X-RAY METHODS FOR
ANALYSIS OF MATERIALS
Nikolay Petkov,
Tyndall, Block B,
E-mail: nikolay.petkov@tyndall.ie
Text Books
- Instrumental Methods of Analysis
- Atkins’ Physical Chemistry
Lecture notes, not really enough but with some
background reading should be fine for the
exam!
Topics
• Introduction
• X-ray diffraction (XRD)
•Crystal structure
•X-ray fluorescence analysis (XRF, EDX)
•X-ray photoelectron spectroscopy (XPS)
INTRODUCTION
Problems:- Destructive
Traditional methods of analysis such as:
Elemental
Atomic Adsorption,
No understanding of their
Mass Spec,
form in the material
Chromatography,
Infra-red spectroscopy,
UV-Vis spectroscopy etc require dissolution into a fluid phase: destructive
Materials or solids analysis has been driven by the requirement to produce nondestructive methods.
They can be described in six categories:Elemental – the atoms present – XRF, EDX
Structural – define atomic arrangement - XRD
Chemical – define the chemical state - XPS
Imaging – what does the morphology look like? – Electron Microscopy
Spectroscopic – energy level transitions – IR
Thermal – effects of heating (sorption) – TGA, DSC
Why to study solid state?
• Solid state includes most of the materials in
that make modern technology possible
• Properties of the solid state differ significantly
from the properties of isolated atoms or
molecules
• The term ‘structure’ takes on a whole new
meaning.
Example: Nanosized metal
particles
Consider complete delocalized
electrons in the metals, special
effect called plasmons
(quantized plasma oscillations,
collective oscillations of the
electron cloud) result is specific
colour of metal particles with
nanosized dimensions.
Challenges in analysis of solids–
compared to molecular analysis
1. Solids have continuum energy states compared to discrete energy
levels in molecular sates.
2. Very high absorptions so that not much energy gets out! Can lead to
adsorption of signal of some elements within the matrix!
3. Saturation effects can not be simply diluted out.
4. Structural differences can often be difficult to resolve.
5. Many of the probes can not be used in simple laboratory
environments.
6. Require complex detection equipment.
7. Standards can be quire difficult to prepare compared to simple
dilution. Phase diagrams and structural change. Matrix effects.
Interaction of the radiation with the matter.
Elastic interactions - in which there is no lost in the energy
Inelastic interactions – with lost of the exciting energy.
Examples of Methods and Interactions with solids
X-ray Diffraction – X-rays in and out but no loss of energy, elastic scattering of xrays
X-ray fluorescence – X-rays in and out but look at x-rays generated by secondary
process, inelastic scattering of x-rays
X-ray Absorption Spectroscopy - X-rays in and out but look for energy losses
X-ray Photoelectron Spectroscopy - X-rays in electrons out (secondary electrons)
Electron microscopy – electrons in and out analyse either transmitted (transmission
EM) or secondary (secondary electrons, inelastic scattering of electrons)
Low Energy Electron Energy Loss Spectroscopy - electrons in and out – energy
analyse the electrons to look at energy loss to give vibrational information
Ultrasound - Sound in and out – use it to analyse for void formation in solids
NMR – radio frequency in and out analyse for energy
loss
No single techniques is capable of providing a complete characterization of a
solid
Bulk vs Surface Analysis
Quantification
Analysing solids is further complicated by
the depth sensitivity.
An ideal technique should be
quantitative e.g.
X-rays deeply penetrate matter so the
analyte depth is high and the analysis is
bulk sensitive rather than surface
sensitive.
Signal height is proportional to the
number of atoms of element or
chemical state present.
However, techniques involving electrons
(even if they are excited by high energy
techniques) are strongly absorbed and so
originate from the surface region. Such are
EDX and XPS.
This can be an advantage since surface
chemistry is of fundamental importance.
However, phase separation and
segregation can give rise to problems.
This is rare in a solid. In molecular
analysis the intensity is usually directly
proportional to the number of molecules
in the analyte. This is because the
systems are frequently dilute – there is
little chance that there are multiple
interactions of other interactions with
primary or secondary radiation. In a
solid exactly the opposite is true. Thus,
solids suffer matrix effects.
