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