Lecture Note for Oct. 23, 2002 (Wed.)





     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:



       Excitation to bound states




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


     At high incident electron energy T > 10B, 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 (Zeff  > 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, Zeff(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 = f0n(K) =(E0n/R)|<n|j exp(iKrj)|0>|2


       K = momentum transfer from the incident electron to the target atom (K = pout pin)


       E0n = Excitation energy


       R = Rydberg energy = 13.6 eV


       n , 0 = final and initial state wave functions  of the target atom


       rj = radial vector of the j-th target electron


     Properties of the GOS


       Optical limit:  lim K 0 f0n(K) = f0n (dipole oscillator strength for photoabsorption)


       Sum rule:  n f0n(K) = N for any K

     PWB cross section is given as an integral of the GOS from the mininum K to the maximum K , both of which are functions of E0n and the incident energy T:



     If the excited state, n , is a continuum state, then 0n is a differential ionization cross section for ionizing an atom to a continuum state with the excitation energy E0n from the ground state


     To get the total ionization cross section, 0n must be integrated over E0n 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:


       Bethe = (4a02R/T)[Mn2 ln(T/R) + Cn],


       Mn2 = f0n(R/E0n)


       Both Mn2 and Cn are indepent of T


     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, , , 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 = mv2/2, 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, Pkl(r) or Pwl(r), and summed over l


     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:


       BE(T) = PWB(T)[T/(T+B+E)]


       E = excitation energy


       BE  PWB at high T


     When inaccurate wave functions are used to calculate PWB, then apply f scaling if more accurate f value is known:


       BEf(T) = BE(T)(faccurate/finaccurate)


     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:


E(T) = CWB(T)[T/(T+E)]


       CWB(T) = Coulomb-wave Born cross section


     As was done for BE, when inaccurate wave functions are used to calculate CWB, then apply f scaling if more accurate f value is known:


       Ef(T) = E(T)(faccurate/finaccurate)


     Since PWB and CWB cross sections are the simplest ones to calculate for excitations, BE, BEf, E, and Ef 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?



      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 Al, Ga, and In (ns2np nsnp2)



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 Rutherford cross section for two free charged particles so that the Pauli exclusion principle is satisfied for the two colliding electrons


     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 (df/dE) must be known


     Accurate df/dE 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 df/dE


     The BED model has been applied successfully to calculate singly differential ionization cross sections (d/dW) of H, He, and H2

     To overcome the lack of reliable df/dE, the BEB model assumes a simple, hydrogenic form of df/dE


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




       S = 4a02N(R/B)2


       N = orbital electron occupation #


       R = Rydberg energy = 13.6 eV


       t = T/B


       u = U/B; U = <p2/2m> = 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 (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 H2 to SF6


     Many examples are available from the public website:




     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