[ 8 . A . ~ Nuclear Physics B42 (1972) 627-643. North-Holland Publishing Company

DIFFERENTIAL CROSS SECTION FOR COHERENT PRODUCTION

OF THREE-PION SYSTEMS ON NUCLEI AT 15.1 GeV/c *

C. B E M P O R A D **, W. BEUSCH, A. C. MELISSINOS ***, E. POLG.~R and D. W E B S D A L E

CERN, Geneva, Switzerland

J. P. D U F E Y ~, K. F R E U D E N R E I C H , R. FROSCH, F. X. G E N T I T and P. M U H L E M A N N

ETH, Ziirich, Switzerland

J. CODLING, J. G. L E E and M. L E T H E R E N Imperial College, London, England

G. B E L L I N I , M. DI C O R A T O and G. VEGNI Istituto di Fisica, Sezione INFN, Milano, Italy

Received 7 January 1972

Abstract: The measured differential cross sections d2g/dt ' dm3T r of three-pion p ro- duction on nine different nuclei (based on 40 000 events) are presented. The co r - rections due to the background of incoherent reactions and due to the limited angular resolution of the experiment are discussed. The comparison of the dif- ferent ial cross sections of coherent three-pion production with the optical model is sat isfactory.

1. I N T R O D U C T I O N

In a r e c e n t a r t i c l e [1] we r e p o r t e d the f i r s t r e s u l t s of an e x p e r i m e n t on c o h e r e n t p r o d u c t i o n of t h r e e - and f i v e - p i o n s y s t e m s on nuc l e i induced by p i o u s of 15.1 G e V / c . T h e e x p e r i m e n t a l me thod and d e t a i l s of the data a n a l y s i s w e r e a l s o d e s c r i b e d .

The a i m of the p r e s e n t p a p e r is twofold: f i r s t l y , to g ive the m e a s u r e d d i f f e r e n t i a l c r o s s s e c t i o n s fo r the r e a c t i o n

* Work supported in part by the Swiss National Science Foundation. ** Permanent address: Istituto di F is ica del l 'Univers i t~ , Pisa , Italy.

*** Permanent address: Universi ty of Rochester , Department of Physics, Rochester New York, USA.

:~ Now at CERN.

628 C . B e m p o r a d et a l . , Pion produc t io~ on nucle i

z - A ~ 7;-7:-~ +A (1)

f o r t h e n i n e t a r g e t n u c l e i A ( i .e . B e , C, A I , Si , T i , Cu , Ag , T a a n d P b ) , a n d f o r s i x d i f f e r e n t i n t e r v a l s of t h e p r o d u c e d t h r e e - p i o n m a s s : s e c o n d l y , to s h o w t h a t t h e s e d i f f e r e n t i a l c r o s s s e c t i o n s c a n b e r e a s o n a b l y w e l l f i t t e d b y t h e o p t i c a l m o d e l .

2. M E A S U R E D D I F F E R E N T I A L CROSS S E C T I O N S

W e p r e s e n t t he m e a s u r e d d i f f e r e n t i a l c r o s s s e c t i o n s d 2 ~ / d t ' d m 3 ~ , b a s e d o n 39 989 e v e n t s t h a t f i t t e d t h e k i n e m a t i c a l h y p o t h e s i s (1). S i n c e t h e r e c o i l of t h e t a r g e t p a r t i c l e w a s no t o b s e r v e d , t h i s i s n o n e - c o n s t r a i n t f i t . E v e n t s h a v e b e e n a c c e p t e d i f t h e f i t p r o b a b i l i t y w a s b i g g e r t h a n 1(~. T h e n u m b e r of s u c c e s s f u l l y f i t t e d e v e n t s a n d a n e s t i m a t e of t h e c o r r e s p o n d i n g n u m b e r of c o h e r e n t e v e n t s o n e a c h t a r g e t a r e l i s t e d in t a b l e 1. T h e d a t a h a v e b e e n c o r r e c t e d f o r a n a l y s i s l o s s , f o r t h e a c c e p t a n c e of t h e a p p a r a t u s , a n d f o r " n o t a r g e t " b a c k g r o u n d , i . e . f o r e v e n t s w h i c h w e r e no t p r o d u c e d in t h e t a r g e t ( t a b l e 1). F u r t h e r m o r e , t h e d i f f e r e n t i a l c r o s s s e c t i o n s a r e d e r i v e d f r o m e v e n t s w i t h t h e i n c i d e n t p i o n m e a s u r e d in t h e m u l t i w i r e p r o p o r t i o n a l c h a m b e r s : t h i s m e a n s t h a t t h e l ow q u a l i t y e v e n t s of r e f . [1] h a v e b e e n e x - c l u d e d .

