Configuration interaction studies of
pairing and clustering in light nuclei
Alexander Volya
Florida State University
DOE support: DE-SC0009883
Trento
September 2016
Questions
• Description of clustering, from theory to experiment
• Geometric structure of alpha clusters, real vs algebraic structures
• Interplay between pairing and clustering
Cluster-nucleon configuration interaction approach
Traditional shell model configuration
m-scheme
| i=
†
|0i ⇠
† †
a1 a2
Cluster configuration
SU(3)-symmetry basis
|channeli = |A {
†
. . . aA |0i
D}
⌘
+
D
| i
•
•
•
•
+
†
|
Di
Boosts, m-scheme, and SU(3) basis
Construction and classification of cluster configurations
Center of mass and translational invariance
Non-orthogonality and bosonic principle
†
†
|0i ⌘
†
|
Di
Center-of-Mass (CM) boosts
and
Shell model, Glockner-Lawson procedure
CM quanta creation and
annihilation (vectors)
Intrinsic
SM state Center-of-mass
state
motion
0
↵
CM angular momentum operator
1s 0d
R↵
0p
0s
Select configuration content of NCSM wave
functions for 4He with ︎Ω = 20 MeV boosted
by 8 quanta (L = 0).
K Kravvaris and A. Volya, Journal of Physics:
Conference Series, Cluster 2016, Napoli.
Cluster configurations
Example: alpha decay with ℓ𝓁=0 from sd shell
21 way to make L=0 T=0 4-nucleon combination
Each nucleon has 2 oscillator quanta, 8 quanta total
In oscillator basis excitation quanta are conserved
We model alpha as 4-nucleons on s-shell (0s)4
Make single SU(3) operator with quantum numbers (8,0)
Cluster coefficient is known analytically Xn⌘0 `
|
n`m (1)
n`m (2) n`m (3)
{z
4⇥2=8 quanta
m-scheme state
n`m (4)
}
$
X
⌘
X n0 `
0
↵
1s 0d
0p
⌘
(8,0):`m
⌘
(8,0):`m
⌘
SU(3) symmetry state
R↵
=
0s
|
0
(R↵ )
↵
{z
} |{z}
n0 `0 m 0
8 quanta
0 quanta
motion of alpha
Volya and Yu. M. Tchuvil’sky, Phys. Rev. C 91, 044319 (2015).
Yu. F. Smirnov and Yu. M. Tchuvil’sky, Phys. Rev. C 15, 84 (1977).
M. Ichimura, A. Arima, E. C. Halbert, and T. Terasawa, Nucl. Phys. A 204, 225 (1973).
O. F. Nemetz, V. G. Neudatchin, A. T. Rudchik, Yu. F. Smirnov, and Yu. M. Tchuvil’sky, Nucleon Clusters in Atomic Nuclei and Multi-Nucleon Transfer Reactions (Naukova Dumka, Kiev, 1988), p. 295.
Center-of-mass recoil correction
Channel of relative motion
0
2
0
1
n2 `2
n1 `1
Translational invariance from recoil
Shell model, Glockner-Lawson procedure
Intrinsic
SM state Center-of-mass
state
vibration
Factorizing center of mass in overlap integral
SM overlap integral (FPC) Translationally invariant part
Spurious CM integral
Recoil factor (inverse of Talmi-Moshinsky coefficient)
↵
Traditional Cluster Spectroscopic Characteristics
=
Recoil Factor
h
D
Cluster Coefficient Fractional Parentage Coefficient
Traditional “old” spectroscopic factors
Expand radial motion in HO wave functions
i
Bosonic nature of 4-nucleon operators
non-orgothogonality
If
is thought of as being a boson then
* For p-shell the result is known analytically 64/45
Effective operators (alphas) are not ideal bosons
Cluster configurations are not orthogonal and not normalized
Orthogonality condition model, new SF
•
•
•
•
Non-orthogonal set of channels (over-complete set of configurations)
