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Microscopic analysis of the 12C(α, γ)16O reaction
Nuclear Physics A430 (1984) 426-444 @ North-Holland Publishing Company
MICROSCOPIC
ANALYSIS
P. DESCOUVEMONT*, Physique 7hhique
OF THE ‘*C(a, y)160
REACI’ION
D. BAYE and P.-H. HEENEN**
er Mafht!mmafique,CP 2.79, UnioersifPLibre de Bruxelles, Brussels, Belgium
Received 28 February 1984 Abstract: Electric transition probabilities in the I60 spectrum, and the %(a, y,,,)‘6O capture cross sections are calculated with antisymmetric wave functions by the generator coordinate method. The influence of bound states on radiative capture is shown to be automatically included in the model. The reduced a-widths of the I60 bound states are discussed, and compared with previous theoretical and experimental estimates. The microscopic E2 capture cross sections to the 0; and 2; states yield an astrophysical S-factor of 0.09 MeV. b at 0.3 MeV. An attempt to treat the El multipolarity by relaxing the long-wavelength approximation leads to a large underestimation of the y-widths. Adopting the experimental y-width and the theoretical reduced a-width of the I ; state provides SE, = 0.30 MeV . b at 0.3 MeV.
1. Introduction The cross sections for the radiative capture of alphas by ‘*C nuclei are important for determining the relative abundances of ‘*C and I60 in the stellar helium-burning process lm3).Th e calculation of the reaction rate ‘) requires a knowledge of these cross sections at very low energies (typically around 0.3 MeV for temperatures of about 2 X 10’ K). Direct measurements are precluded by the extreme smallness of the cross sections at these energies. Accurate experiments have until now been limited to energies larger than 1.4 MeV [refs. “)I. Unfortunately, a marked disagreement still exists between the absolute values of the low-energy data 5-6). The experimental cross sections serve as basic data for different theoretical fits ‘-‘O) which eventually provide extrapolations at stellar energies. However, this procedure meets several difficulties. (i) The extrapolation is made inaccurate by the occurrence of a broad bump in the cross sections in the energy range which can be studied experimentally. This bump is due to the 1; resonance at 9.63 MeV excitation energy in the I60 spectrum. (ii) The relative importance of the El and E2 contributions, which have different energy behaviours, is not known. The data of ref. ‘) establish that the E2 part is small in the energy range dominated by the 9.63 MeV resonance but this result is of limited interest for lower energies. (iii) Still more important, the existence of two bound states just below the LY+ ‘*C threshold makes the estimation of cross sections uncertain. The influence on the El contribution of the 1; state
* Aspirant FNRS. ** Chercheur qualifik
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located 45 keV below the threshold has been discussed by many authors ‘-‘I). A similar effect of the 2: bound state located 245 keV below the threshold on the E2 contribution ‘) has been studied recently 67”). The influence of these bound states depends crucially on some of their properties (like their reduced a-widths in the R-matrix formalism) which cannot be measured directly. An attempt to solve these problems was made recently by Langanke and Koonin “) who calculated the E2 contribution in a semi-microscopic approach. They found that the E2 component is almost negligible, in strong contradiction with the analysis of ref. “). The different quantities which cannot presently be reached by experiments and are inaccurately given by extrapolations can be computed in a microscopic model I’). The generator coordinate method (GCM) has the following advantages [see references in ref. ‘“)I: (i) antisymmetrization is taken into account exactly; (ii) bound and scattering states are treated in a unified manner; (iii) the conservation of the good quantum numbers is treated exactly; (iv) the relative motion of the colliding nuclei is described quantum-mechanically. These properties are made possible in a relatively simple way by the use of Slater determinants defined in the two-centre harmonic-oscillator model. The GCM has been applied to the study of the 160( CX,y)*‘Ne radiative-capture reaction 12*13)for which, unfortunately, no experimental data exist. However, the unified treatment of bound and scattering states provides a test of the accuracy of the model, namely the comparison with the experimental transition probabilities involving bound states and narrow resonances. The main conclusions of refs. 12,13)are the following: (i) antisymmetrization effects are not negligible, even at low energies; (ii) the 160( cy,y)“Ne capture cross section below 1 MeV is about six times larger than the value utilized in astrophysical calculations ‘); (iii) the El contribution is not solely due to a small T = 1 admixture to the wave functions as usually assumed [this theoretical result seems to be confirmed by experimental data about branching ratios for El transitions in the 20Ne spectrum “)I. The aim of the present paper is to apply the same formalism to the 12C((u,y)160 radiative-capture reaction. The situation is, however, less favourable than for the 160( CX,y)“Ne reaction. If the 20Ne spectrum is well described by an (Y+ ‘60g.s.cluster model, the description of I60 by an (Y+ ‘2Cg.s.structure, although qualitatively valid 14*“), is less satisfactory ‘5*‘6).In particular, the I60 ground-state energy is far too low with respect to the cy+ “C threshold. As in ref. 13), the exact energies will have to be reproduced by adjusting a force parameter for each partial wave. The microscopic R-matrix method (MRM) which we use to solve the GCM equations “) has the important advantage of providing naturally reduced a-widths, even for bound states. Besides direct information on E2 capture and partially on El capture, we obtain reliable estimates for the bound-state parameters which influence the cross section at low energies. The present microscopic model will thus question different assumptions and parameters which are encountered in phenomenological 5-‘o) or semi-microscopic “) models.
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The theory is summarized in sect, 2 with emphasis on the influence of bound states on the capture process. In sect. 3, rms radii, quadrupole moments, reduced transition probabilities and reduced a-widths are compared with the available experimental data and theoretical estimates. ‘The different contributions to the capture cross sections are studied in sect. 4. Concluding remarks are presented in sect. 5. 2. Influence of bound states on radiative capture 2.1. CAPTURE
CROSS SECTION
The reduced electric transition probability between two bound states of a nucleus is given by
B(Eh,J:‘-,J;r)=~(~‘~“~~~~~~~(L’~Q~)~2(2Jf+1)/(2Ji+1),
(1)
where A is the order of the (electric) multipole, and J and T denote respectively the total angular momentum and the parity of the initial and final states. In this paper, this probability is calculated utilizing antisymmetric wave functions. The asymptotic behaviour of these wave functions and the c.m. dependence of the JX,” operator are correctly taken into account, as explained in ref. 12). When the initial state J:i is a scattering state at c.m. energy E, the electric radiative-capture cross section towards the final state .!;I is obtained from [ref. “)I a(E, AJ:‘+J;‘)
=[&(A
+l)/A(2A
+ l)!!2h]ti7A+‘B,,,(EA, J?-,J;r),
(2)
where Bcap, is given by (1) with $4” replaced by the J? partial wave of a unit-flux scattering wave function (here and in the following, the colliding nuclei are assumed to have spin zero). In (2), k, is the wave number of the emitted photon. 2.2. REDUCED
WIDTHS
In this paper, the bound and scattering state wave functions are defined in a GCM basis ‘2*‘7).However, the GCM basis states do not have a correct asymptotic behaviour, neither for bound nor for scattering states I’). This problem is solved here by the microscopic R-matrix (MRM) method I’), which furthermore provides a precise definition of the reduced widths. Moreover, as shown hereafter, a contribution of the bound states is automatically included in the scattering wave function. In the MRM framework, the configuration space is divided into two regions. The antisymmetrization is taken into account exactly in the internal region (of radius a), and is neglected outside. Let us consider N basis functions 6? = bI”“( R.) depending on the set of generator coordinate values R,. The 4i”” are defined as Slater determinants, constructed in the two-centre harmonic oscillator model with A, orbitals centred around -A2R,/A and A, around A,R,IA, and projected on _l
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and 7r. In order to simplify c.m. problems, a common harmonic-oscillator 6 is chosen for both clusters. The GCM wave functions are g;F
= ; fW"", n=,
parameter
(3)
where the f: are unknown coefficients. The wave function written as *;F$ = &n&~,&g’“(P) YJ”W,)
(3) can also be (4)
where p is the relative coordinate between the c.m. of both clusters. In (4), .A?is the antisymmetrization projector, &,,. is a c..m. function, and 41 and & are the antisymmetrized internal wave functions of the nuclei. The relative function g” is given by g’“(p) = [256&( with c = A!/A,!A,!(
1 +S,J
1 + &)/(2J+
l)]“‘c
”
f;TJ(p, R,)
(5)
and
TAP, R) = (p/rb2)3’4exp
(-P(P’
+R2)/2b%(&Vb2),
(6)
p being the dimensionless
reduced mass, and i, a spherical Hankel function. In the external region, where antisymmetrization between the clusters is negligible, the operator d can be replaced by c-l; the relative coordinate p has a precise meaning, and $fny’n”;‘” can be factorized in a product of internal and relative wave functions. However, as can be seen from (5), $$yT(p) does not have a correct asymptotic behaviour. Hence, for p larger than a, one defines for scattering states $~$=(p) = i’+’ [n(l +t3,,)(2_l+
1)/W]“’
x~,.,.~,dzyrM(~np)[l~(k~)-
~“O,Udl/k~,
(7)
where I, and 0, are ingoing and outgoing Coulomb wave functions, lJJ” is the collision matrix, and v and k are the relative velocity and wave number, respectively. The collision matrix U”’ and the coefficients firare calculated from the BlochSchrbdinger equation [see ref. “) for the definition of the Bloch operator Y] (H+~-EE)~~~~=~~~~
(8)
and from the continuity of the wave function at the radius a: +ZW
= rVE(a).
(9)
Before proceeding, let us emphasize an important point: in these equations, the MRM radius a must be chosen large enough so that the wave functions have reached their asymptotic behaviour. Then, the physical results are insensitive to the value of a. However, phenomenological R-matrix analyses 7V8S’o*‘1) make use of values of the channel radius which are too small to be taken as the MRM radius. In order to compare our results with these analyses we shall introduce below a second radius, the channel radius ii, smaller than u.
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In order to exhibit the influence of the bound states on the scattering wave function, let us define an orthonormalized set of new basis states Jj”“, depending on the MRM radius u, and related to 4’n”” by the linear transformation &“- = c #;(a)C#J’,M”.
(10)
n
We require that the new basis diagonalizes the Bloch-Schrodinger internal region. The matrix D’” is then given by
equation in the
c D~~(“(a)(~~~IH+Y-~:“(a)J~J,“)p=O, n
(11)
where the subscript P means that the overlap and hamiltonian kernels are calculated over the internal region only. A negative eigenvalue of (11) gives an approximation of a bound-state energy in the J” partial wave. However, it must be noticed that the negative energy ,??fT(a) is not rigorously equal to the exact bound-state energy Ef” given by eq. (16) of ref. I’). Besides, its value depends on the radius a. Numerical examples of the differences between these energies will be given in the next section. In this new basis, the wave functions (3) become * ‘Mm
=
c
f’y~Jy,i’&M~
=
n,i
c
@(+p.
