CHAPTER 2

ELECTRON SCATTERING WITH ATOMS AND

MOLECULES: THEORY AND

APPROXIMATIONS

2.1 Introduction: Present Theoretical Problem

In recent decades, the collisions of electrons with simple atoms, complex many-

electron atoms as well as with various molecules have been investigated by various

workers [1, 2]. We have seen a remarkable renaissance in both experimental and

theoretical activities in the study of electron collision processes with molecules in past

decades [3]. The advances in experimental instrumentations, e.g. improved electron

spectrometers, intense source of spin-polarized electrons, position-sensitive detectors etc

have led to the collection of more accurate cross sections. There are several limitations

for carrying out electron impact collision experiments with reactive radicals, complex

molecules, bigger molecules and even reactive targets. So there is a need for developing

theoretical concepts and models for studying these targets. An exact solution of the

Schrodinger equation is an impractical proposition except for the simplest potentials. In

34

most cases of practical interest, one has to settle for an approximate solution. Thus,

several methods of approximation have come to be devised for tackling various types of

problems in quantum mechanics. Amongst them the high energy approximations viz.,

Born, eikonal and Galuber [4] approximations are applicable to light and simple targets.

Schwinger Variational principle is used for calculating elastic scattering cross sections at

low to intermediate energies [5]. Apart from these there are ab-initio R-matrix

calculations for which special codes have been developed by [6]. The R-matrix

calculations are limited to small systems and low-energy scattering, below 15 eV,

typically below the ionization threshold. To the date electron impact studies have been

limited to ab-initio elastic scattering and excitation calculations using the R-matrix at low

energies. Cross section calculations using a complex optical potential which is also called

Spherical complex optical potential, SCOP [4, 7] are of great interest in view of the

difficulty in experiments as well as the accurate theories like R-matrix. Also, the SCOP

has been successfully employed by many other groups like, Jain and Baluja [8], Jiang and

co-workers [9] and Lee et al. [10] and by our group [11-16]. In this background SCOP is

a feasible alternative to calculate both elastic as well as inelastic scattering cross sections.

The positive side of Complex Optical Potential is that it treats elastic scattering in

presence of all allowed inelastic channels. Inelastic cross section Qinel is not a directly

measurable quantity as such. The only way to compare Qinel is with the sum of all total

excitation and ionization cross sections as measured by different groups or by subtraction

of elastic cross section from the total cross section QT. The elastic cross sections are

available experimentally but again these are obtained by integrating the differential cross

sections and hence experimental Qel has an error of about 20% or more in many cases.

35

Hence, the present objective is to focus on SCOP and try to extend it in some semi-

empirical way so that the information of the ionization which is hidden in the inelastic

cross sections can be brought out. This of course, requires additional assumptions which

are explored fully in our discussions. In this chapter we have examined simple atomic

targets, diatomic molecules and important atmospheric molecules, which have been

unexplored so far in one respect or the other. We have also focused upon extremely new

targets which have never been explored earlier and have no experimental and theoretical

data available for comparisons.

2.2 Spherical Complex optical potential – (SCOP)

To solve the Schrodinger equation, we introduce the total (complex optical)

potential of the system, to be written as,

Vopt = VR + i VI (2.1)

The complex potential introduced in the Schroedinger equation corresponds to loss

of flux in the channels other than elastic channel described by the real potential. The

complex potential describes all the major physical effects in the present targets. The real

part (VR) of the complex potential (Vopt) is the sum of static (Vst), exchange (Vex) and

polarization (Vp) potentials and the imaginary part VI corresponds to the absorption

potential, Vabs. The interaction between electrons and the target is determined by these

potentials. So let us investigate more about these potentials in detail.

36

2.2.1 Target description, basic inputs, Static potential

The static potential Vst is the potential experienced by the incident electron upon

approaching a field of an unperturbed charge cloud of the target atom or molecule. The

Vst (r) at a distance r is given by [3],

2 2

0

1( ) 4 [ ( ') ' ' ( ') ' ']

r

str

ZV r r r dr r r dr

r r

(2.2)

Where ρ(r) is the electronic charge density of the target atom/molecule. For hydrogen

atom we get an exact expression for the wave function and charge density, so that Vst (r)

becomes,

21

( ) 1 r

stV r er

(2.3)

The static potential is real and it is effective only at short distances. For heavier atoms we

cannot find an exact expression for charge density and static potential. So we have to rely

upon various approximate methods. The static charge density may be calculated using the

Roothan Hartree Fock (RHF) wave functions given in terms of the Slater Type Orbitals

(STOs) tabulated in the work of Clementi and Roetti [17]. Later Bunge and

Barrientos[18] tabulated a more accurate version of RHF wave function, but the

improvement has only a marginal effect as far as scattering calculations are concerned.

