The single-particle Berggren basis in structure calculations - PowerPoint PPT Presentation

The single-particle Berggren basis in structure calculations Michigan State University (MSU), Facility for Rare Isotope Beams (FRIB) Kévin Fossez June 11-22, 2018 FRIB, MSU - FRIB workshop: Continuum 2018 Work supported by: DOE: DE-SC0013365 (Michigan State University) DOE: DE-SC0017887 (Michigan State University) DOE: DE-SC0008511 (NUCLEI SciDAC-4 collaboration) NSF: PHY-1403906 FRIB, MSU - Kévin Fossez 1

Nuclei as open quantum systems (OQS) Why do we care about continuum couplings? → In short: quantum systems can break apart. N-2 N-1 N N+1 N+2 Z+1 Z+1 Z+1 Z+1 Z+1 N-2 N-1 N N+1 N+2 Z Z Z Z Z N-2 N-1 N N+1 N+2 E = 0 Z-1 Z-1 Z-1 Z-1 Z-1 New effective N-2 N-1 N N+1 N+2 Z+1 Z+1 Z+1 Z+1 Z+1 scales N-2 N-1 N N+1 N+2 Z Z Z Z Z N-2 N-1 N N+1 N+2 Z-1 Z-1 Z-1 Z-1 Z-1 Coupled by excitations Excitations and decay New paradigm: network of OQS! ⇒ Γ = 0 ⇒ Γ > 0 → Unification of structure and reactions. N. Michel et al. , J. Phys. G 37 , 064042 (2010) FRIB, MSU - Kévin Fossez 2

The quasi-stationary formalism The two possibilities to deal with OQS: ∂ 2 u l ( k , r ) = ( l ( l + 1 ) + 2 m h 2 V ( r ) − k 2 ) u l ( k , r ) . OQS ̵ ∂ r 2 r 2 Time independent u l ( k , r ) ∼ r ∼ 0 C 0 ( k ) r l + 1 . Time dependent + outgoing boundary conditions Real, Resonant, complex energies scattering wfs ̵ E = E 0 − i Γ Rigged Hilbert space h T 1 / 2 = Γ ln ( 2 ) . 2, u l ( k , r ) r →∞ C + ( k ) H + ∼ l ,η ( kr ) + C − ( k ) H − l ,η ( kr ) . C − ( k ) = 0 for bound and decaying states. J.J. Thomson, Proc. London Math. Society, 197 (1884), FRIB, MSU - Kévin Fossez 3 G. Gamow, Z. Physik 51 , 204 (1928), A. F. J. Siegert, Phys. Rev. 56 , 750 (1939)

Resonant and scattering states 0.5 A few definitions: Bound state •Resonant states or Gamow states: poles of the Decaying resonance 0.4 Scattering state S -matrix, i.e. bound states, virtual or antibound u 2 ( r ) (fm − 2 ) 0.3 states and resonances (decaying or capturing). → Discrete energies. 0.2 •Scattering states: nonresonant (continuum) states. → Continuous energies. 0.1 Im ( k ) 0.0 0 5 10 15 20 25 30 r (fm) antibound decaying bound states state resonances Connection between Gamow states and Re ( k ) the (Green function) resolvent’s spec- trum in 1954. capturing resonances subthreshold/virtual resonances R. E. Peierls, Proc. Glasgow Conf. Nucl. Meson Phys., 296 (1954) FRIB, MSU - Kévin Fossez 4

T. Berggren: “What can we do with those states? A basis!” •PhD in 1966 (Lund), groundbreaking work published in 1968: T. Berggren, Nucl. Phys. A 109 , 265 (1968) — On the use of resonant states in eigenfunction expansions of scattering and reaction amplitudes. — •Connection between the Berggren and Mittag-Leffler expansions: T. Berggren and P. Lind, Phys. Rev. C 47 , 768 (1993) — Resonant state expansion of the resolvent. — •Interpretation of the imaginary part of observables: T. Berggren, Phys. Lett. B 373 , 1 (1996) — Expectation value of an operator in a resonant state. — → Many papers based on the Berggren basis nowadays, still spreading. Picture from Symmetry in the world of atomic nuclei by I. Ragnarsson and S. Åberg, Lund University. FRIB, MSU - Kévin Fossez 5

