BETA-STRENGTH FUNCTION IN NUCLEOSYNTHESIS CALCULATIONS Yu.S. Lutostansky, I.V. Panov, and V.N. Tikhonov National Research Center "Kurchatov Institute" Institute of Theoretical and Experimental Physics ITEP

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BETA-STRENGTH FUNCTION IN NUCLEOSYNTHESIS
CALCULATIONSYu.S. Lutostansky, I.V. Panov, and
V.N. TikhonovNational Research Center
"Kurchatov Institute" Institute of Theoretical
and Experimental PhysicsITEP 09.09.2013
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PROCESSES OF NUCLEOSYNTESIS.
Superheavy nuclei
ß-decay
fission
s-process track
r-process track
ß-decay
The tracks of elements synthesis in s (slow)-
and r (rapid)- processes.
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NUCLEOSYNTHESIS OF THE HEAVY NUCLEI
NUCLEOSYNTHESIS OF THE HEAVY NUCLEI in s
(slow) and r (rapid)- processes ? nuclei
withT1/2 ? 1 y. ? T1/2 lt 1 y.
 - predictions.
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I - METHOD r Process equations for the
concentration calculations

• Concentrations n(A,Z) are changing in time (may
  be more than 4000 equations)
•  
• dn(A, Z)/dt ??(A, Z).n(A, Z) ?n?(A,
  Z).n(A, Z) ??n(A1, Z).n(A1, Z)
• ?n?(A1, Z).n(A1,
  Z) ??n(A, Z).n(A, Z)
• ??(A, Z1).n(A, Z1) P?(A, Z1)
  ??(A1,Z1).n(A1,Z1) P1n(A1,Z1)
• ??(A2,Z1).n(A2,Z1)P2n(A2,Z1)
  ??(A3,Z1)n(A3,Z1) P3n(A3,Z1)
• ??(A, Z) Ff (A, Z),
• ?n? and ??n rates of (n,?) and (?,n)
  -reactions, ??ln(2/T1/2) ?-decay rate, P? -
  probability of (A, Z) nuclide creation after
  ?-decay of (A,Z-1) nuclide. Branching
  coefficients of isobaric chains - P1n, P2n, ?3n
  corresponds to probabilities of one-, two- and
  three- neutrons emission in ?- decay of the
  neutron-rich nuclei the total probability of
  the delayed neutrons emission is the sum
  
• Ff (A, Z) describes fission processes
  spontaneous and beta-delayed fission.
• ??(A, Z) - neutrino capturing processes.
• Inner time scale is strongly depends on the
  nuclear reactions rates.

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II. NUCLEOSYNTHESIS WAVE MOVEMENT
Concentrations n? for three time moments
calculated for r-process conditions nn1024
?m-3, ?91. 109K Lutostansky Yu.S., et
al. Sov. J. Nucl. Phys. 1985, v. 42.
s
s
s
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ß-Delayed processes in very neutron-rich nuclei
Delayed neutron emission -(ß, n)
------------------------------------ Multi-neutro
n ß delayed emission - (ß,
kn) ------------------------------------ ß
delayed fission - (ß,f)
GTR
GTR
AR
pigmy-resonances
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Beta Delayed Multi-Neutron Emission
Probability for (ß, 2n) - emission
Probability for (ß, kn) - emission
U, I?(U) energies and intensities in the
daughter nucleus, Wn(U, E) probability of
neutron emission
qi and qf level densities of compound and
final nucleus, ?n(?) transitivity factor
Lyutostansky Yu.S., Panov I.V., and Sirotkin V.K.
The ?-Delayed MultiNeutron Emission. Phys.
Lett. 1985. V. 161B. ?1. 2, 3. P. 9-13.
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BETA-DELAYED NEUTRONS IN NUCLEOSYNTESIS
exp
Calculated abandancies 1 with out (ß,n)-effect
2 with (ß,n)-effect in the relative units
(?109 ?, nn 1024 ??-3). Calc.
Lutostansky Yu., Panov I., et al. Sov. J. Nucl.
Phys. 1986. v. 44.
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BETA-STRENGTH FUNCTION CALCULATIONS-1
COLLECTIVE ISOBARIC STATES
protons
neutrons
G-T - SELECTION RULES ? j 01 ? j 1 j
l1/2 ? j l1/2 ? j 0 j l1/2 ?
jl1/2 ? j 1 j l1/2 ? j l1/2
j l1/2? j l1/2
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BETA-STRENGTH FUNCTION CALCULATIONS-2
MICROSCOPIC DESCRIPTION - 1
The GamowTeller resonance and other
charge-exchange excitations of nuclei are
described in Migdal TFFS-theory by the system of
equations for the effective field



