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A determination of multiple scattering for a negative pion beam Watts, Larry James 1978

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A DETERMINATION OF MULTIPLE SCATTERING FOR A NEGATIVE PION BEAM B.Sc, University of British Columbia, 1971 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES. Department of Physics We accept this thesis as conforming to the required standard by LARRY JAMES WATTS THE UNIVERSITY OF BRITISH COLUMBIA December, 1977 Larry James Watts, 1977 In presenting th i s thes is in pa r t i a l fu l f i lment of the requirements for an advanced degree at the Univers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make it f ree ly ava i l ab le for reference and study. I further agree that permission for extensive copying of th is thesis for scho lar ly purposes may be granted by the Head of my Department or by his representat ives. It is understood that copying or pub l i ca t ion of th is thes is for f inanc ia l gain sha l l not be allowed without my writ ten permission. Department of Phys ics The Univers i ty of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date December 21, 1977 ABSTRACT The multiple Coulomb s c a t t e r i n g of negative pions has s i g n i f i c a n t e f f e c t s on the dose d i s t r i b u t i o n s r e s u l t i n g from pion beams incident on thick targets. The use of negative pions i n radiotherapy requires a detail e d knowledge of the d i s t r i b u t i o n of dose and b i o l o g i c a l e f f e c t . Thus i t i s important to have an accurate d e s c r i p t i o n f o r the l a t e r a l d i s t r i b u t i o n s of pions which r e s u l t from multiple s c a t t e r i n g . I t has been proposed by Fowler and Perkins that these l a t e r a l d i s t r i b u t i o n s are of a Gaussian nature f o r incident p e n c i l beams. In t h i s study an attempt has been made to determine experimentally and t h e o r e t i c a l l y the appropriate value f o r the standard deviation of the Gaussian i n the p e n c i l beam d e s c r i p t i o n . The experimental determination involved placing medical x-ray fi l m s i n a homogeneous water phantom, perpendicular to the beam axis of the M8 biomedical channel at TRIUMF. The d i s t r i b u t i o n s recorded on f i l m for c i r c u l a r l y collimated beams were measured f o r o p t i c a l density and compared to calculated d i s t r i b u t i o n s i n order to extract the p e n c i l beam information. The presence of contaminating electrons and muons as w e l l as the d i f f i c u l t y i n achieving a p a r a l l e l beam complicated the determination of the standard deviation of the Gaussian f o r pions. The experimental determination at the end of a 20.1 cm range i n water i s only 7% greater than the preferred t h e o r e t i c a l c a l c u l a t i o n f o r pions alone. This c a l c u l a t i o n i s based on the f i r s t (Gaussian) term of Moli&re^s theory modified f o r the Fano co r r e c t i o n and energy loss and - i i i -y i e l d s r e s u l t s 20% lower than those of the "standard reference" of Fowler and Perkins. The agreement between the theory f o r pions and the experi-ment for a r e a l beam i n water indicates that the theory presented should be adequate f o r treatment planning c a l c u l a t i o n s . - iv -TABLE OF CONTENTS PAGE TITLE i ABSTRACT i i TABLE OF CONTENTS iv LIST OF TABLES vi LIST OF FIGURES v i i ACKNOWLEDGEMENTS ix 1. INTRODUCTION 1 2. MULTIPLE SCATTER THEORY 6 2.1, Introduction 6 2.2, Pencil Beams 6 2.2.1. Classical Derivation of Gaussian Distribution . . 6 2.2.2. Theory of Moliere 10 2.2.3. Modifications Proposed for Moliere's Theory . . . 14 2.2.4. Application to Thick Scatterers 19 2.3, Finite Size Beams 23 3. EXPERIMENTAL CONSIDERATIONS 30 3.1. Introduction 30 3.2. Film as a Detector 30 3.3. Beamline Characteristics and Setup 34 3.4. Experimental Results 41 3.4.1. Preliminary Experiments , 41 3.4.2, Multiple Scatter Experiments 48 - v -4. DISCUSSION 57 4.1. Introduction 57 4.2. Assumptions i n the Data Analysis 57 4.3. Experimental Factors 59 4.3.1. E f f e c t of Momentum Spread 59 4.3.2. E f f e c t of Contaminating Electrons and Muons . . . 60 4.3.3. E f f e c t of Background Scatter 62 4.3.4. F i l m as a P a r t i c l e Counter 64 4.3.5. E f f e c t of Non-parallel Beam 68 4.4. Comparison with Theory 69 4.5. Comparison to Values i n the L i t e r a t u r e 74 5. CONCLUSIONS 78 BIBLIOGRAPHY 81 APPENDIX A 85 APPENDIX B 92 APPENDIX C 94 - v i -LIST OF TABLES PAGE I FANO CORRECTION u. FOR VARIOUS VALUES OF Z 14 i n II CALCULATION OF WATER EQUIVALENT THICKNESS FOR MATERIALS IN FRONT OF WATER 40 III EXPECTED ACCURACY FOR COMPARING OPTICAL DENSITIES [Dutreix (31)] 45 IV SUMMARY OF PARALLEL FILM MEASUREMENTS 47 V SUMMARY OF MULTIPLE SCATTER EXPERIMENTS PERFORMED 48 VI EXPERIMENTAL VALUES OF a FOR 148 MEV/c WIDE BLADE CASE (AVERAGE OF 2 EXPERIMENTS) 52 VII EXPERIMENTAL VALUES OF a FOR 180 MEV/c WIDE BLADE CASE (AVERAGE OF 2 EXPERIMENTS) 53 VIII EXPERIMENTAL VALUES OF a FOR 180 MEV/c NARROW BLADE CASE (1 EXPERIMENT) 54 IX EXPERIMENTAL VALUES OF a FOR 180 MEV/c WIDE BLADE CASE (1 EXPERIMENT) STRAIGHT EDGE 56 X SUMMARY OF a FOR ALL EXPERIMENTS 56 ms XI COMPARISON OF a WITH CALCULATED CONTRIBUTIONS FROM THE WATER TANK WALL FOR R = 21.5 AND 6 = 1.4 63 XII CALCULATION OF a FROM AXIAL DENSITY AFTER CORRECTION FOR INTFLIGHT INTERACTIONS, 148 MEV/c 67 XIII CALCULATION OF a FROM AXIAL DENSITY AFTER CORRECTION FOR INT-FLIGHT INTERACTIONS, 180.MEV/c 67 XIV COMPARISON OF a FROM THEORY AND EXPERIMENT 71 ms XV COMPARISON OF CALCULATED VALUES FOR a FOR RANGE OF 20.1 CM ^ 75 A l VALUES FOR ENERGY RELATED PARAMETERS AS A FUNCTION OF RANGE FOR PIONS IN WATER 86 A l l SAMPLE CALCULATION OF B~ = C 3R C 4 FOR WATER USING MOLIERE'S f(a 2) EQUATION 90 - v i i -LIST OF FIGURES PAGE 1. MULTIPLE SCATTERING GEOMETRY USED TO DEFINE SCATTERING ANGLE 6 [adapted from Hungerford et a l . (19)] 8 2. MOLIERE DISTRIBUTION PLOTTED AGAINST REDUCED ANGLE G FOR TWO VALUES OF B PARAMETER T 13 3. COMPARISON OF GAUSSIAN TERM WITH NORMALIZED MOLIERE DISTRIBUTION FOR TWO VALUES OF B PARAMETER 17 4. SCHEMATIC DIAGRAM OF SMALL ANGLE SCATTER CONTRIBUTION TO LATERAL DISPLACEMENT FOR THICK SCATTERING MEDIUM 20 5. SCHEMATIC DIAGRAM OF CONTRIBUTION TO THE DISTRIBUTION FOR A POINT AT DEPTH x FROM AN ELEMENT AT THE SURFACE 26 6. CALCULATED DISTRIBUTIONS FOR GAUSSIAN SCATTER WITH CIRCULAR APERTURE FOR SELECTED VALUES OF a' (DESCRIBED IN REDUCED PARAMETERS) 29 7. TYPICAL FILM CHARACTERISTIC CURVES FOR LIGHT AND IONIZING RADIATION (X RAYS OR CHARGED PARTICLES) [adapted from Mees and James (34)] 32 8. SCHEMATIC DIAGRAM OF M8 BIOMEDICAL CHANNEL AT TRIUMF 35 9. EXAMPLE OF RELATIVE PARTICLE FRACTIONS PLOTTED AGAINST MOMENTUM SHOWING EFFECT OF PROTON BEAM STEERING [from Poon (40)] 37 10. SCHEMATIC DIAGRAM OF EXPERIMENTAL SETUP IN PLAN VIEW (MULTIPLE SCATTERING EXPERIMENTS) 39 11. EXPOSURE TEST FOR 148 MEV/c CASE 43 12. EXPOSURE TEST FOR 180 MEV/c CASE 44 13. DENSITOMETER RESPONSE (TEST OF REPRODUCIBILITY) 46 14. VARIATION OF Y PARAMETERS WITH SCATTER PARAMETER a' FOR VALUES OF MAGNIFICATION A 50 15. COMPARISON OF a FOR THEORY (CURVES) AND EXPERIMENT (POINTS) AGAINS?SDEPTH IN WATER 70 16. COMPARISON OF SCATTER PARAMETER FOR THEORY (CURVE) AND EXPERIMENT (POINTS) AS A FUNCTION OF RANGE RELATIVE TO END OF RANGE VALUES 72 17. CALCULATED DISTRIBUTIONS FOR TWO CIRCULAR RADII COMPARING VALUES OF a AT R. = 20,1 cm 76 ms 0 - v i i i -C l . SCHEMATIC DIAGRAM AND PARAMETERS FOR A BEAM CONVERGING TO A POINT 95 C2. PHASE SPACE PLOTS FOR ZERO EMITTANCE BEAMS AT DEPTH x - 0 95 - i x -ACKNOWLEDGEMENTS This project was c a r r i e d out using the f a c i l i t i e s of the Batho Biomedical F a c i l i t y at TRIUMF and the A. Maxwell Evans C l i n i c of the Cancer Control Agency of B r i t i s h Columbia. The author would l i k e to express h i s gratitude to those people associated with both i n s t i t u t i o n s who gave t h e i r assistance during the course of t h i s project. In p a r t i c u l a r , the author would l i k e to thank Dr. R.M. Henkelman f o r many h e l p f u l discussions and Dr. R.O. Kornelsen f o r suggesting and supervising t h i s project, as w e l l as for h i s i n s t r u c t i o n i n r a d i o l o g i c a l physics. The f i n a n c i a l assistance of the B r i t i s h Columbia Cancer Foundation i s g r a t e f u l l y acknowledged. The author would also l i k e to thank his wife, Leslee, f o r her encouragement and patience over the length of t h i s p r o ject. F i n a l l y , while one^s r e l i g i o u s b e l i e f s are e s s e n t i a l l y a personal matter, the author acknowledges h i s own dependence on the sustaining grace of the Eternal God, mediated through His Son, Jesus C h r i s t . 1. INTRODUCTION The use of negative pi-mesons (pions or TT ) in the radio-therapeutic treatment of cancer was f i r s t proposed by Fowler and Perkins (1), in 1961. Since that time, much effort has been expended in the investigation of the advantages of pions over conventional gamma radiation. The two main advantages cited (2,3) are: (i) better dose concentration at the tumour location compared to surrounding tissue; this i s due to the increased energy deposition from charged particles slowing down (Bragg peak effects) and in the case of TT , from the additional charged particles produced i n the "stars" resulting from the nuclear capture of the stopped pions; ( i i ) the reduced dependence on the presence of oxygen' [lower Oxygen Enhancement Ratio (OER)], that results from the high Linear Energy Transfer (LET) character of primary and secondary charged particles at the pion stopping location; i t has been demonstrated that fu l l y oxygenated cells suffer more damage than anoxic cells for the same doses and since tumours are believed to have anoxic regions, i t i s a n t i c i -pated that the differential sensitivity between oxygenated and anoxic cells in the tumour w i l l be reduced at the pion stopping location. The desire to exploit these advantages has led to the incorporation of radiobiology and radiotherapy projects at the three f a c i l i t i e s capable of producing TT beams of sufficient intensity. In fact, some patients have already been treated in preliminary t r i a l s (4,5). At the Batho - 2 -Biomedical F a c i l i t y at TRIUMF, patient treatment is s t i l l approximately one year from realization, but physical measurements and preliminary radiobiology have been going on since June, 1975 (6,7). Before treatment can be carried out, i t is necessary to know in detail the distribution of dose with position, both in depth and lateral extent. While some i n i t i a l treatment and biology can be done using dose distributions determined experimentally in a homogeneous medium, the treatment of many patients, with varying tumour locations and volumes, w i l l require a general method of calculating dose d i s t r i -butions in an inhomogeneous medium. Several factors must be accounted for in the calculation of these dose distributions. One factor that i s required in either a simple empirical calculation [such as that of L i et a l . (8)] or a detailed Monte Carlo treatment [such as those of Armstrong and Chandler (9) or Turner et a l . (10,11)] is a knowledge of the multiple scattering of pions. The concept of a "pencil" beam, i.e. a beam of particles which, i n i t i a l l y , has a unique starting direction and is of zero spatial extent, is used in both of the above schemes for calculating dose distributions. The lateral distribution of pions resulting from the multiple scatter of such pencil beams has been described by Fowler and Perkins (1) to be Gaussian, with symmetry about the incident direction. In addition, Fowler and Perkins indicated that the magnitude of the standard deviation, a, in the Gaussian, was of the order of 1 cm for pions at the end of a 20 cm range i n tissue. For f i n i t e size beams, the multiple scatter w i l l affect the dose primarily near the edges [see for example Hamm et a l . (12)]; in - 3 -the above case within about 2 cm either side of the geometrical edge. This effect i s a major reason for the d i f f i c u l t y in confining the stopping pions to the tumour volume, and the magnitude of these effects creates problems for the treatment of small tumour volumes at these depths [see Fowler and Perkins (1)]. Another situation where the effect of scatter i s important is in the case of inhomogeneities, such as the presence of bone or ai r , in tissue. This situation has been discussed by Santoro et a l . (13) and Hamm et a l . (14,15) for cases where the inhomogeneities are smaller than the pion beam. There is a large effect on the resulting dose distributions due to both the density difference, and the complication of the paths as a result of multiple scattering. In view of the importance of the multiple scattering of pions on the resulting dose distributions, a study was made to determine experimentally the appropriate value of a to be used in the pencil beam description. In addition, theoretical calculations of this parameter were made for comparison with experiment and to provide a description of the variation of a with depth, suitable for use in treatment planning calculations. The experimental determination of a parameter, such as the standard deviation, for a pencil beam of pions from a real f i n i t e size beam is complicated in our case due to two problems: (1) the ina b i l i t y to achieve a parallel beam (2) the presence of contaminating electrons and muons. The effect of these problems on our determination of a is discussed in Chapter 4. - 4 -Medical x-ray f i l m was chosen as our detector i n the experiments due to the following requirements: (1) adequate s e n s i t i v i t y at the i n i t i a l l y low dose rate (2) good s p a t i a l r e s o l u t i o n (3) i n t e g r a t i n g detector system, because of the i n e v i t a b l e f l u c t u a t i o n s i n the proton beam i n t e n s i t y during the early development of the cyclotron. The blackening of the f i l m depends on energy above some threshold being deposited i n each grain. For p a r t i c l e s with a high LET, f i l m responds pr i m a r i l y to the r e l a t i v e number of p a r t i c l e s . We have regarded f i l m as a " p a r t i c l e counter" and i n Chapter 4 give some experimental j u s t i -f i c a t i o n for t h i s assumption. In the experiment, f i l m s , i n a water tank, were placed perpendicular to the beam axis of the M8 biomedical channel at TRIUMF. Measurements were made at two d i f f e r e n t channel momenta with and without water i n the phantom to determine the sca t t e r i n g e f f e c t of water. Water was chosen since i t i s s i m i l a r to some types of ti s s u e i n the pion ranges and dose deposition (since oxygen i s a major component i n t i s s u e and i n water). For the majority of the experiments c i r c u l a r f i e l d s with two d i f f e r e n t r a d i i were used. In addition, one experiment was c a r r i e d out using a beam i n i t i a l l y defined by a sharp s t r a i g h t edge. Collimation of the beam was achieved by placing a sing l e brass collimator i n front of the water tank. The r e s u l t i n g d i s t r i b u t i o n s recorded on the f i l m were measured f o r o p t i c a l density (defined i n Chapter 3) using a scanning densitometer. In order to derive the value of a for the i d e a l i z e d - 5 -p e n c i l beam, the measured d i s t r i b u t i o n s were compared to d i s t r i b u t i o n s calculated f o r beams of the appropriate shape with the assumption that multiple scatter i s of a Gaussian nature. In the next chapter the theory of multiple scattering w i l l be discussed, i n i t i a l l y for the case of p e n c i l beams and t h i n s c a t t e r i n g sections. This theory w i l l then be extended to thick sections and beams of f i n i t e s i z e . In Chapter 3, the experimental setup and evalu-ation of the value of a for the Gaussian p e n c i l beam i s presented. Chapter 4 contains a discussion of the r e s u l t s and the assumptions involved, as w e l l as a comparison of the various t h e o r e t i c a l values to experiment. - 6 -2. MULTIPLE SCATTER THEORY 2.1 Introduction In this chapter a brief description of the Gaussian theory of multiple scatter, based on a classical treatment, w i l l be given, followed by a more exact theory due to Moliere (16,17). Subsequent modifications suggested by Fano (18), Hungerford et a l . (19) and Mayes et a l , (20) are also discussed. This theory, which applies to a pencil beam with normal incidence on thin f o i l s , w i l l then be applied to the case of a thick scattering section and f i n a l l y a real f i n i t e size beam w i l l be discussed. 