Coulomb corrections to elastic pion-nucleus scattering in the eikonal model

ANNALS

OF PHYSICS

121, 285-317 (1979)

Coulomb

Corrections to Elastic Pion-Nucleus in the Eikonal Model JEAN-FRAN~OIS Institut

de Physique,

Scattering

GERMOND*

Neueh&el,

Switzerland

AND COLIN University

College,

WILKIN + London,

United

Kingdom

Received August 24, 1978

The Coulomb force modifies the elastic pion-nucleus scattering amplitude in impact parameter space in three distinct ways. There is an additive phase, a distortion of the pion’s trajectory, and a shift in the effective energy of the pion-nucleus interaction. A formalism is derived for treating these effects when one has no microscopic scattering theory and the method is applied to n* - 12C and r* - %a in the region of the first pion-nucleon resonance. The success of this amplitude analysis suggests that Coulomb corrections are understood well enough, at least at the higher energies, so that neutron distributions may be meaningfully investigated with pions.

1. INTRODUCTION The pion spectrometers at SIN and LAMPF are producing high-precision data on elastic pion-nucleus scattering in the region of the first pion-nucleon resonance [l-4]. Because in this energy domain the elastic scattering of a 7-r-m from a proton is so much stronger than that from a neutron (and vice versa for .rr-), one of the primary hopes of these experiments is to map out the neutron distribution on the surface of the nucleus. Thus comparing the elastic scattering of 130-MeV pions from WZa and 48Ca the w shows a much larger difference than the n+ because of the eight extra neutrons [5]. On the other hand, Coulomb distortion can give rise to differences between the scattering of + and w even on an isoscalar nucleus, such as 40Ca, which are similar in size to the neutron signal in 4sCa. Clearly then our ability to make use of the 4*Ca data will depend to a large extent on mastering at least the dominant Coulomb effects in the data. If we have a semimicroscopic model of pion-nucleus scattering, such as an optical potential or a multiple-scattering theory, the inclusion of Coulomb corrections is in * Supported in part by the Fonds national suisse pour la recherche scientifique. + Supported in part by the Science Research Council.

285 OOO3-4916/79/090285-33$05.00/O Copyright Q 1979 by Academic Press, Inc. All rights of reproduction in any form reserved.

286

GERMOND AND WILKIN

principle straightforward but may be technically difficult. Failing this can we still correct data for Coulomb terms? This problem was attacked many years ago by Bethe [6] for small-angle elastic scattering. By parameterizing the strong interaction pion-nucleus amplitude in the Coulomb-nuclear interference region in terms of one slope parameter he was able to show that the most important change that was introduced was a modification of the phase of the nuclear amplitude. We [7] have previously extended this model by parameterizing the average nuclear amplitude more carefully so as to reproduce the structure of the differential cross section using a representation in terms of the zeros of the scattering amplitude in the complex momentum transfer plane. If, like Bethe, we use the eikonal approximation to find additivity of the phase functions x(b) in impact parameter b space, most of the differences observed between 7r+ and r- scattering by lzC are described. Agreement is not so good for 40Ca which suggests that we examine further mechanisms 181. The additivity of phases neglects the long-range focusing or defocusing of the pion beam by the Coulomb field of the nucleus [9]. Working to first order in the Coulomb parameter this modification can be calculated from the eikonal approximation in terms of a single integral over the nuclear eikonal phase. For strongly absorptive pion-nucleus scattering this can be well approximated by a simple shift in the argument of x(b). A third correction, which can also be reduced to the evaluation of a similar integral, is that due to the energy dependence of the pion-nucleus interaction: the local energy of a n- on the nuclear surface is a few MeV higher than that of a n+ [9-l I]. To compute this contribution reliably we need also the energy dependence of the physical nuclear amplitude. We know this quite well for lzC but more data is desirable on 40Ca, especially for the forward amplitudes. If we neglect any specific energy dependence of the nuclear optical potential, it can be shown that the corrections discussed here are completely equivalent to the leading Wallace [12] corrections to the eikonal approximation to first order in the Coulomb interaction. Inclusion of all three terms explains the vast majority of the charge dependence observed in elastic scattering from lzC and 4oCa. Residual effects may in part be due to slightly different n-p density distributions, a violation of charge symmetry at the nucleon level, a breakdown of the eikonal approximation, dispersive terms, or radiative corrections. In Section 2 we derive the general formalism for Coulomb corrections to elastic scattering and total cross sections in the eikonal framework. The behavior of pionnucleus amplitudes, in particular those of 12C and 40Ca in the first resonance region, are discussed in Section 3. Our calculational techniques with this type of amplitude are described in Section 4 and the results for the two nuclei presented in Section 5. In the conclusions of Section 6 we consider the limitations of our approach, the possibilities for other corrections, and the scope for future work. 2. COULOMB

CORRECTIONS IN THE EIKONAL

For spin zero nuclei the elastic pion-nucleus parameter representation [13]

FORMALISM

scattering amplitude has the impact

EIKONAL

COULOMB CORRECTIONS

F(q) = ik In, J,(qb) b db [1 -

exp(ix(b))],

287 (2.1)

JO

where k is the relativistic center-of-mass three momentum of the incident pion, y the momentum transfer, and b the impact parameter. Despite the complexity of the pion-nucleus dynamics the scattering can always be described by an optical potential VN , though this will in general be complex and energy (and/or momentum) dependent. If, however, we take it to be local, then the eikonal approximation to the phase function x(b) is1 x(b) = j+T [[k* - 2EV,(E ---7;

a.-- 1L’s-m+=l’,(E

: r)]l:* - k] ds

: r) ds,

where E = (k2 t p2)1/2 is the pion total energy, 11its relativistic velocity, and the integration is carried out along the classical trajectory corresponding to the potential V,(E: r). Such a path is hard to visualize for a complex potential so that it is normal to use either the trajectory appropriate to the real part of the potential or to take simply a straight line parallel to the mean of the initial and final momenta. Our basic aim in this paper is to describe the V, V- differences observed in the scattering from isoscalar nuclei. It is therefore very desirable that when we introduce the Coulomb potential we at least keep all the terms linear in this potential in the expression for the scattering amplitude. This is not always possible; in the absence of a complete dynamical theory for V, it is only easy to introduce the Coulomb potential V,(r) corresponding to elastic pion-nucleus scattering. We are therefore neglecting dispersive contributions which come from the excitation of the nucleus by the strong interaction and its deexcitation by the Coulomb. The Coulomb potential is included through the minimal substitution E -+ E -

V,(r)

(2.3)

inside the square root of Eq. (2.2). Expanding this square root and keeping all terms linear in V,

which shows an extra term arising from the energy shift in the Coulomb field. The first term on the right-hand side of Eq. (2.4) cannot yet be identified with the purely nuclear phase because the classical trajectories of a rr- and a 7~- are very different at low energies. Instead of taking a straight line for both we shall compromise by taking the integration along the classical trajectory in the pure Coulomb field. Thus the YT 1 Where there is no confusion the argument E will be suppressed.

