1
3 Crystal Structure Analysis
1. Bragg Equation
2. Single Crystal Diffraction
3. X-ray Powder Diffraction
Crystal Structure Analysis
X-ray diffraction
Electron Diffraction
Neutron Diffraction
Essence of diffraction: Bragg Diffraction
Intensity
Bragg AngleUnit Cell
Crystal (Crystallite) X?ray Powder Pattern
Electron Diffraction Powder Pattern
Crystals, Powders, and Diffraction
0
Polycrystalline Specimen
X-ray Diffraction ? When an X-ray beam
bombards a crystal, the atomic structure of the
crystal causes the beam to scatter in a specific
pattern. This phenomenon, known as X-ray
diffraction, occurs when the wavelength of the
X rays and the distances between atoms in the
crystal are of similar magnitude.
X-ray Diffraction
Laue Equations
crystal ? periodic repetition of identical unit
cells ? diffraction grid
? constructive interference of scattered
waves only in particular directions:
von Laue
Diffraction
λ=α?α h)cos(cosa 0
2
von Laue Diffraction
§ For different incidence directions, the
diffraction patterns are different
λ=γ?γ
λ=β?β
λ=α?α
l)cos(cosc
k)cos(cosb
h)cos(cosa
0
0
0
Laue Diffraction
Single
crystal
Laue′s
Experiments
Laue Images of Zincblende ZnS
(a)show its four-fold axis， (b)show its three-fold axis
Einstein
regards this
experiment
as the most
beautiful one
in physics.
Nobel Prize winner
of 1914
the Bragg Condition of Crystal Diffraction
d
q
l
Strong reflection of the incident wave will occur for
the set of incident angles that satisfy the Bragg
condition: 2dsinθ =nλ
the index n defines the path difference between
waves i & ii when the diffraction occurs … for a given
value of n this path difference is nλ.
Geometry of Bragg Diffraction
Path difference for diffraction of rays from
adjacent planes is 2dhklsinθ, which must
correspond to nλ for constructive interference.
q q
dhklsinqdhklsinq
q q
θ=λ sind2n hkl
Bragg's Law
When x-rays are scattered from a crystal lattice,
peaks of scattered intensity are observed which
correspond to the following conditions:
The angle of incidence = angle of scattering
The pathlength difference is equal to an integer
number of wavelengths.
The conditions for maximum intensity contained
in Bragg's law above allow us to calculate details
about the crystal structure, or if the crystal
structure is known, to determine the wavelength
of the x-rays incident upon the crystal.
3
nλ = 2dsinθ
where n ? order of diffraction
λ ? X-ray wavelength
d ? spacing between layers of atom
θ ? angle of diffraction
Bragg's Law is the fundamental law of x-ray
crystallography.
Bragg's Law The Braggs
British physicists William Henry Bragg
(1862~1942) and William Lawrence Bragg
(1890~1971) won Nobel Physical Prize in 1915
due to their achievements on the Structure
Analysis via X-ray.
Generation of X-rays
Copper anode
Heated
tungsten
filament
electrons
X-rays
-
+
cathode
anode
PD ?
50 kV
§electrons are produced by thermionic emission from a W
filament and are accelerated by a large potential difference
§the high energy electrons (? 50 keV) bombard a metal
target (usually Cu, but can also be Mo)
§X-rays are generated by the interaction between electrons
and target
X-ray Emission Spectrum
§upon collisions the high energy electrons can knock inner core
electrons from the target atoms, leaving vacancies in the
innermost shell (K)§
these vacancies are rapidly filled by electronic transitions from
the other orbitals not all transitions are possible
the wavelengths are characteristic of the
target element
Intensity
Wavelengthlc
Kβ2
Kβ1
Kα1
Kα2 Copper anode:
Kα 1.5418 ?
Kβ 1.3922 ?
