« PreviousContinue »
The first contribution (DT) comes from processes during which the target gets dissociatedwithout producing any new particles:
-the second contribution (PROD) takes care of production processes coming from the nucleons of the target nucleus:
where the "reflection coefficient" (b, si...SA) (compare formulae (3.5)) is related to r(b, si... SA) as follows:
1-7(b, si... SA) = r (b, si... SA).
Hence 1(b, s1... SA) 2 gives (compare the formulae (3.5) of the standard partial wave analysis) the production cross section at the impact parameter b≈ (l+1⁄2)/k with all nucleons frozen at the positions $1, SA.
So, in our model there are three different contributions.
But as long as we construct the profiles of the target nucleus from profiles of elastic scattering, the processes like the one shown in figure 7 (with excited states of the projectile present at intermediate steps) are excluded.
They are the source of the so-called inelastic screening (or inelastic shadowing) phenomenon . In order to include them we have to ascribe some kind of structure to the incident particle. Earlier in these notes we gave some examples of such cases.
To analyze this problem in more detail, one has to link it with diffractive production processes and we shall postpone such a discussion until our analysis of such processes. Here, let us make only the following points:
(i) Diffractive production processes are presumably weak (at least at energies of a few GeV) compared to elastic scattering processes (the cross section is ~10 of elastic cross section). (In fact this is one of the very important questions to be answered by the very high energy experiments of the future: how much cross section goes into diffractive production processes.)
(ii) Nondiffractive processes are presumably not contributing to the inelastic shadowbecause the whole configuration of the target would eventually have to go back to the initial one-a very complex process in which the whole of the nucleus must take part (hence it occurs with small probability).
(iii) Hence "inelastic shadowing" stands a good chance to contribute little (a few percent) to the elastic cross section.
If this is so, then the three contributions to aror discussed above do approximately exhaust the list of processes contributing to elastic scattering.
From our discussion of the components of σTOT (σel, σDT, σPROD) it follows that the measurements of σror may be a good way of finding out whether the inelastic shadowing (or inelastic screening) corrections are important at very high energies: If one computed σTOT from the Glauber model (including all possible effects which the model allows for) and then found a definite discrepancy with experimentally measured Tor-it would very strongly suggest the existence of inelastic shadowing phenomena described above. In fact such an analysis has recently been done for π-d scattering and seems to indicate the existence of such a discrepancy for energies above ~40 GeV .
The remaining important corrections to be discussed (although they are, in principle, included in the algorithm presented above) are: (i) the Coulomb corrections which play an important role in elastic scattering from nuclei of charged hadrons, and (ii) the corrections for the c.m. motion which are important for light nuclei but unimportant for heavy ones.
Let us consider first the Coulomb corrections for heavy nuclei. One can, in principle, use the individual amplitudes which have Coulomb interactions built into them (this very tedious calculation has been done, e.g., in refs. [16, 17], but we shall consider the effects produced by the average Coulomb potential produced by the whole nucleus  which produces almost identical results [16, 17].
We shall assume that, in the high energy limit, the total phase shift is the sum of the Coulomb phase shift (xe, the phase shift one would get if the strong interactions were switched off) and the strong interaction phase shift (x, the phase shift we would get if the Coulomb interactions were switched off; for x. we have the expression x.= Σ; x;.). This assumption is, of course, obvious in potential scattering.
Hence we do not add amplitudes, we add phase shifts. The cross section is
(b) (which is, as a rule, (b) because this is the
The amplitude (3.6) has some simple properties which show that the Coulomb interaction may help us in learning about the real part of the strong interaction phase shift, not well known: we know pretty well the absorption, which is given by dominating process, but not έ (b)).
If the Coulomb interaction is absent (x.=0) the elastic cross section is invariant against the change of sign of . If, however, x.0 some drastic changes may be introduced by changing the sign of which is equivalent to changing the charge of the incident beam of particles. For isospin zero targets (He, 160), if one finds no difference between the elastic cross section for and it implies that there is no real part in the π-nucleus elastic scattering strong interaction phase shift.
