Lecture Note for

VI. INTERACTION OF
ELECTRONS WITH ATOMS

¡¤ Atoms interact with many types of particles:

Electrons, photons, protons, atoms, molecules, etc.

¡¤ When there is enough energy, ionization will
take place and produce secondary (= ejected)
electrons

¡¤ If the secondary electrons have enough
energy, 2nd generation ionization will take place

¡¤ If the secondary electrons from the 2nd
generation ionization have enough energy, 3rd generation ionization will take
place, and so on¡¦

¡¤ Rule of thumb: Only about 1/3 of secondary electrons
have enough energy to cause next generation of ionization

¡¤ The 3rd generation of ionization will have
(1/3)^{3} = 1/27 of the intensity of the 1st generation of ionization

¡¤ Of the many type of collisions,
electron-atom collision is the most useful process for applications

¡¤ For photon-atom interaction, reliable cross
sections can be calculated if good target wave
functions are known

¡¤ For electron-atom collisions, wave functions
for the combined system (= negative ion for a neutral target atom) with at
least one electron in the continuum must be known

¡¤ Two separate approximations are used in all
electron-atom collision theories:

Approx. for
the wave functions of the incident electron, and

Approx. for
the wave functions of the target atom

¡¤ Collision theories
are labeled according to the approximations used for the incident electron

¡¤ There are two approaches to handle
electron-atom collisions:

Weak coupling theory: The interaction energy between the
incident electron and the target electron is treated as a perturbation compared
to the total energies of the atom and the incident electron without interaction

Strong coupling
theory: The interaction energy between the
incident electron and the target atom is treated on equal footing as the
interaction of bound electrons with the target atom nucleus and among the bound
electron themselves

¡¤ Weak coupling theory is valid when the
incident electron energy is high compared to the binding energy of the bound
electron to be excited or ionized

Example: 1st-order Born approx.

Easy to
calculate, but limited validity

¡¤ Strong coupling theory must be used when the
incident electron energy is comparable to the binding energy of the bound electron
to be excited or ionized

Example: Close-coupling, R-matrix theory

Reliable
for slow incident electrons, but difficult to calculate

¡¤ Difficult energy
range for theory:
Intermediate incident electron energy (3 to 10 times the ionization
energy of the target atom).
Unfortunately, this is the range where most electron-atom collision
cross sections peak!

¡¤ Exception: Binary-Encounter-Dipole (BED) and
Binary-Encounter-Bethe (BEB) models produce reliable electron-impact ionization cross sections
for atoms and molecules at all incident electron
energies

¡¤ When the incident electron is slow, it forms
temporary compound states (negative ion), often leading to interference
patterns known as resonances

¡¤ Resonances can be described correctly only
with strong-coupling theories.
Locations (=incident electron energy) and magnitudes of resonances are
strong functions of the theoretical model used

¡¤ Resonances are prominent only near the
thresholds for excitation and ionization

VI-A. First Order Born Approximation

¡¤ Type of electron-atom collisions:

Elastic

Excitation
to bound states

Ionization

Recombination

¡¤ Unlike photon-atom interactions, there are no strict selection rules for electron-atom collisions

¡¤ At high incident electron energy *T*
> 10*B*, where *B* is the binding energy of the target electron
being excited, the 1st-order Born approx. works well

¡¤ There are different
kinds of 1st-order Born cross sections depending on
how to represent the incident electron.
Target wave functions should be as accurate as practical

¡¤ Plane-wave Born approx. (PWBA): Simplest to calculate

Uses plane
waves for the incident electron, before and after collision

Bethe worked out basic features of the PWBA in 1930

¡¤ Coulomb-wave Born approx. (CWBA): More difficult than the PWBA, but much
easier than the DWBA below

Uses
Coulomb waves for the incident electron with appropriate ¡°effective nuclear charge¡±

For collisions with highly charged ions (*Z*_{eff} > 5), the CWBA becomes as
reliable as far more complicated theories

¡¤ Distorted-wave Born approx. (DWBA): Difficult to calculate

Uses
continuum wave functions for the incident electron obtained by solving a
Schrödinger/Dirac equation with an effective potential derived from the charge distribution of
the target atom

This is
equivalent to using an *r*-dependent screened nuclear charge, *Z*_{eff}(*r*)

Compared to the PWBA and
CWBA, 10—100 times more work, but only slightly better results

¡¤ In view of the scaling methods for the PWBA
and CWBA discussed below, it is not necessary to use the DWBA

