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The primary specific ionization in gases of positrons and electrons II Sample, John Thomas 1950

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(is* 6%  THE PRIMARY SPECIFIC IONIZATION IN GASES of POSITRONS AND ELECTRONS I I  by  John Thomas Sample  A Thesis Submitted In P a r t i a l . F u l f i l m e n t Of The Requirements For The Degree Of  MASTER OF ARTS  In The Department of PHYSICS  THE UNIVERSITY OF BRITISH, COLUMBIA October, 19^0  IPARTMENT  THE UNIVERSITY OF BRITISH COLUMBIA VANCOUVER. CANADA OF  P H Y S I C S  October 12,.1950.  Dr. L. W. Dunlap, Librarian, University of B r i t i s h Columbia. Dear Dr. Dunlap: This l e t t e r w i l l c e r t i f y that the thesis of Mr. John Thomas Sample has been c a r e f u l l y studied by the undersigned, and that the thesis meets the required standards and an abstract has been approved by the Department. Yours sincerely,  G. M. Shrum Head of the Department  GMS:1c  G. G. Eichholz Assistant Professor of Physics  . THE PRIMARY SPECIFIC. IONIZATION IN GASES OF POSITRONS. AND ELECTRONS II ' by John Thomas Sample  ABSTRACT By u t i l i z i n g the dependency of the e f f i c i e n c y of a Geiger counter upon the primary i o n i z a t i o n of a counted part i c l e , the minimum primary s p e c i f i c i o n i z a t i o n of electrons i n several gases-has been determined. Gas Helium  The values obtained are:  ion pairs/centimeter/atmosphere 5-47 ± 0-09  Neon  11.6  £ 0.4  Argon  29.8  £ 1.5  Ethylene  40.1.  Ethyl alcohol vapour  61.5 . •+ 1.5  1.0  The dependence of s p e c i f i c i o n i z a t i o n upon the energy of the primary electron has been investigated between the energies of 0.158 and 0.351 Mev". The primary s p e c i f i c i o n i z a t i o n of positrons and electrons of the same energy has been found to be the same to within 1%.  Abstract By u t i l i z i n g the dependency of the e f f i c i e n c y of a Geiger counter upon the primary i o n i z a t i o n of a counted part i c l e , the mimimum primary s p e c i f i c i o n i z a t i o n o f electrons i n several gases has been determined. Gas  The values obtained are:  i o n pairs/centimeter/atmosphere 5.4-7 t 0.0?  Helium Neon  11.6  Argon  29.8 + 1.5  Ethylene  40.1  Ethyl alcohol vapour  6 l . 5 * 1.5  * 0.4  I 1.0  The dependence of s p e c i f i c i o n i z a t i o n upon the energy o f the primary electron has been investigated between the energies of O.158 and 0.351 MeY.  The primary s p e c i f i c  i o n i z a t i o n of positrons and electrons of the same energy has been found to be the same to within 1%.  Acknowledgments This work has been made possible by the award of a National Research Council Bursary to the author, and by the p r i v i l e g e of using some of the equipment o f the Nuclear Techniques Laboratory set up with funds provided by the Defence Research Board. I t i s a.pleasure to acknowledge the guidance given by Dr. J.B. Warren, under .whose supervision t h i s work was carried out.-  ..  The help of Miss L. S i l v e r i n f a b r i c a t i n g the c o u n t e r i s g r a t e f u l l y acknowledged. s  Table of Contents Chapter  Page  I  Introduction  II  R e l a t i v i s t i c Quantum Theory of S p e c i f i c Ionization  III  1  . . . . . .  Experimental V e r i f i c a t i o n of the Theory  . IV  .3 ....  Review of Experimental Methods  V  10 13  Considerations Relating to the Counter E f f i c i e n c y Method . . . . . . . . . . . . .  14  VI  Apparatus  18  VII  Preliminary Investigations  22  VIII  Results and, Conclusions  26  Bibliography  28  Figures 1.  Beam d i s t r i b u t i o n from the Magnetic Analyser.  2.  C i r c u i t diagram of the t r i p l e coincidence mixer.  3.  Graph of equation (16) f o r mixtures of helium and ethylene.  4.  Graph of equation (l6) f o r mixture*of neon and ethylene.  .  Counter e f f i c i e n c y as a function of voltage.  6.  Counter plateau with and without external quenching.  7.  Table of Results.  8.  Variation of p . s . i . with energy.  y  THE PRIMARY SPECIFIC., IONIZATION. IN GASES OF POSITRONS AND ELECTRONS - I I CHAPTER I Introduction The primary s p e c i f i c i o n i z a t i o n , henceforth written p . s . i . , of a p a r t i c l e moving through a gas i s the number of ion pairs per centimeter of path, produced by d i r e c t c o l l i s i o n of the p a r t i c l e with molecules, divided by the pressure of the gas (0°C) i n atmospheres.  Theory predicts a r e l a t i o n between  p . s . i . and v e l o c i t y of the primary p a r t i c l e of the form: p.s.i. o 1 . A  flnM;  2  ) -  +  B?  where r3 i s the v e l o c i t y of the p a r t i c l e as a f r a c t i o n of the v e l o c i t y of l i g h t , and p i s the pressure of the gas i n atmospheres (0°C). Vdepends on the charge of the p a r t i c l e , the density of atomic electrons, and the average i o n i z a t i o n potent i a l of the gas. "B'depends on the average i o n i z a t i o n potential and on the maximum .energy transferred to a molecule i n a c o l l i s ion.  The t h e o r e t i c a l curve of p . s . i . i n a i r versus (3 has a  minimum at  (3 ~  .°6,  corresponding to an energy of 1.5 MeV  for electrons. By measurement of the e f f i c i e n c y o f a Geiger-counter, the p . s . i . of the gas i n the counter can be obtained. been done for several gases.  This has  A magnetic analyser was used to  select electrons o f the energy f o r which the p . s . i . i s a  The values found were 5.47  minimum. 