Paper
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Interpretation of the Wigner Energy as due to RPA Correlations
Authors
Kai Neergaard,Rebecca Robker
Jean-Francois David,Oleg Senkov,Sarah Romans,KOSTAS MAKROPOULOS,Nicholas Luke Seed,behcet alpat,Guy Toner,Rita PANICCIA,vahid niknam,Ayse Palanduz,Julian Bommer,Małgorzata Szereda-Przestaszewska,Antonio Raggi,Istvan Csige,Alexander Grechkin,Robert Armon,Amos Korczyn,Luis Fermin Capitan Vallvey,Harald Böhnel,Pim Groen,Edward (Norm) Dancer,Alfonso V. Ramallo,Ahmed El-Diwany,Anton Hermann,Narayana Battula,Peter Andráš,jane carver,Giancarlo Marconi,Fred Friedman,John Matonic,Valentina Popova,Simon Hernandez-Ortega,Subramaniam C K,Brian Joseph Sheahan,Kai Cai,Paola D'Arrigo,Tom Veldkamp,Antonino Romano,Oleg Parasyuk,Jerzy Lazarewicz,NICOLETTA TESTONI,Christy Ludlow,Luis Campos,Shamez Ladhani,Li Ju,Ewa Szczepanska-Sadowska,Isabelle Arnulf,Igor Krichever,Lisa Wood,Wolf Frommer,Leonid E Popov,Leonidas Bachas,Kevin Brown,L. F. Panchenko,Laura Beebe,alireza emami naeini,Vladimir Kharchenko,David Lide,Seong Mok Jeong,René Botnar,Karl Insogna,Andrii Iftodii,Joel Ouaknine,mohammed cherkaoui,Ismail Sadek,Graeme Pitcher,Barry Ninham,RENZO CARRETTA,Irina Malkina-Pykh,Christoph Weidemann,Erkki Mäkinen,PAOLA MASSARELLI,Roberto Iengo,Ian Stolerman,Paul Lemieux,Luca FERRARI,GEMMA PASCUAL,Feral Temelli,jianfeng feng,Carsten Montzka,Raj Pal Sharma,Robert Randall,Steven Siegel,Ermanno GIORCELLI,thierry jouenne,Catherine Zandonella,Robert Pike,Ramesh Bhat,Christine Chirat,Maria do Carmo de Andrade Nono,Dubravka Cecez-Kecmanovic,Dieter Böning,Hasalettin Deligöz,Leslaw Karpinski,Fernando Oréfice,Kui Ren,James Lowe,Dimitrios Serpanos,Marco A Barranco-Jiménez,Anton Galajinsky,Twan Basten,magda sabbour,Cosme Gay Escoda,Dimitri Pourbaix,kudrat abdulkadyrov,Minho Song,Peter L Borst,James Lindner,David Ranson,Mª Victoria Ribera-Canudas,Daniela Guarnieri,Laurent Chupin,Marko Knezevic,Gustavo Curutchet,Weijia Li,Alexey Grigoryan,Raisa Zorina,Tatiana Kalinina,H Leon Bradlow,Ahmed Noor El Deen,Vladimir Smotlacha,Taiki Todo,Ertugrul Colak,mithat idemen,IRIARTE RAFAEL,Gabriela Nosalova,Michael Lippmann,Ekkard Brinksmeier,MARCELLO IMBRIANI,Karthikeyan S,Fabrizio Minichilli,Enqi Liu,Oto Luthar,Lucia Junqueira,Sergey Kostylev,Hongsheng Chen,Mikhail Nazarkin,Heru Susanto,Warren Grice,Anil Taku,Alexander Pogorelov,Carsten Lutz,Robin Hirsch,Gaspare Carta,Mykhaylo Andreychyn,Hans Hilgenkamp,Balaram Kundu,Yaroslav Bazel,Vitalij Danilchenko,Wolfgang Marquardt,María Ángeles Zulueta,Aleksandr Golenkov,Janet Bradbury,Malgorzata Pawlowska,Boris Shabalkin,Simonetta MASIERI,Amane Koizumi,Houssam Eddine CHAKIR,Arseniy Martinchik,Gilles VARRAULT,Petr Tuzovsky,antonio zainos,Harry Orr,Hakan Ekmekci,Ehsan Elahi Valeem,Matthew Hillwig,Nikolay Kuznetsov,Ulf Janssen,Jitka Langhamrova,Binshan Lin,Andrew Neely,Costas Koumenis,mark Steyvers,Byoungcheon Lee,Chang-Koon Choi,Shunichi Yamashita,Vitaly Konov,Toshiharu Hatanaka,Amir Mir Mohammad Sadeghi,Juan Mateos Guilarte,Jianliang Wang,Svetlana Mustafina,Dr Claudio Brito,VALENTINA PRESUTTI,Yaroslav Kost,wan azlina wan ab karim ghani,parvin pasalar,Graham Jones,Randall Morck,Yu Zuo,Alexander Baranov,Chiara Castelli,Aras Puodziukynas,Kevin Docherty,Mohsen Mirmohammad-Sadeghi,Iwan Prys Williams,Fumio Hirata,Alejandro del Castillo-Rueda,Vladimir ,Ryong Ryoo,Viacheslav Kir'iakov,Russ Van Dissen,Yuanyuan Yang,Lenka Hodačová,Maria Balda,Varvara Dorogova,Elena Medvedeva@gmail.com,William Earnshaw,Wan Kim Seow,Ali Akhavan,Lucio Vegni,Antonio VICINO,Konstantin Yumashev,Enrico Carpaneto,Stefano IURASSICH,ANNA RAMPINI,Hong-Wei HUANG,Valderi Duarte Leite,Valery Biryukov,Bachok M Taib,Katarzyna Blinowska,Otacilio Moreira,Gravel Budenkov,JOSE IVAN SANCHEZ BETANCOURT,Tommaso Maluta,farzin khorvash,Darian Rusu,MARIO INNOCENTI,Ashish Sinha,Gérard SIEST,Josef Pauli,Herbert Hutter,Li-C. 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arXiv:nucl-th/0201009v4 9 May 2002
Interpretation of the Wigner Energy as due to
RPA Correlations
Kai Neerg˚ard
Næstved Gymnasium og HF, Nyg˚ardsvej 43, DK-4700 Næstved, Denmark,
neergard@inet.uni2.dk
Abstract
In a schematic model with equidistant fourfold degenerate single-nucleon levels,
