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The absolute dosimetry of negative pions 1979

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THE ABSOLUTE DOSIMETRY OF NEGATIVE PIONS by KENNETH ROBERT SHORTT B.Sc, McMaster University, 1970 M.Sc, The University of Western Ontario, 1972 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES.. (Department of Physics) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA November 1979 (p) Kenneth Robert,Shortt, 1979 In present ing 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 of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree tha t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r re fe rence and study. I f u r t h e r agree tha t permiss ion f o r ex tens ive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s en t a t i v e s . I t i s understood tha t copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l ga in s h a l l not be a l lowed wi thout my w r i t t e n pe rm iss i on . Department of The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook P lace Vancouver, Canada V6T 1W5 E-6 B P 75-51 1 E ABSTRACT THE ABSOLUTE DOSIMETRY OF NEGATIVE PIONS Soon three centres w i l l be treating cancer with beams of negative pions: LAMPF, Los Alamos, U.S.A.; SIN, V i l l i g e n , Switzerland; and TRIUMF, Vancouver, Canada. In order to understand t h i s new modality, i t w i l l be necessary to compare the re s u l t s of pion therapy to those achieved with conventional means and to compare r e s u l t s among the three centres. Absolute dosimetry i s the basis of th i s comparison. An absolute dose determination has been made for the negative pion beam at TRIUMF. using an i o n i z a t i o n chamber. The r e l a t i o n s h i p required to convert the i o n i z a t i o n per unit mass, J/M, measured by the chamber to dose i n tissue i s D = T, W r F M where W, r and F are calculated q u a n t i t i e s . W i s the average energy expended i n the gas per ion pair produced. Since the W-value for a secondary l i b e r a t e d during pion capture depends upon i t s energy, i t was necessary to average over the energy spectra for the various secondaries. r i s the r a t i o of dose i n the wall material, carbon, to dose i n the gas, either methane or carbon dioxide. Since the pion secondaries have ranges which are of the same s i z e as the cavity of the ion chamber, i t was necessary to e x p l i c i t l y consider those secondaries which emerge from the i i i w a l l w i t h i n s u f f i c i e n t energy to cross the c a v i t y and those pions which form s t a r s i n s i d e the c a v i t y . Thus, r was found to be pressure dependent. The pressure dependence f o r carbon d i o x i d e a r i s e s as f o l l o w s : - as the pressure i s increa s e d , there are more pion captures i n the gas, but si n c e there i s l e s s energy released to charged secondaries per pion capture on oxygen than carbon, the dose i n the gas decreases. Therefore, the c a l c u l a t i o n p r e d i c t e d t h a t , as pressure i s increased, J/M would remain unchanged f o r methane, but would decrease r a t h e r d r a m a t i c a l l y f o r carbon d i o x i d e . F i s the r a t i o of dose i n t i s s u e to dose i n w a l l m a t e r i a l , carbon i n t h i s case. F i s the product of two f a c t o r s : P, which accounts f o r the d i f f e r e n c e i n the stopping pion d e n s i t y f o r t i s s u e compared to carbon, and K, which accounts f o r the d i f f e r e n c e i n energy released per pion capture i n t i s s u e compared to carbon. Experimentally, the i o n i z a t i o n per u n i t mass was measured i n a p a r a l l e l p l a t e chamber as a f u n c t i o n of pressure f o r va r i o u s gases w i t h carbon, aluminum and TE-A150 e l e c t r o d e s . The pressure dependence of J/M measured f o r methane and carbon d i o x i d e w i t h carbon e l e c t r o d e s was compared to the behavior p r e d i c t e d by the c a l c u l a t i o n . Q u a l i t a t i v e l y , the p r e d i c t i o n was confirmed: the i o n i z a t i o n per u n i t mass f o r CC^ decreases more d r a m a t i c a l l y w i t h increased pressure than f o r CH^. Therefore pion capture i n the gas i s s i g n i f i c a n t and the energy released to charged secondaries per pion capture on oxygen i s l e s s than f o r carbon. Q u a n t i t a t i v e l y , the percentage change i s l a r g e r than p r e d i c t e d . The value of J/M ex t r a p o l a t e d to zero pressure and the appropriate values c a l c u l a t e d f o r W and r enabled a determination of the absolute dose i n carbon w i t h an estimated accuracy of ± 5%. The i o n i z a t i o n created w i t h aluminum electrodes was compared to that for carbon i n order to estimate the dose i n aluminum. Aft e r considering the dose contribution from beam contaminants, the value of F required to convert dose i n carbon to dose i n tissue was estimated to be 0.93 ± 0.05. Therefore, the absolute dose i n tissue was determined to an accuracy of ± 7%. In order to increase the accuracy, i t i s necessary to improve the data a v a i l a b l e as input to the c a l c u l a t i o n . V TABLE OF CONTENTS Page ABSTRACT i i - TABLE OF CONTENTS v LIST OF TABLES v i i i LIST OF FIGURES x ACKNOWLEDGEMENT x i i i CHAPTER 1 INTRODUCTION 1 1.1 PION RADIOTHERAPY 1 1.1.1 Goal of Radiotherapy 1 1.1.2' The Advantages of Pion Therapy 3 1.2 PION DOSIMETRY 10 1.2.1 Sign i f i c a n c e of Absolute Dosimetry 10 1.2.2 Ionization Chamber Dosimetry 12 1.2.3 The Approach to Absolute Pion Dosimetry 16- CHAPTER 2 CALCULATION 17 2. 1 INTRODUCTION 17 2.2 THEORY 19 2.3 INPUT DATA 25 2.3.1 Pion Star Data . . . 25 2.3.1.1 Secondary spectra f o r carbon and oxygen . . . 25 2.3.1.2 Fermi-Teller Z-law 33 2.3.1.3 Pion Stopping Density 35 v i TABLE OF CONTENTS (Cont'd) Page 2.3.2 Stopping Power Data 38 2.3.2.1 Stopping power for H and He ions i n the elements H, C and 0 38 2.3.2.2 Bragg a d d i t i v i t y rule 44 2.3.2.3 Stopping power for l i t h i u m 53 2.3.3 W-Value Data 61 2.4 RESULTS OF THE CALCULATION 65 2.4.1 P a r t i t i o n i n g of the Dose Between the Various Event ' Types 65 2.4.2 W and r for CH. and C0„ i n Carbon Walls 69 4 z 2.5 DOSE DEPOSITED BY OTHER PARTICLES «. 75 2.5.1 Electrons and Muons 75 2.5.2 D i r e c t l y Ionizing Pions 77 2.5.3 Neutrons and Gamma-Rays . 79 2.6 SUMMARY 80 CHAPTER 3 EXPERIMENT 81 3.1 INTRODUCTION 81 3.2 APPARATUS 8 2 3.2.1 Beam Delivery System 82 3.2.2 Beam Tune and Dose P r o f i l e s 82 3.2.3 Monitor Chamber 89 3.2.3.1 Construction 90 3.2.3.2 Operation 92 3.2.3.3 C a l i b r a t i o n versus p a r t i c l e f l u x 93 v i i TABLE OF CONTENTS (Cont'd) . Page 3.2.4 P a r a l l e l Plate Ionization Chamber 96 3.2.4.1 Construction 96 3.2.4.2 Gas del i v e r y system 100 3.2.4.3 Temperature monitor 102 3.2.5 E l e c t r o n i c s 103 3.3 RESULTS 106 3.3.1 Saturation C h a r a c t e r i s t i c s 106 3.3.1.1 Theoretical 106 3.3.1.2 Experimental 108 3.3.2 P o l a r i t y E f f e c t 113 3.3.3 J/M Versus Pressure 115 3.4 DISCUSSION 127 3.4.1 Comparison Between Ca l c u l a t i o n and Experiment . . . . 127 3.4.2 Absolute Dose i n Carbon 133 3.4.3 Kerma Factor 139 CHAPTER 4 CONCLUSION 149 REFERENCES 152 APPENDIX A A CHARGE COLLECTOR TO DETERMINE THE STOPPING DISTRIBUTION OF A PION BEAM 162 APPENDIX B DEFINITIONS, EQUATIONS AND CONSISTENCY CHECKS 166 APPENDIX C OXYGEN SPECTRA EXTRACTION 169 APPENDIX D STERNHEIMER DENSITY 'CORRECTION FOR ELECTRON STOPPING ' POWERS 171 BIOGRAPHICAL INFORMATION v i i i LIST OF TABLES Table Page 1.1 Improved Sur v i v a l When Treated with Megavoltage Radiotherapy 2. 2.1 P a r t i c l e and Energy Balance Sheet for Carbon 30 2.2 P a r t i c l e and Energy Balance Sheet for Oxygen 31 2.3 Comparison of Carbon and Oxygen Energy D i s t r i b u t i o n s 32 2.4 Constants Used i n the Dennis Formula for W-Values 62 2.5 Average Energy and Kerma According to P a r t i c l e Type (Carbon) . 66 2.6 Values of W and r for C-CH^ (sphere) 69 2.7 Values of W and r for C-C02 (sphere) 71 3.1 Percentage of the Dose by P a r t i c l e Type 89 3.2 Electrode Materials and Configuration 99 3.3 Gas Density and Purity 102 3.4 L i n e a l Energy at Various Depths 113 3.5 J/M Extrapolated to Zero Pressure f o r Carbon Electrodes . . . 134 3.6 Values Required to Calculate the Dose for CH^ 135 3.7 Values Required to Calculate the Dose for CO2 . 135 3.8 Dose i n Carbon 136 3.9 Evaluation of 7; x r for Carbon Electrodes 138 Wg 3.10 J/M Extrapolated to Zero Pressure for Aluminum Electrodes . .139 W 3.11 Evaluation of — x r for Aluminum Electrodes 144 W3 3.12 J/M Extrapolated to Zero Pressure for TE-A150 Electrodes . . .146 ix LIST OF TABLES (Cont'd) Table Page 4.1 Dose in Carbon and Aluminum 150 B.l Checks of Internal Consistency 1.68 D.1 Parameters Required to Calculate the Density Correction . . . 172 D.2 Electron Stopping Powers 173 X LIST OF FIGURES Figure Page 1.1 Dependence of RBE and OHR- iip'oiv LET ' 6 1.2 Comparison of Depth Dose Curves for y-Rays and Pions 8 1.3 Superposition of P a r a l l e l Opposing F i e l d s f o r y-Rays a n d Pions 9 2.1 Secondary Spectra for Pion Absorption on Carbon 27 2.2 Secondary Spectra for Pion Absorption on Oxygen 28 2.3 Range Ratio f o r Carbon/Methane and Carbon/Carbon Dioxide . . . 37 2.4 Stopping Cross Section for Protons i n Carbon 40 2.5 Stopping Cross Section for Alphas i n Carbon 42 2.6 Stopping Cross Section for Protons i n CO2 46 2.7 Stopping Cross Section for Protons i n CH^ 48 2.8 Stopping Cross Section f o r Alphas i n CO2 50 2.9 Stopping Cross Section for Alphas i n CH^ 52 2.10 Square of the E f f e c t i v e Charge for Lithium Ions 54 2.11 Stopping Cross Section for L i Ions i n Carbon 56 2.12 Stopping Cross Section for L i Ions i n CO2 . . . 58 2.13 Stopping Cross Section for L i Ions i n CH^ 60 2.14 W-Value for Alpha P a r t i c l e s i n CH^ 64 2.15 P a r t i t i o n i n g the Dose Among the Event Types for L i Ions . . . 67 2.16 P a r t i t i o n i n g the Dose Among the Event Types for A l l Secondaries :. 68 2.17 W 3 for C-CH. i n the P a r a l l e l Plate Chamber 73 — 4 Wg r x i LIST OF FIGURES (Cont'd) Figure Page 2.18 W g for C-C02 i n the P a r a l l e l Plate Chamber .74 Wg r 2.19 Electron Stopping Power Ratio Versus Pressure 76 3.1 Block Diagram of Apparatus . . . . . . , • . ; - ; - * - J . ' . ;. ... 83 3.2 Ionization Versus Depth P r o f i l e .v. . . . . . . . 84 3.3 Cross Scans at the Peak 86 3.4 Dose by P a r t i c l e Type 87 3.5 Schematic Diagram of the Monitor Chamber 91 3.6 Transmission Chamber Response Versus Momentum 94 3.7 Transmission Chamber Response Versus Separation 95 3.8 P a r a l l e l Plate Ionization Chamber Construction 98 3.9 Gas Delivery System 101 3.10 Timing Sequence for Reading the Electrometer 104 3.11 Saturation Curves f o r Carbon Dioxide 109 3.12 k Versus Mass Curves at D i f f e r e n t Depths for Methane 110 3.13 k Versus Mass at the Peak I l l 3.14 k Versus Mass at the Plateau 112 3.15 P o l a r i t y Difference Versus Mass 114 3.16 J/M Versus Pressure for Methane 116 3.17 J/M Versus Pressure for Carbon Dioxide 117 3.18 J/M Versus Pressure for A l l the Gases at the Peak with Carbon Electrodes 118 3.19 J/M Versus Pressure for A l l the Gases at the Plateau with Carbon Electrodes 119 3.20 Peak to Plateau Ratio Versus Pressure with Carbon Electrodes . 122 x i i LIST OF FIGURES (Cont'd) Figure Page 3.21 J/M Versus Pressure f o r A l l the Gases at the Peak with Aluminum Electrodes . 123 3.22 J/M Versus Pressure f o r A l l the Gases at the Plateau with Aluminum Electrodes . . . 124 3.23 Peak to Plateau Ratio Versus Pressure with Aluminum Electrodes 125 3.24 J/M Versus Pressure for A i r and TE gas with TE-A150 Electrodes 126 3.25 Factor U(P) Versus Pressure 129 3.26 Comparison of Calculated and Observed Pressure Dependence for J/M 131 x i i i ACKNOWLEDGEMENTS I am indebted to Dr. Mark Henkelman, my thesis supervisor, for his advice and encouragement throughout t h i s i n v e s t i g a t i o n . I am thankful to other members of the Biophysics s t a f f : Dr. Gabriel Lam for h e l p f u l suggestions concerning the experiment and c a r e f u l reading of t h i s thesis; Dr. Jan Nordin for thought-provoking conversations; Mr. Bob Harrison for w r i t i ng the data a c q u i s i t i o n programs; Mr. Bruno Jaggi for maintaining the channel and e l e c t r o n i c s . I am also g r a t e f u l f or the help provided by each s t a f f member with the data a c q u i s i t i o n . I enjoyed the comradeship of fellow graduateistudents Larry Watts, Michael Poon and J u l i e t Brosing. I am g r a t e f u l to Pat B e l l for carefully, typing this manuscript. During t h i s work, I greatly appreciated the moral and s p i r i t u a l support of members of the group. I acknowledge with thanks the recommendations of my advisory committee: Dr. J . Warren, Dr. L.D. Skarsgard, Dr. R.O. Kornelsen, Dr. R.R. Johnson.and Dr. G. Hoffmann, who took the place of Dr. Johnson while he was on sabbatical leave. I am. g r a t e f u l to Dr. H. Biehsel'fbr ' i n i t i a l l y pointing out to me the shortcomings of Bragg-Gray cavity theory and to h i s graduate student, Ms. A. Rubach, for giving me a copy of the f o r t r a n code which she used to perform c a l c u l a t i o n s for -neutron beams s i m i l a r to those I have done for pion beams. I am also g r a t e f u l to Dr. W. Kluge for providing me with the charged p a r t i c l e spectra due to pion capture i n oxygen containing compounds before p u b l i c a t i o n . The f i n a n c i a l support of the National Cancer I n s t i t u t e and the research f a c i l i t y provided by the B.C. Cancer Foundation are g r a t e f u l l y acknowledged. Special thanks are due to my wife, Marilyn, whose love and f a i t h supported me, and to our daughter, Rebecca, who brought me great happiness. I am g r a t e f u l to my parents, who provided my personality with enough determination to carry this work through to completion. 1 CHAPTER 1 INTRODUCTION 1.1 PION RADIOTHERAPY The use of pions for the treatment of cancer was f i r s t proposed by Fowler (1) i n 1961 and i s currently under development as a c l i n i c a l experiment at three centres: TRIUMF, Vancouver, Canada; LAMPF, Los Alamos, U.S.A.; and SIN, V i l l i g e n , Switzerland. It i s anticipated that t h i s new modality w i l l make a s i g n i f i c a n t impact on the cure of deep seated malignancies. However, pion radiotherapy i s considerably more complex than photon radiotherapy i n the production of the r a d i a t i o n , i t s d e l i v e r y , and i t s c h a r a c t e r i z a t i o n . Therefore, i f t h i s new modality i s to be c l i n i c a l l y successful, pion physical dosimetry must be understood. This thesis explores and solves many of the problems of pion dosimetry. In order to appreciate these problems, i t i s necessary f i r s t to understand the goals of radiotherapy and to see how pions are p a r t i c u l a r l y suited to meet these goals. 1.1.1 Goal of Radiotherapy I t i s the goal of radiotherapy to obtain l o c a l control of cancer i n order to cure or p a l l i a t e the patient. The e f f o r t s to a t t a i n this goal are f r u s t r a t e d by those tumours which have c e l l s with p e c u l i a r r a d i o b i o l o g i c c h a r a c t e r i s t i c s (such as radioresistance induced by hypoxia) 2 or which are located at s i t e s involving c r i t i c a l normal tissues (whose low r a d i a t i o n tolerance would lead to unacceptable r a d i a t i o n complications). It i s expected that the a b i l i t y to achieve uncomplicated l o c a l control of cancer would be improved i f i t were possible to increase the e f f e c t i v e tumour dose while maintaining or reducing the e f f e c t i v e dose to the surrounding normal tissue s . This expectation i s based on the dramatic increase i n f i v e year s u r v i v a l observed between 1955 and 1970 which has been a t t r i b u t e d , at l e a s t i n part, to the improved phy s i c a l depth dose c h a r a c t e r i s t i c s of megavoltage X-rays compared to k i l o v o l t a g e X-rays (2) (see table 1.1). The physical properties of negative pions w i l l r e s u l t i n further improvements i n both the phy s i c a l and the b i o l o g i c a l l y e f f e c t i v e dose d i s t r i b u t i o n . Table 1.1: Improved Survival When Treated with Megavoltage Radiotherapy (2) Type or S i t e % 5-Year Survival with % 5-Year Survival with of Cancer Kilovoltage X-Rays (1955) Megavoltage X-Rays (1970) Bladder 0 - 5 2 5 - 3 5 Cervix 35 - 45 55 - 65 Embryonal Cancer of T e s t i s 20 - 25 55 - 70 Hodgkin's Disease 30 - 35 70 - 75 Nasopharynx 20 - 25 45 - 50 Ovary 15 - 20 50 -' 60 Prostate 5 - 1 5 5 5 - 6 0 Retinoblastoma 30 - 40 80 - 85 Seminoma of T e s t i s 65 - 70 90 - 95 T o n s i l 25 - 30 40 - 50 3 1.1.2 The Advantages of Pion Therapy In order to discuss the advantages of pion therapy, i t i s necessary to define terms used commonly to compare d i f f e r e n t r a d i a t i o n types. The qu a l i t y of a r a d i a t i o n f i e l d i s a de s c r i p t i v e term r e f e r r i n g to those features of the s p a t i a l d i s t r i b u t i o n of energy transfers that influence the effectiveness of the r a d i a t i o n . Beams of X-rays, Y - r a v s a n d electrons used ro u t i n e l y i n radiotherapy are of the same low q u a l i t y . On the other hand, pion beam q u a l i t y i s a function of l o c a t i o n i n the pion f i e l d . LET, l i n e a r energy transfer, i s a physical parameter used to specify the beam qua l i t y . For a given p a r t i c l e , LET i s the mean energy l o c a l l y imparted by c o l l i s i o n s with energy transfers l e s s than some s p e c i f i e d value, A, divided by the distance traversed by the p a r t i c l e i n making these tr a n s f e r s . I f A i s set to the maximum kinematically possible, then LET equals stopping power. RBE, r e l a t i v e b i o l o g i c a l effectiveness, of a p a r t i c u l a r r a d i a t i o n i s the r a t i o of the dose of a reference r a d i a t i o n (280 keV X-rays) to the dose of the p a r t i c u l a r r a d i a t i o n (pions) required to a t t a i n the same e f f e c t on a given b i o l o g i c a l system. Radiation induced c e l l death i s produced by a mechanism which i s oxygen dependent. OER, oxygen enhancement r a t i o , i s the r a t i o of the dose required to achieve a s p e c i f i e d e f f e c t under anoxic conditions to the dose required to achieve the same e f f e c t under oxygenated conditions. RBE and OER are dependent upon LET as shown i n figu r e 1.1. This dependence ar i s e s because of matching between the s p a t i a l d i s t r i b u t i o n of i o n i z a t i o n events and the c e l l structure. To understand the advantages of pion therapy compared to ̂ C o Y - r a y s > 60 consider the depth dose curves shown i n figure 1.2. The Co curve exhibits skin-sparing i n the build-up region where the dose r i s e s from 60% r i g h t at the surface to 100% at 5 mm depth. Then the dose f a l l s o f f exponentially 4 to 42% at 14 cm. The pion depth dose curve has three regions: the plateau, the peak and the t a i l . In the plateau, the dose i s deposited by passing high-speed p a r t i c l e s : pions and the accompanying contaminants, muons and electrons. Pions emerging from the channel with the midline momentum of 180 MeV/c, w i l l have a r e s i d u a l k i n e t i c energy of 60 MeV i n the plateau and a stopping 2 power i n water of 3 MeV-cm /g. S i m i l a r l y , contaminant electrons and muons w i l l be minimally i o n i z i n g p r o j e c t i l e s . Except for the small dose deposited by pion i n - f l i g h t nuclear i n t e r a c t i o n s (4), the plateau i s a region of low LET. Right at the surface (which i s not shown i n fi g u r e 1.2) there w i l l be a build-up of knock-on electrons, but no s i g n i f i c a n t skin-sparing build-up of dose (see appendix A or reference (5)). The plateau may be sloped depending on beam focusing. There are two reasons for the increase i n dose and mean LET observed i n the peak. F i r s t l y , as the pions slow down, the stopping power of the medium increases, producing a c h a r a c t e r i s t i c Bragg peak. This increase i n dose i s associated with an increase i n LET. Secondly, when negative pions stop, they are captured by atomic n u c l e i with which they i n t e r a c t . This process i s c a l l e d pion star formation because of i t s appearance i n cloud chambers. Of the 140 MeV pion rest mass, about 28 MeV (for Carbon capture) 3 4 appears i n the form of charged secondaries p, d, t, He, He, and L i , which increases the dose and the LET i n the peak. In the t a i l region most of the dose i s due to contaminating electrons which have a r e s i d u a l k i n e t i c energy of 50 MeV and a minimum stopping power 2 (2 MeV-cm /g). This i s low LET dose. Consequently, the advantages of pion therapy are twofold. F i r s t l y , the. physical dose peak can be made to coincide with the tumour volume as 5 shown i n figu r e 1.3 for the ease of p a r a l l e l opposed f i e l d s . Secondly, the average LET i n the peak region i s higher than i n the plateau and this r e s u l t s i n an increased RBE and-decreased OER for the peak compared with the plateau (6). This provides the a b i l i t y to increase the e f f e c t i v e tumour dose while maintaining or reducing dose to the surrounding healthy tissue. Thus, l o c a l control of cancer with a minimum of treatment-induced s i d e - e f f e c t s ought to be achievable. 6 LET (KeV/M) Figure 1.1: Dependence of RBE and OER upon LET This curve i s c h a r a c t e r i s t i c of various b i o l o g i c a l e f f e c t s on c e l l s . For more information, see reference (3) . Figure 1.2: Comparison of Depth Dose Curves for yRays a n d Pions 60 The Co y-ray depth dose is for 10 x 10 cm f i e l d at 80 cm source to surface distance. The pion depth dose was measured for a midline momentum of 180 MeV/c and 12 cm diameter at the 50% level. The pion curve is shown with an 8 cm shift in order to simulate the treatment of a tumour at 14 cm depth (shown in figure 1.3). 