\chapter{Chapter 1 Introduction}
\chapter{Chapter 2 Collision Physics}
\chapter{Chapter 3 Interaction with Surfaces}

\section{Introduction}
\label{sec:1}
\cite{monograph}.

The interaction of charged particles with matter is determined, for the most
part, by the atomic character of matter. Therefore, particle impact
quantities such as penetration depths scale very nearly with the atomic
(molecular) density of the material. This is true even for metals, for which
the electrons in the solid state show cooperative (plasma) effects absent in
gases and insulating solids. Notable exceptions are crystalline solids in
which channeling (small angle internal reflections) of incident charged
particles along open crystal axes or planes result in increased penetration
depths. In fact, except for low-energy incident charged partidles 
($<$ 100 eV/amu), target molecular species (e.g., CO$_2$) can be
thought of as a sum of their constituents (i.e., 1 C + 2O), referred to as
Bragg's rule. In the bsequent discussions we will ignore any special
properties so that all materials are treated as gases of a given density
when discussing incident particle penetration and energy deposition. The
material properties are introduced latter when describing the fate of the
energy deposited.

The slowing down of a fast particle in a target is described here by
collisions with radomly located atoms. The possible behavior of the incident
and target particles can be described using a Monte Carlo procedure to
simulate the random nature of the stopping process (Robinson 1981). However,
as one is generally incrested in the average behavior of large numbers of
incident and target particles, transport equations (e.g., Boltzmann's
equation) can be used to describe the energy and spatial distributions of
the incident and target particles (Sigmun( 1). In such equations the
randomly located target atoms are treated as a medium parameterized by the
number density of the material. These equations are often treated in two
regimes. At high energies when electronic excitation ad ionzation events
dominate the energy loss, the deflections of the incident particles are
small and the stopping can be thought of as continuous. That is, even though
the stopping power is an atomic quantity [see Eq. (2.11)], written in the
from (dE/dx) it suggests a continuous slowing-down process, very much like a
drag force [see Eqs. (2.12 and 2.14)]. However, for both low incident ion
energies, for which nuclear elastic collisions dominate and very high
velocities, at which nuclear reactions occur, the deflections need to be
incorporated.

In this chapter the physics and chemistry of the irradiation effects are
described. The penetration (or, in solids, implantation) depths and
distributions are discussed first. This is followed by a description of the
motions of the target particles produced by collisions with the incident
particle. The target particles set in motion are generally referred to as
secondaries or recoils. However, it is offen useful to distinguish between
particles struck by the incident particle, the primary recoils, and those
subsequently set in motion, the secondary recoils. Effects produced by the
incident and recoil particles involve the breaking of bonds, and therefore,
the possibility of forming new species and producing changes in the state of
a solid. In a solid, recoil particles near the surface may attain an
appropriate energy and direction of motion to escape from the surface as
ground-state neutrals, excited neutrals, or ions. This results in the
erosion of the surface, a process referred to as sputtering. Sputtering of
neutrals has historically been a means of producing a vapor from solids
having low vapor pressures, and the detection of the ejected ions has been a
means of material analysis referred to as SIMS (Secondary Ion Mass
Spectroscopy) (Benninghoven et al. 1982). Here, of course, we are interested
in how such processes modify materials exposed to plasmas in space and add
new species to the plasma.

In most of the following we will be discussing a solid with a surface. This
surface, however, marks only the point of origin of the cascade of events
produced by the incident particle. For particle bombardment of atmospheres
(planetary, satellite, or cometary) the ``surface'' is roughly that column
density of gas equal to a monolayer in the condensed phase ($\sim 10^{15}$
molecules/cm$^2$) called the exosphere (Chamberlain and Hunten 1987).
Therefore, many of the ideas developed apply to plasma ion interactions with
gravitationally bound gases.

\section{Particle Penetration and Implantation}

Using the stopping powers in the previous chapter, the distance traveled in
a target material (Fig. 3.1) by an incident particle can be calculated in
the continuous slowing-down approximation. In this approximation the
particles all travel the same average distance, called the mean range, 
\={R}, where

\begin{equation}
\mathrm{\bar{R}(E_A) = \int\limits_0^{E_A} dE/|dE/dx|}
\end{equation}
if the target material has a constant density. For an atmosphere of variable
density this is replaced by a mean column density \={N} traversed, using 
$\mathrm{\mid dE/dx\mid = n_B S(E)}$ in Eq. (2.11),

\begin{equation}
\mathrm{\bar{N}(E_A) = \int\limits_{0}^{E_A}dE/S(E), }
\end{equation}
where S(E) is the stopping cross section. Such distances are given in Fig. 
3.2 for a number of particles penetrating H$_{2}$O. The range is often given 
in g/cm$^2$ for fast particles as the mass density of a material is roughly 
proportional to the electron density. In this form the range is similar in 
all materials.

\={R} (or \={N}) given in Eq. (3.1) is the total distance traveled by the
incident particle and not the penetration depth, as the particles may be
deflected significantly at lower energies. In the electronic stopping regime
(see Fig. 2.4), and for heavy ions incident on a low-mass atomic target,
deflections can be safely ignored so that the mean range is also the mean
penetration depth. Whenever deflections are important or the energy losses
in a collision are not small compared to the initial energy, then Monte
Carlo calculations or transport equations (Ziegler et al. 1985) are required
to obtain the penetration depth as well as other properties associated with
the stopping of the incident particles. Although the transport equations are
simple conceptually, they can be difficult to solve. However solutions for
many of the quantities of interest have been accumulated, and references to
a number of reviews are given in the bibliography.

The mean projected range measured parallel to the incident velocity is
written as \={R}$_{\mathrm{p}}$ for the constant density case, or as a
column density, \={N}$_{\mathrm{p}}$. It can be obtained as a solution
to

\begin{equation}
\mathrm{\bar{R}_p (E_A) = [n_B\sigma]^{-1} + \int \frac{d\sigma}{\sigma}
\bar{R}_p (E_A - \Delta E) cos\theta (\Delta E)}
\end{equation}
(see also Appendix 3A). In Eq. (3.3) \={R}$_{\mathrm{p}}$ is seen to be 
equal to the mean-free-path for the first collision, [n$_{\mathrm{B}}
\sigma]^{-1}$ [see Eq. (2.2)], plus the distance traveled following that 
collision. In the first collision the particle has a probability 
(d$\sigma/\sigma$) of losing energy $\mathrm{\Delta E}$ and moving at an 
angle $\mathrm{\theta(\Delta E)}$ to the initial direction, when d$\sigma$ 
is the differential cross section [e.g., $\mathrm{ d\sigma = (d\sigma/d\Omega)
d\Omega}$ in Eq. (2.4) and following]. Therefore, the integral is an average 
over all possible distances traveled \textit{after} the first collision. On 
taking the small $\mathrm{\Delta E}$ limit, with cos$\theta \sim$ 1(i.e., 
no deflections), the projected range, \={R}$_{\mathrm{p}}$, becomes equal 
to the continuous slowing down calculation of the range, \={R}, in Eq. (3.1). 
In Fig. 3.3a and b we also show results for the penetration depth, \={R}$_
{\mathrm{p}}$, obtained by solving Eq. (3.3). When the incident particle 
momentum is large then \={R} $\approx$ \={R}$_{\mathrm{p}}$, but for incident 
electrons \={R} and \={R}$_{\mathrm{p}}$ begin to differ even at high 
velocities. Estimates of the column of mass penetrated, based on the 
discussion following Eq. (2.42), are $\mathrm{(\rho_B \bar{R}_p) \approx 
(E_A/30 MeV)^{1.78} (g/cm^2)}$ for protons (0.2 MeV $< \mathrm{E_A} < 400 
MeV)$ and $\mathrm{(\rho_B \bar{R}_p) \approx  4.6 x l0^{-6} (E_A / 1 
keV)^{1.76} (g/cm^2)}$ for keV electrons (Banks and Kockharts 1973). 
Paretske (1987) uses $\mathrm{\bar{R}_p \approx 40(E_A /1 keV)}$(1 + 0.5 
$\mathrm{(E_A/keV))}$ nm for $\mathrm{\rho_B = 1 g /cm^3}$ for electrons for 
E$_{\mathrm{A}} <$ 20 keV.

As energy is deposited all along the particle's path, \={R}/\={R}$_{\mathrm{p}}$ 
gives a rough estimate of the increase in the energy density deposited
in the material due to deflections. A secondary ion or electron can have a
relatively large ratio of \={R}/\={R}$_{\mathrm{p}}$ for stopping in the
material. This implies that it rattles around a lot before coming to a stop,
but does not travel very far from where it was produced. For a slow
secondary particle stopping in a diffuse medium made up of particles of the
same mass, \={R}$_{\mathrm{p}}$ $\sim $ 3.2 (n$_{\mathrm{B}}$ $\bar{\sigma}_{
\mathrm{d}}$)$^{-1}$ is obtained from Eq. (3.3) using a power law cross
section with n $\rightarrow $ $\infty $ and $\bar{\sigma}_{\mathrm{d}}$ is
the average diffusion cross section in Chapter 2 (R.E. Johnson 1982). That
is, a low energy particle of mass equal to the target atom mass penetrates
of the order of the three mean-free paths. (In a solid, ($\bar{\sigma}_{
\mathrm{d}}$)$^{1/2}$ for low energy atoms may exceed the target atom
spacing further limiting the penetration.)

Because of the statistical nature of atomic collision processes, the
incident particles do not, of course, all stop at \={R}$_{\mathrm{p}}$
(Biersack 1987). The distribution of the implanted particles, F$_{\mathrm{i}%
} $($\mathrm{\vec{p}}$, z), for particles of type i and initial momentum $%
\mathrm{\vec{p}}$ is also obtained from transport equations (Appendix 3A).
For fast ions, the implantation distribution is rougly a narrow gaussian
centered on \={R}$_{\mathrm{p}}$,. The width is referred to as the range
straggling (Winterbon 1986) and is a tabulated quantity (Winterbon 1975;
Ziegler et al. 1985). For low-energy particles the straggling becomes
comparable to the penetration depth and backscattering occurs. As most of
the backscattering occurs in the first few monolayers of the target, the
reflection coefficient can be estimated without solving the full transport
equation (Vukanic and Sigmund 1976). Experimental values of this coefficient
(fraction of energy or particles reflected) for a few materials and incident
ions are given in Fig. 3.4 for laboratory surfaces. In the low energy regime
power law and scaled ``universal'' potentials are often used to describe
elastic collisions (Winterbon et al. 1970) and the solutions for the range
distribution can be scaled using characteristic lengths and energies in
Appendix 2B.

\section{Secondary Particles and the Partitioning of Energy}

On traversing a region of the target material, an incident ion loses energy
by electronic ionizations and excitations as well as by nuclear collisions.
In losing this energy the incident ions produce a variety of secondary
particles. An ionization event can result in the production of an energetic
electron, and a nuclear collision may involve sufficient energy transfer to
set in motion an energetic target atom. These secondary particles, if
produced near the surface, can escape from the material. Otherwise they lose
their energy in the solid, much like the incident particle, producing
additional secondary electrons and atoms. This cascade of events initiated
by an incident ion can also be treated via transport equations. However,
when keeping track of a number of particle types, Monte-Carlo calculations
are often more reasonable. In Fig. 3.5, the ``tracks'' of ionizations and
excitations produced by protons and electrons in condensed H$_{2}$O are
shown. Although a 1 keV electron, for example, has a speed, hence ionization
efficiency, comparable to a 2 MeV proton, the distribution of excitation
energy deposited by protons and electrons in Fig. 3.5 are very different
because the electron is easily deflected, primarily by the target electrons,
whereas the proton is not. In order to determine the effects \textit{on the
solid} of these events, we first consider the partitioning of the energy
deposited.

Ignoring nuclear reactions, it is customary to divide the electronic energy
loss by an ion or electron into the mean energy lost to ionization (\={E}$_{%
\mathrm{i}}$), excitation (\={E}$_{\mathrm{ex}}$) and that going to
subexcitation electrons (\={E}$_{\mathrm{s}}$). Whereas the first two
quantities are clear, the latter is not. Those fast secondary electrons
(often called delta rays) produced by the ion lose energy rapidly to
electronic excitations contributing to \={E}$_{\mathrm{i}}$ and \={E}$_{%
\mathrm{ex}}$. The mean energy of the secondary electrons produced in H$_{2}$%
O or N$_{2}$ by a fast ion or electron is of the order of 50-60 eV. By
additional ionization and excitations, these electrons ``cool '' to energies
below the threshold of the to lowest \textit{electronic} excited state of
the atoms and molecules in the target. The electrons will then ``cool '' by
vibrational or rotational excitation of molecules and by elastic momentum
transfer collisions with atoms. (In a solid, rotational excitations and
elastic collisions are replaced by optical and acoustical phonon
production.) Therefore, \={E}$_{\mathrm{s}}$ is the energy carried by the
electrons that goes \textit{directly} into the atomic and molecular motions
(Platzman 1967; Andersen 1984).

For a nonrelativistic incident particle, the energy loss is
partitioned\smallskip

\begin{equation}
\mathrm{ E_A = N_i \bar{E}_i + N_{ex}\bar{E}_{ex} + N_i\bar{E}_S + \nu(E_A),
= \eta (E_A) + \nu(E_A)}
\end{equation}
where N$_{\mathrm{i}}$, and N$_{\mathrm{ex}}$ are the mean number of 
ionizations and excitations produced during the stopping of particles with 
initial energy E$_{\mathrm{A}}$. $\mathtt{\nu }$(E$_{\mathrm{A}}$) accounts 
for the elastic nuclear energy loss by the \textit{incident} particle, and 
$\eta $(E$_{\mathrm{A}}$) is the total electronic energy. In Fig. 3.6 values 
of $\mathtt{\nu}$ vs. ion energy are given.

The average energy expended by the incident ion that goes into ionization
(electron-hole pair) is W$_{\mathrm{e}}$ $\equiv $ $\eta $(E$_{\mathrm{A}}$%
)/N$_{\mathrm{i}}$, which is approximately E$_{\mathrm{A}}$/N$_{\mathrm{i}}$
for a fast ion or electron. A quite useful and remarkable result is that the
quantity W$_{\mathrm{e}}$ is similar in size for most types of ionizing
radiation (charged particles, X-rays, $\gamma $-rays) absorbed in a given
material as long as the incident energy is large compared to the ionization
energy and the average nuclear elastic energy loss, $\mathtt{\nu }$(E$_{%
\mathrm{A}}$), is negligible for the charged particles (Bischel et al.
1979). Measured values for ions are given in Table 3.1. Dividing W$_{\mathrm{
e}}$ by the ionization energy, I (or in condensed matter, by the band gap, 
E$_{\mathrm{g}}$), and subtracting one indicates that fraction of the energy,
beyond the minimum required, which is expended in producing an ionization
(or electron-hole pairs). In atomic gases, W$_{\mathrm{e}}$ is seen in Table
3.1 to vary from $\sim $1.7 to $\sim $1.8 times I, whereas in molecular
gases and condensed materials it varies from $\sim $ 2.3 to $\sim $ 2.5. The
larger values arise because secondary electrons lose energy more efficiently
to vibrational and phonon excitations than they do to elastic collisions
with gas-phase atoms.