¡¤ Bethe derived many
useful features of the PWBA:

Generalized oscillator strength (GOS): For a given transition (both excitations
and ionization), GOS calculated from the target wave functions before and after
the collision contains all information about the PWB cross section:

GOS = *f*_{0n}(*K*) =(*E*_{0n}/*R*)|<*¥× _{n}*|¥Ò

** K** = momentum
transfer from the incident electron to the target atom (

*E*_{0n}
= Excitation energy

*R* = Rydberg energy = 13.6 eV

*¥×** _{n}* ,

*r*_{j}**
**= radial vector of the

¡¤ Properties of the GOS

Optical limit:
*lim* *K* ¡æ 0 *f*_{0n}(*K*)
= *f*_{0n} (dipole oscillator strength for photoabsorption)

Sum rule:
¥Ò_{n}*f*_{0n}(*K*)
= *N* for any *K*

¡¤ PWB cross section is given as an integral of
the GOS from the mininum ** K** to the
maximum

_{}

¡¤ If the excited state, *¥×** _{n}* , is a continuum
state, then

¡¤ To get the total ionization cross section, *¥ò*_{0n}
must be integrated over *E*_{0n} from the ionization
threshold *B* to infinity

¡¤ Quality of GOS is a strong function of the
wave functions used

¡¤ Once GOS is calculated for an excitation,
cross sections can be calculated for arbitrary *T*

¡¤ Unless the valence shell of a target atom is
strongly affected by electron correlation, a single-configuration Hartree-Fock or equivalent Dirac-Fock wave functions will produce ¡°useful¡± cross
sections at high *T*

¡¤ For high *T*, Bethe
has shown that an integrated cross section (i.e., integrated over angles of the
scattered electron) can be written as:

*¥ò*_{Bethe} = (4*¥ða*_{0}^{2}*R*/*T*)[*M _{n}*

*M _{n}*

Both *M _{n}*

¡¤ ln
(*T*/*R*) comes from the integration over the angles of the incident
electron after the collision, and this is leading term at high *T*

¡¤ Classical collision theory does not lead to
the logarithmic *T* dependence

¡¤ The logarithmic *T* dependence for
dipole allowed transitions is true for any fast, charged incident particles

Examples: *e*^{¡¾}, ¥ì^{¡¾}, ¥ð^{¡¾},
proton, atomic/molecular ions

¡¤ Neutral incident particles and dipole
forbidden transitions of charged particles will not lead to the logarithmic
term in Eq. (56) of the text

¡¤ Ionization cross sections always have
dipole-allowed and forbidden components, and hence have the logarithmic *T*
dependence

¡¤ Another important feature of PWB cross
sections:

Eq. (56) is also valid for fast heavy particles. In this
case, *T* = *mv*^{2}/2, where *m* is the electron mass
(regardless of the type of the incident particle) and *v* is the speed of
the projectile

For
example, a 2-MeV proton has about the same speed as a 1-keV electron. Hence, their cross sections are very
close

¡¤ This conversion is true only if the
projectile is moving much faster than the orbital speed of the target electron

¡¤ The asymptotic form of the PWB cross sections
also exists for singly differential ionization cross sections (= energy
distribution of ejected electrons) and doubly differential ionization cross
sections (angular and energy distributions of ejected electrons) as long as the cross section has been integrated over the
angles of scattered electron

¡¤ The validity of the PWB cross sections for
dipole and spin allowed excitations can be extended to low *T* using the BE scaling described later

¡¤ Coulomb Wave Born
approx. (CWBA) is used when the target atom is an ion

¡¤ CWB cross sections are also valid only for
high *T*. Its validity at low *T*
can be extended using the E scaling described
later

¡¤ Only for H, CWB cross sections can be
expressed in analytic forms

¡¤ For other atoms, Coulomb waves must be expanded
in partial waves, *P _{kl}*(

¡¤ As *T* increases, partial waves must
include more *l*

¡¤ If the same target wave functions are used,
PWB and CWB cross sections become very close as more and more partial waves are
summed in CWBA

¡¤ The MCDF2002 code can calculate PWB cross
sections for excitations between bound states

¡¤ Old DW92 code can calculate CWB and DWB
cross sections in combination with the wave functions from the old DF92 code

¡¤ For neutral targets, PWB, CWB, and DWB cross
sections have the correct shape:
Start from zero, reach maximum at 5-10 times the threshold energy, and
decrease for higher *T*, for both excitations and ionization

