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Home , Attenuation coefficient

Exponential

BAEN-625 Advances in Food Engineering Introduction y Exponential attenuation is relevant primarily to uncharged ionizing radiations y y y Uncharged IRs lose their in relatively few interactions y Charged typically undergo many small collisions, losing their KE energy gradually and Charged 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 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 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 Law of exponential attenuation 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 N L −= μlN ||ln L N o 0 y If N particles are incident N NN lnlnln L −==− 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 2 2 kgmgcm ]/;/[ ρ Approximation of the LEA equation

N )( 2 μμLL )( 3 L −μL 1 μLe +−+−== ... No !2 !3 0.05 L if μL < 0.05 N L −μL 1−≅= μLe No Mean free path or relaxation length 1 ≡ 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 Exponential Attenuation for plural modes of absorption

Partial LAC for process 1 y More than one μ = μ + μ 21 + .... absorption process is μ μ 1 1 2 ++= ... present μ μ y Each event by each N process is totally L = e ( μμ21 ++− ...) L absorbing, producing N o no scattered or − μ L − μ L = 1 eeNN 2 )...)(( secondary particles oL − μL y The total LAC μ is: −=−=Δ ooLo eNNNNN μ μ ()x ()1 −=−=Δ eNNNNN − μL x x Lo μ o o μ Interactions for a single process x alone 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 = oL eeNN ))((

+− μμ21 )( L +− 5)04.002.0(6 = oL eNN = 10)( e ×= 10408.7 5 :absorbed particle ofnumber total ofnumber particle :absorbed 6 5 5 NNN Lo ×=×−=−=Δ 10592.2)10408.710( number absorbed processby 1 and 2 :are μ 02.0 ()NNN 1 10592.2 5 ×=××=−=Δ 1064.8 4 1 Lo μ 06.0 μ 04.0 ()NNN 2 10592.2 5 ×=××=−=Δ 10728.1 5 2 Lo μ 06.0 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 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 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 ,ZE i i=1 μΨ,L = n

∑()Ψi L i=1 The buildup factor, B y Useful in describing a broad-beam attenuation

secondary and scattered primary toduequantity toduequantity primary + scattered and secondary radiation B = primary toduequantity toduequantity primary radiation 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 Mean effective attenuation coefficient

Ψ L −μL ≡= eBe −μ 'L Ψ0 ln B ' μμ−≡ L Buildup factor