---
title: Stopping Power and the Range of Charged Particles
module: Radiation and Applications
moduleNumber: 11
lessonNumber: 1
order: 1101
summary: >
  A heavy charged particle loses energy in a dense sequence of small Coulomb
  collisions with atomic electrons, at a rate the Bethe-Bloch formula fixes from
  the particle's charge and speed and the medium's electron density and mean
  excitation energy. The rate scales as the inverse square of the speed, so most
  energy is deposited at the end of the track in the Bragg peak, and integrating
  the reciprocal rate gives a sharp range. Electrons differ: they also radiate,
  and above a critical energy bremsstrahlung dominates. Fast particles above the
  phase velocity of light in the medium emit Cherenkov radiation.
topics: [Radiation and Applications]
sources:
  - book: Krane
    ref: "Ch. 7 — Detecting Nuclear Radiations; §7.1 Interactions of Radiation with Matter"
  - book: Wong
    ref: "Ch. 4 — Nuclear Collective Motion; §4-1 Interaction of Charged Particles with Matter"
draft: false
---

A charged particle passing through matter is decelerated almost entirely by the
Coulomb force it exerts on the atomic electrons of the medium. Each encounter
transfers a small amount of energy to an electron; over the enormous number of
encounters along even a millimeter of track the losses add to a smooth,
predictable deceleration. Two quantities describe the process: the **stopping
power**, the average energy lost per unit path length, and the **range**, the
total distance the particle travels before stopping. For heavy particles (protons,
alphas, fission fragments, any particle much heavier than the electron) both follow
from a single collision analysis; electrons require a second energy-loss channel,
radiation, and relativistic particles faster than light in the medium add a third,
Cherenkov emission.[^krane-interactions]

## The energy-loss mechanism

A particle of charge $ze$ and speed $v$ interacts with a nearly free atomic
electron at impact parameter $b$. The electron is much lighter, so it recoils
while the heavy particle moves on almost undeflected. The momentum delivered to
the electron is the transverse Coulomb impulse,

$$
\Delta p = \int_{-\infty}^{\infty} F_\perp \, \d t
= \frac{ze^2}{4\pi\epsilon_0}\int_{-\infty}^{\infty}\frac{b}{(b^2 + v^2 t^2)^{3/2}}\,\d t
= \frac{2 z e^2}{4\pi\epsilon_0\, b\, v},
$$

and the energy transferred to a single electron is

$$
\Delta E(b) = \frac{(\Delta p)^2}{2 m_e} = \frac{2 z^2 e^4}{(4\pi\epsilon_0)^2\, m_e v^2\, b^2}.
$$

The energy lost to all electrons in a shell of impact parameters between $b$ and
$b + \d b$, over a path $\d x$ through a medium with electron number density $n$,
is $\Delta E(b)\cdot n\,(2\pi b\,\d b)\,\d x$. Integrating over impact parameters,

$$
-\frac{\d E}{\d x} = \frac{4\pi z^2 e^4\, n}{(4\pi\epsilon_0)^2\, m_e v^2}
\int_{b_\mathrm{min}}^{b_\mathrm{max}} \frac{\d b}{b}
= \frac{4\pi z^2 e^4\, n}{(4\pi\epsilon_0)^2\, m_e v^2}\,\ln\frac{b_\mathrm{max}}{b_\mathrm{min}}.
$$

The logarithm carries the whole result, and its limits are set by physics the
point-charge picture ignores. The **maximum** impact parameter is fixed by the
adiabatic condition: if the collision lasts longer than an atomic orbital period,
the electron follows the field adiabatically and absorbs no net energy, so
$b_\mathrm{max} \approx v/\bar\omega$ with $\bar\omega$ a mean orbital frequency.
The **minimum** impact parameter is set by the largest kinematically allowed
energy transfer, a head-on collision giving the electron speed $2v$, which
translates to $b_\mathrm{min} \approx z e^2/(4\pi\epsilon_0\, m_e v^2)$. This is the
1913 Bohr classical result. The full quantum treatment by Bethe replaces the
orbital frequencies by a single medium constant, the mean excitation energy $I$,
and adds relativistic kinematics.