Using the ionization cross sections given earlier, along with the stopping
powers, the amount of energy deposited per \textit{primary} ionization by an
incident fast charged particle is S$_{\mathrm{e}}$/$\sigma _{\mathrm{I}}$.
This quantity is about twice W$_{\mathrm{e}}$ (v $>$ v$_0$) indicating that, 
on the average, two ionizations result per ionization produced by the incident 
particle. That is, each secondary electron created by the incident ion produces, 
on the average, one additional ionization. Remarkably, this is also true for 
low band gap (2-5 eV) solids. Dividing Eq. (3.4) by E$_{\mathrm{A}}$, the 
fraction of the incident ion's energy which is lost to each process is obtained. 
For a variety of target materials these fractions are similar and nearly 
independent of velocity when $\mathtt{\nu}$(E$_{\mathrm{A}}$) is small. At 
lower ion energies this is not the case since energetic secondary electrons 
are not produced and the charge-exchange cycle is a source of electronic 
energy loss and ionization (Chap. 2). The fractions of energy expended by 
ionization, excitation, and subexcitation processes are given in Table 3.2 
for fast ions and electrons incident on an atomic and a molecular material.

For low energy, heavy ions nuclear elastic energy loss eventually dominates
the stopping (see Fig. 2.4). In a solid one often wants to know the number
of displaced atoms produced by the incident ion after the cascades have
dissipated. In crystalline materials such displacements are referred to as
defects. Because the removal of an atom from a site by a nuclear collision
has similarities with the removal of an electron from an atom by electronic
energy transfer, a quantity W$_{\mathrm{n}}$ is used by analogy with W$_{%
\mathrm{e}}$. W$_{\mathrm{n}}$ is the average nuclear elastic energy
expended per atom displaced by an ion stopping in a solid. That is, W$_{%
\mathrm{n}}$ is equal to $\mathtt{\nu }$(E$_{\mathrm{A}}$) in Eq. (3.4)
divided by the average number of displaced atoms produced by an incident
ion. If the minimum energy required for a displacement is E$_{\mathrm{D}}$,
then one finds, in parallel with the ionization process, that Wn $\approx $
2.5 E$_{\mathrm{D}}$ (Sigmund 1981). E$_{\mathrm{D}}$ is a quantity which is
much larger than the sublimation energy, U, of the solid because energy is
required for the lattice to accommodate the displaced atom in addition to
that energy required to remove the atom from its original site (E$_{\mathrm{D%
}}\approx $ 5U). For an atom ejected from the surface (i.e., sputtering)
then E$_{\mathrm{D}}$ approaches U. Such expressions (W$_{\mathrm{e}}$ and W$%
_{\mathrm{n}}$) allow one to quickly estimate the amount of alteration
(ionization or displacement) produced in a material by an incident charged
particle.

For molecular solids held together by van der Waals forces or hydrogen
bonds, additional effects occur. Since bonds between atoms within the
molecule are much stronger than the bonds between neighboring molecules, the
ions collisionally displace both whole molecules and molecular fragments but
with very different efficiencies. In addition, the \textit{electronic}
excitations produced by the ion \textit{can also result in atomic and
molecular displacements} (defect production )(Perez and Thevenard 1987), a
subject we will return to latter.

The number of recoil atoms set in motion by a particle of initial energy 
E$_{\mathrm{A}}$ slowing down in an atomic gas is determined from a transport
equation [Appendix Eq. (3A.3)]. One of the most useful results of linear
cascade theory (Thompson 1968; Sigmund 1969) is that the total number of
recoil atoms in the target set in motion with energy between E and E + dE by
a particle of enegy E$_{\mathrm{A}}$ is

\begin{equation}
\mathrm{ G(E_A, E) \approx \beta_n \nu (E_A)/E^2, \ \ \ for \ \ \gamma E_A 
\gg E \gg U.}
\end{equation}
This result applies to the power law potentials [power n, Eq. (2.19)], where 
$\mathtt{\nu }$(E$_{\mathrm{A}}$) is the net energy going into nuclear elastic
collisions as in Eq. (3.4), and $\gamma $ is determined from the masses of
the incident and target particles, M$_{\mathrm{A}}$ and M$_{\mathrm{B}}$,
Table 2.1. The value of $\beta _{\mathrm{n}}$ depends weakly on n in the
elastic collision regime. For a cascade of low energy recoils, n $%
\rightarrow $ $\infty $ roughly approximates an exponential interaction
(Sigmund 1969) giving $\beta _{\infty }$ = 6/$\pi ^{2}$ = 0.608, whereas for
n = 2, $\beta _{2}=$ 0.361. Integrating G(E$_{\mathrm{A}}$, E) for all
energies E $>$ E$_{\mathrm{D}}$ gives an estimate of W$_{\mathrm{n
}}$. That is, the number of displaced recoils is ($\beta _{\mathrm{n}}
\mathtt{\nu }$(E$_{\mathrm{A}}$)/E$_{\mathrm{D}}$), or W$_{\mathrm{n}
}\approx $ 1.64 E$_{\mathrm{D}}$ for low energy particles (n $\rightarrow $ $%
\infty $, and $\mathtt{\nu }$(E$_{\mathrm{A}}$) $\approx $ E$_{\mathrm{A}}$%
). Including collisions which lead to replacement of displaced atoms gives
the value 2.5 E$_{\mathrm{D}}$ discussed above.

\section{Spatial Distribution of Deposited Energy}

For a particle of energy E$_{\mathrm{A}}$ incident on a material at an angle 
$\theta _{\mathrm{A}}$ to the normal, the average energy of the particles at
a depth z, E(z), can be estimated using the continuous slowing down
approximation in Eq. (3.2). Writing

\begin{equation}
\mathrm{ \bar{N}(z) = cos\theta_A \int\limits_{E(z)}^{E_A} dE/S(E),}
\end{equation}
where \={N}(z) is the mean column density of material penetrated, E(z) is
obtained for a given \={N}(z). [For a constant density target, \={N}(z) = n$%
_{\mathrm{B}}$z.] Knowing E(z), then the average energy \textit{deposited}
at any depth z is given by dE/dx = $\sum$n$_{\mathrm{i}}$S$_{\mathrm{i}}$
(E(z)), where n$_{\mathrm{i}}$ is the number density of atoms of type i. In
Fig. 3.7a energy deposition profiles are shown for fast ions incident on an
organic material. For such ions it is seen that the region of highest
deposited energy density is at the end of the particle's path. Prior to
this, the ion loses energy gradually and the peak in the energy deposition
profile corresponds to the maximum in (dE/dx)$_{\mathrm{e}}$ in Fig. 2.4. If
the incident ion or electron is deflected significantly, transport equations
(Appendix 3A) have to be solved for the electronic energy, F$_{\mathrm{e}}$(E%
$_{\mathrm{A}}$, cos$\theta _{\mathrm{A}}$, z), or nuclear elastic energy, F$%
_{\mathrm{D}}$(E$_{\mathrm{A}}$, cos$\theta _{\mathrm{A}}$, z) deposited at
depth z by an incident electron or ion with initial energy E$_{\mathrm{A}}$
and direction of motion to the surface normal, cos$\theta _{\mathrm{A}}$.
The subscript D is traditionally used, as the incident particle produces
displacement damage (previous section) according to the amount of nuclear
energy deposited. Hence, the cascade contribution is often referred to as
the damage distribution. A profile for the nuclear and electronic energy
deposited per unit path length is shown in Fig 3.7b. Such a distribution
corresponds to the end of an ion path in Fig. 3.7a.

The energy loss distributions in Fig. 3.7 also give the average energy of
transmitted particles if the thickness of the target is less than 
\={R}$_{\mathrm{p}}$. Such estirmates are of interest for MeV cosmic-ray ions
penetrating an atmosphere, an interstellar molecular cloud, or an ice grain,
and for a plasma ion penetrating Io's exosphere. Because stopping is a
statistical process, the emerging particles do not all have the same energy.
In traversing a very thin layer, $\Delta $x, in which the likelihood of more
than one collision by an incident particle is small, the mean-squared width
in the measured energy loss to the target, $\Delta $E, is written

$$
\mathrm{ (\Delta E - \Delta \bar{E})^2 \equiv \Omega^2 = n_B\Delta x S^{(2)},} 
$$
where S$^{(2)}$ is called the energy straggling cross section [see Eq.
(2.33b), a tabulated quantity, and $\Delta $\={E} is the mean energy loss.

The radial extent of electronic energy loss \textit{about the path} of an
incident ion is determined by the maximum impact parameter which can lead to
an energy transfer and by the radial distribution of secondary particles.
Therefore, the ``track'' produced in a solid by fast incident ions has two
regions each containing about half of the deposited energy. (Brandt and
Ritchie 1974; Fano 1964). The radius of the inner track (``infra-track''),
determined by the Bohr adiabatic cut-off for excitations, goes roughly as r $%
\approx $ v/$\omega $ for fast ions (Chap. 2 and Appendix 2). When v becomes
small, then r is of the order of the atomic size, (i.e., only close
collisions produce excitations). The radius of the outer track
(``ultra-track'') is determined by the mean distance traveled by the fast
secondary electrons (``delta rays''). As the most energetic secondary
electrons are created in close collisions between the incident particle and
an electron, the radial distribution of energy deposited at large radii can
be estimated from the BEA (Chap. 2). It is found that it falls off as
approximately 1/r$^{2}$ for r $>$ v/$\omega$, as indicated by
the measurement in Fig. 3.8, where r is the radial distance from the path of
the ion. As the velocity decreases the track width decreases so that, below
the maximum in the stopping power (see Fig. 2.4), the energy density in the
track is larger for the same (dE/dx).

In the elastic collision regime (low E$_{\mathrm{A}}$) the transport
equations are solved for the deposited energy (damage) distribution
(Winterbon 1975). In Fig. 3.9 results for the projected range and damage
distributions are shown. These were obtained for a power law interaction, n
= 2, which is appropriate over the region of the nuclear stopping maximum.
The results for M$_{\mathrm{A}}$ = M$_{\mathrm{B}}$ in Fig. (3.9) are useful
for describing cascades initiated by a secondary atom. It is seen that the 
\textit{mean} penetration depth of the stopped particle occurs at a larger
depth than that at which the largest number of particles are stopped, and is
more than twice that depth at which most of the energy is deposited.
Therefore, most of the energy of low energy secondaries is deposited close
to the point of production of the secondary, and it is not unreasonable to
treat these low energy cascades as isotropic. The physical size of the
volume directly affected, of course, depends on the density of the medium.
Near the surface of an ice grain this region is of the order of 10 nm, while
near the exobase of an atmosphere it is of order of a few scale heights,
often hundreds of kilometers.

Extensive tables and graphical data have been accumulated for the quantities
discussed above and are listed in the references and bibliography. Because
(dE/dx) changes only slowly with the Z$_{\mathrm{B}}$ of the target
particles, the results given in the previous figures can be applied to
similar target molecules or mixtures of molecules. For example, CO$_{2}$ is
similar to 1.5 CO molecules and CO is like N$_{2}$. O$_{2}$ is also similar
to N$_{2}$ or two H$_{2}$O. etc. Silicates and organics can be thought of as
characteristic of more refractory solids, etc.

Up to this point we have not considered the fate of the energy deposited.
Clearly most of the energy eventually degrades to heat, but during the
degradation process permanent changes, beyond the displacements described
here, can take place. These alterations and their consequences are examined
in the following sections.

\section{Chemical Alterations in the Gas Phase}

Atmospheres are composed of atoms or molecules which have closed-shell
electronic systems and, hence, do not react easily. A principal effect of
ionizing radiations, such as UV photons, is the breaking of this
closed-shell structure, which is then followed by a series of chemical
reactions. These can produce compositional changes in the gas (e.g., CO$_{2} 
$ $\rightarrow $ O$_{2}^{+}$, CO in the Martian atmosphere: Stewart 1972).
Studies of the effect of photons or electrons absorbed in molecular gases
have a long history (Dalgarno 1988 ), and many of the results are applicable
to incident ions. We present here only a few useful results and refer the
readers to texts on atmospheres in the Bibliography.

A typical potential energy diagram for a molecule is shown in Fig. 3.10.
This is seen to consist of both bound state curves having minima and
repulsive curves (monotonically decreasing with increasing separation, R) as
discussed in Chap. 2 (see. Fig. 2.8 and accompanying discussion). Because
the times for producing electronic transitions ($\sim $ 10$^{-16}$s) are
short compared to the vibrational period of the nuclei ( 10$^{-12}$ to 10$%
^{-13}$ s), the nuclei remain very nearly stationary during a change in
electronic state (the Frank-Condon principle). For example, on excitation of
the ground state, X, to the state labeled $\gamma $ in Fig. 3.10, the
molecule goes from one bound state configuration to another. On the other
hand, for the transition from the X state to the state labeled $\alpha $,
the electronic structure changes from a bound state in which the nuclei are
shielded to one in which the nuclei are not fully shielded and is therefore
repulsive. Such an excitation will be followed by dissociation of the
molecule with the nuclei obtaining kinetic energy.

The excited bound state of a molecule will decay by emission of a photon
(luminescence) unless it is `crossed' by a repulsive potential (see Chap. 2)
in which case a nonradiative dissociation of the molecule will occur
[process (3) in Fig. 3.10]. For a large molecule a third possibility exists.
The electronic energy can be distributed among many internal (vibrational)
modes and eventually be dissipated by ejection of low energy species or by
IR photons. Because the non-radiative transitions are fast ($\sim $10$^{-12}$
to 10$^{-13}$ s) compared to photon emission (\symbol{126}10$^{-9}$ s), the
former is the preferred mode of decay. However, in a gas a sufficient number
of photons may be produced to modify the cornposition at some distance from
the ion path, as is the case for cosmic rays penetrating molecular clouds in
the interstellar medium (Sternberg et al. 1988).

On ionization of H$_{2}$, the H$_{2}^{+}$ molecule faces a similar set of
possibilities. Therefore, on absorbing electronic energy the H$_{2}$ gas
will rapidly relax to a gas containing H$_{2}^{+}$ , H$^{+}$, H, and H$_{2}$
all of which may be in excited states. As the excited and ionized species
are open-shell structures, they are also highly reactive. A number of
possible reactions are 

\begin{align} %%%the following doesn't work**
& \mathrm{ H_2^{+} + H_2 \rightarrow H_3^{+} + H \ \ \ \ \ \ \ \ \ \ \ \ a } \notag\\
& \mathrm{ H^{+} + H_2 + M \rightarrow H_3^{+} + M \ \ \ \ \ b } \notag\\
& \mathrm{ H + H + M \rightarrow H_2 + M \ \ \ \ \ \ \ \ \ c } \notag\\
& \mathrm{ H^{+} + H + M \rightarrow H_2^{+} + M \ \ \ \ \ \ d } \notag\\
& \mathrm{ H_2^{+} + e \rightarrow H + H* \ \ \ \ \ \ \ \ \ \ \ \ \ e } \notag\\
& \mathrm{ H_3^{+} + e \rightarrow H_2 + H* \ \ \ \ \ \ \ \ \ \ \ \ f } \notag\\
& \mathrm{ H^{+} + e \rightarrow H* + h\nu \ \ \ \ \ \ \ \ \ \ \ \ \ g }
\end{align}
With the exception of process g, the net effect of these
reactions is a continual reduction in the level of excitation of the gas
with a concomitant production of kinetic energy. The rates for these
reactions depend on the densities of the reacting species and the details of
the potentials. The atomic recombination process (g) requires emission of a
photon and is generally slow compared to processes (e) or (f). Therefore, in
relatively dense gases any H$^{+}$ produced will tend to go through
processes such as (b) or (c) in order to form a molecular ion, which will
then recombine rapidly with ``cool'' electrons [e.g., processes (e) and (f)]
(Bates 1979). In a very low density gas [e.g., diffuse interstellar cloud]
processes such as (g) become important.

