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Energy measurement of pion beams 1976

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ENERGY MEASUREMENT OF PION BEAMS by Takenori Suzuki M.Sc, Tokyo University, 1970 A THESIS SUBMITTED IN PARTIAL FULLFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the Department of Physics We accept t h i s thesis as conforming to the required standard The University of Br i t i s h Columbia October, 1976 © Takenori Suzuki, 1976 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l m a k e i t f r e e l y a v a i l a b l e f o r r e f e r e n c e a n d s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e H e a d o f my D e p a r t m e n t o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f Physics T h e U n i v e r s i t y o f B r i t i s h C o l u m b i a 2075 Wesbrook Place Vancouver, Canada V6T 1W5 D a t e October 6, 1976 A b s t r a c t P i o n e n e r g i e s b e t w e e n 30 MeV a n d 6 0 MeV h a v e b e e n d e t e r m i n e d b y t h r e e m e t h o d s : m a g n e t i c f i e l d , r a n g e a n d t i m e o f f l i g h t . T h e p i o n e n e r g y w a s m e a s u r e d t o a n a c c u r a c y 2 # b y r a n g e a n d 5 % b y t h e t i m e o f f l i g h t , a n d t h e t h r e e m e t h o d s w e r e c o n s i s t a n t w i t h o n e a n o t h e r . T h e p i o n s w h i c h w e r e u s e d w e r e p r o d u c e d b y 5 0 0 M e V p r o t o n s f r o m t h e T R I U M F c y c l o t r o n . T h e p i o n s w e r e f o c u s s e d i n t o a b e a m u s i n g t h e M9 ( o r s t o p p e d ny^j ) c h a n n e l . T h e b e n d i n g m a g n e t s o f t h i s c h a n n e l d e f i n e t h e m o m e n t u m o f t h e b e a m p a r t i c l e s a n d t h u s g i v e o n e m e t h o d o f d e t e r m i n i n g t h e p i o n e n e r g y . D i f f e r e n t i a l a n d i n t e g r a l r a n g e c u r v e s f o r a l u m i n u m a n d c o p p e r w e r e t a k e n a n d t h e d i f f e r e n t i a l c u r v e s w e r e u s e d t o d e t e r m i n e t h e r a n g e . I n t a k i n g t h e d i f f e r e n t i a l c u r v e t h e e f f e c t o f s t o p p i n g p l a t e s w a s e x a m i n e d . I n t h e t i m e o f f l i g h t m e t h o d , t h e l e a d i n g e d g e ; t y p e d i s c r i m i n a t o r was; f o u n d t o g i v e ; a t i m e w a l k , d u e t o t h e d i f f e r e n t e n e r g y - l o s s ; o f p a r t i c l e s i n t h e p l a s t i c s c i n t i - l l a t o r . T h i s h a s a s e r i o u s e f f e c t i n d e t e r m i n i n g t h e p i o n e n e r g i e s . H o w e v e r , t h e c o n s t a n t f r a c t i o n d i s c r i m i n a t o r w a s f o u n d t o g i v e d e p e n d a b l e r e s u l t s a n d t h e m e a s u r e d e n e r g i e s a g r e e d r e a s o n a b l y w e l l w i t h t h o s e o f t h e o t h e r m e t h o d s . - i i i - Energy Measurement of Pion Beams table of contents: page I Introduction A General 1 B The Theory of Energy-Loss 7 (1) The Stopping Power of Charged P a r t i c l e s 7 (2) Range of a Charged P a r t i c l e 9 II Experimental Equipment A Pion Production 12 B M9 Pion Channel 14 C Counters and the Degrader 19 D Electronics 23 III The Basic Experiment for the Measurement of the Range-Energy Relation and Time of F l i g h t A The Range Curve 26 B The Height of Peak i n the D i f f e r e n t i a l Range Curve 30 C The P r i n c i p l e of the Time-of-Flight Method 35 D The Time Walk of the Discriminator 40 IV Results and Discussion A Time of F l i g h t 45 B Range Measurement 49 C The Energy Straggling and Inherent Energy Spread of the Beam 55 D C o n c l u s i o n References - v - l i s t of tables: page 2.1 The s p e c i f i c a t i o n of p l a s t i c s c i n t i l l a t o r s and photo-multiplier used for counters 20 2.2 Components and thicknesses of absorbers used for the degrader 22 3.1 The currents of magnets for the measurement on F-3 33 3.2 The e f f e c t of the stopping plate to the pion energy determination and the comparison of energy between the range and time of f l i g h t method 34 3.3 The energies of pions and muons are measured by the time of f l i g h t method. In order to check the systematic error, two types of discriminators are employed, LRS (leading edge) and C F . (constant fraction) , and two kinds of targets, copper and beryllium. 43 4.1 Magnet currents for the time of f l i g h t and range measurements 46 4.2 The energy i n MeV obtained by the time of f l i g h t and range measurements 46 4.3 The d e t a i l of range c a l c u l a t i o n 53 v i l i s t of figures: page 1.1 Energy-Loss curves for different charged particles in aluminum and copper 10 2.1 M9 pion beam channel 15 2.2 Pion kinetic energy vs. angle in a laboratory system 17 2.3 CH2 target 18 2.4 Configuration of counters and degrader 18 2.5 Scheme for the measurements of the range curve 24 3.1 The differential and integral range curve for 30 MeV pion stopped by Al 27 3.2 The effect of the stopping plate between the third and fourth counter 31 3.3 Circuit diagram of the time of fl i g h t measurement 36 3.4 Timing diagram of TAC and the display of particles on the multi-channel analyzer 36 3.5 Time calibration spectrum (periodic input, 10 nsec) 38 3.6 Time-walk due to three different input pulses of electrons, muons and pions with the same momentum. Since the discriminator level of a leading edge type i s fixed, the outputs shift in time spectrum. 41 - v i i - 3.7 Constant f r a c t i o n of height pick-off 41 4.1 R.F.-referenced time of f l i g h t spectrum for 30, 40, 50 and 60 MeV of pion energies 47 4.2 The range curves of pion i n aluminum (pion energies, 30, 40, 50 and 60 MeV) 50 4.3 The range curves of pion i n copper (pion energies, 30 and 50 MeV) 51 i - v i i i - acknowledgment I am roost g r a t e f u l to Professor D.F. Measday for his very patient explanations to a novice i n the f i e l d , and for his advice on t h i s t h e s i s . I would l i k e to thank members of TINA group, Drs. M. Hasinoff, M. Salomon, J.M. Poutissou and other graduate students for t h e i r help and discussions. The operation of TRIUMF by the cyclotron crew i s greatly appreciated. I would also l i k e to thank my wife, Yoshiko, for her excellent typing. -1- I Introduction A General The energy-loss of charged p a r t i c l e s i n matter has been studied, discussed and reviewed i n many publications. Tables and figures of range-energy r e l a t i o n s which are available i n these summaries are useful to experimental p h y s i c i s t s i n designing and evaluating t h e i r experiment. Though the accuracy i s somewhat worse than a magnetic spectrometer, the energy of charged p a r t i c l e s can be de- fined quite accurately by measuring t h e i r range i n matter. Generally, charged p a r t i c l e s lose t h e i r energy i n passing through matter by e x c i t a t i o n and i o n i z a t i o n of the atoms and molecules. These processes of energy-loss are s i m i l a r for a l l charged p a r t i c l e s , but there are some differences between electrons and heavier charged p a r t i c l e s (muons, pions, protons, ions, e t c ) . Since an electron i s very l i g h t , i t can lose energy by bremsstrahlung r a d i a t i o n . Although pions, muons and electrons are contained i n beam in our experiment, we s h a l l p r i n c i p a l l y discuss the energy- loss of heavy charged p a r t i c l e s i n t h i s work. It w i l l be shown l a t e r that the stopping power of heavy charged p a r t i c l e s depends upon the charge and k i n e t i c energy of the incident p a r t i c l e s and to a le s s e r extent on the material of the stopping medium. If we know the stopping power of one kind of charged p a r t i c l e , then we can apply the s c a l i n g r e l a t i o n of the stopping power to any other kind of heavy charged p a r t i c l e s . In most a r t i c l e s , the -2- s t o p p i n g power and range of protons are discussed^"** Using the s c a l i n g r e l a t i o n , we can c a l c u l a t e the s t o p p i n g powers and ranges o f o t h e r heavy p a r t i c l e s from those o f p r o t o n . The experimental r e s u l t s of the range-energy r e l a t i o n f o r pions produced by the meson f a c i l i t y , TRIUMF, w i l l be d i s c u s s e d i n t h i s work. Although the range o f protons has been measured f o r v a r i o u s energy ranges^ -** few experimental s t u d i e s ^ o f p i o n range have been r e p o r t e d . The p r e s e n t experiment concerns the range o f p i o n i n aluminum and copper. The problem o f the e n e r g y - l o s s o f charged p a r t i c l e p a s s i n g through matter was d i s c u s s e d i n the e a r l y p a r t o f t h i s c e n t u r y . Bohr c o n s i d e r e d the problem by c l a s s i c a l mechanics, then Bethe t r e a t e d the problem s u c c e s s f u l l y by u s i n g quantum mechanics. A s h o r t summary o f the t h e o r y of s t o p p i n g power and range i n matter w i l l be g i v e n i n s e c t i o n B. Reviews of