288

GERMOND

AND

WILKIN

sees the nucleus through a concave lens, the VT- a convex. Most of this focusing or defocusing of the pion beam by the Coulomb interaction is a long-range effect. If the pion wave is strongly absorbed inside the nucleus, it would not be unreasonable to evaluate it using the Coulomb potential due to a point charge Ze, viz., Vgt(r) = Zol/r. The trajectories are, of course, hyperbolas and if we measure distances s along the curves from the point of closest approach, then to first order in Zol r =

(b2

+

s2)1/2

+

n/k

(2.5)

with the Coulomb parameter n defined by n = Zcx/v

(2.6)

Some of our parameterizations of the optical potential will not be black in the interior of the nucleus so that it can then be important to use a finite charge distribution. The classical trajectory of a particle in a general potential V,(r) can easily be found in perturbation theory. To first order in Ve this is

r = (b2+ s2P2 + & cb2 +b2s2)li2 a Jyds’VJ(b2 + sy]]. x [ V&b2+ s2p21 - $s$g

(2.7)

In principle the Coulomb distortion should be included in the three terms on the right-hand side of Eq. (2.4), but the second and third are already proportional to Vc . Inserting Eq. (2.7) into only the first we find (2.8a)

X(b) = Xc@)+ L@), x5&) = XNW + XTD(N + XES(Q,

(2.8b)

xc(b) = - 1 += dz V&h 1’ s--m

(2.8~)

&#I) = - 1v J+dz -cc

(2.8d)

with

XTD@) = - &

V&E : r),

T$ lTrn dz V,(r) VN(r), m

XES(W= J-r dz V,(r) &

[ E vN(kE : “1.

(2.8f)

This defines phases corresponding to the Coulomb potential, the nuclear, the trajec-

EIKONAL

COULOMB

289

CORRECTIONS

tory distortion, and the energy shift. Given a model for the nuclear optical potential V,(E: r) the Coulomb distorted phases can be easily computed from Eq. (2.8). Corrections to the eikonal approximation have been systematically studied by Wallace [12] for the scattering from a local energy-independent potential. In his nonrelativistic formalism this corresponds to the assumption that EV,(E: r) is independent of energy. The leading Wallace correction in VcVN is 1 XWAL@)

=

-

.+m

a

ko2

(

1 +

bb

IS

--m

dz

V&l

V&l,

Under the energy-independence hypothesis this is exactly equivalent to the sum of the trajectory distortion and energy-shift phases of Eq. (2.8). We have, however, preferred to present the more intuitive geometric approach which includes the correct energy behavior. The effect of trajectory and energy distortions by a Coulomb potential can be very easily computed for black disk scattering where

exp[ixN(b)l =0 =I

b < R(E) b > R(E).

(2.10a)

This leads to distorted profiles exp[&(b)]

= 0

b < RedE)

= 1

b > &n(E),

(2.10b)

where the effective radius is given by R&E)

= [l -

V,(R)/kvl R[E -

V&01.

(2.1 I)

This naive limit describes well the gross features of the total cross-section differences observed between r+ and F scattering from light nuclei [I I]. We can, however, do better. Given an ansatz for the nuclear phase xN(E: b), obtained for example from the average of the n-f- and r- data, it is possible to derive the corresponding optical potential through the inversion formula [13] V,(E

: r) = 5

J+I

-*

r/z &

xN[E : (r2 -C zz)l/t],

(2.12)

This allows the explicit dependence upon the optical potential to be eliminated by substituting Eq. (2.12) into the right-hand side of Eq. (2.8).

290

GERMOND

AND

WILKIN

Evaluating first the trajectory distortion

(2.13)

we change the integration variables to z = B cos 4 (2.14)

z’ = B sin 4

so that xdb)

=

-

&

b & jbm dB w

iw’2 d+ V&b2 + (B2 - b2) cos2 c#I)*/~~. (2.15)

In like manner the energy shift yields x&b)

= -$

&

Ib

m dB w

s0

nJ2d$ Vc[(b2 + (B2 - P) cos2 #J2].

(2.16)

Generally V&) varies much more slowly than V,(r). In the extreme case of the black disk model the V, of Eqs. (2.15) and (2.16) are evaluated at B = b. Our subsequent numerical calculations will prove to be much more stable if we take out this black disk limit as an explicit shift of the energy and impact parameter in xN(E: b). We thus redefine h(E:

b> = XNW -

Vc(b): b(l + Vc@)lkv)l + fdb)

(2.17)

+ g&b),

where the residual corrections are f&b) f&b)

= - -?m dB w rrkv I b

b $ I”‘? d$ V&b2 + (B2 - b2) c&

= 4

j=”

&

jbm dB w

95)‘/“],

0

d$ [V&b2

(2.18a) + (B2 - b2) cos2 r,h)‘/“] -

V,(b)].

0

(2.18b) Because V,(b) N l/b is much smoother than xN(b) for large b, it can be easily seen that the residual corrections of Eq. (2.18) decrease much faster with b than the corresponding ones of Eqs. (2.15) and (2.16). As a consequence they lead to much smaller effects than the “asymptotic” corrections characterized by the energy and impact

EIKONAL

COULOMB

parameter shifts of Eq. (2.17). Further integration large b behavior. For example,

.z x

J0

291

CORRECTIONS

by parts may lead to even better

d#, V,[(b” + (B” - b2) co? $,‘:‘].

(2.19)

In Fig. 1 we show the behavior of the Coulomb phase corrections as functions of b for the case of pion scattering from *OCa at 115 MeV using the parameterization of the nuclear amplitude discussed in Section 3. It is clear that the nonasymptotic terms of Eq. (2.18) are much smaller than the asymptotic ones. This suggests that it is permissible to evaluate the former using an approximate potential while keeping the exact form for the asymptotic contributions. One form which is convenient for further integration is [14] V,(r) = Za/(r2 + A2)1:2

(2.20)

which arises from the charge density (2.21) This density only decreases like r5 so that (r2> is not even defined. The amount of charge in the tail is quite small and if we take A2e

(rc2hrge’ .

(2.22)

the bulk of the charge density is reproduced adequately for the evaluation of the small corrections. The 4 integration of Eq. (2.18) then leads to elliptic integrals K and E of the first and second kind [IS] of argument y = (B2 - b2)/(b2 + A2).