If the exciting radiation is turned off, these reactions will continue until
the gas consists of neutral molecules in low-lying excited states or in the
ground state. The excited atoms and molecules unable to react further will
eventually emit radiation or lose their excitation energy collisionally (AB*
+ M $\rightarrow $ AB + M). Therefore, the total energy deposited by the
ionizing radiation eventually appears as photons or kinetic energy of motion
of the molecules.

In a gas of water molecules, more complex reactions can occur. For example,
following ionization the reactions below are possible

\begin{subequations}
\begin{align}
& \mathrm{ H_2 O^{+} + H_2O \rightarrow H_3O^{+} + OH } \notag\\
& \mathrm{ OH^{+} + H_2O \rightarrow OH + H_2O^{+} } \notag\\
& \mathrm{ H^{+} + H_2O + M \rightarrow H_3 O^{+} + M } \notag\\
& \mathrm{ H_2O^{+} + e \rightarrow OH + H } \notag\\
& \mathrm{ H_3O^{+} + e \rightarrow H_2O + H } \notag\\
& \mathrm{ OH + H + M \rightarrow H_2O + M } \notag\\
& \mathrm{ OH + OH + M \rightarrow H_2O_2 + M } \notag\\
& \mathrm{ \ \ \ \ \ \ \ \ \ \ \ \ \rightarrow H_2O + O + M } \notag\\
& \mathrm{ H + H + M \rightarrow H_2 + M } \notag\\
& \mathrm{ H_2O_2 + H \rightarrow 2H_2O }
\end{align}
It is seen from these reactions that, although there are many intermediate
products, the water molecule fragments tend to recompose H$_{2}$O. In the
above, any one of the products on the right-hand side may be in excited
states, and such channels as
\begin{align}
& \mathrm{ H_2O^* \rightarrow H_2 + O} \notag\\
& \mathrm{ \ \ \ \ \ \ \ \ \rightarrow  H + OH}
\end{align}
\end{subequations}
can occur by direct excitation or following electron recombination. In 
addition O$_2$ can be formed from the products in Eq. (3.8a),

\begin{align}
& \mathrm{ O + HO \rightarrow O_2 + H } \notag\\
& \mathrm{ H_2O_2 + OH \rightarrow H_2O + O_2H } \notag\\
& \mathrm{ O + O_2H \rightarrow OH + O_2 } \notag\\
& \mathrm{ H_2O_2 + H \rightarrow H_2O + OH } \notag\\ 
& \mathrm{ O_2H + OH \rightarrow O_2 + H_2O. }
\end{align}
At very high densities of excitation, or when the H$_{2}$ is ``lost'' (e.g., 
by gravitational separation in an H$_{2}$O atmosphere), then the reactions 
in Eq. (3.9) become favored (Yung and McElroy 1977). If O$_{2}$ is formed, 
it is relatively stable, and the gas remains partially decomposed into 
O$_{2}$ molecules and H$_{2}$ molecules. That is, unlike the example of an 
H$_{2}$ gas, which on ``cooling'' eventually reforms the H$_{2}$ molecule, 
it is possible in gases such as H$_{2}$O for ionizing radiation to produce 
stable alterations. In most carbon-containing gases and gas mixtures this is 
also likely. For example, in CO$_{2}$, both CO and O$_{2}$ are produced, 
and in methane, H$_{2}$ and C$_{2}$H$_{6}$ are produced. In such gases
polymerization can also occur for very high doses of radiation as seen in
Table 3.3 (Capone et al. 1983; Delitsky and Thompson 1987; Piscitelli et al.
1988) producing precipitates (Calcagno et al. 1983b; Khare et al. 1984; Sack
et al. 1988).

\section{Solid State Radiolysis}

Processes similar to those described above occur in a condensed gas solid
which, to first order, can be thought of as an extremely high density gas.
However, most of the chemical activity takes place close to the track of the
particle or in the spurs associated with fast secondary electrons because
the chemical species are not very mobile at the very low temperatures in
deep space. In the high excitation density regions of the track fragment
densities can be high so that reactions occur efficiently. Before
considering this, we note that at very low temperatures radicals can be
stored for long periods of time. That is, the reaction process remains
incomplete. On raising the temperature, the ``stored'' radicals can be made
mobile and the reactions proceed. For example, following the production by
extensive photoabsorption of a very high density of radicals in ices at low
temperatures, rapid warming can cause the irradiated material to ``explode''
(d'Hendercourt et al. 1986). This warming may be induced in grains in
molecular clouds by heavy cosmic-ray nuclei (Leger et al. 1985) or
grain-grain collisions (d'Hendercourt et al. 1985). Therefore, the molecular
species originally condensed on the grain may be released into the gas phase
chemically altered.

Quantities, similar to the W values discussed earlier, are often used to
describe the number of new species produced in liquids and solids by
radiation. For example, G$_{\mathrm{i}}$ = number of species of type i
produced per 100 eV of ionizing radiation energy deposited in a solid (Magee
and Chatterjee 1987) which is, roughly, the inverse of W. In the early days
of solid state radiolysis the changes induced in an ice by ionizing
radiations were studied by subliming the solid after irradiation and
collecting the new molecular species produced (Hart and Platzman 1961). This
simulates certain problems in astrophysics. For example, as an ice grain or
a comet approaches a sun, the irradiated, condensed material can release new
species on warming. Results for a few species and a polymer are given in
Table 3.3. All such values depend on the \textit{temperature} of the solid
during irradiation.

A similar set of experiments has been performed for implantation of
energetic ions in various low-temperature frozen gases. For example, low
doses of C atoms in H$_{2}$O at 77 K result in the production of CO
molecules, as does carbon from the solar wind implanted into the lunar rock
(Rocard and Bibring 1982). Energetic ions may be implanted from an external
source or from nuclear reactions induced in the solid (Wolf and R\"{o}ssler
1987; Matsuura 1984). Energetic protons on H$_{2}$O initiate a reaction [ $%
^{16}$O(p, $\alpha $pn) $^{11}$C or $^{16}$O(p, $\alpha $)$^{13}$N] which
results in small quantities of CO and NO in the ice (R\"{o}ssler et al.
1984).

With the development of accurate spectroscopic and electron spin resonance
techniques, measurements of species \textit{in} the solid have also been
made. Radical species, such as H and OH, have been identified in irradiated
ice at very low temperatures (140 K). As the temperature is increased, the
disappearance of these species has also been followed (Hart and Platzman
1961). In ice, as in the gas phase, it is found that water molecules tend to
reconstitute from the dissociated species. On the other hand, if the density
of excitation is high, as in the vicinity of an ion or secondary electron
(delta-ray) track, water has been found to decompose into H$_{2}$ and O$_{2}$%
. In the solid the H$_{2}$ aggregates in the defects (bubbles) formed in the
ice by irradiation, and at a vacuum surface H$_{2}$ is lost readily even at
quite low temperatures. The loss of H$_{2}$ and formation of O$_{2}$ is
similar to processes occurring in an H$_{2}$O atmosphere.

In all condensed gases containing carbon, carbon-carbon bonds form from
dissociated fragments, and in addition CO is a relatively stable product if
oxygen is available (Venkatesan et al. 1987). On irradiation of methane in a
vacuum, H$_{2}$ is formed in the defects and at the surface, as in water
ice, and eventually exits the material (Lanzerotti et al. 1987b). This
leaves behind a polymeric material which is depleted in hydrogen, as shown
in Fig. 3.11. On extensive irradiation or warming, a residue remains which
will be discussed latter. Such residues are also formed on extensive
irradiation of sulfur containing condensed gases such as H$_{2}$S, CS$_{2}$
(Boring et al. 1985) and SO$_{2}$ (Moore 1984; Melcher et al. 1982). In
ammonia ice, H$_{2}$ and N$_{2}$ are formed (Haring et al. 1983, 1984b), in
CO$_{2}$ ice, CO and O$_{2}$ are formed as stable species (Brown et al.
1984), as in the gas phase, and in CO ice, CO$_{2}$ and O$_{2}$ are formed
(Chrisey et al. 1986a, 1989; Haring et al. 1983, 1984a), although very
inefficiently, as well as carbon suboxide (Chrisey et al. 1989). Even in
condensed S$_{8}$ polymeric sulfur is produced, along with fragments (e.g., S%
$_{2}$) (Torrisi et al. 1988; Chrisey et al. 1986b).

Because molecules dissociate, producing energetic fragments, ``hot'' atom
reactions can play an important role in the solid phase (R\"{o}ssler et al.
1984). An example of this is the formation of formaldehyde in irradiation of
a H$_{2}$O, CO$_{2}$ mixture (Pirronello et al. 1982; Haring et al. 1984a).
Pirronello 1985 uses a G $\sim $ 0.004. By using isotopically labeled
species, Pirronello et al. (1982) found that D$_{2}$O$^{16}$ + C$^{13}$O$%
_{2}^{18}$ produced D$_{2}$C$^{13}$O$^{16}$, where the carbon formed the new
species by hot atom insertion into D$_{2}$O$^{16}$. Such reactions can be
important in the solid phase because of the proximity of the species
(Matsuura 1986). Continued irradiation can then lead to a polymeric form of
formaldehyde.

Because the irradiated layers are altered chemically, their spectra can
change dramatically. In some cases, films which are relatively inactive in
the IR before irradiation can exhibit distinct IR features. Irradiated SO$%
_{2}$, of interest at Io, showed SO$_{3}$ in the spectra (Fig. 3.12a) and
irradiation of a hypothetical cometary ice mixture (60\% H$_{2}$O + 10\% NH$%
_{3}$ + 20\% CO + 10\% CH$_{4}$) by MeV protons exhibited a number of
synthesized species (Fig. 3.12b). The spectrum of irradiated ice exhibits a
slight amorphous to crystalline changes due to local agitation of the
lattice as well as the changes in the band structure seen in Fig. 3.13.
Irradiated organics have strong C-H absorptions near 3.5 $\mu $m at low
doses (Fig. 3.14) and eventually become black (Calcagno et al. 1986; Khare
et al. 1989) on long-term bombardment.

We now turn our attention to the removal of material from the surface of the
solid. For radiations with small absorption cross secf\'{\i}ons (e.g., $%
\gamma $-rays, cosmic-ray particles, visible photons, etc.) surface
processes are not important. However, for radiations which have large cross
sections (e.g., UV photons and ions and electrons at velocities $<$ 10 v$_0$), 
the surface will play a role in determining the material alterations.

\section{Sputtering of a Solid}

\subsection{\textit{Nuclear Elastic Collisions}}

Plasma ions incident on a surface can eject atoms and molecules into the gas
phase. This can occur by an increase in the surface temperature enhancing
sublimation. The ejection rate for this process depends non-linearly on the
ion flux, so that at low fluxes considered in Chapter 1 it can be ignored
for a solid target. Of more interest to the fate of surfaces in space is
sputtering. For this process the ejection rate is determined by the energy
deposited by each ion separately, so that the flux of sputtered surface
material is \textit{proportiona}l to the incident ion flux. That is, the
number of target particles ejected per incident ion, called the yield, Y, is 
\textit{independent} of the incident ion flux.

An incident ion collides with atoms near the surface, initiating cascades of
collisions between target particles as shown in Fig. 3.15. For an ion
incident normal to the surface, the struck atoms are directed inward. After
a number of collisions, however, a recoil can be produced near the surface
which has momentum directed outward and, therefore, may exit from the solid.
These multiple collision events are called sputtering when the energy of the
recoils is greater than the background thermal energy. Sputtering due to
cascades of collisions is described below. Superimposed on this are prompt,
``single collision'' ejection processes in which the primary recoil (struck
by the incident ion) is itself ejected (Kelly 1987). This process is
generally a small contribution, but becomes important at grazing incidence,
near threshold, and for thin targets. Finally, at latter times, quasithermal
processes occur during energy dispersal.

\subsection{\textit{Cascade Sputtering}}

When the density of primary recoil atoms set in motion by an incident ion is
small, these recoils deposit their energy and interact with other atoms in
the target independently. If these independent cascades occur close to the
surface, target particles can be displaced from the solid, i.e., sputtering
of the surface occurs. The average number of sputtered particles is
determined by contributions from all of those primary recoils produced close
to the surface (see Fig. 3.15). The yield is then determined as a ratio
between two lengths 
%%%(3.10)
\begin{subequations}
\begin{align}
& \mathrm{ Y \approx \sum\limits_{i}\bar{\Delta z_s}(T_i)/\lambda_c (T_i).} 
\end{align}
Here $\bar{\Delta}$\={z}$_{\mathrm{s}}$ is a `sputter depth'
for a cascade produced by a primary recoil of energy T$_{\mathrm{i}}$ and 
$\lambda_{\mathrm{c}}^{-1}$ is average number of such recoils produced per
unit path length by the incident ion. The ratio is evaluated in Appendix 3B.
Here we note that an estimate is easily obtained using quantities already
discussed. The average $\lambda _{\mathrm{c}}^{-1}$ is proportional to
(dE/dx)$_{\mathrm{n}}$/cos$\theta _{\mathrm{A}}$ divided by a mean energy
transfer, \={T}, where $\theta _{\mathrm{A}}$ is the angle of incidence. As
low energy recoils can have a mean projected range, \={R}$_{\mathrm{P}}$ 
$\approx $ 3.2(n$_{\mathrm{B}}\bar{\sigma}_{\mathrm{d}}$)$^{-1}$ then 
$\bar{\Delta}$\={z}$_{\mathrm{s}}$ is proportional to (n$_{\mathrm{B}}
\bar{\sigma}_{\mathrm{d}}$)$^{-1}$ times the number of `displacements' 
near the surface, which is proportional to \={T}/U (see discussion following 
Eq. (3.5)). Writing (dE/dx)$_{\mathrm{n}}$ = n$_{\mathrm{B}}$S$_{\mathrm{n}}$ 
(Chap. 2), yield has the form 

\begin{align}
&\mathrm{ Y \approx C \frac{S_n}{U\bar{\sigma}_d}(cos \theta_A)^{-1}.}
\end{align}
\end{subequations}
This is seen to be independent of the number density of the target, as both 
lengths depended on the material density. (Thompson 1968; Sigmund 1981). 
Hence, this \textit{form} for the yield applies whether we consider the 
sputtering of a solid surface, in which the binding, U, is produced by the 
atomic or molecular forces, or of an atmosphere in which gravitational 
binding limits escape (Haff et al. 1981a). The proportionality constant, 
C, in Eq. (3.10b) is obtained in the following from the point of view 
outlined by P. Sigmund (1969, 1981).

