Folie 1

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Gross bdecay properties for astrophysical
applications
Karl-Ludwig Kratz
- Institut für Kernchemie, Univ. Mainz,
Germany - HGF VISTARS, Germany - Department of
Physics, Univ. of Notre Dame, USA
2
Nuclear data in astrophysics
What data are needed in nuclear astrophysics ?
(B) Explosive nucleosynthesis e.g.
rp-process, np-process weak and main
r-process

• Quiescent nucleosynthesis
• e.g. H-, He-burning s-process

• nuclear masses (reaction Q-values)
• charged-particle reaction rates
• (e.g. (p,g), (a,g), (a,n))
• neutron capture-rates
• nuclear structure properties
• (e.g. Esp, Jp, C2S)

• nuclear-masses (Qb, Sp, Sn)
• half-lives (T1/2, g.s. , isomers)
• b-delayed quantities (Pp, Pn, Pf)
• neutron capture rates
• neutrino reactions
• nuclear-structure-properties
• (e.g. e2, Esp, J p )

for 10s to 100s of isotopes NEAR b-stability
for 100s to 1000s of isotopes
FAR-OFF b-stability
3
What are the nuclear data needed for?
as input for astrophysical calculations
star evolution, chemical evolution of
Galaxy, specific nucleosynthesis processes
WARNING !
Nuclear data (n.d.) are only ONE set of input
parameters among SEVERAL astrophysics parameter
sets
Depending on mentality of the star-couturier,
nuclear data are considered
important
unimportant
nuclear and astro-parameters of equal
standing n.d. to constrain astro-parameters learn
ing nucl. structure from astro-observables
astro-parameters dominate n.d. just telephone
numbers (too) many (free) parameters n.d.
effects invisible
mathematical
nuclear
astrophysics
4
Basic astronomical question r-process

• Historically,
• nuclear astrophysics has always been
• concerned with
• interpretation of the
• origin of the chemical elements
• from astrophysical and cosmochemical
• observations,
• description in terms of specific
• nucleosynthesis processes
• (already B²FH, 1957).

Solar system isotopic abundances, Nr,?
T91.35 nn1020 - 1028
, Bi
r-process observables
CS 22892-052 abundances
isotopic composition Ca, Ti, Cr, Zr, Mo,
Ru, Nd, Sm, Dy ? r-enhanced
scaled solar r-process
ALLENDE INCLUSION EK-1-4-1
scaled theoretical solar r-process
Zr
Pt
Os
Pb
Cd
Ru
Ba
d
Nd
Sr
Sn
Ga
Pd
Dy
Mo
Gd
Er
Ge
Sm
Ce
Yb
Ir
Hf
Y
Rh
La
Nb
Ag
Ho
Eu
Pr
Tb
Au
Lu
Th
Tm
U
Mass number
Elemental abundances in UMP halo stars
FUN-anomalies in meteoritic samples
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Nuclear models to calculate T1/2 and Pn (I)
Theoretically,
the two gross/ integral b-decay quantities, T1/2
and Pn, are interrelated via their
usual definiton in terms of the so-called
b-strength function Sb(E)
What is that?
a natural adoption of the strength function
concept employed in other areas of nuclear
physics,
e.g. single-particle strength functions,
s-, p-wave neutron strength functions,
multipole strength functions for
photons.
Slc refers to the behavior of the squares of
overlap integrals (g²lc) between two
sets of nuclear wave functions l
represents various states of excitation,
classified by E, Jp, T c refers to the
different reaction / decay channels,
classified by Epart, lpart, rl is the
density of levels l.
Slc ltg²lcgt rl
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Nuclear models to calculate T1/2 and Pn (II)
Application to b-decay
Theoretical definition (Yamada Takahashi,
1972)
Experimental definition (Duke et al., 1970)
b(E)
Sb(E) D-1 ??M(E) ?² ?(E) s-1MeV-1
Sb(E)
s-1MeV-1
f(Z, Qb-E) T1/2
b(E) absolute b-feeding per MeV, f(Z, Qb-E)
Fermi function, T1/2 b-decay half-life.
??M(E) ? average b-transition matrix element
?(E) level density D const.,
determines Fermi coupling constant gv²
1
T1/2 as reciprocal ft-value per MeV
T1/2
? Sb(Ei) x f (Z,Qb-Ei)
0?Ei ?Qb
Sb(E)
6x105
Fermi function
f(Z, Qb-Ei) ? (Qb-Ei)5
3x103
T1/2 sensitive to lowest-lying resonances in
Sb(Ei) Pn sensitive to resonances in Sb(Ei) just
beyond Sn
Qb
same T1/2 !
1
? easily correct T1/2 with wrong Sb(E)
11
EMeV
1 5
10
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Nuclear models to calculate T1/2 and Pn (III)
Before any theoretical approach is applied, its
significance and sophistication should be clear !
In general, 2 groups of models
(2) Models that use an effective nuclear
interaction and solve the microscopic,
quantum-mechanical Schrödinger or Dirac
equation.

