METHODS OF DETERMINATION OF ASYMPTOTIC NORMALIZATION COEFFICIENTS FOR NUCLEAR
ASTROPHYSICS AND IMPORTANT pp-CHAIN AND CNO-CYCLE PROCESSES
Prof. R.Yarmukhamedov
Institute of Nuclear Physics, 100124 Tashkent, Uzbekistan
ASYMPTOTIC NORMALIZATION COEFFICIENTS FOR 3He+α7Be AND p+7Be 8B AND THE DIRECT
3He(α,)7Be AND 7Be(p, ))8B ASTROPHYSICAL S-FACTORS AT SOLAR ENERGIES
S.B. Igamov, K. I Tursunmakhatov and R.Yarmukhamedov
EXPERIMENTAL AND THEORETICAL INVESTIGATIONS OF PROTON- AND
α -PARTICLE-INDUCED NUCLEAR REACTIONS FOR ASTROPHYSICS R.Yarmukhamedov, C.V. Artemov, A. Baykal,
I.Boztosun, N. Burtebayev, V. Jazairov-Kakhramanov, I.Kholbaev, R.J. Peterson and B.S. Yuldashev
KAZAKHSTAN-TURKEY-UZBEKISTAN COLLABORATION
CONTANTS
1. Introduction2. Modified DWBA
3. The 208Pb(6Li, αd) 208Pb Coulomb breakup
4. Modified two-body potential approach for the direct radiative capture A(a,)B reaction 5. Analysis of the direct radiative capture
D( α, )6Li, 3H ( α, )7Li, 3He(α,)7Be and
7Be(p, )8B reactions
6. R-Matrix method for the A(a,)B reaction 7. Analysis of the 12,13C(p, )13,14N reactions
1. Introduction
A reliable estimation of rates of different nuclear
astrophysical processes responsible for the
light elements abundance ( for example, 6,7Li,
Be, B, C, N, O etc.) is one of the most
important problem of the modern nuclear
astrophysics
In this turn, solution of this problem is impossible without obtaining the very-low energy cross sections or
equivalently its the astrophysical S –factors S(E) for such reactions as
2,3
H(
,
)
6,7Li , 3He(
, )7Be , 7Be(p,
)8B ,
12,13
C(p
, )13,14N ,
14N(p
)15O etc.
For example, reliable information on low energy the cross sections for the direct capture 3He(, )7Be
and 7Be(p, )8B reactions at solar energies ( 0 -20 keV)
plays a crucial role for observed abundances of the 7Be
pp-chain
p+p d+e
++
ν
eD+p
3He+
3He+
3He
4He+ p+p
3He
4He
7Be+
7Be+ p
8B +
(
8B
8Be + e
++ ν
e)
(4p
4He
)
(
Q =26.7 MeV, T 10
7K,
E
= 0.5 6 MeV
)
“
hot” pp-chain (CNO - cycle)
12C+ p
13N+
3N
13C+e
++ν
e 13C +p
14N+
14N+ p
15O+
15O
15N+ e
++ν
e 15N+ p
12C+
4He (4p 4He)
A critical analysis of low-energy astrophysical S-factors
data made in work of the authors of E.G. Adelberger et al. [Rev.Mod.Phys. 70(1998)1265)] found out the following facts:
• The extrapolation of measured astrophysical S-factors to lower energies is noticeable ambiguous due to a presence of rather large spread in them.
• The theoretical predictions for S(E) at astrophysically
relevant energies (~ 25 keV) are not always accurate since its depends on input parameters ( for example
The new most accurate measurements of the
astrophysical S-factors S(E) have been performed by the authors T.A.D. Brown et al. (PhRev C76(2007) ) and F. Confortola et al (PR C76(2007) )
for the 3He(,)7Be reaction and
by the authors A.J. Junghans et al (PhRev C68(2003) ) and L.T. Baby et al. (PhRev C68(2003) ) for the 7Be (p, ) 8B
reaction
The recommended values S(0) are
0.5600.017 keVb (F. Confortola et al) and
0.5960.021 keVb (T.A.D. Brown et al) for the 3He(,)7Be reaction
These values of S(0) are considered as the “best” values obtained by the correct analysis of the experimental data
However, there is the discrepancy with the value of
21.40.5(exp)0.6(theor) eVb (A.J. Junghans et al ) and
21.40.5(exp)0.6(theor) eVb (L.T. Baby et al. ) for the 7Be (p, ) 8B reaction
Here there is discrepancy with the values of
S(0)=18.21.8 eVb [G.Tabacaru et al. PhRev(2006))] for the 7Be (p, ) 8B reaction inferred in
the ANC method
The main reason of the discrepancy
Since 1995 many works have been published in which
the radiative capture A+ aB+ reaction rates at stellar energies ( ~ 20 keV) were determined comparatively
in correct way.
