physics department yarmouk university chapter 2: radiation...

© Dr. Nidal M. Ershaidat

Phys. 649: Nuclear Techniques

Physics Department

Yarmouk University

Chapter 2:

Radiation Interactions

Supplements

1



Physics Department

Yarmouk University

Chapter 2: Radiation Interactions

(with Matter)

© Dr. Nidal M. Ershaidat - Nuclear Techniques - Chapter 2: Radiation Interactions 2

Overview

I. IntroductionII. Interaction of Heavy Charged ParticlesIII.Interaction of Fast ElectronsIV.Interaction of Gamma RaysV. Interaction of NeutronsVI.Radiation Exposure and Dose

Introduction


Detectors vs. RadiationsChoice of a detector for a given radiation is essentially based on the type of the radiation and the mechanisms of its interactions with matter.

We shall distinguish between the four categories of radiation mentioned in chapter 1.

But finally, the main distinction is between charged and uncharged radiations.

One of the major differences is their "kind" of interaction with electrons and nuclei of the material they traverse.This implies different "path lengthspath lengths" or characteristic lengths:

Centimeters for the uncharged radiations

Millimeters or micrometers for the charged ones.

2


When gamma rays interact with matter, secondary charged particles are created. These become a nuisance to the detection device.

In contrast, the only way to "measure" neutrons is by detecting the charged "entities" they create when they interact with matter.

In the following chapters we shall discuss criteria for the choice of a detector for a given type of radiation.

But before that we shall discuss the various interaction between all categories of radiation and matter in general.

Gamma rays and NeutronsThe other difference is that the uncharged radiations alter the "matter" structure when penetrating it.

Interaction of Heavy

Charged Particles

A. Nature of the Interaction


Heavy charged particles interact with matter primarily through coulomb forces between their charges and - the negative charge of the orbital electrons within the absorber and - the positive charge of the nuclei.

Interactions with the nuclei (Rutherford scattering for example) occur rarely and are not significant in the response of the detectors.

Interaction – Coulomb Forces

Thus, it is mainly the interaction with electron which predominate.

3


The electrons of the medium "feel" the coulomb

force and they suffer an impulse (∆∆∆∆p = F dt)This impulse, or change of kinetic energy, could be either sufficient to raise the electron to higher energy level (which we call excitation) or to eject this electron completely from its atom (ionization)

Excitation – Ionization

The energy transferred to the electron in these "collisions" are of the order of

Where E and m are respectively the energy and mass

of the heavy charged particle (hcp) and m0 is the

electron mass.

∆∆∆∆E ~ (4××××0.5/3.73) E = (1/1800) E for αααα particles

∆∆∆∆E ~ (4××××0.5/0.938) E = (1/450) E for protons.

∆∆∆∆E=4Em0/m


The primary particle must lose its energy in many such interactions with electrons until it is stoppedstopped. We say that it is absorbed.

The following is a sketch of the "path" followed by an alpha particle in matter. The trajectory is almost linear since the αααα particle does not suffer serious deflection while traversing the matter (except maybe at the end!)

Range

The lines represent the path and their "average" length is what we call the "rangerange" of the charged particles in the material.


Going back to the electrons.

The result of the passage of charged particles is the presence of excitedexcited or ionizedionized atoms.

Choice of the Detector

De-excitation could be a way to detect the type of charged particles and their energies.

Ionization results in "free" electrons and positive ions. These tend to recombine to form neutral atoms again.

In the hcp detectors the recombination ion-electron is suppressed (simply using an appropriate electric field) and the "resulting" current is used to identify the hcp's.

B. Stopping Power

4


We define the linear stopping power S for a charged particle in a given absorber by:

Stopping Power

where

• dE is the differential energy loss within a

material and

• dx is the corresponding differential path

length.

dx

dES −−−−====

dE/dx is also called the "specific" energy loss of

the particle along its track. Sometimes the term "rate" of energy loss is used.

(1)


For particles with a given charge state, S increases as the velocity of the particle decreases.