range energy r e l a t i o n s have been p u b l i s h e d every few y e a r s . The f i r s t e x t e n s i v e summaries were g i v e n by L i v i n g s t o n e and B e t h e l 0 In 1937, they d i s c u s s e d the theory o f s t r a g g l i n g and e n e r g y - l o s s o f charged p a r t i c l e s a f t e r t r a v e r s i n g a g i v e n l e n g t h o f matter. A s h o r t summary a r t i c l e was p u b l i s h e d by T a y l o r } ^ who summarized the 12 s i t u a t i o n up t o 1951. Then A l l i s o n and Warshaw gathered t o g e t h e r a l l o f the s t o p p i n g power data c o v e r i n g low and h i g h e n e r g i e s i n 1953. In 1954, U e h l i n g * ^ p u b l i s h e d a survey on the e n e r g y - l o s s o f heavy charged p a r t i c l e s . He -3- reviewed the arguments on which the e n e r g y - l o s s equations are based and a l s o experimental r e s u l t s . The survey o f the experimental d a t a c o v e r i n g the energy l e s s than 10 14 MeV was done by Whaling i n 1958. A f t e r U e h l ing's summary, some a d d i t i o n a l measurements on a p r o t o n beam up t o 700 MeV have been done. F a n o 1 5 surveyed the l i t e r a t u r e s i n c l u d i n g the a d d i t i o n a l r e s u l t s up to 1963 and reviewed s t o p p i n g power theory i n d e t a i l . S e r r e ^ c a l c u l a t e d the range and the s t o p p i n g power o f v a r i o u s m a t e r i a l s (12 kinds) f o r the wide energy r e g i o n s from 10 MeV t o 30 GeV f o r protons, kaons, pions and muons. The c a l c u l a t e d ranges and s t o p p i n g powers are i n good agreement wi t h experimental v a l u e s t o an accuracy b e t t e r than 1 % i n the energy range from 10 MeV t o 1 GeV. In t h i s t h e s i s we mostly r e f e r t o her r e p o r t f o r numerical v a l u e s o f ranges and s t o p p i n g powers. The most r e c e n t summary appeared i n the American 17 I n s t i t u t e Handbook. In t h i s a r t i c l e are g i v e n the most r e c e n t r e f e r e n c e s , the commonly used formulae and p r i n c i p a l data on the passage o f f a s t charged p a r t i c l e s i n matter. The t a b l e s o f s t o p p i n g powers and ranges are based on c a l c u l a t i o n s u s i n g a c o r r e c t i o n f o r i n n e r s h e l l e l e c t r o n s and the continuous slowing down approximations. Values are g i v e n f o r proton e n e r g i e s between 1 and 1000 MeV. The problem o f e n e r g y - l o s s i n matter i s understood p r e c i s e l y as d i s c u s s e d i n summaries c i t e d above. But, r e c e n t l y , some mi s c e l l a n e o u s e f f e c t s have been p o i n t e d - 4 - out i n the ranges and Bethe's formula. A c c o r d i n g t o the 18 measurements o f Heckman and Lindstrom, the e n e r g y - l o s s r a t e s o f p o s i t i v e p i o n s exceed t h a t o f n e g a t i v e pions by amounts o f 0 t o 60 MeV/cm i n the v e l o c i t y i n t e r v a l 0.051 < £ < 0.178 (200 k e v < E < 2 . 3 MeV). T h i s i s due t o the d i f f e r e n c e i n the Coulomb i n t e r a c t i o n w i t h atomic e l e c t r o n s . Bethe's formula, which i s d e r i v e d by a quantum- mechanical theory based on the f i r s t Born approximation, g i v e s good agreement w i t h the e n e r g y - l o s s experiment o f f a s t p a r t i c l e s but does not e x p l a i n the d i f f e r e n c e i n the range o f o p p o s i t e charged p a r t i c l e s . Such an e f f e c t i s p r e d i c t e d by the second o r d e r Born approximation which 3 g i v e s a term p r o p o r t i o n a l t o Z o f the i n c i d e n t p a r t i c l e s . T h i s h i g h e r o r d e r c o r r e c t i o n o f the s t o p p i n g power, 3 1 9 p r o p o r t i o n a l t o Z, i s d i s c u s s e d by Jackson and McCarthy. 3 They have shown t h a t the Z term a t low v e l o c i t y l e a d s t o the range d i f f e r e n c e f o r p a r t i c l e s o f the same mass, the 20 same i n i t i a l energy, but o f o p p o s i t e charge. I n o k u t i a l s o i n d i c a t e d the departure from Bethe's formula a t low v e l o c i t y i n h i s summary. The time o f f l i g h t method i s a r e l a t i v e l y simple method f o r measuring a b s o l u t e v e l o c i t y o f p a r t i c l e s and u s e f u l over a wide range of p a r t i c l e types and e n e r g i e s . 21 A review o f time i n t e r v a l measurements was g i v e n by Porat who d i s c u s s e d the time i n t e r v a l s i n the range o f p i c o - second t o micro-second. The time o f f l i g h t method has 22-25 been r e p o r t e d f o r measuring c y c l o t r o n beam energy. -5- u s i n g the c y c l o t r o n o s c i l l a t o r as a time standard. The deuteron energy with momentum 1025 MeV/c was measured by the time o f f l i g h t between two s c i n t i l l a t i o n counters and 26 a l s o by the range measurement. For n e u t r a l p a r t i c l e s (neutron), the u s u a l techniques a p p l i e d t o charged p a r t i c l e s such as a magnetic spectrometer and range-energy r e l a t i o n can not be a p p l i e d . However, the time o f f l i g h t method has been a p p l i e d s u c c e s s f u l l y f o r measuring the e n e r g i e s 27-29 o f neutrons. R e c e n t l y the experimental r e s u l t s o f measuring the energy o f p i o n by the time o f f l i g h t have been r e p o r t e d from the meson f a c i l i t i e s 3 0 " 3 2 (SIN, LAMPF, TRIUMF). Those machines can a c c e l e r a t e h i g h c u r r e n t s o f p r o t o n t o produce the h i g h i n t e n s i t y o f p i o n beam. From the meson p r o d u c t i o n t a r g e t , the beam, which i n c l u d e s p i o n s , muons, e l e c t r o n s and o t h e r p a r t i c l e s produced by the n u c l e a r r e a c t i o n , i s e x t r a c t e d t o the experimental area p a s s i n g through the p i o n channel. Each p a r t i c l e c o n t a i n e d i n the beam has the same momentum but the d i f f e r e n t v e l o c i t y due t o the d i f f e r e n t mass. T h e r e f o r e , u s i n g the time o f f l i g h t method, we can i d e n t i f y the p a r t i c l e s and determine the v e l o c i t i e s i n the experimental a r e a . The p r e s e n t r e p o r t w i l l show the range measurements o f p i o n s i n aluminum and copper f o r the energy r e g i o n from 30 t o 60 MeV. The e n e r g i e s o f pions under the same c o n d i t i o n are measured by the time o f f l i g h t method. We s h a l l compare the energy o b t a i n e d from the range w i t h t h a t of the time o f - 6 - f l i g h t . In the measurement o f the time o f f l i g h t , we measure the time d i f f e r e n c e i n a r r i v a l between e l e c t r o n s and mesons which t r a v e l from the p i o n p r o d u c t i o n t a r g e t t o a d e t e c t o r . As e l e c t r o n s , muons and pi o n s w i t h the same momentum have d i f f e r e n t energy l o s s e s i n the p l a s t i c s c i n t i l l a t o r , they have a d i f f e r e n t t r i g g e r i n g time i n the l e a d i n g edge type d i s c r i m i n a t o r . We w i l l d i s c u s s the d i f f e r e n c e i n time p i c k o f f o f d i s c r i m i n a t o r s between the l e a d i n g edge type and the cons t a n t f r a c t i o n i n s e c t i o n 3-D. Since our experiment was done on the M9 channel, where the bending magnets d e f i n e the p a r t i c l e momentum, we s h a l l be a b l e t o u t i l i z e the r e l a t i o n between the s t r e n g t h o f the magnet c u r r e n t s and the p i o n energy. -7- B The Theory o f Energy-Loss The e n e r g y - l o s s o f charged p a r t i c l e s i n matter i s q u i t e w e l l understood. I t i s mainly due t o i o n i z a t i o n and e x c i t a t i o n o f the atoms i n matter. The n u c l e a r i n t e r a c t i o n which has a much s h o r t e r range than the e l e c t r o - magnetic i n t e r a c t i o n may become important when the p a r t i c l e has enough energy t o come c l o s e t o the n u c l e u s . We can f i n d the c l a s s i c a l c a l c u l a t i o n o f the e n e r g y - l o s s i n matter 33-35 i n many elementary books on Nuclear P h y s i c s . (1) The Stopping Power o f Charged P a r t i c l e s o The c l a s s i c a l t h e o r y was developed by Bohr. I t assumes t h a t an i o n w i t h charge Ze and v e l o c i t y v passes w i t h i n a d i s t a n c e b from an e l e c t r o n which i s assumed t o be f r e e and a t r e s t . While the i o n i s a f f e c t e d by the e l e c t r o m a g n e t i c i n t e r a c t i o n w i t h the s t o p p i n g medium, the atomic e l e c t r o n s are assumed t o move ve r y s l o w l y . Since the c o l l i s i o n time i s assumed t o be s h o r t , they a c q u i r e an impulse without changing t h e i r p o s i t i o n . A c c o r d i n g t o t h i s impulse approximation, the momentum a c q u i r e d by the e l e c t r o n must be p e r p e n d i c u l a r t o the t r a j e c t o r y o f the i n c i d e n t p a r t i c l e and can be c a l c u l a t e