Embed Size (px)
Citation preview
Gogny interactions with tensor terms
Marta Anguiano
Departamento de Física Atómica, Molecular y Nuclear (UGR)
First Gogny Conference
Bruyères-le-Châtel, December 2015
Work in colaboration with
I Rémi Bernard
Universidad Autónoma de Madrid (Spain)
I Giampaolo Co’ and Viviana De Donno
Università del Salento (Italy)
I Marcella Grasso
Institut de Physique Nuclèaire, IN2P3-CNRS, UniversitèParis-Sud (France)
I Antonio M. Lallena
Universidad de Granada (Spain)
Outline
Introduction
The method
Results
Perspectives
Outline
Introduction
The method
Results
Perspectives
Outline
Introduction
The method
Results
Perspectives
Outline
Introduction
The method
Results
Perspectives
Motivation
I Tensor force is usually neglected in mean-field methods.I Shell evolution cannot be studied without tensor force (nuclei far
from the stability line).I Crucial in the study of properties of spin and spin-isospin states
(Gamow-Teller and spin-dipole excitations).I Some experiments which indicates the role of the tensor force:
1. Schiffer et al., Phys. Rev. Lett. 92, 162501 (2004):Proton splitting 1h11/2 − 1g7/2 in Z = 50 increases with neutron excess
2. Gaudefroy et al., Phys. Rev. Lett 97, 092501 (2006):Reduction of the neutron splitting 1f7/2 − 2p3/2 in N = 28 going from 49Ca to 47Ar
3. Burgunder et al., Phys. Rev. Lett 112, 042502 (2014):Neutron splitting 2p3/2 − 2p1/2 for N = 20
Motivation
I Tensor force is usually neglected in mean-field methods.I Shell evolution cannot be studied without tensor force (nuclei far
from the stability line).I Crucial in the study of properties of spin and spin-isospin states
(Gamow-Teller and spin-dipole excitations).I Some experiments which indicates the role of the tensor force:
1. Schiffer et al., Phys. Rev. Lett. 92, 162501 (2004):Proton splitting 1h11/2 − 1g7/2 in Z = 50 increases with neutron excess
2. Gaudefroy et al., Phys. Rev. Lett 97, 092501 (2006):Reduction of the neutron splitting 1f7/2 − 2p3/2 in N = 28 going from 49Ca to 47Ar
3. Burgunder et al., Phys. Rev. Lett 112, 042502 (2014):Neutron splitting 2p3/2 − 2p1/2 for N = 20
Motivation
I Tensor force is usually neglected in mean-field methods.I Shell evolution cannot be studied without tensor force (nuclei far
from the stability line).I Crucial in the study of properties of spin and spin-isospin states
(Gamow-Teller and spin-dipole excitations).I Some experiments which indicates the role of the tensor force:
1. Schiffer et al., Phys. Rev. Lett. 92, 162501 (2004):Proton splitting 1h11/2 − 1g7/2 in Z = 50 increases with neutron excess
2. Gaudefroy et al., Phys. Rev. Lett 97, 092501 (2006):Reduction of the neutron splitting 1f7/2 − 2p3/2 in N = 28 going from 49Ca to 47Ar
3. Burgunder et al., Phys. Rev. Lett 112, 042502 (2014):Neutron splitting 2p3/2 − 2p1/2 for N = 20
Motivation
I Tensor force is usually neglected in mean-field methods.I Shell evolution cannot be studied without tensor force (nuclei far
from the stability line).I Crucial in the study of properties of spin and spin-isospin states
(Gamow-Teller and spin-dipole excitations).I Some experiments which indicates the role of the tensor force:
1. Schiffer et al., Phys. Rev. Lett. 92, 162501 (2004):Proton splitting 1h11/2 − 1g7/2 in Z = 50 increases with neutron excess
2. Gaudefroy et al., Phys. Rev. Lett 97, 092501 (2006):Reduction of the neutron splitting 1f7/2 − 2p3/2 in N = 28 going from 49Ca to 47Ar
3. Burgunder et al., Phys. Rev. Lett 112, 042502 (2014):Neutron splitting 2p3/2 − 2p1/2 for N = 20
Motivation
I Tensor force is usually neglected in mean-field methods.I Shell evolution cannot be studied without tensor force (nuclei far
