QRPA calculations for beta-decay i di l iin medium-mass nuclei

P.Sarriguren

Instituto Estructura de la Materia, CSIC, Madrid

E Moya de Guerra E. Moya de Guerra

O. Moreno

R Al R d i R. Alvarez-Rodriguez

A. Escuderos

Int. Workshop on strong, weak, and electromagnetic interactions to probe spin-isospin excitations, ECT*, Trento 2009

QRPA calculations for beta-decay

Motivation : Describe β decay process within a microscopic selfconsistent approachDescribe β-decay process within a microscopic selfconsistent approach

Spin-isospin nuclear properties. (Nuclear deformation)

Exotic/Stable nuclei : Beta-decay / Charge Exchange ReactionsExotic/Stable nuclei : Beta-decay / Charge Exchange Reactions

Limits of applicability of well established microscopic models

Theoretical approach :Deformed HF+BCS+QRPA formalism with Skyrme forces and

residual interactions in both ph and pp channels

Results :Nuclear structure: GT strength distributions (Fe, Pb, A~70)

Nuclear astrophysics: Half-lives of waiting point nuclei in rp-processesNuclear astrophysics Half lives of waiting point nuclei in rp processes

Beta-decay vs. charge exhange reactions

B d Ch h iBeta-decay

• information limited to unstable nuclei

• energy restrictions Q-window

Charge exchange reactions

• stable nuclei

• exotic Inverse kinematics (calibration)energy restrictions, Q window exotic. Inverse kinematics (calibration)

• no energy restrictions

• β-strength (mechanism of reaction)

( )Z,A

(n p)(p,n)Qβ −

(n,p)

(d,2He)(3He,t)

( )Z+1,A

Z+1,N-1 Z,N Z-1,N+1

β-decay

( ) ( )( ) ( )

: , 1,

: , 1,e

e

Z A Z A e

Z A Z A e

β ν

β ν

− −

+ +

→ + + +

→ − + +( )t d htQ M M m T E= − − = +

( ),Z A

( ) ( ): , 1, eEC Z A e Z A ν−+ → − +( )

eparent daughter e eQ M M m T Eβ

ν−+

Qβ −

( )1,Z A+

Decay rate: Fermi’s golden rule

2

0

ln 2 2f

FiV dN

dWtπλ ⎛ ⎞= =⎜ ⎟

⎝ ⎠ h

Density of final states

Matrix elements of the transition0⎝ ⎠

( ) ( ) ( )0 20 01

, ,W

FdN C pW W W Z W dW f Z WdW

λ= − ≡∫Fermi integral ( ) ( ) ( )0 01

0dW ∫

λ(Z,W): Fermi function: Influence of nuclear Coulomb field on electrons

( ) 21fift V− ≈

β-decay

*V g dV g dV Mψ φ φ ψ ψ ψ⎡ ⎤ Θ Θ⎣ ⎦∫ ∫

Matrix elements of the transition

fi f e i f i fiV g dV g dV Mνψ φ φ ψ ψ ψ⎡ ⎤= Θ ≈ Θ =⎣ ⎦∫ ∫( ) /1 1 / 1i p r iφ ⋅

ur rh

r ur rh( ) /

, 1 / 1i p re r e i p r

Vνφ ⋅= = + ⋅ + ≈h h LAllowed approximation

Selection rules

1lj

Allowed approximation : ΔL=0, Δπ = no

• Fermi eν (ΔS=0) ΔJ=0

G T ll ( )

Aii

τ ±∑−t2

−= lj

−tσ

• Gamow-Teller eν (ΔS=1) ΔJ=0,1A

i iiσ τ ±∑(no 0 0 )+ +→

21

+= ljFirst Forbidden ΔL=1, Δπ = yes

• Fermi (ΔS=0) ΔJ=0 1Z NZ N • Fermi (ΔS=0) ΔJ=0,1

• Gamow-Teller (ΔS=1), ΔJ=0,1,2

Ch h ti s f l s t s i t l f

Charge exhange reactions

Charge exchange reactions are a powerful spectroscopic tool for measuring the Fermi and Gamow-Teller strength functions

At high incident energies and forward angles the nuclear states are probed at

Small momentum transfers

At high incident energies and forward angles the nuclear states are probed atsmall momentum transfers.