(2.23)

Explicitly fm(b)

=z

b”yA2

j-=dB

E(Y)+ [E(Y)- K(~)l/y QAB) (b2 + A2)l12

b g&b)

=

c

&

j-u b

B

dB

IE(y)

-

K(.dl

xiv(B)

(B2 - b2)(B” + A7112 ’

BB

’

(2.24a) (2.24b)

It can be seen from Fig. 1 and the other numerical examples discussed in Sections 4 and 5 that the asymptotic energy and impact parameter shifts displayed in Eq. (2.17) represent the vast majority of the Coulomb distortion. These shifts may be combined if we consider the eikonal phase as a function of the angular momentum, 1 = kb - i, rather than b. In this formula the change in I due to the long-range Coulomb distortion

292

GERMOND

AND

WILKIN

ReT(bI

0.05

0

-0.05

0

1

2

3

4

5

6

7

8

b lfml 1. Real (a) and imaginary (b) parts of the Coulomb correction to the profile function I’ for rr--Wa elastic scattering at 115 MeV compared with the purely nuclear amplitude TN estimated using the parameterization of Eq. (3.1) with the data of Ref. [2]. The trajectory distortion (TD) contribution, asymptotic and residual, are shown as solid lines; the energy-shift (ES) terms are dashed lines. The position of the equivalent black disk radius (Rae) is marked. FIG.

of b is exactly compensated by the shift in the local momentum k, a direct consequence of angular momentum conservation. The “asymptotic” correction is therefore only a modification of the energy2 xdE: 4 -

xdE -

VcW + &j/k): 4.

(2.25)

a It has been suggested [16] that only the pion momentum should be displaced and not the energy, thus requiring knowledge of the phase shifts off the mass shell. This question is far from settled but in fact it can only lead to very small differences.

EIKONAL

COULOMB

293

CORRECTIONS

Imrtb) - 0.05

-0.10

-0.15

0

I

I

I

I

1

2

3

4

5

I

I

6

7

8

b (fm1 FIG.

1 (continued)

This simple but efficaceous prescription, which can be derived from the WKB approximation to the phase shifts, has been used very successfully by Jackie, Pilkuhn, and Schlaile [ 161 to investigate 7~* 160 through the resonance region. Having obtained expressions for xTD and xES all the contributions to x can then be inserted into Eq. (2.1) and the scattering amplitudes calculated separately for for vi1 and n-. It is customary [17] to decompose the amplitude into a pure Coulomb term and the rest. There are two schools of thought, one of which [18] suggests that we should remove only the point Coulomb amplitude as opposed to the full amplitude corresponding to a finite charge distribution. Of course, both answers will be the same if no further approximations are made, the “rest” differing in the two cases.

294

GERMOND

AND

WILKIN

We prefer to adhere to the second course so that the “rest” amplitude vanishes when the nuclear potential is zero and in general / Fc 1 < 1FTeBt1 at large angles, which would not follow if only the point amplitude were extracted. Adding and subtracting exp(ix&)) in Eq. (2.1) gives

F(q) = ik j Jo(@)b db [l - vXix&Nl + expGx&))[l - exp(WWl

(2.26a) (2.26b)

= F,(q) + 4.7(q)

which is easily recognized as the eikonal form of the Gell-Mann and Goldberger two-potentials formula [ 193. The Coulomb potential has infinite range and must be handled in the eikonal framework as the limit of a screened potential where in the end the screening radius is allowed to tend to infinity [17]. Thus v,(r)

= .ZI j , {c(‘I!

, C(r’) d3r’,

(2.27)

where PC(r) is the nuclear charge density including the finite sizes of the pion and proton and we shall take the screening function to be (2.28)

C(r) = &a - r).

If a is chosen large on the nuclear scale, then inserting Eq. (2.27) into Eq. (2.11) we can approximate to find xc(b) = -2n

s pc(r’) log ( k 1,‘“”

b’ j ) d3r’

= x84 + xc@). The shielding contribution

(2.29)

is just an additional constant independent of b x*(b) = -2n log(2ku)

leaving a charge phase independent

(2.30)

of a

x0(b) = 2n j pc(r’) log[k ] b - b’ I] d3r’.

(2.31)

For a point charge this latter reduces to

xpt(b)= 2n hdkb)

(2.32)

EIKONAL

COULOMB

295

CORRECTIONS

with a corresponding Coulomb amplitude [14, 171

= - -y

exp [ -2in

[log [+)

T y] -;- ix,]

(2.33)

To first order in n, y is Euler’s constant. For large b the Coulomb phase varies logarithmically with b which causes numerical difficulty in the evaluation of the Bessel transform of Eq. (2.26). However, for values of the impact parameter lying outside the nucleus xpt(b) e x0(b) so that the convergence of the b integration is greatly improved if we take advantage of the analytic form3 given in Eq. (2.33) and write the Coulomb amplitude as

Fdq) = exp(ix,)[F,t(q) + ik SCb dbw[ixdb)l J&r@ x [I - exp[&,(b) - :dbNll/.

(2.34)

The distorted nuclear amplitude of Eq. (2.27) also has this factor of exp(&).

pds) = exp(ixs)[ik Jr Jo(@)b db exp[ix,(b)][l

- exp[ifJb)]]/

(2.35)

0

so that the screening radius does not appear in the final answer for the differential cross section. We can therefore let a + co without affecting any physically measurable quantity. This completes the formalism required to introduce Coulomb corrections into the elastic-scattering data but we should also like to study their effect on the total cross sections. In the absence of the long-range Coulomb force this can be calculated from the imaginary part of the forward amplitude using the optical theorem. Because of the divergence at q = 0 in Eq. (2.33), this is no longer possible. Also we have included correctly only the elastic Coulomb distortion so let us first evaluate the correction to the integrated elastic cross section. This is de$ned experimentally as the cross section for elastic scattering with momentum transfer bigger than some value q, minus the pure Coulomb scattering and minus the Coulomb-nuclear interference, in the limit as q-0.

duo1 __-____uE G lim q+o S[ qs dq’2

deoul dq’2

~h.n dq12

1dq’“.

The integrated elastic cross section is defined such that to first order in n only PM(q) of Eq. (2.26b) contributes. uE =

s

1&(q)j’

dL?.

3 Fc can also be evaluated analytically [14] for the VCof Eq.

(2.37) (2.7).

296

GERMONDAND WILKIN

Inserting the representation of Eq. (2.26a) we find in the eikonal limit uE = -&1

I d% d2b2ewblxc(bd - xc(b2))W- exp(k4b31

X [I - exp(--ifz(b,))] 1 d2q exp[iq . (b, - b2)]. Performing the integral over all momentum (h, - b,) so that oE = 2ir

f

(2.38)

transfer first yields a delta function in

b db 1 1 - exp[$,(b)]j2.

(2.39)

Therefore the Coulomb phase (including the screening) does not affect the integrated elastic cross section at all. This result is well known in the partial wave representation [20]. If we evaluate the Coulomb correction to the total cross section from Eq. (2.39), we clearly underestimate the effect considerably because of the predominance of the inelastic channels. We shall therefore assume that the phase fN(b) is the one to be used in the optical theorem; in the absence of a dynamical theory for the inelastic scattering this cannot be improved upon. Hence or = 47~ b db Re[l - exp[if,(b)]]. I

(2.40)

For later qualitative discussion it is useful to evaluate the effect of adding the Coulomb phase of Eq. (2.8~) in the limit of black disk scattering described by Eq. (2.10a) but with a real part coming from the surface region. exp[ix,(b)]

= B(b - R) + $ipR2 S(b - R).

