Total ionizing dose (TID) estimates calculated for space missions, intended to represent ... total number of liberated e-h pairs is the deposited energy divided by 3.6eV ... experimentally found to produce only half the shift in the flat-band voltage, .... type, where x and t denote the location and time of evaluation of this quantity, ...
L. Edmonds 8/18/14
Dose Conversions between Different Materials 1. Introduction Total ionizing dose (TID) estimates calculated for space missions, intended to represent the radiation environment that will be encountered by electronic devices used in a mission, usually select either Si or SiO2 as the reference material. Sometimes there is a need to know the amount of dose that the same environment would deposit in a different material. This report considers dose conversions from one material to another, but limitations of these conversions are explained first. Basically, all of the limitations taken together reduce to the statement that the impact that a given amount of dose has on the operation of some electronic device depends on the type of radiation that produced the dose. This distinguishes electrons from protons in a natural space environment and also distinguishes X-rays from gamma rays used in the laboratory for TID testing. These limitations are discussed in more detail in Sections 2 through 4. The definition of the type of dose (called the equilibrium absorbed dose) considered here, and the relevant physical mechanisms, are discussed in Sections 5 through 10. Numerical results that include dose conversions between different materials are given in Section 11. 2. Meaning of Absorbed Dose TID is synonymous with absorbed dose and the latter terminology is used here. The absorbed dose in a given material is defined to be the energy per unit mass deposited in the material by ionizing radiation. It should be noted that not all of this energy is converted into ionization. Energy lost by an incident particle is distributed over a number of effects including the creation of phonons or photons, ionization, excitation of electrons, and displacements of atoms within a lattice. Furthermore, many of the electrons liberated by primary collisions (collisions involving the incident particle) have enough energy to produce further ionization. Most of the total ionization is typically produced by these secondary electrons. Like the incident particle, only a portion of the energy lost by secondary electrons is converted into ionization because this energy loss is also distributed over the above effects. A chain of reactions that starts with an incident particle producing first-generation secondary particles, with these particles producing secondgeneration secondary particles, etc., can involve a complicated sequence of interactions. Fortunately there is a simple end result when the number of secondary particles is large enough for cumulative effects to be governed by statistical averages. On average, the total number of e-h pairs created is proportional to the energy lost by the incident particle (we are tacitly assuming charged particle equilibrium here, which will be explained in Section 4). In this context, the total number of e-h pairs includes those produced by all secondary particles. For silicon, the average total number of liberated e-h pairs is the deposited energy divided by 3.6eV (3.6eV is called the W-value for silicon in [1, p. 262]).1 The bandgap of silicon is only about 1.1eV, so roughly one third of the deposited energy is converted into ionization, with the remaining two thirds distributed over the other effects (mostly heat). Similar considerations apply to SiO2 except that we use 18eV instead of 3.6eV [2, p. 91], and the bandgap is about 9eV so about half of the 1
This is true for the types of radiation that are either used for TID testing or are important (from the point of view of TID) components of a natural space environment, but the 3.6eV is not always applicable outside of this context. For example, some single-event effects tests are performed with lasers. These photons do not have enough energy to produce secondary electrons that are energetic enough for further ionization so the number of e-h pairs is calculated from the number of absorbed photons instead of using the 3.6eV conversion.
