Fundamentals of Interactions of Particles in Matter

Fundamentals of interactions of particles in matter

Paulo Martins1,2

1 DKFZ, Division of Biomedical Physics in Radiation Oncology, Heidelberg, Germany 2 IBEB, Faculty of Sciences of the University of Lisbon, Portugal

Physics of Charged Particle Therapy Author DivisionBibliography 7/5/20 | Page2 • Techniques for Nuclear and Particle Physics: A How-to Approach, William R. Leo, Springer-Verlag, 2nd Ed., Ch. 2, pp. 17-53 • Principles of Radiation Interaction in Matter and Detection, Claude Leroy and Pier-Giorgio Rancoita, World Scientific, Ch. 1-2, pp. 19-135 • Physics of Proton Interactions in Matter, Bernhard Gottschalk, in: Proton Therapy Physics, Harald Paganetti, Ch. 2, pp. 20-59

Physics of Charged Particle Therapy | Paulo M. Martins Author DivisionBibliography 7/5/20 | Page3 • Radiation Detection and Measurement, Glenn F. Knoll, Wiley, 4th Ed., Ch. 2, pp. 29-42 • Passage of Particles Through Matter, M. Tanabashi et al. (Particle Data Group), Phys. Rev. D 98, 0300001 (2018) pp. 2-17

Physics of Charged Particle Therapy | Paulo M. Martins Author DivisionGoal 7/5/20 | Page4

Two physics problems in particle radiotherapy:

• Designing beam lines

• Predicting the dose distribution in the patient We need to understand the interactions of particles with matter

Physics of Charged Particle Therapy | Paulo M. Martins Author Division 7/5/20 | Page5

Proton and a photon posterior-oblique beam M. Goitein 2008

Physics of Charged Particle Therapy | Paulo M. Martins Author DivisionAccelerator and beam delivery (passive scattering) 7/5/20 | Page6

M. Goitein 2008

Physics of Charged Particle Therapy | Paulo M. Martins Author DivisionAccelerator and beam delivery (passive scattering) 7/5/20 | Page7 IBA Nozzle (Burr Center Gantry)

B. Gottschalk 2007

Physics of Charged Particle Therapy | Paulo M. Martins Author DivisionOutline 7/5/20 | Page8

• Preliminary notions and definitions • Energy loss of heavy charged particles by atomic collisions • The Bethe-Bloch formula (stopping power) • Fluence, stopping power and dose • Range

Physics of Charged Particle Therapy | Paulo M. Martins January 9, 2009 10:21 World Scientiﬁc Book - 9.75in x 6.5in ws-book975x65˙n˙2nd˙Ed

Introduction 9

January 9, 2009 10:21 World Scientiﬁc Book - 9.75in x 6.5in ws-book975x65˙n˙2nd˙Ed

1.3.1 The Two-Body Scattering

Radiation processes, like the ones resultingIntroduction in energy losses by9 collision, take place January 9, 2009 10:21 World Scientiﬁc Book - 9.75in x 6.5in ws-book975x65˙n˙2nd˙Ed in matter and can1.3.1 beThe consideredTwo-Body Scattering (see following chapters) as two-body scatterings January 9, 2009 10:21 World Scientiﬁc Book - 9.75in x 6.5in ws-book975x65˙n˙2nd˙Ed in which the targetRadiation particle processes, like theis ones almost resulting atin energy rest. losses In by collision, this section,take place we will study the in matter and can be considered (see following chapters) as two-body scatterings kinematics of thesein which the processes target particle and, is almost in at rest. particular, In this section, derive we will study equations the regarding the kinematics of these processes and, in particular, derive equations regarding the maximum energymaximum transfer. energy transfer. Let us considerLet anus consider incident an incident particleIntroduction particle [e.g., [e.g., a proton a (p), proton pion (º), kaon (p), (K), pion etc.] (º), kaon9 (K), etc.] of mass m and momentum p~, and a target particle of mass me [Rossi (1964)] at of mass m andrest. momentum For collision energy-lossp~, and processes, atarget the target particle particle in matter of is usuallymass anme [Rossi (1964)]Introduction at 9 1.3.1 The Two-Bodyatomic electron (see Scattering Fig. 1.1). After the interaction, two scattered particles emerge: rest. For collisionthe former energy-loss with mass m processes,and momentum p~ the00 and target the latter with particle mass me inand matter is usually an momentum p~ . The latter one has the direction of motion (i.e., the direction of the Radiation processes, like0 the ones resulting in energy1.3.1 losses byThe collision, Two-Body take place Scattering atomic electronthree-vector (see Fig.p~ 0) forming 1.1). an After angle µ with the the interaction, incoming particle direction. two scatteredµ is the particles emerge: in matter andangle can at be which considered the target particle (see is followingscattered. The chapters)kinetic energy [see as Eq. two-body (1.12)] of scatterings the former with mass m and momentum p~ 00 Radiationand the processes, latter with like the mass onesm resultinge and in energy losses by collision, take place in which the targetthe scattered particle particle is related almost to its at momentum rest. byIn this section, we will study the 2 2 2 2 4 in matter and can be considered (see following chapters) as two-body scatterings momentumkinematicsp~ of0. these The latterprocesses one and,E hask + m ine thec particular,= directionp0 c + mec derive, of motion equations (i.e.,(1.22) regarding the direction the of the from which we get p in which the target particle is almost at rest. In this section, we will study the three-vectormaximum energyp~ 0) forming transfer. an angle µ with the incoming particle direction. µ is the 2 E + m c2 m2c4 kinematics of these processes and, in particular, derive equations regarding the 2 k e ° e angleLet at which us consider the antarget incident particle particlep0 = is [e.g.,scattered2 a proton. The. (p), kinetic pion (º energy),(1.23) kaon (K), [see etc.] Eq. (1.12)] of ° c ¢ maximum energy transfer. of mass m andThe momentum total energy beforep~, and after a scattering target isparticle conserved. of Thus, mass we havem [Rossi (1964)] at the scattered particle is related to its momentum byLet us considere an incident particle [e.g., a proton (p), pion (º), kaon (K), etc.] 2 2 2 4 2 2 2 2 4 2 rest. ForAuthor collision energy-lossp c + processes,m c + mec = thep00 targetc + m c particle+ Ek + mec in matter is usually an Division 2 2 2of mass2 4m and momentum p~, and a target particle of mass me [Rossi (1964)] at atomic electronPreliminaryand, (see consequently, Fig.p 1.1).E Afterk +notionsm theec interaction,p= andp0 c two definitions+ scatteredm c , particles emerge: (1.22) 7/5/20 | Page9 e 2 2 2 4 2 2 2 4 rest. For collision energy-loss processes, the target particle in matter is usually an the former with mass m andp momentum00 c + m c = pp~c00+andm c theEk, latter with(1.24) mass me and °atomic electron (see Fig. 1.1). After the interaction, two scattered particles emerge: frommomentum which wep~ 0.while get The from latter momentum onepconservation: has the directionp p of motion (i.e., the direction of the Two-body scattering2 2and2 themaximum former with energy mass m andtransfer momentum p~ 00 and the latter with mass me and three-vector p~ 0) formingp~ an00 = anglep~ p~ 0 µ =withp the00 = p incoming+ p0 2 2pp0 cos particleµ. direction.(1.25) µ is the ° ) 2 ° 2 4 angle at which the target particle2 is scatteredEk + m. Theec kineticmomentumm energyec p~ [see0. The Eq. latter (1.12)] one of has the direction of motion (i.e., the direction of the p0 = ° . (1.23) scatteredparticle 2 three-vector p~ 0) forming an angle µ with the incoming particle direction. µ is the the scattered particle is related towithmomentump” its momentumc by ° ¢ angle at which the targetRelation particle kinetic is scattered energy. The momentum kinetic energy [see Eq. (1.12)] of The total energy before andE + afterm c2 scattering= p 2c2 + ism2 conserved.c4, Thus, we(1.22) have k e 0 ethe scattered particle(scattered is related toparticle) its momentum by incomingparticle from which we get2 2 2 4 withmomentump2 p 2 2 2 4 2 p c + m c θ+ mec = p c + m c + Ek + mec 2 2 2 2 4 scatteredelectron 00 Ek + mec = p0 c + mec , (1.22) withmomentump’ 2 2 2 4 2 Ek + mec mec and, consequently,p p0 = p ° from. which we get (1.23) p c2 ° ¢ 2 Fig. 1.1 Incident particle of mass m and momentum p~ emerges with momentum p = p~ ,while 2 2 4 The total energyEnergy before and and momentum2 after2 scattering2 4 are isconserved: conserved.2 2 2 Thus,4 00 | we00| have Ek + mec m c the scattered electronp emergesc + withm momentumc = p0 = p~ 0 .p c + m c E , 2 (1.24) e 00 | | k p = ° . (1.23) ° 0 2 2 2 2 4 2 2 2 2 4 2 c p c + m c + mec = p00 c + m c + Ek + mec ° ¢ while from momentump conservation: p The total energy before and after scattering is conserved. Thus, we have and, consequently,p p 2 2 2 2 2 2 4 2 2 2 2 4 2 p~ 00 = p~ p~ 0 = p00 = p + p0 2pp0pcosc µ+. m c + mec(1.25)= p00 c + m c + Ek + mec 2 2 2 4 2 2 2 4 p°00 c + m c)= p c + m c Ek, ° (1.24) and,° consequently,p p while from momentump conservation:C. Leroy & P.G.p Rancoita, (2009) Radiation Interaction in Matter and Detection, World Scientific 2 2 2 4 2 2 2 4 scatteredparticle2 2 2 p00 c + m c = p c + m c Ek, (1.24) Physicsp~ 00 =of Chargedp~ p~ 0 Particlewithmomentump”= Therapy | Paulop00 =M. Martinsp + p0 2pp0 cos µ. (1.25) ° ° ) while° from momentump conservation: p 2 2 2 scatteredparticle p~ 00 = p~ p~ 0 = p00 = p + p0 2pp0 cos µ. (1.25) withmomentump” ° ) ° incomingparticle withmomentump scatteredparticle θ withmomentump” scatteredelectron incomingparticle withmomentump’ withmomentump scatteredelectron θ withmomentump’ incomingparticle withmomentump scatteredelectron θ Fig. 1.1 Incident particle of mass m and momentum p~ emergeswithmomentump’ with momentum p00 = p~ 00 ,while Fig. 1.1 Incident particle of mass m and momentum p~ emerges with momentum p = p~ ,while | | the scattered electron emerges with momentum p0 = p~ 0 . 00 | 00| the scattered electron emerges with momentum p = p~ . | | 0 | 0|

