Gaute!Hagen!(ORNL) -...
Transcript of Gaute!Hagen!(ORNL) -...
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Coupled-cluster approach to open-shell nuclei Effective interactions and the nuclear shell-model
Lecture 4, TALENT school
Gaute Hagen (ORNL)
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Outline • The nuclear landscape and many-‐body methods • Observa<on of shell structure in nuclei a
paradigm for nuclear theory • Extending CC to open-‐shell nuclei via equa<on-‐
of-‐mo<on techniques • Connec<ng the tradi<onal shell model
(configura<on interac<on) with ab-‐ini<o theory. • Coupled-‐cluster effec<ve interac<ons (CCEI) and
IM-‐SRG valence space interac<ons for the shell model
• Descrip<on of neutron oxygen and carbon isotopes
• Extending CCEI and IM-‐SRG to describe nuclei with protons and neutrons in the valence space.
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Ab initio approaches • Quantum Monte Carlo • Lattice EFT • Configuration interaction/NCSM • Coupled Cluster method • In-Medium SRG • Self-Consistent Green’s Functions
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Mass differences: Liquid drop – experiment. Minima at closed shells.
• Expensive to remove a neutron form a closed neutron shell.
• Signature of magic shell closures for N=2, 8, 20, 28, 50, 82, 128
Bohr & MoYelson, Nuclear Structure.
Shell structure in nuclei
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Nuclei with magic N : • Large separa<on energies • High-‐lying first 2+ exited state • Low B(E2) transi<on strength • Kink/drop in charge radii
E2+
B(E2)
S. Raman et al, Atomic Data and Nuclear Data Tables 78 (2001) 1.
Shell structure in nuclei
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Need spin-‐orbit force to explain magic numbers beyond 20.
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Magic numbers: 2, 8 20, 28, 50,82…
Nobel Prize 1963
Goeppert-‐Mayer Jensen
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Main idea: Use shell gaps as a trunca<on of the model space. Nucleus (N,Z) = Double magic nucleus (N*,Z*) + valence nucleons (N-‐N*, Z-‐Z*) Restrict excita<on of valence nucleons to one oscillator shell. ProblemaBc: Intruder states and core excita<ons not contained in model space. Examples: pf-‐shell nuclei: 40Ca is doubly magic sd-‐shell nuclei: 16O is doubly magic p-‐shell nuclei: 4He is doubly magic
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Traditional shell model
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Traditional shell model
Example: 20 Ne
16O core
Valence space
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Hamiltonian governs dynamics of valence nucleons; consists of one-‐body part and two-‐body interac<on (three-‐body +…) :
Single-‐par<cle energies (SPE) Two-‐body matrix elements (TBME) coupled to good spin and isospin
Annihilates pair of fermions
How does one determine the single-‐par<ce energies (SPE) and two-‐body matrix elements (TBME)?
Shell model Hamiltonian
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Empirical determination of SPE and TBME Determine SPE from neighbors of closed shell nuclei (A) having mass A +1 Determine TBME from fit to empirical data (e.g. excited states, transi<on rates) Accurate descrip<on of nuclei in the area of the nuclear chart where they were fiYed. How to make accurate predic<ons for nuclei beyond the range of where the TBME were fiYed?
