1

Nuclear Reactions – 1/2

DTP 2010, ECT*, Trento

12th April -11th June 2010

Jeff Tostevin, Department of PhysicsFaculty of Engineering and Physical SciencesUniversity of Surrey, UK

2Notes/Resources

http://www.nucleartheory.net/DTP_material/

Please let me know if there are problems.

3The Schrodinger equation

In commonly used notation:

and defining

With

bound states scattering states

4Optical potentials – the role of the imaginary part

5Recall - the phase shift and partial wave S-matrix

and beyond the range of the nuclear forces, then

Scattering states

regular and irregular Coulomb functions

6Phase shift and partial wave S-matrix: Recall

If U(r) is real, the phase shifts are real, and […] also

Ingoing waves

outgoing waves

survival probability in the scattering

absorption probability in the scattering

Having calculate the phase shifts and the partial wave S-matrix elements we can then compute all scattering observables for this energy and potential (but later).

7Ingoing and outgoing waves amplitudes

0

8Semi-classical models for the S-matrix - S(b)

k,

b

for high energy/or large mass,semi-classical ideas are good

kb , actually +1/2

1

b1

absorption

transmission

b=impact parameter

9Eikonal approximation: point particles

Approximate (semi-classical) scattering solution of

assume

valid when high energy

Key steps are: (1) the distorted wave function is written

all effects due to U(r),

modulation function

(2) Substituting this product form in the Schrodinger Eq.

small wavelength

10Eikonal approximation: point neutral particles

1D integral over a straightline path through U at theimpact parameter b

The conditions imply that

and choosing the z-axis in the beam direction

with solution

br

z

Slow spatial variation cf. k

phase that develops with z

11Eikonal approximation: point neutral particles

So, after the interaction and as z

Eikonal approximation to theS-matrix S(b)

S(b) is amplitude of the forward going outgoing waves from the scattering at impact parameter b

theory generalises simply to few-body projectilesMoreover, the structure of the

br

z

12Eikonal approximation: point particles (summary)

b

z

limit of range of finite ranged potential

13Semi-classical models for the S-matrix - S(b)

k,

b

for high energy/or large mass,semi-classical ideas are good

kb , actually +1/2

1

b1

absorption

transmission

b=impact parameter

14Point particle – the differential cross section

Using the standard result from scattering theory, the elastic scattering amplitude is

with is the momentum transfer. Consistent with the earlier high energy (forward scattering) approximation

15

Bessel function

Point particles – the differential cross section

So, the elastic scattering amplitude

Performing the z- and azimuthal integrals

is approximated by

16Point particle – the Coulomb interaction

Treatment of the Coulomb interaction (as in partial wave analysis) requires a little care. Problem is, eikonal phase integral due to Coulomb potential diverges logarithmically.

Must ‘screen’ the potential at some large screening radius

overall unobservable screening phase

usual Coulomb (Rutherford) point charge amplitude

nuclear scattering in the presence of Coulomb

See e.g. J.M. Brooke, J.S. Al-Khalili,

and J.A. Tostevin PRC 59 1560 nuclear phase

Due to finite charge distribution

17Accuracy of the eikonal S(b) and cross sections

J.M. Brooke, J.S. Al-Khalili, and J.A. Tostevin PRC 59 1560

18Accuracy of the eikonal S(b) and cross sections

J.M. Brooke, J.S. Al-Khalili, and J.A. Tostevin PRC 59 1560

19Point particle scattering – cross sections

All cross sections, etc. can be computed from the S-matrix, in either the partial wave or the eikonal (impact parameter) representation, for example (spinless case):

and where (cylindrical coordinates)

etc.

bz

20

Total interaction energy

Eikonal approximation: several particles (preview)

b1

zb2

with composite objects we will get products of the S-matrices

21Eikonal approach – generalisation to composites

Total interaction energy

22Folding models are a general procedure

Pair-wise interactions integrated (averaged) over the internal motions of the two composites

23Folding models from NN effective interactions

)( v)(r )(r dd(R)V 12NN2121 rrRrr BAAB

Single

folding

Single

folding

A R B

)( v)(r d(R)V 2NN22 rRr BB

R B

(r)A (r)B

(r)BOnly ground state densities appear

ABV

BV

Double

folding

Double

folding

24Effective interactions – Folding models

)( v)(r )(r dd(R)V 21NN2121 rrRrr BAAB Double

folding

Double

folding

Single

folding

Single

folding

A1r

R B2r

21 rrR

R B2r

)( v)(r d(R)V 2NN22 rRr BB

2rR

(r)A (r)B

(r)B

ABV

BV

25The M3Y interaction – nucleus-nucleus systems

)( v)(r )(r dd(R)V 12NN2121 rrRrr BAAB Double

folding

Double

folding

A R B

(r)A (r)BABV

M.E. Brandan and G.R. Satchler, The Interaction between Light Heavy-ions and what it tells us, Phys. Rep. 285 (1997) 143-243.

originating from a G-matrix calculation and the Reid NN force

resulting in a REAL nucleus-nucleus potential

26t-matrix effective interactions – higher energies

)( t)(r )(r dd(R)V 12NN2121 rrRrr BAAB Double

folding

Double

folding

A R B

(r)A (r)BABV

M.E. Brandan and G.R. Satchler, The Interaction between Light Heavy-ions and what it tells us, Phys. Rep. 285 (1997) 143-243.

At higher energies – for nucleus-nucleus or nucleon-nucleus systems – first order term of multiple scattering expansion

resulting in a COMPLEX nucleus-nucleus potential

nucleon-nucleon cross section

27Skyrme Hartree-Fock radii and densities

W.A. Richter and B.A. Brown, Phys. Rev. C67 (2003) 034317

28Double folding models – useful identities

proofs by taking Fourier transforms of each element

29Effective NN interactions – not free interactions

|)(| v)(r d(R)V 2NN22 rRr BB

R B2r

2rR (r)B

fk

nuclear

matter

k

include the effect

of NN interaction

in the “nuclear

medium” – Pauli

blocking of pair

scattering into

occupied states

But as E high

Fermi

momentum

freeNNNN vv

),(vNN r

30JLM interaction – local density approximation

fk

nuclear

matter

k

For finite nuclei, what value of density should be used in calculation of nucleon-nucleus potential? Usually the local density at the mid-point of the two nucleon positions

complex and density dependent interaction

R B

2r (r)

xr

31JLM interaction – fine tuning

Strengths of the real and imaginary parts of the potential can be adjusted based on experience of fitting data.

p + 16Op + 16O

J.S. Petler et al. Phys. Rev. C 32 (1985), 673

32JLM predictions for N+9Be cross sections

A. Garcıa-Camacho, et al. Phys. Rev. C 71, 044606(2005)

33JLM folded nucleon-nucleus optical potentials

J.S. Petler et al. Phys. Rev. C 32 (1985), 673

34Cluster folding models – the halfway house

can use fragment-target interactions from phenomenological fits to experimental data or the nucleus-nucleus or nucleon-nucleus interactions just discussed to build the interaction of the composite from that of the individual components.

for a two-cluster projectile (core +valence particles) as drawn

35Cluster folding models – useful identities

proofs by taking Fourier transforms of each element

36So, for a deuteron for example