Weak emitter in a strong adsorbing
Weak emitter as a thin layer at the
matrix. Analyte emission absorbed by surface on strongly absorbing matrix.
matrix and no signal leaves sample.
Analyte
Matrix
How is heat produced ?
Most of the electrons in the incident beam lose energy upon entering a material
through inelastic scattering or collisions with other electrons of the material and
form heat.
In such a collision the momentum transfer from the incident electron to an atomic
electron can be expressed as dp = 2e2 / bv, where b is the distance of closest
approach between the electrons, and v is the incident electron velocity.
The energy transferred by the collision is given by T = (dp)2 / 2m = e4 / Eb2, where
m is the electron mass and E is the incident electron energy, given by E = (1 /
2)mv2.
By integrating over all values of T between the lowest binding energy, Eo, and the
incident energy E, one obtains the result that the total cross section for collision is
inversely proportional to the incident energy E.
A second type of interaction in which
the incident electron can lose its
kinetic energy is an interaction with
the nucleus of a target atom. In this
type of interaction, the kinetic energy
of the projectile electron is converted
into electromagnetic energy.
Interaction of X-rays with matter
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.
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.
incorrect
Symmetry Operations and Elements
A Symmetry operation is an operation that can be performed either
physically or imaginatively that results in no change in the
appearance of an object.
There are 3 types of symmetry operations: rotation, reflection, and
inversion. .
Rotational Symmetry
As illustrated above, if an object can be rotated about an axis and
repeats itself every 90o of rotation then it is said to have an axis of 4fold rotational symmetry. The axis along which the rotation is
performed is an element of symmetry referred to as a rotation
axis. The following types of rotational symmetry axes are possible in
crystals.
•1-Fold Rotation Axis - An
object that requires rotation of
a full 360o in order to restore it
to its original appearance has
no rotational symmetry. Since
it repeats itself 1 time every
360o it is said to have a 1-fold
axis of rotational symmetry.
•2-fold Rotation Axis - If an
object appears identical after a
rotation of 180o, that is twice in
a 360o rotation, then it is said to
have a 2-fold rotation axis
(360/180 = 2). Note that in
these examples the axes we are
referring to are imaginary lines
that extend toward you
perpendicular to the page or
blackboard. A filled oval shape
represents the point where the
2-fold rotation axis intersects
the page.
3-Fold Rotation Axis- Objects that repeat
themselves upon rotation of 120o are said
to have a 3-fold axis of rotational symmetry
(360/120 =3), and they will repeat 3 times
in a 360o rotation. A filled triangle is used
to symbolize the location of 3-fold rotation
axis.
4-Fold Rotation Axis - If an object
repeats itself after 90o of rotation, it will
repeat 4 times in a 360o rotation, as
illustrated previously. A filled square is
used to symbolize the location of 4-fold
axis of rotational symmetry.
6-Fold Rotation Axis - If rotation of 60o
about an axis causes the object to repeat
itself, then it has 6-fold axis of rotational
symmetry (360/60=6). A filled hexagon is
used as the symbol for a 6-fold rotation
axis.
Mirror Symmetry
A mirror symmetry operation is an imaginary operation that can be performed to
reproduce an object. The operation is done by imagining that you cut the object in half,
then place a mirror next to one of the halves of the object along the cut. If the reflection
in the mirror reproduces the other half of the object, then the object is said to have mirror
symmetry. The plane of the mirror is an element of symmetry referred to as a mirror
plane, and is symbolized with the letter m. As an example, the human body is an object
that approximates mirror symmetry, with the mirror plane cutting through the center of
the head, the center of nose and down to the groin.
Center of Symmetry
Another operation that can be
performed is inversion through a
point. In this operation lines are
drawn from all points on the object
through a point in the center of the
object, called a symmetry center
(symbolized with the letter "i"). The
lines each have lengths that are
equidistant from the original
points. When the ends of the lines
are connected, the original object is
reproduced inverted from its original
appearance. In the diagram shown
here, only a few such lines are drawn
for the small triangular face. The right
hand diagram shows the object
without the imaginary lines that
reproduced the object.