from the stability line).I Crucial in the study of properties of spin and spin-isospin states
(Gamow-Teller and spin-dipole excitations).I Some experiments which indicates the role of the tensor force:
1. Schiffer et al., Phys. Rev. Lett. 92, 162501 (2004):Proton splitting 1h11/2 − 1g7/2 in Z = 50 increases with neutron excess
2. Gaudefroy et al., Phys. Rev. Lett 97, 092501 (2006):Reduction of the neutron splitting 1f7/2 − 2p3/2 in N = 28 going from 49Ca to 47Ar
3. Burgunder et al., Phys. Rev. Lett 112, 042502 (2014):Neutron splitting 2p3/2 − 2p1/2 for N = 20
Motivation
I Tensor force is usually neglected in mean-field methods.I Shell evolution cannot be studied without tensor force (nuclei far
from the stability line).I Crucial in the study of properties of spin and spin-isospin states
(Gamow-Teller and spin-dipole excitations).I Some experiments which indicates the role of the tensor force:
1. Schiffer et al., Phys. Rev. Lett. 92, 162501 (2004):Proton splitting 1h11/2 − 1g7/2 in Z = 50 increases with neutron excess
2. Gaudefroy et al., Phys. Rev. Lett 97, 092501 (2006):Reduction of the neutron splitting 1f7/2 − 2p3/2 in N = 28 going from 49Ca to 47Ar
3. Burgunder et al., Phys. Rev. Lett 112, 042502 (2014):Neutron splitting 2p3/2 − 2p1/2 for N = 20
Motivation
I Tensor force is usually neglected in mean-field methods.I Shell evolution cannot be studied without tensor force (nuclei far
from the stability line).I Crucial in the study of properties of spin and spin-isospin states
(Gamow-Teller and spin-dipole excitations).I Some experiments which indicates the role of the tensor force:
1. Schiffer et al., Phys. Rev. Lett. 92, 162501 (2004):Proton splitting 1h11/2 − 1g7/2 in Z = 50 increases with neutron excess
2. Gaudefroy et al., Phys. Rev. Lett 97, 092501 (2006):Reduction of the neutron splitting 1f7/2 − 2p3/2 in N = 28 going from 49Ca to 47Ar
3. Burgunder et al., Phys. Rev. Lett 112, 042502 (2014):Neutron splitting 2p3/2 − 2p1/2 for N = 20
The tensor interaction
I We have proposed different types of finite range tensorinteractions onto D1S and D1M Gogny parametrizations.
I Different fits have been done in order to fix the free parameters ineach case:
1. Adding a the tensor-isospin term, and modifying the strength of thespin-orbit term: D1ST
2. Adding a pure tensor and tensor-isospin terms: D1ST2a, D1ST2b3. Adding a pure tensor, tensor-isospin and modifying the spin-orbit
term: D1ST2c
I With these interactions, we have studied:
1. Binding and single particle energies in HF approximation.2. Excitation states with DRPA and CRPA approximations:
2.1 0− excitation in spherical nuclei2.2 IS and IV states in N = Z and 1+ excitation in N 6= Z nuclei2.3 Magnetic states and charge-exchange excitations
3. Pairing properties using a HF+BCS model.
The tensor interaction
I We have proposed different types of finite range tensorinteractions onto D1S and D1M Gogny parametrizations.
I Different fits have been done in order to fix the free parameters ineach case:
1. Adding a the tensor-isospin term, and modifying the strength of thespin-orbit term: D1ST
2. Adding a pure tensor and tensor-isospin terms: D1ST2a, D1ST2b3. Adding a pure tensor, tensor-isospin and modifying the spin-orbit
term: D1ST2c
I With these interactions, we have studied:
1. Binding and single particle energies in HF approximation.2. Excitation states with DRPA and CRPA approximations:
2.1 0− excitation in spherical nuclei2.2 IS and IV states in N = Z and 1+ excitation in N 6= Z nuclei2.3 Magnetic states and charge-exchange excitations
3. Pairing properties using a HF+BCS model.
The tensor interaction
I We have proposed different types of finite range tensorinteractions onto D1S and D1M Gogny parametrizations.
