Crystal structures, unit cell, symmetry elements
Powder X-Ray Diffraction
INTRODUCTION
X-rays are electromagnetic radiation of
wavelength about 1 Å (10-10 m), which is
about the same size as an atom. They occur in
that portion of the electromagnetic spectrum
between gamma-rays and the ultraviolet. The
discovery of X-rays in 1895 enabled scientists
to probe crystalline structure at the atomic
level.
When certain geometric requirements are met,
X-rays scattered from a crystalline solid can
constructively interfere, producing a diffracted
beam. In 1912, W. L. Bragg recognized a
predictable relationship among several factors.
Fig. 1 Reflection of x-rays from two planes of
atoms in a solid.
The path difference between two waves:
2x = 2dsin(theta)
For constructive interference nλ = 2dsinθ
Bragg equation
X-ray diffraction has been in use in two main
areas, for:1. Fingerprint characterization of crystalline
materials and
2. The determination of their structure.
Each crystalline solid has its unique
characteristic X-ray powder pattern which
may be used as a "fingerprint" for its
identification. Once the material has been
identified, X-ray crystallography may be
used to determine its structure, i.e. how the
atoms pack together in the crystalline state
and what the interatomic distance and angle
are etc. X-ray diffraction is one of the most
important characterization tools used in solid
state chemistry and materials science.
We can determine the size and the shape of
the unit cell for any compound most easily
using the diffraction of x-rays.
The figure below shows the x-ray diffraction
pattern from a single crystal of a layered clay.
Strong intensities can be seen for a number of
values of n; from each of these lines we can
calculate the value of d, the interplanar
spacing between the atoms in the crystal.
Fig. 2 X-ray diffraction pattern from a layered
structure vermiculite clay.
Two reflections from different planes
Lets imagine one piece of solid. That satisfies
Bragg reflection as shown.
We will observe strong diffraction
But if the planes are misaligned
We will observe no diffraction
In any real solid we have a chance orientation. It
would be almost impossible to study diffraction.
However, 99% of all materials are polycrystalline
or can be prepared (by grinding) so as to present
many grains of material. In these some will always
be at the correct alignment.
Those planes at the right orientation will give
strong diffraction
A single crystal, also called monocrystal, is a
crystalline solid in which the crystal lattice of the entire
sample is continuous and unbroken to the edges of the
sample, with no grain boundaries.
Because grain boundaries can have significant effects on
the physical and electrical properties of a material, single
crystals are of interest to industry, and have important
industrial applications. The most notable of these is the
use of single crystal silicon in the fabrication of
semiconductors.
Monocrystals are often made by Czochralski process,
controlled crystallization from the melted material.
The opposite of a single crystal sample is a
polycrystalline sample, which is made up of a number of
smaller crystals known as crystallites. Usually those
crystallites are connected through a amorphous material
to form extended solid.
A galvanized surface (a steel "strong-tie" joiner) showing
visible spangle (of approx. 5mm)
X-ray diffraction provides a 2D image of the
atomic arrangement. Since nλ = 2d sin θ
or sin θ = nλ/2d
The image is a recipricol one.
In a polycrystalline or powder sample though we
do not see one specific arrangement or rotation of
the small crystallites.
If we scan across the image diagonally gives peaks
EXPERIMENTAL SET-UP
Modern machines rotate the x-ray source and the
detector.
detector
source
sample
X-ray sources are cumbersome. To generate
sufficient signal need to generate lots of power i.e
x-rays. Frequently run at 50 kV and 50 mA.
anode
e’s
5V
-ve
50 kV +ve
filament
Moving these are difficult because of the electric
supplies and water used to cool anode to prevent
melting. Older machines moved detector through
2Θ and sample through Θ. This maintains angle
incidence = angle of reflection.
For historical reason we plot intensity versus angle
2Θ
X-RAY LINE BROADENING
When x-rays enter a solid they undergo refraction.
For x-rays this is very small.
But the refraction angle differs from the incidence
angle by only parts per thousand. But because of
this the path length difference slowly varies from
planes deeper into the material. The constructive
interference slowly becomes destructive.
Provided the sample is thick enough (if a sample is
10 μm there are 1 x 10-5/10-10 atom planes = 105)
then all these slightly out of phase reflections will
cancel. Leave just the perfect constructive
interference feature. Shown by rocking curves.
Scan detector across the diffracted beam.
2500
intensity
2000
1500
1000
500
0
44.8
45
45.2
45.4 45.6
45.8
46
46.2
angle 2-theta
Called rocking curve – it is a measure of how
crystalline a material is. A Si wafer rocking curve
could have a rocking curve of 0.1 mrad.
If the samples are thin incomplete cancelling is
observed. The diffracted peaks are not narrow they
become broad. The broadness can be used to
estimate the sample thickness.
Formulism was drawn up by Scherrer:Sample thickness, t
= 0.9 λ/(B cos θ)
B = Bactual = √(B2obs – B2o)
where Bobs = FWHM of reflection
Bo = instrumental FWHM minimum
Small-angle X-Ray scattering.
We have considered that Bragg's Law,
d = /(2 sin),
supports a minimum size of measurement of /2 in a
diffraction experiment
…but does not predict a maximum size.
We have discussed the use of diffraction to measure
crystalline and amorphous structures on the atomic scale,
but clearly, many morphologies are of importance that
have characteristic sizes much larger than the atomic
scale – nano-sized objects.
Guinier was one of the fathers of an outgrowth of
diffraction aimed at large-scale structures in the 1950's.
Bragg's Law predicts that information pertaining to such
nano- to colloidal-scale structures would be seen below
6° 2 in the diffractometer trace. It is possible to design
specialized instruments to measure down to less than
1/1000 of a degree for measurement of up to 1-micron
scale structures using x-rays!
