CONTINUUM STATES IN THE SHELL MODEL

Transcription

1 CONTINUUM STATES IN THE SHELL MODEL Andrey Shirokov Skobeltsyn Institute of Nuclear Physics Lomonosov Moscow State University Collaborators: J. Vary, P. Maris (Iowa State University) A. Mazur, I. Mazur (Pacific National University) INT, Seattle, May 25, 2016

2 Introduction Theory: HORSE (J-matrix) formalism as a natural extension of SM How it works: a model 2-body problem Application to A-body problems within the no-core shell model Summary

3 Lattice QCD vs ab initio nuclear structure: Ideal world Oh!

4 Lattice QCD vs ab initio nuclear structure: Real world Oh!

5 Lattice QCD vs ab initio nuclear structure: Real world Lattice QCD EFT Ab initio nuclear theory Exchange of methods, ideas, etc. Thanks to organizers, I ve learned a lot!

6 Thomas Papenbrock talk, INT, May

7 Thomas Papenbrock talk, INT, May

8 Thomas Papenbrock talk, INT, May We derive similar results for continuum spectrum. Maybe it will be useful for lattice QCD community

9 No-core Shell Model NCSM is a standard tool in ab initio nuclear structure theory NCSM: antisymmetrized function of all nucleons Wave function: = A Y i(r i ) i Traditionally single-particle functions harmonic oscillator wave functions i(r i ) are N max truncation makes it possible to separate c.m. motion Discussed here by James Vary, Angelo Calci, Bruce Barrett

10 No-core Shell Model NCSM is a bound state technique, no continuum spectrum; not clear how to interpret states in continuum above thresholds how to extract resonance widths or scattering phase shifts HORSE (J-matrix) formalism can be used for this purpose Other possible approaches: NCSM+RGM; Gamov SM; Continuum SM; SM+Complex Scaling; All of them make the SM much more complicated. Our goal is to interpret directly the SM results above thresholds obtained in a usual way without additional complexities and to extract from them resonant parameters and phase shifts at low energies. I will discuss a more general interpretation of SM results

11 J-matrix (Jacobi matrix) formalism in scattering theory Two types of L 2 basises: Laguerre basis (atomic hydrogen-like states) atomic applications Oscillator basis nuclear applications Other titles in case of oscillator basis: HORSE (harmonic oscillator representation of scattering equations), Algebraic version of RGM

12 J-matrix formalism Initially suggested in atomic physics (E. Heller, H. Yamani, L. Fishman, J. Broad, W. Reinhardt) : H.A.Yamani and L.Fishman, J. Math. Phys 16, 410 (1975). Laguerre and oscillator basis. Rediscovered independently in nuclear physics (G. Filippov, I. Okhrimenko, Yu. Smirnov): G.F.Filippov and I.P.Okhrimenko, Sov. J. Nucl. Phys. 32, 480 (1980). Oscillator basis.

13 Schrödinger equation: HORSE: H l lm(e,r)=e lm (E,r) Wave function is expanded in oscillator functions: Schrödinger equation is an infinite set of algebraic equations: where H=T+V, 1X n 0 =0 T kinetic energy operator, V potential energy (Hnn l 0 nn 0)a nn0(e) =0.

14 HORSE: Kinetic energy matrix elements: nlm nlm (r) = 1 r R nl(r)y lm (ˆr) T l nn 0 nlm T n0 l 0 m 0 = Z nlm (r)t n0 l 0 m 0(r) d3 r Z = ll 0 mm 0 R nl TR n 0 l dr Kinetic energy is tridiagonal: Note! Kinetic energy tends to infinity as n and n =n, n±1 increases:

15 HORSE: Potential energy matrix elements: For central potentials only Note! Potential energy tends to zero as n and/or n increases: Therefore for large n or n : A reasonable approximation when n or n are large

16 HORSE: In other words, it is natural to truncate the potential energy: This is equivalent to writing the potential energy operator as For large n, the Schrödinger equation takes the form 1X (Tnn l 0 nn 0E)a n 0 l(e) =0, n N +1 n 0 =0

17 This is an exactly solvable algebraic problem! And this looks like a natural extension of SM where both potential and kinetic energies are truncated General idea of the HORSE formalism

18 Asymptotic region n N Schrödinger equation takes the form of three-term recurrent relation: This is a second order finite-difference equation. It has two independent solutions: where dimensionless momentum For derivation, see S.A.Zaytsev, Yu.F.Smirnov, and A.M.Shirokov, Theor. Math. Phys. 117, 1291 (1998)

19 Asymptotic region n N Schrödinger equation: Arbitrary solution a nl (E) of this equation can be expressed as a superposition of the solutions S nl (E) and C nl (E), e.g.: Note that

20 Asymptotic region n N Therefore our wave function Reminder: the ideas of quantum scattering theory. Cross section Wave function δ in the HORSE approach is the phase shift!

