Outline
Exponential Attenuation • Simple exponential attenuation and plural modes of absorption • Narrow-beam vs. broad-beam attenuation • Spectral effects Chapter 3 • The build-up factor F.A. Attix, Introduction to Radiological • The reciprocity theorem Physics and Radiation Dosimetry • Summary
Introduction Simple exponential attenuation • Uncharged particles (photons and neutrons) • The concept is relevant primarily to uncharged ionizing – lose their energy in relatively few large interactions radiation – have a significant probability of passing through • Consider a monoenergetic parallel beam of a very large matter without interactions number N0 of uncharged particles incident – no limiting range perpendicularly on a flat plate of material of thickness L • Charged particles • Assume ideal case where each particle either is – typically undergo many small collisions, losing completely absorbed in a single interaction, producing their kinetic energy gradually no secondary radiation, or passes straight through the – must always lose some or all of their energy entire plate unchanged in energy or direction – range defined by kinetic energy
Simple exponential attenuation Simple exponential attenuation • To find the total change in the number of particles due to absorption in a medium of thickness L:
N L L dN dl N N 0 • Define (1) to be the probability that an individual 0 N particle interacts in a unit thickness of material traversed L eL N0 • If N particles are incident upon dl, the change dN in the number N due to absorption is • The law of exponential attenuation applies to the ideal case of no scattering or secondary radiation in the dN N dl medium (or scattered particles are not counted in NL)
1 Simple exponential attenuation Simple exponential attenuation
• The equation can be replaced by the infinite series • The quantity is the linear attenuation coefficient, 2 3 typically given in units of cm-1 or m-1, and dl is NL L (L) (L) e 1 L ... correspondingly in cm or m N0 2! 3! • Also in use is the mass attenuation coefficient, /r, • If the thickness L is small or absorption is low, L<<1 where r is the density of attenuating medium; units N 2 2 L eL 1 L are cm /g or m /kg N 0 • The quantity 1/ (cm or m) is known as the mean free • For example, for L<0.05 this approximation is valid path or relaxation length of the primary particles. It is within ~0.1% the average distance a single particle travels through an attenuating medium before interacting
Plural modes of absorption Plural modes of absorption • If more than one absorption process is present, then • The law of exponential attenuation we can write that the total linear attenuation N coefficient is equal to the sum of its parts: L e(1 2 )L N or 0 1 2
N N (e1L )(e2L ) where 1 is called the partial linear attenuation L 0 coefficient for process 1, and likewise for the other which demonstrates that the number N of particles processes L penetrating through the slab L depends on the total • Again we assume that each event by each process is effect of all the partial attenuation coefficients totally absorbing, producing no scattered or secondary particles
Plural modes of absorption Example 3.1
• The total number of interactions by all types of • Let = 0.02 cm-1 and = 0.04 cm-1 be the processes is given by 1 2 partial linear attenuation coefficients in the
N N N N N eL slab. Let L = 5 cm, and N = 106 particles. 0 L 0 0 0 and the number of interactions by a single process x How many particles NL are transmitted, and alone is how many are absorbed by each process in x L x the slab? Nx (N0 NL ) N0 (1 e ) where x/ is the fraction of the interactions that go by process x. Note that you need to know total
2 Example 3.1 Example 3.1 • The number of transmitted particles
(12 )L 6 (0.020.04)5 5 • If we do not take into account the total N L N0e 10 e 7.40810 1L 6 6 0.02x5 4 • The number of absorbed particles N1 N0 N0e 10 -10 e 9.5210 6 5 5 Error ~ 10% N 0 NL (10 7.40810 ) 2.59210 2L 6 6 0.04x5 4 • The number of absorbed by processes 1or 2 N2 N0 N0e 10 -10 e 1.81310 0.02 N (N N ) 1 2.592105 8.64104 Error ~ 5% 1 0 L 0.06 • The result for N2 has lower error due to 2 0.04 being closer to N (N N ) 2 2.592105 1.728105 2 0 L 0.06
“Narrow-beam” attenuation “Narrow-beam” attenuation • Exponential attenuation will be observed for a monoenergetic beam of identical uncharged particles • Secondary charged particles should not to be that are “ideal” - absorbed without producing counted as uncharged particles scattered or secondary radiation – charged particles are usually much less penetrating, and • Real beams of particles interact with matter by thus tend to be absorbed in the attenuator processes that may generate either charged or – those that do escape can be prevented from entering the uncharged secondary radiations, as well as scatter detector by enclosing it in a thick enough shield • The total number of particles that exit from the slab • Energy given to charged particles is thus regarded is hence greater than just the surviving primaries as having been absorbed (it is not a part of the • What should be counted by a detector? primary beam anymore)