Bethe Formula

Hans Bethe calculated S, which is given by the so-called "Bethe formula", which is written as:

where ze and v are respectively the charge and

velocity of the charged particle. m0 is the electron

rest mass. N is the number density of the absorber

atoms given by (ρρρρ = density of the material and Muis the Molar mass constant):

BNvm

zek

dx

dE2

0

222)(4 ππππ====−−−− (2)

u

A

MA

NN

ρρρρ==== (3)


Bethe Formula

which we sometimes write as (using ββββ = v/c):

I represents the average excitation and ionization

potential of the absorber. In principle this is a parameter which is experimentally determined for each element. The following empirical formula is

used in order to estimate I:

(((( ))))

ββββ−−−−ββββ−−−−−−−−

ββββ≡≡≡≡ 222

20 1ln

2ln

I

cmZB

)(169.0 eVZI ====

B is given by (Z is the atomic mass number of the

absorber):

−−−−

−−−−−−−−≡≡≡≡

2

2

2

220 1ln

2ln

c

v

c

v

I

vmZB

(5)

(4)

(6)

Bethe Formula

Discussion

5


Calculating the Multiplicative Factor

(((( )))) (((( ))))

(((( )))))(

10062.3

10022.6

511.0

137

1974

44

314

23

2

20

2

20

22

gA

Fg

MAMeV

FMeV

Ncm

cN

cm

ek

u

−−−−−−−− ρρρρ

××××====

ρρρρ××××

ππππ

====ααααππππ

====ππππ h

(((( ))))

ββββ−−−−


ββββρρρρ××××====−−−−

−−−−2

2

2

9.0

3

1

63875ln

)(3062.0

ZZ

gA

cmg

dx

dEin MeV/cm

Exercise: Check the calculationExercise: Check the calculation


Dependence on Energy

The term B (eq. 5) varies slowly with the energy for

a given Z (a given absorber).

Thus the behavior of dE/dx can be inferred from the

behavior of the multiplicative factor. For a given

nonrelativistic particle, dE/dx varies inversely with

v2, i.e. with the energy of the particles.

Heuristically (based on experiment), this behavior could be explained by the fact that "slow" charged particles spend more time in the material and thus the impulse felt by the electron is largest.

The following Figure (Fig. 1) shows the variation of specific energy loss ion air versus energy of the different charged particles (Electrons, muons, ππππmesons, protons, deuterons and alphas)


Dependence on Energy

Figure 1

Log scale


Dependence on zFor different charged particles of same velocity v

the variation of dE/dx is proportional to z2

When comparing different materials as absorbers,

dE/dx depends primarily on the product NZ or ρρρρZ/A

High atomic number, high-density materials will consequently result in the greatest linear stopping power.

6

C. Energy Loss Characteristics


A plot of specific energy loss along the track of a charged particle is known as a Bragg curve.

The difference between the two curves is due to energy straggling.

1. The Bragg Curve

Figure 2: The specific loss energy along an alpha track – Bragg curve


Charge "pickup"Energy at which charge pickup by the ion becomes significant.

Figure 3: dE/dx vs. E for several charged particles in different absorbers

Compare Em in the

different cases.


2. Energy StragglingThe details of microscopic interactions between any The details of microscopic interactions between any

specific particle and matter, vary randomly, the energy specific particle and matter, vary randomly, the energy

loss is a statistical or stochastic process.loss is a statistical or stochastic process.

Therefore, a spread, or straggling (or sprawling) of Therefore, a spread, or straggling (or sprawling) of

energies results after a beam of monoenergetic energies results after a beam of monoenergetic

particles has passed through a given thickness of particles has passed through a given thickness of

absorber.absorber.

The width of this energy distribution gives a measure The width of this energy distribution gives a measure

of energy straggling, which varies with the distance of energy straggling, which varies with the distance

along the particle track.along the particle track.

The following figure (The following figure (Fig. 4Fig. 4) shows a schematic ) shows a schematic

presentation of the energy distribution of initially presentation of the energy distribution of initially

monoenergetic particle at various points along its monoenergetic particle at various points along its

track.track.

