Elastic scattering of pions on light nuclei

Nuclear Physics A174 (1971) 251-272; Not to be reproduced

ELASTIC

by photoprint

SCATTERING

@ North-Holland

Publishing

Co., Amsterdam

or microfilm without written permission from the publisher

OF PIONS ON LIGHT NUCLEI

(I). A study in coordinate space with Kisslinger’s model J. P. DEDONDER Institut

de Physique

NuclPaire,

Division Received

de Physique 2 April

ThPorique t, 91-Orsay

- France

1971

Abstract: We analyze the approximations introduced in Kisslinger’s model and suggest that some of these approximations may be removed without losing the simplicity of the model. We thus obtain good fits for forward scattering at energies close to and above the 3-3 resonance. However, the fits become quite poor at low energies.

1. Introduction The interest in pion-nucleus elastic scattering has been steadily increasing in the past few years. Different approaches ‘) have been introduced to study the experimental data available ‘). However, mcst analyses have used some form of multiple scattering formalism, in which the nuclear target is treated as a system containing elementary subsystems (i.e. nucleons). The interaction of the projectile with the nuclear target is then simply related to the interaction of the projectile with the elementary subsystems. The most successful approach seems to be Kisslinger’s model I’), which has been derived in the framework of Watson’s multiple scattering theory “). One assumes that non-relativistic quantum mechanics is an appropriate tool for the description of the scattering of pions by nuclei. In addition, we make the following assumptions: (i) One may dissociate the nucleon-nucleon interaction from the pion-nucleon interaction. (ii) The pion-nucleon interaction can be described by a potential. (iii) Only two-body forces are important. One neglects, thus, the absorption of pions on a pair of nucleons although it appears to be quite important a process at low energies. In this paper, we shall recall in sect. 2 the derivation of the optical potential using the Kerman-McManus-Thaler method “). In sect. 3 we shall derive Kisslinger’s potential and specify the different approximations introduced. In sect. 4 we present a brief discussion of the wave equation used to calculate the pion-nucleus elastic scattering amplitude. In sect. 5 we drop the A >> 1 approximation and consider nuclear c.m. motion; we suggest then a kinematical correction which allows us to obtain much better fits for forward scattering. We present our conclusions in sect. 6. t Laboratoire

associ6

au CNRS. 251

J. P. DEDONDER

252

2. Pion-nucleus

optical potential

Let us introduce the Hamiltonian H of the pion-nucleus system: H = H,+V-, where Ho = H,+K,. HN

is the Hamiltonian of the target nucleus and K,, the kinetic energy operator of the incident pion. V is the sum of two-body interactions between the incident pion and target nucleons: Y- = $Ji. i=l

Then we know that the pion-nucleus scattering matrix is a solution of the integral equation 9-(E) = V+ ~G,f(E)9-(E), (2.1) G:(E)

=

[E+iE-_O]-l.

(2.2)

If we now iterate eq. (2.1) and sum the terms corresponding to successive scatterings on the same nucleon, we obtain I

= ~ t,(E)+ i=l

~ fi(E)G~(E)~j(E)+

. . .)

i, j=l i#j

(2.3)

where ti(E) = pi+ DiG,f(E)ti(E).

(24

It may then be shown that this operator ti(E), which we shall call the pion-nucleon scattering matrix inside the nucleus, is related to the free pion-nucleon scattering matrix ri(E), which obeys the integral equation

(2.5)

Zi(E) = Vi+ Vigo+(E)ri(E), where g:(E)

so

=

[

_??+I’&- g

-K,

1 -l,

(2.6)

G,f(E)t,(E)*

(2.7)

that

ti(E) = ri(E)+ri(E)g,f(E)

(HN- g}

Since we want to study elastic scattering in the ground state, we may write =
1

Using antisymmetrized

i#j N

target wave functions, we have

(01 C fi(E)IN> = A
(2.8)

n-N ELASTIC SCATTERING

253

if the operator ti(E) is a two-body operator and is diagonal for a11the nucleons of the target except one. Then, = A(Olt(E)IO)+A(A-1)

c (Olt(E)~N)G,+(E-E,))(Nl@)E)IO)+

. . ..