21 Internal region (interaction region) n N Schrödinger equation Inverse Hamiltonian matrix:

22 Matching condition at n=n Solution: From the asymptotic region Note, it is valid at n=n and n=n+1. Hence This is equation to calculate the phase shifts. The wave function is given by where

23 Problems with direct HORSE application A lot of E λ eigenstates needed while SM codes usually calculate few lowest states only One needs highly excited states and to get rid from CM excited states. n 0 are normalized for all states including the CM excited ones, hence renormalization is needed. We need n 0 for the relative n-nucleus coordinate r na but NCSM provides n 0 for the n coordinate r n relative to the nucleus CM. Hence we need to perform Talmi-Moshinsky transformations for all states to obtain n 0 in relative n-nucleus coordinates. Concluding, the direct application of the HORSE formalism in n-nucleus scattering is unpractical.

24 Single-state HORSE (SS-HORSE) Suppose E = E λ : tan (E )= S N+1, (E ) C N+1, (E ) E λ are eigenstates that are consistent with scattering information for given ħω and N max ; this is what you should obtain in any calculation with oscillator basis and what you should compare with your ab initio results.

25 Nα scattering and NCSM, JISP16 E (~,N max )=E A=5 (~,N max ) E A=4 (~,N max )

26 Nα scattering and NCSM, JISP16 E (~,N max )=E A=5 (~,N max ) E A=4 (~,N max )

27 Single-state HORSE (SS-HORSE) Suppose E = E λ : tan (E )= S N+1, (E ) C N+1, (E ) Note, information about wave function disappeared in this formula, any channel can be treated Calculating a set of E λ eigenstates with different ħω and N max within SM, we obtain a set of (E ) values which we can approximate by a smooth curve at low energies.

28 S-matrix at low energies Symmetry property: S( k) = 1 S(k) S(k) =exp2i Hence ( k) = (k), k p E, ' C p E + D( p E) 3 + F ( p E) As k! 0: ` k 2`+1 ( p E) 2`+1 Bound state: Resonance: S (i) b 0 arctan + ik(i) b (k) =k, s k ik (i) b E E b + c E + d( E) 3 + f( E) 5... S (i) r (k) = (i) (k + )(k (k r (i) r ) (i) r )(k + (i) r ) 1 ' arctan ap E E b 2 + cp E + d( p E) , c = a b 2.

29 Universal function " f nl (E) = arctan S (E) % nl $ ' # C nl (E)&

30 S. Coon et al extrapolations S. A. Coon, M. I. Avetian, M. K. G. Kruse, U. van Kolck, P. Maris, and J. P. Vary, PRC 86, (2012) What is λ sc dependence for resonances?

31 f nl (E) = arctan S (E) nl C nl (E) scaling with λ SC = ( m N Ω) / ( 2n + l + 3 2) Limit n!1: n r 2E S nl (q) q p r 0 (n + l/2+3/4) 1 4 jl (2q p n + l/2+3/4) p p r 0 (n + l/2+3/4) 1 4 sin[2q n + l/2+3/4 l/2] C nl (q) q p r 0 (n + l/2+3/4) 1 4 nl (2q p n + l/2+3/4) p p r 0 (n + l/2+3/4) 1 4 cos[2q n + l/2+3/4 l/2] ~ q = 2E Ω q p n + l/2+3/4 = p mn E SC

32 Universal function scaling E cm (MeV) ε = E cm[2(n +1) + l + 3 2] Ω l=2 f N+1,l = - arctan(s N+1,l /C N+1,l ) h = 20 l=2 λ SC = S.Coon et al (ir cutoff) ( m N Ω) / ( N tot + 3 2) 150 N+1 = 5 N+1 = 10 N+1 = 15 N+1 = 20 f nl (degrees) = E [2(N+1)+l+3/2] /h

