Davesne (PDF)

Skyrme N2LO
functionals: first results
on finite nuclei

Skyrme N2LO functionals: first results on
finite nuclei

D. Davesne, P. Becker,
A. Pastore, J. Navarro
Introduction
Infinite matter
calculations

D. Davesne, P. Becker, A. Pastore, J. Navarro

Application to
astrophysics LYVA1
Application to spherical
nuclei

Orsay, October 2017

First results
Conclusion and
perspectives

n

N2LO/N3LO extensions : physical motivation

n

Results in infinite matter

n

Extension of Gogny interaction

n

Application in astrophysics

n

Application to finite nuclei: first results

n

Conclusion

Linear response
formalism

N2LO/N3LO extensions : physical motivation

Skyrme N2LO
functionals: first results
on finite nuclei
D. Davesne, P. Becker,
A. Pastore, J. Navarro
Introduction
Infinite matter
calculations

n Construction of new effective interactions necessary!

Application to
astrophysics LYVA1

n Instabilities experienced with popular interactions (Skyrme,

Application to spherical
nuclei

Gogny)
n Initial idea (Skyrme) : expansion in powers of momentum (k2 )

→ systematic expansion up to kn ... which n???
n N2LO : n = 2 ; N3LO : n = 3 ; ...
2 2
n Gogny: e−r /µ , M3Y : e−µr /µr, ... : SAME kind of expansion

[F. Raimondi et al., Phys.Rev. C84 (2011) 064303]

First results
Conclusion and
perspectives
Linear response
formalism

N2LO/N3LO extensions : physical motivation

Skyrme N2LO
functionals: first results
on finite nuclei
D. Davesne, P. Becker,
A. Pastore, J. Navarro

Finite-range interaction D1S: infinite sum of partial waves.
Introduction

80

E/A [MeV]

60

Infinite matter
calculations

D1S
S-wave
S+P-wave
S+P+D-wave
S+P+D+F-wave

Application to
astrophysics LYVA1
Application to spherical
nuclei
First results

40

Conclusion and
perspectives
Linear response
formalism

20
0
-20
0

0.1

0.2

0.3

0.4

0.5

-3

ρ [fm ]

Only S, P, D and F (` < 4) waves necessary → N3LO good enough

Skyrme pseudo-potential N2LO/N3LO

Skyrme N2LO
functionals: first results
on finite nuclei
D. Davesne, P. Becker,
A. Pastore, J. Navarro
Introduction


1



V(r1 , r2 ) = t0 (1 + x0 Pσ ) + t3 (1 + x3 Pσ )ρα (R)


6
Skyrme (N1LO)



i
h 0

0
1

2
2
+ t1 (1 + x1 Pσ ) k + k + t2 (1 + x2 Pσ ) k · k
2

h
i
1



+ t1(4) (1 + x1(4) Pσ ) (k2 + k0 2 )2 + 4(k0 · k)2


4
Skyrme N2LO




(4)
(4)
0
2
02

+ t2 (1 + x2 Pσ )(k · k)(k + k )
i
 0
h 0

1


+ t1(6) 1 + x1(6) Pσ (k 2 + k2 ) (k 2 + k2 )2 + 12(k0 · k)2 


2
Skyrme N3LO




h
i

0
0
02
(6)
(6)

2 2
2

+ t2 1 + x2 Pσ (k · k) 3(k + k ) + 4(k · k)

n D and F partial waves included
n Gauge invariance
n Also includes:
n spin-orbit term W0
n tensor terms

Infinite matter
calculations
Application to
astrophysics LYVA1
Application to spherical
nuclei
First results
Conclusion and
perspectives
Linear response
formalism

Infinite matter: (S , T ) channels N2LO

Skyrme N2LO
functionals: first results
on finite nuclei
D. Davesne, P. Becker,
A. Pastore, J. Navarro

n Used as a preliminary test before dealing with finite nuclei
n First step: (S , T ) channels

Introduction

n Results compared to BHF calculations from Baldo and al.