Small momentum transfers

Only the central parts of the isovector y peffective interaction contribute to the cross section. The noncentral parts can be neglected.

Multipole expansion. Simple relation between cross sections and beta-strengths.

(p,n) (3He,t)Love and Franey, PRC24, 1073 (1981)

Taddeucci et al, NPA469, 125 (1987)

Osterfeld, Rev Mod Phys 64, 491 (1992)

p

(n,p) (d,2He)

P l d ff

Charge exhange reactions

Projectile distortion effectsFactorization into a nuclear reaction part and a nuclear structure part (exact in PW)

( ) ( )2

2

2

2

0 ,f NN

F i

kd q N J B F i fd k τ τ

σ μπ

⎛ ⎞≈ = →⎜ ⎟Ω ⎝ ⎠h ( )2

,B F i f f T i−→ =

Nuclear structure

( ) ( )2

2

20 ,f NN

GT i

kd q N J B GT i fd k στ στ

σ μπ

⎛ ⎞≈ = →⎜ ⎟Ω ⎝ ⎠h( ) ( )

2,B GT i f f i

μ

β μ−→ = ∑

Volume integralDistortion factor

Improve on

τ( )

( )( )

( ) 0º

DW

PWN τ σττ στ

τ στ θ

σσ

= ( ) ( ) ( 0)NN CJ V qτ στ τ στ= =

Volume integralDistortion factor

στ( )( ) 0ºτ στ θ = Effective NN

interaction at 0 momentum transfer

Eikonal

( )1/3N C A +( )1/3( ) 0expN C xA aτ στ = − +

Gamow-Teller strength: different approaches

Gross theory: Statistical model. Takahashi et al. Prog Theor Phys 41, 1470 (69); 84, 641 (90)

Microscopic models

Large scale shell model calculations: Brown Wildenthal, ADNDT 33, 347 (85)g , , ( )

Strasbourg-Madrid PRC52, R1736 (95); NPA 653(99)439

pnQRPA calculations simplicity, no core, highly excited states

but not all many body correlations taken into accountbut not all many-body correlations taken into account

• Halbeib, Sorensen Spherical, pairing + ph-QRPA NPA98, 542 (67)

• Moeller et al. Nilsson(WS) + BCS + ph-QRPA NPA417, 419 (84); 514, 1 (90)

• Klapdor, Muto, … Nilsson(WS) + BCS + ph(pp)-QRPA ZPA341, 407(92); PRC54, 2972 (96)

• Hamamoto et al Def HF + BCS+ ph-TDA PRC52, 2468 (95)p ( )

• Borzov et al. DFT and continuum QRPA NPA621, 307 (97)

• Bender, Engel, Dobaczewski, Nazarewicz , Colo, …

Sph HFB + ph(pp) QRPA Selfconsistent PRC60 014302 (99); C65 054322 (02)Sph HFB + ph(pp)-QRPA Selfconsistent PRC60, 014302 (99); C65, 054322 (02)

• Ring, Vretenar, Paar, … Relativistic pnQRPA PRL91,262502 (03); PRC75,024304(07)

• Petrovici et al. VAMPIR calculations NPA799, 94 (08)

• This work: Def Skyrme HF + BCS + ph(pp)-pnQRPA separable forces

NPA658; PRC64; PRC79, 044315 (09)

Hartree-Fock method

A A

( ) ( )1 1

,A A

k k lH T k W k l

= < =

= +∑ ∑

How to extract a single-particle potential U(k)

out of the sum of two-body interactions W(k,l)

( ) ( ) ( ) ( ) 011 1

,A A A

resk l kk

W k l UT k U k HkH V< = ==

⎡ ⎤−⎢ ⎥⎣ ⎦+⎡ ⎤= + = +⎣ ⎦ ∑ ∑∑

11 1k l kk <⎣ ⎦

H t F k Hartree-Fock:

Selfconsistent method to derive the single-particle potential

Variational principle: Variational principle:

Search for the best Slater determinant minimizing the energy

Assume that the resulting residual interaction is smallAssume that the resulting residual interaction is small