(2.41)

For a point Coulomb charge, or one whose density is zero beyond the radius R, the total amplitude is approximately

J,(M) F(q) m ikR2 ___ [ qR

-

ipJo$R) ] exp[2ni log(kR)]

- $$ Jo@@)exp [ -2ni (log (-&-) + y)],

(2.42)

where some contributions of order n2 have been dropped from the second term since they are negligible for small qR. 3. THE BEHAVIOR OF THE PION-NUCLEUS SCATTERINGAMPLITUDES In the vicinity of the first pion-nucleon resonance the interaction is so strong that the elastic pion-nucleus scattering resembles that of a fuzzy black disk even for nuclei

EIKONAL

COULOMB

297

CORRECTIONS

as light as 12C. The angular distributions show series of diffraction minima whose depths change dramatically as a function of energy. Note that for spin-zero nuclei the filling-in of these minima cannot be due to spin-flip terms. It was suggested by Gersten [21] and Nichitiu [22] that instead of trying to analyze the amplitude F&q) into partial waves it might be useful to describe some of the features in terms of the complex zeros of the scattering amplitude. Thus a deep first minimum at a value of q2 = Re(qr2) will be due to a zero of F&q”) with Im(qr2) relatively small. In this language the change in the shapes of the minima is due to a rapid change in Tm(q12) as a function of energy. We [23] had earlier found that it was useful to go further and make an explicit parametrization of the amplitude in the form F,(q) = F&9

fi (1 - q2/qi2) exp(-

$zq2),

(3.1)

i=l

where Iw is chosen to be the number of observed or inferred minima (or shoulders) in the measured angular distribution. From the optical theorem t&(O) = z%cr~(l- ip)/47r,

(3.2)

where or and p are the total cross section and relative real part of the forward amplitude, respectively. The Gaussian factor in Eq. (3.1) is purely empirical and may be replaced by another smoothly decreasing function of q if this gives a better fit to the data. The critical assumption is that a, or its equivalent in other parameterizations, is taken to be real so that the phase variation of the amplitude is given by its nearby zeros. In the case of 12C there are typically three zeros and thus nine free parameters at each energy. Since in the main the parameters are determined by different parts of the data, their errors tend to be uncoupled and the values resulting from fits to the data are smooth as a function of energy. On the other hand, at 180 MeV perhaps ten partial waves are significant which means double the number of our parameters. Information over the whole angular range is needed for each parameter so the error matrix is far from diagonal. Since unitarity is not much of a constraint, there are more parameters than the data can support and they are then badly determined. To define an optical potential through Eq. (2.12) we need the phase xN(b) for all b and hence by Eq. (2.1) the amplitude FN(q) for all real positive q. Even if Eq. (3.1) provides a good representation of the observed data, we can expect that other zeros as well as cuts of the scattering amplitude will lie outside the physical domain; the damping factor in Eq. (3.1) represents their influence in the measured region. We shall, however, assume that the parameterization of Eq. (3.1) hold for all q. A consequence of this is that any structure in the potential in b space with a width less than about n/q,,, will not be reproduced reliably. A major uncertainty in determining the parameters is due to the fact that the signs of the Im(qiz) do not affect the average nuclear cross sections. Where there is data at low energy (~70 MeV) the unitarity condition is sufficiently strong to force us to choose Im(q12) positive for both 7r - 4He and rr - 12C [22]. Continuity in energy

298

GERMOND

AND

WILKIN

then allows us to deduce that for 12C Im(qr2) remains positive until about 180 MeV after which it is negative. This behavior is in accord with the strong absorption phase rule we have previously proposed. In the fuzzy black disk model the real part of the amplitude comes from the region of the surface. Neglecting the Coulomb interaction in Eq. (2.42) we have

JdqR) FN(q) m ikR2 ____ [ qR

4 Jo(@) 2

I’

where the value of the relative real part of the forward amplitude p will depend sensitively upon the nuclear surface, but we shall take its value from experiment. Diffuseness of the surface may be included phenomenolgically by multiplying the amplitude F,(q) by a smooth fudge factor G(q) a la Inopin [24]. Now for small p we can incorporate the second term into the first by letting R -+ R = R(l - frip). 1

G(q).

(3.4)

The zeros of the amplitude are thus displaced away from the real axis and in this simplistic model vi E Im(qi2)/Re(qi2) = p

(3.5)

for all the zeros of the amplitude [7]. The argument is similar to the geometric scaling hypothesis [25] used in particle physics where it is assumed that F(q) = ikH(qR)/q2 with H real. The real part of the forward amplitude is known from forward dispersion relations to be positive below resonance, negative above, in qualitative agreement with the behavior found for Im(qr2). More quantitative information comes from the real part of the forward amplitudes deduced from Coulomb-nuclear interference measurements. In Fig. 2 we plot the values of p and y1 for 12C, lsO, and 4oCa deduced from experiments, some of which are as yet preliminary. There are some conclusions which can be drawn immediately. Our phase rule (Eq. (3.5)) is very well satisfied in both sign and magnitude over the range where experiments exist. Extra confirmation of the sign will be found from the results of the Coulomb distortion calculations of Section 5. The results vary smoothly with energy except for v1(40Ca) at 163 MeV but this may have an experimental origin as we discuss later. It appears that p passes through zero at a kinetic energy To only a little below 180 MeV for all three nuclei. On the other hand, it had been assumed, on the basis of forward dispersion relations [31], that To tends to decrease with increasing A and that for 12C it is already down to 165 MeV. The reason for this prediction is that the peak of the pion-nucleus total cross section input though broadening, does move down to lower energy as A is increased [II]. In the evaluation of the dispersion relation some assumptions have to be made about the behavior of the imaginary part of the amplitude in the unphysical region below threshold. Introducing an effective pole there to simulate some of the absorptive

EIKONAL

COULOMB

299

CORRECTIONS

contribution, or performing a second subtraction, it is possible to insist that p passes through zero for 12C at T = 175 MeV to accord with the general trend of the data. Both the original and modified predictions are shown in Fig. 2. When n+ and 7r- data both exist the parameters may be easily determined by separately fitting it away from the forward direction using Eq. (3.1) and averaging the results over the two charge states. This procedure is very stable in that it leads to almost exactly the same values of the mean parameters whether or not the data is corrected for Coulomb distortion. The values of v1 deduced from the SIN [1] and older

-o+

/

,

,

100

150

200

,

,

250

Frc. 2. Plot of the forward relative real part p [Eq. (3.2)] and relative imaginary part of the first zero “I [Eq. (3.511 for elastic scattering of pions from different nuclei as a function of the pion laboratory kinetic energy. a = p(W) [26, 271; b = LJ~(‘“C) [l]; c = vI-(‘“C> [28]; d = ~(“0) [18, 291; e = ~~(~~0) [2, 161: f = #%a) [30]: g = yl(%a) [2, 51. Note that Coulomb corrections have not been applied to the w data of Ref. 1281 (points c). The solid line is a standard forward dispersion relation prediction of [ll, 311 for W; the dashed line results from introducing an effective pole in the unphysical (absorptive) region so as to ensure that p passes through zero at 175 MeV.