Straggling is the (statistical) fluctuation in path length Straggling is the (statistical) fluctuation in path length

for individual particles of the same initial energy.for individual particles of the same initial energy.

7


Schematic Presentation

Fig. 426

Reading Fig. 4The 3 axes represent respectively the path X, the

energy E of the particle and the (z-axis) the energy

distribution f(E,X).

At X=0, E=E0 and the distribution is

a normal (gaussian) one.

In the last portion (near the end of the range, i.e. X = R) of the distribution, the distribution narrows again. This is mainly due to the fact that energy has been greatly reduced.

As the particle moves in the absorber, the distribution becomes wider and wider because of the randomness mentioned earlier (the straggling)

D. Particle Range

28

1. Definition of Ranges

The definition of the range of a charged particle in an absorber depends on the nature of both the particle and the absorber.A classical experiments for measuring this parameter is schematized in Fig. 5-a. Variation of the intensity as a

function of the thickness of the absorber t is shown in Fig. 5-b.

Figure 5-a Figure 5-b

8


Ranges of Alpha ParticlesI(t) is the detected number of alpha particles through the

thickness t of the absorber, I0 (=I(0)) is the number detected

with no absorber at all.

For small values of t, alpha particles lose their energy linearly as they pass through the absorber.

We define the mean range (Rm) as the thickness at

which the initial intensity I0 drops to one-half its value,

i.e.:

For values greater than some value of t, which is the length of the shortest track in the absorbing material, the stopping power of the material increases and the intensity of αααα particles drops rapidly to zero.

[[[[ ]]]]2

1

0

====I

RI m


Mean and Extrapolated Ranges

Rm is the most commonly used definition in tables of

numerical range values.

The range of charged particles of a given energy is thus a fairly unique quantity in a specific absorbing material.

The values of the mean range are important when deciding the active thickness of a detector material used to measure energy of the particles. This thickness should be large enough to stop the particles in the detector.

The extrapolated range is another version of the definition of the range of a charged particle. It's the value obtained when extrapolating the linear portion of the end if the transmission curve to zero.


Ranges of Alpha Particles

The following figures (6, 7 and 8) show the mean rangemean range

of various charged particles in materials of interest in detectors.

Fig. 6 shows the range of

alpha particles in air as a function of its energy.

Figure 6© Dr. Nidal M. Ershaidat - Nuclear Techniques - Chapter 2: Radiation Interactions 32

Ranges of Alpha Particles

bEaR ==== (7)

Fig. 7 shows the calculated range as a function of its

energy for different charged particles.

Fig. 7

The nearly linear behavior of the log-log plot suggests an empirical relation of the form:

ln R = b ln E + ϕϕϕϕ or (a = eϕϕϕϕ )

9


Ranges of Alpha ParticlesFig. 8 shows the range-energy curves calculated

for alpha particles in different materials. Units of the range are given in mass thickness.

Figure 8 © Dr. Nidal M. Ershaidat - Nuclear Techniques - Chapter 2: Radiation Interactions 34

Mass ThicknessThe mass thickness of a given absorber, density ρρρρ of

thickness t, is simply

tρρρρ====ξξξξ (8)


2. Range Straggling

Range straggling is defined as the fluctuation in path length for individual particles of the same initial energy. It's caused by the same stochastic factors that lead to energy straggling.

For protons or alphas, this straggling can amount to few percent of the mean range.


3. Stopping TimeThis is the time needed to completely stop a charged particle in an absorbing material.

If R is the range and <v> the average velocity of a

given charged particle (of total energy E) then the

stopping time is simply T = R/<v>

If the particles were uniformly decelerated then

<v> = v/2 and we have for a non-relativistic particle:

E

m

c

amuMeV.R

cmEc

R

v

RT A

59312

2

2

2

×===

But this is not the case. Generally the charged particles lose energy at a greater rate near the end of their range.

