\chapter{Chapter 1 Introduction}
%\section{Background}
%\section{Plasma Environment}
%\section{Materials}
%\section{Overview of Effects Produced}
%\section{Units}

\chapter{Chapter 2 Collision Physics}
\label{intro} % Always give a unique label

\section{Introduction}
\cite{monograph}.

In order to describe the phenomena discussed in Chapter 1, an
understanding of the transfer of energy between colliding atomic particles
is needed. Such collisions are described by classical or semi-classical
methods with the quantum mechanical effects, such as interference and
diffraction, incorporated as corrections. These methods are employed when
the wavelength associated with the collision is small compared to the
dimensions of the system. This is the same basis for using geometric optics
to approximate the passage of light through a medium. In the problem of
interest, the incident ``radiation'' is a beam of particles and the medium
is the field of a ``target'' atom or molecule. The wavelength of interest is
(h/p), where h is Planck's constant and p is the momentum of the particles.
Comparing this wavelength to an atomic radius, e.g., the Bohr radius, a$_{0}$ 
, a rough criterion for the usefulness of classical or semi-classical
methods can be established: collision energies much greater than a Rydberg
(27.2 eV) for incident electrons and much greater than hundredths of an eV
for incident ions. For ions, atoms, or molecules this criterion is satisfied
for the full energy range of interest in Chapter 1. However, scattering at
small angles (diffraction) is \textit{always} dominated by wave-mechanical
effects, as are regions of interaction in which electronic transitions take
place. 

Before proceeding, certain cross sections will be defined. The total
collision cross section is obtained from an experiment in which a beam of
particles, A, is incident on a target containing atoms, B, and the change in
intensity of the beam is monitored (Fig. 2.1). If the target is ``thin'',
that is, an incident particle is likely to make only a single collision,
then the change in intensity, $\mathrm{\Delta\Phi}$, for a small change in
thickness, $\mathrm{\Delta}$x, is 

\begin{equation}
\mathrm{\Delta\Phi = -\sigma n_B\Phi\Delta x,}
\end{equation}
where n$_{\mathbf{B}}$ is the density of target atoms, B, and $\mathrm{\Phi}$
is the measured intensity [particles/(cm$^{2}$ s)]. In Eq. (2.1) $\sigma $
is the cross section, with units of area, indicating the range of the
interaction between the colliding particles. Such measurements show that the
total cross section varies slowly with velocity, as shown in Fig. 2.2. That
is, atoms have diffuse boundaries and the effective range of the interaction
region changes with velocity. This dependence is quantum mechanical in
nature, as the effective interaction range is determined by scattering at
small angles. Integrating Eq. (2.1), the intensity of unscattered particles
vs thickness, x, for thick samples is 

\begin{equation}
\mathrm{\Phi = \Phi_0 e^{-n_B\sigma x}.} 
\end{equation}
The quantity d$\mathrm{\Phi/\Phi_0 = (n_{B}\sigma) e^{-n_B\sigma x}}$ dx
is the Poisson probability of the first collision occurring between x and 
x + dx, and $\mathrm{(n_B\sigma)^{-1}}$ is the collisional mean free path. \\

Experiments in which the scattered particles are detected (Fig. 2.3) give
more detailed information on atomic interactions. For a beam of atoms A
incident on a target containing atoms B, the angular differential cross
section, $\mathrm{(d\sigma /d\Omega)_A}$, relates the number of
incident particles per unit time scattered into a region of solid angle 
$\mathrm{d\Omega}$ to the incident flux. Similarly, 
$\mathrm{(d\sigma/d\Omega)_B}$ applies to the target particles ejected 
into a region of solid angle. Collecting all particles A (or B) scattered 
into all solid angles is equivalent to integrating $\mathrm{(d\sigma/d\Omega)_A}$ 
over all angles and gives the total scattering cross section $\mathrm{\sigma}$. 
In such experiments, discrimination may also be made between internal changes 
that have occurred (e.g., changes in mass, charge, energy, etc.). Therefore, 
the cross sections determined from these experiments indicate both the nature 
of the interaction between A and B and the probability of a change (transition)
in either A or B. 

In this chapter the nature of the cross section is first described, then the
forces between atomic particles are discussed and are used to calculate
cross sections and transition probabilities. Such calculations are divided
into those methods useful for describing transitions in either fast or slow
collisions. For incident ions and atoms collisions are fast or slow
depending on whether the ratio of the collision time and the characterstic
period of the target, $\mathrm{\tau_c/\tau}$, is less than or greater than one. 
The collision time can be written $\mathrm{\tau_c \sim d/v,}$ where v is the 
relative speed and d is a characteristic dimension associated with A and B, 
e.g., d $\approx$ a$_0$ for outer shell electrons. When describing ionization 
of outer shell electrons, the characteristic period is $\sim 10^{-16}$ s, 
whereas it is $\mathrm{\tau} \sim 10^{-14}$s for molecular vibrational motion 
and $\mathrm{\tau \sim  10^{-13}s}$ for rotational motion. Setting 
$\mathrm{\tau_c \sim \tau}$ these times translate into incident particle 
energies of the order 1 keV/amu, 0.1 eV/amu, and 0.001 eV/amu, where an amu 
is the atomic mass unit. It is useful to know that when the collision time 
is about the same size as the characteristic period, transitions are likely 
and the cross sections are ``large'' (i.e., of the order of the dimensions of 
the colliding particles). For much larger or much smaller collision times the 
cross sections decrease as described below. Many of the details of this 
discussion are given in the Appendix, as are a number of useful expressions. \\

\section{Impact Parameter Cross Sections}

The trajectory of an incident particle, A, interacting repulsively with an
initially stationary target particle, B, is shown in Fig. 2.3. The quantity
b in that figure indicates the closeness of approach and is referred to as
the impact parameter. It is the perpendicular distance between the incident
velocity vector, $\mathrm{\vec{v}_A = \vec{v}}$, and the location of particle 
B (i.e., b $\rightarrow$ R $\mathrm{sin\alpha}$ as z $\rightarrow \infty$). 
The impact parameter also indicates the angular momentum of the colliding 
particles (i.e., $\mathrm{\vert{L}\vert = \vert{M}_A \vec{R} \times \vec{v} 
{\vert} = M_{A} v b)}$ and, along with the relative velocity v (or energy), 
characterizes collisions between spherically symmetric particles. Therefore, 
the probability of a collision is written $\mathrm{P_c(b, v)}$. If the incident 
and/or target particle has a spin or is a molecule then initial orientations 
with respect to the collision axis have to be specified, e.g., $\mathrm{P_c(b, 
\phi, \Omega, v)}$, where $\mathrm{\phi}$ is the azimuthal angle of approach 
and $\mathrm{\Omega}$ is an orientation angle. If all orientations occur then 
$\mathrm{P_c(b, v)}$ is the averaged probability. 

Based on the description leading to Eq. (2.1), the interaction cross section
between A and B is

\begin{equation}
\mathrm{\sigma(v) = 2\pi \int\limits_{0}^{\infty } P_c(b, v)b db.} 
\end{equation}
That is, if A passes through the ring of area 2$\pi$b db about B (Fig. 2.3), 
it will be scattered from the beam with a probability P$_c$, contributing to 
the change in intensity of the beam, $\mathrm{\Delta I}$. In classical 
mechanics, $\mathrm{P_c}$ is zero or one. Therefore, for a collision between 
two spheres of radius $\mathrm{r_A}$ and $\mathrm{r_B}$, $\mathrm{P_c}$ = 0 
for b $\mathrm{\geq r_A + r_B}$, and $\mathrm{\sigma = \pi (r_A + r_B)^2}$. 
For interactions between atomic or molecular particles, the forces are infinite 
in range but the diffraction effect causes the cross section to be finite 
for most interactions (Appendix 2A), as in the measured values shown in Fig. 
2.2. That is, $\mathrm{P_c (b, v) \rightarrow 0}$ as b $\rightarrow \infty$. 

From Fig. 2.3 it is seen that, if the forces between the particles are
independent of their orientations, then those particles passing through the
ring of area 2nb db will be scattered into some angular region 
2$\mathrm{\pi sin \theta_A d\theta_A}$ with unit probability, $\mathrm{P_c}$ = 1. 
The ratio gives the angular differential cross sections,

\begin{equation}
\mathrm{
\left(\frac{d\sigma}{d\Omega _A}\right) \equiv \sigma (\theta_A) = 
\mid\frac{bdb}{\sin\theta_A d\theta_A}\mid,  \left(\frac{d\sigma}
{d\Omega_B}\right) \equiv \sigma(\theta_B) = \mid\frac{bdb}{\sin\theta_B
d\theta_B}\mid,}
\end{equation}
where the simplified notation for azimuthally symmetric cross section, 
$\mathrm{\sigma \theta_A}$ is often used instead of $\mathrm{(d\sigma/d\Omega_A)}$. 
Evaluation of these cross sections requires knowledge of the relationship 
between the angles and b, determined from the forces. 

For \textit{elastic} collisions (no change in internal energy) the energy
and momentum of each particle after the collision is related to the incident
energy and the scattering angle, which can be expressed either in the
laboratory or in the center of mass (CM) system (Goldstein 1950). In Table
2.1 the relationships between laboratory and CM quantities are summarized
for a moving particle A incident on a stationary particle B. The CM
deflection angle vs b, $\chi$(b) (called the deflection function), gives a
cross section 

\begin{equation}
\mathrm{\sigma(\chi) = \left|\frac{bdb}{\sin\chi d\chi}\right|.}
\end{equation}
The calculation of $\mathrm{\chi(b)}$, hence $\mathrm{\sigma(\chi)}$, is
described in the following section and the transformations to the laboratory
scattering cross sections discussed are given in Table 2.1. Because the
collisional (elastic) energy transfer to particle B, T, is simply related to
$\chi $ (Table 2.1), it is often useful to consider the energy transfer
cross section instead of $\mathrm{\sigma(\chi)}$,

\begin{equation}
\mathrm{\frac{d\sigma}{dT} = \frac{4\pi}{\gamma E_A}\sigma(\chi),}
\end{equation}
where the mass factor, $\mathrm{\gamma}$, is defined in Table 2.1. 
Throughout the text we will use $\mathrm{d\sigma [\rightarrow (d\sigma /dT)dT 
\rightarrow \sigma(\chi) 2\pi sin \chi d\chi \rightarrow 2\pi b db]}$
to represent an element of the elastic cross section $\mathrm{\sigma(v)}$. 

In addition to a transfer of kinetic energy to B, there is also a
probability, $\mathrm{P_{0 \rightarrow f} (b, v)}$, that one or 
both particles will experience a change in internal energy (a transition) 
where the subscripts indicate the initial (0) and final (f) states. 
Following Eq. (2.3), for these \textit{inelastic} collisions

\begin{equation}
\mathrm{\sigma_{0\rightarrow f}(v) = 2\pi \int\limits_{0}^{\infty }
P_{0\rightarrow f} (b, v)bdb.}
\end{equation}
The quantity $\mathrm{P_{0\rightarrow f} \rightarrow 0}$, as discussed
earlier, for some $\mathrm{b_{\mathrm{max}}}$. for which the collision time 
($\mathrm{\tau_c \approx b_{max}/v)}$ is of the order of the period
of the electrons. (This is called the Bohr adiabatic cut-off, Appendix 2C.)
Since each inelastic process is associated with an internal energy change, 
Q$_{0\rightarrow f}$, the average inelastic energy lost in the
collision is

\begin{equation}
\mathrm{Q(b) = \sum\limits_f Q_{0\rightarrow f}P_{0\rightarrow f} (b,v).}
\end{equation}
In Eq. 2.8 the sum of probabilities over all processes, including no
internal energy change, is unity $\mathrm{[\sum P_{0\rightarrow f} 
= P_c (b,v) = 1]}$. The calculation of $\mathrm{P_{0\rightarrow f}}$ is
described shortly, and a useful estimate for Q(b) is given in Appendix 2C. 

The above expressions can be used to define the average energy loss per unit
path length for particle A traversing a material consisting of particles B,
an important quantity in Chapter 3. This incident particle may be a cosmic
ray particle traversing an interstellar cloud or an ion in the Jovian
magnetosphere penetrating the icy surface of Europa. In the passage through
a target gas or a solid, A loses energy both to electronic excitations and
to momentum transfer to target nuclei B. Noting that $\mathrm{(n_B\sigma
_{0\rightarrow f})^{-1}}$ is the mean-free path for the occurrence
of an energy loss $\mathrm{Q_{0\rightarrow f}}$, the averaged electronic
energy loss per unit path length in the target is 

\begin{equation}
\mathrm{\left(\frac{dE}{dx}\right)_e = n_B \sum\limits_f Q_{0\rightarrow f}
\sigma_{0\rightarrow f} = n_B 2\pi \int\limits_{0}^{\infty} Q(b)b db.}
\end{equation}
Similarly, if $\mathrm{(n_B (d\sigma/dT)dT)^{-1}}$ is the mean-free
path for transferring an amount of kinetic energy between T and T + dT to a
nucleus B, then the averaged ``elastic'' collisional energy loss per unit
path length is

\begin{equation}
\mathrm{\left(\frac{dE}{dx}\right)_n = n_B \int\limits_{0}^{\infty} T 
\left(\frac{d\sigma}{dT}\right)dT = n_B 2\pi \int\limits_{0}^{\infty}T(b)bdb.} 
\end{equation}
Although the electronic and collisional energy losses occur
concomitantly, they are generally treated separately and additively. Because
they both depend on the material number density, the net energy loss is
generally written

\begin{equation}
\mathrm{\left(\frac{dE}{dx}\right) \approx n_B (S_e + S_n).} 
\end{equation}
The quantity (dE/dx) is referred to as the stopping power of the material,
and the quantities $\mathrm{S_e}$ and $\mathrm{S_n}$ are the electronic
and nuclear elastic (collisional) stopping cross sections. These are
quantities determined by the single collision interaction of A with B and
not properties of the target as a whole. For this reason they apply roughly
to gases, liquids, and solids. Although they are generally measured for ions
incident on neutrals, they can be applied to neutrals injected into a
plasma. Results for S$_{\mathrm{e}}$ are tabulated (see Bibliography), and
simple, generally useful expressions for S$_{\mathrm{n}}$ are available
(Appendix 2B). In Fig. 2.4 we give such a result for hydrogen and oxygen
ions incident on a medium containing H, O, or Si. At high energies most of
the loss is to energy transfer to electrons, and at low energies it is to
energy transfer to the target nucleus. 

