GS Introduction to Medical Physics I Basic Interactions Problem Set 3.1e Solutions

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1 GS-9 - Introduction to Medical Physics I Basic Interactions Proble Set.1e Solutions 1. (J & C 5.15) A MeV photon interacts in a pair process. Calculate the enery of the positron if the electron eeres fro the interaction with enery of. MeV. The enery balance equation is iven by hν - 1. MeV = E + + E - MeV 1. MeV = E + +. MeV E + =.978 MeV. (J & C 5.18) The pair process in lead has a cross section of / at MeV. Find the enery converted into kinetic enery of chared particles when a bea containin photons passes throuh a block of lead of thickness 1. Assue only pair interactions. If we assue only pair interactions, then the enery converted to kinetic enery of chared particles is equal to the enery available for conversion (incident enery inus threshold enery) ultiplied by the nuber of photons interactin (exponential attenuation). Enery available for conversion = MeV 1. MeV = MeV Nuber of photons interactin = N (1 - e -x ). The linear attenuation coefficient can be found fro the cross-section. = =.95 x =.95 = Note that because of the value of x, we need to use the exponential attenuation equation to calculate the nuber of interactions. Nuber of photons interactin = (1 - e -.95 ) = photons Enery converted = MeV =. MeV.

2 . (J & C 5.1) At photon enery of 5 MeV in lead, coherent, Copton, photoelectric, and pair processes all occur. The cross sections in / are iven in coluns,,, and 5 of Table A-i. photons each with enery 5 MeV pass throuh a foil of lead of thickness 1 /. Find the nuber of coherent, Copton, photoelectric, and pair processes. Find the ean enery converted to kinetic enery by each process and so deterine the ean enery transferred for all processes. Copare with the value in the table. The nuber of processes is iven by the nuber of incident photons ultiplied by the cross section for the particular process ultiplied by the thickness of absorber. Cross-sections for each process in lead are iven in J & C Table A-i. N coh = =.7 = 7. photons N inc = =.85 = 85 photons N photo = =.9 = 9. photons N pair = = 7.88 =788 photons To obtain the kinetic enery transferred, consider the followin: For coherent scatter, there is no enery transfer. For Copton scatter, Fi 5-8 ives us the ean fraction of enery transfer, which for 5 MeV photons, is approxiately.. So the enery transfer is iven by = 1.9 MeV For photoelectric effect, all the enery of the incident photon inus the bindin enery of an inner shell electron is transferred to kinetic enery of the photoelectron. For bindin enery, we will use the K-shell value of 88 kev. Consequently, the enery transfer is iven by (5.88) 9. =.1 MeV. For pair production, all of the enery of the incident photon inus the threshold enery of pair production is transferred to kinetic enery of the positron-electron pair. The enery transfer is iven by (5 1.) 788 = 8.99 MeV The total enery transfer is iven by 1.9 MeV +.1 MeV MeV = 5. MeV. The total nuber of interactions is iven by = 1.5. The averae enery transfer per interaction is 5./1.5 MeV =.58 MeV, which copares with the table value of. MeV

3 . (J & C.19) Fro Fiure -7 deterine the nuber of positrons set in otion with eneries between. and.5 MeV when a bea of photons of enery MeV ipines on a foil of lead of thickness. /. The total nuber of pair processes that occur in the lead foil is iven by the crosssection for pair production ultiplied by the nuber of incident photons ultiplied by the thickness of lead (in - ). N = kn x = 18.8 = 5.7 photons photons..97 The aount of kinetic enery available is MeV 1. MeV = MeV. The fraction of available kinetic enery iven to the positrons is.5/18.98 =.1, and fro Fiure -7, the relative probability per fractional enery interval for a fraction of available kinetic enery of.1 is approxiately 1.. Consequently, the nuber of positrons set in otion is iven by N = = 1 (.5.) 18.98

4 5. (J & C.) A slab of carbon thick (density.5 / ) is bobarded by photons, each with enery of MeV. Use data fro Table A-b to deterine the followin: a. Nuber of Copton interactions For MeV photons, the Copton attenuation coefficient is So, for a thickness of, x =.18 = Given the value of x, we ay use the differential for of the attenuation equation. The nuber of Copton interactions is thus iven by N = N µ x =.11 =.11 b. Enery converted to kinetic enery by Copton interactions The averae enery transferred in a Copton interaction at enery MeV is iven by Table A-a to be 1.5 MeV. Consequently the total enery converted to kinetic enery by Copton interactions is iven by MeV = MeV. c. Enery scattered by the Copton process The enery scattered by the Copton process is that fraction of the incident photon enery that is not converted to kinetic enery by the Copton interactions, or.11 ( 1.5) MeV = MeV =. 5 MeV. d. Nuber of pair and triplet processes For MeV photons, the pair attenuation coefficient is So, for a thickness of, x =.11 = The nuber of pair production interactions is thus iven by

5 N = N µ x e. Enery radiated as bresstrahlun =.981 =.981 Fro Table A-b, the ean enery transferred is 1. MeV, while the ean enery absorbed is 15. MeV. Consequently, the ean enery radiated is the difference, or 1.1 MeV. The total nuber of interactions is iven by N = N x, where is the total linear attenuation coefficient. N = N x = N ρx ρ = photons = 7.11 photons The total enery radiated as bresstrahlun is then 1.1 MeV 7.11 = 7.8 MeV. f. Total enery diverted fro the bea The total enery diverted fro the bea is found by ultiplyin the nuber of photons involved in interactions by the photon enery of MeV. So, the total enery diverted fro the bea is 7.11 MeV = 1 MeV.. Total enery converted to kinetic enery The total enery converted to kinetic enery is found by ultiplyin the nuber of photons involved in interactions by the ean enery transferred, or MeV = 11. MeV. h. Total enery radiated. The total enery radiated is the total enery converted to kinetic enery inus the total enery absorbed. The total enery absorbed is found by ultiplyin the incident photon enery by the nuber of photons absorbed. The nuber of photons absorbed is iven by N abs = N abs x, where abs is the linear absorption coefficient. -

6 N abs = N = N = abs ρx ρ abs x = 5.5 photons.11 5 photons - Multiplyin the nuber of photons by MeV, ives us the enery absorbed, which is 5.5 MeV =.9 5 MeV. Thus, the total enery radiated is the total enery converted to kinetic enery, 11. MeV, inus the enery absorbed, 9. MeV, or 7. MeV. i. Make an enery balance. Calculate the enery absorbed usin ab /ρ and copare with the enery converted to kinetic enery. Total enery in bea Enery transitted Enery absorbed Enery re-irradiated = # photons enery per photon =. MeV =. 7 MeV = (# photons - # interactin) enery per photon = ( ). MeV = MeV = MeV = 1.9 MeV = E Copt + E PP + E Bres = ( ) MeV = ( ) MeV =.5 MeV So, Enery transitted + Enery absorbed + Enery re-irradiated = ( ) 7 MeV =. 7 MeV.

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