Volume 196, number 1 PHYSICS LETTERS B 24 September 1987

AMBIGUITY ON THE IMAGINARY POTENTIALS IN THE DIRAC FORMALISM FOR THE ELASTIC AND THE INELASTIC SCATTERING OF NUCLEONS

Jacques RAYNAL Service de Physique Thkorique, CEN-Saclay, F-91191 Gif-sur-Yvette Cedex, France

Received 10 February 1987; revised manuscript received 29 June 1987

The Dirac formalism for the elastic scattering of nucleons has been extended to a coupled channel formalism taking into account

collective excited states. Calculations for the scattering of 500 MeV protons on 40Ca shows large second-order effects on the elastic

scattering. The imaginary scalar potential and the imaginary vector potential can be varied widely without giving so large effects

on the elastic scattering. This large ambiguity is not limited to this case.

The success of the Dirac equation for describing the elastic scattering of nucleons on nuclei at medium energy [ l] incites to extend it to inelastic scattering. The Dirac equation used to describe scattering of nucleons by nuclei is

( fl TaV +B[m+ K(r)1 + V,(r) +$&WV VT(r)1 > v(r) =Ev(r) , (1)

where V,(r), V,(r) and V,(r) are three complex potentials of which the real and the imaginary part are approx- imated by a Woods-Saxon potential but can be replaced by any other form-factor given by a theory. In prin- ciple, this equation is valid only for a target with an infinite mass and m is the rest mass of the nucleon. However,

as the mass of the target is finite, it seems better to use the reduced mass which allows a correct limit at low energy. Anyway, in the computations presented below, fits using the rest mass or reduced mass are about the same, but the parameters of the potentials are different. The vector potential V”(r) and the tensor potential VT(r) include the Coulomb potential; in fact, the tensor potential is reduced here to the Coulomb potential multiplied (at the low energy limit) by the anomalous magnetic moment (p- $z) where z is the charge of the nucleon and p its magnetic moment.

For a strongly deformed target or a target with strong collective states, there is no reason to treat this equation in a different manner than that used since a long time for the Schrodinger equation [2]: the potentials Vs(r), V,(r) and VT(r) are deformed or include creation and annihilation operators of bosons for a spherical nucleus.

The Dirac equation, after separation of large and small components [F(r), - iG( r) ] can be replaced by a Schrodinger equation by elimination of small components. This equation is

[A-I’,(r)-iibVVz(r),V--k2]f(r)=0,

where

’ A%r> 1 -m+~j; [2mV,(r) +2EV,(r) + l?(r) - C(r)1

9(r)=D(r) exp[fiiT/,(r)lm], D(r)=E+m-V,(r)+T/,(r),

but

0370-2693/87/S 03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

(2)

V2(r)=Ln g(r) (3,4)

(596)

7

Volume 196, number I PHYSICS LETTERS B 24 September 1987

f(r)=D(r)-“2F(r) . (7)

The easiest thing to do is to neglect eq. (7), to compute the central potential I’, (r) and the spin-orbit poten- tial VZ( r) and to proceed as for the Schrodinger equation. In some cases, the result is good [ 31, but it can differ considerably from the result of a more complete calculation for which [4] the total wave function has to be written as

where F(r) and G(r) are the radial wave functions of the large and small components and vT describes the target. Using this expression in eq. (1)) we obtain the set of linear equations

G,+(E,-m-r+~)F,-&~ G,=&(r) ,

-fi F,+(E,+m+b+e)G,-&-F&(r),

with

S,(r)=C 3; AJ

(v:+v/)F,+& 7 V+ G, , > 1

~,(r)=x s$ AJ

(Vi-V$)G,+$ K-K

$V+ + $4’:

(9)

(10)

where e, e, v”, are the monopole parts of the potentials V,(r), V,(r), V,(r) and V:, V”,, V!: are their mul- tipole parts of angular momentum a, 3; is the geometrical coefficient which appears in the Schriidinger equa- tion for spin-i particles [ 21, K, is the eigenvalue of 1-a + 1 and E, the total energy of the channel. These equations can be solved by iteration using the Green function built with the regular solution F{(t), G;(r) of the left sides of eq. (9) and their irregular solution F I( r), G)(r), purely outgoing:

+ F:(r) O3 ( >r G;(r) ~ [F](r’)S,(r’)+G:(r’)7’,(r’)] dr’ , (11)

where the term B,O is present only for the incoming channel. Uncoupled regular and irregular solutions are obtained easily from the Schriidinger equation by formulae (2)-(7). The difficulties due to the Coulomb potential which appears as a long-range term z2/r2 in V,(r) and as [ l/(E+m) +Nlm]z/r3 in the spin-orbit potential are overcome using non-relativistic Coulomb functions corrected as for heavy ion scattering [ 51.

