Ebook An introduction to x- ray physics, optics, and applications: Part 2

Part 2 book “An introduction to x- ray physics, optics, and applications” has contents: Compton scattering, photoelectric absorption, absorption spectroscopy, imaging, and detection, refractive and reflective optics, single-crystal and three- dimensional diffraction, diffraction optics,… and other contents.

PART III X-RAY INTERACTIONS WITH MATTER PHOTOELECTRIC ABSORPTION, ABSORPTION SPECTROSCOPY, IMAGING, AND DETECTION 9.1 Absorption coefficients When a beam of light or x rays is incident on a material, photons can be absorbed, causing electrons to be emitted From the point of view of the electron, this process is called photoelectric emission From the point of view of the photon, it is termed photoelectric absorption This knock-out of an electron is the same process discussed in section  4.7 on x-ray fluorescence, although in that case the emphasis was on what FIGURE 9-1 Absorption of happened next if the emitted electron came from a core level a photon can be regarded Absorption can be regarded as causing a fluctuation of the as the fluctuation of the electron energy up to a virtual level, as shown in Figure 9-1 electron energy up to a For an electron to be emitted, the incoming photon must “virtual” electron level have an energy greater than the binding energy of the electron, but it is not necessary that the incoming photon energy match a difference in electron energy levels, as is the case for the emitted photon in x-ray fluorescence, illustrated in Figure 4-3 The rate at which photons will be absorbed by a single atom is given in terms of a cross section σab for absorption, Γab, = σabΨ , (9-1) where Γab,1 is the absorption rate for a single atom, and Ψ is the photon intensity, the number of photons per area per second in the beam For comparison, the rate Γ at which photons would hit a target of area A placed in the beam is Γ = AΨ , (9-2) so that the cross section can be seen to have units of area Cross sections are described in units of barns, where barn = 10−24 cm2 (The unit is a reference to an old saying referring to someone with poor aim as not being able to hit “the broad side of a barn.” A barn is a large cross section in nuclear physics.) 140 ■ Chapter A sense of the dependence of the cross section on photon energy and atomic number can be found from a rough estimate based on a quantum mechanical calculation similar to that of equation 4-6 from section 4.3 on characteristic spectra According to Fermi’s golden rule, the transition probability is proportional to the Hamiltonian matrix element Hif , where i and f refer to the initial and final state, respectively,* σab ∝ Hif (9-3) The initial state has the bound electron and an incident photon, and the final state has the free electron and no photon The matrix element Hif depends on the integral over the initial bound state of the electron (for x rays generally a K or L shell electron), and so depends on the coulomb energy, which, according to equation 4-2 is proportional to Z 2, so that Hif ∝ Z The Hamiltonian for photon absorption includes the photon 1 annihilation operator, which has a normalization factor of ∝ The condition ω U that the kinetic energy of the ejected electron be equal to the difference between the photon energy and the electron binding energy introduces a factor of 1/U in the matrix element after the integration, so that it is expected that Hif ∝ Z 1 ⇒ σab ∝ Z U U U (9-4) Empirically, the atomic cross section for photoelectric absorption depends on the photon energy U and the atomic number Z of the material as U σab ≈ σ o ⎛ o ⎞ Z , ⎝U⎠ (9-5) where Uo = 1 keV (so that U/Uo is a unitless energy), and σo is a constant For the ejection of K shell electrons, σo ≈ 38.8 barns The energy and atomic number dependences in equation 9-5 mean that high-energy photons have a lower cross section and hence are more penetrating, and that high-Z materials stop photons more effectively, as was asserted in chapter 1 EXAMPLE 9-1: CROSS SECTION Nickel has atomic number 28 Silicon has atomic number 14 For photon energies of 10 and 20 keV, estimate the atomic cross section for photoelectric absorption for nickel and silicon *As for section 4.3, the details of the quantum analysis here are not necessary for the rest of the development in this chapter, but the results are again useful More discussion of quantum calculations can be found in the texts listed in the further reading for chapter 4 Photoelectric Absorption, Absorption Spectroscopy, Imaging, and Detection ■ First, we need to check whether K shell electrons can be knocked out by the incoming photons, to determine whether the constant given below equation 9-5 is applicable to this problem For silicon the “K-edge” energy, the minimum energy needed to knock out a K shell electron, is 1.8 keV For nickel, it is 8.3 keV Both photon energies are high enough for both elements Ni at 10 keV: 3 ⎛ 1keV ⎞ U ≈ σo ⎛ o ⎞ Z ≈ (38.8 barns) ⎜ ( 28 )4 ⎟ ⎝U ⎠ ⎝ 10 keV ⎠ σab ≈ 2.4 × 10 barns ≈ 24 kbarns Ni at 20 keV: 3 ⎛ 1keV ⎞ U σab ≈ σo ⎛ o ⎞ Z ≈ (38.8 barns) ⎜ ( 28 )4 ≈ kbarns ⎝U ⎠ ⎝ 20 keV ⎟⎠ As expected, the cross section at 20 keV falls by a factor of compared with that for the same element at 10 keV Si at 10 keV: 3 ⎛ 1ke V ⎞ U σab ≈ σo ⎛ o ⎞ Z ≈ (38.8 barns) ⎜ (14)4 ⎝U ⎠ ⎝ 10 ke V ⎟⎠ ≈1.5 × 103 barns ≈1.5 kbarns The cross section has fallen by a factor of 16 relative to that for nickel at the same energy Si at 20 keV: 3 ⎛ 1keV ⎞ U σab ≈ σ o ⎛ o ⎞ Z ≈ (38.8 barns) ⎜ (14)4 ≈186 barns ⎝U ⎠ ⎝ 20 keV ⎟⎠ As expected, the high-energy photon