Response functions with
Landau residual
interaction
D. Davesne
Introduction
Nuclear matter response functions : principles and
applications with Landau residual interaction
Linear response
Instabilities
Applications
Landau residual
interaction
D. Davesne, A. Pastore, J. Navarro
Université de Lyon, F-69003 Lyon, France,
Université Claude Bernard Lyon 1;
Institut de Physique Nucléaire de Lyon, CNRS/IN2P3,
F-69622 Villeurbanne cedex, France.
Response functions with
Landau residual
interaction
Outline
D. Davesne
Introduction
n Spin-isospin response functions
n Formalism
n Example of the free response function
n Instabilities
n Applications to Landau residual interaction
n
n
n
n
Landau parameters
Quality of approximation
Convergence with `
Estimation of N3LO interaction parameters
n Conclusion
Linear response
Instabilities
Applications
Landau residual
interaction
Response functions with
Landau residual
interaction
Skyrme pseudo potential
Veff = t0 1 + x0 P̂σ δ + t3 1 + x3 P̂σ ρα0 δ
+ 12 t1 1 + x1 P̂σ k02 δ + δk2 + t2 1 + x2 P̂σ k0 δ · k
+ i W0 (σ1 + σ2 ) · k0 δ × k
nh
o
i
+ 21 te 3 σ1 · k0 σ2 · k0 − (σ1 · σ2 ) k02 δ + c.c.
+ 12 to 3 σ1 · k0 δ (σ2 · k) − (σ1 · σ2 ) k0 δ · k + c.c.
D. Davesne
local
Introduction
non local
Linear response
Instabilities
spin-orbit
Applications
tensor
Landau residual
interaction
Skyrme functional (local densities)
ESk
=
XZ
n ρ j
∆ρ
d3 r Ct ρ0 ρ2t + Ct ρt ∆ρt + Ctτ ρt τt + Ct j2t + Cts ρ0 s2t
t=0,1
+Ct∇s (∇ · st )2 + Ct∆s st · ∆st + CtT st · Tt + CtF st · Ft + Ct∇J ρt ∇ · Jt
(0)
(1)
∇j
2
J (2)
+Ct st · (∇ × jt ) + CtJ (Jt(0) )2 + CtJ (J(1)
t ) + Ct
[E . Perlinska et al. Phys. Rev C 69, 014316 (2004)) ]
The coupling constants are fitted on data.
z
X
µν=x




(2) (2) 
Jtµν
Jtµν 



Response function - Formalism
Response functions with
Landau residual
interaction
D. Davesne
We act with an external field
X
j
j
expiqr Θα Θα = 1, σj , τ̂j , σj τ̂j
j
The RPA propagator is the solution of Bethe-Salpeter equation :
(SMI)
GHF (q, ω, k1 )
X Z d3 k
(S’M’I’)
2 (SMI;S’M’I’)
+GHF (q, ω, k1 )
V
(q, k1 , k2 )GRPA
(q, ω, k2 )
(2π)3 ph
(S’M’I’)
0
Linear response
Instabilities
Applications
[J. Navarro et al. , Ann. Phys. 214, 293-340, 1992 ]
GRPA (q, ω, k1 )
Introduction
=
(α,α )
−1
0
with : Vph
(q, k1 , k2 ) = hq + k1 , k−1
1 , (α)|Veff |q + k2 , k2 , (α )i
Landau residual
interaction
Response functions with
Landau residual
interaction
Formalism
D. Davesne
Introduction
Linear response
G1,M,I
RPA (q, ω, k1 )
=
GHF (q, ω, k1 )
Instabilities
Z
d3 k2
(2π)3

X

∗
W (1,I) + W (1,I) [k2 + k2 ] − 2W (1,I) 4π k1 k2
Y1µ
(k̂1 )Y1µ (k̂2 )
1
2
2
2
 1
3
+
GHF (q, ω, k1 )
+
X
h
i 16π (1,I)
∗
W4(1,I) k14 + k24 + 2k12 k22 −
W4 (k13 k2 + k1 k23 )
Y1µ
(k̂1 )Y1µ (k̂2 )
3
µ

X

64π2 2 2 (1,I) X ∗
∗
k k W
Y1µ (k̂1 )Y1µ (k̂2 )
Y1ν
(k̂1 )Y1ν (k̂2 ) G1,M,I
RPA (q, ω, k2 )
9 1 2 4
+
µ
+
Z
d3 k2
4qCI∇J M
(2π)3
r
µ
ν
i
4π h
1
1
k1 YM
