Cross sections as a function of angle and complex phase shifts for the scattering of protons from 12C

[ 22A'c1: I

Nuclear Physics 86 (1966) 130--144; (~) North-Holland Publishing Co., Amsterdam

•

Not to be reproduced by photoprint or microfilm without written permission from the publisher

CROSS

SECTIONS SHIFTS

AS A FUNCTION

OF ANGLE

FOR THE SCATTERING

AND COMPLEX

OF PROTONS

FROM

PHASE

12C

A. C. L. BARNARD, J. B. SWINT t and T. B. CLEGG tt

Bonner Nuclear Laboratories, Rice University, Houston, Texas ttt Received 5 November 1965 Abstract: Absolute differential cross sections as a function of angle for the elastic scattering of protons by 1~C, and for the reactions which occur, were measured at incident proton energies Ep = 2.39, 2.97, 3.97, 4.99, 5.66, 6.18, 6.65, 6.77, 8,07, 8.39, 8.96, 9.39, 9.81 and 11.60 MeV. Complex phase shifts were obtained to represent the elastic scattering cross sections, the total inelastic cross sections and certain polarization data. The phase-shift values at the above discrete energies were used to estimate a continuous energy dependence for the phase shifts in the range 5.0 --< Ep ~ 9.5 MeV, omitting the rapid energy variation due to narrow levels. The estimates were varied slightly to obtain a good fit (except over the narrow levels) to the previously reported cross sections as a continuous function of energy. Finally the narrow levels were included in a calculation of the differential elastic scattering cross section, the total reaction cross section and the polarization at 0lab = 50 °, all as a continuous function of energy. The calculation adequately reproduced the experimental data through most of the energy range. Only one of the states close to Ep = 9.14 MeV was included in the calculations, so that the results differed significantly from the experimental values above Ep = 8.5 MeV. E [ N U C L E A R REACTIONS 12C(p, p), E = 2.4-11.6 MeV; 12C(p, p'), E = 6.6-11.6 MeV; 12C(p, ct), E = 11.6 MeV; measured tr(E; 0). Natural target. 1

1. Introduction S w i n t et al. 1) (in a p a p e r w h i c h will b e r e f e r r e d t o as I) h a v e r e p o r t e d d e t a i l e d m e a s u r e m e n t s o f t h e a b s o l u t e d i f f e r e n t i a l c r o s s s e c t i o n f o r t h e elastic s c a t t e r i n g o f p r o t o n s b y 12C, a n d f o r s o m e o f t h e r e a c t i o n s w h i c h o c c u r . T h e c r o s s s e c t i o n s w e r e m e a s u r e d at fixed angles w i t h t h e e n e r g y i n c r e a s i n g in s m a l l steps, so t h a t t h e s e d a t a m a y be r e f e r r e d to as " c r o s s s e c t i o n s as a c o n t i n u o u s f u n c t i o n o f e n e r g y " . I n t h e p r e s e n t p a p e r , e x p e r i m e n t a l elastic s c a t t e r i n g a n d r e a c t i o n a n g u l a r d i s t r i b u t i o n s at f o u r t e e n " o f f - r e s o n a n c e " e n e r g i e s are r e p o r t e d . T h e s e d a t a will b e r e f e r r e d to as " c r o s s s e c t i o n s at d i s c r e t e e n e r g i e s " . T h e c o m p l e x p h a s e shifts w h i c h fit t h e elastic s c a t t e r i n g a n g u l a r d i s t r i b u t i o n s a n d t h e t o t a l r e a c t i o n c r o s s s e c t i o n s at t h e d i s c r e t e energies were found. At some of the energies polarization data were included among t h e d a t a to b e p a r a m e t e r i z e d b y t h e p h a s e shifts. S m o o t h c u r v e s w e r e d r a w n t h r o u g h t h e p h a s e shifts at t h e d i s c r e t e e n e r g i e s a n d w e r e u s e d to a p p r o x i m a t e t h e c o n t i n u o u s t Now at the University of Florida, Gainesville, Florida. tt National Aeronautics and Space Administration Fellow; now at the University of Wisconsin, Madison, Wisconsin. ttt Work supported in part by the U.S. Atomic Energy Commission. 130