Their parameters are mostly for atoms in ground state. The RHF wave functions, Rnl are

expanded as a finite superposition of primitive radial functions as

37

nl jl jnlj

R S C (2.4)

Where Cjnl are the orbital expansion coefficients and Sjl being the STOs given by,

1exp( )jln

jl jl jlS N r Z r

(2.5)

With,

122

2 !

jln

jl

jl

jl

ZN

n

(2.6)

the normalization factor, njl = principal quantum number, Zjl = orbital exponents and l =

orbital angular momentum quantum number. The orbital expansion coefficients, Cjnl and

the orbital exponents, Zjl are tabulated by the authors [17] from which the electronic

charge density can be obtained.

As an example let us calculate the charge density of the Ne atom. The Cjnl and Zjl

taken from Bunge and Barrientos [18] are used to calculate Sjl and Njl through the

equations 2.5 and 2.6. Thus the radial wave functions for electrons in each orbital, R1s,

R2s and R2p are calculated using the equation 2.4. Then the total charge density for a Neon

atom (Z = 10) can be obtained through,

22 211 2 24

( ) (2 2 6 )s s pr R R R

(2.7)

38

Figure 2.1: Radial charge density curve for Ne atom

Solid curve – D(r) for Ne atom The vertical line marks the outer peak

The radial charge density [D(r) = 4πr2 ρ(r)] for Ne atom is plotted in the figure 2.1. We

can see two distinct peaks, the first one due to the n = 1 and the other due to the n = 2

shells. The quantity Rorb = 0.65a0 is the orbital radius of Ne.

In the work of Cox and Bonham [19], the Hartree-Fock static potential was

represented by an analytical expression involving a sum of Yukawa terms. The potential-

field fitting parameters were determined by least square fits of the radial electron density

or distribution function D(r), using Hartree-Fock and relativistic wave functions. They

have tabulated the parameters for atoms from Z = 1 to Z = 54. The analytical expression

obtained for the static potential is given by,

1

expn

st i i

ZV r r

r (2.8)

39

and for the electronic charge density is given by,

2

1

exp4

n

i i i

Zr r

r

(2.9)

Here γi and λi are the potential field parameters. An improved version of RHF method

which also accounts for the relativistic effects is called the Dirac – Hartree – Fock –

Slater method (DHFS) given by Salvat et al [20]. In this method the authors try to fit the

potential accurately to an analytical expression. This approximation depends on the

parameters for the atomic screening function, which are determined by DHFS self-

consistent data. The analytical expression for the static potential is,

1

expn

st i i

ZV r A r

r (2.10)

Where Ai and αi are the atomic screening parameters, tabulated in their work [21]. The

analytical expressions we have used to calculate the static potential for atoms are from

the works of Bunge and Barrientos [18].

2.2.2 Exchange and Polarization potentials

Exchange Potential

The additional term in the total potential that arises due the exchange of the

incident electron with one of the target electrons is termed as exchange potential. Hara

[21] adopted the „free electron gas exchange model‟ for exchange effect. He considered

the electron gas as a Fermi gas of non-interacting electrons when the total wave function

is anti-symmetrized in accordance with Pauli‟s exclusion principle. The exchange energy

40

is calculated by summing all momentum states up to Fermi level Ef . The Hartree

exchange potential energy is given by

22 1 1 1

, ln2 4 1

ex FV r k k

(2.11)

Where kF is the Fermi wave vector,

23 3Fk r

(2.12)

With

2 2 2F

F

k k I

k

(2.13)

This approximation is usually referred to as „Hara Free Electron Gas Exchange‟

(HFEGE) model. The target electron being bound has a negative energy and the

minimum energy required to make it zero is the ionization energy, which is denoted as I

(in Hartree). Here the approximation of the numerator in equation 2.13 implies that the

electron kinetic energy is given by E + I in the asymptotic region. This may not be a

serious inconsistency since the correct exchange potential is very small at large r, but a

possible way to correct equation 2.13 for large - r behavior is to remove the ionization

