Download So far we have talked about production primarily of gamma rays

So far we have talked about production primarily of gamma rays. These gamma
rays are the result of nuclear processes, in particular, radioactive decay.
Just as important in radiation medicine as production of gamma rays is the
production of x-rays. We will spend the next two lectures talking about how we
produce x-rays.
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Our game plan for this lecture is as follows:
We will first identify what’s needed to produce x-rays.
Next, we will talk about how diagnostic x-rays tubes produce x-rays.
We will then look at the kinds of interactions electrons undergo with x-ray targets to
produce X-rays.
We will not go into great detail into specifics of x-ray production. A lot more of this
is going to be covered in your Medical Physics II and Medical Physics III courses.
Medical Physics II is your imaging physics course and Medical Physics III is your
radiation therapy physics course. In Medical Physics II you will learn how x-rays
are produced in the energy range used for diagnostic imaging whereas in Medical
Physics III you will learn how x-rays are produced in the energy range used for
radiation therapy.
In the next two lectures we are going to be talking about some generalities involved
in the production of X-rays.
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There are basically two ways in which x-rays can be produced for medical
applications. One way to produce x-rays is to take high-energy electrons and force
these electrons to decelerate or deflect in some way. Decelerating electrons will
emit radiation in the form of x-rays; this radiation is called Bremsstrahlung. Those
of you who understand German will recognize that Bremsstrahlung means
“braking” radiation. So the electrons are braking; they are forced to put on the
brake pedal, slow down, and produce x-rays.
The other way to produce x-rays occurs when electrons go from an outer shell of an
atom to an inner shell. When this occurs, they emit radiation called characteristic
radiation.
Both of these are very important ways of producing x-rays in medical applications.
And again, x-rays are produced as a result of electron interactions. So now we have
two kinds of electron interactions that we see are used to produce x-rays.
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Let’s start by talking about Bremsstrahlung.
What do we need in order to get high energy electrons to slow down?
Number one, we need a source of electrons. The source of electrons that we use typically is called a filament. It is a thin wire
through which we run an electric current. The wire gets hot and the electrons are boiled off. This is really a rather low tech
method of producing electrons. So there’s our electron source.
The next step is to get these electrons to go to very high energies. Typically we want these electrons to be in the kiloelectron
volt or mega-electron volt range, that is, thousands or millions of electron volts. There are basically two ways to give these
electrons high energies. You could accelerate the electrons all at once by placing a large potential difference between the
filament and wherever the electrons are heading. The electrons are going to be attracted to a positively charged plate and, with
a high potential difference between the filament cathode and positive anode, the electrons will get up to a pretty high energy.
This is the way we get high energy electrons in a diagnostic x-ray tube. The problem with going to higher energies like the
kinds we need to do radiation therapy is that those high potential differences just really are not safe. You don’t want to have a
potential of several million electron volts sitting in your radiation oncology suite; making that kind of potential difference safe
would be very difficult. What we do instead is we accelerate the electrons in groups. We clump the electrons together and
somehow we pump energy into these clumps, and we get them up to a high energy. The most common way of doing this is
with a linear accelerator. In Medical Physics III you will be going into a lot of details on how a linear accelerator accelerates
electrons. Other devices for accelerating electrons such as the microtron or the van de Graaff generator have not been used
very much for medical applications.
Finally, the last thing you need is a target. You have to stop the electrons or slow the electrons down in some way. Slowing
the electrons down will generate the x-rays.
So, three things are needed to produce x-rays: a source of electrons, an accelerating potential, and a target. When you have all
those three things you can produce x-rays.
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Here’s a diagram of a conventional x-ray tube. We start with a high voltage
accelerating potential. Notice the potential has to be DC because we want to have
one end being negative and one end being positive. We also have a filament
cathode. The electrons are boiled off the filament. The electrons are attracted to a
positively charged anode; when they hit the anode they produce x-rays.
There is a good mnemonic that can help you remember which end is the anode and
which end is the cathode. The word “anode” has the word “and” in it and “and”
means plus. The other one is the cathode. That one is minus by default.
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Let’s look at the electrons that are being produced.
Here we see a plot of tube current as a function of tube voltage for different values
of filament current.
