Statistical ensembles and fragmentation of finite nuclei

Statistical models based on different ensembles are very commonly used to describe the nuclear multifragmentation reaction in heavy ion collisions at intermediate energies. Canonical model results are more appropriate for finite nuclei calculations while those obtained from the grand canonical ones are more easily calculable. A transformation relation has been worked out for converting results of finite nuclei from grand canonical to canonical and vice versa. The formula shows that, irrespective of the particle number fluctuation in the grand canonical ensemble, exact canonical results can be recovered for observables varying linearly or quadratically with the number of particles. This result is of great significance since the baryon and charge conservation constraints can make the exact canonical calculations extremely difficult in general. This concept developed in this work can be extended in future for transformation to ensembles where analytical solutions do not exist. The applicability of certain equations (isoscaling, etc.) in the regime of finite nuclei can also be tested using this transformation relation.

Figure 1

Grand canonical proton and neutron number distributions for fragmenting source $\langle Z_0\rangle =28$, $\langle N_0\rangle =30$ at temperature $T=8$ MeV.

Figure 2

Mass distribution of fragments produced from disassembly of a particular source of mass number 58 and proton number 28, calculated from canonical (black dotted line) and grand canonical (blue dashed line) models for two different temperatures, $T=6$ MeV (left panel) and 8 MeV (right panel). The red solid lines represent the canonical result obtained from the grand canonical model by using Eq. (15).

Figure 3

Variation of average size of the largest cluster ($\langle Z_{\max }\rangle$) with temperature ($T$) for the fragmenting system of charge 28 and mass 58 calculated from canonical (black dotted line) and grand canonical (blue dashed line) models. The red solid lines represent the canonical result obtained from the grand canonical model by using Eq. (15).

Figure 4

Variation of ${\sigma }_n^2$ (red solid line) with temperature ($T$) for the fragmenting system of charge 28 and mass 58.

Figure 5

Multiplicities of $Z=7$ (left panels) and $Z=12$ (right panels) isotopes produced from two fragmenting systems of the same atomic number 28, but different mass numbers 58 (upper panels) and 64 (lower panels), calculated from canonical (black dotted line) and grand canonical (blue dashed line) models. The freeze-out temperature for both the system is $T=$ 8 MeV. The red triangles represent the canonical result obtained from the grand canonical model by using Eq. (15).

Figure 6

Ratios ($R_{21}$) of multiplicities of fragments $(N,Z)$ where mass and charge of the fragmenting system for reaction 1 are 58 and 28 respectively and those for reaction 2 are 64 and 28. The freeze-out temperature for both the fragmenting systems is $T=$ 8 MeV. The left panel shows the ratios as a function of neutron number $N$ for fixed $Z$ values, while the right panel displays the ratios as a function of proton number $Z$ for fixed neutron numbers ($N$) calculated from canonical (black dotted line) and grand canonical (blue dashed line) models. The red triangles represent the canonical result obtained from the grand canonical model by using Eq. (15).