improved theoretical pion-nucleus optical potentials

ANNALS OF PHYSICS 78, 299-339 (1973)

Improved Theoretical Pion-Nucleus Optical Potentials*

R. H. LANDAU,+ S. C. PHATAK, AND F. TABAKIN

Department of Physics, University of Pittsburgh, Pittsburgh, Pennsylvania 15213

Received August 3, 1972

An approach which makes the first order pion-nucleus optical potential theoretically sound is presented. This study should permit higher order improvements to the potential to be more meaningful and the nuclear structure information extracted from pi-nucleus scattering to be more reliable. Based on multiple scattering theory, three optical potentials are constructed and studied in momentum space. These models are the popular Kisslinger potential, the local “Laplacian” potential, and an “improved off-shell potential;” the latter one is derived from absorptive separable pion-nucleon potentials which exactly reproduce on-shell ?TN scattering. By working in momentum space and explicitly including sN resonances and off-shell effects in the definition of the optical potential, the approach described here is capable of handling any number of pi-nucleon partial waves, is applicable over a very wide energy region, is based on a physical model for off-shell behavior, and is extended easily to include higher order effects. The optical potentials are inserted into two different relativistic wave equations to determine the total cross section and elastic differential cross section for pi-nucleus scattering. It is found that the various models for off-shell nN scattering determine significantly different 412 scattering, with the improved off-shell model preferred on theoretical grounds. Also discussed is the importance of properly transforming VN scattering to the pi-nucleus c.m. system, the origin of the shift in the peak position of the n-C total cross section, and the reason for the increased diffractive nature of the differential cross section at 180 MeV.

I. INTRODUCTION

In the near future, extensive and precise data on the scattering of pions from nuclei will be available as a result of the stable, high-flux beams provided by meson factories. At present, we already know that the total and differential cross sections for pions on light nuclei exhibit rather simple features. For example, at intermediate energies [l] the pi-carbon total cross section has only one simple, structureless peak at -140 MeV (pion kinetic energy in the lab). The peak in the pi-nucleon total cross section, in contrast, occurs at the higher energy of 180 MeV and is

* Work supported in part by National Science Foundation. +Present address: Department of Physics, University of British Columbia, Vancouver 8,

British Columbia, Canada.

299 Copyright 0 1973 by Academic Press, Inc. All rights of reproduction in any form reserved.

300 LANDAU, PHATAK, AND TABAKIN

noticeably narrower. Additional simplicity is shown in both the elastic and inelastic differential cross sections where sharp, diffraction-like maxima and minima occur for kinetic energies from 100 to 300 MeV, with maximum diffractive structure at ~180 MeV.

Can basic nuclear structure information, such as the size of neutron and proton distributions, nucleon-nucleon correlations, and nuclear form factors, be extracted from such relatively simple cross-section behaviors ? Clearly, to do so requires a detailed and precise understanding of the reaction mechanism, thereby enabling us to separate out the intrinsic properties of the nucleus from the dynamics of the basic pion-nucleon interaction that is used as a probe. We are aided in this goal by several properties of pion scattering which are distinctly different from corresponding properties of nucleon scattering. First, due to the spinless nature of the pion and its availability in two different charged states, the complete pi-nucleon scattering amplitude is easier to parameterize and to fit to data than is the nucleon- nucleon amplitude. Correspondingly, the ~TN phase shifts are determined accurately over a large energy region in all significant channels-not in just the readily accessible ones as in the NN case. Second, the scattering of pions from nuclei differ from the scattering of nucleons by the existence of TN resonances which make rrN scattering more energy dependent. This should increase the information contained in the energy variation of pi-nucleus scattering and also increase the importance of off-energy-shell scattering.

Many theoretical studies have been made using various descriptions of the pion- nucleus reaction mechanism. Two basic approaches have been adopted. One approach consists of extrapolating the low-energy optical potential of Kisslinger [2], with some adjustments, to the higher energy (Q-g) resonance region [3-61. The other approach consists of using multiple scattering theory directly [7-91. In the cases where the Glauber model is used, its application is simply extended down to the @--8) resonance region.

Unfortunately, by making moderate adjustment of the theoretical parameters, good fits to data have been achieved almost independent of the model adopted. It is unfortunate because a model-independent prediction suggests that only gross nuclear properties (such as nuclear size) and only general features of the pion- nucleon interaction are involved in the reaction. Consequently, we might conclude that little detailed nuclear structure (or pi-nucleon) information can be detected by pi-nucleus scattering. That situation might very well describe the ultimate outcome of further studies. However, before coming to that negative conclusion we believe it is necessary to examine the validity of the various theories in detail and to optimize the reliability of the theoretical input. To begin, we should use a theory valid over a wide energy region rather than simply extrapolating from either low or high energy models. Also, we should be careful to make only physically reasonable approximations and to avoid ad hoc ones.

IMPROVED PION-NUCLEUS POTENTIALS 301

In this paper, we adopt the general Kerman-McManus-Thaler formulation of multiple scattering theory [lo] to discuss the pion-nucleus optical potential. An improved first order optical potential is deduced from pion-nucleon potentials which exactly reproduce on-shell ~TN scattering [ 1 I]. By explicitly including off-shell effects and pion-nucleon resonances in the definition of the optical potential, our approach is made applicable over a very wide energy region. We believe the potential defined here is a considerable theoretical improvement over previous potentials.

Our goal is to make the first order optical potential theoretically sound in the hope that nuclear structure information can be more reliably extracted from data after we include any further modifications to the potential needed to remove discrepancy with the data. Hence, we do not require nor do we adjust our optical potential to fit data, but instead examine and eliminate some ambiguities in the first order theoretical potential.

The definition of the improved optical potential, which explicitly includes high energy pion-nucleon resonant effects, is presented in Section II. The potential is then used in relativistic scattering equations and solved numerically in momentum space (Section III). The total and differential cross sections for elastic rC12 scattering are calculated and the results are presented, compared to other models, and discussed in Section IV and V.

IT. THE OPTICAL POTENTIAL IN MULTIPLE SCATTERING THEORY

The scattering of a pion by a nucleus can be described using multiple scattering theory. The basic picture is that the total scattering is constructed from a sequence of pion collisions with the constituent target nucleons. At intermediate energies many collisions occur, especially in view of the pion-nucleon (Z-N) resonance, and it is convenient to organize the problem by introducing the idea of a theoretical optical potential. The many-body scattering problem is thereby divided into two parts. First, an optical potential is expressed exactly in terms of the basic pi- nucleon collision matrix and nuclear bound state information. Second, the specified theoretical optical potential is inserted into a pi-nucleus wave equation, the solution of which effectively sums an infinite sequence of TN multiple scatterings. Thus, from the exact optical potential and the pi-nucleus wave equation, we can ideally construct the exact predictions of the multiple scattering picture.

Note that the optical potential discussed here is not a phenomenological one simply parameterized and adjusted to fit data. Instead, we speak of a microscopic prediction of pi-nucleus scattering which only employs elementary input.

Fortunately, it often suffices to determine the theoretical optical potential approximately, using only part of the bound state information such as the nuclear


density and pair correlation functions. In this way a many-body problem is reduced to an effective two-body problem and the laborious calculation of corrections to the optical potential is simplified.

A. Theoretical Optical Potential

Let us examine these steps in more detail. The goal is to define an optical potential that can be used in a two-body scattering equation to construct the total pi-nucleus scattering amplitude.

The exact optical potential U is to be inserted in a pi-nucleus scattering equation,

T’ = U + U(1 - Q)(E - K,, - HA + zk)-l T’, (2.1)

whose solution is an exact sum of the multiple scatterings involved in a pi-nucleus collision at the energy E. From T’, we obtain the pi-nucleus collision matrix T = (A/A - 1) T’. (The reader is referred to Kerman, McManus, and Thaler [IO] (KMT) for the basic derivation of the formalism presented here.) HA denotes the nuclear Hamiltonian including nuclear recoil, with all nuclear states understood to be properly antisymmetrized. Kn represents the pion kinetic energy operator and Q = ILo 1 n)(n I is a projection operator for nuclear excited states (0 denotes the ground state). Thus, the intermediate states in (2.1) are restricted by (1 - Q) to be the nuclear ground state only. Consequently, (2.1) is simply a two-body scattering equation for the pi-nucleus system.

The effects of nuclear excitations are incorporated into the definition of the optical potential. The optical potential operator in the meson-nucleus space is shown by KMT to be

U = U(O) + U(O)Q[E - Kv - HA + k-l U, (2.2)

which by virtue of Q, involves only excited nuclear intermediate states. The collision energy E in the pi-nucleus center-of-mass system (c.m.) used here is given by

E = (k,2 +/.9)1/Z + (ko2 + M2)1/2 + E,, , (2.3)

where E, is the nuclear ground state energy (taken to be zero), M is the mass of the nucleus (--am), m is the average nucleon mass, and p is the meson mass, The momenta of the pion and nucleus are E, and -c, , respectively.