Two body interaction: zero range limit of a finite range force

Skyrme effective interactions

Two-body interaction: zero-range limit of a finite range force

• leading term: delta-function with strength t0 and spin exchange x0

• leading finite range corrections t x t x (finite range = momentum dependence)• leading finite range corrections t1 , x1 , t2 , x2 (finite range = momentum dependence)

• short-range spin-orbit interaction with strength W0

Three-body interaction: Short-range density-dependent two-body force t3 x3 αThree body interaction: Short range density dependent two body force t3, x3 , α

( ) ( ) ( ) ( )( )2 21 1 '1V t P k kt P δδ+ + + +ur urur ur

( ) ( ) ( ) ( )( )( ) ( ) ( ) ( )

2 20 0 1 1

2 02

1 '2

1 '

1

'i j i

ij ii j j

i j j

V t x P r r k k

t x P k r r iW k r r

t P r

k

x r

k

σ

σ

σ

σ σ δ

δ

δ

δ= + + + − +

+ + ⋅ − +

−

+ ⋅ × −uur ur ur r uur uur uur ur ur r( ) ( ) ( )

( ) ( )3 31 16 2

i ji j

r rt x P r r α

σ δ ρ⎛ ⎞+

+ + − ⎜ ⎟⎜ ⎟⎝ ⎠

ur urur ur

6 ⎝ ⎠

Parameters (10) fitted using nuclear matter properties and ground state properties of a selected set of nuclei (binding energies, charge radii,…)

Sk3, SG2, SLy4

Skyrme Hartree-FockSchroedinger equationSchroedinger equation

( ) ( ) ( ) ( )( )2

i iq iU r W r i eσ φ φ⎡ ⎤⎢ ⎥−∇ ⋅ ∇ + + ⋅ − ∇× =⎢ ⎥

r uur rur ur u rr

r uh

( ) ( ) ( ) ( )( )2

i iq im r

φ φ∗⎢ ⎥

⎣ ⎦r

( ) ( ) ( ), , qm r U r W r∗r r uur r

Algebraic combinations of the densities [ ], , Jρ τ( ) ( ) ( )q g [ ]

( ) ( ) ( ) ( ) ( ) ( )( ) ( )2 2*

, '

, , , , , , , ', , ,stst i st i i ii i i s

r r s t r r s t J r r s t i r s tρ φ τ φ φ σ φ= = ∇ = − ∇×∑ ∑ ∑r r r ur r ur r r ur ur r

( ) ( ) ( ) ( ) ( ) ( ) ( )2 2

1 1 2 2 1 1 2 2*

1 12 2 1 2 1 22 8 82

qq

t x t x r t x t x rmm r

ρ ρ= + + + + − + + +⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦rh h

rr

( )q

( ) ( ) ( ) ( )( ) ( ) ( ){ }1 1 2 20 0 0 3 3 3

1 12 1 2 2 2 2 1 22 241 1 1

q q p nq t x x t x xU r α α αρ ρ α ρ ρ ρ αρ ρ ρ+ −⎡ ⎤⎡ ⎤= + − + + + + − + + +⎣ ⎦ ⎣ ⎦r

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

21 1 2 2 2 2 1 1 1 1 2 2

21 1 2 2 1 2 1 1 2 2

1 1 12 2 1 2 1 2 3 2 28 8 161 1 13 1 2 1 2

16 8 8

q

qq

t x t x t x t x t x t x

t x t x t t J t x t x J

τ τ ρ

ρ δ

+ + + + + + − + + − + + + ∇⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦ ⎣ ⎦

+ + + + ∇ + − − + +⎡ ⎤⎣ ⎦uur ur ( ),q p coulV r

r

16 8 8 ( )

