deep blue: examining cerenkov radiation through non-traditional media

Deep Blue: Examining Cerenkov Radiation Through Non-traditional Media

A Thesis

Presented to

The Division of Mathematics and Natural Sciences

Reed College

In Partial Fulfillment

of the Requirements for the Degree

Bachelor of Arts

Ian Flower

May 2013

Approved for the Division(Physics)

John Essick

Acknowledgements

I would like to thank my advisor, John Essick for his invaluable guidance in completingthis Thesis. You have been the best advisor I could hope for, and I truly appreciatethe effort you put into getting this project completed.

Mom and Dad, thank you for supporting me all the way. I wouldn’t be who I amtoday with out you both, and I love you dearly. Alison and Liz, thank you for all ofthe wisdom and maturity you have shown me.

To all of the close friends I have had here at Reed, thank you so much! Far toomany of you have meant far too much to me to start listing it all out, but you’vemeant the world to me. Without your support, I would never have made it this far.

Specifically, thank you to J-dorm. Both of my years living with you have beenfantastic, and I couldn’t imagine a better place for me. You have made me so proudover these years, and I am excited about what you can accomplish as you continueyour Reed careers.

Thank you also to Greg Eibel, Melinda Krahenbuhl, and Reuven Lazarus. With-out your help, this experiment would have never taken form.

Table of Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Chapter 1: Cerenkov Radiation . . . . . . . . . . . . . . . . . . . . . . . 31.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.1 Speed Threshold . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.2 Emission as a Cone . . . . . . . . . . . . . . . . . . . . . . . . 41.1.3 The Frank-Tamm Equation . . . . . . . . . . . . . . . . . . . 4

1.2 A Field Oscillator Approach . . . . . . . . . . . . . . . . . . . . . . . 51.3 Predicting a Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Chapter 2: Color Matching . . . . . . . . . . . . . . . . . . . . . . . . . . 152.1 Color Matching Functions . . . . . . . . . . . . . . . . . . . . . . . . 15

2.1.1 Color Vision . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.1.2 RGB Color Matching Functions . . . . . . . . . . . . . . . . . 172.1.3 XYZ Color Matching Functions . . . . . . . . . . . . . . . . . 19

2.2 Color Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.2.1 The xy Color Space . . . . . . . . . . . . . . . . . . . . . . . . 202.2.2 The CIE RGB Color Space . . . . . . . . . . . . . . . . . . . . 222.2.3 Uniform Chromaticity and Hue Angle . . . . . . . . . . . . . . 22

Chapter 3: Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.1 Experimental Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.1.1 Reed Research Reactor . . . . . . . . . . . . . . . . . . . . . . 253.1.2 Collection Tool . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2 Method of Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.2.1 From Power Spectrum to Color . . . . . . . . . . . . . . . . . 263.2.2 From Data to Power Spectrum . . . . . . . . . . . . . . . . . 273.2.3 Prediction of Spectrum . . . . . . . . . . . . . . . . . . . . . . 30

Chapter 4: Data & Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 334.1 Predicted Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.1.1 The Effect of β . . . . . . . . . . . . . . . . . . . . . . . . . . 334.1.2 The Effect of Index of Refraction . . . . . . . . . . . . . . . . 344.1.3 The Effect of Absorption . . . . . . . . . . . . . . . . . . . . . 35

4.2 Observed Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.3 Analysis of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.3.1 From Water . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.3.2 From Cinnamaldehyde . . . . . . . . . . . . . . . . . . . . . . 384.3.3 From Corn Oil . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Chapter 5: Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

Appendix A: ROC Approval . . . . . . . . . . . . . . . . . . . . . . . . . 43

Appendix B: XYZ Color Matching Functions . . . . . . . . . . . . . . . 45

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

List of Figures

1.1 Shock waves in Cerenkov Radiation and water . . . . . . . . . . . . . 41.2 Polarization Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.3 Limit of sin(ωλuλt)/(ωλuλ) . . . . . . . . . . . . . . . . . . . . . . . . 131.4 Integrand Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1 Overall sensitivity curve adapted from Hunt (1991). . . . . . . . . . . 162.2 Cone sensitivity curves adapted from Hunt (1991). . . . . . . . . . . . 162.3 RGB Color Matching Curves adapted from Hunt (1991). . . . . . . . 182.4 The 1931 Standard Observer XYZ Color Matching Functions. . . . . 192.5 The 1931 Standard Observer x,y Chromaticity Diagram. . . . . . . . 202.6 The 1931 Standard Observer x,y Chromaticity Diagram with the CIE

RGB color space shown. . . . . . . . . . . . . . . . . . . . . . . . . . 222.7 The 1976 Uniform Chromaticity Scale Diagram for u′ and v′. . . . . . 23

3.1 Diagram of the experimental set up used. . . . . . . . . . . . . . . . . 263.2 Converting a spectrum to XYZ values . . . . . . . . . . . . . . . . . . 273.3 Calibration Source Spectrum . . . . . . . . . . . . . . . . . . . . . . . 283.4 Efficiency Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.5 Corn Oil Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.6 Diagram of Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . 32

4.1 The Effect of β on Color . . . . . . . . . . . . . . . . . . . . . . . . . 334.2 The Effect of Refractive Index on Color . . . . . . . . . . . . . . . . . 344.3 The Effect of Absorption on Color . . . . . . . . . . . . . . . . . . . . 354.4 Raw Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.5 Collected Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.6 Water Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.7 Comparison of the perceived hues of cinnamaldehyde and water . . . 384.8 Comparison of water and cinnamaldehyde spectra . . . . . . . . . . . 384.9 Corn Oil Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.10 Artists Rendering of Reactor Core in Corn Oil . . . . . . . . . . . . . 39

B.1 The 1931 Standard Observer XYZ Color Matching Functions. . . . . 45

Abstract

In this experiment, we examine possible colors for Cerenkov Radiation. Utilizing theReed Research Reactor, we inserted an aluminum chamber into the γ rich environ-ment near the reactor core and filled it with water, corn oil, and cinnemaldehyde.Spectra taken from the chamber were converted to colors using the 1931 CIE Stan-dard Observer model. The effect of refractive index, particle speed, and absorption onperceived color are examined. These spectra were also compared to those predicted bythe Frank-Tamm formula. It is determined that the most significant changes in colorwill be the result of absorption, though index of refraction can affect both perceivedbrightness and hue.

Introduction

“And in the somber shed where, in the absence of cupboards, the precious particles intheir tiny glass receivers on tables or on shelves nailed to the wall, their phosphorescentbluish outlines gleamed, suspended in the night.”

- Eve Curie (Curie, 1940)

“Wow.”

- Everyone ever

There aren’t a lot of words that can describe a person’s first experience withCerenkov Radiation. On tours, I’ve heard “stunning”, “beautiful”, “magical”, “amaz-ing”, and really it’s all of these. (Though, sometimes the tour groups are left withoutwords and simply gasp.) The glow is captivating to any observer, and when I firstsaw it my Freshman year, it blue my mind. The phosphorescent aura captivated myimagination, and was an instant interest of mine. I can’t guarantee I would still beworking at the Reactor if it weren’t for that glow.

When we give tours, we always have to explain what the glow is, and one facthas always featured prominently in the tours I have given: that replacing the waterwith another substance would change the color’s glow. Excited tour guides wouldsometimes make outlandish claims that it could be purple, or red, or hot pink. WhileI never really believed that a red glow was possible, I always wanted to know whatwas. When it came time to choose a thesis topic, the decision was easy.

Chapter 1

Cerenkov Radiation

1.1 Overview

Accounts and observations of Cerenkov Radiation exist at least as far back as thework of Marie and Pierre Curie. The eerie blue glow that surrounds any submergedradioactive substance has long been a source of interest for curious minds. The firstreal study of its properties, though, was when Mallet observed the spectrum of lightproduced by γ sources in distilled water and carbon disulphide. (Mallet, 1926) Hiswork showed that the effect was very different from fluorescence, the contemporaryhot topic. Separately, Cerenkov began studying this radiation. He showed one couldproduce the light with β rays, and noted a variety of the effect’s other properties.(Cerenkov, 1934) When Cerenkov’s colleagues, Frank and Tamm, proposed the firsttheory of this radiation, he experimentally verified their results. (Frank & Tamm,1937)

These researchers concluded that Cerenkov Radiation is produced by chargedparticles traveling at high speeds through a variety of mediums. Typically, though,high energy charged particles are hard to find. Most observed Cerenkov Radiation,it turns out, is instead a secondary effect of γ radiation. These high energy photons,produced in radioactive decay, are absorbed by electrons in the material, which thenescape the atom and travel at high speeds. This secondary ionization provides mostof the glow surrounding radioactive sources and nuclear reactors.

