Levels in 11B from 7Li(α, α)7Li and 7Li(α, α')7Li∗(0.48)

Nuclear Physics 86 (1966) 481--508; (~) North-Holland Publishing Co., Amsterdam

Not to be reproduced by photoprint or microfilmwithout written permissionfrom the publisher

L E V E L S I N 11B FROM

7Li(o~,~)7Li A N D 7Li(a¢, ~')7Li*(0.48)

R. Y, CUSSON California Institute of Technology, Pasadena, California t Received 12 April 1966 Abstract: The absolute differential cross sections for the reactions 7Li(~, :07Li and 7Li(~, ~')7Li*(0.48) have been measured in the range of energies and angles, 1.6 ~ E~L "< 12.0 MeV, 20° ~ O~L 150°, using thin 7Li targets. Solid state detectors were used to detect the scattered s-particles. A total of 19 scattering anomalies has been observed in this energy range. In the excitation energy range 9.88 ~
NUCLEAR

L

REACTIONS

~Li(~, ~'), E = 1.6-12.0 MeV; measured or(E;

E~,,O).

11B deduced levels, J, z~. Enriched target.

1. Introduction T h e energy level structure o f l i b has been investigated extensively u p to a b o u t 8.92 M e V excitation by studying e l e c t r o m a g n e t i c transitions resulting f r o m 3He b o m b a r m e n t o f 9Be a n d d e u t e r o n reactions 1) with l ° B . The r e a c t i o n 7Li(~, 7 ) t l B * has been used to study the n a r r o w states 2) at 8.92, 9.19 a n d 9.28 M e V in 11B. By l o o k i n g at the d e - e x c i t a t i o n y-ray o f the 0.478 M e V state in 7Li, p r o d u c e d in the r e a c t i o n 7Li+c~ --. 7 L i * ( 0 . 4 8 ) + ~ ~ 7 L i + ~ + y ,

(1)

higher states in 11B, b e g i n n i n g at 9.88 M e V excitation, have been observed 3-5). This r e a c t i o n has been studied u p to E,L ---: 6.0 M e V by Bichsel a n d Bonner. A b o v e 11.4 M e V excitation, i n f o r m a t i o n has been available m o s t l y t h r o u g h n e u t r o n reactions 3 - 7 ) . H i g h - e n e r g y inelastic p r o t o n scattering on 11B has p r o v e d a useful t o o l 8) in energy regions where the level spacing in I~B is o f the o r d e r o f 1 MeV. The stripp i n g r e a c t i o n s 9Be(3He, p ) l l B a n d l ° B ( d , p)11B have been recently r e - e x a m i n e d b y G r o c e et al. 1 o) a n d confirm the existence o f a n a r r o w state at 10.33 M e V excitation, which was first o b s e r v e d w e a k l y b y Elkins ~ ) in the r e a c t i o n l ° B ( d , p)11B. G r o c e et al. r e p o r t e d states at 9.87, 10.33, 10.60, 11.27, 11.46, 11.88, 12.56, 13.3 a n d 14.56 M e V excitation b u t were u n a b l e to o b t a i n spin a n d p a r i t y assignments. t Work supported in part by the Office of Naval Research Contract [Nonr-220(47)]. 481 October 1966

482

R.Y. ctJssoN

The (VLi, c0 channel which opens at 8.67 MeV excitation and the (VLi*(0.48), e') channel which opens at 9.17 MeV provide the only particle decay modes of ~~B up to about 11.4 MeV where several other channels open 6) within a few hundred keV. The present experiment was undertaken in the hope that the relatively simple twochannel problem involved might allow the determination of the spin and parity of some of the levels contained in this energy region. For this purpose, the elastic and inelastic absolute differential cross sections were measured as a function of energy and angle. All the levels previously reported between 9.87 and 11.46 MeV were observed, and possible assignments of spins and parities were obtained.

2. Experimental Method The alpha-particle beam from an electrostatic accelerator was used to bombard thin 7Li(OH) and 7Li2F targets on carbon foil backings. For energies below 3.22 MeV, a 3 MV accelerator, provided with a 14 cm diam scattering chamber, was used. Above BEAM DEFINING (TA) CUP SLITS V F- REGULATION

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Fig. 1. Schematic diagram of target chamber and beam collimation system. This arrangement was used to collect data below E~L = 3.22 MeV. A similar layout was used at the ONR-CIT tandem laboratory with the same cold trap extension and counter collimator. this energy, and up to 12 MeV, the ONR-CIT tandem accelerator was used, with a 28 cm scattering chamber. Fig. 1 shows a sketch of the experimental arrangement in the 14 cm diam target chamber. After electrostatic analysis, the beam energy is known to a few keV, and the tantalum cup slits shown in fig. 1 define an area approxi-

