Deep Blue: Examining Cerenkov Radiation Through Non-traditional Media
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    Deep Blue: Examining Cerenkov Radiation Through Non-traditional Media Deep Blue: Examining Cerenkov Radiation Through Non-traditional Media Document Transcript

    • 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 completing this Thesis. You have been the best advisor I could hope for, and I truly appreciate the 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 am today with out you both, and I love you dearly. Alison and Liz, thank you for all of the wisdom and maturity you have shown me. To all of the close friends I have had here at Reed, thank you so much! Far too many of you have meant far too much to me to start listing it all out, but you’ve meant 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 been fantastic, and I couldn’t imagine a better place for me. You have made me so proud over these years, and I am excited about what you can accomplish as you continue your 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 . . . . . . . . . . . . . . . . . . . . . . . 3 1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.1 Speed Threshold . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.2 Emission as a Cone . . . . . . . . . . . . . . . . . . . . . . . . 4 1.1.3 The Frank-Tamm Equation . . . . . . . . . . . . . . . . . . . 4 1.2 A Field Oscillator Approach . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Predicting a Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Chapter 2: Color Matching . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.1 Color Matching Functions . . . . . . . . . . . . . . . . . . . . . . . . 15 2.1.1 Color Vision . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.1.2 RGB Color Matching Functions . . . . . . . . . . . . . . . . . 17 2.1.3 XYZ Color Matching Functions . . . . . . . . . . . . . . . . . 19 2.2 Color Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2.1 The xy Color Space . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2.2 The CIE RGB Color Space . . . . . . . . . . . . . . . . . . . . 22 2.2.3 Uniform Chromaticity and Hue Angle . . . . . . . . . . . . . . 22 Chapter 3: Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.1 Experimental Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.1.1 Reed Research Reactor . . . . . . . . . . . . . . . . . . . . . . 25 3.1.2 Collection Tool . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.2 Method of Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.2.1 From Power Spectrum to Color . . . . . . . . . . . . . . . . . 26 3.2.2 From Data to Power Spectrum . . . . . . . . . . . . . . . . . 27 3.2.3 Prediction of Spectrum . . . . . . . . . . . . . . . . . . . . . . 30 Chapter 4: Data & Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.1 Predicted Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.1.1 The Effect of β . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.1.2 The Effect of Index of Refraction . . . . . . . . . . . . . . . . 34 4.1.3 The Effect of Absorption . . . . . . . . . . . . . . . . . . . . . 35 4.2 Observed Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
    • 4.3 Analysis of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.3.1 From Water . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.3.2 From Cinnamaldehyde . . . . . . . . . . . . . . . . . . . . . . 38 4.3.3 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 . . . . . . . . . . . . . 4 1.2 Polarization Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3 Limit of sin(ωλuλt)/(ωλuλ) . . . . . . . . . . . . . . . . . . . . . . . . 13 1.4 Integrand Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1 Overall sensitivity curve adapted from Hunt (1991). . . . . . . . . . . 16 2.2 Cone sensitivity curves adapted from Hunt (1991). . . . . . . . . . . . 16 2.3 RGB Color Matching Curves adapted from Hunt (1991). . . . . . . . 18 2.4 The 1931 Standard Observer XYZ Color Matching Functions. . . . . 19 2.5 The 1931 Standard Observer x,y Chromaticity Diagram. . . . . . . . 20 2.6 The 1931 Standard Observer x,y Chromaticity Diagram with the CIE RGB color space shown. . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.7 The 1976 Uniform Chromaticity Scale Diagram for u and v . . . . . . 23 3.1 Diagram of the experimental set up used. . . . . . . . . . . . . . . . . 26 3.2 Converting a spectrum to XYZ values . . . . . . . . . . . . . . . . . . 27 3.3 Calibration Source Spectrum . . . . . . . . . . . . . . . . . . . . . . . 28 3.4 Efficiency Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.5 Corn Oil Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.6 Diagram of Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . 32 4.1 The Effect of β on Color . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.2 The Effect of Refractive Index on Color . . . . . . . . . . . . . . . . . 34 4.3 The Effect of Absorption on Color . . . . . . . . . . . . . . . . . . . . 35 4.4 Raw Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.5 Collected Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.6 Water Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.7 Comparison of the perceived hues of cinnamaldehyde and water . . . 38 4.8 Comparison of water and cinnamaldehyde spectra . . . . . . . . . . . 38 4.9 Corn Oil Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.10 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 the Reed 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 on perceived color are examined. These spectra were also compared to those predicted by the Frank-Tamm formula. It is determined that the most significant changes in color will be the result of absorption, though index of refraction can affect both perceived brightness and hue.