In XRD the atomic scattering factor,
f2 = ne2(1/q),
where q is 4π sin(θ)/λ.
Additionally, the intensity of scattering is known to be
proportional to the number of scattering elements in the
irradiated volume, Np(1/q).
Then, in small-angle scattering we can consider a
generalized rule that describes the behavior of scattered
intensity as a function of Bragg size "d" or "r" that is
observed at a given scattering angle 2q, where r = 1/q.
I(q) = Np(1/q) ne2(1/q)
All scattering patterns in the small-angle regime reflect a
decay of intensity in q and this can be easily described by
considering that at decreasing size scales the number of
electrons in a particle is proportional to the decreasing
volume, while the number of such particles increases
with 1/volume. Then the scattered intensity by equation
(1) is proportional to the decay of the particle volume
with size. This analysis implies that the definition of a
particle, i.e. r, does not necessarily reflect a real domain,
but reflects the size, r, of a scattering element that could
be a component of a physical domain.
The characteristics of materials at these larger size scales
are fundamentally different than at atomic scales. Atomic
scale structures are characterized by high degrees of
order, i.e. crystals, and relatively simple and uniform
building blocks, i.e. atoms. On the nano-scale, the
building blocks of matter are rarely well organized and
are composed of rather complex and non-uniform
building blocks. The resulting features in x-ray scattering
or diffraction are sharp diffraction peaks in the XRD
range and comparatively nondescript diffuse patterns in
the SAXS range.
For example polyethylene sub-micron sized fibrils
High Density Polyethylene showing XRD at high-q,
SAXS at intermediate q and LS (light scattering) at lowq.
One distinct exclusion: Ordered Mesoporous
Materials (OMMs). Somehow show relatively sharp
and intensive X-ray lines, similarly to crystalline solids.
- No atomic arrangement, essentially amorphous but
- Long-range order of mesopores with defined and
controllable pore sizes in the range of 2 – 10 nm.
- Various types of mesophase structures hexagonal,
cubic, etc.
- Hybrid organic/inorganic structures, prepared by selfassembly of surfactants and inorganic molecular
precursors.
100 nm
100 nm
40 nm
Powder XRD of mesoporous solids
6000
(100)
5000
Intensity (cps)
4000
3000
2000
1000
(110)
(200)
0
1
2
3
4
5
2 Theta (degrees)
d(010)
d(110)
c
d(100)
a
b
Show how to calculate the unit cell size from the d(100)
1500
Intensity (cps)
1000
500
0
2
3
4
5
6
7
2 Theta (degree)
Reflections at 2.42, 2.82, 3.68, 3.94, 4.4 and 4.6 degrees
2, give the indexation, if you know that is body centred
cubic and calculate the unit cell size.
Reflection and grazing-incidence geometry for thin
film X-ray analysis
Refection geometry is similar to that of PXRD
geometry; it is usually called (θ – θ) as shown on
the scheme. The X-ray source and the detector are
moved in symmetrical mode thus scanning varying
angels of θ.
In this geometry only planes parallel to the
substrate surface are detectable.
X-ray tube
scanning direction
detector
scanning direction
incident beam
diffracted beam



d
transmitted beam

Substrate
Example: hexagonally ordred silica mesoporous
thin films (MTFs) on silicon wafer.
Case a) synthetically not achievable. Cases b) and
c) show mono-oriented cases where the extinction
of certain reflections suggests preferential
arrangement of mesoporous channels. Compare to
powders where different possible orientations are
possible and the PXRD patterns in (θ – θ) show all
possible reflections.
Problem with this geometry is that the reduced
number of reflections most often hinders the unique
identification of the structure, since larger numbers
of reflections are required to index the structure.
Grazing-incidence diffraction (GID) is a
scattering geometry combining the Bragg condition with
the conditions for X-ray total external reflection from
surfaces when the incident angle of the X-rays is small
enough (typically 0.05 – 3°, depending on the substrate
electron density and the X-ray energy), close to the so
called critical angle c.
In the case of Cu Kα radiation, the critical angle is
o
0.22 for silicon, 0.420 for nickel, and 0.570 for gold.
At this point the surface is not entirely invisible to
the X-rays, but only an evanescent wave penetrates and
scatters from it. These conditions provide superior
characteristics of GID as compared to other diffraction
schemes in the studies of thin surface layers. In order to
perform grazing-incidence diffraction, a highly intense,
parallel, and stable X-ray beam is desirable that can be
obtained only by using Synchrotron radiation. When
coupled with 2D detector (CCD) camera full structural
information can be obtained. Usually the beam is
directed at small angels (grazing) angels close to the c
of the material of interest where both reflected out-ofplane and in-plane reflections can be generated and
recorded by the 2D detector as shown on the scheme.
qz
out of plane diffraction
qx
specularly reflected
beam
CCD
2o
αi
qy
αf
2o
incident beam
mesoporous film
beam stop
substrate
primary beam
sample horizon
in plane diffraction
Example: Determination of the structure of the MTFs by
grazing incidence SAXS (GI-SAXS).
100
3.73 nm
100
010
4.68 nm
The real symmetry is not hexagonal but centered
rectangular because of the contraction normal to
the substrate.
Contraction
dh(010)
dr(110)
dr(110)
ch
cr
cr
ar
dr(200)
dh(100)
ar
ah
bh
br
br
Unique identification of the cubic MTFs is possible
only with this set up. Much of the information is
not detectable in reflection geometry.
3.6 nm
4.3 nm
211
121 21-1
210
200
120 12-1
021
2-1-1
02-1
020
-120 -121
dr(200)