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1 ere is a sample hapter from Letures on Radiation Dosimetry Physis: A Deeper Look into the Foundations of Clinial Protools his sample hapter is opyrighted and made availale for personal use only No part of this hapter may e reprodued or distriuted in any form or y any means without the prior written permission of Medial Physis Pulishing Clik here to order this ook in one of two formats: softover ISBN: $5000 ebook ISBN: $5000 o order y phone, all

2 8 Charged Partile Interations 8 with Matter he photon interations transfer energy to eletrons, and it is these eletrons that deliver dose as they slow down in matter o understand the asi physis involved, we will start with an important derivation for heavy harged partiles that will then e generalized to eventually work for eletrons slowing down in ondensed matter It is in fat remarkale that this extension to eletrons works as well as it does Interations of eavy Charged Partiles In Figure 81, let s onsider a heavy harged partile with kineti energy,, and veloity, v, going very quikly y an atom with an impat parameter, ere we use simple desription of the nuleus, whih is the liquid drop model From this model, we get the nulear radius is R where A is the mass numer nu / A 3 m he radius of the atom is aout a 018βλ from Bohr s theory, where λ is the wavelength of the eletron in an outer shell and β = v/ is the partile speed ompared to the speed of light he radius of the atom is muh larger than the nulear radius It is like a aseall (nuleus) in Camp Randall stadium (atom) Note that there is a veloity dependene to all of this, and this is just order of magnitude validity anyway Zi v, M z i y x R nu z a K e, m L Figure 81 Geometry of a heavy harged partile interating with a single atom Note: this figure is not to sale Mass numer variale, A, here is the numer of protons plus the numer of neutrons, and it is an integer for just this disussion 55

3 56 Letures on Radiation Dosimetry Physis: A Deeper Look into the Foundations of Clinial Protools he heavy and fast partile at the ase of the arrow in Figure 81 has a harge, Z i, and a mass, M i, and a signifiant energy, γm i, suh that 1/ 1v / 1 Notie that fast here means Born approximation he atomi ound eletron, on the other hand, has only a harge, e, a mass, m, and ound energy suh that kineti energy,, roughly equals its potential energy In other words: mv e / ~ / a where a / me, whih gives v / ~ e 4 / ~( 1/ 137), the fine struture onstant, α squared herefore, the inident partile needs to have more kineti energy than this ( > m α ) his implies that only aout 53 kev is needed for the inident partile in this treatment Other generalizations eyond these assumptions will also e possile In general we will e ategorizing reations aording to impat parameter : 1 Soft ollisions ( >> a): Continuous energy loss with many atoms at one interations with oriting eletrons Soft ollisions lead to exitations Soft ollisions are aout half of what is happening ard ollisions ( a): ard ollisions are also near half of the interations, ut large events that an lead to the lieration of other harges with their own trajetories are more rare We will need to arefully aount for this he lierated seondary harges are outer-shell-oriting eletrons ard ollisions give ionizations with resulting exitations 3 Nulear eletri field interations ( R nu ): his results in elasti ollisions (mostly) and remsstrahlung (only 3% for radiation therapy energies) Note that only light partiles, like eletrons, will show any signifiant remsstrahlung 4 Nulear interations ( < R nu ): Nulear reations will only our here Stopping Power and Mass Stopping Power d One of the most important onepts in the physis of dosimetry is stopping power, Stopping power is dx d defined as the rate of energy loss for distane traveled into the medium he mass stopping power,, is then dx otained y dividing stopping power y the density of the material Mass stopping power is usually divided into two terms: one for (soft and hard) ollisional energy losses and the other for radiative (remsstrahlung) energy losses as follows: d d d dx dx dx he units of stopping power and mass stopping power are MeV/m and MeV/(g/m ), respetively Radiative stopping power is the prodution of remsstrahlung eavy harged partiles do not produe muh remsstrahlung: ie, a proton is ~000 times heavier than an eletron and, therefore, times less remsstrahlung is produed hus, if M is muh heavier than an eletron, then we will assume radiative loss is negligile: d dx r 1 ~ 0 ( M ) owever, for eletrons and positrons, this radiative term will not e negligile Later in this hapter, we will disuss the mass stopping power for eletrons and positrons r 0 0 (81) (8) For more disussion aout the Born approximation, visit Evans (1955), page 887 Note we are using slightly different notations: and m and z versus Z Note that this ategorization is general and not just for heavy harged partiles