(12)
I
The unknown coefficients df” are calculated by projecting (8) on Jf”“: d:“(a)=(~j”“l~l~~~~)/(~‘:“(a)-E).
(13)
In the following, it will be convenient to extend the definition of &i”” over the whole space. In order that eq. (12) be valid in the external region, one defines for values of the relative coordinate p larger than a (14) where @i:“(p) is given by (7), and ~$:~,(a) is deduced from ( 10) and from asymptotic form of f$‘,““. The reduced width for the fission into two clusters can be easily obtained calculating the R-matrix in the basis (10). Indeed, applying the formalism of ref. one deduces the R-matrix R’” =C y;,i(a)/[E I
-,+(a)].
the .-by I’),
(15)
The
reduced fission width for each eigenvalue is given by the residue of R’” at (we omit hereafter the index i of the reduced width). In order to compare theoretical values with the experimental data, we must be able to calculate reduced fission width at a radius d different from a. By extension of (15), one
1
@’ our the has
2
y’.(a)
=[256~‘(1+&,)/(2J+
1)][fi2ti/2~m~]
c
n
#;(rr(a)rJ(cS,
R,)
,
(16)
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where the function I’, is given by (6). In this definition of the reduced width, the subscript a is relative to the definition of the MRM wave function (notice that matrix D depends on a, and not 6), while the argument d denotes the channel radius at which the reduced width is calculated. The definition (16) assumes that the antisymmetrization is still negligible at the radius E In fact, in phenomenological analyses, the &values do not completely fulfil this condition. However, if the antisymmetrization effects are small, (16) can be used for comparing our results with these analyses. Besides, for a fixed &value smaller than a, eq. (16) gives stable results as a function of the MRM radius a. One must finally point out that this definition provides straightforwardly the reduced fission width for negative-energy states, as well as for narrow resonances I*). 2.3. BOUND-STATE
CONTRIBUTION
TO THE CAPTURE
CROSS
SECTION
The determination of the GCM-MRM wave functions, as well as the detailed calculation of the matrix elements in (1) and (2) have been reported in ref. I*). In particular, the spurious c.m. motion has been shown to be automatically eliminated in the long-wavelength approximation. If one separates the bound-state contribution in the scattering wave function as 9 JMW= d:“&“” the partial radiative-capture becomes
+ I,!/;%;,,,
(17)
cross section (2), for given transition and multipolarity,
m(E,hJF+J,“t)=[8~(h
+1)(2Jr+l)/hh(2h
+1)!!‘(2Ji+l)]
~ld:i”i(~~f”rll~~II~:i”i) +(IClJfPrll~~II~~~~,,,d)12.
(18)
The bound-state contribution is then easily deduced by neglecting the second term of (18). One obtains the well-known form cR( E, AJ:’ + J;r) =$(I
+6,2)(2Ji+l)
T+“i(E)T,(E, AJF”i J;t) [~-Efi”i+~Jii”i-AJi”i(E)]*+[~~~“i(E)]*’
(19)
iJr=
(20)
where
The fission width rfm(E)
&“-@Q).
and the energy shift A.‘“(E) are given by
-AJ”(E)+$i@‘(E)=y~(~)lJ(k~)/(l-LJ(k~)R~w)
(21)
with L,(ka)
= kaO;(ka)/O,(ka)
The background contribution
= S,(ka) +iP,(ku)
.
Rim is obtained from (15) by removing the bound-state
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contribution. The y-width r, is calculated at the energy E, and is relative to the transition between the bound states J:i and J;(. In ( 19), the calculation of A” and r{= requires a knowledge of Ri”. However, it is interesting to simplify the Breit-Wigner formula (19). Firstly, let us linearize the energy shift: A’“(E)=A:“+(dA’*/dE),(E-E:“). Secondly, we introduce fission width (21) as
an effective reduced width $&(a)
r:=(E) Comparing
=
(22) in order to write the
(23)
2PJ(~~)~:,et&).
(21) and (23) leads to the definition &(a)
= Y%)/ll
-
l;,(~)R612s
(24)
which is slightly energy dependent. In this way, the background contribution R$’ is taken into account in simpler expressions. For the reasons stated above, we must calculate this reduced width for a channel radius E, smaller than the MRM radius a. We thus define &a(~)
= Y’,(a)/]1 -
LJ(~)&~:“~*,
(25)
where r:(a) is given by (16). Since some arbitrariness enters in this definition, we shall calculate &r(ti) by a second method. Assuming a weak channel-radius dependence of the fission width (23), we also define r2,,n(6)=T:“(E)/ZPJ(kcf).
(26)
The effective reduced widths will be considered
as reliable as far as formulas (25) and (26) give close values. A discrepancy would indicate that the channel radius d is too different from a. Finally, it must be noticed that, if one includes the absorption of photons 19) in this theory, the fission width appearing in the denominator of (19) is replaced by the total width r = Tr+ r,, leading to the rigorous Breit-Wigner formula. However, the correction requires much more complicated calculations whereas it is completely negligible. 3. Properties of the a + ‘*C system 3.1. ENERGIES,
mu RADII
AND
QUADRUPOLE
MOMENTS
The calculations are performed with interaction V, [ref. ‘“)I and a spin-orbit force “) (So = 50 MeV . fm’). The oscillator parameter has been chosen intermediate between the values leading to the correct rms radii of a and ‘*C nuclei (b = 1.6 fm). Ten discretization points R, are located between 0.8 and 8.0 fm with a mesh size
P. Descouvemont et al. / “C(cr, ?)I60
433
TABLE 1
Energies, J” 0+I
0+ 3: 2+I :;
Energies
rms radii and quadrupole m
0.6940 0.6400 0.6729 0.6368 0.6504 0.6365
E”
obtained
E’” x
-7.16 -1.11 -1.03 -0.25 -0.05 3.19
are in MeV, lengths
moments
0.0 6.05 6.13 6.9 1 7.12 10.35
in fm and quadrupole
with V, for I60
(?)“2 2.52 2.97 2.56 2.93 2.89 2.9 1 moments
Q 0 0 -4.0 - 14.6 -9.2 -18.0 in e. fm’.
of 0.8 fm. The energies are calculated from eq. (16) of ref. I*) for the bound states, and from the phase shift “) for the 4: resonance. The capture cross section being strongly dependent on the final-state energies, we have allowed the Majorana parameter m to vary with the different states. The m-values displayed in table 1 reproduce the experimental energies 22) within 5 keV. In table 1, we list the rms radius (the rms radius of the proton is taken as 0.8 fm) and the electric quadrupole moment of the bound states and of the 4: resonance. As it has been found in previous microscopic studies 15*‘6),it is difficult to reproduce with the same nuclear interaction the 160 g.s. energy and the energies of the other bound states. In particular, the m-values which give the correct O:, 2:, 4: energies lead to a ground state that is largely overbound with respect to the (Y+ “C threshold. For this reason, the Majorana parameter used for the 0: state is larger than the others. On the contrary, the m-values of the O:, 2; and 4; are nearly equal. The negative-energy levels are treated with their correct asymptotic behaviour, as explained in ref. I*). For the narrow 4: resonance (r, = 30 keV) the rms radius and the quadrupole moment are calculated in the bound-state approximation I*). The rms radius of the g.s. disagrees with the experimental value 22) 2.71 fm. The GCM rms radius is very close to the rms radius (2.48 fm) of the I60 one-centre harmonic oscillator wave function, which represents 96% of the GCM wave function. This one-centre rms radius would be increased by using a larger value of the oscillator parameter. Hence, we have performed a calculation with b = 1.75 1 fm and B, [ref. “)I plus spin-orbit as the nucleon-nucleon interaction. The value S,, = 89 MeV * fm5 reproduces the energy of the 0: state and gives (r*)“’ = 2.74 fm, in agreement with experiment. However, with the BI potential the 0: and 2: states are not bound, even without the repulsive spin-orbit force. This potential can thus not be used to compute radiative-capture cross sections. The intrinsic quadrupole moments I*) of the 2: and 4: levels are 5 1.l and 49.5 e * fm’, respectively, which indicates that these states belong to the same rotational band. The assignment of the 0: state as bandhead is confirmed by the similar values of the rms radius and by the very slight variations of the Majorana parameter required to reproduce the experimental energies of these three states.
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TABLE 2 Reduced E2 transition probabilities (in e* . fm4) in I60 GCM 2:+0; 2:-o; 4;+2; 1;+3;
3.2. ELECTRIC The
TRANSITIONS
2.1 55.1 71.9 17.1
Exp22) 7.8kO.3 67.0 f 7.4 ISI* 52* 18
PROBABILITIES
E2 reduced transition probabilities are compared in table 2 with experimental data 22). They are calculated using (l), with the exception of the B(E2,4:+ 2:) which is deduced from the 4’+2+ partial capture cross section (see subsect. 4.2) by assuming a Breit-Wigner shape around the 4: resonance energy. No effective charge has been introduced in the calculation. The theoretical B(E2) values are systematically smaller than the experimental data, the difference being the largest for the 2: + 0: transition. The best agreement with experiment is obtained for B(E2,2: + 0,‘). As expected for a rotational band, the intrinsic quadrupole moments deduced from the theoretical 4: + 2: and 2: + 0,’ transition probabilities are very close. This result is not confirmed by the experimental data, suggesting that the experimental 4: resonance is a mixing of a a + ‘2Cg.S.structure with more complicated configurations. Results similar to ours have been obtained by Suzuki 14).This author has performed a coupled-channel calculation of the a + “C system including the 2+ and 4+ rotational states of “C in the framework of the orthogonality condition model, a semi-microscopic method which takes the Pauli principle partly into account. The structure of the wave functions obtained by Suzuki is very similar to ours. In both ‘cases, the one-centre component (OpOh) of the g.s. wave functions is largely dominant (more than 90%) and much larger than the OpOhcomponent found in the g.s. wave function of shell-model calculations 24). In the calculation of Brown and Green, the 0;’ and 2: wave functions have larger 2p2h components (22% and 14%) respectively) which probably explain the larger B( E2) values obtained in the shell model 24,25).More puzzling is the good agreement with experiment obtained by Langanke and Koonin I’) since, as in the present model, their g.s. wave function has a 96% one-centre component. Their result is also four times larger than Suzuki’s who used the same model with a larger configuration space. This contradiction seems to be due to an inconsistent use in ref. “) of the orthogonality-condition-model wave functions in the calculation of matrix elements. The a and 12C nuclei having isospin zero, the El transitions are forbidden at the long-wavelength approximation. The non-vanishing dipole-transition probabilities are usually assigned to small T = 1 components in the wave functions. In the a +I60
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?)I60
435
case however, the correct order of magnitude of the El transition probabilities can be reproduced with T = 0 wave functions by expanding the isoscalar part of the El operator to the next order in the photon wave number 13). Moreover, a careful examination of the experimental branching ratios suggests that the weak isospinmixing effects should lead to a matrix element of the isovector part of the El operator which partially cancels out the matrix element of its isoscalar part. For the (Y+ “C system, we have treated the El multipolarity in the same way as in ref. 13). Unfortunately, this procedure fails in the present case. The theoretical B(E1, l;+O:) is equal to 5.4 x lo-*e2 * fm2 while the experimental value 22) is 1.5 low4 e2 * fm2. As for the B(E2,2:+0:), a part of this discrepancy must be due to the fact that the one-centre OpOh component of the g.s. wave function is too large. However, this effect seems insufficient to explain the order of magnitude of the discrepancy. We thus think that the relative importance of isospin-mixing effects in low-lying states is larger for 160 than for 20Ne.