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deposited energy is converted into ionization. Because the number of e-h pairs is proportional to the deposited energy (albeit a different proportionality constant for different materials), absorbed dose is useful information for quantifying the amount of ionization in a given material. However, absorbed dose alone is not enough information for predicting a device response even after a device has been tested for dose effects unless the type of radiation used for testing is the same as the radiation that will be encountered during application. Reasons are given in the next two sections. 3. Prompt Recombination The absorbed dose determines the initial number of e-h pairs but does not determine the number of e-h pairs that survive recombination. In an insulator, such as SiO2, this recombination can be so fast, or prompt, that a significant fraction of the liberated e-h pairs recombine even before the liberated electrons are swept out of the material by an electric field that is present (e.g., via a biasing voltage) in the material. This recombination is more important when the electric field is weak, because electrons remain in the material for a longer time before being swept out of the material by the electric field, so the fraction of e-h pairs that survive recombination increases with increasing field strength. When the electric field is weak enough for this recombination to be important, the recombination can be different for different incident particles that produce the same absorbed dose in the insulating material. The reason is that different incident particles produce different spatial distributions of liberated carriers. In one extreme case, the dose is produced by a large number of incident particles that are individually very lightly ionizing (e.g., electrons). The density of e-h pairs produced by a single incident particle is small enough in this extreme so that the only hole in the vicinity of a given liberated electron is the companion hole (the hole created by the liberation of that electron). Recombination for this extreme case is called geminate recombination. In the opposite extreme, the dose is produced by a relatively small number of incident particles that are individually highly ionizing (e.g., a heavy ion). The density of e-h pairs produced by a single incident particle is so large that a given liberated electron is surrounded by a swarm of holes. Recombination for this extreme case is called columnar recombination. Whether recombination is better described as geminate or columnar is determined by comparing the average spacing between an electron and its companion hole to the average spacing between e-h pairs. If geminate, then recombination is sensitive to the average spacing between an electron and its companion hole but insensitive to the density of e-h pairs. If columnar, then recombination is sensitive to the density of e-h pairs but insensitive to the average spacing between an electron and its companion hole. A typical case is somewhere between geminate and columnar. Absorbed dose does not include the effects of prompt recombination so a separate calculation is needed for this. This is done via a yield function, defined to be the fraction of surviving e-h pairs as a function of the electric field. The absorbed dose and yield function are both needed to predict the response of a device to ionizing radiation. However, the yield function is a different function of electric field for different incident particles as explained in the previous paragraph, so a device response can be different for different incident particles that produce the same absorbed dose in the insulating material. For example, X-rays from a Cu-target X-ray tube were experimentally found to produce only half the shift in the flat-band voltage, compared to the same dose from Co60 gamma rays, in some example MOS capacitors when a weak electric field was present in the oxide during irradiation, but the distinction between radiation sources disappeared when a stronger electric field (> 1 MV/cm) was present during irradiation [3]. This 2
was interpreted as differences in recombination with the differences becoming less important at stronger electric fields, i.e., the yield functions are significantly different at weak fields but nearly the same at stronger fields. Other example comparisons of yield functions for different types of radiations given in [4] show vary large variations between different types of radiations. However, it might be useful to know that these comparisons also show no distinction between 12MeV electrons and Co60 gamma rays. Other examples in [5] compare protons in SiO2 at different proton energies. It was concluded in [5] that at low proton energies (≤ 3MeV) the LET is large enough to produce columnar recombination and this results in a relatively small yield. At high energies (≥ 100MeV) the recombination is geminate and the yield is relatively large. Intermediate energies are between the two extremes but the yield increases with increasing proton energy because the recombination becomes more geminate and less columnar. Although outside the scope of this report, it might be useful to note that there appears to be a consensus that measured TID test results (for most CMOS devices) performed with Co60 gamma rays are conservative (when not accurate) for the purpose of predicting device response in a different application environment that delivers the same dose.
