Cross sections and thermonuclear reaction rates for 51(α, n)54Mn and 51(α, p)54Cr

NUCLEAR PHYSICS A

Nuclear Physics A551 ( 1993) 158-l 72 Nosh-Holland

Cross sections and thermonuclear reaction rates for “V( cy,n)54Mn and 51V(a, p)s4Cr * V.Y. Hansper,

A.J. Morton,

S.G. Tims, C.I.W.

Tingwell,

A.F. Scott and D.G. Sargood

Received 15 June 1992 (Revised 11 August 1992)

Abstract. The cross sections of “V(LU, n?Mn and “V(a, p)s4Cr have been measured in the bombardingenergy ranges of 2.00-9.49 MeV and 5.755-9.590 MeV, respectively. The cross sections, together with cross sections of %Yr(p, y)“Mn and s4Cr(p, n)s4Mn taken from the literature, have been compared with the results of statistical-model calculations. Thermonuclear reaction rates at temperatures appropriate to silicon burning in massive stars and supernovae have been calculated for both reactions.

E

NUCLEAR REACTIONS “‘V(a, n), E = 2-9.49 MeV; “V((Y, p), E = 5.755-9.59 IWeV; measured neutron, proton yields; deduced o(E), the~onucIear reaction rates. Statisticalmodel analysis. Natural targets, activation technique.

1. Introduction Calculations

of nucleosynthesis

in evolving

stars and supernovae

rely heavily

on

nuclear-reaction cross sections computed by means of the statistical model of nuclear reactions. If the statistical-model calculations are to be reliable, they must depend on the use of globally

specified

optical-model

parameters

in the particle

channels

which have been chosen to lead to satisfactory agreement with empirical cross sections for all reactions which are accessible to laboratory measurement. In a previous paper ‘) we drew attention to the key role played in this regard by sets of reactions which involve all four of the proton, neutron, cr-particle and y-ray channels from a given compound nucleus. We now present data for the reactions “V( (Y,p)54Cr and “V((Y, n)54Mn which, together with the 54Cr(p, y)s5Mn and 54Cr(p, n)s4Mn data of Zyskind et al. 2), make up another such set. A partial energy-level diagram relevant Correspondence to: Dr. D.G. Sargood, 3052, Australia. * Supported in part by the Australian 037%94?4/93

School of Physics,

University

Research

(A68830965).

Council

/$ 06.00 0 1993 Elseviet Science Publishers

of Melbourne,

B.V. All rights reserved.

Parkvilie,

Victoria

V. Y. Hamper et al. / Cross sections

to these

four reactions

compilations

of Lederer

is presented

159

in fig. 1. The information

and Shirley 3>, Zhou

2. Experimental

is taken

et al. “) and Wang

from the

et ai. ‘).

details and results

2.1. TARGETS

Three targets of 99.95% pure natural vanadium metal were prepared, two on gold backings of stopping thickness for 10 MeV cu-particles, and the other a selfsupporting transmission target. The backed targets were prepared by evaporation of metallic V from previously outgassed tungsten boats on to gold substrates 0.13 mm thick, at a bell-jar pressure of 2 x lo-’ Torr. The substrates had been etched in aqua regia, rinsed in distilled water and outgassed under vacuum. The targets were transferred under dry argon to a chamber in which their thicknesses were measured by a-particle back-scattering techniques. Alpha particles of incident energies 3.0, 3.5,4.0 and 4.5 MeV, for which

4+

3655 3514 3468 3437 3393

2+

3222 3160 3074 2830 2620

'?I 2+ o+ 2+

3+-

-0

“Mn

2+

5’V+a

0

“Cr + p

(7939)

(10225)

835

o+L-l-

-0

7E--

+ n

(8068)

-0

5/T-

‘*Mn + y Fig. 1. Partial energy-level diagram for the 5’V+a system. The particle-channel threshold energies relative to the ground state of the compound nucleus, 55Mn, are shown in brackets. All energies are in keV. The arrow indicates the y-ray used in the (q n) activation analysis. The data are taken from refs. 3-5).

