What is the Formula Used for Calculating Particle Size?

Scientific Crystallite & Particle Size Estimator (Scherrer Equation)

What is the Formula Used for Calculating Particle Size?

When scientists and researchers ask what is the formula used for calculating particle size, they are most frequently referring to the Scherrer Equation. This mathematical relationship is fundamental in material science and chemistry for estimating the size of extremely small crystals, often referred to as crystallites, using X-ray diffraction (XRD) data.

The core of what is the formula used for calculating particle size lies in the observation that when X-rays interact with a crystalline sample, the resulting diffraction peaks become broader as the crystallite size decreases. This “peak broadening” is the physical basis for the formula. It is essential for nanotechnology, pharmaceuticals, and metallurgy, where controlling the grain size of materials determines their physical properties.

While the Scherrer Equation is the primary answer to what is the formula used for calculating particle size in the context of XRD, other formulas exist for different methods, such as Stokes’ Law for sedimentation or the Brunauer-Emmett-Teller (BET) theory for surface area measurements.

What is the Formula Used for Calculating Particle Size? Mathematical Explanation

The Scherrer Equation is expressed as:

d = (K · λ) / (β · cosθ)

To understand what is the formula used for calculating particle size, we must break down each variable involved in the calculation:

Variable	Meaning	Unit	Typical Range
d	Mean size of the ordered (crystalline) domains	Nanometers (nm)	1 nm to 100 nm
K	Dimensionless shape factor	N/A	0.62 to 2.08 (0.9 is standard)
λ	X-ray wavelength	Nanometers (nm)	0.15406 nm (Cu K-alpha)
β	Full width at half maximum (FWHM)	Radians	0.001 to 0.1 radians
θ	Bragg angle	Degrees/Radians	10° to 80°

It is critical to remember that β (FWHM) must be converted from degrees to radians before using it in what is the formula used for calculating particle size calculations. Failure to do so is the most common error in manual computations.

Practical Examples of Particle Size Calculation

Example 1: Gold Nanoparticles

Suppose you are analyzing gold nanoparticles using a Copper K-alpha X-ray source (λ = 0.15406 nm). You observe a peak at a Bragg angle (θ) of 19.1° with a FWHM of 0.45°. Using what is the formula used for calculating particle size (K=0.9):

• β in radians = 0.45 * (π / 180) = 0.00785 rad
• cos(19.1°) = 0.9449
• d = (0.9 * 0.15406) / (0.00785 * 0.9449) = 18.64 nm

Example 2: Zinc Oxide Thin Film

A sample of ZnO shows a sharper peak, suggesting larger crystals. FWHM is 0.15°, θ is 17.2°, and λ is 0.15406 nm.

• β in radians = 0.15 * (π / 180) = 0.002618 rad
• cos(17.2°) = 0.9553
• d = (0.9 * 0.15406) / (0.002618 * 0.9553) = 55.45 nm

How to Use This Particle Size Calculator

1. Enter the Shape Factor (K): Use 0.9 if you are unsure of the particle geometry.
2. Input the X-ray Wavelength: Most labs use Copper anodes (0.15406 nm), but check your equipment specs.
3. Determine the FWHM: Extract this from your XRD software. It represents the width of the peak at half its intensity.
4. Input the Bragg Angle (θ): Note that many XRD graphs show 2θ. Use half of that value here.
5. Read the Result: The calculator updates in real-time to show the mean crystallite size.

Key Factors That Affect Particle Size Results

Understanding what is the formula used for calculating particle size requires knowing its limitations and environmental factors:

• Instrumental Broadening: The XRD machine itself adds width to the peaks. For accurate results, this “instrumental width” should be subtracted from the measured FWHM.
• Micro-strain: Internal stress in the crystal lattice can broaden peaks, potentially leading to an underestimation of particle size if not accounted for.
• Crystallite vs. Particle: The formula calculates “crystallite size,” which is the size of a single crystal. A “particle” might be composed of multiple crystallites clumped together.
• Shape Factor Variation: While 0.9 is standard, different shapes (like cubes or needles) require different K values ranging from 0.6 to 2.0.
• X-ray Wavelength Accuracy: Minor variations in λ (e.g., using K-beta instead of K-alpha) significantly alter the output of what is the formula used for calculating particle size.
• Small Size Limit: The Scherrer Equation is generally not reliable for particles larger than 100-200 nm because peak broadening becomes too small to measure accurately.

Frequently Asked Questions (FAQ)

Can I use this formula for particles larger than 500 nm?

No, what is the formula used for calculating particle size via Scherrer is best suited for sizes under 100 nm. Beyond that, the broadening is negligible compared to instrumental error.

What happens if I use 2θ instead of θ?

Your result will be incorrect. The formula requires the Bragg angle (θ), which is half of the diffraction angle (2θ) typically reported on XRD plots.

Is crystallite size the same as grain size?

Often they are used interchangeably in nanostructured materials, but technically a grain can consist of multiple crystallites.

Why does the shape factor K change?

K accounts for the mathematical assumptions about the shape of the crystal and how the peak width is defined. 0.9 is the most common average for spherical particles.

Does the formula work for amorphous materials?

No, what is the formula used for calculating particle size requires a crystalline lattice to produce diffraction peaks.

How do I calculate FWHM in radians?

Multiply the value in degrees by π and divide by 180 (Radians = Degrees × 3.14159 / 180).

Can temperature affect the calculation?

Directly, no. But high temperatures can cause crystal growth (sintering), which reduces peak broadening and increases the calculated size.

What is the most common X-ray wavelength?

The Copper K-alpha line at 1.5406 Angstroms (or 0.15406 nm) is the industry standard.

Related Tools and Internal Resources

• Crystallite Size Calculation – A deep dive into XRD analysis techniques.
• X-ray Diffraction Analysis – Understanding Bragg’s Law and peak identification.
• Scanning Electron Microscopy – Comparing XRD results with visual SEM imaging.
• Nanoparticle Characterization – Tools for determining purity and morphology.
• Sedimentation Rate Analysis – Using Stokes’ Law for larger particle measurements.
• Material Science Formulas – A comprehensive library of physics and chemistry constants.