Cross sections of the 21Ne(α, n)24Mg and 22Ne(α, n)25Mg reactions at low energies of astrophysical interest

1

2.A.l

1

Nuclear Physics Not

to be

CROSS SECTIONS

A226 (1974) 493 -5505; @

reproduced

by photoprint

North-Holland

Publishing

Co., Amsterdam

or microfilm without written permission from the publisher

OF THE ‘lNe(a, n)24Mg AND **Ne(a, n)*‘Mg REACTIONS

AT LOW ENERGIES OF ASTROPHYSLCAL H.-B. California

MAK

tt, D. ASHERY

Institute

ttt and C. A. BARNES

of Technology, Received

INTEREST’

9 April

Pasadena,

California,

USA

1974

Abstract: The *‘Ne(cz, n)24Mg reaction has been studied over the energy range E..,. = 1.4-2.57 MeV, by neutron time-of-flight spectroscopy and by the detection of the subsequent y-ray transitions in 24Mg. The astrophysical cross-section factor S(0) was determined and reaction rates were calculated for temperatures up to 5 x IO9 “K. In favorable circumstances, this reaction can provide a significant neutron flux for nucleosynthesis. The absolute normalization of the cross sections obtained in our earlier work on the reaction 2ZNe(a, n)“5Mg has also been verified.

E

NUCLEAR

REACTIONS

21*22Ne(a, n), E < 3 MeV; measured astrophysical S(E). Enriched targets.

a(E).

Deduced

1. Introduction Because of the increasing Coulomb barrier in heavy nuclei, it has been postulated that the nuclei with A 2 60 have been synthesized by neutron capture, either on a long time scale (s-process) or on a short time scale (r-process). Among the various suggestions for the production of the necessary neutron flux, the sequence of reactions, *‘Ne(a, n)24Mg ‘*O(a, n)‘rNe

/T

14N(a, y)18F(efv)‘80<

22Ne(a, n)25Mg

(1)

‘*O(cr, y)22Ne < 22Ne(c(, y)26Mg, has been suggested ‘, ‘). Since the threshold for the reaction “0 ( CL,n)21Ne is 0.70 MeV, this set of reactions is expected to proceed mainly through the ‘*O(cr, y)*‘Ne reaction, and the neutron production would then be mainly from 22Ne(cr, n)25Mg [refs. “*“)I for which the threshold is only 0.48 MeV. Marion and Fowler 5), however, have shown that “Ne can be produced from any 20Ne in the stellar material by the Ne-Na-Mg cycle of reactions in a hydrogen-burning t Supported in part by the National Science Foundation [GP-280271. It Present address: Physics Department, Queen’s University, Kingston, Ontario, Canada. ttt Present address: Physics Department, Tel Aviv University, Ramat Aviv, Tel Aviv, Israel. 493

H. B. MAK

494

et al.

region. Some of the helium produced by hydrogen burning could then produce neutrons by *rNe(cl, n)24Mg in a later helium-burning phase, or convective instabilities between a hydrogen-burning shell and a helium-burning shell could transport *‘Ne to the surface of the helium-burning region to produce neutrons. The cross-section factor for the *rNe(a, n)24Mg reaction is also needed for a detailed study of element synthesis during explosive helium burning “). The *lNe(a, n)24Mg r eaction has been studied previously by Tanner ‘) and, recently, by Haas and Bair “). Our work disagrees strongly with the early work of Tanner, but agrees well with the Haas and Bair work, when the energy-averaged cross sections of the latter work “) are plotted at the center of the (400 keV lab) averaging interval instead of at the upper energy limit of the interval “). There is also satisfactory agreement, for the reaction **Ne(a, n)25Mg, among the cross sections of Ashery 3), those measured in the present work, and those of ref. 4), when the energy-averaged cross sections of ref. “) are plotted 200 keV lab lower in energy.