160

V.Y. Hamper

it was known

the Rutherford

and the pulse-height obtained

spectra

et al. / Cross sections

law holds, were observed were analysed

at a scattering

angle of 145”

in three ways. First, the thickness

from the area of the V peak, the integrated

solid angle determined from the detector aperture distance. A second thickness value was obtained

beam current

was

and the detector

diameter and the target to aperture from a comparison of the V peak

area and the height of the high-energy edge of the spectrum obtained from scattering from a clean Au blank. Finally, an energy thickness was derived from the difference in the positions of the Au edge in the spectra from the Au blank and from the target backing, and this was converted to an absolute thickness by means of the dE/dx tables of Ziegler et al. “). Agreement between the three methods of measurement was at the 4% level. After allowance was made for the 0.25% natural abundance of “V, the adopted values for thickness were (2.17*0.11) x 10” and (4.18*0.21)x 10” 5’V atoms cm-‘. The thicknesses were measured again after the targets had been used for (a, n) excitation-function measurements. No change was observed, at the 4% level. Since only one of the thickness determinations involved integrated beam current, the agreement among the three values for each target constituted a check on our beam-current integration system, at the 4% level. The transmission target was prepared by evaporation of V on to a glass slide coated with 99.999% pure caesium bromide. The vanadium was floated off on distilled water and mounted on a stainless-steel frame. The V peak in a 3 MeV a-particle back-scattering spectrum was flat topped. The full width at half height therefore led to a measure of energy thickness from which suitable energy steps could be derived for (a, p) excitation-function measurements. The absolute thickness was not needed.

2.2. THE

5’V(a,n)54Mn

CROSS

SECTION

The ‘lV(a, n)54Mn cross section was deduced from measurement of the 312 d 54Mn activity and also from measurement of the neutron yield. The gold-backed targets were used for these measurements,

the thicker one for the activation

measure-

ment and the thinner for the neutron-detection measurement. The excitation-function energy steps matched the target thickness at each bombarding energy. The activation measurement involved observation of the 835 keV l+ 0 y-ray transition in 54Cr, which is fed by 100% of 54Mn decays ‘). The target was mounted in a stainless-steel chamber and bombarded by cu-particles delivered by the University of Melbourne 5U Pelletron accelerator. The cY-particle beam was defined by two collimating apertures 4 mm in diameter, 8 cm apart and located 27 cm upstream from the target, then passed through a grounded beam wiper and a -600 V electronsuppressor aperture before entering the chamber which was insulated and acted as a Faraday cup. The beam-line vacuum was maintained at lo-’ Torr. The bombardingenergy range was 5.00-9.49 MeV. The target was bombarded for periods of -4 h y-rays were counted with beam currents of -40 nA of He++ ions and the activation

V. Y. Hamper

for periods

of 4 h during

from the target.

which the beam was stopped

The y-rays

161

et al. / Cross sections

were observed

in a Faraday

cup 6 m upstream

with a 125 cm3 Ge(Li)

detector

located

0.8 cm from the target. The detection efficiency was calibrated in sitta relative to the accumulated target activity at the end of the experiment. The target was then removed from the chamber source.

and its activity was calibrated

The “V(ar, n)54Mn cross section

relative to a standard

is plotted

as a function

s4Mn reference of target-centred

energy in fig. 2, by means of crosses. For the neutron-counting measurement, the target was mounted in a stainless-steel cross located at the centre of a 32 x 32 x 44 cm3 paraffin castle containing six ‘He-filled proportional counters, all feeding a common amplifier ‘). The detection efficiency of this detector is (10.4*0.6)% for neutrons of energy ~1 MeV, falling to 7% at 4 MeV. The effective detection efficiency was determined from the efficiency calibration curve and statistical-model estimates of the neutron energy spectrum at each a-particle bombarding energy. The beam-collimation, electron-suppression and charge-collection details were the same as those for the activation measurement. Precautions taken to minimise the yield of neutrons from the contaminant reaction “C(cy, n)160 included the cleaning of all slits and apertures, the maintenance of a beam-line pressure of
was soldered

a copper

tube

which

also passed

through

a bath

of liquid

E, WV) Fig. 2. Log-linear plot of the cross section of the reaction “Vfa, n)54Mn versus target-centred energy. The points plotted as circles correspond to neutron-detection measurements and those plotted as crosses to “Mn activation measurements. Error bars reflect statistical errors only. The curves represent the results of statistical-model calculations as described in the text.

V. Y. Hansper et al. / Cross sections

162

nitrogen. The cold shield was cooled by a flow of dry nitrogen through the copper tube. A circular hole of diameter 15 mm in the middle of the cylindrical shell was centred factor

on the beam. -100.

