Outline Charged-Particle and • Radiation equilibrium Radiation Equilibria • Charged particle equilibrium • Causes of CPE failure Chapter 4 • Transient CPE • Summary F.A. Attix, Introduction to Radiological Physics and Radiation Dosimetry Introduction Radiation equilibrium • The concepts of radiation equilibrium (RE) and • Consider an extended volume charged-particle equilibrium (CPE) are useful V containing a distributed in relating certain basic quantities radioactive source with a smaller internal volume v • CPE allows the equating of the absorbed dose about a point of interest, P D to the collision kerma Kc and exposure X • Radioactivity is emitted • Radiation equilibrium makes dose D equal to isotropically on average the net rest mass converted to energy per unit • V is required to be large enough so that the maximum distance of penetration d of any emitted ray and its mass at the point of interest progeny (i.e., scattered and secondary rays) is less than the minimum separation s of the boundaries of V and v Radiation equilibrium Radiation equilibrium • Consider a plane T that is tangent to • Radiation equilibrium (RE) exists for the volume the volume v at a point P´, and the v if the following four conditions exist throughout rays crossing the plane per unit area V (in the nonstochastic limit): • In the nonstochastic limit there will be perfect reciprocity of rays of each a. The atomic composition of the medium is homogeneous type and energy crossing both ways, b. The density of the medium is homogeneous due to uniform distribution of the c. The radioactive source is uniformly distributed radioactive source within the sphere S d. There are no electric or magnetic fields present to • This will be true for all possible orientations of tangent planes perturb the charged-particle paths, except the fields associated with the randomly oriented individual atoms around the volume v; therefore, in the nonstochastic limit, for each type and energy of ray entering v, another identical ray leaves • This condition is called radiation equilibrium (RE) with respect to v 1 Radiation equilibrium Radiation equilibrium • As a consequence of radiation equilibrium the energy carried in and that carried out of v are balanced for • In non-stochastic consideration the volume v both indirectly and directly ionizing radiation: can be reduced to infinitesimal dv, then RE exists at the point P Rin u Routu and Rin c Routc • The energy imparted can then be simplified to • Since D d / dm , under condition of radiation equilibrium at a point in a medium, the Q absorbed dose is equal to the expectation • Therefore under RE conditions the expectation value value of the energy released by the radioactive of the energy imparted to the matter in the volume v is d material per unit mass at that point, Q equal to that emitted by the radioactive material in v dm Radiation equilibrium Radiation equilibrium • Presence of homogeneous constant magnetic • The concept of RE has practical importance in the and/or electric field throughout V can make fields of nuclear medicine and radiobiology, radiation field anisotropic where distributed radioactive sources may be • RE will still be satisfied if the flow of radiation is introduced into the human body or other balanced biological systems for diagnostic, therapeutic, or analytical purposes • Consider flows of particles between • The resulting absorbed dose at any given point dv`, dv, and dv``: a+B=A+b depends on the size of the object relative to the • If anisotropy is homogeneous radiation range and on the location of the point everywhere in V, the same balance within the object of flow is held Charged-particle equilibrium Charged-particle equilibrium • Charged particle equilibrium (CPE) exists for the • In many practical cases RE condition is not volume v if each charged particle of a given type satisfied, but cab be adequately approximated if and energy leaving v is replaced by an identical CPE condition exists particle of the same energy entering (in terms of • Consider two general situations: expectation values) 1. distributed radioactive sources • If CPE exists, 2. indirectly ionizing radiation from external sources • RE condition is sufficient for CPE to exist 2 CPE for distributed sources – case 1 CPE for distributed sources – case 2 • For the trivial case of a distributed source emitting • Consider now the case where both charged particles only charged particles, in a system where radiative and relatively more penetrating indirectly ionizing losses are negligible radiation are emitted • The dimension s (the minimum separation of v from Rin u Routu Rin c Routc • Let the distance d be the maximum range of the the boundary) is taken to be greater than the charged particles only, and