Asymptotic normalization coefficient (nuclear vertex constant) and nuclar astrophysics problems

ASYMPTOTIC NORMALIZATION COEFFICIENT (NUCLEAR VERTEX

CONSTANT) AND NUCLAR ASTROPHYSICS PROBLEMS

R. Yarmukhamedov1, S.V. Artemov1, N. Burtebaev 2, S.B. Igamov1, and G.K. Nie1

1. INTRODUCTION

A reliable estimation o f rates o f different nuclear astrophysical processes responsible for the light abundance (for example, 6,7Li, nB e , 7B, C, N, O etc.) is one o f the most actual problem o f the modem nuclear astrophysics [1]. A solution o f this problem is impossible without obtaining the very-low energy cross sections or equivalently its the astrophysical S -factors such reactions as 2,3H(a,y)6,7Li, 3He(a,y)7Be, 7Be(p, y)8B, 1213C(p, y)13,14N, 14N(p, y) 150 , 12C(cc,y) 160 * etc. For example, the cross sections for the 1213C(p, y) 13J4N and 14N(p, y) 150 reactions at solar energies (~20 keV) play a cmcial role in the 13'14N and 15O creation in the CNO cycle [2,3].

Although there is an impressive improvement in our understanding o f these processes made in the past decades [1-10], for example, the discovery that electron neutrinos (v'es ) obtained from the 7B e ( p , Y ) 8B

reaction in the Sun oscillate into v ^ s and/or v Ts [10], some uncertainties in both the extrapolation o f measured cross sections to extremely low energies region and the theoretical predictions for it still exist and it may considerably influence upon the predictions o f the solar model [1].

One o f possible solutions o f this problem is based on the idea that due to strong Coulomb repulsion o f colliding particles and extremely low value o f the binding energy o f the captured particles (proton or a - particle) the direct capture reaction at stellar energies proceeds on the asymptote o f the overlap integral. The only interaction between nuclei in the region is the Coulomb force. The asymptotic tail o f the overlap function is determined by the binding energy and the asymptotic normalization coefficient (ANC), which is the vertex constant (NVC) multiplied by a factor [11]. Therefore considerable amount o f experimental and theoretical studies o f the peripheral proton and a-particle transfer reactions have been performed recently to obtain the AN C ’s for the virtual decays B —» A + p and B —» A + a o f astrophysics interest) (see Refs.[12-16] for example). There the values o f ANC o f nuclei 12,13C+p—>13,14N and 160 * —>12C+a were obtained from analysis o f the peripheral proton transfer reactions 12,13C(3He,d)13,14N [12,13,16], 13C ( 14N, 13C) 14N [14], 12C(7Li,t) 160 * and 12C( 6Li,d) 160 *[15], which were then used for calculation o f the rates o f the nuclear astrophysical reactions 12,13C(p,y) 13,14N [16,17] and 12C(a,y)160*[15].

In this review we present the results o f calculations o f the low-energy astrophysical S-factors for the 12,13C(p,y) 13,14N reactions and the t(a,y)7Li reaction performed within o f the framework o f the R-matrix approach [18-20] and the modified two-body potential approach.

A brief description o f these theoretical approaches is given.

2. MODIFIED TWO - BODY POTENTIAL APPROACH TO THE PERIPHERAL DIRECT

RADIATIVE CAPTURE

a + B + y

In this section we consider the direct radiative capture a + A —» B + y reaction at energies o f astrophysical interest where the a capture proceeds in loosely bound states o f nucleus B in the framework o f the two - body potential model and derive the explicit equation for the direct astrophysical S-factor, S(E), expressed in terms o f the corresponding ANC for the overlap function in the two - body (A + a) - channel [11]. We note that the astrophysical S-factor S(E) is related to the direct cross section <j(E) by

S(E) = Ee2mia(E).

2

.

.

.

.

Here tj = Z AZ ae juAa I k is the Coulomb parameter for the Aa - scattering and k is the relative momentum o f the colliding particles A and a , p Aa is the reduced mass o f the particles A and a , and E - k ! 2 p Aa.

The matrix element o f the reaction considered within the two - body potential approach in the long wavelength approximation has the form [19-20]

M = <1 a a ( r ) | 0 ( r ) | Y k (r ) >, (1) where i Aais the overlap function for the wave function o f the bound state B in the (A + a ) - channel, O is the electromagnetic transition operator and is the scattering wave function o f the colliding particles A and a . The asymptotic behaviour o f the radial overlap function I Aa i B j B ( r ) outside the range o f the nuclear interaction r > rN is given by the relation:

A a 'hİB ( ' ) - c Aa,

w

r (2 )

where Wa.j3 (jc) is the Whittaker function, qB = z Az ae2nAa / kAq is the Coulomb parameter for the B = (A + a) bound state, kAo = yj2juAas Aa , 8 Aa is the binding energy o f the B nucleus in respect to the (A + a ) - channel, rN is the nuclear interaction radius between A and a particles in the (A + a) - bound state, Ib(Jb) is the orbital (total) angular momentum particle a in the nuclear the B = (A + a ) , and CAaA . is the ANC related to the NYC GAa.lbj B for the virtual decay B —» A + a as [11]

G

Ao'JbJb

— _ ^ B +71b

c

Ao'JbJb

(3)