The tube current is the charge per unit time that goes from the cathode to the anode,
that is, the current across the x-ray tube. The amount of radiation that we produce is
going to be proportional to this tube current.
How can we adjust this tube current in order to control the amount of radiation that
we produce? What are the various factors that enter into producing the tube
current? We want to look at those.
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First of all, let’s look at how the current that goes through the filament affects tube
current.
Remember there are two circuits. There is a filament circuit which is used to heat up
the filament. There is the tube circuit in which we are accelerating the electrons.
What happens in the filament? In the filament, the electrons are boiled off. As you
increase the filament current, you increase the number of electrons that are boiled
off. That stands to reason.
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Now, how does the number of electrons that are boiled off affect the tube current?
It does not necessarily result in an increase in the tube current.
Here’s the reason why: We are boiling off electrons. There is a cloud of electrons
in equilibrium around the filament. This cloud of electrons is called a “space
charge.” What happens when more electrons are trying to come off the filament?
You have this negatively charged electron cloud around the filament that’s repelling
the other electrons. It’s harder to boil off more electrons because it takes more
energy to overcome the electrostatic repulsion produced by this space charge. So if
there’s no potential across the x-ray tube, all we have is this space charge hanging
around the filament. Increasing the filament current is not going to affect the tube
current. In fact the tube current is going to be zero because there is nothing to
attract the electrons.
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Let’s now turn on a little bit of accelerating potential. With a small amount of
accelerating potential, we are going to start accelerating electrons over to the
cathode. Now we are going to increase the filament current. What happens now?
More electrons are trying to be boiled off but the tube voltage is low. Not many
electrons are being accelerated to the anode. Consequently, there is still this space
charge hanging around the filament. Changing the filament current when the tube
voltage is relatively low is not going to affect the tube current. This is a region
called the “space charge limited region.”
So for low tube potential, low tube voltage, there is not much you can do to increase
the tube current. That’s the region we are in the lower left of the graph. Observe,
for example, that in the region of tube voltages around 20 kV, the difference in
filament current between 4.1 and 4.2 milliamps doesn’t really affect the tube
current.
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When we now increase the accelerating potential, the space charge effects are
overcome, and we begin pulling electrons away from the cathode. So this is the
region, say at tube voltages in the vicinity of 30 or so kilovolts, where the
relationship between tube voltage and tube current is actually linear. When you
increase the tube voltage, you will also increase the tube current in a linear fashion.
With higher filament currents the rate of increase is greater.
So for constant tube voltage, say 40 kV, we observe that as we increase the filament
current, we increase the tube current.
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What happens if we go to higher voltages, say in the range of 80 kV. For low
filament currents, we have what’s called saturation. The electrons are removed
from the filament as fast as they are boiled off. So any increases in tube voltage
will not change the tube current. The only way to change the tube current when you
are in the saturation region is to change the filament current.
It is important to understand the three regions of voltage:
1. The space charge limited region, in which changing the filament current has no
effect on the tube current;
2. The linear region, in which both changes in tube voltage and filament current
affect the tube current; and
3. The saturation region, in which only changing the filament current will change
the tube current.
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Let’s talk about the x-rays we use in radiographic imaging.
We want a large number of electrons coming out in a very short time. Typical
exposure times are fractions of a second. We want to get a lot of these x-rays to
come out and image the patient before the patient moves. So we need to operate
where there are enough electrons available. We operate above the space charge
limited region and we operate below saturation.
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In this region, the tube current controls the output of the x-ray tube, the photon
fluence and the beam intensity. When we are in this region, we are above space
charge and below saturation, and the tube current is determined by both the tube
potential and the filament current.
Notice, in particular, that if we referred back to the graph on an earlier slide, a 2½%
change in filament current, going from 4.1 milliamps to 4.2 milliamps, results in a
25% change in tube current, going from 350 milliamps to 415 milliamps.
Consequently, the filament current must be very stable to avoid serious fluctuations
in the tube current.
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Let’s go back to this figure. With a tube voltage of somewhere in the vicinity of 60
to 80 kilovolts, a 2½% change in filament current results in a very large change of
25% in tube current. Consequently the filament circuit has to be very stable. When
we talk about diagnostic circuitry you will look at ways to make that circuitry very
stable.