3.3. REDUCED
a-WIDTHS
We present in table 3 the reduced a-widths -y’,(a) [eq. (16)] and &_e( a) [eq. (24)], and the energy shifts [eqs. (20) and (22)] deduced from the MRM calculation at a = 7.2 fm. The effective widths are a factor 3 to 4.5 smaller than &a). The smallness of the 0: and 3; energy shifts lJlr and A: can be explained by their small rms radii and o-widths. Indeed, since the probability densities of these two states are concentrated at low inter&stances, the eigenvalue E’:” of (11) is nearly insensitive to the MRM radius. On the other hand, the proportionality between A’” and y2 [see (21)] explains the small values of A:” and (dA’“/dE), for the 0: and 3; states. For the other states, 21Jn and A:” are more significant. However, it must be pointed out that both shifts are present in (19) with opposite signs, so that their effects nearly cancel out. In table 4, we display the dimensionless reduced a-widths (02 = y&,(d)/ &,, where y$,, = 3R2/2pmNd2 is the Wigner limit) at the channel radius d = 5.4 fm. This
TABLE
a-widths J” 0+
0: 3: 2:
Energies
r:(a) 6.3 6.1 1.1 5.1 6.3 7.3
x x x x x x
IO-’ lo-’ 1o-3 lo-* lo-* lo-*
and energy
1o-5 lo-* 1O-4 lo-* lo-* 10-2
shifts in I60
A'"I
r2...&) 1.4x 1.9 x 2.8 x 1.5 x 1.8 x 2.0 x
3
4.0 x 1o-4 1.7 x 10-l 3.3 x 1oP 1.3 x10-l 1.4x10-’ 1.5x10-’
are in MeV. All the values are calculated
(dA’*/dE) -3.8 -3.2 -4.4 -2.3 -3.2 -2.5
x x x x x x
lO-5 lo-* lo-‘+ 1O-2 lo-’ lo-*
at CI= d = 7.2 fm (see text).
2.0 x lo+ 0.16 0.01 0.11 0.14 0.12
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4
Dimensionless effective reduced a-widths 0: of low-lying I60 levels I”
GCM ‘)
GCM b,
0:
3.9 x lo-4 [3.3 x IO-J] 0.11 3.3 x lo-3 [1.2x10-2] 0.10 0.09
(4.8* 1.0) x IO-’
0;
3; 2: 1; 4;
0.10*0.05 (2*1)x10-3 0.10*0.02 0.09 * 0.02 0.32
Ref. *‘)
Ref. 28)
Ref. 29)
Ref. 6,
O-0.05
0.07
0.08
0.11
0.25?25 0.08
0.14-0.30 O-0.02
0.05 0.03
0.18 0.04
0.68 0.13
0.15-0.27 0.06-0.14 0.25-0.53
0.18 0.025 0.1-0.3
0.26 0.08 0.29
0.6 I 0.24 0.29
Ref. 26)
1.0*0.2 o.19?‘* 0.10 0.29 f 0.03
The data from refs. 28*29)are normalized to 02,(4;) = 0.29 [refs. 6*22)].The value e’,( I ;) = 0.013-0.15 is given in ref. 30). ‘“) S’, obtained from (25) and (16) with a = 7.2 fm and d = 5.4 fm. The bracketed values are calculated with the B, interaction. ‘)
0: obtained
from (26) (see text).
value has been used in most phenomenological analyses. Column (a) is obtained with (25) and (16), and column (b) with (26). The large differences between the experimental data are a consequence of the very indirect measurement methods of reduced widths for bound states. This uncertainty in the experimental results makes the comparison difficult. Since, in the GCM calculation, the antisymmetrization effects at d = 5.4 fm are lower than 7%) we think that the accuracy of the theoretical reduced widths will be at least as good as the experimental one. We discuss now the 0’, value of each state. 0; state: The reduced width of this state is very sensitive to the rms radius. We have calculated 0’, for the two different (r2)“’ values discussed in subsect. 3.1. The first choice (V,, 6 = 1.6 fm) gives f?‘,=3.9 x 10m4 and the second one (B,, b = 1.751 fm) leads to fI’, = 3.3 x 10m3,while the rms radius increases by scarcely 10%. Since the last calculation reproduces at one and the same time the energy and the rms radius of the ground state, the order of magnitude of the corresponding 0: can be considered as reliable. Hence, we think that the indirectly measured experimental values are too large. In particular, the result quoted in ref. “) has been deduced from a model-dependent parametrization of the direct capture cross section. A slight modification in this parametrization could significantly change the e’,(O;‘) value. O:, 27,4: states: The similar reduced a-widths of the 0; and 2; bound states confirm that they belong to the same band. The 4: state being unbound, its e’, is calculated from the phase shift. The GCM calculation for this band is in fair agreement with the results of refs. 26*27).The exderimental 0’, of the 4: resonance is directly obtained from elastic scattering data 6*22).Since our 82,(4:) is in agreement with the data, and since we treat all the states in a unified manner, we believe that the theoretical reduced widths of the 0: and 2; bound states are reasonable. Besides,
i? Descouvemont
they are not inconsistent given by Kettner
et al. / 12C(a, y)160
with most of the experimental
431
data. However,
the &2:)
et al. “) seems to be overestimated.
3; state: Our theoretical lower than for the other
value, although larger than the g.s. one, is significantly states. This relative reduction is confirmed by all the
experimental groups. We assume that, like for the 0: state, the smallness of the 3; rms radius leads to a e’, value too small with respect to the experimental data. A calculation performed with b = 1.751 fm gives (r2)“2 = 2.79 fm and 0”, = 0.012 in agreement with the data of refs. 26-28). The result of ref. 29) is significantly larger. 1; state: The GCM value 0”, = 0.09 obtained for both oscillator parameters agrees nicely with the results of refs. 26,28)and, to a lesser extent, with the data of refs. 6*30). Since this weakly-bound state plays an important role in the radiative-capture process, several theoretical works have been devoted to its study. A shell-model calculation of Stephenson 3’) leads to 6’,( 1;) = 0.08 f 0.04, in nice agreement with our result. From an R-matrix analysis of the ‘*C( CX,a)12C and 12C( (Y,-y)160 reactions and 16N P-decay, Barker ‘) proposes 13’,(1;) = 0.07 - 0.13. An hybrid R-matrix analysis of Koonin et al. “) gives 13’,(1;) = 0.1 S?z:li, in agreement with previous estimates. However, Langanke and Koonin “) have recently obtained 0’,( 1;) = 1.2 x lop3 and 3.1 X 10e3 by fitting two sets of experimental data [ref. ‘) and ref. 6), respectively]. One must notice that the R-matrix analyses of refs. 7,8,“) involve a non-vanishing background Ri” [see (25)], leading to ambiguities in the comparison with the experimental reduced widths. In our calculation, the ratio of y* and & may reach a factor 4.
4. The ?(a, 4.1. E2 CAPTURE
TO THE
y0J)16O radiative capture reactions
GROUND
STATE
The astrophysical S-factor (S = aE exp (27rv), where n is the Sommerfeld parameter) obtained from (2) is depicted in fig. 1 for a = 6.4, 7.2 and 8.0 fm. The curves corresponding to a = 7.2 and 8.0 fm are identical, as expected from a MRM calculation. On the contrary, the use of a = 6.4 fm (dashed line) leads to significantly different values. This confirms that an accurate MRM calculation cannot be performed with radii smaller than 7 fm. Using (17) and (18), the microscopic E2 S-factor is decomposed into its resonant part, due to the tail of the 2: bound state, and its non-resonant one, for radii a equal to 7.2 and 8.0 fm. From fig. 1, several comments can be made: (i) The Breit-Wigner tail of the 2: bound state dominates the E2 capture cross section at stellar energies. (ii) As announced in subsect. 2.2, the decomposition (17) depends on the MRM radius a, while the global result is nearly insensitive to this radius if it is correctly chosen [see ref. ‘*> for details].
438
F? Descouvemont
I
et al. / 12C(cr, y)“jO
2
3
EfMeV)
Fig. 1. (a) Microscopic E2 S-factor for capture to the I60 ground state (solid line: a = 7.2 and 8.0 fm; dash-dotted line: a = 6.4 fm). (b) Resonant part of S,, [first term of (18)] for a = 7.2 fm (solid line) and a = 8.0 fm (dashed line). (c) Same as (b) for the non-resonant part [second term of (18)].
