Transcript Indirect Techniques in Nuclear Astrophysics: The ANC Method

Asymptotic Normalization
Coefficients as an Indirect Technique
for Nuclear Astrophysics
R. Tribble
Texas A&M University
Carpathian Summer School
Sinaia Romania, 2012
What I plan to cover
• Motivation for Asymptotic Normalization
Coefficients (ANCs)
• Proton ANC examples, consistency checks
• ANCs and RIB’s
• Neutron ANC measurements
• More on Spectroscopic Factors & ANCs
• Future prospects at TAMU
What I will not cover
• Astrophysics motivations
– Carbon production and the CNO cycle in early
generation stars
– CNO and HCNO cycles
– HCNO breakout and r-p process
– Production of 18F, 22Na, 26Al, . . .
• Details of techniques used for
measurements
• I am happy to discuss these issues with
you!
Radiative p (a) capture at stellar energies
Classical barrier penetration problem
V
Direct radiative capture
p
E
0

• low energies  capture at large radii
• asymptotic tail dominates
• very small cross sections
ANC: the natural quantity for
charged-particle direct-capture
reactions in nuclear astrophysics
Direct Radiative Proton Capture
 M
2
[S ( E )  Ee2 ]
^
Low energies:
^
M  I ( rBp ) O ( rBp ) (i ) ( rBp )
Integrate over ξ:
Find:
( )
M  A ( B , p , Bp ) O ( rBp ) B ( B ) p ( p ) i ( rBp )
M is:
A
Bp
rB R N
I ( rBp )  C
A
Bp
A 2
capture ( C Bp
)
A
Bp
W  , l 1 ( 2 Bp rBp )
A
2
rBp
Radiative [p(a)] Capture with resonant
and subthreshold states: ANCs
Capture through resonance
V
p1 / 2 1 / 2
M
i
E  E0 
2
resonance
E
Direct capture

A
M  CBp
 subthreshold state
Capture to ground state
through subthreshold state
Interference effects!
B
B
p
p
B

A p
C Bp
p
A
C  AA
A
Bp



A
CBp
1 / 2
M
i
*
E  
2
ANCs in astrophysics
Determining ANCs
- elastic scattering at low energy
- transfer reactions (examples covered today!)
- breakup reactions
Transfer Reaction
Transition amplitude:
A
d
M   (f) I Bp
V I ap
(i )
Standard approach:
I
b
ap
 S ap n d l d j d ( rap )
1
2
d
DW
  S Bpl A j A S apl d jd  l A j A l d jd
d
Transfer Reaction
Transition amplitude:
A
d
M   (f) I Bp
V I ap
(i )
Peripheral transfer:
I
A
Bp
rBp R N
 C
A
Bp
W  , l 1 ( 2 Bp rBp )
A
2
rBp
 l A j A l d jd
d
A
2
d
2
 ( CBpl A j A ) ( Capld jd ) 2
2
d
bBpl A j A bapl
d jd
DW
Practical Issues:
extracting ANCs
•
•
•
•
•
Find a peripheral transfer reaction
Measure angular distribution (abs. c.s.)
DWBA calculation (optical model parameters)
Determine single particle ANCs
Need ANCs for one of the vertices!
 l A j A l d jd
d
A
2
d
2
 ( CBpl A j A ) ( Capld jd ) 2
2
d
bBpl A j A bapl
d jd
DW
Extracting the Information
• DWBA – DWUCK5, PTOLEMY, FRESCO,
...
• Optical Model parameters
– Elastic scattering and/or literature values
A Peripheral Transfer Reaction
• 13C(14N,13C)14N at ~ 10 A MeV
• ‘Elastic’ transfer is special:
𝒅𝝈
𝒅𝜴
13C(14N,14N)13C
∝ 𝑪𝟒
13C(14N,13C)14N
Verifying Peripheral Transfer
• 13C + p  14N
• vary r and a in single-particle binding potential
from 1.1 fm < r < 1.3 fm; 0.5 < a < 0.7
• Checked radial cutoff for change in cross section
4
5
0
5.5
6
7
 vs Rcut(fm)
Consistency check: ANCs from
13C(14N,13C)14N vs 13C(3He,d)14N
13C(14N,13C)14N
13C(3He,d)14N
P1/2 P1/2 dominates
P1/2 P3/2
(C1,1/2)2=18.6(12) fm-1
(C1,3/2)2=0.93(14) fm-1
(C1)2=19.1(9) fm-1 
(C1,1/2)2=18.2(9) fm-1
(C1,3/2)2=0.91(14) fm-1
Test Case
•
16O(p,)17F
- suggested at INT workshop 02/97
- pure direct capture
16O(3He,d)17F
reaction
S factor for 16O(p, )17F
• ANCs 16O(3He,d)17F
(C2)gnd = 1.08 ± .10 fm-1
(C2)ex = 6490 ± 680 fm-1
• Direct Capture data
from Morlock, et. al
Extracting the Information
• DWBA – DWUCK5, PTOLEMY, FRESCO,
...
• Optical Model parameters
– Elastic scattering and/or literature values
– For RIBs, folding model with JLM interaction
Double Folding Model
• Elastic scattering with RIBs to large angles
is very difficult
• Use double folding with JLM(1)
• Renormalization coefficients for loosely
and tightly bound states via p-shell studies
13N(p,)14O
via 14N(13N,14O)13C
13C
@ 15 MeV/u
H2 cryotarget
13N
@12 MeV/u
99% pure, 4 mm dia
Melamine target
Four telescope
system (“the cross”):
DE – PSD 65, 110 mm
E – 500 mm
14N(13N,14O)13C