¡¤ For excitations
(but not ionization) of ions, cross sections at the threshold is finite,
often the maximum, and then decreases at higher *T*

¡¤ The CWB and DWB cross sections reproduce the
correct shape for ion targets, but not the PWB
excitation cross section

¡¤ PWB cross sections always vanish at the
threshold

¡¤ PWB excitation cross sections can be too
large by a factor of two or three at low *T*

VI-B. Scaling of Born Excitation Cross
Sections

¡¤ The performance of the PWB cross sections
for electron-impact excitations can be improved dramatically for dipole and
spin allowed excitations by applying the BE scaling to PWB cross sections:

*¥ò*_{BE}(*T*) = *¥ò*_{PWB}(*T*)[*T*/(*T*+*B*+*E*)]

*E*
= excitation energy

*¥ò*_{BE} ≈ *¥ò*_{PWB}
at high *T*

¡¤ When inaccurate wave functions are used to
calculate *¥ò*_{PWB}, then apply *f* scaling if more accurate *f* value is known:

*¥ò*_{BEf}(*T*) = *¥ò*_{BE}(*T*)(*f*_{accurate}/*f*_{inaccurate})

¡¤ In the past, efforts were made to use a
known *f* value to create an excitation
cross section (e.g., Gaunt factor method)

¡¤ The *f* scaling does not ¡°create¡± an
excitation cross section; it simply alters an
existing cross section

¡¤ For ion targets, E
scaling extends the validity of CWB cross sections to low *T*:

*¥ò*_{E}(*T*) = *¥ò*_{CWB}(*T*)[*T*/(*T*+*E*)]

*¥ò*_{CWB}(*T*) = Coulomb-wave Born cross section

¡¤ As was done for *¥ò*_{BE},
when inaccurate wave functions are used to calculate *¥ò*_{CWB},
then apply *f*
scaling if more accurate *f* value is known:

*¥ò*_{Ef}(*T*) = *¥ò*_{E}(*T*)(*f*_{accurate}/*f*_{inaccurate})

¡¤ Since PWB and CWB cross sections are the
simplest ones to calculate for excitations, BE, BE*f*,
E, and E*f* scalings
offer tremendous savings in computing efforts and unprecedented accuracy

¡¤ Availability of accurate excitation cross
sections is crucial in estimating the contributions of excitation-autoionization

¡¤ The true reason for
the success of the BE and E scalings are still
unknown

¡¤ Related to the orthogonalization
of the plane-waves to the bound electron wave functions?

¡¤ Excitation-autoionization:

When an
inner-shell electron is excited to a bound valence level, the total energy of
the excited atom may be higher than the total energy of its ion. Then, the excited atom is unstable, and
must eventually decay to a lower level either by ejecting an electron (= autoionization) or by emitting a photon (fluorescent
decay). The former is called
excitation-autoionization

¡¤ Excitation-autoionization
(EA) plays an important role in the ionization of many open-shell atoms

¡¤ The most significant EA contributions come
from dipole and spin allowed excitations of inner-shell electrons with the same
principal quantum number as the outermost valence electrons

¡¤ For instance, the EA process almost doubles
ionization cross sections of *n*s^{2}*n*p ¡æ *n*s*n*p^{2})

VI-C. Description of DW92

¡¤ The MCDF2002 code can calculate PWB cross
sections for excitations to bound levels of neutral atoms, but not CWB cross
sections of ions

¡¤ Old DW92 code calculates CWB and DWB cross
sections for excitations of ions, using Dirac-Fock
wave functions generated by the DF92 code

¡¤ The Coulomb-wave part of the DW92 code will
eventually be integrated into the new MCDF2002 code

¡¤ At present, the DW92 code must be used with
the DF92 code for wave functions and the MJ92 code for the integration of
angular varialbles (= Slater coefficients)

¡¤ The DW92 code can also generate PWB cross
sections in partial waves, so that the convergence of CWB cross sections can be
verified as more partial waves are summed

VI-D. Binary-Encounter-Dipole (BED) and
Binary-Encounter-Bethe (BEB) Models

¡¤ Ionization cross sections at intermediate
and low *T* are difficult to calculate because of the difficulty in
representing the correlation between the incident electron and the target
electrons

¡¤ Finding the ¡°exact¡± solution for
electron-impact ionization of H requires thousands of hours of CPU time on
massively parallel, supercomputers

¡¤ The same theoretical method cannot be
extended easily to He and heavier atoms!

