Nucleon Optical Potential in
Brueckner Theory
Wasi Haider
Department of Physics, AMU, Aligarh, India.
E Mail: wasi42001@gmail.com
1.
2.
3.
4.
5.
6.
Introduction.
G-matrix (Effective Interaction).
Nucleon Optical Potential.
Spin-orbit force.
Three nucleon correction.
Conclusions.
1. Introduction:
First Order Microscopic description of the nuclear Collision:
U (k , k )   dpdp k , p G k , p ˆ ( p, p )
Nucleon scattering
Few tens of MeV – 400MeV
Comparison with Empirical Potentials.
Bethe-Brueckner-Goldtone
…theory of Nuclear Matter
PR95,217(1954), Rev. Mod.Phys.30,745(1967), Proc. Roy. Soc. (Lon)
A239,267(1937)
Brieva and Rook, NPA291,317(1977); 307,493(1978)
Jeukenne,Lejeune,Mahaux, Phys. Rep. 25,83(1976)
H.V.von Geramb, in The Interaction between Medium Energy nucleons in
Nuclei (AIP,New York, 1983), Yamaguchi et al.
Recent developments:
Amos et al. Adv. In Nucl. Phys.25, 275 (2000)
Arelleno, Brieva love, Phys. Rev. Lett.63,605 (1989)…
Arlleno and BaugePRC76, 014613 (2007)
All the above approaches : FOLDING of the generalised TWO-Body Infinite
Nuclear Matter effective interaction over the Target ground state
densities.
We briefly discuss the basic formalism, and its successes and in its applications
to finite nuclei.
Failures. Attempts to improve
1. Spin orbit (Direct+Exchange), Central Exchange.
2.Calculations of Three Nucleon effects in the Nucleon optical potential.
Conclusions
2. G-Matrix.
H 0   (Ti  U i )
i
H1   vij   U i
i j
i
Goldstone Perturbation series. First order term:
B.D. Day, Rev. Mod. Phys. 39, 719(1967)
E   Tm 
m
1
 mn v mn  mn v nm

