momentum-space study of pion-nucleus inelastic scattering

I 2-L I Nuclear Physics A226 (1974) 253 -28 1; @ North-Holland Publishing Co., Amsterdam

Not to be reproduced by photoprint or microfilm without written permission from the publisher

MOMENTUM-SPACE STUDY OF PION-NUCLEUS

INELASTIC SCATTERING7

TSUNG-SHUNG H. LEE

Physics Department, University of Pittsburgh, Pittsburgh, Pennsylvania 15260 and

Bartol Research Foundation of The Franklin Institute, Swarthmore, Pennsylvania 19081 w

and

FRANK TABAKIN

Physics Department, Unioersity of Pittsburgh, Pittsburgh, PennsyIvania 15260

Received 22 February 1974

Abstract: The inelastic scattering of intermediate energy pions from lZC is studied using a distorted wave impulse approximation (DWIA). A flexible momentum space formulation is presented which permits examination of a wide variety of non-local pion-nucleus potentials. Inelastic differential cross sections for the 2+(4.43 MeV) and 3-(9.64 MeV) states of 12C are generated from several phenomenological pion-nucleon collision matrices. Form factors for the inelastic transitions are obtained from the collective model and also from microscopic lp-lh (Gillet-Vinh Mau) wave functions. The dramatic role of the (3,3) pion-nucleon resonance, along with an associated pion-nucleus absorption, is illustrated. The inelastic cross sections prove to be somewhat influenced by the off-shell pion-nucleon model, with the effect mostly confined to larger angles, and with changes that closely parallel those found earlier for elastic scattering. Back- angle inelastic cross sections are found to be large by a factor of about 10 compared to prior pion-12C inelastic scattering calculations. This beneficial effect, relative to the available data, is attributed to a corresponding elastic scattering back-angle build-up which arises mainly from transforming the pion-nucleon interaction to the pion-nucleus center of mass frame. The inelastic scattering diffraction patterns are found to depend on the location and tail of the form factor, but much less so on its internal form, which is indicative of a surface direct reaction. Normalization of the collective model form factor to known B(E2) and B(E3) values yields general agreement with the magnitude of the forward angle inelastic 2+ and 3- cross sections. These simple features of calculated pion- ‘*C inelastic scattering encourage further application of the DWIA and of direct reaction strong absorption methods to pion scattering in the resonance region.

1. Introduction

Experiments to be performed at meson factories will hopefully be providing a rich supply of pion-nucleus inelastic scattering data, which has inspired several recent examinations of the associated reaction mechanism ‘-‘). In this paper, we are similarly motivated and therefore consider the case of intermediate energy (100-300 MeV) pion- “C inelastic scattering, for which data are available. A direct interaction

t Work supported by the US National Science Foundation. t+ Present address.

253

254 T.-S. H. LEE AND F. TABAKIN

viewpoint is adopted and the multiple scattering theory for inelastic collisions 43 “) is cast into the form of a standard distorted-wave impulse approximation (DWlA).

The DWIA entails initial and final pion distorted waves (determined from the elastic scattering optical potential) and an inelastic interaction. Inelastic scattering from an initial nuclear state 10) to a final nuclear state (nl is described in the DWIA by a transition amplitude of the form Tri = (xi-‘l(nlUlO>lx$“), where the initial and final pion distorted waves are xi” and xi-‘. These distorted waves describe the motion of the pion as it interacts with the nucleus via an optical potential (OlUlO}; it is further assumed that when the nucleus is excited the same elastic scattering optical potential applies, i.e. (nl U[n) z (OlU(0). The inelastic interaction <nlUlO) induces the nuclear transition. The pion scatters repeatedly from the A constituent nucleons and the operator U can correspondingly be related to the basic pion- nucleon collision matrix ti(o), evaluated in the pion-nucleus c.m. frame at an ap- propriate collision energy IX. The leading term in an expansion for U is a sum U = ~fzlti(co). Multiple scattering theory provides systematic corrections to the above expressions, which is a useful feature provided the corresponding series can be evaluated and converges. Here attention is focused on the above leading term and a variety of “corrections” are not considered. For example, the effect of correlations, of other excited states, of Pauli blocking, of crossing and corrections to the impulse approximation, etc. are all relegated to future studies “). Nevertheless it is encouraging that a simple DWIA apparently incorporates much of the dynamics and provides a reasonable starting point. It remains to be seen whether such corrections can be included with sufficient reliability to warrant the extraction of detailed nuclear structure information. The present study, formulated in a flexible momentum space representation, is hopefully a useful step in that direction.

Even the leading term for U requires study; it is obtained only after making the impulse approximation and then transforming t to the pion-nucleus c.m. system. It is necessary to stipulate the pion-nucleon collision matrix, t, both on and off the energy shell. For that purpose, we have adopted three phenomenological forms; namely, the Kisslinger potential 7), a simple Laplacian model and the bounded off-shell model of Landau and Tabakin (LT) “). These three pion-nucleon models differ dramatically off-shell and are used here to probe the off-shell sensitivity of pion-nucleus scattering. One of the advantages of our momentum space formulation is that future use of improved pion-nucleon interactions, such as the recent Londergan et al. ‘> and Nutt and Wilets ’ “) contributions, involves only slight subroutine alterations.

To construct the optical potential and the inelastic interaction, it is also necessary to stipulate the ground state density and the nuclear wave functions. The ground state density is taken from electron scattering studies, as discussed in ref. 11) for the elastic case. The inelastic interaction (nl UlO) is generated in two ways, first from the collective model by introducing a deformed density “) and then by adopting the lp-lh wave functions of Gillet and Vinh Mau I’). With those form factors the dependence of the inelastic scattering on the nuclear form factors is examined.

PION-NUCLEUS INELASTIC SCATTERING 255

In sect. 2 the pion-nucleus inelastic interaction is discussed along with the re- quisite pion-nucleon models and the nuclear form factors. The explicit construction of the inelastic interaction in momentum space is presented next using the collective (macroscopic) model and also the microscopic lp-lh model of Gillet and Vinh Mau. After expressing the inelastic transition amplitude in terms of momentum space distorted standing-waves (sect. 3), the numerical methods used to make a DWIA calculation in momentum space are outlined. (Similar methods have been indepen- dently developed by Charlton, Robson and Koshel to study finite range and recoil effects in nuclear scattering I”).) Sect. 5 deals with the influence of off-shell matrix elements, of the basic (3,3) resonance and its associated strong absorption, evidence for the surface direct mechanism and a comparison with the available data. Several future problems (and plans for including Coulomb effects) are then discussed.

2. Pion-nucleus inelastic interaction

2.1. PION-NUCLEON MODELS AND THE FORM FACTOR

The inelastic interaction <njUjO) is given in this section in terms of the pion- nucleon interaction and the nuclear form factor, expressed in momentum space. Concise expressions for the optical and inelastic interaction are obtained using P to denote both the momentum p and the spin-isospin variables (m,, m,) for the nucleon, while K denotes both the momentum k and the isospin variable m, for the pion. Ex- pressions involving JdP are understood to represent integration over p and a sum over the associated spin-isospin indices, i.e., Cmsrn, Jdp. The pion and nucleon masses are p and m, respectively.