LEVELS IN llB

483

mately 1 m m 2 at the target position. Electron suppressor apertures are located after the main slits and the stray-beam-catcher aperture to improve the accuracy of the beam integration. Both the target and the beam collection cup were held at + 900 V above ground (the body of the target chamber) in order to keep most of the ejected electrons from hitting the walls of the chamber. This was found to be sufficient to ensure that the beam intensity measured with the target in place was the same to better than 5 ~ as that measured without target, at all energies above 1.5 MeV. We have had some difficulty with target stability under bombardment, especially at the lower energies. In order to decrease the accumulation of hydrocarbon contaminants on the bombarded spot, the target was surrounded with a polished aluminium shield kept at liquid nitrogen temperature (see fig. 1). We have found that this cold shield decreased the rate of contaminant buildup by a factor of 5 or more. It was also necessary to keep the beam intensity as uniform as possible over the bombarded spot, and below 0.05/~A, to prevent the loss of target material due to local overheating. The scattered charged particles were detected by an Au-Si surface barrier counter approximately 200/~m thick with an energy resolution of about 40 keV for 2 MeV alpha particles. The counter collimator subtended an angle of less than 0.5 ° and its angular position with respect to the incident beam was calibrated to better than 0.5 °. The counter angle was adjustable and could be reproduced to an accuracy of about 0.1 °. The configuration of the 28 cm chamber was similar and all the above considerations remained applicable. We have used the method of Dearnley 12) to make the 10-15 #g/cm 2 carbon foils which were used as target backing. The target layer was sometimes evaporated while the carbon was still on the glass and sometimes on the mounted foil. To make the hydroxide targets, 99.99 ~o isotopic 7Li was evaporated under high vacuum on the backing and allowed to form 7LiOH by introducing water vapor into the bell jar. To make the other type of target, 7Li2F was evaporated directly on the backing. All our data were taken with targets satisfying AEt <~ 60 keV, at OiL = 45 ° and EaL 3.1 MeV, with one exception: the data shown in figs. 5 and 6 above 7.5 MeV were taken with a AEt "~ 250 keV target and shifted accordingly. All angular distributions runs were corrected in a similar fashion. The target stability was checked during angular distribution runs by first taking data at the even multiples of 5 ° and then going backwards in angle, but this time taking points on the odd multiples of 5°(lab.). During excitation function runs, the yield at some chosen fixed energy was frequently measured. Fig. 2 shows a typical spectrum obtained with a 7LiOH target. The elastic and inelastic scattering peaks from 7Li are well resolved. This spectrum and all others used to extract data were obtained and treated in the following manner: the pulses from the counter were fed into a low-noise charge-integrating preamplifier located next to the target chamber. The voltage pulses coming from the preamplifier were then amplified to a 40-50 V level and stored in a 200-channel pulse-height analyser. After =

484

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CM ELASTIC DIFFERENTIAL DROSS SECTION TIMES CENTRE OF MASS ELASTIC WAVE NUMBER SQUARED FOR THE REACTION LiT (a,~)Li 7, AT OCM =90 ° vs E~ lab tN MeV

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Fig. 5. Elastic yield curve at 0~L = 60 °. The solid curve represents a smooth line drawn through the data points. The numbered energies correspond to entries in table 1, the list of scattering anomalies. Some of the entries of table 1 are not visible in this graph. They have been observed in other yield curves as explained in the text. The multiplying factor k s makes the cross section dimensionless and removes the reduced de Broglie wave-length dependence. CENTRE OF MASS DIFFERENTIAL CROSS SECTION FOR THE REACTION 7Li((~,~') ?Li* (.48) TIMES THE CM ELASTIC WAVE NUMBER SQUARED AT eL= 60°(8C• ~ 920-+ 2°)

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R. Y. CUSSON

488

the proper amount of charge was integrated, the spectra were recorded on punched paper tape for subsequent plotting and analysis. The spectra were plotted with the

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Fig. 7. Dimensionless inelastic differential cross section for ‘Li (c(, c(’ )7L1xt(U.4nr. 3everar ~~~~~~~ inelastic angular distributions are shown. To provide an estimate of the accuracy with which such data were obtained, two distributions were taken at _& = 6.40 MeV and analysed independently. The results arc shown on the figure. Above _E,L M 4.4 MeV, all the inelastic angular distributions show more or less pronounced minima near 6C.m. = 60” and 120”. This might be due to a direct interaction contribution to the inelastic cross section. The solid line is a smooth curve drawn through the data points.

489

LEVELS IN I1B

help of a computer-operated

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Fig. 8. Dimensionless elastic differential cross section for rLi(e, e)TLi, at various laboratory energies. These angular distributions were obtained by measuring the cross section at intervals of 10 ° in the laboratory starting at forward angles and going towards backward angles. The process was then reversed, but this time the selected angles were taken between the previous ones, to check on target depletion. The solid lines are the results of a least-squares fit to the data using a Coulomb amplitude and Legendre polynomials up to Pn.

490

R.Y. CUSSON

correction i n f o r m a t i o n . The o u t p u t o f this last p r o g r a m was the corrected l a b o r a t o r y yield as a function o f l a b o r a t o r y angles a n d energy. The a n g u l a r distributions were n o r m a l i z e d to one a n o t h e r b y c o m p a r i n g with the 60 ° excitation function o f fig. 6. The 7-ray excitation curve shown in fig. 3 was o b t a i n e d using a 5 × 5 c m scintillator detector. The crystal was placed 18 c m f r o m the target at 90 ° to the b e a m axis a n d viewed the ~-rays t h r o u g h a lucite window. [

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Fig. 9. Dimensionless elastic differential cross section for 7Li(c~,c0rLi from 2.8 to 3.93 MeV laboratory energy.

LEVELS

IN

llB

491

Li and Sherr 5) obtained an absolute cross section for the reaction 7Li(~, cd)VLi* (0.48), using thick target techniques with a probable error of 20 %. We have used their value to normalize our data. The usual normalization in experiments of this type consists in assuming Rutherford scattering at forward angles and low energy (E~L ~ 1 MeV would be required in this case). This method was not found practical here because of the difficulties involved in making accurate low-energy, thin-target measureI

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Fig 10. Dimensionless elastic differential cross section for 7Li(c~,c07Li from 4.07 to 6.70 MeV laboratory energy.

