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
Law of exponential attenuation
y Let (μ.1) be the probability
dN = − μ Ndl
that an individual particle
interacts in a unit
thickness of material
y So, the probability that it
will interact in dl is μdl
y If N particles are incident
upon dl, the change dN in
the number N due to
absorption is:
dN
= − μ dl
N
NL
∫
N =No
L
dN
= − ∫ μ dl
N
t =0
ln N | NN oL = − μ l |0L
ln N L − ln N o = ln
NL
= e − μL
No
NL
= − μL
No
Law of exponential attenuation
NL
− μL
=e
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
[cm / g ; m / kg ]
ρ
Approximation of the LEA
equation
NL
( μL) ( μL)
− μL
= e = 1 − μL +
−
+ ...
No
2!
3!
2
if μL < 0.05
NL
− μL
= e ≅ 1 − μL
No
3
Mean free path or relaxation
length
1
μ
≡
mean free path or relaxation length, [cm or m]
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
absorption process is
present
y Each event by each
process is totally
absorbing, producing
no scattered or
secondary particles
y The total LAC μ is:
μ = μ1 + μ 2 + ....
μ
μ
1 = 1 + 2 + ...
μ
μ
NL
= e − ( μ1 + μ 2 + ...) L
No
N L = N o ( e − μ1 L )( e − μ 2 L )...
Δ N = N o − N L = N o − N o e − μL
Δ N x = (N o − N L )
Interactions for a single process x alone
μx
μ
= N o (1 − N o e − μL ) x
μ
μ
Example 1
y Given:
y μ1 = 0.02 cm-1; μ2 = 0.04 cm-1
y L = 5 cm, No = 106 particles
y Find:
y Particles NL that are transmitted, and
y Particles that are absorbed by each process in the slab
Solution
N L = N o ( e − μ1 L )( e − μ 2 L )
N L = N o ( e − ( μ1 + μ 2 ) L ) = 10 6 e − ( 0.02 + 0.04 ) 5
= 7 .408 × 10 5
total number of particle absorbed :
Δ N = N o − N L = (10 6 − 7 .408 × 10 5 ) = 2 .592 × 10 5
number absorbed by process 1 and 2 are :
0 .02
μ1
5
Δ N 1 = (N o − N L )
= 2 .592 × 10 ×
= 8 .64 × 10 4
0 .06
μ
0 .04
μ2
5
Δ N 2 = (N o − N L )
= 2 .592 × 10 ×
= 1 .728 × 10 5
0 .06
μ
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
unscattered primaries
What should be counted in NL?
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
only valid for simple
absorbing events
y If they reach the detector,
but only the primaries are
content in NL, this
equation is valid
NL
− μL
=e
No
Methods of achieving narrowbeam 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
RL
= 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 ' = μ en
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:
∑ (Ψ ) (μ )
n
μΨ,L =
E ,Z i
i L
i =1
n
∑ (Ψ )
i =1
i L
The buildup factor, B
y Useful in describing a broad-beam attenuation
B=
quantity due to primary + scattered and secondary radiation
quantity due to primary radiation alone
y B = 1 for narrow beam geometry
y B>1 for broad-beam geormetry
For broad-beam
ΨL
− μL
= Be
Ψ0
y For L = 0 (no attenuator between source and
detector)
ΨL
B = B0 ≡
Ψ0
Mean effective attenuation
coefficient
ΨL
− μL
− μ 'L
= Be ≡ e
Ψ0
ln B
μ '≡ μ −
L
Buildup factor