Figure 1 below shows two caloric curves generated by the Purdue Group using SMM. For each point approximately 2000 events were simulated. Temperature, T_break, and excitation energy per nucleon, E^*/nucleon, are in units of MeV.

The size of the system, A₀, was fixed: for green points A₀ = 100, while for red points A₀ = 160. Arrows show the obvious breaks in the respective curves.

A light blue curve shows Fermi Gas behavior, while a dark blue curve shows Ideal Gas behavior.

Note also the red points near E^*/nucleon ~ 9 MeV.

Figure 1:

Figure 2 shows the average multiplicity at a given excitation energy for the same simulation as shown in Figure 1.

Figure 2:

Figure 3 shows the input multiplicity to a canonical simulation in the style of SMM. The input multiplicity serves to generate a X(M_in) parameter which in turn generates a V_f. The output multiplicity agrees with the input multiplicity.

Figure 3:

The output of the fully canonical simulation (energy is conserved in this canonical simulation) is compared to the output of the full SMM simulation in terms of average multiplicity and a caloric curve. In the case of the canonical simulation, the break and backbending in the caloric curve is due solely to the abrupt change in V_f as seen in X(M)_in in the lower part of Figure 3.

Figure 4:

Questions:

• The break shows as a dramatic and discontinuous decrease in T_break. It appears as if there is a change in algorithm from that used at very low multiplicities and E^*/nucleon to calculate partitions, to that used for the higher E^*/nucleon, i.e. the grand canonical approach. Is this what is in fact going on?

• In looking through the SMM code there seems to be several parameters that come into play with the volume:

  • In FRAMIK, the quasi-micro canonical ensemble subroutine, the free volume is determined via: V_f = V₀*((1.+1.44*(SM**0.333333-1.)/(1.17*A0**0.333333))**3-1.), with: 2.le.SM.le.4.
  • In FRAGTE, the grand canonical ensemble subroutine, V_f = V₀*K with K=((1.+1.44*(SM**0.333333-1.)/(1.17*A0**0.333333))**3-1.) with SM=(1.+2.31*(E00-3.5))*A0/100. Here SM is the multiplicity, E00 is the excitation energy and A0 is the size of the fragmenting system.

  With this in mind, the abrupt change in the caloric curve's behavior observed above could be explianed by using FRAMIK until E00 is sufficiently high so that there is a jump in SM when the program switches to FRAGTE. This assumption was tested in the canonical simulations of Figure 3 and 4 above. There it was assumed for the FRAMIK region of E00 that SM was always 2, i.e. for E00 .lt. 5.5 MeV/nucleon. When E00=5.5 MeV/nucleon the switch was made to the expressions in FRAGTE to determine SM, K and V_f. This method reproduced the behavior of the full SMM simulation suggesting the break and backbending in the caloric curve may due to a discontinuty in the treatment of the free volume.

  • It it unclear at this time the precise mechanism for switching from FRAMIK to FRAGTE. More understanding of this mechanism will assist in determining the cause of the break and backbending in the caloric curve of SMM.

• If an algorithm change is not responsible, then what is the mechanism behind the abrupt and discontinuous change in magnitude (ðT_break ~ 1.5 MeV for the A₀ = 100) and shape of these caloric curves?

• Have other simulations show out-lying points such as those seen here near E^*/nucleon ~ 9 MeV for A₀ = 160? If so what is the mechanism responsible for them?

From: J. P. Bondorf et. al., Phys. Rev. C, 58, R27 (1998):

• Seventh paragraph: The total breakup volume is parametrized as V = (1+k) V₀, where V₀ is the compound nucleus volume at normal density and the model parameter k is the same for all channels. The choice of k is motivated by the requirements (a) to avoid overlaps between the fragments and (b) to provide a sufficient reduction of the Coulomb barrier, as seen in the kinetic energy spectra. The entropy associated with the translational motion of fragments is determined by the "free" volume, V_f, which incorporates the excluded volume effects. In general V_f depends on the breakup channel and therefore cannot be fixed to a constant kV₀, as often assumed. In the SMM we parametrized V_f(m) in such a way that is grows almost linearly with the primary fragment multiplicity m or, equivalently, with the excitation energy e^* = E^*/A₀ of the system [12]. According to this parameterization, V_f(m) vanishes for the compound nucleus (m=1) and increases to about 2V₀ at e^* ˜ 10 MeV/nucleon.