(9)

10

37

Simple Calculations

where T is in s and R is in meters and energy E in MeV

These simplified calculations are reasonable for light charged particles, such as alphas and protons over a wide range of energy. These calculations are not valid for electrons because they are relativistic in many cases.

A fair estimation of the stopping time is given by

taking <v> = 0.6 v. The previous equation is modified and we get:

E

mRT A××××××××==== −−−−7

102.1

Using typical range values, stopping times calculated using eq. 9 are a few picoseconds in solids and liquids

and a few nanoseconds in gases. For the fastest-responding detectors these times have to be taken into account.

(9)

E. Energy loss in Thin

Absorbers


Deposited EnergyThe energy deposited by a given charged particle

penetrating a thin absorber of thickness t can be

calculated by:

is the linear stopping power averaged over

the energy of the particle while in the absorber.

tdx

dEE

avg

−−−−====∆∆∆∆ (10)

(((( ))))avgdxdE−−−−

We distinguish two cases when dealing with thin

absorbers:

�Energy loss is small

�Energy loss is not small

If the energy loss is small then the stopping power

does not change much and it can be approximated by

its value at the incident particle energy.© Dr. Nidal M. Ershaidat - Nuclear Techniques - Chapter 2: Radiation Interactions 40

Energy Loss is Small

Values for dE/dx for charged particles in a

variety of absorbing materials can be found in

literature.

Figures 9 and 10 show dE/dx as a function of

energy for several charged particles in silicon

and aluminum.

11

© Dr. Nidal M. Ershaidat - Nuclear Techniques 41

dE/dx in Silicon

Figure 9 © Dr. Nidal M. Ershaidat - Nuclear Techniques - Chapter 2: Radiation Interactions 42

dE/dx in Aluminum

Figure 10

Read carefully and correctly this figure. See Knoll p40.


dE/dx for α's in various absorbers

Figure 11

Combination Range –

Energy Loss

12


Combined EffectsFig. 13 shows the combined effects of particle range and

decrease in dE/dx with increasing energy for protons.

Figure 13

The energy loss is plotted as a function of the proton energy. The range of protons is less than the detector

thickness until E0 = 425 keV.

This is he proton's energy at which the range is exactly equal to the absorber thickness.For E0 > 425 keV, only a portion of the incident energy is deposited and the transmitted proton carries off the remainder


Energy Deposition

Under these conditions the energy deposited in the detector is given by (eq. 10)

Because the stopping power continuously decreases with increasing energy in this region, the deposited energy therefore decreases with further increases in the incident proton energy.

The second curve in Fig. 13 plots the transmitted

energy (Et on the diagram) as recorded by a

second thick detector

tdx

dEE

avg

−−−−====∆∆∆∆

F. Scaling Laws Stopping Power

13


BraggBragg--Kleeman RuleKleeman RuleSome experiments use a given absorber for a given type of particle. Data in literature do not cover all particle-absorber combination.

There are two things to do in such cases:

1) Approximations based on the Bethe formula,

∑∑∑∑

====

i iii

cc dx

dE

NW

dx

dE

N

11

2) Assumptions that the stopping power per atom of compounds or mixtures in additive!

This assumption also known as (aka) Bragg-Kleeman rulecan be written as:

The letter c refers to the compound, i to the ith

element, W is the atom fraction of the ith component. N

represents the atomic density and dE/dx the linear stopping power.

(11)


BraggBragg--Kleeman RuleKleeman Rule

This rule indicates that the stopping power

in a metallic oxide could be obtained from

separate date on the pure metal and in

oxygen.

This kind of calculations should be

conducted with caution since some

measurements for compounds have shown

a stopping power differing by as much as

10-20% from that calculated using eq. 11

Range


Mc is the molecular weight of the compound, ni is the

number of atoms of element i in the molecule.

Range in a compoundThe range of a charged particle in a compound can be estimated provided its range in all constituents elements is known.

In this derivation, it is necessary to assume that the

shape of dE/dx curve is independent of the stopping

medium!