00 LU CO O o LU \ 1 1 1 1 1 1 0 6C> Co 0 - 8 - 0-6 / 0 - 4 P i o n s — 0 - 2 - 0 0 1 1 1 4 6 8 D E P T H (cm) 10 12 14 Figure 1.3: Superposition of P a r a l l e l Opposing Fi e l d s for y-Rays and Pions The tumour was taken to be at the midline of a uniform section 28 cm thick. The dose d i s t r i b u t i o n i s symmetric about the 14 cm depth. 1.2 PION DOSIMETRY 1.2.1 S i g n i f i c a n c e of Absolute Dosimetry It i s the purpose of t h i s project to investigate the absolute dosimetry of the biomedical negative pion beam at TRIUMF. Dose i s defined as the quotient of the energy imparted by i o n i z i n g radiations to a volume element divided by the mass of the element. The measurement of dose involves a s p e c i a l type of l i m i t i n g process, i n that the mass must be small enough so that the measurement may be defined at a point, but large enough that the energy deposition i s not caused by a few in t e r a c t i o n s with large s t a t i s t i c a l uncertainty. An absolute determination of dose requires the use of an instrument which can measure the energy imparted to the mass i n i t s s e n s i t i v e volume without the need for c a l i b r a t i o n i n a known r a d i a t i o n f i e l d . The SI unit of absorbed dose i s the Gray (Gy) = 1 J/kg. The unit used previously was the rad = .01 Gy. An understanding of pion dosimetry contributes to the body of knowledge of physics, radiobiology and medicine. R e l a t i v e l y l i t t l e was . known about the physical dosimetry of pion beams and, therefore, a study of pion dosimetry was thought to be i n t e r e s t i n g and worthwhile for purely academic reasons. As w e l l , t h i s study has placed c e r t a i n demands upon other d i s c i p l i n e s of physics to improve the basic physical data required to carry out the necessary c a l c u l a t i o n s . From the point of view of biology, the a b i l i t y to measure the dose forms the basis of RBE determinations according to RBE = Dose with. 280 keV x-rrays ./, Dose with pions. A f a i l u r e to understand pion dosimetry w i l l lead to misunderstanding the RBE. From a s t r i c t l y pragmatic point of view, a study of pion dosimetry i s not e s s e n t i a l to the i n i t i a t i o n of treatment of patients with pions. As 11 discussed i n the previous paragraph, the pr e d i c t i o n of the e f f e c t of pions on patients w i l l . b e based on b i o l o g i c a l systems whose response to pions w i l l be compared with t h e i r response to conventional therapy. Errors i n dosimetry w i l l have r e c i p r o c a l errors i n RBE, but the product, e f f e c t i v e dose, i s invariant. Even i n the long term, the success of pion therapy compared to conventional therapy w i l l be based upon a c l i n i c a l demonstration of the hoped f o r increase i n s u r v i v a l f o r the same l e v e l of treatment-induced complications. On the other hand, i t i s necessary to understand pion therapy. Such an understanding involves comparisons between the c l i n i c a l e f f e c t s of pions and conventional r a d i a t i o n and between the c l i n i c a l e f f e c t s of the pion beams av a i l a b l e at the d i f f e r e n t l aboratories. A knowledge of pion dosimetry forms the basis of these comparisons, and i s therefore e s s e n t i a l to the understanding of pion therapy. 12 1.2.2 Ionization Chamber Dosimetry In general, three methods are av a i l a b l e f o r dosimetry: i o n i z a t i o n chambers, chemical dosimeters and calorimeters. Due to the r e l a t i v e l y low dose rate (<0.01 Gy/min), only i o n i z a t i o n chamber measurements are p r a c t i c a l . The equation which re l a t e s i o n i z a t i o n i n the gas to dose deposited i n the w a l l i s : D = (J/M) x Wg x r 1.1 wal l where J/M i s the i o n i z a t i o n per unit mass, Wg i s the average energy required to create an ion p a i r i n the gas, and r i s the quotient of dose i n the w a l l divided by dose i n the gas. In order to measure J/M, a l l of the i o n i z a t i o n must be c o l l e c t e d (that i s , saturation must be achieved) and the mass of the gas must be determined. C a l c u l a t i o n i s required to determine the values f o r W and r . For a given p r o j e c t i l e (for example, an alpha p a r t i c l e ) the average energy required to create an ion p a i r i n a p a r t i c u l a r gas i s usually around 30 eV at high p r o j e c t i l e v e l o c i t i e s and i t increases at low v e l o c i t i e s . To determine the average W-value f o r the f r a c t i o n of the dose due to s t a r s , i t i s necessary to average over the energy spectrum of each secondary. S i m i l a r l y , the c a l c u l a t i o n of r f o r the pion s t a r dose also requires a de t a i l e d knowledge of the energy spectra for secondaries since the stopping power of a medium depends upon the p r o j e c t i l e v e l o c i t y . The determination of the value f o r r requires the a p p l i c a t i o n of cavity chamber theory. There are d i f f e r e n t versions of cavity theory depending upon the underlying assumptions. Basic cavity theory (7) was developed by Bragg (8) and modified by Gray (9, 10). For the case of an i n f i n i t e l y small chamber, i n Bragg-Gray theory, r i s the stopping power r a t i o of the w a l l to the gas averaged over the spectrum of secondaries established i n the w a l l . This theory has three assumptions: that primary i n t e r a c t i o n s i n the gas are n e g l i g i b l e , 13 that the spectrum of secondaries established i n the w a l l i s not modified by the gas i n the c a v i t y , and that the primary i n t e r a c t i o n s from which second- aries can enter the cavity i s s p a t i a l l y uniform. The l a s t condition has been embodied into the d e f i n i t i o n of the s p e c i a l l i m i t i n g process required when measuring dose by demanding that the mass be small enough that the dose be defined at a point. The f i r s t two conditions are not v a l i d f o r large c a v i t i e s , where large means a cavity whose size i s comparable to the range of the secondaries which deposit dose i n i t . For such a large cavity, there w i l l be s i g n i f i c a n t pion capture i n the gas. This modifies the t o t a l energy contained i n the spectra of secondaries because the release of energy to charged secondaries due to pion capture i n the gas w i l l , i n p r i n c i p l e , be d i f f e r e n t from that due to pion capture i n the w a l l . As w e l l , the r e l a t i v e d i s t r i b u t i o n of energies i n thfe spectrum for a p a r t i c u l a r type of secondary i s changed because the stopping power of the cavity d i f f e r s i n p r i n c i p l e from that of the w a l l material. Thus, f o r large c a v i t i e s the theory must e x p l i c i t l y consider i n t e r a c t i o n s within the cavity as has been done for neutron i r r a d i a t i o n (11, 12). A rough c a l c u l a t i o n based on the thesis of Henry (13) and the calculated secondary spectra of Guthrie (14) indicated that ion chambers in;.common use (1 to 2 cm^) need to be considered as large compared to the range of the secondaries, and therefore such a de t a i l e d c a l c u l a t i o n of r i s necessary. If the primary r a d i a t i o n were gamma.rays instead of pions, i t would be possible to release the r e s t r i c t i o n on cavity s i z e by s e l e c t i n g a chamber with an atomically matched gas-wall p a i r (for example, ethylene and polyethylene) to which Fano's theorem may be applied. This theorem states that i n a medium of given composition exposed to a uniform f l u x of primary r a d i a t i o n , the f l u x of secondary r a d i a t i o n i s uniform independent of the density of the medium and independent of density v a r i a t i o n s from point to point, provided that the interactions of the primary r a d i a t i o n and the secondary r a d i a t i o n with the atoms of the medium are both independent of density. I t i s a c o r o l l a r y of Fano's theorem that for a cavity with atomically matched gas and w a l l , the stopping power.ratio i s unity-independent o f _ s i z e and.pressure i f the stop- ping power i s independent of density. Unfortunately, ~ in, the pion--ca.se, the stopping power for the pion secondaries depends upon the density of the stopping medium at energies w i t h i n the range of i n t e r e s t . This f a c t has been documented i n a recent survey by Thwaites (15) and i s discussed further i n section 2.3.2.1. Therefore, the atomically matched system employing ethylene and polyethylene would not have the value unity for r . The system employing methane-based TE gas and TE-A150 p l a s t i c i s not atomically matched since the gas and w a l l have d i f f e r e n t r a t i o s of carbon to oxygen atoms, and therefore i t w i l l not have the value unity f o r r . (TE-A150 p l a s t i c i s not ti s s u e equivalent for the same reason.) For pions, there i s no matched gas-wall p a i r presently a v a i l a b l e to which Fano's theorem may be applied. Therefore the r e s t r i c t i o n on cavity s i z e imposed by the simple Bragg-Gray theory cannot be circumvented. Since the range of the pion secondaries i s small compared to the cavity s i z e , the type of cavity theory due to Casewell (16), which e x p l i c i t l y considers i n t e r a c t i o n s w i t h i n the c a v i t y , i s necessary. If the dose i n tissue i s required, a further step i s required: D . = F D „ 1.2 txssue w a l l where F i s loosely c a l l e d the kerma f a c t o r . Kerma i s the quotient of the sum of the i n i t i a l k i n e t i c energies of a l l the charged p a r t i c l e s l i b e r a t e d by i n d i r e c t l y i o n i z i n g p a r t i c l e s i n a volume element divided by the mass of the matter i n that element. The pion i s a d i r e c t l y i o n i z i n g p a r t i c l e while i n f l i g h t because i t i s charged and of s u f f i c i e n t energy to produce i o n i z a t i o n by c o l l i s i o n . The pion i s i n d i r e c t l y i o n i z i n g when captured because i t s i n t e r a c t i o n with the nucleus i s due to the strong force, independent of the pion charge. The kerma factor i s the r a t i o of the dose deposited i n a small element of tissue located i n a block of tissue divided by the dose deposited i n a small element of w a l l material located i n a block of w a l l material when both blocks are exposed to the same fluence of pions. For the pion star dose F i s composed of two parts: F = K x P 1.3 where K corrects f o r the difference i n energy released to charged secondaries by capture i n tissue compared to w a l l by v i r t u e of t h e i r d i f f e r i n g atomic composition, and P corrects f o r the difference i n density of pion stops i n tissu e compared to w a l l by v i r t u e of t h e i r d i f f e r i n g stopping powers. Furthe comments concerning the factor K w i l l be reserved f o r the discussion i n chapter 3. The importance of the fac t o r P w i l l be dealt with i n section 1.2.3 The-..:Approach to Absolute Pion Dosimetry The absolute dosimetry of a negative pion beam was investigated by means of an i o n i z a t i o n chamber. Appropriate values of W and r for two gas-wall combinations (methane and carbon dioxide with carbon electrodes) were calculated for the pion star dose as a function of chamber pressure using the Casewell type of cavity theory, which e x p l i c i t l y considers primary in t e r a c t i o n s i n the gas and perturbation of the secondary spectra by the gas. W and r were shown to be pressure dependent, i n v a l i d a t i n g simple Bragg-Gray cavity theory. Values of J/M were measured for several gas-wall combinations as a function of gas pressure. Measurements i n both the peak and plateau have permitted the e x t r a c t i o n of the pressure depen- dence of e f f e c t s s p e c i f i c to pion stars f or comparison with the c a l c u l a t i o n . The c a l c u l a t i o n and experiment agree q u a l i t a t i v e l y , but the observed dependence of J/M on pressure i s more dramatic than predicted, p a r t i c u l a r l y for oxygen bearing gases. The experimental value of J/M extrapolated to zero pressure was used with the calculated values of W and r i n equation 1.1 to determine the dose i n carbon. The r a t i o of i o n i z a t i o n produced with aluminum and carbon electrodes permitted an evaluation of the dose i n aluminum. The problem of converting dose i n carbon to dose i n ti s s u e i s discussed and the conversion factor i s evaluated. 17 CHAPTER 2 CALCULATION 2.1 INTRODUCTION A c a l c u l a t i o n was performed to determine the average energy expended 0 per i o n i z a t i o n , Wg, i n a gas ca v i t y , and to determine the r a t i o , r , of the energy deposited i n the gas cavity to the energy that would have been deposited i f the cavity had been f i l l e d with w a l l material. The c a l c u l a t i o n was r e s t r i c t e d to the dose deposited by pion stars since t h i s i s the part of the ra d i a t i o n f i e l d that i s uniquely c h a r a c t e r i s t i c of pions and was poorly under- stood. The dose deposited by passing pions and electrons i s better understood, and i s s p e c i f i c to the contamination of the p a r t i c u l a r pion beam and to the po s i t i o n i n the r a d i a t i o n f i e l d . A c a l c u l a t i o n of t h i s type was f i r s t performed by Casewell (16) i n 1964 fo r neutron dosimetry and has been used by several other, investigators to study neutrons (11, 12, 17). The c a l c u l a t i o n evaluates the energy deposited i n a gas f i l l e d cavity by a l l the charged p a r t i c l e s i n t e r a c t i n g i n the cavity. I t assumes that the p a r t i c l e s t r a v e l i n s t r a i g h t l i n e s and lose energy continuously and l o c a l l y . That i s , the e f f e c t s of multiple s c a t t e r i n g , range s t r a g g l i n g and energetic delta rays were not considered. Along with the energy deposited i n the gas, the c a l c u l a t i o n evaluates the i o n i z a t i o n produced by each p a r t i c l e from a knowledge of the W-values f or those p a r t i c l e s . This allows the determination of the average energy expended per i o n i z a t i o n , Wg, to be made. The c a l c u l a t i o n i s then repeated f o r the cavity f i l l e d with an equivalent mass of wa l l material instead of gas. The energy deposited i n the w a l l material i s evaluated i n the same way as i n the case of the gas f i l l i n g , and the r a t i o , r , of the energy i n the wa l l to the energy i n the gas i s computed. The r a t i o r was calculated f o r the dose from pion stars as a function of the pressure of the gas i n the cavi t y , or equivalently as a function of the mass of the gas. This c a l c u l a t i o n i s dependent only on the a r e a l density of the material f i l l i n g the cavity and therefore cannot d i s t i n g u i s h between c a v i t i e s of increased volume and c a v i t i e s with increased pressure. 2.2 THEORY Consider a cavity inside an i n f i n i t e homogeneous phantom (of p h y s i c a l density p) which i s f i l l e d uniformly with stopped pions of density II pions/g. Let the i n i t i a l energy spectrumper pion..for a particular- type of secondary p a r t i c l e be K(e) seeondaries/MeV^pion. Then, the energy spectrum per gram f o r t h i s secondary p a r t i c l e i s given by Np(e) = It K(e) secondaries/MeV-g '2.1 where p refers to the i n i t i a l or production spectrum. Eventually, a summation over p a r t i c l e type i s performed. Consider a surface element located on the cavity boundary represented by dA, i t s unit normal pointing into the cavity. L i s a vector pointing from a unit volume located i n the w a l l to the area dA. 9 i s the angle between dA and L and <j> i s the azimuthal angle. The number of secondaries with energy between e and e + de produced ins i d e a volume element dV located at -L i s given by Np(e) de pdV = Np(e) de pL2sinedLd6d<f> 2.2 Since the production of secondaries i s i s o t r o p i c , the f r a c t i o n of the secondaries o r i g i n a t i n g i n dV t r a v e l l i n g i n d i r e c t i o n L which enter dA ( i f they have s u f f i c i e n t energy to t r a v e l distance L) i s dA cos6. A secondary 4TTLZ of r e s i d u a l energy E at the boundary could have started with energy e^ at L^ or at L^ and so on. That i s , the value of e depends on L for a given E. The t o t a l number of secondaries crossing the area with energy E w i l l involve an i n t e g r a l over L i n expression 2.2. This i n t e g r a l over L can be converted to an i n t e g r a l over e by f i r s t considering that pdL = dE 2.3 S(E) That i s , for an increase i n path length pdL, there w i l l be a change i n the 20 energy of that particle when i t crosses dA of dE = S(E) pdL 2.4 It is important to note that this relationship is true independent of the i n i t i a l energy of the particle e. Therefore, the number of particles crossing dA with energy between E and E + dE in direction L is given by m dE V w / \ j sin8cos6 ,, ,. 0 ,-Np(e) de -. d0 d<j) dA 2.5 J e=I S(E) \ 4TT E For convenience of notation, the equilibrium or slowing-down spectrum is defined as em Nr(E) = ^ Np(e) de 2.6 e=E with units secondaries/MeV-cm2. Then the total number of secondaries of energy between E and E + dE, crossing dA and travelling in direction L is Nr(E) dE S i n / 9 C ° S 9 d6 dd> dA 2.7 4TT Let A be the energy deposited in the cavity by such a secondary. Generally, A depends upon E, cavity geometry and stopping power of the cavity material. The total energy deposited in the cavity is given by integrals over cj>, dA, 6 and E. The integrals are considerably simplified i f the cavity is assumed to.be a sphere of diameter D. In this case, integrations over dA and cj> can be performed to yield /•£max J E=O J e= Energy = D 2 \ \ Nr(E) A(8, E) sin6 cos6 dO dE 2.8 =0 The integral over 9 runs from 0 to TT/2 rather than -IT/2 to TT/2 to exclude particles crossing dA from inside the spherical cavity. These particles w i l l be considered separately. 21 For stoppers (secondaries from the w a l l which stop i n the c a v i t y ) E i s given i m p l i c i t l y by R(E ) = D where R i s the range i n the c a v i t y . it13.x tt19.x Also A = E and DcosO > R(E). The lower bound on cos6 sets an upper bound on 6: 6 = c o s - 1 R(E)/D. Then i n t e g r a t i o n y i e l d s max • 1 J E=0 max E = 7" \ Nr(E) E (D 2 - R 2(E)) dE 2.9 stopper and the i o n i z a t i o n i s given by >r = 4 \ J E=0 max P J = T \ Nr(E) (D 2 - R 2 ( E ) ) dE 2.10 stoppe  4 \ W(E) For crossers ( w a l l secondaries which cross the c a v i t y ) , A i s given i m p l i c i t l y by R(E) = R(E - A) + z where z i s the path length given by Dcosl; In order to have a c r o s s e r , z < R(E) and hence, cosQ < R(E)/D. Therefore, z < R(E) i f R(E) < D z < D otherwise. The i n t e g r a l over 0 can be transformed to an i n t e g r a l over z y i e l d i n g min(R(E),D) r e m r m n \ \ Nr(E) V J E=0 J z=0 E = T r \ Nr(E) \ A(E,z) zdz dE. 2.11 c r o s s e r ° * 1 The t r i a n g u l a r chord length d i s t r i b u t i o n z d z r r e s u l t s from transformation of the i s o t r o p i c angular d i s t r i b u t i o n w i t h i n a sphere. The i o n i z a t i o n f o r crossers i s given by min(R(E) ,D). / * e m /- , J = T \ Nr(E) \ - 7 7 ^ - T9=% zdz dE 2.12 c r o s s e r 2 \ \ I W(E) W(E-A)J J E=0 J z=0 where E/W(E) i s the i o n i z a t i o n that would be produced by completely stopping a p a r t i c l e of energy E i n the gas and (E-A)/W(E-A) i s the corresponding i o n i z a t i o n - f o r the r e s i d u a l energy (E-A). Such a d i f f e r e n t i a l approach i s 22 necessary where W i s a strong f u n c t i o n of energy and A i s l a r g e . This completes the c a l c u l a t i o n of energy and i o n i z a t i o n from secondary p a r t i c l e s produced i n the w a l l . For those charged secondaries which o r i g i n a t e from pion capture i n the c a v i t y m a t e r i a l , the c a l c u l a t i o n of the energy d e p o s i t i o n i s s i m i l a r to that above. By analogy w i t h equation 2.1 we define the i n i t i a l spectrum f o r p ion capture i n the c a v i t y m a t e r i a l to be Np(e) = n 1 K ^ e ) 2.13 The number of secondaries s t a r t i n g i n a volume element dV at -L, w i t h energy between e and e + de and t r a v e l l i n g i n the d i r e c t i o n L to pass through dA (assuming no l o s s due to slowing-down and i s o t r o p i c production) i s 3/• \ •, sin9cos6 ,_ ,n ,, ,, 0 Np(e) de p ^ dL d8 d<j> dA 2.14 The energy deposited i n the c a v i t y by such a secondary, A, i s given i m p l i c i t l y by R(e) = R(e - A) + L 2.15 where R(e) i s the range i n the c a v i t y m a t e r i a l at energy e and L i s the path length t r a v e l l e d i n the c a v i t y . The t o t a l energy deposited i n the c a v i t y i s found by i n t e g r a t i n g over a l l the v a r i a b l e s . Assuming s p h e r i c a l geometry, the i n t e g r a l s of <|> and dA can be done se p a r a t e l y to y i e l d Energy = ^ ^ ^ ^ A(L,e) Np(e) cos9 sin9 dL d6 de 2.16 e 9 L For i n s i d e r s (gas secondaries which remain w i t h i n the c a v i t y ) , A = e and we must have R(e) < L < D cos9. Hence 9 v a r i e s between 0 and 9 m given by • - l R(e) cos — 23 The u p p e r l i m i t on e i s g i v e n i m p l i c i t l y b y R(e ) < D. T h u s , K E- -A x n s x d e r TTD̂ /%£- / * D c o s e J e=0 J 9=0 J L=R(e) (e) cosS s i n e dL dG de 2 . 1 7 P e r f o r m i n g t h e i n t e g r a t i o n on L and t h e n 0 y i e l d s ( 2 D 3 - 3 D 2 R ( e ) + R 3 ( e ) ) eNp(e) de 2 . 1 8 S i n c e t h e i n s i d e r l o s e s a l l o f i t s e n e r g y w i t h i n t h e c a v i t y , t h e i o n i z a t i o n E . • x n s x d e r - f J e=0 xs i n s i d e r 12 P J e=0 ( 2 D 3 - 3 D 2 R(e) + R 3 ( e ) ) W(e) Np(e) de 2 . 