CHARACTERIZATION OF THE STRUCTURE OF SOLIDS

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CHARACTERIZATION OF THE STRUCTURE OF SOLIDS

Three main techniques:

X-ray diffraction Electron diffraction Neutron diffraction

Principles of x-ray diffraction

Single crystal Powder

X-rays are passed through a crystalline material and the patterns produced give information of size and shape of the unit cell

X-rays passing through a crystal will be bent at various angles: this process is called diffraction

X-rays interact with electrons in matter, i.e. are scattered by the electron clouds of atoms

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These conditions are met when the difference in path length equals an integral number of wavelengths, n. The final equation is the BRAGG’S LAW



 2d sin n 

Data are collected by using x-rays of a known wavelength. The position of the sample is varied so that the angle of diffraction changes

When the angle is correct for diffraction a signal is recorded

With modern x-ray diffractometers the signals are converted into peaks

Intensity (a.u.)

2 degrees

(200) (110) (400)

(310)(301)

(600) (411) (002)(611)(321)

Reflection (signal) only occurs when conditions for constructive interference between the beams are met

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TEST

NaCl is used to test diffractometers. The distance between a set of planes in NaCl is 564.02 pm. Using an x-ray source of 75 pm, at what diffraction angle (2) should peaks be recorded for the first order of diffraction (n = 1) ?

Hint: To calculate the angle  from sin , the sin^-1 function on the calculator must be used



 ;2 7.62

3.81

0.066 pm

564.02 2

pm sin 75

sin pm

564.02 2

pm 75

1

d n



 













 2 sin

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The lattice parameters a, b, c of a unit cell can then be calculated

The relationship between d and the lattice parameters can be determined geometrically and depends on the crystal system

Crystal system d_hkl, lattice parameters and Miller indices

Cubic

Tetragonal

Orthorhombic

2

2 2

a l k h

d



 ² 

2

1

2 2

2

1

c l a

k h

d

2 2

2 

 

2 2

1

c l b

k a

h d

2 2

2  



The expressions for the remaining crystal systems are more complex

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THE POWDER TECHNIQUE

An x-ray beam diffracted from a lattice plane can be detected when the x-ray

source, the sample and the detector are correctly oriented to give Bragg diffraction A powder or polycrystalline sample contains an enormous number of small

crystallites, which will adopt all possible orientations randomly

Thus for each possible diffraction angle there are crystals oriented correctly for Bragg diffraction

Each set of planes in a crystal

will give rise to a cone of diffraction

Each cone consists of a set of closely spaced dots each one of which represents a diffraction from a single crystallite

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Formation of a powder pattern

Single set of planes

Powder sample

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Experimental Methods

To obtain x-ray diffraction data, the diffraction angles of the various cones, 2, must be determined

The main techniques are: Debye-Scherrer camera (photographic film) or powder diffractometer

Debye Scherrer Camera

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Powder Diffractometer

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The detector records the angles at which the families of lattice planes scatter (diffract) the x-ray beams and the intensities of the diffracted x-ray beams

The detector is scanned around the sample along a circle, in order to collect all the diffracted x-ray beams

The angular positions (2) and intensities of the diffracted peaks of radiation (reflections or peaks) produce a two dimensional pattern

This pattern is characteristic of the material analysed (fingerprint)

Each reflection represents the x-ray beam diffracted by a family of lattice planes (hkl)

Intensity

2 degrees

(200) (110) (400)

(310)(301)

(600) (411) (002)(611) (321)

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APPLICATIONS AND INTERPRETATION OF X-RAY POWDER DIFFRACTION DATA

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Sample line broadening

*Strain effect - variation in d

- introduced by defects, stacking fault, mistakes

- depends on 2θ

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Scherrer equation

* Determination of size effect, neglecting strain (Scherrer, 1918)

*Thickness of a crystallite L = N d_hkl Lhkl = k λ / (β cosθ),

k: shape factor, typically taken as unity for β and 0.9 for FWHM

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Powder diffraction data from known compounds have been compiled into a database (PDF) by the Joint Committee on Powder Diffraction Standard, (JCPDS)

This technique can be used in a variety of ways

The powder diffractogram of a compound is its ‘fingerprint’ and can be used to identify the compound

‘Search-match’ programs are used to compare experimental diffractograms with patterns of known compounds included in the database

Identification of compounds

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PDF - Powder Diffraction File

A collection of patterns of inorganic and organic compounds

Data are added annually (2008 database contains 211,107 entries)

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Example of Search-Match Routine

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Outcomes of solid state reactions

Product: SrCuO₂?

