Lecture 7Crystal Geometry

and Structure Determination

Today’s lecture

1.Diffractometry: Radiation of fixed wavelength, varying angle: Park the beam and then rotate the sample Output: Intensity vs. angle

2. Laue reflection: Radiation of variable wavelength, fixed angle: each wavelengthdiffracts at different angleOutput: symmetry of spot pattern

Two techniques for structure determination

How Characteristic X-rays are generated??

3

Characteristic X-rays are produced by electron transitionsbetween the electron shells.

X-rays: Characteristic Radiation, K

Target (Anode)

Mo

Cu

Co

Fe

Cr

Wavelength, Å

0.71

1.54

1.79

1.94

2.29

Note that wavelength is typically ranging between 1-2 Å

Two things will be key to explain the phenomenon

1. Model atoms as mirrors: Laws of specular reflection can be applied

2. Apply interference criteria

In-phase rays-Amplify

Out of phase-Dampens

Constructive and destructive interference

In-phase (coherent) monochromatic (single wavelength) incident radiation

Incident Beam

X-Ray Diffraction

Transmitted

Beam

Sample

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

dhk

l

Path Difference =PQ+QR sin2 hkld

i r

P

Q

R

Constructive inteference

sin2 hkldn

Bragg’s law

William Henry Bragg (1862–1942),

William Lawrence Bragg (1890–1971)

Nobel Prize (1915)

A father-son team that shared a Nobel Prize

Interference criteria + crystal structure = set of expected reflections

= Finger print of the crystal structure

{001} in SC reflects at θ{001}

However,

{001} in BCC and FCC does not reflects at θ{001}

Geometric analysis of atom position yields the following selection rules for reflection in cubic crystals

This can be summarised in the form of extinction rule as summarised in the next slide

Courtesy: Sadoway

Extinction Rules

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

X Ray Diffractometer

You do not get indices of plane!!

Output

Diffraction analysis of cubic crystals

sin2 hkld

222 lkh

adhkl

2sin 222

)lkh(

constant

Bragg’s Law:

Cubic crystals

(1)

(2)

(2) in (1) => )(4

sin 222

2

22 lkh

a

Example problem

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

Solving X-Ray pattern

5 steps for the determination of crystal structure

1) Start with 2θ values and generate a set of sin2θ values

2) Normalise the sin2θ values by dividing it with first entry

3) Clear fractions from normalised column: Multiply bycommon number

4) Speculate on the hkl values that, if expressed as h2+k2+l2, would generate the sequence of the “clear fractions” column

5) 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.

Courtesy: Sadoway

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

Courtesy: Sadoway

The (h2 + K2 + l2) derived from extinction rules

SC 1 2 3 4 5 6 8 …

BCC 2 4 6 8 10 12 14 …

FCC 3 4 8 11 12 …

Laue diffraction

2nd Technique

This was proposed by Von Laue, got noble prize in 1914

λ variable, θ fixed

Spot patternCourtesy: Sadoway

Rotational symmetry

(1) (001) Plane of cubic: 4 fold rotational symmetry

(2) (011) plane of cubic: 2 fold rotational symmetry

(3) (111) plane of cubic: 3 fold rotational symmetry

By looking at the symmetry of Laue pattern one can tell which atomic plane of crystal are we looking at

Laue spot pattern

Courtesy: Sadoway