## The Bethe-Bloch formula

The quantum-mechanical stopping power for a heavy charged particle is the
Bethe-Bloch formula,

$$
-\frac{\d E}{\d x}
= \frac{4\pi\, n\, z^2}{m_e c^2\,\beta^2}\left(\frac{e^2}{4\pi\epsilon_0}\right)^{\!2}
\left[\ln\frac{2 m_e c^2 \beta^2 \gamma^2}{I} - \beta^2 - \frac{\delta}{2}\right],
$$

with $\beta = v/c$, $\gamma = (1-\beta^2)^{-1/2}$, and $n = N_A Z \rho / A$ the
electron density of a medium of atomic number $Z$, mass number $A$, and density
$\rho$. Written this way the leading factor is the same combination of constants
that appeared in the classical estimate; the bracket collects the quantum and
relativistic corrections.

> **Definition (Mean excitation energy).** The mean excitation energy $I$ is the
> logarithmic average of the transition energies of the medium's atomic electrons,
> weighted by oscillator strength. It is the single atomic parameter in the
> stopping power and is determined empirically; an adequate approximation is
> $I \approx 10\,Z\ \mathrm{eV}$ for $Z > 16$, rising to $\approx 19\ \mathrm{eV}$
> for hydrogen. Water, the reference tissue-equivalent medium, has
> $I = 75\ \mathrm{eV}$.

Each factor in the formula reads off a distinct dependence:

- **Charge.** The loss scales as $z^2$: an alpha particle ($z=2$) loses energy
  four times as fast as a proton of the same speed. This is why alphas have very
  short ranges.
- **Speed.** The dominant factor is $1/\beta^2$: a slow particle spends more time
  near each electron and delivers a larger impulse. As the particle slows, it
  loses energy faster, which concentrates the deposition at the end of the track.
- **Medium.** The loss is proportional to $n \propto \rho Z/A$. Since $Z/A \approx
  1/2$ for most nuclei (and $\approx 1$ for hydrogen), the **mass stopping power**
  $-\frac{1}{\rho}\frac{\d E}{\d x}$ is nearly independent of the material, which is
  why stopping powers are tabulated per unit mass thickness $\rho\,\d x$.
- **Logarithm.** The bracket grows slowly with energy through $\ln(\beta^2\gamma^2)$,
  the relativistic rise, partly cancelled at high energy by the density-effect
  correction $\delta$ that accounts for the polarization of the medium screening
  distant collisions.

The competition between the falling $1/\beta^2$ prefactor and the rising logarithm
produces a broad minimum. A particle at the minimum is **minimum-ionizing**,
losing about $2\ \mathrm{MeV\,cm^2/g}$ in light materials, near $\beta\gamma \approx 3$.

$$
% caption: Mass stopping power falls as the inverse square of speed, reaches a
% broad minimum near beta-gamma of about three, then rises logarithmically; the
% minimum-ionizing value is close to two MeV per gram per square centimeter.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (8.2,0) node[right, black!70] {speed (log)};
  \draw[->, black] (0,0) -- (0,4.6) node[above, black!70] {mass stopping power};
  % steep 1/v^2 fall, broad minimum, slow relativistic rise
  \draw[acc, very thick]
    (0.5,4.3)
    .. controls (1.2,3.1) and (1.9,2.0) .. (2.8,1.35)
    .. controls (3.6,0.95) and (4.2,0.85) .. (4.9,0.9)
    .. controls (5.9,0.98) and (6.9,1.15) .. (7.8,1.35);
  % minimum marker
  \fill[acc] (4.2,0.85) circle (1.7pt);
  \draw[black, dashed] (4.2,0) -- (4.2,0.85);
  \node[anchor=north, black, font=\scriptsize] at (4.2,-0.05) {minimum ionizing};
  \node[black, anchor=west, font=\scriptsize] at (0.7,3.7) {steep fall};
  \node[black, anchor=south, font=\scriptsize] at (6.6,1.15) {relativistic rise};
\end{tikzpicture}
$$