1 9 F o r s t a r t e r s (gas s e c o n d a r i e s w h i c h e s c a p e f r o m t h e c a v i t y ) we must h a v e R(e) > L . The i n t e g r a l o v e r L h a s l i m i t s 0 t o t h e minimum o f D o f R(e) and t h a t o v e r 6 h a s l i m i t s 0 t o 6 g i v e n by c o s6 = L / D . A f t e r p e r f o r m i n g m m the i n t e g r a t i o n o v e r G, we h a v e s t a r t e r 4 and s t a r t e r 4 max e=0 max 1 Np(e) »min (D ,R(e)) L=0 / * m i n ( D , R ( e ) ) J L=0 2.21 T h i s c o m p l e t e s t he d e r i v a t i o n o f t h e e q u a t i o n s r e q u i r e d t o c a l c u l a t e t h e e n e r g y and i o n i z a t i o n d e p o s i t e d i n t h e c a v i t y due t o s e c o n d a r i e s w h i c h o r i g i n a t e f r o m p i o n s t a r s i n t h e c a v i t y . These q u a n t i t i e s and e q u a t i o n s have b e e n summar i zed i n a p p e n d i x B . I t s h o u l d be n o t e d t h a t t h e i n t e g r a l s f o r s t o p p e r s and c r o s s e r s i n v o l v e d t he s l o w i n g - d o w n s p e c t r u m N r ( E ) as p r o d u c e d A ( L , e ) ( D 2 - L 2 ) dL de 2 . 2 0 e=0 by pion capture i n the w a l l and subsequent slowing-down i n the w a l l . On the other hand, the i n t e g r a l s for s t a r t e r s and i n s i d e r s involve Np(e), the primary spectrum as produced i n the cavity. In the process of c a l c u l a t i n g r , the cavity w i l l a l t e r n a t i v e l y be considered to be f i l l e d with gas and w a l l . For the former, Np(e) w i l l be the spectrum due to gas, and i n the l a t t e r , Np(e) = Np(e) due to w a l l material. 25 2.3 INPUT DATA 2.3.1 Pion Star Data 2.3.1.1 Secondary spectra f o r carbon and oxygen In order to carry out the calc u l a t i o n s outlined i n 2.2, the energy spectra of secondaries, K(e) (equation 2.3), are required. These energy spectra have been studied i n medium energy physics i n an attempt to under- stand the nuclear substructure. The pion-nucleus i n t e r a c t i o n can be considered a two stage process. The f i r s t stage y i e l d s high energy second- ar i e s due to d i r e c t processes such as absorption of the pion on a nuclear p a i r or possibly an alpha p a r t i c l e c l u s t e r . The second stage y i e l d s lower energy secondaries due to "evaporation" of p a r t i c l e s from the r e s i d u a l excited nucleus. Guthrie (14) has used the intranuclear-cascade model to ca l c u l a t e secondary spectra, but this c a l c u l a t i o n places too much energy into charged p a r t i c l e s : 42.7 MeV per pion stopped i n carbon and 36.7 MeV per pion stopped i n oxygen. Various experimenters (18, 19, 20, 21, 22, 23) have measured • spectra f o r some of the secondaries over l i m i t e d energy ranges. Data taken by a Karlsruhe group (24, 25, 26, 27) working at SIN have been used ex c l u s i v e l y i n t h i s c a l c u l a t i o n . These data are comprehensive i n that a l l of the energetic secondaries were measured from the experimental energy threshold up to the kinematical l i m i t . The spectra f o r the d i f f e r e n t p a r t i c l e types are determined simultaneously by a multiparameter datia a c q u i s i t i o n system which records d i f f e r e n t i a l and t o t a l energy loss and t i m e - o f - f l i g h t . Their thin (20 mg/cm2) target data have been corrected for p a r t i c l e loss and energy absorption using the unfolding procedure of Comiso (28). The data f o r secondary spectra from pion absorption on a carbon (CH^) target (24, 25) are shown i n figure 2.1. The data for oxygen have been extracted from preliminary Karlsruhe data (26, 27) for mylar and acetate which have r e l a t i v e capture r a t i o s on carbon and oxygen of 1.7:1 and 1.1:1 respec- t i v e l y (26). The extraction procedure i s outlined i n appendix C and the data are displayed i n figu r e 2 . 2 . The y i e l d of neutrons is-about the same case ; (140) The. y i e l d of charged .par-tieles- from --oxygen ̂ i s considerably reduced compared to carbon, p a r t i c u l a r l y for He and L i ions. P a r t i c l e and energy balance-sheets are given i n tables 2.1 and 2.,2. Each table i s divided i n half:, the upper h a l f records information for the observed p a r t i c l e s and the lower h a l f shows a possible scheme for completing the balance sheet. Consider table 2.1 for carbon. The stated errors of the m u l t i p l i c i t i e s and the average energy per pion stop are 10% and 15 to 20% respec t i v e l y including systematic errors (24). The t o t a l k i n e t i c energy for observed p a r t i c l e s i s 103.5 MeV including neutrons and 27.5 MeV without them. 4.79 neutrons and 2.88 protons are accounted f o r . I f the missing nucleons are assigned to unobserved r e c o i l n u c l e i , then i t i s possible to check whether or not the t o t a l energy sums to the pion r e s t mass. Several of such r e c o i l s have been measured, but since only an estimate i s required, 9 10 then the r e c o i l s were taken to be Be and B. The former i s required to provide a s l i g h t excess of neutrons over protons and the l a t t e r i s expected to be prominent on the basis of capture on a quasideuteron ^ C (TT_, nn) ̂ B. If i t i s assumed that a r e c o i l nucleus c a r r i e s 3 MeV (25) then the unobserved k i n e t i c energy i s 1.32 MeV. If the amount of gamma ray energy i s taken to be 2 MeV for r a d i a t i v e capture and 3 MeV for nuclear e x c i t a t i o n , then 12 K i n e t i c Energy + Gamma Rays + Binding Energy of C - Residual Binding Energy i s 104.77 + 5 + 92.162 - 55.213 giving 146.7 MeV. This i s s l i g h t l y l a r ger than the pion rest mass energy of 139.6 MeV, but i t i s within the l i m i t s of the error. 60 80 100 E N E R G Y (MeV) Figure 2.1: Secondary Spectra for Pion Absorption on Carbon 0 2 0 4 0 6 0 8 0 100 120 140 E N E R G Y (MeV) Figure 2.2: Secondary Spectra f or Pion Absorption on Oxygen oo Now consider table 2.2 for oxygen. The t o t a l observed k i n e t i c energy i s 91.0 MeV including neutrons and 12.0 MeV excluding them. The number of observed p a r t i c l e s accounts for 3,74 neutrons and 1.22 protons. The missing nucleons can be accounted for by assigning them to the r e c o i l 14 n u c l e i . Again, the assignment i s somewhat a r b i t r a r y . N i s expected from capture on a quasideuteron, ^ 0 (TT , nn) ^ N and from capture on an alpha c l u s t e r 0 (TT , nt) C. A study of prompt nuclear X-rays following pion capture (29) has shown that these r e c o i l s predominate, that the t o t a l y i e l d of carbon and nitrogen isotopes i s roughly equal, and that the average r e c o i l energy i s 2.2 MeV for carbon and 0.6 MeV for nitrogen. Since there i s an excess of missing protons, ^Be i s also included. Then, K i n e t i c Energy + Gamma: Rays + Binding Energy of ^ 0 - Residual Binding Energy i s 93.36.+ 5 + 127.621 - 85.722 giving 140.3 MeV, which i s very close to the pion rest mass. Table 2.3 i s a summary to compare carbon and oxygen data. The amount of k i n e t i c energy c a r r i e d away by charged p a r t i c l e s f o r oxygen i s about h a l f that f or carbon. While the t o t a l energy balance i s confirmatory, i t i s not a d e f i n i t i v e test because the energy going into charged p a r t i c l e s i s only a small f r a c t i o n of the t o t a l . Although q u a n t i t a t i v e l y i n c o r r e c t , Guthrie's c a l c u l a t i o n (14) predicted a lower y i e l d of charged secondaries from oxygen than from carbon. The oxygen results<used here have been extracted from preliminary data (26, 30), however, the lower energy into charged p a r t i c l e s from oxygen i s p l a u s i b l e and i s expected to be of t h i s magnitude when the measurements.have been confirmed. The difference appears to be due p a r t l y to the energy going into neutrons and p a r t l y to changes i n the binding energy. Further implications for the kerma factor w i l l be discussed i n chapter 3. Table 2.1: P a r t i c l e and Energy Balance Sheet for Carbon P a r t i c l e Total Number Tot a l K i n e t i c Number of Number of of P a r t i c l e s Energy (MeV) Neutrons Protons Binding Energy Total Binding (MeV) Energy (MeV) n P d He He 6 > 7 L i Totals Missing 2.1 .452 .329 .222 .041 .649 . 165 76 10.51 6.38 3.07 0.64 5.51 1.34 103.45 :t>4 2. 1 .329 .444 .041 1.298 .578 4.790 2.210 .452 .329 .222 .082 1.298 .495 2.878 2.122 0 0 2.225 8.482 7.718 28.296 35.620 0 0 .732 1.883 .316 18.364 5.877 27.172 Be r°B Adjusted Totals .088 .354 .26 1.06 104.77 ? OO .440 1.770 .352 1. 770 58.165 64.751 5. 119 22.922 55.213 Table 2.2: P a r t i c l e and Energy Balance Sheet f or Oxygen P a r t i c l e T o t a l Number Tot a l K i n e t i c of P a r t i c l e s Energy (MeV) n P d He He 6 ' 7 L i T o tal Missing 2.7 .292 .167 .125 .012 .304 .002 79 5.24" 2.39 1.52 .20 2.61 .03. 90.99 Number of Neutrons 2.7 Number of Protons Binding Energy T o t a l Binding (MeV) Energy (MeV) . 167 .250 .012 .608 .007 3.744 5.256 .292 .167 . 125 .024 .608 .006 1.222 5.778 0 0 2.225 8.482 7.718 28.296 35.620 0 0 . .-372 1.060 .093 8.602 .071 10.198 Be 12. 14 N Adjusted Totals .522 .284 .284 1.57 .63 . 17_ 93.36 CN 1.566 1.704 1.988 2.088 1.704 1.988 37.600 92.163 104.660 19.627 26.174 29.723 85.722 Table 2.3: Comparison of Carbon and Oxygen Energy D i s t r i b u t i o n s (MeV) Carbon Oxygen K i n e t i c Energy Neutrons 76 ± 8 79 ± 8 K i n e t i c Energy Charged P a r t i c l e s (observed) K i n e t i c Energy Charged P a r t i c l e s (unobserved) 27.5~S 1.3 28.8 ± 3 12.0 2.4 14.4 ± Gamma Ray Energy 5 ± 3 5 ± 3 Change i n Binding Energy 37 ± 5 42 ± 5 Tot a l 147 ± 10 140 ± 11 2.3.1.2 Fermi-Teller Z-law When pions are captured i n a medium which contains both carbon and oxygen, the charged p a r t i c l e spectra w i l l be given by the sum of the spectra for the i n d i v i d u a l elements weighted:according; to the nuclear capture p r o b a b i l i t y . Fermi and T e l l e r (31) predicted that for a m e t a l l i c compound of the form AnBm, the capture p r o b a b i l i t y r a t i o into the atom A compared to that into the atom B i s nZ^/mZ^. That i s , the capture p r o b a b i l i t y per atom i s proportional to Z. For muons, experimental (32) and t h e o r e t i c a l (33) work has revealed that the capture p r o b a b i l i t y i s not l i n e a r i n Z, but i s proportional to Z n where 0.55 < n < 1.41 and n varies with p o s i t i o n i n the p e r i o d i c table. Recent work at LAMPF (34) using muonic X-rays has shown that a modified Z-law which d i s t r i b u t e s the muons among the atoms (including hydrogen) according to the Z-law and then transfers the muon from hydrogen to the nearest heavy atom gives better agreement with experiment than the Z-law alone. Other than some Russian (35) t h e o r e t i c a l and experimental i n v e s t i g a - tions of pion absorption i n hydrogenous compounds and some early work (36, 37), there has not been as much study of p i o n i c capture as muonic capture. E f f e c t s i n pion capture at the 10 to 20% l e v e l due to d i f f e r e n t chemical and p h y s i c a l composition have been observed (38, 39). Generally these condensed state e f f e c t s seem l e s s prominent with pions than with muons. The Karlsruhe group (26) has found for pion capture on carbon and oxygen i n mylar and acetate that the nuclear capture r a t i o as predicted by chemical analysis and the Z-law agreed within 10% with the r a t i o as measured by the detection of p i o n i c X-rays. 34 Based on t h i s information, the secondary spectra f o r capture i n compounds were evaluated using the Z-law. For hydrogenous compounds i t was assumed that a l l of the pions i n i t i a l l y captured on hydrogen were transferred to adjacent heavy atoms. 2.3.1.3 Pion stopping density In section 2.2 i t was pointed out that the pion stars were taken to be uniformly d i s t r i b u t e d throughout the i n f i n i t e phantom and the cavity material with stopping densities II and II'1 (pion/g) r e s p e c t i v e l y . In general II £ II 1 i f the stopping powers of the phantom and cavity are d i f f e r e n t . Consider a mono-directional stopping pion beam normally incident on a slab-shaped cavity containing an ar e a l density (p g/cm2) of w a l l i n one case and the same are a l density of gas i n the other case. Assume that the pion energy fluence has been selected to provide a uniform stopping density of II pions/g i n the walls and w a l l material f i l l i n g the cavity. When the cavity i s f i l l e d with gas, consider a pion with the correct energy to j u s t cross the gas. Rg(Eg) = p g/cm2. A pion with the same energy EQ w i l l have a range i n w a l l material of.P^CEp). The two ranges are d i f f e r e n t i f the stopping powers are d i f f e r e n t . The number of pions per unit area with energy les s than EQ i s II R ^ E Q ) . Therefore the number of pions per unit area stopping i n the gas i s also II R ^ E Q ) , and hence the average pion stopping density i n the gas i s n 1 = n W / R g ( V 2'22 I f the gas has a higher stopping power than the w a l l , then the pion range w i l l be smaller and the stopping density w i l l be increased. Notice that EQ depends upon the chamber s i z e and gas density. I f the r a t i o of the ranges i s not independent of EQ then the stopping density w i l l be pressure dependent. In order to evaluate t h i s e f f e c t , i t i s necessary to know the range of low energy pions. Since no experimental data were a v a i l a b l e , range versus energy curves for pions were determined by in t e g r a t i n g pion stopping powers derived by s c a l i n g proton data (section 2.3.2). Figure 2.3 shows the r a t i o of the pion ranges f o r carbon to methane and for carbon to carbon dioxide as a function of areal density. The dose deposited by i n s i d e r s and s t a r t e r s needs to be m u l t i p l i e d by this r a t i o . For CO^, the r a t i o i s s l i g h t l y l e s s than unity and changes by only a few percent over the range of pressures used experimentally. For CH^, the r a t i o i s about 1.5 and varies by 20%. Due to the uncertainty i n the value of t h i s r a t i o , the c a l c u l a t i o n s outlined i n section 2.2 were performed with II = IT1 and t h i s e f f e c t was subsequently included as a separate step. The uncertainty arises because of the proton s c a l i n g procedure used to determine the range and- because of the f a i l u r e to include the e f f e c t s of multiple s c a t t e r . For example, the s c a l i n g procedure would predict equal ranges for p o s i t i v e and negative pions, whereas experimentally they d i f f e r by 3% at 1.6 MeV (40). The i n c l u s i o n of multiple s c a t t e r would increase the e f f e c t i v e s i z e of the chamber, thus causing a s h i f t up the a r e a l density axis which would make the magnitude of the r a t i o s closer to the stopping power r a t i o . 10 100 RANGE in GAS (mg/cm 8) H N i-o 33 5 O o ro -̂ 0 9 1000 Figure 2.3: Range Ratio for Carbon/Methane and Carbon/Carbon Dioxide 3 8 2.3.2 Stopping Power Data 2.3.2.1 Stopping power for H and He ions i n the elements H, C and 0 For H and He ions stopping i n the elements H, C and 0, the a n a l y t i c a l f i t s to the experimental data as derived by Andersen and Ziegler- (41, 42) have been used. The stopping power versus energy curve behaves d i f f e r e n t l y i n each of three regions. At low energy,, the FermitThomas p o t e n t i a l has been used to pr e d i c t stopping power proportional to v e l o c i t y . At high energy, Bethe theory with appropriate values f o r the i o n i z a t i o n p o t e n t i a l and s h e l l corrections i s found to be app l i c a b l e . In the i n t e r - mediate energy region, where the stopping power passes through a peak, empirical f i t s have been made to the data. There are discrepancies of the order of 10% between experimental measurements made within the l a s t decade, p a r t i c u l a r l y i n the intermediate energy range. The approach of Andersen and Ziegl e r i m p l i c i t l y denies the a b i l i t y of any one experiment to y i e l d a value closer to the true value than any other experiment. In taking an average over various experimental determinations any/hidden systematic errors between these experiments ought to cancel out. If several independent experiments f i n d agreement on a new value d i f f e r e n t from the present Andersen and Zi e g l e r f i t , then i t would be p r a c t i c a l to modify the constants to f i t the newly accepted value. Before adopting t h i s approach, a l l of the references c i t e d in:the Andersen and Zi e g l e r bibliography (43) and new data (44, 45, 8 7 ) not ava i l a b l e to them were pl o t t e d and compared to t h e i r f i t t e d values. Repre- sentative carbon data i s shown i n figures 2.4 and 2.5. Most of the discrepancy i s i n the region of the peak where the maximum deviation i s 8 % . This discrepancy does not appear to be excessive, and there seems to be no reason to prefer one set of experimental r e s u l t s over the others. Therefore Figure 2.4: Stopping Cross Section for Protons i n Carbon Johansen, Steenstrup and Wohlenberg (46) Bernstein, Cole and Wax (47) (indistinguishable from the Andersen-Ziegler f i t at energies less than 20 keV) O Arkhipov and Gott (48) X Sautter and Zimmerman (49) V Gorodetsky et a l (50) -|- Van Wijngaarden and Duckworth (51) 0 Ormrod, MacDonald and Duckworth (52) A Moorhead (53) T K Fastrup, Hvelplund and Sautter (54) STOPPING CROSS SECTION (lO-l5eV-cm2/atom) o o ro A o> co o ro * o> 0̂ Figure 2.5: Stopping Cross Section f o r Alphas i n Carbon \ Santry and Werner (44) Santry and Werner (45) -|- Sautter and Zimmerman (49) ^fc Van Wijngaarden and Duckworth (51) 0 Moorhead (53) O Porat and Ramavataram (56) /^V Matteson, Chau and Powers (57) [graphite, vapour deposition] X Chu and Powers (58) Br Ormrod and Duckworth (59) 4>- ho the f i t s of Andersen and Zieg l e r were accepted as r e l i a b l e summaries of the experimental stopping powers. The e l a s t i c nuclear s c a t t e r i n g contribution to the stopping power i s small and was not included i n these c a l c u l a t i o n s . In section 1.2.2 i t was pointed out that Thwaites (15) has shown that the stopping power r a t i o f o r vapour to s o l i d i s not unity at low energies. In f a c t , i t may be as large as 1.25 f o r 1 MeV alpha p a r t i c l e s i n hydrocarbons. The Andersen and Ziegler tables do consider such condensed state e f f e c t s f o r alphas by tabulating separate constants f o r gas and s o l i d phases and their'lvalues f o r carbon compare very w e l l with those of Thwaites (15). For protons, Andersen and Z i e g l e r i n d i c a t e the •., r a t i o i s unity, whereas Thwaites indicates a maximum value of 1.1 between 0 and 2 MeV. 2.3.2.2 Bragg a d d i t i v i t y rule The Bragg a d d i t i v i t y rule (64) states that the stopping power of a medium i s the sum of the stopping powers of i t s atomic components and i s independent of both the chemical r e l a t i o n s h i p s between i t s constituents and the p h y s i c a l state of the medium. This r u l e describes (65, 66, 67, 68, 69) a l i m i t e d number of compounds, ions and energy regions, but i t has not been found to be u n i v e r s a l l y valido(70, 71, 72, 73, 74, 75). The" review by Thwaites (15) includes a d d i t i o n a l references to the condensed state e f f e c t s . There i s i n s u f f i c i e n t information i n these references to decide i f the Bragg rule i s v a l i d f o r methane and carbon dioxide. Hence, the stopping power i n H, C and 0 was-calculated (as i n 2.3.2.1), combined according to the Bragg r u l e and compared with the a v a i l a b l e experimental data i n figures 2.6 through 2.9. The f i t i s excellent except for the case of protons i n carbon dioxide where the Bragg r u l e predicts a value 8% higher at the peak than observed by the one a v a i l a b l e experiment. Such agreement j u s t i f i e s the use of the Andersen and Z e i g l e r values and the Bragg r u l e for these two gases, but i t does not v a l i d a t e the rule f o r general use. Figure 2.6: Stopping Cross Section for Protons i n CO O Swint, P r i o r and Ramirez (77) D B r o l l e y and Ribe (78) A P h i l l i p s (79)  • Figure 2.7: Stopping Cross Section for Protons i n CH A Park and Zimmerman (71) X Reynolds et a l (75) O Swint, P r i o r and Ramirez (77) 0 Brolley and Ribe (78) Huges (82) (omitted) -f Sidenius (89)  Figure 2.8: Stopping Cross Section for Alphas i n CO A Bourland, Chu and Powers (83) X Kerr et a l (84) O Rotondi (85) STOPPING CROSS SECTION (lO-"eV-cm2/molecule) 05 Figure 2.9: Stopping Cross Section for Alphas i n CH D Park (68) A Bourland, Chu and Powers (83) X Kerr et a l (84) O Rotondi (85) Whillock and Edwards (87) + Sidenius (89) z^ 53 2.3.2.3 Stopping power for l i t h i u m The experimental data for stopping '-ilthium ions i s very sparse. In carbon there was no data a v a i l a b l e above 150 keV, although recent data by Santry (45) now extends to the peak region. The a n a l y t i c a l f i t f o r carbon to be published (92) ( f a l l of 1979) by Zie g l e r does not extend below 1.4 MeV where the ion becomes p a r t i a l l y charged. In carbon dioxide there i s one experiment which covers the peak (88). In methane an experiment (89) i s a v a i l a b l e below 150 keV and two other experiments (88, 90) i n the peak region disagreed by 25%. A curve of the square of the e f f e c t i v e charge as a func- ti o n of v e l o c i t y was generated by comparing the stopping power for l i t h i u m i n a given material to the stopping power for protons of the same v e l o c i t y as determined by Andersen and Ziegler- arid the Bragg r u l e : dE . _ dx" L l ' vl» G 7.2 = ' 2.23 (G ref e r s to a p a r t i c u l a r compound.) This curve i s shown i n fi g u r e 2.10. Except for minor departures by hydrogen, a l l the gases f a l l on the same curve. At low v e l o c i t y , the a v a i l a b l e data for carbon ind i