41

energy from it. This has led to the introduction of the „Asymptotically Adjusted‟

approximation, (AAHFEGE) which is as above except for the definition of η,

2 2

F

F

k k

k

(2.14)

The exchange potential given in equation 2.15 is used in most of our calculations

especially for the open shell atoms and molecules. A closed shell exchange potential is

also available in the literature from the work of Riley and Truhlar [22]. This potential has

the form,

21, 4

2ex D DV r k E E r

(2.15)

Where the local kinetic energy is

21

2D i stE k V

(2.16)

Polarization Potential

While treating elastic scattering of electrons with atoms or molecules, the effect of

distortion of the target system by the projectile is very important. This transient distortion

of the target by the approaching electron is due to the induced multipole moments and is

attractive in nature. This gives rise to an additional term in the potential energy called

polarization potential. An adiabatic expression for asymptotic polarization potential is

given by,

42

4 62 2

qdpolV

r r

(2.17)

Where we have neglected the higher order multipolar terms. Here αd and αq are

dipole and quadrupole static polarizabilities of the target atom. The potential is attractive

and varies asymptotically as r−4

at r → ∞ and acts at long range. There is singularity at r

= 0. This can be avoided by using a cut-off parameter „rc‟ as follows,

2

2 22

dpol

c

Vr r

(2.18)

This is called „Buckingham‟ polarization potential. Here the system is considered

to be adiabatic, but at high incident energies the response of the target cloud should also

depend on the „speed‟ of the electrons. So at high energies, we have to consider a

dynamic or energy dependent form for the polarization potential. The energy dependent

form of polarization potential is,

421( , )2 2 2 3 2 2 5( ) ( )

rr qdV r kdp

r r r rc c

(2.19)

43

Where rc is the energy dependent cut-off parameter. By using the Born-

approximation [3] we can get an expression for rc as,

38

krc

(2.20)

Here Γ represents the average excitation energy of the atom. The polarization

potential mentioned above is a long range potential. The simple r−4

behavior does not

hold at short distances. Therefore we have to consider the electron correlation effects at

short distances [3].

Correlation polarization potential

Electron correlation refers to everything that is left out of an HF calculation.

Correlation effect is produced, when an electron is far from the target, by the distribution

of inner electrons and the nucleus, resulting in an induced dipole moment [21]. In the

Hartree- Fock approach, electrons are moving in the average Self Consistent Field (SCF)

created by other electrons. Here we have considered the coulomb energy and the Pauli

exclusion principle. Correlation is the correction of this average interaction to allow

electrons to avoid one another in every region of configuration space. For example, when

an electron (bound or in continuum) is sufficiently far from the other electrons in the

closed shell, correlation takes the form of an inner electron (and nucleus) distribution,

resulting in an induced dipole moment.

Thus an approximated local density functional form of the short range (SR)

correlation and the long range polarization potential is given by [21- 24],

44

4

0

02

,( )

,d

corrSR

cp

r

V at r rV r

at r r

(2.21)

Here ro is the point of intersection between the short range correlation partcorr

SRV

and the long range dipole part – αd /2r4. This value may be approximated as the atomic

(or molecular) radius. The short range part of the potential in the above equation is

known as PZ form of the potential, named after the authors Perdew and Zunger[25]. It is

given by,

0.0311ln 0.0584 0.00133 ln 0.0084 , 1

1/ 27 4(1 )( )1 26 3 , 11/ 2 2(1 )

1 2

r r r r at rs S S S S

corr r rV rs sSR at r

Sr rs s

(2.22)

Where the constants are γ = -0.1423, β1 = 1.0529 and β2 = 0.3334 in au. 334 ( )

rs r is

the density parameter, with ρ(r) as the electronic charge density of the target. In figure

2.2 we have displayed the curve for VSR and VLR of Vcp versus the r in ao for Ne atom. The

intersection point ro is 2.245 ao as marked in the curve is used in our calculation for

correlation potential through the equation 2.20. As no single potential can describe the

polarization effect adequately for all targets in a wide range of incident energy, the need

for an alternate method was obvious. Gianturco and his co-workers [24] proposed a

damped energy polarization potential having the form,

45

Figure 2.2 Polarization potential curves of VSR and VLR in Vcp for Ne atom

Solid curve – ; dashed curve – αd /2r4. The vertical line shows intersection point ro

between the two potentials at 2.245ao

[ ( )]42

dV D rpol

r

(2.23)

