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How does one obtain the isoscalar and isovector terms of the nucleus-nucleus interaction potential and what do they signify?

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2 Answers 2

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Optical model potentials originally were invented to describe elastic scattering of a single nucleon (neutron or proton) off a complex nucleus. More recently, the formalism has been modified to include the scattering of heavy ions as well as nucleons. The assumption made in the model is that nuclear matter inside a heavy nucleus is at least partially transparent so that elastic scattering can be treated via a single particle potential and the inelastic (absorptive) channels can be simulated by adding an imaginary part to the single particle potential. Such assumptions clearly have a limited range of validity (generally low to intermediate energies where excitation and/or fragmentation of the target nucleus is not the dominant contributor to the cross-section).

A broad range of potential models have been employed under this heading. These range from purely phenomenological potentials based mostly upon general knowledge (experimental) of the density of nuclear matter inside complex nuclei to mostly derived potentials employing experimental nucleon-nucleon scattering aplitudes folded with single-nucleon densities from theoretical many-body treatments (Hartree-Fock or self-consistent mean field) of nuclear ground state densities. These theoretical effective interaction models of the nuclear ground state are quite good at describing nuclear charge distributions (obtained from analysis of elastic electron scattering from nuclear targets) and other known properties like single-particle separation energies and total nuclear binding energies. This latter approach is referred to as the Impulse Approximation Optical Potential and is mostly derived from experimental data (except for the ground state wave functions) and free of variable parameters. This approach proved quite effective at discriminating between competing models of the nuclear ground state during the early 1980s (see
http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.50.1443)

Isospin dependence of the optical model potential arises from the fact that nucleon-nucleon scattering amplitudes differ for proton-proton, neutron-proton, and neutron-neutron scattering (proton-proton scattering is directly observable, proton-neutron scattering data is inferred from subtracting the P-P amplitudes from the proton-deuteron scattering data, and neutron-neutron scattering is assumed to be equivalent to P-P scattering without the coulomb contribution). In any case, isoscalar refers to the part of the potential that makes no distinction between protons and neutrons in the target (their densities are summed). Isovector means that neutron and proton constituents are treated differently (their densities are subtracted and folded with the isovector part of the N-N scattering amplitude).

For heavy ion scattering the situation is a little more complicated. Here one has to worry about excitation and/or fragmentation of the projectile as well as the target in determining the range of validity of an optical model approach. Nevertheless, so-called double folding techniques have been developed for analyses of elastic heavy ion scattering. Here one calculates a potential by folding the isoscalar and isovector components of the N-N scattering amplitude over the densities of both the projectile and target ground state densities. The isoscalar part depends simply on the sum of neutron and proton densities within both target and projectile while the isovector part makes use of the differences between these densities within both.

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  • $\begingroup$ No i specifically meant nucleus-nucleus. Could we divide the optical potential (say of Woods Saxon form) into isoscalar and isovector terms for heavy ion scattering as well? How does one arrive at the isoscalar and isovector terms? $\endgroup$
    – Ana
    Jan 26, 2016 at 4:53
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    $\begingroup$ @Ana I will edit my answer to address the nucleus-nucleus case when I get a chance. Basically the ideas will be the same as above but now you have to worry about different neutron and proton densities in both the projectile and the target. $\endgroup$ Jan 26, 2016 at 13:45
  • $\begingroup$ @Ana Sorry for the delay, but my edited version is posted. $\endgroup$ Jan 29, 2016 at 15:43
  • $\begingroup$ Thank you for the explanation. Could you provide some references related to the separation between the isoscalar and isovector parts? Also, suppose we use two different projectiles on the same target; then can it be deduced if one projectile is more sensitive to the isoscalar part and one to the isovector part? $\endgroup$
    – Ana
    Jan 30, 2016 at 7:42
  • $\begingroup$ A projectile with even numbers of neutrons and protons on a target with even numbers of neutrons and protons would be mostly sensitive to the isoscalar part. As the neutron/proton ratio deviates from 1 for either the projectile or the target, the role of the isovector part increases. $\endgroup$ Jan 30, 2016 at 15:28
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For nucleus-nucleus it is not common (although it is necessary for reactions such as HI charge-exchange or with exotic neutron/proton rich nuclei) to include isospin-dependent terms in the optical potential since their strength proportional to (Np-Zp)/Ap*(Nt-Zt)/At with the subscripts p, t are the projectile and target, respectively (Lane formalism). For more fundamental origin of isospin-dependent term in nucleus-nucleus, you can construct them from effective NN interaction using double-folding potential. For more detail in isospin-dependent double-folding potential, you can see Brandan and Satchler, Physics Reports 285, 143, 1997 and more specifically, Khoa et al., Nuclear Physics A, 602, 98, 1996 and Loc et al., Physical Review C, 89, 024317, 2014.

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