c a t e d a larger value of Z 2 than for gases. Since there was no carbon data a v a i l a b l e for v > V Q , where VQ i s the Bohr v e l o c i t y = c/137, stopping power data for aluminum were used as well.as measurements of Z 2 (96, 97) for L i ions leaving carbon f o i l s . At a p a r t i c u l a r v e l o c i t y , the former technique gave Z 2 higher than for gases, and the l a t t e r technique gave Z 2 lower. Hence, for v > v^ Z 2 for carbon was taken to be the same as for gases. Stopping powers, based on the e f f e c t i v e charge and proton stopping powers, for the l i t h i u m ions i n carbon, methane and carbon dioxide are shown i n figures 2.11, 2.12 and 2.13. -1 1 1 I 1 — — — - - - CARBON — 1 GASES • i 1 I 10 100 ION VELOCITY/ BOHR VELOCITY Figure 2.10: Square of the E f f e c t i v e Charge for Lithium Ions Figure 2.11: Stopping Cross Section for L i Ions i n Carbon X Santry and Werner (45) -f- Ormrod and Duckworth (59) A Zieg l e r (92) O Hogberg (93) • Pivovar, Nikolaichuk and Rashkovan (94) D Bernhard et a l (95) 95 Figure 2.12: Stopping Cross Section for L i Ions i n CO • A l l i s o n , Anton and Morrison (88)  Figure 2.13: Stopping Cross Section for L i Ions i n CH A l l i s o n , Anton and Morrison (88) X Sidenius (89) - f Teplova et a l (90) I 61 2.3.3 W-Value Data The W-value i s defined by W = E/N 2.24 where N i s the t o t a l number of ions created by completely stopping an ion of energy E i n the gas. As the v e l o c i t y of a heavy ion decreases, the W-value increases, going to i n f i n i t y at the f i r s t i o n i z a t i o n p o t e n t i a l . Based on MacDonald and Sidenius' systematic i n v e s t i g a t i o n (98) of i o n i z a t i o n i n methane for various ions, Dennis (99) has derived an empirical equation to determine the W-value as a function of energy: W(E) = WD 1.035 + A(Z) ( E / M ) ~ n ( Z ) 2.25 P where W i s the W-value f o r f a s t e l ectrons, M i s the ion mass i n amu and p E i s the energy i n keV. The value of n(Z) i s roughly 0.5 so W(E)/W2 i s P l i n e a r i n 1/v (for v < VQ/2). For ions having E > 1000 keV/amu, a d i f f e r e n t equation i s used: ««> " we TETS^bT 2 - 2 6 where b i s chosen to normalize to equation 2.25 at E/M = 1000 keV/amu. -The constants used i n th i s c a l c u l a t i o n are given i n table 2.4. The Dennis equations are i n common use (100, 101, 102, 103). A more recent a n a l y t i c a l f i t based on the same data set (105) removes the dis c o n t i n u i t y i n slope at 1000 keV/amu, but does not change the. values s i g n i f i c a n t l y (102). A comparison of the f i t t e d curve and some experimental data (107, 108, 109, 110) for alpha p a r t i c l e s i n methane i s shown i n figure 2.14. The 6 to 10% discrepancy at low energy between the various experiments i s also t y p i c a l of the agreement obtained with other gases,(109, 111, 112). Table 2.4: Constants Used i n the Dennis Formula f or W-Values Ion Mass (113) (amu) n(Z) A(Z) CH. A(Z) CO. proton alpha l i t h i u m 1.00783 4.00260 7.01600 .04902 .5360 .7630 1301 1.810 3.219 .0814 1.1331 2.0151 W„ = 27.3 for CH. (114) 6 4 Wc = 32.9 for C0„ (114) Figure 2.14: W-Value f o r Alpha P a r t i c l e s i n CH^ O Varma and Baum (107) -4- Bortner and Hurst (108) X. Kuhn and Werba (109) A Jesse (110) So l i d l i n e i s the f i t . of Dennis (9.1) to the data of MacDonald and Sidenius (98) 60 I I I I I I I I I I 001 01 10 10 ENERGY (MeV) 65 2.4 RESULTS OF THE CALCULATION The c a l c u l a t i o n that has been described i n section 2.2 has been performed for a carbon cavity of 2 cm diameter f i l l e d with e i t h e r methane or carbon dioxide at various pressures and 22°C. Appendix B contains a summary of the quantities calculated and the equations used. S e l f - c o n s i s - tency checks b u i l t into the computer program i n d i c a t e that systematic errors i n the computation are less than 0.5%. (2.4.1 P a r t i t i o n i n g the Dose. Between the Various Event Types Figure 2.15 i l l u s t r a t e s how the dose i s p a r t i t i o n e d between the various event types (stoppers, crossers, i n s i d e r s and s t a r t e r s ) f o r l i t h i u m ions released by pion capture i n carbon. At very low pressure, a l l of the dose i s due to crossers o r i g i n a t i n g from pion capture i n the w a l l s . As the gas pressure i s increased, some of the crossers have i n s u f f i c i e n t energy to traverse the cavity and thus become stoppers. At t h i s pressure, there are also a s i g n i f i c a n t number of pion stars which occur i n the gas and whose L i ions have s u f f i c i e n t energy to escape ( s t a r t e r s ) . As the pressure i s further increased, the number of gas stars continues to increase, but ions that once were s t a r t e r s now have i n s u f f i c i e n t energy to escape and become i n s i d e r s . At very high pressures, there w i l l be no crossers and only the cavi t y surface w i l l have stoppers and s t a r t e r s present. The p a r t i t i o n i n g of the dose for l i t h i u m exhibits the most v a r i a t i o n i n the range of chord lengths that are of i n t e r e s t because i t s mean range i s of the order of 1 to 10 mg/cm2. For protons, crossers deposit most of the dose even at the largest chord lengths investigated because the mean range of the protons i s 10 3 - 10 4 mg/cm2. As shown i n table 2.5, 70% of the t o t a l kerma i s due to the Z = 1 secondaries. Therefore fi g u r e 2.16, which shows the p a r t i t i o n i n g 66 of the t o t a l dose among the four event types, shows les s v a r i a t i o n than f o r lithium. I t i s important to note that f o r chambers i n common use (a 2 cm diameter sphere f i l l e d with a i r at one atmosphere has a mean chord length of 1.6 mg/cm2) only 90% of the star dose r e s u l t s from stars i n the w a l l with secondaries having s u f f i c i e n t energy to be crossers. Table 2.5: Average Energy and Kerma According to P a r t i c l e Type (Carbon) P a r t i c l e P d He He L i Production Spectrum (MeV) 23.25 19.39 13.97 15.66 8.49 8.08 Slowing-down Spectrum (MeV) 34.81 30.46 22.60 13.13 11.02 6.33 Energy per Pion Stop (MeV/pion) 10.50 6.38 3.07 .64 5.51 1.34 Fra c t i o n of the Tota l Kerma .383 .233 .112 .023 .201 .049 Figure 2.15: P a r t i t i o n i n g the Dose Among the Event Types for L i Ions cn 1.0, L U CO o o •J < 0.8 0.6 c r o s s e r s • s t o p p e r s • s t a r t e r s * i n s i d e r s > o a 4 1 < a: L L . 0.2| 0.2 All Secondaries 0.5 CHORD LENGTH ( m g / c m 2 ) Figure 2.16: P a r t i t i o n i n g the Dose Among the Event Types for A l l Secondaries 00 69 2.4.2 W and r for CH. and C0„ i n Carbon Walls 4 I If the quotient of the average W-value divided by the fast e l ectron W-value i s defined as u = Wg/We , then equation 1.1 can be rewritten as J 1 1 D w a l l o o-7 — = — x — x — 2.27 M u r W o P Since P^-^ a n d are constants, J/M i s proportional to (ur) ̂ . Table 2.6 gives the calculated values of u, r and (ur) * as a function of pressure for the 2 cm diameter carbon sphere f i l l e d with methane gas at 22°C. Table 2.6: Values of W and r for C-CH (2 cm diameter sphere) Pressure u = Wg/WR r (ur) (100 kPa) P 0 1.0135 .7384 1.3362 0.25 1.0139 .7357 1.3406 0.51 1.0143 .7337 1.3437 1.01 1.0149 .7329 1.3444 2.03 1.0152 .7358 1.3387 4.05 1.0155 .7409 1.3291 8.10 1.0156 .7494 1.3138 For convenience, the c a l c u l a t i o n has been done for a s p h e r i c a l geometry. In order to s i m p l i f y the chamber construction, i t was made of p a r a l l e l plate design. Therefore, the r e s u l t s of the c a l c u l a t i o n f o r the sphere have been compared with the p a r a l l e l p late geometry at equivalent average chord lengths. For a convex body exposed to a uniform i s o t r o p i c f i e l d of s t r a i g h t i n f i n i t e tracks, the mean chord length i s given by the Cauchy theorem (115, 116) to be I = 4V/S 2.28 For the sphere, I = D, where D i s the diameter; and f o r the c y l i n d e r , t = i f r / R » w n e r e h i s the height and R the radius of the c y l i n d e r . The pressure of the sphere, P s, whose 't i s the same as for the cylinder at pressure, P^, i s given by 3h (1 + h/R) D P = — p.„ i < 3 ft i 1_ / -n S T \ X C J For the p a r t i c u l a r experimental geometry which used a cylinder of height 0.614 cm and radius 1.024 cm Ps = 0.576 P c The r e s u l t s of the c a l c u l a t i o n that have been tabulated i n table 2.6 for a 2.0 cm sphere are pl o t t e d i n fig u r e 2.17 as a function of pressure i n an equivalent p a r a l l e l plate chamber. As was discussed i n section 2.3.1.3, there i s a further correction r e l a t e d to the stopping density of pions that must be applied to the i n s i d e r s and s t a r t e r s which i s shown i n fig u r e 2.3. The values of (ur) ^ inc l u d i n g the pion stopping density correction are also shown i n f i g u r e 2.17. The c a l c u l a t i o n was repeated f o r a carbon sphere containing carbon dioxide gas arid the r e s u l t s are displayed i n table 2.7. 71 Table 2.7: Values of W and r for C-C09 (2 cm diameter sphere) Pressure u = Wg/W r (ur) (100 kPa) B 0 1.0129 1.0346 .9542 0.25 1.0129 1.0407 .9486 0.51 1.0129 1.0470 .9429 1.01 1.0127 1.0587 .9327 2.03 1.0124 1.0760 .9180 4.05 1.0120 1.0994 .8988 8.10 1.0116 1.1260 .8780 The uncorrected and density corrected values of (ur) as a function of pressure i n an equivalent p l a t e chamber are p l o t t e d i n figu r e 2.18. The pressure dependence of (ur) * for these gases i s quite d i f f e r e n t . For methane, there i s a 0.6% increase between 0 and 1 atmospheres. Since the W-value i s independent of pressure (= 0.1%), these changes are due to v a r i a t i o n i n the value of r . Further, since the c a l c u l a t i o n treats methane as a pure carbon gas f o r the purpose of pion capture, these changes i n r r e f l e c t the conversion of the secondary spectra from the "equilibrium" spectrum characterizing the crossers to the "production" spectrum character- i z i n g the i n s i d e r s . Carbon dioxide, on the other hand, ex h i b i t s a rapid and continuous decrease of 6^% between 0 and 8 atmospheres. In this case, the conversion from the wa l l "slowing down" spectrum to the gas "production" spectrum i s dramatic because of the rather low kerma for carbon dioxide (16.3 MeV/pion due to i t s oxygen content). In other words, as the pressure increases, a larger f r a c t i o n of the captures occur i n the gas, but since the energy released to charged secondaries per capture i s smaller, the dose decreases. If we consider the dependence of (ur) ^ on pressure a f t e r correcting f o r the perturbation i n pion stopping density, the picture changes. The hydrogen i n methane increases i t s stopping power and therefore increases the density of pion stops i n the gas. This e f f e c t i s large and masks out changes i n (ur) ^ due to changes i n the secondary energy spectra. There i s a net increase i n (ur) ^ of 2.3% between 0 and 8 atmospheres. On the other hand, the density c o r r e c t i o n for CO2 i s less than unity and i t serves to s l i g h t l y enhance the decrease i n (ur) ^ with pressure. There i s a net decrease i n (ur) ^ of 7% between 0 and 8 atmospheres.   75 2.5 DOSE DEPOSITED BY OTHER PARTICLES In t h i s s e c t i o n , the expected pressure dependence of the stopping power r a t i o f o r electrons and passing pions i s c a l c u l a t e d and that f o r neutrons i s discussed. Relative to the length of time and e f f o r t spent i n v e s t i g a t i n g the pion s t a r dose, very l i t t l e was spent i n v e s t i g a t i n g the dose deposited by other p a r t i c l e s i n the beam. The reason for t h i s was pointed out i n section 2.1: i n v e s t i g a t i n g the pion star dose was i n t e r e s t i n g because of i t s uniqueness and i t s possible contribution to a deeper under- standing of cavity theory. The dose from the other p a r t i c l e s i s discussed here to ind i c a t e what tendencies might be observed f o r an experimental chamber necessa r i l y exposed to a l l the beam components. 2.5.1 Electrons and Muons For a l l charged p r o j e c t i l e s moving at r e l a t i v i s t i c v e l o c i t i e s , there i s a density e f f e c t which reduces the mass stopping power for high d e n s i t i e s . In appendix D t h i s Sternheimer density correction has been calculated i n order to determine stopping power for electrons i n carbon dioxide, methane, and carbon at two k i n e t i c energies: 150 MeV and 130 MeV, an average r e s i d u a l electron energy i n the plateau and peak re s p e c t i v e l y (assuming an incident midline momentum of 180 MeV/c). The stopping power r a t i o f o r gas to w a l l i s pl o t t e d i n figure 2.19 as a function of pressure. For both gases the stopping power r a t i o shows a 10% decrease between 0 and 8 atmospheres. Therefore, that part of the i o n i z a t i o n chamber response (J/M) due to the e l e c t r o n contamination will-decrease by 10% (assuming that the electron W-value i s pressure indepen- dent) . There i s also a pressure dependent density e f f e c t f o r the muon 76 Figure 2.19: Electron Stopping Power Ratio Versus Pressure 77 contamination which i s extinguished at depths greater than 5 g/cm2 because the muon r e s i d u a l k i n e t i c energy has f a l l e n below the energy threshold of 0.874 MyC2 = 92.5 MeV (118). 2.5.2. D i r e c t l y Ionizing Pions Next, only the dose deposited by the d i r e c t i n t e r a c t i o n between the pions and the atomic electrons i s considered. The t o t a l dose i n a cavity of a r e a l density pL exposed to a slowing-down spectrum of pions given by Nr(E) as measured at the cavity i s 'E r E s t° P E Nr(E) dE , C J o P L J - _ \ u U C + \ m A Nr(E) ^ d i r e c t \ pL ^ PL E t stop The f i r s t term i s due to pions which stop i n the cavity. E ^ i s defined v * stop i m p l i c i t l y by R ( E g t o p ) = pL where R i s the range. The second term i s the dose due to passing pions. E i s the maximum energy a v a i l a b l e and governs max the width of the stopping peak. The energy deposited by a passer i s A given i m p l i c i t l y by R(E) = R(E - A) + PL. The i n t e g r a l f or passers may be subdivided at energy Ea such that for E > Ea, the s u b s t i t u t i o n A = S(E) • pL i s v a l i d where S(E) i s the stopping power. This gives E a f*Emax D.. , = \ ~ -•: — dE + \ A N N 5 ( E ) dE + \ S(E)Nr(E)dE d i r e c t I p L \ PL \ E J Ea stop The value of E (and hence the r e l a t i v e importance of each i n t e g r a l ) w i l l stop depend upon the are a l density. For the extreme case of (X>2 i n a carbon walled chamber of L = .614 cm and P = 8 atmospheres, E ^_ = 0.92 MeV, r ' stop R ( E g t o p ) = 0.009 g/cm2, Ea = 2.25 MeV and R(Ea) = 0.045 g/cm2. Therefore, at a l l positions of the chamber except at the d i s t a l edge of the stopping d i s t r i b u t i o n , the dose i s dominated by S(E) Nr(E) dE 5 E max Ea and the r a t i o of dose i n w a l l to dose i n gas i s given by the stopping power r a t i o . For example, consider a chamber located i n the plateau at l e a s t a distance R(Ea) (= 0.05 g/cm2) upstream from the leading edge of the pion stopping region. Then Nr(E) = 0 f o r E < E n = Ea and hence, >E 5 max E . mm S C(E) Nr(E) dE s E max S c o (E) Nr(E) dE E . 2 mm Further, sc ( E> s c o 2 ( E ) 1.03 ± 0.005 for E > Ea, and therefore r i s i d e n t i c a l to the stopping power r a t i o independent of the pion energy spectrum, the ar e a l density of the chamber or i t s l o c a t i o n i n the plateau. Next consider the chamber to be located near the dose peak, say 2.5 g/cm2 i n front of the d i s t a l edge of the stopping peak. Then E = 2 2 . 5 MeV and the r e l a t i v e importance of the three i n t e g r a l s i s max r 4%, 6% and 90% respectively. Even though the contribution to the dose from the f i r s t and second i n t e g r a l has started to become s i g n i f i c a n t , the r a t i o of the dose i n carbon to dose i n carbon dioxide does not change from 1.03. The corresponding value f or methane i s 0.74. In conclusion, the r a t i o of dose i n carbon to dose i n gas i s not expected to be s i g n i f i c a n t l y d i f f e r e n t from the stopping power r a t i o . 2.5.3 Neutrons and Gamma.Rays Dose deposited by neutrons a r i s i n g from pion stars i s not l o c a l i z e d and therefore the f r a c t i o n of the dose due to neutrons i s f i e l d s i z e dependent. On the basis of references (119, 120) the neutron dose at the peak f or the f i e l d s i z e used here i s approximately 10% of the charged p a r t i c l e dose (not incl u d i n g electron contamination). The e f f e c t of the neutrons i s to knock on protons and to disrupt n u c l e i i n much the same way as i s done by pion s t a r s . Therefore, as a rough estimate, i t i s assumed that the absolute value and the pressure dependence of ( u r ) - 1 f o r the neutron dose w i l l be the same as that observed f o r the pion s t a r charged p a r t i c l e dose. The gamma ray dose i s one to two orders of magnitude smaller than the neutron dose (119) and has therefore been neglected. 2.6 SUMMARY In t h i s chapter i t was shown that deviations from the simple Bragg- Gray law are expected. That i s , the i o n i z a t i o n per unit mass i s a function of pressure i n the region where the pions stop. S p e c i f i c a l l y , the i o n i z a t i o n per unit mass for carbon dioxide i s expected to decrease rather dramatically as the pressure i s increased, whereas for methane i t i s not expected to change very much. This i s p r i m a r i l y due to the lower y i e l d of charged secondaries from oxygen compared to carbon which becomes evident at higher pressures since more pions are i n t e r a c t i n g d i r e c t l y with the chamber gas. Because t h i s phenomenon i s oxygen s p e c i f i c , a l l oxygen bearing species are expected to e x h i b i t the same decrease i n i o n i z a t i o n per unit mass as the pressure i s increased. The perturbation i n stopping pion density caused by the difference i n stopping power augments the difference i n behavior between methane and carbon dioxide. I t was also shown i n t h i s chapter that the electron contamination present i n a pion beam produces i o n i z a t i o n which w i l l e x h i b i t a pressure dependence due to the Sternheimer density e f f e c t . Since electrons are present i n both the peak and the plateau, :.a decrease i n i o n i z a t i o n per unit mass i s expected for a l l gases i n both the peak and plateau. The decrease i n the t o t a l i o n i z a t i o n per unit mass for oxygen bearing gases ought to be larger i n the peak than i n the plateau since both phenomena are present there. CHAPTER 3 EXPERIMENT 3.1 INTRODUCTION In order to determine the dose using an i o n i z a t i o n chamber, i t i s necessary to measure the i o n i z a t i o n i n a cavity which contains a known mass of s u i t a b l e gas. The equipment and technique used i n making these measure- ments are described i n the f i r s t h a l f of t h i s chapter. This de s c r i p t i o n also includes documentation about the operation of the monitor chambers;'ofv the biomedical channel to which a l l dosimetry i s referenced. In chapter two i t was shown that the i o n i z a t i o n per unit mass i s expected to ex h i b i t a decrease as the pressure i s increased f o r carbon dioxide, but i t ought to remain f a i r l y constant f o r methane. Therefore, i n the second h a l f of t h i s chapter, the measured i o n i z a t i o n per unit mass as a function of pressure i s presented for these gases and several others. Good q u a l i t a t i v e agreement between these measurements and the c a l c u l a t i o n confirms the importance of the p h y s i c a l processes described i n chapter two. 3.2 APPARATUS Figure 3.1 i s a block diagram i n d i c a t i n g the various pieces of apparatus. 3.2.1 Beam Delivery System The experiments d i s c r i b e d here were performed on the M8 channel at TRIUMF (121, 122). The channel i s 8 m i n length and c o l l e c t s pions from the production target at 30° i n the forward d i r e c t i o n above the proton beam. Two 45° bending magnets are used to bring the pion beam h o r i z o n t a l l y into the i r r a d i a t i o n cave. The momentum blades located at the dispersed focus have an acceptance AP/p = 15% FWHM when f u l l y opened (122). Before entering the water i n the dosimetry tank, the beam traverses the end window of the vacuum pipe, the monitor chamber, s c i n t i l l a t o r 1, the multiwire counter, s c i n t i l l a t o r 2, and the front window-of the water tank. A l l of t h i s material amounts to an. equivalent thickness of 2.24 cm of water. 