« PreviousContinue »
For heavy nuclei there are virtually no experiments with good enough resolution to have only pure elastic scattering (in which the target nucleus stays in the ground state). In order to have a genuine elastic scattering one would have to have an energy resolution AE a fraction of an MeV, which for E~20 GeV is still inaccessible. Most experiments (e.g., CERN series-compare [S1]) have poor energy resolution of the incident and outgoing beam (~50 MeV), hence they sum over all nuclear excitations (without producing mesons, however). The cross section for such "inclusive" processes is
This cross section includes, of course, the elastic cross section. The cross section which, upon integration, gives σDT is ᎠᎢ
It is an interesting fact that while do.../dî is very strongly influenced by Coulomb interactions (as we have seen), do DT/d is influenced very little. In order to make this fact more plausible, let us consider a collection of neutrons and protons which do not screen each other. Then, we would have
If we introduce screening there will be, on the average, a certain fraction of nucleons inaccessible to the incident hadron. Hence the above formula can be applied to a certain "effective" number of nucleons. Indeed one can show (compare ref. -the calculation was done with the Coulomb interactions present) that to a good approximation (note that since this formula does not exhibit a forward dip, it is not valid for small ▲)
where σ is an average total hadron-nucleon cross section. If indeed σDT/dî has such a form, the only place where Coulomb interactions enter are in the individual proton amplitudes, |ƒ,(A) |2. But there we know, e.g., from the proton-proton elastic cross section, that Coulomb interactions are important for very, very small momentum transfers only. In any case, they enter incoherently into do Dr/do. These two factors make Coulomb corrections insignificant in do DT/do.
How important are the details of the target nucleus wave function? Not very important. The most important are general characteristics: density distributions (hence possible deformations) but not internal correlations. From the published analyses of hadron-nucleus scattering (see e.g., [S2], , ) one may conclude that:
(i) the shapes of target nuclei are the most important factors determining the cross sections (ii) the internal correlations of nucleons in the nucleus are unimportant for dose/dî or do。 dî. They are of some importance for do Dr/dî (especially at small momentum transfers , ). The confrontation with experiment is impressive. (Compare, e.g., the review article by R. J. Glauber in ref. [S2]).
When we want to discuss light nuclei we have to consider carefully the motion of the center of mass. Take, for example, a deuteron: here taking into account the c.m. motion is trivially accomplished by using the wave functions of the relative motion, ø (r).
In the case of more complicated targets the situation is much more involved and often leads to some serious computational problems. Let us introduce the transverse component of the c.m. vector
Then we can compute the correction factor to M = (M (▲; 81 ... SA)) assuming the wave function to be in the form of a product of the c.m. wave function and the internal wave function.
M= (R(r) | exp (i▲⋅r) | R(r)) (Þ。 (rı′ . . . ra′) | M′ | Þ。 (rı′ . . . ra′) )
This is the corrected amplitude.
Hence if we can factor out the c.m. wave function from the product = II; ; (r;) we can stick to calculating M with Yo but we have to multiply it by a correction factor:
(R (r) | exp (i▲·r) | R (r) )−1.
This can be done explicitly in the case of oscillator potential wave functions (this is partly the reason why they are so popular!). There
® (r) = (A/π3R6) 1/4 exp (− Ar2/2R2)
where R is the size parameter in the Gaussian factor in harmonic oscillator wave functions: exp (-2/2R2). Then
(R (r) | exp (¿▲•r) | ® (r) )−1 = exp (42R2/4A).
When one cannot do this factorization the computations become quite involved (s, are not independent!).
then we shall have b-r instead of b. After changing the variable b=r+b' we get the factor exp (i▲⚫r) in front, and the formula (3.9) follows.
One can write the general formula which takes into account the interdependence of internal coordinates by introducing a Dirac & function into the amplitudes:
This (3) function eliminates redundant excitations of the system of A nucleons. When one cannot factorize the c.m. coordinate and one has to use the above formula the numerical calculations become much more involved (from trivial-they become difficult ).
An illustrative example: the ground state wave function is a Gaussian . The ground state densities and the elementary amplitudes are taken in the form
Then the elastic scattering amplitude (with the c.m. motion correction included) reads
Many general features of the multiple scattering are included in this formula:
(i) If we neglect a, the amplitude becomes purely imaginary (absorptive). A geometrical picture of single-, double-, etc. scattering contributions is as follows:
(ii) With this picture it is easy to establish the existence of diffractive minima, which are filled by the real part of M. (In order to have Re M0 we have to have a0).
(iii) The importance of the c.m. motion correction can be seen from the factor exp (R242/4A). For small A (say A = 2, 3 or 4) it can be a correction of as much as 2 orders of magnitude for A2 0.3 GeV2.
A few concluding remarks about the deuteron target.
A lot of attention was concentrated on the deuteron because it is a very important testing ground for multiple scattering theories (or models).
(i) In experiments (compare deuteron data contained in [S1]), one can clearly see the single and double scattering.
Remark: In fact, this clear distinction between single and double scattering was used to extract the p-nucleon total cross section in y-p production experiments on deuterons (see section 4.2).
(ii) One can also see (again, compare [S1]) how important the deformation of the target is (existence of the D-state in the deuteron ground state). Let us discuss this effect in more detail.
d1, d2 are the Pauli spin operators, and r is the neutron-proton relative coordinate. X1, is the spin function for spin 1 with the magnetic quantum number m. The elastic cross-section is then
which operator, in this approximation, does not depend on spins. So, if not for the S12 term in (3.10), we would have (m | M | m')=0 for mm'. In fact the (m | | m') contributions are indeed the most important but they always lead to a sharp diffractive minimum:
But (m| S12 m')#0 in general (also (m | S12 S12 | m′)#0). This matrix element enters (m | M (▲, s) | m') and results in spin-flip transitions (classically: rotation of the deuteron spin) which have completely different "profiles" than (m | M | m), thus resulting in oscillations which are out of phase with oscillations of (m | M | m) and fill the diffractive minimum: