MHP 2: Radiation Physics Lecture Notes PDF

Lecture Notes MHP 2: RADIATION PHYSICS Introduction Radiation physics is the science of ionizing radiation and its interaction with matter – energy absorbed. Radiation is a form of energy transfer which it does not require a medium of propagation....

Lecture Notes MHP 2: RADIATION PHYSICS Introduction Radiation physics is the science of ionizing radiation and its interaction with matter – energy absorbed. Radiation is a form of energy transfer which it does not require a medium of propagation. Radiation is a process of emitting energy through a medium or space in the form of waves or particles. Ionizing Radiation Two main categories of radiation a. Ionizing radiation (IR): in the electromagnetic (EM) spectrum, waves with shorter wavelength and higher frequency including subatomic particles i.e., alpha, beta, and neutrons are categorized as “ionizing” radiation (IR). This type of radiation is characterized by its ability to excite and ionize atoms of matter with which they interact. (4-25 eV KE or quantum energy range needed to cause a valence electron to escape an atom – ‘ionizing’) *Ionization is a process resulting in the removal of an electron from an electrically neutral atom or molecule. Whenever this happens, it creates an ion pair. *Excitation is a process where an inner electron of an atom receives enough energy raising it to a higher energy level but not enough to leave the atom completely. The atom remains electrically neutral. b. Non-ionizing radiation (NIR): EM waves with longer wavelength and lower frequency are characterized as “non-ionizing” radiation. This type of radiation does not carry enough energy to remove an electron from an atom. Types of ionizing radiation a. Directly ionizing radiation: fast charged particles deliver their energy directly to matter through many small Coulombic interactions along the track. i. Fast electrons: particulate radiation emitted from nuclei (β-rays) or in a charged-particle collision (δ-rays). (Sources: Van de Graaf generator; linear accelerator; betatron; and microtron) ii. Heavy charged particles: particulate radiation obtained from acceleration of proton, deuteron, triton, pions, α-particles, and other heavy charged particles by a Coulomb force field in a Van de Graaf, cyclotron, or heavy- particle linear accelerator; or emitted by some radioactive nuclei. (Sources: Van de Graaf generator; linear accelerator; or heavy charged-particle linear accelerator) b. Indirectly ionizing radiation: ‘two-step process’ uncharged radiation transfer their energy to charged particle which in turn deliver their energy to matter through many small Coulombic interactions along the track. 1  i. Gamma-rays: EM radiation emitted from a nucleus ‘intranuclear’ or in annihilation reaction. (Sources: Radioactive materials i.e., I-131, Co-60, Cs- 137, etc.) ii. X-rays: EM radiation emitted by a charged particle ‘extranuclear’ in changing atomic energy levels (referred to as characteristics or fluorescence x-rays) or in slowing down in a Coulomb force field (referred to as continuous or bremsstrahlung x-rays ‘braking radiation’). X-ray energy ranges are expressed in terms of the generating potential or accelerating potential. (Sources: Medical radiographic machine, cathode ray tube TVs, XRD, XRF, x-ray generators in baggage scanner, etc.) iii. Neutrons: particulate radiation obtained from nuclear reactions ([p,n] reaction or nuclear fission), since they cannot be accelerated electro- statistically. (Source: Nuclear reactors) Description of ionizing radiation field How many rays will strike point P per unit of time? Zero. How large the imaginary sphere should be? It depends on defining the physical quantities (either stochastic or non-stochastic). Characteristics of stochastic quantities: a) Values occur randomly and hence cannot be predicted. However, the probability of any particular value is determined by a probability distribution. b) Defined for finite (i.e., non-infinitesimal) domains only. Its values vary discontinuously in space and time, and it is meaningless to speak of its gradient or rate of change c) Values can each be measured with an arbitrarily small error. d) The expectation value 𝑁! of a stochastic quantity is the mean 𝑁 % of its measured value 𝑁 for an infinite number of measurements. That is, 𝑁! → 𝑁 % as 𝑛 → ∞. Characteristics of non-stochastic quantities: a) Value for non-stochastic quantity can be calculated (for a given condition). b) In general, it is a ‘point function’ defined for infinitesimal volumes => continuous and differential function of space and time, with defined spatial gradient and time rate of change. 