“There are two things to aim at in
life: first, to get what you want;
and, after that, to enjoy it. Only
the wisest of mankind achieve the
second.”
Logan Pearsall Smith,
Afterthought (1931), “Life and
Human Nature”
Chapter 3
Crystal Geometry
and
Structure Determination
Contents
Crystal
Crystal, Lattice and Motif
Miller Indices
Symmetry
Crystal systems
Bravais lattices
Structure Determination
Crystal ?
A 3D translationaly
periodic arrangement
of atoms in space is
called a crystal.
5/87
Cubic Crystals?
a=b=c; ===90
Unit cell description : 1
Translational Periodicity
One can select a small volume of
the crystal which by periodic
repetition generates the entire
crystal (without overlaps or gaps)
Unit
Cell
UNIT CELL:
Unit cell description : 2
The most common
shape of a unit cell is
a parallelopiped.
The description of a unit cell
requires:
1. Its Size and shape
(lattice parameters)
Unit cell description : 3
2. Its atomic
content
(fractional
coordinates)
Size and shape of the unit cell:
Unit cell description : 4
1. A corner as origin
c
2. Three edge vectors {a, b, c}
from the origin define
a CRSYTALLOGRAPHIC
a
COORDINATE

SYSTEM


b
3. The three
lengths a, b, c and
the three
interaxial angles
, ,  are called the
LATTICE PARAMETERS
Unit cell description : 5
7 crystal Systems
Crystal System
Conventional Unit Cell
1. Cubic
a=b=c, ===90
2. Tetragonal
a=bc, ===90
3. Orthorhombic
abc, ===90
4. Hexagonal
a=bc, == 90, =120
5. Rhombohedral
OR Trigonal
a=b=c, ==90
6. Monoclinic
abc, ==90
7. Triclinic
abc, 
Lattice?
A 3D translationally
periodic arrangement
of points in space is
called a lattice.
Crystal
Lattice
A 3D
translationally
periodic
arrangement
of atoms
A 3D
translationally
periodic
arrangement of
points
What is the relation between
the two?
Crystal = Lattice + Motif
Motif or basis: an atom or
a group of atoms associated
with each lattice point
Crystal=lattice+basis
Lattice:
the underlying periodicity of
the crystal,
Basis:
atom or group of atoms
associated with each lattice points
Lattice:
how to repeat
Motif:
what to repeat
Lattice
A 3D translationally periodic
arrangement of points
Each lattice point in a lattice
has identical neighbourhood of
other lattice points.
Lattice + Motif = Crystal
+
Love Lattice
+ Heart
=
=
Love Pattern
Air,
Water
and
Earth
by
M.C.
Esher
Every
periodic
pattern
(and hence
a crystal)
has a
unique
lattice
associated
with it
The six lattice parameters a, b, c, , , 
The cell of the lattice
lattice
+ Motif
crystal
Classification of lattice
The Seven Crystal System
And
The Fourteen Bravais Lattices
TABLE 3.1
22/87
7 Crystal Systems and 14 Bravais Lattices
Crystal System
Bravais Lattices
1. Cubic
P
I
2. Tetragonal
P
I
3. Orthorhombic
P
I
4. Hexagonal
P
5. Trigonal
P
6. Monoclinic
P
7. Triclinic
P
F
F
C
C
P: Simple; I: body-centred; F: Face-centred; C: End-centred
14 Bravais lattices divided into seven crystal
systems
Crystal system
Bravais lattices
1. Cubic
P
Simple cubic
Primitive cubic
Cubic P
Body-centred cubic
Cubic I
I
F
Face-centred cubic
Cubic F
Orthorhombic C
End-centred orthorhombic
Base-centred orthorhombic
14 Bravais lattices divided into seven crystal
systems
Crystal system
Bravais lattices
1. Cubic
P
I
2. Tetragonal
P
I
3. Orthorhombic
P
I
4. Hexagonal
P
5. Trigonal
P
6. Monoclinic
P
7. Triclinic
P
F
?
F
C
C
End-centred cubic not in the Bravais list ?
End-centred cubic = Simple Tetragonal
14 Bravais lattices divided into seven crystal
systems
Crystal system
Bravais lattices
1. Cubic
P
I
2. Tetragonal
P
I
3. Orthorhombic
P
I
4. Hexagonal
P
5. Trigonal
P
6. Monoclinic
P
7. Triclinic
P
F
C
F
C
C
Face-centred cubic in the Bravais list ?
Cubic F = Tetragonal I ?!!!
14 Bravais lattices divided into seven crystal
systems
Crystal system
Bravais lattices
1. Cubic
P
I
2. Tetragonal
P
I
3. Orthorhombic
P
I
4. Hexagonal
P
5. Trigonal
P
6. Monoclinic
P
7. Triclinic
P
F
C
F
C
C
History:
ML Frankenheim
1801-1869
1835:
15
X lattices
1856: 14 lattices
Couldn’t
find his
photo on
the net
Auguste Bravais
1811-1863
1850: 14 lattices
AML120
IIT-D
2012 Civil
Engineers: 13
lattices !!!
Why can’t the FaceCentred Cubic lattice
(Cubic F) be considered
as a Body-Centred
Tetragonal lattice
(Tetragonal I) ?
What is the basis for
classification of lattices
into
7 crystal systems
and
14 Bravais lattices?
UNIT CELLS OF A LATTICE
Nonprimitive cell
If the lattice
points are only
at the corners,
the unit cell is
primitive
otherwise non-
primitive
Primitive
cell
A unit cell of a
lattice is NOT
unique.
Primitive
cell
Unit cell shape
CANNOT be the
basis for
classification of
Lattices
Lattices are
classified on the
basis of their
symmetry
What is
symmetry?
Symmetry
If an object is brought into selfcoincidence after some
operation it said to possess
symmetry with respect to that
operation.
NOW NO SWIMS ON MON
Rotational Symmetries
Z
Angles:
180
120
90
72
60
45
3
4
5
6
8
Fold:
2
Graphic symbols
Crsytallographic Restriction
5-fold symmetry or Pentagonal symmetry is
not possible for crystals
Symmetries higher than 6-fold also not
possible
Only possible rotational symmetries for
periodic tilings and crystals:
2
3
4
5
6
7
8
9…
Reflection (or mirror symmetry)
Translational symmetry
Lattices also have
translational
symmetry
In fact this is the
defining symmetry of
a lattice
Symmetry of lattices
Lattices have
Translational symmetry
Rotational symmetry
Reflection symmetry
Based on the rotational and reflection
symmetry alone (excluding translations)
classification of lattices
 7 types of lattices
 7 crystal systems
Based on the complete symmetry, i.e.,
rotational, reflection and translational
symmetry
 14 types of lattices
 14 Bravais lattices
7 crystal Systems
44/87
Defining
symmetry
Crystal system
Conventional
unit cell
4
Cubic
a=b=c, ===90
1
Tetragonal
a=bc, ===90
3
Orthorhombic abc, ===90
1
Hexagonal
1
Rhombohedral a=b=c, ==90
1
Monoclinic
abc, ==90
none
Triclinic
abc, 
a=bc, == 90, =120
45/87
Tetragonal symmetry
Cubic C = Tetragonal P
Cubic symmetry
Cubic F  Tetragonal I
The three Bravais lattices in the cubic crystal
system have the same rotational symmetry but
different translational symmetry.
Simple cubic
Primitive cubic
Cubic P
Body-centred cubic
Cubic I
Face-centred cubic
Cubic F
Richard P. Feynman
Nobel Prize in Physics, 1965
Feynman’s
Lectures
on Physics
Vol 1
Chap 1
Fig. 1-4
Hexagonal
symmetry
360

