Analytical Methods for Materials

Analytical Methods for Materials Lesson 6 Production & Properties of X-rays Suggested Reading Chapter 1 in Waseda et al. Section 2.1 in Leng Other Reference B.D. Cullity and S.R. Stock, Elements of X-ray Diffraction, 3 rd Edition (Prentice Hall, 2001) Ch. 1 125

Electromagnetic Spectrum 400 nm 700 nm Wavelength (meters) 10-16 10-12 10-8 10-4 10 0 10 4 Gamma rays X-rays ultraviolet infrared radio Frequency (hertz) 10 23 10 19 10 15 10 11 10 7 10 3 Energy (electron-volts) 10 8 10 4 10 0 10-4 10-8 10-12 Figure modeled after http://mrbarlow.files.wordpress.com/2007/09/em_spectrum.jpg 126

Frequency in cycles per second 10-6 Wavelength in nm Visible Light: λ ~6000 Å X-rays: λ ~0.5 2.5 Å Electrons: λ ~0.05 Å nm 10 20 Gamma rays 400 Violet One angstrom One nanometer 10-1 1 X-RAYS 450 500 Indigo Blue 10 15 One micrometer 10 4 Ultraviolet Visible Light Infrared 550 600 Green Yellow Orange 650 10 10 One centimeter One meter 10 9 Short radio waves Short radio waves 700 750 Red 10 5 One kilometer 10 14 Broadcast band Long radio waves Spectrum of electromagnetic radiation as a function of frequency (s -1 ) and wavelength (nm). X-rays are useful b/c λ interplanar spacings 127

X-ray Tube TUNGSTEN FILAMENT (CATHODE) ELECTRONS (ANODE) X-RAYS

X-ray Production Heat a suitable cathode to a high temperature and apply a voltage, V, between the cathode and anode. V - e - Cathode e - Heated below melting point vacuum The electric field (E = V/x, where x is the spacing between the electrodes) pulls electrons with the highest energies out of the cathode and accelerates them towards the anode. + Anode Heat X-rays The electrons rapidly decelerate when they strike the anode. 129

X-ray Production cont d Potential energy of electron: E ev At the anode, electron is accelerated to maximum velocity (v*). Potential energy is converted to kinetic energy. 1 E ev m v 2 * 2 Electrons will either lose some or all of their energy at impact with the anode (target). 1% of the kinetic energy is transferred to atomic and conduction electrons which are excited into higher energy levels. When they revert back to their original energy levels, they emit X-rays. 130

X-ray Production cont d If electrons lose all energy at impact (elastic collision), an X-ray of minimum wavelength is generated: E ev h hc min E max min 34 8 hc (6.62610 )(310 ) SW L 19 ev (1.602210 ) V e hc 1 2 mv* ev e 2 o 12.43 ( A) V This is called the short wavelength limit! (kv) 131

X-ray Production cont d When an incident electron loses a fraction of its energy as opposed to all energy, an X-ray of longer wavelength (λ > λ SWL ) is produced. E e - o h = E 1 E o e - E f =E o -E 1 E f =E o -E 2 h = E 2 This is known as white or continuous radiation! 132

X-ray Emission Spectrum X-RAY INTENSITY (relative units) 6 5 Characteristic 4 Radiation 3 2 1 0 K K 25 20 15 10 Mo (schematic) Continuous Radiation 5 kv 0 1.0 2.0 3.0 SWL hc ev e WAVELENGTH (angstroms) ENERGY All features vary with incident energy (i.e., accelerating voltage) and target material. 133

X-ray Spectrum 70 X-rays result from fast electron irradiation of metal targets (Cu, Co, Cr, Mo) under vacuum. Types of X-rays: Intensity (rel. units) 60 50 40 30 Mo at 35 kv (schematic) K K 1 Characteristic: X-rays from core-level ionization. 20 K K 2 10 Continuous / Bremsstrahlung: white or braking radiation from deceleration of electrons. 0 0.2 0.4 0.6 0.8 1.0 0.70 0.71 0.72 SWL Wavelength (angstroms) 134

Continuous Spectrum Recap: Different electrons lose energy in different ways. Some lose all of their energy in one collision Some will be deflected and lose some of their total energy until all is used up. Result: X-rays can be produced with a wide range of energies and wavelengths. The broad hump occurs where there is the most probable energy loss. 135

