Oblique Basis Approach to Nuclear Structure and Reactions

Transcription

1 Oblique Basis Approach to Nuclear Structure and Reactions Vesselin G. Gueorguiev, Ph.D. Perspectives on the modern shell model and related experimental topics Sat., Feb. 6, 2010 EFES-NSCL workshop 1 workshop support

2 Overview Modeling the interactions b/w nucleons Oblique basis approach to the diagonalization problem Oblique basis sets - some examples Beyond the 2-body interaction Oblique (non-orthogonal) basis for particle transfer reactions in heavy nuclei.

3 Modeling the Interactions: Nuclear Shell-Model Hamiltonian H = #" i a + i a i + # V ijkl a + i a + j a k a l i i, j,k,l Realistic interactions: KB3, GXPF1, FP6D where a i + and ai are fermion creation and annihilation operators, " i and V ijkl are real and V ijkl = V klij = $V jikl = $V ijlk Main problems: Determining the parameters of the interaction - model fitting using input from experiment Doing large model space calculations

4 Nucleon Interaction from QCD (Chiral Perturbation Theory) Chiral perturbation theory (χpt) allows for controlled power series expansion Expansion parameter : $ & % Q " # " #,1 GeV, # - symmetry breaking scale ' ) ( *, Q + momentum transfer, NNN - potentials such as Tucson-Melbourne & Urbana Terms suggested within the Chiral Perturbation Theory V. Bernard, E. Epelbaum, H. Krebs, and Ulf-G. Meiβner Phys. Rev. C 77, (2008). R. Machleidt, D. R. Entem, nucl-th/

5 3 H, 3 He, and 4 He Binding Energy Petr Navratil, V. G. Gueorguiev, J. P. Vary, W. E. Ormand, and A. Nogga, Phys. Rev. Lett. 99, (2007), (nucl-th ). triton β decay A B D. Gazit, S. Quaglioni and P. Navratil Phys.Rev.Lett.103,102502(2009) Along the appropriate curve, all C D -C E points reproduce BE of 3 H & 3 He within 0.1 kev. First values for {C D,C E }=({-1.11,-0.66}, {8.14,-2.02}) by A. Nogga et al, Nucl. Phys. A737, 236 (2004)

6 Along the C D -C E Curve (average) BE deviation is within the accuracy of the kinetic energy for equal mass nucleons m n =m p. "T = T "m m # 26keV (# 37 $ 0.7 $10%3 ) m=(zm p +(A-Z)m n )/A PRC60, (1999) G. P. Kamuntavicius, P. Navratil, B. R. Barrett,G. Sapragonaite, and R. K. Kalinauskas T=3/2 channel has been neglected in the calculations.

7 ab-initio no core with QCD derived nuclear interaction Pushing the limits using the N 3 LO, May need codes with explicit m p & m n. Is the NNN all we need?

8 Effective Interactions in a Finite Model Space H Okubo and Lee-Suzuki transformation method U=e is H eff =UHU -1 H = A p i 2 " + " V ij (q) 2m i=1 A i< j 1 and 2-body interaction terms U=e is H eff = A p i 2 " + " V i1 i 2m 2 (q) V i1...i A (q) i=1 A i 1 <i 2 A " i 1 <...<i A 1, 2, 3 up to A-body interaction terms

9 Effective Hamiltonian in Second Quantized Form H eff = $ " # a + # a # +L+ # A $ k=1 1 $ V (k!) 2 #1 K# 2k a #1 # 1 %...%# k # k+1 %...%# 2k + Ka + #k a Ka #k+1 # 2k Calculating 3-body effective interactions is difficult, however doable, and essential in understanding the structure of light nuclei! P. Navratil and W. E. Ormand, Phys. Rev. C 68, (2003). In principle, A-body effective Hamiltonians can be calculated from realistic interactions, but their applications are yet ahead! Need for exactly solvable models to understand possible implications of the A-body effective interactions!

10 Nuclear Models Multi-particle shell model configurations Harmonic oscillator single-particle basis Configuration mixing (two-body interactions and beyond ) We can build our collective states on this or Group theory models - group chain so(3)=g i G j ; Elliott s SU(3), Symplectic model, Interacting Boson Model (IBM) Lie algebra based models are good because: Guaranteed to be exactly solvable Well defined extension of the operators in product spaces: Co-product Δ(X)=X 1 2 +Y 1 2 (often Y=1) L system =L 1 + L 2 + L 3 + Geometric models - liquid drop model R(θ,ϕ)=R 0 (1+α λµ Y λµ (θ,ϕ)) (β,γ) shape parameters (λ,µ) su(3) irreps.