(k̂1 ) − k2 YM
(k̂2 ) G0,0,I
RPA (q, ω, k2 )
3
n Integration on k1
n Closed system of equations
n Analytical vs numerical solution
Applications
Landau residual
interaction
Response functions with
Landau residual
interaction
Response function - Typical result
n Importance of the tensor on response functions
D. Davesne
n Separation into different channels : (S, M, I)
Introduction
Linear response
Instabilities
Applications
Landau residual
interaction
0.0025
SNM q=kF ρ=ρ0
0.0015
0.001
S
SMI
-1
[MeV fm
-3]
0.002
Uncorrelated
S=0 I=0
S=0 I=1
S=1 M=0 I=0
S=1 M=0 I=1
S=1 M=1 I=0
S=1 M=1 I=1
0.0005
0
0
20
40
60
80
100 120 140 160 180 200
ω [MeV]
Response functions with
Landau residual
interaction
Free response function
D. Davesne
Introduction
Linear response
Instabilities
300
Applications
Landau residual
interaction
ω (MeV)
2
ω = q /2 m
200
100
0
0
1
2
3
q (fm-1)
Excitation of a single nucleon at rest
4
5
Response functions with
Landau residual
interaction
Free response function
D. Davesne
Introduction
Linear response
Instabilities
300
Applications
ω (MeV)
Landau residual
interaction
ω = q2/2 m + q kF/m
200
100
ω = q2/2 m - q kF/m
0
0
1
2
3
4
q (fm-1)
Spreading of the distribution (Fermi momentum)
5
Response functions with
Landau residual
interaction
Free response function
D. Davesne
Introduction
Linear response
Instabilities
Pauli blocking
S (MeV.fm-3)
0.0008
Applications
Landau residual
interaction
0.0004
0
0
50
100
ω (MeV)
q < 2kF
150
200
Response functions with
Landau residual
interaction
Free response function
D. Davesne
Introduction
0.0004
Linear response
Instabilities
Applications
S (MeV.fm-3)
Landau residual
interaction
0.0002
0
0
200
400
600
ω (MeV)
q > 2kF
800
1000
1200
Response function with interactions
Response functions with
Landau residual
interaction
D. Davesne
Introduction
Linear response
Instabilities
Applications
Landau residual
interaction
(S,M,I)
(ω, q) =
χRPA
χHF (ω,q)
1−V(ω,q)χHF (ω,q)
n Collective modes : as soon as 1 − V(ω, q)χHF (ω, q) , 1
n Reshaping : microscopic physical interpretation (π, ρ, ∆)
Response function with interactions
Response functions with
Landau residual
interaction
D. Davesne
(S,M,I)
(ω, q) =
χRPA
χHF (ω,q)
1−V(ω,q)χHF (ω,q)
Introduction
Linear response
n Collective modes : as soon as 1 − V(ω, q)χHF (ω, q) , 1
Instabilities
n Reshaping : microscopic physical interpretation (π, ρ, ∆)
Applications
Landau residual
interaction
Instabilities
An instability in the functional causes an infinity in the response function :
1/χSMI (ω = 0, q) = 0
D. Davesne
Introduction
Linear response
0.6
S=0 M=0 I=0
S=0 M=0 I=1
S=1 M=1 I=0
S=1 M=0 I=1
S=1 M=1 I=1
ρsat=0.16 fm-3
0.5
Instabilities
Applications
Landau residual
interaction
0.4
ρ [fm-3]
Response functions with
Landau residual
interaction
0.3
0.2
0.1
0
0
1
2
q [fm-1]
3
4
Instabilities in the different spin/isospin channels (S,M,I) for T22.