The nuclear stopping cross section, $\mathrm{S_n}$, is also related to the
diffusion cross section or momentum transfer cross section, $\mathrm{\sigma_d}$ 
(Banks and Kocharts 1973; Johnson 1982). The change in the parallel component 
of momentum of a particle A due to a collision with B involving no electronic 
excitations is written 
$\mathrm{(\Delta \vec{p}_A)_{//} = mv(1 - cos\chi)}$, where m is the reduced 
mass in Table 2.1. Therefore, the drag force on a particle A traversing a 
target containing particles B is the collision rate times this momentum loss,

\begin{equation}
\mathrm{(\Delta\vec{F_A})_{//}) = \int(\Delta\vec{p_A})_{//} vn_B(2\pi bdb),}
\end{equation}
which we write as $\mathrm{(\Delta \mathrm{\vec{F}}_A)_{//} = n_B mv^2 \sigma_d}$, 
where

\begin{equation}
\mathrm{\sigma_d = 2\pi\int\limits_{0}^{\infty}[1-cos\chi(b)]bdb = 
2\pi\int\limits_{-1}^{1}(1-cos\chi)\sigma(\chi)dcos\chi.}
\end{equation}
Using the expression for T from Table 2.1, $\mathrm{S_n}$ and $\mathrm{\sigma_d}$ 
are related by

\begin{equation}
\mathrm{S_n = \frac{\gamma E_A}{2}\sigma_d.}
\end{equation}
That is, the drag force is related to the stopping power, i.e., 
$\mathrm{(\Delta F_A)_{//} = (1 + }$ \ \ \ \ \ \ %
$\mathrm{M_B/M_A)(dE/dx)_n}$. The electronic 
component adds to this. The elastic component of the force is also the force a 
streaming plasma applies to neutrals, which is of interest, for example, where 
the Jovian plasma torus intercepts the corona of Io. Because the factor (1 - 
cos$\mathrm{\chi}$) goes to zero as $\mathrm{\chi}$ goes to zero, both 
$\mathrm{S_n}$ and $\mathrm{\sigma_d}$, unlike $\mathrm{\sigma}$, can be 
calculated classically (Appendix 2A). In order to give a feeling for the 
relationship between the quantities we have been discussing, the deflection 
function, energy transfer cross section, the $\mathrm{S_n(\sigma_d)}$ are given 
in Fig. 2.5 for a repulsive force which approximates the S$^+$ + O collision. 

The cross sections have been defined without reference to the forces between
the particles. These forces produce both the deflections and the transitions
(inelastic processes). However, it has become customary to separate the
discussion of these two effects. The deflections (elastic energy loss) are
considered first and then, after a description of the forces, the inelastic
effects are considered. 

\section{Elastic Collisions}

In the CM system, the two colliding particles follow equivalent 
trajectories and the collision can be described as the scattering of a 
particle of reduced mass, m, by a stationary center of force (B stationery 
and $\mathrm{\theta_A \rightarrow \chi}$ in Fig. 2.3). Therefore, the 
angular momentum, which is conserved, is

%---------(2.15a)(2.15b)
\begin{subequations}
\begin{align}
& \mathrm{L = mvb = mR^{2}\dot{\alpha_R}.} \notag\\
\end{align}
Rearranging Eq. (2.15a) and integrating over time gives the CM 
scattering angle  
\begin{align}
& \mathrm{\chi(b) = \pi - \int\limits_{-\infty }^{\infty} \dot{\alpha}_R dt  
                  = \pi - vb \int\limits_{-\infty}^{\infty}\frac{dt}{R^2}.}
\end{align}
\end{subequations}
Assuming the interaction potential V(R) depends only on the separation R,
energy conservation is expressed as

%---------(2.16a)(2.16b)
\begin{subequations}
\begin{align}
& \mathrm{E = \frac{m\dot{R}^2}{2} + \frac{L^2}{2mR^2} + V(R).} \notag\\
\end{align}
Rearranging this gives a radial velocity, 
\begin{align} 
& \mathrm{\dot{R} = \pm v [1 - b^2 / R^2 - V/E]^{1/2},}
\end{align}
\end{subequations}
which goes to zero at the distance of closest approach R$_0$. Writing
$\mathrm{|dt| = |dR/\dot{R}|}$, Eq. (2.15b) becomes

\begin{equation}
\mathrm{\chi(b) = \pi - 2b \int\limits_{R0}^{\infty}\frac{dR}{R^2}
\left[1- b^2/R^2 - V/E \right]^{-1/2},} 
\end{equation}
This expression can be integrated analytically for a few potentials of the
form V(R) = $\mathrm{A/R^n}$ (Goldstein 1950) but otherwise is treated by
the simple numerical procedure, given in Appendix 2B, used to obtain the
results in Fig. 2.5b. \\

For fast incident ions [ratio V/E small in Eq. (2.17)], the deflections are
generally small, and it is useful to replace the expression for 
$\mathrm{\chi}$(b) by an impulse approximation. That is,

\begin{equation}
\mathrm{\chi(b) \approx \frac{\int\limits_{-\infty}^{\infty} F_\bot dt}{mv} = 
-\frac{d}{db}\left[\frac{1}{2E}\int\limits_{-\infty}^{\infty} VdZ\right],}
\end{equation}
in which a straight line trajectory is assumed, i.e., R$^2$ = b$^2$ + Z$^2$, 
with Z = vt and R$_0$ = b. For power law potentials (V = $\mathrm{C_n/R^n)}$ 

\begin{equation}
\mathrm{\chi(b) \approx a_n V(R_0)/E,}
\end{equation}
where $\mathrm{a_n}$ is given in Appendix 2B. This shows explicitly that
the deflection is determined primarily by the nature of the potential
\textit{near the distance of closest approach} of the colliding particles.
Therefore, it is not necessary to know the potential accurately at all R in
order to obtain an estimate of the deflection function as long as V/E $\ll $
1. Since V is of the order of electron volts, this criterion is satisfied
for a large number of astrophysical problems. 

From Eq. (2.19) the quantity $\mathrm{\chi}$E is seen to depend only on R$_0$.
Using Eq. (2.19) in Eq. (2.5) the modified cross section x sin $\mathrm{\chi \sin
\chi \sigma (\chi)}$, given in terms of $\chi$E, is \textit{independent
of the energy} E for small angles. [In the lab frame use $\mathrm{\theta_A}$
$\mathrm{sin\theta_A \sigma(\theta_A)}$ in terms of 
$\mathrm{\theta_A E_A}$.] Therefore experimental measurements of angular 
differential cross sections can be directly converted into the useful power-law 
potentials which are applicable over limited ranges of R. It is seen in Fig. 2.6 
that n (Eq. 2.19) goes to unity as $\mathrm{\chi}$E gets large (i.e., 
$\mathrm{R_0 \rightarrow  0}$) as it should for the repulsive interaction 
between atoms. Also, as $\chi$E decreases (i.e., R$_0$  increases) n increases; 
that is, the potential becomes steeper. \\

In Fig. 2.7 are schematic drawings of the deflection functions for a
predominantly repulsive potential and a long-range attractive plus
short-range repulsive potential. Based on Eq. (2.19) these follow the form
of the potential. For the repulsive potential $\chi $ is always positive,
attaining a maximum value of $\mathrm{\pi}$ for head-on collisions [b = 0, 
Eq. (2.17)]. For the potential with a minimum $\chi $ is negative at large b 
but again reaches $\mathrm{\pi}$ at b = 0. Because the detector collects both 
negative and positive angles, it is seen in Fig. 2.7 that more than one value 
of the impact parameter contributes to scattering at a number of angles for 
such potentials. Since $\chi $ also goes through a minimum, $\mathrm{\chi_r}$,
at the impact parameter labeled $\mathrm{b_r, d\chi /db}$ becomes zero,
and the classical cross section in Eq. (2.5) becomes infinite. This large 
\textit{enhancement} in the scattering probability is similar to the effect
that produces rainbows in the scattering of light from water droplets;
hence, $\mathrm{\chi_r}$ is called the rainbow angle. When $\mathrm{\chi_r < pi}$ 
only one impact parameter contributes for $\mathrm{\chi > \chi_r}$. For angles 
$\mathrm{\chi < \chi_r}$ three impact parameters contribute with the same value 
of cos$\chi$ in Eq. (2.5). Using a Lenard-Jones form for the potential, 
V = $\mathrm{C_{2n}/R^{2n} - C_n /R^n}$, the rainbow angle is obtained from 
Eq. (2.19),

\begin{equation}
\mathrm{\chi_r \approx \frac{-1}{4E} \frac{a_n C_n)^2}{a_{2n} C_{2n}} = - 
\frac{a_n^2}{a_{2n}}\frac{\mid V_{min}\mid}{E}.}
\end{equation}
Therefore, knowledge of the depth of the potential, $\mathrm{{\vert}V_{min}
\vert}$, gives the angle at which the cross section is strongly enhanced. 
For large E this angle is often negligibly small. 

As the collision energy becomes small, $\mathrm{\chi_r}$ can become much
larger than $\mathrm{\pi}$, so that the two particles may orbit each other 
before separating. In fact, for very small velocities there is a particular 
impact parameter for which the particles orbit continuously (i.e., 
$\mathrm{d^2 R/dt^2 = 0 at dR/dt = 0}$). Using the attractive part of 
potential above along with Eq. (2.16b), this impact parameter is

\begin{equation}
\mathrm{b_0 \approx \left(\frac{n}{2}\frac{C_n}{E}\right)^{1/n} \bigg/
\left(\frac{n-2}{n}\right)^{(n-2)/2n}}
\end{equation}
Therefore, for b $<$ b$_0$, \textit{the interaction times can 
be long} because the particles orbit, whereas for b $>$ b$_0$, 
the particles simply scatter. Because the long interaction times enhance the 
likelihood of low probability processes, b$_{0}$ is very useful for estimating 
inelastic ion-molecule cross sections for astrophysics (McDaniel et al. 1970). 

Near the rainbow angle (or when orbiting occurs) a number of impact
parameters contribute to the scattered flux at a given observation angle;
hence, interference phenomena occur and classical cross section estimates of
$\mathrm{\sigma(\chi)}$ are not valid. If the angular resolution is not large,
the simple addition of the contributions to the scattered flux from each
impact parameter gives an adequate representation of the cross section. That
is,

\begin{equation}
\mathrm{\sigma(\chi) \approx \sum\limits_{i}\left|\frac{bdb}{\sin\chi d\chi}
\right|_{b=b_i}},
\end{equation}
where b$_{\mathrm{i}}$ are all those impact parameters giving the same value 
of cos$ \chi $ [i.e., $\mathrm{\chi \rightarrow \pm (\chi + 2\pi q)}$, q an
integer]. In Appendix 2A, we briefly review the wave-mechanical description
of the elastic scattering cross section. The results have parallels in light
scattering, a subject familiar to the astronomy community. In the following
section we describe the nature of the interaction potential V(R).

\section{Interaction Potentials}

The primary force determining the behavior of colliding atoms or molecules
is the Coulomb interaction. This acts between each of the constituent
electrons and nuclei of the colliding particles and, therefore, the
description of the interaction between such particles would appear to be
straightforward. However, the Pauli principle must be applied, and each of
the constituents moves relative to the center of mass of its parent atom or
molecule. As this motion is superimposed on the overall collisional motion,
the description of a collision can be complex even for the simplest atoms.
Rather than solve the complete many-body system, the interaction potential
between the atomic centers is averaged over the motion of the electrons.
This separation of the motion of the electrons and nuclei, based on their
huge differences in mass, is referred to as the Born-Oppenheimer separation
(Torrens 1972; Johnson 1982). 

Since the behavior of the electrons during a collision will depend on the
relative motion of the nuclei, interaction potentials are generally
calculated in two limiting cases (Bates 1962; Massey 1979). If the collision
of two atoms is fast relative to the motion of the electrons (v $\gg $ v$_{
\mathrm{e}}$), then the electronic distribution is static during the
collision, except for abrupt changes or transitions, which occur when the 
particles are at their closest approach. These transitions reflect the 
ability of an atom to absorb (emit) energy when exposed to the time-varying 
Geld of a moving particle (Appendix 2C) in the same way that atoms absorb or 
emit photons. Before and after the transition, the potentials are determined 
from the separate atomic charge distributions. 

In the opposite extreme (slow collisions, v$\ll $v$_{\mathrm{e}}$) the
electrons adjust continuously and smoothly to the nuclear motion, returning
to their initial state at the end of the collision. This is an adiabatic
process, as the electrons do not gain or lose energy. That is, even though
the molecules may be deflected and change kinetic energy, their initial
\textit{electronic state} does change, i.e., it is an ''elastic'' collision.
During the collision the electron distribution evolves from one in which
electrons are attached to separate centers Into a distribution at small
separations in which the electrons are shared by the two centers, a covalent
distribution. Therefore, for every possible initial state for each atom
there is a corresponding adiabatic potential, resulting in a complex
potential diagram like that shown in Fig. 2.8. Such potentials also
determine the ability of the two particles to form a molecule. 

For collisions which are nearly adiabatic the nuclear motion can induce a
transition between states. This is often described as a ''transition between
potential curves,'' because these curves indicate the energy levels of the
system at each R. Such transitions generally occur at well-defined
internuclear separations at which the atomic character of the wave function
gives way to the molecular, covalent character (Olson 1980). 

Since the electrons in different shells have very different velocities, the
separation into fast and slow collisions allows the orbitals to be treated
separately. For instance, when a collision is fast with respect to the
outer-shell electrons, it may be adiabatic with respect to the inner-shell
electrons. Therefore, the inner-shell electrons return to their initial
state only screening the interaction between the nuclei and, hence, play a
passive role in the collision. As a point of reference, it is useful to
remember that a nucleus with a speed equivalent to an electron in the ground
state of a hydrogen atom (2.19 x 10$^{8}$ cm/s) has an energy of about 25
keV/amu. Therefore, a 25 keV He$^{+}$ or even a 100 keV O$^{+}$ colliding
with an H or O atom is considered a slow collision. In addition, the orbital
speed of an electron in the target atom scales to the speed of an electron
in a ground state of hydrogen using the effective nuclear charge, Z$\acute{}$
(Clementi and Roetti 1974), which accounts for screening of nucleus. 

Interaction potentials between two atoms, such as those in Fig. 2.8, can be
written as a sum of the nuclear repulsion and the averaged electronic
energy, $\varepsilon_{\mathrm{j}}$(R),

\begin{equation}
\mathrm{ V_j(r) = \frac{Z_A Z_B e^2}{R} + [\varepsilon_j(R) - \varepsilon_j^0].} 
\end{equation}
In this expression j labels the electronic state, Z$_{\mathrm{A}}$ and Z$_{
\mathrm{B}}$ are the nuclear charges, and $\varepsilon_{\mathrm{j}}^{0}$ is 
the value of the electronic energy, $\varepsilon _{\mathrm{j}}$(R), as R 
$\rightarrow \infty$. Each state, j, is associated with a pair of atomic 
states at large R as indicated in Fig. 2.8. In the electrostatic limit (v$\gg$
v$_{\mathrm{e}}$) the electronic energy, $\varepsilon _{\mathrm{j}}$, is a 
sum of the electronic energies of the separated atoms, ($\varepsilon_
{\mathrm{Aj}}^{0}$ + $\varepsilon _{\mathrm{Bj}}^{0}$) = $\varepsilon _
{\mathrm{j}}^{0}$, and the averaged interaction of the electrons on each atom 
with the electrons and nucleus of the other atom, which we write 
V$_{\mathrm{j}}^{\mathrm{e}}$. That is

$$\mathrm{V_j(R) \approx \frac{Z_A Z_B e^2}{R} + V_j^e}$$
with V$_{\mathrm{j}}^{\mathrm{e}}$ written as
$$
\mathrm{V_j^e(R) = - Z_B e^2 \int \frac{\rho_{Aj}(\vec{r}_A)}{|\vec{R}-\vec{r}_A|}
d^3 r_A - Z_A e^2 \int \frac{\rho_{Bj}(\vec{r}_B)}{|\vec{R}+\vec{r}_B|}
d^3 r_B} $$
\begin{equation}
\mathrm{ + e^2 \int \frac{\rho_{Aj}(\vec{r}_A)\rho_{Bj}(\vec{r}_B)}
{|\vec{R}-\vec{r}_A+\vec{r}_B|} d^3 r_A dr_B.}
\end{equation}
Here the $\rho_{\mathrm{Aj}}$ and $\rho_{\mathrm{Bj}}$ are the atomic
densities for the electrons on atoms A and B. The evaluation of 
V$_{\mathrm{j}}^{\mathrm{e}}$ is treated in texts on electrostatics 
(Jackson 1963). \\

In the adiabatic limit (v $\ll $v$_{\mathrm{e}}$), $\varepsilon _{\mathrm{j}
} $(R) in Eq. (2.23) is calculated at each R from the electronic wave
function or an approximation to it (Bates 1962). Because the deflection
function is determined primarily by a narrow region of R for each b (or $
\chi $), [viz: Eq. (2.19)], the behavior of V$_{\mathrm{j}}$(R) for various
regions of R is described bekow. 