with the low-Z material has the smallest cross section EXAMPLE 9-2: ABSORPTION RATE FOR A SINGLE ATOM Find the rate of photoelectric absorption and the power loss for a single nickel atom in a beam of 10 keV photons with an intensity of 1.6 mW/cm2 I = Ψ U ⇒Ψ = I 1.6 × 10 −3 W/cm ph ≈ ≈1012 U 10 × 103 eV 1.6 × 10 −19 J cm i s ph eV 141 142 ■ Chapter Hence the rate at which photons are absorbed is ⎛ ph 10−24 cm ph ⎞ ≈ 2.4 × 10 −8 , (2.4 × 10 Γ1, ab =Ψ σab ≈ ⎜ 1012 barns) ⎟ cm i s ⎠ s barn ⎝ and the rate at which power is lost from the beam is ⎛ 1.6 × 10−19 J ⎞ ⎛ −8 ph ⎞ − 23 Ploss = U Γ ≈ ⎜ 10 × 103 eV ⎜⎝ 2.4 × 10 ⎟ ≈ × 10 W ⎟ s ⎠ eV ph ⎝ ⎠ This is not an easy way to detect a single nickel atom Generally, one is interested in the absorption from a collection of atoms in a material From the calculations of example 9-2, the change in power if a beam passes through a material with Natom atoms is ΔP = − N atom Ploss, = − N atom U Γ = −NatomUΨ σab = − N atom Iσab (9-6) The loss of intensity if a beam passes through a material of thickness Δz is then ⎧N ⎫ ΔP N = − atom Iσab = −Iσab ⎨ atom ⎬ Δz A A ⎩ AΔz ⎭ = − (I)σab {ρ atom } Δz ΔI = (9-7) The atomic density, atoms per unit volume, ρatom can be computed from ρ atom = ρ NA , Mm (9-8) where ρ is the mass density, Mm is the molar mass, and NA is Avogadro’s number Considering an infinitesimal slice of material, dI = − Iσab ρatom dz (9-9) The resulting depth dependence is exponential, I = Ioe − µabz , (9-10) where Io is the incident intensity, and the constants are combined into an absorption coefficient μab, µab = ρ atomσab = ρNA σ Mm ab (9-11) Absorption coefficients are usually tabulated without the density factor, as the density can depend on the detailed preparation of the material, but rather as the mass absorption coefficient μab /ρ, µab N A = σab , ρ Mm (9-12) Photoelectric Absorption, Absorption Spectroscopy, Imaging, and Detection ■ so that none of the factors are preparation dependent The intensity can also written in terms of an absorption length μab−1, I = I oe − z µab −1 (9-13) Thus the absorption length is the thickness of material through which the x-ray beam can pass before falling to about one-third of its original intensity The transmission T through a thickness L of material is the ratio of intensity out to the intensity in, T= I = e − µab L Io (9-14) EXAMPLE 9-3: ABSORPTION Nickel has density of 8.9 g/cm3 For nickel and a photon energy of 20 keV, estimate the absorption coefficient, the mass absorption coefficient, the absorption length, and the transmission through a 30 μm thick foil The atomic density is ⎛ 8.9 g ⎞ ⎛ 6.02 × 10 23 atoms ⎞ ρ N A ⎝ cm3 ⎠ ⎝ atoms mole ⎠ ρ atom = ≈ ≈ 9.1× 10 22 g Mm cm 58.7 mole The absorption coefficient is atoms ⎞ ⎛ 10− 24 cm ⎞ ≈ 272 cm −1 (3 × 103 barns) ⎜ µab = ρ atomσab ≈ ⎛ 9.1× 10 22 ⎝ ⎝ barn ⎟⎠ cm ⎠ The absorption length is µab −1 = 1 ≈ 3.7 ì 10 cm = 37 àm àab 272 cm −1 The mass absorption coefficient is µab 273 cm −1 cm ≈ ≈ 30.6 g g ρ 8.9 cm The transmission is T =e − µ ab L ≈e cm ⎞ ⎛ (30 µm) − ⎜ 273 10 −4 ⎝ µm ⎟⎠ cm ≈ 44% 143 ■ Chapter 104 Total attenuation with coherent scattering (cm2 g–1) 100 Incoherent scattering Coherent scattering 10–4 Photoelectric absorption 10–3 10–2 10–1 100 101 Photon energy (MeV) FIGURE  9-2 Mass attenuation coefficients for water From  D Pfeiffer, Requirements for Medical Imaging and X- Ray Inspection, chap 31, Handbook of Optics, 3rd  ed., vol 5, McGraw-Hill, 2010 9.2 Attenuation versus absorption The energy dependence of the photoelectric absorption coefficient is shown in Figure 9-2 for water In addition to photoelectric absorption, photons can be removed from the x-ray beam by incoherent and coherent scattering, which will be discussed in 1,E+04 μtot /ρ (cm2 g–1) 144 Z = 10 Z = 30 Z = 50 Z = 70 Z = 90 1,E+02 1,E+00 1,E–02 1,E–03 1,E–01 1,E+01 1,E+03 1,E+05 U (MeV) FIGURE 9-3 Comparison of mass attenuation coefficients for a variety of atomic numbers as a function of photon energy in megaelectronvolts From I Akkurt et al., Journal of Quantitative Spectroscopy and Radiative Transfer 94, nos 3–4 (1 September  2005): 379–85 Copyright Elsevier Photoelectric Absorption, Absorption Spectroscopy, Imaging, and Detection ■ chapters 10 and 11 The total attenuation coefficient is given by the sum of the coefficients for absorption, scatter, and pair production, µ tot = µab + µs, coh + µs, incoh + µ P (9-15) Pair production (generation of an electron and a positron) is not possible at a photon energy below U = 2Ueo, twice the rest mass energy of the electron, about 1 MeV, an energy higher than in most of the applications discussed in this text Pair production is included in Figure 9-3 9.3 Index of refraction As was seen in equation 5-44, the electric field amplitude of a plane wave can be written as E = Re { Eoei(κ z −ω t ) } (9-16) For convenience we will leave the time dependence and the real part as understood, and write E = Eoeiκ z (9-17) If, instead of being in vacuum, the wave is traveling in a material of index of refraction n, the wavevector κ becomes κ= 2π n, λ (9-18) where λ is the vacuum wavelength The index of refraction is complex and is generally written n = − δ + iβ (9-19) As will be seen shortly for β (and in chapter 11 for δ ), β and δ are both small, and |n| ~ Because the refractive index is close to 1, x rays mostly not refract when passing through materials The result is expected, as it implies that shadow images, such as that in Figure 1-1, are sharp The consequences for refractive optics are discussed in section 12.1 The real part of the index of refraction is less than 1, as will be computed in section 11.1 The consequences for phase velocity and for reflectivity and reflective optics are discussed in sections 11.3, 11.9, and 12.2 Concentrating now on the absorption term, and using equation 9-19 in the equation for the field gives E = Eoe iκ z = Eoe i 2π (1 − δ + iβ )z λ = Eoe i 2π 2π (1−δ )z − βz λ e λ (9-20) The intensity is then I= cε cε cε E = EE * = ⎡⎢ Eo2 2 ⎣2 ⎤ ⎛ e− ⎥⎦ ⎜⎝ 2π βz ⎞ λ ⎛ − 4λπ β z ⎞ = I [ ] o ⎜e ⎟⎠ ⎟⎠ , ⎝ (9-21) 145 146 ■ Chapter where Io is the incident intensity.