Radiation Interactions with Matter: Energy Deposition Biological effects are the end product of a long series of phenomena, set in motion by the passage of radiation through the medium. Image removed. 1 Interactions of Heavy Charged Particles Energy-Loss Mechanisms ? The basic mechanism for the slowing down of a moving charged particle is Coulombic interactions between the particle and electrons in the medium. This is common to all charged particles ? A heavy charged particle traversing matter loses energy primarily through the ionization and excitation of atoms. ? The moving charged particle exerts electromagnetic forces on atomic electrons and imparts energy to them. The energy transferred may be sufficient to knock an electron out of an atom and thus ionize it, or it may leave the atom in an excited, nonionized state. ? A heavy charged particle can transfer only a small fraction of its energy in a single electronic collision. Its deflection in the collision is negligible. ? All heavy charged particles travel essentially straight paths in matter. Image removed. [Tubiana, 1990] 2 Image removed. W is the energy required to cause an ionization Image removed. 3 Maximum Energy Transfer in a Single Collision The maximum energy transfer occurs if the collision is head-on. Image removed. Fig. 5.2 in Turner J. E. Atoms, Radiation, and Radiation Protection, 2 nd ed. New York: Wiley-Interscience, 1995. Assumptions: ? The particle moves rapidly compared with the electron. ? For maximum energy transfer, the collision is head-on. ? The energy transferred is large compared with the binding energy of the electron in the atom. ? Under these conditions the electron is considered to be initially free and at rest, and the collision is elastic. Conservation of kinetic energy: 2 1 MV 2 = 2 1 MV + 2 1 2 1 mv 2 1 Conservation of momentum: MV = MV 1 + mv 1 . Q max = 2 1 MV 2 - 2 1 MV = 2 1 2 )( 4 mM mME + , Where E = MV 2 /2 is the initial kinetic energy of the incident particle. 4 Q max values for a range of proton energies. Except at extreme relativistic energies, the maximum fractional energy loss for a heavy charged particle is small. Maximum Possible Energy Transfer, Q max , in Proton Collision with Electron Proton Kinetic Maximum Percentage Energy E Q max Energy Transfer (MeV) (MeV) 100Q max /E 0.1 0.00022 0.22 1 0.0022 0.22 10 0.0219 0.22 100 0.229 0.23 10 3 3.33 0.33 10 4 136 1.4 10 5 1.06 x 10 4 10.6 10 6 5.38 x 10 5 53.8 10 7 9.21 x 10 6 92.1 2 max )( 4 mM mME Q + = 5 Single Collision Energy Loss Spectra Image removed. Fig. 5.3 in [Turner]. ? The y axis represents the calculated probability that a given collision will result in an energy loss Q. ? N.B., the maximum energy loss calculated above for the 1 MeV proton, of 21.8 keV is off the scale. ? The most probable energy loss is on the order of 20 eV. ? N.B., energy loss spectra for fast charged particles are very similar in the range of 10 – 70 eV. ? Energy loss spectra for slow charged particles differ, the most probable energy loss is closer to the Q max . 6 Stopping Power ? The average linear rate of energy loss of a heavy charged particle in a medium (MeV cm -1 ) is of fundamental importance in radiation physics, dosimetry and radiation biology. ? This quantity, designated –dE/dx, is called the stopping power of the medium for the particle. ? It is also referred to as the linear energy transfer (LET) of the particle, usually expressed as keV μm -1 in water. ? Stopping power and LET are closely associated with the dose and with the biological effectiveness of different kinds of radiation. Stopping powers can be estimated from energy loss spectra. ? The “macroscopic cross section”, μ, is the probability per unit distance of travel that an electronic collision takes place. ? The reciprocal of μ is the mean distance of travel or the mean free path, of a charged particle between collisions. ? Stopping power is the product of the macroscopic cross section and the average energy lost per collision. - dx dE = μQ ave Example: The macroscopic cross section for a 1-MeV proton in water is 410 μm -1 , and the average energy lost per collision is 72 eV. What are the stopping power and the mean free path? The stopping power, - dx dE = μQ ave = 410 μm -1 x 72 eV= 2.95 x 10 4 eV μm -1 The mean free path of the 1-MeV proton is 1/μ = 1(410 μm -1 ) = 0.0024 μm = 24 ?. [Atomic diameters are of the order of 1 ? to 2 ?.] 