a conventional isovector pairing force and a symmetry force, the RPA correlation
energy rises almost linearly with the isospin T , thus producing a Wigner term in
accordance with the empirical proportionality of the symmetry energy to T (T + 1).
Key words:
symmetry energy, Wigner energy, isovector pairing force, symmetry force, isobaric
invariance, RPA, Goldstone mode
PACS-numbers: 21.10.Dr, 21.60.Jz, 27.40.+z
Nearly symmetric nuclei have an extra binding, the so-called Wigner energy,
that is not described by the quadratic symmetry term in the semi-empirical
mass formula [1]. This is explained in various ways in the literature. Counting
the even bonds among supermultiplet degenerate nucleon orbitals, Wigner
estimates that the isospin-dependent part of the interaction energy in the
ground state of a doubly even nucleus is proportional to T (T + 4), where T
is the isospin [2]. Talmi proves that for seniority-conserving forces such as the
pairing force acting in a single j-shell, this part of the interaction energy is
proportional to T (T + 1) [3], and Bohr and Mottelson point out that this
isospin-dependence arises in general from an interaction proportional to the
scalar product of the isospins of the interacting nucleons [4]. A symmetry en-
ergy proportional to T (T + 1) also fits the empirical masses well [5]. Myers
and Swiatecki attribute the extra binding of nearly symmetric nuclei to the
interaction of neutrons and protons in overlapping orbitals [6]. Shell model
calculations with realistic forces are succesful in reproducing the measured
binding energies [7], whereas with Skyrme forces, no Wigner term appears in
Hartre-Fock-Bogolyubov calculations and only a small one in Hartree-Fock
calculations [8]. However, by invoking a particular isoscalar pairing force that
Preprint submitted to Elsevier Preprint
Interpretation of the Wigner Energy as due to
RPA Correlations
Kai Neerg˚ard
Næstved Gymnasium og HF, Nyg˚ardsvej 43, DK-4700 Næstved, Denmark,
neergard@inet.uni2.dk
Abstract
In a schematic model with equidistant fourfold degenerate single-nucleon levels,
a conventional isovector pairing force and a symmetry force, the RPA correlation
energy rises almost linearly with the isospin T , thus producing a Wigner term in
accordance with the empirical proportionality of the symmetry energy to T (T + 1).
Key words:
symmetry energy, Wigner energy, isovector pairing force, symmetry force, isobaric
invariance, RPA, Goldstone mode
PACS-numbers: 21.10.Dr, 21.60.Jz, 27.40.+z
Nearly symmetric nuclei have an extra binding, the so-called Wigner energy,
that is not described by the quadratic symmetry term in the semi-empirical
mass formula [1]. This is explained in various ways in the literature. Counting
the even bonds among supermultiplet degenerate nucleon orbitals, Wigner
estimates that the isospin-dependent part of the interaction energy in the
ground state of a doubly even nucleus is proportional to T (T + 4), where T
is the isospin [2]. Talmi proves that for seniority-conserving forces such as the
pairing force acting in a single j-shell, this part of the interaction energy is
proportional to T (T + 1) [3], and Bohr and Mottelson point out that this
isospin-dependence arises in general from an interaction proportional to the
scalar product of the isospins of the interacting nucleons [4]. A symmetry en-
ergy proportional to T (T + 1) also fits the empirical masses well [5]. Myers
and Swiatecki attribute the extra binding of nearly symmetric nuclei to the
interaction of neutrons and protons in overlapping orbitals [6]. Shell model
calculations with realistic forces are succesful in reproducing the measured
binding energies [7], whereas with Skyrme forces, no Wigner term appears in
Hartre-Fock-Bogolyubov calculations and only a small one in Hartree-Fock
calculations [8]. However, by invoking a particular isoscalar pairing force that
Preprint submitted to Elsevier Preprint
breaks geometric symmetries, Satula and Wyss obtain a significant Wigner en-
ergy in approximately number-projected Bogolyubov calculations [9]. In finite-
temperature Bogulyubov calculations with a Yamaguchi force, R¨opke et al.