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Single reference coupled cluster theory
Islands of inversion
Neutrinoless double-‐beta decay 76Ge
How to extend CC to open-‐shell nuclei? Answer: EOM and coupled-‐cluster effec<ve interac<ons (CCEI)
• Most nuclei are open-‐shell. • Shell structure changes in
neutron rich nuclei. • Near degeneracy in very
neutron rich nuclei
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Going beyond closed shell nuclei with equation-of-motion techniques
Remember in coupled-‐cluster theory we deal with the similarity transformed Hamiltonian:
H = e�THeT
With T being the cluster amplitude (sum of 1p-‐1h, 2p-‐2h, 3p-‐3h, … excita<ons:
Structure of similarity-‐transformed Hamiltonian aoer the CCSD equa<ons are solved
0p0h 1p1h 2p2h 0p0h 1p1h 2p2h
We can diagonalize in a sub-‐space of n-‐par<cle-‐m-‐hole to access other regions of Fock space
H
we can obtain excited states for the closed-‐shell systems by diagonalizing in a sub-‐space 1p-‐1h and 2p-‐2h excita<ons
�HRA
µ
�C|�0i = !uR
Aµ |�0i
H RAµ =
X
ia
rai a†aai +
1
4
X
ijab
rabij a†aa
†bajai
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Going beyond closed shell nuclei with equation-of-motion techniques
Par<cle aYached equa<on-‐of-‐mo<on theory for nuclei : J. R. Gour et al PRC 74 024310 (2006)
�HRA±1
µ
�C|�0i = !uR
A±1µ |�0i
RA+1µ =
X
a
raa†a +1
2
X
iab
rabi a†aa†bai RA�1
µ =X
i
riai +1
2
X
ija
raija†aajai
We can access the Fock space by diagonalizing in sub-‐space of one-‐par<cle (hole) and two-‐par<cle-‐one-‐hole (one-‐par<cle-‐two-‐hole) excita<ons:
A± 1 H
The right eigenvectors are given by: A± 1
Note that since the similarity transformed Hamiltonian is non-‐Hermi<an we need to compute the corresponding leo eigenvectors if we want to compute expecta<on values and transi<on densi<es, i.e.
hOi = hLµ|O|Rµi
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Computing nuclei with A+1 Example: proton-halo state in 25F
States dominated by 3p-‐2h excita<ons
Excita<on spectra of 25F obtained with par<cle-‐aYached equa<on-‐of-‐mo<on techniques. Here we included some 3p-‐2h excita<ons as well.
Zs. Vajta et al. PRC 89, 054323 (2014)
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Going beyond closed shell nuclei with equation-of-motion techniques
Two-‐par<cle aYached equa<on-‐of-‐mo<on theory for nuclei : G. Jansen et al PRC 2011, G. Jansen (2013), J. Shen and P. Piecuch J. Chem. Phys (2013) 2PA-‐EOM compares well with NCSM for spectra in 6Li (P. Navra<l private communica<on)
�HRA±2
µ
�C|�0i = !uR
A±2µ |�0i
We can access the Fock space by diagonalizing in sub-‐space of two-‐par<cle (hole) and three-‐par<cle-‐one-‐hole (one-‐par<cle-‐three-‐hole) excita<ons:
A± 2 H
RA+2µ =
1
2
X
ab
raba†aa†b +
1
6
X
iabc
rabci a†aa†ba
†cai
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Computing nuclei with A+2: Example Fluorine-26 G. Jansen et al PRC 2011, J. Shen and P. Piecuch J. Chem. Phys (2013)
Experimental spectra in 26F compared with phenomenological USD shell-‐model calcula<ons and coupled-‐cluster calcula<ons
A. Lepailleur et al PRL (2012)
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Valence cluster expansion: connecting ab-initio approach with the shell-model
P. Navra<l et al, PRC 55, 573 (1997) A. Lisetskiy et al. ,PRC 78, 044302 (2008), PRC 80, 023315 (2009)
• Need to solve for the core (Ac), and the Ac+1, Ac+2,… neighboring nuclei.
• Project the Ac+1, Ac+2, … wavefunc<ons onto the valence space via a similarity transforma<on to obtain an effec<ve Hamiltonian
H =X
i<j
✓(pi � pj)
2mA+ V
(i,j)NN
◆+
X
i<j<k
V(i,j,k)3N
We start from the intrinsic Hamiltonian with NN and 3N forces:
Write the intrinsic Hamiltonian as a valence cluster expansion:
HA = HA,Ac +HA,Ac+1 +avX
k=2
HA,Ac+k
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• Define a model space P and complement space Q