2 m,n

v is the realistic two-body inter-nucleon potential. v is
replaced by the effective interaction, g-matrix:
v  g ( w)  v  v(Q / e) g ( w)
v
g
Calculation of g is summing all the infinite ladder diagrams and it amounts to
solving the Schrödinger equation between two particles in presence of all other
nucleons.
 rs  rs  (Q / e) g (w)rs
rs (r1, r2 )  r (r1 )s (r2 )  r, s
 rs  rs  (Q / e)v rs
d 3kQ(k , K rs ) ik.( r  r) 
k (r , r )  
e
3
(2 ) e(k , K rs )
K:0-6.0fm-1; KF:0.5-2.0fm-1, L=0-6, Four Coupled states
U (k , k F ) 
 kj g (w) kj
j k F
100
Imaginary Potential (MeV)
KF
Real Potential (MeV)
KF
0
50
0
-50
-100
Nuclear Matter Real Potential
V-14
-150
0
1
2
3
-1
kinc(fm )
4
-20
-40
-60
Nuclear Matter Imaginary Potential
v-14
5
-80
0
1
2
3
-1
Kinc(fm )
4
5
3. Nucleon Optical Potential:
We define the radial dependence of the g-matrix such that the
nuclear matter optical potential is reproduced ie:
rs g (r ) rs  rs v rs
1
U (r1 , E )    (r2 ) g ( r1  r2 ,  ( (r1  r2 )), E )d 3 r2
2
c
opt
c
D
1
   (r1 , r2 ) g ( r1  r2 ,  ( (r1  r2 )), E ) j0 (k r1  r2 )d 3r2
2
c
Ex
0
0
-10
-5
-10
W(r) (MeV)
V(r) (MeV)
-20
Ep (MeV)
-30
10
20
50
100
150
200
300
-40
40
p- Ca
-50
V-14
-60
Ep (MeV)
-15
-20
10
20
50
100
150
200
300
40
p- Ca
V-14
-25
-30
0
2
4
6
8
0
2
r(fm)
4
6
8
r (fm)
1.5
Ep (MeV)
0.3
40
p- Ca
V-14
10
20
50
100
150
200
300
0.5
10
20
50
100
150
200
300
0.2
Wso(r) (MeV)
1.0
Vso(r) (MeV)
Ep MeV
0.1
0.0
-0.1
40
p- Ca
-0.2
V-14
0.0
-0.3
0
2
4
6
r (fm)
8
10
0
2
4
r (fm)
6
8
0
0
-2
-5
m*
*
no m
-4
-10
W(r) (MeV)
-6
W(r) (MeV)
m*
*
no m
-8
-10
-15
40
p- Ca 200 MeV
-20
-12
40
p- Ca 30 MeV
-14
-25
-16
0
2
4
r(fm)
6
8
0
2
4
6
8
r(fm)
Effective mass correction in the calculated Imaginary part.
3
2
10
2
10
2
10
10
1
10
10
10
0
d/d (mb/sr)
10
10
d/d (mb/sr)
d/d (mb/sr)
3
3
10
1
0
10
1
10
0
v-14
V-14
V-14
p-
118
p- Sn 39.6MeV
116
p- Sn 39.6MeV
10
10
-1
20
40
60
80
100
10
-1
20
120
40
3
10
2
10
10
10
d/d (mb/sr)
d/d (mb/sr)
10
1
0
p-
10
10
80
CM (deg)
100
120
40
60
80
100
120
CM (deg)
Predictions
0
124
Sn 39.6 MeV
Sn 39.6 MeV
60
20
1
p-
40
-1
Differential crosssection: p-Sn
Isotopes at 40 MeV.
v-14
-1
20
120
2
V-14
10
100
Sn 39.6MeV
3
10
122
80
CM (deg)
CM(deg)
10
60
120
-1
20
40
60
80
CM (deg)
100
120
IIT, BARC, AMU
collaboration
1.0
1.0
1.0
116
Sn 39.6 MeV
p-
118
p-
0.5
0.0
0.0
0.0
AY()
0.5
-0.5
40
60
80
100
120
Sn 39.6 Mev
-0.5
-0.5
-1.0
20
120
Sn 39.6MeV
0.5
AY()
AY()
p-
-1.0
20
40
60
80
100
120
Cm(deg)
CM(deg)
-1.0
20
40
60
80
100
120
CM (deg)
1.0
p-
122
1.0
Sn 39.6 MeV
Polarisation
Predictions for pSn Isotopes at 40
MeV
0.5
0.0
AY()
AY()
0.5
0.0
IIT, BARC, AMU
Collaboration
-0.5
-0.5
v-14
-1.0
20
p-
40
60
80
CM (deg)
100
120
-1.0
20
124
Sn 39.6 MeV
40
60
80
CM (deg)
100
120
Neutron Elastic scattering
from Fe56, Y89 and Pb208 at
96 MeV
Uppsala, AMU
Collaboration
PRC77, 024605 (2008)
0
Conclusions:
-2
1. Neutron and proton
scattering is nicely
reproduces.
3. Spin-orbit is not well
determined.
4. Binding energy of
Nuclear matter is not
reproduced.
E/A (Energy Per Nucleon)
2. Calculated imaginary
part is large.
-4
-6
HJ(BB)
-8
-10
-12
V14(LP)
-14
-16
-18
V14(BB)
Remedial steps:
1. Spin-orbit part.
2. Three-body effects
-20
0.4
0.6
0.8
1.0
1.2
1.4
-1
kf(fm )
1.6
1.8
2.0
4. Nucleon Spin – Orbit Potential:
Vs.o. (r )  Const.
The earliest formula is by Blin-Stoyle:
where the constant is the first term of a series,
Given by Greenlees et al. We show that the series
is not rapidly convergent and we are able to
calculate the Direct part exactly.
V (r )   
D
s.
Using
We get
1 d
r dr
 

 
D
 ( r2 ) g s .o . L.S N ( r2 ) d r2

N
N
 1  
      
L.S  (r1  r2 ) X ( p1  p2 ).( s1  s2 ), x  (r2  r1 )
2
  D   
1
V ( r )    ( r1  x ) g so xXp1 .s1dx
2
D
s .o .
 
