1. Charged-Particle Interactions in Matter Coulomb-Force Interactions Stopping Power Range

2. Introduction Charged particles lose their energy in a manner that is distinctly different from that of uncharged radiations (x- or  -rays and neutrons). An individual photon or neutron incident upon a slab of matter may pass through it with no interactions at all, and consequently no loss of energy. Or it may interact and thus lose its energy in one or a few “catastrophic” events. 2

3. Introduction (cont.) By contrast, a charged particle, being surrounded by its Coulomb electric force field, interacts with one or more electrons or with the nucleus of practically every atom it passes. Most of these interactions individually transfer only minute fractions of the incident particle’s kinetic energy, and it is convenient to think of the particle as losing its kinetic energy gradually in a friction like process, often referred to as the “continuous slowing-down approximation” (CSDA). 3

4. Introduction (cont.) Charged particles can be roughly characterized by a common pathlength, traced out by most such particles of a given type and energy in a specific medium. 4

5. called the range, ℜ. Introduction (cont.) Note that because of scattering, all identical charged particles do not follow the same path, nor are the paths straight, especially those of electrons because of their small mass. Because of the multitude of interactions undergone by each charged particle in slowing down, its pathlength tends to approach the expectation value. The expectation value of pathlength that would be observed as a mean for a very large population of identical particles is 5

6. Types of Charged-Particle Coulomb-Force Interactions Charged-particle Coulomb-force interactions can be simply characterized in terms of the relative size of the classical impact parameter b vs. the atomic radius a, as shown in the following figure. The following three types of interactions become dominant for b >> a, b ~ a, and b << a, respectively: – “Soft” collisions – Hard (or “knock-on”) collisions – Coulomb-force interactionswiththeexternal nuclear field 6

7. Important parameters in charged-particle collisions with atoms: a is the classical atomic radius; is the classical impact parameter 7

8. “Soft” Collisions (b >> a) When a charged particle passes an atom at a considerable distance, the influence of the particle’s Coulomb force field affects the atom as a whole, thereby  distorting it,  exciting it to a higher energy level, and sometimes ionizing it by ejecting a valence electron. The net effect is the transfer of a very small amount of energy (a few eV) to an atom of the absorbing medium. 8

9. “Soft” Collisions (cont.) Because large values of b are clearly more probable than are near hits on individual atoms, “soft” collisions are by far the most numerous type of charged-particle interaction, and they account for roughly half of the energy transferred to the absorbing medium. 9

10. Hard (or “Knock-On”) Collisions (b ~ a) When the impact parameter b is of the order of the atomic dimensions, it becomes more likely that the incident particle will interact primarily with a single atomic electron, which is then ejected from the atom with considerable kinetic energy and is called a delta (  ) ray In the theoretical treatment of the knock-on process, atomic binding energies have been neglected and the atomic electrons treated as “free” 10

11. Hard Collisions (cont.)  -rays are of course energetic enough to undergo additional Coulomb-force interactions on their own. Thus a  -ray dissipates its kinetic energy along a separate track (called a “spur”) from that of the primary charged particle. 11

12. Hard Collisions (cont.) The probability for hard collisions depends upon quantum-mechanical spin and exchange effects, thus involving the nature of the incident particle. Hence, as will be seen, the form of stopping- power equations that include the effect of hard collisions depend on the particle type, being different especially for electrons vs. heavy particles. Although hard collisions are few in number compared to soft collisions, the fractions of the primary particle’s energy that are spent by these two processes are generally comparable. 12

13. Hard Collisions (cont.) It should be noted that whenever an inner-shell electron is ejected from an atom by a hard collision, characteristic x rays and/or Auger electrons will be emitted just as if the same electronhadbeenremovedbyaphoton interaction. Thus some of the energy transferred to the medium may be transported some distance away from the primary particle track by these carriers as well as by the  -rays. 13

14. Coulomb-Force Interactions with the External Nuclear Field (b << a) When the impact parameter of a charged particle is much smaller than the atomic radius, the Coulomb-force interaction takes place mainly with the nucleus. This kind of interaction is most important for electrons (either + or -) in the present context, so the discussion here will be limited to that case. 14

15. Interactions with the External Nuclear Field (cont.) In all but 2 – 3% of such encounters, the electron is scattered elastically and does not emit an x-ray photon or excite the nucleus It loses just the insignificant amount of kinetic energy necessary to satisfy conservation of momentum for the collision Hence this is not a mechanism for the transfer of energy to the absorbing medium, but it is an important means of deflecting electrons 15

16. Interactions with the External Nuclear Field (cont.) The differential elastic-scattering cross section per atom is proportional to Z ². This means that a thin foil of high- Z material may be used as a scatterer to spreadoutanelectronbeamwhile minimizingtheenergylostbythe transmitted electrons in traversing a given mass thickness of foil. 16

17. Interactions with the External Nuclear Field (cont.) In the other 2 – 3% of the cases in which the electron passes near the nucleus, an inelastic radiative interaction occurs in which an x-ray photon is emitted The electron is not only deflected in this process, but gives a significant fraction (up to 100%) of its kinetic energy to the photon, slowing down in the process Such x-rays are referred to as bremsstrahlung, the German word for “braking radiation” 17

18. 18

19. Interactions with the External Nuclear Field (cont.) This interaction also has a differential atomic cross section proportional to Z ², as was the case for nuclear elastic scattering. Moreover, it depends on the inverse square of the mass of the particle, for a given particle velocity. Thus bremsstrahlung generation by charged particlesotherthanelectronsistotally insignificant. 19

20. Stopping Power The expectation value of the rate of energy loss per unit of path length x by a charged particle of type Y and kinetic energy T, in a medium of atomic number Z, is called its stopping power, ( dT / dx ) Y,T,Z. Stopping power is typically given in units of MeV/cm or J/m. Dividing the stopping power by the density  of the absorbing medium results in a quantity called the mass stopping power ( dT /  dx ), typically in MeV cm 2 /g or J m 2 /kg. 20

21. Stopping Power (cont.) Stopping power may be subdivided into: “collision stopping power” and “radiative stopping power” Collision stopping power is the rate of energy loss resulting from the sum of the soft and hard collisions, which are conventionally referred to as “collision interactions”. Radiative stopping power is that owing to radiative interactions. Energy spend in radiative collisions is carried away from the charged particle track by the photons, while that spent in collision interactions produces ionization and excitation contributing to the dose near the track. 21

22.  Stopping Power (cont.) The mass collision stopping power can be written as where subscripts c indicate collision interactions, s being soft and h hard  dT  dT s  dT h    dx  c   dx  c   dx  c 22

23. .  .  ln   ln I    The Hard-Collision Term for Heavy Particles (cont.)        Zz  A    dT   dx  c in which I is the mean excitation potential of the struck atom, (Appendices B.1 and B.2 list some I -values according to Berger and Seltzer) Z is the atomic number of the medium, A is the atomic mass of the medium, z is the atomic number of the incident charge particle and  =v/ c. 23

24. Dependence on the Stopping Medium There are two expressions influencing this dependence, and both decrease the mass collision stopping power as Z is increased The first is the factor Z / A outside the bracket, which makes the formula proportional to the number of electrons per unit mass of the medium The second is the term –ln I in the bracket, which further decreases the stopping power as Z is increased 24

25. Dependence on the Stopping Medium (cont.) The term –ln I provides the stronger variation with Z The combined effect of the two Z - dependent expressions is to make ( dT /  dx ) c for Pb less than that for C by  40-60 % within the  -range 0.85-0.1, respectively. 25

26. Mass collision stopping power for singly charged heavy particles, as a function of  or of their kinetic energy T 26

27. Dependence on Particle Charge The factor z 2 means that a doubly charged particle of a given velocity has 4 times the collision stopping power as a singly charged particle of the same velocity in the same medium. For example, an  -particle with  = 0.141 would have a mass collision stopping power of 200 MeV cm 2 /g, compared with the 50 MeV cm 2 /g shown in the figure for a singly charged heavy particle in water. 27

28. Dependence on Particle Mass There is none All heavy charged particles of a given velocity and z will have the same collision stopping power 28

29.  r BcmT 20  0    Mass Radiative Stopping Power Only electrons and positrons are light enough to generate significant bremsstrahlung, which depends on the inverse square of the particle mass for equal velocities. The rate of bremsstrahlung production by electrons or positrons is expressed by the mass radiative stopping power ( dT /  dx ) r, in units of MeV cm 2 /g, which can be written as. where the constant  0 = 1 / 137 ( e 2 / m 0 c 2 ) 2 = 5.80  10 -28 cm 2 /atom, T is the particle kinetic energy in MeV, and ̅ B r is a slowly varying function of Z and T.  dT  N A Z 2   dx  r A 29

