Elastic and inelastic scattering of 14.1 MeV neutrons from C, Mg, Si and S

Nuclear Physics 53 (1964) 177N203; ( ~ North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or roierofilm without written permission from the publisher

E L A S T I C AND I N E L A S T I C S C A T T E R I N G

OF 14.1 MeV NEUTRONS FROM C, Mg, Si and S R. L. CLARKE and W. G. CROSS Chalk River Nuclear Laboratories, Chalk River, Ontario, Canada Received 25 November 1963

Abstract: The differential cross sections for the elastic scattering of 14.1 MeV neutrons and for inelastic scattering to the first-excited state have been measured for C, Mg, Si and S. Elastic and inelastic neutrons were separated by their flight times. Scattering to higher states was observed but a differential cross section was obtained only for the 9.6 MeV state of C12. Good optical model fits to the elastic scattering results for Mg, Si and S were found. The same optical model parameters were then employed in a distorted wave Born approximation calculation which gave a good fit to the shape of the angular distributions for scattering to the first-excited states of these nuclei. The fits to the magnitude of this scattering yieldedvalues of the mean-square quadrupole distortions of 0.62, 0.43 and 0.32 for Mg, Si and S, respectively, in good agreement with values derived from electromagnetic measurements on these states. For carbon, both the elastic and inelastic scattering results show a definite disagreement between experiment and theory.

1. Introduction The elastic scattering o f 14 MeV neutrons provides a test for the applicability of the optical model and is a useful source o f data for determining the parameters of this model. Such scattering has been investigated by a number o f groups 1-13). Bjorklund and Fernbach 14) and others is) have calculated the angular distributions using a set of optical-model parameters which varied smoothly with mass number. The agreement obtained between theory and experiment is good for nuclei heavier than calcium but is less successful for lighter elements, such as A1. Observations of the angular distribution of 14 MeV neutrons scattered inelastically from carbon have been made by Anderson et al. 16) and others 17-2o) and for heavier nuclei by Tesch 13) and Stelson 21). Time-of-flight techniques were used to separate neutrons scattered from individual levels. The present experiment was undertaken primarily to extend the data on inelastic scattering to other light nuclei where good separation of states is possible, with the object of comparing the results with the predictions of a direct-interaction model. The nuclei used, C 12, Mg 24, Si 28 and S 32, are all even. Since, with the possible exception of carbon, the first-excited states of these nuclei are probably collective, the relevant calculations of direct-interaction inelastic scattering become relatively simple. Such calculations have been made, for neutrons, by Glendenning 22) and by Pinkston and Satchler 23). Levinson and Banerjee 24) have calculated the scattering of protons from carbon using a shell model for the carbon nucleus and a specific 177 April 1964

178

R. L. CLARKE AND W . G. CRO88

nucleon-nucleon interaction. At 14 MeV and for a very light nucleus like carbon, the scattering of incident protons and neutrons would not be expected to be very different, and this expectation has been confirmed 16) for both elastic scattering and scattering to the first-excited state of C 12. In the present work the differential cross sections for elastic scattering and for inelastic scattering to the first-excited states of C, Mg, Si and S have been measured. The inelastic scattering results are comlJared with similar results obtained for protons of about the same energy. The elastic distributions have been fitted with the optical model and the inelastic distributions on the distorted wave Born approximation, using the same optical-model parameters. G o o d agreement between theoretical and experimental results has been obtained for both elastic and inelastic scattering from Mg, Si and S. For carbon, however, both the calculated elastic scattering at back angles and the calculated inelastic scattering are in disagreement with experimental results.

2. Apparatus and Procedure The spectrum of scattered neutrons resulting from incident 14.1 MeV neutrons was measured by a time-of-flight technique. The apparatus, shown schematically in fig. 1, was similar in principle to that described previously 25). Neutrons from the T(d, n)He 4 reaction were produced by 120 keV deuterons striking a thick tritium-Zr target. The measured time was that between detection of an alpha particle from the reaction and detection of the corresponding neutron in a counter about 3 m away. This delayed coincidence requirement between the neutron and alpha counters in effect defined a narrow " b e a m " of 14.1 MeV neutrons, at about 88 ° to the deuteron beam. The width of the neutron beam was determined by the geometry of the alpha counter. The alpha counter and scattering sample were mounted so that they could be rotated together around the axis of the deuteron beam, while the neutron counter remained fixed, thus varying the scattering angle 0. The alpha detector was a piece of plastic scintillator, 0.25 m m thick, mounted directly on a 56 AVP photomultiplier. The neutron beam had an approximately Gaussian profile in the scattering plane with a full width at half maximum of either 5 ° or 10° (depending on the aperture used for the alpha counter) and had an energy spread of about 150 keV. The neutron detector was also of plastic scintillator, 10 cm thick, in contact with an R.C.A. 7046 photomultiplier. A second neutron detector was used as a monitor. Its bias was sufficiently high to be insensitive to any D D neutrons produced. A steel collimator and heavy shielding (shown in part in fig. 1) around the neutron counter reduced its sensitivity to neutrons and gamma rays coming from positions away from the scatterer. A copper and wolfram "shadow-bar" was mounted between the target and the neutron counter to reduce the flux of neutrons going directly from target to counter. The scattering samples were sheets of natural elements, approximately 10 × 10 cm

ELASTIC AND INELASTIC NEUTRON SCATYERING

!179

and of such thickness that about 10 ~ of the incident neutrons interacted with them. They were supported on a light brass frame, with their plane usually at 45 ° to the incident neutron beam, and 14 cm from the target. The neutron beam was covered by the sample while virtually none of it struck the support. 7046

PLASTIC SCINTILLATOR

I

,.,....~] SCATTERER

IBUI

t_ SOORCE NEOTRON W,.Co ' I ALP.A

SHADOW BAR 319

SCINTILLATOR

crrl

Fe SHIELDING

Fig. 1. A r r a n g e m e n t o f time-of-flight a p p a r a t u s . T h e deuteron b e a m is n o r m a l to the plane o f the figure a n d strikes a t r i t i u m target at the position m a r k e d n e u t r o n source. T h e scatterer a n d alphaparticle c o u n t e r are m o u n t e d together as a unit w h i c h c a n rotate in t h e p l a n e o f the figure a b o u t t h e axis o f the deuteron beam.

Neutron counter

Start ator

J --

Alpha counter

Io=i Fig. 2. Block d i a g r a m m e o f the time-of-flight circuits.

A block diagramme of the circuits used to measure flight times is shown in fig. 2. The output pulses of the time-to-pulse-height converter were displayed on a 100channel analyser, the width of each channel corresponding to between 0.3 and 1.36 nsec, depending on the energies of the neutrons under investigation. These pulses

180

R.L.

CLARKE

AND

W.

G. CROSS

were gated by slow coincidences between the outputs of side-channel discriminators used to select the alpha and neutron pulse-heights. The discrimination level of the neutron side channel was varied f r o m 1 to 7 MeV, again depending on the neutron energies being studied. Compensation was used 2~) for the apparent time shifts caused by differences in the heights of neutron pulses. With a neutron counter bias of 5 MeV or more, the time resolution of the whole system (full width of a peak at half maximum) was 2 nsec for 14 MeV neutrons, limited principally by the flight time of neutrons through the neutron scintillator. This resolution corresponds to about 900 keV at 14 MeV. gO

I

I

I

I

I

70

60~

Z 50~

o= 40~

~

20--

I0 -- /

o/ o

En

I ~

I 2o

t

30

MeV

I

4o

I

50

CALCULATED ,",T

I

60

I ro

~o

CHANNELS

Fig. 3. Comparison of observed and calculated differences between the flight times of 14 MeV neutrons scattered at known angles from hydrogen and those going directly to the neutron counter. The solid points are for hydrogen in polyethylene while the open circles are for a plastic scintillator. One channel corresponds to 1.36-t-0.02 nsec. The inset scale shows the energy of the scattered neutrons.

