Linear response functions in infinite matter with Skyrmetype

Linear response functions in infinite matter with Skyrme-type interactions Nguyen Van Giai Institut de Physique Nucléaire Université Paris-Sud, Orsay 30 May, 2012 ESNT 2012 1

Outline of talk: 1. Self-consistent mean field and RPA linear response 2. Skyrme interaction and RPA in infinite medium 3. Closed form expressions of response functions 4. Odd-moment sum rules and instabilities 5. Conclusion 30 May, 2012 ESNT 2012 2

1. Self-consistent mean field and RPA linear response Ø Why are we using the Random Phase Aproximation? Energy Density Functionals (EDF) are built primarily for predicting ground state properties of nuclear systems, via Hartree-Fock or more general mean field approaches Ø An important class of excitations consists of small amplitude vibrations Ø RPA is the small amplitude limit of TDHF, and the p-h interaction is uniquely derived from the EDF: Ø Ø The sssssss 30 May, 2012 The RPA eigenmodes reflect the Instabilities of the mean field ESNT 2012 3

Ø Advantages: - study ground states and excitations of nuclear systems within one framework (EDF) Ø - the s. c. RPA enables one to evaluate various energy-weighted sum rules without calculating detailed response functions. Ø Ø Drawbacks: l In atomic physics one has theoretically well established EDF, but not in nuclear physics where applications are done mostly with Skyrme-type EDF Ø Why are sum rule methods useful? Ø One can evaluate the global properties of nuclear excitations without extensive calculations (cf. , O. Bohigas, A. M. Lane, J. Martorell, Phys. Rep. 51, 267 (1979); E. Lipparini, S. Stringari, Phys. Rep. 175, 103 (1989) 30 May, 2012 ESNT 2012 4

Ø - In this presentation we concentrate only on homogeneous nuclear matter in the framework of Skyrme-type EDF, because analytical expressions can be obtained. Ø - Many RPA studies of finite nuclei have been performed (Skyrme forces, Gogny forces, realistic interactions, …), they will be discussed in this workshop. 30 May, 2012 ESNT 2012 5

General form of Skyrme-type interaction Central and spin-orbit Tensor 30 May, 2012 ESNT 2012 6

The free (HF) retarded p-h propagator Ø (p-h) Green’s function Ø Ø In the presence of p-h residual interactions the excitations are described by the RPA Green’s function: 30 May, 2012 ESNT 2012 7

The p-h interaction Momentum transfer along Oz 30 May, 2012 ESNT 2012 8

Response functions HF RPA Strength distributions 30 May, 2012 ESNT 2012 9

Explicit form of RPA response functions The special structure of Skyrme V_ph allows to obtain in closed form for infinite medium: - Central interaction: C. Garc’ia-Recio, J. Navarro, NVG, L. L. Salcedo, ANN. Phys. (N. Y. ) 214, 293 (1992) - S. O. interaction: J. Margueron, NVG, J. Navarro, Phys. Rev. C 74, 015805 (2006) - Tensor interaction: D. Davesne, M. Martini, K. Bennaceur, J. Meyer, Phys. Rev. C 80, 024314 (2009); - Erratum Phys. Rev. C 84, 059904 (2011). e Sum rules: A. x Pastore et al. , . ar. Xiv: 1204. 0336 t 30 May, 2012 ESNT 2012 10

Generalized Lindhard functions (1) 30 May, 2012 ESNT 2012 11

Generalized Lindhard functions (2) 30 May, 2012 ESNT 2012 12

RPA response function in terms of Skyrme parameters 30 May, 2012 ESNT 2012 13

nn Longitudinal RPA response HF response Skyrme force: Sk. M 30 May, 2012 ESNT 2012 14

Transverse RPA response HF response Skyrme force: Sk. M 30 May, 2012 ESNT 2012 15

Inclusion of spin-orbit interaction The previous expression (p. 13) becomes: : 30 May, 2012 ESNT 2012 16

Effect of spin-orbit interaction - - ----- S=0 - - - S=1 SLy 230 b J. Margueron, NVG, J. Navarro, Phys. Rev. C 74, 015805 (2006) 30 May, 2012 ESNT 2012 17

Sum Rules (1) Excitation operator 30 May, 2012 aaaa ESNT 2012 18

Sum Rules (2) Linear energy-weighted Cubic energy-weighted C. Garcia Recio et al, Ann. Phys. (N. Y. ) 214, 293 (1992) 30 May, 2012 ESNT 2012 19

Sum Rules (3) Inverse energy-weighted Tensor force: see Pastore’s talk 30 May, 2012 ESNT 2012 20

Polarizability Me. M Sk. M Mean energies : 30 May, 2012 ESNT 2012 21

Mean energies (Sk. M) 30 May, 2012 ESNT 2012 22

Break down of sum rule if poles! EWSR Response function Force: T 44 A. Pastore, D. Davesne, Y. Lallouet, M. Martini, K. Bennaceur, J. Meyer, ar. Xiv: 1204. 0336 Instabilities in nuclei? 30 May, 2012 ESNT 2012 23

Summary -The Skyrme EDF allows one to carry out detailed analytical calculations of nuclear responses in homogeneous medium - one has a practical tool for evaluating strong points/defects of various parametrizations -- present studies of effects of tensor component on responses are timely 30 May, 2012 ESNT 2012 24

Thank You! 30 May, 2012 ESNT 2012 25