K
M
L
Kβ2Kβ1Kα1Kα2
X-ray Generation Using a Synchrotron
High intensity X-rays can be generated using a particle
accelerator such as a synchrotron: charged particles
(electron or positrons) are accelerated round a circle and
emit radiation tangentially. A particular wavelength can be
selected from the continuous spectrum of X-rays generated.
Synchrotron radiation: §tunable
§intense
X-rays
X-rays
beam
Synchrotron Radiation
More intense X-rays at shorter wavelengths mean higher
resolution & much quicker data collection.
4
X-ray Generators ? The Synchrotron
European Synchrotron
Radiation Facility
Grenoble, France
Electrons (or positrons) are
released from a particle
accelerator into a storage ring.
The trajectory of the particles is
determined by their energy and
the local magnetic field. Magnets
of various types are used to
manipulate the particle
trajectory. When the particle
beam is “bent”by the magnets,
the electrons (or positrons) are
accelerated toward the center of
the ring. Charged particles
moving under the influence of an
accelerating field emit
electromagnetic radiation, and
when they are moving at close to
relativistic speeds, the radiation
emitted includes high energy x-
ray radiation..
Schematic Illustration
of an X-ray Diffractometer
X-RAY
SOURCE
CRYSTAL
DETECTOR
PATH OF
DETECTOR
An experimental
arrangement for using Bragg
diffraction to determine the
structure of single-crystal
samples is illustrated
schematically left
Since it is usually difficult
to move the X-ray source the
sample itself is rotated with
respect to the source and
when the sample is moved
through an angle θ the
detector is moved through an
angle 2θ.
The wavelength of the x-ray
source is well known so by
measuring the angles at
which strong diffraction
peaks (i.e. strong detector
signal) occur the spacing of
particular planes can be
determined.
Type Tube Specimen Receiving
Slit
r1 r2
Brag -Brentano
q:2q
Fixed Varies as q Varies as 2q Fixed =r1
Brag -Brentano
q:q
Varies as q Fixed Varies as q Fixed =r1
Seeman -Bohlin Fixed Fixed Varies as 2q Fixed variabl
e
Texture
Sensitive
(Ladel)
Fixed Varies as q
processes about
a
Varies as 2q Fixed variabl
e
* Generally fixed, but can rotate about a or rock about goniometer axis.
Common Mechanical
Movement in Powder
Diffractometers
Schematic Illustration of X-ray Diffraction
To obtain nearly monochromatic x-rays, an x-ray tube is
used to produce characteristic x-rays. Matched filters are
used in the x-ray beam to optimize the fraction of the
energy which is in the Kα line.
θ-2θ geometry
The Bragg-Brentano diffractometer is the dominant
geometry found in most laboratories. In this system, if
the tube is fixed, this is called θ-2θ geometry. If the
tube moves (and the specimen is fixed), this is called θ :
θ geometry. The essential characteristics are:
(1) The relationship between θ (the angle between the
specimen surface and the incident x-ray beam) and 2θ
(the angle between the incident beam and the
receiving slit detector) is maintained throughout the
analysis.
(2) r1 and r2 are fixed and equal and define a
diffractometer circle in which the specimen is always
at the center.
Radiation Method
White Laue: stationary single crystal
Monochromatic Powder: specimen is polycrystalline,
and therefore all orientations are
simultaneously presented to the beam
Rotation, Weissenberg: oscillation
De Jong-Bouman: single crystal rota tes
or oscillates about chosen axis in path of
beam
Precession: chosen axis of single crystal
precesses about beam direction
5
§ Diffraction can occur whenever Bragg's law is
satisfied. With monochromatic radiation, an
arbitrary setting of a single crystal in an x-ray
beam will not generally produce any diffracted
beams. There would therefore be very little
information in a single crystal diffraction
pattern from using monochromatic radiation.
§ This problem can be overcome by continuously
varying λ or θ over a range of values, to satisfy
Bragg's law. Practically this is done by:
(1)using a range of x-ray wavelengths (i.e. white
radiation), or
(2)by rotating the crystal or, using a powder or
polycrystalline specimen.