29.8  t 1.5,  40.1  t 1.0,  * 0.09,  11.6  i  0.4,  and 61.5 .£ 1.5 respectively f o r helium,  neon, argon, ethylene, and ethyl alcohol vapour. The v a r i a t i o n of p . s . i . with energy has been investigated f o r energies between 0.158  MeV and O.351 MeV.  The shape  of the curve of p . s . i , versus energy i s similar to that of the t h e o r e t i c a l curve. The e f f i c i e n c y of a counter was found to be the same to within 0.25%  f o r electrons and positrons o f the same energy,  as predicted by theory.  3 CHAPTER I I R e l a t i v i s t i o Quantum Theory o f r s p e c i f i o Ionization 1.  Bohr (1) obtained relativist!© and n o n - r e l a t i v i s t i o  expressions for the energy l o s t per centimeter by a charged part i c l e passing through matter*According to these expressions, energy loss per centimeter varies d i r e c t l y as the density of atomic electrons and the square of the charge on the p a r t i c l e , and inversely as the square of the v e l o c i t y of the p a r t i c l e . In quantum mechanical theories the dependence on electron dens i t y and .charge i s the same, but the dependence on v e l o c i t y i s somewhat d i f f e r e n t . 2.  A n o n - r e l a t i y i s t i c quantum-mechanical r e l a t i o n was  derived by Bethe ( 2 ) . This may be expressed: s = 0.285  / 2TTNZ,z e4 \ In ( 42 mv 2  mv I 2  2  )  (1)  I  where s = p.s.i. N = number of atoms/unit volume. 2 = atomic ..number of the target atom, ez = charge of incident p a r t i c l e e » electronic charge m = electron mass v = v e l o c i t y of incident p a r t i c l e I • average i o n i z a t i o n potential of the electrons of the target atom. 3.  A true r e l a t i v i s t i o quantum mechanical treatment was  f i r s t presented by Bethe (3), an outline of which i s given  4, below.  More detailed derivations have been given by M i l l e r  (4), and Bethe and Fermi. (5). A free electron, with energy E, momentum p may be described by the Dirac eigenfunction  (2)  f ( p ) exp i (p . R - Et) n"  T =  where #>(p) has four components ^ , -—, 4^, dependent only upon — •  momentum and spin orientation, not upon p o s i t i o n vector R.  From  the eigenfunctions before and a f t e r a c o l l i s i o n with an atom,, the charge and current densities are constructed ( M i l l e r , (4)}: P = -e Y t p ^ y t p ) «= -eA.o(p, l) exp i [(p^pl) .R-(E-El)t7 _» S p  j = - e c a ^ p ) exp i 1  np^pM.R  - (E-Sl)t^  (?)  where c = v e l o c i t y of l i g h t  ao=  f*(pl)f(p)  a = - ^ ( p ) ^ f(p) 1  c ot = Dirac v e l o c i t y operator E  1  and p  1  are respectively the. energy and, momentum o f the  incident electron a f t e r the c o l l i s i o n . From these the retarded, scalar and vector potentials of the associated electromagnetic  f i e l d are computed.  Then, f o r the  p r o b a b i l i t y of a c o l l i s i o n i n which the incident electron i s scattered into s o l i d angle  d<*J  at angle©", and the atom i s  raised from the ground state (energy E , eigenfunction Y ) to 0  0  the nth state (energy En, eigenfunction T " n ) , the Born approximation gives: d#n-= 4e4 f l E S d c o f(p=pl)2«( - l) /C^ 1  2  E  E  I  fTn(r) •  exp i - - 3 « 2  [(p^p ^] 1  5 The i n t e g r a l i n (4) i s extended over the coordinates of the atomic electrons.  The index j r e f e r s to the j  atomic  electron. a© introduces the e f f e c t of the scalar p o t e n t i a l , «T .  ^  that of the vector p o t e n t i a l . .The term ( E - E ) / © ( E n - E ^ e 1  2  2  2  i s due to retardation. I f the atom receives energy small compared to mc ,  the  2  rest energy of the electron, Schrfldinger rather than Dirac eigenfunotions  be used f o r 7^ andYn.  may  Even then the e f f e c t  of the vector p o t e n t i a l must not be neglected  since the i n t e r •p  action cross-section would then be too large by a factor  E mc  a  approximately. For small scattering angles, a  =1,  0  a = v, c  and,  taking the v e l o c i t y of atomic electrons as small compared to that of the incident electron,  Tn where -* t  ^ j  T  V.  d r  - -  ^ V o =_L_  (S -E ) n  that of the coordinates, of the j  a  0  (J>)  signifies- the matrix element of the v e l o c i t y ,  on  rn  0  Also, for small s c a t t e r i n g angles, we may  atomic electron. expand the exponen-  t i a l function.in (4) and neglect terms containing higher powers than the f i r s t . Jn - i f~Tn = i x 0  n  I  s  x  o n  «S  Then the i n t e g r a l i n (4) i s : f (P-P  /P P  1  1  * En-Eov). r,-] To  - En-Bo  at  p /  (6)  the e l e c t r i c dipoie moment corresponding to the transpp  ii tt ii oo nn  o o .—* .—*  n, n, hence hence (x©n| (x©n| , except for a f a c t o r , i s the  t i c a l transition probability.  op-  Geometric considerations (p^p -) 3  2  _ (En-Ep) o~z~  (p=pl - E n - E o p ) ""E  2  give:  •. 2("p -E(En-Eo)l (l-oos &) + (En-E ) m c o s ^ + ~ c2 P 2  2  2  e  2  - 2{p -En-Eo}(2E -m c ) lo2~" 2  2  2  2  4  (l-oos^)+(En-Eo)  2 4 xffl"2l+--E p 2  (7)  2  (7) and (6) are next inserted i n (4) and the expression grated over the angle & .  2  intej  The upper l i m i t of integration,  i s chosen small enough that expansion of the exponential  i n (4)  i s allowed, large enough that the expressions (7) depend p r i n c i p a l l y on &  t  not on the transferred energy En-E .  