A determination of the flux of energized particles across the surface
requires the solution of the transport equation for the flow of moving atoms
in a material. However, the number of recoil atoms set in motion by an
incident particle is directly proportional to the nuclear elastic energy
deposited so that the sputter yield has the very general form\smallskip

\begin{equation}
\mathrm{ Y(E_A, cos\theta_A) \approx F_D(E_A, cos\theta_A, 0) \Lambda.}
\end{equation}
Here F$_{\mathrm{D}}$(E$_{\mathrm{A}}$, cos$\theta_{\mathrm{A}}$, 0) is 
the net elastic collision energy deposited per unit path length at
the surface (Fig. 3.7b) by an incident ion of energy E$_{\mathrm{A}}$ and
direction of motion to the surface normal indicated by cos$\theta _
{\mathrm{A}}$. $\Lambda $ is a sputter efficiency which is estimated as 
follows. Using Eq. (3.5), the number of recoils set in motion is proportional 
to the collisional energy deposited. Therefore, the mumber of recoils 
\textit{per unit depth} set in motion in the surface region with energy 
between $\varepsilon^{\prime}$ and $\varepsilon ^{\prime}$ + d$\varepsilon
^{\prime}$ is

\begin{equation}
\mathrm{ \beta_n F_D (E_A, cos\theta_A, 0)/\epsilon^{{\prime}2}.} 
\end{equation}
In order for one of these recoils to escape from the material, it must have 
the appropriate energy and direction of motion to be able to overcome the
binding, and it must make no further collisions while exiting. The latter
criterion means that the particle no longer participates in the cascade of
collisions initiated by the incident ion. For a recoil of energy $%
\varepsilon ^{\prime }$ and direction of motion cos$\theta ^{\prime }$ to
the surface normal, we write the probability of escape as P$_{\mathrm{es}}$ 
($\varepsilon^{\prime}$ , cos$\theta^{\prime}$) and the mean path length
for escape as $\mathrm{\lambda}_{\mathrm{d}}$($\varepsilon^{\prime}$
)cos$\theta ^{\prime }$. Here $\mathtt{\lambda }_{\mathrm{d}}$($\varepsilon
^{\prime}$) is the mean free path for an energy transfer collision by a
recoil particle.

The number of particles ejected is obtained by multiplying the number of
recoils per unit depth set in motion in Eq. (3.12) by the escape depth and
the probability of escape. Assuming the recoils move isotopically in the
solid, a very rough assumption (see Fig. 3.9b), and using $\beta_{\infty}$
= 6/$\pi^2$ for the low energy recoils, the contribution to sputter yield
from energies $\epsilon^{\prime}$ to $\epsilon^{\prime}$ + d$\epsilon^{\prime}$ 
and angular region d$\Omega^{\prime}$ is %%%(3.13)

\begin{subequations}
\begin{align}
& \mathrm{ \lbrace\frac{6}{\pi^2}\frac{ F_D(E_A,cos\theta_A,0)}
{\epsilon^{\prime 2}}d\epsilon^{\prime}\frac{d\Omega^{\prime}}{4\pi}\rbrace 
P_{es}(\epsilon^{\prime},cos\theta^{\prime})\lambda_d(\epsilon^{\prime}) 
cos\theta^{'}. } 
\end{align} 
The factor $\mathrm{d}\Omega^{\prime}/4\pi$ accounts for the assumed 
isotropic distribution of recoils. The total yield is obtained by integrating 
over energy and angle, and $\lambda_{\mathrm{d}}$ is replaced by an averaged
value $\lambda_{\mathrm{d}}$. In determining P$_{\mathrm{es}}$ the forces
at the surface of a solid are often represented by a planar barrier of
energy U if the atomic binding is large and the yields small, as for
refractory solids (Thompson 1968). Therefore,

\begin{align}
& \mathrm{P_{es}}(\epsilon^{\prime},\mathrm{cos}\theta^{\prime}) = 
\begin{cases} 1, & \epsilon^{\prime}\mathrm{cos}^2\epsilon^{\prime} \ge U,
                   \mathrm{cos}\theta^{\prime}\ge 0, \\
              0, & \mathrm{otherwise}. 
\end{cases}
\end{align}
On integration of Eq. (3.13a) subject to the conditions in Eq. (3.13b), 
the sputter factor in Eq. (3.11) is 

\begin{align}
 & \mathrm{ \Lambda_p = \frac{3}{4\pi^2}\frac{\bar{\lambda_d}}{U},}
\end{align}
\end{subequations}
where the p implies planar binding condition. In obtaining this expression the 
upper limit on the energy of the recoils, $\epsilon^{\prime}$ = $\gamma$
E$_{\mathrm{A}}$ (Chap. 2), was assumed to be much larger than U.

If the surface is very rough on an atomic scale (Sigmund 1969), if the
yields are large as for condensed gases (O'Shaughnessy et al. 1988b), or if
sputtering is from an atmosphere (McGrath and Johnson 1987), then
``spherical'' binding conditions are more appropriate. That is,

\begin{subequations}
\begin{align}
& \mathrm{P_{es}}(\epsilon^{\prime}, \mathrm{cos}\theta^{\prime}) = 
 \begin{cases} 
   1, & \varepsilon^{\prime} \ge U, \quad \mathrm{cos}\theta^{\prime}\ge 0, \\
   0, & \mathrm{otherwise}. 
 \end{cases}
\end{align}
giving
\begin{align}
& \mathrm{ \Lambda_s = 2\Lambda_p.} 
\end{align}
\end{subequations}
For \textit{macroscopically} \textit{rough} surfaces (e.g., a regolith of a
moon) the sticking of a sputtered species on a nearby surface and the
variety of incident angles will be considered in Chapter 4.

For the sputter ejection of many layers of a solid U is roughly equivalent
to the sublimation energy. The mean free path for an energy transfer
collision can be estimated $\lambda _{\mathrm{d}}$ $\approx $ $\varepsilon
^{\prime }$(d$\varepsilon ^{\prime }$/dx)$^{-1}$, where (d$\varepsilon
^{\prime }$/dx) is the stopping power of the recoils of energy $\varepsilon
^{\prime }$. As the ejection is dominated by low-energy particles the
average can be written using Eq. (2.14) as $\bar{\lambda}_{\mathrm{d}}$ $%
\sim $ (n$_{\mathrm{B}}\bar{\sigma}_{\mathrm{d}}$/2)$^{-1}$, where $\bar{%
\sigma}_{\mathrm{d}}$ is an average diffusion cross section. As the quantity
F$_{\mathrm{D}}$ in Eq. (3.13a) is also proportional to the number density n$%
_{\mathrm{B}}$, Sigmund writes F$_{\mathrm{D}}$ $\approx $ $\alpha $(dE/dx)$%
_{\mathrm{n}}$ = $\alpha $n$_{\mathrm{B}}$S$_{\mathrm{n}}$, where S$_{%
\mathrm{n}}$ is the nuclear elastic stopping power of the incident ion and $%
\alpha $ is a parameter which has to be determined. The yield now has the
``standard'' form for planar binding which is

\begin{equation}
\mathrm{ Y \approx \frac{3}{2\pi^2}\frac{\alpha S_n}{\bar{\sigma}_d U}
\approx 0.042\alpha S_n/(U \dot{A}^2),}
\end{equation}
in which Sigmund used $\bar{\sigma}_{\mathrm{d}}$ $\sim$ 3.6 A$^{2}$ for 
metals, which is of the order gas phase cross sections in the eV range for 
atoms (Yee and Dalgarno 1987). This result is like that in Eq. (3.10b), but 
more recent work suggests a smaller $\bar{\lambda}_{\mathrm{d}}$ (larger 
$\bar{\sigma}_{\mathrm{d}}),$ of the order of a monolayer of material when 
considering solids (Kelly 1987; Cui et al. 1988).

The factor $\alpha $ in Eq. (3.15) accounts for the fraction of S$_{\mathrm{n%
}}$ participating in the cascade of collisions near the surface as indicated
in Fig. 3.15. Therefor $\alpha $ accounts for: the dependence of F$_{\mathrm{%
D}}$ on angle of incidence; the fact that the cascades do not fully develop
at the surface and are not located at the point of production of the primary
recoil, the deflection of the incident atom in the surface region, and that
fraction of the energy of the secondaries lost to inelastic processes [$\eta 
$ in Eq. (3.4)] (Sigmund 1981).

Anderson and Bay (1981) have summarized the extensive data for incident keV
ion sputtering of metals and semi-conductors which have relatively large
binding energies $\sim $ 2-6 eV. In Fig. 3.16 are shown data for various
ions sputtering carbon (graphite) at normal incidence. Such data can be
organized using the expression in Eq. (3.15), in which case $\alpha $ is
determined by fitting the data. In Fig. 3.17 empirical values for $\alpha $
for keV ions at normal incidence are given for use with Eq. (3.15) with the
assumed $\bar{\sigma}_{\mathrm{d}}$ above. Quite remarkably, at normal
incidence, $\alpha $ is primarily a function of the mass ratio. As the
assumptions in the above model are very rough, the empirical values of $%
\alpha $ in Fig. 3.17 can be used with Eq. (3.15) to estimate yields (to
within a factor of two) when data is not available.

For the important molecular condensed gases (H$_{2}$O, CO, SO$_{2}$, S$_{8}$%
, etc) and for certain other interesting molecular compounds (e.g., Na$_{2}$%
S) (Chrisey et al. 1986a, b, 1988) Eq. (3.11) can also be used to estimate
the total yield at low temperatures to within a factor of two. The yield in
molecules removed per ion incident is obtained by using the material
cohesive energy (molecular sublimation energy), $\alpha $ from Fig. 3.17,
and dividing the expression in Eq. (3.11) by the number of atoms per
molecule. This is equivalent to using the atomic $\bar{\sigma}_{\mathrm{d}}$
as a size for each atom (roughly equivalent to replacing $\bar{\sigma}_{%
\mathrm{d}}\approx $ n$_{\mathrm{B}}^{-2/3}$ the molecular size). For
tightly bound \textit{molecular} materials (e.g., SiO$_{2}$, Fig. 3.16b)
with low yields, the above procedure also gives roughly correct yield values
when surface charging is controlled (Betz and Wehner 1983).

We have concentrated above on what is commonly referred to as back
sputtering: particles leaving a surface with motion opposite to the incident
ion. For thin samples (e.g., the Io corona, ice grains) sputtering is also
produced on transmission. The expression in Eq. (3.11) still applies, but F$%
_{\mathrm{D}}$ is the energy deposited per unit path length at the \textit{%
exiting} surface (i.e., $\alpha $ $\approx $ 1 if the ion has not lost much
energy). In this case the ``single collision'' ejecta, not included in Eq.
(3.15), play a larger role (Chap. 4).

The dependence of $\alpha $, hence the yield, on the angle of incidence is
shown in Fig. 3.18 for back sputtering, indicating that for most ion/target
combinations Y varies as a higher power of cos$\theta _{\mathrm{A}}^{-1}$ at
small angle (Oliva-Florio et al. 1987), going through a maximum and
decreasing again at large values of $\theta _{\mathrm{A}}$. This dependence (%
$\sim $cos$\theta _{\mathrm{A}}^{-1.6}$) is due to the forward directedness
of the cascade of collisions in the surface region (see Fig. 3.15; Sigmund
1969). The maximum (at $\sim $ 80$^{\circ }$) is associated with total
reflection from the surface and surface roughness. Of course, the planetary
regoliths and interplanetary grains have very ``rough'' surface (Chap. 4).

\subsection{\textit{High (dE/dx) Sputtering}}

The above discussion applies when the density of atoms set in motion is low,
as when light ions are incident. For high (dE/dx)$_{\mathrm{n}}$ the
cascades in Fig. 3.15 can overlap and an energized cylindrical region is
produced about the ion's track. The region so ``heated'' is offen referred
to as a ``thermal'' or ``diffusive'' \textit{spike} (Vineyard 1976). The
diffusion equations for dissipation of the energy in the spike are
considered in Appendix 3C. Ejection occurs, per incident particle, due to
the local energy density, $\varepsilon $, and its gradient produces a volume
force (Johnson et al. 1989b). The yield of atoms or small molecules \textit{%
subliming} from the cylindrical spike produced by a single ion is
written

\begin{equation}
\mathrm{ Y_s = \int\limits_{0}^{\infty}dt\int\limits_{0}^{\infty}
\pi dr^2\Phi [\varepsilon_s(r,t)/n_B U],}
\end{equation}
where $\Phi$ is the flux of atoms or molecules from a surface due to the 
surface energy density, $\varepsilon_{\mathrm{s}}$, which depends on the 
radial distance from the track of the ion, r, and the time, t, after the 
passage of the ion. Using the escape criterion [e,g., Eq. (3.13b) or 
(3.14a)] this yield is

\begin{equation}
\mathrm{ Y_s \approx c_s [l_s (dE/dx)_n /U]^2 g(x_0), \ \ x_0 = n_B U/
\varepsilon_s (0,0) }
\end{equation}
(Evatt and Johnson 1980; Sigmund and Claussen 1981), In this expression the 
constant c$_{\mathrm{s}}$ and the function g(x$_{0}$) depend on the local 
energy spectrum (Appendix 3C) and the escape criterion, and l$_{\mathrm{s}}$ 
is the average thickness of a monolayer of material: l$_{\mathrm{s}}$ $\approx$ 
n$_{\mathrm{B}}^{-1/3}$. Because the yield in Eq. (3.17) depends on the 
\textit{meterial density} via (dE/dx)$_{\mathrm{n}}$, spick effects are 
important only in a condensed medium.

The function g(x$_{0}$) in Eq. (3.17) depends on the initial energy density
at the track center, $\varepsilon $(0,0). For a hot narrow spick the yield
is quadratic in the stopping power as g(x$_{0}$) approaches unity for U $\ll 
$ $\varepsilon $(0,0). This implies that (dE/dx)$_{\mathrm{n}}$ $\gg $ Un$_{%
\mathrm{B}}$ $\pi $r$_{0}^{2}$, where r$_{0}$ is the initial mean radius of
the deposited energy distribution.

At large (dE/dx)$_{\mathrm{n}}$ a volume of material may be ejected due to
the rapid material expansion (pressure pulse) about the ion track (the
gradient of $\varepsilon $), leaving a small ctater (Carter 1983). This
means large whole molecules can be ejected into space from the solid state
(Sundqvist 1989). In this region the yield eventually becomes proportional
to (dE/dx)$_{\mathrm{n}}^{3}$ for fast ions (Johnson et al. 1989b). Because
random motion is detrimental to organic molecules, this mechanism will
dominate for large molecule ejection. Calculating a net yield over a broad
range of (dE/dx)$_{\mathrm{n}}$, it is common to sum the various
contribution. For metals sputtered by fission fragments with high (dE/dx)$_{%
\mathrm{n}}$, a nonlinear dependence on (dE/dx)$_{\mathrm{n}}$ is needed
measured (Andersen and Bay 1981). Because of the dependence on U, nonlianear
yield are also found for weakly bonded materials, such as the condensed gas
solids, even for light incident ions. However, because such materials are
not good conductors, the electronic contribution to the energy deposition is
also important, as discussed shortly.

\subsection{\textit{Theshold Yield}}

The behavior of the sputter yield at low ion bombardment energy is much less
certain. This region is of considerable interest for erosion of grains by
shocks in the interstellar medium (kimura 1962; Draine et al. 1983). Below
some incident ion energy (see Appendix 2A and 2B), the diffusion cross
section becomes a constant, and the nulear stopping cross section S$_{%
\mathrm{n}}$ = $\sigma _{\mathrm{d}}$ ($\gamma $ E$_{\mathrm{A}}$/2) [Eq.
(2.14)] is linear in E$_{\mathrm{A}}$. Therefore, the expressions in Eqs.
(3.10b) and (3.15) also become roughly lenear in E$_{\mathrm{A}}$ (Sigmund
1981). At very low energies, a threshold for ejecting a surface atom occurs
when the maximum energy transfer is of the order of the surface binding
energy. This can be approximated by replacing $\gamma $E$_{\mathrm{A}}$ by ($%
\gamma $E$_{\mathrm{A}}$-U) in the expression for yield. As the mean
projected range, \={R}$_{\mathrm{P}}$(E$_{\mathrm{A}}$), of the incident
particle also becomes comparable to the mean sputter depth $\bar{\Delta}$\={z%
}$_{\mathrm{s}}$ in Eq. (3.10a), the incident ion can be thought of as
creating a hot spot (Appendix 3C) about \={R}$_{\mathrm{P}}$(E$_{\mathrm{A}}$%
).

