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(1)NIC-IX Satellite WS 2006
- Institut für Kernchemie, Univ. Mainz, Germany- HGF VISTARS, Germany- Department of Physics, Univ. of Notre Dame, USA
Karl-Ludwig Kratz
Gross –decay propertiesfor astrophysical applications
(2)NIC-IX Satellite WS 2006
Nuclear data in astrophysics
What data are needed in nuclear astrophysics ?
(A) Quiescent nucleosynthesis e.g. H-, He-burning; s-process
• nuclear masses (reaction Q-values)
• charged-particle reaction rates
(e.g. (p,), (,), (,n))
• neutron capture-rates
• nuclear structure properties
(e.g. Esp, J, C2S)
for 10’s to 100’s of isotopes NEAR -stability
(B) Explosive nucleosynthesis e.g. rp-process, p-process;
“weak” and “main” r-process
• nuclear-masses (Q, Sp, Sn)
• half-lives (T1/2,; g.s. , isomers)
• -delayed quantities (Pp, Pn, Pf)
• neutron capture rates
• neutrino reactions
• nuclear-structure-properties
(e.g. 2, Esp, J …)
for 100’s to 1000’s of isotopes FAR-OFF -stability
(3)NIC-IX Satellite WS 2006
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 parametersamong SEVERAL astrophysics parameter sets
Depending on “mentality of the star-couturier”, nuclear data are considered
unimportant
astro-parameters dominate
n.d. just “telephone numbers”
(too) many (free) parameters
n.d. effects invisible
important
nuclear and astro-parameters of equal standing
n.d. to constrain astro-parameters
“learning” nucl. structure from astro-observables
mathematical nuclearastrophysics
(4)NIC-IX Satellite WS 2006
-2
0
2
4
6
8
10
12
14
16
38 40 42 44 46 48
Mass number
scaled theoretical solar r-processscaled solar r-process
Nb
Zr
Y
SrMo
Ru
Rh
Pd
Ag
Ba
La
Ce
Pr
Nd
Sm
Eu
Gd
Tb
Dy
Ho
Er
Tm
Yb
Lu
Hf
Os
Ir
Pt
Au
Pb
ThU
GaGe
CdSn
Elemental abundances in UMP halo stars
r-process observables
Solar system isotopic abundances, Nr,
“FUN-anomalies” in meteoritic samples
isotopiccomposition Ca, Ti, Cr, Zr, Mo, Ru, Nd, Sm, Dy ↷ r-enhanced
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).
[‰
]
ALLENDE INCLUSIONEK-1-4-1
Mass number
CS 22892-052 abundances
T9=1.35; nn=1020 - 1028
Basic astronomical question: r-process
, Bi
(5)NIC-IX Satellite WS 2006
Nuclear models to calculate T1/2 and Pn – (I)
Theoretically,
the two gross/ integral -decay quantities, T1/2 and Pn, are interrelated via their usualdefiniton in terms of the so-called
-strength function [S(E)]
What is that?
… a natural adoption of the strength function concept employed in other areasof nuclear physics,
e.g.: single-particle strength functions, s-, p-wave neutron strength functions, multipole strength functions for photons.
Sc = <²c>
Sc refers to the behavior of the squares of overlap integrals (²c) between two sets of nuclear wave functions:l represents various states of excitation, classified by E, J, T;c refers to the different reaction / decay channels, classified by Epart, lpart,… is the density of levels .
(6)NIC-IX Satellite WS 2006
1,E -01
1,E +00
1,E +01
1,E +02
1,E +03
1,E +04
1,E +05
1,E +06
1 2 3 4 5 6 7 8 9 10
1
103
106
Nuclear models to calculate T1/2 and Pn – (II)
Application to -decay:
“Theoretical” definition (Yamada & Takahashi, 1972)
S = D-1 · M(E) ² · (E) [s-1MeV-1]
M(E) average -transition matrix element (E) level density D const., determines Fermi coupling constant gv²
“Experimental” definition (Duke et al., 1970)
S(E) =b(E)
f(Z, Q-E) · T1/2
[s-1MeV-1]
b(E) absolute -feeding per MeV,f(Z, Q-E) Fermi function,T1/2 decay half-life.