(iii) The interference between both contributions is found constructive at all energies for a = 8.0 fm, but destructive at energies lower than 0.5 MeV for a = 7.2 fm. Hence the constructive or destructive nature of this interference has no absolute meaning. As a MRM calculation is not valid for radii below 7 fm, we do not present the decomposition for such values. Because of the disagreement between the theoretical and experimental r,.(2:+ 0:) (see subsect. 3.2), the E2 S-factor presented in fig. 1 is underestimated at the vicinity of the threshold. For the sake of providing a reliable estimate, we have multiplied the first term of the r.h.s. of (18) by 1.9, so that the y-width in (19) reproduces the experimental value. This correcting factor is necessary to compensate the lack of flexibility of the present GCM wave function. Introduction of other configurations in the GCM basis space would probably reduce the importance of this factor. The capture cross section obtained in such a way is presented in fig. 2 for a MRM radius equal to 7.2 fm. The dashed line corresponds to a = 8.0 fm. The stability with
l? Descouvemont
et al. / “C(ct,
2
1
y)160
439
3
EiMeV)
Fig. 2. (a) Renormalized E2 S-factor (see text) for capture to the I60 ground state (solid line: a = 7.2 fm; dashed line: a = 8.0 fm). (b) Experimental fit proposed by Kettner et al. 6, for the E2 capture. (c) Semi-microscopic E2 calculation of Langanke and Koonin I’).
respect to a is lost but the difference between both results remains small at stellar energies. At E = 0.3 MeV, we find S&0.3) = 0.09 MeV * b. Notice that the E2 S-factor can be accurately parametrized by formula (19) using the parameters given in table 1 for a = 7.2 fm. We have also plotted by Kettner
in fig. 2 the E2 part of the S-factor
et al. “) from an analysis
has an energy
dependence
similar
of their experimental
parametrization data. Their
obtained E2 S-factor
to our result, but is a factor of 2.5 larger. At first
sight this difference is related to the &2:) employed by these authors which is 10 times larger than our value and more than 5 times larger than the other experimental results (see table 4). One should therefore expect from (19) a similar ratio for the capture cross section near threshold since we have used the experimental r-width for the 2: state. To make a deeper connection with the parametrization of Kettner et al, we have fitted a Breit-Wigner formula using e”, (5.4) given in table 4. We have not changed the energy shift 1” and we have adjusted the shift parameters A:+ and (dA*+/dE), in order to reproduce our E2 S-factor from 0 to 1.5 MeV. The = +0.35. Taking all the factors values obtained are A:* - 0.14 MeV and (dA*+/dE), into account this leads in (19) to a denominator (E + 0.129 - 0.35 E)* for the resonant capture cross section. A similar expression can be obtained in the fit of Kettner et al. by linearizing the low-energy dependence of S,( kii) which is a fair approximation. With all the factors introduced in ref. 6), this leads to a denominator (E +0.214 +
440
l? Descouvemont
ef al. /
“C(a, yJL60
Both denominators differ by a factor 4 at E = 0.3 MeV reducing the discrepancy between the capture cross sections. The main difference between both expressions arises from the change in the sign of the derivative of the energy shift. The negative derivative obtained by Kettner et al. is due to the properties of the Coulomb wave functions and the absence of background I?;*. In our calculation, the non-negligible effect of the background, already partly introduced in &, leads to a positive derivative. This result indicates once more how difficult it is to compare different R-matrix analyses. The absence of background and the choice of too small a channel radius in the analysis of Kettner et al. lead to reduced a-widths which are not directly comparable with microscopic results. A detailed comparison with the theoretical study of Langanke and Koonin “) is not possible, since these authors do not separate the resonant and direct E2 captures. The small value SE2(0.3) = 0.0054 MeV +b can be understood if one assumes a destructive interference of the resonant and non-resonant parts (supposed to be of similar importance) of the E2 capture cross section. Since Langanke and Koonin do not give their 82,(27) value, it is not possible to evaluate their resonant contribution. Anyhow, we think that the inconsistency discussed in subsect. 3.2 may strongly affect the values of the astrophysical factor in their model. 0.42E)*.
4.2. E2 CAPTURE
TO THE 2; STATE
The
E2 S-factor towards the 2: excited state of I60 is presented in fig. 3b, without any normalization factor. The astrophysical factor is almost constant below 2.5 MeV and is dominated by the 4: resonance between 2.5 and 3.5 MeV. Our results are too small by a factor of two with respect to the experimental data of Kettner et aL “). Tire a- and r-widths have been deduced from the Breit-Wigner shape of the peak around the 4: resonance. The a-width of the 4: state is equal to 30 keV, in nice agreement with experiment, but the theoretical r-width is smaller than the experimental value (see subsect. 3.2). The S-factor extrapolated by Kettner et al. at E = 0.3 MeV for the ‘*C(cu, y3)160 reaction is S(0.3) = 12* 2 keV * b, while our value is 3.3 keV - b. If one takes into account the factor two mentioned above, the remaining difference is probably due to the &(2:) values, ,which influence the direct capture towards the 2: state% Anyhow, our theoretical calculation and the experimental estimate of ref. 6, show that the capture cross section is significantly smaller to the 2: excited state than to the ground state. 4.3. El CAPTURE The
TO THE GROUND
STATE
much-too-small r-width that we have obtained for the 1; state (see subsect. 3.2) leads to a large underestimation of the El capture cross section. This underestimation is not particular to our treatment of the El multipolarity. Using a different
P. Descouuemont et al. / “C(CY,?)I60
441
b)
1
2
3
EIMeV)
I
2
3
Fig. 3. (a) E2 [see fig. 2(a)] and El (see text) S-factors for capture to the I60 ground state. The experimental data are from ref. 5, (full circles) and ref. 6, (open circles). (b) E2 S-factor for capture to the 2: state. The experimental data are from ref. 6).
approach, leading also to non-vanishing El transitions, Koonin et al. 8, and Langanke and Koonin I’) have been obliged to scale the direct El cross section by a very large effective dipole strength. In order to compare our results with phenomenological analyses, we have calculated the resonant part of the El cross section by taking the experimental r,(l;) value and the theoretical reduced a-width and energy shifts given in table 3. The El S-factor obtained by using (19) with a = 7.2 fm is depicted in fig. 3a. Note that, up to 1 MeV, the same curve can be fairly reproduced with (19) and d = 5.4 fm. To this end, one must use 0’,( 1;) given in table 4, and the energy shift (22) calculated with A;-=0.16 MeV and (dA’-/dE), =0.35. At 0.3 MeV, we find &,(0.3)= 0.30 MeV * b. Langanke and Koonin ‘I) have obtained a very similar value by fitting the Miinster data (SE, = 0.34) while the value deduced from their fit of the Caltech data is a factor of two smaller. The result obtained by Kettner et al. “) (SE, = 0.25) is in nice agreement with our value. Since these recent values are larger than all previous estimates 5*7-1o),th e reaction rate usually used in astrophysics ‘) should be reconsidered. Our S,, value estimated at 0.3 MeV is probably less reliable than SEz since in this case, the direct and resonant contributions are included. However, with the reason-
442
I? Descouwmont et aL / ‘*C(a,
Y)‘~O
able assumption that the resonant part of SE, dominates the capture cross section at stellar energies, our estimation, based on microscopic R-matrix parameters, is probably not far from the correct value. The influence of the broad I; resonance at 9.63 MeV, which cannot be studied in our model, is most likely small at low energies [see fig. 14 of ref. “)I.
5. Conclusion The generator coordinate method provides a unified description of bound and scattering states. Using this property, we have shown that the influence of bound states on capture cross sections is automatically included in the MRM. This influence is of course also present in other variants of the GCM, but the similarity between the MRM and conventional R-matrix analyses enables one to go deeper into the comparison between microscopic and phenomenological results. Starting from a two-body nucleon-nucleon interaction, adapted to each partial wave, we have calculated a- and y-widths of the (Y+ ‘*C system without adjustable parameters. Unfortunately, one cannot presently describe correctly the electric properties of all the states of the unified system with such a basic method. The main conclusions which can be drawn from our work concern the general trends of these electric properties and the validity of the a-widths used in phenomenological analyses. It appears clearly that the coherence of the experimental information on the a-widths is not very satisfactory. These widths are fitted on different processes (elastic scattering, direct capture); a background contribution from other states is sometimes included, sometimes not. Moreover, R-matrix analyses are usually performed at much-too-small radii. The use of pure Coulomb wave functions and the neglect of antisymmetrization are questionable for the a + ‘*C system at a radius as small as 5.4 fm. The choice of a larger radius, around 7 fm, would improve the validity of the phenomenological analyses and facilitate the comparison with microscopic calculations. In our study, the only state for which the a-width is experimentally well defined is the 4: state since it is not bound. The very good agreement that we have obtained with the experimental 82,(4;‘) enables us to discriminate between the experimental data for the 0: and 2: states, some of them being much too large. For bound states the experimental determinations of reduced a-widths are so indirect that the values obtained are probably not meaningful for the radiativecapture problem. The E2 S-factor that we have obtained after an empirical correction of the B(E2,2; + 0:) value should be reliable. Its value at stellar energies, although smaller than the value of Kettner et al., is non-negligible and influences the a-capture rate at 0.3 MeV. Our result contradicts the smallness of the E2 S-factor of Langanke and Koonin, which could be due to an inconsistency in their semi-microscopic model. The problem raised by E 1 transitions is more complicated. The correct order of magnitude of the El transition probabilities cannot be obtained with T = 0 wave
P. Descouvemont et al. / ‘*C(a, y)160
443
functions by relaxing the long-wavelength approximation, like for the (Y+ I60 system. We have therefore calculated the resonant part of the El S-factor by using the experimental r-width and the MRM reduced width and energy shift. In this way, we have taken the background into account but not the effect of the 1; resonance. The value obtained at 0.3 MeV agrees with the results of Kettner et al. and their fit by Langanke and Koonin. These evaluations indicate that the total S-factor at stellar energies is probably significantly larger than usually assumed in astrophysical calculations. Our work can be improved in two ways. The enlargement of the GCM basis will certainly bring the B(E2,2:+ 0:) closer to the experimental value. Also, to obtain agreement for the r-width of the 1; state necessitates the inclusion of isospin impurities in the GCM basis. This introduction is certainly much more complicated. However, both improvements seem necessary to make possible a direct comparison between microscopic results and the experimental data.
