The 26Al(p, γ)27Si reaction at low stellar temperature

Nuclear Physics ASS6 (1993) North-Holland

123-135

NUCLEAR PHYSICS A

The 26Al(p, y)27Si reaction

at low stellar temperature

A.E. Champagne Department of Physics and Astronomy, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, USA and Triangle Universities Nuclear Laboratory, Duke University, Durham, NC 27706, USA

B.A. Brown Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, MI 48824, USA

R. Sherr Department of Physics, Princeton University, Princeton, NJ 08544, USA

Received 17 August (Revised 9 December

1992 1992)

Abstract: Shell-model calculations have been used to predict the locations of states in “Si which are analogous to well-studied states in *‘Al. From this, we have determined the resonance properties of the known states in “Si near the 26A1 + p threshold. The resulting thermonuclear reaction rate is uncertain by about a factor of ten at low temperatures, but it appears that the Z6Al(p, y)“Si reaction is too slow to destroy a significant amount of Z6AI at these temperatures.

1. Introduction Diffuse

emission

of 1809 keV gamma

rays from the interstellar

medium

of our

galaxy has been attributed to a large equilibrium abundance (~3Ma) of 26Al [refs. I-“)] which has a half-life of 7.2 x lo5 y. Based upon the available thermonuclear reaction rates, there is strong circumstantial evidence which suggests that 26Al is produced at low temperatures which are characteristic of e.g. asymptotic giant branch (AGB) stars or Wolf-Rayet stars ‘). However, no published model calculations for these or other sources succeeds in producing more than a fraction of the observed abundance “). This situation may imply that what is observed is the result of contributions from several different types of sources. Alternately, the theoretical difficulty with producing enough 26Al may be a result of incomplete nuclear-reaction input: Although most of the relevant reaction rates have been accurately determined, Correspondence to: Dr. A.E. Champagne, Carolina

at Chapel

0375-9474/93/%06.00

Hill, Chapel

Department Hill, NC 27599, USA.

@ 1993 - Elsevier

Science

of Physics and Astronomy,

Publishers

B.V. All rights reserved

University

of North

A.E. Champagne et al. / ‘“Al(p, y)27Si

124

the reaction which destroys 26A1, namely 26Al(p, y)“Si, has not been studied in sufficient detail. In equilibrium, 26A1 will exist not only in its ground state, but in a thermal population of excited states ‘). However, at low temperatures, only the ground state will make a significant contribution to the reaction rate. Direct (p, y) 8*9)have been limited to center-of-mass energies in excess of I&,,. = measurements 189 keV and therefore have not probed the energy region of interest for lowtemperature nucleosynthesis. Because the MgAl cycle is not closed 10211)any new, low-energy resonances which would be strong enough to increase the 26A1destruction rate would make it more difficult to produce enough 26A1with existing stellar models. Measurements of the 28Si(3He, a)“Si reaction by Schmalbrock et al.“) and of the 27A1(3He, t)“Si reaction by Wang et al. 13) have located four states between the proton-capture threshold in 27Si and the lowest observed (p, y) resonance (fig. 1).

0.3

7466

4-J =Al+

p

7426 7436

t 47LEVELS

2163

957 761

7+

1

3'

1+

5+

0

27Si Fig. 1. Level structure of *‘Si near the 26A1+ p threshold [from ref. “)I. The locations and widths of the Gamow peak at various temperatures (in units of IO9 K) are shown to the right. Excitation energies are in units of keV. Spins are listed as 2J.

A.E. Champagne

None of these states were observed

et al. / 26Al(p, y)“Si

in a recent measurement

125

14)of the 26A1(3He, d)*‘Si

reaction because of contaminant states in the region of interest. By assigning analog states in 27Si to three known J” =4’ or y’ states in “Al, Wang et al. 13) concluded that only one of these four new states could be formed by s-wave capture. To make these assignments, they used the relative shifts in excitation energy of the four known mirror pairs with J” 3 3’ [as compiled by Endt “)I to constrain the magnitude of the shifts at the higher energies of interest. However, these known mirrors range in excitation energy from 2.2 to 4.5 MeV and therefore do not form an adequate systematic basis to predict mirror pairs at E, = 7.5-8 MeV. To illustrate this point, we show in fig. 2 and table 1 the shifts AE = E,(“Si) - E,(27A1) for established positive-parity mirror pairs. Also displayed in fig. 2 are level shifts computed using a single-particle potential model 16) with a core consisting of 26A1(5t). The calculations show that level shifts in the region of interest may be as large as -650 keV whereas Wang et al. 13) assumed AE = -100 keV. However, the observed shifts lie near or above the Id 5,2 curve in fig. 2. This situation may result from both an absence of 2~,,~ strength as well as to parentage in higher-lying states in 26A1 (which perforce leads to larger binding and higher Coulomb energy). The nature of the level shifts in A = 27 appear to be more complicated than what can be calculated in a naive model. However, using a more realistic calculation of these shifts to predict resonance properties in “Si is probably more reliable than