492

R.Y. CUSSON

ments, with an alpha beam and a LiOH target. We have fitted our inelastic angular distributions with expansions in Legendre polynomials. Although the coefficients CL of the Legendre polynomials PL proved somewhat unstable for L > 1, as Lmax was increased up to 6, the coefficient Co could be obtained reliably. Fig. 4 shows the energy dependence of this coefficient. The overall normalization constant of all the data was adjusted to make the coefficients Co fall on the curve obtained from the 7-ray measure-

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LEVELS IN liB

493

ment (fig. 3). On a relative scale, the normalization is accurate to approximately 5 %. The error flags in fig. 4 indicate the average fluctuation of Co as Lmax was increased in integral steps from 1 to 7. Including the 20 % probable error on the Li and Sherr measurement, the overall absolute accuracy of our measurement is about 22 % for the elastic cross section and 26 % for the inelastic cross section. Figs. 5 and 6 show the centre-of-mass elastic and inelastic excitation functions (taken at 60 ° laboratory angle), multiplied by the square of the centre-of-mass, elastic-scattering wave number to make the measurement dimensionless. The solid line is a smooth line drawn through the data points. Fig. 7 shows several typical inelastic angular distributions and figs. 8-11 show all the elastic angular distributions measured. 3. Extraction of Resonance Parameters

By inspection of figs. 3-11, many peaks and valleys can be observed in the experimental cross sections; these are listed in table 1. The level density is probably not high enough for the cross section to be described by Ericson's theory of fluctuations; this is especially true below the neutron thresholds where our analysis will be concentrated. As has been pointed out above, only two particle channels are open below E~L = 4.38 MeV (E x = 11.464 MeV in 11B); these are VLi(g.s.)(S ~ = ~ - ) + ~ ( S ~ = 0+),

(2a)

which will be referred to as E, and 7Li*(0.48)(S~ = ½ - ) + e ( S ~ = 0+),

(2b)

which will be referred to as I. We have chosen to attempt the extraction of resonance parameters from the experimental data through the R-matrix formalism, as a test of the practicability of handling two-channel reactions in this way. The expression for the many-channel scattering amplitude given by Lane and Thomas 13) in their equation (1X.2.6) has been specialized to the two-channel case for the present analysis. Before we attempt the resonance analysis, it is of some interest to determine how much information is required if we wish to extract the complete scattering amplitudes from the data and not just resonance parameters. The number of partial waves contributing at a given energy to the scattering amplitude is limited by the centrifugal and Coulomb barriers. In R-matrix theory, the effect of these barriers is given, in part, by the penetration factor which is shown in fig. 12 for a radius Ro of 6 fm in dimensionless form. We shall give reasons later for using such a large value of Ro. For a narrow resonance involving one L-value, the width is given by F = 2PL72, where 72 (hz/mR~)O 2. The quantity h2/MR 2 = 0.451 MeV in the present case. The quantity 02 is dimensionless and equal to 1 for a square well. The figure shows a graph of the penetration factor in the elastic channel, for various L-values as a func=

494

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LEVELS IN l i B

495

t i o n of the incident energy. This graph m a y also be used to find the p e n e t r a t i o n factor in the inelastic channel. The inelastic p e n e t r a t i o n factor at incident energy E is equal to the elastic p e n e t r a t i o n factor evaluated at E - 0 . 7 5 2 MeV. A t E,~, = 4.0 MeV, the elastic p e n e t r a t i o n factor is 0.18 for L = 4 a n d 0.029 for L = 5. A s s u m i n g time reversal invariance, one finds that 46 i n d e p e n d e n t functions of the energy m u s t be I0

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Fig. 12. Penetration factor for ~ on 7Li at R0 = 6 fin. The penetration factor plotted here is dimensionless. For a narrow resonance involving one L-value, the width is given by/~ = 2PL~'L 2, ~i, ~ = (h2/MRo2)O 2, where t~2/MRo ~ = 0.45 in this case. 03 is dimensionless and equal to unity for a square well. The energy scale is given for the elastic channel.

given to specify the energy a n d a n g u l a r dependence for the t w o - c h a n n e l p r o b l e m of eq. (2) t r u n c a t e d at L = 4. O u r data are n o t accurate a n d complete e n o u g h to allow a least-squares fit with that m a n y parameters. We have used a simpler approach, which is to follow Lane a n d T h o m a s a n d parametrize their R - f u n c t i o n by expressing it as a many-level expansion. A finite n u m b e r of levels m a y be considered explicitly a n d

496

R . Y . CUSSON TABLE 2

List o f resonance p a r a m e t e r s for a seven-level fit to the observed cross section for the reactions 7Li(cq ~)VLi a n d 7Li(~, cd)TLi*(0.48) below the l ° B + n threshold at E~t, = 4.38 M e V ID no.

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Er lab

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4.0 0.04 0.0 4.6 1.0 1.5 0.0

0.0 1.0 1.1 0.0 0.0 1.1 0.0

1.195 1.590 1.656

1.88 2.50 2.60

1.935 2.25 2.61

3.03 3.54 4.10

E,t . . . . (MeV)

1 2 3 5' 4 5 6

2.195 1.590 1.656 1.71 a) 1.935 3.26 b) 2.61

}~½+ {-+ {,~+

a) This level is included to generate a b a c k g r o u n d near E~L = 2.5 MeV. b) This level is shifted heavily because the b o u n d a r y condition is given for level 3. ¢) The observed width is smaller t h a n / ' t o t due to the effect o f dSe/dE.

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TOTAL INELASTIC CIM CROSS SECTION FOR THE REACTION ?Li (a,a') 7Li ( 4 8 ) times ELASTIC C.M. WAVE NUMBER SQUARED

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LEVELS

IN

497

lib

the effect of the other levels (an infinite n u m b e r o f them) m a y be lumped into a socalled b a c k g r o u n d term. We did not include b a c k g r o u n d terms in the R-function, but we did include levels with a very large width in order to imitate b a c k g r o u n d terms having a slowly varying energy dependence. F o r the case given by eqs. (2), we find that four parameters are sufficient to specify a level with a spin and parity J~, J >___3, namely the resonance energy E~ and the reduced widths in the elastic and inelastic

ELASTIC DIFFERENTIAL C.M. CROSS SECTION FOR 7 L i ( e , a ) 7 L i at 0¢.m.:159" times CM. WAVE NUMBER SQUARED SOLID LINE IS COMPUTATION, POINTS WERE MEASURED

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Eac.rn(MeV) Fig. 14. Comparison of the model-program calculation with experiment for the elastic cross section at 159°. The solid line is the model program calculation using the parameters listed in table 2. The points are experimental values• The dotted line is a smooth curve drawn through the data points.