With the damping factor D [ρ(r)] obtained through the charge density ρ(r) which

brings the potential to zero at r = 0. They represented the correlation potential in the

form,

2 2

3 3 2

( / )

2( )

r k Zd q

Vcp

r dc

(2.24)

corr

SRV

46

Where αd and αq are the electric dipole and quadrupole polarizabilities respectively. Z is

the atomic number of the target, dc is the cut-off parameter obtained by matching VCP

with at the orbital radius of the atom.

In conclusion we can say that for high energies (typically above 100 eV), the

dynamic polarization potential Vdp given in equation 2.19 may be used. While, at

intermediate energies (typically below 100 eV) the correlation potential shown in

equation 2.22 has to be considered in the calculations.

Zhang Polarization model

Zhang et al [26] have proposed a new approach for polarization potential such

that the potential have the correct asymptotic form αd /2r4 at large r values and approach

corr

SRV in the near-target region, which is parameter-free and is given by

2 2 22( )pol

dVr r

co

(2.25)

Where rco can be determined by letting 4

, (0) / 2 ( 0)corr

Z pol d co SRV r V r . In this

way ensures that VZ, Pol (r) equals corr

SRV at the origin and approaches corr

SRV in the near

target region. Furthermore, it contains some multipole and non-adiabatic corrections in

the intermediate region and smoothly approaches the correct asymptotic form for large r.

2.2.3 Absorption Potential

The imaginary part VI of the optical potential (Vopt) from equation 2.1 corresponds

to the „absorption‟ potential. The imaginary part corresponds to removal or absorption of

47

the incident particles into the inelastic channels. The idea of complex optical potential

was first introduced in nuclear physics [27] and later adopted to electron - atom

collisions.

There are several absorption potential models employed in the electron – atom

scattering theory from the literature, particularly the works of Reitan [28], Green et al

[29] and Staszewska et al [30, 31]. The imaginary part (Vabs), accounts for the total loss

of scattered flux into all the allowed channels of electronic excitation and ionization. In

the present work we have used the modified version [32] of the original model potential,

given by Staszewska et al [30, 31], which is a function of electronic charge density ρ(r),

and the local kinetic energy. It is a quasi-free, Pauli-blocking, dynamic absorption

potential (Vabs) given in au, as

12

( )abs loc eeV r v

(2.26)

Here, vloc is the local speed of the incident electron, and ζee denotes the average

total cross section of the binary collision of the incident electron with a target electron.

This is rewritten as a simple non-empirical formula [31] as,

2 28( ) ( 2 )( )1 2 332 10

TlocV r p k A A A

abs Fk EF i

(2.27)

The local kinetic energy of incident electron is obtained from

48

( )loc i R i st ex pT E V E V V V

(2.28)

At high energies only the first term is important in equation 2.27. Also, 23 3 ( )Fk r is

the Fermi wave vector magnitude and,

35

1 2

3 2 2(5 3 )

2 2 2 2( )

2 2 5 / 2(2 2 )2 22 (2 2 )

3 2 2 2( )

kFA

k p kF FA

p kF

k pFA k p

Fp k

F

(2.29)

Where, p2 = 2Ei, kF is the Fermi wave vector and ∆ is an energy parameter. Further, θ(x)

is the Heaviside step-function, such that θ(x) = 1 for x ≥ 0, and is zero otherwise. The

dynamic functions A1, A2 and A3 given above depend differently on ρ(r), I, ∆ and Ei. The

parameter ∆ is assumed to be fixed in the original model and determines a threshold

below which Vabs = 0, and the ionization or excitation is prevented energetically. Here we

have modified ∆ such that at impact energies close to the ionization threshold I, the

excitations to the discrete states also take place, but as Ei increases valence ionization

becomes dominant, together with the possibility of ionization of the inner electronic

shells. In the range of intermediate energies the Vabs shows a rather excessive loss of flux

into the inelastic channels.

For example in the case of atomic O (figure 2.3), at 100 eV the starting point of

this attractive potential is 0.85 ao. In figure 2.3 we also show the charge distribution

49

Figure 2.3: Radial charge density and absorption potential for O atom at 100 eV

Solid curve – (D(r) /10); dash - Vabs with Δ = I = 13.6 eV; dot - Vabs with Δ = 15.07 eV.

