4), shown as the solid and dashed line, respectively.

Figure 8

Level density of 2${}^{+}$ states in ${}^{40}$Ca extracted from the ($p,p^{\prime }$) data (filled symbols) compared to model predictions: BSFG-Rauscher [13] (dashed line), BSFG–von Egidy [41] (solid line), HF-BCS [15] (dotted line), and HFB [16] (triangles; the connecting line is a guide for the eye only).