## The Bragg curve and range

Because $-\d E/\d x$ rises as the particle slows, the energy deposited per unit
length is small where the particle is fast and reaches a sharp maximum just before
it stops. A plot of ionization density against depth is the **Bragg curve**, and
its peak is the **Bragg peak**. Beyond the peak the particle has so little energy
that the loss falls abruptly to zero, so the particle stops within a narrow depth
interval.

$$
% caption: The Bragg curve of ionization versus depth: a slow rise while the
% particle is fast, a sharp peak as the inverse-square factor takes over near the
% end of the track, and an abrupt cutoff at the range.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (7.8,0) node[right, black!70] {depth $x$};
  \draw[->, black] (0,0) -- (0,4.4) node[above, black!70] {ionization per length};
  \draw[acc, very thick]
    (0.2,1.05)
    .. controls (2.0,1.2) and (3.7,1.55) .. (4.7,2.05)
    .. controls (5.4,2.5) and (5.75,3.9) .. (5.95,3.9)
    .. controls (6.12,3.9) and (6.28,0.5) .. (6.5,0.12)
    -- (6.85,0.06);
  \draw[black, dashed] (6.45,0) -- (6.45,1.0);
  \node[anchor=north, black!70] at (6.45,0) {range $R$};
  \node[acc, anchor=south east, font=\scriptsize] at (6.05,3.9) {Bragg peak};
  \node[black, anchor=west, font=\scriptsize] at (1.4,1.2) {plateau};
\end{tikzpicture}
$$

The range is the path length over which the particle loses all its energy. In the
**continuous-slowing-down approximation** (CSDA), which ignores the small
statistical fluctuations, it is the integral of the reciprocal stopping power,

$$
R(E_0) = \int_0^{E_0}\left(-\frac{\d E}{\d x}\right)^{-1}\d E.
$$

Two particles of the same charge in the same medium have ranges related by a
simple scaling. Since the stopping power depends on speed, not energy, and
$E = \tfrac12 M v^2$ nonrelativistically, the range of a particle of mass $M$ and
charge $z$ at speed $v$ obeys $R \propto M/z^2$ at fixed $v$. A range-energy
relation measured for protons therefore predicts the range of any other heavy ion:

$$
R_2(E) = \frac{M_2\, z_1^2}{M_1\, z_2^2}\, R_1\!\left(\frac{M_1}{M_2}E\right).
$$

An alpha and a proton of the same energy per nucleon have nearly equal speed;
the alpha, with $z^2 = 4$ and $M = 4$, has $R_\alpha = (4/1)(1/4)\,R_p = R_p$ at the
same speed but, at the same total energy, a much shorter range because it is slower.

$$
% caption: Range grows steeply with energy and, at fixed energy, is far shorter
% for alphas than protons because the alpha carries twice the charge and moves
% more slowly; both curves steepen at high energy.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (7.8,0) node[right, black!70] {energy $E$};
  \draw[->, black] (0,0) -- (0,4.4) node[above, black!70] {range $R$};
  % proton range (longer)
  \draw[acc, very thick]
    (0.3,0.2) .. controls (2.5,0.8) and (4.5,2.2) .. (7.2,4.1);
  \node[acc, anchor=south east, font=\scriptsize] at (7.1,3.9) {protons};
  % alpha range (shorter, steeper start)
  \draw[black, very thick, densely dashed]
    (0.3,0.1) .. controls (3.0,0.4) and (5.5,1.1) .. (7.4,2.1);
  \node[black, anchor=north west, font=\scriptsize] at (6.6,1.55) {alphas};
\end{tikzpicture}
$$