1.1.1 Speed Threshold

Not just any charged particle can emit Cerenkov Radiation. This seems intuitivelyobvious: charged particles are around us all the time, and yet we only see CerenkovRadiation in exceptional circumstances, like near reactor cores. The predominantreason for this fact is a speed threshold for the charged particle, which is set by

vT =c

nω(1.1)

where vT is the threshold speed of the particle, c is the speed of light in a vacuumand n is the index of refraction of the medium at hand at a particular frequency ω.

4 Chapter 1. Cerenkov Radiation

In other words, the particle must be traveling faster than the phase speed of light inits medium in order to produce radiation. This is one of the reasons it is frequentlylikened to the equivalent of a sonic boom for light.

1.1.2 Emission as a Cone

The other reason Cerenkov Radiation is often likened to a sonic boom is that thelight produced by the fast-moving charged particle with velocity v is generated alongthe walls of a cone. Specifically, the radiation is emitted at an angle that satisfies

cos θ =c

vnω=vTv. (1.2)

Here, θ is the angle between the direction of propagation and the direction of radi-ation. This relation is frequently referred to as the Cerenkov Relation. Note thatthis relation carries with it the speed threshold. If the velocity of the particle is lessthan vT then the right side of Equation 1.2 is greater than one, and is not physicallyrealizable.

Figure 1.1: Comparison of shock waves in water and in Cerenkov Radiation. Adaptedfrom Jelley (1958).

To better visualize this effect, imagine a boat traveling in a flat lake. As it moves,it perturbs the surface, and those perturbations spread out in concentric circles. Aslong as the boat travels slower than the velocity of these waves, the rings never cross,and never form a crest. However, if the boat surpasses the wave speed, a cone-shapedcrest forms behind it, like shown in Figure 1.1. The Cerenkov cone could very wellbe considered the electromagnetic equivalent of this cone, in three dimensions.

1.1.3 The Frank-Tamm Equation

When the first full theory of Cerenkov Radiation was put forth by Frank and Tammin 1937, it included a prediction for the power P emitted by a single charged particle:

P =e2vµ0

4π

∫v>vT

[(1− c2

v2n2ω

)ω

]dω (1.3)

1.2. A Field Oscillator Approach 5

This equation, known as the Frank-Tamm Equation, gives total the power emitted inradiation by a particle of charge e (the electron charge) traveling at speed v througha medium with index refraction nω. The shape of the spectrum produced by a singleparticle, then, is determined by the term in square brackets. The index of refractionterm accounts for the dependence of the spectrum on the material, and the ω virtuallyguarantees that more energy will be emitted at higher energies than at lower energies.Note that if this equation were to integrate over all frequencies, it would be catas-trophic. The particle would be emitting an infinite amount of energy every second.Fortunately, it is limited to the frequencies for which the speed threshold conditionholds, which is to say the frequencies at which nω are appropriate (nω > c/v).

1.2 A Field Oscillator Approach

While the original Frank-Tamm approach of directly solving Maxwell’s Equations isstill valid, some modern approaches are more comprehensible, yield the same results,and offer better insight into the mechanism of Cerenkov Radiation. In this section, Iwill walk through the Field Oscillator Approach, which relies on Maxwell’s equations,and an expansion of the vector potential of a moving charged particle in terms ofcavity modes. This derivation closely follows the one in Razpet & Likar (2010). Asimilar approach is taken for dipole radiation in Razpet & Likar (2009).

The approach is based on the expression for energy stored in the electric andmagnetic fields:

H =1

2

∫ (ε0 ~E

2 +1

µ0

~B2

)dV, (1.4)

where this integral is evaluated over a large cubical volume V.In this expression, we will replace ~E and ~B with the vector and scalar potentials

~A and Φ.

~E = −∂~A

∂t− ~∇Φ

~B = ~∇× ~A (1.5)

Also, we will work in the Coulomb gauge where ~∇ · ~A = 0. This then allows us tosplit the electric field ~E into a longitudinal, ~El, and a transverse component ~Et where

~El = −~∇Φ , ~Et = −∂~A

∂t. (1.6)

noting

~∇× ~El = ~∇×(−~∇Φ

)= −~∇× ~∇Φ = 0,

~∇ · ~Et = ~∇ ·

(−∂

~A

∂t

)= − ∂

∂t

(~∇ · ~A

)= 0.


Then, we can expand Equation 1.4.

H =1

2

∫ (ε0 ~E

2 +1

µ0

~B2

)dV

=1

2

∫ (ε0( ~Et + ~El)

2 +1

µ0

~B2

)dV

=1

2ε0

∫~E2t dV + ε0

∫~El · ~EtdV +

1

2ε0

∫~E2l dV +

1

2µ0

∫~B2dV (1.7)

We note that the second integral in Equation 1.7 vanishes as follows:∫~El · ~EtdV =

∫ (−~∇Φ

)·

(−∂

~A

∂t

)dV =

∂

∂t

∫~∇ ·(

Φ ~A)

dV

where we have used the fact that ~∇ · ~A = 0. From the divergence theorem,∫~∇ ·(

Φ ~A)

dV =

∮ (Φ ~A)· d~s

where d~s is a surface element on the large cubical volume V. Given that the radiation

fields we are considering vanish at large distances,∮ (

Φ ~A)· d~s = 0.

Additionally, the third integral in Equation 1.7 is not relevant to our work. Start-ing with Gauss’ law and using our expression for ~E in the Coulomb gauge,

ρ

ε= ~∇ · ~E = ~∇ ·

(−∂

~A

∂t− ~∇Φ

)= − ∂

∂t

(~∇ · ~A

)− ~∇2Φ = −~∇2Φ.

It is well known that, for a given ρ (~r, t), this expression for Φ yields the instan-taneous Coulomb potential. Since the instantaneous potential does not contribute toradiation, we can neglect the integral

∫~E2l dV . Without these two terms, Equation 1.7

can be rewritten as

H =ε02

∫~E2t dV +

1

2µ0

∫~B2dV. (1.8)

The next step, then, is to find ~A, which we can do by expanding it as a Fourierseries. The components of this expansion represent standing waves in an imaginedcubical cavity of volume V and side length L,

~A (~r, t) =1√V ε0

∑λ,i

qλi (t) ~Aλi (~r) . (1.9)

where λ represents the cavity mode for the Fourier series term. In order to obeyour chosen boundary conditions, this expansion includes two sets of plane wave basisfunctions represented by i = 1, 2.

~Aλ1 =

√2

nλ~eλ cos

(~kλ · ~r

), ~Aλ2 =

√2

nλ~eλ sin

(~kλ · ~r

)(1.10)


with the vectors ~eλ and ~kλ being the polarization and wave vectors, respectively. Forthese plane waves, we will take periodic boundary conditions, that is A(0) = A(L).This means the end points of these waves are free to move, with the quantizationrequirement for kλ given by:

~kλ =

(2π

Lnx,

2π

Lny,

2π

Lnz

)(1.11)

where nx, ny, nz are positive integers.Using this expansion, we can actually rewrite Equation 1.8. Looking at the electric

field portion first,

~Et = −∂~A

∂t

= − 1√V ε0

∑λ,i

qλi ~Aλi

~E2t =

1

V ε0

∑λ,i

∑µ,j

qλiqµj( ~Aλi · ~Aµj) (1.12)

Where the orthogonality condition for our basis functions is∫V

~Aλi · ~AµjdV = δλµδijV. (1.13)

If we plug Equation 1.12 and 1.13 into the first integral of Equation 1.8, it willjust pick out the terms where µ = λ and i = j, yielding a factor of V , so we get∫

V

~E2t =

1

ε0

∑λ,i

q2λi. (1.14)

Meanwhile, a similar treatment can be applied to the magnetic field.

~B = ~∇× ~A

=1√V ε0

~∇×∑λ,i

qλi ~Aλi

=1√V ε0

∑λ

(qλ1 ~Aλ1 − qλ2 ~Aλ2)× ~kλ

~B2 =1

V ε0

[∑λ

qλ1

(~Aλ1 × ~kλ

)−∑λ

qλ2

(~Aλ2 × ~kλ

)]2=

1

V ε0

∑λ

[q2λ1

(~Aλ1 × ~kλ

)2− 2qλ1qλ2

(~Aλ1 × ~kλ

)·(~Aλ2 × ~kλ

)+ q2λ2

(~Aλ2 × ~kλ

)2].


where

(a× b) · (c× d) = (a · c) (b · d)− (a · d) (b · c) .