According to the KMT analysis of multiple scattering theory 4), the first order pion-nucleus inelastic interaction is obtained in momentum space by forming matrix elements of the operator U(E) = (A - 1)7(E) using products of nuclear eigenstates l@J and pion plane waves I&$ = Ik, m,),

(nK’J UjOK) = U,,,(k’mi, km,)

= (A- l)(Gn, k’m$(E)j@, , km,). (1)

Here z(E) is the scattering operator for a pion striking a bound nucleon. Because the struck nucleons are embedded in a nucleus, the operator z(E) differs dynamically from the free pion-nucleon scattering operator t(m). Here E is the pion-nucleus collision energy and cc) is, so far, an unspecified energy.

The first step in evaluating the optical potential and the inelastic interaction (1) is to relate the operator z(E) to the free pion-nucleon collision matrix t(w). Although the (3, 3) resonance lifetime is quite short compared with the time scale of nucleon motion, corrections to the impulse approximation z x t have recently been estimated by Eisenbesg “) to be roughly 20 % of the leading term. One possible way of generating improved convergence, would be to choose w to minimize corrections; that program


has not yet been performed. Instead, a physically motivated choice of o is made later and it is hoped that choice provides a reasonable start. Clearly, further study of corrections to the impulse approximation and of Pauli blocking, etc. are needed.

Adopting the impulse approximation and introducing momentum space nuclear wave functions, the inelastic interaction can be written explicitly as an integral over nucleon momentum

Here 4 = k’ - k is the momentum transfer, and

is the overlap function which physically measures the extent to which a nucleon in state (p, m,m,) in the nucleus can be scattered to (p’, mirn:), while the nucleus is excited from the ground state Idi,) to the excited state [Up,).

These expressions for the inelastic interaction can be simplified further using the property that the nuclear size is much larger than the pion-nucleon interaction “range”. The overlap function F”‘(p - q, p) is consequently a sharply peaked function of p relative to the t-matrix. Therefore it is reasonable to use an average pion-nucleon collision matrix and to factor t out of the integration over nucleon momentum rl, 1 “).

With this “factorization approximation”, a separation of the inelastic interaction into a product of pion and nuclear dynamics is made

u,o(K’, K) = (A - 1) C <PO-q, mL mi; ~‘lt(~o)lPo~ m, m,; OP~sm~~, m,m,((l). (4 m’,m’, msmc

Here P~~~~~,, m,m,(q) = S F”‘? m .,,,,, m,m,(p - q, p)dp is the nuclear transition form factor for a given spin-isospin transition. The average nucleon momentum p. is chosen by assuming that the nucleons in the nucleus are “frozen” during the collision and hence p. = - k,/A (here k, is the incident pion momentum as seen in the pion- nucleus c.m.). Using this average nucleon momentum our choice for the average

collision energy c0 is cue = dki + p2 + dm2 + (ko/A)2, ignoring a small binding term. This factorization approximation is then corrected ‘I) subsequently for the effect of nucleon motion by averaging the on-shell pion-nucleon t-matrix over the nuclear ground state wavefunction.

The pion-nucleon t-matrix in eq. (4) is defined in the pion-nucleus c.m. To use the free pion-nucleon scattering data as input to calculate eq. (4), a transformation of the pion-nucleon collision matrix from the pion-nucleon c.m. to pion-nucleus c.m. must


be defined. The transformation is assumed to be of the form

<k PO--ll+h)lk PO> = Y<~‘l’t(W~>. (5) Here rc’ and IC are the relative momenta in the pion-nucleon c.m., c& is the corresponding pion-nucleon collision energy. The on-shell momenta IC, K’, the coefficient y, and the collision energy &, are determined using a Lorentz transformation. The on- shell relation is then extended to transform the off-shell momenta, as described in ref. II). In the “frozen” nucleon approximation, y is defined as

’ = E~(~)E~~(~‘>EN(~)EN(~‘) 1 l E,(k)E,(k’)EN(k/A)EN(k’/A)

(6)

The transformations defined above only relate the magnitude of the momenta in different frames. However, as pointed out by several authors 11,15), a proper transformation of collision angles (k - k = cos fl,, and 12 - 12’ = cos f3,,) is also necessary. The relation between cos 9,, and cos enN leads to a significant mixing of partial wave scattering amplitudes. The mixing of the partial waves is defined by a relation between the Legendre functions for cos 8,, and cos I&,,

(7)

The coefficients d,, are given in table 1 of ref. ‘l). A somewhat modified version of this table has recently been suggested by Miller and Phatak (private communication).

Eqs. (5)-(7) relate t(o,) to the c.m. collision matrix l(&,). This step corresponds to including “recoil” effect 13). Now the pion-nucleus transition potential can be constructed, once the c.m. collision matrix 2(&e) is completely defined.

Since a complete dynamical theory of the pion-nucleon interaction is not available, various phenomenological pion-nucleon ?(GO) matrices have been used in pion- nucleus scattering calculations. To extract detailed nuclear structure information from pion-nucleus scattering, it is very important to examine the extent to which pion-nucleus scattering is affected by different off-shell pion-nucleon interaction models.

To make such a study, three off-shell models for the pion-nucleon collision matrix are used in our calculations. The first two models are the Kisslinger model:

<7c’&%&rc> = a(&_J + b(fQc’ * K,

and the Laplacian model:

(8)

<Ic’I?(6&c> = u’(c&) + b’(&&c’ - 7c)“. (9)

Here, the parameters (a, b, a’ and b’) are determined from the S and P pion-nucleon phase shifts. The third phenomenological form, I(GO), is obtained by solving an inverse scattering problem for an absorptive separable model “). For that case, one


has an off-shell collision matrix for each eigenchannel a(& 1,j) of the form

where g, is given in ref. “). These three models (8)-(10) are exactly the same on the energy shell, but they differ dramatically off-shell as shown in fig. 1. (Recall that on the scale in fig. 1 p(q) is sharply peaked.) Londergan et al. “) have recently generated real g,(rc) functions from an inverse scattering solution for a simple coupled-channels pion-nucleon model. Their g(rc) functions are smooth and nicely bounded and hence offer another reasonable candidate for future applications. The Kisslinger, Laplacian and LT models are used later to test the sensitivity of pion-nucleus inelastic scattering to pion-nucleon off-shell effects.

To complete the definition of Un,-,, the nuclear transition form factors p*‘(q) of eq. (4) will now be constructed. It is easy to show that the transition form factor is the Fourier transform of the nuclear transition density matrix

and we have

The next task is to construct the transition density matrix using various nuclear models.

2.2. COLLECTIVE MACROSCOPIC MODEL

Edwards and Rost “) have extracted a simple inelastic interaction from the density- dependent Kisslinger optical potential. The basic procedure is to use a macroscopic picture of a deformed density and expand that density about a spherical distribution. The corresponding collective macroscopic wave functions are of the form Yoo(@) and Y&(84) for the ground and excited states, respectively; the orientation of the ellipsoidal nucleus with respect to the space-fixed frame is given by the angles 04. These collective macroscopic wave functions describe the 2+ and 3- states of “C as (K = 0) rotational motion of a deformed nucleus (for the present DWIA calculation the description as a surface vibration is equivalent).