CERN [28] data agree quite well especially when it is realized that the CERN points come from 7~- scattering which have shallower minima below resonance than the 7-l. because of the Coulomb distortion which we have not removed. Another striking fact is the similarity in the behavior of the different nuclei shown in Fig. 2. Between 115 and 240 MeV all the measurements of p and v1 lie in a narrow band about 0.1 in height. Isolated points measured on a variety of other nuclei also fall within this

300

GERMOND

AND

WILKIN

band. This is a little surprising in that we might expect p and vi to decrease as ,4 increases. The situation is far less clear for the second minimum mainly because of experimental difficulties. The cross section in this region is very low leading to problems of counting rates. For 12C at 226 MeV [l] there appears even to be a double-minimum structure, which may be of experimental origin, but which we shall have to treat as a real effect. At low energies the second minimum comes at large angles which are at the edge of the measured region. Though the positions of these minima are quite well defined, the depths are often ill-determined so that with the present amount of data we cannot draw useful trajectories for v2 . There is indeed information [l, 21 that v2 is very small near 115 MeV for both 12C and 160 but not Ya. Optical model fits [32] in the 12C case suggest that the second zero crosses the axis twice at low energy as well as in the resonance region such that the strong absorption phase rule might still be valid around 180 MeV. Extra evidence for this comes from the highprecision LAMPF data [3] on silicon at 162 MeV where within quite small error bars v2 = v, and both lie in the band of Fig. 2. The total cross sections of both 7~+ and n- in 12C are quite well measured [l l] but comparable figures for 40Ca have not yet been published. We are therefore forced to interpolate the results from other nuclei. For example, the maximum value of the average pion-nucleus total cross section is rather well reproduced by 132A2j3 mb [ll, 331. The 4oCa values obtained in this way are in qualitative agreement with Glauber-model calculations based upon nN or ~TTOI inputs. This introduces some uncertainty which at 115 MeV must be added to that of p. Of course, if we knew the differential cross section accurately we could deduce the total cross section from the optical theorem. The uT obtained in this way from the preliminary small-angle LAMPF data [30] have large error bars and a rather suspect energy dependence. Uncertainties in the normlization and angular resolution of the SIN data make the method as yet inappropriate there also. We have in fact renormalized this data so that our parameterization, when extrapolated to zero degrees, agrees with the optical theorem using the interpolated o’T. Choosing a form other than Gaussian in Eq. (3.1) would then lead to a slightly different normalization. The finite angular resolution may account for our normalization factors, Ren of Tables I and 11 being in general above one. The slope parameter a is well defined at all energies assuming it is real. In a strong absorption model it might be better to replace by a(1 - ip) but since the second zero may not satisfy this rule well, we have refrained from so doing. Values of the different parameters deduced from the fits to the rzC and 40Ca SIN data are given in Tables I and II. Some of the signs of the higher Im(qi2) are ambiguous though the choice shown might lead to a marginally lower x2. In Fig. 3 we show the real and imaginary parts of the impact parameter amplitudes (profile functions) (3.6) rdb) = 1 - exp[ixd@l for 40Ca at 163 MeV deduced from our parameterization

of Eq. (3.1) with three,

EIKONAL

COULOMB CORRECTIONS TABLE

301

I

Parameters for the rr - ‘%I Amplitude.” UT (mb) Energy WV)

Ren

Particle

148

694 0.15 I .09*

nf 71x+.‘T-

162

688 0.065 1.052*

226

585 -0.205 l .097*

qa2(fm ?)

a (fm? -~ 0.912 0.907 0.910

a2 (fm+J Real Imag

qz2 (fm-2J Real lmag

Real

lmag

1.335 1.358 1.342

0.204 0.199 0.202

3.378 3.435 3.398

0.202 0.170 0.188

5.9* 5.9* 5.9’

-0.92* - 0.92* - 0.92’

n-7 ?iiT+,,71-

0.892 0.984 0.936

1.388 I .401 1.396

0.118 0.122 0.118

3.612 3.659 3.630

0.255 0.317 0.282

6.26 6.03 6.11

- 0.88 - 1.06 ~0.84

7T* xP+;‘T-

1.120 1.045 1.081

1.603 1.592 1.597

7.57 7.67 7.45

0.48 0.15 ~-0.16

P

a The fit uses Eq. (3.1) on the held fixed, as were the values of correspond to fitting separately affect typically the last decimal

-0.284 -0.257 -0.272

5.87 5.38 5.66

-0.41 -0.28 -0.33

data of Refs. [I, 2, 51. The parameters marked with an asterisk were or and p, leaving the normalization factor Ren free. The three rows the nT and X- and then the combined values. The statistical errors quoted and must be overshadowed by the systematic errors. TABLE

II

Parameters for the rr - %a Amplitude.” CT (mb)

qp2(fm-“)

Energy (MeV) ._ -

Ren

Particle

a (fm2)

Real

115.5

1518 0.21 1.037

7rk xX+.:7-

1.982 2.085 1.991

0.584 0.597 0.591

0.135 0.117 0.129

1.724 1.696 1.717

130.0

1535 0.16 1.026

,+ nn-.:x-

2.190 I .956 2.082

0.582 0.610 0.598

0.095 0.090 0.095

1.787 1.863 1.844

0.347 0.211 0.270

3.20’ 3.216 3.20

0.10* 0.10* 0.10*

163.3

1523 0.05 1.143

nn-rf-.vI’

2.126 2.014 2.042

0.597 0.636 0.626

0.085 0.034 0.03 1*

2.070 2.090 2.088

0.206 0.072 0.098

3.67* 3.67* 3.67*

0.27* 0.27* 0.27*

241.0

1385 -0.21 0.975

n+ TXnfix-

1.966 2.073 2.000

0.705 0.705 0.704

4.91* 4.91* 4.91*

0.40 ~ 1.02 m-O.48

a2 (h-2)

P

lmag

Real

-__

-0.157 -0.145 -0.149

2.360 2.452 2.393

qa2 (fm ‘) Jmag Real Imag ..-- --_. ~~~~ 0.295 3.05* 0.10* 0.272 3.05* 0.10* 0.289 3.05” 0.10*

-0.463 -0.433 -0.438

a The fit uses Eq. (3.1) on the data of Refs. [l, 2,5]. The parameters marked with an asterisk were held fixed, as were the values of or and p, leaving the normalization factor Ren free. The three rows correspond to fitting separately the n+ and rr- and then the combined values. The statistical errors affect typically the last decimal quoted and must be overshadowed by the systematic errors.

