On the scattering of pions by deuterons

Nuclear Pl;ysics 29 (1962) 680---686; ~ ) North-Holland Publisking Co., Amstere~zm Not to be A~t,~luced by photoprint or microfilm without writteu permis~on from the publisher

ON THE SCATTERING

OF PIONS

BY DEUTERONS

A I J . A D I RAMAKRISHlqAN, V. DEVANATHAN and K. V E N K A T E S A N t

University o/ Madras, Madras-25, India Received 10 July 196I The scattering of pions b y deuterons has been studied under the impulse approTim~tion and explicit expressions for the cross-sections for the elastic, inelastic and charge exchange scatterings have been obtained using the Chew-Low amplitude for the pion-nucleon scattering. Numerical calculations have been carried out a t various energies and compaxison has been made with the available experimental results.

Abstract:

1. I n t r o d u c t i o n

Fernbach, Green and Watson 1) and Rockmore s) have studied earlier the scattering of pious by deuterons using the impulse approximation which involves the neglect of internucleon potential, multiple scattering and off-the-energy shell scattering of pion by nucleon. The correction due to the internucleon potential has been investigated by Rockmore 2) and is found to be less than 10 %. But there seems to be a controversy as regards the multiple scattering. The multiple scattering effect has been initially studied by Brueckner 8) and he finds a considerable reduction in the cross-section of about 50 ~o at 30° and at 135 MeV. However, Rockmore points out that the neglect of off-theenergy-shell matrix elements in multiple scattering is certainly invalid. He has carried out the calculations at only one energy (85 MeV) using pure impulse approximation and finds an excellent agreement with the experimental results of Rogers and Lederman ~). The elastic scattering of pious by deuterons has also been studied by Bransden and Moorhouse 5) employing the variational method and the numerical results are presented at various incident pion energies from 85 to 378 MeV. In the expression they have derived, they are able to identify the terms corresponding to multiple scattering and show that the inclusion of those terms alters the cross-section by less than 5 % even at the most favourable angles and energies in the energy range considered. Their numerical results agree well with the experimental results of Rogers and Lederman l), Arase et ~g. s) and Pewitt d ~g. ~). The investigations of Rockmore and Bransden and Moorhouse seem to show that the impulse approximation is certainly an approximation valid in the t Senior Research Fellow of the Atomic Energy Commi~ion. 680

ON

THE

SCATTERING

OF

PION$

BY

DEUTERONS

681

energy region 85 to 300 MeV. So we thought it worth while to carry out the calculations using the Chew-Low amplitude s) for the pion-nucleon scattering and present numerical results for the elastic, inelastic and charge-exchange scattering of pions by deuterons at various incident pion energies 85, 140, 195 and 250 MeV. The recent experiment on the elastic scattering of pious by deuterons at 142 MeV by Pewitt etag. and further investigations at higher energies will clearly show how far the impulse approximation is reliable. The numerical results that we have obtained agree well with the experimental values of Rogers and Lederman for elastic and inelastic scattering of positive pions by deuterons at 85 MeV; but the fit is not so good in the case of elastic scattering at 140 MeV. Earlier Green 9) has reported an impulse approximation calculation for elastic scattering of charged pious by deuterons at 135 MeV and he has also found too large values for the cross-sections as compared with the then available experimental result of Arase eta/. As a result, he has made some sceptical remarks about the validity of the impulse approximation at the energy 135 MeV. Unfortunately, there are no experimental data at present at higher energies available for comparison and until experimental results are obtained at such higher energies, nothing can be definitely said about the validity of the impulse approximation in the energy range 140 MeV and above. The impulse approximation has also been used for analysing the various other nuclear processes 10) including the photo-production of pions xLxz) and hence the investigation of the energy range in which the impulse approximation is valid assumes a greater importance. 2. Differential C r o s s - S e c t i o n s

The Chew-Low amplitude s) for the pion-nucleon scattering in the static approximation is given by t(2, 1) = 2_~ ~ P~(2,1)hffi,

(1)

OJ a ~ l

where P

(2, 1) :

=

e"

sin

J ~ and J= are the projection operators for the isobaric spin and angular momentum states, eo the meson energy, ~ its momentum and 8~ the p-wave phase shifts. In the above a stands for any one of the four states 33, 31, 13, 11 in Fermi's notation. The above amplitude is used to obtain the cross-sections for the following

682

ALLADI RAMAKRISHNANet al.

processes: (i) ~ + + D -+ ~ + + D (elastic), (ii) ~ + + D - + ~ + + D , ~ + + p + n (elastic+inelastic), (iii) ~ + + D -+ r # + p + p (char ge~exchange). Following the procedure a d o p t e d in ref. x,), the differential cross-sections for the various processes can be written as follows:

(~)elastlc ~---(2~)-'°~*[l(a+c)l'+~l(b+d)"]'d""

(2)