Anderson and Bay (1981) have summarized the data for the threshold regime.
In Fig. 3.19 are given the threshold energies, E$_{\mathrm{t}}$, vs. M$_{%
\mathrm{B}}$/M$_{\mathrm{A}}$. These approach U/$\gamma $ at large M$_{%
\mathrm{B}}$/M$_{\mathrm{A}}$, as discussed above, but approach U[8(4/$%
\gamma $)$^{1/3}$] at small M$_{\mathrm{B}}$/M$_{\mathrm{A}}$. Writing e$%
^{\prime }$ = E$_{\mathrm{A}}$/E$_{\mathrm{t}},$ with E$_{\mathrm{t}}$, from
Fig. 3.19, the semi-empirical expression for the yield applicable for e$
^{\prime }$ $<$ 10 is

\begin{subequations}
\begin{align}
& \mathrm{ Y(e^{\prime}) \approx Q(M_A, M_B, U) f_t(e^{\prime})} \notag\\
\end{align}
with
\begin{align}
& \mathrm{ f_t(e^{\prime}) \approx 0.0085(e^{\prime})^{1/4}(1-{e^{\prime}}^{-1})
^{7/2}.}
\end{align}
\end{subequations}
This exhibits a slower dependence on E$_{\mathrm{A}}$ than suggested above, 
and Q $\approx $ 0.75 M$_{\mathrm{B}}\gamma^{5/3}$ gives yields within a factor 
of 2 for most measured ion-target combinations, with M$_{\mathrm{B}}$ in amu.

\subsection{\textit{Energy Spectra}}

The sputtering regimes discussed above (separate cascades and cylindrical
spike) exhibit rather different velocity spectra also. The number of
particles ejected with energy between E and E + dE and angle to the surface
normal $\theta $ is written

\begin{equation}
\mathrm{ Y(E, cos\theta) = Yf(E, cos\theta).}
\end{equation}
Using the description of moving recoils in Eq. (3.5) along with the exit 
criteria, P$_{\mathrm{es}}$ in Eq. (3.13b) and Eq. (3.14a), the energy 
distributions for the independent cascades (low energy density) with planar 
or spherical binding are

\begin{subequations}
\begin{align}
& \mathrm{ f^p(E, cos\theta) \approx \frac{4UE}{(E+U)^3} cos\theta} 
\end{align}
\begin{align}
& \mathrm{ f^s (E, cos\theta) \approx \frac{2U}{(E+U)^2} cos\theta} 
\end{align}
normalized over the integration range (0 $\le$ cos$\theta$ $\le$ 1; 
0 $\le$ E $<$ $\infty$). These ex-pressions exhibit the characteristic 
1/E$^2$ dependence at high energies (Thompson 1968) based on the recoil energy 
distribution in Eq. (3.5) and observed in many experiments (Andersen and Bay 
1981). Depending on the angle of incidence a ``single collision'' component 
adds to this at large E and this component exhibits preferential ejection 
angles (Betz and Wehner 1981).

For the case of a high density of cascades, the calculated energy
distribution for the spike with cylindrical geometry and spherical binding
(Appendix 3C)
\begin{align}
& \mathrm{ f_s^s (E, cos\theta) = \frac{4U^2}{(E+U)^3} g(x_0, E) cos\theta, 
x_0 = n_B U/\varepsilon_s(0,0)}
\end{align}
\end{subequations}
gives a lower average energy. For a narrow spike for which g(0, E) $\rightarrow 
$1 as x$_{\mathrm{0}}$ $\rightarrow$ 0, the sputtered-particle energy dependence 
goes as 1/E$^{3}$ at large E. For a spike with a significant initial width, 
r$_{\mathrm{0}}$, the dependence on E at large E is much more rapid, decaying
roughly exponentially with E. At very high (dE/dx)$_{\mathrm{n}}$, rapid
ejection also occurs due to an impulsive expansion about the track again
modifying the energy spectra of the ejecta. This regime has been less
severely tested by experiment. Because of the rough additivity of the linear
and spike contributions to the yield [Eq. (3.17)], the forms above are often
used to fit experimental data with an effective U as a parameter (Reimann et
al. 1984; Boring et al. 1984a, b; Haring et al. 1984c).

\section{Sputtering of Multicomponent Materials}

For a material consisting of different atomic species (e.g., metal alloy or
molecular solid) the sputtering process is more complicated (Betz and Wehner
1983). For a porous mineral regolith, Hapke (1986) shows that the adsorption
of the ejected atoms on neighboring grains controls the net loss rate. That
is, sputtering of each species in their absorbed state and not their state
in the mineral is mportant.

In the spike regime for a laboratory solid the individual constituents will
evaporate independently according to their separate sublimation rates (Roth
1983). For a given average spike temperature, T, 

\begin{equation}
\mathrm{ Y_1^{\prime} /Y_2^{\prime} \approx (c_1/c_2)(M_2/M_1)^{1/2} 
exp[(U_2 - U_1)/kT].}
\end{equation}
Here Y$_1^{\prime}$ and Y$_2^{\prime}$ are the actual yields of the two 
species occurring with concentrations c$_{1}$ and c$_{2}$ [i.e., 
c$_{1}$ = n$_{1}$/(n$_{1}$ + n$_{2}$) where n$_{\mathrm{i}}$ is a number 
density]. It is seen that the more weakly bound and lightest species sputter 
more efficiently. (The binding energies are concentration-dependent, but 
using the separate U is generally sufficient.)

For a low cascade density, a number of detailed models have been published
in order to describe the sputtering of mixtures (Sigmund 1981; Betz and
Wehner 1983). When the masses are nearly equal, then the earlier expressions
apply, 
\begin{subequations}
\begin{align}
& \mathrm{ Y_1^{\prime}/Y_2^{\prime} \approx \frac{c_1 Y_1}{c_2 Y_2} \approx 
\frac{c_1 U_2}{c_2 U_1}} 
\end{align}
and
\begin{align}
& \mathrm{ Y_1^{\prime} \approx \frac{c_1}{U_1}[c_1 U_1 Y_1 + c_2 U_2 Y_2], } 
\end{align}
where the quantities U$_{\mathrm{i}}$Y$_{\mathrm{i}}$ are the yield times 
sublimation energy for the pure substance. For the nonequal mass case there 
is a weak dependence on mass,
\begin{align}
& \mathrm{ Y_1^{\prime} /Y_2^{\prime} = \frac{c_1}{c_2} \left(\frac{M_2}{M_1}
\right)^{2/n} \left(\frac{U_2}{U_1}\right)^{1-(2/n)},} 
\end{align}
where n is the power law dependence for the low energy recoil collisions. 
As stated earlier, the low energy collisions are reasonably well approximated 
by n $\rightarrow $ $\infty$, in which case Eq. (3.22c) is equivalent to Eq. 
(3.22a). However, small isotopic fractionation effects have been seen for 
elemental substances (U$_{1}$ = U$_{2}$) (Watson and Haff 1979). In minerals, 
isotopic fractionation of the heavy elements (Mg, Si, Ca, etc.) is such that 
the lighter isotope is lost preferentially in agreement with Eq. (3.22c), 
but for the lighter constituents (e.g., O) the heavier isotope is lost 
preferentially, at the few percent level in both cases (Haff et al. 1981b).

Because the relative yields of the separate constituents may differ, their
concentration in the surface region will eventually become different from
that in the bulk. After long-term irradiation, the surface concentrations
will become such that the species eject stoichiometrically according to
their \textit{bulk} concentrations. That is, as the material is modified
down to the full depth of penetration of the ions, removal of a small amount
of surface allows ions to penetrate into and modify an equivalent amount of
fresh material. Therefore, one is, in effect, always removing fresh
material, and the ratio of the yields approaches (see Eq. (4.1lb))

\begin{align}
& \mathrm{ Y_1^{\prime}/Y_2^{\prime} \rightarrow (c_1/c_2) bulk.}
\end{align}
\end{subequations}
After large fluences, the bulk concentrations can be used along with an 
effective (empirical) binding energy ($\sim$U1, $\sim$U2 $\sim$U$_
{\mathrm{eff}}$) to estimate the yield. Care must be taken in using such 
expressions as extensively bombarded surfaces are offen chemically altered 
in the penetrated region (Betz and Wehner 1983; Roth 1983). Therefore SiO$_2$ 
does not sputter as Si and O atoms, but rather as Si, O, SiO, O$_{2}$, and 
SiO$_{2}$, with the most volatile component (O$_{2}$) being lost preferentially.

In general, ion bombardment displaces atoms and, hence, would tend to
convert crystalline materials into amorphous materials down to the
penetration depth of the ion. However, the residual energy deposited by the
ion agitates the atoms and molecules in the solid. This results in ion beam
enhanced diffusion in which the more volatile species reach the vacuum
interface (Arnold and Mazzolli 1987), and are ejected as discussed earlier
(e.g., Na from silicates). It is also possible that in an amorphous material
the residual heat can be sustained long enough to form crystalline regions
if the thermal diffusivity of the material is low. Therefore, materials with
high thermal diffusivity will generally become amorphous under bombardment,
while those with low thermal diffusivity can form crystallites. Evidence for
the latter in ice was found (Benit et al. 1987). In addition, in molecular
materials such as SiO or SiO$_{2}$, the preferential loss of O (as O$_{2}$)
can result in the formation of Si crystallites within the solid after
long-term irradiation (Walters et al. 1988).

\section{Mixing, Adhesion, and Desorption}

When gases condense on a surface, at submonolayer to few monolayer coverage,
an incident ion may cause mixing with the surface or desorption of the
absorbed molecules, depending on how these molecules are struck by the ion.
For example, ``knock-on'' collisions of the ion and an atom of the absorbed
gas can drive the atom into the substrate, a process used to intentionally
modify surfaces. The surface can also be modified simply by bombarding it
with those ions (atoms) which one would like to implant. The incident
particles penetrate and stop in the material, as discussed earlier, changing
the character of the surface (Roth 1983), a process used in the electronics
industry and used to reduce surface corrosion. A region of the order of the
penetration depth of the ion will become a new mix of the original
components and the bombarding particles. If, in addition, the electronic
structure is disrupted, then reactions can be induced. If the new species
produced are volatile, they may be lost rapidly (chemi-sputtering).

These process occur also when an ion or an electron traverses the interface
between two materials (metals, semi-conductors, or insulators). What is
found experimentally is that the adhesion between two materials in physical
contact generally is increased by ion and electron bombardment (Werner et
al. 1983). This occurs when the adhesion forces are weak (van der Waals)
contact forces, and binding is enhanced by mixing and the formation of
covalent bonds between atoms on the two surfaces (Ingemarsson et al. 1989).
Such enhanced adhesion may be important, for example, in the formation of
Brownlee particles from cometary grains (see Fig. 4.6) for which the icy
mantles are lost (Johnson 1985b; Johnson and Lanzerotti 1986).

An incident slow ion can also eject absorbed surface gases. This desorption
can be described in the same manner as the sputtering process if account is
taken of the mass differences of the surface atoms and absorbed species. A
rough estimate of the cross section for desorption can be obtained from the
sputtering yield expressions just discussed. For \textit{submonolayer}
coverage

\begin{equation}
\mathrm{ \sigma_{De} \sim Y\bar{\sigma}_d \sim \frac{3}{2\pi^2} 
\frac{\alpha S_n}{U},}
\end{equation}
where U is the binding energy of the species absorbed on the substrate and 
$\bar{\sigma}_{\mathrm{d}}$ is assumed to describe the collision between 
substrate and adsorbed atoms. In this expression ($\alpha $S$_{\mathrm{n}}$) 
is calculated for the energy deposition in the substrate and the yield is 
given by ($\sigma_{\mathrm{De}}$N), where N is the coverage of the surface 
expressed as a column density. As N increases, both U and ($\alpha 
$S$_{\mathrm{n}}$) gradually become characteristic of a solid made up of the 
adsorbed species. For thicknesses greater than the sputter depth, the substrate 
does not play a role and the standard sputtering expressions apply.

\section{Electronically Induced Desorption}

Ionizing radiations (photons, electrons, and ions) have been shown to eject
molecules condensed on a substrate (Tolk et al. 1983; Brenig and Menzel
1985; Stuhlen and Knotek 1988). For submonolayer to few monolayer
thicknesses, the condensed gas is said to be desorbed by the radiation a
process of interest in interstellar clouds (Greenberg 1973). For thick
condensed-gas layers the ejection process is referred to as electronic
sputtering, discussed in the next section.

The electronic energy deposited in a solid by an ion, electron, or photon is
dissipated via the processes described earlier. Most of the energy is
eventually converted to heat and is finally radiated in the IR or thermally
conducted to a cooler region. Higher energy photons (luminescence) may also
be emitted because of electronic and vibrational transitions in a molecular
solid. In addition, ejection of atoms or molecules can result from the
changed electronic binding produced by the excitations. This occurs when the
electrons are excited to repulsive states or to states which relax to a
repulsive state as in Fig. 3.10. In such a process the electronic energy
deposited in the solid is converted to atomic motion (Menzel and Gomer 1964;
Redhead 1964). If the repulsive state is excited in the surface layer or
between a surface atom (molecule) and its substrate, then desorption can
occur. Depending on the nature of the electronic states excited, the
repulsive ejection can lead to the desorption of either an ion or a neutral.
In general, neutrals are the dominant ejected species, although most
experimental work has been on the ejection of the easier to detect ionic
species. The dominance of neutral species occurs because the ions are more
strongly attracted to the surface (e.g., polarization or image charges) and
ions exiting the surface can be neutralized by electron capture. In general,
the average repulsive energies transferred to the neutrals are small. For
example, for H$_{2}$O on TiO$_{2}$ (Avouris et al. 1986), OH is observed
with $\sim $ 0.25 eV, whereas ions are typically ejected with energies 
$>$ 1 eV.

A number of general statements can be made about the importance of the
substrate in the desorption process. Electronic energy deposited in the
metallic substrate is quickly dissipated, and in many instances the metallic
substrate acts as a sink for energy initially deposited in the adsorbed
layer, \textit{decreasing} the efficiency of the desorption process. On the
other hand, for an insulating substrate, electronic excitation produced near
the surface may result in a repulsive relaxation which can \textit{enhance}
the ejection of adsorbed species. Therefore electronically stimulated
desorption occurs much more efficiently on an insulating than on a metallic
substrate. Further, the different substrates can result in different ratios
of ions to neutrals ejected. Metallic surfaces, for example, can donate
electrons, neutralizing exiting ions. Such processes are very sensitive to
the nature of the surface and the adsorbed species (Madey 1986).