2.2 Pencil Beams 2.2.1 Classical Derivation of Gaussian Distribution The spreading out of an i n i t i a l l y collimated beam of particles, incident at a point on a plane scattering medium, is primarily due to lateral scatter attributable to Coulomb interactions with the medium. In the case of a single elastic interaction, this results in the well known Rutherford formula for the distribution of classical particles with angle. The formula is given by Bethe and Ashkin (21) as da z Z e d Q 16T 2sin 4 (9/2) where ze and Ze are the charges of the incident particles and target nucleus respectively, T i s the kinetic energy of the incident particles (units of MeV and MeV/c w i l l be used throughout for energy and momentum respectively), and dQ i s the element of solid angle at angle 6. This expression is correct for the case where the incident particle mass is much less than the mass of the nucleus. - 7 -In the small angle approximation (sin 6 = 9 and cos 6 = 1 ) and substituting pv/2 for T (where p is momentum, v is velocity) this expression reduces to f(6) d6 = 8TT z 2 Z 2 e 4 / (p 2v 26 3) d6. (1) 2 Classifying pions as heavy charged particles requires using Z rather than Z (Z+l) which applies in the case of incident electrons (17,18). This case of small angle scatter with 6, the polar angle with respect to the incident direction, i s shown in figure 1. For a f o i l thin enough for the incident particles to suffer several interactions without significant energy loss, the distribution of emerging particles w i l l have azimuthal symmetry about the incident normal direction. As argued by Bethe and Ashkin (21), the distribution with respect to the angle 6 should be given by a Gaussian distribution, i.e. P(6) d6 = (26/<62>) exp (-82/<62>) d6. (2) 2 Here <^6 ^  i s the mean square angle for multiple scatter which can be evaluated by the following formula <e 2 > = m a x N e z f ( e ) de dx (3) 0 mxn for a f o i l with thickness Ax and N scatterers per unit volume. Substi-tuting for f(6) from eq. (1) we get <62> = (8TrNz 2Z 2e 4 / (p 2v 2) ) Ax ln (6 /6 . ), (4) N ' r max min where 6 and 6 . allow for corrections to the Rutherford formula due max mm to the f i n i t e size of the nucleus and the shielding of the nucleus by - 8 -Y FIGURE 1 MULTIPLE SCATTERING GEOMETRY USED TO DEFINE SCATTERING ANGLE 6 [adapted from Hungerford et a l . (19)] - 9 -the atomic electrons. The selection of the limits 0 and 0 . has been max mm discussed by Bethe and Ashkin and only the f i n a l expression w i l l be quoted, < 0 ^ = [4TT e 4 N q z 2 Z 2 / ( A p 2v 2)] t In [4rr Z 4 / 3 Z 2 N q t fi2/(A m^v 2)] ( 5 ) = X 2 In [4TT Z 4 / 3 Z 2 N t n 2/(A m 2 v 2 ) ] c o e where N has been replaced by N q p/A, N q i s Avogadros number. A = atomic weight of f o i l in grams. 3 p = density in g/cm . m = mass of electron, e t = pAx in g/cm2. The angle x c i s defined (21) such that there is on the average one scattering, in thickness t, of angle greater than x c -Preston and Koehler (22) have used equation ( 5 ) for the case 2 A / 3 1 / 3 of protons with Z replaced by Z ( Z + 1 ) and Z replaced by Z ( Z+l). Fano ( 1 8 ) points out that this is only an order of magnitude correction for the effect of inelastic collisions with the atomic electrons, since i t assumes that the Rutherford scattering formula is accurate at a l l angles. He has proposed a modification to Moliere's theory which is discussed i n section 2.2.3 and adopted in our calculations. The d i f f i c u l t i e s in choosing suitable values for 0 , 0 . ° max min and in taking account of inelastic electron collisions led us to investigate the theory of Moliere. This theory, while being more complicated to evaluate, has been demonstrated to give good agreement - 10 -with experiment for "thin" f o i l s as shown by Hungerford et a l . (19) and Mayes et a l . (20),. Although we give the complete formula according to Moliere, we w i l l use only the f i r s t (Gaussian) term in the expansion for calculations, due to the complexities of thick scattering sections. This Gaussian approximation for lateral scatter has also been made in deriving the pencil beam parameter, a, from the measured distributions of f i n i t e size non-parallel beams, because of the mathematical complexity for more exact descriptions. 2.2.2 Theory of Moli&re The theory of small angle multiple scattering by fast charged particles has been reviewed by Scott (17). While there are several theories that can give accurate results, the theory of Moliere, described by Bethe (16), is the best known and i s relatively simple to evaluate using the functions described i n Bethe's a r t i c l e . The mathematical description of the succession of single scatterings experienced by a fast charged particle traversing a thin scatterer (this assumes no energy loss) can be most easily described using Hankel (Fourier-Bessel) transforms. This allows the convolutions associated with each successive scattering event to be treated by a single multiplication in the transformed domain. Another way of describing the process (16) uses the Boltzmann transport equation, but use i s also made of these transforms to simplify the solution in terms of an ordinary differential equation. The details are found in Bethe (16) and Scott (17) and we merely quote the results. Moliere's theory for small angles involves some assumptions as listed by Hungerford et a l . (19): - 11 -(i) small angle approximation, i.e. sin 9 = 9 , cos 0 = 1 and integrals from 0 to TT are replaced by integrals from 0 to » ( i i ) Thomas-Fermi screening by atomic electrons ( i i i ) absence of spin effects (iv) absence of scattering by atomic electrons Moliere uses a modified form of the Rutherford formula, but as Bethe (16) points out, the strength of Moliere's theory l i e s in the fact that the scattering i s described by a single parameter, the screening angle x 3. (analogous to 9mj_n) • Also the distribution function f(0) i s independent of the shape of the single scattering differential cross section, except that i t goes over to the Rutherford law at large angles. The distribution in 0 is the inverse transform of the multi-plication in the Fourier-Bessel domain described earlier and following Bethe i s given by f(9) 0d9 = AdX JAXj) explk y 2 (-b + In [k y 2])] ydy. (6) J O Moliere expands this integral as a power series to give f(0) 9d0 = 0 r d0 r [2 exp [-0 r 2] + B"1 f ( 1 ) ( 0 r ) + B~ 2 f ( 2 ) ( 0 r ) + . . .] (7) 'CO where f U ' ( 9 ) r (n!) -1 J Q(0 ru) exp [-%u2] (5$u2 In [V i 2 ] ) n udu. J, 0 zero order Bessel function. u 0 r XB~h = 0 / ( x c B3*). B i s defined by B-lnB = b. b = In ( X 2 / X a 2) - -1544. C 3. 2 A 2 2 2 2 Y = 4ire N t z Z / (A p v ) , described e a r l i e r , c o X a 2 = X Q 2 f(« 2) = X 0 2(1.13 + 3.76 a 2 ) . a = zZe 2 / (tiv) = zZ / (137 6). X 0 = h / ( p r Q ) . -8 -1/3 r ^ = 0.468 x 10 Z cm, Thomas-Fermi radius of the atom. g,p = v e l o c i t y , momentum of incident p a r t i c l e . Bethe (16) has discussed the s i g n i f i c a n c e of the power se r i e s expansion, and states that only the f i r s t three terms are required to achieve an accuracy of 1%. The f i r s t term i s Gaussian, the second goes over to the si n g l e s c a t t e r i n g formula at large angles, while the t h i r d i s a c o r r e c t i o n with no simple i n t e r p r e t a t i o n . Using tables given i n Bethe (16), f ( 8 ) i s p l o t t e d vs 6 r for two values of B ranging near the extremes i n our case (fi g u r e 2). When we substitute numerical values f o r the constants i n the above formula we f i n d , X c 2 = .1569 z 2 Z 2 t / (A p V ) , (8) 2 where t i s i n grams/cm , A i s i n grams and pg i s i n MeV, X 2 / x 2 = 8838.4 z 2 Z 4 / 3 t / (A 3 2 (1.13 + 3.76a2)), (9) C SL b = In ( X 2 / x 2 ) ~ -1544 = B - l n B. (10) c a 2 2 Scott (17) gives an approximate formula, accurate to .5% f o r x c / x a 2 5 2 2 from 10 to 10 (where x / x i s representative of the mean number of C 3. c o l l i s i o n s i n thickness t ) . B = 1.153 + 2.583 l o g 1 0 ( x ^ / x / ) • (11) - 13 -(D r-or < o z < 1.0 2.0 3.0 REDUCED ANGLE 9r FIGURE 2 MOLIERE DISTRIBUTION PLOTTED AGAINST REDUCED ANGLE 6 R FOR TWO VALUES OF B PARAMETER - 14 -2.2.3 Modifications Proposed for Moli&re's Theory As stated, Moliere's original theory neglected the scattering by atomic electrons, but Fano (18) has calculated the contribution of inelastic collisions with the atomic electrons i n a form compatible with Moli&re's theory. He states that i t is incorrect to simply replace 2 Z with Z(Z+1), because the actual cross sections depart at small angles from the Rutherford formula. Instead of rising to i n f i n i t y , the cross sections are cut off with different cutoffs for elastic and inelastic collisions. Fano gives a recipe: 2 (i) i f the incident particles are electrons replace Z by Z(Z+1) and replace Moli&re's b by b+B" where B" = (Z+l) - 1 (ln [0.160 Z _ 2 / 3 (1+3.33 Z/(137g))]-u f a); ( i i ) i f the incident particles are heavy charged particles leave 2 Z unaltered but replace Moliere's b by b+B" where B" = Z _ 1 (ln [1130 Z" 4 / 3 B 2 / ( l - B 2 ) ] - u. - h B 2 )• (12) in Here is the integral over the incoherent scattering function and Fano gives values for u. as found in table I. ° in TABLE I FANO CORRECTION u. FOR VARIOUS VALUES OF Z in z u. in 1 -3.6 3 -4.6 8 -5.0 82 -6.3 - 15 -We have adopted Fano's suggestion for calculating B and this results in different values for B than those calculated by others, such as Bichsel (23) (details in Appendix A). Another suggested modification to Moli&re's theory is given by Hungerford et a l . (19), who point out that the Rutherford cross section applies to the centre of mass frame, and as such the values for momentum p, velocity 3, and angle 6 are cm. values. They point out, that in the small angle approximation, the r e l a t i v i s t i c cross section i s , to f i r s t order, equal to the Rutherford cross section with cm. angles and velocities. They derive a new expression for 2 A c X 2 = .157 z 2 Z 2 t / (A e 2 (p 3 ) 2) (rad 2). c rcm cm 2 They give a complicated expression for e but i t turns out that e = P^a^/P and therefore the corrected expression is given by X c 2 = .157 z 2 Z 2 t / (A ( p ^ 3 c m ) 2 ) . 2 If one ignores the correction to 3 in the evaluation of Moliere's B (which should be a small effect) then to f i r s t order <e2,> / <e2> = (x 2) B / (x 2 B) x 'corr ' x ' v x c "corr ' v x c ' lab cm or as shown by Highland (24) ( < e 2 > c o r / 2 = « « 2 > ) h C 1 + M P 2 I ( EP V » ' where M , E are the mass and total energy of the incident particle and P P - 16 -Mfc is the mass of the target atom. Calculations indicate, that over the range of pion energies that we are dealing with, in tissue equivalent materials, the corrections amount to only 1% to 1.5% and have been neglected. However, they may be added later i f desired. A more significant modification proposed was that of Mayes et a l . (20) who suggested, that on the basis of the experimental results for various particles, targets, and energies, a better f i t of the theory 2 to the data could be effected by changing Moliere's f ( a ) equation from f ( a 2 ) - 1.13 + 3.76 a 2 to f ( a 2 ) = 0.59 + 3.44 a 2 . They s t i l l , however, have a discrepancy, especially for low Z, that 2 2 <(e ^ e X p / ^9 ~^th *"s aPP r o x i m a t e-'-y 1*06 - 1.08. Since they have not applied Fano's correction, which i s particularly important for low Z materials, their proposal should be held in abeyance pending more refined experiments (25). We have calculated values for B in Appendix A on the 2 basis of both f ( a ) equations. In view of the uncertainty of the validity of their modification, we have elected to use only Moli&re's results. As mentioned earlier, in view of the complexities involved in the transition to thick scattering sections and f i n i t e size beams, only the f i r s t (Gaussian) term w i l l be retained, using for the mean square angle <e2> = x c 2 B. (13) In order to see the size of this approximation we have plotted in figure 3 the Gaussian term and the complete distributions for the two values of B - 17 -FIGURE 3 COMPARISON OF GAUSSIAN TERM WITH NORMALIZED MOLIERE DISTRIBUTION FOR TWO VALUES OF B PARAMETER - 18 -used f o r f i g u r e 2. Here we have normalized a l l d i s t r i b u t i o n s f or 6 = 0. r I t i s seen that the Gaussian appears as the l i m i t as the number of c o l l i s i o n s goes to i n