Explicit treatment of internucleon P- and D-states in the calculation of the nucleon optical potential

Nuclear Physics A426 (1984) 92-108 © North-Holland Publishing Company

E X P L I C I T T R E A T M E N T O F I N T E R N U C L E O N P- A N D D - S T A T E S IN THE CALCULATION OF THE NUCLEON OPTICAL POTENTIAL A.M. KOBOS*, W. HAIDER** and J.R. ROOK

Nuclear Physics Laboratory, Oxford, UK Received 28 October 1983 (Revised 5 March 1984) Abstract: Usual calculations of the nucleon optical potential using the Brueckner t-matrix treat the internucleon P- and D-states in a nuclear matter approximation. We use the method of Kisslinger to obtain the potential when P- and D-states are treated more precisely. We show phenomenologically that the effect of an explicittreatment of P-states is small, consistent with calculations indicating that the total P-state contribution to the t-matrix is small. This result provides additional verification that the Brueckner method is reasortably satisfactory. The potential obtained from an explicit treatment of the D-states is such that, although large, it can be reduced approximately to the form usually used. We show that the error in this approximation is not large though non-negligible when comparing calculated cross sections with experimental data. We also show that the P- and D-states corrections do not substantially aitect phenomenological fits to experimental data.

1. Introduction

T h e use o f the f o l d i n g m o d e l 1,2) with a t - m a t r i x c a l c u l a t e d using B r u e c k n e r t h e o r y s-6) to give a n u c l e o n - n u c l e u s o p t i c a l p o t e n t i a l has p r o v e d very successful in a c c o u n t i n g for n u c l e o n scattering cross sections. In this m e t h o d the B r u e c k n e r t - m a t r i x is first c a l c u l a t e d in infinite n u c l e a r m a t t e r for a r a n g e o f m o m e n t a a n d densities, t h e n the d e p e n d e n c e o f t on the s e p a r a t i o n o f the n u c l e o n p a i r is o b t a i n e d b y u s i n g slightly v a r y i n g m e t h o d s 2,7,s). F i n a l l y the o p t i c a l p o t e n t i a l for a p a r t i c u l a r finite n u c l e u s is o b t a i n e d b y f o l d i n g the t - m a t r i x with the n u c l e a r density. S u c h a f o l d i n g o n l y p a r t i a l l y allows t h a t the n u c l e a r d e n s i t y is n o t c o n s t a n t in a finite nucleus, as a s s u m e d in t h e n u c l e a r m a t t e r a p p r o a c h ; effects o f the finite size o f the n u c l e u s have not b e e n t a k e n into a c c o u n t in these calculations. T h e success o f the a b o v e a p p r o a c h suggests that o t h e r finite-size effects are not large b u t this can o n l y b e e s t a b l i s h e d b y e x p l i c i t c a l c u l a t i o n . T h e p u r p o s e o f this p a p e r is to e x a m i n e o n e such finite-range effect, i.e. that a s s o c i a t e d with the fact t h a t the i n t e r n u c l e o n i n t e r a c t i o n is n o t restricted o n l y to relative S-states as is i m p l i c i t l y a s s u m e d in the a b o v e m e t h o d ~-6). T h e p i o n - n u c l e o n i n t e r a c t i o n in a n a p p r o p r i a t e energy range is well k n o w n to b e d o m i n a t e d b y the i n t e r a c t i o n in relative P-states 9). This l e a d s to the w e l l - k n o w n * On leave from Institute of Nuclear Physics, Cracow, Poland. ** Now at the Aligarh Muslim University, Aligarh (UP), India. 92