4. Dose Enhancement and Equilibrium Absorbed Dose Dose calculations are relatively simple only when charged-particle equilibrium (CPE) can be assumed, which is explained as follows. Energy deposited in a small volume element of material can be divided into three categories. The first is the energy deposited by the incident radiation. The second category is the energy carried into the volume element by secondary particles entering the volume element after being created outside the volume element. The third category is a negative contribution and is the energy carried out of the volume element by secondary particles leaving the volume element. The second and third categories can each be subdivided by separating secondary electrons from secondary photons (e.g., bremsstrahlung photons). CPE is defined by the condition that the energy carried into a volume element by secondary electrons subtracts out with the energy leaving the volume element by secondary electrons. Stated another way, secondary electrons that are stopped in the volume after creating ionization within the volume are replaced by new secondary electrons created (directly or indirectly) by the incident radiation, so that secondary electrons leave the volume at the same rate that they enter. Note that this alternate statement (that secondary electrons are created within the volume at the same rate that they are stopped in the volume) will be satisfied if nearly all secondary electrons created in the volume are stopped in the volume, but this sufficient condition is not necessary for CPE (a more detailed discussion of CPE is given in the appendix). The dose calculated from the assumption of CPE is called the equilibrium absorbed dose. CPE applies only when two conditions are satisfied. The first is that the incident radiation must be nearly homogeneous over spatial distances equal to the average range of the secondary electrons. The second condition is that the location within the material, at which the dose estimate is intended to apply, be sufficiently deep in the material so that the distance between this point and the material boundary is at least as large as the average range of the secondary electrons. If these conditions are not satisfied, e.g., if the location of interest is too close to a boundary separating dissimilar materials, the actual absorbed dose will not be equal to the equilibrium absorbed dose. The ratio of the actual dose to the equilibrium dose is called the interface absorbed dose enhancement factor, or simply the dose enhancement factor.
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The dose enhancement factor is another distinction between different types of radiations if the location of interest is near the interface separating dissimilar materials. The reason is that the materials might appear to be only slightly dissimilar from the point of view of one type of radiation, and very dissimilar from the point of view of another type of radiation. For example, looking ahead to Table IA (explained later), we find that the mass energy absorption coefficient in silicon is almost twice as large as in silicon dioxide for 20KeV photons (a representative photon energy for some X-ray machines), but the materials are very similar from the point of view of 1MeV photons (a representative energy for gamma rays from a Co60 source). We can therefore expect the dose enhancement factor near a Si/SiO2 interface to be different for photons produced by an X-ray tester compared to photons produced by a Co60 source. Experimental measurements reported in [3] support this assertion. Another issue regarding a Co60 source is that some of the most energetic secondary electrons can have ranges between two- and three-tenths of a centimeter in materials such as Si or SiO2. CPE will not apply to a sample of such material having much smaller dimensions unless a shield of similar material is inserted between the sample and the Co60 source. Without CPE, the dose enhancement factor becomes an essential part of the calculation of absorbed dose [2, Section 8.3.3]. 5. Intensity Vector for Ballistic Energy Transport A particle is classified here as ballistic if its kinetic energy is large enough to be distinguishable from thermal energy and, if the particle is charged, the kinetic energy is also much larger than the potential energy associated with an electrostatic potential. A convenient description of a radiation field consisting of such particles is the differential directional particle intensity (the phrase “particle intensity” distinguishes this from another type of intensity that has the dimensions of power per area). Any selected type of ballistic particles (e.g., secondary electrons with energies in some arbitrarily selected energy interval) has its own differential directional particle intensity function. Let f(x, t, n, E) denote this quantity for a selected particle type, where x and t denote the location and time of evaluation of this quantity, the unit vector n is a direction of particle travel, and E is the particle energy (if the selected particle type specifies an energy interval then f is zero if E is outside that interval). We define f(x, t, n, E) by the condition that f(x,t,n,E)ΔEΔΩ is the rate per area that particles of the selected type, having energies within a small increment ΔE of E and having trajectories within a small solid angle increment ΔΩ of the direction n, cross a surface element located at x at time t and oriented to be normal to n. From this function we can construct an intensity vector for the selected particle type, denoted j(x,t) and defined by j ( x, t )