The desorption rate is expressed in terms of a cross section, as in the
previous section. For \textit{submonolayer} coverage, the yield per incident
ion, electron or photon is (N$\sigma _{\mathrm{De}}$). Roughly, $\sigma_{
\mathrm{De}}\approx \sigma _{\mathrm{ex}}$ P$_{\mathrm{D}}$ where 
$\sigma_{\mathrm{ex}}$ is the cross section for excitation to the repulsive
(desorbing) state and P$_{\mathrm{D}}$ is the probability of desorption.
This can be much less than one because of energy dissipation to the
substrate. For example, whereas valence excitation cross sections for 
$\sim$ 100 eV electrons are $\sim$10$^{-16}$ cm$^{2}$, cross sections for 
ion desorption for such electrons are $\gtrsim$ 10$^{-20}$ cm$^{2}$ and for
neutral desorption $\gtrsim$ 10$^{-18}$ cm$^{2}$ (Madey 1986). $\sigma_
{\mathrm{De}}$ for neutrals can approach the electronic excitation cross
section. For CO molecules adsorbed on a metallic surface by van der Waals
forces (physisorbed), the desorption cross section is close to the cross
section for electronic excitation of a free CO molecule to an internal
repulsive state (Leung et al. 1977), of the order of 10$^{-16}$ cm$^{2}$ for
electrons with energies of the order 50 eV. For low energy ions charge
exchange may also excite a surface repulsive state (Baragiola 1984).

An oxygen (hydrogen) atom attached chemically to a metallic surface
(chemisorbed) requires the excitation of the metal-oxygen (metal-hyrogen)
bond to a repulsive state. When either of these species is adsorbed on an
insulating surface, $\sigma _{\mathrm{De}}$ will be larger.
Photon-stimulated desorption cross sections are estimated in the same way
and are, therefore, much smaller than those for electrons and ions, because
UV cross sections for excitation to a repulsive state are smaller. For
desorption of ionic species, $\sigma _{\mathrm{De}}\sim $ $\sigma _{\mathrm{%
ex}}$ P$_{\mathrm{D}}$(1 - P$_{\mathrm{N}}$), where P$_{\mathrm{N}}$ is the
neutalization probability. Here $\sigma _{\mathrm{ex}}$ is the excitation to
a repulsive curve that separates to an ionic state (Brenig and Menzel 1985).
Not only is P$_{\mathrm{N}}$ close to one for many surfaces, but the
excitation cross section $\sigma _{\mathrm{ex}}$ is smaller than the cross
sections for valence electron excitations leading to neutral desorption.
Therefore, ion desorption cross sections are smaller, requiring more
energetic excitatation (e.g., shake-up excitations or inner shell
excitations). For example, for neutral CO on a metal substrate the
metallic-CO bond can be excited at about 5 eV, leading to desorption
(Feulner et al. 1981), whereas states leading to ion desorption are
typically $\sim $ 15 eV (Avouris and Walkup 1989).

\section{Sputtering of a Solid: Electronic Stimulation}

Electronic states excited in the solid by photons, electrons or ions can
result in the energetic repulsion of neighboring atoms, as exemplified by
desorption from and defect formation in insulators (Brown 1987). These are
the surface and bulk ramifications of the same processes. When the thickness
of the layer of adsorbed gases, N, increases beyond some depth, the
``substrate'' becomes unimportant and the ejection process is called
electronic sputteing. This occurs, for instance, following the production of
a track of electron-hole pairs produced by a fast ion see Fig. 3.20.

\subsection{\textit{Linear Yields}}

If a repulsive state is produced at the surface of a molecular solid, atoms
are generally ejected. If this state occurs below but close to the surface
of a molecular solid, the energized atoms produced can transfer their energy
collisionally to other target molecules, creating a mini-cascade of
collisions, as indicated in Fig. 3.20b, which can cause molecules to be
sputtered from a weakly bound solid. These low energy cascades can be
incorporated into our earlier description of sputtering. That is, the yield
is written as in Eq. (3.11),

\begin{subequations}
\begin{align}
& \mathrm{ Y \approx F_e (E_A, cos\theta_A,0)\Lambda_e,} 
\end{align}
where F$_{\mathrm{e}}$ is the electronic energy deposited in the surface 
regime (e.g., Fig 3.7b) and $\Lambda _{\mathrm{e}}$ the efficiency factor 
for electronic sputtering. The yield can also be conveniently described in 
terms of the individual energizing processes (Johnson and Brown 1982),
\begin{align}
& \mathrm{ Y = \sum\limits_{i}\Delta \bar{z}_s (\Delta E_i)/\lambda_i }
\end{align}
\end{subequations}
as in Eq. (3.10a). Here $\lambda_{\mathrm{i}}^{-1}$ is the mean-free path for 
excitation of an electronic process which will lead to a repulsive energy 
$\Delta $E$_{\mathrm{i}}$ and $\Delta $\={z}$_{\mathrm{s}}$($\Delta 
$E$_{\mathrm{i}}$) is the average sputter depth (Appendix 3B) for this 
$\Delta $E$_{\mathrm{i}}$. As typical outer-shell repulsive energies (Chap. 2) 
of the order of one to a few eV, are comparable to or larger than the cohesive 
energies of the condensed gas solids, Table 3.4, electronic sputtering occurs 
from such solids. In semi-conductors and room temperature insulators, which 
have much larger cohesive energies, multiple excitations or inner shell 
excitations are required (Brenig and Menzel 1985).

Computer simulations of the cascade of collisions following a few eV
repulsive energy input in a condensed-gas solid (Cui et al. 1988, 1989)
indicate the ejection occurs from the surface layer and

\begin{subequations}
\begin{align}
& \mathrm{ \bar{\Delta z_s}(\Delta E) \approx 0.15 l_s \frac{\Delta E - U}{U}}
\end{align}
with the average layer spacing 1$_{\mathrm{s}}$ $\sim $ n$^{-1/3}$. If the 
$\Delta $E$_{\mathrm{i}}$ in Eq. (3.24b) are all roughly the same for the 
principal electronic excitations produced, then
\begin{align}
& \mathrm{ Y \approx 0.15 \left(\frac{\bar{\Delta E}-U}{U}\right) l_s /\lambda_e.}
\end{align}
\end{subequations}
Here $\mathrm{\lambda }_{\mathrm{e}}^{-1}$ is the mean free path for producing 
an excitation near the surface. For incident electrons and fast ions, ionization 
is the dominant electronic excitation process (Table 3.2), so that 
$\mathrm{\lambda}_{\mathrm{e}}^{-1}$ $\approx$ F$_{\mathrm{e}}$/W$_{\mathrm{e}}$, 
[roughly W$_{\mathrm{e}}^{-1}$(dE/dx)$_{\mathrm{e}}$ cos$\theta_{\mathrm{A}}^{-1}$].
Therefore the efficiency factor in Eq. (3.24a) is

\begin{equation}
\mathrm{ \Lambda_e \approx 0.15f_e l_s / U,}
\end{equation} 
where f$_{\mathrm{e}} $ ($\approx $ $\bar{\Delta}$\={E}/W$_{\mathrm{e}}$) is 
the fraction of the electronic energy deposited that goes into repulsive decay 
(nuclear motion) (Brown and Johnson 1986; Schou 1987). Results for the sputtering 
of N$_{2}$and O$_{2}$ by fast protons and helium ions in Fig. 3.21a are close to
linear at low (dE/dx)$_{\mathrm{e}}$. The size of f$_{\mathrm{e}}$ is $\sim $
0.1 for N$_{2}$ and $\sim $ 0.2 for O$_{2}$ which is consistent with
incident electron results (Ellegaard et al. 1986, Pedrys et al. 1989). Since
condensed gases at the few monolayer level are generally very nonuniform,
the yields are enhanced, as ``islands'' of material can be ejected by a
single ion or electron.

The expression in Eq. (3.25b) applies also to yields for photon-stimulated
sputtering (Harrison and Schoen 1967; Cheng and Johnson 1988; Stuhlen and
Knotek 1988). In this case, $\mathrm{\lambda }_{\mathrm{e}}^{-1}$ is the
target number density times the photoabsorption cross section to the lowest
repulsive or to an excited state which lies above it and f$_{\mathrm{e}}$ 
$\sim $ $\bar{\Delta}$\={E}/h$\nu $ in Eq. (3.26). The excitation cross
sections in the UV are of the order of 10$^{-17}$ to 10$^{-18}$ cm$^{2}$.
Therefore, using $\bar{\Delta}$\={E} $\sim $ 2eV, the photo-sputtering yield
is of the order of 10$^{-2}$ to 10$^{-3}$.

\subsection{\textit{Nonlinear Yields}}

The yields for small molecules at the higher (dE/dx)$_{\mathrm{e}}$ shown in
Fig. 3.21a and 3.2lb go roughly as (dE/dx)$_{\mathrm{e}}^{2}$ if the
penetration depth is `large' due to the overlap of the energized regions at
high excitation densities. Therefore the form Eq. (3.17) has been useful for
extrapolating experimental results.

\begin{equation}
\mathrm{ Y_s \approx C_e [f_el_s(dE/dx)_e / U]^2,}
\end{equation}
where f$_{\mathrm{e}}$ is the fraction of the deposited electronic energy 
contributing to sputtering (Rook et al. 1985; Gibbs et al. 1988). In Table 3.4 
we summarize much of the data available for the electronic sputtering of 
condensed gases. At higher (dE/dx)$_{\mathrm{e}}$ a steeper, nearly cubic, 
dependence on (dE/dx)$_{\mathrm{e}}$ occurs due to the implusive volume 
expansion of the material around the track (Johnson et al. 1989b, 1990).

Shown in Fig. 3.22a is the sputtering yield of H$_{2}$O at low temperature,
which is, roughly, quadratic in (dE/dx)$_{\mathrm{e}}$, down to very low
(dE/dx)$_{\mathrm{e}}$. Following ionization, the electron recombination
processes in Eq. (3.8a) produce an energetic H or H$_{2}$ (e.g., H$_{3}
$O$^{+}$ $\ $+ e $\rightarrow $ H$_{2}$O + H, H$_{2}$O$^{+}$ + e 
$\rightarrow$ OH + H or O + H$_2$). An ($\sim $ 1 eV) H or H$_{2}$ produced 
at the surface will escape, but it is very inefficient at ejecting additional 
H$_2 $O molecules which are bounded by $\sim $0.52 eV. Therefore, more than
one excitation is required for sputtering H$_{2}$O at low S$_{\mathrm{e}}$.
At very high S$_{\mathrm{e}}$, a somewhat steeper dependence on S$_{\mathrm{e}}$ 
is seen (Cooper and Tombrello 1984; Benit et al. 1987). Large molecules,
which are also damaged by the energy deposited (e.g., organic molecules),
respond primarily to the impulsive, roughly (dE/dx)$_{\mathrm{e}}^{3}$,
volume ejection caused by the pressure pulse (Hedin et al. 1987; Fenyo et
al. 1989; Cui and Johnson 1989). The leucine data in Fig. 3.21b, which are
the yields of \textit{whole} leucine molecules ejected, exhibit this.

In Fig. 3.22b the results for sputtering of water ice over a broad range of
ion energies are presented for incident protons and oxygen-like ions. What
is clearly seen is the dependence of S$_{\mathrm{n}}$ at low energies and on
S$_{\mathrm{e}}$ at high energies. In Fig. 3.23 is shown the dependence of
the yield on the angle of incidence, $\theta _{\mathrm{A}}$, for CO at 10 K
(Brown et al. 1984). These results are well approximated by

\begin{equation}
\mathrm{ Y(\theta_A) \approx Y(0)cos\theta_A^{-1.6}}.
\end{equation} 
Similar results have been found for the other condensed gases in the 
quadratic sputtering regime (Gibbs et al. 1988; Johnson 1989a), and appear 
to confirm the spike character of the ejection.

Of interest on planetary satellites are the ejection energies of the
sputtered molecules. Sample energy distributions are shown in Fig. 3.24.
These energies are much lower, on the average, than those predicted in Eq.
(3.20a) if U is the sublimation energy. Such differences are also seen for
sputtering of condensed gases when the nuclear stopping dominates the energy
loss (e.g., 50 keV A$_{\mathrm{r}}^{+}$ in Fig. 3.24). The measured energy
distributions appear to be more closely described by the spherical binding
condition Eq. (3.20b). Because of deficiencies in the models at small E an
empirical ``effective'' U is often used in Eq. (3.20a) to fit the data
(Reimann et al. 1984; Haring 1984c).

For incident electrons, nonlinear yields are also expected when the
excitation densities are comparable to those for the ions. Because of the
deflection of incident electrons, however, F$_{\mathrm{e}}$ is not very
close to (dE/dx)$_{\mathrm{e}}$ (Schou, 1987). Because UV photons typically
produce single excitations, materials which do not exhibit a linear
sputtering regime at a given temperature will \textit{not} be sputtered
efficiently by UV photons. Energetic photons, however, can produce inner
shell excitations or multiple excitations. It is also possible for radicals
or other new species to be stored in the solid and be acted upon by
subsequent photons, as will be discussed, or ejected on warming (d'
Hendecourt et al.1986), therefore the water ice yields for MeV protons at T 
$>$ 110 K may be linear (Brown et al. 1980b).

\section{Track Production}

An early ramification of highly ionizing particles penetrating insulators
was the use of tracks to identify particles (Fleischer et al. 1975). These
tracks are damage paths in a solid and are often directly abservable in an
electron microscope, but were initially studied using chemical etching. It
is assumed that the incident ion produces a large number of defects and
displacement per unit path length in the solid. If this damage density is
sufficiently high, the track can be chemically etched to enhance its
visibility. In nuclear physics, the track characteristics are associated
with a particle type and energy. Because the cosmic rays and solar energetic
particle spectra contain highly ionizing particles, tracks can be seen in
materials collected from space (Bradley et al. 1984; Taylor 1982). In
addition, energetic protons and alpha particles, which do not produce tracks
efficiently, can induce nuclear fragmentation, thereby producing track
forming particles in the solid. In Fig. 3.25 are indicated the energy and
particle types for which tracks appear in typical insulators.

The process of track formation is thought to occur when the level of primary
ionization close to the ion's path is so high that all the molecules in the
track core are ionized. This would lead to an explosive outward repulsion of
the nuclei of the solid (a pressure pulse), an extreme case for the
repulsive decay described earlier. That multiple excitations are required
can be inferred from the fact that the materials of interest in Fig. 3.25
are room temperature insulators with sublimation energies $\sim $5 eV and
even larger bulk displacement energies, E$_{\mathrm{D}}$. Therefore, in a
low-temperature condensed gas the effect of a highly ionizing particle can
be dramatic. Because the secondary electrons produce ionization at some
distance from the ion's path, the primary ionization rate ($\sim $ n$_{%
\mathrm{B}}\sigma _{\mathrm{I}}$) most closely correlates with the onset of
track formation in Fig. 3.25. This quantity has a dependence similar to
(dE/dx)$_{\mathrm{e}}$ at large velocities.

\section{Chemical Effects in Sputtering}

The yields described above are measured at low temperatures, where they have
been found to be independent of the initial temperature of the condensed gas
solid. These yields become temperature-dependent when the temperature is
increased, as seen in Fig. 3.26. In all cases shown, the loss of molecules
due to sublimation during the measurements is subtracted from the total loss
of material in order to obtain a sputtering yield, (i.e., molecules ejected
per incident ion). At higher temperatures, sublimation will eventually
dominate sputtering and ion bombardment can enhance the ejection rate
(Lanzerotti and Brown 1983).

