3.3 DWBA

Let consider the transfer reaction

$$A+b\rightarrow a+B\,\,\,\,(A=a+v,\, B=b+v),$$ (37)

When the coupling to intermediate channels is weak, it is reasonable to evaluate the transition amplitude in Born Approximation (BA). In the case of rearrangement reactions there are several ways to describe the interaction between the different fragments, one for each partition. For example, if we choose to describe the scattering in terms of the nuclei of the entrance partition, the projectile target interaction will be written as

V_Ab=V_vb+U_ab (38)

The interaction V_vb is the the potential which binds the v valence particle to the b core. In general, it will be described as a real potential (for which we use the notation V). The potential U_ab is the optical potential describing the scattering between b and v. It will typically contain both real and imaginary parts (we use the letter U). In this representation, known as prior form, the transition amplitude for the transfer process is given by

$$\langle\chi_{\beta}^{(+)}\phi_{a}\phi_{B}\vert V_{vb}+U_{ab}-U_{\alpha}\vert\chi_{\alpha}^{(+)}\phi_{A}\phi_{b}\rangle$$ T_prior =

$$\int d\mathbf{R}_{\alpha}d\mathbf{R}_{\beta}\chi_{\beta}^{(-)}(\m... ...mathbf{R}_{\beta},\mathbf{R}_{\alpha})\chi_{\alpha}^{(+)}(\mathbf{R}_{\alpha}),$$ = (39)

with

$$I_{\beta\alpha}(\mathbf{R}_{\beta},\mathbf{R}_{\alpha})=(\phi_{a}\phi_{B}\vert V_{vb}+U_{ab}-U_{\alpha}\vert\phi_{A}\phi_{b})$$ (40)

Analogously, for the exit channel: V_aB=V_av+U_ab. In this case, the expression (39) reduces to:

$$T_{\textrm{post}}=\langle\chi_{\beta}^{(+)}\phi_{a}\phi_{B}\... ...{ab}-U_{\beta}\vert\chi_{\alpha}^{(+)}\phi_{A}\phi_{b}\rangle.$$ (41)

where, $U_{\beta}$ is the optical potential describing the elastic scattering in the exit channel.

In either prior and post form the differential cross section is calculated as:

$$\frac{d\sigma}{d\Omega}=\frac{\mu_{\beta}\mu_{\alpha}}{(2\pi... ...{\alpha}}\right)\vert T(\mathbf{k}_{\beta},\mathbf{k}_{\alpha})$$ (42)

According the previous expressions a basic ingredient required to calculate the transfer amplitude in the prior DWBA approximation is are the internal wave functions for the initial ( $\phi_{A},\,\phi_{b}$) and final ( $\phi_{a},\,\phi_{B}$) nuclei. In this scheme, the valence particle v is bound to the b core to give the composite B. In the simplest picture, the valence particle particle can be considered a pure single-particle state. This means that, within in this extreme model, there is only one possible configuration of the core and the valence particle to give the nucleus B and thus, the wave function for this nucleus can be written as:

$$\phi_{B}^{JM}(\xi,\mathbf{r})=\left[\phi_{b}^{I}(\xi)\otimes\varphi_{\ell sj}(\mathbf{r})\right]_{JM}$$ (43)

In a more realistic model, however, the state of the composite contains components of many single-particle states coupled to all possible core states and thus, the wave function $\phi_{B}^{JM}(\xi,\mathbf{r})$ is built as a superposition of the form:

$$\phi_{B}^{JM}(\xi,\mathbf{r})=\frac{1}{\sqrt{n_{B}}}\sum_{I\... ..._{b}^{I}(\xi)\otimes\varphi_{\ell sj}(\mathbf{r})\right]_{JM},$$ (44)

where the coefficients $A_{\ell sj}^{IJ}$ are the so called coefficients of fractional parentage (cfp) or spectroscopic amplitudes, and their square moduli $S_{\ell sj}^{IJ}=\vert A_{\ell sj}^{IJ}\vert^{2}$ the spectroscopic factors. The spectroscopic factor $S_{\ell sj}^{IJ}$ can be regarded as the probability of finding the valence particle v in a single particle state $\ell,\, s,\, j$ coupled to the core with spin I. The quantity n_B is the number of nucleons (or clusters!) in the composite system that are identical to that transferred. The factor $1/\sqrt{n_{B}}$ is introduced just for convenience.