The experimental values of d2~/dt ' dm3~ are given in table 2 for each

Tab le 1 T a r g e t p r o p e r t i e s and da ta co l l ec t ed wi th each t a r g e t .

T a r g e t A v e r a g e T h i c k n e s s a) Dens i ty Tota l n u m b e r " C o h e r e n t "No t a r g e t " nuc leus A of e v e n t s b ) e v e n t s " c) con tamin .

(em) (g/cm 3) (%)

Be 9 2.55 1.85 4497 3555 0.2 C 12 1.50 1.68 4342 3452 0.3 A1 27 0.75 2.70 1795 1349 0.4 Si 28 0.26 2.35 8837 6760 8.4 Ti 48 0.25 4.49 4126 2888 0.9 Cu 64 0.125 8.87 3571 2345 1.1 Ag 108 0.076 10.47 3904 2295 1.7 T a 181 0.036 16.48 5020 2529 3.0 Pb 207 0.0505 11.31 3897 1792 3.3

39989 26865

a) The t a r g e t s had a c y l i n d r i c a l shape . T h e i r d i a m e t e r was 5.8 cm wi th the excep t ion of the Si t a r g e t , which was an a s s e m b l y of f ive s i l i con d e t e c t o r s hav ing a usefu l d i a m e t e r of 1.4 cm.

b) N u m b e r of even ts g iv ing a s u c c e s s f u l k i n e m a t i c a l f i t for the hypo thes i s of r e a c t i o n (1). T h i s n u m b e r a l so inc ludes i n c o h e r e n t even t s .

c) N u m b e r of even t s ob ta ined a f t e r hav ing appl ied a t ' - c u t c o r r e s p o n d i n g to Qc - 3.6, w h e r e c is the hal f d e n s i t y r a d i u s of the nuc leus g iven by eq. (12) and Q is the t h r e e - m o m e n t u m t r a n s f e r to the nuc leus . The t ' - v a l u e of the cut is a p p r o x i m a t e l y in the f i r s t m i n i m u m of the d i f f e r e n t i a l c r o s s sec t ion . The s a m e cut has been used in re f . [1].

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632 C . B e m p o r a d et al . , Pion production on nuclei

Tab le 2 Si t a r g e t

D i f f e r en t i a l c r o s s s e c t i o n d 2cr/d t' dm 37r (mb /G e V3),

Bfn Mass interval (GeV)

1.0- 1.2 0.9-1.1 1.1-1.3 1.3-1.5 1.5-1.7 1.7-1.9

1 1229 1145 921 413 124 43 2 1065 949 1033 255 206 115 3 1022 838 867 264 173 90 4 755 638 738 245 156 88 5 642 562 534 252 126 89

6 485 419 423 171 101 52 7 382 305 333 154 73 41 8 233,0 201.0 173,7 106,00 50,8 44,0 9 150,4 119,9 140,1 61,9 32.7 28.5

10 95,7 66.2 94,5 55.5 18.4 17.0

11 65,5 50.5 63,1 22.6 17.7 17,5 12 39.5 37.7 32,2 23,3 11.7 8.4 13 33.4 17.8 32.4 16.6 13.1 10.4 14 28.2 19.7 25.8 15,3 9.1 9,6 15 20,3 14.4 24.8 6.4 9.3 3.0

16 22.2 20.7 13.1 12.0 8.0 5.6 17 23.1 19.4 19.7 11.9 6.2 3.9 18 22,5 16,6 16,6 8,9 8.8 3.7 19 10,7 11.3 12.2 7,3 6.6 2.4 20 11.9 7.1 13.2 ii.i 8.4 2.8

21 7.8 5.4 6.1 7.6 3.5 3.7 22 8.0 5.1 9.9 5.8 4.3 3.0 23 7.0 5.0 7.5 3.7 5.0 1.9 24 3,6 3.6 5,3 2,2 1,8 1,8 25 4,5 2.3 6.0 4,2 1.7 3,5