I Different fits have been done in order to fix the free parameters ineach case:
1. Adding a the tensor-isospin term, and modifying the strength of thespin-orbit term: D1ST
2. Adding a pure tensor and tensor-isospin terms: D1ST2a, D1ST2b3. Adding a pure tensor, tensor-isospin and modifying the spin-orbit
term: D1ST2c
I With these interactions, we have studied:
1. Binding and single particle energies in HF approximation.2. Excitation states with DRPA and CRPA approximations:
2.1 0− excitation in spherical nuclei2.2 IS and IV states in N = Z and 1+ excitation in N 6= Z nuclei2.3 Magnetic states and charge-exchange excitations
3. Pairing properties using a HF+BCS model.
The tensor interaction
I We have proposed different types of finite range tensorinteractions onto D1S and D1M Gogny parametrizations.
I Different fits have been done in order to fix the free parameters ineach case:
1. Adding a the tensor-isospin term, and modifying the strength of thespin-orbit term: D1ST
2. Adding a pure tensor and tensor-isospin terms: D1ST2a, D1ST2b3. Adding a pure tensor, tensor-isospin and modifying the spin-orbit
term: D1ST2c
I With these interactions, we have studied:
1. Binding and single particle energies in HF approximation.2. Excitation states with DRPA and CRPA approximations:
2.1 0− excitation in spherical nuclei2.2 IS and IV states in N = Z and 1+ excitation in N 6= Z nuclei2.3 Magnetic states and charge-exchange excitations
3. Pairing properties using a HF+BCS model.
The tensor interaction
I We have proposed different types of finite range tensorinteractions onto D1S and D1M Gogny parametrizations.
I Different fits have been done in order to fix the free parameters ineach case:
1. Adding a the tensor-isospin term, and modifying the strength of thespin-orbit term: D1ST
2. Adding a pure tensor and tensor-isospin terms: D1ST2a, D1ST2b3. Adding a pure tensor, tensor-isospin and modifying the spin-orbit
term: D1ST2c
I With these interactions, we have studied:
1. Binding and single particle energies in HF approximation.2. Excitation states with DRPA and CRPA approximations:
2.1 0− excitation in spherical nuclei2.2 IS and IV states in N = Z and 1+ excitation in N 6= Z nuclei2.3 Magnetic states and charge-exchange excitations
3. Pairing properties using a HF+BCS model.
The tensor interaction
I We have proposed different types of finite range tensorinteractions onto D1S and D1M Gogny parametrizations.
I Different fits have been done in order to fix the free parameters ineach case:
1. Adding a the tensor-isospin term, and modifying the strength of thespin-orbit term: D1ST
2. Adding a pure tensor and tensor-isospin terms: D1ST2a, D1ST2b3. Adding a pure tensor, tensor-isospin and modifying the spin-orbit
term: D1ST2c
I With these interactions, we have studied:
1. Binding and single particle energies in HF approximation.2. Excitation states with DRPA and CRPA approximations:
2.1 0− excitation in spherical nuclei2.2 IS and IV states in N = Z and 1+ excitation in N 6= Z nuclei2.3 Magnetic states and charge-exchange excitations
3. Pairing properties using a HF+BCS model.
The tensor interaction
I We have proposed different types of finite range tensorinteractions onto D1S and D1M Gogny parametrizations.
I Different fits have been done in order to fix the free parameters ineach case:
1. Adding a the tensor-isospin term, and modifying the strength of thespin-orbit term: D1ST
2. Adding a pure tensor and tensor-isospin terms: D1ST2a, D1ST2b3. Adding a pure tensor, tensor-isospin and modifying the spin-orbit
term: D1ST2c
I With these interactions, we have studied:
1. Binding and single particle energies in HF approximation.2. Excitation states with DRPA and CRPA approximations:
2.1 0− excitation in spherical nuclei2.2 IS and IV states in N = Z and 1+ excitation in N 6= Z nuclei2.3 Magnetic states and charge-exchange excitations
3. Pairing properties using a HF+BCS model.
The tensor interaction
I We have proposed different types of finite range tensorinteractions onto D1S and D1M Gogny parametrizations.
