Bethe–Bloch Energy Loss Calculator

Introduction: what this calculator estimates

When a charged particle such as a proton, alpha particle, muon, pion, or other heavy ion passes through matter, it interacts electromagnetically with the electrons in the medium. Those repeated interactions gradually remove kinetic energy from the projectile by ionizing and exciting atoms. The average energy lost per unit distance is called the stopping power and is commonly written as −dE/dx. In detector physics, medical physics, radiation transport, and shielding work, stopping power tells you how fast a particle slows down and how much energy it deposits along the way.

This page implements a simplified form of the Bethe–Bloch equation to estimate stopping power for heavy charged particles over a broad range of energies, from non-relativistic motion up into the moderately relativistic regime. The calculator reports three outputs that are useful in slightly different ways. It gives a mass stopping power in MeV·cm²/g, which lets you compare materials without immediately tying the answer to density. It also gives a linear stopping power in MeV/cm, which is the quantity most people picture when they think about energy loss per centimeter of path length. Finally, it provides a simple range estimate, meant as a quick reality check for how far the particle might travel before losing its kinetic energy.

That final range number is intentionally approximate. The calculator estimates range by dividing the current kinetic energy by the current linear stopping power, which assumes the stopping power stays constant as the particle slows. Real tracks do not behave that way. As the particle loses speed, stopping power often rises and eventually produces the familiar Bragg peak near the end of the path. So the range output is best read as an order-of-magnitude shortcut, not as a replacement for a full transport calculation or measured range table.

How to use the calculator and what each input means

To use the tool, enter the particle properties first and then the target material properties. All entries must be positive, and the units matter because the relativistic terms and material constants are combined directly. If one number is off by a factor of a thousand, the answer can shift dramatically, so it is worth checking both values and units before interpreting the result.

1. Particle charge number z: this is the projectile charge in units of the proton charge. A proton has z = 1 and an alpha particle has z = 2. In the simplified model, stopping power scales roughly with z², so charge matters a great deal.
2. Particle mass m (MeV/c²): use the rest mass in high-energy units. Typical examples are proton ≈ 938.272 MeV/c², muon ≈ 105.658 MeV/c², and alpha ≈ 3727.38 MeV/c².
3. Kinetic energy T (MeV): enter the kinetic energy only, not the total relativistic energy. This input sets the particle speed through the γ and β factors.
4. Material atomic number Z and mass number A: these describe the target medium in the standard Bethe–Bloch combination Z/A. For silicon, Z = 14 and A ≈ 28.085 is a common value.
5. Mean excitation energy I (eV): this is the characteristic energy scale that enters the logarithm. Typical values are tens to hundreds of eV. Silicon is about 173 eV, water is about 75 eV, and lead is about 823 eV.
6. Density ρ (g/cm³): density converts the mass stopping power into a linear stopping power. Air is around 0.0012 g/cm³, silicon about 2.33 g/cm³, and lead about 11.34 g/cm³.

Decimal inputs are accepted everywhere, including the material mass number A. That is useful because tabulated atomic weights and effective material constants are often not integers. If you are modeling a compound or mixture, it is common to use effective values taken from a trusted reference rather than forcing the material into a single elemental approximation.

Formula used in this simplified Bethe–Bloch model

The calculation begins by converting the entered kinetic energy T and particle mass m into the usual relativistic factors. The Lorentz factor is γ = 1 + T/m, and the speed parameter is β = √(1 − 1/γ²). Here β is the particle speed divided by the speed of light. Those terms control both the kinematics of collisions and the strong low-velocity rise associated with the 1/β² factor.

The next step is to compute the maximum kinetic energy that can be transferred to an electron in a single collision. In this implementation that quantity is written as:

Once T_max is known, the calculator evaluates the stopping-power expression below. The result is treated as a mass stopping power, and then density is used to convert it into MeV/cm.

The numerical constants and unit conventions used here are straightforward. The code uses K = 0.307075 MeV·mol⁻¹·cm² and m_ec² = 0.511 MeV. The excitation energy I is entered in eV and converted internally to MeV by multiplying by 10⁻⁶. The calculator then multiplies the mass stopping power by the entered density ρ to obtain the linear stopping power in MeV/cm.

Worked example: a 200 MeV proton in silicon

Suppose you want a quick estimate for a 200 MeV proton traveling through silicon. Enter z = 1, m = 938 MeV/c², T = 200 MeV, Z = 14, A = 28 or 28.085, I = 173 eV, and ρ = 2.33 g/cm³. The calculator then builds the relativistic factors from the proton energy, evaluates the logarithm term, and converts the result into both mass and linear stopping power.