The first order optical potential operator is simply

U’O’ = (A - 1) 7(E), (2.4)

where A denotes the number of target nucleons and T(E) is the bound collision matrix for a pi meson scattering from a bound nucleon. The basic interaction between the incident pion and all the nucleons in the nucleus is given by a sum of


pi-nucleon potentials, V = C: vi . As a consequence of nuclear anti-symmetry, hver, it is possible to express T(E) in terms of only the two-body potential v acting between the pion and any one of the target nucleons. U(O), therefore, *t&s (A - 1) T where the bound collision matrix is

T(E) = v + v(E - Km - HA + k-l T(E). (2.5)

Note that T(E) is a many-body operator as a consequence of the appearance of HA in the propagator, which of course represents the effect of nuclear binding.l

If the incident pion energy is sufficiently high compared to the binding of a si rmucleon, it appears reasonable to approximate T(E) using an impulse approximation. Essentially, we neglect nuclear binding or equivalently assumes that the n - N collision is much more rapid than the time scale associated with motion of the nucleon within the nucleus. It is known that caution is required in applying the impulse approximation near a resonance [ 131. However, in the rr -- Iv ease the (Q-8) resonance width (120 MeV) still corresponds to a rapid collision on the time scale associated with nucleon motion in the nucleus. Therefore, we adopt the usual impulse or rapid approximation although it is clear that further sturdy of the validity of this approximation is desirable. Such future studies are perhaps possible using the momentum space results of this paper.

Thus, a collision matrix t(w) is introduced to represent the free rr - N collision via the potential v:

t(w) = v + v(w - K, - KN + k-l t(w), (2.6)

where KN denotes the nucleon kinetic energy operator. The binding of the single nucleon is not included in (2.6). It is easy to show that T(E) and t(w) are exactly related by

T(E) = t(w) + t(w) [ 1 1

E - H.4 - K, + ic - u - KN - K, + ie 1 T(E). (2.7)

The impulse approximation consists of taking T to be the free 7~ - N collision matrix, T(E) N t(w), a step which allows us to make contact with the experimental information available from r - N scattering, i.e., the phase shifts. Note that the matrix elements of the free rr - N collision matrix, (It’, 3’ / t(w)/ E, a), depend on the initial and final pion momenta (c’, R), the nucleon momenta (fi’, p’), and on an energy variable w. Probably the best way to choose w when scattering from a

1 In Watson’s multiple scattering formulation [12], the expression for r(E) includes a projection operator Q, in addition to the one in U, (2.2). Here we use the KMT version in which + is given by (2.5) with unrestricted intermediate states. The choice of 7 alters the expression for higher order terms in U.


bound nucleon would be to use (2.7) to pick an w that makes the impulse approximation valid. Our particular choice for w, called o, , will be given in the next section.

The first order optical potential in the impulse approximation is, therefore, @en by

U(O) = (A - 1) t(OJO). (2.8)

Equation (2.8) expresses the optical potential as an operator in the pi-nucleus c.m. system. Correspondingly, we need to find the free 7r - N collision matrix t(w) in the pi-nucleus c.m. Pi-nucleon scattering information, however, is always given in the 7~ - N cm. system. The required transformation will be discussed later, see Section II D.

B. First Order Potential

Although the free 7~ - N collision matrix t(w) refers only to a pion and a sit& nucleon, this two-body operator is imbedded in a many-nucleon Hilbert space; i.e., the nucleon occurs in a nuclear bound state. We, therefore, introduce meson- nucleus states 1 kn), where k labels the meson’s momentum R (in the pi-nucleus c.m.) and the meson’s isospin (m,); n labels the properly antisymmetrized nuclear state (including recoil energy). In addition, the spin and isospin dependence of the n - N interaction can be expressed in terms of spin and isospin operators, 0,. For example, 0, is given by the usual operators 1, 6 * G, i * i and (+ * i)(G * 7i) where + and i are the single nucleon and meson isospin operators). We isolate the spin and isospin dependence of t(w) by writing

6% i-f I t(w) I It, $1 = c (It’, li’ I t,(w) I It, $1 0, . (2.9) (Y Using that decomposition and meson-nucleus states, the first order optical

potential in momentum space for elastic scattering is

(ii’0 1 U(O) 1 LO) = (A - 1) c j- (rt’, p” j t&J I It, 5) F,(p”, j) dp’ dj’ (2.10) 0:

or

(It’ I Vi! I z> = (A - 1) c j- Cc’, j - 4’ I t&J I k 9) Ex’,(p’ - B, $1 44 (I

(2.11)

where 4’ = p - t% is the three-momentum transferred to the pi meson by the nucleus. The step from (2.10) to (2.11) is a consequence of the momentum- conserving delta functions introduced by the collision matrix elements. The overlap,


Fa,( j’, p’), is defined using the momentum space nuclear ground state wave function, #( a, a2 Y--*7 m:

The matrix elements of the nucleon spin and isospin operators are included in this F, , however, the meson isospin operators in 0, are still present. Hence, F, is an operator in the meson isospin space, and is a function of the nucleon quantum numbers m, and m7 for a single nucleon. We will use that fact later in forming the suitably averaged optical potential.

The evaluation of the overlap functions F, involves a constraint that the initial nucleus has a total momentum of -,& in the pi-nucleus c.m. For light nuclei, proper treatment of the nuclear center-of-mass motion is a difficult matter; that problem is often avoided by releasing (2.12) from such constraints and simply integrating over the nucleon momenta as independent variables. The effect of c.m. motion will be included in the nuclear form factor defined later (2.38).

For the first order potential an additional simplification is to assume that t, can be taken outside of the integral in (2.11) and evaluated at some average nucleon momentum called j&. The optical potential is then a product of free x -- N amplitudes and nuclear form factors pa(q),

(6’ I vi? I R) = (A - 1) c (rt’, i-4 - 4 j t,(wJ I R, PO> par(q). (2.13)

p.(q) is an operator in meson isospin space defined by (2.12) and

pa(q) = j Fdii - 6 ii> 4. (2.14)

For cy = 0, the form factor ~~(4) is recognized as the Fourier transform of the nuclear density.

A reasonable choice of w0 can now be made based on the idea that w, represents the collision energy of the r - N collision as seen in the pi-nucleus c.m. For some value of the nucleon momentum p’, that collision energy is

0, = (ko2 + p2y2 + (p” + m2y2. (2.15)

Later we will make the special assumption that the nucleons are either “frozen” in the target nucleus, and, hence, each nucleon has momentum j = --&,/A (in the pi-nucleus c.m.) or each nucleon moves within the nucleus and, hence, must be averaged over a distribution of p”s.

The factorization of the optical potential (2.1 l), into the form (2.13) is based on the observation that compared to (I?, j?? - 4’ [ t j R, j!), the overlap function Fa( p’ - q, 3) is a sharply peaked function of the nucleon momentum p’ (the


“average” momentum 3 should actually be the peak value). The rapid fall-off of F, is a consequence of the nuclear size being large compared to the range of the n - N interaction. Of course, we should really evaluate (2.11) and correctly integrate over the motion of the nucleon in the nucleus as specified by the overlap function. Actually, the approximation in (2.13) can be improved upon somewhat by backstepping a bit and averaging t,(wJ over just the momentum distribution of nucleons within the nucleus, and then multiplying it by the form factor. This procedure, which would be correct for Gaussian forms of F, , is discussed in more detail in Section II G.

The final expression obtained for the optical potential, (2.13), is seen to be simply products of the r - N collision matrix and nuclear form factors. The form factors measure the ability of the nucleus to absorb momentum 4’ and still remain in the ground state. The collision matrix determines the extent to which the n - N collision can transfer momentum Cj to a single nucleon at a collision energy w, . As a consequence of the 7~ - N transition matrix being complex, the potential, U(O), always has an imaginary part which means that some pi-nucleus collisions result in a loss of flux from the elastic channel, i.e., some recoil nucleons have momenta that cannot be accommodated by the final ground state.2 In the forward direction, E’ = c, the imaginary part of U(O) is proportional to the total n - N cross section; Eq. (2.13) represents an improvement over strictly forward direction estimates for pi-nucleus absorption.

Now let us examine the spin and isospin dependence of the first order optical potential so that it is clear what average over 7r - N amplitudes is required to construct U(O).

The dependence of the nuclear form factors pa on the meson-nucleon operators 0, must first be considered. For 0, = 1, the corresponding form factor PO(q) is simply related to the proton (m, = ++) and neutron (m7 = -4) density, for each direction of nucleon spin (m,):

P~,.~,G) = J e-iaGpp,,,,,(i.) &. (2.16)

After including the 0, = i * i term in (2.13), we find the optical potential is given by a sum over nucleon spin and isospin quantum numbers, for each meson isospin,

<It’, ml I UZ’ I 6, ml>

= (A - 1) C CC’, A - G; mmm, I to + i - % I k ho ; wnflm,) pm,.&j). %.%

The meson isospin label m, is f I for r*, respectively. (2.17)

2 We have not included the actual nuclear absorption of a pion on two nucleons-this is a second order process of the type not included here.


For nuclei with total spin zero, the spin-flip (6 . k) and spin-isospin flip (6 . l;)(i . ?) terms do not contribute to U,, (O). Also, we assume now that the density is independent of nuclear spin and has the same shape for neutrons (n) and protons (p), i.e., we take

Pn = Pnr +pni

= 2Pn,

= W/A) p (2.18)

and

PP = PDT +P,l

= 2P,T

= (Z/A) p, (2.19)

where N is the number of neutrons in the nucleus and Z the number of protons. For 7~* scattering on such nuclei, the optical potential can be expressed in terms of neutron and proton transition matrices,

or in terms of isospin + and $+ z= - N transition matrices [14],

(P, 37+ 1 u($’ / &, 77’)

= (A - 1) p(G) CL’, $0 - 4’ I (2NPA) t,/z + ((N + 3Z)/3A) t,/, I I;, $0:. (2.21)

The rr- scattering case is obtained from (2.21) by interchanging N and Z. We note here that for the interesting case of Cl3 the spin-flip terms would

contribute to the first order optical potential.

