Lawrence Livermore National Laboratory Coupled

Transcrição

Lawrence Livermore National Laboratory
Coupled-channels
Neutron Reactions on Nuclei
Ian Thompson
with: Gustavo Nobre, Frank Dietrich, Jutta Escher (LLNL)
and: Toshiko Kawano (LANL), Goran Arbanas (ORNL)
Lawrence Livermore National Laboratory, P. O. Box 808, Livermore, CA 94551!
This work performed under the auspices of the U.S. Department of Energy by
Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344
LLNL-‐PRES-‐437553
Channel Couplings in Neutron-nucleus Collisions
Neutrons incident on Spherical Nuclei
Scidac Project UNEDF
Use mean-field models with RPA excited states
Use real effective interactions
Calculate inelastic cross sections to all RPA states
Calculate transfer cross sections to all
one-nucleon-transfer states
•  Predict Reaction cross sections
•  Predict Optical Potentials: Nonlocal & Local-equivalent
• 
• 
• 
• 
LLNL-‐PRES-‐437553 and 438113!
2
1: UNEDF project: a national 5-year SciDAC collaboration
Target
A = (N,Z)
UNEDF:
VNN, VNNN…
Structure Models
Methods: HF, DFT,
Ground state
Excited states
Continuum states
RPA, CI, CC, …
KEY:
UNEDF Ab-initio Input
User Inputs/Outputs
Exchanged Data
Related research
Transition
Density [Nobre]
Transition Densities
Veff for
scattering
UNEDF Reaction Work
Folding
Eprojectile
[Escher, Nobre]
Transition Potentials
Deliverables
Residues
(N’,Z’)
HauserFeshbach
decay chains
[Ormand]
Partial
Fusion
Theory
[Thompson]
Inelastic
production
Compound
emission
Preequilibrium
emission
Neutron escape
[Summers,
Thompson]
Global optical
potentials
LLNL-‐PRES-‐437553 and 438113!
Voptical
Coupled
Channels
[Thompson, Summers]
Two-step
Optical
Potential
or
Elastic
S-matrix
elements
Resonance
Averaging
[Arbanas]
Optical Potentials
[Arbanas]
3
Diagonal Density
90
Densities for Zr
From M. Dupuis’ calculations
0.2
!Total
!n - !p
!p
!n
0.18
0.16
-3
density [fm ]
0.14
Example of diagonal
Density for 90Zr
0.12
0.1
0.08
0.06
RPA
0.04
0.02
0
0
1
2
3
4
5
r [fm]
6
7
8
9
10
Folding of densities with n-n interaction  Transition potentials
LLNL-‐PRES-‐437553 and 438113!
4
Nuclear Excited States from Mean-field Models
 
 
Mean-field HFB calculations using SLy4 Skryme functional
Use (Q)RPA to find all levels E*, with transition densities from the g.s.
30
28
28
26
26
24
24
22
22
Excitation energy (MeV)
RPA levels in Zr
30
20
18
16
14
12
10
8
20
18
16
14
12
10
8
6
6
4
4
2
2
0
1
2
3
4
5
6
Spin of state
QRPA states in 90Zr
90
90
Particle!hole levels in Zr
0
Collaboration with
Chapel Hill: Engel
& Terasaki
7
8
9
10
0
Uncorrelated
particle-hole states
0
1
2
3
4
5
6
Spin of state
7
Correlated p-h
states in HO basis
8
9
10
30
28
26
24
22
20
18
16
14
12
10
8
6
4
2
0
0
2
4
6
8
10
Correlated p-h
states in 15 fm box
Neutron separation energy is 9.5 MeV.
Above this we have discretized continuum.
LLNL-‐PRES-‐437553 and 438113!
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Transition densities to Transition potentials
Diagonal folded potential
Off-diagonal couplings
RPA transition potentials from the gs to states E* < 10 MeV
10
HFB diagonal folded potential
KD optical potential
0
0
Vf0(r) (MeV)
Real central potential V(r)
20
!20
!10
90
n + Zr
at 40 MeV
!40
All potentials real-valued
!60
0
1
2
3
4
5
6
Radius (fm)
7
8
9
10
!20
0
2
4
Radius r
6
(fm)
8
10
Natural parity states only: no spin-flip, so no spin-orbit forces generated.