2  c) Value is equal to, or based upon, the expectation value of a related stochastic quantity, if one exists. Most common and useful quantities for describing ionizing radiation and its interaction with matter, except in microdosimetry, are all non-stochastic. In general one can assume that a “constant” radiation field is strictly random with respect to how many rays arrive at a given point per unit area and time interval. It can be shown that the number of rays observed in repetitions of the measurement (assuming a fixed detection efficiency and time interval, and no systematic change of the field versus time) will follow a Poisson distribution. For large numbers of events this may be approximated by the normal (Gaussian) distribution. If 𝑁! is the expectation value of the number of rays detected per measurement, the standard deviation of a single random measurement 𝑁 relative to 𝑁! is equal to: % 𝜎 = +𝑁! ≅ +𝑁 (2.1) and the corresponding percentage standard deviation is "##$ "## "## 𝑆= %! = ≅ √%( (2.2) &%! The approximation of 𝑁! by the mean value 𝑁 % in equation (2.1) and (2.2) is necessary because 𝑁! is unknown but can be approached as closely as desired by the mean value 𝑁 % of 𝑛 measurements, 𝑁! → 𝑁 % as 𝑛 → ∞. It is useful to know how closely 𝑁 % is likely to approximate 𝑁! for a given number of measurements 𝑛. This information is conveyed by the standard deviation of the mean value 𝑁 % relative to 𝑁! : $ % ( % 𝜎) = =. *! ≅. * (2.3) √* and the corresponding percentage standard deviation is "##$ " "## "## "## 𝑆) = %! = ≅ √*%( = (2.4) &*%! &%# The standard deviation in equations (2.1) to (2.4) are based exclusively upon the stochastic nature of radiation fields, not taking account of instrumental or other experimental fluctuations. An estimate of the precision of any single random measurement 𝑁 made by a radiation detector should be determined from the data of 𝑛 such measurements by means of the equation: $ " % )- ] % 𝜎 ≅ [*+" ∑*,."(𝑁, − 𝑁 (2.5) instead of equation (2.1). Here 𝑁, is the value obtained in the ith measurement. An estimate of the precision of the mean value 𝑁 % of n measurements should likewise be obtained from the experimental data by $ " % )- ] % 𝜎 ) ≅ [*(*+") ∑*,."(𝑁, − 𝑁 (2.6) in place of equation (2.3). It should also be pointed out that the expectation value of the measurements is not necessarily the physically correct value, and in fact will be if the 3 measuring instrument is improperly calibrated or is otherwise biased. 𝑁! is merely the % approached as 𝑛 → ∞. value of 𝑁 Activity The following set of count readings was made in a gradient-free 𝛾-ray field, using a suitable detector for repetitive time periods of one minute: 18 500; 18 410; 18 250; 18 760; 18 600; 18 220; 18 540; 18 270; 18 670; 18 540. a) What is the mean value of the number of counts? b) What is its standard deviation (SD)? c) What is the theoretical minimum SD of the mean? d) What is the actual SD of a reading? e) What is the theoretical minimum SD of a single reading? Simple description of ionizing radiation fields by non-stochastic quantities: a) Fluence refers to the expectation value of rays passing through an infinitesimal area at the point of interest ‘spatial distribution’. (Units: 𝑚+- or 𝑐𝑚+- ) 1%! Φ= 12 (2.7) b) Fluence rate or flux density refers to the increment fluence over an infinitesimal time interval at the point of interest ‘temporal distribution’. (Units: 𝑚+- ∙ 𝑠 +" or 𝑐𝑚+- ∙ 𝑠 +" ) 13 1 1% 𝜑= 14 = 14 < 12! = (2.8) c) Energy fluence refers to the radiant energy carried by all the expectation value of rays crossing an infinitesimal area surrounding the point of interest. Radiant energy is the energy of particles transmitted, transferred, or received excluding rest energy. (Units: 𝐽 ∙ 𝑚+- or 𝑒𝑟𝑔 ∙ 𝑐𝑚+- ) 15 Ψ = 12 (2.9) d) Energy fluence rate or energy flux density refers to an increment in energy fluence over an infinitesimally small-time interval at the point of interest. (Units: 𝐽 ∙ 𝑚+- ∙ 𝑠 +" or 𝑒𝑟𝑔 ∙ 𝑐𝑚+- ∙ 𝑠 +" ) 16 1 15 𝜓= 12 = 14 depend on the direction of incidence of the rays striking it => better if we express the non-stochastic quantities 4  above as a function of the KE or QE of rays E, polar angle θ, and azimuthal angle, β – differential distribution. Radioactivity and Modes of Radioactive Decay How is Ionizing