This is the third of three lectures on cavity theory.

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In this lecture, we are going to go over what is meant by charged particle equilibrium and look at the dose and kerma when you have charged particle equilibrium. But since true charged particle equilibrium is very difficult to achieve, one deals mainly with something called “transient charged particle equilibrium.” We are going to look at a new definition for particle fluence and see how that leads to the Bragg-Gray cavity theory. And then we will briefly talk about the Spencer-Attix cavity theory, which is a modification of the Bragg-Gray theory that is used in all of our dosimetry protocols.

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We’ve presented the definition of charged particle equilibrium before, but let’s look at it again.

It exists in a volume if each charged particle of a given type and energy leaving the volume is replaced by an identical particle entering the volume. Again, I just want to stress that this is a statistical concept, and we must always deal with the expectation values. And if you are not particularly comfortable with the concept of expectation values for a large number of interactions, we can say that the expectation value approaches the mean value. In fact, in a lot of our calculations we look at mean values rather than doing the integration because you get quite good results doing that.

If you go back to the equations we looked at in previous lectures, if charged particle equilibrium exists in a region or volume, that simply says the energy carried out by charged particles is equal to the energy carried in by charged particles. You have an equality and you can make use of that a lot.

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This is shown schematically here using our volume diagrams.

The volume marked with a lower case v is the volume that we have been talking about. It’s a small volume but now we are going to surround it by a larger volume. In order to have charged particle equilibrium exist in this small volume, you must have a larger volume around it. So in the figure, the larger volume contains the smaller one and the boundaries of upper case V and lower case v are separated by at least the maximum distance of penetration of any secondary charged particles.

e1 is the track of a secondary charged particle produced at point P1 at the edge of the volume lower case v. It just has enough energy to exit lower case v and enter the volume upper case V, but not leave upper case V. This particle is compensated by the particle whose track is e3 produced at point P3 in upper case V, entering lower case v, and stopping at the edge of lower case v. So we are always thinking about distances surrounding our point of interest, the distance being equal to the maximum range of secondary electrons set in motion.

If you are dealing with neutrons then you might be dealing with other secondary particles, protons, for example, nevertheless, the concept is the same.

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We are going to identify some conditions such that if these conditions are satisfied throughout the larger volume, then charged particle equilibrium will exist in the smaller volume.

The first condition requires that the medium be homogenous in both density and composition. That is, the medium both in the large volume and the small volume. You don’t want any changes in the medium or in its density.

The second condition is that there exists a uniform field of photons. (Here we will just talk about photons, and, as I said, you can apply this to neutrons.) The photons are coming in and interacting in the larger volume V. You want negligible attenuation of these photons in traversing this volume.

The final condition is that there are no inhomogeneous electric or magnetic fields present. In most cases we won’t worry about that. But you don’t want anything that is going to disturb the electrons.

Note that it is not necessary that the secondary particles be isotropic; we just drew them that way in the picture for clarity. They can be going off in any direction.

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If we have charged particle equilibrium, then the radiant energy of charged particles entering our volume of interest is equal to the radiant energy of charged particles exiting the volume.

If we now substitute this equality into the expression for energy imparted, we see that under conditions of charged particle equilibrium, the energy imparted is simply the energy of uncharged particles entering the volume minus the energy of uncharged particles exiting the volume plus any changes in mass.

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Let’s now recall our energy definitions, but this time under conditions of charged particle equilibrium.

First of all, as we showed on the previous slide, the energy imparted is the radiant energy of the photon entering the volume minus the radiant energy of the photons exiting the volume. The radiant energy of the photon entering the volume is hν1, while the radiant energy of the photons exiting the volume is hν2 + hν3. The quantity T’, the radiant energy of charged particles exiting the volume, and ultimately yielding a photon with energy hν4, is compensated for by another T’ entering the volume from a photon interaction outside the volume.

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Let’s now take the expression for energy transferred, that is, the total energy received by charged particles, and insert it into the expression for energy imparted under charged particle equilibrium.