“Narrow-beam” attenuation “Narrow-beam” attenuation
• The scattered and secondary uncharged particles can • Real attenuation coefficient must be numerically either be counted in N , or not larger than the value of any corresponding L effective attenuation coefficient ´ that is observed • If they are counted, the exponential attenuation under broad-beam attenuation conditions equation becomes invalid in describing the variation of N vs. L: case of broad-beam attenuation • There are two general ways of achieving narrow- L beam attenuation: • If scattered or secondary uncharged radiation – Discrimination against all scattered and secondary reaches the detector, but only the primaries are particles that reach the detector, on the basis of particle counted in NL, the exponential attenuation equation energy, penetrating ability, direction, coincidence, is valid: case of broad-beam geometry but narrow- anticoincidence, time of arrival (for neutrons), etc. beam attenuation – Narrow-beam geometry, which prevents any scattered or secondary particles from reaching the detector
3 Narrow-beam geometry Narrow-beam geometry
• The shield is assumed to stop all radiation incident upon it except that passing through its aperture • If it allows any leakage, it may be necessary to put a supplementary shield around the detector that allows entry of radiation at angles 0 – Lead is the usual shielding material for x- or -rays, especially where space is limited • Achieving narrow-beam geometry (“good” – Iron and hydrogenous materials are preferable for fast geometry) is not difficult experimentally neutrons • Used to obtain tabulated values of
Broad-beam attenuation Ideal broad-beam attenuation • The attenuator must be thin enough to allow the escape of • Any attenuation geometry other than narrow-beam – all the uncharged particles resulting from first interactions by geometry is called broad-beam geometry the primaries • The concept of an ideal broad-beam geometry is – all the x-rays and annihilation -rays emitted by secondary more difficult to define, and is experimentally less charged particles that are generated by primaries in the accessible attenuator • In ideal broad-beam geometry every scattered or • Multiple scattering is excluded from this ideal case secondary uncharged particle strikes the detector, but • If we have ideal broad-beam geometry, and the detector only if generated in the attenuator by a primary that responds in proportion to the radiant energy of all the particle on its way to the detector, or by a secondary primary, scattered, and secondary uncharged radiation, charged particle resulting from such a primary then we have a case of ideal broad-beam attenuation
Ideal broad-beam attenuation Ideal broad-beam attenuation • is often used as an approximation to the effective • For this case we can write an exponential equation: en attenuation coefficient ´ for thin absorbing layers in R L een L broad-beam attenuation R0 • It is referred to as the “straight-ahead approximation”: the scattered and secondary particles are supposed to – R0 is the primary radiant energy incident on the detector – RL is the radiant energy of uncharged particles striking the continue straight ahead until they strike the detector detector when the attenuator is in place • The approximation is often not accurate even for thin – L is the attenuator thickness (must remain thin enough to absorbers, but the true ´ is often not known allow escape of all scattered and secondary uncharged particles) • It is adequate in calculating photon attenuation in the
– en is the energy-absorption coefficient wall of an ionization chamber made of low-Z material
4 Broad-beam attenuation Types of geometries and attenuations • Narrow-beam geometry: only primary strikes the detector, is • Practical broad-beam geometries usually are not ideal observed for monoenergetic beams
– Some of the scattered and secondary radiation that is • Narrow-beam attenuation: Only primaries are counted in NL, supposed to reach the detector fails to arrive - this loss of is observed for monoenergetic beams radiation can be called out-scattering (particles S ) 1 • Broad-beam geometry: at least some scattered and secondary – Similarly, in-scattering is defined as the arrival at the radiation strikes the detector detector of scattered and secondary uncharged particles that are generated in the attenuator by primaries that are not • Broad-beam attenuation: scattered and secondary radiation is aimed at the detector (particles S2) counted in NL, `< is observed Ideal broad-beam • Ideal broad-beam geometry: every scattered or secondary geometry may be uncharged particle generated by primary strikes the detector simulated if in-scattered • Ideal broad-beam attenuation: ideal broad beam geometry and particles compensate for the detector response ~ to the radiant energy striking; ` en out-scattered