0e.m. =

channels 7L, E, 7L+2,E, ]~L',I° W h e n J = ½, only one L-value occurs in the elastic channel so that the n u m b e r o f parameters is reduced to three. Alternately, we can specify the three reduced widths by giving Ftot, F J F E, (o = tg-I(yL+Z,E/TL, E). A p r o g r a m to calculate the R-function as a function o f energy was written for the I B M 7094 computer; the necessary input data include the n u m b e r o f levels ( ~ 10), their spins and parities, and the four parameters mentioned above for each level. Eqs. (VII.1.6b) and (VII.2.6) of Lane and T h o m a s were used to calculate the scattering amplitudes and cross sections as a function o f energy and angle. Subroutines to c o m p u t e C o u l o m b wave functions and Clebsch-Gordan coefficients were used.

498

R . Y . CUSSON

A one-level fit to the 10.60 M e V level (level 4 in table 2) was first a t t e m p t e d since its width is small. M a i n l y because o f the r a t h e r strong L = 3 interference effect visible in fig. 9, J= = ~+ soon a p p e a r e d as the best choice. Other levels were then a d d e d a n d their p a r a m e t e r s were adjusted by c o m p a r i n g the calculated cross sections with the m e a s u r e d ones for various values o f these parameters. The p o i n t o f vanishing returns I

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ELASTIC DIFFERENTIAL C.M. CROSS SECTION

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FOR7Li(a,a)TLi at Oc.m.: 90: times C.M. WAVE NUMBER SQUARED

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{

{{/

{

1.0

12

14

'1.6

ENTA, VALOES

{{{{{

1.8

20

22

24

26

Z 8

:50

Ea c m ( M e V )

Fig. 15. C o m p a r i s o n o f the m o d e l - p r o g r a m calculation with experiment, for the elastic cross section

at 0c.m. -- 90°. The solid line was computed using the parameters listed in table 2. The points are experimental values. As can be seen only the qualitative features can be reproduced. The discrepancy is mostly due to our lack of knowledge about the background phase shifts which are not "hard sphere" phase shifts. was reached with a seven-level m o d e l whose p a r a m e t e r s are given in table 2. Figs. 13-15 show some results o f the c o m p u t a t i o n c o m p a r e d with the experimental data.

4. Results In this section we discuss first in some detail the spin assignment possibilities for the six a n o m a l i e s visible below the ~°B + n threshold. N o spins higher t h a n ~ were considered. We then discuss briefly the energy region f r o m the n e u t r o n threshold up to 16.31 M e V excitation, the highest energy at which d a t a were obtained. A s u m m a r y o f

LEVELS

the results is contained

IN

499

llB

in table 1. We have listed there all the scattering

anomalies

or

“bumps” observed in this experiment. Some of these are very weak; others are inferred to account for broad maxima in the elastic and inelastic yield curves. 4.1. THE 9.88 MeV LEVEL

Our best fit to this level, using the model program is J” = 3’. We can give some simple arguments to explain why this is a preferred value. It is well known that the resonant elastic cross section, at resonance, is proportional to [rg(25+ 1)1/G while the inelastic cross section goes like f,r,(25+ 1)/r,. From fig. 4, showing the total inelastic cross section, we may write (3) where S = + for the present case. This yields the quadratic equation x(1 -x)(2J+ where x = rr/rr,

which has the following

J=

1) x 0.64, alternative

(4) solutions

for each J-value:

3 x1 = 0.5+0.26,/-

1

s 0.20

5 0.12

0.09

4 0.07

x2 = 0.5 -0.264

- 1

0.80

0.88

0.91

0.93.

i

(5)

The complex solution for J = + could be taken as ruling out J = ), or it could be an indication either that the normalization of the data is incorrect or that there is a substantial background contribution to the total inelastic cross section at the resonance. Taking x1 = x2 = 0.5 gives the maximum k2a,/4n = 0.125 for J = $. This value is barely within our error limit for the measured cross section. The assumption of a substantial background is unattractive because the penetration factor is small for all channels at this energy (see fig. 12). If we take J = $, x = 0.5, we find that the model program predicts a sizable positive interference at t3L = 159”, for which there is no evidence in fig. 14. When we considered models with J 2 4, we found that the second solution x2 always produced a prominent elastic interference at 159”; hence the x2 solutions were rejected. Next we consider the penetration factor. We have

Using the second solution of (5) and taking rT = 0.15 MeV,,,. from table 1, we list the values of 6’ for the remaining J” values, 3 5 J 5 4 in table 3. On the basis of table 3, in order not to have too large a value of Q2, our choice is narrowed down to J” = 3’. It might be argued that if most of the wave function for that level were con-

500

R. Y. CUSSON

centrated near the nuclear surface, the Wigner limit (02 = 1.5) might be exceeded. But, if we take into account the energy derivative of the level shift, we find that the observed width Fxob~ of a resonance is bounded from above by the inequality

FTo,s < [2Pc/(dS~/dE)]E= E. . . . . . ,

(8)

where c is the channel for which 02 --, oo. In this case, d S J d E was found large enough to give all the assignments of table 3 except ~+, too small a value of FTo~. Indeed, it was in the process of fitting this resonance that we were forced to increase the radius TABLE 3