3.2.2 Beam Tune and Dose P r o f i l e s A 500 MeV proton beam at 10 to 12 yA was used to produce pions i n the 10 cm beryllium target at s t a t i o n T2. The channel tune //10 provided a midline momentum of 180 MeV/c. This p a r t i c u l a r target and momentum are used commonly. The electron contamination was measured at the entrance to the water tank by a t i m e - o f - f l i g h t technique (123, 124) and i t was found to be 28 ± 2% of the p a r t i c l e f l u x . The muon contamination was previously found to be 10% of the f l u x (123). The i o n i z a t i o n versus depth p r o f i l e for t h i s tune i s shown i n fig u r e 3.2. The data was measured using an aluminum walled p a r a l l e l p late ion chamber f i l l e d with methane gas at a pressure of 250 kPa. The chamber had a gap of 0.619 cm and a diameter of 2.049 cm. The depth axis was GAS SYSTEM SCINTILATORS 8 M WPC BEAM PORT MONITOR PROBE WATER TANK RESET ADC ^CONTROLLER |e THERMOMETER - 1 0 0 V C 0 M P U T E R £ TELETYPE SCALER Figure 3.1: Block Diagram of Apparatus  c a l i b r a t e d i n centimetres of water by comparing a stopping density d i s t r i b u t i o n of FWHM = 1 cm measured with a d i f f e r e n t i a l range telescope, whose p o s i t i o n had been c a l i b r a t e d by d i r e c t measurement, to the corres- ponding i o n i z a t i o n curve measured with a nominal 1 mm gap. Such a c a l i b r a t i o n procedure assumes that the i o n i z a t i o n and stopping curves are coincident and that the e f f e c t i v e centre of the chamber i s located r i g h t at the front surface of the p a r a l l e l plate chamber. This l a t t e r assumption i s v a l i d i n the plateau region where the r a d i a t i o n i s p r i m a r i l y normally-incident, but not i n the peak region where the pion star products are i s o t r o p i c . Here, the e f f e c t i v e centre of the chamber i s s h i f t e d towards the centre of mass of the gas ( 1 - 2 mm). The e f f e c t of t h i s s h i f t i s small enough that i t can be ignored. The positions of measurement i n the peak and plateau are indicated by arrows i n f i g u r e 3.2. Orthogonal cross scans measured at the peak are shown i n f i g u r e 3.3. It can be seen that the f u l l width at 80% i s 9.9 cm v e r t i c a l l y and 8.6 cm h o r i z o n t a l l y . A f i e l d which i s large compared with the 2 cm diameter c o l l e c t o r was chosen so that the dose which would o r d i n a r i l y be provided by secondaries from the side walls w i l l instead be replaced by dose from secondaries o r i g i n a t i n g i n the front and back face. The 5%% horn v i s i b l e on the h o r i z o n t a l scan i s due to second order aberrations i n the beam opt i c s . In order to extract the dependence of J/M on pressure for the star dose alone, i t i s necessary to separate the dose into i t s various components according to p a r t i c l e type: electron dose, muon dose, dose due to d i r e c t i o n i z a t i o n by pions, star dose, and dose due to i n - f l i g h t i n t e r a c t i o n s . The separation for t h i s tune i s shown i n f i g u r e 3.4. Consider the plateau region f i r s t . The percentage dose due to i n - f l i g h t i n t e r a c t i o n s was taken from the experiment of Nordell et a l (125), i i i — I — i — i — I I I I — i — i — i — i — r 0-4h 0 2 h O o L _ _ L _ J I I I I 1 I I I I I I L -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 O F F - A X I S D I S T A N C E ( c m ) O F F - A X I S D I S T A N C E ( c m ) V E R T I C A L H O R I Z O N T A L Figure 3.3: Cross Scans at the Peak T 1 r 5 10 15 2 0 2 5 3 0 35 DEPTH (cm) Figure 3.4: Dose by Particle Type oo whose values were about twice as large as calculated by Monte Carlo techniques (126). This l a t t e r reference (126) was used to f r a c t i o n the res t of the plateau dose between the d i r e c t pion dose and the dose due to contaminating muons plus electrons. The e + u contamination i n the c a l c u l a t i o n was 23% + 14% = 37%, which i s comparable to tune #10 with 28% + 10% = 38%. The contaminant dose was further divided between electrons and muons using the c a l c u l a t i o n of A l s m i l l e r (127) as a guide to the shape of the electron curve, the c a l c u l a t i o n of Turner (126) as a guide to the shape of the muon curve, and the experimentally determined r a t i o of electrons to muons. This completes the separation of the dose i n the plateau. This picture i s consistent with what was expected on the basis of combining the measured p a r t i c l e f l u x with estimated stopping powers and including the measured (124) i n - f l i g h t i n t e r a c t i o n at 1.6%/cm. The dose i n the peak region can be separated between d i r e c t pion i o n i z a t i o n and s t a r s . The star dose is^ proportional to the pion stopping density which was determined by the charge c o l l e c t o r technique. This method (5) i s described i n appendix A. On the basis of the data from Turner (126) and Nordell (125) f o r narrow momentum b i t e s , the percentage of the t o t a l dose which i s at t r i b u t e d to, stars was set to 50% at the star dose peak. F i n a l l y , the star dose plus contaminant doses were subtracted from the t o t a l to y i e l d the d i r e c t i o n i z a t i o n dose. This method of f r a c t i o n - ing the dose y i e l d s the correct r a t i o f o r the energy contributed by passing pions divided by the energy contributed by st a r s . The f r a c t i o n of the dose by p a r t i c l e type at both i r r a d i a t i o n positions i s given i n table 3.1. The dose due to neutrons i s included with the stars and i s responsible f o r the small star dose component extending into the t a i l region. Table 3.1: Percentage pions (direct) i n - f l i g h t stars electrons muons i f the Dose by P a r t i c l e Type Plateau Peak (%) (%) 63 ± 5 37 ± 8 11 + 4 46 ± 10 19 ± 3 12 ± 2 7 ± 2 5 ± 2 This - completes the de s c r i p t i o n of the beam tune and dose p r o f i l e s . Those parameters of t h i s p a r t i c u l a r beam which could be measured d i r e c t l y were combined with c a l c u l a t i o n s and experiments from the l i t e r a t u r e to provide as complete and accurate a char a c t e r i z a t i o n as possible at present. Microdosimetric measurements (128) are consistent with t h i s d e s c r i p t i o n . 3.2.3 Monitor Chamber A l l experiments on the M8 channel requiring a proton production current of more than 1 yA are normalized to the i o n i z a t i o n produced i n a transmission monitor chamber. This chamber was an i n t e g r a l part of t h i s experiment, and i t w i l l be used to determine the patient dose during c l i n i c a l t r i a l s . It was e s s e n t i a l therefore that t h i s device and i t s operation be thoroughly understood. 3.2.3.1 Construction The monitor chamber i s of the transmission type with nine aluminizedfmylar•foil electrodes of thickness 0.81 mg/cm2 arranged as shown i n figu r e 3.5. There are s i x chambers i n t o t a l . The ones l a b e l l e d "B" and "T" are of symmetric design and are i d e n t i c a l . "T" was the one used throughout t h i s experiment. During patient i r r a d i a t i o n "B" w i l l be used as a backup monitor. F o i l "Q", which has been etched to make one chamber i n each quadrant, i s used to monitor beam-drift. Each f o i l was stretched between two snap together rings with an open space of 18 cm diameter. The d e f l e c t i o n at the centre of a t i g h t l y stretched f o i l did not exceed 1 mm when supporting a weight of 10 g. E l e c t r i c a l contact to the f o i l was made by a copper wire soldered to the r i n g and then to a BNC connector at the casing. The rings were held i n place by a set of three t e f l o n i n s u l a t o r s which rode on guide posts and which determined the r e l a t i v e spacing between f o i l s . The c a l i b r a t i o n change as a r e s u l t of complete disassembly was t y p i c a l l y 1%. The monitor was mounted on the pole faces of the l a s t quadrupole, Q5, by means of e l e c t r i c a l l y i n s u l a t i n g nylon fee t . t e f lon s p a c e r binding post h.v. s ignal JUL > B my la r snap- toge the r r i ngs 5 c m ground b a c k u p t ransmiss ion quad ran t ground Figure 3 . 5 : Schematic Diagram of the Monitor 92 3.2.3.2 Operation The chamber was a i r - f i l l e d and open to the atmosphere with two 2 mm diameter ports. The pressure was measured with a p r e c i s i o n mercury barometer (Princo) located near the c o n t r o l console. The temperature was monitored by a surface probe thermistor (YSI T2630) attached to the aluminum casing. Chamber readings were corrected to 22° C and standard pressure (1 atmosphere) i n the data a c q u i s i t i o n program according to 273+T 1 equivalent monitor counts = s p e c i f i e d monitor counts x x 273+22 p(atmospheres) The temperature varied between 26.3 and 31.9°C and the pressure v a r i e d between 742 and 767 mm during a four month experimental period. The chamber output current was integrated by a d i g i t a l current integrator (Ortec model 439). The read sequence i s described i n s e c t i o n 3.2.5. T y p i c a l l y , f o r a proton current of 10 yA, 20 pulses of 10 ^ C/pulse were counted i n 9% seconds (2.1 x 10 ^ A). There was no observable p o l a r i t y e f f e c t . An i n v e s t i g a t i o n of saturation for pions at low rates (0.1 x 10 ^ A) and for X-rays at high rates (16 x 10 ^ A) indicated that the saturation current was 0.25% higher than the current obtained at the usual operating voltage (100 Volte). When the channel was used with p o s i t i v e pions, an absorber was required to stop the low energy proton contamination which otherwise produced about 100 times the i o n i z a t i o n caused by the passing pions, muons, and electrons. 93 3.2.3.3 Calibration versus particle flux Some preliminary dosimetry measurements had been taken with the channel tuned at midline momentum 148 MeV/c. When the momentum was changed to 180 MeV/c as a routine, i t became necessary to investigate the momentum dependence of the monitor chamber response per incident particle in order to compare with the earlier measurements. At that time, the primary proton current was of the order 1,yA and so the dead time correction for coincidences in scin t i l l a t o r s 1 and 2 was less than a few percent. The chamber response per 1.2 coincidence versus midline momentum ( f u l l momentum bite) is shown in figure 3.6. The response per coincidence has a peak at 110 MeV/c. As the momentum i s decreased from the peak, the electron fraction of the beam increases (123) and since the electrons have a lower stopping power than the pions, the response per particle decreases. On the other hand, as the momentum is increased from the peak, the pion fraction of the beam increases, and since the pions have a lower stopping power, the higher their momentum, the chamber response per particle decreases again. The calibration of the chamber depends on the separation between the scin t i l l a t o r s and the chamber. This i s caused by pion i n - f l i g h t decay to muons, most of which come off at a laboratory angle of about 14°. These muons can miss the scintillators,'causing the•chamber response per 1.2 coincidence :to increase (see figure 3 . 7 ) . ' • " • It i s not possible to test the dose rate response of the chamber over a very large range of rates because at proton currents less than 1 uA, ele c t r i c a l leakage in the monitor is a significant fraction of the signal,, and at proton currents larger than 1 uA dead time for the 1.2 coincidence is excessive. The dose rate response was linear for 100 keV X-rays up to the maximum rate investigated which was equivalent to a 75 yA proton current. Figure 3.6: Transmission Chamber Response Versus Momentum 95 Figure 3.7: Transmission Chamber Response Versus Separation 96 It has been noticed at the termination of high current tests (Ip = 100 yA) that the monitor appears to be detecting a radioactive background which decays with a h a l f l i f e of the order of 10 minutes. This could be due to the b u i l d up of r a d i a t i o n induced positron emitters 1 5 0 (Th = 2.1 m) , 1 3N (Th = 10.0 m) or llC (Th = 20.5 m) i n s i d e the monitor chamber. This decaying s i g n a l i s i n s i g n i f i c a n t i n comparison to the beam s i g n a l , but i t i s s i g n i f i c a n t l y larger than the usual chamber leakage. This completes the discussion of the monitor chamber. Throughout the experiments the chamber operated r e l i a b l y and was reproducible at the ± 1% l e v e l . 3.2.4 P a r a l l e l Plate Ionization Chamber The p a r a l l e l plate chamber design was selected for the measurement of dose i n the phantom because of i t s ease of construction and f l e x i b i l i t y i n exchanging electrodes. As well as eliminating leakage from the high voltage to;the c o l l e c t o r , the guard ri n g reduces the problem of e l e c t r i c f i e l d rounding at the edges and thereby gives a well defined volume. 3.2.4.1 Construction A schematic cross section of the ion chamber i s shown i n f i g u r e 3.8. The outer, casing i s made of aluminum and i s held together by s i x brass rods. The casing, could be disassembled to exchange or remove the spacer rings which had thicknesses of 5 mm and 10 mm i n order to provide nominal gaps of 1, 6 or 11 mm. Electrodes of various materials and sizes were a v a i l a b l e as shown i n table 3.2. The electrodes were thick enough that a l l secondaries from stars i n the i n s u l a t o r (except for the most energetic protons) would be stopped i n the electrodes with the r e s u l t that the cavity viewed a slowing down spectrum Figure 3.8: P a r a l l e l Plate Ionization Chamber Construction 1. Spring loaded contact to H.V. electrode 2. H.V. plate 3. Polystyrene insu l a t o r for H.V. plate 4. Aluminum face plate 5. Nylon screw 6. Rubber 0-ring 7. Body rin g 8. Space ri n g (5 mm) 9. Gas flow groove; gas i n l e t and outlet are not shown 10. Back plate 11. Guard ri n g 12. C o l l e c t o r 13. C o l l e c t o r screw 14. Polystyrene insulator for signa l plate 15. Connector plate 16. Water proof conduit This view of the e l e c t r i c a l connections i l l u s t r a t e s the p r i n c i p l e of the method for e l e c t r i c a l feed through. In a c t u a l i t y , both the signal and high voltage leads were located at the same r a d i a l distance as shown here for the H.V. lead. The lead (not shown) which made contact between the guard r i n g and connector plate was spring loaded.  99 which i s c h a r a c t e r i s t i c of the electrode material only. The outer diameter of the guard extended to where the dose was less than 50%. E l e c t r i c a l connections were made through a water proof conduit fixed to the back pl a t e . The connectors were BNC type with ceramic-to-metal seals and rubber o-rings (Kings E l e c t r o n i c s M39012/24-0002). The high voltage lead was spring loaded and the connection to the s i g n a l lead was made with a tapered pin. Care was taken to f i l l the space around the leads with polystyrene i n s u l a t i o n to prevent arcing around the high voltage lead and to avoid creating any secondary chamber within the case. In order to eliminate a large p o l a r i t y e f f e c t , a 0.18 mm thick s t r i p of c e l l u l o s e acetate i n s u l a t i o n was inserted into the space between the guard r i n g and c o l l e c t o r . Without i t , the p o l a r i t y e f f e c t was large because the electrometer held the c o l l e c t o r at approximately +100 mV with respect to the guard r i n g , thus creating a secondary ion chamber. Table 3.2: Electrode Materials and Configuration (the gap and diameter are independently selected) Material Carbon Density (g/cm3) 1.786 ± .015 Impurities Gap (mm) .06% (130) 6.14 ± .02 Diameter** (mm) 20.47 ± .03 TE A150 plastic Aluminum 1.125 ± .007 2.714 ± .017 * 6.37 ± .02 .5% (131) 6.19 ± .02 1.17 ± .04 11.17 ± .05 21.03 ± .03 20.49 ± .03 50.47 ± .04 10.53 ± .03 * TE A150 plastic is a mixture widely used in neutron dosimetry whose elemental composition has recently been documented (129): (percent mass) H 10.33, C 76.93, N 3.30, 0 6.94, F 1.14 and Ca 1.37. ** Effective diameter is the mean of the collector diameter and the guard ring opening. 100 3.2.4.2 Gas de l i v e r y system The gas i n l e t and outlet were connected by grooves so that the gas was forced to flow from-one side of the chamber to the other passing over the c o l l e c t o r . Although the chamber was not used with continuous flow, t h i s arrangement provided e f f i c i e n t f l u s h i n g . Rapid approach to the equilibrium pressure was provided by V ' p o l y f l o tubing coupled to connectors threaded to the back plate and located alongside the e l e c t r i c a l connectors. A schematic diagram of the system i s shown i n fi g u r e 3.9. The gas c y l i n d e r s , gauges and valves were located near the c e n t r a l console about 8 m from the chamber. The system was usually evacuated when not i n use. This was p a r t i - c u l a r l y important f o r the carbon electrodes which were very e f f i c i e n t at adsorbing gases. Before taking data with a new gas, the system was a l t e r n a t e l y f i l l e d and evacuated three times. Readings were usually taken s t a r t i n g at low pressures. This procedure was varied occasionally to look for systematic errors, which were never observed. Three 6" diameter p r e c i s i o n gauges (Matheson) on the outlet port were used to measure pressure i n the ranges 0 - 760 mm absolute, 0 - 3 0 psig and 0 - 100 psig. Corrections for gauge n o n - l i n e a r i t y were made a f t e r c a l i b r a t i o n using a mercury manometer. In order to determine the absolute pressure, the atmospheric pressure was measured with a p r e c i s i o n mercury barometer (Princo). The uncertainty i n pressure was ± 1% except at the ." • lowest pressure (15 kPa) where i t was ± 2%. The various gases used, t h e i r densities (132) at 22°C and the manufacturer's stated purity are shown i n table 3.3. REGULATOR GAS CYLINDER GAUGES MANOMETER © © © • EXHAUST VALVE BLEED VALVE POLYFLO TUBING VACUUM PUMP T ^IONIZATION CHAMBER EXHAUST Figure 3.9: Gas Delivery System 102 Table 3.3: Gas Density & Pur i t y Gas CH 4 C 4 H 1 0 CO 2 0 2 Density (mg/cm3) at 22°C and 1 atmosphere 0.6633 2.416 1.8296 1.3224 Composition or Pur i t y commercial 99.0% instrument grade 99.5% bone dry 99.8% medical 99.6% 1.6507 pr e p u r i f i e d 99.998% N 2 1.1573 oxygen free 99.998% N20 a i r 1.8296 1.1965 medical 98% 80% N 2 + 20% 0 2 TE 1.0416 65.88% CH4 + 31.2% C0 2 + 2.92% N 2 by p a r t i a l pressure 3.2.4.3 Temperature monitor The gas temperature ins i d e the ca v i t y was taken to be the same as the water tank which was monitored with a thermistor probe (YSI T2635). Over a four month period, the temperature varied between 24 and 28°C. The i o n i z a t i o n i n s i d e the cavity was corrected to 22°C according to J(22°C) = J(T) x 273 + T 273 + 22 The mass of gas i n the cavity at 22°C was taken to be the product of volume, density and absolute pressure (atmospheres). 103 3.2.5 E l e c t r o n i c s There were three voltage sources used for various ranges: 0 to 1000 Volts - b a t t e r i e s , 100 to 3000 Volts - Power Designs high voltage c a l i b r a t e d d.c. power source, model 1570, and 500 to 5000 Volts - CPS p r e c i s i o n HV power supply, model 5001. Voltages were measured where necessary with a d i g i t a l multimeter, e i t h e r DANA model 4300 or FLUKE model 8000A. The charge from the p a r a l l e l plate chamber was determined by means of Keithley electrometers, e i t h e r model 616 or model 610C, with range -9 s e l e c t o r set to 10 Coulomb f u l l scale and the m u l t i p l i e r switch chosen appropriately. These two instruments agreed with each other to ± and were therefore used interchangeably. They have been compared with other s i m i l a r devices during pion dosimetry workshops and the agreement with the mean has been better than ± 1%. The electrometer was used i n the charge c o l l e c t i o n mode i n order to obtain adequate s e n s i t i v i t y at the low dose rates a v a i l a b l e and to average over beam fl u c t u a t i o n s during i r r a d i a t i o n , since the beam i s d i f f i c u l t to turn o f f . Consequently, the dosimetry system was operated i n a dynamic charge c o l l e c t i o n mode. The voltage output from the e l e c t r o - meter, which was proportional to the charge c o l l e c t e d , was read by a minicomputer (Data General NOVA 2/10) through a chain c o n s i s t i n g of an i n v e r t i n g three times a m p l i f i e r , a multiplexer and a 10-bit analogue to d i g i t a l converter (GEC E l l i o t ADC 1201) (see figure 3.1). The pulsed output from the d i g i t a l current integrator used to read the transmission chamber was fed to a preset s c a l e r which also reported to the minicomputer. The timing sequence for reading these devices i s shown schematically i n f i g u r e 3.10. The electrometer was released from zero ELECTROMETER ANALOGUE VOLTAGE 0 + MONITOR PULSES Figure 3.10: Timing Sequence for Reading the Electrometer B i s the 3 second time delay required to avoid transients C i s the elapsed time used to re j e c t readings taken at too low dose rates 105 and charge c o l l e c t i o n i n i t i a t e d by a command to the solenoid reset device. There was a three second delay i n order to avoid transients associated with t h i s zero release. Immediately a f t e r the a r r i v a l of the next pulse from the d i g i t a l current integrator, the electrometer and the r e a l time clock were read. Immediately a f t e r the l a s t of a predetermined number of pulses was received, the electrometer and clock were read again. The difference between the two electrometer readings divided by the s p e c i f i e d number of monitor pulses gave the charge ( i n units of the ADC) c o l l e c t e d per monitor pulse. The t o t a l elapsed time was used to r e j e c t measurements taken when the average i n t e n s i t y was l e s s than 80% of the expected i n t e n s i t y . The dosimetry system i s described f u l l y i n (139). This system exhibited a l k % decrease i n s e n s i t i v i t y over f u l l * s c a l e d e f l e c t i o n . This n o n - l i n e a r i t y was corrected manually. The absolute value of the normalized charge was c a l i b r a t e d i n the following manner. With the beam o f f , the electrometer was zeroed and released. The beam was turned on and the automatic control of the beam stopper was used to terminate the i r r a d i a t i o n a f t e r the required number of monitor pulses. A graph of the c o l l e c t e d charge versus number of monitor counts,has a.slope (charge per monitor count) which was compared to the computer determined value (ADC units per monitor count). The o v e r a l l uncertainty i n the normalized absolute charge i s ± 1.0% f o r the electrometer c a l i b r a t i o n , ± 0.8% f o r the ADC-computer system c a l i b r a t i o n and ± 1.0% f o r the transmission chamber r e p r o d u c i b i l i t y which gives ± 1.6% when added i n quadrature. 106 3 . 3 RESULTS 3 . 3 . 1 Saturation C h a r a c t e r i s t i c s 3 . 3 . 1 . 