Radiation Produced? Ionizing radiation can be produced by disintegration process (𝛼, 𝛽 + , and 𝛽 8 disintegrations, and electron-capture transitions), particle accelerator, annihilation reaction, and nuclear reaction. Radioactivity is a process by which an unstable nucleus (parent) decays into a new nuclear configuration (daughter) that may be stable or unstable. If the daughter is unstable, it will decay further through a chain of decays until a stable configuration is attained. Radioactive decay involves a transition from the quantum state of the parent to a quantum state of the daughter. The energy difference between the two quantum states is called the decay energy. The decay energy is emitted either in the form of EM radiation or KE of the reaction products. All radioactive processes are governed by the same formalism based on: (a) characteristic parameter called the decay constant λ and (b) activity A(t) defined as λN(t) where N(t) is the number of radioactive nuclei at time t. Specific activity (𝑎) is the parent’s activity per unit mass. 9(4) ;%(4) ;%& 𝑎= : = : = 9 (2.11) where 𝑁9 is the Avogadro’s number and A is the atomic mass number. Activity represents the total number of disintegrations (decays) of parent nuclei per unit time. The SI unit of activity is the Becquerel (Bq) (1 Bq = 1 s +" ). The older unit of activity is the Curie (Ci) (1 Ci = 3.7 × 10"# s +" ), originally defined as the activity of 1 g of Ra-226. ;' Decay of radioactive parent P into stable daughter D: 𝑃 [\ 𝐷 The rate of depletion of the number of radioactive parent nuclei 𝑁< (𝑡) is equal to the activity 𝐴< (𝑡) at time t 1%( (4) 14 = −𝐴< (𝑡) = −𝜆< 𝑁< (𝑡) (2.12) >( (4) =>( (4) ? ∫%((#) >( = − ∫# λ< dt (2.13) where 𝑁< (0) is the initial number of parent nuclei at time t = 0. The number of radioactive parent nuclei 𝑁< (𝑡) as a function of time t is: 𝑁< (𝑡) = 𝑁< (0)𝑒 +;( 4 (2.14) 5  e+@? is the decay factor, the fraction of radioactive atoms remaining after time t. When the exponent in the decay factor is small, that is less than approximately 0.1, the decay factor may be approximated by 1 − λt. The activity of the radioactive parent 𝐴< (𝑡) as a function of time t is: 𝐴< (𝑡) = 𝜆< 𝑁< (𝑡) = 𝜆< 𝑁< (0)𝑒 +;( 4 = 𝐴< (0)𝑒 +;( 4 (2.15) where 𝐴< (0) is the initial activity at time t = 0. Half-life, T$ is the time required for the radionuclide to decay to 50% of its initial % activity level. It is generally used to describe the rate of decay as it allows you to ! interpret how stable a radionuclide is or how quickly it will decay at T = T! and N = "#. " The half-life and decay constant of a radionuclide are related as: #.B7C T$ = @ (2.16) % Radioactive decay is a process by which unstable nuclei reach a more stable configuration. There are four main modes of radioactive decay: (a) alpha decay; (b) beta decay (beta plus, beta minus, and electron capture); (c) gamma decay (pure gamma decay and internal conversion); and (d) spontaneous fission. a) Alpha decay: occurs mainly in heavy nuclei (when neutron to proton ratio is too low) and emits an energetic alpha particle (He-4 ion); the atomic number Z of the parent decreases by 2; and the mass number A of the parent decreases by 4. (not isobaric) 9 9+E D 𝑃 → D+- 𝐷 + E- 𝐻𝑒 (2.17) Best known example of alpha decay is the transformation of Ra-226 into Rn-222 with a half-life of 1600 y. --B --- FF 𝑅𝑎 → FB 𝑅𝑛 + 𝛼 b) Beta-minus decay: occur in neutron-rich radioactive parent nucleus transformed a neutron into a proton; an electron and anti-neutrino, sharing the available energy, are ejected from the parent nucleus; the atomic number Z of parent is increased by 1; and the mass number A of the parent remains the same. (isobaric) 9 9 D P → D8" D + 𝑒 + + 𝜈%! (2.18) An example of beta minus decay is the transformation of Co-60 into Ni-60 with a half-life of 5.26 y. B# B# + -G Co → -F Ni + 𝑒 + 𝜈%! Beta-plus decay: occur in a proton-rich radioactive parent nucleus transforming a proton into a neutron; a positron and neutrino, sharing the available energy, are ejected from the parent nucleus; the atomic number Z of the parent is decreased by 1; and the mass number A of the parent remains the same. (isobaric) 9 9 D P → D+" D + 𝑒 8 + 𝜈! (2.19) An example of a beta plus decay is the transformation of N-13 into C-13 with a half-life of 10 min. "CG N → "CB C + 𝑒 8 + 𝜈! Electron capture decay: parent nucleus capture its own electron from K- or L- shell and emits monoenergetic neutrino; a proton transforms into a neutron; the 6  atomic