 60 o
6
“Fig. 1-4 is an invented arrangement for ice, and
although it contains many of the correct features of the
ice, it is not the true arrangement. One of the correct
features is that there is a part of the symmetry that is
hexagonal. You can see that if we turn the picture
around an axis by 120°, the picture returns to itself.”
Correction:
Shift the
box
One suggested
correction:
But gives H:O = 1.5 : 1
http://www.youtube.com/watc
h?v=kUuDG6VJYgA
The errata has been accepted by
Michael Gottlieb of Caltech and the
corrections will appear in future
editions
Website www.feynmanlectures.info
QUESTIONS?
Miller Indices 1
Miller Indices of directions and
planes
William Hallowes Miller
(1801 – 1880)
University of Cambridge
Miller Indices 2
Miller Indices of Directions
z
1. Choose a point on the direction
as the origin.
2. Choose a coordinate system
with axes parallel to the unit
cell edges.
y 3. Find the coordinates of
another point on the direction in
terms of a, b and c
x 1a+0b+0c
1, 0, 0
4. Reduce the coordinates to smallest integers. 1, 0, 0
5. Put in square brackets [100]
Miller Indices 3
z
y
x
Miller indices of a
direction represents only
the orientation of the line
corresponding to the
direction and not its
position or sense
[100]
All parallel directions have the
same Miller indices
Miller Indices of
Directions (contd.)
z
OA=1/2 a + 1/2 b + 1 c
Q
A
z
1/2, 1/2, 1
[1 1 2]
y
y
O
P
x
PQ = -1 a -1 b + 1 c
-1, -1, 1
__
[111]
x
-ve steps are shown as bar over the number
Miller Indices 4
Miller indices of a family of symmetry related directions
uvw = [uvw] and all other directions related to
[uvw] by the symmetry of the crystal
[001]
Tetragonal
Cubic
[010]
[010]
[100]
100 cubic
= [100], [010],
[001]
[100]
100
= [100], [010]
tetragonal
z
Miller Indices for planes
1. Select a crystallographic
coordinate system with origin not
on the plane
2. Find intercepts along
axes in terms of respective
lattice parameters
O
x
y
1 1 1
3. Take reciprocal 1 1 1
4. Convert to smallest integers in
the same ratio
1 1 1
5. Enclose in parenthesis
(111)
Miller Indices for planes (contd.)
z
z
E
Plane
ABCD
OCBE
origin
O
O*
intercepts 1
∞ ∞
reciprocals 1 0 0
A
B
O
O*
y
D
x
C
x
Miller
Indices
(1 0 0)
Zero
represents that
the plane is
parallel to the
corresponding
axis
1 -1
∞
1 -1 0
_
(1 1 0)
Bar
represents
a negative
intercept
Miller indices of a plane
specifies only its orientation in
space not its position
z
E
A
All parallel planes have
the same Miller Indices
_ _ _
(h k l )  (h k l )
B
O
y
D
C
x
(100)
_
(100)  (100)
Miller indices of a family of symmetry related planes
{hkl }
= (hkl ) and all other planes related to
(hkl ) by the symmetry of the crystal
All the faces of the cube
are equivalent to each
other by symmetry
Front & back faces: (100)
Left and right faces: (010)
Top and bottom faces: (001)
{100} = (100), (010), (001)
Miller indices of a family of
symmetry related planes
Cubic
z
Tetragonal
z
y
y
x
{100}cubic = (100), (010), (001)
x
{100}tetragonal = (100), (010)
(001)
Some IMPORTANT Results
Weiss zone law
Not in the textbook
Condition for a direction [uvw] to be parallel to a plane or lie in
the plane (hkl):
hu+kv+lw=0
True for ALL crystal systems
CUBIC CRYSTALS
[111]
[hkl]  (hkl)
(111)
Angle between two directions [h1k1l1] and [h2k2l2]:
cos 
h1h2  k1k2  l1l2
h12  k12  l12 h22  k22  l22
C
dhkl
Interplanar spacing
between ‘successive’ (hkl)
planes passing through the
corners of the unit cell
cubic
d hkl

