• A relativistic charged particle interacts with the electrons and nuclei close to its trajectory via electromagnetic effects.  A reasonable approximation to the result can be obtained using simple electrostatics.

 
• The time the particle is close enough to a "target" electron or nucleus to affect it is sufficiently short that it is usually reasonable to use the impulse approximation.  This assumes that the target does not move significantly during the collision, so that the integrated effect parallel to the direction of motion of the "projectile" particle is zero, and momentum transfer is purely transverse.  It is easiest also to assume that a small enough amount of energy is transferred in each interaction that the target particle remains non-relativistic.

 
• The momentum transfer in an individual interaction then depends on the mass and charge of the target particle, the charge and velocity of the projectile particle and the impact parameter of the collision.

 
• Almost all energy is lost to atomic electrons, which are more numerous and much lighter than the nuclei, despite their lower charge.

 
• The overall energy loss of a particle passing through a slice of matter is then obtained by integrating over the distribution of impact parameters involved.  Conservation of energy, atomic and quantum effects limit the range of impact parameter over which the integral is physically meaningful.

 
• Further corrections are required for two forms of screening:
• The "shell correction" is a small adjustment due to the screening of inner atomic electrons by outer ones.  (It can often be ignored.)
• The "density effect" is caused by polarisation of the medium as a whole, and reduces the effective field of the particle at large impact parameters.  It is important in denser materials (hence the name) and for large values of the relativistic &gamma of the projectile.

where	is the mean energy loss in passing through thickness Δx,
	,
	me is the mass of the electron,
	Z and A are the atomic number and mass of the medium being traversed,
	z is the charge (in units of electronic charge) of the projectile particle,
	β and γ are the usual relativistic parameters,
	ρ is the density of the medium,
	I0 is its mean ionisation energy,
	ε is the shell correction  and
	δ is the density effect.

(For an alternative formulation and description, with different notation, see the BriefBook.)

The energy loss therefore only depends on the incoming particle's velocity, and not directly on its mass, as is shown in the figure below.

Energy loss in argon, as a function of particle mass and momentum;  the vertical scale gives the relative increase above the minimum of ionisation.
(With apologies for the poor quality of the figure!)

The energy loss therefore shows four regions:

1. a rapid decrease proportional to 1/β² at lower velocities (mostly off the left of this plot);

 
2. a minimum at E ~ 3 Mc² (i.e. γ ~ 3);

 
3. a slow logarithmic "relativistic rise", proportional to ln(γ)

 
4. a plateau as ionisation is limited by the density effect.
• for very low density materials, such as gases, the energy loss eventually levels out at 50% above minimum when γ reaches ~1000,
• for solids, the plateau is only 10% above minimum, and may be reached with γ as low as 10.