• S.B . Igamov,R. Yarmukhamedov. Phys.At.Nucl. (1995) • A.M. Mukhamedzhanov et al.[ Phys. Rev. (1995)
• S.B. Igamov et al.Phys. Atom. Nucl. (1997) • A.Azhari et al. Phys.Rev. Lett. (1999)
• A.Ashari et al.Phys.Rev. (1999)
• A.Sattarov et al. Phys.Rev.C60(1999)035801
• A.M. Mukhamedzhanov et al. Nucl.Phys. (2003) • G.Tabacaru et al. Phys.Rev. (2006)
The main idea of these works is the fact that a reliable value of the nuclear vertex constant (Gl) (NVC) for the
virtual decay B A+a can be used as an input information for the A(a, )B reaction.
The NVC(or ANC) for the virtual decay B A+a
is a fundamental characteristic of the nucleus B
and determines the probability of finding a particles A and a
in the (A+a)-configuration at the distance out nuclear interaction.
In other hand, the NVC is determined by the dynamics of the strong interaction
Information about the two-body (Aa)
nuclear potential
G
l(C
l) the form of nuclear potential
where C is the asymptotic normalization coefficient for a radial overlap function of the nuclear B in
2
.
Modified DWBA for determination of ANC for
the A+aB
To determine a value of ANC for the A + p B and
A + B of astrophysical interest, in recent years
considerable amount of experimental and theoretical
works for the peripheral one-particle ( proton and -particle) transfer reactions A(x,y)B
( B=(A+a) and x=(y+a), where a=proton or -particle)
were performed
within the DWBA approach (see for example:
S.V. Artemov et al. Phys.At.Nucl (1996);A.Azhari et al. Phys.Rev. Lett. (1999) ;A.Ashari et al.Phys.Rev.
A+(y+a)
(A+a) +y
The main mechanism of this reaction in DWBA is
However, the DWBA approach used is the zero- or first
order perturbation approximation over the Coulomb
But as it was shown in
• G.V. Avakov et al. Sov.J.Nucl.Phys. (1986)
• Sh.S.Kajumov et al. Z.Phys. (1990)
• R.Yarmukhamedov. Phys.At. Nucl. (1997)
when the residual nuclei B are formed in weakly
bound states, this assumption is not guaranteed
for the peripheral proton ( -particle) transfer
reactions of astrophysical interest . So , the
obtained values of ANC's for the astrophysical
application may not have the necessary
In this case an inclusion of all orders
( the first, second and higher orders)
of the power expansion in a series over V
Cis
required in the transition operator for the DWBA
cross section calculations.
They can be described
by the following diagrammes
)
,
(
~
2 2 2
E
G
G
R
d
d
Aa ya
R
2= 1.03 – 1.17
for the
13C(
3He,d)
14N and
13C(
14N,
13C)
14N
reactions and ANC (C
r= C/R)
for
14N
13C + p
would be used for the calculation of S(E)
for the
13C(p,)
14N reaction
performed by A.M.Mukhamedzhanov at al.
[Nucl.Phys.(2003)]
3. The
208Pb(
6Li, αd)
208Pb
Coulomb
breakup
6
Li ---
α+d
S.B. Igamov and R.Yarmukhamedov.
Nucl.Phys. A673(2000)509
4. Modified two-body potential
approach for the direct
radiative capture
A(a,)B
reaction
S.B. Igamov, R. Yarmukhamedov .