We adopt the same procedure, inserting the deformed density into (4) and (ll), to construct inelastic interactions for more general choices of the pion-nucleon interaction [eqs. (8)-(lo)]. A deformed density is described by taking an effective nuclear radius R as a function of coordinate angles (0’4’) defined with respect to body-fixed principal axes of the nucleus, i.e. R = Ro(l+&j?JY~o(B’@)), with 2 the body-fixed symmetry axis and pr the usual deformation parameter. The density p(v, R) viewed in the space fixed frame can be expanded in powers of Bt

P(F) = P(C Ro(l+ 7 Bt r,o(Gq> = P(C Ro) + l$ PlwNo(Qo (12)


where the deformed density is

FL(T) = Ro 8& p(r, R)IR=R~ - (13)

The density p(r, R,), and hence F$(Y), is then determined from electron scattering information using a standard form for p[see ref. “)I. Using the addition theorem

Y,,(W@) = (4n/(2Zf 1))” ‘&YI,(E’)YL,(@$) and the simple (Yoo(Q4), Y&W)>

description of the nuclear states, we find that the “nuclear form factor” is

with

prM(q) = ediq ’ "Y&(P)pr s

-& FI(T)dY. (15)

For “C we take the neutron and proton densities to be the same; also no nucleon spin occurs in the deformed p. Correspondingly, the nucleon spin-isospin variables msm, are not altered by p and (14) includes the + &,,.m,, 2&_, factor. Substituting (14) into (4) and summing over nucleon spin and isospin rnirn:, we obtain the inelastic interaction for the collective model

u IM, OO(K’, K E) = +(A- 1) c <pa--q, m, m,; K’lt(oo)lpo, in, m,; K>prM(q). (16) msm,

To facilitate momentum space numerical calculations, eq. (16) will be reduced to a multipole series. The pion-nucleon collision matrix in (16), because of (14), is already diagonal in spin-isospin space. Thus only spin-isospin non-flip terms in t contribute. The corresponding multipole decomposition of the spin-isospin non-flip term is

“,c, <PO - 4 m, m,; K’lt(oo)lpo, k, m,; 0 + C f&t’, k, wO)y;I~#?)&&), (17) s ? imi

where f,(k, k’, coo) is the spin-isospin averaged pion-nucleon collision matrix in the pion-nucleus c.m. To decompose the nuclear form factor pr”(q), the plane wave is expanded and the integration over angle P can be completed. Then, we get

x (4x)%/(21+ 1)(2E1 + 1)(21, + 1) ii k i) F$f,‘,(k’, k), (Ha)

with a “radial form factor” defined by

v-w The radial form factor F$i affects the extent to which the pion-nucleus relative angular momentum I2 is changed to I, when I = l2 - I, is the angular momentum

260 T.-S. I% LEE AND F. TABAKIN

transferred to the nucleus. Note that free and not distorted radial waves appear in (18b); distorted waves will enter later.

Substituting (IS) into (16) and summing over the m quantum numbers, we get

with

Much of the angular momentum coupling is isolated into

Expressions (19) and (20) give the inelastic amplitude in the plane wave impulse approximation, along with the relevant selection rules. For the collective model eq. (20) explicitly states how the pion-nucleon I-wave interaction causes excitation of the nucleus to the state JAW), while the pion-wave is scattered from an orbital angular momentum state I to f’. Similar forms for the inelastic operator can now be obtained from a microscopic nuclear model.

2.3. MICROSCOPIC MODEL

The inelastic interaction (4) can also be generated from a microscopic picture of nuclear motion as coherent superpositions of lp-lh states. To use such (Gillet- Vinh Mau) wave functions, it is helpful to first &press the transition form factor, ~~‘(~), in terms of single nucleon transition form factors F”@(a). In second quantized language, that relationship is simply

P *0 m,s,,,p., m.,,,,(a) = ; CB<~~lb.*baI~o)F~~,,=,n~~,(q),

a,

where F@ is determined by the single particle wave function (b,

F$sZZ., ~~~~(~) = s dr eWiq * ‘+z(r, ml mI)#b(r, m, yn,).

IHere ct denotes the orbital quantum nmbers [a,l,j,mj%, m&. Substituting into (4), we get a microscopic pion-nucleus inelastic interaction


In contrast to the collective model case (16), the single nucleon form factor is not in general diagonal in spin-isospin space, hence pion-nucleon spin-flip and isospin-flip terms can now contribute to U,,, . For convenience, we decompose the inelastic interaction into a sum of spin-independent and spin-dependent terms. Each term also contains isospin-flip and isospin-nonflip effects. In order to perform this decomposition, we separate “) the pion-nucleon spin-dependent interaction from the spin-independent term

In the above equation, ZI and fl are operators in isospin space

i,(k’, k) = & 7 pdz+ l)h:,=,+,(k’, k)+ Z_&++(k’, k)], (25)

47cJE(Z+ 1) ‘t,(k’, k) = (2z+ f)’ T PT[fi,Tj=l++(k’, k)-f,FJ=,-+(k’, k)],

where the standard isospin projection operators are P,=, = f(l -I * z), and P,=, = *(2+ Z . z);f,TJ is the usual pion-nucleon amplitude, suitably Fermi averaged.

Substituting eq. (24) into eq. (23) and performing straightforward angular momentum algebra 5), the microscopic pion-nucleus transition potential for the T = 0,

2+ and 3- excited states of 12C can be decomposed into coupling potentials:

= (A- 1) C ~;,@‘)3&@)( - 1)” (;, Im

_“, ;) [@(k’, k)+ @(k’, k)]. (27)

I’m’

Here Ui() and D{<’ are the spin-independent and spin-dependent terms, respectively. The spin-independent term is

where ?r (k’, k) is the spin-independent ith wave pion-nucleon interaction which has been averaged over nucleon isospin. The corresponding spin-independent transition density E;(r) is expressed in terms of the Gillet-Vinh Mau RPA coefficients X,, and Y,, by


Similarly, the spin-dependent term for the normal-parity states (2” and 3-) is

where Ir(k’, k) is the spin-dependent, itI1 wave pion-nucleon interaction, which has been averaged over nucleon isospin. The spin-dependent transition density is expressed in terms of the RPA coefficients as

The tensor operator ?‘,(Y,, C) is defined in ref. 16); it depends on orbital and spin angular momentum.

We have constructed pion-nucleus transition potentials from the collective model and also from microscopic (GV) nuclear wavefunctions. Now, we can examine the pion-nucleus transition matrix using these transition potentials.

3. Inelastic amplitude aml cross sections

In this section, we first present the multipole decomposition of the inelastic amplitude in momentum-space. This decomposition in momentum-space is then used to discuss our numerical method for calculating inelastic scattering cross sections. Related methods are developed in ref. 13).

