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4 Transition Amplitudes

Greens function methods may also be used to solve the coupled equations, and furthermore suggest various approximations which simplify the solution methods in many special cases of interest. First, therefore, we present in this section several exact results using T-matrix integrals derived from Greens function analysis. In the following section 5 we examine various consequent approximations that are often still accurate. If the full wave function $\Psi$ were known, then the scattering amplitude fi for the $i_0\rightarrow i$ reaction may be found from the T-matrix by the equivalence
$\displaystyle T_{i_0i} = - \frac{2 \pi \hbar^2}{\mu_i} f_i$     (20)

so that, written in terms of the transition amplitudes, the expression for the cross section becomes
$\displaystyle d\sigma_i(\theta_i)/d\Omega = \frac{\mu_i\mu_{i_0}}{(2\pi\hbar^2)...
...i}{k_{i_0}} ~\frac{n_{pi_0}!n_{ti_0}!}{n_{pi}!n_{ti}!} ~ \vert T_{i_0i}\vert ^2$     (21)

Expressions for the T transition amplitudes may be derived by using either plane waves or distorted waves in the exit channel. In addition, for transfer reactions where the channel Hamiltonians are different in the initial and final channels, we have a further choice of using either post or prior forms of the coupling. The post form uses the form of Hi for the exit channel, and the prior form the Hi0 from the entrance channel. The plane-wave post matrix element is
Ti0i = $\displaystyle \langle \phi_{pi}\phi_{ti}e^{i{\bf k}_i{\bf R}_i} \vert H_i-E \vert\sum_j \phi_{pj}\phi_{tj}\psi_j^{(+)} \rangle$ (22)
  = $\displaystyle \langle \phi_{pi}\phi_{ti}e^{i{\bf k}_i{\bf R}_i} \vert V_i \vert\sum_j \phi_{pj}\phi_{tj}\psi_j^{(+)} \rangle$ (23)

where the (+) superscript in $\psi_j^{(+)}$ indicates that they are found with plane incoming waves in the i0 channel. A prior matrix element uses the Hamiltonian Hi0 of the initial channel. Direct substitution in Eq. (22), however, affords no simplifications, so we insert it in the matrix element for the time-reversed reaction, and derive
$\displaystyle T_{i_0i}
= \langle \sum_j \phi_{pj}\phi_{tj}\psi_j^{(-)} \vert V_{i_0} \vert
\phi_{pi_0}\phi_{ti_0}e^{i{\bf k}_{i_0}{\bf R}_{i_0}}\rangle$     (24)

where the (-) superscript in $\psi_j^{(-)}$ indicates that it has an incident plane wave along ${\bf k}_{i_0}$ and incoming spherical waves e-ikiRi in all channels. Distorted-wave expressions may be found by replacing the exponential factors on the left sides by one-channel scattering waves ( $\chi_i^{(+)}({\bf R}_i)$ on the right sides and $\chi_i^{(-)}({\bf R}_i)$ on the left), found with some distorting potential Wi by $ [T_i + W_i - E_i]\chi_i({\bf R}_i)=0$. The distorted-wave post matrix element is then
$\displaystyle T_{i_0i}
= \langle \phi_{pi}\phi_{ti}\chi_i^{(-)} \vert V_i-W_i \vert\sum_j \phi_{pj}\phi_{tj}\psi_j^{(+)} \rangle$     (25)

and the equivalent prior form is
$\displaystyle T_{i_0i}
= \langle \sum_j \phi_{pj}\phi_{tj}\psi_j^{(-)} \vert V_{i_0}-W_{i_0} \vert
\phi_{pi_0}\phi_{ti_0}\chi_i^{(+)}\rangle$     (26)

The distorting potential Wi may be real or complex without affecting the validity of these matrix elements. All these four expressions are so far identical, and exactly equivalent to solving the coupled equations directly and using Eq. (20).
next up previous
Next: 5 Distorted Wave Born Up: Methods of Direct Reaction Previous: 3 Model Schrödinger Equation
Prof Ian Thompson 2006-02-08