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The XRD Pattern Calculator computes the d-spacing and diffraction angle (2θ) for specific crystal planes in cubic systems. By entering the lattice parameter, X-ray wavelength, and Miller indices, you can predict exactly where diffraction peaks will appear in an X-ray diffraction pattern. This is essential for phase identification, indexing diffraction patterns, and understanding crystal structure.
X-ray diffraction pattern analysis is the primary method for identifying crystalline phases in materials. Each crystalline material produces a unique set of d-spacings and relative intensities, serving as a fingerprint for identification. The International Centre for Diffraction Data (ICDD) maintains a database of over 900,000 diffraction patterns for reference.
For cubic crystal systems, the d-spacing is calculated from the lattice parameter and Miller indices:
$$d_{hkl} = \frac{a}{\sqrt{h^2 + k^2 + l^2}}$$
The diffraction angle is then obtained from Bragg's law:
$$2\theta = 2\arcsin\left(\frac{n\lambda}{2d_{hkl}}\right)$$
Combining these equations:
$$\sin\theta = \frac{n\lambda\sqrt{h^2 + k^2 + l^2}}{2a}$$
For a valid diffraction peak, sin θ must be ≤ 1. The ratio $$\sin^2\theta$$ is proportional to $$(h^2 + k^2 + l^2)$$, which provides a systematic method for indexing cubic patterns. Allowed reflections depend on the lattice type: SC allows all (hkl), BCC requires h+k+l = even, and FCC requires h,k,l all odd or all even.
A 2θ angle of 0 indicates that the reflection is geometrically impossible (sin θ > 1) for the given wavelength and d-spacing. Peaks at lower 2θ correspond to larger d-spacings. For silicon (a = 5.43 Å) with Cu Kα, the first allowed peaks are (111) at 28.4°, (220) at 47.3°, (311) at 56.1°. The pattern of allowed and forbidden reflections identifies the lattice type.
Inputs
Results
d = 5.43/√3 = 3.135 Å, 2θ = 2×arcsin(1.5406/(2×3.135)) = 28.44°
Inputs
Results
d = 4.05/√4 = 2.025 Å, 2θ = 2×arcsin(1.5406/(2×2.025)) = 44.74°
Measure 2θ values for all peaks, convert to sin²θ, and divide each by the smallest sin²θ value. The resulting ratios should be integers or simple fractions corresponding to h²+k²+l². For SC: 1,2,3,4,5,6,8... For BCC: 1,2,3,4,5,6,7... (h+k+l even only). For FCC: 3,4,8,11,12,16... (all odd or all even hkl).
Systematic absences occur due to destructive interference from atoms at centering positions. In BCC, the body-center atom cancels reflections where h+k+l is odd. In FCC, face-center atoms cancel reflections where h,k,l are mixed (some odd, some even). These absences are structure-factor zeros and help identify the lattice type.
Cu Kα (1.5406 Å) is the most common for routine phase identification. Mo Kα (0.7107 Å) provides access to more reflections and is preferred for single-crystal work. Co Kα (1.7890 Å) reduces fluorescence from iron-containing samples. Choose based on your sample composition and angular range needs.
This calculator uses the cubic d-spacing formula only. For tetragonal, hexagonal, orthorhombic, and lower-symmetry systems, the d-spacing equations involve additional lattice parameters and angles. The Bragg's law portion remains the same regardless of crystal system.
Peak intensity depends on the structure factor (atomic positions and scattering factors), multiplicity (number of equivalent planes), Lorentz-polarization factor, temperature factor (Debye-Waller), absorption, and preferred orientation (texture). This calculator provides peak positions only; intensity calculations require knowledge of the full crystal structure.
Most powder diffractometers scan from 5° to 140° 2θ. The practical upper limit is set by decreasing peak intensity at high angles (due to thermal vibrations and scattering factor decay) and increasing peak overlap. The lower limit is set by the direct beam and low-angle scattering artifacts.
Measure the d-spacings and relative intensities of all peaks, then search the ICDD PDF (Powder Diffraction File) database for matching patterns. Software like JADE, HighScore, or Match! automates this search-match process. At least 3-5 strongest peaks must match for reliable identification.
The sum h²+k²+l² determines d-spacing for cubic systems. Not all integer values are possible: 7, 15, 23, 28... cannot be expressed as a sum of three squared integers. The sequence of observed h²+k²+l² values identifies the cubic lattice type and lattice parameter simultaneously.
Smaller crystallites produce broader peaks (described by the Debye-Scherrer equation). Crystallites below about 100 nm show measurable broadening with laboratory instruments. Below 5 nm, peaks become so broad they may overlap significantly. Peak broadening analysis (Williamson-Hall or Warren-Averbach methods) separates size and strain contributions.
A powder sample contains randomly oriented crystallites, so all possible Bragg angles are simultaneously satisfied. The resulting pattern is a plot of diffracted intensity versus 2θ, showing peaks at positions determined by d-spacings. Each peak corresponds to a set of crystal planes (hkl), and the pattern is characteristic of the crystal structure.
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