These  0  conditions are compatible with each other. Integration of (4) under these conditions gives: (1§J0) = 2 TT 4 r x > f m 2p (l-oos• & ) - o p 1 4  e  J  °  where W (  mv21y '  '  - 2 7Te mvHRy '  '  = £ m  2  2  2  B  I  m2C«a-i«)Si  *-  TZ"J (8) J  (En-Eo)z(l-  (1-cos <%> ) i s the energy, from  a  conservation  laws, transferred to a free electron when the incident electron i s scattered through an angle The energy l o s s per centimeter i s obtained from (8) by summing over a l l possible states n (including the continuum)j multiplying each term by the energy transferred, E n - E o . .. gives: - dE = 2 TTQ^ Z f i n 2mv W . - /3 ] dx" mv I WTl=F) J 2  2  2  '  This (9)  1  where E i s the average e x c i t a t i o n energy of the target atoms. The p . s . i . i s related to -dE by dx  7 s - -1 dE ) , where d£\ i s that part of dE producing VT dx / ion 3x/ion dx i o n i z a t i o n ((8) summed only over states i n the continuum), and Vo i s the average energy expended i n creating an ion p a i r . Then (Bethe (2), Williams (6)): s - 0.285  2 7Te N Z f In 42mv - ln(l-£ )- ( 3 j nv2l I I 4  2  2  (10)  2  where I i s the average i o n i z a t i o n potential of atomic electrons. The underlying physical assumptions used to obtain (10) are as follows:.. a)  The energy transferred to the atom i s much l e s s than the  rest energy of an electron. (Cloudr?chamber pictures show few high energy secondaries!. b)  The scattering angle & ranges., only over values for which  the expansion of the exponential i n (4) i s v a l i d .  (This assum-  ption i s also necessary i n approximating the scalar and vector potentials),. c)  The v e l o c i t y of atomic electrons i s small compared to that  of incident electrons. 4.  Williams (6) u s e s t h e method of impact parameters  and c l a s s i c a l r e l a t i v i t y to obtain a r e l a t i v i s t i c term to be added, to equation. (1). (10).  In t h i s manner he obtains equation  The necessary physical assumptions are s i m i l a r to those  given above: a)  Energy transferred i n a c o l l i s i o n i s l e s s than W, where  I«W b)  «  (Imc )^ 2  The momentum of o r b i t a l electrons must be much l e s s than  that of incident electrons.  8 Williams estimates that 45% of -dE i s expended i n cEx  i o n i z a t i o n , and that the number of ejected electrons with energy greater than 21 i s n e g l i g i b l e . Since Williams obtains equation (10) by applying c l a s s i c a l r e l a t i v i t y , experimental v e r i f i c a t i o n of equation (10) i s not a v e r i f i c a t i o n of r e l a t i v i s t i c quantum.theory, but only of c l a s s i c a l r e l a t i v i t y theory, which has been v e r i f ied for charged p a r t i c l e s at l e a s t . 5.  The r e l a t i v i s t i c terms i n (10) have l i t t l e effect at  v e l o c i t i e s smaller than 0.°c.  However, as v increases, instead  of decreasing monotonically as i n the n o n - r e l a t i v i s t i o case, s, should pass through a minimum and increase as v—«• c.  Equating  ds to zero leads to the r e l a t i o n : •d(S  ln/42mc2)+ l n / _ x \ = 1 I ' l - x ' 1-x K  (11)  7  where x =• {2* , (3> being the value of {3 when s has i t s minimum value.  Graphical solution of ( l l ) gives values of the k i n e t i c  energy at the minimum ranging from 1.5 MeV 1.2 MeV  for lead.  0.5 MeV  and, 5  6.  for hydrogen to  Equation (10) i s almost constant between  MeV.  Although equation (10) has been applied to electrons,  i t applies s t r i c t l y only to heavy p a r t i c l e s , f o r two a)  reasons:  For electrons, there i s an, appreciable contribution from  c o l l i s i o n s with scattering angles greater than @° . b)  Since incident and atomic electrons are s i m i l a r p a r t i c l e s ,  exchange forces must contribute to the i n t e r a c t i o n . these into account, Bethe (7) obtains:  Taking  -dE ) = 2 ^ 9 % ^ dx/ion mv2  (In mv T . ( /I-yP - l + ^ 2 ) a ^ 2 ? 1 2l2(l-/2*) J (12) 2  2  l n  + 1  where T i s the r e l a t i v i s t i c k i n e t i c energy of the incident electron.  From the graphical representation of equation (12)  for a i r (Reference ( 7 ) ) , a minimum occurs at 1 MeV, dE being. . dx 10% greater than the minimum at energies of 0.45 MeV and 4.5MeV. The minimum,for lead occurs at about.0.8.MeV. 7.  Equations (1), (10), and (12) are similar at low  energies, and may be applied to energies as low as 0.2 MeV. Below t h i s energy,, p . s . i . should deorease sharply, due to the i n e f f i c i e n c y of energy transfer processes (velocity of the incident p a r t i c l e comparable to that of the o r b i t a l (K) electrons) At energies greater than lJMeV, (10) and (12) are s t i l l  similar,  increasing slowly as energy increases. At 10 MeV, both indicate that the p ^ s . i . should be a factor 2 greater than the minimum, markedly d i f f e r e n t from the c l a s s i c a l value, nearly zero. 8.  I f the exchange forces between a positron and an  electron d i f f e r from those between 2 electrons, the energy loss per centimeter-of positrons would not be given by equation (12). However, equation (9) should apply to positrons as well as i t does to electrons, and experiments cannot as yet d i s t i n g u i s h between equations (9) and (12) (or t h e i r p . s . i . analogues). 9.  I, the average i o n i z a t i o n p o t e n t i a l , i s d i f f i c u l t to  compute accurately, since the wave-functions atoms are unknown. I » (13.5 Z)  of multi-electron  Bloch (8) suggests the approximation:  eV.  (13)  10 CHAPTER I I I  .-  Experimental V e r i f i c a t i o n of the Theory 1.  Although equation (12)  should be applied to p . s . i . o f a  electrons a l l investigators have used/semi-empiriffial form of equation s =  (10): A (In K + In  -(3*)  (14)  where A and. K are dependent upon material and p a r t i c l e . 