X-ray Diffraction Crystallography
κρυσταλλος krustallos “hard ice”
low quartz (α-SiO2) Crystals • plane faces • straight line edges • point vertices
• constant interfacial angles • rational intercepts
• 3-D periodic internal lattice structure κρυσταλλος krustallos “clear ice”
low quartz (α-SiO2) A Chronology of Crystallography Classical antiquity Greco-Roman thinkers – Nature of matter, polyhedral geometry, κρυσταλλος 1611 Johannes Kepler – Hexagonal snow crystals, hcp and ccp spheres 1600s René Descartes, Robert Hooke, Christiaan Huygens – Speculations on periodic spheroid packing in crystals 1669 Nicolaus Steno (Niels Stensen), 1688 Domenico Gugliemini, and 1772 Jean-Baptiste Louis Romé de l’Isle – Law of Constant Interfacial Angles 1783 Abbé René-Just Haüy – Law of Rational Indices, “molécules intégrantes,” unit cells 1839 William Hallowes Miller – stereographic projection, Miller indices 1849 Auguste Bravais – Lattice theory 1883 William J. Pope and William Barlow – Speculations on atomic and ionic sphere-packing in crystals. 1890 Evgraf Stepanovich Federov, 1892 Arthur Moritz Schoenflies, and 1894 William Barlow (all three independently) – Space group theory 1883 Paul Heinrich Ritter von Groth – Chemical and optical crystallography 1895 Wilhelm Conrad Röntgen – X-rays 1912 Walther Friedrich, Paul Knipping, and Max von Laue – X-ray diffraction 1913 William Henry and William Lawrence Bragg – X-ray crystal structures Discovery of X-Rays 1895
Wilhelm Konrad Röntgen 1845-1923 Early X-ray photographs Anna Bertha Ludwig Röntgen (above) Albert von Kölliker (below) h p://en.wikipedia.org/wiki/Wilhelm_R%C3%B6ntgen Founding Fathers of X-ray Crystallography
Röntgen Laue The Braggs
Discovery Discovery of X-ray First X-ray of X-rays diffraction by crystals crystal structures 1895 1912 1913
Nobel Laureates in Physics 1901, 1914, 1915 The Discovery of X-Ray Diffrac on by Crystals An der Ludwig-Maximilians-Universität München im Januar 1912:
Paul Peter Ewald, a Ph.D. candidate in Arnold Sommerfeld’s Ins tute of Theore cal Physics working on a thesis concerned with visible light refrac on by crystals:
If crystals consist of regular, periodic arrangements of atoms, the interatomic spacings and unit cell dimensions should be of the order of 10-8 cm. 3 M r mu acell = NA ρm
Max Von Laue, a professor in Sommerfeld’s Ins tute: If X-rays are waves they should have wavelengths of the order of 10-8 cm, as had been es mated in Wilhelm Röntgen’s Ins tute of Experimental Physics, and they should manifest diffrac on effects with crystals.
a(cosν1 − cosµ1 ) = hλ
b(cosν 2 − cosµ2 ) = kλ c(cosν 3 − cosµ3 ) = l λ 75 Years of X-Ray Diffrac on and Crystal Structure Analysis (1987)
ZnS zinc blende sphalerite
CuSO4•5H2O blue vitriol chalcanthite ZnS zinc blende sphalerite
CuSO4•5H2O blue vitriol chalcanthite The Laue, Friedrich, and Knipping XRD aparatus Walther Friedrich, Paul Knipping, and Max von Laue (1912)
X-ray diffraction from CuSO4•5H2O crystals
(a) The first single-crystal X-ray diffraction pattern (b) Pulverized sample crystal (c) Longer crystal-to-film distance
(d) Shorter crystal-to-film distance
Figure copied from W.L. Bragg, The Development of X-ray Analysis, 1972. X-ray diffrac on by a crystal
h p://www.chem.ufl.edu/~itl/2045/ma er/FG11_039.GIF Crookes and Coolidge X-Ray Tubes
h p://en.wikipedia.org/wiki/X-ray_tube Genera on of X-rays kV accelera on, mA current
h p://pubs.usgs.gov/of/2001/of01-041/htmldocs/images/xrdtube.jpg Crystallographic Diffraction
Laue diffraction from a three-dimensional abc lattice grating
ˆ ˆ a(cos!1 " cosµ1 ) = a i(s " s0 ) = h#
b (cos! 2 " cosµ2 ) = b i(sˆ " sˆ0 ) = k#
c (cos! 3 " cosµ3 ) = ci(sˆ " sˆ0 ) = l #
Max von Laue, Walther Friedrich, and Paul Knipping (1912).