One note that the spectroscopic factor Z Aa.t j is determined by the formula [11]

oo

Z Aa;,Bj B = \ d r r 2I 2Aa, BjB( r ) . (4) 0

In the standard two - body potential model calculation the unknown radial overlap function I AaA j (r) is approximated by a model function as following

1/2

^ ActJbJ'b ~ ^ A a ; l B j B bJb (5)

Herz,(pnl j (r) is the single-particle bound state wave function o f the bound (A + a) state, which satisfies the radial Schrodinger equation with the phenomenological Woods -

Saxon potential and has asymptotic behaviour for r > rN as

V A a ' h h ^ ~ bAajisjB

W-rlBlBA \l2^l K A a r ')

(6)

where b

Zq\IbJb is the single - particle ANC and n is the number o f nodes, for example, for the direct capture

p + A —» B + y reaction the lion's share o f the dependence o f the cross sections c ( E ) (or S(E)) on the r0 and a parameters come from the single -particle wave function ÇnıBj-B ( r ) , which leads to the strong (r0, a ) - dependence o f the function Sj j (E ). So, these parameters cannot be fixed unambiguously. The theoretical

r

[23,24], show a noticeable sensitivity to the form o f used potentials [24]. So the problem o f reliable determination o f the (r0, a ) is still discussed by many authors ( see Refs.[25, 26,12] for example). Then, the astrophysical S-factor S(E) for the direct capture a + A —» B + / reaction within the two - body potential approach is reduced to

S ( E ) = Z Z A a , , Bj B S , B j B { E ) .

Jb

Here, we assume that only one l B is admitted in the expansion Eq.(6) and

(7)

s

,

J

( E ) =

I

J

)

(8)

A

is the single-particle astrophysical S-factor and its explicit form is determined by the expression Eq.(l) and corresponding formulas o f Ref.[19], where X is multipole order o f the electromagnetic transition. According to Ref. [19], the function j ^ ( E ) contains the radial overlap integral

oo

= W ^ A a ; n W , ( rV l A. ^ . M , (9) where I=A(A -1) for the electric (magnetic) transition, ^iAay^a(r) mis the radial wave function for the A a - scattering in the initial state, and lAa{ j Aa) is the orbital (total) angular momentum o f the relative motion of particles A and a in the initial state. The bound state wave function, is determined by the solution o f the radial Schrödinger equation. The solution o f this equation is found for the given quantum numbers n, l B and j B as well as geometric parameters o f rQ and a and with the depth to fit the binding energy 8 A . For the guarantee o f self-consistency, the same form o f the potential is used for the calculation o f the Aa - scattering wave function y/} . (r ).

Thus the overall normalization o f the cross section o f the direct capture a + A —» B + y reaction in the standard two - body potential approach is expressed in terms o f the spectroscopic factors Z Aa.t j , which are the only unknown quantities in Eq (7).

For each fixed value o f lB and j B the spectroscopic factor Z AaA . is usually found from analysis o f experimental cross sections on different one-particle transfer reactions (for example, A(d, n)B and A(3He,d)B) and the radiative capture reactions a + A —» B + / ,

by means o f normalization o f the calculated cross sections to the experimental ones. But the empirical values o f the factor Z Aa.t j obtained in such a way depend strongly on model parameters, the radius r0 and the diffuseness a o f the Woods - Saxon potential, which are not measured directly. For example, for the direct capture p + A —» B + / reaction the lion's share o f the dependence o f the cross sections g(E) (or S(E)) on the r0 and a parameters comes from the single -particle wave function which in turn leads to the strong (r0, a ) - dependence o f the function S t j (E ) . So, these parameters can not be fixed unambiguously. The theoretical predictions o f the spectroscopic factor Z AaA • , based on different restrictions on structure of nucleus B show a noticeable sensitivity to the form o f used potentials. So the problem o f obtaining o f reliable value o f Z Aa.t j is still discussed by many authors.

But, situation is simplified when one considers the peripheral direct radiative capture a + A —» B + y reaction in which at rather low energies o f colliding particles the direct capture proceeds to weakly bound states o f the residual nucleus B. In this case, similar to what has been proposed in [22] for an analysis o f the

transfer reaction cross sections, it is expedient to express the astrophysical S-factor S(E), given by Eq.(7), in the term o f ANC's, CAa.t j , rather than the spectroscopic factors, Z Aa.t j • Then, by using the relation [11,22]

z

c

2

A a 'JBJB the Eq.(7) can be rewritten in the following form

S ( E ) - ( Z c zAırj t h )R,B {E,bM İ B j B ),

In (12) , as it is shown in the calculation the ratio o f the r.h.s. for electric transitions does not depends on j B. It is seen from here that the overall normalization o f the astrophysical S-factor for the peripheral direct capture a + A —» B + / reaction is determined by the value o f the ANC corresponding to the (A + a) -configuration in the nucleus B. Consequently, if the A N C s are known, they allow one to calculate the direct astrophysical S-factor, S(E), o f the reaction under consideration at stellar energies.