+2o*- {Re a R e c + I m a I m c + ~ ( R e b R e d + I m b Im d))]

(4)

(d~)chare exe,ane = (2~)-'o,*[lel'+l/l'-{(31el*+ltl').~], with 27~

a = - - cos 0(2e ia,' sin 8 ~ + e ~a*' sin ~sz), oq 2~ b = - - sin 0(e ~a,* sin 8 ~ - - e iatL sin 8al), oJq 2r~ c = - - cos 0(2e ~a*, sin ~ + e 3o,q

*at, sin ~31+4e~a,, sin ¢Sza+2e ~an sin ~n),

d = 2~ sin 0(e ~.~ sin cS~--e iast sin ~Saz+2e ~at* sin ¢Sza--2e ~Sn sin 811), 3oq 2~12 e -- - cos O(2e*a*, sin 8 ~ + e ia,, sin 831--2e fslt sin ~za--e ian sin 611), 3o~q / -- - 3of 8-

- -

sin O(e ia~s sin 8 ~ - - e ia~, s i n ~31--e iazs s i n ~13+e ~an sin 8n), arctg

+arctg

--2 arctg

t • m 1--~Pz ko arctg

+arctg

--2 a r c t g x ~ l / ,

1- 01 ko

,

where 0). k02 = z--~a~fl--cos 2x Here, 0 is the angle between the incoming and the outgoing pions. The overlap

ON THE SCATTERING OF PIONS BY DEUTERONS

683

integrals ~ and ~ - are obtained using the Hulth~n wave function for the deuteron. The values of the constants 0t, fl and Pl are taken as follows: =

0.3274,

fi =

2.068,

px :

1.231.

In processes (ii) and (iii), plane waves have been used for the final states and the integration over the final relative momentum of the nucleons has been carried out using the closure approximation. It will be instructive to compare the cross-sections for the deuteron with the free nucleon cross sections

(d~).+÷p~,++p = (2~)-*~*(lal*+ IbiS),

(5)

(2~)-~o~*(Icl*+ Idl~),

(0)

(2~)-*~*(IeI*-4-III*).

(7)

(~)~r++n~r++n

We find that in the case of the sum of the elastic and inelastic scattering b y deuteron, the cross-section is larger than the sum of the flee-proton and the free-neutron cross-sections. In the case of charge exchange scattering, to which only the neutron contributes, we find that the cross-section is reduced b y the presence of the other particle (proton). This effect is to be attributed to the Pauli exclusion principle. In the former case (elastic and inelastic scattering b y deuteron) this effect is shrouded b y the presence of the interference terms. The overlap integrals 8 and ~- are decreasing functions of both energy and angle. Hence at high energies and at large angles, the differential cross-section for the process (ii) approaches the sum of the freeproton and the free-neutron cross-sections and for the process (iii) the crosssection approaches that of the charge-exchange scattering of n+ b y free neutrons.

3. Numerical

Results

and

Discussion

The differential cross-sections for the elastic, inelastic and charge-exchange scattering of pions b y deuterons have been calculated numerically using eqs. (2), (3) and (4) at various incident pion energies: 85, 140, 195 and 250 MeV. It m a y be noted that the expressions (2), (3) and (4) have to be evaluated in the c. m. system of the pion-nucleon system (and not the pion-deuteron system), since in the impulse approximation, the incident pion "sees" only the individual nucleons and the interaction is described as the sum of the individual scattering amplitudes. The results are given in tables 1--3 in the laboratory system to facilitate direct comparison with the experimental data. Only the

ALLADI R A M a ~ I S H N A N

$t G/~.

dominant ~u phase shifts have been taken into account in the numerical calculation and in the case of elastic scattering, the effect of including the other p-wave phase shifts have also been studied and as shown in figs. I and 3, the effect is to reduce the cross-sections slightly. TABLE 1 T h e differential cross section dffld.O for t h e elastic scattering of ~+ b , d e u t e r o n s in u n i t s of m b rsr 0o

85 140 195 250

12.62 45.19 68.71 38.44

30 °

7.568 23.66 31.17 15.13

60 °

1.747 4.342 4.737 1.480

90 °

0.6359 1.340 1.311 0.4756

120 °

1.039 1.974 1.660 0.610

150 °

180 °

1.388 2.503 2.008 0.7131

1.495 2.641 2.186 0.735~

TABLE 2 T h e differential cross section d a l d Q for t h e s u m of elastic a n d inelastic scatterin__g of ~+ b y d e u t e r o n in u n i t s of m b / s r ~ - ~ . . ~ b Pimz energy (MeV) 85 140 195 250

.

angle

0°

12.62 45.19 68.71 38.44

30 °

60 °

90 °

120 °

150 °

8.104 27.54 40.18 21.75

2.844 9.754 L4.43 7.857

1.674 5.853 8.794 4.884

2.928 10.08 14.99 8.333

4.263 14.63 21.82 12.08

180"