302

GERMOND

AND

WILKIN

Rertb)

0 0

I

2

3

L

5

6

7

b lfml

0

I

2

3

L

5

6

7

b IfmJ

FIG. 3. Real (a) and imaginary (b) parts of the profile function [Eq. (3.7)] for ,PCa elastic scattering at 163 MeV obtained by applying to the data of Ref. [2] the parameterization of Eq. (3.1) with 1,2, or 3 zeros, The position of the equivalent black disk radius and shape of the corresponding profile function are shown.

EIKONAL

COULOMB CORRECTIONS

303

two, and one zeros. In suppressing a zero qiz we modify the slope parameter but keep it real a + a + 2 Re[l/qi2].

(3.7)

As the number of zeros is increased the 7~- 40Ca amplitude is described well out to larger values of q2 and the magnitude and wavelength of oscillations in b space reduced. Providing we are interested in integrals of I’,(b) times smooth functions of b then the (probably) unphysical oscillations cause no problems but for this reason we must avoid using a point Coulomb potential in calcuIating the distortion terms. However, when F,(b) comes close to unity the real part of the corresponding phase xN(b) will change very fast with b. This defect of our parameterization can cause numerical problems in the evaluation of Eqs. (2.15) and (2.16) but this is not present in Eq. (2.18).

4. APPLICATION

OF THE THEORY TO 12C AND 4oCa

The electromagnetic form factors of light nuclei such as 12C and 40Ca can be very well parameterized by a shell model form S(q) = (1 - q2/qa2)(l - q2/qa2) exp(-BW.

(4.1)

We must however modify the values of/F? to reflect the sizes of both proton and pion.

8” = PO”+ 3K~v2> + +-,“>I.

(4.2)

Taking the pion and proton rms radii as 0.8 fm we obtain for 12C [34] /I2 = 0.775 fm2,

qa2 = 3.37 fm-2,

qbe = co

(4.3a)

qa2 = 1.40 fm-2.

qh2 = 4. I6 fm-2.

(4.3b)

and for 40Ca [35] p” = 1.178 fm2, The germane Coulomb exponential integral

phase of Eq. (2.31) can then be evaluated in terms of an

xAb> = x&b) + n 1~ [$$-I x Cp&

+

~-~~ pbo2

+ exp [ - G] (1 - ba/4p2) P434qa2%2Ii .

(4.4)

The form factor S(q) also defines a Coulomb potential through 2za m V,(r) = io(sr) m) I ~ 0

&*

(4.5)

304

GERMOND AND WILKIN

Using the parameterization function

of Eq. (4.1) this can be integrated in terms of an error

V&> = +

erf (+)

X

- +

exp (-

-&)

3(1 - r2/6p2) 2P2qa2qb2 -

1

[

(4.6)

In Section 3 we motivated the parameterization of the pion-nucleus scattering amplitude in terms of its zeros as in Eq. (3.1). With a Gaussian factor this can be easily Fourier transformed to yield for three zeros the impact parameter phase exp[ix&)]

= 1-

uc14;aip)

[do + d,x + 4x2 + d3xy

(4.7)

with x = b2/2a and do = 1 - c1 + 2c2 - 6c,, dl = cl - 4c, + 18c, ,

(4.8a)

d2 = c2 - 9c, , d3 = c3

with

c1 = wq12 + l/q22+ l/q,21? a

c =

wq12q22

2

” = [a3qlz:.“y$l

+ l/q22492 + llq32q121 , a2

(4.8b)

.

5. RESULTS FOR lzC AND “Ca In Fig. 4 we show the predictions for the 7r+, rr- total cross section difference for 12C and 40Ca expressed as percentages of the average cross sections. The agreement with the experimental measurements [ll] on 12C is quite satisfactory but results of 40Ca measurements are not yet available. It is likely though that our predictions in the latter case may overestimate the effect slightly in the same way as they do for the shift in position of the diffracton minimum. It can be seen from Fig. 4 that as A increases, the trajectory distortion becomes larger and dominates the energy shift, mainly because the peaks in the average cross section are then very wide. The residual corrections of Eq. (2.18) contribute generally less than fr ‘A for 40Ca and even less for 12C.

EIKONAL

305

COULOMB CORRECTIONS

15

IO

F

I%)

5

0 200

250

Tn IMeV)

FIG. 4. Percentage total cross section difference for V, 71+ interacting with leC and %a as a function of the laboratory kinetic energy. The errors in the ‘*C data of Ref. [I I] include a contribution from the uncertainty in the value of p. The effect of only including the asymptotic trajectory distortion is the solid curve; the long dashed curve includes also the asymptotic energy shift; the addition of the residual terms of Eq. (2.18) yields the short dashed curve. The introduction of chargesymmetry-violating terms through Eq. (5.5) leads to the dot-dashed line.

The same trend of the increasing importance of the trajectory distortion for larger A may be observed in the corresponding differential cross sections of Figs. 5 and 6 and in the percentage cross-section difference of Fig. 7. It is only on the linear scale of this latter representation that we can show the small effects of the residual Coulomb corrections. Note that even for 4oCa the largest term is still that due to the additive Coulomb phase. The theory reproduces well the available data on 12C but for 4oCa, although we explain the vast majority of the charge dependence, there are some discrepancies especially at the lowest energy. This may be due in part to errors in the prdiminary data that we are using in the comparison. In this context it is important to note the excellent accord at 130 MeV [5] compared to that at 115 MeV [2], the data being taken by different groups. Any error in the experimental data induces errors in the fit parameters that we use in the calculation of Coulomb corrections at that energy and also, through the energy dependence, at other energies. It is unfortunate that the behavior of this correction in the vicinity of the second minimum at 241 MeV depends upon that at the edge of the measured region at 115 MeV! In order to ensure continuity in energy at 163 MeV we have takena value for v1 which is a factor of 2

306

GERMOND

AND

WILKIN

IO2

1

t,

IO'

i

\

.++++ ‘t!!t,i;t \\vtir \fi i

*

!

