9-1

Today the structure determinations etc are all computer -assisted

It is instructive however for you to do a simple structure by hand

Rocksalt Structure

Quite common in nature

KCl, NaCl, MgO

Typical XRD scan

d spacing

5 %7 %

29 %32 %

…100 %

…

32 counts40 counts

173 counts192 counts

…590 counts

…

20.32 °20.66 °21.07 °28.12 °

…32.88 °

…

1234…9…

RelativeIntensity

AbsoluteIntensity

2θangle

Peak No

λ = 2 d sinθ

λ/(2 sinθ) = d

X-ray wavelength used:CoKα = 1.7889 Å

5.07 Å4.98 Å4.89 Å3.68 Å

…3.16 Å

…

9-2

9-3

A comparison of the X-ray powder diffraction patterns of NaCl (bottom) and KCl (top). Peaks in the KCl diffraction pattern are labeled with Miller indices, h k l, indicating the set of lattice planes responsible for that diffraction peak. The KClpeaks are shifted to lower angles relative to the NaCl pattern due to the larger cubic unit cell of KCl.

9-4

Systematic absences

Sometimes particular hkl reflections are "missing"

this can be useful

let's see why and how

‘Systematic absences’ in XRDWe have seen that in a PXRD experiment a crystalline solid gives peaks at certain angles (2θ) only. These peaks occur when there is constructive interference between the X-ray scattered from the crystal. As given by Bragg’s law.

However some of the “expected” peaks could be absent due to additional symmetry in the crystal

Eg. For a BCC crystal, X-rays scattered from the atom at the centre of the cube can interfere can interfere with those scattered by the corner atoms.

9-5

Systematic absences for BCC & FCC

BCC - Reflections absent if h+k+l = ODD

FCC - Absent UNLESS h,k,l are all ODD or all EVEN

(100) (110) (111) (200) (210) (211) (220)…………

(100) (110) (111) (200) (210) (211) (220)…………

9-6

More:

Consider alpha - Iron which is boby centre cubic

Reflections from the 100 planes would be expected

but the reflection from the body centre atom is 180 out of phase with these, (midway between planes)

in a bulk crystal there are equal numbers of corner andbody centre atoms and so the beams diffracted by each cancel exactly

In contrast the 200 beam is present because there are noatoms between the 200 planes

9-7

Lattice Type Rule for reflection to be observed

Primitive P None

Body Centred, I hkl: h+k+l =2n

Face Centred, F hkl: hkl either all odd or all even

Side Centred hkl: h + k = 2n

Rhomohedral hkl: -h + k + l = 3mor (h-k + l) =3n

also see lect 16: Bravais Latices 9-8

For a systematic absence two conditions must be simultaneously met

1.

1. waves are exactly 180 degrees out of phase2. Waves must be exactly the same amplitude

2. is determined by the scattering factor

Good example: rocksalt structure

"face centre cubic"

so using Table above hkl must be all odd or even for reflectionto be observed:

so : 110 reflection is systematically absent 9-9

But look at the 111 planes

Some have Na on them some have Cl

and they are interleaved : looks OK

but the two planes have different scattering powers

X rays are scattered by electrons

so in the series KCl KBr KF KI

the intensity of the 111 for KCl is zero

the trend in intensity is:

KCl < KF < KBr < KI 9-10

Phase purity.In a mixture of compounds each crystalline phase present will contribute to the overall powder X-ray diffraction pattern. In preparative materials chemistry this may be used to identify the level of reaction and purity of the product. The reaction between two solids Al2O3 and MgO to form MgAl2O4 may be monitored by powder X-ray diffraction.

9-11

PXRD line-broadening and crystallite size

Several effects could change PXRD linewidths: crystallite size, overlap of peaks, microstrain, lattice and stacking faults

‘Particle’ ‘Particle’

= one crystallite = five fused crystallites

NB : PXRD measures crystallite size!!9-12

Determination of crystallite size

The effect of crystallite size on peak width in PXRD

(a) Instrumental broadening

(b) 1μm particles

(c) 100 nm

(d) 10 nm

(e) 5 nm

In order to observe sharp diffraction maxima in the PXRD pattern, the crystallites need to be of sufficient size to ensure that slightly away from the 2Θ maximum, destructive interference occurs.

9-13

9-14

XRD and the nano-world: see page 159

In order to produce a nice sharp diffraction pattern you need many many planes toReflect from.

The more planes there are, the more rigid is the Bragg condition. Peaks are sharper and Easier to resolve.

For small particles the peaks broaden. From the broadening you can get a fix on the Particle size. Let’s see how.

Scherrer (1918) first observed that small crystallite size could give rise to peak broadening. He derived a well-known equation for relating the crystallite size to the peak width, which is called the Scherrerformula: t = Kλ/(B cosθ)where t is the averaged dimension of crystallites; K is the Scherrer constant

9-15

Scherrer Formula

T -- crystallite thicknessλ -- wavelengthθ -- Bragg AngleBm , Bs – Peak widths of sample and Standard in Radians

Bs - Also compensates for instrumental broadening

ϑλ

cos9.0

22sm BB

t−

=

180degπϑϑ ×=rad

Paul Scherrer

bismuth zinc niobate nanoparticles

example

SEM and TEM

9-16

9-17

ok enough of X-rays for now. You will learn more in nano2100

Next….particle sizes and surface areas