Calculator
Formula used
This calculator uses a Bethe-style expression for the mass stopping power:
- K = 0.307075 MeV·mol−1·cm2, with Z, A for the target.
- β and γ come from β or kinetic energy and mass.
- Tmax is the maximum energy transfer to an electron.
- δ and C are optional density and shell corrections.
- zeff is an optional effective-charge heuristic for slow ions.
How to use this calculator
- Select a material preset or enter custom target values.
- Enter projectile charge and choose energy or beta input.
- Provide projectile mass in amu or MeV/c².
- Keep δ and C as zero unless you have corrections.
- Optionally set thickness to estimate total energy loss.
- Press Calculate to view results above this form.
- Use CSV or PDF buttons to export the summary.
Example data
| Scenario | Mass stopping power (MeV·cm²/g) | Linear stopping power (MeV/cm) |
|---|---|---|
| Proton 150 MeV in Silicon | 4.3914 | 10.2275 |
| Alpha 40 MeV in Aluminum | 136.3862 | 368.2428 |
| Carbon ion 200 MeV/u in Water | 160.9749 | 160.9749 |
What stopping power measures
Stopping power quantifies how fast a charged particle loses energy while traversing matter. The calculator reports mass stopping power (MeV·cm²/g) and linear stopping power (MeV/cm). Mass form removes density, letting you compare materials fairly, while linear form multiplies by density to estimate local energy loss.
Where the Bethe-style model applies
The Bethe framework is most reliable for moderately relativistic projectiles where ionization dominates. Typical useful regions include protons from tens of MeV to several GeV and heavy ions above a few MeV per nucleon. At very low velocity, additional effects can reduce accuracy, so treat results as estimates.
Key inputs and their impact
Target parameters Z, A, density ρ, and mean excitation energy I set the baseline interaction strength. The ratio Z/A often falls near 0.4–0.5 for many solids, directly scaling stopping power. I is usually tens to hundreds of eV; increasing I generally decreases stopping power through the logarithmic term.
Velocity terms: β and γ
The dominant scaling is roughly proportional to 1/β², while the logarithmic term grows with γ². As a result, stopping power decreases from low energy toward a broad minimum (often around 1–2 MeV·cm²/g for minimum-ionizing singly charged particles), then slowly rises at higher energies before density corrections moderate the rise.
Maximum electron energy transfer
The model uses Tmax, the maximum energy a single collision can transfer to an electron. For light projectiles at high γ, Tmax can reach MeV scales, strengthening the logarithmic term. For heavy projectiles, the mass ratio suppresses Tmax, shifting the balance of the bracketed expression.
Corrections: density effect and shell term
The density effect correction δ reduces stopping power at high energies by accounting for medium polarization that screens distant interactions. The shell correction C addresses atomic binding at lower energies, effectively reducing the logarithmic gain. When you lack measured values, setting δ and C to zero is a conservative starting point.
Effective charge for slow ions
Slow, highly charged ions can capture electrons and behave as if their charge is reduced. The effective-charge option applies a common heuristic to shrink z toward a velocity-dependent value, which can prevent unrealistically large 1/β² growth. This is most helpful for heavy ions below a few percent of light speed.
Thickness and energy-loss estimate
The thickness field multiplies linear stopping power by x to estimate energy loss assuming stopping power stays constant along the path. This is a good approximation for thin layers or small fractional energy loss. For thicker absorbers, stopping power changes as energy decreases, producing Bragg-peak behavior that requires stepwise integration.
FAQs
1) What is the difference between mass and linear stopping power?
Mass stopping power is energy loss per areal density (MeV·cm²/g). Linear stopping power multiplies by density to give MeV/cm, which is useful for thickness-based energy loss estimates.
2) Which particle types does this calculator suit best?
It is designed for charged particles where ionization dominates, such as protons, alphas, and heavy ions. It is not intended for neutrons or photons, which lose energy through different mechanisms.
3) Why can the result become small or negative?
At low energy, the Bethe-style bracket may drop due to model limits or overestimated corrections. If this happens, use higher energies, enable effective charge for ions, or treat the output as outside validity.
4) How should I pick the mean excitation energy I?
Use published I values for the material when possible. Typical I ranges are about 60–100 eV for light materials and several hundred eV for heavier elements. Small changes in I can shift results noticeably.
5) Does the thickness calculation include Bragg peak effects?
No. It uses a constant stopping approximation, so it is most accurate for thin targets or small energy losses. For thick targets, you should integrate stopping power as energy decreases.
6) When should I enable effective charge?
Enable it for slow heavy ions or when you suspect charge exchange is significant. It reduces the effective projectile charge at low velocity, which can improve realism and stabilize the 1/β² scaling.
7) What units should I use for density and thickness?
Enter density in g/cm³ and thickness in cm. The calculator then produces linear stopping power in MeV/cm and estimates energy loss in MeV using those consistent units.