Inelastic scattering of protons around 500 MeV on 40Ca has been studied with the Schrodinger equation [ 61

and the Dirac formalism [ 71. Avilable data [ 81 include the elastic cross section up to 30”) polarization UP to 35’ and Q [ 91 up to 21”) the cross section and analysing power on the first 3- state up to 30”; there are also cross sections and analysing powers on a second 33, on two 2+ and a S- state.

The optical parameters of the Dirac formalism are not very well defined: very different values give equivalent fits. This is particularly the case of the imaginary part of the scalar and the vector potential. The inclusion of the observable Q in the x2 lead to a very different depth of the imaginary potentials [ 71. To study this ambiguity

8

Tab

le 1

Aut

omat

ic s

earc

hes

on 9

par

amet

ers

wit

h fi

xed

dept

h o

f the

im

agin

ary

vect

or p

oten

tial

Wv

give

n on

the

fir

st li

ne,

on t

he e

last

ic c

ross

sec

tion

(15

4 va

lues

) an

d th

e po

lari

zati

on (

134

valu

es)

give

s th

e p

aram

eter

s li

sted

in

the

9 fo

llow

ing

line

s. P

olar

izat

ion

data

are

wei

ghte

d by

a r

atio

5 i

n th

e se

arch

. E

xper

imen

tal

valu

es o

f Q

(3

2 va

lues

) ar

e no

t ta

ken

into

acc

ount

. A

uto

mat

ic s

earc

h o

n 4

def

orm

atio

ns

wit

h fi

xed

opti

cal

para

met

ers,

tak

ing

into

acc

ou

nt

only

the

cro

ss s

ecti

on o

f th

e 3

( 68

valu

es)

and

its

anal

ysin

g p

ow

er (

67

valu

es)

wit

h a

wei

ght

5, g

ives

the

def

orm

atio

ns

and

the

y 2

list

ed b

elow

.

Par

amet

er

IV,

0.00

-2

0.0

0

-40

.00

-6

0.0

0

-80

.00

-

100.

00

- 12

0.00

-

140.

00

- 16

0.00

-

180.

00

IV,

-74

.72

-4

3.4

9

10.9

9 23

.30

60.2

3 96

.00

125.

51

155.

70

185.

89

223.

60

R~

1.03

55

1.03

26

1.03

37

1.04

07

1.05

70

1.06

96

1.06

00

1.05

31

1.04

57

1.06

39

A~

0.57

8 0.

577

0.57

5 0.

572

0.57

0 0.

567

0.56

1 0.

559

0.57

0 0.

562

V,

- 18

6.67

-2

04

.58

-2

22

.01

-2

41

.66

-2

69

.41

-2

94

.14

-2

88

.95

-2

76

.22

-2

57

.52

-2

12

.40

R

s 1.

1389

1.

1142

1.

0933

1.

0736

1.

0520

1.

0382

1.

0381

1.

0408

1.

0526

1.

0388

A

s 0.

668

0.66

6 0.

664

0.66

2 0.

661

0.66

6 0.

690

0.72

9 0.

743

0.71

2

Vv

145.

01

152.

05

158.

97

167.

61

181.

44

193.

33

186.

70

174.

12

158.

69

127.

66

Rv

1.08

87

1.07

90

1.07

00

1.06

01

1.04

79

1.03

98

1.03

90

1.04

27

1.05

77

1.02

98

Av

0.64

8 0.

639

0.63

4 0.

630

0.62

7 0.

635

0.66

7 0.

718

0.74

1 0.

693

a~, o

p~

1142

10

64

1070

10

92

1064

98

4 91

4 10

37

1429

25

10

P~ o

~ 65

4 62

4 59

8 56

2 50

1 43

2 45

7 57

8 88

7 88

8 Q

,~ o

pt

2339

15

57

961

499

209

231

309

374

433

1127

]~

0.36

53

0.37

29

0.38

21

0.39

08

0.40

56

0.41

11

0.45

57

0.46

21

0.40

93

0.66

28

fls~

0.33

11

0.27

73

0.35

96

0.14

53

0.28

53

0.18

97

0.20

25

0.40

12

0.46

61

0.06

07

fl~

0.