E f ( x, t , nˆ, E ) dE nˆ d . 4 0
(1)
The magnitude of j(x,t) is the rate per area of the net flow of energy through a surface element oriented so that the direction of j(x,t) is normal to the surface. Net flow means that energy flow in the reverse direction (the direction opposite to the direction of j(x,t)) is subtracted from energy flow in the forward direction, but the larger flow defines the direction of j (the forward direction) via (1). The rate of energy flow through a surface element having arbitrary orientation is the area of the surface element multiplied by the component of j(x,t) in the direction of the surface normal unit vector. The surface integral of j(x,t) over a closed surface is the net rate of energy 4
flow out of the volume enclosed by the surface (when the surface normal vector that defines the surface integral is an outer normal, which is the usual convention when using the divergence theorem that will be needed later). It is possible for a net outward flow to be positive if, for example, j refers to bremsstrahlung photons and such photons are created by electrons contained within the volume. However, it is more convenient to refer to a net inward flow for reasons that will be seen later. This is the negative of the surface integral of j. The significance of j can be seen as follows. Select a volume element ΔV of the material. Let E denote the sum of energies of all particles of the selected type contained within this volume element. Depending on how inclusive or exclusive the selected particle type is, it may be possible for collisions in the material to produce energy exchanges between particles of the selected type and particles outside the selected type. In this case there are three contributions to the rate of change of E. The first is the rate that collisions transfer energy from particles outside the selected type to particles of the selected type. The second is the rate that collisions transfer energy from particles of the selected type to particles outside the selected type. The sum of these two rates will be denoted ‒Lnet where Lnet is the net loss (loss minus gain) rate and is positive when energy loss of the selected particle type exceeds the energy gain. The remaining contribution to the rate of change of E is the net inward flow of energy into the volume from outside the volume. It was concluded at the end of the previous paragraph that this contribution is the negative of the surface integral of j. The result is E Lnet j dS t
where the surface integral is over the closed boundary of the volume element ΔV. The divergence theorem states that this surface integral is the volume integral, over the volume ΔV, of the divergence of j. Taking ΔV to be small enough so this divergence can be regarded as constant and factored out of the integral, the above equation becomes E Lnet j V . t
Dividing by ΔV gives
Lnet j t
(2a)
where ԑ is the energy of the selected particle type per unit mass of the material, ρ is the mass density of the material (so ρԑ is the energy per unit volume), and Lnet is the net energy loss (loss minus gain) rate of the selected particle type, via energy exchanges with other particle types, per unit volume. In all applications in later sections, the incident radiation will be constant in time and the radiation field associated with any selected type of ballistic particles will be constant in time. Therefore the left side of (2a) is zero if the selected particle type consists of ballistic particles. The only stored energy that is considered to be increasing with time is the deposited energy discussed later in Section 7. For ballistic particles under steady-state conditions we have 5
j Lnet
(ballistic particles ) .
(2b)
6. Quasi-Homogeneity A radiation field cannot be exactly homogeneous if it deposits energy in a material. Exact homogeneity implies that the intensity vector j(x) for a selected particle type (explained in Section 5) does not depend on x and therefore has a zero divergence. However, a radiation field can be quasi-homogeneous within some volume of material in the sense that the dose is approximately uniformly distributed throughout that volume. For example, energy deposited by a proton can be described in terms of a linear energy transfer (LET) of the proton in a selected material. A nonzero LET means that the proton energy necessarily changes as it moves through the material. However, if this change in energy is a small enough fraction of the initial energy, the LET is nearly constant as the proton travels through the material. This is the meaning of quasi-homogeneity as applied to protons. For another example, energy deposited by gamma rays is proportional to the gamma-ray intensity, which necessarily changes if gamma rays are absorbed. However, if this change in intensity is a small enough fraction of the initial intensity, the intensity is nearly constant throughout the material. This is the meaning of quasihomogeneity as applied to gamma rays. In the more general context of an intensity vector, quasihomogeneity in some volume of material does not mean that the divergence of the intensity vector is zero. It means that the divergence of the intensity vector is nearly uniform within that volume. 