PROCESSES OF NUCLEOSYNTESIS .

BETA-STRENGTH FUNCTION

IN NUCLEOSYNTHESIS CALCULATIONS

YuS Lutostansky IV Panov and VN Tikhonov

National Research Center Kurchatov Institute

Institute of Theoretical and Experimental Physics

ITEP ndash 09092013

PROCESSES OF NUCLEOSYNTESIS

The tracks of elements synthesis in s (slow)- and r (rapid)- processes

β-decay

β-decay

r-process track

s-process track

fission

Superheavynuclei

NUCLEOSYNTHESIS OF THE HEAVY NUCLEI

NUCLEOSYNTHESIS OF THE HEAVY NUCLEI in s (slow) and r (rapid)- processes ndash nuclei withT12 1 y О ndash T12 lt 1 y + ‑ predictions

I - METHOD r ndashProcess equations for the concentration calculations

Concentrations n(AZ) are changing in time (may be more than 4000 equations)

dn(A Z)dt = ndash (A Z)n(A Z) ndash n(A Z)n(A Z) + n(A+1 Z)n(A+1 Z) +

+ n(Andash1 Z)n(Andash1 Z) ndash n(A Z)n(A Z) +

+ (A Zndash1)n(A Zndash1) times P(A Zndash1) + + (A+1Zndash1)n(A+1Zndash1) times P1n(A+1Zndash1)+

+ (A+2Zndash1)n(A+2Zndash1)timesP2n(A+2Zndash1) + (A+3Zndash1)n(A+3Zndash1) times P3n(A+3Zndash1)+

+ (A Z) + Ff (A Z)

n and n mdash rates of (nγ) and (γn) -reactions =ln(2T12) mdash-decay rate P - probability of (A Z) nuclide creation after ndash-decay of (AZ-1) nuclide Branching coefficients of isobaric chains - P1n P2n Р3n corresponds to probabilities of one- two- and three- neutrons emission in ndash- decay of the neutron-rich nuclei the total probability of the delayed neutrons emission is the sum

Ff (A Z) describes fission processes mdash spontaneous and beta-delayed fission

(A Z) - neutrino capturing processes

Inner time scale is strongly depends on the nuclear reactions rates

k

knn PP

II NUCLEOSYNTHESIS WAVE MOVEMENT

Concentrations

nА=

for three time moments calculated for r-processconditions

nn=1024 сm-3

Т9=1= 109K

Lutostansky YuS et alSov J Nucl Phys 1985 v 42

z

)ZA(ns

s

s

β-Delayed processes in very neutron-rich nuclei

Delayed neutron emission -(β n)

------------------------------------Multi-neutron β ndash delayed emission - (β kn)------------------------------------β ndash delayed fission - (βf)

GTR

AR

GTR

ldquopigmyrdquo-resonances

Beta ndash Delayed Multi-Neutron Emission

Probability for (β 2n) - emission

U I(U) ndash energies and intensities in the

daughter nucleus Wn(U E) ndash probability of neutron emission

nBU

0infn

nfnn

)U(Г)U(q2dE)BEU(q)E(T

)BEU(q)E(T)EU(W

qi and qf ndash level densities of

compound and final nucleusТn(Е) mdash transitivity factor

Probability for (β kn) - emission

Lyutostansky YuS Panov IV and Sirotkin VK ldquoThe -Delayed MultindashNeutron Emissionrdquo Phys Lett 1985 V 161B 1 2 3 P 9-13

Q

B

Q

nkn

kn

kn

dUdEEUWUIP0

)()(

Q

i

i

Q

B ijlm tot

fi

n

dEESEQZf

dEESEQZf

P n

0

2

)()1(

)()1(2

BETA-DELAYED NEUTRONS IN NUCLEOSYNTESIS

Calculated abandancies 1ndash with out (βn)-effect 2 ndash with (βn)-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

exp

BETA-STRENGTH FUNCTION CALCULATIONS-1 COLLECTIVE ISOBARIC STATES

G-T - SELECTION RULES Δ j =0plusmn1Δ j =+1 j =l+12 rarr j =lndash12 Δ j =0 j =l+12 rarr j=l+12Δ j = ndash1 j =lndash12 rarr j =l+12 j =lndash12rarr j =lndash12

neutrons

protons

BETA-STRENGTH FUNCTION CALCULATIONS-2

MICROSCOPIC DESCRIPTION - 1The GamowndashTeller 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