2.  Only f o r hydrogen has p . s . i . been accurately c a l c u l -  ated.-' Williams. (6) 1  atomic hydrogen.  gives the value 2.5 f o r the minimum i n  Assuming, a value of twice that f o r molecular  hydrogen, t h i s i s i n f a i r agreement with the value 5-2 obtained by Williams and Terroux (12),  though considerably lower than  the value 6.2 obtained by Danforth and Ramsey (13), obtained by Cosyns  and 6.0  (i4).  Williams (16)  found that equation (1)  applied to  gave  hydrogen^results within 10% of the experimental values f o r electrons with <? « 0.5, 3.  0.75,  O.96.  Range energy relationships derived from integration  of equations (1)  and (9)  can be f i t t e d empirically to exper-  imental data with good r e s u l t s f o r heavy p a r t i c l e s . 15).  (Reference  "Straggling" makes t h i s method d i f f i c u l t to apply to  electrons. 4.  T has been determined by no means other than measure-  ments of p . s . i . and dE, so that checks by two d i f f e r e n t „dx iments are not as yet possible. 5.  C l a s s i c a l l y , p . s . i . i s expected to vary as 1 . v 2  exper-  Instead, Williams and Terroux (12) f i n d , f o r @ between 0.5 and O.96, the r e l a t i o n s = 5.2 Q  -  0  ,  2  for hydrogen  s = 22 (3  -  0  :  2  for oxygen  That the exponent i n a f i t of this kind i s l e s s than 2, i n d i c ates the e f f e c t of a term such as I n / l  6. Hereford  )  l-  Of many investigators, only Corson and Brode (17) and (18) confirm the r e l a t i v i s t i c Increase i n p . s . i .  Corson and Brode show three points beyond the minimum l y i n g close to a curve of form (14).  One point near the minimum and three  on the low energy side also l i e close to the curve.  Hereford  shows three points beyond the minimum confirming the increase. S i x points near the minimum d e f i n i t e l y . l i e below the theoreti c a l value for hydrogen ( I = 13.5 ev") from equation (10). Hazen (19) concludes that the p . s . i . of mesons i n a i r r i s e s only one fourth as r a p i d l y as expected-from theory.  In another  paper (20) he confirms the r e l a t i v i s t i c increase i n p . s . i . of cosmic ray electrons, but i d e n t i f i c a t i o n of p a r t i c l e s i s not certain.  Measurements of grain density along cosmic ray tracks  in. photographic emulsions do not indicate an increase i n dens i t y i n the r e l a t i v i s t i c region (26).  This could mean that p.  s . i . does not increase i n t h i s region, or that an unknown effect i n the photographic process prevents an increase i n density. 7.  Results quoted i n the l i t e r a t u r e show considerable  divergence,  i n d i c a t i n g that the accuracy of the cloud-chamber  method has been overestimated.  The shape of the t h e o r e t i c a l  curve (14) lias been v e r i f i e d f a i r l y w e l l , though i n cases where quantitative-theoretical r e s u l t s are available, agreement i s not good* -  CHAPTER IV Review of Experimental Methods 1.  . Most of the p . s . i . values to date are from cloud-  chamber data. method.  Brode (21) has f u l l y discussed the cloud-chamber  In d i r e c t measurements of p . s . i . j the chamber i s ex-  panded at regular i n t e r v a l s , only sharp tracks being used. t h i s way,  In  secondary ions are not d i s t i n c t , forming a c l u s t e r  around the primary ions.  The p . s . i . i s given by the number of  clusters per centimeter divided by the post-rexpansion pressure i n atmospheres.  The energy of the p a r t i c l e i s determined from  the curvature of i t s track i n a magnetic f i e l d .  Inaccuracy  arises from estimation of pressure, temperature, concentration of water vapour and alcohol vapour, and track length, and from s t a t i s t i c a l variation.  References (6), ( 1 2 ) , (16), ( 1 7 ) , ( 1 9 ) ,  (20), and (21) r e f e r to the clouds-chamber method. 2.  The p . s . i . of the gas i n a Geiger counter can be  determined by measuring the e f f i c i e n c y of the counter (Chapter V).  Few investigators have as yet applied t h i s method, to  measurement of p . s . i . , but i t has been used by van A l l e n to estimate the energy of cosmic ray p a r t i c l e s , or a l t e r n a t i v e l y to i d e n t i f y p a r t i c l e s .  With the exception of Hereford (18) and  Curran and Reid (22) a l l investigators using the counter method to measure p . s . i . have done so f o r cosmic rays only, separating neither p a r t i c l e s nor energies.  References (10),(12), (14),  (18), and (22) r e f e r to the counter e f f i c i e n c y method. 3.  Although Powell has measured the v a r i a t i o n i n grain  density along tracks i n photographic emulsions * t h i s method serves as yet only to i d e n t i f y p a r t i c l e s .  14 CHAPTER V Considerations Relating'to. the Counter E f f i c i e n c y Method. 1.  The. r e l a t i o n between counter e f f i c i e n c y and p . s . i .  depends upon the a)  assumptions:  If.an ion pair i s created anywhere within the volume of  the counter, a count i s recorded. b)  The counter i s not photo-sensitive, that i s , photons from I n i t i a t e  atoms excited, by the passage of a f a s t p a r t i c l e do not^indicate a Geiger discharge. ing  This assumption  i s reasonable, since quench-  agents absorb photons, and counter wails can be made to  have a high work function. If  s i s the p . s . i . i n ion pairs per centimeter per atmos-  phere, and p (atmospheres) i s the gas pressure i n the counter, then spdx i s the p r o b a b i l i t y of forming at least one ion pair in distance dx (centimeters). The p r o b a b i l i t y of forming no ion of  pairs i n dx i s then (l-spdx).  