4071

0.

4074

0.

4151

0.

4231

0.

4295

0.

4406

0.

4957

0.

4953

0.

4326

0.

8069

fl

~ 0.

4360

0.

3357

0.

2854

0.

3142

0.

2391

0.

2380

0.

3866

0.

4425

0.

1012

a~t ~

o~

3895

34

70

3002

25

75

2240

18

58

1775

21

29

2521

23

00

Pc, ~

o~

2891

27

29

2301

18

36

1777

12

59

1334

24

21

2880

20

56

Q~t

~o~

2439

16

25

1003

51

9 22

0 22

3 31

6 37

8 41

5 10

81

O'in

el 28

03

2616

25

93 "

" 23

78

1812

13

68

1313

13

29

1297

21

00

anal

ysin

g p

ow

er

279

267

269

284

364

368

398

354

263

674

,.<

c3

t-

,q

,O

o¢

Volume 196, number 1 PHYSICS LETTERS B 24 September 1987

10 ~ t_ u1

E 10 a q0Ca ( P , p , ) ~Oca

b ~o ~ ~\ L~9~ MeV

l o !.

l o o

lO -1

I I l O - e I \ \ 10• 20. 30. qO.

0cm

Fig. 1. Extrema on the elastic cross section obtained with the imaginary vector potential Wv ranging from 0 to - 160 MeV. Experimental errors are multiplied by 10.

1 . 0

0 . 5

0 . 0

- 0 . 5

- 1 . 0

p - '~°C~

Li9 ~] MeV P o l a r i s a ~ _ i o n i j .L

10. 20. 30. qO.

8o~

Fig. 2. Extrema on the elastic polarization obtained with the imaginary vector potential Wv ranging from 0 to - 160 MeV. Real experi- mental errors are shown.

10

Volume 196, number 1

1 ,0

0 , 5

0 . 0

- 0 , 5

PHYSICS LETTERS B

l

.#,r/

p - ~°Ca

El:92 Me V

I #..x "{ IJ ';~ ,,z . S j I ,

,if iv/ i\" ' ': t /

;,V /

_ _ k l , , = - 2 0 bleV

__k lv= - 8 0 MeV

@ ...... Hv=- l_qO kleV

- 1 . 0 I I I 10. 20. 30. q0.

o,

Fig. 3. Some results for the observable Q with the parameters of table 1.

24 September 1987

and to appreciate the importance of the observable Q, we choose the same geometry for the two imaginary potentials, we introduce a weight 5 for the polarization and we neglect Q in the X 2. These searches were done with the proton reduced mass, with and without an anomalous magnetic moment fixed to the value 2, and a charge distribution with and without diffuseness, the geometrical parameters of the charge being the ones of the real vector potential. These four series of searches give equivalent results. The one with anomalous magnetic moment and no charge diffuseness is presented in table l, for fixed depths of the imaginary vector potential Wv ranging from 0 MeV to - 180 MeV by steps of 20 MeV. Maximum and minimum values of the cross section and the polarization for Wv ranging from 0 MeV to - 160 MeV are shown in figs. 1 and 2. The imaginary central potential V,(r) of the equivalent Schr6dinger equation stays almost constant whereas the imaginary spin-orbit potential changes its sign, this effect being compensated by variation of radii and diffusenesses. Some results for Q are given in fig. 3; they are not good because Q was not included in the search. The best agreement is for Wv from - 8 0 MeV to - 120 MeV. But, even Q does not determine Wv: a search with the cross section and Q with weight 10 and without polarization gives for Q a z 2 of 5 t at Wv = - 100 MeV but only 82 at W= - 40 MeV and 75 at Wv= - 1 4 0 MeV; the fit of the polarization is very bad. Consequently, Q is more restrictive than the polarization for the imaginary potentials, but the ambiguity is still very large.

