Diffraction

Difrakce
X-ray scattering
Interaction of X-rays with samples
• Scattering
– Important – because no lenses for imaging
– Need understanding of how atomic structure
affects the scattering of materials
• Absorption
Electromagnetic waves.
• Ei = EOiexp[2πiν(t-x/c)] = EOi[cos2πν(t-x/c)+isin2πν(t-x/c)]
Ei is field, t time, x position - from origin O. i = √-1
– Ei represented as complex vector.
• Field accelerates a charged particle with frequency ν.
– Max. acceleration as particle passes node max Ei.
– Thus (electron) particle displacement π/2 from Ei.
• The accelerating orbital electron initiates a second
electromagnetic wave with a 2nd phase change of π/2.
EOi
Ei
Elastic Scattering
•
•
•
•
•
Coherent, Thompson.
No loss of energy (λ unchanged).
Phase changes exactly π.
Dominates diffraction.
Mechanism to be discussed a little later.
Coherent Scattering
Incident
X-ray photon
Low energy photons
only scattering, no ionization
Scattered
X-ray photon
Thompson scatter.
• Consider the electric field at an observation
position r:
Ei = EOi exp[2π iν(t - r/c) – iπ]
• Thompson scattered wave (Ed):
Ed = (1/r) EOi (e2/mc2) sin φ
– mass, m in denominator → nuclei unimportant
− φ is the angle between electron acceleration and
direction of scatter
Another implication of the
Thompson Equation
• Integration over all directions
– For crystal of
• A typical size
• Perfect condition
– Shows ≤ 2% of x-rays diffracted
• Weak data
Polarization
• Assuming polarized x-rays
Electrons oscillate ⊥ EOi and || EOi‘s oscillation
Along direction of electron oscillation, no Ed
Orthogonal to e- oscillation: Ed ∝ (1/r) EOi
• Non-polarized x-rays = sum of two orthogonallypolarized waves.
• Partially polarized: weighted sum
– Most experimental sources
– Degree of polarization depends on apparatus, not
structure
• Correct as if data collected with sin φ = 1.0.
Compton Scattering
•Inelastic, Incoherent.
•In collision w/ e− (rarely
nucleus)...
•Fraction of energy
transferred
–Change in wavelength
•Phase unpredictable.
→ weak background nonBragg scatter.
•Usually a nuisance,
requiring correction
E Energy of incident photon
E’ -”- scattered photon
mc2 mass equivalent energy
of 1electron
Photoelectric effect
Example
•
•
•
•
•
Photon with energy 40keV
enters
Photoelectron from K-shell
with energy (40-33.2)=6.8keV
exits
Electron from M- to K-shell
Characteristic radiation at
(33.2-0.6)= 31.6KeV in
a random direction.
The Atom now has positive
charge
Iodine
Energy levels
K -33.2keV
L -4.3keV
M -0.6keV
KLM
X-rays Fluorescence.
•
•
•
•
Excited atoms return to minimal state.
Release series of energy quanta.
Sometimes of x-ray energy.
Characteristic lines.
Diffraction
Diffraction
No lenses available for X-ray
Constructive interference
Two coherent wave sources produce a diffraction
image on a distant screen. Coherent sources are
produced by passing waves from a single source S
through two slits, S1 and S2, of size of the order of
the wave’s wavelength. According to Huygens’
principle that every point on a wave front behaves
like a point source, the slits produce two coherent
sources.
Interference I
Bright
r2
S1
S
d
θn
r1
Bright
S2
Bright
λ=wavelength
d=distance between slits
S=slits
r=optical paths of wave fronts
n=diffraction number
r1 − r2 = d ⋅ sin θ n = nλ
Interference II.
d
n=2
n=1
n=0
n = -1
n = -2
λ
grating
screen
In a diffraction grating for visible light, constructive interference
between light rays passing through slits of the grating leads to
light intensity ONLY at certain locations on the screen:
Interference III.