5
Response functions with
Landau residual
interaction
Instabilities
D. Davesne
Introduction
An instability in the functional causes an infinity in the response function :
Linear response
1/χSMI (ω = 0, q) = 0
Instabilities
Applications
0.0012
0.6
S=0 M=0 I=0
S=0 M=0 I=1
S=1 M=1 I=0
S=1 M=0 I=1
S=1 M=1 I=1
-3
ρsat=0.16 fm
0.5
Landau residual
interaction
q = 1.33 fm-1
0.001
0.0008
S (MeV-1fm-2)
ρ [fm-3]
0.4
0.3
0.0006
0.0004
0.2
•
0.0002
0.1
0
0
0
1
2
q [fm-1]
3
4
5
0
50
100
w (MeV)
Instabilities in the different spin/isospin channels (S,M,I) for T22.
150
200
Response functions with
Landau residual
interaction
Instabilities
D. Davesne
Introduction
An instability in the functional causes an infinity in the response function :
Linear response
1/χSMI (ω = 0, q) = 0
Instabilities
Applications
0.0016
0.6
S=0 M=0 I=0
S=0 M=0 I=1
S=1 M=1 I=0
S=1 M=0 I=1
S=1 M=1 I=1
-3
ρsat=0.16 fm
0.5
Landau residual
interaction
q = 1.50 fm-1
0.0014
0.0012
0.001
S (MeV-1fm-2)
ρ [fm-3]
0.4
0.3
0.0008
0.0006
0.0004
0.2
•
0.0002
0.1
0
-0.0002
0
0
1
2
q [fm-1]
3
4
5
0
50
100
w (MeV)
Instabilities in the different spin/isospin channels (S,M,I) for T22.
150
200
Response functions with
Landau residual
interaction
Instabilities
D. Davesne
Introduction
An instability in the functional causes an infinity in the response function :
1/χ
SMI
Linear response
Instabilities
(ω = 0, q) = 0
Applications
Landau residual
interaction
0.004
-1
q = 1.70 fm
0.6
S=0 M=0 I=0
S=0 M=0 I=1
S=1 M=1 I=0
S=1 M=0 I=1
S=1 M=1 I=1
-3
ρsat=0.16 fm
0.5
0.0035
0.003
S (MeV-1fm-2)
ρ [fm-3]
0.4
0.3
0.0025
0.002
0.0015
0.2
0.001
•
0.1
0.0005
0
0
0
1
2
q [fm-1]
3
4
5
0
50
100
w (MeV)
Instabilities in the different spin/isospin channels (S,M,I) for T22.
150
200
Response functions with
Landau residual
interaction
Instabilities
D. Davesne
Introduction
An instability in the functional causes an infinity in the response function :
1/χ
SMI
Linear response
Instabilities
(ω = 0, q) = 0
Applications
Landau residual
interaction
0.025
-1
q = 1.80 fm
0.6
S=0 M=0 I=0
S=0 M=0 I=1
S=1 M=1 I=0
S=1 M=0 I=1
S=1 M=1 I=1
-3
ρsat=0.16 fm
0.5
0.02
S MeV-1fm-2)
ρ [fm-3]
0.4
0.3
0.015
0.01
0.2
•
0.005
0.1
0
0
0
1
2
q [fm-1]
3
4
5
0
50
100
w (MeV)
Instabilities in the different spin/isospin channels (S,M,I) for T22.
150
200
Response functions with
Landau residual
interaction
Instabilities
D. Davesne
Introduction
An instability in the functional causes an infinity in the response function :
1/χ
SMI
Linear response
Instabilities
(ω = 0, q) = 0
Applications
Landau residual
interaction
0.08
-1
q = 1.82 fm
0.6
S=0 M=0 I=0
S=0 M=0 I=1
S=1 M=1 I=0
S=1 M=0 I=1
S=1 M=1 I=1
-3
ρsat=0.16 fm
0.5
0.07
0.06
0.05
S (MeV-1fm-2)
ρ [fm-3]
0.4
0.3
0.04
0.03
0.02
0.2
•
0.01
0.1
0
-0.01
0
0
1
2
q [fm-1]
3
4
5
0
50
100
w (MeV)
Instabilities in the different spin/isospin channels (S,M,I) for T22.
