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Cavity Theory I Bragg-Gray Theory Averaging of Stopping Powers Bragg-Gray Theory • The basis for cavity theory is contained in the following equation: dT / dx c t dT D dx (MeV/g) t c 10 dT 1.602 10 dx Gy c Bragg-Gray Theory (cont.) • If a fluence of identical charged particles of kinetic energy T passes through an interface between two different media, g and w, as shown in A in the following diagram, then one can write for the absorbed dose on the g side of the boundary dT dx Dg c, g T and on the w side, dT dx Dw c,w T (A) A fluence of charged particles crossing an interface between media w and g. (B) A fluence of charged particles passing through a thin layer of medium g sandwiched between regions contain medium w. Bragg-Gray Theory (cont.) • Assuming that the value of is continuous across the interface (i.e., ignoring backscattering) one can write for the ratio of absorbed doses in the two media adjacent to their boundary Dw dT / dxc,w Dg dT / dxc, g Bragg-Gray Theory (cont.) • Bragg and Gray applied this equation to the problem of relating the absorbed dose in a probe inserted in a medium to that in the medium itself • Gray in particular identified the probe as a gas- filled cavity, whence the name “cavity theory” • The simplest such theory is called the Bragg-Gray (B-G) theory, and its mathematical statement, the Bragg-Gray relation, will be developed next Bragg-Gray Theory (cont.) • Suppose that a region of otherwise homogeneous medium w, undergoing irradiation, contains a thin layer or “cavity” filled with another medium g, as in B in the following diagram • The thickness of the g-layer is assumed to be so small in comparison with the range of the charged particles striking it that its presence does not perturb the charged-particle field • This assumption is often referred to as a “Bragg- Gray condition” (A) A fluence of charged particles crossing an interface between media w and g. (B) A fluence of charged particles passing through a thin layer of medium g sandwiched between regions contain medium w. Bragg-Gray Theory (cont.) • This condition depends on the scattering properties of w and g being sufficiently similar that the mean path length (g/cm2) followed by particles in traversing the thin g-layer is practically identical to its value if g were replaced by a layer of w having the same mass thickness • Similarity of backscattering at w-g, g-w, and w-w interfaces is also implied Bragg-Gray Theory (cont.) • For heavy charged particles (either primary, or secondary to a neutron field), which undergo little scattering, this B-G condition is not seriously challenged so long as the cavity is very small in comparison with the range of the particles • However, for electrons even such a small cavity may be significantly perturbing unless the medium g is sufficiently close to w in atomic number Bragg-Gray Theory (cont.) • Bragg-Gray cavity theory can be applied whether the field of charged particles enters from outside the vicinity of the cavity, as in the case of a beam of high-energy charged particles, or is generated in medium w through interactions by indirectly ionizing radiation • In the latter case it is also assumed that no such interactions occur in g Bragg-Gray Theory (cont.) • All charged particles in the B-G theory must originate elsewhere than in the cavity • Moreover charged particles entering the cavity are assumed not to stop in it Bragg-Gray Theory (cont.) • A second B-G condition, incorporating these ideas, can be written as follows: – The absorbed dose in the cavity is assumed to be deposited entirely by the charged particles crossing it • This condition tends to be more difficult to satisfy for neutron fields than for photons, especially if the cavity gas is hydrogenous, thus having a large neutron-interaction cross section Bragg-Gray Theory (cont.) • The heavy secondary charged particles (protons, -particles, and recoiling nuclei) also generally have shorter ranges than the secondary electrons that result from interactions by photons of quantum energies comparable to the neutron kinetic energies • Thus we see that the first B-G condition is the more difficult of the two to satisfy for photons and electrons, while the second B-G condition is the more difficult to satisfy for neutrons Bragg-Gray Theory (cont.) • For a differential energy distribution T (particles per cm2 MeV) the appropriate average mass