Giant dipole resonance built on hot rotating nuclei produced during evaporation of light particles from the ${}^{88}\mathrm{Mo}$ compound nucleus

High-energy giant dipole resonance (GDR) $\gamma$ rays were measured following the decay of the hot, rotating compound nucleus of ${}^{88}\mathrm{Mo}$, produced at excitation energies of 124 and 261 MeV. The reaction ${}^{48}\mathrm{Ti}$ + ${}^{40}\mathrm{Ca}$ at 300 and 600 MeV bombarding energies has been used. The data were analyzed using the statistical model Monte Carlo code gemini++. It allowed extracting the giant dipole resonance parameters by fitting the high-energy $\gamma$-ray spectra. The extracted GDR widths were compared with the available data at lower excitation energy and with theoretical predictions based on (i) The Lublin-Strasbourg drop macroscopic model, supplemented with thermal shape fluctuations analysis, and (ii) The phonon damping model. The theoretical predictions were convoluted with the population matrices of evaporated nuclei from the statistical model gemini++. Also a comparison with the results of a phenomenological expression based on the existing systematics, mainly for lower temperature data, is presented and discussed. A possible onset of a saturation of the GDR width was observed around $T=3$ MeV.

Figure 1

Schematic view of the experimental setup in the ${}^{48}\mathrm{Ti}$ + ${}^{40}\mathrm{Ca}\to {}^{88}\mathrm{Mo}$* experiment.

Figure 2

Correlations between the fast and slow components of CsI(Tl) scintillation signal. The 2D plot shows a very good separation of protons and alpha-particle event distributions. The data are from the ${}^{48}\mathrm{Ti}$ + ${}^{40}\mathrm{Ca}$ reaction at the bombarding energy of 300 MeV.

Figure 3

Angular distribution (in the laboratory) of the evaporation residues at 300 MeV (red) and 600 MeV (black) beam energy. Dashed lines represent emitted residues (gemini++ calculations), while continuous lines correspond to the residues which fulfill experimental detection conditions (gemini++ calculations with event-by-event filtering when residue is entering phoswich detectors).

Figure 4

(a) 2D plot of the $gA$ parameter $(\Delta E$ energy deposit) and ToF (time of flight) dependence for one of the 32 phoswich detectors, with indicated regions for different reaction products [e.g., particles, evaporation residues, fission fragments (f)f, quasifission fragments (qf), etc.]. (b) 2D plot of the $\gamma$-ray energy $\left(E_{\gamma }\right)$ in HECTOR versus the ToF for heavy fragments in the phoswich detector. The data are for the 300 MeV reaction.

Figure 5

Time-of-flight (ToF) spectrum for the HECTOR detectors, measured for the ${}^{48}\mathrm{Ti}$ + ${}^{40}\mathrm{Ca}$ reaction, at the beam energy of 300 MeV. Dashed lines define the gating region for $\gamma$-ray selection.

Figure 6

Measured $\gamma$-ray energy-spectra obtained while gating on the HECTOR ToF (full squares) and on the residues after subtracting the cosmic-ray and Bremsstrahlung contributions (open circles); (a) 300 MeV data, (b) 600 MeV data.

Figure 7

The experimental alpha-particle (left) and proton (right) energy spectra for the beam energies of 300 MeV (top) and 600 MeV (bottom); full circles. They correspond to measurements with the detectors placed at the mean angle of $\theta$ = ${47}^{\circ }$. The full lines show the results of the statistical code gemini++ model calculations.

Figure 8

Left panel: The population matrices, calculated using the gemini++ code, of all nuclei through which the evaporation process of ${}^{88}\mathrm{Mo}$ proceeds, with an imposed condition that in the evaporation chain the residuum is reached. Two-dimensional spectra show the numbers of such nuclei at any given spin and temperature in a logarithmic scale for the reactions (a) at 300 MeV and (b) at 600 MeV. Right panel: Analogous to the left panel, but with an additional condition: Only the nuclei emitting a $\gamma$ ray with $E_{\gamma }>11$ MeV (i.e., from the GDR decay) are considered.

Figure 9

A comparison of the $\gamma$-ray spectra from the ${}^{48}\mathrm{Ti}$ + ${}^{40}\mathrm{Ca}$ reaction, at the beam energies of 300 and 600 MeV, with the results of the gemini++ fit (see text).