¡¤ For applications to other atoms and molecules,
we must find more practical methods

¡¤ Most existing ¡°practical¡± methods are either
empirical, or have limited validity

¡¤ An exception is the BED model developed by
Kim and Rudd [PRA **50**, 3954 (1994)]

¡¤ The BED model
combines the asymptotic form of the Bethe cross
section [Eq. (56)] for high *T* with a modified
form of the Mott cross section

¡¤ The Mott cross section is the exact solution
for the collision of two free electrons

¡¤ The Mott cross section is a modification of
the

¡¤ Although the Mott cross section is valid at
all *T*, it cannot be used without modification because the Mott cross
section becomes infinite for the collision of electrons with zero-kinetic
energy

¡¤ Also, the Mott cross section is for two free
electrons, not for one free and one bound electrons

¡¤ The BED model is for the energy distribution
of secondary electrons ejected by electron-impact ionization

¡¤ To use the BED model, the continuum dipole
oscillator strength (d*f*/d*E*) must be
known

¡¤ Accurate d*f*/d*E*
for each orbital in an atom or molecule is not easy to measure or calculate

¡¤ Photoionization
cross sections with data on the energy of ejected photoelectrons are needed to
find such d*f*/d*E*

¡¤ The BED model has been applied successfully
to calculate singly differential ionization cross sections (d*¥ò*/d*W*)
of H, He, and H_{2}

¡¤ To overcome the lack of reliable d*f*/d*E*, the BEB model
assumes a simple, hydrogenic form of d*f*/d*E*

¡¤ The BEB model offers a simple analytic
formula that uses data from the ground state only:

_{}

*S* = 4*¥ð**a*_{0}^{2}*N*(*R*/*B*)^{2}

*N*
= orbital electron occupation #

*R*
= Rydberg energy = 13.6 eV

*t*
= *T*/*B*

*u* = *U*/*B*; *U* = <*p*^{2}/2*m*>
= orbital kinetic energy

¡¤ The BEB cross section for each orbital is
summed over the occupied orbitals to obtain the total
ionization cross section

¡¤ The first logarithmic term comes from the
leading term of the Bethe cross section

¡¤ The rest of the terms in [¡¦] come from the
Mott cross section

¡¤ All terms in the BEB formula are based on
rigorous derivations except for the denominator *t* + *u* + 1

¡¤ This denominator was introduced by Burgess
(1960) to scale down PWB cross sections at low *T*

¡¤ His justification: The effective energy between the
incident and target electrons is not *T* but *T* + the potential
energy of the bound electron (*U* + *B*). In the threshold units, *T* + *U*
+ *B* ¡æ (*T* + *U* + *B*)/*B* = *t* + *u* +
1

¡¤ Originally, *T* in the denominator was
introduced to normalize the cross section for each incident electron

¡¤ The ¡°Burgess¡± denominator can be altered to
adapt the BEB formula for different collision conditions

¡¤ For instance, for a singly charged ion
target, use

*t* +
(*u *+ 1)/2

¡¤ For an orbital whose principal quantum
number *n* > 2, use

*t* +
(*u* + 1)/*n*

¡¤ Accurate total ionization cross sections
have been calculated and verified for C, N, O, Al, Ga,
and In by using the BEB model for direct ionization and the BE scaling for
excitation-autoionization

¡¤ The BEB model has been very successful in
reproducing total ionization cross sections of many neutral and singly charged
molecules, from H_{2} to SF_{6}

¡¤ Many examples are available from the public
website:

*http://physics.nist.gov/ionxsec*

¡¤ Production of
multiply charged ions

Multiply
charged ions can either be produced by several electrons being ejected from the
same orbital, or by ejecting a deep inner-shell electron followed by a series
of Auger decay

¡¤ There are no effective *ab**
initio* theory for ejecting several electrons from
the same orbital, because these electrons are tied together by strong
correlation

¡¤ On the other hand, multiple
ionization through the Auger decay can be handled by the BEB model and
many other theories

¡¤ If the inner-shell hole is deep, the hole
may be filled by fluorescent decay rather than by the Auger decay

¡¤ The rate for fluorescent decay can be
calculated by the MCDF2002 code

¡¤ The rate for Auger decay is more complicated
to calculate, and the present MCDF2002 code cannot calculate the rate. (An old version of the MCDF2002 code
did, and this capability will eventually be included in the code)

¡¤ The ratio between the fluorescent decay and
Auger decay is called the fluorescence yield