d by a p p l y i n g Gauss' theorem t o a c y l i n d e r w i t h the r a d i u s b. The momentum g i v e s the c l a s s i c a l k i n e t i c energy l o s t by the i o n a t an impact parameter b. Then, the r a t e o f e n e r g y - l o s s per u n i t path l e n g t h , -dE/dx, i s g i v e n by i n t e g r a t i n g the c l a s s i c a l k i n e t i c energy over b from a minimum t o a maximum. Thus, the s t o p p i n g power o f -8- the absorbing medium was o b t a i n e d i n the c l a s s i c a l form. 9 10 Bethe ' developed the theory d i s c u s s e d above by c o n s i d e r i n g quantum theory, r e l a t i v i s t i c e f f e c t s and s h e l l e f f e c t s . The Bethe's equation i s as f o l l o w s : where t h e r e are n atoms wi t h atomic number Z i n the u n i t volume o f the absorber, and i s the s h e l l c o r r e c t i o n o f the i - t h s h e l l . S ince Bethe's equation i s o b t a i n e d by u s i n g the Born approximation f o r the c o l l i s i o n p r o c e s s , i t can be a p p l i e d when the v e l o c i t y o f the i n c i d e n t p a r t i c l e i s l a r g e compared t o the v e l o c i t y o f the atomic e l e c t r o n s . Bethe's equation depends on the v e l o c i t y and charge number of i n c i d e n t p a r t i c l e s . In o r d e r f o r the equation dE/dx(m^,Z)=dE/dx(m 2,Z) t o h o l d f o r two k i n d s o f i n c i d e n t p a r t i c l e s o f d i f f e r e n t mass but the same charge, the f o l l o w i n g r e l a t i o n i s needed; where and T 2 are the k i n e t i c energy o f two ki n d s o f i n c i d e n t p a r t i c l e s . From t h i s r e s u l t , i f the s t o p p i n g power f o r protons i s known, then the s t o p p i n g power o f the p a r t i c l e w i t h mass M v and one e l e c t r i c a l charge i s o b t a i n e d (1.1) (1.2) - 9 - by s h i f t i n g the v a l u e s f o r proton by a f a c t o r M /M (=0.150 x p f o r p i o n ) . S i m i l a r l y , i f p a r t i c l e s w i t h d i f f e r e n t Z are c o n s i d e r e d , the e n e r g y - l o s s o f a p a r t i c l e w i t h charge Z^ i s 2 s h i f t e d by a f a c t o r (Z^/Z2) from the case o f a p a r t i c l e w i t h charge Z 2. For the case o f an alpha p a r t i c l e and a 2 proto n , (Ztf/Z ) =4. In the r e f e r e n c e 6, the experimental ir s t o p p i n g powers f o r proton e n e r g i e s o f 0.05 t o 12 MeV are g i v e n . Using the r u l e t o c o n v e r t the proton s t o p p i n g power to a p i o n s t o p p i n g power, the s t o p p i n g power f o r p i o n e n e r g i e s o f 0.05 t o 10 MeV i s shown i n F i g . 1.1 f o r the case o f aluminum and copper. I f the s t o p p i n g power o f a standard absorber a t a gi v e n energy i s known f o r a c e r t a i n p a r t i c l e , then, f o r any absorber under the same energy c o n d i t i o n , the s t o p p i n g power of any absorber can be d e r i v e d from the r a t i o T h i s i s c a l l e d the r e l a t i v e s t o p p i n g power and i s used t o c a l c u l a t e the t h i c k n e s s o f d i f f e r e n t m a t e r i a l s r e l a t i v e t o the standard absorber. In t h i s r e p o r t , t h i s r e l a t i o n w i l l be used t o c a l c u l a t e the r e l a t i v e t h i c k n e s s o f p l a s t i c s c i n t i l l a t o r t o aluminum and copper a b s o r b e r s . Q = = Const. (1.3) (2) Range o f a Charged P a r t i c l e S i n c e the equation f o r the e n e r g y - l o s s o f charged CSI e 2 ° 10 £ LUI x TO -a r o 1 ^ 1. Pions iip aluminum 10"3 102 Protons inaluminibm 10"1 1.0 10 Kinetic energy , MeV 10' i O I F i g . 1.1 Energy-Loss curves f o r d i f f e r e n t charged p a r t i c l e s i n aluminum and copper. - 1 1 - p a r t i c l e s i s o b t a i n e d i n the l a s t s e c t i o n ( 1 ) , i t i s p o s s i b l e t o i n t e g r a t e from the i n i t i a l i n c i d e n t energy t o the energy a t r e s t t o f i n d the t o t a l range o f t r a v e r s i n g p a r t i c l e s i n matter. We have where the continuous slowing-down approximation i s a p p l i e d t o the i n t e g r a t i o n . Since the s t o p p i n g power depends on the p a r t i c l e v e l o c i t y and charge, we can o b t a i n the range o f p a r t i c l e s from the known range f o r a c e r t a i n p a r t i c l e by a s c a l i n g r u l e . I t i s d i f f i c u l t t o use the equation o f s t o p p i n g power t o get the range a t low energy, s i n c e the theory has been o b t a i n e d by u s i n g the approximation t h a t the i n c i d e n t p a r t i c l e s have h i g h enough e n e r g i e s t o assume t h a t the atomic e l e c t r o n s are a t r e s t , as d i s c u s s e d i n the l a s t s e c t i o n . T h i s g i v e s a l i t t l e u n c e r t a i n t y t o the i n t e g r a t e d range. In o r d e r t o a v o i d t h i s u n c e r t a i n t y , the i n t e g r a l can be separated i n t o two energy r e g i o n s , i . e . , 0 < E < E j = l MeV and E ^ < E < C E q . Then, f o r the low energy 17 r e g i o n , the experimental data shown i n F i g . 1.1 can be used. For the h i g h energy r e g i o n , the s t o p p i n g power equation can be used s u c c e s s f u l l y . In the r e f e r e n c e 16 t o which we w i l l mostly r e f e r i n t h i s work, the experimental r e s u l t s o f 36 Barkas have been employed f o r the low energy r e g i o n . (1.4) -12- I I Experimental Equipment A Pion P r o d u c t i o n The e x t r a c t e d proton beam from the TRIUMF c y c l o t r o n can be v a r i e d i n energy c o n t i n u o u s l y from 150 t o 520 MeV. In t h i s experiment, the 500 MeV proton beam was p r o v i d e d by the c y c l o t r o n and the c u r r e n t was kept i n the range of 1 t o 5 nano-amps. The c y c l o t r o n r a d i o frequency i s 23.04 MHz which corresponds t o a 43.403 n s e c - p e r i o d . When the R.F. s i g n a l was f e d t o the Time t o Amplitude Converter (TAC, ORTEC 437A) as the stop s i g n a l f o r the time o f f l i g h t measurement, every second p u l s e o f the R.F, s i g n a l was erased t o produce two peaks i n the time s p e c t r a f o r a s i n g l e type o f p a r t i c l e . In t h i s experiment, the b e r y l l i u m t a r g e t o f 10 cm t h i c k n e s s was used mainly f o r the meson p r o d u c t i o n t a r g e t because o f the h i g h p r o d u c t i o n r a t e . S i n c e , f o r a t a r g e t of low atomic number, the p r o d u c t i o n r a t e o f e l e c t r o n s i s s m a l l , the copper p r o d u c t i o n t a r g e t was employed t o compare the peak p o s i t i o n o f e l e c t r o n s i n a time-delay spectrum w i t h the case o f b e r y l l i u m t a r g e t . The r e s u l t and d i s c u s s i o n w i l l be g i v e n i n the s e c t i o n 3-D. A l s o a CH 2 t a r g e t was used t o measure the e f f e c t o f the t h i c k n e s s o f a s t o p p i n g p l a t e between the t h i r d and f o u r t h counter ( F i g . 2.4). The p i o n s , muons and e l e c t r o n s produced a t the T2 o t a r g e t pass through the M9 channel which i s a t 135 t o the d i r e c t i o n o f the proton beam ( F i g . 2.1). We can choose p o s i t i v e o r n e g a t i v e pions by changing -13- the p o l a r i t y o f the magnets. Since the p o s i t i v e p i o n 37 p r o d u c t i o n i s l a r g e r than the n e g a t i v e p i o n p r o d u c t i o n , p o s i t i v e p ions were employed i n our experiment. Furthermore, n e g a t i v e pions are captured by n u c l e i a t the end o f t h e i r range and the r e a c t i o n produces p a r t i c l e s and photons. As these p a r t i c l e s produce a background s i g n a l , the range curve f o r a p o s i t i v e p i o n i s g e n e r a l l y c l e a n e r than t h a t f o r a n e g a t i v e p i o n . -14- B M9 Pion Channel The l a y o u t o f M9 channel i s shown i n F i g . 2.1. The f i x e d p a r t o f the M9 channel c o n s i s t s o f f i v e quadrupole magnets, two bending magnets and the c o l l i m a t i n g s l i t . The s l i t i s s e t between the f i r s t bending magnet and the t h i r d quadrupole magnet so t h a t we can a d j u s t the beam s i z e . I t has 30 cm x 30 cm beam s i z e when wide open and i n our e x p e r i - ment we used 2 cm width o f h o r i z o n t a l s l i t and 10 cm width o f v e r t i c a l s l i t . A f t e r the f i f t h quadrupole magnet, the meson beam e n t e r the experimental area through a vacuum window. In the experimental a r e a , t h e r e are two p l a c e s f o r s e t t i n g c o u n t e r s . The f i r s t p o s i t i o n r i g h t a f t e r the f i f t h quadrupole magnet i s c a l l e d F-2, (the second f o c u s ) . The second p o s i t i o n i s behind the C o r v a l l i s magnet and c a l l e d F-3, (the t h i r d f o c u s ) . When F-3 i s used, the t r i p l e t quadrupole magnet i s i n s t a l l e d i n the p l a c e o f F-2 t o focus the beam down to F-3. In t h i s experiment, both p o s i t i o n s were used because o f the experimental schedule o f o t h e r groups. From the p o i n t o f view o f the s e p a r a t i o n o f p a r t i c l e s i n the time-delay spectrum, F-3, which has long d i s t a n c e from the t a r g e t , i s b e t t e r than F-2. But the f l u x a t F-2 i s l a r g e r than t h a t a t F-3. The p i o n energy i s d e f i n e d by the two bending magnets and the p r o f i l e o f beam i s shaped by the e i g h t quadrupole magnets. The e f f e c t s on beam energy o f v a r y i n g the c u r r e n t 38 f o r the each magnet have been c a l c u l a t e d . T h e r e f o r e , i n o r d e r t o d e f i n e the momentum o f the p i o n s , we can s e t the Radiation area Proton beam F i g . 