Fig. 1.1 Incident particle of mass m and momentum p~ emerges with momentum p = p~ ,while 00 | 00| the scattered electron emerges with momentum p = p~ . 0 | 0| January 9, 2009 10:21 World Scientiﬁc Book - 9.75in x 6.5in ws-book975x65˙n˙2nd˙Ed

10 Principles of Radiation Interaction in Matter and Detection

Equation (1.25) can be rewritten taking into account Eq. (1.23):

2 E + m c2 m2c4 (E + m c2)2 m2c4 2 2 k e ° e k e e January 9, 2009 10:21p00 = p + World Scientific2 Book - 9.75in x 6.5in2p cos µ ws-book975x65˙n˙2nd˙Ed2 ° , ° c ¢ ° s c which becomes, after substituting p00 obtained from Eq. (1.24) and squaring both sides of that equation, 10 Principles of Radiation Interaction in Matter and Detection E p2c2 + m2c4 = E m c2 + pc cos µ (E + m c2)2 m2c4, k ° k e k e ° e Equation (1.25) can be rewritten taking into account Eq. (1.23):q from which wep get 2 E + m c2 m2c4 (E + m c2)2 m2c4 2 2 k e ° e k e e p00 = p + 2 2p cos µ 2 ° , ° c ¢ 2 ° 2s c Ek +2Ekmec 2 2 2 2 4 which becomes, afterpc cos substitutingµ p00 obtained2 from= Eq.m (1.24)ec + and squaringp c + bothm c , sides of that equation, s Ek p and, finally,E byp2 squaringc2 + m2c4 = bothE m sidesc2 + pc ofcos theµ (E equation+ m c2)2 wem2 canc4, derive the expression for k ° k e k e ° e Author q the kineticfrom which energy wepDivision getPreliminaryEk of the scatterednotions and target definitions particle, i.e., 7/5/20 | Page10 E2 +2E m c2 4 2 2 pc cos µ k k e = m 2c2m+ec pp2c2cos+ m2cµ4, TwoE -=body scattering2 e and maximum energy transfer. (1.26) ks Ek 2 m c2 + p2c2 p+ m2c4 p2c2cos2 µ and, finally, by squaringKinetic bothenergy sides ofe the of the scattered equation target we can particle: derive the° expression for the kinetic energy Ek of the scattered≥ target particle, i.e., ¥ The kinetic energy E of the recoilingp4 2 2 target particle is the amount of transferred k 2mec p cos µ Ek = 2 . (1.26) energy in the interaction.m c2 + Fromp2c2 + Eq.m2c4 (1.26),p2c2cos we2 µ note that the maximum energy e ° transfer Wm is for µ = 0, i.e., when a head-on collision occurs. For µ = 0, Eq. (1.26) The kinetic energy E of≥ the recoilingp target particle¥ is the amount of transferred Maximumk energy transfer W is for q = 0 (head-on collision): becomes:energy in the interaction. From Eq. (1.26), wem note that the maximum energy transfer Wm is for µ = 0, i.e., when a head-on collisionp occurs.2c2 For µ = 0, Eq. (1.26) becomes: Wm = . (1.27) 1 2 12 2 2 2 2 2 2 4 mec + p (cm /me) c + p c + m c Wm = 2 2 . (1.27) 1 2 1 2 2 2 2 2 4 2 mec + 2 (m /me) c + p c + m c From Eq. (1.10), the incoming particle energy Ei isp given by From Eq. (1.10), thePhysics incoming of Charged Particle particle Therapy energy | Paulo M. MartinsEi isp given by

2 2 22 2 4 2 2 2 4 Ei =Emi∞c==m∞pcc +=m c p. c + m c . p We canWe can rewrite rewrite Eq. Eq. (1.27) (1.27) as: as: p 2 1 2 2 2 me me ° Wm =2mec Ø ∞ 1+ +2∞ . (1.28)1 m m 2 ∑2 2 2 m∏e me ° Wm =2mec Ø ≥∞ ¥1+ +2∞ . (1.28) Massive particles (e.g., proton§,K,º etc.) are particlesm whose massesm are much larger than the electron (or positron) mass me,i.e.,∑ ≥ ¥ ∏ Massive particles (e.g.,m protonm ( 0§.511,K, MeV/cº etc.)2). are particles whose masses are much ¿ e º largerFor than massive the particles, electron at su± (orciently positron) high energies mass‡, i.e., whenme,i.e., the incoming momentum p is m m ( 0.511 MeV/c2). m2e p¿ c, º ¿ me For massive particles, at su±ciently high energies‡, i.e., when the incoming momen- The rest mass of the proton is 938.27 MeV/c2. § º For instance, this condition is satisfied by an incoming º with momentum 36 GeV/c or an tum p‡ is ¿ incoming proton with momentum 1.7TeV/c. ¿ m2 p c, ¿ me The rest mass of the proton is 938.27 MeV/c2. § º For instance, this condition is satisfied by an incoming º with momentum 36 GeV/c or an ‡ ¿ incoming proton with momentum 1.7TeV/c. ¿ January 9, 2009 10:21 World Scientific Book - 9.75in x 6.5in ws-book975x65˙n˙2nd˙Ed

January 9, 200910 10:21Principles World of Radiation Scientiﬁc Book Interaction - 9.75in x 6.5in in Matter and Detection ws-book975x65˙n˙2nd˙Ed

Equation (1.25) can be rewritten taking into account Eq. (1.23):