f i n i t y . Since i n general we w i l l be dealing with a very thick section t h i s i s a reasonable approximation. As mentioned e a r l i e r , the theory of Moliere, as o u t l i n e d , holds for t h i n f o i l s where the energy loss i s n e g l i g i b l e . In radio-therapy we are dealing with cases where the amount of s c a t t e r i n g material i s very large. In f a c t , we have the case f o r negative pi-mesons where the region of i n t e r e s t extends from t h e i r r e l a t i v i s t i c incidence at the surface to the point where they have l o s t a l l t h e i r energy and come to r e s t , causing a nuclear d i s i n t e g r a t i o n following capture. It i s appropriate at t h i s point to summarize the modifications to Moliere's theory that have been discussed and to note which of them w i l l be used. (i ) Fano's co r r e c t i o n f or c a l c u l a t i o n of B—used, discussed i n Appendix A. ( i i ) Centre of mass corrections—-not used. 2 ( i i i ) f ( a ) equation c o r r e c t i o n s — c a l c u l a t e d but not used. (iv) Gaussian angular approximation—used with mean square angle of X 2 B. A c (v) Energy loss c o r r e c t i o n — u s e d , discussed i n the next section and i n d e t a i l i n Appendix A. - 19 -2.2.4 Application to Thick Scatterers In the case of radiotherapy we are not primarily interested in the angular distributions after multiple scatter from a thin section, but in the radial distribution at some point in the f i n i t e range of a pion. We w i l l follow in general the treatment of Fowler and Perkins (1), Preston and Koehler (22,26) and Carlsson and Rosander (27). Referring to figure 4, the case of interest i s the lateral distribution at a plane located at depth s in the medium, for a mono-energetic beam of particles entering at x = 0 with a total range RQ (no straggling). If the thickness dx is sufficiently thick to produce a Gaussian angular distribution we w i l l argue that the lateral d i s t r i -bution w i l l also be Gaussian in the small angle approximation. Since each increment in range results in a Gaussian distribution of angle, then two successive increments also result in a Gaussian, with mean square angle increased by twice the mean square deviation that one increment would give, i.e., the mean square deviations are additive. (Equivalently the convolution of two Gaussians is a Gaussian with 2 2 2 <^0 )>T = ^0 + <^0 .) The contribution of a thin scattering section, at the surface, to the distribution of later a l displacement at s, would be the projection of this angular distribution magnified by the distance from the surface to the plane, s. This implies a Gaussian distribution in lateral displacement for a Gaussian angular distribution at the surface. For a series of scatterers between the surface and s, each with a Gaussian distribution in angle, we seek a description of the contributions of each to the resultant mean square lateral displacement <^ r X at a depth s. In general, for uncorrelated parameters a and b with a functional - 20 -Surface Plane of Range interest pions - u d 5 ^ R c F * ^» • * > 5 dx |dr FIGURE 4 SCHEMATIC DIAGRAM OF SMALL ANGLE SCATTER CONTRIBUTION TO LATERAL DISPLACEMENT FOR THICK SCATTERING MEDIUM - 21 -relationship y = f(a,b) the resultant standard deviation is given by [see Bevington (28)] a 2 = (a y/da) 2 a 2 + Oy/db)2 a 2 y a b In the case of small angle scatter (figure 4) dr - d6 (s-x) i.e. r - 9 (s-x), 2 2 2 2 2 and therefore a = (s-x) a. + ( - 0 ) a , r 6 x which for Gaussian distributions results in <r 2> = (s-x) 2 < 0 2 > + 0 2 <x2>. 2 2 2 2 Assuming (s-x) <^ 0 y to be much greater than 0 <(x >^, which is reasonable since we assume no straggling in x and each angle 0 i s small, then the 2 contribution to <V >^ for a thin section dx at depth x is given by d<r2> = (s-x) 2 d<0 2 > 2 and therefore <^ r >^ = S (s-x) 2 d<0 2 >. (14) '0 The radial distribution i s given by a Gaussian N(r,s) = ( T r<r 2>) - 1 exp (-r2/<r2>) 2 2 where ( r ) i s a function of s. In order to evaluate <^ r ) we substitute from equations (8) and (13) for ^0 )> to give <r2>- (s-x) 2 [.157 z 2Z 2 B / (A p 2g 2)] dx (15) '0 where t has been replaced by dx. From figure 4 we see that s-x = R-Rp, - 22 -where R i s the residual range. Substituting this in equation (15) and changing the limits of integration we get, <r 2> = 0 (R-Rp)2 [.157 z 2 Z 2 B / (A pV)] dR = K R 0 ( R - V 2 E B / C p V ) ] dR. *F (16) For heavy charged particles p3 can be well described as a function of the residual range R by a simple power law, C2 Pf3 = Cj_ R with the numerical evaluation of the constants given in Appendix A. Now we must find a suitable expression for B . One approach is to average B over the entire range and give i t a constant value and then integrate equation (16). This is undesirable in our case since we w i l l be dealing with pions of varying ranges R Q . We have found that B can also be well approximated as a simple function of the residual range, as shown in Appendix A, to be B = C3R where B indicates we have averaged over the various atomic constituents of the scatterer. Thus equation (16) may be written <r2> rR 0 (R-Rp)2 K (Gyc^) R~ 2 C2 + C4 dR, *F = K' R 0 ( R - V 2 R _ 2 C ' d R » *F ^ - 23 -and integrated to yield <r 2> = ° ,P 3-2c' 3-2c' 1 -(3-2c') (1-c') Ro + (3-2c') (l-2c') *F -i2 R 0 i 3 - 2 c ' ( 3 - 2 c ' ) (3-2c') (l-2c') " (17) This i s evaluated numerically in Appendix A. We have chosen to write the expression for the Gaussian as N(r,x) = (2Tr(a )*) 1 exp [-r 2/ (2(a ) 2)], where a m g = a(x) and x = R^ -R^ ,. This i s normalized in two dimensions (18) and = (<(r >^/2) 2, corresponding to Fowler and Perkins, DA proj We have so far been considering pencil beams of particles. In the next section we w i l l extend the Gaussian theory [equation (18)] to f i n i t e f i e l d sizes, with the aim of the eventual extraction of the pencil beam value for a from the distributions that result for f i n i t e fields. 2.3 Finite Size Beams Before dealing with experimental conditions, the description of multiple scattering that we have developed w i l l be extended to some fi n i t e fields of simple shape. In the case of proton scattering, the transition to large fields has been investigated for beam cross sections of regular (22) and arbitrary geometry (27). The methods involve the convolution in two dimensions of the pencil beam (Gaussian) description with the given - 24 -f i e l d shapes and result in numerical computations for the distributions at depth in the scatterer. Preston and Koehler (22) considered three primary geometrical shapes: circular, rectangular and in f i n i t e half plane. For the f i r s t two shapes they primarily dealt with special cases, almost exclusively with the intensity on the axis. Only in the case of the i n f i n i t e straight edge do they calculate the general intensity off the axis. They do, however, demonstrate the problem of small f i e l d sizes and significant multiple scattering and give a formula of interest in our case. The intensity on the axis at a depth x, with standard deviation (x), for a circular collimated beam of radius i s given by 1(0,x) = 1 - exp [-r c 2 / ( 2 ( a m g ) 2 ) ] . (19) This decrease in central axis intensity i s due to the scattering out of the beam from the axis. This becomes an important consideration for the case of pions at the end of their range for small treatment volumes, as indicated in Chapter 1. We have settled on a circular beam shape for the greatest part of this study as a result of a number of considerations which include: (i) symmetry ( i i ) ease of machining ( i i i ) analogy to pencil beams. The application to an i n f i n i t e straight edge i s outlined in Appendix B. For the case of an incident uniform and parallel beam, the problem of determining the distribution, at a depth behind a circular - 25 -opening, can be regarded as the convolution (in two dimensions) of a Gaussian (multiple scatter) distribution with the shape of the opening, projected to the depth of interest. The geometrical arrangement i s presented in figure 5 for the contribution from an element Q on the surface to the distribution at a point P (with position (r,8,x) in cylindrical polar coordinates). The distribution at P is the sum of contributions Q from the entire circular cross section at the surface. Mathematically this can be written ''X) = f f f(r,8,x) = | I A(r',(j>,0) G(r P,9 + <|>,x) r'dr'dcj), (20) where for our case A(r',$,0) = 1 r < r , and G(r ,6 + c^x) = ( 2 T T ( 0 J 2 ) " 1 exp[-r 2 / ( 2 ( a ) 2 ) ] , 2 2 2 where r^ = r + r' - 2rr' cos(8+cj>). Circular symmetry allows us to choose 8 = 0 and simplify the integral to r2TT f(r,x) = r c (2T T ( O ) 2 ) - 1 exp[(-r 2-r' 2+2rr ,cos (())/(2(a )2)]r'dr,d(|) (21) Q ms ms where a = o (x) as before, ms ms We have decided, that rather than evaluate f(r,x) using equation (21), we would follow Moliere's example and use the well known fact [see, for example, Bracewell (29)] that convolution in one domain i s equivalent to multiplication in the Fourier transformed domain. This technique is frequently used in data processing (e.g. for seismic geophysical data) as a time saving step. - 26 -FIGURE 5 SCHEMATIC DIAGRAM OF CONTRIBUTION TO THE DISTRIBUTION FOR A POINT AT DEPTH x FROM AN ELEMENT AT THE SURFACE - 27 -The use of a circular collimator and the resultant circular symmetry allows the conventional two-dimensional Fourier transform to reduce to the Fourier-Bessel transform [see Bracewell (29)] F(s) = 2TT f(r) J 0(2irrs) r dr, (22) and the inverse transform i s given by f(r) = 2TT F(s) J Q(2Trrs) s ds. (23) As may be easily shown (29), the transform, F c ( s ) , of a c i r c l e of radius r i s given as c e F c(s) = (r c/s) J 1(2Trr cs), where is the f i r s t order Bessel function and where f (r) = 1 r < r . c c The transform of a Gaussian such as that of the multiple scattering g(r) = ( 2 T r a 2 ) - 1 exp(-r 2/(2a 2)), i s given by G(s) = exp ( - 2 T T 2 O 2 S 2 ) Using these facts, the transform of f(r,x) denoted by F^ i s given by 2 2 2, F F = G(s) F C ( S ) = (r c/s) J 1 ( 2 T r r c s ) exp(-2Tr a s ) Thus using equation (23), the radial distribution f(r,x) is given by f(r,x) = 2rrr 2 2 2 J g ^ T r r s ) J^(2irr cs) exp(-2Tr a s ) ds, (2A) which involves only a single integration. The computation of this - 28 -expression is accomplished using an integration routine (QINF) available on the UBC computer and the Bessel functions are evaluated using poly-nomial approximations l i s t e d in Abramowitz and Stegun (30). The calculation using equation (24) is approximately twice as fast as that using equation (21). The evaluation of equation (24) gives good agree-ment for r = 0 with the values calculated using equation (19). It also may be shown to reduce to the appropriate form for f(r,0) using properties of integrals of Bessel functions found in reference (30), although the numerical computation of equation (24) for small values of a requires some care. It should be noted that for a - 0, f(r,0) = 1 for r < r c as expected and the normalization integral i s given by f oo 2 f(r,x) 2irrdr = irr h  C Using this calculation for f(r,x) we have shown, in figure 6, the resultant distributions that i l l u s t r a t e the effects of multiple scattering in relation to the collimator hole radius. We have plotted f (r') versus r'" for various values of a' using the reduced parameters a' = cr/r c and r' = r / r c > This reduction in parameters allows the convenient plotting of radial distributions without reference to any particular value of a or radius . This section has assumed that the incident beam of particles is parallel and spatially uniform in intensity. As w i l l be discussed in the next chapter, the preliminary analysis of the experimental data had indicated that the beam was not parallel. However, for some restricted conditions discussed in Appendix C, i t i s possible to make a modification in equation (24) that permits i t to be used in these cases also [equation (C-l)]. i — 1 r— 1 1 1 r 1.0 2.0 3.0 RADIAL DISTANCE r' FIGURE 6 CALCULATED DISTRIBUTIONS FOR GAUSSIAN SCATTER WITH CIRCULAR APERTURE FOR SELECTED VALUES OF a* (DESCRIBED IN REDUCED PARAMETERS) - 30 -3. EXPERIMENTAL CONSIDERATIONS 3.1 Introduction In this chapter we w i l l discuss the relevant details of the experimental setup for determining the value of a in the multiple scattering formalism. We begin by looking at film as a detector, then discuss the characteristics of the M8 biomedical channel at TRIUMF along with the experimental setup and then present the results of the film measurements. 