( ) ( )012 qqW Wr ρ ρ= ∇ + ∇

urr r uruu

Axially symmetric deformed nuclei

Expansion into eigenfunctions of deformed harmonic oscillator

( ) ( ) ( ) ( ) ( ) ( )1/ 2 1/ 2, , , ,i i

i

i ii q i iR q q r z e r z eπ ϕ ϕσ χ χ σ χ σ

− +Ω + Λ − Λ+ −

⎡ ⎤Φ = Φ + Φ⎣ ⎦ur

Expansion into eigenfunctions of deformed harmonic oscillator

( ) ( ) ( ) ( )ieR r z

φ

φ σ ψ ψ χ σΛ

Λ=ur

2 2 2 21 1( , )2 2 zV r z M r M zω ω⊥= +

( ), ( ) ( ) ( )2r zn nR r zαφ σ ψ ψ χ σπ Σ=

/ 2 / 2( ) 2 ( )r N e Lηψ β η ηΛ Λ Λ − Λ=2 2 ( ) 2 ( )r r rn n nr N e Lψ β η η⊥=

21/ 2 / 2( ) ( )z z zn n z nz N e Hξψ β ξ−=

Optimal basis to minimize truncation effects ( ) ( )1/ 21/3 1/32 2

o z zMβ ω ω β β⊥ ⊥⎡ ⎤= =⎢ ⎥⎣ ⎦

2⎛ ⎞N= 12 major shells 2

z z

q ω βω β

⊥ ⊥⎛ ⎞= = ⎜ ⎟

⎝ ⎠

( ) ( ) { }i∑ur ur( ) ( ) { }, , , , , , ,

i

ii q r zR q C R n nα α

α

σ χ φ σ αΦ = = Λ Σ∑

Pairing correlations in BCS approximation

B d ( ) 0+ +∏BCS ground state ( )0

0BCS i i i ii

u v a aϕ + +

>

= +∏

( )H G+ + + ++∑ ∑( ) ''0 ' 0

k k k k kk k k kk kk

H a a a a G a a a aε + + + +

> >

= + −∑ ∑

Variational equation constrained by the expectation value of particle number

BCS eqs. Number eq. Gap eq.22 i

iv N=∑

0k k

kG u v

>

Δ = ∑0k >

( )22 21 1 ; 2

ii i i

i

ev E eE

λ λ⎡ ⎤−

= − = − + Δ⎢ ⎥⎣ ⎦

Fixed gaps taken from phenomenology

[ ]1 ( 2, ) 4 ( 1, ) 6 ( , ) 4 ( 1, ) ( 2, )8n B N Z B N Z B N Z B N Z B N ZΔ = − − − + − + + +

Constrained HF+BCS

Energy vs. deformation• HF: Single solution (Slater determinant of lowest energy)gy)

• Constrained HF: Minimization of the HF energy under the constraint of holding the nuclear deformation fixed

Energy as a function of the deformationgy f f f m

More than one local minimum at close energy

Shape Coexistence

M f ld ( l l b ) R d l (G ) RP

QRPA

Mean field (single particle basis) Residual interaction (GT) QRPA

Phenomenological approach:

Mean field (s.p. basis) and residual interactions independently chosen

Selfconsistent approach:

• The underlying mean field calculation must be consistent

• The residual interaction used in QRPA must be derived from the same force that determines the mean field: Second derivatives

Theoretical approach: Residual interactions

( )2

' '1 2 1 2 1 2

1 1 ( 1) 1 ( 1)s s t th

EV r rδσ σ τ τ δ− −⎡ ⎤ ⎡ ⎤= + − ⋅ + − ⋅ −⎣ ⎦ ⎣ ⎦∑uur uur ur uur ur ur

r r

Particle-hole residual interaction consistent with the HF mean field

( )2

' '1 ( 1) ( 1)s s t t EV r rστ δσ σ τ τ δ− −= − − ⋅ ⋅ −∑uur uurur uur ur ur

( )1 2 1 2 1 2' ' 1 ' ' 2

1 ( 1) 1 ( 1)16 ( ) ( )ph

sts t st s t

V r rr r

σ σ τ τ δδρ δρ

⎡ ⎤ ⎡ ⎤+ +⎣ ⎦ ⎣ ⎦∑ ur ur

( )

( )

