The 12C(α, γ)16O reaction at stellar energies

Nuclear Physics A410 (1983) 334-348 @ North-Holland Publishing Company

THE ‘*C(a,yf60

REACTION K. LANGANKE*

AT STELLAR ENERGIES+ and SE. KOONIN

W.K. Kellogg Radiation Laboratory, California Institute of Technology, Pasadena, California 91125, USA

Received 7 June 1983 Abstract: The most recent experimental data on the %(a, y)i60 reaction at energies E, m s 3 MeV have been analysed to determine the astrophysical S-factor. The absolute E2 capture cross section was calculated in a microscopically based direct capture model, while the El cross section was described by the hybrid R-matrix model. We find that the E2 rate is less than 4% of the El rate at astrophysically important energies. The two sets of data yield S-factors at E,., = 300 keV of 0.15 MeV . b and 0.34 MeV . b.

1. Introduction The3a+ ‘*C and the ‘*C((w,y)r60 reactions are the main processes of stellar helium burning. Their relative rates determine the ‘*C/l60 ratio at the end of helium burning, which is of key importance for the subsequent stellar evolution and nucleosynthesis of heavier elements ‘). In contrast to the 3a + ‘*C process, the “C(a, y)i60 reaction rate at stellar energies is still rather uncertain. Since the most effective energy (E,.,. = 300 keV at T = 5 x 10’ K) is far below the Coulomb barrier, a direct measurement of the required cross section is impossible with existing techniques. On the other hand, the extrapolation to low energies of cross sections measured in the laboratory for E c.m.b 1.4 MeV, as is usual in nuclear reactions of astrophysical interest, is particularly unreliable here. This is because there are two 160 bound states (spins and parities l- and 2+ at E,, = 7.12 MeV and E,, = 6.92 MeV, respectively), which are just below the ‘*C+ a-threshold at E,, = 7.16 MeV, and may therefore substantially influence the stellar rate. Further complications arise from the broad l- state in 160 at E,, = 9.63 MeV, which dominates the range of energies that can be investigated experimentally. Until recently, it was generally assumed that the main contribution to the total cross section at E,.,. = 300 keV was due to El radiation. This hypothesis was supported by a theoretical estimate of the E2 radiation which, however, assumed ’ Supported in part by the National Science Foundation [PHY82-07332 and PHY79-236381. * On leave from the University of Miinster. 334

K. Langanke, SE. Koonin / “C(a, y)160

335

direct capture only 3). The calculation agreed with the (T~JG~~ ratios measured for energies above E,.,. = 2 MeV and predicted a negligible E2 contribution at stellar energies. Models which included the influence of the l- bound state, based on the El capture data from ref. 4), estimated the astrophysical S-factor to be S(300 keV) 0.08-o. 14 MeV . b [refs. 4-6)]. This picture has been challenged recently by a measurement ‘) of the total ‘*C(cy, y)i60 cross section for E,.,. = 1.4-3 MeV which found values some 40% higher than the El data of ref. “). A simple extrapolation of this new measurement gives S(300 keV) = 0.43 MeV * b [ref. 7)], more than three times the previous value. Furthermore, in this analysis the E2 cross section was found to be comparable to the El at stellar energies, contributing some 40% of the total at 300 keV. The determination of the astrophysical S-factor given in ref. ‘) can be criticized on several points. First, relative El and E2 contributions were determined only by fitting the energy dependence of the total ‘*C((w,y)160 cross section for E,.,. = 1.4-3 MeV, rather than from the less ambiguous angular distributions. Since previous measurements of the angular distributions have determined ~ui/(~u~0.01-0.35 for E,.,. - 2-3 MeV [ref. 4)] and the influence of the l- and 2+ states at 7.12 MeV and 6.92 MeV on the total cross section is very uncertain, the extrapolation of both (+E2/~E1 and the S-factor to 300 keV is on less than firm ground. Furthermore, the method of extrapolating the cross sections ignored any background contributions from high-lying states, which previous analyses have shown to be important. In this paper we present an analysis aimed at reducing the uncertainties in the El and E2 contributions to the ‘*C(a, y)160 cross section. We must, of course, model both components. We treat the E2 process in a microscopically founded potential model which simultaneously reproduces the scattering phase shifts of ref. ‘) and the properties of the bound state at E,, = 6.92 MeV, which is known to have a strong ‘*C+(u component. With these constraints both below and above the ‘*C+a-threshold, we are able to calculate the absolute E2 cross section for capture to the ground state for E,,. s 3 MeV and hence determine the El component of the data in ref. ‘) by simple subtraction. An analogous microscopic study of the El radiation would be extremely complicated, since the El transition can only occur via small T = 1 admixtures in the states at E,, = 7.12 MeV and 9.58 MeV. We have therefore chosen to describe the El cross section with the hybrid R-matrix formalism of ref. ‘). Since the 160 ground-state wave function used in our microscopic investigation of the E2 capture discussed above differs slightly from that used in ref. 5), we have also redone the hybrid R-matrix analysis of the El data of ref. 4). Our paper is organized as follows. In sect. 2 we present our microscopic treatment of the E2 capture. In sect. 3 we briefly review the hybrid R-matrix model for the El transition. The results of our analysis of the data of refs. 4*7)are presented and discussed in sect. 4.