    • Introduction “And in the somber shed where, in the absence of cupboards, the precious particles in their tiny glass receivers on tables or on shelves nailed to the wall, their phosphorescent bluish 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 with Cerenkov 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 without words and simply gasp.) The glow is captivating to any observer, and when I first saw it my Freshman year, it blue my mind. The phosphorescent aura captivated my imagination, and was an instant interest of mine. I can’t guarantee I would still be working 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 fact has always featured prominently in the tours I have given: that replacing the water with another substance would change the color’s glow. Excited tour guides would sometimes make outlandish claims that it could be purple, or red, or hot pink. While I never really believed that a red glow was possible, I always wanted to know what was. 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 the work of Marie and Pierre Curie. The eerie blue glow that surrounds any submerged radioactive substance has long been a source of interest for curious minds. The first real study of its properties, though, was when Mallet observed the spectrum of light produced by γ sources in distilled water and carbon disulphide. (Mallet, 1926) His work showed that the effect was very different from fluorescence, the contemporary hot topic. Separately, Cerenkov began studying this radiation. He showed one could produce 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 first theory of this radiation, he experimentally verified their results. (Frank & Tamm, 1937) These researchers concluded that Cerenkov Radiation is produced by charged particles 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 then escape the atom and travel at high speeds. This secondary ionization provides most of the glow surrounding radioactive sources and nuclear reactors. 1.1.1 Speed Threshold Not just any charged particle can emit Cerenkov Radiation. This seems intuitively obvious: charged particles are around us all the time, and yet we only see Cerenkov Radiation in exceptional circumstances, like near reactor cores. The predominant reason 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 vacuum and 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 in its medium in order to produce radiation. This is one of the reasons it is frequently likened 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 the light produced by the fast-moving charged particle with velocity v is generated along the walls of a cone. Specifically, the radiation is emitted at an angle that satisfies cos θ = c vnω = vT v . (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 that this relation carries with it the speed threshold. If the velocity of the particle is less than vT then the right side of Equation 1.2 is greater than one, and is not physically realizable. Figure 1.1: Comparison of shock waves in water and in Cerenkov Radiation. Adapted from 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. As long 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-shaped crest forms behind it, like shown in Figure 1.1. The Cerenkov cone could very well be 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 Tamm in 1937, it included a prediction for the power P emitted by a single charged particle: P = e2 vµ0 4π v>vT 1 − c2 v2n2 ω ω dω (1.3)
    • 1.2. A Field Oscillator Approach 5 This equation, known as the Frank-Tamm Equation, gives total the power emitted in radiation by a particle of charge e (the electron charge) traveling at speed v through a medium with index refraction nω. The shape of the spectrum produced by a single particle, then, is determined by the term in square brackets. The index of refraction term accounts for the dependence of the spectrum on the material, and the ω virtually guarantees 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 condition holds, 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 is still valid, some modern approaches are more comprehensible, yield the same results, and offer better insight into the mechanism of Cerenkov Radiation. In this section, I will 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 of cavity modes. This derivation closely follows the one in Razpet & Likar (2010). A similar approach is taken for dipole radiation in Razpet & Likar (2009). The approach is based on the expression for energy stored in the electric and magnetic fields: H = 1 2 0E2 + 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 to split 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.
    • 6 Chapter 1. Cerenkov Radiation Then, we can expand Equation 1.4. H = 1 2 0E2 + 1 µ0 B2 dV = 1 2 0(Et + El)2 + 1 µ0 B2 dV = 1 2 0 E2 t dV + 0 El · EtdV + 1 2 0 E2 l dV + 1 2µ0 B2 dV (1.7) We note that the second integral in Equation 1.7 vanishes as follows: El · EtdV = − Φ · − ∂A ∂t dV = ∂ ∂t · ΦA dV where we have used the fact that · A = 0. From the divergence theorem, · ΦA dV = ΦA · ds where ds is a surface element on the large cubical volume V. Given that the radiation fields we are considering vanish at large distances, ΦA · ds = 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 = · E = · − ∂A ∂t − Φ = − ∂ ∂t · A − 2 Φ = − 2 Φ. 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 to radiation, we can neglect the integral E2 l dV . Without these two terms, Equation 1.7 can be rewritten as H = 0 2 E2 t dV + 1 2µ0 B2 dV. (1.8) The next step, then, is to find A, which we can do by expanding it as a Fourier series. The components of this expansion represent standing waves in an imagined cubical cavity of volume V and side length L, A (r, t) = 1 √ V 0 λ,i qλi (t) Aλi (r) . (1.9) where λ represents the cavity mode for the Fourier series term. In order to obey our chosen boundary conditions, this expansion includes two sets of plane wave basis functions represented by i = 1, 2. Aλ1 = √ 2 nλ eλ cos kλ · r , Aλ2 = √ 2 nλ eλ sin kλ · r (1.10)
    • 1.2. A Field Oscillator Approach 7 with the vectors eλ and kλ being the polarization and wave vectors, respectively. For these 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 quantization requirement for kλ given by: kλ = 2π L nx, 2π L ny, 2π L nz (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 0 λ,i ˙qλiAλi E2 t = 1 V 0 λ,i µ,j ˙qλi ˙qµj(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 will just pick out the terms where µ = λ and i = j, yielding a factor of V , so we get V E2 t = 1 0 λ,i ˙q2 λi. (1.14) Meanwhile, a similar treatment can be applied to the magnetic field. B = × A = 1 √ V 0 × λ,i qλiAλi = 1 √ V 0 λ (qλ1Aλ1 − qλ2Aλ2) × kλ B2 = 1 V 0 λ qλ1 Aλ1 × kλ − λ qλ2 Aλ2 × kλ 2 = 1 V 0 λ q2 λ1 Aλ1 × kλ 2 − 2qλ1qλ2 Aλ1 × kλ · Aλ2 × kλ + q2 λ2 Aλ2 × kλ 2 .
