An analytical formula is derived for the amplitude for transfer of a nucleon in quasi-elastic reactions between heavy ions. The derivation takes advantage of the semiclassical conditions found in peripheral collisions between heavy ions. The relative motion of the two nuclei is treated classically and the transfer amplitude is calculated by a perturbation method. Under the approximation of small overlap between the nuclear potentials, the semiclassical amplitude is reduced to a surface integral. This can be calculated analytically by using Hankel function forms for the bound-state wavefunctions and by approximating the actual orbit by a constant velocity orbit tangential to it at the distance of closest approach. These approximations seem reasonable in strong absorption conditions. Corrections to the formula of the amplitude are evaluated. The analytical form of the amplitude exhibits an exponential behaviour as a function of the distance of closest approach. The decay constant of the exponential is given explicitly and it is found to be an important parameter of the reaction. Kinematical conditions for maximum transfer are derived which relate the incident energy to the reaction Q-value. The physical interpretation of the amplitude is discussed. In the case of proton transfer, the effect of Coulomb potential results in a shift of the binding energy of the proton. With this prescription we still obtain the same form of the transfer amplitude for both neutrons and protons. The formula for the semiclassical tranfer amplitude is used to calculate angular distributions within a simplified formalism derived from the distorted wave Born approximation (DWBA). The reactions considered are 208Pb(16O,l50)209Pb, 26Mg(11B,10B) 27Mg and 34S(32 S, 33S)33S for neutron transfer and 208Pb(16O, 15N)209Bi for proton transfer. It is found that the shapes of the present angular distributions agree with full DWBA calculations but the magnitude of the former depends on whether the distance of closest approach is that of the initial channel, thefinal channel or some average of the two. Conditions for the selective population of definite states are discussed in relation to the reaction Q-value, energy and initial and final states involved. It is found that an inversion of the selectivity with respect to the spins of the initial and final state occurs when the energy of relative motion at distance of closest approach equals the reaction Q-value. An approximate formula for the angle-integrated cross section has also been derived.
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