336

K. Langanke,

SE. Koonin / “C(a,

y)160

2. Potential model for the E2 cross section

In this section we calculate the absolute ‘*C((u, y)160 E2 transition from the I= 2 12C+a partial wave to the 160 ground state in a microscopically founded potential model. Three ingredients are necessary: the I= 2 continuum wave function for ‘*C+(u scattering, the E2 transition operator, and the 160 ground-state wave function. We consider each of these in turn. 2.1. THE I= 2 CONTINUUM

WAVE

FUNCTION

We start from the observation 9, that the 2’ state at E,, = 6.92 MeV, as well as the I = 2 scattering states at low energies, can be adequately described by a many-body wave function of the form PI=&) = d{cpolcpIc=Ogl=z(r)],

(2.1)

where cpa and cp&=’are fixed internal wave functions for the a-particle and the 12C ground state. These can be approximated as the SU(3) (0,O) and (0,4) configurations. respectively, with a common oscillator parameter 9). The 16-body antisymmetrizer is d. As in the resonating group method lo), for a given many-body hamiltonian the wave function of relative motion gl&) can be determined by solving the Schroedinger equation in that part of Hilbert space spanned by many-body wave functions of the form (2.1). In our study, however, we have chosen to calculate glz2(r) by solving a one-body Schroedinger equation of relative motion similar to that of the orthogonality condition model ‘l*l*): l(l+ 1)A2 v2

gl=z(r) = 0 ,

(2.2)

where V((r) is a local, energy-independent potential. The non-local Pauli projector A ensures that the relative wave function is orthogonal to the set of Pauli-forbidden states. Apart from generating the correct number of nodes in g,(r) required by the orthogonality condition, its main influence is to damp the relative wave function in the internal region. The importance of A for nucleus-nucleus interactions is discussed in ref. 13). If, as in our calculation, the internal wave functions cpa, cpc are described by pure harmonic oscillator shell-model wave functions with a common oscillator parameter 6, the Pauli forbidden states are given exactly by the radial harmonic oscillator wave functions u!,(r) with oscillator constant p2 = b2/p, where F = 3 is the reduced mass “). Hence the Pauli operator A in eq. (2.2) for our calculation is known exactly. We have adopted b = 1.7 fm, the weighted mean of those values which minimize the binding energies of the a-particle and the i*C ground state when calculated with the effective Brink-Boeker force Bl [ref. i4)]. In principal, the potential VI(~) in (2.2) can be determined by a two-step folding relation from the non-local, energy-dependent RGM potential, so that eq. (2.2) is

K. Langanke, SE. Koonin / “C(a, y)160

337

a reformulation of the RGM equation. The advantage of eq. (2.2) over the RGM equation is that the solutions g*(r) are orthogonal to each other and therefore can be interpreted as wave functions of relative motion. Since it is unlikely that all microscopic details of the ‘?+a system are important for the capture process, we have chosen to adjust Vl to the experimental data rather than to extract it from a microscopic calculation. In particular, we assume that V((r) is the sum of nuclear ( VN) and Coulomb (V,) terms. For the former, we take a gaussian form, 2