~2

0

2

Ef7AI)

Fig. 2. A = 27 energy

4

h

x

(MeV)

shifts [defined as AE = Ex(27Si) - E,(“Al)]. The solid curves predictions derived from a single-particle potential model.

are theoretical

126

A.E. Champagne

et al. / ‘“Al(p, TABLE

Known

25,

states E,( *‘Si)

s(AE) ‘)

exp “)

talc b,

(kev)

1,

talc ‘)

talc *)

(kev)

exp (kev)

(kev)

(kev)

844 3680

912 3709

781 3540

825 3650

780 3564

-24

3, 3, 3,

1014 2982 3957

1264 2780 4027

957 (2866) 3804

1048 2951 3919

974 2871 3835

-17 -5 -31

5, 5, 5, 5, 5,

0 2735 4410 4812 5248

0 2708 4139 4939 5320

0 2648 4289 4704 5062

0 2748 4417 4782 5210

-44 2664 4301 4685 5090

44 -16 -12 19 -28

71 7,

2211 4580

2326 4665

2164 4475

2248 4573

2161 447 1

9,

3004

3025

2910

2969

2913

-3

4510

4584

4447

4491

4385

62

1,

11, “) b, ‘) d, and ‘)

1

A = 27 mirror

E,( *‘Al)

y)z7Si

(kev)

1

3 4

Experimental values from ref. I’). Calculated using the Wildenthal sd interaction ‘s). Calculated using Ormand and Brown interaction 19) for the shift between “Si and *‘Al. Corrected values (as described in the text) using 8, = 20 (35), sZ = -55 (15), Ed = -66 (45) y = -12.4 (3.9). Defined as E,(*‘Si),,,En(27Si)_Ic.

simply calculating these properties outright: The former procedure is a straightforward perturbation of the known spectrum of 27Al rather than a construction of 27Si from scratch. To account for the possibility of mixed configurations, we initially used shell-model computations of neutron and proton occupation numbers in the various T = 5, positive-parity states of 27Al to find the fractional numbers of Id,,,, Id 3/2 and 2S1/2 neutrons which are changed into protons for 27Si. These were multiplied by the computed single-particle shifts (in fig. 2) to find the total shift. Because this technique ignores any parentage in excited states of 27Si, a second type of calculation was performed. Fractional parentage coefficients were used to calculate level shifts for states in which a neutron (proton) of the appropriate j was coupled to various states in 26Al. The weighted-average shift was then determined. While preferred theoretically, it was impractical to carry out this sort of calculation with the necessary completeness. Furthermore, both methods ignore shifts caused by nuclear forces. Consequently, we present here a hybrid model from which we ultimately obtain the resonant and nonresonant contributions to the *‘Al(p, y)“Si reaction rate.

127

A. E. Champagne et al. / Z6AI(p, yjz7Si

2. Model for level shifts with the shell-model code The positive-parity kVe~S Of 27A1 were calculated OXBASH “) using USD matrix elements for the W-interaction in the 2sld model ix). The predicted excitation energies for the states in 27Al with known mirror space states

are listed

in table

1 along

with experimental

values.

