Statistical approach of nuclear multifragmentation with realistic nuclear equation of state

In CTM, DasGupta_book ; Das it is assumed that statistical equilibrium is attained at freeze-out stage and population of different channels of disintegration is solely decided by statistical weights in the available phase space. The calculation is done for a fixed mass and atomic number, freeze out volume and temperature. In a canonical model Das , the partitioning is done such that all partitions have the correct $A_{0},Z_{0}$ (equivalently $N_{0},Z_{0}$). The canonical partition function is given by

where the sum is over all possible channels of break-up (the number of such channels is enormous) satisfying $N_{0}=\sum N\times n_{N,Z}$ and $Z_{0}=\sum Z\times n_{N,Z}$; $\omega_{N,Z}$ is the partition function of the composite with $N$ neutrons & $Z$ protons and $n_{NZ}$ is its multiplicity. The partition function $Q_{N_{0},Z_{0}}$ is calculated by applying a recursion relation Chase . From Eq. (1), the average number of composites with $N$ neutrons and $Z$ protons can be expressed as,

The partition function of a composite having $N$ neutrons and $Z$ protons is a product of two parts: one is due to the the translational motion and the other is the intrinsic partition function of the composite:

where $V$ is the volume available for translational motion, which can be expressed as $V=V_{f}-V_{0}$ , where $V_{0}$ is the normal volume of nucleus with $Z_{0}$ protons and $N_{0}$ neutrons. $z_{N,Z}(int)$ is the internal partition function, the proton and the neutron are fundamental building blocks, thus $z_{1,0}(int)=z_{0,1}(int)=2$, where 2 takes care of the spin degeneracy. For ²H, ³H, ³He, ⁴He, ⁵He and ⁶He, $z_{N,Z}(int)=(2s_{N,Z}+1)\exp[-\beta E_{N,Z}(gr)]$ where $\beta=1/T,E_{N,Z}(gr)$ is the ground-state energy of the composite and $(2s_{N,Z}+1)$ is the experimental spin degeneracy of the ground state. Excited states for these very low-mass nuclei are not included.

Fig. 2 represents the mass distribution obtained from CTM calculation with conventional liquid drop model and more realistic compressible liquid drop approach with Sly5 parameters at three different temperatures 3 MeV, 5 MeV and 8 MeV from disassembly of $Z_{0}$=75, $A_{0}$=168 at constant free-out volume 6$V_{0}$. The intermediate mass fragment (IMF) multiplicity ($M_{IMF}$) is also an important observable in nuclear multifragmentation process which is measured both experimentally and theoretically in many situations Peaslee ; Ogilvie ; Tsang ; Ogul ; Mallik3 ; Mallik16 ; Mallik19 . The IMF multiplicity ($M_{IMF}$) dependence with temperature (from disassembly of $Z_{0}$=75, $A_{0}$=186 at constant free-out volume 6$V_{0}$), freeze-out volume (for the same fragmenting system $Z_{0}$=75, $A_{0}$=168 but at constant temperature 5 MeV) and neutron to proton ratio of the fragmenting system (from disassembly with constant $Z_{0}$=75 at constant temperature 5 MeV and free-out volume 6$V_{0}$) are presented in Fig. 3. From Fig. 3, it can be verified that, the effect of nuclear binding on intermediate mass fragment multiplicity is more significant at higher temperature and freeze-out volume i.e. the conditions at which the system breaks more. For a given temperature and freeze-out volume, the IMF multiplicity is strongly modified for the fragmentation from very isospin asymmetric systems due to significant production of neutron rich nuclei. Fig. 4 represents the isotopic distribution of some selective lower (Helium), intermediate (Carbon, Oxygen, Silicon and Calcium) and heavy (Nickel) mass fragments originated from the multifragmentation of $Z_{0}$=75, $A_{0}$=168 system at temperature 5 MeV and free-out volume 6$V_{0}$. As in IMF region, neutron rich isotopes are more bound in compressible liquid drop approach with Sly5 parameters compared to conventional liquid drop model, hence the production of neutron rich isotopes of Carbon, Oxygen, Silicon and Calcium is significantly enhanced.