* Therefore, by comparison with equation 9-10, the absorption coefficient μab is related to the complex part of the index of refraction β by µab = 4π β λ (9-22) The index of refraction can also be written in terms of a dimensionless quantity called the atomic scattering factor† f = f1 − i f2 (9-23) as n = − ρ atom re λ ( f1 − if ), 2π (9-24) where re is the classical electron radius, which is a somewhat arbitrary constant obtained from setting the rest mass of the electron equal to the potential energy of a charge at a distance re, M ec = q2e q2e ⇒ re = 4πεore 4πεoM ec (9-25) Thus, by comparison with equation 9-18, β = ρ atom reλ f2 2π (9-26) Combining equations 9-11, 9-22, and 9-26, gives σab = 2reλ f (9-27) EXAMPLE 9-4: INDICES For nickel and a photon energy of 20 keV, estimate the imaginary parts of the index of refraction and the atomic scattering factor First, we need the wavelength λ = part of the index of refraction, β = hc 12.4 keV i Å ≈ ≈ 0.6 Å to compute the complex U 20 keV 0.6 ì 108 cm àab ≈ ⎜ ⎟⎠ ( 272 cm ) ≈1.3 × 10 As ⎝ 4π 4π *There is a sign convention to be careful of here In equation 9-16 the exponential is written with a positive i, in which case the imaginary part of the index of refraction must be taken as positive as well The other convention, in which both are negative, is also seen Either is fine, but they should not be mixed Taking one positive and one negative will result in an unphysical expression in which the intensity grows with distance instead of decaying † The positive choice of sign convention for the imaginary part of the index of refraction forces the negative choice for the imaginary part of the atomic scattering factor 330 ■ Solutions to Chapter 11 So = àab 2.3 ì 106 We also need the Debye-Waller roughness factor, 4π ΔR DW ≈ Ro θc 5.80E−03 16π 2 z rms θ ≈ 2.6 × 103 Roθ The results are given in the table λ2 θB Ro ΔRDW Net R θ Δ β θA 5.28E−03 6.54E−06 0.0E+00 2.6E−03 1.00 0.07 0.93 5.28E−03 6.54E−06 2.30E−06 8.6E−04 2.7E−03 0.59 0.04 0.55 6.45E−03 −7.2E−06 2.7E−03 0.0E+00 0.17 0.02 0.15 6.45E−03 −7.2E−06 2.30E−06 2.8E−03 8.3E−04 0.16 0.02 0.14 For the smaller angle, the reflectivity including β has dropped 40% compared with the case without absorption Chapter 12 a) Compute the focal length for a copper lens with a negative 1 cm radius of curvature at 10 keV (use the parameters for the problems from chapter 11) Lf = R/2 R −1cm ≈ ≈ ≈ 290 m n − −2δ ( −1.7 × 10−5 ) b) Compute the focal length for a compound optic with a radius of −0.1 cm, and 100 lenses L′f = R − 0.1cm ≈ ≈ 0.29 m N 2δ (100)( −1.7 × 10−5 ) c) Estimate the smallest possible focal spot from this optic If it were diffraction limited it would be quite small: Rspot ≈ λ Lf (1.24 ì 1010 m)(0.29 m) 0.02 àm 2R (10−3 m) An elliptical mirror is made from copper to be used at 10 keV If the beam to be focused is 2 mm thick, and the source is 3 m away, how long must the mirror be? L≈ y × 10−3 m ≈ ≈ 0.34 m The distance to the mirror is irrelevant θc (5.9 × 10−3 ) What is the expected focal spot size at 10 keV from a polycapillary optic with a focal length of 20 mm and a channel size of μm θc ≈ Up 30 eV ≈ ≈ mrad U 10 keV wspot ≈1.3Lf θc + w ≈1.3(20 × 10−3 m)(3 × 10−3 ) + µm ≈ 80 µm Solutions to Chapter 13 ■ What is the focal length of a micropore optic with a radius of curvature of 1 m? Lf = R = 0.5 m Chapter 13 A face-centered cubic (fcc) crystal has atoms per unit cell: at a corner and at the center of the three nearest faces: (0, 0, 0), (0, 1/2, 1/2), (1/2, 0, 1/2), and (1/2, 1/2, 0) Show that the structure factor for an fcc crystal is F = f (1 + (−1)h+ k + (−1) h+l + (−1) k+l ), when h, k, l are all even or all odd ! ! ! The lattice is cubic, so, as in example 13-5, u1 = uxˆ , u2 = uyˆ , u3 = uzˆ, and ! 2π G= (hxˆ + kyˆ + lzˆ) The four atom locations are as follows: for the one at the u ! ! ! ! corner, r1 = (0u1 + 0u2 + 0u3 ) = 0, and for the three faces, u ! ! ! 1! ! u u r2 = ⎛ 0u1 + u2 + u3 ⎞ = ( yˆ + zˆ), r!3 = ( xˆ + zˆ), and r4 = ( xˆ + yˆ ) Thus ⎝ 2 ⎠ 2 ! ! 2π ! ! ! ! u (hxˆ + k yˆ + lzˆ) i (yˆ + zˆ) = π (k + l), G i r3 = π (h + l), and G i r1 = 0, G i r2 = u ! ! iπ G i r4 = π (h + k) Since e = cos(π) + i sin (π) = −1, the structure factor is FFCC (h, k, l) = ∑ ! ! f j ei G i rj = f (1 + eiπ(h+ k) + eiπ(h+l) + eiπ(k+l) ) j = f (1 + (−1) (h+ k) + (−1) (h+l ) + (−1) (k+l) ) = { 4f if h, k, l are all even or all odd otherwise A polycrystalline sample of an fcc crystal with a unit cell cube edge of 0.3 nm is irradiated with 10 keV photons a) What is the reciprocal lattice constant (the length of the edge of the reciprocal lattice cube)? g= 2π 2π ≈ ≈ 2.1 Å −1 ≈ 21 nm −1 u 3Å b) Find the diameters of all the diffraction rings on a 300 mm diameter detector placed 100 mm on the source side of the sample (the detector has a hole to hc 12.4 keV i Å ≈ ≈1.24 Å allow the incident beam to pass through) λ = U 10 keV The calculations are shown in the table Because the detector is on the source side of the sample, we will see rings only for 2θ > 180o The ring radius will then be R = L tan (π − 2θ ) Sufficient angle for