N

(2.9) Let us introduce the projection operator on the ground state P, = lO)(Ol and the operator

T,,(E)= A+ , so that we finally obtain the two equations

T,,(E) = uoo(E){lfG,+(E)T,o(E~l~ UOO(E) = (A - l)
(2.11)

1

1 -(A - l)(l - ~o)G~(~)~(~)

IO>*

(2.12)

Thus, the eiastic pion scattering problem has been reduced to the problem of the scattering by a potential. This potential contains all the necessary information on nuclear structure. 3. Kisslmger’s model This model, which has been derived to study the elastic a-nucleus scattering in the low-energy range, is readiIy found from Watson’s theory by introducing a few approximations. Coherent scattering approximation. In other words, we neglect all virtual nuclear excited states, so that: U”(E) = (A - l)(OJt(E)~O>. (3.1) Impulse approximation. We drop the second term of eq. (2.7) so that

4(-Q = r&),

(3.2)

where E is the total energy of the n-nucleus system and E is the energy of the incident pion. This means that we suppose that the pion-nucleon scattering matrix is not modified in nuclear matter. This is certainly valid in the high-energy domain where one may neglect Pauli correlations as well as the binding energies of the nucleons. It has been shown however by Goldberger and Watson “) that the impulse approximation is not valid in the vicinity of the resonance and that the free pion-nucleon scattering amplitude has to be corrected. One supposes A B 1. Thus eq. (2.11) which, using (2.10) and (3.1), may be written

= A+ AG,+(E) ‘2 becomes



(Ol9-(E)lO> = A(1 +G,+tE))-

(3.3)

(3.4)

J. P. DEDONDER

254

This is obviously terms of order

an excellent

approximation

l/A, but it is dubious

for most nuclei,

for light nuclei

since one neglects

(one should

remark

that most

studies of the rc-nucleus interaction have been made for light nuclei: 4He, “C, r60). Kinematical approximations. Let qi and qf be the momenta of the pion in the pionnucleus c.m. system (which will be referred to, from now on, as ACMS). Neglecting nuclear c.m. motion (we shall use independent-particle model wave functions which are not translationally invariant), the optical matrix element may be written:

s

= A dkl
(3.5)

where q = qi-qf is the momentum transfer and poo(kl +q, k,) is the probability that a nucleon has momentum k, in the initial state and k, +q in the final state: ,ooo(k, +q, k,) =

s

dk, . . . dk,, @;(k, +q, k, . . ., k&D&,

k,, . . . k,).

(3.6)

In order to compute the optical-potential matrix element, we introduce a few assumptions. We consider scattering with a momentum transfer small compared to the mean nucleon momentum in the target. In this case, the kinematics is close to that of free pion-nucleon scattering in the forward direction: 3qi)ACM w
(ki + 4, qrlr(sJlki

7 kl)*

One then supposes that this matrix element is a slowly varying function tum k, 50 that it can be taken out of the integral (3.5). We thus obtam (4fl

of momen-

uoo(EA)14i).4CM = Az*CM(cli 34f 3 EA)P((Ih

(3.7)

where 44) is the nuclear

form factor,

=

s

dk, Poo(k, + 4, k,)

and

rACM(qi >(If 3&A) = TAcM(qi >qf 3 k, 9EA)IC~ =O

=

(k,

>

qflT(EA)lqi 3kl)ACMlkLI =o

ZACM((Iiy qf, eA) is the free pion-nucleon

scattering matrix element where the sum over the spin and isospin coordinates has already been done. This approximation (as well as the impulse approximation) should be quite good

for high-energy processes such that (qi( and )qfl >> k, where k, is the Fermi momentum of the nucleus. However, for intermediate-energy pions one cannot expect it to hold (we have taken k, = 0 in the lab system as is usually done). We now have to compute the matrix element r ACM(qi, qf, E”) from the pion-nucleon elastic scattering amplitude in the two-body pion-nucleon c.m.s. (which will be noted 2 CMS). Using invariant amplitudes, we get t

+ We show the dependence nucleon scattering amplitude.

on energy

of the amplitude

because

we need, in fact, the off-shell

pion-

n-N ELASTIC SCATTERING

where Ed is the pion energy in the 2 CMS, Ez the the pion energy in the ACMS, and EA the nucleon One is now left with the problem of determining EA. This can be dealt with using the invariants of A; = (Ed+&)’

=

255

nucleon energy in the 2 CMS, sA. energy in the ACMS. the quantities q2, q;, .Q, E,, E*, the problem 4):

(Wl+E,)‘-k;,

where I is the incident pion energy (momentum) in the lab system. Let EI be the total energy of the nuclear target in the ACMS, one has El x AE,.

