UNEDF meetingThe nuclear Energy Density Functional approach

NSCL/MSUOctober 12-13 2006

Standard Skyrme functional. I

New density/momentum dependent terms, tensor, ...

1. Witek - general introduction, tensor, time-odd, density dependence, surface effects…

2. Alex - tensor3. Thomas- some limitations of Skyrme4. Scott - the DME at the HF level starting from mow-momentum

interactions

Towards the Universal Nuclear Energy Density Functional

ρ0

r r ( )= ρ0

r r ,

r r ( )= ρ

r r στ;

r r στ( )

στ∑ isoscalar (T=0) density ρ0 = ρn + ρp( )

ρ1

r r ( )= ρ1

r r ,

r r ( )= ρ

r r στ;

r r στ( )

στ∑ τ isovector (T=1) density ρ1 = ρn − ρp( )

v s 1r r ( )= ρ

r r στ;

r r σ 'τ( )

σσ 'τ∑ σσ 'σ τ

v s 0r r ( )= ρ

r r στ;

r r σ 'τ( )

isovector spin densityσσ 'τ∑ σσ 'σ

r j T

r r ( )=

i2

isoscalar spin density

r ∇ '−

r ∇ ( )ρT

r r ,

r r '( ) r

r '=r r

t J T

r r ( )=

i2

r ∇ '−

r ∇ ( )⊗ v s T

r r ,

r r '( ) r

r '=r r

τT

r r ( )=

r ∇ ⋅

r ∇ 'ρT

r r ,

r r '( ) r

r '=r r

r T T

r r ( )=

r ∇ ⋅

r ∇ '

r s T

r r ,

r r '( ) r

r '=r r

current density

spin-current tensor density

kinetic density

kinetic spin density

HTr r ( )= CT

ρρT2 + CT

s sT2 + CT

ΔρρTΔρT + CTΔsr s T Δ

r s T

+CTτ ρTτT − jT

2( )+ CTT r

s T ⋅r T T −

t J T

2( )+ CT∇J ρT

r ∇ ⋅

r J T +

r s T ⋅

r ∇ ×

r j T( )[ ]

Etot =

h2

2mτ0 + H0

r r ( )+ H1

r r ( )

⎡

⎣ ⎢

⎤

⎦ ⎥ ∫ d3r Total ground-

state HF energy

Local densitiesand currents

+ pairing…

Example: SkyrmeFunctional

Walter Kohn: Nobel Prize in Chemistry in 1998

Construction of the functional:E. Perlinska et al.Phys. Rev. C 69, 014316 (2004)

Justification of the standard Skyrme functional: LGI

V

r r 1′,

r r 2′;

r r 1,

r r 2

⎛ ⎝ ⎜ ⎞

⎠ ⎟

For simplicity, consider non-local, but velocity-independent interaction (and let us disregard for a moment spin and isospin degrees of freedom):

The corresponding HF interaction energy contains both direct term and exchange term:

E int =12

d3r1′∫ d3r2′d3r1d

3r2Vr r 1′,

r r 2′;

r r 1,

r r 2

⎛ ⎝ ⎜ ⎞

⎠ ⎟ ×

ρr r 1,

r r 1′

⎛ ⎝ ⎜ ⎞

⎠ ⎟ ρ

r r 2,

r r 2′

⎛ ⎝ ⎜ ⎞

⎠ ⎟ − ρ

r r 2,

r r 1′

⎛ ⎝ ⎜ ⎞

⎠ ⎟ ρ

r r 1,

r r 2′

⎛ ⎝ ⎜ ⎞

⎠ ⎟ ⎡

⎣ ⎢ ⎤ ⎦ ⎥

Let us now consider a local gauge transformation of the HF wave function:

where φ is a real function. The corresponding density matrix reads:

In general, the interaction energy is not invariant with respect to such transformation. However, it is invariant for local interactions. Now it is easy to check that the following combinations of local densities are gauge invariant:

′ Ψ = exp i φr r j( )

j=1

A

∑⎧ ⎨ ⎪

⎩ ⎪

⎫ ⎬ ⎪

⎭ ⎪ Ψ

′ ρ r r ,

r ′ r ( )= exp i φ

r r ( )− φ

r ′ r ( )[ ]{ }ρ

r r ,

r ′ r ( )

ρTτT − jT2

r s T ⋅

r T T −

t J T

2

ρT

r ∇ ⋅

r J T +

r s T ⋅

r ∇ ×

r j T( )

A simplification; often used

Justification of the standard Skyrme functional: DME

E int =

12

d3rd3 ′ r ∫ Vr r −

r ′ r ( ) ρ

r r ( )ρ r

′ r ( )− ρr r ,

r ′ r ( )ρ r

′ r ,r r ( )[ ]