In the moments space representation, the lowest order pion-nucleus inelastic amplitude 5, is

Tzo(kb , k, , E) = f$ jdki jdkz x&)*(WJ,o(k, 3 k,)&‘(k,), (30)

where k, and kb are initial and final pion-nucleus relative momenta in the pion- nucleus c.m. system. We assume that the elastic optical potential U,, for the nuclear final excited state I@*> is equal to U,, . The distorted waves 1 x,“,) are calculated from the elastic optical potential U,,(E) by solving relativistic scattering equations

and

(31b)

where Kxn is the relativistic pion-nucleus relative motion kinetic energy operator,


and E, is the nuclear excitation energy. Using a spherical ground state density for “C, the elastic pion- 12C optical po en t tial 11) is a central potential. Therefore, it is clear from (31) that xi:’ can be expanded as

X!c%) = ; x~,~~(k,)Y,~(ff,)Y,(R,), (32)

where xi,:o)(k) is the radial wavefunction of orbital angular momentum 1. Substituting eq. (32) and the multipole decomposed U,,, [eq. (19) or eq. (27)] into eq. (30) and performing the integrations over angles k, and k,, we obtain

To perform the remaining double integration numerically, using an extended matrix inversion method ’ 7), it is convenient to express the distorted wave /xi:‘> in terms of distorted standing waves /@to) and I@,“,), which are defined by

and

Here, the principal value operator P in the propagator indicates that I@‘> and ]GL> are distorted-waves with standing-wave boundary conditions. We will proceed to construct distorted waves Ix&) from ]@E;L). From eqs. (31a) and (34a) we can deduce a relation between I$) and ]@EO), which is

x!:‘(k,) = @:o(kl>- in s

dk; X:z’(k$(E -E,~,)R,,(k~ , k, , E). (35)

Here R,, is the elastic R-matrix which is determined from the elastic optical potential. Performing the partial-wave decomposition of eq. (35), we achieve our goal and get a partial-wave relation

Similarly, we get a relation between I&‘) and ]@kO)

&!:(ka) = 1

1 +2ip,Rl(kb, kb , E) @$o(kJ. (37)

Here, PE = (U~~)(W~~l)l~,=k,, is the density of states at energy E.


Substituting (36) and (37) into eq. (33), the inelastic transition matrix becomes a function containing only the distorted standing waves,

T,::$G,, ko) = 1 1

1+2@&,(k;, , 6,) 1+2&R&o, k,)

The main difficulty in evaluating the transition matrix is to perform the double integration (39), since the radial wavefunctions (at, and @h, contain a “singular” principal value part P(E- He)-’ [ see eq. (34)]. In the next section, we will present a numerical method to resolve this problem.

Once the transition matrix is determined, the cross section can be calculated in a standard way. Choosing &,l@ and setting kb in the x-z plane of the coordinates (x-u-z), the transition matrix becomes

T IM, o&b, k, , E) = ; ; J(21, + 1)(21, + 1) (; 1.7

; _IM)

P,J”(cos e)TI',:l(k;, , k,). (40)

The unpolarized inelastic differential cross section is

do -= (2~)4E,(ko)E,(ko)E12(k~)~“tk~)

da

3 c ,TIM, ,O(cos e)j’

[K&J + W&,)1 C&(kb) + 4,(kbll ko M (41)

Eqs. (38)-(41) will be used in our numerical calculation to study the pion-nucleus inelastic scattering. The detailed numerical methods will be discussed next.

4. Distorted wave method in momentum space

A numerical method for performing the distorted wave calculation in momentum space is presented in this section. Our numerical procedures are the following: (i) solve the elastic R-matrix equation by extending the matrix inversion method of Haftel and Tabakin r7) to complex potentials; (ii) show that the distorted standing- waves @Eke (k) and @kko(k) can be directly obtained from the inverse matrices, (iii) reduce the double integration involved in obtaining the transition matrix, eq. (39), to a simple matrix multiplication which can be evaluated.


To solve the R-matrix equation, multipole decompositions of R and of the elastic optical potential UOO are used to obtain one-dimensional integral equations in momentum space, obtained in the “prior” form

Choosing N Gaussian grid points and calling the (N+ 1)th grid point kN+ 1 = k,,

the above integral equation becomes an (Nf 1) dimensional matrix equation [see refs. “9 “)I. The Rz matrix can then be obtained from the matrix equation by inverting a finite matrix sI. For the half-on-shell R-matrix, the solution for the Ith partial- wave is

N+l

&(k, 3 Q = R~(k~~ $3 k) = C 9: ‘(k~+ I f k~~u~(k~~ 9 k)- (44 m=l

Here Ff 1 is the inverse of sI, which is determined by the optical potential

@C(ki , km) = (4, f U!“‘(k, , k)Kz), (44)

where W,m is related to the Gaussian grid points w,,, by

2 k;

ii E-Efk,) wm W, =

- 2 lim 7c (

k”-“‘)( 5 w,k; )

k’+ko E-E(k’) Z=I k&k,2 172 = Nfl.

Applying the same procedure to the “post” form of the R-matrix equation, we get

Here, the $I matrix is similar to, but slightly different from, F2 (44), namely

~~(k,, ki) = S,i+ W, Uj”(k, ) ki).

From either (43) or (45), one can calculate the on-shell R,(kO, k,), which is also needed in evaluating the transition matrix (39). The inverse matrices %;I and 9, * -’ will ahso be used to obtain the distorted standing waves.

We spare the reader additional details [see ref. “)] and simply assert that the above inverse matrices FL-* and @;’ can indeed be used directly to construct the distorted standing-waves within integrals of the type given by eq. (39)?. This important feature, which is demonstrated in ref. 5), allows us to use numerical solutions of the R-matrix

equation to compute integrals involving distorted waves. Using that property, the

* One can show that the r-space radial distorted waves can be expressed as a Schliimilch expansion @t,kc(+) = ~~~~(~~~~)~~-I(~~~, k,), which demonstrates the close relationship batwean the momentum space method described here and that discussed by Robson and Koshei 13).


double integration in eq. (39) reduces to an easily evaluated double sum

Therefore, the inverse matrices grml and 3;’ are now direct input for the distorted wave calculation.

With this numerical method, we can now calculate the pion-nucleus inelastic scattering transition matrix for either nonlocal or local pion-nucleus optical potentials. For the cases studied in this work 16 Gaussian grid points prove generally to be enough to get sufficient accuracy.

5. Results and discussion

The inelastic scattering of 120-280 MeV pions was calculated for the excited 2+(4.43 MeV) and 3-(9.64 MeV) states of “C using the momentum space distorted wave method (sect. 4). The flexibility of the momentum space numerical method permits us to perform exlensive calculations using various nonlocal inelastic interactions, U,, , which can differ from each other in the choice of pion-nucleon off-shell interaction. By comparing the results calculated using different models of the pion- nucleus inelastic interaction, we are able to study the pion-nucleon off-shell effect on pion-nucleus inelastic scattering. Separately, we study the reaction mechanism and explore the extent to which the pion can be used to distinguish between different nuclear wavefunctions. The results are also compared to available data 19).