26 5,8 3,9 6,2 6.0 4,0 2,1 27 5,10 4,14 6.00 2,70 1,89 2,00 28 3,57 1,01 6,10 1.20 2.21 1.56 29 3,44 2,45 2,75 1,50 2,.84 0,38 30 2,50 0,94 2.98 2,70 0,69 0,73

31 2,26 0.45 2.55 3.06 1,00 1,77 32 1.25 1.32 1.06 2.61 1.87 1.70 33 0.91 2.14 1.02 1.26 0.63 1.00 34 0.59 0.82 1.00 0.61 1.76 0.97 35 0.86 0.80 1.61 0.59 1.49 0.94

36 i.ii 1.95 1.26 0.29 0,88 0.92 37 1.35 1.90 0.61 0.28 0.85 0.59 38 1.32 1.48 0.59 0.28 0.55 0.29 39 0.26 0.72 0.29 0.27 0.27 0.56 40 1.00 1.05 1.13 1.04 0.00 0.55

Error 0.4582 0.4952 0:4284 0.4127 0.4215 0.4436 factor

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636 C. Bemporad el al . , Pion production o~ nuclei

T a b l e 2 T a t a r g e t

Differential cross section d2(~/dt' din37 (mb/GeV3).

Bin Mass interval (GeV)

1.0-1.2 0.9-1.1 1.1-1.3 1.3-1.5 1.5-1.7 1.7-1.9

- i - ~ 2 5 4 7 0 2 4 . . . . 4 9 6 7 . . . . . 1 3 8 4 - 3 2 - 1 0 2 5783 5322 4668 1863 322 376 3 4437 3925 3670 906 204 253 4 2685 2297 2218 869 277 219 5 1395 1136 1258 489 178 90

6 858 671 718 247 138 34 7 515 343 480 217 78 43 8 354 284 313 179 74 43 9 415 451 302 125 66 65

i0 348 306 296 145 47 26

11 254 223 259 88 71 37 12 178 158 159 49 40 30 13 129 123 95 69 44 25 14 93 88 92 30 31 26 15 86,8 68.3 77.6 31.5 26.4 27.5

16 75.2 63.4 84.5 40.4 41.2 17.3 17 46.9 23.3 68.7 28,0 23.3 24.3 18 35.1 19.1 51.9 21.6 27.0 25.7 19 29.1 35.8 38.6 31.8 23.1 12.1 20 30.3 27.5 20.7 25.9 Ii.0 9.2

21 21.0 23.2 25.8 14.4 12.6 4.4 22 15.5 12.2 22.8 17.6 8.0 10.5 23 27.7 18.0 20.0 18.7 13.5 4.0 24 21.9 13.1 24.7 8.9 9.3 11.5 25 13.9 8.3 10.9 5.2 6.9 3.7

26 18.2 4.1 24.7 9.9 6.8 5.3 27 14.3 19.7 15.0 9.5 11.3 8.3 28 5.18 7.36 9.55 16.77 0.00 4,91 29 4.87 1.71 7.76 11.84 4.61 3.16 30 6.23 3.43 11.95 4.32 4.32 1.53

31 9.22 6.52 7.25 2.70 2.83 5.90 32 8.81 3.10 5.71 4.05 1.43 2.86 33 9.81 1,50 13.62 6.46 2.65 4.15 34 11,19 5,82 7.95 0.00 2.57 2,69 35 1.30 2.83 1.30 3.70 1.20 3,91

36 1.37 1.37 2.43 3.59 2.43 2.64 37 7.71 6.68 4.83 4.62 3.60 1.23 38 3.70 2.50 2.40 5.60 0.00 2.40 39 1.17 0.O0 2.34 3.31 1.17 1.17 40 3.61 5.03 3.32 1.04 1.04 2.37

E r r o r 0.8969 0,9692 0.8385 0.8078 0.8250 0.8682 f a c t o r

C. B e m p o r a d et a l . , P ion produc t ion on nucle i 637

Table 2 Pb t a rge t

Di f fe ren t ia l c r o s s s ec t ion d2ff /dt ' dm3~ (mb/GeV3).