The background coincidence rate, in the absence of a scatterer, was approximately constant over the region of the pulse-height spectrum displayed and could be accounted for almost entirely by accidental coincidences between alpha-particle pulses and neutron-counter pulses arising from the operation of the accelerator. The ratio of true-to-background coincidences was therefore inversely proportional to the source intensity, so that it was necessary to limit severely the neutron output and to count for long periods. The deuteron beam current was controlled automatically throughout the run to hold the neutron output constant. The background for each run was taken from parts of the spectrum where there was no true yield, for example, at flight times just shorter than those of the fastest neutrons. It was confirmed that the background in such regions was the same with and without a scattering sample.

ELASTIC AND INELASTIC NEUTRON SCATTERING

181

The time scale of the system was measured in three ways; (1) by inserting known cable lengths to delay artificial pulses put into the system at the photomultiplier outputs, (2) by changing the length of the flight path with the 14.1 MeV neutron beam aimed directly at the neutron counter and with the shadow-bar removed, and (3) by observing the flight times of neutrons scattered by hydrogen through known angles, in arrangements described below under the measurement of neutron counter I

I

I

I

I

I

I

I.(l--

~

rUAS 1MeV

O.B

0.6

0.4

0,2 I.) ~

w >

I

0

I

I

I

I

O.e

Q~ 0.(

/

BIAS S;S IVle,V

0.~

0

0

I

4

•

6

NEUTRON

I

B

I

I0

1

12

I

14

16

ENERGY MeV

Fig. 4. The relative efficiency of the neutron counter as a function of energy, for two different values of bias. The curves are calculated and are normalized to the experimental points.

sensitivities. An example of the agreement obtained between methods (1) and (3) is shown in fig. 3. The time dispersion of the system was first measured to be 1.36 nsec/channel, using delay cables and artificial pulses. The measured flight times of neutrons scattered by hydrogen were then compared with the times calculated. The times AT actually plotted in fig. 3 are the differences in flight times between neutrons scattered through a certain angle and those going directly from the source to the neutron counter. The straight line in the figure corresponds to agreement between calculated and observed times. The relative sensitivity of the neutron detector to neutrons of different energies

182

R. L. CLARKE AND W . O. CROSS

was determined by measuring the yields of neutrons scattered by hydrogen at known angles and assuming that this scattering is isotropie in the C.M. system. Two arrangements were used. In the first, the hydrogen was in a polyethylene sheet, 0.5 x I0 x 10 cm, supported edge-on to the neutron beam, with its centre 25 em from the neutron source. The scattering angle was changed, just as for other samples, in order to change the energies of the scattered neutrons. The spread in scattering angle over the scatterer was less than 3 °. Consequently the energy spread of neutrons scattered from hydrogen was small enough that they were clearly resolved from those scattered elastically from carbon, at angles corresponding to scattered neutrons of below 12 MeV. At angles for which the hydrogen peak overlapped that for inelastic scattering to the 4.4 MeV state in C 12, a correction for the latter peak was made, based on the previously measured ratios of the elastic and inelastic peaks in carbon. In the second method, a thin sheet of plastic scintillator, also edge-on to the neutrons and viewed by an R.C.A. 6342 photomultiplier, was used as the scatterer. Recoil protons in this "scattering counter" were used instead of the alpha particles to give the starting time signal of the time-of-flight system. The bias in the scattering counter was low enough to detect all recoil protons that corresponded to neutrons scattered into the neutron counter. The scattering counter was rotated, just as scattering samples were, to give various known scattering angles. This second method of calibration has the advantages over the first that the background of accidental coincidences is lower (the neutron source strength can therefore be increased and the calibration done more quickly), and that neutrons scattered from carbon do not produce coincidences. Typical sensitivity curves measured by both these methods are shown in fig. 4, for two choices of the bias setting of the neutron counter. A bias o f between 5 and 7 MeV has the advantage that the efficiency of the neutron counter is almost constant over the energy range most often of interest in these scattering experiments. The efficiency curves of fig. 4 were calculated from the hydrogen cross section and counter bias. Because this calculation did not take into account the effects o f multiple scattering in the scintillator, the calculated absolute efficiencies were not sufficiently accurate, so the curves shown have been normalized to the measured values. The relative efficiencies, however, are in good agreement with experiment. For each element the relative counting rate in the elastic and each inelastic peak was measured for scattering angles at 5 or 10 degree intervals. The yields were corrected for the relative efficiency of the neutron counter, for geometrical factors, for angular resolution and for absorption and. double scattering in the scatterer 10). Angular resolution and double scattering corrections were significant only for the elastic distributions and were less than 10 % except near the first minima of the angular distributions. The relative yields were then converted to differential cross sections by measuring the cross section for elastic scattering at one angle as follows. With no scatterer and no shadow bar, the ratio of neutron counts (not coincidences) to alpha counts, Rn/R,, was measured. The scatterer was then mounted at a scattering

ELASTIC AlqD INELASTIC NEUTRON SCATTERING

183

angle near the second maximum of the elastic angular distribution (e.g. at about 60° for S) and the ratio of delayed coincidences in the elastic peak to alpha counts Rc/r, was measured. Provided that the scatterer completely covers the neutron beam, the differential elastic cross section at this angle is then given by

o(O)-Rdr~ G eo R./R, Na, e,

(1)

Here N is the number of scattering atoms per cm 2 in the scatterer, f], is the solid angle subtended by the alpha counter at the tritium target, ~0/es is the ratio of the efficiencies of the neutron counter for the direct and scattered neutrons, and G is a geometrical factor that corrects for the fact that the scatterer is at a slightly different distance from the neutron counter than the neutron source is. The angle 12~ was determined by counting alpha particles from an Am T M source of known strength, placed at the position of the target. For Mg, Si and S, the direct and scattered neutrons differed in energy by only about 4 ~ and the ratio of efficiencies was unity. This method of measuring the absolute differential cross section is similar to the transmission method of measuring attenuation cross sections, in that neither the source strength nor the absolute efficiency of the neutron counter need be known separately. 3. Results

Examples of time-of-flight spectra are shown in figs. 5, 7, 9 and 11. The energy of the scattered neutrons increases in going from left to right, the right hand peak corresponding to elastic scattering. The peak due to gamma rays from inelastic scattering is well beyond the right-hand end of the time scale. Although for Mg and Si the peaks for the ground state and first-excited states overlapped (as shown in figs. 7 and 9) analysis into two peaks (as illustrated for Mg in fig. 7) was helped by the fact that the shapes, widths and peak separations were known. Both peaks were assumed to have the same shape and width as the peak obtained with no scatterer, when the neutron beam was directed at the neutron counter. This assumption was confirmed for the elastic peak for all scatterers, and for the inelastic peak in the case of carbon, where the peaks were almost completely separated. In all cases the low-energy tail of the elastic peak must be subtracted from the peak of the first-excited state. At low scattering angles where the elastic peak is very strong, the error thereby introduced becomes large and limits the minimum angle at which inelastic scattering can be measured. In fig. 7 the low-energy side of the Mg 2+ inelastic peak is distorted by neutrons scattered inelastically from Mg 26. The correction of the cross section for these neutrons is described later. 3.1. C A R B O N

Two examples of time-of-flight spectra are shown in fig. 5. In the upper spectrum the peaks correspond to the ground state and to the excited states of C x2 at 4.43 and

184

it.

L. C L A R K E

AND

W.

G.

