Simulation of pre-compound reaction

The precompound stage of nuclear reaction is considered until nuclear system is not an equilibrium state. Further emission of nuclear fragments or photons from excited nucleus is simulated using an equilibrium model.

Statistical equilibrium condition

In the state of statistical equilibrium, which is characterized by an eqilibrium number of excitons $n_{eq}$, all three type of transitions are equiprobable. Thus $n_{eq}$ is fixed by $\omega_{+2}(n_{eq},U) = \omega_{-2}(n_{eq},U)$. From this condition we can get

$$n_{eq} = \sqrt{2gU}.$$ (28.1)

Level density of excited (n-exciton) states

To obtain Eq. ($\ref{PCS1}$) it was assumed an equidistant scheme of single-particle levels with the density $g \approx 0.595 aA$, where $a$ is the level density parameter, when we have the level density of the $n$-exciton state as

$$\rho_{n}(U) = \frac{g(gU)^{n-1}}{p!h!(n-1)!}.$$ (28.2)

Transition probabilities

The partial transition probabilities changing the exciton number by $\Delta n$ is determined by the squared matrix element averaged over allowed transitions $<\vert M\vert^{2}>$ and the density of final states $\rho_{\Delta n}(n,U)$, which are really accessible in this transition. It can be defined as following:

$$\omega_{\Delta n}(n,U)=\frac{2\pi}{h}<\vert M\vert^{2}>\rho_{\Delta n}(n,U).$$ (28.3)

The density of final states $\rho_{\Delta n}(n,U)$ were derived in paper [2] using the Eq. ($\ref{PCS2}$) for the level density of the $n$-exciton state and later corrected for the Pauli principle and indistinguishability of identical excitons in paper [3]:

$$\rho_{\Delta n = +2}(n,U)=\frac{1}{2}g\frac{[gU - F(p+1,h+1)]^2} {n+1} [\frac{gU - F(p+1,h+1)}{gU - F(p,h)}]^{n-1},$$ (28.4)

$$\rho_{\Delta n = 0}(n,U)=\frac{1}{2}g\frac{[gU - F(p,h)]}{n} [p(p-1) + 4ph + h(h-1)]$$ (28.5)

and

$$\rho_{\Delta n = -2}(n,U)=\frac{1}{2}gph(n-2),$$ (28.6)

where $F(p,h)=(p^2 + h^2 + p -h)/4 - h/2$ and it was taken to be equal zero. To avoid calculation of the averaged squared matrix element $<\vert M\vert^2>$ it was assumed [1] that transition probability $\omega_{\Delta n = +2}(n,U)$ is the same as the probability for quasi-free scattering of a nucleon above the Fermi level on a nucleon of the target nucleus, i. e.

$$\omega_{\Delta n =+2}(n,U)=\frac{<\sigma(v_{rel})v_{rel}>}{V_{int}}.$$ (28.7)

In Eq. ($\ref{PCS7}$) the interaction volume is estimated as $V_{int}=\frac{4}{3}\pi(2r_c + \lambda/2\pi)^3$, with the De Broglie wave length $\lambda/2\pi$ corresponding to the relative velocity $<v_{rel}>=\sqrt{2T_{rel}/ m}$, where $m$ is nucleon mass and $r_c = 0.6$ fm.

The averaging in $<\sigma(v_{rel})v_{rel}>$ is further simplified by

$$<\sigma(v_{rel})v_{rel}> =<\sigma(v_{rel})><v_{rel}>.$$ (28.8)

For $\sigma (v_{rel})$ we take approximation:

$$\sigma(v_{rel})=0.5[\sigma_{pp}(v_{rel})+\sigma_{pn}(v_{rel}]P(T_F/T_{rel}),$$ (28.9)

where factor $P(T_F/T_{rel})$ was introduced to take into account the Pauli principle. It is given by

$$P(T_F/T_{rel})=1 - \frac{7}{5}\frac{T_F}{T_{rel}}$$ (28.10)

for $\frac{T_F}{T_{rel}} \leq 0.5$ and

$$P(T_F/T_{rel})=1 - \frac{7}{5}\frac{T_F}{T_{rel}}+ \frac{2}{5} \frac{T_{F}}{T_{rel}}(2 - \frac{T_{rel}}{T_F})^{5/2}$$ (28.11)

for $\frac{T_F}{T_{rel}} > 0.5$.

The free-particle proton-proton $\sigma_{pp}(v_{rel})$ and proton-neutron $\sigma_{pn}(v_{rel})$ interaction cross sections are estimated using the equations [4]:

$$\sigma_{pp}(v_{rel}) = \frac{10.63}{v^2_{rel}}-\frac{29.93}{v_{rel}}+42.9$$ (28.12)

and

$$\sigma_{pn}(v_{rel}) = \frac{34.10}{v^2_{rel}}-\frac{82.2}{v_{rel}}+82.2,$$ (28.13)

where cross sections are given in mbarn.

