A semi-rigorous method Modified single center additivity rule msc-ar for

calculating various total cross sections

Minaxi Vinodkumar

Department of Physics and Astronomy, The Open University,

Milton Keynes, MK7 6AA, UK.

V. P. & R. P. T. P. Science College, Vallabh Vidyanagar – 388 120, INDIA

Outline of the talk

Why this work? Theoretical Methods Employed

SCOP & CSP-ic and DM formalism Theory Results

Summary & Conclusion Thanks

Why this work ? Applications of e-atom / molecule CS to,

atmospheric sciences (ozone, climate change etc.) plasma etching understanding & modeling plasmas in fusion devices In radiation physics (medical science) etc.

Electrons: an effective source Difficulty in performing experiments

Expensive Limitation to targets Time consuming

Limitations to accurate theoretical methods: Slow and tedious calculations Limitation to energy Limitation to targets

Need for simple, reliable and fast calculations

Complex Optical PotentialComplex Optical Potential

RealReal ImaginaryImaginary

StaticStatic ExchangeExchange PolarizationPolarization

AbsorptionAbsorption

HaraHaraRHF WFRHF WF BuckinghamBuckingham

Energy DependentEnergy Dependent

Modified ModelModified Model

Short RangeShort Range Long RangeLong Range

Final Form of the Complex Optical Potential

Vopt = Vst + Vex + Vpol + i Vabs

Formulation of the Complex Optical Potential, Vopt = VR + iVI

SCOP Method

Various Model Potentials

Real Potentials

Static Potential:The potential experienced by the incident

electron upon approaching a field of an undisturbed target charge cloud.

The static charge density can be calculated using HF wave functions given in terms of STO.

Cox and Bonham gave analytical expression for static potential involving the sum of Yukawa terms

rr

ZrV i

n

iist

exp1

Hara adopted free electron gas exchange model. He considered the electron gas as a Fermi gas of non interacting electrons when the total wave function is antisymmetrised in accordance with Pauli’s exclusion principle.

Polarisation Potential: This potential arises due to the transient distortion produced in the target charge cloud due to the incoming incident electron.

We used the correlation polarization potential at low energy and dynamic polarization potential given by Khare et al at high energies.

Exchange Potential: This potential arises due to exchange of the incident electron with one of the target electrons. It is short ranged potential.

Where rc is the energy dependent cut off parameter.

Absorption Potential : This potential accounts for the removal or absorption of incident particles into inelastic channel. The imaginary part of the absorption potential accounts for the total loss of the scattered flux into all the allowed channels of electronic excitation and ionization.

We use the quasi free, Pauli blocking, dynamic absorption potential given by Staszewska which is function of charge density local kinetic energy and the raadial distance r. We have modified the absorption potential to account for screening of inner electrons by the outer ones.

322

2

c

dp

rr

rrV

SCOP method1,2 for QT

1 K N Joshipura et al, J Phys. B: At. Molec. Opt. Phys., 35 (2002) 42112 K N Joshipura et al, Phys. Rev. A, 69 (2004) 022705

1. Formulate Schrödinger eqn using the SCOP

2. Solve this eqn numerically to generate the complex phase shifts using the “Method of Partial Waves”

3. Obtain the Qel and Qinel (Vibrationally & rotationally elastic)

)()()( iinelieliT EQEQEQ Then the QT is found through,

SCOP method continued…

The grand TCS, QTOT is, )()()( irotiTiTOT EQEQEQ

Present energy range From ionization threshold to 2keV

Various Additivity Rules

Simple Additivity Rule (AR): The total cross section for a molecule AB is given by

Q(AB) = Q(A) + Q(B)

This is crude approximation and works for few molecules with larger separation between the atoms.

Modified Additivity Rule(MAR): The individual cross sections are modified to incorporate the molecular properties such as structure and ionization energy and the polarizability of the target.

)()(1

MQAQQ Pol

n

iSRT

Various Additivity Rules

Single Center Approach (SC): Additivity methods do not take into account the bonding between the molecules. Single center approach takes into account the bonding of the atoms.

The molecular charge density which is major input for obtaining the total cross section.

For the diatomic molecule AB, the simplest additivity rule for the charge density of the molecule is

BAAB

This again does not include the bonding of atoms in molecule.

For the hydride AH, the charge density is made single center by expanding the charge density of lighter H atom at the centre of heavier A atom for e.g. C, N or O.

Various Additivity Rules

RrrRr HAAH ,,

For the polyatomic complex molecules, we use group additivity rule MSC-AR. The number of centres and their position will depend on the structure of the molecule.

