Evaporation model

The Weisskopf treatment is an application of the detailed balance principle that relates the probabilities to go from a state \(i\) to another \(d\) and viceversa through the density of states in the two systems:

\[P_{i \rightarrow d} \rho(i) = P_{d \rightarrow i} \rho(d)\]

where \(P_{d \rightarrow i}\) is the probability per unit of time of a nucleus \(d\) captures a particle \(j\) and form a compound nucleus \(i\) which is proportional to the compound nucleus cross section \(\sigma_{\mathrm{inv}}\). Thus, the probability that a parent nucleus \(i\) with an excitation energy \(E^*\) emits a particle \(j\) in its ground state with kinetic energy \(\varepsilon\) is

(252)\[ P_j(\varepsilon) \mathrm{d}\varepsilon = g_j \sigma_{\mathrm{inv}}(\varepsilon) \frac{\rho_d(E_{\mathrm{max}} - \varepsilon)}{\rho_i(E^*)} \varepsilon \mathrm{d}\varepsilon\]

where \(\rho_i(E^*)\) is the level density of the evaporating nucleus, \(\rho_d(E_{\mathrm{max}} - \varepsilon)\) that of the daugther (residual) nucleus after emission of a fragment \(j\) and \(E_{\mathrm{max}}\) is the maximum energy that can be carried by the ejectile. With the spin \(s_j\) and the mass \(m_j\) of the emitted particle, \(g_j\) is expressed as \(g_j = ( 2 s_j + 1 ) m_j / \pi^2 \hbar^2\).

This formula must be implemented with a suitable form for the level density and inverse reaction cross section. We have followed, like many other implementations, the original work of Dostrovsky et al. [DFF59] (which represents the first Monte Carlo code for the evaporation process) with slight modifications. The advantage of the Dostrovsky model is that it leds to a simple expression for equation (252) that can be analytically integrated and used for Monte Carlo sampling.

Cross sections for inverse reactions

The cross section for inverse reaction is expressed by means of empirical equation [DFF59]

(253)\[ \sigma_{\mathrm{inv}}(\varepsilon) = \sigma_g \alpha \left( 1 + \frac{\beta}{\varepsilon} \right)\]

where \(\sigma_g = \pi R^2\) is the geometric cross section.

In the case of neutrons, \(\alpha = 0.76+2.2A^{-\frac{1}{3}}\) and \(\beta = (2.12 A^{-\frac{2}{3}} - 0.050)/\alpha\) MeV. This equation gives a good agreement to those calculated from continuum theory [BW52] for intermediate nuclei down to \(\varepsilon \sim 0.05\) MeV. For lower energies \(\sigma_{\mathrm{inv},n}(\varepsilon)\) tends toward infinity, but this causes no difficulty because only the product \(\sigma_{\mathrm{inv},n}(\varepsilon)\varepsilon\) enters in equation (252). It should be noted, that the inverse cross section needed in (252) is that between a neutron of kinetic energy \(\varepsilon\) and a nucleus in an excited state.

For charged particles (p, d, t, ³He and \(\alpha\)), \(\alpha = (1+c_j)\) and \(\beta = -V_j\), where \(c_j\) is a set of parameters calculated by Shapiro [Sha53] in order to provide a good fit to the continuum theory [BW52] cross sections and \(V_j\) is the Coulomb barrier.

Coulomb barriers

Coulomb repulsion, as calculated from elementary electrostatics are not directly applicable to the computation of reaction barriers but must be corrected in several ways. The first correction is for the quantum mechanical phenomenoon of barrier penetration. The proper quantum mechanical expressions for barrier penetration are far too complex to be used if one wishes to retain equation (252) in an integrable form. This can be approximately taken into account by multiplying the electrostatic Coulomb barrier by a coefficient \(k_j\) designed to reproduce the barrier penetration approximately whose values are tabulated [Sha53].

\[V_j = k_j \frac{Z_j Z_d e^2}{R_c}\]

The second correction is for the separation of the centers of the nuclei at contact, \(R_c\). We have computed this separation as \(R_c = R_j + R_d\) where \(R_{j,d} = r_c A_{j,d}^{1/3}\) and \(r_c\) is given [ASIP94] by

\[r_c = 2.173 \frac{1 + 0.006103 Z_j Z_d}{1 + 0.009443 Z_j Z_d}\]