    • 8 Chapter 1. Cerenkov Radiation where (a × b) · (c × d) = (a · c) (b · d) − (a · d) (b · c) . Employing this, the summed terms become (Aλ1 × kλ) · (Aλ1 × kλ) = A2 λ1k2 λ − (Aλ1 · kλ)2 = A2 λ1k2 λ, (Aλ2 × kλ) · (Aλ2 × kλ) = A2 λ2k2 λ − (Aλ2 · kλ)2 = A2 λ2k2 λ, (Aλ1 × kλ) · (Aλ2 × kλ) = (Aλ1 · Aλ2) k2 λ − (Aλ1 · kλ) (Aλ2 · kλ) = (Aλ1 · Aλ2) k2 λ. The square of the magnetic field then becomes B2 = 1 V 0 λ q2 λ1A2 λ1k2 λ + (Aλ1 · Aλ2) k2 λ + q2 λ2A2 λ2k2 λ . Then, plugging this into the second integral of Equation 1.8 and applying the orthogonality condition, Equation 1.13, the cross term disappears, giving V B2 dV = 1 0 λ,i q2 λik2 λ. (1.15) Combining these results, we can rewrite Equation 1.8. H = 0 2 E2 t dV + 1 2µ0 B2 dV = 1 2 λ,i ˙q2 λi + 1 2 0µ0 λ,i q2 λik2 λ = 1 2 λ,i ( ˙q2 λi + c2 q2 λik2 λ) = 1 2 λ,i ( ˙q2 λi + ω2 λq2 λi), (1.16) where the speed of light in vacuum is c = 1√ 0µ0 . Clearly, then, if we are able to obtain an expression for the qλi, we will be able to compute the energy stored in the field, and thus the power irradiated. To obtain this expression, we start with the familiar Maxwell Equation × B = µ0J + 1 c2 ∂E ∂t . (1.17) Using the definitions of E and B from Equation 1.5, this becomes × × A = µ0J − 1 c2 ∂2 A ∂t2 − 1 c2 ∂Φ ∂t .
    • 1.2. A Field Oscillator Approach 9 Noting, for a charge distribution ρ moving at velocity v, J = ρv and × × A = − 2 A + · A = − 2 A our expression becomes 2 A − 1 c2 ∂2 A ∂t2 = −µ0ρv + 1 c2 ∂Φ ∂t . We then use Equation 1.10 to expand this equation. 1 √ 0V λ,i qλi 2 Aλi − 1 c2 1 √ 0V λ,i ¨qλiAλi = −µ0ρv + 1 c2 ∂Φ ∂t . (1.18) We can simplify this a little bit by evaluating 2 Aλi using the identities · (ue) = e · u , (ue) = u × e as follows. 2 Aλ1 = 2 √ 2eλ cos kλ · r = · √ 2eλ cos kλ · r − 2 √ 2eλ cos kλ · r = √ 2 cos kλ · r kλ · eλ kλ − k2 λeλ = √ 2 cos kλ · r k2 λeλ = −k2 λAλ1. The same is true for i = 2, reducing Equation 1.18 to 1 √ 0V λ,i qλik2 λAλi + 1 c2 1 √ 0V λ,i ¨qλiAλi = µ0ρv − 1 c2 ∂Φ ∂t . The presence of Aλi terms under sums on the left side of this prompts a use of the orthogonality condition, Equation 1.13. To do this, we multiply both sides by Aµj and integrate over the volume V to obtain 1 √ 0V qµjk2 µV + 1 c2 1 √ 0V ¨qµjV = V µ0ρv · Aµj − 1 c2 Aµj ∂Φ ∂t dV. The second term in the integral on the right is equal to 0, which leaves just
    • 10 Chapter 1. Cerenkov Radiation 1 √ 0V qµjk2 µV + 1 c2 1 √ 0V ¨qµjV = V µ0ρv · AµjdV. To clear this up even further, we can multiply everything by c2 0 V , exploiting the relations 0 = 1 µ0c2 and c2 k2 µ = ω2 µ and then rewrite our µ, j as λ, i. qλiω2 λ + ¨qλi = 1 √ V 0 V ρv · AλidV. (1.19) In the case of Cerenkov radiation, ρe represents a moving point charge, so we can write 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 + ω2 λqλ1 = √ 2 nλ √ V 0 e0(v · eλ) cos(kλ · re(t)) ¨qλ2 + ω2 λqλ2 = √ 2 nλ √ V 0 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 appearing in these equations. In order to simplify the expression v·eλ, we can break it apart into perpendicular 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
    • 1.2. A Field Oscillator Approach 11 component lies along eλb, which is still perpendicular to kλ, but is also perpendicular to v. Since this component is perpendicular to the particle’s motion, its contribution to 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 + ω2 λqλ1 = √ 2 nλ √ V 0 e0(−v sin γ) cos(kλ · re(t)), ¨qλ2 + ω2 λqλ2 = √ 2 nλ √ V 0 e0(−v sin γ) sin(kλ · re(t)). (1.22) These equations describe driven harmonic oscillators. The equations look pretty messy, but for the most part the terms on the right sides of these equations are constants. The mechanics all lie in the terms kλ · re(t) = ωλnλv cos γ c t This means the right-hand sides of Equation 1.22 represent driven oscillators with driving frequency ωd = ωλnλv cos γ c . (1.23) We suddenly have pretty simple solutions for qλ1,2 so long as we’re willing to put up with some ugly constants. In fact, the exact solutions for Equation 1.22 are qλ1 = −α cos(ωλuλt) − cos(ωλt) ω2 λ(1 − u2 λ) qλ2 = +α sin(ωλuλt) − sin(ωλt) ω2 λ(1 − u2 λ) (1.24) Where we introduce two factors, uλ and α to clean things up. uλ ≡ ωd ωλ = nλv cos γ c , α ≡ √ 2e0v sin γ nλ √ V 0 (1.25) At this point, we can take a brief pause. Soon we will use these expressions for qλ1,2 to finally solve for the power emitted by Cerenkov radiation, but first we can address these important results. What we’ve found is that the charged particle, as it’s moving, will excite certain modes of the vector potential. The exact mode excited is all carried in the definition of uλ above. A given particle traveling at speed v through a medium of index of refraction nλ will drive the resonant modes that correspond to light at angle γ. That is, at speed v, the associated driving frequency ωd will resonantly excite the cavity