V&J=

V0exp

I I -k

(2.3)

,

while for the latter we take the Coulomb potential of a homogeneously sphere of radius rc = 3.55 fm: Vc(r)= 12”

I

l/r

forrar,

(3 -r2/rz)/2rc

forr
charged

(2.4)

The values V. = -126.92 MeV and ro = 2.6 fm give an I= 2 “C+a bound state at &.,. = -0.2448 MeV, very nearly equal to the experimental value of E,.,. = -0.2449 MeV [ref. *‘)I; this potential also reproduces the phase shifts of ref. 8). It does not, however, reproduce the I = 2 resonance at E,.,. = 2.68 MeV, which has a small a-width and so is not well described by an ansatz like (2.1). The quadrupole moment of the 2+ bound state agrees within 10% with the experimental value [ref. “)I extracted from the transition strength between the I = 2’ bound state and the I = 0’ level at 6.06 MeV if one assumes that both states are members of a rotational band. Potentials with similar parameter sets also give excellent fits to the “C+ (Yphase shifts in other partial waves (see sect. 3). 2.2. THE

E2 TRANSITION

OPERATOR

E2 transition from an I = 2 scattering state 401-zto the 160 ground state (P~.~., is governed by a matrix element of the electric quadrupole operator: The

(c~~=alQzl~g.s.>

(2.5)

,

with 16 Qz=$e

1

(1-7,,)rfY2(?i)

i=l

.

(2.6)

Here, 7i is the isospin operator of the ith nucleon, and ri its position. This matrix element can readily be evaluated if the 160 ground-state wave function has a form similar to that of (2.1): (p&r) = ~b,cp~~"hkO(rN,

(2.7)

where cp= and q&=:=”are the same internal wave functions as defined above. Such an ansatz for (P~.~.is reasonable, since the class of many-body wave functions which can be factored in this way includes the harmonic oscillator shell-model ground state, which is known to be a good description of the 160 ground state.

338

K. Langanke, SE. Koonin / “C(cx, y)160

Since the initial and final states in eq. (2.5) both have total isospin T = 0, only the isoscalar part of Q2, denoted by 6,) contributes. This operator can be written as A II ?. 1 Qz= Q2,u + Q2.c + Q2.r (2.8) 6,,, act only on the internal coordinates of the a-particle where &, nucleus. The isoscalar relative quadrupole operator is

&

= &er2Y2(;)

.

and the 12C

(2.9)

Using eqs. (2.1) and (2.7), eq. (2.5) can be rewritten as (cp~=zlQ2I~n.s.>= (~ol~~=:=Ogr=21~~.2~l~rr~~=:=Oh~=o) = 1 (g,~2~Uf;2)(cppcp’,=ou~=2~~‘2~~~,cp~=oU~~o)(Ufi~o~h,~0)) n,n’

(2.10)

where in the last step we have expanded the relative wave functions g, h into radial harmonic oscillator wave functions and have used the fact that [Q2, d] = 0. In order to evaluate the 16-body matrix element in (2.10) we assume 2n +2~2n’ without loss of generality. By inserting the identity 16) (2.11)

where the sum is over all 16-body wave functions which can be factored into a relative wave function and internal parts for the a-particle and the 12C nucleus, the 16-body matrix element in eq. (2.10) can be transformed into r6)

+(u~=2162.,~u~)(cp~(p~=ou~~~l~~~~=o~~~o)}.