backscattering requires large (hkl), but if (hkl) is too large, it is not possible to satisfy Bragg’s law with the given wavelength Thus there are only two diffraction rings 331 F = + (−1)h + k + (−1)h + l + (−1)k + l h k l h2 + k + l 0 0 0 F is zero; no ring 1 1 d= u h2 + k + l ∞ 1.73 sinθ B = λ 2d 2θ L tan(π−2 θ) 0.0 0.0 0.36 41.9 Forward directed (so does not hit detector) 0 4 1.50 0.41 48.8 Forward 1 2 0 10 1 11 0.90 0.69 86.5 Forward 2 12 0.87 0.72 91.4 Forward 14 0 16 0.75 0.83 111.5 253.7, misses detector 2 17 17 1 18 3 18 3 19 0.69 0.90 128.5 125.7 20 0.67 0.92 135.1 99.6 21 3 22 2 24 0.61 1.01 Not allowed  Solutions to Chapter 13 ■ c) The grain size is 50 nm What is the width of the rings due to grain-size broadening? For the smallest ring: d 135° ⎞ 0.67 Å Δθ d = ⇒ Δ2θ = tan θ ≈ tan ⎛ ⎝ ⎠ 500 Å tanθ Wgrain Wgrain ≈ 6.5 mrad ≈ 0.37° For the other ring: 128° ⎞ 0.69 Å Δ2θ = tan ⎛ = 5.7 mrad = 0.33° ⎝ ⎠ 500 Å d) What is the largest sample size that will give a ring width small enough to observe the grain-size broadening in part c if the source is 200 mm from the sample? Following equation 13-79, but in the case of backscatter, W⎞ Θ ⎛ rring ⎞ ⎛ = arctan ⎜ tan ⎛ π − 2θ B ± ⎞ ± ⎟ 2θ = arctan ⎜ ⎝ ⎝ ⎠ 2L ⎠ ⎝ L ⎟⎠ Θ W ≈ 2θ B ± ± cos (π − 2θ B ), 2L W Assuming the z sample size broadening must be less than the smallest grain size broadening from part c, where W is the sample size The global divergence is Θ ≈ Δ2θ size ≈ Wsample L cos (π − 2θ B ) + ⇒Wsample < Wsample z < Δ2θ grain Δ2θ grain 5.7 × 10−3 rad ≈ ≈ 0.6 mm 1 cos (2θ B ) + cos (128°) + 200 mm 100 mm z L Silicon is diamond cubic A diamond cubic crystal has eight atoms per unit cell in two groups of four: the first group includes a corner and the center of the three nearest faces: (0, 0, 0), (0, 1/2, 1/2), (1/2, 0, 1/2), and (1/2, 1/2, 0) The second group is displaced 1/4, 1/4, 1/4 from those four: (1/4, 1/4, 1/4), (1/4, 3/4, 3/4), (3/4, 1/4, 3/4), and (3/4, 3/4, 1/4) Show that the structure factor for the (100), (200) and (300) planes is zero Following the pattern of problem 13-1, there are eight terms in the structure factor, but they occur in two groups of four Because the structure factor can be expressed as a product, both terms must be nonzero for diffraction to be observed 333 334 ■ Solutions to Chapter 13 π π ⎛ i(h+ k+l ) i (3h+3k+l) ⎞ iπ(h+ k) iπ(h+l) iπ(k+l) 2 + e + e + e + e + e ⎟ Fdc = f ⎜ ⎜ ⎟ π π i (3h+ k+3l) i (h+3k+3l) ⎜⎝ + e ⎟⎠ +e π ⎛ i (h+ k+l) ⎞ iπ(h+ k) = f ⎜1 + e + eiπ(h+l) + eiπ(k+l) ) ⎟⎠ (1+e ⎝ π ⎛ i (h+ k+l) ⎞ = Ffcc ⎜ + e ⎟⎠ ⎝ ⎧ 8f if (h, k, l are all even) AND h + k + l = 4n ⎪ = ⎨ 4f (1 ± i ) if h, k, l all odd ⎪ else ⎩ The (100) and (300) planes are not all even or all odd The (200) plane does not satisfy h + k + l = 4n Chapter 14 A powder diffractometer uses 10 keV x rays What is the radius of the Ewald sphere? 2π 2π REwald = ≈ ≈ 5.1Å −1 This can be compared with the reciprocal lattice λ 1.24 Å length of g ≈ 2.1 Å−1 for a crystal with lattice constant of u = Å in problem 13-2a A face-centered cubic (fcc) single crystal with lattice constant u = Å is irradiated with a bremsstrahlung source producing x rays of energy ranging from to 40 keV The x-ray beam is incident in the–y direction The detector is 30 mm from the sample (on the side away from the source) Find the wavelength for the diffraction from the (111) plane and the (x, y) coordinate of the spots on the detector For the (hkl) = (111) plane, the plane spacing is d = u h +k +l 2 ≈ 3Å ≈1.7 Å ! 2π 2π Following example 14-1, using κ f = − yˆ + (hxˆ + kyˆ + lzˆ), where u λ ! ! 2π 2ku κ f i κ f = ⎛ ⎞ yields λ = For the (111) plane ⎝ λ ⎠ h + k2 + l λ= 2ku 2(1)(3 Å) ≈ 2 ≈ Å Then applying Bragg’s law as usual gives 2 h + k + l +1 +1 ⎛ 2Å ⎞ ⎛ λ⎞ θ = sin −1 ⎜ ⎟ ≈ sin −1 ⎜ ≈ 35° So the radius for forward scatter is ⎝ 2d ⎠ ⎝ 2(1.7Å) ⎟⎠ Solutions to Chapter 14 ■ R = L tan (2θ) ≈ (30 mm) tan (70.5°) ≈ 85 mm In this case instead of a ring we will get a single spot, so we also need the azimuthal angle, ⎛k ⎞ l ϕ = tan −1 ⎜ z ⎟ = tan −1 ⎛ ⎞ ≈ 45° ⎝ ⎠ k ⎝ kx ⎠ The location on the detector is thus (R cos(ϕ), R sin(ϕ)) ≈ (60, 60) mm A bcc crystal in a θ-2θ diffractometer run with Cu Kα radiation has a peak for the (111) planes at 2θ = 40° a) What is the d spacing of the crystal? λ≈ 12.4 keV i Å ≈1.55 Å Then the plane spacing from Bragg’s law is keV d= 1.55 Å λ ≈ ≈ 2.266 Å sin θ sin (20°) b) The sample is strained by 1% (the lattice constant increases by 1%) What is the new peak angle if the strain is parallel to the (111) planes? This changes the spacing in the plane but doesn’t change the d spacing between the planes, or the diffraction angle c) If the strain is perpendicular to the planes? ⎛ λ ⎞ ⎛ 1.55 Å ⎞ = sin −1 ⎜ = 39.6° d ′ = (1.01) d = 2.289 Å ⇒ 2θ ′ = 2sin −1 ⎜ ⎟ ⎝ 2d ′ ⎠ ⎝ (2.289 Å) ⎟⎠ The angle changed by 0.4°, so broadening effects from sample and beam sizes need to be small to measure the strain Compute the Darwin width for diffraction from a (111) silicon crystal at 8 keV The structure factor is given in problem 13-3 π ⎛ i (h+ k+l) ⎞ F (h, k, l) = FFCC ⎜ + e ⎟⎠ = fo (1 − i), so that ⎝ F = (4fo ) Silicon has a lattice constant of 0.54 nm, so the plane spacing is d= u h2 + k + l ≈ 5.4 Å ≈ 3.1 Å, and the Bragg angle is ⎛ 1.55 Å ⎞ ⎛ λ⎞ ≈14° θ = sin −1 ⎜ ⎟ ≈ sin −1 ⎜ ⎝ 2d ⎠ ⎝ (3.1 Å) ⎟⎠ sin (θ ) ≈ 0.16 Å −1 , and, from λ Figure 13-4, the scattering factor is about 11 The Darwin width is then The factor proportional to the momentum transfer is 335 336 ■ Solutions to Chapter 15 F ⎛ + cos 2θ ⎞ Δθ = 2re λ ⎜ ⎟ ⎝ ⎠ πVc sin2θ (11) ⎛ + cos (14°) ⎞ ≈ (2.8 × 10−15 m)(1.55 × 10−10 m)2 ⎜ ⎟ ⎝ ⎠ π (5.4 × 10−10 m)3 sin (28°) ⎛ 180 deg ⎞ ⎛ 60 ⎞ ⎛ 60 sec ⎞ arcsec 34 àrad (34 ì 106 ) ⎜ ⎝ π ⎟⎠ ⎜⎝ deg ⎟⎠ ⎝ ⎠ Chapter 15 A zone plate with 40 open zones is designed for 10 nm radiation, with a focal length of 50 cm a) What is the zone plate diameter? The number of zones in the zone plate area is NZ = 2Nopen = 2(40) = 80 The radius of the outermost zone is then RN = NZ λ L f ≈ 80 (10 × 10−9 m)(50 × 10−2 m) ≈ 630 µm, Z so the diameter is 1.3 mm b) What is the focal spot size? ( Rspot ≈1.22wN ≈1.22 RN Z − RN Z ≈ Z −1 ) ≈1.22 1.22 (10 × 10−9 m)(50 × 10−2 m) 80 λL f NZ ≈ 4.8 µm c) Where is the third-order image of an object placed 40 cm from the zone plate? Lf = L= Lf The focal length can be used in the thin-lens equation, 1 ≈ ≈ 29 cm 1 − − L f z 50 cm 40 cm The image is 29 cm from the zone plate, 69 cm from the object A multilayer film is composed of alternating 6 nm layers of C, with a plasma energy of 29 eV, and 6 nm layers of W, which has a plasma energy of 80 eV It is to be used to diffract 0.3 keV x rays What is the required incident angle? Solutions to Chapter 15 ■ First, the wavelength is λ= hc 12.4 × 10−10 m i keV ≈ ≈ 4.1 × 10−9 m U 0.3 keV If we ignored the index of refraction, the Bragg angle would have been ⎛ 4.1 nm ⎞ ⎛ λ ⎞ ≈ 9.9° θ B′ = sin −1 ⎜ ≈ sin −1 ⎜ ⎟ ⎝ 2d M ⎠ ⎝ (12 nm) ⎟⎠ The individual index decrements are δC = Up, Mo ⎛ 29 eV ⎞ ≈ ⎜ ≈ 0.005 U2 ⎝ 0.3 × 10 eV ⎟⎠ Up,2 Mo ⎛ 80 eV ⎞ and δ W = ≈ ⎜ ≈ 0.036, U2 ⎝ 0.3 × 103eV ⎟⎠ so that the average index is δ= dCδ C + dWδ W 6(0.005) + 6(0.036) ≈ ≈ 0.028 dM 12 The real Bragg angle is thus ⎛ ⎞ ⎛ ⎞ λ 41 Å ≈ sin −1 ⎜ ≈10.2° θ B = sin −1 ⎜ ⎟ ⎝ 2d M (1 − δ ) ⎠ ⎝ (120 Å) (1 − 0.028) ⎟⎠ The incident angle can be found from Snell’s law, θo = cos −1 ((1 − δ ) cos θ B ) ≈ cos −1 ((1 − 0.028)cos (10°)) ≈17° For these soft x rays, the incident angle needs to be nearly twice that calculated from the usual Bragg’s law 337 INDEX absorption, 7, 139, 140, 144, 157, 174; coefficient of compounds and broadband radiation (including example), 147–48; coefficients including example, 139–41; compared to Compton scatter, 164; cross-section (including example), 129n, 140–41, 185; depth in fluorescence samples, 64–65; edges, 148–49; and filtering, 151–52; in high harmonic generation, 131–32; and the index of refraction (including example), 145–47; length, 83, 143, 155, 157; rate for a single atom, 141–43; spectroscopy (including EXAFS and XANES and XRS), 149–51; in two-level system, 127–29; versus attenuation, 144–45; in x-ray tubes, 83–84, 102–4 See also contrast (including example); detectors; imaging Advanced Light Source (ALS), 125 Advanced Photon Source (APS), 125 angular intensity, 70–71; example, 71–72, 74; linear, 72–73 See also photon angular intensity anodes, 45, 82, 95; bremsstrahlung radiation from a thick anode, 101; bremsstrahlung radiation from a thin anode, 98–99; comparison for fluorescence, 64; in an ion chamber detector, 157; rotating, 83 See also copper; local divergence; silver; tungsten atomic energy levels, 34, 46 See also characteristic emission atomic scattering factor, 146, 175, 178–79 See also Kramers-Kronig relations atoms: density, 99–100; ionized, 34; motion, 36; two-level, 129n, 130 See also atomic energy levels; characteristic emission; diffraction; nuclear decay; plasma; x-ray fluorescence (XRF) attenuation, 144 See also absorption; scatter Auger constant, 52 Auger emission, 52 autocollimator, 224 autoradiography, 15 background (including examples), 56–64, 98, 223, 288, 295 bandwidth (including examples), 76–78, 120, 151, 215, 286–87, 294, 296, 297 beam divergence See divergence; global divergence; local divergence beam hardening, 148 Bessel functions, 120, 285 blackbody radiation, 8, 33–35, 128, 129, 129n blur, 80, 81, 84, 159, 198, 206, 208, 216, 253 See also diffraction resolution; radiography body-centered cubic (bcc), 242, 243, 265 Boltzmann’s constant, 33, 41, 128 Bragg, William Henry, Bragg, William Lawrence, Bragg angle, 238, 251, 258, 260, 265, 266, 272, 288, 289, 290, 291, 294, 296, 297 See also diffraction Bragg geometry, 268 Bragg’s law, 80, 237–38; compared to Ewald sphere construction, 256, 265; energy spread from, 252 See also diffraction bremsstrahlung, 34, 95, 97, 98, 102–3; background in x-ray fluorescence (XRF), 57; conversion efficiency, 101–2; emission rate (including example), 102–5; relativistic (including example), 114–17; and spectral shaping, 105–6; spectrum from a thick anode, 101; spectrum from a thin anode, 98–99 brightness, 75–77, 118–20; apparent, 83; for a broadband source, 77; for a narrow emission line, 77–78 See also heel effect brilliance, 78, 126, 135, 215; apparent, 83; example, 78 broadband radiation, 76–77, 147, 148, 264, 294 See also bremsstrahlung; monochromatic radiation capillary optics, 212 cathode, 44–45, 82; cold, 84 cathode ray tubes, 3, CdZnTe detector, 157 channeling radiation, 135 characteristic emission, 34, 44; line width, 53–54; or radiation, 45; rate, 50–52; spectra, 48–49 See also Moseley’s law; x-ray fluorescence (XRF) classical electron radius, 146, 178, 184, 228, 272 coherence, 84–86, 92, 125–26; in a double-slit experiment, 88, 89; length, 91; requirement for in phase imaging, 92; spatial, 86–89; temporal, 90–91; time, 91; for zone plates, 282 coherent diffraction imaging, 271 coherent scatter, 144, 165, 174–75; cross section (including example), 183–87; from a single electron, 227–29; and the index of refraction (including example), 175–78, 195–99; relativistic cross section, 187 See also atomic scattering 340 ■ Index coherent scatter (cont.) factor; diffraction; Kramers-Kronig relations; phase