• Models where the physical quantity of interest
• is given by a polynomial or some other algebraic
• expression.

• parameters adjusted to exp. data
• describes only a single nucl. property
• no nuclear wave functions
• no insight into underlying SP structure

• provides nuclear wave functions
• within the same framework,
• describes a number of nucl.
  properties
• (e.g. g.s.-shape Esp, Jp , log(ft),
  T1/2 )

Examples
Examples Kratz-Herrmann Formula (1973) Gross
Theory (1973) New exponential law for
T1/2(b) (Zhang Ren 2006) T1/2(b-)
from GTGR known log(ft)s (Kar,
Chakravarti Manfredi 2006)
FRDMQRPA (1997 2006) Self-consist. Skyrme-HFB
QRPA (Engel et al. 1999) Large-Scale Shell
Model (Martinez-P. Langanke 1999,
2003) Density-Functional Finite-Fermi
System (Borzov et al. 2003) PN-Relativistic
QRPA (Niksic et al. 2005)

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Nuclear models to calculate T1/2 and Pn (IV)

• Simple statistical approaches
• assumptions

• b-decay energy is large (Qb ? 5 MeV)
• high level density
• Sb(E) is a smooth function of E (e.g. Sbconst.
  Sb ? r (E))
• is insensitive to nature of final
  states
• does not vary significantly for
  different types of nuclei (ee, o-mass, oo).

The Kratz-Herrmann Formula, applied to Pn values
? Sb(Ei) x f (Z,Qb-Ei)
with Sbconst.
b
Sn?Ei ?Qb
(Qb Sn)
Pn
Pn ? a
? Sb(Ei) x f (Z,Qb-Ei)
(Qb C)
C?Ei ?Qb
a, b as free parameters, to be determined by a
log-log fit to known Pn-values C
is a cut-off parameter (? pairing-gap
in b-decay daughter)
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Nuclear models to calculate T1/2 and Pn (V)
From Pfeiffer, Kratz Möller, Prog. Nuclear
Energy 41 (2002) 39-62
Parameters from fits to known Pn-values
full line
dashed line
as a kind of joke
-b
T1/2 ? a (Qb-C)
Parameters from fit to known T1/2 of n-rich nuclei
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Nuclear models to calculate T1/2 and Pn (VI)
NO joke !
in 2006, two examples for big steps BACKWARDS

• K. Kar, S. Chakravarti V.R. Manfredi
• arXiv astro-ph/0603517 v1
• Beta-decay rates (115 lt A lt 140) for r-process
• nucleosynthesis

• X. Zhang Z. Ren PRC73, 014305
• New exponential law for b decay half-lives
• of nuclei far from b-stable line

we have discovered a new exponential law for
T1/2(b)as a function of neutron number
the xth re-invention of the Gross Theory !
log10 T1/2 a x N b
shell model results indicate that the GT
strength distribution.. can be taken as a
Gaussian.
authors give fit parameters for a and b, for
(I) different Z-regions (II) allowed
b-decay (III) first-forbidden b-decay
(IV) second-forbidden b-decay

• GT strength distributes among 3 different types
  of
• final states
• discrete low-lying states with known log fts
• discrete states above with unknown strengths
• a part of the GT giant resonance (GTGR).