Intensity of the Continuous Spectrum from Target V I cs AiZV where, Am, = constants i = current Z = atomic number = accelerating voltage m Higher atomic number (Z) elements and high voltages are required to generate high intensity X-rays 136

The Characteristic Spectrum What if incident e - s have enough energy to knock inner shell electrons from their orbits (i.e., to cause core-level ionization)? This leaves the atom in an excited state. An electron from a higher energy level can fall into the lower energy state emitting an X-ray with a characteristic energy/wavelength. 137

Origin of the Characteristic Spectrum M K L E e W K Incident electron E e M L K W M W L Ejected K-shell electron W K M L K Hole in L-shell Energy Liberated X-RAY (energy) (E) Photoelectron Hole in K-shell K X-ray 138

X-ray Energy Equal to the difference between the energy of the incident electron and the binding energy of the electrons in their shells/orbitals. M L K Hole in L-shell Energy Liberated X-RAY (energy) (E) Characteristic X-ray Emission Exray WLWK Type of X-ray will depend upon orbitals 139

K and K Radiation Hole in M-shell M L K Hole in L-shell X-RAY (energy) (E) M L K X-RAY (energy) (E) K X-ray L K K X-ray M K X-ray radiation is characteristic of the energy liberated during the electron transition 140

The Characteristic Spectrum Shells and sub-shells have different energy levels L K K K L M Different types of characteristic radiation! L III (2.52 kev) II (2.63 kev) K (20keV) K 1 L III K K 2 L II K I NOTE: there is no transition between the L I subshell and the K-shell K 1, K 2, K 1, the strongest of the K set, are typically used for diffraction. 141

Energy-level diagram for molybdenum showing all allowed electron transitions N V N IV N III N II N I Energy (log scale) M V M IV M III M II M I L III L II L I α 1 β 1 γ 1 L I series L II series L III series L III = 2.52 kev L II = 2.63 kev α 2 β 3 γ 2 K K series K = 20 kev 142

70 60 Mo at 35 kv (schematic) K α The K α line is referred to as the K α doublet Intensity (rel. units) 50 40 30 K α1 K α1 is twice as strong as K α2 K α(avg) = ½ (2 K α1 + K α2 ) 20 K β K α2 10 E E L L III II K K 20 2.52 17.48 kev, 0 0.2 0.4 0.6 0.8 1.0 0.70 0.71 0.72 SWL Wavelength (angstroms) K1 K 2 o 0.709A 20 2.63 17.37 kev, 0.714A o 143

Intensity of the Characteristic Spectrum from the Target V K I Bi( V V where I intensity i tube current V applied voltage voltage to eject K-shell electrons n constant 1.6 K ) n 144

10 6 10 5 Excitation energy increases with atomic number of the target. D. Brandon and W.D. Kaplan, Microstructural Characterization of Materials, Wiley, New York (1999), p. 73. E K,L,M (ev) 10 4 10 3 10 2 K L M 10 1 1 10 100 Atomic Number (Z) Variation of excitation energy to eject electrons from inner shells with atomic number. 145

Commonly Used X-ray Wavelengths (THIS IS IMPORTANT!) Copper Bearden Holzer et al. Cobalt Bearden Holzer et al. Anodes (1967) (1997) Anodes (1967) (1997) Cu K α1 1.54056Å 1.540598 Å Co K α1 1.788965Å 1.789010 Å Cu K α2 1.54439Å 1.544426 Å Co K α2 1.792850Å 1.792900 Å Cu K β1 1.39220Å 1.392250 Å Co K β1 1.62079Å 1.620830 Å Molybdenum Bearden Holzer et al. Chromium Bearden Holzer et al. Anodes (1967) (1997) Anodes (1967) (1997) Mo K α1 0.709300Å 0.709319 Å Cr K α1 2.28970Å 2.289760 Å Mo K α2 0.713590Å 0.713609 Å Cr K α2 2.293606Å 2.293663 Å Mo K β1 0.632288Å 0.632305 Å Cr K β1 2.08487Å 2.084920 Å Values from Cullity (1956) and Bearden [Rev. Mod. Phys. 39 (1967) 78-124] usually quoted. However, they are incorrect. Bearden s values (1967) are reprinted in International Tables for X-Ray Crystallography and most XRD textbooks. The most recent values are from Hölzer et al. [Phys. Rev. A 56 (1997) 4554-4568]. Adapted from the viewgraphs of Scott A Speakman, Ph.D., http://prism.mit.edu/xray 146