velocity (including example); reflectivity and reflection coefficients (including examples) collimation: in synchrotrons, 125–26; with polycapillary optics, 216 collimator, 21 Compton scatter, 163–65; cross section, 165–66; and energy loss example, 165; inverse Compton sources (including examples), 166–68 See also scatter Compton wavelength, 165 computed tomography (CT), 24, 92, 148, 160–61 confocal systems, 218–19 constructive interference, 86–87, 91; in diffraction, 230, 232–33; in free electron lasers, 134; in gratings, 274; in wigglers, 124; in zone plates, 282 contrast (including example), 92, 152–54, 159, 171, 276; edge contrast, 289; phase-contrast, 271; subject contrast, 153; with scatter, 169–70 See also imaging copper, 39, 47, 49, 51–52, 53, 61, 82; anodes, 83, 151–52 core atomic energy levels See atomic energy levels Cornell High Energy Synchrotron Source (CHESS), 108 correlation length, 198 Crookes, William, crystal optics, 56, 288–89; and curved crystals (including examples), 294–98; doubly curved crystal (DCC) optics, 295 See also multilayer optics (including examples) crystal planes (including example), 235–37 cyclotrons, 108, 109–10; cyclotron frequency, 110 See also synchrotrons Darwin width (including example), 272–73 Debye, Peter, 6, 40 Debye length, 40; example, 40; fluctuations and the Debye length, 42; screening and the Debye length, 41–42 Debye-Waller factor, 248; Debye Waller-like factor, 196 detectors, 20–22, 156–60; charge injection device (CID) detectors, 158; classifications, 156; detective quantum efficiency (DQE), 157, 160; detector quantum efficiency (DQE), 157; development, 4; digital imaging detectors, 159; direct detectors, 158, 179; indirect detectors, 156; ion chamber detectors, 156–57; quantum detector efficiency (QDE), 157; silicon drift detectors, 158 diffraction, 4, 6, 80–81, 135, 227, 237, 261, 271, 288; from amorphous materials, 250–51; from a chain of atoms, 231–33; diffraction-enhanced imaging, 289; diffraction-limited spot size, 205, 285; dynamical (including examples), 271–73; experiments, 80; Fourier transform relationships, 230–31; indexing a diffraction pattern, 243–44; intensity, 244–46; Laue, 264, 268, 269; multiple anomalous, 270; from a pair of electrons, 229–230; powder (including example), 238–40, 243; ringing, 82; single-crystal (including examples), 264–68 See also Bragg’s law; coherent scatter; diffraction optics; diffraction resolution; filtering; phase problem (in crystallography); protein crystallography; reciprocal lattices (including example); scatter; structure factor (including example); transverse momentum transfer coefficient diffraction optics, 274 See also crystal optics; gratings (including examples); multilayer optics (including examples) diffraction resolution: and the effect of angular broadening (including example of peak broadening), 251–52; and the effects of crystal size, 249–50; and the effects of mosaicity, 246, 258; and the effects of thermal vibrations, 247–49; and energy spread, 252–53; and global divergence and aperture size, 253; and local divergence, 253–54 diffractometer, 238, 239, 257 direct detectors, 158, 179 divergence, 116–17, 121, 134, 215, 261–62, 271 See also global divergence; local divergence DNA analysis, 10 Doppler broadening, 53 Doppler shift, 124, 166; examples, 167–68 dose, 153, 154, 155, 159, 170 See also imaging doubly curved crystal (DCC) See diffraction optics DQE (detective quantum efficiency) See detectors: detective quantum efficiency (DQE) Drude model, 175 See also coherent scatter dynamic range, 159 EDS (energy dispersive spectroscopy) See x-ray fluorescence (XRF) Einstein A coefficient, 127 Einstein B coefficient, 128 electrons: classical electron radius, 146, 178, 184, 228, 272; electron bunching, 110, 133–35; knockout, 139; loss of energy of in a cyclotron, 109–10; in nuclear decay, 13, 14; in plasmas, 33–34; radiation from accelerated, 95–99, 108–17; rest mass energy, 6, 35, 46, 112, 145, 146, 187; slightly bound electrons and phase response (including example), 180–82; transition to lower energy states, 55–56; transition rules, 49; and unoccupied “trap states,” 20–21 See also atomic energy levels; bremsstrahlung; characteristic emission; Drude model; photoelectric emission; x-ray tubes emission See bremsstrahlung; characteristic emission; emission lines, example of plasma emission line wavelength emission lines, example of plasma emission line wavelength, 35 See also characteristic emission energy levels See atomic energy levels energy-dispersive x-ray spectroscopy (EDS), 55 ESRF (European Synchrotron Research Facility), 125 EUV (extreme ultraviolet), 4, 131, 132, 275, 287 Index ■ Ewald sphere construction, 256–57, 264 EXAFS (extended x-ray absorption fine structure), 151 FELs See free electron lasers (FELs) Fermi’s golden rule, 49, 140 film, 158 filtering, 151, 160, 223 fine structure constant, 116 Fourier transform, 27, 53, 98, 151, 159, 160, 196, 224, 230–31, 234, 242, 249, 251, 270, 271, 277 free electron lasers (FELs), 133–35 See also x-ray lasers free electron modeling See coherent scatter; Drude model frequency doubling, 131 Fresnel reflection coefficient See reflectivity and reflection coefficients (including examples) Fresnel zone plates See zone plates (including examples) gamma camera, 21 Gaussian distributions, 23, 53–54 Geiger counter, 157 global divergence, 79–80, 212, 215, 217; from aperture size, 253; in diffraction, 261–62; effect on diffraction resolution, 253 gratings (including examples), 274–78 half-life (including examples), 10–14 Hamiltonian, 52, 140 heel effect, 83–84 