Crystal Size Calculator using Scherrer Equation

An essential tool for materials scientists and chemists for the calculation of crystal size from XRD data. Input your diffraction peak parameters to estimate nanoscale crystallite dimensions accurately.

Result Visualization

Chart showing the relationship between FWHM (β) and the calculated crystal size. Note how size decreases as the peak becomes broader.

What is the Calculation of Crystal Size using Scherrer Equation?

The calculation of crystal size using Scherrer equation is a fundamental technique in X-ray diffraction (XRD) used to estimate the average size of crystalline domains, or crystallites, within a solid material. When crystallites are smaller than approximately 100-200 nanometers, the diffraction peaks in an XRD pattern become broader. The Scherrer equation provides a direct mathematical relationship between this peak broadening and the crystallite size. This method is crucial for researchers in nanotechnology, materials science, and geology who need to characterize nanoscale materials without resorting to more complex techniques like Transmission Electron Microscopy (TEM). It offers a quick, non-destructive way to gain insights into the material’s microstructure.

The Scherrer Equation Formula and Explanation

The formula is a cornerstone for anyone performing a calculation of crystal size. It is expressed as:

D = (K * λ) / (β * cos(θ))

Understanding each variable is key to an accurate calculation.

Variables for the Scherrer Equation

Variable	Meaning	Unit (Auto-Inferred)	Typical Range
D	Average Crystallite Size	Nanometers (nm) or Ångströms (Å)	1 – 200 nm
K	Scherrer Constant (Shape Factor)	Dimensionless	~0.89 for spherical, ~0.94 for cubic
λ	X-ray Wavelength	Nanometers (nm) or Ångströms (Å)	0.15406 nm (Cu Kα), 0.07107 nm (Mo Kα)
β	Peak Broadening (FWHM)	Radians	0.05 – 5 degrees (before conversion)
θ	Bragg Angle	Degrees	5 – 70 degrees

The term ‘β’ represents the line broadening at half the maximum intensity (FWHM) due to crystallite size, after correcting for instrumental broadening. A core part of the XRD analysis is ensuring this value is in radians for the calculation.

Practical Examples

Example 1: Zinc Oxide (ZnO) Nanoparticles

A researcher synthesizes ZnO nanoparticles and records an XRD pattern using a Cu Kα source. A prominent peak is observed at 2θ = 36.2°. After fitting, the FWHM (β) is determined to be 0.45 degrees. The goal is the calculation of crystal size.

• Inputs:
  • K = 0.89 (assuming spherical crystallites)
  • λ = 0.15406 nm
  • β = 0.45 degrees
  • θ = 36.2 / 2 = 18.1 degrees
• Calculation Steps:
  1. Convert β to radians: 0.45 * (π / 180) ≈ 0.00785 rad
  2. Calculate cos(θ): cos(18.1°) ≈ 0.9505
  3. Apply Scherrer Equation: D = (0.89 * 0.15406) / (0.00785 * 0.9505) ≈ 18.4 nm
• Result: The average crystallite size is approximately 18.4 nm.

Example 2: Gold (Au) Nanocrystals

An XRD analysis of gold nanocrystals is performed. The (111) peak appears at 2θ = 38.2° with an FWHM of 0.25 degrees.

• Inputs:
  • K = 0.9
  • λ = 0.15406 nm
  • β = 0.25 degrees
  • θ = 38.2 / 2 = 19.1 degrees
• Calculation Steps:
  1. Convert β to radians: 0.25 * (π / 180) ≈ 0.00436 rad
  2. Calculate cos(θ): cos(19.1°) ≈ 0.9449
  3. Apply Scherrer Equation: D = (0.9 * 0.15406) / (0.00436 * 0.9449) ≈ 33.6 nm
• Result: This successful calculation of crystal size using Scherrer equation indicates an average crystallite size of about 33.6 nm. For more complex structures, you might need a Lattice Parameter Calculator.