XRD: “Rocking”Curve Scan
Vary orientation of ?k relative to sample normal while
maintaining its magnitude.
How? “Rock”sample over a very small angular range.
Resulting data of Intensity vs. theta (θ, sample angle)
shows detailed structure of diffraction peak being
investigated.
ik fk
k?
“Rock”Sample
k? Sample normal
XRD: Rocking Curve Example
Rocking curve of single crystal GaN around (002)
diffraction peak showing its detailed structure.
16.995 17.195 17.395 17.595 17.795
0
8000
16000
GaN Thin Film
(002) Reflection
Intensity (Counts/s)
theta (deg)
Rocking Curves
assessing crystal quality
To estimate the crystal quality a crystal is rotated
through θ with the counter set at a known Bragg
angle, 2θ. The resulting intensity versus θ curve is
known as a rocking curve. The width of the rocking
curve is a direct measure of the range of
orientation on mosaic spread present in the
irradiated area of the crystal, as each sub grain of
the crystal will come into orientation as the crystal
is rotated. For a film that isn't truely epitaxial,
the width of a rocking curve of the layer peak will
be a measurement of the quality of the layer.
Rotating Crystal Method
In the rotating crystal method, a
single crystal is mounted with an
axis normal to a monochromatic x-
ray beam. A cylindrical film is
placed around it and the crystal is
rotated about the chosen axis. As
the crystal rotates, sets of lattice
planes will at some point make the
correct Bragg angle for the
monochromatic incident beam, and
at that point a diffracted beam will
be formed.
The reflected beams are located
on the surface of imaginary
cones. When the film is laid out flat,
the diffraction spots lie on
horizontal lines.
Laue Method
§ The Laue method is mainly used to determine the
orientation of large single crystals. White radiation is
reflected from, or transmitted through, a fixed crystal.
§ The diffracted beams form arrays of spots, that lie on
curves on the film. The Bragg angle is fixed for every set
of planes in the crystal. Each set of planes picks out and
diffracts the particular wavelength from the white
radiation that satisfies the Bragg law for the values of d
and θ involved. Each curve therefore corresponds to a
different wavelength. The spots lying on any one curve
are reflections from planes belonging to one zone. Laue
reflections from planes of the same zone all lie on the
surface of an imaginary cone whose axis is the zone axis.
§ Experimental
§ There are two practical variants of the Laue method, the
back-reflection and the transmission Laue method.
6
Back-reflection Laue Methods
§ In the back-reflection
method, the film is placed
between the x-ray source
and the crystal. The beams
which are diffracted in a
backward direction are
recorded.
§ One side of the cone of
Laue reflections is defined
by the transmitted beam.
The film intersects the
cone, with the diffraction
spots generally lying on an
hyperbola.
Transmission Laue Methods
§ In the transmission Laue
method, the film is
placed behind the crystal
to record beams which
are transmitted through
the crystal.
§ One side of the cone of
Laue reflections is
defined by the
transmitted beam. The
film intersects the cone,
with the diffraction spots
generally lying on an
ellipse.
Stoe IPDS Image Plate Diffraction System
single crystal
size < 0.5 mm
Chemical Crystallography??
Single Crystal Analysis
Single Crystal X-ray Diffraction
Single crystal x-ray diffraction is a kind of method by put
a crystal in the beam, observing what reflections come out
at what angles for what orientations of the crystal with
what intensities.
§Advantage: You can learn everything to know about the
structure.
§Disadvantages: You however may not have a single
crystal. It is time-consuming and difficult to orient the
crystal. If more than one phase is present, you will not
necessarily realize that there is more than one set of
reflections.
Single Crystal X-ray Diffraction (Cont.)
§ Primary application is to determine atomic
structure (symmetry, unit cell dimensions, space
group, etc.,).
