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“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”
Contents
Crystal
Crystal, Lattice and Motif
Miller Indices
Crystal systems
Bravais lattices
Symmetry
Structure Determination
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 description : 1
The description of a unit cell requires:
1. Its Size and shape (lattice parameters)
2. Its atomic content
(fractional coordinates)
Unit cell description : 3
Size and shape of the unit cell:
1. A corner as origin
2. Three edge vectors {a, b, c} from the origin define
a CRSYTALLOGRAPHIC COORDINATE
SYSTEM
3. The three lengths a, b, c and the three interaxial angles , , are called the LATTICE PARAMETERS
a
b
c
Unit cell description : 4
A 3D translationally periodic arrangement of points
Each lattice point in a lattice has identical neighbourhood
of other lattice points.
Lattice
Crystal System Bravais Lattices
1. Cubic P I F
2. Tetragonal P I
3. Orthorhombic P I F C
4. Hexagonal P
5. Trigonal P
6. Monoclinic P C
7. Triclinic P
P: Simple; I: body-centred; F: Face-centred; C: End-centred
7 Crystal Systems and 14 Bravais Lattices
The three cubic Bravais lattices
Crystal system Bravais lattices
1. Cubic P I F
Simple cubicPrimitive cubicCubic P
Body-centred cubicCubic I
Face-centred cubicCubic F
7 crystal SystemsUnit Cell Shape Crystal System
1. a=b=c, ===90 Cubic
2. a=bc, ===90 Tetragonal
3. abc, ===90 Orthorhombic
4. a=bc, == 90, =120 Hexagonal
5. a=b=c, ==90 Rhombohedral OR Trigonal
6. abc, ==90 Monoclinic
7. abc, Triclinic
Why half the boxes are empty?
E.g. Why cubic C is absent?
Crystal System Bravais Lattices
1. Cubic P I F
2. Tetragonal P I
3. Orthorhombic P I FC
4. Hexagonal P
5. Trigonal P
6. Monoclinic P C
7. Triclinic P
?
14 Bravais lattices divided into seven crystal systems
Crystal system Bravais lattices
1. Cubic P I F C
2. Tetragonal P I
3. Orthorhombic P I F C
4. Hexagonal P
5. Trigonal P
6. Monoclinic P C
7. Triclinic P
14 Bravais lattices divided into seven crystal systems
Crystal system Bravais lattices
1. Cubic P I F C
2. Tetragonal P I
3. Orthorhombic P I F C
4. Hexagonal P
5. Trigonal P
6. Monoclinic P C
7. Triclinic P
Couldn’t find his photo on the net
1811-1863
Auguste Bravais
1850: 14 lattices1835: 15 lattices
ML Frankenheim 1801-1869
26th July 2013: AML120 Class
Sem I 2013-2014
13 lattices !!!
IIT-Delhi
X1856: 14 lattices
History:
Primitivecell
Primitivecell
Non-primitive cell
A unit cell of a lattice is NOT unique.
If the lattice points are only at the corners, the unit cell is primitive otherwise non-primitive
UNIT CELLS OF A LATTICE
Unit cell shape CANNOT be the basis for classification of Lattices
Primitivecell
UNIT CELLS OF A LATTICE A good after-class question from the last class:If we are selecting smallest possible region as a unit cell, why can’t we select a triangular unit cell?Unit cell is a small volume of the crystal which by periodic repetition generates the entire crystal (without overlaps or gaps)
Why can’t the Face-Centred Cubic lattice
(Cubic F) be considered as a Body-
Centred Tetragonal lattice (Tetragonal I) ?
If an object is brought into self-coincidence after some operation it said to possess symmetry with respect to that operation.
Symmetry?
Lattices also have translational symmetry
Translational symmetry
In fact this is the defining symmetry of a lattice
If an object come into self-coincidence through smallest non-zero rotation angle of then it is said to have an n-fold rotation axis where
0360n
=180
=90
Rotation Axis
n=2 2-fold rotation axis
n=4 4-fold rotation axis
Crsytallographic Restriction
5-fold symmetry or Pentagonal symmetry is not possible for Periodic TilingsSymmetries higher than 6-fold also not possible
Only possible rotational symmetries for lattices
2 3 4 5 6 7 8 9…
The group of all symmetry elements of a crystal except translations (e.g. rotation, reflection etc.) is called its
POINT GROUP.