References 1) W.A. Fowler, G.R. Caughlan and B.A. Zimmerman, Ann. Rev. Astron. Astrophys. 5 (1967) 525; 13 (1975) 69 2) C.A. Barnes, in Advances in nuclear physics, ed. M. Baranger and E. Vogt, vol. 4 (Plenum, New York, 1971) p. 133 3) C. Rolfs and H.P. Trautvetter, Ann. Rev. Nucl. Part. Sci. 28 (1978) 115 4) R.J. Jaszczak and R.L. Macklin, Phys. Rev. C2 (1970) 2452 5) P. Dyer and CA. Barnes, Nucl. Phys. A233 (1974) 495 6) K.U. Kettner, H.W. Becker, L. Buchmann, J. Giirres, H. Krawinkel, C. Rolfs, P. Schmalbrock, H.P. Trautvetter and A. Vheks, Z. Phys. A308 (1972) 73 7) F.C. Barker, Austral. J. Phys. 24 (1971) 777 8) SE. Koonin, T.A. Tombrello and G. Fox, Nucl. Phys. A220 (1974) 221 9) J. Humblet, P. Dyer and B.A. Zimmerman, Nucl. Phys. A271 (1976) 210 10) T.A. Tombrello, SE. Koonin and B.A. Flanders, in Essays in nuclear astrophysics, ed. C.A. Barnes, D.D. Clayton and D.A. Schramm (Cambridge University Press, New York, 1982) p. 233 11) K. Langanke and S.E. Koonin, Nucl. Phys. A410 (1983) 334 12) D. Baye and P. Descouvemont, Nucl. Phys. A407 (1983) 77 13) P. Descouvemont and D. Baye, Phys. Lett. 127B (1983) 286 14) Y. Suzuki, Prog. Theor. Phys. 55 (1976) 1751; 56 (1976) 111 15) Y. Fujiwara, H. Horiuchi, K. Ikeda, M. Kamimura, K. Kato, Y. Suzuki and E. Uegaki, Prog. Theor. Phys. Suppl. 68 (1980) 29 16) M. Libert-Heinemann, D. Baye and P.-H. Heenen, Nucl. Phys. A339 (1980) 429 17) D. Baye, P.-H. Heenen and M. Libert-Heinemann, Nucl. Phys. A291 (1977) 230 18) D. Baye and P. Descouvemont, Nucl. Phys. A419 (1984) 397 19) A.M. Lane and R.G. Thomas, Rev. Mod. Phys. 30 (1958) 257 20) A.B. Volkov, Nucl. Phys. 74 (1965) 33 21) D. Baye and N. Pecher, Bull. Cl. SC. Acad. Roy. Belg. 67 (1981) 835 22) F. Ajzenberg-Selove, Nucl. Phys. A375 (1982) 1 23) D.M. Brink and E. Boeker, Nucl. Phys. A91 (1967) 1 24) G.E. Brown and A.M. Green, Nucl. Phys. 75 (1966) 401 25) T. Erikson, Nucl. Phys. A170 (1971) 513 26) H.M. Loebenstein, D.W. Mingay, H. Winkler and C.S. Zaidins, Nucl. Phys. A91 (1967) 481
444
F? Descouoemoni
et al. / ‘*C(a,
y)160
27) F. F’iihlhofer, H.G. Ritter, R. Bock, G. Brommundt, H. Schmidt and K. Bethge, Nucl. Phys. Al47 (1970) 258 28) F.D. Becchetti, E.R. Flynn, D.L. Hanson and J.W. Sunier, Nucl. Phys. A305 (1978) 293 29) F.D. Becchetti, D. Overway, J. JHnecke and W.W. Jacobs, Nucl. Phys. A344 (1980) 336 30) C. Wemtz, Phys. Rev. C4 (1971) 1591 31) G.J. Stephenson, Astrophys. J. 146 (1966) 950
MICROSCOPIC
ANALYSIS
P. DESCOUVEMONT*, Physique 7hhique
OF THE ‘*C(a, y)160
REACI’ION
D. BAYE and P.-H. HEENEN**
er Mafht!mmafique,CP 2.79, UnioersifPLibre de Bruxelles, Brussels, Belgium
Received 28 February 1984 Abstract: Electric transition probabilities in the I60 spectrum, and the %(a, y,,,)‘6O capture cross sections are calculated with antisymmetric wave functions by the generator coordinate method. The influence of bound states on radiative capture is shown to be automatically included in the model. The reduced a-widths of the I60 bound states are discussed, and compared with previous theoretical and experimental estimates. The microscopic E2 capture cross sections to the 0; and 2; states yield an astrophysical S-factor of 0.09 MeV. b at 0.3 MeV. An attempt to treat the El multipolarity by relaxing the long-wavelength approximation leads to a large underestimation of the y-widths. Adopting the experimental y-width and the theoretical reduced a-width of the I ; state provides SE, = 0.30 MeV . b at 0.3 MeV.
1. Introduction The cross sections for the radiative capture of alphas by ‘*C nuclei are important for determining the relative abundances of ‘*C and I60 in the stellar helium-burning process lm3).Th e calculation of the reaction rate ‘) requires a knowledge of these cross sections at very low energies (typically around 0.3 MeV for temperatures of about 2 X 10’ K). Direct measurements are precluded by the extreme smallness of the cross sections at these energies. Accurate experiments have until now been limited to energies larger than 1.4 MeV [refs. “)I. Unfortunately, a marked disagreement still exists between the absolute values of the low-energy data 5-6). The experimental cross sections serve as basic data for different theoretical fits ‘-‘O) which eventually provide extrapolations at stellar energies. However, this procedure meets several difficulties. (i) The extrapolation is made inaccurate by the occurrence of a broad bump in the cross sections in the energy range which can be studied experimentally. This bump is due to the 1; resonance at 9.63 MeV excitation energy in the I60 spectrum. (ii) The relative importance of the El and E2 contributions, which have different energy behaviours, is not known. The data of ref. ‘) establish that the E2 part is small in the energy range dominated by the 9.63 MeV resonance but this result is of limited interest for lower energies. (iii) Still more important, the existence of two bound states just below the LY+ ‘*C threshold makes the estimation of cross sections uncertain. The influence on the El contribution of the 1; state
* Aspirant FNRS. ** Chercheur qualifik
FNRS. 426
I? Descouvemoni et al. / ‘*C(q y)160
427
located 45 keV below the threshold has been discussed by many authors ‘-‘I). A similar effect of the 2: bound state located 245 keV below the threshold on the E2 contribution ‘) has been studied recently 67”). The influence of these bound states depends crucially on some of their properties (like their reduced a-widths in the R-matrix formalism) which cannot be measured directly. An attempt to solve these problems was made recently by Langanke and Koonin “) who calculated the E2 contribution in a semi-microscopic approach. They found that the E2 component is almost negligible, in strong contradiction with the analysis of ref. “). The different quantities which cannot presently be reached by experiments and are inaccurately given by extrapolations can be computed in a microscopic model I’). The generator coordinate method (GCM) has the following advantages [see references in ref. ‘“)I: (i) antisymmetrization is taken into account exactly; (ii) bound and scattering states are treated in a unified manner; (iii) the conservation of the good quantum numbers is treated exactly; (iv) the relative motion of the colliding nuclei is described quantum-mechanically. These properties are made possible in a relatively simple way by the use of Slater determinants defined in the two-centre harmonic-oscillator model. The GCM has been applied to the study of the 160( CX,y)*‘Ne radiative-capture reaction 12*13)for which, unfortunately, no experimental data exist. However, the unified treatment of bound and scattering states provides a test of the accuracy of the model, namely the comparison with the experimental transition probabilities involving bound states and narrow resonances. The main conclusions of refs. 12,13)are the following: (i) antisymmetrization effects are not negligible, even at low energies; (ii) the 160( cy,y)“Ne capture cross section below 1 MeV is about six times larger than the value utilized in astrophysical calculations ‘); (iii) the El contribution is not solely due to a small T = 1 admixture to the wave functions as usually assumed [this theoretical result seems to be confirmed by experimental data about branching ratios for El transitions in the 20Ne spectrum “)I. The aim of the present paper is to apply the same formalism to the 12C((u,y)160 radiative-capture reaction. The situation is, however, less favourable than for the 160( CX,y)“Ne reaction. If the 20Ne spectrum is well described by an (Y+ ‘60g.s.cluster model, the description of I60 by an (Y+ ‘2Cg.s.structure, although qualitatively valid 14*“), is less satisfactory ‘5*‘6).In particular, the I60 ground-state energy is far too low with respect to the cy+ “C threshold. As in ref. 13), the exact energies will have to be reproduced by adjusting a force parameter for each partial wave. The microscopic R-matrix method (MRM) which we use to solve the GCM equations “) has the important advantage of providing naturally reduced a-widths, even for bound states. Besides direct information on E2 capture and partially on El capture, we obtain reliable estimates for the bound-state parameters which influence the cross section at low energies. The present microscopic model will thus question different assumptions and parameters which are encountered in phenomenological 5-‘o) or semi-microscopic “) models.
428
P. Descouwmonr et al. / “C(a,
y)160
The theory is summarized in sect, 2 with emphasis on the influence of bound states on the capture process. In sect. 3, rms radii, quadrupole moments, reduced transition probabilities and reduced a-widths are compared with the available experimental data and theoretical estimates. ‘The different contributions to the capture cross sections are studied in sect. 4. Concluding remarks are presented in sect. 5. 2. Influence of bound states on radiative capture 2.1. CAPTURE
CROSS SECTION
The reduced electric transition probability between two bound states of a nucleus is given by
B(Eh,J:‘-,J;r)=~(~‘~“~~~~~~~(L’~Q~)~2(2Jf+1)/(2Ji+1),
(1)
where A is the order of the (electric) multipole, and J and T denote respectively the total angular momentum and the parity of the initial and final states. In this paper, this probability is calculated utilizing antisymmetric wave functions. The asymptotic behaviour of these wave functions and the c.m. dependence of the JX,” operator are correctly taken into account, as explained in ref. 12). When the initial state J:i is a scattering state at c.m. energy E, the electric radiative-capture cross section towards the final state .!;I is obtained from [ref. “)I a(E, AJ:‘+J;‘)
=[&(A
+l)/A(2A
+ l)!!2h]ti7A+‘B,,,(EA, J?-,J;r),
(2)
where Bcap, is given by (1) with $4” replaced by the J? partial wave of a unit-flux scattering wave function (here and in the following, the colliding nuclei are assumed to have spin zero). In (2), k, is the wave number of the emitted photon. 2.2. REDUCED
WIDTHS
In this paper, the bound and scattering state wave functions are defined in a GCM basis ‘2*‘7).However, the GCM basis states do not have a correct asymptotic behaviour, neither for bound nor for scattering states I’). This problem is solved here by the microscopic R-matrix (MRM) method I’), which furthermore provides a precise definition of the reduced widths. Moreover, as shown hereafter, a contribution of the bound states is automatically included in the scattering wave function. In the MRM framework, the configuration space is divided into two regions. The antisymmetrization is taken into account exactly in the internal region (of radius a), and is neglected outside. Let us consider N basis functions 6? = bI”“( R.) depending on the set of generator coordinate values R,. The 4i”” are defined as Slater determinants, constructed in the two-centre harmonic oscillator model with A, orbitals centred around -A2R,/A and A, around A,R,IA, and projected on _l
P. Descouvemont
et al. / “C(cy, y)“O
429
and 7r. In order to simplify c.m. problems, a common harmonic-oscillator 6 is chosen for both clusters. The GCM wave functions are g;F
= ; fW"", n=,
parameter
(3)
where the f: are unknown coefficients. The wave function written as *;F$ = &n&~,&g’“(P) YJ”W,)
(3) can also be (4)
where p is the relative coordinate between the c.m. of both clusters. In (4), .A?is the antisymmetrization projector, &,,. is a c..m. function, and 41 and & are the antisymmetrized internal wave functions of the nuclei. The relative function g” is given by g’“(p) = [256&( with c = A!/A,!A,!(
1 +S,J
1 + &)/(2J+
l)]“‘c
”
f;TJ(p, R,)
(5)
and
TAP, R) = (p/rb2)3’4exp
(-P(P’
+R2)/2b%(&Vb2),
(6)
p being the dimensionless
reduced mass, and i, a spherical Hankel function. In the external region, where antisymmetrization between the clusters is negligible, the operator d can be replaced by c-l; the relative coordinate p has a precise meaning, and $fny’n”;‘” can be factorized in a product of internal and relative wave functions. However, as can be seen from (5), $$yT(p) does not have a correct asymptotic behaviour. Hence, for p larger than a, one defines for scattering states $~$=(p) = i’+’ [n(l +t3,,)(2_l+
1)/W]“’
x~,.,.~,dzyrM(~np)[l~(k~)-
~“O,Udl/k~,
(7)
where I, and 0, are ingoing and outgoing Coulomb wave functions, lJJ” is the collision matrix, and v and k are the relative velocity and wave number, respectively. The collision matrix U”’ and the coefficients firare calculated from the BlochSchrbdinger equation [see ref. “) for the definition of the Bloch operator Y] (H+~-EE)~~~~=~~~~
(8)
and from the continuity of the wave function at the radius a: +ZW
= rVE(a).