Employing this, the summed terms become

(Aλ1 × kλ) · (Aλ1 × kλ) = A2λ1k

2λ − (Aλ1 · kλ)2 = A2

λ1k2λ,

(Aλ2 × kλ) · (Aλ2 × kλ) = A2λ2k

2λ − (Aλ2 · kλ)2 = A2

λ2k2λ,

(Aλ1 × kλ) · (Aλ2 × kλ) = (Aλ1 · Aλ2) k2λ − (Aλ1 · kλ) (Aλ2 · kλ) = (Aλ1 · Aλ2) k2λ.

The square of the magnetic field then becomes

~B2 =1

V ε0

∑λ

[q2λ1A

2λ1k

2λ + (Aλ1 · Aλ2) k2λ + q2λ2A

2λ2k

2λ

].

Then, plugging this into the second integral of Equation 1.8 and applying theorthogonality condition, Equation 1.13, the cross term disappears, giving

∫V

~B2dV =1

ε0

∑λ,i

q2λik2λ. (1.15)

Combining these results, we can rewrite Equation 1.8.

H =ε02

∫~E2t dV +

1

2µ0

∫~B2dV

=1

2

∑λ,i

q2λi +1

2ε0µ0

∑λ,i

q2λik2λ

=1

2

∑λ,i

(q2λi + c2q2λik2λ)

=1

2

∑λ,i

(q2λi + ω2λq

2λi), (1.16)

where the speed of light in vacuum is c = 1√ε0µ0

.

Clearly, then, if we are able to obtain an expression for the qλi, we will be able tocompute the energy stored in the field, and thus the power irradiated.

To obtain this expression, we start with the familiar Maxwell Equation

~∇× ~B = µ0~J +

1

c2∂ ~E

∂t. (1.17)

Using the definitions of ~E and ~B from Equation 1.5, this becomes

~∇×(~∇× ~A

)= µ0

~J − 1

c2∂2 ~A

∂t2− 1

c2~∇∂Φ

∂t.


Noting, for a charge distribution ρ moving at velocity ~v,

J = ρ~v

and

~∇×(~∇× ~A

)= −~∇2 ~A+ ~∇

(~∇ · ~A

)= −~∇2 ~A

our expression becomes

~∇2 ~A− 1

c2∂2 ~A

∂t2= −µ0ρ~v +

1

c2~∇∂Φ

∂t.

We then use Equation 1.10 to expand this equation.

1√ε0V

∑λ,i

qλi~∇2 ~Aλi −1

c21√ε0V

∑λ,i

qλi ~Aλi = −µ0ρ~v +1

c2~∇∂Φ

∂t. (1.18)

We can simplify this a little bit by evaluating ~∇2 ~Aλi using the identities

~∇ · (u~e) = ~e · ~∇u , ~∇ (u~e) = ~∇u× ~e

as follows.

~∇2 ~Aλ1 = ~∇2(√

2~eλ cos(~kλ · ~r

))= ~∇

(~∇ ·(√

2~eλ cos(~kλ · ~r

)))− ~∇2

(√2~eλ cos

(~kλ · ~r

))=√

2 cos(~kλ · ~r

)((~kλ · ~eλ

)~kλ − k2λ~eλ

)=√

2 cos(~kλ · ~r

)k2λ~eλ

= −k2λ ~Aλ1.

The same is true for i = 2, reducing Equation 1.18 to

1√ε0V

∑λ,i

qλik2λ~Aλi +

1

c21√ε0V

∑λ,i

qλi ~Aλi = µ0ρ~v −1

c2~∇∂Φ

∂t.

The presence of ~Aλi terms under sums on the left side of this prompts a use of theorthogonality condition, Equation 1.13. To do this, we multiply both sides by ~Aµjand integrate over the volume V to obtain

1√ε0V

qµjk2µV +

1

c21√ε0V

qµjV =

∫V

(µ0ρ~v · ~Aµj −

1

c2~Aµj ~∇

∂Φ

∂t

)dV.

The second term in the integral on the right is equal to 0, which leaves just


1√ε0V

qµjk2µV +

1

c21√ε0V

qµjV =

∫V

µ0ρ~v · ~AµjdV.

To clear this up even further, we can multiply everything by c2 ε0V

, exploiting therelations ε0 = 1

µ0c2and c2k2µ = ω2

µ and then rewrite our µ, j as λ, i.

qλiω2λ + qλi =

1√V ε0

∫V

ρ~v · ~AλidV. (1.19)

In the case of Cerenkov radiation, ρe represents a moving point charge, so we canwrite it out again using

ρe(~r) = e0δ3(~r − ~re(t)),

where δ3 is the three dimensional Dirac delta. If we also plug in our expression for~Aλi coming from Equation 1.10, Equation 1.19 becomes

qλ1 + ω2λqλ1 =

√2

nλ√V ε0

e0(~v · ~eλ) cos(~kλ · ~re(t))

qλ2 + ω2λqλ2 =

√2

nλ√V ε0

e0(~v · ~eλ) sin(~kλ · ~re(t))(1.20)

Figure 1.2: Polarization vector components for the field oscillator approach. (Razpet& Likar, 2010)

At this point, we need to take a closer look at the polarization vector, ~eλ appearingin these equations. In order to simplify the expression ~v ·~eλ, we can break it apart intoperpendicular components. We define the particle’s motion to lie along the z -axis,so that Figure 1.2 shows the components we choose. The first component lies along~eλa, which is perpendicular to ~kλ and lies in the plane formed by ~v and ~kλ. The other


component lies along ~eλb, which is still perpendicular to ~kλ, but is also perpendicularto ~v. Since this component is perpendicular to the particle’s motion, its contributionto ~v ·~eλ vanishes, and we are left with only the contribution from ~eλa. Thus, we find:

~v · ~eλ = −v sin γ (1.21)

which turns Equation 1.20 into

qλ1 + ω2λqλ1 =

√2

nλ√V ε0

e0(−v sin γ) cos(~kλ · ~re(t)),

qλ2 + ω2λqλ2 =

√2

nλ√V ε0

e0(−v sin γ) sin(~kλ · ~re(t)).(1.22)

These equations describe driven harmonic oscillators. The equations look prettymessy, but for the most part the terms on the right sides of these equations areconstants. The mechanics all lie in the terms

~kλ · ~re(t) =ωλnλv cos γ

ct

This means the right-hand sides of Equation 1.22 represent driven oscillators withdriving frequency

ωd =ωλnλv cos γ

c. (1.23)

We suddenly have pretty simple solutions for qλ1,2 so long as we’re willing to putup with some ugly constants. In fact, the exact solutions for Equation 1.22 are

qλ1 = −αcos(ωλuλt)− cos(ωλt)

ω2λ(1− u2λ)

qλ2 = +αsin(ωλuλt)− sin(ωλt)

ω2λ(1− u2λ)

(1.24)

Where we introduce two factors, uλ and α to clean things up.

uλ ≡ωdωλ

=nλv cos γ

c, α ≡

√2e0v sin γ

nλ√V ε0

(1.25)

At this point, we can take a brief pause. Soon we will use these expressions forqλ1,2 to finally solve for the power emitted by Cerenkov radiation, but first we canaddress these important results.

What we’ve found is that the charged particle, as it’s moving, will excite certainmodes of the vector potential. The exact mode excited is all carried in the definitionof uλ above. A given particle traveling at speed v through a medium of index ofrefraction nλ will drive the resonant modes that correspond to light at angle γ. Thatis, at speed v, the associated driving frequency ωd will resonantly excite the cavity


mode ωλ = ωd. Then, with uλ = ωdωλ

= 1, the speed threshold comes from the rest ofEquation 1.25.

cos γ =c

nλv= 1.

since cos γ can be at most 1, v can be at minimum cnλ

.Thus, just from the definition of qλ1,2, we arrive at two of the important charac-

teristics of Cerenkov radiation.Inserting Equation 1.24 into Equation 1.16, we get

H =e20v

2

ε0V

∑λ

sin2 γ

n2λ

((1 + u2λ)(1− cos((1− uλ)ωλt))

ω2λ(1− u2λ)2

+sin(ωλt) sin(ωλuλt)

ω2λ(1 + uλ)2

)(1.26)

Where we can convert this sum over the modes into an integral over dNλ. Thisquantity represents the number of modes per frequency interval dωλ per solid angledΩ = 2πd(cos γ). Integrated over all of the solid angles and frequencies, this integralcovers all of the possible cavity modes. The number of modes per solid angle andfrequency interval is

dNλ =n3λω

2λV dωλdΩ

(2πc)3.