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When we are dealing with fluoroscopy we keep the x-rays going for a period of
time, and we operate at a lower filament current. Here we are in the saturation
region. So the intensity of our x-rays can be adjusted by adjusting the filament
current.
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Let me digress for a little bit and talk about sinusoidal behavior. The electric power that we get
out of the wall is alternating current.
Do all of you know the story about Westinghouse versus Edison? Westinghouse was the one
who favored alternating current; Edison favored direct current. One of the advantages of
alternating current is that alternating current electric power can be transmitted over long
distance; if you want to use direct current, you needed power generation very close to where you
are using it. Edison envisioned a whole bunch of power stations in downtown Manhattan when
he did his direct current demonstration. Edison was so opposed to alternating current that he
advocated the use of alternating current in the development of the electric chair because he
thought that people are going to know that alternating current is dangerous, and therefore they
will go with direct current. That is the reason why electric chairs were operated on alternating
current.
Obviously, Westinghouse eventually won out. We use alternating current.
The electric power that we get is alternating current. Ultimately we are going to have to rectify
it and make it direct current.
Let’s talk a bit about how voltage and current depend on time. They are characterized by
sinusoidal behavior. This leads us to discuss sinusoidal behavior.
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Here is a graph depicting sinusoidal behavior. We can describe the voltage as a
function of phase angle as a sine function whose value ranges from 0 to a maximum
value, then down to 0, then to the negative of the maximum value, and then back up
again, and so forth. However, in an x-ray circuit, we will wind up reversing the
negative portion, rectifying it so it only goes in one direction.
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Let us calculate the average voltage over half a cycle. To determine the average
voltage we integrate the voltage, which is V0 sin θ, over half a cycle, and divide it
by the integral of dθ over the half cycle 0 to π.
When we do the integration of sin θ dθ, we get -cos θ, and when we integrate dθ we
get θ. So the definite integral in the numerator is V0 times -cos θ evaluated at π
minus the same quantity evaluated at 0, and the definite integral in the denominator
is equal to π. So the average voltage over a half cycle is equal to 2 divided by π
times the maximum voltage.
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There are some quantities associated with sinusoidal behavior, such as power, that
go as the square of the voltage. So we also want to look at the square of the voltage.
Let’s calculate the root mean square voltage.
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The root mean square voltage is the square root of the average value of the square of
the voltage. The square of the voltage is the quantity V0 sin θ squared. We now
integrate over a half cycle, that is, from 0 to π, the quantity V0 squared sin θ squared
dθ and divide by the integral from 0 to π of dθ. After we do the integration, we get
V0 squared over π times ½ of θ minus sin θ cos θ, evaluated between π and 0.
I won’t go over the integral. I’ll ask you to recall your kindergarten calculus or look
the integral up in your integral tables, or use something like Wolfram alpha.
So when we go through the mathematics, we find that the root mean squared
voltage is 1 over the root square root of 2 times the maximum voltage.
These are just some of the things to keep in mind with sinusoidal behavior.
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To summarize, the average value is 2 over π times the maximum value, or
approximately 0.64 times the maximum value. The root mean square value is 1
over the square root of 2 times the maximum value, or 0.71 times the maximum
value.
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So the voltage in an x-ray tube is going to be given by V0 sin θ. If we are below the
saturation region, the current is going to be nearly linearly proportional to the
voltage, so that the x-ray output is going to be roughly proportional to V2. This is
something to keep in mind when we have this behavior.
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Once we have the electrons accelerated, we need to have the electrons interact with
the target to produce x-rays. We want to look at three kinds of interactions of
electrons with target material.
One kind of interaction is the interaction of accelerated electrons with outer-shell or
valence electrons. These interactions involve collisional energy transfer. This type
of interaction produces more electrons, but no x-rays.
The accelerated electrons can also interact with inner-shell electrons. Interactions
with inner-shell electrons also produce electrons. These interactions are also
collisional interaction, but because we are kicking out inner-shell electrons, we can
also produce characteristic x-rays.
Finally we have interactions of the electrons with the nucleus in which the electron
is actually deflected by the nucleus. These interactions produce Bremsstrahlung.
We need to keep in mind these three kinds of interactions with the target.