• Construct a similarity transformation such that
• This defines an effective
Hamiltonian in the P-space which exactly reproduces n eigenvalues of the full Hamiltonian:
. . e2 e1
. . en+2 en+1
PHP
PHQQHQ
QHP
Q(X�1HX)P = 0
PHQ
P�X�1HX
�P | e↵
k i = ek| e↵k i
Effective interactions from similarity transformations
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Bloch-Brandow effective interaction (Lee-Suzuki similarity transformation)
Upk = h↵P | ki
| e↵k i ⌘ P | ki
h↵P |He↵ |↵P 0i =dX
k=1
h↵P | e↵k iekh e↵
k |↵P 0i
h↵P |He↵ |↵P 0iLS =dX
k=1
h↵P | kiekh↵P 0 | ki
Write the effective Hamiltonian in spectral representation which reproduces d eigenvalues of the full Hamiltonian:
Effective eigenvectors are defined as (Bloch-Brandow):
This gives the effective Hamiltonian in the P space:
Where are the matrix elements of the inverse of the matrix U with matrix elements:
h↵P | ki
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Coupled cluster effective interaction
Solve for the Ac+2 problems via two-particle attached equation-of-motion coupled-cluster (Talks by P. Piecuch and G. Jansen)
To obtain Heff we can either project the left or the right solutions onto the P-space:
Using the right eigenvector projections we obtain CCEI:
We can hermitize HCCEI by symmetric orthogonalization procedure (I. Mayer Int. J. Quantum Chem 90, 63 (2002))
⇥S†S
⇤1/2H
ACCEI
⇥S†S
⇤�1/2
h↵P |HA,Ac+2e↵ |↵P 0i =
dX
k=1
h↵P |RA,Ac+2k iekh↵P 0 |RA,Ac+2
k i
h�0|LAc+2µ H = !µh�0|LAc+2
µ
HRAc+2µ |�0i = !µR
Ac+2µ |�0i
| e↵k i ⌘ P |RA,Ac+2i
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2v 1q1v 2q 3p1h 4p2h2v 1q1v 2q 3p1h 4p2h
8
28
20
50
0p3/2
0d5/2
1s1/2
0d3/2
0p1/2
0g9/2 0f5/2
1p3/2
1p1/2
0f7/2
h
p
v
q
H(s = 0) ! H(1)
Open-shell systems
Separate p states into valence states (v) and those above valence space (q)
Define Hod to decouple valence space from excitations outside v
Tsukiyama, Bogner, Schwenk, PRC (2012) Bogner et al, PRL (2014).
IM-SRG: Valence-Space Hamiltonians
H(s) = U(s)HU †(s) ⌘ Hd(s) +Hod(s) ! Hd(1)
⌘I (s) =⇥Hd(s), Hod(s)
⇤
dH(s)
ds= [⌘(s), H(s)]
⌘(s) ⌘ (dU(s)/ds)U†(s)
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Open-shell systems
Separate p states into valence states (v) and those above valence space (q)
Core physics included consistently (absolute energies, radii…)
Inherently nonperturbative – no need for extended valence space
Non-degenerate valence-space orbitals
2v 1q1v 2q 3p1h 4p2h2v 1q1v 2q 3p1h 4p2h
8
28
20
50
0p3/2
0d5/2
1s1/2
0d3/2
0p1/2
0g9/2 0f5/2
1p3/2
1p1/2
0f7/2
h
p
v
q
H(s = 0) ! H(1)
Heff
IM-SRG: Valence-Space Hamiltonians
Tsukiyama, Bogner, Schwenk, PRC (2012) Bogner et al, PRL (2014).
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Nuclear forces from chiral effective field theory [Weinberg; van Kolck; Epelbaum et al.; Entem & Machleidt; …]
CE CD
Similarity renormaliza<on group for nuclear forces: S. Bogner, R. J. Furnstahl and A. Schwenk Prog. Part. Nucl. Phys. 65, 94 (2010)
Hebeler, Holt, Menendez, Schwenk, Ann. Rev. Nucl. Part. Sci. in press (2015)
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Coupled-cluster effective interaction in practice
• Obtain excited states of Ac+ 1 and Ac+ 2 from PA-EOMCCSD(2p1h) and 2PA-EOMCCSD(3p1h)
• The Ac+1 Hamiltonian is diagonal and given by the Ac+1 lowest eigenvalues
• Are results sensitive to the choice of left/right eigenvector projections for A+2?
• How do we choose the d “exact” A+2 wavefunctions? Ø Largest overlap with model
space Ø Lowest energies
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CCEI: Application to the oxygen chain
G. R. Jansen, J. Engel, G. Hagen, P. Navra<l, A. Signoracci, PRL 113, 142502 (2014).
• Start from chiral NN(N3LOEM) + 3NF(N2LO) interac<ons SRG evolved to 2.0fm-‐1
• Model space size Nmax = E3max = 12, hw = 20MeV
P
Q
Q
p3/2 f7/2 d3/2 s1/2 d5/2 p1/2 p3/2 s1/2
...