We find that the energy imparted is equal to the energy transferred plus the radiant energy of photons produced by nonradiative processes exiting the volume minus the radiant energy of photons exiting the volume.

The radiant energy of photons produced by nonradiative processes exiting the volume is hν2, and the radiant energy of photons exiting the volume is hν2 + hν3, so the energy imparted under charged particle equilibrium is the energy transferred minus the radiant energy of photons produced by radiative processes exiting the volume, which is hν3.

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Finally we note that the net energy transferred is equal to the energy transferred minus the radiative losses from particles originating in the small volume, and, under charged particle equilibrium, we can relate the net energy transferred to the energy imparted.

Working this out is going to be a bit complicated, and requires some understanding of charged particle equilibrium.

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Recall that Rur are the radiative losses from particles originating in v, and (Rout)u

nonr

is the energy of photons leaving v, except those that are radiative. In a moment we will see how that will work out.

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Under charged particle equilibrium conditions, the following equation holds as long as the volume is small enough that all radiative-loss photons created in the volume can escape. That is in our little volume, if an electron loses energy by a radiative loss, it gives off Bremsstrahlung. It is important that the Bremsstrahlung escapes the volume completely.

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The condition of charged particle equilibrium is needed to ensure that for every radiative loss contributing to Ru

r outside of v, there is a similar radiative event occurring inside v. The volume must then be small enough that the radiative photon gets out of v to be counted in (Rout)u.

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The assumption is made as long as hv2 escapes from v, we are in good shape. We have simplified things a little bit in this diagram.

An electron is going to come into this volume and that electron in the volume undergoes a radiative loss. A photon is given off with energy hv2 and it may create secondary radiations. But we make the assumption for an electron entering the volume v, there is always an electron with similar energy going out, and undergoing a similar radiative loss outside of the volume. Again we can make that assumption if we have a large number of events and look at the average values. We further make the assumption that the energy of the Bremsstrahlung photon, hv2, is equal to hv1. You can then put that into the equation and you will find that under charged particle equilibrium, the radiant energy of photons exiting the volume is equal to the sum of the energy from nonradiative interactions plus the energy from radiative interactions.

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Now, under charged particle equilibrium we can make some important simplifications.

The net energy transferred under charged particle equilibrium is the energy imparted in the volume and if we shrink the volumes down, we can take the differentials. The derivative of the net energy transferred per unit mass is equal to the derivative of the energy imparted per unit mass. That is, under conditions of charged particle equilibrium the absorbed dose is equal to the collision kerma. That is the important equality we want to make at this point.

When you have charged particle equilibrium at a point in the medium, the absorbed dose is going to be equal to the collision kerma.

But that is only under the conditions of charged particle equilibrium.

Note that the equality of dose and collision kerma is true irrespective of radiative losses. Radiative losses become important at energies of about 1 MeV. Below 1 MeV you really don’t have to worry about radiative losses, but above 1 MeV, they start to become important. By the time you start dealing with 4 MeV x-rays, radiative losses become significant. So the point where radiative losses in dosimetry come in is right in the range we use for therapy.

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So the equality of dose and collision kerma under conditions of charged particle equilibrium is important because it relates absorbed dose, a quantity we can measure with a great deal of accuracy, to collision kerma, a quantity we can calculate. This equality, in fact, gives us a way for checking out some of our calculations. If we measure absorbed dose and make the equality with collision kerma, we can see how good our calculations are.

This perhaps is the most important result that comes out of it in terms of everyday dosimetry.

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And another aspect of the equality is that it allows the ratio of the doses in two media to be calculated. If you have the same photon fluence, if charged particle equilibrium holds, then the ratio of the dose in one medium to the dose in another is going to be equal to the ratio of collision kermas, which, in turn, equals the ratio of the mass energy absorption coefficients.

Often in dosimetry we wish to relate the dose in one medium to the dose in another medium, for example, dose in soft tissue or dose in bone to dose in water. We see here that under conditions of charged particle equilibrium, the ratio of doses is simply equal to the ratio of mass energy absorption coefficients.