\subsection{\textit{Short-Range Potentials}}

For close collisions (R $\ll$ r$_{\mathrm{A}}$, r$_{\mathrm{B}}$ where 
r$_{\mathrm{A}}$ and r$_{\mathrm{B}}$ are the atomic radii) nuclear 
repulsion dominates. The electrons screen this repulsive interaction, 
giving an effective potential

\begin{equation}
\mathrm{ V_j(R) \approx \frac{Z_A Z_B e^2}{R} + \Phi(R/\bar{a}),} 
\end{equation}
where \={a} is the screening length and $\mathrm{\Phi}$ the screening 
function. Considerable effort has been expended determining $\mathrm{\Phi}$
and \={a} for many-electron atoms and, therefore, experimental results
should be used when possible. For fast collisions, the electrostatic
calculation in Eq. (2.24) can give a reasonable estimate. Even simpler, 
$\mathrm{\Phi}$ is often approximated as exp(- R/\={a}) in scattering 
calculations (Everhart et al. 1955). Since the electrons in different 
shells have different screening lengths, a better approximation for a bare 
ion on a light atom is to assume the ith shell has a screening constant 
determined by the electron removal (ionization) energy I$_{\mathrm{i}}$, 
a$_{\mathrm{i}} \sim $ a$_{\mathrm{0}}$ (I$_{0}$/I$_{\mathrm{i}}$)$^{1/2}$, 
where I$_{0}$ and a$_{0}$ are the ground-state, hydrogen atom ionization 
potential and radius. Then, Z$_{\mathrm{B}}\mathrm{\Phi} \sim \sum$ 
N$_{\mathrm{i}}$ exp( - r/a$_{\mathrm{i}}$) where N$_{\mathrm{i}}$ is the 
number of electrons in the ith shell. 

For collisions between larger (Z $>$ 10) atoms, the screening is
often estimated by Thomas-Fermi (electron-gas) approximation in the Appendix
2B (Torrens 1972). Expressions for $\mathrm{\Phi}$, the energy transfer cross
section, d$\sigma $/dT, and the nuclear stopping cross section, S$_{\mathrm{n%
}}$, have been given by Lindhard et al. (1968) (Appendix 2B). Even though
such static potentials are intended for close collisions, universal
semi-empirical functions based on this have been obtained which roughly
describe elastic collisions in the repulsive regime even down to very low
energies. These are also given in Appendix 2B, and a useful screening
constant for organizing the many data sets (Ziegler et al. 1985)
is

\begin{equation}
\mathrm{\bar{a} = 0.885 a_0 /(Z_A^{0.23} + Z_B^{0.23})}.
\end{equation}

\subsection{\textit{Long-Range Interactions}}

For slow collisions even small disturbances can lead to deflections;
therefore the interaction potential at large separations (R $\gg $ r$_{%
\mathrm{A}}$, r$_{\mathrm{B}}$) is of interest. The electronic distributions
of each of the colliding particles are slightly distorted by the presence of
the other so that the long-range interaction potential is written in powers
of R,

\begin{equation}
\mathrm{ V_j(R) = \sum C_n / R^n.} 
\end{equation}

Generally, only the largest term (lowest power of n) is kept and, therefore
the power-law expression for cross section, Eq. (2.19), can be used. The
lead terms in Eq. (2.27) can be obtained from the electrostatic interaction
in Eq. (2.24) by expanding the denominators in powers of R. For ion-ion
collisions the lead term in Eq. (2.27) is n = 1, C$_{1}$ = Z$_{\mathrm{A} 
}^{\prime }$Z$_{\mathrm{B}}^{\prime }$e$^{2}$, which is the Coulomb
interaction with Z$_{\mathrm{A}}^{\prime }\rightarrow $ Z$_{\mathrm{A}}$ - N$ 
_{\mathrm{A}}$ (the net ionic charge). For ion collisions with a neutral
molecule having a dipole moment $\mu _{\mathrm{B}}$ (e.g., a water molecule)
the lead term is n = 2, C$_2$ = - Z$_{\mathrm{A}}\mu_{\mathrm{B}}$e cos$ 
\theta $, where cos$\theta$ is the angle between the internuclear axis and
the dipole moment. If the neutral molecule does not have a dipole moment
(e.g., O$_{2}$, N$_{2}$) the ion-quadrupole interaction dominates (n = 3).
For two colliding neutral molecules having dipole moments, the dipole-dipole
interaction (n = 3) dominates, and so on. 

For colliding atomic particles the electrostatic multipole-moments of the
charge distribution are zero and for some molecules (e.g., H$_{2}$) the
moments are small. How\'{e}ver, the field of the other particle distorts the
charge distribution inducing such moments. The lead-term for ion-neutrals is
the ion-induced dipole interaction (n = 4) C$_{4}$ = - $\alpha _{\mathrm{B}}$%
(Z$_{\mathrm{A}}$e)$^{2}$/2, where $\alpha _{\mathrm{B}}$ is the
polarizability of the neutral particle B. For collisions between two
neutrals there is no average field at B due to A or vice versa, but
instantaneous fluctuations in charge density on either atom lead to
short-lived fields. In this case the lead term is the
(induced-dipole)-(induced-dipole) interaction, the well-known van der Waals
interaction used to describe the behavior of realistic gases. This
interaction (n = 6) is attractive [i.e., C$_{6}$ negative in Eq. (2.27)],
and coefficients have been obtained from experiment. In Table 2.2 we give
values for the dipole moments and the polarizabilities for a number of atoms
and molecules in their ground states. These coefficients depend on the
initial state and, therefore, differ considerably for collisions with an
excited atom or molecule. 

\subsection{\textit{Intermediate-Range Potentials and Charge Exchange}}

When R is the order of the atomic radii (r$_{\mathrm{A}}$, r$_{\mathrm{B}}$%
), the electron clouds on the two interacting particles overlap. In this
region the distortion of the charge clouds on each center becomes too large
to treat as simply a perturbative polarization of the separated electronic
distributions. 

The overlap of charge on the two centers allows for the possibility of
charge-exchange collisions (e.g., H$^{+}$ + O$_{2}$ $\rightarrow $ H + O$%
_{2}^{+}$). For such collisions the potentials before and after can be
drastically different, as seen in Fig. 2.9 for S$^{2+}$ + O $\rightarrow $ S$%
^{+}$ + O$^{+}$, a process of considerable interest in the Io plasma torus.
Depending on the final states of O$^{+}$ and S$^{+}$, the initial and final
state potential energy curves may cross. Therefore, electron transfer
between S$^{2+}$ and O can occur, with little change in the total electronic
energy, at the crossing points. This transfer gradually results in a net
change in the final electronic energy when the particles separate from each
other. The likelihood of charge exchange occurring at the crossing point is
proportional to the size of the overlap of the charge distributions of the
electron on each center (Rapp and Francis 1962). 

For intermediate R the covalent (electron exchange) interaction can be
examined by considering the H$^{+}$ + H (i.e., H$_{2}^{+}$) system. For this
colliding pair the electron transfer H$^{+}$ + H $\rightarrow $ H + H$^{+}$
occurs \textit{without} a net change in internal energy. The two identical
states at large R (H + H$^{+}$ and H$^{+}$ + H) mix at smaller R because of
electron sharing, forming both attractive and repulsive potentials. These
are seen in Fig. 2.10 and can be understood by considering the wave function
at large R. If the wave functions placing the electron on centers A and B in
identical states of the hydrogen atom are $\phi _{\mathrm{A}}$ and $\phi _{
\mathrm{B}}$, then the linear combinations appropriate to the H$_{2}^{+}$
molecule are $\psi_{\mathrm{g,u}}$ $\approx$($\phi _{\mathrm{A}}\pm \phi_{
\mathrm{B}}$)/$\sqrt{2}$. The labels g and u refer to symmetric and anti
symmetric (gerade and ungerade) states of H$_{2}^{+}$ (Herzberg 1950). The
forms of these wave functions are such that the electronic binding energy
decreases in the u state, as the electron is excluded from the region R/2,
while the reverse is true for the g state. Therefore, in the scattering of
protons by hydrogen, half the collisions will be along the repulsive
potential (u) and half along the attractive potential (g). The difference in
energy between these states is the exchange energy which determines the
behavior of charge-exchange collisions. The exchange energy decreases with
increasing R like the overlap of wave functions for the electron on A and on
B, which decays like exp(- R/a$_0$) for R $>$ a$_0$. 
Because the covalent interaction decays exponentially, it is eventually
dominated at very large R by the long-range, power-law potentials discussed
earlier. 

Molecular orbital potentials for H$_{2}^{+}$ can also be constructed for
each excited state of H as well as for one electron in the field of two
identical nuclei \textit{and} other electrons. In Fig. 2.11 we give a
diagram showing the general behavior of these one electron, molecular
orbital binding energies, $\varepsilon _{\mathrm{j}}$(R) (Herzberg 1950).
This quantity does not include the nuclear repulsive term in Eq. (2.23), so
we can follow its behavior down to R = 0. The diagram in Fig. 2.11
correlates the one electron states of the separated centers (R$\rightarrow $
$\infty $) with those of the united atom (R $\rightarrow $ 0) indicating how
the electronic binding energy changes with R. The states are labeled by
their symmetry (g or u) under inversion and by the component of electronic
angular momentum along the internuclear axis ($\mid \mathrm{M}_1\mid$ = 
0, 1, 2, $\dots$, called $\sigma$, $\pi$, $\delta,\dots$). 

To obtain total interaction potentials for a particular pair of atoms, the
net binding energy of each electron, determined from the correlation diagram
and the Pauli principle, is combined with the nuclear repulsive potential in
Eq. (2.23). One finds that two hydrogen atoms (H$_2$) interact via a
singlet state ($\sigma_{\mathrm{g}}$ls)$^2$ $^1\mathrm{\Sigma_g}^{+}$
, which is attractive at intermediate R, and a triplet state 
($\sigma_{\mathrm{g}}$ls) ($\sigma_{\mathrm{u}}$1s)$3\mathrm{\Sigma}_
{\mathrm{g}}^{+}$, which is repulsive. Therefore, in describing collisions 
between two H atoms, 3/4 are repulsive and 1/4 attractive. Two helium atoms 
(He$_2$) interact via a ($\sigma_{\mathrm{g}}$ls)$^2$ ($\sigma_{\mathrm{u}}
$ls)$^{2}$ $^{1}\mathrm{\Sigma_g^+}$ ground repulsive state. The lowest state 
for two oxygen atoms is 
($\sigma_{\mathrm{g}}$ls)$^2$ ($\sigma_{\mathrm{u}}$1s)$^2$ 
($\sigma_{\mathrm{g}}$2p)$^2$ ($\pi_{\mathrm{u}}$2p)$^4$ 
($\pi_{\mathrm{g}}$2p)$^4$ ($\sigma_{\mathrm{g}}$2s)$^2$ 
$^3\mathrm{\Sigma}_{\mathrm{g}}^{-}$. In Fig. 2.8 the $\mathrm{\Sigma}$, 
$\mathrm{\Pi}$, $\mathrm{\Delta}$, . . . represent the net angular momentum 
along the internuclear axis. These states are doubly degenerate (i.e., the 
angular momentum vector can be pointed either way) except for the 
$\mathrm{\Sigma}$ states which are, therefore, also labeled (+ or -) according 
to their behavior under reflection about a plane containing the nuclei. 

The molecular orbital diagrams are extremely useful for determining which
transitions are likely to occur. Following the molecular orbital states at
large R into small R will result in a number of curve crossings, as seen in
the potential diagram for certain states of He$_{2}^{2+}$ in Fig. 2.12. In a
fully adiabatic process, for which the electrons \textit{totally} adjust to
one another, crossings are avoided for states of the same total symmetry.
Therefore, the crossings in the molecular orbital diagram indicate at which
R the character of the wave function is changing (i.e., the transition
regions for slow collisions). The ($\sigma _{\mathrm{u}}$1s)$^{2}$ $%
^{1}\Sigma _{\mathrm{g}}^{+}$ state of He$_{\mathrm{g}}^{2+}$ is seen to
make a series of crossings as both of the electrons are promoted to (2p)
electrons in the united atom limit. Therefore, half of the collisions for 
He$^{2+}$ on He will proceed along an attractive $^{1}\mathrm{\Sigma}_
{\mathrm{u}}^{+}$ for which transitions \textit{are unlikely} at low 
velocities and half along a strongly repulsive $^{1}\mathrm{\Sigma}_
{\mathrm{g}}^{+}$ state for which transitions are \textit{very likely} to 
occur if the distance of closest approach is small enough. Such \textit
{electron promotion} occurs in all colliding systems as there are two atomic 
orbital states at large R for every atomic orbital state at small R 
(Fig. 2.10), as pointed out by Fano and Licthen (1965). 

Molecular orbital diagrams are also constructed for an electron in the field
of two positive centers having different effective charges Z$_{\mathrm{A}%
}^{\prime }$ and Z$_{\mathrm{B}}^{\prime }$. The importance of promotion and
the covalency of the interactions depends on the similarity of these
charges. For instance, for a proton interacting with a nitrogen atom (NH$%
^{+} $ potential) the ground state potentials at intermediate R are well
described by a single electron shared by an N$^{+}$ core and a proton. These
have binding energies I$_{\mathrm{N}}$ = 14.5 eV and I$_{\mathrm{H}}$ = 13.6
eV giving effective charges Z$^{\prime }$=$\sqrt{\mathrm{I/13.6eV}}$ which
are similar (Z$_{\mathrm{N}}^{\prime }$= 1.03, Z$_{\mathrm{H}}^{\prime }$ =
1). Therefore, the lowest $\Sigma $ state potentials of NH$^{+}$ resemble
those of H$_{2}^{+}$ in Fig. 2.10 at intermediate R, except there is a small
energy difference as R $\rightarrow $ $\infty $. The similarity is even
stronger for H$^{+}$ + O for which the ionization potentials are very close
in value. The covalent nature of such states results in an exchange
interaction like that for H$_{2}^{+}$ , written as

\begin{equation}
\mathrm{\Delta} \varepsilon \approx \mathrm{A} \mathrm{exp(- R/\bar{a})}, 
\end{equation}
where the effective radius is \={a} = a$_{0}$/(Z$_{\mathrm{A}}^{\prime }$ 
+ Z$_{\mathrm{B}}^{\prime }$ ). Smirnov (1964) evaluates A. 

This confusion of potentials presents a problem. For example, even though
the O$^{+}$ + O (O$_{2}^{+}$ system) has a lowest lying attractive (bound)
state, $^{2}\mathrm{\Pi}_{\mathrm{g}}$, the collision of an oxygen ion with an
oxygen atom involves seven different potential curves each with a different
multiplicity (Fig. 2.8). In a detailed calculation these potentials are not
averaged; rather the elastic collision cross sections for each potential are
averaged [viz. Eqs. (2.22) and 2A.8)] (Yee and Dalgarno 1987). However, when
calculating integrated quantities such as S$_{\mathrm{n}}$ [Eq. (2.11)] a
net repulsive potential is often used at intermediate R. Based on the
electron exchange interaction discussed above, the form, V = Ae$^{\mathrm{-R/
\bar{a}}}$ referred to as a Born-Mayer potential, is used in particle
penetration calculations and is essentially an extension of the short-range
repulsive interaction [Eq. (2.25)]. The ``universal'' potential (Appendix
2B) has also proven useful for describing the \textit{average} repulsive
energy transfer cross sections at \textit{intermediate and short range} for
many electron systems. 