Then A; = (Q+E$)’

= (Am+&,)‘-k;.

Thus A 2

= L &4 =

m2+2msL+p2

A*

A qz

A2m2+2AmeL+pz

1 +,

where q2(qA) is the 2 CMS(ACMS) + momentum, and z.4CM(4i

P

4f

9

&A)= -2n

Since EI >> E*, one finally gets (4fl

’ %_f

uoo(EA)lqi> =-

&A

q2

If we go to the Chew-Low static limit (p/m + 0), as is done in Kisslinger’s original work, we find

=

-

‘f &A

qA, (

qi’4f

7 &A

P(qi-qf)*

(3.10)

4:

Auerbach et al. lcs Id) have considered the form ++

this approximation may be justified for very small angles, although the replacement of q2 * q;/qi by qi * qf/qj in the free elastic pion-nucleon scattering amplitude should yield quite different results for non forward scattering. We feel that, in order to be consistent with the approximation introduced, one should use (3.9). One must bear in mind that in all these calculations we have neglected nuclear cm. motion and considered only small momentum transfer. Moreover, we have completely neglected the nucleon momentum in the target by setting k, to zero in the lab system. We feel one should look carefully into this approximation. + Ql = /%I = 14. ++It should be noted that in refs. Ia) through lc) all the authors have neglected the transformation from the lab system to the ACMS which, if not important for 12C, is not negligible for 4He.

J. P. DEDONDER

256

Furthermore, it should be noted that one needs, in fact, the off-shell pion-nucleon scattering amplitude to evaluate the optical potential. Thus one should use a model that gives these effects correctly. The Chew-Low “) model appears to be quite appropriate for such calculations and one should try to evaluate the optical potential with it. Free pion-nucleon scattering. To determine completely the optical potential for elastic scattering one has to specify the free pion-nucleon scattering amplitude. The most striking features of Ic-nucleon scattering are the P-wave character of low-energy scattering and the presence of the P-wave 3-3 resonance at T, NN195 MeV. In the energy range of interest (O-300 MeV), where only S- and P-waves are important, the elastic z-N scattering amplitude is (in the n-N c.m.s., i.e. 2 CMS) f(k,k’)

= 2k[{.)‘P”(k,k’)+c,,Pt(k,k’)}~~+{ol,,P*(k,k’)+a,,Pt(k,k’)}Ttl + ; [% T*+a, T+l,

where

(3.13)

*k = lkl = lk’l, u2J,

and

2T

=

exp

is2J,

2Th)

sin

82J,

2Thh

$T = exp id&&& sin 82T(&&

are respectively the P- and S-wave partial amplitudes, with

E,‘= (k2 +p2)*. P,(k, k’) = 3$ (2k * k’- ia * (k’ A k)) and P,(k, k’) = 5

{k * k’+ iu - (k’ h k)}

are the projection operators on the states of angular momentum J = 3 and J = + T+(T+) is the isospin projection operator on the state of total isospin +(+). For a

doubly closed-shell nucleus, the spin-flip terms give no contribution. Otherwise, for a J = 0 nucleus the exact contribution of the spin-flip terms is very small and disappears

in the Chew-Low

static limit. Thus, we obtain i

(q2. ~4;4; ,e2)=a+pq, 42

*

* q;,

where c( = L q2

(3A-2Z)a,+2Zcr,

3A

’

p = J_ (3A-2ZX2%+~l)+2%, 4:

+2a,,),

(3.14)

n-N ELASTIC SCAmERING

dz =

257

1421 = ltd.