The temperature dependence observed in Fig. 3.26 is a result of the onset of
solid-state chemical reactions between species produced by the incident
radiation (Brown et al. 1984). Such reactions were discussed earlier, and
the mass spectra measurements of the ejected species confirm that chemical
processes occur, (deVries et al. 1984a). In Fig. 3.27 are shown the signals
for the ejection of masses 20 (D$_{2}$O), 32 (O$_{2}$) and 4 (D$_{2}$) from
an initially pure D$_{2}$O ice. These are plotted vs irradiation time for
1.5 MeV Ne$^{+}$ ions incident on the solid ($\sim $ 10 K). The irradiation
time is expressed as a fluence, the number of ions incident per unit area,
so that the results are independent of beam intensity at low beam
intensities.

The D$_{2}$O signal in Fig. 3.27 rises ``immediately'' from zero to a
constant value showing no fluence dependence. The radicals produced can
react to form species which are more weakly bonded to the solid and are
ejected by subsequent ions, as indicated by the fluence dependence of the O$%
_{2}$ and D$_{2}$ yields. Whereas the O$_{2}$ yield rises from zero
lineraly, the D$_{2}$ yield rises promptly, indicating formation and
ejection occur in a single event. It then also exhibits a slow steady rise
due to solid-state chemical process. In all these measurements, when the
beam is turned off, the signal rapidly decays to zero. When it is turned
back on again, the signal immediately returns to that value which it had
prior to being turned off. This indicates that the changes produced in the
ice by the ions at the observed temperature are \textit{permanent} over time
periods of the order of minutes to hours. We also note that O$_{2}$ is
readily formed in the solid, so that permanent decomposition of the ice is
taking place.

In Fig. 3.28 the temperature dependence of the D$_{2}$ and O$_{2}$ yields
are given at a fluence of $\sim $ 10$^{15}$ ions/cm$^{2}$ . (Such plots, of
course, are fluence-dependent.) This clearly indicates that the observed
temperature dependence in Fig. 3.26 is due to formation of new species.
Using the measured activation energies, $\Delta $E$_{\mathrm{a}}$, the yield
can roughly be written

\begin{equation}
\mathrm{ Y(T) \approx Y(0) + c_1 [l_s (dE/dx)_e / U]^n exp(- \Delta E_a/kT)}
\end{equation}
where Y(0) is the low temperature yield [e.g., Eq. (3.27)] and c$_{1}$ and n 
are constants. It is found that $\Delta $E$_{\mathrm{a}}$ $\approx $ 0.05 to 
0.07 eV for O$_{2}$ ejection from H$_{2}$O with n $\approx $ 1 (Brown et al. 
1980b; Reimann et al. 1984).

For water ice, the end-state species are more easily ejected than the
original molecules. That is, D$_{2}$ is attached to ice with an energy of
the order of 0.02 eV and O$_{2}$ is attached with energies of the order of
0.10 eV, whereas H$_{2}$O requires 0.52 eV for removal. In the sputtering
process, radicals (e.g., OD) produced by an incident ion and stored in the
ice at low temperature are also ejected (Haring et al. 1983; Bar Nun et al.
1985). Even though radicals are produced more efficiently than O$_{2}$ [see
Eqs. (3.8) and (3.9)], they form a much smaller fraction of the yield
because of the higher attachment energies to the lattice, consistent with
the dependence on U in Eq. (3.27). This selectivity in the ejection process
due to surface bonding enhances the likelihood of formation and ejection of
species having saturated bonds and low attachment energies (e.g., O$_{2}$
and D$_{2}$). Therefore, ions, which have even higher removal energies than
radicals, exhibit very low yields ($\sim $ 10$^{-3}$ to 10$^{-4}$
ion/neutral).

In Fig. 3.29 are shown for 1 keV energy ions incident on ice, the yields of
the various masses of neutrals which are ionized and then detected. At these
energies, collisional interactions are the dominant dissociation and
ejection processes. However, after the dissociation occurs, the subsequent
reactions are very nearly independent of the dissociation mechanism. We
therefore find chemical alterations similar to those in the electronic
sputtering of H$_{2}$O, but because of the more efficient ejection of
material, a larger fraction of radicals is observed (Haring et al. 1983,
1984a, b; Bar-Nun et al. 1985), after correcting for fragmentation due to
ionization.

The mass 20 signal in Fig. 3.28 is seen to be independent of temperature up
to temperatures close to the point where sublimation becomes important, and
is the dominant species ejected at the lower temperatures. At these
temperatures electron spin resonance measurements on an irradiated ice
clearly indicate the presente of radicals in the solid as discussed (Hart
and Platzman 1961). Further, warming the sample makes these radicals mobile
and reactive so that the signal disappears $\sim $150 K. In this regard the O%
$_{2}$ signal in Fig. 3.27 exhibits two separate regions of increasing
signal. One of them is predominant at low temperatures and one at high
temperatures. The slopes of these transients vs. the inverse of the
temperature are displayed in Fig. 3.30. The activation energies indicated by
these plots are similar for the two processes, but the onsets in fluence and
temperature are quite different. The increase at low temperature is due to O$%
_{2}$ forming in the track without significant diffusion. (The O$_{2}$ so
formed is then ejected by a subsequent ion.) As the track radii are of the
order of 1 nm, the observed saturation of this component at a fluence of
about 10$^{14}$ ions/cm$^{2}$ is understandable. The second rise is
predominantly due to the formation of O$_{2}$ from dissociated species
produced by two separate incident particles. Its \textit{onset} influence
occurs when the tracks begin to overlap.

In the second process above, the dissociated species would tend to favor
recombination to form ice [e.g., see reactions in Eqs. (3.8)]. However, at
all temperatures the D$_{2}$ signal in Fig. 3.27 saturates at high fluence
and then declines. Further, the size of the signal depends on the \textit{%
thickness of the sample}. As the incident ion passes through the sample and
stops in the substrate, the thickness dependence implies that D$_{2}$ comes
from all depths. Hence, D is being preferentially depleted from the film via
the diffusion of D$_{2}$ to the surface and subsequent loss. Strazzulla et
al. (1988) recently related the loss of D$_{2}$ (H$_{2}$) to film damage as
indicated by increased light scattering (Brown et al. 1978; Johnson et al.
1985). The preferential depletion of D (H) assures the formation of O$_{2}$ 
\textit{rather than} the reconstitution of the D$_{2}$O (H$_{2}$O)
molecules. This is confirmed by the fact that the rate of \textit{decrease}
in the D$_{2}$ signal in Fig. 3.27 is close to the rate of \textit{increase}
of O$_{2}$ at the higher temperatures. O$_{2}$ once formed also diffuses to
the surface, where it is ejected (Reimann et al. 1984; Benit et al. 1988).
For ions which stop in a thick ice layer, an equilibrium level of O$_{2}$
and D$_{2}$ ejection is obtained (Bar-Nun et al. 1985; Chrisey et al. 1986a).

Even more dramatic changes in the sputtering rates occur when ices
containing either carbon or sulfur are bombarded. The yields for CO ejected
by keV ions (Chrisey et al. 1986a; Haring et al. 1984a), electrons (Schou et
al. 1985), and MeV ions (Chrisey et al. 1989) are strongly
fluence-dependent. In addition, the mass spectra indicate that O atoms are
lost (via CO$_{2}$, O$_{2}$, O) more rapidly than\ the carbon atoms: $\sim $%
10\% surplus in the rate of loss of O (as O$_{2}$ and CO$_{2}$) over that of
C. At very high fluences the residual carbon (or carbon suboxide (Chrisey et
al. 1989)) is observable as a dark residue with a low sputtering yield.
Carbon suboxide is also apparently produced with UV irradiation (Anicich et
al. 1989). Similar observations have been made for CH$_{4}$, other organics,
and CO$_{2}$. Sulfurous residues have been formed after extensive
irradiation of CS$_{2}$, SO$_{2}$, and H$_{2}$S (Boring et al. 1985) and SO$%
_{3}$ and, probably sulfur suboxide, formed from SO$_{2}$ (Moore 1984;
Boring et al. 1983, 1984a). A few percent of the initial carbon remains
behind as a residue when CO is bombarded. This amount is typical of most of
the condensed gases except for CS$_{2}$ and organics such as CH$_{4}$. For
these systems, very little sputtering occurs for incident light ions and the
condensed gas is rapidly polymerized and then transformed into a refractory
sputter-resistant solid (Calcagno et al. 1983b).

The process of polymerization and residue formation can be understood quite
simply even if the details cannot be described very accurately. As molecules
such as CH$_{4}$ are dissociated, carbon atoms are able to form C-C bonds,
and H$_{2}$ is formed as in the case of water ice (i.e., H2 and O-O bonds).
A very simple example is 2CH$_{4}$ $\rightarrow $ C$_{2}$H$_{4}$ + 2H$_{2}$
or C$_{2}$H$_{6}$ + H$_{2}$. For incident light ions, the larger molecules, C%
$_{2}$H$_{4}$ or C$_{2}$H$_{6}$, sputter much less efficiently than CH$_{4}$
but the H$_{2}$ is lost easily. Therefore the region penetrated by the
radiation is driven to a new mixture of H$_{2}$, CH$_{4}$, C$_{2}$H$_{4}$, C$%
_{2}$H$_{6}$, and radicals in our simplified picture.

In solid methane two additional effects occur. First, H$_{2}$ and H are not
very mobile in CH$_{4}$. Therefore, the H$_{2}$ ejected comes primarily from
the surface layer until sufficient alteration of the ice occurs (Fig.
3.31a), after which loss of H$_{2}$ occurs from all depths penetrated by the
ion. Second, whereas O$_{2}$ (or H$_{2}$O$_{2}$, O$_{3}$) saturate the
ability of O to bond, in a carbon (or sulfur) containing material polymers
can form. Therefore, as hydrogen is continuously lost, first a polymeric
material and, eventually, a predominantly carbon material is formed, as
suggested in Fig. 3.11. This carbonized material becomes, locally, graphite
or diamond-like. Therefore, the presence of a vacuum-solid interface insures
an irreversible change in the material. This reverses the gas-phase
equilibrium process which led originally to the formation of CH$_{4}$ and
``permanently'' separates the C and H. At low (dE/dx)$_{\mathrm{e}}$ and low
fluences, $< \Phi t >$ ions per unit area, the
statistical nature of the interactions [see Eq. (2.2)] is such that the
fraction of the carbon converted into sputter-resistant (refractory)
material can be written (Calcagno et al. 1985a) as [1 - exp( - $\sigma _{%
\mathrm{x}}\langle \Phi $t$\rangle $)]. In this expression the ``cross
linking'' cross section, $\sigma _{\mathrm{x}}$, is roughly, $\sigma _{%
\mathrm{x}}$ $\approx $ (S$_{\mathrm{e}}$/W$_{\mathrm{x}}$) with W$_{\mathrm{%
x}}$ $\approx $ 130 eV. For large fluences W$_{\mathrm{x}}$ gradually
increases. Rather than express the alterations as a cross section times a
fluence, it has become customary in radiation effects to use doses (D$_{%
\mathrm{M}}$, in eV/molecule) times the G-values (see Table 3.3; alterations
per 100eV deposited). Therefore the fraction converted is [1 - exp( - D$_{%
\mathrm{M}}$G)] where D$_{\mathrm{M}}$ = S$_{\mathrm{e}}$ $\langle \Phi $t$%
\rangle $ and G = W$_{\mathrm{x}}^{-1}$ which is G $\approx $ 0.6/100eV for
linking of carbon at low doses. This is a G value typically found for 2(CH$%
_{4}$) $\rightarrow $ (C$_{2}$H$_{6}$) + H$_{2}$ for incident $\gamma $-ray
(Horigome et al. 1978; Johnson 1989d).

Condensed-gas films irradiated at low temperatures deteriorate in quality
(Johnson et al. 1985) because of the formation of defects (interior
surfaces) which allow H$_{2}$ to be formed efficiently (Strazzulla et al.
1988). On a stable surface the defects eventually create a path to the full
penetration depth of the ions. The refractory film produced by continued
sputtering or on warming, although locally dense, is riddled with large
voids, forming fairy-castle-like materials (Foti et al. 1984). This means
that to the full depth of penetration of the ion, any newly created
volatiles have access to the vacuum. It is of more than passing interest,
therefore, that polymeric materials may have been identified at Halley
(Huebner 1987) and that cometary dust (debris) is an underdense, fluffy
material at the submicron level (Kissel et al. 1986). However, the relevance
of these organics to space is still controversial (Chyba and Sagan 1988;
Wdowiak et al. 1989; Strazzulla and Johnson 1989).

Since all ionizing radiations cause such changes, it is important to make
quantitative distinctions between the radiation types. By examining the
(dE/dx)$_{\mathrm{e}}$ dependence of the sputtering processes, Brown,
Lanzerotti and coworkers have shown that there are distinct linear and
nonlinear regimes. Effects occuring in the linear regime are similar for all
ionizing irradiations depending only on the energy absorption cross section
as discussed. For the higher (dE/dx)$_{\mathrm{e}}$, nonlinear regimes this
is \textit{not} the case.

Since the threshold dose for the onset of H$_{2}$ loss given in eV deposited
per molecule in Fig. 3.31b becomes \textit{constant} at low (dE/dx)$_{%
\mathrm{e}}$ the process is linear. Therefore, very energetic photons and
GeV cosmic rays can also produce the modifications in CH$_{4}$ described
above (Johnson 1989d). The size of this dose is $\sim $60eV/mol, a value
close to the average energy lost per primary event produced by the incident
ion. The temperature dependent contribution of the sputtering yields by
protons also appears to be linear at some fluence, Eq.(3.29) (Brown et al.
1980b).

The alteration of materials also depends on its ability to store radicals
over those periods of time of interest in astrophysics. Whereas in the low
flux, ion beam experiments at low temperature and high vacuum the diffusion
and recombination of radicals was found to be slow on the time scale of the
experiments (typically hours), loss of precursor radicals may still occur
for the very low fluxes over the long storage times expected in
astrophysical applications. Quantitative use of these data, therefore,
requires care. Further, data obtained from discharge or very high fluxes of
particles or photons cannot be used because heating of the solid is likely
to have occurred, enhancing the reactivity. Finally, as the species most
readily lost is hydrogen, back reactions of stored radical with available
hydrogen (low energy adsorbed H$_{2}$ or high energy implanted H) can
inhibit material alteration particularly when photons are the stimulating
flux.

\section{Appendix to Chapter 3}

\subsection{\textit{A Transport Equations and the Recoil Energy
Distribution}}

\let\rememberthechapter=\thechapter
\renewcommand{\thechapter}{3A}
\setcounter{equation}{0}

In describing the statistics of the energy loss by fast ions or electrons in
solid, Monte Carlo calculations are made in which the probabilities are
determined by the cross sections in Chapter 2. Alternatively, transport
equations, described in the text, are solved. The average collision
probability is d$\sigma $/$\sigma $, where d$\sigma $ is that part of the
total cross section, $\sigma $, associated with a given range of deflections
and energy transfers. Because the classical value of $\sigma $ is infinite
for long-renge forces, Eq. ((3.3) is often rewritten as\smallskip

\begin{equation}
\mathrm{ 1 = n_B \int d\sigma [\bar{R}_p(E_A) - cos\theta \bar{R}_p(E_A - 
\Delta E)]}, 
\end{equation}
where $\theta $ is the deflection associated with energy transfer $\Delta$E. 
In this form, when $\Delta$ E$\rightarrow$ 0 (i.e., d$\sigma $ becomes large) 
then substituting \={R}$_{\mathrm{p}}$(E$_{\mathrm{A}}$ - $\Delta $E) 
$\approx$ \={R}$_{\mathrm{p}}$(E$_{\mathrm{a}}$) + (d\={R}/dE$_{\mathrm{A}}$)
$\Delta$ E and cos$\theta $ $\approx$ 1, gives, \={R}$_{\mathrm{p}} 
\rightarrow$\={R} the continuous slowing-down result in Eq. (3.1). Analytic 
solutions to Eq. (3A.l) are obtained using cross sections for the power-law 
potentials in Chapter 2; i.e., \={R}$_{\mathrm{p}}$(E$_{\mathrm{A}}$) $\propto$ 
E$_{\mathrm{A}}^{\mathrm{2/n}}$ (Sigmund 1981; R.E. Johnson 1982).