150
200
Response functions with
Landau residual
interaction
Sum rules
The energy-weighted sum-rules :
Mp(α) (q)/N = −
D. Davesne
1
πρ
+∞
Z
dωωp Imχα (ω, q)
0
p=0
ω→0:
χ(α) (ω, q) ≈ −2ρ
+∞
X
(ω)2p M−(2p+1) (q)/N
p=0
Channel (1,0,1)
0.9
integral
formula
integral (tensor)
formula (tensor)
0.8
0.7
M-1 (MeV)
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.5
1
Linear response
Instabilities
+∞
X
χ(α) (ω, q) ≈ 2ρ (ω)−(2p+2) M2p+1 (q)/N
ω→∞:
Introduction
1.5
q (fm-1)
2
2.5
3
Applications
Landau residual
interaction
Response functions with
Landau residual
interaction
Sum rules
The energy-weighted sum-rules :
Mp(α) (q)/N = −
D. Davesne
1
πρ
Z
+∞
Introduction
dωωp Imχα (ω, q)
Linear response
0
Instabilities
ω→∞:
χ
(α)
(ω, q) ≈ 2ρ
Applications
+∞
X
−(2p+2)
(ω)
Landau residual
interaction
M2p+1 (q)/N
p=0
ω→0:
χ(α) (ω, q) ≈ −2ρ
+∞
X
(ω)2p M−(2p+1) (q)/N
p=0
Channel (1,0,1)
0.9
0.6
S=0 M=0 I=0
S=0 M=0 I=1
S=1 M=1 I=0
S=1 M=0 I=1
S=1 M=1 I=1
ρsat=0.16 fm-3
0.5
0.7
0.6
M-1 (MeV)
0.4
ρ [fm-3]
integral
formula
integral (tensor)
formula (tensor)
0.8
0.3
0.5
0.4
0.3
0.2
0.2
0.1
0.1
0
0
0
1
2
q [fm-1]
3
4
5
0
0.5
1
1.5
q (fm-1)
2
2.5
3
Response functions with
Landau residual
interaction
A new fitting protocol!!!
Lyon-Saclay protocol improved : linear response included
D. Davesne
Introduction
n We add the constraint from IM at each iteration
Linear response
n No extra CPU time required!
Instabilities
Applications
We compare SLy5 (old) and SLy5* (new) (K. Bennaceur et al.)
Landau residual
interaction
0.5
-3
ρpole [fm ]
0.4
0.3
0.2
S=0 M=0 I=0
S=0 M=0 I=1
S=1 M=0 I=0
S=1 M=0 I=1
S=1 M=1 I=0
S=1 M=1 I=1
0.1
0
S=0 M=0 I=0
S=0 M=0 I=1
S=1 M=0 I=0
S=1 M=0 I=1
S=1 M=1 I=0
S=1 M=1 I=1
SLy5
1
2
-1
q [fm ]
3
SLy5*
4
0
1
2
-1
q [fm ]
3
4
By construction the new functional is now stable in the spin channel!
n Problem of transferability (nucleus = finite system!) (cf V. Hellemans,
P.H. Heenen, T. Duguet, A. Pastore, M. Bender, K. Bennaceur, D.D.,
ArXiv 2013)
Response functions with
Landau residual
interaction
Applications of this approach
D. Davesne
Introduction
Different possibilities
Linear response
n Can be used for different systems (neutron matter) : done (even at finite
Landau residual
interaction
n Can be used with different Skyrme potentials : density dependent
terms : S. Goriely, N. Chamel, M. Pearson, M. Martini, A. Pastore, J.
Navarro, D.D., to be submitted).
