Analyses of 49.6 MeV proton scattering from the ground and first excited states of 64Zn and 114Cd

2.E

I I

Nuclear Physics A101 (1967) 17--50; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

A N A L Y S E S OF 49.6 MeV P R O T O N SCATTERING F R O M THE G R O U N D A N D FIRST EXCITED STATES OF 64Zn A N D 114Cd V. R. W. E D W A R D S

Nuclear Physics Laboratory, Oxford, England Received 8 M a y 1967

Abstract: Recent m e a s u r e m e n t s o f 50 M e V p r o t o n scattering f r o m the g r o u n d a n d first excited states o f ~ Z n a n d 11~Cd are analysed using the optical model, the strong coupling a p p r o x i m a tion (SCA) and the distorted wave Born a p p r o x i m a t i o n ( D W B A ) . T h e results are used to throw light on optical-model p a r a m e t e r systematics in the energy range 30-78 MeV, to estimate the basic accuracy o f optical-model cross-section predictions and to assess the relative merits o f the SCA and D W B A m e t h o d s o f analysing inelastic scattering.

I. Introduction

In this paper we analyse some recent measurements 1) of 50 MeV proton scattering from the ground and first excited states of 6 4 Z n and ~ 1 4 C d . The principal aim of this work is to contribute to the understanding of the behaviour of the opticalmodel parameters in the energy range 30-78 MeV. The systematic trends of the proton, optical-model parameters with energy and nuclear mass have been established 2) in the energy range 9-22 MeV. Attempts to extend this understanding to higher energies were initially frustrated by the poor quality of the experimental data, although a few simple analyses were attempted. Recently excellent data suitable for a many-parameter analysis have become available, and thorough optical-model studies have been performed at 30 MeV [refs. 3-6)], 40 MeV [refs. v- 9)] and 78 MeV [ref. 1o)]. There are also a number of other mediumenergy analyses which are less interesting for establishing parameter systematics because they concern light nuclei 11-14). In this paper we aim also to test the strong coupling approximation 15) (SCA) generalisation of the optical model, which allows both elastic and inelastic scattering to be predicted. The only previous SCA analyses at 50 MeV are those of Fannon et al. i~) on 12C and of Craig et al. 12) on 12C and Si. Inelastic scattering from more typical optical-model nuclei has been studied at 30 MeV by Ridley et al. 16), and at the same energy Cole et al. 13) have examined scattering from Si. The DWBA method has also been used to analyse inelastic scattering at 55 MeV [ref. 17)] and at 40 MeV [ref. 7)J. 17

18

V. R. W. EDWARDS

2. Optical-model studies 2.1. P R O C E D U R E

The optical potential used in this work had the form

V(r) = Vc(r ) - Uf(r, ru, au)-iWvf(r, rwv, awv)--iWog(r, two, awo) (h] 2

+ \n~c/

Uso h(r, ruso, auso) l- ~.

Here Vc(r) is the Coulomb potential due to a uniformly charged sphere of radius 1.25 A +, U, Wv, WD and Uso the depths of the real, volume absorption, surface absorption and spin-orbit potentials, respectively, and m~ the pion mass. The form factors f, 0 and h are defined by

f(r, rx, ax) =

1 + exp \

ax

IA

g(r, rx, ax) = -4ax ; f(r, r~, a~), h(r, r~, a~) = -rl d f(r, r~, ax). The quantity minimised in the fitting was the rms percentage deviation p between the theoretical cross sections th(01) and the experimental cross sections ex(0i) defined by ~ex(0i)- th(0i)] z,

where N is the number of data points and 01 the angles at which measurements were taken. This criterion has two advantages over the commonly used Z2 and A tests. (i) It has a more direct meaning as a measure of quality of fit. A comparison of p with the rms error on the data gives an easily assimilated picture of the shortcomings of the optical model. (ii) It gives equal weight to all the points and fits therefore tend to be of more uniform quality throughout the angular range. It also leads to parameters which do not depend on arbitrary error assignments and which give a better representation of the average characteristics of the nucleus. When the optical model was introduced, the poor precision on many of the experimental points necessitated some form of weighting. With modern techniques and "normal" targets, relative errors in elastic cross-section measurements do not usually exceed the ~ 5 % uncertaintities of the optical model itself. The Z2 and A'

Volume o n l y - e q u a l start Separate real and imaginary form factors

Volume o n l y - Fricke and Satchler start

Surface only

Fixed form factor test

E0 E1

F0 Fx

G

1.092

44.43

0.903

0.843

0.799

0.819

0.810 0.810

(0.650) (0.650)

au

(1.16)

1.113

(0.75)

0.834

6.87

(0)

8.80

15.61

15,02

14.01

14.81

11.75 11.87

14.41 14.64

Wv

(1.37)

1.388

1.201

1,234

rU

rU

rU rv

(1.250) (1.250)

rwv

(0.63)

0.716

0.731

0,652

aU

aU

0.737 0.737

(0.650) (0,650)

awv

1.70

11.43

(0)

(0)

(0)

(0)

(0)

1.78 1.79

0,05 0.01

WD

(1.37)

0.844

rU rU

(1.250) (1.250)

rWD

(0.63)

0.969

awv awv

(0.470) (0.470)

awo

(6.04)

(5.5)

(6.0)

8.83

(4.50)

2.22

(4.50)

(4.50) 4.55

(4.50) 4.25

Uso

1.000

0.814

0.932

0.782

0.809

0.861

0.829

0.886 0.880

0.894 0.877

7.3

11,7

6.7

6.0

6.4

6.9

7.7

5.9 5.9

32,1 31.5

Normalization N P(%)

Bracketed quantities were left fixed. Constraints are recorded in the appropriate columns. F o r cases A - - F r U = r u s o and a U = aus o. F o r case G r u s o = 1.064 and aus o = 0.738.

42.51

45.52 not run

42.21 1.154 0.785 no improvement on freeing Uso

1.166

41.04

1.184

39.97

Do D1

1.169

Volume o n l y - e q u a l start C o m m o n real and imaginary form factors

Co CI

1.144 1.168

(1.250) (1,250)

rc

39.89

41.19 Volume and surface absorption 41.19

B0 BI

30.99 31,30

U

Fixed Perey form factor

Type of search

A0 Ax

Identifier

TABLE 1 Optical-model parameters for ~ Z n fits

Surface only

Fixed form factor test

F

G

0.852 0.839 (0.75)

1.147 1.171 (1.16)

0.769

0.787

7.51

(0)

11.56

11.53

12.34

10.16

15.22

Wv

(1.37)

1.304

1.241

rv

(0.63)

0.660

0.845

aU

0.748

(0.650)

(1.250) rU

awv

rwv

au s o ::: 0.738.

Bracketed quar~tities were left fixed. Constraints are recorded in the a p p r o p r i a t e columns. F or cases A

44.22

39.73

43.67

Volume only Satchler start

E

Fricke and

34.36

Volume only - equal start Separate real and imaginary form factors

D 1.245

1.238

35.06

Volume only - equal start C o m m o n real and imaginary form factors

C

0.779

1.238

(0.650)

(1.250)

Volume and surface absorption 35.40

au

ru

B

34.54

U

Fixed Percy form factor

Type of search

A

Identifier

TABLE 2 Optical-model p a r a m e t e r s for x~4Cd fits

(1.37)

0.946

rU

(1.250)

rWD

(0.63)

0.986

awv

(0.470)

aWD

(6.04)

8.03

7.80

4.33

4.84

4.68

4.14

1.083

0.954

0.996

1.005

1.014

1.032

1.119

14.5

16.4

l 1.6

12.2

12.3

11.7

29.5

Normalization USO N P(°/o)

F r U ~ rus o and a U = a us o. F o r case G r u s o -- 1,064 and

1.50

10.01

(0)

(0)

(0)

1.063

0.02

WD

PROTON

21

SCATTERING

tests remain necessary for "difficult" targets or for the analysis of mixed elastic and inelastic or cross-section and polarization data. Search codes which employ the A criterion do not require modification for use with the p test. It is sufficient to assign 1 ~ errors to all the experimental points. I

]01 - - I F

I

I

I

I

f

~

f

I

,-T~I

I

]~

I

I

642,', (op ) , _49 6 McV

o

u - 30.09 M~-V

ru " 1,25 fm

~qV= 14.41

HcV

rWV= 1.25 frn awv=O.65 f m

WD= 0.05 MeV Us0~.~,5 McV

r~b= 125 fm aWD=0 47 fm rdSOI 1.25 f m %50=0.65 frn

~

'MCd (p,p)

u = 34.54 McV

ru :1.25 fm

a u : 0 . 6 5 fm

Wv= i5,22 McV WD" 0.02 McV

rwT-I,25 f m rw0"125 fm

~Uwv-O.65 fm awl ) ' 0 . 4 7 f m

U5~ 4.14 L

Ou = 0 6 5 f m

t

I ,

~ ,

ruso=l.25 f m

aus0"0.65 fm

~

9~

,

I

,

!

t

30 0 0 120 crn. SCATTER!NO ANGLE (Degrees)

150

Fig. 1. Optical-model fits made to the r4Zn and 114Cd elastic cross sections with a fixed Perey f o r m f a c t o [ (fit A ) .