T1/2 as reciprocal ft-value per MeV
T1/2 = S(Ei) x f (Z,Q-Ei)
0Ei Q
11
1
f(Z, Q-Ei) (Q-Ei)5
S(E)
E*[MeV]
Q
Fermi function
T1/2 sensitive to lowest-lying resonances in S(Ei)Pn sensitive to resonances in S(Ei) just beyond Sn
↷ easily “correct” T1/2 with wrong S(E)
same T1/2 !
1 5 10
1
3x103
6x105
(7)NIC-IX Satellite WS 2006
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:
(1) “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
Examples:Kratz-Herrmann Formula (1973)Gross Theory (1973)·
·New exponential law for T1/2(+) (Zhang & Ren; 2006)T1/2(-) from GTGR + known log(ft)’s (Kar, Chakravarti & Manfredi; 2006)
·
(2) Models that use an effective nuclear interaction and solve the microscopic, quantum-mechanical Schrödinger or Dirac equation.
• provides nuclear wave functions• within the same framework, describes a number of nucl. properties (e.g. g.s.-shape; Esp, J, log(ft), T1/2 … )
Examples:
FRDM+QRPA (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)
(8)NIC-IX Satellite WS 2006
Nuclear models to calculate T1/2 and Pn – (IV)
(1) Simple “statistical” approaches
assumptions:• -decay energy is large (Q ≳ 5 MeV)• high level density• S(E) is a smooth function of E (e.g. S=const.; S (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
Pn =
CEi Q
S(Ei) x f (Z,Q-Ei)
S(Ei) x f (Z,Q-Ei)
SnEi Q
with S=const.
Pn ≃ a (Q – Sn)
(Q – C)
b
a, b as “free parameters”, to be determined by a log-log fit to known Pn-valuesC is a “cut-off parameter” (↷ pairing-gap in -decay daughter)
(9)NIC-IX Satellite WS 2006
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
Region Lin. regression Least-squares fit
29 Z 43
47 Z 57
29 Z 57
a [%] b r² a [%] b red. ²
88.2 4.1 0.81 106 5.5 8140 0.6
84.4 3.9 0.86123 4.7 5741 0.5
81 4.7 7821 0.3
85.2 4.0 0.83
… as a kind of “joke”:T1/2 ≃ a (Q-C)
Lin. regression Least-squares fita [ms] b r² a [ms] b red. ²
2.74E06 4.5 0.72 7.07E05 4.0 1.1E045.33E05 0.4
dashed line full line
Parameters from fit to known T1/2 of n-rich nuclei
-b
(10)NIC-IX Satellite WS 2006
Nuclear models to calculate T1/2 and Pn – (VI)
… NO joke !in 2006, two examples for big steps BACKWARDS :
(I) X. Zhang & Z. Ren; PRC73, 014305“New exponential law for + decay half-lives of nuclei far from -stable line”
“…we have discovered a new exponential law for T1/2(+)…as a function of neutron number…”
log10 T1/2 = a x N + b
authors give fit parameters for a and b,for (I) different Z-regions (II) allowed +-decay (III) first-forbidden +-decay (IV) second-forbidden +-decay
↷ finally “a simple and accurate formula” emerges:
log10 T1/2 = (c1Z + c2) N + c3Z + c4
(II) K. Kar, S. Chakravarti & V.R. Manfredi;arXiv: astro-ph/0603517 v1“Beta-decay rates (115 < A < 140) for r-processnucleosynthesis”
… the xth re-invention of the Gross Theory !“… shell model results… indicate that the GT strengthdistribution.. can be taken as a Gaussian.”