The distribution of particles of type i and initial momentum $\mathrm{\vec{p}}$ 
``stopped'' between z and z + dz is given by a Boltzman equation in which
all the collisions of the moving particles are with target particles which
are stationery,

\begin{subequations}
\begin{align}
& \mathrm{ -cos\theta_A\frac{\partial}{\partial z}F_i(\bar{p},z) = n_B \int 
d\sigma [F_i(\vec{p},z) - \hat{p}\cdot\hat{p}^{\prime}F_i(\vec{p}^{\prime},z)],} \notag\\
\end{align}
where d$\sigma $ = d$\sigma $($\mathrm{\vec{p}}$,$\mathrm{\vec{p}}^{^{\prime }}$)
and cos$\theta _{\mathrm{A}}$ is the direction of $\mathrm{\vec{p}}$
relative to the z axis. In this form the contribution to particles stopping
between z and z + dz is described by the differences between those that
change their momentum in this layer from $\mathrm{\vec{p}}$ to $\mathrm{\vec{p}}
^{^{\prime }}$ and those that do not. Therefore, the distribution
functions for all particles having momenta less than p must be obtained in
order to find F$_{\mathrm{i}}$ ($\mathrm{\vec{p}}$, z). The average
penetration depth obtained from F$_{\mathrm{i}}$ ($\mathrm{\vec{p}}$, z) is
equal to \={R}$_{\mathrm{p}}$ as determined from Eq. (3A.1). An equation
similar to Eq. (3A.2a) can be constructed to describe the distribution in
the energy deposited in the material.

Numerical solutions to Eq. (3A.2a) for the penetration (implantation)
distribution are required. However, if the average energy loss in a
collision is small compared to the CM energy and the deflections are small 
($\mathrm{\vec{p}}$ $\cdot $ $\mathrm{\vec{p}}^{^{\prime }}\approx 1$, for
electronic energy loss Q), then, using the energy-loss cross section 
(i.e., d$\sigma$ = (d$\sigma$/dQ)dQ), Eq. (3A.2a) can be rewritten in the
form

\begin{align}
& \mathrm{- cos\theta_A\frac{\partial}{\partial z} F_i(\bar{p}, z) = n_B \int 
\frac{d\sigma}{dQ} dQ Q \frac{\partial F_i}{\partial E}(\vec{p}},z)
= \left(\frac{dE}{dx}\right)\frac{\partial F_i}{\partial E} ({\vec{p},z).} \notag\\
\end{align}
This is the continuous slowing-down result as z and E (or p) are related by 
(dE/dx) so that F$_{\mathrm{i}}$($\mathrm{\vec{p}}$,z) = $\delta $(z - \={R} 
cos$\theta _{\mathrm{A}}$). That is, all the particles stop at the same depth, 
\={R} cos$\theta _{\mathrm{A}}$. This form for F$_{\mathrm{i}}$ describes
the stopping of fast ions in the electronic energy loss regime and very
heavy slow ions stopping in a material consisting of much lighter atoms. At
the other extreme, if an incident particle loses most of its forward
momentum in a single collision (i.e., $\mathrm{\vec{p}}$ $\cdot$ 
$\mathrm{\vec{p}}^{^{\prime }}\approx 0$), the contribution from scattered 
particles, F$_{\mathrm{i}}$ ($\mathrm{\vec{p}}$,z) in Eq. (3A.2a), can be 
ignored and

\begin{align}
& \mathrm{ -cos\theta_A \frac{\partial}{\partial z}F_i(\bar{p},z) = n_B 
\sigma F_i (\vec{p}, z).}
\end{align}
\end{subequations}
Letting $\lambda$ = (n$_{\mathrm{B}}\sigma $)$^{-1}$, the mean free path
the first collision, then F$_{\mathrm{i}}$($\mathrm{\vec{p}}^{\prime }$, z)
= F$_{\mathrm{i}}$($\mathrm{\vec{p}}$, 0) exp ( - z/$\lambda $cos$\theta
_{\mathrm{A}}$). This is the penetration distribution in Eq. (2.2). Such a
result for F$_{\mathrm{i}}$ is appropriate for neutron and photon
penetration, and is not a bad estimate for low energy electrons or slow
light ions in a material of heavy atoms replacing $\sigma$ by 
$\sigma_{\mathrm{d}}$, the diffusion cross section of Chapter 2.

In Eq. (3.5) we gave the energy distribution of target atoms set in motion
by a primary target recoil of energy E$_{\mathrm{A}}$ stopping in a gas
composed of atoms of the same mass, which is the basis for many of the
results obtained on multiple collision phenomena in Chapters 3 and 4.
Calling G(E$_{\mathrm{A}}$, E)dE the number of recoils produced with energy
between E and E + dE by a particle of energy E$_{\mathrm{A}}$ and the same
mass, then

\begin{equation}
\mathrm{ G(E_A, E) = (d\sigma/dE)/\sigma + \int [(d\sigma/dT)/\sigma] 
[G(E_A - T, E) + G(T, E)] dT.}
\end{equation}
Here (d$\sigma$/dT)/$\sigma$ is the probability
that in the first collision the particle E$_{\mathrm{A}}$ loses energy T.
Therefore, the first term is the probability of producing a recoil in the
appropriate energy range, and the next term is a sum of the subsequent
recoils produced by the two colliding species, integrated over all possible
energy transfers. This is solved subject to the boundary condition that G(T,
E) = 0 for T $\leq$ E. For the simple hard sphere collision (Chapter
2) (d$\sigma $/dT)/$\sigma $ = ($\gamma $E$_{\mathrm{A}})^{-1}$ with $\gamma 
$ = 1 for equal mass, the result is G(E$_{\mathrm{A}}$, E) = E$_{\mathrm{A}}
$/E$^{2}$ as can be verified by substitution in Eq. (3A.3). For realistic
classical cross sections, $\sigma $ $\rightarrow $ $\infty ,$ so Eq. (3A.3)
is written in the form of Eq. (3A.l). Sigmund (1969) has shown that, for the
power law potentials in Chapter 2, (d$\sigma $/dT) = (C$_{\mathrm{n}}$/E$_
{\mathrm{A}}^{\mathrm{n}}$)/T$^{\mathrm{1+n}}$, the lead term is G(E$_
{\mathrm{A}}$, E) $\approx $ $\beta _{\mathrm{n}}$(E$_{\mathrm{A}}$/E$^{2}$)
with $\beta _{\mathrm{\infty }}$ = 6/$\pi ^{2}$.

\let\thechapter=\rememberthechapter

\subsection{\textit{B Cascade Sputtering}}

\let\rememberthechapter=\thechapter
\renewcommand{\thechapter}{3B}
\setcounter{equation}{0}

In Eq. (3.10a) and Eq. (3.24b) are expressions for sputtering in a solid in
which an amount of kinetic energy, $\Delta $E$_{\mathrm{i}}$, is imparted to
the atoms or molecules either directly via nuclear elastic collision or
indirectly, via repulsive electronic relaxation (electronic sputtering).
Defining the yield from an event at depth z in the solid as Y($\Delta 
$E$_{\mathrm{i}}$,z), then

\begin{equation}
\mathrm{ \bar{\Delta z_s}(\Delta E_i)= \int\limits_{0}^{\infty} dz Y(\Delta 
E_i, z).}
\end{equation}
Therefore, the average yield when such events occur randomly with depth 
(but with a very small probability of overlapping) is

\begin{equation}
\mathrm{ Y(\Delta E_i) = \bar{\Delta z_s}(\Delta E_i)/\lambda_i,}
\end{equation}
where $\lambda_{\mathrm{i}}^{-1}$ is the average spacing of the excitations 
(Johnson and Brown 1982). Y($\Delta $E$_{\mathrm{i}}$, z), hence 
$\bar{\Delta}$\={z}$_{\mathrm{s}}$, is calculated using the escape 
probabilities defined for Eq. (3.13a) and the recoil spectrum above. That is, 
a recoil atom of energy $\varepsilon^{^{\prime }}$ generated near z can escape 
if it produces no additional recoils while exiting the solid. If the recoils 
are produced isotopically

$$
\mathrm{ \bar{\Delta z_s} \approx \int\limits_{0}^{\infty} dz 
\int\limits_{0}^{\gamma E_A} \int\limits_{-1}^{1} \int\limits_{0}^{2\pi} 
\frac{1}{4\pi} G(\Delta E_i, \varepsilon^{\prime})\cdot\Theta (\bar{\lambda}_d 
cos\theta - z)}
$$

\begin{equation}
\mathrm{ P_es(\varepsilon^{\prime},cos\theta)d\phi dcos\theta 
d\varepsilon^{\prime}}.
\end{equation}
Here G is the number of recoils produced by an event of energy $\Delta 
$E$_{\mathrm{i}}$ [Eqs. (3.5) and (3A.3)] and $\Theta $(x) is a step function, 
indicating that the depth of origin of an escaping recoil is less than the 
mean-free path for a collision. Finally, P$_{\mathrm{es}}$ is the planar or 
spherical binding condition [e.g., either Eq. (3.13b) or (3.14a)]. For 
$\Delta $E$_{\mathrm{i}}\gg $ U, this integrates to

\begin{equation}
\mathrm{ \bar{\Delta\bar{z}_s}(\Delta E_i) \approx \left(\frac{\beta_n 
\nu(\Delta E_i)\bar{\lambda_d}}{2U}\right) \cdot %\lbrace 
\begin{cases} \frac{1}{4}, & planar  \\ 
              \frac{1}{2}, & spherical  
\end{cases} }
\end{equation}
Sigmund (1969) uses a factor of two reduction in G because of the presence of 
a surface. This gives the result in Eq. (3.15) and the sputtered energy
distribution in Eq. (3.20a) is obtained by not carrying out the integral
over $\varepsilon^{\prime}$.

\let\thechapter=\rememberthechapter

\subsection{\textit{C Spike Sputtering}}

\let\rememberthechapter=\thechapter
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\setcounter{equation}{0}

If the energy density deposited by the incident ion is high, then, following
the initial cascade of collisions, energy is dispersed by diffusion. Calling
the kinetic energy density at time t and position $\mathrm{\vec{r}}$, $%
\varepsilon $($\mathrm{\vec{r}}$, t), then

\begin{equation}
\mathrm{ \bigtriangledown(\kappa(\varepsilon)\nabla \varepsilon) - 
\varepsilon /\tau = \partial\varepsilon/\partial t}
\end{equation}
describes this process. Here $\kappa $($\varepsilon$) is the diffusivity,
which in the gas phase is determined by the elastic collision cross sections
in Chapter 2 and $\tau $ is a dissipation time. Analytic solutions are
useful for $\varepsilon $ in an infinite material assuming either spherical
or cylindrical symmetry and $\kappa $($\varepsilon $) = K$\varepsilon^
{\mathrm{n}}$ (Vineyard 1976; Johnson and Evatt 1980; Sigmund and Claussen
1981; Johnson 1989a). For a solid, the surface is approximated in two ways.
If the escape rate of molecules is large, the effect of the surface on 
$\varepsilon$ is ignored; if it is very small, the surface is treated as an
energy reflector. Using the value of $\varepsilon $ at the surface, 
$\varepsilon_{\mathrm{s}}$, the flux of material, $\Phi $, is obtained
assuing a form for the energy distribution about $\varepsilon _{\mathrm{s}}$
. If the molecules are in local thermodynamic equilibrium, then the
sublimation flux can be obtained from the vapor pressure, P(T$_{\mathrm{S}}$), 
at the surface: $\varepsilon _{\mathrm{s}}$ $\approx $ (3/2)n$_{\mathrm{B}}$
kT$_{\mathrm{s}}$ and $\Phi $(T$_{\mathrm{s}}$) $\approx $ P(T$_{\mathrm{s}}$) 
exp (- U/kT$_{\mathrm{s}}$)/(2$\pi $M$_{\mathrm{B}}$kT$_{\mathrm{s}}$)$^{1/2}$. 
For a randomly deposited spherical or point energy input of size $\Delta 
$E$_{\mathrm{i}}$ with $\tau $ $\rightarrow $ $\infty $, $\bar{\Delta}
$\={z}$_{\mathrm{s}}$ of Eq. (3B.2) is

\begin{equation}
\mathrm{\bar{\Delta z_s}(\Delta E_i) = \frac{1}{2} \int d^3 r \int\limits_{0}
^{\infty} dt\Phi [T_s (\vec{r}, t)] \propto \Delta E_i^{5/3}.}
\end{equation}
For a narrow cylindrically symmetric energy deposition (closely spaced 
$\Delta$E$_{\mathrm{i}}$) about the path of an ion

\begin{equation}
\mathrm{ Y = \int d^2r \int\limits_{0}^{\infty} dt\Phi(T_s) \propto 
(\Delta E_i /\lambda_i)^2 \propto (dE/dx)^2}
\end{equation}
as $\varepsilon_{\mathrm{s}}$ (hence T$_{\mathrm{s}}$) depends on the number 
of excitations per unit path length $\lambda_{\mathrm{i}}^{-1}$. The latter 
result has been used in the text to discuss the measured yields Eq. (3.16). 
The incident angle dependence, Eq. (3.28), has been confirmed for condensed-gas 
solids (Johnson 1989a). The energy distribution of ejecta can be obtained 
letting Y $\rightarrow $ Y(E) and $\Phi $(T$_{\mathrm{s}}$) $\rightarrow$ 
$\Phi $(T$_{\mathrm{s}}$)$\cdot$[E$^{1/2}$ exp( - E/kT$_{\mathrm{s}}$)
/$\pi^{1/2}$(kT$_{\mathrm{s}}$)$^{3/2}$] in Eq. (3C.3). If the initial radius 
of the energy deposition is small then, using planar binding, Y(E) $\propto$ 
E/(E + U)$^{4}$ for the cylindrical case, and for the nonoverlapping point 
sources Y $\propto$E/(E + U)$^{10/3}$ The latter is not very different 
from the cascade result E/(E + U)$^{3}$. The spike yields have been heavily 
discussed in the sputtering literature (Sigmund 1981; Kelly 1987) but not 
extensively verified. However, they provide a useful way of thinking about the
low-energy contribution to the ejecta. Leger et al. (1985) use a spike model
to discuss interstellar grain erosion. At very high (dE/dx) the \textit{%
gradient} in $\varepsilon $ gives a momentum to a volume at the surface. For
normal incidence this ejects a hemi-spherical volume with radius
proportional to (dE/dx) (Feny\"{o} et al. 1989; Johnson et al. 1989b; Cui
and Johnson 1989).

\let\thechapter=\rememberthechapter

\end{document}