X-ray Diffraction

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 – 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) hp://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 1901, 1914, 1915 The Discovery of X-Ray Diffracon by Crystals An der Ludwig-Maximilians-Universität München im Januar 1912:

Paul Peter Ewald, a Ph.D. candidate in ’s Instute of Theorecal Physics working on a thesis concerned with visible light refracon 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 Instute: If X-rays are waves they should have wavelengths of the order of 10-8 cm, as had been esmated in Wilhelm Röntgen’s Instute of Experimental Physics, and they should manifest diffracon effects with crystals.

a(cosν1 − cosµ1 ) = hλ

b(cosν 2 − cosµ2 ) = kλ c(cosν 3 − cosµ3 ) = l λ 75 Years of X-Ray Diffracon 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 diffracon by a crystal

hp://www.chem.ufl.edu/~itl/2045/maer/FG11_039.GIF Crookes and Coolidge X-Ray Tubes

hp://en.wikipedia.org/wiki/X-ray_tube Generaon of X-rays kV acceleraon, mA current

hp://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 scaering in a crystal

Roger Pynn, Neutron Scaering: A Primer. Physics Today, Special Supplement, Jan. 1985. hp://la-science.lanl.gov/lascience19.shtml Neutron, X-ray, and electron scaering

Roger Pynn, Neutron Scaering: A Primer. Physics Today, Special Supplement, Jan. 1985. hp://la-science.lanl.gov/lascience19.shtml Some Elementary Opcs 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 transmied in straight lines through the two holes. This is proved as oen as the experiment is tried by observing how great the base IK is in fact and deducing by calculaon how great the base NO ought to be which is formed by the direct rays. Further it should not be omied 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)

“diffracon” from Lan diffringere “to break into pieces”

monochromac light

hp://www.faculty.fairfield.edu/jmac/sj/sciensts/grimaldi.htm hp://en.wikipedia.org/wiki/Diffracon Single Slit Diffracon

monochromac

hp://media-2.web.britannica.com/eb-media/97/96597-004-4602C228.jpg Double Slit Diffracon

monochromac

hp://media-2.web.britannica.com/eb-media/96/96596-004-1D8E9F0F.jpg Fraunhofer single and double slit diffraction

hp://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/imgpho/doubsli.gif

� = 700 nm

� = 400 nm

Wave/Parcle 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

hp://www.answers.com/topic/photon-2 hp://www.windows2universe.org/physical_science/magnesm/photon.html Huygens’ wavelets principle

Huygens (1678). Traité de la Lumière.

Every point on a propagang wavefront acts as the source of a spherical secondary wavelet that has the same wavelength and speed of propagaon as the primary wavefront, such that at some later me the propagang wavefront is the envelope surface tangent to the secondary wavelets. Diffraction according to Huygens’ wavelets principle 1st order diffracon

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

hp://en.wikipedia.org/wiki/Diffracon 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 Moon Sinusoidal surface wave

Ripples from a pebble dropped into a sll pond Construcve and destrucve interference of water surface waves

Near Cook Strait, South Island, NZ

hps://www.flickr.com/photos/brewbooks/309494512/in/photostream/ Wave Interference and Diffracon

−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° hp://physics-animaons.com/Physics/English/waves.htm Sinusoidal traveling waves

Transverse wave Longitudinal wave

Water wave hp://physics-animaons.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 Opcs. 3rd ed. : McGraw-Hill Book Co., Inc. Plane-wave wavefronts Wavefronts are surfaces of constant phase.

hp://en.wikipedia.org/wiki/Plane_wave hp://gemologyproject.com/wiki/index.php?tle=Image:Wavefront.png Projecon and cross-secon views of a sinusoidal plane wave

crest crest trough trough

hp://www.school-for-champions.com/science/waves.htm Incident plane wave Scaered spherical waves

Physics Today, Special Supplement, Jan. 1985.

hp://la-science.lanl.gov/lascience19.shtml

Propagaon 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 Opcs. 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 funcon 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 properes

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 Electromagnec 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 Superposion and interference of sinusoidal transverse waves of equal wavelength

hp://media-2.web.britannica.com/eb-media/95/96595-004-16C2DCAD.gif Superposion 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 superposion 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 Electromagnec Wave Nature of X-Rays Electromagnec Waves

hp://hyperphysics.phy-astr.gsu.edu/hbase/HFrame.html hp://hyperphysics.phy-astr.gsu.edu/hbase/HFrame.html Electric field E at a point at r from a charge q that experiences an acceleraon 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 radiaon from an accelerated charge