? finally a simple and accurate formula emerges
admitted problems centroid of GTGR
? from Bertsch Esbensen (1987) width of
GTGR ? free parameter !
log10 T1/2 (c1Z c2) N c3Z c4
useful to experimental physicists for
analyzing b-decay data.
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Nuclear models to calculate T1/2 and Pn (VII)
(2) QRPA type, microscopic models
Recent review by J. Engel Proc. Workshop on The
r-Process Seattle (2004) World Scientific
Among recent theoretical schemes
Some methods emphasize global applicability,
others self-consistency, and still others the
comprehensive inclusion of nuclear correlations.
None of the methods includes all important
correlations, however.
(2.1) FRDM QRPA
(2.2) Self-consistent Skyrme-HFB QRPA
Macroscopic-microscopic mass model
FRDM Schrödinger equation solved in QRPA GT
force with standard choice for GT
interaction latest version includes ff-strength
from Gross Theory.
Skyrme interaction SKO ? reasonable reproduction
of energies and strengths of GT resonances
strength of T0 np pairing adjusted to fit
known T1/2

• disadvantages only spherical shape
• only GT
• only n-magic (N50, 82, 128)
• Skyrme interaction not good
• enough to makedecisive
• improvement
• advantage self-consistency

?GT 23 MeV/A

• disadvantage not self consistent
• advantages global model for all shapes and
• types of nuclei
• large model space

? T1/2 shorter than those from FRDM QRPA
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Nuclear models to calculate T1/2 and Pn (VIII)
(2.3) Large-scale Shell Model
(2.4) Density Functional HFB QRPA
shell-model code ANTOINE restricted, but
sufficiently large SP model space, with residual
interaction split into (I) monopole part (II)
renormalized G-matrix component monopole
interaction tuned to reproduce exp.
spectra admitted, that truncated space may still
miss some correlations.
density-functional / Greens-function-based model
finite-Fermi-systems theory not quite
selfconsistent, but with well-developed
phenomenology.

• disadvantages
• only n-magic nuclei (N50, 82, 126)
• only GT-decay
• only spherical.

• disadvantage
• only spherical nuclei

• advantages
• several essential correlations included
• treatment of ee and odd-p isotopes.

• advantages
• all types of nuclei (ee, o-mass, oo)
• includes ff-strength microscopically.

? T1/2 even shorter than those of SC-HFB QRPA
? T1/2 (in particular with ff) short
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Nuclear models to calculate T1/2 and Pn (IX)
(2.5) Fully consistent relativistic pn-QRPA
use of new density-dependent interaction in
relativistic Hartree-Bogoliubov calculations of
g.s. and particle-hole channels finite-range
Gogny D1S interaction for T1 pairing
channel inclusion of pn particle-particle
interaction.
Conclusions
J. Engel it is argued on the basis of a
measurement of a strength distribution (i.e. N82
130Cd) that the transitions at N82 calculated by
the shell model, HFB QRPA and
Density-functional FFS are too fast. this
will force the other groups to go back
and examine their calculated strength
distributions.

• disadvantages
• only spherical ee nuclei
• Ni half-lives overestimated by factor 10
• (spherical QRPA normalized to
• deformed 66Fe40 ! )
• our model predicts that 132Sn is stable
• against b-decay
• (exp. T1/240 s Qb3.12 MeV).

P. Möller there is no correct model in
nuclear physics. Any modeling of
nuclear-structure properties involves approximatio
ns to obtain a formulation that can be solved,
but that retains the essential features of the
true system.

• advantages
• theoretical T1/2 reproduce the exp. data
• for Fe, Zn, Cd, and Te
• sufficiently large model space.