(9)
Before proceeding, let us emphasize an important point: in these equations, the MRM radius a must be chosen large enough so that the wave functions have reached their asymptotic behaviour. Then, the physical results are insensitive to the value of a. However, phenomenological R-matrix analyses 7V8S’o*‘1) make use of values of the channel radius which are too small to be taken as the MRM radius. In order to compare our results with these analyses we shall introduce below a second radius, the channel radius ii, smaller than u.
430
k? Descouuemonr
et aL / “C(a,
y)160
In order to exhibit the influence of the bound states on the scattering wave function, let us define an orthonormalized set of new basis states Jj”“, depending on the MRM radius u, and related to 4’n”” by the linear transformation &“- = c #;(a)C#J’,M”.
(10)
n
We require that the new basis diagonalizes the Bloch-Schrodinger internal region. The matrix D’” is then given by
equation in the
c D~~(“(a)(~~~IH+Y-~:“(a)J~J,“)p=O, n
(11)
where the subscript P means that the overlap and hamiltonian kernels are calculated over the internal region only. A negative eigenvalue of (11) gives an approximation of a bound-state energy in the J” partial wave. However, it must be noticed that the negative energy ,??fT(a) is not rigorously equal to the exact bound-state energy Ef” given by eq. (16) of ref. I’). Besides, its value depends on the radius a. Numerical examples of the differences between these energies will be given in the next section. In this new basis, the wave functions (3) become * ‘Mm
=
c
f’y~Jy,i’&M~
=
n,i
c
@(+p.
(12)
I
The unknown coefficients df” are calculated by projecting (8) on Jf”“: d:“(a)=(~j”“l~l~~~~)/(~‘:“(a)-E).
(13)
In the following, it will be convenient to extend the definition of &i”” over the whole space. In order that eq. (12) be valid in the external region, one defines for values of the relative coordinate p larger than a (14) where @i:“(p) is given by (7), and ~$:~,(a) is deduced from ( 10) and from asymptotic form of f$‘,““. The reduced width for the fission into two clusters can be easily obtained calculating the R-matrix in the basis (10). Indeed, applying the formalism of ref. one deduces the R-matrix R’” =C y;,i(a)/[E I
-,+(a)].
the .-by I’),
(15)
The
reduced fission width for each eigenvalue is given by the residue of R’” at (we omit hereafter the index i of the reduced width). In order to compare theoretical values with the experimental data, we must be able to calculate reduced fission width at a radius d different from a. By extension of (15), one
1
@’ our the has
2
y’.(a)
=[256~‘(1+&,)/(2J+
1)][fi2ti/2~m~]
c
n
#;(rr(a)rJ(cS,
R,)
,
(16)
I? Descouvemont
et al. / “C(a,
431
y)160
where the function I’, is given by (6). In this definition of the reduced width, the subscript a is relative to the definition of the MRM wave function (notice that matrix D depends on a, and not 6), while the argument d denotes the channel radius at which the reduced width is calculated. The definition (16) assumes that the antisymmetrization is still negligible at the radius E In fact, in phenomenological analyses, the &values do not completely fulfil this condition. However, if the antisymmetrization effects are small, (16) can be used for comparing our results with these analyses. Besides, for a fixed &value smaller than a, eq. (16) gives stable results as a function of the MRM radius a. One must finally point out that this definition provides straightforwardly the reduced fission width for negative-energy states, as well as for narrow resonances I*). 2.3. BOUND-STATE
CONTRIBUTION
TO THE CAPTURE
CROSS
SECTION
The determination of the GCM-MRM wave functions, as well as the detailed calculation of the matrix elements in (1) and (2) have been reported in ref. I*). In particular, the spurious c.m. motion has been shown to be automatically eliminated in the long-wavelength approximation. If one separates the bound-state contribution in the scattering wave function as 9 JMW= d:“&“” the partial radiative-capture becomes
+ I,!/;%;,,,
(17)
cross section (2), for given transition and multipolarity,
m(E,hJF+J,“t)=[8~(h
+1)(2Jr+l)/hh(2h
+1)!!‘(2Ji+l)]
~ld:i”i(~~f”rll~~II~:i”i) +(IClJfPrll~~II~~~~,,,d)12.
(18)
The bound-state contribution is then easily deduced by neglecting the second term of (18). One obtains the well-known form cR( E, AJ:’ + J;r) =$(I
+6,2)(2Ji+l)
T+“i(E)T,(E, AJF”i J;t) [~-Efi”i+~Jii”i-AJi”i(E)]*+[~~~“i(E)]*’
(19)
iJr=
(20)
where
The fission width rfm(E)
&“-@Q).
and the energy shift A.‘“(E) are given by
-AJ”(E)+$i@‘(E)=y~(~)lJ(k~)/(l-LJ(k~)R~w)
(21)
with L,(ka)
= kaO;(ka)/O,(ka)
The background contribution
= S,(ka) +iP,(ku)
.
Rim is obtained from (15) by removing the bound-state
432
I? Descouvemont et al, / “C(a,
y)160
contribution. The y-width r, is calculated at the energy E, and is relative to the transition between the bound states J:i and J;(. In ( 19), the calculation of A” and r{= requires a knowledge of Ri”. However, it is interesting to simplify the Breit-Wigner formula (19). Firstly, let us linearize the energy shift: A’“(E)=A:“+(dA’*/dE),(E-E:“). Secondly, we introduce fission width (21) as
an effective reduced width $&(a)
r:=(E) Comparing
=
(22) in order to write the
(23)
2PJ(~~)~:,et&).
(21) and (23) leads to the definition &(a)
= Y%)/ll
-
l;,(~)R612s
(24)
which is slightly energy dependent. In this way, the background contribution R$’ is taken into account in simpler expressions. For the reasons stated above, we must calculate this reduced width for a channel radius E, smaller than the MRM radius a. We thus define &a(~)
= Y’,(a)/]1 -
LJ(~)&~:“~*,
(25)
where r:(a) is given by (16). Since some arbitrariness enters in this definition, we shall calculate &r(ti) by a second method. Assuming a weak channel-radius dependence of the fission width (23), we also define r2,,n(6)=T:“(E)/ZPJ(kcf).
(26)
The effective reduced widths will be considered
as reliable as far as formulas (25) and (26) give close values. A discrepancy would indicate that the channel radius d is too different from a. Finally, it must be noticed that, if one includes the absorption of photons 19) in this theory, the fission width appearing in the denominator of (19) is replaced by the total width r = Tr+ r,, leading to the rigorous Breit-Wigner formula. However, the correction requires much more complicated calculations whereas it is completely negligible. 3. Properties of the a + ‘*C system 3.1. ENERGIES,
mu RADII
AND
QUADRUPOLE
MOMENTS
The calculations are performed with interaction V, [ref. ‘“)I and a spin-orbit force “) (So = 50 MeV . fm’). The oscillator parameter has been chosen intermediate between the values leading to the correct rms radii of a and ‘*C nuclei (b = 1.6 fm). Ten discretization points R, are located between 0.8 and 8.0 fm with a mesh size
P. Descouvemont et al. / “C(cr, ?)I60
433
TABLE 1
Energies, J” 0+I
0+ 3: 2+I :;
Energies
rms radii and quadrupole m
0.6940 0.6400 0.6729 0.6368 0.6504 0.6365
E”
obtained
E’” x
-7.16 -1.11 -1.03 -0.25 -0.05 3.19
are in MeV, lengths
moments
0.0 6.05 6.13 6.9 1 7.12 10.35
in fm and quadrupole
with V, for I60
(?)“2 2.52 2.97 2.56 2.93 2.89 2.9 1 moments
Q 0 0 -4.0 - 14.6 -9.2 -18.0 in e. fm’.
of 0.8 fm. The energies are calculated from eq. (16) of ref. I*) for the bound states, and from the phase shift “) for the 4: resonance. The capture cross section being strongly dependent on the final-state energies, we have allowed the Majorana parameter m to vary with the different states. The m-values displayed in table 1 reproduce the experimental energies 22) within 5 keV. In table 1, we list the rms radius (the rms radius of the proton is taken as 0.8 fm) and the electric quadrupole moment of the bound states and of the 4: resonance. As it has been found in previous microscopic studies 15*‘6),it is difficult to reproduce with the same nuclear interaction the 160 g.s. energy and the energies of the other bound states. In particular, the m-values which give the correct O:, 2:, 4: energies lead to a ground state that is largely overbound with respect to the (Y+ “C threshold. For this reason, the Majorana parameter used for the 0: state is larger than the others. On the contrary, the m-values of the O:, 2; and 4; are nearly equal. The negative-energy levels are treated with their correct asymptotic behaviour, as explained in ref. I*). For the narrow 4: resonance (r, = 30 keV) the rms radius and the quadrupole moment are calculated in the bound-state approximation I*). The rms radius of the g.s. disagrees with the experimental value 22) 2.71 fm. The GCM rms radius is very close to the rms radius (2.48 fm) of the I60 one-centre harmonic oscillator wave function, which represents 96% of the GCM wave function. This one-centre rms radius would be increased by using a larger value of the oscillator parameter. Hence, we have performed a calculation with b = 1.75 1 fm and B, [ref. “)I plus spin-orbit as the nucleon-nucleon interaction. The value S,, = 89 MeV * fm5 reproduces the energy of the 0: state and gives (r*)“’ = 2.74 fm, in agreement with experiment. However, with the BI potential the 0: and 2: states are not bound, even without the repulsive spin-orbit force. This potential can thus not be used to compute radiative-capture cross sections. The intrinsic quadrupole moments I*) of the 2: and 4: levels are 5 1.l and 49.5 e * fm’, respectively, which indicates that these states belong to the same rotational band. The assignment of the 0: state as bandhead is confirmed by the similar values of the rms radius and by the very slight variations of the Majorana parameter required to reproduce the experimental energies of these three states.