number Z of the parent decreases by one; and the mass number A remains the same. The resulting shell vacancy is filled with electron from higher orbital shell, leading to the emission of fluorescence x-ray. Also, this is a process that competes with beta-plus disintegration. (isobaric) 9 D P + +"#e → 9 D+" D + 𝜈! (2.20) An example of nuclear decay by electron capture is the transformation of Be-7 into Li-7. GE Be + +"#e → &C Li + 𝜈! c) Gamma decay: nuclear transformation in which an excited parent nucleus, generally produced through alpha decay, beta minus or beta plus decay, attains its ground state through emission of one or several gamma rays. The atomic number Z and the mass number A do not change in gamma decay. In most alpha and beta decays, the daughter de-excitation occurs instantaneously, so that we refer to the emitted gamma rays as if they were produced by the parent nucleus. If the daughter nucleus de-excites with a time delay, the excited state of the daughter is referred to as a metastable state and process of de- excitation is called an isomeric transition. 9I 9 D D → D D+𝛾 (2.21) Examples of gamma decay are the transformation of Co-60 into Ni-60 by beta minus decay, and transformation of Ra-226 into Rn-222 by alpha decay. Internal conversion: nuclear transformation in which the nuclear de-excitation energy is transferred to an orbital electron (usually K-shell). The electron is emitted from the atom with a kinetic energy equal to the de-excitation energy less the binding energy. The resulting shell vacancy is filled with a higher-level orbital electron and the transition energy is emitted in the form of characteristic photons or Auger electrons. 9 9 D D∗ → D D8 + 𝑒 + (2.22) An example for both the emission of gamma photons and emission of conversion electrons is the beta minus decay of Cs-137 into Ba-137 with a half-life of 30 y. d) Spontaneous fission: nuclear transformation by which a high atomic mass nucleus spontaneously splits into two nearly equal fission fragments. Two or four neutrons are emitted during this process. In practice, spontaneous fission is only energetically feasible for nuclides with atomic mass above 230 u or 𝑍 - ⁄𝐴 ≥ 235. The spontaneous fission is a competing process to alpha decay; the higher is A above U-238, the more prominent is the spontaneous fission in comparison with the alpha decay and the shorter is the half-life for spontaneous fission. Activity 1. A sealed capsule contains 5550-MBq (150 mCi) Rn-222 (𝑡$ = 3.8 𝑑). (a) What is the Rn % activity 38 days later? (b) How many Rn atoms have decayed? (c) How many moles of He have been produced from alphas during 38 days? 7 2. If 1.0 MBq (27 mCi) of I-131 is needed for a diagnostic test, and if 3 days elapse between shipment of the radioiodine and its use in the test, how many Bq must be shipped? To how many mCi does this correspond? Charged Particle Interaction with Matter How is Ionizing Radiation Interacts with Matter? As the energetic electron traverses with matter, it undergoes Coulomb interactions with absorber atoms, i.e., with (a) atomic orbital electrons and (b) atomic nuclei. Through these collisions the electrons may: (i) lose their KE (collision and radiative loss) and (ii) change the direction of motion (scattering). Energy losses are described by stopping power. Meanwhile, scattering is described by angular scattering power. o Stopping power: expectation value of the rate of energy loss per unit of path length as it travels through a medium. Units: MeV/cm or J/cm o Radiation yield: total fraction of the initial KE of charged particle that is emitted as EM radiation while the particle slows down and comes to rest. o Range: expectation value of the path length the charged particle follows until it comes to rest. The collision between the incident electron and an absorber atom may be: a. Elastic: the incident electron is deflected from its original path but with no energy loss occurs b. Inelastic: with orbital electron, the incident electron is deflected from its original path and loses part of its KE. In an inelastic collision with nucleus, the incident electron is deflected from its original path and loses part of its KE in the form of bremsstrahlung. The type of inelastic interaction that an electron undergoes with a particular atom of radius a depends on the impact parameter b of the interaction. a) Soft collision (for b ≫ a): most probable type of interaction; the incident electron excites atom (as a whole) to a higher energy level or ionize atom by ejection of valence electron; atom receives small amount of its KE (few %). 