Lattice Parameters Calculation

[Carol]

Alattice constant describes the spacing between adjacent unit cells in a crystal structure. The unit cells or building blocks of the crystal are three dimensional and have three linear constants that describe the cell dimensions. The dimensions of the unit cell are determined by the number of atoms packed into each cell and by how the atoms are arranged. A hard-sphere model is adopted, which allows you to visualize atoms in the cells as solid spheres. For cubic crystal systems, all three linear parameters are identical, so a single lattice constant is used to describe a cubic unit cell.

Identify the space lattice of the cubic crystal system based on the arrangement of the atoms in the unit cell. The space lattice may be simple cubic (SC) with atoms only positioned at the corners of the cubic unit cell, face-centered cubic (FCC) with atoms also centered in every unit cell face, or body-centered cubic (BCC) with an atom included in the center of the cubic unit cell. For example, copper crystallizes in an FCC structure, while iron crystallizes in a BCC structure. Polonium is an example of a metal that crystallizes in a SC structure.

Find the atomic radius (r) of the atoms in the unit cell. A periodic table is an appropriate source for atomic radii. For example, the atomic radius of polonium is 0.167 nm. The atomic radius of copper is 0.128 nm, while that of iron is 0.124 nm.

Calculate the lattice constant, a, of the cubic unit cell. If the space lattice is SC, the lattice constant is given by the formula a = [2 x r]. For example, the lattice constant of the SC-crystallized polonium is [2 x 0.167 nm], or 0.334 nm. If the space lattice is FCC, the lattice constant is given by the formula [4 x r / (2)1/2] and if the space lattice is BCC, then the lattice constant is given by the formula a = [4 x r / (3)1/2].

Pearl Lewis has authored scientific papers for journals such as "Physica Status Solidi," "Materials Science and Engineering" and "Thin Solid Films" since 1994. She also writes an education blog entitled Simple Science in Everyday Life. She holds a doctorate from University of Port Elizabeth.

Our goal is to make science relevant and fun for everyone. Whether you need help solving quadratic equations, inspiration for the upcoming science fair or the latest update on a major storm, Sciencing is here to help.

Materials Project computed data is currently generated using a technique known as Density Functional Theory (DFT) with the PBE exchange-correlation functional. This results in some well-understood, systematic differences from experiment. Typically, this means that our computed lattice parameters will over-estimate experimental lattice parameters by 2-3% on average. Note that layered materials (any material where van der Waals bonding might be significant) will have larger errors in their inter-layer distance. Finally, note that these lattice parameters are nominally at 0 K, and do not take thermal expansion into account.

Note that lattice parameters on Materials Project are often given as their primitive cell, if you want the conventional lattice parameters make sure to download the CIF file in the "conventional" setting.

Band gaps will be systematically under-estimated by a large degree when using PBE (see our documentation). Spin-orbit coupling is also not included. The electronic band structures on Materials Project are most useful for seeing the shapes of the bands and the character of the gap (e.g. indirect, direct, between what symmetry points, etc.), the absolute magnitude of the band gaps are only useful for trends between different materials.

Better computational techniques can give results with smaller systematic errors, and we're constantly evaluating using some of these better techniques with the Materials Project. The trade-off here is that Materials Project tries to calculate properties for 100,000s of materials, and so using these better techniques is not always practically possible due to their computational cost.

With this context, to answer the question of "which should I use?", the question depends on what you want to use it for. If you want to know the "true" value, always defer to high-quality X-ray diffraction (bearing in mind the experimental value might be affected by grown-in strain, impurities, the temperature the measurement is taken at and other factors). However, if you want to do additional calculations with PBE, it's often easier to start from the previously-computed geometry. The computed geometry is also useful for examining differences between materials (e.g. varying composition) and also for materials where high-quality experimental data has not been acquired.

Likewise, for band gap, I would always defer to the experimental value, but of course there are also experimental issues too; experimentally, the optical gap is usually what is measured (e.g. via photoluminescence), there might be defect levels, finite temperature effects, excitonic effects, unintentional doping, Moss-Burstein shifts, etc., you might be only measuring the direct gap, whereas computationally you're predicting the fundamental gap (strictly speaking, the "Kohn Sham gap" using traditional DFT, which is another very important but subtle point). So there's no easy answer for which is better. The computational picture might give you a better picture for how a hypothetical pristine material might behave, but is typically most useful for trends and comparisons between similar materials.

