Isospin effect on the liquid-gas phase transition for finite nuclei

In CTM, DasGupta_book ; Das it is assumed that statistical equilibrium is attained at freeze-out stage. 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

$\displaystyle Q_{N_{0},Z_{0}}$

$\displaystyle=$

$\displaystyle\sum\prod\frac{\omega_{N,Z}^{n_{N,Z}}}{n_{N,Z}!}$

(1)

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,

$\displaystyle\langle n_{N,Z}\rangle$

$\displaystyle=$

$\displaystyle\omega_{N,Z}\frac{Q_{N_{0}-N,Z_{0}-Z}}{Q_{N_{0},Z_{0}}}$

(2)

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:

$\displaystyle\omega_{N,Z}=\frac{V}{h^{3}}(2\pi mT)^{3/2}A^{3/2}\times z_{N,Z}(int)$

(3)

Here, $V$ is the volume available to the clusters and free nucleons for translational motion. Assuming that the clusters can not overlap, the available volume can be expressed as $V=V_{f}-V_{ex}$ , where $V_{f}$ is the freeze-out volume and $V_{ex}$ is the excluded volume, which is considered as constant and equal to the normal nuclear volume of fragmenting system with $Z_{0}$ protons and $N_{0}$ neutrons Bondorf1 ; Das . In this work the freeze-out volume is kept constant at $6V_{0}$ as fragmentation from central collision reactions at intermediate energies are considered. $z_{N,Z}(int)$ 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, $z_{N,Z}(int)=(2s_{N,Z}+1)\exp[-E_{N,Z}(gr)/T]$ where $E_{N,Z}(gr)$ 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 (as the excitation energies of these excited states are too high).
However, intermediate energy heavy-ion reactions occur at finite temperatures, and the produced clusters are no longer in their ground state. The free energy is a more comprehensive quantity that includes both energy (binding as well as excitation) and entropy contributions. So, the internal partition function of clusters with $Z\geq$3 can be expressed as,

	$\displaystyle z_{N,Z}(int)$	$\displaystyle=$	$\displaystyle\exp^{-\frac{F_{N,Z}}{T}}$		(4)
		$\displaystyle=$	$\displaystyle\exp^{-\frac{B_{N,Z}+E^{*}_{N,Z}-TS_{N,Z}}{T}}$

where, $B_{N,Z}$, $E^{*}_{N,Z}$ and $S_{N,Z}$ are the ground state binding energy, excitation and entropy respectively, of the fragment with $Z$ protons and $N$ neutrons. In conventional CTM approach as well as in most of the other statistical models of multifragmentation, for determining the internal partition function of nuclei with $Z\geq$3 the liquid-drop formula is used for calculating the binding energy and the remaining part of free energy is taken from the Fermi-gas model. In a recent work Mallik25 , the internal partition function of the CTM is upgraded by the semi-microscopic approach where the Helmholtz free energy of a nucleus with $N$ neutrons and $Z$ protons can be decomposed as Mallik_NuclAstro1 ,

$$F_{N,Z}=F^{bulk}+F^{surf}+F^{coul}.$$

(5)

where the bulk part $F^{bulk}$ is originated from bulk nuclear matter at baryonic density $\rho_{c}=\rho_{c,n}+\rho_{c,z}$ ($\rho_{c,n}$ and $\rho_{c,z}$ are neutron and proton density respectively) and isospin asymmetry $\delta_{c}=(\rho_{c,n}-\rho_{c,z})/\rho_{c}=\frac{N-Z}{N+Z}$ occupying a finite spatial volume $V_{c}=(N+Z)/\rho_{c}$. The baryonic density with isospin asymmetry $\delta_{c}$ is approximated Gulminelli2015 to the corresponding saturation density ($\rho_{0}$) of symmetric nuclear matter at finite asymmetry according to:

$\displaystyle\rho_{c}(\delta_{c})=\rho_{0}\bigg{(}1-\frac{3L_{sym}\delta_{c}^{2}}{K_{sat}+K_{sym}\delta_{c}^{2}}\bigg{)}.$

(6)

The above expression is obtained from the assumption of first order derivative of the binding with respect to baryonic density equal to zero papa and consideration up to the curvature term in generalized liquid drop approach Ducoin1 . The bulk part of the Helmholtz free energy given by,

$\displaystyle F^{bulk}=V_{c}\bigg{[}-\frac{2}{3}\sum_{q=n,z}\xi_{c,q}+\sum_{q=n,z}\rho_{c,q}\eta_{q}+v(\rho_{c},\delta_{c})\bigg{]}$

(7)

where $\xi_{c,n}$ and $\xi_{c,z}$ are the kinetic energy density of the nucleus due to neutron ($q=n$) and proton ($q=p$) contribution respectively, which can be expressed as $q=n,p$,

$\displaystyle\xi_{c,q}$

$\displaystyle=$

$\displaystyle\frac{3h^{2}}{2\pi m^{*}_{c,q}}\bigg{(}\frac{2\pi m^{*}_{c,q}T}{h^{2}}\bigg{)}^{5/2}F_{3/2}(\eta_{c,q})$

(8)

with $\eta_{c,q}=F^{-1}_{1/2}\bigg{\{}\bigg{(}\frac{2\pi m^{*}_{g,q}T}{h^{2}}\bigg{)}^{3/2}\rho_{c,q}\bigg{\}}$, $F_{1/2}$ and $F_{3/2}$ are the Fermi integrals. The expression of potential energy per particle that can be adapted to different effective interactions and energy functionals is given by:

	$\displaystyle v(\rho_{c},\delta_{c})$	$\displaystyle=$	$\displaystyle\sum_{k=0}^{N}\frac{1}{k!}(v^{is}_{k}+v^{iv}_{k}\delta_{c}^{2})x^{k}$		(9)
		$\displaystyle+$	$\displaystyle(a^{is}+a^{iv}\delta_{c}^{2})x^{N+1}\exp(-b\frac{\rho_{c}}{\rho_{0}}),$

where $x=\frac{\rho_{c}-\rho_{0}}{3\rho_{0}}$, $a^{is}=-\sum_{k\geq 0}^{N}\frac{1}{k!}v^{is}_{k}(-3)^{N+1-k}$ and $a^{iv}=-\sum_{k\geq 0}^{N}\frac{1}{k!}v^{iv}_{k}(-3)^{N+1-k}$. $N=4$ and $b=10ln2$ are chosen for this model. This value of $b$ leads to a good reproduction of the Sly5 functional which is used for the numerical applications presented in this paper. The model parameters $v_{k}^{is(iv)}$ can be linked with a one-to-one correspondence to the usual EoS empirical parameters Margueron2018 , via:

	$\displaystyle v^{is}_{0}$	$\displaystyle=$	$\displaystyle E_{sat}-t_{0}(1+\kappa_{0})$
	$\displaystyle v^{is}_{1}$	$\displaystyle=$	$\displaystyle-t_{0}(2+5\kappa_{0})$
	$\displaystyle v^{is}_{2}$	$\displaystyle=$	$\displaystyle K_{sat}-2t_{0}(-1+5\kappa_{0})$
	$\displaystyle v^{is}_{3}$	$\displaystyle=$	$\displaystyle Q_{sat}-2t_{0}(4-5\kappa_{0})$
	$\displaystyle v^{is}_{4}$	$\displaystyle=$	$\displaystyle Z_{sat}-8t_{0}(-7+5\kappa_{0})$		(10)