However, instead of integrating over dΩ, we can integrate over duλ using

dΩ = 2πd(cos γ)

= 2πd

(uλc

nλv

)=

2πc

nλvduλ

dNλ =n2λω

2λV dωλduλ(2πc)2

.

Rewriting Equation 1.26 as an integral over dNλ as defined above, we get

H =e20vµ0

4π2

∫ ∞0

dωλ

∫ nλv/c

0

sin2 γ

[(1 + u2λ)

(1 + uλ)2(1− cos((1− uλ)ωλt))

(1− uλ)2

]+ sin2 γ

[sin(ωλt) sin(ωλuλt)

(1 + uλ)2

]duλ (1.27)

For the second term in the integrand of Equation 1.27, we consider what wouldhappen at large t. As shown in Figure 1.3, the term sin(ωλuλt)/(ωλuλ) approachesa δ function centered around ωλuλ = 0 of area π. Using this information, we canrewrite that term of Equation 1.27 as


Figure 1.3: Visual depiction of the limit of sin(ωλuλt)/(ωλuλ) for large t. The plotsshow this value as a function of uλ for increasing values of t. As t increases, the plotbecomes more and more peaked approaching a δ function. The area under each curveis π.

∫ ∞0

sin2 γ

(1 + uλ)2sin(ωλt)ωλuλπδ(ωλuλ)duλ

Which is just equal to zero. Without this term, we can rewrite Equation 1.27 as

H =e20vµ0

4π2

∫ ∞0

dωλ

∫ nλv/c

0

sin2 γ

((1 + u2λ)

(1 + uλ)2(1− cos((1− uλ)ωλt))

(1− uλ)2

)duλ (1.28)

Figure 1.4: Visual depiction of the limit shown in Equation 1.29. The plots show theterm in the limit as a function of uλ for increasing values of t. As t increases, theplot becomes more and more peaked, approaching a δ function. The area under eachcurve is ωλπt.

As shown in Figure 1.4, the integrand simplifies at large t because

limt→∞

(1− cos((1− uλ)ωλt))(1− uλ)2

= ωλπtδ(uλ − 1). (1.29)

The delta function will then pick out uλ = 1, but we also need to deal with thesine term. By the definition of uλ, we have


cos γ =cuλvnλ

,

sin2 γ = 1− c2u2λv2n2

λ

. (1.30)

Using this expression, and plugging in the delta function, we reduce Equation 1.28to

H =e20vµ0

4π2

∫ ∞0

dωλ

∫ nλv/c

0

(1− c2u2λ

v2n2λ

)((1 + u2λ)

(1 + uλ)2ωλπtδ(uλ − 1)

)duλ

H =e20vµ0

4π

∫ ∞0

(1− c2

v2n2λ

)tωλdωλ. (1.31)

Note that the integral ωλ only covers frequencies where nλv/c > 1. In order toget the radiated power, we need only take the time derivative of this.

P =dH

dt=e20vµ0

4π

∫ ∞0

(1− c2

v2n2λ

)ωλdωλ (1.32)

which is exactly the Frank-Tamm equation.

1.3 Predicting a Spectrum

The Frank-Tamm Equation has been shown to accurately predict the spectrum of asingle charged particle. However, producing an accurate model for the glow surround-ing a nuclear reactor is much harder. Around a reactor, the spectrum of CerenkovRadiation would depend on the spectrum of charged particles present. In such anenvironment, charged particles can be produced in a huge variety of ways. A fewexamples are:

1. Free electrons resulting directly from fission.

2. β particles emitted by the spectrum of fission products their daughters.

3. β particles emitted by decaying core components.

4. Pair production from the γ flux.

But far more significant than any of these sources would be electrons that absorb anyof the γ radiation emitted from the same sources listed above. Further, these photonscould have interacted via Compton scattering before being absorbed, which wouldhave to be accounted for. Beyond that, the directionality associated with CerenkovRadiation might mean some sources contribute more to the overall spectrum thanothers.

In the end, a full theoretical treatment of such a spectrum is beyond the scopeof this experiment. If a β spectrum could be taken, it could be transferred into aCerenkov spectrum. To my knowledge, no such spectrum exists.

Chapter 2

Color Matching

2.1 Color Matching Functions

2.1.1 Color Vision

It is clear from Equation 1.3 that the light produced by Cerenkov Radiation is aspectrum. When viewed from the top of a pool of water, the spectrum appears blue.Intuitively, this seems obvious: more energy is being emitted at shorter wavelengthsthan at higher ones, so the color should be at the very least, bluish. To understandexactly how these spectra would look, though, requires knowledge of color vision. Forthe most part, this chapter draws upon the contents of Hunt (1991).

The simplest model is to consider the overall sensitivity of the eye to light ofdifferent wavelengths. Figure 2.1 shows the sensitivity curve that represents overallperceived brightness. From the curve, it is clear that peak sensitivity occurs around550 nm (green), and drops off by 400 nm or 700 nm (blue and red, respectively).Obtaining this plot is done by asking participants to match a color of a given wave-length with white light by brightness. This curve only represents brightness, though.In order to deal with color, more information is needed.

In humans, color vision is governed by rod and cone cells in the eye. For practicalpurposes, the contribution to color vision by rod cells can be ignored for optical stimuligreater than 10−2 cd/m2. For reference, the night sky is around 10−3 cd/m2, whereas the the moon is around 2.5 cd/m2. Since Cerenkov Radiation is bright enough, weneed only worry about the cone cells.

Cone cells come in three varieties, usually dubbed β, γ, and ρ. Each cone detectslight over a different range of wavelengths, so that between the three types of cones,the entire range of human vision is covered. Approximate sensitivity curves are shownfor each of these in Figure 2.2. An overly simple, but illustrative model, is to think ofthese three types of cones as representing the three primary colors of human vision.In this simple model, β cones would represent blue, γ cones would be green, and ρcones would be red. If you were to stare at a strongly yellow light, the γ and ρ coneswould activate. Looking at a turquoise light would instead cause the β and γ conesto activate and so on.

While this model is not completely accurate, it hints at an important fact: every

16 Chapter 2. Color Matching

Figure 2.1: Overall sensitivity curve adapted from Hunt (1991).

Figure 2.2: Cone sensitivity curves adapted from Hunt (1991).

visible color can be expressed by the activation of these types of cones. If two spectraproduce the same responses on all three cones, then they will physically look thesame. For example, the light reflected off of an orange flower petal could be a fullspectrum of wavelengths. When a photo of the same flower is viewed on a computerscreen, the color is reproduced by a combination of just red, green, and blue. Becausethe cones in your eye produce the same response in both cases, the colors will appearidentical.

Because color vision relies on only three varieties of cones, we should be able toexpress any color as set of three numerical values. The goal of color matching is todetermine these three numerical values for any given spectrum of light. The idealvalues would be the exact responses of the three types of cones, but the curves inFigure 2.2 are not known to sufficient precision.

2.1. Color Matching Functions 17

2.1.2 RGB Color Matching Functions

The standard model for matching spectra to colors was established by the Interna-tional Commission on Illumination (abbreviated as the CIE, from their French name)in 1931. The model combined the results of two separate color matching experimentsperformed in the late 1920s. In both experiments, participants were shown a testcolor to match. On an adjacent screen, the participants could control an additivemixture of red, green and blue light. They then adjusted the mixture until the colorsmatched exactly. From this, the researchers could determine how much red, green,and blue light respectively were required to match any given wavelength of light. Onechallenge the CIE faced was that the two experimenters used different light sourcesfor their red, green, and blue lamps. To homogenize the results, each data set wasconverted to what would have been obtained if they had mixed precise wavelengths.Their chosen wavelengths were:

Red = 700.0 nmGreen = 546.1 nmBlue = 435.8 nm

The first thing they discovered was that equal luminances of red, green, and bluelight did not make white light. For a variety of reasons, lights of the same luminancedo not appear to have the same intensity to an observer. In order to make whitelight, you have to mix red, green, and blue equal in perceived intensity, not physicalintensity. To account for this, new units were proposed so that one “Red Unit” ofred, one “Green Unit” of green and one “Blue Unit” of blue would mix to make whitelight. These units were defined as:

1 of R = 1 cd/m2at 700.0 nm,

1 of G = 4.5907 cd/m2at 546.1 nm,

1 of B = 0.0601 cd/m2at 435.8 nm,

(2.1)

The second thing they discovered, though, was that certain colors could not bematched exactly using their three chosen colors. Instead, for certain wavelengths oflight, color had to be added to the test color to make it matchable. By convention,these cases were treated as though they required “negative” amounts of that color.For example, if 500 nm light, when mixed with 10 Red Units of red was matched by39 Green Units of green and 20 Blue Units of blue, it could be said that:

1 unit of 500 nm plus 10 of R is matched by 39 of G plus 20 of B

or alternatively, employing the negative convention:

1 unit of 500 nm is matched by -10 of R plus 39 of G plus 20 of B.