1 2 1 2 1 2' ' 1 ' ' 2

2 2

( 1) ( 1)16 ( ) ( )1 24 2 2

phsts t st s t

V r rr r

t t k t k t r rα

σ σ τ τ δδρ δρ

ρ σ σ τ τ δ

= ⋅ ⋅

⎡ ⎤= +⎢ ⎥

∑ ur ur

uur uurur uur ur ur( )0 1 2 3 1 2 1 2 1 2 4 2 216 3F Ft t k t k t r rρ σ σ τ τ δ= − − + − ⋅ ⋅ −⎢ ⎥⎣ ⎦

Separable forcesAverage over nuclear volume

0, 12 ( 1) , ph ph K

GT GT K K K K KK

V t a aν ππν

χ β β β σ ν σ π+ − + + +−

= ±

= − = =∑ ∑

( )20 1 2 33

3 1 18 2 6

phGT Ft k t t t

Rαχ ρ

π⎧ ⎫= − + − +⎨ ⎬⎩ ⎭

( )0, 1

2 ( 1) , pp pp KGT GT K K K K

KV P P P a aν π

πν

κ ν σ π++ + + +

= ±

= − − =∑ ∑

pnQRPA with separable forces

Phonon operator , K K

K K K K K KK

n pX A Y A Aω ωω γ γ γ γ γ

γ

α αΓ+ + + + +⎡ ⎤= − =⎣ ⎦∑

pnQRPA equations

0 0, 0K K Kω ω ωΓ Γ+= =

⎡ ⎤pnQRPA equations0 0 0 0,

K K K KKA H Aγ ω γ ωφ φ ω φ φΓ Γ+ +

+ + + +

⎡ ⎤ =⎣ ⎦

⎡ ⎤0 0 0 0,K K K KKH A Aω γ ω γφ φ ω φ φΓ Γ+ + + +⎡ ⎤ =⎣ ⎦

( ) ( )

A B X XB A Y Y

ω⎛ ⎞⎛ ⎞ ⎛ ⎞

=⎜ ⎟⎜ ⎟ ⎜ ⎟− −⎝ ⎠⎝ ⎠ ⎝ ⎠

( ) ( ) ( )( )

, ' ' , ' ' ' ' ' '

, ' ' ' ' ' '

, ' ' phpn p n p n pn p n p n p n p n p n

pppn p n p n p n p n p n

A pn p n V u v u v v u v u

V u u u u v v v v

ε ε δ= + + +

+ +

( ) ( ), ' ' , ' ' ' ' ' ' , ' ' ' ' ' 'ph pp

pn p n pn p n p n p n p n p n pn p n p n p n p n p nB V u v v u v u u v V u u v v v v u u= + − +

pnQRPA with separable forces

Transition amplitudes 0 KK K M ωω β ±

±= m

( )M X Yω ω ω∑ % ( )M X Yω ω ω∑ %( )K K KM q X q Yω ω ωπν πν πν πν

πν− = +∑ % ( )K K KM q X q Yω ω ω

πν πν πν πνπν

+ = +∑ %

q u v q v uπν πν πν ν σ πΣ Σ Σ% , , K K K Kq u v q v uπν ν π πν ν π ν σ πΣ Σ Σ= = =

2( )B GT I M K I M K∑B(GT) in the lab. system ( )

f

f f f i i iM

B GT I M K I M Kμμ

σ= ∑

2 1I +

In terms of intrinsic matrix elements

( ) { }2,0

2 1 ( 1)16 1

I I K IMK K M K K

K

IIMK D Dφ φπ δ

+ − +−

+= + −

+

In terms of intrinsic matrix elements

220 0 0 0 1 1 0( ) 2K K KB GT δ φ σ φ δ φ σ φ= +,0 0 0 0 ,1 1 0( ) 2

f f fK K KB GT δ φ σ φ δ φ σ φ++

Qualitative behaviour of Matrix Elements

2qp

Skyrme HF + BCS + QRPA (ph,pp)ph

+ def

Redistribution and

reduction of the strengthpp

Smooth occupations

New excitations

Fragmentation

Selfconsistent mean field with Skyrme forces

Suitable for extrapolations to unknown regions

2qp / TDA / RPA Skyrme force

ω

B(GT) (2qp) Vres =0

B(GT) (QTDA) Yω=0

Role of deformation

nsphpsph pprol nprolpobl nobl

38 36

Allowed GT transitionsΔπ=0

74KrZ=36 N=38

Role of residual interactions

2qp

ph

pp

• Redistribution of the strengthph : shift to higher energiespp : shift to lower energiespp : shift to lower energies