2.1 M9 pi o n beam channel Shielding A Experimental area Bi Q W; -Bending magnet -Quadrupole magnet -Window location Qc I Pion focus,F-2 Pa Corvallis I Pion beam Pion focus,F-3 c u r r e n t o f each magnet a c c o r d i n g t o the t a b l e 4.1. Then, i n o r d e r t o get the optimum beam o f mesons, we can a d j u s t the c u r r e n t s o f the second bending magnet and the e i g h t quadrupole magnets w h i l e h o l d i n g f i x e d the c u r r e n t o f the f i r s t bending magnet. The energy d e f i n e d by the magnets was checked by u s i n g + 39 the p i o n produced from the r e a c t i o n , p+p—» TC +d. The p i o n k i n e t i c energy as a f u n c t i o n o f angle i s shown i n F i g . 2.2 40 f o r the proton energy from 300 t o 600 MeV. The i n i t i a l p i o n energy of the M9 channel has t o be o b t a i n e d by c o n s i d e r i n g the t h i c k n e s s of the t a r g e t as shown i n F i g . 2.3, To c a l c u l a t e the mean energy o f the beam we assume t h a t the pions are produced a t the middle o f the CHj t a r g e t . The e n e r g y - l o s s o f 500 MeV protons i n the CH 2 t a r g e t (1 x 1 x 0.5 cm) i s 2.9 MeV and a t the middle o f the t a r g e t i t has l o s t a h a l f o f 2.9 MeV. As the e n e r g y - l o s s i s not so l a r g e and w i t h i n the u n c e r t a i n t y o f beam energy, i t i s assumed t h a t the p i o n i s produced by the r e a c t i o n o f 500 MeV proton a t the c e n t e r o f the t a r g e t . Thus, the pions produced 41 0 have an energy 30.9 MeV a t 135 as shown i n F i g . 2.2. Then the p i o n energy i s reduced by the amount o f 1.5 MeV i n the t a r g e t . T h e r e f o r e the pions produced a t the middle o f t a r g e t have an energy 29.4 MeV a t the s u r f a c e o f t a r g e t . Thus when we a d j u s t the magnets t o o b t a i n a nominal 30 MeV p i o n beam produced by the pp—»7L +d r e a c t i o n , the magnet c u r r e n t s a c t u a l l y correspond t o a p i o n energy o f 29.4 MeV. -17- Lab Trt MeV 320 240 160 80 0 0 Tp = 6do MeV P + P 11+4 40 80 120 Lab 9 D e g r e e s ) P i g . 2.2 P i o n k i n e t i c energy v s . angle i n a l a b o r a t o r y system. -18- 1 cm 0 5 cm Proton beam 45° Pion beam, M9 Channel F i g . 2.3 CH 0 t a r g e t Mylar window Degrader c Si S 2 S3, b 4 M9 Channel Sj; Scintillators >Pion beam Stopping plate 8.50 i 0,05 m H F-2 13.2U0.10 m F-3 F i g . 2.4 C o n f i g u r a t i o n of counters and degrader -19- C Counters and the Degrader A f t e r emerging from the vacuum p i p e of the M9 channel, which has the seven magnets i n s i d e o f the r a d i a t i o n area, the mesons come out i n t o the experimental area c a l l e d F-2. The vacuum i s enc l o s e d by mylar window w i t h 0.025 mm t h i c k - ness. As e x p l a i n e d i n the s e c t i o n B, t h e r e i s another experimental area F-3 a f t e r the t r i p l e t quadrupole and C o r v a l l i s magnet. In the experimental a r e a , f o u r s c i n t i - 42 l l a t i o n counters w i t h a p l a s t i c s c i n t i l l a t o r s (NE102A) were s e t t o measure the range and time o f f l i g h t . NE102A has the f a s t decay time o f 2.4 nsec and the r a t i o H:C atoms o f 1.104. A RCA 8575 p h o t o m u l t i p l i e r was used and has an anode p u l s e r i s e time o f 2.5 ns. The l a y o u t o f the f o u r counters i s shown i n E l g . 2.4 and the c r o s s - s e c t i o n and the t h i c k n e s s o f p l a s t i c s c i n t i l l a t o r s are shown i n t a b l e 2.1. Each p l a s t i c was wrapped i n b l a c k v i n y l tape to exclude l i g h t . The t h i c k n e s s o f the b l a c k v i n y l tape, 180 microns, must a l s o be c o n s i d e r e d when c a l c u l a t i n g the p i o n range. A s m a l l s i z e o f the p l a s t i c s c i n t i l l a t o r (5 cm x 5 cm) was used e s p e c i a l l y f o r the second counter t o d e f i n e the s m a l l beam p r o f i l e i n f r o n t o f the degrader. Since the beam spreads out a f t e r p a s s i n g through the absorber, the p l a s t i c s c i n t i l l a t o r s o f t h e t h i r d and f o u r t h counter were made q u i t e l a r g e . The f o u r t h counter which gave a n t i - c o i n c i d e n c e s i g n a l had a c r o s s - s e c t i o n o f 20 cm x 20 cm. The c r o s s - s e c t i o n o f the f i r s t counter (10 cm x 10 cm) was l a r g e r than t h a t o f the second counter (5 cm x 5 cm). The -20- NO THICKNESS SIZE MATERIAL PHOTO MUL S l 0.64  C m 10x10 C T n 2 PLASTIC; NE 102A DENSITY? 3 1.03 g/cm H/C; 1.104 RCA 8575 S 2 0.16 5x5 S 3 0.16 15x15 S4 0.32 20x20 Table 2.1 The s p e c i f i c a t i o n o f p l a s t i c s c i n t i l l a t o r s and p h o t o - m u l t i p l i e r used f o r c o u n t e r s . -21- c o i n c i d e n c e s i g n a l o f the f i r s t and second counter was used t o d e f i n e the number o f incoming p i o n s . The area o f the f i r s t counter i s l a r g e r than t h a t o f the second counter so t h a t the beam i s d e f i n e d almost e n t i r e l y by the second counter o n l y . The degrader, which was s e t between the second and t h i r d counter, has s i x p l a t e s o f absorber which can be moved independently by a i r p i s t o n s . I t can v a r y the t o t a l range 2 2 from 0 t o 21.3 g/cm f o r copper and from 0 to 13.9 g/cm f o r aluminum. Each p l a t e o f absorber has a c r o s s - s e c t i o n 18 cm x 18 cm square. The t h i c k n e s s e s o f the p l a t e used f o r the degrader are shown i n t a b l e 2.2. These t h i c k n e s s e s were measured by a micrometer which was a c c u r a t e t o 2.5 micron. The p l a t e s were measured a cm o r so i n a t the f o u r c o r n e r s and the average o f these numbers was taken. I t was found t h a t a l l p l a t e s had a c o n s t a n t t h i c k n e s s w i t h i n 0.2 % d e v i a t i o n ( t a b l e 2.2). The p l a t e s were made o f commercially a v a i l a b l e a l l o y s and the composition i s shown i n t a b l e 2.2. A l l aluminum p l a t e s and t h i n copper p l a t e s had a w e l l - f i n i s h e d s u r f a c e and were p o l i s h e d w i t h f i n e sand paper to c l e a n the s u r f a c e b e f o r e the experiment. The t h i c k copper p l a t e s w i t h 1.1 cm t h i c k n e s s were ground by a l o c a l machine shop so t h a t the t h i c k n e s s was uniform t o w i t h i n 30 micron. -22- PLATE NO COPPER ALUMINUM COPPER 99.5 % LEAD ZINC IRON 0.5 % DENSITY 8.93 g/cm 3 MAGNESIUM 1.0 % SILICON 0.6 % CHROMIUM 0.25 % COPPER 0.25 % ALUMINUM 97.9 % DENSITY 2.70 g/cm 3 1 0.273 ± 0.0012 g/cm 3 0.218 ± 0.0005 g/cm 3 2 0.728 ± 0.0013 0.443 ± 0.001 3 1.388 ± 0.0027 0.897 ± 0.001 4 2.890 ± 0.0034 1.770 ± 0.001 5 5.787 ± 0.005 3.537 ± 0.002 6 10.241 ± 0.011 6.992 ± 0.007 TOTAL 21.307 ± 0.013 13.857 ± 0.008 Table 2.2 Components and t h i c k n e s s e s o f absorbers used f o r the degrader. -23- D E l e c t r o n i c s The schematic diagram o f the e l e c t r o n i c s f o r the range measurements i s shown i n F i g . 2.5. In t h i s experiment, both the d i f f e r e n t i a l range curve and i n t e g r a l curve were taken. In o r d e r t o take the d i f f e r e n t i a l c urve, the counts o f c o i n c i d e n c e i n the f i r s t , second and t h i r d counter w i t h the a n t i - c o i n c i d e n c e o f the f o u r t h counter were normalized by the counts o f c o i n c i d e n c e i n the f i r s t and second c o u n t e r . The c o i n c i d e n c e o f the f i r s t and second counter g i v e s the number of p a r t i c l e s which pass through both c o u n t e r s . Most o f these p a r t i c l e s are expected t o have emerged from the M9 channel. These p a r t i c l e s are e l e c t r o n s , p i o n s and muons. Each s i g n a l from f o u r d e t e c t o r s i s fed t o each d i s c r i m i n a t o r . The output from the f i r s t , t h i r d and f o u r t h counter are f e d to the l e a d i n g edge type d i s c r i m i n a t o r (LRS quad-rdiscriminator) . The c o n s t a n t f r a c t i o n d i s c r i m i n a t o r (ORTEC 463) i s used o n l y f o r output from the second counter. The d i f f e r e n c e between the l e a d i n g edge type and c o n s t a n t f r a c t i o n d i s c r i m i n a t o r w i l l be d i s c u s s e d i n the f o l l o w i n g c h a p t e r . The output from the LRS q u a d - d i s c r i m i n a t o r i s a r e c t a n g u l a r p u l s e o f which the width i s v a r i a b l e w i t h the amplitude o f 0.8 v o l t . The width o f output from the f i r s t and t h i r d d i s c r i m i n a t o r i s 20 nsec wide and from the f o u r t h d i s c r i m i n a t o r 200 nsec wide. The output o f c o n s t a n t f r a c t i o n has a width l e s s than 10 nsec. The wide r e c t a n g u l a r p u l s e i s used f o r the f o u r t h output which i s the a n t i - c o i n c i d e n c e s i g n a l i n o r d e r t o o v e r l a p completely the s i g n a l from another d i s c r i m i n a t o r . S;, Scintillators Si n Lj,Light guides Dj, Delay E.G.G.DP463 Pj, Photomultipliers LRS DISCRI. L2 Degrader C F . DISCRI. ORTEC 463 L3 Stopping plate * E.G.G. , AND LRS DISCRI. * Counters PRESET , i to I F i g . 