NERS555: Introduction to Radiological Physics and Radiation Dosimetry

Quiz 7/Exam 1/Assignment 4 (double-weight) Winter 2017

Revision: March 7, 2017 Alex Bielajew, 2927 Cooley, bielajew@umich.edu

This is a set of instructions, please read it carefully and completely. If any clarifications are needed, pleaseask questions in class.

This is a ”Special Event” for NERS 555, one that will reinforce the material covered up to the start of SpringBreak. Teaching through reinforcement works, in my experience. Also, I obsess about preparing studentsfor their future exams related to this subject matter. If the teaching sticks, as is my goal, it will benefit yougreatly in the future.

Today (March 6) we will have a double-weight Quiz, lasting no more than to the end of class time, 7:30 pm.You may turn it in earlier if you like. This Quiz is modeled on questions of the length and format that youwill see on future exams. Please bring at least 10 sheets of blank paper to class today.

This ”Special Event” will entail a 3-part process that is described below:

1) An in-class Quiz today. I am encouraging you NOT to study for this Quiz. It will be graded according toexam standards, and this grade will reflect what you would receive for such an exam if you had not studied.This grade will NOT count toward your course grade. It is for your information only. We shall try our bestto have this graded by March 8. You may pick up this graded Quiz from me directly during office hoursthat day (March 8) between 10 am and 4 pm. That would be a great time to ask questions about how yourin-class Quiz was graded, or any other course-related question.

2) At the end of class today, I will post this Quiz (identical version) online as take-home Exam 1. This willalso be graded according to examination standards. You should take this Exam after studying for it on yourown, and take it on your own within a contiguous 2 hour time frame of your choosing. It is closed-book andclosed-notes Exam. (Please consider these constraints as Honor Code requirements.) The grade you receivefor this WILL count as a double-weight Quiz and will reflect the grade you would receive if you had studiedfor the Exam very efficiently. The grade you receive, if higher than any of your previous Quizzes, will replaceup to two previous Quizzes. This portion is to be turned in before class on Monday March 13. We shall tryour best to have this graded by March 15. You may pick up this graded Exam 1 from me directly during officehours that day (March 15) between 10 am and 4 pm. That would be a great time to ask questions about howyour Exam 1 was graded, or any other course-related question. Item 2) is worth up to 4% of your course grade.

3) The final portion is to use this Quiz (identical version) as double-weight Assignment 4 (open-book andopen notes). There is no time constraint other than the due date: before class on March 20. I shall considerthis educational experiment a failure, if I have not raised your understanding of this material to the 100%level. I am counting on you to help me achieve this goal. Item 3) is worth 22.5% of your course grade.

I will be pleased to answer any questions at the start of class regarding this process or any other course-related material.

Solutions are given in blue italic font.

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1 Fundamental concepts

1. What is meant by radiation equilibrium? Why is this concept important in dosimetry?

The discussion of this is given in Lecture 2, Slides 45-48. There is a definition of translational andisotropic radiation equilibrium.

A region of space, V , is considered to be in a state of translational radiation equilibrium if that radiationfield is constant in time, and any measurement of fluence is invariant with respect to translations ofthe detector within V .

A region of space, V , is considered to be in a state of isotropic radiation equilibrium if that radiationfield is constant in time, and any measurement of fluence is invariant with respect to translations ofthe detector from within V and invariant with respect to rotations of the detector in V .

This is important to dosimetry because it means that an ideal detector’s measurement of fluence isindependent of its shape and orientation, in a region of radiation equilibrium.

2. Give two real-life examples where radiation is nearly in a state of equilibrium.

Three examples are given in Lecture 2 Slide 48.

An isotropic source may be produced at the center of a very large sphere with a uniform surface source,or uniform volume source would produce an isotopic radiation equilibrium at its center.

A monodirectional source may be produced by a point source moved a very long distance away from thedetector.

A uniform source with angular dependence may be produced by a very large plane source (of someuniform thickness with a source distributed through its volume. Self-shielding will cause an angulardependence in this case.

3. Without equations, describe the major result of Fano’s theorem.

This is stated in Lecture 2 Slide 52.