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Here’s a picture of the three kinds of interactions.
First, we have the interactions with outer shell electrons. These are multiple
interactions with electrons with low binding energies. Most of the energy
transferred to these electrons ultimately goes into heat. An electron typ0ically gives
up about 33 eV per ion pair produced, so that a 100 keV electron can create up to
3,000 ionizations.
Next, we have interactions with core-shell electrons. These interactions give rise to
characteristic x-rays once an outer-shell electron fills the vacancy left behind from
the ionization of the inner-shell electron. We know that the energy of the
characteristic x-ray is the difference in binding energies between the outer-shell
electron and the inner-shell electron.
And finally, we have interactions with the nucleus in which the electron is deflected.
The electron gives off most of its energy in the form of Bremsstrahlung x-rays, for
which the energy is going to be somewhere between zero and the total kinetic
energy of the electron.
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The interaction of incident electrons with valence electrons is the dominant
interaction. This interaction is a Coulomb interaction with loosely bound valence
electrons. There is a relatively small amount of energy transfer. The ejected
electron doesn’t really travel very far and it is typically absorbed in the target or in
the housing of the x-ray tube.
We don’t really need to worry much about this type of interaction, other than this
interaction with the valence electrons produces a lot of heat and we do have to
worry about heat production when we are designing an x-ray tube.
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The energy transferred from the incident electrons is ultimately deposited as heat.
Notice that for electrons incident at 100 keV, 99% of the energy shows up as heat.
Only 1% of the incident energy produces x-rays. We are looking at a very, very
inefficient method of producing x-rays, and we have to figure out ways handle the
large amount of heat that is produced.
If you are working with an x-ray tube, you will find that you are always keeping an
eye on tube loading. For example, take our Philips AcQSim CT scanner that we use
to acquire images for patient planning. We can’t acquire scans that take more than
100 seconds. Part of the reason for this time limit is the concern with overheating
the tube. If you operate any type of diagnostic x-ray unit for a long period of time,
it’s going to shut off because so much more heat is being generated than is being
dissipated. You have to let the tube cool down and then continue.
So cooling of the x-ray target is a major issue because this process is so inefficient.
At the energies we are considering, most of the interactions are these collisional
interactions that give rise to low-energy electrons that give rise to heat.
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Most of the interactions are low energy transfer – the electron interacts with a large
number of atoms. These electrons can be looked at as incident electrons that lose
small amounts of energy; these electrons continuously slow down and lose energy
with roughly the same rate as they are slowing down. We can approximate this by
what’s called a “continuous slowing down approximation” (CSDA), which basically
says that the energy loss process is a continuous process. In fact the loss of energy
is a discrete process, but this energy loss occurs so rapidly and so often in the
electrons’ path that we can approximate it as a continuous slowing down. We are
going to talk about CSDA and collisional energy transfer a lot more later on in the
course when we talk about charged particle interactions.
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For the point of view of producing x-rays however these collisional interactions
aren’t very useful. They produce a lot of heat but not much in the way of x-rays.
What is more important are the interactions with core electrons. If the incident
electron has sufficient energy to overcome the binding energy of this inner shell
electron, we can eject a core electron. The outer shell electron then fills the vacancy
and produces characteristic x-rays. Generally, we are most interested in Kcharacteristic x-rays because those are the hot ones with the highest energy. Those
are most interesting from the point of view of imaging purposes.
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The characteristic x-ray has energy equal to the difference in binding energies of the
initial and final shell. It is fairly straightforward how you do one of these
calculations to find the energy of characteristic x-rays. You just have to know what
the target of the electrons is. As long as you know what the target is, you can
readily calculate what energies of characteristic x-rays are coming out. And notice
that K characteristic x-rays can come from L-shells and M-shells and these shells
are also going to have sub-shells as well depending on the azimuthal quantum
number of the electron.
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Here’s an example. A common material that we use for targets in x-ray tubes is
tungsten – you’ll learn why later. If we know the binding energies of the various
shells of tungsten, we can calculate what the characteristic x-ray energies are. We
have K-shell binding energy of 69.525 keV; we have LI, LII, and LIII at around 12,
11½, and 10 keV. So based on the initial and final energy states, we can see what
the different emission lines are.