A pulser

The cold shield feeding

as well as a scaler provided

reduced

the preamplifier

carbon

build-up

on the target by a

of one of the proportional

the data for calculation

of dead-time

were always less than 6%. The “V( M, n)54Mn excitation

function

counters

corrections. was measured

These over

the energy range 2.00-7.14 MeV. The upper limit of this energy range corresponded to emissions of neutrons of the maximum energy of our detection-efficiency calibration Beam currents were typically 30 nA of He++ ions and charge per point 20 PC. At each energy, the neutron yield was measured with the beam striking the target and also with the beam striking a clean gold blank. The structure of the yield function obtained by subtraction of the gold blank yield from the “V yield function, at low energies, did not correspond to that for 13C(cu, n)“O alone. However, a superposition of published yields of 13C(q n) [ref. ‘)I, “O(cr, n) [ref. “)I and “O( (Y,n) [ref. ‘“)I was found which did reproduce this structure within errors. This contaminant yield was assumed to come from the target and the sum of it and the yield from the gold blank was subtracted from the 5’V(a, n) measurement yield at all energies. It was found that statistical-model calculations for these three contaminant reactions predicted the averaged energy dependence of the literature data well, and we therefore used normalised statistical-model estimates for o-particle energies above those of the literature. This overall contaminant subtraction was considered reliable at the 25% level and represented 50% of the raw measured yield at EC, = 4.5 MeV, falling to 30% at E, = 5 MeV, 10% at E, = 5.7 MeV and (5% for .& > 6 MeV. The ‘*V(cy, n) contribution was estimated to be (0.5% and was neglected. The resultant “V( cr, n)54Mn cross section is plotted in fig. 2 by means of circles. Error bars in fig. 2 reflect statistical errors only. The error in the cross-section scale is estimated to be 7% for the activation measurement and 14% for the neutron-counting measurement. The main sources of error were y-ray-detection efliciency 2%, neutron-detection efficiency 12%, target thickness 5% and beamcurrent

2.3. THE

integration

4%.

5’V(a,p)SJCr CROSS SECTION

The 5’V( (Y,p)?Zr cross section was measured with the transmission target mounted in a stainless-steel chamber 60 cm in diameter and maintained at a pressure of 1.5 x lo-’ Torr. The beam-collimation and electron-suppression arrangements were the same as those for the (q n) measurements. A Faraday cup mounted 3 cm behind the target was electrically connected to the target and both were held at +600 V relative to the chamber. This provided efficient suppression of secondary electrons from the target. The excitation function was measured over the bombarding-energy range 5.755-9.590 MeV. Beam currents were typically 20 nA of He++ ions and charge per point 50 PC. Protons were detected by means of a counter telescope mounted

V. Y. Hamper et al. / Cross sections

5.5 cm from the target

in the 123” direction

telescope consisted of a transmission particles but thin enough to transmit model calculations a depletion depth detector

outputs,

predicted sufficient

which

163

corresponded

to 125” c.m. The

detector, thick enough to stop 10 MeV (Yall protons of energies for which statistical-

a significant yield, and a surface-barrier detector with to stop all protons of interest. The sum of the two

gated by the surface-barrier

detector

output,

gave a pure proton

spectrum, an example of which is shown in fig. 3. As may be seen in fig. 4, the output of the transmission detector was dominated by elastic-scattering peaks, the most prominent being that due to vanadium. Over the bombarding-energy range 5.755-7 MeV, the a-particle scattering followed the energy dependence of the Rutherford law and it was possible to measure the “V(a, p)54Cr cross section by direct comparison with the V(a, a) Rutherford cross section, without reference to target thickness, detector solid angle, or integrated beam current. The (a, a) measurement also served to calibrate the combination of these three quantities for use with higher bombarding energies. A pulser fed pulses to both of the detector preamplifiers and to a scaler, and this provided the data needed to calculate dead-time and pile-up corrections. These corrections never exceeded 4% in either the gated proton spectrum or the transmission detector spectrum. The only contaminant reactions making significant contributions to the proton spectrum were 27A1((Y,P)~“S’1 and 14N( (Y,p)“O. To deal with the 27A1 contamination

II

35:

11

“V(a, p)‘*Cr

1

I

I

I

I

I

I’

E_ = 7.447 MeV

30-

PI

25 _^

PZ 14N

5.0

6.0

Proton energy (MeV) Fig. 3. Energy spectrum for protons detected by the counter telescope. The peaks due to “V(a, pO), p), which are (a, P,), (a, PZ). (01, ~3.4) and (a, ps .h,7) are identified. The only two peaks due to “Al(q resolved, are also identified. The vertical arrow marks the position at which the contribution from 14N(a, p,J would appear.

164

V.