Level densities

The simplest and most widely used level density based on the Fermi gas model are those of Weisskopf [Wei37] for a completely degenerate Fermi gas. We use this approach with the corrections for nucleon pairing proposed by Hurwitz and Bethe [HB51] which takes into account the displacements of the ground state:

(254)\[ \rho(E) = C \exp{\left( 2 \sqrt{a(E-\delta)} \right)}\]

where \(C\) is considered as constant and does not need to be specified since only ratios of level densities enter in equation (252). \(\delta\) is the pairing energy correction of the daughter nucleus evaluated by Cook et al. [CFdLM67] and Gilbert and Cameron [GC65] for those values not evaluated by Cook et al.. The level density parameter is calculated according to:

\[a(E,A,Z) = \tilde{a}(A) \left \{ 1 + \frac{\delta}{E} [1 - \exp(-\gamma E)] \right\}\]

and the parameters calculated by Iljinov et al. [IMB+92] and shell corrections of Truran, Cameron and Hilf [TCH70].

Maximum energy available for evaporation

The maximum energy \(\varepsilon_j^{\mathrm{max}}\) available for the evaporation process (i.e. the maximum kinetic energy of the outgoing fragment \(j\)) is computed like

\[\varepsilon_j^{\mathrm{max}} = \frac{(M_i + E^*)^2 + M_j^2 - M_d^2}{2 (M_i + E^*)} - M_j\]

where \(M_i\), \(M_d\), and \(M_j\) are the nuclear masses of the compound, residual, and evaporated nuclei respectively. The minimum energy \(\varepsilon_j^{\mathrm{min}}\) of the evaporation fragment depends also on the Coulomb barrier \(Q_j\) and and the paring correction \(\delta\) which increasing effective residual mass :math:`(M_f = M_i + E^* - M_j - Q_j - delta):

\[\varepsilon_j^{\mathrm{max}} = \frac{(M_i + E^*)^2 + M_j^2 - M_f^2}{2 (M_i + E^*)} - M_j\]

Total decay width

The total decay width for evaporation of a fragment \(j\) can be obtained by integrating equation (252) over kinetic energy

\[\Gamma_j = \hbar \int_{V_j}^{\varepsilon_j^{\mathrm{max}}} P(\varepsilon_j) \mathrm{d}\varepsilon_j\]

This integration can be performed analiticaly if we use equation (254) for level densities and equation (253) for inverse reaction cross section. Thus, the total width is given by

\[\begin{split}\begin{array}{lll} \Gamma_j = \frac{g_j m_j R_d^2}{2 \pi \hbar^2} \frac{\alpha}{a_d^2} & \times & \Biggl \lgroup \biggl \{ \left(\beta a_d - \frac{3}{2}\right) + a_d (\varepsilon_j^{\mathrm{max}} - V_j) \biggr \} \exp{\left\{-\sqrt{a_i(E^*-\delta_i)}\right\}} + \nonumber \\ & & \biggl \{ (2\beta a_d - 3) \sqrt{a_d (\varepsilon_j^{\mathrm{max}} - V_j)} + 2 a_d (\varepsilon_j^{\mathrm{max}} - V_j) \biggr \} \times \nonumber \\ & & \exp{ \left\{ 2 \left[ \sqrt{a_d(\varepsilon_j^{\mathrm{max}} - V_j)} - \sqrt{a_i(E^* - \delta_i)} \right] \right\} } \Biggr \rgroup \end{array}\end{split}\]

where \(a_d = a(A_d,Z_d,\varepsilon_j^{\mathrm{max}})\) and \(a_i = a(A_i,Z_i,E^*)\).

GEM model

As an alternative model we have implemented the generalized evaporation model (GEM) by Furihata [Fur00]. This model considers emission of fragments heavier than \(\alpha\) particles and uses a more accurate level density function for total decay width instead of the approximation used by Dostrovsky. We use the same set of parameters but for heavy ejectiles the parameters determined by Matsuse et al. [MAL82] are used.