3.2 Film as a Detector The mention of film for use in quantitative measurements of radiation fields usually evokes feelings such as those described by Dutreix (31) with regard to film dosimetry, "Film dosimetry does not usually raise great enthusiasm among physicists, and conversely, i t sometimes generates too much confidence among radiotherapists". However, as she goes on to point out, "Experience shows that f i l m dosimetry i s an excellent practical method particularly for high energy photons and electrons". The prime value is in relative, rather than absolute measurements. In the detection of heavy charged particles the standard technique i s to use nuclear emulsions and analyze the resulting particle tracks with high magnification microscopes. This method was used in the early detection of pions and determination of the frequency and energy spectra of particles emitted from pion stars (32). This technique has also been used in microdosimetric studies on pion beams (33), however, we w i l l be using measurements of optical density to determine - 31 -the lateral distributions of pions in a water phantom. A detailed description of film and the photographic process, including sensitometry, is given by Mees and James (34). We w i l l define only the term optical density and give a description of characteristic curves in this discussion. The formation of latent image centres in the silver halide grains, by the deposition of energy (through excitation and ionization loss of passing charged particles), and subsequent chemical development produce grains of metallic silver in the emulsion layer. These blackened grains absorb and scatter incident light and the macroscopic effect is to attenuate the light intensity exponentially such that (35), X t = To exP(-nV°-Here I q refers to the incident intensity, I T the transmitted intensity, x is the emulsion thickness, n is the number of blackened grains per unit volume and is the total cross section for scattering and absorption of light. The ratio I t / I is called the transmittance T, and the optical density (D or O.D.) is defined by D = l o g 1 0 (1/T). A plot of the net density (total minus background) against the relative exposure is called the characteristic curve. The curve for x rays and charged particles is typically a straight line through the origin for low densities whereas for light i t has a characteristic "toe" at low densities (see figure 7). This has been attributed to the fact that a single quantum of x-radiation or a single charged particle can render a grain developable. - 32 -i 1 1 r RELATIVE EXPOSURE FIGURE 7 TYPICAL FILM CHARACTERISTIC CURVES FOR LIGHT AND IONIZING RADIATION (X RAYS OR CHARGED PARTICLES) [adapted from Mees and James (34)] - 33 -The role of film in radiation dosimetry has been reviewed by Dudley (36) who l i s t s some of the advantages of film as (i) wide dose range (sensitivity) ( i i ) large time scale (integrating detector) ( i i i ) large areas (iv) high spatial resolution. He also l i s t s some disadvantages as (i) dependence on LET ( i i ) dependence on dose rate ( i i i ) dependence on environment and processing. The dependence on dose rate i s not a problem in the range of dose rates at which we are operating. The environment and processing problems can be minimized by careful handling with regard to temperature and humidity and with the use of automated film processors to reduce the nonuniformity of development. The chief disadvantage appears to be the variation of response with respect to LET as shown by Tochilin et a l . (37). The films that they used (with the exception of nuclear emulsion) showed l i t t l e or no increase in optical density with depth, even when located in the Bragg peak of a charged particle beam, where the specific ionization (as measured with a parallel plate ionization chamber) was increasing rapidly with depth. From these observations they concluded that each particle deposits enough energy to render a fi l m grain developable and any extra energy deposited (due to increased LET) i s wasted. Hence for a given dose, as the LET increases the dose sensitivity decreases, - 34 -where dose sensitivity i s inversely proportional to the exposure required to produce a given density. Tochilin et a l . state that at the energies in their study, film response was primarily a record of particle flux. Similar results using more modern emulsions have been quoted by Dutreix (31). The previous multiple scattering experiments (22,27) employed a diode as the detector. However, for the low dose rates available at the time of our experiments, a si l i c o n diode was shown to be un-suitable. In order to get the spatial resolution required we elected to use medical x-ray film (Kodak XM2). The use of film allowed for very efficient use of available beam time, in that a l l the information at one depth could be collected on one film. In addition the film holder design allowed the exposure of up to three films for a particular experiment, i f desired. In view of the observation of Tochilin et a l . mentioned earlier, we treated the films as "particle counters". The effects of LET dependence should be reduced by placing the films perpendicular to the incident beam direction, so that each point on the f i l m i s subject to approximately the same LET spectrum. A benefit in using film as a particle counter was that i t allowed direct comparison with the theory, since the theoretical calculations produced number d i s t r i -butions rather than dose distributions. 3.3 Beamline Characteristics and Setup The M8 biomedical beamline at TRIUMF has been described by Harrison (38,39) and Henkelman et a l . (7). It consists of an achromatic system of nine electromagnets comprising two dipoles, five quadrupoles, - 35 -Q - Quadrupole Focussing Magnet B - Dipole Bending Magnet S - Sextupole Magnet PION R A D I O T H E R A P Y B E A M LINE AT TRIUMF FIGURE 8 SCHEMATIC DIAGRAM OF M8 -BIOMEDICAL CHANNEL AT TRIUMF - 36 -and two sextupoles over a length of 7.5 meters (figure 8). In the middle of the third quadrupole there is a dispersion plane and two movable blades for selecting a spread of momentum (from ± 1.5% at half maximum, to ± 6.7% -^ -) . As in any pion channel there is the inevitable contamination of muons (y) and electrons (e). This contami-nation has been investigated using a time of fl i g h t system (6,40,41). The muons arise from the decay of pions i n - f l i g h t , whereas the electrons are produced predominantly in the target, due to pair production by the y rays which arise from the decay of neutral pions produced along with the charged pions. The relative contribution of contaminants to the beam flux has been shown to be dependent on target type and channel momentum (40). In addition, i t appears that the relative steering of the proton beam on the target has an effect on the relative yield of TT and the e contamination. This i s thought to be due to greater or lesser thicknesses of target for the secondary beam to traverse and is a significant effect as shown by figure 9. Another effect of importance i s the loss of ir from the beam due to in-flight interactions. This has been measured in a water phantom using integral range curves (42) employing large plastic sc i n t i l l a t o r s and yields a value of 1.62%/cm. In order to reduce the effects of multiple scatter before the patient treatment area, the beam i s transported from the target in a vacuum system that extends up to the last quadrupole. After the vacuum window there i s a transmission ionization chamber, then a variable length of air path (- 1 meter, depending on beam tunes) before the experimental apparatus which i s mounted on a motor driven table. The - 37 -FIGURE 9 EXAMPLE OF RELATIVE PARTICLE FRACTIONS PLOTTED AGAINST MOMENTUM SHOWING EFFECT OF PROTON BEAM STEERING [from Poon (40)] - 38 -physical layout i s depicted in figure 10 and the thicknesses and conversions to equivalent depth in water are list e d in table II. The film-holding apparatus consisted of three Perspex film envelopes, each capable of holding an 8 x 10 inch "ready pack" film. These envelopes were then suspended from a frame attached to the water phantom. The frame allowed adjustment of the films to any depth in the phantom (except for the f i r s t 2.0 cm from the front). This setup for the multiple scattering experiments i s illustrated in figure 10. The repositioning error for the film envelopes has been estimated to be about ± .1 cm and the reproducibility in setting the depths and lateral position of the frame as ± .2 cm. For simplicity in sharing the water phantom with other experiments a single collimator was located just before the water tank, rather than a two collimator system. The collimator was brass, two inches thick and allowed the use of inserts with hole diameters up to 7/8 inch. The repositioning error of the collimator system was estimated to be ± .3 cm. The presence of scattering material, such as sc i n t i l l a t o r s , etc,, before the collimator and the wall of the water tank after the collimator, makes the attainment of a parallel beam by collimation clearly impossible in the present setup. A request was made of the beam tuning group for a broad, spatially uniform beam, as nearly parallel as possible. However, when dealing with charged particle beams of f i n i t e emittance (43) i t i s theoretically impossible to achieve a completely parallel beam. In addition, the focussing and defocussing characteristics of magnetic lenses make achieving even a practically parallel beam d i f f i c u l t . It turned out that the beam envelope was essentially parallel in one direction (vertical, y 1) Transmission chamber - > — Beam axis Vacuum window Collimator Scintillators / \ Multi-wire proportional counter K F M mNU K Film A envelope Film frame Water tank FIGURE 10 SCHEMATIC DIAGRAM OF EXPERIMENTAL SETUP IN PLAN VIEW (MULTIPLE SCATTERING EXPERIMENTS) - 40 -TABLE II CALCULATION OF WATER EQUIVALENT THICKNESS FOR MATERIALS IN FRONT OF WATER SOURCE THICKNESS cm DENSITY , 3 g/cm THICKNESS , 2 g/cm Vacuum Window 0.024 1.19 0.029 Transmission Chamber 0.008 1.19 0.010 T. Chamber Cover 0.015 1.19 0.018 Scintillators 0.635 1.032 0.655 Covers for Scint. 0.090 1.19 0.107 MWPC 0.010 1.19 0.012 Plastic Tank 1.240 1.19 1.476 2 Total thickness in g/cm 2.307 Multiply by S., /S u _ v J 3 Perspex ^ 0 0.971 Total water equivalent 2 thickness in g/cm 2.240 - 41 -and divergent in the perpendicular (horizontal, x') direction. The divergence, while not negligible, appeared to be small enough to satisfy the conditions of Appendix C. 3.4 Experimental Results Measurements were carried out at two channel momenta: the f i r s t series at 148 MeV/c corresponding to a pion range in water of 13.5 cm and the second series, three months later, at 180 MeV/c cor-responding to a pion range in water of 22.3 cm. The beam tune for the 180 MeV/c experiments was derived by scaling the magnet settings for the 148 MeV/c tune by the ratio 180/148. This should give some cor-respondence between beam tunes, but due to the decrease in electron contamination at higher momentum (22% at 180 MeV/c versus 43% at 148 MeV/c) and the time separation between experiments, i t i s advisable to regard them as two completely different beam tunes. This section w i l l be divided into two subsections, f i r s t the preliminary and ancillary experiments followed by the multiple scattering experiments. 3.4.1 Preliminary Experiments Experiments that f a l l into this category are exposure tests, calibration curves, and films parallel to the beam direction. Exposure tests were carried out at each midline momentum (with "wide blades", i.e. ± 6.7% ^ -) by exposing films at different depths in the water phantom for a series of "monitor counts". The "monitor counts" refer to ionization collected in the transmission - 42 -i o n i z a t i o n chamber, mounted on the l a s t quadrupole and used as a beam monitor. The current was i n t e g r a t e d and converted to output p u l s e s , each corresponding to a f i x e d amount of charge, which were counted by a preset s c a l e r . The accumulation of a preset number of counts auto-m a t i c a l l y terminated the exposure by causing the beam stop to be i n s e r t e d . The f i l m s were developed i n a standard medical x-ray auto-processor w i t h i n one to two days a f t e r exposure. I t should be pointed out that the f i l m processor was changed between the 148 MeV/c experiment and the 180 MeV/c experiment. O p t i c a l d e n s i t y measurements were obtained using a scanning densitometer (Kipp and Zonen, model DD 691-E) w i t h a l i g h t s l i t a p p r o x i -mately ,02 x .4 cm i n s i z e . The densitometer s i g n a l was a m p l i f i e d and recorded on a s t r i p c h a r t recorder (Hewlett-Packard, model 680-M). D e n s i t i e s f o r a l l experiments included base and fog d e n s i t i e s . Graphs of f i l m d e n s i t y a g a i n s t r e l a t i v e exposure are p l o t t e d f o r the 148 MeV/c and 180 MeV/c runs i n f i g u r e s 11 and 12 r e s p e c t i v e l y . They show the l i n e a r curve p r e d i c t e d (34) i n the d e n s i t y range we have used. The i n t e r c e p t corresponds to the background d e n s i t y of the f i l m and the d i f f e r e n c e between the two backgrounds i s probably due to emulsion changes and d i f f e r e n t processing c o n d i t i o n s . The d i f f e r e n t slopes f o r d i f f e r e n t depths w i l l be discussed i n s e c t i o n 4.3.4. D u t r e i x (31) gives some f i g u r e s on the expected accuracy when comparing o p t i c a l d e n s i t i e s and these are l i s t e d i n t a b l e I I I . 1.6 >-CO z LU Q 1.2 • 1 " Position Depth in o n T i l m water (cm) | x - 2.4 x - 11.3 • .o 500 1000 1500 MONITOR COUNTS 2000 FIGURE 11 EXPOSURE TEST FOR 148 MEV/c CASE _| — i 1 — r Position Depth in on film _ j l _ l I — 500 1000 1500 2000 M O N I T O R C O U N T S FIGURE 12 EXPOSURE TEST FOR 180 MEV/c CASE - 45 -TABLE III EXPECTED ACCURACY FOR COMPARING OPTICAL DENSITIES [Dutreix (31)] Comparison of Optical Density Expected Accuracy On same film 2% On films processed simultaneously 3% On films processed separately (identical processing condition) 5% On films of different batches ? This may explain the variation of background between experiments, and points out the need for care in using film of the same batch when making comparisons between films. Each multiple scattering experiment (next section) used films from the same box so that the variations of optical density from film to film would be expected to be less than 5%. The relative measurements of the distributions on each film could be expected to show less than 2% variation without taking account of densitometer variations. In order to test the reproducibility of the densitometer we periodically scanned a calibrated test strip (diffuse density) obtained from Kodak. The densities measured on our densitometer were not numerically equal to those given with the test strip due to the differing geometries of light collection (34). However, the requirement of r e l i -a b i l i t y , as shown in figure 13, i s well satisfied in the lower density range. These are results taken over a time span of a few months and - 46 -— t 1 1 I i 0.4 0.8 1.2 CALIBRATION DENSITY FIGURE 13 DENSITOMETER RESPONSE (TEST OF REPRODUCIBILITY) - 47 -the indicated ranges represent the minimum and maximum values measured. As a result of this data, we elected to use an exposure of 2100 monitor counts (measured O.D. approximately 1.2) for our multiple scatter experiments to stay in the linear portion of the densitometer response. In addition to the exposure tests, we exposed three films parallel to the beam direction. These films were located approximately 3 to 4 cm inside the water phantom and were in fact angled approximately 5° from the central axis of the beam to try to overcome problems due to air gaps surrounding the film. Dutreix (31) has pointed out the need for care i n exposing films parallel to 20—25 MeV electron beams and i t i s assumed that the same problems apply here. In general, the results exhibit a level plateau at the beginning of the film, followed by a rise in O.D. near the pion stopping peak, followed by a decline to a lower level again after the peak. The results are summarized in table IV. TABLE IV SUMMARY OF PARALLEL FILM MEASUREMENTS Momentum MeV/c Exposure Monitor Counts Ap/p % Optical Density (including base and fog) Plateau Peak Past Peak 148 1500 6.7 0.75 0.85 0.46 180 2100 6.7 1.02 1.04 0.60 180 2100 2.0 0.85 0.92 0.44 The increase in O.D. (plateau to peak) ranges from a low of 2% to a high of 13%. In spite of the associated problems mentioned by Dutreix and the - 48 -LET dependency, the interpretation of these results indicate the behaviour of the film to be primarily as a "particle counter". 