Symmetry-adapted bases for ab initio structure and reaction theory - - PowerPoint PPT Presentation

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Symmetry-adapted bases for ab initio structure and reaction theory - - PowerPoint PPT Presentation

May 27, 2023 489 likes •698 views

Symmetry-adapted bases for ab initio structure and reaction theory Alexis Mercenne, Kristina Launey, Tomas Dytrych, Jerry Draayer Department of Physics & Astronomy, Louisiana State University, Baton Rouge, LA, 70803 Jutta Escher

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SLIDE 1

Symmetry-adapted bases
 for ab initio structure and reaction theory

Alexis Mercenne, Kristina Launey, Tomas Dytrych, Jerry Draayer

Department of Physics & Astronomy, Louisiana State University, Baton Rouge, LA, 70803

Jutta Escher

Lawrence Livermore National Laboratory, Livermore, California 94550, USA

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SLIDE 2
  • Symmetry-adapted no-core shell model (SA-NCSM)

Based on NCSM:

  • Spherical harmonic oscillator basis
  • Distributions of nucleons over shells
  • Ab initio (no restrictions for interactions …NN, NNN, non-

local,…) New features in SA-NCSM: NCSM with symmetry-adapted (SA) basis (reorganization of model space): SU(3)-coupled basis states or Sp(3,R)-coupled basis states Model space selection (truncation) – physically relevant + exact center-of- mass factorization! Equal to NCSM in complete-Nmax model space

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SLIDE 3

Total HO quanta: NCSM: Nmax determines the size of model space SA-NCSM: keeps track of Nx, Ny, Nz SU(3) basis states: With spin:

Launey et al., Prog. Part. Nucl. Phys. 89 (2016) 101 Dytrych et al., Phys. Rev. Lett. 111 (2013) 252501 LSU code (LSU3shell): sourceforge.net/projects/lsu3shell

  • SA-NCSM: SU(3)-scheme basis

SU(3) basis states: unitary transformation from m-scheme Gives information about important deformed configurations

max max max

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SLIDE 4
  • SA-NCSM: SU(3)-scheme basis

How is the SU(3)-scheme basis constructed? ➢ Intuitive way: Considering 2 particles: Tedious… not used for many-particle system Reduced SU(3) CG ➢ For fast basis construction, use of Gel’fand patterns For a single shell!

Draayer et al., “Representations of U(3) in U(N)”, Comp. Phys. Commun. 56 (1989) 279

spin isospin

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SLIDE 5

SU(3) tensors of NN interaction nrns (λ μ)

N3LO (Nmax=6) ħΩ =11 MeV

initial state (i) final state (f)

NN SU(3) Tensors NN in SU(3) basis jj-coupled NN

  • SA-NCSM: SU(3)-scheme interaction

Equivalent to m-scheme Matrices are smaller and sparser

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SLIDE 6
  • SA-NCSM with SU(3) scheme: Examples

0! 1! 2! 3! 4! 5! E

x!(MeV)!

E xp.! 1/2+! 3/2+! NNLO

  • pt!

<2>8,!<3>9! (9/2)-! (7/2)-! ħΩ=15!MeV! Ne! 19Ne

SA-NCSM (selected model space): 50 million SU(3) states Complete model space: 1000 billion states

1 2 3 4 5 6

Energy (MeV)

  • Expt. SA-NCSM

0+ 2+ 2+ 2+ 5.2-8.4 W.u. 4W.u.

24Ne SRG-N3LO, 2fm–1 ħΩ=15MeV <2>6 1 2 3 4 5 6 7 8

Energy (MeV)

  • Expt. SA-NCSM

0+ (2+) 2+ 0+ ground state: rrms(m)=2.89 fm

32Ne N2LOopt ħΩ=15MeV <2>4

2+ 0+

NNLOopt <2>10 Robert Baker, PhD student, LSU

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SLIDE 7

Symplectic Sp(3,R) basis: Symmetry-adapted: SU(3), Sp(3,R) Describes deformation Equilibrium shape

  • SA-NCSM: Sp(3,R)-scheme basis

Unitary transformation from SU(3) scheme Find eigenvectors/ eigenvalues of the second-order Sp(3,R) Casimir invariant for each SU(3) irrep