2. Experimental

procedure

A bunched and chopped, singly charged 4He beam from the ONR-CIT tandem accelerator was used to bombard a neon gas target, enriched to 50.9 % (by number) in *lNe. The gas pressure was measured by a mercury manometer, and was kept at approximately 50 mm of mercury. The gas cell had a nickel entrance foil 0.584 mg . cm-* thick, which was measured by observing the shift of the r9F(p, ay)160 resonance at 873 keV when the foil was inserted in the beam path, and also by determining the energy loss of 2.32 MeV a-particles in the foil with a magnetic spectrom-

2.557 *‘Ne+a

-n

Fig. 1. Energy

level diagram

of 24Mg.

(~1, n) CROSS

SECTIONS

495

eter. The straggling of the a-beam in passing through the foil was measured to be about 46 keV. When combined with the energy loss in the neon gas, this gives a beam energy-resolution of approximately 110 keV. Since, in most astrophysical situations, the thermal distribution of energies averages the yield over sharp resonances, this energy spread is not a serious limitation for astrophysical purposes. The neutron groups, n, and n, , leading to the ground state and the 1.369 MeV state of 24Mg (fig. 1) were detected by means of the neutron time-of-flight technique, using a 12.7 cm diameter by 5.1 cm thick ‘Pilot B’ plastic scintillator. A typical neutron time-of-flight spectrum is shown in fig. 2. The distance from the target to the neutron

6oo_

Neutron

Time-of-Flight

at Em = 2.36

Spectrum

MeV, 8,,,

q

.I

25”

Y 1

400 -

i

p . "0

I>

s

b 200

\ . '.

0

40

80

CHANNEL

120

-

Jl

-1 I '0&-'f ~..v_y.#w.. . I 160 200

NUMBER

Fig. 2. A typical neutron time-of-flight spectrum. The time scale is approximately 1 ns per channel; n,, n, and n3 indicate the neutron groups from the “Ne(a, n)24Mg reaction.

detector was varied from 16 cm to 101 cm in such a way that the no and n1 groups were always clearly resolved from the neutron groups from the reaction “Ne(a, n) 25Mg. The neutron groups leading to the 4.12 MeV and 4.23 MeV states of 24Mg had too low an energy to be detected efficiently by the time-of-flight method. Instead, the y-rays emitted by these levels were observed by a 12.7 cm diameter by 10.2 cm thick NaI(T1) scintillator, placed at 55” with respect to the beam line, and at a distance of 1.4 cm from the target. The full-energy-peak efficiencies of this crystal for detecting the emitted 1.37 MeV and 2.8 MeV y-rays were determined with calibrated 6oCo and “*TI sources. The full-energy-peak efficiency for detecting 4.2 MeV y-rays was determined by detecting the 4.44 MeV y-ray from the ‘Be(a, ny)“C reaction in coincidence with the neutron group populating the 4.44 MeV state in “C.

496

H. B. MAK et al. 24000

I

I

I

I

I

Gamma Rays Associated EC,= 2.10 MeV

_

i $7

16000

I With

I

Beam

I Burst

&$J MeV

MeV

2.8 MeV

I

5

20

60 1

I

24001

I.46

100 1 I Background

140 I Gammo

I Rays

160 I

I

1

MeV

. .

I600

..

2.61 MeV

I

20

I

60

too

CHANNEL

1

140

I

I

180

NUMBER

Fig. 3. A typical y-spectrum from (a) the reaction *‘Ne(cc, n)**Mg and (b) room background. The spectra were obtained by routing the y-ray counts to separate sections of the multichannel analyser memory by two single-channel analysers, one set to bracket the prompt y-rays, and one set to include a known portion of the time-inde~ndent background in the y-ray time-of-fight spectrum.

A pulsed-beam was also used for detecting the y-rays in order to minimize the 1.46 MeV 40K and 2.61 MeV 208Tl backgrounds from natural radioactivity in the walls of the laboratory. Fig. 3 shows a typical y-ray spectrum. The angular distributions of the y-ray transitions were fitted satisfactorily by a sum of Legendre polynomials of the form, da = dSZ -

Ao+A,QzF,(cos

8)+A,Q,P,(cos

@),

t-4

where the Q, are correction factors arising from the finite size of the detector. In the present experimental set-up, A,Q, is negligible (IA4Q4/A01 < 0.001) for all of the y-ray transitions, and, since the detector was placed at 59, the second term could also be ignored. Thus, A0 is directly proportional to the measured yield at 55”.