While

differences

are

occasionally as large as 200 keV, there is no ambiguity in identifying the appropriate model state. The energies of the 27Si analogs (also listed in table I) were obtained using the global empi& isospin-nonconserving interactions of Ormand and Brown i9). The differences between predicted and actual energies are as large as 148 keV, but multiparticle, state-dependent binding-energy effects have not been included at this point. As it is not possible to use the actual ‘6A1 parents with their correspondingly different binding effects, we assume that there will be a shift for each orbital averaged over the many parents that will be roughly constant for the various states. We also include a dependence on binding (excitation) energy which we assume to be I-independent for the sake of simplicity (in fact the 2s dependence is steeper than for the ld orbitals). A final correction to account for mixed configurations was made by first calculating the fraction of 2s,jz, ld,,, and Id,,, orbitals describing the odd neutron in 27Al. For each orbital we ascribe a residual energy shift ezj. The correction factor for the calculated energy shift for state i can be expressed as cS(AE)~ = &,a\“+ ~cx:i)+ ~,a$‘)+ yEi(“7Al) , where ozj is the mixing fraction and y is an empirical parameter. The parameters &zj and y were determined by a least-squares fit to the level shifts for the states listed in table 1 (in table 1 and throughout this article, orbitals will be referred to by 25,, i.e. the nth state with spin J). The r.m.s. deviation between the actual shifts and our final predictions is 26 keV. Because the correction factor 6(AE), does not seem to possess an obvious dependence on E, or J, we used these same parameters to predict the locations of the low-lying 26Al+p resonances. At the higher energies

of interest,

the increased

level density

makes it difficult to

assign a unique model orbital to each level (fig. 3). However, by using the expected accuracy of the predicted excitation energy (about 5200 keV) and the known spin information, for the ambiguous cases it is possible to limit assignments to one of two orbitals (as summarized in table 2). Analog shifts were calculated via the procedure outlined above. To be conservative, each predicted energy shift was assigned a systematic uncertainty of ct50 keV (i.e. double the r.m.s. error at lower excitation energies). The corresponding states in *‘Si whose excitation energies fall within our predicted ranges are listed in table 2. For each state in “Al, there are several proposed, mutually exclusive, analog assignments. In other words, with each known state near the ‘6Al+p threshold, we can associate several possible model configurations (listed in table 3). This ambiguity will naturally lead to some uncertainty in our final calculation of the reaction rate.

A.E. Champagne et al. / ‘6AI(p, y)27Si

128

a-

3

cJ+

(3

-Kw R.774 R3R7

5+

AlR?

3-

9-

5,

-ikus6R1.1n d

7

R97A -

R3117

CI

I?174

5

R1.14

9

,5_gp

23M3~(J~, 7997

9

_zaa7qq5

iulL(9.

,L

7wlo 7RSR

(5 71. 3 "3

77w=

g

+

f3-71

7771

s+

-ggp77~f3q7_,,*

5+

757R 75sn

3

7477 7443

9

75n4E1fi

7.

Q

7467

,9 a

EXPERIMENT SHELL MODEL Fig. 3. Levels in 27A1 which could be analogous to astrophysically interesting states in “Si. On the left side are known states tabulated in ref. 15). The right side shows shell-model predictions for positive-parity states. Spins are denoted by 2J.

3. Calculation

3.1. NARROW

of the reaction

rate

RESONANCES

The thermonuclear reaction rate is the product of the cross section CTand center-ofmass velocity v, averaged over a maxwellian velocity distribution: (au)= For isolated,

narrow

($)“2(kr)P3’2 resonances, (au)=

where w-y is the resonance

1 a(E)

exp (-E/kT)

the rate may be approximated

(-$+)3’2 h*wy exp (-E,/kT) , strength

defined

(2J,-+1) w=(2Ji+1)(2J,+1)’

by Y=

r,r, l-

.

dE. by

et al. / ‘“Al(p,

A. E. Champagne

y)27Si

129

TABLET

Predicted E, ( 27Al) “) (kev) 7578 7660 7677 7679 7721 7798 7806 7948 7997 8037 8043 8065

25 “)

Possible configuration

(7511) (3,5) (779) (357) (9Hll) 9 (5-9) (3,5)

mirror

states

shell-model assignment 5 1 13Y9,

3, 9,,7,, 5 1z 7 10,711 9s lL,9,,9, 9,>9s 71, T712 513.514r 7,,>7,,, 3 10

Possible analog state in “Si ‘)

b,

9,

“) Ref. I’). “) The first value listed is preferred on the basis of excitation ‘) Excitation energies from ref. I’).

