E. N agy
гг JK
3 9 5 3Z
K F K I - 7 1 - 8 3
M O M EN TU M ESTIMATION O F ELECTRONS TRAVERSING HEAVY MEDIA
IN H O M O G E N E O U S M AGNETIC FIELD II.
( Ш я т ^ т а п S ^ c a d e m y o f S c i e n c e s
CENTRAL RESEARCH
INSTITUTE FOR PHYSICS
KÖZPONTI "p./X.
't/
KÖNYVTARA *
BUDAPEST
MOMENTUM ESTIMATION OF ELECTRONS TRAVERSING
HEAVY MEDIA
IN HOMOGENEOUS MAGNETIC FIELD I I .
E. Nagy
H igh E n e rg y P h y a i c a D i v i s i o n C e n t r a l R e e e a r c h I n a t i t u t e o f P h y s i c s
B u d a p e s t
ABSTRACT
The m ethod p ro p o s e d i n r e f . [1] h a s b een t e s t e d e x p e rim e n t a l l y b o t h on m o n o ch ro m atic p o s i t r o n beam t r a v e r s i n g a h eav y l i q u i d b u b b le ch am b er and on "y r a y s p ro d u c e d i n h ig h e n e r g y i n t e r a c t i o n
o f K* m esons w i t h t h e n u c l e i o f a h eav y l i q u i d b u b b le ch am b er.
РЕЗЮМЕ
Метод» предложенный в р аботе [ I ] , был проверен экспериментально с помощью пузырьковой камеры, наполненной тяжелой жидкостью, облученной монохроматическим позитронным пучком, а также с помощью гамма-квантов, рожденных в К+ взаимодействиях с ядрами.
KIVONAT
Az [1] m unkában j a v a s o l t m ódszer k í s é r l e t i e l l e n ő r z é s é t v é g e z tü k e l neh éz f o l y a d é k k a l t ö l t ö t t b u b o ré k k a m rá b a b e s u g á r z o t t p o z i t r o n n y a lá b o n é s K* m a g -k ö lc s ö n h a tá s b a n k e l e t k e z e t t f o to n o k o n .
1 . INTRODUCTION
I n r e f . [1] a new m ethod was p r o p o s e d f o r t h e d e t e r m i n a t i o n o f th e momentum o f f a s t e l e c t r o n s b e in g d e t e c t e d i n h e a v y m e d ia , e . g . i n h e a v y l i q u i d b u b b le ch am b er. I n t h i s p a p e r we p r e s e n t some e x p e r i m e n t a l r e s u l t s o b ta in e d b y th e p r o p o s e d m eth o d . I n Sec. 2 we b r i e f l y r e c a l l th e b a s i c fo r m u la e t o b e u sed an d i n t r o d u c e a n a p p r o x im a tio n w h ic h f a c i l i
t a t e s c o n s i d e r a b l y t h e a p p l i c a t i o n . S e c . 3 i s th e n d e v o te d t o t h e d e s c r i p t i o n o f th e e x p e r i m e n t a l p r o c e d u r e a n d to th e p r e s e n t a t i o n o f th e r e s u l t s . Some c o n c l u s i o n s on t h e a p p l i c a b i l i t y o f o u r method a r e a l s o draw n i n S e c . 3»
2 . THE DESCRIPTION OF THE METHOD
As u s u a l t h e co m p o n en ts o f t h e e l e c t r o n ’ s th ree-m o m en tu m to be d e te r m in e d a r e r e p l a c e d b y t h e f o l l o w i n g new p a r a m e t e r s :
a / " d i p ” (X ) , i . e . th e a n g le b e tw ee n t h e e l e c t r o n ’ s i n i t i a l d i r e c t i o n and t h e p la n e p e r p e n d i c u l a r t o th e m a g n e tic f i e l d B , .we c a l l i t / х у / p l a n e .
b / " a z im u th " ( ) , i . e . t h e a n g le b e tw e e n t h e p r o j e c t e d i n i t i a l d i r e c t i o n o f th e e l e c t r o n o n to t h e / х у / p l a n e and th e x a x i s c h o s e n a r b i t r a r i l y i n t h i s p l a n e .