8  b) Hard collision (for b~a): interaction probability is different for different particles; the incident electron interacts primarily with single atomic electron, then the electron gets ejected with a considerable KE; the ejected delta-ray dissipates energy along its track; either characteristic x-ray or Auger electrons are produced. c) Coulomb interaction (for b ≪ a): important for high-Z materials and high energies; most important for electrons in cases; the incident electron will undergo a radiation collision with the atomic nucleus and emit a bremsstrahlung photon with energy between 0 the incident electron KE; for antimatter only, in- flight annihilation may occur wherein two photons are produced d) Nuclear interaction (b < a): for heavy-charged particles; inelastic interaction with nucleus; one or more individual nucleons may be driven out of nucleus in intra-nuclear cascade process; highly excited nucleus decays by emission of nucleons and gamma-ray. Gamma- and X-rays Interaction with Matter How is Ionizing Radiation Interacts with Matter? Ionizing photon radiation is classified into four categories: a) Characteristic x-ray: results from electronic transitions between atomic shells b) Bremsstrahlung x-ray: results mainly from electron-nucleus Coulomb interactions c) Gamma ray: results from nuclear transitions d) Annihilation quantum (annihilation radiation): results from positron-electron annihilation In penetrating an absorber medium, photons may experience various interactions with the atoms of the medium, involving the (a) absorbing atom as a whole, (b) nuclei of the absorbing medium, and (c) orbital electrons of the absorbing medium. Interactions of photons with nuclei may be: (a) direct photon-nucleus interactions (photodisintegration) and (b) interactions between the photon and the electrostatic field of the nucleus (pair production) Photon-orbital electron interactions are characterized as interactions between the photon and either a loosely bound electron (Compton effect, triplet production) or a tightly bound electron (photoelectric effect). o A loosely bound electron is an electron whose binding energy 𝐸K to the nucleus is small compared to the photon energy ℎ𝜈 (𝐸K ≪ ℎ𝜈). An interaction between a photon and a loosely bound electron is considered to be an interaction between a photon and a free (unbound) electron). o A tightly bound electron is an electron whose binding energy 𝐸K is comparable to, larger than, or slightly smaller than the photon energy ℎ𝜈. For a photon interaction to occur with a tightly bound electron, the 𝐸K of the electron must be of the order of, but slightly smaller, than the photon energy (𝐸K ≤ ℎ𝜈). An 9  interaction between a photon and a tightly bound electron is considered an interaction between a photon and the atom as a whole. As far as the photon fate after the interaction with an atom is concerned there are two possible outcomes: o Photon disappears (i.e., is absorbed completely) and a portion of its energy is transferred to the light charged particles (electrons and positrons in the absorbing medium). o Photon is scatter and two outcomes are possible: § The resulting photon has the same energy as the incident photon and no light charged particles are released in the interaction. § The resulting scattered photon has a lower energy than the incident photon and the energy excess is transferred to a light charged particle (electron). The light charged particles produced in the absorbing medium through photon interactions will either deposit their energy to the medium through Coulomb interactions with orbital electrons of the absorbing medium (collision loss) or radiate their KE away through Coulomb interactions with the nuclei of the absorbing medium (radiative loss). The following are the five types of interactions with matter by x- and 𝛾-ray photons which must be considered in radiological physics: a) Photoelectric effect: a photon of relatively low energy (< 1 𝑀𝑒𝑉) transfer all of its energy to a tightly bound electron in an inner shell. The electron is then ejected from the parent atom. o Photon disappears o Almost always the K-shell electron (ejected electron = photoelectron) o Dominates at lower photon energy range (~𝑘𝑒𝑉 range); Z-dependent b) Compton effect: an elastic collision between a photon and an electron in which only part of the photon energy is transferred