Level densities and thermodynamical quantities of heated ${}^{93-98}\mathrm{Mo}$ isotopes

Level densities for ${}^{93-98}\mathrm{Mo}$ have been extracted using the (${}^3\mathrm{He}$,$\alpha \gamma$) and (${}^3\mathrm{He}$,${}^3\mathrm{He}$${}^{\text{'}}\gamma$) reactions. From the level densities thermodynamical quantities such as temperature and heat capacity can be deduced. Data have been analyzed by utilizing both the microcanonical and the canonical ensemble. Structures in the microcanonical temperature are consistent with the breaking of nucleon Cooper pairs. The $\mathsf{S}$ shape of the heat capacity curves found within the canonical ensemble is interpreted as consistent with a pairing phase transition with a critical temperature for the quenching of pairing correlations at $T_c~0.7-1.0$MeV.

Figure 1

Charged ejectile spectra for ${}^{93-98}\mathrm{Mo}$ in coincidence with γ-rays, labeled by the product nuclei. The arrows indicate the neutron-separation energy $B_n$.

Figure 2

γ-ray multiplicity evaluated by Eq. (1) versus excitation energy. The individual spectra are labeled by the product nuclei. Solid and dashed lines represent (${}^3\mathrm{He}$,α) and (${}^3\mathrm{He}$,${}^3\mathrm{He}$${}^{\text{'}}$) reactions, respectively.

Figure 3

Normalization procedure of the experimental level density (data points) of ${}^{97}\mathrm{Mo}$. The data points between the arrows in the upper panel are normalized to known levels at low excitation energy (histograms). In the lower panel they are normalized to the level density at the neutron-separation energy (open triangle) using a Fermi-gas extrapolation (line).

Figure 4

Level densities at the neutron-separation energy. The unknown level density of ${}^{94}\mathrm{Mo}$ (open circle) is estimated from the slope of the data points of the odd-mass molybdenum isotopes.

Figure 5

Level density of nine molybdenum isotopes. The histograms represent levels from spectroscopy [39]. A straight line is drawn from these levels to the level density at the neutron-separation energy that is determined by average neutron-resonance spacings. The line represents the constant-temperature level-density formula (see text).

Figure 6

Normalized level densities for ${}^{93-98}\mathrm{Mo}$. The open and filled circles are data from the (${}^3\mathrm{He}$,α) and (${}^3\mathrm{He}$,${}^3\mathrm{He}$${}^{\text{'}}$) reactions, respectively.

Figure 7

Experimental entropy for ${}^{93,94}\mathrm{Mo}$ (upper panel) and ${}^{97,98}\mathrm{Mo}$ (lower panel) as function of excitation energy E.

Figure 8

Model calculation for nuclei around ${}^{96}\mathrm{Mo}$. The four panels show the free energy F, the entropy S, the thermal excitation energy $\langle E\rangle$, and the heat capacity $C_V$ as a function of temperature T. The arrow at $T_m~0.9$ MeV indicates the local maximum of $C_V$ where the pair-breaking process takes place in the even-even system. The same parameter set is used for even-even (solid lines), odd (dashed lines), and odd-odd systems (dash-dotted lines).

Figure 9

Calculated level density of ${}^{98}\mathrm{Mo}$ (solid line) and ${}^{97}\mathrm{Mo}$ (dashed line) as function of average excitation energy $\langle E\rangle$. The big open circles and squares are experimental level-density anchor points from Ref. 41. The small filled and open circles are experimental data points measured with the (${}^3\mathrm{He}$,α) and (${}^3\mathrm{He}$,${}^3\mathrm{He}$${}^{\text{'}}$) reactions, respectively for the two isotopes.

Figure 10

Observed heat capacity as functions of temperature in the canonical ensemble for the even ${}^{94,96,98}\mathrm{Mo}$ (upper panel) and odd ${}^{93,95,97}\mathrm{Mo}$ (lower panel) nuclei.

Figure 11

Critical temperature for the quenching of pair correlations for ${}^{93-98}\mathrm{Mo}$ isotopes.

Figure 12

Entropy extracted at excitation energies of 1 and 7 MeV as a function of neutron number N (left panel) and number of available neutrons in the model (right panel) for odd-even (open circles) and even-even (filled circles) molybdenum isotopes.