3 Model Schrödinger Equation

The Schrödinger equation for the model wave function $\Psi_{\rm model}$ at total energy E is

$$[E - H_{\rm opt}(E)] \Psi_{\rm model} = 0$$ (14)

We use the expansion of Eq. (1) in square-integrable states $\phi$ of the participating nuclei. Using for each pair i of nuclei the `channel optical Hamiltonians' H_i defined above (Eqs. 12 or 13), Eq. (14) becomes

$$\sum_i^N [T_i({\bf R}_i) + V_i -E_i] \phi_{pi}\phi_{ti} \psi_i({\bf R}_i) = 0$$ (15)

where in each channel there is an asymptotic kinetic energy E_i = E - e_pi-e_ti, corresponding to momentum ${\bf k}_i= (2\mu_i E_i / \hbar^2)^{1/2}$. Operating on Eq. (15) to the left by $\langle \phi_{pj}\phi_{tj} \vert$, we obtain (after swapping the i and j indices)

$$[T_i + U_i -E_i] \psi_i({\bf R}_i) + \sum_j V_{ij} \psi_j({\bf R}_j) = 0$$ (16)

$$\mbox{with } V_{ij} \equiv \langle \phi_{pi}\phi_{ti} \vert H_m - E \vert\phi_{pj}\phi_{tj}\rangle$$

Here, the Hamiltonian H_m in the coupling operator V_ij takes one of the forms of Eq. (13): choosing m=i is called the post form, and choosing m=j the prior form; the two alternatives should give the same results. (If i and j describe different inelastic states of the same nuclei, then ${\bf R}_i\equiv {\bf R}_j$, we can use orthogonality of $\phi_{ni}$ and $\phi_{nj}$, and the coupling V_ij is just a local function of ${\bf R}_i$:

$$V_{ij}({\bf R}_i) = \langle \phi_{pi}\phi_{ti} \vert V_i \vert\phi_{pj}\phi_{tj}\rangle \ .)$$ (17)

This set of coupled equations (16) must be solved with boundary conditions at large radii consisting of an incoming plane wave in channel (say) i₀, and outgoing spherical waves in all other channels:

$$\psi_{ii_0}({\bf R}_i) =_{R_i\rightarrow\infty} \delta_{i_0i} e^{i{\bf k}_{i_0}\cdot{\bf R}_i} + f_{ii_0}(\theta) e^{ik_iR_i}$$ (18)

Since $\phi_{pi}$, $\phi_{ti}$, $\phi_{pj}$ and $\phi_{tj}$ are all antisymmetrised internally, the cross section depends on the number of identical nucleons (or nucleonic clusters) that may be transferred in the reaction (see Austern [1, §3.5] or Satchler[2, §2.11.3.2]). If we define n_pi and n_ti as the numbers of nucleons (or clusters) within respectively the projectile and target states i that are identical to the transferred set, then the scattering amplitude f_ii0 for each channel i directly gives the cross section in that channel as

$$\frac{d\sigma_i(\theta)}{d\Omega} = \frac{k_i}{\mu_i}\frac{\mu_{i... ..._0}} \frac{n_{pi_0}!n_{ti_0}!}{n_{pi}!n_{ti}!} ~ \vert f_{ii_0}(\theta) \vert^2$$ (19)

(neglecting a possible elastic Coulomb amplitude in the incoming channel i₀). Partial wave expansions for the $\Psi_i$, and corresponding expressions for the $f_{ii_0}(\theta)$ will be presented in section 6.