exp  C

14
13
O
1
N1
2
14

C13 N 3
2 

C1
 
2
  14
   N 14 O
  b13 3 b13 1
  C12 N12

 C14 N
1
13

C1
2
  14
14
 b13 N 1 b13 O 1
 C12 N12
2


 DW
1 1
 1212

2


 DW
1 3
 1212







C2 = 29.0 ± 4.3 fm-1
ANC for 13N + p  14O
S-factor for 13N(p,)14O
Extracting the Information
• DWBA – DWUCK5, PTOLEMY, FRESCO,
...
• Optical Model parameters
– Elastic scattering and/or literature values
– For RIBs, folding model with JLM interaction;
renormalization coefficients from p-shell studies
• Checks done for some cases with Coupled
Channel codes
= studied at TAMU/Rez
32Ar
33Ar 34Ar
35Ar
36Ar
30Cl
31Cl
32Cl
33Cl
34Cl
35Cl
33S
34S
27S
28S
29S
30S
31S
32S
25P
26P
27P
28P
29P
30P
31P
23Si
24Si
25Si
26Si
27Si
28Si
29Si
30Si
22Al
23Al
24Al
25Al
26Al
27Al
28Al
= planned with T-REX
21Al
31Ar
20Mg 21Mg 22Mg 23Mg 24Mg 25Mg 26Mg 27Mg
19Na 20
Na 21Na 22Na 23Na 24Na
16Ne 17
Ne 18Ne 19Ne
20Ne 21Ne 22
Ne 23Ne
14F
15F
16F
17F
18F
19F
12O
13O
14O
15O
16O
17O
18O
11N
12N
13N
14N
15N
9C
10C
11C
12C
13C
14C
8B
9B
10B
11B
7Be
8Be
9Be
6Li
7Li
8Li
15C
Neutron ANCs
• Neutron transfer in a heavy-ion reaction is
highly peripheral
• Sensitive to neutron ANC
• Whittaker function replaced by Hankel
function in exterior of overlap integral
• For proton direct capture, convert neutron
ANCs to proton ANCs by charge symmetry
ANC’s for 7Be + p  8B
from mirror reaction 13C(7Li,8Li)12C
• separate p1/2 and p3/2
• Fits ANC’s
7Li + n  8Li:
• C2(p3/2) = .384±.038 fm-1
• C2(p1/2) = .048±.006 fm-1
 7Be + p  8B:
• C2(p3/2) = .405±.041 fm-1
• C2(p1/2) = .050±.006 fm-1
p3/2
p1/2
= studied at TAMU/Rez
21Al
32Ar
33Ar 34Ar
35Ar
36Ar
30Cl
31Cl
32Cl
33Cl
34Cl
35Cl
33S
34S
27S
28S
29S
30S
31S
32S
25P
26P
27P
28P
29P
30P
31P
23Si
24Si
25Si
26Si
27Si
28Si
29Si
30Si
22Al
23Al
24Al
25Al
26Al
27Al
28Al
= planned with T-REX
= studied at TAMU
31Ar
20Mg 21Mg 22Mg 23Mg 24Mg 25Mg 26Mg 27Mg
19Na 20
Na 21Na 22Na 23Na 24Na
16Ne 17
Ne 18Ne 19Ne
20Ne 21Ne 22
Ne 23Ne
14F
15F
16F
17F
18F
19F
12O
13O
14O
15O
16O
17O
18O
11N
12N
13N
14N
15N
9C
10C
11C
12C
13C
14C
8B
9B
10B
11B
7Be
8Be
9Be
6Li
7Li
8Li
15C
Spectroscopic Factors vs ANCs
• S.F. is not invariant under unitary