5.1. OFF-SHELL EFFECT

The pion-nucleon off-shell effect on pion-nucleus inelastic scattering is studied in this section by comparing the inelastic differential cross section calculated using three different off-shell pion-nucleus inelastic interactions U,,, . These three transition potentials are separately constructed from the Kisslinger, Laplacian and LT off-shell models of the pion-nucleon t-matrix. In each case, the same 2+(4.43 MeV) collective nuclear form factor [eq. (15)] is used. To probe the off-shell effect on the excitation mechanism, only the inelastic interaction U,, is varied while the distorted waves xzO used in evaluating the inelastic transition matrix (~~O]Uno]~k+o) are all generated from the same elastic optical potential ‘I) (using the LT model to define the elastic optical potential U,, off-the-energy shell). The off-shell behavior of these three pion-nucleon, off-shell models is illustrated in fig. 1. We see that the Kisslinger and Laplacian models are divergent at high momenta, while the Landau-Tabakin model is designed to fall-off to zero. The pion-nucleus inelastic differential cross sections (do/da) calculated from these three off-shell models are compared in fig. 2,


LAPLACIAN_,j / IO - I ./

“>

: ./

: ./ e

3

2 6

x I- O

0 0 200 400 600 800

?f-N OFF-SHELL MOMENTUM K (M&W

Fig. 1. Three half-off-shell (K’ f K = ~a) models of the pion-nucleon collision matrix in the pion- nucleon c.m. system. The dashed-dot curve is for the Kisslinger model: the dashed curve is for the

Laplacian model; the solid curve is for the Landau-Tabakin model.

4o” 80” 120° 40” 80” 120°

8 cm.

Fig. 2. Pion-nucleon off-shell effect on the pion-nucleus inelastic scattering, only U,,, is altered. The three curves are the pion-l*C inelastic differential cross sections for the 2+ state calculated using three different pion-nucleon off-shell models which are the Landau-Tabakin (solid curve), the

Laplacian (dashed curve), and the Kisslinger (dashed-dot curve).

at various pion kinetic energies. One can see that off-shell variations in U,,, yield

negligible changes in the region around the first diffraction maximum 9 M 30”.

This small off-shell effect in the region of small scattering angles is probably due to

the fact that these three off-shell models are almost identical for small off-shell mo-

menta (fig. 1). At larger angles, the depth of the second diffraction minimum at


0 x 62” and the magnitudes of the second diffraction maximum are different for different choices of the off-shell models. The importance of the off-shell effect at large angle parallels that found for elastic scattering ll). The significant off-shell effect is mostly in the region of large momentum transfer and is apparently caused by the tremendous differences between the three pion-nucleon off-shell interactions at high momenta (fig. 1).

Another way to isolate off-shell dependence, which is perhaps more consistent, is to obtain both the inelastic interaction U,, and the optical potential U,, from the same pion-nucleon model. Such calculations have also been done with essentially the same results. The off-shell dependence appears mainly at large angles and closely parallels the elastic scattering changes. The latter point, of a strong elastic to inelastic relationship, suggests that strong absorption ideas are applicable near the (3,3) resonance region 2 “).

From the comparison of do/d9 for different off-shell interactions, it is seen that for large angle results it is of importance to use a physically reasonable pion-nucleon interaction. Since the LT case has desirable bounded high energy behavior, we have ad- vocated its use here; other bounded pion-nucleon models have recently been developed and are viable candidates for future application ‘*r’).

5.2. REACTION MECHANISM

In this subsection we study the dependence of pion-nucleus inelastic scattering on the shape of the nuclear form factor FJ(r). Our overall goal is to learn whether the nuclear wavefunctions can be explored using pion-nucleus inelastic scattering. We also wish to test the applicability of simple surface direct reaction ideas. Only the collective rotational model of the 2+(4.43 MeV) level of 12C is considered in this analysis of the reaction mechanism.

5.2.1. Resonance ejjiect. The main special feature of pion-nucleus scattering is that the incident pion can form a (3,3) P-wave resonance with individual nucleons in the nucleus. To see the dramatic effect of this resonance on nuclear excitation, the pion-nucleus inelastic differential cross sections were calculated using a pion-nucleon interaction omitting the (3, 3) eigenchannel from U,,, and comparing with the results using the full pion-nucleon interaction. The results obtained for energies near the resonance (EL NN 180 MeV) are shown in fig. 3.

The important role of the (3,3) resonance is shown in fig. 3. Without the resonance

in unO, the cross section is reduced by a factor of almost 100 and a much steeper fall- off with angle appears. Clearly, both the magnitude and the back-angle behavior is dominated by the (3, 3) resonance, as is true for the elastic case also. Another role played by the resonance can be seen in the change in the shape of doidS as the incident pion kinetic energy varies. It is shown in fig. 9 that in the energy range between 120-280 MeV both the data and the calculational results are highly diffractive near the resonance energy EL w 180 MeV. The reason for this feature is probably that as the resonance energy is approached the contributions of the real and imaginary


I I I I I I

20’ 40' 60' 80' 100" 120°

8 c.m.

Fig. 3. Pion-nucleon resonance effect on the pion-nucleus inelastic scattering. The dashed curve is the pion-% inelastic differential cross section for the 2+ state calculated without including the pion- nucleon (3,3) resonance force. The solid curve is the result calculated using the full pion-nucleon

interaction.

parts of the pion-nucleus inelastic scattering amplitude are in phase as functions of the scattering angles. Hence, the diffractive minimum is correspondingly deeper. Similar behavior was seen in the elastic scattering case ‘I). Also, near the resonance energy, the (3,3) resonance generates a strong pion-nucleus absorption and hence a close relationship between earlier elastic scattering and the present inelastic scattering results is expected and seen 2o-22).

5.2.2. Evidence for a surface reaction mechanism. To understand detailed features of pion-nucleus inelastic scattering, let us first consider the partial-wave contributions to the inelastic differential cross section do/dL?. In fig. 4, we compare do/da calculated at EL = 180 MeV using I,,, = 5, 8, and 11, where I,,, is the maximum value of the pion-nucleus relative angular momentum included in evaluating the inelastic transition T-matrix [eq. (40)]. The main feature seen in fig. 4 is that the magnitudes of do/d9 for the forward angles are mainly built up by including the higher partial waves 5 6 I s 8. This large contribution from the higher partial waves indicates that the pion-nucleus scattering is a “surface” reaction 20-z2).

The diffractive pattern of do/da is also caused mostly by the higher partial waves. Comparing dc/d!J for I,,,,, = 8 and I,,, = 11 in fig. 4, we see that the higher partial waves 8 5 2 5 11 tend to give more diffractive structure at large angles. On the other


EL = 180 MeV

I I / / I I I

20” 40’ 60° 80” loo0 I 2o”

6 c.m.

Fig. 4. The contributions to the pion-lzC inelastic differential cross section from the low and high partial waves are compared. The LM is the maximum value of the pion-nucleus relative angular momentum included in the calculation of the pion- lzC inelastic transition matrix for the 2+ state

hand, the dc/dR for only six partial waves (I S 5) is almost out of phase with the da/dQ found using I,,,,, = 8. This indicates that the low partial waves tend to wash out the large angle diffractive structure. Hence, it is natural to expect that the do/dQ will be more diffractive if the nuclear form factor has a longer tail or is located at a larger distance from the nuclear center.

