Engineering & Science
Bethe-Bloch Mass Stopping Power (-dE/rho*dx) & Proton Bragg Peak Formula
Calculate collisional mass stopping power S_mass = 0.307075 * (Z/A) * (z^2 / beta^2) * (LN((2 * m_e * c^2 * beta^2) / ((1 - beta^2) * I_eV)) - beta^2) in MeV*cm^2/g for protons and alpha particles in tissue/water.
Why cancer proton therapy dumps its dose in a sharp Bragg Peak right at the tumor!
Unlike X-ray photons that attenuate exponentially from the skin inward, heavy charged particles (protons z = 1, alpha/helium ions z = 2, carbon ions z = 6) lose energy via Coulomb collisions with atomic electrons governed by Hans Bethe and Felix Bloch's relativistic stopping-power formula: -dE / (rho * dx) = K * (Z / A) * (z^2 / beta^2) * (LN((2 * m_e * c^2 * beta^2 * gamma^2) / I_eV) - beta^2), with K = 0.307075 MeV*cm^2/g and electron rest energy m_e * c^2 = 511,000 eV!
Look at the 1 / beta^2 prefactor in liquid water / soft tissue (Z/A = 0.5551, mean excitation potential I = 75 eV): an energetic clinical proton entering the body at beta = 0.50 (~145 MeV) deposits a gentle 5.57 MeV*cm^2/g across healthy entrance tissue; as the proton slows down to beta = 0.20 (~19 MeV) near the tumor depth, 1 / beta^2 surges and stopping power jumps nearly FIVE-FOLD to 26.85 MeV*cm^2/g—carving out the famous therapeutic Bragg Peak! And an alpha particle (z = 2) at beta = 0.50 deposits z^2 = 4x more (22.29 MeV*cm^2/g)!
| Projectile in Water/Tissue (Z/A = 0.5551, I = 75 eV) | Charge z & Speed beta | Mass Stopping Power (MeV*cm^2/g) | Clinical / Shielding Regime |
|---|---|---|---|
| Fast Entrance Proton (~145 MeV) | z = 1, beta = 0.50 | 5.57 MeV*cm^2/g | Low entrance skin dose |
| Slowing Proton Near End-of-Range | z = 1, beta = 0.20 | 26.85 MeV*cm^2/g (4.82x surge!) | Rising edge of Bragg Peak! |
| Alpha Particle / He-4 Ion | z = 2, beta = 0.50 | 22.29 MeV*cm^2/g (Exact z^2 = 4x!) | High-LET dense ionization track |
Bethe-Bloch Mass Stopping Power -dE/(rho*dx) in MeV*cm^2/g
Uses gamma^2 = 1 / (1 - beta^2) and guards against non-relativistic shell-capture breakdown where the log argument <= 1.
=IF(OR(B2 <= 0, B3 <= 0, B3 >= 1, B4 <= 0, B5 <= 0, ((2 * 511000 * (B3 ^ 2)) / ((1 - (B3 ^ 2)) * B5)) <= 1), "Outside Bethe-Bloch Velocity Domain", ROUND(0.307075 * B4 * ((B2 ^ 2) / (B3 ^ 2)) * (LN((2 * 511000 * (B3 ^ 2)) / ((1 - (B3 ^ 2)) * B5)) - (B3 ^ 2)), 2))Linear Stopping Power LET = -dE/dx = S_mass * Density_g_cm3 (MeV/cm)
Multiplies mass stopping power by absorber mass density B6 (g/cm^3, 1.00 for water; 2.70 for aluminum spacecraft hull).
=IF(OR(B2 <= 0, B3 <= 0, B3 >= 1, B6 <= 0), "Outside Bethe-Bloch Velocity Domain", ROUND(B6 * 0.307075 * B4 * ((B2 ^ 2) / (B3 ^ 2)) * (LN((2 * 511000 * (B3 ^ 2)) / ((1 - (B3 ^ 2)) * B5)) - (B3 ^ 2)), 2))Try the example
Use the original values or change them to test your understanding.