No density dependence. Direct terms only: no exchange contributions. (Yet.)
LLNL-‐PRES-‐437553 and 438113!
6
Cross Sections for Excited States
30
30
!
!
5
20
10
10
0
200
0
200
150
150
!
3
100
50
0
75
!
1
50
25
60
0
40
50
0
75
!
1
50
25
60
0
40
+
4
20
20
0
150
0
150
100
!
3
100
Inelastic cross section (mb)
Inelastic cross section (mb)
5
20
+
4
100
+
2
50
50
0
0
40
+
2
40
+
20
0
+
0
0
10
20
Uncorrelated p-h
0
20
30
0
0
10
20
30
RPA Correlated states
LLNL-‐PRES-‐437553 and 438113!
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Reaction Cross Sections with Inelastic Couplings
  (Q)RPA Structure Calculations for n,p + 40,48Ca, 58Ni, 90Zr and 144 Sm
  Couple to all excited states, E* < 10, 20, 30, 40 MeV
  Find what fraction of σR corresponds to inelastic couplings
Not Converged yet!  E* < 50, 60, 70, …?
LLNL-‐PRES-‐437553 and 438113!
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Summed inelastics => Reaction Cross Section
Use ‘doorway model’:
these inelastic cross sections are sum of
escape and compound-nucleus production rates.
!R (mb)
200
Couplings to/from g. s. only
L = 0, 2
L = 0, 2, 4
Elab = 10 MeV
100
40
p + Ca
Elab = 20 MeV
100
300
Effects of Couplings
between States
80
Elab = 20 MeV
200
100
0
60
40
0
2
p + Ca 4
6
8
Partial Wave
Total Reaction Cross!sections
1000
40
20
0
800
0
1
2
3
4
5
6
Optical Model
CC; QRPA E* < 10 MeV
CC; QRPA E* < 30MeV
7
Partial Wave
600
!R (mb)
!R (mb)
0
σR(L)
!R (mb)
120
Convergence with E*
But: long way from
reaction cross section
from optical model
400
200
0
10
15
20
25
30
35
40
Elab (MeV)
LLNL-‐PRES-‐437553 and 438113!
9
Pick-up Channel: Deuteron Formation
40Ca(d,d)
elastic scattering
N. Keeley and R. S. Mackintosh*
showed the importance of including
pick-up channels in coupled reaction
channel (CRC) calculations.
*Physical Review C 76, 024601 (2007)
Physical Review C 77, 054603 (2008)
d
LLNL-‐PRES-‐437553 and 438113!
10
Many Transfer Channels! NdN
These give large contributions
to the reaction cross sections.
There are many nucleons in the target
that can be picked out to make a deuteron.
300
Neutrons
Finite
Well
N=2n+L
With spin-orbit force
0g
0g7 / 2
nocc(j)
Sum
Closed
shells
(10)
(2)
50
40
50
(6)
(4)
!
(8)
!
(4)
!
(2)
!
(6)
!
!
(2)
!
(4)
!
38
32 !
28
20
16
1d5 /2
0g9 /2
Protons
3!" !
!
1p
0f
2!" !
!
1s
0d
1!" !
!
0p
!
!
!
!
!
0!"
!
0s
nL
N!"
!
!
!
!
0 f5 /2
!
0 f7 /2
!
!
!
0d 3/2
1s1/2 !
!
0d5 /2
!
0 p1/2 !
0 p3/2 !
!
!
!
!
0s1/2 !
nL !
2 j +1
!
!
!
!
!
!
!
1p1/2
1p3/2
!
j
!
!
(2)
!
!
LLNL-‐PRES-‐437553 and 438113!
BG optical potl
QRPA inelastic
Inel+transfers
crcn!da.reac
20
14
8 !
6
2
" 2 j +1
!
Reaction cross section (mb)
Harmonic
Oscillator
200
90
n + Zr
40 MeV
100
8
2
0
0
1
2
3
4
5
6 7 8 9 10 11 12 13 14 15
Partial Wave
!
Effect
depends on binding energy and
Size of bound state wave functions.
These are given by the mean-field model.
11
Comparison with Experimental Data
Good description of
experimental data!
There is still
possibility for
improvements
.