Based on the Fermi gas model, the level density function is expressed as

(255)\[\begin{split} \rho(E) = \left\{ \begin{array}{ll} \frac{\sqrt{\pi}}{12} \frac{e^{2\sqrt{a(E-\delta)}}}{a^{1/4}(E-\delta)^{5/4}} & \rm{for} \quad E \geq E_x \\ \frac{1}{T} e^{(E-E_0)/T} & \rm{for} \quad E < E_x \end{array} \right.\end{split}\]

where \(E_x = U_x + \delta\) and \(U_x = 150/M_d + 2.5\) (\(M_d\) is the mass of the daughter nucleus). Nuclear temperature \(T\) is given as \(1/T = \sqrt{a/U_x} - 1.5U_x\), and \(E_0\) is defined as \(E_0 = E_x - T(\log{T} - \log{a}/4 - (5/4)\log{U_x} + 2 \sqrt{aU_x})\).

By substituting equation (255) into equation (252) and integrating over kinetic energy can be obtained the following expression

\[\begin{split}\Gamma_j = \frac{ \sqrt{\pi} g_j \pi R_d^2 \alpha}{12 \rho(E^*)} \times \left\{ \begin{array}{ll} \{I_1(t,t) + (\beta+V)I_0(t)\} & \rm{for} \quad \varepsilon_j^{\mathrm{max}} - V_j < E_x \\ \{I_1(t,t_x)+I_3(s,s_x)e^s+ & \\ (\beta+V)(I_0(t_x)+I_2(s,s_x)e^s)\} & \rm{for} \quad \varepsilon_j^{\mathrm{max}} - V_j \geq E_x . \end{array} \right.\end{split}\]

\(I_0(t)\), \(I_1(t,t_x)\), \(I_2(s,s_x)\), and \(I_3(s,s_x)\) are expressed as:

\[\begin{split}\begin{array}{lll} I_0(t) & = & e^{-E_0/T} (e^t -1) \\ I_1(t,t_x) & = & e^{-E_0/T} T \{(t - t_x + 1)e^{t_x} - t -1 \} \\ I_2(s,s_x) & = & 2\sqrt{2} \biggl \{ s^{-3/2} + 1.5 s^{-5/2} + 3.75 s^{-7/2} - \nonumber \\ & & (s_x^{-3/2} + 1.5 s_x^{-5/2} + 3.75 s_x^{-7/2}) \biggr \} \\ I_3(s,s_x) & = & \frac{1}{2\sqrt{2}} \Biggl [ 2 s^{-1/2} + 4 s^{-3/2} + 13.5 s^{-5/2} + 60.0 s^{-7/2} + \nonumber \\ & & 325.125 s^{-9/2} - \biggl \{ (s^2 - s_x^2) s_x^{-3/2} + (1.5s^2 + 0.5s_x^2) s_x^{-5/2} + \nonumber \\ & & (3.75s^2 + 0.25s_x^2) s_x^{-7/2} + (12.875s^2 + 0.625s_x^2) s_x^{-9/2} + \nonumber \\ & & (59.0625s^2 + 0.9375s_x^2) s_x^{-11/2} + \nonumber \\ & & (324.8s^2 + 3.28s_x^2) s_x^{-13/2} + \biggr \} \Biggr ] \end{array}\end{split}\]

where \(t = (\varepsilon_j^{\mathrm{max}}-V_j)/T\), \(t_x = E_x/T\), \(s = 2 \sqrt{a(\varepsilon_j^{\mathrm{max}}-V_j -\delta_j)}\) and \(s_x = 2 \sqrt{a(E_x - \delta)}\).

Besides light fragments, 60 nuclides up to \(^{28}\)Mg are considered, not only in their ground states but also in their exited states, are considered. The excited state is assumed to survive if its lifetime \(T_{1/2}\) is longer than the decay time, i. e., \(T_{1/2}/\ln{2} > \hbar/\Gamma_j^*\), where \(\Gamma_j^*\) is the emission width of the resonance calculated in the same manner as for ground state particle emission. The total emission width of an ejectile \(j\) is summed over its ground state and all its excited states which satisfy the above condition.

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