This is the last of our four introductory lectures. We still have some loose ends, and
in today’s lecture, we will try to tie up some of these loose ends.
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We’re going to cover a variety of topics today. We’ll talk about relating wavelength
of radiation to the photon energy, that is, relate the wave nature of electromagnetic
radiation to the particulate view of electromagnetic radiation.
We’ll look at the various components of the electromagnetic spectrum.
We’ll talk a bit about how we refer to energies within the electromagnetic spectrum.
We’ll look at a little bit of relativity and be able to relate kinetic energy to
relativistic energy and look at rest mass in terms of energy.
Finally, we are going to look at exponential behavior, which is a behavior common
in radiation medicine, and identify quantities like half-life and average life.
So, there are a several loose ends that we will try to tie up in this lecture, and not
everything is going to be related.
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The first loose end is characterization of the quantum nature of radiation. We have
been talking about electromagnetic radiation as waves. We all know that from the
quantum theory that photon radiation has a dual nature; we can look it as either
waves or as particles.
There are a number of particle properties that a photon radiation beam will have.
We can relate the energy of the photons to the frequency and/or wavelength by the
relationship E = h or E = hc/λ. E is the photon energy and h is Planck’s constant,
which is 6.63  10 -34 Joule-seconds. The quantity  is the frequency of the waves,
λ is the wavelength, and c is the speed of light, which equals 3  108 meters per
second; c is really 3.00, or more precisely, 2.9979  108 meters per second.
In some circumstances we are going to look on electromagnetic radiation as a beam
of particles. We call these particles photons. Photons have no mass and photons
carry energy. The energy is going to be related to the frequency or wavelength of
the electromagnetic radiation.
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Let’s connect wavelength to energy.
We often express the energy in electron volts, which is a very common unit of energy. An electron
volt is the energy obtained by an electron accelerated through a potential of one volt.
We often express wavelengths in Angstrom units. It is very useful at times to relate the wavelength
in Angstrom units to the energy in electron volts.
So let’s plug in some numbers. The constant h, we said, is 6.63  10-34 Joule-second and c is 3 X 108
meters per second. So if we express the wavelength in meters, the conversion to Joules is 1.989 
10-25. That’s a very hard number to remember, and it is not very useful because we rarely express
wavelengths in meters.
But if we now convert units, we divide that 1.989  10-25 by 1.6  10-19 to convert Joules to electron
volts and multiply by 1010 Angstroms per meter, we find the conversion factor is a very easy number
to remember, 12. 4. So if you can express the wavelength in Angstrom and divide the wavelength
into 12.4 you will have the energy in kiloelectron volts.
I’m giving you this equation because it is an easy one to remember as there will be times when you
want to convert a wavelength to energy or vice versa and just remember that is energy in kiloelectron
volts is equal to 12.4 divided by the wavelength in Angstrom. That’s a useful relationship.
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Let’s look at some conventions for expressing energy. When do we use electron
volts and when do we use volts? When do we use kiloelectron volts and when to we
use kilovolts?
Electron volts, kiloelectron volts, and megaelectron volts are quantities used to
express energy. They are typically used to describe an electron beam or a proton
beam, which are nearly monoenergetic beams. These units are also used to describe
a monoenergetic photon beam.
Kilovolts peak and megavolts peak are quantities used to represent the maximum
energy of photons in a polyenergetic photon spectrum.
Volts, kilovolts, and megavolts are quantities that describe electrical potential. We
use these quantities to describe the accelerating potential in an X-ray machine or in
a linear accelerator.
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Let me give you an example of this. We will talk of an 18 MV photon beam exiting
a linear accelerator. That’s actually a nominal energy, a name given to describe the
energy of the beam. We could just as well have called them blue photons as well as
18 MV photons. It is just a name to differentiate the photons from a 6 MV photon
beam from a linac. But the key point in labeling an 18 MV linac beam is that the
electrons in the linac are accelerated to approximately 18 million electron volts of
energy. I said “approximately” because the actual acceleration depends on the
design of the linac. These electrons can be removed from the accelerator as an 18
million electron volt electron beam. So the electrons having been accelerated to 18
million electron volts can be removed as an electron beam or we can take these
electrons, have them impinge on a target and generate an 18 MV X-ray beam. The
beam really consists of a whole spectrum of X-rays that comprise the polyenergetic
beam emanating from the linac. In fact, the average energy is typically about 1/3 of
that. So roughly, the average photon energy is 6 MeV, but we say we have an 18
MV X-ray beam. Again it’s a name – maximum energy will be around 18 million
electron volts, mean energy will be roughly 6 million electron volts, but calling it 18
MV describes the energy spectrum.