Reorganization of model space: “bin” SU(3) basis states into Sp(3,R) symplectic irreps

Launey, Dytrych, Draayer, Prog. Part. Nucl. Phys. 89 (2016) 101

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SLIDE 8
  • SA-NCSM with Sp(3,R) scheme: Examples

12 Sp(3,R) irreps Single Sp(3,R) irrep

JISP16, Nmax = 6, hΩ = 20 MeV

B(E2) e2fm4

Preliminary

10 20 30 40 50 60 70 80 90

/2 1 /2 0 /2 1 /2 1 /2 2 /2 2 /2 1 /2 3 /2 1 /2 2 /2 2 /2 3 /2 1 /2 2 /2 2 /2 3 /2 1 /2 3 /2 1 /2 3 /2 1 /2 1 /2 1 /2 1 /2 1

robability Amplitude (%)

1+ 3+ 2+

N=0 N=2 N=4 N=6 N=8 N=10

(95% of wave functions)

Probability Amplitude (%)

0p-0h equilibrium shape + SU(3) configurations up to Nmax=12

  • cf. Launey’s talk
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SLIDE 9

SA-NCSM:

  • SU(3)-coupled basis – fast construction (Gel’fand patterns)
  • NN interaction SU(3) tensors – generated once per interaction
  • Hamiltonian –
  • Wigner-Eckart theorem … reduced matrix elements (rme’

s)

  • Decoupling to single-shell tensors Tn1n2n1n1 -> Tn2 x Tn1n1n1
  • Important pieces of information … single-shell rme’

s

Current limit p-shell sd-shell pf-shell

NCSM SA-NCSM

Important pieces of information (memory requirement)

p

  • s

h e l l s d

  • s

h e l l pf-shell

  • Sp(3,R)-coupled basis – fast construction (in selected spaces)
  • Hamiltonian – matrices of small dimension; eigenvectors solved on a laptop
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SLIDE 10
  • Resonating Group Method

Cluster wave function: For 2 clusters:

  • Antisymmetrizer does not act on r
  • Set of basis vectors which are not
  • rthogonal between each other
  • Very relevant to unify structure

and reaction

Unknown

1-cluster 2-clusters 3-clusters intrinsic function relative motion

Y .C. Tang et al, Physics Report 47 (1978) 167

  • S. Quaglioni and P

. Navratil Phys. Rev. C 79, 044606 (2009)

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SLIDE 11
  • Resonating Group Method

Hill-Wheeler equations: What we need: ▪ Interaction ▪ 1-body and 2-body density matrices (OBDME,TBDME)

Gives information on the structure of the tar

Antisymmetrizer: Eigenvectors have SU(3) symmetry

  • Figs. from S. Quaglioni and P

. Navratil Phys. Rev. C 79, 044606 (2009)

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SLIDE 12
  • Resonating Group Method

Non orthogonality is short range: for Introduce an orthogonalized version of Hill-Wheeler equations: Can be solved with R-matrix method Translationally invariant equation using Talmi-Moshinsky transformation

Calculation can become numerically challenging:

  • 1. The inversion of the norm
  • 2. The TM transformation

Some applications combining RGM + structure approach : ➢NCSM+RGM ➢NCSMC ➢GSMCC ➢SA-NCSM + RGM ➢SA-NCSMC ?

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SLIDE 13
  • Resonating Group Method with SU(3)-scheme basis (benchmarks)

Nmax =12 N3LO SU(2) SD OBDMEs SU(3) SD OBDMEs

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SLIDE 14
  • Resonating Group Method with SU(3)-scheme basis

Target wave function: Relation to partial-wave channels l [or SU(2)]: Expansion in terms of composite shapes

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SLIDE 15
  • Resonating Group Method with SU(3)-scheme basis

Target wave function: Relation to partial-wave channels l [or SU(2)]: Expansion in terms of composite shapes SU(3)-scheme Hecht and Suzuki

  • Norm kernel is diagonal in the SU(3) basis
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SLIDE 16
  • Resonating Group Method with SU(3)-scheme basis

Relation to partial-wave channels l [or SU(2)]: Expansion in terms of composite shapes SU(3)-scheme

  • Transformation

based on U(A)xU(3) Hecht and Suzuki

  • Norm kernel is diagonal in the SU(3) basis

Q 3=t

Q t ! Q 2!

