Multiplicity distributions of intermediate mass fragments in the thermodynamic model

Multiplicity distributions of intermediate mass fragments (IMFs) seen in intermediate energy heavy ion collisions have been of great interest in the last ten years. The distributions of single intermediate sized elements are close to Poissonian and have been subjected to widely different physical interpretations. We calculate the multiplicity distributions in a thermodynamic model whose physical basis is equal populations in phase space, which implies equilibrium statistics. We found distributions of a single element are to a good approximation Poissonian, and this result is attributed to the small emission probability of a given IMF among all the possible channels. We also found significant deviation from a Poisson distribution for a range of IMFs, and suggest strong correlations as an explanation. The systematics of the multiplicity distributions are explored in both one-component and two-component models. In the one-component model, sudden changes for the multiplicity distributions are found at the phase transition region. In the two-component model, the multiplicity distributions have more subtleties. Results from the current model are compared with available data, and good agreements are found. New signals for the multiparticle correlations are also suggested.

Figure 1

Specific heat $C_V$ as a function of temperature for a system of total mass $A=200$. The density of the fragmentation source is fixed at $\rho =0.05{\mathrm{fm}}^{-3}$. The peak at $6.5\mathrm{MeV}$ is defined as the phase transition temperature $T_b$ in the current one-component canonical model.

Figure 2

The multiplicity distribution $P\left(M\right)$ of a single cluster with mass $k=6$ as a function of multiplicity of this cluster $M$ in the one-component canonical model. The fragmentation source has 200 particles, and density at fragmentation is $0.05{\mathrm{fm}}^{-3}$. The phase transition temperature for this source is at $T_b=6.5\mathrm{MeV}$. The full, dashed, and dot-dashed lines are for temperatures of $T=6$, 6.5, and $7\mathrm{MeV}$, respectively.

Figure 3

Same as Fig. 2. The multiplicity distribution $P\left(M\right)$ of single clusters, $k=6$, as a function of multiplicity $M$ in the one-component canonical model, in a log plot. The left, middle, and right panels are for temperatures of $T=6$, 6.5, and $7\mathrm{MeV}$, respectively. In all three panels, the canonical results are shown with square symbols and the Poisson distribution with the same mean is shown with a solid line. It is clear that the canonical result for a single cluster follows closely Poisson distribution, except at large multiplicities. However, significant differences show up at large multiplicities. There is a change of the super-Poissonian distribution to a sub-Poissonian distribution when the temperature changes from below the phase transition temperature to above the phase transition temperature.

Figure 4

The Poisson test parameter $\lambda \left(M\right)$ for the multiplicity distribution of single clusters, $k=6$, as a function of multiplicity $M$ in the one-component canonical model. See Eq. (14) for the definition of $\lambda \left(M\right)$.By definition, a Poisson distribution will have $\lambda \left(M\right)\equiv 1$ for all $M$. The Poisson test parameter shows the difference in the multiplicity distribution as the temperature changes from below to above $T_b$. Right around the phase transition temperature, the Poisson test parameter is closest to unity for a large range of multiplicities, while below and above the phase transition temperature, the Poisson test parameter is far from unity for most multiplicities.

Figure 5

The multiplicity distribution $P\left(M\right)$ of intermediate mass fragments, $k=6–40$, as a function of multiplicity $M$ in the one-component canonical model. The fragmentation source has total mass 200, and its density is fixed at $0.05{\mathrm{fm}}^{-3}$. The left, middle, and right panels correspond to temperatures below, at, and above the phase transition temperature. The Poisson distribution with the same mean is plotted together with the corresponding canonical distribution. Clearly, the deviation from the Poisson distribution is visible even in this linear plot. The mean multiplicity for a range of clusters is considerably higher than that for a single cluster.

Figure 6

The Poisson test parameter for the multiplicity distribution of a range of clusters, $k=6–40$, in the one-component canonical model. The system size and breakup volume are the same as in Fig. 5.

Figure 7

The multiplicity distribution of ${}^7\mathrm{Li}$ and $\mathrm{Li}$(all) at $T=5.2\mathrm{MeV}$, in the two-component model with $A_{sys}=200$ and $Z_{sys}=86$, in a log scale. The data symbols are the results from the canonical model while solid lines indicate the corresponding Poisson distribution with the same mean.

Figure 8

The Poisson test parameter for multiplicity distribution of single clusters, $Z=3$, $A=5$,7,9, at three different temperatures $T=4.0$, 4.6, and $5.2\mathrm{MeV}$. The system is $(A_{sys}=200,Z_{sys}=86)$ and the phase transition temperature is $T_b=4.6\mathrm{MeV}$ for this system. The line at $\lambda \left(M\right)=1$ is for the exact Poisson distribution.