2.5 Scheme f o r the measurements o f the range curve The signal for the time of f l i g h t i s the output of the coincidence between the f i r s t and second counter. I t i s important to define the distance from the pion production target to measure the time of f l i g h t . Therefore, i f the coincidence s i g n a l i s fed to the time-to-amplitude converter (TAC, ORTEC 437A) we have to know which signal of two counters i s a s t a r t or stop s i g n a l . In order to make sure that the second discriminator signal always triggers the coincidence s i g n a l , the second discriminator i s adjusted to give the narrow pulse. Furthermore, i t i s important to use the constant f r a c t i o n discriminator to feed the s t a r t or stop pulse to the time-to-amplitude converter instead of the leading edge type discriminator. The e f f e c t s of these discriminators i n the time of f l i g h t w i l l be discussed i n the following chapter. The R.F. si g n a l i s employed as a stop s i g n a l , because the frequency of the R.F. i s too high to use the si g n a l as a s t a r t s i g n a l for the TAC. The output of the TAC i s analyzed by the multi-channel analyzer (Northern S c i e n t i f i c Model, NS 900) and the time delay spectrum of the analyzer i s typed out by a t e l e p r i n t e r . The count of the (1,2) coincidence i s used as the preset count i n the dual counter/timer (ORTEC 715) and (1,2,3,4) coincidence i s counted. This w i l l be discussed i n d e t a i l i n the following chapter. When the count rate i s high, 10 4 of preset count i s employed but, when i t i s low, a h a l f of 4 10 i s employed. -26- I I I The B a s i c Experiment f o r the Measurement o f the Range- Energy R e l a t i o n and Time o f F l i g h t There a re s e v e r a l methods t o measure the momentum and energy o f charged p a r t i c l e s . In our experiment, the energy o f p i o n was d e f i n e d by t h r e e methods, the range- energy r e l a t i o n , the time o f f l i g h t and the bending magnet c u r r e n t . In t h i s c h a p t e r , the b a s i c i d e a and fundamental problem i n measuring the range and the time o f f l i g h t w i l l be d i s c u s s e d . A The Range Curve When charged p a r t i c l e s pass through matter, they l o s e energy by i o n i z a t i o n . I f we know the range i n matter, the energy o f charged p a r t i c l e s can be o b t a i n e d . In our e x p e r i - ment, the energy o f a p o s i t i v e p i o n i s determined by i t s range i n aluminum and copper. In the s e c t i o n 2-D above, the method o f measurement o f range has been shown b r i e f l y . The c o i n c i - dences (1,2,3,4) and (1,2,3) g i v e the d i f f e r e n t i a l curve and i n t e g r a l curve, r e s p e c t i v e l y . The number e n c l o s e d w i t h parentheses s i g n i f i e s the counter i n F i g . 2.5 and the bar on the head o f the number 4 d e s i g n a t e s t h a t i t i s an a n t i - c o i n c i d e n c e s i g n a l from the f o u r t h c o u n t e r . The example o f the d i f f e r e n t i a l and i n t e g r a l range curve i s shown i n F i g . 3.1. These data were taken u s i n g a C l ^ p i o n p r o d u c t i o n t a r g e t under the c o n d i t i o n t h a t the pi o n s were stopped i n 2 a 0.16 g/cm t h i c k s c i n t i l l a t o r , the h o r i z o n t a l s l i t was c l o s e d t o 2 cm and the absorber used was aluminum. F i g . 3.1 The d i f f e r e n t i a l arid i n t e g r a l range curve f o r 30.MeV p i o n stopped by A l . With no absorber and perfect geometry, one might expect that the counts of (1,2,3,4) and (1,2,3) would equal 0 and 4 4 10, respectively, f o r the case of (1,2)=10. But as shown i n F i g . 3.1, (1,2,3,4)=80 and (1,2,3)=9400 without the absorber. This means that 600 p a r t i c l e s out of 10, which f i r e the f i r s t and second counter, go aside to miss the t h i r d counter. As the absorber thickness increases, the d i f f e r e n t i a l curve shows c l e a r l y two peaks. One of them i s the pion peak and the other the muon peak. These peaks r e s u l t from the stopping of pions and muons i n the s c i n t i - l l a t o r of the t h i r d counter. So, as the matter which stops the p a r t i c l e s increases, these peaks get much higher. The pion peak i s higher than the muon peak due to the higher percentage of pions i n the beam. The pions and muons have d i f f e r e n t ranges because the two p a r t i c l e s have the same momentum but d i f f e r e n t energies (the muon mass i s 105.7 2 2 MeV/c and the pion mass i s 139.6 MeV/c ). We also observe an electron peak but the peak po s i t i o n i s o f f scale due to the large range. The d i s t r i b u t i o n of two peaks should be Gaussian. When a beam of p a r t i c l e s loses energy by i o n i z a t i o n , the p a r t i c l e s do not a l l stop a f t e r passing through the same thickness of material. The p r o b a b i l i t y of a p a r t i c l e stopping within dR of R, P(R)dR of a p a r t i c l e , i s given by p ( R ) d R = ^ e x P i - ^ - } d R -29- where Rg i s the mean range obtained by integration over the average energy-loss (Eq. 1.4) or, equivalently, from the posi t i o n of peak on the d i f f e r e n t i a l range curve. In Pig. 3.1, the i n t e g r a l range curve shows two slopes around the peaks of the d i f f e r e n t i a l curve. The f i r s t and second slope give the range of pion and muon, respectively. The mean range of the p a r t i c l e s can be obtained from the in t e g r a l curve. In the figu r e , we f i n d that the pion step has i t s half value at R S = RQ* which i s the pos i t i o n of the peak of the d i f f e r e n t i a l curve. By drawing a tangent at the steepest point of the pion slope and obtaining the inte r s e c t i o n of the tangent with the R-axis, we f i n d the extrapolated r a n g e * 0 ' 4 3 R„ vt. r» which i s given by The difference R e x t r ~ R n * s d e f i n e d as the straggling para- meter S. Since i t i s clea r from Pig. 3.1 that both the i n t e g r a l and d i f f e r e n t i a l curve give the same range, we w i l l use the d i f f e r e n t i a l curve to define the range of p a r t i c l e i n matter by using Gaussian curve f i t t i n g . (3.2) - 3 0 - B The Height of Peak i n the D i f f e r e n t i a l Range Curve In the l a s t section i t was noted that the p a r t i c l e s which are counted i n the d i f f e r e n t i a l range (1,2,3,4) must be counted i n s c i n t i l l a t o r #3 yet not be counted i n s c i n t i - l l a t o r #4. These pions must therefore stop some way into s c i n t i l l a t o r #4 (or i n the wrapping material). In our 2 experiment a t h i n p l a s t i c s c i n t i l l a t o r (0.16 g/cm ) was used for the t h i r d counter to reduce the absorption of the pion. I t was expected that the width of peak i n the range curve might be narrower by using the th i n s c i n t i l l a t o r for the t h i r d counter without any absorber between the t h i r d and fourth counter. The width of the peak i s caused mostly by the i n i t i a l energy spread of beam and the stragglings in the degrader and p l a s t i c s c i n t i l l a t o r s contribute a l i t t l e to the width. I f there i s an absorber behind the t h i r d counter, the energy straggling i s increased and the width of peak seems to be wider. In order to see the e f f e c t of the thickness such an' absorber behind the t h i r d counter, the range curves were taken for three cases; (1) no absorber, (2) with 0.8 mm thickness aluminum absorber, (3) with 1.5 mm thickness aluminum absorber. For t h i s t e s t , 30 MeV pions were employed and the CHj target was used at the T2-target. The range curves are shown i n F i g . 3.2. This experiment was performed on the F-3 p o s i t i o n behind the C o r v a l l i s magnet. The experimental arrangement i s shown i n F i g . 2.4 and the currents of bending magnets and guadrupole magnets - 3 1 - 0 1 • 2 Absorber thickness ( cm,Al) F i g . 3.2 The e f f e c t o f the s t o p p i n g p l a t e between the t h i r d and f o u r t h counter. are g i v e n i n t a b l e 3.1. The r e s u l t s are shown i n t a b l e 3.2. F i g . 