Bragg Ewald Construction (k+G on circle)
2θ_Bragg
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d_hkl
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|F(hk)|²
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Bragg Cond.
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All Reflections (Bragg condition status)
| h | k | d (Å) | 2θ (°) | |F|² | Bragg Condition |
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Diffraction Intensity I(2θ) – High resolution (sharp peaks)
📊 Bragg peak data table
| # | h | k | 2θ (°) | Intensity | d (Å) |
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Powder Diffraction Pattern & Crystallite Size (Sherrer)
ℹ️ Peaks are broadened with angle-dependent FWHM.
Crystallite size D = Kλ / (β·cosθ) for first 5 peaks (K=0.94).
0.70
0.30
3.5
Combined Pattern (Weighted Sum)
Separated Phases (Blue: Phase 1, Orange: Phase 2) with Weights
Grazing Incidence Diffraction (GIXRD)
EDXRD: Intensity vs Energy (fixed 2θ)
2D Reciprocal Space & Brillouin Zone (clipPolygon)
a*
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b*
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|k|
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Ewald R
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Bragg Pts
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Sym. Path
Γ-X-M-Γ
Band Structure E(k) along symmetry path (2D)
📊 Model: Free Electron (E ∝ k²)
3D Reciprocal Space | Drag to rotate | SC/BCC/FCC/HCP/Diamond + Advanced
3D Lattice
SC
Structure Factor Rule
All hkl allowed
Symmetry Points
Γ, X, M, R
1st-4th Brillouin Zones | Mouse wheel zoom | Adjustable opacity | Vectors (red) | Paths (cyan)
Density of States (3D band model)
3D Band Structure along high-symmetry path (active model)
300 K
350 K
Debye‑Waller factor exp(-2W)
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Peak broadening FWHM (deg)
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Intensity reduction I(T)/I₀
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Selected peak (h,k)
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Thermal broadening of a Bragg peak
Normalized intensity vs temperature
📋 Active Bragg peaks:
2D Reciprocal Space Intensity Map |F(hk)|² (non‑negative)
0.0%
Strain Effect on Diffraction Pattern
Original a (Å)
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Strained a (Å)
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Peak Shift (Δ2θ)
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Δd/d (%)
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FWHM vs 2θ (always ≥ 0)
Rocking Curve (Intensity vs θ for selected peak)
Arrhenius Plot: ln(I/I₀) vs 1/T
Fermi Surface in 3D k-space | Drag to rotate
Model
Free Electron
EF (eV)
1.50
kF (Å⁻¹)
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Points
0
1.00
2.50
Loaded CIF Structure (Direct Lattice) | Drag to rotate | Scroll to zoom
Name
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Atoms
0
a, b, c (Å)
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α, β, γ (°)
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Phonon Dispersion ω(k) along symmetry path
Model: Harmonic (Nearest Neighbor)
SC/BCC/FCC only
Effect of Point Defects on Diffraction Pattern
Vacancy %
0%
Substitution %
0%
Subst. atom
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Total intensity reduction
0%
ℹ️ Defects modify structure factor: F_defect = F_clean * (1 - vacancy% - substitution%) + substitution_factor.
Band structure is modified using virtual crystal approximation.
Heat Capacity (Cv) and Entropy (S) vs Temperature
Cv at 300 K (J/mol·K)
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S at 300 K (J/mol·K)
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Debye temp (K)
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ℹ️ Cv and S are calculated from DOS using harmonic approximation.
Cv(T) = ∫ DOS(E) * (E/kB T)^2 * exp(E/kB T) / (exp(E/kB T)-1)^2 dE, S(T) = ∫ DOS(E) * [E/T * f(E) + kB * ln(1 + exp(-E/kB T))] dE.
30
80
● Pattern 1 ● Pattern 2
Compare Two Diffraction Patterns (different θ)
Parameter Animation (θ or λ)
Current Diffraction Spectrum
| h | k | d (Å) | 2θ (°) | |F|² |
|---|
| Rank | h | k | 2θ (°) | |F|² | Relative Intensity |
|---|
3D Band Structure (Multiple Bands)
3D Diffraction: hkl Reflections with Extinction Rules
3D Reciprocal Space Intensity Map |F(hk)|²
0
Peak Fitting: Blue=Data, Red=Fitted, Green=Residual
| Param | Value | Error |
|---|
Bragg's Law: 2d sinθ = nλ
d = interplanar spacing, θ = glancing angle
Laue condition: k + G = k' , |k'| = |k|
Ewald sphere: |k+G| = |k|
Structure Factor: F(G) = Σ fⱼ e^{iG·rⱼ}
Intensity ∝ |F|², basis positions rⱼ
Reciprocal Lattice vectors: a* = 2π (b×c)/V , etc.
BCC: h+k+l odd → F=0; FCC: mixed parity → F=0; Diamond: additional condition; HCP special extinctions
Band models: Free: E∝k² ; NFE: E = k² + V₀cos(2k) ; Tight-Binding: E = -2(cos kx+cos ky)
Sherrer: D = Kλ / (β·cosθ) with K≈0.94
Debye‑Waller: I(T) = I₀ exp(-2W) with 2W ∝ T/θ_D²
Fermi Surface: E(k) = EF
Surface of constant energy in k-space
Phonon Dispersion: ω(k) from dynamical matrix
D(q) = (K/m) Σ (1 - cos(q·R)) · (R̂ R̂ᵀ)
Defect Structure Factor: F_def = Σ fⱼ (1 - c_v - c_sub) e^{iG·rⱼ} + c_sub Σ f_sub e^{iG·rⱼ}
Heat Capacity: Cv = ∫ DOS(E) · (E/k_B T)² · exp(E/k_B T) / (exp(E/k_B T)-1)² dE
Entropy: S = ∫ DOS(E) · [E/T · f(E) + k_B · ln(1 + exp(-E/k_B T))] dE