Crystals- Diffraction

X-rays

electromagnetic waves with wavelength between 0.1 and 10 Angstroms

Poynting vector (+ equation)

vector describing the direction of energy projection (E x B)

Wave vector

similar to a reciprocal lattice vector, expressed in relation to the reciprocal lattice vectors of a given crystal

Wave number (equation)

k = 1/wavelength

Coherent scattering/Thompson scattering

incident photon interacts with electron with no energy loss and no phase change (elastic process)

Incoherent scattering/Compton scattering

electron absorbs incident energy, energy is emitted at different energy and different phase

X-ray generation

bombardment of a metal target with high energy electrons (over 35 keV)

White radiation (continuous background)

a continuous wavelength spectrum of x-rays with the minimum wavelength, minimized with metal foil filters

Characteristic radiation

peaks of intensity from transitions in the system of atomic electrons of specific wavelength and characteristic for a given element

Bragg law

2dsin(theta) = wavelength

Value of diffraction patterns

contain information about the strength and spatial arrangement of scatterers (atoms, ions, etc)

Why are there no diffraction maxima/Bragg peaks in silica glass?

no periodicity, no long-range order, short-range order features hidden

Ray path difference (+ equation)

the phase difference between coherently scattered waves from consecutive sheets of atoms (MPN = 2dsin(theta) = wavelength)

Scattering vector (equation) (incident wave vector = k0, diffracted vector = k1)

q = 4pi/wavelength*sin(theta) = k1-k0

Forbidden reflections

missing peaks due to perfectly destructive interference

Resolution limit

sin(theta) less than 1, so d greater than wavelength/2

Ewald sphere

shows all possible scattering results with k0, k1, q, and theta

Bragg condition in vector notation

k1-k0 = g(hkl) = ha* + kb* + lc*

Magnitude of g(hkl) in Bragg geometry

|g(hkl)| = 1/d = 2sin(theta)/wavelength

Limiting sphere

sphere representing the maximum extent of reciprocal lattice points that can be detected for a given wavelength, contains the Ewald sphere

Structure factor (F)

sum amplitude of the x-ray waves scattered for planes (hkl) considering all interference phenomena of the scattered waves

Structure factor equation

F(hkl) = sum(A*exp(i*phase)) = sum(f_a*exp{2pi*i[hu+kv+lw]})

Intensity-structure factor relation

I(hkl) = |F(hkl)|^2 = F(hkl)F*(hkl)

f_a

atomic scattering factor (amplitude of scattered wave), proportional to the number of electrons surrounding the scattering atom/ion

increasing theta causes f_a to…

increase

Why do we calculate structure factor?

tells us if there is perfect destructive interference (F = 0)

Systematic absences

perfect destructive interference/extinctions