Broad-beam geometries Broad-beam geometries
The narrow radiation The attenuating material beam enters through a is arranged in spherical small hole and impinges shells surrounding the detector; the beam is on the attenuating layers made large enough to of material inside of a irradiate the attenuators (hypothetical) spherical- fully shell detector
• In this case the out-scattered rays such as S1 generated in the attenuator upstream of the detector, but not striking it, tend to be • Practically all scattered rays (S ) originating in the attenuator will 1 compensated by in-scattered rays such as S originating strike the detector, regardless of their direction (except 180) 2 elsewhere in the attenuator • A deep well-type detector could roughly approximate this geometry • This simulates ideal broad-beam geometry at least as closely as • This setup approaches ideal broad-beam geometry any arrangement that relies on such compensation
Broad-beam geometries Broad-beam geometries A plane beam that is • For a diverging beam the observed attenuation would infinitely wide compared to the effective maximum range be exaggerated by a loss of intensity in proportion to of scattered and secondary the inverse square of the distance from the source radiation, and incident • The detector receives no back-scattered radiation, since perpendicularly on similarly there is no material behind it wide attenuating plates • The irradiated attenuator subtends a solid angle at the • The detector is kept as close as possible to the attenuator to allow detector of only about 2 radians, as compared to 4 laterally out-scattered rays such as S1 to be maximally replaced by radians for “b” in-scattered rays such as S 2 • The smaller the subtended solid angle, the poorer the • In practice the detector is kept stationary, and the attenuating slabs “coupling” between the detector and the attenuator, and are added in sequence of increasing thickness (u z) the less scattered radiation will reach the detector
5 Broad-beam geometries Broad-beam geometries A detector that may be positioned If a smaller beam size is used in at a variable depth x from the front geometries “c” and “d”, out-scattered surface of a large mass of solid or rays S1 are less fully compensated by liquid medium, designed to in-scattered rays (S2), and the response simulate the attenuating properties of the detector to scattered radiation of the human body (a phantom) decreases relative to its response to primary radiation • Uncharged particle (usually photon) beams of various cross- • The effective attenuation coefficient observed at a given depth sectional dimensions are directed perpendicularly on the phantom, will be closer to and the detector response is measured vs. depth • This trend is even more accentuated by moving the detector a • The resulting function, the “central-axis depth-dose” of the beam, distance d away from the attenuators for a specified SSD is used in radiotherapy treatment planning • The larger the ratio d/w (for beam width w large enough to cover • If the beam and tank were vary wide, the attenuation function the detector), the closer the setup is to narrow-beam geometry observed would be similar to that in geometry “c”
Broad-beam attenuation examples Broad-beam geometries – narrow-beam attenuation
en – ideal broad-beam attenuation A point source and a detector ` – broad-beam attenuation are immersed in an infinite homogeneous medium (e.g., positive slope water) and separated by a variable distance • The effect of attenuation can be separated from that of the inverse square law by comparing the detector response in the medium with that in vacuum for the same distance 60 203 • Note that out-scattered rays like S1 are compensated by in- Broad-beam attenuation of (a) Co (1.25 MeV) and (b) Hg scattered rays like S2, but additional backscattered rays such as (0.279 MeV) gamma rays as a function of distance from a point S3 may also strike the detector, especially when it is close to the source in an infinite water medium (geometry “f”) source and the primary (source) energy is low
Spectral effects Spectral effects • Scattered • Energy dependence of detector response will affect observed particles have attenuation in broad-beam geometries due to difference in lower energy and sensitivity to primary vs. scatter affect attenuation • For narrow-beam attenuation the spectral effect will be observed beam curves the most at for poly-energetic beams broadening shallower depth narrow • In general, for the differential energy-fluence spectrum L(E) (in beam • Even for narrow- J/m2 keV) reaching the detector through attenuator thickness L, beam attenuation and the narrow-beam attenuation coefficient E,Z - need to define the slope changes a mean value: if the beam is Emax L EE,Z dE E0 poly-energetic ,L Emax (spectrum) L E dE E0