\section{Inelastic Collisions}

When the interacting particles are moving, as in a collision, then the
forces described in the preceding sections become dynamic and the
time-dependent fields produce changes in the internal state of the molecule.
The motion of each of the constituent particles of the molecules can be
characterized by a period $\tau$ and a mean radial extent from the center
of mass r($\sim$\={a}). The interaction with a passing particle of
velocity v at a distance b from the center of a target atom or molecule is
said to be adiabatic if $\tau_{\mathrm{c}}\gg \tau$, where $\tau_{\mathrm{
c}}$ is the larger of b/v and r/v. For such collisions, electronic
transitions are unlikely (b$_{\mathrm{max}}\approx $ v/$\omega $, where $
\omega $ = 2$\pi $/$\tau $ is referred to as the Bohr adiabatic cut-off).
For t, $\tau _{\mathrm{c}}\ll \tau $ , the interaction times are short and
again transitions are unlikely. Therefore, inelastic cross sections
generally go through a broad maximum when $\tau _{\mathrm{c}}$ $\approx $ $
\tau $, as discussed in the introduction to this chapter. The collision time
gives an uncertainty in the energy during the collision of $\mathrm{\Delta E}
\approx$ h/$\tau_{\mathrm{c}}$, i.e., a width to the state. Therefore, the
maximum in the cross section occurs when $\mathrm{Delta}$E equals the 
excitation energy $\hbar\omega$. This is referred to as the Massey criterion.

Many of the approximations used to describe such transitions incorporate
only the initial and final states. Such models can be divided into two
categories: strong interactions, for which the evolution of the electronic
states during the collision is important (e.g., curve crossing trans\'{\i}t%
\'{\i}ons); and weak interactions, for which the initia! charge distribution
can be considered static. This parallels the division used above to
calculate the interaction potentials. The first case applies to velocities
comparable to or smaller than the speeds of the constituent particles of the
target and is generally associated with the intermediate range interactions.
The second case corresponds to large velocities. If, in addition, the
collision is a distant collision, (b $\gg $ r) small momentum transfer, the
target atom or molecule must be viewed as a whole and the constituent
particles are often treated as bound oscillators of frequency $\omega$. 
These oscillators are then excited by the time-dependent field of
the passing particle (Appendix 2C). On the other hand, for close collisions,
(b $\ll $ r) large momentum transfer, the incident particle can be thought
of as interacting with each of the constituents of the target molecule
separately. This is referred to as the binary-encounter limit. 

In the following we first calculate the large momentum transfer (close
collision) contribution to the inelastic energy-loss cross section, and then
the Bethe-Born approximation for the two contributions. Following this
strong interactions at intermediate range such as charge exchange are
considered. 

\subsection{\textit{Binary-Encounter Approximation}}

In the often-used binary-encounter approximation (BEA), the incident
particle is assumed to make a close collision with only one of the
constituents of the target molecule. Therefore, the stopping power [Eqs.
(2.10) and (2.11)] can be approximated as a sum of contributions from each
of the target molecule constituents, electrons and nuclei,

\begin{equation}
\mathrm{ S = S_e + S_n \approx \sum N_j \int Q d\sigma_j(Q) + \sum\limits_k\int
T d\sigma_k(T).} 
\end{equation}
The cross section d$\sigma_{\mathrm{j}}$ (Q) is the elastic energy transfer
cross section to the electrons in orbit j with population N$_{\mathrm{j}}$
and d$\sigma _{\mathrm{k}}$(T) is the energy transfer cross section to
nucleus k in the target molecule. For a close collision between charged
particles 

\begin{equation}
\mathrm{d\sigma_k(T) = 2\pi \frac{(Z_A Z_k e^2)^2}{M_k v^2}\frac{dT}{T^2}.}
\end{equation}
Using Eq. (2.30) in Eq. (2.29) the nuclear stopping cross section [Eq.42.12b)]
is
%---------(2.31a)(2.31b)
\begin{subequations}
\begin{align}
& \mathrm{ S_n \approx \sum\limits_k 2\pi\frac{Z_AZ_ke^2)^2}{M_kv^2}
\ln(\gamma_kE_A/T_k),} \notag\\ 
\end{align}
where $\gamma_{\mathrm{k}}$E$_{\mathrm{A}}$ is the maximum energy transfer
to nucleus k and T$_{\mathrm{k}}$ is a ``minimum'' energy
transfer due to the screening by the other constituent particles (Appendix
2B). This is the general form for S$_{\mathrm{n}}$ (see Fig. 2.4).

S$_{\mathrm{n}}$ can be rewritten so that energy transferred to the center
of mass of the target molecule and that energy transferred to vibrational
and rotational excitation of the molecule are separated. That is, energy
transfer is assumed to occur predominantly in close collisions of the ion
with one of the atoms of mass M$_{\mathrm{k}}$ in the molecule of total mass
M. The deflection of the \textit{incident} particle is determined by this
binary collision and the direction of motion of the center of mass of the
\textit{target molecule} is along the direction of the impulse to M$_{%
\mathrm{k}}$. Of the total energy transfer T, an amount (M$_{\mathrm{k}}$%
/M)T goes into motion of the center of mass and the remainder, (1 - M$_{%
\mathrm{k}}$/M)T, goes into relative motion of the components. If the latter
energy is greater than the dissociation energy for M$_{\mathrm{k}}$, the
molecule is likely to dissociate, but if it is less, then it all goes into
vibration and/or rotation of the molecule. For a diatomic molecule, if $\phi
$ is the angle between the direction of the impulse given to M$_{1}$ and the
orientation of the axis between M$_{1}$ and M$_{2}$, then T$_{\mathrm{vib}}$
= (M$_{2}$/M)T cos$^{2}\phi $ and T$_{\mathrm{rot}}$ = (M$_{2}$/M)T sin$%
^{2}\phi $. For random orientation one-third of the internal energy goes
into vibration and two-thirds into rotation, consistent with the number of
modes available. The binary collision model breaks down if the important
energy transfers occur with the impact parameter comparable to the bond
lengths.  

Whereas the BEA can be used to calculate the total energy transfer to nuclei
in molecules or solids, it describes only the high momentum transfer
collisions of an ion with the electrons. Since the energy transfer cross
section to the electrons on B can also be described by the coulomb
interaction between each electron and the incident ion A, d$\sigma _{\mathrm{
j}}$ (Q) has the same form as d$\sigma _{\mathrm{k}}$(T) in Eq. (2.29) with
T $\rightarrow $ Q, $\gamma $E$_{\mathrm{A}}$ $\rightarrow $ 2 m$_{\mathrm{e}
}$v$^{2}$, Z$_{\mathrm{B}}$ $\rightarrow $ 1, and M$_{\mathrm{K}}$ 
$\rightarrow $ m$_{\mathrm{e}}$ in Eq. (2.30). The quantity S$_{\mathrm{e}}$
then also follows from Eq. (2.31a),

\begin{align}
& \mathrm{ S_e \approx \sum\limits_j N_j 2\pi \frac{(Z_A e^2)^2}{m_e v^2} 
\ln\left(\frac{2m_e v^2}{I_j}\right),} 
\end{align}
\end{subequations}

where I$_{\mathrm{j}}$ is the ionization potential of the j electrons. It is
seen that both S$_{\mathrm{e}}$ and S$_{\mathrm{n}}$ vary as (lnE$_{\mathrm{A
}}$)/E$_{\mathrm{A}}$ at high energies. The large difference in magnitudes
between S$_{\mathrm{e}}$ and S$_{\mathrm{n}}$, exhibited in Fig. 2.4 are due
to the \textit{difference in mass of the particle receiving the energy
transfer}, M$_{\mathrm{k}}$ or m$_{\mathrm{e}}$ in Eq. (2.31a) and
(2.31b).

\subsection{\textit{Bethe-Born}}

The calculation of the energy loss to the electrons also requires a
description of the binding conditions, often estimated by treating the
electrons as classical oscillators (Appendix 2C). The appropriate quantum
mechanical calculation at high velocities is to use the first Born
approximation (Bethe and Jackiw 1968). In the Bethe estimate the large and
small momentum transfers (close and distant collisions) can be separated
(Appendix 2D). The large momentum transfers are equivalent to the BEA
estimate, and the small momentum transfers are equivalent to \textit{%
photon-like} dipole excitations of the target by the time-dependent field of
the passing particle. Remarkably, in determining S$_{\mathrm{e}}$, the forms
for these two contributions are identical so that the final expression is
not sensitive to the separation between close and distant collisions. The
expression often used for an incident ion of charge Z$_{\mathrm{A}}$
is

\begin{equation}
\mathrm{ Se \equiv \sum\limits_f\int(Q_{0\rightarrow f})d\sigma_{0\rightarrow f}
\approx \frac{4\pi(Z_A e^2)^2}{m_e v^2} Z_B\left[ln\frac{2m_e v^2}{\bar{I}}-1-
C/Z_B\right] }
\end{equation}
(Inokuti et al. 1981). Here Q$_{\mathrm{0\rightarrow f}}$ = $\varepsilon _{
\mathrm{f}}^{0}$ - $\varepsilon _{\mathrm{0}}^{0}$, \={I} is called an
average ionization potential, and C is a factor associated with the shell
structure of the target atom or molecule giving a small correction to the
expression. Values of \={I} and C have been extracted from experiment and
are tabulated (see Bibliography). The expression in Eq. (2.32) for S$_{
\mathrm{e}}$ is about twice the BEA result in Eq. (2.3 lb), indicating that
for incident ions distant \textit{and} close collisions contribute equally
at high velocities. As the velocity decreases, close encounters are
eventually required to produce an excitation, so the BEA applies using an
appropriate interaction potential (Johnson and Gooray 1977). The quantity 
(2m$_{\mathrm{e}}$v$^2$/\={I}) in Eq. (2.32) is often given as a ratio of
impact parameters (b$_{\mathrm{max}}$/b$_{\mathrm{min}}$) (Appendix 2C).

The Bethe-Born method can also be used to estimate the total ionization
cross section $\sigma _{\mathrm{I}}$ (Inokuti 1971) and the straggling cross
section S$_{\mathrm{e}}^{(2)}$, a quantity we will discuss in the following
chapter. The lead terms at large v are

%%%(2.33a)(2.33b)
\begin{subequations}
\begin{align} 
& \mathrm{ \sigma_I \equiv \sum\limits_f \int d\sigma_{0\rightarrow f} \approx 
\frac{4\pi(Z_A e^2)^2}{m_e v^2} Z_B\left(\frac{2m_e \bar{r}^2}{3\hbar^2}\right) 
ln\left(\frac{2m_e v^2}{\bar{I}}\right), } \notag\\
\end{align}
where f implies all final states leading to an ionization and $\bar{r}^2$ 
is the mean-squared radius of the initial state, which is tabulated for most 
atomic species, and 
\begin{align}
& \mathrm{ S_e^{(2)} \equiv \sum\limits_f\int(Q_{0\rightarrow f)^2 d}\sigma_
{0\rightarrow f} \approx 4\pi (Z_A e^2)^2 Z_B.}
\end{align}
\end{subequations}
S$_{\mathrm{e}}^{(2)}$, which weighs large energy changes heavily, is given by
the BEA calculation at high velocities as close collisions dominate, and is
approximately a constant. On the other hand, the ionization cross section, $%
\sigma _{\mathrm{I}}$, which, unlike S$_{\mathrm{e}}$ and S$_{\mathrm{e}%
}^{(2)}$, is not weighted by the energy transfers, is determined primarily
by the dipole excitations (Appendix 2D). Because of this, $\sigma _{\mathrm{I%
}}$, has a high energy dependence which parallels that of the stopping cross
section, i.e., (ln E$_{\mathrm{A}}$)/E$_{\mathrm{A}}$, as seen in Fig.
2.13a. This has the very useful consequence that the \textit{ionization rate}
in a gas or solid due to a fast incident ion is \textit{roughly proportional}
to S$_{\mathrm{e}}$. For fast (neutralatom)-(neutral-atom) collisions the
dipole excitations are small contributors and BEA gives reasonable estimates.

Cross sections for incident electron ionization (Fig. 2.13a) have the same
high energy dependence as those for incident protons\textit{\ at the same
value of v}. On the other hand, at low incident velocities an energy
threshold is seen for electron impact ionization. For these same v the ions
have considerable energy and the cross sections decrease more slowly. As the
ionization cross sections described have a smooth dependence on v,
semi-empirical expressions (Table 2.3) have been developed based on the
forms given above (Green and McNeal 1971). The electron impact cross
sections are also approximated by such expressions (Jackman et al. 1977) and
averaged over a speed distribution given by the plasma temperature. These
rate constants (k$_{\mathrm{I}} = \bar{\sigma_\mathrm{I}\mathrm{v}}$, e.g.,
Fig.2.13b) are used to determine the ionization rate of neutrals ejected
into a plasma in Chapter 4. 

\section{Two-State Models: Charge Exchange}

For collision velocities comparable to or smaller than the speed of the
constituents of the target molecules (v $<$ v$_{\mathrm{e}}$),
the details of the interaction potentials can control the transition
probabilities. When the coupling between neighboring states is strong, a
number of models are used to calculate these probabilities based on the
impact parameter cross section in Eq. (2.7) and the transition probabilities
in Appendix 2D. A requirement for strong coupling \textit{in this velocity
range} is that the energy difference between the states at some point in the
collision, $\varepsilon _{\mathrm{f}}$(R)-$\varepsilon _{\mathrm{0}}$(R), is
small compared to the uncertainty in the energy of the states during the
collision: $\Delta$ E $\approx  \hbar/(\Delta$R$_{\mathrm{x}}$/v),
where $\Delta $R$_{\mathrm{x}}$ is the extent of the transition region. For
two strongly interacting states the transition probabilities and the angular
differential cross sections exhibit interference and are oscillatory in
velocity and angle. Here we consider the integrated cross sections. 

Approximate models for the two-state impact parameter cross section in Eq.
(2.7) can be written in the form

\begin{equation}
\mathrm{ \sigma_{0\rightarrow f} \approx \bar{P}_{0\rightarrow f}(\pi b_x^2)} 
\end{equation}
(Appendix 2D). Here b$_{\mathrm{x}}$ is that impact parameter giving the
onset for a transition, and $\bar{\mathrm{P}}_{0\rightarrow f}$ is the
averaged transition probability for impact parameters less than 
b$_{\mathrm{x}}$. 

For symmetric-resonant charge exchange (e.g., H$^{+}$ + H$\rightarrow $ H + H
$^{+}$ or O$_{2}^{+}$ + O$_{2}$ $\rightarrow $ O$_{2}$ + O$_{2}^{+}$) the
average probability of a transition (exchange) is 1/2, as the initial and
final states are identical (i.e., Q$_{0\rightarrow f}$ = $\varepsilon _{
\mathrm{f}}^{0}$ - $\varepsilon _{\mathrm{0}}^{0}$ = 0). This is described in
Appendix 2D. As the time for exchanging an electron is $\tau _{\mathrm{x}
}\approx $ h/$\Delta \varepsilon $, where $\Delta \varepsilon $ is the
exchange energy in Eq. (2.28), then at some large value of b, $\tau _{
\mathrm{x}}\gg \mathrm{\tau }_{\mathrm{c}}$ so that the transition
probability goes to zero. Since $\Delta \varepsilon $(R) increases
exponentially with decreasing R, that impact parameter at which charge
exchange begins is

\begin{equation}
\mathrm{ b_x = b_{ct} \approx \bar{a}[B(v) - ln v]. }
\end{equation}
Here B is a slowly varying function of
v [see Eq. (2D.12)] and \={a} is the screening constant in Eq. (2.28). Using
b$_{\mathrm{ct}}$ in Eq. (2.34), it is seen that the charge exchange cross
section increases slowly with decreasing velocity at intermediate
velocities. At high velocities (v $>$ v$_{\mathrm{e}}$) the
exchange of an electron from a stationary atom to a fast ion requires a
significant change in the momentum of the electron. Therefore, \={P}$_{
\mathrm{0}\rightarrow \mathrm{f}}$ goes to zero rapidly ($\sim $ v$^{-12}$
for H$^{+}$ + H). The net cross section over many orders of magnitude in E$_{
\mathrm{A}}$ is shown in Fig. 2.13a and is compared to ionization cross
sections. It is seen that at large energies ionization dominates in a plasma
whereas at low energies charge exchange dominates. 