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energy dependence of the slowly varying part of the collision matrix diagonal element. The part of the matrix element which gives sharp resonances was approximated by a single-level form, except in one case where the overlap of states of the same spin and parity was very strong. It was found that up to Ep = 8.5 MeV a reasonable fit could be obtained, as a continuous function of energy, to the elastic differential cross sections (the data in I), the total reaction cross sections and the polarization at 0~ab = 50 °. Above Ep = 8.5 MeV the inclusion of only one of the states near Ep = 9.14 MeV (see I) caused the calculated results to differ significantly from the experimental data.

2. Experimental Methods and Results The experimental equipment and methods of measuring the absolute differential cross sections were exactly as described in I. Angular distributions were measured at incident proton energies Ep = 2.39, 2.97, 3.97, 4.99, 5.66, 6.18, 6.65, 6.77, 8.07, 8.39, 8.96, 9.39, 9.81 and 11.60 MeV. With two exceptions, these energies are away from I

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133

PROTON SCATTERING FROM X2C

n a r r o w resonances (the energies 8.07 and 8.96 MeV are quite close t o the resonances at 8.17 and 9.14 MeV). The differential cross sections at the discrete energies for the elastically scattered protons are shown in fig. 1. The cross sections for the inelastic scattering reaction 12C(p, p')12C* (Q = - 4 . 4 3 MeV) were measured at 6.65 M e V and the six highest energies and are shown i n fig. 2. The worst counting statistics were + 4 . 0 ~ for the elastic and _ 6.0 ~ for the inelastic case so that the overall accuracy is estimated as + 4 . 4 ~ for the elastic and + 6 . 3 ~ for the inelastic cross sections. I

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Fig. 3. Absolute differential cross sections as a function of angle for the lsC(p, p")l~C* (Q = -7.66 MeV) and z2C(p, ~0)aB reactions at Ep = 11.60 MeV. The dots are the experimental data, and the crosses are repeated points. Errors indicate statistical uncertainties only. The solid curve was calculated from the Legendre series in table 1. W h e n the Ep = 11.60 MeV angular distribution was being measured some cross sections for the inelastic scattering reaction 12C(p, p")12C* (Q = - 7 . 6 6 MeV) and the 12C(p, ~o)9B reaction were obtained and are shown in fig. 3. The data o f fig. 2 have been fitted to a series o f Legendre polynomials, the necessary coefficients being shown in table 1. The fits were carried out for 5, 7 and 9 polynomials. The n u m b e r o f polynomials was judged adequate if adding two more improved the

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Fig. 4. Absolute differential cross sections for the elastic scattering of protons by lzC as a function of incident proton energy (lab system) at the angles shown (c.m. system). The points are a condensation of the data in I. The solid curves were calculated as described in the text. The break in the curves around 5.35 MeV is to indicate the uncertainty of the resonating 6a- phase shift in this region.

136

A.C.L. BARN,ARDet aL

fit only m a r g i n a l l y . T h e L e g e n d r e series t a k e n to represent the d a t a are p l o t t e d as solid lines in fig. 2. T h e t o t a l cross section for inelastic scattering (Q = - 4 . 4 3 M e V ) was e v a l u a t e d f r o m aT = 4nAo, Where Ao is the coefficient o f the z e r o - o r d e r p o l y n o m i a l . E x c e p t at Ep = 11.60 MeV, this was also t a k e n to be the t o t a l r e a c t i o n cross section, since n o o t h e r r e a c t i o n g r o u p s were observed. A t 11.60 M e V the inelastic scattering (Q = - 7 . 6 6 M e V ) t o t a l cross section a n d an e s t i m a t e o f the 12C(p, ~ o ) 9 B t o t a l cross section were i n c l u d e d in the t o t a l r e a c t i o n cross section. T h e t o t a l cross sections a t the discrete energies are given in table 1. I n fig. 4 a c o n d e n s a t i o n is m a d e o f the elastic-scattering d a t a (as a c o n t i n u o u s funct i o n o f energy) p r e s e n t e d in I. E v e r y t h i r d p o i n t is p l o t t e d except over the s h a r p resonances. I