434
I? Descouvemont
et al. / 12C(a,
y)160
TABLE 2 Reduced E2 transition probabilities (in e* . fm4) in I60 GCM 2:+0; 2:-o; 4;+2; 1;+3;
3.2. ELECTRIC The
TRANSITIONS
2.1 55.1 71.9 17.1
Exp22) 7.8kO.3 67.0 f 7.4 ISI* 52* 18
PROBABILITIES
E2 reduced transition probabilities are compared in table 2 with experimental data 22). They are calculated using (l), with the exception of the B(E2,4:+ 2:) which is deduced from the 4’+2+ partial capture cross section (see subsect. 4.2) by assuming a Breit-Wigner shape around the 4: resonance energy. No effective charge has been introduced in the calculation. The theoretical B(E2) values are systematically smaller than the experimental data, the difference being the largest for the 2: + 0: transition. The best agreement with experiment is obtained for B(E2,2: + 0,‘). As expected for a rotational band, the intrinsic quadrupole moments deduced from the theoretical 4: + 2: and 2: + 0,’ transition probabilities are very close. This result is not confirmed by the experimental data, suggesting that the experimental 4: resonance is a mixing of a a + ‘2Cg.S.structure with more complicated configurations. Results similar to ours have been obtained by Suzuki 14).This author has performed a coupled-channel calculation of the a + “C system including the 2+ and 4+ rotational states of “C in the framework of the orthogonality condition model, a semi-microscopic method which takes the Pauli principle partly into account. The structure of the wave functions obtained by Suzuki is very similar to ours. In both ‘cases, the one-centre component (OpOh) of the g.s. wave functions is largely dominant (more than 90%) and much larger than the OpOhcomponent found in the g.s. wave function of shell-model calculations 24). In the calculation of Brown and Green, the 0;’ and 2: wave functions have larger 2p2h components (22% and 14%) respectively) which probably explain the larger B( E2) values obtained in the shell model 24,25).More puzzling is the good agreement with experiment obtained by Langanke and Koonin I’) since, as in the present model, their g.s. wave function has a 96% one-centre component. Their result is also four times larger than Suzuki’s who used the same model with a larger configuration space. This contradiction seems to be due to an inconsistent use in ref. “) of the orthogonality-condition-model wave functions in the calculation of matrix elements. The a and 12C nuclei having isospin zero, the El transitions are forbidden at the long-wavelength approximation. The non-vanishing dipole-transition probabilities are usually assigned to small T = 1 components in the wave functions. In the a +I60
l? Descouvemonr
et al. / “C((r,
?)I60
435
case however, the correct order of magnitude of the El transition probabilities can be reproduced with T = 0 wave functions by expanding the isoscalar part of the El operator to the next order in the photon wave number 13). Moreover, a careful examination of the experimental branching ratios suggests that the weak isospinmixing effects should lead to a matrix element of the isovector part of the El operator which partially cancels out the matrix element of its isoscalar part. For the (Y+ “C system, we have treated the El multipolarity in the same way as in ref. 13). Unfortunately, this procedure fails in the present case. The theoretical B(E1, l;+O:) is equal to 5.4 x lo-*e2 * fm2 while the experimental value 22) is 1.5 low4 e2 * fm2. As for the B(E2,2:+0:), a part of this discrepancy must be due to the fact that the one-centre OpOh component of the g.s. wave function is too large. However, this effect seems insufficient to explain the order of magnitude of the discrepancy. We thus think that the relative importance of isospin-mixing effects in low-lying states is larger for 160 than for 20Ne.
3.3. REDUCED
a-WIDTHS
We present in table 3 the reduced a-widths -y’,(a) [eq. (16)] and &_e( a) [eq. (24)], and the energy shifts [eqs. (20) and (22)] deduced from the MRM calculation at a = 7.2 fm. The effective widths are a factor 3 to 4.5 smaller than &a). The smallness of the 0: and 3; energy shifts lJlr and A: can be explained by their small rms radii and o-widths. Indeed, since the probability densities of these two states are concentrated at low inter&stances, the eigenvalue E’:” of (11) is nearly insensitive to the MRM radius. On the other hand, the proportionality between A’” and y2 [see (21)] explains the small values of A:” and (dA’“/dE), for the 0: and 3; states. For the other states, 21Jn and A:” are more significant. However, it must be pointed out that both shifts are present in (19) with opposite signs, so that their effects nearly cancel out. In table 4, we display the dimensionless reduced a-widths (02 = y&,(d)/ &,, where y$,, = 3R2/2pmNd2 is the Wigner limit) at the channel radius d = 5.4 fm. This
TABLE
a-widths J” 0+
0: 3: 2:
Energies
r:(a) 6.3 6.1 1.1 5.1 6.3 7.3
x x x x x x
IO-’ lo-’ 1o-3 lo-* lo-* lo-*
and energy
1o-5 lo-* 1O-4 lo-* lo-* 10-2
shifts in I60
A'"I
r2...&) 1.4x 1.9 x 2.8 x 1.5 x 1.8 x 2.0 x
3
4.0 x 1o-4 1.7 x 10-l 3.3 x 1oP 1.3 x10-l 1.4x10-’ 1.5x10-’
are in MeV. All the values are calculated
(dA’*/dE) -3.8 -3.2 -4.4 -2.3 -3.2 -2.5
x x x x x x
lO-5 lo-* lo-‘+ 1O-2 lo-’ lo-*
at CI= d = 7.2 fm (see text).
2.0 x lo+ 0.16 0.01 0.11 0.14 0.12
436
i? Descouwmont
et al. / “C(a,
TABLE
y)160
4
Dimensionless effective reduced a-widths 0: of low-lying I60 levels I”
GCM ‘)
GCM b,
0:
3.9 x lo-4 [3.3 x IO-J] 0.11 3.3 x lo-3 [1.2x10-2] 0.10 0.09
(4.8* 1.0) x IO-’
0;
3; 2: 1; 4;
0.10*0.05 (2*1)x10-3 0.10*0.02 0.09 * 0.02 0.32
Ref. *‘)
Ref. 28)
Ref. 29)
Ref. 6,
O-0.05
0.07
0.08
0.11
0.25?25 0.08
0.14-0.30 O-0.02
0.05 0.03
0.18 0.04
0.68 0.13
0.15-0.27 0.06-0.14 0.25-0.53
0.18 0.025 0.1-0.3
0.26 0.08 0.29
0.6 I 0.24 0.29
Ref. 26)
1.0*0.2 o.19?‘* 0.10 0.29 f 0.03
The data from refs. 28*29)are normalized to 02,(4;) = 0.29 [refs. 6*22)].The value e’,( I ;) = 0.013-0.15 is given in ref. 30). ‘“) S’, obtained from (25) and (16) with a = 7.2 fm and d = 5.4 fm. The bracketed values are calculated with the B, interaction. ‘)
0: obtained
from (26) (see text).
value has been used in most phenomenological analyses. Column (a) is obtained with (25) and (16), and column (b) with (26). The large differences between the experimental data are a consequence of the very indirect measurement methods of reduced widths for bound states. This uncertainty in the experimental results makes the comparison difficult. Since, in the GCM calculation, the antisymmetrization effects at d = 5.4 fm are lower than 7%) we think that the accuracy of the theoretical reduced widths will be at least as good as the experimental one. We discuss now the 0’, value of each state. 0; state: The reduced width of this state is very sensitive to the rms radius. We have calculated 0’, for the two different (r2)“’ values discussed in subsect. 3.1. The first choice (V,, 6 = 1.6 fm) gives f?‘,=3.9 x 10m4 and the second one (B,, b = 1.751 fm) leads to fI’, = 3.3 x 10m3,while the rms radius increases by scarcely 10%. Since the last calculation reproduces at one and the same time the energy and the rms radius of the ground state, the order of magnitude of the corresponding 0: can be considered as reliable. Hence, we think that the indirectly measured experimental values are too large. In particular, the result quoted in ref. “) has been deduced from a model-dependent parametrization of the direct capture cross section. A slight modification in this parametrization could significantly change the e’,(O;‘) value. O:, 27,4: states: The similar reduced a-widths of the 0; and 2; bound states confirm that they belong to the same band. The 4: state being unbound, its e’, is calculated from the phase shift. The GCM calculation for this band is in fair agreement with the results of refs. 26*27).The exderimental 0’, of the 4: resonance is directly obtained from elastic scattering data 6*22).Since our 82,(4:) is in agreement with the data, and since we treat all the states in a unified manner, we believe that the theoretical reduced widths of the 0: and 2; bound states are reasonable. Besides,
i? Descouvemont
they are not inconsistent given by Kettner
et al. / 12C(a, y)160
with most of the experimental
431
data. However,
the &2:)
et al. “) seems to be overestimated.
3; state: Our theoretical lower than for the other
value, although larger than the g.s. one, is significantly states. This relative reduction is confirmed by all the
experimental groups. We assume that, like for the 0: state, the smallness of the 3; rms radius leads to a e’, value too small with respect to the experimental data. A calculation performed with b = 1.751 fm gives (r2)“2 = 2.79 fm and 0”, = 0.012 in agreement with the data of refs. 26-28). The result of ref. 29) is significantly larger. 1; state: The GCM value 0”, = 0.09 obtained for both oscillator parameters agrees nicely with the results of refs. 26,28)and, to a lesser extent, with the data of refs. 6*30). Since this weakly-bound state plays an important role in the radiative-capture process, several theoretical works have been devoted to its study. A shell-model calculation of Stephenson 3’) leads to 6’,( 1;) = 0.08 f 0.04, in nice agreement with our result. From an R-matrix analysis of the ‘*C( CX,a)12C and 12C( (Y,-y)160 reactions and 16N P-decay, Barker ‘) proposes 13’,(1;) = 0.07 - 0.13. An hybrid R-matrix analysis of Koonin et al. “) gives 13’,(1;) = 0.1 S?z:li, in agreement with previous estimates. However, Langanke and Koonin “) have recently obtained 0’,( 1;) = 1.2 x lop3 and 3.1 X 10e3 by fitting two sets of experimental data [ref. ‘) and ref. 6), respectively]. One must notice that the R-matrix analyses of refs. 7,8,“) involve a non-vanishing background Ri” [see (25)], leading to ambiguities in the comparison with the experimental reduced widths. In our calculation, the ratio of y* and & may reach a factor 4.
4. The ?(a, 4.1. E2 CAPTURE
TO THE
y0J)16O radiative capture reactions
GROUND
STATE
The astrophysical S-factor (S = aE exp (27rv), where n is the Sommerfeld parameter) obtained from (2) is depicted in fig. 1 for a = 6.4, 7.2 and 8.0 fm. The curves corresponding to a = 7.2 and 8.0 fm are identical, as expected from a MRM calculation. On the contrary, the use of a = 6.4 fm (dashed line) leads to significantly different values. This confirms that an accurate MRM calculation cannot be performed with radii smaller than 7 fm. Using (17) and (18), the microscopic E2 S-factor is decomposed into its resonant part, due to the tail of the 2: bound state, and its non-resonant one, for radii a equal to 7.2 and 8.0 fm. From fig. 1, several comments can be made: (i) The Breit-Wigner tail of the 2: bound state dominates the E2 capture cross section at stellar energies. (ii) As announced in subsect. 2.2, the decomposition (17) depends on the MRM radius a, while the global result is nearly insensitive to this radius if it is correctly chosen [see ref. ‘*> for details].