transformation (ANC is invariant)
• In principle, ANC is an observable, S.F. is not
• In practice, today both are ‘extracted quantities’
• Clearly S.F. is interesting
• As Furnstahl and Schwenk have reported:
Nuclear observables such as binding energies and cross sections
can be directly measured. Other physically useful quantities, such
as spectroscopic factors, are related to measured quantities by a
convolution whose decomposition is not unique and so we refer
here to the extracted quantities as ‘non-observables’.
[arXIV: 1001.0328 (nucl-th), 2010]
• Need absolute S.F. for s-wave (n,) direct capture
• A new approach to extracting S.F.’s
Spectroscopic Factors
• What is S for configuration A + n in B?
• Define overlap function:
I (r )  A ( A ) n ( N ) | B ( A ,  N ; r ) 
• Model independent definition:
S  N  I (r ) | I (r ) 
An Example: (7Ligs + n)2+ 8Ligs

S   drr 2 I 2 (r )
0
• Single particle approach:
I (r )  S1/ 2 (r )
0  r  6 fm
96%
tail determines the ANC
Asymptotic Region - I
• Single particle overlap function for r > RN
I(lj ) (r) 
 K(lj )(lj ) (r)
r  RN
(lj ) (r) 
b(lj ) i h (i r )
r  RN
•
(1)
l
Model independent definition:
r  RN
I (lj ) (r ) 
 C(lj ) i hl(1) (i r )
  2m An B ,
An
 B  mA  mn  mB
An
Asymptotic Region - II
• Asymptotic Normalization Coefficient
C(lj )  K(lj ) b(lj )
• Typical approach, assume for all r
I lj (r )  K ljn( lj ) (r )



Slj   drr I (r )  K ( lj )  dr r  (r )  K ( lj )
2 2
lj
0
2
2
0
2
lj
2
Usual DWBA Analysis for (d,p)
[Note that this works also for Adiabatic Wave Approximation]
• Cross section for A(d,p)B

DW
 M  
2
()
f
I
B
An
V  pn
( )
i

2
• With the single particle approximation
d
()
() 2
 S  f  An (nr lj ) V  pn i 
d
S is the normalization (i.e. spectroscopic,
scaling, . . .) factor
Modified DWBA Analysis
• Zero Range form for M
M  0 S
1/ 2
   An 
()
f
()
i

• With explicit internal and external parts
M  0 S
1/ 2
ˆ (b)  bM
ˆ )
(M


where have used:   bhl (ir )
Modified DWBA Analysis
• Very peripheral transfer 
M  0 S
1/ 2
ˆ    CM
ˆ
bM

0

C is independent of b, S = S(b)
• Less peripheral, both parts contribute
M  0 S
1/ 2
ˆ (b)  bM
ˆ )
(M