Vpnω is an external charge-exchange field dpn

1 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 nucleonndashnucleon interaction in the particlendashhole and the particlendashparticle 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 dpn

1 = 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 rarr Eλ = --------------------------------------------------------------------------The selfcosistent microscopic theory used for the beta-strength function calculations

np

nppnnppnqpn ГVeV

np

hnppnnp

hpn ГV

np

nppnnppn Гd 11

np

nppnnppn Гd 22

22


MICROSCOPIC DESCRIPTION - 2For the GT effective nuclear field system of equations in the energetic λ-representation has the form [Migdal Gaponov]

Local nucleonndashnucleon δ-interaction Γω in the Landau-Migdal form used

Г = С0 (f0prime + g0prime σ1σ2) τ1τ2 δ(r1- r2)

where coupling constants of f0prime ndash isospin-isospin and g0prime ndash spin-isospin quasi-particle

interaction with L = 0 ------------------------------------------------------------------------------------Matrix elements MGT where χλν ndash mathematical deductions

where nλ and ελ are respectively the occupation numbers and energies of states λ---------------------------------------------------------------------------------------------

2121

21

21

2 VAM GT

G-T selection rules Δ j =0plusmn1Δ j =+1 j=l+12 rarr j =lndash12

Δ j =0 j=lplusmn12 rarr

j=lplusmn12Δ j = ndash1 j=lndash12 rarr j=l+12 j =lndash12rarr j =lndash12

G-T values are normalized in FFST

2iM

Constants f0prime and g0prime are the phenomenological parameters

i

qi ZNeM )(322

Standard sum rule for στ-excitations i

i ZNM )(32

Effective quasiparticle charge is the ldquoquenchingrdquo parameter of the theory01802 qe


MICROSCOPIC DESCRIPTION - 3RESONANCE STRUCTURE OF BETA-STRENGTH FUNCTION

The Bright-Wigner form for E gt Sn

1 Discrete structure of beta-strength function Partial function Ci(old variant)

2 Resonance structure of beta-strength function

Partial function )(ES i 22)ω( ii

i

AtildeE

Atilde

=

Гi value up to Migdal is Г = ndash 2 Im [sum (ε + iI)]

and Г = ε | ε | + βε3 + γ ε2 | ε | + O(ε4)hellip where 1

F

Гi(i) = 0018 Ei2 МэВ

)(ES i =

)(

21

EiiM

E

)(ES i )(ES i

71GeExp Krofcheck D et al Phys Rev Lett 55 (1985) 1051- - - Borzov I Fayans S Trykov E Nucl Phys А 584 (1995) 335- Borovoi A Lutostan- sky Yu Panov I et al JETP Lett 45 (1987) 521

Yu V Gaponov and Yu S Lyutostansky Sov J Phys Elem Part At Nucl 12 528 (1981)

Sn

BETA-STRENGTH FUNCTION FOR 127Xe

1 - Breaking line ndash experimental data (1999) M Palarczyk et al Phys Rev 1999 V 59 P 500

2 ndash Solid red line TFFS calculations with еq= 09 3 - Solid black line ndash calculations with еq= 08 YuS Lutostansky NB Shulgina

Phys Rev Lett 1991 V67 P 430

Dependence from eg

QUENCHING EFFECT for 127Xe

1 - Breaking line ndash experimental data M Palarczyk et al Phys Rev 59 (1999) 500

2 - line ndashTFFS calculations with еq= 09 YuS Lutostansky and VN Tikhonov Bull Russ Acad Sci Phys 76 476 (2012)

3 - - - - ndashTFFS calculations with еq= 08

YuS Lutostansky and NB Shulgina Phys Rev Lett 67 (1991) 430

QUENCHING EFFECT ndash EXPERIMENT

Standard sum rule for στ-excitations

For G-T beta-strength function

In FFST [Migdal] theory

For experimental data sum rule Σ B(GT) = must be = 3(N ndash Z)