A visual representation of these coefficients is shown in Figure 2.3. The coefficientsof R, G, and B can be read directly off the plots for a given wavelength. These


Figure 2.3: RGB Color Matching Curves adapted from Hunt (1991).

coefficients are usually referred to as tristimulus values. Notice that on the plot,nearly every wavelength has at least one negative component. The only exceptionsare at the wavelengths chosen to represent R, G, and B. This reflects the fact that itis impossible to create the color of a pure wavelength by mixing other wavelengths oflight.

Mixing colors using tristimulus values is incredibly simple, though. If you havetwo colors, matched by the tristimulus values R1,G1,B1 and R2,G2,B2, the mixtureof the colors would be matched by R1+R2,G1+G2,B1+B2. Mixing different amountsof different colors is also easy. Let’s say you’re mixing some quantity, α of color Awith some quantity β of color B such that:

1 unit of A is matched by RA of R plus GA of G plus BA of B.1 unit of B is matched by RB of R plus GB of G plus BB of B.

Then,

α units of A is matched by αRA of R plus αGA of G plus αBA of B.β units of B is matched by βRB of R plus βGB of G plus βBB of B.

and

α units of A plus β units of B is matched by

αRA + βRB of R plus αGA + βGB of G plus αBA + βBB of B. (2.2)

Given a complete spectrum of wavelengths and associated strengths, the generalform of this can be written out.

1 unit of C is matched by∑

cλrλ of R plus∑

cλgλ of G plus∑

cλbλ of B. (2.3)

where rλ, gλ, bλ are the functions shown in Figure 2.3. The coefficients cλ are therelative intensities of the spectrum that represents the color X at the wavelength

2.1. Color Matching Functions 19

λ. Because these are essentially the weighting functions used to match spectra totristimulus values, they are most commonly referred to as color matching functions.If we had functional forms of both the spectrum and the color matching functions,we could replace the sums in Equation 2.3 with integrals over the visible spectrum.

2.1.3 XYZ Color Matching Functions

Perhaps the best way to think about the R, G, B tristimulus values discussed in theprevious section are as a set of basis vectors. Every visible color appears somewherein the space spanned by these vectors, and we can use the color matching functions toreduce any spectrum to its R, G, B coordinates. But like any three dimensional space,we are free to select a new set of orthonormal basis vectors if it suits us. This sectiondiscusses the XYZ tristimulus values defined by the CIE 1931 Standard Observer.The color matching functions for these values are shown in Figure 2.4.

Figure 2.4: The 1931 Standard Observer XYZ Color Matching Functions.

The first goal of the new matching functions was to create a set of tristimulusvalues that were always positive for visible colors. This was done by selecting a setof values that were each a linear combination of the R, G, B values:

X = 0.49 R + 0.31 G + 0.20 B,

Y = 0.17697 R + 0.81240 G + 0.01063 B,

Z = 0.00 R + 0.01 G + 0.99 B.

(2.4)

Another goal with the creation of the XYZ tristimulus values was to easily repre-sent the brightness of a given color. To do this, the coefficients for Y in Equation 2.4were set so their ratios were the same as in Equation 2.1. The effect of this is togive the color matching function the same shape as the sensitivity curve shown inFigure 2.1.

The final constraint placed on the XYZ values was to make it so that white wasrepresented by the point where X = Y = Z. The white in the RGB system had been


defined by R = G = B, so the coefficients for each value in Equation 2.4 were chosento sum to 1. So, if a spectrum has R = G = B, then it will also have X = Y = Z.

In this system, the tristimulus values no longer correspond to specific wavelengthsof light. However, the entire visible color space now lies in the area where X, Y, Z > 1.Additionally, we can easily compare the perceived brightness of two spectra solely bycomparing their Y coordinates.

2.2 Color Spaces

2.2.1 The xy Color Space

Both the XYZ and RGB tristimulus value sets represent a three dimensional spacethat contains all of the visible colors. However, in order to display the visible colorspace on a page, a two dimensional space is required. The easiest way to do this isto normalize the XYZ values into a new set of values defined by:

x =X

X + Y + Z, y =

Y

X + Y + Z, z =

Z

X + Y + Z(2.5)

From these definitions, it is clear that x+y+z = 1, so only two of the three valuesis linearly independent. Conventionally, the values x and y are chosen, and the pairdefine the two dimensional color space shown in Figure 2.5. This diagram is usuallycalled the CIE x,y chromaticity diagram. Note that only colors where X, Y, Z > 1are shown, as these represent the entire visible color space. By restricting the spaceto two dimensions, all that has been lost is some measure of intensity: all points inthe diagram have the same X + Y + Z value.

Figure 2.5: The 1931 Standard Observer x,y Chromaticity Diagram.

Despite missing the third dimension, the diagram in Figure 2.5 is full of interestingfeatures.

2.2. Color Spaces 21

1. All of the pure wavelengths lie along the upper edge of the space.

2. All of the purple hues lie near the bottom boundary, (between the reds andblues) therefore it is frequently referred to as the “purple boundary.”

3. Any point not contained by this diagram represents an “imaginary” color, mean-ing no spectral stimulus can produce it.

4. Any mixture of two colors lies along the straight line connecting those colors inthe diagram.

This last point naturally extends to a mixture of three or more colors. In anycase, the area contained by the points representing those colors contains all of thepossible mixtures of those colors.


2.2.2 The CIE RGB Color Space

Figure 2.6: The 1931 Standard Observer x,y Chromaticity Diagram with the CIERGB color space shown.

The color space representing the wavelengths specified in Equation 2.1 can beshown on the x,y chromaticity diagram as in Figure 2.6. In this diagram, the pointE refers to the white point. The triangle shown connects the three wavelengths to bemixed: 435.8 nm, 546.1 nm, and 700.0 nm. Every point within the triangle representsa color which can be created by mixing the three wavelengths.

Note that a large portion of the diagram lies outside this triangle, though. Eachof these colors lies outside the CIE RGB color space, meaning it cannot be createdby mixing the three colors that define that color space. Among the colors outsidethe space are all of the other pure wavelengths. Before, this was explained by havingnegative R, G, or B values. For the most part, though, colors outside the space canbe approximated by colors inside the space. This is the basis for computer screens.For every color they can’t represent using their red, green and blue, they approximateit to the nearest possible color.

2.2.3 Uniform Chromaticity and Hue Angle

A major flaw with the xy color space diagram shown in Figure 2.5 is the distributionof distinct colors. Even the humblest observer will note that the diagram devotesmuch more space to teal than any of the other colors. In fact, if you were to calculatethe distance between colors that were “equally different” from each other, you wouldfind these distances are extremely non-uniform across the diagram. In other words,if you started in the teal region and tried to find a color that was very distinctlydifferent, you would have to move much further than if you were starting from theorange or blue region.

2.2. Color Spaces 23

Figure 2.7: The 1976 Uniform Chromaticity Scale Diagram for u′ and v′.

This represents a flaw in the xy color space in general: it is extremely non-uniform.In order to correct this, the “CIE 1976 uniform chromaticity scale diagram” wasdeveloped using the two coordinates:

u′ = 4x/(−2x+ 12y + 3)

v′ = 9y/(−2x+ 12y + 3)(2.6)

The diagram for these coordinates is shown in Figure 2.7. Using these new coor-dinates, it is possible to obtain a simple measure of how different to hues are fromeach other. This measure is called the “hue-angle” and is obtained by:

huv = arctan[(v′ − v′n)/(u′ − u′n)] (2.7)

Where v′n and u′n are the coordinates of a suitably chosen “reference white.” Toget an idea of what this image represents, imagine a conical color space. In this colorspace, “hue” is represented by the angle around the z -axis, so that by traveling afull 2π around the axis, you pass through every color. The radial coordinate thenrepresents saturation, where colors close to the origin are paler, and colors furtheraway are richer. Distance along the z -axis represents brightness. In this system, theangle between two colors represents the difference between the two hues.

On top of being a useful quantitative measure, many popular image processingprograms can take advantage of these angles. Adobe Photoshop, for example, cantranspose entire images, or just sections of them, by any given hue angle. Given aphotograph of the reactor at power, and a hue angle, the new colors can be visualizedin a more realistic way.

