Multiplicity derivative: A new signature of a first-order phase transition in intermediate-energy heavy-ion collisions

Measurement of $M$, the total multiplicity, for central collision between comparable mass heavy ions can provide a signature for first-order phase transition. The derivative of $M$ with respect to $E^{*}/A$, where $E^{*}$ is the excitation energy in the center of mass and $A$ is the total mass of the dissociating system, is expected to go through maximum as a function of $E^{*}$. Theoretical modeling shows that this is the energy where the specific heat $C_v$ maximizes, which typically happens at the first-order phase transition. The measurement of total $M$ is probably feasible in more than one laboratory.

Figure 1

Variation of multiplicity $M$ (left panels) and $dM/dT$ (right panels) with temperature (bottom $x$ axes) and excitation per nucleon (top $x$ axes) from the CTM calculation for fragmenting systems having $Z=82$ and $N=126$ (top panels). Bottom panels represent the same but for a hypothetical system of one kind of particle with no Coulomb interaction but the same mass number ($A=208)$. $E^{*}$ is $E-E_0$, where $E_0$ is the ground-state energy of the dissociating system in the liquid drop model whose parameters are given in Ref. [7].

Figure 2

Same as Fig. 1 but the fragmenting systems are $Z=28$ and $N=30$ (top panels) and $A=58$ (bottom panels).

Figure 3

Variation of $dM/dT$ (red solid lines) and $C_v$ (green dashed lines) with temperature from CTM for fragmenting systems having $Z=82$ and $N=126$ (left panel) and for hypothetical systems of one kind of particle with no Coulomb interaction of mass number $A=208$. To draw $dM/dT$ and $C_v$ in the same scale, $C_v$ is normalized by a factor of $1/50$.

Figure 4

Same as in Fig. 3, but the fragmenting systems are $Z=28$ and $N=30$ (left panel) and $A=58$ (right panel).

Figure 5

Variation of entropy (blue dashed lines) and $dM/dT$ (red solid lines) with temperature from CTM for fragmenting systems having $Z=82$ and $N=126$ (top panel) and for hypothetical system of one kind of particle with no Coulomb interaction of mass number $A=208$ (bottom panel). To draw $S$ and $dM/dT$ in the same scale, $S$ is normalized by a factor of $1/20$ for $Z=82$ and $N=126$ system and $1/50$ for hypothetical system of one kind of particle.

Figure 6

Effect of secondary decay on $M$ (left panel) and $dM/dT$ (right panel) for fragmenting systems having $Z=28$ and $N=30$. Red solid lines show the results after the multifragmentation stage (calculated from CTM), whereas blue dashed lines represent the results after secondary decay of the excited fragments.

Figure 7

Variation of intermediate-mass fragment (IMF) multiplicity $M_{\mathrm{IMF}}$ (left panels) and first-order derivative of IMF multiplicity $d{M}_{\mathrm{IMF}}/dT$ (right panels) with temperature from CTM calculation for fragmenting systems having $Z=82$ and $N=126$. Variation of $C_v$ with temperature ($T$) is shown by green dashed line in right panel. To draw $d{M}_{\mathrm{IMF}}/dT$ and $C_v$ in the same scale, $C_v$ is normalized by a factor of $1/100$.