30. Mass Radiative Stopping Power (cont.) ̅ B r has a value of  16 / 3 for T << 0.5 MeV, and  6 for T = 1 MeV (roughly),  12 for 10 MeV, and  15 for 100 MeV ̅ B r Z 2 is dimensionless 30

31.  The massradiativestoppingpoweris proportional to N A Z 2 / A, while  the mass collision stopping poweris Mass Radiative Stopping Power (cont.) proportional to N A Z / A, the electron density. Thus their ratio would be expected to be proportional to Z. 31

32. Mass Radiative Stopping Power (cont.) The equation also shows proportionality to T + m 0 c 2, or to T for T >> m 0 c 2. The corresponding energy dependence of the collision stopping power is not obvious from its formula, but can be seen in the following diagram Above T = m 0 c 2 it varies only slowly as a function of T. Thus the ratio of radiative to collision stopping powers will be roughly proportional to T at high energies. 32

33. Mass radiative and collision stopping powers for electrons (and approximately for positrons) in C, Cu, and Pb 33

34.  TZdxdT r   / Mass Radiative Stopping Power (cont.) The ratio of radiative to collision stopping power is often expressed in the form: in which T is the kinetic energy of the particle, Z is the atomic number of the medium, and n is a constant variously taken to be 700 or 800 MeV.  dT /  dx  c n 34

35. Mass Radiative Stopping Power (cont.) The figure shows the stopping power trends vs. energy and Z The collision stopping power is relatively independent of Z, so any ratio of ( dT /  dx ) c for one medium to that for another is only weakly dependent on T Also, above 1 MeV the variation of ( dT /  dx ) c itself vs. T is very gradual. 35

36.  0,so For heavier particles ( dT /  dx ) r ( dT /  dx ) = ( dT /  dx ) c almost exactly. Mass Radiative Stopping Power (cont.) The total mass stopping power is the sum of the collision and radiative contributions: dT  dT  dT   dx   dx  c   dx  r Along with its parts, dT /  dx is tabulated as a function of T for a given stopping medium and type of charged particle; in Appendix E, for electrons. 36

37. Radiation Yield The radiation yield Y ( T 0 ) of a charged particle of initial kinetic energy T 0 is the total fraction of that energy that is emitted as electromagnetic radiation while the particle slows and comes to rest. For heavy particles Y ( T 0 )  0. For electrons the production of bremsstrahlung x- rays in radiative collisions is the only significant contributor to Y ( T 0 ). For positrons, in-flight annihilation would be a second significant component, but this has customarily been omitted in calculating Y ( T 0 ). 37

38. Radiation Yield (cont.) We define y ( T ) as  dT /  dx  r  dT /  dx  y  T   38

39. Restricted Stopping Power The restricted stopping power is that fraction of the collision stopping power that includes all the soft collisions plus those hard collisions resulting in  rays with energies less than a cutoff value . The mass restricted stopping power in MeV cm 2 /g, will be symbolized as ( dT /  dx ) . An alternative and very important form of restricted stopping power is known as the linear energy transfer, symbolized as L . 39

40.  and Restricted Stopping Power (cont.) The usual units for L  are keV/  m, so that   dT  2  10   dx    If the cutoff energy  is increased to equal T max [ T /2 for electrons, T for positrons], then  dT  dT    dx     dx  c L   L  40

41.  dT   2    2    G  ,       k  ln     dx    2 I / m 0 c   Restricted Stopping Power (cont.) The calculation of ( dT /  dx )  for heavy particles gives (in MeV cm 2 /g).  dT  2m 0 c 2  2  2 2C  2   Z  For electrons and positrons this quantity is given by the following equation, in which  T / m 0 c 2 and  / T.  2C   Z  2 41

42. Range Range may be defined as follows: – The range  of a charged particle of a given type and energy in a given medium is the expectation value of the pathlength p that it follows until it comes to rest (discounting thermal motion). A second, related quantity, the projected range, is defined thus: – The projected range of a charged particle of a given type and initial energy in a given medium is the expectation value of the farthest depth of penetration t f of the particle in its initial direction. 42

43. Illustrating the concepts of pathlength p and farthest depth of penetration, t f, for an individual electron. p is total distance along the path from the point of entry A to the stopping point B. Note that t f is not necessarily the depth of B. 43