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Fig. 5. Estimates of total cross sections for the 12C(p, p')12C* (Q = -4.43 MeV) reaction as a function of incident proton energy, based on the differential cross sections at six angles presented in I, are plotted as dots. The crosses are the total cross sections from table 1, based on the angular distributions in the present paper, and "on-resonance" data from ref. 2). The solid curve is the total reaction cross section as a continuous function of energy, calculated as described in text. T h e t o t a l cross sections for inelastic scattering (Q = - 4 . 4 3 M e V ) were e s t i m a t e d as a c o n t i n u o u s f u n c t i o n o f energy in the range 6.7 < Ep < 11.5 MeV, using the d a t a at six angles in I. T h e results are s h o w n in fig. 5, t o g e t h e r with the t o t a l cross sections d e r i v e d f r o m the d a t a at discrete energies in the present p a p e r a n d s o m e further ang u l a r d i s t r i b u t i o n s at " o n - r e s o n a n c e " energies 2).

3. Phase Shift Analysis T h e c o m p l e x phase shifts (fi + i?), usually expressed in terms o f a real phase shift 6 a n d a d a m p i n g p a r a m a t e r ~, were o b t a i n e d b y the m e t h o d described in the a p p e n d i x . T h e d i a g o n a l elements o f the collision m a t r i x at some p a r t i c u l a r energy were w r i t t e n UtJ = eZ'(~'J+~r'J) = ~ e 2'~'~,

J = l+__½,

137

PROTON SCATTERING FROM 12C

where, because of the unitarity of the matrix, the damping parameter a s describes a b s o r p t i o n o f t h e i n c i d e n t p a r t i a l w a v e (l, J ) i n t o all o p e n . r e a c t i o n c h a n n e l s . T h e t o t a l r e a c t i o n c r o s s s e c t i o n is t h e n g i v e n b y 7z a r = ~-~ ~ [ ( 2 / + 1 ) - ( l +

1)(~+) 2 - l(~t-')2].

T h e e x p r e s s i o n s f o r t h e d i f f e r e n t i a l c r o s s s e c t i o n a n d p o l a r i z a t i o n a r e well k n o w n a n d will n o t b e r e p e a t e d h e r e . TABLE 2 Complex phase shifts which represent the data Ev (MeV)

6o ~o

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d2~2-

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2.39

121.73

-- 7.25

-- 9.40

0

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2.97

110.00

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95.39

4.99

85.58

5.66

83.00

6.18

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6.65

79.17

6.77

88.51

8.07

71.90 0.92 74.60 0.92 74.23 0.85 71.00 0.87 72.65 0.97 78.54 0.67

1.0 1.0 1.0 1.0 1.0 1.0 1.0

8.39 8.96 9.39 9.81 11.60

1.0

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--19.78 1.0

--19.19 1.0

--21.23 1.0

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-- 39.30 0.88 --35.50 0.83 - - 34.45 0.79 -- 30.60 0.68 --30.91 0.59 --37.15 0.64

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20.04 1.0

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-- 10.62 1.0 -- 13.97 1.0 -- 15.52 1.0 -- 14.23 1.0 --12.30 0.86 --12.35 0.83 --15.92 0.71 --18.40 0.72 -- 9.69 0.78 --27.62 0.62

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0.68 1.0 0.70 1.0 0.34 1.0 1.10 1.0 4.20 0.99 7.60 0.99 12.30 1.0 --4.50 0.96 4.40 0.96 4.26 0.78