3.4.2 Multiple Scatter Experiments In the multiple scatter experiments we used the setup illustrated in figure 10. Collimator aperture sizes of .50 cm radius and 1.125 cm radius were used. A number of experiments were performed for both hole sizes and various depths in water (inside tank wall to film plane). These are summarized in table V. TABLE V SUMMARY OF MULTIPLE SCATTER EXPERIMENTS PERFORMED Momentum Ap/p Hole Radius Depths in Water Number of MeV/c % cm cm . cm cm Experiments 148 6.7 1.125 2.4 11.3 — 2 180 6.7 1.125 7.55 15.1 20.1 2 180 6.7 0.50 7.55 15.1 20.1 2 180 2.0 1.125 7.55 15.1 20.1 1 180 2.0 0.50 7.55 15.1 20.1 1 180 6.7 Straight edge 7.55 15.1 20.1 1 For each experiment successive runs were made with and without water in the tank, to determine the scattering effect of the water. After development each film was scanned on the densitometer in two perpendicular directions, coinciding with the x',y' directions of the beam mentioned earlier. This resulted in four densitometer scans for - 49 -each depth for each experiment. For the c i r c u l a r apertures l i s t e d i n table V there were 88 p r o f i l e s i n a l l to be analyzed. Rather than d i g i t i z e each p r o f i l e , a convenient way of describing the r e s u l t i n g d i s t r i b u t i o n s i n o p t i c a l density was sought. The procedure chosen was to take the f u l l width of the p r o f i l e at the 75%, 50% and 25% l e v e l s of the net density d i s t r i b u t i o n ( t o t a l minus background, determined from each p r o f i l e ) . These values were then divided by the diameter of the aperture which resulted i n normalized parameters Y , .Q and ^25" These parameters were compared with t h e o r e t i c a l values for d i s t r i b u t i o n s calculated using equation ( 2 4 ) for various values of a'. Preliminary attempts to analyze the data indicated that i t was not correct to assume that the beam was p a r a l l e l . As mentioned i n section 3 . 3 the beam diverges i n one d i r e c t i o n , r e s u l t i n g i n an approxi-mately e l l i p t i c a l shape f o r the d i s t r i b u t i o n s at depth. A f i r s t approximation f o r t h i s s i t u a t i o n ( d e t a i l s are given i n section 4 . 3 . 5 and Appendix C) can be made by assuming a point source divergence. This assumes that each densitometer p r o f i l e i s the r e s u l t of scatter from a c i r c l e with a magnified (or decreased) radius. This magnification at depth i s described by an a d d i t i o n a l parameter, A. I t should be noted that i n t h i s d e s c r i p t i o n , the value of A f o r the two axes of the e l l i p s e on the same f i l m i s not expected to be the same. Using equation (C-l) with parameters A and a', new values f o r Y^r., Y ^ Q , Y25 were calculated and i n f i g u r e 14 these parameters are pl o t t e d against af for three values of A. I t can be seen that the v a r i a t i o n of the Y parameters with a' i s complicated but there are at - 50 -FIGURE 14 VARIATION OF Y PARAMETERS WITH SCATTER PARAMETER a*' FOR VALUES OF MAGNIFICATION A - 51 -l e a s t two regions which can be distin g u i s h e d . For the region of a' < .4, the main e f f e c t i s due to the divergence while i n the region f o r a' > .6, the increasing l a t e r a l scatter (a) causes the e f f e c t of divergence on A to be masked. The e f f e c t s of A on cr* are discussed l a t e r i n Chapter 4. In order to search the two parameter space i n a systematic fashion, a computer program was w r i t t e n based on a subroutine c a l l e d CURFIT (28), to determine the values of A and a' that gave best agreement with the experiment. The o u t l i n e of the program i s as follows: ( i ) Assume A, cr', c a l c u l a t e Y,-Q, Y 2 5 " ( i i ) Compute chi-square parameter with experimental point. ( i i i ) Step i n a new d i r e c t i o n u n t i l chi-square decreases by calcu-l a t i n g new A, a' and hence new Y^,_, Y,.Q, ^ 2 5 * ( i v ) When v a r i a t i o n i n chi-square i s l e s s than some number, l i k e 10% of previous value, stop search. The r e s u l t s of the determinations of cr , a and a are given i n w wo ms tables VI, VII and VI I I . Here cr i s the value of a determined f o r ' w water i n the tank, a i s the value f o r no water i n the tank and a ' w o ms the resultant multiple scatter value. Note that i n the tables cr' has been corrected f o r the respective hole sizes to give a, e.g. a = a ' r . r w w c The averages and t o t a l u ncertainties are calculated using formulas given by Bevington (28) a = E(a i/v. 2) / E ( l / v . 2 ) Aa 2 = ( E d / v . 2 ) ) " 1 . This has the e f f e c t of giving more weight to the values with the smaller - 52 -TABLE VI EXPERIMENTAL VALUES OF a FOR 148 MEV/c WIDE BLADE CASE (AVERAGE OF 2 EXPERIMENTS) RADIUS DEPTH PROFILE a a a AVERAGE a w wo ms ms cm cm cm cm cm cm 1.125 2.4 x' .133±.004 .124±.004 .048±.015 .052±.011 y' .172±.003 .163±.004 .055±.015 1.125 11.3 x' .542±.013 .340±.005 .422±.017 .4301.016 y* .630±.040 .396±.006 .490±.052 - 53 -TABLE VII EXPERIMENTAL VALUES OF a FOR 180 MEV/c WIDE BLADE CASE (AVERAGE OF 2 EXPERIMENTS) RADIUS DEPTH PROFILE a a a AVERAGE a w wo ms ms cm cm cm cm cm cm 1.125 7.55 x' .3061.004 .2051.003 .2271.006 .2201.005 y' .341±.004 .2701.004 .2081.008 15.1 x' .680±.068 .3621.006 .5761.080 .5821.059 y' .727±.069 .4271.009 .5881.086 20.1 x' 1.075±.129 .4311.005 .9851.141 .9771.127 y r 1.096±.253 .5621.012 .9411.295 0.50 7.55 x' .303±.024 .2141.004 .2151.034 .2281.028 y' .352±.036 .2411.007 .2571.050 15.1 x 1 .756±.134 .4121.004 .6341.160 .6321.141 y' .761±.241 .4341.040 .6251.295 20.1 x' .984±.lll .4971.050 .8491.132 .8431.117 y' .997±.209 .5741.033 .8191.256 - 54 -TABLE VIII EXPERIMENTAL VALUES OF a FOR 180 MEV/c NARROW BLADE CASE (1 EXPERIMENT) RADIUS DEPTH PROFILE a a a AVERAGE a w wo ms ms cm cm cm cm cm cm 1.125 7.55 x' .308±.006 .2061.005 .2291.009 .2181.007 y' .324±.005 .2531.006 .2021.011 15.1 x' .7131.096 .3781.009 .6051.113 .6061.099 y' .7581,163 .4501.020 .6101.203 20.1 x' 1.1091.179 .4591.011 1.0101.197 .9571.135 y' 1.0671.156 .5551.033 .9111.184 0.50 7.55 x' .2661.037 .2031.004 .1701.058 .2251.015 y f .3401.005 .2511.015 .2291.015 15.1 x' .6581.011 .3841.086 .5341.063 .5721.048 y r .7581.008 .4301.107 .6241.074 20.1 x' .9951.124 .4481.186 .8881.167 .8661.112 y' 1.0081.120 .5461.080 .8471.152 - 55 -uncertainties, Here the uncertainties, are the values returned by the program. These values of v^ r e f l e c t how steep the chi-square surface i s i n the v i c i n i t y of the minimum. This depends to some extent on the value of the cutoff chosen i n searching the two parameter space as w e l l as the uncertainties of the Y parameters input to the program. Some of the effects on the determination of a w i l l be discussed i n Chapter 4, a i s calculated by a = [ a 2 - ( a ) 2 f 2 ms w wo — 1 2 2 J ' and the uncertainty by Acr = a [ ( A a a ) + ( A a a ) ] 2 . J J ms ms w w wo wo In addition to the c i r c u l a r f i e l d data, one experiment using a straight edge aligned along the y' dir e c t i o n was carried out. The results of this experiment, using equation (B--4) for the determination of a , are given i n table IX, An ov e r a l l summary of the a values for a l l f i e l d J ms sizes i s given i n table X, A discussion of the assumptions involved i n the analysis w i l l be given i n the next chapter, however, i t i s w e l l to note that the results at 1 8 0 MeV/c for the three f i e l d sizes are i n substantial agreement with the exception of the deepest depth where there appears to be a systematic increase i n a with f i e l d s i z e . - 56 -TABLE IX EXPERIMENTAL VALUES OF a FOR 180 MEV/c WIDE BLADE CASE (1 EXPERIMENT) STRAIGHT EDGE DEPTH cr w a wo a ms cm cm . cm cm 7.55 .302 ± .017 .195 ± .018 .231 ± .027 15.1 .724 + ,035 .344 ± .036 .637 ± .044 20.1 1.178 ± .053 .510 ± .036 1.062 ± .061 TABLE X SUMMARY OF a FOR ALL EXPERIMENTS ms MOMENTUM MeV/c RANGE IN TANK cm DEPTH cm a ms cm r = 0.50 cm c r = 1.125 cm c Straight Edge 148 11.3 2.4 0.052 ± .011 11.3 0.430 ± .016 180 20.1 7.55 0.226 ± .013 0.219 ± .010 0.231 ± .027 15.1 0.578 ± .045 0.588 ± .051 0.637 ± .044 20.1 0.855 ± .081 0.968 ± .093 1.062 ± .061 4. DISCUSSION 4.1 Introduction In this chapter the assumptions made in the analysis of the experiments are presented, followed by a discussion of the experimental factors involved. These experimental factors affect both the analysis and the overall significance of the experiments and consist of the following: (1) effect of momentum spread, (2) effect of contaminating electrons and muons, (3) effect of background scatter, (4) film as a particle counter, (5) effect of non-parallel nature of beam. These factors are discussed point by point and then the theory presented in Chapter 2 and Appendix A i s compared to the experimental results. Finally, the values of a for both theory and experiment are compared to theories presented in the literature. 4.2 Assumptions in the Data Analysis As mentioned in Chapter 3, the procedure for analyzing the optical density profiles of the films was based on a comparison of the three experimental points Y 7 5, Y 5 Q, Y 2 5 on the profile, with those from calculated distributions based on equation (C-l) for values of the two parameters, A and a * . The following assumptions are involved in using the calculated distributions to analyze the experimental data. - 58 -(1) In both cases with and without water in the tank, the d i s t r i -butions are assumed to result from a simple Gaussian convolved with a circular aperture. This implies that the value of a i s the same at the three levels (i.e. 75%, 50%, 25%) of the resulting distribution. This assumption i s violated by the effects of momentum spread and contamination which, involve sums of Gaussian terms (discussed i n sections 4.3.1, 4.3.2). In addition, the assumption i s made that the background scatter (from before and after the collimator) i s also Gaussian (or may be approximated by a Gaussian). Also, as mentioned i n Chapter 2, the multiple scatter i s assumed to be Gaussian in 2 2 1^  order that a may be calculated as [a - (a ) ] 2 (discussed ms 3 w wo in section 4.3.3). (2) The calculated distributions are number distributions and the assumption i s made that the film chosen behaves as a particle counter, particularly i n the direction perpendicular to the beam axis (discussed i n section 4.3.4). (3) It i s assumed that any non-parallel behaviour of the beam can be accounted for in the magnification parameter, A. This assumes that i t i s possible to separate the effect of f i n i t e emittance into a point source term (zero emittance) described by the parameter A and a Gaussian term included i n the back-ground scatter (discussed in section 4.3.3). Finite emittance is defined as the area in phase space occupied by the beam, but in our case the effects are regarded as due only to a f i n i t e source size. The observed divergence in one direction means that an e l l i p t i c a l , rather than circular shape results - 59 -at depth, which involves some error in the evaluation of A (discussed i n section 4.3.5). The last assumption made is that the point source is far enough from the collimator for the divergence to be small i n angle, to allow the simple correction derived in Appendix C. The extent to which the experimental factors influence these assumptions i s given in the next section as each factor is discussed in detail. 4.3 Experimental Factors 4.3.1 Effect of Momentum Spread This effect includes the effect of range straggling, but since the momentum spread is much larger only the case of momentum spread w i l l be discussed. The experimental data for the 180 MeV/c experiments with large momentum spread (wide blades, table VII) does not show any significant difference from that for a small momentum spread (narrow blades, table VIII). In order to see i f this was reasonable, calculations were made to see what effects could be expected for two shapes of the momentum spectrum: (1) Gaussian and (2) Rectangular. These spectra N(p) were chosen to have equal areas and the same widths, i.e. ± 7% ^ at half maximum. The calculations used equation (17) [numerically (A-5)] for finding a as a function of depth x for the ranges r Q ( P ) i n the 180 MeV/c case. The resultant distribution for each of the two spectra was compared with that for a monoenergetic beam with the range 20.1 cm at three points; where the monoenergetic beam was predicted to have values of 75%, 50%, and 25% of the peak. As expected, the resultant - 60 -distribution, which was the sum of Gaussian terms, was not Gaussian. However, at each level i t was possible to assign a value of a to give agreement. In doing this, the resulting deviation of a from the average value was less than 1% and the average values themselves were within 1% of the single momentum case, except for the depth corresponding to the mean range of 20.1 cm. At this depth the spread of momentum leads to a calculated reduction in the value of a by 10% for the assumed Gaussian and 8% for the rectangular momentum spectrum. One possible reason for not observing this difference was that the narrow blade case was not truly mono-energetic, having a momentum spread of approximately ± 2%. However, calculations for this case indicate that there should s t i l l be a 7% difference between the wide and narrow blade Gaussian cases. This 7% spread is within the limits of the estimated error (tables VII and VIII), so i t is not surprising that no systematic difference i s observed between the two cases. 