Scattering Cross-Section Calculator computes differential and total cross-sections for a particle scattering off a hard sphere, visualizing results for quantum studies.
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Formulas Used in Scattering Cross-Section Calculator
The calculator uses the following formulas for s-wave scattering off a hard sphere:
Wave Number:
\\[
k = \frac{\sqrt{2 m E}}{\hbar}
\\]
s-wave Scattering Amplitude:
\\[
f(\theta) \approx -a
\\]
Differential Cross-Section:
\\[
\frac{d\sigma}{d\Omega} = |f(\theta)|^2 \approx a^2
\\]
Total Cross-Section:
\\[
\sigma = 4 \pi |f(\theta)|^2 \approx 4 \pi a^2
\\]
Where:
\\( m \\): Particle mass (kg)
\\( E \\): Incident energy (J, converted from eV)
\\( \hbar \\): Reduced Planck constant (\\( 1.0545718 \times 10^{-34} \, \text{J·s} \\))
\\( a \\): Sphere radius (m)
\\( k \\): Wave number (m⁻¹)
\\( f(\theta) \\): Scattering amplitude (m)
\\( \frac{d\sigma}{d\Omega} \\): Differential cross-section (m²/sr)
\\( \sigma \\): Total cross-section (m²)
Example Calculations
Example 1: Electron Scattering
Input: Mass = 9.11e-31 kg, Energy = 0.1 eV, Radius = 1e-10 m
\\[
E = 0.1 \times 1.60217662 \times 10^{-19} \approx 1.602 \times 10^{-20} \, \text{J}
\\]
\\[
k = \frac{\sqrt{2 \cdot 9.11 \times 10^{-31} \cdot 1.602 \times 10^{-20}}}{1.0545718 \times 10^{-34}} \approx 5.126 \times 10^9 \, \text{m}^{-1}
\\]
\\[
ka \approx 5.126 \times 10^9 \cdot 10^{-10} \approx 0.5126
\\]
\\[
\frac{d\sigma}{d\Omega} \approx (10^{-10})^2 = 10^{-20} \, \text{m}^2/\text{sr}
\\]
\\[
\sigma \approx 4 \pi \cdot 10^{-20} \approx 1.257 \times 10^{-19} \, \text{m}^2
\\]
Result: Differential Cross-Section: 1e-20 m²/sr, Total Cross-Section: 1.257e-19 m²
Example 2: Heavy Particle
Input: Mass = 1e-27 kg, Energy = 0.1 eV, Radius = 1e-10 m
\\[
E = 0.1 \times 1.60217662 \times 10^{-19} \approx 1.602 \times 10^{-20} \, \text{J}
\\]
\\[
k = \frac{\sqrt{2 \cdot 10^{-27} \cdot 1.602 \times 10^{-20}}}{1.0545718 \times 10^{-34}} \approx 1.697 \times 10^9 \, \text{m}^{-1}
\\]
\\[
ka \approx 1.697 \times 10^9 \cdot 10^{-10} \approx 0.1697
\\]
\\[
\frac{d\sigma}{d\Omega} \approx (10^{-10})^2 = 10^{-20} \, \text{m}^2/\text{sr}
\\]
\\[
\sigma \approx 4 \pi \cdot 10^{-20} \approx 1.257 \times 10^{-19} \, \text{m}^2
\\]
Result: Differential Cross-Section: 1e-20 m²/sr, Total Cross-Section: 1.257e-19 m²
Example 3: Larger Radius
Input: Mass = 9.11e-31 kg, Energy = 0.1 eV, Radius = 5e-10 m
\\[
E = 0.1 \times 1.60217662 \times 10^{-19} \approx 1.602 \times 10^{-20} \, \text{J}
\\]
\\[
k = \frac{\sqrt{2 \cdot 9.11 \times 10^{-31} \cdot 1.602 \times 10^{-20}}}{1.0545718 \times 10^{-34}} \approx 5.126 \times 10^9 \, \text{m}^{-1}
\\]
\\[
ka \approx 5.126 \times 10^9 \cdot 5 \times 10^{-10} \approx 2.563
\\]
\\[
\frac{d\sigma}{d\Omega} \approx (5 \times 10^{-10})^2 = 2.5 \times 10^{-19} \, \text{m}^2/\text{sr}
\\]
\\[
\sigma \approx 4 \pi \cdot 2.5 \times 10^{-19} \approx 3.142 \times 10^{-18} \, \text{m}^2
\\]
Result: Differential Cross-Section: 2.5e-19 m²/sr, Total Cross-Section: 3.142e-18 m²
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