In a coupled channel calculation including the 3- state with a deformation of the order of 0.4, the fit of the inelastic scattering is quite good but the Z 2 of the elastic scattering increases strongly. In fact, a search on 9 parameters on this coupled channel case, starting from one set of parameters listed in table 1 and including only the elastic scattering data in the X 2 leads to a Z 2 almost 20% lower than the one obtained without the 3- state. In order to see how much the inelastic scattering result depends on the optical model, searches were made for each potential of table 1 on different deformations for the real and the imaginary part of the scalar and the vector potential; the Z 2 includes only the inelastic cross section and the analysing power with a weight 5. The deformations and the partial Z 2 are given in table 1. The maximum and minimum of the cross section

11

Volume 196. n u m b e r I PHYSICS LETTERS B 24 Sep tember 1987

c_ tn

.13 E

10 8

'~°Ca ( p , p , ) '*°Ca / ~ . Lt9'7 Me V

3- a L 3 "736 Me V 10 1

10 o

10-2

10-2 I I I 10. 20. 30. ~0.

Oc.

Fig. 4. Ex t rema on the inelas t ic cross sect ion. Exper imen ta l errors are mu l t ip l i ed by 10.

1.0

0.5

0.0

-o.s ~OCa (p,p') ~OCa L~9~ Me V

3- aL 3. "]36 MeV Rnalysin 9 Power

- 1 . O [ t 10. 20. 30.

Oee

Fig. 5. Ex t rema on the inelast ic analys ing power. Real expe r imen ta l errors are shown.

riO.

[2

Volume 196, number 1 PHYSICS LETTERS B 24 September 1987

and the analysing power for Wv ranging from 0 MeV to - 160 MeV are shown in figs. 4 and 5. There is no just if icat ion to use a different deformation for each potential; it was done only to see how good

a fit can be obtained. Anyway, if the geometrical parameters are the same for all the potentials, there is an arbitrariness in the use of 4 deformations because the scalar and the vector transi t ion form-factors can be mul- tiplied by an arbitrary phase with no effect on the result. For the other levels, if reorientat ion terms or coupling between different states can affect the inelastic scattering as much as the inclusion of the 3 - state affected the elastic scattering, they seem insufficient to explain the difference between the two 3 - states or the two 2 + states. Forward experiments on the second 3 - state are quite well reproduced with a mixture of first and second deriv- ative in the t ransi t ion form-factor where the second derivative is the most important; but the backward data are still out of phase. The two 2 + are reproduced with a small admixture of second derivative. Furthermore, if the states are not very collective, there must be a term AS= 1 in the interaction. The form-factor can be the first or the second derivative of the potential. For such a term, the geometrical coefficients in the two equations (10) are opposite. Its effects should be large at high energy: in fact it damps the analysing power. Analysis with four deformations: AS= 0 and AS= 1, first and second derivative gave good fits; but it is possible that the sec- ond derivative cancels some inadequacy of the optical potential. The AS= 1 terms have to be justified by a

microscopic analysis. Similar results have been obtained also on 48Ca at 500 MeV and on 4°Ca at 800 MeV. If the requirement

on the Z 2 for the elastic scattering (with only this channel) is released up to the value obtained when including the 3 state with the best optical model parameters obtained previously for the elastic scattering only, the ambi- guity on these parameters becomes very large. For some of these sets of parameters, the second-order effects of the inclusion of the 3 - state can act to obtain a better Z 2 than with the elastic scattering alone. Inelastic scattering does not seem to be able to resolve this ambiguity. The measurement of Q is more restrictive but does not give a precise value of the parameters.

Referenes

[ 1 ] L.G. Arnold, B.C. Clark, R.L. Mercer and P. Schwandt, Phys. Rev. C 23 (I 981 ) 1949. [2] Y. Tamura, Rev. Mod. Phys. 37 (1965) 679. [3] J. Raynal and H.S. Sherif, Proc. Sixth Intern. Symp. on polarisation phenomena in nuclear physics (Osaka, 1985), J. Phys. Soc.

Japan Suppl. 55 (1986) 922. [4] J. Raynal, Use of the optical model for the calculation of neutron cross sections below 20 MeV (Paris, 1985), NEANDC 222 u. [5] J. Raynal, Phys. Rev. C 23 (1981) 2571. [6] K.K. Seth et al., Phys. Lett. B 158 (1985) 23. [ 7 ] B.C. Clark, R.L. Mercer and P. Schwandt, Phys. Lett. B 122 (1983) 211. [8] K.K. Seth and C. Glashausser, private communication. [ 9 ] R.J. Glauber and P. Osland, Phys. Lett. 47 ( 1981 ) 1811.

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