If we have a cobalt machine, the energy of the photons are emitted from a cobalt
source are 1.25 MeV. So the 1.25 MeV designates a mono-energetic photon beam.
The phrase 18 MV, using the units MV rather than MeV, designates a poly-energetic
photon beam. We could have used MVp, but as a matter of convention, we don’t.
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Let’s look at other radiations in the electromagnetic spectrum.
When we talk about radio waves we usually characterize the radio waves in terms of
frequency. The AM frequency band goes from 550 kHz all the way up to 1500 kHz,
whereas the FM band ranges from 87.9 MHz to 107.9 MHz.
Microwaves are characterized by either frequency or wavelength. 450 MHz
microwaves are the frequency that has sometimes been used in hyperthermia; that
is, the use of heat for treatment. 3,000 MHz microwaves are produced by the power
supply in the linear accelerator. Now, if we look at the wavelength that corresponds
to that frequency we find that wavelength is roughly about a centimeter. In fact a
cross sectional dimension of the wave guide in the linear accelerator is about 1
centimeter. So that gives us some hint as to the design of the linac. If we want
something to have waves of roughly 1 centimeter traveling down this wave guide
we need a cross section of about a centimeter.
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Now we will look at infrared, visible and ultraviolet light. Here we characterize the radiation in terms of its wavelengths,
which are typically expressed in Angstrom or nanometers. We really should be using nanometers, but I think that out of force
of habit, physicists tend to use Angstrom.
And finally, when we talk about X-rays, we have the following energy ranges: For diagnostic imaging, energies are in the
kilovolt (kVp) range, mammography is 30 to 40 kVp; x-rays used in conventional imaging are about 70 to 140 kVp, and xrays used in CT scanners are about 120 kVp.
For radiation therapy, we actually have some low energy therapy called Grenz rays, an old-fashioned therapy device, less that
20 kVp. Going up in energy, we have contact X-rays. You may have seen some of these units but those devices are fairly old.
They operate in the energy range 20 to 50 kVp. Superficial X-rays go from 50 to 150 kVp, orthovoltage are in the range 150 to
300 kVp. Finally megavoltage x-rays, which is the most common radiation used in radiation therapy these days, are in the
range 1 MV to around 50 MV, with the most common energies in radiation therapy in the range of 6 MV to about 20-25 MV.
So this gives you kind of a feel for the energies that we use.
Here’s a question. What are the implications of selection of energy in diagnostic x-ray imaging?
It turns out, and we’ll see this in more detail later in the course, that lower energy X-rays provide a lot more contrast. In
mammography, for example, you are trying to look at very small calcifications in essentially a fatty matrix. So it turns out that
the low energy X-rays are needed for breast imaging.
As you increase the energy of the x-rays, you increase the penetrating ability. You wind up with less contrast with higher
energies but you might want to penetrate thicker body parts, delivering lower dose to the patient at the higher energies, so we
go up in the range of 70 to 140 kVp. With the higher energies, there is less attenuation of the x-ray beam, so for the same
amount of radiation reaching the image receptor, the higher energies deliver less radiation dose. The specific energies we use
in imaging are selected based on the nature of the examinations.
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Now we’ll talk a bit about relativity.
We know that in classical physics, kinetic energy is one-half mass times velocity squared. You all
got that in your kindergarten physics class. What happens when we go to high kinetic energy? In
radiation oncology we use electrons with energies of 1 MV, or 10 MV, or something like that. If we
try to use the classical equation for kinetic energy, we find that we will have electron velocities that
exceed the speed of light. That’s clearly a “No No.” We are not allowed to do that. As the electrons
increase in energy the speed of travel of the electrons starts reaching the speed of light so we need to
start worrying about relativity. The increase in energy of an electron when we are at these higher
energies really comes from an increase in the mass of the electron. Mass increases so we never
really get to the speed of light. The energy of the electron now is given by mc2 where m is now the
relativistic mass, not the rest mass.