1
0.8
1
S=0 M=0 I=0
S=0 M=0 I=1
S=1 M=0 I=0
S=1 M=0 I=1
S=1 M=1 I=0
S=1 M=1 I=1
ρsat [fm-3]
1.6*ρsat [fm-3]
BSK17
0.8
S=0 M=0 I=0
S=0 M=0 I=1
S=1 M=0 I=0
S=1 M=0 I=1
S=1 M=1 I=0
S=1 M=1 I=1
ρsat [fm-3]
1.6*ρsat [fm-3]
BSK21
ρ [fm-3]
0.6
ρ [fm-3]
0.6
0.4
0.4
0.2
0.2
0
0
0
0.5
1
1.5
2
q [fm-1]
2.5
3
3.5
4
0
0.5
1
1.5
2
q [fm-1]
2.5
Instabilities
Applications
temperature : A. Pastore, J. Navarro., D.D., to be submitted)
3
3.5
4
Applications of linear response formalism : higher order terms
Response functions with
Landau residual
interaction
D. Davesne
Introduction
(F. Raimondi, B.G. Carlsson, J. Dobaczewski)
Linear response
Instabilities
Applications
(4)
V̂Sk
=
+
+
+
h
i
1 (4)
t1 (1 + x1(4) Pσ ) (k2 + k0 2 )2 + 4(k0 · k)2
4
t2(4) (1 + x2(4) Pσ )(k · k0 )(k2 + k0 2 )
i
h
te(4) (k2 + k0 2 )Te (k0 , k) + 2(k · k0 )To (k0 , k)
"
#
1
to(4) 5(k · k0 )Te (k0 , k) − (k2 + k0 2 )To (k0 , k) ,
2
where
Te (k0 , k)
=
~ 2 ),(1)
3(~
σ1 · k0 )(~
σ2 · k0 ) + 3(~
σ1 · k)(~
σ2 · k) − (k0 2 + k2 )(~
σ1 · σ
To (k , k)
=
~ 2 ), (2)
3(~
σ1 · k0 )(~
σ2 · k) + 3(~
σ1 · k)(~
σ2 · k0 ) − 2(k0 · k)(~
σ1 · σ
0
Landau residual
interaction
Applications of linear response formalism : higher order terms
(F. Raimondi, B.G. Carlsson, J. Dobaczewski)
Response functions with
Landau residual
interaction
D. Davesne
Introduction
(4)
V̂Sk
=
+
+
+
(6)
V̂Sk
=
+
+
+
h
i
1 (4)
t1 (1 + x1(4) Pσ ) (k2 + k0 2 )2 + 4(k0 · k)2
4
t2(4) (1 + x2(4) Pσ )(k · k0 )(k2 + k0 2 )
h
i
te(4) (k2 + k0 2 )Te (k0 , k) + 2(k · k0 )To (k0 , k)
#
"
1
to(4) 5(k · k0 )Te (k0 , k) − (k2 + k0 2 )To (k0 , k) ,
2
t1(6)
h
i
(1 + x1(6) Pσ )(k2 + k0 2 ) (k2 + k0 2 )2 + 12(k0 · k)2
2
h
i
t2(6) (1 + x2(6) Pσ )(k · k0 ) 3(k2 + k0 2 )2 + 4(kk0 )2
 2


2

 (k + k0 )2

(6) 
0
2
02
0
0
0 2

t e 
+ (k · k )  Te (k , k) + (k + k )(k · k )To (k , k)
4


 2
 (k + k0 2 )2


+ (k · k0 )2  To (k0 , k) + (k2 + k0 2 )(k · k0 )Te (k0 , k)
to(6) 
4
New parameters are constrained : contribution to EoS, Landau parameters, ...
Linear response
Instabilities
Applications
Landau residual
interaction
Applications of linear response formalism : higher order terms
Response functions with
Landau residual
interaction
D. Davesne
Introduction
Linear response
n Energy per particle
Instabilities
(E/A)(4)
=
(E/A)(6)
=
i
9 h (4)
3t + (5 + 4x2(4) )t2(4) ρkF4
280 1
i
2 h (6)
3t1 + (5 + 4x2(6) )t2(6) ρkF6
15
n Landau parameters
n Contribution up to ` = 3 for central terms and ` = 2 for tensor terms
n Important because of universality of Landau parameters : can be
deduced from ANY interaction!!!
n First step towards a complete treatment of N3LO : response with
Landau residual interaction :
X
h
i
h
i k2
Vph =
S12 P`
f` +f`0 (τ1 ·τ2 )+ g` + g0` (τ1 · τ2 ) (σ1 ·σ2 )+ h` + h0` (τ1 · τ2 ) 12
2
kF
`
where S12 ≡ 3(k̂12 · σ1 )(k̂12 · σ2 ) − (σ1 · σ2 ) and P` ≡ P` (k̂1 · k̂2 ).
