exponential attenuation - moreira.tamu.edu attenuation.… · simple exponential attenuation y...
TRANSCRIPT
Exponential attenuation
BAEN-625 Advances in Food Engineering
Introduction
Exponential attenuation is relevant primarily to uncharged ionizing radiations
PhotonsNeutrons
Uncharged IRs lose their energy in relatively few interactionsCharged particles typically undergo many small collisions, losing their KE energy gradually
Photon and Charged Particle
A photon has a large probability of passing straight through a thick layer of material without losing any energy
It has no limiting ‘range’ through matter
A charged particle must always lose some or all energy
It has a range limit as it runs out of KE
Photons penetrate much farther through matter than charged particles
Above 1 MeV this difference gradually decreases
10 MeV phontons
10 MeV electrons
20 cm x 20 cm square concrete
Simple exponential attenuation
Monoenergetic parallel beamIdeal case (simple absorption, no scattering or secondary radiation)Large # No of uncharged particles incident perpendicularly on a flat plate of material of thickness L
Scattering and backscateering
Simple exponential attenuation
AssumeEach particle either is completely absorbed in a single interaction, producing no secondary radiation, orPasses straight through the entire plate unchanged in energy or direction
Law of exponential attenuation
L
o
L
o
LoL
LNN
L
t
N
NN
eNN
LNNNN
lN
dlN
dN
dlN
dNNdldN
L
o
L
o
μ
μ
μ
μ
μ
μ
−
==
=
−==−
−=
−=
−=
−=
∫∫
lnlnln
||ln 0
0
Let (μ.1) be the probability that an individual particle interacts in a unit thickness of materialSo, the probability that it will interact in dl is μdlIf N particles are incident upon dl, the change dN in the number N due to absorption is:
Law of exponential attenuation
L
o
L eNN μ−=
Linear attenuation coefficient
Linear attenuation coefficient, μ
Or attenuation coefficientAlso refereed as ‘narrow-beam attenuation coefficient’When divided by density r of the attenuating medium, the mass attenuation coefficient is obtained:
]/;/[ 22 kgmgcmρμ
Approximation of the LEA equation
LeNN
LLLeNN
L
o
L
L
o
L
μ
μ
μμμ
μ
μ
−≅=
<
+−+−==
−
−
1
0.05 L if
...!3)(
!2)(1
32
Mean free path or relaxation length
It is the average distance a single particle travels through a given attenuating medium before interactingIt is the depth to which a fraction 1/e (~ 37%) of a large homogeneous population of particles in a beam can penetrate
1≡
μmean free path or relaxation length, [cm or m]
Exponential Attenuation for plural modes of absorption
More than one absorption process is presentEach event by each process is totally absorbing, producing no scattered or secondary particlesThe total LAC μ is:
( ) ( )μμ
μμ
μμ
μμ
μμμ
μ
μ
μμ
μμ
xLoo
xLox
LooLo
LLoL
L
o
L
eNNNNN
eNNNNN
eeNN
eNN
−
−
−−
++−
−=−=Δ
−=−=Δ
=
=
++=
++=
1
)...)((
...1
....
21
21 ...)(
21
21
Interactions for a single process x alone
Partial LAC for process 1
Example 1
Given:μ1 = 0.02 cm-1; μ2 = 0.04 cm-1
L = 5 cm, No = 106 particlesFind:
Particles NL that are transmitted, andParticles that are absorbed by each process in the slab
Solution
( )
( ) 5522
4511
556
5
5)04.002.0(6)(
10728.106.004.010592.2
1064.806.002.010592.2
:are 2 and 1 processby absorbednumber 10592.2)10408.710(
:absorbed particle ofnumber total10408.7
10)(
))((21
21
×=××=−=Δ
×=××=−=Δ
×=×−=−=Δ
×=
==
=+−+−
−−
μμμμ
μμ
μμ
Lo
Lo
Lo
LoL
LLoL
NNN
NNN
NNN
eeNN
eeNN
Narrow-beam attenuation of UR
Real beams of photons interact with matter by processes that may generate
Charged or uncharged secondary radiationsScattering primaries either with or without a loss of energySo, the total number of particles that exit from the slab is greater than the unscattered primaries What should be counted in NL?
Secondary charged particles
Not to be counted as uncharged particlesThey are less penetrating, and thus tend to be absorbed in the attenuatorThose that escape can be prevented from entering the detector by enclosing it in a thick enough shieldSo, energy given to charged particles is absorbed and does not remain part of the uncharged radiation beam
Scattered and secondary uncharged particles
Can either be counted or notIf counted this equation is not valid because it is only valid for simple absorbing eventsIf they reach the detector, but only the primaries are content in NL, this equation is valid
L
o
L eNN μ−=
Methods of achieving narrow-beam attenuation
Discrimination against all scattered and secondary particles that reach the detector, on the basis of
Particle energyPenetrating abilityDirectionTime of arrivalEtc
Narrow-beam geometry, which prevents any scattered particles from reaching the detector
Narrow-beam geometry
Broad-beam attenuation of UR
Any attenuation geometry in which some primary rays reach the detectorIn 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 primaryRequirements
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
Requires the detector to respond in proportion to the radiant energy of all the primary, scattered, and secondary uncharged radiation incident upon it
L
o
L eneRR μ−=
Ro = the primary radiant energy incident on the detector when L =0RL = radiant energy of uncharged particle striking the detector when
the attenuator is in placed, L is the attenuator
Different types of geometries and attenuations
Narrow-beam geometry – only primaries strike the detector; μ is observed for monoenergetic beamsNarrow-beam attenuation – only primaries are counted in NL by the detector, regardless of wheather secondariesstrike it; μ is observed for monoenergetic beamsBroad-beam geometry – other than narrow-beam geometry; at least some scattered and secondary radiation strikes the detectorBroad-beam attenuation – scattered and secondary radiation in counted in NL by the detector
μμ <'Effective attenuation coefficient
Different types of geometries and attenuations
Ideal Broad-beam geometry – every scattered and secondary uncharged particle that is generated directly or indirectly by a primary radiation strikes the detectorIdeal Broad-beam attenuation – in this case
enμμ ='Energy-absorption coefficient
Absorbed dose in detector
It is a function of the energy fluence ΨThe narrow-beam attenuation coefficient will have a mean value:
( ) ( )
( )∑
∑
=
=Ψ
Ψ
Ψ= n
iLi
n
iiZELi
L
1
1,
,
μμ
The buildup factor, B
Useful in describing a broad-beam attenuation
B = 1 for narrow beam geometryB>1 for broad-beam geormetry
aloneradiation primary toduequantity radiationsecondary and scattered primary toduequantity +
=B
For broad-beam
For L = 0 (no attenuator between source and detector)
LL Be μ−=ΨΨ
0
00 Ψ
Ψ≡= LBB
Mean effective attenuation coefficient
LB
eBe LLL
ln'
'
0
−≡
≡=ΨΨ −−
μμ
μμ
Buildup factor