Reduced inelastic width for the 9.88 MeV level in ~IB for various Jn S~ 012

~+ 1.5

~6.5

~+ 66

~6.5

~+ 66

~> 100

~+ > 100

,~> 100

for the penetrabilities, from the usually accepted value s) of 3.8--4.8 fm, to the value 6 fm used here. It was found very difficult to maintain Fr/FT: and Fro,, s simultaneously, when using a small radius, even when the Wigner limit was very badly exceeded, due to the effect mentioned above. However, we have found that the potential phase shifts resulting from a radius of 6 fm produced too large a background cross section. Thus, we were forced to use a smaller radius to compute them. The value used in figs. 13-15 was 3.8 fm. This procedure may perhaps be justified on physical grounds, following some calculations made by Vogt 14). Vogt has shown that, using a Saxon-Woods potential having a surface thickness of 1 fm, the penetration factor was effectively increased by a factor of nearly 8 over the value predicted when using a square well with the same average radius. Yet the non-resonant phase shifts for a Woods-Saxon potential having an average radius Ro and a square well with radius Ro, are similar. We shall return later to the question of background scattering in connection with possible direct interaction effects. Our analysis for this level suggests that J~ = ~:+ is the most likely assignment, although J~ = 1± cannot be rigorously excluded. 4.2. THE 10.26 AND 10.32 MeV LEVELS We consider these two anomalies together because the ratio of the width to the separation F/D is of order unity. The c.m. energies where these anomalies appear are E, = 1.59 and 1.656 MeV corresponding to the laboratory energies 2.50 and 2.60 MeV. In fig. 3, which shows the ?-ray yield, only one anomaly is visible at 2.50 MeV. In fig. 5, which shows the elastic 90 ° c.m. yield, only one anomaly is visible again, but this time around 2.65 MeV. The experimental points in fig. 14 show a considerably higher yield on the low-energy side of the 10.32 MeV level. Since it was possible to reproduce this asymmetry numerically using two levels (see table 2 and figs. 13-15), we believe there is not much doubt about the presence of at least two levels close together in energy. The 10.32 MeV level was first observed, rather weakly, in the reac-

LEVELS IN llB

591

tion X°B(d, p)llB by Elkins 1~) with a total width of 54+- 17 keV in the c.m. system and more recently by Groce et al. ~o). In figs. 13-15, we see that the fit to these levels is, in general, rather poor and only the qualitative features ale reproduced. Table 2 lists the parameters used in obtaining figs. 13-15. Because of the strong asymmetry of the observed peak at E,L = 2.6 MeV, the parity of this level must be negative. A positive parity resonance decays, in this case, with odd L-values which cannot create the interference, at 90 °, which is necessary to give asymmetrical line shapes. Fig. 15 shows that the calculated asymmetry has a shape similar to the observed one but the calculated magnitude is much smaller. This means that the background phase shifts are different from those predicted by Coulomb and potential scattering. We have not found any convenient model which would give suitable background phase shifts. For example, the elastic background due to a square well of fixed depth and radius has a strong energy dependence and produces resonances of its own; thus, it is unsuitable. Even so, there were few pairs of assignments which gave a line shape even qualitatively similar to that observed at 159 °. For the 2.5 MeV level, j ~ = ~1± , ~3 ± and for the 2.6 MeV level, J~ = ~ , -~- were choices of similar value. The pair j~ = ~5÷, J~ = -~- for the two levels gave rather poorer agreement. For the 2.6 MeV level, J~ > ~ - was excluded by barrier penetration considerations, and J < ~2 gave too small a cross section at 159 °. 4.3. T H E 10.60 M e V L E V E L

After target thickness corrections, we find the resonance energy to be E,L = 3.032+--0.01 MeV and the width to be FT,,b = 110+__10 keV. In the c.m. system, these values are respectively 1.93 and 0.070 MeV. Our value for the excitation energy in a lB is in good agreement with the recent measurement of Groce et al. ~o) who find Ex = 10.594+-0.012 MeV. From the elastic angular distributions given in figs. 8-11, it is not difficult to see that the elastic-yield curve at 0.... = 140.8 ° shows no sign of the resonance. This angle is a zero of P3(cos 0). This effect will be produced if, in the elastic channel, the observed flux is almost entirely due to an L = 3 resonating amplitude interfering with a larger non-resonant amplitude. An obvious way to accomplish this is to have a s t r o n g l y i n e l a s t i c r e s o n a n c e w i t h J ~ = 7 + , ~+. Since j ~ = -~ + would have to decay with L = 5 in the inelastic channel, we may exclude it because the penetration factor is too small (fig. 12). The model program results for J~ = 7+ can be seen in figs. 13-15. The agreement with the data is good. The yield in the elastic channel is somewhat sensitive to the type of background used, and to the value of FE/FT. Some backgrounds were used for which FE/FT = 0.3 gave a reasonable value of the 159 ° cross section. For these backgrounds, the resonant yield at 140.8 ° was small. For the model given in table 2, the angle where the zero resonant yield occurs is shifted slightly, but the general behaviour of the computed elastic and inelastic angular distributions (not shown here) is in good agreement with the data. (At these higher energies, the in-

502

R.Y. CUSSON

elastic angular distributions could be measured more reliably). The models built on j~ = 3 + , ~ s + would offer no explanation for the strong F-wave elastic interference since they can decay by P-waves in the elastic channel. Furthermore, j~ =~3 + yields a total elastic cross section which is too small by about a factor o f 2, while ~+ fails in the same respect by about 25 ~ . Thus ½+ is preferred and I + is a possible, t h o u g h unlikely, alternative. 4.4. THE 11.0 MeV ANOMALY This anomaly shows up only as a b r o a d peak in the total inelastic cross section at E~L ~- 3.6 MeV. The other yield curves do not show a peak at this energy (figs. 5 and 6). It is thus possible that m a n y b r o a d levels are overlapping in that region. For this reason, o u r fit to this anomaly only indicates that it may be a state. We have found that the non-resonant total inelastic cross section up to E~L = 3.2 MeV could be fitted rather well assuming a large P- and D-wave inelastic amplitude. (An S-wave inelastic amplitude requires a D-wave in the elastic channel, and would be expected to rise more slowly with increasing b o m b a r d i n g energy than the P-wave inelastic amplitude, which also requires a P-wave in the elastic channel.) A b r o a d state with J~ - 1+ / ' T = 4.0 M e V . . . . . Fe/F I = 1, was located at E~. . . . = 1.71 M e V to nearly saturate the inelastic P-wave channel. With FT = 4.0 MeV, it also served the purpose of maintaining the total inelastic yield at higher energy. We then looked for a spin and parity assignment to superimpose on the b r o a d ½+ level, which would fit the elastic 159 ° and 90 ° c.m. yields, while providing the correct total inelastic cross section. The only levels having a D-wave inelastic amplitude are I - and I - - Of these, I - seemed to give a somewhat better overall fit to the elastic yield curves and angular distributions and to the 11.27 MeV state which we shall discuss next. 4.5. THE 11.27 MeV STATE This state was first identified by Groce et al. lO) at Ex = 11.266+0.007 MeV. Since it has a negligible inelastic width (see fig. 6), it has escaped detection in previous w o r k on the inelastic scattering. The elastic yield curve at 159 ° c.m. (fig. 14) shows a large peak at E~L ~ 4.1 MeV, E~ .... ~ 2.61 MeV. We thus expect to find a large value for the spin o f that level. The elastic yield shows only a small symmetrical b u m p at 0 .... = 90 ° (figs. 5 and 15), indicative of a positive parity level (no interference at 0 .... = 90°). M a n y choices for spin and parity were tested with the model program. Models with J < 3 could not reproduce the large 159 ° yield and so were rejected. Models with J~ = 5 - , ½or 92- showed rather p r o n o u n c e d 90 ° interference and were also rejected. The three remaining possibilities J~ = ~s + , 27 + , z9 + were examined in somewhat more detail. The assignment 2s+ suffers f r o m two diseases; its 159 ° cross section is too small, and if it has any P-wave component, the line shape is incorrect at 159 ° (a dip on the low-energy side). The two remaining choices, ~7+ and 9+ seem equally promising, the