T a b l e 2 a n d fig. 6 s h o w t h e v a l u e s o f t h e p h a s e s h i f t s a n d d a m p i n g p a r a m e t e r s n e c e s s a r y t o fit t h e e l a s t i c s c a t t e r i n g d i f f e r e n t i a l c r o s s s e c t i o n s a t t h e f o u r t e e n d i s c r e t e energies, the total reaction cross sections where measured and some polarization data. T h e e l a s t i c s c a t t e r i n g a n g u l a r d i s t r i b u t i o n s c a l c u l a t e d f o r t h e s e v a l u e s a p p e a r as s o l i d a n d d a s h e d c u r v e s i n fig. 1. P o l a r i z a t i o n d a t a f r o m v a r i o u s s o u r c e s a t e n e r g i e s c l o s e t o s o m e o f t h o s e u s e d h e r e a r e s h o w n i n fig. 7. T h e d a s h e d c u r v e s w e r e c a l c u l a t e d f r o m t h e p a r a m e t e r s i n t a b l e 2.

138

A.C.L.

BARNARD

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Fig. 7. P o l a r i z a t i o n d a t a as a f u n c t i o n o f angle at energies close to s o m e o f t h o s e used in the p r e s e n t e x p e r i m e n t . T h e e x p e r i m e n t a l d a t a are t a k e n f r o m the following sources: the solid curve at 5.66 a n d 6.18 M e V f r o m ref. ~), t h e d o t s at 6.18, 6.77, 7.99 a n d 8.66 M e V f r o m ref. 5) a n d the d o t s at 6.60, 8.60 a n d 11.70 M e V f r o m ref. a).

4. Comparison with Some Other Results Differential cross sections up to approximately Ep = 5 MeV were measured by Reich, Phillips and Russell 3). The phase shifts obtained by those authors were modified slightly by Tombrello, Barloutaud and Phillips 4) to fit limited polarization data,

a.c.L. BARNARDet aL

140

in addition to the cross sections. Moss and Haeberli 5) have made measurements o f both differential cross sections and polarization as a function of angte at 8 energies between 4.66 and 8.66 MeV, and have obtained the complex phase shifts which parameterize these data. In general, the cross sections and complex phase shifts reported in this paper agree well with those in refs. 3- 5), although there are some differences. The c5~- phase shift used to describe the Wisconsin data 5) at 4.66 and 5.04 MeV is about 10 ° higher than that obtained by the earlier Rice groups 3,4), and the current work confirms this higher value. 5. Calculation of Cross Sections and Polarization as a Continuous Function of Energy

To take account of a sharp resonance in data otherwise varying slowly with energy~ the collision matrix may be split into a single-level part and a "background" part 6): U = U ° + U 1. For the problem discussed in this paper the background part U o must be non-diagonal to account for the slowly-varying absorption. Only the diagonal elements of U o enter the expression for the elastic scattering cross sections, and these elements were approximated by U°J(E) = a~e 2i~'J. Smooth curves (not shown) were drawn through the isolated points on fig. 6 and were used as a first trial for the continuous energy dependence of the parameters and 6. In order to obtain a good fit to the data shown in fig. 4 these smooth curves were varied somewhat. The solid curves Shown on fig. 6 are the values finally used. In the resonant part U 1 the approximation R ° • L° = 0 was used (although this choice cannot be entirely correct since it implies'that U o is diagonal). Then the diagonal element is approximately =

= at e

[1 d- a f ( e 2i#'J -- 1)].

Provided that a =< 1 and a =< 1 this form guarantees that U~eU¢c* =< 1 so that the unitarity condition cannot be violated by the diagonal element alone. Some other forms which did not make this guarantee were discarded. The resonance phase shifts fl] were approximated by flf = arctg •

:

½_~F E o ' E

where F is real and independent of energy (i.e. the level shift and the energy dependence of the width were neglected for the narrow levels) and Eo is the resonance energy. In the above expression for Uc~, the term outside the square parentheses describes the slowly varying processes and the inside term describes the rapidly varying processes due to a single level in the compound nucleus. The definitions of "slowly" and "rapidly varying" are obviously somewhat arbitrary and, correspondingly, some effects could be included in either term. Terms of the outside type can simply b e regarded as representing the effects of those levels not represented by terms of the:

PROTON

SCATTERING

FROM

141

12C

inside type. In the present calculation (5.0 < Ep < 9,5 MeV) the levels whose effects were represented by the outside terms were (a) the known levels below the energy range of the calculation; (b) the known and unknown levels above the energy range of the calculation; (c) the strongly overlapping J~ = ~+ levels at Ep = 5.3 and 6.6 MeV, because of the possible importance of the interference terms between them, and because of the large width of the upper state. Thus the outside terms were taken from the solid curves on fig. 6. The cross sections calculated are shown as solid curves on figs. 4 and 5. In fig. 4 a break in the solid curves around 5.35 MeV emphasizes the uncertainty of the resonating complex phas e shift. In fig. 6 a the uncertainty of 62 is clearly seen at the 5.30 MeV

r

+ 1 -

l

T

T

z o

~

.0

-

,,r

~

T

l1

~.

"

N -J O ta

~ .; '-

L

5 INCIDENT

L__

6 PROTON

t

T ENERGY:

MeV

8 (lab)

9

10

Fig. 8, P o l a r i z a t i o n a t 0xab = 50 ° as a c o n t i n u o u s f u n c t i o n o f energy. T h e p o i n t s a r e the e x p e r i m e n t a l v a l u e s f r o m ref. ~), a n d t h e s o l i d c u r v e w a s c a l c u l a t e d u s i n g t h e s o l i d c u r v e s o n fig. 6. E r r o r s i n d i c a t e s t a t i s t i c a l u n c e r t a i n t i e s only.

resonance. In fig. 8 the calculated polarization at 0~ab = 50 ° is shown as a solid curve, compared to the data of ref. 5).

6. Revised Level Parameters

Obtaining the level parameters f o r the wide J~ = ½+ state a t E p = 6.35 MeV is complicated by the presence of the Ep = 5.3 MeV state of the same spin and parity, which is overlapped by the wider state. The values Of the 6~- phase shift between Ep = 4.5 and 4.9 MeV and between Ep = 5.6 and 7.8 MeV (that is, excluding the region of the 5.3 MeV state but including the rest of the 6.35 MeV state) were fitted to an expression of the form 6~ = q~f + arctg

72 p, E;~+ A t - E

to determine some of the parameters of the wider state. A radius R = 4.8 fm was

142

A.C.L. BARNARDet aL

used. Since the ~2 parameter shows no dip corresponding to the state at 6.35 MeV, the partial reaction width for this state was taken to be zero. The parameters for the 6.35 MeV state, and estimates of the parameters of the 5.30 MeV state, were then used to calculate the resonance phase shift from 4.5 to 6.2 MeV. A simple two-level formula was used for the resonant part of the phase shift: (

fl; = a r c t g

½F

T22p 2

Eo-E +

E~+A2-E!

The 62 phase shifts calculated are shown as a dashed curve in fig. 6a. In table 3 the level parameters used for these and other states are given. Note that only one of the levels close to Ep = 9.14 MeV (see I) was included in the calculation, since the spin and parity of the other are unknown. Thus the fit above Eo = 8.5 MeV is rather poor. TABLE 3 Level parameters in 1aN Ep

Ex

(MeV) a)

(MeV)

4.80 c) 5.30 5.88 6.35 7.53 8.16 9.13

6.37 c) 6.83 7.37 7.79 8.89 9.47 10.37

jrt

I1

alj

(keV) a) ~+ c) ~+ ~~+ ½~½-

12 c) 74 70 1720 250 30 84

Ez

(MeV) b)

)'z S

~;~2/(3i~/2~R2)

(MeV)

1.0 c) 0.10 1.00 0.66 0.58 0.90

6.87

0.04

0.014

8.86

0.9

0.31

a) Energies in the laboratory system b) Relative to ground state of 13N e) From ref. 3).