The rest energy is m0c2, in which m0 is the rest mass of the electron, but now the mass increases with
the velocity. So one of the problems with electrons is that if we’re dealing with relativistic energy,
we have an increase in mass and the energies we deal with are relativistic.
This is one of the reasons we don’t use a cyclotron to accelerate electrons. A cyclotron is a device
that accelerates charged particles in a magnetic field. However, in order for a cyclotron to work, the
mass of the accelerated particle has to be constant. For protons used in radiation therapy, the mass is
constant, but because of relativistic effects, for electrons in radiation therapy, the mass is not
constant.
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So the mass increases with the velocity. Beta is the velocity divided by the speed of
light. The mass is equal to the rest mass divided by the square root of 1 – β2. This
falls out of relativity.
So what is the kinetic energy? The kinetic energy is the total energy, mc2, minus the
rest energy m0c2. So it is m0c2 times 1 over the square root of (1 – β2) - 1. That’s the
kinetic energy.
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Now what we are going to find is that in the limit of small velocity, the nonrelativistic limit, small values of beta, we can actually show that the kinetic energy
is one-half times the rest mass times v2. Note that 1 over the square root of (1 – β2)
- 1 can be approximated as 1 + ½β2 – 1. Remember how to do a Taylor series
expansion for small β from first grade calculus. So this quantity in the parenthesis
is ½β2.
So the kinetic energy is m0c2 x ½ β2. The c’s cancel out and the kinetic energy then
is ½ the rest mass times v2. Kinetic energy is ½ times the rest mass times v2 and
total energy is relativistic mass times c2.
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This is a very important table for you to look at. It has beta and total mass for
various energies. Notice that for a 100 keV electron, beta is equal to 0.5 and the
mass is 1.2 times the rest mass of the electron. So even with a 100 keV electron, an
energy we obtain in an X-ray tube, we are starting to worry a little bit about
relativistic effects because the electron mass is somewhat more than the rest mass.
When we get to a 1 MeV electron, not even the energy we have in a linear
accelerator, we find beta is 0.94 and the mass of the electron is 3 times the rest
mass. For a 10 MeV electron, beta is 0.999 and the electron mass is over 20 times
the rest mass.
So electrons, especially those we use in radiation therapy, are relativistic.
For 100 MeV protons, beta is 0.43, and the mass is 1.11 times the rest mass. Typical
proton energies in radiation therapy are in the range 100 – 200 MeV, so we start
looking at relativistic issues for protons, but we don’t really have to worry too much
about relativistic effects there. For electrons, however, we have to worry about
relativity.
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Now, note two important mass-energy relationships. These are numbers you will
have to probably memorize. There will not be a lot of memorization in this class.
You don’t have to memorize a lot of numbers, but just because you use certain
things a lot and you don’t want to have to stumble through your notes to look up
some numbers, here are a few you need to memorize. You need to remember that
the rest mass of an electron is 0.511 MeV. You are going to use this number so often
that it will be useful to remember it. Also, remember that one atomic mass unit is
931.5 MeV. Again, that is a number you will use a lot, so it wouldn’t hurt to
remember that one as well.
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The last thing I want to look at today, and it is actually going to take a little while, is
exponential behavior. We are going to see exponential behavior in a large number
of applications in this course.
Basically the idea is if there is a quantity that changes by a certain factor in an
interval, whether that interval is time or whether that interval is distance, the
behavior is exponential.
Here are some examples of exponential behavior: Cell growth is exponential. Cell
kill is exponential. Radioactive decay, the build-up of radioactive material and
attenuation of photons are all events that exhibit exponential behavior.
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What is common to all of these is that they are stochastic events. As we pointed out
in the first lecture, stochastic events are controlled by laws of probability – they are
not deterministic. An isolated event is probabilistic, if it is a stochastic event.