Chapter 3

Method

3.1 Experimental Apparatus

3.1.1 Reed Research Reactor

The data for this experiment was taken at the Reed Research Reactor situated onthe Reed College Campus. It is a TRIGA Mark II Nuclear Reactor with a maximumlicensed thermal output of 250 kW. TRIGA type reactors use uranium zirconiumhydride fuel and are designed to operate at relatively low temperatures. Like manyTRIGA type reactors, the Reed Research Reactor employs low (20%) enriched fuel.The Reed Research Reactor core lies at the bottom of a 25 foot, open pool of filtered,demineralized water. The large tank offers an unparalleled view of the core and theCerenkov Radiation it produces.

The β spectrum in the area around the core would be comparable to any otherlow enriched, uranium based reactor. For this reason, the results of this experimentwould be nearly identical if carried out any other such facility.

3.1.2 Collection Tool

The experimental tool consists of an aluminum chamber which can be filled by avariety of fluids and placed near the Reactor Core. The chamber has a capacity ofjust over a liter, and the lid of the chamber can be removed to change out the fluids.The outer walls of the chamber are 1.5” thick aluminum. Aluminum was used becauseit is resistant to corrosion, light weight, and has a short half life. Because the chamberwill be close to the reactor, the aluminum will activate, but the half life of Al-28 isonly 2.24 minutes. (National Nuclear Data Center, 2013) After approximately onehour in the pool, nearly all of the radioactive aluminum will have decayed, and thechamber can be safely removed from the tank. The bottom of the chamber is asheet of 0.125” thick aluminum held in place by aluminum bolts. The bottom is keptthinner than the walls so that radiation from the core can enter the chamber. Asdiscussed earlier, the primary cause of Cerenkov Radiation is free radicals producedby gamma radiation. The chamber was suspended on a rigid aluminum rod into thereactor pool so that it hung approximately 1 foot off the top of the reactor core.

26 Chapter 3. Method

Figure 3.1: Diagram of the experimental set up used.

Bolted to the interior ceiling of the chamber is a Planar Irradiance Collectorproduced by HOBI Labs. This sort of detector is designed to collect light over the180 in front of it. The HOBI Labs model is explicitly designed to interface withwater. The light incident on the collector is directed into a 600 µm diameter fiberoptic cable. The other end of the 10 meter cable was attached to an Ocean OpticsUSB2000+ Spectrometer. The spectrometer then interfaced with a computer runningthe SpectraSuite software, which was used to collect spectral data. Each spectrumwas taken as ten 60 second counts averaged together.

For more information, the approval request sent to the Reed Research Reactoroperations committee regarding this experiment is included as an appendix.

3.2 Method of Analysis

3.2.1 From Power Spectrum to Color

Chapter 2 covers the theory behind converting a power spectrum to a set of colorcoordinates. For each spectrum generated in this experiment, the following approachwas used to convert it to a perceived color.

First, each spectrum was converted to XYZ coordinates using the color matchingfunctions shown in Figure 2.4 and listed in Appendix B. The form is the same as inEquation 2.3, but using the x, y, and z matching functions instead.

X =780∑

λ=380

cλx , Y =780∑

λ=380

cλy , Z =780∑

λ=380

cλz (3.1)

where cλ represents the value of the power spectrum at that wavelength and the sum

3.2. Method of Analysis 27

is conducted over 5 nm intervals. Intervals of 5 nm were used to correspond with CIEpublished values for x, y, and z. According to Hunt, 5 nm intervals provide sufficientprecision for all applications.

The extreme case of this would be to imagine the sums as integrals. In thatcase, the X, Y, and Z coordinates would represent the areas under the product of thespectra and the matching functions. The process is illustrated in Figure 3.2. In theillustration, the matching functions are shown in the top left, an example spectrumin the bottom left, and the product of the two is shown on the right. The X, Y, Zcoordinates are the areas of the shaded regions shown.

Figure 3.2: Illustration of the process used to convert spectra to tristimulus values.Top left: The XYZ Color Matching Functions. Bottom Left: An example spectrum.In this case, it is the Cerenkov spectrum produced by an electron of β = 0.8 in water.Right: The product of the spectrum and color matching functions. The areas of theshaded regions correspond to the XYZ tristimulus values.

To image the colors, we must convert from XYZ to RGB. The matrix used toconvert from XYZ to the RGB used by Mathematica is:

RGB

=[M] XY

Z

,[M]

=

0.412387 0.212637 0.01933060.357591 0.715183 0.1191970.18045 0.0721802 0.950373

(3.2)

Each set of RGB coordinates was then normalized so that the greatest coordinatewas equal to 1. So, for example, if the coordinates were R = 1, G = 2, B = 4, thenew coordinates would be R = 0.25, G = 0.5, B = 1. This step ensures that the hueis being compared, instead of brightness. For the relative brightness of the glow, theY coordinate by itself is listed.

3.2.2 From Data to Power Spectrum

Raw data collected by the spectrometer is not yet ready to be converted into a color.In order to transform this data into a form compatible with Equation 2.3, it must go


through some processing. The general form of this is:

(1)

(2)

(3)

(4)

(5)

(6)

Raw Data↓

Correct for Detector Efficiency↓

Average Over 5 nm Bands↓

Subtract Background Light↓

Factor in Absorption↓

Visible Spectrum

This process is visually illustrated in Figure 3.6.

Correcting for Detector Efficiency

Figure 3.3: Plot of the manufacturer given calibration results for the Oriel InstrumentsQuartz Tungsten Halogen Lamp and the best fit curve used.

The first step in analyzing the spectral data is accounting for the efficiency of thesetup. Because the detector is more sensitive to some wavelengths than others, thecollected data spectrum isn’t exactly the spectrum of light seen. The efficiency of thedetector, cable, light collector and any extraneous factors can be corrected easily byusing a known light source. In this case, that light source is an Oriel InstrumentsQuartz Tungsten Halogen Lamp. Operating at 3000 K, it provided a clear blackbody spectrum to account for.

Because we know that black body radiation takes the spectral form:

P(λ) = C1

λ51

eα/λ − 1(3.3)

for some values of C and α. Using manufacturer calibration results for the light sourceand Mathematica’s built in form fitting function, the values of C and α were foundto be:


C = 1970.53 Wµm5 , α = 4.50165 µm (3.4)

The plot of Equation 3.3 using these values are shown alongside the manufacturer’sresults in Figure 3.3. This plot represents the exact spectrum of the light source.

The spectrum of the light source was then taken using the full set up of thisexperiment. The bottom of the chamber was removed, and the detector pointedtowards the source. Using this spectrum and the black body spectrum, the efficiency(η) of the set up could be determined by:

η(λ) =Measured

Black Body(3.5)

Figure 3.4: Efficiency curve obtained for the experimental set up and Oriel Instru-ments Quartz Tungsten Halogen Lamp.

The efficiency curve this produces is shown in Figure 3.4. This efficiency curvecan then be applied to collected spectra to account for the sensitivity of the wholeset up:

Actual Spectrum =Collected Spectrum

Efficiency Curve(3.6)

Subtracting Background and Averaging

After correcting for the efficiency of the set up, the spectra need to be sorted intothe 5 nm bands used in color matching. To do this, we average the data points inthe interval surrounding each wavelength used in Equation 3.1. So, the value of cλ inthat equation would be the average of the data points in the region [λ− 2.5, λ+ 2.5].

Then, we need to subtract any background light in the system. For each spectrumtaken with the Reactor at full power, a spectrum was taken with it shutdown. Toobtain the spectrum due solely to the Cerenkov Radiation, the shutdown spectrumwas subtracted from the at power spectrum.


Figure 3.5: Absorption spectrum for corn oil (solid line) adapted from Vijayan et al.(1996). The units along the vertical axis are cm−1. The absorption spectrum forwater (dashed line) is also shown, adapted from Smith & Baker (1981).

Factoring in Absorption

Viewed through meters of the material, though, the spectrum looks different, becausedifferent frequencies of light attenuate differently in matter. Values for the absorptioncoefficient of water and corn oil are shown in Figure 3.5 over the visible spectrum.The attenuation coefficient relates to the power spectrum via the equation,

I = I0 exp(−αx) (3.7)

where I is the measured intensity through the medium, I0 would be the measuredintensity without the medium, α is the attenuation coefficient in units of inversedistance and x is the thickness of material between observer and source.

Over short distances, the attenuation in most transparent fluids is minimal. How-ever, over meters of the fluid, the attenuation can build up to significantly alter theperceived color. For example, in water, the attenuation coefficient is much higherover the red region of the spectrum than in the blue region. As a result, the red lightwill be attenuated more than the blue light, and the color will appear bluer than itwould without all of the additional water.

After factoring in the absorption in the material, the resulting spectrum is readyto be converted directly into a color via Equation 3.1.

