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 $s_p$ is the spin transfer of the projectile $I_p$, $s_t$ is the spin transfer of the target $I_t$, and $\ell$ is the orbital angular momentum transfer. These coupled operators are defined following in Bohr & Mottelson, Vol. 1, section 1A-5c, as

  $$(F_{\lambda_1} G_{\lambda_2})_{\lambda \mu} = \sum_{\mu_1 \mu_2}... ...a_2 \mu_2 \vert \lambda \mu \rangle F_{\lambda_1\mu_1} G_{\lambda_2\mu_2} \ ,$$ (13)

applied for the case of $F_{s_t} = [\ell,s_p]s_t$ and $G_{s_t}=s_t$. The overall ${\bf S}$ operator is a monopole (scalar), for which the tensor product is thus of the kind

  $$(F_{\lambda} G_{\lambda})_{00} = (2\lambda+1)^{-1/2} \sum_{\mu} (-1)^{\lambda-\mu} ~ F_{\lambda\mu} G_{\lambda-\mu} \ .$$ (14)

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:

  $$\langle j_f m_f \vert \hat{O}_{\lambda \mu} \vert J_i m_i \rangle... ...j_f m_f \rangle ~ \langle j_f \vert\vert \hat{O}_\lambda \vert\vert j_i\rangle$$ (15)

The matrix elements of this operator are

$$\langle(LI_p)J,I_t;J_TM_T \vert ~{\bf S}([\ell,s_p]s_t,s_t) ~\vert (L'I_p')J',I_t';J_TM_T \rangle$$

$$= (-1)^{s_t+J_T+J'+I_t} \left\{ \begin{array}{ccc} J' & I_t' & J_T ... ...rac{1}{\sqrt{4\pi}} ~ \hat{\ell} \hat{L} \langle L 0 \ \ell 0 \vert L'0 \rangle$$

$$\times \langle I_p\vert\vert s_p\vert\vert I_p'\rangle \langle I_t\vert\vert s_t\vert\vert I_t'\rangle$$ (16)

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 from data files.

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

• the projectile diagonal reduced matrix element $\langle I_p\vert\vert s_p=0\vert\vert I_p\rangle = \hat{I_p}$ is supplied by Fresco,
• the factor $1/\sqrt{4\pi}$ is omitted above,
• and an additional symmetric factor of $((2I_t'+1)(2I_t+1))^{1/4}$ is supplied to allow the monopole radial form factors to have their physical values for any target spin.
• The forward and reverse form factors should be identical for Hermitian couplings.

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

• the target diagonal reduced matrix element $\langle I_t\vert\vert s_t=0\vert\vert I_t\rangle = \hat{I_t}$ is supplied by Fresco,
• the factor $1/\sqrt{4\pi}$ is omitted above,
• and an additional symmetric factor of $((2I_t'+1)(2I_t+1))^{1/4}$ is supplied to allow the monopole radial form factors to have their physical values for any target spin. The forward and reverse form factors should be identical for Hermitian couplings.