Deep Blue: Examining Cerenkov Radiation Through Non-traditional Media
Deep Blue: Examining Cerenkov Radiation Through Non-traditional Media
The Division of Mathematics and Natural Sciences
In Partial Fulﬁllment
of the Requirements for the Degree
Bachelor of Arts
Approved for the Division
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.
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.
“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)
- 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.
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,
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
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 =
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
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:
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
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 = −
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 = −
× El = × − Φ = − × Φ = 0,
· Et = · −
· A = 0.
6 Chapter 1. Cerenkov Radiation
Then, we can expand Equation 1.4.
0(Et + El)2
t dV + 0 El · EtdV +
l dV +
We note that the second integral in Equation 1.7 vanishes as follows:
El · EtdV = − Φ · −
· 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,
= · E = · −
− Φ = −
· A − 2
Φ = − 2
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
t dV +
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) =
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.
eλ cos kλ · r , Aλ2 =
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:
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 = −
V 0 λ,i
˙qλi ˙qµj(Aλi · Aµj) (1.12)
Where the orthogonality condition for our basis functions is
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
Meanwhile, a similar treatment can be applied to the magnetic ﬁeld.
B = × A
V 0 λ
(qλ1Aλ1 − qλ2Aλ2) × kλ
qλ1 Aλ1 × kλ −
qλ2 Aλ2 × kλ
λ1 Aλ1 × kλ
− 2qλ1qλ2 Aλ1 × kλ · Aλ2 × kλ + q2
λ2 Aλ2 × kλ
8 Chapter 1. Cerenkov Radiation
(a × b) · (c × d) = (a · c) (b · d) − (a · d) (b · c) .
Employing this, the summed terms become
(Aλ1 × kλ) · (Aλ1 × kλ) = A2
λ − (Aλ1 · kλ)2
(Aλ2 × kλ) · (Aλ2 × kλ) = A2
λ − (Aλ2 · kλ)2
(Aλ1 × kλ) · (Aλ2 × kλ) = (Aλ1 · Aλ2) k2
λ − (Aλ1 · kλ) (Aλ2 · kλ) = (Aλ1 · Aλ2) k2
The square of the magnetic ﬁeld then becomes
λ + (Aλ1 · Aλ2) k2
λ + q2
Then, plugging this into the second integral of Equation 1.8 and applying the
orthogonality condition, Equation 1.13, the cross term disappears, giving
Combining these results, we can rewrite Equation 1.8.
t dV +
λi + c2
λi + ω2
where the speed of light in vacuum is c = 1√
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 +
Using the deﬁnitions of E and B from Equation 1.5, this becomes
× × A = µ0J −
1.2. A Field Oscillator Approach 9
Noting, for a charge distribution ρ moving at velocity v,
J = ρv
× × A = − 2
A + · A
= − 2
our expression becomes
= −µ0ρv +
We then use Equation 1.10 to expand this equation.
¨qλiAλi = −µ0ρv +
We can simplify this a little bit by evaluating 2
Aλi using the identities
· (ue) = e · u , (ue) = u × e
Aλ1 = 2
2eλ cos kλ · r
2eλ cos kλ · r − 2
2eλ cos kλ · r
2 cos kλ · r kλ · eλ kλ − k2
2 cos kλ · r k2
The same is true for i = 2, reducing Equation 1.18 to
¨qλiAλi = µ0ρv −
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
µ0ρv · Aµj −
The second term in the integral on the right is equal to 0, which leaves just
10 Chapter 1. Cerenkov Radiation
µ0ρv · AµjdV.
To clear this up even further, we can multiply everything by c2 0
, exploiting the
relations 0 = 1
µ0c2 and c2
µ = ω2
µ and then rewrite our µ, j as λ, i.
λ + ¨qλi =
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)),
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
e0(v · eλ) cos(kλ · re(t))
¨qλ2 + ω2
e0(v · eλ) sin(kλ · re(t))
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
e0(−v sin γ) cos(kλ · re(t)),
¨qλ2 + ω2
e0(−v sin γ) sin(kλ · re(t)).