1
  A ( r1 ) l1 .s1 / r1
2
where

  D  
A(r1 )r1 / r1    ( r1  x ) g s.o. xdx
Greenlees makes a Taylor series expansion of A(r1):
(2v  2)
d 2v
A(r )  4 
I 2v  4
 (r )
dr
v 0 (2v  3)!

where
I N   x N g ( x)dx,
The First Term of the series is
2v
2 v 1
d

2
v
d

2v
   2v 
dr
r dr 2v 1
4 d
A(r ) 
I4
3
dr
We have done model calculation of the first Two terms of the above series
and find that the second term is quite large. Thus the first term alone is
not enough. We calculate the whole series without making any
approximation about the short range nature of the effective interaction
2.0
15
SPIN-ORBIT POTENTIAL
40
p- Ca 30 MeV
Spin-Orbit potential
First Order
First +second
1.5
Vs.o.(r) (MeV)
Vso(r) (Arb. scale )
10
5
Vso Direct(New)
Vso Direct(old)
Vso Exchange
1.0
0.5
0
0.0
0
2
4
6
8
10
r (fm)
A slight rearrangement of the
expansion gives the results of
Scheerbaum
0
2
4
6
8
10
r (fm)
1
d
2
A(r )  4 { j1 (kx) g ( x) x dx}
k
dr
Thus we are able to calculate the Direct part of the
microscopic spin-orbit part exactly.
5. Three-body terms:
Considerable efforts to Cal. The effect of Higher order terms in the Binding energy
of Nuclear Matter: Bethe, Rajaraman, Day: (Three-body give: -5.0 MeV)
Only two efforts made for the Optical potential: Kidwai, WH.
Three hole-line Diagrams:
Faddeyev:
T=T(1)+T(2)+T(3),
T(3)= g12 – g12(Q/e)[ T(1) + T(2)]
We introduce a three-body wave function in
Coordinate space:
T (3)   g12 (3)
Z (3)  (Q / e) g 23  (Q / e) g13  (Q / e) g 23Z (1)  (Q / e) g13Z ( 2)
Where Z (i )     (i )
; Main task is to Cal. Fns Z(i)
Three-Body Functions:
(Q / e) g ij
Two types: (1) The 3rd Nucleon
is in ground state, and
(2) The Third nucleon is also in
the Excited State.
(Q / e) gij e
i k .rij  2 P.Rij 
 k , P (rij )e
2iP.Rij 
and
Accordingly we differentiate.
(Q / e) gij e
i k .rij  2 P.Rij 
  k , P (rij )e
2iP.Rij 
NPA 504, 323 (1989)
Two Approx. Methods:
Bethe, Day.
 ijB (rij ) 
 k , P (rij )
j0 (krij )
 k , P (rij )
 (rij ) 
j0 (krc )
D
ij
The assumption is that the defect functions are independent of k,P.
Singlet s-sate.
We see that Bethe’s
approximation is
justified.
Day’s approximation
gives similar results.
U 3  8 2  2 12 g 23Z 1 (r12 , r23 , r13 )dr12dr23dr13
 8  12 g13Z (r12, r23 , r13 )dr12dr23dr13
2
2
2
Results:
KF= 1.4fm-1 (Nuclear Interior)
En (MeV
30.0
80.0
Re U2
-54.243
-58.076
Im U2
Re U3
Im U3
-1.566
-10.817
-14.348
-11.535
+0.793
+1.703
Re U3ReU2 ImU3/ImU2
0.132
0.099
-0.253
-0.079
KF = 0.90 fm-1 (Nuclear Surface)
30.0
-14.184
-0.661
-1.321
+0.164
0.047
-0.124
80.0
-14.015
-2.662
-1.001
+0.429
0.036
-0.081
The results using Day’ approximation is very similar, and hence we do not quote
them here.
• Conclusions:
1. Satisfactory agreement with Nucleon scattering data.
2. The exchange parts of the nucleon optical potential should be
treated more carefully.
3. Calculation of three-body effects should be improved.
4. The calculated potentials depend sensitively on the point nucleon
densities used. Hence the approach can be used to study neutron
skin in nuclei.
Thank you.