C. OR-Energy-Shell Character

The first order optical potential, as defined in momentum space (2.21), is a function of meson momenta, (6’, E) and of some average rr - N collision energy w, (2.15). Note that this equation defines off-energy-shell (1 E’ / # j k’ j # / I& 1) as well as on-shell (1 I$’ 1 = 1 i 1 = 1 k, 1) matrix elements of the optical potential. The collision matrix is off-shell since the kinematics for scattering from bound and free nucleons differ and for dynamical reasons.

There are important reasons for needing to know the off-shell matrix elements


of U(O). First, determination of the correction to the impulse approximation (2.7) and the second order term of the optical potential expansion (2.2) require knowing U(O) off-shell. For example, the evaluation of d U = U - U(O),

(6’0 j AU [ ZO) = (E’O &4 - I)-’ U’O’[(E - HA - K= + k)-l

- (E - K,, - KN + i+‘] U’O’

+ U’O’Q[E - HA - K,, - QU’O’Q + ie]-l U’O’ 1 ffO>, (2.22)

requires U(O) off-shell. Feshbach [15] and others [16] have discussed methods for treating these corrections, which are important since they depend not only on the nuclear density but also on the dynamical correlations between bound nucleon pairs. It is clear from (2.22), however, that knowledge of U(O) both on and off- energy-shell is also required before reliable evaluation of d U is possible. One of the aims of our work is to provide a better U(O) to be used in (2.22) for future work.

Second, the off-shell properties of U are also required for the potential term of a Lippmann-Schwinger integral equation,

Solution of (2.23) requires that (E’O 1 U I &O) be stipulated not only on-shell but also off-shell. However, in order to determine the off-shell pi-nucleus potential we must know the off-shell TT - N collision matrix. A major concern of this work, consequently, is to develop a first order optical potential which has physically reasonable off-shell behavior so that it can be used reliably in (2.22) and (2.23). Perhaps then, nuclear structure information can be more reliably extracted by analysis of experi- ments, expecially when using the first order potential.

D. Transformation of n - N Collision Matrix

In our expression for the pi-nucleus optical potential (2.21), the collision matrix t(q) describes the T - N collision as seen in the pi-nucleus c.m. In this section we express that collision matrix in terms of the usual collision matrix defined in the pi-nucleon c.m., f(&,), which can be more directly related to phase shifts. However, the transformation from the T -N to the pi-nucleus frames is ambiguous. For the corresponding scattering amplitudes (i.e., the on-energy-shell collision matrices) there is no difficulty, but we require the transformation suitable for the off-shell collision matrices. In lieu of a complete theory of meson-nucleon dynamics, it is extremely difficult to define a consistent means of transforming from one frame to another for off-shell collision matrices, We resolve that difficulty by basing our transformation on those used for the on-shell amplitudes and simply applying them to the off-shell transition matrices.


The transformations of the collision matrix from the r - N to the pi-nucleus c.m. systems are of the form

Here i? and i refer to the meson momenta in the pi-nucleus c.m., while I?’ and li denote the relative momenta in the pi-nucleon c.m. system. If for the transformation, we assume the nucleon is initially “frozen” in the moving target nucleus, we would have jO = --&,/A, where E, is the actual (on-shell) pi-nucleus c.m. momentum. For on-shell r - N scattering the energy variable wO would then be the corresponding n - N collision energy

where

w - -s&J + ML/4 o- (2.25)

J%(P) = (P” + p2Y!“,

E,(p) = (p” + rny2.

The corresponding on-shell collision energy GO and on-shell momentum K, in the n - N c.m. are determined using the invariance of the four-vector product s = (P, + PN)” (or equivalently a Lorentz transformation):

-2 cog =s

= [&r(‘d + ~N(‘%)]~ (2.26) = [E&J + E,(k,/A)]2 - k,2( 1 - l/,4)2.

We choose, as our model, to relate the off-energy-shell momenta K’ and K in the rr - N cm. to their corresponding off-shell momenta in the rr-nucleus c.m., k’ and k, by these same relations (2.25) and (2.26).

For on-shell scattering, Lorentz invariance of probability [13] and again the approximation of the nucleon being frozen in the nucleus, determine y to be

The corresponding relation for the off-shell momenta is

E,(K) EJK’) EN(~) EN(K’)

I’ = [ E,,(k) E&t’) E,(k/A) E,(k’/A) 1 1,‘2

. (2.28)

In addition to the factor y and the expression (2.26) used to get K, K’ and K, ,

the scattering angles in the pi-nucleus c.m. (k . &’ = cos O,,,) and the n - N c.m.


(I? * R = cos e,,,) are to be related. For on-shell scattering, the invariance of the four vector product I = (Ppitial - Piina1)2 determines the relation n

cos enN = tco2 - ko2

KO 2 + $ cos o,, . (2.29)

A corresponding relation for off-shell scattering is

cos e&J = E,(K) &(K’) - E,,(k) &(k’) I kk’ cos o

KK’ KK’ nil * (2.30)

It is clear that the transformations are not unique. but we believe (2.24)-(2.30) are all reasonable choices.+ Actually, as a result of computation, we have seen that the freedom is choosing y ((2.27), (2.28)) and the angles ((2.29), (2.30)) have very little effect on the elastic scattering cross sections. Therefore, from a practical viewpoint any combination of these are essentially equivalent.

We have found, however, that it is very important to make (some) proper transformation of the angles from the 7~ - N to the pi-nucleus frames. For example, the change in angles given by either (2.30) or (2.29) has a large effect on the differential pi-nucleus cross section especially at wide angles. When the partial wave expansion of t (nucleus frame) is expressed in terms of that of tl (nucleon frame) the main effect of the angle transformation is a mixing of 1 values,

P,(cos d,,,) = P,(a + b cos fl,,,)

= c dzz~P&os en,), Z’<Z

(2.31)

where the coefficients d,,, in general are easily found, and are given in Table I for the first four waves and the transformation (2.30). As a result of (2.31), the P-wave

TABLE I

Transformation Coefficients drr, for Legendre Polynomials

\

I’ 0 1 2 3 I

0 1 0 0 0 1 2 (3a* + ; - 1)/2

6 0 0 3ab

(5ub2 + 5d - 3u)/2 (3b8 - 36 + 15u2b)/2 5:Lz 0

3 b3

See Eq. (2.31) for definition of a and b.

+ These relations in effect define an analytic continuation of cos Bn~ into an unphysical region. This unphysical region appears to be an unavoidable consequence of the kinematics for scattering from bound and free nucleons differing.


resonance in the 7r - N c.m. contributes to the I’ = 0, 1 waves of the TI - N collision matrix t(w) when transformed to the pi-nucleus c.m. Not all authors have included this important effect in their calculations; some have simply set d,,, = 6,,, in (2.31) [3-51.

An objection to the use of this type of transformation has recently been raised by Faldt [ 171 on the grounds that it can lead to what appears to be unphysical parts to the optical potential (in the coordinate space representation) which create rather than absorb pions. We feel this objection is not valid for the nonlocal potential determined by (2.21). These optical potentials by their nonlocal nature, conserve flux on a global scale and not on a local scale. That is, for a nonlocal potential the appropriate continuity equation tells us that it is proper for flux to be created in one region of space, absorbed in another, without upsetting total balance. Furthermore, we verify in our calculations that no model violates unitarity in any partial wave, and, hence, all models considered here correctly yield absorption and no creation of mesons.

E. Partial Wave Decomposition

To solve for the pi-nucleus scattering amplitude by means of the integral equation (2.23), it is necessary to make a partial wave decomposition of the optical potential and scattering amplitude. The optical potential is built from the rr -- N collision matrix t(qJ and from the nuclear form factor p(q). The collision matrix t(q) is deduced from tl(&,) using the previously described transformations, (2.24)- (2.31); whereas, I(G,) is to be calculated from knowledge of the tabulated 7~ - N phase shifts in each eigenchannel. Therefore, to decompose U(O), we need to expand both f(G&,) and the form factor p(q).

The partial wave decomposition of the spin-averaged collision matrix i is given in the rr - N cm. system for each meson isospin state (labeled by I) as

(2’ 1 iI 1 2) = c (j f +)(K’ / $j(i&) i K) P,(t’ a c). (2.32) Z.i=htt

For on-energy-shell scattering i& is related to the rr - N phase shift 6 and absorption parameter 7 in each eigenchannel (I, I, j) in the usual way [14]:

For off-energy-shell scattering different models are used to relate (K’ 1 f;(G)1 K’)

to (K, / fL(C&)i K,); these are discussed in the next section. The 14 TNeigenchannels corresponding to S, P, D, and F wave scattering are included in our 2.

The nuclear form factor PO(q) can also be expanded in partial waves:

p,(&’ - 2) = ; p#’ I k) Pl(L’ . A). (2.34) l-0


This expansion can be combined with the one for t(w& using (2.21) and (2.24-2.31), to form the partial wave expansion of the optical potential in the pi-nucleus c.m.

<k’ I U:’ I 6 = $ c (2L + 1) (k’ I U,(q) I k) P,(F;’ - ii). (2.35) L

Here U, is a combination of the I = 4 and 3 meson-isospin contributions,

(2.36) This expression for the optical potential in the Lth pi-nucleus partial wave includes the effects of transformation and of combining the collision matrix with the form factor. Note that the nuclear form factor mixes all the n - N partial waves into each pi-nucleus L value. Thus, the (g-8) 7r - N resonance will contribute to, and in fact dominates, all relative partial waves of pi-nucleus scattering. The angle transformation factor d,,~ enters (2.36) as an additional mixing effect.