In recent years considerable amount o f experimental and theoretical studies devoted to the peripheral one - particle (proton and a-particle) transfer reactions were performed within the DWBA approach to determine the A N C s for the A + p—»B and A + a —» B o f astrophysics interest (see Refs.[12-16] for example). However, the DWBA approach is the first order perturbation approximation over the Coulomb polarization operator Ac in which the transition operator Ac sandwiched by the initial- and final state wave functions is assumed to be small. But as it was shown in Refs.[13,27-29] when the residual nuclei B are formed in weakly bound states, this assumption is not guaranteed for the peripheral proton a-particle) transfer reactions and, so the obtained values o f A N C s for the astrophysical application may not have the necessary accuracy. In this case an inclusion o f all orders (the first, second and higher orders) o f the power expansion in a series over Ac is required in the transition operator for the DWBA cross section calculations [13,28].

However, the required ANC for the a + A —» B can also be obtained from experimental astrophysical S - factor, Sexp(E), o f the direct a + A —» B + / capture reaction at extremely low energies if a measurement o f Sexp(E) is done with sufficient precision. In this case the equation (11) can be used as an independent source o f information on astrophysical interest value o f the ANC for the a + A —» B. To this end the magnitude

RAa-iBj B{E) (12) must be considered as a function o f the single-particle ANC bAa.j j (= bAa;i j (r0,a)) for each fixed value o f energy E from the range E min < E < £ max in which the direct radiative capture mainly occurs. Therefore, all of the (rQ, a ) - dependence o f the direct capture astrophysical S-factor, S(E), enters also through the single-particle ANC i>Aa-lBjB(r0, a ) . The latter can be found from variation in a physically acceptable range (about ± 10% in respect to their standard values) o f the geometrical parameters (r0 a n d a ) o f the Woods-Saxon potential for the bound (A+ a ) state o f the nucleus B.

In this case peripheral character for the direct capture reaction capture a + A —» B + / must be conditioned by

^ lB ) = f ( E ) (13)

as a function o f the bAa.t j energies E from the range E min < E < £ max in which the direct radiative capture mainly occurs, and by

c 2 . AajsjB

S(E) R lB (E ’ b A a j Bj B )

const (14)

for each fixed o f E nd the function o f Rj ( E , b AaA . ) from (13). Fulfillment (or at weak violation within the experimental errors for the Sexp(E)) o f the conditions (13) and (14) allows one to obtain a valuable information about the experimental value o f the ANC ( C ^ . )2 for the A + a —» B by using Sexp(E) instead o f the S(E) in the r.h.s. o f Eq.(14):

(Cexp .2 £eXP(£) ₍₁₅₎

Then the obtained from Eq.(17) value o f the ANC, ( C ^ 7 ) 2, together with the condition (13) can be used for calculation o f S(E) stellar energies E<Emin by the expression:

)

2

R,B (E.bAa.jB h ).

(16) Obtained in such way values (C ^ p7 .

Jb

) and the S(E) at stellar energies E can be considered as an "indirect measurement" o f the ANC (or NVC) for the A + a —» B and o f the astrophysical S-factor for the direct capture a + A —» B + y reaction at E<Emin, including E=0.

The developed modified two-body potential approach can be applied for an analysis nonresonance (direct) radiatiove capture a + A —» B + y reaction for which a dominant contribution into the astrophysical S-factor comes from the external region. As there are resonance contributions ito the cross section it is necessary to carry out an analysis with taking into account resonances amplitudes. In section 4 we briefly present the basis stage o f the R-matrix approach following Refs. [18-20]

3. ANALYSIS OF THE T + a -> 7Li+y REACTION

In this section, to determine the ANC's values for the 7Li—>t+a the reanalysis o f the experimental astrophysical S-factors, Sexp(E), for the the t+a —» 7Li+y reaction populating the ground and first excited (E =0.478 MeV; 1^= 1/2") states is carried out on the basis o f relation (11) and the conditions (13) and (14) as well as o f the relations (15) and (16). For these reactions, the value o f 1B (B=7Li) is taken to be equal to 1 and the value o f jB is taken to be equal to 3/2 (1/2) for the ground (first excited) state o f 7Li, while lat=0, 2 for the E l - transition and lat= l for the E2 -transition. The experimental data have been obtained by many authors with absolute experimental uncertainty A being within A =14-=-25%. But, Recently, C.R. Brune et al [30] performed rather precise measurement o f the astrophysical S-factor in the energy range o f 55< E< 1189 keV with A= 6-rl0%. So in our analysis we naturally use the Sexp(E) measured in [30].

The Woods-Saxon potential split on a parity (/ - dependence) with the spin-orbit term proposed by the authors o f Refs. [31] is used here for the calculations o f both the bound state radial wave function

q>niBh (raJ) and the scattering wave function y/l . (rat) .

The test o f the peripheral character o f the t+ a —» 7Li+ y reaction for the energy range o f 55<E< 1189 keV has been made by means o f verifying the conditions (13) and (14) and by changing the geometric parameters (radius r0 and diffuseness a ) o f the adopted Woods-Saxon potential using the procedure o f the depth adjusted to fit the binding energies. The calculation shows that for each energy E the lion's share o f a dependence o f the function S ( E ) on the parameters r 0 and a comes mainly from the single-particle ANC b(=b(r0,a)). Here and below for simplicity all indexes specifying the singe-particle ANC, the quantum numbers specifying the ANC's, NVC's and spectroscopic factors as well as the functions Rt (E , b Aa.; . ) and

S . (E )have been dropped. It should be noted that, if one varies only one parameter r0 or a fixing other