We made a number of fits with various combinations of constraints using pure volume absorption, pure surface absorption and a mixture of both. The opticalmodel parameters obtained from these searches for 6 4 Z n a n d l lgCd are presented in tables 1 and 2, respectively. It proved difficult to avoid false minima in searches which employed a large number of free parameters. We found that the most reliable procedure was to perform the searches in three phases; (i) vary only the well depths, (ii) vary the well depths and the diffusenesses and (iii) vary all the parameters.

22

v.R.w.

EDWARDS

In searches on the 64Zn data, Uso showed a tendency to go to zero, and it was therefore fixed for the main part of each search at 4.5, 5.5 or 6.0 MeV, depending on the value found in the corresponding 114Cd fits. At the end of the search Uso was freed and allowed to take its optimum value. These values showed considerable

101

l0 °

10-1

ta-

c~

163

.

I

0

I

I

I

.SO

~

I

I

60

t

I

I

90

.

I

.

.

.

.

t~

.

.

I

120

I

I

150

c.m. SCATTERING ANGLE (Degrees) F i g . 2. T h e i n c r e a s e d d a m p i n g o f t h e 6~Zn o p t i c a l - m o d e l d i s t r i b u t i o n s w h i c h o c c u r s as t h e d i f f u s e n e s s p a r a m e t e r s a u = a w v - - aWD = a u s o a r e r a i s e d f r o m 0.4 f m t o 1.0 fro, w h i l e t h e o t h e r p a r a m e t e r s are held at the values given above the figure. The successive curves are displaced downwards by one cycle of the ordinate scale.

variation for different types of fit and are not reliable. In table 1, the fits in which Uso was fixed are identified by a subscript 0 and those for which it was free by a subscript 1. This problem was not encountered with 114Cd, and the results for this nucleus were obtained with Uso free from the start. The only piece of experimental data which was discarded in the fitting was the 49 ° elastic point for 64Zn, which contributed over half the total p2 when it was included.

23

PROTON SCATTERING

2.2. FITS M A D E W I T H A F I X E D P E R E Y F O R M F A C T O R (FIT A)

Perey z) f o u n d that the scattering of 9-22 MeV protons from a range of nuclei could be fitted with fixed f o r m factors having the parameters ru = rwv = r w o = rus o = 1.25 fm, au = awv = auso = 0.65 fm and awD = 0.47 fro. These form factors gave p o o r fits (p ~ 30 ~o) to the present data, as will be seen from fig. 1 and the first lines of tables 1 and 2. The fits are not surprising as previous analyses at 30 [ref. 5)] and 40 MeV [ref. 7)] have shown that the Perey form factors are less satisfactory at medium energies. I01

I

i

i

I

I

I

I

I

1

I

i

I

u : 41.1g NeV

49.6 MeV ru - 1.144 fm clu- 0.8[0 frn

Wv :

E175 MeV

rwv = ru

awv~0.737~m

WD =

178

McV

rwD= ru

awD = awv

Uso= 450 M~V

qJso=ru

%so = au

I

~Zn ( p , p )

,o° g 1 % 6d' cz7

2 2

162

:

163

F i g . 3. O p t i c a l - m o d e l

!5.4o M~V

Wv: 10,16 M~V

rwv = ru

a,wv:0748 fm

Wl): 1.06 McV Lk:,o=4,6B Me"/

rwl) = ru ruso: ru

awo - awv auso = a u

i

p

i l i i i I i I i 30 60 90 120 crn, SCATTERING ANGLE (D~grecs)

i

i 150

fits m a d e t o t h e " 4 Z n a n d 114Cd e l a s t i c c r o s s s e c t i o n s w i t h v a r i a b l e

form

factors

subject to the constraints r U ~ r w v ~ rWD ~ r u s o , a U ~ a u s o, a w v - - aWD. Very similar fits were obtained with pure v o l u m e absorption using b o t h separate a n d c o m m o n real a n d i m a g i n a r y f o r m factors. T h e curve for 114Cd is displaced d o w n w a r d s by one cycle o f the ordinate scale. 2.3. FITS M A D E W I T H A V A R I A B L E F O R M F A C T O R A N D A M I X T U R E O F S U R F A C E A N D V O L U M E A B S O R P T I O N (FIT B) The main

defect of the angular

distributions

are that they are insufficiently damped

obtained

at backward

with the Perey form

angles. The damping

factor

of optical-

TABLE 3

t°Ni

tONi, 5~Fe, ~ F e , 65Cu

~4Zn, la4Cd

Ce, 2°~Bi

"°Ni

~"Ni, 51Fe, ~6Fe, ~ C u

~Z n, n~Cd

~'~Ni,2°9Bi

4.0

50

78

30

40

50

57

Nuclei

30

Energy (MeV)

1.28

1.16

1.16

(1.20)

1.23

1.14

1.17

(1.20)

ru

0.762

0.848

0.756

0.713

0.57

0.837

0.728

0.693

aU

rU

1.27

1.46

1.659

rwv

aU

0.656

0.643

0.556

awv

1.22

0.895

1.03

1.14

rWD

0.72

0.978

0.743

0.767

awD

rU

ru

rU

1.048

rU

rU

rU

1.049

rUs o

au

au

au

0.713

aU

au

aU

0.693

aus o

volume

surface

Data fitted

cross sections

cross sections

cross sections

polarizations and cross sections

cross sections

cross sections

cross sections

polarizations and cross sections

Type of a bs orpt i on

present work 2G)

Preliminary analysis only

7)

Jo)

present work

7)

~)

Ref.

11'1Cd results are from FrickeSatchler start s~Zn paranacters were obtained with Uso fixed

6~Zn pa ra m eter s were obtained with Uso fixed

Comments

S u m m a r y of m ea n geometrical p a r a m e t e r s found in optical model fits to prot on scattering from m e d i u m and heavy nuclei in the energy range 30-78 MeV

1.11

1.16

1.15

•W,Fe, ,~SNi, OONi 5'~Co, Cu, '2°Sn, 2°spb

5SNi, 9°Zr, e°sPb

,54Fe, 58Ni, 59Co, 6°Ni, 68Zn, 9OZr, 12°Sn, 2°8pb

G4Zn, lt~Cd

30

40

40

50

1.19

(1.20)

~SNi, 5°Co, 6°Ni

30

1.19

aaFe, ~r'Ni, G°Ni 59Co, Cu, 12°Sn, 2°sPb

30

0.794

0.764

0.734

0.760

0.699

0.689

rU

1.37

1.34

1.32

1.24

rU

0.742

0.639

0.606

0.618

0.691

aU

rU

rwv

rwv

rwv

rwv

rU

awv

awv

awv

awv

awv

0.696

rU

1.047

1.070

rU

1.065

rU

aU

0.775

0.687

aU

0.699

C;U

cross sections

polarizations and cross sections '~4Znp a r a m e t e r s were obtair, cd with Uso fixed

present work

0)

8)

s)

polarizations and cross sections polarizations and mixture cress sections of volume and polarizations and surface cross sections