A

A

ca.w' 1

4~+Q ~=4

a A

N D si 3e C

LU S TE R S

231 Fi g. 2. The S

U ( 3) r xoupl i ng t r ansf or m

at i on f or eq. (19) . The t ri angl es represent S U ( 3) coupl i ng.

are t ot al l y sym

m et r i c so t hat .s+d act s onl y on t he t rue cl ust er_f unct i ons. The c. m

.

exci t at i ons r angef r omQ

s = 0 (t he nonspur i ous com ponent of ~~t o Q z = k, w her e

k i s t he t ot al num

ber of har m

  • ni c osci l l at or exci t at i ons i n t he cl ust er-l i ke f unct i on .

S i nce c. m

. exci t at i on f unct i ons w

i t h di f f erent val ues of Q

2 are or t hogonal t o each

  • t her and si nce t he overl aps of bot h cl ust er-l i ke and t rue cl ust er f unct i ons are di agonal

i n t he S

U ( 3) quant umnum ber s, (~u), t he over l aps <~"~~ar e si m

pl y rel at ed t o t he cor r espondi ng <~" ~~` ~

~ep[ c ' N

~x

l x~~~~c~~xQ

  • »t avl ~ =

C A

  • 4I Q

<~ckN ~xt zo>x~l ~ypl c~~xQ

  • l x~>~

A

+

~

Q

!

C

A

  • 4\ Q

~ C 4~Q

~ ~ ~t pa~~xt z~ol xx . ~' 1~~~~N

~xQ ~o~xx' r~' 1~

x u( ( z~~~xQ ~ox~~xQ 2o) ; ( ~~r ax~) ) U ( ( ~~~xQ t oX ~uxQ 2o) ; ( ~~~~( ~) ) .

(20) [ I t has beenassum

ed f or si m

pl i ci t y t hat t he cor e st at es ( ~, ~~and (~, ~~) carry t hesam

e num ber of osci l l at or quant a. ]

I n t he S

U ( 3) st r ong- coupl ed si ngl e channel appr oxi m at i on [ based on a si ngl e

cor e st at e (~. ~~] , t he above rel at es t he nor m al i zat i on coef f i ci ent s of t he cl ust er-

l i ke st at es, ~

. t K r t o t he nor m

al i zat i on coef f i ci ent s of t he t rue cl ust er st at es, N

ß

x"~ (t he nor m

al i zed f unct i ons are 1~P " and N

~`", respect i vel y) . I n t hi s case eq. (20)

becom es

1 __

1 ] 1A

  • 41Q

r

[ N Q

~112

[ ~ a, ~ ~

s AA

Q !

A

  • 4 Q

'

4 Q

~

1

+ Q ~t

Q t ! Q Z! ~ A ~

C A )

c~' 1 [ N Q I ~, ~] z ~2( ( ~~~xQ

t ax~~xQ Zor , ( z~~~x~o». 4~+Q , =4

(21)

H ence N

4xa1 i s det er m

i ned f r omt he nor m

  • f t he cl ust er-l i ke st at e i f t he nor m

al i zat i on const ant s f or st at es Q

t < Q

are know

  • n. I f t he cor e st at e i s t hat of ans-d shel l nucl eus

i n

i t s gr ound- st at e conf i gurat i on,

  • e. g.,

t he cl ust er f unct i on w

i t h Q= 8+k cor r esponds t o a shel l -m

  • del f unct i on of k-uni t s of osci l l at or exci t at i on. I t s nor m

Target wave function:

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SLIDE 17
  • Resonating Group Method with SU(3)-scheme basis

Target wave function: Relation to partial-wave channels l [or SU(2)]: Expansion in terms of composite shapes SU(3)-scheme Wave function OBDME Cross section

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SLIDE 18
  • Conclusion

➢Taking advantage of SA-NCSM in the RGM to reach heavier nuclei ➢Current work: SU(3) symmetry; next: use of Sp(3,R) ➢Reactions of interest:

  • n + alpha (benchmark)
  • Ne isotopes (intermediate mass)
  • Ca isotopes (medium mass)
  • 23Al(p,γ)24Si (important for X-ray burst nucleosynthesis)