And we are going to show now that if you assume that an event is stochastic, when
you have a large number of these events, the probability of that event occurring is
going to be governed by exponential behavior.
Let’s use an example. The example is we have n nuclei of radioactive material.
Will a given nucleus decay if we watch it? We don’t know. It is stochastic. We
cannot predict that a particular nucleus decays, but we can determine the probability
that say x of these nuclei will decay assuming the probability that any one nucleus
will decay is p. So what is the probability that x of these nuclei will decay assuming
that the probability of any one nucleus decaying is p?
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Each decay event is an independent event. The fact that one nucleus decays has
nothing to do with the fact that another nucleus might decay.
The probability of any one event occurring is governed by the binomial distribution.
So the probability is given by this quantity: n factorial divided by the quantity x
factorial times n - x factorial, all of this times p to the x power times 1 – p to the n-x
power. That’s the binomial distribution. And in your statistics class you are going
to see this binomial distribution occurring.
We are going to start with this binominal distribution. And, now we’ll take the limit
of the binomial distribution for a large number of events.
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We want to look at the deterministic quantities that result from sampling a large
number of stochastic quantities. All we are able measure are the deterministic
quantities.
Let’s take a very large number of radioactive nuclei. n is very, very much greater
than 1. The first thing we are going to do is take this 1 – p to the (n – x) power and
separate it into factors: (1 – p) to the minus x power and (1 – p) to the n power.
Everything else is going to be the same.
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Now let’s look at this second factor, n factorial over (n – x) factorial. This quantity
can be expanded to be n times (n – 1) times (n – 2) times a lot of other factors times
(n – x + 2) times (n – x + 1). Now we are going to make the assumption that n is
very, very large, very much larger than x. If that were true, each of these factors is
approximately equal to n, so that this quantity n factorial over (n – x) factorial is
roughly equal to n raised to the x power.
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If we let μ be the expectation number of decays, that’s n times p, the probability of
one decay times the number of decays, we have terms in p-x, px, and nx, so that the
probability can be written as 1 over x factorial multiplied by μx times (1 – p)-x times
(1 – p)n .
Now remember that p and x are very small and n is large. So this quantity, (1 – p)-x
is roughly equal to 1 + px, which is approximately equal to 1, so let’s get rid of it.
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Let’s look at the limit of 1 – p for very small p. We write n as μ divided by p and
take the limit of (1 – p)n as p approaches 0. The limit as p approaches 0 of 1- p to
the nth power, remembering that p is small, is the limit 1 - p to the 1 over p power
and raised to the μ power. This quantity is simply 1 over e to the μ power. This is a
definition of e, that is, the limit as p goes to 0 of 1 – p raised to the 1 over p power.
Consequently the limit as p approaches zero of 1 – p to the nth power is e-μ.
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So in the limit of small p the probability is e-μ divided by x factorial times μx, and μx
is equal to exlog μ.
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Let’s look at μ. The quantity μ is a deterministic quantity. It is the mean value of
the stochastic variable x, the number of decay events. But the decay events occur
over a time interval. So the number of decay events is going to be proportional to
the time, μ is equal to λt. The quantity λ is the mean number of decay events per
unit of time.
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We want to look at how these decay events are distributed over time. We want to
know how many nuclei are present at any specific time, that is, how many have not
decayed. So what is the probability that an event has not happened. What is the
probability that x is equal to 0.
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We look at the limit of the probability as both p approaches 0 and x approaches 0.
This limit is equal to e-λt. That is the probability that decay has not occurred, that is,
that we still have a nucleus.
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The expected number of radioactive nuclei present is equal to the original number
present multiplied by the probability that no interaction has occurred. So the
number present is equal to the original number N0 times the quantity e-λt, which tells
us that no interaction has occurred.
This is an example of exponential decay. What we have determined, then, is that if
the probability of an individual nuclear decay event is stochastic, then the expected
number of nuclei that have not undergone a decay is given by exponential behavior.