4)

cross sections

26

V. R. W . E D W A R D S

model distributions can often be increased by raising the absorption potential or by using an imaginary radius which is larger than the real radius 18). The first approach is clearly unsuccessful here as both W v and WDwere free to vary in the fitting, while the second approach may require more parameters than is desirable. For the present data, a third method was found effective, namely increasing the diffuseness parameters. Fig. 2 shows how the damping varies for 6 4 Z n a s a U = awv = awD = avs o is raised from 0.4 fm to 1.0 fm, while the other parameters are fixed at the values given in line A o of table 1. The increase in the slope of the angular distributions which also occurs as au is raised can be partly compensated for by variation of the other parameters. This effect was exploited in a second search in which U, Wv, WD, Uso, ru, au and awv were varied and the other parameters were subject to the constraints rwv = rwD ----- ruso = ru, aWD = awv and auso = au. Fig. 3 shows that the fits are greatly improved (p ~ 6 ~ for 6 4 Z n and ~ 12 ~o for 114Cd). The parameters obtained for the two nuclei are similar, except that the 114Cd radius parameter is 10 ~ larger than that for 64Zn and consequently the real well depths differ considerably. The real and imaginary diffusenesses are significantly larger than the Perey values. Table 3 compares the present findings on this point with those of previous analyses in the range 30-78 MeV. It is clear that the increase in nuclear diffuseness is a general phenomenon in this energy range. Analyses 18) at ~ 180 MeV show that the diffuseness decreases again at higher energies. 2.4. FITS MADE WITH PURE VOLUME ABSORPTION (FITS C, D AND E) Several fits to the data made with pure volume absorption gave results which were almost indistinguishable from those obtained above. Two of these searches were started with equal real and imaginary geometry parameters, the first search being subject to a constraint which maintained this equality, while the second was not. For 64Zn, the additional free parameters in the second search led to a small improvement in p. The values obtained f o r p from x 1 4 C d w e r e almost identical in the two cases, although there was a considerable change in the imaginary diffuseness. The differences observed in the second search between the real and imaginary form factors of 64Zn were in qualitative agreement with those noted by Fricke and Satchler (their mean values for volume absorption were ru = 1.16 fm, rwv = 1.46 fm, av = 0.756 fro, awv -- 0.643 fm), however our radius parameters showed less difference and our diffuseness parameters more than their values. There is undoubtably an ambiguity here, as both an increase in the real diffuseness or a difference between the real and imaginary radii can be used to produce the damping effects necessary to fit the cross sections. To test this point a third volume only search was started with the mean Fricke and Satchler geometry. It will be seen from tables 1 and 2 that the difference between the real and the imaginary radii increased for both nuclei. For 64Zn the difference in the diffuseness parameters decreased substantially. The 114Cd diffusenesses still showed

27

PROTON SCATTERING

a large difference, a fact that may be connected with the anomalously high value of Uso, which is almost twice as large as in the other volume only fits. For 64Zn, the Fricke and Satchler start yielded fits which were slightly worse than those obtained in the second equal start, volume only search (fit D). For 114Cd, the fits obtained with the Fricke and Satchler start were better than even those obtained with a mixture of volume and surface absorption. i01

i

i

i

i

i

i

i

I

I

I

l

i

l

l

64Zn(p,p) 49,0 MeV u = 45.52. McV

p~

ru : I,II5 fm au=0.854 fm

W 0=[154HeY rw0"0.844 Ow6-0.969fm Uso:55McV ruso" ru %s0" °u

i0 0

o

uJ

c~ iO_ I

f'~ f f ~

"'Cd(p,p)496 H~V'.....

0

i0-2

US~ 8.05 NeV

i03 0

30

rus0" 1,171 f m

6

0

OL~0=0.859 f m

I 0

150

crn SCATTERING ANGLE (Degrees) Fig. 4. Optical-model fits made to the 64Zn and 11~Cd elastic cross sections with pure surface absorption. The curve for 114Cd is displaced downwards by one cycle of the ordinate scale. 2.5. FITS MADE WITH P U R E SURFACE ABSORPTION (FIT F)

Fig. 4 shows that the fits obtained with pure surface absorption are markedly inferior to the volume only fits. It will be noticed from tables 1 and 2 that the program attempts to mimic the effects of volume absorption by decreasing the imaginary radius and increasing the imaginary diffuseness until it is slightly larger than the radius parameter. The geometrical parameters obtained for 64Zn and 114Cd were very

28

V.R.W. EDWARDS

similar and differed from the mean surface absorption parameters of Fricke and Satchler (ru = 1.17 fro, rwD = 1.03 fro, a u = 0.728 fm and awD = 0.743 fro) in showing a greater disparity between the real and imaginary radii. Also the diffusenesses, particularly the imaginary diffusenesses, were considerably larger. 2.6. T H E E F F E C T O F S M E A R I N G O N T H E ~4Cd FITS

It will be noticed that the l a 4 C d fitS are considerably worse than the corresponding fits. This appears to be due to the greater effects of smearing on the l I~Cd angular distributions, as over half the total pZ comes from six points in the vicinity of the first minimum. The optical-model program which we employed corrects for smearing, but these effects are very large for the first minimum and the uncertainty in the effective smearing aperture is too great to permit the corrections to be made with sufficient accuracy. This point was further investigated by dividing the angular range in two and making separate fits to the forward and backward angle data. It will be seen from tables 4 and 5 that the 11~Cd backward fit is of comparable quality to that for 64Zn, but that the forward-angle fit is relatively poor. 64Zn

TABLE 4 Comparison of volum ~ - only fits made with equal real and imaginary form factors to different angular ranges of the H~Cd cross-section data Angular range of data fitted

U

Wv

Uso

r

a

N

P ~o

Pall 1

Pall 2

10- 72

33.82

11.46

6.43

1.248

0.752

1.112

13.4

46.6

24.0

74-146

33.29

12.01

4.10

1.254

0.757

0.935

5.5

13.7

13.7

10-146

35.06

12.33

4.84

1.238

0.787

1.019

12.3

TABLE 5 C o m p a r i s o n of volume only fits made with separate real and imaginary form factors to different angular ranges of the 64Zn cross section data A n g u l a r range of data fitted

U

Wv

Uso

ru

au

rwv

awv

N

P °/oo

Pall l ( ~ )

10- 63

41.17

11.30

(4.5)

1.166

0.791

1.234

0.873

0.928

4.1

10.7

65-141

41.45

15.19

(4,5)

1.166

0.848

1.234

0.699

0.712

4.4

11.5

10-141

41.04

15.02

(4.5)

1.166

0.843

1,234

0.652

0.810

6.4

These separate, forward-angle and backward-angle fits are also of interest because they allow the meaningfulness of the optical-model parameters to be assessed. If optical-model fits were just empty parameterisations unrelated to the basic physics of nuclear scattering, one would expect different parameters to be obtained from the forward- and backward-angle data. In fact, almost identical values are obtained

34.15

37.84

Volume o n l y - equal start C o m m o n real and im ag ina ry form factors

Volume only - equal start Separate real and imaginary form factors

Surface only

C

D

F 1.115

1.212

1.244

1.232

ru

0.871

0.816

0.780

0.764

au

14.06

12.93

9.55

Wv

Constraints are recorded in the ap pr opr iat e columns. In all cases r U =:: rus o and a U : : a us o.