2 2 2 4 2 2 2 4 10 2 2 PrinciplesEk + ofm Radiationec Interactionmec in Matter and(E Detectionk + mec ) mec p00 = p + ° 2p cos µ ° , January 9, 2009 10:21 World2 Scientiﬁc Book - 9.75in x 6.5in2 ws-book975x65˙n˙2nd˙Ed ° c ¢ ° s c whichEquation becomes, (1.25) can after be substituting rewritten takingp00 intoobtained account from Eq. Eq. (1.23): (1.24) and squaring both sides of that equation, 2 E + m c2 m2c4 (E + m c2)2 m2c4 2 2 k e ° e k e e p00 = p + 2 2p cos µ 2 ° , E p2c2°+ m2c4 c=¢ E m c2°+ pc cossµ (E + cm c2)2 m2c4, k ° k e k e ° e which becomes, after substituting p00 obtained from Eq.q (1.24) and squaring both fromsides which of that wep equation, get

2 2 2 2 2 4 E +2Ekm2Introductionec 2 2 2 4 11 Ek p pcc +cosmµc = k Ekmec + pc cos= mµ c(2E+k + mpe2cc2)+ mm2ecc4,, ° E2 e ° p s k q from which we get p and, finally, by squaring both2 sides of2 the equation we can derive the expression for Eq. (1.27) becomes Ek +2Ekmec 2 2 2 2 4 the kinetic energypc cosEkµof the scattered2 = targetmec particle,+ p c + i.e.,m c , s Ek 2m c4p2cosp2 µ and, finally, by squaring both sides ofW the equatione pc we canE derive. the expression for Ek = m 2 i . (1.26) 2 2 2º 2 4º 2 2 2 the kinetic energy Ek of them scatteredec + targetp c + particle,m c i.e., p c cos µ 4 2 2 ° In the extreme relativistic case,≥ ap massive2mec p cos particleµ ¥ can transfer all its energy to the The kinetic energyEkE=k of the recoiling target2 particle is the. amount of(1.26) transferred m c2 + p2c2 + m2c4 p2c2cos2 µ target electronenergy in in the a head-on interaction.e collision, From Eq. (1.26), i.e., we a° proton note that can the maximum be stopped energy by interacting transfer Wm is for µ = 0,≥ i.e., whenp a head-on¥ collision occurs. For µ = 0, Eq. (1.26) with an electron.The kinetic At energy lowerEk of energies the recoiling† target,i.e.,when particle is the amount of transferred becomes:energy in the interaction. From Eq. (1.26), we note that the maximum energy 2 2 2 transfer Wm is for µ = 0, i.e., when a head-on collisionp mc occurs. For µ = 0, Eq. (1.26) Wm = . (1.27) becomes: 1 2 1 p2 2 c, 2 2 2 4 Author 2 mec + 2 (m /me) c + p c + m c DivisionPreliminary notionspø2c2 andm edefinitions 7/5/20Wm| =Page11 . (1.27) From Eq. (1.10), the incoming1 m c2 + particle1 (m2/m energy) c2 + Epi2ispc2 given+ m2c4 by the maximum energy transfer2 e [see2 Eq.e (1.27)] by particles with m m is appro- Two-body scattering2 and2 2 maximum2 4 energy transfer e From Eq. (1.10), the incomingEi particle= m∞c energy= pEicisp+ givenm c by. ¿ ximated by 2 2 2 2 4 We can rewrite Eq.Since: (1.27)E as:i = m∞c = ppc + m c . 2 We can rewrite Eq. (1.27) as: p 2 2 p 1 2 2 2 me me ° Wm =2meWc Øm∞ 1+2mec +2∞ . (1.28) 2 1 2 2 2 º me m mmce °m Wm =2mec Ø ∞ 1+∑ ≥ +2¥ ∞ . ∏ (1.28) m ≥ m ¥ Massive particles (e.g., proton§,K,∑ º≥etc.)¥ are particles∏ whose masses are much and, because p = mØ∞c,wehave:For particles � ≫ � largerMassive than the particles electron (e.g., (or proton positron)§,K,º massetc.) areme particles,i.e., whose masses are much larger than the electron (or positron) mass m ,i.e.,2 Øe 2 m me (2 0.511 MeV/c ). 2 2 2 Wm m 2¿mm(ec0º.511 MeV/c2=2). mec Ø ∞ . (1.29) º¿ e º 1 Ø2 For massive particles, at su±ciently high energies‡, i.e.,(valid whenfor the2�� incoming≪ �) momen- For massive particles, at su±ciently high energies° , i.e., when the incoming momentum p is ‡ For instance,tum ap is proton ofPhysics 10 of Charged GeV Particle Therapy has | Paulo a M. Lorentz Martins factor ∞ 10 and Ø 1. Thus, its 2 2 m º º maximum energy transfer is W p100m MeV.c, m p ¿ c,m º¿ me e 2 §TheThe rest rest mass mass of of the the proton proton is is 938938.27.27 MeV/c MeV/c2. . § º º ForFor instance, instance, this condition condition is is satisfied satisfied by byan incoming an incomingº withº momentumwith momentum36 GeV/c36 or GeV/c an or an ‡ ‡ ¿ ¿ incomingincoming proton proton with with momentum momentum 1.7TeV/c.1.7TeV/c. 1.3.2 The Invariant Mass¿¿

The four-momentum of a particle with rest mass‡ m0 is deﬁned as E q˜ = , p~ . c µ ∂ The scalar product between two four-momentaq ˜ andq ˜0 is an invariant§§ quantity and is given by (e.g., Section 38 of [PDB (2008)])

EE0 q˜ q˜0 = p~ p~ 0. (1.30) · c2 ° · The invariant mass of a particle is related to the scalar product of its four- momentum by: q˜ q˜ = q2 · E2 = p~ p~ c2 ° · E2 = p2 c2 ° 2 2 = m0c For instance, this condition is satisﬁed by an incoming º with momentum 36 GeV/c or an † ø incoming proton with momentum 1.7TeV/c. ø ‡The rest mass is the mass of a body that is isolated (free) and at rest relative to the observer. §§The invariant mass or intrinsic mass or proper mass or just mass is the mass of an object that is the same for all frames of reference. Author DivisionOutline 7/5/20 | Page12

• Preliminary notions and definitions • Energy loss of heavy charged particles by atomic collisions • The Bethe-Bloch formula (stopping power) • Fluence, Stopping Power and Dose • Range

Physics of Charged Particle Therapy | Paulo M. Martins Author January 9, 2009 10:21DivisionEnergy World Scientiﬁc Book loss - 9.75in x 6.5inof heavy charged ws-book975x65˙n˙2nd˙Ed particles by atomic collisions 7/5/20 | Page13 Bohr’s calculation – the classical case Electromagnetic Interaction of Radiation in Matter 33

Fig. 2.1 The impact parameter b is the minimum distance between the incoming particle and the target by which it is scattered. maximum transferable energyThe from the impact incident particle parameter to atomic electrons, b is± theis minimum distance between the incoming the correction for the density-eÆect, and finally U is the term related to the non- participation of electrons of inner shells (K, L, ...) forparticle very low incoming and kinetic the target by which it is scattered energies (i.e., the shell correction term). The number of electrons per cm3 (n) of the traversed material is given by (ZΩN)/A [Eq. (1.40)], where Ω is the material density in g/cm3, N is the Avogadro numberC. (seeLeroy Appendix & P.G. A.2), RancoitaZ and A are, the(2009) Radiation Interaction in Matter and Detection, World Scientific atomic number (Sect. 3.1) and atomic weight (see page 14 and Sect. 1.4.1) of the material, respectively.Physics The atomic of Charged number ParticleZ is the Therapy number | ofPaulo protons M. Martins inside the nucleus of that atom. The minus sign for dE/dx, in Eq. (2.1), indicates that the energy is lost by the particle. For a heavy particle, the collision energy-loss, dE/dx, is also referred to as the stopping power. In Sect. 1.3.1, we have derived the expression of the maximum energy transfer Wm for the relativistic scattering of a massive particle onto an electron at rest. Usually, because the maximum energy transfer is much larger than the electron binding energy, this latter can be neglected. Thus, the value of Wm is given by formula (1.28), or, in most practical cases, by its approximate expression given in Eq. (1.29): we will make use of this latter equation in the present chapter. Once the approximate expression of Wm is used, Eq. (2.1) (i.e., the energy-loss formula) can be rewritten in an equivalent way as dE 4ºnz2e4 2mv2∞2 ± U = ln Ø2 , (2.2) ° dx mv2 I ° ° 2 ° 2 ∑ µ ∂ ∏ or dE 4ºnz2e4 = L, (2.3) ° dx mv2 where L is a dimensionless parameter called the stopping number, which contains the essential physics of the process. In Eq. (2.3), L is given by the term in brackets in Eq. (2.2). As discussed later, the stopping number can be modified by adding Author January 9, 2009 10:21 World Scientific Book - 9.75in x 6.5in ws-book975x65˙n˙2nd˙Ed DivisionEnergy loss of heavy charged particles by atomic collisions 7/5/20 | Page14