1 T h e o r e t i c a l One of the experimental d i f f i c u l t i e s involved with i o n i z a t i o n chamber measurements i s to ensure that a l l of the ion pa i r s are c o l l e c t e d . Complete c o l l e c t i o n (saturation) i s hampered by ion recombination which i s of two types: i n i t i a l and general. I n i t i a l recombination occurs where the ions formed i n the track of a si n g l e i o n i z i n g p a r t i c l e meet and recombine. The amount of i n i t i a l recombination i s determined by the ion density along each separate track and i s independent of the dose rate. Thus, at low pressures, i n i t i a l recombination i s important for the high LET tracks only, whereas at high pressures, i t i s important for both high and low LET tracks. Once the i n i t i a l track structure has been o b l i t e r a t e d by thermal d i f f u s i o n and i o n i c d r i f t , only general recombination remains. In general recombination,: ions formed'-'by- the d i f f e r e n t i o n i z i n g p a r t i c l e s recombine as they d r i f t towards opposite electrodes. Therefore, general recombination increases with dose rate. In t h i s experiment, the dose rate was low enough that general recombination was n e g l i g i b l e . According to Boag ( 1 3 3 ) , the general recombination e f f i c i e n c y of ion c o l l e c t i o n i n an a i r - f i l l e d p a r a l l e l - p l ate chamber i s given by . 1 (1 + " r ? 2) 3 . 1 where E, = li — Jq", M = 36.7v/p", P i s the pressure i n atmospheres, d i s the plate separation i n cm, V i s the applied voltage i n V o l t s , and q i s the - 3 - 1 ion production rater i n esu cm sec . For the carbon chamber f i l l e d with a i r and placed at the peak dose rate provided by a proton current of 12 uA, 107 -3 - 3 - 1 q < 9.2 x 10 P esu cm sec . I f the loss due to general recombination i s to be kept less than 1%, £ < .246 and hence, V > 36.7 P (.614) 2 9.2 x 1 0 _ 3 P (.246)" 1 > 5.4 P At the maximum pressure of 7% atmospheres, the applied voltage must be greater than 40 V o l t s . Since t h i s was e a s i l y attained f o r a l l gases investigated, general recombination was n e g l i g i b l e . In some cases i t was not possible to eliminate i n i t i a l recombination e n t i r e l y , and so some method of extrapolation to saturation was required. Based on a theory of Ja f f e (134) as modified by Xanstra (135), Boag (133) has derived the expression - = T + F 3.2 i I E where i i s the current measured at e l e c t r i c f i e l d strength E (volts/cm), I i s the saturation current a v a i l a b l e at i n f i n i t e voltage and b i s a constant f o r a given type of r a d i a t i o n . This can be rearranged to g i v e - i = I - b l 3.3 or f = ^ = 1 - b ^ 3.4 It has already been stated that f, the c o l l e c t i o n e f f i c i e n c y , must be independent of dose rate for i n i t i a l recombination. Therefore, the parameter b i n 3.4 must be inversely proportional to dose rate, and t h i s i s inconvenient. Since the quantity being studied here was J/M, the charge c o l l e c t e d per unit mass per monitor pulse, then i n equation 3.3, i i s replaced by J/M and b l by a constant k giving J/M ••= (J/M) s - k (J/ME ) 3.5 During the experiments, saturation was investigated by measuring J/M at d i f f e r e n t f i e l d strengths, E, and p l o t t i n g J/M versus J/ME. This was extrapolated to J/ME = 0, that i s E very large, to y i e l d the saturation value, (J/M) g, and the slope, k. Similar procedures are i n common use for neutron beams (136), alpha p a r t i c l e s (137) and heavy ions (60). 3.3.1.2 Experimental Saturation was studied for the various gases according to the technique outlined above. In figure 3.11, the values of J/M are plotted against J/ME (E i s the f i e l d strength i n V/cm) for the case of aluminum electrodes and carbon dioxide gas. As the pressure i s increased, the c o l l e c t i o n e f f i c i e n c y at a p a r t i c u l a r voltage decreases. This i s manifested by an increase i n the value k. At high pressure, i t i s more d i f f i c u l t to f i t the data to a s t r a i g h t l i n e . General recombination causes an a d d i t i o n a l decrease i n e f f i c i e n c y at the low voltage end which makes the l i n e s bend downwards. Ionization by c o l l i s i o n at high voltages causes an increase i n current which eventually leads to sparking. This l a t t e r problem has been encountered with ion chambers used to study f i s s i o n r e c o i l s (104) and i t may be diminished by rounding of chamber parts, p a r t i c u l a r l y at the electrode edges (106). In f i g u r e 3.12, the value of k i s plotted versus mass with chamber po s i t i o n as a parameter for the case of carbon electrodes and methane gas. The steepness of the k versus mass curve increases with depth i n the phantom because the mean LET increases with depth as shown i n table 3.4. Some ion chamber systems (81) designed to measure mean LET are based on t h i s dependence of k upon LET. The values of k are plotted as a function of mass for a number of gases i n fi g u r e 3.13 for the peak p o s i t i o n and fi g u r e 3.14 for the plateau p o s i t i o n . If the e l e c t r i c f i e l d i s 100 V/cm, then k i s numerically the percentage increase i n current required to achieve saturation. The data 109 0 2 4 6 8 I 1 1 1 T J / M E d o - " C - c m / g - V - D C I ) Figure 3.11: Saturation Curves f or Carbon Dioxide 1 r plateau ' 14-1 cm o proximal peak 18 4cm A peak 20-6cm + distal peak 23-7cm * Figure 3.12: k Versus Mass Curves at D i f f e r e n t Depths f o r Methane  Figure 3.14: k Versus Mass at the Plateau on these graphs are averages over several runs with d i f f e r e n t electrodes. For k > 10 V/cm, the uncertainty i s ± 1.3 V/cm, between k = 2 and 10 V/cm, i t i s ± 1.0 V/cm and for k < 2 V/cm, it\is,.±'0.5 V/cm. Table 3.4: L i n e a l Energy at Various Depths Po s i t i o n Depth Mean L i n e a l Energy (128) (cm) (MeV/um) plateau 14.1 7 proximal peak 18.4 29 peak 20.6 41 d i s t a l peak 23.7 50 3.3.2 P o l a r i t y E f f e c t The magnitude of the i o n i z a t i o n per unit mass i s larger for negative p o l a r i t y than p o s i t i v e . This p o l a r i t y e f f e c t . i s - l e s s than:± 0.8% at pressures greater than one atmosphere. The p o l a r i t y difference i s plotted as a function of mass i n fi g u r e 3.15. At pressures le s s than one atmosphere, the difference i n J/M increases dramatically because the charge deposited i n the electrodes due to the pions stopping there i s a s i g n i f i c a n t f r a c t i o n of the i o n i z a t i o n produced i n the gas. This e f f e c t i s larger i n the peak than i n the plateau as expected. In order to f i n d the true i o n i z a t i o n i n the gas, the average over p o s i t i v e and negative p o l a r i t y was taken. For some gases, J/M was measured-at negative p o l a r i t y only and h a l f the average p o l a r i t y difference was subtracted to y i e l d the true i o n i z a t i o n . 114 10 8 5 4 Q o = 0 l 5 14 plateau 12 10 8 6 4 2 peak _L 10 15 MASS (mg) 20 25 Figure 3.15: P o l a r i t y Difference Versus Mass Data i s shown for a l l the gases measured. 115 3.3.3 J/M Versus- Pressure • The saturation value of J/M plotted against- pressure-is shown fo r methane and carbon dioxide i n figures 3.16 and 3.17 re s p e c t i v e l y . These data were taken with carbon electrodes at negative p o l a r i t y . The c i r c l e s i n d i c a t e values of J/M made by the extrapolation technique (equation 3.5). The tri a n g l e s i n d i c a t e values of J/M made at f i x e d f i e l d strengths (of the order 100 to 200 V/cm) which were subsequently corrected to saturation using k values determined i n the f i r s t experiment. The-error bars shown--in these figures represent the r e l a t i v e ' uncertainty i n J/M. The uncertainties were assigned by considering best and worst f i t s to the saturation data. At low pressure, the uncertainty i s large because of fluctuations i n the ion chamber leakage. T y p i c a l l y , the leakage was ± 10 to 20 fA. This i s about 1% of the s i g n a l a v a i l a b l e from most gases at atmospheric pressure f o r a 10 yA proton current from the cyclotron. The uncertainty i n J/M passes through a minimum between 1 and 3 atmospheres. At high pressures, the slope of the saturation curves , i s large and somewhat uncertain because of the e f f e c t s discussed i n section 3.3.1„arid t h i s r e s u l t s i n increased error bars. For the values measured at f i x e d voltage, the error bar i s the sum i n quadrature of the s t a t i s t i c a l uncertainty i n the reading and the error i n the k value. The agreement between the c i r c l e s and t r i a n g l e s f o r methane at pressures above one atmosphere indicates that the day-to-day r e p r o d u c i b i l i t y i s about ± 2%, which i s within the error bar. After the-average, was taken ,over p o l a r i t y and s-random flu c t u a t i o n s were smoothed, the values of J/M f o r carbon electrodes were pl o t t e d f o r a l l the gases i n figures 3.18 and 3.19 f o r the peak and plateau r e s p e c t i v e l y . The uncertainty i n the average value at zero pressure i s t y p i c a l l y ± 2%. 116 PRESSURE (100 kPa) Figure 3.16: J/M Versus Pressure for Methane 4 T <*>0 0 plateau I I l i i i i 2 4 6 8 PRESSURE (100 kPo) Figure 3.17: J/M Versus Pressure for Carbon Dioxide 2 4 PRESSURE (100 kPa) Figure 3.18: J/M Versus Pressure for A l l the Gases at the Peak with Carbon Electrodes 119 120 The value of J/M for a l l oxygen bearing gases decreases with pressure i n both the peak and plateau. The rate of decrease i s f a s t e r i n the peak than i n the plateau as indicated by the value of the pressure at which the curves for oxygen bearing gases cross over the one for nitrogen. This fact can be seen more c l e a r l y i n figure 3.20 where the i o n i z a t i o n i n the peak i s divided by that i n the plateau and the peak to plateau r a t i o i s p l o t t e d against pressure f o r the various gases. At zero pressure, a l l of the i o n i z a t i o n i s produced by crossers which ori g i n a t e i n the carbon walls. Since there are no gas s p e c i f i c contributions to the i o n i z a t i o n , the cavity i s an i d e a l Bragg-Gray cavity and the peak to plateau r a t i o i s a constant independent of the type of gas f i l l i n g . The spread i n values of J/M between the various gases at zero pressure i s ± 2% and gives an i n d i c a t i o n of the uncertainty i n the value of the peak/plateau r a t i o for any p a r t i c u l a r gas. After correcting! for saturation and p o l a r i t y , the data for aluminum electrodes are displayed i n figures 3.21 .and 3.22 for the peak and plateau respectively. In most cases, the shape of the curve f o r a p a r t i c u l a r gas i s the same whether the data was acquired with carbon electrodes or with aluminum. The peak to plateau r a t i o i s shown i n figure 3.23. Once again, the J/M curves for oxygen bearing gases exhibit a dramatic dependence on pressure. The peak to plateau r a t i o extrapolated to zero pressure i s lower for aluminum than carbon. This i s rel a t e d to the kerma factor and w i l l be discussed i n section 3.4.3.. Besides the-usual measurements-in. the peak and-plateau, a measurement of J/M versus pressure with aluminum electrodes and carbon dioxide was made i n the region of the electron t a i l (35.6 cm depth). The absolute change i n J/M i s s m a l l " b u t the percentage change i s larger than expected from the Sternheimer density e f f e c t . The implication of such a r e s u l t i s discussed i n section 3.4.1. Data for a i r and TE gas was taken with TE-A150 p l a s t i c electrodes. The averaged values a f t e r smoothing are plotted i n fig u r e 3.24. There i s l i t t l e d ifference i n absolute value f o r these curves and those with carbon electrodes. Figure 3.20: Peak to Plateau Ratio Versus Pressure with Carbon Electrodes 123 Figure 3.21: J/M Versus Pressure f o r A l l the Gases at the with Aluminum Electrodes 124 Figure 3.23: Peak to Plateau Ratio Versus Pressure with Aluminum Electrodes 126 4 6 8 PRESSURE (100 kPa) Figure 3.24: J/M Versus Pressure for A i r and TE gas with TE-A150 Electrodes 127 3.4 DISCUSSION 3.4.1 Comparison Between Ca l c u l a t i o n and Experiment This comparison deals with i o n i z a t i o n chambers having carbon electrodes and being f i l l e d with methane or carbon dioxide gas. For a beam cons i s t i n g only of the charged secondaries released during pion capture, the c a l c u l a t i o n of chapter 2 predicts that the i o n i z a t i o n per unit mass i s a function of pressure due to pion captures i n the gas and perturbations of the secondary f l u x by the cavity. Between 0 and 7 atmospheres, the calculated change i s -7% for CO2 (see figure 2.18) and +2% for CH^ (see figure 2.17). The decrease for CO^ i s p r i m a r i l y due to the release of le s s k i n e t i c energy to charged secondaries per pion capture i n oxygen than i n carbon. In p r a c t i c e , i o n i z a t i o n i n the c h a m b e r gas i s produced not only by the charged star secondaries, but also by contaminating electrons and muons, passing pions and s t a r neutrons. Of these other p a r t i c l e s , f u r t h e r c a l c u l a t i o n i n chapter 2 indicated that o n l y i o n i z a t i o n s by e l e c t r o n s arenexpectedl - t o G e x h i b i t a pressure dependence, and t h i s i s of the order of -10% f o r e i t h e r gas b e t w e e n 0 and 7 atmospheres (see figure 2.19). The observed change i n J/M measured at the peak and plateau res p e c t i v e l y i s -37% and -27% f o r C 0 2and -3% and -2% f o r CH^ (see figures 3.18 and 3.19). For CO^, t h i s i s about .four'times as large as cal c u l a t e d . Besides this, discrepancy i n magnitude f o r CO^, an experiment i n the electron t a i l revealed a change of -23%. Increasing the magnitude of the Sternheimer e f f e c t to account f o r t h i s would give a percentage change i n the plateau which exceeds that i n the peak, and so there must be a systematic e f f e c t acting on the whole beam which has yet to be discovered. 128 Consider the equation [U(P)]" 1 (g. F (P) + g w + g c F (P) )' = J/M(P) 3.6 where the g / s (see tables 3.6 and 3.7) are f r a c t i o n s of the t o t a l i o n i z a t i o n due to electrons, muons plus passing pions and pion s t a r s ; the F's are functions of pressure normalized to unity at P = 0 and represent the pressure dependence of the i o n i z a t i o n f or e l e c t r o n s , ; F e , and s t a r s , F g ; U(P) i s a systematic correction factor under consideration; and J/M(P) i s the measured i o n i z a t i o n per unit mass also normalized to unity at P = 0. There are equations of t h i s form for measurements i n the peak, plateau and electron t a i l . In p r i n c i p l e , since g g = 0 i n the electron t a i l , one could solve t h i s equation for U(P). This was not done because values of g and g are f a i r l y uncertain i n the e l e c t r o n t a i l . e y . -,- Instead, J/M(P) measured i n the peak and plateau were used to solve for U(P) and F (P) simultaneously. When the value of U(P) i s substituted back i n t o the equation for the e l e c t r o n t a i l , reasonable agreement. (± 2%) i s obtained. The values of U(P) are p l o t t e d i n figure 3.25. For CH^, U(P) i s independent of pressure w i t h i n ± 0.5% i n d i c a t i n g that there i s no systematic correction factor f or t h i s gas. For C0,y, U(P) i s independent of pressure up to 2 atmospheres, a f t e r which i t increases to 1.22 at P = 700 kPa. I t i s i n t e r e s t i n g to speculate what causes t h i s phenomenon. It has long'been recognized that i n i t i a l recombination i s a problem with electronegat-iye gases. (76) . F a i l u r e to c o l l e c t a l l the i o n i z a t i o n due to i n i t i a l recombination seems u n l i k e l y because sound and c a r e f u l methods were used to f i n d the saturation current. Another possible explanation i s that the electron W-value i s a function of pressure f o r the electronegative, gases. Such gases are capable of electron trapping according to the Figure 3.25: Factor U(P) Versus Pressure 130 process: 0 2 + e~ + S ^ > 0~ + S* + energy 3.7 In the case of pure oxygen, S i s another 0^ molecule and the energy appears as t r a n s l a t i o n or i n t e r n a l energy of the molecule. The W-value increases because energy which may have produced i o n i z a t i o n ends up as e x c i t a t i o n . Since the attachment c o e f f i c i e n t increases l i n e a r l y with pressure for E/P < 3V/cm-torr 091)-* the W-value would also be pressure dependent. This process i s not re l a t e d to i n i t i a l recombination,.-arid instead would apply to the delta rays and t e r t i a r y electrons l i b e r a t e d by high and low LET p a r t i c l e s a l i k e . Whether or not electron trapping causes t h i s systematic s c a l i n g factor f or CO^ i s not c r u c i a l to the argument. I t i s not a s i g n i f i c a n t e f f e c t below 200 -kPa.- The extracted pressure dependence, F g ( P ) , and the calculated value, G s(P), are compared i n f i g u r e 3.26. Q u a l i t a t i v e l y the c a l c u l a t i o n and the experiment y i e l d s i m i l a r pictures i n that the CH^curve i s f a i r l y f l a t and the CO^ decreases. The percentage difference between C0 2 and CH^ at 700 kPa was calculated to be 8%, whereas experimentally i t i s 16%. The uncertainty i n at 700 kPa was estimated to be ±-'A% by considering d i f f e r e n t values for the g^' s and allowing J/M to change by ± 2%. The value f o r F e(P) w a s taken to be correct; Since the extracted pressure dependence, F^CP), changes more quickly with pressure than predicted, G g(P), i t i s important to consider what aspects of the c a l c u l a t i o n could be a l t e r e d and what e f f e c t t h i s a l t e r a t i o n would have on the agreement. C l e a r l y , the i n c l u s i o n of dose due to heavy ions would make G g(P) a stronger function of pressure since the range of the heavier ions i s small compared with the s i z e of the chamber. This was i l l u s t r a t e d f o r L i ions and protons i n figures 2.15 and 2.16. 131 I ' I I I I I 0 2 4 6 8 P R E S S U R E (100 kPa) Figure 3.26: Comparison of Calculated Observed Pressure Dependence for J/M 132 Such an e f f e c t would be more important for oxygen, which has 2.4 MeV missing out of 14.4 MeV, than for carbon, which has 1.3 MeV missing out of 28.8 MeV. Since most of these heavy ions w i l l be stoppers or i n s i d e r s , only t h e i r energy i s important to the c a l c u l a t i o n and not t h e i r stopping power. The i n c l u s i o n of the dose due to undetected heavy ions would improve the agreement between F (P) and G (P). s s Star neutrons have not been dealt with e x p l i c i t l y . I t i s reasonable to assume that high energy neutron i n t e r a c t i o n w i l l produce a spectrum of charged secondaries s i m i l a r to that produced by pion capture. Since the number of neutrons released per pion i s about the same for both carbon and oxygen, and since the decrease of J/M with pressure calculated for the charged secondaries i s due to le s s energy a v a i l a b l e per pion capture on oxygen than carbon, i n c l u s i o n of some neutron dose would tend to make G g(P) for CC^ less pressure dependent than already calculated, thus increasing the discrepancy between the c a l c u l a t i o n and experiment. Even i f the values for W and r were calculated f or the neutrons, there s t i l l remains the problem of how to f r a c t i o n a t e the dose between neutrons and charged p a r t i c l e s . An increased understanding of pion capture and intra-molecular transfer would be useful to c a l c u l a t e a more accurate value of G (P). . But, s even complete transfer from the C atom to one of the two 0 atoms within CC^ would make only minor changes i n G g(P). There remains an unresolved quantitative discrepancy between the c a l c u l a t i o n and experiment. The q u a l i t a t i v e agreement does confirm the physical e f f e c t s which were included i n the c a l c u l a t i o n of chapter 2. These ef f e c t s stated that there i s s i g n i f i c a n t pion capture i n the gas and that pion capture on oxygen y i e l d s l e s s k i n e t i c energy to charged secondaries than capture on carbon. 133 3.4.2 Absolute Dose i n Carbon In t h i s section, measured values of J/M for carbon dioxide and methane are used along with calculated values of W and r to determine the dose i n carbon. The approach taken was to extrapolate J/M to P = 0 where the non-Bragg-Gray nature of the cavity i s i n s i g n i f i c a n t . A l terna- t i v e l y , J/M at P = 100 KPa could have been used with appropriate values f o r W and r . The former approach s u f f e r s because of a la r g e r uncertainty i n the extrapolation procedure f o r J/M. The l a t t e r s u f f e r s because of the larger discrepancy between the calculated and measured W and r curves. Table 3.5 gives the values of J/M extrapolated to zero pressure f o r carbon electrodes with a l l the gases used experimentally. The t o t a l dose i s calculated according to 8 i % 1 V 1 J 3.8 where g. i s the f r a c t i o n of the i o n i z a t i o n due to the i ^ source: l e l ectron, muon, passing pion or pion star; r i s the r a t i o of dose i n carbon to dose i n gas f o r the pion s t a r s , and i t i s the stopping power r a t i o f o r the electrons, muons and passing pions; and the other terms have been defined previously. The value of g^ was determined f o r the sake of completeness from the value of i n table 3.1 ( f ^ i s the f r a c t i o n of the dose due to the i * " * 1 source) according to: W e — r = f D 3.9 W6 8 i W r i i T T @ ± Values of (W/W^)^, r ^ and g^ i n peak and plateau taken from various graphs and tables are c o l l e c t e d for CH. and C0_ i n tables 3.6 and 3.7 respectively. 