7260, 7276, 7324, 7341 7383, 7428, 7436, 7468 7324, 7341, 7383 5‘341, 7383, 7428, 7436, 7468 7468, 7532, 7557 7468, 7532, 7557 7557, 7592, 7652 7741, 783: 7741,783l 7690, 7702, 7789, 7792 7792,7831,7893 7577,7592,7652

energy

Here Jf, Ji and J, refer to the spins of the final state, target and projectile, respectively, and r,, l’,, and r are the proton and gamma-ray partial widths and the total width, respectively. At low energies (temperatures), I’, > r, and the resonance strength reduces to w y = UT,. In the present case, we are dealing with resonances at E,,, = 4,68,93 and 128 keV, all of which qualify as low energy, isolated and narrow. The spin statistical factor is determined by the choice of model orbital. Proton widths were derived from sand d-wave spectroscopic factors calculated with our shell-model wavefunctions (table 3) and the familiar relation r, = Sr,,, , where S is the spectroscopic factor and r,., is the proton width of a (fictitious) pure single-particle state. These widths were obtained from a calculation of the cross section for resonant scattering from a Coulomb plus Woods-Saxon potential. The resulting resonance strengths are listed in table 3 for the four resonances of interest as well as for the lowest observed resonances. The possible g-wave orbitals yield strengths which are negligibly small because of barrier-penetrability considerations. With the exception of the 188 keV resonance, particular choices of orbitals yield strengths that are consistent with experiment. However, the 277 and 367 keV resonances seem to require the same orbital, gs. Thus it is clear that we cannot predict a particular resonance strength, but rather a range of values that includes the measured value. We have not accounted for any possible p-wave resonances amongst the states of interest. The two known negative-parity mirrors display small, positive energy shifts [where AE = E,( 27Si) - E,( 27Al)] which would be expected if these states were predominantly hole states in character. At higher energies, there are three negativeparity states in *‘Al which warrant scrutiny: With a positive energy shift, the 7477 keV

130

A.E. Champagne et al. / 26Al(p, y)“Si TABLE 3 Predicted

W2’Si) “1

EC

m.

(kcv)

(keV)

7468

4

26AI + p resonance

Shell-model 2J “)

configuration

5 12

68

10

9 x 1o-3

oY(lit)

cl.8

“)

x lo-=

“)

~2.3 x lo-l3

“)

0.12 8.5 x 1O-3 0.034, 0.037

2.9 x10-79

11,

0.26, 0.72

2.7 x lo-”

0.12

2.9 x lo-l5

8.5 x 1O-3

2.0 x lo-l6

0.034, 0.037

2.2 x lo-l3 <1.9x

310 710

1592

(eV

7 II

710 71,

93

W(ca1c) ?

9,

9, 7557

S

assignment

I

7532

strengths

0.12

10FOd)

3.4x&

711

8.5 x 1O-3 7 x 10-4, 0.063

128

9s 3 10

7 x 10-4, 0.063

2.4x 1O-8

188

98 3 10

7 x 10-4, 0.063

1.6 x 1O-6

238

98 7 11

2.3 x IO-l3 5.3 x lo-” C5.7 x 1o-6 d), 12.3 x lo+

7652 7702

7 7741

277

7192

328

7831

367

11:

(9,ll)

(9,ll)

‘)

5.5 (9) x 1o-5 8.5 x 1O-3

4.3 x lo+

0.010

5.0 x 1o-6

0.045, 0.13

9,

7 x 10m4, 0.063

9, 7 11

7 x 10-4, 0.021 8.5 x lo-’

0.014

10 (5) x 1o-6 2.9 (3) x 1O-3

1.9x10-3 2.0 x 1o-3 1.9 x 1o-4 2.2 x 1o-4

7,

0.010

9;-

7 x 10-4, 0.063

0.042

9,

7 x 10-4, 0.021

4.5 x 1o-3

0.069 (7)

“) Ref. “). ? ‘)

WYcalc= (Wr&+(Wrp),d. Ref. 9), unless otherwise

noted.

d, Ref. 13). ‘)

Ref. 14).

state (2J” =7-) could be analogous to any of the states of interest. However, a positive shift would be consistent with a hole state rather than a single-particle state and this would imply a weak (p, y) resonance. On the other hand, both the 7900 keV [( 5,7)-l and 7935 keV [ (9, ll))] states could shift down into the region of interest and appear as strong p-wave resonances in *‘Si. Although our calculation ignores this possibility, it does predict that the lowest three resonances are either s- or d-wave and that the I28 keV resonance is either s-wave or negligible. Therefore, unless these states are found to have pathologically large I = 1 spectroscopic factors, any p-wave resonance strengths should fall within the range of our predictions. The thermonuclear reaction rate has been calculated for all possible, self-consistent combinations of model orbitals for the lowest four resonances. In all cases, the

A.E. Champagne 4

keV resonance

was found

et al. / Z6Al(p, y)“7Si

to be insignificant.