Bragg reflection from families of parallel hkl lattice planes
# d & 2d sin! = n" , 2 hkl sin! = " , 2d sin! = " hkl $% n '( nhnk nl
William Henry and William Lawrence Bragg (1913). (Father and son)
Integrated Bragg reflection intensities
2 2 2 E" 2 # e & 3 # vxtal & , *µ(t0+t1) / 1 # 1 1 2 & 2 ! = = kAL p Fhkl = 2 ) e dv % + cos 22( Fhkl % ( % ( .+vxtal 1 I0 $ mc ' $ Vcell ' - 0 sin22 $ 2 2 '
Charles G. Darwin (1914). (Grandson of the author of the theory of evolution) Crystallographic Diffraction
X-Rays Electromagnetic radiation beam hc ⎧λ = 1Å Ultra-short wavelength light E = hν = , ⎨ λ ⎩E = 12.4 keV Scattered by atomic electron densities
X 1 1 ⎛ sinθ ⎞ ⎪⎧ 0 ≤ h < ∞ f h = F − r , h = = 2 hkl , a ( ) ⎣⎡ρa( )⎦⎤ ⎜ ⎟ ⎨ X d ⎝ λ ⎠ Z f 0 hkl ⎩⎪ a ≥ a >
Electrons Negative particle beam
h h 1 2 ⎧E ~ 100 keV deBroglie matter-waves λ = = , E = me v , ⎨ p me v 2 ⎩λ ~ 0.1Å Scattered by atomic electrostatic potentials 2 X e −1 1 ⎛ 2me e ⎞ Za − fa (h) fa (h) = F ⎡φa (r)⎤ ≈ ⎣ ⎦ 4πε ⎜ h2 ⎟ 2 0 ⎝ ⎠ h
Neutrons Neutral particle beam T ~ 300 K h h 1 2 3 ⎧ deBroglie matter-waves λ = = , E = 2 mn v = 2 kBT , ⎨ p mn v ⎩λ ~ 1.5 Å Scattered by atomic nuclei
(point scatterers ⇔ -independent scattering lengths)
−1 b c, a = F ⎡ρa(r)⎤ b c ≈ 5 fm bc, H = −3.74 fm b c, D = +6.67 fm ⎣ ⎦ b i, H = 25.3 b i, D = 4.04 Neutron, X-ray, and electron sca ering in a crystal
Roger Pynn, Neutron Sca ering: A Primer. Physics Today, Special Supplement, Jan. 1985. h p://la-science.lanl.gov/lascience19.shtml Neutron, X-ray, and electron sca ering
Roger Pynn, Neutron Sca ering: A Primer. Physics Today, Special Supplement, Jan. 1985. h p://la-science.lanl.gov/lascience19.shtml Some Elementary Op cs Principles Wave Interference and Diffraction “When the light is incident on a smooth white surface it will show an illuminated base IK notably greater than the rays would make which are transmi ed in straight lines through the two holes. This is proved as o en as the experiment is tried by observing how great the base IK is in fact and deducing by calcula on how great the base NO ought to be which is formed by the direct rays. Further it should not be omi ed that the illuminated base IK appears in the middle suffused with pure light, and at either extremity its light is colored.” Francesco Maria Grimaldi, S.J. (1613-1665)
“diffrac on” from La n diffringere “to break into pieces”
monochroma c light
h p://www.faculty.fairfield.edu/jmac/sj/scien sts/grimaldi.htm h p://en.wikipedia.org/wiki/Diffrac on Single Slit Diffrac on
monochroma c
h p://media-2.web.britannica.com/eb-media/97/96597-004-4602C228.jpg Double Slit Diffrac on
monochroma c
h p://media-2.web.britannica.com/eb-media/96/96596-004-1D8E9F0F.jpg Fraunhofer single and double slit diffraction
h p://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/imgpho/doubsli.gif
� = 700 nm
� = 400 nm
Wave/Par cle Duality light wave energies photon momenta hc h m v E = !ω = hν = p = !k = = 0 λ λ v2 1− 2 c
photon wave packets
h p://www.answers.com/topic/photon-2 h p://www.windows2universe.org/physical_science/magne sm/photon.html Huygens’ wavelets principle
Huygens (1678). Traité de la Lumière.