collision stopping power in the cavity medium g is Tm ax dT 0 dx dT ΦT c, g m Sg Tm ax 0 T dT 1 Tm ax dT Dg T dx dT 0 c, g Bragg-Gray Theory (cont.) • Likewise, for a thin layer of wall material w that may be inserted in place of g, Tm ax dT 0 dx dT ΦT c,w m Sw Tm ax 0 T dT 1 Tm ax dT Dw T dx dT 0 c,w Bragg-Gray Theory (cont.) • Combining these two equations gives for the ratio of absorbed dose in w to that in g, which the B-G relation in terms of absorbed dose in the cavity: Dw m S w m S g w Dg m S g Bragg-Gray Theory (cont.) • If the medium g occupying the cavity is a gas in which a charge Q (of either sign) is produced by the radiation, Dg can be expressed (in grays) in terms of that charge as Q W Dg m e g where Q is in coulombs, m is the mass (kg) of gas in which Q is produced, and (̅W/e)g is the mean energy spent per unit charge produced (J/C) Bragg-Gray Theory (cont.) • By substitution, we obtain the B-G relation expressed in terms of cavity ionization: Q W Dw m S gw m e g • This equation allows one to calculate the absorbed dose in the medium immediately surrounding a B- G cavity, on the basis of the charge produced in the cavity gas, provided that the appropriate values of the various parameters are known Bragg-Gray Theory (cont.) • So long as m S gw is evaluated for the charged- particle spectrum T that crosses the cavity, the B-G relation requires neither charged- particle equilibrium (CPE) nor a homogeneous field of radiation • However, the charged-particle fluence T must be the same in the cavity and in the medium w where Dw is to be determined Bragg-Gray Theory (cont.) • If CPE does exist in the neighborhood of a point of interest in the medium w, then the insertion of a B-G cavity at the point may be assumed not to perturb the “equilibrium spectrum” of charged particles existing there, since by definition a B-G cavity satisfies the B-G requirements • Thus a B-G cavity approximates an evacuated cavity in this respect • The presence of an equilibrium spectrum of charged particles allows some simplification in estimating T Bragg-Gray Theory (cont.) • The medium w surrounding the cavity of an ionization chamber is ordinarily just the solid chamber wall itself, and one often refers to the B-G theory as providing a relationship between the doses in the gas and in the wall Corollaries of the Bragg-Gray Relation • Two useful corollaries of the B-G relation can be readily derived from it • The first relates the charge produced in different gases contained in the same chamber, while the second relates the charge in the same gas contained by different chamber walls First Bragg-Gray Corollary • A B-G cavity chamber of volume V with wall medium w is first filled with gas g1 at density 1, then with gas g2 at density 2 • Identical irradiations are applied, producing charges Q1 and Q2, respectively • The absorbed dose in gas g1 can be written as Q1 W D1 Dw m S g1 e 1V w 1 and the dose in gas g2 as Q2 W D2 Dw m S g2 e 2V w 2 First B-G Corollary (cont.) • The ratio of charges therefore becomes Q2 2V W / e1 m S w 2 g Q1 1V W / e2 m S w 1 g which reduces to the first B-G corollary: Q2 2V W / e 1 m S gg12 Q1 1V W / e 2 First B-G Corollary (cont.) • This equation does not depend explicitly upon the wall material w, implying that the same value of Q2/Q1 would be observed if the experiment were repeated with different chamber walls • This is true as long as the spectrum T of charged particles crossing the cavity is not significantly dependent on the kind of wall material • For example, the starting spectrum of secondary electrons produced in different wall media by - rays is the same if the -energy is such that only Compton interactions can occur First B-G Corollary (cont.) • Although different wall media modify the starting electron spectrum somewhat differently as the electrons slow down, the resulting equilibrium spectrum that crosses the cavity in different thick-walled ion chambers is sufficiently similar that Q2/Q1 is observed to be nearly independent of the wall material in this case Second Bragg-Gray Corollary • A single gas g of density is contained in two B-G cavity chambers that have thick walls (exceeding the maximum charged-particle range), and that receive identical irradiations of penetrating x- or -rays, producing CPE at the cavity • The first chamber has a volume V1 and wall material w1, the second has a volume V2 and wall w2 Second B-G Corollary (cont.) • The absorbed dose