What happens when x-rays encounter matter? Adapted from Fig. 4-7 in B.D. Cullity and S.R. Stock, Elements of X-ray Diffraction, 3 rd Edition, Prentice Hall, Upper Saddle River, NJ, 2001 x Fluorescent X-rays Scattered X-rays Incident X-rays Transmitted X-rays Absorbing material Heat Electrons Absorbed X-rays Thompson unmodified (coherent) Compton modified (incoherent) Compton recoil electrons Auger electrons Photoelectrons X-rays are attenuated Part of incident X-ray energy is lost as they travel through a material 147

Absorption Modes of X-ray Attenuation Caused by electronic transitions within an atom. X-rays transfer their energy to the material, requiring re-emission of a new X-ray (just like electron beams). Scattering X-ray is deflected from its original path with or without energy loss. Scattering occurs in all directions; thus, energy in the scattered beams does not appear in the transmitted beam. 148

Nice explanation of absorption & fluorescence http://www.youtube.com/watch?v=1v3bmo1wdqe 149

Absorption - 1 When a photon comes in, is destroyed, and an electron is excited from a lower energy level to a higher energy level. hν o L 3 L 2 L 1 e - Incident X-ray Transmitted X-ray Incident X-ray Photon from X-ray source (characteristic) K N in Absorber dx N out Absorption of a photon C k 3 conc. of electrons/cm that will be excited by x-rays const. of proportionality related to probability of excitement 150

Absorption - 2 When a photon comes in, is destroyed, and an electron is excited from a lower energy level to a higher energy level. hν o L 3 L 2 L 1 e - Incident X-ray Transmitted X-ray Incident X-ray Photon from X-ray source (characteristic) K N in Absorber dx N out Absorption of a photon Let N probability of absorption Ck # photons # electrons/gram dv dv volume of absorber 11dx density of absorber 151

Absorption - 3 Incident X-ray Transmitted X-ray N in Absorber dx N out Nin # photons entering absorber N Nout # photons exiting absorber N Ndx N Nout Nin N Ndx N Ndx dx 0; N dn dn Ndx dn x N N e dx Says how # photons changes as they pass through an absorber. Assumes α is proportional to N and dx. 152

Absorption - 4 Now consider an absorbing material with a proportionality constant α = μ. I o How does I f depend on I o? I I f x x I f e Io How can we use this? Incident X-ray Absorber dx ΔI Transmitted X-ray Recall, inside of absorber: F(, ) concentration, C I f x e Io I f ln, x I o Example: We can use this principle to ID a material. Expose it to radiation with a variety of λ s; to determine μ, and conc. 153

Absorption - 5 I o Change in X-ray intensity caused by absorption I I f x di I dx Incident X-ray Absorber x Transmitted X-ray linear absorption coeff. It is a function of and. ΔI a portion of the incident beam is removed 154

I I o di I dx ln I xc at x0, I I, thus C lni I( x) I e o x o (Beer s law) If a beam of X-rays travels a distance x=1/μ through a solid, intensity will be decreased to 36.78% of its original value. Since depends upon and, we can re-write this relationship as: I I e o ( / ) x / is a constant known as the mass absorption coefficient. There are tabulated / values for each element. 155

For alloys or compounds w w w 1 2 n 1 2 n where: w x = weight fractions of elements (/) x = elemental mass absorption coefficients ALL MATERIALS ABSORB 156

Energy (erg) [ 10-8 ] 4 3 2 1 0 Wavelength dependence of / Ni Binding energy for inner shell electrons k Z 3 3 If X-ray photon energy is high enough to eject an inner shell electron, μ/ρ will suddenly increase. / (cm 2 /g) 400 300 200 100 0 k Z 3 3 K 0 0.5 1.0 1.5 2.0 Wavelength (Å) K absorption edge k Z 3 3 This is the absorption edge UP TO THE ABS. EDGE Long λ (low E) X-rays are easily absorbed Short λ (high E) X-rays penetrate deeply and are not easily absorbed 157

There are many absorption edges Absorption edges correspond to the binding energies for different types of inner shell electrons k Z 3 3 B.D. Cullity and S.R. Stock, Elements of X-ray Diffraction, 3 rd Edition, Prentice Hall, Upper Saddle River, NJ, 2001 158

