The average excitation energy

We devote the remainder of this section to explain how average gamma and residual energies are derived when no gamma energy distributions are given.

For continuum two-body reactions the kinematics is treated very much like that of discrete two-body reactions in the previous section. One difference is that the angular distribution is now assumed to be isotropic, so that (11) simplifies to (12). The other difference is that the energy of the reaction Q is an average value, dependent upon the average excitation energy of the residual nucleus, and both of these quantities are to be determined.

When the x-particle in (11) plays the rôle of the residual nucleus, then the mass ratios in (9) take the form

$$\alpha_{{\rm res}}^{} {\frac{{m_{\rm targ}m_{\rm yo}}}{{(m_{\rm yi}+ m_{\rm targ})^2}}} \beta_{{\rm res}}^{} {\frac{{m_{\rm yo}}}{{m_{\rm yi}+ m_{\rm targ}}}} \gamma_{{\rm res}}^{} {\frac{{m_{\rm yi}m_{\rm res}}}{{(m_{\rm yi}+ m_{\rm targ})^2}}}$$ (18)

Then, equation (12) provides us with a relation between the average kinetic energy to the residual $\langle$E_res$\rangle$ and the average energy of the reaction  $\langle$Q$\rangle$,

$$\langle \rangle \alpha_{{\rm res}}^{} \gamma_{{\rm res}}^{} \beta_{{\rm res}}^{} \langle \rangle$$ (19)

In terms of the energy Q₀ due to the mass difference (1) and the excitation levels W_targ and W_res of the target and residual, the average energy $\langle$Q$\rangle$ of the reaction is given by

$$\langle \rangle \langle \rangle$$ (20)

It therefore follows from energy conservation (2) that for a stationary target we have

$$\langle \rangle \langle \rangle \langle \rangle$$ (21)

Upon solving equations (19) and (21) for the unknowns $\langle$Q$\rangle$ and $\langle$E_res$\rangle$, we find that

$$\langle \rangle {\frac{{1}}{{1 - \beta_{\rm res}}}} \alpha_{{\rm res}}^{} \gamma_{{\rm res}}^{}$$ (22)

and

$$\langle \rangle {\frac{{1}}{{1 - \beta_{\rm res}}}} \alpha_{{\rm res}}^{} \gamma_{{\rm res}}^{} \beta_{{\rm res}}^{} \beta_{{\rm res}}^{} \langle \rangle$$ (23)

We conclude from (22) and (20) that the average excitation level  W_res of the residual is given by

$$\langle \rangle {\frac{{1}}{{1 - \beta_{\rm res}}}} \langle \rangle \alpha_{{\rm res}}^{} \gamma_{{\rm res}}^{}$$ (24)

For continuum decay the order of operations in the endep code is as follows. We begin by calculating the average energy $\langle$E_yo$\rangle$ of the secondary particle by using the integral (16). We then use (24) to compute $\langle$W_res$\rangle$, the average excitation level of the residual. It sometimes happens that this calculation produces a negative result. If there is no further decay, we set $\langle$W_res$\rangle$ = 0 and endep prints a warning. Otherwise, the endep code keeps the negative $\langle$W_res$\rangle$ and just prints a warning message. Finally, the average kinetic energy of the residual is obtained from energy conservation (20) and (21) as

$$\langle \rangle \langle \rangle \langle \rangle$$ (25)

If there is no further particle emission, the average energy to the gammas, $\langle$E_$\gamma$$\rangle$, is taken to be $\langle$E_$\gamma$$\rangle$ = $\langle$W_res$\rangle$. Otherwise, the average kinetic $\langle$E_res$\rangle$ and excitation $\langle$W_res$\rangle$ energies are used in the modelling of the continuum decay process, as described in the next section.