Every point on a propaga ng wavefront acts as the source of a spherical secondary wavelet that has the same wavelength and speed of propaga on as the primary wavefront, such that at some later me the propaga ng wavefront is the envelope surface tangent to the secondary wavelets. Diffraction according to Huygens’ wavelets principle 1st order diffrac on
AA’ is tangent to wavefronts of secondary wavelets from adjacent sources that differ by one wavelength.
http://www.britannica.com/ebc/art-3156 Thomas Young’s drawing to explain the results of his two-slit interference experiment in 1803
h p://en.wikipedia.org/wiki/Diffrac on Double Slit Interference of Light Waves
http://physics.weber.edu/carroll/honors-time/doubleslit.htm Huygens-Fresnel construction for Young’s two-slit experiment
d ≈2.5 λ
Constructive interference in the directions with s i n θ m = m λ d http://commons.wikimedia.org/wiki/Image:Two-Slit_Diffraction.png Fraunhofer (far field) diffraction condition
π −θ 2
bright d sinθ = mλ ⎪⎫ 1 ⎬ m = 0, ±1, ± 2,... dark d sin m θ = ( + 2 )λ ⎭⎪ Path length difference Δ L at scattering angle θ
π −θ 2
ΔL = L − L′ = d sinθ
⎧ nλ ⇒ constructive interference π ⎪ − θ ΔL = 1 2 ⎨ ⎛ ⎞ ⎪ ⎜ n + ⎟ λ ⇒ destructive interference ⎩ ⎝ 2⎠
Some Elementary Principles of Wave Mo on Sinusoidal surface wave
Ripples from a pebble dropped into a s ll pond Construc ve and destruc ve interference of water surface waves
Near Cook Strait, South Island, NZ
h ps://www.flickr.com/photos/brewbooks/309494512/in/photostream/ Wave Interference and Diffrac on
−1 θ2 = sin (2λ d) sinθn = nλ d −1 d ≈ 3.5λ θ1 = sin (λ d)
sinθ1 ≈ 0.29 θ0 = 0 θ1 ≈ 16°
sinθ2 ≈ 0.57
θ2 ≈ 35° h p://physics-anima ons.com/Physics/English/waves.htm Sinusoidal traveling waves
Transverse wave Longitudinal wave
Water wave h p://physics-anima ons.com/Physics/English/waves.htm Sinusoidal surface waves and wave interference
Near Cook Strait, South Island, NZ Plane waves passing through slits
http://www.physics.gatech.edu/gcuoUltrafastOptics/OpticsI/lectures/OpticsI-20-Diffraction-I.ppt Wave interference near the Strait of Gibraltar Transverse wave in a homogeneous medium
λ
Spherical wave Plane wave near field far field amplitude 1/ R constant amplitude
∝ 2 intensity amplitude intensity = energy area−1 me−1 2 intensity 1/∝ R ∝ F.A. Jenkins and H.E. White (1957). Fundamentals of Op cs. 3rd ed. New York: McGraw-Hill Book Co., Inc. Plane-wave wavefronts Wavefronts are surfaces of constant phase.
h p://en.wikipedia.org/wiki/Plane_wave h p://gemologyproject.com/wiki/index.php? tle=Image:Wavefront.png Projec on and cross-sec on views of a sinusoidal plane wave
crest crest trough trough
h p://www.school-for-champions.com/science/waves.htm Incident plane wave Sca ered spherical waves
Physics Today, Special Supplement, Jan. 1985.