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Unfortunately, charged particle equilibrium does not exist in many situations in dosimetry.

We need to be aware when it breaks down, what to do about it, and how it may affect the theories we deal with.

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When you have high energy photon beams, charged particle equilibrium can only occur after full buildup has been achieved. If you think about it that is a reasonable statement. You are looking at the effect of the secondary electrons and how far they go, and it’s the secondary electrons that build up in the buildup region. So you must get those to some kind of equilibrium before you can have charged particle equilibrium. That is one condition in the buildup region for high energy photons charged particle equilibrium doesn’t exist.

If the attenuation of the photon beam is significant in the distance traveled by the electrons set in motion by the photons, then it’s impossible to get charged particle equilibrium. Remember a few moments ago, we said that a requirement for charged particle equilibrium was there would be no photon attenuation over the range of the secondary electrons. Well, if you have a 10 MeV photon beam it’s attenuated about 7% in the range of its secondary electrons. The 7% is a large number so that charged particle equilibrium cannot exist. Down at about 1 MeV it is 1% and that’s a bit better to get by with, but at 10 MeV you certainly can’t neglect the attenuation of the photon beam. So, for high-energy photons, we don’t really have charged particle equilibrium.

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So we introduce a concept called transient charged particle equilibrium, which allows us to go ahead and use the theories and work on the equations. We simply define transient charged particle equilibrium at all points within a region in which the absorbed dose is proportional to the collision kerma. Remember that under charged particle equilibrium, we can derive quite rigorously that the dose equals the collision kerma. But when charged particle equilibrium breaks down we are fortunate there is a region in which the dose may not be equal to the collision kerma, but it is proportional to the collision kerma.

And if you look at the standard curve that plots both collision kerma and depth dose, when you get into the attenuation part here beyond the buildup region, you will find that the two curves become parallel. In this region, where the two curves are parallel, we have transient charged particle equilibrium. In this particular slide we are at low enough energy that the radiative losses can be ignored, so we have made kerma equal to collision kerma. But one could have radiative losses and it would look about the same.

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In the plot of dose versus depth, the absorbed dose is non-zero at the surface due to back-scattered electrons and photons. The depth dose increases as the charged particle fluence builds up and reaches a maximum at the point where the increase becomes balanced by the decrease due to attenuation of the photon beam. The dose maximum occurs at approximately the point where the dose and kerma curves cross over. But these curves will cross over and somewhere beyond that is where transient charged particle equilibrium exists.

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At a depth equal to the maximum distance that the secondary charged particles starting at the surface can penetrate, the dose curve becomes parallel to the kerma curve, and hence transient charged particle equilibrium exists.

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So in a typical depth dose curve, the dose starts out not quite zero at the surface but pretty low and increases to the maximum value. The kerma starts at some value on the surface; there is no build up. Kerma at the surface is called K0 and as we have said earlier, we have assumed that in this case radiative losses are negligible so the kerma is equal to the collision kerma. The kerma at the surface is K0 and it attenuates with depth, but with a reasonable approximation it will be exponential with some effective attenuation coefficient.

We go to rmax , which is the range of the secondary electrons. Beyond that you get transient charged particle equilibrium and you get the proportionality relationship between dose and collision kerma.

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Roesch suggested the following relationship between dose and collision kerma. In a region in which there is transient charged particle equilibrium, dose is equal to collision kerma multiplied by an exponential term. He then expanded this exponential term, keeping only the linear term in depth, and he called the multiplicative factor beta. In his equation, μ’ is an effective attenuation coefficient and x bar is the mean distance of electrons transporting their kinetic energy in the beam’s direction.

Don’t worry too much about that but understand that now under transient charged particle equilibrium, dose is proportional to collision kerma, and the proportionality constant is called beta. It’s the ratio of absorbed dose to collision kerma at a given depth; beta is always greater than 1.

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That is on the diagram we see that at depths beyond the cross over, the dose curve always above the collision kerma. You will never find collision kerma coming out on top, kerma is always below the dose curve.