I Different fits have been done in order to fix the free parameters ineach case:
1. Adding a the tensor-isospin term, and modifying the strength of thespin-orbit term: D1ST
2. Adding a pure tensor and tensor-isospin terms: D1ST2a, D1ST2b3. Adding a pure tensor, tensor-isospin and modifying the spin-orbit
term: D1ST2c
I With these interactions, we have studied:
1. Binding and single particle energies in HF approximation.2. Excitation states with DRPA and CRPA approximations:
2.1 0− excitation in spherical nuclei2.2 IS and IV states in N = Z and 1+ excitation in N 6= Z nuclei2.3 Magnetic states and charge-exchange excitations
3. Pairing properties using a HF+BCS model.
The tensor interaction
I We have proposed different types of finite range tensorinteractions onto D1S and D1M Gogny parametrizations.
I Different fits have been done in order to fix the free parameters ineach case:
1. Adding a the tensor-isospin term, and modifying the strength of thespin-orbit term: D1ST
2. Adding a pure tensor and tensor-isospin terms: D1ST2a, D1ST2b3. Adding a pure tensor, tensor-isospin and modifying the spin-orbit
term: D1ST2c
I With these interactions, we have studied:
1. Binding and single particle energies in HF approximation.2. Excitation states with DRPA and CRPA approximations:
2.1 0− excitation in spherical nuclei2.2 IS and IV states in N = Z and 1+ excitation in N 6= Z nuclei2.3 Magnetic states and charge-exchange excitations
3. Pairing properties using a HF+BCS model.
The tensor interaction
I We have proposed different types of finite range tensorinteractions onto D1S and D1M Gogny parametrizations.
I Different fits have been done in order to fix the free parameters ineach case:
1. Adding a the tensor-isospin term, and modifying the strength of thespin-orbit term: D1ST
2. Adding a pure tensor and tensor-isospin terms: D1ST2a, D1ST2b3. Adding a pure tensor, tensor-isospin and modifying the spin-orbit
term: D1ST2c
I With these interactions, we have studied:
1. Binding and single particle energies in HF approximation.2. Excitation states with DRPA and CRPA approximations:
2.1 0− excitation in spherical nuclei2.2 IS and IV states in N = Z and 1+ excitation in N 6= Z nuclei2.3 Magnetic states and charge-exchange excitations
3. Pairing properties using a HF+BCS model.
The tensor interaction
I We have proposed different types of finite range tensorinteractions onto D1S and D1M Gogny parametrizations.
I Different fits have been done in order to fix the free parameters ineach case:
1. Adding a the tensor-isospin term, and modifying the strength of thespin-orbit term: D1ST
2. Adding a pure tensor and tensor-isospin terms: D1ST2a, D1ST2b3. Adding a pure tensor, tensor-isospin and modifying the spin-orbit
term: D1ST2c
I With these interactions, we have studied:
1. Binding and single particle energies in HF approximation.2. Excitation states with DRPA and CRPA approximations:
2.1 0− excitation in spherical nuclei2.2 IS and IV states in N = Z and 1+ excitation in N 6= Z nuclei2.3 Magnetic states and charge-exchange excitations
3. Pairing properties using a HF+BCS model.
The tensor interaction
I We have proposed different types of finite range tensorinteractions onto D1S and D1M Gogny parametrizations.
I Different fits have been done in order to fix the free parameters ineach case:
1. Adding a the tensor-isospin term, and modifying the strength of thespin-orbit term: D1ST
2. Adding a pure tensor and tensor-isospin terms: D1ST2a, D1ST2b3. Adding a pure tensor, tensor-isospin and modifying the spin-orbit
term: D1ST2c
I With these interactions, we have studied:
1. Binding and single particle energies in HF approximation.2. Excitation states with DRPA and CRPA approximations:
2.1 0− excitation in spherical nuclei2.2 IS and IV states in N = Z and 1+ excitation in N 6= Z nuclei2.3 Magnetic states and charge-exchange excitations