(2.12)

The sum in eq. (2.12) can be reduced drastically by using 2n + 2 2 2n’ and the fact that the antisymmetrizer conserves harmonic-oscillator energy. Consequently, the first term has to vanish, since the only configuration for the a-particle which would conserve the energy is the a-particle ground state which, of course, has no electromagnetic quadrupole moment. In the second term, contributions come from only those 12C configurations which have the same oscillator shell-model energy as the 12C ground state, but have internal spin I = 2. The only possible configuration is the SU(3) description of the first excited state in 12C, denoted in the following by cp&=*.Furthermore, n’ is restricted to 2n’ = 2n +2. In the last term energy and

K. Langanke, S.E. Koonin / 12C(a, y)160

339

angular momentum conservation require L = 0 and N = n’. The matrix elements with 2n + 2 c 2n’ can be evaluated similarly, if the order of the antisymmetrizer and the quadrupole operator is interchanged in eq. (2.12). Hence, eq. (2.5) can be finally rewritten as

with (cpolcp~=ouffo~d~~,cp~=ou~~o) /.&an<= I=0 1=2 I ((P&c ufI 14wP~=“d=2)

for2n’G2n+2 for 2n’>2n +2.

(2.13b)

The 16-body overlap matrix elements required in eq. (2.13a), ((p~(p~~“~2uf;o’2~~~cp,cp~~ou~~o’2), are given in ref. l’). The matrix element can be related to the experimentally known lifetime (65*9 fs) kJ~=“l~2.cl~~=2) of the 2’ state at E,, = 4.43 MeV in ‘*C. Finally, the matrix element involving radial wave functions can easily be evaluated numerically. Since the wave function for the 160 ground state is short-ranged and 02,r connects only adjacent harmonic oscillator shells, the sum over II, n’ is restricted to only a few values. 2.3. THE I60 GROUND

STATE

Since the E2 transition matrix element will depend strongly on the 160 groundstate relative wave function, we have constrained this wave function by the following three requirements: (i) hrco must have the appropriate number of radial nodes (n = 2); (ii) the binding energy relative to the ‘*C+(u threshold [7.162, ref. “)I must be reproduced; (iii) the lifetime of the 2’ state of E,, = 6.92 MeV for decay to the ground state [6.6*0.4 fs, ref. “)I must be reproduced when this transition is calculated from eq. (2.5) using the bound state of the I= 2 potential described in subsect. 2.1 above. These three constraints are satisfied by the only bound state of a Vlzo potential whose Coulomb part is given by eq. (2.4) and whose nuclear part is of the form (2.3) with V. = -121.56 MeV and r. = 2.08 gm; this bound state is calculated from eq. (2.2). The appropriate number of nodes is automatically ensured by the Pauli projector A. The bound-state energy is calculated to be E,.,, = -7.160 MeV. In calculating the lifetime of the 2+ state the interference between the I = 0 and I = 2 terms in eq. (2.13) forces us to adopt a sign for the matrix element (rp~=“(&c]~~=2). We take it to be negative, consistent with the theoretical intrinsic quadrupole moment of the ‘*C ground state and the experimental quadrupole moment of the 2+ excited state at E,, =4.43 MeV in ‘*C [ref. ‘*)I. With this choice, our 160

340

K. Langanke, S.E. Koonin / ‘*C(a, y)160

ground-state wave function gives a lifetime for the 160 2+ state of 6.51 fs. If we had adopted a positive sign for (~p~~~&~~~~~), the calculated lifetime would be 6.45 fs. There is little sensitivity to this sign since the second term in eq. (2.13a) is much larger than the first. Our 160 ground-state wave function has a probability of 94.4% to be in the harmonic oscillator shell-model ground state, and probabilities of 5.1% and 0.5% to have the relative cluster wave function excited for 2 or 1 quanta, respectively. All other components have probabilities less the 0.1% . 2.4. THE E2 CROSS SECTION

By constraining the I60 ground state and the I= 2 12C+(Y scattering as discussed above, we expect to have a reasonable description of the E2 transition matrix elements at low 12C+ (Yenergies, especially those energies of astrophysical importance, which are only some 500 keV above the 6.92 MeV 2’ bound state. The E2 capture cross section is given by (2.14) where E,., and u are the c.m. energy and relative velocity of the (Y+ 12C channel and E, = 7.162 MeV + E,.,. is the energy of the emitted photon. At energies below the Coulomb barrier, where the cross section drops rapidly with decreasing energy, it is more convenient to discuss the astrophysical S-factor, rather than the cross section: SK,.)