a
h 2  k 2 l 2
z
E
B
O
O
d100  a
x
(100)
d1 1 0 
x
a
2
Summary of Notation convention for Indices
[uvw]
Miller indices of a direction (i.e. a set of
parallel directions)
(hkl)
Miller Indices of a plane (i.e. a set of
parallel planes)
<uvw>
Miller indices of a family of symmetry
related directions
{hkl}
Miller indices of a family of symmetry
related planes
In the fell clutch of circumstance
I have not winced nor cried aloud.
Under the bludgeonings of chance
My head is bloody, but unbowed.
From "Invictus" by William
Ernest Henley (1849–1903).
Some crystal structures
Crystal
Lattice
Motif
Lattice
parameter
a=3.61 Å
Cu
FCC
Cu 000
Zn
Simple Hex Zn 000,
a=2.66
Zn 1/3, 2/3, 1/2 c=4.95
Q1: How do
we determine
the crystal
structure?
X-Ray Diffraction
Sample
Incident Beam
Transmitted Beam
Diffracted
Beam
X-Ray Diffraction
≡ Bragg Reflection
Sample
Incident Beam
Transmitted
Beam
Braggs Law (Part 1): For every diffracted
beam there exists a set of crystal lattice
planes such that the diffracted beam
appears to be specularly reflected from this
set of planes.
X-Ray Diffraction
Braggs’ recipe for Nobel prize?
Call the diffraction a reflection!!!
X-Ray Diffraction
Braggs Law (Part 1): the diffracted beam
appears to be specularly reflected from a set of
crystal lattice planes.
Specular reflection:
Angle of incidence
i
=Angle of reflection

(both measured from the
plane
plane and not from
the normal)
r

The incident beam, the reflected beam and the
plane normal lie in one plane
X-Ray Diffraction
r
i

dhkl

Bragg’s law (Part 2):
n  2dhkl sin 
r
i


 
P
R
dhkl
Q
Path Difference =PQ+QR  2d hkl sin 
i
r


P
R
Q
Path Difference =PQ+QR  2d hkl sin 
Constructive inteference
n  2dhkl sin 
Bragg’s law
Two equivalent ways of stating Bragg’s Law
n  2dhkl sin 
1st Form
d hkl
 2
sin 
n
d nh,nk ,nl
d hkl


2
2
2
n
(nh)  (nk)  (nl)
a
   2dnh nk nl sin 
2nd Form
Two equivalent ways of stating Bragg’s Law
n  2dhkl sin 
nth order
reflection from
(hkl) plane
   2dnh nk nl sin 