In case of C2H6 molecule, we identify two scattering centers at the center of each carbon atom. The charge density of all three hydrogen atoms is expanded at the centre of Carbon atom and the total charge density is then renormalised to get total number of electrons in the molecule.

When diatomic molecule is formed by covalent bonding there is partial migeration of charge across the either atomic partners.

RrHN

AqHNr

AN

AqANRr HAAH ,,

The Complex Scattering Potential-ionization contribution, CSP-ic method1,2 for Qion

1 K N Joshipura et al, J Phys. B: At. Molec. Opt. Phys., 35 (2002) 42112 K N Joshipura et al, Phys. Rev. A, 69 (2004) 022705

SPU-VVNCSP-ic Method

In CSP-ic method the main task is to extract out the total ionization cross section from the total inelastic cross section.

exciioniinel QEQEQ )()(

The first term on RHS is total cross section due to all allowed ionization processes while the second term mainly from the low lying dipole allowd transistions which decreases rapidly at high energies.

The Complex Scattering Potential-ionization contribution, CSP-ic method1,2 for Qion

1 K N Joshipura et al, J Phys. B: At. Molec. Opt. Phys., 35 (2002) 42112 K N Joshipura et al, Phys. Rev. A, 69 (2004) 022705

)()( iioniinel EQEQ

)(

)()(

iinel

iioni EQ

EQER

The CSP-ic originates from the inequality,

Now we will define a ratio,

Using R(Ei ) we can determine the Qion from Qinel

This method is called CSP-ic

SPU-VVNCSP-ic Method

U

U

aU

CCUR

ln1)( 2

1

U

U

aU

CCUf

ln)( 2

1

This ratio proposed to be of the form, 1 – f (U),

whereI

EU i

pi

pip

i

i

EEfor

EEatR

IEat

ER

,1~

,

,0

)(Above ratio has three conditions to satisfy:

where subscript ‘p’ denotes the value at the peak of Qinel

SPU-VVN

& ;

CSP-ic method continued…

This is the method of CSP-ic. Using this energy dependant ratio R ,Qion from Qinel can be extracted.

Results

Figure 1: e – CH41

1 K N Joshipura et al, Phys. Rev. A, 69 (2004) 022705

SPU-VVN

Results Continued…

Figure 2: e – O3 at 100 eV1

1 K N Joshipura et al, J Phys. B: At. Mol. Opt. Phys. 35 (2002) 4211

Results Continued…

Figure 3: Plasma molecules

102 103

0

1

2

3

4

5

6

7 Present BEB Christophorou Poll Nishimura

e - CF4

Qio

n (Å2 )

Ei (eV)

Summary & Conclusion on SCOP & CSP-ic

Results on most of the molecules studied shows satisfactory agreement with the previous investigations where ever available.

First estimates of the Qion for many aeronomic, plasma & organic molecules are also done. We believe it to be reliable from our previous results.

Advantages

Quantum mechanical approximation

Calculating different CS from the same formalism

Simple and fast method

First initiation to extract Qion from Qinel

Disadvantages

Spherical approximation

Lower energy limit of ~10eV

Semi empirical method to find Qion

DM Formalism

The original concept of Deustch and Meark formalism was developed for the calculation of the atomic ionization cross sections. It was then modified for the molecular targets.

In DM formalism only direct ionization processes are considered. That is prompt removal of a single electron from the electron shell by the incoming electron therefore it is not possible to distinguish between single and multiple ionization where inner shell ejection occurs.

The semiclassical formula used for the calculation of the ionization cross sections for the atoms was given as

ufrg nlnlln

nl 2

,

2nlr

nlg

Where is the mean square radius of the (n,l) subshell and are the appropriate weighing factors given by Deustch et al. F(u) is the energy dependence of ionization cross section while zeta depends on the orbital angular momentum quantum number of the atomic electrons.

DM formalism can easily be extended to the case of molecular ionization cross section provided one carries out a Mulliken or other molecular orbital population analysis which expresses the molecular orbitals in terms of atomic orbitals.

ufrg jjj

j 2

Where summation is now carried out over molecular orbitals j.

10 100 10000

1

2

3

4

5

TIC

S (

Å2 )

Ei (eV)

Present Qion

Chatham Qion

Orient Qion

Nishimura Qion

Jain Qion

Kim Qion

e - CH4

Thanks

Professor K N Joshipura Bobby, Chetan, Bhushit and Chirag

Department of Physics, Sardar Patel University, Vallabh Vidyanagar 388 120, India

Professor N J MasonDirector, CeMOS, Open University, Milton Keynes, United Kingdom

Professor Jonathan TennysonHead, Dept of Physics & Astronomy, University College London, United Kingdom