4 Z I t has been assumed that the W-value f o r electrons i s not a function of energy, and therefore W/Wfi = 1.00 i n both the peak and plateau. 134 Table 3.5: J/M Extrapolated to Zero Pressure f o r Carbon Electrodes Gas W* ' J/M (peak) J/M (plateau) Ratio (eV) (nC/g - DCI) CH. 4 c o 2 N 2 °2 TE axr 27.10 1.666 1.114 1.496 32.80 0.982 0.662 1.483 34.65 0.930 0.618 1.505 N 20 32.55 1.012 0.679 1.490 30.83 1.090 0.735 1.483 29.2 1.281 0.863 1.484 33.73 1.004 0.675 1.487 26.3 1.005 C 4 H 1 Q 23.2 1.731 * W-values are from ChristophorOu (86) except f o r a i r (61) and TE gas (62), Table 3.6: Values Required to Calculate the Dose f o r CH^ Peak Plateau W/Wg r g ± r g ± electrons 1.00 0.591 0.15 0.587 0.24 muons and passing pions stars 1.05 0.740 0.40 0.740 0.66 1.014 0.738 0.45 "0.738 0.11 Table 3.7: Values Required to Calculate therDose for CO2 Peak Plateau W/Wg r g ± r g ± electrons 1.00 0.768 0.16 0.762 0.25 muons and passing pions stars 1.05 1.031 0.40 1.031 0.65 1.013 1.035 0.45 1.035 0.10 The rather high value of W/WQ = 1.05 for passing pions i s an p average made by comparing preliminary measurements of D i c e l l o (55) f o r 78 MeV pions i n a i r , nitrogen and argon to the accepted values for electrons. The dose i n carbon calculated using equation 3.8 i s summarized i n table 3.8. Table 3.8: Dose i n Carbon (yGy/DCI) Methane Carbon Dioxide Peak 33.2 32.8 Plateau 22.0 21.7 The agreement between the dose values measured with the d i f f e r e n t gases i s excellent. The r e l a t i v e uncertainty i s ± 2% i n the measurement of J/M and ± 1% i n the electron W-value giving a t o t a l of ± 2.2%. The absolute uncertainty i s more d i f f i c u l t to assess: 1.6% systematic uncertainty i n determining the absolute normalized current (see section 3.2.5). 1.7% uncertainty i n measuring J/M due to d a i l y v a r i a t i o n i n r e p r o d u c i b i l i t y and extrapolation. This i s l e s s than the r e l a t i v e uncertainty mentioned above since the monitor chamber r e p r o d u c i b i l i t y i s already included i n the 1.6%. 2.0% uncertainty i n the mass determination, about 0.4% f o r the volume and the remainder for the pressure. Dose CH Dose CO 1.012 1.014 137 1.0% uncertainty i n the absolute value of Ŵ . 3.9% (peak) i * ^ w f J> uncertainty i n the value of ^ g^ TT~ r ^ irom 4.2% (plateau) _J equation 3.8. An uncertainty of 3% i s assigned to the absolute value of r f o r both the star products and the passing pions. For the electrons, r i s uncertain by 2%. The uncertainty i n W-value i s 1% for the s t a r products and W 4% for the passing pions. Then the product of — r has 3 an uncertainty for the s t a r products of 3.2%, for the passing pions of 5.0% and for the electrons of 2.0%. These are weighted by the respective g^ values to give 1.5 + 2.0 + 0.3 = 3.8% i n the peak and 0.3 + 3.3 + 0.5 = 4.1% i n the plateau. There i s a 1% uncertainty due to the values of g^ which adds i n quadrature to give 3.9% i n the peak and 4.2% i n the plateau. Therefore the t o t a l absolute uncertainty i s ± 5.1% i n the peak and ± 5.3% i n the plateau. W The value of — x r for a p a r t i c u l a r gas has been evaluated 3 using t h i s dose c a l i b r a t i o n : | — - _ < j - we , (g- , r j ) c y co2 w 3 gas T s \ gas gas These values are found i n table 3.9. A carbon walled i o n i z a t i o n chamber f i l l e d with any one of these gases and used to measure J/M extrapolated to zero pressure w i l l y i e l d the dose i n carbon according to: D . = ^ x WR x [\- x r ) 3.11 carbon M g 3 g V Wg Jg Note that 77- x r i s larger by 1.012 ± 0.005 i n the peak than i n the B w plateau. The absolute uncertainty i n — x r i s about 4% due to the B i n i t i a l c a l i b r a t i o n . I f the chamber i s used at atmospheric pressure, there i s an a d d i t i o n a l uncertainty of the order of ± 2% due to the non- Bragg-Gray e f f e c t s shown gr a p h i c a l l y i n figure 3.26. W Table 3.9: Evaluation of 77- x r f o r Carbon Electrodes Peak Plateau CH. 0.731 0.725 4 C0 2 1.025 1.008 N 2 1.024 1.022 N 20 1.002 0.990 0 2 0.982 0.966 TE 0.882 0.869 a i r 0.975 0.961 A 1.249 C 4 H 1 0 0.822 139 3.4.3 Kerma Factor In t h i s section, the problem of determining the dose i n a material of i n t e r e s t on the basis of the dose measured i n a d i f f e r e n t material i s discussed. F i r s t l y , a comparison of the measurements made with aluminum electrodes to those made with carbon electrodes enables an evaluation of the dose i n aluminum to be done. Then a discussion of how to convert the carbon dose to tissue dose i s presented. Table 3.10 gives the values of J/M extrapolated to zero pressure for the aluminum electrodes. The average peak to plateau r a t i o f o r aluminum i s 1.435 ± 0.008 and the corresponding value f o r carbon (from table 3.5) i s 1.490 ± 0.008. The difference between these two r a t i o s indicates that there i s a small difference i n the kerma factor f o r these two materials. Table 3.10: J/M Extrapolated to Zero Pressure for Aluminum Electrodes Gas J/M peak J/M plateau Ratio (nC/g - DCI) CH. 1.706 1.178 1.448 4 C0 2 1.045 0.725 1.441 N 2 1.005 0.703 1.430 N 20 1.076 0.751 1.433 0 2 1.130 0.790 1.430 TE 1.362 0.955 1.426 a i r 1.097 0.762 1.440 140 For the passing pions, muons and electrons, the dose i n aluminum i s related to the dose i n carbon by the stopping power r a t i o s A 1 D A 1 , = Tr , y , e D C 3 U TT,- , y , e gC T T , y , e T T , y , e For the case of stars due to stopped pions the r e l a t i o n s h i p i s D A 1 = F D C = P K D° 3.13 s s s F i s the kerma fa c t o r given by the product of P and K. The perturbation factor i s P =S /S . The r a t i o of the pion stopping density f o r TT TT aluminum/carbon i s the same as the stopping power r a t i o . This i s an approximate r e l a t i o n s h i p which breaks down for very low energy pions stopping i n t h i n gas targets because the stopping power r a t i o i s no longer independent of energy. In such a case, P = ^ w a ] _ ] / ^ g a s a s s n o w n i n section A l C A l 2.3.1.3. The s t a r energy factor i s K = k /k where k i s the energy released to charged secondaries per pion capture i n aluminum plus that f r a c t i o n of the energy, released to a l l neutrons i n 'the stopping f i e l d - which i s deposited at the point of i n t e r e s t . The charged secondaries and .neutrons- aire'handled differently.-because a f i e l d which has a diameter o f 2 cm ,is large enough to,..provide an -equilibrium spectrum of -charged p a r t i c l e s at i t s centre, but most-of-the energy given.to neutrons w i l l escape. For the i n - f l i g h t pion capture, the relevant equation i s A l A l . A l AI n o k D f - -c-E- - c - D i n o k TT where n i s the number density of atoms and a i s the cross s e c t i o n a l area for pion capture i n f l i g h t . As a rough estimate, a i s taken to be 141 proportional to the geometric cross section and therefore A l A l n a jr C C n a TT I 12 \ 3 v27 ' 1.16 The t o t a l dose i n aluminum at the peak i s ,A1 A l e p ^ s / s A 1 e f C + e ,A1 ,A1 \ f C + IT C S 3.15 where the superscript on f r e f e r s to the f r a c t i o n of the dose i n carbon and the subscript r e f e r s to dose from electrons. Hence, the i o n i z a t i o n per unit mass for aluminum electrodes divided by that f o r carbon electrodes i s ( J / M ) J ( J / M ) ' A l ,A1 f C + V ,A1 f C + TT F* 1 f C J 3.16 A l In order to f i n d F , i t i s necessary to evaluate A l A l , g A l , g 3.17 142 3.18 For the passing pions, muons and electrons, the value of g^ for aluminum i s given by 1 1 (J/M)^ 1' g Since only the i o n i z a t i o n due to stars i s changing, t h i s i s simply a remormalization. For s t a r s , g i s given by A l , g = _ A l , g _ A l , g _ A l , g & s TT & y e To complete the evaluation of equation 3.17, i t i s assumed that the change i n secondary spectra from carbon to aluminum does not s i g n i f i c a n t l y a l t e r W/W„ and that, since rCarbon f o r s t a r s i s not s i g n i f i c a n t l y d i f f e r e n t 3 gas . -. Aluminum _ r , from the stopping power r a t i o , the value f o r r g a s f o r s t a r s l s found - i ^ - -i • Carbon , _ u _ . * t Aluminum-Carbon by multiplying r g a s by t h e stopping power r a t i o /S^ C C Then using values of f ^ from table 3.1, g^ from tables 3.6 and 3.7, C (10 from table 3.9, g s c s c p _ TT s A 1 = s A 1 y IT 1.14 ± 0.01 (80) s A 1 e s c e 0.948 ± 0.001 (61) ( J / M ) A 1 1.063 ± 0.023 (peak) and ^ — = (tables 3.5 and 3.10) ( J / M ) g 1.101 ±0.028 (plateau) 143 y i e l d s an average value of (v) 1.128 ± 0.008 (peak) 3.20 A 1 1.122 ± 0.008 (plateau) This error bar does not include the uncertainty introduced by the A l assumptions concerning W/W and r above. p s For the peak, the kerma factor f o r the dose due to stars i s found .by solving 3.16, AT F? = 1.00 ± 0.03 The error bar i s smaller tiian anticipated because the values of f ^ are coupled. The star energy r a t i o f o r th i s f i e l d s i z e i s found from 3.13 as • A l K = — = 1.14 ± 0.03 kL The t o t a l dose i n aluminum i s given by A l = C. /1^63\ UT T \ 1.128/ 31.1 pGy/DCI There i s a 2.3% uncertainty for the value i n brackets which, when added i n quadrature with the 5.1% uncertainty i n y i e l d s a t o t a l uncertainty f o r A l of ± 5.5%. The values of 1 3.21 (10 (0 A l g g are found i n table 3.11 and have an uncertainty of ± 4.6%. A s i m i l a r procedure ca r r i e d out for the plateau gave A l A l . A l n a k F c - - c - i - 1 ' 7 +~ ° ' 5 n o k TT The error Bar i s large because of the uncertainty i n the value f. for k A 1 i n - f l i g h t i n t e r a c t i o n . Using the value of from above, A l A l * n a IT — — = 1.5 ± 0.5 which i s i n agreement with the value 1.2 n a TT estimated above. The dose i n the plateau i s »f = 4 (itri) - ^ A l I hi \ with an estimated uncertainty of ± 5.9%. The values are calculated f o r the plateau according to equation 3.21 and displayed i n table 3.11. W Table 3.11: Evaluation of — r for Aluminum Electrodes Peak Plateau CH. 0.673 0.672 4 C0 2 0.907 0.902 N 2 0.893 0.880 N 20 0.888 0.877 0 2 0.893 0.880 TE 0.782 0.769 a i r 0.840 0.834 For a beam with no contamination, that i s f = f = 0, y then expression 3.20 becomes A l IT and hence 3.16 s i m p l i f i e s to A l k A l + 3.22 (J/M)? If the r a t i o of J/M values were experimentally extrapolated to zero f i e l d s i z e i n order to eliminate the dose due to neutrons, then solving 3.22 secondaries for pion capture i n aluminum compared to carbon. This completes the discussion of the determination of the dose i n aluminum and of the kerma factor required to convert pion star dose i n carbon to an equivalent dose i n aluminum. Before discussing the conversion of dose i n carbon to dose i n t i s s u e , a b r i e f discussion of the measurements with TE-A150 p l a s t i c w i l l be given. Consider table 3.12 which indicates the values of J/M extrapolated to zero pressure f o r TE-A150 p l a s t i c electrodes. The peak to plateau r a t i o for TE p l a s t i c f a l l s between the values f o r carbon and aluminum. As w e l l , the r a t i o s f o r i o n i z a t i o n with TE-A150 p l a s t i c compared to carbon are not s i g n i f i c a n t l y d i f f e r e n t from unity: peak value 1.012 (TE gas) and 0.971 ( a i r ) ; plateau value 1.024 (TE gas) and 0.987 ( a i r ) . Therefore, for would give the r a t i o of the energy released to charged 146 the average energy made l o c a l l y a v a i l a b l e per incident pion i s about the same for TE-A150 p l a s t i c as for carbon. This equality a r i s e s i n s p i t e of the fact that TE-A150 p l a s t i c contains 7% oxygen for which a s i g n i f i c a n t l y lower energy release to charged secondaries than f o r carbon has been demonstrated. Consideration of the neutron dose and the other components TE—A150 C of TE-A150 p l a s t i c may make up the d e f i c i t . Since k /k i s TE-A150 approximately unity, F is' approximately the stopping power r a t i o f o r TE-A150 p l a s t i c over carbon, and hence the dose i n TE-A150 p l a s t i c i s equal to the dose i n carbon m u l t i p l i e d by the stopping power r a t i o . The use of TE-A150 p l a s t i c f o r pion dosimetry i s a questionable p r a c t i c e . Since i t contains f i v e elements, f i v e times the measurements of charged p a r t i c l e spectra must be performed, or a l t e r n a t i v e l y a s p e c i a l e f f o r t made to s p e c i f i c a l l y measure the spectra for t h i s material. Further- more, since i t i s a mixture, the pion capture process may be d i f f i c u l t to determine due to the e f f e c t s of mechanical v a r i a t i o n s (39, 38). This material i s improperly l a b e l l e d " t i s s u e equivalent" for pions because i t does not have the correct r a t i o of carbon to oxygen atoms. Without th i s s p e c i a l advantage of tissue equivalence there i s no reason for i t s continued use i n pion dosimetry. Table 3.12: J/M Extrapolated to Zero Pressure f o r TE-A150 Electrodes J/M peak J/M Plateau Ratio (nC/g - DCI) TE 1.296 0.884 1.466 a i r 0.975 0.666 1.464 147 The remaining problem to be discussed i s how to convert the dose i n carbon (or aluminum) to the dose i n muscle. Equations 3.12, 3.13 and 3.14 need to be evaluated i n order to do t h i s . The crux of the problem 11 s s uc C i s to assign a value to the quantity k /k . There are two aspects of t h i s evaluation about which l i t t l e i s known: the contribution to the s t a r dose by neutrons which may make t h i s r a t i o f i e l d s i z e dependent, and the t o t a l energy released to charged p a r t i c l e s due to pion capture i n t i s s u e . There are two parts of the l a t t e r problem: lack of knowledge of the energy released to charged p a r t i c l e s per pion capture on the elements composing t i s s u e , and lack of knowledge of the capture process i t s e l f . In order to provide an estimate of the conversion f a c t o r , consider tissue to contain only oxygen and carbon. The Z-law gives capture p r o b a b i l i t i e s of 0.86 and 0.14 on oxygen and carbon r e s p e c t i v e l y . Assume that s t a r neutrons deposit 5.75 MeV/pion l o c a l l y . This i s 20% of the energy released to charged secondaries i n the case of carbon and i s t y p i c a l of the value expected for the large f i e l d s izes required for therapy. Also assume that the same amount of energy i s a v a i l a b l e from neutrons i n the case of t i s s u e . Then the s t a r energy r a t i o i s TF k (28.77 x 0.14) + (14.36 x 0.86) + 5.75 n ,, _ _ _ = = U.D4 k 28.77 +5.75 28.77 MeV and 14.36 MeV are the energies released per pion capture including that which goes to heavy undetected ions f o r carbon and oxygen respectively (see tables 2.1 and 2.2). The conversion factor f or carbon TE C to tissue i s the same as for equation 3.15 with S /S = 1.12 (61) M e e , and S T E / S C = 1.11 (80, 63): Tf TT 1.12 x 0.12 + 1.11 x 0.42 + 1.11 x 0.64 x 0.46 = 0.93 148 By allowing the star energy for oxygen to vary within reasonable l i m i t s , the uncertainty was estimated to be ± 0.05. This concludes the discussion of kerma. The absolute dose i n tissue at the peak for t h i s p a r t i c u l a r beam tune i s 30.7 yGy/DCI with an uncertainty of ± 7.4%. CHAPTER 4 CONCLUSION In order to perform an absolute dose determination f o r the TRIUMF biomedical negative pion beam, an i o n i z a t i o n chamber was used. The i o n i z a t i o n i n a carbon walled cavity was studied as a function of pressure both t h e o r e t i c a l l y and experimentally. Because the secondaries released by pion capture have ranges of the order of the s i z e of the cavit y , the theory used was one which e x p l i c i t l y considers pion stars which o r i g i n a t e within the cavity and pion secondaries which emerge from the walls with i n s u f f i c i e n t energy to cross the cavity. This c a l c u l a t i o n predicted that because pion capture on an oxygen nucleus y i e l d s less energy than capture on carbon, there would be a decrease i n the i o n i z a t i o n per unit mass as the pressure was increased f o r carbon dioxide f i l l i n g , but not for methane. Q u a l i t a t i v e l y , t h i s p i c t u r e was confirmed by the experiment. Due to the lack of quantitative agreement between the theory and experiment f o r the pressure dependence, the dose was determined by extrapolation of the i o n i z a t i o n per unit mass to zero pressure. For t h i s p a r t i c u l a r tune, the doses i n carbon and aluminum are summarized i n table 4.1. The error bar estimates the uncertainty of the accuracy. The error p r e c i s i o n i s less than h a l f of these-values. Tables of factors were 150 determined-to-be able -to-convert an i o n i z a t i o n reading to • dose for .carbon or aluminum walled chambers f i l l e d with a number of common gases. Table 4.1: Dose i n Carbon and Aluminum Dose i n Carbon Dose i n Aluminum 10~ 6 Gy/DCI 10~ 6 Gy/DCI peak 33.0 ± 1.7 31,1 ± 1.7 plateau 21.9 ± 1.2 21.4 ± 1.2 The kerma correction factor required to convert the dose,*due to pion stars alone, i n carbon to the dose i n aluminum was estimated to be 1.00 ± 0.03 f o r the peak. An estimate of the kerma cor r e c t i o n f a c t o r required to convert the dose, due to stars and contaminants, i n carbon to the dose i n tissue was 0.93 ± 0.05. In order to make an accurate evaluation of t h i s f a c t o r , there i s a need f o r a de t a i l e d knowledge of the pion capture process and for further experiments concerning the spectra of energetic secondaries emitted i n pion stars i n t i s s u e . Further dosimetry experiments are recommended i n two areas. In order to investigate the non-Bragg-Gray nature of an i o n i z a t i o n chamber, the i o n i z a t i o n per unit mass should be studied as a function of cavity gap while keeping the pressure f i x e d . In this way, the Sternheimer density correction f o r the electron stopping power and the hypothesized dependence of the W-value on pressure would be eliminated. I t i s expected that the dependence of the i o n i z a t i o n per unit mass on chamber s i z e would be less dramatic than was i t s dependence on pressure. The second area of needed research concerns the problem of sorting out the s t a r dose into i t s charged p a r t i c l e and neutron f r a c t i o n s . A l C A c a r e f u l study of the r a t i o k /k as a function of f i e l d s i z e may enable this to be done. This experiment could y i e l d r e s u l t s more quickly by using a proportional counter which can operate i n a t i m e - o f - f l i g h t mode. In one experiment the response could be studied as a function of f i e l d s i z e with the chamber set to look at only those events coincident with pion a r r i v a l . The experiment could be repeated i n an :untimed mode. The timed mode i s A l C expected to show the lar g e s t change i n k /k as the f i e l d s i z e i s varied. Another area for research not d i r e c t l y r e l a t e d to dosimetry concerns the electron trapping mechanism for electronegative gases which was hypothesized to be responsible f o r an apparent increase i n W-value with increased pressure. Such an e f f e c t would also cause pion trapping to be pressure dependent. I t might be possible to use pionic X-rays to learn about the pion capture process and about the electron trapping process by studying gas mixtures containing an electronegative gas at varying pressure The understanding of pion dosimetry has been furthered by the work discussed here, but the. f i e l d of pion dosimetry i s s t i l l young and much work remains to be done. This thesis has attempted to focus on the p r i n c i p l e s of measurement. 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Systematic stopping cross section measurements with low energy ions i n gases. Det Kongelige Danske Videnskabernes Selskab Matematisk-fysiske Meddelelser 39 #4, 1 (1974) 90. Teplova, Ya. A. et a l . Slowing down of multicharged ions i n s o l i d s and gases. Soviet Physics JETP 1_5, 31 (1962) 158 106. Boag, J.W. and Seelentag, W.W. A general saturation curve for an i o n i z a t i o n chamber f i l l e d with nitrogen at pressures up to 8 atmospheres. Phys. Med. B i o l . 20, 624 (1975) 107. Varma, M.N. and Baum, J.W. Energy dependence of W for alpha p a r t i c l e s i n N 2 , C02, CH4, Ar, H2 and Rossi-type tissue-equivalent gases. Phys. Med. B i o l . 23, 1162 (1978) 108. Bortner, T.E. and Hurst, G.S. Ionization of pure gases and mixtures of gases by 5 MeV alpha p a r t i c l e s . Phys. Rev. 93^, 1236 (1954) 109. Kuhn, H. and Werba, T. Measurements of the energy expenditure for the production of an ion pair i n tissue equivalent gas for heavy p a r t i c l e s . i n Proceedings of the Third Symposium on Neutron Dosimetry i n Biology and Medicine, ed. G. Burger, pp. 85-96. Luxembourg: Euratom (1977) 110. Jesse, W.P. Alpha p a r t i c l e i o n i z a t i o n i n argon-methane mixtures and the energy dependence of the ion pair formation energy. Phys. Rev. 174, 173 (1968) 111. Rohrig, N. and Colvett, R.D. Measurements of W for protons, helium-4 ions and carbon ions i n tissue equivalent gas. Brookhaven National Laboratory Report 23691 to be published i n Radiation Research (1979) 112. Boring, J.W., Strohl, G.E. and Woods, F.R. Total i o n i z a t i o n i n nitrogen by heavy ions of energies 25 to 50 keV. Phys. Rev. 140, A1065 (1965) 113. Wapstra, A.H. and Bos, K. Atomic Data and Nuclear Data Tables 19, 177 (1977) 114. Whyte, G.N. Energy per ion p a i r for charged p a r t i c l e s i n gases. Radiation Research 18, 265 (1963) 115. K e l l e r e r , A.M. Considerations on the random t r a v e r s a l of convex bodies and solutions for general c y l i n d e r s . Radiation Research 47, 359 (1971) 116. Birkhoff, R.D. et a l . The determination of LET spectra from energy proportional pulse.height measurements. I: Track length d i s t r i b u t i o n s i n c a v i t i e s . Health Physics .18, 1 (1970) 117. Sternheimer, R.M. and P e i e r l s , R.F. General expression for the density e f f e c t for the i o n i z a t i o n loss of charged p a r t i c l e s . Phys. Rev. B _3, 3681 (1971) 118. Kim, Y.S. Density e f f e c t i n dE/dx of f a s t charged p a r t i c l e s traversing various b i o l o g i c a l materials. Radiation Research _56, 21 (1973) 119. S c h i l l a c i , M. and Roeder, D. Dose d i s t r i b u t i o n due to n and y r e s u l t i n g from I T - capture i n t i s s u e . Phys. Med. B i o l . 