h p://la-science.lanl.gov/lascience19.shtml
Propaga on of a water surface wave
Δx c = Δt 2
λ
y1 = f (x − ct) ⎫ ⎪ ′ Δx y1 = f ⎡(x + Δx) − c(t + Δt)⎤ ⎬ c = ⇒ Δx = c Δt ⎣ ⎦ Δt = f ⎡ x − ct + Δx − cΔt ⎤ = y ⎪ ⎣( ) ( )⎦ 1 ⎭ ∂y ∂ f ∂u ∂u The composite function chain rule: y = f (u(x)) ⇒ = = f ′(u) ∂x ∂u ∂x ∂x ∂y ∂y ⎫ = f ′(x − ct) , = −c f ′(x − ct) ∂x ∂t ⎪ !##"##$ !###"###$ ⎪ 2 2 Slope at x Velocity at t ⎪ ∂ y 1 ∂ y ⎬ = 2 2 x2 c2 t 2 ∂ y ∂ y 2 ⎪ !∂ ##"#∂#$ 2 = f ′′(x − ct) , 2 = c f ′′(x − ct) The wave equation !∂x##"##$ !∂t###"###$ ⎪ Curvature at x Acceleration at t ⎭⎪ F.A. Jenkins and H.E. White (1957). Fundamentals of Op cs. 3rd ed. New York: McGraw-Hill Book Co., Inc. From a wave function to the wave equation
Given a general form for the wave function for a traveling wave, y = f (u(x, t)) , u = (x − ct) , application of the chain rule for differentiation gives ⎧ ∂y d f ∂u ∂u = = f ′ u = f ′ x − ct ⎪ ∂x du ∂x ( ) ∂x ( ) ⎨ ∂y d f ∂u ∂u ⎪ = = f ′ u = −c f ′ x − ct ⎩ ∂t du ∂t ( ) ∂t ( ) ⎧ ∂2 y ∂u ⎪ 2 = f ′′(u) = f ′′(x − ct) ⎨ ∂x ∂x ⎩⎪ and thence the wave equation, ∂2 y 1 ∂2 y = . ∂x2 c2 ∂t 2
The dispersion relations for a wave function Dispersion relations describe the interrelationship of the wave properties: wavelength �, period �, speed of propagation c, frequency �, and wavenumber k. y = f (x, t) = f (x − ct)
c = λ τ ω = 2πv = 2π τ = 2π c λ k = 2πσ = 2π λ = 2π c τ λ ⎛ 2π ⎞ ⎛ ω ⎞ ω c = = λν = ⎜ ⎟ ⎜ ⎟ = τ ⎝ k ⎠ ⎝ 2π ⎠ k
y = f (x − ct) = f ⎣⎡x − (ω k)t ⎦⎤ = f ⎣⎡(kx −ωt) k⎦⎤ Sinusoidal wave func on ψ =ψ (t, r)
i t k r ⎧ 2π t0 τ = ωt0 ⎡ (ω − i −ϕ0 ) ⎤ ψ = Re ψ 0e =ψ 0 cos(ωt − k i r −ϕ0 ) , ϕ0 = ⎨ ⎣ ⎦ 2 r kr ⎩ π 0 λ = 0 Wave proper es
i t k r ⎧ 2π t0 τ = ωt0 ⎡ (ω − i −ϕ0 ) ⎤ ψ = Re ψ 0 e =ψ 0 cos(ωt − k i r −ϕ0 ) , ϕ0 = ⎨ ⎣ ⎦ 2π r λ = kr ⎩ 0 0 amplitude ψ 0 =ψ max(t, r) maximum wave displacement phase ϕ = ϕ t, r (rad) argument of the 2�-periodic sinusoidal ( ) π ψ = ψ cosϕ = ψ sin⎛ − ϕ ⎞ wave function 0 0 ⎝ 2 ⎠ wavelength λ = cτ L distance crest-to-crest and trough-to-trough period τ = λ c T time per wave cycle
–1 speed c = λ τ = λν L T speed of wave propagation frequency ν = 1 τ = c λ T –1 wave cycles per unit time
–1 wavenumber σ = 1 λ L wave cycles per unit distance angular frequency ω = 2π τ = 2πν (rad) T –1 phase angle cycles per unit time angular wavenumber k = 2π λ = 2πσ (rad) L–1 phase angle cycles per unit distance Electromagne c wave in space- me
E = Re ⎡E ei(ωt−kir−ϕ0 ) ⎤ = E cos ωt − k i r −ϕ ⎣ 0 ⎦ 0 ( 0 ) π ϕ = 0 2
Jens-Als Nielsen & Des McMorrow (2001). Elements of Modern X-Ray Physics. New York: John Wiley & Sons. Superposition of two equal-frequency, equal-amplitude waves
Constructive interference of waves almost in phase
Destructive interference of waves almost 180° out of phase
http://www.phy.ntnu.edu.tw/ntnujava/index.php Superposition of two equal-frequency, unequal-amplitude waves
in phase 180º out of phase crest on crest crest on trough and and trough on trough trough on crest
http://scienceworld.wolfram.com/physics/ConstructiveInterference.html Superposition of two equal-frequency, equal-amplitude waves Superposi on and interference of sinusoidal transverse waves of equal wavelength
h p://media-2.web.britannica.com/eb-media/95/96595-004-16C2DCAD.gif Superposi on of any number of equal-wavelength sinusoidal (sine and/or cosine) waves gives a sinusoidal resultant wave.