The complete group of all symmetry elements including translations of a crystal is called its SPACE GROUP
Point Group and Space Group
Classification of lattices
Based on the space group symmetry, i.e., rotational, reflection and translational symmetry
14 types of lattices 14 Bravais lattices
Based on the point group symmetry alone (i.e. excluding translational symmetry 7 types of lattices
7 crystal systems
Crystal systems and Bravais Lattices
Classification of Lattices
37/87
7 crystal Systems
Cubic
Defining Crystal system Conventional symmetry unit cell
4
A single
3
1
A single
A single
none
Tetragonal
Orthorhombic
Hexagonal
Rhombohedral
Triclinic
Monoclinic
a=b=c, ===90
a=bc, ===90
abc, ===90
a=bc, == 90, =120
a=b=c, ==90
abc, ==90
abc,
A 3D translationally periodic arrangement of atoms
Crystal
A 3D translationally periodic arrangement of points
Lattice
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
Feynman’s Lectures on Physics Vol 1 Chap 1 Fig. 1-4
“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.”
Hexagonal symmetry
o606
360
Correction: Shift the box
One suggested correction:
But gives H:O = 1.5 : 1
http://www.youtube.com/watch?v=kUuDG6VJYgA
The errata has been accepted by Michael Gottlieb of Caltech and the
corrections will appear in future editions
Website www.feynmanlectures.info
Miller Indices of directions and planes
William Hallowes Miller(1801 – 1880)
University of Cambridge
Miller Indices 1
1. Choose a point on the direction as the origin.
2. Choose a coordinate system with axes parallel to the unit cell edges.
x
y 3. Find the coordinates of another point on the direction in terms of a, b and c
4. Reduce the coordinates to smallest integers. 5. Put in square brackets
Miller Indices of Directions
[100]
1a+0b+0c
z
1, 0, 0
1, 0, 0
Miller Indices 2
y
zMiller indices of a direction represents only the orientation of the line corresponding to the direction and not its position or sense
All parallel directions have the same Miller indices
[100]x
Miller Indices 3
x
y
z
O
A 1/2, 1/2, 1
[1 1 2]
OA=1/2 a + 1/2 b + 1 c
P
Q
x
y
z
PQ = -1 a -1 b + 1 c-1, -1, 1
Miller Indices of Directions (contd.)
[ 1 1 1 ]
_ _
-ve steps are shown as bar over the number
Miller indices of a family of symmetry related directions
[100]
[001]
[010]
uvw = [uvw] and all other directions related to [uvw] by the symmetry of the crystal
cubic100 = [100], [010],
[001]tetragonal
100 = [100], [010]
CubicTetragonal
[010][100]
Miller Indices 4
5. Enclose in parenthesis
Miller Indices for planes
3. Take reciprocal
2. Find intercepts along
axes in terms of respective
lattice parameters
1. Select a crystallographic
coordinate system with origin not
on the plane
4. Convert to smallest integers in
the same ratio
1 1 1
1 1 1
1 1 1
(111)
x
y
z
O
Miller Indices for planes (contd.)
origin
intercepts
reciprocalsMiller Indices
AB
CD
O
ABCD
O
1 ∞ ∞
1 0 0
(1 0 0)
OCBE
O*
1 -1 ∞
1 -1 0
(1 1 0)_
Plane
x
z
y
O*
x
z
E
Zero represents
that the plane is parallel to
the corresponding
axis
Bar represents a negative intercept
Miller indices of a plane specifies only its orientation in space not its position
All parallel planes have the same Miller Indices
AB
CD
O
x
z
y
E
(100)
(h k l ) (h k l )_ _ _
(100) (100)_
Miller indices of a family of symmetry related planes
= (hkl ) and all other planes related to (hkl ) by the symmetry of the crystal
{hkl }
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)
{100}cubic = (100), (010), (001)
{100}tetragonal = (100), (010)
(001)
Cubic
TetragonalMiller indices of a family of symmetry related planes
x
z
y
z
x
y
Some IMPORTANT Results
Condition for a direction [uvw] to be parallel to a plane or lie in the plane (hkl):
h u + k v + l w = 0
Weiss zone law
True for ALL crystal systems
Not in the textbook
CUBIC CRYSTALS
[hkl] (hkl)
Angle between two directions [h1k1l1] and [h2k2l2]:
C
[111]
(111)
22
22
22
21
21
21
212121coslkhlkh
llkkhh
dhkl
Interplanar spacing between ‘successive’ (hkl) planes passing through the corners of the unit cell
222 lkh
acubichkld
O
x(100)
ad 100
BO
x
z
E
2011
ad
[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
Summary of Notation convention for Indices
In the fell clutch of circumstanceI have not winced nor cried aloud.Under the bludgeonings of chanceMy head is bloody, but unbowed.From "Invictus" by
William Ernest Henley (1849–1903).
