00 Foundations

Proton range (Bragg–Kleeman)

R = α E^p in water, with p ≈ 1.77 and α ≈ 0.0022 cm·MeV^{−p}.

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Simulation

Proton range (Bragg–Kleeman) — Change the numbers; the scene follows.

Where it works

Proton gantry

Patient / Bragg peak

Along the proton path in tissue, from the snout to the distal Bragg peak.

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Formula

R=\alpha E^{p},\quad p\approx 1.77,\quad \alpha\approx 0.0022\,\mathrm{cm\,MeV}^{-p}

Variables

E · Kinetic energyMeVα · Bragg–Kleeman αcm·MeV⁻ᵖp · Exponentd · Depthcm

Results

R
Range in water
26.0163cm
R−d
Residual range
6.0163cm
E_res
Residual energy
87.4492MeV

Curve

Explanation

R=\alpha E^{p},\quad p\approx 1.77,\quad \alpha\approx 0.0022\,\mathrm{cm\,MeV}^{-p}

What it means

The Bragg–Kleeman rule is the power-law fit to CSDA proton range in a given material. In water, 80 MeV protons stop near 5 cm (ocular / shallows), 160 MeV near 17 cm, 200 MeV near 26 cm — the numbers every proton physicist quotes from memory. Differentiating gives the residual-range relation used to pull a spread-out Bragg peak (SOBP) from a pristine peak. This is a working relation in Radiation physics.

Where it is used

Clinically it sits on the Proton gantry — Patient / Bragg peak. Along the proton path in tissue, from the snout to the distal Bragg peak. Radiation physics lives at the x-ray target, the linac head, and inside the patient: how a photon is born, how it scatters, and how it dies. Use these relations before you trust a spectrum, a wall, or a kV-versus-MV contrast argument.

Proton gantry · Open this machine

How to use it

Enter kinetic energy E in MeV. Optionally edit α and p for a non-water medium (scale α by 1/ρ and a weak Z dependence). Read range in cm water and the residual energy at a stated depth. Change one input and watch the curve and the simulation follow.

Symbols

EKinetic energy200 MeV
αBragg–Kleeman α0.0022 cm·MeV⁻ᵖ
pExponent1.77
dDepth20 cm

Worked example

200 MeV, α=0.0022, p=1.77 → R ≈ 25.5 cm water. Residual energy at 20 cm is the E that has range 5.5 cm ≈ 90 MeV. In numbers: E = 200 MeV (Kinetic energy); α = 0.0022 cm·MeV⁻ᵖ (Bragg–Kleeman α); p = 1.77 (Exponent); d = 20 cm (Depth) → R = 26.0163 cm; R−d = 6.0163 cm; E_res = 87.4492 MeV.

Typical values give

R = 26.0163cm
R−d = 6.0163cm
E_res = 87.4492MeV

Where it comes from

The displayed formula is the working relation. R = α E^p in water, with p ≈ 1.77 and α ≈ 0.0022 cm·MeV^{−p}. Usual reference: Bragg & Kleeman / ICRU 78. Derive it in the specialty lesson, then return here to pin the numbers.

Reference: Bragg & Kleeman / ICRU 78

Assumptions & limits

CSDA continuous-slowing-down range, not the distal 90% clinical range. Ignores range straggling (~1% of R), inelastic nuclear loss (~1%/cm in water), and heterogeneity. Clinical TPS uses Monte Carlo or measured IDs.

Pitfalls

Do not mix free-electron Compton kinematics with photoelectric-dominated kV imaging. Check keV versus MeV, and never treat a spectrum as one photon. CSDA continuous-slowing-down range, not the distal 90% clinical range. Ignores range straggling (~1% of R), inelastic nuclear loss (~1%/cm in water), and heterogeneity. Clinical TPS uses Monte Carlo or measured IDs.

Keep this

Photons do not deposit dose; the electrons they set in motion do. CSDA continuous-slowing-down range, not the distal 90% clinical range. Ignores range straggling (~1% of R), inelastic nuclear loss (~1%/cm in water), and heterogeneity. Clinical TPS uses Monte Carlo or measured IDs.

In this specialty

Radiation physics