qrpa calculations of stellar weak decay...
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QRPA calculations of stellar weak decay rates
Instituto de Estructura de la Materia
CSIC, Madrid, Spain
International School of Nuclear Physics. 36th Course. Nuclei in the Laboratory and in the Cosmos. Erice-Sicily. September 16-24, 2014
P. Sarriguren
E. Moya de Guerra,
R. Alvarez-Rodriguez, O. Moreno
Universidad Complutense Madrid
Problem : Weak-decay rates determine late stages of stellar evolution.
Experimental extrapolations or theoretical predictions:
Reproduce exp information on T1/2 and BGT under terrestrial conditions.
Theoretical approach : Deformed HF+BCS+QRPA formalism with Skyrme forces and residual
interactions in both ph and pp channels.
Results : Weak-decay rates at various (,T) in stellar scenarios pf-shell nuclei. Main constituents of stellar core in presupernova
formations: Sc, Ti, V, Mn, Fe, Co, Ni, Zn isotopes.
BGT measured in laboratory (Charge exchange reactions) and comparedwith benchmark Shell Model calculations.
Neutron-deficient waiting-point isotopes (Ni-Sn) at (,T) typical of rp-process.
Neutron rich Zr-Mo isotopes: r process.
Weak-decay rates
Weak decay rates
/( ) /( )2 1( ) , 2 1 i iE kT E kTii i
i
JP T e G J eG
Nuclear structure
Phase space factors :
2 2
eff
( )ifk kA
kV
g f tB G ig
T
Initial states thermally populated
11/ 2 ,ln 2ln 2 ( ) ifii
iffT P T
DTB
ECif if if
are different from laboratory (Pi, cEC)
if if
+, EC : - :
. . ( 0) 1i g sP T
2 21( ) ( , ) 1 ( )cECif if e ifQ F Z S S Q d
2 2
11( ) ( 1, ) 1 1 ( )ifQ
if if p ifQ F Z S S Q d
1
exp / 1ee
SkT
Distribution functions
Se: Fermi-Dirac distributionSp=S
Phase space factors
2
1if p d i f
e
Q M M E Em c
2 2
11( ) ( 1, ) 1 1 ( )ifQ
if if e ifQ F Z S S Q d
111
2 222
2 Ko
KC
LE
K L Lq q Bg gB q Neutrino energyg Radial components of the e-wf at r=0B Exchange and overlap corrections
Hartree-Fock method
How to extract a single-particle potential U(k)
out of the sum of two-body interactions W(k,l)
1 1
,A A
k k lH T k W k l
011 1
,A A A
resk l kk
W k l UT k U k HkH V
Hartree-Fock: Selfconsistent method to derive the single-particle potential
Variational principle: Search for the best Slater det. minimizing the energy
Assume that the resulting residual interaction is small
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 t1 , x1 , t2 , x2 (finite range = momentum dependence)
• short-range spin-orbit interaction with strength W0
Short-range density-dependent two-body force t3, x3 ,
Skyrme effective interactions
2 20 0
3 3
1 1
2 02
1 1 '2
1 '
1
1 16 2
'i j i
ij
i j
ii j j
i j
i j
j
V
r rt x P r
t x P r r k k
t x P k r r iW k r r
t P r
k
x r
r
k
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 equation
2
2i iq iU r W r
m ri e
, , qm r U r W r
Algebraic combinations of the densities
2 2
1 1 2 2 1 1 2 2*
1 12 2 1 2 1 22 8 82
t x t x r t x t x rmm r
1 1 2 20 0 0 3 3 3
21 1 2 2 2 2 1 1 1 1 2 2
21 1 2 2 1 2 1 1 2 2
1 12 1 2 2 2 2 1 22 241 1 12 2 1 2 1 2 3 2 28 8 161 1 13 1 2 1 2
16 8 8
q q p n
q
q
q
q
t x x t x x
t x t x t x t x t x t x
t x t x t
U r
t J t x t x J
,q p coulV r
012 qqW Wr
, , J
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 t
Optimal basis to minimize truncation effects
N= 12 major shells
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 e
2 2 2 21 1( , )2 2 zV r z M r M z
, ( ) ( ) ( )2r z
i
n neR r z
/ 2 / 2( ) 2 ( )r r rn n nr N e L
21/ 2 / 2( ) ( )
z z zn n z nz N e H
1/ 21/3 1/32 2
o z zM
2
z z
q
, , , , , , ,i
ii q r zR q C R n n
Pairing correlations in BCS approximation
BCS ground state
Variational equation constrained by the expectation value of particle number
BCS eqs. Number eq. Gap eq.