LEVELS IN 11B

503

,7+ giving somewhat too small a cross section and the 9 + producing the opposite effect (especially at 90 ° c.m.). However, the background due to the 11.0 MeV anomaly can be adjusted in both cases to give good fits. Each of the two possibilities has a special appeal. A level with J~ = 7 + at 11.28 MeV excitation energy could supply S-wave flux into the opening neutron channel at 11.46 MeV. A threshold cusp might then be seen there 15) which would explain the scattering anomaly, which occurs at that energy (see figs. 3 and 6). On the other hand, a level with J~ = z9 + requires an L = 5 decay in the inelastic channel. Thus we would have an obvious explanation for the lack of inelastic yield for that level, since the penetration factor would then be very small. 4.6. THE SCATTERING ANOMALIES FROM 11.46 A N D 16.31 MeV

We have pointed out earlier that the opening of several new channels, near 11.4 MeV excitation in 11B, makes our model program inadequate above this energy since it is a two-channel model. However, for the sake of completeness, we now summarize qualitatively whatever can be concluded about this energy region without performing detailed numerical calculations. In table 1 we show all the irregularities that can be seen in figs. 5 and 6, which show the elastic and inelastic yield at 0,L = 60 ° versus the laboratory bombarding energy. The first column is an identification number; the other columns list the laboratory energy at which the structure is observed, the centre-of-mass width and a code referring to the curve in which this structure is observed. The excitation energy in 11B is given, along with the excitation energy of levels found in previous work, which may be correlated with the ones seen here. Finally, the last column shows the spin and parity assignments proposed here with comments. It is probable that the anomalies listed in this table do not all represent states in 11B. For example, items 7, 11, 14, 15, 18 all coincide within their width with the thresholds for forming the first five excited states of 10 B plus a neutron. Since tests exist 15) within R-matrix theory to determine the effects of S-wave neutron thresholds (these effects are sometimes known to be large 16)), it might be prudent to investigate this point before building nuclear models involving these anomalies. Before we discuss table 1, we make some comments on the general features of the obseived cross section at these higher energies. The behaviour of the cross section is rather puzzling in this many-channel region, apart from possible threshold effects. The elastic cross section shows strong backward peaking (figs. 8-11) while the inelastic angular distributions (fig. 7) seem to have diffraction-type maxima and minima, whose positions have a smooth energy dependence. We may speculate that this behaviour is due to the presence of a large direct interaction amplitude. Whaling 17) has observed the protons from the reaction 7Li(~, p)l°Be (E~L ~ 10 MeV) and finds that they are strongly forward peaked, although some of the maxima, for the yield-versus-energy excitation function at OpL 10 °, do coincide with known states in 1lB. Thus, in that case at least, both compound and direct reactions are present. If we assume that the elastic scattering of alphas on

504

R.

Y.

CUSSON

‘Li can take place by a triton exchange mechanism, then the even and odd L amplitudes tend to be decoupled. This effect can be simulated by considering scattering from two square wells, one being a direct potential and the other being an exchange potential. It is then possible to select these potentials so that the elastic cross section acquires a strong backward peak. The inelastic scattering might also be understood as an exchange mechanism if we do not assume that the exchanged triton is captured in the ground state of the triton-alpha system. In addition, there is another source of inelastic scattering which could also be important; it is a transition from the ground state to the first excited state of ‘Li via the E2 matrix element, which is known to be large 18*19). Although the Coulomb excitation amplitude is small, there is, in principle, no objection to using the nuclear force to make the transition. Since a direct amplitude contains many J-values, it may interfere with most resonances, with the result that the (total or differential) inelastic cross section may show bumps that are displaced in energy from the resonance position. For this reason, the level identification may be unreliable. This should be kept in mind in the next few paragraphs where we discuss our findings in this higher-energy region. We begin with anomalies number 7 and 8 of table 1. In fig. 5 (90’ c.m. elastic), they are not resolved, and all we see is a small bump at about 11.5 MeV excitation (E,, Z 4.44 MeV) and a dip about 110 keV higher. In fig. 6 (92” c.m. inelastic), they are again unresolved and only a large peak can be seen around 11.5 MeV. In fig. 3 (total inelastic), they are barely resolved. If we assume that bump number 7 (11.49 kO.035 MeV) is a level and not a threshold effect, then the shape of the 90” elastic yield curve indicates that one of the two is probably of positive parity, since it is similar to the shape observed for the 11.27 MeV level. As we shall see later, the level at 10.89 MeV in ’ 'C is a suitable candidate for the analogue of a level in “B around 11.46 MeV. Since we do not know which of the two levels should be of positive parity, the entry in table I has double brackets to indicate the rather speculative nature of the assignment. We also note that it is not unreasonable to identify the level at 11.68 rt 0.1 MeV, reported by Mehta et al. ‘) with the 11.60 level discussed here since the neutron yield curve could be shifted due to the very strong resonance observed thereat EaLz 7.0 MeV. Level number 9 in the list is well resolved in the inelastic yield curve and has been observed in other reactions (see table 1). In fig. 5, however, it appears as an interference dip in the 90” elastic cross section. From this, we may infer that this level has negative parity. Next we consider items 10 and Il. Level 10 is included in order to account for the broad maximum in the elastic cross section in the region 5 MeV 5 EaL 5 6 MeV. Item 11 represents only a small dip in the elastic cross section above the threshold for “B*(O .717 2 1 ‘)+n. However, no state similar to item 10 has been reported in any other reaction, while a bump at 12.18 MeV excitation has been rejected as a state by Mehta et al, ‘) because they believe it may be a neutron threshold effect in their counter. We believe that between 12.0 and 12.2 MeV excitation there is at least one state, width unknown.