to the electron. o The electron may be ejected (ejected electron = recoil electron) o The photon with reduced energy is scattered o Dominates at intermediate photon energy range (~1 𝑀𝑒𝑉); Z-independent and depends on the electron density c) Pair production: when a photon (with 𝐸 ≥ 1.022 𝑀𝑒𝑉) passes close to a high-Z nucleus, the photon interacts with the Coulomb field of the nucleus, gets absorbed, and a positron-electron pair each having some KE is produced. o The positron eventually combines with an electron o The two particles annihilate each other converting mass back to energy thus producing two 0.511 MeV gamma-rays travelling in opposite direction. o Dominates at higher photon energies; Z-dependent d) Rayleigh (coherent) scattering: inelastic collision with weakly bound outer-shell electron. This interaction is elastic wherein the photon does not lose its energy but redirected only at a small angle and the atom moves just enough to conserve momentum. Photon scattering angle depends on Z and energy 10  e) Photonuclear interaction: occurs when photon with energy exceeding few MeV (~10 𝑀𝑒𝑉) enter and excites nucleus which in turn emits proton or neutron. Radioactivation occurs when stable nuclei are transformed into radioactive species by bombardment of photons with sufficiently high energy Quantities for Describing the Interaction of Ionizing Radiation with Matter Three non-stochastic quantities in describing the interaction of the radiation field with matter (in terms of expectation values for the infinitesimal sphere at the point of interest): (1) KERMA, (2) Absorbed dose, and (3) Exposure. KERMA (Kinetic Energy Released per unit Mass of Absorber) is the expectation value of the energy transferred (stochastic quantity) to charged particles per unit of mass at P in V, including radiative-loss energy (charged particles – bremsstrahlung or in-flight annihilation) but excluding the energy passed from one charged particle to another. KERMA can be expressed in units of erg/g, rad, or J/kg (1 Gy = 1 J/kg). 1L)* 𝐾= 1M (2.23) Absorbed dose is the expectation value of the energy imparted (stochastic quantity) to matter per unit mass at a point. Absorbed dose can be expressed in units erg/g, rad, or J/kg (1 Gy = 1 J/kg). 1L 𝐷 = 1M (2.24) Absorbed dose rate at a point P and time t is given by: 1N 1 1L 𝐷̇ = 14 = 14 100 keV – 2 MeV 20 > 2 MeV – 20 MeV 10 > 20 MeV 5 Effective dose (𝐸 = 𝑤O 𝐷; 𝑤O = tissue weighting factor) is a measure of the effect of a particular type of radiation on organs or tissues. Effective dose can be expressed in 11  units of sievert (Sv). E considers the radiosensitivities of different tissues or organs. 𝑤O are multipliers used for radiation protection purposes to account for the different sensitivities of different organs and tissues to the induction of stochastic effects of radiation. Exposure is defined as the dQ quotient of the absolute value of the total charge of the ions of one sign produced in air when all electrons liberated by photons in air of mass dm are completely stopped in air. The roentgen (R) is the customary and more commonly encountered unit of exposure. It is defined as the exposure that produces, in air, one esu of charge of either sign per 0.001293 g of air (i.e., the mass contained in 1 cm3 at 760 Torr, 0℃) irradiated by the photons. 1P 𝑋 = 1M (2.26) Exponential Attenuation In beam attenuation, each incident uncharged particle can either be: (a) absorb by the matter; (b) scattered; (c) produce secondary charged particles; or (d) penetrate the matter without interacting. The most important parameter used for characterization of x-ray or gamma-ray penetration into absorbing media is the linear attenuation coefficient 𝜇. The 𝜇 depends on the energy ℎ𝜈 of the photon beam and the atomic number Z. The linear attenuation coefficient may be described as the probability per unit path length that a photon will have an interaction with the absorber. The attenuation coefficient 𝜇 is determined experimentally using the so-called narrow beam geometry technique that implies a narrowly collimated source of monoenergetic photons and a narrowly collimated detector. 