3. Pairing properties using a HF+BCS model.
Our Hartree-Fock (HF) approximation
I We considere as effective nucleon-nucleon interaction afinite-range two-body force of the type:
V(~r1, ~r2) =
6∑p=1
Vp(~r1, ~r2)Op(1, 2) + VSO(~r1, ~r2) + VDD(~r1, ~r2) + VCoul(~r1, ~r2)
I Op(1, 2) indicates 1, ~τ1 · ~τ2, ~σ1 · ~σ2, ~σ1 · ~σ2 ~τ1 · ~τ2, S12, S12 ~τ1 · ~τ2.I VSO and VDD, terms of zero-range (like the corresponding terms
in Skyrme-like forces)I We solve, in coordinate space, a set of equations of the type:
− ~2
2mk∇2
1 φk(~r1)+ U(~r1)φk(~r1) −∫
d3r2W(~r1, ~r2)φk(~r2) = εkφk(~r1)
I Hartree (Direct) termI Fock (Exchange) term
Our Hartree-Fock (HF) approximation
I We considere as effective nucleon-nucleon interaction afinite-range two-body force of the type:
V(~r1, ~r2) =
6∑p=1
Vp(~r1, ~r2)Op(1, 2) + VSO(~r1, ~r2) + VDD(~r1, ~r2) + VCoul(~r1, ~r2)
I Op(1, 2) indicates 1, ~τ1 · ~τ2, ~σ1 · ~σ2, ~σ1 · ~σ2 ~τ1 · ~τ2, S12, S12 ~τ1 · ~τ2.I VSO and VDD, terms of zero-range (like the corresponding terms
in Skyrme-like forces)I We solve, in coordinate space, a set of equations of the type:
− ~2
2mk∇2
1 φk(~r1)+ U(~r1)φk(~r1) −∫
d3r2W(~r1, ~r2)φk(~r2) = εkφk(~r1)
I Hartree (Direct) termI Fock (Exchange) term
Our Hartree-Fock (HF) approximation
I We considere as effective nucleon-nucleon interaction afinite-range two-body force of the type:
V(~r1, ~r2) =
6∑p=1
Vp(~r1, ~r2)Op(1, 2) + VSO(~r1, ~r2) + VDD(~r1, ~r2) + VCoul(~r1, ~r2)
I Op(1, 2) indicates 1, ~τ1 · ~τ2, ~σ1 · ~σ2, ~σ1 · ~σ2 ~τ1 · ~τ2, S12, S12 ~τ1 · ~τ2.I VSO and VDD, terms of zero-range (like the corresponding terms
in Skyrme-like forces)I We solve, in coordinate space, a set of equations of the type:
− ~2
2mk∇2
1 φk(~r1)+ U(~r1)φk(~r1) −∫
d3r2W(~r1, ~r2)φk(~r2) = εkφk(~r1)
I Hartree (Direct) termI Fock (Exchange) term
Our Hartree-Fock (HF) approximation
I We considere as effective nucleon-nucleon interaction afinite-range two-body force of the type:
V(~r1, ~r2) =
6∑p=1
Vp(~r1, ~r2)Op(1, 2) + VSO(~r1, ~r2) + VDD(~r1, ~r2) + VCoul(~r1, ~r2)
I Op(1, 2) indicates 1, ~τ1 · ~τ2, ~σ1 · ~σ2, ~σ1 · ~σ2 ~τ1 · ~τ2, S12, S12 ~τ1 · ~τ2.I VSO and VDD, terms of zero-range (like the corresponding terms
in Skyrme-like forces)I We solve, in coordinate space, a set of equations of the type:
− ~2
2mk∇2
1 φk(~r1)+ U(~r1)φk(~r1) −∫
d3r2W(~r1, ~r2)φk(~r2) = εkφk(~r1)
I Hartree (Direct) termI Fock (Exchange) term
Our Hartree-Fock (HF) approximation
I We considere as effective nucleon-nucleon interaction afinite-range two-body force of the type:
V(~r1, ~r2) =
6∑p=1
Vp(~r1, ~r2)Op(1, 2) + VSO(~r1, ~r2) + VDD(~r1, ~r2) + VCoul(~r1, ~r2)
