Experimental and theoretical investigations of proton-and α -particle-induced nuclear reactions for astrophysics

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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

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ASYMPTOTIC NORMALIZATION COEFFICIENTS FOR 3He+α7_{Be AND p+}7_Be 8_{B AND THE DIRECT}

3He(α,)7Be AND 7Be(p, ))8_{B ASTROPHYSICAL} S-FACTORS AT SOLAR ENERGIES

S.B. Igamov, K. I Tursunmakhatov and R.Yarmukhamedov

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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

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CONTANTS

1. Introduction

2. 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

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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

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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(



,



)

6,7

Li , 3He(



_,₎7

Be , 7Be(p,

 ₎8

B ,

12,13

C(p

_,₎13,14

N ,

14

N(p

₎15

O 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

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pp-chain

p+p  d+e

+

ν

e

D+p 

3

He+

3

He+

3

He 

4

He+ p+p

3

He

4

He 

7

Be+ 

7

Be+ p

8

B +

(

8

B 

8

Be + e

+

+ ν

e

)

(4p 

4

He

)

(

Q =26.7 MeV, T  10

7

K,

E

= 0.5  6 MeV

)

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“

hot” pp-chain (CNO - cycle)

12

C+ p

13

N+

3

N

13

C+e

+

+ν

e 13

C +p

14

N+ 

14

N+ p

15

O+ 

15

O

15

N+ e

+

+ν

e 15

N+ p

12

C+

4

He (4p  4He)

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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

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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

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The recommended values S(0) are

0.5600.017 keVb (F. Confortola et al) and

0.5960.021 keVb (T.A.D. Brown et al) for the 3He(,)7_Be_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

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21.40.5(exp)0.6(theor) eVb (A.J. Junghans et al ) and

21.40.5(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)=18.21.8 eVb [G.Tabacaru et al. PhRev(2006))] for the 7Be (p, ) 8B reaction inferred in

the ANC method

The main reason of the discrepancy

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Since 1995 many works have been published in which

the radiative capture A+ aB+ 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)

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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

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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

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2 .

Modified DWBA for determination of ANC for

the A+aB

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.

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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

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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

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In this case an inclusion of all orders

( the first, second and higher orders)

of the power expansion in a series over V

C

is

required in the transition operator for the DWBA

cross section calculations.

They can be described

by the following diagrammes

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)

,

(

~

2 2 2







E

G

R

d

Aa ya





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R

2

= 1.03 – 1.17

for the

13

C(

3

He,d)

14

N and

13

C(

14

N,

13

C)

14

N

reactions and ANC (C

r

= C/R)

for

14

N 

13

C + p

would be used for the calculation of S(E)

for the

13

C(p,)

14

N reaction

performed by A.M.Mukhamedzhanov at al.

[Nucl.Phys.(2003)]

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3. The

208

Pb(

6

Li, αd)

208

Pb

Coulomb

breakup

6

Li ---

 α+d

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S.B. Igamov and R.Yarmukhamedov.

Nucl.Phys. A673(2000)509

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4. Modified two-body potential

approach for the direct

radiative capture

A(a,)B

reaction

S.B. Igamov, R. Yarmukhamedov .

Nucl.Phys.(2007)

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5. Analysis of the direct radiative

capture

D( α, )6Li, 3H ( α, )7Li,

3He(α,)7Be

and

7Be(p, )8B

reactions

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α+ t7Li(gs)

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α

3

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Exp. data C2 _(C*2₎ (fm-1₎ |G|2 _(|G*|2₎ (fm) S_tot(0) (keVb) Refs. 3He(,)7_Be[1,2,3] 23.191.37 (15.73 1.02) 1.11 0.07 (0.75 0.05) 0.6100.037 0.545 0.017 0.530 0.042 0.596 0.021 Our Our [1,2] [3] [4] 3He(,)7_Be[5] 20.891.75 (14.66 1.36) 0.99 0.08 (0.690.06) 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]

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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

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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

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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 r_c ), 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].

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The total amplitude and cross section are

).

,

;

(

)

,

;

(

) ( ; exp exp ) ( ; ; c Aa DC J J sl c Aa a r J J sl J J sl

r

C

E

M

r

C

E

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   





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7. Analysis of the

12

C(p,)

13

N and

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7. Perspective

(Kazakhstan-Turkey-Uzbekistan-USA collaboration)

Title: “Experimental and Theoretical

Investigations of Proton- and

-Particle-Induced radiative capture Reactions for

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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(

,)

6

Li,

6

Li(p,)

7

Be and

12,13

C(p,)

13,14

N

at energy range E100 keV

(ii) Theoretical interpretation of obtained experimental

data and their reliable extrapolation to the stellar energy region (E25 keV), including E=0.

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EXPERIMENTAL FACILITIES

The TR-2 Accelerator (TAEK, Turkey),

E

_p

= 300 – 1800 keV ,

E=50 – 350 keV

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The Van de Graff generator EG-2 Sokol

(RIAP, Tashkent-Uzbekistan) E=300-1100

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The UKP-2-1 Accelerator(INP,Kazakhstan)

proton and -particle beam with energies

E= 300 - 600 keV

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The project is devoted to the