Another motivation… Let us consider a local two-body interaction The HF interaction energy is:

V (r r −

r ′ r )

ρ

r r ,

r ′ r ( ) ≈ ρ

r q ( )ir s ⋅

r j

r q ( )+

12

s2 τr q ( )−

14

Δ r q ρ

r q ( )

⎡ ⎣ ⎢

⎤ ⎦ ⎥ ,

r q =

r r +

r ′ r

2, r s =

r r −

r ′ r

ρ

r r ,

r ′ r ( )2

≈ ρ2 r q ( )− s2 ρ

r q ( )τ r

q ( )−r j 2

r q ( )−

14

ρr q ( )Δ r

q ρr q ( )

⎡ ⎣ ⎢

⎤ ⎦ ⎥

In practice, the density matrix is strongly peaked around r=r’ (cf. TF expression!). Therefore, one can expand it around the mid-point:

The Skyrme functional was justified in such a way in, e.g., •Negele and Vautherin, Phys. Rev. C5, 1472 (1972); Phys. Rev. C11, 1031 (1975)•Campi and Bouyssy, Phys. Lett. 73B, 263 (1978)

However, the parameters derived in such a way do not reproduce the nuclear bulk properties precisely enough. Hence, the density matrix expansion should be used as a guiding principle, but the ectual parameters should be adjusted phenomenologically.

However,the Skyrme functional does not have to be related to any given effective two-body force!Actually, many currently used nuclear energy functionals are not related to a force.

The origin of SO splitting can be attributed to 2-body SO and tensor forces, and 3-body force

R.R. Scheerbaum, Phys. Lett. B61, 151 (1976); B63, 381 (1976); Nucl. Phys. A257, 77 (1976); D.W.L. Sprung, Nucl. Phys. A182, 97 (1972); C.W. Wong, Nucl. Phys. A108, 481 (1968)

The maximum effect is in spin-unsaturated systems

Discussed in the context of mean field models:Fl. Stancu, et al., Phys. Lett. 68B, 108 (1977); M. Ploszajczakand M.E. Faber, Z. Phys. A299, 119 (1981); J. Dudek, WN, and T. Werner, Nucl. Phys. A341, 253 (1980); J. Dobaczewski, nucl-th/0604043

and the nuclear shell model:T. Otsuka et al., Phys. Rev. Lett. 87, 082502 (2001); Phys. Rev. Lett. 95, 232502 (2005)

Spin-Orbit and Tensor Force(among many possibilities)

j<εF

2, 8, 20

j>Spin-saturated systems

j<

j>εF

28, 50, 82, 126

Spin-unsaturated systems

J. Dobaczewski, nucl-th/0604043

acts in s and d states ofrelative motion

acts in p states

SO densities(strongly depend on shell filling)

T. Otsuka et al. Phys. Rev. Lett 87, 082502 (2001)

J. Dobaczewski, nucl-th/0604043

SM, Gogny

Skyrme-DFT

However: it is not trivial to relate theoretical s.p. energies to experiment.

J. Dobaczewski and J. Dudek, Phys. Rev. C52, 1827 (1995)M. Bender et al., Phys. Rev. C65, 054322 (2002).

very poorly determinedvery poorly determined Can be adjusted to the Landau parameters

•Important for all I>0 states (including low-spin states in odd-A and odd-odd nuclei)

•Important for terminating (maximally aligned) states

•Impact beta decay•Influence mass filters (including odd-even mass difference)•Limited experimental data available

G. Stoitcheva et al., Phys. Rev. C73, 061304(R) (2006)

Density dependence

Momentum dependence of effective mass

Coulomb correlation energy

Pairing term

Can dynamics be incorporated directly into the functional?

Microscopic LDM and Droplet Model Coefficients: PRC 73, 014309 (2006)

Collective potential V(q)

Universal nuclear energy density functional is yet to be developed

Surface symmetry energy crucial

Different deformabilities!

Different deformabilities!

P.H. Heenen et al., Phys. Rev. C57, 1719 (1998)

Shell effects in metastableminima seem to be under control.

Important data needed to fixthe deformability of the NEDF:

• absolute energies of SD states• absolute energies of HD states

Advantages:

• large elongations• weak mixing with ND structures

From Qualitative to Quantitative!

Microscopic Mass Formula(can we go below 500 keV?)

Goriely, ENAM’04 Reinhard 2004

Challenges:•need for error and covariance analysis (theoretical error bars in unknown regions)•a number of observables need to be considered (masses, radii, collective modes)•only data for selected nuclei used