[ 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.

## Adiabaticity

## 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*+D*T
*for a projectile of energy *E *has a cross section *d*__s__(*E, T*) =* *(*d*__s/__*d*__T__)
*d*__T. __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 *b*_{min}* *and
if *b*_{max} 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 *T*_{max}=2*mv*^{2}.
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*^{.}2p*bdbdx.
*Thus, the stopping cross section is:

but, from the previous discussion:

Thus we arrive to the Bloch formula:

### Bethe formula

### Corrections

Relativistic and

Polarization

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:

## Homework

1.
An H_{2} molecule with an equilibrium internuclear distance of 1
Å 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*.