LEVELS IN llB

505

Level number 12 at 12.55 MeV excitation is well resolved in the inelastic channel, but appears highly unsymmetrically in the elastic scattering. I f this is an interference effect, then this level must be of negative parity. Items 13 and 14 are again considered together. Level 13 appears to be about 300 keV wide, but its proximity to a neutron threshold could distort it considerably. I f this were the case, we might identify it with the level at 13.15 __0. I0 observed by Mehta et al. 7). Since it only appears as a dip in the elastic cross section, it should also have negative parity. I t e m 14 was observed as a strong peak in the plot of X2 versus energy for a sixth-order Legendre polynomial fit to the inelastic cross section (this fit is not presented here). A small kink in the elastic cross section can also be seen at E~L ~ 7.20 MeV. Items 15 and 16 are unresolved in fig. 6 (inelastic 60 ° lab ~ 92 ° c.m.) and only a dip is observed in fig. 5 (elastic 90 ° c.m.) at 14.05 MeV excitation. No other broad level has been reported at 13.63 MeV excitation but Groce et al. have observed a broad state at 13.3 MeV. I f this 13.3 MeV level were responsible for the peak we observe at 13.63 MeV, one would have to explain why it is shifted so much. Possibly the presence of a nearby threshold ~°B(2.18)+ n at 13.61 MeV could account for the shift. We have identified level number 17 at 14.69 MeV with the level at 14.56 MeV reported by Groce et al. The discrepancy of 0.13 MeV could be due to the paucity of data in fig. 6 in that energy region. Item 18 again happens to coincide with a neutron threshold but since it has been observed in other reactions 7), although with a different apparent width, it might represent a level. •Finally, item 19 represents a small peak in the 92 ° inelastic yield at about 15.73 MeV excitation. This level falls within the error limit of the 15.88 +_0.2 MeV level of Mehta et al. 7). It now appears clear that only the narrow levels can be located and identified readily. It is probable that many broad overlapping levels are present and this plus the possible direct-amplitude interference makes it difficult to tabulate the level structure in this energy region.

5. Comparison of the Levels of UB and llC As pointed out above, the level structure of 11B appears rather complex above 11.46 MeV excitation, the energy of the first threshold for neutron decay. Since it is not known, at the present time, to what extent these neutron-decay channels can distort the various observed particle spectra, we shall limit our comparison to excitation energies below 11.46 MeV. Although the assignments proposed here are not in all cases unique, it is still interesting to compare them with our present knowledge about 1C in this energy region. Fig. 16 shows a scale diagram for the energy levels of ~1B and 11C. The lower ener-

506

R.Y. CUSSON

gy part o f this figure is adapted f r o m Olness et al. ~). The level sequence starting at 9.28 M e V excitation in aaC is adapted f r o m Overley and W h a l i n g 20). The levels in 1B a b o v e 9.28 M e V excitation were discussed above. A total o f 14 states in ~tB are reported between 6.81 and 11.23 M e V excitation. As an empirical tool for finding analogue states, we subtract 0.52 M e V f r o m all these

P'46

1:.27 (9/2+7/2 +) ~- .... ~ !0.89 ~ " - . j I0 69

•,,,:~ 7 / / / / / / / / / , I 0 60

_i326

7/2 +

10 52

9 88

+ 9 28 5/2 7/2+

7/2 + (5/2~5/2)

.(--)

~'~-.,. 9 28

5/# ~ . ~

99z

8666 7Li+a

rO08 ~, 9 74

( 3/2 +) ~-~.

-919

(9/2 + )

857

<< 5/2-

799

3/2 +

870(5/2,+7/2 + } ~'~.":'-~.~ .866 N, \ 5 / 2 + 7 / 2 +<_9/2" - . . " - . . 843

""~'.~.

75Q (5/2 +) (3/2 +)

730 (318,5•2) +

7.544 Be

~ "-... 6 9 0

.676 6 81(l/2'3/2)~/'2~

+o

5•2 +

649 [ 3 / 2 , 5 1 2 , 7 / 2 / " 6 55 -<52/2÷

5 ~3

0/2,3/2)-

481 (5/2~ 5•2-)

446

5/2- - ~ ~

Z 14

I/2

452

5/2-

2.00

(i/2-)]

I

jTr = 3/2-]

J~ 3/z-j HB

II C

Fig. 16. Comparison of the level diagrams of a~B and 11C. The lower energy part of this diagram is adapted from Olness et al. 1). Additional assignments are due to Earwaker 22). We have connected with dashed lines the probable analogue states. All the spin assignments in laB, including those discussed in this paper, are consistent with those for the analogue levels in n c , for the case where they are known in both nuclei. The nearly identical level spacing for the two nuclei, which has previously been observed at lower energy, continues up to at least 11 MeV.