S
so
φ
detector
ψ
λ
a
crystal lattice
k = s/ λ,
k =vlnový vektor
s=jednotkový vektor
Laue equations
hλ = a(cos ψ1 – cos φ1) = (s.a – s0.a)
kλ = a(cos ψ2 – cos φ2) = (s.b – s0.b)
lλ = a(cos ψ2 – cos φ2) = (s.c – s0.c)
Original von Laue Formulation
of X-Ray Diffraction
s=ñ-jednotkový vektor
Bragg’s Law
2dhkl sin θ = nλ
• Correlates X-ray wave length, λ,
interplanar spacing, d, and
reflection angle, θ.
BD = d sin Θ
A
Θ
DC = d sin Θ
B
C
dhkl
BD + DC = n λ
D
Bragg rovnice:
2dhkl sin Θ = n λ
Reálný Prostor
h,k,l (uzlové roviny)
Reciproký prostor
h,k,l (mřížkové body)
Vektor reciproké mřížky :
Hhkl = ha* + kb* + lc*
a*,b*, c* - bázové vektory reciproké mřížky
h, k, l – „Miller Indexy“ roviny
Reciproký vektor je kolmý na osnovu hkl rovin
Délka reciprokého vektoru je 1/d. d – mezirovinná vzdálenost
Reciproká mřížka (opakování)
Laue-ho podmínky pro difrakci
Laue rovnice
hλ = (s.a – s0.a)
kλ = (s.b – s0.b)
lλ = (s.c – s0.c)
k = s/ λ,
Hhkl = ha*+kb*+lc*
skalární součin a.Hhkl = h
Spojením obou rovnic
dostáváme jednoduchou
rovnici pro difrakci:
Laue rovnice
h = (k.a – k0.a)
k = (k.b – k0.b)
l = (k.c – k0.c)
a.Hhkl = h
b.Hhkl = k
c.Hhkl = l
k - ko =Hhkl
k – směr rozptýleného záření
k0 – směr dopadajícího záření
H – vektor reciproké mřížky
h,k,l –indexy roviny odpovídající vektoru H
Difrakce. Geometrická interpretace
k - ko = Hhkl
=Hhkl
Ewald Construction
Ewald Construction for XRD
Ewald Construction
ELECTRON DIFFRACTION PATTERNS
A SPOT PATTERN REPRESENTED A PARTICULAR PLANE OF THE
RECIPROCAL LATTICE PASSING THROUGH OF THE POINT 000
Hhkl = ha* + kb* = lc*
a*, b*, c* are axial vectors,
h,k,l are point indices
INDEXING OF DIFFRACTION PATTERN
Intenzita
Starting Assumptions
For x-rays, electrons, and neutrons incident on a crystal,
diffraction occurs due to interference between waves scattered
elastically from the atoms in the crystal.
If we treat the incident waves as plane waves and the electrons as
ideal point scatterers, the scattered waves are spherical waves.
We will assume they are also isotropic.
Physical Model for X-ray Scattering
Consider two parallel plane waves scattered elastically from two
nearby electrons A and B in a solid material:
v
k
A
P
v
k′
v
ρ
O
B
ψ incident = φe
vv
i ( k ⋅r −ωt )
v
v
elastic scattering: k ′ = k
ψ scattered =
φf
R
e
v
i ( k ′⋅rv ′ −ωt )
f = scattering power of electron
Phase Difference Between the Waves
For the spherical waves scattered from electrons A and B :
ψA =
φf
RA
e
v
i ( k ′⋅rv ′ −ωt )
ψB =
φf
RB
e
v
i ( k ′⋅rv ′ −ωt + Δ )
RA= position of detector relative to A
RA = position of detector relative to B
r = position of B relative to A
Δ= phase difference between ΨA and ΨA
v r
v r r v v r v
Δ = − k ⋅ r + k ′ ⋅ r = r ⋅ (k ′ − k ) = r ⋅ Δk
So the wave scattered
from the j-th electron is:
Rj – position of atom j
relative to A
ψj =
φfj
Rj
scattering
vector
e
v
r v
i ( k ′⋅rv ′ −ωt + r j ⋅Δk )
Sum of Scattered Waves
Thus the total scattered wave at the detector is:
ψ=
φfj
∑R
all el .
j
e
v v
r v
i
(
r
i ( k ′⋅r ′ −ωt )
j ⋅Δk )
e
≅
φfj
R
e
v v
i ( k ′⋅r ′ −ωt )
∑e
r v
i ( r j ⋅Δk )
all el .