To complete the definition of the first order optical potential, the explicit density p(r) and the off-shell 7~ - N collision matrix must still be chosen. The density p(r) is chosen to be of the form used to parameterize electron scattering [I 81,

p(r) = POW + ~WGYI exp[--W,J21. (2.37)

The corresponding form factor is

p(4) = [l - 4wmW(2 + 341 expHwJW1,

where 01 = (A - 4)/6.

(2.38)

In using (2.37) and (2.38) we choose u,, and a& so as to include the effect of nuclear center-of-mass motion but not to include the proton’s finite size. The proton’s spatial extension is (conceivably) included in the experimental rrN amplitude we use as input to our theory.3 The a’s fitted by electron scattering are related by [18]

where up2 = 2/3(r2), (E 213 (0.76F)2),

a cm = 1.66F.

(2.39)

8 We thank M. Stemheim for bringing this point to our attention.


After removing the proton’s contribution we obtain

a ch = 1.59F.

The density is normalized such that p(q = 0) = 1. The expansion of p(q) using (2.34) yields

where

pz(k’ / k) = (exp(-x)/(2 + 301))[(2 + 3a - 2aoc)(21 t 1) h(z)

+ 24 L(z) + (I + 1) &+,c4)1,

z = +kk’a:h ,

x = a(k” + k’2) a:h ,

(2.40)

and &(z) is a modified spherical Bessel function of the first kind [19].

F. Models of the TT - N Collision Matrix

A basic ingredient of the pi-nucleus optical potential is the off-energy-shell TN collision matrix. This is clearly shown in (2.13). Unfortunately, without a complete dynamical theory of the rrN interaction the off-shell behavior is unknown and has to be hypothesized, with each choice of this behavior leading to a different pi- nucleus optical potential. Kisslinger [2] originally generalized the low energy expansion of the on-shell transition matrix, to take it off-shell, and used the form,

(2’ 1 f(cG,)/ I?) = a(&) + b(li&) rZ’ f d, (2.41)

where a(&) describes on-shell rrN S-wave scattering and Ko2b(&), the P waves. When combined with the nuclear form factor, this f gives the familiar nonlocal coordinate space potential

V(F) = (27r)3[a’(S,) p(F) - b’(GO)V . (p(r) B)], (2.42)

where a’ and b’ are related to a and b by the angle transformation (2.29). The nonlocal 9 * pv P-wave term leads to a surface peaking.

Studying the form (2.41), we note several shortcomings in the use of the potential (2.42) at high energies. First, only S and P TN waves are included; second, there is no explicit resonant behavior in f as a function of K or K’. The basic reason for these shortcomings is that the off-shell behavior in the variables K and K’ does not reflect the physical occurence of resonances, or other known properties, in the Z-N scattering amplitude.

If the TN collision matrix is hypothesized to depend only on momentum transfer q2 = (12’ - I?)~ as

(2’ I &&)I 4 = a@,> + bG,) q2, (2.43)

595/78/2-2


a local potential of the form

V(F) = (243[a”(B,) p(F) - b”V”pQ)] (2.44)

is obtained. Here the Laplacian term gives a surface peaking with (a + 2K02b)

describing on-shell TN&wave scattering and --2~,~b, P wave. Even though a local potential is generally more appealing than a nonlocal one, we see on theoretical grounds that especially for pions the optical potential should be nonlocal. In our momentum space calculations we actually generalize the form (2.43) to include D and F-wave TN scattering

(K’ 1 f(&)l 2) = a(&) + b@,) q2 + C(k) q4 + d(&,) q6. (2.45)

Since this T matrix yields an optical potential of the form (V2), p(u) it will often be referred to as a Laplacian. In all cases, the coefficients, a, b, c, and dare determined by the experimental phase shifts and absorption parameters so that on-shell all forms agree with the scattering amplitude (2.33).

We see that the transition matrices (2.43) or (2.45) which yield a Laplacian potential possess many of the same (undesirable) features as Kisslinger’s hypothesis, (2.43). First, there is no explicit resonant behavior in tl as a function of Z or 2’. For the intermediate energies considered here, the nN amplitude is too energy- dependent to be represented well by a function of only q2 and correspondingly a, b, c, and d must vary rapidly with energy. Furthermore, this form of the T matrix diverges quadratically as r7 or ri’ go far off-shell- it was designed to give a local potential and does not manifestly generate the off-shell behavior expected from a more physical model. A more physical collision matrix would also include resonant effects in its off-shell K and K’ dependence and, thus, provide a natural cutoff of any divergence.

The third pi-nucleus optical potential studied here is based on a simple physical model of the ~TN collision matrix. The model consists of a complex separable potential, in each eigenchannel (I, I, j), which exactly reproduces the tabulated on-shell rrN scattering and has asymptotic Regge behavior [l I]. From the tabulated phase shifts and absorption parameters [20], we have previously constructed such forms by solving the inverse scattering problem. The advantage of this collision matrix is threefold. First, it has a dependence on the ri and 2’ variables which reflects the dynamics of the physical (g-Q) resonance or more generally of all the S, P, D, and F waves; second, it has a reasonable fall off for very large rZ and I? (it goes to zero); and, finally, it has the analyticity properties appropriate to an actual Schriidinger equation. The separable TN transition matrix is given for each eigenchannel n = (Z, 1, j) by [l l]


-10 I I1 1 / I,, I / I I / I I / 0 500 1000 1500 2000 2500

P c.m. (MeV/c )

FIG. 1. A pi-nucleon (TN) absorptive separable potential g(p) constructed from complex TN phase shifts. Upper Graph: the absorption parameter 7 and phase shift 6 as a function of TN c.m. momentum for the P33 channel. Lower Graph: the real (Reg) and imaginary (Img) parts of the separable potential function calculated from these r)‘s and 6’s. N.B. the momentum scale is greatly compressed. Details of the calculation and other potentials are given in Ref. [ll].

The functions g,(K) are constructed from the complex phase shifts. A typical form of these phase shifts and the real and imaginary part of the corresponding g, is shown in Fig. 1 for the 1 = 1, I = Q, j = $ channel; the potentials for other channels are in Ref. [ll]. We assert that this procedure for obtaining f offers an improved, physically motivated, albeit nonunique, means of defining the off-shell dependence on the K’ K variables. We have also constructed optical potentials with completely energy-independent forms of this collision matrix,

and (K’ 1 f(b)/ K) = &(K’) &(K)/DT[ij(K’)]. (2.48)

The results were almost identical. In Fig. 2 we compare the half off-energy-shell (K’ # K = Ko) P-wave collision

matrices predicted by the Kisslinger, Laplacian, and separable forms. The on-shell momentum Kg is 100 MeV/c. At low momenta (near shell) all three agree. However, at higher momenta the Kisslinger form for the transition matrix diverges linearly;


LAPLACIAN

KISSLINGER

SEPARABLE _-----___

uo 200 400 600 600

T-N OFF-SHELL MOMENTUM K (MeV/c)

FIG. 2. Three models for the half off-energy-shell (x’ # K = KJ P-wave, pi-nucleon collision matrix in aN c.m. When used in the construction of a pi-nucleus optical potential, the solid curve generates the Kisslinger potential; the long-dashed curve generates a Laplacian (local) potential; and the short-dashed curve (constructed from complex separable potentials) generates the improved off-shell potential. Note, the separable curve eventually falls smoothly to zero.

whereas, the Laplacian form diverges quadratically. The T matrix calculated with the separable potential, however, exhibits structure at intermediate momenta, which is related to TN resonances and thresholds; whereas, at very high momenta it decays smoothly to zero.

Of the three collision matrices defined here, one is preferred on the basis of having proper off-shell behavior. The question now is: does it really matter how we define the r - N collision matrix when it is used only in the construction of the pi-nucleus optical potential. The usual claim is that the form factor p(q) in (2.13) is sharply peaked compared to the collision matrix and consequently damps out off-shell K # K’ effects. (p(q) is sharply peaked relative to t because the nuclear size is appreciably larger than the n - N interaction range.) Offhand, it seems difficult to anticipate the effect of even slight differences in the off-shell potential since the Born approximation is not valid and it is necessary to use U” in a relativistic wave equation to calculate the net effect on the scattering. Clearly, much of the off-shell differences between the models based on (2.41), (2.43), and (2.46) will be damped by the p(q) factor and it is not clear if what remains will be significant.

Actually, there are very significant differences between the Kisslinger optical potential and other models which becomes most evident in momentum space: the Kisslinger potential is singular as K = K’ + co, the others are not.4 This is a

4 There are also related singularities in the coordinate space Kisslinger potential, see e.g. Ref. [21].


consequence of the exponential term in the form factor (2.38), exp[-(2 - 2)” a2/4] only damping out the large K and K’ behavior of the collision matrix when ri’ i 2. The Laplacian potential, fortuitously, vanishes in the r7’ = ri -+ cc limit since its assumed off-shell collision matrix vanishes then.

G. Fermi Motion

So far we have considered the nucleons to be “frozen” in the target nucleus. An approximate way of correcting for this was noted in Section II B and is to use a collision matrix f which has been initially averaged over the probability of finding a nucleon with momentum 9 in the nuclear ground state. The probability is given by the nuclear overlap function FO( p’, a), (2.12). The basic assumption used is

F&p - CA f> = Fo(F, i9 p(q)> (2.49)

which is valid for Gaussian forms of F0 (within the integral in (2.11)) but is surprisingly successful for other forms [22]. This procedure permits us to maintain the factored form of the optical potential (2.13) except now the collision matrix has been “folded” into a momentum distribution.

