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Next: 7 Channel couplings Up: Methods of Direct Reaction Previous: 5 Distorted Wave Born


6 Partial-wave expansions

The total wave function is expanded in partial waves using a coupling order such as
$\displaystyle {\bf L}+ {\bf J}_p = {\bf J};\ \ \ {\bf J}+ {\bf J}_t = {\bf J}_T ,$     (30)

which may be defined by writing
$\displaystyle \psi_{i J_T}^{M_T} =
\left \vert ~(L~J_p )J, J_t ; J_T \right\rangle$     (31)

where Jp = projectile spin, Jt = target spin, L = orbital partial wave, and JT = total system angular momentum. The set {i, (L Jp )J, Jt ; JT } will be abbreviated by the single variable $\alpha$. Thus, in each partition the partial wave expansion of the wave function is
  $\textstyle \psi_{i J_T}^{M_T}
\left ( {\bf R}_i , \xi_p , \xi_t \right )=
\sum_...
...i_{J_p} ( \xi_p )~\phi_{J_t} ( \xi_t )~
i^L Y_L^M ({\hat {\bf R}_i} )
\nonumber$    
  $\textstyle f^{~i J_T}_{(LJ_p )J,J_t} (R_i)/{R_i}
\left\langle LM J_p \mu_p \vert JM_J \right\rangle
\left\langle JM_J J_t \mu_t \vert J_T M_T \right\rangle$   (32)

here $ \xi_p$ and $ \xi_t$ are the internal coordinates of the projectile and target, and
$\displaystyle f^{~i J_T}_{(LJ_p )J,J_t} (R)
\equiv f _\alpha (R)$     (33)

are the radial wave functions. The (optional) iL 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 - Jp - Jt. The coupled partial-wave equations are of the form
    $\displaystyle \left [ E_{i} - T_{i L } {(R_i)}- U_i {(R_i)}\right ]
f_\alpha {(R_i)}= \nonumber$  
    $\displaystyle \sum_{\alpha' , \Gamma> 0 }
{ i ^ {L' - L} ~
V^\Gamma_{\alpha :\alpha' } {(R_i)}f_{\alpha'} {(R_i)}}$  
  + $\displaystyle \sum_{\alpha' ,i' \neq i } i ^ {L' - L} ~
{\int_ 0 ^ {R_m }
{ V_{\alpha :\alpha' }( R_i , {R_{i'}} )
f_{\alpha'} (R_{i'})
d R_{i'} }}$ (34)

where the partial-wave kinetic energy operator is
$\displaystyle T_{i L} (R_i) = - {\hbar^2 \over {2 \mu_i}} ~
\left ( {d^2 \over dR_i^2} - {{L(L+1)} \over R_i^2} \right ) ,$     (35)

Ui (Ri) is the diagonal optical potential with nuclear and Coulomb components, and Rm is a radius limit larger than the ranges of Ui (Ri) 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 Ri > Rm of
$\displaystyle f_\alpha {(R_i)}=
{i \over 2} \left [ \delta_{\alpha\alpha_0 }
H^...
...i}} ( k_i R_i )
- S_{\alpha\alpha_0 }
H^{(+)}_{L {\eta_i}} ( k_i R_i )
\right ]$     (36)

where $ H^{(-)}_{L \eta}$ and $ H^{(+)}_{L \eta}$ are the Coulomb functions with incoming and outgoing boundary conditions respectively, and
$\displaystyle \eta _i = {{2 \mu_i} \over \hbar^2} ~~~
{ Z_{pi} Z_{ti} e^2 \over 2 k _i }$     (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
    $\displaystyle f_{m' M' : mM}^{ii_0} (\theta) =
\delta_{\alpha\alpha_0} F_c (\theta) +$  
    $\displaystyle \sum_{LL'JJ'J_T }
\langle L0 J_p m \vert Jm\rangle\langle Jm J_t M \vert J_T M_T\rangle$  
    $\displaystyle \langle L' M_{L'} J'_ p m' \vert J' M_{L'} + m'\rangle
\langle J' M_{L'} +m' J'_ t M' \vert J_T M_T\rangle$  
    $\displaystyle {4 \pi \over k_{i_0}} %% \sqrt {\frac{k'}{\mu'}\frac{\mu}{k}}
...
...a_L - \sigma_0 )}
e^{i( \sigma'_ {L'} - \sigma'_ 0 )}\sqrt {{2L+1 \over 4 \pi}}$  
    $\displaystyle \left ( {i \over 2} \right )
\left [ \delta_{\alpha',\alpha} - S^{J_T}_{\alpha'\alpha} \right ]
~ Y_{L'}^{m' +M' -m-M}(\theta,0)$ (38)

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

are the Coulomb phase shifts and the Coulomb amplitude Fc is
$\displaystyle F_c ( \theta ) = - {\eta_{i_0} \over 2k_{i_0}} ~~
{ \exp (-2 i \eta_{i_0} \ln(\sin \theta /2)) \over \sin^2 \theta /2}\ .$     (40)

The corresponding differential cross section is
$\displaystyle {d \sigma_{ii_0}(\theta) \over d \Omega}$ = $\displaystyle \frac{k_i}{\mu_i}\frac{\mu_{i_0}}{k_{i_0}}
\frac{n_{pi_0}!n_{ti_0}!}{n_{pi}!n_{ti}!}
{1 \over (2J_p + 1)(2J_t + 1) } ~$  
  x $\displaystyle \sum_{m' M' m M} \!\!\!
\left \vert f_{m' M' : mM}^{ii_0} (\theta) \right \vert ^2 .$ (41)

The spherical tensor analysing powers Tkq 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
$\displaystyle (\tau_{kq})_{m m''} =
\sqrt{2k+1}\langle J_p m k q \vert J_p m''\rangle,$      

we have
$\displaystyle T_{kq} (\theta)$ = $\displaystyle {Tr ({\bf f} \tau_{kq} {\bf f}^+)} \over
{Tr ({\bf f} {\bf f}^+)}$  
  = $\displaystyle \hat{k} \!\!\!\!\!
\sum_{m' M' m M} \!\!\!\!\! f_{m' M' : m M}^* (\theta)
\langle J_p m k q \vert J_p m''\rangle
f_{m' M' : m'' M} (\theta)$  
    $\displaystyle \div \sum_{m' M' m M}\vert f_{m' M' : mM} (\theta) \vert ^2$  


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Next: 7 Channel couplings Up: Methods of Direct Reaction Previous: 5 Distorted Wave Born
Prof Ian Thompson 2006-02-08