“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

Crystal systems

Bravais lattices

Symmetry

Structure Determination

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

Crystal ?

5/87

Cubic Crystals?

a=b=c; ===90

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

7 crystal SystemsCrystal System Conventional Unit Cell

1. Cubic a=b=c, ===90

2. Tetragonal a=bc, ===90

3. Orthorhombic abc, ===90

4. Hexagonal a=bc, == 90, =120

5. Rhombohedral a=b=c, ==90 OR Trigonal

6. Monoclinic abc, ==90

7. Triclinic abc,

Unit cell description : 5

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

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

A 3D translationally periodic arrangement of points

Each lattice point in a lattice has identical neighbourhood

of other lattice points.

Lattice

+

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

Classification of lattice

The Seven Crystal SystemAnd

The Fourteen Bravais Lattices

22/87

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

TABLE 3.1

14 Bravais lattices divided into seven crystal systems

Crystal system Bravais lattices

1. Cubic P I F

Simple cubicPrimitive cubicCubic P

Body-centred cubicCubic I

Face-centred cubicCubic F

Orthorhombic CEnd-centred orthorhombicBase-centred orthorhombic

14 Bravais lattices divided into seven crystal systems

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

?

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 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

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 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

2012 Civil Engineers: 13

lattices !!!

AML120IIT-D

X

1856: 14 lattices

History:

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?

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

Lattices are classified on the

basis of their symmetry

What is symmetry?

If an object is brought into self-coincidence after some operation it said to possess symmetry with respect to that operation.

Symmetry

NOW NO SWIMS ON MON

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 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)

Lattices also have translational symmetry

Translational symmetry

In fact this is the defining symmetry of a lattice

Symmetry of lattices

Lattices have

Rotational symmetry

Reflection symmetry

Translational symmetry

classification of lattices

Based on the complete symmetry, i.e., rotational, reflection and translational symmetry

14 types of lattices 14 Bravais lattices

Based on the rotational and reflection symmetry alone (excluding translations)

7 types of lattices 7 crystal systems

44/87 7 crystal Systems

Cubic

Defining Crystal system Conventional symmetry unit cell

4

1

3

1

1

1

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,

45/87 Tetragonal symmetry Cubic symmetry

Cubic C = Tetragonal P Cubic F Tetragonal I

The three Bravais lattices in the cubic crystal system have the same rotational symmetry but different translational symmetry.

Simple cubicPrimitive cubicCubic P

Body-centred cubicCubic I

Face-centred cubicCubic F

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]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

Tetragonal

Miller 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=2.66

c=4.95

Q1: How do we determine the crystal structure?

Incident Beam Transmitted Beam

Diffra

cted B

eam

Sample

DiffractedBeam

X-Ray Diffraction

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!!!

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

Q

R

Constructive inteference

sin2 hkldn

Bragg’s law

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

X-raysCharacteristic Radiation, K

Target

Mo

Cu

Co

FeCr

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

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-ray tube

detector

Crystal monochromat

or

X-ray powder diffractometer

The diffraction pattern of austenite

Austenite = 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 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

2sin 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.

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