' с / » th e i n i t i a l c u r v a t u r e o f th e t r a c k p r o j e c t e d o n to t h e / х у / p l a n e , w h ic h , w hen m easu red i n cm“ 1 , i s c o n n e c te d t o p , t h e a b s o l u t e v a lu e o f th e e l e c t r o n ’ s t h r e e momentum th r o u g h t h e w e l l known fo r m u la :
2
О. зв P o --- -
р . C O S A
Н еге t h e m a g n e tic f i e l d , s t r e n g t h s h o u ld b e p u t i n kG auss when p i s m e a su re d i n M eV/c.
P a r a m e te r / а / c a n b e e s t i m a t e d s e p a r e t e l y w h i l s t t h e e s ti m a t i o n o f po and Ф i 3 made i n th e sam e p r o c e d u r e , which i n t r o d u c e s a c o r r e l a t i o n b e tw e e n t h e i r e s t i m a t e d v a l u e s .
2 . 1 . 1 . ESTIMATION OF THE D IP
F o r t h e b e a t e s t i m a t e o f X o u r m ethod y i e l d s th e f o ll o w i n g v a l u e :
w ith
tgX o = ( / £2+4a2 - b ) / ( 2 a ) ;
Д / 2
6X о = vc o s 2 X0 tgXQ b 2+4a2;
a = s . g Cl) i * i j
*i * 1 ^ s j ' zi
b - g 4 / s., - z. д 4Ц Zj = Ьх - b 2 t h
121
W here i s t h e p r o j e c t e d a r c l e n g t h o f t h e i m easu red p o i n t , z^ i t s m e a su re d z - c o o r d i n a t e , and i s t h e i n v e r s e c o v a r ia n c e m a t r i x b e in g th e sum o f two t e r m s , one r e s u l t i n g fro m t h e Coulomb s c a t t e r i n g , th e o t h e r fro m th e m easu rem en t e r r o r :
( gU1) i j " ^ ° c b )±j + ( o S L ) = <d± V •
i s d i s t a n c e o f th e 1ъ p o i n t from t h e e l e c t r o n t r a j e c t o r y / s e e P i g . 1 / -
I t i s i n t e r e s t i n g t o com pare e q u . / 2 / w ith th e r e s u l t o f th e c l a s s i c a l m eth o d s [ 2 , 3 ] :
- 3 -
tg x 0 = a / b i
6 X Q => c o a 2 \ Ql / Б ^
/3 /
The two r e s u l t s a r e e q u i v a l e n t p r o v i d e d t h a t I n th e l a t t e r c a e e one r e p l a c e s = <d'L d'> , w h e re d^ I s d e f i n e d i n F i g . 1 . H o w ev er,
i n p r a c t i c e one u s u a l l y u s e s
( • * ) „ - d t ( * , - О
w here XQ i s t h e r a d i a t i o n l e n g t h o f th e l i q u i d and Sj^ i s d e f i n e d i n F ig . 1 . Equ / 4 / i s c o r r e c t i n c a s e o f e q u . / 2 / b u t i t s u s e i n t r o d u c e an a p p r o x i m a t i o n / a t l e a s t i n p r i n c i p l e / f o r v a l u e s o b ta in e d b y e q u . / 3 / . C l e a r l y , t h i s a p p r o x im a tio n i s b e t t e r when a p p r o c h e s to z e r o , w here e q u . / 2 / an d / 3 / c o i n c i d e . A ls o , i f >> о , , t h e tw o m ethods g iv e p r a c t i c a l l y t h e same r e s u l t s .
4
2 . 2 . E s t i m a t i o n o f Pq and Ф.