5.2.3. Dependence on the nuclear form factor. Having examined the pion-nucleus inelastic scattering mechanism, we will now investigate the extent to which a pion can distinguish between various shapes of the nuclear form factors, i.e. nuclear wavefunctions. The form factor is characterized mainly by the location of its peak and its shape. Therefore, the surface nature of the reaction and the sensitivity of the pion- nucleus inelastic scattering to nuclear structure is readily studied by examining the change induced in do/dO by shifting the form factor’s peak, and varying the inner and outer slopes of the form factor.

In fig. 5b we show the effect of dramatically shifting the location of the form factor peak. The dashed curve is the do/da calculated using a form factor which is located at a smaller radius (obtained by F,(r) + F,(r+0.5 fm) shown in fig. 5a. It is clear that for a smaller radius the first diffractive minimum is shifted to a larger angle. This change is, of course, due to the sensitivity of the diffractive “surface” reaction to the “optical radius”.

Another important effect of shifting the location of the form factor is that the large angle da/dQ is less diffractive (the second minimum is shallower). This is due to the fact that a smaller-radius form factor includes less higher partial wave scattering and the large angle do/dQ is consequently less diffractive. For the same reason, the magnitude of the forward do/da is smaller.


E, = 120 MeV

tl I I I I I

4o” 80” 120”

fm e c.l-0

Fig. 5a. Fig. 5b.

Fig. 5. (a) The solid curve is the original collective 2’ form factor. The dashed curve is the shifted form factor (F(r) -+ F(r+0.5 fm)). (b) The corresponding inelastic differential cross sections calcu-

lated using the form factors in (a) are compared.

To examine the sensitivity of do/dQ to the slope of the form factor (or its shape), we multiply the form factor by an exponential function to generate form factors which have the same peak position, but have greatly different shapes; these altered form factors are shown in fig. 6a. In fig. 6b we show the results for form factors with different slopes at small distances (r 5 1.8 fm). It is clear from fig. 6 that only the back angle scattering is affected by the internal shape of the form factor. This indicates that most of the pions are “absorbed” before they reach the interior of the nucleus. However, the results do show a slight dependence on the internal shape. In particular the back angle do/d!2 is larger if the magnitude of the form factor is smaller in the interior of the nucleus.

In figs. 7a and 7b, the results found using form factors of different external or “surface” slopes are compared. One sees once more that the pion-nucleus inelastic scattering is predominantly a “surface” reaction, since the induced surface effect seen in fig. 7 is very large. One can see from fig. 7 that for a longer-tailed form factor do/dP


EL= 120 MeV

fm 8 cm.

Fig. 6a. Fig. 6b.

Fig. 6. (a) The solid curve is the original collective 2+ form factor. The dashed (dashed-dot) curve is the form factor generated by multiplying the collective 2+ form factor for distances I 5 a,, = 1.8 fm by e-8(*-& (eS(r-ao)), where /I = 0.36 fm-r. (b) The corresponding inelastic differential cross sec-

tions calculated using the form factors in (a) are compared.

is more diffractive, because more higher partial wave scattering is involved. The magnitude of doIdS in the forward angles and the position of the diffraction minimum are also very dependent on the outside slope of the form factor. For a steeper form, dc/dS1 is smaller at forward angles and the second diffractive minimum is shifted to a larger angle. IIence, the effect of reducing the outside slope of the form factor is similar to the effect of shifting the peak of the form factor to a smaller radius.

It should be noted that the variations in form factor illustrated in figs. 5-7 are dramatic and go beyond what is permitted from knowledge of nuclear density. They are used here only to determine the sensitivity of cross sections and clearly reveal only a slight (fig. 6b) dependence on internal and a great (fig. 7b) dependence on external form. That result is indicative of a surface reaction. In conclusion, the pion- nucleus inelastic scattering is quite sensitive to the peak position and the surface slope of the form factor. A more diffractive pattern in do/d&l generally reflects either a longer-tailed form factor or a form factor peaked at a larger radius. The interior


0.1 - 1

IO k /“\ J’ : EL= 120 MeV

I F \i 260

0.1

0.01 I -. tl I I I IU -- 2 3 4 5 4o” 60” 120”

fm e c.m. Fig. 7b. Fig. 7b.

Fig. 7. (a) The solid curve is the original collective 2+ form factor. The dashed (dashed-dot) curve is the form factor generated by multiplying the collective 2 + form factor for distances I 2 a, = 2.2 fm by e-B(r-ao) (es@-ao)), where /? = 0.36 fm-‘. (b) The corresponding inelastic differential cross sec-

tions calculated using the form factors in (a) are compared.

structure of the nuclear form factor plays a small role, and that only for large angle

scattering.

5.3. COMPARISON TO DATA

In this subsection, we compare our theoretical pion-nucleus inelastic scattering

results to the presently available data I’) for the “C 2+(4.43 MeV) and 3-(9.64 MeV)

excited states. For each excited state, cross sections are calculated using three different

nuclear form factors. The first two form factors are separately constructed from the

collective or “macroscopic” model [eq. (15)] and the “microscopic lp-lh model”

[eqs. (28) and (29)] of Gillet and Vinh Mau r2). The results calculated from these

two “physical” wavefunctions are compared with each other and with the data, in

order to see to what extent the GV and the collective nuclear models can be distin-


TABLE 1

Parameters for the nuclear form factors

Collective Gillet-Vinh Mau rotational spin- spin-

model independent dependent

Artificial model

2 + (4.43 Me V) state

dfm3 A(fme2) B(fme3) C(fmT4) a(fm)

3- (9.64 Me V) state

f&m-“) A(fm-*) B(fmm3) C(fm- 4, a(fm)

0.01489 0.0035 0.0035 0.0156 -0.17662 5.52 -3.54 0.5

0 0 0 0 0.38279 -0.0677 0.246 0.1 1.59 1.644 1.644 1.59

0.01489 0.0112 0.0112 0.0107 -0.17622 0 0 -0.2466

0 -0.545 0.178 0 0.38270 0 0 0.365 1.59 1.644 1.644 1.59

The nuclear form factors for the excited states 2+ (4.43 MeV) and 3 - (9.64 MeV) or rzC are defined as Fi(r) = po(Ar2+Br3+Cv4)e-rain’.

guished by pion-nucleus inelastic scattering data. The third form factor is not con-

structed from a physical wavefunction but is artificially designed to further investigate

the sensitivity of the pion-nucleus inelastic scattering to nuclear structure.

5.3.1. Normalization of nuclear form factors. To compare the results using

different nuclear wavefunctions, the collective and the artificial form factors are both

normalized to have the same electric transition rates [B(E2) for 2+ state, and B(E3)

for 3- state] as that of the GV microscopic form factor. Using this normalization,

the deformation parameters pZ, p3 for the collective rotational model prove to be

approximately equal, pZ w j3s M 0.55. The normalization and the parameters of the

artificial form factor are presented in table 1. As a consequence of using this co~lllllon

physical normalization, we find that the magnitude of the first diffraction maximum

is approximately the same for all of these form factors. This similarity suggests that

the pion-nucleus scattering process might be directly related to nuclear structure in-

formation such as is contained in B(E2) and B(E3) values. Similar observations are

well known for other reactions, for example for proton inelastic scattering 23).

