Hello Lana Agbaria

The process of determine the unit cell of a structure, from XRD powder pattern, is called Indexing. To fully understand how to do it, you may have to understand the very basics of Crystallography.

Each of the peaks in your pattern corresponds to a crystallographic plane, like (100), (110), (111), etc. Each of these planes has a interplanar distance, d, which can be considered as the distance between that plane and the origin of the unit cell.

Consider the (100) plane: It represents the distance between the origin of the unit cell and one of the "faces" of the cell. This distance is the length of the axis a of the cell, and generally is represented as d(100). So, the interplanar distance of (010), d(010), is the length b, and d(001) is the length of c axis.

And now you have the three axis' length, and what to do with them? If you consider the Braggs' Law:

n. lambda = 2 . d . sin(theta)

It relates interplanar distance d with the angle theta. Which means that you can determine what will be the position of each crystallographic plane in your pattern. So, if you have a unit cell, cubic, with lattice parameter a=b=c=4.2 Angstrom, you can easily determine what will be the position of the planes (100) (010), (111), (221), etc. Which means that you can do the opposite. Given some peaks in your pattern, you can go "backwards", and find what planes they are, and then find what unit cell you have. This is a difficult process, that can be done with softwares EXPO2014, FOX, Crysfire, etc.

Feel free to reply if you need help with the use of these softwares, or any other you might find.

Best regards,

Ricardo Tadeu

You need to fit your XRD peaks using a peak function (Voigt, Pearson IV etc.) to determine their center, from which you can calculate the d-spacings using the Bragg Law. Then depending on the unit cell of your material (cubic, hexagonal etc.) you can use specific relations between the d-spacings and the lattice parameters (3 distances and 3 angles). You can average your results as different peaks would probably give you slightly different results. Another method would be to use full pattern fitting, but it would be more complicated.

Determining the precise cell parameters (such as lattice constants aaa, bbb, ccc, and angles α\alphaα, β\betaβ, γ\gammaγ) of a phase in powder diffraction involves several steps. Here's a general approach:

1. Collect High-Quality Diffraction Data

• Sample Preparation: Ensure that your sample is well-prepared, with a uniform and fine powder to avoid issues like preferred orientation or large particle effects.
• Instrument Settings: Use high-resolution settings on your XRD instrument to minimize peak broadening and to accurately capture the positions and intensities of the diffraction peaks.

2. Peak Identification and Indexing

• Identify Peaks: Determine the positions (2θ values) of the diffraction peaks in your pattern. Use software to accurately locate the peak maxima.
• Indexing: Assign Miller indices (hkl) to the identified peaks. This step involves matching the observed d-spacings (calculated from peak positions) to theoretical ones based on possible crystal structures. Software such as X'Pert HighScore, Jade, or TOPAS can help with this.

3. Initial Lattice Parameter Estimation

• Bragg's Law: Use Bragg's Law, nλ=2dsin⁡θn\lambda = 2d\sin\thetanλ=2dsinθ, to calculate the interplanar spacings (d-spacings) from the peak positions.
• Cell Parameter Estimation: From the d-spacings and the indexed Miller indices, you can estimate the unit cell parameters using the appropriate equations for different crystal systems (cubic, tetragonal, orthorhombic, etc.).

4. Refinement of Cell Parameters

• Le Bail Refinement: Perform a Le Bail fit where the peak positions are refined without considering the intensities, allowing for more accurate determination of the cell parameters.
• Rietveld Refinement: Use Rietveld refinement for a more precise determination of the cell parameters. In this method, the entire diffraction pattern is modeled, including both peak positions and intensities, and the fit to the observed pattern is iteratively improved. The Rietveld method accounts for factors such as peak shapes, background, and instrument broadening.
• Software Tools: Programs like FullProf, GSAS, or TOPAS are commonly used for Rietveld refinement. They allow you to input an initial model of the crystal structure, which is then refined against the observed data.

5. Verification and Validation

• Consistency Check: Ensure that the refined cell parameters are consistent with the crystal symmetry and the known structure of the phase.
• Comparison with Literature: Compare your refined parameters with reported values in the literature or databases to validate the results.

6. Error Analysis

• Uncertainty Estimation: The refinement software typically provides standard deviations or error estimates for the refined parameters. This allows you to assess the precision of your measurements.
• Check for Systematic Errors: Ensure that there are no systematic errors in your experiment (such as sample displacement or instrument misalignment) that could affect the accuracy of the cell parameters.

By following these steps, you can accurately determine the precise cell parameters of a phase from powder diffraction data.