D(r) = 4πr2ρ(r) for comparison. At 500 eV (not shown in figure 2.3) the starting point is

0.25 ao. This amounts to a rather excessive loss of scattered flux into inelastic channels

i.e., into higher values of Qinel. This is partly also due to neglect of finer aspects in the

Vabs, like different transition probabilities for different states etc. The potential Vabs also

penetrates into the region of inner electronic shells, which are harder to be excited or

ionized. This absorption potential has a very good analytical facility, and it does have a

general predictive capacity, therefore, in order to employ this potential at medium

energies it is necessary to rectify it through ∆ > I. Hence we choose the ∆ parameter to be

a slowly varying function of Ei, around the value of I.

At high energies of course the potential Vabs and the cross section Qinel decrease in

strength. For a further clarification on this point, in figure 2.3 we give a graphical plot of

the radial charge density, 4πr2ρ(r) of the target, together with the attractive potential Vabs,

50

for two choices (a) ∆ = I, and (b) ∆ as a continuous variable around I. This figure

explains the behavior of the absorption potential as stated above. In most of the targets

investigated by us so far, the choice (b) results into a better accord with most of the

measured data. This theoretical procedure, although approximate, makes the absorption

potential physically acceptable.

Let us see how the Δ (Ei) can be made to vary with respect to incident energy Ei.

Here in below equation 2.30, Δmin, β and Δmax are three input parameters. The value of

Δmin is chosen to be 0.8 I for many targets (except H atom and H2 molecule) and Δmax is

generally at or near the ionization threshold I of the target. Thus,

Δ (Ei) = Δmin + β (Ei − Δmax) (2.30)

The β can be calculated by taking Δ (Ei) = Δmax at Ei = EP, where Ep is the energy at

which Qinel is maximum. So our Δ (Ei) will vary from Δmin to Δmax in small energy interval

and after Ei = EP it will become constant as Δ (Ei) becomes Δmax. As for example consider

the behaviour of Δ (Ei) for N atom, one of our targets in figure 2.5. We have kept Δmin to

be at 0.8 I which is equal to 11.64 eV, also close to first excitation threshold of N atom.

Further, Δmax = 14.53 eV i. e. the ionization Potential (IP) of Nitrogen atom. So our Δ (Ei)

will vary in small interval with respect to energy Ei and it will become constant after Ei =

Ep. The Ep of the N atom is found to be 70 eV (See figure 2.4). Here in the graph of

figure 2.5 we can visualize the behaviour of Δ (Ei) with respect to incident energy Ei.

Using this methodology we have varied our Δ (Ei) for all targets which are studied here.

51

10 100 1000

0.0

0.5

1.0

1.5

2.0

2.5

Ep

e-N

Qin

el(Å

2)

Ei (eV)

Qinel

Figure 2.4 Qinel for Electron impact on Nitrogen atom

Figure 2.5 Variation of Δ with respect to incident energy Ei for N atom

52

2.3 Methodology for molecules

The success of our theory for atomic targets inspired us to embark upon the study

of more important and more complex targets, molecules. The method discussed above

may be used to calculate theoretical cross section for electron scattering from atoms and

molecules.

2.3.1 High energy Additivity rules

Consider a molecule as a target of electron scattering. For high energy cases the

individual atomic cross sections are modified to include some of the molecular

properties. This can be related to molecular structure and/or its ionization potential.

Based on the molecular properties introduced in the simple Additivity Rules, different

versions of Modified Additivity Rules (MAR) are in described in available literature. The

MAR used to calculate the total ionization cross sections of molecules [33] can be

described briefly as follows. Ionization potential is one of the most important properties

of a target atom or a molecule. Especially when we calculate the total ionization cross

section of the molecule, it has much more importance than other parameters. So the

individual atomic inelastic cross sections are calculated after replacing the ionization

potentials of the constituent atoms by the ionization potential of the molecule. As for

example, the total cross section for O2 molecule by MAR method is the sum of the total

cross sections of individual O atom calculated using the ionization potential of O2

molecule. Then by using CSP-ic method as described in the further sections, we estimate

the Qion in the MAR. The geometry (bond length, bond angle etc) will also play an

important role in the evaluation of the cross section. This method, with the degree of

reliability associated with it, might enable us to estimate ionization cross sections for

53

bigger biological molecules. There are other variants of MAR in the literature. In one of

the approaches by Deutsch-Maerk group [34], the authors use a parameter (or function),

which is multiplied to the individual atomic cross sections to get the TCS (Q׳ = x Q).