I will give a quick answer from my experience, but typically you want to use the computed lattice parameter. The reason for this is that otherwise you induce strain in your computed cell, but in general this is a red flag anyways. You really want your computed lattice parameter to match if possible which might involve using a different functional or even things such as DFT+U corrections.

I am actually not sure of any circumstance where you would want to use the experimental lattice constant over the calculated one. Maybe someone can list a reason. Also keep in mind the experimental lattice constant is taken at some experimental conditions, which are not represented in the calculation (frozen geometry, 0K, etc).

It is well known that very few DFT approximations (methods, functional, pseudopotential, basis set, etc.) give good lattice values. And when we said good, we mean in agreement with experimental ones (by the way, one of the path to test your simulation results is comparing with experimental ones).

You use the experimental lattice parameter to optimize your unit cell. After the optimization you can get the calculated lattice constant to use for subsequent calculations that don't involve relaxations.

Although we endeavor to make our web sites work with a wide variety of browsers, we can only support browsers that provide sufficiently modern support for web standards. Thus, this site requires the use of reasonably up-to-date versions of Google Chrome, FireFox, Internet Explorer (IE 9 or greater), or Safari (5 or greater). If you are experiencing trouble with the web site, please try one of these alternative browsers. If you need further assistance, you may write to he...@aps.org.

Electron-doped SrTiO3 (where the dopant can be Nb or La) has been widely investigated for both its fundamental interest in condensed matter physics and for industrial applications. Its electronic properties are closely related to the Ti-O bonding states in the SrTiO3 crystal. To further develop and control these properties, it is crucial to understand the factors controlling the change in lattice parameters and local lattice distortion upon doping with various atoms. Herein, we report the changes in lattice parameters and local lattice distortion in Nb- and La-doped SrTiO3 single crystals, investigated by in-plane x-ray diffraction and first-principles calculations. The lattice parameter of Nb- and La-doped SrTiO3 single crystals increased with dopant concentration. The broad intensities around the Bragg peak observed in the in-plane x-ray-diffraction experiments indicated that the local lattice expansion and contraction, or local lattice distortions, in the crystal were caused by the dopant atoms. First-principles calculations similarly showed that the lattice expansion and local lattice distortions in the SrTiO3 crystals were caused by doped Nb and La atoms. Atoms surrounding Schottky pairs of O and Sr vacancies were displaced both away from and towards the vacancies, resulting in a reduction in the lattice expansion of donor-doped SrTiO3. Donor atoms and Schottky pairs thus play an important role in determining the lattice parameters of SrTiO3 crystals. These fundamental structural analyses provide a useful basis to further investigate the electronic conductivity of electron-doped SrTiO3.

(a) Linear and (b) log-scale intensities of out-of-plane and in-plane XRD patterns obtained for a pure SrTiO3 single crystal. Reflections of the out-of-plane and in-plane XRD patterns are for 002 and 020 SrTiO3, respectively. Rescaled data are shown in the inset at the upper right corner of (b).

In-plane XRD patterns obtained from 020 reflections of (a) Nb- and (b) La-doped SrTiO3 single crystals. The red and black arrows in (a) and (b) indicate the intensities for the crystal lattice contraction and expansion, respectively. The rescaled data for Nb- and La-doped SrTiO3 are shown in (c) and (d), respectively. The red-line XRD patterns in all the images are from the nondoped SrTiO3 single crystal. The insets in the upper right corners of (c) and (d) show magnified images of the regions around the peaks of nondoped SrTiO3.

(a) Fitting profile obtained using Gaussian functions for the experimental XRD pattern of 0.02-at.-% Nb-doped SrTiO3. The dotted black, orange, and blue lines indicate the Gaussian functions used for the main, expansion, and contraction intensities, respectively. Gaussian curves were numbered as labeled. (b, c) The contraction (b) and expansion (c) intensity ratios calculated from the fitting results in Fig. S2 of Supplemental Material [34], as a function of dopant concentration. Horizontal axes in (b) and (c) are log-scale, except for the values for nondoped SrTiO3 shown as a black diamond on the left side of each graph. (d) Ratios of the contraction intensities as a function of the FWHM of the rocking curves. Gray dashed lines in (b), (c), and (d) are included to aid comparison between Nb-doped, La-doped, and nondoped SrTiO3.

3a8082e126