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

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

• 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- N max model space

• SA-NCSM: SU(3)-scheme basis Total HO quanta: NCSM: N max determines the size of model space SA-NCSM: keeps track of N x , N y , N z SU(3) basis states: max max With spin: max SU(3) basis states: unitary transformation from m-scheme Gives information about important deformed configurations LSU code (LSU3shell): sourceforge.net/projects/lsu3shell Dytrych et al., Phys. Rev. Lett. 111 (2013) 252501 Launey et al., Prog. Part. Nucl. Phys. 89 (2016) 101

• SA-NCSM: SU(3)-scheme basis How is the SU(3)-scheme basis constructed? ➢ Intuitive way: spin isospin Considering 2 particles: Reduced SU(3) CG Tedious… not used for many-particle system ➢ 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

• SA-NCSM: SU(3)-scheme interaction SU(3) tensors of NN interaction n r n s ( λ μ ) jj-coupled NN N3LO (N max =6) NN SU(3) Tensors NN in SU(3) basis ħ Ω =11 MeV initial state (i) final state (f) Equivalent to m -scheme Matrices are smaller and sparser

• SA-NCSM with SU(3) scheme: Examples Ne ! NNLOopt 19 Ne 5 ! <2>10 (7/2)- ! 4 ! (9/2)- ! x ! (MeV) ! NNLO 3 ! opt ! <2>8, ! <3>9 ! E 2 ! 3/2+ ! 1 ! 1/2+ ! 0 ! ħΩ =15 ! MeV ! E xp. ! N2LOopt SA-NCSM (selected model space): 24 Ne SRG-N3LO, 32 Ne ħΩ =15MeV 2fm –1 6 2 + 8 ħΩ =15MeV <2>4 50 million SU(3) states <2>6 7 5 0 + 6 2 + Energy (MeV) 4 Energy (MeV) 2 + Complete model space: 1000 billion states 5 3 4 2 + ground state: 2 3 r rms(m) =2.89 fm 5.2-8.4 2 1 (2 + ) W.u. 4W.u. 2 + 1 0 + 0 0 + 0 + Expt. SA-NCSM 0 Expt. SA-NCSM Robert Baker, PhD student, LSU

• SA-NCSM: Sp(3,R)-scheme basis Symmetry-adapted: Symplectic Sp(3,R) basis: SU(3), Sp(3,R) Describes deformation Equilibrium shape 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 Unitary transformation from SU(3) scheme Launey, Dytrych, Draayer, Prog. Part. Nucl. Phys. 89 (2016) 101

• SA-NCSM with Sp(3,R) scheme: Examples 0p-0h equilibrium shape + SU(3) configurations up to N max =12 90 (95% of wave functions) Probability Amplitude (%) 1+ 3+ 2+ 80 robability Amplitude (%) 70 60 50 40 30 20 10 0 /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 N=0 N=4 N=6 N=8 N=10 N=2 cf. Launey’s talk JISP16, N max = 6, h Ω = 20 MeV B(E2) e 2 fm 4 Preliminary Single Sp(3,R) irrep 12 Sp(3,R) irreps

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 Tn 1 n 2 n 1 n 1 -> Tn 2 x Tn 1 n 1 n 1 • Important pieces of information … single-shell rme’ s • Important pieces of information (memory requirement) NCSM SA-NCSM pf-shell pf-shell sd-shell l l e h s - d s p-shell Current l l e h s - p limit Sp(3,R)-coupled basis – fast construction (in selected spaces) • Hamiltonian – matrices of small dimension; eigenvectors solved on a laptop •

intrinsic function relative motion • Resonating Group Method 1-cluster 2-clusters Cluster wave function: 3-clusters For 2 clusters: Unknown • Antisymmetrizer does not act on r • Set of basis vectors which are not orthogonal between each other • Very relevant to unify structure and reaction Y .C. Tang et al, Physics Report 47 (1978) 167 S. Quaglioni and P . Navratil Phys. Rev. C 79 , 044606 (2009)

• Resonating Group Method Hill-Wheeler equations: Antisymmetrizer: Eigenvectors have SU(3) symmetry What we need: ▪ Interaction ▪ 1-body and 2-body density matrices (OBDME,TBDME) Gives information on the structure of the tar Figs. from S. Quaglioni and P . Navratil Phys. Rev. C 79 , 044606 (2009)

• 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 ?

• Resonating Group Method with SU(3)-scheme basis (benchmarks) SU(2) SD OBDMEs SU(3) SD OBDMEs N3LO N max =12

• 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

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

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 . • Resonating Group Method with SU(3)-scheme basis 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 ange f r om Q 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 oni 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 . Target wave function: 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 ot 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 Relation to partial-wave i n t he S U ( 3) quant um num 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 < ~" ~ ~` ~ channels l [or SU(2)]: - 4 I Q ~ep[ c ' N l x~~~~c~~xQ o»t avl ~ = <~ckN ~xt zo>x~l ~ypl c~~xQ ol x~>~ ~x A C A A - 4\ Q ~ C + ~ ! 4 ~Q ~ ~ ~t pa~~xt z~ol xx . ~' 1~~~~N Expansion in terms of composite shapes Q ~xQ ~o~xx' r~' 1~ SU(3)-scheme A A Q t ! Q 2! C Q 3=t ca .w' 1 4~+Q ~=4 x u( ( z~~~xQ 2o) ; ( ~~r ax~) ) U 2o) ; ( ~~~~( ~) ) . ~ox~~xQ ( ( ~~~xQ t oX ~uxQ (20) [ I t has been assum ed f or si m pl i ci t y t hat t he cor e st at es ( ~, ~~ and (~, ~~) carry t he sam e num ber of osci l l at or quant a . ] • Norm kernel is diagonal in the SU(3) basis 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- • Transformation based on l i ke st at es, ~ 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 K U(A)xU(3) (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 A - 4 1Q r 1 __ s AA [ ~ a, ~ ~11 2 ~ [ N Q - 4 Q ' 4 Q ~ Q ! A 1 I ~, ~] z ~2( ( ~~~xQ t ax~~xQ Zor , ( z~~~x~o » . Z! ~ A ~ c~' 1 [ N Q C A ) + Q t ! Q Q ~t 4~+Q , =4 (21) Hecht and Suzuki H ence N 4xa1 i s det er m i ned f r om t he nor m of 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 an s-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 odel 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

• 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 OBDME function Cross section

• 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) 23 Al(p, γ ) 24 Si (important for X-ray burst nucleosynthesis) •