Here x can be some scaling parameter (or function) that depends on the geometry of the

molecule [35].

2.3.2 Single Centre and Multiple Centre approaches

A major part of the present work is based on the single-centre approach to

calculate the molecular charge density, which is the major input for obtaining the total

and ionization cross sections of the molecules. We consider the charge density and static

potential for the molecule in question, expanded at a single centre, which is essentially

the geometrical centre of mass of the system. However, if the molecule has one very

heavy and the other relatively light atom (nucleus) then the expansion of the charge

density and potential is done through the centre of the heavier atom. In this process the

geometry, ionization potential and charge distribution are taken into account.

Diatomic molecules

Let us begin with the example of a hydride molecule, AH. Here A (e.g. C, O, F,

N, Al, Si etc.) is relatively a heavier atom, and hence the charge density of hydrogen

atom is expanded at the nucleus of A. Also, from the work of Bader [35] we can see that

the atomic charge densities in a molecule are not simply additive. If ρA is the charge

density of free atom A and ρB is that of free atom B, then, to a first approximation,

ρAB = ρA + ρB (2.31)

54

This is a crude approximation. It doesn't incorporate overlap or bonding. A simple

correction to the above was introduced and ρAH for a hydride AH was obtained

previously in our group [32, 36, and 37]. In that work electron impact cross sections were

calculated by simple method. In the quoted calculations, the charge density of H atom in

an AH hydride with bond length R is expanded to atom A [38, 39]. This quantity, as a

function of distance r from the nucleus of A is given [32, 36] as under

for (2.32)

and

for (2.33)

This result is based on the Bessel function expansion of the exponentials [39]. The

charge density thus obtained for H atom in the AH molecule was added to that of A atom

to get the total charge density of the hydride molecule. This resultant approximate charge

density was employed [32, 36] in the molecular scattering calculations. When a diatomic

molecule is formed by covalent bonding there is a partial migration of electronic charge

(sharing of electrons) across both the atomic partners. This is in accordance with the

electro-negativity of the two atoms. For instance, we know that in the HF molecule, the

electron charge from the hydrogen atom is migrated partly towards the F atom, giving a

small negative charge to the latter and a corresponding fractional positive charge to the H

atom. If this small charge transferred is q(A) then H loses q(A), while F gains q(A) of

55

charge. We can write a general expression for the charge density ρAH of a molecule AH

as thus,

(2.34)

where N(A) is the number of electrons in the atom A and N(H) = 1 for hydrogen. The

q(A) values for many hydrides are tabulated by Bader [35]. For example, the value,

called bond charge, for q(A) = - 0.760 au. for the HF molecule. Similarly, in the case OH

molecule, the bond charge q(A) = -0.585 au is migrated towards the O atom from the H,

giving a partial negative and partial positive charge to O and H atom respectively. The

same procedure can be applied to any diatomic molecule. Thus, for a diatomic molecule

AB, we write the following general expression,

(2.35)

where, fA and fB are the weighing factors that arise due to partial charge migration in

covalent bonding. Here, N(AB)=N(A)+N(B). Of course now the codes for molecular

wave functions have been developed for higher order of accuracy, but the above method

retains simplicity and gives reasonably good single centre charge densities. For a

diatomic molecule AB, we can expand the charge density of each atom from the centre of

mass of the system if A and B are comparable. For the homo-nuclear diatomic molecule

A2, we can expand the potential as well as charge density at the mass center from the half

bond length,

RAA of the A2 molecule.

56

Radial charge density is now available accurately in the Quantemol R – matrix

package also.

Simple polyatomic molecules

In case of simple polyatomic molecules, the concept of Single Centre Approach is

retained and we adopt a similar procedure as discussed for the diatomic targets. Here we

have to find the position of the centre of mass of the system using the geometry of the

molecule i.e. bond length, bond angle etc. Then the charge density and static potential of

each atom is expanded at the centre of mass from the respective atoms. After appropriate

addition as in equation (2.35), we normalize the expression to the total number of

electrons in the molecule. In this way we obtain a spherical charge density and potential,

which are approximate though, bear a dependence on molecular properties. This density

and the potential serve as input for our SCOP formalism for scattering calculations.