## Range straggling

The CSDA range is an average. The actual number of collisions and the energy lost
in each fluctuate, so identical particles stop at slightly different depths. The
distribution of stopping points is nearly Gaussian, and its width is the **range
straggling**. For heavy particles the fractional straggling $\sigma_R/R$ is small,
a few tenths of a percent to about one percent, because thousands of collisions
average out; the sharp Bragg peak survives. Electrons straggle far more, since a
single collision can remove a large fraction of a light particle's energy and can
also scatter it through a large angle, blurring the concept of a definite range.

$$
% caption: Identical heavy particles stop within a narrow band about the mean
% range; the number stopped per depth interval is a narrow Gaussian whose width
% is the straggling, a small fraction of the range.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (7.8,0) node[right, black!70] {depth $x$};
  \draw[->, black] (0,0) -- (0,4.2) node[above, black!70] {fraction stopped};
  % integral transmission falling to zero at mean range
  \draw[acc, very thick]
    (0.3,3.7) -- (4.4,3.7)
    .. controls (4.9,3.7) and (5.0,0.2) .. (5.6,0.15)
    -- (7.2,0.12);
  \draw[black, dashed] (5.0,0) -- (5.0,3.7);
  \node[anchor=north, black!70] at (5.0,0) {mean range};
  % straggling width bracket
  \draw[black] (4.5,0.7) -- (5.5,0.7);
  \draw[black] (4.5,0.6) -- (4.5,0.8);
  \draw[black] (5.5,0.6) -- (5.5,0.8);
  \node[black, anchor=south, font=\scriptsize] at (5.0,0.75) {straggling};
\end{tikzpicture}
$$

## Electrons: collisional and radiative loss

Electrons lose energy by the same ionizing collisions, but with two differences.
First, the projectile and target have equal mass, so a single collision can
transfer up to half the electron's kinetic energy, and the electron scatters
sharply and follows a tortuous path; the "range" becomes the mean penetration
depth rather than the path length. Second, an accelerating charge radiates, and
the deflection of a light electron in the strong field near a nucleus produces
**bremsstrahlung** (braking radiation). The total stopping power is the sum of a
collisional and a radiative term,

$$
-\frac{\d E}{\d x} = \left(-\frac{\d E}{\d x}\right)_\mathrm{coll}
+ \left(-\frac{\d E}{\d x}\right)_\mathrm{rad}.
$$

The collisional term follows a Bethe formula modified for the electron's
indistinguishability from the target electrons. The radiative term grows with
both energy and atomic number, approximately

$$
\frac{(-\d E/\d x)_\mathrm{rad}}{(-\d E/\d x)_\mathrm{coll}} \approx \frac{E\,Z}{700\ \mathrm{MeV}},
$$

with $E$ the electron kinetic energy in MeV. Radiative loss dominates above the
**critical energy** $E_c$, the energy at which the two terms are equal. A useful
parametrization for solids is $E_c \approx 610\ \mathrm{MeV}/(Z + 1.24)$; for lead
($Z = 82$) it is about $7\ \mathrm{MeV}$, for water about $78\ \mathrm{MeV}$. Below
$E_c$ an electron ionizes; above it, it mainly radiates, and the emitted photons
seed the electromagnetic cascades used in calorimetry.

$$
% caption: For electrons the collisional loss dominates at low energy and grows
% only logarithmically, while the radiative loss rises linearly with energy; they
% cross at the critical energy, above which bremsstrahlung takes over.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (8.0,0) node[right, black!70] {electron energy $E$};
  \draw[->, black] (0,0) -- (0,4.4) node[above, black!70] {energy loss per length};
  % collisional: slowly rising log
  \draw[acc, very thick]
    (0.4,1.2) .. controls (2.5,1.6) and (4.5,1.85) .. (7.6,2.1);
  \node[acc, anchor=south, font=\scriptsize] at (2.3,1.5) {collisional};
  % radiative: linear rise from origin
  \draw[black, very thick, densely dashed]
    (0.4,0.15) -- (7.6,3.9);
  \node[black, anchor=south east, font=\scriptsize] at (7.4,3.7) {radiative};
  % crossing = critical energy
  \fill[black!70] (5.0,2.6) circle (1.7pt);
  \draw[black, dashed] (5.0,0) -- (5.0,2.6);
  \node[anchor=north, black!70, font=\scriptsize] at (5.0,-0.05) {critical energy};
\end{tikzpicture}
$$