Bin M a s s in te rva l (GeV)

1 . 0 - 1 . 2 0 . 9 - 1 . 1 1 . 1 - 1 . 3 1 . 3 - 1 . 5 1 . 5 - 1 . 7 1 . 7 - 1 . 9

1 8548 8490 5687 1454 514 104 2 5380 4218 5385 1529 699 289 3 4213 3597 3672 906 325 238 4 2631 1833 2017 657 229 103 5 1207 1121 867 378 117 68

6 705 567 651 146 185 76 7 470 358 403 165 116 91 8 397 432 317 138 94 27 9 524 454 427 186 65 51

10 348 238 270 135 65 40

11 265 145 238 72 91 42 12 81 75 95 72 46 29 13 89.8 69.6 97.6 32.8 16.6 8.7 14 125.3 96.4 93.1 54.5 28.1 12.1 15 76.1 88.6 67.1 36.3 48.4 23.5

16 69.0 56.0 36.5 34.2 14.0 18.4 17 63.2 45.9 66.1 45.2 33.2 13.8 18 22.1 24.1 29.0 30.5 21.8 3.2 19 30.1 26.0 39.9 26.0 11.8 12.4 20 38.6 41.6 40.8 19.3 25.2 0.0

21 36.5 33.3 41.6 26.3 26.8 16.5 22 19.7 19.7 26.4 15.1 2.4 10.7 23 21.4 18.7 32.7 11.9 7.4 10.2 24 20.5 13.6 24.0 9.1 14.0 4,9 25 36.6 15.2 35.1 24.0 9.0 4,7

26 11.8 10.1 6.7 8.4 2.1 4,5 27 13.7 9.4 10.6 6.1 6.1 2,2 28 15.1 4.9 18.7 9.7 13.9 0,0 29 6.0 2.3 13.7 9.4 9.5 0,0 30 12.3 8.6 15.0 7.3 3.7 1.9

31 11.7 6.1 14.9 15.8 5.5 0.0 32 3.8 6.4 8.8 10.2 3.5 1.8 33 3.5 1.9 6.8 5.0 6.8 1.8 34 10.4 5.3 6.7 6.4 6.6 0.0 35 1.7 0.0 3.2 1.5 4.7 5.0

36 6.5 3.6 7.9 1.5 4.7 0.0 37 8.0 5.2 5.1 5.9 1.4 3.1 38 3.05 4.96 2.93 0.00 0.00 1.53 39 2.97 3.22 2.85 1.36 1.49 0.00 40 1.45 1.69 7.01 1.33 2.78 1.45

E r r o r f a c to r 1.0118 1.0934 0.9459 0.9112 0.9307 0.9794

638 C. Bemporad et al . , Pion production on nuclei

t a r g e t n u c l e u s and for s i x i n t e r v a l s of the p r o d u c e d t h r e e - p i o n m a s s (width of i n t e r v a l 0.2 GeV, c e n t r e s at 1, 1.1, 1.2, 1.4, 1.6 and 1.8 GeV). F o r each t a r g e t n u c l e u s the t ab l e has 40 t ' - e n t r i e s . We have c h o s e n b in s of c o n s t a n t w id th in Q± = ~ l t ' 1. Such b in s have a s i m p l e r e l a t i o n to the e x p e r i m e n t a l a n g u l a r r e s o l u t i o n ; m o r e o v e r , the n u m b e r of e v e n t s p e r bin (and t h e i r s t a t i s t i c a l e r r o r ) v a r i e s l e s s than in the c a s e of b in s wi th c o n s t a n t wid th in t ' . The b in width in Q L i s 15 M e V / c . Wi th th i s c h o i c e , the u p p e r l i m i t s of b i n s 1, 2, 3 . . . . a r e (0.015) 2, (0.03) 2 , (0.045) 2 . . . . (G e V /c ) 2 and t h e i r c e n - t r a l v a l u e s (0.0075) 2, (0.0225) 2, (0.0375) 2 . . . . ( G e V / c ) 2.

The s t a t i s t i c a l e r r o r s on d 2 ~ / d t ' dm3~ a r e not t a b u l a t e d . T h e y can be c o m p u t e d us ing the r e l a t i o n

ld2<, 1 a \ d t ' d r n 3 J = f (2) V tit ' dm3~ Q± c e n t r a l

1

w h e r e f i s a c o n s t a n t [in un i t s of (mb)~ /GeV ] for a g iven t a r g e t and a g iven m a s s i n t e r v a l . It can be found at the b o t t o m of each c o lumn .