(cc, n) CROSS

497

SECTIONS

3. Results and discussion Twenty-two angular distributions of the unresolved n, + n, groups were obtained, covering the energy range EC.,,. = 1.5 - 2.57 MeV in 60 keV steps. At most bombarding energies, differential cross sections for the neutrons were measured at O_,, = 27”, 58”, 78”, 99”, 119” and 138”. The angular distributions were fitted with sums of Legendre polynomials of the forms

(3) ~

=

i~oBi Pi(COSe).

(4)

In no case was the ratio A,/B, different from unity by more than 6 %, and the average of A,, and B, was used in calculating the cross sections. The uncertainties in the absolute values of the cross sections for the no and n1 groups were estimated I

I

I

I

I

I

I

I

I

I

I I 14

18

22

I

26

E,,(MeV) Fig. 4. Excitation function for the reaction *lNe(cc, n)24Mg. The solid curve is a fit of the smoothly rising part of the yield curve (i.e., omitting the unresolved cluster of resonances near EC,,. = 1.9 MeV) to the expression for 0 discussed in the text.

498

H. B. MAK et al.

were to be 1.5%, except for the two lowest energy points, where the uncertainties estimated to be 20 %. The cross sections for populating the 4.12 MeV and 4.23 MeV states were found to be an order of magnitude smaller than those populating the ground and 1.37 MeV states, and their unce~ainties are estimated to be 15 y0-20 %. Fig. 4 shows the excitation function for the “Ne(ar, nfz4Mg reaction, summed over the ground state and first three excited states. The present experimental results can be used to obtain the cross section factor S(E), defined by the relation ‘)

S(E) (_ (g]

a(E) =E

exp

~

where Eo is the “Gamow energy” The smoothly rising part of the yield curve was parametrized by the expression given by Fowler and Hoyle lo), fmax i? = 1 o,, 1=0

(6)

CL = 4rc~izo(2Z-t l)Prj?I(T’,/T)l.

(7)

where

The quantity PI results from averaging over the shape resonances of the compound nucIeus, and is expected to be of order unity ’ “). The penetration factor P, is defined here as P, = (@-l-G:)-‘, where Fl and G, are the regular and irregular Coulomb wave functions, respectively. The factor P, was evaluated for a reaction radius of 5.8 fm as suggested by Michaud et al. r’). In the energy range of the present experiment, experiment, (r,/r), M 1 for all I-values of interest. The quantity iz, is defined by A, = ~(2~~~)i where M is the reduced mass of the system and Vo = 75 MeV, as suggested by ~ichaud et al. i ‘). S ince the ii, are negligibly small for I >= 4, only Ivalues 6 3 were included in the sum. If we make the additional approximation that ~~(~“~~)~ is the same for all /-values in the sum, and insert the appropriate numerical values, we obtain

where E is in MeV. This expression was used to fit the smoothly rising yield curve, extrapolated under the cluster of unresolved resonances near EC_,. = 1.9 MeV, by varying the parameter fir,/r. For minimum x2, /3r,/r = 1.16. The cross section factor S(0) was then found, by extrapolating to zero energy, to be 2.4 x lo9 MeV * b. This value is in good agreement with a theoretical estimate by Reeves “), who obtained a value of S(0) = 2.45 x IO9 MeV . b (for a slightly smaller radius than we have used here). The yield curve obtained in the present measurements disagrees strongly in both

(CX,n) CROSS

SECTIONS

499

magnitude and energy dependence with that obtained by Tanner ‘). It seems possible that a considerable part of the yield in this early work was from the contaminant reaction 13C(~, n)r60. The present ‘lNe(cc, n)24Mg measurements agree well with the results of Haas and Bair “) when the energy-averaged cross sections of ref. “) are plotted at the energy corresponding to the center of the averaging interval. [In ref. 4), they are plotted at an energy corresponding to the top of the interval “).I The agreement between the two sets of data is further improved if the effective energies discussed in the appendix are used instead of the mean energies. The excitation functions from both sets of data agree well in overall energy dependence with that to be expected from the kind of parameterization given by eq. (8), except for the two lowest points of ref. “) which lie above such an extrapolation, and the five highest points of ref. “) which lie slightly below the extrapolation. Fowler, Caughlan and Zimmerman l3 ) have found that a good fit to both sets of data can be found by parameterizating the quantity S(E), of eq. (5), as

