1. V. Nuclear Reactions Topics to be covered include:
Two-potential formula
Distorted Wave Born Approximation
Coupled-Channels
Applications
Eikonal Theory
General References:
1) Rodberg and Thaler, “The Quantum Theory of Scattering,” Chapter 12.
2) Roy Glauber, in “Lectures in Theoretical Physics,” Vol. 1, Interscience,
Boulder 1958 series, starting on page 337.

2. In previous chapters we have covered elastic scattering in detail
In previous chapters we have covered elastic scattering in detail. Here I focus on methods used to calculate inelastic scattering from nuclei and single-nucleon transfer reactions. I will not get bogged down in the intense amount of angular momentum algebra required in order to do real calculations. My goal is to show the structure of the scattering equations and give you a flavor of how the reaction calculations are done.
I will start with the two-potential formula in formal scattering theory which provides the basis for many applications. P A
projectile
Target nucleus

3. Two-potential formula:

4. Two-potential formula:

5. DWBA:

6. DWBA:
Apply the DWBA to inelastic scattering where the target nucleus is excited from the g.s. to some excited state F* and the projectile remains in its g.s. The nuclear physics notation for this reaction is A(p,p’)A*. For a single particle excitation: p n
1d3/2
2s1/2
1d5/2
1p1/2
1p 3/2
1s1/2
18O P +

7. DWBA:

8. 16O(p,p’)16O* at 800 MeV fixed target
DWBA:
16O(p,p’)16O* at 800 MeV fixed target
PRL 43, 421 (1979).
Angular position of first peak indicates the
L transfer to the nucleus. For 0+ ground-states,
this gives the state’s Jp.

9. DWBA:
116,124Sn(p,p’) at 800 MeV
PRL 42, 363 (1979)

10. DWBA:
12C,208Pb(p,p’) at 800 MeV
Phys. Rev. C 18, 1436 (1978)

11. DWBA for single nucleon transfer:
Here we consider reactions where the incoming projectile either sheds a nucleon which then binds to the target or picks up a nucleon from the target. The first is called a stripping reaction and the second a pick-up reaction.
Examples of stripping reactions: AZ(d,p)A+1Z*, AZ(3He,d)A+1(Z+1)*
Examples of pickup reactions: AZ(p,d)A-1Z*, AZ(3He,a)A-1Z*
where (*) indicates that the final-state nucleus can be in an excited energy level.
The DWBA amplitude for a pick-up reaction (see cartoon) is obtained as follows: p C n
Entrance channel p C n
Exit channel

12. DWBA for single nucleon transfer:

13. DWBA for single nucleon transfer:
In the next few slides I show some examples of single-nucleon transfer reactions with DWBA shape predictions, and amplitude fits using the spectroscopic factor. These are (d,p) stripping reactions where a neutron is injected into the target nucleus into one of its states. The DWBA t-matrix has the same form as above where the single particle shell model state is the one receiving the transferred neutron.

14. DWBA for single nucleon transfer:
Proton stripping reaction to various final
states in 29P at beam energy of 35 MeV
Phys. Rev. C 13, 1367 (1976)
DWBA CC p
28Si

15. DWBA for single nucleon transfer:
Proton dropped into d5/2 levels

16. DWBA for single nucleon transfer:
Proton dropped into f7/2 and p3/2 levels

17. Coupled-Channels:

18. Coupled-Channels: 12C(p,p’) at 800 Mev PRL 40, 1547 (1978) 4+ DWBA 2+0+
DWBA
Coupled-Channels
Coupled-channels allows multiple pathways to get to the final-state. These become important when the coupling strength increases, e.g. highly deformed nuclei or with strong vibrational excitations.

19. Coupled-Channels: Coupled-channels DWBA 176Yb(p,p’) at 800 MeV
Phys. Rev. C 22, 1168 (1980)
The multiple pathways, or scattering amplitudes interfere which can significantly shift the diffractive patterns.

20. Coupled-Channels: DWBA Coupled-channels 154Sm(p,p’) at 800 MeV
Phys. Rev. C 22, 1168 (1980)

21. Eikonal approximation – the Glauber model:
The eikonal approximation is a high energy, forward scattering limit solution of the Schrodinger equation, which provides an intuitive connection between the scattering potential and the cross section. Consider a plane wave scattering from a spherical potential V:
Incident
plane wave z a
Roy Glauber

22. Eikonal approximation – the Glauber model:

23. Eikonal approximation – the Glauber model:
Incident
plane wave z k b

24. Eikonal approximation – the Glauber model:

25. Eikonal approximation – the Glauber model:
This has the form of an incident plane wave which passes along a straight line through the potential at some impact parameter and undergoes a phase shift which depends on the “profile” of the scattering potential.

26. Eikonal approximation – the Glauber model:
King Glauber I

27. Eikonal approximation – the Glauber model:
First-order p + 16O optical potential obtained by folding the free-space N+N t-matrix with the nuclear matter density estimated from the measured charge density; including Coulomb and spin-orbit potentials in the Glauber eikonal approximation. From
Varma and Zamick, Phys. Rev. C 16, 308 (1977).

28. Eikonal approximation – the Glauber model:
Varma and Zamick, Phys. Rev. C 16, 308 (1977).

29. This concludes Chapter V:
Nuclear Reactions