	$\displaystyle v^{iv}_{0}$	$\displaystyle=$	$\displaystyle E_{sym}-\frac{5}{9}t_{0}[(1+(\kappa_{0}+3\kappa_{sym})]$
	$\displaystyle v^{iv}_{1}$	$\displaystyle=$	$\displaystyle L_{sym}-\frac{5}{9}t_{0}[(2+5(\kappa_{0}+3\kappa_{sym})]$
	$\displaystyle v^{iv}_{2}$	$\displaystyle=$	$\displaystyle K_{sym}-\frac{10}{9}t_{0}[(-1+5(\kappa_{0}+3\kappa_{sym})]$
	$\displaystyle v^{iv}_{3}$	$\displaystyle=$	$\displaystyle Q_{sym}-\frac{10}{9}t_{0}[(4-5(\kappa_{0}+3\kappa_{sym})]$
	$\displaystyle v^{iv}_{4}$	$\displaystyle=$	$\displaystyle Z_{sym}-\frac{40}{9}t_{0}[(-7+5(\kappa_{0}+3\kappa_{sym})]\ ,$		(11)

In order to do that, in addition to Sly5 EoS, the semi-microscopic cluster functional of the CTM described in Section II is determined from NL3 Lalazissis and SGII Nguyen EoS also. Isoscalar and isovector parameters used in this paper for three different EoS are mentioned in Table-1. Nuclear phase transition from heavy-ion reactions at intermediate energies can be explained as the competition between the surface energy and excitation energy. The surface tension effect tries to accumulate more nucleons together (i.e. prefers liquid phase) on the other side excitation of the nucleus tries to break it in to free nucleons and small composites (i.e. prefers gas phase). Here surface energy of a fragment is kept identical (as described in the last part of Section-II), but the excitation energy of different fragments at a given temperature obtained from SGII (NL3) EoS are higher (lower) compared to that of Sly5 EoS. This bulk excitation energy per nucleon of the fragment with $Z$ protons and $N$ neutrons (at baryonic density $\rho_{c}$ and isospin asymmetry $\delta_{c}$) is determined from the expression,

	$\displaystyle\frac{E^{*}_{N,Z}(\rho_{c},T)}{A}$	$\displaystyle=$	$\displaystyle\frac{1}{{\rho_{c}}}\sum_{q=n,z}\bigg{[}\frac{3h^{2}}{2\pi m^{*}_{c,q}}\bigg{(}\frac{2\pi m^{*}_{c,q}T}{h^{2}}\bigg{)}^{5/2}F_{3/2}(\eta_{c,q})\bigg{]}$		(16)
		$\displaystyle-$	$\displaystyle\frac{t_{0}}{2}\bigg{(}\frac{\rho_{c}}{\rho_{0}}\bigg{)}^{2/3}\bigg{[}(1+\kappa_{0}\frac{\rho_{c}}{\rho_{0}})f_{1}(\delta_{c})$
		$\displaystyle+$	$\displaystyle\kappa_{sym}\frac{\rho_{c}}{\rho_{0}}f_{2}(\delta_{c})\bigg{]}$

where, $t_{0}=\frac{3\hbar^{2}}{10m}\big{(}\frac{3\pi^{2}\rho_{0}}{2}\big{)}^{2/3}$ is kinetic energy per nucleon in symmetric matter at saturation and the functions $f_{1}(\delta_{c})=(1+\delta_{c})^{5/3}+(1-\delta_{c})^{5/3}$ and $f_{2}(\delta_{c})=\delta_{c}[(1+\delta_{c})^{5/3}-(1-\delta_{c})^{5/3}]$ Margueron2018 ; Mallik_NuclAstro1 . For example, the temperature dependence of excitation energy per nucleon of ¹²C and ²⁰O fragments for these three considered EoS are shown in Fig. 2. Hence, due this change in excitation, the peak of the specific heat per particle at constant volume ($C_{v}/A_{0}$) as well as multiplicity derivative normalised by fragmenting system mass number $\frac{d(M/A_{0})}{dT}$ (which represents the temperature at which nuclear liquid-gas phase transition occurs) obtained from CTM calculation with SGII (NL3) EoS shifted towards lower (higher) temperature side compare to that of Sly5 EoS. This is displayed in Fig. 3 and 4 for the two fragmenting systems $A_{0}$=72, $Z_{0}$=30 and $A_{0}$=186, $Z_{0}$=75 respectively.