Bohr’s calculation –34 the classicalPrinciples of case Radiation Interaction in Matter and Detection

Fig. 2.2 Incoming fast particle of charge ze scattered by an atomic electron almost at rest: for small energy transfer, the particle trajectory is not deﬂected. The incoming fast particle of charge ze is not deflected by an atomic electron correction terms which also depend on the particle velocity v, charge number z, almostatomic at target rest number forZ smalland excitation energy energy transferI. Equation (2.1) and its equivalent expression given by Eq. (2.2) are also termed Bethe–Bloch formula. In literature, the C. Leroy & P.G. RancoitaBethe–Bloch, (2009) formula Radiation may also be Interaction found without in theMatter shell correctionand Detection, term. In World this Scientific latter form, it describes the energy-loss of a charged massive particle like a proton Physics of Charged Particle Therapy | Paulowith M. Martins kinetic energy larger than a few MeV (e.g., see Sect. 2.1.1.2 and Section 27.2.1 of [PDB (2008)]). Let us discuss an approximate derivation, i.e., without entering into complex calculations, of the energy-loss formula following closely previous approaches [Fermi (1950); Sternheimer (1961); Fernow (1986)]. In this way, the physical meaning of the terms appearing in the formula and their behavior as a function of incoming velocity become more evident. We restrict ourselves to cases where only a small fraction of the incoming kinetic energy is transferred to atomic electrons, so that the incoming particle trajectory is not deviated. Now, we introduce the impact parameter b describing how close the collision is (see Fig. 2.1): b is the minimal distance of the incoming particle to the target electron. In general, large values of b correspond to the so-called distant collisions, conversely small values to close collisions. Both kinds of collisions are important for determining the average energy-loss [Eq. (2.1)], the energy straggling (i.e., the energy-loss distribution) and the most probable energy-loss. When a particle of charge ze interacts with an electron almost at rest ‡‡, we assume that, to a ﬁrst approximation, the electron will emerge only after the particle passage so that we can consider the electron essentially at rest throughout the interaction. This way, for symmetry reason (see Fig. 2.2), the transferred momentum I will be almost along the direction perpendicular to the particle trajectory. In ? addition, the order of magnitude of the maximum strength of the Coulomb force acting along the perpendicular direction F is ze2/b2. Thus, the maximum electric ? º ‡‡An electron is almost at rest, when its velocity is much smaller than the incoming particle velocity v. Author DivisionEnergy loss of heavy charged particles by atomic collisions 7/5/20 | Page15 Bohr’s calculation – the classical case

The transferred momentum � will be almost along the direction perpendicular to the particle trajectory

More momentum is transferred � ≈ �� /� to the electron, the longer the �� = ∫ � �� ∼ particle stays in its vicinity �� where the interaction time is ≈ inversely proportional to the incoming particle velocity and directly proportional to b

Physics of Charged Particle Therapy | Paulo M. Martins Author DivisionEnergy loss of heavy charged particles by atomic collisions 7/5/20 | Page16 Bohr’s calculation – the classical case

Relationship between the impact parameter b and the transferred energy W Kinetic energy : � 2�� ∼ = 2� ��

Distant collisions are soft ones, while close collisions allow large transfers of kinetic energy (energy transferred to recoiling nuclei can be usually neglected with respect to the one from recoiling electrons – ratio 10-4)

Physics of Charged Particle Therapy | Paulo M. Martins 22 2. Passage of Radiation Through Matter

The inelastic collisions are, of course, statistical in nature, occurring with a certain quantum mechanical probability. However, because their number per macroscopic pathlength is generally large, the fluctuations in the total energy loss are small and one can meaningfully work with the average energy loss per unit path length. This quantity, often called the stopping power or simply dEldx, was first calculated by Bohr using classical arguments and later by Bethe, Bloch and others using quantum mechanics. Bohr's calculation is, nevertheless, very instructive and we will briefly present a simpli- fied version due to Jackson [2.1] here.

2.2.1 Bohr's Calculation - The Classical Case Consider a heavy particle with a charge ze, mass M and velocity v passing through some material medium and suppose that there is an atomic electron at some distance b from the particle trajectory (see Fig. 2.2). We assume that the electron is free and initially at rest, and furthermore, that it only moves very slightly during the interaction with the heavy particle so that the electric field acting on the electron may be taken at its initial position. Moreover, after the collision, we assume the incident particle to be Author DivisionEnergy loss of heavy charged particles by atomic collisions Ej ú= 7/5/20 | Page17 essentially undeviated from its original path because of its much larger mass me)' This is one reason for separating electrons from heavy particles! Bohr’s calculation – the classical case W.R. Leo, (1994) Techniques for Nuclear and Particle Physics Experiments, Springer-Verlag e Let n be the density of electrons, ," -----Z--- --ill-i'b then the energy lost to all the electrons I ú=úf = located at a distance between b and I .. x \ M.ze \! Fig. 2.2. Collision of a heavy charged particle with an atomic b + db in a thickness dx is: 1,- ______úL= electron 4�� −�� = Δ� � � �� = �� Let us now try to calculate the energy gained by the electron by finding the momentum impulse it receives from colliding with the heavy particle. Thus where � �� = � 2�� are the number of electrons encountered by the particle along a path dx at impact parameter between b and b + db (2,.16) Physics of Charged Particle Therapy | Paulo M. Martins

where only the component of the electric field E1. perpendicular to the particle trajectory enters because of symmetry. To calculate the integral JE1. dx, we use Gauss' Law over an infinitely long cylinder centered on the particle trajectory and passing through the position of the electron. Then

(2.17)

so that 2ze2 1=-- (2.18) bv

and the energy gained by the electron is

/2 2z2e4 L1E(b) =--= (2.19) 2 me m e v 2 b 2 Author DivisionEnergy loss of heavy charged particles by atomic collisions 7/5/20 | Page18 Bohr’s calculation – the classical case

The overall energy loss by collisions calculated by integrating over the range of the impact parameter is:

�� 4�� 4�� − = = ln ��

Physics of Charged Particle Therapy | Paulo M. Martins Author DivisionEnergy loss of heavy charged particles by atomic collisions 7/5/20 | Page19 Bohr’s calculation – the classical case

Limits: �� ≃ �̅ where �is the characteristic mean frequency of excitation of electrons; and � ≃ is the collision time that cannot exceed the typical time period associated with bound electrons. This is the principle of adiabatic invariance - the perturbation must not be adiabatic, otherwise no energy is transferred.

Physics of Charged Particle Therapy | Paulo M. Martins Author DivisionEnergy loss of heavy charged particles by atomic collisions 7/5/20 | Page20 Bohr’s calculation – the classical case

Limits: ��ℎ � ≃ � if we introduce the mean excitation energy � = ℎ�̅

Physics of Charged Particle Therapy | Paulo M. Martins Author DivisionEnergy loss of heavy charged particles by atomic collisions 7/5/20 | Page21 Bohr’s calculation – the classical case

Limits: ℎ � ≃ , where Pecm is the electron momentum in the CMS 2�

In the classical approach, the wave characteristics of particles are neglected and this is valid as long as the impact parameter b is larger than the de Broglie wavelength of the electron in the CMS of the interaction

Physics of Charged Particle Therapy | Paulo M. Martins Author DivisionEnergy loss of heavy charged particles by atomic collisions 7/5/20 | Page22 Bohr’s calculation – the classical case

Limits: ℎ � ≃ �� = �� ⟹ � ≃ . 2��

The electron mass is much smaller than the mass of the incoming particle and the CMS is approximately associated with the incoming particle. The electron velocity in the CMS is opposite and almost equal to the vparticle.