Infinite matter
calculations
Application to
astrophysics LYVA1

15
10
5
0
-5
-10
-15
0

BHF
SLy5
NLO
N2LO

-5

-5

-10

-10

-15

-15

-20

-20

-25

-25

-30

-30
0

0.1

0.2
ρ [fm-3]

0.3

0

0.1

0.2
ρ [fm-3]

0.3

Application to spherical
nuclei
First results

(S=1,T=1)

15
10
5
0
-5
-10
-15
0

(S=0,T=1)

(S=1,T=0)

(S=0,T=0)

(1997)

Conclusion and
perspectives
Linear response
formalism

Infinite matter: (S , T ) channels N3LO

Skyrme N2LO
functionals: first results
on finite nuclei

40
35
30
25
20
15
10
5
0
0
-5
-10
-15
-20
-25
-30
-35
-40
-45

BHF
N1LO
N2LO
N3LO

0

0.2

0.4

0.6

0.8

0

0.2

ρ [fm-3]

n Agreement up to ρ = 0.8 fm−3
n Exploration of a new parameter space

0.4

0.6

ρ [fm-3]

0.8

Introduction
Infinite matter
calculations

(S=1,T=1)

40
35
30
25
20
15
10
5
0
0
-5
-10
-15
-20
-25
-30
-35
-40
-45

Application to
astrophysics LYVA1
Application to spherical
nuclei
First results
Conclusion and
perspectives

(S=0,T=1)

(S=1,T=0)

(S=0,T=0)

D. Davesne, P. Becker,
A. Pastore, J. Navarro

Linear response
formalism

Infinite matter: (S , T ) channels M3Y

Skyrme N2LO
functionals: first results
on finite nuclei
D. Davesne, P. Becker,
A. Pastore, J. Navarro
Introduction

30

Infinite matter
calculations

20

20

10

10

0

0

0

0

(S=1,T=1)

40

30

-10

-10

-20

-20

-30

-30

-40
0

0.2

0.4

0.6

0
0.2
-3
ρ [fm ]

0.4

0.6

-40
0.8

n M3Y takes into account nuclei and (S , T ) channels: both are not

incompatibles

Application to
astrophysics LYVA1
Application to spherical
nuclei

(S=0,T=1)

(S=1,T=0)

(S=0,T=0)

M3Y
40

First results
Conclusion and
perspectives
Linear response
formalism

Infinite matter: (S , T ) channels Gogny

Skyrme N2LO
functionals: first results
on finite nuclei
D. Davesne, P. Becker,
A. Pastore, J. Navarro

Not possible...

Introduction

20

3 Gaussians
D1M-fit
D1M

30
20
10

10

(S=1,T=1)

(S=0,T=0)

30

Infinite matter
calculations

40

40

-10

-20

-20

-30

-30

-40
0

0.2

0.4

0.6

... except with a third gaussian

0
0.2
-3
ρ [fm ]

0.4

0.6

-40
0.8

(S=0,T=1)

(S=1,T=0)

0

-10

Application to spherical
nuclei
First results
Conclusion and
perspectives

0
0
0

Application to
astrophysics LYVA1

Linear response
formalism

Determination of the three ranges

Skyrme N2LO
functionals: first results
on finite nuclei
D. Davesne, P. Becker,
A. Pastore, J. Navarro

Physical meaning of a range :
n Yukawa potential: related to masses (770, 490, 140 MeV)
n Gaussian potential??? → definition via the self-energy

Introduction
Infinite matter
calculations
Application to
astrophysics LYVA1
Application to spherical
nuclei
First results
Conclusion and
perspectives

0.25

0.2

Example:

RF/H Yukawa

mρ = 770 MeV
→ µ−1
Y =0.256 fm

RF/H Gogny

RF/H

0.15

0.1

0.05

0
0

0.5

1

1.5

2

2.5

3

→ R(µ−1
Y )=0.228 =R(µG )
→ µG = 0.475 fm

x

→ ranges: µ1 = 0.475 fm, µ2 = 0.746 fm , µ3 = 1.964 fm

Linear response
formalism