Difracción de RX

Difracción de RX
History: Wilhelm Conrad Röntgen
Wilhelm Conrad Röntgen discovered 1895 the X-rays. 1901 he was
honoured by the Noble prize for physics. In 1995 the German Post
edited a stamp, dedicated to W.C. Röntgen.
The Principles of an X-ray Tube
X-Ray
Cathode
Fast electrons
Anode
focus
The Principle of Generation the
Characteristic Radiation
Photoelectron
M
Emission
Kα-Quant
L
K
Electron
Lα Quant
Lα-Quant
Kβ-Quant
The Generating of X-rays
energy levels (schematic) of the electrons
M
Intensity ratios
Kα1 : Kα2 : Kβ = 10 : 5 : 2
L
K
Kα1
Kα2
Kβ1
Kβ2
The Generating of X-rays
Anode
(kV)
Wavelength, λ [Angström]
Kα1 : 0,70926
Mo
C
Cu
20,0
9,0
Kα2 : 0,71354
Kβ1 :
Kß-Filter
Zr
0,08mm
0,63225
Kα1 : 1,5405
,
Kα2 : 1,54434
Ni
0,015mm
Kβ1 : 1,39217
Co
Kα1 : 1,78890
7,7
Kα2 : 1,79279
β :
Kβ1
Fe
Fe
0,012mm
1,62073
,
Kα1 : 1,93597
7,1
Kα2 : 1,93991
Kβ1 :
1,75654
Mn
0 011mm
0,011mm
The Generating of X-rays
Emission Spectrum of a
Molybdenum X-Ray Tube
Bremsstrahlung = continuous spectra
characteristic radiation = line spectra
ρ
Eo
Centro difusor
ρ
Eo
ρ
E
2θ
1 – Caso simple: électron solo :
x
ρ
i
ρ
Eo
O
r
y
Onda plana
2θ
ρ
E
P
z
Onda esférica
Caso real.- estructura periódica.
O
Na
Cl
A
A
d00
1
2
2
P(x,y,
( ,y,
z)
Si, P lejos del cristal:
⇒ INTERFERENCIA en ONDAS PLANAS
La intensidad difractada esta determinada por fenómenos de
dispersión y modulada por un mecanismo de interferencia ( constructivas y
d t
destructivas)
destructivas
ti
).
Luz
Rayos X
El t
Electrons
Neutrons
→
→
→
→
Dispersión de Rayleigh
Dispersión de Thompson
Di
Dispersión
ió electrostática
l t
táti
Dispersión nuclear
En todos los casos,
casos, on se utiliza un factor de dispersión,
dispersión
p
,f, q
que es
proporcional a la amplitud dispersada por un átomo.
átomo.
Centro dispersor
e-
2θ
16
Dispersión elastica
elastica::
Conservación de la energía del fotón
fotón,,
λ=constante
λ=
constante..
Interférencias :
Suma de amplitudes (amplitud et fase) :
Σa
Dispersión inelástica
inelástica:: No conservación de λ.
N hay
No
h interferencias
i t f
i :
Suma de intensidades : Σ|a|2
Consecuencia : Amplificación del fenómeno de dispersión en un cristal.
cristal.
Σa ∝ Na
Σ |a|2
I ∝ N2a2
I ∝ N |a|2
17
Como cambia el factor de dispersión con Z y el ángulo
Ca (Z=20)
Al0 (Z=13)
C (Z=6)
20
15
10
5
0
0
Sinθ
Sin
θ/λ = 1/2d (nm-1)
1
2
(x10-1)
Factor de estructura :
Fhkl = Σj fj exp[2πi (huj + kvj + lwj )]
Ihkl ∝ F2hkl
rj : uj
vj
wj
21
Factor de estructura y redes de BRAVAIS :
Tipo de celda :
Primitiva
:
Todas los planos hkl estan presentes
presentes..
Centrada cuerpo :
condición de existencia:
existencia: h+k+l = par
100,, 200,
100
200, 010,
010, 020,
020, 001,
001, 002,
002, 110,
110, 1-10
10,, 111,
111, 210,
210, …
Centrada en las caras
caras:: condición de existencia:
existencia: h,k,l con la misma
paridad.. (todos pares o todos nones).
paridad
nones).
100,, 200,
100
200, 010,
010, 020,
020, 001,
001, 002,
002, 110,
110, 1-10
10,, 111,
111, 210,
210, …
22
Efecto de tamaño de dominio cristalino.
cristalino.
Granos grandes
Difractograma
Granos pequeños
Difractograma
El ancho de los picos de difracción está relacionado con el tamaño promedio de los cristales D
por medio de la fórmula de Scherrer :
.
Ver cálculo numérico
D=
λ
cos θ ∆2 − ∆2inst .
∆ : ancho del pico (en
(
radianes
radianes)) medido
en 2θ.
: ancho instrumental(~
instrumental(~ 0,2°),
∆inst
inst..
medido a partir de un polvo de grano
grande..)
grande
∆
∆inst
.