= &.,.c@,.,.)

exp {27rrl) ,

(2.15)

where n = 12e2/hv is the Sommerfeld parameter. The E2 S-factor calculated with our potential model is shown by the solid curve in fig. 1. The strong increase in SE2 for E,.,. + 0 is caused by the 2’ bound state in I60 at E, = 6.92 MeV (E,.,. = -0.245 MeV). Nevertheless, our calculated E2 S-factor at energies below E,.,. = 3 MeV is overall very small; we find SE2 (300 keV) = 0.0054 MeV * b. This value is therefore not in agreement with that given in ref. ‘), SE2 (300 keV) = 0.2 MeV - b. As can be seen in fig. 1, the S-factor has a zero at E,.,. = 1.15 MeV, reflecting a sign change in the matrix element (2.5) at this energy. This zero is a consequence of the antisymmetrized 16-body wave functions in eq. (2.5), as can be demonstrated by calculating the E2 S-factor when antisymmetrization is neglected; that is, by replacing the matrix elements (cp,rp&=:=‘u !,==” I&lq,cp c=:=“u !,T”), ((Pa(P~=oOU~~0.2~~~~P(P~=oOU~~0.2 ) in (2.13) by 0 or 1, respectively. This “non-microscopic” S-factor, shown by the dashed curve in fig. 1, does not have any zeros for E2 S-factor is also clearly E _,,. =s4 MeV. Note that even this “non-microscopic” smaller than that given in ref. ‘), by about a factor 10 at 300 keV.

K. Langanke, S.E. Koonin / ‘?(a,

Ecm

y)160

341

[MeVl

Fig. 1. Microscopic (solid line) and “non-microscopic”

(dashed line) S-factor for E2 capture.

3. Synopsis of the hybrid model for El capture Since El transitions between T = 0 states are forbidden, the %(ar, y) El cross section at low energies is believed to result from T = 1 admixtures in the l- states of 160 at E,, = 7.12 MeV and 9.63 MeV and in the continuum between them. Since the, nature of these T = 1 admixtures is unknown, a microscopic treatment of the El capture similar to that of the E2 given above is very difficult. However, the experimental El cross section is strongly correlated with the I = 1 elastic (Y-scattering data; both are dominated by the broad 9.63 MeV resonance. This fact is the basis of the hybrid R-matrix analysis in which the El capture is the coherent sum of contributions from the 9.63 MeV (and higher-lying) states and from the 7.12 MeV state. The former are computed from elastic scattering wave functions determined from a potential which fits the experimental l- 12C+ar phase shifts and from an energy-independent effective dipole strength, d, which represents the amount of T = 1 admixture. The contribution of the 7.12 MeV state is parametrized by a pole in the R-matrix. The location of this pole is known, while its residue is related to the a-width of the 7.12 MeV state (yo1,7.12),which is known to be small. When this model is fitted to the experimental El data with d and 7/=,7.l2as adjustable parameters, excellent fits to the experimental 12C@, y) El cross sections of ref. 4,

342

K. Langanke, SE. Koonin / ‘*C(a, -y)160

are obtained 5), with constructive interference occurring between the 7.12 and 9.63 MeV contributions at energies below the latter resonance. Our analysis of the experimental El data of refs. 4*7)uses the “hybrid” R -matrix formulation of ref. 5), although differing from it slightly in that the scattering and bound states we use have a firmer microscopic foundation. For completeness, we briefly review the essentials of the hybrid R-matrix approach. The elastic CY+ “C phase shift S is related to the elastic scattering R-function, R,,, by 19) tan(S-c)=lf(PRiPS.