A.M. Kobos et al. / Explicit treatment

93

Kisslinger 10) potential for the pion-nucleus optical potential. This potential involves terms depending on the momentum of the incident pion and on derivatives of the density of nucleons in the nucleus. In this paper we examine the application of this method to nucleon scattering, In sect. 2 we obain the Kisslinger potential by a method which shows its close connection with the usual 2,4.5) standard methods used in nucleon scattering. We are also able to extend the method to nucleon-nucleon scattering in D-states. In sect. 2 also we show how the P- and D-state terms in the optical potential may be reduced to potentials of the usual form. Sect. 3 is devoted to analyses of the results obtained from the use of potentials obtained by considering explicitly the relative P- and D-states. We first discuss the size of the contribution of the various partial waves to the calculated internucleon t-matrix. The largest contribution comes from the S-states, the total effect of the P-states is very small but the D-states contribute substantially. The smallness of the P-state interaction leads to a correspondingly small effect on nucleon-nucleus scattering and the confirmation of this we obtain by comparison with experimental data is a useful check on the validity of the whole Brueckner approach. The size of the D-state term in the t-matrix would at first sight suggest a large effect on nucleon-nucleus scattering. However, it turns out that the structure of the optical-model equation is such that the overall effect of the large D-state term on nucleon scattering cross sections is small but not quite negligibly so. 2. Formalism 2.1. GENERAL FORMULATION

We let U be the optical potential, 4~ be the corresponding wave-function and p be the density of nucleons in the nucleus. Then following Watson ~l) we have

U(r)4~(r) -=~-~

e""r(q, q') e-~'"~(r ') d~',

(1)

where d~" denotes an integration over all the variables except r and 1 T(q, q') = (2~)3/2 I e-i(q-q')'"t(q' q ' ) o ( r ' ) d r '

(2)

where t is the internucleon t-matrix in momentum space. We assume that t can be written t(q, q') = s(q - q') + p ( q - q')q. q' + d ( q - q')½[3(q • q') - q2q,2].

(3)

The separation in eq. (3) corresponds to the parts of t arising respectively from S-, P- and D-states. Eqs. (2) and (3) are inserted into eq. (I) to give U. Corresponding to s, p and d in eq. (3) we have Us, Up and Ua so that

u= Us+Up+U,,.

(4)

94

A.M, Kobos et al. / Explicit treatment

The derivation of the explicit formulae follows Kisslinger ~o). Consider first the term s. We obtain

uT~4,(r)= (2"/'r) 1 7/2 .I[ e/q'"

e - i ' " r ' ~ b ( r ') e-i(q-~')'Zs(q -

q')p(z) d r ,

(5)

which is easily seen to be

U~(r)4~(r) = q~(r) V~(r),

(6a)

V,(r) = f s(r - z)p(z) dz, J

(6b)

where

and

s(x) =

'I

(2~r) 3/z

s( Q) e iO"~ dO

(7)

is the S-wave part of t in coordinate space. We use the same symbols for quantities defined in momentum and coordinate space. Corresponding to p in eq. (3) we obtain

Up(r)~(r) =

1 (2~)7/2 f ei¢ •, e-i¢,.,,~b(r,) e-iCq-¢,). Zp(q _ q,)q. q'p(z) d~"

'I

- (2~.)7/z

(8)

V,(e iq'')" V,,(eTiq"")rb(r ') e-i(g-q')'~p(q- q')p(z) dr, (9)

which after integration by parts becomes

~r f e '~" " r e-iq', r' e-i(q-¢,) (2~r)7/2

"~p(q -- q')p(Z)" V,,(b(r') dr.

(10)

Comparing (5) and (10), similarly to eq. (6) for the S-state terms, Up(r)~(r) = - V . [ Vp(r)V ~b(r)],

(11)

Vp(r) = ~ p(r - z)p(z) dz,

(12)

where

and p in coordinate space is defined analogously to s in eq. (7). This is essentially the result of Kisslinger Jo). The term d in eq. (3) is handled similarly and finally we obtain

Ua(r)qb(r) = V2[ Va(r)V2 ~b(r)]

(13)

f Va(r) = J d(r - z)p(z) dz,

(14)

with

where d is also defined similarly to s in eq. (7).

A.M. Kobos et al. / Explicit treatment

95

2.2. SITUATION IN NUCLEAR MATI'ER In nuclear matter the potentials V~, Vp and Vd are independent of position while ~b is a plane wave with m o m e n t u m say k. Thus from eqs. (6), (1 I) and (13) we obtain Un.m. = Vs + k2Vp +k4Vd = p f [s(x) + kEp(x) + k4d(x)] dx.