Heisenberg uncertainty, 53, 79, 205, 227, 249, 250 See also diffraction: diffraction-limited spot size; lifetime Hertz, Heinrich, high harmonic generation See x-ray lasers imaging: coherent diffraction imaging, 270; and contrast, 152–54; and dose, 154; in-line phase imaging, 92–93; monochromatic beams and medical imaging, 298; and noise, 154–56; photon counting imaging systems, 158 See also absorption; computed tomography (CT); detectors; nuclear medicine; phase imaging; radiography; single-photon emission computed tomography (SPECT) incoherent scatter See Compton scatter incoherent source See coherence index of refraction, 145–46, 175–77, 181; correction for in multilayer optics, 290–92; and phase imaging, 276; and phase velocity, 179; and reflection, 191–93; and Snell’s law, 187; and surface roughness, 196 See also absorption; coherent scatter; refractive optics indexing a diffraction pattern See diffraction indirect detectors, 156, 158 insertion devices, 121–25; undulators, 123–25; wigglers, 121–23 intensity, 68–69; example, 69–70 See also angular intensity; diffraction: intensity; photon intensity inverse geometry, 84 ion chamber, 156 ions See plasma: highly ionized isotopes, 13; neutron-poor isotopes, 14 isotropic radiation, 15, 20, 51, 69–70, 72, 79, 187 K edge, 141 See also atomic energy levels K shell, 49, 64, 140–41, 150 See also atomic energy levels k space, 256 See also reciprocal lattices (including example) Kα emission See characteristic emission Kirkpatrick Baez mirror, 209 Klein-Nishina cross section, 187 See also Compton scatter Kramer’s constant, 101 Kramers-Kronig relations, 182–83, 270 lasers, 158; diode lasers, 131; free-electron lasers (FELs), 133–35; HeNe lasers, 131; ruby lasers, 131; visible light lasers, 131, 131n See also x-ray lasers lattices See diffraction; reciprocal lattices (including example) Laue geometry, 268 Lepton number, 13n l’Hôpital’s rule, 232 Liénard-Wiechert potentials, 97 lifetime, 11, 53–54, 91, 130–31, 158 line widths, 53–54; examples, 54 Liouville’s theorem, 215 lithography, 80 lobster eye optics See micropore (MCP) optics local divergence, 80–82, 126, 215–16; effect on coherence, 88–89; effect on diffraction resolution, 254; effect on resolution, 81; effect of source size, 81; example, 82 See also heel effect Lorentz contraction, 124 Lorentz distribution, 23, 53 Lorentz factor, 112, 117, 134 Lyman α, 35 Maxwell equations, 95, 120, 175, 190 metastable emitters and half-life (including examples), 10–13 Michelson-Morley experiment, microbunching, 133–34 See also electrons: electron bunching; free electron lasers (FELs) microcalorimeter, 158 micropore (MCP) optics, 219–20, 223 minimum detectable limit (MDL), 63; MDL comparison example, 61, 63–64 mirrors See reflective optics modulation transfer function (MTF), 159 monochromatic radiation, 76, 125, 295; and medical imaging, 298 See also broadband radiation monochromators, 56, 288–89 Monte Carlo simulations, 224 mosaicity, 246, 251, 258 See also diffraction resolution: and the effects of mosaicity 341 342 ■ Index Moseley’s law, 47 multilayer optics (including examples), 289–94 multiple isomorphic replacements, 270 multiwire detector, 157, 158 National Light Source (NLS), 113 neutrinos, 13n neutrons, 13, 14, 154; fast neutrons, 14 Nevot- Croce correction (including example), 197–99 noise, 22–23, 27, 59, 60, 154–55, 160; thermal noise, 158 See also photons: photon statistics; quantum noise; radiography; x-ray fluorescence (XRF) nuclear decay, 13–14 nuclear medicine, 10, 14–16; as functional imaging, 10; goal, 14 See also pinholes Nyquist frequency theorem, 159 optics, 201; optics simulations, 224–25 See also diffraction optics; pinholes; reflective optics; refractive optics parametric radiation, 135 particle-induced x-ray emission (PIXE), 55, 98 Patterson difference map, 270 phase imaging, 92, 276–78 phase problem (in crystallography), 270 phase velocity (including example), 179–80 phosphor, 3, 156, 157, 158–59 See also indirect detectors photoelectric absorption See absorption photoelectric emission, 55, 139 See also absorption photomultiplier tube (PMT), 21, 156, 220 photon angular intensity, 73; example, 74; example of photon angular intensity density, 76; linear, 75 photon intensity, 73; example, 73–74 See also photon angular intensity photons, 5; photon detection and scatter rejection (including examples), 20–22; photon statistics (including examples), 22–24; zero-energy “virtual” photons, 102 See also photon intensity pinholes, 15–16; examples, 16–18 pixels, 26, 154, 157, 159, 278 Planck’s constant, 5, 33, 205 planes See crystal planes (including example) plasma, 33, 41–42; energy level of bound states in, 34–35; formed by cosmic radiation, 38; highly ionized, 131; laser-generated, 131n; very hot, 35–37 See also Debye length; plasma frequency; x-ray lasers plasma frequency, 37–40, 177, 192, 214 Poisson ratio, 297 Poisson statistics, 22–24 polarization, 176, 184–85, 288–89 polycapillary optics (including examples), 213–19, 262 positron emission tomography (PET), 24n proportional counter, 157 protein crystallography, 269–70 protons, 11, 13, 14, 34, 98 QDE (quantum detector efficiency) See detectors: quantum detector efficiency (QDE) quantum mechanics, 5; Fermi’s “golden rule” concerning, 49, 140 quantum noise, 22, 23, 153, 159 See also noise; photons: photon statistics quantum numbers See atomic energy levels quarks, 13 radial distribution function, 150 radiation: channeling, transition, or parametric radiation, 135 See also blackbody radiation; bremsstrahlung; characteristic emission; dose; synchrotron radiation