Tensor Force, Rearrangement & Symmetry Energy

“3rd International Symposium on Nuclear Symmetry Energy” NSCL/FRIB, East Lansing, Michigan. July 22-26, 2013

Isaac Vidaña CFC, University of Coimbra

I will show that …

ü  Tensor force plays a crucial role for both Esym(ρ) & L(ρ)

ü  Negligible contribution of rearrangement term to Esym(ρ) & L(ρ)

In preparation…

based on: Phys. Rev. C 84, 062801 ( R) (2011)

using: ü  BHF with Av18 + UIX

ü  Hellmann-Feynman Theorem

Symmetry Energy & Tensor Force

“Analysis of the contribution of the different terms of the NN force to Esym and L”

In collaboration with:

C. Providência A. Polls

BHF approximation of ANM

"   Bethe-Goldstone Equation Partial sumation of pp ladder diagrams

ü  Pauli blocking ü  Nucleon dressing

"   Energy per particle

Infinite sumation of two-hole line diagrams

€

EA

(ρ,β) =1A

2k 2

2mτ

+12

Re Uτ ( k )[ ]

%

& '

(

) *

k≤kFτ

∑τ

∑

€

G ω( ) =V +V Qω − E − E ' + iη

G ω( )

€

Eτ (k) =2k 2

2mτ

+ Re Uτ (k)[ ]

€

Uτ (k) = k k ' G ω = Eτ (k) + Eτ ' (k ')( )

k k '

Ak '≤kFτ '

∑τ '∑

€

Un ~U0 +Usymβ

€

Usym =Un −Up

2β

BHF nucleon mean field in ANM

Symmetry potential

Isospin splitting of mean field in ANM

€

Up ~U0 −Usymβ

G-matrix gives access to in-medium NN cross sections

€

σττ ' =mτ*mτ '

*

16π 242J +14π

Gττ '→ττ 'LL 'SJ 2

, ττ '= nn, pp,npLL 'SJ∑

Brueckner-Hatree-Fock:

ü  gives total energy

ü  does not gives separately neither <T> nor <V> because it does not provide the correlated many-body wave function

But

However

Hellmann-Feynman theorem can be used to calculate <V>

Hellmann-Feynman theorem

€

dEλ

dλ=ψλ

d ˆ H λdλ

ψλ

ψλ ψλ

Proven independently by many-authors: Güttinger (1932), Pauli (1933), Hellmann (1937), Feynman (1939)

§  Writting the nuclear matter Hamiltonian as:

€

ˆ H = ˆ T + ˆ V §  Defining a λ-dependent Hamiltonian:

€

ˆ H λ = ˆ T + λ ˆ V

è

€

ˆ V =ψ ˆ V ψψ ψ

=dEλ

dλ$

% &

'

( ) λ=1

H. Hellmann R. P. Ferynman

ENM ESM Esym L

<T> 53.321 54.294 -0.973 14.896

<V> -34.251 -69.524 35.273 51.604

Total 19.070 -15.230 34.300 66.500

§  Kinetic contribution to

ü  L: smaller than FFG one (~29.2 MeV) Figure from Xu & Li, arXiv:1104.2075v1 (2011)

ü  Esym: very small and negative in contrast to FFG result (~14.4 MeV) è strong isospin dependence of short range NN

correlations

§  Potential contribution to

ü  L: ~78% of the total L

ü  Esym: almost equal to total Esym

Kinetic and Potential energy contributions

(S,T) ENM ESM Esym L

(0,0) 0 5.600 -5.600 -21.457

(0,1) -29.889 -23.064 -6.825 -17.950

(1,0) 0 -49.836 49.836 90.561

(1,1) -4.362 -2.224 -2.138 0.450

<V> -34.251 -69.524 35.273 51.604

Spin-Isospin channel & partial wave decomposition

ENM ESM Esym L

3S1 0 -45.810 45.810 71.855

3D1 0 -0.981 0.981 -3.739

Σrest up to J=8 -34.251 -22.733 -11.518 -16.512

ü  Largest contribution from S=1, T=0 channel

ü  Similar T=1 channel contributions to ENM and ESM which almost cancel out in Esym

ü  Main contribution from 3S1-3D1 p.w. (not present in NM)

In agreement with previous BHF calculations

Vsym ρ( ) = 1A

12

Re Un (k,β =1)!

"#$

%

&'

(

)*

k≤kFn

∑ −1A

12

Re Uτ (k,β = 0)!

"#$

%

&'

(

)*

k≤kFτ

∑τ=n,p∑

Note that in these works:

NM correlation energy SM correlation energy

§  Zuo et al., (1999): Av14 & Paris §  Dieperink et al., (2003): Reid93

≠ VNM( ) ≠ VSM( )

Few words on the NN and NNN forces used …

§  Argonne V18 (Av18) NN potential

€

Vij = Vp (rij )Oijp

p=1,18∑ Oij

p=1,14 = 1,σ i ⋅σ j( ),Sij,

L ⋅S,L2,L2

σ i ⋅σ j( ),

L ⋅S( )

2"#$

%&'⊗ 1, τ i ⋅

τ j( )"# %&

€

Oijp=15,18 = Tij ,

σ i ⋅ σ j( )Tij ,SijTij , τ zi + τ zj( )[ ]

§  Urbana IX (UIX) NNN potential

€

VijkUIX =Vijk

2π +VijkR

€

Vijk2π : Attractive Fujita-Miyazawa force

π

π

€

Vijk2π = A Xij ,X jk{ } τ i ⋅

τ j , τ j ⋅ τ k{ } +

14Xij ,X jk[ ] τ i ⋅

τ j , τ j ⋅ τ k[ ]