C = C(b) AND S = S(b)
Modified DWBA Analysis
• Measure the ANC for a system
• Find a reaction that is less peripheral
• Measure the transfer and calculate C(b)
• Compare to measured ANC to get b
• Find S from S = C2/b2
14C(d,p)15C
Relative Cross Section
14
Reaction studies
C(d,d)14C Ed=60 MeV
14C(d,p)15C
1E+24
1E+23
1E+22
1E+21
1E+20
0
5
10
15
20
25
30
35
40
s1/2 gs
Lab Angle (degrees)
14
C(d,p)15C Ed=60 MeV
Differential Cross
Section (mb/sr)
1.00E+02
1.00E+01
gs
d5/2 740 keV
1.00E+00
d5/2 740 keV
1.00E-01
0
5
10
15
Lab Angle (degrees)
20
25
Ed=17 MeV
Issues for Future
• Reaction Theory
– new theoretical formulations
• Reaction Mechanism
– when do we need to go beyond first order
• Optical Model Parameters
– can we develop a global version of ‘folding model’
potentials for complicated projectiles
• Structure Theory
– can S.F.’s and ANC’s be reported from theory
The Texas A&M Upgrade:
Accelerated RIBs
at Intermediate Energies
CI Upgrade (tasks)
•
•
•
•
T-REX
Re-activate K150 (88”) cyclotron (I)
Build ion guides and produce RIBs (II)
Accelerate RIBs in K500 cyclotron (III)
Project deliverables (‘DOE speak’):
Use K150 stand-alone and as a
driver for secondary rare-isotope
beams that are accelerated with
K500 cyclotron
TASK 1 – Done
• Refurbish K150 (88”) cyclotron

[new: power supplies, vacuum pumps, rf system, control system]
• Move existing high frequency ECR source
from K500

• Connect beam lines to experimental areas

and use K150 for experiments
T-REX
TASK 2
• Develop light-ion guide (helium based)
(based on successful JYFL design)
T-REX

Light Ion Guide Concept
Glaser lens
proton beam
~ 1mbar
~10-6mbar
Target Chamber
He ~ 300 – 500 mbar
~10-2mbar
to CBECR
Faraday Cup
In collaboration with Dr. J. Arje - Jyvaskyla
T-REX
Light Ion Guide Test Setup
T-REX
TASK 2
• Develop light-ion guide (helium based)
(based on successful JYFL design)
• Develop heavy-ion guide (helium based)
(collaboration with Argonne Lab)
T-REX


Gas Catcher – ANL Design
T-REX
Gas Catcher parts
T-REX
Gas Catcher – 11/11/2011
T-REX
TASK 2
• Develop light-ion guide (helium based)
(based on successful JYFL design)
T-REX

• Develop heavy-ion guide (helium based)
(collaboration with Argonne Lab); use
‘BigSol’ as frontend