§ Older methods used a stationary crystal with
"white x-ray" beam (x-rays of variableλ) such
that Bragg's equation would be satisfied by
numerous atomic planes.
§ Modern methods (rotation, Weissenberg,
precession, 4-circle) utilize various combination
of rotating-crystal and camera setup to overcome
limitations of the stationary methods
X-rays Diffraction
For perfect crystals, I(2 θ)
consists of delta functions
(perfectly sharp scattering).
For imperfect crystals, the
peaks are broadened.
For liquids and glasses, it is a
continuous, slowly varying
function.
X-ray diffraction works on the principle that x-rays form
predictable diffraction patterns when interacting with a
crystalline matrix of atoms.
7
STADI-P Stoe Powder diffractometer
powder sample
in glass capillary
Chemical Crystallography ??
Powder Analysis
powder
Powder X-ray Diffraction
Measuring samples consisting of a collection of many
small crystallites with random orientations.
Powder XRD is used routinely to assess the purity and
crystallinity of materials
Each crystalline phase has a unique powder diffraction
pattern
Measured powder patterns can be compared to a
database for identification
§Advantages over Single Crystal Diffraction
It is usually much easier to prepare a powder sample.
You are guaranteed to see all reflections.
Information from Powder XRD
Phase purity
–both qualitative and quantitative
Crystallinity
–amorphous content, particle size and strain
Unit cell size and shape
–from peak positions
20 30 40 50 60 70 800
10000
20000
30000
40000
50000
20 30 40 50 60 70 800
50
100
150
200
20 30 40 50 60 70 80200
400
600
800
1000
1200
1400
1600
1800
2000
Amorphous Polycrystalline Crystalline
The Debye-
Sherrer
Camera
8
The Debye-Sherrer Camera
L2180R4 =pi??θ pi?=θ 180R4 L2
θ is diffraction angle, R is radii of camera, 2L is the distance of
every pair of arcs in the image
Debye Scherrer Camera
§ A very small amount of powdered material is
sealed into a fine capillary tube made from
glass that does not diffract x-rays. The
specimen is placed in the Debye Scherrer
camera and is accurately aligned to be in the
center of the camera. X-rays enter the camera
through a collimator.
§ The powder diffracts the x-rays in accordance
with Braggs law to produce cones of diffracted
beams. These cones intersect a strip of
photographic film located in the cylindrical
camera to produce a characteristic set of arcs
on the film.
Debye-Scherrer Camera
Can record sections on these cones on film or some
other x-ray detector
–Simplest way of doing this is to surround a capillary
sample with a strip of film
–Can covert line positions on film to angles and
intensities by electronically scanning film or
measuring positions using a ruler and guessing the
relative intensities using a “by eye”comparison
1916
X-ray powders
diffraction
Powder Diffraction Film
§ When the film is removed from the camera, flattened
and processed, it shows the diffraction lines and the
holes for the incident and transmitted beams.
§ There are always two arcs in the x-ray beams Kα and
Kβ, this causes the highest angle back-reflected arcs to
be doubled. From noting this, it is always clear which
hole is for the transmitted beam and which is for the
incident beam in the film.
Dutch post stamp, 1936, memorizing
Peter Josephus Wilhelmus Debye and
his Nobel prize.
§ The schematic shows the Debye cones that intersect
the film in the camera, and how diffractions are
measured on the film to determine the d-spacings for
the reflections measured.
Debye-Scherrer powder camera photographs of gold
(Au), a Face centered cubic structure that exhibits a
fairly simple diffraction
Debye-Scherrer powder camera photographs of Zircon
(ZrSiO4). Zircon is a fairly complex tetragonal
structure and this complexity is reflected in the
diffraction pattern.
9
Measurement of Debye-Scherrer
Photographs
Film from powder
camera laid flat.