To generalize we use this same derivation for any stochastic process. The
expectation number of something happening is stochastic, independent of whatever
else goes on. We are saying that the probability that something has not occurred is
governed by exponential behavior. So that for radioactive decay we make the
assumption that a decay process is independent of anything. Radioactive decay is
independent of pressure, temperature, solubility; it is a strictly stochastic event.
Consequently we have exponential behavior for radioactive decay.
The probability that a photon will interact with a target atom is a stochastic event.
Consequently from this observation, we derive the fact that as a photon beam
penetrates through absorbing material, the intensity of the beam or the number of
photons that come through is going to be governed by exponential behavior.
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We began with a stochastic approach and demonstrated that exponential behavior is
the non-stochastic, or deterministic, average of a stochastic event.
So the change in the number of a quantity, whether it is number of radioactive
nuclei, number of photons, number of cells, is going to be equal to some
proportionality constant times the number present times the interval, whether it’s a
time interval or a spatial interval.
This is the equation; ΔN is equal to ±λ N Δt. If we have a plus sign, then we have
growth; if we have a minus sign, then we have a reduction. This is the mathematical
description of exponential behavior.
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Let us now look at the differential equation, dN is equal to ± λ N dt, or dN over N is
equal to ± λ dt. If we now integrate this expression, we get that N is equal to N0 e to
the power ± λt, plus λt if we have exponential growth, minus λt if we have
exponential decay.
There is a very important way to look upon this equation, for example, the equation
for exponential decay, and interpret the proportionality constant. In the case of
exponential decay, the transformation constant λ can be looked upon as the fraction
of nuclei decaying, that is, dN over N, in a time interval dt. If we are looking at
exponential attenuation, where the equation is now dN over N equals –μdx, the
linear attenuation coefficient μ is the fraction of photons interacting in a given
thickness of absorbing material. This is a very important way to look at exponential
behavior. We will come back to this later in this course when we talk about
radioactive decay and when we talk about interaction of photons with matter.
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When we have exponential behavior such as exponential decay occurring, we can identify a time
when the number is equal to half the initial number. That time interval is called a half-life. For
exponential attenuation, we have a thickness required to reduce the beam intensity to half its initial
value, and we call this thickness a half-value layer, or a half-value thickness.
When t is equal to t½, N will be equal to ½N0. Let’s figure out how to determine that time interval.
Taking logarithms of both sides of the exponential equation gives us that the logarithm of N over N0
is equal to minus λt. So when N is equal to ½N0, the log of ½ is equal to minus λ times the half-life.
So the half-life is the logarithm of 2 divided by λ.
Just in passing, I need you to know that I use the abbreviation “log” to represent natural logarithms.
Some of you may have seen “ln” for that abbreviation, reserving the symbol “log” for base ten
logarithms. But we never use base ten logarithms in this course. So log means natural logarithms.
Since the natural logarithm of 2 is 0.693, the half-life is always equal to 0.693 divided by the
transformation constant λ.
Very often we will talk about half-lives rather than transformation constants. I think a half-life gives
us a more intuitive feel for what’s going on with radioactive material.
The half-life of a cobalt 60 radioactive source is about 5.25 years. So we know the activity of a
cobalt source, five years from now the activity is going to be half; about 10½ years from now the
activity will be one-quarter of what we have now.
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Another quantity we use occurs when for N to be equal to N0 over e. That time is
the average life. So the logarithm of N0 over e divided by N0 is the logarithm of 1
over e, and is equal to minus the average life. Because the logarithm of e equals 1,
the average life will be equal to 1 over λ, or the mean path is equal to 1 over μ.
So those are two things that we are really going to see a lot of when we are dealing
with exponential behavior. We are going to be revisiting this several times in this
course. In particular when we talk about radioactive decay and in particular when
we talk about photon interactions.
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The final item is the plot of exponential behavior. If we plot exponential behavior
on linear graph paper, we are going to find that it is represented by a curved line
starting at 1.0 (the decimal point does not show up very well here) and never
reaching zero.
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Very often we will plot exponential behavior on semi-log paper. The horizontal axis
is linear, and the vertical axis is logarithmic. Here we see one decade and we can
plot multiple decades which you should be fairly used to doing that on semi-log
paper and Cartesian paper.
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