Transcript Calculation of ionization cross sections of free radicals

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
SCOP Method

Formulation of the Complex Optical Potential, Vopt = VR + iVI
Complex Optical Potential
Real
Short Range
Static
Imaginary
Long Range
Exchange
RHF WF
Hara
Polarization
Buckingham
Energy Dependent
Final Form of the Complex Optical Potential
Vopt = Vst + Vex + Vpol + i Vabs
Absorption
Modified Model
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
Z
Vst r   
r
n

i 1
i
exp i r 
Exchange Potential: This potential arises due to exchange
of the incident electron with one of the target electrons. It is
short ranged potential.
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.
V p r   
r
d r 2
2

2 3
c
r
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.
SCOP method continued…
SCOP method1,2 for QT
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)
Then the QT is found through,
The grand TCS, QTOT is,
QT (Ei )  Qel (Ei )  Qinel (Ei )
QTOT ( Ei )  QT ( Ei )  Qrot ( Ei )
Present energy range  From ionization threshold to 2keV
1K
N Joshipura et al, J Phys. B: At. Molec. Opt. Phys., 35 (2002) 4211
2 K N Joshipura et al, Phys. Rev. A, 69 (2004) 022705
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.
n
QT   QSR ( A)  QPol ( M )
i 1
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
  AB    A   B
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
 AH r, R   A r    H r, R
When diatomic molecule is formed by covalent bonding there is
partial migeration of charge across the either atomic partners.
 AH r , R  
N  A   q  A
N H   q A
 A r  
 H r , R 
N  A
N H 
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.
CSP-ic Method
The Complex Scattering Potential-ionization
contribution, CSP-ic method1,2 for Qion
In CSP-ic method the main task is to extract out the total
ionization cross section from the total inelastic cross section.
Qinel (Ei )  Qion (Ei )  Qexc
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.
1K
2
N Joshipura et al, J Phys. B: At. Molec. Opt. Phys., 35 (2002) 4211
K N Joshipura et al, Phys. Rev. A, 69 (2004) 022705
SPU-VVN
CSP-ic Method
The Complex Scattering Potential-ionization
contribution, CSP-ic method1,2 for Qion
The CSP-ic originates from the
inequality,
Now we will define a ratio,
Qinel ( Ei )  Qion ( Ei )
Qion ( Ei )
R( Ei ) 
Qinel ( Ei )
Using R(Ei ) we can determine the Qion from Qinel
This method is called CSP-ic
1K
2
N Joshipura et al, J Phys. B: At. Molec. Opt. Phys., 35 (2002) 4211
K N Joshipura et al, Phys. Rev. A, 69 (2004) 022705
SPU-VVN
CSP-ic method continued…
Above ratio has three conditions to satisfy:
SPU-VVN
 0, at E i  I

R( Ei )   R p , at Ei  E p
~ 1, for E  E
i
p

where subscript ‘p’ denotes the value at the peak of Qinel
This ratio proposed to be of the form, 1 – f (U),
E
where U  i
I
ln U 
ln U 
 C2
 C2


; f (U )  C1 
& R(U )  1  C1 

U  a U 
U  a U 
This is the method of CSP-ic.
Using this energy dependant ratio R ,Qion from Qinel can be extracted.
Results
Figure 1: e – CH41
14
e - CH4
Upper Curves
Present QT
Jain QT
Zecca QT
Lower Curves
Prestnt Qion
Chatham Qion
Nishimura Qion
Present Qexc
Nakano QNDiss
Kanik Qexc
12
10
8
2
TCS (Å )
SPU-VVN
6
4
2
0
1
10
2
10
Ei(eV)
1
K N Joshipura et al, Phys. Rev. A, 69 (2004) 022705
10
3
Results Continued…
Figure 2: e – O3 at 100 eV1
16
Total Cross Sections in Å 2
14
12
58%
of QT
10
42%
of QT
8
6
71%
of Qinel
4
29%
of Qinel
2
0
1
QT
Qel
Qinel
Qion
 Qexc
K N Joshipura et al, J Phys. B: At. Mol. Opt. Phys. 35 (2002) 4211
Results Continued…
Figure 3: Plasma molecules
e - CF4
7
Present
BEB
Christophorou
Poll
Nishimura
6
2
Qion (Å )
5
4
3
2
1
0
10
2
10
Ei (eV)
3
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
   gnl rnl 2 nl f u 
n,l
Where r 
is the mean square radius of the (n,l) subshell
and g 
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.
2
nl
nl
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.
   g  r   f u 

2
j
j
j
j
Where summation is now carried out over molecular orbitals j.
Thanks
Professor K N Joshipura
Bobby, Chetan, Bhushit and Chirag
Department of Physics, Sardar Patel University, Vallabh Vidyanagar 388 120, India
Professor N J Mason
Director, CeMOS, Open University, Milton Keynes, United Kingdom
Professor Jonathan Tennyson
Head, Dept of Physics & Astronomy, University College London, United Kingdom