Y.Hamper

el

al. / Cross sections

l”‘~l’~“l~~~~l”~‘l’~~~l’~~~~

50000 - Alpha particle backscattering spectrum E. = 7.0 MeV V_

40000 !$

S U

30000~ 20000 0 Fulser

lOOOO-C

0

BrCs

-

It_._-: II I I I II I I II I I I I I / I I I7

O,,

100

200

300

400

500

600

700

Channel No. Fig. 4. Pulse-height spectrum to V and various contaminants

from the transmission detector. Alpha-particle elastic-scattering in the target are labelled. None of the contaminants identified to the proton spectrum.

peaks due contributed

we made an ‘7Al target of the same thickness as the “V target and used it in a measurement of the 27Al(a, p) excitation function at the same energies as for 5’V( (Y,p). This excitation function then provided the ratio of the yields of unresolved ‘7Al peaks in the 5’V spectrum to the yields of the resolved ‘7Al peaks. Since no 14N peak was resolved at any energy this procedure could not be adopted for 14N and regions of the proton spectrum which could be affected by 14N( (Y,p) yield were excluded from the analysis. The use of a single detector at 125” c.m. is strictly valid only if the reaction always proceeds with either s- or p-wave cY-particles or s- or p-wave protons. To obtain an estimate of the relative importance of reactions with d- or higher-order waves in both channels, we first used statistical-model transmission coefficients for all IS 10 to obtain relative partial cross sections corresponding to individual l-wave pairs. The fraction of the cross section which involved d- or higher-order waves in both channels

rose from 13% at E, = 6 MeV to 30% at E, = 8 MeV and 40% at E, = 9 MeV.

These are the fractions for which angular-distribution effects could be important. However, calculations of the compound-nucleus level density by means of the Fermi back-shifted gas formula, which is incorporated in the statistical-model code, indicated that, at all bombarding energies, our target thickness straddled >400 levels with J” which could lead to reactions with significant yields and angular distributions involving Pz4(cos 0) terms. For measurements made with our target there would therefore be averaging over many angular distributions, W( 0) = C,, a,P,,(cos O), with positive and negative values of the u, combining to smooth out the resultant angular distributions. We obtained an estimate of the overall angular-distribution effect on our measurements by averaging over assumed representative functions,

et al. / Cross sections

V. Y. Hamper

W( 125”) = 1 f [IR,(cos

125”)j + ]P,(cos

mon as the other. The transmission not important. siderably

Although

higher

less: we therefore assumption

125”)(], with one sign assumed

coefficients

some individual

corrections regard

than

165

showed that higher-order

angular

that of our

distributions

W(125”),

our W( 125”) as a reasonable

that one sign is twice as common

twice as com-

others

terms were

would involve would

representative

as the other significantly

con-

certainly estimate.

be The

underestimates

the degree of cancellation we should expect from the addition of so many angular distributions, and we are confident that our estimate of the magnitude of the overall angular-distribution effect is a conservative one. This procedure led to the result that the effect of the higher Z-wave reactions on the cross section measured at 125” was ~5% at every point in the measurement, for Em < 7.5 MeV and ~8% for E, > 7.5 MeV. The cross sections of “V(a, pO), (q p,), (q pl), (a, p3+) are plotted as functions of target centred energy in fig. 5. Error bars reflect statistical errors only. The error in the cross-section scale is considered to be less than 9%, the main contributors being angular-distribution corrections 8%, uncertainty in the channel number

7.0 E. WV)

6.0

8.0

9.0

7.0 8.0 E. (MeW

6.0

7.0

8.0 E. WeV)

9.0

9.0

E, WV)

Fig. 5. Log-linear plots of cross section versus target-centred energy corresponding to the highest-energy proton groups for the reaction “V(a, p)?r. Error bars reflect statistical errors only. The curves represent the results of statistical-model calculations as described in the text.

166

defining

V. Y. Hamper

the low-energy

to which the cu-particle

et af. / Cross sections

limit of the a-particle

scattering

scattering

to follow the Rutherford

was known

peak 3%, and the accuracy law 2%.

3. AnaIysis The cross-section data have been compared with the predictions of the statisticalmodel code HAUSER* [ref. “)I modified as described in ref. ‘*) to permit application of width-fluctuation corrections at all energies. The statistical-model calculations were made with the global optical-model parameters of Becchetti and Greenlees 13) and McFadden and Satchler 14), and also with the modified parameters of Tims et al. r5) which correspond to the global parameters with the imaginary surface well depth reduced to 6 MeV in the proton channel and 4 MeV in the neutron channel and the imaginary volume well in the a-particle channel replaced by an imaginary surface well of depth 10 MeV. The results of the global-parameter calculations are shown by the full lines in figs. 2 and 5, and those of the modified-parameter calculations by dashed lines. The reactions 54Cr(p, y)55Mn and 54Cr(p, n)54Mn pass through the same compound nucleus as do “V(cl,, p)54Cr and “V(cu, n)54Mn, and we carried out the calculations for them also, comparing the results with the data of Zyskind ef al. “>. The results of the comparisons for all four reactions are shown in fig. 6, which is a plot of the ratio of theoretical to experimental cross section as a function of energy. For 5’V(a, p)54Cr, the comparison is limited to the sums of those proton groups which could be reliably extracted in different energy ranges. The comparisons with global-parameter calculations are plotted as diamonds and with modified-parameter calculations as crosses. Error bars reflect statistical errors in the experimental data and are shown