Under these conditions the range in the compound is given by:

(((( ))))∑∑∑∑====

iiii

cc

RAn

MR (12)

14


(Range) Bragg-Kleeman rule

In the absence of data for all constituent elements

in a compound estimates can be made based on a

semiempirical formula, call Bragg-Kleeman rule as

well,

where ρρρρ and A represent density and atomic

weight and the subscripts 0 and 1 refer to different

absorbing materials!

01

10

0

1

A

A

R

R

ρρρρ

ρρρρ≅≅≅≅ (13)

The choice of A1 and A0 is crucial for the accuracy

of the estimate. They have to be chosen as close to

each other as possible.

57

Generalization

Integration of the Bethe Formula (eq. 2) gives:

For two different types of charged particles, a (za,

ma) and b (zb, mb) we thus have:!

(((( )))) (((( ))))vFz

mvR

2≅≅≅≅ (15)

which means that we can deduce the range for

particle a, knowing that of particle b. The

technique is to compute the velocity of particle a,

then find a particle (b) whose range at the same

velocity is known in the absorbing material.

(((( )))) (((( ))))vRzm

zmvR b

ab

baa 2

2

≅≅≅≅ (16)

G. Fission Fragments


Fission Fragments

One major difference of between fission fragments and alpha particles (which we considered as the heavy charge particles till now) is the fact that these fragments are much heavier and much charged!

Because these fragments are stripped of many of their electrons their effective large charge results in a specific energy loss greater than that encountered with the radiation we have been discussing till now.

But, because the initial energy is also very high however, the range of a typical fission fragment is

approximately half that of a 5 MeV alpha particle (Fig. 8 – Chapter 1)

15


Fragment Kinetic Energy

Figure 8 (Chapter 1)61

An important feature of the fission fragment track is the fact that the specific energy loss

(-dE/dx) decreases as the particle loses energy in the absorber.

This behavior is the result of the continuous decrease in the effective charge carried by the fragment as its velocity is reduced.

The pickup of electrons begins immediately at the start of the track, and therefore the factor z in Bethe Formula continuously drops. The

resulting decrease in –dE/dx is large enough to

overcome the increase that normally accompanies a reduction in velocity.

Pickup EffectPickup Effect

H. Secondary Electron

Emission from Surfaces

63

As charged particles lose their kinetic energy during the slowing-down process, many electrons from the absorber acquire a sufficient impulse to travel a short distance from the original track of the particle

Other electrons may have large kinetic energies and thus large velocities. These fast electrons may be energetic enough to produce ionization. Delta rays is the name used for this type of electrons (The term was coined by J.J. Thomson)

These electrons could have small kinetic

energy (order of few eV) and such electrons

are unable to produce any new ionization.

Delta Electrons

16


Some of these electrons can escape the surface of the absorber and they are called secondary electrons.

Other electrons may have large kinetic energies and thus large velocities. These fast electrons may be energetic enough to produce ionization.

A single heavy ion such as a fission fragment can produce hundreds of these escaping secondaries, whereas alphas might produce 10 or fewer secondaries.

Secondary Electrons

Delta rays is the name used for this type of electrons (The term was coined by J.J. Thomson)


Secondary Electrons

1



Physics Department

Yarmouk University

Chapter 2: Radiation Interactions

(with Matter)


This Lecture

I. IntroductionII. Interaction of Heavy Charged ParticlesIII.Interaction of Fast ElectronsIV.Interaction of Gamma RaysV. Interaction of NeutronsVI.Radiation Exposure and Dose

Interaction of Fast

Electrons


A. Energy Loss and RangeFast electrons lose their energy at a lower rate than heavy charged particles.

Their path in absorbers is tortuous while that of heavy charged particles can be considered as linear.A series of tracks from a source of monoenergetic electrons might appear as in the sketch below:

Figure 14


Bethe FormulaThe main reason for this behavior is that the electrons has equal mass to that of the orbital electrons with which they interact.