131

For the three remaining

resonances

at 68, 93 and 128 keV, respective orbital assignments of 7 ,“, 7,, and 3 ,0 produced a lower bound on the rate. An upper bound was obtained with s-wave resonances at 68 (9,) and 128 (9*) keV. Here the 93 keV resonance was also found to be unimportant. The resulting analytic expressions for these rates are N,(av),,,

= 4.67 x 10m’0T,3’2 exp (-0.789/

Tq)

+3.71 x 10-XT;3’2 exp (-1.079/T,), NA((TU)high= 3.54 x 10-sT,3’2 +3.87x

exp (-0.789/

lo-‘T;“‘exp

Ty)

(-1.485/T,),

where T9is the temperature in units of 10” K. The first term in each limit represents the contribution of the 68 keV resonance. The second terms represent the 93 (low) and 128 keV (high) resonances. In addition, the rate for the higher-energy resonances [from ref. “)I is N,(av)=8.97T~3’2exp(-2.191/Ty)+473T~3~2exp(-3.220/Ty) + 77631 T9exp (-3.9441 These rates are shown

TV).

in fig. 4 and are listed in table 4.

Fig. 4. The Z6Al(p, y)*‘Si reaction rate as a function of temperature. The present results are represented by the solid curves. The shaded area denotes the uncertainty in our calculations at low temperatures. The dashed and dotted curves are the rates tabulated by Caughlan and Fowler “1 and the upper limit of Wang et al. 14),respectively.

1.27

2.99 1.69 7.48

7.47 E - 17

8.89 8.67

4.38

3.50

7.36

1.98 E -09

1.09 E -08

4.58

1.40

“) Ref. ‘).

1.00

9.00 8.85

1.28 E -08

b, Ref. *I).

‘) Ref. 13).