I Op(1, 2) indicates 1, ~τ1 · ~τ2, ~σ1 · ~σ2, ~σ1 · ~σ2 ~τ1 · ~τ2, S12, S12 ~τ1 · ~τ2.I VSO and VDD, terms of zero-range (like the corresponding terms
in Skyrme-like forces)I We solve, in coordinate space, a set of equations of the type:
− ~2
2mk∇2
1 φk(~r1)+ U(~r1)φk(~r1) −∫
d3r2W(~r1, ~r2)φk(~r2) = εkφk(~r1)
I Hartree (Direct) termI Fock (Exchange) term
Our Hartree-Fock (HF) approximation
I We considere as effective nucleon-nucleon interaction afinite-range two-body force of the type:
V(~r1, ~r2) =
6∑p=1
Vp(~r1, ~r2)Op(1, 2) + VSO(~r1, ~r2) + VDD(~r1, ~r2) + VCoul(~r1, ~r2)
I Op(1, 2) indicates 1, ~τ1 · ~τ2, ~σ1 · ~σ2, ~σ1 · ~σ2 ~τ1 · ~τ2, S12, S12 ~τ1 · ~τ2.I VSO and VDD, terms of zero-range (like the corresponding terms
in Skyrme-like forces)I We solve, in coordinate space, a set of equations of the type:
− ~2
2mk∇2
1 φk(~r1)+ U(~r1)φk(~r1) −∫
d3r2W(~r1, ~r2)φk(~r2) = εkφk(~r1)
I Hartree (Direct) termI Fock (Exchange) term
Fit of the tensor terms: D1ST interaction
v6,b(r) = v6,AV18(r)(
1 − e−br2)
V6(q)S12(q) =∫
d3reiq·rv6(r)S12(r) = −4π∫
drr2j2(qr)v6(r)S12(r)
Fit of the tensor force: D1STEnergies of the first 0− states
12.0
14.0
16.0
18.0
20.0
8.0
10.0
12.0
0.0 0.4 0.8 1.2 1.64.0
6.0
8.0
10.0
3.0
5.0
7.0
9.0
3.0
5.0
7.0
0.0 0.4 0.8 1.2 1.6
2.0
4.0
6.0
ω[M
eV]
0−
b [fm−2] b [fm−2]
12C
16O
40Ca
48Ca
90Zr
208Pb
I Fit for 16O:E(0−)=10.96MeV
I D1STb=0.6 fm−2,W0=134 MeV
I D1MTb=0.25 fm−2,W0=122.5 MeV
Phenomenological RPA with LM + v6,b(r)
Fit of the tensor force: D1ST2a and D1ST2b
I Experimentally, the difference between the energies of thesingle-particle neutron 2p3/2, 1f7/2 states increases from 40Ca to48Ca: O. Sorlin and M.-G. Pourquet, Prog. Part. Nucl. Phys. 61, 602 (2008)
18 20 22 24 26 28 30
3.0
3.4
3.8
4.2
4.6
5.0
5.4
5.8
18 20 22 24 26 28 30
GAP(N
=28)[M
eV]
N N
(a) (b)
Ca
SLy5T
SLy4
SLy5T (−αT)
D1S(−T)
D1S
D1ST
M. A et al., Phys. Rev. C86, 054302 (2012)
Fit of the tensor force: D1ST2a and D1ST2b
N. Onishi and J.W. Negele, NPA301 (1978), 336
Vtensor(~r1,~r2) = (VT1 + VT2 Pτ12) S12 e−(r1−r2)2/µ2
T
=
[(VT1 +
12
VT2
)+
12
VT2 τ (1) · τ (2)
]S12 e−(r1−r2)
2/µ2T
D1ST2a → neutron 1f splitting in 48Ca and 0− state of 16O:
VT1 = −135 MeV, VT2 = 115 MeV
D1ST2b→ N = 28 neutron gap increase from 40Ca to 48Ca asobtained in HF calculations with the SLy5T force and 0− state of 16O:
VT1 = −182 MeV, VT2 = 102 MeV
Fits of the tensor force: D1ST2c
Following the strategy of Zalewski et al., Phys. Rev. C77, 024316 (2008):
D1ST2c→ neutron 1f splitting in 40Ca, 48Ca and 56Ni
1. First, we fit the splitting 1f in 40Ca by modifying the spin-orbitparameter WLS,
2. second, we fit the splitting 1f in 48Ca adjusting the like-particlepart of the Gogny tensor term and, VT1 + VT2,