LEVELS IN llB

507

energies and look for nearby states of 1~C, having approximately the resulting energies. The results are shown in table 4. Beginning with the 9.88 MeV level in ~XB, we associate with it a level at 9.28 MeV excitation in 1~C reported once by Cerineo 6) but not seen by Overley in the elastic scattering of protons by 1°B. Our present assignment for this level would then be ~+, which would imply a D-wave decay for it, in the channel 1°B + p at about 600 keV. Since this level should have a large width in the channel 7Be+ c~, it would not have been seen by Overley because its Wigner limit width for decay into ~°B + p is about 13 keV. Overley could not see states below Ep ~ 700 keV, which had Fv/F.r <~ 0.05. TABLE 4 C o m p a r i s o n of the energy levels of UB and 11C

Ex(UB)

E,,(UB)-- 0.52 MeV

E,,(nC)

AE

6.81 7.30 7.99 8.57 8.92 9.19 9.28 9.88 10.26 10.32 10.60 11.0 11.27 11.46

6.29 6.78 7.47 8.05 8.40 8.67 8.76 9.36 9.74 9.80 10.09 10.48 10.75 10.94

6.35 6.90 7.50 8.10 8.43 8.66 8.70 9.28 9.74

0.06 0.12 0.03 0.05 0.03 -0.01 -0.06 -0.08 0.0

10.08

-0.01

10.69 10.89

-0.06 -0.05

One of the members of our doublet at 10.26 and 10.32 MeV excitation in 11B should be associated with 9.74 MeV anomaly in 11C. Overley has reported difficulties in fitting his observed cross section in the region of the 9.74 MeV level. The dotted line below the 10.08 MeV level in fig. 16 indicates his conjecture to explain the discrepancy. The existence of this doublet, and if it exists, which state in ~~C is the analogue of the other member of the doublet in t ~B, remains uncertain. It is gratifying that our most definite assignment J~ = ~+ for the 10.60 MeV level in ~IB finds its analogue in the 10.08 MeV level in ~ C , which is also reported to be ½+ by Overley. We find no analogue in ~~C for the 11.0 MeV state in 1~B. However, because of its large width, this level would be very difficult to identify in ~1C in all the experiments which have been carried out. The 11.27 MeV level in 11B should be the analogue of the 10.69 MeV state in tiC. 9 + in 11B)are consistent. The spin assignments for these two levels (9+ in laC; -~+, ~We have previously pointed out that the anomaly at 11.46 MeV might possibly be a threshold effect. However, the existence of a level at 10.89 MeV in ~xC suggests

508

R, V. CUSSON

the existence of its analogue around 11.4 MeV in 1lB. The 11.46 MeV anomaly might thus instead be just this analogue. Thus, we may conclude that the level structures of the two mirror nuclei a ~C and 1B continue to be similar up to at least I 1 MeV excitation. 6. Conclusion

We conclude by emphasizing that it is probable that more detailed information about the energy and angle dependence above the neutron threshold of the two reactions studied here might well allow additional useful restriction on the spin and parities of the levels found there. Preliminary calculations 2 3) to fit the excited level sequence beginning at 6.81 MeV excitation in 11B using the Nilsson model seem to indicate that non-axially-symmetric average potentials should be considered or that rotation-vibration couplings are important. This work will be described elsewhere. We gratefully acknowledge the help of the staffof the Kellogg Radiation Laboratory, without whose collaboration this work would have been impossible. We wish to thank in particular, Professor C. A. Barnes who originally suggested this experiment and who supplied constant guidance throughout this work, Professor T. Lauritsen with whom we have had many enlightening discussions on models for light nuclei and Professor W. Whaling for help in comparing our results with previous published work. We thank Barbara Zimmerman and the staff of the Booth Computing Center for their assistance in performing the many computer calculations involved in this work. Finally, we acknowledge the essential financial assistance of a Navy contract research assistantship and recently, a fellowship from General Atomic Division of General Dynamics Corporation. References l) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23)

J. W. Olness, E. K. Warburton, D. E. Alburger and J. H. Becker, Phys. Rev. 139 (1965) 512 L. L. Green, G. A. Stephens and J. C. Willmott, Proc. Phys. Soc. 79 (1962) 1017 H. Bichsel and T. W. Bonner, Phys. Rev. 108 (1957) 1025 N. P. Heydenburg and G. M. Temmer, Phys. Rev. 94 (1954) 1252 C. W. Li and R. Sherr, Phys. Rev. 96 (1954) 389 F. Ajzenberg-Selove and T. Lauritsen, Nuclear Physics 11 (1959) 1 M. K. Mehta, W. E. Hunt, H. S. Plendl and R. H. Davis, Nuclear Physics 48 (1963) 90 A. B. Clegg, Nuclear Physics 38 (1962) 353 T. Lauritsen and F. Ajzenberg-Selove, Energy levels of light nuclei, Nuclear data sheets, Sets 5 and 6 (National Academy of Sciences-National Research Council, Washington D.C. 1962). D. E. Groce, J. H. McNally and W. Whaling, Bull. Am. Phys. Soc. 8 (1963) 486 M. M. Elkind, Phys. Rev. 92 (1953) 127 G. Dearnaley, Rev. Sci. Instr. 31 (1960) 197 A. M. Lane and R. G. Thomas, Revs. Mod. Phys. 30 (1958) 257 E. Vogt, Phys. Lett. 1 (1962) 84 W. E. Meyerhof, Phys. Rev. 129 (1963) 692 H. W. Newson et al., Phys. Rev. 108 (1957) 1294 W. Whaling, private communication C. M. Chesterfield and B. M. Spicer, Nuclear Physics 41 (1963) 675 P. H. Stelson and. F. K. McGowan, Nuclear Physics 16 (1960) 92 J. C. Overley and. W. Whaling, Phys. Rev. 128 (1962) 315 S. G. Nilsson, Mat. Fys. Medd. Dan. Vid. Selsk. 29, No. 16 (1955) L. Earwaker, private communication R. Y. Cusson, thesis, California Institute of Technology, (1965) unpublished