Figure 9

The effect of simple decay on the multiplicity distribution of IMFs. We have plotted the before and after decay multiplicity distributions of single element $\left(Z=3\right)$, and IMFs $\left(3⩽Z⩽20\right)$ in the left and right panels, respectively. The system is $(A_{sys}=200,Z_{sys}=86)$ at $T=5.2\mathrm{MeV}$. The decay probability is assumed to be 10%.

Figure 10

The mean and variance of multiplicity distributions change as a function of decay probability in the two-component model. Here we have used the same multiplicity distribution of a single element $\left(Z=3\right)$ and IMFs $\left(3⩽Z⩽20\right)$ as shown in Fig. 9, but the resulting linear relation is universal. The mean $⟨M⟩$ is defined in Eq. (17), and the scaled variance $\widetilde{{\sigma }^2}$ is defined in Eq. (20).

Figure 11

The Poisson test parameter $\lambda \left(M\right)$ for the multiplicity distribution of a single element, $Z=3$, as a function of multiplicity $M$. The effect of simple decay will not alter the Poisson test parameter. The system is $(A_{sys}=200,Z_{sys}=86)$ at $T=5.2\mathrm{MeV}$.

Figure 12

The effects of decay and feeding on the multiplicity distribution of element $Z=3$. The left panel shows the multiplicity distributions in linear scale, while the right panel shows the Poisson test parameter for the different multiplicity distributions. The line labeled “original” is the distribution without any consideration of either decay or feeding; the line labeled “smp decay” includes only the effect of decay processes; the line labeled “complete” is the exact result which includes both decay and feeding; the line labeled “idp model” assumes that the production of clusters are independent of each other [see Eq. (15)]. In general, inclusion of feeding processes reduces the effect of decay and produces results closer to the “original” distribution. Such decay and feeding processes will not change the systematics of the Poisson test parameter. Also (not quite easy to see in the left panel), the “idp model” yields values that are up to three significant digits as good as the “complete” calculation for low multiplicities $M<10$.

Figure 13

Multiplicity distribution of a single cluster $k=6$ as a function of system size at fixed temperature $T=7\mathrm{MeV}$ in the one-component model.

Figure 14

Multiplicity distributions of a range of clusters $k=6–40$ are plotted as a function of system size at fixed temperature $T=7\mathrm{MeV}$ in the one-component model. Only multiplicities in the range of $10⩽M<20$ are plotted here.

Figure 15

Multiplicity distribution of $Z=3$ is plotted as a function of system size at fixed temperature of $5.2\mathrm{MeV}$ in a two-component model. The charge number of the fragmentation source is chosen so that the total system isospin asymmetry is as close to $\delta =0.1$ as possible.

Figure 16

Multiplicity distribution of IMFs $3⩽Z⩽20$ is plotted as a function of system size at fixed temperature of $5.2\mathrm{MeV}$ in a two-component model. The charge number of the source is chosen so that the total system isospin asymmetry is as close to $\delta =0.1$ as possible. Only lines with multiplicity in the range $7⩽M⩽17$ are plotted.

Figure 17

Multiplicity distribution, for single element $Z=3$, is plotted as a function of system size in a two-component model. We have assumed 15% decay probability for all Li isotopes and the temperature of the fragmentation source is fixed at $T=4.6\mathrm{MeV}$. This result roughly corresponds to the multiplicity distribution in the range $500⩽E_T⩽1000\mathrm{MeV}$ in the left panel of Fig. 4 in [1].

Figure 18

The Poisson test parameter, for single element $Z=3$, is plotted as a function of system size. We have assumed 15% decay probability for all Li isotopes and the temperature of the fragmentation source is fixed at $T=4.6\mathrm{MeV}$. This result roughly corresponds to the right panel of Fig. 4 in [1], with the range $500\mathrm{MeV}⩽E_T⩽1000\mathrm{MeV}$. Note, our definition of Poisson test parameter $\lambda \left(M\right)$ scaled out the obvious factor of mean multiplicity, see Eq. (14).

Figure 19

Multiplicity distribution, for a range of IMFs $3⩽Z⩽20$, is plotted as a function of system size for a two-component system. We have assumed 25% decay probability for all Li isotopes and the temperature of the fragmentation source is fixed at $T=3.5\mathrm{MeV}$. This result roughly corresponds to the range of $550⩽E_T⩽1100\mathrm{MeV}$ in the top panel of Fig. 3 in [3].

Figure 20

The Poisson test parameter, for a range of IMFs $3⩽Z⩽20$, is plotted as a function of system size for a two-component system. The IMF selection and $E_T$ correspondence are the same as in Fig. 19. We have assumed 25% decay probability for all Li isotopes and the temperature of the fragmentation source is fixed at $T=3.5\mathrm{MeV}$.