T h e s e p a r a m e te r s a r e e s t i m a t e d u s i n g th e maximum l i k e l i h o o d m ethod w i t h t h e l i k e l i h o o d f u n c t i o n :
± ( ро'Ф) = f e x p | " 1 П± [аСв)] g f p Пj [a ( s ) ] . f [ a ( s ) ] Л [a f s ) ] , P
w h e r e
/ 5 /
y i ” Y iCaCs)] * .m b e in g t h e m easu red у - c o o r d i n a t e and
У i 5 УW ‘ Í s i
1 dT s i n 'T
f po e a ( t ) d t + ф
оJ J
О
/ 6 i s th e so c a l l e d Coulomb mean, d e te r m in e d b y t h e c o n t i n u o u s l y v a r y i n g i o n i s a t i o n l o s s and b y t h e ra n d o m ly v a r y in g b r e m s s t r a h l u n g l o s s . Po exp [ a ( t )] d e s c r i b e s th e v a r i a t i o n o f t h e c u r v a t u r e and f o r a moment we t o o k i n t o a c c o u n t o n ly t h e e f f e c t o f b r e m s s t r a h l u n g due t o w h ic h th e i n i t i a l e n e r g y , E0 o f t h e e l e c t r o n i s a b s o rb e d b y an a b s o r b - t l o n c o e f f i c i e n t « an d b e co m e s:
E ( t ) = E е - а ^ - я й p = c s t . p “ 1 e
i s th e e l e c t r o n ’ s i n i t i o l momentum
O f c o u rs e = 0 a n d t i s a m o n o to n o u s ly i n c r e a s i n g f u n c t i o n . The i n t e g r a n d o f e q u . / 5 / i s th u s f a c t o r i s e d , t h e f i r s t f a c t o r g i v i n g th e p r o b a b i l i t y t h a t y® ’ s a r e o b s e r v e d once s i s g i v e n , th e sec o n d f a c t o r , on th e o t h e r h a n d , s ta n d s f o r th e p r o b a b i l i t y o f a p a r t i c u l a r t r a j e c t o r y . The l i k e l i h o o d f u n c t i o n i s th e n o b t a i n e d by summing o v e r a l l p o s s i b l e t r a j e c t o r i e s , t h i s i s m eant b y th e sym bol
As b e f o r e ,
N e x t we make t h e f o l l o w i n g a p p r o x im a tio n :
- 5 -
/ 7 /
The l a s t f a c t o r i s known t o b e [ 4 ] b s - 1 ot О
/ 8 /
w i t h b 1
Xo ln 2
an d aо a
s Q b e i n g th e maximum t r a c k l e n g t h .
5 ( s ) i s a w e l l d e f i n e d t r a j e c t o r y , i n o u r a p p r o x im a tio n a s t r a i g h t l i n e :
“(s) s “o f '
О/9/
T h is a p p r o x im a tio n i s n e c e s s a r y , s in c e t h e c a l c u l a t i o n o f t h e o r i g i n a l i n t e g r a l ( 5 ) c a n n o t be c a r r i e d o u t a n a l i t i c a l l y a n d th e num er
i c a l c a l c u l a t i o n u s u a l l y r e q u i r e s c o n s i d e r a b l e c o m p u te r t i m e . On th e o t h e r h an d t h e p a r a m e t r i s a t i o n o f a ( s ) w ith o n l y one p a r a m e t e r ( c . f . e q u . / 9 / ) i s j u s t i f i e d i n c a s e s , w h ere onüy few p o i n t s a r e m easu red on t h e t r a c k s , and t h u s th e m e asu re d c o o r d i n a t e s d o n o t c a r r y en o u g h i n f o r m a t i o n on a ( s ) . M o re o v e r, a s c a n b e e a s i l y s e e n u s i n g t h e p ro b a b i l i t y f u n c t i o n / 8/ , ä ( s ) ( f u n c t i o n / 9/ ) i s t h e mean o f a l l t r a j e c
t o r i e s o i ( s ) .
S u b s t i t u t i n g e q u ./ 8 / and / 9 / i n t o e q u . / 7 / , and t a k i n g i n t o a c c o u n t i n f i r s t o r d e r t h e d e f o r m a t i o n o f th e t r a j e c t o r y d u e t o th e i o n i s a t i o n l o s s , we g e t
w h ere
- 6 -
t
cos(*-{)[siCt)Jíe As± + sln(*-S)[ci(t)]«e ü£ii lf + 6 ^ ( slí>o)3_ po s o . “o 2 . 2d
w it h e = _ _ , л - — ;
к= Ö 3 5
о о
d b e in g t h e d e n s i t y o f th e l i q u i d , and
Si ( t ) = I S iS ^ - d t ; C i ( t ) - I d t .