𝑥: represents total thickness of the absorber 𝑥 ) : represent the thickness variable 12  A slab of absorber material of thickness 𝑥 decreases the detector signal intensity from 𝐼(0) to 𝐼(𝑥). A layer of thickness 𝑑𝑥 ) reduces the beam intensity by 𝑑𝐼 and the fractional reduction in intensity, −𝑑𝐼⁄𝐼 is proportional to 𝜇 and 𝑑𝑥 ). The fractional reduction in intensity is given as: 1Q − Q = 𝜇𝑥 (2.27) It follows that, 𝐼(𝑥) = 𝐼(0)𝑒−𝜇𝑥 (2.28) This is analogous to equation (2.29) which the law of exponential attenuation. This applies either for the ideal case (simple absorption, no scattering or secondary radiation) or where scattered or secondary particles may be produced but are not counted in transmitted rays, 𝑁R. %+ %, = 𝑒 +SR (2.29) For a homogeneous medium 𝜇 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 and one gets the standard exponential relationship valid for monoenergetic photon beams (equation 2.28). For 𝑥 = 𝐻𝑉𝐿, Q(T) Q(#) = 0.5 (2.30) Several thicknesses of special interest are defined as parameters for monoenergetic photon beam characterization in narrow beam geometry: o Half-value layer (𝐻𝑉𝐿" or 𝑥"/- ): absorber thickness that attenuates the original intensity to 50%. o Mean free path (𝑀𝐹𝑃 or 𝑥̅ ): absorber thickness which attenuates the beam " intensity to ! = 36.8%. o Tenth-value layer (𝑇𝑉𝐿 or 𝑥"/"# ): absorber thickness which attenuates the beam intensity to 10%. In addition to the linear attenuation coefficient 𝜇 other related attenuation coefficients and cross sections are in use for describing photon beam attenuation: (a) mass attenuation coefficient (𝜇/𝜌); (b) atomic cross section (𝜇/𝑛° ); and (c) electronic cross section (𝜇/𝑍𝑛° ). 𝑛° is the number of atoms per volume of absorber with density 𝜌 and atomic mass 𝐴. Two types of attenuation: 1) Narrow beam attenuation is a type of attenuation that counts only the primary photons striking the detector, make used of narrow beam geometry which prevents scattered or secondary particles reaching the detector, and consider the real attenuation coefficient, 𝜇. 2) Broad beam attenuation is a type of attenuation that counts the secondary or scattered uncharged radiation reaching the detector, but only if generated in the attenuator by a primary particle on its way to the detector or charged particle resulting from such a primary; make used of broad-beam geometry which at least 13  some scattered or secondary radiation strikes the detector; and effective attenuation coefficient, 𝜇) is less than the real attenuation coefficient. Reciprocity theorem states that reversing the position of the source and detector within an infinite homogeneous media does not change the amount of radiation detected. This is useful in calculating the attenuation of radiation emitted in nonhomogeneous media if either: the primary rays dominate, or the generation and propagation of scattered rays is not strongly dissimilar in different media. X-ray Filtration and Beam Quality Fluorescence x-rays are produced when an inner shell electron in an atom is removed as a result of electron capture or internal conversion, photoelectric effect, and hard collision with charged-particle. The minimum energy required to ionize the inner shell electron is the electron binding energy with respect to that shell. The vacancy is filled by an electron from higher shell. The transition is accompanied by the emission of an x- ray photon called fluorescence x-rays. When an electron passes near the nucleus, an inelastic radiative interaction occur in which x-ray is emitted. The electron is not only deflected but gives a significant fraction of its kinetic energy to the photon. Such x-rays are referred to as bremsstrahlung x- rays. Unfiltered x-ray beam contains both fluorescence (characteristic of the target) and bremsstrahlung x-ray. The principle of using an x-ray filtration is to remove photon at energies where attenuation coefficient is largest. It can also use to narrow the spectral distribution progressively; where beam is hardened by filtration. The objective is to reduce radiation without compromising the image quality. High Z- filters provide strong filtering action but cause discontinuities at shell binding energies while lower Z-filters smooth out the resulting spectrum. Combination filters are preferred as it is capable of narrowing spectrum to any desired degree while preserving more of the x-ray output. It is positioned in descending order of Z, going in the direction of the rays to remove fluorescence x-rays originating in higher-Z filter upstream from it. 14  Beam can be specified by either: spectra, attenuation curves, and half-value layers (HVL). HVL1 is defined as the thickness required to reduce the exposure by half, in narrow-beam geometry. HVL2 is the thickness necessary to reduce it by half again under the same conditions. 15