10-l

8,, 0

15

30

In-j 45

#

-

tdeg 1 n 60

I5

ec,,, I+)ldagl FIG. 5. Elastic n+ and T- scattering from leC at (a) 148 MeV, (b) 162 MeV, and (c) 226 MeV, the data being taken from Ref. [l]. The effect of the Coulomb phase is shown as the dashed curve, with the trajectory distortion added as the dot-dashed and including also the energy shift as the solid curve. At 162 MeV the points with 0 > 133” were normalized by demanding continuity with smaller angles.

lower than that given by a straight fit to the 40Ca measurements and this leads to the prediction of a very sharp diffraction minimum for T+ scattering which we are unable to eliminate by any reasonable variation of the parameters. It should be noted that the new LAMPF [3] measurements at 162 MeV on (natural) silicon do show such a sharp minimum. Some of the uncertainty in our input parameters would be reduced by having data over a wider angular range especially near the forward direction. Also Chabloz [36] finds that a better representation of the new large-angle data on 12C at 162 MeV is achieved with the Gaussian in Eq. (3.1) replaced by a (1 + aq2)-N factor though this leads to large oscillations in the profile function. We have previously noted that the first minima are deeper for ST+below resonance

EIKONAL

40

COULOMB

307

CORRECTIONS

60

5 (continued)

FIG.

and for rr- above. From the general energy dependence of V shown in Fig. 2 this implies that vI+ < vl-. In the fuzzy black disk model of Eq. (2.42), working to first order in II and neglecting an overall phase factor, the additive Coulomb phase leads to an amplitude F(q) = ikR2 [q

-

+MqW

2

’

+ +

J&R)]

(5.1)

For small p and n and away from the forward direction we can incorporate the last two terms with the first by defining an effective radius R=[l-p++]R as in Eq. (3.4). Consequently in this strong absorption parts of the zeros are displaced by

(5.2)

model the relative imaginary

dv, = vi- - vi+ = 8 ’ n l/(jl,J2,

(5.3)

308

GERMOND

ecm

AND

WILKIN

(n')ideg)

FIG. 5 (continued)

where thej,,* are the zeros of the Bessel function. In particular Au, = 0.54 I n 1.

(5.4)

The depths of the diffraction minima in Figs. 5 and 6 are not greatly changed by the trajectory distortion but the energy shift is important and this works in opposition to the Coulomb phase. To see this we take the energy shift on the nuclear surface which overestimates this contribution. dv = 2zol a3 -JnE---

1.2jnj R

(5.5)

The combination of Eqs. (5.4) and (5.5) explains the sign of the cross section difference at the first minimum but the model is too simple to reproduce quantitatively the values of& - 0.08 found for 40Ca and 0.04 for r2C.

EIKONAL

COULOMB

309

CORRECTIONS

.ecm I K- 1(deg 1

FIG. 6. Elastic W+ and n- scattering from Ta at (a) 115 MeV, (b) 130 MeV, (c) 163 MeV, and (d) 241 MeV, the I30-MeV data being taken from Ref. [5] and the rest coming from Ref. [3]. The effect of the Coulomb phase is shown as the dashed curve, with the trajectory distortion added as the dot-dashed curve and including also the energy shift as the solid curve.

The differential cross section corresponding da

J&R) ’ + Ji?J- [ kR2 qR

to Eq. (5.1) is

1 [(~W22

- F)

Jo(qR)12.

(5.6)

It can then be seen that the correction due to the Coulomb phase should vanish at the zeros of J,,(qR) and the m+ and n- cross sections should be tangential at these points. The theoretical predictions, shown in Figs. 5 and 6 as dashed lines, obey this rule quite well for the first zero and in Fig. 7 it explains the broad valley (peak) at 26” (18’) at 115 (241) MeV. For 12C this is the dominant Coulomb term so that the trend can then be seen directly in the data.

310

GERMOND

\

AND

WILKIN

\ Ib)

\

IO’ c

\ t

.

-I

ecm ldldeg) 0

10-2

0

'

' 15

'

' 30

'

15 ' ' 45

30 ' 60

'

45 ' 75

'

60 ' 90

'

75 ' ' 105

i i

90 ' '111111 I20

'

'

'

'

Q,,, In’ I tdeg) FIG.

6 (continued)

In Tables I and II we show, in addition to the best overall T+/ST- parameters used in the calculation of the figures, values deduced from fitting the 7r+ and w cross sections separately. In the case of 40Ca the Coulomb correction displaces the position of the first minimum in the right direction but by up to 2 % too much in angle. If we define effective black disk radii individually for n+ and r-, this would imply R+ > R- and could be understood if the proton distribution in 40Ca were spread out a little further than that of the neutron. Hartree-Fock calculations [37] for this nucleus do suggest such a behavior with (rp2)1/z roughly 18 % bigger than (r,2)1/2 and a much increased skin thickness. To make a rough estimate of the consequences of this for pion scattering we have fit the nuclear amplitude at 163 MeV with the optical limit of Glauber theory and, assuming in the resonance region that the effective ?r+p amplitude is three times the r-p, calculated the difference in the 4oCa amplitude to be expected using the neutron-proton density difference of the Orsay group [37]. This

EIKONAL

0

10

20

30

311

COULOMB CORRECTIONS

40 ec,,, In’1 ldeg I

FIG. 6 (continued)

led to a reduction in (TTat this energy of 0.7 % and decreased R, compared to R,, by about 0.5 %. Though in the right direction its significance is marginal. We must also take seriously the possibility of a charge symmetry violation (CSV) at the nucleon level. In a recent measurement [38] of the +d and n-d total cross sections differences of up to 5 % were found after the data was corrected for Coulomb distortion. It appears that the A,, resonance occurs at a lower kinetic energy for 7~bp scattering than n-n so that at 163 MeV the imaginary part of the input parameters in a Glauber model could be 4 “/, stronger for n-1 4oCa scattering than 7~- 40Ca. Such a change would reduce da, and (R, - R,,) by about I o/, and effectively eliminate the final discrepancy. The CSV contribution may be estimated more elegantly if we note that the corrected deuterium data [38] may be very well represented by

ImLL - L+J

m -0.0217T

aTa

Im[.L + f,+J 2

I.

(5.7)

312

GERMOND

AND

WILKIN

IOLL , , , , , , , , , , , , , , , , , , , , , , _ Id1

10'

IO0 _

E&, 104.

0

'

"1 IO

20

0 '

"1

30

In-1 tdegl

20

K)

40 9c,,,

FIG.

'1'

30

40

80 -

'

50

60

70

80

J

ln+Jldegl

6 (continued)

Up to a possible additive real constant, which would be zero if the CSV is associated with this single resonance, the same form must be true for the real part of the amplitude also. We shall now make the ad hoc assumption that the formula is valid for all q and not just in the forward direction

f-(q) - f+(q) w

-0.0217+

a

(5.8)

I

Since in the Glauber model the x(b) are linear functionals of such f’s (neglecting correlations), the CSV results in an extra contribution to the eikonal x,&b)

= f0.0109

+ T&

[kX(E : b)]

(5.9)

where the minus sign refers to TT-. In form this is somewhat similar to the Coulomb energy shift but opposite in sign; at resonance it corresponds to a change of &2 MeV

EIKONAL

COULOMB CORRECTIONS

I

I IO

I M

I 10

I 20

i'

I

I

I

I

30

40

50

60

I 30

I 40

313

I 50

I 60

I 70

I 70

I 60

I 60

II\ 901

I 90

, I IO0 110

I 100

I10

FIG. 7. Percentage difference between elastic x- and in+ cross sections of ‘Wa at (a) 115 MeV and (b) 241 MeV, the data being taken from Ref. [2]. The effect of the Coulomb phase is shown as the long-dashed curve, with the asymptotic trajectory distortion as the dot-dashed curve, including the asymptotic energy shift as the short-dashed curve. The solid curve reflects also the residual distortions. No estimate has been made of the effect of experimental resolution on these predictions.