)(3)( 2

0

β

max

ZNedEES q

E

i

i ZNM )(32

)(3)(max

0

β ZNdEESE

i

ii EGTB )(

Ideal Emax= infin

71Ge

127Xe

)(3

)(

ZN

GTBi

i

INTERACTION CONSTANTS

For the (ττ) coupling constant f0 the value f0

= 135 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

For (στ) coupling constant g0 value g0

= 122 plusmn 004 received from comparison of calculated energy differences between GTR and the low-lying ldquopigmyrdquo-resonance with the experimental data for nine Sb isotopes [K Pham J Jaumlnecke D A Roberts et al Phys Rev C 51 (1995) 526]

Three main parameters of FFST theory eq f0

g0

are taken from exp and calc data comparison--------------------------------------------

eq ndash from ldquoquenchingrdquo effectf0

and g0 ndash from energy

splitting data

BETA - DELAYED FISSION

Beta ndash Delayed Fission Calculations

Q

0 i

i

Q

0 i tot

fi

f

dE)E(S)EQZ(f

dEГ

Г)E(S)EQZ(f

PProbabilities - Pβf

4

)E(Г)EE(

)E(Г)E(M

2

C)E(S

22

ii

i2i

N

Beta Strength function

Г(Е) widths approximation Г(Е) = αE2 + βE3 + hellip where α asymp 1εF and β laquo α so we used only the first term

As Гf laquo 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 ndash Delayed Fission Calculations

YuS Lutostansky VI Liashuk IV Panov ldquoInfluence of the delayed fission on production of transuranium elements in the explosive nucleosynthesisrdquo Preprint ITEP 90-25 1990 Moscow

Dubnium Beta ndash Delayed Fission Calculations

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)

I Panov Yu Lutostansky F-K Thielemann

2013

Factor of the concentration losing in Prompt-process

Np β-delayed fission probabilities

MODEL DESCRIPTION OF Sβ(E) - 1

Calculated (circles ndash ) and experimental () dependencies of the relative energy y(x)=Δ(EGTR-EAR)Els from the dimensionless value x=EEls Black circles () connected by line ndash calculated values for Sn isotopes

Mat model developed for the approximate solutions of equations of the FFST theory by the quasi-classical method -----------------------------2 new parameters

E = EF(n) ndash EF(p) =

Els ndash average energy of the spinndashorbit splitting

313112

0

000 80)(])2(1[

3

2])(1[

1)(

AAcAbEExxAcxg

gbbxfg

E

EEy ls

ls

AgraveETHAtildeOgraveETH

Wignerrsquos SU(4) super-symmetry restoration in the heavy nuclei

A

ZNEF

3

4

MODEL DESCRIPTION OF Sβ(E) ndash 2 T12 calculations

1988

The dependence of r-process duration time on mass A-value under different external conditions curve 1) ndash constant nn=1026 cm-3 T=15 109K 2) ndash the same nn T=1109K 3) ndash dynamical calc

with ρ0=2105 gcm5 T=1109K [(t) = 0 ехp (-tH) Т(t)= Т0 ехр(-t3H)]

β-decay time

Fermi-function

YuS Lutostansky and I V Panov Astron Letters 14 no 2 (1988) 168

i

Q

i dEEQZfEST

0

)(121 )()(

E

L dUUSUEUUUZFf1

22 )()(1)(

Time of new nuclei synthesis

M2GTR asymp 3 (N-Z) eq

2

M2IAS asymp

E

GT and IASResonances

in Sβ(E )-function

Neutrino capturing

QE

eeeA dEEZFESEp

c

gE

043

2

)()()(

Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

PROCESSES OF NUCLEOSYNTESIS

The tracks of elements synthesis in s (slow)- and r (rapid)- processes

β-decay

β-decay

r-process track

s-process track

fission

Superheavynuclei









+ (A Z) + Ff (A Z)