(16a) (16b)

This result is apparent also from eq. (3). In nuclear matter we must have q = q' and also q ~ k, leading directly to eqs. (16) using the standard folding model 1.2). In finite nuclei the particle states are not eigenstates of m o m e n t u m so that q and q' are not necessarily equal and the effect of this is to introduce terms involving the derivatives of p, as in subsect. 2.1. Our derivation above has contained an approximation in that s, p and d are not just functions of q - q' but depend also on the position in the nucleus. A dependence on z can be included without alteration but more usually 2) the dependence is on R = ½(r + z). Such a dependence cannot be included easily in the above formalism so we shall consider formally just a dependence on z. However, in the subsequent calculations we shall assume that s, p and d depend on x and R. With this approximation we see that the usual method 4) of calculating U is to use the quantity in square brackets in eq. (16b) as the t-matrix and to fold this t-matrix with the density p. Note that in this method k depends on r. This neglects the terms involving the derivatives of the potentials in eqs. (11) and (13) and it is just the effect of these derivative terms which is of interest in this paper.

2.3. FORMAL DISCUSSION OF EQS. (ll) AND (13) Since the effect of effects separately and of eq. (16), but taking only the P-state term,

P- and D-states turns out to be fairly small we discuss the include all the other partial waves as in the approximation k and q to be the same, as already discussed. Thus, isolating eq. (l l) gives for the function q~ the equation h2 -2--~ V2~b + Vs~b- V • (VpV~b)= E~,

(17)

where Vs now includes all partial waves except P-states by the approximation of eq. (16). This equation could be solved directly but it is also easily reduced, without approximation, to the standard form o f the optical model involving however an auxiliary wave function u, eq. (19), rather than the true wave function 4~. In place o f eq. (17) we use - ( h 2 / 2 m + V p ) ~ 2 (~ - V Vp . V ~) -~- V s = E ~ .

(18)

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A.M. Kobos et al. / Explicit treatment

We put

~b = Xu,

(19)

X = (1 +2mVp/h2) -~/2

(20)

where is chosen to eliminate terms in Vu. Note that as r ~ o o , X ~ 1 and ~b~ u. Thus the scattering matrix elements obtained from u are the same as those obtained form ~b. From eqs. (18), (19) and (20) the equation for u is -

h2 [ h2V2X m , ] 2m V2u + u 2m X ~--~X ( V v p ) E - x 2 V s - x 2 E + E =Eu

(21)

which is of the usual form of the optical-model equation with the quantity in square brackets as the effective optical potenial, say Ve~. From eq. (21) we see that P-state interactions lead directly to a potential Ven of the type usually assumed in phenomenological analyses. Thus phenomenological fitting of experimental data cannot determine any special effects of P-state interactions without further physical assumptions concerning the form of the potentials. Eq. (18) may also be reduced approximately to the usual form 4), involving now the true wave-function ~b, by neglecting the term involving V Vp. Put V2~b = - k 2 q ~ ,

(22)

where k is a function of r. Then eq. (18) becomes, neglecting V Vp, hE

- ~m VZ Ch+ ( V, + k 2 Vp)dp = Eqb ,

(23)

and hence 2m

k2=-~

( E - Vs - k 2 Vp).

(24)

The potential V~ + k2Vp is that usually 4) used, the self-consistency implied by eq. (24) is essentially included in the t-matrix calculation 4). Thus the term involving Vp appearing in eq. (23) is already included in the usual 4) calculations. The size of the term in eq. (18) which is neglected in obtaining eq. (24), and hence in the usual calculations, can be estimated by taking Iv61 = k ~ .

(25)

From eq. (18) the neglected term is V Vp. V~b while the term usually included is k 2 Vp~b from eq. (23). The ratio of these terms is vv~. v6

1

- 2ka'

(26)

where a is a diffuseness parameter and to obtain the particular form in (26) we have used a Woods-Saxon form for Vp. Taking a = 0.5 fm and k the value corresponding

97

A.M. Kobos et al. / Explicit treatment

to the nuclear surface at a bombarding energy of 30 MeV we find that the ratio of the terms in (26) is about 0.5. This implies that unless Vp itself is negligible the approximation leading to eq. (23) is inadequate and the more correct form, eq. (21) should be used. 2.4. D-STATE Treating now the D state by a method similar to that in subsect. 2.3 for the P-state, the optical-model equation becomes, from eq. (13), h2

---V24, + V~b +~72(Va~724,)=E4,, 2m

(27)

where V~ now includes approximately all the partial waves except D-states. We are not able to reduce this equation to the usual form of optical-model equation exactly but we may proceed approximately. As in subsect. 2.3 we define k ( r ) by V24, = -k24,.