Definition of the s.p. Berggren basis The Berggren basis: → Single particle basis including bound states, decaying resonances and scattering states. u ℓ ( k n )∣ + ∫ L + dk ∣ u ℓ ( k )⟩ ⟨ ˜ u ℓ ( k )∣ = ˆ ∣ u ℓ ( k n )⟩ ⟨ ˜ ∑ 1 ℓ, j . S( = 0 ) n ∈( b , d ) Im ( k ) Im ( k ) bound states decaying resonances Re ( k ) Re ( k ) L − ( = 0 ) discretized continuum L + Cauchy’s residue in momentum space theorem FRIB, MSU - Kévin Fossez 6

How to generate a s.p. Berggren basis? The truth. A few options are available to generate a Berggren basis: Realistic interaction Many-body Effective perturbation theory interaction HF potential Woods-Saxon Ab initio Many-body One-body potential method model One-body Jost Sph. Bessel density function functions Berggren basis: {∣ pole/scat , n , ℓ, j , m t ⟩} FRIB, MSU - Kévin Fossez 7

Spherical Bessel functions ∂ 2 ψ l ( k , r ) = ( l ( l + 1 ) The easy way to go: − k 2 ) ψ l ( k , r ) ∂ r 2 r 2 •Analytical solutions, regular at the origin j ℓ ( r ) (first kind) √ and ideally ℓ -dependent. φ ℓ ( kr ) = π krj ℓ ( kr ) 2 •Extended into the complex plane using a recurrence rela- ∞ dr φ ℓ ( kr ) φ ℓ ( k ′ r ) = δ k , k ′ tion (NIST 10.51(i)) accurate for ℓ < 7: ∫ 0 f n + 1 ( z ) + f n − 1 ( z ) = 2 n + 1 f n ( z ) . pole z 0.0 •Expand the s.p. Schrödinger eq.: Im( k ) (fm − 1 ) ̵ ) + ( V ( k 0 , k 0 ) V ( k 1 , k 0 ) -0.1 h 2 2 m ( k 2 ) ( c 0 V ( k 1 , k 1 )) ( c 0 ) = E ( c 0 ) 0 0 V ( k 0 , k 1 ) k 2 0 c 1 c 1 c 1 1 -0.2 •Physical states do not depend on the basis. 0.0 0.5 1.0 Re( k ) (fm − 1 ) FRIB, MSU - Kévin Fossez 8

The Jost function method(s) There are in fact two methods: •Searching the zeros of the outgoing Jost function for poles. •Directly integrating the “Jost functions” from r = 0 to r → ∞ . ∂ r 2 u ℓ,η ( k , r ) = ( ℓ ( ℓ + 1 ) ∂ 2 + 2 m h 2 V ( r ) − 2 η k + k 2 ) u ℓ,η ( k , r ) with u l ( k , r ) ∼ r ∼ 0 C 0 ( k ) r l + 1 . ̵ r 2 r ⎧ ⎪ F ℓ,η ( z ) ∓ iG ℓ,η ( z ) for η ≠ 0 ⎪ ℓ,η ( z ) = ⎨ •Solutions at large distances (Hankel functions): H ± ⎪ ⎪ z [ j ℓ ( z ) ∓ n ℓ ( z )] for η = 0 ⎩ •General solution (linear combinaison): u ℓ,η ( k , r ) = C + ( k ) H + ℓ,η ( kr ) + C − ( k ) H − ℓ,η ( kr ) (at large r ) ⇒ u ℓ,η ( k , r ) = C + ( k ) u + ℓ,η ( k , r ) + C − ( k ) u − ℓ,η ( k , r ) ℓ,η ( k , r ) and the coefficients? How can be obtain u ± R. G. Newton, Scattering Theory of Waves and Particles , Springer-Verlag, New-York (1982, 2nd ed.) [p.341], FRIB, MSU - Kévin Fossez 9