7. Discussion The dashed curves on fig. 6a are the phase shifts that would be expected on the basis of the known level scheme of 13N. Where no known levels of a given spin and parity are nearby, hard sphere phase shifts are plotted. Dispersion theory calculations are shown for cases where the effects of known levels would be expected to be appreciable. It is seen that the phase shifts necessary to fit the experimental data differ significantly from the dashed curves, so that the scattering of protons by ~2C is not properly explained by the known 13N levels in the energy range of the experiment. In I it was reported that the resonance at Ep = 9.14 MeV is caused by two closely spaced levels. It is well known that the general shape of the anomaly is that of a ~ resonance, so that probably the stronger state has this spin and parity. In the present calculations the weaker state has not been included, since its spin and parity are presently unknown. The contrast between the good fits obtained through most of

PROTON SCATTERING FROM 12C

143

the energy range and the p o o r fits over the 9.14 MeV anomaly is confirmation of the existence of the weaker state. The strong overlap of the two states makes the establishment of definite spins and parities very difficult. On the basis of the shell model, a J " = ~+ state would be expected in this region, but at the present this assignment to the weaker state cannot be confirmed. Appendix EXTRACTION OF PHASE SHIFTS FROM EXPERIMENTAL DATA

I f the complex phase shifts are written (6 + i7) then the cross section at the ith angle may be written in terms of 2 N parameters ,T, oxp = f , ( 6 , 6 2 . . .

aN

l

""

For instance for partial waves through l = 3, N = 7 so that cross section data at 14 angles would give 14 equations like that above. The equations are nonlinear and can not be solved directly. However if a set of trial parameters 6~°) • • • ,,Nx(°)"(°)~l• • • 7~°) can be found with a corresponding cross section a~°) then the original problem can be replaced by the problem of reducing the quantity e~ = aioxp-a} °) to zero for all i. Small changes A6 and A 7 in the parameters produce a change Aa} °) in a} °) which change can be approximated by a Taylor series in which only linear terms are retained. Then if A6 and A7 could be ideally chosen ei =

(~f'l

A6x+

(~f'~

A62-.F

""

.(Of'i A?I

.-[- "" ".

(A.I)

I f perfect measurements of ei at 2N angles were available, the simultaneous linear equations (A. 1) could be solved for the 2N parameters (A6, AT). Thus the non-linear problem has been linearized with the aid of the trial set of parameters. Although the symbol tr is used above, the 2N pieces of experimental data could be any mixture of cross section and polarization data. In practice, more than 2N pieces of imperfect data are available and the problem is changed slightly. I f e is regarded as a dependent variable and the partial derivatives as independent variables, the problem is to find the values of the (A6, A~) such that an equation of the form (A. 1) is a best fit to the experimental e v Using the least-squares criterion the object is to minimize a quantity such as

(e, oxp -

e, ~,~)2.

i

The quantity actually used was [Pjexp--Pjcalc~ .

o,°.,

_

2

,,,.., :,

where the factor 3 was inserted because the polarization data were approximately three times less accurate than the cross-section data.

144

A.C.L. lIAR.NARDet aL

References 1) 2) 3) 4) 5)

J. B. Swint, A. C. L. Barnard, T. B. Clegg and J. L. Weil, Nuclear: Physics 86 (1966) 119 J. B. Swint, M.A. Thesis, Rice University (1964), unpublished C. W. Reich, G. C. Phillips and J, L. Russell, Jr., Phys. Rev. 104 (1956) 143~ T. A. TombreUo, R. Barloutaud and G. C. Phillips, Phys. Rev. 119 (1960) 761 S. J. Moss, Ph.D. Thesis, University of Wisconsin (1961), unpublished; S. J. Moss and W. Haeberli, to be published 6) A. M. Lane and R. G. Thomas, Revs. Mod. Phys. 30 (1958) 321 7) T. B. Clegg, A. C. L. Barnard and J. B. Swint, Nucl. Instr. 40 (1966) 45 8) L. Rosen, P. Darriulat, H. Faraggi and A. Garin, Nuclear Physics 33 (1962) 458