    • 12 Chapter 1. Cerenkov Radiation mode ωλ = ωd. Then, with uλ = ωd ωλ = 1, the speed threshold comes from the rest of Equation 1.25. cos γ = c nλv = 1. since cos γ can be at most 1, v can be at minimum c nλ . 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 = e2 0v2 0V λ sin2 γ n2 λ (1 + u2 λ)(1 − cos((1 − uλ)ωλt)) ω2 λ(1 − u2 λ)2 + sin(ωλt) sin(ωλuλt) ω2 λ(1 + uλ)2 (1.26) Where we can convert this sum over the modes into an integral over dNλ. This quantity represents the number of modes per frequency interval dωλ per solid angle dΩ = 2πd(cos γ). Integrated over all of the solid angles and frequencies, this integral covers all of the possible cavity modes. The number of modes per solid angle and frequency interval is dNλ = n3 λω2 λV dωλdΩ (2πc)3 . However, instead of integrating over dΩ, we can integrate over duλ using dΩ = 2πd(cos γ) = 2πd uλc nλv = 2πc nλv duλ dNλ = n2 λω2 λV dωλduλ (2πc)2 . Rewriting Equation 1.26 as an integral over dNλ as defined above, we get H = e2 0vµ0 4π2 ∞ 0 dωλ nλv/c 0 sin2 γ (1 + u2 λ) (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 would happen at large t. As shown in Figure 1.3, the term sin(ωλuλt)/(ωλuλ) approaches a δ function centered around ωλuλ = 0 of area π. Using this information, we can rewrite that term of Equation 1.27 as
    • 1.2. A Field Oscillator Approach 13 Figure 1.3: Visual depiction of the limit of sin(ωλuλt)/(ωλuλ) for large t. The plots show this value as a function of uλ for increasing values of t. As t increases, the plot becomes more and more peaked approaching a δ function. The area under each curve is π. ∞ 0 sin2 γ (1 + uλ)2 sin(ωλt)ωλuλπδ(ωλuλ)duλ Which is just equal to zero. Without this term, we can rewrite Equation 1.27 as H = e2 0vµ0 4π2 ∞ 0 dωλ nλv/c 0 sin2 γ (1 + u2 λ) (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 the term in the limit as a function of uλ for increasing values of t. As t increases, the plot becomes more and more peaked, approaching a δ function. The area under each curve is ωλπt. As shown in Figure 1.4, the integrand simplifies at large t because lim t→∞ (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 the sine term. By the definition of uλ, we have
    • 14 Chapter 1. Cerenkov Radiation cos γ = cuλ vnλ , sin2 γ = 1 − c2 u2 λ v2n2 λ . (1.30) Using this expression, and plugging in the delta function, we reduce Equation 1.28 to H = e2 0vµ0 4π2 ∞ 0 dωλ nλv/c 0 1 − c2 u2 λ v2n2 λ (1 + u2 λ) (1 + uλ)2 ωλπtδ(uλ − 1) duλ H = e2 0vµ0 4π ∞ 0 1 − c2 v2n2 λ tωλdωλ. (1.31) Note that the integral ωλ only covers frequencies where nλv/c > 1. In order to get the radiated power, we need only take the time derivative of this. P = dH dt = e2 0vµ0 4π ∞ 0 1 − c2 v2n2 λ 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 a single charged particle. However, producing an accurate model for the glow surround- ing a nuclear reactor is much harder. Around a reactor, the spectrum of Cerenkov Radiation would depend on the spectrum of charged particles present. In such an environment, charged particles can be produced in a huge variety of ways. A few examples 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 any of the γ radiation emitted from the same sources listed above. Further, these photons could have interacted via Compton scattering before being absorbed, which would have to be accounted for. Beyond that, the directionality associated with Cerenkov Radiation might mean some sources contribute more to the overall spectrum than others. In the end, a full theoretical treatment of such a spectrum is beyond the scope of this experiment. If a β spectrum could be taken, it could be transferred into a Cerenkov 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 a spectrum. 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 wavelengths than at higher ones, so the color should be at the very least, bluish. To understand exactly how these spectra would look, though, requires knowledge of color vision. For the 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 of different wavelengths. Figure 2.1 shows the sensitivity curve that represents overall perceived brightness. From the curve, it is clear that peak sensitivity occurs around 550 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 practical purposes, the contribution to color vision by rod cells can be ignored for optical stimuli greater than 10−2 cd/m2 . For reference, the night sky is around 10−3 cd/m2 , where as the the moon is around 2.5 cd/m2 . Since Cerenkov Radiation is bright enough, we need only worry about the cone cells. Cone cells come in three varieties, usually dubbed β, γ, and ρ. Each cone detects light 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 shown for each of these in Figure 2.2. An overly simple, but illustrative model, is to think of these 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 ρ cones would be red. If you were to stare at a strongly yellow light, the γ and ρ cones would activate. Looking at a turquoise light would instead cause the β and γ cones to 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 spectra produce the same responses on all three cones, then they will physically look the same. For example, the light reflected off of an orange flower petal could be a full spectrum of wavelengths. When a photo of the same flower is viewed on a computer screen, the color is reproduced by a combination of just red, green, and blue. Because the cones in your eye produce the same response in both cases, the colors will appear identical. Because color vision relies on only three varieties of cones, we should be able to express any color as set of three numerical