only on the points

plotted

4. Thermonuclear Thermonuclear

reaction

as crosses. reaction rates

rates were calculated

(uu)“=(8/~~)“2(~~)-

3’2

u(E)E

by means

of the equation

exp (-E/kT)

dE

of Fowler et al. 16), where M and E are the reduced mass and c.m. energy, and the superscript zero indicates that the target is in its ground state. For values of r(E) corresponding to energies outside our experimental range we used the results of HAUSER* calculations with the modified parameters, normalised to our data at the appropriate end of the energy range. For “V(cr, n)s4Mn, the experimental data we used were averages of the cross sections based on the neutron-detector and activation measurements over the energy range common to both, those based on the neutron-detector measurement at lower energies, and on the activation measurement at higher energies. The contribution of calculated cross sections to the reaction-rate integral was 2% for T9 = 4.5, 15%

V. Y. Hamper et

of./

Cross

sections

167

, , , , , , , , , , , , , , , , , , m-0.6 +(a.

I I

I

p)“Cr f

, -0.4

I

::

I

5.0 E,(MeV)

’

I

_

-Ox5

3.0 E, (MeVf Fig. 6. Plots of the ratio of calculated cross section to measured cross section for the four reactions discussed in the text. The points plotted as diamonds correspond to comparisons with theoretical cross sections obtained with the global parameters of refs. 13*IJ) and the crosses to comparisons with cross sections obtained with the modified parameters of ref. I’). The error bars refer to statistical uncertainties in the experimental data and are shown only on the points plotted as crosses. The horizontal lines correspond to agreement between theory and experiment at the factor of 1.5 level. To avoid congestion on the page, the data points for “V(ru, pl and for “V( LY,nf at E, < 6.2 MeV have been added in threes before being compared with theory. For “V(cr, p)“%Jr, the ratio refers only to the sum of those proton groups whose yields could be reliably extracted: these are identified by the labels below the plotted points for the energy ranges bounded by the dashed lines. The crosses in the ‘“Wp, y) and 5’Cr(p, n) plots have been joined with straight lines to distinguish them more readily from the diamonds.

for

‘T..= 3 and 8, 25% for

T9 = 2.5 and

10, 50% for Ty = 2 and essentially

100% for

T,=s 1, where T9 is the temperature in units of lo9 K. For ‘rV(cr, p)?Zr, the experimental data at each energy corresponded to the proton groups which could be reliably extracted, and the partial cross sections so obtained were multiplied by ~~~~~~~ Upartinl as calculated with HAUSER*4. This calculated ratio was typically 1.5. The contribution of statistical-model cross sections to the reaction-rate integral was 7% at TV= 5,27% at T9 = 3.5 and 10,65% at T9 = 2.5 and essentially 100% for T,c 1.5. The rates were converted to stellar rates, (au>*, corresponding to target nuclei with a thermal distribution of excited states, by means of the ratio {~u}*/(~u~’ from

168

Woosley number,

V. Y. Hamper

et al. I’). The stellar

rates

et al. / Cross sections

per mole,

NA(au)*,

where

are listed in table 1 along with the experimental

et al. “), the theoretical

of Woosley

values of Woosley

and Hoffman

our stellar

al. 20). The parametrisation =

rates according

formulae

to the scheme

of Woosley

et

are

(2.k + 1)(2Jn+ 1) (2Jt-t

for 5’V( CY,n)54Mn,

(cz, n) values of Roughton

et al. “) and the values in the compilation

r9).