One of the important aspect of intermediate energy heavy ion reactions study is to reduce the uncertainty in nuclear equation of state. Isoscaling Tan ; Mallik23 ; Tsang1 ; Botvina1 ; Chaudhuri4 ; Colonna ; Raduta ; Mallik5 ; Mallik8 and isobaric yield ratio Mallik5 ; Mallik8 ; Huang ; Tsang2 ; Ma1 methods are commonly used to search the precise nuclear equation of state. In this work, the effect of nuclear binding on isoscaling and isobaric yield ratio are investigated in the framework of CTM. It has been observed both experimentally and theoretically that the ratio of yields from two different reactions(having different isospin asymmetry), exhibit an exponential relationship as a function of the neutron($N$) and proton($Z$) number and this is termed as ’isoscaling’. Two fragmentation reactions ”$1$” and ”$2$” at a given energy are being considered whose fragmenting systems have different mass $A_{01}$ and $A_{02}$ ($A_{02}>A_{01}$) but same charge $Z_{1}=Z_{2}=Z_{0}$.

Where $\alpha$ and $\beta$ are the isoscaling parameters and $C$ is a normalization factor. Fig. 5 represents, the isoscaling ratios obtained from CTM calculation at constant temperature $T=5$ MeV and freeze-out volume $V_{f}=6V_{0}$ for two fragmenting sources with identical atomic number $Z_{0}=75$ but different mass number $A_{01}=168$ and $A_{02}=186$. The isoscaling ratio $R_{21}$ is plotted as function of the neutron number ($N$) for $Z=2$, $4$, $6$ and $8$ in the left panel whereas the right panel displays the ratio as function of the proton number ($Z$) for $N=2$, $4$, $6$ and $8$. The lines are the best fits of the calculated $R_{21}$ ratios to Eq. 18. From Fig. 5, it can be concluded that CTM calculation with more realistic semi-microscopic binding and excitation also shows isoscaling behavior but with the inclusion of realistic semi-microscopic binding, the magnitude of the isoscaling parameters $\alpha$ and $\beta$ decreased.
The isobaric ratio of yields Huang of two different types of fragments having same mass number $A$ but different isospin asymmetry $I=N-Z$ and $I^{{}^{\prime}}=N^{{}^{\prime}}-Z^{{}^{\prime}}$ originating from the fragmenting system is given by,

The quantity $R[I^{{}^{\prime}},I,A]$ shows linear behavior with $A^{2/3}$ for $I$=1 and $I^{\prime}$=-1 from conventional liquid drop approach Mallik8 . Fig. 6 confirms that, for semi-microscopic realistic binding this linear behavior also holds but for a given fragment mass number, production of neutron rich isotopes is more with realistic binding compare to conventional liquid drop approach, hence the isobaric yield ratio value is less. This reduction is more for fragmentation from more isospin asymmetric system ($Z_{0}=75$ and $A_{0}=186$) compare to other one ($Z_{0}=75$ and $A_{0}=168$).

The excited fragments produced in the multifragmentation stage decay to their stable ground states. They can $\gamma$-decay to shed energy but may also decay by light particle emission to lower mass nuclei. Hence an evaporation model with the same realistic semi-microscopic binding and excitation is developed. Emissions of $n,p,d,t,^{3}$He and ⁴He particles are considered. Particle-decay widths are obtained using Weisskopf’s evaporation theory Weisskopf . To study the cluster functional effect on cold fragments for isospin sensitive observables like isotopic distribution, isoscaling and isobaric yield ratio, two separate cases are considered-(a) CTM calculation described in II-A followed by evaporation model Mallik1 ; Mallik_Thesis with binding from conventional liquid drop model and excitation from Fermi gas model and (b) CTM calculation described in II-B followed by evaporation model with realistic binding and excitation. The results are presented in Fig. 7, which confirms relative enhancement of multiplicities of very neutron rich isotopes and reduction of isoscaling parameters as well as isobaric yield ratio due to introduction of new cluster functional exist strongly even after secondary decay.