Pauli exclusion principle
Matching procedure, asymptotic normalization, connection to observables
No agreement with experiment on absolute scale
Resonating group method
New spectroscopic factor
Sum of all new SF from all parent states to a given final
state equals to the number of channels
R. Id Betan and W. Nazarewicz Phys. Rev. C 86, 034338 (2012)
S. G. Kadmenskya, S. D. Kurgalina, and Yu. M. Tchuvil’sky Phys. Part. Nucl., 38, 699–742 (2007).
R. Lovas et al. Phys. Rep. 294, No. 5 (1998) 265 – 362.
T. Fliessbach and H. J. Mang, Nucl. Phys. A 263, 75–85 (1976).
H. Feschbach et al. Ann. Phys. 41 (1967) 230 – 286
Alpha clustering in sd-shell nuclei
USDB interaction [5]
(8,0) configuration
• Old SF are small
• Old SF decrease with A
[1] T. Carey, P. Roos, N. Chant, A. Nadasen, and H. L. Chen, Phys. Rev. C 23, 576(R) (1981).
[2] T. Carey, P. Roos, N. Chant, A. Nadasen, and H. L. Chen, Phys. Rev. C 29, 1273 (1984).
[3] N. Anantaraman and et al., Phys. Rev. Lett. 35, 1131 (1975).
[4] W. Chung, J. van Hienen, B. H. Wildenthal, and C. L. Bennett, Phys. Lett. B 79, 381 (1978).
[5] B. A. Brown and W. A. Richter, Phys. Rev. C 74, 034315 (2006)
Alpha cluster spectroscopic factors in 24Mg
Experimental results
Theoretical calculations in SD shell
100
SF
10-1
E. S. Diffenderfer, et.al Phys. Rev. C 85, 034311 (2012).
01
L=0
10-2
SF
10-3
10
-3
10
-1
10-2
10
SF
L=2
21
10
-3
L=4
41
-2
10-3
SF
10
L=6
-1
61
10-2
10-3
SF
10
-1
L=8
82
Rotational Structures
10-2
10-3
0
5
10
15
20
25
E [MeV]
The sd valence space is considered with USDB interaction the operator is
Building cluster channels with realistic alpha
SD-shell nuclei
JISP interaction
1
1
0.9
0.9
0.8
0.8
0.7
0.7
SF
SF
Effective sd-shell interaction USDB
0.6
0.5
0.6
0.5
0.4
0.4
Nmax=0
Nmax=2
Nmax=4
Nmax=6
0.3
0.2
20
25
30
A
35
Nmax=0
Nmax=2
Nmax=4
Nmax=6
0.3
0.2
40
20
25
30
A
35
40
6+2
6+251
16
p-sd shell model
6+1
5-1
14
0+3
4+2
4+1
2+2
1-2
1-1
2+1
3-1
0+2
4+2
2+3
0+3
4+1
2+2
2+1
1-1
0+2
3-1
Excitation Energy [MeV]
6+1
SU(3) configurations
12
For positive parity
10
8
6
4
2
0+1
Spectroscopic factor S
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
Hamiltonian from
E. K. Warburton and B. A. Brown, Phys. Rev. C 46 (1992) 923
Y. Utsuno and S. Chiba, Phys. Rev. C83 021301(R) (2011)
0.1
Theory
0
0
Experiment
0+1
(new)
Rotational structure in 16O
60
50
exp
SM
02 21
41
62
J(J+1)
40
30
20
10
0
Full p-sd shell calculations, included operators are
SF(alpha)
0.3
02 21
41
62
0.2
0.1
0
This Work
0.8
0.54
0.22
* Yamada, et. al Clusters in nuclei, Vol 2, (Springer-Verlag, 2012) page 229.