7. A Continuity Equation Consider the case where energy stored per unit mass, denoted ԑ, is a sum over all forms of energy including: thermal, the binding energies of excited electrons, the binding energies of freed electrons, the kinetic energies of ballistic secondary electrons, atomic binding energy associated with displacement damage, and energy stored in the electromagnetic field (if photons are present). Because total energy is conserved, the loss term in (2a) is zero if j includes all forms of energy flow. The result is jtotal ( x ) ( x, t ) t where jtotal(x) is the total intensity vector. Its magnitude is the rate per area of the net flow of energy, including all modes of energy flow, through a surface element oriented normal to the intensity vector (see Section 5). The total intensity can be expressed as the sum jtotal jHC jEC jIP jSE jS
where the above terms are defined by: jHC ≡ Heat conduction vector. It is proportional to the gradient of the temperature. The negative divergence is the rate of change of thermal energy per unit volume associated with heat conduction. jEC ≡ Energy flow vector from electric (drift-diffusion) currents. It is the current density vector multiplied by the electrostatic potential. The negative divergence is the Joule heating rate per unit volume. 6
jIP ≡ Intensity vector for the ballistic transport of the incident particles (see Section 5). The negative divergence is the rate per unit volume that energy is lost by the incident particles. Some of this energy might be transferred to a different particle type (adding to the divergence of the corresponding intensity vector) with the remaining energy added to the energy per unit volume stored in the material. jSE ≡ Intensity vector for the ballistic transport of secondary electrons (see Section 5). The negative divergence is the rate per unit volume that energy is lost by the secondary electrons. jSγ ≡ Intensity vector for secondary photons. These are photons produced either by bremsstrahlung or Compton scattering. The negative divergence is the rate per unit volume that energy is lost by the secondary photons. Most of this lost energy will be transferred to second- or higher-generation secondary electrons, which adds to the divergence of the secondary electron intensity vector. The above list is not exhaustive but is assumed to include all types of energy transport that are relevant to macroscopic TID. One type of energy transport not included in the above list occurs when a high-energy proton collides with the nucleus of a resident atom and creates nuclear fragment recoils (possibly even releasing energy associated with fission), with the recoils themselves producing ionization. This could be important to studies of micro-dose, produced locally by a single ionizing particle, but it is assumed here that this is a negligible contribution to estimates of macroscopic dose, which is a more global quantity that is summed over all ionizing particles. Combining the above equations gives
jHC jEC jIP jS jSE . t
The energy deposited in the material by radiation is defined here to be the change in stored energy that would occur without heat conduction or drift-diffusion currents. In other words, the deposited energy per unit mass, denoted ԑdep, is defined by
jIP jS jSE dep . t
For steady-state irradiation, the stored energies per unit volume of photons or ballistic electrons are constant in time and do not contribute to the rate of change on the right, so the deposited energy need not include these modes of energy storage. More generally, the only contribution to the rate of change is from energy stored in the last stage of the sequence of energy transfers. For all intermediate stages of energy storage, under steady-state conditions, there is a balance between energy flow, energy gain, and energy loss, as indicated in (2b), so the stored energy is constant in time. However, energy stored in the last stage does not have a loss or a flow, only a gain, so this stored energy increases with time. For example, energy stored in the last stage includes ionization energy. Ionization energy has no loss mechanism associated with the energy deposition process because, by convention, carrier recombination is regarded as a material response that occurs after energy deposition and is separate from energy deposition. Ionization energy has no energy flow mechanism associated with the energy deposition process because, by convention, drift-diffusion currents are regarded as material responses that occur after energy deposition and are separate from energy deposition. Interpreting the absorbed dose D to be the 7
energy stored in the final stage per unit mass, i.e., stored energy summed over all modes of energy storage that have no loss or flow mechanisms, the time integral of the above equation can be written as D
1 J IP J S J SE
(3)
where each upper case J is the time integral of the corresponding lower case j. Charged-particle equilibrium (CPE) is defined by the condition that JSE has a zero divergence, so (3) reduces to D
1 J IP J S
(CPE ) .