1.66 E+02 2.10 E+02

1.24 E +02 1.70 E+02 1.70 E +02 1.70 E +02

l.l2E+02

6.46 1.24 E+02

8.27 1.24 E +02

8.27

4.84 4.84 8.27

E + 01 E +Ol

4.84

E - 04 E - 04 E - 04 E -04

7.82

1.30 E -01

-08 8.39

E E

1.33 E -08

-02

6.82 E -02

-02 8.47

E E -08

3.21

1.37

1.38

1.30

8.00

7.00

E +Ol E + 01 E + 01

3.02 2.33

2.33 2.33 E +Ol

E +00 E +Ol E +Ol E +Ol

E - 02

1.37

4.27

2.46

1.36

E

f02 2.07 E +02

1.56

1.08 E+02

6.64

3.43

E - 01 E -01 E+OO E +00 E+OO E +Ol E +Ol E +Ol

1.33 E -01

6.75

7.35

6.96 E -04

1.35 E -08

E+OO E + 01 E +Ol E + 01

1.03

4.15 Iz+oo

5.03

4.15

-03

- 04 - 03

-04

-05

-06

- 08 - 07

-09

-11

- 23 - 17 - 14

3.19 E -02

3.48

E - 02 E - 02 E -01 E - 01

E E E E E E E E E E E E E

‘1

1.40E-02

5.64

2.03

6.04

1.28

1.47

3.42

5.44

5.00

2.01

2.22

3.71

1.98

5.37

2.07 E + 00 8.28

E -01 E+OO 4.15 I?+00

4.31 E-03

6.00

6.69 1.76

5.97

1.76

5.97

1.76

8.28 E + 00

5.60

1.24 E -08

E -04 E -04 E - 04 E - 04

8.28 E + 00

4.72

1.14 E -08

1.06 E -03

5.00

E - 04

4.49

4.50 E - 01

E - 01 E - 01 E -01 E - 01 E - 01 E+OO

3.73

1.52

E

7.35 1.50

7.77

-02

5.97

2.68

-08

1.01

E

8.45 E - 09

1.65 E -04

3.50

4.97 E - 05

E - 09

4.00 E - 01

E - 01 E - 01

1.50 E-01

1.50

1.66 E-04

6.39

1.15 E -05

3.00

7.78

7.77

1.30 E -04

5.50 E - 09

5.81 E -06

2.80 E - 01

E -02 E - 02 E -01 E -01 E +00

3.20

E - 02 E - 02 E -01 E - 01 E+OO E+OO

3.75 E -02

3.74

1.23 E -02

3.74

9.63

4.58 E - 09

E - 06

4.02 E - 03 1.67 E -02

1.66 E -02

6.75

6.71

2.73

6.75

1.66 E-02

2.60 E -01

E - 01

E - 04 l.O8E-03

6.71 E -03

3.67

1.18 E -06

2.40

4.38

2.79

4.58 E - 07

2.20 E -01

E - 04 E - 03 E - 03

2.27 2.42

7.39

2.40

E - 04 E - 03 E -03

7.26

2.40 E - 03

- 05 - 05 - 05 - 05

-05

1.32 2.57

1.98

E -09 E - 09 E - 09

1.29 E -09

-01

E - 08 E -07

E

1.57

1.80

7.26

3.53 E -05

1.78 E -04

1.73 E -04

E -04 E - 04

-06

-06

-07

-08

- 13 - 12 - 11 - 10 - 09

1.73

E E E E E E

4.01

E E E E E E E E

5.62

3.01

3.48

2.96

E - 06 E -05

2.10

4.00

5.81

5.88

3.61

1.08

1.31 E -07

1.89 E -08

E - 23 E - 18

1.05 E-15

1.51

4.44

2.83

2.56

8.68

8.91

2.07 E - 09

- 23 - 17 - 14 - 13 - 12 - 10

2.83 E -05

2.56

- 10 - 09 - 08 - 06

E E E E E E

Y

Literature

1.83 E -06

3.93

2.00 E - 01

1.60 E -01

E - 01

1.27

2.47 E - 10

1.20 E - 01

- 11 - 11 - 10 - 10 - 10 E - 08 E - 07

9.96 E - 09

E E E E E

E E E E

E -10 E -09 E - 08 E - 06 E - 05

5.11

1.12

2.97

1.79E-11

3.62

1.00 E -01

9.00

E - 13 E - 12

5.06

1.56 E -09

3.35 E-12

5.63

E - 02 E - 02

8.00

1.64

E ~ 14 1.31 E-11

8.44 1.24E-11

1.52

E - 13

7.26

6.178-14

7.00 E - 02

4.88

1.20

2.58

9.23

high

9.45

E -24 E - 19 E - 16 E - 15

low

Sum

1.90E-13

9.37

1.01 E -13

E - 15

1.82

4.18

4.88

6.00 E - 02

1.20

E - 16 E - 15

1.35 3.45

E - 45 E - 29 E -21

8.40 3.30

1.67

- 23 - 17 - 14 - 13 - 12 - 10

7.27

E E E E E E

1.42 E - 16

2.58

1.64E-18

9.23

1.22 E -24

3.42 E - 19

high

(cm’ SK’ mole-‘)

E, 2 188 keV “)

NA(m)

5.00E-02

E -25 E - 21

low

E,<188keV

4

rates for *6Al(p, y)“Si

4.00 E - 02

E - 02

3.00

non-res

2.00 E - 02

T9

Reaction

TABLE

A. E. Champagne

3.2. NON-RESONANT

et al. / 26Al(p,

y)27Si

133

CAPTURE

At any given temperature, the 26Al(p, r)27Si reaction may proceed via direct capture (DC) and through the tails of high-energy or sub-threshold resonances. In general, these non-resonant contributions are insignificant if there are local resonances at stellar energies, as is the case for “Si. For example, the fact that an s-wave resonance at 4 keV has no significant influence on the reaction rate would seem to imply that resonances at -28 and -36 keV may also be neglected. Consequently, we have ignored any contributions from resonance tails. However, we have calculated the DC rate using the single-particle potential model of Rolfs ‘O), but with realistic, diffuse potentials (rather than square wells) for the incident particle and final bound state. In this model, the bound states are treated as single-particle states. We then multiply the cross sections to individual final states by our calculated spectroscopic factors to obtain the total DC cross section. Because our calculation is limited to positive-parity states, we cannot treat El capture to any negative-parity states. Fortunately, there are few negative-parity bound states in *‘Si and none below E, = 4055 keV. Therefore, we do not expect that the omission of these states will lead to a gross underestimate of the DC cross section. From this calculation, we have obtained the following approximate analytic expression for the DC rate: N,(av),,c This contribution about T9= 0.02.