Applications
Landau residual
interaction
Response with Landau residual interaction
Response functions with
Landau residual
interaction
D. Davesne
Introduction
Linear response
Instabilities
Main objectives :
Applications
Landau residual
interaction
n Obtain response functions up to ` = 3 for central terms and ` = 2 for
tensor terms
n Study the convergence with `
n (Astro)physical interest
n Validity of Landau approximation??? (q = 0 partially relaxed!)
n Possibility to determine an order of magnitude for N3LO parameters?
(impossible for initial Skyrme interaction but...)
Response functions with
Landau residual
interaction
Response with Landau response functions
D. Davesne
Introduction
Linear response
Validity of Landau approximation (pure neutron matter, ab initio
calculations) :
Instabilities
Applications
Landau residual
interaction
0.005
(a)
-3
(b)
0.004
-1
0.004
-Im χ(ν)/π [MeV fm ]
Landau
RPA, k=0.1
RPA, k=0.5
-1
-3
-Im χ(ν)/π [MeV fm ]
0.005
0.003
0.002
0.001
0
0.4
0.8
ν
1.2
1.6
2
0.003
0.002
0.001
0
0.4
Important reshaping of response functions for q = kF .
0.8
ν
1.2
1.6
2
Response with Landau response functions
Convergence with ` (SNM, D1ST, ρ = 0.16 fm−3 and q/kF = 0.1) :
Response functions with
Landau residual
interaction
D. Davesne
Introduction
Linear response
Instabilities
0.002
a)
HF
lC=0
lC=1; lT=0
b)
c)
lC=2; lT=1
lC=3; lT=2
d)
-1
-3
-Imχ(q,ω)/π [MeV fm ]
0.003
0.001
0.003
0.002
0.001
0.003
0.002
e)
f)
0.001
0
5
10
0
5
10
ω [MeV]
Convergence of response functions reached with `C = 1 and `T = 0 excepted
for (S, M, I) = (1, 0, 0) and (1, 1, 0).
Applications
Landau residual
interaction
Response with Landau response functions
Convergence with ` (SNM, D1ST, ρ = 0.16 fm−3 and q/kF = 0.5) :
Response functions with
Landau residual
interaction
D. Davesne
Introduction
Linear response
Instabilities
0.002
Applications
a)
b)
c)
d)
e)
f)
Landau residual
interaction
-1
-3
-Imχ(q,ω)/π [MeV fm ]
0.003
0.001
0.003
0.002
0.001
0.003
0.002
0.001
0
20
40
60
20
40
60
ω [MeV]
Convergence of response functions reached with `C = 1 and `T = 0 excepted
for (S, M, I) = (1, 0, 0) and (1, 1, 0).
Response with Landau response functions
Convergence with ` (SNM, M3Y P2, ρ = 0.16 fm−3 and q/kF = 0.5) :
Response functions with
Landau residual
interaction
D. Davesne
Introduction
Linear response
Instabilities
0.002
Applications
a)
b)
c)
d)
e)
f)
Landau residual
interaction
-1
-3
-Imχ(q,ω)/π [MeV fm ]
0.003
0.001
0.003
0.002
0.001
0.003
0.002
0.001
0
20
40
60
20
40
60
ω [MeV]
Convergence of response functions reached with `C = 1 and `T = 0 excepted
for (S, M, I) = (1, 0, 0) and (1, 1, 0).