(28)

u = ( 1 + ~2 m - k2Va)dp

(29)

Then put

giving, from eq. (27), ---2m V2u + u

1 +(2m/h:)k2Va + E

= Eu,

(30)

and again this is an equation of the standard form with the quantity in square brackets as an effective potential Vefr. As before u and 4, have the same scattering matrix elements. Unlike eq. (21) we can only solve eq. (30) if k is known but for this we need to know the solution. This suggests we try an iterative solution. From eq. (27) an approximate solution is k 2 = 2 m / h Z ( E - Vs - k 4 Va) = 2m/h:(E - U),

(31) (32)

when 0 is the potential usually 4) used. Inserting this in eq. (30) gives Ve, = O. (33) Thus in this approximation the usually used 0 appears as the first iteration to the potential for u, where u and 4, are related by eq. (29). This already indicates the reason for the success of approaches 1-8) neglecting explicit D-state effects when, as we shall see, the D-state contribution is not very small. It is natural now to continue the iteration process but because of the occurrence of derivatives higher than the second in eq. (27) we have done this in a rough approximation. This approximation is dictated entirely by the requirement that we

98

A.M. Kobos et al. / Explicit treatment

can use the standard optical-model programme ~2) to perform the calculations. We do not claim that the neglected terms are small. From eqs. (28), (29) and (30) we obtain 2m V2u = - - ~ - (E - Vs)~b - - [ 1 +-~T2mk2Vd]kE~+4~V2[l

(34)

+--~2mk2Vd],

(35)

where terms involving gradients have been dropped. Thus this approximation (35) is only going to give us an indication of the likely validity of the steps leading to eq. (33). A second estimate of k is obtained from eqs. (34) and (35) by replacing k inside the square brackets by the first iteration, eq. (32), and then the new value of k used in eq. (30) to give a second estimate of u and hence of the scattering matrix elements. Arguments similar to those leading to eq. (23) for P-states may be followed for D-states also. The approximate potential included in the usual calculations is Vs + k 4 Va. A similar development to that at the end of subsect. 2.3 indicates that for D-states also the neglected terms are not small. It is in fact the argument leading to eq. (33) which ensures that the usual method is satisfactory albeit in terms of the function u and not ~b. 3. Results

We have used a standard method 2,4) for calculating the t-matrix, the internucleon potential is that of Hamada and Johnston 13). We call this type of calculation "on-shell". We first consider the relative size of the total contribution to the optical potential from various values of the relative internucleon orbital angular momentum L using the "on-shell" procedure. To illustrate the results we consider the centre of a nucleus, density p = 0.185 fm -3, Fermi momentum kF = 1.4 fm -1. The momentum of the incident nucleon is k and figs. 1 and 2 show the k-dependence of the potential for various L. Consider first the lower values of k, say k < 2.5 fm -~. It is immediately apparent that the S-state is largest, the P-state is quite small, especially for the real part and that the D-state is large, being for example about 50% of the S-state contribution for k = 2.0 fm -1. As is also apparent from the results of ref. 4), we find that the small contribution from P-states is a consequence of cancellation between the individual partial waves making up the total P-state contribution. As an example of the resulting values of Vs, Vp and Vd, eqs. (6), (12) and (14), we show in figs. 3 and 4 the potentials obtained for the scattering of 30.3 MeV protons from 4°Ca. In fig. 3 the smallness of Re Vp and the comparatively large size of Re Vd are noticeable, as discussed above, and we also note that the surface structure of the real part of the total optical potential U arises almost entirely from the S-state contribution. In

99

A.M. Kobos et al. / Explicit treatment l

I

l

I

I

I

I

I

l

20

....

~°~°~'~°~'~'~.~.