(4)
8. Energy Loss Classifications and Approximations Recall that the time-integrated intensity vector J representing a given particle type has the property that its negative divergence is the energy lost per unit volume for that particle type as indicated in (2b). Collisions that change the direction but not the energy of a particle (e.g., elastic scattering of electrons, or coherent scattering of photons such as Rayleigh scattering) do not contribute to the divergence. The energy loss of the incident particles is a sum of several contributions according to
J IP J IP dep J IP SE J IP S
(5a)
where the terms on the right are explained as follows. The first term on the right, representing deposited energy, is the energy lost by the incident particle when producing excitation or creating secondary electrons that do not have enough energy for additional ionization so that this energy is deposited locally in the material. The second term on the right is the energy lost by the incident particle when producing secondary electrons that do have enough energy for additional ionization so that this energy is transferred from one mode of energy transport to another mode of energy transport. The third term on the right is the energy lost by the incident particle when producing photons so this is also a transfer of energy from one mode of energy transport to another mode of energy transport. Note that if the incident particle is a photon and undergoes a Compton scattering so that the incident photon is replaced by an electron and another photon, this other photon is classified as a secondary photon. Any photon created by bremsstrahlung is also classified as a secondary photon. Similarly, for secondary photons we have
JS JS dep JS SE JS IP JS SE
(5b)
where the first two terms have the same meaning as in (5a) but with secondary photons substituting for incident particles. Physical mechanisms that contribute to these terms include the photoelectric effect and Compton scattering. The third term on the right side of (5b) is an energy loss that has a negative value. It is the negative of the energy gained by the secondary photons 8
from the incident particles. Similarly, the last term is the negative of the energy gained by the secondary photons from the secondary electrons. Physical mechanisms that contribute to this term include bremsstrahlung. Any term in (5) containing a right arrow is positive if not zero and any term containing a left arrow is negative if not zero. In fact, one of the terms in (5b) is the negative of one of the terms in (5a). It is evident that the energy gained by secondary photons from the incident particles is the energy lost by the incident particles to the secondary photons so
J S
IP J IP S .
(5c)
More generally,
J A B JB A
(6)
for any two particle classifications denoted above as A and B. However, there is a subtle issue that should be discussed regarding the equation
JS SE JSE S
in general, or the equation
JIP SE JSE IP
for the special case in which the incident particles are photons. These equations state that the energy given up by a photon to create a secondary electron is the energy gained by the electron from the photon. This is correct if the energy of the electron is defined by its position in an energy level diagram so the energy includes binding energy in addition to kinetic energy. However, an alternate convention assigns only kinetic energy to the electron. With this alternate convention, the energy given up by the photon will be greater than the energy gain (the kinetic energy) of the electron, and a correction term would be needed in the above equations to account for binding energy that will eventually be seen as fluorescence. We use the former convention here so no correction term is needed in the above equations but it should be understood that the energy of a mobile electron as defined here is the band-gap energy plus the kinetic energy. This issue is not important when the electron energy is much greater than the band-gap energy. Combining (4) with (5) gives D
1 J IP dep J IP SE J S dep J S J S
SE
SE
(CPE ) .
(7a)
The next step utilizes an approximation. The energy locally deposited by photons is the energy transferred to those secondary electrons that did not receive enough energy to produce further ionization. This can be a significant amount of energy transfer from the photons if there is a large quantity of photons that transfer this amount of energy, which could be the case if the incident 9
particles are photons with energies selected for this purpose. However, these incident photons have relatively low energies and will not create a significant number of secondary photons. Higher-energy incident photons produce a spectrum of secondary photon energies with only a small portion of this spectrum capable of producing local energy deposition. The approximation used here is that energy deposition by secondary photons, the third term on the right side of (7a), can be neglected. We now utilize another approximation. Consider a sample of material that is being exposed to incident radiation. The approximation neglects the secondary electrons created within the sample by secondary photons that were also created within the sample. This approximation omits the fourth term on the right side of (7a). This approximation can be used only for those cases in which the physical dimensions of the sample are not too large. Recall that CPE requires the material thickness (or net thickness if the sample is surrounded by an over-layer or shield of similar material to produce CPE) is not too small. An obvious question is whether it is possible to be both, not too small and not too large. This question is particularly concerning for the example in which SiO2 is exposed to gamma rays from a Co60 source because it was noted in Section 4 that the ranges of the longest