‘bJb

one, then changes become stronger. But if one varies r0 or a with the condition b=b(r0,a) =const, then there is a week dependence o f the . (E) function on r0 or a. We vary r0 and a in the ranges (r0 in 1.62-1.98 fm and a in 0.63-0.77 fm or 2.3 < b < 3.1 fm 1/2) in respect to the standard values (r0 =1.80 fm and a =0.70

fm [31]). The "residual" (r0 , ^-dependence o f the single- particle astrophysical S-factor, S l . (E ) , on r0 and a for each b(r0,a) -const turns out to be extremely weak up to about « 2%. So, the S t j (E ) is a rapidly varying function o f b with the extremely weak "residual" (r0 , ^-dependence for each b - c o n s t . However, for each fixed experimental point o f energy E a quantity R(E,b) depends weakly (up to ± 8.5 %) on the variation o f b, and its "residual" (r0 , ^-dependence on r0 and a for each b -const is also extremely week (up to ± 2 % ). Besides, it should be emphasized that the uncertainty in the R(E,b) values is decreased with a decrease o f the energy E. It follows from here that the condition (13) is satisfied for the considered reaction within the uncertainties not exceeding the experimental errors o f Sexp (E). This circumstance allows us to test the condition (14), which is no less essential for the peripheral character o f these reactions.

The performed calculation shows that the values o f the C 2 are also weakly dependent on the b -value (2.3 < b < 3.1 fm 1/2), which corresponds to the parameters o f the adopted Woods-Saxon potential r0 ranging from 1.62-1.98 fm and a in the range o f 0.63-0.77 fm. However, the values o f the spectroscopic factors, Zat and Z*at corresponding to the (a +t)-configuration for 7Li(g.s.) and 7Li(0.478 keV), respectively, change strongly (about a factor o f 1.7).

For each experimental point o f E (E=Ej i= l-17) from the interval o f 55 < E< 1189 keV using the corresponding experimental astrophysical S-factor in the r.h.s. o f the relation (15) instead o f the S(E) and the central values o f R(E, b) corresponding to the adopted values o f the parameters rQ and a the values o f the ANC's are obtained for the a + t—» 7Li(g.s.) and a + t—>7L i(0.478 MeV). The results o f the ANC's, (C ^ p) 2 and(C;;xp)2 for the seventeen experimental points o f energy E are displayed in Fig. 1. The uncertainties pointed in this figure correspond to those found from (15) (averaged square errors (a.s.e.), which includes both the statistic experimental errors in the corresponding experimental astrophysical S-factor and the aforesaid uncertainty in the R(E, b) , and the systematic uncertainty o f 6 % [30] added to the a.s.e.). It is seen from Fig. 1 that the ratio in the r.h.s. o f the relation (15) practically does not depend on the energy E although absolute values o f the corresponding experimental astrophysical S-factors for the reactions under consideration depend noticeably on the energy and change by up to about 1.6 times in changing E from 55 k eV to 1189 keV.

This fact allows us to conclude that the energy dependence o f the experimental astrophysical S-factors [30] is well determined by the calculated function R(E,b) and, hence, the corresponding experimental astrophysical S-factors can be used as an independent source o f getting information about the ANC's for the a + t—>7Li(g.s.) and a + t—>7L i(0.478 MeV).

The weighted means o f the ANC-values obtained from the data presented in Fig. 1 for the a + t—>7Li(g.s.) and a + t-> ?L i(0.478 MeV) are equal to ( C exp) 2 -1 2 .7 4 ± 1.10 fm 1 and (C*'xp) 2 =9.00± 0.90 fm 1 . The corresponding values o f the NVC's are | G Qxp | 2 - 0.60± 0.05 fm and | G*^xp | 2 - 0.42 ±0.04 fm.

The equation (16) and the weighed means o f the ANC's obtained can be used for calculating the t+a —» 7Li+ y astrophysical S-factor for capture to the ground and first excited states as well as the total astrophysical S- factor at stellar energies (E< 50 keV). At first, we tested again the fulfilment o f the condition (13) in the same way as it is done above for E > 50 keV. Similar results for E > 50 keV are also observed for a dependence o f the R(E, b) function on the single particle ANC, b, at stellar energies o f E< 50 keV. The results o f extrapolation o f the astrophysical S-factor for six values o f E (E - 0, 10, 20, 30, 40 and 50 keV) obtained by us are displayed in Fig. 2. In this figure circles and squares correspond to the astrophysical S- factors S ex(E) and Sg s (E) for the capture to the excited (0.478 MeV) and ground states o f 7L i , respectively, triangles correspond to the total astrophysical S - factor S(E) The closed symbols in the Fig.2 are experimental data from [30], while the open symbols are the result o f extrapolation o f the S-factor at the above mentioned energies. The solid lines present our calculations performed with the standard values o f the geometric parameters ro-1 .8 0 fm and a -0 .7 0 fm both for the bound (a + t) state and for a t -scattering state. As it is seen from Fig2, the equation (16) allows us to perform a correct extrapolation o f the corresponding astrophysical S-factor at stellar energies practically in an independent way when the corresponding ANC values are known. For a comparison, in Fig.2 the results o f Ref. [32] (dashed lines) and Ref.[33] (dotted lines) are also displayed. The figure shows that at extremely low energies a discrepancy between the present results and that o f Ref.[32] at E <150 keV occurs. The obtained value o f S(0)-0.0974 ±0.0100 keVb is in a good agreement with that od S(0)-0.10 ±0.02 keVb recommended in Ref.[34].

