Transcript Document

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 – 09.09.2013
PROCESSES OF NUCLEOSYNTESIS.
Superheavy
nuclei
β-decay
s-process track
β-decay
r-process track
The tracks of elements synthesis in s (slow)- and r (rapid)- processes.
fission
NUCLEOSYNTHESIS OF THE HEAVY NUCLEI
NUCLEOSYNTHESIS OF THE HEAVY NUCLEI in s (slow) and r (rapid)- processes
 – nuclei withT1/2  1 y. ; О – T1/2 < 1 y.; + - predictions.
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(A+1, Z).n(A+1, Z) +
+ n(A–1, Z).n(A–1, Z) – n(A, Z).n(A, Z) +
+ (A, Z–1).n(A, Z–1) × P(A, Z–1) + + (A+1,Z–1).n(A+1,Z–1) × P1n(A+1,Z–1)+
+ (A+2,Z–1).n(A+2,Z–1)×P2n(A+2,Z–1) + (A+3,Z–1)n(A+3,Z–1) × P3n(A+3,Z–1)+
+ (A, Z) + Ff (A, Z),
n and n — rates of (n,γ) and (γ,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, Р3n 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:
Pn   Pkn
k
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.
II. NUCLEOSYNTHESIS WAVE MOVEMENT
Concentrations:
nА=  n(A, Z)
s
z
for three time
moments
calculated
for r-process
conditions:
s
nn=1024 сm-3,
Т9=1.= 109K
s
Lutostansky Yu.S.,
et al.
Sov. J. Nucl. Phys.
1985, v. 42.
β-Delayed processes in very neutron-rich nuclei
Delayed neutron emission (β, n)
-----------------------------------Multi-neutron β – delayed
emission - (β, kn)
-----------------------------------β – delayed fission - (β,f)
GTR
GTR
AR
“pigmy”-resonances
Beta – Delayed Multi-Neutron Emission
Probability for (β, 2n) - emission:
f
Q
   f (Z  1, Q  E )S  ( E ) 
i
P2 n 
B2 n 
ijlm
dE
tot
Q
  f (Z  1, Q  E )S  ( E )dE
i
0
i
Probability for (β, kn) - emission:
Pkn 
Q Qkn
  I  (U )W (U , E )dUdE
n
Bkn 0
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,
Тn(Е) — transitivity factor
Tn (E)q f ( U  E  Bn )
Wn ( U, E)  UB
n
 Tn (E' )q f (U  E'Bn )dE'2q i (U)Г  (U)
0
Lyutostansky Yu.S., Panov I.V., and Sirotkin V.K. “The -Delayed Multi–Neutron Emission.”
Phys. Lett. 1985. V. 161B. №1. 2, 3. P. 9-13.
BETA-DELAYED NEUTRONS IN NUCLEOSYNTESIS
exp
Calculated abandancies: 1– with out (β,n)-effect; 2 – with (β,n)-effect; in the relative units
(Т=109 К, nn =1024 см-3).
Calc.: Lutostansky Yu., Panov I., et al. Sov. J. Nucl. Phys. 1986. v. 44.
BETA-STRENGTH FUNCTION CALCULATIONS-1:
COLLECTIVE ISOBARIC STATES
protons
neutrons
G-T - SELECTION RULES : Δ j =0;±1
Δ j =+1: j =l+1/2 → j =l–1/2
Δ j =0: j =l+1/2 → j=l+1/2
Δ j = –1: j =l–1/2 → j =l+1/2
j =l–1/2→ j =l–1/2
BETA-STRENGTH FUNCTION CALCULATIONS-2:
MICROSCOPIC DESCRIPTION - 1
The Gamow–Teller resonance and other charge-exchange excitations of nuclei are
described in Migdal TFFS-theory by the system of equations for the effective field:

V pn  eqV pn   Г np,
np  pn

h
V pnh   Г np,

np
pn

1
d 1pn   Г np,

np pn
2

2
d pn
  Г np,

np pn
pn
pn
pn
pn
where Vpn and Vpnh are the effective fields of quasi-particles and holes, respectively;
Vpnω is an external charge-exchange field; dpn1 and dpn2 are effective vertex functions that
describe change of the pairing gap Δ in an external field;
Γω and Γξ are the amplitudes of the effective nucleon–nucleon interaction in, the particle–hole
and the particle–particle channel;
ρ, ρh, φ1 and φ2 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:
ελ → Eλ =  2  2
-------------------------------------------------------------------------The selfcosistent microscopic theory used for the beta-strength function calculations.
BETA-STRENGTH FUNCTION CALCULATIONS-3:
MICROSCOPIC DESCRIPTION - 2
For the GT effective nuclear field, system of equations in the energetic λ-representation
has the form [Migdal, Gaponov]:
G-T selection rules:
Δ j =0;±1
Δ j =+1: j=l+1/2 → j =l–1/2
Δ j =0: j=l±1/2 → j=l±1/2
Δ j = –1: j=l–1/2 → j=l+1/2
j =l–1/2→ j =l–1/2
where nλ and ελ are, respectively, the occupation numbers and energies of states λ.
--------------------------------------------------------------------------------------------Local nucleon–nucleon δ-interaction Γω in the Landau-Migdal form used:
Г = С0 (f0′ + g0′ σ1σ2) τ1τ2 δ(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 : M 2 
GT
   A  V


1 2
1 2
G-T M
2
i values
1 2
where χλν – mathematical deductions
1 2
 M  e 3( N  Z )
 M i2  3( N  Z )
2
i
are normalized in FFST:
2
q
i
Standard sum rule for στ-excitations:
i
Effective quasiparticle charge eq2  0.8  1.0 is the “quenching” parameter of the theory.
BETA-STRENGTH FUNCTION CALCULATIONS-1:
MICROSCOPIC DESCRIPTION - 3
RESONANCE STRUCTURE OF BETA-STRENGTH FUNCTION
1. Discrete structure of beta-strength function. Partial function:
(old variant)
Si (E ) = Ci
1
E
M
( i , E )
2
i
2. Resonance structure of beta-strength function.
Partial S i (E ) function:
Sn
Si (E ) = Si (E )
Ãi
( E  ωi ) 2  Ã i2
The Bright-Wigner form for E > Sn
Гi value up to Migdal is: Г = – 2 Im [∑ (ε + iI)]
and Г =  .ε | ε | + βε3 + γ ε2 | ε | + O(ε4)…, where    F1
Гi(i) = 0,018 Ei2 МэВ
71Ge
Yu. V. Gaponov and Yu. S.
Lyutostansky, Sov. J.
Phys. Elem. Part. At.
Nucl. 12, 528 (1981).
Exp. Krofcheck D., et al.
Phys. Rev. Lett. 55 (1985)
1051.
- - - Borzov I. Fayans S.,
Trykov E. Nucl. Phys. А.
584 (1995) 335.
Borovoi A., Lutostansky Yu., Panov I., et al.
JETP Lett. 45 (1987) 521
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 еq= 0.9 ;
3 - Solid black line – calculations with еq= 0.8 : Yu.S. Lutostansky, N.B. Shulgina.
Phys. Rev. Lett. 1991. V.67. P. 430;
QUENCHING EFFECT for
127Xe
1 - Breaking line – experimental data: M. Palarczyk, et. al. Phys. Rev. 59 (1999) 500;
2 - line –TFFS calculations with еq= 0.9;
Yu.S. Lutostansky, and V.N. Tikhonov. Bull. Russ. Acad. Sci. Phys. 76, 476 (2012).
3 - - - - –TFFS calculations with еq= 0.8:
Yu.S. Lutostansky, and N.B. Shulgina. Phys. Rev. Lett. 67 (1991) 430;
QUENCHING EFFECT – EXPERIMENT
M
Standard sum rule for στ-excitations:
i
Emax
For G-T beta-strength function:
 3( N  Z )
2
i
 S ( E) dE  3 ( N  Z )
β
0
Ideal Emax= ∞
Emax
In FFST [Migdal] theory:
S
0
For experimental data sum rule:
 B(GT ) i
β
( E ) dE  eq2  3 ( N  Z )
Σ B(GT) =  B(GT ) i Ei must be = 3.(N – Z).
i
i
3( N  Z )
127Xe
71Ge
INTERACTION CONSTANTS
For the (ττ) 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].
112-124Sn (3He,
t) reaction
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
For (στ) 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].
BETA - DELAYED FISSION
Beta – Delayed Fission Calculations
Q
Probabilities - Pβf :
Pf 
Гf
  f (Z, Q  E)S (E) Г
0
i
i
Q
dE
tot
i
f
(
Z
,
Q

