4. Stopping

1. Introduction & Collision Kinematics ] 2. Cross Sections ] 3. Potentials & Collision Dynamics ] [ 4. Stopping ] 5. Ranges & Radiation Damage ] 6. Sputtering ]

Inelastic Collisions (incomplete)


An atomic projectile can loose energy in an elastic collision, as we have seen, by transferring it to a target atom.  In addition, there may be an inelastic energy transfer, where energy goes into internal degrees of freedom.  These can be electronic and, in molecules, vibrational or rotational.  Usually the most significant are the electronic energy transfers, and they are the most studied.  Inelastic collisions can be endothermic or endothermic depending on whether energy is lost or gained.  Only the first kind can occur in collisions between ground-state atomic particles. Electronic transfer, without energy transfer, occurs in collisions with no inelastic energy loss, called resonant.

          Electronic excitations can be single or multiple, and involve outer (valence) or inner electrons.  A particular form of electronic excitation is ionization, where the electron is removed from the atom. 



Curve Crossings of Potential Energy Curves

Adiabatic maximum rule

Electronic Stopping Power - high velocities (non-relativistic)

A collision producing energy loss between T and T+DT for a projectile of energy E has a cross section ds(E, T) = (ds/dT) dT.  For a thin layer of material containing N atoms per unit volume1,


S(E) is called the stopping cross section.  The quantity NS(E) = dE/dx  is called the stopping power.  This means power to stop, dE/dx  is a stopping force, since dE/dx = Fdx/dx  = F.


Bohr formula

Niels Bohr classical description considers that electronic excitations occur due to binary Coulomb (or Rutherford) scattering between the projectile's nucleus and each of the Z electrons of the target atom.  The energy transfer of a projectile of charge z to an electron of mass m, charge e, due to a collision with impact parameter b is:


Note that T does not depend on the projectile velocity but only on its velocity and on its charge. Since the charge goes squared, the energy is the same for negative or positive particles in this approximation.  Integrating T (b) presents two problems: the integral diverges for bmin and if bmax is allowed to go to infinity.  The first problem is resolved by artificially limiting b to the value it would give the maximum T(b) allowed by energy conservation. For a head on (b=0) collision with a projectile of mass >> m, the energy transfer is Tmax=2mv2.  Incorporating this value in T(b) one obtains


Thus, the classical Rutherford energy transfer would be limited to:



For distant collisions (large impact parameters) we need to take into account that the electron moves during the collision, adjusting adiabatically to the perturbation.  An estimate of this effect can be obtained by assuming that no excitations occur if the collision time, b/v is larger than the period of a classical orbit, 1/w.  This means that the adiabatic cutoff is


We can now obtain the stopping cross section.  For N target atoms per unit volume, each with Z electrons, the number of electrons at distance b, b+db in a layer of thickness dx of a material of density N is: dn=NZ.2pbdbdx. Thus, the stopping cross section is:


but, from the previous discussion:


Thus we arrive to the Bloch formula:


Bethe formula


Relativistic and


Shell corrections

Charge exchange

Electronic Stopping Power - low velocities

Energy Straggling

Energy losses have fluctuations.  Classically, this is due to a range of impact parameters in each collision, and a statistics of multiple collision.  Quantum mechanically, even a single collision to a particular scattering angle may have a range of inelastic energy losses.  All these effects contribute to energy straggling, the fluctuation in the energy loss.  If the collisions are independent and distributed according to Poisson statistics, the energy straggling is given by:



1.    An H2 molecule with an equilibrium internuclear distance of 1 dissociates into two protons in a collision with an inelastic energy transfer of 35 eV.  The two protons are detected along the projectile direction, meaning one was emitted forward and another backward in the center of frame.  Calculate the energy of each proton in the lab system taking into account the Coulomb repulsion between fragments.

2.    Assume that the Bohr radius separate close collisions from hard collisions.  Calculate and plot the ratio of stopping power due to close and distant collisions to the total stopping power.  Give the asymptotic value for high velocities.

3.    You want to test the assumption that there is a density effect (the stopping cross section S depends on N) when the energy loss in a volume given by dx and the radius of excitations (the adiabatic cutoff) equals 20 eV per target atom in that volume.  Use the Bohr-Bloch theory to calculate when this will happen as a function of z, Z, and v.

Copyright 2003: Ral A. Baragiola