In addition, electron-nuclear interactions, which can abruptly change the electron direction sometimes occurThe following Bethe formula for fast electrons gives the specific energy loss due to ionization and excitation (these loses are called "collisional loses" )

(((( ))))(((( ))))

(((( )))) )

(2

22

22

22

220

20

118

11

1122ln12

ln2

ββββ−−−−−−−−++++ββββ−−−−++++

ββββ++++−−−−ββββ−−−−−−−−


ββββππππ====

−−−−

I

Evm

vm

ZNe

dx

dE

c (17)


Radiative LosesAnother difference between fast electron and heavy charged particles is that fast electrons may lose energy by radiative processes as well as by coulomb interactions.

These radiative loses can take the form of bremsstrahlung or electromagnetic radiation, which can emanate from any position along the electron track.The linear specific energy loss through this radiative process is (given in the classical me theory)

(((( ))))

(((( ))))

−−−−

++++αααα====

−−−−

3

42ln4

12

022

0

4

cm

E

cm

eZZEN

dx

dE

r

(18)

where αααα (~ 1/137) here is the fine structure constant.

2

B. Electron Range and

Transmission CurvesStopping Power


1. Absorption of monoenergetic electrons

In Fig. 15 an attenuation experiment for a source of monoenergetic electrons is shown

The left part of the curve shows that there is a loss of some electrons from the beam even for small thicknesses of the absorber. This is mainly due to the scattering of these electrons inside the material

Figure 15


RangeA rapid and gradual drop to zero is observed.

Electrons which penetrate the greatest absorber thickness are those whose initial direction changed least in their path through the absorber.

The total path length for fast electrons is considerably greater than the distance of penetration along the initial velocity vector.

The range here is less definite than for heavy charged particles.

The range is taken from a transmission curve, Fig. 14 by extrapolating the portion of the curve to zero and represents the thickness of the absorber required to ensure that almost no electrons can penetrate the entire thickness!


Range-Energy PlotsFig. 16 shows a typical range-energy plot for fast electrons. Two absorbers are considered, Si and NaI.

Note that when using the mass thickness as the unit for thickness, values at the same electron energy are similar for materials with widely different physical properties or atomic number.

Figure 16


Electrons vs. hcpFor equivalent energy, the specific energy loss of electrons is much lower than that of heavy charged particles (because of the mass difference), so their path length in typical absorbers is hundreds of times greater.

A rough estimate of electron ranges is about 2 mm

per MeV in low density materials and about 1 mm per

MeV in materials of moderate density

The most well-known references for stopping power and range of electrons and positrons in elements and compounds are:

1. NBSIR Report 82-2550-A (Berger and Seltzer)2. ICRU Report 37

3


This exponential behavior is completely empirical and has no fundamental basis whatsoever as does the exponential attenuation of gamma rays!

2. Absorption of Beta Particles

The energy distribution of beta particles is continuous, thus their transmission curve is different from that of monoenergetic electrons!

In a majority of cases of beta emitters, the curve has a near exponential shape (on a log-scale, the shape is thus linear as in Fig. 17)

Figure 17© Dr. Nidal M. Ershaidat - Nuclear Techniques - Chapter 2: Radiation Interactions 14

Absorption CoefficientAn absorption coefficient n is defined by:

(((( )))) tneItI −−−−==== 0(19)

where

I0 = counting rate without absorber

I(t) = counting rate with absorber of thickness t

If t is in g/cm2 then n has for units cm2/g


Absorption CoefficientThe absorption coefficient n correlates well with the endpoint energy of the beta emitter for a specific absorbing material.

Figure 18


Backscattered electrons do not deposit all their energy and thus can have a significant effect on the detector response.

3. BackscatteringBackscattering is the phenomenon due to the fact that the electrons often undergo large-angle deflections along their track may finish their "trip" in the detector material by re-emerging again from the surface through which they entered.

Backscattering is more pronounced for low-energy electrons and for absorbers of large atomic number.

These electrons escape detection entirely.


Fraction of Backscattered Electrons

Figure 19

C. Positron Interactions

4


Therefore, the tracks of positrons in an absorber are the same for equal initial energies.

