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valence shell excitations in even-even spherical nuclei within microscopic model Ch. Stoyanov Institute for Nuclear Research and Nuclear Energy Sofia, Bulgaria

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The model Hamiltonian

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Woods-Saxon potential

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Spin-orbital term

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Coulomb potential

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Constant pairing

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Separable force and multipole expansion

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Central forces

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Spherical case Nguyen Van Giai, Ch. Stoyanov, V. V. Voronov, Phys. Rev. C 57 1204 (1998) Contribution of F 0 (r):

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Landau-Migdal form of the Skyrme interaction Nguyen Van Giai, Sagawa, H., Phys. Lett B106 (1981) 379

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cutoff radius R Introducing the coefficientand the p-h matrix elements Gauss integration formula with abscissas and weights {r k, w k }.

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The radial integrals can be calculated using a N-point integration Gauss formula with abscissas and weights r k, ω k Thus the residual interaction can be represented as a sum of N separable terms. Where are the single-particle radial matrix elements of the multipole operators: Separable Form Nguyen Van Giai, Ch, Stoyanov and V.V.Voronov., Phys. Rev. C57,1204 (1998)

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Quasiparticle RPA (collective effects)

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JJm denote a single-particle level of the average field for neutrons (or protons) TThe neutron […] λμ means coupling to the total momentum λ with projection μ: TThe quantity is Clebsch-Gordon coefficient BBogoliubov linear transformation Quasiparticle RPA (2) (quasiboson approximation)

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Phonon properties Phonons are not only collective Collective many amplitudes Non-collective a few amplitudes Pure quasi-particle state only one amplitude Diverse Momentum and Parity J π spin-multipole phonons The interaction could include any kind of correlations (particle-particle channel) LARGE PHONON SPACE

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Quasiparticle RPA (3) (collective effects)

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Harmonic vibrations To avoid Pauli principle problem

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Microscopic description of mixed- symmetry states in nearly spherical nuclei Chavdar Stoyanov and N. Lo Iudice

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Microscopic description of mixed- symmetry states in nearly spherical nuclei

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Introduction Low-lying isovector excitations are naturally predicted in the algebraic IBM-2 as mixed symmetry states. Their main signatures are relatively weak E2 and strong M1 transition to symmetric states. A. T. Otsuka, A.Arima, and Iachello, Nucl.Phys. A309, 1 (1978) B. P. van Isacker, K.Heyde, J.Jolie et al., Ann. Phys. 171, 253 (1986)

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Definitions The low-lying states of isovector nature were considered in a geometrical model as proton- neutron surface vibrations. is in-phase (isoscalar) vibration of protons and neutrons. is out-of-phase (isovector) vibration of protons and neutrons. A. A.Faessler, R. Nojarov, Phys. Lett., B166, 367 (1986) B. R. Nojarov, A. Faessler, J. Phys. G, 13, 337 (1987)

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Review paper N. Pietralla, P. von Brentano, and A. F. Lisetskiy, Prog. Part. Nucl. Phys. 60, 225 (2008).

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Microscopic calculations Within the nuclear shell model A. F. Lisetskiy, N. Pietralla, C. Fransen, R. V. Jolos, P. von Brentano, Nucl. Phys. A677, 1000 (2000) Within the quasi-particle-phonon model (QPM) N. Lo Iudice and Ch. Stoyanov, Phys. Rev. C 62, 047302 (2000) N. Lo Iudice and Ch. Stoyanov, Phys. Rev. C 65, 064304 (2002)

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Definition In order to test the isospin nature of 2 + states the following ratio is computed: This ratio probes: 1.The isoscalar ((2 + )<1) and 2.The isovector (B(2+)>1) properties of the 2 + state under consideration

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The dependence of M1 and E2 transitions on the ratio G (2) /k 0 (2) in 136 Ba.

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Structure of the first RPA phonons (only the largest components are given) and corresponding B(2 + ) ratios for 136 Ba B (2 + )

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The values of B(2 + ) for 144 Nd

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Explanation of the method used The quasi-particle Hamiltonian is diagonalized using the variational principle with a trial wave function of total spin JM Where ψ 0 represents the phonon vacuum state and R, P and T are unknown amplitudes; ν labels the specific excited state.

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Explanation of the method used (2) Taking into account the fermionic structure of the phonon operator, their commutation relations read The first term corresponds to the ideal boson approximation The second one takes into account the internal fermionic structure of the phonons The second term is important in considering multi- phonon states which may violate the Pauli principle

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Explanation of the method used (3) The normalization condition for states reads

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Energies and structure of selected low-lying excited states in 94 Mo. Only the dominant components are presented.

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94 Mo level scheme. /low-lying transitions/

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E2 transitions connecting some excite states in 94 Mo calculated within QPM.

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M1 transitions connecting some excite states in 94 Mo calculated within QPM.