Rotoinversion
Combinations of rotation with a center of symmetry
perform the symmetry operation of rotoinversion. Objects
that have rotoinversion symmetry have an element of
symmetry called a rotoinversion axis. A 1-fold
rotoinversion axis is the same as a center of symmetr.
•3-fold Rotoinversion - This involves
rotating the object by 120o (360/3 = 120),
and inverting through a center. A cube is
good example of an object that possesses
3-fold rotoinversion axes. A 3-fold
rotoinversion axis is denoted as
(pronounced "bar 3"). Note that there are
actually four axes in a cube, one running
through each of the corners of the cube. If
one holds one of the axes vertical, then
note that there are 3 faces on top, and 3
identical faces upside down on the bottom
that are offset from the top faces by 120o.
Combinations of Symmetry Operations
As should be evident by now, in three dimensional objects,
such as crystals, symmetry elements may be present in several
different combinations. In fact, in crystals there are 32 possible
combinations of symmetry elements. These 32 combinations
define the 32 Point Groups.
Thus, this crystal has the
following symmetry elements:
•1 - 4-fold rotation axis (A4)
•4 - 2-fold rotation axes (A2), 2
cutting the faces & 2 cutting the
edges.
•5 mirror planes (m), 2 cutting
across the faces, 2 cutting
through the edges, and one
cutting horizontally through the
center.
•Note also that there is a center
of symmetry (i).
The symmetry content of this
crystal is thus: i, 1A4, 4A2, 5m
The space groups in three dimensions are made from combinations of the 32
crystallographic point groups with the 14 Bravais lattices which belong to one of 7
lattice systems. This results in a space group being some combination of the
translational symmetry of a unit cell including lattice centering, the point group
symmetry operations of reflection, rotation and improper rotation (also called
rotoinversion), and the screw axis and glide plane symmetry operations. The
combination of all these symmetry operations results in a total of 230 unique space
groups describing all possible crystal symmetries
Crystal family
Triclinic
Monoclinic
Orthorhombic
Tetragonal
Hexagonal
Cubic
Total: 6
Required
Crystal system symmetries of point groups
point group
None
2
1 twofold axis
of rotation or 1 3
mirror plane
3 twofold axes
of rotation or 1
twofold axis of 3
rotation and two
mirror planes.
1 fourfold axis
7
of rotation
1 threefold axis
Trigonal
5
of rotation
1 sixfold axis of
Hexagonal
7
rotation
4 threefold axes
5
of rotation
7
32
space groups
bravais lattices Lattice system
2
1
Triclinic
13
2
Monoclinic
59
4
Orthorhombic
68
2
Tetragonal
7
18
1
Rhombohedral
1
Hexagonal
36
3
Cubic
230
14
7
27
There are at least eight methods of naming space groups. Some of these methods can
assign several different names to the same space group, so altogether there are many
thousands of different names.
•International symbol or Hermann-Mauguin notation. describes the lattice and some
generators for the group. It has a shortened form called the international short
symbol, which is the one most commonly used in crystallography, and usually consists
of a set of four symbols. The first describes the centering of the Bravais lattice (P, A, B,
C, I, R or F). The next three describe the most prominent symmetry operation visible
when projected along one of the high symmetry directions of the crystal. These symbols
are the same as used in point groups, with the addition of glide planes and screw axis,
By way of example, the space group of quartz is P3121, showing that it exhibits primitive
centering of the motif (i.e., once per unit cell), with a threefold screw axis and a twofold
rotation axis. Note that it does not explicitly contain the crystal system, although this is
unique to each space group (in the case of P3121, it is trigonal).