Figure 21

The multiplicity gated to ungated particle yield ratio $R_k\left(M\right)$, for a single component system at around the phase transition temperature region. The three panels from left to right are for the same system of $A_{sys}=200$ particles at different temperatures $T=5$, 6, and $7\mathrm{MeV}$, while the phase transition temperature $T_b=6.5\mathrm{MeV}$. For IMF in the range of $6⩽k⩽40$, the mean multiplicities at the three temperatures are 0.12, 1.55, and 12.72, respectively.

Figure 22

The multiplicity gated to ungated element yield ratio $R_k\left(M\right)$, for a single component system, with $A_{sys}=200$ and $T=7\mathrm{MeV}$.

Figure 23

The multiplicity gated to ungated charge yield ratio $R_Z\left(M\right)$ in a two-component model. The total system is $A_{sys}=200$, $Z_{sys}=86$ at $T=4\mathrm{MeV}$. IMFs are defined as clusters with $3⩽Z⩽20$, and the mean IMF multiplicity is $⟨M⟩=6.45$ in this system.

Figure 24

Same at in Fig. 23. The multiplicity gated to ungated charge yield ratio in a two-component model where the IMF range selection is varied. The system is $A_{sys}=200$, $Z_{sys}=86$ at $T=4\mathrm{MeV}$. The different lines correspond to different IMF ranges; the higher cutoff is from $Z⩽10$ to $Z⩽40$, while the lower cutoff is fixed at $Z⩾3$.

Figure 25

Specific heat $C_V$ and scaled variance $\widetilde{{\sigma }^2}$ as a function of temperature in the one-component canonical model. The temperature at maximum specific heat is defined as the phase transition temperature $T_b$ in the canonical model. For a system of size $A_{sys}=200$ and density $\rho =0.05{\mathrm{fm}}^{-3}$, the phase transition temperature is at $T_b=6.5\mathrm{MeV}$. The peak for maximum IMF correlations is at $T=6.4\mathrm{MeV}$. IMFs are defined as clusters of size $6⩽k⩽40$.

Figure 26

The temperatures for maximum specific heat and maximum correlation are plotted as a function of system size in the one-component model. The density of the source is at $\rho ∕{\rho }_0=1∕3$.

Figure 27

The scaled variance is plotted as a function of temperature in the phase transition region. Different lines correspond to different IMF selections. The change of the variance for a single cluster $k=6$ is quite small across the critical temperature, while the change is significantly enhanced when including a large range of clusters.

Figure 28

The scaled variance as a function of temperature and system size for the one-component canonical model. The shade of darkness represents the value of the scaled variance and the correspondence is shown by the inset scale. The contour regions are drawn in steps of 0.05 in scaled variance, while contour lines are drawn in steps of 0.15 starting from $\widetilde{{\sigma }^2}=0.275$. The contour region corresponding to $\widetilde{{\sigma }^2}=0.6$ is darkened while that for $\widetilde{{\sigma }^2}=0.45$ is hatched. This result may be compared with Fig. 3 in [1] in the transverse energy range $700⩽E_T⩽1400\mathrm{MeV}$.

Figure 29

Contour plot of the scaled variance as a function of temperature and charge of the source $Z_{sys}$, where system size is fixed $A_{sys}=110$. The dark colored band indicates the region with $\widetilde{{\sigma }^2}=0.82$, corresponding to the variance measured by Moretto et al. [3], in $\mathrm{Ar}+\mathrm{Au}$ at $110\mathrm{MeV}$/nucleon. The hatched band indicates the region with $\widetilde{{\sigma }^2}=0.75$, which is the decay corrected value for the variance. Each contour band corresponds to a change in variance by 0.035 , while contour lines are also drawn in steps of 0.015 starting at $\widetilde{{\sigma }^2}=0.35$. The temperature obtained here is around $3\mathrm{MeV}$, if the isospin asymmetry of the source is around $\delta =0.15$. This temperature is slightly lower than $3.5\mathrm{MeV}$ assumed from the multiplicity distribution result in Fig. 19 in Sec. 4.

Figure 30

Contour plot of the scaled variance as a function of temperature $T$ and charge of the source $Z_{sys}$, where system size is fixed $A_{sys}=220$. The dark colored band indicates the region with $\widetilde{{\sigma }^2}=0.62$, corresponding to the variance measured in Fig. 4 in [3], in $\mathrm{Ar}+\mathrm{Au}$ at $110\mathrm{MeV}$/nucleon. The hatched band indicates the region with $\widetilde{{\sigma }^2}=0.5$, which is the decay corrected value for the variance. Each contour band corresponds to a change in variance by 0.02, while contour lines are drawn in steps of 0.06 starting from $\widetilde{{\sigma }^2}=0.31$. The temperature we find here is around $3.5\mathrm{MeV}$ if the isospin asymmetry of the source is estimated as $\delta =0.15$. This temperature is in agreement with that from multiplicity distribution results in Sec. 4.