in the wall of the first chamber, adjacent to its cavity, can be written as en Dw1 K c w1 CPE w1 Q1 W D1 m Sw1 m S g w1 V1 e g g • A similar expression can be written for the absorbed dose in the second chamber Second B-G Corollary (cont.) • The ratio of the two ionizations in the two chambers is V2 en / w w1 Q2 S 2 m g Q1 V1 en / w 1 m S w2 g where the constancy of (̅W/e)g for electron energies above a few keV allows its cancellation Second B-G Corollary (cont.) • A further simplification of the final factor to w1 m S w2 can be made only if the charged-particle spectrum T crossing the cavity is the same in the two chambers • If such a cancellation of stopping powers thus eliminates g from the equation, the same value of Q2/Q1 should result irrespective of the choice of gas Second B-G Corollary (cont.) • A similar expression can be obtained for neutron irradiations in place of photons by substituting kerma factors Fn for the mass energy-absorption coefficients: Q2 V2 Fn w2 m S gw1 W / e1 Q1 V1 Fn w1 m S g W / e 2 w2 • The ratio ̅W/e may have to be retained here if w1 and w2 differ sufficiently to produce heavy charged-particle spectra that have somewhat different ̅W/e values even in the same gas Spencer’s Derivation of the Bragg-Gray Theory • Consider a small cavity filled with medium g, surrounded by a homogeneous medium w that contains a homogeneous source emitting N identical charged particles per gram, each with kinetic energy T0 (MeV) • The cavity is assumed to be far enough from the outer limits of w that CPE exists • Both B-G conditions are assumed to be satisfied by the cavity, and bremsstrahlung generation is assumed to be absent Spencer’s Derivation of the B-G Theory (cont.) • The absorbed dose at any point in the undisturbed medium w where CPE exists can be stated as CPE Dw K w NT0 (MeV/g) where 1 MeV/g = 1.602 10-10 Gy Spencer’s Derivation of the B-G Theory (cont.) • An equilibrium charged-particle fluence spectrum eT (cm-2 MeV-1) exists at each such point, and the absorbed dose can be written in terms of this spectrum as T0 dT Dw dx dT e T 0 w where (dT/dx)w has the same value as the mass collision stopping power for w, in the absence of bremsstrahlung generation Spencer’s Derivation of the B-G Theory (cont.) • The value of eT that satisfies the integral equation formed by setting these two equations equal is N e T dT / dxw • The equilibrium spectrum for an initially monoenergetic source of charged particles is directly proportional to the number released per unit mass, and is inversely proportional, at each energy T T0, to the mass stopping power in the medium in which the particles are allowed to slow down and stop Spencer’s Derivation of the B-G Theory (cont.) • The following diagram is a graph of the equilibrium spectrum of primary electrons that result for this equation when it is applied (twice) to the example of two superimposed sources of N electrons per gram each, one emitting at T0 = 2 MeV and the other at T0 = 0.2 MeV, in a water medium • This is not a realistic spectrum, however, as -ray production has been ignored Example of an equilibrium fluence spectrum, eT = N/(dT/dx), of primary electrons under CPE conditions in water, assuming the continuous-slowing-down approximation Spencer’s Derivation of the B-G Theory (cont.) • Since the same equilibrium fluence spectrum of charged particles, eT, crosses the cavity as exists within medium w, the absorbed dose in the cavity medium g can be written as T dT / dx g e dT T Dg T dx dT N 0 dT / dx dT 0 0 0 g w Spencer’s Derivation of the B-G Theory (cont.) • The ratio of the dose in the cavity to that in the solid w is then Dg 1 T0dT / dxg 0 dT / dxw dT m S w g Dw T0 which is the same as the B-G relation, considering Spencer’s added assumptions of monoenergetic starting energy T0, charged-particle equilibrium, and zero bremsstrahlung Spencer’s Derivation of the B-G Theory (cont.) • The equivalence of S , as employed here, m w g w to the reciprocal of S as defined in m g Dw m S w m S g w Dg m S g may not be immediately obvious, and will be explained in the next section Spencer’s Derivation of the B-G Theory (cont.) • The foregoing Spencer treatment of B-G theory can be generalized somewhat to accommodate bremsstrahlung generation by electrons and its subsequent escape • The dose in the medium w can be rewritten as Dw K c w NT0 1 Yw T0 CPE where (Kc)w is the collision kerma and Yw(T0) is the radiation yield for medium w