Introduction to NUSHELLX and transitions

Transcription

1 Introduction to NUSHELLX and transitions Angelo Signoracci CEA/Saclay Lecture 4, 14 May 213

2 Outline 1 Introduction 2 β decay 3 Electromagnetic transitions 4 Spectroscopic factors 5 Two-nucleon transfer/ other reaction model inputs

3 Outline 1 Introduction 2 β decay 3 Electromagnetic transitions 4 Spectroscopic factors 5 Two-nucleon transfer/ other reaction model inputs

4 Review Level schemes covered in Lecture III 1 Calculate energies and wavefunctions with lpe option of shell 2 NUSHELLX provides output in *,lp,*.lpe,*.lpt, and *.eps files From the wavefunction, can determine many properties of interest 1 β decay 2 Electromagnetic transitions 3 Spectroscopic information 4... Only need to know the appropriate operator Like the Hamiltonian, need effective operators in the reduced model space Requires consistent technique for all operators See Lecture VI for more information Typically, phenomenology guides operators in shell model For example, computation of electromagnetic transitions Effective charges in the sd shell e p = 1.5e e n =.5e

5 Review Level schemes covered in Lecture III 1 Calculate energies and wavefunctions with lpe option of shell 2 NUSHELLX provides output in *,lp,*.lpe,*.lpt, and *.eps files From the wavefunction, can determine many properties of interest 1 β decay 2 Electromagnetic transitions 3 Spectroscopic information 4... Only need to know the appropriate operator Like the Hamiltonian, need effective operators in the reduced model space Requires consistent technique for all operators See Lecture VI for more information Typically, phenomenology guides operators in shell model For example, computation of electromagnetic transitions Effective charges in the sd shell e p = 1.5e e n =.5e

6 Review Level schemes covered in Lecture III 1 Calculate energies and wavefunctions with lpe option of shell 2 NUSHELLX provides output in *,lp,*.lpe,*.lpt, and *.eps files From the wavefunction, can determine many properties of interest 1 β decay 2 Electromagnetic transitions 3 Spectroscopic information 4... Only need to know the appropriate operator Like the Hamiltonian, need effective operators in the reduced model space Requires consistent technique for all operators See Lecture VI for more information Typically, phenomenology guides operators in shell model For example, computation of electromagnetic transitions Effective charges in the sd shell e p = 1.5e e n =.5e

7 One-body transition densities One-body transition densities (OBTD) 1 Provide the information to calculate matrix elements of one-body operators 2 Compact, convenient form 3 Different in J-scheme and m-scheme (only present J-scheme here) 4 Different in isospin and proton-neutron formalism (see Lecture 3 for definitions) Only present simpler proton-neutron equations in lecture Given a one-body tensor operator of rank λ O λ = ij i O λ j a i a j in a reduced model space with unique values of l, j One-body transition densities between many-body states OBTD(i, j, k, k, λ) = { } ( 1) J k +J k +λ j j Jk J k λ j i j j J k k A 1 Ψ k a i Ψ k Ψ k a j Ψ k Product of 6j coefficients and reduced matrix elements Related to coefficients of fractional parentage (CFP) (full derivation not included)

8 One-body transition densities Write reduced matrix element between many-body states as a product of 1 OBTD 2 Reduced matrix elements between single particle states Ψ k O λ Ψ k = ij OBTD(i, j, k, k, λ) i O λ j Formalism looks complex, but greatly simplifies the many-body problem NUSHELLX calculates all OBTD between selected initial and final states With den option of shell In appropriate formalism (only proton-neutron shown here) Reduced single particle matrix elements Calculated by dens executable Automatically run by NUSHELLX in many cases (B(E2), B(GT ), etc.)

9 One-body transition densities Write reduced matrix element between many-body states as a product of 1 OBTD 2 Reduced matrix elements between single particle states Ψ k O λ Ψ k = ij OBTD(i, j, k, k, λ) i O λ j Formalism looks complex, but greatly simplifies the many-body problem NUSHELLX calculates all OBTD between selected initial and final states With den option of shell In appropriate formalism (only proton-neutron shown here) Reduced single particle matrix elements Calculated by dens executable Automatically run by NUSHELLX in many cases (B(E2), B(GT ), etc.)