5.3.2. Excitation of the 2+ state. The inelastic differential cross sections calculated

using B(E2) normalized wavefunctions for the excited 2+ (4.43 MeV) state are presented

in fig. 9 and compared with the data. Three different curves are calculated using

three different shapes of the form factors shown in fig. 8. Since the collective form

factor does not have a spin-dependent term, for the sake of equivalent comparison,

we have also omitted the spin-dependent term of the GV form factor. The small

effect of the spin-dependent term [eq. (29)] will be discussed later.


fm

8. Three shapes of the 2+ form factors are compared. The parameters of the Gillet-Vinh Mau (GV) collective rotational (COL) and artificial (ART) form factors are given in table 1.

Fig.

From fig. 9 we see that inside the second diffraction minimum 0 w 62”, the theoret-

ical results calculated (without adjusting any parameters) for these three nuclear

form factors are very close to each other and to the data. This result suggests that the

lowest order distorted wave theory is adequate to describe the pion-nucleus inelastic

scattering in the region of small momentum transfer.

Although the magnitudes of drr/dQ for these three form factors are very similar at

small angles 0 5 62”, the position of their second diffraction minima are different.

For all the pion kinetic energies, the diffraction minima are systematically shifted

to a large angle as the shape of the form factor is varied from the collective to the

GV form. This trend of changing the minimum position is, of course, mainly due to

the corresponding shifts of the peak position. (The peaks of these three form factors

are indicated in fig. 8.) Compared to the data, the position of the second diffraction

minimum is perhaps best described by the microscopic GV form factor.

At large angles 8 > 62”, da/d&J for these three form factors are very different in

magnitude and in the diffraction pattern. Since the collective form factor is peaked

at larger distance and has a longer tail (fig. S), the large-angle da/da for the collective

model is therefore more diffractive than the results found for the other two form

factors. For the same reason, the large angle da/dQ for the artificial form factor is

more diffractive than that of the GV form factor.

The magnitude and the diffractive pattern of da/d9 calculated using the collective

form factor is comparable to the data. However, the diffraction minima predicted


I I I I I T I I t , ! , ,-- , 40D 80’ 120” 4o” 00” 120°

9 c.m.

Fig. 9. The inelastic differential cross sections calculated using the three 2+ form factors (shown in fig. 8) are compared to the data 19).

using the collective form factor are at angles that are too small compared to the data, especially for the third minimum. Perhaps the results for the collective model can be improved if use of a smaller nuclear radius could be justified which would shift the peak of the collective form factor closer to the nuclear center; the diffractive minima then would be correspondingly shifted to a larger angle and yield a better fit to the data.

The large angle da/da results for the GV form factor are generally too large and less diffractive than required by the data. The reason for this behavior is that the tail of the GV form factor falls off too fast and therefore lacks sufficient contributions from higher partial waves, which are essential for producing the diffraction pattern at large angles. The inclusion of a small component of higher orbitals in the microscopic 2+ excited state wavefunction could sufficiently elongate the tail of the form factor, and might therefore improve the fit to the pion-nucleus inelastic scattering data.

PION-NUCLEUS INELASTIC SCATTERING

1

3

I 2 3 4

Fermi

277

Fig. 10. Three shapes of the 3- form factors are compared. The parameters of the Gillet-Vinh Mau (GV), collective rotational (COL) and artificial (ART) form factors are given in table 1.

5.3.3. Excitation of the 3- state. The quality of the fit to the data for the 3- (9.64 MeV) state is similar to the 2+(4.43 MeV) case. The shapes of the three form factors used in the calculation are shown in fig. 10. The artificial form factor used for this case is located at a larger distance from the nuclear center and has the longest tail. The theoretical results are compared to the data in fig. 11.

Inside the first minimum 6’ 5 66”, the three theoretical results are very similar, and the fit to the data is quite good. This again means that the da/d8 inside the lirst minimum is not very sensitive to the details of the nuclear structure and can be satisfactorily described by the lowest order distorted wave theory and the B(EL) normalized wavefunctions.

The relationship between the location of the first diffraction minimum and the form factor peak is the same as that of the 2+ state. Compared to the other two form factors (see fig. lo), the GV form factor has its peak at the smallest distance; correspondingly the first diffraction minimum occurs as expected at a larger angle.

The sensitivity of the large angle do/dQ to the surface shape of the form factor is also the same as that of the 2+ state. Comparing the tail part of these three form factors in fig. 10, we see that the GV form factor falls off to zero faster than the other two form factors. Consequently, the large angle do/da for the GV form factor is less diffractive compared to that of the other two form factors. To have a better fit to the data, a small component of higher orbital configurations is apparently also needed in the microscopic wavefunction of the 3- state.


230 E,= 150 MeY

FlI( I II 1 -_-.

0.01 , / , ’ , , , / -_

4o” 80” 120” 40’ 80’ 120” 8 c m.

Fig. 11. The inelastic differential cross sections calculated using the three 3- (9.64 MeV) form factors (shown in fig. 10) are compared to the data 19).

5.3.4. Spin-dependent eflect. In the previous calculations the spin-dependent term

of the GV form factor was not included. Here, we study the extent to which the do/da will be changed when the pion-nucleon spin-dependent interaction is included.

As shown in eq. (26), the spin-dependent term is mainly determined by the pion- nucleon P-wave interaction. The contributing spin-dependent term proves therefore to be momentum-dependent. Consequently, the spin-dependent effect is expected to be significant at large angles [as suggested by Nishiyama and Ohtsubo in their plane wave case “)I.

In fig. 12, we compare do/da calculated using the GV form factor with and without the spin-dependent term. One can see that the spin-dependent effect is negligible at angles smaller than the first minimum. Beyond the first minimum, the magnitude of do/d0 is increased by including the spin-dependent effect, but is not changed very significantly.

The major effect of the spin-dependent term is to fill in the minimum, yielding a value that is closer to the data. However, the position of the minimum is not shifted by the spin-dependent interaction.


EL= 150MeV

I I I I i I I I Fi I I I I I I 40” 80” 120” 4o” 80” 12o0

8 c.m.

Fig. 12. Spin-dependent effect on the pion-nucleus inelastic scattering. The solid curve is the result calculated only including the spin-independent force. The dashed curve is the result calculated including both the spin-dependent and spin-independent forces. The Gillet-Vinh Mau form factor is used

in constructing the pion-nucleus inelastic interaction.

5.4. CONCLUSIONS

Momentum-space distorted wave impulse approximation has been used to study the reaction mechanism for pion-nucleus inelastic scattering at energies near the (3, 3) resonance. We found that the nucleus is mainly excited by the pion-nucleon (3,3) resonance when the medium-energy pion is inelastically scattered from the nucleus. Because of this strong resonance force, the diffractive pattern of pion-nucleus inelastic differential cross sections are highly energy dependent, and are most diffractive near the (3, 3) resonance. We also show that the pion-nucleus inelastic scattering near the resonance is predominantly a “surface” reaction; the nuclear interior can only be explored by pions at large angles. Hence the ideas in ref. ‘“) concerning strongly absorbed particles are clearly applicable.