48.95

36.73

U

Volume and surface a bs o r ption

Type o f search

B

Identifier

TABLE 6

1.250

rU

rU

/'WV

0.713

au

0.798

awv

14.81

1.65

WD

0.927

rU

rWD

Optical-model pa ra mete rs for i14Cd fits w i t h o u t the extreme forward angle points (10°-27 °)

0.935

aWV

awo

8.191

4.815

4.403

4.29

USO

0.622

0.805

0.810

0.908

N

8.8

6.0

6.3

5.5

P

30

V. R. ~,V. ED~,VARDS

except for Uso, which one would not expect to be well determined by cross-section data alone, and for the normalisation. The change in this latter quantity is puzzling. Similar differences also occur between the values found in different types of fits in tables 1 and 2. It would appear that the normalisation is not well determined by the optical model in this energy range. As a further test of the meaningfulness of the parameters, we examined the quality with which the forward- and backward-angle 101

i

l

i

l

~

i

i

i

ll4Cd (p,p).496 M~'V

m

f

l

i

i

FIT B

u = 56.73 MeV

ru= 1232 fm o u - 0.704 fm

Wv = 9.55 MeV Wo : 1.65 MeV Llso:4.29 MeV

rwv= ru rw~= r u rus0-r u

~tv: 0.798 frn OwD =

Owv

aus0= o u

I0 c

,.=, 114Cd (p, p) 49.6 MeV FIT F

.'~ o

5

Id 2 u = 4895 MeV WD= !481 HeY

ru=lil5 fm a u ' O . B l l f r h ~D'O.g27fm cIw~0.935 fm

US~ 8]9

rus0 -nj

MeV

OlJs0- o u

id 3 L 0

30 cm

60 SCAITENNG

90 120 ANGLE (Degrees)

150

Fig. 5. Optical-model fits B and F made to the 114Cd elastic cross-section data without the forwardangle points (10-27°). In order to avoid confusion, curve F has been displaced downwards by one cycle of the ordinate scale. The improved fits are a consequence of the removal of the experimental points m o s t badly affected by the smearing. The two other fits (C and D / w e r e almost identical to B.

parameters fitted the whole of the data, the normalisation being left fixed at the values found in the partial fits. In tables 4 and 5, the p-value obtained is listed under the column Pan 1" Except for the 114Cd, forward-angle case, the fits are moderately good. Allowing the normalization to take the value which gives the optimum fit to

PROTON

31

SCATTERING

the whole data somewhat improved the fit of the 114Cd forward-angle parameters, as will be seen from columnpa . 2 in table 4. The four most interesting 114Cd searches (B, C, D and F) were repeated without the extreme forward-angle points (10°-27°), and the results are given in table 6. It will be seen that these fits are slightly better than the corresponding 64Zn fits. The best results were obtained with a mixture of surface and volume absorption. However, pure volume absorption is little worse. Surface absorption gives p 2 values more than twice as large as those obtained with volume absorption. Fig. 5 contains a plot of fits B and F. Fits C and D resemble B very closely. 4 0 1 1 1 1 1 1 1 1 1 1 1 1 1 ] I I I CLASS

P% Q \ 50

°~\

\

ck "'".

......

,ncreas~ in p r e l a t i w to minimum

0

I

1.0

I

I

CLASS

B

CLASS

C

\ i

2

~: > . . ~

25°/o

A

64Zn

114Cd

~-"-rj

_/0 PilNIMUM

[ 12

I

" ~"~....,,:,~ .~ " y

I

I II

I

I

ru

t

I

I

I

I 13

I

~L___

Hm)

Fig. 6. Plot o f p versus r u for the three classes o f U - r u ambiguity tests made on ~4Zn and 114Cd. 2.7. S T U D Y O F T H E U - - r U A M B I G U I T Y

We made a detailed study of the U - r u ambiguity at these higher energies. Without such studies it is not possible to make meaningful comparisons between different analyses. Moreover, the U - r U ambiguity is the principal source of uncertainty in the determination of the optical-model parameters. The method used was to fix ru at various values and vary other parameters to obtain the best fit to the data for each case. Pure volume absorption was employed in these tests with identical real and imaginary form factors for 114Cd and separate

32

V. R. W . E D W A R D S

ones for 6 4 Z n . We performed three classes of test in which the free parameters were, respectively (A) U and Wv, (B) U, Wv and the diffuseness, (C) (for 64Zn only) U, W v the diffuseness and rwv. For 114Cd only, Uso was varied in addition to the above parameters. It will be seen from fig. 6, which shows the variation of p with ro, that parameters other than U and ru also play a part in the ambiguity. This is particularly noticeable I01

I

I

I

I

I --I

64

I

I

I

I

I

I

I

I

I

I

Zn ( p , p ) 49.6 MeV

f~(z)

......

ru "

le5

......

ru -

1.166f m

ru

,.o5 Ir~ (x)

:

(Y)

i0 c

,.'5,

2 = 10-'

i6z

I

I

I 30

I

&

I 60

cm. SCATTERING

I

I

I 90

ANGLE

l

I

I 120

150

(Degrees)

Fig. 7. Three fits made to the e~Zn elastic cross section with different values o f r U. These curves correspond to the points X, Y, Z on fig. 6, and show that the main effect of increasing r U is to d a m p the diffraction pattern m o r e strongly and to move it in to smaller angles.

at ru ~ 1.25 fm for 6 4 Z n , where freeing rwv revealed a subsidiary minimum. For 114Cd at ru ~ 1.10 fm, variations in the common diffuseness parameter halved the p-value. The effect that changing r u has on the angular distributions is shown in fig. 7 for 64Zn. The larger values of ru d a m p the diffraction pattern more strongly and move it into smaller angles.

PROTON

33

SCATTERING

In all cases the variation of U was found to be accurately fitted by a U r U = K law, where n and K are constants. Fig. 8 shows a plot of log~o U versus lOgl0ru for a typical set of results (64Zn, class A). The values obtained for n and K are listed in table 7. It will be seen that n decreases as the number of additional free parameters is raised. This is to be expected, because these parameters assist in maintaining the ambiguity and consequently U needs to vary less for a given change in rv. The larger i

1

,

~

i

,

i

i

r

i

t

,

,

,

180

i

"60

1'50iL

t

1400

J

I

0.02

I

0.014

I

I

0.06

L

0.;8

I

0.10

0 i2

0.14i

log Io ru

Fig. 8. A plot of log10 U against log10 ru obtained in the class A U - r U ambiguity test on ~Zn. The straight line which fits the points so well corresponds to the equation Uru3-03 -- 65.4.

values of n obtained for ~ 14Cd are probably caused by the constraint rwv = re. The results for n should be compared with the value of 3 given at high energies by the Born approximation and with the figure of ~ 2 observed at lower energies. Figs. 9 and 10 show the variations in the other parameters. These variations appear to be very complicated. The only behaviour which is common to the two nuclei is the decrease of a v with increasing r u. This phenomenon has also been observed 5) at 30 MeV. The variation of Wv for ~14Cd is of interest because it is fitted well by a linear law with a slope constant of - 8 9 MeV/fm for class A and - 6 0 MeV/fm for

34

V. R. W . EDWARDS

TABLE 7 Values o f n a nd K K (MeV) • fm ~ 6~Zn

a~4Cd

64Zn

~4Cd

class A

3.03

3.58

65.4

75.3

class B

2.99

3.26

64.9

70.4

class C

2.78

62.8

6O

,

2CLA,S,S' A&B

t

- - - CLASS B - - CLASS C

> 40

LI

1.0

0.9 =

20

U

0

t >~ 50 IE

0.8

/

....

CLASS A

---

CLASS B

--

CLASSC /

1.0

. . . . CLASS B l!I " . --CLASSC~,

/ t /' /

o.8~ v

I

z0

0.6

~,

O

¼

12 0.8

I.o

N

/

0,7

v

.

. . . . CLASS B -CLASS C 0.0

H

i.2 ru

(fm)

0.8 [

11.5

1.5 ru

(fro)

Fig. 9. The v a r i a t i o n of the o p t i c a l - m o d e l p a r a m e t e r s with r U o b t a i n e d in the /.dr u a m b i g u i t y tests on 6~Zn. The arrows m a r k the points at which there is a 25 ~ d e t e r i o r a t i o n in the quality of fit relative to t h a t o b t a i n e d with the o p t i m u m va l ue of r U.

class B. This b e h a v i o u r is a result o f the c o n s t r a i n t rwv = ru. F o r 64Zn a similar decrease takes place for the lower values o f r v , however the rate o f fall o f Wv increases as the numlzer o f constraints is reduced. The change in b e h a v i o u r for 6 4 Z n ol~served at radii > 1.2 fm is a consequence o f the subsidiary m i n i m u m . This m i n i m u m is characterized by extremely low values o f the i m a g i n a r y radius ( < 0.7 fm). It will be noticed that IPv a n d awv are forced up to c o m p e n s a t e for this effect. The rv scale on the graphs is m a r k e d by arrows at the radii which c o r r e s p o n d to a 25 )~o deterioration in the quality o f FLt. The m a x i m u m v a r i a t i o n of the p a r a m e t e r s observed within

35

PROTON SCATTERING

these limits was used to calculate the uncertainties in the optical-model parameters listed in table 8. The figures for 114Cd are based on the class B curves and those for 6 4 Z n o n the class C curves. It should be noted that these estimates are lower limits. There are other sources of uncertainty beside the U - r u ambiguity. In addition the choice of a 25 ~ change in p is arbitrary. 12

t

i ....