L

--"

"

~

°°'°°°°°°,

H

f °°°'°o°°°

J °°°°°oo°

........... ° ° ° ° ° ° ° ° ' ~ o ° ° °/o ° o ° ° ° ° ° °

-2C 0

/

/

/

/ ~

-4.0

/

b g

--...... ...... -.....

/

/ / /

"60

/ /

-80

/

/

/ I

0

/

/

/

/

I

1

° ........

°°

D

k F = 1.4 fm -I p =0.185 fm -3

/

/

Z

° ° ° ° ° ° ° ° ° ° ° o° ° ° . . . . .

/

S - state P-state D - state F - state G - state

(L = 0) ( L = 1) (L = 2l (L = 3) (L = t~)

- . - H - s t a t e (L = 5}

I

I

I

2

I

3 k

I

I

t

4

(fro -1)

Fig. 1. Dependence of the real part of the optical potential on k for individual relative internucleon orbital states. fig. 4 we note how the surface structure of the imaginary part of the total potential arises from S-states. More detailed examination shows that in both cases the structure comes from the 3S, contribution. We m a y also note in fig. 4 that the P-state contribution to the imaginary part is relatively not so small as for the real part. We now turn to the discussion of the theoretically calculated cross sections and polarisations and the comparison with experimental data. We consider mainly the scattering o f 30.3 MeV protons from 4°Ca for which extensive experimental data ~4.1s) are available. The radial dependence of the various components of the t-matrix and the resulting optical potential are calculated as in ref. 2). The spin-orbit term is included as in ref. ~6) and then kept throughout the formalism of sect. 2. Exchange terms are added as in ref. 2). We study separately the special effects of the P- and D-state contributions, discussing first the P-state. We have considered this in two

100

A•M. Kobos et al. I

/ Explicit treatment

I

i

i

~--~---

~'~.'.: .....

,, --,.,.........

:E:

\..--\
.-g p

¢3 ._E

g -2o

G

S

•••..• •...

x

..... • ,,

= 0,185 fm -1

--- S - state -.-- P-state ...... D - s t a t e F - sta}e .... G - state ..... H - state

" .....

....

..........

•\

k F = 1./, fm -1 -10

i

\

(L = O) (L= 1) (L = 2) (L = 3) (L =/,) (L = 51

"\ "\. \. x.\. "x. %.% "%

g

-p

Z

'

-3C

~

'

z'

' k

~

'

~'

'

(fro -1)

Fig. 2. Imaginary parts correspondingto fig. 1• ways, firstly in subsect. 3.1 with phenomenological potentials, and secondly in subsect. 3.2 using the calculated potentials Throughout the following discussion the symbol Os will be used to mean the total potential left after the P- or D-state part is calculated explicitly. It is obtained in practice in the standard way 2) ignoring the P- or D-state potentials according to which is considered explicitly• ,

I

i

lC '. . . .

c

I

4 ° C o ( p , p ) 30.3 HeV P

.,-"'"

.....................

/~

-.~.?:.-::.-.:r..=.'•-=-

D •,,o""°"

¢'Y

/z

~/J f/11 -t<

/

.... ..... ......

S- sfat~ P-st~e D-store l

'

½

'

'

,

6

r (fro)

Fig• 3• The real parts of the potentials Vs, Vp and

Va

and U for the scattering of 30.3 MeV protons from

4°Ca"

I01

A.M. Kobos et al. / Explicit treatment i

i

i

~

i

1

~°Ca(p,p] 30.3MeV ( ............

:.s

...............

........

/

-0

,/ -

~

-.... ..... ....

/ / ~

•

I

I

2

I

I

L, r (frn)

I

Total S- state P- state 13-state

I

6

I

8

Fig. 4. Imaginary parts corresponding to fig. 3.

3.1. PHENOMENOLOGICAL TREATMENT OF P-STATE Previous work ,7) has established that a reasonable fit to the experimental cross sections ,4) and polarisations 15) can be obtained in terms of the usual type of phenomenological optical potential. We write this potential as UBG. The argument leading to eq. (21) shows that we can only perform phenomenological calculations which distinguish between S- and P-state contributions if we assume definite radial shapes for the potentials Up, of eq. (4), and Os. Recollect that as is now to be understood as including the optical potential resulting from all partial waves other than P-states. For the purely phenomenological discussion of this subsection we make simple assumptions for the radial shapes of O~ and Ve consistent with the requirement that U = 0~ + Up is approximately equal to UaG through eq. (16b). Thus we put (]s = cUaG ,

(36)

1--c Vp = - - ~ U a c ,

(37)

where k2 = 2 m

-~- (E - UB6).