4.3,2 Effect of Contaminating Electrons and Muons The contaminating electrons and muons, due to their independent scattering, should affect the resulting scattering distributions, particularly since i t i s assumed that the different particle types contribute to film blackening on the basis of number. Since i t was impossible to separate the scattering effects of each particle in our experiments, a calculation was made to try to estimate the effects of the different particle types. Due to the fact that electrons are such light particles, - 61 -c a l c u l a t i o n s of electron penetration i n a th i c k scattering medium are very d i f f i c u l t ; the quantitative assessment of the stopping d i s t r i -butions involves Monte Carlo treatments or solutions of the transport equation, both beyond the scope of t h i s discussion. Thus, we have ar r i v e d at a procedure f o r i n f e r r i n g the values of a f o r the electrons. The c a l c u l a t i o n s assumed the case of a s i n g l e momentum with each p a r t i c l e type represented by a Gaussian.scattering d i s t r i b u t i o n weighted according to the f r a c t i o n by number determined experimentally for the beamline (40), The procedure involved c a l c u l a t i n g the values of a f o r the pions and muons at each depth using equations of the form of equation (17), The differe n c e between the experimental values of a m g determined f o r the r = 1.125 cm case and the pion and muon values c r calculated were a t t r i b u t e d e n t i r e l y to the sca t t e r i n g of the electrons. The values of a f o r the electrons determined at each of the three positions (at the 75%, 50%, 25% l e v e l s , calculated using cr^) were d i f f e r e n t since again a sum of Gaussian terms was involved. The v a r i a t i o n about the average a for the three pos i t i o n s was t y p i c a l l y 3% to 10% and t h i s i s larger than f o r the case of momentum spread. The values required f o r the standard deviation, a, of the electron s c a t t e r i n g are generally larger than the cal c u l a t e d pion values. Simple c a l c u l a t i o n s f or electrons [based on the Continuous Slowing Down Approximation (CSDA) data i n reference (44)] y i e l d values too small to account for the whole differ e n c e between experiment and the pion and muon c a l c u l a t i o n s . Since accurate values of a for electrons are d i f f i c u l t to c a l c u l a t e , we have not attempted to separate t h e i r c o n t r i -bution from the t o t a l measured value of o. - 62 -4.3.3 E f f e c t s of Background Scatter This includes a l l factors which are common to both the "water i n " and "water out" d i s t r i b u t i o n s . These are p r i m a r i l y s c a t t e r i n g from structures before and a f t e r the collimator but also included i s the f i n i t e emittance part of the beam as discussed i n section 4.2. It may be e a s i l y shown [ e s p e c i a l l y using the convolution-m u l t i p l i c a t i o n properties of Fourier transforms (29)] that the convolution of two Gaussians, with standard deviations and o^, i s simply another 2 2 2 Gaussian with o" = ai + a2 * ^ n u s ^ t n e n o water and multiple 2 2 2 sca t t e r i n g d i s t r i b u t i o n s are both Gaussian then a = (a ) + (a ) 6 w ms wo 2 2 2 or as we have used i t (a ) = a - (a ) . We assume that the no water ms w wo d i s t r i b u t i o n accounts f o r a l l e f f e c t s except the desired measurement. Our assumption i s that t h i s no water d i s t r i b u t i o n i s given by the convolutions of e f f e c t s that are Gaussian or may be approximated by a Gaussian and hence the t o t a l d i s t r i b u t i o n i s Gaussian. The two contributions, beam emittance and scatter from before the collimator, can be treated together since, as Banford (43) points out, scatter changes a zero emittance beam to a f i n i t e emittance beam. Under some s p e c i a l i z e d assumptions f o r a f i n i t e emittance beam (uniform density of e l l i p t i c a l shape i n phase space) the beam p r o f i l e i n one 2 2 d i r e c t i o n can be described by 1 - x /a for y = 0. This can be f i t t e d by a Gaussian with some adjustment i n the value of a at the three measurement l e v e l s . The v a r i a t i o n i s approximately 10% from the average value of a. The e f f e c t of t h i s v a r i a t i o n on cr w i l l be presented a f t e r the following discussion. - 63 -The last consideration i s that of scatter from the water tank wall after the collimator. If the thickness of the Perspex wall i s replaced by the equivalent thickness in water (table II), i.e. 1.43 cm, then the effect of this scatter can be calculated for the various film depths by modifying equation (16), If one changes the limits to R Q - 6 and R Q for the lower and upper limits respectively, then one obtains on integration <r > = 3-2c' (3-2c' 1 - 1 - ^ ~ 3-2 c' + 2 (3-2c') (2-2c«) *F 1 - 6_ R , 2-2c' - 1 + (3-2c') d-2c') *F R , 1 -R0! l-2c' (25) This i s evaluated, using values of K' and c' given in Appendix A, for R Q = 21.5 cm (20.1 + 1.4) and 6 = 1.4 cm for the various depths 2 J>-(+ 1.4 cm) of the experiment in table XI. Here a = (<(r )>/2) 2, and 0 i s the value for scatter from the water tank only. TABLE XI COMPARISON OF a WITH CALCULATED CONTRIBUTIONS wo FROM THE WATER TANK WALL FOR R» = 21.5 AND 6=1.4 Depth (+1.4) a w t (Eq. 25) a (minimum-maximum) wo cm cm cm 8.95 ,136 .203 - .270 16.5 .261 .362 - .450 21.5 . v .343 .43 - .58 - 64 -The values for cr ^  and a from table XI can be used in conjunction with wt wo the values of a (table X). to determine the effect of the 10% variation ms in a^ e ( f i n i t e emittance) on g^ R. Considering only the variation in O f e > • ) 2 = Co. ) 2 + (o J wo fe wt 2 2 2 i t may easily be shown that i f (cr  (a f  cr ) then 2 2 Acr (a ) - (a . ) Aa-ms _ wo wt f e a , .2 ac ms (a ) fe ms This results in variations for a m g generally less than the 10% variation proposed for O f e , especially at the two largest depths i n the experiment. It thus appears that the Gaussian approximation i s well established and 2 2 J' l i t t l e error should result in determining a using [a - (a ) ] 2 . • ms • w wo J 4.3.4 Film as a Particle Counter It was stated in section 3.2 that on the basis of the data of Tochilin et a l . (37) film would be considered to be a particle counter. In our experiments we are interested in the radial distributions measured from films placed perpendicular to the beam axis. There are situations, such as that for a pure monoenergetic beam, where the radial distributions at any depth (except where TT star products are important) should not depend on whether film behaves as a particle counter, since approximately the same LET spectrum should be present at any radial position. The case where the effects of the particle counter assumption on the radial distributions should be most noticeable is for a beam with a large momentum spread and having contaminants with significantly different scattering from the pions. The effect of momentum spread, as indicated in section 4.3.1 for the 180 MeV/c experiments, does not seem to cause any experimentally detectable difference in the value of o"ms determined - 65 -for large or small momentum spreads. In this case, while i t is d i f f i c u l t to determine the scattering of the electrons (section 4.3.2), the derived values of a for the electrons are larger than those calculated for the pions. Thus, i t appears that in this case where the effects of the particle counter assumption should be most noticeable, i f there i s any non-compliance with the assumption, i t does not greatly affect the radial distribution. However, we would s t i l l l i k e to see how closely the film behaves as a particle counter. We have three sets of experimental data that lend support for the assumption of film as behaving as a particle counter, on the basis of the overall film darkening with depth. They are : (i) parallel films ( i i ) exposure tests ( i i i ) axial density from multiple scatter data. The films parallel to the beam axis were mainly simple i n i t i a l attempts to measure the variation in fil m blackening with depth, and no special precautions were taken to eliminate air gaps. In spite of the problems mentioned by Dutreix (31), which make using the numbers for the film densities of l i t t l e value, the generally uniform response with depth lends support for considering the film to behave as a particle counter, rather than as a dosimeter. In the exposure test films, which were done without collimation, there appears to be some inconsistency in the film darkening with depth. In the 148 MeV/c case (figure 11) the O.D. at the peak (x = 11.3 cm) i s greater than the O.D. at the front (x = 2.4 cm). In the 180 MeV/c test - 66 -(figure 12) the O.D. decreases steadily as depth increases. The lack of collimation and non-uniform intensity, which causes different densities at different areas on the film, may account for some of this discrepancy due to the scattering out from the higher density areas into the lower density areas to give a more uniform density at depth. This explanation appears to be lik e l y since the results based on the collimated multiple scatter data (discussed next) are consistent for both momenta. As mentioned, the multiple scattering data can also be analyzed to see how closely film behaves like a particle counter. If equation (19) is used, which relates the intensity on the axis to the radius of the aperture, r , and the standard deviation of the scatter, a, and corrections are made to the net O.D. (total minus background) for the loss of particles due to in-flight interactions, i t i s possible to calculate a value for a based on the optical density on the axis only. Data for the 148 MeV/c case (including a run of r £ = 0.50 cm not analyzed for lateral distribution) with the corrections made is given in table XII. This calculation has also been done for the 180 MeV/c case and i s given i n table XIII. The shallowest depth and largest f i e l d size i s used as the point of reference and normalization in each case. Comparison with the values of i n tables VI, VII and VIII shows that the values for a are, within 10% to 15%, in agreement for the two methods, From the results of the variation of fil m darkening with depth i t would appear that the film may be assumed, within 10% to 15%, to be a particle counter. The effect that this has on the radial - 67 -distribution i s certainly much less than this limit, as witnessed by the good agreement for the wide and narrow blade cases discussed earlier. Thus, our determination of a from the radial distributions should not be restricted by this 15% limit on the validity of the particle counter assumption. TABLE XII CALCULATION OF a FROM AXIAL DENSITY AFTER CORRECTION FOR IN-FLIGHT INTERACTIONS, 148 MEV/c Depth Net Density Corrected, Normalized a [Equation (19)] Net Density cm cm r =1.125cm c r =0.50cm c r =1.125cm c r =0.50cm c r =1.125cm c r =0.50cm c 2.4 .99 .90 1.00 .91 < .34 .23 11.3 .72 .27 .85 .32 .59 .57 TABLE XIII CALCULATION OF a FROM AXIAL DENSITY AFTER CORRECTION FOR IN-FLIGHT INTERACTIONS, 180 MEV/c Depth Net Density Corrected, Normalized a [Equation (19)] Net Density cm cm r =1.125cm r =0.50cm r =1.125cm r =0.50cm r =1.125cm r =0.50cm c c c c c c 7.55 .92 .62 1.00 .674 < .34 .33 15.1 .62 .20 .767 .248 .66 .66 20.1 .36 .10 .491 .137 .97 .92 - 68 -4.3.5 Effect of Non-parallel Beam In this section only the point source approximation is discussed since the assumption of the separation of the f i n i t e emittance from the divergence has been discussed i n section 4.2 and 4.3.3. As mentioned in Chapter 3, the beam envelope is observed to diverge in the x' direction and remain approximately parallel in the y* direction, although the focus seems to be far enough from the film to satisfy the small angle criterion. At points downstream from the collimator, the resulting beam w i l l be somewhat e l l i p t i c a l in shape. It is hard to estimate quantitatively the effect this may have in determining a * but as may be seen in tables VI, VII,and VIII there is a systematic tendency for values of a and a to be lower for the J J w wo x' profile (major axis of ellipse) than for the y 1 profile (minor axis of ell i p s e ) . The explanation of this observation involves consideration of the effect of assuming circular symmetry in the f i t t i n g procedure and the effect of variations in the magnification parameter A, on the scattering parameter a ' . Along the major axis of the e l l i p t i c a l distribution there i s less scatter contribution from off the axis than for a circular d i s t r i -bution of the same radius. Thus, i f one assumes a circular shape in the f i t t i n g procedure, the value of the magnification parameter A determined w i l l be an underestimate of the actual value. Conversely, along the minor axis of the ellipse the magnification (A) w i l l be overestimated.. The effect of these errors in A for estimating a ' can be seen from figure 14 to depend on the region of a ' that is involved. Generally for o ' below about .5, the effect of underestimating A is to - 69 -underestimate a' since both the and Y^Q tend i n this direction, although the goes i n the opposite direction. Similarly a' i s overestimated when A is overestimated. In the region of a' greater than .5, the effect of underestimating A is for a' to be overestimated and vice versa for A being overestimated. Within the limits of the estimated error for a r, this argument tends to explain the observations of cr and a for the x', y r data given in tables VI, VII and VIII. w wo ' The effect on cr i s d i f f i c u l t to assess, although i t is possible to ms » o r conceive of situations which reverse the systematic difference between x', y' directions. Averaging the values of a m g for the two directions as has been done should result in a better estimate of a ms 4.4 Comparison with Theory The theoretical calculations of a for pions, based on the ms 2 values for K' and c f given in Appendix A for the two f(a ) equations, are evaluated using equation (17). These values, as well as the results of the experiments, are given in table XIV. Comparison of the values in table XIV shows that, with the exception of the peak value of the 148 MeV/c case and the peak value for r c = 0.50 cm in the 180 MeV/c case, the experimental values are greater than the theoretical values. This i s what has been observed by Mayes et a l . (20), but even with their suggested correction the situation i s not altered significantly. Clearly the uncertainty in the experiment i s such that their proposal cannot be commended over that of Moliere's original theory. In order to better show the variation 2 of a with depth, the theoretical curves [Moliere's f(a )] are plotted ms against depth in figure 15. Also shown are the experimental values, E o Z o < > LU Q O rr < o z 1.6 1.2 .8 Field size r c - 1.125 cm r c -0 .50 cm straight edge (  Range R0(cm) 11.3 20.1 • • O o 20.0 FIGURE 15 COMPARISON OF a FOR THEORY (CURVES) AND ms EXPERIMENT (POINTS) AGAINST DEPTH IN WATER - 71 -with error bars for the r = 1.125 cm case only. An alternate form of c J presentation due to Preston and Koehler (22) y which incorporates a l l the data, is shown in figure 16. Here a/a^ i s plotted against x/R_, RQ ^ 0 where c r R is the value given by