The b e s t e s t i m a t e s o f p о and ф a re t h e s o l u t i o n s o f th e sy ste m o f e q u a t i o n s :
3PX 3PX _ 3P±
3 ^ = Эф“ " 3 ^ ~ ° • / U /
I n s p e c t i n g e q u . / 3 0 / one can h o w e v e r r e c o g n i s e t h a t t h e maximum l i k e l i hood m ethod can o n l y b e a p p l i e d , when
s o > F~ о ' / 1 2 /
i . e . when t h e t r a c k l e n g t h i s l o n g e n o u g h . S in c e i n many p r a c t i c a l c a s e s c o n d i t i o n / 12/ i s n o t f u l f i l l e d , we e l a b o r a t e d two m eth o d s f o r th e c a s e
So < IT ‘
О
i / The f i r s t m ethod c a n be a p p l i e d , i f t h e r e a r e s u f f i c i e n t p o i n t s on th e t r a c k i n o r d e r t o c a r r y i n f o r m a t i o n f o r th e t r u e v a lu e o f a Q The e x p r e s s i o n " s u f f i c i e n t " h a s a v e ry q u a l i t a t i v e m eaning d e p e n d in g on th e l i q u i d , t h e d i s t r i b u t i o n o f th e p o i n t s on t h e t r a c k , e t c . I n p r a c t i c e t h e m ethod i s not a p p l i c a b l e w i t h l e s s t h a n s i x m e asu re d p o i n t s .
I f t h e r e a r e enough p o i n t s , f o r some f i x e d v a l u e s o f a Q one r e s o l v e s th e s y s te m o f e q u a t i o n s
э р.
ü l
Эф о / 1 3 /
- 7 -
The o b ta in e d s o l u t i o n s h a v e t o be w e ig h te d by e x p |- j x
n i ( ao ' po ' * ) g i j nj ( “o ' p0 ' ♦ ) ] ' t h e P r o b a b i l i t y o f t h e
" f i t ” , m u ltip lie d by t h e p r o b a b i l i t y á p r i o r i o f t h e p a r a m e te r s [5 ] , u s u a l l y by я / «0 , s Q/ . The b e s t e s t i m a t e o f t h e p a r a m e t e r s ,
P0 , ♦. 0.o i s t h e i r a v e r a g e w e ig h te d i n t h i s w ay.
i i / On th e o t h e r h a n d , i f t h e m easured p o i n t s do n o t c o n t a i n i n f o r m a tio n on aQ / e . g . t h e i r num ber i s n o t s u f f i c i e n t / t h e f i r s t f a c t o r o f th e w e ig h t i . e . t h e p r o b a b i l i t y o f th e f i t d o e s n o t d ep en d on aQ , and t h e w e ig h te d a v e r a g e t u r n s o u t t o be a lw a y s th e same f o r a l l t r a c k s . I n many c a s e s t h i s c a n g iv e r i s e t o a s y s t e m a t i c b i a s , e . g . to a s y s t e m a t i c o v e r e s tim a te o f th e e l e c t r o n e n e r g y , a s c a n be e a s i l y p r o v e d . T h e r e f o r e i n t h i s c a s e / p r a c t i c a l l y a lw ay s when t h e num ber o f m easu red p o i n t s i s l e s s th a n s i x / ,w e p ro p o s e t o ch o o se a Q ra n d o m ly a c c o r d in g t o d i s t r i b u t i o n ( 8 ) an d to s o l v e e q s . / 13/ w i t h t h i s v a l u e .
T h is p r o c e d u r e c l e a r l y i n c r e a s e s t h e e r r o r o f t h e p a r a m e te r s / i n th e c a s e o f g a u s s i a n d i s t r i b u t i o n t h e e r r o r becom es /2 * tim e s l a r g e r / , b u t a v o id s s y s t e m a t i c b i a s e s .