Physics of Charged Particle Therapy | Paulo M. Martins Author DivisionEnergy loss of heavy charged particles by atomic collisions 7/5/20 | Page23 Bohr’s calculation – the classical case

Substituting:

− = ln = ln

�� 4�� 2�� − = ln ��

Physics of Charged Particle Therapy | Paulo M. Martins Author DivisionEnergy loss of heavy charged particles by atomic collisions 7/5/20 | Page24 Bohr’s calculation – the classical case

Using the value of the maximum energy transfer Wm:

�� 4�� 2�� − = ln �� (1 − �)

This is essentially Bohr's classical formula. It gives a reasonable description of the energy loss for heavy particles such as the a-particle or heavier nuclei. However, for lighter particles, e.g. the proton, the formula breaks down because of quantum effects.

Physics of Charged Particle Therapy | Paulo M. Martins Author DivisionOutline 7/5/20 | Page25

• Preliminary notions and definitions • Energy loss of heavy charged particles by atomic collisions • The Bethe-Bloch formula (stopping power) • Fluence, stopping power and dose • Range

Physics of Charged Particle Therapy | Paulo M. Martins January 9, 2009 10:21 World Scientiﬁc Book - 9.75in x 6.5in ws-book975x65˙n˙2nd˙Ed

Introduction 11

Eq. (1.27) becomes W pc E . m º º i In the extreme relativistic case, a massive particle can transfer all its energy to the target electron in a head-on collision, i.e., a proton can be stopped by interacting with an electron. At lower energies†,i.e.,when m2 p c, Author ø me DivisionThe Bethe-Bloch formula (Stopping Power) the maximum energy7/5/20 | transferPage26 [see Eq. (1.27)] by particles with m m is appro- ¿ e ximated by Bethe relativistic formula (energy-loss formula) �� 4�� 2�p �2 − = W 2mlnc2 − � �� m�º e �mc(1 − �) ≥ ¥ and, because p = mØ∞c,wehave: Ø2 W 2m c2 =2m c2Ø2∞2. (1.29) m º e 1 Ø2 e �� 4��° 2�� For instance, a proton of− 10 GeV= has a Lorentzln factor ∞ − 102� and Ø 1. Thus, its º º maximum energy transfer is��W �100� MeV.� 1 − � m º

Physics of Charged Particle Therapy | Paulo M. Martins 1.3.2 The Invariant Mass

The logarithmic term increases quadratically with �� = (��)/�

Physics of Charged Particle Therapy | Paulo M. Martins Author DivisionThe Bethe-Bloch formula (Stopping Power) 7/5/20 | Page28 Features:

• Energy loss depends on ln(bmax/bmin); 1/bmin and bmax increase with bg

• 1/bmin -> enhancement of the maximum transferable kinetic energy

• bmax -> dilation of the maximum impact parameter • With increasing particle energy, d-rays are emitted along the trajectory • Cylindrical region surrounding the particle path is enlarged • Emission of d-rays is responsible for the difference between the energy lost by the particle in the medium and the actually energy deposited. • With increasing particle energy, the deposited energy approaches an almost constant value (Fermi Plateau) - depends on absorber size and density

Physics of Charged Particle Therapy | Paulo M. Martins Author DivisionThe Bethe-Bloch formula (Stopping Power) 7/5/20 | Page29 Energy-loss formula (rewritten) classical electron radius = = 0.1535

�� 2�� − = 0.1535 ln − � − − �� 1 − � 2 2

Mean excitation energy: � 7 � = 12 + , � < 13 = 9.76 + 58.8 �., � ≥ 13 � � �

Physics of Charged Particle Therapy | Paulo M. Martins Author DivisionThe Bethe-Bloch formula (Stopping Power) 7/5/20 | Page30 Mass stopping power (radiotherapy energy 3-300 MeV) � 1 �� MeV = = 0.3072 ln − � � � �� g/cm � = � � = � � -> fraction by weight of the ith element for mixtures � � A 10 MeV proton loses about the same amount of energy in 1 g/cm2 of copper as in 1 g/cm2 of aluminum or iron

Physics of Charged Particle Therapy | Paulo M. Martins Author DivisionThe Bethe-Bloch formula (Stopping Power) 7/5/20 | Page31 Mass stopping power for positive muons in copper as a function of bg = p/Mc. µ- illustrate the “Barkas effect” – dependence of the stopping power on projectile charge at very low energies dE/dx in the radiative region is not simply a function of b M. Tanabashi et al. (Particle Data Group), (2018) Phys. Rev. D 98, 030001

Physics of Charged Particle Therapy | Paulo M. Martins Author DivisionThe Bethe-Bloch formula (Stopping Power) 7/5/20 | Page32 The Barkas effect

Barkas Formula:

Heavier ions at low energies collect electrons from surrounding material thus rapidly decreasing its zeff

Schwab (1991) PhD thesis, University of Giessen

Physics of Charged Particle Therapy | Paulo M. Martins Author DivisionThe Bethe-Bloch formula (Stopping Power) 7/5/20 | Page33 Energy dependence • At non-relativistic energies, dE/dx is dominated by the overall 1/b2 factor and decreases with increasing velocity until about v = 0.96 c, where a minimum is reached (MIPs). • The minimum value of dE/dx is almost the same for all particles of the same charge. • As the energy increases beyond this point, the term 1/b2 becomes almost constant and dE/dx rises again due to W.R. Leo, (1994) Techniques for Nuclear and Particle Physics Experiments, Springer-Verlag the logarithmic dependence.

Physics of Charged Particle Therapy | Paulo M. Martins Author DivisionThe Bethe-Bloch formula (Stopping Power) 7/5/20 | Page34 Mean energy loss rate in liquid hydrogen, gaseous helium, carbon, aluminum, iron, tin, and lead.

• Similar (slow decrease) rates of energy loss • Density-effect correction for Helium – d(bg) • Broad minima drops from bg = 3.5 to 3.0 as Z goes from 7 to 100 • Most relativistic particles (e.g., cosmic-ray muons) have mean energy loss rates close to the minimum (minimum ionizing particles - MIPs)

M. Tanabashi et al. (Particle Data Group), (2018) Phys. Rev. D 98, 030001

Physics of Charged Particle Therapy | Paulo M. Martins Author DivisionThe Bethe-Bloch formula (Stopping Power) 7/5/20 | Page35 The shell and density correction • The density effect arises from the fact that the electric field of the particle tends to polarize the atoms along its path • Electrons far from the path are shielded form the full electric field intensity and the collisions contribute less to the energy loss • Depends on the velocity of the particle and the density of the material (the higher both variables the stronger is the effect) W.R. Leo, (1994) Techniques for Nuclear and Particle Physics Experiments, Springer-Verlag

Physics of Charged Particle Therapy | Paulo M. Martins Author DivisionThe Bethe-Bloch formula (Stopping Power) 7/5/20 | Page36 The shell and density correction • The shell effect arises when the velocity of the incident particle is comparable or smaller than the orbital velocity of the bound electrons. • At such energies, the assumption that the electron is stationary with respect to the incident particle is no longer valid and the Bethe-Bloch formula breaks down. • The correction is very small! W.R. Leo, (1994) Techniques for Nuclear and Particle Physics Experiments, Springer-Verlag

Physics of Charged Particle Therapy | Paulo M. Martins Author DivisionOutline 7/5/20 | Page37

• Preliminary notions and definitions • Energy loss of heavy charged particles by atomic collisions • The Bethe-Bloch formula (Stopping Power) • Fluence, stopping power and dose • Range

Physics of Charged Particle Therapy | Paulo M. Martins Physics of Proton Interactions in Matter 25

speed v or momentum p, given its kinetic energy E.* Radiotherapy protons are somewhat relativistic (their speed is of order c ! 300 106 m/sf! 1 t/ns), so the relevant equations (see any textbook on special relativity) are as follows: v pc β ≡ = (2.1) c Em+ c2

22 222 (Em+=cp)(cm)(+ c ). (2.2) From these, if we frst defne a reduced kinetic energy, E τ≡ (2.3) mc2 we can derive several useful equations whose relativistic ( ≫ 1) and non- relativistic ( ≪ 1) limits are obvious at a glance:

2 τ + 2 β = 2 τ (2.4) ()τ + 1

2 2 (pc)(=+τ 2)mc E (2.5) τ + 2 pv = E. (2.6) τ + 1 26 The last quantity, pv, occurs frequently in MCSProton theory. Therapy For E Physics = 160 MeV we fnd β = 0.520 (speed just over half the speed of light), pc = 571 MeV, and pv = 297 MeV. The fuence rate is the time derivative of the fuence*: 2.2.2 Fluence, Stopping Power,dΦ prandoton Doses Φ! ≡ . (2.8) dt 2 Suppose a beam of protons slows downcm s and eventually stops in a water tank. TheAt stopping any given power depth is the26 werate may at which be concerned a single proton with loses the number kinetic energy: of Protonprotons, Therapy their Physics individual rate of energy loss, or the total rate at which they deposit energy dE MeV in the water. Let x (cm) beS displacement≡− . along the beam direction(2.9) and y (cm) be transverse displacement.The fuence rate dxis the cmtime derivative of the fuence*: Author The fuence, Division, is a quantity which depends on position in the water tank. The mass stopping powerΦ Fluence, is stopping stopping power power“corrected”! anddΦ for dosepr density:otons 7/5/20 | Page38 Φ ≡ 2 . (2.8) It is de26fned as the number of protons, during dta givencm exposures Proton Therapy or treatment, Physics crossing an infnitesimalDefinitionsSd element1ME of areaeV dA normal to x.† The stopping≡− power is the rate at which a single proton (2.10)loses kinetic energy: ρρdx g/cm2 dN proton s Fluence dE Me*V The fuence rate is the timeΦ derivative≡ of Sthe2≡− f. uence : . (2.7) (2.9) where ρ (g/cm3) is the local density of dAthe stoppingcm medium.dx cm For instance, the mass stopping powers of air and water dareΦ similar,proton whereass the stopping TheFluence mass stopping rate power! is stopping power “corrected”† for density: power* of air is about a thousand timesΦ less≡ than that2 of .water. The physical (2.8) In radiotherapy physics, kinetic energy is oftendt denotedcm s E, unlike particle physics where E absorbed dose, D, at some point in a radiation feld is the energy absorbed per usually stands for total (kinetic + rest) energy. Sd1ME eV † Whether in the beam line or the patient, radiotherapy≡− protons are always directed within (2.10) unit targetThe mass. stopping In SI power Massunits, is stopping the rate atpower which a single proton 2loses kinetic energy: a few degrees of the x axis. Other types of radiationρρ requiredx g/ a cmmore general defnition of fuence. J whereDose ρ (g/cmD3) is the. localdE densityMeV of the stopping medium.(2.11) For instance, kgS ≡− . (2.9) the mass stopping powers dx of air cmand water are similar, whereas the stopping (B. Gottschalk)† H. Paganetti 2012 The special unit of dosepower used of inair radiation is about a therapy thousand is timesthe Gray: less than1 Gy that ≡ 1 J/kg.of water. The physical To give aThe rough mass idea stopping ofPhysicsabsorbed the of Chargedpower numbers, dose Particle ,is D Therapy,stopping at a some course| Paulo M. point Martins power of protonin a “corrected” radiation radiotherapy feld for is density:the might energy absorbed per consist of ≈ 70 Gy to ≈ unit1000 target cm3 of mass. target In volumeSI units, given in ≈ 35 fractions (2 Gy/ Sd1ME eV session). However, a single whole- body dose of 4 Gy is lethal (with a probabil- (2.10) ≡− 2 J ity of 50%) even with good medical care.ρρ To putdx thisg/ intoDcm perspective,. assum- (2.11) kg ing the typical thermal power radiated by an adult weighing 80 Kg is 100 W, a lethal where dose of ρ ionizing(g/cmThe3) specialis radiation the unitlocal correspondsof density dose used of in tothe radiation the stopping amount therapy medium. of is thermal the Gray:For instance, 1 Gy ≡ 1 J/kg. energy giventhe mass off in stopping 3 s!To Ionizing give powers a rough radiation of idea air of andis the nasty waternumbers, stuff. are a similar, course of whereas proton radiotherapy the stopping might 3 An earlier,power arguably of air consistis more about ofconvenient ≈a 70thousand Gy to unit≈ 1000 times of cm physical lessof target than absorbed volume that of givendose water. isin the≈† 35The fractions physical (2 Gy/ session). However, a single whole- body dose of 4 Gy is lethal (with a probabil- rad: 1 Gyabsorbed = 100 rad. dose OlderD oncologists frequently hedge, saying “centiGray” ,ity , of at 50%) some even point with in good a radiation medical care.feld To is putthe this energy into perspective,absorbed per assum - (cGy) instead of rad. unit target mass.ing the In typical SI units, thermal power radiated by an adult weighing 80 Kg is 100 W, a lethal dose of ionizing radiation corresponds to the amount of thermal J 2.2.3 Energy Lost vs.energy Energy given Deposited off in 3 s! IonizingD .radiation is nasty stuff. (2.11) An earlier, arguably more convenientkg unit of physical absorbed dose is the It is good to rememberrad :that 1 Gy the = 100 energy rad. Older lost by oncologists a proton frequently beam exceeds hedge, the saying “centiGray” energy absorbedThe special locally unit(cGy) by of instead thedose patient usedof rad. inor radiationwater phantom. therapy A is fraction the Gray: of the1 Gy ≡ 1 J/kg. beam’s energyTo give goes a rough into idea neutral of the secondaries numbers, (aγ-rays course and of neutrons),proton radiotherapy which might consist of ≈ 70 Gy to ≈ 1000 cm3 of target volume given in ≈ 35 fractions (2 Gy/ 2.2.3 Energy Lost vs. Energy Deposited * session).⋅ However, a single whole- body dose of 4 Gy is lethal (with a probabil- Sometimes Φ is written φIt. Inis the good early to literature remember fuence that rate the is energycalled “f lostux.” by a proton beam exceeds the † The oftenity used of 50%)quantity even ρ dx with (g/cm good2) is the medical areal density care. or To simply put thethis “grams into perspective,per square assum- centimeter”ing of the an typicalelementenergy of thermal stopping absorbed powermedium locally radiated of thickness, by the by patientdx an. It adultis theor water thicknessweighing phantom. of a80 slab Kg A fractionis 100 W, of the of stoppinga lethal material dose timesbeam’s ofits density. ionizing energy To determine radiation goes into it experimentally, neutral corresponds secondaries one to usually the (γ-rays amount measures, and neutrons), of thermal which instead, mass divided by area. energy given off in 3 s! Ionizing radiation is nasty stuff. ⋅ An earlier, *arguably Sometimes Φmore is written convenient φ. In the early unit literature of physical fuence rate absorbed is called “f ux.”dose is the † The often used quantity ρ dx (g/cm2) is the areal density or simply the “grams per square rad: 1 Gy = 100centimeter” rad. Older of an oncologists element of stopping frequently medium of hedge, thickness, saying dx. It is the“centiGray” thickness of a slab (cGy) instead ofof stoppingrad. material times its density. To determine it experimentally, one usually measures, instead, mass divided by area.

2.2.3 Energy Lost vs. Energy Deposited It is good to remember that the energy lost by a proton beam exceeds the energy absorbed locally by the patient or water phantom. A fraction of the beam’s energy goes into neutral secondaries (γ-rays and neutrons), which

⋅ * Sometimes Φ is written φ. In the early literature fuence rate is called “fux.” † The often used quantity ρ dx (g/cm2) is the areal density or simply the “grams per square centimeter” of an element of stopping medium of thickness, dx. It is the thickness of a slab of stopping material times its density. To determine it experimentally, one usually measures, instead, mass divided by area. Physics of Proton Interactions in Matter 27

may deposit their energy some distance away (for instance, in the shielding of the treatment room). Energy is conserved, of course, but only if we take the region of interest large enough and include the very small fraction that may go into changing the energy state of target molecules.