k

knn PP


Concentrations

nА=


nn=1024 сm-3

Т9=1= 109K


z

)ZA(ns

s

s




GTR

AR

GTR






nBU

0infn

nfnn


)BEU(q)E(T)EU(W





Q

B

Q

nkn

kn

kn

dUdEEUWUIP0

)()(

Q

i

i

Q

B ijlm tot

fi

n

dEESEQZf

dEESEQZf

P n

0

2

)()1(

)()1(2




exp



neutrons

protons










np

nppnnppnqpn ГVeV

np

hnppnnp

hpn ГV

np

nppnnppn Гd 11

np

nppnnppn Гd 22

22








2121

21

21

2 VAM GT





2iM


i

qi ZNeM )(322


i ZNM )(32








i

AtildeE

Atilde

=



F


)(ES i =

)(

21

EiiM

E

)(ES i )(ES i



Sn





Dependence from eg











)(3)( 2

0

β

max

ZNedEES q

E

i

i ZNM )(32

)(3)(max

0

β ZNdEESE

i

ii EGTB )(

Ideal Emax= infin

71Ge

127Xe

)(3

)(

ZN

GTBi

i








g0




splitting data



Q

0 i

i

Q

0 i tot

fi

f

dE)E(S)EQZ(f

dEГ

Г)E(S)EQZ(f


4

)E(Г)EE(

)E(Г)E(M

2

C)E(S

22

ii

i2i

N











2013








313112

0

000 80)(])2(1[

3

2])(1[

1)(

AAcAbEExxAcxg

gbbxfg

E

EEy ls

ls



A

ZNEF

3

4


1988



β-decay time

Fermi-function


i

Q

i dEEQZfEST

0

)(121 )()(

E

L dUUSUEUUUZFf1

22 )()(1)(



2

M2IAS asymp

E


in Sβ(E )-function

Neutrino capturing

QE

eeeA dEEZFESEp

c

gE

043

2

)()()(

Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24









+ (A Z) + Ff (A Z)