How to Use This Crystal Size Calculator

1. Enter Scherrer Constant (K): Start by inputting the shape factor. A value of 0.9 is a common and safe assumption for most calculations.
2. Input X-ray Wavelength (λ): Enter the wavelength of your X-ray source. Select the correct unit (nm or Å). The calculator defaults to 0.15406 nm for Cu Kα radiation.
3. Provide FWHM (β): Enter the Full Width at Half Maximum of your diffraction peak. Ensure you have selected the correct unit, either degrees or radians. This value should be corrected for instrumental broadening for highest accuracy.
4. Enter Bragg Angle (θ): Input the Bragg angle, θ. **Crucially**, this is half of the 2θ value you read from your diffractogram.
5. Interpret Results: The calculator automatically performs the calculation of crystal size and displays the result in nanometers. It also shows intermediate values like FWHM in radians for verification.

Key Factors That Affect Scherrer Equation Calculations

• Instrumental Broadening: Every diffractometer adds some broadening to the peaks. For an accurate calculation, this instrumental contribution must be measured (using a standard like LaB₆) and subtracted from the experimental peak width.
• Microstrain: Lattice strain within the crystallites can also cause peak broadening, leading to an underestimation of the crystal size. The Scherrer equation does not account for this, which is a major limitation.
• Crystallite Shape (K-Factor): The shape factor K changes with the geometry of the crystallites. While 0.9 is a good estimate, the actual value can vary, introducing a systematic error.
• Peak Overlap: If multiple diffraction peaks are close together, it can be difficult to accurately determine the FWHM of an individual peak, affecting the final result. Proper XRD analysis software can help deconvolve overlapping peaks.
• Crystallite Size Distribution: The Scherrer method provides a volume-weighted average size. If the sample has a very wide distribution of sizes, the result may not be representative of the number-average size.
• Background Noise: A high background signal can make it difficult to accurately measure the peak’s true maximum intensity and width, which is critical for the FWHM value. Good sample preparation for XRD is essential.

Frequently Asked Questions (FAQ)

1. Why must I use radians for FWHM in the formula?

The Scherrer equation is derived from principles of diffraction physics where angles are naturally expressed in radians. Using degrees will produce a result that is incorrect by a factor of (180/π). Our calculator handles this conversion for you automatically.

2. What is the difference between crystallite size and particle size?

A ‘particle’ can be an agglomerate of several smaller ‘crystallites’. A crystallite is a single, ordered domain of a crystal. The Scherrer equation measures the size of these individual crystallites, not the overall particle size.

3. What does “instrumental broadening” mean?

It’s the inherent broadening of a diffraction peak caused by the X-ray instrument itself (e.g., non-ideal optics, slit sizes). This must be subtracted from your measured peak width to isolate the broadening caused by the small crystallite size.

4. My calculated size seems too small. What could be wrong?

If you have not corrected for instrumental broadening or if your sample has significant microstrain, both effects will increase the FWHM and lead to an artificially small calculated crystal size. This is a common pitfall in the calculation of crystal size using Scherrer equation.

5. Can I use this calculator for any diffraction peak?

Yes, but it’s best to use a strong, well-defined peak that is isolated from others. Peaks at lower 2θ angles are often broader and may give more reliable results. Using multiple peaks can provide an average size and information about anisotropic (direction-dependent) crystal shapes.

6. What is a typical value for the Scherrer Constant (K)?

0.9 is widely used as a general approximation, especially for spherical or near-spherical crystallites. For precise work, K can be calculated based on the crystal’s Miller indices and shape, but this is rarely done in routine analysis.

7. How accurate is the Scherrer equation?

It is an estimation method. Accuracy is typically within 20-30% and is highly dependent on correcting for instrumental broadening and the absence of microstrain. It is best used for comparing relative sizes between samples rather than determining an absolute value.

8. Why do I need to divide my 2θ value by two?

The Scherrer equation uses the Bragg angle (θ), which is the angle between the incident X-ray beam and the diffracting crystal planes. XRD instruments measure the angle between the incident and diffracted beams, which is 2θ. Therefore, you must always divide the instrument’s output angle by two.