Thus, it follows from here that the overall normalization o f the astrophysical S-factors at stellar energies for the reactions under consideration is mainly determined by the ANC - values for the a + t—>7Li(g.s.) and a + t—>7L i(0.478 MeV), which can be determined rather well from an analysis o f the corresponding experimental astrophysical S-factor [30] a model independent way, and the values o f the ANC's allow us to perform correct extrapolation o f the astrophysical S-factors for the direct radiative capture t+a —» 7Li+y reaction at stellar energies, including E=0.

4. R-MATRIX APPROACH

The R-matrix method assumes that the space o f interaction o f the colliding nuclei is divided into two regions: the internal region (with radius rc ), where nuclear force becomes important, and the external region, where the interaction between the nuclei is governed by the Coulomb force only [35].

At first let us consider the elastic scattering b+c—>b+c and the elastic scattering S-matrix element in channel a is given by [35,36]

5_{a a} = e

İ / R ' - I A ' - B ' + İ P 'İ E ) ] ’

(17)

where R[ is the R-marix for the /th partial wave, Ai is the Thomas shift, B t is the energy - independent R- matrix boundary condition constant, and Pi is the penetration factor:

P,(E)

________

_________

Gt(E,rc) + F?(E,rc)

(18)

Here G[(E,rc) and F[(E,rc) are the singular (at the origin) and regular solutions o f the radial Schodinger equation with a pure Coulomb potential at E>0, i.e.

= G , ( E , r c) + i F , ( E , r c ) G i ( E , r c ) ~ iF l ( E , r c ) ’

(19)

where is the so-called hard sphere scattering phase shift. The elastic scattering amplitude is given by the sum o f two terms

T /v/v — \ — S A /v/v — Tr

(pot) a a

+ 7)

( r )

where T ^ ,) ( T ^ J ) is the potential (resonance ) scattering amplitude, and the has the form as

= ~ 2 i e

P,(E)

R l - [ A l ( E ) - B l + iPl ( E ) ]

(20)

where the R-matrix is

R , ( E ) = y Yca

x E c X - E

(21)

Here E cx are the poles o f the R-matrix and ycx is the reduced width o f the Xth level. If the energy o f the subthreshold bound state is very close to threshold and incident energy E—»0 then we can use the one-level R-matrix approximation (A,=A,0), which leads to the expression as

rr ( r) _ 1 a a l i e 2i(<j)l -c>)l E cK Pi( E ) YcA0______________ E - i A ' W - B ' + i P ' i E ) ] / 2^ ’ (22)

Usually the poles o f the R-matrix don’t coincide with the poles o f the S-matrix. But, if we choose the boundary condition Bl=A[(E^), where the resonance energy E ^ is defined as the solution o f the equation E cx-E-Ai(E)=0, so that

(23)

i.e., the higher order terms o f the Teylor’s expansion being neglected. Then the expression (22) can be reduced to the form as

E,, -

( E ) - E * ( E < ” - E)(

lie

■2

____

E (r)

P ,( E ) r X

where the effective reduced width is given by

rcA„ =■

/ X

\ + y zCÂ[d A, (E )/d E] E_E(t)

The ‘observed partial width is determined by the relation

r cio;/( £ M ) = 2P/ ( £ M ) ^ .

For an inelastic scattering c—> c ’ the resonance amplitude has the form

(24) (25) (26) rr ( r )

_

lie

E (r) - E - iP!(E)]y

Now we can extend the R-matrix approach with inclusion o f photon process. Although the treatment o f photons is different from that o f heavy particles the qualitative role played by photon is similar in nuclear reaction, since photon processes are usually included in the theory o f nuclear reactions by analogy with usual particle processes. For example, the dispersion theory for nuclear reactions going through a compound nucleus state is extended to photon by means o f including a damping width for photon processes. This is plausibly by the Bohr mechanism o f the compound nucleus, which decays by competition through various channels, including photon emission. Thus, assuming that the internal region involves large enough contributions to the matrix collision for the radiative capture a + A —» B + y reaction, we may assert that

rr ( r) —

L a y

l ie

i((f>i-CF)l P r ( E ) y c, k ^ n rr„

E ^ - E - i P , ( E ) y X ’

(28)

where y ^ is the y-ray reduced width and X is the multipole transition.

Let us consider a particle-photon cross section proceeding through an isolated resonance with taking into account spins o f the particles. Let us write 1; (lf) for the relative angular moment for particles A and a in the entrance channel (the bound B=(A+ a ) state ) and Ji(Jf) for the spin for the resonance (final) nucleus. In this case, the resonance amplitude (28) in the one-level and one-channel approximations can be written as [35]

r p ( r )

1 ay^sl^ {J j-

= -ie

(£ )]'

where T j ~ ,sl J is the total T - width, T x .su f (E) is the particle (a) partial width o f the resonant state o f B in the (A+ a )-channel and E Xy.U j (E) is the y-width for the decay B —>B+y. Energy dependence o f the partial widths is determined by the relations

r

=

l p iSE ) r l ,

2 e

+1n 2

y

y

yxnx - j j f 0 ^ 0 •

Experimental particle-and y-widths for the given set o f the quantum numbers slJi and /L/^are

j-iexp _

Xn :slJ,

(E

(r) p-iexp _

1Y ~

r

The amplitude o f the radiative direct capture in the long-length wave approximation can be written as

where _ / 0 \ 3 / 2 . l i +X-l f+\ 1 A + l / 2 ( ^ a e / i \ / l % A e \ 1 a r M j j f ~ vz / 1 e k " Aa + ^ 1 _ / l

-(DC)

(2 1 + 1)!!