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The r-process waiting-point nucleus 130Cd
...obtain a physically consistent picture!
T1/2, Q?, E(1), I?(1), log ft
Q?
Sn
7.0
8.9
J?1
?g7/2, ?g9/2
2QP
4QP
1.2
2.9
free choice of combinations
T1/2(GT) 233 ms 1130 ms 76 ms 246 ms
low E(1) with low Qb high E(1) with low
Qb low E(1) with high Qb high E(1) with high Qb
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Shape of Nr,? abundance peak
rising wing 122ltAlt130
Deficiencies explained by

• neutrino induced reactions ?
• Qian,Haxton et al. (1997)
• waiting-point concept breaks down ?
• Martinez-P. Langanke (1999)
• nuclear structure below 132Sn not understood ?
• Kratz et al. (since 1993)

• importance of ng7/2 ? pg9/2 GT
• position of ng7/2 SP state
• nd3/2 rel. to nh11/2
• spin-orbit splitting n3p3/2 - n3p1/2
• n2f7/2 - n2f5/2
• p2p3/2 - p2p1/2
• p1f7/2 - p1f5/2
• N82 shell quenching

QRPA (Nilsson, Woods-Saxon, Folded Yukawa) OXBASH
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Level systematics of the lowest 1 state in
neutron-rich even-mass In isotopes
OXBASH (B.A. Brown, Oct. 2003)
Experimental
1 2181
1 2120
(new)
1 1173
1731 keV
1 1382
1 688
(old)
Reduction of the TBME (1) by 800 keV
1 243
3 0
3 389
3 0
3 0
3 473
124In75
126In77
128In79
1- 0
1- 0
130In81
130In81
Dillmann et al., 2003
Configuration 3 nd3/2 ? pg9/2
Configuration 1 ng7/2 ? pg9/2
Configuration 1- nh11/2 ? pg9/2
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Beta-decay odd-mass, N82 isotones
nSP states in N81 isotones
S1n3.98MeV
S1n3.59MeV
S1n5.246MeV
S1n2.84MeV
EMeV
2648
ng7/2
2643
7/2
2637
7/2
67 4.1
45 4.25
2607
24 4.5
7/2
ng7/2
88 4.0
2565
89 4.0
P1n25 P2n45 P3n11
P1n39 P2n11 P3n 4.5
P1n29 P2n 2
S1n1.81MeV
Pn4.4
Pn9.3
P4n 8.5 P5n 1
908
1/2
1/2
814
1/2
728
601
1/2
536
ns1/2
524
3/2
472
3/2
414
3/2
331
3/2
282
nd3/2
0
nh11/2
Ib log(ft)
11/2-
11/2-
11/2-
11/2-
2.3 6.3
0.9 6.4
1.2 6.3
0.6 6.4
0.5 6.45
123Mo81
125Ru81
131Sn81
127Pd81
129Cd81
42
44
46
50
48
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Effects of N82 shell quenching
change of T1/2 ?
19
Possible effect of shell quenching
Nilsson potential gradual reduction of l2-term
EMeV
L2 standard
20 red.
40 red.
60 red.
10 red.
ng7/2
3549
3327
ng7/2
ng7/2
3027
912keV
684keV
ng7/2
2806
379keV
2648
2643
2637
7/2
ng7/2
2607
7/2
7/2
ng7/2
2565
nd3/2
2497
199keV
T1/24.6/6.15ms
T1/214.4/17.3ms
T1/22.0/2.85ms
1771
3/2
T1/241.4/48.4ms
T1/2157ms
1.96MeV
1057
1299keV
3/2
643keV
3/2
650
3/2
536
472
3/2
414
319keV
3/2
331
3/2
nd3/2
282
nh11/2
11/2-
11/2-
11/2-
11/2-
0
131Sn81
123Mo81
50
125Ru81
127Pd81
129Cd81
42
44
46
48
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Beta-decay of 129Ag isomers
Separation of isomers by fine-tuning of laser
frequency
pp1/2
30
pg9/2
158ms
70
46ms
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Terrestrial and stellar half-lives of
odd-mass N82 waiting-point isotopes
Isotope Experiment
QRPA(GTff))
T1/2(stellar)
T1/2(stellar)
T1/2(pg9/2) T1/2(pp1/2)
T1/2(pg9/2) T1/2(pp1/2)
131In 280ms 350ms 300ms
157ms 477ms 253ms
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129Ag 46ms 158ms
80ms 43ms 140ms 72ms
47
127Rh ------ -----
------ 14.4ms 25.4ms
17.7ms
45
125Tc ------ -----
------ 4.60ms 4.45ms
4.5ms
43
123Nb ------ -----
------ 2.01ms 1.91ms
1.98ms
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) Nuclear masses ADMC,2003 ETFSI-Q
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Astrophysical consequences