ranged secondary electrons is slightly greater than twotenths of a centimeter, so CPE requires this thickness or more. A thickness between two-tenths of a centimeter and one- or two-centimeters will satisfy CPE and it is argued below that this will also satisfy the not-too-thick requirement. The argument notes that the mean free path between collisions for the incident gamma rays (having a representative energy of about 1MeV) is about seven centimeters,2 so the number of secondary photons will be smaller than the number of incident photons. Also, the average energy of secondary photons is more than half of the incident photon energy [1, p. 184], i.e., more than 500KeV, and the mean free path between energytransferring collisions for photons having this energy in SiO2 is about 5cm. Therefore, most of these photons, which are fewer in number than the primary photons, will leave the sample without an energy-transferring collision. We conclude that the not-too-thick and not-too-thin conditions are both satisfied if the sample of material under test is not more than a couple of centimeters thick and the net thickness (material plus shield) is at least several tenths of a centimeter. Omitting the third and fourth terms on the right side of (7a) for reasons explained in the previous two paragraphs, and rewriting the last term via (6) gives D
J IP dep J IP SE J SE S ( thin sample with CPE ) . 1
(7b)
An alternate expression for the dose is exact whether the sample is thick or thin (but if it is not thin enough for quasi-homogeneity the dose will not be uniform within the sample) and does not even require CPE. This is obtained by substituting the expansions in (5), with a similar
2
Mean free path can be estimated as follows. Select a collision type such as energy-transferring (e.g., Compton scattering), or scattering without energy transfer (e.g., Rayleigh scattering), or both types lumped together. Define a corresponding attenuation coefficient to be the cross section per unit volume (per-atom cross section multiplied by the number of atoms per unit volume) for the incident particle to have a collision of the selected type. This coefficient is also equal to the expected number of collisions of the selected type per unit distance of travel for one incident particle. Therefore the mean free path between collisions of the selected type is the reciprocal of the corresponding attenuation coefficient. These coefficients can be found in [7].
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expansion for secondary electrons, into (3) and then using (6) to substitute for each term having a left arrow. The result is D
1 J IP dep J S J SE dep . dep
(7c)
If the incident particles are X-ray or gamma ray photons, the first two terms on the right can be neglected compared to the third so the dose is the energy deposited by the secondary electrons. However, even if all secondary electrons stop in the sample, the energy deposited by them is still not equal to all of the energy that was given to them because some of the latter energy was removed from the secondary electron population via bremsstrahlung. X-rays and gamma rays are discussed in more detail in the next section.
9. Equilibrium Absorbed Dose Calculation Methods for X-Rays and Gamma Rays When the incident particles are photons, various types of interactions are represented by corresponding cross sections. The goal of this section is to relate the various terms in (5) and (7) to these cross sections so that the terms can be estimated from data available in the literature. However, it is first necessary to clarify the meaning of incident particle. Two types of photon interactions are considered. One type changes the direction but not the energy of the photon. Examples include Rayleigh scattering. All of these types of interactions will be lumped together in a category that will be called coherent scattering. The other type of interaction has an energy transfer (e.g., the photoelectric effect or Compton scattering). Any interaction of this type will be called an energy-transferring interaction. If an incident photon undergoes a coherent scattering, the resulting photon is classified here as an incident photon that was deflected. Therefore, a photon that is classified as an incident photon can be either, a deflected photon or a photon that had no collisions at all. With this classification convention, coherent scattering does not contribute to the divergence of JIP because no energy was removed from the population of incident particles. If an incident photon has an energy-transferring interaction, all particles (electrons and photons) that result from the interaction are classified as secondary particles. A. The Meaning of an Event Cross Section The goal is to relate the various terms in (5) and (7) to cross sections published in the literature and the first step is a brief review the significance of event cross sections in general. For any type of reaction done by any kind of incident particle having energy E, let σ(E, n) denote the corresponding cross section assigned to some target, such as a molecule. The unit vector n in the argument allows for the possibility that the cross section might depend on the directional orientation of the target relative to the incident particles. The event cross section satisfies the condition that the expected (statistical average) reaction rate, or expected event rate, is given by Reaction Rate
4
f ( x, nˆ, E ) ( E , nˆ ) d
where f is the differential directional particle intensity discussed at the beginning of Section 5, and the location of evaluation x is the location of the target. Cross sections are regarded as isotropic in the applications considered here so the above equation reduces to 11
Reaction Rate ( x, E ) ( E )
(8a)
where φ is the omnidirectional flux defined by
( x, E )
4
f ( x, nˆ, E ) d .