The pattern of lines on a
photograph (left figure)
represents possible values of
the Bragg angles which
satisfy Bragg’s equation:
hklhkl sind2n θ=λ
Powder X-ray Diffraction
(Powder diffraction film)
The distance S1 corresponds to
a diffraction angle of 2θ. The
angle between the diffracted
and the transmitted beams is
always 2θ. We know that the
distance between the holes in
the film, W, corresponds to a
diffraction angle of θ = pi . So we
can find θ from:
W2
S1pi=θ
)WS1(2 2?pi=θor
We know Bragg's Law: nλ = 2dsinθ and the
equation for inter-planar spacing, d, for cubic
crystals is given by:
where a is the lattice parameter this gives:
From the measurements of each arc we can now
generate a table of S1, θ and sin2θ.
222hkl lkh
ad
++
=
)lkh(a4sin 22222 ++λ=θ
Interpretation of Powder Photographs
First task is to familiarize ourselves with these patterns.
The three most common structures are called face-centered
cubic (FCC), body-centered cubic (BCC) and hexagonal close-
packed (HCP).
Powder patterns of three
common types of simple
crystal structures.
(a) Face-centered cubic
(b) Body-centered cubic
(c) Hexagonal close-
packed
Cubic Structures
§For a cubic structure only one quantity is involved,
the cell edge or the lattice parameter , we have
Not all values of h2 + k2 + l2, which we shall call N, are
possible. Numbers such as 7, 15, 23, 28, 60 are said to
be forbidden.
For small values of N, the values of h, k and l are
easily deduced.
)lkh(a4sin 22222 ++λ=θ
Tetragonal and Hexagonal Structures
§ For tetragonal structures, we have
For hexagonal structures, or trigonal
structures referred to hexagonal axes
)lcakh(a4sin 22
2
22
2hkl
2 ++λ=θ
)lcakhkh(a4sin 22
2
22
2hkl
2 +++λ=θ
10
Indexing a diffraction pattern means assigning
Miller indices hkl to each value of d
If we know the unit cell, we can assign hkl
values to each d value using:
Similarly, if we know hkl values, we can
calculate the unit cell
However, often we don’t know hkl or the unit
cell… .
2
2
2
2
2
2
2 c
l
b
k
a
h
d
1 ++=
Indexing a Diffraction Pattern
2dsinθ = λ
The split of the XRD lines:
The symmetry of structure
decreased, the lines
increased.
h k l h 2 + k 2 + l 2 h k l h2 + k2 + l 2
1 0 0 1 2 2 1, 3 0 0 9
1 1 0 2 3 1 0 10
1 1 1 3 3 1 1 11
2 0 0 4 2 2 2 12
2 1 0 5 3 2 0 13
2 1 1 6 3 2 1 14
2 2 0 8 4 0 0 16
In a cubic material, the largest d-spacing that can be
observed is 100=010=001. For a primitive cell, we count
according to h2+k2+l2
Note: 7 and 15 impossible
Note: we start with the largest d-spacing and work down
Note: not all lines are present in every case
How Many Lines Are Possible? Observable diffraction
peaks
222 lkh ++
Ratio
Simple
cubic
SC: 1,2,3,4,5,6,8,9,10,11,12..
BCC: 2,4,6,8,10, 12… .
FCC: 3,4,8,11,12,16,24….
222hkl lkh
ad
++
=
λ=θ nsind2
Systematic Absences and Centering
The presence of a centered lattice leads to the
systematic absence of certain types of peak in
the diffraction pattern
For I centered lattices:
h + k + l = 2n for a line to be present
For an F centered lattice:
h + k =2n, k + l = 2n and h + l = 2n
For a C centered lattice:
h + k =2n
Ex: An element, BCC or FCC, shows diffraction peaks at
2θ: 40, 58, 73, 86.8,100.4 and 114.7.
Determine:(a) Crystal structure? (b) Lattice constant?
(c) What is the element?