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Let’s talk now about an alternative formulation of particle fluence; in an earlier lecture, we talked about particle fluence and the number of particles crossing a sphere of unit cross section dA. One can look at it slightly differently and this formulation helps us certainly in cavity theory.

The alternative formulation for particle fluence is that it equals the sum of the particle track lengths in a volume, divided by the volume. And that simply has you sum the track lengths in the volume divided by the volume itself and that’s an alternative definition of fluence.

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We’ll see how we can make use of that alternative formulation.

In this case, we will be talking about electrons; we are going to apply them in cavity theory. Consider an electron passing through a slab of area dA and thickness t. We will ignore creation of secondary electrons by this electron and any Bremsstrahlung losses. The path length in the volume of interest is going to be t divided by cosine theta.

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Now the electron mass collisional stopping power (S/ρ)col gives the energy lost to the electrons in the material per unit path length. That’s the definition of the collisional stopping power.

The energy deposited in this slab is given by the energy lost to the electrons per unit path length multiplied by its path length.

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In this simple case the particle fluence, that is, the path length per unit volume, is simply given as 1 over dA cos θ, giving us an expression for the energy deposited in terms of the fluence. We see that the energy deposited is the fluence multiplied by the cross sectional area times the density times the thickness times the collisional mass stopping power.

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Dividing this expression by the mass of the slab, we get dose in the slab as the energy deposited by the mass. We can put it back into that equation and you get a very nice equation. That is the dose equals the particle fluence multiplied by the mass collisional stopping power. This is analogous to the relationship between kerma and dose under conditions of charged particle equilibrium. This important relationship between fluence and dose holds for arbitrary volumes and fluence in any direction.

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The derivation assumes that radiative photons escape from the volume of interest and the secondary electrons are absorbed on the spot. We need to be aware that the latter condition does not necessarily hold, but, in conditions when we have charged particle equilibrium, the result is still valid because energy transported out of the volume by collisional electrons is replaced by energy transported into the volume by similar collisional electrons coming into the volume.

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Let’s consider the fluence of charged particles of energy T crossing an interface between two media. In this case the interface is between a medium w and a medium g. We normally talk about w as the wall and g as the gas, but it could be the interface between any two materials.

So we have a particle of energy T and it crosses this interface between the medium w and g.

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The fluence in w on the left of the interface is the same as the fluence in g on the right of the interface. Hence we can write that the dose to g equals the fluence times the mass collisional stopping power in g. The dose to w equals the fluence times the mass collisional stopping power in w.

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We can now write the ratio of the dose in w to the dose in g as the ratio of the mass collisional stopping powers.

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Now let’s do a simple little thought experiment and place another slab of w on the other side of the slab of g. We will now shrink the slab of g so that we have an otherwise homogeneous medium, w. which contains a very thin layer of cavity filled with another medium g.

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We have made it so thin we have had some certain conditions that we assume.

The first condition is that the thickness of the layer of g is assumed to be so small in comparison with the range of charged particles crossing it, that its presence does not perturb the charged particle field. That is one of the very basic assumptions made in Bragg-Gray theory. That is, when you introduce a small cavity or slab of other material, the presence of this slab does not interfere with the charged particle distribution in energy or direction.

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The second condition is that the absorbed dose in the cavity is assumed to be deposited entirely by the charged particles crossing the cavity. This condition is valid for gas-filled cavities in photon beams.

What that means is we are allowing no interactions to take place within the cavity itself in which energy can be deposited. So all these particles are going to cross the cavity depositing energy in the cavity, and we’re not allowing any energy to be deposited by electrons that were set in motion by photon interactions taking place inside the cavity. That’s a valid assumption when you have a very small cavity which is filled by a much lower density material than the material surrounding it. This condition is valid for gas-filled cavities in photon beams, where the density of the gas in the cavity is orders of magnitudes less than the density in the wall.

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The previous relationships were based on fluence from mono-energetic sources. What happens if we have a polyenergetic distribution of charged particles?