4 8 Charged Partile Interations with Matter 57 Collisional Mass Stopping Power for eavy Charged Partiles Returning to Figure 81, onsider the momentum impulse as we integrate in time the rate of momentum hange (Gaussian units here): F dp 1 e E vb (83) dt owever, only the eletri field part is important, sine v/ ~ α For the geometry given aove, and for a given, the kiks in x and y are given y the following: Ze vt Ex (84) v t 3 / E y Ze v t One then integrates from infinity to + infinity in time while notiing that the kik in E x vanishes eause it is an odd funtion herefore, only the y-kik omes into the momentum hange: 3 / (85) P P dt ee () t y Ze v Ze v 0 y d( vt) ( vt) 3 / (86) his is like an impulse with P y_initial = 0 he energy lost is simply otained then y (ΔP) / m to arrive at the following very important result for the interation with a single eletron: ( P) ( ) m Ze 1 mv 4 (87) Notie that this energy loss is independent of the inident partile mass, M i, even though a large mass is assumed, given that the inident partile does not hange diretions in this approximation Also notie that the energy loss is proportional to 1/v his is a very important result that remains dominant as we proeed further owever, that was only for a given, ut in our prolem, we have a whole range of values herefore, we need to integrate over the full range of the impat parameter, Aording to Figure 8, the numer of eletrons in the differential annulus of volume (dx) (πd) is [(πd) (dx)] ρ (N A z/a) he differential energy loss experiened y the partile having an impulse with this annulus is then the following: 4 Ze 1 d ( d) ( dx) N Az / A mv (88) Note that the small Z is the material atomi numer here, opposite from most ooks

5 58 Letures on Radiation Dosimetry Physis: A Deeper Look into the Foundations of Clinial Protools d Zi, M v i z y x Figure 8 Geometry of a heavy harged partile interating with a differential ring of atoms dx Rearranging and noting that we need to integrate over all possile values: 4 d NAz Ze dx 4 A mv 0 ( d / ) (89) he ritial issue is how to perform the integral Sine with integrating from zero to infinity, the stopping power would diverge Instead of the full integral, we will then take the integral from a min to, whih results in the following: 4 d NAz Ze dx 4 A mv ln min (810) Aording to the Equation (88), if goes to zero, then d will rise without any limit owever, there is a ertain imum for energy loss and that ours with a head-on ollision So we need to hoose a minimum for whih gives the imum magnitude for energy loss After derivations of energy transfers with ound eletron shell states in quantum mehanis, and some ompliated Bessel funtion math, the stopping power is given y the following: where, d dx n Z ze 4 B mv 4 ln /, B min 3 mv ( 1 13) Ze i (811) (81) ere, is the average motion frequeny of eletrons, and n is the numer density of atoms Note and rememer from earlier hapters that ( 1) / M and 1/ 1 For more disussion regarding this topi, see JD Jakson s Classial Eletrodynamis, Chapter 13 he fator 113 is from 113 = /exp(0577), and 0577 is Euler s onstant