Finally, the optical potential is given by: <4rl uoo(8A)lqi) = Cs + l4i * clf)P(Qi -4f),

(3.15)

where according to Kisslinger’s original work Ia) s = Sk C--Q,

2x s‘4

t = fK =--

211 BP &A

(3.16)

or using Auerbach’s calculations I’) q.4 __ s = s*, = - 2n - - a, &A (I2

(3.17) Because we need the off-shell pion-nucleon scattering amplitude, we shall suppose it has the same form as for on-shell scattering and we shall always use on-shell partialwave amplitudes although this is probably not a very good approximation (Soft-pion theorems cannot be applied, in fact, because we need optical potential matrix elements very far away from the energy shell to solve the integral equation (2.11)). 4. The wave equation What one has to do now is to solve (3.4) with the optical potential (3.15) i.e., one has to solve an integral equation. This is in itself a difficult problem and, in general, one transforms to coordinate space and solves the associated Schrodinger equation. To do this one has to compute the Fourier transform of the optical potential. This is simple if one assumes the coefficients s and t to be slowly varying functions of the momenta (i.e., factorization approximation); one then gets PN(%(~) = +I uoolrp) = A{s&%+)-

rV(&)Vrp(r))19

(4.1)

where p(r) is the nuclear density and q(r) the pion wave function. The contribution of the P-wave gives rise to two terms appearing one as an effective-mass term and the other, depending on the density gradient, as a surface term which should be particularly important for light nuclei. Thus, one should have to solve the equation { -V2+2VN(r)}(p(r)

= 2flGcp(r),

(4.2)

where p is the n-nucleus reduced mass and T, the kinetic energy of the pion in the ACMS. However, since pions even in the energy range considered, are relativistic

J. P. DEDONDER

258

particles, Kisslinger modified slightly this equation to take into account, roughly, relativistic effects. He used the following wave equation: (A + ~~)~(~) = 28, ~~(~)~(~)*

at least

(4.3)

This is equivalent to the replacement of the reduced mass j,Iin (4.2) by the pion total energy &A. Auerbach et al. Ia) have preferred to introduce the Klein-Gordon wave equation where the optical potential is treated as a fourth component (in analogy to the Coulomb potential which is the fourth component of the four-vector potential in electromagnetism). Then the pion wave function satisfies the equation (-vz

+$%(y)

=

(&A--

vCVc(r)-

vi/N(r))2v)(y)-

(4.4)

One then drops the I’; term assuming it is small compared to 2~~ V,(F), although there is little theoretical justification for this assumption. Thus they have to solve the following equation: ( - v2 + fi’)(P(r) =

(&A-

~C(r))2dr)

-

udr)y

(4.5)

where

5. Some improvements of the model 5.1. REMOVAL OF THE A >> 1 ASSUMPTION CM. MOTION

AND

CONSIDERATION

OF NUCLEAR

(i) Since we want to study the elastic pion scattering on light nuclei, we shall first drop the A > 1 assumption. We thus have to solve eq. (2.11) with @‘(&A) = (A - 1)(O12(a.&o), which yields the optical potential matrix element: <4flUoo(EA)l@i>

=

(A-1)(s+t4i

’ 4&(4i_9f)*

P-1)

The elastic n-nucleus scattering amplitude is then given by (2.10). Then, as we have remarked earlier, the optical potential matrix elements are evaluated for independent-particle models (in general, the harmonic oscillator model). Thus one completely neglects the nuclear c.m. motion because such models give rise to wave functions which are not translationally invariant. It has been shown, however, by S. Gartenhaus and C. Schwartz ‘) that one can introduce a unitary transformation which allows one to take into account the effects due to this motion. Let K and K’ be the nuclear momenta in the initial and final states, eq. (3.5) becomes then <4f, K’(‘O’(&A)l~, qi> = (A-l)6(K’+4r_K_qi)

I ir dki6(K-‘,) l

X[@g(k,+qi-q,,

’ i=l

kz,. . .) kA) (5.2)

n-N ELASTIC SCATTERING

259

As we have already done, we use the factorization approximation thus have to compute the form factor: s(q)

=I

+q, k2 3 . . .> k,J@&

fi dkia(K-kJ@,*(kr

, k,,

(eq. 3.7) and we

. . ., k,,),

(5.3)

Transfor~ng to coordinate space and applying the Gartenhaus-Schwartz formation, we obtain:

trans-

i=l

with 4 =

4i--4f.