Spencer’s Derivation of the B-G Theory (cont.) • The equations for the doses are changed to T0 dT Dw dx dT e T 0 c,w and T0 dT Dg dx dT e T 0 c, g where (dT/dx)c,w and (dT/dx)c,g are the mass collision stopping powers in media w and g, respectively Spencer’s Derivation of the B-G Theory (cont.) • The equilibrium fluence, as given by N e T dT / dxw remains unchanged; hence one can rewrite Spencer’s statement of B-G theory in the following form to take account of bremsstrahlung Dg 1 T0 dT / dx c , g Dw T0 1 Yw T0 0 dT / dxw dT m S w g Averaging of Stopping Powers • For the special case treated by Spencer, the spectrum of primary charged particles crossing the cavity is known, being given by N e T dT / dxw • The evaluation of m S w in g Dg 1 T0 dT / dxg dT m S w g Dw T0 0 dT / dxw is seen to be a simple average of the ratio of stopping powers throughout the energy range 0 to T0, apparently unweighted by eT Averaging of Stopping Powers (cont.) • In fact the fluence weighting is implicit, as can be seen by applying Spencer’s assumption to Tm ax dT 0 dx dT ΦT c, g m Sg Tm ax 0 T dT 1 Tm ax dT Dg T dx dT 0 c, g Averaging of Stopping Powers (cont.) • Setting Tmax = T0 for the upper limit of integration, assuming CPE and the absence of bremsstrahlung, this equation becomes T0 dT e dT dx g T 0 1 T0 dT Sg e e dT T dx g m T0 T dT e 0 0 T N T0 dT / dx g Dg e 0 dT / dx dT e w Averaging of Stopping Powers (cont.) • A similar equation can be written for m̅Sw • The mean mass-stopping-power ratio m S w can then g be obtained as shown in Dg 1 T dT / dxg dT m S w 0 g Dw T0 0 dT / dxw through the application of CPE Dw K w NT0 (MeV/g) which clearly depends on the existence of an equilibrium spectrum Averaging of Stopping Powers (cont.) • Since the Spencer B-G treatment was limited to only a single starting energy (T0) of the charged particles, it will be useful to extend it to distributions of starting energies, such as are generated by photons in a statistically large number of Compton events • Consider a homogeneous source of charged particles throughout medium w, emitting a continuous distribution of starting energies: Let NT0 charged particles of energy T0 to T0 + dT0 be emitted per gram of w and per MeV interval, where 0 T0 Tmax • Assume that CPE exists, and that bremsstrahlung may be produced and it escapes Averaging of Stopping Powers (cont.) • The absorbed dose in w is given by CPE Dw Kc w NT0 T0 1 Yw T0 dT0 Tmax T0 0 while the dose in the cavity medium g is Tmax dT T0 Dg dT0 dT e dx c , g T0 0 T 0 T Tmax dT / dx c, g T0 NT dT0 dT T 0 0 T 0 dT / dx 0 w Averaging of Stopping Powers (cont.) • Thus for a continuous distribution of charged-particle starting energies the ratio of absorbed doses in cavity and wall is given by Tmax dT / dx c, g T0 Dg 0 NT dT0 0 dT / dx dT 0 w m Sw g 0 NT T0 1 Y T0 dT0 T Dw max 0 g where the double bar on m Sw signifies integration over the T0 distribution, as well as over T for each T0-value Averaging of Stopping Powers (cont.) • Where CPE does not exist in the vicinity of the cavity, mean stopping powers can be calculated as an average weighted by the differential charged- particle fluence distribution T crossing the cavity • Thus in general the mean stopping-power ratio for a B-G cavity can be expressed as 1 Tmax dT 0 T dx c, g dT Sg Dg mS m g dT w 1 Tmax Sw Dw 0 T dx c,w dT m Averaging of Stopping Powers (cont.) • Since collision stopping powers for different media show similar trends as a function of particle energy, their ratio for two media is a very slowly varying function • This allows the preceding equation to be reasonably well approximated through simple estimation Averaging of Stopping Powers (cont.) • For example, one may first determine the average energy ̅T of the charged particles crossing the cavity: Tmax T 0 TTdT 1 Tmax T TdT Tmax 0 0 T dT and then look up the tabulated mass collision stopping powers for the media in question at that energy Averaging of Stopping Powers (cont.) • For an equilibrium spectrum resulting from charged particles of mean starting energy ̅T0, the stopping powers may be looked up at the energy ̅T0/2 for a crude (but often adequate) estimate of the mean stopping-power ratio required for the B-G relation • The average starting energy ̅T0 of Compton-effect electrons can be calculated from tr T h e e

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posted: | 2/17/2012 |

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