6 8 Charged Partile Interations with Matter 59 ale 81: Maximum Energy ransferred for Various Situations* Partile Value of ' For eletrons ' = / For positrons For a heavy harged partile with mass M For a heavy harged partile with mass M, whih M>>m and <<M ' = *Note that for positrons, the imum energy transferred to an eletron is twie what it is for eletron to another eletron hat is eause the partiles are different, and the seondary is not simply the one with the lower energy 1 ( M / ) 1( Mm) /( m/ ) m mv 1 Let s separate the ollisional mass stopping power into two terms: one for hard ollisions and one for soft ollisions We will start out with an aritrary energy utoff,, that will separate the two y an impat parameter or, equivalently in this ase, an energy If σ s and σ h are ross setions for soft and hard ollisions, respetively: d ds dh NAz dx dx dx A min d s d d d h d d (813) he quantity ' here is the energy transferred y the fast harged partile to eletrons he quantity here is somewhat aritrary, ut it should e aout the value of the minimum energy at whih the eletron is ejeted for hard ollisions ale 81 illustrates the values of ' Soft Collision Mass Stopping Power for eavy Charged Partiles If the energy transferred, ', is less than the ionization potential, only exitations an our If it is larger, then ionizations an our, ut they will also have exitations as well, and the energy must e larger than he quantity I is defined as the mean exitation potential of the asoring medium; it inludes oth exitations and ionizations See Attix (004), Appendix B1 and B here is a lot of unertainty to I, ut at least it is in the logarithm, whih redues the sensitivity to that unertainty I is approximately proportional to z, the atomi numer of the medium Derived using the Born approximation (β >> zz/137)), then the soft ollision mass stopping power is the following: ds NAz Z m rm dx 0 A ln I ( 1 ) z Z m ( 01535) ln A I ( 1 ) material partile (814) he partile s kineti energy is muh greater than an eletron s potential (orital) energy

7 60 Letures on Radiation Dosimetry Physis: A Deeper Look into the Foundations of Clinial Protools ere the quantity 0 r mn A (MeV/(g/m )) For ease later, we an define the following: z Z k ( 01535) A he units of k are MeV/(g/m ) In fat, very important dependenies are in k he z A (815) term is a property of Z only the material, and is a property of only the inident partile, and it is this part that produes the Bragg peak he aove applies to eletrons, positrons, and heavy harged partiles Soft ollisions are most, or at least half, of the ollisions for radiation therapy energies ard Collision Mass Stopping Power for eavy Charged Partiles A hard ollision is often further defined as when an eletron is ejeted with a onsiderale fration of the imum energy transferale, hese reoil eletrons are alled delta-rays or knok-on eletrons, and they an e seen in ule hamers he differential hard ollision ross setion per eletron for heavy harged partiles is given y the following: d rm Z h d 1 ( / ) 0 ( ) (816) here is some dependene on partile spin for this differential hard ollision ross setion per eletron We will use the form for zero spin partiles (like alpha, pion, ), ut it will apply to spin ½ partiles (like protons, eletrons, muon, ) provided ' << M hen the hard ollision mass stopping power for << ' is the following: dh NAz dx A d h d d k d/ ( / d ) k ln( / ) ( / m k ln( / ) ax )( ) (817) Now with omining Equations (813), (814), and (817), mass stopping power for heavy harged partiles an e written as the following: d ds dh m k dx dx dx I ln k ( 1 ) ln( / ) (818)

8 8 Charged Partile Interations with Matter 61 ale 8: Magnitudes for Parameters Related to igh-energy Partile Speed Relative to the Speed of Light β p β ln(β /(1 β )) ln(β /(1 β )) β With a it of math, and realling that for a heavy harged partile m mv, the Equation 1 (818) simplifies as follows: d dx m k ln I (819) ( 1 ) here is also a more handy form for mass ollisional stopping power for spin ½ and ' << M as follows: d dx z Z A (03071) ln ln I 1 medium partile (80) ere, mean exitation potential, I, has units of ev Note that the energy dependene in the rakets of Equation (80) is weak (see ale 8) Shell Corretion he omplex motion of the orital eletrons is aounted for with the shell orretion When the partile veloity is less than the orital veloity of the eletrons in that shell, then those eletrons do not partiipate signifiantly in ollisions with the partile he shell orretion fator is written as C/z, and it (mostly) dereases the mass ollisional stopping power y a small amount as follows: d dx m k ln I( 1 ) C / z he shell orretion is a funtion of the partile veloity and the atomi numer of the medium See Attix (004), page 17, Figure 83 for the magnitude and proton kineti energy dependene of the shell orretion, as an example Note that it is more important at lower energies Dependene of the Stopping Power on the Medium he fator z/a has a value near 05 ±005 for most elements, dropping to lower values at higher z ydrogen-1 has the highest value of 1, whih is why it is used to slow or shield fast harged partiles See appendix B1 in Attix (004), page 57, for z/a values he term lni also makes high-z materials have lower stopping power Also, the shell orretion generally dereases the stopping power (81)