The s t a t i s t i c a l n o r m a l i z a t i o n e r r o r fo r the c r o s s s e c t i o n on a g iven t a r g e t i s about 2% and the p o s s i b l e s y s t e m a t i c e r r o r of t he n o r m a l i z a t i o n (which is the s a m e fo r a l l t a r g e t s ) a m o u n t s to 10% ( u n c e r t a i n t y of the g e o - m e t r i c a l a c c e p t a n c e and of the e f f i c i e n c y of the k i n e m a t i c a l f i t) .

The d i f f e r e n t i a l c r o s s s e c t i o n s of t a b l e 2 r e p r e s e n t a s u p e r p o s i t i o n of c o h e r e n t and i n c o h e r e n t c o n t r i b u t i o n s . The a r e s o m e w h a t a f f e c t e d by the l i m i t e d a n g u l a r r e s o l u t i o n and by an e x p e r i m e n t a l b i a s a g a i n s t i n c o h e r e n t e v e n t s . In the nex t two s e c t i o n s we wi l l g ive the f a c t o r s r e l e v a n t to the c o r - r e c t i o n of t h e s e e f f e c t s and we a l s o wi l l p r e s e n t our e s t i m a t e of t h e i r i n - f l u e n c e .

3. C O R R E C T I O N F O R THE L I M I T E D A N G U L A R R E S O L U T I O N O F THE E X P E R I M E N T

The e x p e r i m e n t a l r e s o l u t i o n in t ' o r in the e q u i v a l e n t q u a n t i t i e s Qj_ o r 0 r e l a t e d by

2 _ p 2 0 2 t ' : - Q ± ~- (3)

p l a y s an i m p o r t a n t r o l e fo r the i n t e r p r e t a t i o n of the d a t a with the o p t i c a l m o d e l (p i s the m o m e n t u m of the i n c i d e n t p ion and 0 the s c a t t e r i n g a n g l e ; bo th a r e in the lab. s y s t e m ) .

T h e e x p e r i m e n t a l a n g u l a r r e s o l u t i o n A0 h a s been d e t e r m i n e d by the Mon te C a r l o m e t h o d o u t l i n e d in re f . [1] to have the v a l u e 1.2 m r a d . The e r r o r on th i s v a l u e i s e s t i m a t e d a t 0.2 m r a d . A0 i s c o n s t a n t wi th v a r y i n g 0, t hus c a u s i n g a v a r i a b l e e x p e r i m e n t a l r e s o l u t i o n in t ' .

The m e t h o d of c o r r e c t i n g fo r the l i m i t e d a n g u l a r r e s o l u t i o n i s the fo l lowing . L e t 0 T be the t r u e s c a t t e r i n g a n g l e of an e v e n t in the l a b o r a t o r y s y s t e m , i . e . the a n g l e b e t w e e n the t r u e m o m e n t a of i n c i d e n t p ion and o u t g o i n g t h r e e - p i o n s y s t e m . It then can be shown that the p r o b a b i l i t y to m e a s u r e a s c a t t e r i n g ang le 0 i s e x p r e s s e d by

C. Bernporad et al., Pion production on nuclei 639

R ( O ) = R 0 0 exp - - (4) ( 0)2 2(2 '

with the n o r m a l i z a t i o n cons t an t R 0 g iven by

77

f n(o) dO : 1 . (5) 0

T h i s r e s o l u t i o n func t ion then has to be folded into the t h e o r e t i c a l a n g u l a r d i s t r i b u t i o n N(O) [which wil l be g iven by the op t ica l model ] to ob ta in the c o r r e c t e d n u m b e r N'i(O i) of even t s in b in i with the bin edges Oia and Oib:

Ni(Oi) = N(O ) R(O-~? ) dO dO . (6) 0 Oia

4. INCOHERENT SUBTRACTION

The b a c k g r o u n d of i n c o h e r e n t r e a c t i o n s may be w r i t t e n as

d2~I dr ' dm3~ - I0(m31r) eBt' PS . (7)

I0 (m3u) i s a s c a l i n g f ac to r and B deno te s the s lope of the d i f f e r e n t i a l c r o s s s e c t i o n on hydrogen . The two func t ions P and S have been i n t r o d u c e d to d e s c r i b e the dev i a t i on f r o m the p u r e exponen t i a l f o r m , due to t h e o r e t i c a l r e a s o n s (P) and to e x p e r i m e n t a l r e a s o n s (S).