Response functions with
Landau residual
interaction
Response with Landau response functions
D. Davesne
Introduction
Linear response
Order of magnitude for N3LO parameters from Landau parameters :
Instabilities
Applications
n From Gogny D1ST :
Order
Skyrme
N2LO
N3LO
t0
-1177
-1600
-1746
t1
368
739
908
t2
-220
259
632
t3
5478
7340
8057
Landau residual
interaction
(4)
t1
0
-103
-221
(4)
t2
0
-133
-393
(6)
t1
0
0
5.5
(6)
t2
0
0
12
n From Nakada M3Y P2 :
Order
Skyrme
N2LO
N3LO
t0
-1342
-1627
-1734
t1
330
702
993
t2
122
370
673
t3
6350
7414
7713
(4)
t1
0
-103
-305
(4)
t2
0
-69
-279
(6)
t1
0
0
9.3
(6)
t2
0
0
9.7
Response functions with
Landau residual
interaction
Response with Landau response functions
D. Davesne
Introduction
Linear response
Instabilities
Order of magnitude for N3LO parameters from Landau parameters :
Applications
n From Gogny D1ST :
Order
Skyrme
N2LO
N3LO
t0
-1177
-1600
-1746
t1
368
739
908
t2
-220
259
632
t3
5478
7340
8057
Landau residual
interaction
(4)
t1
0
-103
-221
(4)
t2
0
-133
-393
(6)
t1
0
0
5.5
(6)
t2
0
0
12
ρsat
0.165
0.168
0.17
(E/A)ρsat
-9.4
-14.9
-16.9
K∞
178
185
213
n From Nakada M3Y P2 :
Order
Skyrme
N2LO
N3LO
t0
-1342
-1627
-1734
t1
330
702
993
t2
122
370
673
t3
6350
7414
7713
(4)
t1
0
-103
-305
(4)
t2
0
-69
-279
(6)
t1
0
0
9.3
(6)
t2
0
0
9.7
ρsat
0.164
0.162
0.164
(E/A)ρsat
-12.4
-15.5
-16.12
K∞
218
208
219
Response functions with
Landau residual
interaction
Response with Landau response functions
D. Davesne
Order of magnitude for N3LO parameters from Landau parameters :
Introduction
n From Gogny D1ST :
Linear response
Instabilities
Order
Skyrme
N2LO
N3LO
t0
-1177
-1600
-1746
t1
368
739
908
t2
-220
259
632
t3
5478
7340
8057
(4)
t1
0
-103
-221
(4)
t2
0
-133
-393
(6)
t1
0
0
5.5
(6)
t2
0
0
12
ρsat
0.165
0.168
0.17
(E/A)ρsat
-9.4
-14.9
-16.9
K∞
178
185
213
n From Nakada M3Y P2 :
Order
Skyrme
N2LO
N3LO
t0
-1342
-1627
-1734
t1
330
702
993
t2
122
370
673
t3
6350
7414
7713
(4)
t1
0
-103
-305
(4)
t2
0
-69
-279
(6)
t1
0
0
9.3
(6)
t2
0
0
9.7
ρsat
0.164
0.162
0.164
(E/A)ρsat
-12.4
-15.5
-16.12
Important points :
n saturation obtained!!!
n PNM EoS above SNM EoS
n t0 , t1 , t2 , t3 modified when higher order included
(2)
(4)
(6)
n ti : ti : ti = 100 : 10 : 1
n ρsat , (E/A)ρsat and K∞ converge towards a reasonable value!
K∞
218
208
219
Applications
Landau residual
interaction
Collaborators
Response functions with
Landau residual
interaction
D. Davesne
Collaborators "from Lyon" :
A. Pastore, M. Martini, J. Meyer, R. Jodon, K. Bennaceur...
... and many others from other labs!!!
Introduction
Linear response
Instabilities
Applications
Landau residual
interaction
Special thanks to A. Pastore and J. Navarro !!!
Bibliography
n
n
n
n
n
n
n
n
A. Pastore, K. Bennaceur, D. Davesne, J. Meyer, Int.J.Mod.Phys. E21,1250040, (2012)
A. Pastore, D. Davesne, Y. Lallouet, M. Martini, K. Bennaceur, J. Meyer, Phys.Rev. C85, 054317, (2012)
A. Pastore, M. Martini, V. Buridon, D. Davesne, K. Bennaceur, J. Meyer, Phys.Rev. C86, 044308, (2012)
A. Pastore, D. Davesne, K. Bennaceur, J. Meyer, V. Hellemans arXiv:1210.7937(2012)
A. Pastore, T. Duguet, K. Bennaceur, D. Davesne, J. Meyer, V. Hellemans, M. Bender and P.-H. Heenen
submitted
A. Pastore, J. Navarro, D. D. : Response functions from Laudau residual interaction to be submitted
A. Pastore, J. Navarro, D. D. : Response functions for asymmetric nuclear matter to be submitted
A. Pastore, J. Navarro, D. D. : papers in preparation!!!