At very low velocities the interaction potential at large R can control the
collision. An ion and neutral ($<$ 10's eV) can orbit because of
the long-range polarization potential: n = 4 (McDaniel et al. 1970). From
Eq. (2.21) the orbiting occurs at an impact parameter b$_0 \propto$ v
$^{-1/2}$, which varies faster than b$_{\mathrm{ct}}$, in Eq. (2.35) and
eventually exceeds b$_{\mathrm{ct}}$ as v $\rightarrow $ 0. Therefore, the
cross section grows as v$^{-1}$ at very low velocity. For ion-ion charge
exchange (e.g., S$^{2+}$ + S$^{+}$ $\rightarrow $ S$^{+}$ + S$^{2+}$) the
repulsive force dominates at low energies so that the cross section goes to
zero (see Fig. 2.16c). 

Charge exchange between nonsymmetric systems (e.g., S$^{+}$ + O in the Io
plasma torus or H$^{+}$ + O in the Saturnian magnetospheric plasma) requires
a net change in the electronic energy, albeit a small change. Such cross
sections will exhibit a maximum, as discussed earlier. For large velocities,
when the uncertainty in the energy $\Delta $E is much greater than the state
separation, Q$_{\mathrm{0}\rightarrow \mathrm{f}}$, the cross section
behaves like the symmetric resonant cross section (e.g., b$_{\mathrm{x}}$
given by Eq. (2.35) and \={P}$_{\mathrm{0}\rightarrow \mathrm{f\ }}\approx $
1/2; see H$^{+}$ + H$_{2}$O in Fig. 2.13a). At velocities for which $\Delta $
E is comparable to or smaller than Q$_{\mathrm{0}\rightarrow \mathrm{f}}$
the ``transition'' generally occurs in a narrow range of internuclear
separations $\mathrm{\Delta} $R$_{\mathrm{x}}$ about a particular value, 
R$_{\mathrm{x}}$, and the transition probability is a rapidly varying 
function of v. These transitions are treated according to whether the 
potentials cross (Fig. 2.9) or do not cross (Olson 1980). 

For the curve crossing case, bx in Eq. (2.34) is set equal to the crossing
point, R. For impact parameters less than b$_{\mathrm{x}}$ the colliding
particles pass through the point R$_{\mathrm{x}}$ twice, on the approach and
exit (Fig. 2.14). At each passage the average transition probability is P$_{
\mathrm{0f}}$. After the collision a change in state will have taken place
if a transition occurred on the first passage but not the second [p$_{
\mathrm{0f}}$(1 - p$_{\mathrm{0f}}$)] and vice versa [(1 - p$_{\mathrm{0f}}$
)p$_{\mathrm{0f}}$]. Therefore, \={P}$_{\mathrm{0f}}$ = 2p$_{\mathrm{0f}}$(l
- p$_{\mathrm{0f}}$), and Eq. (2.34) becomes

\begin{equation}
\mathrm{ \sigma_{0f}^{LZS} \approx 2p_{0f} (1 - p_{0f})\pi R_x^2.} 
\end{equation}
The Landau-Zener-Stueckleberg expression for is p$_{\mathrm{0f}}$ is

\begin{equation}
\mathrm{ p_{0f} \approx 1 - exp(-\tau_c/\tau). }
\end{equation}
Here $\tau _{\mathrm{c}}$ is the collision time, the time spent in the
interaction region $\mid\Delta$R$_{\mathrm{x}}/\mathrm{(dR/dt)}\mid$, and 
$\tau$ the electron exchange time $\mid\pi\Delta \varepsilon$(R)/$\hbar
\mid^{-1}$. These are evaluated at R = R$_{\mathrm{x}}$ with $\Delta 
\varepsilon $(R) as in Eq.(2.28) and $\Delta$R$_{\mathrm{x}}$ $\approx 
\mid 2\Delta \varepsilon$(R)/[d($\varepsilon_{\mathrm{f}}$ - $\varepsilon_0$)
/dR]|, where $\varepsilon_{\mathrm{f}}$ and $\varepsilon_{0}$
are the electronic energies of the states (e.g., S$^{2+}$ + O = S$^{+}$ + O$
^{+}$ in Fig. 2.9). Extensive measurements have been made for doubly charged
ions on neutrals, for which Olson et al. (1971) fit a form for $\Delta
\varepsilon $ [see Eq. (2.28)] for use with Eq. (2.37),

$$
\mathrm{ \Delta \varepsilon(R) \approx (I_A I_B)^{1/2}(R/\bar{a})exp(-0.86R/\bar{a})} 
$$

\begin{equation}
\mathrm{\bar{a} = a_0(Z_A^{\prime} + Z_B^{\prime})^{-1}\ with \ \ 
Z^{\prime} = (I/13.6eV)^{1/2}.} 
\end{equation}
Such forms have been used to estimate cross sections for the Io plasma torus
(Johnson and Strobel 1982; Brown et al. 1983b; McGrath and Johnson 1989) and
they join to the symmetric-resonant-like result [i.e., Eq. (2.35)] at high
velocities. 

For the noncrossing case an effective transition region, R$_{\mathrm{x}}$,
exists at the separation for which the interaction, $\Delta \varepsilon$
(R), is approximately half the spacing between states $\mid\varepsilon_
{\mathrm{f}}$(R) - $\varepsilon_{0}(\mathrm{R})\mid$.
Demkov (1964) gives an expression for the transition probability

\begin{equation}
\mathrm{\bar{P}_{0\rightarrow f} \approx (1/2) sech^2\{(\pi \textit{\={a}}/2\hbar)|
[\varepsilon_f(R) - \varepsilon_0(R)]/\dot{R}|_{R_x}\}.}
\end{equation}
with \={a} as given above. In evaluating \={P}$_{0\rightarrow f}$ 
in Eq. (2.39) the energy difference $\mathrm{[\varepsilon_f(R) - \varepsilon_0(R)]}$ 
is often set equal to the initial energy difference and 
$\mathrm{\mid dR/dt \mid \approx v,}$ giving an extremely simple 
expression for $\bar{\mathrm{P}}_{0 \rightarrow \mathrm{f}}$, requiring only 
the electron binding energies. Applying Eq. (2.34), b$_{\mathrm{x}}\approx$ 
R$_{\mathrm{x}}$ and, again, the low-velocity results join smoothly onto the 
symmetric-resonant-like result at high velocities where the energy difference 
Q$_{\mathrm{0}\rightarrow \mathrm{f}}$ ,f becomes un\'{\i}mportant.Figure 2.15 
from Rapp and Francis (1962) shows the relationship between the symmetric 
and nonsymmetric charge exchange for differing $\mathrm{Q}_{0\rightarrow f}$. 

The expressions above have been developed for transfer of a single s
electron but are often used more generally. For other cases the matrix
elements may have prefactors determined by the spin and orbital angular
momentum of the states (Matic et al. 1980; McGrath and Johnson 1989). The
expressions above can still give rough estimates if \={P}$_0
\rightarrow \mathrm{f}$ is multiplied by the fraction of states associated
with the incident ion and atom which have the same \textit{molecular}
identification as the exiting ion and atom states. For one electron transfer
in which net spin is conserved (the so-called Wigner-Witmar rule), these
factors and estimates of $\Delta \varepsilon $ have been used in the impact
parameter equations to calculate the cross sections in Fig. 2.16 and
estimates of sulfur and oxygen charge transfer cross sections in the Io
torus are given in Table 2.4. 

At very low velocities, when orbiting occurs, the cross sections for both
the crossing and noncrossing cases may again increase \textit{if the
transitions are exothermic}. That is, even though the transition probability
is small for any one pass through the transition region, after many passes
the accumulative effect can lead to a significant probability for a change
in the electronic state. At very low speeds this can occur only if internal
energy is released. This principle is extensively used by chemists to
estimate low-temperature reactions rates. The cross section for such
reactions (e.g., ion-molecule reactions, Langevin cross section) is obtained
by setting bx in Eq. (2.34) to the orbiting radius, b0, Eq. (2.21), as descr%
\symbol{126}bed earlier (McDaniel et al. 1970). The value of \={P}$_{\mathrm{%
0}\rightarrow \mathrm{f}}$ is then estimated statistically for those final
states available energetically. As a rough estimate, for N available final
states, \={P}$_{\mathrm{0}\rightarrow \mathrm{f}}$ $\sim $ N$^{-1}$ for each
possible transition. 

Although we have primarily discussed charge-exchange collisions and only
collisions involving ions and atoms, these procedures also apply to
molecular collisions and to all varieties of internal changes including
molecular reactions. However, the approximations given here and in Appendix
2 should be used only as a guide when more accurate results are not
available.

\section{Stopping Cross Section: Summary}

In the above we have considered the individual collisional processes which
will determine the behavior of a fast particle penetrating a gas, liquid, or
solid. Here we examine the relative importance of the events described for
determining the overall energy loss rate of the fast particle. In Fig. 2.17
the total stopping cross section, S $\equiv $ n$_{\mathrm{B}}^{\mathrm{-1}}$
(dE/dx) [see Eq. (2.11)], and the separate contributions to S are shown for
protons losing energy to H$_{2}$O. 

First, the molecular effects are small, so that the stopping cross section
for H$_{2}$O is very nearly a sum of 2H's and an O from Fig. 2.4. The most
important difference would be that the ionization potential is lower for H$
_{2}$O. Secondly, it is seen that the separate contributions, S$_{\mathrm{e}
} $ and S$_{\mathrm{n}}$, dominate the total stopping in very different
velocity regions, as discussed earlier. At high velocities S$_{\mathrm{e}}$
and S$_{\mathrm{n}}$ both decay as (ln E$_{\mathrm{A}}$)/E$_{\mathrm{A}}$
but their magnitudes differ by the ratio of the mass of the electron to that
of the nucleus. That is, even though in a close collision an ion transfers
energy more efficiently to a nucleus, cross sections are dominated by the
large impact parameters, for which energy is transferred more efficiently to
the lighter elections [Eqs. (2.31a) and (2.31b)]. 

At high velocities ionization is the dominant process with, roughly, a 10\%
contribution from excitations but charge exchange dominates S$_{\mathrm{e}}$
at lower velocities (see Fig. 2.13a). Energy loss due to charge exchange is
a two-step process differing, therefore, from the other contributions which
are single collision. That is, on traversing a gas, ions capture an electron
and later are ionized again. The capture and loss at low velocities can be
described in the solid state as a drag force on the ion in which the
electron cloud is distorted (polarized) by the passing ion increasing the
electron density in the vicinity of the ion (i.e., the covalent effect
discussed earlier). The drag force produced is such that (dE/dx) $\propto$
v. Lindhard and Scharff (1961) give an expression for the S$_{\mathrm{e}}$
at low v (v $<$ Z$_{\mathrm{A}}^{2/3}$ v$_0$) based on the
Thomas-Fermi model of the atom,

\begin{equation}
\mathrm{ S_e \approx \xi_e 8\pi e^2 a_0\frac{Z_A Z_B}{Z}\frac{v}{v_0} = 
\xi_e \frac{Z_A Z_B}{Z}\frac{v}{v_0}(1.92 \times 10^{-14} eV cm2).}
\end{equation}
In this expression v$_0$ is the velocity of an electron on the hydrogen atom 
(2.19 $\times $ 10$^8$ cm/s), Z is the average charge [Z$^{1/3}$ = 
(Z$_{\mathrm{A}}^{2/3}$ + Z$_{\mathrm{B}}^{2/3}$)$^{1/2}$] used in the Thomas-
Fermi screening length (Appendix 2B) and $\xi _{\mathrm{e}}\approx$ 
Z$_{\mathrm{A}}^{1/6}$. 

Because the values of the stopping cross section, S, given in data tables
are \textit{measurements}, the charge exchange cycle is always included.
This also means that the tabulated values of S are \textit{not} necessarily
those for the ion initially incident on a solid or on a gaseous envelope of
an object, it may be larger or smaller (Crawford 1989). Imagining the
incident ion to be in a variety of charge states during its passage through
even a relatively thin material, these stopping cross sections are referred
to as equilibrium-charge-state stopping cross sections. Average charge
states for ions \textit{exiting} a sample are given in Fig. 2.18. We see
that ions become neutralized at low velocities and fully stripped at high
velocities. The thickness of material at which the equilibrium charge state
is achieved is determined by the atomic density of the material and the
average charge capture, $\sigma _{\mathrm{ct}}$, and loss, $\sigma _{\mathrm{
I}}$, cross sections i.e., [($\sigma _{\mathrm{ct}}$ + $\sigma _{\mathrm{I}}$
)/n$_{\mathrm{B}}\sigma _{\mathrm{ct}}\sigma _{\mathrm{I}}$)]. This
thickness corresponds to hundreds of monolayers of material at high velocity
(v $\gg $ v$_{\mathrm{e}}$) and 10's of monolayers at intermediate
velocities (v $\approx $ v$_{\mathrm{e}}$). 

A number of semi-empirical expressions for the effective equilibrium charge
state, Z$_{\mathrm{eff}}$, have been given. For example (Sh\'{\i}ma et al.
1985),

\begin{equation}
\mathrm{ Z_{eff} \approx Z_A (1 + X^{-1/0.6})^{-0.6},\ \ \ X \equiv 3.86 
\sqrt{E_A/M_A}/Z_A^{0.45} }
\end{equation}
with E$_{\mathrm{A}}$ in MeV and M$_{\mathrm{A}}$ in
amu. At large velocities, estimates of S$_{\mathrm{e}}$ are made using Z$_{%
\mathrm{eff}}$ in place of Z$_{\mathrm{A}}$ in Bethe-Born expression for a
bare incident ion in Eq. (2.32). In fact, as I in Eq. (2.32) depends
primarily on the target properties, \textit{heavy ion values} of S$_{\mathrm{
e}}$ and $\sigma _{\mathrm{I}}$, can be estimated from measured \textit{%
proton values} by multiplying by (Z$_{\mathrm{eff}}$)$^{2}$. 