Two d i f f e r en t f o r m s of P have a c t u a l l y been used. F i r s t l y ,

and s econd ly ,

P ~ 1 , (8)

QF ~FF/ for Q ~< 2Q F ,

P = 1 for Q > 2QF , (9)

w h e r e Q2 = Q 2 + [ ( r n 2 m 2 ) / 2 p ] 2 and w h e r e QF is the F e r m i m o m e n t u m of the n u c l e o n s in the n u c l e u s . T h i s second f o r m a c c o u n t s for the s u p p r e s s i o n of i n c o h e r e n t r e a c t i o n s a t low t ' v a l u e s impose d by the P a u l i p r i n c i p l e [2].

S is a c o r r e c t i o n f ac to r due to a~t ' dependen t a c c e p t a n c e of the e x p e r i - m e n t a l a r r a n g e m e n t for i n c o h e r e n t r e a c t i o n s . The r e a s o n for th is c o r r e c - t ion is that the r e c o i l i n g nuc l eon .~s on ly s e e n by the a n t i c o i n c i d e n c e c o u n t e r s a r o u n d the t a r g e t if it is a p r o t o n l e a v i n g the t a r g e t with a m o m e n t u m b i g g e r than a c e r t a i n va lue ( m i n i m u m d e t e c t a b l e r e c o i l m o m e n t u m ) . T h i s m e a n s that for t ' -~ 0, w h e r e the r e c o i l i n g p r o t o n has v e r y low m o m e n t u m , an i n - c o h e r e n t event on a p ro ton is a l m o s t n e v e r a n t i c o i n c i d e d , w h e r e a s at l a r g e t ' the p r o t o n is e j e c t ed with a l a r g e m o m e n t u m and the event is an t i co inc ided .

640 C. Bernporad et al., Pion production on nuclei

(The g e o m e t r y of the a n t i c o i n c i d e n c e c o u n t e r s a t the t a r g e t i s g iven in f ig. 1 of re f . [1]; the p r o p e r t i e s of the t a r g e t s a r e l i s t e d in t a b l e 1.) Our e s t i m a t e of th i s c o r r e c t i o n u s e s the fo l lowing s i m p l i f y i n g a s s u m p t i o n s :

(i) n e u t r o n s and p r o t o n s a r e e q u a l l y d i s t r i b u t e d in the n u c l e u s , (ii) the a m p l i t u d e for t h r e e - p i o n p r o d u c t i o n i s the s a m e on n e u t r o n s and

on p r o t o n s , ( i i i ) the p r o t o n (or neu t ron ) does not i n t e r a c t in the t a r g e t a f t e r i t s

e j e c t i o n f r o m the n u c l e u s , (iv) the m i n i m u m d e t e c t a b l e r e c o i l m o m e n t u m for ou r a r r a n g e m e n t of

a n t i c o i n c i d e n c e c o u n t e r s i s 100 M e V / c , (v) the fac t tha t the nuc l eon i s bound in the n u c l e u s i s n e g l e c t e d .

U n d e r t h e s e a s s u m p t i o n s the c o r r e c t i o n f a c t o r i s g iven by the f r a c t i o n ~ ( t ' , rn 3~) of p r o t o n s l e a v i n g the t a r g e t wi th a m o m e n t u m b i g g e r than the m i n i m u m d e t e c t a b l e r e c o i l m o m e n t u m :

A - $ ( t ' , m 3 v ) Z A - Z S = A ' A - '< S --< 1 , (10)

wi th A be ing the n u m b e r of n u c l e o n s and Z the n u m b e r of p r o t o n s . T h e f r a c t i o n ~ ( t ' , rn 3~) c l e a r l y i s a l s o a func t ion of the g e o m e t r i c a l t a r g e t d i m e n s i o n s and of the t a r g e t m a t e r i a l ( i .e . of A and the d e n s i t y ; s e e t ab l e 1). F o r the r a n g e of p r o t o n s in the t a r g e t m a t e r i a l the r e s u l t s of the c a l c u l a - t i o n s of A r o n [3] have been used . Range s t r a g g l i n g h a s been n e g l e c t e d .