Figure 10

The experimental, $Y_{\exp }\left(E_{\gamma }\right)$, and fitted, $L\left(E_{\gamma }\right)$, GDR strength functions for (a) 300 and (b) 600 MeV beam energies.

Figure 11

Distribution of the nuclei in the evaporation process as functions of the nuclear temperature reached after the $\gamma$ decay and the emitted $\gamma$-ray energy; calculated with the code gemini++ for the data at the beam energies of (a) 300 MeV and (b) 600 MeV.

Figure 12

Temperature distributions of the number of nuclei after the GDR emission. The calculations were performed using the gemini++ code with experimental conditions imposed (see the text for details). The blue (dotted) and red (solid) histograms represent the data at the beam energies of 300 and 600 MeV, respectively. The vertical solid lines indicate the average values of the distributions. The dashed and dot-dashed vertical lines show corresponding quartiles used for estimation of temperature uncertainties.

Figure 13

Average mass $\langle A\rangle$ $(z$ axis) of a nucleus emitting GDR $\gamma$ rays during the decay of the ${}^{88}\mathrm{Mo}$ compound nucleus produced at (a) 300 MeV and (b) 600 MeV beam energies as functions of temperature and spin. Calculations were performed using the statistical code gemini++.

Figure 14

The nuclear-shape probability distributions for ${}^{88}\mathrm{Mo}$ at temperatures $T=1$, 2, 3, and 4 MeV and spin $40\hslash$ in panels (a), (b), (c), and (d), respectively. The energy minimization is performed over ${\alpha }_{40},{\alpha }_{60}$, and ${\alpha }_{80}$ and the results are projected onto the $(\beta ,\gamma )$ plane.

Figure 15

The calculated GDR strength functions at various temperatures and spins for the ${}^{88}\mathrm{Mo}$ nucleus, averaged using the shape probability distributions shown in Fig. 14, with ${\Gamma }_0=5$ MeV.

Figure 16

(a) The evaporation widths ${\Gamma }_{\mathrm{EV}}$ calculated using the gemini++ code at various temperatures. (b) Intrinsic width of the GDR: a constant ${\Gamma }_0\left(T\right)={\Gamma }_0(T=0)=5$ MeV (dashed line) and one dependent on the evaporation width ${\Gamma }_{\mathrm{EV}}$, in the form of ${\Gamma }_0\left(T\right)=5$ MeV + $2{\Gamma }_{\mathrm{EV}}$ (solid line).

Figure 17

Left: Comparison of the GDR strength functions extracted from the experimental data and the effective model values based on the macroscopic LSD model with TFM [Eq. (15)], for the beam energies of (a) 300 MeV and (b) 600 MeV. The theoretical GDR strength functions are estimated with a constant (dashed line) and a temperature-dependent (solid line) intrinsic GDR width ${\Gamma }_0$. Right: Analogous to the left panel comparison of experimentally extracted GDR strength functions to the effective model values based on the PDM [Eq. (24)] for beam energies of (c) 300 MeV and (d) 600 MeV.

Figure 18

Temperature dependence of the GDR width obtained as the FWHM of the strength functions. The experimental values (full squares) are shown together with the model predictions by the LSD model with TFM (triangles) and the PDM (circles). The results of the calculations based on the LSD model with TFM by using either ${\Gamma }_0=5$ MeV (full triangles) or ${\Gamma }_0\left(T\right)$ including the lifetimes of the compound nucleus and of nuclei produced in the evaporation process (open triangles) are presented. The experimentally extracted GDR widths are bounded in the shaded areas determined by the width of the experimental temperature distributions and the vertical error bars. The straight lines connecting the theoretical predictions are drawn to guide the eyes.

Figure 19

Measured GDR widths as a function of temperature for nuclei in the mass region of ${}^{88}\mathrm{Mo}$ (full circles) compared with predictions obtained using the expression proposed in [53] based on the experimental systematics (open symbols). The sets of data are selected to correspond to the angular momentum interval $\left(18\text{–}25\right)\hslash$. The data have the following origins: ${}^{86}\mathrm{Mo}$ from [10], ${}^{92}\mathrm{Mo}$ from [54], and ${}^{100}\mathrm{Mo}$ from [11].