At very high velocities relativistic effects associated with the electron
density in the target alter the form for S$_{\mathrm{e}}$ in Eq. (2.32)
although ionization is still the dominant energy loss process. Ignoring C,
the term in brackets is replaced by [ln (2m$_{\mathrm{e}}$v$^{2}$/\={I} [1 -
(v/c)$^{2}$]) - (v/c)$^{2}$ ]. As v $\rightarrow $ c the In term increases
so that S$_{\mathrm{e}}$ increases, as seen in Fig. 2.19, because of
emission of radiation (Brehmstrallung). Calling $\beta $\ = v/c, a useful
form for the electronic stopping power of an ion, Z$_{\mathrm{A}}$, is
(Turner 1986) 

$$
\mathrm{ \left|\frac{dE}{dx}\right|_e = 5.10 \times 10^{-19}eV cm^2 Z_{eff}^2
Z_B n_B [G(\beta) + ln(I_0/\bar{I})]/\beta^2 } 
$$

\begin{equation}
\mathrm{G(\beta) = ln\left[\frac{2mc^2}{I_0}\frac{\beta^2}{(1-\beta^2}\right] 
-\beta^2 = ln [7.52\times 10^4 \beta^2/(1-\beta^2)] - \beta^2},
\end{equation}
where I$_{0}$( = 13.59 eV) is the ionization energy of the hydrogen atom. 
In Eq. (2.42) Z$_{\mathrm{B}}$n$_{\mathrm{B}}$ is the total electron density 
of the material. Because the electron density is roughly proportional to the 
mass density for heavy elements, results are often presented as $\rho_
{\mathrm{B}}^{-1}$(dE/dx)$_{\mathrm{e}}$ (= S$_{\mathrm{e}}$/M$_{\mathrm{B}}$) 
as in Fig. 2.19. To estimate \={I} in eV ($\approx$ 19, Z$_{\mathrm{B}}$ = 1; 
$\approx$ 11.2 + 11.7 Z$_{\mathrm{B}}$, 2 $\le$ Z$_{\mathrm{B}}$ 
$\>$ 13, $\approx$ 52.8 + 8.71 Z$_{\mathrm{B}}$, Z$_{\mathrm{B}}$
> 13). For a molecular medium the (dE/dx)$_{\mathrm{e}}$ can be
added for each atomic constituent (called Bragg's rule). This is equivalent
to calculating a new effective \={I} (1n \={I} = $\sum $Z$_{\mathrm{i}}$ ln
\={I}$_{\mathrm{i}}$/$\sum $Z$_{\mathrm{i}}$; where Z$_{\mathrm{i}}$ and
\={I}$_{\mathrm{i}}$ are the nuclear charge and \={I} for each atomic
constituent: for H$_{2}$O I $\approx$ 74.6 eV). In the
condensed state the effective \={I} in Eq. (2.42) is about 20\% larger. In
addition, at very high velocity the material density modifies the G above.
Note that although (dE/dx)$_{\mathrm{e}}$ becomes small at high velocities,
it is because the energy loss events are less probable. Each event is
energetic, however, resulting in ionizations. 

At high energies (many MeV), nuclear reactions can also be induced. For
example, in the close collision of a GeV proton with a nucleus, a proton or
neutron can be ejected by a direct collision. Since the mean free path for
nuclear inelastic collisions by GeV protons in many materials is $\sim $%
100gm/cm$^{2}$ with the proton giving up approximately half of its energy
this contribution to the mass stopping is $\sim $ 5 MeV cm$^{2}$/gm. This is
comparable in size to the radiation losses. Because the incident proton
experiences an impulse, the resulting deflections also limit its penetration
into the material. The ejected species carry off some of the energy and the
``damaged'' nucleus may decay (e.g., emit a $\gamma $-ray, $\alpha $%
-particle, etc.) releasing additional energy. Except for the binding energy
of the ejected nucleon, the energy lost by the incident proton is mostly
converted into ionization of the target. Because a variety of nuclear
processes can be induced, this component is harder to quantify in a simple
way (Ryan and Draganic 1986). 

\section{Appendix to Chapter 2}

\subsection{\textit{A Wave Mechanics: Elastic Scattering}}

%%%appendix equation number
\let\rememberthechapter=\thechapter
\renewcommand{\thechapter}{2A}
\setcounter{equation}{0}

In wave mechanics the scattering of the particle is replaced by scattering
of a wave. In the CM system (see Table 2.1) the differences from and
similarities to the quantities considered in the text are shown below. The
beam of particles incident on a target Is replaced by a plane wave, exp [i$
\mathrm{\vec{K}}\cdot \mathrm{\vec{R}}$ - $\omega $t)], where K = 2$\pi $/$
\lambda $, $\lambda $ the wavelength, $\mathrm{\hbar \vec{K}}$ gives the
momentum, and $\hbar \omega =\hbar^2$ K$^2$/2 m is the energy. When
this incident wave is scattered, as in light scattering, the outgoing wave
at very large distances from the scattering center has the general
form

\begin{equation}
\mathrm{ \left[exp(i\vec{K}\cdot\vec{R})+f(\chi)\frac{e^{iKR}}{R}\right] 
e^{-i\omega}t. }
\end{equation}
The plane wave is the unscattered portion and the second term is the
scattered part. The magnitude of the wave scattered into an angular region
about $\chi$ is given by $\mid\mathrm{f}(\chi)\mid^2$ /R$^2$ where f($\chi$) 
is referred to as the scattering amplitude and R is the distance from the 
scattering center. Therefore, the differential cross section (probability 
of scattering into a unit sold angle about $\chi$) is

\begin{equation}
\sigma(\chi) = |\mathrm{f}(\chi)|^{2}.
\end{equation}
That is, the scattering problem reduces to solving the Schr\"{o}edinger
equation subject to the boundary condition at large R given by Eq. (2A.1). 

To draw analogies with the classical calculation of cross section, it is
customary to express the plane wave in terms of multiple moments,

\begin{equation}
\mathrm{exp(i\vec{K}\cdot\vec{R}) = \sum\limits_{\ell=0}^{\infty} i^{\ell}
(2\ell + 1)P_{\ell}(cos\chi)j_{\ell}(KR).} 
\end{equation}
The j$_{\ell}$(KR) are the spherical Bessel functions which have the
asymptotic form j$_{\ell}$(KR) $\rightarrow $ sin (KR - $\ell \pi $/2)KR.
The P$_{\ell }$ are the Legendre polynomials, where $\ell $ labels the
variqus moments ($\ell $ = 0, spherical; $\ell $ = 1, dipole; $\ell $ = 2
quadrupole; ...) and is the angular momentum index [L$^2$ = $\ell (\ell
+ 1)\hbar^2$]. Therefore, $\ell$ replaces the impact parameter

\begin{equation}
\mathrm{b \rightarrow \sqrt{\ell(\ell + 1)\hbar}/mv\approx(\ell + 1/2)K^{-1}.} 
\end{equation}

Upon intersecting a scattering center each spherical wave,
j$_{\ell}$, experiences a phase shift ($\eta_{\ell}$) becoming sin (KR - $
\ell \pi$ /2 + $\eta_{\ell}$)/KR in the asymptotic region. Substituting
Eq. (2A.3) and the asymptotic forms for j$_{\ell}$ into the radial part of
Eq. (2A.1) 

\begin{equation}
\mathrm{f(\chi) = \frac{1}{2Ki} \sum\limits_{\ell =0}^{\infty}
(2\ell + 1)[exp(2i\eta_{\ell}) - 1] P_{\ell}(cos\chi).}
\end{equation}
Substituting this expression for f($\chi$) into Eq. (2A.2), the scattering
cross section can be reduced to the determination of the phase shifts. A
useful semi-classical estimate for the phase shift has a form like that in
light scattering. Since $\hbar$ K = h/$\lambda $ is a momentum, the phase
shift is related to the accumulated change in the wavelength (momentum) due
to the presence of the target atom,

\begin{equation}
\mathrm{ \eta_{\ell} \approx \eta (b) = \frac{2}{\hbar }\left[ \int
\limits_{R_0}^{\infty}p(R)dR-\int\limits_b^{\infty}p_0(R)dR\right]  \approx 
\frac{1}{2\hbar v} \int\limits_{-\infty}^{\infty}V(R)dZ. }
\end{equation}
The radial momentum p(R) = m $\dot{R}$ is
obtained from Eq. (2.16b), and p$_0$(R) is the same quantity in the
absence of the potential, V(R) = 0. Therefore, the shift in phase is due to
the interaction potential which acts like a refracting medium. 

The total cross section, obtained using Eq. (2A. 5) and by integrating over
angle in Eq. (2A. 2), becomes

\begin{equation}
\mathrm{\sigma = 2\pi \int\limits_{-1}^{1}|f(\chi)|^2 dcos\chi 
= \frac{8\pi}{K^2}\sum\limits_{\ell =0}^{\infty} (\ell + 1/2) sin^2 \eta_{\ell} 
\approx 2\pi \int\limits_{0}^{\infty} 4 sin^2 \eta(b)bdb}.
\end{equation}
It is seen that the expression for $\sigma $ is in the form of the \textit
{impact parameter cross section} in Eq. (2.3) \'{\i}f the collision probability 
is P$_{\mathrm{c}}$(b) $\approx $ 4 sin$^{2}$ $\eta $(b). At large impact
parameters (or $\ell $ ) the phase shift $\eta $(b) and the scattering
probability go to zero and, therefore, the cross section can be \textit{
finite}. However, because of diffraction, the ``probability'' P$_{\mathrm{c}
} $, above has an average value of 2 at small b and not unity as in the
classical model, clearly showing that $\sigma $ is a nonclassical quantity.

In evaluating the sum in Eq. (2A.5) and substituting into Eq. (2A.2)
constructive interference occurs at those impact parameters which produce
\textit{classical} scattering into the angular region $\chi $ as determined
by Eq. (2.17). Therefore, the classical trajectories are like the light rays
in geometrical optics. In this spirit the angular differential cross section
can be approximated as a sum of contributing amplitudes and their
corresponding phases

\begin{equation}
\mathrm{\sigma(\chi) \approx \mid\sum\limits_{i}\{\sigma[\chi(b_i)]\}^{1/2} 
exp [i\alpha(b_i)]\mid^2.} 
\end{equation}
This replaces the expression in Eq. (2.22), where $\sigma[\chi$
(b$_{\mathrm{i}}$)] is the classical value $\mid \mathrm{bdb}/sin \chi d\chi
\mid_{\mathrm{b=b}_{\mathrm{i}}}$ and exhibits interference between 
contributions to scattering into angle $\chi $ from different impact 
parameters. The net phase factor $\alpha$(b) is $\alpha$(b) = A/$\hbar \pm 
\pi/4 + \varepsilon$, where $\varepsilon$ is a constant, and A = [2$\hbar\eta$
(b) - L$\chi$] is the difference in the classical action between an
undeflected particle and a scattered particle. Therefore, for the sum in Eq.
(2A.5), the regions of constructive interference are those for which the
difference in this \textit{classical action is a minimum} ($\partial A/
\partial$ b = 0), i.e., the classical trajectories. Now $\chi$ and $\eta$
(b) are related by

\begin{equation}
\chi =\frac{2}{\mathrm{K}}\frac{\mathrm{\partial \eta (b)}}{\mathrm{
\partial b}}. 
\end{equation}
This is equivalent to
the expression for $\chi$(b) in Eq. (2.17) using the form for $\eta$(b)
given in Eq. (2A.6). Therefore, if only one region of impact parameter
contributes in Eq. (2A.8), the \textit{semiclassical} result for ${{}\sigma }
$($\chi $) is \textit{identical} to the \textit{classical} result. When more
than one region contributes at a given scattering angle, as in the rainbow
scattering described earlier, $\sigma $($\chi $) is oscillatory. When (V/E $
\ll $1) then the classical deflection estimate in Eq. (2.18) is obtained.

Using Eq. (2A.6) in Eq. (2A.5) for $\eta _{\ell }$ and an approximation to P$
_{\ell }$(cos $\chi $) in which $\ell $ (i.e., b) is a continuous variable,
a wave mechanics estimate of f($\chi $) which is valid for small V/E can be
obtained

\begin{equation}
\mathrm{ f(\chi) \approx -\frac{m}{2\pi \hbar^2}\int d^3 R V(R)
e^{-i\Delta\vec{p}\cdot\vec{R}/\hbar}, }
\end{equation}
The scattering amplitude is seen to be a Fourier transform of the interaction 
potential, which is the frequently used first Born approximation for elastic 
collisions (Appendix 2B). 

For wavelengths small compared to the dimensions of the particles (K$^{-1} 
\ll$ d), the integral in Eq. (2A.7) can be estimated: the Massey-Mohr
approximation to $\sigma$ in Appendix 2B. For power law potentials with 
n $>$ 2, $\sigma$ is finite and the cross section decreases
monotonically with increasing energy as shown in Fig. 2.2. A similar
evaluation of the diffusion cross section, $\sigma _{\mathrm{d}}$, (or S$_{
\mathrm{n}}$) produces the \textit{same} form as the classical expression
[Eq. (2.13)] at small wavelengths, because of the exclusion of scattering at
$\chi$ = 0. Therefore, classical expressions for \textit{the diffusion cross
} and \textit{the nuclear stopping cross section} are accurate over a broad
range of incident ion energies, allowing their use for ion penetration of
gases and solids. 

When the wavelength is \textit{large} compared to the dimensions of the
particles (K$^{-1}\gg $ d), then only the lowest $\mathtt{\ell }$ values
contribute in Eq. (2A.5). At very low energies often only the first term is
needed.

\begin{equation}
\mathrm{\sigma(\chi) = \mid f(\chi)\mid^2 \approx \frac{1}{K^2} sin^2 \eta_0.}
\end{equation}
This expression represents the same region of \textit{impact parameter} as
above. However, because the wavelength is large, an increase of $\ell $ from
zero to one in Eq. (2A.4) makes b $\approx $ K$^{-1/2}$ $\gg $d, the size of
the colliding particles. In this limit it is seen that the differential
cross section is \textit{independent of angle for any potential}, a result
obtained classically only for the collision of spheres. Because of its
simplicity, such an isotropic scattering cross section is often used for low
energy collisions, even when the large wavelength criterion does not
rigorously apply. The integrated cross section in Eq. (2A.7) becomes 
$\sigma$ $\approx$ (4$\pi$/K$^2$) sin$^2$ $\eta_0$ for this case,
which is also equal to the diffusion cross section, $\sigma_{\mathrm{d}}$,
for large wavelengths. At long wavelengths, $\eta_0 \rightarrow$ const
$\cdot$ K and sin$^2 \eta_0$ $\propto$ K$^2$, so that $\sigma$
and $\sigma_{\mathrm{d}}$, are independent of E$_{\mathrm{A}}$ and 
S$_{\mathrm{n}}$ $\propto$ E$_{\mathrm{A}}$ (Appendix 2B).

\let\thechapter=\rememberthechapter

\subsection{\textit{B Elastic Collision Expressions: Summary}}

Some expressions are summarized for the classical and wave mechanical
elastic collision quantities discussed in the text and Appendix 2A (Johnson
1987).