CROSS

7.6 MeV. In the lower spectrum (taken at a different scattering angle) the width of each time channel was increased about three times as compared with the upper spectrum, in order to see neutrons scattered to higher excited states. At all scattering 800

I

700

I

I

F

I

I

J

I

r

t 60

i 70

r so

t so

-4.43

CARBON ( -13.5 °

600

0.41±.01 nS/CHANNEL 500

400

3OO

¢0

°

I

I

I

I

t

F0

6OO

-,oi,-?

400

o

to

~ . 20

J ~o

E 40 ANALYZER

t ~o

too

CHANNEL

Fig. 5. Time-of=flight spectra for scattering from carbon at two different angles. The energY of the scat. tered neutrons increases from left to right. The arrows show the calculated positions of peaks corresponding to known states o f C is. In the upper diagramme each channel represents 0.41 nsec. In the lower diagranlme the time scale is compressed to show scattering to the higher states o f C 1~.

angles, the position of the left-hand peak corresponded to an excitation energy of 9.7__.0.1 MeV. Since the 9.63 MeV level is the only one close to this energy, this peak was assumed to be predominantly from the 9.63 MeV state, although its measured width of about 400 keV (after allowance had been made for instrumental

ELASTIC AND INELASTIC NEUTRON SCATTERING

185

resolution) was greater than the 30 keV width given for this state 27). It is possible that this peak is affected by the very broad 10.1 MeV state as) of C 12. Fig. 6 shows the angular distribution of neutrons scattered from the ground state and three excited states of C 12. The errors on the points shown in this and in the other angular distributions (figs. 8, 10 and 12) are from counting statistics and from uncertainties in the corrections applied for angular resolution and double scattering, IOOC

=

I

I

I

I

1

I

I

I

c**=o.

m ~-

-~ -

~=o

I rl

20"

40"

I

O0*

I

IlO"

I

tOO"

C.M, $CATT~Rt~G

I

~20"

II

¢10°

I

BO°

180*

AMGLE

Fig. 6. The differential cross sections for the scattering of 14 MeV neutrons to the ground state (solid circles), 4.43 MeV state (triangles), 7.6 MeV state (squares) and 9.6 MeV state (open circles)of C11. The solid curve is the calculated optical-model distribution, including the calculated contribution from compound elastic scattering. The dashed curve is the distribution omitting compound nuclear scattering. and do not include an uncertainty of about 7 % in the absolute normalization o f the cross sections. At backward angles where the cross section is low, the principal source of error is uncertainty in the background. The curves drawn are theoretical distributions and are discussed later. The measured distribution for the elastic scattering agrees within the errors with the results of N a k a d a et al. 2) and of Coon et al 4). The distribution observed for the first-excited state agrees with the results of Anderson et al. t6) and of Tesch 13). We have extended the angular distribution down to 13.5 ° , where our data indicate that the cross section for this state levels out, in contrast to the general rising trend at small angles observed both in the neutron scattering data o f these and other 19) authors, and in the 14 MeV proton inelastic scattering results of Peelle 2s). In the proton inelastic scattering

186

R. L. CLARKE ANO W. G. CROSS

results of Nagahara 29) at 14.3 MeV, on the other hand, the shape of the angular distribution at low angles is similar to that shown in fig. 6. The cross section for the 7.66 MeV state was measured only at angles below 40 °. At higher angles this peak was too weak compared with the background for us to obtain an angular distribution. Rethmeier et al. 2o) have reported differential cross sections of about 50 mb/sr at angles of 10°, 20 ° and 40o and have pointed out that their values are much larger than the cross sections for 14 to 18 MeV protons obtained by Peelle 2s). Our cross sections for this state, which agree with the results of Bouchez et al. 3o) at 22 °, are over ten times smaller than those of Rethmeier et al. and are very similar to Peelle's proton results. Other measurements on neutron inelastic scattering give only upper limits to the cross section for this state. In view of the close similarity observed in this energy range between neutrons and protons scattered to the ground, 4.43 and 9.63 MeV states of C 12, it would be surprising if there were a large difference in scattering to the 7.6 MeV state. Measurements of neutrons going to the 7.6 and 9.6 MeV states may be complicated by the reaction C 12(n, ¢x)Be 9 followed by neutron emission. If, as has been suggested a 1), some of these reactions go through the 6.76 MeV state of Be 9, the resulting neutrons would have energies between 4 and 5 MeV. These neutrons would have about the same. energy as those produced at small scattering angles by inelastic scattering to the 9.6 MeV state of C 12, while they would have energies comparable to those of neutrons scattered at large angles to the 7.6 MeV state. We were not able to identify any neutrons from this reaction, and have estimated, on the basis of a HauserFeshbach calculation, that their number should be insignificant. The distribution of neutrons from the 9.6 MeV state exhibits less forward peaking than that from the 4.4 MeV state, although the cross section decreases in the backward hemisphere. The angular distribution found by Heyman et al. ts) is similar to ours but our cross sections are about 40 % smaller than theirs. The angular distribution found by Peelle 2s) for 16.7 MeV protons scattered to this state is also very similar. It is possible that there is a contribution to this peak from neutrons scattered to the broad 10.1 MeV state of C ~2. However, if this state has zero spin, as has been suggested 32), neutrons scattered to it would be expected to have an angular distribution strongly concentrated in the forward direction, quite unlike the observed distribution. Integration of the elastic differential cross section over all angles gives a value of 0.73_+0.07 b, in agreement with the difference of the total and non-elastic cross sections, 0.69+0.02 b. This comparison is shown in table 1 along with the corresponding comparisons for Mg, Si and S. For the latter three distributions the theoretical curve has been used to extrapolate the experimental distribution to small angles where no experimental points were measured. The agreement obtained in all instances with the more accurate values a6-47) of atot- ant is evidence of the accuracy of the normalization of the differential cross sections. The integrated cross section found for scattering to the first-excited state was

187

ELASTIC AND I N E L A S T I C NEUTRON SCATTERING

209 + 25 mb, in agreement with the values 220_+ 25 and 220 + 30 obtained by Anderson et al. 1~) and by Tesch ~3). This is also to be compared with the cross section measured by Beneviste et al. 33) (249+28 rob) and by Deuchaxs and Dandy a4) (290_+50 rob) for the production of 4.4 MeV gamma rays. As nearly all the higher states that can be excited by 14 MeV neutrons axe believed to decay almost completely by alpha emission 3s), these cross sections would be expected to be nearly equal. TABLE 1 Integrated cross sections for elastic and non-elastic scattering

Element

la(O)d¢,o experiment (b)

~ t -(b) -~ne

C Mg Si S

0.734-0.07 0.824-0.10 0.754-0.07 0.744-0.08

0.694-0.02 0.804-0.03 0.774-0.07 0.784-0.05

~a(O)dco calculated (b)

s) b) c) a)

0.69 0.79 0.77 0.76

s-a) Apply only to measurements of total cross sections. s) Refs. 87-,2). b) Refs. 87, ~. ,o, ,a). e) Ref. 87). ~) Refs. ,4, 48,,7). s) Refs. ,7).

~ne experiment (b)

0.604-0.02 e) 0.994-0.02 f) 1.064-0.06 ~) 1.144-0.04 t)

a) Refs. 87,,o).

ane calculated (b)

0.59 0.98 1.06 1.11

e) Refs. 8B,,,-4e).