The e r r o r s on t h e p a r a m e te r s c o n s i s t o f two te r m s , t h e f i r s t b e i n g th e u s u a l one o b ta in e d a t f i x e d aQ fro m t h e maximum l i k e l i h o o d m eth o d , th e se c o n d i s due t o th e f l u c t u a t i o n o f aQ »
6p =
w here
(Да0 ) 2 = | w( a i )
w(a
2) ^ ( a - a ^ - a 2 ) a 2 da^ do^ da . / 1 5 / T h is l a s t i n t e g r a l i s b e s t t o c a l c u l a t e b y Monte C a r lo m eth o d . The r e s u l t i s q u o te d i n T a b le 1 :- 8 -
T a b le 1
a 0 . 0 .2 0 .4 0 .6 0 .8 1 .0 1 .2 1 .4 1 .6 1 .8
(Act) 2 0 . 0 .2 4 0 .6 8 1 .1 3 1 .6 8 2 .2 3 2 .4 6 2 .7 6 3 .0 2 3 -4 7
3 . EXPERIMENTAL RESULT
3 . 1 T e s t on a m onochrom atic p o s i t r o n beam
I n o r d e r t o v e r i f y e l e c t r o n p ro g ra m s , D .M o r e i l e t h a s i n i t i a t e d t h e i r r a d i a t i o n o f t h e h eav y l i q u i d b u b b le cham ber BP3 a t S a c l a y by a m onochrom atic p o s i t r o n beam o f momentum /4 6 4 - 1 5 / MeV/ с . The beam mo
mentum was c a l i b r a t e d by m e a s u rin g s to p p in g p r o to n s i n s i d e t h e cham ber, w h ich h ad th e same momentum a s t h e p o s i t r o n s a t th e e n t r a n c e . The r a d i a
t i o n l e n g t h o f t h e l i q u i d was 18 cm.
Each p o s i t r o n t r a j e c t o r y was th e n m easu red i n 12 p o i n t s d i s t r i b u te d u n if o r m ly fro m th e b e g in n in g o f t h e t r a c k u n t i l t h a t i t r o t a t e d more th a n 6 0 ° . I n 80% o f th e t r a c k s th e t o t a l m easu red l e n g t h f u l f i l l e d th e c o n d i t i o n b s Q > 1 . I f n o t , we w ere p r a c t i c a l l y a lw ay s l e f t w ith more th a n 6 p o i n t s , so we u sed m ethod i / .
I n F i g . 2 t h e c a l c u l a t e d momentum d i s t r i b u t i o n c an b e s e e n .
F ig . 2
- 9 -
The s l a s h i n d i c a t e s th e t r u e e n e r g y . E v e n ts to w a rd s lo w e r e n e r g i e s a r e m a in ly d ue to p o s i t r o n s w hich h a v e l o s t e n e r g y a t th e e n t r a n c e / b e f o r e th e f i r s t m easu red p o i n t / . On t h e o t h e r h a n d , th e lo n g t a i l to w a rd s h i g h e r e n e r g i e s i s p a r t l y a c o n se q u e n c e o f o u r p r o c e d u r e , w h ich i n f a c t g iv e s e s t i m a t i o n on 1 /p i n s t e a d o f p . T h is c a n be v e r i f i e d on t h e 1 /p d i s t r i b u t i o n / F i g . 3 / , w hich i s i n f a c t n e a r e r t o a g a u s s i a n o n e .
F i g . 3
T h e r e f o r e , i n o r d e r t o e s t i m a t e th e a v e r a g e r e l a t i v e e r r o r o f o u r momen
tum d e t e r m i n a t i o n we u sed t h e h a l f w id th o f t h e 1 /p d i s t r i b u t i o n :
I n F i g . 4 we p l o t th e d i s t r i b u t i o n o f t h e r e l a t i v e e r r o r c a l c u l a t e d f o r e ac h i n d i v i d u a l p o s i t r o n . T h is e s t i m a t i o n з е е т з t o be c o r r e c t in c o m p arin g i t w i t h th e a v e r a g e v a lu e g iv e n a b o v e .
- l ó
i i g . 4
3 . 2 T e a t on тт° p r o d u c t i o n
As a se c o n d t e s t o f o u r e l e c t r o n p ro g ra m we u s e d у - r a y s fro m t h e K* N X* N* ir+ тг” ir° r e a c t i o n s t u d i e d i n th e CEEN 1 m h e a v y l i q u i d b u b b le cham ber C 6]• The l i q u i d c o m p o s itio n was 60%
IJg and 4Q% C i^ B r w i t h r a d i a t i o n l e n g t h 25 cm and d e n s i t y 0 ,8 3 g /c m ^ . The e v e n ts w ere t a k e n i f two and o n ly two у r a y s w ere s e e n a s s o c i a t e d w i t h a 3 -p r o n g i n t e r a c t i o n .