“…GT strength distributes among 3 different types offinal states:(a) discrete low-lying states with known log ft’s;(b) discrete states above with unknown strengths;(c) a part of the GT giant resonance (GTGR).”
admitted “problems”:centroid of GTGR ↷ from Bertsch & Esbensen (1987)width of GTGR ↷ free parameter !
“…useful to experimental physicists for analyzing+-decay data.”
(11)NIC-IX Satellite WS 2006
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 comprehensiveinclusion of nuclear correlations. None of the methods includes all important correlations, however.”
(2.1) FRDM + 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.
disadvantage: not “self consistent” advantages: global model for all shapes and
types of nuclei; large model space
VGT = GT : _ · +
GT = 23 MeV/A
(2.2) Self-consistent Skyrme-HFB + QRPA
Skyrme interaction SKO ↷ reasonable reproduction of energies and strengths of GT resonances; strength of T=0 pairing “adjusted” to fit known T1/2
disadvantages: only spherical shape;only GT;only -magic (N=50, 82, 128);Skyrme interaction not goodenough to make…decisive improvement
advantage: self-consistency
↷ T1/2 shorter than those from FRDM + QRPA
(12)NIC-IX Satellite WS 2006
Nuclear models to calculate T1/2 and Pn – (VIII)
(2.3) Large-scale Shell Model
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 somecorrelations.
disadvantages:only -magic nuclei (N=50, 82, 126);only GT-decay;only spherical.
advantages:several essential correlations included;treatment of ee and odd- isotopes.
↷ T1/2 even shorter than those of SC-HFB + QRPA
(2.4) Density Functional HFB + QRPA
density-functional / Greens-function-based model + finite-Fermi-systems theory;not quite selfconsistent,but with well-developed phenomenology.
disadvantage:only spherical nuclei
advantages:all types of nuclei (ee, o-mass, oo);includes ff-strength microscopically.
↷ T1/2 (in particular with ff) short
(13)NIC-IX Satellite WS 2006
Nuclear models to calculate T1/2 and Pn – (IX)
(2.5) Fully consistent relativistic -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 T=1 pairing channel;inclusion of particle-particle interaction.
disadvantages:only spherical ee nuclei;Ni half-lives overestimated by factor ∼ 10(spherical QRPA “normalized” to deformed 66Fe40 …! );“… our model predicts that 132Sn is stableagainst -decay…”(exp.: T1/2=40 s ; Q=3.12 MeV).
advantages:“…theoretical T1/2 reproduce the exp. datafor Fe, Zn, Cd, and Te…”;sufficiently large model space.
Conclusions
J. Engel“… it is argued on the basis of a measurement of astrength distribution (i.e. N=82 130Cd) that thetransitions at N=82 calculated by the shell model,HFB + QRPA and Density-functional + FFS are too fast.…this will force the other groups to go back andexamine their calculated strength distributions.”
P. Möller“…there is no “correct” model in nuclear physics.Any modeling of nuclear-structure properties involvesapproximations … to obtain a formulation that can besolved…, but that “retains the essential features” of the true system.”
(14)NIC-IX Satellite WS 2006
The r-process “waiting-point“ nucleus 130Cd
Q
7.0 8.9
2.9
2QP
4QP
J=1+
{g7/2, g9/2}
Sn
T1/2, Q, E(1+), I(1+), log ft
1.2
...obtain a physically consistent picture!