values. The goal of color matching is to determine these three numerical values for any given spectrum of light. The ideal values would be the exact responses of the three types of cones, but the curves in Figure 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 experiments performed in the late 1920s. In both experiments, participants were shown a test color to match. On an adjacent screen, the participants could control an additive mixture of red, green and blue light. They then adjusted the mixture until the colors matched exactly. From this, the researchers could determine how much red, green, and blue light respectively were required to match any given wavelength of light. One challenge the CIE faced was that the two experimenters used different light sources for their red, green, and blue lamps. To homogenize the results, each data set was converted to what would have been obtained if they had mixed precise wavelengths. Their chosen wavelengths were: Red = 700.0 nm Green = 546.1 nm Blue = 435.8 nm The first thing they discovered was that equal luminances of red, green, and blue light did not make white light. For a variety of reasons, lights of the same luminance do not appear to have the same intensity to an observer. In order to make white light, you have to mix red, green, and blue equal in perceived intensity, not physical intensity. To account for this, new units were proposed so that one “Red Unit” of red, one “Green Unit” of green and one “Blue Unit” of blue would mix to make white light. These units were defined as: 1 of R = 1 cd/m2 at 700.0 nm, 1 of G = 4.5907 cd/m2 at 546.1 nm, 1 of B = 0.0601 cd/m2 at 435.8 nm, (2.1) The second thing they discovered, though, was that certain colors could not be matched exactly using their three chosen colors. Instead, for certain wavelengths of light, 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 by 39 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 coefficients of R, G, and B can be read directly off the plots for a given wavelength. These
    • 18 Chapter 2. Color Matching 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 exceptions are at the wavelengths chosen to represent R, G, and B. This reflects the fact that it is impossible to create the color of a pure wavelength by mixing other wavelengths of light. Mixing colors using tristimulus values is incredibly simple, though. If you have two colors, matched by the tristimulus values R1, G1, B1 and R2, G2, B2, the mixture of the colors would be matched by R1 +R2, G1 +G2, B1 +B2. Mixing different amounts of different colors is also easy. Let’s say you’re mixing some quantity, α of color A with 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 general form of this can be written out. 1 unit of C is matched by cλ¯rλ of R plus cλ¯gλ of G plus cλ ¯bλ of B. (2.3) where ¯rλ, ¯gλ, ¯bλ are the functions shown in Figure 2.3. The coefficients cλ are the relative 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 to tristimulus 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 the previous section are as a set of basis vectors. Every visible color appears somewhere in the space spanned by these vectors, and we can use the color matching functions to reduce 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 section discusses 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 tristimulus values that were always positive for visible colors. This was done by selecting a set of 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 2.4 were set so their ratios were the same as in Equation 2.1. The effect of this is to give the color matching function the same shape as the sensitivity curve shown in Figure 2.1. The final constraint placed on the XYZ values was to make it so that white was represented by the point where X = Y = Z. The white in the RGB system had been
    • 20 Chapter 2. Color Matching defined by R = G = B, so the coefficients for each value in Equation 2.4 were chosen to 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 wavelengths of 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 by comparing 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 space that contains all of the visible colors. However, in order to display the visible color space on a page, a two dimensional space is required. The easiest way to do this is to 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 values is linearly independent. Conventionally, the values x and y are chosen, and the pair define the two dimensional color space shown in Figure 2.5. This diagram is usually called the CIE x,y chromaticity diagram. Note that only colors where X, Y, Z > 1 are shown, as these represent the entire visible color space. By restricting the space to two dimensions, all that has been lost is some measure of intensity: all points in the 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 interesting features.
    • 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 and blues) 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 in the diagram. This last point naturally extends to a mixture of three or more colors. In any case, the area contained by the points representing those colors contains all of the possible mixtures of those colors.
