ipmliqdrop.doc 8-27-97

Independent Particle Motion in Atoms

It should be mentioned that there is an important question of whether the

idea of particles moving approximately INDEPENDENTLY in a potential makes

sense at all. In atoms, it is easy to argue in the affirmative, since there

the ACTUAL potential felt by each electron is dominated by that due to the

nucleus itself. The effect of electron screening does not change things very

much, (except for affecting the order of levels). Since the Coulomb

interactions have long range, the FORCES between electrons (which vary as 1/$%

r^{2}$) are relatively weak, and we do not have many true collisions. Thus

it makes good sense to speak of electron ORBITS, which is, of course, the

idea of an independent particle model.

Independent particle motion in solids

Electrons feel interaction due to nuclei and other electrons. In metals the free electron model works quite well. The Pauli principle plays a crucial role here. The effective potential felt by an electron can be obtained using the method of pseudopotentials. The attraction near a nucleus is largely

canceled by the presence of the other electrons. It can be shown that electrons are bound, but that the average potential is quite smooth.

Independent particle motion in nuclei

Naively, the prospects for obtaining independent particle motion in nuclei

appear not to be favorable. Here there is no common center of force, and

also, the internucleon forces are known to be strong and shortranged. One

would, therefore, expect that there should be frequent COLLISIONS between

nucleons. One can, of course, always define an AVERAGE potential. The

question is whether the ACTUAL potential felt by a nucleon is well described

by this average, or whether the FLUCTUATIONS in the potential are large.

Indeed, until about 1950, when the evidence for nuclear independent particle

motion became overwhelming, there was general belief that the nucleus is

really a strongly coupled system, something like a liquid drop. After the

discovery of the nuclear shell model, which gave clear indication that the

IPM is valid after all, (at least at low energies), it was indeed pointed

out by Weisskopf that the Pauli principle, by forbidding scattering into

states already occupied by other nucleons, helps to "save" IPM, but we will

see that this is not the whole story!

In addition to the Pauli principle, the nature of the internucleon forces

themselves may well help to make the IPM valid. These interactions may well

not be additive, on account of the underlying quark structure of nucleons.

(There may be some rough analogy here with the three body interactions

between atoms which arise from polarization effects. The two body term is

the well- known VanderWaals interaction. The three body interaction between

atoms is known to play a small but noticable role in determining the

saturation density of atomic matter.) Thus, besides the well-known NN

interactions, there are probably also NNN interactions. The NN interactions

are mainly attractive and the NNN interactions mainly repulsive. This is

roughly equivalent to having the NN interactions depend linearly on the

nucleon density. Thus the effective NN interaction in the nuclear interior

is much weaker than in free space. Thus the nucleons move independently in

the interior and interact only when at the surface, where the density is

lower.

One of the key elements in the success of the nuclear shell model is the long nucleon mean path in nuclear matter. That certainly contradicts the conventional wisdom held in the 1930's. But people in the 1930's also knew about the Pauli principle, and many perturbation calculations done at that time took it into account. Yet with the interactions used at that time, there must have been strong attractive correlations. The introduction of a short range repulsion in the 1950's certainly helped to make the shell model more understandable, but we still don't understand it in simple terms, even now in 1997. Probably the role of density dependence, or possibly three-body interactions, in reducing attractive correlations is still not quite as clear as it should be. It would be nice if the validity of the nuclear shell model were understood as well as, say, the validity of the free-electron model of metals. This would seem to be a crucial component of the bridge between nuclear forces and nuclear structure!