Particle Size Calculator (Scherrer Equation)
An essential tool for estimating nanoscale crystallite size from X-ray Diffraction (XRD) data.
Dimensionless value, typically 0.9 for spherical crystallites.
Unit: nanometers (nm). Default is Cu Kα radiation.
Full Width at Half Maximum of the diffraction peak. Ensure this is corrected for instrumental broadening.
This is the Bragg angle θ, NOT 2θ. The value should be in degrees.
The crystallite size D is calculated as (K * λ) / (β * cos(θ)), where β and θ are in radians.
Crystallite Size vs. FWHM
What is Particle Size Calculation using the Scherrer Equation?
The particle size calculation using scherrer equation is a widely used method in X-ray diffraction (XRD) and crystallography to estimate the size of sub-micrometer crystallites in a solid material. It is important to note that the equation provides the size of the coherently scattering domains, known as crystallites, which may be smaller than the overall particle size. A single particle can be composed of multiple smaller crystallites. This technique is fundamental in materials science, chemistry, and nanotechnology for characterizing nanomaterials.
This calculation is not for determining the size of large grains but is specifically limited to the nanoscale, typically for crystallites smaller than 0.1 to 0.2 µm (100-200 nm). Anyone working with powder XRD data for nanomaterials, from researchers to quality control engineers, will find this calculation essential. A common misunderstanding is confusing crystallite size with particle size; the Scherrer equation estimates the former, providing a lower bound for the latter.
Scherrer Equation Formula and Explanation
The Scherrer equation is a simple yet powerful formula that connects the broadening of an XRD peak to the size of the crystallites. The fundamental principle is that smaller crystallites cause broader diffraction peaks.
The formula is expressed as:
D = (K * λ) / (β * cos(θ))
For a precise particle size calculation using scherrer equation, it is crucial to understand each variable.
| Variable | Meaning | Unit | Typical Range |
|---|---|---|---|
| D | Mean crystallite size | nanometers (nm) | 1 – 200 nm |
| K | Scherrer constant (shape factor) | Unitless | 0.89 – 1.0 (0.9 is common for spheres) |
| λ | X-ray wavelength | nanometers (nm) | 0.15406 nm for Cu Kα |
| β | Full Width at Half Maximum (FWHM) | Radians | Must be converted from degrees |
| θ | Bragg diffraction angle | Radians | Must be converted from degrees (input on the calculator is in degrees) |
For more advanced analysis, check out our Williamson-Hall plot calculator, which can also account for lattice strain.
Practical Examples
Example 1: Zinc Oxide (ZnO) Nanoparticles
A researcher analyzes a ZnO nanoparticle sample and gets a prominent XRD peak.
- Inputs:
- Shape Factor (K): 0.9
- X-ray Wavelength (λ): 0.15406 nm (Cu Kα)
- Bragg Angle (θ): 18.04 degrees
- Peak FWHM (β): 0.4 degrees
- Calculation Steps:
- Convert β to radians: 0.4 * (π / 180) ≈ 0.00698 rad
- Convert θ to radians: 18.04 * (π / 180) ≈ 0.3148 rad
- Calculate cos(θ): cos(0.3148) ≈ 0.9509
- Apply the formula: D = (0.9 * 0.15406) / (0.00698 * 0.9509)
- Result: D ≈ 20.8 nm. The average crystallite size is approximately 20.8 nanometers.
Example 2: Ceria (CeO2) Nanoparticles with Broader Peak
Another sample of ceria shows a much broader peak, indicating smaller crystallites.
- Inputs:
- Shape Factor (K): 0.9
- X-ray Wavelength (λ): 0.15406 nm (Cu Kα)
- Bragg Angle (θ): 14.2 degrees
- Peak FWHM (β): 1.5 degrees
- Calculation Steps:
- Convert β to radians: 1.5 * (π / 180) ≈ 0.02618 rad
- Convert θ to radians: 14.2 * (π / 180) ≈ 0.2478 rad
- Calculate cos(θ): cos(0.2478) ≈ 0.9693
- Apply the formula: D = (0.9 * 0.15406) / (0.02618 * 0.9693)
- Result: D ≈ 5.47 nm. The much broader peak corresponds to a significantly smaller crystallite size. A correct particle size calculation using scherrer equation is key to understanding these properties. For more details on sample handling, see our guide on XRD sample preparation.
How to Use This Scherrer Equation Calculator
Using this calculator is a straightforward process:
- Enter the Shape Factor (K): Start with the default of 0.9, which is suitable for spherical or unknown crystallite shapes. Adjust if you have information about the crystallite morphology.