•Reduction of the strength

Stable nuclei in Fe-Ni mass region: Theory vs experiment

Main constituents of stellar core in presupernovaeMain constituents of stellar core in presupernovae

Comparison with :

exp. (n,p), (p,n)

SM calculations

GT properties: Test of QRPAGT properties: Test of QRPASM: NPA 653, 439 (1999)

QRPA: NPA 716, 230 (2003)

Constrained HF+BCS

Medium mass proton rich nuclei: Ge, Se, Kr Sr

Waiting point nuclei in rp-processes

Shape coexistenceShape coexistence

Large Q-values

Isotopic chains approaching the drip lines

Energy vs. deformation Shape coexistenceBeyond full Shell Model

Medium mass proton rich nuclei: Ge, Se, Kr SrConstrained HF+BCS

Waiting point nuclei in rp-processes

Shape coexistenceShape coexistence

Large Q-values

Isotopic chains approaching the drip lines

Beyond full Shell ModelEnergy vs. deformation Shape coexistence

Medium mass proton rich nuclei: Ge, Se, Kr SrGT strength distributionsGT strength distributions

Medium mass proton rich nuclei: Ge, Se, Kr SrGT strength distributionsGT strength distributions

Dependence on deformationNPA 658, 13 (1999)

β−decay: Nuclear Structure: Deformation

Gamow-Teller strength: Gamow Teller strength: Theory and Experiment

oblate prolateprolateoblate

p

Total absorption spectroscopy

74Kr76Sr

oblate

prolate

Exp: Poirier et al. PRC69, 034307 (2004) Exp: Nacher et al. PRL92, 232501 (2004)

Gamow-Teller strength: Theory and Experiment

High resolution spectroscopyExp: Piqueras et al. EPJA 16, 313 (2003)

SLy4

QEC= 5.070 MeV

Gamow-Teller strength: Theory and Experiment

SLy4

Half-lives: Theory and Experiment

( ) ( ) ( )0 20 01

,,W

f Z W pW W W dWZ Wβ λ± ±= −∫( )2

21 /eff

/A V ECg gT f I Iβ β

+− +∑ phase space Coulomb effecteff1/ 2 6200

f

f iI

T f I Iβ β= ∑1 11

2 22 2

2 K LC

LKE

K Lf q BgqBgπ ⎡ ⎤= + +⎣ ⎦L

phase space

Neutrino energy

Coulomb effect

( ) ( )/ 0 74 /g g g g= Neutrino energyRadial components of the bound state e-wf at the origin

Exchange and overlap corrections

( ) ( )eff bare/ 0.74 /A V A Vg g g g=

EPJA 24, 193 (2005)

Good agreement with experiment:Reliable extrapolations

Exotic Nuclei : Nuclear AstrophysicsQuality of astrophysical models depends critically on the quality of input (nuclear)

Exotic Nuclei play a relevant role in explosive events

Very limited experimental information ⇒ Network calculations based on model predictions

rp-process: Proton capture reaction rates orders of magnitude faster than the mp tin β+ d scompeting β+ -decays

Waiting point nuclei: When the dominant p-capture is inhibited: the reaction flow waits for a slow beta-decay to proceedf y p

5758

5

Ru (44)Rh (45)Pd (46)Ag (47)

Waiting points

5455

56

Sr (38)Y (39)

Zr (40)Nb (41)

Mo (42)Tc (43)Waiting points

45464748

49505152

535455

G (32)As (33)

Se (34)Br (35)Kr (36)Rb (37)

Sr (38)

272829303132333435363738394041424344Ga (31)

Ge (32)

Schatz et al. Phys. Rep. 294, 167 (1998)

Weak decay rates in rp-process

X-ray bursts: Generated by thermonuclear runaway in the hydrogen rich environment of an accreting neutron star hydrogen-rich environment of an accreting neutron star, which is fed from a binary red giant companion.