2θ
Existen tres tipos de esfuerzos internos σI, σII, σIII:
σi
Valeur
promedio
σI : Valor promedio → Desplazamiento de picos
(esfuerzos de tensión→ desplazamiento hacia ángulos pequeñosθ)
σII : Distancias grandes
grandes→
→ Ensanchamiento simétrico de picos
σIII : Distancias cortas→
cortas→ Ensanchamiento asimetricos de picos.
picos.
β = 4εtanθ
β = ancho (en radianes)
ε = deformación (∆l/l)
Powder Diffraction Diffractogram
Bragg law
nλ=SQ+QT
nλ=2d
λ 2d sinθ
i θ
Como indenficar una fase conocida
conocida.. ficha JCPDS .
λ
ref..
ref
Indexación
dhkl
Indicador
d confiaza
de
fi
Compuesto
Tipo de red
G.E..
G.E
Densidad
calculada
Parametros
Intensidad
número de
formules
Biblio.
+ información
hardware
Standard
interno
29
Bragg‐Brentano Diffractometer
y A scintillation counter may be used as detector instead of film to yield exact intensity data.
y Using automated goniometers step by step scattered intensity may be measured and stored digitally.
The Bragg‐Brentano gg
Geometry
Detector
Tube
q
focusingcircle
Sample
measurement circle
2q
The Bragg‐Brentano gg
Geometry
Antiscatterslit
Divergence
g
slit
Monochromator
Detectorslit
Tube
Sample
Esquema de funcionamiento del
difractometro.:
difractometro
.:
muestra
fuente
RX
Círculo de
desplazamiento del
monocromador y del
detector.
.
θ
.
haz difrac
ctado
2θ
}
Circulo de
focalisación
primario
monocromador
Kβ
ángulo α fijo
Kα
detector
47
49
Comparison Bragg‐Brentano Geometry versus Parallel y
Beam Geometry
Bragg-Brentano
Geometry
Parallel Beam Geometry
generated by Göbel Mirrors
Parallel Beam Geometry with Parallel‐Beam
Geometry with
Göbel Mirror
Göbel
mirror
Detector
Soller Slit
Tube
Sample
“Grazing Incidence X‐ray g
y
Diffraction”
Soller slit
Tube
Sample
Measurement circle
Detector
“Grazing
Grazing Incidence Diffraction
Incidence Diffraction” with Göbel Mirror
Soller slit
Göb l mirror
Göbel
i
Tube
S
Sample
l
Measurement circle
Detector
Powder Pattern and Structure
y The d‐spacings of lattice planes depend on the size of the elementary cell and Th d
i f l tti l
d
d th i f th l
t ll d determine the position of the peaks.
y The intensity of each peak is caused by the crystallographic structure, the position of the atoms within the elementary cell and their thermal vibration.
i i f h i hi h l
ll d h i h
l ib i
y The line width and shape of the peaks may be derived from conditions of measuring and properties ‐ like particle size ‐ of the sample material.
Phase Identification & Lattice Constant D
i
i
Determination
A) Ph
Phase Identification Id tifi ti 1) Compare the observed spectrum to a library (JCPDS) containing diffraction patterns of known compounds 2) Index the peaks from known lattice constants B) Lattice Constant Determination 1) Lattice Constant Refinement a) Least squares refinement of individually fit peaks b) Whole pattern fitting uto de g
2)) Autoindexing
a) Internal standard critical b) Unidentified impurities significantly reduce p
y
the probability of success Structure Determination
1) Determine the Crystal System and Lattice Constants
(By indexing the peaks in the diffraction pattern) 2)) Identify
Id if the
h Space
S
G
Group
(Using the systematic absences) 3) Determine an Approximate Structure
a) Empirical Methods
b) Ab Initio Approach
4) Refine the Approximate
pp
Structure
(Using the Rietveld refinement method) 5) Test and compare alternative models
Sources of Peak Broadening
1) Instrumental Broadening
a) Nonideal optics
b) Wavelength
W l
th dispersion
di
i
c) Specimen transparency
d) Misalignment
2)) Finite Crystallite
ll Size
3) Strain
a) Uniform strain (shifts peaks) b) Nonuniform strain (broadens peaks) 4) Extended Defects
a)) Stacking
g faults
b) Antiphase boundaries
In short ‐ XRD information are taken from:
In short.
XRD information are taken from:
y peak position
y peak intensity y peak broadening
y scaling factor
amount
dimension of the elementary cell
content of the elementary cell
strain/crystallite size
quantitative phase Neutron Powder Diffraction
1) Neutrons are scattered by nuclei instead of electrons.
This leads to a completely different scattering power distribution for the elemets. This
is extremely useful for determining positions of light atoms in compounds containing
both light and heavy atoms. light and heavy atoms 2) The absorption coefficients for neutrons are much smaller than they are for
x‐rays.
y
This results in much greater penetration depths for neutrons (i.e. 25 mm depth for
neutrons compared with 25 mm depth for x‐rays). This is advantageous in reducing
errors such as matrix effects and preferred orientation, but can be disadvantageous in that larger samples are needed. are needed 3) Scattering factors do not decrease with 2θ. This increases the information content of the high angle region of the diffraction
pattern. 4) Resolution is typically lower than found in high resolution x‐ray diffraction.
This can be problematic if there is considerable peak overlap. On
overlap On the other hand, peak
hand peak
shapes are more easily modeled.