Example: the ground state of the ²⁰⁹Bi nucleus can modeled to a good approximation as a valence proton coupled to the core ²⁰⁸Pb. Due to the double close shell nature of the core, the valence proton can be regarded as a nearly pure single-particle state, occupying the 1h_9/2 orbital. Then, we have: I=0, J=9/2, $(s,\ell,j)=(1/2,\,5,\,9/2)$ and $S_{\ell sj}^{IJ}\approx1$. Moreover, as there is only one particle with this configuration, n_B=1 in this case.

Notice that the integral $I_{\beta\alpha}(\mathbf{R}_{\beta},\mathbf{R}_{\alpha})$ involve the overlap between the composite and core wave functions. Using the expansion (44) the integral on the core internal variables $\xi$ can be explicitly performed giving just unity by normalization.

$$\left(\phi_{B}^{JM}(\xi,\mathbf{r}),\phi_{b}^{I}(\xi)\right)... ...B}}}\sum_{\ell j}A_{\ell sj}^{IJ}\varphi_{\ell sj}(\mathbf{r})$$ (45)

The bound wave functions $\varphi_{\ell sj}(\mathbf{r})$ obey the Schrodinger equation³:

$$\left[T+V_{vb}(\mathbf{r})+\epsilon-E\right]\varphi_{\ell sj}(\mathbf{r})=0\,,$$ (46)

where $\epsilon$ is the binding energy of the valence particle.

The information required by FRESCO to construct the wave functions is provided in the section of form factors, which corresponds to the namelist &overlap/ in the fortran 90 version. However, the cfp's and the valence-target and core-target potentials are given in the couplings section, through the &coupling/ namelist.

It is important to note that the calculation of the transition amplitude involves the integration in the channel coordinates $\mathbf{R}_{\alpha}$ and $\mathbf{R}_{\beta}$ (see Fig. 3), which, after the integration on the angular coordinates, becomes a integral in $R_{\alpha}$ and $R_{\beta}$. Then, the coupled channels equations becomes:

$$\left[E_{\alpha}-T_{\alpha L}(R_{\alpha})-U_{\alpha}(R_{\alpha})\right]f_{\alpha}(R_{\alpha}) \int_{0}^{R_{m}}U_{\alpha\beta}(R_{\alpha},R_{\beta})f_{\beta}(R_{\beta})dR_{\beta}$$ =

$$\left[E_{\beta}-T_{\beta L}(R_{\beta})-U_{\beta}(R_{\beta})\right]f_{\beta}(R_{\beta}) \int_{0}^{R_{m}}U_{\beta\alpha}(R_{\alpha},R_{\beta})f_{\alpha}(R_{\alpha})dR_{\alpha}\,,$$ = (47)

The integrals

$$S(R_{\beta})=\int_{0}^{R_{m}}U_{\beta\alpha}(R_{\alpha},R_{\beta})f_{\alpha}(R_{\alpha})dR_{\alpha}$$

are evaluated in steps of HCM and up to RMATCH. Previously, Fresco has to evaluate and store the function $U_{\beta\alpha}(R_{\alpha},R_{\beta})$. It results numerically advantageous to perform the change of variables: $R_{\beta}\rightarrow D_{\beta\alpha}=R_{\beta}-R_{\alpha}$ and thus the functions that are actually stored are $U_{\beta\alpha}'(D_{\beta\alpha},R_{\beta}).$ The variable $D_{\beta\alpha}$ is discretized in intervals of HNL, from CENTRE-RNL/2 to CENTRE+RNL/2, i.e., range of RNL centred at CENTRE.

Figure 3: Relevant coordinates for the description of the transfer reaction