Similar Effects - DifferencesThe coulomb forces constitute the major mechanism of energy loss for both electrons and positrons and for heavy charged particles.

The impulse and energy transfer for particles of equal mass are about the same.

One big difference between e- and e+ is the annihilation radiation which could be produced when positrons are present in matter. The

resulting 0.511 MeV photons are very penetrating compared with the range of positrons and can lead to the deposition of energy far from the original positron track.

IV. Interaction of Gamma

Rays


These processes lead to a partial or complete transfer of the energy of gamma rays to electrons.

Three Major ProcessesInteraction of gamma rays with matter is essentially due to three processes:� Photoelectric Effect,� Compton Effect,� Pair Production.

In the photoelectric effect and in pair production, the photon disappears and all its energy is transferred to an electron (in the first one) and or the energy is converted to

"matter" creating a pair e+e-.

In Compton effect, part of the energy of the electron is transferred to an electron and the resulting photon has a larger wavelength than the incident photon


Review

Photoelectric Effect

Compton Effect

Pair Production


All Processes

Figure 20


Fig. 21 shows a transmission experiment of monoenergetic gamma-rays.

B. Gamma Rays Attenuation1. Attenuation Coefficients

Figure 21

5


Each of the three processes removes the gamma-ray photon from the beam either by absorption or by scattering away from the detector direction and can be characterized by a fixed probability of occurrence per unit path length in the absorber.

µµµµ = ττττ(photoelectric) + σσσσ (Compton) κκκκ(pair)

Linear Attenuation Coefficient

µµµµ is called the linear attenuation coefficient.

The number of transmitted photons is given by:

(((( )))) teItI µµµµ−−−−==== 0(20)

where

I0 = number of photons without an absorber

If t is un g/cm2 then n has for units cm2/g© Dr. Nidal M. Ershaidat - Nuclear Techniques - Chapter 2: Radiation Interactions 26

We also define the mean free path of gamma rays, λλλλ as the average distance traveled in the absorber before an interaction takes place. It's a statistical average given by:

Mean Free path

and Eq. 20 can be written as:

Typical values for λλλλ ranges from few mm to tens of cm in solids for common gamma-ray energies.

µµµµ========λλλλ

∫∫∫∫

∫∫∫∫

∞∞∞∞µµµµ−−−−

∞∞∞∞µµµµ−−−−

1

0

0

dxe

dxex

t

t

(21)

(((( )))) λλλλ−−−−==== teItI 0 (22)


This it is the same for water present ion liquid or vapor form.

It is practical to use the mass attenuation coefficient definedρρρρ

Mass Attenuation Coefficients

The linear attenuation coefficient varies with density of the absorber, even though the absorber material is the same.

For a given gamma-ray energy, the mass attenuation coefficient does not change with the state of the material!

ρρρρ

µµµµ====ηηηη (23)


The mass attenuation coefficient for a compound or a mixture of elements can be calculated using the following equation:

Case of Compounds

where wi is the weight fraction of element i in the compound or mixture.

∑∑∑∑

ρρρρ

µµµµ====

ρρρρ

µµµµ

i i

i

c

w (24)


Using the mass attenuation coefficient ρρρρt, the attenuation law of gamma rays takes the form:

2. Absorber Mass Thickness

The mass thickness ρρρρt (units g/cm2) is now the significant parameter that determines the attenuation of gamma rays

(25)(((( )))) (((( )))) teItI ρρρρρρρρµµµµ−−−−==== 0

Absorber thicknesses are measured using mass thickness because it is a more physical quantity.


The previous discussion and definitions suppose a so-called "good geometry" setup.

3. Buildup Factor

A good geometry (or narrow beam) setup is one in which the gamma rays are collimated to a narrow beam before striking the detector.

(26)(((( )))) (((( )))) (((( )))) teEtBItI ρρρρρρρρµµµµ−−−−γγγγ==== ,0

If there is no collimation used, the attenuation law is modified as:

B(t,Eγγγγ) is called the buildup factor and is introduced simply as a correction factor (it is unitless).