2.2.4 The Fundamental Equation How does physical absorbed dose relate to fuence and stopping power? Suppose dN protons pass through an infnitesimal cylinder of cross sectional area dAPhysics and thickness of Proton dx Interactions. In the cylinder in Matter 27 energy −×()dE/dx dx× dN D ≡ = mass ρ××dA dx may deposit their energy some distance away (for instance, in the shielding or of the treatment room). Energy is conserved, of course, but only if we take the region of interest large enoughS and include the very small fraction that may go into changing the energyD = Φ state of target molecules. (2.12) ρ

Dose equals2.2.4 fTheuence Fundamental times mass stoppingEquation power. Proton therapy calculations, whether beam line designAuthor or dose reconstruction in the patient, usually How does physicalDivision Fluence, absorbed stopping dose relate power to fuence and anddose stopping power? begin with this formula7/5/20 in| onePage39 form or another. However, it is not convenient Supposeto use SI dNunits protons throughout. pass through Gray (= an J/Kg) inf nitesimalis fne, but cylinder S/ρ is invariably of cross sectional in MeV/(g/cmarea dA 2and), square thickness metersThe dx is. Infundamental far the too cylinder large an equation area, and one proton is far too few. Therefore, let Φ = 1 Gp/cm2, where Gp ≡ gigaproton ≡ 109 protons, energy −×()dE/dx dx× dN and let S/ρ = 1 MeV/(g/cm2). DAfter≡ appropriate= conversions such as 1 MeV = 0.1602 × 10–12 J we fnd mass ρ××dA dx or S 2 D = 0.G1602 Φ y F in Gp/cm (2.13) ρ S D = Φ S/r in Mev/(g/cm2) (2.12) with Φ in Gp/cm2 and S/ρ in MeV/(g/cm2) as usuallyρ tabulated. Another useful form is found by taking the time derivative of Equation 2.13, Dose equals fuence times mass stopping power. Proton therapy! calculations,! expressing fuence rate in terms of proton current density (iAp //==Ne AeΦ), assumingwhether a current beam density line design of 1 nA/cm or dose2, and reconstruction converting units. in We the f patient,nd usually begin with this formula in one form or another. However, it is not conve- (B. Gottschalk) H. Paganetti 2012 nient to use SI units throughout. Gray (= J/Kg) is fne, but S/ρ is invariably ip S Gy in MeV/(g/cm2), squarePhysics meters ofD !Charged= Particle is far Therapy too | Paulo large M. Martins an area, and one (2.14)proton is far A ρ s too few. Therefore, let Φ = 1 Gp/cm2, where Gp ≡ gigaproton ≡ 109 protons, and let S/ 1 2MeV/(g/cm2). After appropriate2 conversions such as 1 MeV with ip/A in nA/cmρ = and S/ρ in MeV/(g/cm ). For a current density of = 0.00330.1602 nA/cm × 210 and–12 J S we = 5f ndMeV/(g/cm2) (170 MeV protons in water), we fnd ! D 0.G017 y/sG 1 y/min, a typical radiotherapyS rate. Typical targets have areas of several cm2 and there are variousD = 0.G1602 ineffΦcienciesy involved (discussed (2.13) ρ next), so we have already shown that the proton current entering the treatment headwith orΦ in“nozzle” Gp/cm must2 and be S/ ρof in the MeV/(g/cm order of nA.2) as usually tabulated. In usingAnother the last useful few formulas, form is found we must by taking remember the time the distinction derivative betweenof Equation 2.13, ! ! absorbedexpressing dose (what fuence we are rate interested in terms ofin) proton and the current tabulated density stopping (iAp // power==Ne AeΦ), assuming a current density of 1 nA/cm2, and converting units. We fnd

ip S Gy D! = (2.14) A ρ s

2 2 with ip/A in nA/cm and S/ρ in MeV/(g/cm ). For a current density of 0.0033 nA/cm2 and S = 5 MeV/(g/cm2) (170 MeV protons in water), we fnd D! 0.G017 y/sG 1 y/min, a typical radiotherapy rate. Typical targets have areas of several cm2 and there are various ineffciencies involved (discussed next), so we have already shown that the proton current entering the treatment head or “nozzle” must be of the order of nA. In using the last few formulas, we must remember the distinction between absorbed dose (what we are interested in) and the tabulated stopping power Physics of Proton Interactions in Matter 27

energy −×()dE/dx dx× dN D ≡ = mass ρ××dA dx or S D = Φ (2.12) ρ Dose equals fuence times mass stopping power. Proton therapy calculations, whether beam line design or dose reconstruction in the patient, usually begin with this formula in one form or another. However, it is not convenient to use SI units throughout. Gray (= J/Kg) is fne, but S/ρ is invariably in MeV/(g/cm2), square meters is far too large an area, and one proton is far too few. Therefore, let Φ = 1 Gp/cm2, where Gp ≡ gigaproton ≡ 109 protons, and let S/ρ = 1 MeV/(g/cm2). After appropriate conversions such as 1 MeV = 0.1602 × 10–12 J we fnd S D = 0.G1602 Φ y (2.13) ρ with Φ in Gp/cm2 and S/ρ in MeV/(g/cm2) as usually tabulated. Author Another useful form is foundDivisionFluence, by taking thestopping time derivative power ofand Equation dose 2.13, 7/5/20 | Page40 ! ! expressing fuence rate in terms of proton current density (iAp //==Ne AeΦ), assuming a current density of 1Dose nA/cm rate2, and and converting beam current units. We fnd

ip S Gy 2 D! = ip/A in nA/cm (2.14) A ρ s 2) 2 2 S/r in MeV/(g/cm with ip/A in nA/cm and S/ρ in MeV/(g/cm ). For a current density of 0.0033 nA/cm2 and S = 5 MeV/(g/cm2) (170 MeV protons in water), we fnd D! 0.G017 y/sG 1 y/min, a typical radiotherapy rate. Typical targets have areas of several cm2 and there areFor various a 170 inefMeVfciencies proton involved beam in (discussed water: 2 next), so we have already shown• thatcurrent the proton density current of 0.0033 entering nA /cmthe treatment head or “nozzle” must be of• theS = order 5 MeV/(g/cm of nA. 2) In using the last few formulas, we must⟹ remember�̇ = 0.017 the distinctionGy/min (typical between radiotherapy rate) absorbed dose (what we are interested in) and the tabulated stopping power (B. Gottschalk) H. Paganetti 2012

Physics of Charged Particle Therapy | Paulo M. Martins Author DivisionFluence, stopping power and dose 7/5/20 | Page41 Dose rate and beam current • current density of 0.0033 nA/cm2 • Targets with several cm2 require proton currents entering the treatment head in the order of nA • The tabulated stopping power is somewhat greater that the absorbed dose (a fraction goes into neutral secondaries – g-rays and neutrons ranging further) • The dose rate must never be used to estimate the dose delivered to the patient!! (dosimeters needed!) (B. Gottschalk) H. Paganetti 2012

Physics of Charged Particle Therapy | Paulo M. Martins Physics of Proton Interactions in Matter 29

1.0 0.9 0.8 0.7 0.6 D

MO 0.5 f 0.4 0.3 0.2 0.1 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 m100 / d100

FIGURE 2.6

Dependence on relative modulation of fMOD, the relative dwell time of the deepest step, for a typical set of range modulators. d100 is the depth of the distal corner of the SOBP. m100 is the distance between the corners. Author DivisionAccelerator and beam delivery (passive scattering) 7/5/20 | Page42

Call that, the fractional dwell time of the frst (deepest) step, fMOD. Putting everything together we fDosend, averaged rate and over beam one currentmodulator cycle, i ! p ⎛ S ⎞ Gy 〈〉Df= ε BP fMOD ⎜ ⎟ (2.15) A ⎝ ρ ⎠ s

2 2 with ip (incident proton current)e = efficiency in nA, (single A in scattering cm , and ~ 0.05;S/ρ in double MeV/(g/cm scattering). ~ 0.45) Strictly speaking S/ρ is the mass stopping power in water at the water tank entrance, but you can usefBP the= peak incident-to-entrance energy ratio because of pristine energy BP ~ loss 3.5 (energy in the independent) beam spreading system isfMOD small = fractional by design. dwell time of the deepest step (range modulation) It remains to determine fMOD as a function of modulation. For zero modulation, fMOD = 1. Otherwise, we need to design a set of modulators (1). Figure 2.6 shows fMOD as a function of relative modulation for a typical beam spreading system. The shape of the curve comes mainly from the shape of the Bragg(B. Gottschalk) H. Paganetti 2012 peak. It is little affectedPhysics by detailsof Charged Particle of Therapythe beam | Paulo M. Martinsspreading system, so Figure 2.6 can safely be used for rough estimates. Dependence on modulation is nonlinear. Therefore Eq. 2.15 implies that, whereas dose rate is strictly proportional to inverse area, it is not proportional to thickness (extent in depth) because the deepest Bragg peak already delivers a considerable dose to the entire volume whatever the irradiation strategy (passive or scanning).