S(E) = S(0) exp [-cTE-/IE~]~

(9)

TABLE 1 Reaction

rate for 2’Ne(cc, n)24Mg

at various Reaction

TS

experimentala)

0.02 0.04 0.06 0.08 0.10 0.15 0.20 0.30 0.40 0.50 0.60 0.80 1.00 1.50 2.00 3.00 4.00 5.00

5.93 x 10-55

1.09 2.71 1.25 2.56 6.50 1.63 2.84 1.29 9.75 2.58 2.94 8.63 2.13 5.18 1.82 1.35 4.86

x x x x x x x x x x x x x x x x x

10-39 IO-= 1O-27 lo-= IO-r9 lo-I5 10-r’ 10-a 10-7 lo- 5 10-a 1O-2 10’ 10’ lo4 105 lo5

temperatures rate (cm” . set-

in units of set-’

per mole/cm3

1 . mole-‘)

corrected for contributions by excited states of Z’Ne b,

x lo-55 1.09 x 10-39 2.72x10-= 1.25x10-27 2.56 x 1O-24 6.49 x lo-r9 1.63 x 1O-‘5 2.86 x lo-r1 1.32 x IO-’ 1.01 x10-6 2.70 x 10-s 3.20 x 1O-3 9.70 x 10-z 2.51 x 10’ 6.49 x 10’ 2.68 x IO4 2.42 x lo5 1.09 x 106 5.98

The temperature Ts is in units of 10°K. “) Least squares adjustment of experimental data yields S = 3.42 x lo9 exp (-0.518E-0.143E’). Includes resonance effects near 1.90 MeV. b, Least squares adjustment to S = 3.48 x lo9 exp (-0.624E) and normalized partition function = 1+2.5 exp (-6.33/T,). Includes resonance effects near 1.90 MeV. See ref. r3).

H. B. MAK et al.

500

S(0) = 3.42~ lo9 MeV . b, t( = 0.518 MeV-I, and /I = 0.143 MeVm2. The value of CLwas constrained to have the same value as obtained from a statistical model calculation [carried out without the assumption that j?r,/r is the same for all I-values]. The change in S(0) from the value quoted earlier in the paper is typical of the kind of change obtained from various different parameterizations of the data, and may indicate the level of uncertainty in this quantity. The astrophysical rates derived from this parametrization are given in table 1. The total neutron cross section of magnesium shows strong resonances near E,, = 2.65 MeV 14), which is in the stellar thermal region for 21Ne(cr, n)24Mg. Although the cross sections which we have extrapolated to obtain S(0) are averages over compound-nucleus levels, the value of S(0) obtained in this way might seriously underestimate the astrophysical cross section if one of the compound nucleus levels in the thermal region should have an anomalously large g-particle width. Even with the values obtained here for S(O), 2.4 to 3.4 x lo9 MeV . b, it appears that the “Ne (c(, n)24Mg reaction could be a significant source of neutrons in the appropriate astrophysical environment ‘). with

TABLE2

Reaction rate for 22Ne(c(,n)25Mg at various temperatures in units of set-’ Reaction rate (cm3. see-’ . mole-‘)

7-9

experimental

0.04 0.06 0.08 0.10 0.15 0.20 0.30 0.40 0.50 0.60 0.80 1.00 1.50 2.00 3.00 4.00 5.00

per mole/cm3

“)

2.01 X lo-” 9.32 X 1O-9 7.13x10-7 1.91 x lo- 5 2.19~10-~ 6.09 x 1O-2 1.22 x 10 3.03 x 102 1.33 x 104 1.16 x lo5 4.65 x 10’

corrected for contributions by excited states of 22Ne b, 4.36 X 1O-6s 6.50 x 1O-45 7.93 x 10-35 8.94 x IO-=’ 1.96 x 1O-2o 5.31 x 10-16 1.86 x lo-” 9.11 x10-9 7.12x10-’ 1.94 x 10-s 2.29 x 1O-3 6.59 x lo-’ 1.46 x 10 4.03 x 102 2.19 x IO4 2.27 x 105 1.06 x lo6

The temperature T9 is in units of 109K. “) Least squares adjustment of experimental data yields S = 2.88 X lo9 exp (-0.533B-0.116E’). No correction for threshold effects. “) Least squares adjustment to S = 2.94x lo9 exp (-0.6366). Corrected for threshold effects. See ref. 13).