where Vpn and Vpnh are the effective fields of
quasi-particles and holes, respectively Vpn? is
an external charge-exchange field dpn1 and dpn2
are effective vertex functions that describe
change of the pairing gap ? in an external field
G? and G? are the amplitudes of the effective
nucleonnucleon interaction in, the particlehole
and the particleparticle channel ?, ?h, f1 and
f2 are the corresponding transition densities.
-------------------------------------------------
--------------------------


Effects associated with change of the pairing gap
in external field are negligible small, so we set
dpn1 dpn2 0, what is valid in our case for
external fields having zero diagonal elements
Migdal. Pairing effects are included in the
shell structure calculations e? ? E?
-------------------------------------------------
------------------------- The selfcosistent
microscopic theory used for the beta-strength
function calculations.

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BETA-STRENGTH FUNCTION CALCULATIONS-3
MICROSCOPIC DESCRIPTION - 2
For the GT effective nuclear field, system of
equations in the energetic ?-representation has
the form Migdal, Gaponov
G-T selection rules ? j 01 ? j 1
jl1/2 ? j l1/2 ? j 0 jl1/2 ?
jl1/2 ? j 1 jl1/2 ? jl1/2
j l1/2? j l1/2

where n? and e? are, respectively, the occupation
numbers and energies of states ?. ----------------
--------------------------------------------------
---------------------------
Local nucleonnucleon d-interaction G? in the
Landau-Migdal form used ?? ?0 (f0' g0'
s1s2) t1t2 d(r1- r2) where coupling constants
of f0' isospin-isospin and g0' spin-isospin
quasi-particle interaction with L 0.
-------------------------------------------------
-----------------------------------

Constants f0' and g0' are the phenomenological
parameters.
Matrix elements MGT
where ???
mathematical deductions
G-T values are normalized in FFST
Standard sum rule for st-excitations
Effective quasiparticle charge
is the quenching parameter of the theory.
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BETA-STRENGTH FUNCTION CALCULATIONS-1
MICROSCOPIC DESCRIPTION - 3
RESONANCE STRUCTURE OF BETA-STRENGTH FUNCTION
1. Discrete structure of beta-strength function.
Partial function Ci (old
variant)

2. Resonance structure of beta-strength
function. Partial function


The Bright-Wigner form for E gt Sn
Sn
?i value up to Migdal is ? 2 Im ? (e
iI) and ? ? .e e ße3 ? e2 e
O(e4), where
?i(?i) 0,018 Ei2 ???

• Exp. Krofcheck D., et al. Phys. Rev. Lett. 55
  (1985) 1051.
• - - Borzov I. Fayans S., Trykov E. Nucl. Phys.
  ?. 584 (1995) 335.
• Borovoi A., Lutostan- sky Yu., Panov I.,
  et al. JETP Lett. 45 (1987) 521

71Ge
Yu. V. Gaponov and Yu. S. Lyutostansky, Sov. J.
Phys. Elem. Part. At. Nucl. 12, 528 (1981).
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BETA-STRENGTH FUNCTION FOR 127Xe
Dependence from eg
1 - Breaking line experimental data (1999) M.
Palarczyk, et. al.