We have parametrised

NA(uu)*

NA is Avogadro’s

1)(2J, + 1)

and

=L2J:r+l)Wp+ 1) AcrAp

NA(au)*

(250,+1)(23,+1) x Tc2r3 exp

[

[A,A.,Iii2~exp[-~l

9

TABLE

Stellar rates, N,(c+u)*, 5’V(~, this work

ref. ‘s) “)

for “V(a,

C’V((Y, p)Wr

ref. “)

ref. 19) 4.OE - 24

8.178-24

4.228 - 24

6.98E - 20

3.258 - 20

7.10E-

15

9.27E-

12

1.2

1.41E-09

1.4

6.38E - 08

1.5

3.16E-07

1.6

1.34E - 06

2.90E - 15 3.5E-

14

P)‘~C~ (in cm3 s-’ mol-‘)

n)54Mn

0.5

1.0

(2)

1

r~)~~hiInand “V(a,

0.6 0.8

1

A -- T;,3(l+BT9+CT;+DT:)

3.60E - 12

5.4E - 12

this work

ref. “)

2.58E-22

9.18E-23

1.92E-19

7.188-20

ref. 19) 7.1E-24

1.83E - 15

6.59E-

16

6.61E-

2.17E-

13

13

5.4E-

14

4.11E-11 l.O3E-09 1.4E-08

1.30E-07

2.OE - 07

4.28E - 09

1.31E-09

1.7E-09

1.65E-06

6.50E - 07

8.8E -07

1.278-04

5.88E -05

1.63E-08 1.90E - 07

1.8

1.65E-05

2.0

1.37E-04

2.3E-05

6.49E - 05

2.5

8.498 -03

3.3E -03

4.71E-03

3.0

1.82E-01

1.2E-01

l.l4E-01

3.5 4.0

2.03 E + 00 1.43E+Ol

1.8E+oo lSE+Ol

1.37l?+oo l.OOE+Ol

4.5

7.348+01

8.8E+ol

5.19lz+01

5.0

2.888+02

3.6E+02

2.05E+

6.0

2.51E+O3

3.5E+03

1.79E+03

1.4E-01 1.2EfOl

02

7.0

1.25E+04

1.9E+04

8.89E+03

8.0

4.15E+O4

6.2E + 04

2.978+04

9.0

l.o2E+05

1.5E+05

7.328+04

10.0

2.00E+05

3.OE+05

1.43E+05

“) The values listed are those for NA(ou)’

9.1E-05

2.4E + 02 9.2E+03

1.6E+05

3.328-03

1.67E-03

4.2OE-02

2.24E - 02

3.238-01

1.80E-01

1.74E + 00

l.OlE+OO

7.15E+OO

4.30E+OO

6.67E + 01

4.30E+Ol

3.52E+O2

2.418+02

1.23E+O3

8.87E+O2

3.14E+O3

2.368+03

6.33E + 03

4.89E+03

from ref. Ia) multiplied

by (~v)*/(~Tv)’

1.6E-03 1.6E-01 3.9E + 00 2.4E+02

4.3E+03

from ref. I’).

V. Y. Hamper et al. / Cross sections

for 51V(~, p)54Cr, where perature-dependent state;

the J’s, A’s and

partition

Q is the Q-value

functions,

of the inverse

169

G’s are nuclear

spins,

and the superscript exoergic

reaction,

masses

and tem-

zero refers to the ground in MeV; and T is given by

,

r = 4.2487(Z’,Z:,&”

where the Z’s are atomic numbers and 2 is the reduced mass in the cy-particle channel. For the (cu, n) reaction A-F are all free parameters, but for the ((Y, p) reaction A has a defined value which reflects the effects of Coulomb-barrier penetration at low energies and the only free parameters are B, C and D. We were unable to obtain a satisfactory fit to the “V(cu, p) rates with a single set of parameters and present one set for 0.5 < T,< 1 in which r and A have their defined values, and another set for I< T,< 10 in which r and A have been treated as free parameters. The parameters for both reactions are listed in table 2. They reproduce the rates of table 1 to within 14%.

5. Discussion

5.1. CROSS

SECTIONS

From fig. 6 it is clear that HAUSER”4, with either the global parameters of refs. 13,14)or the modified parameters of ref. 15), predicts the “V(a, n) and 5’V( LY,p) cross sections to within a factor of 1.5 over most of the energy range of the data. The global parameters lead to somewhat better agreement than do the modified parameters for the (a, p) reaction at energies >7 MeV, but even the modifiedparameter predictions are still comfortably inside the factor of 2 level of reliability required for nucleosynthesis flow calculations. On the other hand, the modified

TABLE 2 Parameters

for fits to stellar reaction

“V( LY,r~)‘~Mn 0.5 < r,<

10

rates

“V( (Y, p)Wr 0.5 < i-q < 1

lCT,
7

84.43

84.44

51.97

A B

2.102 x lolX -7.088 x lo-’

56.71 -1.876x 10-l

20.11 -1.836 x 10-l

C

D E F

7.726 x 10m3 -2.506

x 10m4

2.639 1.318

1.727 x 10-l -3.945

x 10-l

1.771 x lo-* -6.341

x lo-’

