N may be the number of photon interactions in a given volume or the number of radioactive disintegrations in a given time. Expectation value:

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1 DESCRIPTION OF IONIZING RADIATION FIELDS (Chapter 1 p5-19) Stochastic Vs Non-Stochastic Description Raiation interaction is always stochastic to some egree. o entities whether photons or charge particles are iscrete. o the iniviual behavior of particles has some level of ineterminacy. o It is quite tricky to efine erivatives for stochastic quantities. o photon interactions are ranom spatially an raioactive ecay is ranom temporally. Both can be escribe by Poisson processes. A non-stochastic quantity is continuous in space an time. o can be efine through an expectation value of a stochastic quantity. o erivatives can be efine at a point Be N i is the result of the i th measurement of a set of n measurements. Then the average result is: ( n N i ) i N mean n N may be the number of photon interactions in a given volume or the number of raioactive isintegrations in a given time. Expectation value: N e N as n See the figure on page 8 of the course notes for the last hour. With this trick, we can make non-stochastic quantities out of stochastic ones. In the limit n we can shrink the volume or the time interval to a point an the expectation value is still efine. 1

2 Fluence, φ N e is the expectation value of the number of particles crossing a sphere of maximum cross-sectional area a about a point P. N e P Area a The (omniirectional) particle fluence at P is efine as:: φ N e a particles unit area

3 Chilton s Definition of Fluence li is the tracklength of particle i through volume V l i Volume, V P Chilton's efinition of fluence is quite interesting. It is the sum of the pathlengths of particles passing through a volume V. Having a large number of particles an shrinking the volume to a point P woul obtain the expectation value of fluence. N i l i φ 1 as N an V 0 V Although Chilton prove this efinition for straight paths it also shoul apply for curve paths. Fluence Rate, φ & & φ particles φ t unit area unit time Note: Attix uses the variable ϕ, rops the ot an calls this variable flux ensity. Raiant Energy, R The raiant energy is the expectation of the energy (not incluing rest-mass energy) carrie by N e particles. Usually in units of [J]. for monoenergetic particles: R T N e Raiant Energy for Particles with Mass (where T kinetic energy) R hv N e Raiant Energy for Photons Note: Attix uses E instea of T. 3

4 Energy Fluence, ψ ψ R a energy unit area where, a, is again the cross-sectional area of a small sphere aroun P. applies for monoenergetic particles: ψ Tφ Energy Fluence for Particles with Mass ψ hvφ Energy Fluence for Photons energies for particles are normally given in electron-volts or ev. It represents the amount of kinetic energy gaine by a singly charge particle in accelerating through a potential of 1 Volt: T e V. Energy Fluence Rate, ψ& ψ& ψ t t R a energy unit time unit area applies for monoenergetic particles: ψ & T & ϕ For Particles with Mass ψ & hv & ϕ For Photons Differential Distributions Differential istributions are ensity functions in the phase space of a particle. there are eight variables which escribe the phase space (state) of a particle: type (this inclues species, rest energy, charge, spin, etc.) position in three space (e.g. x,y,z) irection of motion (e.g. polar angles, β ) kinetic energy (T for particles with mass, hν for photons) time position, irection, energy an time are continuous variables that can be thought of as specifying a 7 -imensional phase space. The istribution functions are assume to be ifferentiable, such that they can be efine for each, for all, or for any combination of these phase space variables. 4

5 e.g. a ifferential quantity with respect to position, irection, energy an time woul be the ifferential particle fluence: φ ( x y, z,, β, T, t), 7 φ x y z β T t this istribution contains all of the information that can be known about the raiation fiel. if we restrict ourselves to looking at a point P then a very useful ifferential quantity, which completely specifies the raiation fiel at P, is the ifferential fluence rate. Differential Fluence Rate 3 φ φ (, β, T, t) φ (, β, T ) β T the ifferential fluence rate is usually expresse per unit soli angle, Ω. soli angle units are [steraian] [sr] Ω sin β 5

6 6 β β r r sin sin Ω Attix FIGURE 1.: Polar coorinates. The element of soli angle is Ω. ( ) ( ) β φ φ sin /,,, T T & & Ω the ifferential fluence rate (being continuous) may be integrate over the entire phase space to yiel the fluence rate: ( ) T T T T β φ φ π β π sin, max Ω & & x y z

7 Energy Spectrum φ (T) efines the spectrum of (particle) fluence rate: & & φ π π φ ( T ) & φ 0 β 0 ( Ω, T ) sin β T & T max φ & φ T o For particles with kinetic energy T the spectrum of energy fluence rate, ψ& (T), is efine as: ( T ) T & φ ( T ) ψ & T max ( T ) T T & φ ( T )T ψ & & T max T o ψ T o The same equations woul apply to photons if hν replaces T. Suppose we ha the following spectrum of fluence rate. It will turn out that this is a non-physical energy spectrum. 7

8 we can see from this spectrum that: & a φ T 0 < T < T T ψ& & φ ψ& ( ) ( ) ( T ) b ( T < T < T max) ( T ) a ( 0 < T < T ) a constant b constant ( T ) bt ( T < T < T max ) The spectrum of energy fluence rate looks like this: a Particle Energy, T [kev] The fluence rate is: & T a T φ 0 T + T T a ln T / 0 + b T which is unefine (or rather infinity)! max bt ( ) ( max T ) ψ& at + T T max 0 T btt b ' at + ( T max T ) The energy fluence rate is finite(i.e. oes exist). 8

9 note that because of continuity at T T' that b T a : Hence a has the units [ ( )] or [ J /( cm s) ]. ( T T ) a ψ& a T + max T J / cm s kev since ψ& has the units of energy fluence rate 9

10 Angular Distribution & φ & φ β 0 T 0 Ω π T max ( ) & φ (, T ) sin β T the above is the ifferential fluence rate per unit angle expresse in particles/(unit area. unit time. unit raian) one can also efine the ifferential fluence rate per unit soli angle: & φ & φ T max particles ( Ω) ( Ω T ) T Ω T 0 φ, unit area unit time unit sr in many problems in raiation physics the istribution is inepenent of the azimuthal angle β : & φ ( ) π sin & φ ( Ω) for & φ ( Ω) inepenent of β Planar Fluence, φ p the planar fluence is the number of particles crossing a fixe plane per unit area of the plane. Alternative efinition: The planar fluence is the component of the (omniirectional) fluence perpenicular to the plane of interest. 10

11 Incoming X-rays Raiation penetrating the etector Same planar fluence, but larger omniirectional fluence Scattere X-rays for the spherical etector N incoming N scattere i.e. more scattere particles entering the etector cos l scattere l incoming i.e. same average path length the (omniirectional) fluence for the spherical etector woul be: φ scattere φincoming cos for the flat etector N scattere N incoming i.e. same number of scattere particles entering l incoming the etector l scattere i.e. longer average path length cos 11

12 the (omniirectional) fluence for the flat etector woul be: φ scattere φincoming cos therefore, both spherical an flat etectors woul etect the increase in omniirectional fluence ue to scattering. However,if the raiation oes not penetrate the etector (energy etector), the flat etector respons to planar fluence: Incoming X-rays Scattere X-rays if the raiation is non-penetrating the flat etector is responing to planar fluence whereas the spherical etector woul still be responing to (omniirectional) fluence. the formal efinition of planar fluence rate is given by: & φ p π π T max 0 β 0 & T 0 φ Ω (, T ) cos sin β T 1