E
)
S
(E)dE




0 i
Beta Strength function:
S (E) 
CN
M i2 (E i )

2 i
Г( E )
Г 2 (E)
(E i  E) 
4
2
# Г(Е) widths approximation: Г(Е) = α·E2 + β·E3 + …
where α ≈ 1/εF and β « α, so we used only the first term.
# As Гf « Гn 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 Pβf is from barrier energy Bf .
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.
Dubnium Beta – Delayed Fission Calculations
I. Panov,
Yu. Lutostansky,
F.-K. Thielemann
2013
Upper panel: the neutron beta-delayed emission probabilities Pβdn (dashed line), beta-delayed fission
probabilities Pβdf (line) and number of delayed neutrons per one decay (in percents) In (dotted line)
for isotopes of Dubnium (Z=105); down panel: total energy of beta-decay Qβ (line), neutron
separation energy Sn (dashed line) and fission barriers (bold line) for the same isotopes (in MeV).
Factor of the concentration losing in Prompt-process
Np β-delayed fission
probabilities
MODEL DESCRIPTION OF Sβ(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) =

4
N Z
EF
3
A
Els – average energy of
the spin–orbit splitting
Wigner’s SU(4)
super-symmetry
restoration in the
heavy nuclei
Calculated (circles – ○) and experimental (■) dependencies of the relative energy y(x)=Δ(EGTREAR)/Els from the dimensionless value x=E/Els. Black circles (●) connected by line – calculated
values for Sn isotopes.
y
1  b g0
E ÃÒÐ  E ÀÐ
2
 ( g0  f 0) x  b
[1  c( A) x 2 ]1 ; x  E / Els ; b  [1  (2 A) 1/ 3 ]; c( A)  0.8 A1/ 3
Els
g0 x
3
MODEL DESCRIPTION OF Sβ(E) – 2. T1/2 calculations 1988.
Time of new nuclei synthesis
β-decay time:
Q
1
1/ 2
T
   S ( i ) ( E )  f ( Z , Q  E ) dE
i
0
Fermi-function:
E
f   F (Z ,U )U U 2  1( E  U ) 2 S L (U ) dU
1
The dependence of r-process duration time on mass A-value under different external conditions:
curve 1) – constant nn=1026 cm-3, T=1.5 109K; 2) – the same nn, T=1.109K; 3) – dynamical calc.
with ρ0=2.105 g/cm5, T=1.109K [(t) = 0 . ехp (-t/H), Т(t)= Т0 . ехр(-t/3H)].
Yu.S. Lutostansky, and I. V. Panov. Astron. Letters. 14, no 2 (1988) 168.
E
Neutrino capturing
GT and IAS
Resonances
in Sβ(E )-function

g A2
 ( E )  . 3 4
c
E Q
p
e
Ee S  ( E ) F ( Z , Ee ) dE
0
M2GTR ≈ 3 (N-Z) eq2
M2IAS ≈ (NZ)