The liquid gas phase transition of nuclear matter is connected to the isospin asymmetry Ducoin . To study the effect of isospin asymmetry on these nuclear phase transition signatures for finite nuclei, CTM calculation with semi-microscopic cluster functional (with Sly5 EoS only) is performed for two sets of disintegrating system-(i) fixed $Z_{0}$=30 but three different $A_{0}$=72, 66 and 60 which are expected to be formed in central collision of ⁴⁸Ca+⁴⁸Ca, ⁴⁰Ca+⁴⁸Ca and ⁴⁰Ca+⁴⁰Ca reactions respectively (by assuming $25\%$ pre-equilibrium emission in each reaction) and (ii) fixed $Z_{0}$=75 but three different $A_{0}$=186, 177 and 168 which are expected to be formed in central collision of ¹²⁴Sn+¹²⁴Sn, ¹¹²Sn+¹²⁴Sn and ¹¹²Sn+¹¹²Sn respectively (by assuming $25\%$ pre-equilibrium emission in each reaction). Results for set (i) and (ii) are displayed in Fig. 5 and 6 respectively. For both sets of disintegrating systems, peak of both $C_{v}/A_{0}$ and $\frac{d(M/A_{0})}{dT}$ is almost independent of isospin asymmetry. The dependence of phase transition temperature with isospin asymmetry (neutron to proton ratio of the fragmenting system) obtained from both $C_{v}/A_{0}$ and $\frac{d(M/A_{0})}{dT}$ is displayed in Fig. 7. Fig. 7 concludes that the dependence of transition temperature on isospin asymmetry is very less in heavy ion reactions with finite nuclei.

The excited fragments produced in the multifragmentation stage decay to their stable ground states before reaching the detector. Hence in experiments, total multiplicity will be modified. Effect of secondary decay on experimentally accessible signature $\frac{d(M/A_{0})}{dT}$ is examined for fixed $Z_{0}$=30 but three different $A_{0}$=72, 66 and 60. This is presented in Fig. 8. For each fragmenting systems different isospin asymmetries the peak position in multiplicity derivative with respect to temperature remains unchanged even after the secondary decay and the peaks become sharper.
The intermediate mass fragment (with atomic number $3{\leq}Z{\leq}20$) multiplicity ($M_{IMF}$) is another key observable in nuclear reactions at intermediate energies, which has been studied both experimentally and theoretically in various scenarios Peaslee ; Ogilvie . The first order derivatives of the intermediate mass fragment multiplicity and light fragment (with atomic number $Z$=$1$ and $2$) multiplicity with respect to temperature ($\frac{d(M_{IMF}/A_{0})}{dT}$ and $\frac{d(M_{LMF}/A_{0})}{dT}$ respectively) are studied from the CTM with Sly5, NL3 and SGII EoS for the same two disintegrating systems (i) $A_{0}$=72, $Z_{0}$=30 and (ii) $A_{0}$=186, $Z_{0}$=75 (displayed in Fig. 9 and 10 respectively). For both observables, a phase transition signal is observed, and the temperatures at which the maxima of $\frac{d(M_{IMF}/A_{0})}{dT}$ and $\frac{d(M_{LMF}/A_{0})}{dT}$ occur are very close to the peak of $\frac{d(M/A_{0})}{dT}$ and $C_{v}/A_{0}$ for each of these three EoS.
The effect of isospin asymmetry of the fragmenting system on $\frac{d(M_{IMF}/A_{0})}{dT}$ and $\frac{d(M_{LMF}/A_{0})}{dT}$ is also examined within the CTM framework using the Sly5 EoS. This analysis considers same two sets of systems as before: (i) the same $Z_{0}$=30 but different $A_{0}$=60, 66 and 72 and (ii) the same $Z_{0}$=75 but different $A_{0}$=168, 177 and 186 (displayed in Fig. 11 and 12 respectively). These two figures indicate that, similar to the specific heat and total multiplicity derivative, the peak positions of both the intermediate mass fragment and light fragment multiplicity derivatives are almost independent of isospin asymmetry.