C

<

Es>

( E , rc )G, (E,rc W

_ ^ l/+V2

(2

J

)

X

P,1/2 (E)C ,yinW(Mf J is\liJ f V\ ( l, I f ),

1 00

+

(ltl f ) = — \drr

w

w

(2* Aar) Flt(E,r) _ G;,(£ ,r )

(2KAarc) F X E , r c)

G , ( E , r c)

One notes that the direct radiative capture amplitude is normalized in terms o f the ANC,

CA l .

^ay\slf JtJf X ~ ^ay\sliJiJf + ^ay\XsltJ f *

Then the R-matrix radiactive cross section to the state o f nucleus B with spin Jf is given by [37]

a j f ( E ) = T Ja Jij f ( E ^

(34)

2

a

,(* ) =

n

2 J, +1

k 2

(2J a

\

5. ANALYSIS OF THE 12C(P,y)13N AND 13C(P, y)14N REACTIONS

In this section we present the results o f the analysis o f the 12C(p,y)13N and 13C(p, y)14N reactions using the information about the previously obtained ANC’s for 13,14N nuclei in the (p+12,13C)- channels and Eqs. (1), (34) and (35).

The 12C(p,y)13N experimental data for the ground state 13 N are taken from the latest work [37] together with our data, which were measured at beam energies Ep=365-1065 keV with rather high precision (less than 10 %) in he electrostatic tandem accelerator UKP-2-1 at the Institute o f Nuclear Physics in Kazakhstan. An analysis o f the astrophysical S-factor S(E) for the 12C(p,y)13N reaction at energies E < 2.5 MeV is performed taking into account the contributions o f the two resonances with energies E l^ b =457 keV (.P=l/2 + ), and

Eİab= 1699 keV (J7t= 3 /2 ') , direct capture and their interference.

The measured ANC Cn} (here and below for simplicity the indexes speciflng the p 12C - channel has been dropped) is taken from [13,39] and

C ii= l.43 ± 0.09 fm 1/2 (|Gn|2=0.34± 0.04 fm). The resonance parameters for the partial proton and y-ray widths corresponding to the first and second resonance states o f 13N found by different authors prove to be rather different (see [40-43]). Therefore, these parameters and the resonant energies are varied by means of fitting to experimental data to minimize %2. The fitted values o f these parameters are given in Table 1. In Fig.3 the energy dependence o f the fitted S(E) (solid line) and the Sexp(E) are displayed, which corresponds to the channel radius rc=5.0 fm providing a minimum %2 equal to 8.5 for the energy region with E < 1.0 MeV.A comparison o f the calculated astrophysical S-factor S(E) with the experimental value Sexp(E) shows that at E > 1.7 MeV agreement can be obtained with the experimental data as the contributions from highly excited resonances states o f 13N are approximated by the two background (El-and Ml-)resonance transitions. The parameters o f these background resonances are presented in Table 1 also (the fifth and eighth lines). The results o f our calculations o f the astrophysical S-factor at the stellar energies E=0 and 25 keV are S(0)=1.60 keVb and S(25)= 1.75 keVb. One notes that our result for S(25 keV) differs noticeably from that o f S(25)=1.33±0.15 keVb [44], 1.45 ± 0.20 keVb [38] and 1.54±0.08 keVb [45]. It should be noted that a value o f S(25)in Refs [38,44,45] has been also obtained within the R-matrix approach without taking into account the information about the NVC (or ANC), while our result is obtained at fixed value. Perhaps, that is one o f the possible reasons the observed discrepancy between our results o f the calculation and those o f obtained in Refs [38,44,45] for S(25 keV).

We also analysed the low-energy experimental astrophysical S-factors for the 13C(p, y)14N reactions [46] taking into account contributions from both resonant and nonresonant (direct) captures. There are two low-lying resonances in 14N at 2 ^ = 4 1 7 .9 keV (J7:=2 ) and £ ^ = 5 1 1 .4 keV (.T=r). The first resonance is very narrow and is not in the astrophysical important energy region, while the second one dominates the capture

• • , _ J F K \ (

cross section in the energy region near E 2 and down. An accurate allowance for the direct (nonresonant) contributions is done by means o f fixing values o f A N C ’s for 14N in the(13C+p)-channel, which are presented in Table 2 and are obtained independently from the analysis o f peripheral proton transfer 13C( 3He,d)14N reaction [12] taking into account the three-body Coulomb dynamics o f the transfer mechanism [13]. We find that rc=5.0 fm provides the best fit for the transition to the ground and first five excited states o f 14N. The best description of experimental data on the astrophysical S - factor is obtained at the proton width 37.14 keV (E, 1) and 408.57 keV (0 ,1) for the first and fifth resonances. These values agree with the results o f work [46].