Moliere's f ( a ) equation when x = R_, 0 again only the error bars for the = 1.125 cm case are shown. TABLE XIV COMPARISON OF a FROM THEORY AND EXPERIMENT ms Momentum MeV/c Range in Tank cm Depth cm a (Theory) ms cm c r (Experiment) cm Moliere Mayes r =0.50cm c r =1.125cm c Straight Edge 148 11.3 2.4 .042 .043 .052±.011 11.3 .508 .523 .4301.016 180 20.1 7.55 .179 .184 .226±.013 .219+.010 .2311.027 15.1 .544 .560 .578±.045 .5881.051 .6371.044 20.1 .904 .930 .8551.081 .9681.093 1.0621.061 The agreement is not too bad, when one considers that the theoretical value is for pions only and the experiment includes contri-butions due to electrons and muons, as well as star products at those depths where pions are stopping. When calculating the lateral d i s t r i -bution of pions, for purposes of dose distributions in treatment planning, i t thus seems reasonable to assume that the lateral distributions of the electrons and muons are the same as for the pions as L i et a l . (8) have done. Indeed, when one is calculating the dose distributions, the contributions of electrons and muons to the dose are a much smaller fraction of the total than their fractions by numbers. (See for example COMPARISON OF SCATTER PARAMETER FOR THEORY (CURVE) AND EXPERIMENT (POINTS) AS A FUNCTION OF RANGE RELATIVE TO END OF RANGE VALUES - 73 -Turner et a l . (10) where the assumed relative numbers are comparable to the 180 MeV/c case.) As observed in Chapter 3, there appears to be a systematic increase in a with f i e l d size for the peak depth in the 180 MeV/c ms case (table X or XIV). In order to decide i f there i s a reasonable explanation for this difference, we re-assess the data analysis for the various fi e l d s . Consideration of the case of circular fields shows that for large values of a r , the central axis intensity is considerably reduced (see for example figure 6). In the case of r ^ = 0.50 cm at the peak depth (x = 20.1 cm) the value of a ' is on the order of 2 which results in the distribution being very spread out. This means that the determi-nation of the Y parameters, especially ^25* is very sensitive to v a r i -ations in the background film density. A means of checking the r e l i a b i l i t y of the determination of a ' i s to examine the consistency of the values of A resulting from the f i t t i n g procedure. One way of doing this i s to assume that the value of the magnification, A, i s correct for the shallowest depth (smaller c r r , sharp edge), then calculate the value of the focal distance f (Appendix C). Using this value for f, values of A for the succeeding depths are calculated for the same experimental conditions. Comparing the calculated values with the values given by the f i t t i n g program indicated that for = 0,5 cm the A values for the two methods do not agree well, whereas for r £ = 1.125 cm the agreement was much better. Thus i t would appear that the values for r £ = 1.125 cm, (and r c = 0.50 cm at shallow depths) because they have smaller values of a ' , resulting in steeper distributions, are less sensitive to variations in the background fi l m density and provide a more .reliable - 74 -value for A and thus a . Matching the aperture size to the value of a ( t r ' - l ) should be expected to give sufficient information without overly reducing the signal to noise ratio at any of the levels chosen for analysis. In the case of the straight edge experiment we have used a different method of analysis. There is no term to correct for divergence so that i t has been assumed that there i s either no divergence or that the projection of the edge at the film remains a straight edge, except for scatter effects. This i s not expected to be a particularly good assumption since there w i l l be some spread at depth due to the observed divergence. One possible correction that can be made i s to assume that the value of the magnification, A, for the = 1.125 cm case i s valid for the straight edge and divide the straight edge value of a m g by A. Using a value of A = 1.2, which is typical for the wide blade case at x = 20.1 cm depth, this reduces a from 1.062 cm to 0.885 cm, which i s r ' ms in substantial agreement with the values for the circular apertures. Another problem arises due to the d i f f i c u l t y in achieving a uniform intensity distribution for a beam that is uncollimated and of large spatial extent. It is therefore argued that the values for the r c = 1.125.cm case are more reliable and should be considered to represent the best experimental values. However, a l l values w i l l be included in the compari-son with the published values. 4.5 Comparison to Values in the Literature Most references to calculations of pion scattering have been to the work of Fowler and Perkins (1), whose values were used in both dose calculating schemes (8,11,12) mentioned in Chapter 1. Curtis and - 75 -Raju (45) have also given a formula using different numerical coefficients than Fowler and Perkins. These two formulas for the standard deviation in the multiple scattering Gaussian, for x = R Q, are given in table XV, as well as that of our theoretical derivation presented in Chapter 2. When comparison i s made to the experimental data i t would appear that the values of Curtis and Raju are too large and that those of Fowler and Perkins are probably too large also, although there is the single point for the straight edge experiment which would overlap their value of a ms TABLE XV COMPARISON OF CALCULATED VALUES FOR cf FOR RANGE OF 20.1 CM ms Source K" c" Calculated Values of o = K"R„ C" (R_ = 20.1 cm) ms T0 0 Fowler and Perkins (1) 0.07 0.92 1.11 cm Curtis and Raju (45) 0.0763 0.95 1.32 cm Equation (A-5) 0.045 1.00 0.90 cm 3 a = (<r2>/2)^ evaluated for R_ = 0 ms r The theory presented in Chapter 2 gives better agreement with the experiment than either of the other theories. In order to demonstrate whether the differences between the theories have practical significance, we have calculated the radial distributions for two circular fields of ra d i i , 1.0 cm and 5.0 cm at a depth of 20,1 cm, and these are plotted in figure 17. These are the resultant distributions for a parallel beam of uniform intensity over - 76 -i — — r .8 Source of (Tms rc= 1.0 cm > Eqn(A-5) Referenced) Reference(45) rc= 5.0 cm 2.0 4.0 6.0 RADIAL DISTANCE (cm) 8.0 FIGURE 17 CALCULATED DISTRIBUTIONS FOR TWO CIRCULAR RADII COMPARING VALUES OF a AT R. = 20,1 cm ms 0 - l i -the, area of each f i e l d at zero depth. The differences are significant and since the theory presented in this work gives better agreement with experiment, i t i s suggested that this theory be adopted at least until more experimental work is done. In any event, the results indicate that multiple scattering w i l l greatly affect the shape of the dose distribution. When treating patients, i t has been proposed to use a fixed channel momentum in the range of 160 MeV/c to 200 MeV/c to reduce the electron contamination and increase the pion flux. It w i l l be necessary in this approach to add bolus (absorber) to adjust the stopping peak to the tumour location. Assuming a mean momentum of 180 MeV/c or a range of 22,3 cm of water, this would imply that f i e l d sizes less than 5 cm radius w i l l begin to be strongly affected by the multiple scattering of the pions. Thus, while one of the hoped for advantages of pions i s better dose concentration at the peak, i t appears that multiple scatter w i l l severely restrict the size of tumours for which this applies. If i t i s desired to treat small f i e l d sizes at shallow depths using a fixed midline momentum, i t w i l l be necessary to collimate the beam as close to the end of the bolus as possible. - 78 -5. CONCLUSIONS In this study the theory of multiple scatter has been examined and measurements have been made using medical x-ray film for a beam of negative pions incident on a thick water phantom. Calculations have been presented for the multiple scatter based on the Gaussian term of Moliere^s theory, with modification for the Fano correction as well as energy loss. This calculation gives values for the standard deviation of the Gaussian lateral distribution which are 20% lower than values from the previous calculation of Fowler and Perkins (1). The contamination of the pion beam by electrons and muons makes i t d i f f i c u l t to derive the pion scattering from the experiments, since i t i s assumed that the contaminants contribute to the films by number. Since the magnitude of electron scattering for thick sections i s d i f f i c u l t to calculate, the separation of the pion scattering from the total i s uncertain, and this has not been done. However, when the best value of a for the pencil beam [derived from the f i n i t e size beam measurements (r = 1.125 cm)] is compared with that of our theory for pions alone, the experimental value i s greater than the theoretical by only 7%. It i s possible to correct the theory for centre of mass effects amounting to about 1% and an additional correction of 3% has been proposed by Mayes et a l , (20) but the vali d i t y of this i s uncertain at the present time. The agreement between the measured and calculated values for depths less than the f i n a l range would indicate that equation (A-5) should be adequate for treatment planning. This equation is derived from the theory presented in Chapter 2 [equation (17)] with values for - 79 -the constants K* and c' determined i n Appendix A and is reproduced below: <r2> = 0.004064R 1.99855 0 1 - 4.002904 + 1381.373180 - 1378.370276 r^U.99855 R0 R, 2 i^-where a = [<r >/2] 2 with R^, = RQ - x. A simplification of the above formula arises in the case of water as pointed out in Appendix A, such that the expression for <^r >^ becomes <r2> = . 0 0 4 1 R Q 2 1 - 4 ^ l 2 3 - 2 In R , which gives results of sufficient accuracy. In addition, the agreement between theory and experiment indicates that i t should be valid to assume that the contaminating electrons and muons scatter the same as pions when calculating dose distributions. Due to the uncertainty of the LET dependency of the film, the electron contamination, as well as a possible f i e l d size effect, i t i s recommended that measurements be made with some other detector (such as a sil i c o n diode). There are problems with detectors which behave more like dosimeters than particle counters, since at depths near the pion stopping peak i t w i l l be necessary to account for the broadening effect to the dose distribution by the pion star products. However, alternate measurements may allow resolution of the apparent f i e l d size effect and comparison of the results with those of film may give further indication of the accuracy of film. This may allow the use of film in relative - 80 -measurements, p a r t i c u l a r l y i n the i n v e s t i g a t i o n of the e f f e c t s of inhomogeneities. In the a p p l i c a t i o n of pions for radiotherapy, our r e s u l t s i n d i c a t e that the stopping d i s t r i b u t i o n s should be l e s s dispersed than previously predicted. However, i t has also been demonstrated that the e f f e c t s of multiple s c a t t e r i n g are s i g n i f i c a n t . The s i z e of these e f f e c t s i n d i c a t e s that i f absorber i s used to s h i f t the depth of the stopping peak then c o l l i m a t i o n of the pion beam should be done a f t e r the absorber i f p o s s i b l e . In summary, the r e s u l t s of the experiment are a f i r s t attempt to determine the multiple s c a t t e r i n g of pions from the complicated case of a f i n i t e s i z e , n o n ^ p a r a l l e l beam, with a d d i t i o n a l problems due to star products and contaminating electrons and muons. The agreement with the theory presented appears to be s u f f i c i e n t to allow the use of t h i s theory i n c a l c u l a t i o n s f or radiotherapy, at l e a s t u n t i l more experimental determinations are completed. - 81 -BIBLIOGRAPHY 1. P.H, FOWLER and D.H. PERKINS, "The Possibility of Therapeutic Applications of Beams of Negative Tr-Mesons". Nature, 189, 524-528 (1961). 2. P.H. FOWLER, "1964 Rutherford Memorial Lecture: Tr-Mesons versus Cancer?". Proceedings of the Physical Society (London), 85, 1051-1066 (1965). 3. M.R. RAJU and C. RICHMAN, "Negative Pion Radiotherapy: Physical and Radiobiological Aspects". 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TURNER, "Monte Carlo Treatment of Multiple Coulomb Scattering i n Pion-Beam Dose Calculations". Journal of Applied Physics, 46_, 4445-4452 (1975). 13. R.T. SANTORO, R.G. ALSMILLER and K.C. CHANDLER, "Calculation of the Effects Caused by Bone Present in Phantoms Irradiated by Negatively Charged Pions". Medical Physics, I, 303-310 (1974). 14. R.N, HAMM, H.A. WRIGHT and J.E. TURNER, "Effects of Tissue Inhomo-geneities on Dose Patterns in Cylinders Irradiated by Negative Pion Beams". ORNL-TM-5088 Report (Oak Ridge National Laboratory, October, 1975). 15. R.N. HAMM, H.A. WRIGHT and J.E. TURNER, "Effects of Tissue Inhomo-geneities on Dose Patterns in Cylinders Irradiated by Negative-Pion Beams". Physics in Medicine and Biology, 21, 982-987 (1976). 16. H.A. BETHE, "Moliere's Theory of Multiple Scattering". Physical Review, 89, 1256-1266 (1953). 17. W.T. SCOTT, "The Theory of Small-Angle Multiple Scattering of Fast Charged Particles". Reviews of Modern Physics, 35_, 231-313 (1963). 18. U. 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Roesch, editors (Academic Press, New York, 1968), pp. 157-228. 24. V.L. HIGHLAND, "Some Practical Remarks on Multiple Scattering". Nuclear Instruments and Methods, 129, 497-499 (1975). 25. "Pion and Muon Multiple Coulomb-Scattering Experiment on EPICS Beam Line (Exp. 325)" i n "Medium Energy Physics Program", compiled by O.B. Van Dyck and E.D. Dunn, Los Alamos Scientific Laboratory Progress Report LA-6938-PR (1977), pp. 79-80. - 83 -26. M.R. RAJU, J.T. LYMAN, T. BRUSTAD and C.A. TOBIAS, "Heavy Charged-Particle Beams". In "Radiation Dosimetry, Volume 3", second edition, F.H. Attix and E. Tochilin, editors (Academic Press, New York, 1969), pp. 151-199. 27. J. CARLSSON and K. ROSANDER, "Effects of Multiple Scattering on Proton Beams in Radiotherapy". Physics in Medicine and Biology, 18, 633-640 (1973). 28. P.R. BEVINGTON, "Data Reduction and Error Analysis for the Physical Sciences" (McGraw-Hill, New York, 1969). 29. R.M. BRACEWELL, "The Fourier Transform and i t s Applications" (McGraw-H i l l , New York, 1965). 30. M. ABRAMOWITZ and I.A. STEGUN, editors, "Handbook of Mathematical Functions" (Dover, New York, 1965). 