5. COMPARISON WITH THE O P T I C A L MODEL

The o p t i c a l m o d e l (or G l a u b e r m o d e l in the o p t i c a l l i m i t ) d e s c r i b e d in r e f . [1] h a s b e e n u s e d to i n t e r p r e t the d i f f e r e n t i a l c r o s s s e c t i o n s of t a b l e 2. In th i s m o d e l the c o h e r e n t d i f f e r e n t i a l c r o s s s e c t i o n i s e x p r e s s e d a s

d 2 a c e Bt ' A 2 IF( t ' , m37r)l 2 d t ' dm3v - C0(rn3v) , (11)

t o

w h e r e C0(rn37 r) i s a s c a l i n g f a c t o r and w h e r e the f o r m f a c t o r F , m o d i f i e d by the a b s o r p t i o n of the i n c o m i n g and the ou tgo ing p a r t i c l e w a v e s in n u c l e a r m a t t e r , i s a l s o a func t ion of the m o d e l p a r a m e t e r s a l , e l , a2, e2 , c and a (the s u b s c r i p t 1 r e l a t e s the q u a n t i t i e s to the i n c i d e n t p ion , the s u b s c r i p t 2 to the ou tgo ing t h r e e - p i o n s y s t e m ; the a i deno te the t o t a l c r o s s s e c t i o n of the m e s o n in q u e s t i o n on n u c l e o n s ; ot i i s the r a t i o of r e a l to i m a g i n a r y p a r t of the f o r w a r d e l a s t i c s c a t t e r i n g a m p l i t u d e of p i o n s and of the t h r e e - p i o n s y s t e m on n u c l e o n s , r e s p e c t i v e l y ; c and a a r e the two p a r a m e t e r s of the F e r m i d i s t r i b u t i o n of the n u c l e a r dens i ty ) . F o r the c o m p a r i s o n with the d a t a the m o d e l p a r a m e t e r s have been f ixed a t the fo l lowing v a l u e s :

¢Yl = 25.4 mb , e 1 = - 0 . 1 , a 2 = 2 5 m b , e 2 = 0 , 1

c = 1.12 A~ fm , a = 0.545 f m , B = 10 (G e V /c ) -2 (12)

T h e p r e s e n t v a l u e of a 2 has been b a s e d on the r e s u l t s o b t a i n e d by f i t t i ng the A - d e p e n d e n c e of the i n t e g r a t e d c r o s s s e c t i o n for c o h e r e n t p r o d u c t i o n [1].

C. Benzporad et al . , Pion production on nuclei 641

1< m~,~<1.2 G~

dr dmrw °

Ti

10

. . . . . . . . . .

0 (105 0.1 0.15' 0.2

I

Fig. 1. Differential cross section d2o-/dt ' dm3n for different target nuclei and for produced masses in the interval 1.0 - 1.2 GeV.

T h i s m e a n s that only the s c a l i n g f a c t o r s C0(m3~) and 10(m37r) have been a d j u s t e d by the fit. The model c u r v e a c t u a l l y f i t ted was the s u m of e x p r e s - s i o n s (7) and (11), which was s m e a r e d out u s i n g eqs. (4) and (6), in o r d e r to a ccoun t for the e x p e r i m e n t a l a n g u l a r r e s o l u t i o n . In what c o n c e r n s the i n c o h e r e n t c o n t r i b u t i o n , the f o r m without the s u p p r e s s i o n due to the P a u l i p r i n c i p l e (8) was c l e a r l y f avou red by the f i ts . Al l f i g u r e s p r e s e n t e d in this p a p e r a r e g iven for th is fo rm.

642 C.Bernporad et al . , Pion production on nuclei

The b e s t fit c u r v e s and the e x p e r i m e n t a l po in t s a r e shown in fig. 1 for a l l the t a r g e t e l e m e n t s and for a p r o d u c e d t h r e e - p i o n m a s s be tween 1 and 1.2 GeV. The mode l c u r v e s d e s c r i b e the e x p e r i m e n t a l d i s t r i b u t i o n s f a i r l y wel t a p a r t f r o m a d i s c r e p a n c y in the r e g i o n of the f i r s t dip and the second m a x i m u m . In p a r t i c u l a r , the s e c o n d a r y s t r u c t u r e s in the d i s t r i b u t i o n s of the haevy t a r g e t e l e m e n t s T a and Pb a r e wor th m e n t i o n i n g ; they show up in the da ta and in the mode l c u r v e s at the s a m e t ' - v a i u e s . The s m a l l d e c r e a s e of the mode l c u r v e in the f i r s t few b ins is due to the a n g u l a r r e s o l u t i o n folded in to the t h e o r e t i c a l cu rve . The effect of the r e c o i l c o r r e c t i o n is i l l u s t r a t e d for the d i s t r i b u t i o n s of Be and Pb: c u r v e I i n c l u d e s the r e c o i l c o r r e c t i o n , w h e r e a s c u r v e II r e p r e s e n t s a pu re exponen t i a l f o rm of the i n c o h e r e n t b a c k g r o u n d .

