Rietveld methodology (Rietveld, 1969) has had a very important impact on a broad scientific community as it is possible to address many problems concerning crystalline materials (Young, 1993).The Rietveld Method is firmly established its position at the centre of structural physics, chemistry and materials science. This rigorous approach can also be applied to give the most accurate quantitative phase analyses possible. The Rietveld method for structural refinement of powder diffraction data has been developed over the last four decades and has proved indispensable in solving crystal structures. The process involves minimising the difference between a crystallographic model and experimental data, via a least squares refinement; such intricate modelling of the height, width and position of Bragg reflections in an X-ray diffraction pattern can yield a lot of information about the crystal structure of a material. In 1964-1966 the need to refine crystal structures from powder where the peaks were too much overlapped and Peak separation by least squares fitting (gaussian profiles) cannot be used for severe overlapping. Therefore in 1967 - First refinement program by H. M. Rietveld, single reflections + overlapped, no other parameters than the atomic parameters was used. In 1969 - First complete program with structures and profile parameters distributed 27 copies (ALGOL) was used. Then by 1972 - FORTRAN version was distributed worldwide. By 1977 Wide acceptance lead it to extend to X-ray data.

Hugo Rietveld realized that if a pattern could be modeled, the fit between a computed pattern and observed data could be optimized.

The Rietveld Method involves:

- Recording the diffracted intensities at several thousand equal increments
- Developing a crystal structure model – i.e. symmetry, atomic positions, unit-cell size, site occupancies
- Varying these and other parameters by least squares refinement to get best fit between calculated and observed data

Peak shape and diffractometer set-up are taken into account.

The Rietveld Method is generally perform for solving an unknown crystal structure, calculating the amount disorder or mixing on a Wyckoff site, Quantitatively determining the percentages of different phases in your sample and determining the crystallite sizes in your samples.

**GOAL OF THE RIETVELD METHOD**

To minimize the residual function using a non-linear least squares

Algorithm given by

where c - scale factor, I^{calc}

Thus refine the crystal structure of a compound (cell parameters, atomic positions and Debye-Waller factors).

RIETVELD METHOD IN PRACTICE BASIC REQUIREMENTS

- Accurate diffraction data
- A reasonable starting structural model
- Space group symmetry
- Approximate atomic positions
- model may be from: isostructural materials, theoretical simulations, high-resolution atomic imaging
- A Rietveld refinement program: GSAS (Larson and von Dreele), Fullprof (Rodriguez – Carvajal), Others: BGMN (Bergmann), DBW (Wiles and Young), LHPM-Rietica (Hunter), MAUD (Lutterotti), Rietan (Izumi), Simref (Ritter), TOPAS (Bruker)
- Must be possible to fit peak shapes
- Q range and resolution demands dictated by structural complexity
- Data from lab instruments should be used with caution for structure determination
- Neutron data are usually necessary for occupancy determination

Rietveld Analysis of XRD patterns can be utilized for many purposes

- It can identify strain present in samples (single or multiphase)
- It can determine crystal size from peak broadening (single or multiphase)
- It can be used to quantify relative fraction of phases in a multiphase sample
- It can detect the quantity of the glassy nature of any crystalline phase undergone partial glass transition