CASE

A

z

~.o

+

4,

6O

0.9

"

t"l

)

'

1

2O

/

20

//

10 >

10" ~ , t~

[2 ru

,

I4

1.5

;

Urn?

,+

112. 13 FLI (f,-n)

ll.4

Fig. 10. T h e v a r i a l i ~ n of the o p t i c a l - m o d e l p a r a m e t e r s with r U o b t a i n e d in the U - r u a m b i g u i t y tests on r e C d . The a r r o w s m a r k the points at which there is a 25 % de t e ri ora t i on in the qua l i t y of fit relative to t h a t o b t a i n e d with the o p t i m u m value of r u.

TABLE 8 U n c e r t a i n t i e s in the o p t i c a l - m o d e l p a r a m e t e r s o b t a i n e d with pure v o l u m e a b s o r p t i o n

Parameter U Wv Uso rU au rwv awv N

64Zn M e a n value and errors

114Cd M e a n value and errors

4 1 . 0 4 + 5 . 7 - - 4 . 3 MeV 1 5 . 0 + 0 . 4 3.8 MeV

35.1 + 8 . 7 - - 6 . 4 MeV 12.3-; 5.7--4.7 MeV 4 . 8 4 + 1 . 9 5 - - 1 . 3 5 MeV 1.238" 0.079 0 . 0 9 9 f m 0 . 7 8 7 + 0 . 0 9 3 0.054 fm rU at; 1.014 " 0 . 2 i - 0 . I 1

1.166+0.041 --0.049 fm 0 . 8 4 3 + 0 . 0 2 3 0.033 fm 1 . 2 3 4 + 0 . 0 6 1 - - 0 . 0 0 4 fm 0 . 6 5 2 + 0 . 0 0 7 - - 0 . 1 0 fm 0 . 8 0 9 + 0 . 0 8 6 0.015

36 2.8. A V E R A G E

V. R. GEOMETRY

"~'.

EDWARDS

PARAMETERS

At the lower energies it has been a fruitful practice to obtain sets of m e a n metrical parameters suitable for use over a wide range of energies and nuclei. few analyses have been performed at 30-78 M e V to establish a set which can be with any confidence at these energies, however two groups of workers 7,9) proposed tentative average geometries for use at 40 MeV. In the hope that at

i01

'

~

6 j~Zn i p , p ;i

49.6 ,

MeV , ~

~

,

,

-1-

u = 42.51

McV

r u = [16 f m

Wv = 6.87 = i.70

MeV MeV

rwv= 1.57 fm qWv=0.53 fm rWD=I57Inn CW~=0.63 fm

,LIs~6.0~HcVI tI

rjso~lO64fmQUSO'0.738f m

I(}'

"

,

geoToo used

have least

,

cl u = 0.75 frn

114Cd ( p ' P) 49.6 MeN

o

16'--,

;LZ

u = 4422 ~ WV= 7.51 I~V

ru = 1.16 fm q j - 0.75 frn ~ k . , , ~ , N rw¢,. 1.57 frn Owv-- 0.65fro "x2/"O.o'~

Wo - 1.50

McV

rw~= 1.57 f m OwD-- 0.65 f m

Uso'6b4

McV

ruso:l.O641m%so'OT~fm

,

;

,

,

~

'

,

,

,

"~

x

3 0 90 120 cm SCATTERING ANGLE (Degrees)

,

,

150

F i g . l 1. O p t i c a l - m o d e l fits m a d e to the 64Zn and 11~Cd elastic cross sections with the fixed g e o m e t r y and the v a l u e o f Uso p r o p o s e d b y F r i c k e et al. 9). T h e curve for uaCd is displaced d o w n w a r d s b y o n e cycle o f the ordinate s c a l e .

one of these parameter sets w o u l d be useful at 50 MeV, we tried t h e m on our o w n data. The results obtained with the geometry of ref. 7) were poor (p = 21 ~o for 64Zn and 2 0 ~ for 1~4Cd). The results obtained with the parameters o f ref. 9) (r U = 1.16, a u = 0.75, rwv = rwD = 1.37, awv = aWD = 0.63, ruso = 1.064, auso ---- 0.738, Uso = 6.04) are d o c u m e n t e d in the last lines of tables 1 and 2 and are plotted in

37

PROTON SCATTERING

fig. 11. It will be seen that these fits are only slightly worse than the best fits obtained with variable form factors, particularly for 64Zn (p = 7 %). Almost certainly, the l l4Cd results would have been comparable to those for 64Zn but for the effects of smearing already discussed. These findings together with those of Fricke e t al. 9) suggest that this form factor is satisfactory over the range 30-50 MeV.

lO I

i

I

/

l

e4Zn (p,p).49.6 McV

I0° BOTH NO COULOMB EXCITATION

-I

2 o

I0 •

c~ LJ

°o-'.~

' ~ ELASTIC ONLY \NO COULOMB eae

o of(32

~,~,kBOTH WITH COULOMB EXCiTATiON

164

0

3'0

s

r

go

I

I

~o

I

i

*~o

]

I

I

i~o

cm. SCATTERING ANGLE (Degrees) Fig. 12. S C A fits to the 64Zn elastic cross section. C a p t i o n s beside each curve indicate w he t he r C o u l o m b ex citation was included, and w h e t h e r both the elastic a nd inelastic d a t a or the elastic da t a alo ne were fitted. Successive curves are displaced d o w n u a r d s by one cycle of the o r d i n a t e scale.

3. SCA analyses 3.1. P R O C E D U R E

The 64Zn and 11~Cd nuclei are two typical vibrational nuclei whose ground states have rms distortions /3 = 0.25 [ref. 19)] and /~ = 0.20 [ref. 10)]. The first excited levels are respectively at 995 and 556 keV, and both are 2 +, one-phonon, quadrupole

38

V.

R.

%V. EDWARDS

vibrations. In ref. 1), scattering from these levels was measured at the same angles as the elastic scattering. Here we attempt to fit both the elastic and inelastic data using SCA theory. To reduce the computing time for the analyses, the geometrical parameters were left fixed at the values found in the equal start, volume only, optical-model searches (fit C for ll4Cd and fit D O for 64Zn). Even with only the four free parameters U,

[0l

--T----'I~

I

i

I

I

I

I

t

:'}_rid(p,p) 49.6

I

I

i

I

I

I

.~.v

I

I00i A ' ~

®®% ""

~ '

E o o

.. D~

L/

°"

~

i ~t IJ ~"

'~.,'/

\

' °%

EXCITATION

,,,,~, ELASTICONLY ~ , EXCITATION

-t,_x -',, NO COULOMB

".

".

..~'\

~L..~Z,~,, BOTHWITH

•

~,
~.COULOMBEXCITATIO i tl,

.

'6' i

i

/

I~) 4

o

I

I

~ _

30

I

i

<;o

I ~--

I

~o

'

I

Jo

I

&..---J

~o

cm SCATTERING ANGLE (Dcgrees) Fig. 13. S C A fits to the m C d elastic cross section. See c a p t i o n to fig. 12.