33 How it works Model problem: nα scattering by Woods-Saxon potential J. Bang and C. Gignoux, Nucl. Phys. A, 313, 119 (1979). UV cutoff of S. A. Coon, M. I. Avetian, M. K. G. Kruse, U. van Kolck, P. Maris, and J. P. Vary, PRC 86, (2012) to select eigenvalues: = p m nucl ~ (N max /2)

34 Model problem

35 Model problem

36 Model problem 1 ' arctan ap E E b 2 + cp E + d( p E) , c = a b 2.

37 Model problem

38 Model problem

39 nα scattering: NCSM, JISP16 E (~,N max )=E A=5 (~,N max ) E A=4 (~,N max ) s = ~ (N max /2).

40 nα scattering: NCSM, JISP16 E (~,N max )=E A=5 (~,N max ) E A=4 (~,N max )

41 nα scattering: NCSM, JISP16 E (~,N max )=E A=5 (~,N max ) E A=4 (~,N max )

42 nα scattering: NCSM, JISP16 E (~,N max )=E A=5 (~,N max ) E A=4 (~,N max )

43 nα scattering: NCSM, JISP16 E (~,N max )=E A=5 (~,N max ) E A=4 (~,N max ) 1 ' arctan ap E E b 2 + cp E + d( p E) , c = a b 2.

44 nα scattering: NCSM, JISP16 E (~,N max )=E A=5 (~,N max ) E A=4 (~,N max ) s = ~ (N max /2).

45 nα scattering: NCSM, JISP16 E (~,N max )=E A=5 (~,N max ) E A=4 (~,N max )

46 nα scattering: NCSM, JISP16 E (~,N max )=E A=5 (~,N max ) E A=4 (~,N max )

47 nα scattering: NCSM, JISP16 E (~,N max )=E A=5 (~,N max ) E A=4 (~,N max )

48 nα scattering: NCSM, JISP16 E (~,N max )=E A=5 (~,N max ) E A=4 (~,N max ) 1 ' arctan ap E E b 2 + cp E + d( p E) , c = a b 2.

49 nα scattering: NCSM, JISP16 E (~,N max )=E A=5 (~,N max ) E A=4 (~,N max ) s = ~ (N max /2).

50 nα scattering: NCSM, JISP16 E (~,N max )=E A=5 (~,N max ) E A=4 (~,N max )

51 nα scattering: NCSM, JISP16 E (~,N max )=E A=5 (~,N max ) E A=4 (~,N max ) 0 arctan s E E b + c E + d( E) 3 + f( E) 5...

52 Coulomb + nuclear interaction V Sh = V Nucl + V Coul, r apple R 0 ; 0, r > R 0. R 0 R Nucl. tan = W R 0(j,F ) W R0(n Sh,F ) tan W R 0(j,G ) W R 0(n,G ) tan Sh. d h i R 0(j,F )= j (kr) F (,kr) dr j (kr) d dr h i F (,kr), = µz 1Z 2 r=r k 0 = Z 1 Z 2 r µc 2 2E SS-HORSE: tan (E )= W R 0(n,F )S 2N+2, (E )+W R 0(j,F )C 2N+2, (E ) W R 0(n,G )S 2N+2, (E )+W R 0(j,G )C 2N+2, (E ). Scaling at N +1 `(E )= r 2E ~ : arctan F`( (E ), 2 p E /s) G`( (E ), 2 p E /s). Same idea as discussed by Gautam Rupak on May 20

53 pα scattering: NCSM, JISP16 E (~,N max )=E A=5 (~,N max ) E A=4 (~,N max )

54 pα scattering: NCSM, JISP16 E (~,N max )=E A=5 (~,N max ) E A=4 (~,N max )

55 pα scattering: NCSM, JISP16 E (~,N max )=E A=5 (~,N max ) E A=4 (~,N max )

56 pα scattering: NCSM, JISP16 E (~,N max )=E A=5 (~,N max ) E A=4 (~,N max )

57 pα scattering: NCSM, JISP16

58 Summary SM states obtained at energies above thresholds can be interpreted and understood. Parameters of low-energy resonances (resonant energy and width) and low-energy phase shifts can be extracted from results of conventional Shell Model calculations A message: looks like that at large quanta only kinetic energy is important. Neglecting potential energy in higher model subspaces can save a lot of memory and computer resources.

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