Specifically, our previous on-shell i matrices are simply replaced by the folded ones,

(Ko 1 fnn(&> 1 ‘d -+ J di F&L $)(‘%[ijl 1 ~nn[~o(P)l 1 ‘%[~I)~

Ko( p’), C&(3) and ij are related through the invariant s,

(2.50)

cigji)2 = s

= p2 + m2 + 2E,,(k,) E,(p) - 2R, * p’ (2.51)

= (-%(Ko) + EN(‘%))~~ (2.52)

Note that (2.50) is a two-dimensional integration over I p’ j and the angle & .$. We have used the shell model value for F,(p,p), which is appropriate to the

nuclear density employed (2.37),

= & [l + y (~4~1 exp[-W21, (2.53)

with a = 1.66F. This method of including Fermi motion produces averaged values of the input coefficients a(&), b(&), etc. in (2.41)-(2.45); the other steps used to define the optical potential are unaltered. The procedure we adopt is simple and only indicates the possible significance of including internal nucleon motion.


With this procedure for Fermi averaging (also called folding or smearing) we complete the definition of Uj,“’ and can proceed to determine the full pi-nucleus elastic scattering amplitude.

III. CALCULATION OF PI-NUCLEUS SCATTERING AMPLITUDE

To calculate the pi-nucleus c.m. scattering amplitude with various definitions of the pi-nucleus optical potential, the integral equation (2.23) must be solved numerically. For numerical solution it is convenient to adopt standing wave boundary conditions (or principal value integrations) and to introduce the decomposition of the optical potential into partial waves, (2.35) and (2.36). Then the integral equation to be solved, in the pi-nucleus c.m. is

The intermediate pi-nucleus energy E(k) is defined relativistically,

E(k) = E,(k) + E,(k),

where

E,,(k) = (k2 + $)1/Z,

E,(k) = (k2 + M2)li2. (3.la)

The usual nonrelativistic Schriidinger equation is simply (3.1) with energies defined nonrelativistically

E,(k) = WtL,

E,(k) = k2/2M. (3.lb)

The on-shell values are k’ = k, and E = E(k,). The on-shell R-matrix yields both the real and imaginary parts of the pi-nucleus

phase shifts (the convention for pi-nucleus partial waves differ by 2.rr2 from the convention for TN waves (2.33))

tan UM = --2kA&) UWE(kJ x (k, I WWdl W. (3.2)

Here RL and, hence, the phase shifts 6, are complex since there is absorption in pi-nucleus scattering. From these complex phase shifts the pi-nucleus c.m.


scattering amplitude is constructed by summing the contributions from the various partial waves,

f(O) = (A//j - 1) c (2L + 1) exp[is~~ko;~osin *L(ko) PL(d’ . rz). (3.3) L

The differential cross section for elastic pi-nucleus scattering is then

-g- (9 = I.m)l’,

and the elastic and total cross sections are given by

= $ go (2L + I) I exp(2iSL) - 1 i2,

(3.4)

(3.5)

oTOT(ko) = p Imf(B = 0) 0

= uEL -t %NEL

= -f$ T (2L + l)(l - Re exp(2iSL)). (3.6)

To solve (3.1) for the complex n-nucleus phase shift we use a simple numerical procedure to evaluate the principal value integral and ignore the contribution of the Coulomb potential to U, . Consequently, the integral equation and the optical potential should be well-behaved in momentum space and the matrix inversion method described by Haftel and Tabakin [23] can be applied. The only modification needed is to treat complex potentials, and consists of dividing both R, and U, into their real and imaginary parts and writing (3.1) as a set of coupIed equations,

(k’ I kR I kJ = (k’ I UL~ I k,)

+$pj m dk k2[(k’ I UL~ I Wk I hR I k,) - (k’ I u: I k)(k I RL’ I Ml 0 E - E(k)

(3.;a)

(k’ I R; I kJ = <k’ I U,’ I kJ

+;pj m dk k%k’ I UL~ I kW I RL’ I kJ + <k’ ; U,’ I kW I JbR I Ml 0 E - E(k)

(3.;b)


Equation (3.7) can be converted to matrix equations and solved numerically using the coupled channels case given in Ref. [23]. Treatment of the pi-nucleus scattering problem in momentum space provides the great flexibility that is required to resolve many of the questions raised in Section II. Different kinds of off-shell behavior can be treated, any number of ?TN partial waves can be included, and various kinematic and dynamic effects are easy to study. The price paid for that flexibility is to ignore the Coulomb force, which can be included at least approximately in future work.

Equation (3.1) represents only one possible choice of the relativistic pi-nucleus scattering equation. Another possibility, which is more often used [3], is an approximate version of the Klein-Gordon equation. The approximations used follow from the steps.

(-Vz + /F’) $ = (E - U)2 y5 N (E2 - 2EU) #, (3.8)

where ,Z = PM/& + M) =p, is the pi-nucleus reduced mass. Written as an integral equation with standing wave bound condition, (3.8) leads us to an R-matrix equation of the form (3.1), with the replacements E(k) -+ k2/2E and E -+ k,,z/2E. With these replacements, solving (3.1) is equivalent to solving the approximate Klein-Gordon equation (3.8).

Direct application of the matrix inversion technique in momentum space with the Kisslinger potential, (2.41) and (2.42), leads to unstable results. The instability is a consequence of the singular nature of the Kisslinger optical potential as discussed in Section II F. Specifically, in each partial wave of (3.7) the potential U,(k ] k) approaches a nonzero constant as k -+ co; whereas, our matrix inversion technique requires this matrix element to vanish. We can introduce arbitrary cutoffs to the upper limit of the momentum integral but then the numerical results are quite sensitive to the choice of cutoff. Furthermore, this introduction of an arbitrary cut-off means that we are no longer treating the original Kisslinger potential. Dedonder [5] has introduced such cutoffs in his momentum space study of pi-nucleus scattering and his results indeed do deviate from those found for the Kisslinger potential in coordinate space.

Fortunately, we had Sternheim and Auerbach’s code Abacus-M available5 in order to correctly treat the singular Kisslinger potential in coordinate space. As a check, we reproduced Sternheim and Auerbach’s published results for the Kisslinger potential [3], and furthermore found that Abacus-M and our matrix inversion program yielded the same numerical results for the Laplacian potential (2.43). In using Abacus-M we turned off the Coulomb force and used input rrN parameters equivalent to those used in our matrix inversion program.

6 We thank M. Sternheim, L. Kisslinger, R. Eisenstein and J. Penkrot for assistance with Abacus.

A.

Angular Momentum 1

Im Tl

T-C, LAPLACIAN

JO.. I..” . . . . . ...’

.g’: - -,- - -1 : s+

- ,“‘i

i

G

: a, \

:.. ‘\ I

‘\ ‘I

0.2 2

3.6

A?=3

\ /

\ ,124

I I2cJMeV 2 - 150 3 - 160 4’. 200 5-230 6 - 260 7 -280 8 -300 9 - 400

IO-500

B.

Re Ta

FIG. 3. A. Partial wave decomposition of o ror. The contribution of different pi-nucleus partial waves to the total n-C cross section. The arrow indicates the nuclear surface in the semi- classical picture (kR = 2 + l/2). The cross sections (calculated with improved optical potential) are multiplied by ke/4n at each pion lab. energy in order to separate the curves. B. Argand plot for the first five pi-nucleus partial wave amplitudes, Tr = (7 exp(2iS)-1)/2i, calculated with a Iaplacian potential. The Re TZ is the abscissa and the Im Tr is the ordinate. Scale marks are indicated for the ten energies listed; at zero energy all curves converge onto the unitarity circle (dashed curve at bottom).


The optical potentials obtained using the local (2.45) or separable (2.47) forms for the pi-nucleon interaction are nonsingular and were, therefore, solved in momentum space by matrix inversion of (3.7). These potentials are quite smooth and long-ranged in coordinate space (range ~2.5F). Consequently, only a few grid points were needed to calculate phase shifts with sufficient precision for the partial wave sum to be accurate; e.g., 16-24 Gauss points yielded phase shifts accurate in the fourth place.

The number of partial waves that contribute significantly to pi-nucleus scattering increases with collision energy. The maximum contribution is found to come from the nuclear surface region, corresponding to L N k,R, where R is the nuclear radius (~2.58’). In Fig. 3A, the contribution of each z--nucleus partial wave to the total cross section is displayed as a function of L and of collision energy. Additional insight into the pi-nucleus interaction is provided by the pi-nucleus phase shifts which are given in Fig. 3B in the form of an Argand diagram 1201. It is clear from Fig. 3, that the interaction peaks in the nuclear surface region and is highly absorptive in the nuclear interior.

In our calculation, the input TN scattering amplitudes are determined by inter- polating upon the tabulated complex rrN phase shifts (CERN Theoretical Fit [20]). The phase shifts below those tabulated values are calculated using the scattering lengths given in Table I of Ref. [14], and beyond the tabulated values we have determined the complex phase shifts from Regge pole fits to higher energy rrN scattering. This latter technique is discussed in Ref. [ll]. In all cases these amplitudes are then folded over the momentum distribution of nucleons within the nucleus, employing the procedure outlines in Section II G.

We now use the procedures discussed here to see how sensitive pi-nucleus scattering is to the form of the optical potential. We use various off-shell definitions for the pi-nucleon interaction, along with various mappings to the pi-nucleus c.m. frame, various versions of the scattering equations, etc., and determine their influence on the predictions of multiple scattering theory.

