Doctoral Thesis - presentation

Petr Veselý

ÚČJF MFF UK 18 July 2009

Main points of the presentation

• SkyrmeHF+SeparableRPA Model

• Codes and technical details

• Presentation of the numeric results –

IV-GDR, IS-GQR, spin-M1

SHF+SRPA model I

Principal Skyrme Hamiltonian - some time-odd terms (s2, s.∆s, s2ρα) are missing… It is usually used for the description of electric giant resonances – E1, E2…

For the M1 description all the spin-density dependent terms is necessary to include – we use the full Skyrme functional…

We keep the Galilean invariance – see the terms like (ρτ – j2),…

SHF+SRPA model II

time dependent formulation:

perturped many-body w.f. we need to specify it…

linear regime – small time-dependent perturbations

Mean field Hamiltonian: static g.s. + time-dependent response

Three steps

SHF+SRPA model IIIThree steps

1

1

ˆ ˆ ˆ ˆˆ ˆ,ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ[ , ] ,

k k k k

k k ph k k k

Q Q TQ T Q

P i H Q P TP T P

+ −

+ −

= =

= = = −

( ) cos( )( ) sin( )

k k

k k

q t q tp t p t

ωω

==

I) macroscopic step:

II) microscopic step:

III) merging step:

perturped w.f. via scaling

perturped w.f. via Thouless theorem

we equal scaling&Thouless variatins

SHF+SRPA model IV

Final RPA equations:

where e.g.

0 ' ' ''

ˆ ˆ ˆ ˆ1/ 2 { }kk k k kk k kkk

H h X X Y Yκ η= + +ĺ

T-even T-odd

RPA spectrum

SHF+SRPA model VStrength function:

difficult calculation

definition

simplier calculation by Cauchy theorem

unperturbed 2qp strength

contribution of residual inter.

SHF+SRPA model VIChoice of the input operator Q & P

Depending what excitation spectra (giant resonances of which multipolarity) we intend to calculate we choose:

Electric transitions

Magnetic transitions

time-even

time-odd

time-odd

time-even

the generalized momentum partner to Q

surfacevolume contribution

Codes & technical details I

Skyax – numeric implementation of the Skyrme Hartree-Fock equations, with additional BCS pairing calculation, both HF+BCS solved iteratively within the gradient method

Skyax_me – next step, it calculates the matrix elements of the input operators Q, P and the X, Y operators in the Hartree-Fock basis, the sequence of steps depends on the electric or magnetic type of the transition

Skyax_srpa – final step, it constructs the SRPA matrix, calculates the corresponding strength function and additionally solves the RPA solutions

The role of the author of the thesis – to construct the subroutines for the magnetic input and transition operators and their correct implementation into the structure of Skyax_me and Skyax_srpa codes…

The axially deformed single-particle basis is represented on the 2D grid with the mesh size 0.7 fm.

Codes & technical details II

Independent and self-consistent calculation of the axial deformation due to the constraint condition to the quadrupole moment Q2.

Dependence of the isoscalar dipole mode (the CM spuriousity) on the s.p. basis and the grid size…

The grid size 0.7 fm is good choice. Most important is to take as big configuration space as possible…

Spurious peak at 2-3 MeV – divided from the region of the giant resonances…

Codes & technical details III

Convergence of the strength function due to number of input operators is quite fast…

Comparison of SRPA with full RPA possible only in light nuclei and smaller config. space

Numeric results I

E1 (T=1) giant resonance in 150Nd

experimentP.Carlos et al., NPA 172, 437 (1971)

JANIS database

with time-odd current

)( 13

1 µµ YrYr +

without time-odd current

P.Carlos et al., NPA 172, 437 (1971)B.L.Bergman et al, RMP 47, 713 (1971)A.V.Varlamov et al., Atlas of Giant R., INDC(NDS)-394, 1999

Numeric results I

E1 (T=1) giant resonance in 150Nd

Numeric results II

The physical mechanisms of widths:

• deformation

• Landau fragmentation (dumping) (interaction with the nearby 2qp poles)

• continuum

• contribution by complex configurations (higher phonon excit.)

We cannot implement into our model – instead look at the role of the Lorentzian width ∆

Numeric results III

• Variation of width depending on ∆ is small – dominant part of Γ by deform. & Landau

• deformation makes not more than 40 % of the total Γ

Numeric results IVIn the study of the spin-M1 resonance we notice the essential role of the spin-density terms in the collective shifts.

Without the terms s2, s.∆s, s2ρα, (s.T-J2) we get no any collective shift of the M1 strength!

necessity of using the full Skyrme energy functional!

input operators: s1µ,r2s1µ,r2Y2µ

Numeric results V

The spin-M1 resonance consists of proton & neutron bumps.

In the experimental data (Laszewski Phys. Rev. Lett. 61,1710 (1988)) we don’t see the two-bump structure but only one dominant (?isovector?) peak.

Numeric results VI

High variability of results with respect to the used parametrization.

All the Skyrme parametrizations were fixed due to calculations with the “principal” Skyrme Hamiltonian. The parameters essential for the spin-density terms are actually random quantities…

input operators: s1µ,r2s1µ,r2Y2µ

List of collaborators

1) Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Moscow region, 141980, Russia2) Technical Universiy of Dresden, Institute for Analysis, D-01062, Dresden, Germany3) Institute of Particle and Nuclear Physics, Charles University, CZ-18000 Praha 8, Czech Republic4) Institute of Theoretical Physics II, University of Erlangen, D-91058, Erlangen, Germany

V.O. Nesterenko 1)

W. Kleinig 2) J. Kvasil 3) , P. Veselý 3)

P.-G. Reinhard 4)