k

knn PP


Concentrations

nА=


nn=1024 сm-3

Т9=1= 109K


z

)ZA(ns

s

s




GTR

AR

GTR






nBU

0infn

nfnn


)BEU(q)E(T)EU(W





Q

B

Q

nkn

kn

kn

dUdEEUWUIP0

)()(

Q

i

i

Q

B ijlm tot

fi

n

dEESEQZf

dEESEQZf

P n

0

2

)()1(

)()1(2




exp



neutrons

protons










np

nppnnppnqpn ГVeV

np

hnppnnp

hpn ГV

np

nppnnppn Гd 11

np

nppnnppn Гd 22

22








2121

21

21

2 VAM GT





2iM


i

qi ZNeM )(322


i ZNM )(32








i

AtildeE

Atilde

=



F


)(ES i =

)(

21

EiiM

E

)(ES i )(ES i



Sn





Dependence from eg











)(3)( 2

0

β

max

ZNedEES q

E

i

i ZNM )(32

)(3)(max

0

β ZNdEESE

i

ii EGTB )(

Ideal Emax= infin

71Ge

127Xe

)(3

)(

ZN

GTBi

i








g0




splitting data



Q

0 i

i

Q

0 i tot

fi

f

dE)E(S)EQZ(f

dEГ

Г)E(S)EQZ(f


4

)E(Г)EE(

)E(Г)E(M

2

C)E(S

22

ii

i2i

N











2013








313112

0

000 80)(])2(1[

3

2])(1[

1)(

AAcAbEExxAcxg

gbbxfg

E

EEy ls

ls



A

ZNEF

3

4


1988



β-decay time

Fermi-function


i

Q

i dEEQZfEST

0

)(121 )()(

E

L dUUSUEUUUZFf1

22 )()(1)(



2

M2IAS asymp

E


in Sβ(E )-function

Neutrino capturing

QE

eeeA dEEZFESEp

c

gE

043

2

)()()(

Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24


Concentrations

nА=


nn=1024 сm-3

Т9=1= 109K


z

)ZA(ns

s

s




GTR

AR

GTR






nBU

0infn

nfnn


)BEU(q)E(T)EU(W





Q

B

Q

nkn

kn

kn

dUdEEUWUIP0

)()(

Q

i

i

Q

B ijlm tot

fi

n

dEESEQZf

dEESEQZf

P n

0

2

)()1(

)()1(2




exp



neutrons

protons










np

nppnnppnqpn ГVeV

np

hnppnnp

hpn ГV

np

nppnnppn Гd 11

np

nppnnppn Гd 22

22








2121

21

21

2 VAM GT





2iM


i

qi ZNeM )(322


i ZNM )(32








i

AtildeE

Atilde

=



F


)(ES i =

)(

21

EiiM

E

)(ES i )(ES i



Sn





Dependence from eg











)(3)( 2

0

β

max

ZNedEES q

E

i

i ZNM )(32

)(3)(max

0

β ZNdEESE

i

ii EGTB )(

Ideal Emax= infin

71Ge

127Xe

)(3

)(

ZN

GTBi

i








g0




splitting data



Q

0 i

i

Q

0 i tot

fi

f

dE)E(S)EQZ(f

dEГ

Г)E(S)EQZ(f


4

)E(Г)EE(

)E(Г)E(M

2

C)E(S

22

ii

i2i

N











2013








313112

0

000 80)(])2(1[

3

2])(1[

1)(

AAcAbEExxAcxg

gbbxfg

E

EEy ls

ls



A

ZNEF

3

4


1988



β-decay time

Fermi-function


i

Q

i dEEQZfEST

0

)(121 )()(

E

L dUUSUEUUUZFf1

22 )()(1)(



2

M2IAS asymp

E


in Sβ(E )-function

Neutrino capturing

QE

eeeA dEEZFESEp

c

gE

043

2

)()()(

Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24




GTR

AR

GTR






nBU

0infn

nfnn


)BEU(q)E(T)EU(W





Q

B

Q

nkn

kn

kn

dUdEEUWUIP0

)()(

Q

i

i

Q

B ijlm tot

fi

n

dEESEQZf

dEESEQZf

P n

0

2

)()1(

)()1(2




exp



neutrons

protons










np

nppnnppnqpn ГVeV

np

hnppnnp

hpn ГV

np

nppnnppn Гd 11

np

nppnnppn Гd 22

22








2121

21

21

2 VAM GT





2iM


i

qi ZNeM )(322


i ZNM )(32








i

AtildeE

Atilde

=



F


)(ES i =

)(

21

EiiM

E

)(ES i )(ES i



Sn





Dependence from eg











)(3)( 2

0

β

max

ZNedEES q

E

i

i ZNM )(32

)(3)(max

0

β ZNdEESE

i

ii EGTB )(

Ideal Emax= infin

71Ge

127Xe

)(3

)(

ZN

GTBi

i








g0




splitting data



Q

0 i

i

Q

0 i tot

fi

f

dE)E(S)EQZ(f

dEГ

Г)E(S)EQZ(f


4

)E(Г)EE(

)E(Г)E(M

2

C)E(S

22

ii

i2i

N