S in c e t h e s e two p h o to n s w ere p ro d u c e d w i t h a h ig h p r o b a b i l i t y i n a tt° d e c a y , o u r f i r s t aim was a t t h e r e c o n s t r u c t i o n o f th e
two p h o to n i n v a r i a n t mass d i s t r i b u t i o n .
The num ber o f t h e m easu red p o i n t s on t h e e l e c t r o n an d p o s i t r o n t r a j e c t o r i e s v a r i e d fro m 3 t o l o /T a b le 2 / . We h av e t a k e n t h e p o i n t s on t h e t r a c k u n t i l t h a t i t h a s n o t b e e n c u rv e d more th a n 6 0 ° .
T ab ló 2
Number o f p o i n t s on t r a c k s
3 4 5
" T...
6 7 8 9 l o 1
Number o f t r a c k s
231 351 53 163 62 25 76 77 lo 4 4
.For 3 p o i n t t r a c k s th e momentum was d e te r m in e d by c a l c u l a t i n g t h e r a d i u s o f a c i r c l e c o n n e c tin g t h e 3 p o i n t s and th e n s im p ly c o r r e c t ing i t f o r b r e m s s tr a h lu n g and i o n i s a t i o n l o s s . A lm ost i n e v e r y c a s e , w here b s o tu r n e d o u t t o b e s m a l l e r th a n 1 / i n 6358 o f t h e t r a c k s / , we a p p l i e d o u r m ethod i i / i . e . we h av e c h o se n a Q ra n d o m ly . The e f f e c t i v e mass d i s t r i b u t i o n o b t a i n e d i n t h i s way i s shown i n F i g . 5 . The tt°
12
p e a k i s l o c a t e d on t h e c o r r e c t p l a c e . The b a c k g ro u n d t o t h e p e a k con
s i s t s m a in ly o f e v e n t s , w here th e y ’ s w ere n o t p ro d u c e d i n a s i n g l e d e c a y .
F o r th e s a k e o f c o m p a riso n i n F i g . 6 t h e same e v e n ts a r e shown,
F i g . 6
w here i n t h e c a s e o f b s Q < 1 we a p p l i e d m ethod i / . The a v e r a g e e r r o r seem s t o be s l i g h t l y l e s s b u t th e p e a k i s s l i g h t l y s h i f t e d to w a rd s h i g h e r e n e r g i e s , j u s t a s e x p e c te d .
The same m a t e r i a l e n a b le d u s t o v e r i f y o u r e s t i m a t i o n c o n c e r n in g Ф a s w e l l . N am ely, we c o u ld com pare t h e c a l c u l a t e d d i r e c t i o n o f e v e r y Y r a y w i t h t h a t o b ta in e d by th e m easurem ent o f th e i n t e r a c t i o n p o i n t / t h e o r i g i n o f th e у r a y s / an d t h e у r a y v e r t e x . S in c e th e m easurem ent e r r o r o f t h e c o o r d i n a t e s i n q u e s t i o n i s u s u a l l y s m a l l , t h e d i r e c t i o n d e te r m in e d by th e i n t e r a c t i o n p o i n t and t h e у v e r t e x c a n be c o n s id e r e d a s " t r u e " . I n F i g . 7 we p l o t th e d i s t r i b u t i o n o f th e
- 13 -
q u a n t i t y
F i g . 7
. " t r u e " .m easured A , = -it---
w here о ^ i s t h e e s t i m a t e d m easu rem en t e r r o r . I f t h e e s t i m a t i o n i s c o r r e c t , t h e q u a n t i t y A^ s h o u ld h ave a g a u s s i a n d i s t r i b u t i o n o f u n i t
h a l f w i d t h . The o b s e r v e d d i s t r i b u t i o n f u l f i l e t h i s c r i t e r i o n f a i r l y w e l l . We h a v e o b ta in e d a s i m i l a r d i s t r i b u t i o n f o r t h e d i p , A •
As a l a s t s t e p we com pare o u r d i p e s t i m a t i o n / e q u / 2 / / w i t h t h e c l a s s i c a l one / e q u / 3 / / i n F ig . 8 . No s i g n i f i c a n t d i f f e r e n c e can be o b s e r v e d , w h ich i s e x p la i n e d m a in ly by t h e f a c t , t h a t th e m e a su re ment e r r o r s o f t h e z - c o o r d i n a t e s a r e r a t h e r b i g .