The resonance e n erg ies^ (i=l and 2) and radiative widths for transitions to each the ground from the excited bound states were varied within the intervals determined in Ref. [46] and are given in Table 3. We use the A N C ’s to calculate the astrophysical S-factors for each transition. They are given in the 6th column o f Table 2. The results of the fit are displayed in Figs. 4 and 5. The value obtained for the total astrophysical

S-factor S(0)=7.60 keVb This is in excellent agreement with the result found in the R ef [46] and [47] despite the differences in the analysis.

Thus, it follows from here that, the A N C ’s determined from the peripheral proton transfer 13C( 3He,d)14N reaction [12] provides good fits to the astrophysical S-factors o f the 13C(p, y)14N reactions at extremely low energies populating to the ground and fifth excited states o f the nucleus 14N.

6. CONCLUSION

1. The modified two-body potential approach for analysis o f the astrophysical S- factor S(E) for the peripheral direct radiative capture a+A->B+y reaction of astrophysical interest is proposed in which two additional conditions of verification o f the peripheral character o f the a+A->B+y reaction are also formulated.

2. The proposed approach allows one to reduce to minimum the model dependence o f the calculated direct astrophysical S-factor, S(E), on the geometric parameters (radius r0 and diffuseness a) o f the adopted Woods-Saxon potential, which is usually used for calculations both for the two-body (A+a) bound state and for the Aa-scattering state.

3. The proposed approach has been applied to the analysis o f the experimental astrophysical S-factor, Sexp(E), for the t+a —>7Li+ y reaction.

4. It is demonstrated that the experimental astrophysical S-factors o f the reactions under consideration can be used as an independent source o f the information about the ANC's (or NVC's) for a-particle removed from the 7Li.

5. The obtained values o f the ANC's were used for the estimation o f values for the astrophysical S-factor S(E) at solar energies.

6. New estimations for the rates o f the 12,13C(p,y)13,14N reactions have been obtained within the R-matrix approach using information about A N C ’s for the 13,14N the ( 12,13C+p)- configuration.

7. REFERENCES

1. E.G. Adelberger et al. Rev.Mod.Phys. 70(1998)1265.

2. C.Rolfs, W.S.Rodney. Cauldrons in the cosmos, University o f Chicago Press, Chicago, IL, 1988.

3. R.K.Ulrich, in : C.A. Barnes, D.D. Clayton, D.N. Schramm (Eds.), Essays in Nuclear Astrophysics, Cambridge Univ.Press, Cambridge, UK, 1982, p. 301. 4. D.N.Schramn , R.V.Wagoner. Ann.Rev.Nucl.Sci.27(1977)37.

5. R.V.Wagoner. Astrophys.J.Suppl.Ser. 18(1969)247. 6. R.G.H.Robertson et al. Phys.Rev.Lett.47(1981)1867.

7. K.M.Nollett, M.Lemoine, D.N.Scharmm. Phys.Rev.C56(1997)l 144. 8. J.N.Bahcall. Neutrino Astrophysics. Cambridge University Press. 1989. 9. S.B.Igamov , R.Yarmukhamedov. Nucl.Phys.A673(2000)509.

10. Q.R.Ahmad et al. Phys.Rev.Lett.87(2001)071301.

11. L.D.Blokhintsev, I.Borbely, E.I.Dolinskii. Sov.J.Part.Nucl.8(1977)485. 12. S.V. Artemov et al. Phys.Atom.Nucl. 59(1996)428.

13. R. Yarmukhamedov. Phys.Atom. Nucl.60( 1997)910. 14. L. Trache et al. Phys.Rev.C58(1998)2715.

15. C.R. Brune et al. Phys.Rev. Lett. 83(1999)4025. 16. Bern et al. Phys.Rev.C62(2000)024320.

constant for the vitual decay 13N —>12C +p and the astrophysical S-factor o f the 12C(p,y)13N reaction at extremely low energies. Proc. o f the 3th Inter. Conf. Nuclear and Radiation Phys., 4-7 June 2001 ,Almaty , Kazakhstan. Part 3 , p.p. 259-268. 18. A. M. Mukhamedzhanov et al. Nucl.Phys. A725(2003)279.

19. K.H. Kim, M.H. Park, and B.T. Kim, Phys. Rev. C 35 (1987) 363. 20. R. G.H. Robertson et al. Phys. Rev. Lett. 47 (1981) 1867. 21. P. Mohr et al. Phys. Rev. C 48 (1993) 1420.

22. S.A. Goncharov et al. Sov.J.Nucl.Phys. 35(1982)383. 23. S.Cohen and D.Kurath. Nucl.Phys.73( 1965)1; A 101(1967)l. 24. N.K. Timofeyuk. Nucl.Phys. A632(1998)19.

25. H.T.Fortune et al. Phys.Rev.l79(1969)1033. 26. C.Rolfs. Nucl.Phys.A217(1973)29.

27. G.V.Avakov et al. Sov.J.Nucl.Phys.43(1986)524.

28. Sh.S.Kajumov, A.M.Mukhamedzhanov, R.Yarmukhamedov, and I.Borbely. Z.Phys.A336( 1990)297.

29. V. Artemov et al.Izv.RAN.Seriya Fiz.66(2002)55.

30. C.R.Brune, R.W.Kavanagh, and C.Rolfs.Phys.Rev.C50( 1994)2205.

31. V.I. Kukulin, V.G. Neudatchin, and Yu. F. Smirnov, Nucl.Phys. A245 (1975) 429. 32. S.B. Igamov, T.M. Tursunmuratov and R. Yarmukhamedov. Phys.At.Nucl. 60(1997)1126. 33. K.M. Nollett.Phys.Rev.C63(2001)054002.

