Bethe-Bloch Formula: How Charged Particles Lose Energy

Key takeaways

• The Bethe-Bloch formula gives the mean energy a fast charged particle loses per unit path in matter. The Particle Data Group (PDG) calls it the Bethe equation and says it holds for βγ from about 0.1 to 1000, with an accuracy of a few percent.
• Stopping power falls roughly as 1/β² at low speed, reaches a broad minimum near βγ of 3 to 3.5 and then creeps upward in the relativistic rise.
• A minimum-ionising particle in silicon loses 1.664 MeV cm²/g, which is 3.876 MeV/cm. A 300 µm layer takes a mean of about 116 keV.
• The most probable loss in that layer is lower, about 84 keV, because a few rare hard collisions drag the mean up. That is why detector designers quote the most probable value.
• Below βγ of roughly 0.1 the formula needs shell, Barkas and Bloch corrections. Electrons and positrons need a separate treatment.

The Bethe-Bloch formula tells you how much energy a fast charged particle deposits per centimetre of material. For a muon at its energy-loss minimum in silicon the answer is 1.664 MeV cm²/g, or 3.876 MeV/cm. In a 300 µm silicon sensor that is a mean of 116 keV, yet the most probable deposit is near 84 keV, or about 23,000 electron-hole pairs.

This page is the hub for our detector-physics pieces. It sets out the formula, each symbol, the shape of the curve and its limits, then works one example through. The spread of single events around the mean is covered in our guide to the Landau distribution and the piece on energy straggling in silicon.

The formula and what each term does

The PDG chapter on the passage of particles through matter writes the mean mass stopping power as Eq. 34.5 (PDG Review of Particle Physics, 2024, section 34). In plain text it reads as follows.

-dE/dx = K z² (Z/A) (1/β²) [ ½ ln( 2 mec² β²γ² Wmax / I² ) - β² - δ(βγ)/2 ]

The result has units of MeV cm²/g. Multiply by the density in g/cm³ to get the linear rate in MeV/cm. The prefactor sets the scale, 1/β² drives the fall at low speed and the logarithm drives the rise at high speed.

The PDG gives W_max for a heavy particle as 2m_ec²β²γ², divided by 1 + 2γm_e/M + (m_e/M)². It adds that dropping that denominator, as older texts do, can err by more than 6 percent for a pion in copper at 100 GeV.

The shape of the curve: fall, minimum and rise

The PDG describes three regimes. At low speed the stopping power falls with an exponent between 1.7 and 1.5, a little slower than a pure 1/β² because the logarithm grows too. It reaches a broad minimum and then rises again. The minimum sits at βγ of about 3.5 for Z = 7 and 3.0 for Z = 100.

Particles near that minimum are called minimum-ionising, or mips. The PDG gives cosmic-ray muons as the standard example. The table below shows the whole shape for muons in silicon, taken from the PDG muon energy-loss table for silicon.

Between 20 MeV and the minimum the loss drops by more than half. From the minimum to 1 GeV it rises only 8 percent. The PDG attributes about two thirds of that rise to the explicit β²γ² term in the logarithm. The remaining third comes from W_max, which keeps growing and lets rare, violent collisions into the average.

Those rare collisions matter. The PDG warns that the mean is weighted by single events with large losses, so the most probable loss in a real detector is well below the Bethe value. For muons crossing 0.64 cm of plastic scintillator it gives the ratio of most probable to mean rate as 0.69, 0.57, 0.49, 0.42 and 0.38 at 0.01, 0.1, 1, 10 and 100 GeV.

Why heavy particles stop with a bang: the Bragg peak

A proton or ion that slows in matter climbs the left side of that same curve. The 1/β² term grows as the speed drops, so the particle deposits most of its energy in the final stretch of its path. This sharp maximum before the particle stops is the Bragg peak. Robert Wilson published the case for using fast protons in radiology in 1946, and proton therapy uses the peak to place dose inside a tumour.

The table evaluates Eq. 34.5 for protons in liquid water, with I = 79.7 eV and Z/A = 10/18.015 = 0.5551. No shell or Barkas correction is applied, which keeps it inside the PDG range of validity down to 10 MeV.

A 10 MeV proton loses energy 8.4 times faster than a 150 MeV proton in the same water. The particle spends its last few MeV in a very short distance, so the dose spikes. At much lower energy the curve finally turns over. The PDG tabulates a maximum near 120 keV for protons in copper.