(a, n) CROSS

501

SECTIONS

4. Cross section of the ‘*Ne(a, n)25Mg reaction Since the neutrons from the reaction ‘*Ne(c(, n)*‘Mg also showed up prominently in the time-of-flight spectra of the present work, because of the large amount of *‘Ne in our gas target, it was possible to compare the cross sections for this reaction with those measured earlier by Ashery “). In addition, several runs were made with “Ne of high iso t o p’IC p urity in the gas target, at energies where the cross sections given by Ashery are not strongly energy dependent, to check the overall cross section scale. As in the work described above, and also in the work described in ref. 3), the energy resolution was about 100 keV. In general, these new measurements of the **Ne(Cr, n)“Mg cross sections agreed within about 15 % with those of Ashery “1. The new “Nefcr, nJz5Mg measurements and those of Ashery agree quite we11 with those of ref. 4>, when the energy-averaged cross sections of ref, “> are attributed to the energy at the center of the averaging interval. As in the case of the ‘lNe(a, n) 24Mg reaction, the Caltech cross sections are slightly higher than the Oak Ridge cross sections at the higher energies (E,.,, 2 3 MeV), and somewhat lower at the lower energies (E % 1.8 MeV). A parameterization of the combined data, of the form S(E) = S(0) exp (-c&--BE”), yields So = 2.88 x IO9 MeV . b, 01= 0.533 MeV-I, and /3 = 0.116 MeVe2 [ref. 13)]. The rates calculated from this parameterization are given in table 2. The authors would like to thank Prof. W. A. Fowler for many helpful discussions, and B. A. Zimmerman for assistance with the calculations. The help of F. Mann during some of the runs is appreciated. One of us (C.A.B.) would like to thank the Nordisk Institut for Theoretisk Atomfysik for the award of a guest professorship, during which the preparation of this paper was completed.

Appendix ESTIMATION SECTION

OF THE EFFECTIVE

ENERGY

FOR AN ENERGY-AVERAGED

CROSS

1x1 the measurement of nuclear reaction cross sections for astrophysical purposes, it is usually necessary to employ targets of considerable thickness in order to extend the measurements to low energies. The use of such a “semi-thick target” has the desirable property, from the astrophysical point of view, of averaging the cross section over closely spaced compound-nucleus levels, just as the thermal distribution of velocities averages the cross section over narrow levels in astrophysical environments. Some care is necessary, however, in assigning effective energies to the averaged cross sections obtained in this way, because the excitation function (curve of cross section versus energy) is usually a steep function of bombarding energy and becomes rapidly steeper as the energy is lowered. Any error in the effective energy produces an amplified error in the extrapolated cross section at astrophysical (tow) energies. Similar care is

502

H. B. MAK et al.

necessary when thin-target cross sections are averaged over an appreciable energy interval, to obtain “smoothed” cross sections, which are then to be used for extrapolation to low energies. An approximate method for determining the effective energy, E,,, , for such energy-averaged cross sections is discussed below as case 1. A closely related problem in determining E,,, arises, even for thin targets, if the incident particles have an appreciable spread in energy, which situation we discuss below as case 2. Although we do not consider in detail the situation in which energy averaging arises both from the use of “semi-thick” targets and from a distribution in beam energy, for reasons of brevity, the techniques described below may be extended to cover this situation without difficulty. Case 1; Energy averaging produced by a “semi-thick” target. For a target of appreciable thickness, the number of reactions per incident particle of laboratory energy Eb, is Y(E,)