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92 Zr

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92 Zr Contribution of N and Z in the 2+ QRPA phonons

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Energy and phonon structure in 92 Zr.

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E2 and M1 transitions connecting excited st. in 92 Zr

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QPM, EXP and SM g-fact. of low-lying excited st. in 92 Zr

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QRPA Results for N=80 isotones Two-quasi-particle proton states, entering into the first 2 + excitations

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Energy and structure of selected low-lying excited states

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The N=80 isotones N. Pietralla et al., Phys. Rev. C 58, 796 (1998). G. Rainovski, N. Pietralla et al., Phys. Rev. Lett. 96, 122501 (2006). T. Ahn, N. Pietralla, G. Rainovski et al., Phys. Rev. C 75, 014313 (2007). K. Sieja et al., Phys. Rev. C, v. 80 (2009) 054311.

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Experimental results

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Fermi energy as a function of the mass number

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Occupation probabilities

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Results on QRPA level

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QPM Results for N=80 isotones 134 Xe 136 Ba 138 Ce 134 Xe 138 Ce

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N=84: Experimental results

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N=84: theoretical description N. Pietralla et al., Phys. Rev. C 58, 796 (1998). G. Rainovski, N. Pietralla et al., Phys. Rev. Lett. 96, 122501 (2006). T. Ahn, N. Pietralla et al.,Phys. Rev. C 75, 014313 (2007).

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Two quasiparticle poles

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N=84: theoretical description

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Comparison to the experiment

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Conclusions There are two modes in the low-lying quadrupole excitations – isoscalar and isovector one. The properties of these two modes are close to IBM-2 symmetric and mixed-symmetry states. The coupling of the modes leads to variety of excited states. There are well pronounced regularities of E2 and M1 transitions connecting the states. The spin degree of freedom pays a dominant role in some states, such as second 1 +.

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Recent experimental results Sn PRL 98, 172501 (2007) PRL 99, 162501 (2007) PRL 101, 012502 (2008) LoI Phys. Lett. B 695, 110 (2011).

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Experimental and theoretical B(E2) values for the Sn isotopes reported from Ref.[5]. The dashed and solid curves represent the results from shell model calculations using different cores (for details see Ref.[5]).

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Calculations A. Ansari, Phys. Lett. B 623, 37 (2005). A. Ansari and P. Ring, Phys. Rev. C 74, 054313 (2006). J. Terasaki, Nucl. Phys. A 746, 583c (2004). N. Lo Iudice, Ch. Stoyanov, and D. Tarpanov PRC 84, 044314 (2011)

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Selected proton s. p. states around the Fermi energy

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Selected neutron s. p. states around the Fermi energy

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Experimental values of B(E2, g.s. -->2 + 1 ) and calculated neutron gaps in tin isotopic chain

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B(E2) through the Sn isotopic chain without and with quadrupole pairing.

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Calculated versus Experimental energies of 2+1 states. The data are taken from [19].

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QPM versus experimental B(E2). The data are taken from [4, 19]

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B(E2) through the Sn isotopic chain without and with quadrupole pairing.

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Sn neutron s. p. states

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104-112 Sn B(E2; g. st. 2 + 1 ) [e 2 b 2 ]

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Mass number Calculation B(E2) e 2 b 2 % EWSR Exp. 1 PRL 99 (2007) Exp. 2 PRL 101 (2008) Exp. 3 PRL 98 (2007) Percent of Z in the str. of 2 + 1 104 0.144 2.4 % ---4.44 106 0.214 3.4 % 0.2400:195 (39)6.4 108 0.234 3.7 % 0.2300:222 (19)7.2 110 0.269 4.2 % 0.240 0:220 (0:022) 8.1 1120.274 4.4 % 0.2408.6 104-112 Sn B(E2; g. st. 2 + 1 ) [e 2 b 2 ]

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Quasiparticle composition of the 21+ state in two typical Sn isotopes.

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Percent of 2 + 1 phonon in the str. of 2 + 1 state Mass number B(E2) e 2 b 2 Percent 102 0.078 97 % 104 0.171 93 % 106 0.248 91 % 108 0.255 92 % 110 0.255 94 % 112 0.260 96 %

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106 Sn

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Mass number Separable Skyrme Skyrme potential Experiment Energy [MeV] B(E2) [e 2 b 2 ] Energy [MeV] B(E2) [e 2 b 2 ] Energy [MeV] B(E2) [e 2 b 2 ] 1081.2310.2831.2060.2051.2060.222 (19) 1061.2350.2561.2060.1941.2060.195 (39) 1041.2660.1921.2600.1841.260---

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Conclusions There are two modes in the low-lying quadrupole excitations – isoscalar and isovector one. The properties of these two modes are close to IBM-2 symmetric and mixed-symmetry states. The coupling of the modes leads to variety of excited states. There are well pronounced regularities of E2 and M1 transitions connecting the states.

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