Worked example: a minimum-ionising particle in 300 µm of silicon

Silicon sensors in tracking detectors are typically a few hundred micrometres thick, as in our overview of silicon pixel sensors. The table works through a 300 µm layer step by step. Every input comes from the PDG atomic and nuclear properties page for silicon or from the PDG chapters, and every output is our arithmetic.

The two most probable values differ by about 6 percent. The PDG detector chapter quotes 3.7 fC, or 23,000 electrons, for a minimum-ionising particle in 300 µm of silicon (PDG section 35, particle detectors). The simple Landau-Vavilov expression in the passage chapter gives a slightly lower figure. The PDG also notes that folding in experimental resolution usually moves a measured peak higher. We use the PDG charge as the headline value and show the formula result beside it.

The pair count uses 3.65 eV per pair, which is the PDG average at room temperature. The band gap is only 1.12 eV. Phonons take part of the energy, so the average cost per pair is about three times the gap. You can run this calculation for your own particle and material in the ScienceShot Bethe-Bloch energy loss calculator. It reports the mean rate, the Landau most probable loss and the straggling in 16 materials.

Minimum ionisation in common materials

Per gram, minimum-ionisation losses change little from material to material. Per centimetre they differ widely, because density does most of the work. The PDG shows the mass value falling slowly with atomic number.

Lead is worth a second look. Its mass stopping power is only 57 percent of water’s, but its density is more than eleven times higher. Each centimetre of lead therefore removes 12.74 MeV against 1.98 MeV for water, a ratio of 6.4.

Where the formula stops working

• Above βγ of about 1000 radiative losses start to matter, and the stopping power is no longer a function of βγ alone (PDG, section 34.2.3).
• Below βγ of about 0.1 the projectile speed approaches that of atomic electrons. Adding shell, Barkas and Bloch terms keeps the treatment accurate to about 1 percent down to β of 0.05, which is about 1 MeV for protons.
• Between β of 0.01 and 0.05 the PDG says there is no satisfactory theory, and fitted formulae are used. Below about 0.01 the Lindhard description of electronic stopping takes over.
• Heavy ions need extra terms for higher-order coupling and finite target size. These can change the loss by a factor of two or more for the heaviest nuclei.
• Electrons and positrons are a separate case. The PDG gives them their own section, because the incoming electron is identical to the target electron in Møller scattering and the maximum energy transfer differs.

Thin absorbers raise a further issue. A few collisions cannot average out, so the real distribution is asymmetric and a single event can stray far from the mean. That is the subject of the Landau and straggling pieces linked above.

Cosmic-ray muons are a convenient source of minimum-ionising particles for testing sensors. Cosmic rays are also a hazard for electronics, as we describe in our piece on cosmic rays and silent data corruption.

Who wrote the formula, and when

Hans Bethe derived the quantum-mechanical stopping formula in 1930, using the Born approximation. Felix Bloch extended it in 1933 with a correction for slower particles. The table gives the references.

Textbooks join the two names. The PDG itself says Bethe equation and calls the low-energy term the Bloch correction.

Sources

• S. Navas et al. (Particle Data Group), Review of Particle Physics, Phys. Rev. D 110, 030001 (2024), section 34, Passage of particles through matter
• Particle Data Group, Review of Particle Physics 2024, section 35, Particle detectors at accelerators (semiconductor detectors)
• PDG, Atomic and nuclear properties of silicon
• PDG, Muon energy loss table for silicon
• PDG, Atomic and nuclear properties of liquid argon
• PDG, Atomic and nuclear properties of liquid water
• PDG, Atomic and nuclear properties of polyvinyltoluene
• PDG, Atomic and nuclear properties of lead
• H. Bethe, Zur Theorie des Durchgangs schneller Korpuskularstrahlen durch Materie, Annalen der Physik 397, 325 (1930)
• H. Bethe, Bremsformel für Elektronen relativistischer Geschwindigkeit, Zeitschrift für Physik 76, 293 (1932)
• F. Bloch, Zur Bremsung rasch bewegter Teilchen beim Durchgang durch Materie, Annalen der Physik 408, 285 (1933)
• R. R. Wilson, Radiological use of fast protons, Radiology 47, 487 (1946)