The integrated cross section for scattering to the 9.6 MeV state was 45 + 10 rob, somewhat lower than the values given by the results of Singletaxy and Wood x7) (96 mb) or of Heyman et al. is) (71 mb). The 4.4 and 9.6 MeV states together account for between 40 and 50 ~ of the total non-elastic cross section 36) of 600 rob. TABLE 2 Cross sections to individual states

Target

C 12

Mg 8' Si 28 S88

-- Q (MeV)

al = Io a(0)dto (mb)

at/ane

4.43

2094-20

0.35

9.6

454-10

0.08

1.37 1.78 2.24 4-6

1684-25 109± 16 644-10 504-20

0.17 O. 11 0.06 0.05

s) Ref. xe). b) Ref. ts). e) Ref. 88). h) Measured at 16.7 MeV.

ai theory (rob)

200

168 104 65

a) Ref. 2,).

Other results 14 MeV ~ 14 MeV neutrons protons

2204-30 t) 2204-4-25 b)

2254-8 e) 2154-10 d)

96 e) 71 4)

51-4-2 e,h)

54-4-9 b) 88-4-12 b) e) Ref. 17).

t) Ref. 18).

153 g) 110 g) 57 s)

g) Ref. 88).

188

R. L. CLARKE AND W. G. CROSS

Comparison of these integrated cross sections with those of other investigators, for both neutrons and protons of 14 MeV, are shown in table 2, along with similar results for Mg, Si and S. 3.2. M A G N E S I U M

Time-of-flight spectra for Mg, at laboratory scattering angles of 45 ° and 120° are shown in fig. 7. The peaks are from the ground state and first-excited states of Mg 24. Inelastic scattering to the levels of Mg 24 at 4.12 and 4.24 MeV would have given a peak near the left hand end of the spectrum. At some scattering angles a peak at the correct position was observed, but at all angles out to 120 ° its intensity was less than one quarter that of the 1.37 MeV peak. I00

~o

o o ~loc

~ Q=O Q -- - 1.37 ,~, { ~ ~ r

~

i

Mg AT

120"

I Q =-1.57 0=0 ,.J-'-1.83

L.[

Mg AT 40

50

0

IO

20

50

ANALYSER CHANNEL Fig. 7. Examples o f time-of-flight spectra for Mg. T h e curves s h o w the estimated shapes a n d positions o f the separate p e a k s for the g r o u n d state a n d 1.37 M e V state o f M g s'. T h e d a s h e d curve is the backg r o u n d for n o scatterer. T h e c h a n n e l width is 0.48 nsec.

Fig. 8 shows the angular distributions obtained for scattering to the ground and first excited states. The curves are again theoretical distributions. For Mg, Si and S, the corrections to the experimental points for angular resolution and double scattering made near the first minimum of the distribution were too large for the results to be accurate. At other angles, uncertainties in these corrections were usually smaller than the statistical errors. The normalization of the angular distribution to absolute cross sections was not made in the manner used for C, Si and S (described previously) but is based on the elastic differential cross section at 65 ° which is taken to be 60 rob/st from the work of Berko et al. 5) and of Cross and Jarvis lo).

ELASTIC AND INELASTIC NEUTRON SCATTERING

189

N o correction to elastic scattering was made for the presence of Mg 25 and Mg 26, which together constitute about 21 ~ of the scatterer, as their differential cross sections are expected to be very similar to that of Mg 24. For scattering to the 1.37 MeV state of Mg 24, a correction to the cross section was made for inelastic scattering from Mg 2s and Mg 26, since neutrons scattered to the 1.61 MeV state of Mg 25 and 1.83 MeV state of Mg 26 were only partly resolved from

_

\

!

I I I MAGNESIUM

-

.

_---

-

~-

t

40*

20"

40'

60"

80"

I00"

|

120=

t tt

t40=

160"

C.M. SCATTERING ANGLE

Fig. 8. The differential cross sections for elastic scattering from Mg and for scattering to the 1.37 MeV state of Mg24. The curves are theoretical, calculated as described in the text. those scattered to the first-excited state of Mg 24. Since the differential cross sections for this latter state are quite similar for 14 to 15 MeV protons 4s, 49) and 14 MeV neutrons, it was assumed that this is also true for the interfering states in Mg 2s and Mg 26. The 17 MeV proton cross sections of Schrank et al. 50) were therefore used in making the correction which amounted to about 5 ~ . Other low-lying states of Mg 25 are not expected to interfere, since they are excited about 10 times less strongly than the 1.61 MeV state, by both 17 MeV protons and 15 MeV deuterons 51). A feature of the cross section for scattering to the 1.37 MeV state is that at all angles greater than 90 ° it is comparable in magnitude with the elastic cross section. The integrated cross section for the 1.37 MeV level is 168-t-25 mb, about 17~o of the

190

R.

L. C L A R K E A N D W .

O . CROSS

total non-elastic cross section. This is similar to the values found for inelastic, scattering of protons to this level; 158 mb at 14 MeV and 112 mb at 18 MeV (refs. 4s, 52), respectively). The angular distributions are also similar for neutrons and protons, as is shown in fig. 13. 3.3. S I L I C O N

Fig. 9 shows the spectrum for Si measured at a laboratory angle of 61 °. The lines indicate the peak positions that would correspond to the known s3) states of Si 28. Fig. 10 shows the angular distribution of neutrons leading to the ground state and first-excited state at 1.78 MeV. Inelastic scattering from Si 29 and Sis°, which together comprise about 8 ~o of natural Si, would not be expected to distort the angular I 7(

I

I

I

I

I

SILICON El • 6 1 "

4. Io

+ q"

i

--4 I

.J

I

z :£

S0--

0.53

x~ .

ser./¢mmd

3O

,,ltt

z o

1111

'°-

"t'

t

\ I

0

20

40

ANALYSER

60

Io

I °°* 100

CHANNEL

Fig. 9. T h e time-of-flight s p e c t r u m for silicon at a scattering angle o f 61 °. A t the right h a n d part o f t h e figure the ordinates are reduced five times. T h e arrows s h o w t h e calculated flight times for n e u t r o n s scattered to the k n o w n s t a t e s o f Si".

distribution of the first-excited state of Si 28 by more than a few percent. The curves shown are theoretical. The integrated cross sections for elastic scattering and for scattering to the first excited state are shown in tables 1 and 2. In fig. 9 there is also clear evidence for excitation of one or both of the states at 4.61 and 4.97 MeV and some for one of the states near 6.9 MeV, the differential cross sections for these two peaks being about 3.6 mb/sr and 2.4 rob/st, respectively. Excitation of the 6.28 MeV state at this angle is comparatively weak. N o angular distribution measurements were made on these states.

ELASTIC A N D INELASTIC NEUTRON SCATTERING

19:1

3.4. SULPHUR A n example o f a time-of-flight spectrum f r o m S is s h o w n in fig. 11, for a scattering angle o f 44.6 ° . The peaks are due to elastic scattering, inelastic scattering t o the firstexcited state o f S 32 at 2.24 M e V and to t w o or m o r e unresolved levels between 4 and 6 MeV. The lines s h o w the expected positions o f peaks f r o m k n o w n 53) states I000,

I

I

I

I

I

I

--

SILICON

IOC

-t

I0-0

~

(~'

20"

40 +

60e C.M.

Q • - I . 7 8 MeV

80e

SCATTERING

I00"

120"

"-<

140"

160 e

ANGLE

Fig. 10. The differential cross sections for elastic scattering from Si and for scattering to the 1.78 MeV state of Si". The curves are theoretical, calculated as described in the text. o f S a2. Scattering to other isotopes o f S is not expected to contribute m o r e than a few percent to these peaks. At this angle, no scattering to the second-excited state at 3.78 M e V was observed, as is to be expected if this is a zero-spin state s+, ss). Fig. 12 s h o w s the angular distributions for neutrons leading to the ground and first-excited states. The elastic distribution agrees at forward angles with the results o f St. Pierre et al. 7), but at backward angles our points are lower. Other measurements o f this cross section have been m a d e by Elliot 3), Tesch 13) and Strizhak et aL 12). In fig. 15 is s h o w n the angular distribution going to states between 4 and and 6 M e V . The integrated cross section for this group is 50_+ 20 mb.

