6 Partial-wave expansions

The total wave function is expanded in partial waves using a coupling order such as

$${\bf L}+ {\bf J}_p = {\bf J};\ \ \ {\bf J}+ {\bf J}_t = {\bf J}_T ,$$ (30)

which may be defined by writing

$$\psi_{i J_T}^{M_T} = \left \vert ~(L~J_p )J, J_t ; J_T \right\rangle$$ (31)

where J_p = projectile spin, J_t = target spin, L = orbital partial wave, and J_T = total system angular momentum. The set {i, (L J_p )J, J_t ; J_T } will be abbreviated by the single variable $\alpha$. Thus, in each partition the partial wave expansion of the wave function is

$$\psi_{i J_T}^{M_T} \left ( {\bf R}_i , \xi_p , \xi_t \right )= \sum_... ...i_{J_p} ( \xi_p )~\phi_{J_t} ( \xi_t )~ i^L Y_L^M ({\hat {\bf R}_i} ) \nonumber$$

$$f^{~i J_T}_{(LJ_p )J,J_t} (R_i)/{R_i} \left\langle LM J_p \mu_p \vert JM_J \right\rangle \left\langle JM_J J_t \mu_t \vert J_T M_T \right\rangle$$ (32)

here $\xi_p$ and $\xi_t$ are the internal coordinates of the projectile and target, and

$$f^{~i J_T}_{(LJ_p )J,J_t} (R) \equiv f _\alpha (R)$$ (33)

are the radial wave functions. The (optional) i^L factors are included to simplify the spherical Bessel expansion of the incoming plane wave. The wave function $\psi$ could also have been defined using the `channel spin' representation $\psi = \vert L, (J_p J_t)S; J_T\rangle$, which is symmetric upon projectile $\rightarrow$ target interchange except for a phase factor (-1)^{S - J_p - J_t}. The coupled partial-wave equations are of the form

$$\left [ E_{i} - T_{i L } {(R_i)}- U_i {(R_i)}\right ] f_\alpha {(R_i)}= \nonumber$$

$$\sum_{\alpha' , \Gamma> 0 } { i ^ {L' - L} ~ V^\Gamma_{\alpha :\alpha' } {(R_i)}f_{\alpha'} {(R_i)}}$$

$$\sum_{\alpha' ,i' \neq i } i ^ {L' - L} ~ {\int_ 0 ^ {R_m } { V_{\alpha :\alpha' }( R_i , {R_{i'}} ) f_{\alpha'} (R_{i'}) d R_{i'} }}$$ + (34)

where the partial-wave kinetic energy operator is

$$T_{i L} (R_i) = - {\hbar^2 \over {2 \mu_i}} ~ \left ( {d^2 \over dR_i^2} - {{L(L+1)} \over R_i^2} \right ) ,$$ (35)

U_i (R_i) is the diagonal optical potential with nuclear and Coulomb components, and R_m is a radius limit larger than the ranges of U_i (R_i) and of the coupling terms. The $V^\Gamma_{\alpha :\alpha' } {(R_{i'})}$ are the local coupling interactions of multipolarity $\Gamma$, and the $V_{\alpha :\alpha' }( R_i , {R_{i'}} )$ are the non-local couplings between mass partitions that arise from particle transfers. The equations (34) are in their most common form; they become more complicated when non-orthogonalities are included by the method of section 8.1. For incoming channel $\alpha_0$, the solutions $f_\alpha {(R_i)}$ satisfy the boundary conditions when R_i > R_m of

$$f_\alpha {(R_i)}= {i \over 2} \left [ \delta_{\alpha\alpha_0 } H^... ...i}} ( k_i R_i ) - S_{\alpha\alpha_0 } H^{(+)}_{L {\eta_i}} ( k_i R_i ) \right ]$$ (36)

where $H^{(-)}_{L \eta}$ and $H^{(+)}_{L \eta}$ are the Coulomb functions with incoming and outgoing boundary conditions respectively, and

$$\eta _i = {{2 \mu_i} \over \hbar^2} ~~~ { Z_{pi} Z_{ti} e^2 \over 2 k _i }$$ (37)

is the Sommerfeld parameter for the Coulomb wave functions. In terms of the S-matrix elements $S_{\alpha\alpha_0 }$, and for coupling order of Eq. (30), the scattering amplitudes for transitions to projectile & target m-states of m, M to m', M' are

$$f_{m' M' : mM}^{ii_0} (\theta) = \delta_{\alpha\alpha_0} F_c (\theta) +$$

$$\sum_{LL'JJ'J_T } \langle L0 J_p m \vert Jm\rangle\langle Jm J_t M \vert J_T M_T\rangle$$

$$\langle L' M_{L'} J'_ p m' \vert J' M_{L'} + m'\rangle \langle J' M_{L'} +m' J'_ t M' \vert J_T M_T\rangle$$

$${4 \pi \over k_{i_0}} %% \sqrt {\frac{k'}{\mu'}\frac{\mu}{k}} ... ...a_L - \sigma_0 )} e^{i( \sigma'_ {L'} - \sigma'_ 0 )}\sqrt {{2L+1 \over 4 \pi}}$$

$$\left ( {i \over 2} \right ) \left [ \delta_{\alpha',\alpha} - S^{J_T}_{\alpha'\alpha} \right ] ~ Y_{L'}^{m' +M' -m-M}(\theta,0)$$ (38)

where

$$\sigma _L = \arg \Gamma(1+L+i\eta_i)$$ (39)

are the Coulomb phase shifts and the Coulomb amplitude F_c is

$$F_c ( \theta ) = - {\eta_{i_0} \over 2k_{i_0}} ~~ { \exp (-2 i \eta_{i_0} \ln(\sin \theta /2)) \over \sin^2 \theta /2}\ .$$ (40)

The corresponding differential cross section is

$${d \sigma_{ii_0}(\theta) \over d \Omega} \frac{k_i}{\mu_i}\frac{\mu_{i_0}}{k_{i_0}} \frac{n_{pi_0}!n_{ti_0}!}{n_{pi}!n_{ti}!} {1 \over (2J_p + 1)(2J_t + 1) } ~$$ =

$$\sum_{m' M' m M} \!\!\! \left \vert f_{m' M' : mM}^{ii_0} (\theta) \right \vert ^2 .$$ x (41)

The spherical tensor analysing powers T_kq describe how the outgoing cross section depends on the incoming polarisation state of the projectile. If the spherical tensor $\tau_{kq}$ is an operator with matrix elements

$$(\tau_{kq})_{m m''} = \sqrt{2k+1}\langle J_p m k q \vert J_p m''\rangle,$$

we have

$$T_{kq} (\theta) {Tr ({\bf f} \tau_{kq} {\bf f}^+)} \over {Tr ({\bf f} {\bf f}^+)}$$ =

$$\hat{k} \!\!\!\!\! \sum_{m' M' m M} \!\!\!\!\! f_{m' M' : m M}^* (\theta) \langle J_p m k q \vert J_p m''\rangle f_{m' M' : m'' M} (\theta)$$ =

$$\div \sum_{m' M' m M}\vert f_{m' M' : mM} (\theta) \vert ^2$$