I f pv(x) i s the p r o b a b i l i t y  producing^ no ion. pairs i n distance x, then: p (x+dx) = p ( x ) p ( d x ) = p (x)(l-spdx) o  o  dp Hence p  0  0  p (x+dx)-p (x) = -p (x).spdx  88 0  - e"  6  0  0  0  s p x  Then, i f £ i s the average path length of p a r t i c l e s traversing the counter, the e f f i c i e n c y  (the p r o b a b i l i t y that a p a r t i c l e  w i l l induce a discharge) i s given by: = l-e- P s X  I f two gases are present i n the counter, t h i s becomes: >« l - e -  U  (  s  l l p  +  S  21  (15.)  P  I f T i s the absolute temperature at which p i and p  2  are  measured, we have, from (14T):  3 + £2 s ? 58 l i ••JL. • I l n ( l - e )l . Pi JL 273 Pi  (16)  1  where p^, p£ are now measured i n centimeters of mercury.  If6  i s measured for several values of ££, and the quantity on the Pi r i g h t of (16) i s plotted against p_2, a straight l i n e should •Pi may he calculated. (16)  r e s u l t , from which s^ and  i s also  i n suitable form f o r a "least-squares" f i t . 2.  In measurements of 6., electrons of a selected  energy pass through the counter under investigation (counter 1) into another, (counter 2). I f N2 electrons are recorded, by counter 2, the p r o b a b i l i t y o f any one-of these i n i t i a t i n g a count from counter 1 i s 6.  From A i t ken (9), the. number of  counts, c, i n counter 1, coincident with counts i n counter 2, obeys a binomial..distribution: p(c) = N with mean c  e  2  £°(1-  6) 2-o N  » 6N2 and standard deviation  6*~=  {N 6(12  •  Hence € i s given by..e = 0 + JNze 41- a)}^ N N • 2  2  = o i f c  ( l - o ))^  N*2 iW  2  - W J  z  2  In a t y p i c a l case, c • lj>,000, N Then  6~= 3.1  z  (17)  10**  5  2  = 20,000  = 0.4% of 6 .  I f a background n2 i s to be subtracted from N2, a further deviation i s introduced. This i s approximately 6~2- =  N  22..  I f t h i s i s to be of the same magnitude as the  2  deviation i n the example above, ng. N2 to achieve.  <tf.l8.  This i s not d i f f i c u l t  3.  To f i n d the values of 6 f o r which s t a t i s t i c a l v a r i a -  tions have the l e a s t effect i n determining the r i g h t hand side of equation (l£), l e t x » s i p = - l n ( l - £ ) Then  -  putting A €  8  a£ -£  =  1  1 . N %  / e (l.  2  £7  ) /  ,  From this i t i s obvious that 6 should be as  small as possible. To f i n d the values of 6 at which a measurement of £ i s most sensitive to a difference 6 =l-e"  i n the value of s i p , take  x  —X  'y  A.6 = e  ^ x  ' - (1- e  x  From t h i s i t i s apparent that (= should be as small as possible. 4.  The "dead-times" (time of i n s e n s i t i v i t y following  a c t i v a t i o n by a count) of counters and electronics necessitate correction of equation (17). follows: xi,x  The problem may be phrased as  Two counters with counting rates Ni,N , dead-times 2  respectively, activate separate channels of a c o i n c i d -  2  ence mixer of resolving time Tr, dead-time t .  The i n d i v i d u a l  counting rates and the coincidence rate c are recorded by scalers of dead-time y i , y , y j respectively. 2  I f counter 2  has a background rate n , what i s the e f f i c i e n c y 2  i n terms of N that y  lt  1}  N, 2  o, x^, x , T, t, 3£.,.y, y^, and n , given 2  y , y? > x i , x 2  of counter 1  2  2  2  > t > f ?  Since yi>X]_, scaler 1 introduces the only loss i n . counting rate N i . of each minute.  C i r c u i t 1 i s "dead" f o r the f r a c t i o n N i y i  Hence the "true r a t e "  i s given by:  or  N-^ = N i 1  s i m i l a r l y Ng^ = N 1-N y 2  2  2  Since t < y ^ , the "true" coincidence rate, c , i s given by 1  c  1  =  o l-cy  5  "Accidental'* coincidences occur because of f i n i t e resolving time, the rate n n  c  c  being,given approximately by:  - 2(N -c)(N -c)r 1  2  Coincidences w i l l be l o s t when counters 1 and 2 are activated by non-coincident counts. i s determined  Since xj_, x > t , the loss 2  by the counter dead times.  To a near approxim-  ation, the time during which coincidences cannot be recorded i s given by (Nj-c) X j + (N -c) x . 2  A c t u a l l y this time i s  2  diminished by overlapping o f these terms, but i n our case the second term i s n e g l i g i b l e , and .hence the overlapping correction. Neglecting the second term then, c  s u f f e r s a further correct-  1  ion: c  =  1 1  c 1-lN-j-cjX! 1  From equation ( 1 7 ) : -. «ii N !-n  e  2  2  . o . -2(Ni-c)(N -c)T Il-oy3)q-(N -o)x 7 n ~ n 1-N y2 2  1  1  2  2  Of a l l the corrections, only 1 - ( % - C ^ a n d n  2  are appreciable.  Then: £ =  o N -n 2  11 2  + (Ni-c)xi7  (18)  CHAPTER VI Apparatus 1.  A description of the vacuum chamber, wedge magnet-  analyser, and other apparatus has been given by L. S i l v e r ( 1 0 ) . 2.  The electron source used was 2 m i l l i c u r i e s of Radium  D.-E. precipitated from acid solution onto platinum f o i l . Emax of the electrons from Radium E i s 1.17 MeV. from Ra D and oi-rays  Soft (3-rays  from Ra F are eliminated by the analyser.  The positron source used i n one experiment was a f o i l of Cu^ from the Chalk River p i l e . 3.  4  Emax of the positrons i s .66 MeV.  To a i d i n energy selection, and to reduce the 2^-ray  i n t e n s i t y from the source, i f any, a lead .baffle was Inserted just before the exit window of the analyser. 