B(E2)
0+(1)
0+(2)
0+(3)
Yamada*
0.6
0.8
0.2
02 21 41
0++p-sd shell model calculation
7 2
4+
6
+
p-sd effective6hamiltonian
E. K. Warburton
and B. A. Brown, Phys. Rev. C 46 (1992) 923
5
62
Y. Utsuno and S. Chiba, Phys. Rev. C83 021301(R) (2011)
4
3
2
1
0
0
2
4
6
8
10
Ex [MeV]
12
14
16
6+2
6+251
16
6+1
5-1
14
0+3
4+2
4+1
2+2
1-2
1-1
2+1
3-1
0+2
4+2
2+3
0+3
4+1
2+2
2+1
1-1
0+2
3-1
Excitation Energy [MeV]
6+1
12
10
8
6
4
2
0+1
Spectroscopic factor S
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
Theory
0
0
Experiment
0+1
(new)
28Si
48Cr
1
1
0+1
0+2
+
0.8 03
0+4
0+5
0.6
0.6
SF
SF
0+1
+
02
0+3
0.8 0+
4
0+5
0.4
0.4
0.2
0.2
0
0
-1
0
1
pairing strength
2
0
1
2
pairing strength
3
Classic Example: 24Mg, 8 nucleons in sd-shell
Realistic
(6,2)
(8,4)
(5,1)
Pairing (T=1)
(6,5)
(2,4)
(3,2)
(1,3)
(2,1)
(9,2)
(10,0)
(7,0)
(4,3)
(2,7)
(7,3)
(8,1)
(8,4)
(7,3)
(6,2)
(8,1)
(7,0)
(10,0)
(9,2)
(4,6)
Random
(7,0)(8,4)
(8,1)
(7,3)
(4,6)
(1,0)
(1,6)
(0,5)
(0,2)
(1,6)
(0,5)
(2,4)
(5,1)
(0,8)
(3,5)
(4,0)
(3,2)
(0,8)
(3,5)
(1,3)
(4,0)
(5,1)
(2,1)
(0,2)
(6,2)
(4,3)
USD realistic interaction
Ground state J=T=0
breakdown in SU(3) irreps
24Mg
phase diagram
Contour plot of invariant correlational entropy showing a phase diagram as a function
of T=1 pairing (λT=1) and T=0 pairing (λT=0); three plots indicate phase diagram as
a function of non-pairing matrix elements (λnp) . Realistic case is λT=1=λT=0 =λnp=1
Invariant Correlational Entropy
• Parameter-driven equilibration (pairing strength)
• Averaged density matrix
• ICE
Advantages
•Basis independent
•Explore individual quantum states
•Needs no heat bath
•No equilibration, thermalization and particle number
conservation issues.
•Probe sensitivity of states to noise in external parameter(s)
•Phase transitions -> peaks in ICE
24Mg
phase transitions
How deformation can enhance paring.
Quadrupole-quadrupole interaction on single j-level
• Exchange terms result in
attractive pairing interaction
• Terms scale as 1/Ω where
Ω=2j+1
• Pairing is the strongest
particle-particle interactions
A. Volya, Phys. Rev. C. 65, (2002) 044311.
Acknowledgements:
Thanks to: K. Kravvaris, V. Zelevinsky, Yu. Tchuvil’sky, J. Vary, T Dytrych
Funding: U.S. DOE contract DE-SC0009883. Publications:
K Kravvaris and A. Volya, Journal of Physics: Conference Series, Cluster 2016, Napoli.
A. Volya and Y. M. Tchuvil'sky, Phys.Rev.C 91, 044319 (2015); J. Phys. Conf. Ser. 569,
012054 (2014); (World Scientific, 2014), p. 215.
M. L. Avila, G. V. Rogachev, V. Z. Goldberg, E. D. Johnson, K. W. Kemper,
Y. M. Tchuvil'sky, and A. Volya, Phys. Rev. C 90, 024327 (2014).
A. Volya, Phys. Rev. C. 65, (2002) 044311.
A. Volya and V. Zelevinsky, Phys. Lett. B574, (2003) 27-34.