3.2 shows t h a t the peak w i t h A l absorber i s h i g h e r than the case without the absorber and peak p o s i t i o n s s h i f t a c c o r d i n g t o the t h i c k n e s s o f the absorber behind the t h i r d c ounter. The t o t a l ranges o f the t h r e e cases ( c o r r e c t i n g f o r the a d d i t i o n a l absorber) are g i v e n i n t a b l e 3.2 and agree v e r y w e l l each o t h e r . A l s o the energy o b t a i n e d from the range i s c o n s i s t a n t w i t h the energy from the measurement o f the time o f f l i g h t . T h e r e f o r e , from the energy p o i n t o f view, i t does not matter whether the absorber behind the t h i r d counter i s added o r not. But from the p o i n t o f view o f measurement, as seen from F i g . 3.1, the s t o p p i n g r a t e w i t h the absorber i s g r e a t e r than the s t o p p i n g r a t e without the absorber. I t i s a l s o c l e a r from the t a b l e 3.2 t h a t t h e r e i s not a g r e a t change i n the f u l l - w i d t h half-maximum even w i t h the aluminum absorber. As the measurements f o r t h i s t h e s i s were made w i t h a s m a l l p r o t o n c u r r e n t (between 1 and 5 nA), a s t o p p i n g p l a t e was used i n our range experiment t o improve the count r a t e ; as a compromise the t h i c k n e s s chosen was 0.8 mm o f aluminum p l a t e . -33- MAGNET CURRENT* (Amp.) MAGNET CURRENT* (Amp.) B l 245 .5 Q 4 168 .9 B2 247.0 178.6 168.1 Q6 205.3 163.7 167.4 Q 3 128.7 Q 8 119.0 Table 3.1 The currents of magnets for the measurement on F-3. * The numbers in the table actually show the voltage which is an output of the amplifier, 0-50 mV->0-5 V. It amplifies the voltage across the shunt, 50 mV-500 A,+0.25%, which i s installed in series with the magnet. Thus the reading of the amplifier gives directly the amount of magnetic current. THICKNESS OF ^ ^ ^ A B SORBE R 000 mm 0.8 mm 1.5 mm PEAK POSITION 2 . (g/cm ) 3.59+0.24 3.38*0.22 3.29+.0.23 RANGE OF ALUMINUM INCLUDING THE THICKNESS OF STOPP- ING PLATE . . 2. j ., ' (g/cm ) 3.62±0.24 3.51±0.22 3.51*0.23 HALF WIDTH OF P E A K (g/cm 2) 1.30 1.20 1.32 CORRECTION FROM PLASTIC AND VINYL COVER,IN ALUMINUM *™GE > (g/cm 2) 1.19 1.29 1.29 F: ••: . •• •', TOTAL RANGE IN ALUMINUM RANGE • '; 2 •?>•"!:• (g/cm ) 4.81±0.25 4.80±0.23 4.80±0.24 ENERGY OBTAINED FROM ALUMINUM RANGE i (MeV) 29.3±0.9 29.3±0.8 29.3±0.8 ENERGY OBTAINED FROM THE TIME OF FLIGHT METHOD l" (MeV) 30.02±0.87 Table 3.2 The e f f e c t o f the st o p p i n g p l a t e t o the p i o n energy d e t e r m i n a t i o n and the comparison o f energy between the range and time of f l i g h t method. C The P r i n c i p l e of the Time-of-Flight Method The M9 meson beam channel i s shown i n F i g . 2.1. The 500 MeV proton s t r i k e s the T2 target and produce pions (as well as heavier ions such as deuterons, alphas, e t c . ) . Some of the charged pions decay into muons; the neutral pions a l l decay into two gamma rays which immediately create electron-positrons pairs i n the surrounding material. A l l these p a r t i c l e s , emitted at 135°, form into the M9 beam. As a l l these events occur very quickly, the time structure of the M9 beam i s always related to the time structure of the proton beam and thus to the R.F. of the accelerator. In our experiment, signals of the R.F. and the s c i n t i l l a t i o n counter were employed to measure the v e l o c i t y of p a r t i c l e s . The electronics for the time of f l i g h t i s shown i n F i g . 3.3. We used the time-to-amplitude converter (TAC, ORTEC 437A) and multi-channel analyzer (N.S. 900) to measure the time of f l i g h t of p a r t i c l e s from the T2 production target to the s c i n t i l l a t i o n counter i n the experimental area. In th i s type of time-interval measurement, the output voltage of the TAC i s l i n e a r l y proportional to the time difference t 2 - t ^ = 4 t , where the a r r i v a l of the s t a r t pulse and the stop pulse i s at t ^ and t 2 , respectively. The timing diagram (Fig. 3.4) shows the s t a r t and stop, the time difference counted by the TAC and the place of each p a r t i c l e on the MCA. Since the high frequency signal of the R.F. can not be accepted as the s t a r t pulse, the R.F. signal i s fed to (Cf. Fjg. 2.5) Li ' D 2 DIS'CRI E.G.G. AND START i PION 43.4 ns R.F. MECL3I PRESCALER STOP 86.8 ns ORTEC 437 TAC MCA NS900 PRINTER F i g . 3.3 C i r c u i t diagram of the time of f l i g h t measurement. PROTON T2 O k— 43.4 ns START - (COUNTER) STOP L_ (R.F.) TAC 86 £ ns A H ©2 r OUT PUT VOLTAGE OF TAC DISPLAY OF MCA x C D (—4 LU X •43.4ns 7fr CHANNEL NUMBER V=0 TIME TIME F i g . 3.4 Timing diagram of TAC and the display of p a r t i c l e s on the multi-channel analyzer. -37- the stop and the signal of the s c i n t i l l a t i o n counter to the s t a r t as shown i n F i g . 3.4. Consequently, the order of p a r t i c l e s on the MCA i s reversed so that the electron peak appears a f t e r the pion and muon peaks. The R.F. has a 43.4 nsec r e p e t i t i o n time but i n our experiment a scale of two was added to the R.F. pulses, thus providing a stop pulse every 86.8 nsec. Therefore, there are two peaks for each p a r t i c l e i n the time-delay spectrum. Since these two peaks have a time difference of 43.4 ns, t h i s gives natural c a l i b r a t i o n for the MCA. We also cal i b r a t e d the MCA using the time c a l i b r a t o r (ORTEC 462). The c a l i b r a t i o n r e s u l t s are shown i n F i g . 3.5. The periodic input of 10 ns sign a l i s fed to TAC. (The time c a l i b r a t o r produces a p a i r of pulses separated by 10, 20, 40 and 80 ns; the s p e c i f i c a t i o n s of the instrument indicate that the error i s +10 psec for the 10 nsec i n t e r v a l and 0.005 % for the other intervals.) As shown i n F i g . 3.5, a l l peaks are almost equally separated and each channel corresponds to 178 psec. Since the MCA shows good l i n e a r i t y , the uncertainty of the MCA i s n e g l i g i b l e i n our experiment. In order to calculate the v e l o c i t y of p a r t i c l e s , we have to know the distance from the T2 to the counter which gives the s t a r t signal to the TAC and, also, the time difference i n a r r i v a l at the counter between the electron and another p a r t i c l e . Assuming that the electrons t r a v e l with the velo- c i t y of l i g h t , the time of f l i g h t f or electron from T2 to the counter i s known. Then, adding the time difference between 15 xicr 10 in *-> c Z3 O o 0 -i . L — A 2 ^ A 2 — * A 2 —$K A, si A 1,2 =10 nsec Ai =56.2 channels A2 =56.3 channels _!_L 100 200 300 400 Channel number F i g . 3.5 Time c a l i b r a t i o n spectrum ( p e r i o d i c i n p u t , 10 nsec) 500 -39- the p a r t i c l e and e l e c t r o n , the time o f f l i g h t f o r the p a r t i c l e i s c a l c u l a t e d . T h i s time g i v e s the v e l o c i t y o f the p a r t i c l e . -40- 44 D The Time Walk o f the D i s c r i m i n a t o r In the time o f f l i g h t measurement, the problem of time walk a r i s e s from the d i s c r i m i n a t o r . Since t h i s g i v e s s e r i o u s d i f f e r e n c e i n the p a r t i c l e energy, we have to choose the s u i t a b l e d i s c r i m i n a t o r . The time walk o f d i s c r i m i n a t o r i s caused by the d i f f e r e n t e n e r g y - l o s s o f p a r t i c l e s i n matter. I f we have a beam i n c l u d i n g p i o n s , muons and e l e c t r o n s w i t h the same momentum, e l e c t r o n s have a s m a l l e r e n e r g y - l o s s than p i o n s and muons i n the p l a s t i c s c i n t i l l a t o r . The examples o f d e t e c t o r p u l s e and outputs o f l e a d i n g edge type d i s c r i m i n a t o r are shown i n F i g . 3.6. C o n s i d e r i n g the energy- l o s s o f each p a r t i c l e w i t h the same momentum around 100 MeV/c, the p u l s e 1, 2 and 3 i n F i g . 3.6 correspond to e l e c t r o n s , muons and pions, r e s p e c t i v e l y . Suppose we employ the l e a d i n g edge t r i g g e r type d i s c r i m i n a t o r w i t h the f i x e d d i s c r i m i n a t o r l e v e l , each p a r t i c l e has the d i f f e r e n t t r i g g e r i n g time as shown i n F i g . 3.6. Since these d i f f e r e n c e s g i v e a wrong time o f f l i g h t , the e n e r g i e s o f p a r t i c l e s are o v e r e s t i m a t e d . In o r d e r to a v o i d the time walk, a c o n s t a n t f r a c t i o n d i s c r i m i n a t o r was used i n our experiment. The c o n s t a n t f r a c t i o n d i s c r i m i n a t o r i s b e t t e r than a l e a d i n g edge t i m i n g d i s c r i m i n a t o r because i t has the same t r i g g e r i n g time, independent o f the p u l s e amplitude and r i s e time. The 45 46 p r i n c i p l e ' i s i l l u s t r a t e d i n F i g . 3.7. The d e t e c t o r p u l s e i s delayed (a) and a f r a c t i o n o f the undelayed p u l s e (b) i s s u b t r a c t e d from i t ( c ) . The a t t e n u a t e d p u l s e e x a c t l y c a n c e l s the d e l a y e d and i n v e r t e d p u l s e a t the f r a c t i o n Out put of photomultiplier Time F i g . 