9 6 Letures on Radiation Dosimetry Physis: A Deeper Look into the Foundations of Clinial Protools Auto-normalized Dose Auto-normalized Depth 1 Figure 83 Dose depth urves for heavy ions along the entered axis of a road eam Note that the Bragg peak is due to the 1/β in k Mass Collisional Stopping Power Dependene on Partile Veloity he term 1/β is dominant at low energies It auses the mass ollisional stopping power to sharply inrease as the partile slows down he result is the so-alled Bragg Peak (see Figure 83) When β = v/ 1, the energy inreases quikly as the speed reeps up against he lina aelerating mirowave avity uses this fat to keep the partile in phase with the phase veloity of the standing or traveling mirowave he avity is loaded with periodi arriers to slow the phase veloity elow Also, when β 1, the 1/β term has little influene, ut the β /(1 β ) term inreases See ale 8 he kineti energy of a partile is diretly proportional to its rest mass here is no mass dependene on the heavy harged partile stopping power herefore, any heavy partile with the same veloity and harge will have the same stopping power, ut the satter and range straggling ould e different Mass Collisional Stopping Power Dependene on Partile Charge At low energies (speeds β < 01), there is an effetive harge, Z *, to e used instead of the harge, Z, eause of the attahment of the inident partile s eletrons igher-energy partiles are more likely to e fully ionized See Anderson (1984), page 1, Figure 4 he Z fator means that partiles with multiple harges have a muh higher mass ollisional stopping power than singly harged partiles For example, if, then: p p d d dx 4 dx (8) his fat an e used to otain stopping powers for any heavy harged partile from a tale of mass ollisional stopping powers for protons For this purpose, do the following steps: 1 Look up or alulate β for partile x with kineti energy, x Look up or alulate the proton kineti energy, p, for the same β 3 Look up the mass stopping power for a proton with kineti energy p 4 Multiply the mass stopping power for a proton y ( / * Z Note that x Zp) Z p * is unity, and Z x is the effetive harge on partile x at its speed

10 8 Charged Partile Interations with Matter 63 primary > / e Inident eletron Bound eletron e seondary < / = / Figure 84 Kineti energy after a hard ollision y an eletron By onvention, the primary harge has more energy than the seondary harge It should e emphasized that through the deades, the stopping power formula has een progressively refined One of the most important refinements was to ure the high-speed (v approahing ) limit As explained in Jakson (1999), page 636, the very fast partile auses so muh ionization from relativisti effets that it selfshields its harge from the plasma effet of the ionizations it reates he time sale assoiated with this shielding is manifest y the plasma frequeny quantity that appears in this limit Yet, the fundamental relations overing most of the important physis for most appliations and for most partiles are desried y the simple derivation at the start of hapter, and this fat is remarkale in the history of modern physis Eletron and Positron Interations In hard ollisions y eletrons, one annot tell whih was the primary or whih was the seondary y onvention, the one with the highest energy (that is, the faster eletron) is the primary herefore, the imum kineti energy transferale is the amount that the lower-energy eletron an have (Figure 84) owever, in the positron ase, we do know whih partile is whih, and we annot reassign the primary lael, so the imum kineti energy transferred is (look at ale 81) herefore, the imum kineti energy transferred is / for eletrons, and it is for positrons he differential ross setion for the eletron hard ollisions whih desries the ollision etween two free eletrons was derived y Möller (193) as follows: d h r m d ( ) m m 1 (83) Note that z dependene here is impliit in β via mass stopping power he differential ross setion for positron-eletron ollisions is even more omplex, and it was derived y Bhaha (1936)

11 his onludes the first half of Chapter 8 of Letures on Radiation Dosimetry Physis: A Deeper Look into the Foundations of Clinial Protools Phone to order the omplete ook Or lik here to order this ook in one of two formats: softover ISBN: $5000 ebook ISBN: $5000