Low-‐lying states in 17O as a func<on of A. These energies defines the single-‐par<cle energies of Heff
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CCEI: Application to the oxygen chain
G. R. Jansen, J. Engel, G. Hagen, P. Navra<l, A. Signoracci, PRL 113, 142502 (2014).
Comparison between coupled-‐cluster effec<ve interac<on (CCEI) and “exact” coupled-‐cluster calcula<on with inclusion of perturba<ve triples Λ-‐CCSD(T) (Talk by R. BartleY).
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Coupled-cluster effective interactions for the shell model: Oxygen isotopes
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Coupled-cluster effective interactions for the shell model: Carbon isotopes
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Benchmarking different methods: Binding energies oxygen isotopes
In-‐medium SRG S. Bogner et al, Phys. Rev. LeY. 113, 142501 (2014)
Coupled-‐Cluster EffecBve InteracBons G. R. Jansen et al, Phys. Rev. LeY. 113, 142502 (2014)
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Benchmarking different methods: Spectra in 22,23,24O
In-‐medium SRG S. Bogner et al, Phys. Rev. LeY. 113, 142501 (2014) Hebeler, Holt, Menendez, Schwenk, Ann. Rev. Nucl. Part. Sci. in press (2015)
Coupled-‐Cluster EffecBve InteracBons G. R. Jansen et al, Phys. Rev. LeY. 113, 142502 (2014)
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Extending CCEI to nuclei with protons and neutrons in valence space
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CCEI for nuclei with protons and neutrons in valence space: Ne and Mg isotopes
0
1
2
3
4
5
Ene
rgy
[MeV
]
1/2+5/2+
3/2+
9/2+
7/2+
1/2+5/2+
3/2+
9/2+
3/2+
7/2+
19Ne
0+
2+
4+
0+
2+
4+
20Ne
1/2+
5/2+
3/2+
5/2+
7/2+
3/2+9/2+7/2+9/2+
1/2+
(5/2+)
(3/2+)
(3/2+)
(3/2+)
25Ne
0+
2+
4+
2+
0+
2+
2+
26Ne
3/2+
1/2+
7/2+5/2+
5/2+
3/2+11/2+1/2+5/2+3/2+9/2+
(3/2+)
(1/2+)
27Ne
CCEI
Exp.
0
1
2
3
4
5
Ene
rgy
[MeV
]
0+
2+
4+
2+
3+
0+
2+
4+
2+
22Mg
CCEI
Exp.
3/2+
5/2+
1/2+3/2+
7/2+
5/2+9/2+7/2+
7/2+
3/2+
5/2+
7/2+
1/2+
9/2+(3/2+)
(3/2+)
1/2+
23Mg
CCEI
Exp.
0+
2+
2+
3+
4+
0+
2+
4+2+
24Mg
CCEI
Exp.
1/2+
3/2+
5/2+5/2+
3/2+1/2+7/2+7/2+3/2+5/2+9/2+5/2+5/2+9/2+7/2+9/2+
5/2+
1/2+
3/2+
7/2+
5/2+
1/2+7/2+3/2+
9/2+
5/2+9/2+3/2+
25Mg
CCEI
Exp.
0+
2+0+
2+
2+3+
3+4+
0+
2+
2+
0+
3+
4+2+3+
26Mg
![Page 33: Gaute!Hagen!(ORNL) - nucleartalent.github.ionucleartalent.github.io/Course2ManyBodyMethods/doc/pub/cc/pdf/...Coupled-cluster approach to open-shell nuclei E"ective interactions and](https://reader031.fdocuments.in/reader031/viewer/2022022609/5b90a7ea09d3f2b6628c7cf0/html5/thumbnails/33.jpg)
Benchmarking different methods in 24F
IM-‐SRG: L. Caceres et al arXiv:1501.01166 (2015)
0 250 500 750
1000 1250 1500 1750 2000 2250 2500 2750 3000 3250 3500
Exci
tatio
n En
ergy
(keV
)
24F Level Scheme
CCEI3+2+
1+4+0+3+
2+1+5+2+
EOM<CCSD3+
2+
1+0+
4+
3+
5+2+
Exp.(3+)
(2+)
1+
(4+)(3+)
(2+,4+)(1+,2+)
IM<SRG3+2+
1+
0+4+
3+
2+