Figure copied from Compton and Allison (1935). X-Rays in Theory and Experiment. All the light in the universe is electromagnec radiaon generated by accelerated moon of electrical charges. em radiaon pulse from a momentarily accelerated charge

r ≥ rRR′ = rA = c(tAB + tBC )

r ≤ rQQ′ = rB = ctBC em radiaon wave from a harmonically oscillang charge r ≥ rRR′ = rA = c(tAB + tBC )

r ≤ rQQ′ = rB = ctBC

em pulse from a momentary charge acceleraon

em wave from a sinusoidal charge oscillaon em radiaon pulse from an accelerated charge

Ÿ At rest at t0 Ÿ Accelerated from t0 to t1 Ÿ 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 radiaon from an accelerated charge

Maximum radiaon perpendicular to the acceleraon direcon. Zero radiaon in the acceleraon direcon Electromagnec 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

hp://physics-animaons.com/Physics/English/down.htm hp://www.youtube.com/watch?v=4CtnUETLIFs Electromagnec wave - side view

Ex = E0 cos(ωt − kz) Ey = 0 Ez = 0

Bx = 0 By = cB0 cos(ωt − kz) Bz = 0

hp://en.wikipedia.org/w/index.php?tle=File%3AElectromagnecwave3Dfromside.gif Electromagnec wave - 3D view

Ex = E0 cos(ωt − kz) Ey = 0 Ez = 0

Bx = 0 By = cB0 cos(ωt − kz) Bz = 0 Z

hp://en.wikipedia.org/w/index.php?tle=File%3AElectromagnecwave3D.gif Sinusoidal electric field oscillaons induce in-phase magnec field oscillaons, and vice versa, and propagate electromagnec radiaon.

hp://www.monos.leidenuniv.nl/smo/basics/images/wave.gif hp://www.opcs.arizona.edu/Wright/images/wave_anim.gif Electromagnec 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 hp://www.opcs.arizona.edu/Wright/images/wave_anim.gif The oscillaons of the electric and magnec fields in an electromagnec 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 electromagnec ⎧ω = 2πν = 2π c λ inducon ⎨ 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 hp://www.opcs.arizona.edu/Wright/images/wave_anim.gif Walter Kauzmann (1957). Quantum : An Introducon. New York: Academic Press. e and m wave fields from an electric dipole oscillator

For a vercal oscillaon, horizontal radiaon is maximal. hp://physics.stackexchange.com/quesons/20331/understanding-the-diagrams-of-electromagnec-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 illustraon of the electric component of a monochromac, linearly polarized, em plane wave

E

B

c

hp://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 Opcs. 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 oscillaons will be three dimensional. Waves propagang from monopole and dipole oscillators

hp://www.acs.psu.edu/drussell/demos/rad2/mdq.html em waves from an electric dipole oscillator

For a vercal oscillaon, horizontal radiaon is maximal. hp://www.ryerson.ca/~kantorek/ELE884/EMdipole.gif Sinusoidally oscillang lines of force in the electric field of an oscillang electric charge

For vercal oscillaon, horizontal radiaon is maximal. Oscillang electric field around an oscillang electric dipole

hp://www.physics.upenn.edu/courses/gladney/phys151/lectures/lecture_apr_07_2003.shtml em wavesOscillang electric dipole radiator from an electric dipole oscillator

For a vercal oscillaon, horizontal radiaon is maximal.

hp://en.wikipedia.org/wiki/File:DipoleRadiaon.gif em waves from an electric dipole oscillator

For a vercal oscillaon, horizontal radiaon is maximal. http://commons.wikimedia.org/wiki/File:Dipole.gif e and m wave fields from an electric dipole oscillator

For a vercal oscillaon, horizontal radiaon is maximal. hp://physics.stackexchange.com/quesons/20331/understanding-the-diagrams-of-electromagnec-waves Electromagnec radiaon from an electric dipole oscillator ! E! B

! E ! ! c c

Oscillang em fields E and B propagang with velocity c hp://www.physics.upenn.edu/courses/gladney/phys151/lectures/lecture_apr_07_2003.shtml e and m waves from an electric dipole oscillator

hp://it.stlawu.edu/~jahncke/clj/cls/104/104Links.html X-Ray Scaering, Interference, and Diffracon Electromagnec radiaon scaering

IR-vis-UV X-ray wavelengths wavelengths

⎧ coherent, Rayleigh Thomson ⎪ elastic scattering scattering em radiation ⎪ } scattering ⎨ ⎪ incoherent, Raman Compton scattering ⎩⎪ inelastic } scattering

Thomson scaering of an X-ray wave via driven oscillaon 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°

Elasc scaering of an electromagnec wave

hp://www.gly.uga.edu/schroeder/geol3010/3010lecture09.html Compton scaering of an X-ray photon by an electron

Inelasc photon-electron parcle collision hp://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 absorpon ~0.15% inelasc, Compton scaering ~0.15% elasc, Laue-Bragg scaering

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)