(3.1)

au

Here, P, S and p are the penetration function, the shift function, and the hard-sphere phase shift, respectively. Since the nucleus-nucleus potential is non-negligible beyond the channel radius, the quantities P, S and cp must be calculated using the inward continuation of the regular and irregular I = 1 Coulomb functions 20), rather than with the latter functions alone. This inward integration is, of course, done using eq. (2.2). The El capture cross section is related to the capture R-function R,, by “) (TEl

=

6~ TP k

2

R I 1 - (S +$R,,

(3.2)

I ’

where k is the wave number for relative ‘2C+cu motion. In the hybrid R-matrix approach both R-functions are split into two parts: R,, =Rf;+R:;‘2’,

R,,=R;;+R:;12’,

(3.3)

where the quantities with superscript “(7.12)” arise from the bound state at E,, =.7.12 MeV, while the terms with superscript “(0)” arise from a potential description of the broad I= 1 resonance at E,, = 9.63 MeV. The bound-state contributions to the R-functions are described by the usual pole associated with a single level 19): @;12’ = R:;12

=

y:,7.12/(E:7.‘2’

~c+7.12Jrv;1.12/(Ei?*)

-E)

,

-E)

,

(3.4)

where 7/u,7.12is the reduced (Y-width of the bound state. The formal El y-ray width for the decay of this state to the 160 ground state, rY,7.12, is related to the experimentally measured y-decay width, f t:.i2, by 20) (3.5) The pole energy EL7.12’is chosen so that the resonant part of the phase shift in (3.1) is equal to $r at the observed position of the bound state.

K. Langanke, SE. Koonin / ‘*C(a, y)160

343

EC-,, [ MeVl Fig. 2. Fit to the experimental ‘*C+a I = 1 phase shifts of ref. s) using a gaussian potential form factor with v,, = -123.65 MeV, rD= 2.6 fm. The energy scale of the data has been adjusted by 20 keV as described in ref. ‘).

describe the quantities with superscript “(0)” we have used the Coulomb (2.4) and a nuclear potential of the gaussian form (2.3); the parameters V. and r. were then adjusted so that eq. (2.2) reproduces the I = 1 elastic phase shifts of ref. ‘) for E,.,.G 3 MeV. Fig. 2 shows the fit obtained with V. = -123.65 MeV, ro= 2.6 fm. These parameters are very similar to those which describe the I= 2 partial wave (see sect. 2) or fit the I= 4 phase shifts (V. = -129.0 MeV, r. = 2.6 fm). These very similar parameter sets would not, by themselves, lead to a strong parity splitting of the (Y- 12C bands with the bandheads at E,, = 6.06 MeV (Oc) and E, = 9.63 MeV (17. Such a splitting, however, is due to the parity dependence of the Pauli repulsion embodied in the projector A in eq. (2.2) [ref. t3)], and is therefore an essential feature of the nucleus-nucleus interaction. We calculate R?i from the potential phase shifts by inverting eq. (3.1). We then use the formulas of ref. ‘I) to express the El cross section in terms of the lpotential scattering state and the 160 ground-state wave function of sect. 2 and determine RF! by inverting an equation similar to (3.2). Note that the El cross section in this model does not vanish, since the ratios of the masses of the (Y-particle and the 12C nucleus are non-integral. To fit the magnitude of the experimental El cross sections it is necessary to scale the quantity R L: by an effective dipole strength d [ref. ‘)I. Finally we have performed standard x2 fits to the experimental data of To

potential

344

K. Langanke,

S.E. Koonin / ‘*C(a,

y)160

refs. 4,7) varying the parameters ya,7.12 and d. In detail, for a given yo1,7.12, Er.‘2’ was calculated and then f,,7.,2 from eq. (3.5). To calculate S and R,, at the negative energies required, their values at small positive energies were extrapolated using a 10th degree polynomial; the estimated accuracy of this procedure is better than 5 digits for those negative energies of interest (E,.,. 3 -0.3 MeV). We have used an extremely accurate program to calculate the Coulomb functions which reproduces the exact values 22) to more than 10 digits in the parameter ranges of interest.

4. Data analysis and discussion Fig. 3 shows the best fit to experimental El data of ref. 4), obtained with the parameters d = 47.1 and 3/a,7.12 = -0.0294 MeV”2. The value for x2 obtained for this fit with 22 degrees of freedom (24 experimental data points and 2 free parameters) was x2 = 24.9. The differences of the parameters d and ya,7.12 from the values found in ref. *) (d = 8 and ym,7.12= -0.36 MeV1’2) are due to the different wave functions used to describe the I60 ground state.