34. C.Angulo. Nucl.Phys.A656( 1999)3.

35. A.M. Lane, and R.G. Thomas. Rev.Mod.Phys. 30(1958)257.

36. J.M. Blatt and V.F. Weisskopf. Theoretical Nuclear Physics (Springer-Veriag), New York, 1979).

37. F.C. Barker and T. Kajino. Aust.J.Phys. 44(1991)369. 38. C.Rolfs and R.E. Azuma, Nucl.Phys. A227(1974)291.

39. S.V. Artemov et al. Izv.RAN(Bull. Russia Acad.Sci.) Ser. Fiz. 66(2002) 60. 40. F.Ajzenberg-Selove. Nucl.Phys. A 152(1970)1.

41. F.Riess et al. Phys.Rev. 176(1968)1140.

42. S.Hinds,and F.C. Barker. Aust. J. Phys.45(1992)749. 43. F.Ajzenberg-Selove, Nucl.Phys. A523( 1991)1. 44. D.F.Hebbard and J.L. Vogl. Nucl.Phys. 21(1960)652. 45. F.C.Barker and N.Ferdous Aust.J.Phys.33(1980)691. 46. J.D. King et al. Nucl.Phys. A567( 1994)354.

47. A.M. Mukhamedzhanov et al. Nucl.Phys. A725(2003)279. 48. R.J. Peterson, J.J. Hamili. Nucl. Phys. A362 N1 (1981) 163. 49. F.Ajzenberg - Selove. Nucl.Phys. A 449 (1986) 1.

Fig. 1. The values o f the ANC's, (C^xp ) 2 and(C*,exp ) 2 , for the a + t—>7Li at all experimental energies E;. The square (circle) symbols are for the ground (excited (E* =0.478 MeV)) state o f 7Li.

Fig. 2. The astrophysical S - factors for the t+a —» 7Li(g.s)+y (square), the t+ a —» 7Li(0.478 MeV)+y (circles) and t+a —» 7Li (g.s.+ 0.478 M eV )+7Li (triangles) reactions. The filled symbols are experimental data taken from [30] and the blank points are our results. The solid lines are the results o f our calculation with the standard values o f r 0 =1.80 fm anda =0.70 fm, the dashed lines are the results o f Ref.[32] and the dotted lines are the results o f Ref.[33].

Fig. 3. The astrophysical S - factor for the 12C+p —» 13N+y reaction. The square (filled) points are experimental data taken from [38] (our data).

Fig. 4. The astrophysical S - factor for the 13C+p —» 14N+y reaction. The experimental data from [18].

Fig. 5. The astrophysical S - factor for the 13C+p —» 14N+y reaction. The experimental data from [18].

Table 1. The table o f the T - widths for the 12C+p —» 13N+y reaction Resonance I T ( k e V ) n ( e V ) J * , ER(MeV) Transition 1 4 36.5İ0.9, 33.7+1.8 [40] 0.67 [40] - , 0.422 33.9±0.9, 36.0±1.8 [41] 0.45İ0.05 [41] 2 _{31.7İ0.8 [43] 0.50İ0.04 [43]} El _{35.0 (our) 0.65 (our)} 100. (our) 900.0 (our) 5.0 3~ 62.0İ4.0 [43] 0.64 [43] - , 1-57 60.0 (our) 0.40 (our) 2 M l 100. (our) 3000. (our) 5.0

Table 2. Asymptotical normalization coefficients for 14N —>13C+p E*, MeV (J”,T) 0.0 (1+, 0) 0.0 (1+, 0) 0.0 (1+, 0) 2.313 (0+, 1) 3.948 (1+, 0) 4.915 (O', 0) 5.106 (2“, 0) 5.691 ( r , 0) 5.834 (3“, 0) nlj ip 1/2 + İP3/2 ip 1/2 İP3/2 ip 1/2 ip 1/2 2Si/2 1^5/2 2Si/2 1^5/2 C2 , fm"1 14.47 13.78 0.69 9.78 1.67 10.50 0.451 6.53 [48] 0.169 [48]

Table 3. The table o f the T ^ 1* - widths for the 13C+p —» 14N+y reaction J 71 E r (MeV) j 71 ^ (M e V ) r r ^ - eV [491 [461 [471 This work T; 0.517 T; 0.000 9.9 ± 2 .5 9.1 ± 1.30 9.10 9.10 0+; 2.313 0.17 ± 0 .0 5 0.22 ± 0.04 0.22 0.22 T; 3.948 1.56 ± 0 .4 0 1.53 ± 0.21 1.53 1.53 0-; 4.915 0.23 ± 0.06 0.262 ± 0.043 0.260 0.30 2'; 5.106 0.03 ± 0.02 0.075 ± 0 .0 2 5 0.085 0.085 T; 5.691 0.43 ± 0 .1 2 0.61 ± 0 .1 4 0.63 0.60 O'; 1.244 T; 0.000 46 ± 12 46 ± 12 56.0 56.0 T; 3.948 0.46 ± 0.14* 0.56 0.56 2'; 5.106 0.37 ± 0.14* 0.23 0.23 T; 5.691 0.23 ± 0.20* 0.23 8.1