18, 821 (1973) 159 91. Christophorou, L.G. "Atomic and molecular r a d i a t i o n physics." London: Wiley Interscience p. 492-500 (1971) 92. Z i e g l e r , J.F. personal communication concerning "Heavy ions: stopping powers and ranges." Volume 5 of "The stopping and ranges of ions i n matter." New York: Pergamon Press to be published (1979) 93. Hogberg, G. E l e c t r o n i c and nuclear stopping cross sections i n carbon for l i g h t mass ions of 4.5 to 46 keV energy. Phys. Stat. Sol. (b) 48, 829 (1971) 94. Pivovar, L.I., Nikolaichuk, L.I. and Rashkovan, V.M. Passage of l i t h i u m ions through condensed targets. Soviet Physics JETP ^0, 825 (1965) 95. Bernhard, F. et a l . Stopping cross sections of L i + ions with energies from 30 to 100 keV i n various target materials. Phys. Stat. Sol. 35, 285 (1969) 96. Stocker, H. and Berkowitz, E.H. Atomic charge state r a t i o s of ^Ll ions from 5.8-16.4 MeV. Can. J . of Phys. _49, 480 (1971) 97. Marion, J.B. and Young, F.C. "Nuclear reaction analysis graphs and tables." Amsterdam: North Holland (1968) 98. MacDonald, J.R. and Sidenius, G.. The t o t a l i o n i z a t i o n i n methane of ions with 1 « z $ 22 at energies from 10 to 120 keV. Phys. L e t t . 28A, 543 (1969) 99. Dennis, J.A. Computed i o n i z a t i o n and kerma values i n neutron i r r a d i a t e d gases. Phys. Med. B i o l . 18, 379 (1973) 100. Turner, J.E., Wright, H.A. and Hamm, R.N. Estimated W-values for negative pions i n N2 and Ar. Health Phys. J29, 792 (1975) 101. Turner, J.E. et a l . Estimated W-values for negative pions i n t i s s u e - equivalent gas, CO2 and N 2. Health Physics 32, 300 (1977) 102. D e l a f i e l d , H.J^ and Harrison, K.G. Ionization measurements and the d e r i v a t i o n of Wri> i n acetylene and,carbon dioxide i r r a d i a t e d with neutrons. Phys. Med. B i o l . _24, 271 (1979) 103. Edwards, A.A. and Dennis,, J.A. The c a l c u l a t i o n of charged p a r t i c l e fluence and LET spectra for the i r r a d i a t i o n of b i o l o g i c a l l y s i g n i f i c a n t materials by neutrons. Phys. Med. B i o l . _20, 395 (1975) 104. Loosemore, W.R. and K n i l l , G. The c o l l e c t i o n e f f i c i e n c y of a mean current f i s s i o n i o n i s a t i o n chamber. Atomic Energy Research E s t a b l i s h - ment at Harwell Report AERE-R3677 (1961) 105. Dennis, J.A. and Edwards, A.A. National Radiological Protection Board Report M20 (1975) 160 120. Brenner, D.J. and Smith, F.A. Dose and LET d i s t r i b u t i o n s due to n and y emitted from stopped TT~. Phys. Med. B i o l . 22̂ , 451 (1977) 121. Lang, H. and Harrison, R.W. The biomedical beam l i n e c o n t r o l system at TRIUMF. TRIUMF Report TRI-I-75-2 (1975) 122. Henkelman, R.M. et a l . Tuning of the biomedical negative pion beam l i n e at TRIUMF. Nucl. Inst. Meth. _155, 317 (1978) 123. Pbon, M.N.C. Optimization studies of the TRIUMF biomedical pion beam. M.Sc. Thesis, University of B r i t i s h Columbia (1977) 124. Skarsgard, L.D. et a l . Physical and r a d i o b i o l o g i c a l properties of the negative pi-meson beam at TRIUMF. IAEA-SM-212/65 (1977) 125. Nordell, B. et a l . Determination of some parameters f o r pion radio- biology studies. Phys. Med. B i o l . _22, 466 (1977) 126. Turner, J.E. et a l . The computation of pion depth dose curves i n water and comparison with experiment. Radiation Research _52, 229 (1972) 127. A l s m i l l e r , R.G., Barish, J . and Dodge, S.R. Energy deposition by high energy electrons (50 to 200 MeV) i n water. Preprint ORNL-TM-4419 (1974) 128. Ito, A. and Henkelman, R.M. Microdosimetry of the pion beam at TRIUMF. submitted to Radiation Research (1979) 129. Goodman, L- Density and composition uniformity of A-150 t i s s u e - equivalent p l a s t i c . Phys. Med. B i o l . 23_, 753 (1978) 130. Personal communication with Canadian Stackpole regarding graphite grade SR-30 131. Personal communication with Wilkinson regarding aluminum grade Alcan 2S #1100 132. . American I n s t i t u t e '• of-' Physics Handbook-, • ed.--D.E-. -Gray.- .New- York:-. McGraw-Hill (1972) 133. Boag, J.W. Ionization chambers. Chapter 1 of "Radiation dosimetry, Volume II Instrumentation," ed. F.H. A t t i x and W.C. Roesch. New York: Academic Press (1966) 134. J a f f e , G. Zur Theorie der Ionisation i n Kolonnen I. Annalen der Physik 42, 303 (1913) 135. Xanstra, H. A short method for determining the saturation current according to J a f f e ' s theory of column i o n i s a t i o n . AERE-TRANS 876 tra n s l a t o r J.B. Sykes (1961) 136. D e l a f i e l d , H.J. personal communication (1978) 161 137. Mustafa, S.M. and Mahesh, K. C r i t e r i o n f o r determining saturation current i n p a r a l l e l plate i o n i z a t i o n chambers. Nucl. Inst. Meth. 150, 549 (1978) 138. International Commission on Radiation Units and Measurements. ICRU Report 16, "Linear Energy Transfer." Washington (1970) 139. Lam, G.K.Y., Henkelman, R.M. and Harrison, R.W. An automated dose mapping system f or the TRIUMF biomedical pion beam. Phys. Med. B i o l . 23, 768 (1978) 140. K l e i n , U. Measurement of neutrons emitted following the absorption of stopped negative pions i n the b i o l o g i c a l l y relevant n u c l e i l^C, 1 4N and L 60., Ph.D. Thesis, Karlsruhe University (1978) DO NOT COPY APPENDIX A : P H Y S . M E D . B I O L . , 1978, V o l . 23 , N o . 3, 4 9 5 - 4 9 8 . P r i n t e d i n G r e a t B r i t a i n Scientific Note A Charge Collector to Determine the Stopping Distribution of a Pion Beam K . R. SHORTT, M.SC. and R. M. H E N K E L M A N , PH.D. Batho Biomedical Facility, TRIUMF, University of British Columbia, Vancouver, Canada V6T 1W5 Received 31 October 1977, in final form 15 December 1977 1. Introduction Because of the variation in quality throughout the radiation field produced by a beam of negative 7t mesons, the specification of the field by dose alone is not adequate for a determination of the biological effectiveness of the radiation. Microdosimetric measurements of various types have been used to provide supplementary information about the quality of pion radiation fields (Lucas, Quam and Raju 1969, Amols, Dicello and Lane 1976, Richman 1976). It is anticipated that such measurements will be useful for the prediction of the variable biological effectiveness of the pion radiation. In a pion radiation field most of the densely ionising radiation is generated by the charged secondary particles from the pion stars and by the pions at the extreme ends of their tracks. The energy from the charged-particle secondaries is deposited close to the pion star; 50% within 0-3 mm of the star (Henry 1973). It is therefore expected that the spatial distribution of the densely ionising component of a pion radiation field will be well represented by the spatial distribution of the pion stars. We have therefore investigated the use of a charge collector as a means of measuring the depth distribution of pion stars. 2. The detectors and measurements Charge collectors have been used to measure the net charge deposition pattern in electron beams (Laughlin 1965, Van Dyk and MacDonald 1972). A detector of the type used by Van Dyk and MacDonald (1972) is shown in fig. 1. It consists of a polystyrene disc 100 mm in diameter by 1 mm thick which is coated with Aquadag and connected to the central lead of a coaxial cable. The complete assembly is covered with a close fitting 1 mm thick polystyrene insulator which is itself coated with Aquadag and connected to the sheath of the coaxial cable. The complete assembly is housed in, a polystyrene water- proof housing. Charge which is deposited in the inner collector or out to a thickness half way through the 1 mm insulating layer will either leak to the innermost Aquadag surface or will induce a charge of opposite sign onto that surface and thus, in either case, produce a current of one unit charge on the 0 0 3 1 - 9 1 5 5 / 7 8 / 0 0 0 3 - 0 4 9 5 $01 .00 © 1978 T h e I n s t i t u t e o f P h y s i c s DO NOT COPY K. R. Shortt and R. M. Henkelman central lead of the coaxial cable. This geometry results in an effective collection volume of 101 mm diameter by 2 mm thick. The detector was suspended in a water phantom by a remote controlled three-dimensional scanner. Measurements at shallow depths were made in air with various thickness of polystyrene absorber in front of the charge collector. The depths in polystyrene were converted to the equivalent depths in water Fig. 1. The construction of the polystyrene charge collector showing the two conducting layers and the cable connections. using density and stopping power corrections. The detector was irradiated in the TRIUMF biomedical pion beam (Harrison and Lobb 1973, Henkelman, Skarsgard, Lam, Harrison and Palcic 1977) with a 148 MeV/c beam which was composed by number of 50% pions, 10% muons and 40% electrons. The momentum acceptance of the beam was measured to be ± 7 % Apjp F W H M . The beam had lateral dimensions of 4 cm x 4 cm F W H M and so was completely intercepted by the charge collector. The flux of the incident beam was monitored with a transmission ionisation chamber which had been calibrated to measure particle flux at low flux rates. The experiment was performed using an incident flux of 107 particles s - 1. The charge from the collector was integrated using a Keithley Model 610C electrometer. The electrometer was zeroed to give a null reading when the beam was off. Multiple readings were made at each depth and the mean and standard error were calculated. 3. Results The amount of charge collected per incident particle as a function of the effective depth in water to the centre of the collector is shown in fig. 2. The error bars are the standard deviations of 10 measurements. A subsequent experiment showed a similar distribution of measured values. The measure- ments show a collection of positive charge at depths which are less than 1 cm corresponding to the knock out of negative electrons from the medium by the incident pions. The mean energy of 8-rays from fast pions is expected to be of the order of 1 MeV. Coaxial cable' DO NOT COPY Charge Deposition by Stopping Pions There is then little net charge deposition until a depth of 10 cm where a net deposition of negative charge is observed extending to approximately 17 cm. This corresponds to the charge deposited by the stopping pions. The solid curve in fig. 2 is a differential range curve measured with a plastic scintillator range telescope at low beam intensity for the same pion beam tune. The differential range curve is arbitrarily-normalised and the baseline is adjusted to a -3 •o I • 1 , , ^ 0 5 10 15 20 Depth in water (cm) F i g . 2. T h e n e t c h a r g e c o l l e c t e d a s a f u n c t i o n o f d e p t h i n a p h a n t o m i r r a d i a t e d i n t h e p i o n b e a m . T h e s o l i d c u r v e i s a d i f f e r e n t i a l r a n g e c u r v e m e a s u r e d f o r t h e s a m e t u n e w h i c h h a s b e e n s c a l e d t o p a s s t h r o u g h t h e d a t a . D a t a p o i n t s a r e s h o w n w i t h s t a n d a r d e r r o r s o f t h e m e a n . match the charge collection curve. However, the agreement in shape and range between the two measurements confirms that the charge collector is measuring the pion stopping distribution. Furthermore, the area under this stopping peak above the interpolated background corresponds to — 0-9 ± 0-2 elementary charges per stopping pion. The net collection of positive charge before and after the stopping peak may not be significant. The dashed line which has been drawn through the data points has no statistical significance. In the region between 1 and 10 cm in depth there will be deposition of negative charge by in flight capture of pions and depletion of negative charge by the knock out of secondary electrons by the incident 148 MeV electrons of the beam. Delta rays from these high energy electrons could have ranges comparable to the dimensions of the phantom. Depending on the relative magnitude of these two effects, a net charge collected of either sign could be explained. Since very few of the electrons are expected to stop in a range of up to 25 cm, they are not generally detected by this type of measurement. 4. Discussion It has been possible using a charge collector to measure the stopping distribu- tion of negative pions even in the presence of a large electron contamination. It is anticipated that measurements made in a pion beam with low electron contamination would show a reduced background which would in turn improve the ability to localise the pion stars even where the stopping distribution is 19 DO NOT COPY Charge Deposition by Stopping Pions extended in depth. An electron knock on region at shallow depths has also been detected, although it is unlikely that this will correspond to any significant dose build-up since most of the entrance dose is directly deposited by the incident charged particles. It is anticipated that the ability to measure the stopping distribution in high intensity pion beams will provide another means of characterising the high LET component of the pion radiation field and thus be of predictive value for biological isoeffect. We wish to thank Dr. J. C. F. MacDonald for the loan of the charge collector used in this experiment. K. R. S. is a research student of the National Cancer Institute of Canada. REFERENCES AMOLS, H., DICELLO, J., and LANE, T., 1976, in Proc. Fifth Symp. on Midrodosimetry, EUR5452 d-e-f (Euratom) pp. 911-928. HARRISON, R. W., and LOBB, D. E. , 1973, IEEE Trans. Niccl. Sci., NS -20, 1029-1031. HENKELMAN, R. M., SKARSGABD, L. D., LAM, K. Y., HARBISON, R. W., and PALCIC, B., 1977, Int. J. Radiat. Oncol. Biol. Phys., 2, 123-127. HENBY, M. I., 1973, M.Sc. Thesis, University of British Columbia. LAUGHLTN, J . S., 1965, in Symp. on High-energy Electrons, Montreux, Ed. A. Zuppinger and G. Poretti (Berlin: Springer-Verlag) pp. 11-16. LUCAS, A. C , QUAM, W. M., and RAJTJ, M., 1969, EG & G Technical Paper No. S-54-TP. RICHMAN, C , 1976, Radiat. Res., 66, 453-471. VAN DYK, J. , and MACDONALD, J . C. F., 1972, Radiat. Res. 50, 20-32. ,166 APPENDIX B: DEFINITIONS OF QUANTITIES, EQUATIONS AND CONSISTENCY CHECKS This appendix i s a c o l l e c t i o n of various terms and r e l a t i o n s h i p s required by the c a l c u l a t i o n discussed i n chapter 2. stopping power: S(e) = — - 7 — with units MeV-cm2/g pdx range: ) s o J 0 R(E) = \ 1T7~T d e w i t n units g/cm2 production spectrum: /„\ „ ^ / \ . , secondaries Np(E) = n Kp(e) with units MeV-g where II i s the pion stopping density with units pion stops/g and Kp(e) i s the number of secondaries per pion stop with . ' . secondaries energy between e and e + de with units —: rr-rr 6 1 7 pion stop-MeV slowing down spectrum: (equilibrium spectrum) •E , - I. \ M A X „ , s . , . secondaries Nr(E) = 'ZTZ7\ \ Np(e) de: with units KB, ^ MeV-cm2 e=E The t o t a l number of secondaries crossing unit area i s given by Nr(E) dE = \ R(e) Np(e) de J E=0 e m E=0 Check #1 tests the value of the r a t i o of these two i n t e g r a l s , R, and i s found i n column 5 of table B . l . 167 Bragg-Gray i n t e g r a l : kerma: •5; max BG = \ S(E) Nr(E) dE with units MeV/g For an i n f i n i t e s i m a l c a v i t y , the dose deposited i s given by the stopping power of the cavity medium averaged over the spectrum of secondaries crossing the cavity from the w a l l . I f the cavity i s f i l l e d with w a l l , that i s i t has the same spectrum of secondaries and same stopping power as the w a l l , then Fano's theorem concludes that the dose deposited i n a l l c a v i t i e s regard- les s of t h e i r s i z e or pressure i s the same as given by the Bragg- Gray i n t e g r a l , or a l t e r n a t i v e l y , the kerma. Check #2 tests the r a t i o D ,/K and i s found i n column 6 of table B . l . The t o t a l t o t a l dose i n column 3 i s the mean over eight d i f f e r e n t a r e a l d e n s i t i e s and the fact that the rms deviation i s small i s another test of check #2. max K = \ e Np(e) de with units MeV/g In the c a l c u l a t i o n , kerma has been r e s t r i c t e d to the energy deposited by charged secondaries released by the pion-nucleus i n t e r a c t i o n . In t h i s case, kerma i s the value of the dose that would be deposited at the centre of a large block of material which i s exposed to a stopping density of 1 pion/g where K MeV are released per pion stop. The t o t a l dose i n the slowing down spectrum i s given by the Bragg- Gray i n t e g r a l which equals the kerma for w a l l f i l l i n g the cavity. Check #3 tests the r a t i o BG/K and i s found i n column 7 of table B . l . 168 Table B . l i l l u s t r a t e s the r e s u l t of the c a l c u l a t i o n for a sph e r i c a l cavity containing w a l l material. The averages are taken over eight d i f f e r e n t a r e a l densities increasing by factors of two from 2.44 mg/cm2 to 312 mg/cm2. This i s equivalent to a 2 cm diameter sphere when f i l l e d with carbon dioxide at pressures between 0.25 and 32 (xlOO kPa) Table B . l : Checks of Internal Consistency #1 #2 #3 Secondary Kerma Tot a l Dose BG Integral R D t o t a l / K BG/K (MeV/g) (MeV/g) (MeV/g) (%) (%) (%) P 10.505 10.501 ± .004 10.544 ± .006 -.00 -.04 .38 d 6.383 6.379 ± .002 6.411 ± .006 .01 -.06 .43 t 3.074 3.071 ± .001 3.091 ± .006 .05 -.10 .56 He 3 .640 0.638 ± .001 0.643 ± .002 .15 -.22 .44 He 4 5.510 5.491 ± .009 5.525 ± .013 .26 -.35 .26 L i 1.337 1.330 ± .005 1.338 ± .002 .28 -.56 .09 The table indicates that systematic errors are less than %% and are therefore n e g l i g i b l e . APPENDIX G: EXTRACTION OF OXYGEN SPECTRA The secondary spectra f or pion capture i n oxygen can be extracted from the data for mylar and acetate which exhibit d i f f e r e n t capture r a t i o s f o r .carbon to oxygen. Independent measurement of the secondary spectra f o r carbon l e d to the adoption of the following extraction procedure f o r oxygen. For each p a r t i c l e type and each energy, l e t the true y i e l d s be a^, a^, a^ and a^ for capture on carbon, oxygen, acetate and mylar res p e c t i v e l y . Let the measured y i e l d s be b = a + a for carbon C l b, = a, + 3 for acetate C.2 A A and b M a^ + y for mylar C.3 where a, g, and y are errors i n the measurement. Since the capture r a t i o (carbon:•oxygen) i s 1.1:1 for acetate and 1.7:1 for mylar (27), 1.1 an + 1.0 an = 2.1 a A C.4 C 0 A and 1.7 a + 1-.0 a = 2.7 4 ^ C.5 Eliminating a^, 2.1 a^ - l . l a ^ = 2 . 7 a ^ - 1.7 a^ C.6 Substituting C l , C.2 and C.3 into C.6 and rearranging gives D'= 0.6 b - + • 2; 1 h - 2.7• b • = 0.6 a + 2.1 3 - 2.7 y C A M Minimizing the sum of the squares of the errors (a, g, and y) i s achieved by assigning 0.6 D 01 = : ; (0.6) 2 + (2.1) 2 + (2.7) 2 and so on. Then a^ can be obtained from e i t h e r C.4 or C.5 as a 0 = 2 , 1 ( b A ' 3 ) " 1 , 1 ( b C " a ) C ' 7 or a Q = 2.7 ( b M - y) - 1.7 ( b c - a) C,8 The values obtained for the y i e l d s from carbon by t h i s e x traction procedure were approximately 1% higher than the values measured i n the experiment by Mechtersheimer (25). Therefore, both the carbon and oxygen y i e l d s as extracted by the above procedure were normalized by this r a t i o to maintain consistency with the carbon data which had been used extensively i n e a r l i e r c a l c u l a t i o n s . 171 APPENDIX D: STERNHEIMER DENSITY CORRECTION TO THE STOPPING POWER OF ELECTRONS The c a l c u l a t i o n of the Sternheimer density correction i s based on reference (117) and the notation i s from reference (118). The Bethe- Bloch formula f o r electrons i s given by: = 0^1535 / Z \ r £ n { ( 2 T + 4 ) M 2 } + F - 6 - 2 l n / l ) + 27.63l] dx (v/c) X A / m = .511 MeV/c2 v = electron speed c = photon speed . T = reduced k i n e t i c energy = T/M </T-N> = 2 W. Z./A. where W. i s the f r a c t i o n a l weight of the i * " ^ element N A / i = j i x x x . - • • • -| - ( 2 T + 1) £n2 F = kinematics term = 1 - ( v / c ) 2 + 2 £n(x/2) + ( T + l ) 2 ^ 1 ^ = adjusted mean e x c i t a t i o n energy (eV) given by £„<!> •- ( j ' w ; I u !,)/<!> <5 = density c o r r e c t i o n = 4.606 X + C X = l o g 1 0 (3/(1 - = l o g 1 0 (P/MC) C = -2 £n{ <I> /(28.804 , /p <Z/A> )} - 1 p = density (g/cm3) Values of I for hydrogen, carbon and oxygen were taken to be 18.7, 78.0 and 89.0 eV respectively (138). 172 Table D.1: Parameters Required to Calculate the Density Correction Parameter Carbon Dioxide Methane Carbon < Z / A ) > 1/2 5/8 1/2 log < I > 4.453 3.785 4.357 <I> (eV) 85.85 44.05 78.0 p (g/cm3) 1.8296 x 10 3 .6633 x 10' 3 1.77 at 1 atmosphere and 22% C Electrons of momentum 180 MeV/c, have k i n e t i c energy of 179.5 MeV and a range i n water of roughly 93 cm (61). In the plateau (depth -~15 cm) the r e s i d u a l energy i s about 150 MeV and i n the peak (depth *~ 22 cm) i t i s about 130 MeV. Using t h i s to evaluate T , F and X gives dE dx .1535 <Z/A)> [ K - 6 - 2 In <I>| and <5 = 4.606 X - 2 ln{ <£> /(28.804 sjp <Z/A> ) } - 1 .where K 42.350 and X = 2.407 in:.the peak and K = 42.778 and X = 2.469 i n the plateau. This has been used to c a l c u l a t e the stopping powers shown below. Table D.2: Electron Stopping Powers ' (MeV-cm2/g) Pressure - Carbon Dioxide Methane (atmospheres) Peak Plateau Peak Plateau 0 2.566 2.599 3.336 3.377 1 2.496 2.508 3.196 .-3.210 2 2.443 2.454 3.130 3.144 4 2.390 2.401 3.063 -3.077 8 2.337 2.348 2.997 3.011 These can be converted to stopping power r a t i o s with respect to carbon by noting that the stopping powers f or carbon i n the peak and plateau are 1.970 and 1.981 MeV-cm2/g re s p e c t i v e l y .

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