    • 22 Chapter 2. Color Matching 2.2.2 The CIE RGB Color Space Figure 2.6: The 1931 Standard Observer x,y Chromaticity Diagram with the CIE RGB color space shown. The color space representing the wavelengths specified in Equation 2.1 can be shown on the x,y chromaticity diagram as in Figure 2.6. In this diagram, the point E refers to the white point. The triangle shown connects the three wavelengths to be mixed: 435.8 nm, 546.1 nm, and 700.0 nm. Every point within the triangle represents a color which can be created by mixing the three wavelengths. Note that a large portion of the diagram lies outside this triangle, though. Each of these colors lies outside the CIE RGB color space, meaning it cannot be created by mixing the three colors that define that color space. Among the colors outside the space are all of the other pure wavelengths. Before, this was explained by having negative R, G, or B values. For the most part, though, colors outside the space can be 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 approximate it 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 distribution of distinct colors. Even the humblest observer will note that the diagram devotes much more space to teal than any of the other colors. In fact, if you were to calculate the distance between colors that were “equally different” from each other, you would find 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 distinctly different, you would have to move much further than if you were starting from the orange 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” was developed 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 from each other. This measure is called the “hue-angle” and is obtained by: huv = arctan[(v − vn)/(u − un)] (2.7) Where vn and un are the coordinates of a suitably chosen “reference white.” To get an idea of what this image represents, imagine a conical color space. In this color space, “hue” is represented by the angle around the z-axis, so that by traveling a full 2π around the axis, you pass through every color. The radial coordinate then represents saturation, where colors close to the origin are paler, and colors further away are richer. Distance along the z-axis represents brightness. In this system, the angle between two colors represents the difference between the two hues. On top of being a useful quantitative measure, many popular image processing programs can take advantage of these angles. Adobe Photoshop, for example, can transpose entire images, or just sections of them, by any given hue angle. Given a photograph of the reactor at power, and a hue angle, the new colors can be visualized in 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 on the Reed College Campus. It is a TRIGA Mark II Nuclear Reactor with a maximum licensed thermal output of 250 kW. TRIGA type reactors use uranium zirconium hydride fuel and are designed to operate at relatively low temperatures. Like many TRIGA 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 the Cerenkov Radiation it produces. The β spectrum in the area around the core would be comparable to any other low enriched, uranium based reactor. For this reason, the results of this experiment would 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 a variety of fluids and placed near the Reactor Core. The chamber has a capacity of just 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 because it is resistant to corrosion, light weight, and has a short half life. Because the chamber will be close to the reactor, the aluminum will activate, but the half life of Al-28 is only 2.24 minutes. (National Nuclear Data Center, 2013) After approximately one hour in the pool, nearly all of the radioactive aluminum will have decayed, and the chamber can be safely removed from the tank. The bottom of the chamber is a sheet of 0.125” thick aluminum held in place by aluminum bolts. The bottom is kept thinner than the walls so that radiation from the core can enter the chamber. As discussed earlier, the primary cause of Cerenkov Radiation is free radicals produced by gamma radiation. The chamber was suspended on a rigid aluminum rod into the reactor 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 Collector produced by HOBI Labs. This sort of detector is designed to collect light over the 180◦ in front of it. The HOBI Labs model is explicitly designed to interface with water. The light incident on the collector is directed into a 600 µm diameter fiber optic cable. The other end of the 10 meter cable was attached to an Ocean Optics USB2000+ Spectrometer. The spectrometer then interfaced with a computer running the SpectraSuite software, which was used to collect spectral data. Each spectrum was taken as ten 60 second counts averaged together. For more information, the approval request sent to the Reed Research Reactor operations 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 color coordinates. For each spectrum generated in this experiment, the following approach was used to convert it to a perceived color. First, each spectrum was converted to XYZ coordinates using the color matching functions shown in Figure 2.4 and listed in Appendix B. The form is the same as in Equation 2.3, but using the ¯x, ¯y, and ¯z matching functions instead. X = 780 λ=380 cλ ¯x , Y = 780 λ=380 cλ ¯y , Z = 780 λ=380 cλ ¯z (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 CIE published values for ¯x, ¯y, and ¯z. According to Hunt, 5 nm intervals provide sufficient precision for all applications. The extreme case of this would be to imagine the sums as integrals. In that case, the X, Y, and Z coordinates would represent the areas under the product of the spectra and the matching functions. The process is illustrated in Figure 3.2. In the illustration, the matching functions are shown in the top left, an example spectrum in the bottom left, and the product of the two is shown on the right. The X, Y, Z coordinates 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 the shaded regions correspond to the XYZ tristimulus values. To image the colors, we must convert from XYZ to RGB. The matrix used to convert from XYZ to the RGB used by Mathematica is:   R G B   = M   X Y Z   , M =   0.412387 0.212637 0.0193306 0.357591 0.715183 0.119197 0.18045 0.0721802 0.950373   (3.2) Each set of RGB coordinates was then normalized so that the greatest coordinate was equal to 1. So, for example, if the coordinates were R = 1, G = 2, B = 4, the new coordinates would be R = 0.25, G = 0.5, B = 1. This step ensures that the hue is being compared, instead of brightness. For the relative brightness of the glow, the Y 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
    • 28 Chapter 3. Method 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 Instruments Quartz Tungsten Halogen Lamp and the best fit curve used. The first step in analyzing the spectral data is accounting for the efficiency of the setup. Because the detector is more sensitive to some wavelengths than others, the collected data spectrum isn’t exactly the spectrum of light seen. The efficiency of the detector, cable, light collector and any extraneous factors can be corrected easily by using a known light source. In this case, that light source is an Oriel Instruments Quartz Tungsten Halogen Lamp. Operating at 3000 K, it provided a clear black body spectrum to account for. Because we know that black body radiation takes the spectral form: P(λ) = C 1 λ5 1 eα/λ − 1 (3.3) for some values of C and α. Using manufacturer calibration results for the light source and Mathematica’s built in form fitting function, the values of C and α were found to be:
    • 3.2. Method of Analysis 29 C = 1970.53 Wµm5 , α = 4.50165 µm (3.4) The plot of Equation 3.3 using these values are shown alongside the manufacturer’s results 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 this experiment. The bottom of the chamber was removed, and the detector pointed towards 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 curve can then be applied to collected spectra to account for the sensitivity of the whole set 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 into the 5 nm bands used in color matching. To do this, we average the data points in the interval surrounding each wavelength used in Equation 3.1. So, the value of cλ in that 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 spectrum taken with the Reactor at full power, a spectrum was taken with it shutdown. To obtain the spectrum due solely to the Cerenkov Radiation, the shutdown spectrum was subtracted from the at power spectrum.