2.3 Stopping Protons slow down in matter, mainly through myriad collisions with atomic electrons. In collisions at a given distance a proton loses more energy, the Author DivisionAccelerator and beam delivery (passive scattering) 7/5/20 | Page43

(R. Slopsema) H. Paganetti 2014

Physics of Charged Particle Therapy | Paulo M. Martins Author DivisionFluence, stopping power and dose 7/5/20 | Page44 Dose rate and beam current

• For zero modulation, fMOD =1 • The dose rate is proportional to inverse area, but not to thickness of the modulator • The deepest BP already delivers considerable dose to the entire volume (passive or active) (B. Gottschalk) H. Paganetti 2014

Physics of Charged Particle Therapy | Paulo M. Martins Author DivisionAccelerator and beam delivery (passive scattering) 7/5/20 | Page45

(R. Slopsema) H. Paganetti 2014

Physics of Charged Particle Therapy | Paulo M. Martins Author DivisionOutline 7/5/20 | Page46

• Preliminary notions and definitions • Energy loss of heavy charged particles by atomic collisions • The Bethe-Bloch formula (stopping sower) • Fluence, stopping power and dose • Range

Physics of Charged Particle Therapy | Paulo M. Martins Author DivisionRange 7/5/20 | Page47

• Range depends on the type of material, particle type and its energy • Range can be determined by passing a beam of particles at the desired energy through different thicknesses of the material and measure the ratio of transmitted to incident particles

Physics of Charged Particle Therapy | Paulo M. Martins Author DivisionRange 7/5/20 | Page48 • As the range is approached the ratio drops. W.R. Leo, (1994) Techniques for Nuclear and Particle Physics Experiments, Springer-Verlag • The ratio does not drop immediately to the background level, but slopes down due to the non-continuous energy loss (statistical). • Two identical particles with the same initial energy will not in general suffer the same number of collisions and energy loss. • A measurement with an ensemble of identical particles shows a statistical distribution of ranges (range straggling), Typical range number-distance curve centered about a mean value (mean range).

Physics of Charged Particle Therapy | Paulo M. Martins Author DivisionRange 7/5/20 | Page49 • Mean range: midpoint on the descending W.R. Leo, (1994) Techniques for Nuclear and Particle Physics Experiments, Springer-Verlag slope where half of the particles are absorbed

• Extrapolated or pratical range: the point at which the curve drops to the background level (tangent to the curve at the midpoint and extrapolating to the zero-level) – all particles are absorbed

Typical range number-distance curve

Physics of Charged Particle Therapy | Paulo M. Martins Author DivisionRange 7/5/20 | Page50 • Mean Range: � � = ∫ �� • Approximate pathlength travelled and ignores MCS • The strait-line range is smaller than the total zigzag • MCS effect is small for heavy charged particles

• Semi-empirical formula: • � � = � � + ∫ �� • Tmin = minimum energy at which dE/dx is valid • � � = empirically determined constant

Calculated range curves of different W.R. Leo, (1994) Techniques for Nuclear and heavy particles in aluminum Particle Physics Experiments, Springer-Verlag

Physics of Charged Particle Therapy | Paulo M. Martins Author DivisionRange 7/5/20 | Page51 • Linear on the log-log scale • � ∝ �

• The stopping power is dominated by the b-2 term, • -dE/dx −d�/�� ∝ � ∝ �

• Integrating: � ∝ � • A more accurate fit: � ∝ �.

Calculated range curves of different W.R. Leo, (1994) Techniques for Nuclear and heavy particles in aluminum Particle Physics Experiments, Springer-Verlag

Physics of Charged Particle Therapy | Paulo M. Martins 34 2. Passage of Radiation Through Matter

where we have assumed that the density of plastic scintillator is about 1.03 g/cm 3 (see Table 7.1 for example). Thus one should expect to see a peak in the signal pulse height spectrum coming from the counter, since muons of energy greater than about 300 MeV will deposit about the same amount of energy in the counter. This implies also that cosmic ray muons might also be used for calibration purposes.

Example 2.2 A beam of 600 MeV protons can be "lowered" in energy by passing it through a block of material such as copper and then "cleaned" using a series of analyzing magnets. What thickness of copper would be required to lower the average energy of this beam to 400 MeV? To find the thickness for a given energy change, we must invert dEldx and integrate over energy. Thus,

-1 Author 500 dE DivisionRange L1x= - J -( ) dE . 7/5/20 | Page52 600 dx Practical example: A beam of 600 MeV protons can be "lowered" in energy by passing it In general, this must be integrated numerically. If we use a simple rectangular integra- through a block of material suchtion aswithcopper energy intervalsand then of 20 "cleaned"MeV and dEl dxusing evaluateda series in the middleof analyzing of each inter- magnets. What thickness of copperval, we wouldfind a thicknessbe required of to lower the average energy of this -1 beam to 400 MeV? 1 dE 1 dE Range (MeV) L1x = L1E ( - ) p dx p dx

600 - 580 1.768 11.31 �� 580- 560 1.791 11.17 Δ� = − �� 560- 540 1.815 11.02 �� 540- 520 1.841 10.86 520- 500 1.870 10.69 Simple rectangular integration 500-480 1.901 10.52 480-460 1.934 10.34 with energy intervals of 20 MeV 460-440 1.971 10.15 and dEldx evaluated in the 440-420 2.012 9.94 420-400 2.056 9.73 middle of each interval L1Xtotal = 105.73 g/cm 2 = 11.88 cm W.R. Leo, (1994) Techniques for Nuclear and Particle Physics Experiments, Springer-Verlag Had we made an even simpler one step calculation taking the dEldx at 500 MeV and Physics of Charged Particle Therapy | Paulo M. Martins solving for L1x, we would have found L1x = 106.1 g/cm 2 = 11.92 cm, which, in fact, is not very different! Note that we are dealing with mean energy losses here. The energy of the protons leaving the other side of the copper degrader will, in fact, be distributed in energy as a Gaussian with 400 MeV as the mean (see Sect. 2.6). To produce a monoenergetic beam of 400 MeV protons now would require selecting out the 400 MeV protons in the peak of the distribution. This can be done using a magnetic field to "bend" the outgoing particles and keeping only those deflected at the correct angle. Author DivisionRange 7/5/20 | Page53 Bragg-Kleeman rule For the same particle in different materials:

� � � = � � � Range of heavy charged particles in liquid (bubble chamber) hydrogen, helium gas, carbon, iron, and lead.

M. Tanabashi et al. (Particle Data Group), (2018) Phys. Rev. D 98, 030001

Physics of Charged Particle Therapy | Paulo M. Martins Author DivisionRange 7/5/20 | Page54 • Proton range–energy relation around the clinical regime for four useful materials.

• At a given energy, the range expressed in g/cm2 is greater (the stopping power is lower) for heavy materials.

(B. Gottschalk) H. Paganetti 2014

Physics of Charged Particle Therapy | Paulo M. Martins Author DivisionRange 7/5/20 | Page55

• Range–energy relation in Be, Cu, and Pb and a test of cubic spline interpolation. • Log-log graph for three materials over an extended region. • Lines eventually curve. From 3-300 MeV (relevant energies) they are nearly straight, and for Be, Cu, and Pb, nearly parallel. • Good candidates for cubic spline interpolation. • Only thirteen input values for each material at 0.1, 0.2, 0.5, 1, ..., 500, 1000 MeV are required.

(B. Gottschalk) H. Paganetti 2012

Physics of Charged Particle Therapy | Paulo M. Martins