IV. RESULTS

In addition to using experimental input obtained from pion-nucleon and electron-nucleus scattering data, our theoretical input consists of using various off-shell models to define the first order optical potential. Solving the pion-nucleus scattering equation is then a means of summing higher orders of multiple scattering, with the single scattering contribution given simply by the first order optical potential itself. Our results consist of the pion-nucleus differential and total cross sections calculated in momentum space according to the procedures described in Section III. The purpose in making these calculations is to see how sensitive pi-


nucleus scattering is to the particular theoretical choices made in defining the first order optical potential. Before presentation of our results, we emphasize that no attempt is made to fit the pi-nucleus cross sections by adjusting free parameters. In line with this view, we do not make extensive comparisons with the data, but employ ~T-C elastic scattering data of Binon et al. [I] primarily to indicate where future theoretical improvements are required. Hopefully, development of a theoretically improved first order optical potential will ultimately permit us to correctly include higher order corrections and then to produce not only the desired agreement with experiment but also some basic nuclear structure information.

A. Importance of T-matrix Transformation

We begin by observing the effect of using the “proper” transformation (2.29)- (2.31) of the pi-nucleon collision matrix from the pi-nucleon to pi-nucleus cm. systems. As discussed in Section II D the pi-nucleon partial waves get mixed in the transformation; for example, the P33 resonance maps into both the S and P wave parts of the T-matrix when viewed in the nucleus c.m. system. At the energies considered here (pion lab kinetic energies of 100-300 MeV), this mapping has a significant and quite sizeable effect on the optical potential and elastic scattering of pi’s on carbon.

In Fig. 4A the angular distribution of 120 MeV pions scattered from C, calculated with the “standard” Kisslinger model (i.e., & = S,,, in (2.31)), is compared to that predicted by a “modified” Kisslinger model (i.e., including the proper mapping defined by (2.29)). In Fig. 4B there is a similar comparison for scattering at 180 MeV, for a model derived from separable n - N potentials. Use of the proper mapping increases back-angle scattering by an order of magnitude and consequently increases agreement with large-angle data. This trend is observed at all energies with all models, although it is usually most pronounced with the Kisslinger model. We note, however, that this “proper” mapping shifts the position of the second peak to smaller angles, thereby causing data near this one region to be matched more successfully in some cases with the “standard” mapping. Nevertheless, we believe on theoretical grounds that a transformation which mixes the partial waves is physically correct and, hence, proceed to present results only for calculations employing the correct transformation defined by Eqs. (2.29) and (2.31). By including the proper transformation, our Kisslinger and Laplacian models are modified versions of the ones used by other workers [3-51.

It has been demonstrated (Fig. 4) that it is important to include the effect of transforming from the pi-nucleon to pi-nucleus c.m. system. However, our calculations showed that the various ways of defining the transformation ((2.27) or (2.28), (2.29) or (2.30)) lead to essentially identical results. We conclude, therefore, that it is not crucial to resolve the off-shell ambiguities involved in defining the

El

6 c.m.

FIG. 4. A comparison of the effect on pi-nucleus scattering of two different ways of transforming the rN scattering amplitudes from the pi-nucleon to pi-nucleus c.m. systems. One is the “proper” transformation which mixes partial waves, and the other is the standard & = &r,, or l:l, mapping. A. Angular distribution of 120 MeV pions elastically scattered from carbon, calculated with Kisslinger model. Modified Kisslinger model, -.-.--; standard Kisslinger model, - - - - -. B. Similar comparison at 180 MeV calculated with an optical potential derived from separable nN potentials: proper mapping, -; 1:l mapping, ----.


proper mapping, but it is very important to include this basic transformation effect in some reasonable way.

B. Comparison of da/d&J for Various Models

A priori, it is not obvious that theoretical optical models which differ only in the off-energy-shell behavior assumed for their underlying pi-nucleon collision matrix should predict different pi-nucleus scattering. Since the nuclear interior appears rather absorptive to a pion, most of the interaction occurs at the nuclear surface; consequently, the fine details of the pi-nucleon interaction may be masked. (In Fig. 3A, we have already seen that the major contribution to pi-nucleus scattering is indeed from angular momenta which correspond to grazing collisions. Also in Fig. 3B, the absorptive nature of the nuclear interior is seen for the corresponding low partial waves.)

Another reason for expecting that the off-shell pi-nucleon matrix elements will not play a significant role in pi-nucleus scattering, is based on the nuclear size being considerably larger than the pi-nucleon interaction range, As a consequence, the nuclear form factor will be sharply peaked relative to the pi-nucleon collision matrix in the optical potential, (2.21), and the momenta variables will be kept effectively on-shell.

Despite these apparently sound physical arguments, the off-shell choice made for the pi-nucleon collision matrix in fact does effect the pi-nucleus scattering in a significant, albeit not dominant, fashion. We have found that the first few pi-nucleus partial waves are very dependent on the optical model used and intermediate partial waves proved to be slightly model-dependent. The complete pi-nucleus scattering, constructed from all partial waves, is subsequently found to be significantly dependent on the choice made for the underlying pi-nucleon interaction. Apparently, since the scattered pions only interact in the small surface region (and correspondingly for short times) they may get far off their energy shell.

These observations are based on comparing our results found using the Kisslinger, Laplacian and improved off-shell optical potentials. For convenience, we refer to the optical potential obtained using the separable pi-nucleon collision matrix as the improved off-shell optical potential. Note that it is only the pi- nucleon interaction that is separable; the optical potential involves the nuclear form factor and is consequently not separable. In making these comparisons, the same values for the pi-nucleon phase shifts are used and the Coulomb force is ignored in all cases. Although the Kisslinger cases include only S and P waves, the other cases include higher (D and F) waves, whose effect will be discussed later. All cases shown here also include the Fermi motion of the nucleons as discussed in Section II G.

Figures 5 and 6 show a comparison of the pure nuclear differential cross sections predicted by the Kisslinger, Laplacian (local) and improved off-shell optical


potentials. The experimental data of Binon et al. [l] is also shown; Coulomb scattering contributes significantly only to the very small angle scattering at these high energies. The empirical input consists solely of the experimental pi-nucleon phase shifts and the nuclear density (2.37), as measured by electron scattering.

We see (Fig. 5) that even after modification of the Kisslinger model to include the proper transformation of partial waves, the Kisslinger model does not predict as much large angle scattering as is predicted by the improved off-shell optical

0 60” 120’ 180’ 0 60’ I260 180”

8 c.m.

FIG. 5. Differential cross sections in cm. system of pions elastically scattered from carbon as calculated with the modified-Kisslinger potential and the improved off-shell (derived from separable TN) potentials. The indicated energies are pion kinetic energies in the lab, and the data are from Ref. [l]. The input parameters are from electron-nucleus and pi-nucleon scattering with the latter folded into a distribution of Fermi momenta. Derived from separable ?rN, -; modified Kisslinger model, -. -. - .


IO” c

* IO2 z

1

IO3

IO _ \ 120MeV 2 \ F IO :

,d ‘\

, o \

[

:,

\ $6 , h * ’ 2 t

‘t IO-? :

L \ 1, 260

IO - \ ‘\

it4 I.,.

‘. .- f

10’ +

‘I 1-

280

102- ‘\

I I ' 'L]. .

d I I .I I I I '\A 1-I I I" 0 60’ 120’ 180” 0 60’ 120’ 180’

8 cm.

FIG. 6. Same as Fig. 5 except calculated with the Laplacian model and improved off-shell model (solid curve). Derived from separable TN, -; Laplacian (local) model, - - - - .

potential, which is derived from a separable nN interaction. This latter potential predicts back scattering comparable with experiment, which might be significant if higher order corrections are correspondingly small at large angles. The predictions of the Laplacian model, although quantitatively different from those of the improved off-shell model (Fig. 6), are roughly of similar size for intermediate angles (30-lOO”), with the Laplacian predicting less back-angle scattering.

The fall-off of the calculated second maximum in Figs. 5 and 6, especially for our high energy calculations, is somewhat steeper than the data and also steeper than those calculated by Sternheim and Auerbach [3]. Use of their small value for the radius for carbon (uCh = ISF and acm = 1.5F in (2.37)) in place of the correct


value (1.59F, 1.66F) would also lead us to a broader second peak, but at the expense of not using a density compatable with electron scattering measurements. (As noted earlier, the second peak is also shifted inwards slightly by our partial wave transformation procedure, a procedure not employed in Ref. [3].) We feel it is more profitable to attempt to understand these discrepancies with the data rather than to remove them by arbitrarily adjusting what should be known parameters. Accordingly, we offer two possible explanations. First, as already calculated by several authors for nucleon-nucleus scattering [15, 161, the inclusion of nucleon-nucleon correlations in the second order optical potential tends to broaden the second peak, and, thus, if included here would probably improve the predictions. Second, as suggested by Lee and McManus [6], the poor match with data may be caused by permanent nuclear deformation in the ground state of carbon.

For forward-angle scattering, note from Fig. 6 that the Laplacian case and the improved off-shell optical model generally produce very nearly equal (to about 10 %) cross sections. That fact indicates that these cases will produce similar total cross sections. Compared to the Kisslinger case shown in Fig. 5, both the Laplacian and improved off-shell optical models differ (by 225 %) from the Kisslinger model at small angles, with most of the difference seen at low energies. As we will see in Section III B, these differences in forward scattering are significant.

It is clear from our results that differences in the elastic scattering cross sections are produced by changing the choice of off-shell pi-nucleon collision matrix. These differences occur at large angles as discussed and also can effect the smaller angle regions. The cases of a local pi-nucleon interaction and the separable pi-nucleon interaction yield similar pi-nucleus results, but the Kisslinger case proves to be substantially different. Since the separable pi-nucleon amplitude has been constructed explicitly to represent improved off-shell behavior, we consider it theoretically preferable over the other choices discussed here. (Actually, the scattering is not sensitive to the details, such as wiggles, in the TN separable T matrix; other matrices with the same magnitude near shell and which fall to zero or remain constant far off-shell give very similar results.)