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Notice that N-shell electrons going to fill a K-shell vacancy are called Kβ2 x-rays.
I’m not sure what the origin of this terminology. The N-shell electrons have a very
low binding energy so that the energy of the Kβ2 x-rays is about 69 keV. Electrons
from the MIII going to the K shell give rise to Kβ1 x-rays. I think these names must
be historical. The energy emitted as the characteristic x=-rays is 67.2 keV. There
are three times as many of those x-rays as Kβ2 x-rays. The important characteristic
x-rays result from the LIII to K transitions. Those are the Kα1 with energy 59.3 keV
and those have a much higher energy. Notice also that the Kα3 x-rays do not exist.
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We do not have LI to K transitions, which involve a change in ℓ of 0, and that’s
quantum mechanically forbidden. Some of the transitions are going to be
forbidden; most of them will occur.
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Notice also that we have a relatively complex spectrum of characteristic x-rays. So
as long as the energy of the incident electrons exceeds 69.5 keV we will see Kcharacteristic x-rays coming out of the x-ray tube. So if the potential on the x-ray
tube is only 50 kV potential, we are not going to see these characteristic x-rays
being emitted. If the potential is 60 kV, we are not going to see them. If the
potential is 70 kV, we are going to start seeing them.
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These characteristic x-rays are very important in the diagnostic x-ray region. Notice
that as much as 30% of the energy in a 100 kV beam is characteristic x-rays. So at
the low end of energies they are very important. 100 kV is a typical tube potential.
When you go up to 200 kV we have very little in the way of characteristic x-rays;
almost everything is Bremsstrahlung. When we get into the megavoltage range
almost nothing comes from characteristic x-rays.
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Now, a very interesting thing happens in mammography. Typically in
mammography we use a molybdenum target. The idea is we want essentially a
mono-energetic beam from our target, so we would like a lot of characteristic xrays. Characteristic x-rays from a molybdenum target are in the range of 17 to 20
keV. Most of the photon interactions when we are in that energy range are what we
call photoelectric absorption; we will learn about that a little bit later. The reason
that we want these very low energy photons that have a lot of photoelectric
absorption is that photoelectric absorption is very sensitive to small changes in
atomic number. What are we looking for in a mammogram? We are looking for
calcifications. Calcifications have higher atomic number than soft tissue, and we
are going to see that the probability of an interaction goes as the cube of an atomic
number. So if we can get very low energy x-rays, such as 20 keV, that turns out to be
an excellent energy to look for small calcifications in tissue. So whereas most of
the targets in x-ray tubes that operate at more typical x-ray energies, say 70 to 140
kV, are typically tungsten targets, when we are looking at mammography tubes we
are going to typically have molybdenum targets more likely than tungsten.
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The final thing I want to talk about today is a concluding comment about these interactions with
core electrons. We will look at the Bremsstrahlung in our next lecture. If these characteristic xrays resulting from interactions with core electrons are low in energy relative to the
Bremsstrahlung spectrum, we usually consider these characteristic x-rays to be a contaminant.
For example, if we produce low-energy L shell characteristic x-rays, we would rather not have
those low-energy x-rays. If the characteristic x-rays are high in energy relative to the
Bremsstrahlung spectrum, they are usually considered to be beneficial.
How do we get rid of low-energy x-rays in an x-ray tube? We typically filter them out by
selectively attenuating these low-energy x-rays. By the use of filters, we get rid of both lowenergy characteristic x-rays, which are a contaminant, as well as the low-energy component of
the Bremsstrahlung spectrum. The low-energy x-rays don’t penetrate very far. From an imaging
point of view, we want the x-rays, at least some of the x-rays to penetrate through the patient and
reach the image receptor; otherwise, we are not going to see anything if everything is absorbed
in the patient. So a lot of these really low-energy x-rays will be absorbed by the patient. To get
rid of them, we will typically put in some kind of attenuating material that will selectively
attenuate the low energy x-rays. These filters will also attenuate the high-energy x-rays but not
as much.
Also when we use orthovoltage x-rays for therapy purposes, nowadays rarely, we also put in
some sort of filtration. Again, we want to get rid of the low-energy and keep the higher-energy
x-rays.
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