For a small sample, the distances Rj are all essentially the same (≈ R). Thus
we see that constructive and destructive interference between the scattered
waves that reach the detector is due to the atomic sum. The detector location
is determined by the scattered wave vector k and thus Δk .
We can see:
•
General wave motion
•
Sum represents Amplitude
Atomic scattering factors
Atomový faktor
Rozptyl na všech elektronech jednoho atomu
Z
sin(θ)/λ
Summing Over Lattice points
Now assume a crystal whose lattice has base vectors a, b, c , with a
total number of atoms along each axis M, N, and P, respectively:
Thus the amplitude of the total wave at the detector is proportional to:
ψ∝
∑e
v
i ( r j ⋅Δk )
M −1 N −1 P −1
= ∑∑∑ e
v v v v
i[( ma + nb + pc )⋅Δk ]
m =0 n =0 p =0
all atoms
Replacing n1n2n3 with the familiar hkl, we see by inspection that these
three conditions are equivalently expressed as:
v r
Δk = H hkl = ha * + kb* + lc *
The sum of the scattered x-rays from the crystal was found to be composed of
two components: sum over the lattice and sum over the one unit cell
A∝
∑f
all atoms
j
e
r v
i ( r j ⋅Δk )
=
∑ ∑f
lattice basis
j
e
r r
i 2π ( r j ⋅ H hkl )
Strukturní Faktor Fhkl
Strukturní faktor je suma rozptylových schopností (vln) všech atomů v
základní buňce (součet komplexních čísel, vektorový součet)
Fhkl = ∑ f j e
r r
i 2π ( r j ⋅ H hkl )
r r
rj ⋅ H hkl = (hx j + ky j + lz j )
j
f –atomový (rozptylový) faktor
Kde reciproký vektor a polohy všech atomů v buňce jsou:
r
v* v*
v
*
H hkl = ha + kb + lc
v
r
v
rj = x j a + y j b + z j cv
Structure factor
• The structure factor F(hkl) is (vector) sum of
the scattering f of each atom in the unit cell
Im
b
F(hkl)
Re
O
a
v
F (hkl ) = ∑ f j exp{2πi(hx j + ky j + lz j )}
j
Intenzita Ihkl
Intenzita je druhá mocnina (čtverec) absolutní hodnoty strukturního faktoru.
Intenzita je reálné (ne komplexní číslo).
Informace o fáze strukturního faktoru zaniká.
I hkl = Fhkl ⋅ F
*
hkl
Intenzita závisí také na experimentálních podmínkách:
Polarizace rtg. záření, rychlostí pronikání vektoru difrakční sférou (Lp faktor).
Absorpční faktor. Extinkční faktor. Integrální intenzita.
Difrakční
snímek
Laue Metoda
Laue Pattern
Prášková metoda
Záznam
Práškový záznam
3000
110
I
2000
200
1000
0
111
30
40
211
220
50
2Θ
Braggova rovnice:
60
70
310
80
2d sin Θ = n λ
Scattering by elements of electron
density
• F(r*) = ΣNj=1 Aj exp 2πi r*·rj
• Let rj be center of infinitely small element of electron
density, ρ.
• Consider total scattering:
– F(r*) = ∫V ρ(r)exp 2πir*·r dr
• Right-hand side ≡ FT(ρ).
• Structure determination:
– measure amplitude
– determine phase throughout (continuous) function, F(r*)
– compute inverse FT Æ electron density:
ρ(r) = T-1[F(r*)] = V* ∫V*F(r*)exp -2πi r*·rdr*
Atomic Factor eqn.(2)
• fat ≡ atomic scattering factor ≡ FT isolated atom (later):
• Depends on # electrons, thermal vibration.
– Tabulated theoretical or experimental values.
• Can be approximated by spherically symmetric Gaussian.
– But usually more sophisticated description
• Scattering ≡ FT(molecule)