10 One-body transition densities Write reduced matrix element between many-body states as a product of 1 OBTD 2 Reduced matrix elements between single particle states Ψ k O λ Ψ k = ij OBTD(i, j, k, k, λ) i O λ j Formalism looks complex, but greatly simplifies the many-body problem NUSHELLX calculates all OBTD between selected initial and final states With den option of shell In appropriate formalism (only proton-neutron shown here) Reduced single particle matrix elements Calculated by dens executable Automatically run by NUSHELLX in many cases (B(E2), B(GT ), etc.)

11 Outline 1 Introduction 2 β decay 3 Electromagnetic transitions 4 Spectroscopic factors 5 Two-nucleon transfer/ other reaction model inputs

12 Introduction Primary method of decay for ground states of nuclei accessible with CI theory β ± decay converts a nucleus with (Z, A) to (Z 1, A) β + decay can also proceed by electron capture (EC) Mediated by weak interaction in the form of W bosons Will only deal with so-called allowed decays 1 No angular momentum carried away by electron and neutrino 2 Not an actual selection rule- just dominant mode (by orders of magnitude) 3 Two types: Fermi and Gamow-Teller decay Q-value for β ± decay (ignoring electron binding energies) δ nh =.782MeV x = 2 for β + x = BE(A, Z f ) BE(A, Z i ) δ nh m ec 2 x where mass difference between neutron and Hydrogen atom for electron capture and β

13 Fermi and Gamow-Teller decay O(F ±) = A t a± O(GT ±) = a=1 A σ at a± a=1 a index refers to all nucleons in the many-body system Fermi decay 1 S = ( J = ) 2 T = (recall from Lecture II: isospin treated as good quantum number) B k,k (F ±) = Ψ k O(F ±) Ψ k 2 = [T k (T k + 1) T zk (T zk ± 1)]δ Jk J k δ Mk M k δ Tk T k Only connects isobaric analogue states! Gamow-Teller decay 1 S = 1 ( J = 1) 2 T =, 1 (isospin is good quantum number) B k,k (GT ±) = Ψ k O(GT ±) Ψ k 2 2J k + 1 = ij [M k,k (GT±)]2 2J k + 1 i σt ± j OBTD(i, j, k, k, 1)

14 β ± decay Half-life t 1/2 typically quoted in terms of ft values given by ft k,k 1/2 = C B k,k (F ±) + (g A /g V ) 2 B k,k (GT ±) where f = phase space factor depending on Q-value C = constant with the value 617(4) g A = axial vector coupling constant g V = vector coupling constant In QCD, g A = g V In nuclear physics, typically use value fit to neutron decay g A /g V = 1.261(8)

15 Comparison of experimental M(GT ) in sd shell to empirical interactions 1 3 M(GT) USDB M(GT) USDB experiment 2 1 experiment 2 1 M(GT) USDA M(GT) USD M(GT) USDA M(GT) USD On left: bare coupling constants From neutron decay On right: Least squares fit As in QCD, g A /g V 1 experiment theory theory fit 1 Courtesy of Alex Brown; from W.A. Richter et al., Phys. Rev. C 78, 6432 (28)

16 Outline 1 Introduction 2 β decay 3 Electromagnetic transitions 4 Spectroscopic factors 5 Two-nucleon transfer/ other reaction model inputs

17 Introduction Most common method of decay for low-lying excited states of nuclei Decay occurs within two states of same nucleus Mediated by electromagnetic interaction in the form of γ rays Sum of electric and magnetic multipole operators of tensor rank λ O = λµ [O(Eλ) µ + O(Mλ) µ] O(Eλ) µ = r λ Yµ λ (ˆr)e tz O(Mλ) µ = [ 2g λ(2λ + 1)r λ 1 l tz µ N λ + 1 [Y λ 1 ] l ] λ µ + gt s z [Y λ 1 s ] Free nucleon values: e p = 1 g l p = 1 g s p = e n = g l n = g n p = 3.826

18 Reduced transitions As described above Ψ k O λ Ψ k = ij OBTD(i, j, k, k, λ) i O λ j Like the example for β decay, reduced matrix elements easy to find Calculated by dens executable i O(E λ) j i O(Mλ) j With the necessary selection rules, refer to reduced transition probability B B(X λ; k k) = J k O(X λ) J k 2 2J k + 1 where X = E or M