3.2.3 Prediction of Spectrum

In order to predict colors, the spectra were predicted based on Equation 1.3. (TheFrank-Tamm Equation) Specifically, the equation can be rewritten as

dP

dω=e2vµ0

4π

(1− c2

v2n2ω

)ω (3.8)

All of the frequency dependence lies to the right of this equation, so the shape ofthe spectrum can be given entirely by


ω

(1− c2

v2n2ω

). (3.9)

Note that the constants would be important for determining the quantitativepower output by Cerenkov Radiation, but do not factor into color matching otherthan overall brightness. In terms of comparing Cerenkov Radiation in two differentmaterials, these constants can be safely ignored. So then for a given material, thefunctional form for the index of refraction is fed into Equation 3.9 along with a

value for β =v

c. For example, an electron traveling at β = 0.8 through water

produces a spectrum of the shape shown in Figure 3.2. To match this spectrum usingEquation 3.1, we simply choose the values of the spectrum at each 5 nm increment.

For water, Daimon & Masumura (2007) gives

n2λ = 1 +

5.68403 ∗ 10−1λ2

λ2 − 5.10183 ∗ 10−3+

1.72618 ∗ 10−1λ2

λ2 − 1.82115 ∗ 10−2

+2.08619 ∗ 10−2λ2

λ2 − 2.62072 ∗ 10−2+

1.13075 ∗ 10−1λ2

λ2 − 1.06979 ∗ 101. (3.10)

For cinnamaldahyde, Rheims et al. (1997) gives

nλ = 1.57008 + 0.01523 λ−2 + 0.00084 λ−4. (3.11)

Note that refractive index for water is given in the form of a Sellmeier equation,while the refractive index for cinnamaldehyde is given as a Cauchy equation. Whilethey have different forms, both are accurate over the visible spectrum, so they bothwork fine for this experiment.


Figure 3.6: Illustration of data analysis process.

Chapter 4

Data & Analysis

4.1 Predicted Results

4.1.1 The Effect of β

For a single charged particle creating Cerenkov Radiation, the spectrum of light isrelated to the speed of the particle. This is clear in Equation 3.9, where the speed v

is prominent in defining the shape of the spectrum. In essence, the value of β =v

cdetermines the strength of the 1/n2 contribution.

Moreover, the speed of the particle can interact with the speed threshold in Equa-tion 1.1. For a given material with refractive index n(λ), a particle traveling at speedβ can only produce light over the wavelengths where Equation 1.1 is satisfied. In thevisible spectrum, where refractive indexes are essentially always lower in the higherwavelength regions, then those regions will be the first to be cut off. Effectively, thereare “threshold” regions for β where light is only being emitted in parts of the visiblespectrum.

For a charged particle traveling above this threshold region in water, the effectsof increasing β are shown in Figure 4.1. Clearly, in the image, the color changessomewhat dramatically with the speed of the particle. Particles traveling at relativelylow speeds produce much richer blues than faster moving particles.

Since the β spectrum will only change if we were to redesign the reactor itself,this effect can’t be controlled. It does, however, interact with the index of refractionas discussed in the following section.

Figure 4.1: The effect of particle speed on perceived color in water. Particle speedincreases to the right. Mathematica RGB coordinates are displayed below each color.The leftmost color represents a β of 0.7525, and each square incrementally increasesβ by 0.0025.

34 Chapter 4. Data & Analysis

4.1.2 The Effect of Index of Refraction

Figure 4.2: The effect of refractive index on perceived color. The top image is forβ = 0.755, the bottom is for β = 0.8. From left to right, the fluids are water,glrycerol, ethylene glycol, cinnemaldihyde, and a theoretical non-dispersive mediumwith n(λ) = 1.5. Mathematica RGB values are shown for each color. All refractiveindexes pulled from Polyanskiy (2012)

When changing the medium of travel, the first optical property to examine is theindex of refraction. As with the β fraction, index of refraction features prominently inEquation 3.9. When switching between materials, their colors change as is shown inFigure 4.2. The most obvious reason for a change is that if the shape of the spectrumof n(λ) changes, then the shape of the Cerenkov spectrum will change. A differentshape to the spectrum means a different color.

For values of β sufficiently high, the effect of changing the index of refractionbecomes increasingly minimal. For lower values of β, though, the effect can be quitestrong. The reason for this is the interaction between β and n in Equation 3.9:

ω

(1− c2

v2n2ω

).

In this equation, the most dramatic changes in c2

v2n2ω

(and therefore the shape of

the spectrum) will occur when the fraction is near 1, when the v is near cnω

, the speedthreshold. In other words, as the particle nears the speed threshold, the change incolor accelerates. A change to β of 0.0005 will more significantly impact the colorspectrum when the particle is traveling at a speed near the threshold than it will ifthe particle is traveling much faster.

The same effect occurs when changing index of refraction, but in reverse. As aparticle’s speed approaches the threshold from above for a fixed index of refraction,βnω

changes more dramatically. In the same way, as the index of refraction approachesthe threshold from below, the fraction changes more dramatically. Because the indexof refraction depends on wavelength, its distance from the threshold also depends

4.1. Predicted Results 35

on the wavelength. Then, near that threshold, the power emitted at a particularwavelength will vary more dramatically at wavelengths where the index of refractionis closer to 1

β.

That being said, the effect of index of refraction on perceived color appears to besomewhat minimal. Figure 4.2 shows a whole range of different refractive indexes anddispersions, and yet all of the colors are some shade of blue. The main barrier is that itis extremely rare to find a fluid with a refractive index that increases with wavelengthin the visible spectrum. In fact, even a fluid with constant index of refraction wouldyield a blue color, because of the dominating ω term in Equation 3.9.

4.1.3 The Effect of Absorption

Figure 4.3: The effect of absorption on perceived color. The leftmost block in eachrow is the perceived color of the Cerenkov spectrum in water without absorption.The top two rows represent particles traveling at β = 0.755, while the bottom tworows represent particles traveling at β = 0.8. Rows 1 and 3 show the effect ofwater’s absorption spectrum, where each block represents an additional 1 meter ofabsorption. Rows 2 and 4 show the effect of corn oil’s absorption spectrum, whereeach block represents an additional 0.02 meters of absorption.

As noted in the previous section, factoring in changes to refractive index can reallyonly create different shades of blue. When factoring in absorption however, drasticallydifferent colors can be achieved. Figure 4.3 shows the effect of the absorption spectrumof water and corn oil. A functional form for the refractive index for corn oil could


not be found, so the image shows the effect of corn oil’s absorption coefficients on theCerenkov spectrum in water. While this means the colors depicted are not completelyaccurate for corn oil, they demonstrate the effect of two different absorption spectraon the same source. Corn oil’s Cerenkov spectrum likely also appears blue, however,like all of the materials shown in the previous section, so the color won’t be far off.

The absorption coefficients for water increase dramatically towards the long endof the visible spectrum. This sharp rise increases absorption in the red portion of thespectrum, and the resulting color is shifted towards blue. As more water is placedbetween the viewer and the source, the more the color shifts towards a deep blue.

Meanwhile, corn oil has greater absorption in the high frequency region of thevisible spectrum. This is shown in Figure 3.5. As a result, corn oil absorbs moreblue than red or green, and the overall color shifts away from blue. The absorptioncoefficients are also generally much stronger than in water, so it only takes a fewcentimeters of oil to turn the original Cerenkov spectrum green.

If a viewer was interested in changing the perceived color of the glow aroundreactors, this would by far the easiest way to do it. While refractive indexes oftransparent fluids tend to be similar, their absorption spectra can vary widely.

4.2 Observed Spectra

Figure 4.4: Collected spectra for water with the reactor at power (blue) and shutdown (red). The Cerenkov curve is very visible.

In this experiment, we observed Cerenkov Radiation through three materials:water, corn oil and cinnamaldehyde. Figure 4.4 shows what the directly measuredspectra looked like. After performing all of the efficiency correction and backgroundsubtraction, we obtained the Cerenkov Spectra shown in Figure 4.5. These spectrarepresent our major results. Clearly, the three materials produce different glows, andin the next section we will analyze the differences.

4.3. Analysis of Results 37

Figure 4.5: Spectra associated with Cerenkov Radiation in water (dashed), corn oil(dotted), and cinnamaldehyde (dashed).

4.3 Analysis of Results

4.3.1 From Water

Figure 4.6: Perceived color result for water, factoring in absorption. The leftmostcolor comes directly from the collected spectrum. Each subsequent color adds 1meter of absorption. At the Reed Research Reactor, there is approximately 6 metersof water above the core.