IO’

I

I

-

Caltech

IO’

-

1,

2 13

I

3 Ec,.,,[

Fig. 3. Hybrid

R-matrix

I

data

Me:1

fit to the experimental El cross section4) -0.0294 Me?‘*.

using d =47.1,

ya.7.12=

K. Langanke, S.E. Koonin / “C(a, y)160

IO2-

IO’

345

I

Munster

data

-

Fig. 4. Fit to the experimental %(a, y)160 cross section of ref. ‘) using the parameters d = 55.8 and ya,7,12 = -0.0468 MeV”’ in the hybrid R-matrix approach. The E2 contributions were calculated as discussed in sect. 2 (see fig. 1).

The same procedure was also applied to the El data of ref. ‘), obtained by subtracting the E2 cross section calculated in sect. 2 from the total experimental capture cross section. The best fit was obtained with d = 55.8 and yu,7.12= -0.0468 MeV”*. The x2 value for 36 degrees of freedom (all data below E,.,. = 3.1 MeV have been considered) was x2 = 97.5. The fit to the total ‘*C((Y,y)r60 cross section of ref. ‘), including the E2 cross section as calculated in sect. 2, is shown in fig. 4. In all of these calculations, a channel radius of 5.3 fm was adopted. As shown in ref. 23), variation of the channel radius or the inclusion of an energy dependence in the effective dipole strength does not significantly change the results obtained. An essential feature of our model for E2 capture is the constraint that the lifetime of the 2+ state at E,, = 6.92 MeV be correctly reproduced. We therefore expect a reasonable description of the r*C(a, y)160 E2 cross sections at the low energies (EC.,,,.= 300 keV) of astrophysical interest. The validity of our E2 cross sections can be checked at higher energies by comparing our calculated ~u~/(~n~ ratios with those extracted from the angular distributions measured in ref. 4). This comparison is indeed a test of our absolute u E2 cross sections, since our E2 and El calculations

346

K. Langanke, SE. Koonin / “C(a, y)160

0

3

I

Ec,[MeV? Fig. 5. Comparison of the ratios uE1/uE2 calculated in the present study to the experimental data of ref. 4).

are independent of each other and the latter is adjusted to fit the pure El data of the same experiment, from which the experimental ratios ~~~~~~~ have been determined. Our calculated ratios (T&(T~~, together with the experimental values, are plotted in fig. 5. Even for I?,.,. = 2-3 MeV the magnitude of the calculated E2 cross sections agree fairly well with the experimental data. With increasing energy, however, the predicted E2 cross section is lower than the measured data. At energies E,,. =0.8-l .6 MeV the ratio (T&(T~~ is smaller than 0.01, associated with the zero in the E2 cross section (see fig. 1). Otherwise, g.EZ/~E1 is rather monotonic in this energy range, as neither the I = 1 nor the I= 2 partial waves have resonances here. The turn-over of the ratio at E,.,. = 0.3 MeV is associated with the influence of the l- bound state at E,, = 7.12 MeV. Finally we have calculated the total S-factor (2.15) for the %(a, y)160 reaction at energies EC.,. c 3.5 MeV, with the El contribution fitted to the data of refs. 4*7) (see fig. 6). At the “most effective” astrophysical energy we find S(300 keV) = 0.15 MeV * b for the Caltech data 4). This result is in agreement with the original 3-level R-matrix fit of Dyer and Barnes 4, S(300 keV) = 0.14 MeV * b, although a completely different method of analysis has been used. This value is also in agreement with the previous studies 5*6)based on the data of ref. 4). The slight differences from the calculated S-factor given in ref. ‘), S (300 keV) = 0.1 are due to the different descriptions of the 160 ground-state and continuum wave function. Our fit to the Munster data ‘) gives S (300 keV) = 0.34 MeV * b, consistent with the value S (300 keV) = 0.43$:: MeV * b given in ref. 7>.However, our calculation