Charles W. Bunn (1964). Crystals: Their Role in Nature and in Science. New York: Academic Press A superposi on of three equal-wavelength, equal-amplitude sinusoidal waves with different phases
a case in which \\\\\/
Henry S. Lipson (1970). Crystals and X-Rays. London: Wykeham Publications. Picturing the Electromagne c Wave Nature of X-Rays Electromagne c Waves
h p://hyperphysics.phy-astr.gsu.edu/hbase/HFrame.html h p://hyperphysics.phy-astr.gsu.edu/hbase/HFrame.html Electric field E at a point at r from a charge q that experiences an accelera on a z
a r a = rˆ × a × rˆ E ⊥ ( ) α
q y
x q r E = − (rˆ × a) × rˆ , rˆ = , r = r c2r r q E = − asinα , E = E , a = a c2r em radia on from an accelerated charge
Figure copied from Compton and Allison (1935). X-Rays in Theory and Experiment. All the light in the universe is electromagne c radia on generated by accelerated mo on of electrical charges. em radia on pulse from a momentarily accelerated charge
r ≥ rRR′ = rA = c(tAB + tBC )
r ≤ rQQ′ = rB = ctBC em radia on wave from a harmonically oscilla ng charge r ≥ rRR′ = rA = c(tAB + tBC )
r ≤ rQQ′ = rB = ctBC
em pulse from a momentary charge accelera on
em wave from a sinusoidal charge oscilla on em radia on pulse from an accelerated charge
At rest at t0 Accelerated from t0 to t1 Constant velocity from t1 to t2
c(t1 − t0 )
c(t2 − t0 )
R.M. Eisberg (1963). Fundamentals of Modern Physics. New York: John Wiley and Sons, Inc. em radia on from an accelerated charge
Maximum radia on perpendicular to the accelera on direc on. Zero radia on in the accelera on direc on Electromagne c Waves
x
z E0 = c H0 y
Ex = E0 cos(ωt − kz) Ey = 0 Ez = 0
H x = 0 H y = cH 0 cos(ωt − kz) H z = 0
2π c ω hc h ω = 2πν, k = ν = , c = λν = Eγ = %ω = hν = , pγ = %k = !######λ#"##λ #####$k !######"λ######$λ classical wave properties Plank-Einstein-deBroglie wave-particle properties
h p://physics-anima ons.com/Physics/English/down.htm h p://www.youtube.com/watch?v=4CtnUETLIFs Electromagne c wave - side view
Ex = E0 cos(ωt − kz) Ey = 0 Ez = 0
Bx = 0 By = cB0 cos(ωt − kz) Bz = 0
h p://en.wikipedia.org/w/index.php? tle=File%3AElectromagne cwave3Dfromside.gif Electromagne c wave - 3D view
Ex = E0 cos(ωt − kz) Ey = 0 Ez = 0
Bx = 0 By = cB0 cos(ωt − kz) Bz = 0 Z
h p://en.wikipedia.org/w/index.php? tle=File%3AElectromagne cwave3D.gif Sinusoidal electric field oscilla ons induce in-phase magne c field oscilla ons, and vice versa, and propagate electromagne c radia on.