Since nuclear wavefunctions are explicitly needed in this distorted wave calculation, some relationships between the pion-nucleus inelastic scattering and nuclear structure were established. These relationships are mainly: (i) at angles smaller than the first diffraction minimum, the pion-nucleus inelastic scattering is insensitive to details of


the nuclear structure; (ii) the location of the diffraction minimum is sensitive to the location of the nuclear form factor peak (the first diffraction minimum is located at a smaller angle when the form factor is peaked at a larger distance from the nuclear center); (iii) the large angle do/d!2 is mostly sensitive to the outside surface shape of the nuclear form factor (the large angle do/dQ is more diffractive for a longer-tailed form factor); (iv) the inner surface shape of the form factor can only be detected by measuring the back angle scattering. Hence, detailed nuclear structure information is mainly revealed in pion-nucleus inelastic scattering in the region well beyond the first dtj?raction minimum. Since we also find that the pion-nucleon off-shell effect is also signi$cant at large angles, it is therefore necessary to use a physically reasonable pion-nucleon o#-shell model to unfold this information. The nonlocal separable LT model includes the resonance and is bounded, but it is by no means a best model, Use of better, nondivergent off-shell models of the pion-nucleon interaction is apparently desirable for future calculations.

One satisfying result of this work is a build-up of the back-angle cross section by a factor of about 10 compared to earlier DWIA calculations of Edwards and Rost “). This build-up, which is also found in the elastic case, most likely arises from the use of angle transformation [eq. (7)] as discussed by Kisslinger and Tabakin 15),

In comparing with the available data, we find that the theoretical results fit the data quite well at small angles, but are not good beyond the first minimum. A better fit can be obtained, not only by using a better off-shell model or improved nuclear form factors, but also by using better distorted waves. The distorted waves can be refined by improving the pion-nucleus elastic optical potential. This indicates that the optical potential must be improved to precisely fit the elastic data, in order to ensure a fit to the inelastic data.

Some other possible future improvements of the optical potential are to include correlation and nuclear excitation corrections, and the effect of the Pauli principle “). The momentum-space distorted wave method developed in this work is flexible and can be used for any bounded nonlocal or local form of pion-nucleus potential. Hence, the method can readily be extended to include such effects, which is the main advan- tage of the momentum-space formulation. Another possible application of this formulation is to study pion-nucleus charge-exchange reactions. The Coulomb force is not significant for intermediate energy pions scattering from light nuclei, and has been ignored in our theoretical study of pion- 12C inelastic scattering. However, in order to extend our method to study the low-energy pion-nucleus scattering and the scattering of pions from heavy nuclei, we plan to properly include the Coulomb force in our momentum-space distorted wave method using an approach recently developed by Vincent and Phatak 24).

Based on the study of the pion-nucleus inelastic scattering presented in this work, we are encouraged to make further studies and to apply direct reaction, strong absorption ideas to the forthcoming data. Ultimately, we hope that such methods might be used to extract new nuclear structure information. The forthcoming pion-nucleus


scattering data could then be used to confirm and perhaps improve our understanding of nuclear structure.

The authors deeply appreciate the constructive and generous help provided by Dr. R. M. Drisko. Dr. G. M. Miller made several very important suggestions for which we are grateful. We would also like to thank Drs. R. H. Landau, S. Phatak, C. M. Vincent, S. Pittel and N. Austern for helpful discussions. Use of the University of Pittsburgh Computer Center is also acknowledged.

References

1) H. K. Lee and H. McManus, Nucl. Phys. Al67 (1971) 257; C. Rogers and C. Wilkins, Nuovo Cim. Lett. 1 (1971) 575; C. Wilkin, Spring School on pion interactions at low and medium energies, Geneva, 1971; R. Guardiola, to be published

2) T. Nishiyama and H. Ohtsabo, Prog. Theor. Phys. 48 (1972) 188; Y. Futami and T. Kohmura, Prog. Theor. Phys. 48 (1972) 1563

3) G. W. Edwards and E. Rost, Phys. Rev. Lett. 26 (1971) 785; G. W. Edwards, Ph. D. thesis, Univ. of Colorado, 1972

4) A. K. Kerman, H. McManus and R. M. Thaler, Ann. of Phys. 8 (1959) 564 5) T. S. H. Lee, Ph. D. thesis, Univ. of Pittsburgh, 1973 6) H. A. Bethe, Phys. Rev. Lett. 30 (1973) 105;

N. R. Nath, H. J. Weber and J. M. Eisenberg, Phys. Rev. C8 (1973) 2488; C. B. Dover and R. H. Lemmer, Phys. Rev. C7 (1973) 2312; R. H. Landau and M. McMillan, Phys. Rev. C8 (1973) 2094; J. B. Cammarata and M. K. Banerjee, Phys. Rev. Lett. 31 (1973) 610; Proc. Int. Summer School on pion-nucleus multiple scattering, Los Alamos, 1973, and references therein

7) L. S. Kisslinger, Phys. Rev. 98 (1955) 761 8) R. H. Landau and F. Tabakin, Phys. Rev. D5 (1972) 2746;

M. G. Piepho and G. E. Walker, Bull. Am. Phys. Sot. 17 (1972) 589 9) J. T. Londergan and E. J. Moniz, Phys. Lett. 45B (1973) 195;

J. T. Londergan, K. W. McVoy and E. J. Moniz, to be published 10) W. P. Nutt and L. Wilets, to be published 11) R. H. Landau, S. C. Phatak and F. Tabakin, Ann. of Phys. 78 (1973) 299 12) V. Gillet and N. Vinh Mau, Nucl. Phys. 54 (1964) 321 13) D. Robson and R. D. Koshel, Phys. Rev. C6 (1972) 1125;

L. A. Charlton and D. Robson, Bull. Am. Phys. Sot. 17 (1972) 508; L. A. Charlton, Phys. Rev. C8 (1973) 146

14) D. S. Koltun, Advances in nuclear physics, vol. 3, ed. M. Baranger and E. Vogt (Plenum Press, New York, 1969)

15) G. Faldt, Phys. Rev. C5 (1972) 400; L. S. Kisslinger and F. Tabakin, Phys. Rev. C9 (1974) 188; E. Kujawski and G. A. Miller, preprint

16) D. M. Brink and G. R. Satchler, Angular momentum (Oxford Univ. Press, London, 1968) 17) M. I. Haftel and F. Tabakin, Nucl. Phys. Al58 (1970) 1 18) M. I. Haftel, Ph. D. thesis, Univ. of Pittsburgh, 1969 19) F. Binon, P. Duteil, J. P. Garron, J. Gorres, L. Hugon, J. P. Peigneux, C. Schmit, M. Spighel and

J. P. Stroot, Nucl. Phys. B17 (1970) 168 20) N. Austern and J. S. Blair, Ann. of Phys. 33 (1965) 15 21) N. Austern, Direct nuclear reaction theories (Wiley, NY, 1970) 22) J. B. Blair, Lectures in theoretical physics, vol. 1 (Univ. of Colorado Press, 1966) 23) G. R. Satchler, 2. Phys. 260 (1973) 209 24) C. M. Vincent and S. C. Phatak, private communication, and to be published

momentum-space study of pion-nucleus inelastic scattering

Documents