This last expression is the A-particle form factor

We now suppose that the nucleons in the nucleus are completely uncorrelated so that the form factor F(q) breaks into the product of A terms:

~Gz>= P

(dy)[P(g-y

(5.4)

where p(q) is the one-partide form factor. In the case of the 4He with the configuration (1~)~ and using harmonic oscillator wave functions we get F(q)

= exp - 5,

where v’=

A -v A-l

and v is the harmonic oscillator strength parameter. In the case of carbon or oxygen nuclei, if one considers small momentum transfers, eq. (5.4) reduces to, using harmonic oscillator wave functions:

flk) =

$1exp (- f;),

(I- F

(5.5)

2 being the number of protons (Z = A/2), v’ = (A/A - 1)~ and we have 52-4

v
22

where is the mean square radius of the nucleus. The optical potential is thus given by expression (5.1) when we replace p(q) by S(q).

J. P. DEDONDER

260

(ii) ResuIts. lomb potential:

In these calculations we have solved eq. (4.5) neglecting the Cou(A + 42&9

= uq(r).

Following Auerbach et al., we neglected the corrections due to the transformation from the lab system to the ACMS; so that in fact we have solved the equation

where a and p are given by (3.14). TABLE 1

The parameters Energy

,h, and bl calculated from Donnachie’s

phase-shift

analysis

Re b.

Im bc

Re b,

Im bl

-1.78 -1.52 -1.46 -1.11 -1.10 -0.65 -0.40

1.08 0.68 0.58 0.44 0.36 0.32 0.28

6.00 7.80 8.05 7.95 5.20 0.15 -2.05

0.26 1.26 1.90 5.26 8.50 1.35 2.55

(MW

24 60 69.5 120.0 153.0 200.0 280.0 Units are fm3.

The parameters be and b,, which are identical for all three nuclei (4He, 12C, i60), are calculated from Donnachie’s phase-shift analysis “) and their values appear in table 1. The density, calculated for the harmonic oscillator well, is equal to

PC9=

2 1)’ ‘{l+_5(Z-2)vr2]t exp (-2). 2

;

The results of the calculations appear in figs. 1-8 (we present calculation for “He and “C, since for 12C and 160 the results are quite similar). We can see that the results are in general rather good. The removal of the two approximations mentioned in this section allows us to obtain better overall fits using the theoretical parameters. As expected, the effects are especially important for the 4He f Dropping the A > 1 approximation,

we have to replace A by (A-l).

~~----*----L-

-c’Z

120

140

160 8"

Fig. 1. n--r2C differential cross sections at Tz = 69.5 MeV. The dotted curve is obtained with the original Auerbach calculations (eq. 5.6). The dash-dot curve is obtained with the removal of the A > 1 approximation and the plain curve is obtained by removing the A >> 1 approximation and considering nuclear cm. motion. The rms radius is
0 01

n

T-E= 69.5 Me\!

00,

1

’

’

I,,

160 a” Fig. 2. Same as fig. 1 for Tn = 120 MeV [data from ref. 2r)1.

L..-LLL_L_I._I_L~_~-~6 20 40 60

1

E

N

Me\'

Fig. 3. Same as fig. 1 for 7’, = 200 McV [data from ref. 2’)].

-r-c.,7. ‘I; =200 du-

dla

._

Fig. 4. Same as fig. 1 for T, = 280 MeV [data from ref. “)I.

n-N ELASTIC SCATTERING

-v c

263

264

J. P. DEDONDER

265

n-N ELASTIC SCAlTERING

nucleus. However, we cannot pretend that we represent the experimental data for large momentum transfer. As for total cross sections, we obtain very similar results with or without the two approximations we mentioned (we see, however, that if we dropped only the A > 1 approximatlon we would obtain much larger values especially around the 3-3 resonance). It should be noted that in any case the cross sections obtained are quite different from the experim~n~l ones. The theoretical values of b,,’ and bi’ compared with fitted values of bo and bl n-‘2C

__.-

Re b’.

Energy (MeV) 69.5 120.0 200.0 280.0

60 “) 153 “)

Im W1 -

-0.37 -2.65 -4.72 -1.61

-4.94 -5.40 -0.59 -l-1.40

The values between parentheses

Energy (MeV)

Re b’l

Im b’O

2.99(0.93) 8.75(3.14) 12.52(8&l) 4.24(4.48)

11.93( 6.81) 12.40( 9.24) - 0.17(-0.39) - 3.98(-1.69)

are those of ref. Id).

