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Appendix V : KIND=1 Spin Transfer Couplings

The definition of the KIND=1 spin transfer couplings is not given in the Computer Physics Reports article, so in versions after March 1998 these are redefined for IP3=0 or 1, and new Racah algebra factors included.


We want to calculate the coupling interactions of the monopole operator $ {\bf S}([\ell,s_p]s_t,s_t) $, where sp is the spin transfer of the projectile Ip, st is the spin transfer of the target It, and $\ell$ is the orbital angular momentum transfer. These coupled operators are defined following in Bohr & Mottelson, Vol. 1, section 1A-5c, as

\begin{displaymath}
(F_{\lambda_1} G_{\lambda_2})_{\lambda \mu}
= \sum_{\mu_1 \...
... \lambda \mu \rangle F_{\lambda_1\mu_1} G_{\lambda_2\mu_2} \ ,
\end{displaymath} (1)

applied for the case of $F_{s_t} = [\ell,s_p]s_t$ and Gst=st. The overall ${\bf S}$ operator is a monopole (scalar), for which the tensor product is thus of the kind
\begin{displaymath}
(F_{\lambda} G_{\lambda})_{00}
= (2\lambda+1)^{-1/2} \sum_{\mu} (-1)^{\lambda-\mu} ~ F_{\lambda\mu}
G_{\lambda-\mu} \ .
\end{displaymath} (2)

This differs from common definitions (eg of Satchler) by a factor of $ (-1)^\lambda (2\lambda+1)^{-1/2} $.


Reduced matrix elements are defined everywhere in FRESCO by:

\begin{displaymath}
\langle j_f m_f \vert \hat{O}_{\lambda \mu} \vert J_i m_i \r...
...~
\langle j_f \vert\vert \hat{O}_\lambda \vert\vert j_i\rangle
\end{displaymath} (3)


The matrix elements of this operator are

    $\displaystyle \langle(LI_p)J,I_t;J_TM_T \vert ~{\bf S}([\ell,s_p]s_t,s_t) ~\vert (L'I_p')J',I_t';J_TM_T \rangle$  
  = $\displaystyle (-1)^{s_t+J_T+J'+I_t}
\left\{ \begin{array}{ccc}
J' & I_t' & J_...
...rac{1}{\sqrt{4\pi}} ~ \hat{\ell} \hat{L} \langle L 0 \ \ell 0 \vert L'0 \rangle$  
    $\displaystyle \times \langle I_p\vert\vert s_p\vert\vert I_p'\rangle \langle I_t\vert\vert s_t\vert\vert I_t'\rangle$ (4)

In using KIND=1 couplings with IP3=0, the first line of these factors is generated automatically.
The product of the reduced matrix elements for the intrinsic nuclear states, $\langle I_p\vert\vert s_p\vert\vert I_p'\rangle \langle I_t\vert\vert s_t\vert\vert I_t'\rangle$, has to be included explicitly in the factor FSCALE, or in the radial form factors. The radial shapes have to be read in using Cards 16.7.

With IP3=2, implying jlmP input for target-only couplings,

With IP3=3, implying a similar input for projectile-only couplings,


next up previous
Next: Appendix VI : Search Up: Fresco Previous: Appendix IV : FORTRAN
Ian Thompson 2011-09-08