    • 30 Chapter 3. Method 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 for water (dashed line) is also shown, adapted from Smith & Baker (1981). Factoring in Absorption Viewed through meters of the material, though, the spectrum looks different, because different frequencies of light attenuate differently in matter. Values for the absorption coefficient 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 measured intensity without the medium, α is the attenuation coefficient in units of inverse distance 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 the perceived color. For example, in water, the attenuation coefficient is much higher over the red region of the spectrum than in the blue region. As a result, the red light will be attenuated more than the blue light, and the color will appear bluer than it would without all of the additional water. After factoring in the absorption in the material, the resulting spectrum is ready to 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. (The Frank-Tamm Equation) Specifically, the equation can be rewritten as dP dω = e2 vµ0 4π 1 − c2 v2n2 ω ω (3.8) All of the frequency dependence lies to the right of this equation, so the shape of the spectrum can be given entirely by
    • 3.2. Method of Analysis 31 ω 1 − c2 v2n2 ω . (3.9) Note that the constants would be important for determining the quantitative power output by Cerenkov Radiation, but do not factor into color matching other than overall brightness. In terms of comparing Cerenkov Radiation in two different materials, these constants can be safely ignored. So then for a given material, the functional 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 using Equation 3.1, we simply choose the values of the spectrum at each 5 nm increment. For water, Daimon & Masumura (2007) gives n2 λ = 1 + 5.68403 ∗ 10−1 λ2 λ2 − 5.10183 ∗ 10−3 + 1.72618 ∗ 10−1 λ2 λ2 − 1.82115 ∗ 10−2 + 2.08619 ∗ 10−2 λ2 λ2 − 2.62072 ∗ 10−2 + 1.13075 ∗ 10−1 λ2 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. While they have different forms, both are accurate over the visible spectrum, so they both work fine for this experiment.
    • 32 Chapter 3. Method 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 is related 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 c determines 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 speed β can only produce light over the wavelengths where Equation 1.1 is satisfied. In the visible spectrum, where refractive indexes are essentially always lower in the higher wavelength regions, then those regions will be the first to be cut off. Effectively, there are “threshold” regions for β where light is only being emitted in parts of the visible spectrum. For a charged particle traveling above this threshold region in water, the effects of increasing β are shown in Figure 4.1. Clearly, in the image, the color changes somewhat dramatically with the speed of the particle. Particles traveling at relatively low 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 refraction as discussed in the following section. Figure 4.1: The effect of particle speed on perceived color in water. Particle speed increases to the right. Mathematica RGB coordinates are displayed below each color. The leftmost color represents a β of 0.7525, and each square incrementally increases β 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 for β = 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 medium with n(λ) = 1.5. Mathematica RGB values are shown for each color. All refractive indexes pulled from Polyanskiy (2012) When changing the medium of travel, the first optical property to examine is the index of refraction. As with the β fraction, index of refraction features prominently in Equation 3.9. When switching between materials, their colors change as is shown in Figure 4.2. The most obvious reason for a change is that if the shape of the spectrum of n(λ) changes, then the shape of the Cerenkov spectrum will change. A different shape to the spectrum means a different color. For values of β sufficiently high, the effect of changing the index of refraction becomes increasingly minimal. For lower values of β, though, the effect can be quite strong. The reason for this is the interaction between β and n in Equation 3.9: ω 1 − c2 v2n2 ω . In this equation, the most dramatic changes in c2 v2n2 ω (and therefore the shape of the spectrum) will occur when the fraction is near 1, when the v is near c nω , the speed threshold. In other words, as the particle nears the speed threshold, the change in color accelerates. A change to β of 0.0005 will more significantly impact the color spectrum when the particle is traveling at a speed near the threshold than it will if the particle is traveling much faster. The same effect occurs when changing index of refraction, but in reverse. As a particle’s speed approaches the threshold from above for a fixed index of refraction, β nω changes more dramatically. In the same way, as the index of refraction approaches the threshold from below, the fraction changes more dramatically. Because the index of 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 particular wavelength will vary more dramatically at wavelengths where the index of refraction is closer to 1 β . That being said, the effect of index of refraction on perceived color appears to be somewhat minimal. Figure 4.2 shows a whole range of different refractive indexes and dispersions, and yet all of the colors are some shade of blue. The main barrier is that it is extremely rare to find a fluid with a refractive index that increases with wavelength in the visible spectrum. In fact, even a fluid with constant index of refraction would yield 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 each row 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 two rows represent particles traveling at β = 0.8. Rows 1 and 3 show the effect of water’s absorption spectrum, where each block represents an additional 1 meter of absorption. Rows 2 and 4 show the effect of corn oil’s absorption spectrum, where each block represents an additional 0.02 meters of absorption. As noted in the previous section, factoring in changes to refractive index can really only create different shades of blue. When factoring in absorption however, drastically different colors can be achieved. Figure 4.3 shows the effect of the absorption spectrum of water and corn oil. A functional form for the refractive index for corn oil could