get at low temperature in the local density approximation a contribution from
neutron-proton pairing to the binding energies in the A = 40 isobaric chain
with a maximum at N − Z = −1 and a quite irregular dependence on the
asymmetry [10].
The RPA is the leading order correction to the Hartree-Fock-Bogolyubov ap-
proximation. I have therefore calculated the RPA correlation energy from the
schematic Hamiltonian
H = H0 − GP † · P + κ
2 T 2, H0 = ∑
kστ
ǫka†
kστ akστ .
In this expression, the index kστ labels orthonormal nucleon orbitals, and akστ
are the corresponding annihilation operators. kσ takes the values k and k so
that the orbital k is obtained from the orbital k by time-reversal, and τ is
‘n’ for a neutron orbital or ‘p’ for a proton orbital. P is the pair annihilation
isovector, and T denotes the total isospin. The former has the coordinates
Px = (−Pn + Pp)/√2, Py = −i(Pn + Pp)/√2, Pz = Pnp,
Pτ = ∑
k
akτ akτ , Pnp = ∑
k
(akpakn + aknakp)/√2.
The single-nucleon energy ǫk takes Ω equidistant values separated by η, and
G and κ are coupling constants.
To describe states with a given number Av = ∑
kστ a†
kστ akστ of valence nucleons
and a given isospin, I employ the Routhian
R = H − λAv − µTz .
It is just for convenience that Tz is chosen here as the isospin-coordinate to be
constrained. Since H is isobarically invariant, one could equivalently constrain
the projection of T on any axis in isospace. Following Marshalek [11] I base
the RPA on the Hartree-Bogolyubov (not Fock) self-consistent state derived
from R. This is the Bogolyubov vacuum that minimizes E0 − λ〈Av〉 − µ〈Tz 〉,
where
E0 = 〈H0〉 − |∆|2
G + κ
2 〈T 〉2, ∆ = −G〈P 〉.
At the minimum one has 〈Tx〉 = 〈Ty〉 = 0. For large values of µ a product of
neutron and proton BCS states is expected. Since this state is invariant under
2
ergy in approximately number-projected Bogolyubov calculations [9]. In finite-
temperature Bogulyubov calculations with a Yamaguchi force, R¨opke et al.
get at low temperature in the local density approximation a contribution from
neutron-proton pairing to the binding energies in the A = 40 isobaric chain
with a maximum at N − Z = −1 and a quite irregular dependence on the
asymmetry [10].
The RPA is the leading order correction to the Hartree-Fock-Bogolyubov ap-
proximation. I have therefore calculated the RPA correlation energy from the
schematic Hamiltonian
H = H0 − GP † · P + κ
2 T 2, H0 = ∑
kστ
ǫka†
kστ akστ .
In this expression, the index kστ labels orthonormal nucleon orbitals, and akστ
are the corresponding annihilation operators. kσ takes the values k and k so
that the orbital k is obtained from the orbital k by time-reversal, and τ is
‘n’ for a neutron orbital or ‘p’ for a proton orbital. P is the pair annihilation
isovector, and T denotes the total isospin. The former has the coordinates
Px = (−Pn + Pp)/√2, Py = −i(Pn + Pp)/√2, Pz = Pnp,
Pτ = ∑
k
akτ akτ , Pnp = ∑
k
(akpakn + aknakp)/√2.
The single-nucleon energy ǫk takes Ω equidistant values separated by η, and
G and κ are coupling constants.
To describe states with a given number Av = ∑
kστ a†
kστ akστ of valence nucleons
and a given isospin, I employ the Routhian
R = H − λAv − µTz .
It is just for convenience that Tz is chosen here as the isospin-coordinate to be
constrained. Since H is isobarically invariant, one could equivalently constrain
the projection of T on any axis in isospace. Following Marshalek [11] I base
the RPA on the Hartree-Bogolyubov (not Fock) self-consistent state derived
from R. This is the Bogolyubov vacuum that minimizes E0 − λ〈Av〉 − µ〈Tz 〉,
where
E0 = 〈H0〉 − |∆|2
G + κ
2 〈T 〉2, ∆ = −G〈P 〉.
At the minimum one has 〈Tx〉 = 〈Ty〉 = 0. For large values of µ a product of
neutron and proton BCS states is expected. Since this state is invariant under
2
100%