314

GERMOND AND WILKIN

in kx independent of 2. This is less than the Coulomb shift for 40Ca but similar to that for lzC. This is borne out in the calculations of the total cross-section differences shown in Fig. 4. The overall effect of the CSV term is not large but is comparable to the remaining discrepancies between the theory and data. It is, however, dangerous to trust the detailed values for it because of the reliance upon Eq. (5.8) which has not been tested away from the forward direction.

6. CONCLUSIONS

Our aim in this paper has been to find a pion-nucleus amplitude which is reasonably free from Coulomb distortion but the Coulomb distortion does help us in determining the parameters of this amplitude. The best known example of this is the determination of the real part of the forward amplitude from Coulomb-nuclear interference measurements [6] but we show that the behavior of the n+, rr- cross-section difference in the vicinity of the first diffraction minimum pins down the sign and magnitude of the imaginary part of the first complex zero of the scattering amplitude. Experimental results confirm quite well the phase rule for this quantity which follows from a naive strong absorption mode. There are limitations to our approach. We have not seen fit to calculate the phase correction to higher order in (20~) and not all terms of order 01have been included. The Coulomb distortion has been introduced only in the elastic channel though it is possible to Coulomb excite the nucleus and deexcite it through the strong interactions. This dispersive correction to the elastic TA amplitude will have opposite sign for 7r+ and 7~- scattering but to calculate its size would require a theory which parameterizes correctly all the inelastic transitions! Without a microscopic theory we have to assume that in the absence of Coulomb forces the n* A amplitudes would be identical for isoscalar nuclei even though there is evidence from the total cross sections of deuterium [38] that there is a slight violation of charge symmetry in pion-nucleon scattering. This may be parameterized as a charge dependence of a few MeV in the mass and width of the d resonance. We have made simple estimates using Glauber theory of how this might reflect on the nuclear case and the effect, though small, is not negligible especially for the light nuclei. The deuteron is guaranteed to be charge symmetric but we have to assume the same is true for 12Cand “OCa. The small difference in the neutron and proton distributions predicted by nuclear theory for 40Ca could be partly responsible for some of the residual differences that we find. The 44*4sCaand 13C contaminants in the natural calcium and graphite targets have a similar, though smaller, effect but they may lead to some filling-in of deep diffraction minima. Radiative corrections [39] are small, especially for the cross-section difference, on the 1 % level. We have through Eq. (2.2) solved approximately a Schrodinger equation with relativistic kinematics but it could be argued that we should have more properly introduced the potentials as fourth components of world vectors in a Klein-Gordon equation. Since the strong interaction optical potential that we have used is purely

EIKONAL

COULOMB

CORRECTIONS

315

empirical and energy dependent, the only change this engenders is a term quadratic in the Coulomb potential in Eq. (2.3). This Vc2 term is known to cause theoretical problems, the l/r2 asymptotic behavior getting confused with the centrifugal barrier term generating cuts in the angular momentum plane [40]. Since we are only working to first order in (Zol), we can neglect the problem but this would not be safe for lowenergy scattering from zo8Pb. At each pion energy we have implicitly fit a local optical potential V, to the data so by construction V, will be energy dependent. However, in many of the commonly used optical potentials [41,42] some of the energy dependence of the on-shell amplitudes arises from a momentum dependence of the potential. These two types of potential will be subject to different Coulomb corrections in the inner region. This may not be too important near resonance since the middle of the nucleus is black. We can investigate the periphery explicitly in the case of the Kisslinger [41] momentumdependent potential. By going into the local representation of the Schrodinger equation [43] it can be seen that the Coulomb corrections of order VcVN are exactly the same as those of the corresponding Laplacian local potential. Thus in the lowdensity limit on the edge of the nucleus the momentum-dependent potential feels the same Coulomb shift as the local potential. This argument is not, however, simple to generalize to potentials with a finite-range nonlocality. It has been suggested [16] that Coulomb corrections could be more accurately computed using partial wave rather than impact parameter amplitudes. It is of course easy to project out phase shifts 6, from the parameterized amplitude of Eq. (3.1) but any small error in one of the parameters can induce much larger errors in many of the 6, . The simplest Coulomb correction comes from assuming additivity of the Coulomb and nuclear phase shifts. The asymptotic part, which in the present application is dominant, can be included by displacing only the energy as in Eq. (2.25). This prescription may be derived by approximating [16, 441 the WKB form for the total phase shift. However, even small errors in the estimates of the 6, will show up at large angles because of the cancellations among the Legendre polynomials when many partial waves are important. The method does not therefore avoid the possible limitation of the eikonal approximation to small angles. In fact summing approximations to the WKB phase shifts leads in’b space to the first Wallace correction [12] to the eikonal limit which accounts for some of the curvature in the semiclassical trajectories. It is therefore not surprising that the Karlsruhe [16] program and our own yield similar results. Further refinement of the theory are required if it is deemed necessary to obtain the residual inner corrections to the partial waves. This could for example be done by fitting a square well potential to each phase shift and correcting them to first order in Zol through the Auvil formula [45]. This destroys the simplicity of the scheme. We have shown that the dominant Coulomb corrections to pion elastic scattering from light nuclei can be quite accurately obtained in terms of the physical nuclear amplitudes. In this way we expect to be able to remove the Coulomb distortion from elastic n*-4*Ca data in a model-independent way. In general the corrections will be different for the different calcium isotopes and the 40Ca results are only used to verify 595/121/1-2-21

316

GERMOND AND WILKIN

the method. Of course, the interpretation of the resultant bare n-48Ca amplitudes in terms of neutron and proton distributions requires a more specific scattering theory. For 12C the Coulomb phase is the largest effect in the r+, rr-- difference, whereas for 40Ca the trajectory distortion is very significant. In a deformed black disk model the Coulomb phase induces no n+, n- difference for the excitation of the nuclear surface though the trajectory distortion certainly will. Data for the scattering to the first three levels of 12C show much smaller charge effects than for elastic scattering. We are presently extending our analysis to encompass the Coulomb effects in a DWIA treatment of inelastic transitions. This could prove important because of the suggestion [46] that the excitation of the surface levels with 7~+and n- might be a sensitive test of the relative neutron-proton deformation.

ACKNOWLEDGMENTS We are very grateful to the members of the Neuchltel-GrenobldIN and KarlsruheGrenobleSIN groups for providing us with their results prior to publication and to many valuable discussions concerning their interpretation. Useful comments and correspondence with C. Dover, W. H. Dragoset, P. Huguenin, R. J. Lombard, H. 0. Meyer, H. Pilkuhn, C. S&nit, and B. Zeidman are also gratefully acknowledged.

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