    • 36 Chapter 4. Data & Analysis not be found, so the image shows the effect of corn oil’s absorption coefficients on the Cerenkov spectrum in water. While this means the colors depicted are not completely accurate for corn oil, they demonstrate the effect of two different absorption spectra on 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 end of the visible spectrum. This sharp rise increases absorption in the red portion of the spectrum, and the resulting color is shifted towards blue. As more water is placed between 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 the visible spectrum. This is shown in Figure 3.5. As a result, corn oil absorbs more blue than red or green, and the overall color shifts away from blue. The absorption coefficients are also generally much stronger than in water, so it only takes a few centimeters of oil to turn the original Cerenkov spectrum green. If a viewer was interested in changing the perceived color of the glow around reactors, this would by far the easiest way to do it. While refractive indexes of transparent 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 shut down (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 measured spectra looked like. After performing all of the efficiency correction and background subtraction, we obtained the Cerenkov Spectra shown in Figure 4.5. These spectra represent our major results. Clearly, the three materials produce different glows, and in 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 leftmost color comes directly from the collected spectrum. Each subsequent color adds 1 meter of absorption. At the Reed Research Reactor, there is approximately 6 meters of water above the core. The spectrum collected for Cerenkov Radiation through pure water is shown in Figure 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 the gamma spectrum near a reactor core. Nakashima et al. (1971) showed that the number 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 will be exponentially more photons close to the threshold than of higher energy. As a result, there will be generally more electrons with speed near the threshold than of higher energy. The overall color, then, should be dominated by “low” energy electrons, which is what we see.
    • 38 Chapter 4. Data & Analysis Figure 4.7: Comparison of the perceived hues of cinnamaldehyde and water. On the left 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 in Figure 4.7. As is pretty clear in the image, the produced color is extremely similar to that of water. This is unsurprising, as the shapes of the two refractive index curves are not dramatically different. However, cinnamaldehyde’s index of refraction is around 1.62 in the visible spectrum, compared to water’s 1.33. This overall increase does not change the hue, but does change the brightness of the glow. To show this, the spectrum obtained for cinnamaldehyde is compared with the spectrum 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 were full, the intensity would have likely been even larger. Figure 4.8: Comparison of water and cinnamaldehyde spectra. The solid line shows the magnitude of the cinnamaldehyde spectrum over the visible region, while the dashed 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 color matching 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 be 1.26 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 leftmost color comes directly from the collected spectrum. Each subsequent color adds 1 meter of absorption. At the Reed Research Reactor, there is approximately 6 meters of 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 dominates the color of the glow. By the time the light hits the detector, it has already passed through a few centimeters of oil, and is already somewhat green. The underlying blue glow 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 and corn 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’s rendering of the reactor in a pool of corn oil. This is shown in Figure 4.10. Note that this image is not precise by any means. It is simply an approximation based on the hue 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 the original image.
    • Chapter 5 Conclusion The purpose of this experiment was to examine the potential colors of Cerenkov Radiation as perceived around a nuclear reactor. To this end, we took spectra of this radiation produced in water, corn oil, and cinnamaldehyde. Through analyzing these spectra and predicting spectra from the Frank-Tamm equation, we came to the following conclusions. • The shape of the refractive index curve over the visible spectrum can affect the perceived hue of that Cerenkov spectrum. Every refractive index we used yielded a perceived hue that was clearly blue. Some variation was apparent between blues, but for the most part they were all blue. These results are shown 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 materials that absorb more at lower wavelengths produce more green/red light. These effects can be dramatic. These results are shown in Figure 4.3. • The measured spectrum for water produced a perceived color that matches visual 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. This was 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 as a 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 0.0014 385 0.002236 0.000064 0.01055 590 1.0263 0.757 0.0011 390 0.004243 0.00012 0.02005 595 1.0567 0.6949 0.001 395 0.00765 0.000217 0.03621 600 1.0622 0.631 0.0008 400 0.01431 0.000396 0.06785 605 1.0456 0.5668 0.0006 405 0.02319 0.00064 0.1102 610 1.0026 0.503 0.00034 410 0.04351 0.00121 0.2074 615 0.9384 0.4412 0.00024 415 0.07763 0.00218 0.3713 620 0.85445 0.381 0.00019 420 0.13438 0.004 0.6456 625 0.7514 0.321 0.0001 425 0.21477 0.0073 1.03905 630 0.6424 0.265 0.00005 430 0.2839 0.0116 1.3856 635 0.5419 0.217 0.00003 435 0.3285 0.01684 1.62296 640 0.4479 0.175 0.00002 440 0.34828 0.023 1.74706 645 0.3608 0.1382 0.00001 445 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
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