4.  Figure ( l ) i s a plot of i n t e n s i t y versus displace-  ment, horizontal and v e r t i c a l , obtained by moving (across theexit window at a distance of 0.8 cm.) a counter accepting p a r t i c l e s through a 3  inch hole.  From t h i s the focussing  32  of the analyser was assumed -to be s a t i s f a c t o r y . J?.  The counters (Reference (23')) to contain the i n -  e f f i c i e n t gas mixtures were of two types.  Some were made from  square brass tubing, about 2 cm. by 2 cm.  Others were c y l i n d -  r i c a l brass stock bored out and.machined f l a t to augment attaching windows.  2-4 milligram per square centimeter mica  was cemented over 3_ inch diameter holes on two sides of each lbcounter. A f t e r evacuation, the anode wires were "flashed" at white heat and held at red heat f o r ^30 minutes. ;  Pressures  were measured with a mercury manometer and t r a v e l l i n g t e l e s cope (accuracy i 0.002 cm.). recorded.  Room temperature  at the time  A d i f f u s i o n time of 12 hours was allowed.  was  Path  length was obtained by d i r e c t measurement before the counters were completed.  A small correction (0.05 cm.)  f o r sag of the  windows i s added. The second counter was a standard "bell-counter  11  f i l l e d with, 9 centimeters of argon and 1 centimeter of ethyl alcohol.  This counter was very nearly 100% e f f i c i e n t , with a  plateau slope of 3.5% per 100 v o l t s over a range of 150 v o l t s . I t was operated at a few v o l t s above threshold to reduce the p r o b a b i l i t y of "double counts".  A c y l i n d r i c a l lead s h i e l d  1 inch thick reduced the background rate of t h i s counter to about 7 counts, per minute. 6.  To reach the sensitive volume of the i n e f f i c i e n t  counter, p a r t i c l e s must pass through two mica windows, approximately 5 milligrams per square centimeter. the energy loss of 1 MeV per square centimeter.  From Fermi (11),  electrons i n a i r i s 1.7 MeV per gram Assuming a stopping number r e l a t i v e to  a i r of 1.5 for mica, the energy l o s s i s 0.013 MeV,  or about  one percent. 7.  Since energy i s l o s t i n the window of the counter,  there i s some p r o b a b i l i t y that an electron w i l l be ejected from the window into the counter, causing a discharge when electro*  the primary^would  not i t s e l f do so.  About 7000 eV (1 the 2"  loss calculated abojvje) i s l o s t i n the window, and, according to Williams ( 6 ) , about 45% of t h i s produces i o n i z a t i o n .  Also  very few electrons are ejected with k i n e t i c energy greater than 21.  Assuming ah average k i n e t i c energy of ;1.5I> and a  value of,80 (reference 15) v o l t s f o r I (that of a i r ) we have, for the number of. secondaries ejected by one primary: n = 7000 x 0.45 (1.5+1)80  = 16  (19)  These sixteen electrons have energy about 120 ev", a range i n which energy i s r a p i d l y given up to l a t t i c e v i b r a t i o n s .  We  conclude that the p r o b a b i l i t y o f one of these being h i t close enough to the inner window surface to escape i s n e g l i g i b l e . There remains to consider the few high energy secondaries (a number so small that contribution to energy loss i s considered n e g l i g i b l e i n a l l theories).  In a t o t a l of 1800 centimeters of  track i n a i r , ( p . s . i . -22) Williams and Terroux (12) found evidence of only 51 secondary electrons of energy greater than 10,000 ev. (even at t h i s energy few electrons w i l l leave the window, since ..escape).  most w i l l have a d i r e c t i o n unfavourable to  Hence the f r a c t i o n of i o n i z i n g c o l l i s i o n s i n which  the secondary electron has energy greater than 10,000 ev. i s : n  1  =  51 - 0.0013 1800 x 22  Then from (19),  we expect about 0.0013 x 16 = 0.021 energetic  secondaries per primary electron.  These secondaries w i l l have  effect only i n those cases when the primary electron does not i n i t i a t e a discharge, a f r a c t i o n ( l - £ ) .  I f £- = 0.75, t h i s  contribution to w i l l be 0.25 x 0.021 = .005, or 0.7% of 6, of the order of magnitude of the standard deviation.  The actual  contribution w i l l probably be much smaller than t h i s , since  only a few of the energetic secondaries w i l l be produced i n a d i r e c t i o n and p o s i t i o n favourable to entering the counter. 8.  Two 2000 v o l t regulated power supplies are  connected  to potentiometers i n the head amplifier chassis, allowing continuous v a r i a t i o n of counter voltages.  The head amplifiers  are single stage plus a cathode follower, (the c i r c u i t diagram si * 1  given i n reference ( 1 0 ) ) .  The output pulses, from the amplif-  i e r s activate separate channels of the coincidence mixer, the i n d i v i d u a l and coincidence counts being recorded by three Dynatron model 200 A decade scalers.  A Gossor model 1035  double-beam oscilloscope i s used to monitor the head amplifier outputs. Figure (2) mixer.  i s a c i r c u i t diagram of the—coincidence  Only, one channel, (A) i s shown.  exactly s i m i l a r .  Channels B and C are  When properly adjusted, the mixer, i s i n -  sensitive to pulse height v a r i a t i o n over a range from 5 v o l t s to 100 v o l t s .  Except i n the case of the positron experiment,  the mixer was operated as a double coincidence by putting one pulse i n two  channels.  Each day pulses from a pulse generator were fed to a l l three mixer channels to check loss of counts by the mixer or scalers.  