(8b)
It is important to note that this omnidirectional flux is the same as the flux traditionally used to quantify a beam intensity (the rate per area that particles cross a surface perpendicular to the beam) when the beam consists of unidirectional particles. It is equally important to note that if the particles are not unidirectional then the omnidirectional flux, used to calculate a reaction rate via (8a), is not the same as the rate per area that particles cross some selected surface. B. Definition and Significance of the Total Attenuation Coefficient The first quantity considered here, denoted μ(E), is the cross section per unit volume (a permolecule cross section multiplied by the number of molecules per unit volume in the target material) for a collision by a photon with energy E. This is the cross section per unit volume for coherent collisions plus the cross section per unit volume for energy-transferring collisions. The significance of μ is seen when considering a flat wafer-shaped material sample, with area A and thickness T, below a source of photons all having energy E and traveling in the z-direction, which is called the downward direction and is perpendicular to the wafer. Let I0 denote the timeintegrated radiation intensity just above the wafer. The intensity is power per area so I0 is energy per area transported across the top wafer boundary by the incident radiation. Let Iuncol(z) denote the time integrated intensity, at the depth z, consisting only of those incident photons that had no scattering collisions prior to reaching this depth. A property that is unique to un-collided photons is that the photons that contribute to Iuncol(z) are both unidirectional and monoenergetic. Therefore Iuncol(z) = E Φuncol(z, E) where Φuncol (z, E) is the omnidirectional fluence of un-collided photons, which is the time integral of the omnidirectional flux defined by (8b). Now consider a slice of the wafer between the depths z and z+Δz for some small Δz. The collision cross section for this slice is the volume times the cross section per unit volume, which is A μ Δz. We therefore have A ( E ) z I uncol (z) E uncol ( z, E ) A ( E ) z E [ Number of Collisions ] .
(9)
Note that each collision in the slice removes a photon from the un-collided population. The amount of energy removed from this population is the far right side of (9). This removed energy is A Iuncol(z) ‒ A Iuncol(z+ Δz) so I uncol (z) I uncol (z z ) ( E ) z I uncol (z) .
Continuing the analysis by taking the limit as Δz → 0 will ultimately lead to I uncol (z) I 0 exp ( E ) z
12
(10)
which is exact regardless of whether z is large or small. Unfortunately, an exact result analogous to (10) does not apply to other photon categories, as the next example will show. C. Definition and Significance of the Attenuation Coefficient without Coherent Scattering The next quantity considered is the same as the previous except that the cross section does not include coherent scattering events. It is the cross section for an energy-transferring collision summed over all possible types of energy-transferring interactions. This cross section per unit volume (a per-molecule cross section multiplied by the number of molecules per unit volume in the target material) is denoted μet here, with the subscript representing energy-transferring events, and should not be confused with the energy transfer coefficient denoted μtr and discussed in the next subsection. The significance of μet is seen when considering the same wafer of material exposed to the same radiation that was discussed in the previous subsection but now the intensity considered, denoted IIP(z), includes all incident photons, i.e., un-collided as well as coherently scattered photons. The photons that contribute to IIP(z) all have the same energy E but, unfortunately, they are not unidirectional. Therefore, it is not true that IIP(z) is E multiplied by the omnidirectional fluence of incident photons, so the first equality in (9) will not apply if the set of un-collided photons is replaced by the set of all incident photons. An approximation is needed. The approximation that is consistent with quasi-homogeneity is a no-shadowing approximation, which requires that the wafer not be too thick. This approximation assumes that all collision targets in the wafer are exposed to essentially the same incident particle radiation, which is the radiation that enters the top of the wafer. This approximation is accurate when the wafer thickness T is small enough to satisfy μT