Peak conditions of ρ=10 6-7 g.cm-3 and T=1-3 GK are reached

Mechanism is the rp capture process

Nuclear models should be able to describe at least the experimental information (Half-lives and GT strength distributions) available under terrestrial conditions

Decay rates λ (ρ,T)• Effect of T: Thermal population of excited states in the parent • Effect of T: Thermal population of excited states in the parent nucleus

• Effect of ρ and T: Atoms are completely ionized. Electrons form d l d a degenerate plasma obeying Fermi-Dirac distribution: Continuum

electron capture becomes possible.

Weak decay rates in rp-process

( )/( ) /( )2 1 E kT E kTJ +∑ ∑

Weak interaction rates

( )/( ) /( )2 1 , 2 1 i iE kT E kTii i

i i

J e G J eG

λ λ − −+= = +∑ ∑

Thermal population of excited states

72,74KrKr76,78Sr

Weak decay rates in rp-process

( )/( ) /( )2 1 , 2 1 i iE kT E kTii i

J e G J eG

λ λ − −+= = +∑ ∑

i iG

( )ln 2 B Tρλ λ= = Φ∑ ∑ ( ),i iff

ifif

fDB Tρλ λ= = Φ∑ ∑

Nuclear structure Phase space factor

( () )B B GT B F= +

p

( () )if if ifB B GT B F= +2 2

( ) 1 k kAg f iB TG⎛ ⎞⎜ ⎟ ∑

eff

( )2 1

k kA

ki Vif

g f t iJ

B Tg

G σ ±= ⎜ ⎟+ ⎝ ⎠∑

( ) ( )/ /( ) ( )eff bare/ 0.74 /A V A Vg g g g=

Ph

Weak decay rates in rp-process

cECif if if

β +

Φ = Φ + Φ

Phase space

f f f

( )2 21( ) ( , ) 1 ( )cECif if e ifQ F Z S S Q dνω

ω ω ω ω ω ω ω∞

⎡ ⎤Φ = − + − +⎣ ⎦∫ll

( )2 2

11( ) ( 1, ) 1 1 ( )ifQ

if if p ifQ F Z S S Q dβνω ω ω ω ω ω ω

+

⎡ ⎤ ⎡ ⎤Φ = − − − + − − −⎣ ⎦ ⎣ ⎦∫

( )2

1if p d i f

e

Q M M E Em c

= − + −

Se,p,ν (ω): Distribution functions (inhibit/enhance the phase space available)

S S 0

( ) ( )1

exp / 1ee

SkTω μ

=− +⎡ ⎤⎣ ⎦

Se: Fermi-Dirac distributionSp=Sν=0

( , )e Tμ ρ

Weak decay rates in rp-process

6 -310 g cmρ = 10 g cmρ

PLB 680, 438 (2009)

Shape dependence of GTdistributions in neutron-deficient Hg, Pb, Po isotopes

• Triple shape coexistence at low excitation 6+

...

(MeV)Triple shape coexistence at low excitation energy

• Search for signatures of deformation on 2+

4+

6+

1their beta-decay patterns 0+

0+

e-e-γ

0+

186Pb0

Shape dependence of GTdistributions in neutron-deficient: Pb isotopes

B(GT) strength distributionsB(GT) strength distributions

• Not very sensitive to : Skyrmeforce and pairing treatmentp g

• Sensitive to : Nuclear shape

Signatures of deformation

l t

g

PRC 72, 054317 (2005), PRC 73, 054302 (2006)

prolate

oblate

h lspherical

QEC

Conclusions

Theoretical approach based on a deformed Skyrme HF+BCS+QRPA

Used to describe spin-isospin nuclear properties (GT & M1) in stable and exotic nuclei in various regions along the nuclear chart exotic nuclei in various regions along the nuclear chart

Find

• Good agreement with experimentGood agreement with experiment

• GT strength distributions (Fe-Ni, p-rich A=70, 2νββ emitters)

• Half-lives (waiting point nuclei)Half lives (waiting point nuclei)

• Spin M1 strength distributions (rare earths & actinides)

• 2νββ matrix elementsββ

• Sensitivity to the nuclear deformation

• Signatures of deformation in GT strength distributions in A=70 and neutron-deficient Pb isotopes

• Suppression mechanism in 2νββ matrix elements