1st order
reflection from
(nh nk nl) plane
e.g. a 2nd order reflection from (111)
plane can be described as 1st order
reflection from (222) plane
X-rays
Characteristic Radiation, K
Target
Mo
Cu
Co
Fe
Cr
Wavelength, Å
0.71
1.54
1.79
1.94
2.29
Powder Method
 is fixed (K radiation)
 is variable – specimen consists of
millions of powder particles – each
being a crystallite and these are
randomly oriented in space – amounting
to the rotation of a crystal about all
possible axes
Powder
diffractometer
geometry
Strong
intensity
Diffracted
beam 2
i



plane
t
r
Transmitted
sample
beam
Intensity
Incident beam
21
Zero
Diffracted intensity
beam 1
X-ray
detector
21
22
2
Crystal
monochromator
detector
X-ray
tube
X-ray powder diffractometer
The diffraction pattern of austenite
Austenite = fcc Fe
Bcc crystal
x

/2
d100 = a
z
100 reflection= rays
reflected from
adjacent (100) planes
spaced at d100 have a
path difference 
y
No 100 reflection for bcc
No bcc reflection for h+k+l=odd
Extinction Rules: Table 3.3
Bravais Lattice
Allowed Reflections
SC
All
BCC
(h + k + l) even
FCC
h, k and l unmixed
h, k and l are all odd
DC
Or
if all are even then
(h + k + l) divisible by 4
Diffraction analysis of cubic crystals
Bragg’s Law:
  2d hkl sin 
(1)
Cubic crystals
d hkl 
(2)
a
h2  k 2  l 2
2
sin

(2) in (1) =>
2
4a 2
(h 2  k 2  l 2 )
constant
2
2
2



sin
 (h k l )
2
h2 + k2 + l2
SC
1
100
2
110
3
111
111
4
200
200
5
210
6
211
FCC
BCC
DC
110
111
200
211
7
8
220
220
9
300, 221
10
310
11
311
311
12
222
222
13
320
14
321
220
220
310
311
222
321
15
16
400
17
410, 322
18
411, 330
19
331
400
400
400
411, 330
331
331
Crystal Structure
Allowed ratios of Sin2 (theta)
SC
1: 2: 3: 4: 5: 6: 8: 9…
BCC
1: 2: 3: 4: 5: 6: 7: 8…
FCC
3: 4: 8: 11: 12…
DC
3: 8: 11:16…
Ananlysis of a cubic diffraction pattern

19.0
22.5
33.0
39.0
41.5
49.5
56.5
59.0
69.5
84.0
sin2
0.11
0.15
0.30
0.40
0.45
0.58
0.70
0.73
0.88
0.99
p sin2 h2+k2+l2 p sin2 h2+k2+l2
p=18.87
p=9.43
1.0
1.4
2.8
3.8
4.1
5.4
6.6
6.9
8.3
9.3
1
2
3
4
5
6
8
9
10
11
sc
2
2.8
5.6
7.4
8.3
10.9
13.1
13.6
16.6
18.7
2
4
6
8
10
12
14
16
18
20
bcc
p sin2 h2+k2+l2
p=27.3
2.8
4.0
8.1
10.8
12.0
15.8
19.0
20.1
23.9
27.0
3
4
8
11
12
16
19
20
24
27
fcc
This is an
fcc crystal
Ananlysis of a cubic diffraction pattern contd.

19.0
22.5
33.0
39.0
41.5
49.5
56.5
59.0
69.5
84.0
h2+k2+l2
3
4
8
11
12
16
19
20
24
27
hkl
a
111
200
220
311
222
400
331
420
422
511
4.05
4.02
4.02
4.04
4.02
4.04
4.03
4.04
4.01
4.03
Indexing of
diffraction patterns
2
4
a
(h  k  l )  2 sin2 

2
2
2
The
diffraction
pattern is
from an fcc
crystal of
lattice
parameter
4.03 Å
Education is an admirable
thing, but it is well to
remember from time to
time that nothing that is
worth knowing can be
taught.
-Oscar Wilde
A father-son team that shared a Nobel
Prize
William Henry Bragg (1862–1942),
William Lawrence Bragg (1890–1971)
Nobel Prize (1915)
One of the
greatest scientific
discoveries of
twentieth century
Max von Laue, 1879-1960
Nobel 1914
Two Questions
Q1: X-rays waves or particles?
Father Bragg: Particles Son Bragg: Waves
“Even after they shared a Nobel Prize in 1915,
… this tension persisted…”
– Ioan James in Remarkable Physicists
Q2:Crystals: Perodic arrangement of atoms?
X-RAY DIFFRACTION: X-rays are waves and
crystals are periodic arrangement of atoms
If it is permissible to
evaluate a human discovery
according to the fruits which
it bears then there are not
many discoveries ranking on
par with that made by von
Laue.
-from Nobel Presentation Talk