438
F? Descouvemont
I
et al. / 12C(cr, y)“jO
2
3
EfMeV)
Fig. 1. (a) Microscopic E2 S-factor for capture to the I60 ground state (solid line: a = 7.2 and 8.0 fm; dash-dotted line: a = 6.4 fm). (b) Resonant part of S,, [first term of (18)] for a = 7.2 fm (solid line) and a = 8.0 fm (dashed line). (c) Same as (b) for the non-resonant part [second term of (18)].
(iii) The interference between both contributions is found constructive at all energies for a = 8.0 fm, but destructive at energies lower than 0.5 MeV for a = 7.2 fm. Hence the constructive or destructive nature of this interference has no absolute meaning. As a MRM calculation is not valid for radii below 7 fm, we do not present the decomposition for such values. Because of the disagreement between the theoretical and experimental r,.(2:+ 0:) (see subsect. 3.2), the E2 S-factor presented in fig. 1 is underestimated at the vicinity of the threshold. For the sake of providing a reliable estimate, we have multiplied the first term of the r.h.s. of (18) by 1.9, so that the y-width in (19) reproduces the experimental value. This correcting factor is necessary to compensate the lack of flexibility of the present GCM wave function. Introduction of other configurations in the GCM basis space would probably reduce the importance of this factor. The capture cross section obtained in such a way is presented in fig. 2 for a MRM radius equal to 7.2 fm. The dashed line corresponds to a = 8.0 fm. The stability with
l? Descouvemont
et al. / “C(ct,
2
1
y)160
439
3
EiMeV)
Fig. 2. (a) Renormalized E2 S-factor (see text) for capture to the I60 ground state (solid line: a = 7.2 fm; dashed line: a = 8.0 fm). (b) Experimental fit proposed by Kettner et al. 6, for the E2 capture. (c) Semi-microscopic E2 calculation of Langanke and Koonin I’).
respect to a is lost but the difference between both results remains small at stellar energies. At E = 0.3 MeV, we find S&0.3) = 0.09 MeV * b. Notice that the E2 S-factor can be accurately parametrized by formula (19) using the parameters given in table 1 for a = 7.2 fm. We have also plotted by Kettner
in fig. 2 the E2 part of the S-factor
et al. “) from an analysis
has an energy
dependence
similar
of their experimental
parametrization data. Their
obtained E2 S-factor
to our result, but is a factor of 2.5 larger. At first
sight this difference is related to the &2:) employed by these authors which is 10 times larger than our value and more than 5 times larger than the other experimental results (see table 4). One should therefore expect from (19) a similar ratio for the capture cross section near threshold since we have used the experimental r-width for the 2: state. To make a deeper connection with the parametrization of Kettner et al, we have fitted a Breit-Wigner formula using e”, (5.4) given in table 4. We have not changed the energy shift 1” and we have adjusted the shift parameters A:+ and (dA*+/dE), in order to reproduce our E2 S-factor from 0 to 1.5 MeV. The = +0.35. Taking all the factors values obtained are A:* - 0.14 MeV and (dA*+/dE), into account this leads in (19) to a denominator (E + 0.129 - 0.35 E)* for the resonant capture cross section. A similar expression can be obtained in the fit of Kettner et al. by linearizing the low-energy dependence of S,( kii) which is a fair approximation. With all the factors introduced in ref. 6), this leads to a denominator (E +0.214 +
440
l? Descouvemont
ef al. /
“C(a, yJL60
Both denominators differ by a factor 4 at E = 0.3 MeV reducing the discrepancy between the capture cross sections. The main difference between both expressions arises from the change in the sign of the derivative of the energy shift. The negative derivative obtained by Kettner et al. is due to the properties of the Coulomb wave functions and the absence of background I?;*. In our calculation, the non-negligible effect of the background, already partly introduced in &, leads to a positive derivative. This result indicates once more how difficult it is to compare different R-matrix analyses. The absence of background and the choice of too small a channel radius in the analysis of Kettner et al. lead to reduced a-widths which are not directly comparable with microscopic results. A detailed comparison with the theoretical study of Langanke and Koonin “) is not possible, since these authors do not separate the resonant and direct E2 captures. The small value SE2(0.3) = 0.0054 MeV +b can be understood if one assumes a destructive interference of the resonant and non-resonant parts (supposed to be of similar importance) of the E2 capture cross section. Since Langanke and Koonin do not give their 82,(27) value, it is not possible to evaluate their resonant contribution. Anyhow, we think that the inconsistency discussed in subsect. 3.2 may strongly affect the values of the astrophysical factor in their model. 0.42E)*.
4.2. E2 CAPTURE
TO THE 2; STATE
The
E2 S-factor towards the 2: excited state of I60 is presented in fig. 3b, without any normalization factor. The astrophysical factor is almost constant below 2.5 MeV and is dominated by the 4: resonance between 2.5 and 3.5 MeV. Our results are too small by a factor of two with respect to the experimental data of Kettner et aL “). Tire a- and r-widths have been deduced from the Breit-Wigner shape of the peak around the 4: resonance. The a-width of the 4: state is equal to 30 keV, in nice agreement with experiment, but the theoretical r-width is smaller than the experimental value (see subsect. 3.2). The S-factor extrapolated by Kettner et al. at E = 0.3 MeV for the ‘*C(cu, y3)160 reaction is S(0.3) = 12* 2 keV * b, while our value is 3.3 keV - b. If one takes into account the factor two mentioned above, the remaining difference is probably due to the &(2:) values, ,which influence the direct capture towards the 2: state% Anyhow, our theoretical calculation and the experimental estimate of ref. 6, show that the capture cross section is significantly smaller to the 2: excited state than to the ground state. 4.3. El CAPTURE The
TO THE GROUND
STATE
much-too-small r-width that we have obtained for the 1; state (see subsect. 3.2) leads to a large underestimation of the El capture cross section. This underestimation is not particular to our treatment of the El multipolarity. Using a different
P. Descouuemont et al. / “C(CY,?)I60
441
b)
1
2
3
EIMeV)
I
2
3
Fig. 3. (a) E2 [see fig. 2(a)] and El (see text) S-factors for capture to the I60 ground state. The experimental data are from ref. 5, (full circles) and ref. 6, (open circles). (b) E2 S-factor for capture to the 2: state. The experimental data are from ref. 6).
approach, leading also to non-vanishing El transitions, Koonin et al. 8, and Langanke and Koonin I’) have been obliged to scale the direct El cross section by a very large effective dipole strength. In order to compare our results with phenomenological analyses, we have calculated the resonant part of the El cross section by taking the experimental r,(l;) value and the theoretical reduced a-width and energy shifts given in table 3. The El S-factor obtained by using (19) with a = 7.2 fm is depicted in fig. 3a. Note that, up to 1 MeV, the same curve can be fairly reproduced with (19) and d = 5.4 fm. To this end, one must use 0’,( 1;) given in table 4, and the energy shift (22) calculated with A;-=0.16 MeV and (dA’-/dE), =0.35. At 0.3 MeV, we find &,(0.3)= 0.30 MeV * b. Langanke and Koonin ‘I) have obtained a very similar value by fitting the Miinster data (SE, = 0.34) while the value deduced from their fit of the Caltech data is a factor of two smaller. The result obtained by Kettner et al. “) (SE, = 0.25) is in nice agreement with our value. Since these recent values are larger than all previous estimates 5*7-1o),th e reaction rate usually used in astrophysics ‘) should be reconsidered. Our S,, value estimated at 0.3 MeV is probably less reliable than SEz since in this case, the direct and resonant contributions are included. However, with the reason-
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able assumption that the resonant part of SE, dominates the capture cross section at stellar energies, our estimation, based on microscopic R-matrix parameters, is probably not far from the correct value. The influence of the broad I; resonance at 9.63 MeV, which cannot be studied in our model, is most likely small at low energies [see fig. 14 of ref. “)I.
5. Conclusion The generator coordinate method provides a unified description of bound and scattering states. Using this property, we have shown that the influence of bound states on capture cross sections is automatically included in the MRM. This influence is of course also present in other variants of the GCM, but the similarity between the MRM and conventional R-matrix analyses enables one to go deeper into the comparison between microscopic and phenomenological results. Starting from a two-body nucleon-nucleon interaction, adapted to each partial wave, we have calculated a- and y-widths of the (Y+ ‘*C system without adjustable parameters. Unfortunately, one cannot presently describe correctly the electric properties of all the states of the unified system with such a basic method. The main conclusions which can be drawn from our work concern the general trends of these electric properties and the validity of the a-widths used in phenomenological analyses. It appears clearly that the coherence of the experimental information on the a-widths is not very satisfactory. These widths are fitted on different processes (elastic scattering, direct capture); a background contribution from other states is sometimes included, sometimes not. Moreover, R-matrix analyses are usually performed at much-too-small radii. The use of pure Coulomb wave functions and the neglect of antisymmetrization are questionable for the a + ‘*C system at a radius as small as 5.4 fm. The choice of a larger radius, around 7 fm, would improve the validity of the phenomenological analyses and facilitate the comparison with microscopic calculations. In our study, the only state for which the a-width is experimentally well defined is the 4: state since it is not bound. The very good agreement that we have obtained with the experimental 82,(4;‘) enables us to discriminate between the experimental data for the 0: and 2: states, some of them being much too large. For bound states the experimental determinations of reduced a-widths are so indirect that the values obtained are probably not meaningful for the radiativecapture problem. The E2 S-factor that we have obtained after an empirical correction of the B(E2,2; + 0:) value should be reliable. Its value at stellar energies, although smaller than the value of Kettner et al., is non-negligible and influences the a-capture rate at 0.3 MeV. Our result contradicts the smallness of the E2 S-factor of Langanke and Koonin, which could be due to an inconsistency in their semi-microscopic model. The problem raised by E 1 transitions is more complicated. The correct order of magnitude of the El transition probabilities cannot be obtained with T = 0 wave
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functions by relaxing the long-wavelength approximation, like for the (Y+ I60 system. We have therefore calculated the resonant part of the El S-factor by using the experimental r-width and the MRM reduced width and energy shift. In this way, we have taken the background into account but not the effect of the 1; resonance. The value obtained at 0.3 MeV agrees with the results of Kettner et al. and their fit by Langanke and Koonin. These evaluations indicate that the total S-factor at stellar energies is probably significantly larger than usually assumed in astrophysical calculations. Our work can be improved in two ways. The enlargement of the GCM basis will certainly bring the B(E2,2:+ 0:) closer to the experimental value. Also, to obtain agreement for the r-width of the 1; state necessitates the inclusion of isospin impurities in the GCM basis. This introduction is certainly much more complicated. However, both improvements seem necessary to make possible a direct comparison between microscopic results and the experimental data.
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