We now define the spectrum-averaged mass collisional stopping power in the cavity medium. That is we are going to average over the energy distribution of secondary electrons. Fortunately, we do not have to do these integrations; someone had done them for us and there are tables of them. But years ago if you wanted your average stopping powers you would have to calculate them yourself and do the integration.

Nowadays, in the back of standard textbooks, you can find tables of these average mass collisional stopping powers.

Combining these equations, we get the Bragg-Gray relationship in terms of the absorbed dose in the cavity, namely.

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If we look at the wall of the chamber, we get the same sort of spectrum-averaged mass collisional stopping power in the wall.

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Combining these equations, we can relate the dose to the wall as the dose to the gas multiplied by the ratio of spectrum-averaged mass collisional stopping powers. The bars over the S indicates that the average is taken over the energy spectrum of secondary electrons.

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If the cavity is filled with a gas in which a charge Q is produced by radiation, ion pairs are going to be created. Then, using the definition of W/e, we have that the dose in the gas is the charge divided by the mass times W/e. That is sort of the basic equation in the Bragg-Gray cavity.

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But the equation we obtained is for the dose in the gas; what we really want is the dose in the medium surrounding the gas. We are not particularly interested in the dose in the gas, but the dose in the medium is related to the dose in the gas by the ratio of the collision stopping powers.

Note that these equations make use of the unrestricted collisional stopping powers. The effects of knock-on electrons, that is, the secondary electrons produced by the electron-electron interactions, are included in these stopping powers. Note that when we calculate the mean collisional mass stopping power, we integrate over the fluence spectra for primary electrons only.

The dose in the medium equals the charge that you measure per unit mass in the gas cavity times W/e, times the ratio of collisional mass stopping powers, and this is really is the Bragg-Gray relation expressed in the terms of the cavity ionization chamber. And if you are only doing Bragg-Gray cavity theory this is the equation you will use.

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Note that the theory requires charged-particle equilibrium of at least the secondary electrons in the cavity, the knock-on electrons, since this is required in order to use the relationship between dose, fluence, and stopping powers. If there is also charged-particle equilibrium for the primary electron spectrum, certain computational shortcuts are possible. We won’t go into that but you can do that but at least the theory requires that for the knock-on electrons, that is, the secondary electrons, we have charged-particle equilibrium.

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What has always surprised people is that the Bragg-Gray cavity theory works at all. Because some of the assumptions that we make, such as the neglect of the spectrum of the knock-on electrons you might think the Bragg-Gray theory wouldn’t work at all. But it works remarkably well. However, it could be improved upon, and the individuals who did the most important work of improving on the Bragg-Gray cavity theory were Spencer and Attix.

A small historical digression here: Bragg was a British scientist who first suggested the theory; he didn’t develop it, but wrote a short paper in Nature, a British scientific journal, in the 1920-30’s. Gray was a radiation physicist in England who took Bragg’s idea and developed the theory. If you want the basic papers you go back to Gray’s papers before World War II, in which he developed the theory using slightly different notation from what we use today.

Gray became a well-known medical physicist and really was responsible in many ways in introducing the rad or absorbed dose concept into the field. He visited MD Anderson once and we had a conference here in the 1960’s and he was the featured speaker.

Spencer and Attix were two Americans who were working in the 1950’s at the National Bureau of Standards. That agency is now known as the National Institute of Science and Technology, or NIST. Spencer and Attix decided they could improve on the Bragg-Gray theory. Spencer remained at NIST and did a lot of calculations, Attix became a medical physicist; he worked at NIST, and he worked for the Naval Research Lab in Washington. D.C. on dosimetry and ended up at Wisconsin in their medical physics program as a professor.

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The Spencer-Attix theory was in much better agreement with the experimental results and this has been the accepted procedure ever since. People always sort of try and modify the Bragg-Gray and Spencer-Attix theories. There are other theories in the literature, which we won’t go into.

The Spencer-Attix theory has persisted now for a long time and it’s the basis for all dosimetry protocols.

The theory still requires the two Bragg-Gray conditions to hold. In fact, Spencer-Attix theory is more stringent because it assumes the entire secondary electron spectrum is not disturbed by the cavity.

