Xrd Grain Size Calculator

Scherrer Equation:

\[ D = \frac{K \lambda}{\beta \cos \theta} \]

Shape Factor (K):

unitless

Wavelength (λ):

FWHM (β):

radians

Angle (θ):

degrees

Grain Size (D):

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1. What is the Scherrer Equation?

The Scherrer equation is used in X-ray diffraction (XRD) to estimate the average size of crystalline grains in a material. It relates the size of sub-micrometer particles to the broadening of peaks in a diffraction pattern.

2. How Does the Calculator Work?

The calculator uses the Scherrer equation:

\[ D = \frac{K \lambda}{\beta \cos \theta} \]

Where:

\( D \) — Average grain size (nm)
\( K \) — Shape factor (typically 0.9 for spherical crystals)
\( \lambda \) — X-ray wavelength (nm)
\( \beta \) — Line broadening at FWHM (radians)
\( \theta \) — Bragg angle (degrees)

Explanation: The equation accounts for the inverse relationship between peak broadening and crystallite size, with correction for the diffraction angle.

3. Importance of Grain Size Calculation

Details: Grain size affects material properties like strength, ductility, and electrical conductivity. XRD provides a non-destructive method to estimate crystallite size in polycrystalline materials.

4. Using the Calculator

Tips: Enter wavelength in nm (Cu Kα is 0.15406 nm), FWHM in radians, and angle in degrees. All values must be positive (angle between 0-90°).

5. Frequently Asked Questions (FAQ)

Q1: What is the typical value for K?
A: The shape factor K is typically 0.9 for spherical crystals, but can range from 0.62 to 2.08 depending on crystal shape and definition of size.

Q2: How to measure FWHM (β)?
A: FWHM should be measured after subtracting instrumental broadening, typically from a standard sample. Convert from degrees to radians for calculation.

Q3: What are the limitations of this equation?
A: The Scherrer equation assumes size broadening dominates and doesn't account for strain broadening. It's most accurate for sizes <100 nm.

Q4: Why convert angle to radians?
A: The trigonometric functions in programming languages typically use radians, so conversion is necessary for accurate calculation.

Q5: Can this be used for all crystal systems?
A: Yes, but results may vary in accuracy depending on crystal anisotropy. Multiple peaks should be analyzed for best results.