6 The buildup factor The buildup factor • For energy fluence arriving at the detector behind a medium • The concept of buildup factor B is useful in quantitative thickness L, and the narrow-beam attenuation coefficient description of broad-beam attenuation
• It can be applied with respect to any specified geometry, L L attenuator, or physical quantity (number of particles, dose, etc.) Be 0 • The general definition can be written as • When L=0 (i.e., no attenuator between source and detector), B
quantity due to primary becomes equal to B0 L/0=1 for most broad-beam geometries scattered and seconday radiation B • For geometry “d”, when the detector is at the phantom surface quantity due to primary radiation alone (depth L = 0), backscattered rays will still strike it
• Hence > and so B > 1 even for L = 0 • B=1 for narrow-beam geometry L 0 0 • In that case B0 is called the backscatter factor • B>1 and is depth-dependent for broad-beam geometry
The buildup factor The buildup factor
water lead • An alternative concept to the buildup factor is the mean effective attenuation coefficient, which can be defined through energy fluence ratio as: L Be L eL 0 or, solving for, ln B • Build-up factors for other quantities show similar behavior of increase with depth L • Rate of increase depends on material (Z) and photon energy
The buildup factor The reciprocity theorem
The simplest formulation of the reciprocity theorem for the attenuation of any kind of radiation:
< < Reversing the positions of a • Comparison of exposure buildup factor B and mean point detector and a point effective attenuation coefficient for a plane beam of 1-MeV source within an infinite -rays in water homogeneous medium does • Mean effective attenuation coefficient ` is not as strongly not change the amount of dependent on depth L as B radiation detected
7 The reciprocity theorem The reciprocity theorem
• If P and Q are different with respect to their • The theorem is not exact in dissimilar media, scattering and/or attenuating properties, the except for primary rays transmission of primary rays still remains the same, left or right • It is still useful in calculating the attenuation • However, the generation and/or transmission of a of radiation in dissimilar or nonhomogeneous scattered ray may differ media, if – If the scattered ray is absorbed more strongly in – either the primary rays dominate medium Q than in P, all else being equal, it is more – or the generation and propagation of scattered likely to reach the detector in case b than in case a, rays is not strongly dissimilar in the different since its path length in medium Q is longer in case a media
The reciprocity theorem The reciprocity theorem • Mayneord extended the reciprocity theorem to the case where • As a corollary to this theorem, one can state that: the source and detector were both extended volumes: – If S and V contain identical, uniformly distributed total activities, they will each deliver to the other the same average absorbed dose – The integral dose in a volume V due to a -ray source uniformly • Furthermore: distributed throughout source volume S is equal to the integral dose that would occur in S if the same activity density per unit mass were – If all the activity in S is concentrated at an internal point P, then the dose distributed throughout V at P due to the distributed source in V equals the average dose in V resulting from an equal source at P • This can be exact with respect to the dose resulting from • This latter statement can be taken a step further to say: primary rays only, unless V and S are parts of an infinite – The dose at any internal point P in S due to a uniformly distributed source homogeneous medium throughout S itself is equal to the average absorbed dose in S resulting • The theorem as stated is only true from the same total source concentrated at P if the mass energy-absorption • This relationship, though exact only in an infinite homogeneous medium, or for primary radiation, is nevertheless practically useful in calculation of coefficients are the same for internal dose due to distributed sources in the body (MIRD tables) the materials in S and V
Summary • Simple exponential attenuation equation is valid only for mono-energetic beams in narrow-beam geometry • Broad-beam geometry is more achievable in realistic experiments • Broad-beam attenuation equation can be effectively used, taking into account energy-dependent detector response • The build-up factor and mean effective attenuation coefficient are used for quantitative description of broad- beam attenuation • The reciprocity theorem is practically useful in calculation of internal dose due to distributed sources in the body
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