Inelastic convergence
when coupling up to all
open channels
LLNL-‐PRES-‐437553 and 438113!
12
Several Projectiles, Targets and Energies
  Multiple targets
  More Proton data exists for
reaction cross sections
(normalised to nuclear area)
Elab = 30 MeV
r0 = 1.2 fm
2.0
J. F. Turner et al., 1964
R. F. Carlson et al., 1975
J. F. Dicello et al., 1970
400
1.5
1/3 2
40
p+ Ca
"R/!(r0 A )
!R (mb)
800
0
1.0
!R (mb)
0.5
800
neutron as projectile
R. F. Carlson et al., 1994
0.0
48
p+ Ca
400
proton as projectile
1.5
transfers
J. J. Menet et al., 1971
J. F. Turner et al., 1964
T. Eliyakut!Roshko et al., 1995
800
58
p+ Ni
400
0
10
15
20
25
Elab (MeV)
30
35
40
Inel: dash
+transfer:
dash-dotted
+tr+nono:
short-dash
optical model:
solid
"R/!(r0 A )
!R (mb)
1/3 2
0
optical
potential
1.0
inelastic
0.5
0.0
50
70
90
1/3 2
110
130
!(r0 A )
LLNL-‐PRES-‐437553 and 438113!
13
Non-Orthogonality and Fraction of σR
Behaviour of non-orthogonality is
sensitive to changes of the deuteron
potential:
Better definition
needed!
Using the Daehnick et al.§ potential
for the deuteron.
Using Johnson-Soper* prescription:
Vd(R)=Vn(r)+Vp(R)
 Coupling to 90Zr(n,d,n) channel gives a large increment,
approaching to the optical model calculation.
 Non-Orthogonality has an additional effect.
αCC < αCC+CRC and αCC+CRC+NO
*Physical
§Physical
Review C 1, 976 (1970)
Review C 21, 2253 (1980)
LLNL-‐PRES-‐437553 and 438113!
14
Elastic Angular Distributions
#!
,
• Provide complementary information on reaction mechanisms
• Are sensitive to the effective interaction used
#!
#!
*
#!
#
#!
!
-".-#
#!
+
,!
/D'@E-E@)F&4GDB&H&+!789
#!
+
#!
*
#!
#
#!
!
-".-#
,!
//&0123&45&6&+!&789
/1/
/1/&:&;<
/1/&:&=>?@!(AB?C
/1/&:&;<&:&=>?@!(AB?C
!
,
"!
#!!
(
!$%&' )
Our approach predicts a variety of
reaction observables.
Data provides constraints on the
ingredients.
#"!
;5'?E-E?)B&F65A&G&+!78D
/0&1234$56&7(-86&'9.&:1)
;<;&=&>1
;<;&=&>1&=&:24?!(@A43
;<;&=&>1&=&:1B&/0&5C&DA5@8
;<;&=&>1&=&:1B&/0&5C&DA5@8&5?-&D-
!
"!
#!!
#"!
(
!$%&' )
Density-dependent effective interaction:
•  Resulting coupling potentials improve large-angle behavior,
still need improvements for small angles.
•  Work in progress to treat and then test UNEDF Skyrme
functionals.
LLNL-‐PRES-‐437553 and 438113!
15
Optical Potentials
Define: The one-channel effective interaction to generate all the
previous reaction cross sections
  Needed for
•  direct reactions: use to give elastic wave function
•  Hauser-Feshbach: use to generate reaction cross sections =
Compound Nucleus production cross sec.
  In general, the ‘exact optical potential’ is
•  Energy-dependent
•  L-dependent, parity-dependent
•  Non-local
  Empirical:
•  local, L-independent, slow E-dependence
•  fitted to experimental elastic data
LLNL-‐PRES-‐437553 and 438113!
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Two-Step Approximation
We found we need only two-step contributions
•  These simply add for all j=1,N inelastic & transfer states:
VDPP = ΣjN V0j Gj Vj0.
Gj = [En - ej – Hj]-1 : channel-j Green’s function
Vj0 = V0j : coupling form elastic channel to excited state j
•  Gives VDPP(r,r’,L,En): nonlocal, L- and E-dependent.