2013








313112

0

000 80)(])2(1[

3

2])(1[

1)(

AAcAbEExxAcxg

gbbxfg

E

EEy ls

ls



A

ZNEF

3

4


1988



β-decay time

Fermi-function


i

Q

i dEEQZfEST

0

)(121 )()(

E

L dUUSUEUUUZFf1

22 )()(1)(



2

M2IAS asymp

E


in Sβ(E )-function

Neutrino capturing

QE

eeeA dEEZFESEp

c

gE

043

2

)()()(

Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24





nBU

0infn

nfnn


)BEU(q)E(T)EU(W





Q

B

Q

nkn

kn

kn

dUdEEUWUIP0

)()(

Q

i

i

Q

B ijlm tot

fi

n

dEESEQZf

dEESEQZf

P n

0

2

)()1(

)()1(2




exp



neutrons

protons










np

nppnnppnqpn ГVeV

np

hnppnnp

hpn ГV

np

nppnnppn Гd 11

np

nppnnppn Гd 22

22








2121

21

21

2 VAM GT





2iM


i

qi ZNeM )(322


i ZNM )(32








i

AtildeE

Atilde

=



F


)(ES i =

)(

21

EiiM

E

)(ES i )(ES i



Sn





Dependence from eg











)(3)( 2

0

β

max

ZNedEES q

E

i

i ZNM )(32

)(3)(max

0

β ZNdEESE

i

ii EGTB )(

Ideal Emax= infin

71Ge

127Xe

)(3

)(

ZN

GTBi

i








g0




splitting data



Q

0 i

i

Q

0 i tot

fi

f

dE)E(S)EQZ(f

dEГ

Г)E(S)EQZ(f


4

)E(Г)EE(

)E(Г)E(M

2

C)E(S

22

ii

i2i

N











2013








313112

0

000 80)(])2(1[

3

2])(1[

1)(

AAcAbEExxAcxg

gbbxfg

E

EEy ls

ls



A

ZNEF

3

4


1988



β-decay time

Fermi-function


i

Q

i dEEQZfEST

0

)(121 )()(

E

L dUUSUEUUUZFf1

22 )()(1)(



2

M2IAS asymp

E


in Sβ(E )-function

Neutrino capturing

QE

eeeA dEEZFESEp

c

gE

043

2

)()()(

Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24




exp



neutrons

protons










np

nppnnppnqpn ГVeV

np

hnppnnp

hpn ГV

np

nppnnppn Гd 11

np

nppnnppn Гd 22

22








2121

21

21

2 VAM GT





2iM


i

qi ZNeM )(322


i ZNM )(32








i

AtildeE

Atilde

=



F


)(ES i =

)(

21

EiiM

E

)(ES i )(ES i



Sn





Dependence from eg











)(3)( 2

0

β

max

ZNedEES q

E

i

i ZNM )(32

)(3)(max

0

β ZNdEESE

i

ii EGTB )(

Ideal Emax= infin

71Ge

127Xe

)(3

)(

ZN

GTBi

i








g0




splitting data



Q

0 i

i

Q

0 i tot

fi

f

dE)E(S)EQZ(f

dEГ

Г)E(S)EQZ(f


4

)E(Г)EE(

)E(Г)E(M

2

C)E(S

22

ii

i2i

N











2013








313112

0

000 80)(])2(1[

3

2])(1[

1)(

AAcAbEExxAcxg

gbbxfg

E

EEy ls

ls



A

ZNEF

3

4


1988



β-decay time

Fermi-function


i

Q

i dEEQZfEST

0

)(121 )()(

E

L dUUSUEUUUZFf1

22 )()(1)(



2

M2IAS asymp

E


in Sβ(E )-function

Neutrino capturing

QE

eeeA dEEZFESEp

c

gE

043

2

)()()(

Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24



neutrons

protons










np

nppnnppnqpn ГVeV

np

hnppnnp

hpn ГV

np

nppnnppn Гd 11

np

nppnnppn Гd 22

22








2121

21

21

2 VAM GT





2iM


i

qi ZNeM )(322


i ZNM )(32








i

AtildeE

Atilde

=



F


)(ES i =

)(

21

EiiM

E

)(ES i )(ES i



Sn





Dependence from eg











)(3)( 2

0

β

max

ZNedEES q

E

i

i ZNM )(32

)(3)(max

0

β ZNdEESE

i

ii EGTB )(

Ideal Emax= infin

71Ge

127Xe

)(3

)(

ZN

GTBi

i








g0




splitting data



Q

0 i

i

Q

0 i tot

fi

f

dE)E(S)EQZ(f

dEГ

Г)E(S)EQZ(f


4

)E(Г)EE(

)E(Г)E(M

2

C)E(S

22

ii

i2i

N











2013








313112

0

000 80)(])2(1[

3

2])(1[

1)(

AAcAbEExxAcxg

gbbxfg

E

EEy ls

ls



A

ZNEF

3

4


1988



β-decay time

Fermi-function