Nucl.Phys.(2007)
5. Analysis of the direct radiative
capture
D( α, )6Li, 3H ( α, )7Li,
3He(α,)7Be
and
7Be(p, )8B
reactions
α+ t7Li(gs)
α
3Exp. data C2 (C*2) (fm-1) |G|2 (|G*|2) (fm) Stot(0) (keVb) Refs. 3He(,)7Be[1,2,3] 23.191.37 (15.73 1.02) 1.11 0.07 (0.75 0.05) 0.6100.037 0.545 0.017 0.530 0.042 0.596 0.021 Our Our [1,2] [3] [4] 3He(,)7Be[5] 20.891.75 (14.66 1.36) 0.99 0.08 (0.690.06) 0.535 0.040 0.519 0.045 [6] [5] R-matrix method 14.36(14.36) 0.68(0.68) 0.510 0.040 [7] Two body cluster
potential method 0.516 [8]
References
1.F.Confortola et al. Phys.Rev.C75(2007)065803. 2.D.Bemmerer et al. Phys.Rev.C75(2007)065803.
3.B.S.Nara Singh et al.Phys.Rev.Lett.93(2004)262503 4.T.A.D.Brown et al. Phys.Rev.C75(2007)065803.
5.J.L.Osbore et al. Nucl.Phys. A419(1984)115 6.S.B. Igamov et al. Phys.At.Nucl.60(1997)1126
7.P.Descouvemont et al At. Data and Nucl. Data Tables. 88(2004)203 8.P.Mohr et al.Phys.Rev.C48(1993)1420
9.T.Kajino. Nucl.Phys. A460(1986)559.
10.K.M. Nollet.Phys.Rev.C63(2001)055402 11.K.Langanke.Nucl.Phys.A457(1986)351
Thus, the modified two- body potential approach
for the direct radiative capture reaction
(IYa. Nucl.Phys(2007) )
allows to determine the ANC’s
,
which can then be used for the reliable
extrapolation of the astrophysical factors at solar
energies
6.R-matrix approach for the radiative
A(a,)B capture reaction
The R-matrix method assumes that the space of interaction of the colliding nuclei is divided into two
regions: the internal region (with radius rc ), where nuclear forces are important, and the external region, where the interaction between the nuclei is governed by the Coulomb
force only A. M. Lane and R.G. Thomas [ Rev.Mod.Phys. 30(1958)257].
The total amplitude and cross section are
).
,
;
(
)
,
,
,
;
(
) ( ; exp exp ) ( ; ; c Aa DC J J sl c Aa a r J J sl J J slr
C
E
M
r
C
E
M
M
f i i f i i f i f
i f i f J J J J(
E
)
(
E
),
. ) 1 2 )( 1 2 ( 1 2 ) ( 2
; 2 i f i i f i sl J J sl A a i J J M J J J k E
7. Analysis of the
12C(p,)
13N and
7.
Perspective
(Kazakhstan-Turkey-Uzbekistan-USA collaboration)
Title: “Experimental and Theoretical
Investigations of Proton- and
-Particle-Induced radiative capture Reactions for
The main goal of the proposal is to measure
the cross sections (the astrophysical
S-factors) in a region of extremely low
energies for the following different reactions:
(i) d(
,)
6Li,
6Li(p,)
7Be and
12,13C(p,)
13,14N
at energy range E100 keV
(ii) Theoretical interpretation of obtained experimental
data and their reliable extrapolation to the stellar energy region (E25 keV), including E=0.
EXPERIMENTAL FACILITIES
The TR-2 Accelerator (TAEK, Turkey),
E
p= 300 – 1800 keV ,
E=50 – 350 keVThe Van de Graff generator EG-2 Sokol
(RIAP, Tashkent-Uzbekistan) E=300-1100
The UKP-2-1 Accelerator(INP,Kazakhstan)
proton and -particle beam with energies
E= 300 - 600 keV
The project is devoted to the
experimental and theoretical
investigations of direct radiative
capture reactions at extremely low
energies ( 400 keV) which are of
paramount importance in
7.CONCLUSION
1. The modified two-body potential approach and the
R-matrix method are a good tool for the analysis of the direct and resonant radiative capture
reaction a+A B+ and they can be served as an independent source of getting the information about the ANC for the a+AB.
2. The modified two-body potential approach proposed
can be used also for verification of the accuracy of the DWBA calculation for the ANC for the
3. The further experimental and theoretical
investigation of the D(α,)
6Li,
6