Transcript Trento_1 - NT¤ at NuclearTheory.net

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Nuclear Reactions – 1/2
DTP 2010, ECT*, Trento
12th April -11th June 2010
Jeff Tostevin, Department of Physics
Faculty of Engineering and Physical Sciences
University of Surrey, UK
Notes/Resources
http://www.nucleartheory.net/DTP_material/
Please let me know if there are problems.
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The Schrodinger equation
In commonly used notation:
and defining
bound states
With
scattering states
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Optical potentials – the role of the imaginary part
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Recall - the phase shift and partial wave S-matrix
Scattering states
and beyond the range of the nuclear forces, then
regular and irregular Coulomb functions
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Phase shift and partial wave S-matrix: Recall
If U(r) is real, the phase shifts
are real, and […] also
Ingoing
outgoing
waves
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).
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Ingoing and outgoing waves amplitudes
0
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Semi-classical models for the S-matrix - S(b)
b=impact parameter
for high energy/or large mass,
semi-classical ideas are good
k, 
kb  , actually  +1/2
b

1
absorption
transmission
1
b
Eikonal approximation: point particles
Approximate (semi-classical) scattering solution of
assume
valid when
small wavelength
 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.
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Eikonal approximation: point neutral particles
 imply that
The conditions
Slow spatial variation cf. k
and choosing the z-axis in the beam direction
phase that develops with z
with solution
b
r
z
1D integral over a straight
line path through U at the
impact parameter b
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Eikonal approximation: point neutral particles
So, after the interaction and as z
Eikonal approximation to the
S-matrix S(b)
S(b) is amplitude of the forward
going outgoing waves from the
scattering at impact parameter b
b
r
Moreover, the structure of the
theory generalises simply to few-body projectiles
z
Eikonal approximation: point particles (summary)
b
z
limit of range of
finite ranged
potential
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Semi-classical models for the S-matrix - S(b)
b=impact parameter
for high energy/or large mass,
semi-classical ideas are good
k, 
kb  , actually  +1/2
b

1
absorption
transmission
1
b
Point 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
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Point particles – the differential cross section
So, the elastic scattering amplitude
is approximated by
Performing the z- and azimuthal  integrals
Bessel
function
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Point particle – the Coulomb interaction
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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 usual Coulomb
nuclear scattering in the presence
(Rutherford) point
screening phase
of Coulomb
charge amplitude
nuclear
phase
Due to finite charge
distribution
See e.g. J.M. Brooke, J.S. Al-Khalili,
and J.A. Tostevin PRC 59 1560
Accuracy of the eikonal S(b) and cross sections
J.M. Brooke, J.S. Al-Khalili, and J.A. Tostevin PRC 59 1560
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Accuracy of the eikonal S(b) and cross sections
J.M. Brooke, J.S. Al-Khalili, and J.A. Tostevin PRC 59 1560
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Point 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):
etc.
and where (cylindrical coordinates)
b
z
Eikonal approximation: several particles (preview)
b1
Total interaction energy
b2
with composite objects we will
get products of the S-matrices
z
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Eikonal approach – generalisation to composites
Total interaction energy
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Folding models are a general procedure
Pair-wise interactions integrated (averaged) over
the internal motions of the two composites
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Folding models from NN effective interactions
Double
folding
VAB
Single
folding
VAB (R)   dr1  dr2  A (r1 )  B (r2 ) v NN (R  r2  r1 )
A
R
B
 B (r)
 A (r)
VB (R)   dr2  B (r2 ) v NN (R  r2 )
VB
Only ground state densities appear
R
B
 B (r)
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Effective interactions – Folding models
Double
folding
VAB
Single
folding
VB
VAB (R)   dr1  dr2  A (r1 )  B (r2 ) v NN (R  r1  r2 )
A
 A (r)
r1
R
r2
R  r1  r2
B
 B (r)
VB (R)   dr2  B (r2 ) v NN (R  r2 )
R
R  r2
r2
B
 B (r)
The M3Y interaction – nucleus-nucleus systems
Double
folding
VAB
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VAB (R)   dr1  dr2  A (r1 )  B (r2 ) v NN (R  r2  r1 )
A
R
 A (r)
B
 B (r)
originating from a G-matrix calculation and the Reid NN force
resulting in a REAL nucleus-nucleus potential
M.E. Brandan and G.R. Satchler, The Interaction between Light Heavy-ions and what it tells us,
Phys. Rep. 285 (1997) 143-243.
t-matrix effective interactions – higher energies
Double
folding
VAB
VAB (R)   dr1  dr2  A (r1 )  B (r2 ) t NN (R  r2  r1 )
A
 A (r)
R
B
 B (r)
At higher energies – for nucleus-nucleus or nucleon-nucleus
systems – first order term of multiple scattering expansion
nucleon-nucleon cross section
resulting in a COMPLEX
nucleus-nucleus potential
M.E. Brandan and G.R. Satchler, The Interaction between Light Heavy-ions and what it tells us,
Phys. Rep. 285 (1997) 143-243.
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Skyrme Hartree-Fock radii and densities
W.A. Richter and B.A. Brown, Phys. Rev. C67 (2003) 034317
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Double folding models – useful identities
proofs by taking Fourier transforms of each element
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Effective NN interactions – not free interactions
VB (R)   dr2  B (r2 ) v NN (| R  r2 |)
R
r2
R  r2
B
 B (r)
Fermi
momentum
k
kf
nuclear
matter

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include the effect
of NN interaction
in the “nuclear
medium” – Pauli
blocking of pair
scattering into
occupied states
 v NN (  , r)
But as E  high
v NN  v free
NN
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JLM interaction – local density approximation
complex and density
dependent interaction
k
kf
nuclear
matter

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 rx
R
B
r2
 (r)
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JLM interaction – fine tuning
Strengths of the real and imaginary parts of the potential
can be adjusted based on experience of fitting data.
p + 16O
J.S. Petler et al. Phys. Rev. C 32 (1985), 673
JLM predictions for
N+9Be
A. Garcıa-Camacho, et al. Phys. Rev. C 71, 044606(2005)
cross sections
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JLM folded nucleon-nucleus optical potentials
J.S. Petler et al. Phys. Rev. C 32 (1985), 673
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Cluster folding models – the halfway house
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for a two-cluster projectile (core +valence particles) as drawn
can use fragment-target interactions from phenomenological
fits to experimental data or the nucleus-nucleus or nucleonnucleus interactions just discussed to build the interaction of
the composite from that of the individual components.
Cluster folding models – useful identities
proofs by taking Fourier transforms of each element
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So, for a deuteron for example
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