7 Calculations of Stopping Power In 1913, Niels Bohr derived an explicit formula for the stopping power of heavy charged particles. Bohr calculated the energy loss of a heavy charged particle in a collision with an electron, then averaged over all possible distances and energies. Image removed. Fig. 5.4 in [Turner]. Image removed. Fig. 5.5 in [Turner]. The Bethe Formula for Stopping Power. Using relativistic quantum mechanics, Bethe derived the following expression for the stopping power of a uniform medium for a heavy charged particle: - dx dE = 22 422 0 4 β π mc nezk ? ? ? ? ? ? ? ? 2 2 22 )1( 2 ln β β β I mc . k o = 8.99 x 10 9 N m 2 C -2 , (the Boltzman constant) z = atomic number of the heavy particle, e = magnitude of the electron charge, n = number of electrons per unit volume in the medium, m = electron rest mass, c = speed of light in vacuum, β = V/c = speed of the particle relative to c, I = mean excitation energy of the medium. ? Only the charge ze and velocity V of the heavy charged particle enter the expression for stopping power. ? For the medium, only the electron density n is important. 8 Tables for Computation of Stopping Powers If the constants in the Bethe equation for stopping power, dE/dX, are combined, the equation reduces to the following form: ]ln)([ 1008.5 2 231 ev IF nzx dx dE ?=? ? β β MeV cm -1 where, 2 2 26 1 1002.1 ln)( β β β β ? ? = x F Image removed. Table 5.2 in [Turner]. 9 Conveniently,….. For a given value of β, the kinetic energy of a particle is proportional to the rest mass, Table 5.2 can also be used for other heavy particles. Example: The ratio of kinetic energies of a deuteron and a proton traveling at the same speed is 2 2 1 2 1 2 2 == p d p d M M VM VM Therefore the value of F(β) of 9.972 for a 10 MeV proton, also applies to a 20 MeV deuteron. 10 Mean Excitation Energies Mean excitation energies, I, have been calculated using the quantum mechanical approach or measured in experiments. The following approximate empirical formulas can be used to estimate the I value in eV for an element with atomic number Z: I ≈ 19.0 eV; Z = 1 (hydrogen) I ≈ 11.2 eV + (11.7)(Z) eV; 2 ≤ Z ≤ 13 I ≈ 52.8 eV + (8.71)(Z) eV; Z > 13 For compounds or mixtures, the contributions from the individual components must be added. In this way a composite lnI value can be obtained that is weighted by the electron densities of the various elements. The following example is for water (and is probably sufficient for tissue). nlnI = N ∑ i i Z i lnI i, Where n is the total number of electrons in the material (n = Σ i N i Z i ) When the material is a pure compound, the electron densities can be replaced by the electron numbers in a single molecule. Example: Calculate the mean excitation energy of H 2 O Solution: I values are obtained from the empirical relations above. For H, I H = 19.0 eV, for O, I O = 11.2 + 11.7 x 8 = 105 eV. Only the ratios, N i Z i /n are needed to calculate the composite I. Since H 2 O has 10 electrons, 2 from H and 8 from O, the equation becomes lnI = 312.4105ln 10 81 0.19ln 10 12 =+ xx giving I = 74.6 eV 11 Stopping power versus distance: the Bragg Peak ]ln)([ 1008.5 2 231 ev IF nzx dx dE ?