2.4 Other theoretical methods: the R-Matrix approach

In many collision problems there exits an internal region where the interaction is

complicated so that the solution cannot be found exactly and an external region where the

interaction is relatively simple and where often analytic solution exists. For example, in

the case of electron collisions with atoms and ions, at short distances the effective

interaction potential contains complicated electron exchange and correlation terms, while

at large distance it contains only the Coulomb potential and (or) other terms falling off as

inverse power of the distance between the electron and atomic nucleus.

The R-matrix method was originally introduced by Wigner and co-workers [40] in

1946, in the theory of nuclear physics. This method was then modified to include a wide

range of atomic processes including electron-atom scattering [41] and later extended to

57

treat scattering by diatomic molecules [6]. R-matrix method takes advantage of the fact

that the dynamics of the projectile - target system depends on the distance „r‟ between the

colliding particle and the target by dividing the complete range 0 ≤ r < ∞ of the radial

coordinate „r‟ into an internal region 0 ≤ r ≤ a and an external region a ≤ r < ∞. The two

regions are linked by joining the internal and external wave function at r = a.

2.5 Rotational excitation in polar molecules

In the case of molecules having a permanent dipole and quadrupole moments, the

non-spherical potential and the resulting cross section also need to be considered. Even

though molecules exhibit anisotropic and non-spherical potential, the corresponding cross

sections are more appreciable at low energies. Molecules with a permanent dipole

moment will exhibit a long range dipole potential, at large r. The corresponding inelastic

processes will include strong rotational excitation. For a linear rigid rotator molecule, the

rotational excitation cross section (Qrot) can be calculated using the Born approximation

for the excitation, J → J ׳ = J + 1, given by [41, 42].

2.5.1 Dipole potential – a realistic cut-off model

Let us consider the molecule to be a rigid rotator exhibiting a long range dipole

potential. For a molecule of dipole moment D

, the dipole potential is usually given by

the asymptotic point dipole model [42],

2

ˆ

r

rDrV

rD

(2.36)

This simple form equation (2.36) leads to overestimation of the rotational cross

sections. In reality the dipole potential should go to zero at the origin. Therefore in the

58

present work, we have introduced a modified dipole potential by defining a cut-off

parameter, inside which the potential varies linearly with radial distance and outside

which it takes the usual asymptotic form. Thus

(2.37)

An interesting choice for rd would be the magnitude of dipole moment D

in au.

Now, we have [43] calculated analytically the differential cross section for rotational

excitation from initial rotational quantum number J to final value J‟ (J‟ = J ± 1) by

applying the Born approximation to VDM, and this is given by an analytical expression

2

46

2 )()(1

12

'

5

'48),( d

d

d

d

i KrCosKr

KrSin

rKJ

JD

k

kE

d

d

(2.38)

Here k and k‟ are the magnitudes of initial and final wave-vectors, while K is the

magnitude of the inelastic wave-vector transfer. The total rotational cross section Qrot (Ei)

with J = 0 and J‟ = 1 is then obtained by numerical integration over the scattering angle θ,

at different energies Ei. The results of this model potential are shown for some polar

molecular targets in the last section of chapter 4. In figure 2.6 the point dipole potential

[42] and our model [43] are compared with each other.

3

2

0DM d

d

d

D rV r r

r

D rr r

r

59

0 1 2 3 4 5

-30

-20

-10

0

VD

P

R

Point Dipole

Realistic Dipole

Figure 2.6 A comparison of the realistic dipole potential model with point

dipole model

2.6 Grand total cross sections

The total cross section, QT, as we have defined in previous section 1.3.3, is the

spherical part of the grand total cross section, QTOT. For molecules with a permanent

dipole moment, rotational excitation cross section (Qrot) should be added to the total cross

section (QT). Here QT is the spherical part („sp‟) and Qrot is the non-spherical part („nsp‟)

of the grand total cross section. Thus grand total cross section (QTOT) is defined by

QTOT (Ei) = Qsp + Qnsp (2.39)

So

QTOT (Ei) = Qsp + Qnsp (2.40)

Here we have not included the vibrational excitation of the molecule. Considering

the present energy range (~ 15 eV to 2000 eV) and comparing with other processes,

60

vibrational excitation contribution is negligible [11]. So our present QTOT is vibrationally

elastic, and equation (2.39) is reliable when the incident energy is above 30 eV or so.