K Y. Hamper et at. / Cross sections

170

parameters lead to the better results for 54Cr(p, y) and s4Cr(p, n), particularly at the lower end of the energy range, which dominates in reaction-rate calculations for T, -=c5, the temperature

at which nuclear

statistical

When averaged over structure, the results obtained are in agreement with experiment to within a factor range, for all four reactions,

and this constitutes

equilibrium

is established

21).

with the modified parameters of 1.7 over the whole energy

a level of overall

agreement

which

is superior to that of the global parameters. This feature of the modified parameters, significant improvement on the low-energy results of the global parameters for proton-induced reactions with only marginal change for cu-particle induced reactions, has been observed before, e.g. Tims et al. “), and adds support to our strengthening belief that the modified parameters lead, in general, to better overall values for reaction cross sections in the energy range of importance for nucleosynthesis

calculations,

5.2. T~ERM~NUCLEAR

than do the global

REACTION

parameters.

RATES

Our “V(cu, n)54Mn reaction rates are in mild disagreement with those of Roughton et al. 18) for high T9 and in serious disagreement for low T9. Roughton et al. measured the activation of a target of stopping thickness as a function of tY-particle energy. In terms of y(E), the thick target yield per incident o-particle, the reaction-rate integral becomes “) 02

(cTD)O= (8/kh)'/2(k~)--3/2

I

0

Y(E)

& [-ME)

exp (-E/WI

dE,

where E(E) is the atomic stopping power of the target. To obtain values of y(E) for energies outside their experimental range, E, = 5.369-14.244 MeV, they used analytic functions to extrapolate from their data. For Ea > 14.244 MeV an exponential function was quadratic function For 6 < T,c 10 This lies outside

used and, for E, from 5.369 MeV to the neutron threshold, a lx). their rates are consistently higher than ours by a factor of 1.5. the limits of combined errors. In this temperature range their

reaction-rate integrals and ours are dominated by the experimental data, suggesting an experimental rather than analytical cause for the discrepancy. Roughton et al. found it necessary to introduce an annulus held at -300 V in front of their target to suppress secondary electrons dislodged from the target by the cu-particle bombardment. They make no mention, however, of a beam wiper to shield this annulus from any possible halo of a-particles scattered by the perimeter of their beam collimator. This leaves open the possibility of halo-induced secondary electrons produced at the annulus being driven by the -300 V suppressor potential on to the target, thereby leading to an underestimate of the beam current and a corresponding overestimate of the yield per incident a-particle. Our experience with different designs of electron-

171

b! Y. Hamper et al. / Cross secfions

suppression

systems

suggests

that this could

largely

account

disagreement with our rates, for high TV. Inthe low-temperature range the cause of the disagreement cal. For T,< 6 the low-energy and it is significant

extrapolations

that as the temperature

for the factor appears

play an increasingly

is reduced

the reaction

of 1.5

to be analytiimportant

part

rates of Roughton

et al. fall much more rapidly than do ours, and are smaller than ours by factors of 6, 23 and 260 at T,= 2, 1.5 and 1, respectively. We believe that these very large discrepancies are caused by their low-energy extrapolation function seriously underestimating y(E) for low values of 6 This belief is supported by the fact that even just the 50% of our Tg= 2 rate which is attributable directly to our experimental o(E) data, exceeds their totai by a factor of 3. Further, from our data and our statistical-model extrapolation, we determined that, for T,=2, the integrand is peaked at an effective interaction energy E0 = 4 MeV with a full width at l/e height of ,4.& = 1.7 MeV, and that >98% of the integral arises from energies <(E,,+ A&) = 5.7 MeV. Since the lowest bombarding energy in the work of Roughton et al. was 5.349 MeV their Ty= 2 rate was heavily influenced by their extrapolation function which clearly underestimates the yield and, as their lower-temperature rates indicate, does so by an increasingly greater factor as the energy is further reduced. Our lower-temperature rates also are heavily dependent on an extrapolation but it is an extrapolation with all the sound physical basis of the statistical model. On two previous occasions we have found the low-temperature rates of Roughton et al. I*) to be lower than ours by large factors. For T9= 1 the factor for 48Ti( Ly,n)s’Cr was 330 [ref. “)I and for 54Fe(Lu, P)‘~CO was 5000 [ref. “)I. By contrast we were in substantial agreement with their low-temperature rates for 54Fe(a, n)57Ni [ref. “)I and 59Co(a, n)“Cu [ref. 15)]. These two reactions, however, have large negative Q-values with the result that good experimental yields were obtainable at energies down to the neutron threshold, and low-energy extrapolations were not involved in the reaction-rate calculations. On the other hand, the low-energy yields of 4”Ti(q n)5’Cr, 51V(cz, n)54Mn and 54Fe(ar, p)“Co were limited by barrier penetration effects and the low-energy extrapolations played an important part. This body of evidence leads us to suspect that the function used by Roughton et al. in ref. *‘) for low-energy extrapolations always seriously underestimates the yield. This has implications