The Jost function method(s) The first method: searching the zeros of the outgoing Jost function for poles. •Integrate from zero to r = R , then matching conditions: ⎧ ⎪ dr ( C + ( k ) H + ℓ,η ( kR ) + C − ( k ) H − ℓ,η ( kR )) = du ℓ ( k , R ) ⎪ d ⎨ ⎪ dr ⎪ C + ( k ) H + ℓ,η ( kR ) + C − ( k ) H − ℓ,η ( kR ) = u ℓ ( k , R ) ⎩ → The differentiability of u ℓ ( k , r ) is not ensured for outgoing states ( C − ( k ) = 0)! •Definition of the outgoing and incoming Jost functions: ℓ ( k , r ) du ℓ ( k , r ) ℓ ( k , r ) − u ℓ ( k , r ) du ± J ± ℓ ( k ) = W ( u ± ℓ ( k , r ) , u ℓ ( k , r )) = u ± . dr dr → No r -dependance by def., we only need to vary k to get: J + ℓ ( k ) = 0 and hence the differentiability. Basically a search of zeroes for outgoing states (poles)! R. G. Newton, Scattering Theory of Waves and Particles , Springer-Verlag, New-York (1982, 2nd ed.) [p.341], FRIB, MSU - Kévin Fossez 10

The Jost function method(s) The second method: directly integrating the “Jost functions” from r = 0 to r → ∞ . “Jost function”: Initial condition: ℓ,η ( kr ) 2 µ ∂ ℓ,η ( r , k ) = ∓ i F ± ℓ,η ( 0 , k ) = 1 ∂ r F ± h 2 V ( r ) u ℓ ( r , k ) k H ∓ ̵ Wave function: u ℓ ( r , k ) = 1 2 (F + ℓ,η ( r , k ) H + ℓ,η ( kr ) + F − ℓ,η ( r , k ) H − ℓ,η ( kr )) •Start the integration of F + ℓ,η ( r , k ) at r = 0, compute the wave function, iterate, etc. r →∞ F + ℓ,η ( r , k ) = J + ℓ,η ( k ) , and obviously: C ± ( k ) = 1 2 J ± ℓ,η ( k ) . •The connection with the Jost function: lim Very simple method that gives the wave function and ANCs simultaneously, but you need to know where are the poles beforehand. R. G. Newton, Scattering Theory of Waves and Particles , Springer-Verlag, New-York (1982, 2nd ed.) [p.341], FRIB, MSU - Kévin Fossez 11 H. Masui et al. , Prog. Theor. Exp. Phys. 2013 , 123A02 (2013)

Normalization: exterior complex-scaling method Rigged Hilbert space norm, regularization methods: Resonant states: N 2 = ⟨ ˜ ∞ u ℓ,η ∣ u ℓ,η ⟩ = ∫ ℓ,η ( r ) = 1 dr u 2 0 Rigged Hilbert space norm Scattering states: C + ( k ) C − ( k ) = 1 2 π •Several possibilities to regularize the integral: Ya. B. Zel’dovich, uniform and exterior complex- scaling (UCS,ECS). Exterior complex-scaling: ℓ,η ( r ) + ( C + ( k )) 2 ∫ ℓ,η ( kr )) 2 = I R + ( C + ( k )) 2 ∫ N 2 = ∫ dr ( H + dx ( H + ℓ,η ( k [ R + xe i θ ])) R ∞ ∞ 2 e i θ dr u 2 0 R 0 FRIB, MSU - Kévin Fossez 12

The Coulomb and centrifugal barriers Just a note about long-range terms in the Hamiltonian: •The effect of the centrifugal barrier (1 / r 2 ) can be included exactly when using ℓ -dependent spherical Bessel basis states. •Including the effect of the Coulomb barrier (1 / r ) requires Hankel functions in the complex plane (Only two codes published so far?). It is, of course, always possible to go around the problem and directly integrate the centrifugal and Coulomb barrier in the Schrödinger eq., but for an inevitable loss of accuracy in sensitive calculations ( i.e. , reactions, some atomic physics problems). I. J. Thompson et al. , J. Comp. Phys. 64 , 490509 (1986) [see FRESCO], FRIB, MSU - Kévin Fossez 13 N. Michel, Comp. Phys. Comm. 176 , 232 (2007).