“free choice” of combinations:
low E(1+) with low Q
high E(1+) with low Q
low E(1+) with high Q
high E(1+) with high Q
T1/2(GT) 233 ms1130 ms 76 ms 246 ms
(15)NIC-IX Satellite WS 2006
Shape of Nr, abundance peak rising wing 122<A<130
solar r abundances
“short“ T1/2
• 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 g7/2 g9/2 GT position of g7/2 SP stated3/2 rel. to h11/2
spin-orbit splitting 3p3/2 - 3p1/2
f7/2 - f5/2
p3/2 - p1/2
f7/2 - f5/2
N=82 shell quenching
QRPA (Nilsson, Woods-Saxon, Folded Yukawa)OXBASH
Deficiencies explained by :
(16)NIC-IX Satellite WS 2006
Reduction of the TBME (1+)by 800 keV
OXBASH(B.A. Brown, Oct. 2003)
3+ 0 3+ 3893+ 03+ 0 3+ 473
1- 0 1- 0124In75
126In77
130In81
130In81
128In79
1+ 243
1+ 688
1+ 1173
1+ 2120 1+ 2181(new)
1+ 1382
(old)
Experimental
Level systematics of the lowest 1+ statein neutron-rich even-mass In isotopes
Configuration 3+ : d3/2 g9/2
Configuration 1+ : g7/2 g9/2
Configuration 1- : h11/2 g9/2
17
31
ke
V
Dillmann et al., 2003
(17)NIC-IX Satellite WS 2006
0
0,5
1
1,5
2
2,5
3
Beta-decay odd-mass, N=82 isotones
0h11/2
282d3/2
s1/2
g7/2
524
2565 26077/2+ 7/2+ 7/2+
1/2+
1/2+
1/2+1/2+
3/2+3/2+
3/2+3/2+
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
89%4.0
88%4.0
67%4.1
45%4.25
24%4.5
g7/22648 2643 2637
601
331
728
414
814
472
908
536
S1n=5.246MeV S1n=3.98MeV S1n=3.59MeV
Pn=4.4% Pn=9.3%P1n=29%P2n= 2%
P1n=39%P2n=11%P3n= 4.5%
P1n=25%P2n=45%P3n=11%
131Sn8150
129Cd81127Pd81
125Ru81123Mo81
48 46 44 42
E*[MeV]
SP states in N=81 isotones
P4n= 8.5%P5n= 1%
Ilog(ft)
S1n=2.84MeV
S1n=1.81MeV
(18)NIC-IX Satellite WS 2006
Effects of N=82 „shell quenching“
g 9/2
g 9/2
i13/2
i13/2
p1/2
f5/2
p 1/2
p 3/2
p3/2
f7/2
f7/2
h9/2
h 11/2
h 11/2
g 7/2g 7/2 d 3/2
d 3/2
s1/2
s 1/2
d 5/2
d 5/2
g 9/2g 9/2
f5/2f5/2
p1/2
p 1/2
h 9/2 ;f 5/2
N/Z
112
70
40
50
82
126
B. Pfeiffer et al.,Acta Phys. Polon. B27 (1996)
100% 70% 40% 10%
Strength of ℓ 2-Term
5.0
5.5
7.0
6.5
6.0
Sin
gle
– N
eutr
on E
nerg
ies
(Uni
ts o
f h 0)
• high-j orbitals (e.g. h11/2)• low-j orbitals (e.g. d3/2)• evtl. crossing of orbitals• new “magic” numbers / shell gaps (e.g. 110Zr70, 170Ce112) 40 58
change of T1/2 ?
(19)NIC-IX Satellite WS 2006
0
0,5
1
1,5
2
2,5
3
3,5
4E*[MeV]
h11/2
d3/2
g7/2
131Sn8150 129Cd81127Pd81
125Ru81123Mo81
48 46 44 42
g7/2
11/2- 11/2- 11/2- 11/2-
3/2+3/2+ 3/2+ 3/2+
3/2+
3/2+
3/2+
d3/2
0
282 331414 472 536
650
1057
1771
24972565 2607 2648 2643 26372806
3027
3327
3549
g7/2
g7/2
g7/2
7/2+ 7/2+ 7/2+ g7/2
319keV
643keV
1299keV 1.96MeV
912keV684keV
379keV
199keV
T1/2=157msT1/2=41.4/48.4ms
T1/2=14.4/17.3ms
T1/2=4.6/6.15ms
T1/2=2.0/2.85ms
L2 standard 10% red. 20% red. 40% red. 60% red.