(38)

The parameter c is a free parameter to be obtained by comparing the calculated cross sections and polarisations with experimental data.

A.M. Kobos et aL / Explicit treatment

102

~'°Co (p, p) ~'°Co 30.3 HeV 10

cr (e) Or(e) oO°.

~'°'°°"

,

;t-k; / / ",.

BG, C=I.0

10-I

L 0

I

60

\

°x N . . / l

I \

\

i

i

I

I

.4

--"x " /

- - - BG, £ =0.8 . . . . BG, C =0.5 ..... Refitted, C = 0.86 80

.,'"

I

I

120

i/ I

"'.-./

I

160

e~r. Fig. 5. Dependence on c, eqs. (36) and (37), of the cross sections for proton scattering from 4°Ca at

30.3 MeV. In fig. 5 we show the dependence of the calculated cross sections on the parameter c, keeping UBc as in ref. JT). We note that for c > 0.8 the cross sections are rather independent of c but start to differ appreciably for smaller values of c. We then tried to establish whether, with plausible assumptions, analysis of the experimental data could provide any estimate of the value of c. We used a standard search procedure ~s) as used for fitting optical potentials. I f the value of c alone as obtained by fitting we find c = 1 since UBc already gives a good fit to the data. However, we may use UBG to give just the radial shapes of the potentials through eqs. (36), (37). Thus we first tried to obtain a fit to the data treating c and all the depths of the potentials in UBo as variable parameters, retaining the geometrical parameters of ref. ~7). We found that the best fit was always obtained with c within 15% of unity. This method still prejudices the fits to be in the region of c = 1 so we also tried varying both the strengths and shapes of the potentials making up UsG and treating these strength and radial shape parameters as well as c as parameters to be fitted to experimental data. As also shown in fig. 5 we found a good fit with c = 0.86. However another good fit with c = 1 and different remaining parameters was also obtained. We thus conclude that the minimum of X2, when c is the parameter, is very shallow and analysis of this data can only show that c > 0.85. This result is, however, dependent on our assumptions concerning the form of the potentials

A.M. Kobos et al. / Explicit treatment

103

although we would argue that we have used the best available. The result c > 0.85 is roughly consistent with the values o f the potentials shown in figs. 3 and 4. For example, from fig. 3, c ~ 1 and, from fig. 4, c ~ 0 . 8 . In this subsection we have arbitrarily assumed the same value of c for the real and imaginary parts of the potential but our result suggests that the size of the calculated P-state contribution, figs. 3 and 4, is consistent with that resulting from fitting experimental data.

3.2. T R E A T M E N T O F P-STATE WITH C A L C U L A T E D t

We have used the method o f sect. 2 to obtain the potentials Us and Up from the calculated t-matrices and compared the resulting cross sections and polarisations with experimental data. As is common 2) we multiply the calculated real and imaginary parts of the central and spin-orbit potentials by corresponding normalisation factors A~t, A~, A~°, A]"°, denoted collectively by k. The same normalisation parameters k were used for the potentials Us and U~. For a satisfactory theory we require that all the k should be close to one. The parameters k were obtained by fitting the experimental data as above. For the calculations reported in this paper the quality of fits to the experimental data are such that the values of A]fl and A]° are not separately well determined. For this reason we arbitrarily assume throughout that A~° - A~ . We may further note that even with this restriction A~t° is not well determined. The first row of table 1 gives the values of k required to give the best fit to the cross sections and polarisations when there is no explicit consideration of P-states (nor D-states), i.e. the theory of ref. 4). The second row gives similar values when P-states are considered explicitly using the formulae of sect. 2. We see that in both cases A~t is close to one as required while the value of AR° is not well determined. The value of A~ indicates that the calculated imaginary potential is too large, a well-known property of calculations of this kind. The explanation for this has been given 20) as arising from an effective mass factor not included in the results shown in figs. 2 and 4. Bearing this factor in mind our value of A~ is perhaps acceptable. We note that the fit to the polarisation data is not extremely good t7), possibly indicating some deficiency in the calculated potential in this respect. Figs. 6 and 7 show the comparison of the experimntal data ~4.15)with --

s.o.