3. finally, we use the 56Ni to fit the neutron-proton contribution of thetensor term, VT2 .1
WLS = 103 MeV fm5, VT1 = −135 MeV, VT2 = 60 MeV
D1MT2c → following the same procedure, and using D1M asstarting point we have fit another interaction
WLS = 95 MeV fm5, VT1 = −175 MeV, VT2 = 40 MeV
1M. Grasso and M. A, Phys. Rev. C88, 054328 (2014)
Fits of the tensor force: D1ST2c
Following the strategy of Zalewski et al., Phys. Rev. C77, 024316 (2008):
D1ST2c→ neutron 1f splitting in 40Ca, 48Ca and 56Ni
1. First, we fit the splitting 1f in 40Ca by modifying the spin-orbitparameter WLS,
2. second, we fit the splitting 1f in 48Ca adjusting the like-particlepart of the Gogny tensor term and, VT1 + VT2,
3. finally, we use the 56Ni to fit the neutron-proton contribution of thetensor term, VT2 .1
WLS = 103 MeV fm5, VT1 = −135 MeV, VT2 = 60 MeV
D1MT2c → following the same procedure, and using D1M asstarting point we have fit another interaction
WLS = 95 MeV fm5, VT1 = −175 MeV, VT2 = 40 MeV
1M. Grasso and M. A, Phys. Rev. C88, 054328 (2014)
Fits of the tensor force: D1ST2c
Following the strategy of Zalewski et al., Phys. Rev. C77, 024316 (2008):
D1ST2c→ neutron 1f splitting in 40Ca, 48Ca and 56Ni
1. First, we fit the splitting 1f in 40Ca by modifying the spin-orbitparameter WLS,
2. second, we fit the splitting 1f in 48Ca adjusting the like-particlepart of the Gogny tensor term and, VT1 + VT2,
3. finally, we use the 56Ni to fit the neutron-proton contribution of thetensor term, VT2 .1
WLS = 103 MeV fm5, VT1 = −135 MeV, VT2 = 60 MeV
D1MT2c → following the same procedure, and using D1M asstarting point we have fit another interaction
WLS = 95 MeV fm5, VT1 = −175 MeV, VT2 = 40 MeV
1M. Grasso and M. A, Phys. Rev. C88, 054328 (2014)
Bulk properties of spherical nuclei
6.0
7.0
8.0
9.0
14
16
22
24
28 40
48
52
60
48
56
68
78
90 100
114
116
132
208
6.0
7.0
8.0
9.0
-6.0
-4.0
-2.0
0.0
2.0
A
∆E
E/A
[MeV
]E/A
[MeV
]
D1S
D1M
D1S
D1ST exp
D1M
D1MT exp
O Ca Ni Zr Sn Pb
(a)
(b)
(c)
∆E = 100ED1αT − ED1α
ED1α
α ≡ S,M
s.p. energies of spin–orbit partnersProtons
-4.0
-2.0
0.0
2.0
-4.0
-2.0
0.0
2.0
14
16
22
24
28 40
48
52
60
48
56
68
78
90 100
114
116
132
208
-4.0
-2.0
0.0
2.0
A
∆s[M
eV]
D1S
D1M
O Ca Ni Zr Sn Pb
1p
1d
1f
(a)
(b)
(c)
s = εl−1/2 − εl+1/2
Neutrons
-4.0
-2.0
0.0
2.0
-4.0
-2.0
0.0
2.0
14
16
22
24
28 40
48
52
60
48
56
68
78
90 100
114
116
132
208 -4.0
-2.0
0.0
2.0
A
∆s[M
eV]
D1S
D1M
O Ca Ni Zr Sn Pb
1p
1d
1f
(a)
(b)
(c)
∆s = sD1αT − sD1α
Discrete RPA: 0− states
exp D1S D1ST D1M D1MT
12C 18.40 19.63 14.42 18.83 15.2716O 10.96 13.95 10.94 13.08 10.96