= n/;_AE@)(dE,dx)-l b

dE,

(A-1)

where n is the number of target nuclei per unit volume, dE/dx is the energy loss per unit path length in the target material, and o(E) is the (energy-dependent) reaction cross section. (For consistency, all energies are in the laboratory system.) For the present purposes, we shall assume that a(E) is a smooth average over possible “finestructure” compound nuclear resonances, if these are present in the reaction. For simplicity, we also assume that the thickness of the target AE (in energy units) is small enough that dE/dx can be treated as a constant and evaluated at Eb-$AE with little error. (Strictly speaking, it is the functionf(E) = c(E)(dE/dx)-l which is being averaged over the interval AE, and the analyses given below remain valid, with the substitution off(E) for o(E), if it is not permissible to take (dE/dx)-’ outside of the integral.) We wish to determine an effective energy, Eelf , defined by 4%)

= <4E))

= (l/AE)~;_$E)dE. b

(A.2)

If o(E) is a constant or varies linearly with energy over the interval AE, we obtain immediately the well-known result, E,,, = E,,-3AE. More generally, if a(E) is a known function of energy, E,,, may be determined from eq. (A.2), without difficulty, at least in principle. Since the energy dependence of a(E) is usually not known apriori, we must choose a form for c(E), guided by the experimental data and whatever theoretical knowledge is at our disposal. In the region near or above the top of the Coulomb barrier, where a(E) varies rather slowly with energy, we might choose to approximate o(E) by a simple polynomial in Eand require that o(E), integrated over AE, should fit the experimental data. However, it is usually not difficult to determine the correct form for o(E) in this region of the excitation function, since the yields are usually large

(CL, n) CROSS SECTIONS

enough

to permit

the use of rather

thin

targets,

503

for which

the approximation

.F“eff = Eb - fAE is adequate. As the energy is lowered below the top of the Coulomb barrier, we quickly reach a situation in which the partial width for the incident particles, rin, is small compared with the width, r, of the compound-nucleus levels. The dominant energy dependence in c(E) will then be that of the Coulomb barrier penetration factor for the ingoing particles. On this simple model (the black nucleus model), c(E) K E-*z(21+1)(F:+G:)-5 1 for spinless

interacting

nuclei,

(A.3)

or, more generally,

v(E) a; E-“C(25f1)x J

c (F;+G,2)+. s 1=/J-sj

(A.9

In these expressions, I is the orbital angular momentum, J is the compound nuclear angular momentum, S is the incident channel spin, and F, and G, are the regular and irregular Coulomb functions, respectively evaluated at the nuclear radius R = r,(At+At). Admittedly, there is some uncertainty in the choice of ro, which influences the over-all steepness of the excitation function. A simple technique, which we have found useful for determining E,, on several occasions, is to plot log a(E), as given by eq. (A.3) or eq. (A.4), versus laboratory bombarding energy, and then to approximate the local energy dependence of o(E) as an exponential characterized by an e-folding energy, E,, which varies slowly with bombarding energy. With this approximation,

we obtain

from eq. (A.2), EefF = E,-f(x)AE

In [x(1 -e-x>-1]),

= E,-AEjx-’ TABLE Al The function f(x) x

0.05 0.10

0.50 1.00 2.00 3.00 4.00 5.00 7.00 10.00

f(x) 0.498

0.496 0.479 0.459 0.419 0.383 0.351 0.323 0.278 0.230

GW

H. B. MAK et al.

504

where x = AE/E,. Some representative values of j(x) are given in table Al. From as x -+ 0, as one might expect. table Al, we see immediately that E;,,, -+ E,-3AE this expression is already within 1 o/0of the full expression for For x = l,f(x)-‘lnx: f(x) at x = 5. As a numerical example, let us consider the reaction ‘lNe(a, n)24Mg. For r. = 1.4, the expected e-folding energy (in lab units) varies form about 85 keV at E = 1.5 MeV to approximately 660 keV at E = 4 MeV. For an energy averaging interval of 400 keV, we find that E,,, is 68 keV higher than the center of the interval, at E = 1.5 MeV, but only IO keV higher at E = 4 MeV. These shifts have a significant effect on the overall steepness of CT(E) and, consequently, on the choice of pg. If on the other hand, r. is considered to be well determined from other experiments, the procedure outlined here could be used to search for other kinds of energy dependence in o(E), or systematic errors in the experiment. Case 2: Thin target. Energy averaging produced by incident energy spread. If the beam of particles incident on the target material has a spread in energy comparable with the energy interval in which the cross section changes by a significant fractional amount, we again measure an energy-averaged cross section. We shall assume, as in case 1, that dE/dx changes by a small enough fraction, over the interval in energy spanned by the incident beam, that it may be removed from the integral, and we are left with an average of a(E). (Again, it is not difficult to include the (dE/dx)-r in the averaging procedure, if necessary.) We have