• ...mainly resulting from new nuclear structure
  information
• better understanding of formation and shape of,
  as well as r-process matter flow
• through the A130 Nr,? peak
• no justification to question waiting-point
  concept
• (Langanke et al., PRL 83, 199 Nucl. Phys. News
  10, 2000)
• no need to request sizeable effects from
  n-induced reactions
• (Qian et al., PRC 55, 1997)

? r-process abundances in the Solar System and in
UMP Halo stars... ...are
governed by nuclear structure!
short T1/2
long T1/2
Nuclear masses from AMDC, 2003 ETFSI-Q Normalize
d to Nr,? (130Te)
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Lets come back to global calculations of gross
b-decay properties
only model that can calculate on a
macroscopic-microscopic basis all types of
nuclei (nearly) all nuclear shapes g.s. and
odd-particle excited-states decays
mass models FRDM (ADNDT 59, 1995) ETFSI-Q (PLB
387, 1996) QRPA model pure GT (ADNDT 66,
1997) GT ff (see above URL
http//t16web/moeller/publications/rspeed2002.html
ADNDT, to be submitted KCh Mainz
Report (unpubl.), URL www.kernchemie.uni-mainz.de
)
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T1/2 and Pn calculations in 3 steps (I)
Typical example

• FRDM /ETFSI-Q
• ? Qb, Sn, e2
• Folded-Yukawa wave fcts.
• QRPA pure GT
• with input from mass model
• potential Folded Yukawa
• Nilsson (different ?, m )
• Woods-Saxon
• pairing-model Lipkin-Nogami
• BCS

Sn
Qb
(2) as in (1) with empirical spreading of SP
transition strength, as shown in experimental
Sb(E)
(3) as in (2) with addition of first-forbidden
strength from Gross Theory
note effect on Pn !
25
T1/2 and Pn calculations in 3 steps (II)
Another spherical case
and a typical deformed case
note effect on T1/2 !
Note low-lying GT-strength ff-strength
unimportant!
26
Experimental vs. theoretical b-decay properties
T1/2, Pn gross b-strength properties
from FRDM QRPA
Requests (I) prediction / reproduction of
correct experimental number (II)
detailed nuclear-structure understanding
? full spectroscopy of
key isotopes, like 80Zn50 , 130Cd82.
Total Error 3.73
Total Error 5.54
Pn-Values
Half-lives
QRPA (GT)
QRPA (GT)
QRPA (GTff)
QRPA (GTff)
(P. Möller et al., PR C67, 055802 (2003))
Total Error 3.52
Total Error 3.08
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Effects of T1/2 on r-process matter flow
T1/2 (GT ff)

• Mass model ETFSI-Q
• all astro-parameters kept constant
• r-process model
• waiting-point approximation

r-matter flow too slow
r-matter flow too fast
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Conclusion
Despite impressive experimental and theoretical
progress, situation of
nuclear-physics data
for explosive nucleosynthesis calculations
still unsatisfactory !

• better global models

with sufficiently large SP model space, for all
nuclear shapes (spherical, prolate, oblate,
triaxial, tetrahedral,)
and all nuclear types (even-even, odd-particle,
odd-odd)

• more measurements

masses gross b-decay properties level
systematics full spectroscopy of selected
key waiting-point isotopes