i

Q

i dEEQZfEST

0

)(121 )()(

E

L dUUSUEUUUZFf1

22 )()(1)(



2

M2IAS asymp

E


in Sβ(E )-function

Neutrino capturing

QE

eeeA dEEZFESEp

c

gE

043

2

)()()(

Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24










np

nppnnppnqpn ГVeV

np

hnppnnp

hpn ГV

np

nppnnppn Гd 11

np

nppnnppn Гd 22

22








2121

21

21

2 VAM GT





2iM


i

qi ZNeM )(322


i ZNM )(32








i

AtildeE

Atilde

=



F


)(ES i =

)(

21

EiiM

E

)(ES i )(ES i



Sn





Dependence from eg











)(3)( 2

0

β

max

ZNedEES q

E

i

i ZNM )(32

)(3)(max

0

β ZNdEESE

i

ii EGTB )(

Ideal Emax= infin

71Ge

127Xe

)(3

)(

ZN

GTBi

i








g0




splitting data



Q

0 i

i

Q

0 i tot

fi

f

dE)E(S)EQZ(f

dEГ

Г)E(S)EQZ(f


4

)E(Г)EE(

)E(Г)E(M

2

C)E(S

22

ii

i2i

N











2013








313112

0

000 80)(])2(1[

3

2])(1[

1)(

AAcAbEExxAcxg

gbbxfg

E

EEy ls

ls



A

ZNEF

3

4


1988



β-decay time

Fermi-function


i

Q

i dEEQZfEST

0

)(121 )()(

E

L dUUSUEUUUZFf1

22 )()(1)(



2

M2IAS asymp

E


in Sβ(E )-function

Neutrino capturing

QE

eeeA dEEZFESEp

c

gE

043

2

)()()(

Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24








2121

21

21

2 VAM GT





2iM


i

qi ZNeM )(322


i ZNM )(32








i

AtildeE

Atilde

=



F


)(ES i =

)(

21

EiiM

E

)(ES i )(ES i



Sn





Dependence from eg











)(3)( 2

0

β

max

ZNedEES q

E

i

i ZNM )(32

)(3)(max

0

β ZNdEESE

i

ii EGTB )(

Ideal Emax= infin

71Ge

127Xe

)(3

)(

ZN

GTBi

i








g0




splitting data



Q

0 i

i

Q

0 i tot

fi

f

dE)E(S)EQZ(f

dEГ

Г)E(S)EQZ(f


4

)E(Г)EE(

)E(Г)E(M

2

C)E(S

22

ii

i2i

N











2013








313112

0

000 80)(])2(1[

3

2])(1[

1)(

AAcAbEExxAcxg

gbbxfg

E

EEy ls

ls



A

ZNEF

3

4


1988



β-decay time

Fermi-function


i

Q

i dEEQZfEST

0

)(121 )()(

E

L dUUSUEUUUZFf1

22 )()(1)(



2

M2IAS asymp

E


in Sβ(E )-function

Neutrino capturing

QE

eeeA dEEZFESEp

c

gE

043

2

)()()(

Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24







i

AtildeE

Atilde

=



F


)(ES i =

)(

21

EiiM

E

)(ES i )(ES i



Sn





Dependence from eg











)(3)( 2

0

β

max

ZNedEES q

E

i

i ZNM )(32

)(3)(max

0

β ZNdEESE

i

ii EGTB )(

Ideal Emax= infin

71Ge

127Xe

)(3

)(

ZN

GTBi

i








g0




splitting data



Q

0 i

i

Q

0 i tot

fi

f

dE)E(S)EQZ(f

dEГ

Г)E(S)EQZ(f


4

)E(Г)EE(

)E(Г)E(M

2

C)E(S

22

ii

i2i

N











2013








313112

0

000 80)(])2(1[

3

2])(1[

1)(

AAcAbEExxAcxg

gbbxfg

E

EEy ls

ls



A

ZNEF

3

4


1988



β-decay time

Fermi-function


i

Q

i dEEQZfEST

0

)(121 )()(

E

L dUUSUEUUUZFf1

22 )()(1)(



2

M2IAS asymp

E


in Sβ(E )-function

Neutrino capturing

QE

eeeA dEEZFESEp

c

gE

043

2

)()()(

Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24





Dependence from eg











)(3)( 2

0

β

max

ZNedEES q

E

i

i ZNM )(32

)(3)(max

0

β ZNdEESE

i

ii EGTB )(

Ideal Emax= infin

71Ge

127Xe

)(3

)(

ZN

GTBi

i








g0




splitting data



Q

0 i

i

Q

0 i tot

fi

f

dE)E(S)EQZ(f

dEГ

Г)E(S)EQZ(f


4

)E(Г)EE(

)E(Г)E(M

2

C)E(S

22

ii

i2i

N











2013








313112

0

000 80)(])2(1[

3

2])(1[

1)(

AAcAbEExxAcxg

gbbxfg

E

EEy ls

ls