distance s > d maximum range d of the particles Distributed source of charged Distributed source of charged particles and photons (neutrons) • Conditions a through d are satisfied particles only • If all ofQ the four conditions a - d • Only CPE exists in this case are satisfied, both RE and CPE • RE is not attained since rays exist (they are identical for this case) escaping from the volume v are not replaced, Routu Rin u CPE for distributed sources – case 2 CPE for distributed sources – case 2 • The equation for the expectation value of the energy imparted • Now assume that the size of the volume V occupied R R Q in u outu by the source is expanded so that distance s • Since the indirectly ionizing rays are so penetrating increases to being greater than the effective range of that they do not interact significantly in v, is equal the indirectly ionizing rays and their secondaries to the kinetic energy given to charged-particles by • This transition will cause the (Rin)u term to increase the radioactive source in v, less any radiative losses until it equals (Rout)u in value by those particles while in v • RE will be restored • The average absorbed dose in v is thus D / m • The energy imparted would be transformed into that for CPE condition for RE: CPE for distributed sources – case 3 CPE for distributed sources • A distributed source emitting penetrating • The calculation of the absorbed dose is straightforward for either of these limiting cases indirectly ionizing radiation (CPE or RE) • Achieving CPE will also require that RE is • Intermediate situations are more difficult to deal attained with, i.e., when the volume V is larger than necessary • The following equations are applicable to achieve CPE in v, but not large enough for RE • In that case some fraction of the energy of the and indirectly ionizing radiation component will be absorbed, and it is relatively difficult to determine what that fraction is 3 CPE for indirectly ionizing CPE for indirectly ionizing radiation from external sources radiation from external sources • A volume V contains a smaller volume v • These conditions are similar to those of RE, except for: • The boundaries of v and V are required to be separated by at least the maximum distance of penetration of any secondary – Condition c: uniform field of radiation replaces the uniform radioactive source charged particle present R R • If the following conditions are satisfiedin throughoutc out Vc , CPE – The separation of boundaries of v and V are required to be at will exist for the volume v: least the maximum distance of penetration of any secondary a. The atomic composition of the medium is homogeneous charged particle, rather than that of the most penetrating b. The density of the medium is homogeneous radiation (indirectly ionizing) c. There exists a uniform field of indirectly ionizing radiation – The last condition d has been shown to be a sufficient (rays are only negligibly attenuated passing through the medium) substitute for the requirement of a complete absence of d. No inhomogeneous electric or magnetic fields are present electric or magnetic fields CPE for indirectly ionizing CPE for indirectly ionizing radiation from external sources radiation from external sources • Analogy with the distributed source case: since the • Homogeneous anisotropy, together with a uniform medium in which the charged particles can slow down throughout V radiation field is uniform, the number of charged (as guaranteed by the first two conditions) is sufficient to particles produced per unit volume in each energy produce CPE for the volume v interval and element of solid angle will be uniform • Example: a simplified case of straight charged-particle • However, since neutron and photon interactions tracks, all emitted at angle q through identical interactions at generally result in anisotropic angular distributions different points of secondary and scattered radiations, these • Three charged particles e1-e3 will particles are not emitted isotropically as in the case contribute the same kinetic energy as e alone would, if its entire track of radioactive point sources 1 remained inside of v • This anisotropy will be homogeneous throughout V • Thus CPE exists inside of v CPE for indirectly ionizing CPE for indirectly ionizing radiation from external sources radiation from external sources r Ru hv1 Since e1 n originated in V; • Relating tr and for CPE conditions: Routu hv2 the resultant radiative loss can non-r R non-r 0 be outside of V n R R R r out u tr outu outu u • However, under those same conditions we may Under CPE e1 and e2 also assume that any radiative interaction by a are equivalent charged particle after it leaves v will be replaced (4.4b) by an identical interaction inside