Fixed gaps taken from phenomenology
0
0BCS i i i ii
u v a a
''0 ' 0
k k k k kk k k kk kk
H a a a a G a a a a
22 ii
v N0
k kk
G u v
22 21 1 ;
2i
i i ii
ev E eE
1 ( 2, ) 4 ( 1, ) 6 ( , ) 4 ( 1, ) ( 2, )8n B N Z B N Z B N Z B N Z B N Z
2' '
1 2 1 2 1 2' ' 1 ' ' 2
2 20 1 2 3 1 2 1 2 1 2
1 ( 1) ( 1)16 ( ) ( )1 2 4 2 2
16 3
s s t tph
sts t st s t
F F
EV r rr r
t t k t k t r r
2
' '1 2 1 2 1 2
' ' 1 ' ' 2
1 1 ( 1) 1 ( 1)16 ( ) ( )
s s t tph
sts t st s t
EV r rr r
Residual interactions
Particle-hole residual interaction consistent with the HF mean field
Separable forcesAverage over nuclear volume
0, 12 ( 1) , ph ph K
GT GT K K K K KK
V t a a
0, 1
2 ( 1) , pp pp KGT GT K K K K
KV P P P a a
20 1 2 33
3 1 18 2 6
phGT Ft k t t t
R
pnQRPA with separable forces
Phonon operator
pnQRPA equations
, K K
K K K K K KK
n pX A Y A A
0 0, 0K K K
Transition amplitudes 0 KK Kt M
K K KM q X q Y
K K KM q X q Y
, , K K K Kq u v q v u
Single-particle states from deformed Skyrme Hartree-Focku, v: Occupation amplitudes from BCS pairing
B(GT) in the lab. system 0 0 1i i f fI K I K K
2 2
0 0 0 1 1 1( ) 0 2 0B GT t t
Stable nuclei in Fe-Ni mass region: Theory vs experiment
Main constituents of stellar core in presupernovae. Comparison with :
• exp. (n,p), (p,n)
• SM calculations
GT properties: Test of QRPASM: NPA 653, 439 (1999)
QRPA: NPA 716, 230 (2003)
Weak decay rates in pf-shell nuclei
exp: charge exchange reactions
(n,p)
Accumulated GT strength
2( )ifB GT f t iQRPA
Shell Model
A.L. Cole PRC 86, 015809 (2012)
P.S. PRC87, 045801 (2013)
Calculations for several isotopes:
Sc, Ti, V, Fe, Mn, Ni, Co, Zn
Weak decay rates in pf-shell nuclei
( ,, ) ifiif
ifT P T TB
2 21( ) ( , )
1 ( )
cECif if
e if
Q F Z
S S Q d
1
exp / 1ee
SkT
2
1if p d i f
e
Q M M E Em c
Si burning
Ia Supernova
Electron Capture from e- plasma
Weak decay rates in pf-shell nuclei
P.S. PRC87, 045801 (2013)(Q<0) large: sensitive low-lying exc.
Weak decay rates in pf-shell nuclei
Q small
Exotic Nuclei : Nuclear Astrophysics
rp-process: Proton capture reaction rates are orders ofmagitude faster than the competing -decays.
X-ray bursts: Source of intense X-ray emissions generated by thermonuclear runaway in the H-rich environment of an accreting n-star, fed from a binary red giant companion.
Nucleosynthesis mechanism: rp capture process
Waiting point nuclei: When thedominant p-capture is inhibited, the reaction flow waits for a slowbeta-decay to proceed further.
Time scale
Isotope abundances
Light curve profile
272829303132333435363738394041424344
45464748
49505152
535455
56
5758
5
Ga (31)Ge (32)As (33)
Se (34)Br (35)Kr (36)Rb (37)
Sr (38)Y (39)
Zr (40)Nb (41)
Mo (42)Tc (43)
Ru (44)Rh (45)Pd (46)Ag (47)
Constrained HF+BCS
Medium mass neutron deficient isotopes
Waiting point nuclei in rp-processes
Shape coexistence
Beyond full Shell Model
Gamow-Teller strength distribution
Calculations for
Ni, Zn, Ge, Se,
Kr, Sr, Zr, Mo,
Ru, Pd, Cd, Sn
(N=Z and neighbors)
Gamow-Teller strength distribution
Gamow-Teller strength: Theory and Experiment
ISOLDE: Total absorption spectroscopy
Exp: Poirier et al. PRC69, 034307 (2004) Exp: Nacher et al. PRL92, 232501 (2004)
+/EC half-lives: Theory and Experiment
Good agreement with experiment:
Reliable extrapolations to high and T
eff bare/ 0.74 /A V A Vg g g g
22
1 /eff1/ 2
/6200
A V EC
f
g gT f i
2 2
11( ) ( , )ifQ
if ifQ F Z d
1 11
2 22 2
2 K LEC
K LK LB qgq Bg
Weak decay rates in rp-process
78Sr
74Sr
Competition +/EC P.S. PRC83, 025801 (2011)
Deformed pn-QRPA
B(GT) and T1/2: Good agreement with
experiment
Weak decay rates in rp-process
78Sr
74Sr
76Sr (WP)
Weak decay rates in rp-process
70Kr(Large Q)
74Kr(Small Q)
P.S. Phys. Rev. C 83, 025801 (2011)
Weak decay rates in rp-process
70Kr(Large Q)
74Kr(Small Q)
72Kr (WP)
Zr-Mo neutron-rich isotopes
100-116Zr104-120Mo
Zr-Mo B(GT)
Zr-Mo B(GT)
Zr-Mo Half-lives
• Triple shape coexistence at low excitation energy
• Search for signatures of deformation on their beta-decay patterns
0+
186Pb
0+
0+
2+
4+
6+
e-e-
...
0
1
(MeV)
Shape dependence of GTdistributions in neutron-deficient Hg, Pb, Po isotopes
Shape dependence of GTdistributions in neutron-deficient: Pb isotopes
B(GT) strength distributions
• Not very sensitive to : Skyrmeforce and pairing treatment
• Sensitive to : Nuclear shape
prolate
oblate
spherical
QEC
Signatures of deformation
PRC 72, 054317 (2005), PRC 73, 054302 (2006)
Shape dependence of GTdistributions in neutron-deficient Hg, Po isotopes
Conclusions
Nuclear structure model (deformed Skyrme HF+BCS+QRPA)
o Nuclear structure in different mass regions, astrophysicalapplications.
o Reproduce half-lives and main features of GT strengthdistributions extracted from beta-decay and/or chargeexchange reactions.
o Weak-decay rates at (,T) typical of astrophysical scenarios:
pf-shell nuclei (presupernova): EC rates from QRPA comparable quality to benchmark SM calculations.
Neutron-deficient WP nuclei (rp-process): EC/+ compete
Neutron-rich Zr-Mo isotopes (r process).