(222)60.709057.35114.7
(310)50.590350.2100.4
(220)40.472143.486.8
(211)30.353836.573
(200)20.2352958
(110)1 0.1172040
(hkl)theta2theta q2sin 222 lkh ++
a =3.18 ? , BCC, à W
11
Particle size
The width of the peaks in a powder pattern
contain information about the crystallite size in
the sample (and also the presence of
microstrain)
Scherrer equation
t ? mean size of crystallites
K ? constant, roughly 1: depends on shape of
crystallites
B ? width of reflection in radian
θ
λ=
cosB
Kt
Effect of Particle Size in X-ray Diffraction
Scherrer
Equation
2 θ →
B
2 q2 2 q1
2 qB
Smaller Crystallite
Relative Intensity
2 θ→
B
2 q2 2 q1
2 qB
Larger Crystallite
Relative Intensity
BcosB
9.0t
θ
λ=
Bcost
9.0B
θ
λ=
or
Nanocrystal X-ray Diffraction Types of Diffraction Experiment
X-ray
-Routinely used to provide structural information on
compounds and to identify samples
-Used with both powder and single crystal samples
-X-rays produced in the home lab or using synchrotrons
-Can also be used to examine liquids and glasses
Electron diffraction
–primarily used for phase identification, and unit cell
determination on small crystallites in the electron microscope
-also used for gas phase samples
Neutrons
–useful source of structural information on crystalline
materials, but expensive
-Also useful for spectroscopy and structure of liquids/glasses
-good for looking at light atoms
–sensitive to magnetic moments
HRTEM Image and ED Pattern of
Tetrapod CdS Nanocrystals
Low Energy Electron Diffraction (LEED)
Provides information about surfaces
This is a LEED pattern from a
surface of Si. A bulk X-ray
diffraction pattern would show
only the strong spots on
defining the hexagonal pattern.
The additional fainter spots
contain important information
about the way in which Si
atoms re-arrange themselves at
the surface of Si, i.e. the surface
has a slightly different
structure from the bulk. These
measurements are very
important to device
manufacturers, since the
surface of silicon is used in so
many technological applications.
12
4 Crystal Chemistry
close packed structures
octahedral and tetrahedral holes
basic structures
Inorganic Crystal Structures
All crystal structures may be
described in terms of the unit cell and
atomic coordinates of the contents
Many inorganic structures may be
described as arrays of space filling
polyhedra - tetrahedra, octahedra, etc.
Many structures - ionic, metallic
covalent - may be described as close
packed structures.
Close Packing Concept
2D close-packed layer ?
Most efficient way to fill
space
Random arrangement
of atoms (hard,
neutral spheres)
Close Packed Structures - Metals
Most efficient way of packing equal sized
spheres. In 2D, have close packed layers
Coordination
number (CN) = 6.
This is the
maximum
possible for 2D
packing.
Can stack close packed (CP) to give 3D structures.
Three Dimensional Packing
The simplest arrangement is to place a second
layer of spheres on top of the first layer.
Two Main Stacking Sequences:
If we start with one close packed layer, two
possible ways of adding a second layer (can
have one or other, but not a mixture) :
13
Two Main Stacking Sequences:
If we start with one close packed layer, two
possible ways of adding a second layer (can
have one or other, but not a mixture) :
Three-dimensional Packing of Spheres (Atoms)
hexagonal close
packing (HCP)
cubic close
packing (CCP)
hexagonal close
packing (HCP)
(Be, Mg, Zn, Cd, Ti,
Zr, Ru ...)
cubic close packing
(CCP or FCC)
(Cu, Ag, Au, Al, Ni,
Pd, Pt ...)
Common Unit Cells for HCP and CCP Body centred cubic unit cell (BCC) and its
lattice point representation
fractional counting of atoms
with respect to the content
of a unit cell !
(Fe, Cr, Mo, W, Ta, Ba ...)