Possible effect of “shell quenching”
Nilsson potential; gradual reduction of l2-term
(20)NIC-IX Satellite WS 2006
127Ag
p1/2
g9/2
T1/2(m)=(15860) ms
T1/2(g)=(46 ) ms-9+5
129mAg 82g9/2p1/2129gAg 82
Beta-decay of 129Ag isomers
Separation of isomersby fine-tuning of laser frequency
p1/2
g9/2
30%
70%158ms
46ms
(21)NIC-IX Satellite WS 2006
Isotope Experiment QRPA(GT+ff)*)
T1/2(g9/2) T1/2(p1/2) T1/2(g9/2) T1/2(p1/2)T1/2(stellar) T1/2(stellar)
131In 280ms 350ms 300ms 157ms 477ms 253ms
129Ag 46ms 158ms 80ms 43ms 140ms 72ms127Rh ------ ----- ------ 14.4ms 25.4ms 17.7ms125Tc ------ ----- ------ 4.60ms 4.45ms 4.5ms123Nb ------ ----- ------ 2.01ms 1.91ms 1.98ms
*) Nuclear masses: ADMC,2003 & ETFSI-Q
Terrestrial and stellar half-lives of odd-mass N=82 waiting-point isotopes
49
47
45
43
41
(22)NIC-IX Satellite WS 2006
...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 -induced reactions
(Qian et al., PRC 55, 1997)
Astrophysical consequences
r-process abundances in the Solar System and in UMP Halo stars... ...are governed by nuclear structure!
Nuclear masses from
AMDC, 2003
ETFSI-Q
Normalized to Nr, (130Te)
„short“ T1/2 „long“ T1/2
(23)NIC-IX Satellite WS 2006
Let’s come back to global calculations of gross -decay properties…
… only model that can calculate on a macroscopic-microscopic basisall types of nuclei(nearly) all nuclear shapesg.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)
(24)NIC-IX Satellite WS 2006
“Typical example”:
note: effect on Pn !
T1/2 and Pn calculations in 3 steps – (I)
(1) FRDM /ETFSI-Q↷ Q, Sn, 2
Folded-Yukawa wave fcts.
QRPA pure GT with input from mass model potential: Folded Yukawa
Nilsson (different , ) Woods-Saxonpairing-model: Lipkin-Nogami
BCS
(2) as in (1) with empirical spreading of SP transition strength, as shown in experimental S(E)
SnQ
(3) as in (2) with addition of first-forbidden strength from Gross Theory
(25)NIC-IX Satellite WS 2006
Another “spherical” case:
note : effect on T1/2 !
…and a typical “deformed” case:
Note: low-lying GT-strength; ff-strength unimportant!
T1/2 and Pn calculations in 3 steps – (II)
(26)NIC-IX Satellite WS 2006
Total Error = 5.54
Total Error = 3.52
Total Error = 3.73
Total Error = 3.08
Pn-ValuesHalf-lives
(P. Möller et al.,PR C67, 055802 (2003))
Experimental vs.theoretical-decay properties
T1/2, Pn gross -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.
QRPA (GT)
QRPA (GT+ff)
QRPA (GT)
QRPA (GT+ff)
(27)NIC-IX Satellite WS 2006
T1/2 : 3
r-matter flow too slow r-matter flow too fast
Effects of T1/2 on r-process matter flow
Mass model: ETFSI-Q- all astro-parameters kept constant
r-process model: “waiting-point approximation“
T1/2 x 3
T1/2 (GT + ff)
(28)NIC-IX Satellite WS 2006
Conclusion
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
massesgross -decay propertieslevel systematicsfull spectroscopy of selected “key“ waiting-point isotopes
Despite impressive experimental and theoretical progress, situation of