192

R. L. CLARKE AND W . (3. CROSS

80

I

I

I

I

I

1

SULPHUR

70

O = 44.6

°

60

g

6

4O

J

2O

0

I0

20

3£)

40

50

Anolyser

60

70

80

9O

KX)

Chonnel

Fig. 11. The time-of-flight spectrum for sulphur, at a scattering angle o f 44.6 °. The arrows show the calculated flight times for neutrons scattered to the k n o w n states o f S 8s. The channel width is 0.41 nsec.

r

[

I

I

I

_

SULPHUR

Q=O

*I IO'~-

. J

Q = - 2.24MeV

~__

,I o~

[]

I

4O' C.M.

SCATTERING

ANGLE

Fig. 12. The differential cross sections for elastic scattering from S and for scattering to the 2.24 MvV state o f S s~. The curves arc theoretical, calculated as described in the text.

ELASTIC AND INELASTIC NEUTRON SCATTERING I

193

4. Discussion 4.1. E L A S T I C S C A T T E R I N G

The elastic scattering results of the present experiments have been compared with optical model calculations. The computer programmes for this and for other calculations described below were developed by J. M. Kennedy and D. MePherson. The potential used has the form 14) (2)

U = V f ( r ) + i W . f ( r ) + iWsg(r ) + Vs. ,,

where l+exp

f(r)=

g(r) = exp

-

Vo.t = Vs

-

r

f ( r ) a " I.

The nuclear radius R, is given by R = 1.25 A ~ fro. The surface thickness parameters a and b and the four well-depth parameters were chosen to fit both the measured differential cross sections and the total and non-elastic cross sections; the latter was fitted to the difference of the calculated reaction and compound elastic cross sections. As the non-elastic and total cross sections have been measured more accurately than the differential cross sections, only sets of parameters that gave good agreement with measured values of o'tot and ane were considered. The comparison with the total elastic and non-elastic cross sections is shown in table 1, for the parameters also giving the closest agreement with the elastic angular distribution. TABLE 3 Optical-model parameters for elastic scattering Element

RAi

a

b

V

Wv

Ws

V8

C

1.25

0.50

0.35

--50

--6

--10

+6

Mg, Si, S

1.25

0.70

1.10

--44

--2

-- 7

+6

All distances are in fro, potentials are in MeV.

An automatic search programme was used to obtain parameters giving a leastsquares fit to each angular distribution. Each experimental point was given a weight proportional to [A(0)] -2 where A(O) is the experimental error. For Mg, Si and S, a single set of parameters was then selected to give the best overall fit to the three distributions. The parameters for carbon were determined independently. The parameters giving the best agreement with experiment are listed in table 3. They differ only slightly from those used by Bjorklund and Fernbach i4) for 14 MeV neutron

194

R.L.

CLARKE AND W. O. CROSS

scattering and by Wong et aL 56) for the scattering of 24 MeV neutrons from heavier nuclei. The calculated differential cross sections are shown in figs. 6, 8, 10 and 12. Although both a surface and a volume term were used for the imaginary part of the potential, almost as good agreement with experiment was obtained for Mg, Si and S, using either only a surface absorption term (with W, = - 9 MeV) or predominantly volume absorption. The main effect of the spin-orbit potential is to remove deep minima in the calculated angular distributions near 140° . At angles below 90 ° this term makes virtually no difference. The incident neutrons were assumed to be unpolarized 57). An estimate of the contribution of compound elastic scattering was made by a statistical calculation, using the expression given, for example, by Lane and Thomas (eq. (3.18) of ref. 5s)), which takes into account the requirements of angular momentum conservation. Transmission functions were calculated from the optical model, with the same parameters that were used to match the shape-elastic scattering. For Mg, the parameters in the expression used for the level densities were chosen to match the known levels s3-59) in Mg 24, Na 24 and Ne 2x. The total compound elastic cross section for Mg was thus estimated to be 3 mb and the differential cross section between 90 ° and 140° was 0.23 mb/sr, which can be considered negligible in comparison with experimental values. For Si and S the calculated compound elastic cross section was just over half as large as for Mg. For these three nuclei the general agreement between the calculated and experimental elastic distributions is comparable to that obtained for heavier nuclei 14). Closer agreement can be obtained by adjusting the parameters of each nucleus separately. Scattering to the first-excited state is sufficiently strong, particularly in the case of Mg, that coupling between elastic and inelastic scattering may distort the elastic angular distribution appreciably. If coupled equations were used in the calculation 60), the parameters required to fit the experimental results might be significantly different from those given in table 3. For carbon, the parameters chosen were again required first to give reasonable total and absorption cross sections and then to give the best fit to the angular distribution. Table 3 lists the parameters for the curve shown in fig. 6. As was found for Mg, Si and S, an almost equally good fit was obtained using only a surface term for the imaginary potential. The agreement with the experimental angular distribution is not very good at angles greater than 100°, as has also been observed in previous attempts to calculate proton scattering from carbon 24). With parameters similar to those given by Nodvik et al. for protons ~1), the elastic differential cross section is reproduced much more closely, but the calculated absorption cross section is between 300 and 400 mb (as was found for calculations 61) on protons) as compared with the experimental value for the non-elastic neutron cross section of 600___30 mb (refs. 36,44,46, 47)). The cross section for compound elastic scattering was estimated in the manner described for Mg, except that the resulting value was multiplied by (trreaet--O'dir)/trreaot,

ELASTIC AND INELASTIC NEUTRON SCATTERING

195

where areact is the calculated reaction cross-section and trdir is the sum of the experimental cross-sections for the 4.4 and 9.6 MeV states. For carbon almost all the possible final states are known, and in the expression for the competition among final states the integration in the denominator (of eq. (3.8) of ref. 5a)) was replaced by a summation over the final states of C 12 and Be 9. In fig. 6, the dashed curve shows the differential cross section calculated without the compound nuclear cross section, while the solid curve includes this contribution. It is possible that only a few levels in the compound nucleus are excited. However, even if only compound levels of spin Jr were excited - a circumstance that would enhance decay to the ground state of C 12 - the calculated compound elastic cross section would only be increased by a factor of about 1.5, as compared with the value calculated for a statistical distribution of compound state spins. It therefore appears very unlikely that compound nuclear effects can account for the difference between the calculated and observed elastic distributions. It is possible that much of this difference arises from coupling between elastic scattering and the strongly excited inelastic scattering to the 4.43 MeV state. Evidence that the possible excitation of only a few compound nuclear levels does not strongly effect the angular distribution for carbon is given by comparison of the distribution of fig. 6 with that of Anderson et aL 16). Although the latter results were obtained for neutrons for which the energy varied with scattering angle from 13.6 to 14.7 MeV, the angular distribution is (except in the angular region below 30 °) in agreement with the present results. Further evidence is given by the facts that the differential cross sections (a) are very similar for neutrons and protons in this energy range and (b) change slowly and regularly with energy 2s, 2 9 ) for the scattering of protons of energies from 14 to 18 MeV. 4.2. INELASTIC SCATTERING TO FIRST-EXCITED STATES It has been shown theoretically by Pinkston and Satchler 23) and confirmed experimentally for protons by Cohen and Rubin 62) that at small angles the cross section for inelastic nucleon scattering leading to a given level is related to B ( E - l ) , the electric/-pole transition probability between that level and the ground state. Since for transitions from the first-excited 2 + state of even nuclei B ( E - 2 ) i s strongly enhanced by collective effects, as compared with the calculated single particle transition probability, neutron scattering to these 2 + states would be expected to have a relatively large cross section. This expectation is borne out by the spectra of scattered neutrons shown in figs. 5, 7, 9 and 11. Similarly, higher collective states should also be excited, although with a smaller cross section. As 3- states have been relatively strongly excited by protons 63), deuterons 64), alpha particles 65) and electrons 66), it is expected that they would also be strongly excited by fast neutrons. This is observed for the 9.6 MeV level of C 12 and possibly for a 3 - state in S. In fig. 13 is shown a comparison of the present neutron results with the inelastic scattering of 14 MeV protons to the first-excited states of Mg 24, Si 2s and S 32, taken from the results of Oda et al. 4a). While there are differences in the detailed angular

196

R.L.