The mean relative kinetic energy $T_{rel}$ is needed to calculate $<v_{rel}>$ and the factor $P(T_F/T_{rel})$ was computed as $T_{rel}=T_{p}+T_{n}$, where mean kinetic energies of projectile nucleons $T_p = T_F +U/n$ and target nucleons $T_N = 3T_F/5$, respecively.

Combining Eqs. ($\ref{PCS3}$) - ($\ref{PCS7}$) and assuming that $<\vert M\vert^{2}>$ are the same for transitions with $\Delta n = 0$ and $\Delta n = \pm 2$ we obtain for another transition probabilities:

$$\begin{array}{c} \omega_{\Delta n =0}(n,U)= \\ =\frac{<\sig... ...}]^{n+1} \frac{p(p-1) + 4ph +h(h-1)}{gU - F(p,h)} \end{array}$$ (28.14)

and

$$\begin{array}{c} \omega_{\Delta n = -2}(n,U)= \\ =\frac{<\... ...1)}]^{n+1} \frac{ph(n+1)(n-2)}{[gU - F(p,h)]^2}. \end{array}$$ (28.15)

Emission probabilities for nucleons

Emission process probability has been choosen similar as in the classical equilibrium Weisskopf-Ewing model [5]. Probability to emit nucleon $b$ in the energy interval $(T_b, T_b+dT_b)$ is given

$$W_{b}(n,U,T_b) = \sigma_{b}(T_b)\frac{(2s_b+1)\mu_b}{\pi^2 h^3} R_b(p,h) \frac{\rho_{n-b}(E^{*})}{\rho_n(U)}T_b,$$ (28.16)

where $\sigma_{b}(T_b)$ is the inverse (absorption of nucleon $b$) reaction cross section, $s_b$ and $m_b$ are nucleon spin and reduced mass, the factor $R_b(p,h)$ takes into account the condition for the exciton to be a proton or neutron, $\rho_{n-b}(E^{*})$ and $\rho_n(U)$ are level densities of nucleus after and before nucleon emission are defined in the evaporation model, respectively and $E^{*}=U-Q_b-T_b$ is the excitation energy of nucleus after fragment emission.

Emission probabilities for complex fragments

It was assumed [1] that nucleons inside excited nucleus are able to "condense" forming complex fragment. The "condensation" probability to create fragment consisting from $N_b$ nucleons inside nucleus with $A$ nucleons is given by

$$\gamma_{N_b}=N^3_b(V_b/V)^{N_b -1}=N^3_b(N_b/A)^{N_b -1},$$ (28.17)

where $V_b$ and $V$ are fragment $b$ and nucleus volumes, respectively. The last equation was estimated [1] as the overlap integral of (constant inside a volume) wave function of independent nucleons with that of the fragment.

During the prequilibrium stage a "condense" fragment can be emitted. The probability to emit a fragment can be written as [1]

$$W_{b}(n,U,T_b) =\gamma_{N_b}R_b(p,h) \frac{\rho(N_b, 0, T_b... ..._b+1)\mu_b}{\pi^2 h^3} \frac{\rho_{n-b}(E^{*})}{\rho_n(U)}T_b,$$ (28.18)

where

$$g_b(T_b)=\frac{V_b(2s_b+1)(2\mu_b)^{3/2}}{4\pi^2 h^3}(T_b+Q_b)^{1/2}$$ (28.19)

is the single-particle density for complex fragment $b$, which is obtained by assuming that complex fragment moves inside volume $V_b$ in the uniform potential well whose depth is equal to be $Q_b$, and the factor $R_b(p,h)$ garantees correct isotopic composition of a fragment $b$.

The total probability

This probability is defined as

$$W_{tot}(n,U) =\sum_{\Delta n =+2,0,-2}\omega_{\Delta n }(n,U) + \sum_{b=1}^{6}W_b(n,U),$$ (28.20)

where total emission $W_b(n,U)$ probabilities to emit fragment $b$ can be obtained from Eqs. ($\ref{PCS16}$) and ($\ref{PCS18}$) by integration over $T_b$:

$$W_{b}(n,U)=\int_{V_b}^{U-Q_b} W_b(n,U,T_b)dT_b.$$ (28.21)

Calculation of kinetic energies for emitted particle

The equations ($\ref{PCS16}$) and ($\ref{PCS18}$) are used to sample kinetic energies of emitted fragment.

Parameters of residual nucleus.

After fragment emission we update parameter of decaying nucleus:

$$\begin{array}{c} A_f=A-A_b; Z_f=Z-Z_b; P_f = P_0 - p_b; \\ E_f^{*}=\sqrt{E_f^2-\vec{P}^2_f} - M(A_f,Z_f). \end{array}$$ (28.22)

Here $p_b$ is the evaporated fragment four momentum.