2.7 Complex Scattering Potential –ionization contribution (CSP-ic)

Going back to complex spherical potential, the inelastic channels in electron

atom/molecule scattering contain discrete excitations and ionizations, so that we can

partition the total inelastic cross section as per equation (1.8).

( )n

inel ion exc

n

Q Q A Q (2.41)

The second term in equation (2.40) arises mainly from the electronic excitations

having thresholds below the Ionization threshold I. The corresponding total excitation

cross sections become small progressively above I. Hence, as the incident energy

increases the second term in equation (2.40) dominates over the first, so that the

calculated inelastic quantity Qinel can be employed to derive the total ionization cross

section Qion. The inelastic cross section Qinel is not a directly measurable quantity in a

single experiment, but in view of the equation 2.40, we write presently

Qinel (Ei) ≥ Qion (Ei) (2.42)

At incident energies above I, the ionization processes begin to play a dominant

role due to the availability of infinitely many open channels of scattering, while

excitation corresponds to discrete finite channels. In an attempt to separate out the

ionization contribution in the cumulative inelastic scattering, we have introduced an

approximation [11-16] that starts with a ratio function,

61

(2.43)

Obviously R = 0 when Ei ≤ I. For a number of standard atomic – molecular

targets like Ne, Ar, O2, CH4, H2O, etc., for which several experimental ionization cross

section data-sets are known accurately the ratio is seen to be rising steadily as the energy

increases above the threshold, and approaching unity at high energies. Thus,

R(Ei)=0, for Ei ≤ I (2.44a)

R (Ei) = Rp, for Ei = Ep (2.44b)

R (Ei) ≈1, for Ei >> Ep (2.44c)

Here, Ep marks the incident energy at which our calculated Qinel reaches the

maximum and Rp = 0.7 stands for the value of R at the peak. The assumed value of Rp

corresponds to 70% contribution of ionization in the Qinel. The choice Rp = 0.7 is based

on the general observation that at energies close to the peak of the ionization cross

section the contribution of the target Qion is about 70-80 % of the total inelastic cross

sections Qinel. Though this is the general trend true for most targets, we find (from our

calculations compared with other data) that for lighter atomic/molecular systems H, He,

H2, the ratio seems to be ∼ 0.6. Even for heavier targets, the value of Rp (as 0.7 or 0.8)

finds an uncertainty for the average value 0.75. This is still smaller than the usual

experimental error of about 10-15 %. The higher limit of the ratio i.e. 80 % is observed

only in the targets having very high ionization potentials like Ne (I = 21.56 eV) [32].

Most of the targets discussed in this work however have the ionization potentials ranging

from ∼ 10 to ∼ 14 eV. Hence we have selected the lower limit of 70 %. We emphasize

ion i

i

inel i

Q ER E =

Q E

62

that the present choice Rp = 0.7 preserves the generality of our method. The quantum

mechanical origin of the present value of Rp seems to be the fact that gradually above the

threshold the transitions to continuum start dominating, since they correspond to an

infinite number of open scattering channels. The value of ratio R cannot be calculated in a

rigorous manner. It was observed by Antony et al. [15] based on R matrix calculations,

that even the largest electronic excitation cross section in LiH is about three orders of

magnitude smaller than the elastic cross sections. Thus the electronic excitation channels

are weak and the above assumption for the ratio is valid here.

Now, for the actual calculation of Qion from our Qinel we need R (Ei) as a

continuous function of energy Ei. Therefore, as in [20], we define a dimensionless

variable U = Ei/I, and express the ratio function as

R (Ei ) = 1 - f (U) =

U

U

aU

CC

ln

)(1 2

1

(2.45)

Equation (2.45) involves dimensionless parameters C1, C2, and a, which can be

determined from the three conditions stated in equations (2.44a-c). Having thus obtained

the parameters, we calculate Qion using equations (2.44, 2.45).

So the method CSP-ic can be well described by equations 2.40-2.45. Based on this

methodology we have calculated total cross sections for electron impact on various

targets shown in table 1.1. The calculations, results, discussions and conclusions for these

targets are discussed in forthcoming chapters.

63

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