Significance of Absorption Absorption limits the sample thickness and x-ray wavelengths that can be used. Example: Analysis of Fe E FeK = 7.109 kev ( = 1.74 Å) CuKα X-rays ( = 1.54 Å): μ/ρ is too high CoKα X-rays ( = 1.79 Å): μ/ρ is lower; X-rays absorbed less Absorption removes part of the incident beam / (cm 2 /g) 1.00 Fe k Z CuKα 1.54 3 3 Fe-K absorption edge 1.74 1.25 1.50 1.75 2.00 Wavelength (Å) CoKα 1.79 159

Good aspects/uses of absorption X-ray Filters We can conduct XRD experiments without adversely influencing our health by using X-ray absorbing shields (e.g., lead aprons, leaded glass, etc.). We can also utilize absorption to improve our XRD experiments Filtering of unwanted X-rays. 160

NO FILTER FILTER 70 60 Copper (schematic) CuK α 1.54 Ǻ 70 60 Copper (schematic) Ni 1.49 Ǻ Intensity (rel. units) 50 40 30 Intensity (rel. units) 50 40 30 CuK α 1.54 Ǻ NiK α absorption edge 20 20 10 CuK β 1.38 Ǻ 10 CuK β 1.38 Ǻ 0 0 SWL Wavelength (angstroms) SWL Wavelength (angstroms) In this case, the Ni filter primarily absorbs the K β X-rays. It absorbs less K α X-rays leaving us one kind of X-rays for analysis. 161

General Rules for Filtering If we place a thin foil with atomic number (Z-1) in a beam of X-rays generated with a target of atomic number Z, the K β peak of the target radiation will be almost completely absorbed and the background intensity around the K α peak will also be attenuated (i.e., reduced). BEFORE K K 7.5 1.0 K K AFTER 500.0 1.0 162

Filters for common target metals Table 1. Filters to Supress Kβ Radiation. Filter thickness for a I( K) 500 I( K ) 1 Incident Beam* in trans. beam I( K) I( K) trans Target 2 Filter I( K) mg/cm ; inches I( K) incident Mo Zr 5.4 77 ; 0.0046 0.29 Cu Ni 7.5 18 ; 0.0008 0.42 Co Fe 9.4 14 ; 0.0007 0.46 Fe Mn 9.0 12 ; 0.0007 0.48 Cr V 8.5 10 ; 0.0006 0.49 * Intensity ratio at the target [1]. This ratio outside the X-ray tube will be changed somewhat by the differential absorption of K and K by the tube window. [1] International Tables for Crystallography, Ed. A.J.C. Wilson, Vol. A-C (Kluwer, 1995) Adapted from B.D. Cullity and S.R. Stock, Elements of X-ray Diffraction, 3 rd Edition, Prentice Hall, Upper Saddle River, NJ, 2001 163

Bad aspects/uses of absorption X-ray Fluorescence Short λ, high energy x-rays can generate secondary characteristic x-rays within the specimen/target. (Process is identical to how electrons generate x-rays) Energy loss produces fluorescent X-rays with lower energies and longer λ than the characteristic radiation. L 3 L 2 e - L 3 L 2 Characteristic secondary X-rays L 1 L 1 hν o hν 2 Incident X-ray Photon from X-ray source (characteristic) K K Fluorescence hν 3 Absorption of a photon Relaxation and Emission of a photon 164

X-ray Fluorescence cont d Fluorescent x-rays radiate in all directions. These X-rays will be out of phase with incident x-rays; thus they will not reinforce the diffracted characteristic x-rays. Fluorescent radiation produces an undesired distributed background, which can be large enough to obscure the diffraction effects that you are after. 165

X-ray Fluorescence cont d Most of the loss in X-ray intensity results from fluorescent absorption. Want to avoid fluorescence, minimize absorption of the incident beam. Select X-radiation for with close to minimum, but on the long side (i.e., higher ) of the absorption edge. 166

For analysis of steel: Example CuK x-radiation (=1.54 Å) is less than ideal for steels and other iron alloys (E FeK =7.109 kev 1.743 Å). CoK x-radiation (=1.789 Å) is just on the long side of the K Fe edge and will yield sharp fluorescence-free diffraction patterns from steel. What about MoK x-radiation? 167

Start reading Chapters 2 and 6 in Waseda. 168