Wv, Uso and/7, the program took over 30 min on the Harwell Atlas Computer to do one cycle of the search. Most of the work was without Coulomb excitation. Including it increased the computing time by ~ 30 }/o. Four different types of SCA fit were made as described below. The resulting parameters are presented in tables 9 and 10, and the theoretical cross sections are compared with the experimental data in figs. 12-15. In all these calculations, both the real and imaginary potentials were

15.02

15.02

50.00

42.98

41.04

41.04

inelastic only no Coulomb excitation

elastic and inelastic cross sections fitted simultaneously with Coulomb excitation

D W B A comparison run no Coulomb excitation

optical-model run D O (for comparison)

C

O

E

F

13.63

18.45

14.13

40.72

elastic only no Coulomb excitation

B

14.12

Wv

43.88

U

elastic and inelastic cross sections fitted simultaneously no Coulomb excitation

Type of search

A

ldentilicr

4.50

4.50

4.42

1.68

4.69

3.99

Uso

0.02

0.243

0.222

0.133

0.222

fi

0.244

0.232

flel'fcctlye

Parameters obtained for coupled channel fits to ~4Zn data

TABLE 9

0.809

0.00613

0.9128

0.837

0.8346

0.9162

N

6.4

10.6

6.7

11.9

Pel

23.3

21.9

19.0

22.6

Plnel

17.3

18.1

Pall

36.17

34.70

37.78

35.89

35.06 35.06

elastic only no C o u l o m b excitation

inelastic only no C o u l o m b excitation

elastic and inelastic cross sections fitted simultaneously with C o u l o m b excitation

D W B A comparison run no C o u l o m b excitation

optical-model run C (for c o m p a r i s o n )

B

C

D

E

F

u

elastic and inelastic cross sections fitted simultaneously no C o u l o m b excitation

Type of s e a r c h

A

Identifier

TABLE 10

12.34

12.34

10.33

10.43

11.86

10.21

Wv

4.84

4.84

4.35

2.07

510

3.96

Uso

0.02

0.192

0.170

0.111

0.170

//

0.198

0.162

/~rf~0.,,o

P a r a m e t e r s obtained for coupled channel fits to m C d data

1.014

0.0121

1.154

1.304

1.006

1.189

N

12,3

14.5

12.4

15.4

Pc1

18.0

11.6

10.7

13.5

Plnel

13.1

14.5

P~n

PROTON

41

SCATTERING

deformed, but the spin-orbit potential was spherical. Coupling terms up to second order in/7 were employed.

"e

/A J'3

~'Zn

lOI I'- BOTH I seA

(p,p')_490 M~v

z ~ 995 k~v STATE

\

I'°-...i i i\ ,_#'\ /

c/\..

I~ELASr~C~ "

\

NO COUL ~ ".

~

,oc,o i. co

".

zE

~'''0

~o

•

Q°o I

o I

-

•

\

~...

"~.

~,

..

I~)4

t /

0

" o I

I

I

50

!

I

60

I

I

I

90

I

I

I--I

120

~'__

150

c.rn. SCATTERING ANGLE (Degrees)

Fig. 14. S C A a n d D W B A fits m a d e to the G~Zn i n e l a s t i c cross section. C a p t i o n s b e s i d e each c u r v e i n d i c a t e w h e t h e r S C A o r D W B A w a s used, w h e t h e r C o u l o m b e x c i t a t i o n w a s i n c l u d e d a n d w h e t h e r b o t h the elastic a n d i n e l a s t i c d a t a o r the i n e l a s t i c a l o n e w e r e fitted. S u c c e s s i v e c u r v e s are a r b i t r a r i l y displaced.

42

v . R . W, EDWARDS

i

i

'7

i

i

IMCd (p p ) 4 9 6 MeV

IOI LSCA \ [NO COUL

2 + 556 keV STATE

I

SCA A

.'.

QOO

E IO-i o=

INO COUL

F--

,o

g (o

g:

152

L,,_

A

t

16 4 0

;o

go

;o

'

120 cr'n SCATTERING ANGLE (Degrees)

150

Fig. 15. SCA and D W B A fits m a d e to the 114Cd inelastic cross section. See c a p t i o n to fig. 14. 3.2. FITS M A D E S I M U L T A N E O U S L Y TO T H E E L A S T I C A N D I N E L A S T I C CROSS SECTIONS WITHOUT COULOMB EXCITATION

In the first search, the elastic and inelastic data were fitted simultaneously. The normalization was adjusted to give the best fit to just the elastic data. The p criterion was again used, and since there were the same number of elastic and inelastic points,

PROTON SCATTERING

43

the elastic and inelastic data had equal weights in the fitting. For both 64Zn and 114Cd the fits to the elastic data were worse than those obtained with the ordinary optical model, which is not surprising when one remembers that the same parameters are being used here to fit both the elastic and the inelastic data. For 114Cd the inelastic fit was slightly better than the elastic fit. For 64Zn the inelastic fit is markedly inferior, the disagreement being particularly bad at extreme forward and backward angles. The parameters obtained are very similar to those found in the optical-model fits. (For ease of comparison, these latter results are included as the last lines of tables 9 and 10.) The decrease in Wv is the most significant change. This is caused by the fact that the excitation of the first inelastic level is explicitly calculated in the SCA approach, while in the optical model it is included with the other non-elastic processes which are represented by the imaginary potential. The size of the change in Wv depends on fl and on the position of the first level, as well as on other factors. Here, it will be seen, the lower energy of the first level of 1i 4Cd leads to a considerably larger change in Wv for that nucleus even though its fl-parameter is 30 % lower than that of 64Zn. 3.3. SEPARATE ELASTIC AND INELASTIC FITS MADE WITHOUT COULOMB EXCITATION The fact that the SCA elastic fits are worse than the optical-model fits suggest that somewhat different parameters are required to fit the elastic and inelastic cross sections in a coupled-channels calculation. This is a point of some importance since any significant difference in excess of that expected from the different proton energies would indicate that the SCA theory was unsound. A test of this type does not seem to have been made by any previous worker. Lines B and C of tables 9 and 10 contain the results obtained in separate fits to the elastic and inelastic data. The elastic fits were performed in the same manner as the simultaneous fits. With the inelastic fits, it was necessary to fix either ~ or the normalization since both affect the absolute value of the cross section. The more accurate procedure would have been to fix the normalization. However, little error is caused and much time is saved by adopting the alternative approach. (The value/~ changes only slightly from the value found in the simultaneous fits.) The effective value of fl was extracted by assuming that the inelastic cross section was proportional to/~2 and is quoted in a separate column in the tables. Almost identical values of U and Wv were obtained from the elastic and inelastic fits. In both cases the elastic fits yielded lower values of U and higher values of Uso than did the ordinary optical model, while the reverse was true of the inelastic fits, which shows that no significance should be attached to the changes which coupling induces in these parameters. Since previous SCA analyses are weighted strongly in favour of the elastic data, U has always been observed to decrease when the coupling is introduced. The most interesting result obtained in these tests was that the elastic/%values were

44

V . R. W . E D W A R D S

very much smaller than the reliable figures previously obtained from quadrupole moment and transition rate data. The p-values obtained for the elastic only SCA fits are ~ 0.2 % higher than those obtained with the ordinary optical model. Small differences in the accuracy of the numerical methods used by the two programmes and the fact that it was not practical to take the SCA searches right up to convergence are probably responsible. 3.4. THE EFFECT OF COULOMB EXCITATION The/~-values obtained in these fits are lower than those obtained by other methods. A number of authors 7,21) have shown that the neglect of Coulomb excitation usually leads to low values of/3. Coulomb excitation also has some effect on angular distributions, particularly at forward angles and at low energies. These considerations led us to try a simultaneous fit to the elastic and inelastic data with Coulomb excitation included. The results are given in line D of tables 9 and 10. It will be seen that the fits are slightly improved and that the/%values are higher by 9 % for 64Zn and by 11% for ~14Cd than the values found in search A. Our final values for/3, namely 0.243 for 64Zn and 0.192 for 114Cd, are somewhat lower than the accepted values of 0.25 and 0.20, respectively. For 64Zn the fit is of similar quality to that obtained using the SCA, but for 114Cd the D W B A fit is significantly worse. In both cases somewhat higher values of/~ were obtained from the DWBA. 3.5. COMPARISON WITH DWBA The results of a D W B A fit to the inelastic data without Coulomb excitation are given in line E of tables 9 and 10. Because no D W B A code was available, these calculations were made with the SCA program using a distortion parameter of only 0.02 and assuming that the inelastic cross section was proportional to ,8z. 4. Discussion

