How to calculate mosaicity?

Mosaicity, a critical parameter for characterizing crystalline materials, is primarily calculated by analyzing the full width at half maximum (FWHM) of a rocking curve obtained through X-ray diffraction. This measurement provides a direct insight into the angular disorientation of crystallites within a material.

Understanding Mosaicity

Mosaicity refers to the slight angular misorientation between small, coherent diffracting blocks, often called "mosaic blocks," within a larger single-crystal or polycrystalline material. Imagine a perfect crystal as a perfectly aligned stack of bricks; a mosaic crystal would have these bricks tilted slightly relative to each other. This disorientation, even if subtle, significantly impacts the material's physical properties and its interaction with X-rays.

How to Calculate Mosaicity

The most common and precise method for determining mosaicity involves X-ray diffraction, specifically using a technique known as a rocking curve measurement.

The Rocking Curve Method

To calculate the mosaicity angle, follow these steps:

1. Perform an Omega (ω) Scan: In an X-ray diffractometer, position the detector at a fixed two-theta (2θ) angle corresponding to a specific diffraction peak of the crystal. Then, rotate the crystal sample around its omega (ω) axis, which is typically perpendicular to the diffraction plane, while continuously measuring the intensity of the diffracted X-rays.

2. Utilize a Wide Open Detector Window: It is crucial to use a wide open window in front of the detector during this scan. This ensures that the entire diffracted beam, regardless of its angular spread caused by the mosaicity, is collected. This integration prevents underestimation of the true angular spread.

3. Generate the Rocking Curve: The recorded intensity as a function of the crystal's rotation angle (ω) forms a plot known as a rocking curve. The width of this curve is directly related to the angular spread of the mosaic blocks.

4. Determine the Full Width at Half Maximum (FWHM): Analyze the rocking curve to find its FWHM. The FWHM is the width of the diffraction peak measured at half of its maximum intensity. This value quantifies the angular spread of the diffracting crystallites.

5. Calculate Mosaicity Angle: In this specific experimental setup (Omega scan with a wide-open detector window), half of the measured FWHM directly represents the angle of the rocking blocks in the mosaicity structure.

   $$ \text{Mosaicity Angle} = \frac{\text{FWHM (of the rocking curve)}}{2} $$

   Note: The FWHM is typically measured in degrees or radians.

Key Components Explained:

• Rocking Curve: A plot of diffracted X-ray intensity versus the crystal's rotation angle (ω) around a diffraction peak. Its width is sensitive to the crystal's quality and the angular spread of its domains.
• Omega (ω) Scan: A type of diffraction scan where the sample is rotated while the detector remains stationary at a specific 2θ position. It isolates the crystal's orientation effects.
• FWHM (Full Width at Half Maximum): A standard metric for the width of a peak. In the context of rocking curves, a broader FWHM indicates higher mosaicity (more disoriented blocks), while a narrower FWHM signifies a more perfect crystal structure with less angular spread.
• Wide Open Detector Window: Essential for collecting all diffracted radiation from the mosaic blocks. If the window were narrow, it would selectively collect only a portion of the spread, leading to an artificially narrow FWHM and an inaccurate mosaicity calculation.

Importance of Mosaicity

Mosaicity is a fundamental characteristic that influences various material properties and applications:

• Crystal Quality Assessment: It is a key indicator of the structural perfection of single crystals. Lower mosaicity generally implies higher crystal quality.
• Device Performance: In semiconductor devices, optical components, or X-ray optics, high mosaicity can degrade performance by scattering light or X-rays rather than reflecting or transmitting them efficiently.
• Diffraction Efficiency: For applications involving X-ray diffraction, such as monochromators or neutron scattering experiments, controlled mosaicity can enhance diffraction efficiency.
• Material Processing Insights: Changes in mosaicity can reveal information about crystal growth conditions, post-growth treatments (like annealing), or induced stress and defects within the material.

Practical Steps for Mosaicity Measurement

For researchers and engineers, measuring mosaicity typically involves these practical considerations:

1. Sample Preparation: Ensure the crystal surface is clean and flat for accurate alignment.
2. Diffractometer Setup: Use a high-resolution X-ray diffractometer equipped with a goniometer capable of precise angular movements.
3. Peak Selection: Choose a strong, well-defined diffraction peak (e.g., a (00n) reflection for epitaxial layers or a fundamental reflection for bulk crystals).
4. Initial Alignment: Carefully align the crystal to find the exact peak position for the selected reflection.
5. Rocking Curve Scan: Set up an ω scan with appropriate angular step sizes and scan range to fully capture the diffraction peak. Confirm the detector's receiving slit or window is wide enough.
6. Data Analysis: After acquiring the rocking curve data, fit a suitable function (e.g., Gaussian, Lorentzian, or pseudo-Voigt) to the peak to precisely determine its FWHM.
7. Calculation: Apply the formula: Mosaicity Angle = FWHM / 2.

By diligently following this procedure, the mosaicity of a crystalline material can be accurately determined, providing valuable insights into its structural integrity.