## Cherenkov radiation

A charged particle moving through a dielectric of refractive index $n$ faster than
the phase velocity of light in that medium, $v > c/n$, emits a coherent shock front
of visible light, **Cherenkov radiation**. The condition is a threshold on speed,

$$
\beta > \beta_\mathrm{th} = \frac{1}{n},
$$

and below it no Cherenkov light appears. Above threshold the wavefronts from the
particle's successive positions add constructively along a cone whose half-angle
satisfies

$$
\cos\theta_c = \frac{1}{\beta n},
$$

the optical analogue of a supersonic Mach cone. The number of photons radiated per
unit path length and per unit wavelength interval is the Frank-Tamm result,

$$
\frac{\d^2 N}{\d x\,\d\lambda} = \frac{2\pi\alpha}{\lambda^2}\left(1 - \frac{1}{\beta^2 n^2}\right),
$$

with $\alpha$ the fine-structure constant. The $1/\lambda^2$ weighting makes the
light bluish. Because the threshold and cone angle depend only on $\beta$,
measuring the presence and opening angle of the cone determines a fast particle's
speed, and combined with a momentum measurement it identifies the particle's mass.

$$
% caption: Above the speed threshold a particle outruns its own light and the
% spherical wavefronts pile up on a cone; the half-angle set by the cosine of one
% over beta times the index measures the particle speed.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % particle track
  \draw[->, black, very thick] (0.3,2.0) -- (7.4,2.0);
  \node[black!70, anchor=north, font=\scriptsize] at (7.0,1.95) {particle path};
  % expanding wavefronts (circles of growing radius)
  \foreach \cx/\rr in {1.2/0.35, 2.6/0.9, 4.0/1.45, 5.4/2.0} {
    \draw[black] (\cx,2.0) circle (\rr);
  }
  % cone envelope (Cherenkov wavefront) as two tangent lines
  \draw[acc, very thick] (6.6,2.0) -- (1.0,3.9);
  \draw[acc, very thick] (6.6,2.0) -- (1.0,0.1);
  % cone angle
  \node[acc, anchor=west, font=\scriptsize] at (5.2,2.6) {cone angle};
  \draw[black, ->] (6.3,2.0) arc (180:158:0.9);
\end{tikzpicture}
$$

> **Result.** For a heavy charged particle the stopping power is the Bethe-Bloch
> $z^2/\beta^2$ times a slowly varying logarithm, concentrating the deposited
> energy in a Bragg peak at the end of a sharp range. Electrons add a radiative
> channel that overtakes ionization above the critical energy, and any particle
> exceeding $c/n$ radiates Cherenkov light on a cone whose angle measures its
> speed. The same $-\d E/\d x$ that stops the particle is the signal every
> ionization detector collects, taken up
> [in the detector lesson](/nuclear-physics/radiation-matter-applications/radiation-detectors).

The remaining uncharged radiations, photons and neutrons, do not ionize
continuously; they travel until a single event removes them, and their attenuation
follows an exponential rather than a Bragg curve. That is the subject of the
[next lesson](/nuclear-physics/radiation-matter-applications/photon-neutron-interactions).

[^krane-interactions]: **Krane**, _Introductory Nuclear Physics_, §7.1 (Interactions
of Radiation with Matter), and **Wong**, _Introductory Nuclear Physics_, §4-1. The
Bohr impulse derivation, the Bethe-Bloch formula with the mean excitation energy
$I$, the Bragg curve and CSDA range, electron radiative loss and the critical
energy, and the Cherenkov threshold and cone angle. Tabulated stopping powers and
ranges (the PSTAR/ASTAR/ESTAR databases and mean excitation energies) are
maintained by NIST, [https://physics.nist.gov/](https://physics.nist.gov/).