%

& '

(

) *

cyclic∑

€

Xij =Y mπ rij( ) σ i ⋅ σ j + T mπ rij( )Sij

€

Y (x) =e−x

x1− ex

2( )

€

T(x) = 1+3x

+3x 2

"

# $

%

& ' e−x

x1− ex

2( )2

€

VijkR = B T 2 rij( )T 2 rjk( )

cyclic∑

€

VijkR : Repulsine & Phenomenological

€

UNNeff r ij( ) = VUIX r i,

r j , r k( )n r i,

r j , r k( )d3 r k∫

Reduced to an effective density-dependent 2BF

Oijp=1,3 =1,

σ i ⋅σ j( ) τ i ⋅

τ j( ),Sij

τ i ⋅τ j( )

Contributions from different terms of the NN force

ü  Largest contribution from tensor components

ü  Contributions from other terms negligible (e.g. charge symmetry b re a k i n g t e r m s ) o r almost cancel out

§  Esym: 36.056 (Total: 34.4)

§  L: 69.968 (Total: 66.5)

In agreement with other calculations …

§  Sammarruca (2011): DBHF

Large contribution of π to Esym

§  A. Li & B.A. Li (2011):

Vsym ≈12E

Vtensor2 (r)

Brown & Machleidt, PRC 50, 1731 (1993)

Summarizing …

§  Main contribution to Esym and L from < V >. Kinetic energy contribution < T > to Esym very small and negative.

§  Partial wave & spin-isospin channel decompositon of Esym & L è major contribution from (S=1, T=0) channel.

§  Dominant effect of the tensor force: critical role in the determination of Esym(ρ).

Symmetry Energy & Rearrangement

using again:

ü  BHF with Av18 + UIX

“Analysis of the contribution of the rearrangement term to Esym and L”

Hugenholtz-Van Hove theorem

N. M. Hugenholtz L. Van Hove

EA(ρ)+ ρ

∂ E / A( )∂ρ

Ω

=∂E∂A Ω

= εF (ρ) è EAρ0( ) = εF (ρ0 )

at saturation

§  Original HvH theorem

§  Generalized HvH theorem

EA(ρ)+ ρ

∂ E / A( )∂ρ

Ω

= xnεn + xpεp =12(1+β)εn + (1−β)εp$% &'

Satpathy & Nayak, PRL 51, 1243 (1983)

Powerful tool to check thermodynamical consistency

It is well known that BHF:

ü  Violates HvH theorem by about 20 MeV

BHF & Hugenholtz-Van Hove theorem

ü  T h e r m o d y n a m i c a l consistency restored b y i n c l u d i n g r e a r r a n g e m e n t contribution to the single-particle energy

Rearrangement contribution of the s.p. energy

E = nτ i2ki

2

2m+

12Ω

nτ 'in τ '' jτ 'iτ '' j G τ 'iτ '' j

Aτ 'iτ '' j∑

τ i∑

§  BHF total energy

§  Single-particle energy BHF potential (UBHF,τ(k))

Rearrangement (Urearr,τ(k))

(infinite # of terms) +

12Ω

nτ 'iτ 'iτ '' j∑ nτ '' j

δ τ 'iτ '' j G τ 'iτ '' jA

δnτk

ετk =δEδnτk

=2k2

2m+

1Ω

nτ 'i τ 'iτk G τ 'iτkA

τ 'i∑

we evaluate the s.p. energy including Urearr at the Fermi surface

δδnτkFτ

~ ∂∂kFτ

∂kFτ∂ρτ

δρτδnτkFτ

=kFτ3ρτ

1Ω

∂∂kFτ

Using:

ετkFτ ~2kFτ

2

2m+

1Ω

nτ 'i τ 'iτkFτ G τ 'iτkFτ Aτ 'i∑

+kFτ3ρτ

12Ω2 nτ 'i

τ 'iτ '' j∑ nτ '' j

∂ τ 'iτ '' j G τ 'iτ '' jA

∂kFτ

Urearr,τ(kFτ)

UBHF,τ(kFτ)

Identifying ετkτ with the Fermi energy the HvH theorem is fulfilled

Rearrangement & Symmetry Energy Using:

Δµ = µn −µp =2ρ∂(E /Ω)∂β

EΩ

ρ( ) = eo(ρ)+ esym (ρ)β 2

è Δµ =4ρβesym (ρ) = 4βEsym (ρ)

Taking: µτ ≡ ετkFτ =2kFτ

2

2m+UBHF (kFτ )+Urearr (kFτ )

Esym ρ( ) = 14β

2kFn2

2m−2kFp

2

2m

"

#$$

%

&''+ UBHF

n (kFn )−UBHFp (kFp )( )+ Urearr

n (kFn )−Urearrp (kFp )( )

(

)**

+

,--

Free Fermi Gas BHF Rearrangement

Rearrangement contribution to the single-particle potential

ü  Almost constant w i t h i s o s p i n asymmetry

ü  Very similar for n e u t r o n s & protons

ρ0 ρ0 FFG UBHF Urearr Total 2BF Esym 0.240 16.7 19.5 -0.4 35.8

L 33.5 25.3 4.3 63.1 2+3BF Esym 0.187 14.4 20.6 -0.7 34.3

L 29.2 35.8 1.5 66.5

Rearrangement contribution to

Esym& L

ü  Negligible contribution to both Esym & L

Summarizing …

§  Rearrangement contribution to neutron & proton s.p. potentials:

§  Negligible contribution of rearrangement term to both Esym(ρ) & L(ρ)

ü  Almost constant with β ü  Similar for neutrons and protons