@%td= (a(E)> = j n(E)@+%

(A.7)

where the integral is over the incident energy spread and we have assumed that [n(E)dE = 1. In n(E) is a constant over some interval AE, and zero elsewhere, we have the same situation as case 1. A more interesting situation arises when the incident beam particles must pass through an appreciable layer of inert material, before bombarding the target, such as the foil window of a gas target. In this case n(E) is the energystraggling distribution for the given window. We consider explicitly only the case where the inert material is thick enough that the straggled energy distribution can be approximated with good accuracy as a Gaussian function. Thus

n(E)dE = J2&, ~

exp [-(E -&,,)*/2~*]dE,

(A.9

where E,,, is the mean energy of the beam incident on the target material and K is the standard deviation of the distribution. As is well known, the full width at half maximum (FWHM) of such a distribution is 2.35 K. We now assume as, in case 1, that the cross section can be approximated locally as an exponential function,

o(E) = 4&J expC(E- %)I-Kl,

(A.9

(EL,n) CROSS SECTIONS

505

where E, is the (local) e-folding energy. If we now assume that K < Em, we obtain from eq. (A.7), Eerf = E, f K2j2E,. fA.lO) As a numerical exampie, we consider values of E’, of 8.5 keV and 660 keV, as in Case 1, and a straggled ~~~uss~a~~ energy distribution with FWHM = 46 keV (K = 19.6 keV). We find from eq. (A.10) that I& is shifted upwards from E, by 2.26 keV for E, = 85 keV, and by 0.29 keV for E, = 660 keV. These energy shifts are small compared with the uncertainty in the energy loss in the foil, and have been ignored in the present experiment. A small correction has been made in the present experiment for the thickness of the gas target (LIE e 80 keV). The determination of ESff when we have energy averaging arising both from the use of a semi-thick target and from a spread in beam energy can be carried out by using the e-folding technique, by first integrating ~~~){d~/d~)-’ over the target thickness AE, as in case 1, and then integrating over the distribution in incident energy, as in case 2. References 1) E. M. Burbidge, G. R. Burbidge, W. A. Fowler and F. Hoyle, Rev. Mod. Phys. 29 (1957) 547 2) A. 0. W. Cameron, Astronom. J. 65 (1960) 485 3) D. Ashery, Nucl. Phys. Al36 (1969) 481 4) F. X. Haas and J. K. Bair, Pbys. Rev. C7 (1973) 2432 5) J, B. Marion and W. A. Fowler, Astrophys. J. 125(1957) 221 6) R. G. Couch and W. D. Arnett, Astrophys. J. 178 (1972) 771 7) N. W. Tanner, Nuci. Phys, 61 (1965) 297 X> J. K. Bair, private comnlunication 9) W. A. Fowler, G. R. Caughian and B. A. Zimmerman~ Amt. Rev. Astron. and Astrophys. 5 (1967) 525 IO) W. A. Fowler and F. Hoyle, Astrophys. J. Suppi. no. 91,9 ($964) 251 11) G. Michaud, L. Scherk and E. Vogt, Fhys. Rev. C1 (1970) 864 12) H. Reeves, Astrophys. J, I46 (1966) 447 13) W. A. Fowler, G. R. Caugblan and 3. A. Zimmerman, Ann. Rev. Astron. and Astrophys., to be published 14) J. R. Stein, D. W. Goldberg, B. A. Maguna and R. Wiener-Chasman, Neutron cross sections, BNL 325, 2nd ed., Suppf. no. 2 (Brookhaven National Laboratory, Associated Universities, Inc. 1964)