31. A, DUTREIX, *'Film Dosimetry". In "Radiation Dosimetry", American Association of Physicists in Medicine 1976 Summer School, University of Vermont (1976). 32. P.H, FOWLER and V.M, MAYES, "The Capture of r r " Mesons in Oxygen and in Other Nuclei". Proceedings of the Physical Society (London), 92, 377-389 (1967). 33. N.F. KEMBER, F.A. SMITH and A.G., PERRIS, "An Approach to Microdosimetry in a T f T " Meson Beam using Nuclear Emulsions". Physics in Medicine and Biology, 20, 918-925 (1975). 34. C.E.K. MEES and T.H. JAMES, editors, "The Theory of the Photographic Process", third edition (Macmillan, New York, 1966). 35. R.L. DIXON and K.E. EKSTRAND, "Heuristic Model for Understanding X-Ray Film Characteristics". Medical Physics, 3, 340-345 (1976). 36. R.A. DUDLEY, "Dosimetry with Photographic Emulsions". In "Radiation Dosimetry, Volume 2", second edition, F.H. Attix and W.C. Roesch, editors (Academic Press, New York, 1966), pp. 325-387. 37. E. TOCHILIN, B.W. SHUMWAY and G.D. KOHLER, "Response of Photographic Emulsions to Charged Particles and Neutrons". Radiation Research, 4_, 467-482 (1956). 38. R.W. HARRISON, "A Beam Transport System for the Medical F a c i l i t y at TRIUMF". M.Sc. Thesis, University of Victoria (1972). 39. R.W. HARRISON and D.E. LOBB, "A Negative Pion Beam Transport Channel for Radiobiology and Radiation Therapy at TRIUMF". IEEE Transactions on Nuclear Science, NS-20, 1029-1031 (1973). 40, M, P00N, "Optimization Studies of the TRIUMF Biomedical Pion Beam". M.Sc. Thesis, University of Bri t i s h Columbia (1977). - 84 -41. H. APPEL, V. BOHMER, G, BUCHE, W, KLUGE and H. MATTAY, "iT-Beam Studies Using Time-rof-Flight Methods". Atomkernenergie, 27, 177-180 (1976). 42. M.R. RAJU, E. LAMPO, S.B. CURTIS and C, RICHMAN, "Dosimetry of Tr~^Mesons Using Silicon Detectors and Plastic Scintillators". Physics in Medicine and Biology, _16, 599-610 (1971). 43. A.P. BANFORD, "The Transport of Charged Particle Beams" (E. and F.N. Spon, London, 1966). 44. "Studies in Penetration of Charged Particles in Matter". U. Fano, Chairman (National Academy of Science, National Research Council Publication No. 1133, Washington, 1964), p. 249. 45. S.B. CURTIS and M.R. RAJU, "A Calculation of the Physical Characteristics of Negative Pion Beams — Energy-Loss Distribution and Bragg Curves". Radiation Research, 34, 239-255 (1968). 46. M.I. HENRY, "Dose Calculations Relating to the Use of Negative Pi-Mesons for Radiotherapy". M.Sc. Thesis, University of British Columbia (1973). - 85 -APPENDIX A NUMERICAL EVALUATION OF CONSTANTS FOR MEAN  SQUARE LATERAL DISPLACEMENT <r > [Equation (17)] A.l Determinations of Constants in Equation Relating Momentum and Residual Range (pg = C]R C 2) The data used for this determination i s given in table VI of Henry (46), This i s range-energy data for pions in water scaled from proton values given by Bichsel (23). Table Al contains the values of 2 2 R,T, pg and g over the ranges of interest. The values of pg and g are calculated using the following formulas 2 pg = mc [y - 1/YL 2 where mc = rest mass of pion = 139.6 MeV, Y = T/(mc2) + 1, and g 2 = 1 - [ 1 / Y 2 ] . The data was fi t t e d to a power curve and the values determined for the coefficients C^ and are given below. The correlation coefficient was 0,99997 indicating a good f i t for the range of .6 cm to 30 cm of water 2 (since p = 1 g/cm for water, distances are given in cm). Thus for pions in water pg = C ^ 2 C x = 26.45527, C 2 = 0.5411887. - 86 -TABLE Al VALUES FOR ENERGY RELATED PARAMETERS AS A FUNCTION OF RANGE FOR PIONS IN WATER RANGE ENERGY P6 B 2 cm MeV MeV .6 10.468 20.206 .1346 1.0 13.898 26.538 .1729 1.4 16.754 31.713 .2028 2.4 22.81 42.416 .2612 3.4 27.92 51.186 .3056 5.5 36.98 66.215 .3750 7.6 44.82 78.746 .4270 10.8 55.57 95.317 .4884 14.0 65.39 109.92 .5362 18.0 76.79 126.33 .5838 22.0 87.5 141.29 .6221 26.0 97.69 155.16 .6539 30.0 107.5 168.23 .6808 - 87 -A.2 Determination of Constants in Equation Relating Moliere's B and Residual Range (B = C3R 4) One can calculate the value of Moliere's B parameter quite easily using equation (9) and either equation (10) or (11). The value 2 of B i s seen to depend on Z, A and t of the medium and z, 3 of the 2 incident particle. Making the substitution of Z(Z+1) for Z and 1/3 4/3 Z (Z+l) for Z , one can reproduce the values of B tabulated by 2 Bichsel (23). (A in grams, t in g/cm .) Moliere's theory i s valid for thin f o i l s or no energy loss. However, we are interested in thick scattering sections of compounds such as water and tissue. It is possible to find suitable expressions 2 for 3 as a function of residual range R, but one i s s t i l l l e f t with the problem of what value of the thickness t should be used. Our solution to this problem is to find the thickness t at each range such 2 that the value of 3 changes by only 1% in traversing thickness t. 2 The procedure was to f i r s t find an expression for 3 as a function of R. We decided on the form In (3 2) = X 1 + X 2 InR + X g (InR) 2. (A-l) Using the values in table Al for pions in water we find X x = -1.752155, X 2 = 0.498911, X 3 = ^0.027780, a 2 2 2 These values gave good agreement since £(3^ - 3^ ) was less than - 88 --5 A 2 2 7 x 10 where 3^ i s the value of 3 calculated using equation (A-l) for each value of R in table A l . The next step was to find the value of t at each value of R in table A l . To do this we differentiated equation (A-l) to give (1/B2) dg2/dR = X2/R + (2X3/R) In R, (A-2) 2 2 equating df3 with A3 and dR with AR = t we solved for t such that 2 2 A3 / f 3 = .01 and arrived at t = AR = (.01 R) / (X 2 + 2X3 In R). (A-3) Using equation (A-3) and the values of R in table Al we were able to 2 tabulate t as a function of R (table A l l ) . (R, t in cm since p = 1 g/cm .) Since we also wanted to incorporate Fano's correction [equation (12)] using values of u.^ in table I we then had to solve i n B - In B = In [x// (x Q 2f («2)) ] -..1544 + B' = In 8838.4 z 2 Z 4 / 3 t A(1.13 3 2 + 3.76 z 2Z 2/(137) 2) - .1544 + Z ^ d n [1130 Z 4 / 3 ( 3 2/(l - 3 2 ) ) ] - u. - Js3 2). (A-4) i n When dealing with compounds such as water we then evaluated equation (A-4) for each element at each residual range R making the substitution of t^ = e^t where i s the fraction of the element in the compound by weight. When we had values for B^ at each R we then formed the quantity J = Z e ± (Z.2/A.) B. and divided by <Z2/A> where i - 89 -<Z2/A> = E e± QZ±2/k±) { i which gives for each value of R B~ = J / <Z2/A> . The next step was then to approximate B as a function of the residual range R. Again a power law was found to give good agreement, 2 with a correlation coefficient of 0,99995 using Moliere's f(a ) equation. In the case of water the coefficients in the equation B = CgR1^, are given by C 3 = 9,888340, C. = 0.080927. 4 An example for water i s given in table A l l . A.3 Numerical Evaluation of Equation (17) An evaluation of equation (17) based on these constants, 2 2 2 using Moliere's f (a ) equation where we replace Z /A in K by <(Z /A^ > (since a compound) yields K* = .008122 and c' = 0.500725 and thus <r 2> = 0.004064 R 1 ' 9 9 8 5 5 1 - 4.002904 R, - 1378.370276 R„ + 1381.373180 5L Ro 1.99855 , (A-5) and again a = (<r >/2)' TABLE A l l SAMPLE CALCULATION OF B = C,R 4 FOR WATER USING MOLIERE'S f (a ) EQUATION HYDROGEN OXYGEN RANGE cm t cm 2/ 2.. 2. X c /X Q f(a ) B" BH fco 2/ 2,, 2. xc /xG f ) B" Bo J B 0,6 .0114 .00128 73.4156 8.7018 15.5901 .01012 542.2527 .9161 9.2860 34.7203 9.4770 1.0 .0200 .00224 100.2708 8.9783 16.2179 .01776 753.5909 .9507 9.6926 36.2345 9.8903 1.4 .0292 .00327 124.8120 9.1597 16.6442 .02593 946.6396 .9734 9.9716 37.2730 10.1738 2.4 .0533 .00596 176,9393 9.4596 17.3336 .04734 1357.6684 1 .0109 10.4131 38.9180 10.6228 3.4 .0789 .00883 223.9035 9.6564 17.7919 .07007 1728.2041 1 .0355 10.7068 40.0123 10.9215 5,5 .1361 .01523 314.7878 9.9316 18.4438 .12087 2445.5927 1 .0699 11.1269 41.5772 11.3486 7.6 .1968 ,02202 399.7635 10.1224 18.8978 .17478 3116.8202 1 .0937 11.4192 42.6660 11.6458 10.8 .2945 .03296 523.0492 10.3394 19.4104 .26155 4091.0911 1 .1208 11.7466 43.8860 11.9788 14.0 .3974 .04447 642.8971 10.5069 19.8044 .35293 5038.6160 1 .1418 11.9970 44.8193 12.2336 18.0 .5320 .05953 790.4997 10.6765 20.2004 .47247 6205.7157 1 .1630 12.2471 45.7518 12.4881 22.0 ,6724 .07524 937.6287 10.8174 20.5281 .59716 7369.2237 1 .1806 12.4533 46.5207 12.6980 26.0 .8179 .09152 1085.0648 10.9392 20,8096 .72638 8535.3611 1 .1958 12.6294 47.1776 12.8773 30.0 .9679 .10831 1233.3389 11.0470 21.0573 .85959 9708.1559 1 .2093 12.7838 47.7536 13.0345 - 9 1 -2 The evaluation of K' and c' using the suggested f ( a ) equation of Mayes et a l . ( 2 0 ) gives values K' = . 0 0 8 6 7 5 , c' = . 5 0 2 7 9 2 . These values when inserted i n equation ( 1 7 ) allow c a l c u l a t i o n of a ms for comparison with values using equation (A-5) as shown i n table XIV. 2 We decided to use only Moliere's f ( a ) equation and hence have only presented equation (A-5) f o r c a l c u l a t i n g cr . In the case of water 2 using Moliere's f(a ) equation, the value of c' i s close enough to .5000 that a simple approximation may be made before the i n t e g r a t i o n which y i e l d s equation ( 1 7 ) . This allows an alte r n a t e form that i s s u f f i c i e n t l y accurate f o r our purposes, <r 2> = . 0 0 4 1 R Q 2 1 - 4 3 - 2 In V R, (A -6) This converges to . 0 0 4 1 RQ as Rp goes to zero as i s expected. I t should" be noted that t h i s equation applies only f o r pions i n water and f o r other p a r t i c l e s or materials i t i s necessary to follow the above procedure f o r the determination of K' and c' and subsequent s u b s t i t u t i o n i n equation ( 1 7 ) In addition, since we have been dealing with the s p e c i a l case of water 2 with p = 1 g/cm , both R and t have been given i n cm r e s u l t i n g i n a being 2 2 given i n cm. Obviously i f p f 1 g/cm then with R and t measured i n g/cm , a i s given i n g/cm APPENDIX B EXTENSION OF PENCIL BEAM SCATTER TO INFINITE STRAIGHT EDGE We would l i k e to c a l c u l a t e the scatte r contribution at a point (x,y) from a point at ( x ' , y ? ) , where the multiple s c a t t e r i n g d i s t r i b u t i o n i s given by a Gaussian, G(x-x% y-y') = ^ T r a 2 ) " 1 exp[- (x-x')^/ (2a^) - (y-y') V(2a^)] . For an incident p a r a l l e l beam the l a t e r a l d i s t r i b u t i o n at a point (x,y,z) i s given by the sum of a l l contributions i n the beam p r o f i l e , i . e . , t \ 2 / / o 2,. f(x,y,z) = N(x»,y') G(x-x'.y-y') dx'dy'. ( B - l ) Here a = a(z). I f we collimate the beam by blocking out the l e f t h a l f plane, i . e . a collimator i s aligned along the y' axis we define N(x*,y') = 1 x' > 0. Therefore equation (B-l) becomes f(x,y,z) = dy' ( 2 T r a 2 ) ^ exp[-(x-x')^/(2a z)] exp[-(y-y')'/(2a^)]dx', (2iT)"^a ^jxplXx-x') 2/^ 2)^' 1exp[-(y-y') z/(2a^)]dy' J -co , » N.2 / /o_2^ (27r)"°5a""1 exp[-(x-x*) z/(2a^)]dx^ • 1, 2 2 = (TT)' X /a/I exp(-v ) dv, (B-2) 2 2 2 where we have made the s u b s t i t u t i o n v = (x-x') / (2a ) Therefore f(x,z) = % ± (TT) ^ fx/a/2~ exp(-v ) dv. (B-3) 0 Here the plus sign i s for x>0 and minus for x<0 and obviously for x=0, f(x,z) = ,5. For comparison with experiment we are interested in the cases where f(x,z) = .75, and f(x,z) = .25. Using formulas given in reference (30) for these types of integrals we find for f(x,z) = ,75 , x/a = .6742 f(x,z) = .25 , x/a = -.6742 Thus x?r. - x 2 5 = 1.348 a. (B-4) - 94 -APPENDIX C MODIFICATION TO ALLOW FOR DIVERGENCE In experiments w i t h charged p a r t i c l e beams i t i s d i f f i c u l t , i f not i m p o s s i b l e , to get p a r a l l e l beams. Our p r e l i m i n a r y a n a l y s i s showed that t h i s was tru e i n our beam a l s o , so we undertook a f i r s t order approximation to take account of divergence or convergence. R e f e r r i n g to f i g u r e C l we see the case of a converging beam, i n f a c t , convergent to a p o i n t a d i s t a n c e f from the l o c a t i o n of a c o l l i m a t o r of rad i u s r . c In the case of a p a r a l l e l beam we v i s u a l i z e the l a t e r a l d i s t r i -b u t i o n a t depth as the sum of p e n c i l beams, where each p e n c i l beam spreads out due to m u l t i p l e s c a t t e r i n g given by a Gaussian. For a p a r a l l e l beam the angle a = 0 and the summation at depth f takes p l a c e i n the plane perpendicular to the beam. In the case of a beam convergent to a p o i n t a t depth f , each p e n c i l has a unique angle a ' < a and each p e n c i l has a Gaussian d i s t r i -b u t i o n p e r p e n d i c u l a r to t h i s angle, say f ( r ' ) . I f we measure, w i t h a f i l m say, d i s t r i b u t i o n s i n a plane perpendicular to a = 0, we w i l l measure c o n t r i b u t i o n s from each p e n c i l beam f ( r ' ) a t the r a d i a l d i s t a n c e r = r ' cos ct r '. The maximum angle i s a and thus f o r s m a l l enough values of a the d i s t r i b u t i o n measured perpendicular to a = 0 w i l l show l i t t l e smearing due to summing c o n t r i b u t i o n s over a range of r ' f o r a given r . We set a l i m i t on a such that cos a = .99 or a to be about 8°. For a depth, x, l e s s than f , the beam has r a d i u s r c * = r £ (1-x/f) and e f f e c t i v e l y we are d e a l i n g w i t h a beam of smaller r a d i u s . However, - 95 -Focal plane FIGURE C l SCHEMATIC DIAGRAM AND PARAMETERS FOR A BEAM CONVERGING TO A POINT FIGURE C2 PHASE SPACE PLOTS FOR ZERO EMITTANCE BEAMS AT DEPTH x = 0 - 96 -the summation procedure i s the same as i f we had a p a r a l l e l beam. If we define A = 1 - x/f then i f we replace by A r c we should have the proper c a l c u l a t i o n i n equation (24), except that the number per u n i t area has increased. If we assume a uniform density across the cross s e c t i o n a l area then the number of p a r t i c l e s remains the same but the area has decreased so we 2 need to multiply by the fac t o r 1/A . Thus to correct equation (24) f o r a converging beam under these assumptions, we replace by A r £ and _2 multiply by A , i . e . 2 '°° f ( r , x ) = (27rAr c/A ) J (2irrs) J.(2TrAr s) e x p [ - 2 T r 2 a 2 s 2 ] ds. (C-l) 0 l c Obviously t h i s holds f o r a diverging beam where f = - f and hence A > 1. The point source requirement means we are s t i l l dealing with a zero emittance beam instead of a r e a l f i n i t e emittance beam, but the phase space diagram i s a l t e r e d as shown i n fi g u r e C2. The ph y s i c a l i n t e r p r e t a t i o n of A i s as the magnification, i d e n t i c a l with point source l i g h t o p t i c s . 

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