$\chi$(b) = $\pi$ - 2b $\int\limits_{\mathrm{R}_{\mathrm{0}}}^{\infty }$ $
\frac{\mathrm{dR}}{\mathrm{R}^{\mathrm{2}}}\left( 1-\frac{\mathrm{b}^{
\mathrm{2}}}{\mathrm{R}^{\mathrm{2}}}-\frac{\mathrm{V}}{\mathrm{E}}\right)
^{-1/2}\approx -\frac{1}{\mathrm{2E}}\frac{\mathrm{d}}{\mathrm{db}}
\int\limits_{-\infty }^{\infty }\mathrm{V(R)dZ\smallskip \smallskip }$

$\eta (\mathrm{b})=\frac{\mathrm{p}_{\mathrm{0}}}{\hbar}\left[
\int\limits_{\mathrm{R}_{\mathrm{0}}}^{\infty }\left( 1-\frac{\mathrm{b}^2}
{\mathrm{R}^2}-\frac{\mathrm{V}}{\mathrm{E}}\right)^{1/2}\mathrm{dR-}\int
\limits_{\mathrm{b}}^{\infty}\left( 1-\frac{\mathrm{b}^2}{\mathrm{R}^2}
\right)^{1/2}\mathrm{dR}\right],
\hspace{0.2in}\mathrm{p}_0\mathrm{=mv\smallskip \smallskip }$

$\hspace{0.2in}\hspace{0.1in}\approx \frac{-1}{2\hbar \mathrm{v}}
\int\limits_{-\infty }^{\infty }\mathrm{V(R)dZ\smallskip }$

L$\chi $(b) = (2$\hbar $b) $\frac{\partial \eta \mathrm{(b)}}{\partial
\mathrm{b}}$, \hspace{0.1in}L = p$_{_{0}}$b\smallskip \smallskip

$\sigma $($\chi $) = $\frac{\gamma \mathrm{E}_{\mathrm{A}}}{4\pi }\frac{%
\mathrm{d\sigma }}{\mathrm{dT}}$(B initially stopped), S$_{\mathrm{n}}$ = $
\frac{\gamma \mathrm{E}_{\mathrm{A}}}{2}\sigma _{\mathrm{d}}\smallskip
\smallskip \smallskip \smallskip $

\paragraph{\textit{Impulse Estimates} (V/E $\ll $ 1)\hspace{4.3in}\hspace{
4.3in}}

For potentials of the form V = C$_{\mathrm{n}}$/R$^{\mathrm{n}}$,\medskip

\[ \mathrm{\chi(b) \approx a_n V(b)/E, \hspace{0.1in}
a_n = \pi^{1/2} \Gamma \left(\frac{n+1}{2}\right)/\Gamma \left(\frac{n}{2}
\right) \stackrel{n\rightarrow\infty}{\longrightarrow} (\pi n/2)^{1/2}} \] 

$\eta $(b) $\approx \frac{\mathrm{-L\chi (b)}}{\mathrm{2\hbar (n-1)}}
\smallskip \hspace{0.3in}\smallskip \hspace{4.3in}\hspace{4.3in}\hspace{0.3in
}\hspace{0.3in}\hspace{4.3in}$

[$\mathrm{\Gamma} \mathrm{(x+1)=x\Gamma (x),\hspace{0.1in}\Gamma (1)=1,
\hspace{0.1in}\Gamma (1/2)=\pi }^{\mathrm{1/2}}$; gamma function-tabulated]
\smallskip\smallskip

\textrm{T}$\approx \mathrm{(a}_{\mathrm{n}}\mathrm{V(b))}^{\mathrm{2}}
\mathrm{/(M}_{\mathrm{B}}\mathrm{v}^{\mathrm{2}}\mathrm{/2)\smallskip
\smallskip }$

$\frac{\mathrm{d\sigma }}{\mathrm{dT}}\approx \frac{\pi }{\mathrm{n}}\mathrm{
A}_{\mathrm{n}}^{\mathrm{2}}\mathrm{(\gamma E}_{\mathrm{A}}\mathrm{)}^{
\mathrm{1/n}}\mathrm{T}^{\mathrm{1+1/n}}\mathrm{;\hspace{0.1in}A}_{\mathrm{n}
}\mathrm{=}\left( \frac{\mathrm{2M}_{\mathrm{A}}}{\mathrm{M}_{\mathrm{A}}
\mathrm{+M}_{\mathrm{B}}}\mathrm{a}_{\mathrm{n}}\mathrm{C}_{\mathrm{n}
}\right) ^{\mathrm{1/n}}\smallskip \smallskip $

$_{\mathrm{n}}$ $\approx \frac{\pi }{\mathrm{n-1}}\mathrm{A}_{\mathrm{n}}^{
\mathrm{2}}\mathrm{(\gamma E}_{\mathrm{A}}\mathrm{)}^{\mathrm{1-2/n}}\mathrm{
,\hspace{0.1in}n>2.\smallskip \smallskip \smallskip \smallskip }$
\\
\\
\\
For the potential V=(Z$_{\mathrm{A}}$Z$_{\mathrm{B}}$e$^{2}$)exp(-$\beta$
R)/R

\[ \chi(b) \approx \frac{(\mathrm{Z_A}\mathrm{Z_B}e^2\beta)}{\mathrm{E}}
\mathrm{K}_1(\beta b) \stackrel{b\rightarrow\infty}{\longrightarrow} 
\left(\frac{\pi\beta \mathrm{b}}{2}\right)^{1/2} \frac{V(b)}{\mathrm{E}}\]

\[ \eta(b) \approx - \frac{(\mathrm{Z}_{\mathrm{A}}\mathrm{Z}_{\mathrm{B}}
\mathrm{e}^{2})}{\mathrm{\hbar v}}\mathrm{K}_0\mathrm{(\beta b)
\stackrel{b\rightarrow\infty}{\longrightarrow}}
\frac{\mathrm{-L\chi(b)}}{\mathrm{2\hbar (\beta b)}} \]

(K$_{\mathrm{n}}$ are the modified Bessel Functions - tabulated).

\paragraph{\textit{Massey-Mohr}}

This gives an estimate to the total elastic collision cross section when the
tail of the potential can be fitted to a power potential C$_{\mathrm{n}}$/R$
^{\mathrm{n}}$. \={b} is obtained by setting $\mid \eta$(\={b})
$\mid \approx$ 1/2 (R.E. Johnson 1982), \\

$\mathrm{\sigma \approx 2\pi \bar{b}^2 \left[ 1+\frac{1}{2n-4}\right],
\ \ \ \ \bar{b} \approx \left|\frac{2a_n C_n}{(n-1)\hbar v}\right|^{1/(n-1)}, 
\ \ \  n > 2,}$

a$_{\mathrm{n}}$ given above.

\paragraph{\textit{Born Approximation}[Eq. (2A.10)]}

For the potential V = (Z$_{\mathrm{A}}$Z$_{\mathrm{B}}$e$^{2}$) exp (- $
\beta $R)/R,\smallskip \smallskip

$\frac{\mathrm{d\sigma }}{\mathrm{dT}}\approx $ $\pi $A$^{2}$/($\gamma $E$_{
\mathrm{A}}$)(T + T$_{0}$)$^{2}$, \hspace{0.1in}A = $\left( \frac{\mathrm{2M}
_{\mathrm{A}}}{\mathrm{M}_{\mathrm{A}}\mathrm{+M}_{\mathrm{B}}}Z_{\mathrm{A}
}Z_{\mathrm{B}}e^{2}\right) $, \hspace{0.1in}T$_{0}$ = $\frac{\mathrm{
(\hbar \beta )}^{\mathrm{2}}}{\mathrm{2M}_{\mathrm{B}}}\smallskip
\smallskip $

$\sigma \approx $ $\pi $A$^{2}$/T$_{0}$ ($\gamma $E$_{\mathrm{A}}$ + T$_{0}$
)\smallskip \smallskip \smallskip

S$_{\mathrm{n}}$ $\approx \frac{\mathrm{\pi A}^{\mathrm{2}}}{\mathrm{(\gamma
E}_{\mathrm{A}}\mathrm{)}}\left[ \ln \left( \frac{\mathrm{\gamma E}_{\mathrm{
A}}}{\mathrm{T}_{\mathrm{0}}}+1\right) -\frac{\mathrm{\gamma E}_{\mathrm{A}}
}{(\mathrm{\gamma E}_{\mathrm{A}}+\mathrm{T}_{\mathrm{0}})}\right]
\smallskip \smallskip \smallskip \smallskip $

\paragraph{\textit{Thomas Fermi (Lindhard)\protect }}

These are historically important forms used in the literature on particle
penetration.

Scaling quantities;\smallskip \smallskip

\hspace{0.6in}length: a$_{\mathrm{TF}}$ = 0.8853a$_{0}$/(Z$_{\mathrm{A}
}^{2/3}$ + Z$_{\mathrm{B}}^{2/3}$)\smallskip $^{1/2}\smallskip $

\hspace{0.6in}energy: $\varepsilon $ = ($\gamma $E$_{\mathrm{A}}$)a$_{
\mathrm{TF}}$/2A, \hspace{0.1in}(A as above)\smallskip \smallskip

\hspace{0.1in}energy transfer: t = $\varepsilon ^{2}$T/($\gamma $E$_{\mathrm{
A}}$).\smallskip \smallskip \smallskip

$\hspace{0.1in}\frac{\mathrm{d\sigma }}{\mathrm{dt}}\mathrm{\approx \pi a}_{
\mathrm{TF}}^{\mathrm{2}}\mathrm{\xi /4t}^{\mathrm{4/3}}\mathrm{[1+\xi }^{
\mathrm{2/3}}\mathrm{t}^{\mathrm{4/9}}\mathrm{]}^{\mathrm{3/2}}\mathrm{,
\hspace{0.1in}\xi \approx 2.62\smallskip \smallskip \smallskip }\hspace{4.3in}$ \\

S$_{\mathrm{n}}$ $\approx \frac{9\pi }{2}\frac{\mathrm{A}^{\mathrm{2}}}{
(\gamma E_{\mathrm{A}})}$\{1n $\left[ \mathrm{B+(1+B}^{2}\mathrm{)}^{1/2}
\right] $ - B/(1 + B$^{2}$)$^{1/2}$\}, \hspace{0.1in}B = $\xi ^{1/3}$ $
\varepsilon ^{4/9}\smallskip \smallskip \smallskip $

\paragraph{\textit{Lenz-Jensen Potential}}

Gives a better low energy dependence for S$_{\mathrm{n}}$ than does the
Thomas Fermi form above. The potential in Eq. (2.25) is\smallskip \smallskip

V = $\frac{\mathrm{Z}_{\mathrm{A}}\mathrm{Z}_{\mathrm{B}}}{\mathrm{R}}$ e$
^{2}\mathrm{\Phi}$(R/a$_{\mathrm{TF}}$)\smallskip \smallskip

$\mathrm{\Phi}$ = (1 + b$_{1}$ y + b$_{2}$y$^{2}$ + b$_{3}$y$^{3}$ + b$_{4}$y$^{4}$)
e$^{\mathrm{-y}}\smallskip \smallskip $

y= (9.67R/a$_{\mathrm{TF}}$)$^{1/2}\smallskip \smallskip $

b1 = 1, \hspace{0.1in}b2 = 0.3344, \hspace{0.1in}b3 = 0.0485, \hspace{0.1in}
b4 = 0.002647\medskip \smallskip

\paragraph{\textit{Universal Potential}}

This potential and the corresponding stopping cross section are the most
recent improvement in scaling and fitting to laboratory data for atomic
collisions (Ziegler et al. 1985). It is close to the Lenz-Jensen potential,
but provides a somewhat better fit and should be used in most cases for
describing the average repulsive interaction.

\paragraph{\textit{Scaling Quantities}}

\hspace{0.6in}length: a$_{\mathrm{U}}$ = 0.8853a$_{0}$/(Z$_{\mathrm{A}
}^{0.23}$ + Z$_{\mathrm{B}}^{0.23}$)\smallskip \smallskip

\hspace{0.6in}energy: $\varepsilon $ = ($\gamma $E$_{\mathrm{A}}$)a$_{
\mathrm{U}}$/2A, (A as above)\smallskip \smallskip

energy transfer: t = $\varepsilon ^{2}$T/($\gamma $E$_{\mathrm{A}}$
).\smallskip \smallskip

The potential in Eq. (2.25) is 

V(R) = $\frac{\mathrm{Z}_{\mathrm{A}}\mathrm{Z}_{\mathrm{B}}\mathrm{e}^{2}}{
\mathrm{R}}$ $\mathrm{\Phi} $(R/a$_{\mathrm{U}}$)\smallskip \smallskip

$\mathrm{\Phi}$(y) = 0.181 exp ( - 3.2x) + 0.5099 exp ( - 0.9423x)\smallskip
\smallskip

\hspace{0.5in}+ 0.2802 exp( - 0.4029x) + 0.02817 exp (- 0.2016x)\smallskip
\smallskip

S$_{\mathrm{n}}$ = 2$\pi $ $\frac{\mathrm{A}^{\mathrm{2}}}{\gamma \mathrm{E}
_{\mathrm{A}}}$ [2$\varepsilon $s$_{\mathrm{n}}$ ($\varepsilon $)]\smallskip
\smallskip

s$_{\mathrm{n}}$ ($\varepsilon $) is the ``reduced'' stopping cross
section\smallskip \smallskip \hspace{0.2in}\hspace{0.2in}\hspace{0.2in}
\hspace{0.2in}$\smallskip \smallskip $

\hspace{0.2in}[2$\varepsilon $s$_{\mathrm{n}}$($\varepsilon $)] = $\lceil
\ln \varepsilon ,\hspace{0.2in}\varepsilon > 30$

$\hspace{0.9in}\hspace{0.05in}\hspace{0.02in}\lfloor $ln (1 + 1.138$
\varepsilon$)/(1 + 0.0132$\varepsilon^{-0.787}$ + 0.196$\varepsilon^{0.5}$, 
$\varepsilon$ $<$ 30 

$\hspace{0.5in}\frac{\mathrm{d\sigma }}{\mathrm{dt}}$ =$\frac{\mathrm{\pi a}
_{\mathrm{u}}^{\mathrm{2}}}{2}\frac{\mathrm{f(t}^{\mathrm{1/2}}\mathrm{)}}{
\mathrm{t}^{\mathrm{3/2}}}$; f($\varepsilon $) = $\frac{\mathrm{d}}{\mathrm{d
}\varepsilon }$[$\varepsilon $s$_{\mathrm{n}}$($\varepsilon $)]\medskip

\paragraph{\textit{Quadrature}}

In order to integrate the deflection function and, hence, obtain
differential cross sections for any potential, a simple quadrature is often
used.\medskip

g(x) $\equiv \frac{\mathrm{b}}{\mathrm{R}_{\mathrm{0}}}\left[ \frac{\mathrm{
1-x}^{\mathrm{2}}}{\mathrm{1-(b/R}_{\mathrm{0}}\mathrm{)}^{\mathrm{2}}
\mathrm{x}^{\mathrm{2}}\mathrm{-(V(R}_{\mathrm{0}}\mathrm{/x)/E}}\right]
^{1/2}\medskip \smallskip $

$\chi $(b) $\approx $ $\pi \left[ 1-\frac{\mathrm{1}}{\mathrm{m}}
\sum\limits_{\mathrm{i=1}}^{\mathrm{m}}\mathrm{g(x}_{\mathrm{i}}\mathrm{)}
\right] ,\hspace{0.2in}$x$_{\mathrm{i}}$ = cos[(2i - 1)$\pi $/4m]\smallskip

$\eta $(b)$\approx \frac{\mathrm{2L}}{\hbar }\frac{\pi }{(\mathrm{2m+1)}}
\sum\limits_{\mathrm{j=1}}^{\mathrm{m}}\sin ^{2}\left( \frac{\mathrm{j\pi }}{
\mathrm{2m+1}}\right) (\mathrm{g(x}_{\mathrm{j}}\mathrm{)}^{-1}-1)],\hspace{
0.2in}\mathrm{x}_{\mathrm{j}}=\cos \left[ \frac{\mathrm{j\pi }}{\mathrm{2m+1}
}\right] \smallskip $

(m is the number of terms used in the calculation).\medskip \smallskip

\subsection{\textit{C Classical Oscillator}}

%%%appendix equation number
\let\rememberthechapter=\thechapter
\renewcommand{\thechapter}{2C}
\setcounter{equation}{0}