4.1. SUMMARY We consider three topics arising from this work; (i) the relative merits of the DWBA and SCA methods of analysing inelastic scattering, (ii) the basic accuracy of the optical model and its generalization and (iii) the conclusions that can be drawn from this work about the systematics of the optical-model parameters for proton scattering in the energy range 30 78 MeV. 4.2. COMPARISON OF DWBA AND SCA An SCA analysis of inelastic scattering from a 2 + state requires ~ 100 times as much computing time as the corresponding DWBA analysis. The present work consumed 40 h of time on the Harwell Atlas computer. Three of the arguments which have been advanced to justify the use of the SCA despite its inefficiency are (i) that

PROTON SCATTERING

45

the D W B A is inaccurate for large distortions; (ii) that the D W B A cannot be used where coupling between several inelastic levels has to be considered and (iii) that the SCA allows one to calculate tile effect of the inelastic channels on the elastic scattering. It was hoped that optical-model parameters obtained from the SCA approach would show a smoother variation fi'om nucleus to nucleus as a consequence of the removal of the fluctuating effects of the strongest inelastic channels. The first of these arguments originated from ref. i9) and was the consequence of a subtle error which was discussed by later workers 22). There is only one case on record 23) of disagreement between SCA and DWBA, and that concerns a light nucleus and a low proton energy at which there was a strong resonance for one of the partial waves. The theoretical considerations in ref. 23) which show that SCA and DWBA calculations MAY differ in terms of O(fl 4) do not constitute a closed argument. The extensive check fits made in ref. 22) make it clear that the D W B A does not show any tendency to fail at large distortions when it is applied to good optical-model nuclei. The comparison in tables 9 and 10 between the DWBA and SCA further confirms that the two approaches are of comparable accuracy. The/~-values obtained with the DWBA are both ~ 10 ~ higher than those for the SCA calculation. The somewhat better fits to the inelastic angular distributions which were obtained from the SCA are not surprising as the parameters used in this model were adjusted to fit the inelastic cross section. If a similar fitting procedure had been applied to the D W B A calculations, these fits would have been considerably improved. So far as the second argument is concerned, the only multiple coupling problems which are of much interest are (i) the multiple excitation process for two-phonon, quadrupole states and (ii) the excitation of the group of low-lying states which arises in odd-mass vibrational nuclei from the coupling of the first quadrupole state of the core to the last nucleon. The first of these still cannot be treated by DWBA, but the second can be handled with sufficient accuracy for most cases within the framework of the core-excitation model 24, 25). In most recent work which has employed the SCA approach, the sole reason for preferring it to the D W B A has been the third argument, despite the fact that there has never been an experimental test of the validity of the SCA predictions for the effect that the inelastic coupling has on the elastic scattering. The discovery in the present work that the value of fi which gives the best fit to the elastic cross section is only about half as large as the actual distortion parameter of the nucleus and consequently corresponds to only a quarter of the inelastic coupling which one would expect, throws doubt on this aspect of SCA theory. A possible explanation may lie in an unfortunate ambiguity which allows the optical-model parameters to absorb some of the effects of the coupling. Apart from the obvious choice Wv, which is known to represent coupling effects well from the ordinary optical model, Uso is also a possible culprit because the damping effects

46

V. R. W . E D W A R D S

p r o d u c e d by the c o u p l i n g are very similar to those which can be p r o d u c e d by increasing Uso. This similarity is p r o b a b l y c o n n e c t e d with the fact t h a t the s p i n - o r b i t effects also enter t h r o u g h c o u p l e d equations. These ideas are s u p p o r t e d by the o b s e r v e d trends in the fitting. F o r 114Cd ' b o t h Wv a n d Uso are larger for the elastic only fit t h a n they are for the s i m u l t a n e o u s fit to the elastic a n d the inelastic data. F o r 64Zn, Uso is significantly larger for the elastic only fit, while W v has a p p r o x i m a t e l y the same values for b o t h types o f fit. Nevertheless these results m u s t cast suspicion on S C A t h e o r y since, if it gave a g o o d a c c o u n t o f the coupling, one w o u l d expect this a c c u r a t e d e s c r i p t i o n to d o m i n a t e over the c r u d e r i m a g i n a r y potential representation or the casual similarity between s p i n - o r b i t effects a n d the inelastic coupling. 4.3. THE ACCURACY OF OPTICAL-MODEL, CROSS-SECTION PREDICTIONS It has only recently b e c o m e possible to answer the question " h o w accurate are o p t i c a l - m o d e l , cross-section p r e d i c t i o n s ? " E a r l y e x p e r i m e n t a l results were of p o o r precision a n d the low energies used in m o s t experiments m e a n t that c o m p o u n d elastic scattering could n o t be neglected. I n the present w o r k neither o f these c o m p l i c a t i o n s exist, a n d the use o f the p criterion leads to a simple i n t e r p r e t a t i o n o f the quality o f fit. I n table 11, we c o m p a r e the rms percentage d e v i a t i o n p between t h e o r y a n d e x p e r i m e n t o b t a i n e d in our best fits with the rms e x p e r i m e n t a l errors. In m o s t cases p is a b o u t twice the rms error. T h e m o s t n a t u r a l e x p l a n a t i o n for this discrepancy is t h a t the quality o f fit is limited by the accuracy o f the optical m o d e l and its S C A generalization. T h e c o l u m n s h e a d e d E in table 11 c o n t a i n an estimate o f this latter q u a n t i t y o b t a i n e d b y assuming t h a t p2 is equal to the sum o f the squares o f the theoretical a n d e x p e r i m e n t a l errors. It w o u l d seem that the elastic, cross-section p r e d i c t i o n s are accurate to a b o u t 5 ~o a n d the inelastic ones to 10-20 ~ . These figures are necessarily tentative since there m a y be sources o f experimental error o f which we were n o t aware. TABLE 11 Comparison between the quality of fit and the rms experimental errors of the present work Elastic cross section

Inelastic cross section

run

p(~o)

%rms error

E(~o)a)

run

p(~)

~4Zn

table 1 run Bo

5.9

2.8

5.2

table 9 run C

19.0

3.4

18.7

I14Cd

table 6 run B

5.5

3.5

4.2

table 10 run C

10.7

5.9

8.9

Nucleus

%rms error E(~o)a)

a) E is the effective error arising from the optical model and was calculated by assuming that p2 is equal to the sum of E 2 and the square of the rms experimental error. It is difficult to c o m p a r e our findings on this subject with those o f the other a u t h o r s because they use the Z2 or A criterion. I n ref. 6), however, a fit criterion similar to

47

PROTON SCATTERING

our own is used, all the elastic cross-section points being equally weighted with 3 o/ /o errors. The effective p-value m a y be obtained from the Xz value quoted in that paper using the expression

where N is the n u m b e r of experimental points. In table 12, we c o m p a r e the effective p with the rms experimental error for the seven nuclei 56Fe, 58Ni, 6°Ni, 59Co, 635Cu, lZ°Sn and 2°8pb. The picture that emerges is similar to our findings. The somewhat better fit for Cu reflects the fact that there are only 29 data points for this nucleus, while there are ten free parameters in the optical-model potential. I f these data are excluded, the rms error from the optical model is 4.0 ~ . The apparent greater accuracy of the optical model for 208pb m a y be due to a small over-estimate of the experimental error. Even a 20 ~ over-estimate would be equivalent to an opticalm o d e l error of 2.7 ~ . TABLE 12 Comparison between the quality of fit and the rms experimental errors for Satchler 6) Nucleus

56Fe 5SNi 6°Ni 59Co 63.5Cu 12°Sn 2°sPb

Z2

234 349 164 189 28 326 163

N

75 75 75 72 29 75 72

A Effective p°/o

B rms error

5.3 6.5 4.4 4.9 2.9 6.3 4.5

2.7 3.2 3.0 3.2 3.0 4.6 4.5

C v' A2-B 2

4.6 5.7 3.2 3.7 4.3

4.4. OPTICAL-MODEL PARAMETER SYSTEMATICS The present work helps to complete the documentation of the transition from p r e d o m i n a n t surface absorption to predominant volume absorption which occurs as the p r o t o n energy is raised. This question has already been thoroughly studied for lzC by F a n n o n e t al. 1,), where it was found that pure volume absorption gave rather worse fits at 30 MeV than pure surface absorption, while the reserve was true at 40 and 50 MeV. It has been observed by a n u m b e r of authors that when a mixture of surface and volume absorption is used light nuclei usually require a larger p r o p o r t i o n of volume absorption than do heavy nuclei. For this reason it was not expected that the behaviour of lzC would be typical. Prior to the present work, the available evidence on the behaviour of heavier nuclei suggested that volume absorption was becoming important at 40 MeV, but that it did not predominate until the p r o t o n energy was in excess of 80 MeV. Thus at 30 MeV, surface absorption was f o u n d 5) markedly superior to volume absorption. At 40 MeV, equally g o o d fits were obtained 7) with surface and volume absorption. How-

48

v.R.W.