Re bb

Im 6’*

-4.29(-2.63) -3.56(-1.09)

0.12(-1.65) -3.59(-1.02)

Re b’,

Im Fl

10.87(5.96) 8.47(5.27)

1.96(0.167) 12.98(0.18)

“) Values between parentheses are those of ref. 2b). b, Values between parentheses are those of ref. lc). X-l60 Energy (MeV)

Re 6’0

Im W0

87.5 “)

--5.51(-1.12)

-1.02(0.28)

“) Values between parentheses 5.2. A BETTER

Re &I

12.93(6.49)

Im b’l

4.55(1.46)

are those of ref. Ia).

KINEMATICAL

APPROXIMATION

We shall now use the suggestion we made at the end of sect. 3 and we shall use the form given in eq. (3.12) for the optical potential. We may thus write:

Since one has the relationship

3. P. DEDONDER

,>,’

”

-

i

:. ‘.. . .._ --..._.

_.__._... .____.I.. i __.I

_._.--~

i

-.-.

n-N ELASTIC

SCA’lTERlNG

267

1 8 i-

I

100

200

I

Total

12

cross

nC

300

I

sections

Fig. 13. Total cross sections for n-J2C scattering, same description as in fig. 9 [data from ref. *‘)].

L

20Cl-

40 o-

60

80

u-

40

60

80

100

120

140

160 8”

160

Fig. 14. n-“He elastic differential cross sections at T, = 60 MeV. Same description as in tig. 9 [data from ref. Zb)].

20

269

n-N ELASTIC SCATTERING

j

$ 7D

;

I

-...._ ..

.-

.I

5 di

J. P. DEDONDER

270

We obtain finaliy <4rl uoo(e.4)14i>

=

(A-l){s’+

t’4i

’ 4f)P((li_4fh

with

Thus we solve eq. (4.5), dropping the Coulomb potential as we did before, which gives (A +&&)

= Q+)

(5.7)

with

=

b; =

(A -

I)(

-

d

% P(+P@)

+ bi V~~(r)b(~))~7

W)

4”q2 /3, qA

b;=E

E-

{l- ($2]b;.

We see that the parameters bb and b; are essentially dieerent from b. and br ; moreover, the theoretical values are slightly different for the different nuclei studied (because of the transformation from the lab system to the ACMS). We thus present values at several energies for three nuclei (4He, r2C ,r60) in table 2 and compare them to the parameters derived by a minimi~tion procedure used to fit the experimental data. We see that the introduction of this kinematical transformation may help us to understand why different authors 1c,2b) found negative values for the imaginary part of 6, and also why they found values quite different from those expected from the theory for Re b,, Re b, and Im b,. We have to note here that it may be sometimes hazardous to speak of theoretical parameters, because the on-shell free pion-nucleon phase shifts are not well determined in the low-energy domain (O-100 MeV); more specifically, the real part of the S-wave phase shift seems to be almost undetermined below 150 MeV, in an energy domain where the contribution of the pion-nucleon S-wave phase shift is very important for the pion-nucleus scattering amplitudes. We shall remark here that, for T, 5 150 MeV, in the pion-nucleus S-wave the absorption is always very small and sometimes violates unitarity. This is probably related either to the inaccuracy of pion-nucleon scattering parameters or to the approximations involved in the derivation of Kisslinger’s potential or to both.

n-N ELASTIC SCATTERING

271

The results of the calculations with the transformed parameters 6; and b; appear in figs. 9-16 where they are compared to the calculations + with b. and bl. It appears that we obtain a much better agreement with the experimental data for forward scattering, especially close to and above the 3-3 resonance. This may be explained by the fact that our approximations are now quite consistent and we know that our model, as well as Glauber’s theory, should work for rather high energy and small momentum transfer scattering. We have shown, in fig. 13, the total cross sections for x-“C scattering. The fit remains quite poor and it should be noted that the maximum of the cross section is shifted close to T, x 100 MeV whereas the zero of the real part of the scattering amplitude is close to 150 MeV.