4.751 16.31 ~ 24.34 13.47

TABLE 3 T h e differential cross section dff/dQ for t h e charge exchange scattering of :r~ b y d e u t e r o n s in u n i t s of m b / s r

85 140 195 250

0°

30 °

60*

900

120 °

150"

180"

0 0 0 0

0.3865 1.878 3.417 2.153

0.3708 1.471 2.371 1.397

0.2826 1.058 1.644 0.9315

0.4489 1.740 2.760 1.563

0.6488 2.530 3.990 2.277

0.7252 2.829 4.460 2.544

At 85 MeV, there is a good agreement between the numerical results that we have obtained and the experimental data of Rogers and Lederman. In figs. I and 2 are indicated the experimental results of Rogers and Lederman for the elastic and inelastic scattering along with our theoretical results and the agreement is satisfactory. However, in the case of charge exchange scattering, the values that we have obtained are much lower than that of Rogers and Lederman at large angles and that m a y be due to the neglect of s and other p-wave phase shifts. Fig. 3 represents the elastic scattering at 140 MeV. The experi-

ON

THE

SCAT'rKRING

OF

PIOlqS

]BY

68~

DEUTERONS

't~lfI = 6

3"

5

5

,.E,

o

3~"

B~

9~

,~o0 , ~

t

18o°

deuterons a t 85 MeV. The solid curve represents t h e theoretical result obtained t ~ n g into account only the domln~nt ~ = phase' shift. The dotted curve is obtained taking into account all t h e p-wave phase shifts. The experimental points are those of Rogers and Lederman, =

60"

9(~"

i e 120

150 =

IBO"

0 LAB Fig. 2. Differential cross-section in the laboratory system for the sum of elastic and inelastic scattering of y~+ b y deuterons. The experimental points are those of Rogers and L e d e r m ~

0 LAB P i g . ] . ] ~ f f e r e n t i a l cross-section i n t h e l a b o r a t o r y system for the elastic scattering of ~+ by

= I

4

i

•

,

b~

'

I

;

i

~ I

|

30"

6o"

90'

120"

150"

~80•

e LAB

Fig. 3. Differential c ~ o n in the laboratory system for the elastic scattering of ~E+ b y deuterons s t 140 MeV. The solid curve represents the theoretical result obtained talri~g into account only the domin~tnt ~ phase shift. The dotted curve is obtained linking into account all the p-wave phase shifts. The experimental points are those of Pewitt et al.

686

ALLADI RAMAKRISHNAN $~ ~ .

mental points marked therein are the results of Pewitt et ~g. The theoretical cross-sections have to be corrected for the Coulomb scattering, the effect of which is, as shown b y Rockmore, to depress the cross-sections at small angles. The dotted lines in figs. 1 and 3 are obtained after including the other p-wave phase-shifts. Unfortunately there are no experimental data to compare with for the inelastic and charge-exchange scattering at 140 MeV. The recent experiment of Pewitt el al. is on the elastic scattering alone and that too at only one energy viz., 140 MeV. The continuance of experimental investigations at higher energies is strongly suggested for it is hoped that these experiments will clearly indicate the range of validity of the impulse approximation. We are grateful to Dr. S. K. Srinivasan and Mr. G. Ramachandran for stimulating discussions. References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10)

11)

12)

S. Fernbach, T. A. Green and K. M. Watson, Phys. Rev. 80 (1951) 1084 R . M . Rockmore, Phys. Rev. 10S (1957) 256 K. A. Brueckner, Phys. Rex,. 89 (1953) 834; 90 (1953) 715 K. C. Rogers and L. M. Lederman, Phys. Rev. 105 (1957) 247 B. H. Brandsen and R. G. Moorhouse, Nuclear Physics 6 (1958) 310 E. Arase, G. Goldhaber and S. Goldhaber, Phys. Rev. 90 (1953) 160 E. G. Pewitt eta/., Proc. 1960 Ann. Int. Conf. on High Energy Physics a t Rochester (1960) p. 196 G. F. Chew and F. E. Low, Phys. Rev. 101 (1956) 1570 T. A. Green, Phys. Rev. 90 (1953) 161 G. F. Chew, Phys. Rev. 80 (1950) 196; H. P. Noyes, Phys. Rev. 81 (1951) 924; R. F. Riley, Nuclear Physics 13 (1959) 407 G. F. Chew and H. W. Lewis, Phys. Rev. 8 4 (1951) 779; M. Lax and H. Feshbach, Phys. Rev. 88 (1952) 509; S. Penner, Phys. Rev. 105 (1957) 1113 V. D e v a n a t h a n and G. Ramachandran, Nuclear Physics 23 (1961) 312; A. Ramakrishnan, V. D e v a n a t h a n and G. Ramachandran, Nuclear Physics 24 (1961) 163