XRD data processing Scherrer formula for calculating grain size

XRD Grain Size Analysis

Many people want to calculate the size of the grain.

In fact, our professional terminology is not called grain size, but “subcrystalline size”, which characterizes not the diameter of a particle. Let's put it this way: powder consists of many “particles”, each particle is made up of many “grains” aggregated together, and a grain is made up of many “single cells” stitched together. The size of a crystal block measured by X-rays is the size in the direction of the index of the diffracting plane; if there are M cells in this direction and the spacing of the crystal planes in this direction is d, then the measured size is Md. If there are N cells in a certain direction (HKL), and the spacing of the crystal planes is d1, then the size in this direction is Nd1. It can be seen from this that the size of a crystal block measured by different diffracting planes may not necessarily be the same.

If this grain is a complete, defect-free grain, it can be considered as one test unit. However, if the grain has defects, it is not a test unit, and the units separated by defects are called “subcrystals”. For example, if a grain consists of two small grains that pass through a subcrystalline boundary (called a subcrystal), then it is not the size of the grain that is measured, but the size of the subcrystal.

Why do so many people prefer to use the term “grain size” instead of the specialized interpretation? It's all because of “nanomaterials”. Nanograins are originally very small, generally can be regarded as a nanograin in a nanograin no longer exists in the subcrystalline, but a complete grain, therefore, the term subcrystalline size has been applied to the nanograins of the “particle size” up. In fact, there is a national standard for the characterization of particle size and particle size distribution of nanomaterials, which needs to be measured by the “small angle scattering” method. For example, Beijing Iron and Steel Research Institute has been doing this for a long time. However, for one thing, not many places to do small-angle scattering, and do it is particularly troublesome (now better, especially for light can be some of the automatic), so few people to do. Besides, the “particle size” calculated from the diffraction peak width is always so small, so why not? Privately, I think that some people are trying to steal the concept. Over time, people have come to accept it.

Example of Scherrer's formula for calculating grain size of XRD samples

Our common Scherrer's formula is: D=Kλ/(βcosθ)

K is a constant; λ is the X-ray wavelength; β is the half-height width of the diffraction peak; θ is the diffraction angle. The value of constant K in the above equation is related to the definition of β. When β is the half-width height, K takes 0.89; when β is the integral width, K takes 1.0.

However, in practice, how to obtain the above parameters from an ordinary XRD spectrum to calculate the grain size there are still the following problems: using XRD to calculate the grain size must be deducted from the instrument broadening and stress broadening effects. How to deduct the effects of instrumental and stress broadening? Under what circumstances can this step be simplified?

A: When the grain size is less than 100 nm, the stress-induced broadening can be neglected compared to the broadening caused by the grain size. At this point, the Scherrer formula applies. However, when the grain size is large to a certain extent, the broadening caused by stress is more significant, and at this time, the broadening caused by gravitational force must be considered, and the Scherrer formula is no longer applicable.

When we calculate the grain size, we generally use a low angle diffraction line, if the grain size is large, can be replaced by a higher diffraction angle diffraction line. Scheller's formula applies to the range of 1-100 nm, grain size is less than 1 nm greater than 100 nm, the use of Scheller's formula is not very accurate, when the grain size of 30 nm when the calculation of the most accurate results. At the same time, Scheller's formula is only suitable for spherical particles, for cubic particles, the constant K should be changed to 0.943, and the half-height and width should be converted to radian system, i.e., [(β ÷ 180) × 3.14].

The graph below shows that Jade 5.0 reads a grain size of 264(A°) which is 26.4 nm.

The data available over here is the X-ray wavelength λ = 0.15405 nm , the half-height width β = 0.332, and 2θ = 36.159. Here's how we calculated it:

The value calculated by myself is very close to the value calculated with the software.