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Exponential Attenuation Outline Introduction Simple Exponential Attenuation Simple Exponential Attenuation Simple Exponential At 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 -1 -1 typically given in units of cm or m , 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 NL (1 2 )L coefficient is equal to the sum of its parts: e N0 or 1 2 L L N N (e 1 )(e 2 ) 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 L the slab? N (N N ) x N (1 e ) x x 0 L 0 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
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