Phys. Rev. 1999. V. 59. P. 500 2
Solid red line TFFS calculations with ?q 0.9
3 - Solid black line calculations with ?q
0.8 Yu.S. Lutostansky, N.B. Shulgina.
Phys. Rev. Lett. 1991. V.67. P. 430
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QUENCHING EFFECT for 127Xe
1 - Breaking line experimental data M.
Palarczyk, et. al. Phys. Rev. 59 (1999) 500 2 -
line TFFS calculations with ?q 0.9
Yu.S. Lutostansky, and V.N. Tikhonov. Bull.
Russ. Acad. Sci. Phys. 76, 476 (2012). 3 - - -
- TFFS calculations with ?q 0.8
Yu.S. Lutostansky, and N.B. Shulgina. Phys. Rev.
Lett. 67 (1991) 430
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QUENCHING EFFECT EXPERIMENT
Standard sum rule for st-excitations For G-T
beta-strength function
In FFST
Migdal theory For experimental data sum
rule S B(GT) must
be 3.(N Z).

Ideal Emax 8
127Xe
71Ge

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INTERACTION CONSTANTS
For the (tt) coupling constant f0/ the value f0/
1.35 was used, taken from comparison of
calculating energy splitting between the analog
and anti-analog isobaric states (IS) with the
experimental data for the large number of nuclei
Gaponov, Lutostansky 1970 - 1972.
Three main parameters of FFST theory eq, f0/,
g0/ are taken from exp. and calc. data
comparison. --------------------------------------
------ eq from quenching effect f0/
and g0/ from energy splitting data
112-124Sn (3He, t) reaction
For (st) coupling constant g0/ value g0/ 1.22
0.04 received from comparison of calculated
energy differences between GTR and the low-lying
pigmy-resonance with the experimental data for
nine Sb isotopes K. Pham, J. Jänecke, D. A.
Roberts, et al., Phys. Rev. C 51 (1995) 526.
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BETA - DELAYED FISSION
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Beta Delayed Fission Calculations
Probabilities - Pßf
Beta Strength function
?(?) widths approximation ?(?) a?E2
ß?E3 where a 1/eF and ß a, so we
used only the first term. As ?f ?n so
neutron emission dominates when this
energetically possible. Sub-barrier fission
probabilities in the daughter nucleus are small
to gamma decay of exited states (barrier was
taken in standard parabolic form). Main
dependence of Pßf is from barrier energy Bf .
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Neptunium Beta Delayed Fission Calculations
Yu.S. Lutostansky, V.I. Liashuk, I.V. Panov.
Influence of the delayed fission on production
of transuranium elements in the explosive
nucleosynthesis. Preprint ITEP 90-25. 1990
Moscow.
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Dubnium Beta Delayed Fission Calculations
I. Panov, Yu. Lutostansky, F.-K. Thielemann 2013
Upper panel the neutron beta-delayed emission
probabilities Pßdn (dashed line), beta-delayed
fission probabilities Pßdf (line) and number of
delayed neutrons per one decay (in percents) In
(dotted line) for isotopes of Dubnium (Z105)
down panel total energy of beta-decay Qß (line),
neutron separation energy Sn (dashed line) and
fission barriers (bold line) for the same
isotopes (in MeV).
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Factor of the concentration losing in
Prompt-process
Np ß-delayed fission probabilities
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MODEL DESCRIPTION OF Sß(E) - 1
Mat. model developed for the approximate
solutions of equations of the FFST theory by the
quasi-classical method. -------------------------
---- 2 new parameters ?E EF(n) EF(p)
Els average energy of the spinorbit
splitting
Wigners SU(4) super-symmetry restoration in the
heavy nuclei
Calculated (circles ?) and experimental ()
dependencies of the relative energy
y(x)?(EGTR-EAR)/Els from the dimensionless value
x?E/Els. Black circles (?) connected by line
calculated values for Sn isotopes.
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MODEL DESCRIPTION OF Sß(E) 2. T1/2 calculations
1988.
Time of new nuclei synthesis
ß-decay time Fermi-function
The dependence of r-process duration time on mass
A-value under different external conditions
curve 1) constant nn1026 cm-3, T1.5 109K 2)
the same nn, T1.109K 3) dynamical
calc. with ?02.105 g/cm5, T1.109K ?(t) ?0 .
??p (-t/?H), ?(t) ?0 . ???(-t/3?H).
Yu.S. Lutostansky, and I. V. Panov. Astron.
Letters. 14, no 2 (1988) 168.
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Neutrino capturing
En
GT and IAS Resonances in Sß(E )-function
n
M2GTR 3 (N-Z) eq2 M2IAS (N-Z)