3.6 Time-walk due to th r e e d i f f e r e n t i n p u t p u l s e s o f e l e c t r o n s , muons and p i o n s w i t h the same momentum. Sinc e the d i s c r i m i n a t o r l e v e l o f a l e a d i n g edge type i s f i x e d , t h e outputs s h i f t i n time spectrum. a, Inverted and delayed anode pulse b, Attenuated anode pulse c, Resulting zero- crossing pulse d, Current gate e, Zero-crossing trigger F i g . 3.7 Constant f r a c t i o n o f h e i g h t p i c k - o f f . -42- phase point on the delayed pulse. The time of f l i g h t experiment was done at the F-3 focussing position with 50 MeV pions i n order to see the e f f e c t of the time walk of a discriminator. The constant f r a c t i o n discriminator (ORTEC 463) and the LRS leading edge type discriminator were employed. The r e s u l t s are shown i n table 3.4. I t i s apparent that the energy of LRS discriminator gives higher energy than that of the C F . discriminator. From the time-delay spectrum of the MCA, there i s 1.0 nsec time difference between LRS and C F . discriminator i n the time of f l i g h t of the pion from the T2 to the counter. In the case of muon, t h i s difference i s about 0.8 nsec. Since the energy-loss of 30 MeV pion i n the p l a s t i c s c i n t i l l a t o r (0.76 MeV) i s larger than that of 50 MeV pion (0.56 MeV), the time difference for 30 MeV pion increases to 2.0 nsec. In order to define the meson v e l o c i t y by the time of f l i g h t method, we have to know the peak po s i t i o n of the pions and electrons on the time-delay spectrum. As seen i n F i g . 4.1, the height of the electron peak comes down as the energy of pion increases. Since the contamination of electron i s increased by using the target with a large atomic number, we compare the energy difference between the beryllium target and copper target. The r e s u l t s are shown i n table 3.1. Though the height of the electron peak i n case of copper i s almost twice as high as that i n case of beryllium, we do not detect any difference i n the r e l a t i v e COPPER TA RGET BERYLLIUM TARGET LRS C F . LRS C F . K.E. (MeV) 54.67 ± 2.3 51.51 ± 1.91 55.22 ± 2.31 50.93 ± 1.94 MOMENTUM (MeV/c) 135.3 + 3.3 130.7 + 2.8 136.1 ± 3.3 129.8 ± 2.9 BETA 0.695 + 0.008 0.682 + 0.008 0.697 ± 0.009 0.680 ± 0.008 TIME* (nsec) 57.13 ± 0.54 58.18 ± 0.45 56.97 ± 0.53 58.39 ± 0.45 K.E. (MeV) 63.91 ± 3.8 59.91 ± 3.1 62.79 ± 3.6 59.33 ± 3.0 MOMENTUM (MeV/c) 132.8 ±4.8 127.6 ± 4.0 131.4 ± 4.6 126.9 ± 3 . 9 BETA 0.782 ± 0.01 0.769 ± 0.01 0.778 ± 0.01 0.767 ± 0.01 * TIME (nsec) 50.80 ± 0.58 51.61 + 0.50 51.02 ± 0.56 51.74 ± 0.48 * Time of f l i g h t between the pion production target (T2) and counter. Table 3.3 The energies of pions and muons are measured by the time of f l i g h t method. In order to check the systematic error, two types of discriminators are employed, LRS (leading edge) and C F . (constant f r a c t i o n ) , and two kinds of targets, copper and beryllium. -44- p o s i t i o n s o f the e l e c t r o n and p i o n peaks. Thus, t h e r e i no d i f f e r e n c e i n the energy d e t e r m i n a t i o n between the t a r g e t s . -45- IV R e s u l t s and D i s c u s s i o n The measurements o f the time o f f l i g h t and range were made on F-3. We chose f o u r d i f f e r e n t e n e r g i e s , 30, 40, 50 and 60 MeV f o r the case o f aluminum absorber and two e n e r g i e s , 30 and 50 MeV w i t h the copper ab s o r b e r s . The magnet c u r r e n t s f o r each energy are shown i n t a b l e 4-1. A Time o f F l i g h t The time o f f l i g h t s p e c t r a o f f o u r e n e r g i e s a r e shown i n F i g . 4.1. The experimental e l e c t r o n i c s and p r i n c i p l e o f the time o f f l i g h t have been d i s c u s s e d i n the l a s t c h a p t e r . We can see two p i o n peaks on the spectrum and a l s o muon and e l e c t r o n peaks except f o r the 60 MeV case. The s e p a r a t i o n of two p i o n peaks corresponds t o the R.F. p e r i o d o f 43.40 nsec. Knowing the d i s t a n c e from the p i o n p r o d u c t i o n t a r g e t t o the counter, (8.50+0.05 m), we can o b t a i n the p i o n energy. In the energy c a l c u l a t i o n , we assume t h a t the e l e c t r o n v e l o c i t y i s the v e l o c i t y of l i g h t , because o f i t s l a r g e momentum.' T h i s assumption g i v e s n e g l i g i b l e e f f e c t t o the energy c a l c u l a t i o n o f p i o n . For example, e l e c t r o n s c o n t a i n e d i n the beam w i t h 97 MeV/c, i n which pions have a K.E. o f 30 MeV, have a v e l o c i t y o f 0.99999 (=@). F i g . 4.1 shows t h a t the beam c o n s i s t s mainly o f p i o n s w i t h a few muons and e l e c t r o n s . Though we can expect muons from the decay o f p i o n s , most o f the muons i n the muon peak come from the p r o d u c t i o n t a r g e t . Because the peak has a v e l o c i t y correspond- i n g t o muons wit h the momentum determined by the channel. -46- ENERGY, MeV 29.4 39.5 50.5 59,0 MOMENTUM, MeV/c 95.4 112.3 129.2 141.2 MAGNET, Amp B l 247.4 291.2 335.0 366.0 B2 252.6 297.0 342.0 373.0 Q l 192.0 226.0 260.0 284.0 Q 2 159.5 187.8 216.0 235.0 Q 3 129.2 152.0 175.0 191.0 Q4 169.1 199.0 229.0 250.0 Q 5 178.7 210.0 242.0 265,0 Table 4.1 Magnet c u r r e n t s f o r the time o f f l i g h t and range measurements. MAGNET ENERGY TIME OF FLIGHT RANGE OF A l RANGE OF Cu 29.4 PION 30.9 ± 0.9 29.8 ± 0.8 30.1 ± 1.2 MUON 37.4 ± 1.6 36.1 ± 1.3 36.4 + 1.1 39.5 41.5 ± 1.5 39.6 + 1.1 50.5 52.4 ± 2.2 49.7 ± 1.1* 50.1 ± 1.4** 50.3 i 1.8 59.0 60.6 ± 2.9 58.6 ± 1.0 59.0 ± 1.4 * F i t t e d w i t h the Gaussian d i s t r i b u t i o n . ** F i t t e d w i t h the extreme v a l u e d i s t r i b u t i o n . Table 4.2 The energy i n MeV ob t a i n e d by the time o f f l i g h t and range measurements. 0 100 0 8 xic/* 30 MeV Pion 200 -47- 300 400 500 7C .x103 n 7L . 40 MeV * * t • '* 3 x102 ..e • « « 0 *-> c o o x10H 50 MeV 0 2 1 \ x10 0 x10H 60 MeV n. 1 x10 . 7t 100 200 300 Channel number 400 500 F i g . 4.1 R.F.-referenced time of f l i g h t spectrum f o r 30,40,50 and 60 MeV o f p i o n e n e r g i e s . The decay muons would appear between the muon and e l e c t r o n peak but the experimental s p e c t r a do not show c l e a r e v i d e n c e f o r such muons. The p i o n peak has a time spread. In our time o f f l i g h t measurements, the R.F. s i g n a l o f the a c c e l e r a t o r i s used as the stop p u l s e a g a i n s t the s t a r t p u l s e o f the second counter. The proton beam a t the p i o n p r o d u c t i o n t a r g e t has a bunch o f about 4 nsec. T h i s i s the p r i n c i p a l c o n t r i b u t o r t o the time spread as measured i n our method. The peak width a t h a l f maximum i s 3.6 nsec and n e a r l y equals the proton beam time spread. A l s o the energy r e s o l u t i o n (-|r-) and momentum r e s o l u t i o n (-p—) o f the 30 MeV p i o n are 8.2 % and 4.5 % a t FWHM, r e s p e c t i v e l y . In o r d e r t o c a l c u l a t e the peak p o s i t i o n i n the time o f f l i g h t spectrum, the data p o i n t s are f i t t e d w i t h a Gaussian 47 d i s t r i b u t i o n . The v a l u e o f the measured p o s i t i o n i s w r i t t e n as X + ry where # = =- i s the standard e r r o r o f the mean and n i s the number o f data p o i n t s . The curve f i t t i n g program i s w r i t t e n by u s i n g the LQF s u b r o u t i n e which i s one o f the l e a s t squares f i t t i n g program i n the UBC computer 48 l i b r a r y program. The r e s u l t s a re shown i n t a b l e 4.2. The en e r g i e s have been d e f i n e d w i t h a d e v i a t i o n o f 3 t o 5 -49- B Range Measurement The range curves for A l and Cu are shown i n F i g . 4.2 and 4.3, respectively. The energies obtained from ranges are shown i n table 4.2. The experimental data are f i t t e d with a Gaussian d i s t r i b u t i o n as discussed i n the l a s t section. The curves of the Gaussian d i s t r i b u t i o n and the data are i l l u s t r a t e d i n F i g . 4.2 and 4.3 by a s o l i d l i n e and dots, respectively. In order to calculate the energy of pion, we have to know the thickness and content of the absorber i n d e t a i l . The thicknesses of the metal of the degrader are given by the peak p o s i t i o n of the f i t t e d curve with the Gaussian d i s t r i b u t i o n . Elements contained i n the absorber are shown i n table 2.2. Since, i n both the aluminum and copper absorber, 99.5 % of the constituents can be considered as elements with si m i l a r atomic numbers, the e f f e c t s from the d i f f e r e n t elements can be neglected i n the range c a l c u l a t i o n . We have to count the thickness of p l a s t i c s c i n t i l l a t o r s and also the windows which keep a vacuum i n the M9 channel. The thicknesses of p l a s t i c s c i n t i l l a t o r are shown i n table 2.1. Each p l a s t i c i s covered by black v i n y l tape, with a thickness of 180 microns, i n order to keep out l i g h t . The vacuum window i s made of mylar of 0.25 mm thickness. The 3 density of p l a s t i c s c i n t i l l a t o r (NE 102A) i s 1.032 g/cm and i t s t o t a l thickness including the f i r s t , second and t h i r d 2 counter i s 0.983 + 0.1 g/cm. Assuming the density of the 3 2 black v i n y l tape and mylar as 1.0 g/cm,,we get 0.037 g/cm 5 0 1 1 ' • 1 , 1 1 1 1 1 ' ' I > - I — " - • • * 1 1 1 xicr j 1-3.37 •A 6.47 • 30 MeV A 40 MeV o 50 MeV x 60 MeV — Gaussian distri — Extreme value DUtion distribution \ r 1 I 7 .12 ic-10.27 \ \ \\ \ 13.64 184 • . ' A i i i i i I 0 ' A * • I i i or ' 7 / v / 0 * O O 0 0 0 A A A A \ v \ 0 5 1 0 . 