6. Conclusions

The removal of the A >> 1 approximation and the proper consideration of nuclear c.m. motion have allowed us to improve the theoretical results for the elastic differential cross sections of pions on light nuclei. The kinematical correction suggested in sect. 3 [eqs. (3.18) and (3.19)] appears to give much better results than before for small momentum transfer. However, the model does not give good fits for total cross sections and it seems clear that one should not expect to fit backward scattering with such a model. As a matter of fact most of the approximations introduced are (as in Glauber’s theory) valid at high energies and small momentum transfer. We shall show in a forthcoming paper that off-shell effects are very important for n-nucleus scattering, especially below the 3-3 resonance. It is thus clear that off-shell pion-nucleon scattering effects must be precisely studied. We feel that one should use the Chew-Low model for pion-nucleon scattering because it does describe the 3-3 resonance and one may hope that it gives correctly off-shell effects. One should thus calculate the optical potential more precisely. Specifically, we think that one should drop the impulse approximation (3.2) and the static approximation (3.7). Kinematics should also be carefully studied. One will also have to study the validity of the coherent scattering approximation (3.1). Finally we feel that the resolution of the integral equation for the T-matrix in momentum space representation is a more appropriate method since it allows one to suppress several dubious approximations. The author would like to thank C. Schmit for having suggested this investigation and his continued guidance during the course of this work. He is greatly indebted to Dr. C. B. Dover and Professor F. Becker for their comments and suggestions. t In these calculations, we have suppressed the A B 1 approximation motion.

and considered nuclear c.m.

272

J. P. DEDONDER

References 1 a) L. S. Kisslinger, Phys. Rev. 98 (1955) 761 b) W. F. Baker, H. Byfield and J. Rainwater, Phys. Rev. 112 (1958) 1773 D. M. Fleming and M. M. Sternheim, Phys. Rev. 162 (1967) 1683 d) E. H. Auerbach and M. M. Sternheim, Phys. Rev. Lett. 25 (1970) 1500 e) J. P. Maillet, C. Schmit and J. P. Dedonder, Nuovo Cim. Lett. 1 (1971) 191 f) M. Krell and S. Barmo, Nucl. Phys. B20 (1970) 461 g) C. Schmit, Nuovo Cim. Lett. 4 (1970) 454 h) C. Wilkin, Nuovo Cim. Lett. 4 (1970) 491 i) K. Bjornenak, J. Finjord, P. Osland and A. Reitan, Nucl. Phys. B22 (1970) 179 j) T. E. 0. Ericson and M. P. Lecher, Hadron-nucleus forward dispersion relations, CERN report 69-30 k) J. Beiner and P. Huguenin, 2. angew. Math. Phys. 18 (1967) 912 1) M. M. Block and D. Koetke, Nucl. Phys. BS (1968) 451 2 a) M. E. Nordberg and K. F. Kinsey Jr., Phys. Lett. 20 (1966) 692 b) K. M. Crowe, A. Fainberg, J. Miller and A. S. L. Pearsons, Phys. Rev. 180 (1969) 1349 c) Yu. A. Bugadov, P. F. Ermolov, E. A. Kushnirenko and V. I. Moskalev, JETP (Sov. Phys.) 15 (1962) 824 d) W. F. Baker, J. Rainwater and R. E. Williams, Phys. Rev. 112 (1958) 1763 e) R. M. Edelstein, W. F. Baker and J. Rainwater, Phys. Rev. 122 (1961) 252 f) F. Binon, P. Duteil, J. P. Garron, J. Gorres, J. Hugon, J. P. Peigneux, C. Schmit, M. Spighel and J. P. Stroot, Nucl. Phys. B17 (1970) 168 3) K. M. Watson, Phys. Rev. 89 (1953) 575; 105 (1957) 1388 4) A. K. Kerman, H. McManus and R. M. Thaler, Ann. of Phys. 8 (1959) 551 5) K. M. Watson and M. L. Goldberger, Collision theory (Wiley, New York, 1964) p. 683 6) C. F. Chew and F. E. Low, Phys. Rev. 101 (1956) 1571 7) S. Gartenhaus and C. Schwartz, Phys. Rev. 108 (1957) 482 8) A. Donnachie, in Particle interactions at high energies, eds. T. W. Priest and L. L. J. Wick (Oliver and Boyd, Edinburgh, 1967) c) E. H. Auerbach,