19 Reduced transitions B(X λ; k k) = Ψ k O(X λ) Ψ k 2 2J k + 1 [M k,k (X λ)]2 2J k + 1 Reduced matrix element is symmetric between initial and final state Reduced transition probability depends on direction B(X λ; k k ) = 2J k + 1 2J k + 1 B(X λ; k k) Most common for shell model B(E2) (also provides quadrupole moment) B(M1) (also provides magnetic moment) Experimentally, B(E 2) transitions are usually published For Coulomb excitation: B(E2; ) B(E2; ) For γ decay: B(E2; ) B(E2; )

20 Comparison of experimental M(E2) in sd shell to empirical interactions 2 4 experiment (e fm 2 ) free nucleon M p effective M p USDA (e fm 2 ) A p A n USD (e fm 2 ) A p USDB (e fm 2 ) A n USDB (e fm 2 ) Phenomenological effective charges for the sd shell (e p 1.5e, e n.5e) 2 Courtesy of Alex Brown; from W.A. Richter et al., Phys. Rev. C 78, 6432 (28)

21 Comparison of experimental M(M1) in sd shell to empirical interactions 3 experiment M(M1) USDB M(M1) USDB 1 experiment experiment 4 M(M1) USDA M(M1) USDA M(M1) USD M(M1) USD 3 2 On left: Free nucleon g-factors On right: Least squares fit g l p = 1.16, g l n =.9 g s p = 5.15, g s n = 3.55 Less affected than charge theory theory fit 3 Courtesy of Alex Brown; from W.A. Richter et al., Phys. Rev. C 78, 6432 (28)

22 Outline 1 Introduction 2 β decay 3 Electromagnetic transitions 4 Spectroscopic factors 5 Two-nucleon transfer/ other reaction model inputs

23 Spectroscopic factors Common notation- not employed by NUSHELLX Briefly mentioned in Lecture 1 before developing formalism Basis-independent, but not observable Spectroscopic probability matrices S +ij k Ψ A a i Ψ A+1 k Ψ A+1 k a j ΨA and S ij k Ψ A a j ΨA 1 k ΨA 1 k a i Ψ A Spectroscopic factors found from tracing spectroscopic probability matrices In reduced model space, recover typical definitions SF + k Ψ A+1 k a i ΨA 2 and SF k ΨA 1 k a j Ψ A 2

24 Effective single particle energies (ESPE) Common notation- not employed by NUSHELLX Basis-independent, but not observable Requires summation over particle and hole states as described by Baranger 4 Solution to eigenvalue problem h cent ψ cent i = e cent i ψi cent, where hij cent S +ij k (E A+1 k E ) + S ij k (E E A 1 k ) k H A+1 k H A 1 In reduced model space, recover ei cent = ɛ = SF + k (E A+1 k E ) + k H A+1 k ) k H A 1 SF k (E E A 1 Are ESPE observable? Can they be defined in absolute terms? 5 4 M. Baranger, Nucl. Phys. A 149, 225 (197) 5 See T. Duguet and G. Hagen, Phys. Rev. C 85, 3433 (212)

25 Notation in NUSHELLX Written explicitly with dependence on isospin C 2 S + k = TT z T T ztt z 2 Ψ A+1 k a i ΨA k 2 (2J + 1)(2T + 1) C 2 S k Sum of spectroscopic strength for an orbit i ESPE in reduced model space = TT z T T ztt z 2 Ψ A k a i ΨA 1 k 2 (2J + 1)(2T + 1) (2j i + 1) = 2J k + 1 2J + 1 C 2 S + k + k H A+1 k H A 1 C 2 S k (2j i + 1)e cent i = (2j i + 1)ɛ = k H A+1 2J k + 1 2J + 1 C 2 S + k E + k + Consistent treatment is required k H A 1 C 2 S k E k

26 Comparison of experimental C 2 S in sd shell to empirical interactions 6 6 C 2 S experiment C 2 S USDA C 2 S USD C 2 S USDB Good agreement between all interactions and with experiment Conditions?