The spectrum collected for Cerenkov Radiation through pure water is shown inFigure 4.6. This color is good agreement with the colors shown in Figures 4.1 and 4.3.It also agrees well with observer intuition of the actual color of the reactor’s glow.

Of particular note is that this color seems to represent a fairly low β fraction,as shown in Figure 4.1. Some explanation of this can be given by examining thegamma spectrum near a reactor core. Nakashima et al. (1971) showed that thenumber of photons falls off roughly exponentially with the energy of the particle.This means that no matter where the speed threshold lies for a substance, there willbe exponentially more photons close to the threshold than of higher energy. As aresult, there will be generally more electrons with speed near the threshold than ofhigher energy. The overall color, then, should be dominated by “low” energy electrons,which is what we see.


Figure 4.7: Comparison of the perceived hues of cinnamaldehyde and water. On theleft is the color obtained from the cinnamaldehyde spectrum, on the right is water.

4.3.2 From Cinnamaldehyde

The spectrum collected for Cerenkov Radiation through cinnamaldehydeis shown inFigure 4.7. As is pretty clear in the image, the produced color is extremely similar tothat of water. This is unsurprising, as the shapes of the two refractive index curves arenot dramatically different. However, cinnamaldehyde’s index of refraction is around1.62 in the visible spectrum, compared to water’s 1.33. This overall increase does notchange the hue, but does change the brightness of the glow.

To show this, the spectrum obtained for cinnamaldehyde is compared with thespectrum obtained for water in Figure 4.8. When obtaining the spectrum for cin-namaldehyde, the tool was only half full (due to limitations in acquiring the sub-stance), and yet the spectrum is clearly stronger than that of water. If the tool werefull, the intensity would have likely been even larger.

Figure 4.8: Comparison of water and cinnamaldehyde spectra. The solid line showsthe magnitude of the cinnamaldehyde spectrum over the visible region, while thedashed line shows that of water.

More quantitatively, the Y value of each spectrum was calculated using Equa-tion 2.3. Recall that the Y value represents overall brightness because its colormatching function exactly matches the overall sensitivity curve shown in Figure 2.1.For the spectra shown in Figure 4.8, the Y value of cinnamaldehyde was found to be1.26 times greater than that for water, despite the tool being only half full for that

4.3. Analysis of Results 39

measurement.Note that absorption is not factored into any of the results with cinnamaldehyde

because absorption data was not available for it in the visible spectrum.

4.3.3 From Corn Oil

Figure 4.9: Perceived color result for corn oil, factoring in absorption. The leftmostcolor comes directly from the collected spectrum. Each subsequent color adds 1meter of absorption. At the Reed Research Reactor, there is approximately 6 metersof water above the core.

The spectrum collected for Cerenkov Radiation through corn oil is shown in Fig-ure 4.9. In this case, the absorption spectrum shown in Figure 3.5 quickly dominatesthe color of the glow. By the time the light hits the detector, it has already passedthrough a few centimeters of oil, and is already somewhat green. The underlying blueglow means that through a pool of corn oil, the glow appears lime green.

By using the formulation set forth in Equation 2.7, the hue angles for water andcorn oil at 6 meters of absorption were calculated to be:

Corn Oil : huv = 13.8733 Water : huv = 67.4291

For these equations, the reference white used was Mathematica’s R = G = B = 1.A photograph of the core was then adjusted using this hue angle to produce an artist’srendering of the reactor in a pool of corn oil. This is shown in Figure 4.10. Note thatthis image is not precise by any means. It is simply an approximation based on thehue angles measured.

Figure 4.10: Qualitative rendering of the reactor core if it were in a pool of corn oil.Based on the difference in hue angle between water and corn oil. On the right is theoriginal image.

Chapter 5

Conclusion

The purpose of this experiment was to examine the potential colors of CerenkovRadiation as perceived around a nuclear reactor. To this end, we took spectra ofthis radiation produced in water, corn oil, and cinnamaldehyde. Through analyzingthese spectra and predicting spectra from the Frank-Tamm equation, we came to thefollowing conclusions.

• The shape of the refractive index curve over the visible spectrum can affectthe perceived hue of that Cerenkov spectrum. Every refractive index we usedyielded a perceived hue that was clearly blue. Some variation was apparentbetween blues, but for the most part they were all blue. These results areshown in Figure 4.2.

• The most significant changes to color are the result of optical absorption. Mate-rials that absorb more at higher wavelengths produce bluer light, and materialsthat absorb more at lower wavelengths produce more green/red light. Theseeffects can be dramatic. These results are shown in Figure 4.3.

• The measured spectrum for water produced a perceived color that matchesvisual observation of the Reed Research Reactor.

• The overall magnitude of the refractive index can substantially affect the per-ceived brightness of the radiation. Higher refractive indexes produce more ra-diated power over the entire spectrum, which creates a brighter color. Thiswas shown predominantly in the spectrum obtained through cinnamaldehyde.These results are shown in Figure 4.8.

• Corn oil was shown to produce a dramatically different color of light than water,mostly due to its absorption. The light as seen through corn oil is perceived asa bright lime green. A comparison of these colors is shown in Figure 4.10.

Appendix A

ROC Approval

44 Appendix A. ROC Approval

Appendix B

XYZ Color Matching Functions

Figure B.1: The 1931 Standard Observer XYZ Color Matching Functions.

46 Appendix B. XYZ Color Matching Functions

λ (nm) x y z λ (nm) x y z

380 0.001368 0.000039 0.00645 585 0.9786 0.8163 0.0014385 0.002236 0.000064 0.01055 590 1.0263 0.757 0.0011390 0.004243 0.00012 0.02005 595 1.0567 0.6949 0.001395 0.00765 0.000217 0.03621 600 1.0622 0.631 0.0008400 0.01431 0.000396 0.06785 605 1.0456 0.5668 0.0006405 0.02319 0.00064 0.1102 610 1.0026 0.503 0.00034410 0.04351 0.00121 0.2074 615 0.9384 0.4412 0.00024415 0.07763 0.00218 0.3713 620 0.85445 0.381 0.00019420 0.13438 0.004 0.6456 625 0.7514 0.321 0.0001425 0.21477 0.0073 1.03905 630 0.6424 0.265 0.00005430 0.2839 0.0116 1.3856 635 0.5419 0.217 0.00003435 0.3285 0.01684 1.62296 640 0.4479 0.175 0.00002440 0.34828 0.023 1.74706 645 0.3608 0.1382 0.00001445 0.34806 0.0298 1.7826 650 0.2835 0.107 0.450 0.3362 0.038 1.77211 655 0.2187 0.0816 0.455 0.3187 0.048 1.7441 660 0.1649 0.061 0.460 0.2908 0.06 1.6692 665 0.1212 0.04458 0.465 0.2511 0.0739 1.5281 670 0.0874 0.032 0.470 0.19536 0.09098 1.28764 675 0.0636 0.0232 0.475 0.1421 0.1126 1.0419 680 0.04677 0.017 0.480 0.09564 0.13902 0.81295 685 0.0329 0.01192 0.485 0.05795 0.1693 0.6162 690 0.0227 0.00821 0.490 0.03201 0.20802 0.46518 695 0.01584 0.005723 0.495 0.0147 0.2586 0.3533 700 0.011359 0.004102 0.500 0.0049 0.323 0.272 705 0.008111 0.002929 0.505 0.0024 0.4073 0.2123 710 0.00579 0.002091 0.510 0.0093 0.503 0.1582 715 0.004109 0.001484 0.515 0.0291 0.6082 0.1117 720 0.002899 0.001047 0.520 0.06327 0.71 0.07825 725 0.002049 0.00074 0.525 0.1096 0.7932 0.05725 730 0.00144 0.00052 0.530 0.1655 0.862 0.04216 735 0.001 0.000361 0.535 0.22575 0.91485 0.02984 740 0.00069 0.000249 0.540 0.2904 0.954 0.0203 745 0.000476 0.000172 0.545 0.3597 0.9803 0.0134 750 0.000332 0.00012 0.550 0.43345 0.99495 0.00875 755 0.000235 0.000085 0.555 0.51205 1. 0.00575 760 0.000166 0.00006 0.560 0.5945 0.995 0.0039 765 0.000117 0.000042 0.565 0.6784 0.9786 0.00275 770 0.000083 0.00003 0.570 0.7621 0.952 0.0021 775 0.000059 0.000021 0.575 0.8425 0.9154 0.0018 780 0.000042 0.000015 0.580 0.9163 0.87 0.00165

References

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deep blue: examining cerenkov radiation through non-traditional media

Technology

perceived color

color vision

color matching15

color spaces

rgb color matching functions

rgb color matching curves

xy color space

cie rgb color space