A

ZNEF

3

4


1988



β-decay time

Fermi-function


i

Q

i dEEQZfEST

0

)(121 )()(

E

L dUUSUEUUUZFf1

22 )()(1)(



2

M2IAS asymp

E


in Sβ(E )-function

Neutrino capturing

QE

eeeA dEEZFESEp

c

gE

043

2

)()()(

Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24











)(3)( 2

0

β

max

ZNedEES q

E

i

i ZNM )(32

)(3)(max

0

β ZNdEESE

i

ii EGTB )(

Ideal Emax= infin

71Ge

127Xe

)(3

)(

ZN

GTBi

i








g0




splitting data



Q

0 i

i

Q

0 i tot

fi

f

dE)E(S)EQZ(f

dEГ

Г)E(S)EQZ(f


4

)E(Г)EE(

)E(Г)E(M

2

C)E(S

22

ii

i2i

N











2013








313112

0

000 80)(])2(1[

3

2])(1[

1)(

AAcAbEExxAcxg

gbbxfg

E

EEy ls

ls



A

ZNEF

3

4


1988



β-decay time

Fermi-function


i

Q

i dEEQZfEST

0

)(121 )()(

E

L dUUSUEUUUZFf1

22 )()(1)(



2

M2IAS asymp

E


in Sβ(E )-function

Neutrino capturing

QE

eeeA dEEZFESEp

c

gE

043

2

)()()(

Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24








g0




splitting data



Q

0 i

i

Q

0 i tot

fi

f

dE)E(S)EQZ(f

dEГ

Г)E(S)EQZ(f


4

)E(Г)EE(

)E(Г)E(M

2

C)E(S

22

ii

i2i

N











2013








313112

0

000 80)(])2(1[

3

2])(1[

1)(

AAcAbEExxAcxg

gbbxfg

E

EEy ls

ls



A

ZNEF

3

4


1988



β-decay time

Fermi-function


i

Q

i dEEQZfEST

0

)(121 )()(

E

L dUUSUEUUUZFf1

22 )()(1)(



2

M2IAS asymp

E


in Sβ(E )-function

Neutrino capturing

QE

eeeA dEEZFESEp

c

gE

043

2

)()()(

Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24



Q

0 i

i

Q

0 i tot

fi

f

dE)E(S)EQZ(f

dEГ

Г)E(S)EQZ(f


4

)E(Г)EE(

)E(Г)E(M

2

C)E(S

22

ii

i2i

N











2013








313112

0

000 80)(])2(1[

3

2])(1[

1)(

AAcAbEExxAcxg

gbbxfg

E

EEy ls

ls



A

ZNEF

3

4


1988



β-decay time

Fermi-function


i

Q

i dEEQZfEST

0

)(121 )()(

E

L dUUSUEUUUZFf1

22 )()(1)(



2

M2IAS asymp

E


in Sβ(E )-function

Neutrino capturing

QE

eeeA dEEZFESEp

c

gE

043

2

)()()(

Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24






2013








313112

0

000 80)(])2(1[

3

2])(1[

1)(

AAcAbEExxAcxg

gbbxfg

E

EEy ls

ls



A

ZNEF

3

4


1988



β-decay time

Fermi-function


i

Q

i dEEQZfEST

0

)(121 )()(

E

L dUUSUEUUUZFf1

22 )()(1)(



2

M2IAS asymp

E


in Sβ(E )-function

Neutrino capturing

QE

eeeA dEEZFESEp

c

gE

043

2

)()()(

Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24








313112

0

000 80)(])2(1[

3

2])(1[

1)(

AAcAbEExxAcxg

gbbxfg

E

EEy ls

ls



A

ZNEF

3

4


1988



β-decay time

Fermi-function


i

Q

i dEEQZfEST

0

)(121 )()(

E

L dUUSUEUUUZFf1

22 )()(1)(



2

M2IAS asymp

E


in Sβ(E )-function

Neutrino capturing

QE

eeeA dEEZFESEp

c

gE

043

2

)()()(

Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24


1988



β-decay time

Fermi-function


i

Q

i dEEQZfEST

0

)(121 )()(

E

L dUUSUEUUUZFf1

22 )()(1)(



2

M2IAS asymp

E


in Sβ(E )-function

Neutrino capturing

QE

eeeA dEEZFESEp

c

gE

043

2

)()()(

Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24


2

M2IAS asymp

E


in Sβ(E )-function

Neutrino capturing

QE

eeeA dEEZFESEp

c

gE

043

2

)()()(

Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

PROCESSES OF NUCLEOSYNTESIS .

Documents

Transcript of PROCESSES OF NUCLEOSYNTESIS .