When xc=0, 1-e-(b) cos § (b) and e−t) sin § (b) go to zero for b>R (R is the radius of the target). They have, in general, quite different shapes, however-hence Im M and Re M oscillate differently. They are out of phase and since (b) is small in general, | Im M | > | Re M.
If, however, xe0, the situation may change dramatically: x. (b) may 'stabilize' the arguments of cos (...) and sin (...): § (b) decreases; however, the Coulomb phase shift x. (b) ~(Ze2/v) ln (kb) increases with b. This last expression is the Coulomb phase shift produced by a point charge. If Xc+ varies around n, the situation is more or less the same as in the case x=0 (scattering of neutral particles). If, however, x.+ stabilizes around (n+2), the roles of real and imaginary parts may be interchanged: Re M may become large and Im M small.
The Conclusion: The Coulomb interactions for large nuclei are, in general, important for all angles and momentum transfers.
In order to compute the amplitude one has to bear in mind that at large b, x. (b) behaves like a Coulomb phase shift produced by a point charge and hence diverges logarithmically. But we do know the analytic expression for the Coulomb scattering amplitude of point-like charges:
and hence we get the convergent expression for the complete amplitude by adding and subtracting a Coulomb point charge amplitude:
db bJ, (Ab) [1 − exp (ix.”(b))]+ik [* db bJo(Ab) exp [ix.” (b)+ix.(b)]
· Mc (P) +ik
db bJo (Ab) [exp (ixeP (b)) — exp (ixe (b)) (1 − r(b))].
This last integral has no divergences anymore (although x. (b) and x. (b) both diverge logarithmically at large b). In general x. (b) has to be computed numerically
Hence, for large b, xc (b)→xc2 (b) and the integral for M converges.
Let us construct x. (b) in the case of A large (a large target nucleus). We assume (for the sake of simplicity) the independent particle model wave function of the nucleus:
exp [ix. (b)]= (1 − f('sp(s) (b−s)^ —— exp [−4 ƒ d's p(s) y (b−s)].
(In order to perform a careful limiting procedure A→∞, one should keep eix) under the integral sign of the expression for M . Generalizing slightly (allowing for different neutron and proton profiles and densities) we have (N number of neutrons, Z-number of protons):
When yn (b) and y,(b) are very sharp compared with pn (s) and pp(s) we have
ix. (b)≈-Nyn (0) În (b) — ZYp (0) pp (b).
db exp (iò. b) y (b), when y (b) is very sharp compared to 1/8 (hence we limit our
selves to forward scattering processes) we can approximate y (b) ≈y (0) d‹2) (b),
where a, and a, (the ratio of the real to the imaginary part of the forward forward scattering amplitude) are defined by fn.p(0) = (i+an.p) kon.p/4π where on.p are the total cross section for scattering on either neutron or proton. (Incidentally, one can define an optical potential — (1/v) ƒ‰ dz Vopt(b,z) =x. (b) which is equivalent to our multiple scattering description).
From this expression (3.7) one can see that the interplay of xc(b) and §(b) is, in this optical limit, determined by the size and sign of an and a,. Some calculations were done  with p(r): Po(1-exp [(r-R)/c]). For 208 Pb, R-6.5 fm and c=0.523 fm. The densities pa (b), pp (b) were obtained by integrating p(r) over z. The parameters of x. (b) were taken from proton-nucleon scattering cross sections. For an ap=-0.33, σn=0,=38.9 mb (these parameters are resonable for ~20 GeV protons), one gets the following table
Note that 527.85, 61⁄2π=9.42. This means that near to the nuclear boundary sin (x.(b)+§(b)) is large. It is amusing that numerically Im M with proper x. is approximately the same as Re M without xe! This is true for 208 Pb. In general one gets all kinds of intermediate situations. In any case, the influence of the Coulomb interaction is very important "everywhere" as the figure below (see ) for a 208 Pb target and incident neutral-, positive-, and negative- particles which interact strongly as 20 GeV nucleons.