27 Exotic behavior Enhanced stability near the neutron dripline for oxygen isotopes New shell closures at N = 14 and N = 16 Evaluate typical truncation procedures in isotopic chain Must understand effect of experimental limitations Truncate Baranger sum rule in reduced model space ɛ trunc trunc (SF + k E + k + SF k E k ) k trunc In terms of spectroscopic strength SF ± In terms of excitation energy E ± k (SF + k + SF k ) k

28 Exotic behavior Enhanced stability near the neutron dripline for oxygen isotopes New shell closures at N = 14 and N = 16 Evaluate typical truncation procedures in isotopic chain Must understand effect of experimental limitations Truncate Baranger sum rule in reduced model space ɛ trunc trunc (SF + k E + k + SF k E k ) k trunc In terms of spectroscopic strength SF ± In terms of excitation energy E ± k (SF + k + SF k ) k

29 Exotic behavior Enhanced stability near the neutron dripline for oxygen isotopes New shell closures at N = 14 and N = 16 Evaluate typical truncation procedures in isotopic chain Must understand effect of experimental limitations Truncate Baranger sum rule in reduced model space ɛ trunc trunc (SF + k E + k + SF k E k ) k trunc In terms of spectroscopic strength SF ± In terms of excitation energy E ± k (SF + k + SF k ) k

30 Evolution of single particle shell structure A O neutron orbits -2 ESPE (MeV) d 3/2 ε F 1s 1/2 d 5/2 Exotic oxygen isotopes N = 16 shell closure at A = A Isotope E 2 + (th.) 1 E 2 + (exp.) 1 SF /+ ef ESPE Characterization 2 O /.34. Open-shell 22 O / Closed-subshell 24 O / Good closed-shell

31 Truncation in spectroscopic strength A. Signoracci, T. Duguet, unpublished Error (MeV) d 3/2 1s 1/2 d 5/2 2 O # of states included missing strength (%) log 1 [S trunc ] Open-shell both addition and removal channels required

32 Truncation in spectroscopic strength A. Signoracci, T. Duguet, unpublished Error (MeV) d 3/2 1s 1/2 d 5/2 22 O # of states included missing strength (%) log 1 [S trunc ] Closed-subshell 5 kev effect from lowest state of secondary channel Spin-orbit splitting affected by over 1 MeV by exclusion of secondary channel

33 Truncation in spectroscopic strength A. Signoracci, T. Duguet, unpublished Error (MeV) d 3/2 1s 1/2 d 5/ O # of states included missing strength (%) log 1 [S trunc ] Good closed-shell 5 kev effect from secondary channel Need to include all SF.1 for desired precision

34 Error based on truncation 24 O d 3/2 E trunc Legend (MeV) S trunc In all experiments, truncation in excitation energy and SF is necessary Effect on d 3/2 ESPE is 6 kev for E trunc 3 or SF trunc.4

35 Error based on truncation 24 O e F ESPE E trunc Legend (MeV) S trunc Fermi gap can vary by 1 MeV for reasonable experimental conditions

36 Outline 1 Introduction 2 β decay 3 Electromagnetic transitions 4 Spectroscopic factors 5 Two-nucleon transfer/ other reaction model inputs

37 Two-nucleon transfer Especially important decay mode near driplines Cannot reduce nuclear structure component to a simple number Requires the use of two-body transition operators (TBTD) Formalism not presented here Too complicated and less commonly utilized by practitioners No examples in tutorial sessions Similar to procedure for one-nucleon transfer (spectroscopic factor) Output in *.tna file displays two-nucleon transfer amplitudes Typically used as input in reaction code calculation Interference effects are important Must be consistent in phase convention for wavefunctions

38 Reaction model calculations Common quantities utilized in reaction calculations 1 Spectroscopic factors (calculated with den option of shell ) 2 Two-nucleon amplitudes (calculated with den option of shell ) 3 Single particle energies Determine effective single particle energies by Baranger formula Determine Skyrme, HO, or Woods-Saxon mean field SPE with dens executable 4 Single particle wavefunctions (calculated with dens executable) Windows version includes executable for coupled-channels reaction code FRESCO Many, many more possibilities Learn by trial and error Ask for more information relevant to your research