Exponential attenuation BAEN-625 Advances in Food Engineering Introduction y Exponential attenuation is relevant primarily to uncharged ionizing radiations y Photons y Neutrons y Uncharged IRs lose their energy in relatively few interactions y Charged particles typically undergo many small collisions, losing their KE energy gradually Photon and Charged Particle y A photon has a large 10 MeV phontons probability of passing straight through a thick layer of material without losing any energy y It has no limiting ‘range’ through matter y A charged particle must always lose some or all energy 10 MeV electrons y It has a range limit as it runs out of KE y Photons penetrate much farther through matter than charged particles y Above 1 MeV this difference gradually decreases 20 cm x 20 cm square concrete Simple exponential attenuation y Monoenergetic parallel beam y Ideal case (simple absorption, no scattering or secondary radiation) y Large # No of uncharged particles incident perpendicularly on a flat plate of material of thickness L Scattering and backscateering Simple exponential attenuation y Assume y Each particle either is completely absorbed in a single interaction, producing no secondary radiation, or y Passes straight through the entire plate unchanged in energy or direction tionLaw ofttenuaxponential a e y Let (μ.1) be the probability dN = −μNdl that an individual particle dN = −μdl interacts in a unit N thickness of material N L dN L = − μdl y So, the probability that it ∫N ∫ NN= o t =0 will interact in dl is μdl lnNN |=L −μ lL | N o 0 y If N particles are incident N lnNN− ln = lnL =μL − upon dl, the change dN in L o N the number N due to o N absorption is: L = e − μL N o Law of exponential attenuation N L = e−μL No Linear attenuation coefficient Linear attenuation coefficient, μ y Or attenuation coefficient y Also refereed as ‘narrow-beam attenuation coefficient’ y When divided by density r of the attenuating medium, the mass attenuation coefficient is obtained: μ cm[/;/]2 g m2 kg ρ tion ofximaAppro the LEA tionequa N ()μLL2 () μ 3 =L e =−μL 1 −μ L + −... + No 2 ! 3 ! if Lμ < 0.05 N =L e−μL ≅1 μ − L No Mean free path or relaxation length 1 ≡ mean free path or relaxation length, [cm or m] μ y It is the average distance a single particle travels through a given attenuating medium before interacting y It is the depth to which a fraction 1/e (~ 37%) of a large homogeneous population of particles in a beam can penetrate or tion fExponential Attenua plural modes of absorption Partial LAC for process 1 y More than one μ = μ1+ μ 2+ .... absorption process is μ μ =1 1 +2 ... + present μ μ y Each event by each N process is totally L = −e μ( 1 + μ 2 ...) + L absorbing, producing N o or no scattered − μ L − μ L N=( N e )(1 2 e )... secondary particles L o − μL y LAC The total μ is: NΔ = No − N L = N o − o N e μ μ NΔ = N() − Nx = N()1 −− μ NL x e x o L μ o o μ x alonefor a single process Interactions Example 1 y Given: -1 -1 y μ1 = 0.02 cm ; μ2 = 0.04 cm 6 y L = 5 cm, No = 10 particles y Find: y Particles NL that are transmitted, and y Particles that are absorbed by each process in the slab Solution − μ1L − μ 2 L NL= N( o e )( e ) −()μ1 + μ 2 L6 ( 0 .− 02 + 0 . 04 ) 5 NL= ( N o e )= 10e 7 .= 408 × 5 10 total number of particle absorbed : 6 5 5 Δ =NNN( 10 −o 7 = L . 408 − 10 × ) = 2 . 592 × 10 absorbednumber by process 1 and 2 are : μ 0 . 02 ΔNNN =() −21 . = 592 ×5 108 × . 64 =4 10 × 1 o L μ 0 . 06 μ 0 . 04 ΔNNN =() −22 . = 592 ×5 101 × . 728 =5 10 × 2 o L μ 0 . 06 Narrow-beam attenuation of UR y Real beams of photons interact with matter by processes that may generate y Charged or uncharged secondary radiations y Scattering primaries either with or without a loss of energy y So, the total number of particles that exit from the slab is greater than the What should be counted in N ? unscattered primaries L Secondary charged particles y Not to be counted as uncharged particles y They are less penetrating, and thus tend to be absorbed in the attenuator y Those that escape can be prevented from entering the detector by enclosing it in a thick enough shield y So, energy given to charged particles is absorbed and does not remain part of the uncharged radiation beam Scattered and secondary uncharged particles y Can either be counted or not y If counted this equation is not valid because it is N only valid for simple L = e−μL absorbing events N y If they reach the detector, o but only the primaries are content in NL, this equation is valid Methods of achieving narrow- beam attenuation y Discrimination against all scattered and secondary particles that reach the detector, on the basis of y Particle energy y Penetrating ability y Direction y Time of arrival y Etc y Narrow-beam geometry, which prevents any scattered particles from reaching the detector Narrow-beam geometry Broad-beam attenuation of UR y Any attenuation geometry in which some primary rays reach the detector y In ideal broad-beam geometry every scattered or secondary uncharged particle strikes the detector, but only if generated in the attenuator by a primary particle on it ways to the detector, or by a secondary charged particle resulting from such a primary y Requirements y Attenuator be thin to allow the escape of all uncharged particles resulting from first interactions, plus the X-rays emitted by secondary charged particles Broad beam geometry Broad beam geometry y Requires the detector to respond in proportion to the radiant energy of all the primary, scattered, and secondary uncharged radiation incident upon it R L = e−μen L Ro Ro = the primary radiant energy incident on the detector when L =0 RL = radiant energy of uncharged particle striking the detector when the attenuator is in placed, L is the attenuator Different types of geometries and attenuations y Narrow-beam geometry – only primaries strike the detector; μ is observed for monoenergetic beams y Narrow-beam attenuation – only primaries are counted in NL by the detector, regardless of wheather secondaries strike it; μ is observed for monoenergetic beams y Broad-beam geometry – other than narrow-beam geometry; at least some scattered and secondary radiation strikes the detector y Broad-beam attenuation – scattered and secondary radiation in counted in NL by the detector μ'< μ Effective attenuation coefficient Different types of geometries and attenuations y Ideal Broad-beam geometry – every scattered and secondary uncharged particle that is generated directly or indirectly by a primary radiation strikes the detector y Ideal Broad-beam attenuation – in this case μ'= μen Energy-absorption coefficient Absorbed dose in detector y It is a function of the energy fluence Ψ y The narrow-beam attenuation coefficient will have a mean value: n Ψ μ ∑()i L ()EZ, i i=1 μΨ,L = n ∑()Ψi L i=1 The buildup factor, B y Useful in describing a broad-beam attenuation quantity due to primary + scatteredradiation and secondary B = quantity due to radiationprimary alone y B = 1 for narrow beam geometry y B>1 for broad-beam geormetry For broad-beam Ψ L = Be−μL Ψ0 y For L = 0 (no attenuator between source and detector) ΨL BB=0 ≡ Ψ0 tion ttenuae aMean efectivf coefficient Ψ L=Be−μL ≡ −μ e'L Ψ0 ln B μ≡' μ − L Buildup factor.