for all their low-temperature

(N, n) and (cu, p) rates except for those relating

to (LX,n) reactions with large negative Q-values. The rates of ref. 17), which were calculated from cross sections generated by a statistical-model code employing square-well potentials and omitting width-fluctuation corrections, are consistently smaller than ours at all temperatures, for both reactions. For 51V((u, n) they are small by a factor which is constant at 1.4 for 10 > I;> 4, then rises steadily to 2.6 at T, = 1 before falling again to 1.9 at T, = 0.5. For “V(LU, p) they are small by a factor which rises steadily from 1.3 at T, = 10 to 3.3 at Tg= 1.5before falling again to 2.8 at TV= 0.5. The status of ref. I’) as a source of reaction rates which are reliable to within a factor of 2 is therefore weakened by

V. Y. Hamper et al. / Cross sections

172

the present results. However, with the modified parameters theoretical

reaction

fig. 6 lends weight to our contention is a reliable source of cross sections

that HAUSER* on which to base

rates.

References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19)

20) 21) 22) 23)

V.Y. Hansper, C.I.W. Tingwell, S.G. Tims. A.F. Scott and D.G. Sargood, Nucl. Phys. A504 (1989) 605 J.L. Zyskind, J.M. Davidson, M.T. Esat, M.H. Shapiro and R.H. Spear, Nuci. Phys. A301 (1978) 179 CM. Lederer and V.S. Shirley, Table of isotopes t Wiley, New York, 1978) C. Zhou, E. Zhou, X. Lu and J. Huo, Nucl. Data Sheets 48 (1986) 11 G. Wang, J. Zhu and J. Zhang, Nucl. Data Sheets 50 (1987) 255 J.F. Ziegler, J.P. Biersack and U. Littmark, Stopping powers and ranges in all eiements, vol. 1 (Pergamon, Oxford, 1985). C.I.W. Tingwell, V.Y. Hamper, S.G. Tims, A.F. Scott and D.G. Sarwood, Nucl. Phys. A480 (1988) 162 K.K. Sekharan, A.S. Divatia, M.K. Mehta, S.S. Kerekatte and K.B. Nambiar, Phys. Rev. 156 (1967) 1188 J.K. Bair and F.X. Haas, Phys. Rev. C7 (1973) 1356 J.K. Bair and H.B. Willard, Phys. Rev. 128 (1962) 299 F.M. Mann, Hanford Eng. Devel. Lab. report HEDL-TME-7680 ( 1976), unpublished A.J. Morton, S.G. Tims, A.F. Scott, V.Y. Hamper, C.I.W. Tingwell and D.G. Sdrwood, NucI. Phys. A537 (1992) 167 F.P. Becchetti and C.W. Greenlees, Phys. Rev. 182 (1969) 1190 L. McFadden and R. Satchler, Nucl. Phys. 84 (1966) 177 S.G. Tims, C.I.W. Tingwell, V.Y. Hansper, A.F. Scott and D.G. Sargood, Nucl. Phys. A483 (1988) 354 W.A. Fowler, G.R. Coughlan and B.A. Zimmerman, Annu. Rev. Astron. Astrophys. 5 (1967) 525 S.E. Woosley, W.A. Fowler, J.A. Holmes and B.A. Zimmerman, Caltech preprint OAP-422 (1975), unpubIished N.A. Roughton, T.P. fntrator, R.J. Peterson, C.S. Zaidins and C.J. Hansen, At. Data Nucl. Data Tables 28 (1983) 341 S.E. Woosley and R.D. Hoffman, Tables of reaction rates for nucleosynthesis for charged particle and weak interactions (6s 2~39): version 86.3 (University of California, Santa Cruz, 1986), unpublished S.E. Woosley, W.A. Fowler, J.A. Holmes and B.A. Zimmerman, At. Data Nucl. Data Tables 22 (1978) 371 SE. Woosley, W.D. Arnett and D.D. Clayton, Astrophys. J. Suppl. 26 (1973) 231 N.A. Roughton, M.J. Frits, R.J. Peterson, C.S. Zaidins and C.J. Hansen, Astrophys. J. 205 (1976) 302 S.G. Tims. A.J. Morton, C.I.W. Tingwell, A.F. Scott, V.Y. Hansper and D.G. Sargood, Nucl. Phys. A524 (1991) 479