Some crystal structures
Crystal Lattice Motif Lattice parameter
Cu FCC Cu 000 a=3.61 Å
Zn Simple Hex Zn 000, Zn 1/3, 2/3, 1/2
a=2.66c=4.95
Incident Beam
X-Ray Diffraction
Transmitted Beam
Diffra
cted
BeamSample
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.
≡ Bragg Reflection
X-Ray Diffraction
Braggs’ recipe for Nobel prize?
Call the diffraction a reflection!!!
“The important thing in science is not so much to obtain new facts as to discover new ways of thinking about them”.
W.L. Bragg
Braggs Law (Part 1): the diffracted beam appears to be specularly reflected from a set of crystal lattice planes.
Specular reflection:Angle of incidence =Angle of reflection (both measured from the plane and not from the normal)
The incident beam, the reflected beam and the plane normal lie in one plane
X-Ray Diffraction
i
plane
r
sin2n
dhkl
sin2 hkldn
sin2 nlnknhd
n
d
nlnknh
ad hkl
nlnknh
222,,
)()()(
Two equivalent ways of stating Bragg’s Law
1st Form
2nd Form
sin2 hkldn sin2 nlnknhd
nth order reflection from (hkl) plane
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
Two equivalent ways of stating Bragg’s Law
Powder Method
l 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
21Incident beam Transmitted
beam
Diffracted
beam 1
Diffracted
beam 2X-ray detector
Zero intensity
Strong intensity
sample
Powder diffractometer geometry
i plane
r
t21 22 2
Inte
nsi
ty
x
y
zd100 = a
100 reflection= rays reflected from adjacent (100) planes spaced at d100 have a path difference
/2
No 100 reflection for bcc
Bcc crystal
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
DC
h, k and l are all oddOr
if all are even then (h + k + l) divisible by 4
Diffraction analysis of cubic crystals
sin2 hkld
µq2sin ++ 222 )lkh(constant
Bragg’s Law:
222 lkh
adhkl
Cubic crystals
(1)
(2)
(2) in (1) =>
)(4
sin 2222
22 lkh
a
h2 + k2 + l2 SC FCC BCC DC
1 100
2 110 110
3 111 111 111
4 200 200 200
5 210
6 211 211
7
8 220 220 220 220
9 300, 221
10 310 310
11 311 311 311
12 222 222 222
13 320
14 321 321
15
16 400 400 400 400
17 410, 322
18 411, 330 411, 330
19 331 331 331
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…
19.022.533.039.041.549.556.559.069.584.0
sin2
0.110.150.300.400.450.580.700.730.880.99
2468
101214161820bcc
h2+k2+l2
123456891011sc
h2+k2+l2 h2+k2+l2
348
11121619202427fcc
This is an fcc crystal
Ananlysis of a cubic diffraction patternp sin2
1.01.42.83.84.15.46.66.98.39.3
p=9.43
p sin2
2.84.08.1
10.812.015.819.020.123.927.0
p=27.3p sin2
22.85.67.48.310.913.113.616.618.7
p=18.87
a
4.054.024.024.044.024.044.034.044.014.03
hkl
111200220311222400331420422511
19.022.533.039.041.549.556.559.069.584.0
h2+k2+l2
348
11121619202427
Indexing of diffraction patterns
The diffraction pattern is
from an fcc crystal of
lattice parameter
4.03 Å
Ananlysis of a cubic diffraction pattern contd.
ql
=++ 22
2222 sina4)lkh(
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
William Henry Bragg (1862–1942), William Lawrence Bragg (1890–1971)
Nobel Prize (1915)
A father-son team that shared a Nobel Prize
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
"However, the data which really helped us to obtain the structure was mainly obtained by Rosalind Franklin".
-Francis Crick
Rosy, of course, did not directly give us her data. For that matter, no one at King's realized they were in our hands.
J.D. Watson