CLARKE

AND

W.

G.

CROSS

distributions, the general shapes and particularly the absolute values of the differential cross sections are very similar. A comparison of the scattering of 14 MeV protons and neutrons to the first excited state of C 12 has been given by Anderson et aL 16). Further comparison of the integrated cross sections for these levels is given in columns 3, 6 and 7 of table 2.

I

I

-1-

I

I

I 1

40-20-I0-6->

4d--

~i_ 2(1-E G

-$32 ~

~+_

I

20*

I

40*

I

6G"

I

80*

I

I00*

I

120"

l

14G* 160e

C.M. SCATTERING ANGLE

Fig. 13. Comparison between the inelastic scattering of 14 MeV neutrons and protons to the first excited states of Mg ~4, Si ~s and S82. The solid curves represent the proton results of Oda et aL 48) while the points show the present results for 14.1 MeV neutrons. For SP 8 all cross sections have been multiplied by 2. The dashed curve shows the calculated angular distribution of Glendenning s2), normalized to match the experimental results for Mg 2..

Inelastic scattering to the first-excited states of these four nuclei was calculated with the distorted wave Born approximation. The real potential Vf(r) of eq. (2) was modified by making the nuclear radius R a function of angle, according to

R(o) = RG[I+#2 Y2G(0)]

(3)

for a nucleus with a permanent quadrupole deformation and

R(O, qS) = R G [ I + /.L~ 2 , Y2u(O,qg)]

(4)

for the quadrupole vibrations of a spherical nucleus. In the latter case, let//2 denote the expectation value of ~(~2,) 2. The quantity//2, thus introduced for both rotations and vibrations, does not affect the inelastic angular distributions but does affect the magnitude of the inelastic cross section which is proportional to R~ V2//2. The para-

ELASTIC AND INELASTIC NEUTRON SCATTERING

197

meters V and Ro were taken to be the same as those used to match the elastic scattering, so that flz was the only free parameter available to match the magnitude of the cross section to the first-excited state. In calculatiiag inelastic scattering, the real and imaginary parts of the potential were taken to have an energy dependence 67) V = Vo+½E and Ws = Wo-½E, where E is the neutron energy in MeV. For carbon, Vo = - 57 MeV, W o = =- 3 MeV and V¢', = 1 - ½ E MeV, while for Mg, Si and S, 1Io = - 5 1 MeV, W o = 0 and Wv was taken to have a constant value of - 2 MeV. For the incident 14 MeV neutrons the potentials are thus the same as those (shown in table 3) giving good agreement with elastic scattering results. The other opticalmodel parameters are those given in table 3, except that no spin-orbit term was used. It is not expected that this would affect significantly either the angular distribution or the absolute value of the cross section. Compound nuclear scattering to the first-excited state was estimated as described previously for the ground state. For Mg, Si and S, the calculated compound nuclear scattering has no significant effect on the inelastic angular distribution, and changes only slightly the value of f12 required for normalizing the calculated cross section to the experimental data. The agreement between the calculated and experimental angular distribution shown in figs. 8, 10 and 12, is excellent for Mg and reasonable for Si and S. It has been shown by Glendenning 22) that, with the assumption that the interaction is concentrated in the surface region, the shape of the inelastic angular distribution depends primarily on the nuclear radius and the spins and parities of the initial and final states and is relatively insensitive to the details of the potential well. In fig. 13, the dashed curve shows the angular distribution calculated by Glendenning for a radius similar to those of Mg and Si. Since this curve agrees almost as well with the experimental results as does the angular distribution calculated here on the basis of both surface and volume absorption, we conclude that a comparison of angular distributions alone yields only very limited information on the nuclear potential. The comparison of the absolute cross sections with theory is more significant. A value for the deformation parameter f12 can also be derived from measured B(E-2) probabilities, as determined from heavy-ion Coulomb excitations 68-7 o), high energy electron scattering 71) or resonance fluorescence measurements of lifetimes 7z- 76). On the assumption that the first-excited state of an even nucleus is represented by the irrotational motions of an incompressible, uniformly-charged liquid with a sharp boundary, flz isrelated to B(E--2) and to the collective enhancement factor B(E-2)/ B(E-2)s.p. to the first order by f12_

B(E-2)

(3ZeR2/4r:)2

_

4rr B ( E - 2 ) 5Z 2 B(E-2)s.p."

(5)

In table 4 are shown the values of f12 used to normalize the calculated neutron cross sections to the experimental results. These values are compared with those derived,

198

R. L. CLARKE AND W. O. CROSS

according to eq.(5), from electromagnetic measurements, for which the results are given in columns 2 and 3. In calculating/~2, the electromagnetic radius of the nucleus was taken to be 1.2A+ fm. For Mg, Si and S, the agreement between the f12 parameters from neutron scattering and from electromagnetic measurements is within the uncertainties of the experiments and of the correctness of the factor in eq. (5) relating/~ to B ( E - 2 ) . These values of f12 are also in agreement with calculations 77) based on the Nilsson model. TABLE 4 Comparison of fl+ from neutron scattering a n d electromagnetic measurements Nucleus C is

Mg 2'

Si *s

S s+

z mean (10 -~a see)

B(E-- 2) e2× 10 -4s era.4

f12 Electromagnetic

0.534-0.11 s) 0 . 6 5 ± 0 . 1 2 b)

0.62 0.56

19.0 4-2.0 s) 14.0 ± 4 . 0 o)

fll Neutrons 0.82

0.034±0.007 a) 0.054 e) 0.065 f)

0.61 0.71 0.54 0.68 0.75

0.62

0.025 ± 0.005 a) 0.025 e) 0.046 ~)

0.44 0.40 0.36 0.36 0.48

0.43

0.38

0.32

6.0 4-1.1 s) 7.3 4-2.2 ~)

1.6 4-0.2 s)

I) Ref. 7o). b) Ref. 71). e) Weighted m e a n of refs. 7s-Ts). a) Ref. 6s) a n d private communication from authors, e) Ref. 69). h) Ref. 7+). l) Ref. 7s). J) Ref. 78).

~) Ref. 70).

a) Ref. 7a).

The value of/~2 for Mg 2+ derived from neutron scattering results on the basis of the distorted wave Born approximation is considerably larger than that derived on the strong-absorption diffraction model from the inelastic scattering of 42 MeV alpha particles 7s). At least part of this difference is probably due to the large nuclear radius used in the latter analysis, some of which can be ascribed to the alpha particle 7s). For scattering to the first-excited state of C t2, the predicted angular distribution differs considerably from that observed, as shown in fig. 14. Calculations 2+) on proton angular distributions show a similar discrepancy in this energy region. The calculated compound nuclear scattering to this state is almost isotropic (with a differential cross section of about 4 mb/sr) and hence decreases the ratio of the calculated differential cross section at forward angles to that near 90 °, thereby making the discrepancy between theoretical and experimental angular distributions worse. It is not expected that the possibl,~ excitation of relatively few compound states would

ELASTIC AND

INELASTIC NEUTRON

199

SCATTERING

alter the angular distribution very much, for the reasons given in the discussion of elastic scattering. The value offl2 for carbon, shown in the last column of table 4, was chosen to match the integrated cross section to the 4.43 MeV state, and is larger than the fie derived f r o m electron scattering 71) and resonance fluorescence 7z) by a factor of 1.4. 100

-_

I

I

I c '2 I

1

I

I

I

-

60

o

40

20

I0

4 m

~

2 D

)o

I

20 °

I

40 °

I

I

60 ° C.M.