Resolving time of the mixer was measured by means  of a t r i p l e pulse generator giving three pulses of variable The r e s u l t , 0.66 micro-seconds,  spacing i n time.  agreed with  that from counting accidental coincidences and solving for T i n N acc. = 2NlN T. 2  CHAPTER. VII Preliminary Investigations During the i n i t i a l testing, and i n the course of the f i r s t actual runs, several f a c t s , pleasing and otherwise, were discovered: 1.  Counters of the same f i l l i n g but d i f f e r e n t path  length gave the same value,within standard deviation, for the right side of equation (16), showing that path length estimates were correct. 2.  When corrected for dead-time l o s s , a counter  filled  with 9 centimeters of argon and 1 centimeter of alcohol was 99.9% e f f i c i e n t (expected value 99.99%).  From t h i s i t would  seem there are no unforseen losses of coincidences. 3.  Diaphragms, ^  inch diameter, placed over both  windows of the i n e f f i c i e n t counter, had no s i g n i f i c a n t effect on e f f i c i e n c y .  Since these r e s t r i c t possible v a r i a t i o n i n path  length, i t i s permissible to use the d i r e c t l y measured path length rather than c a l i b r a t e the counter with known gases. 4.  Although accuracy c r i t e r i a derived i n (Gh.V, 3) show  that 6 should be very small, t h i s could not be attained i n practice.  A certain pressure must be maintained i n the counter  for proper Geiger operation and the p . s . i . of most quenching agents i s large. 5.  Values of € used ranged from 0.6j> to 0.85.  Figures (3) and (4) show plots of equation (16) f o r  mixtures o f helium and ethylene, and neon and ethylene.  The  fact that a straight l i n e i s obtained j u s t i f i e s the assumption that photons do not count, and corroborates the c a l c u l a t i o n of  the contribution to e f f i c i e n c y from the counter windows. 6.  When several anomalously low values, of e were  measured, investigation showed € to be a smooth function o f the voltage applied to the counter.  This i s plotted, with the  Geiger plateau, i n figure (5).  From t h i s plot i t seems that  a r i s e i n e f f i c i e n c y , not multiple counts, accounts f o r the plateau slope over most of the plateau of counters with these particular f i l l i n g s .  I t was necessary to increase the voltage  u n t i l e f f i c i e n c y no longer increased appreciably to obtain consistent values f o r the right side of ( l 6 ) .  This r e s t r i c t e d  s t i l l further the possible values of £, since the counters required higher, concentrations of quenching.agent to operate, at higher voltages.  I t seems l i k e l y that t h i s effect i s  caused by negative i o n formation (a molecule captures a secondary electron.).  Presumably negative ions do not i n i t i a t e a  discharge, or at least delay the discharge beyond the coincidence resolving time.  The p r o b a b i l i t y of negative i o n formation  i s a property of the gas, and decreases as E increases, where p E i s e l e c t r i c f i e l d strength and p i s the t o t a l pressure of gas i n the counter ( ^ f ) . 7.  Counter l i f e and plateau length varied with f i l l i n g .  Most f i l l i n g s had l i f e longer than the experiment (about 10^ counts), although some were shorter. to 500 v o l t s .  Plateaus varied from £0 v  Those shorter than 100 v o l t s were not useful,  since <£ was s t i l l increasing at the end of the plateau.  Those  of $00 v o l t s were not duplicable, and d i d not remain as great as t h i s f o r very long.  In almost a l l cases, i t was necessary  to "age" the counters, that i s , increase the voltage i n steps of 25 v o l t s , allowing 10^  counts between each step.  I f this  was not done, a s u f f i c i e n t l y high voltage could not be reached. 8. and (7),  To overcome the d i f f i c u l t i e s l i s t e d i n sections two quench amplifiers were b u i l t *  These, upon being  triggered by a Geiger pulse feed back a 350 second negative pulse to the counter.  (6)  v o l t , 300  micro-  Plateaus of a t y p i c a l  counter with and without t h i s quenching action are shown i n figure (6).  Unfortunately, t h i s c i r c u i t i s sensitive to  input pulse height, i n that the time-delay between input and output pulse could be as great as 2 micro-seconds.  Since the  coincidence resolving time i s 0.66microseconds, coincidences w i l l be l o s t , unless the input pulses to .each amplifier are of the same height.  This i s d i f f i c u l t to achieve, since the  i n e f f i c i e n t counter must be operated at the top of i t s plateau, the b e l l counter at the bottom. 9.  For the same reason, a quench unit i-^"} capable of  quench pulses up to 800  v o l t s was b u i l t .  I t too introduced  a.time-delay, though not one dependent on.input pulse.  Hence  a time delay device i n the b e l l counter c i r c u i t might enable use of t h i s unit.  Pure inert gases could then be investigated  d i r e c t l y , since a pulse of such size can quench pure argon counters. 10.  In measuring the p . s . i . of positrons, i t was  found  that a n n i h i l a t i o n r a d i a t i o n contributed so strongly to the background that the method as so f a r discussed was not usable. By placing another two-windowed counter i n front of the  i n e f f i c i e n t counter, and measuring the t r i p l e coincidence rate, and the double coincidence rate of the front counter and b e l l counter, the e f f i c i e n c y of the middle counter can be found. I t i s the r a t i o of the two rates.  