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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