- Set the X-ray Wavelength (λ): The default is 0.15406 nm, the standard for a Cu Kα X-ray source. Change this value only if your XRD instrument uses a different source (e.g., Mo, Co).
- Input the Peak FWHM (β): Obtain this value from your XRD software. It’s the “Full Width at Half Maximum” of your diffraction peak of interest. The value must be in degrees. Our calculator will handle the conversion to radians.
- Input the Bragg Angle (θ): This is the angle of the peak maximum. Your XRD data will likely show a `2θ` value. You must divide this value by two to get `θ` before entering it into the calculator. For example, if your peak is at 36.08 degrees 2θ, you enter 18.04 degrees for θ.
- Interpret the Results: The calculator instantly provides the average crystallite size (D) in nanometers. It also shows intermediate calculations like the FWHM and Bragg angle in radians to ensure transparency in the particle size calculation using scherrer equation.
Key Factors That Affect Scherrer Equation Results
- Instrumental Broadening: Every XRD instrument adds some broadening to the peaks. For accurate results, this instrumental broadening should be subtracted from the measured FWHM before using the equation.
- Microstrain: Non-uniform lattice strain within the crystallites can also cause peak broadening. The Scherrer equation does not account for this, which can lead to an underestimation of the crystallite size. Techniques like Williamson-Hall plots are needed to separate size and strain effects.
- Crystallite Shape (K-Factor): The shape factor K varies with the crystallite’s geometry. While 0.9 is a common approximation, the actual value can differ, introducing a level of uncertainty.
- Peak Selection: Choosing a clear, well-defined peak that doesn’t overlap with others is crucial for obtaining an accurate FWHM value.
- Data Quality: A low signal-to-noise ratio in the XRD data can make it difficult to determine the FWHM accurately, directly affecting the calculation.
- Crystallite Size Distribution: The Scherrer equation provides a volume-weighted average size. It doesn’t reveal the full distribution of sizes within the sample. For more on this, explore our article on nanomaterial characterization.
FAQ about the Particle Size Calculation using Scherrer Equation
- 1. Why must FWHM and Bragg angle be in radians?
- The Scherrer formula is derived from principles of diffraction and trigonometry where angles are naturally expressed in radians. Using degrees directly will produce an incorrect result. Our calculator automatically converts the degree inputs for your convenience.
- 2. What is the difference between crystallite size and particle size?
- A “particle” can be an agglomerate of several smaller crystalline domains, called “crystallites.” The Scherrer equation measures the size of these individual crystallites, not the overall particle. Therefore, the crystallite size is often a lower limit for the actual particle size.
- 3. What are the limitations of the Scherrer equation?
- It is only applicable for crystallites up to about 200 nm. It also doesn’t account for peak broadening caused by lattice strain, which can lead to inaccurate size estimations. For a robust analysis, separating size and strain effects is recommended.
- 4. How do I correct for instrumental broadening?
- You must measure the FWHM of a peak from a strain-free standard material with large crystallites (like LaB6). This value (β_instrumental) is then subtracted (often in quadrature) from your sample’s measured FWHM (β_measured).
- 5. Can I use any peak in my XRD pattern?
- It’s best to use a strong, symmetric, and non-overlapping peak. Using multiple peaks and comparing the results can give you a better sense of the overall crystallite size and shape anisotropy.
- 6. Does the calculator handle 2θ values?
- No, you must input the Bragg angle, which is θ (half of the 2θ value from your diffractogram). This is a common point of error in manual calculations.
- 7. What if my material is amorphous?
- The Scherrer equation is not applicable to amorphous materials, as they do not produce sharp diffraction peaks. They instead show broad humps in the XRD pattern.
- 8. How accurate is the particle size calculation using scherrer equation?
- It provides an estimation. Due to assumptions (like the K-factor) and uncorrected factors (like strain), the result should be considered an approximation of the true crystallite size. For high-precision work, more advanced methods are necessary.
Related Tools and Internal Resources
To continue your materials analysis journey, explore these related resources:
- Williamson-Hall Plot Calculator: An advanced tool to separate crystallite size and microstrain effects.
- Guide to XRD Sample Preparation: Learn best practices for preparing your samples for analysis.
- Understanding Crystallography: A primer on the basics of crystal structures.
- Materials Analysis Services: Discover how our experts can assist with your characterization needs.
- Top 5 Nanomaterial Characterization Techniques: A broad overview of methods including XRD, SEM, and TEM.
- Contact Us: Have questions? Reach out to our team of experts.