h p://www.monos.leidenuniv.nl/smo/basics/images/wave.gif h p://www.op cs.arizona.edu/Wright/images/wave_anim.gif Electromagne c Waves
E x z E c B B y 0 = 0
Ex = E0 cos(ωt − kz) Ey = 0 Ez = 0
Bx = 0 By = cB0 cos(ωt − kz) Bz = 0
2π c ω hc h ω = 2πν, k = ν = , c = λν = Eγ = %ω = hν = , pγ = %k = !######λ#"##λ #####$k !######"λ######$λ classical wave properties Plank-Einstein-deBroglie wave-particle properties h p://www.op cs.arizona.edu/Wright/images/wave_anim.gif The oscilla ons of the electric and magne c fields in an electromagne c wave mutually induce one another.
Ex (out-of-plane) ⎧ E E cos t kz • By x = 0 (ω − )
⎪ Ex ⎨ Ey = 0 ⎪ • ⎩ Ez = 0 wavefont
⎧ Bx = 0 By • ⎪ ⎨ By = cB0 cos(ωt − kz) ⎪ B = 0 Faraday • ⎩ z electromagne c ⎧ω = 2πν = 2π c λ induc on ⎨ x B • ⎩ k = 2π λ y points along a plane wave
z z ⎧ Eγ = !ω = hν = hc λ ⎨ y x p k h ⎩ γ = ! = λ y h p://www.op cs.arizona.edu/Wright/images/wave_anim.gif Walter Kauzmann (1957). Quantum Chemistry: An Introduc on. New York: Academic Press. e and m wave fields from an electric dipole oscillator
For a ver cal oscilla on, horizontal radia on is maximal. h p://physics.stackexchange.com/ques ons/20331/understanding-the-diagrams-of-electromagne c-waves A propagating em wave alters the condition of physical free space. An em plane wave can be imagined as a sinusoidal oscillation of the density or closenessof perpendicular electric-field and magnetic-field lines of force.
⎧ Ex = E0 cos(ωt − kz) ⎪ ⎨ Ey = 0 ⎪ ⎩ Ez = 0
⎧ Bx = 0 ⎪ ⎨ By = cB0 cos(ωt − kz) ⎪ ⎩ Bz = 0 ⎧ ω = 2πν = 2π c λ ⎨ ⎩ k = 2π λ
⎧ Eγ = !ω = hν = hc λ ⎨ p k h ⎩ γ = ! = λ
Figure adapted from Francis W. Sears (1958). Optics. Reading, Mass.: Addison Wesley Publishing Co. An illustra on of the electric component of a monochroma c, linearly polarized, em plane wave
E
B
c
h p://en.wikipedia.org/wiki/Plane_wave Electromagnetic Plane Wave
Oscillating electric and magnetic force fields propagating as sinusoidal waves, in-phase in time and space, and mutually perpendicular to one another and to the direction of propagation.
A monochromatic, linearly-polarized, coherent light beam illustrated as a bundle of light rays.
http://www.cs.brown.edu/stc/outrea/greenhouse/nursery/physics/emwave.html Mechanical models for 3-D electron oscillators
Spheres of uniform charge density with total charge q and mass m
anisotropic
Classical electron radius γ Electrostatic potential energy ⎫ 2 ⎪ 2 E = qφ(r) = q(q r) = e re ⎪ e ⎬ re = 2 Relativistic mass - energy mec 2 ⎪ E = mec ⎭⎪
John Strong (1958). Concepts of Classical Op cs. San Francisco: W.H. Freeman & Co. Mechanical models for 3-D oscillators
If a electron is driven to oscillate by an unpolarized em wave, the electron oscilla ons will be three dimensional. Waves propaga ng from monopole and dipole oscillators
h p://www.acs.psu.edu/drussell/demos/rad2/mdq.html em waves from an electric dipole oscillator
For a ver cal oscilla on, horizontal radia on is maximal. h p://www.ryerson.ca/~kantorek/ELE884/EMdipole.gif Sinusoidally oscilla ng lines of force in the electric field of an oscilla ng electric charge
For ver cal oscilla on, horizontal radia on is maximal. Oscilla ng electric field around an oscilla ng electric dipole
h p://www.physics.upenn.edu/courses/gladney/phys151/lectures/lecture_apr_07_2003.shtml em wavesOscilla ng electric dipole radiator from an electric dipole oscillator
For a ver cal oscilla on, horizontal radia on is maximal.