C. Total Cross Sections and Forward Scattering

In Fig. 7 the predictions of the modified-Kisslinger, Laplacian, and improved off-shell models for the elastic, inelastic, and total cross sections are displayed. In all cases the effects of folding are included and any possible Coulomb effects are ignored. Note that results obtained using the improved off-shell and Laplacian models differ mainly at lower energies; whereas, both of these models differ significantly from the modified Kisslinger model over the entire range. This is not at all surprising since the Laplacian and improved potentials are different but well behaved in momentum space; whereas, the Kisslinger potential is actually divergent


200-

L

01 ' ' ' ' 3 I ' ' I ' ' ' 0 100 200 300

FIG. 7. ~4’: derived from separable rrN potential, -; modified Kisslinger model, -.-, -.-; and Laplacian (local) model, ----- . The n-C elastic, inelastic, and total cross sections as a function of pion lab kinetic energy, predicted by the modified Kisslinger, the Laplacian, and the improved off-shell potentials. Input parameters are theoretical and rrN amplitudes are folded. NB, the calculations for the Kisslinger potential are made with Abacus-M.

(Section 11 F). The relative magnitudes of the predicted total cross sections are approximately in the same ratios as the relative magnitudes of the 0” differential cross sections; as percentages these differences are small and were consequently difficult to observe on the previous semilog plots (Figs. 5 and 6).

From the magnitudes of these calculated total cross sections, it follows that a pion striking carbon at these energies experiences at least two multiple scatterings. The sum of all single pi-nucleon scatterings contributes 647~7) + 6o(n-n) to the total cross section, which is about twice the calculated cross sections at resonance. Double and higher orders of multiple scattering must therefore be important and act to reduce this large single scattering contribution. Single scattering corresponds to the Born term of (2.23), Uit); whereas, full solution of the integral equation is a means of including the higher order multiple scatterings. It is, therefore, essential that one solves the full integral equation to include all higher order scattering, especially near a resonance. The considerable difference between the single scattering prediction and the result found including all multiple scatterings shows that the multiple scattering series (in the impulse approximation) and with only elastic intermediate states will converge slowly, if at all.

The physical reason for the sizeable contribution of multiple scattering terms is simply that the mean free path of pions in the nuclear medium becomes quite short at collision energies for which the pi-nuclear cross section is sizeable. In

59j/78/2-3


this case, multiple scatterings make the nuclear interior a rather good absorber of pions, and, consequently, scattered pions are observed mostly from collisions with the relatively few surface nucleons.

We can also notice in Fig. 7 that the location of the experimental peak in uTOT cTn = 130-140 MeV) is essentially obtained correctly using either the improved off-shell or the Laplacian models. The peak obtained using the modified Kisslinger model occurs at too low an energy. This experimental peak is shifted approximately 40-50 MeV downward in energy compared to the peak in a(rJV); theoretical explanations or calculations of this shift appear in several studies [7, 8, 24,251. Here we find that various definitions of the optical potential lead to different values for the downward shift. Once again, therefore, although the pions are strongly absorbed by the nucleus, their assumed off-energy-shell behavior has important consequences which indicates that very precise total cross section measurements may well yield valuable information.

To help understand this shift in the peak total cross section and our results for forward scattering, it is useful to recall that the general partial wave decomposition of the forward scattering amplitude for two particles in their c.m. system is

Imf(B = 0) = l/k C (21 + 1) sin2 6,) (4.1)

Ref(r3 = 0) = I/k c (21 + 1) sin 6, cos 6,) 1

(4.2)

where 6, is the (neal) phase shift for the Zth partial wave. By use of the optical theorem, we also have

uTOT = 4r/k2 C (21+ 1) sin2 6,. 1

(4.3)

Note that (4.1) and (4.2) have a l/k factor; whereas, the total cross section includes a 1 /k2 factor.

By simple application of (4.1)-(4.3) to the pi-nucleon and pi-nucleus scattering problems, we now can explain the downward shift as a direct consequence of these factors and the broadening of the pi-nucleus amplitude.

First, we note that a slight “downward shift” occurs even in the pi-nucleon case. Pi-proton scattering below 300 MeV is almost completely dominated by the I = 1 partial wave due to the isospin 8, spin 8 P-wave resonance. At T, = 195 MeV (EC, = 1238 MeV) the phase shift for this channel passes very rapidly through 90” [20]. Correspondingly, since this one wave dominates the scattering, the Ref(0) = 0 at T, = 195 MeV, as is predicted by (4.2). However, because of the l/k factor in the expression for Imf(O), (4.1), the Imf(0) has its maximum shifted downwards slightly to T, N 187 MeV. Since the expression for the total cross section (4.3) has an additional l/k factor in front, it is shifted down still further to T, N 180 MeV.


For pions scattering from a nucleus, a similar analysis of the cross section applies, except for some additional physics. First, as demonstrated by the inapplicability of the single scattering approximation for the r-C total cross section, the pions are scattered frequently by several nucleons. This multiple scattering (or equivalently, the passage of the pion through a dispersive, absorptive medium) tends to diminish the energy variation offW,(0) and, thus, leads to a broadening. Although the multiple scattering does broaden peaks in Im,f$(O), it does not necessarily shift them.

4- ,- \ \

\

c 8-

2 b-

s - 4

E -

2-

,--

I’

,‘,$s.q

e

/

1’

: Im f,-,(O)

I’ ,’

Oh 0 100 200 300

T LAa (MeV)

FIG. 8. Folded separable sN -. not folded - - - - - . L Scott et al. (+C) and l Binon et al. The real and imaginary parts of the pure nuclear pion-carbon forward scattering amplitude calculated with a potential derived from separable ?rN potentials. For an isospin zero nucleus like C, both w and rr+ Coulomb-nuclear interference measurements determine the same nuclear amplitude and therefore both types of data are displayed. The solid curve has the input rrN amplitudes folded in a Fermi-motion distribution, the dashed curve is calculated with no folding.

In Figs. 7 and 8, we see this tendency of multiple scattering to give a broader energy variation. The lower part of Fig. 8 demonstrates that the Imfnc(0) has a rather broad peak with a maximum occurring at approximately the same energy as the maximum in ImfWP(0), -187 MeV. The major cause of the broadening is


the occurrence of the nuclear form factor in UAg) and the higher order multiple scatterings obtained when solving (2.23). In addition, including nucleon Fermi motion also broadens and lowers the peak slightly. This effect is shown in Fig. 8, from which it is clear that folding contributes, but is not the major cause of the broadening.

For pi-nucleus scattering the Im&(O) is considerably broader than the corresponding pi-nucleon amplitude Imfnp(0). Multiplication of Imf$(O) by l/k to form the total pi-nucleus cross section (Fig. 7), therefore, leads to a considerably larger shift (-40 MeV) in the peak position than was found in the corresponding step for the pi-nucleon cross section (there a shift of ~10 MeV occurred). We conclude that the shifting of the peak in the rrC total cross section is simply a consequence of the broadening caused by multiple scattering and is not an exotic effect [24].

This simple reason for the downward shift also clarifies why use of Glauber’s completely on-energy-shell multiple scattering theory can also predict the peak in trot correctly [8, 251; namely, there is no need to generate a shift in Imy,,-(0); all we need is a broadening effect.

Additional insight into the pi-nucleus scattering is provided by examining the real part of the pi-nucleus forward scattering amplitude, Re&(O). In the upper part of Fig. 8, the results for Refnc(0) calculated using the improved off-shell optical potential with and without folding are presented. The experimental points are measurements of Coulomb-nuclear interference in the forward direction [26]. Aside from the respectable agreement with data, notice that the calculated Re&-(0) crosses zero at about 162 MeV, which is considerably down from the energy at which the corresponding pi-nucleon quantity Refr,(0) equals zero, i.e., at -195 MeV.

Thus, although the imaginary part of the pi-nucleus amplitude, Imf,,-(0), does not have an appreciable shift, the real part Ref$(O) does. However, this shift in the zero-point of Re&(O) is not at all surprising. First, the kinematical transformation to the rr-nucleus c.m. and the averaging of the scattering amplitudes over neutrons and protons already brings the zero point down by -7 MeV. Second, in ZC scattering no one partial wave completely dominates the scattering, and no single partial wave resonates while all other waves remain constant. Instead, we see from (4.2) that the energy at which Ref,o(O) crosses zero is determined by the sum of many partial waves. Note that the P33 r-nucleon resonance contributes to, and in fact dominates all of the pi-nucleus partial waves. From the partial wave decomposition of the optical potential (2.36), it is clear that the P33 channel tends to be the major contributor to each pi-nucleus wave; our calculations confirm this remark. Also, in the extended sense of a resonance applicable to strong absorption (as discussed in Ref. [20]), all of the pi-nucleus partial waves resonate. (Also see the Argand diagram in Fig. 3.) Consequently, no one partial wave completely dominates the sum (4.2) and the location of a zero in Ref,&O) can easily be shifted from the free pi-nucleon value simply because of the many phase shifts


and weightings occurring in the sum. For example, a simple case, where the zero in Ref,,(O) would shift while the peak in Imf,,c.O) would not, occurs if f(0) consists of a resonant part plus a constant background term.