• Collect RIBs and inject into CBECR


TASK 3
• Transport RIBs to K500 cyclotron
• Inject and accelerate
• Transport accelerated beam to
experimental areas
T-REX
Transport Line
T-REX
Ion-Guide Cave
T-REX
Change from Original Plan
• Accelerate H- and D- for LIG
• Have ~100% extraction efficiency
40.69at Jyväskylä
Dee tank
IRON
42.34
• Using ion source
built
R
Curv R
x
y
x
y
35 mA proton
35.19
-24.20 at -62.00
-736.00 1,045.00
• Extracted  25
beam
30 MeV
H- Ion
Source
y-axis
Einzel
Lens
36
37
38130.00
39
40
6-Way Cross for
41
Turbo Pumps &
80.00
Diagnostics 42
43
44
45
30.00
Isolation 46
Valve
47
48
49 -20.00
50
90o Dipole
Magnet
51
52
53 -70.00
54
55
56
ECR2
T-REX
K150
Cyclotron
-120.00
35.08
35.00
35.02
35.24
35.85
37.14
39.46
Carbon
43.11
Stripper Foil
48.21
54.86
63.15
73.18
85.09
Stripper Foil
99.09
Mechanism
115.54
134.81
157.35
Extraction
183.94
Dipole
215.44Magnet
252.72
296.88
62.00
62.00
-62.00
-62.00
-62.00
62.00
62.00
-62.00
-114.00 -1,273.00
736.00 -1,045.00
114.00 38"1,273.00
Extraction
-736.00 Radius
1,045.00
43.52
129.72
129.72
5.72
5.72
-27.50
-27.50
96.50
H96.50
-27.50
-5.88
56.32
141.32
79.12
-5.88
139.00
-92.80
-70.00
161.80
139.00
Cross at Center
x
H+ y
0.00
62.00
0.00
-62.00
-114.00
-71.00
470.00
736.00
-1,273.00
-816.00
-671.00
-1,045.00
67.72
67.72
56.32
60.62
114.72
141.32
-92.80
-47.10
-32.60
-70.00
62.00
-62.00
96.50
-27.50
0.00
0.00Beam
1
1
1
1
1
1
1
Efficiencies for RIB estimates
• LIG – calculated from Jyväskylä experience
and measurements from test setup
• HIG system
BigSol – about 30% collection efficiency
gas cell – 25%
• Transfer and charge-breeding from 4 - 7%
• Transport to K500 and extraction – 11%
T-REX
Projected Beam Intensities from LIG after K500
(p,n)
Max. Energy
Intensity
Product
MeV/A
particles/s
27Si
57
6 x 103
50Mn
45
2 x 104
54Co
45
6 x 103
64Ga
45
4 x 104
92Tc
35
4 x 104
106In
28
4 x 104
108In
28
3 x 104
110In
26
6 x 104
Assuming 14 mA beam, realistic LIG, CBECR,
transport and K500 extraction efficiencies
T-REX
Examples of reaccelerated beams produced in DIC:
Isotope
t1/2>100ms
Calculations by
G. Souliotis
T-REX
Max. Energy
MeV/u
Neutron rich
9
Li
11
Be
22
O
24
Ne
32
Mg
38
S
40
S
42
S
42
Ar
44
Ar
46
Ar
62
Fe
60
Cr
45
45
40
40
40
36
32
29
39
38
35
38
32
Proton rich
7
Be
8
B
11
C
14
O
22
Mg
23
Al
27
P
62
Ga
64
Ga
60
70
63
70
57
60
62
47
45
2.0-4.
Intensity
Pps
1.7-3.4106
0.7-1.4106
2.0-4.0104
0.5-1.0104
1.3-2.6104
2.5-5.0105
0.5-1.0105
1.8-3.6103
3.3-6.6105
0.9-1.8105
1.8-3.6104
1.9-3.8104
0.5-1.0103
0.5-1.0106
1.2-2.4106
1.3-2.6106
0.7-1.4105
3.1-6.3104
1.2-2.4103
1.0-2.0103
2.1-4.3102
0.9-1.9104
Science accessible with the TAMU upgrade
[long(er) term]
• Nuclear Astrophysics – indirect techniques
• Nuclear Structure – transfer reactions, 
spectroscopy, …
• Fundamental Interactions – trapping expts.
• Dynamics and Thermodynamics – N/Z
degrees of freedom
T-REX
T-REX
[TAMU Reaccelerated Exotics]
T-REX
Collaborators
• T. Abdulla, A. Banu, C. Gagliardi, J. Hardy, V. Iacob, M.
McCleskey, E. Simmons, A. Alharby, B. Roeder, A.
Spiridon, G. Tabacaru, L.Trache – Cyclotron Institute,
Texas A&M University
• J. Aysto, A. Saastamoinen, A. Jokinen – Univ. of
Jyvaskyla
• P. Woods, T. Davinson, G. Lotay – Univ. of Edinburgh
• D. Jenkins, M. Bentley – Univ. of York
• L. Achouri – LPC Caen
• P. Bem, V. Burjan, V. Kroha, E.Simeckova, J. Vincour –
INP, Czech Republic
• F. Carstoiu – IAP, Romania