• First compute γ = 1 + T/m ≈ 1 + 200/938 ≈ 1.213.
• Then compute β² = 1 − 1/γ² ≈ 0.321, so β ≈ 0.566.
• Use β and γ in the T_max expression to estimate the largest single-collision energy transfer to an electron.
• Insert those values into the simplified Bethe–Bloch bracket to get a mass stopping power of the order of a few MeV·cm²/g.
• Multiply by ρ = 2.33 g/cm³ to obtain a linear stopping power on the order of roughly 10 MeV/cm.
• Finally estimate the range as T divided by the linear stopping power to get a rough stopping distance in centimeters.

The exact number you see may differ somewhat from a textbook table because this calculator intentionally omits several standard corrections and because published values of A and I can vary slightly by source. That is normal. The point of the worked example is to show the scale of the answer and the role of each input, not to reproduce a reference database digit for digit.

Reference table for typical magnitudes

The following examples are illustrative only, but they help anchor intuition. They show how large charge, low energy, or dense materials can raise stopping power sharply, while light and fast projectiles in thin media can travel much farther.

Limitations, assumptions, and when not to use this result directly

This calculator is designed for learning, quick estimates, and parameter exploration. It uses a simplified Bethe–Bloch expression and therefore has important limitations that should be kept in mind before using the output in a report, simulation chain, or design decision.

• No density-effect correction (δ): at high energies, polarization of the medium reduces the effective electric field seen at large distances. Leaving out δ can overestimate the relativistic rise.
• No shell corrections: at lower particle speeds, electron binding and atomic structure become more important, so the free-electron approximation becomes less reliable.
• No heavy-ion charge-state model: very heavy ions may not behave like fully stripped projectiles through the entire path, so the effective charge can differ from the nominal nuclear charge.
• Range is a constant-stopping-power estimate: real range calculations integrate 1/(dE/dx) over energy and naturally capture the Bragg peak.
• Not intended for electrons or positrons: light particles need a different treatment because radiative losses and identical-particle effects matter much more.

If you need high-precision answers for detector optimization, shielding certification, dosimetry, or clinical planning, you should compare against validated databases or full transport tools such as NIST stopping-power tables or Monte Carlo simulations. This page is best treated as a compact estimate and teaching aid.

How to interpret the output and build intuition

A useful way to read the result is to separate the trends into particle effects and material effects. On the particle side, stopping power scales roughly with z², so doubling charge can raise energy loss by about a factor of four in the simplest picture. The 1/β² dependence also means slow particles lose energy more aggressively than fast ones. This is why tracks can deposit a great deal of energy near the end of their path.

On the material side, the combination Z/A and the mean excitation energy I shape the logarithm, while the density ρ directly scales the linear stopping power. A denser medium converts the same mass stopping power into a larger energy loss per centimeter. That, in turn, shortens the rough range estimate. So if you keep the particle fixed and increase density, you should expect the reported MeV/cm to go up and the estimated stopping distance to go down.

A quick sanity check is to compare your answer with familiar scales. For many solids, minimum-ionizing heavy particles often sit at a few MeV·cm²/g. If you obtain a huge value at very high energy, remember that the missing density-effect correction can exaggerate the relativistic rise. If you obtain a negative or non-finite value, the cause is usually an unrealistic excitation energy, an extremely low kinetic energy, or another unit mismatch that pushes the logarithm outside a physical regime.

Practical notes for choosing inputs

For elemental materials, Z and A are easy to locate in standard data tables. The subtle input is usually the mean excitation energy I. That value is essential because it sits inside a logarithm, so a poor choice can move the output in a noticeable way. For compounds and mixtures such as water, air, plastic scintillator, or tissue-equivalent materials, the fully correct treatment uses effective values. If you have a trusted table for the compound, use that effective data directly rather than substituting a single element that only vaguely resembles the material.

It is also helpful to remember the sign convention. In derivations, dE/dx is negative because the particle is losing energy as x increases. In practical use, stopping power is often quoted as a positive magnitude. This calculator follows that practical convention and reports positive values for the size of the energy loss.

Mini FAQ

Why does the calculator ask for mass in MeV/c²?

Using MeV/c² for mass is standard in particle and nuclear physics. With that convention, kinetic energy in MeV combines cleanly with rest mass to produce γ = 1 + T/m without additional conversion factors cluttering the calculation.

Can I use this for electrons?

Not reliably. Electrons and positrons require extra terms and a different loss model. They are light enough that radiative processes and identical-particle effects become important in regimes where heavy-particle Bethe–Bloch works well.

Why is the range only an estimate?

Because dE/dx changes as the particle slows. A better range model integrates the reciprocal stopping power over energy, often using tabulated data or a richer theoretical model. The simple ratio T/(dE/dx) is still useful as a quick first pass, especially when you want to compare trends across different particles or materials.