11 Group Theory Models rank A n su(n+1) B n so(2n+1) C n sp(2n) D n so(2n) 1 su(2) pairing so(3)~su(2) sp(2) ~su(2) so(2) ~u(1) 2 su(3) Elliott so(5) pnpairing sp(4) ~so(5) so(4) ~su(2) su(2) 3 su(4) Wigner so(7) o(8) FDSM sp(6) Rowe & Rosensteel so(6)~su(4) 4 su(5) so(9) sp(8) so(8) Evans, FDSM

12 The Challenge... Nuclei display unique characteristics: Single-particle Features Pairing Correlations Deformation/Rotations Closed Shell Pairing Rotations Vibrations Closed Shell

13 We usually understand a problem H = H 0 H = H 0 + V H = H 1 (") + H 2 (#) +... Transition from mode one when it is: exactly solvable (symmetry at play) perturbative regime What about more than one exactly solvable part beyond the perturbative regime? to mode two should occur. Two or more different sets of basis states could be employed to understand the problem

14 Important Exactly Solvable 2-body Nuclear Interactions Quadrupole-Quadrupole interaction, Elliott s SU(3) model: H ˆ = "N # $ 2 Q% Q Pairing interaction: H ˆ = &" # (a + #$ a #$ + a + a ) #% ' G& a + #% #$ a + #% a (% a ($ # #(

15 SU(3) Symmetry Breaking in the pf-shell nuclei Low Energy States SU(3) Structure Contribution J= 6 J= 4 J= 2 J= 0 Realistic spin-orbit (l s) single particle energy splitting! C2 of SU(3) Low Energy States SU(3) Structure, No Spin-Orbit Interaction Turn off the s.p.e. spin-orbit splitting! Coherent state structure in 48 Cr using Kuo-Brown-3 interaction. (l s) V. G. Gueorguiev, J. P. Draayer, and C. W. Johnson PRC63,14318(2001) Contribution J= 6 J= 4 J= 2 J= C2 of SU(3)

16 Eigenvalue Problem in an Oblique Basis Spherical basis states e i SU(3) basis states E α Overlap matrix g ˆ g = $ & % e i e j e i E " E # e j E # E " ' $ ) = & 1 µ ') ( % µ + 1( The eigenvalue problem H" = E" # H ˆ $ " ˆ = E g ˆ $ " ˆ

17 Example of an Oblique Basis Calculation: 24 Mg We use the Wildenthal USD interaction and denote the spherical basis by SM(#) where # is the number of nucleons outside the d 5/2 shell, the SU(3) basis consists of the leading irrep (8,4) and the next to the leading irrep, (9,2). Model Space SU3 (8,4) SU3+ (8,4)& (9,2) GT100 SM(0) SM(1) SM(2) SM(4) Full Dimension (m-scheme) % Visualizing the SU(3) space with respect to the SM space using the naturally induced basis in the SU(3) space. SU(3) basis space SM space SU(3) basis space SM space SM(2) & SU3+ SM(4) & SU3+

18 Main Contribution % Overlaps With The Exact Eigenvectors For 24Mg SM(2) SU(3) SM(2)+SU(3) SM(4) SM(4)+SU(3) Eigenvectors V. G. Gueorguiev, W. E. Ormand, C. W. Johnson, and J. P. Draayer PRC65,024314(2002)

19 Three-Mode Systems J-Pair Modes Gpp 4 Gpn 4 Gnn Dpp 2 Dpn Spp 0 Spn 2 Dnn 0 Snn pp pn nn Closed Shell Core S & D pairs Three-Mode Systems s & d bosons Oblique basis using states constructed from physically significant S and D pairs. O(6) Interacting Boson Model Full shell-model calculations to obtain the spectrum of near closed shell nuclei. U(5) Multi-Mode SU(3)

20 Evaluation Steps Matrix elements ( H and g ) H and g g=uu T (Cholesky) (U -1 ) T HU -1 (UΨ)=E (UΨ) Eigenstates (Lanczos) SM basis (spherical) m- scheme SU(3) basis (cylindrical) O and g E> Expectation values and matrix elements of operators <O> and <E 1 O E 2 > ε i and <j 1 j 2 J'T' V j 3 j 4 J''T''>

21 Algebraic Models Standard pairing is actually an SU(2) RG-model. Proton-neutron T=1 pairing is SO(5) RG-model. Generalized Richardson-Gaudin models are new interesting set of exactly solvable algebraic models. rank A n su(n+1) B n so(2n+1) C n sp(2n) D n so(2n) 1 su(2) pairing so(3)~su(2) sp(2) ~su(2) so(2) ~u(1) 2 su(3) Elliott so(5) pnpairing sp(4) ~so(5) so(4) ~su(2) su(2) 3 su(4) Wigner so(7) o(8) FDSM sp(6) Rowe & Rosensteel so(6)~su(4) 4 su(5) so(9) sp(8) so(8) Evans, FDSM

