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3 Model Schrödinger Equation

The Schrödinger equation for the model wave function $\Psi_{\rm model}$ at total energy E is
$\displaystyle [E - H_{\rm opt}(E)] \Psi_{\rm model} = 0$     (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' Hi defined above (Eqs. 12 or 13), Eq. (14) becomes
$\displaystyle \sum_i^N [T_i({\bf R}_i) + V_i -E_i] \phi_{pi}\phi_{ti} \psi_i({\bf R}_i) = 0$     (15)

where in each channel there is an asymptotic kinetic energy Ei = E - epi-eti, 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)
    $\displaystyle [T_i + U_i -E_i] \psi_i({\bf R}_i) + \sum_j V_{ij} \psi_j({\bf R}_j) = 0$ (16)
  $\textstyle \mbox{with }$ $\displaystyle V_{ij} \equiv \langle \phi_{pi}\phi_{ti} \vert H_m - E \vert\phi_{pj}\phi_{tj}\rangle$  

Here, the Hamiltonian Hm in the coupling operator Vij 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 Vij is just a local function of ${\bf R}_i$:
$\displaystyle V_{ij}({\bf R}_i) = \langle \phi_{pi}\phi_{ti} \vert V_i \vert\phi_{pj}\phi_{tj}\rangle \ .)$     (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:
$\displaystyle \psi_{ii_0}({\bf R}_i) =_{R_i\rightarrow\infty} \delta_{i_0i} e^{i{\bf k}_{i_0}\cdot{\bf R}_i}
+ f_{ii_0}(\theta) e^{ik_iR_i}$     (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 npi and nti 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 fii0 for each channel i directly gives the cross section in that channel as
$\displaystyle \frac{d\sigma_i(\theta)}{d\Omega} = \frac{k_i}{\mu_i}\frac{\mu_{i...
..._0}}
\frac{n_{pi_0}!n_{ti_0}!}{n_{pi}!n_{ti}!} ~ \vert f_{ii_0}(\theta) \vert^2$     (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.
next up previous
Next: 4 Transition Amplitudes Up: Methods of Direct Reaction Previous: 2 Elimination of the
Prof Ian Thompson 2006-02-08