- 14 -
F i g . 8
I n c o n c lu s io n l e t us sum m arize th e r e s u l t s o f o u r e x p e rim e n t a l v e r i f i c a t i o n . Our d i r e c t i o n e s t i m a t i o n seem s t o be c o r r e c t . The a v e r a g e r e l a t i v e e r r o r o f t h e e n e r g y e s t i m a t i o n i s 27% i n a l i q u i d o f 18 cm r a d i a t i o n l e n g t h . /T h e c o r r e s p o n d i n g v a l u e o b ta in e d by th e SPIRAL m ethod [1] i s v e r y n e a r t o t h i s o n e / . B e s id e t h e s e f a c t s we s h o u ld l i k e t o p o i n t o u t t h e s p e c i a l v i r t u e o f o u r method m a in ly o f p r a c t i c a l use» i t r e j e c t s a v e r y few p e r c e n t / i n g e n e r a l 1- 2%/ o f t r a c k s a s u n e s ti m a b le .
I n o u r o p in io n th e m ain p a r t o f u n c e r t a i n t y i n o u r e n e r g y e s t i m a t i o n i s du e t o th e a p p ro x im a tio n made i n d e r i v i n g e q u / 7 / » T h e re f o r e we h ave t r y e d t o a p p ro x im a te b e t t e r th e a c t u a l t r a j e c t o r y a s th a n w ith a l i n e a r one by i n t r o d u c i n g more p a r a m e t e r s . H ow ever, imme-
- 1 5 -
d i a t e l y we were f a c e d w ith two d i f f i c u l t i e s : 1 . The new p o c e d u re r e q u ire s more m easured p o i n t s on t h e t r a c k , and 2 . an o r d e r o f m ag n itu d e more c o m p u te r tim e . So i t c a n n o t b e u se d when a g r e a t num ber o f e v e n ts h a s t o b e t r e a t e d .
I sh o u ld l i k e t o e x p r e s s my d e e p g r a t i t u d e t o P r o f . A.
L a g a r r ig u e f o r h i s warm h o s p i t a l i t y a t t h e O rs a y L a b o r a t o i r e d e 1*
A c c é l é r a t e u r L i n é a i r e , w here a p a r t o f t h i s w ork was c a r r i e d o u t . I am i n d e b t e d to D r. D. M o r e i l e t , who k i n d l y l e n t me th e e x p e r i m e n t a l m a t e r i a l on p o s i t r o n t r a c k s an d th u s e n a b le d me t o c a r r y o u t a v e r i f i c a t i o n o f t h e p ro p o s e d m e th o d . I am a l s o i n d e b te d t o him f o r u s e f u l d i s c u s s i o n s . A se c o n d v e r i f i c a t i o n o f t h e m ethod was c a r r i e d o u t on an e x p erim en tal m a t e r i a l k i n d l y sen d me b y R. A rn o ld , w h ich i s g r e a t l y a k n o tflo d g e d .
R e f e r e n c e s
[1] £ . Nagy KFKI 7 0 -1 0 HEP
[2] D. M o r e ile t LAL 1190, O rs a y , 1968.
[3 ] B. M aniukov, P .S h ly a p n i k o v , Dubna P I 0-4255> 1969.
[4] B. R o s s i , H igh E n e rg y P a r t i c l e s
[5 ] C. P a s c a u d , T h e se LAL 1 1 7 1 , O rsa y , ISPS?.
Annexe I .
[6 ] P a r i s - B ergen - S t r a s b o u r g - M adrid C o l l a b o r a t i o n . To be p u b lis h e d i n P h y s ic s L e t t e r s .
C i . 7
K ia d ja a K ö z p o n ti F i z i k a i K u ta tó I n t é z e t F e l e l ő s k ia d ó s K is s D ezső , a KFKI N ag y en er
g i á j ú F i z i k a i Tudományos T a n á csá n ak e ln ö k e Szakm ai l e k t o r : P .S Í j a p n y i k o v
N y e lv i l e k t o r : ' T e l b i s z F e re n c
P é ld á n y sz á m : 320 T ö rz ssz á m : 6253 K é s z ü lt a KFKI s o k s z o r o s í t ó üzem ében B u d a p e s t, 1 9 7 2 . j a n u á r hó