K. Langanke, S.E. Koonin / ‘*C(a, y)160 I

I

‘(

------

Munster

347 I

data

I

I

I

I

2

3

Ecm

[ MN1

Fig. 6. Astrophysical S-factors for the ‘*C(a, y) reaction as extracted in the present study from the experimental data of ref. 4, (solid line) and ref. ‘) (dashed line). The S-factor for E2 capture was calculated as described in sect. 2. The S-factor for El capture was determined from the experimental data within the hybrid R-matrix model.

predicts that the astrophysical S-factor is dominated by El capture, while the authors of ref. ‘) predict comparable El and E2 contributions. The higher values of the total S-factor extracted from the Munster data arise partly from an overall enhancement relative to the Caltech data 4, of approximately 40%. As the 150% increase in S-factor of the Munster data over the Caltech data markedly changes the predicted C/O ratio at the end of helium burning and hence the subsequent nucleosynthesis, it is clear that the r2C((u, y)160 cross section should be remeasured to resolve the discrepancies between the two sets of data. It would also be useful to extend the angular distribution measurements to energies E,.,. i 2 MeV, from which one could deduce the importance of the E2 contribution at stellar energies. The authors are grateful to C.A. Barnes; William A. Fowler and T.A. Tombrello for many stimulating discussions. References 1) C.A. Barnes, in Advances in nuclear physics, ed. M. Baranger and E. Vogt, vol. 4 (Plenum, New York, 1971) p. 133

2) C.A. Barnes, in Essays in nuclear astrophysics, ed. CA. Barnes, D.D. Clayton and D.N. Schramm (Cambridge University Press, New York, 1982) p. 193

348 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23)

K. Lnnganke, S.E. Koonin / 12C(a, y)160

T.A. Tombreiio, quoted in ref. 4). P. Dyer and CA. Barnes, Nucl. Phys. A233 (1974) 495 S.E. Koonin, T.A. Tombreilo and G. Fox, Nucl. Phys. A220 (1974) 221 J. Humblet, P. Dyer and B.A. Zimmerman, Nucl. Phys. A271 (1976) 210 K.U. Kettner, H.W. Becker, L. Buchmann, J. Gorres, H. Krlwinkel, C. Roifs, P. Schmalbrock, H.P. Trautvetter and A. Vlieks, Z. Phys. A308 (1982) 73 C. Jones, G.C. Phillips, R.W. Harris and E.H. Beckner, Nucl. Phys. 37 (1962) 1 Y. Suzuki, Prog. Theor. Phys. 55 (1976) 1751 K. Wildermuth and Y.C. Tang, A unified theory of the nucleus (Vieweg, Braunschweig, 1977) S. Saito, Prog. Theor. Phys. 41 (1969) 705; S. Saito, S. Okai, R. Tamagaki and M. Yasuno, Prog. Theor. Phys. 50 (1973) 1561 H. Friedrich, Phys. Reports 74 (1981) 209 H. Friedrich and K. Langanke, Phys. Rev. C (1983), to be published D.M. Brink and E. Boeker, Nucl. Phys. A91 (1967) 1 F. Ajzenberg-Selove, Nucl. Phys. A375 (1982) 1 Y. Suzuki, Prog. Theor. Phys. 56 (1976) 111; S. Okabe, Nucl. Phys. A 404 (1983) 179 H. Horiuchi, Prog. Theor. Phys. 51(1974) 7%5 W.J. Vermeer, M.T. Esat, J.A. Kuehner, R.H. Spear, A.M. Baxter and S. Hinds, Phys. Lett. 122B (1983) 23 A.M. Lane and R.G. Thomas, Rev. Mod. Phys. 30 (1958) 257 C.H. Johnson, Phys. Rev. C7 (1973) 561 R.F. Christy and I. Duck, Nucl. Phys. 24 (1961) 89 Handbook of mathematical functions, ed. M. Abramowitz and I. Segun (National Bureau of Standards, Appl. Math. Series, 1964) T.A. Tombrello, S.E. Koonin and B.A. Flanders, in Essays in nuclear astrophysics, ed. C.A. Barnes, D.D. Clayton and D.N. Schramm (Cambridge University Press, New York, 1982) p. 233