1 5 20 Absorber thickness (gm/ Cm 2,AD . 4.2 The range curves of p i o n i n aluminum( pi o n energies,30,40,50 and 60 MeV) x ^ 3.84—nl A / \a /* \ )MeV I* \ • \ 11.74 i L 50 Me \l • • A » • 1 » /A A / A • i i i n peak • i i i A A A i i • i 0 0 5 10 15 Absorber thickness (g/ Cm 2, Cu) F i g . 4.3 The range curves of p i o n i n copper( p i o n energies,30 and 50 MeV), 20 for these absorbers. The error caused from the assumption 2 of the density as 1.0 g/cm might be ne g l i g i b l e i n comparison with the t o t a l thickness due to the small number. Thus, the 2 thickness of absorber besides the metal i s 1.02+0.1 g/cm. The r a t i o of the number H versus C i s 1.104 i n the p l a s t i c s c i n t i l l a t o r . The stopping powers of CĤ ^ are calculated by using the numbers of stopping power i n the reference 16 and show i n table 4.3. Assuming that the 2 thickness of 1.02 g/cm obtained above i s made of CE^ ^, we can cal c u l a t e the equivalent thickness of CHj ^ for the aluminum and copper range curves (table 4.3). Since the M9 channel has a vacuum of less than 0.05 mm Hg, we can neglect the slowing down of charged p a r t i c l e s by a i r inside the channel. A f t e r emerging from the vacuum pipe, the charged p a r t i c l e s s t i l l have to go through a i r of 1 m or so, which has a density of 1.29 mg/cm. This contributes to the slowing down of charged p a r t i c l e s . The e f f e c t 2 2 corresponds to 0.15 g/cm i n aluminum range and 0.17 g/cm in copper range. For 50 and 60 MeV range curves of aluminum, the data 49 are also f i t t e d with the extreme value d i s t r i b u t i o n , because 2 "y^ - t e s t for the curve f i t t i n g with the Gaussian d i s t r i b u t i o n i s worse than that of the extreme value d i s t r i b u t i o n . The pro b a b i l i t y density function i s given by P(*)^pH^-exP(-^)} „.x, PION ENERGY STOPPING POWER OF C H 1 . 1 EQUIVALENT RANGE OF C K U 1 RANGE OBTAINED FROM RANGE CURVES TOTAL RANGE** Al Cu - - A l .Cu A l Cu MeV 30 2 " Mev-cm /g 4.443 g/cm 1.29*0.10 g/cm2 1.49±0.10 / 2 g/cm 3.40*0.23 g/cm2 3.84*0.37 / 2 g/cm 4.95*0.23 g/cm2 5.86*0.37 40 3.723 1.28+0.10 6.47±0.37 8.01*0.37 50 3.230 L.26±0.10 1.44±0.10 10.12+0.41 11.64*0.41 13.71+0.75 10.27*0.55 11•74+0•Iz • 11.79*0.55 60 2.945 1.25+0.10 13.64*0.43 15.15*0.43 L3.84±0.6G 15.35*0.60 * Range curves are f i t t e d with the Extreme value d i s t r i b u t i o n . ** The thickness of stopping plate behind the t h i r d counter are considered. Table 4.3 The d e t a i l of range c a l c u l a t i o n . The curves f i t t e d w i t h two d i s t r i b u t i o n s are shown i n P i g . 4.2. I t i s c l e a r t h a t the extreme v a l u e d i s t r i b u t i o n g i v e s good f i t t i n g around the peak and the peak p o s i t i o n s h i f t s a l i t t l e from the Gaussian case. The e n e r g i e s d e f i n e d by two d i s t r i b u t i o n s a re shown i n t a b l e 4.2. 2 The aluminum p l a t e w i t h 0.220 g/cm t h i c k n e s s was i n s t a l l e d t o stop pions between the t h i r d and f o u r t h counter. A h a l f o f the t h i c k n e s s o f the p l a t e i s c o n s i d e r e d t o c o n t r i b u t e t o the range. For the copper range c u r v e s , we employed the 2 copper p l a t e w i t h 0.728 g/cm t h i c k n e s s behind the t h i r d c o unter. The t o t a l ranges shown i n t a b l e 4.3 are g i v e n by the summation o f ranges i n aluminum (or co p p e r ) , CH^ ^ and a i r . In o r d e r t o o b t a i n the energy o f the pions from the range, the range-energy t a b l e o f r e f e r e n c e 16 i s used, and the r e s u l t s are shown i n t a b l e 4.2. Thus, the range curves g i v e the p i o n e n e r g i e s w i t h i n an e r r o r o f °± 3 %. -55- C The Energy Straggling and Inherent Energy Spread of the Beam The straggling parameter S i s discussed i n the section 3-A. From F i g . 3.1, the value of S can be obtained and i s 2 about 0.59 g/cm i n aluminum range. I t i s d i f f i c u l t to get the exact value of the straggling parameter because the beam contains pions, muons and electrons, - the R-axis for the pions can not be defined e a s i l y . The straggling parameter i s also given by S^J^-at, where o< i s given by Eq. 3.1 which i s employed to f i t the d i f f e r e n t i a l curve. 2 From the c a l c u l a t i o n of curve f i t t i n g , 0C* =0.55 g/cm, then 2 S=0.69 g/cm. This i s close to the number obtained appro- ximately i n F i g . 3.1. The difference between the two values of S may come from the uncertainty i n choosing the R-axis. The straggling of the range of charged p a r t i c l e s due to the fluctuations of the i o n i z a t i o n process i s calculated t h e o r e t i c a l l y by sternheimer. / For the case of 30 MeV pions, the straggling parameter i n aluminum i s equal to 0.19 2 g/cm. The difference between the straggling of range obtained from the experiment and the t h e o r e t i c a l straggling parameter mentioned above must be mainly attributed to the energy spread of the pion beam. Thus, the energy spread of the / 2 2 p i o n s , Y ^ (R - RQ) } t i s equal to 0.53 g/cm. The f l u - 34 ctuation i n range and i n energy-loss i s relat e d . For a 2 small thickness dR and a small energy-loss dE, ( A E )^R S B (~) 2(4 R 2 ) d E where E 2 ) d R = ^ ( E - E Q ) 2 > and ( 4 R 2 ) d E = 2 <^(R-RQ) > . According to t h i s r e l a t i o n , the energy spread -56- o f p i o n beam, -/ < ( E - E Q ) 2 > y i e l d s 1.9 MeV, i d A E / E = 6 . 3 % (AP/P= 3 . 2 %) f o r the 30 MeV o f the p i o n beam. These val u e s are s m a l l e r than the v a l u e s o b t a i n e d by the time o f f l i g h t spectrum. -57- D C o n c l u s i o n The v a r i o u s s y s t e m a t i c e f f e c t s l i m i t the accuracy o f the energy d e t e r m i n a t i o n . We d i s c u s s e d the time walk o f the l e a d i n g edge type and c o n s t a n t f r a c t i o n d i s c r i m i n a t o r i n the l a s t c h a p t e r . Then, we c o u l d reduce the time walk by u s i n g the constant f r a c t i o n d i s c r i m i n a t o r . In the time of f l i g h t measurement, the time walk o f the d i s c r i m i n a t o r i s the most s e r i o u s s y s t e m a t i c e r r o r i n the energy d e t e r m i - 51 n a t i o n . The s p e c i f i c a t i o n of the constant f r a c i o n d i s c r i - minator c l a i m s t h a t the walk i s l e s s than 0.51 nsec. I f t h e r e i s s m a l l time walk such as 0.1 nsec, the energy determined by the time o f f l i g h t w i l l be reduced by 0.2 (0.6) MeV f o r the 30 (60) MeV p i o n . In the energy c a l c u l a t i o n by the range method, t h e r e i s the u n c e r t a i n t y i n the range- energy t a b l e d We can expect t h a t i t i s b e t t e r than 0.5 %. I t has been shown t h a t t h e r e are t h r e e independent methods t o determine the p i o n energy. The c a l c u l a t e d magnetic c u r r e n t s o f the M9 p i o n channel choose the p a r t i c l e s w i t h the same momentum but d i f f e r e n t energy. The time of f l i g h t and range methods a c t u a l l y work as a c a l i b r a t i o n o f p i o n energy d e f i n e d by the magnets o f the M9 c h a n n e l . In t h i s work, we have shown t h a t t h r e e independent methods f o r measuring p i o n e n e r g i e s are c o n s i s t a n t w i t h one another w i t h i n the s t a t i s t i c a l e r r o r and p i o n e n e r g i e s can now be measured w i t h c o n f i d e n c e t o an accuracy 2 % by range and 5 % by the time of f l i g h t method. We can improve the accuracy of the time o f f l i g h t -58- measurement by u s i n g a stop s i g n a l from a counter i n s t e a d o f the R.F. s i g n a l o r by u s i n g a chopped beam w i t h a sharp bunch. In the range-energy d e t e r m i n a t i o n one might be a b l e t o improve the geometry o f the f o u r counters somewhat. - 5 9 - r e f e r e n c e s : 1 W.H. Barkas and S. von F r i e s e n , Nuovo Cimento 19, 41 (1961) 2 U.P. Z r e l o v and G.D. S t o l e t o v , S o v i e t P h y s i c s JETP 9, 461 (1959) 3 R. Mather and E. Segre, Phys. Rev. 84, 191 (1951) 4 C.J, Bakker and E. Segre, Phys. Rev. 81, 489 (1951) 5 H. 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