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
This means the right-hand sides of Equation 1.22 represent driven oscillators with
ωλnλv cos γ
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)
λ(1 − u2
qλ2 = +α
sin(ωλuλt) − sin(ωλt)
λ(1 − u2
Where we introduce two factors, uλ and α to clean things up.
nλv cos γ
, α ≡
2e0v sin γ
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
= 1, the speed threshold comes from the rest of
cos γ =
since cos γ can be at most 1, v can be at minimum c
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
(1 + u2
λ)(1 − cos((1 − uλ)ωλt))
λ(1 − u2
λ(1 + uλ)2
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
However, instead of integrating over dΩ, we can integrate over duλ using
dΩ = 2πd(cos γ)
Rewriting Equation 1.26 as an integral over dNλ as deﬁned above, we get
(1 + u2
(1 + uλ)2
(1 − cos((1 − uλ)ωλt))
(1 − uλ)2
(1 + uλ)2
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
(1 + uλ)2
Which is just equal to zero. Without this term, we can rewrite Equation 1.27 as
(1 + u2
(1 + uλ)2
(1 − cos((1 − uλ)ωλt))
(1 − uλ)2
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
(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 =
γ = 1 −
Using this expression, and plugging in the delta function, we reduce Equation 1.28
(1 + u2
(1 + uλ)2
ωλπtδ(uλ − 1) duλ
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.
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
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
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.
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
. For reference, the night sky is around 10−3
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
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,
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
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.
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.
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.
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
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 + Y + Z
, y =
X + Y + Z
, z =
X + Y + Z
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
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
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)
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.
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
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.
cλ ¯x , Y =
cλ ¯y , Z =
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:
, M =
0.412387 0.212637 0.0193306
0.357591 0.715183 0.119197
0.18045 0.0721802 0.950373
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:
Correct for Detector Eﬃciency
Average Over 5 nm Bands
Subtract Background Light
Factor in Absorption
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
eα/λ − 1
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
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:
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
Actual Spectrum =
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
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 −
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 =
. 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
λ = 1 +
5.68403 ∗ 10−1
λ2 − 5.10183 ∗ 10−3
1.72618 ∗ 10−1
λ2 − 1.82115 ∗ 10−2
2.08619 ∗ 10−2
λ2 − 2.62072 ∗ 10−2
1.13075 ∗ 10−1
λ2 − 1.06979 ∗ 101
For cinnamaldahyde, Rheims et al. (1997) gives
nλ = 1.57008 + 0.01523 λ−2
+ 0.00084 λ−4
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.
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 =
determines the strength of the 1/n2
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
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 −
In this equation, the most dramatic changes in c2
(and therefore the shape of
the spectrum) will occur when the fraction is near 1, when the v is near c
, 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,
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
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
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
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
• 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.
Cerenkov, P. (1934). Visible emission of clean liquids by action of gamma radiation.
Proceedings of the USSR Academy of Sciences, 2, 451.
Curie, E. (1940). Marie Curie. New York, NY: Garden City Publishing Co.
Daimon, M., & Masumura, A. (2007). Measurement of the refractive index of distilled
water from the near-infrared region to the ultraviolet region. Applied Optics, 46,
Frank, I., & Tamm, I. (1937). Proceedings of the USSR Academy of Sciences, 14,
Griﬃths, D. (1999). Introduction to Electrodyanamics. Upper Saddle River, NJ:
Hiscocks, P. D. (2008). Measuring light.
Hunt, R. W. G. (1991). Measuring Colour, Second Edition. Chichester, West Sussex:
Jelley, J. V. (1958). Cerenkov Radiation and its Applications. New York, NY: Perg-
Mallet, L. (1926). Comptes Rendus de l’Acadmie des Sciences, 183, 274.
Nakashima, Y., Minato, S., Kawano, M., Tsujimoto, T., & Katsurayama, K. (1971).
Gamma-ray energy spectra observed around a nuclear reactor. Journal of Radiation
Research, 12, 138–147.
National Nuclear Data Center (2013). Interactive chart of nuclides.
Polyanskiy, M. (2012). Refractive index database.
Razpet, N., & Likar, A. (2009). The electromagnetic dipole radiation ﬁeld through
the hamiltonian approach. European Journal of Physics, 30, 1435–1446.
Razpet, N., & Likar, A. (2010). Cerenkov radiation through the hamiltonian ap-
proach. American Journal of Physics, 78, 1384–1392.
Reed Research Reactor (2011). Training manual.
Rheims, J., Koser, J., & Wriedt, T. (1997). Refractive-index measurements in the
near-ir using an abbe refractometer. Measurement Science and Technology, 8(6),
Shabbar, A. (2012). Cerenkov Radiation from Beta-Decay. Undergraduate thesis,
Smith, R. C., & Baker, K. S. (1981). Optical properties of the clearest natural waters
(200-800nm). Applied Optics, 20, 177–184.
Vijayan, J., Slaughter, D. C., & Singh, R. P. (1996). Optical properties of corn
oil during frying. International Journal of Food Science and Technology, 31(4),
Watson, A. A. (2011). The discovery of cherenkov radiation and its use in the detec-
tion of etensive air showers.