Announcements

1)Revised Lab timings: 1-3 PM (all groups)

2) Quiz 1, 28th Jan 2014, Tuesday 7:30 PM, WS 209, WS 213

Chapter 3

Crystal Geometry and

Structure Determination

RecapLattice, Motif/Basis

Crystal = Lattice + Motif

e.g. Brass, diamond, ZnS

Miller indices of direction: components of vector w.r.t to basis vector a, b and c

Miller Indices of directions and planes

William Hallowes Miller(1801 – 1880)

University of Cambridge

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

Courtesy: H Bhadhesia

Courtesy: H Bhadhesia

Courtesy: H Bhadhesia

Crystallographically equivalent planes

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

CUBIC CRYSTALS

[hkl] (hkl)

Angle between two directions [h1k1l1] and [h2k2l2]:

C

[111]

(111)

22

22

22

21

21

21

212121coslkhlkh

llkkhh

Some IMPORTANT Results

Weiss zone law

True for ALL crystal systems

Not in the textbook

• If a direction [uvw] lies in a plane (hkl) then

• uh+vk+wl = 0

[uvw

]

(hkl)

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

How do we determine the structure of a piece of crystalline solid?

You can probe the atomic arrangements by X-ray diffraction

(XRD)

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

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

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

X Ray Diffractometer

You do not get indices of plane!!

Cu target, Wavelength = 1.5418 Angstrom

2θ

44.48

51.83

76.35

92.90

98.40

121.87

144.54

Unknown sample, cubic

Determine:1)The crystal structure2)Lattice parameter

5 step program for the determination of crystal structure

1)Start with 2θ values and generate a set of sin2θ values2)Normalise the sin2θ values by dividing it with first entry3)Clear fractions from normalised column: Multiply by common number4) Speculate on the hkl values that, if expressed as h2+k2+l2, would generate the sequence of the “clear fractions” column5) Compute for each sin2θ /(h2+k2+l2) on the basis of the assumed hkl values. If each entry in this column is identical, then the entire process is validated.

2θ Sin2θ Sin2θ/Sin2θ1 Clear fractions

(hkl)? sin2θ /(h2+k2+l2)

44.48 0.143 1.00 3 111 0.0477

51.83 0.191 1.34 4 200 0.0478

76.35 0.382 2.67 8 220 0.0478

92.90 0.525 3.67 11 311 0.0477

98.40 0.573 4.01 12 222 0.0478

121.87 0.764 5.34 16 400 0.0477

144.54 0.907 6.34 19 420 0.0477

William Henry Bragg (1862–1942), William Lawrence Bragg (1890–1971)

Nobel Prize (1915)

A father-son team that shared a Nobel Prize

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

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

X-raysCharacteristic Radiation, K

Target

Mo

Cu

Co

FeCr

Wavelength, Å

0.71

1.54

1.79

1.94

2.29