I

dt' dm3Tt Ta target

.1 GeV

10C

10

1.7, .9

I , ~ i ill J'l i if, I . . . . 0 .05 (11 0.15 0.2

[rl

Fig. 2. Differential cross section d2(y/dt ' drn3y for different mass intervals and for the target nucleus Ta.

C. Bernporad et al., Pion production on nuclei 643

A f u r t h e r i n d i c a t i o n of the goodness of the op t ica l mode l l i e s in the fact tha t the s c a l i n g f a c t o r s Co(m 3~) ob ta ined f rom the f i ts shown in fig. 1 do not depend on the t a r g e t n u c l e u s wi th in the s t a t i s t i c a l e r r o r s of the fit. T h i s is in a g r e e m e n t with the r e s u l t s p r e s e n t e d in ref . [1], w he r e the c o m p a r i s o n b e t w e e n mode l and da ta has been done u s ing the c o h e r e n t p r o d u c t i o n c r o s s s e c t i o n s i n t e g r a t e d o v e r t ' .

F ig . 2 i l l u s t r a t e s the c o m p a r i s o n be tween opt ica l mode l and data for T a in the f ive m a s s b in s of width 0.2 GeV and c o v e r i n g the r e g i o n f r o m 0.9 to 1.9 GeV. A l so h e r e the a g r e e m e n t is qu i te r e m a r k a b l e . The f i r s t d i f f r a c t i o n peak b e c o m e s s m a l l e r with i n c r e a s i n g m a s s as the p h y s i c a l l i m i t tmi n a p p r o a c h e s the f i r s t m i n i m u m . Th i s is well r e p r o d u c e d by the model .

T h e r e r e m a i n s s o m e d i s c r e p a n c y be tween the fit and the da ta in the t r a n s i t i o n r e g i o n be tween the f o r w a r d peak and the i n c o h e r e n t s lope. Th i s d i s c r e p a n c y is m o s t m a r k e d for low m a s s and heavy nuc l e i (see top of f igs . 1 and 2). While it is h a r d to see how an e x p e r i m e n t a l b i a s o r b a c k g r o u n d could have such a dependence on A and m3n , and e x p e r i m e n t a l o r i g i n of the d i s c r e p a n c y cannot be exc luded with c e r t a i n t y . It is a l so p o s s i b l e that the op t i ca l mode l n e e d s s o m e r e f i n e m e n t fo r th is r e g i o n of r e l a t i v e l y l a r g e t ' ~. In t e r m s of c o n t r i b u t i o n to the i n t e g r a t e d c o h e r e n t p r o d u c t i o n c r o s s s e c t i o n , th i s d i s c r e p a n c y has , howeve r , a v e r y s m a l l effect .

We wish to thank Drs . L. Le~n iak and J. P e r n e g r for t he i r c r i t i c a l r e a d i n g of the m a n u s c r i p t .

RE FERENCES

[1] C. Bemporad, W. Beusch, A. C. Melissinos, E. Polg~tr, D. Websdale, J. D. Wilson, J. P. Dufey, K. Freudenreich, R. Frosch, F. X. Gentit, P; Mtthlemann, J. Codling, J. G. Lee, M. Letheren, G. Bellini, M. de Corato and G. Vegni: Nucl. Phys. B33 (1971) 397.

[2] C.A. Engelbrecht, Phys. Rev. 133B (1964) 988; M. L. Goldberger and K. M. Watson, Collision theory (Wiley, 1967), 3rd ed., chapter 11.

[3] W. A. Aron, UCRL-1325 (1951).

P re l imina ry calculations with the optical model including the Coulomb phase (accounting for the elastic scattering of the ingoing and outgoing part icle in the nuclear Coulomb field) show that this effect fills in the f i rs t minimum in the model curves of the heavy target nuclei. On the other hand, it does not explain the devia- tion in the region of the second maximum.