22 The Pairing Problem Exactly solvable cases: H " = %# j n j $ c jj' A + j A j' j % jj' $, A j + = %($) j$m a + + jm a j$m. Constant pairing c jj =G with non-degenerate single particle energiesε j ε j R. W. Richardson, Phys. Lett. 5, 82 (1963). Separable pairing c jj =c j c j with degenerate single particle energies ε j =ε j F. Pan, J. P. Draayer, W. E. Ormand, Phys. Lett. B 422, 1 (1998). T=1 proton-neutron pairing as Richardson-Gaudin model with nondegenerate single particle energies, J. Dukelsky, V.G. Gueorguiev, P. Van Isacker (PRL96,072503(2006)). Extended pairing - A-body interaction F. Pan, V. G. Gueorguiev, J. P. Draayer, Phys. Rev. Lett. 92, (2004). m>0

23 H eff = $ " # a + # a # +L+ # Extended Pairing Model A $ k=1 1 $ V (k!) 2 #1 K# 2k a #1 # 1 %...%# k # k+1 %...%# 2k + Ka + #k a Ka #k+1 # 2k Nilsson single particle energies ε jm play the role of 1-body effective Hamiltonian that takes into account the nuclear deformation due to the Q Q interaction. Simplifying assumption: equal coupling between different configurations: V "1 " 2 = V "1 " 2 " 3 " 4 = V "1 K" 2A = G The RESULT is Exactly solvable Hamiltonian: ˆ H = + 1 #" i n i $ G# b i b j $ G# # Kb + ik b ik+1 Kb i2k, b + i = a + + i& a i' i ij A + b (k!) 2 i1 k= 2 i 1 %...%i 2k F. Pan, V. G. Gueorguiev, J. P. Draayer, Phys. Rev. Lett. 92, (2004).

24 5 pairs in 10 levels e 1 =1.179 e 2 =2.650 e 3 =3.162 e 4 =4.588 e 5 =5.006 e 6 =6.969 e 7 =7.262 e 8 =8.687 e 9 =9.899 e 10 =10.20

25 Extended Pairing for Nuclei Zero BE Reference Pb Yb Sn Zero BE Reference Nilsson levels using nuclear deformation (Audi & Wapstra (1995)). Pauli blocking for odd A nuclei. Set the scale of the single particle energies from near closed shell system (Nilsson BE is 3/4 E filling, Ring & Schuck) V. G. Gueorguiev, Feng Pan and J. P. Draayer, European Physical Journal A25(s1)515(2005)

26 BE Relative to the 208 Pb core

27 Extended Pairing log(g(a)) Pb Yb Sn G(A) ~ Has nice and interesting systematic behavior! The Extended Pairing model gives reasonable results Many-body interactions beyond two- & three- body! " Dim(A) V. G. Gueorguiev, Feng Pan and J. P. Draayer, European Physical Journal A25(s1)515(2005)

28 Other problems that can benefit from Non-Orthogonal Basis

29 Transfer Reactions for Deformed Nuclei Using Sturmian Basis States for axially deformed nuclei: Bohr-Mottelson rotor model for deformed nuclei; Rotor plus particle/hole system; " # =$ % & core K, E # = E core K + ' % ; Deformed Woods-Saxon potential for the single particle states. Transfer reactions on deformed nuclei: Distorted Wave Born Approximation (DWBA) within the Optical Model Potential (OMP) theory: d"(j i K i # J f K f ;$) = %%(a $ v $ c nlj v ) 2 d" DW nlj ; a $ - Coriolis band mixing amplitude, v $ - BCS occupation number, lj & $ = c v nlj ' nlj - all ' nlj have the same energy ( $! n Sturmian Basis provides proper wave function tails B. L. Andersen, B. B. Back, and J. M. Bang, Nucl. Phys. A 147, 33 (1970). V. G. Gueorguiev, P. D. Kunz, J. E. Escher, and F. S. Dietrich (nucl-th/ )

30 The circle indicates the region of agreement b/w the linear and the exact numerical treatment Neutron separation

31 Norm convergence as function of the model space. " # = c v nlj $ nlj - all $ nlj have the same energy % #!

32 Model space convergence of the Sturmian basis

33 V. G. Gueorguiev, P. D. Kunz, J. E. Escher, and F. S. Dietrich (nucl-th/ )

34

35 Summary QCD derived effective interactions and effective O-L-S methods suggest A-body interactions terms The oblique basis build on the essential modes can help in the understanding of heavy nuclei. In studies of nuclear reactions non-orthogonal basis may resolve the tail problem inherited in the HO wave function.