Introduction

FRESCOX is a Coupled-Channels program that can have finite-range transfer interactions among any number of mass partitions, and any number of nuclear excitations in each partition.

This is accomplished by calculating and storing the kernels of the non-local interactions, and then solving the coupled-channels set iteratively. At each iteration the stored kernels are used to integrate the wave functions to generate the source terms for the next iteration. Pade approximants can be used if the iterations diverge because of strong couplings.

The program can also generate local couplings for the rotational or single-particle excitations of either the projectile or the target in any partition, the local form factors for zero-range or local-energy-approximation can also be used. These various local interactions may also be included iteratively (to give multistep DWBA), or alternatively the first few coupled channels may be blocked together and solved by exact coupled-channels methods.

Special treatment is provided for any long ranged Coulomb multipoles, using James Christley's coupled-Coulomb functions CRCWFN, and interpolation in partial waves is also possible.

The nonlocal kernels for single-particle transfers are calculated first at a much smaller number of interpolation points, and then expanded when necessary to calculate the source terms by integrating

$$S(R _ f ) = \int _ 0 ^ {R _ {match}} K _ {fi} (R _ f , R _ i ) u(R _ i ) dR _ i$$ (1)

where RMATCH and HCM, the step size, are given in the &Fresco namelist. Since the kernel function $K_{fi}(R_f,R_i)$ is usually rapidly varying with $D_{fi} = R_{f} - R_{i}$ (especially with heavy-ion reactions), and only slowly varying with $R_{f}$ (if $D_{fi}$ is constant), FRESCOX calculates and stores the function $K_{fi}'(R_{f},D_{fi})$ at intervals of RINTP in $R_{f}$, and intervals of HNL in $D_{fi}$. The $D_{fi}$ range considered is CENTRE-RNL/2 to CENTRE+RNL/2, i.e. range of RNL centred at CENTRE, and FRESCOX later suggests improved values for RNL & CENTRE. The HNL reflects to physical variation of $K_{fi}'$ with $D_{fi}$, and can be a fraction of HCM (for heavy ion reactions) or a multiple of HCM (for light ion reactions especially with `prior' interactions). If HNL is a submultiple of HCM, the program anticipates the interpolation of $u(R_{i})$ in the equation above, and only stores an effective kernel function at intervals of HCM.

The new variable MTMIN in the FRT version controls the method used to calculate the form factors of the transfer kernels. If the L-transfer value (estimated by $\ell_f + \ell_i$) is larger than or equal to MTMIN, then the m-dependent expressions for the spherical harmonics are used explicitly. Otherwise, the angular dependencies of the initial and final transfer states are transformed into those of the channel variables $R_{i}$ & $R_{f}$ using Moshinsky's solid-harmonic transformation, as given for example in Austern et al. Phys. Rev. 133 pp B 3 - 16. By careful control of the numerical approximations, the second method can be extended to find heavy-ion transfer form factors, but the maximum transferred angular momentum L is still limited by numerical cancellation errors arising from the finite roundoff errors in the computer. With 8-byte word lengths, the unit roundoff is approximately $10^{-15}$, and this limits the transferred L to 6 or less, so the default value of MTMIN is set to 6. The Moshinsky method is much faster when it is accurate, so the default value of MTMIN has been set as high as practical.

A general description of the formalisms used is given in I.J. Thompson, Coupled Reaction Channels Calculations in Nuclear Physics, Computer Physics Reports, 7 (1988) pp 167 - 212.