44. CSDA Range, ℜ CSDA Experimentally the range can be determined (in principle) for an optically transparent medium such as photographic emulsion by microscopically following each particle track in three dimensions, and obtaining the mean pathlength for many such identical particles of the same starting energy 44

45. CSDA Range (cont.) A closely similar but not identical quantity is called the CSDA range, which represents the rangeinthecontinuousslowingdown approximation. In terms of the mass stopping power,  1 the CSDA 0 where T 0 is the starting energy of the particle If dT /  dx is in MeV cm 2 /g and dT in MeV, then  CSDA is thus given in g/cm 2 45

46. CSDA Range (cont.) For all practical purposes  CSDA can be taken as identical to the range  as defined earlier. Their small and subtle difference is due to the occurrence of discrete and discontinuous energy losses. The effect is expected to make the CSDA range slightly underestimate the actual range, by 0.2% or less for protons and by a somewhat greater (but undetermined) amount for electrons. 46

47.  CSDA  0T0T 1 CSDA Range (cont.) The following figure gives the CSDA range  CSDA for protons in C, Cu, and Pb. (  CSDA ) for carbon can be approximately represented (  5%) in g/cm 2 by for proton kinetic energies 1 MeV < T 0 < 300 MeV Because of the decrease in the stopping power with increasing atomic number, the range (in mass/area) is greater for higher Z. 415 670 1.77  47

48. 48

49. CSDA Range (cont.) The range of other heavy particles can be obtained from a proton table by recalling that: – All particles with the same velocity have kinetic energies in proportion to their rest masses. – All singly charged heavy particles with the same velocity have the same stopping power. – Consequently the range of singly charged heavy particles of the same velocity is proportional to their rest mass, since a proportional amount of energy must be disposed of. 49

50.  M M z CSDA Range (cont.) In general the procedure for finding the CSDA range of a heavy particle of rest mass M 0 and kinetic energy T 0 is to enter proton CSDA range tables at a proton energy T 0P = T 0 M 0P / M 0, where M 0P is the proton’s rest mass If the tabulated proton CSDA range is  PCSDA, the other particle’s range  CSDA is then obtained from 0 P CSDA P 2 0   CSDA 50

51. CSDA Range for electron CSDA Ranges for electron are tabulated in Appendix E. 51

52. Projected Range The projected range is most easily visualized in terms of flat layers of absorbing medium struck perpendicularly by a beam of charged particles. One counts the number of incident particles that penetrate the slab as its thickness is increased from zero to  (or to a thickness great enough to stop all the incident particles). 52

53. Projected Range (cont.) The following figure shows typical graphs of the number of particles penetrating through slabs of varying thickness t. All particlesareassumedtobe monoenergeticandperpendicularly incident. 53

54. 54

55. Projected Range (cont.) In the figure practically no reduction in numbers of particles is observed until the projected range is approached, where a steep decrease to zero occurs. The value of t beyond which no particles are observed to penetrate is called t max, the maximum penetration depth. For a proton or heavier particle this is only slightly less than the maximum pathlength, since t max represents those particles which happen to suffer little scattering. The range  (the mean value of the pathlength) is generally not more than 3% greater than for protons. 55

56. Electron Range The electron CSDA range and the projected range are calculated the same as for heavy charged particles. However, it should be evident from pane c that these quantities are of marginal usefulness in characterizing the depth of penetration of electrons (or positrons). Scattering effects, both nuclear and electron- electron, cause the particles to follow such tortuous paths that t f ( t ) is smeared out from very small depths up to t = t max  2. 56

57. 57

58. Electron Range (cont.) For low-Z media, t max is comparable to  ( or  CSDA ), which is a convenience in the practical application of range tables.  increases as a function of Z, similar to the that seen for protons. However, acorrespondingincreaseinthe incidence of nuclear elastic scattering also takes place and tends to make and t max roughly independent of Z for electrons and positrons. 58

59. Table 8.4 Comparison of Maximum Penetration Depth t max with CSDA Range for Electrons of Energy T 0 59

60. Electron Range (cont.) The final column in this table gives the ratio t max /  CSDA, which decreases from about 0.85 to 0.48 as Z goes from 13 to 79. This ratio shows very little energy dependence in the range 50  T 0  150 keV. This trend is continued at higher energies as well, judging from the calculations of Spencer, which are excerpted in the following table. 60

61. Table 8.5. t max /  CSDA for a Plane Perpendicular Source of Electrons of Incident Energy T 0 61