radioactive decay, 10 radioactive iodine-125, 14 radiography, 14; medical radiography, 84; morphological imaging, 10; quantum noise in, 22 See also autoradiography; computed tomography (CT); imaging; nuclear medicine Rayleigh scatter, 179 See also coherent scatter reciprocal lattices (including example), 233–35; reciprocal lattice vectors, 234, 235 reflective optics: array optics, 219–23; capillary optics, 211–13; elliptical optics (including examples), 206–9; and energy filtering, 223; microchannel plate optics, 220; micropore (MCP) optics, 219–20, 223; and optics metrology, 223–24; polycapillary optics (including examples), 213–19, 262; Wolter optics, 209–10 reflectivity and reflection coefficients (including examples), 190–92, 193–95; specular, 196; and surface roughness (including example), 195–99 refractive optics, 7, 145, 189, 206 relativistic effects, 97, 112–17, 164, 187 See also Compton scatter: inverse Compton sources; Lorentz contraction; synchrotron radiation relativity, 5, 179 retarded time, 97 rocking curve, 246 roentgen (R), 154 roentgen equivalent man (rem), 154 Röntgen, Wilhelm Conrad, 3–4 Rowland circle, 294; and beam divergence, 261–62; proof that the angle of incidence is always θ B on the Rowland circle, 260–61; texture and strain measurements with the θ-2θ diffractometer, 262–64; and the θ-2θ diffractometer, 257–59 Rydberg energy, 34, 46 SASE See self-amplified spontaneous emission (SASE); x-ray lasers scatter: effect on contrast in radiography, 168–70; fraction, 168–69, 170, 171–72; rejection in nuclear medicine, 20, 22; rejection in radiography, 171–72 See also coherent scatter; Compton scatter scintillator, 20, 21 screening See atomic energy levels; Debye length self-amplified spontaneous emission (SASE), 130, 135 Index ■ semiconductor detectors, 157; high-Z semiconductors, 157, 158 sensitivity: energy sensitivity, 22, 157; in nuclear medicine, 16–17; in x-ray fluorescence, 64 Siegbahn, Kai M., signal-to-noise ratio (SNR), 60–63, 155, 159–60, 170 silicon pin diode or drift detectors, 158 silver, 158 single-photon emission computed tomography (SPECT), 24–27, 68, 73, 76, 160 Snell’s law, 187–88 SNR See signal-to-noise ratio (SNR) soller slit, 261 source size See local divergence; x-ray tubes: heat load SPECT See single-photon emission computed tomography (SPECT) specular reflectivity, 196 spontaneous emission See x-ray lasers Spring-8, 125 Stanford Linear Accelerator FEL, 135 statistics See minimum detectable limit (MDL); Poisson statistics; quantum noise; signal-to-noise ratio (SNR) streak artifacts, 26 structure factor (including example), 242–43 surface profilometry, 223 synchrotron radiation, 92, 108, 112–13; power, 113–16; spectra (including example), 117–21 synchrotrons, 14, 113, 117; beamlines on, 4, 287; and collimation and coherence, 125–26; and insertion devices, 121–25 See also Advanced Light Source (ALS); Advanced Photon Source (APS); ESRF (European Synchrotron Research Facility); Spring-8 Talbot distance, 278 technetium: metastable (99mTc), 11; and nuclear β decay, 13–14 Tesla, Nikola, thermal diffuse scattering, 249 Thomson scatter, 179 See also coherent scatter tomography See computed tomography (CT) tomosynthesis, 160–61 transition radiation, 135 transition rules, 49 transmission, 14, 38, 143, 152, 171, 214, 271, 278, 292 transverse momentum transfer coefficient: in diffraction, 229; in reflectivity, 191; in surface roughness, 196 tubes See capillary optics; x-ray tubes tungsten, 46, 52, 64, 82, 99, 104 undulator, 124–25, 133–34, 135 uranium, 11; bombardment of with fast neutrons, 14 voltage See Debye length; detectors; x-ray fluorescence (XRF); x-ray tubes: voltage wavelength-dispersive x-ray fluorescence or spectroscopy (WDXRF or WDEDS), 56, 289 W–boson, 13n white-beam diffraction geometry See diffraction: Laue wiggler, 121–23 Wolter optics, 209–10 XANES See absorption: spectroscopy (including EXAFS and XANES and XRS) x-ray astronomy, 4, 35, 209, 294 x-ray fluorescence (XRF), 4, 8, 10, 15, 55, 57–61, 68, 98, 139, 149, 218; examples, 58–59; lead in toys example, 62–63; micro- (MXRF), 218; quantitative, 65; wavelength-dispersive, 289 See also minimum detectable limit (MDL); sensitivity; signal-to-noise ratio (SNR) x-ray lasers: and high-harmonic generation (including example), 131–33; and the problem of providing a laser cavity, 130; and stimulated and spontaneous emission, 127–30; three classes, 128 See also free electron lasers (FELs) x-ray optics See optics x-ray spectrometry or spectroscopy (XRS) See absorption: spectroscopy (including EXAFS and XANES and XRS) x-ray tubes, 44–45; design, 82–84; field emission cathodes, 44; heat load, 83, 102; source size, 82; voltage, 44–45, 47, 50–51, 84, 99, 101, 105–6 See also anodes; cathode x rays: definition, 4–6; discovery, 3–4; importance, 4, 6–8 XRF See x-ray fluorescence (XRF) Young’s double-slit experiment, 86, 87 Young’s modulus, 248 zone plates (including examples), 279–88; amplitude zone plates, 287; phase zone plates, 287 343 ... 9 -22 , and 9 -26 , gives σab = 2reλ f (9 -27 ) EXAMPLE 9-4: INDICES For nickel and a photon energy of 20  keV, estimate the imaginary parts of the index of refraction and the atomic scattering factor... Detectors, chap 61, both in M Bass, C DeCusatis, J Enoch, V. Lakshminarayanan, G Li, C MacDonald, and V N Mahajan, and E Van Stryland, eds., Handbook of Optics, 3rd ed., vol 5, McGraw-Hill, 20 10... Jr, and John M Boone, The Essential Physics of Medical Imaging, 3rd ed., Lippincott Williams & Wilkins, 20 12 W Gibson and P Siddons, Introduction to X -ray Detectors, chap 60, and A Couture, Advances