80*

I

100"

SCATTERING

I

120"

I

140 )

I

160"

1800

ANGLE

Fig. 14. The differential cross sections for inelastic scattering of 14 MeV neutrons to the 4.43 MeV of Cxm.The curves are theoretical, calculated as described in the text. The solid curve includes the effect of compound nuclear scattering, while the dashed curve omits this.

state

These discrepancies in both the shape and absolute value of the differential cross section m a y arise from failure to take account of the strong coupling between elastic scattering and inelastic scattering to this state. The effect should be less important for Mg, Si and S, for which scattering to the first-excited state represents a smaller fraction of the total non-elastic cross section. For carbon some of the discrepancy m a y also arise f r o m the general inadequacy of the optical model for such a light nucleus, but no conclusion can be reached until the use of coupled equations has been thoroughly explored. 4.3. EXCITATION OF HIGHER STATES

Carbon. The strong forward peaking noted in scattering to the 7.6 MeV state of C 12 is in agreement with the proton results of Peelle 2s) and consistent with the prediction, on the basis o f a 0 + assignment for this state, of a small cross section and a forward peaked angular distribution similar to that found for elastic scattering. The ratio of the cross sections for scattering to the 3 - state at 9.6 MeV and the 2 + state at 4.43 MeV is about one quarter, in qualitative agreement with the comparable ratios found for high energy electron scattering in heavier nuclei 66) and for alpha particle scattering 6s). These results imply a ratio of the octupole to quadrupole deformation

R . L . CLARKE AND W. O. CROSS

200

parameters fla/fl2 of the order of one half. In the case of C 12, however, it is not at all certain that the 9.6 MeV state is collective. Magnesium. Scattering to one or both of the 2 +. and 4 + states near 4.2 MeV in Mg 24 was observed. The relatively small yield (compared with that of the lowest 2 + state) is consistent with collective-model direct-interaction predictions, if the second 2 + state is not a single-phonon rotational state. These states have also been observed in the inelastic scattering of 14 MeV protons ,8) and 15 MeV deuterons 78). Silicon. At 35 ° the inelastic peak corresponding to a mean Q of - 4 . 6 MeV was observed to have a cross section about half that of the lowest 2 ÷ state of Si 2s - a considerably higher ratio than was found for the 4 + state of 1V[g2.. It is expected that the 0 + state 5,) at 4.93 MeV was only weakly excited and that most of this peak was due to the 4+ state 54) at 4.61 MeV. This state was also excited in the inelastic scattering of 14.2 MeV protons ,8). The observed cross section for neutrons was several times larger than that calculated for compound nuclear scattering, using the Hauser-Feshbach model. The other inelastic peak observed (fig. 9) appears to be due to one or more of the three states between 6 and 7 MeV. The assignments of the two states near 6.9 MeV are not known. A level in this region was prominently excited by 185 MeV protons so). It therefore, seems likely that at least one of these states has collective character. I000 I

SULPHUR

100 I

,ol

O = - 5 MeV

r t-

LI O'~!o

I 20=

i 40 ° 60 ° 80 = I00= 120= 140= 160" C.M. SCATTERING ANGLE

Fig. 15. The differential cross sections for neutrons scattered inelastically from S, corresponding in energy to states in Saa between 4 and 6 McV. The curve has no theoretical significance.

ELASTIC AND INELASTIC NEUTRON SCATTERING

201

Sulphur. The

broad peak corresponding t o Q ~ - 5 MeV (fig. 11) has an angular distribution peaked in the forward direction, as shown in fig. 15. It could be attributed to some or all of the known states of S 32 at 4.29, 4.47, 4.70 and 5.01 MeV. Some of these states were also seen as an unresolved group in the inelastic scattering of 187 MeV electrons 7 x) and 185 MeV protons 80). For neutrons, the total cross section for exciting this group is comparable to that for the 2 + state at 2.24 MeV, just as was found for excitation by 42 MeV alpha particles 8x) of a state near 5.0 MeV. This state has a spin s4, 82) of 3 and negative parity 81), and its relatively strong excitation by alpha particles suggests that it is collective. Hauser-Feshbach calculations indicate that a substantial fraction of the neutron peak we observed could arise from compound nuclear scattering to the four states. 5. Conclusions The angular distributions of 14 MeV neutrons elastically scattered from Mg, Si and S are reasonably well described by the optical model. Inelastic scattering leading to the first-excited states of these nuclei can be described by the distorted-wave directinteraction approximation, and the observed cross sections are consistent, within the experimental errors, with those derived from the value of B(E-2) for these states, as determined from electron scattering, Coulomb excitation and lifetime experiments. This good agreement supports the ~asefulness of inelastic scattering of medium energy nucleons as a means of determining nuclear deformation parameters. Both the magnitude and general shape of the angular distributions for these states are similar for excitation by 14 MeV neutrons and 14 MeV protons. The angular distribution for elastic scattering from carbon is fitted only in tile forward hemisphere. At backward angles we have not obtained agreement with the observed distribution using parameters that also give agreement with the total non-elastic cross section. For inelastic scattering to the first excited state of C 12, the experimental observations are in disagreement with both the angular distribution and magnitude calculated on the distorted wave Born approximation, using the optical model parameters found for elastic scattering and the B ( E - 2 ) determined from lifetime measurements. The cause of these disagreements may be the incorrectness of treating elastic and inelastic scattering independently or the inadequacy of the optical model for very light nuclei. Compound nuclear effects do not appear to contribute significantly to these discrepancies. Some higher excited states in C, Mg, Si and S have been observed by neutron scattering, and what little is known of the angular distributions of them is in qualitative agreement with direct interaction theory. We are indebted to D. McPherson and E.W. Vogt for many valuable discussions. We are also grateful to J. M. Kennedy and D. McPherson for developing the computer programme used for all the optical and statistical model calculations.

202

R.L. CLARKE AND W. G. CROSS References

1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) 26) 27) 28) 29) 30) 31) 32) 33) 34) 35) 36) 37) 38) 39) 40) 41) 42) 43) 44) 45) 46) 47) 48) 49)