Background i s eliminated by  a  t h i s method.  When Ra E source i s used, the double and t r i p l e  coincidence methods give the same value of & , confirming the v a l i d i t y of the corrections applied to the double coincidence method.  CHAPTER VIII Results and Conclusions 1. (3)  A "least squares" f i t of the points shown i n figure to equation (16)  gives the valued 5.4-7  ±.09  and 39.7  +  0.9  ion pairs per centimeter per atmosphere f o r helium and ethylene respectively. 11.6  t 0.4-  .-The points of figure (4)  and 43.7  t 2.8  give the solutions  for neon and ethylene respectively.  The larger standard deviations of the second set of values are due to poorer s t a t i s t i c s . for ethylene i s 40.1 2.  *  The weighted mean of the two values  1.0.  Using these values, the  p . s . i . values of argon and (16).  ethyl alcohol were found by d i r e c t solution of equation Two such values were found i n each case, 29.1 argon, 61.4 29.8  * 1.5  and 6l.7 and 61.5  and 30.6  for  f o r ethyl alcohol, giving values - 1.5  respectively.  Only small pressures  of argon could be used, since the counters contained ethylene for quenching and helium to r a i s e the pressure, making the e f f i c i e n c y too high f o r accuracy.  This explains the difference  i n values f o r argon. 3.  The table (figure (7))  those of other investigators. exception of alcohol.  compares these r e s u l t s with Agreement i s f a i r , with the  Both experimental values f o r alcohol  are much larger than the values 33» obtained by extrapolating through points of p . s . i . versus number of electrons, and  42  obtained by adding the p.s.i.. of chemical constituents. 4.  Figure (8)  i s a plot of I n ( l - 6)(proportional to  p.s.i.) versus energy of electrons. A c l a s s i c a l "1"  curve,  27  obtained by equating the value f o r the lowest energy to k , i s shown for comparison.  Unfortunately, the counter broke down  before the.minimum of the curve was reached, and time d i d not permit another run.. j>.  The e f f i c i e n c y o f a counter, by the t r i p l e coincid-  ence method (Ch.VII,. 10) was found to be . 7 8 6 f o r positrons of . 5 MeV energy and . 7 8 4 f o r electrons of the same energy. The difference between these values is. less than the standard deviation.  Hence, to the accuracy of the measurement, p . s . i .  i s the same for positrons and electrons.  o  ;  z  3  f  P o s i t i o n of Detector, centimeters.  Figure 1.  Beam d i s t r i b u t i o n from the Magnetic Analyser.  15  . \ /0H IOOH  I 33  K.  +325*  K  ^  IT.  fig. CIRCUIT  2  DIAGRAM  OF TRIPLE  COINCIDENCE MIXER  Figure J .  Graph of equation ethylene.  for mixtures of helium and  fo  so bo  70  90  <fo  100  110 1x0  2k -JL JL  Figure 4.  273  130 1+0  \so ibo  i n (l-fr) pl  Graph of equation (16) f o r mixtures of neon and ethylene.  110 1  Voltage  Figure j5-  Counter e f f i c i e n c y as a function of voltage.  Voltage  Figure 6.  Counter plateau with and without external quenching.  p.s.i. Gas  Author  Helium  5.47 t  Neon  11.6  0.09  * 0.4  Others  6.6  19  5.9  14  12.6  25  29.4  14  29.8  22  Argon  29.8 + -1.5  Ethylene  40.1  + 1.0  -  Ethyl Alcohol Vapour  61.5 I 1.5  79  Figure 7.  Table of Results.  Reference  22  1-bO  ln(l-e) Z-5-5-  /•45  1+0  1-35  0-1  O-Z  0-3  Energy, MeV  Figure 8.  V a r i a t i o n of p . s . i . with energy.  '  Ob  O-  28  Bibliography 1.  Bohr, N., P h i l . Mag. 2 5 , 1 0 , P h i l . Mag. 3 0 , 5 8 1 ,  (1913). (1915).  2.  Bathe, H.A.,  Ann. der Physik, 5 , 3 2 5 ,  3.  Bethe, H.A.,  Z e i t s . f. Physik, 7 6 , 2 9 3 ,  4.  M i l l e r , Chr., Z e i t s . f. Physik, 7 0 , 7 8 6 , Ann.  (1930). (1932). (1931).  der Physik, 14, 5 3 1 , ( 1 9 3 2 ) .  5.  Bethe, H.A.,  and Fermi, E., Z e i t s . f . Physik, 7 7 ,  6.  Williams, E.J., Proc. Roy. S o c , A 1 3 9 , 1 6 3 ,  7.  Bethe, H.A.,  8.  Bloch, F., Z e i t s . f . Physik, 8 1 , 3 6 3 ,  9.  Aitken, A.C., S t a t i s t i c a l Mathematics, p.49.  296,  (1932).  (1933).  Handbuch der Physik, p.523*  1 0. S i l v e r , L.M., M.A.  (1933).  Thesis, University of B r i t i s h Columbia,  April, 1949. 11. Fermi, B., Nuclear Physics, p.31. 12.  Williams, E.J., and Terroux, F.R., Proc. Roy. Soc. A 1 2 6 , 289,  13.  Danforth, W.E.,  (1930).  -  and Ramsey, W.B.,  Phys. Rev., 4 9 , 8 5 4 ,  (1936).  14. Cosyns, M., B u l l . Tech. Ass. Ing. Brux., 1 7 3 - 2 6 5 , 15.  Bethe, H.A.,  and Livingston, M.S.,  261-276,  (1936).  Rev. Mod. Phys. 9 ,  (1937).  16.  Williams, E.J.', Proc. Roy. Soc. A 1 3 5 ,  17.  Corson, D.R.,  18.  Hereford, F.L., Phys. Rev., 7 4 , 5 7 4 ,  19.  Hazen, W.E.,  108,  (1931).  and Brode, R.B., Phys. Rev., 5 3 , 7 7 3 , (1948).  Phys. Rev., 6 5 , 2 5 9 , (1944).  (1938).  67,  20. Hazen, W.E.,  Phys. Rev.,  269, (1945).  21. Brode, R.B.,  Rev. Mod. Phys., 11, 222, (1939). Nature, 160,  866,  22. Curran, S.G.,  and Reid, J.M*,  (1937).  23. S i l v e r , L.M.,  and Warren, J.B., Rev. S c i . Inst., 21, 95,  (1950). 24. Healey, R.H.,  and Reed, J.W.,  Electrons i n Gases, 25. Skramstad, H.K.,  The Behaviour of Slow p.23.  and Loughridge, D.W.,  50,  Phys. Rev.,  677, (1936). -26. Edinburgh Conference on Elementary P a r t i c l e s , Nature,  165, 54, (1950). 27.  Gooke-Yarborough, E.H., F l o r i d a , CD., J.S.I.,  26,  124,  (1949).  and Davey,  C.N.,  .  '  

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