CHARACTERIZATION OF THE STRUCTURE OF SOLIDS Three main techniques: X-ray diffraction Electron diffraction Neutron diffraction Single crystal Powder Principles of x-ray diffraction 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

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

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

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 dhkl, lattice parameters and Miller indices Cubic Tetragonal Orthorhombic The expressions for the remaining crystal systems are more complex

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

Formation of a powder pattern Single set of planes Powder sample

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

Powder Diffractometer

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 Each reflection represents the x-ray beam diffracted by a family of lattice planes (hkl) This pattern is characteristic of the material analysed (fingerprint) Intensity 2 degrees (200) (110) (400) (310) (301) (600) (411) (002) (611) (321)

APPLICATIONS AND INTERPRETATION OF X-RAY POWDER DIFFRACTION DATA

Sample line broadening *Strain effect - variation in d - introduced by defects, stacking fault, mistakes - depends on 2θ

Scherrer equation * Determination of size effect, neglecting strain (Scherrer, 1918) *Thickness of a crystallite L = N dhkl Lhkl = k λ / (β cosθ), k: shape factor, typically taken as unity for β and 0.9 for FWHM

Identification of compounds The powder diffractogram of a compound is its ‘fingerprint’ and can be used to identify the compound Powder diffraction data from known compounds have been compiled into a database (PDF) by the Joint Committee on Powder Diffraction Standard, (JCPDS) ‘Search-match’ programs are used to compare experimental diffractograms with patterns of known compounds included in the database This technique can be used in a variety of ways

PDF - Powder Diffraction File A collection of patterns of inorganic and organic compounds Data are added annually (2008 database contains 211,107 entries)

Example of Search-Match Routine

? Outcomes of solid state reactions Product: SrCuO2? Pattern for SrCuO2from database Product: Sr2CuO3? Pattern for Sr2CuO3from database

Phase purity When a sample consists of a mixture of different compounds, the resultant diffractogram shows reflections from all compounds (multiphase pattern) Sr2CuO2F2+ Sr2CuO2F2+ + impurity *

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 1 Crystal class comparison of the diffractogram of the unknown compound with diffractograms of known compounds (PDF database, calculated patterns) 2 Indexing Assigning Miller indices to peaks 3 Determination of lattice parameters Bragg equation and lattice parameters Cubic system

Selected data from the NaCl diffractogram 2 () h,k,l 27.47 111 31.82 PROBLEM NaCl shows a cubic structure. Determine a (Å) and the missing Miller indices ( = 1.54056 Å). ? ? Selected data from the NaCl diffractogram 2 () h,k,l 27.47 111 31.82 ? 45.62 56.47 222

a (Å) Use at least two reflections and then average the results Å (222) Miller Indices A

F I P Systematic Absences Conditions for reflection 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) Conditions for reflection F i.e indices are all odd or all even I P No conditions

to either the correct lattice(s). 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 1 1 0 1 1 1 2 0 0 2 1 0 2 1 1 2 2 0 3 1 0 3 1 1 y

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

Derivation of