A layer of atoms placed in
the spaces between the first
layer of atoms gives rise to
the body-centered cubic
unit cell. The first and
third layer line up. Here
68% of the unit cell is filled
and 32% is the spaces
between spheres.
Examples are iron,
chromium, and all group
1A elements.
No matter what type of packing, the
coordination number of each equal size
sphere is always 12
Packing Fraction
The fraction of space which is occupied by atoms is
called the “packing fraction”, η, for the structure.
space available
atoms by occupied space? =
74.023r216
r34
4? 3
3
=pi=
pi
×=
For cubic close packing:
The spheres have been packed together
as closely as possible, resulting in a
packing fraction of 0.74.
14
Primitive Cubic Packing
a = 2r
a3 = 8r3
No. of atoms = (8 x 1/8) = 1
52.06r8
r34
3
3
=pi=
pi
=η
Simple Cubic Packing
Coordination number = 6
packing efficiency 52%
1. A layer of atoms
placed directly over
another layer of atoms
will give rise to the
simple cubic unit cell.
Here 52% of the unit
cell is filled with the
atoms and 48% is the
spaces between atoms.
Two-Dimensional Packing
§ What is the most efficient way to arrange
circles on a plane surface?
Square packing
coordination number = 4 Close packingcoordination number = 6
XY
Y
Y
Y XY
YY
Y
YY
Metals usually have one of three structure
types:FCC, HCP or body centered cubic (BCC).
The reasons why a particular metal prefers a
particular structure are still not well understood.
FCC HCP ? BCC
Unit Cell of Body-centered Close Packing
Tungsten
Body-centered packing
Hexagonal Close Packing
(HCP)
Coordination number = 12
packing efficiency 74%
15
Cubic Close Packing (CCP) Close Packed Ionic Structures
Ionic structures - cations (+ve) and anions (-ve)
In many ionic structures, the anions, which are
larger than the cations, form a close packed
array and the cations occupy interstitial holes
within this anion array.
Two main types of interstitial site :
Tetrahedral : CN = 4
Octahedral: CN = 6
Two Kinds of Interstitials Holes
in Close Packing Structures
Tetrahedral T+ Tetrahedral T?Octahedral O
Two kinds of Holes
Octahedral Hole Tetrahedral Hole
C.N. = 3
C.N. = 4
C.N. = 6
cubic hole
cuboctahedralhole
16
Fractional Coordinates
Used to locate atoms within unit cell
1. 0, 0, 0
2. ?, ?, 0
3. ?, 0, ?
4. 0, ?, ?
Note: atoms are in
contact along face
diagonals (close packed)
Octahedral Sites
Coordinate ?, ?, ?
Distance = a/2
Coordinate 0, ?, 0 [=1, ?, 0 ]
Distance = a/2
In a face centered cubic anion array, cation octahedral
sites at: ? ? ?, ? 0 0, 0 ? 0, 0 0 ?
Can divide the FCC unit cell into 8 ‘minicubes’
by bisecting each edge; in the center of each
minicube is a tetrahedral site
Location and Number of Octahedral
Holes in a FCC (CCP) Unit Cell
Z = 4
(number of atoms in the
unit cell)
N = 4
(number of octahedral
holes in the unit cell)
So 8 Tetrahedral Sites in a FCC
4 anions per unit cell at: 000, ??0, 0??,
?0?
4 octahedral sites at: ???, 00?, ?00, 0?0
4 tetrahedral T+ sites at: ??? , ???,
???,???
4 tetrahedral T? sites at: ???, ???, ???
???
A variety of different structures form by
occupying T+ 、 T? and O sites to differing
amounts: they can be empty, part full or full.
Can also vary the anion stacking sequence -
CCP or HCP
Sizes of Interstitials
FCC
Spheres are in contact along face
diagonals
octahedral site, bond distance = a/2
radius of octahedral site = (a/2) ? r
tetrahedral site, bond distance = a√3/4
radius of tetrahedral site = (a√3/4) ? r