40Ca 10.78 12.22 9.57 11.56 9.6048Ca 8.05 14.10 11.63 12.85 11.26
208Pb 5.28 8.27 7.93 8.24 7.92
M. A et al. Phys. Rev. C83 (2011) 064306
Exp. values:A. Heusler et al. Phys. Rev. C75 (2007) 024312
http://www.nndc.bnl.gov/
Discrete RPA: 1+ excitations in N 6= Z nuclei
-3.0
-2.0
-1.0
0.0
1.0
14 22 24 28 48 52 60 114 116 132 208
-3.0
-2.0
-1.0
0.0
1.0
A
[MeV
][M
eV]
D1S
D1M
O Ca Sn Pb
(a)
(b)
ωtt − ωnn (ωtt − ωtn) + (ωt − ωn)ωtt − ωtn
ωt − ωn
First 1+ state in 208Pb
E(1+1 ) B(M1)1[MeV] [µ2
n]exp 5.85 2.0
D1S 7.80 5.08D1ST 4.76 2.41D1M 6.50 2.33
D1MT 4.82 1.80
M. A et al. Phys. Rev. C83 (2011) 064306
Continuum RPA: Magnetic dipole response in O isotopes
0 60 120 180
0.000
0.002
0.004
0.006
0 5 10 15 20
0
2
4
6
8
10
0 5 10 15 20
0
2
4
6
8
10
0 10 20 30 40 50
0.00
0.04
0.08
B(M
1)↑(µ
2 N)
B(M
1)↑(µ
2 N)
ω (MeV) ω (MeV)
22O
28O
16O
24O
nntntt
(a) (b)
(c) (d)
main p-h excitation⇒ [(ν1d3/2)(ν1d5/2)−1]
Continuum RPA: Magnetic dipole response in Ca isotopes
0 10 20 30 40
10-2
100
102
0 10 20 30 40
B(M
1)↑(µ
2 N)
ω (MeV) ω (MeV)
52Ca48Ca
(a) (b)
main p-h excitation⇒ [(ν1f5/2)(ν1f7/2)−1]
E(1+): 10.23 MeV (EXP) 10.15 MeV (D1S) 8.56 MeV (D1ST)
Charge exchange excitations: 1+ GT and SQ
0
5
10
15
0.0
0.4
0.8
0
5
10
15
20
0
1
2
3
0 10 20 30
0
20
40
60
80
0 10 20 30 40 50 60 70
0
5
10
15
20
D1M
D1MT2c
exp
48Ca(p, n)48Sc 48Ca(p, n)48Sc
90Zr(p, n)90Nb 90Zr(p, n)90Nb
208Pb(p, n)208Bi 208Pb(p, n)208Bi
ΓGT−
1+(ω)(M
eV−1)
ΓSQ−
1+(ω)(×
100MeV
−1fm
4)
ω (MeV) ω (MeV)
(a)
(b)
(c)
(d)
(e)
(f)
V. De Donno et al., Phys. Rev. C90, 024326 (2014)
Splitting in 40Ca, 36S and 34Si: N = 20 isotones
Tensor induced and pure spin-orbit effects
From 40Ca to 36S (tensor) From 36S to 34Si (spin orbit)Splitting D1S D1S
2p 13% 43%D1ST2a D1ST2a
2p 40% 39%D1ST2c D1ST2c
2p 27% 42%
Reductions of the neutron 2p splitting.
M. Grasso and M. Anguiano, Phys. Rev. C92, 054216 (2015)
Interplay between tensor force and pairing correlations
0
3
6
9
12
114 120 126 132 138 144 150 156
0
2
4
6
8
〈(∆N)2〉 p
〈(∆N)2〉 p
HFB
HF+BCS
HFB
HF+BCS
A
N = 82
(a)
(b)
D1S
D1ST2a
Pairing field in HFB: VCentral + VSO + VCoul.
Pairing field in BCS: VCentral
Interplay between tensor force and pairing correlations
0.0
0.2
0.4
0.6
0.8
1.0
134 140 146 152
-10
-8
-6
-4
v2
ǫ(M
eV)
N = 82
A
(a)
(b)
1g7/2
2d5/2
1g7/2
2d5/2
D1S
D1ST2aProton Levels
Tensor force?
neutron 1h11/2
A new fit for the tensor interaction
Vtensor(~r1,~r2) = VT S12 e−(r1−r2)2/µ2
T + VTτ τ (1) · τ (2) S12 e−(r1−r2)2/µ2
Tτ
1. A more general fit, with five parameters
WLS VT VTτ µT µTτ
2. How? Using the procedure following to obtain D1ST2c, andadding two observables more.
Energy of the first 0− state of 16O and 1+ GT of 48Ca