Based on this analysis of pi-nucleus forward scattering, we can now understand why the on-shell Glauber theory can, on the one hand, correctly predict the position of the maximum in ‘~ror , while failing to determine the correct energy at which Refmc(0) = 0 (at least in the Glauber calculations of Refs. [8] and [25]). The reason is that the total cross section peak is simply a broadening effect with no shift in Im&-(0) required. On the other hand, Glauber theory predicts that since the input pi-nucleon amplitudes all have RefvK(0) = 0 at T,, N 195 MeV, the corresponding pi-nucleus amplitude Refmc(0) should also have its zero at approximately the same energy. To get a shift down in Ref,,(O), and, therefore, to explain the Coulomb-nucleus interference results, it seems necessary to deal with an off- shell theory such as discussed here!

0 SO" 120" 180" 60" 120" 180" 60" 120" 180°

8 c.m. FIG. 9. n-C scattering calculated with separable TN, - ---, Re f(0); -, Imf(0).

The angular dependence of the real and imaginary parts of the r--C c.m. scattering amplitude f(0) as calculated with the improved off-shell model for different pion kinetic energies. The “+” and “-” signs indicate the sign of f(8) for each peak. At -162 MeV, Ref(0 = 0) = 0, whereas at 170 MeV, Ref(8) and Imf(0) are closely in phase.


D. Difractive Nature of do/dQ

The foregoing analysis of the pi-nucleus forward scattering amplitude can also be extended to provide insight into the diffractive nature of the differential cross section. We can now show that the vanishing of Re fTc(0) at T, N 162 MeV noticeably affects the differential cross section and, in particular, is related to the apparent increase in the diffractive nature of da/dQ at T, = 180 MeV. To understand this assertion, let us examine the angular dependence of the real and imaginary parts of fnc(0) for different pion kinetic energies. Shown in Fig. 9 are some results, where the + and - signs indicate the sign of f(e) for each peak. First notice that the Imf(0) varies only slightly as a function of energy. Its minimum and maximum shift very gradually to smaller angles with increasing energy as expected for diffractive scattering which depends chiefly on qR, . The energy variation of Ref(0) is more interesting. As the energy increases from 150 MeV, the entire pattern moves in rather quickly, causing the first (positive) peak to completely disappear at T, N 162 MeV-that is where Ref(0) = 0. After this point the variation is less rapid, although still quicker than the imaginary part’s,.

A significant consequence of this behavior is that the relative phase of Ref(B) and Imf(@ is very sensitive to energy. In fact, we see that the increased depth of minima and height of maxima in du/ds2 at 180 MeV is predominantly a consequence Ref(B) and Imf(@ being roughly in phase at this energy and not, as has been thought [27], of the amplitude being relatively imaginary. Furthermore, on the basis of this analysis we predict that the differential cross section for pions on carbon would be even more “diffractive,” i.e., have sharper features, at T, ‘v 170 MeV than at 180 MeV. We hope this type of prediction will serve as a sensitive test of the reaction mechanism.

E. Eflect of Fermi Motion Inclusion

We have already seen in Fig. 8 that the approximate inclusion of the nucleon’s internal motion within the nucleus (Fermi motion) has a noticeable effect upon the predicted rrC forward scattering amplitude. In Fig. 10, this effect is further demonstrated for a total and differential ~TC cross section calculated from the improved off-shell optical model, which is derived from a separable pi-nucleon interaction. In both figures, we note somewhat better agreement with experiment occurs when the input TN collision matrices are folded into, or averaged over, the momentum distribution of nucleons within the nucleus. Since our folding procedure is only approximate, the significance of Fig. 10 is that it does show the importance of and the changes introduced by the inclusion of Fermi motion. For a more quantitative conclusion, and a more reliable optical model, a better treatment of nucleon motion is perhaps required.


8 c.m.

FIG. 10. n-C from separable TN: folded, -; not folded, - - - -. The effect on calculated V-C scattering of folding the input ?rNamplitudes over the internal (Fermi) motion of the nucleons. The upper graphs is for total cross sections and the lower graph for a typical angular distribution.

F. Importance of rrN, D, and F Waves and Wave Equation

The singular nature of the Kisslinger potential in momentum space required that our calculations with this model be performed in coordinate space with the program Abacus-M [3]. This code solves the approximate Klein-Gordon equation discussed in Section III for a pi-nucleus potential containing only S and P Z-N waves. In contrast, the calculations presented here for the other models represent solutions in momentum space of the Schrodinger equation with relativistic kinematics for potentials which contain S, P, D, and F TN waves. For the type of comparisons made in this paper these differences in rrN waves and equations are not important, although they are significant if detailed comparisons with data are to be made, especially at the higher energies.

Specifically, the pi-nucleon D and F waves first affect the z-C differential cross section (by about 10 %) at pion kinetic energies of -280 MeV; at 400 MeV these waves affect the first “break” at 40” by ~50 ‘%. At 500 MeV the D and F waves are quite significant and affect the structure of do/dQ at most angles.


The differences in cross sections predicted by a relativistic Schrodinger equation and an approximate Klein-Gordon equation also are energy-dependent. At nonrelativistic energies the two predict the same scattering; at the higher energies considered here (~280 MeV) the difference is roughly the same size as the experimental error bars-except for large angle scattering (> 100’) where the results differ by a factor of two at a maximum. At even higher energies, the TC cross section falls so rapidly with angle that the difference is deemphasized and and becomes noticeable only at the breaks, where it is on the order of 20-30 %.

The differences generated by using alternate wave equations probably are not significant at large angles, because other effects can produce even larger changes in the back angle scattering; for example, corrections to the first order optical potential. The differences in du/dQ at small angles produced by changing the choice of relativistic wave equation probably represent a basic ambiguity in using a potential model.

V. SUMMARY AND CONCLUSION

The optical potential which describes pion-nucleus scattering is constructed in first order from the elementary off-energy-shell pi nucleon transition matrix and the nuclear density. We have constructed and studied in momentum space three such optical potentials. These models are the popular Kisslinger potential, the local Laplacian potential, and an improved off-shell potential derived from absorptive separable TN potentials. At this stage, the ability of one potential to give better fits to the experimental data is not considered to be particularly significant because many corrections have been omitted which could change the predictions of the models. It is significant, however, that the various off-shell models yield different cross sections and that the cross sections are sensitive to the nuclear radius. Therefore, to improve the theory so as to determine nucleon- nucleon correlations or to finely probe the matter density of the nucleus, a reasonable choice of first order optical model needs to be made. Since these optical models differ in the off-energy-shell behavior assumed for the pi-nucleon collision matrix, the choice should be based on the most physically reasonable off-shell pi-nucleon interaction.

Both the Laplacian and Kisslinger potentials are derived from hypothesized TN collision matrices which diverge far off shell and do not display behavior expected from a physical model (e.g., resonances). Even after including the natural cutoff provided by the nuclear form factor, the Kisslinger potential is singular in momentum space (the Laplacian accidently is not). The improved off-shell optical potential derived from separable TN potentials, however, has an off-shell behavior which reflects the dynamics of physical resonances, thresholds, etc., has a reasonable fall off in momentum space, and possesses analytic properties appropriate to a

IMPRqVED PION-NUCLEUS POTENTIALS 337

Schrbdinger equation. It clearly is our choice for the most physically reasonable off-shell model of the three and, we feel, is a considerable theoretical improvement over previous potentials.

We have also found that with the proper transformation of rrN scattering to the pi-nucleus center-of-mass system, the theoretical optical potential yields the back angle scattering very well, with a small discrepancy at the second diffraction maximum which probably would be corrected if nucleon-nucleon correlations were included [I 5, 161. Furthermore, we find precision total cross section measurements may represent a good test of different models and are sensitive to the nuclear size. In fact, the observed shift in the pi-carbon total cross section is dependent on the off-shell model and given correctly only by the Laplacian and improved off- shell models. This shift is then interpreted as caused solely by the width of the P33 TN resonance being broadened by multiple scattering within the nucleus. We also have shown that the increased diffractive nature of the differential cross section at 180 MeV is predominantly caused by the real and imaginary parts of the scattering amplitude being in phase, and that the scattering is probably more

“diffractive” at around 170 MeV. These results of our present calculation tend to encourage us to make further

application of the improved off-shell potential, for example, to other nuclei [28], to inelastic scattering [29], to pionic atoms, and to other phenomena [30]. Further- more, since the approximate inclusion of Fermi motion is important it should be handled correctly. This appears fairly straight forward in momentum space and with the improved off-shell potentiaL5 The Coulomb interaction, however. may be more difficult to include correctly in momentum space. Since nucleon-nucleon correlations appear important, a calculation of the second order optical potential also seems worthwhile. (An effort is underway to obtain a useful coordinate space space representation of the improved off-shell optical potential, which should aid in application to these problems.)

In general, then, we feel that it is possible and important to make the first order optical potential theoretically sound and that the biggest ambiguity, related to the off-shell pion behavior, can be effectively removed. Many of the other theoretical uncertainties proved not important.

Hence, we believe that our study of the first order optical potential is a step toward using pion-nucleus scattering to learn about nuclear structure.

ACKNOWLEDGMENTS

It is a pleasure to thank several of our Pittsburgh colleagues for their advice and helpful com- ments. In particular we wish to acknowledge stimulating and illuminating conversations with

5 A first step toward correcting several basic approximations in the optical model has been made along these lines by Schmit [4].


Dr. C. M. Vincent, Dr. L. Kisslinger, Dr. R. Drisko, Dr. R. Eisenstein, and Dr. P. Barnes. One of the authors (F. T.) also wishes to thank Dr. L. Heller for his hospitality at Los Alamos Scientific Laboratory, where part of this manuscript was prepared.

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improved theoretical pion-nucleus optical potentials

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