=? ? β β MeV cm -1 ? At low energies, the factor in front of the bracket increases as β → 0, causing a peak (called the Bragg peak) to occur. ? The linear rate of energy loss is a maximum as the particle energy approaches 0. Distance of penetration dE dx – single particle parallel beam Rate of energy loss along an alpha particle track. ? The peak in energy loss at low energies is exemplified in the Figure, above, which plots -dE/dx of an alpha particle as a function of distance in a material. ? For most of the alpha particle track, the charge on the alpha is two electron charges, and the rate of energy loss increases roughly as 1/E as predicted by the equation for stopping power. ? Near the end of the track, the charge is reduced through electron pickup and the curve falls off. 12 Image removed. Fig. 24.6 in Hall, Eric J. Radiobiology for the Radiologist, 5 th ed. Philadephia PA: Lippincott Williams & Wilkins, 2000. 13 Stopping Power of Water for Protons ]ln)([ 1008.5 2 231 ev IF nzx dx dE ?=? ? β β MeV cm -1 What is needed to calculate stopping power, - dE/dX? n the electron density z the atomic number lnI the mean excitation energy For protons, z = 1, The gram molecular weight of water is 18.0 g/mole and the number of electrons per molecule is 10. One m 3 of water has a mass of 10 6 g. The density of electrons, n, is: n = 6.02 x 10 23 molecules/mole x moleg mg /0.18 10 36 ? x 10 e - /molecule = 3.34 x 10 29 electrons/m 3 As found above, for water, ln I ev = 4.312. Therefore, eq (1) gives ]31.4)([ 170.0 2 ?=? β β F dx dE MeV cm -1 At 1 MeV, from Table 5.2, β 2 = 0.00213 and F(β) = 7.69, therefore, ]31.469.7[ 00213.0 170.0 ?=? dx dE = 270 MeV cm -1 The stopping power of water for a 1 MeV proton is 270 MeV cm -1 14 15 Mass Stopping Power ? The mass stopping power of a material is obtained by dividing the stopping power by the density ρ. ? Common units for mass stopping power, -dE/ρdx, are MeV cm 2 g -1 . ? The mass stopping power is a useful quantity because it expresses the rate of energy loss of the charged particle per g cm -2 of the medium traversed. ? In a gas, for example, -dE/dx depends on pressure, but –dE/ρdx does not, because dividing by the density exactly compensates for the pressure. ? Mass stopping power does not differ greatly for materials with similar atomic composition. ? Mass stopping powers for water can be scaled by density and used for tissue, plastics, hydrocarbons, and other materials that consist primarily of light elements. For Pb (Z=82), on the other hand, -dE/ρdx = 17.5 MeV cm 2 g -1 for 10-MeV protons. (water ~ 47 MeV cm 2 g -1 for 10 MeV protons) **Generally, heavy atoms are less efficient on a g cm -2 basis for slowing down heavy charged particles, because many of their electrons are too tightly bound in the inner shells to participate effectively in the absorption of energy. Range The range of a charged particle is the distance it travels before coming to rest. The range is NOT equal to the energy divided by the stopping power. Table 5.3 gives the mass stopping power and range of protons in water. The range is expressed in g cm -2 ; that is, the range in cm multiplied by the density of water (ρ = 1 g cm -3 ). Like mass stopping power, the range in g cm -2 applies to all materials of similar atomic composition. A useful relationship….. For two heavy charged particles at the same initial speed β, the ratio of their ranges is simply )( )( 2 1 β β R R = 2 2 1 1 2 2 Mz Mz , where: R 1 and R 2 are the ranges M 1 and M 2 are the rest masses and z 1 and z 2 are the charges If particle number 2 is a proton (M 2 = 1 and z 2 = 1), then the range R of the other particle is given by: R(β) = 2 z M R p (β), where R p (β) is the proton range. 16 Image removed. Table 5.3 in [Turner]. Figure 5.7 shows the ranges in g cm -2 of protons, alpha particles, and electrons in water or muscle (virtually the same), bone, and lead. For a given proton energy, the range in g cm -2 is greater in Pb than in H 2 O, consistent with the smaller mass stopping power of Pb. Image removed. Fig. 5.7 in [Turner]. 17