If the macroscopic cross sections within a region of space and its (internal) sources are proportional todensity, the fluence is independent of how the density changes.

4. Discuss the important consequences of the combination of Fano’s theorem and radiation equilibrium.

This is stated in Lecture 2 Slide 52.

The implications are that a fluence detector’s composition and density changes do not change a mea-surement of fluence, as long as the medium and the detector’s atomic composition are closely matched.

2

2 Basic interaction physics

The relevant material for this section is scattered through Lecture 3, Slides 14–93.

1. Ignoring photonuclear processes, what are the 3 most important photon interactions to the generalarea of medical physics?

See Slide 14: 1) pair production in the nuclear field, 2) incoherent (Compton) interaction, 3) photo-electric interaction.

In what energy range is each dominant?

See Slide 28: In low-Z materials, photoelectric dominates below about 20 keV, pair dominated aboveabout 30 MeV, and Compton dominates in between.

What is the atomic number dependence of each?

Photoelectric: Z4 (Slide 25), Compton Z (Slide 20), and pair Z2 (Slide 17)

2. Ignoring electronuclear processes, what are the 2 most important electron interactions to the generalarea of medical physics?

See Slide 33: 1) Møller scattering, 2) bremsstrahlung emission. A good case may be made for Rutherfordscattering as well.

In what energy range is each dominant?

For low-Z materials, bremsstrahlung dominates starting at about 100 MeV and Møller below. See Slide81. Rutherford is active at all energies, though the effects more dramatic at lower energies.

What is the atomic number dependence of each?

Z for Møller, Z2 for brem, and Rutherford.

3. Ignoring electronuclear processes, what are the 3 most important positron interactions to the generalarea of medical physics?

See Slide 33: 1) Bhabha scattering, 2) bremsstrahlung emission, 3) annihilation. A good case may bemade for Rutherford scattering as well.

In what energy range is each dominant?

For low-Z materials, bremsstrahlung dominates starting at about 100 MeV and Bhabha below. See Slide81. Annihilation dominates when the positron loses almost all its energy and is at rest. Rutherford isactive at all energies, though the effects more dramatic at lower energies.

What is the atomic number dependence of each?

Z for Bhabha and annihilation, Z2 for brem and Rutherford.

4. A 50 MeV photon enters a thick Pb slab. Draw a representative picture of the entire life of this photonand all of its daughter particles. Make sure that you show all the interactions described above.

The easy way to do this is refer to the picture already given,

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5. Repeat the previous question but with a 1 MeV photon. What processes can not occur?

Pair production can not occur. Since there are no positrons, there is also no Bhabha or Annihilation.

6. Consider the following diagram. The interaction points are labeled by uppercase letters. For eachinteraction type, name the interaction process (e.g. Compton) and give the atomic number dependentsof each cross section (e.g. Z/atom). What is the approximate energy dependence of each [(e.g. E−1

γ ].

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Interaction name or type Z dependence (/atom) Energy dependence

A pair Z2 logEγ

B Compton Z 1/Eγ

C Photoelectric Z3 1/E3γ

D Coherent Z2 none

E bremsstrahlung Z2 logEe±

F Møller Z 1/Ee−

G Bhabha Z 1/Ee+

H Rutherford Z2 not given in notes

I annihilation Z none

7. Describe, in words, what the Compton interaction is.

Compton scattering is the name given to the deflection of a photon caused by a free (at rest, unbound)electron, producing a lower-energy scattered photon, and a recoil electron.

Provide a drawing illustrating it.

5

8. For the Compton interaction, show that the energy of the incoming photon, Eγ , and the energy of thescattered photon, E′

γ , are related to the scattering angle of the scattered photon, θ, by the relation:

1

E′

γ

−1

Eγ=

1

mec2(1− cos θ)

where, mec2 is the rest mass energy of the electron.

Conservation of energy and momentum implies:

E +mc2 = E1 +mc2γ

E

cn̂ =

E1

cn̂1 +m~veγ ,

Making the previous equations dimensionless, and some reorganization results in:

α+ 1− α1 = γ

αn̂− α1n̂1 = ~βγ ,

where the α’s are photon energies divided my the electron rest mass energy. Squaring both equations andsubtracting them eliminated the electron from the equations, by exploiting the relationship γ2(1−β2) =1.

