“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 3Crystal Geometry

and Structure Determination

ContentsCrystal

Crystal, Lattice and Motif

Miller Indices

Crystal systems

Bravais lattices

Symmetry

Structure Determination

A 3D translationaly periodic arrangement of atoms in space is called a crystal.

Crystal ?

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 most common shape of a unit cell is a parallelopiped.

Unit cell description : 2UNIT CELL:

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

Lattice?A 3D translationally periodic arrangement of points in space is called a lattice.

A 3D translationally periodic arrangement of points

Each lattice point in a lattice has identical neighbourhood

of other lattice points.

Lattice

Classification of latticeThe Seven Crystal System

And

The Fourteen Bravais Lattices

Crystal System Bravais Lattices

1. Cubic P I F2. Tetragonal P I3. Orthorhombic P I F C4. Hexagonal P5. Trigonal P6. Monoclinic P C7. 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 lattices1. Cubic P I F

Simple cubicPrimitive cubicCubic P

Body-centred cubicCubic I

Face-centred cubicCubic F

Orthorhombic CEnd-centred orthorhombicBase-centred orthorhombic

15/87

Cubic Crystals?

a=b=c; ===90

7 crystal SystemsUnit Cell Shape Crystal System

1. a=b=c, ===90 Cubic2. a=bc, ===90 Tetragonal3. abc, ===90 Orthorhombic4. a=bc, == 90, =120 Hexagonal5. a=b=c, ==90 Rhombohedral OR Trigonal6. 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 F2. Tetragonal P I3. Orthorhombic P I F

C4. Hexagonal P5. Trigonal P6. Monoclinic P C7. Triclinic P

?

End-centred cubic not in the Bravais list ?

End-centred cubic = Simple Tetragonal

2a

2a

14 Bravais lattices divided into seven crystal systems

Crystal system Bravais lattices1. Cubic P I F C2. Tetragonal P I3. Orthorhombic P I F C4. Hexagonal P5. Trigonal P6. Monoclinic P C7. Triclinic P

Now apply the same procedure to the FCC lattice

Cubic F = Tetragonal I ?!!!

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

1811-1863Auguste 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

End of Lec 3 (Lec 1 on crystallography)

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) ?

What is the basis for classification of lattices

into 7 crystal systems

and 14 Bravais lattices?

Lattices are classified on the

basis of their symmetry

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

Rotational Symmetries

Z180 120 90 72 60

2 3 4 5 6

45

8

Angles:

Fold:

Graphic symbols

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…

Symmetry of lattices

Lattices have

Rotational symmetry

Reflection symmetry

Translational symmetry

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,

38/87

Tetragonal symmetry Cubic symmetry

Cubic C = Tetragonal P Cubic F Tetragonal I

End of Lec4 on 30.07.2013Lec2 on crystallography

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

+

Love PatternLove Lattice + Heart =

=

Lattice + Motif = Crystal

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

crystal

+ Motif

Richard P. Feynman

Nobel Prize in Physics, 1965

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

QUESTIONS?

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] tetragonal100 = [100], [010]

CubicTetragonal

[010][100]

Miller Indices 4

End of Lec 05 31/07/2013Lec 3 on crystallography

Lec 6

Lec 4 on crystallography

Miller Indices Continues

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

A B

CD

O

ABCDO

1 ∞ ∞1 0 0

(1 0 0)

OCBEO*

1 -1 ∞ 1 -1 0

(1 1 0)_

Plane

x

z

yO*

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

A B

CD

O

x

z

y

E

(100)

(h k l ) (h k l )_ _ _

(100) (100)

_

Y

Z

X

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

dhklInterplanar spacing between ‘successive’ (hkl) planes passing through the corners of the unit cell

222 lkhacubic

hkld

O

x(100)

ad 100

BO

x

z

E

2011ad

[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).

End of Lecture 6Lec 4 on crystallography

Some crystal structuresCrystal Lattice Motif Lattice

parameterCu FCC Cu 000 a=3.61 Å

Zn Simple Hex Zn 000, Zn 1/3, 2/3, 1/2

a=2.66c=4.95

Q1: How do we determine the crystal structure?

Incident Beam Transmitted Beam

Diffracte

d Beam

Sample

DiffractedBeam

X-Ray Diffraction

Incident Beam

X-Ray Diffraction

Transmitted Beam

Diffract

ed

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

X-Ray Diffraction

i

r

dhkl

Bragg’s law (Part 2):

sin2 hkldn

i

r

Path Difference =PQ+QR sin2 hkld

P

Q

R

dhkl

Path Difference =PQ+QR sin2 hkld

i r

P

QR

Constructive inteference

sin2 hkldn

Bragg’s law

sin2n

dhkl

sin2 hkldn

sin2 nlnknhd

nd

nlnknh

ad hklnlnknh

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

X-raysCharacteristic Radiation, K

TargetMoCuCoFe Cr

Wavelength, Å0.711.541.791.942.29

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

21Incident beam Transmitted

beam

Diffracted

beam 1

Diffracted

beam 2 X-ray detector

Zero intensity

Strong intensity

sample

Powder diffractometer geometry

i plane

r

t21 22 2

Inte

nsity

X-ray tube

detector

Crystal monochromat

or

X-ray powder diffractometer

The diffraction pattern of austeniteAustenite = fcc Fe

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 AllBCC (h + k + l) evenFCC h, k and l unmixed

DCh, k and l are all odd

Orif all are even then

(h + k + l) divisible by 4

End of Lec 706/08/13

Lec 5 on crystallography(Last lecture)

Diffraction analysis of cubic crystals

sin2 hkld

µ2sin 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 DC1 1002 110 1103 111 111 1114 200 200 2005 2106 211 21178 220 220 220 2209 300, 221

10 310 31011 311 311 31112 222 222 22213 32014 321 3211516 400 400 400 40017 410, 32218 411, 330 411, 33019 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

a4.054.024.024.044.024.044.034.044.014.03

hkl111200220311222400331420422511

19.022.533.039.041.549.556.559.069.584.0

h2+k2+l2348

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.

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

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

Watson, Crick and Wilkins

Nobel Prize, 1962

Rosalind Elsie Franklin 

"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

Google Doodle on 25th July, 2013

Franklins birth centenary 25 July 2020

Rosalind Elsie Franklin 

25 July 1920 – 16 April 1958