TABLE 1 Values o f k required to give a 'best' fit to experimental data of 4°Ca(p, p) at 30.3 MeV

standard P-state D-state

A~

A~

Aho. -xl .....

0.96 0.95 1.05

0.56 0.56 0.52

0.59 0.91 0.52

104

A.M. Kobos et aL / Explicit treatment

4°Co (p, p) 4°Ca 30.3 kleV

loi a, (O) ~R('01

I

.-f

r.,,, ' 1'

I

I

I

I~ ~

----

10

I

0

'

I

On- shelf standard 0n-shell P-rS:~it~e~xpticif

I

40

I

I

i

I

80

i

120

I

160

e~m Fig. 6. Fits obtained with and without explicit P-state effects to cross sections for proton scattering from 4°Ca at 30.3 MeV.

1.0

tl I

~°co (p, p) 30.3 MeV

",\

I\~

0.5

Y,, \ I t !

V m ....

-0.5

On- shell standard On - she[[ P - state explicit

½ I

I

40

80

0~m

I

I

120

160

Fig. 7. Fits to polarisation corresponding to fig. 6.

A.M. Kobos et al. / Explicit treatment

105

our calculated results. We show calculations corresponding to the explicit treatment of P-states and also those obtained using the standard method 2.4). The fits to the data are good in both cases. There is a marginal improvement, about 20%, in the value of X 2 using explicit P-state terms with no additional free parameters. It is not clear that this by itself is significant. Figs. l and 2 show that the magnitude of the P-state terms, particularly the imaginary part, becomes large at higher momenta, say bombarding energies >150 MeV. However, there is a well-known difficulty 2L) in calculating the optical potential at these energies from Brueckner theory. Thus a successful analysis in terms o f the calculated potentials alone is not possible. We have recently suggested 22) that the calculated central imaginary part is satisfactory near 200 MeV bombarding energy while the real central and spin-orbit components are not satisfactorily given by the calculation. We assume the correctness of this suggestion in the following discussion. Thus we cannot expect to use successfully the P-state terms discussed in this paper for the real and spin-orbit parts but, from the results of ref. 22), we may use our theory for the imaginary part to investigate whether the large P-state contribution seen in fig. 2 gives rise to appreciable effects. Thus we have considered proton scattering from 4°Ca at 182 MeV. The experimental data or cross section and polarisation are given in refs. 2J,23). For reasons just described we follow the procedure of ref. 22) and use a phenomenological potential for the real central and complex spin-orbit parts and use the calculated potential only for the central imaginary part. We first consider the importance of including the P-state effect for the imaginary part, retaining the same potential for all the other parts. We use the phenomenological potential of ref. 22) but the central imaginary part is calculated firstly as in ref. 22), i.e. the usual or standard way, and then including the P-state effect. We adopt the normalising factor k~ = 1.05 from ref. 22). The cross sections are compared in fig. 8 and we note that, as at 30 MeV, the P-state effect is small. We then investigated whether, having completely refitted the parameters of the phenomenological part of the potential and also k~ (its value becomes 1.12), retaining the calculated P-state central imaginary part gives a better fit than obtained in ref. 22) without the P-state effect. The resulting cross section is shown in fig. 8 and in fact the value o f the total X 2 is improved by 20% mainly as a result of an improved fit to the polarisation. This is a similar improvement to that found at 30 MeV but again it is unclear whether taken by itself it is really significant. However, taken with the improvement found at 30 MeV we believe it is significant.