(α+ 1− α1r − |α+ 1− α1|2 = γ2(1− β2)

After cancellation and organization and dividing by αα1:

1

α1

−1

α= 1− cos θ ,

that relates the scattered photon energy to its scattering angle.

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3 Fluence, energy fluence, and Dose

1. What is the ICRU definition of fluence?

From Lecture 2, Slide 4: The ICRU definition of fluence is:

Φ = lim∆a→0

∆N

∆a

wheresymbol meaning∆a the cross sectional area of a small sphere∆N the number of particles that enter the sphere

2. What is the definition of planar fluence?

From Lecture 2, Slide 41: Planar fluence is the number of particles that strike a surface irrespective ofdirection.

3. What does Chilton’s theorem on fluence state?

From Lecture 2, Slide 9: Chilton’s expression for fluence is:

where

symbol meaning∆V the volume of arbitrary shapeN the number of particles that pass through ∆V∆li the chord length of the i’th particle track through ∆V

4. A fluence, differential in energy, E, has the form:

ϕ(E) =A

E3Emin ≤ E ≤ Emax

and zero everywhere else. A is a constant.

(a) What is the total fluence?

Φ =

∫ Emax

Emin

dEA

E3=A

2

(

1

E2min

−1

E2max

)

(b) What is the energy fluence, differential in energy?

ψ(E) = Eϕ(E) =A

E2Emin ≤ E ≤ Emax

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(c) What is the total energy fluence?

Ψ =

∫ Emax

Emin

dEA

E2= A

(

1

Emin

−1

Emax

)

5. This question has been eliminated.

6. This question has been eliminated.

7. This question has been eliminated.

4 Fluence and planar fluence detectors

See the attached for the solution.

Imagine a radiation field in vacuum with particles going in one direction only. The particle fluence is Φ0, asmeasured by a spherical fluence detector with radius a. Now, imagine that you have a disk-shaped fluencedetector. It has radius a and depth d.

1. Using directly Chilton’s definition of fluence,

Φ =

∑Ni=1

∆li∆V

show that the detector measures a fluence Φ0, when its symmetry axis (axis of rotational symmetry)is aligned with the direction of particles in the field. (θ = 0). You may assume that there are enoughparticles in the field to allow you to evaluate the sum above as an integral.

2. Using directly Chilton’s definition of fluence, show that the detector measures a fluence Φ0, when itssymmetry axis (axis of rotational symmetry) is perpendicular to the direction of particles in the field.

(θ = π/2). Hint:∫ π/2

−π/2 dφ cos2 φ = π/2.

3. Without calculation, what would you expect if the fluence detector has an angle θ0 with respect to thefield direction?

Now, imagine that a detector is an ideal planar fluence detector.

What does this detector measure for

1. θ = 0?

2. θ = π/2?

3. θ at an arbitrary angle?

5 Kerma

This question has been eliminated.

8

6 Finite detectors in non-equilibrium fields

See the attached for the solution.

The fluence in vacuum due to a point source that emits N particles is:

Φ(r) =N

4πr2

1. Imagine that an ideal disk-shaped fluence detector with finite radius a and negligible thickness, iscentered a distance r from the source and is perpendicular to it. It measures a fluence ΦM(r). Showthat,

ΦM(r) = Φ(r)r2

a2log

(

1 +a2

r2

)

.

Hint: It’s much easier to use the Chilton definition of fluence.

2. Is the fluence continuous and finite everywhere along z? Prove your result.

3. Show that

limz→∞

φ(z) −→N0

4πz2[1 + k(a/z)2 + ...]

Determine the constant k.Hint:

limǫ→0

log(1 + ǫ) ≈ ǫ− ǫ2/2

when ǫ≪ 1.

What does this mean?

4. Adapt your result for the first part to account for attenuation in the air between the radiating disk andthe measurement point. Assume that the attenuation is a constant characterized by the attenuationconstant µ. Note, just state the resulting integral and do not attempt to solve it. Assuming µa≪ 1 andµz ≪ 1, attempt a first order correction to the first parts of this solution that accounts for attenuation.

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