Pattern for SrCuO₂from database

Product: Sr₂CuO₃?

Pattern for Sr₂CuO₃from database

CuO

2SrCO₃  SrCuO²

CuO3

Sr₂

?

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When a sample consists of a mixture of different compounds, the resultant diffractogram shows reflections from all compounds (multiphase pattern)

Phase purity

Sr₂CuO₂F_2+

Sr₂CuO₂F_2+ + impurity

*

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Determination of crystal class and lattice parameters

X-ray powder diffraction provides information on the crystal class of the unit cell (cubic, tetragonal, etc) and its parameters (a, b, c) for unknown compounds

Indexing Assigning Miller indices to peaks

1

Determination of

lattice parameters Bragg equation and lattice parameters

2

 ² ² ²

2 2 2

4 h k l

sin   a  

Cubic system

Crystal class comparison of the diffractogram of the unknown

compound with diffractograms of known compounds (PDF database, calculated patterns)

3

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PROBLEM

NaCl shows a cubic structure. Determine a (Å) and the missing Miller indices ( = 1.54056 Å).

2 () h,k,l

27.47 111

31.82 ^?

45.62 ^?

56.47 222

Selected data from the NaCl diffractogram

? ?

 ² ² ²

2 2 2

4 h k l

sin   a  

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  ₅_.₆₃₈

2 473 . sin 56

4

12 541

. 1 sin

4 ²

2 2

2 2 2

2 



 



 



 h k l a

Use at least two reflections and then average the results

(222) Å

a (Å)

Miller Indices

 ² ² ²

2 2 2

4 h k l

sin   a  

A

 ² ² ²

2 A h k l

sin    

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⁵ ⁶³⁸ ⁰ ⁰¹⁸⁶⁷

4

54056 1

4 ²

2 2

2

. . .

A a 

 

 

 ² ² ²

2

l k

A h

sin    

82

31 2  .

4 026 01867 4

0

822 31

2

 .  .

sin . ^h² ^ ^k² ^^l² ^ ⁴^^²⁰⁰^

62

45

2  . ₈ ₀₅₂ ₈

01867 0

622 45

2

 .  .

sin .

^h² ^ ^k² ^^l² ^ ⁸^²²⁰

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Systematic Absences

Conditions for reflection



^h² ^ ^k ²

 

^, ^h² ^ ^l²

 

^, ^k ² ^ ^l ²



^ ²ⁿ ^(even^number)

F

i.e indices are all odd or all even

I



^h² ^ ^k² ^ ^l²



^ ²ⁿ

P No conditions

For body centred (I) and all-face centred (F) lattices restriction on reflections from certain families of planes, (h,k,l) occur. This means that certain reflections do not appear in diffractograms due to ‘out-of-phase” diffraction

This phenomenon is known as systematic absences and it is used to identify the type of unit cell of the analysed solid. There are no systematic absences for primitive lattices (P)

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Considering systematic absences, assign the following sets of Miller indices to either the correct lattice(s).

Lattice Type

Miller Indices P I F

1 0 0 Y N N

1 1 0 Y Y N

1 1 1 Y N Y

2 0 0 Y Y Y

2 1 0 Y N N

2 1 1 Y Y N

2 2 0 Y Y Y

3 1 0 Y Y N

3 1 1 y N Y

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Autoindexing

Generally indexing is achieved using a computer program.

This process is called ‘autoindexing’

Input: •Peak positions (ideally 20-30 peaks)

•Wavelength (usually =1.54056 Å)

•The uncertainty in the peak positions

•Maximum allowable unit cell volume

Problems: •Impurities

•Sample displacement

•Peak overlap

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Derivation of ² ²₂  ² ² ²

4 h k l

sin   a  



  2d sin

2

2 2

a

l k

h d



 ² 

2

1

2 2

2

l k

4a h

l k

2a h

l k

h a 1

l k

h a 1

d l

k h

d a











 



 



 

2 2 2

2 2

2

;

 





sin2

sin