For a detector which responds to a broad range of gamma-ray energies, the buildup factor for thick slab absorbers tend to be equal to the thickness of the absorber measured in units of mean free path of the incident gamma rays.

6

IV. Interaction of

Neutrons


Neutrons being neutral (as γγγγ's) the cannot interact in matter by means of the coulomb force.Neutrons can also travel through many centimeters of matter without any type of interaction and thus can be totally invisible to a detector of common size!

A. General Properties

When a neutron interacts in matter it is with a nucleus of the absorbing material, i.e. through a nuclear interaction.

The neutron either vanishes and is replaced by one or more secondary radiations, or else its energy and direction are changed significantly.


The probability of interaction of neutrons change dramatically with energy.

The neutrons, from this point of view, are divided into two categories:

The cadmium cutoff energy

� Slow neutrons and� Fast neutrons

The limit between these categories is at about 0.5

eV which corresponds to the abrupt drop in

absorption cross section in cadmium.

This limit value is called the cadmium cutoff energy


B. Slow Neutron Interactions


C. Fast Neutron Interactions


Neutrons are characterized by the fact that the probability per unit length for any of their interaction mechanisms is a constant for neutrons of fixed energy.

D. Neutron Cross Sections

Cross-section (symbol σσσσ) per nucleus is used to express this probability for each type of interaction

Thus nuclear species have an elastic scattering

cross section (σσσσsc.), a radiative capture cross

section (σσσσr), etc …

7


When multiplying the cross section by the number of nuclei per unit volume involved in

an interaction (N) one gets the macroscopic cross section

Macroscopic Cross Section

ΣΣΣΣΣΣΣΣ = = NN σσσσσσσσΣΣΣΣ has the dimensions of inverse length. Thus it translates the probability per unit length for an interaction to occur described by the

"microscopic" cross section σσσσ.

A total macroscopic cross section ΣΣΣΣtotrepresents the probability per unit length that any interaction process can occur and we write:

ΣΣΣΣ = ΣΣΣΣscatter + ΣΣΣΣrad. capture + … (27)


ΣΣΣΣtot has the same significance for neutrons as the linear absorption coefficient for gamma rays.

Mean Free Path

By analogy with gamma rays, the mean free

path is defined by: λλλλ = 1/ΣΣΣΣtot

(28)(((( )))) ttoteItIρρρρΣΣΣΣ==== 0

In solid materials, λλλλ for slow neutrons may be of the order of a centimetera centimeter or less, whereas for fast neutrons, it is normally tens of tens of centimeterscentimeters.


In general neutrons are not narrowly

collimated, i.e. their detection falls in the category of "bad geometry" experiments type!

Good/Bad Geometry

Eq. 28 is no more valid for the calculation of attenuation. More sophisticated neutron transport calculation, involving simulations and computers, are used for this purpose.

Neutronics is the branch of nuclear physics which is concerned with everything related to neutrons and especially to the "calculation" of their interaction cross section with matter.


For neutrons with single energy or fixed v, the

product vΣΣΣΣ defines the interaction frequency (in s-1

or Hz) for the process for which ΣΣΣΣ is the macroscopic cross section.

Neutron Flux – Reaction Rate

Generalization for an energy-dependent flux ϕϕϕϕ(r,E)

and cross section ΣΣΣΣ(E) is given by:

The number of reactions per unit volume per time

unit or is given by: n(r) v ΣΣΣΣ, where n(r) is the

neutron number density at position r.

(29)( )Σϕ=⇒ rdensityratereaction

(30)(((( )))) (((( ))))dEEErdensityratereaction ∫∫∫∫∞∞∞∞

ΣΣΣΣϕϕϕϕ====0

,

The neutron flux (dimensions L-2T-1) ϕϕϕϕ(r) is thus

n(r)v and:


Next Lecture

Chapter 3

Counting Statistics

and Error Prediction

physics department yarmouk university chapter 2: radiation...

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