Common crystal structures
Fd3m space group
Fm3m space group
Diamond structure, Si and Ge
Rock salt structure, metals
Double Slit and Diffraction Grating.
a pattern of
dark and
bright
fringes
At C all wavelengths arrive
in phase and interfere
constructively to produce a
“central image”
At some other point P which is at a distance
L from one slit and L + nλ from the other (λ is
some specific wavelength present in the light
beam; n is an integer) there is also
constructive interferences and a bright fringe
appears with the color pertaining to that
specific wavelength. At intermediate points
distant L and L + (2n + 1) (λ/2), destructive
interference occurs for that wavelength λ.
X-ray diffraction
Bragg’s Law
The path difference between two waves:
2x = 2dsin(theta)
For constructive interference nλ = 2dsinθ
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 and his son recognized a
predictable relationship among several factors.
Another view of Bragg’s law
When an X-ray beam hits an atom, the electrons around the atom start to
oscillate with the same frequency as the incoming beam. In almost all
directions we will have destructive interference. However the atoms in a
crystal are arranged in a regular pattern, and in a very few directions we
will have constructive interference. Hence, a diffracted beam may be
described as a beam composed of a large number of scattered rays
mutually reinforcing one another.
vector product
X-ray diffraction has been in use in two main areas:
1) Fingerprint characterization of crystalline materials –
powder XRD
2) The determination of crystal structure e.g. identification of
atomic position of crystals - single crystal XRD analysis.
But also…
1) Strain and lattice mismatch in crystals
2) Orientation of thin films – grazing incidence XRD, rocking
curves
3) Low angle X-ray scattering – form and shape of polymers,
meso-materials, colloidal particles.
Single Crystal
Polycrystalline
Powder XRD
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. Every lab in materials science has an XRD instrument.
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.
Bragg-Brentano Geometry (θ - 2θ)
detector
source
θ
θ
θ
sample
Monochromator
Divergence slit
Antiscatter slit
soller slit
Detector slit
Tube
Which Information does a Powder
Pattern offer?
Primary information
1) peak position – d-spacing
2) peak intensity – structure factors
3) peak broadening – size of a crystallite
Secondary information - calculated
•dimension of the elementary cell
•content of the elementary cell
•strain/crystallite size
•quantitative phase amount
In the kinematical approximation for diffraction, the intensity of a diffracted beam is
given by:
where
and
is the wavefunction of a beam scattered a vector
is the so called structure factor which is given by:
,
Here, rj
is the position of an atom j in the unit cell, and fj is the scattering power of
the atom, also called the atomic form factor. The sum is over all atoms in the unit
cell. It can be shown that in the ideal case, diffraction only occurs if the scattering
vector
is equal to a reciprocal lattice vector
.
The structure factor describes the way in which an incident beam is scattered by the
atoms of a crystal unit cell, taking into account the different scattering power of the
elements through the term fj. Since the atoms are spatially distributed in the unit cell,
there will be a difference in phase when considering the scattered amplitude from
two atoms. This phase shift is taken into account by the complex exponential term.
The atomic form factor, or scattering power, of an element depends on the type of
radiation considered. Because electrons interact with matter though different
processes than for example X-rays, the atomic form factors for the two cases are not
the same.
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 crystallite 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. But if the crystallite is small , below 100 nm there are only 1000
planes and the effect of canceling is not so pronounced which will lead to
broadening of the reflection.
Scherrer equation
crystallite size = 0.9 λ/(B cos θ)
B = = √(B2obs – B2o)
where Bobs = FWHM of reflection
Bo = instrumental FWHM minimum
Systematic absences
Some types of unit cell give characteristic and easily recognizable pattern
lines. For example, in a cubic primitive lattice of unit cell dimensions a the
spacing is given by the equation:
sin θ = (h2 + k2 + l2)1/2 λ/2a
The reflections arte than predicted by substituting the values h, k, l:
{h,k,l}
{100} {110} {111} {200} {210} {211} {220}
h2 + k2 + l2
1
2
3
4
5
6
8
The 7 (and 15…) is missing because the sum of the squares of three
integers cannot be 7. Therefore the pattern has systematic absences that
are characteristic of the cubic P lattice.
For FCC, h,k,l all even or all odd are present
For BCC, h+k+l = odd are absent