In detail:
VDPP(r,r’,L,En) = ΣjN V0j(r) GjL(r,r’) Vj0(r’) = V + iW
•  Quadratic in the effective interactions in the couplings Vij
•  Can be generalised to non-local Vij(r,r’) more easily than CCh.
•  Treat any higher-order couplings as a perturbative correction
Tried by Coulter & Satchler (1977), but only some inelastic states included
LLNL-‐PRES-‐437553 and 438113!
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Previous examples of Non-local Potentials
  Coulter & Satchler NP A293 (1977) 269:
Real Part
Imaginary Part
LLNL-‐PRES-‐437553 and 438113!
18
Calculated Nonlocal Potentials V(r,r’) now
Real
Imaginary
7
6
5
4
3
2
1
0
-1
-2
7
6
5
4
3
2
1
0
-1
-2
9
8
7
6
5
4
3
2
1
0 0
4
3
2
1
6
5
7
8
1
0.5
0
-0.5
-1
-1.5
-2
-2.5
-3
-3.5
8
7
6
9
4
3
2
1
0 0
1
2
3
4
5
6
7
8
8
7
9
6
5
4
3
2
1
0 0
1
2
3
4
5
6
7
8
9
1
1
0
L=9
-1
0
-2
-1
-3
-2
-4
-3
-5
-4
-5
9
5
4
3
2
1
0
-1
-2
-3
-4
-5
-6
9
1
0.5
0
-0.5
-1
-1.5
-2
-2.5
-3
-3.5
9
L=0
4
3
2
1
0
-1
-2
-3
-4
-5
-6
8
7
6
5
4
3
2
1
0 0
1
2
3
4
5
6
7
8
9
LLNL-‐PRES-‐437553 and 438113!
19
Low-energy Equivalents: Vlow-E(r) =
∫ V(r,r’) dr’
Imaginary
Real
15
5
Increasing L
0
Imag Vlow!E (MeV)
VlowE (MeV)
10
5
0
!5
!5
Increasing
L
!10
0
2
4
6
Radius (fm)
8
10
!15
0
2
4
6
Radius (fm)
8
10
See strong L-dependence that is missing in empirical optical potentials.
LLNL-‐PRES-‐437553 and 438113!
20
Comparison of (complex) S-matrix elements
1
Koning!Delaroche optical potential
Becchetti!Greenless optical potential
CRC + NONO
0.5
2
1
0
33
8
Imag(SL)
888
9 9
9
44
55
0
7
7
6
6
!0.5
5
77 5
01
2 0
1
3 2
4 3
66
4
10
1010
11
11
11
12
12
13
14
15
16
17
18
19
20
Comparison
of CRC+NONO
results
with
Empirical
optical potls
(central part).
See more rotation
(phase shift).
Labeled by partial wave L
!1
!1
!0.5
0
Real(SL)
0.5
1
Room for
improvements!
LLNL-‐PRES-‐437553 and 438113!
21
Exact equivalents: fitted to S-matrix elements
Fit real and imaginary shapes of an optical potential
to the S-matrix elements.
2
Imag part of fitted optical potential (MeV)
Real part of fitted optical potential (MeV)
20
30 MeV
36 MeV
39 MeV
40 MeV
0
!20
!40
!60
!80
0
1
2
3
4
5
6
Radius (fm)
7
8
9
10
0
!2
!4
!6
30 MeV
36 MeV
39 MeV
40 MeV
!8
!10
0
2
4
6
Radius (fm)
8
10
Again: too much attraction at short distances
LLNL-‐PRES-‐437553 and 438113!
22
Further Research on Optical Potentials
1. 
2. 
3. 
4. 
Compare coupled-channels cross sections with data
Reexamine treatment of low partial waves: improve fit?
Effect of different mean-field calculations from UNEDF.
Improve effective interactions:
• 
• 
• 
Spin-orbit parts  spin-orbit part of optical potential
Exchange terms in effective interaction  small nonlocality.
Density dependence (improve central depth).
5.  Examine effect of new optical potentials:
• 
• 
Are non-localities important?
Is L-dependence significant?
6.  Use also ab-initio deuteron potential.
7.  Do all this for deformed nuclei
(Chapel Hill is developing a deformed-QRPA code)
LLNL-‐PRES-‐437553 and 438113!
23

Lawrence Livermore National Laboratory Coupled

Transcrição

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