Treating the electrons on an atom or molecule as classical oscillators gives
the energy loss forms discussed in the text. For distant collisions, low
momentum transfer, the bound electrons are excited by the time varying field
of the passing particle (Jackson 1963). The motion of an electron
oscillating in a field $\mathrm{\vec{E}}_{\mathrm{j}}$(t) is

\begin{equation}
\mathrm{ m_e \vec{\ddot{r}}_j + \Gamma_j \vec{\dot{r}}_j + m_e\omega_j^2
\vec{r}_j = -e\vec{E}_j(t), } 
\end{equation}
where $\omega_{\mathrm{j}}$ is the binding frequency and $\mathrm{\Gamma}_{\mathrm{j}}$. 
is a damping constant. Writing the energy transfer to the electron as 
Q$_{\mathrm{j}}$(b) = $\int\limits_{-\infty }^{\infty}$ $\mathrm{\vec{r}}_
{\mathrm{j}}$( - e$\mathrm{\vec{E}}_{\mathrm{j}}$(t))dt, it is straight 
forward to show that, as $\Gamma_{\mathrm{j}}$ $\rightarrow$ 0,

\begin{equation}
\mathrm{ Q(b) \rightarrow \frac{\pi}{m_e}\mid e\vec{E}_j(\omega_j)\mid^2, }
\end{equation}
where $\mathrm{\vec{E}}_{\mathrm{j}}(\omega)$ is the Fourier transform 
$\mathrm{\vec{E}}{\mathrm{j}}$(t). \\

Describing $\mathrm{\vec{E}}_{\mathrm{j}}$ in Eq. (2C.1) as the field
associated with a screened Coulomb potential, V = (Z$_{\mathrm{A}}$e$^{2}$
/R)e$^{\mathrm{-\beta }^{\prime }\mathrm{R}}$ and assuming straight line
trajectories, (R$^{2}\approx $ b$^{2}$ + v$^{2}$t$^{2}$), the energy
transfer becomes

\begin{equation}
\mathrm{ Q_j(b) = \frac{2(Z_A e^2)^2}{m_e v^2}\frac{1}{b^2}\left[(\beta_j
^{\prime}b)^2 K_1^2(\beta_j^{\prime}b)+\left(\frac{\omega_j b}{v}\right)^{2}
K_0^2(\beta_j^{\prime} b)\right], }
\end{equation}
where ($\beta_{\mathrm{j}}^{\prime}$)$^{2}$ = ($\beta^{\prime }$)$^{2}$ + 
($\omega_{\mathrm{j}}$/v)$^{2}$ and K$_{1}$ and K$_{0}$ are modified Bessel 
functions. This energy transfer goes through a maximum and behaves 
asymptotically at high and low velocities as

\begin{equation}
\mathrm{ Q_j(b) \rightarrow \frac{2(Z_A e^2)^2}{m_e v^2}\frac{1}{b^2} \begin{cases}  
(\beta^{\prime}b)^2 K_1^2(\beta^{\prime}b) \ \ \ for \ \ \ \beta^{\prime} \neq 0;  
& \omega_j b/v \ll 1. \\ 
\pi\left(\frac{\omega_j b}{v}\right)exp(-2\omega_j b/v); & \omega_j b/v \gg 1.
\end{cases}}
\end{equation}

In these expressions the screening of the collision is a result of \textit{
both} the screening in the potential, via $\beta^{\prime}$[ = \={a}$^{-1}$
in Eq. (2.25)], and dynamic screening due to the motion of the electron, via
$\omega _{\mathrm{j}}$. The limit ($\omega_{\mathrm{j}}$b/v $\gg$ 1) is
the adiabatic limit discussed in the text giving an effective b$_{\mathrm{max
}}$ $\approx$ v/$\omega_{\mathrm{j}}$, the Bohr adiabatic radius. Writing
the electronic stopping power as in Eqs. (2.9) and (2.29)

\begin{equation}
\mathrm{ S_e = \sum\limits_j N_j 2\pi \int \limits_{b_{\min}}^{\infty}Q_j(b)bdb, } 
\end{equation}
where N$_{\mathrm{j}}$ is the number of electrons of frequency $\omega_
{\mathrm{j}}$. Substituting Eq. (2C.3) into Eq. (2C.5), the expression for 
S$_{\mathrm{e}}$ is exactly integrable and depends on the lower limit. 
Setting b$_{\mathrm{min}}$ equal to the wavelength in the center of mass 
of the electronincident-ion system ($\hbar/m_{\mathrm{e}}$v), 
then

\begin{equation}
\mathrm{ S_e \approx 4\pi\frac{(Z_A e^2)^2}{m_e v^2}\sum N_j\left[ln\left(
\frac{1.123m_e v}{\beta_j^{\prime}\hbar}\right)-(1/2)\left(\frac{\beta^{\prime}}
{\beta_j^{\prime}}\right)^2\right]. }
\end{equation}
When the direct screening is dominant (i.e., $\beta _{\mathrm{j}}^{\prime}$
\ $\rightarrow $ $\beta ^{\prime }$ when v is large), then the screening
constant determines the effective low-energy cut-off,

\begin{subequations}
\begin{align}
& \mathrm{ S_e \approx 2\pi \frac{(Z_A e^2)^2}{m_e v^2}Z_B\left[ln\left
(\frac{2m_e v^2}{Q_{min}}\right) -1\right]; \ \ \ \ Q_{min} \approx 1.59 
(\beta^{\prime}\hbar^2/m_e } \notag\\
\end{align}
as for collisions of neutrals. On the other hand when $\beta^{\prime}$ = 0 
(i.e., no direct screening, only dynamic screening) then
\begin{align}
& \mathrm{ S_e \approx 4\pi \frac{(Z_A e^2)^2}{m_e v^2} \sum N_j \left[ln
\left(\frac{2m_e v^2}{\hbar\omega_j}\right) - 0.577\right].}
\end{align}
\end{subequations}
This has the same form as the Bethe-Born result in
Eq. (2.32).

\let\thechapter=\rememberthechapter

\subsection{\textit{D Wave Mechanics: Inelastic Collisions}}

%%%appendix equation number
\let\rememberthechapter=\thechapter
\renewcommand{\thechapter}{2D}
\setcounter{equation}{0}

The important Born scattering amplitude for a transition from an initial
state (0) to a final state (f) is given by

\begin{equation}
\mathrm{f^{(1)}(\chi) = \frac{-m}{2\pi \hbar^2}\int e^{-i\vec{K}_f\cdot\vec{R}} 
V_{f0}(R)e^{i \vec{K}_0\cdot \vec{R}} d^3 R.} 
\end{equation}
This leads to a cross section having a form like that in Eq. (2A.2)

\begin{equation}
\sigma_{0\rightarrow\mathrm{f}}^{(1)} (\chi) = \frac{\mathrm{K}_{\mathrm{f}}}
{\mathrm{K}_0} \mid \mathrm{f}_{\mathrm{0}\rightarrow \mathrm{f}}^{(1)}\mid^2,
\end{equation}
where $\hbar $K$_{\mathrm{f}}$ and $\hbar $K$_{0}$
are the final and initial momentum and cos$\chi $= \^{K}$_{\mathrm{f}}\cdot$
\^{K}$_{0}$. In Eq. (2D.1), V$_{\mathrm{f0}}$ (R) is the interaction
potential averaged over the final and initial states involved in the
transition given as incoming and outgoing plane waves. For the Born
approximation for elastic scattering, the final and initial electronic
states are the same, hence, the potential is that for the initial state,
V(R), and exp [i($\mathrm{\vec{K}}_{0}$ - $\mathrm{\vec{K}}_{\mathrm{f}}$) $
\cdot \mathrm{\vec{R}}$] = exp (- i$\Delta \mathrm{\vec{p}}$\textrm{\ }$
\cdot \mathrm{\vec{R}}$/$\hbar $), yielding the results in Eq. (2A.10). 

For a fast collision of an incident ion with a neutral (e.g. Bethe and
Jackiw 1968; Inokuti et al. 1981) Bethe approximated the cross section above
as

\begin{equation}
\mathrm{d\sigma_{0 \rightarrow f}^{(1)} \approx \frac{2\pi (Z_A e^2)^2}{m_e v^2}
\frac{dQ}{Q2} Z_B \mid M_{0f}(Q)\mid^{2},}
\end{equation}
where M$_{\mathrm{0f}}$(Q) is the matrix element of the ion interaction with 
an electron on the atom. This is often written in terms of

\begin{equation}
\mathrm{F}_{\mathrm{0f}}(Q) = [(\varepsilon_{\mathrm{f}} - \varepsilon_{0})/
\mathrm{Q}]\cdot \mathrm{Z}_{\mathrm{B}}\mid \mathrm{M}_{0\rightarrow\mathrm{f}}
(\mathrm{Q})\mid^{2},
\end{equation}
where F$_{\mathrm{0f}}$(Q) is called the generalized oscillator strength of 
the electrons on the atom (e.g., by analogy with model in Appendix 2C) and 
Q = $\Delta$ p$^2$/2m$_{\mathrm{e}}$. Here $\Delta $p is not the momentum
transfer to a single electron, as in the BEA method, but to the atomic
system B as a whole. Hence, even for very small momentum transfers those
electronic transitions which require significant internal energy changes
\textit{can} occur, albeit with small probability. The generalized
oscillator strength has the property that

\begin{equation}
\sum\limits_{\mathrm{f}}\mathrm{F}_{\mathrm{0f}}(\mathrm{Q}) = \mathrm{Z}_
{\mathrm{B}}. 
\end{equation}
Therefore, the \textit{average} effect of all the electronic transitions 
is like the BEA. 

For large Q, Bethe showed that

\begin{equation}
\mathrm{| M_{0\rightarrow f}(Q)|^2 \approx \begin{cases} 0, & 
\varepsilon_f^0 - \varepsilon_0^0 \neq Q \\
1, & \varepsilon_f^0 - \varepsilon_0^0 = Q 
\end{cases}} 
\end{equation}

giving the classical BEA result discussed earlier. This means that the 
momentum transferred to B is, with high probability, equal to that 
transferred to a single electron raising it to an excited state. At low 
Q, on the other hand, F$_{\mathrm{0f}}$(Q) is approximately the 
\textit{dipole} oscillator strength which determines the polarizability 
of an atom. Using these expressions to calculate the stopping power, one 
finds the remarkable result that the form of the stopping power is the 
same at high and low Q. Therefore, the total, given in Eq. (2.32), is 
about twice the BEA result in Eq. (2.31 b).

For the impact parameter formulation of the cross section in Eq. (2.7) the
calculation of the transition probability is straightforward. The wave
equation is treated as a time-dependent equation due to the motion of the
nuclei, i.e., $\mathrm{\vec{R}}$(t), and one often uses the straight line
trajectory [R$^{2}$(t) $\approx $ v$^{2}$t$^{2}$ + b$^{2}$]. When only two
states are involved, the wave function is written as

\begin{equation}
\mathrm{\psi}(t) \approx  \mathrm{C}_{0}(t)\mathrm{\psi}_{0} + 
\mathrm{C}_{\mathrm{f}}(t)\mathrm{\psi}_{\mathrm{f}},
\end{equation}
where $\mathrm{\psi}_{0}$ and $\mathrm{\psi}_{\mathrm{f}}$ correspond to 
the separate states associated with $\varepsilon _{0}^{0}$ and $\varepsilon_
{\mathrm{f}}^{0}$ described earlier and the C are coefficients which give 
the likelihood of being in either state as a function of time. Now the wave 
equation, i$\hbar \partial\mathrm{\psi}$/$\partial $t = H$\mathrm{\psi}$, 
can be reduced to two coupled equations for the coefficients,

$$
i \hbar \frac{\mathrm{\partial \mathrm{C}}_{0}}{\mathrm{\partial t}} = 
\mathrm{C}_{0}\varepsilon_{0}^{0} + \mathrm{C}_{\mathrm{f}} \mathrm{V}_
{\mathrm{f0}} 
$$

\begin{equation}
i \hbar \frac{\mathrm{\partial \mathrm{C}}_{\mathrm{f}}}{\mathrm{\partial t}} = 
\mathrm{C}_{\mathrm{0}} \mathrm{V}_{\mathrm{f0}} + \mathrm{C}_{\mathrm{f}}
\varepsilon_{\mathrm{f}}^0,
\end{equation} 
where V$_{\mathrm{f0}}$ is an interaction potential [e.g., $
\Delta \varepsilon $ of Eq. (2.28)]. These equations can be numerically
integrated to obtain C$_{0}$ and C$_{\mathrm{f}}$ at t $\rightarrow $ $
\infty $ (e.g., McGrath and Johnson 1989). However, simple expressions are
often useful.\\

If at t $\rightarrow$ -$\infty$, C$_{0}$(t) = 1 and C$_{\mathrm{f}}$(t) =
0, then the transition probability in Eq. (2.7) is

\begin{equation}
\mathrm{P}_{\mathrm{0}\rightarrow \mathrm{f}} = \mid \mathrm{C}_{\mathrm{f}}
(\infty)\mid^2.
\end{equation}
The Born approximation is the case $\mid C_{\mathrm{f}}(\textit{t})\mid 
\ll$ 1, so that Eq. (2D.8) yields

\begin{equation}
\mathrm{P_{0\rightarrow f} \approx \left|\frac{1}{\hbar }
\int\limits_{-\infty}^{\infty} V_{f0} e^{i\omega_{f0}t}dt\right|^2,} 
\end{equation}
where $\hbar \omega_{\mathrm{f0}}$ = $\varepsilon _{\mathrm{f}}^{0}$ - $
\varepsilon _{\mathrm{0}}^{0}$. Using a straight line trajectory this is
also equivalent to the interaction with a classical oscillator in Appendix
2C. Equation (2D.10) used in Eq. (2.7) also yields the Bethe-Born result
above for the integrated cross sections. \\

For strong coupling C$_{\mathrm{f}}$ is not small. Equation (2D.8) can
describe charge transfer when the momentum change of the electron is small.
For symmetric resonance charge transfer $\varepsilon _{0}^{0}$ = $
\varepsilon _{\mathrm{f}}^{0}$ and the solutions to Eq. (2D.8)
give

$$\mathrm{ P_{0\rightarrow f} = sin^2\Delta\eta_{0f}}$$ 

\begin{equation}
\mathrm{\Delta \eta_{0f} = \frac{1}{\hbar} \int\limits_{-\infty
}^{\infty } V_{0f}dt.}
\end{equation}
Using $\Delta \varepsilon $ in Eq. (2.28) for V$_{\mathrm{f0}}$, $\sigma_
{\mathrm{0}\rightarrow \mathrm{f}}$ can be integrated numerically. Firsov 
(e.g. Olson 1980) usefully approximates the integral by defining 
b$_{\mathrm{ct}}$ to be that value of b for which $\Delta \eta_{\mathrm{0f}} 
\sim 1/\pi$. This becomes, roughly,

\begin{equation}
\mathrm{\Delta \varepsilon (b_{ct})(2\pi b_{ct}\bar{a})^{1/2}/\hbar v \sim 1/\pi.}
\end{equation}
Using the form for $\Delta \varepsilon$ in Eq. (2.28) gives estimates of 
b$_{\mathrm{ct}}$, and $\sigma_{\mathrm{0}\rightarrow \mathrm{f}}$ for Eqs.
(2.35) and (2.34). Smirnov (1964) gives a general very accurate expression
for this cross section. \\

When $\varepsilon _{0}^{0}$ $\neq $ $\varepsilon _{\mathrm{f}}^{0}$
numerical integration is required to obtain P$_{\mathrm{0}\rightarrow
\mathrm{f}}$. However, the general form P$_{\mathrm{0}\rightarrow \mathrm{f}%
} $(b) $\approx $ 2P$_{\mathrm{0}\rightarrow \mathrm{f}}$ sin$^{2}\Delta
\eta _{\mathrm{0f}}$ has been rued by a number of authors to calculate $%
\sigma _{\mathrm{0}\rightarrow \mathrm{f}}$. Expressions for crossing and
noncrossing states are given in the text.

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