Anderson, Gardner, Nakada and Wong, Phys. Rev. 110 (1958) 160 Nakada, Anderson, Gardner and Wong, Phys. Rev. 110 (1958) 1439 J. O. Elliot, Phys. Rev. 101 (1956) 684 Coon, Davis, Felthauser and Nicodemus, Phys. Rev. I U (1958) 250 Berko, Whitehead and Groseelose, Nuclear Physics 6 (1958) 210 K. Yuasa, J. Phys. See. Japan 13 (1958) 1248 St. Pierre, Machwe and Lorrain, Phys. Rev. 115 (1959) 999 Anderson, Gardner, McClure, Nakada and Wong, Phys. Rev. 115 (1959) 1010 L. A. Rayburn, Phys. Rev. 116 (1959) 1571 W. G. Cross and R. G. Jarvis, Nuclear Physics 15 (1960) 155 F. G. J. Percy and P. Lorrain, Bull. Amer. Phys. See. 5 (1960) 18 Strizhak, Bobyr and Grona, JETP (Soviet Physics) 14 (1962) 225 K. Tesch, Nuclear Physics 37 (1962) 412 F. E. Bjorklund and S. Fernbach, Phys. Rev. 109 (1958) 1295, Univ. of California Report U.C.R.L. 4926-T (1957) unpublished; Bjorklund, Fernbach and Sherman, Phys. Rev. 101 (1956) 1832 W. S. Emmerich and R. M. Sinclair, Phys. Key. 104 (1956) 1399; Lukyanov, Orlov and Turovtsev, Nuclear Physics $ (1958) 325, 35 (1962) 71 Anderson, Gardner, McClure, Nakada and Wong, Phys. Rev. U l (1958) 572 J. B. Singletary and D. E. Wood, Phys. Rev. 114 (1959) 1595 Heyman, J~r6mie, Kahane and Sene, J. Phys. Rad. 21 (1960) 380 Bobyr, Grona and Strizhak, JETP (Soviet Physics) 14 (1962) 18 Rethmeier, Jonker, Rodenburg, Hovenier and van der Meulen, Nuclear Physics 38 (1962) 322 P. H. Stelson, private communication N. K. Glendenning, Phys. Rev. 114 (1959) 1297 W. T. Pinkston and G. R. Satehler, Nuclear Physics 27 (1961) 270 C. A. Levinson and M. K. Banerjee, Ann. Phys. 3 (1958) 67 R. L. Clarke, Can. J. Phys. 39 (1961) 957 B. Johansson, Nucl. Instr. 1 (1957) 274 Dunbar, Pixley, Wentzel and Whaling, Phys. Rev. 92 (1953) 649 R. W. Peelle, Phys. Rev. 105 (1957) 1311 Y. Nagahara, J. Phys. See. Japan 16 (1961) 133 Bouchez, Dubus, Duclos, Hamilton, Perrin and Quivy, Comptes Rend. 254 (1962) 2744; Bouchez, Duclos and Perrin, Nuclear Physics 43 (1963) 628 Barjon, Flamant, Perehereau and Rode, Nuclear Physics 36 (1962) 247 Cook, Fowler, Lauritsen and Lauritsen, Phys. Rev. 111 (1958) 567 Beneviste, Mitchell, Schrader and Zenger, Nuclear Physics 19 (1960) 448 W. M. Deuchars and D. Dandy, Prec. Phys. See. A75 (1960) 855 F. Ajzenberg-Selove and T. Lanritsen, Nuclear Physics U (1959) 1 E.R. Graves and R. W. Davis, Phys. Rev. 97 (1955) 1205 Coon, Graves and Barshall, Phys. Rev. 88 (1952) 562 Bratenahl, Peterson and Steering, Phys. Rev. 110 (1958) 927 J. P. Conner, Phys. Rev. 109 (1958) 1268 C. F. Cook and T. W. Bonner, Phys. Rev. 94 (1954) 651 Poss, Salant, Snow and Yuan, Phys. Rev. 87 (1952) 11 Fossan, Walter, Wilson and Barshall, Phys. Rev. 123 (1961) 209 J. F. Vervier and A. Martegani, Phys. Rev. 109 (1958) 947 MacGregor, Ball and Booth, Phys. Rev. 108 (1957) 726 V. 1. Strizhak, Sov. J. Atom. Energ. 2 (1957) 72 V. M. Gorbachev and L. B. Poretskii, Sov. J. Atom. Energ. 4 (1958) 259 N. N. Flerov and V. M. Talyzin, J. Nucl. Energ. 4 (1957) 529 Oda, Takeda, Takano, Yamazaki, Hu, Kikuchi, Kobayashi, Matsuda and Nagahara, J. Phys. See. Japan 15 (1960) 760 Matsuda, Nagahara, Oda, Yamamuro and Kobayashi, Nuclear Physics 27 (1961) 1

ELASTIC A N D 50) 51) 52) 53) 54) 55) 56) 57) 58) 59) 60) 61) 62) 63) 64) 65)

66) 67) 68) 69) 70) 71) 72) 73) 74) 75) 76) 77) 78) 79) 80) 81) 82)

INELASTIC

NEUTRONSCATTERING

203

Schrank, Warburton and Daehnick, Phys. Rev. 127 (1962) 2159 A. G. Blair and E. W. Hamburger, Phys. Rev. 122 (1961) 566 P. C. Gugelot and P. R. Phillips, Phys. Rev. 101 (1956) 1614, quoted in ref. 50) P. M. Endt and C. van tier Leun, Nuclear Physics 34 (1962) 1 C. Broude and H. E. Gove, Ann. of Phys., to be published Wakatsuki, Hirao, Okada and Miura, Prog. Theor. Phys. 24 (1960) 918 Wong, Anderson, McClure and Walker, Phys. Rev. 128 (1962) 2339; Stuart, Anderson and Wong, Phys. Rev. 125 (1962) 276 R. B. Perkins and J. E. Simmons, Phys. Rev. 124 (1961) 1153; P. J. Pasma, Nuclear Physics 6 (1958) 141 A. M. Lane and R. G. Thomas, Revs. Mod. Phys. 30 (1958) 257 C. T. Hibdon, Phys. Rev. 124 (1961) 500 B. Buck, Phys. Rev. 130 (1963) 712 Nodvik, Duke and Melkanoff, Phys. Rev. 125 (1962) 975 B. L. Cohen and A. (3. Rubin, Phys. Rev. 111 (1958) 1568 K. Matsuda, Nuclear Physics 33 (1962) 536 B. L. Cohen and R. E. Price, Phys. Rev. 123 (1962) 283; Jahr, Miiller, Oswald and Schmidt-Rohr, Z. Phys. 161 (1961) 509 A. L Yavin and G. W. Farwell, Nuclear Physics 12 (1959) 1; McDaniels, Blair, Chert and Farwell, Nuclear Physics 17 (1960) 614; Crut, Swectman and Wall, Nuclear Physics 17 (1960) 655 Crannell, Helm, Kendall, Oeser and Yearian, Phys. Rev. 123 (1961) 923 Melkanoff, Moszkowski, Nodvik and Saxon, Phys. Rev. 101 (1956) 507 H. E. Gove and C. Broude, in Prec. Second Conf. on Reactions between Complex Nuclei (John Wiley and Sons, New York, 1960) p. 57 Alkhazov, Grinbcrg, Gusinskii, Erokhina and Lemberg, JETP (Soviet Physics) 35 (1959) 736 Andreyev, Grinbcrg, Erokhina and Lembcrg, Nuclear Physics 19 (1960) 400 R. H. Helm, Phys. Rev. 104 (1956) 1466 Rasmussen, Metzger and Swarm, Phys Rev. 110 (1958) 154 S. Ofer and A. Schwartzchild, Phys. Rev. Lett. 3 (1959) 384 Metzger, Swarm and Rasmussen, Nuclear Physics 16 (1960) 568 Arns, Sund and Wiedenbeck, Phys. Rev. Lett. 2 (1959) 50 N. N. Delyagin and V. S. Shpinel, Aead. Sci. U.S.S.R. Bull. 22 (1958) 855 H. E. Gove, in Prec. Int. Conf. on Nuclear Structure, Kingston, Canada (University of Toronto Press, 1960) p. 438 J. S. Blair, G. W. Farwell and D. K. McDaniels, Nuclear Physics 17 (1960) 641 J. L. Yntema and B. Zeidman, Phys. Rev. 114 (1959) 815 H. Tyr6n and Th. A. J. Marls, Nuclear Physics 6 (1958) 446 J. S. Blair, in Prec. Int. Conf. on the Nuclear Optical Model (Florida State University, Tallahassee, Florida, 1959) p. 116 J. S. Blair, Phys. Rev. 115 (1959) 928