3.3. D-STATE C O N T R I B U T I O N

As we have not solved eq. (27) exactly, all we can do is to see whether the approximation of eq. (33), which implies the usual optical model for u rather than ~b, is satisfactory. We do this by comparing the second iteration, eq. (35), with the first-order iteration, eq. (33), and with experimental data. We use calculated potentials throughout and introduce parameters k as above. Fig. 9 shows a comparison

106

A.M. Kobos et

aL/

Explicit treatment

10

~°(a (p,p) ~Ca 182 HeY

o(e)

dR(e) 10

10-z

.... ........ ,

P -state (refitted) P -state Standard Experiment

2'o 0~.~

Fig. 8. Cross sections obtained with and without explicit P-state effects for 182 MeV proton scattering from 4°Ca. of the cross sections obtained from the first- and second-order iterations using the values of k in the first row of table I, these are obtained by fitting to experimental cross sections ~4,J5) and polarisations. We recall from eq. (33) that the first iteration implies the standard method of refs. ~-6) but with the wave function u rather than ~b. It can be seen that there is some difference between the curves in fig. 9 and hence that the explicit D-state contribution is not entirely negligible even after the transformation of eq. (33). We then tried to refit the data retaining the second-order D-state contribution, ¢q. (35), but treating the parameters k as free to fit the data. We were able to obtain satisfactory fits with the values of k in the third row of table I. The resulting cross sections are indistinguishable from those given by the continuous line in fig. 9. We thus see that a slight renormalisation of the potentials yields practically the same fit as that given by the usual calculation. Hence it is unlikely that the effects discussed here could be determined empirically. The extent of the renormalisation can be seen by comparing the first and third rows of table I. both of which give equivalent fits to the data. Again the renormalisation A~° o f the spin-orbit potential is not well determined at this level of quality of fits.

A.M. Kobos et at / Explicit treatment

4°Ca (p,p) ~°Ca 30.3 MeV

10

(0) ~R(O)

/

/

.,i

.,\ .,/

J

---

10-1

107

I

I

40

I

° l

0n-shell sfondord 0n-shell D - s t o l e explicit I

1

80

J

120

I

I

160

Fig. 9. Cross sections obtained from first- and second-order D-state potentials, eqs. (32)--(35), compared with proton scattering from 4°Ca at 30.3 MeV.

4. Conclusions

The first major result we have obtained is the consistency between the calculated small contribution to the t-matrix from P-states and the empirical result that with certain plausible assumptions the P-state contribution to the optical model is small, in eq. (36) c/> 0.85. This provides additional support for the supposition that using first-order Brueckner theory, as in ref. 4) for example, is essentially correct in the energy regions considered here. This result is by no means obvious in view of the nuclear matter and angle-average assumptions made in the calculation of t. A second result, arising from eq. (21), is that it is not possible to determine the P-state effects purely phenomenologically. With certain additional assumptions we do, however, find that there is a slight improvement in the fir to the data both at high and low energies when P-states are included explicitly. The results shown in figs. la and 2 indicate clearly that, unlike the situation for P-states, the D-state contribution to the t-matrix is large. In view of this it is very important that the resulting optical model, eq. (27), can be reduced approximately to an equation of the standard form, eq. (30), involving the usual optical model, eq. (33), adn the transformed wave function u, eq. (29). Thus in this approximation there are no D-state effects on the scattering matrix elements no matter how strong

108

A.M. Kobos et al. / Explicit treatment

the D-states might be. We have, however, shown that the errors in the approximation leading to eq. (33) are not entirely negligible, see fig. 9. Thus when comparing calculated potentials with experimental data the normalising factors obtained from the first and third lines of table 1 should be noted. It must also be remembered that certain terms have been neglected in deriving the second iteration, eq. (35), and we do not claim that those neglected terms are small. In comparing phenomenological potentials with experimental data our calculations seem to indicate that explicit D-state effects are negligible and that only a renormalisation of the potential is required (last line of table 1). In some applications, e.g. distorted-wave Born approximation, the actual wave function is important. From eqs. (19) and (29) we see that the wave functions calculated using our effective optical potentials differ from the true wave function ~b. At the centre of the nucleus the true wave function ~b is, from eqs. (19) and (29), about 16% bigger than the wave function obtained using the effective potentials. Virtually all the increase arises from the D-state effect. This implies that a similar increase should be applied when using phenomenological potentials. One of us W.H., would like to thank the Royal Society for a grant. We also thank the Science and Engineering Research Council for the use of the CRAY-1 computer. References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) l l) 12) 13) 14) 15) 16) 17) 18) 19) 20)

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