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The basic idea of the Spencer-Attix theory is explicitly to take into account all knock-on electrons above a certain energy threshold, which we will call delta. The theory then considers all other energy losses as local. So it divides the electron spectrum up into two regions. One region is the part of the electron spectrum with energy above the threshold delta, which then gets put back into the primary spectrum, with all electrons with energies below delta resulting in local energy loss.

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Spencer-Attix theory considers the electron fluence to include primary electrons as well as knock-on electrons with energies above delta. And the integrals for the dose now start at delta. So Spencer and Attix changed the Bragg-Gray theory. Rather than integrating over the energy spectrum from 0 to a maximum value of energy, now the integrals go from delta to the maximum energy value.

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Also, rather than using the unrestricted mass collisional stopping powers, Spencer-Attix theory now uses the restricted mass collisional stopping powers. Set a value on delta and it tells you the energy in keV at which you start the integrations.

So when you do the integrations you consider only energy losses creating electrons below energy Δ, because the energy lost to higher energy electrons is explicitly accounted for by the presence of these electrons in the spectrum. Below the value of delta you are interested in the energy loss, but above this energy you consider the electrons to be primary electrons.

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Look now at the equations, but don’t be too worried about them. Rather than integrating unrestricted mass collisional stopping power over energies from zero to the maximum energy value, we now integrate restricted mass collisional stopping power over energies from delta to the maximum energy value. This ratio is going to be dependent on the value of delta, but we’ll see in a moment the specific value of delta doesn’t really matter too much.

There is a track end term because you must to take into account track ends. That is not a big correction; it’s for completeness to take into account.

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The value of Δ is traditionally taken as the energy of an electron whose range is equal to the mean path length in the cavity. A more physical choice would be based on the mean energy needed for a knock-on electron created in the cavity to escape from it. We don’t generally think of it that way, the way as most people think about it is the top one. Δ is traditionally taken as the energy of an electron whose range is equal to the mean path length in the cavity. You have got a cavity of 2 or 3 millimeters in diameter, which is the about the size of most ion chambers. An electron with a range of a few millimeters is going to be about 10 keV. Fortunately, in practical situations, the theory is relatively insensitive to the choice of delta.

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A value of 10 keV is often used for convenience and the continuous slowing down approximation range of a 10 keV electron is about 2 mm. So that is generally what is used in the calculations.

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You do have to take into account the energy deposited by those particles whose energies fall below delta. And it does represent a significant fraction of the energy deposited and it points out they should be the same because the fluence is the same.

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The track end has a term a lot like this but you do not to really be concerned about it because the calculations of the stopping powers will tell you if it is included. Both evaluated at the cutoff energy Δ is roughly equal to the number of stoppers per unit mass.

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The nice thing about Spencer-Attix cavity theory is that it does not require charged-particle equilibrium to apply, as long as the cavity does not disturb the electron fluence. So the Bragg-Gray cavity conditions exist for Spencer-Attix those two conditions still apply, but you don’t need charged-particle equilibrium because the effect of knock-on electrons is considered explicitly and the assumption of local energy deposition is accurate on its own without invoking charged-particle equilibrium. And the Spencer-Attix theory includes an explicit dependence, through the choice of Δ, on the cavity size. So if you have a real small or large chamber you can in fact see if it’s going to make a difference.

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And I think the last slide simply shows it makes rather little difference.

These are calculations of water to air stopping power ratios at a depth of more than halfway into the range of the electrons. The incident electron energies are on the left. The top row indicates the cutoff values in keV of the Spencer-Attix theory, and the column on the right are the Bragg-Gray values. You see that if you choose cutoff values that are rather high, you get values that are fairly close to the Bragg-Gray values, but if we go down to very low cutoff energies, we see the difference between Bragg-Gray and Spencer-Attix values is about 2%.

The effect is not large but it is important enough in dosimetry and certainly for standards labs to take this into account. When theory is being compared with very accurate dose measurements using calorimetry or chemical dosimetry, we see that the Spencer-Attix approach is more accurate than the Bragg-Gray.