= 1.53 x 109T~‘.75 exp (-23.19/

to the reaction

rate is also negligible

Ti”)

.

for temperatures

greater than

4. Conclusions At temperatures T,s 0.08, our present rate for the 26Al(p, y)27Si reaction does not differ dramatically from what has been tabulated by Caughlan and Fowler 2’) (fig. 4 and table 4), which was based upon different resonance strengths (note that there are minor differences between their tabulated rate and their analytic expression for the rate). However, we believe that our l-2 order-of-magnitude uncertainty at low temperatures is generally representative of the quality of estimates of this rate. Our upper limit for Tg> 0.03 is l-2 orders of magnitude lower than that of Wang et cdL3), primarily because they assumed proton widths for their resonances which were equivalent to single-particle estimates whereas we made use of our calculated spectroscopic factors in determining widths. For temperatures T,>O.l, we recommend the experimentally derived rate of Vogelaar et al. “) rather than the tabulated 2’) rate. For Tg< 0.1, the large uncertainty in the rate does not necessarily result in an equally large uncertainty in the net abundance of 26A1 because its production rate is comparable to a typical burning lifetime. In other words, the abundance of 26A1 builds up slowly enough so that only a small fraction can be destroyed before burning ceases. We illustrate this situation in a schematic fashion

134

A.E. Champagne et al. / 26Ai(p, -y)27Si

for the case of hot-bottom burning in intermediate-mass AGB stars. In these stars energy is provided by thin hydrogenand helium-burning shells. The latter shell burns in a series of pulses and can produce “5Mg via the sequence 14N(~, ~)‘8F(@‘)‘“O(a, Y)22Ne(a, n)25Mg. Between pulses, we assume that the “Mg will be mixed into the envelope where it will be converted into 26A1 via a (p, y) reaction. As the star ages, the temperature at the base of the envelope will increase and the interpulse period will decrease (a consequence of an increasing core mass). Production of 26Al will occur if this temperature exceeds T9 = 0.04. In fig. 5 we show the net amount of 26Al {expressed as a fraction of the initial 25Mg abundance) produced as a function of time. This calculation was performed at constant temperature and density ( p = 1 g/cm3, mass fraction of hydrogen = 0.7) and therefore ignores the effect of convection in the envelope. However, it illustrates the result of an uncertain ‘(jAI(p, y)27Si rate: For Tg= 0.04, this rate is small compared to the beta-decay rate and hence there is an insignificant unce~ainty in the (small) amount of 26Al produced. Its abundance grows almost linearly with time because the beta-decay rate is much slower than the production rate. At T,=O.M, 26Al is produced more rapidly, but destruction is also more efficient and this causes the abundance curve to flatten out with time. Here the interpulse period is on the order of IO3 yr [ref. “)I after which the unce~ainty in the 26Al abundance is only 14%. However, at T,=O.O8,where the interpulse period drops to a few hundred years,

TIME (lOgsec)

Fig. 5. The amount of 26A1 (expressed as a fraction of the initial as a function of burning time for temperatures T, = 0.04, 0.06 and to upper and lower limits (which correspond to lower and upper reaction), respectively. Constant temperature and

“Mg abundance) converted into “Si 0.08. The dashed and solid lines refer limits on the rate of the “Afjp, y)*‘Si density are assumed.

A. E. Champagne et al. 1 Z6Al(p, y)*‘Si

the final abundance

of 26Al depends

more heavily

135

on the 26Al(p, y) rate (which is

uncertain by a factor of 4). The abundance quickly reaches a maximum at the point where all of the initial “Mg is exhausted. The amount of 26Al surviving past this time depends solely on its destruction rate. When convection is included, the uncertainty in 2”Al production will be less dramatic (simply because the material will spend an appreciable fraction of the burning time at lower temperatures), but will still amount to at least a factor of 2. It appears that the “Al(p, y)27Si reaction rate is low enough to insure that *‘Al will survive at low stellar temperatures. Although we have shown that reasonably accurate predictions of the 26A1abundance may be made despite the large uncertainty in its destruction rate, more accurate results are clearly desirable. Experimentally determined analog assignments would be useful, but an improved 26Al(3He, d)*‘Si measurement is obviously called for. Work in this direction is in progress. We would like to thank M. Betterton and Z.Q. Mao for their help with these computations. Also, we would like to thank B.H. Wildenthal for his calculation of several spectroscopic factors. This work was supported in part by the U.S. Department of Energy (contract #DE-FG05-88ER40442) and in part by the National Science Foundation (contract #PHY90-17077 and #PHY91-04414). References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22)

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