EDWARDS

ever, at 78 MeV for Bi and Ce, predominantly surface absorption was required ~o). The same authors found that these nuclei at 150 MeV were best fitted by pure volume absorption. Our findings conflict with the 78 MeV work, but are consistent with that at 30 and 40 MeV. N o t only are our volume absorption fits very much better (a factor of 4 in p2 for 64Zn and 2 for ~14Cd), but the geometrical parameters obtained with surface absorption show that the program is attempting to mimic the effects of volume absorption. We are thus led to the conclusion that volume absorption l~ecomes predominant not at energies in excess of 80 MeV but somewhere between d0 and 50 MeV. Table 3 contains a summary of the mean geometrical parameters found in opticalmodel fits to proton scattering from medium and heavy nuclei in the energy range 30-78 MeV. The data are divided into three classes according to the form of imaginary potential. For the case of pure surface absorption, it will be seen that our findings fit in well with the 30 and 40 MeV work and establish a smooth trend in the parameters; rwD appears to decrease with energy while both awo and au increase. The 78 MeV results do not fit into this pattern. The 30 MeV data are unfortunately based on a single nucleus. This is not due to any lack of experimental data, but just to the fact that the published analyses employ either a mixed imaginary form or a fixed geometry. With pure volume absorption a trend is descernible between 30 and 50 MeV; au increases fairly rapidly with energy, while rwv decreases tending to become equal to the real radius. The parameter awv increases very much less rapidly than au in contrast to surface absorption where the imaginary diffuseness has the faster rate of rise. The 57 MeV results are consistent with these tendencies, but the different constraints make interpretation difficult. It would seem from the fact that r U and au lie close to the mean 50 MeV values of ru and rwv and au and awv, respectively, that the parameters change little in the range 50-57 MeV. Analyses which use a mixture of volume and surface absorption are unlikely to show systematic trends in the geometrical parameters because volume absorption at these energies leads to imaginary radii which are larger than the real radii, while the converse is true of surface absorption. The interpretation of the results is further complicated by the differences in the constraints. However there again seems to be a tendency for au to increase with energy. Because this work is based on different groups of nuclei, the findings are necessarily tentative. The only work to date for optical-model fits to scattering from a given nucleus at several energies in the range 30-50 MeV is that of Fannon et al. ~ ) on ~zC and that of Craig et al. 12) on 12C and 288i. The only systematic trend observed by the first workers was a slow increase in au with energy for pure surface absorption. For pure volume absorption even this trend was absent. The constraints used by these workers and the lightness of 12C may be to blame. For 28Si, using mixed absorption, Craig et al. observed a significant increase in au = awv and rwo and a decrease in awo. The present work is not completely satisfactory as a guide to optical-model systematics. The discrepancy in the normalization of the experimental data noted in

PROTON SCATTERING

49

ref. 1) meant that this quantity had to be left free in the fitting. The fluctuations observed in this quantity undoubtedly induce errors in the optical-model parameters. A second criticism is that the nuclei used in this work have ground states which are strongly vibrational. Such nuclei can be expected to have larger diffuseness parameters for the ordinary optical model than stationary spherical nuclei, because this model is sensitive to the nuclear matter distribution averaged over time and angle. Since the SCA calculations take the vibrations into account explicitly, the diffuseness parameters used there should not be enhanced. These considerations suggest that it may be bad practice to fix the SCA geometrical parameters at the values obtained with the ordinary optical model. However, the effect may be too small to matter. Certainly the 64Zn diffuseness parameters are not greater than the corresponding 14Cd diffusenesses for all the classes of search made in tables 1 and 2 even though 6 4 Z n has a much larger/3. Elastic polarization measurements have recently been performed for 64Zn. An optical-model analysis of these data 2s) confirms many of the results reported here. The author would like to thank Drs. A. D. Hill and R. C. Barrett for the use of their optical-model search code 29) and Dr. A. D. Hill for the use of his SCA program. We would also like to express our appreciation for the fine service provided by the SRC Atlas Computing Laboratory, Harwell and to thank SRC for providing the research grant which supported this work. References l) V. R. W. Edwards e t al., Nuclear Physics A93 (1967) 370 2) F. G. Perey, Phys. Rev. 131 (1963) 745; L. Rosen e t al., Ann. of Phys. 34 (1965) 96 3) D. W. Devins, H. H. Forster and G. G. Gigas, Nuclear Physics 35 (1962) 617 4) R. C. Barrett, A. D. Hill and P. E. Hodgson, Nuclear Physics 62 (1965) 133 5) G. W. Greenlees and G. J. Pyle, Phys. Rev. 119 (1966) 836 6) G. R. Satchler, Nuclear Physics A92 (1967) 273 7) M. P. Fricke and G. R. Satchler, Phys. Rev. 139 (1965) B567 8) E. N. Blumberg et al., Phys. Rev. 147 (1966) 812 9) M. P. Fricke, E. E. Gross, B. J. Morton and A. Zucker, Phys. Rev., submitted 10) C. Rolland e t al., Nuclear Physics 80 (1966) 625 11) J. A. Fannon, E. J. Burge, D. A. Smith and N. K. Ganguly, Nuclear Physics 9"/ (1967) 263 12) R. M. Craig et al., Nuclear Physics 83 (1966) 493 13) R. K. Cole, C. N. Waddell, R. R. Dittman and H. S. Sandhu, Nuclear Physics 75 (1966) 241 14) J. K. Dickens, D. A. Haner and C. N. Waddell, Phys. Rev. 129 (1963) 743 15) D. M. Chase, L. Wilets and A. R. Edmonds, Phys. Rev. llO (1958) 1080 16) B. W, Ridley, J. F. Turner and J. C. Kerr, Rutherford High Energy Laboratory Report, N I R L / R / 81 (11964) p. 40 17) K. Yagi et al., Phys. Lett. 10 (1964) 186 18) A. Johansson, U. Svanberg and P. E. Hodgson, Ark. Fys. 19 (1961) 541 19) B. Buck, Phys. Rev. 130 (1963) 712 20) D. M. Brink, Prog. Nuclear Physics 8 (1960) 99 21) M. Sakai and T. Tamura, Phys. Lett. 10 (1964) 323; G. C. Pramila, R. Middleton and T. Tamura, Nuclear Physics 61 (1961) 448

50 22) 23) 24) 25) 26)

V. R. W. EDWARDS

F. Perey and G. R. Satchler, Phys. Lett. 5 (1963) 212 A. P. Stamp and J. R. Rook, Nuclear Physics 53 (1964) 657 F. Perey, R. J. Silva and G. R. Satchler, Phys. Lett. 4 (1963) 25 R. D. Lawson and J. Uretsky, Phys. Rev. 108 (1957) 1300 A. D. Hill, private communication of unpublished analysis of 57 MeV proton scattering data of ref. 37) 27) I. Nonaka et al., J. Phys. Soc. Japan 17 (1962) 1817 28) V. E. Lewis, Nuclear Physics A101, No. 2 (1967) 29) A. D. Hill, R. C. Barrett and A. E. Forest, Atlas Computer Laboratory, Harwell, Nuclear Physics Program Library Report 1