Stopping Power and the Range of Charged Particles

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.¹

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,

and the energy transferred to a single electron is

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

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 $b_{\max }\approx v/\bar{\omega }$ 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 $b_{\min }\approx z{e}^2/(4\pi {ϵ}_0{m}_e{v}^2)$. 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,

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.

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$-\frac{1}{\rho }\frac{\mathbf{d}E}{\mathbf{d}x}$ is nearly independent of the material, which is why stopping powers are tabulated per unit mass thickness $\rho \mathbf{d}x$.
• 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 $2\mathrm{MeV}\mathrm{c}{\mathrm{m}}^2/g$ in light materials, near $\beta \gamma \approx 3$.

The Bragg curve and range

Because $-\mathbf{d}E/\mathbf{d}x$ 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.

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,

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 $E=\frac{1}{2}M{v}^2$ 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:

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.

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.

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,

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

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.

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,

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

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,

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.

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.