h p://en.wikipedia.org/wiki/File:DipoleRadia on.gif em waves from an electric dipole oscillator
For a ver cal oscilla on, horizontal radia on is maximal. http://commons.wikimedia.org/wiki/File:Dipole.gif e and m wave fields from an electric dipole oscillator
For a ver cal oscilla on, horizontal radia on is maximal. h p://physics.stackexchange.com/ques ons/20331/understanding-the-diagrams-of-electromagne c-waves Electromagne c radia on from an electric dipole oscillator ! E! B
! E ! ! c c
Oscilla ng em fields E and B propaga ng with velocity c h p://www.physics.upenn.edu/courses/gladney/phys151/lectures/lecture_apr_07_2003.shtml e and m waves from an electric dipole oscillator
h p://it.stlawu.edu/~jahncke/clj/cls/104/104Links.html X-Ray Sca ering, Interference, and Diffrac on Electromagne c radia on sca ering
IR-vis-UV X-ray wavelengths wavelengths
⎧ coherent, Rayleigh Thomson ⎪ elastic scattering scattering em radiation ⎪ } scattering ⎨ ⎪ incoherent, Raman Compton scattering ⎩⎪ inelastic } scattering
Thomson sca ering of an X-ray wave via driven oscilla on of an electron spherical wave spherical ⎛ e2 ⎞ E i ωt −δ / 0 e ( ) E = −⎜ 2 ⎟ plane wave plane ⎝ mc ⎠ r 2θ = + 90° / iωt E = E0e
spherical wave 2θ = − 45°
Elas c sca ering of an electromagne c wave
h p://www.gly.uga.edu/schroeder/geol3010/3010lecture09.html Compton sca ering of an X-ray photon by an electron
Inelas c photon-electron par cle collision h p://hyperphysics.phy-astr.gsu.edu/hbase/quantum/imgqua/compton.gif Interactions of � ≈ 1 Å X-rays within a typical 100 µm protein crystal
~98% transmission ~1.7% photoelectric absorp on ~0.15% inelas c, Compton sca ering ~0.15% elas c, Laue-Bragg sca ering
Murray,…, & Garman (2005). “… the X-ray dose absorbed by macromolecular crystals.” J. Synchrotron Rad. 12, 268. Crystallographic Diffraction
Laue diffraction from a three-dimensional abc lattice grating
ˆ ˆ a(cosν1 − cosµ1 ) = a i(s − s0 ) = hλ
b (cosν 2 − cosµ2 ) = b i(sˆ − sˆ0 ) = kλ
c (cosν 3 − cosµ3 ) = ci(sˆ − sˆ0 ) = l λ
Max Laue, Walther Friedrich, and Paul Knipping (1912).
Bragg reflection from families of parallel hkl lattice planes
⎛ d ⎞ 2d sinθ = nλ , 2 hkl sinθ = λ , 2d sinθ = λ hkl ⎝⎜ n ⎠⎟ nhnk nl
William Henry and William Lawrence Bragg (1913). (Father and son)
Integrated Bragg reflection intensities
2 2 2 Eω 2 ⎛ e ⎞ 3 ⎛ vxtal ⎞ ⎡ −µ(t0+t1) ⎤ 1 ⎛ 1 1 2 ⎞ 2 ρ = = kAL p Fhkl = 2 λ e dv ⎜ + cos 2θ⎟ Fhkl ⎜ ⎟ ⎜ ⎟ ⎢∫vxtal ⎥ I0 ⎝ mc ⎠ ⎝ Vcell ⎠ ⎣ ⎦ sin2θ ⎝ 2 2 ⎠
Charles G. Darwin (1914). (Grandson of the author of the theory of evolution)