Electron impact excitation

and

its application to plasma modeling

Reetesh Kumar Gangwar Department of Physics, IIT Roorkee

Roorkee–247667, INDIA email : reeteshkr@gmail.com

)},()F ..., , ,(Α{

)1(),()F ..., , ,(

a

b

a,

b,

1NkN21

1NkN21

a

b

+Φ×

+−+Φ=+

−→

DWrela

bDWrel

bRDWba NUVT

µ

µ

∑−

+−==

++

N

jNrZV

11

1

1Nj rr

Relativistic Distorted Wave Theory

Interaction potential

relativistic distorted wavefunctions of projectile electron in initial (final) state distortion potential

antisymmetrization operator

Relativistic target atom wavefunction in the initial (final) state

T-matrix

Distortion potential

The scattering amplitude:

The differential cross section (DCS):

The total cross section:

ΩΩ

= ∫ ddd

Totσσ

),µ,M;J,µ,M(JTkk

π)(),µ,M;J,µ,Mf(J iiifffDWfi

i

fiiifff →= 22

2

1221 ∑+

=ff,ii µ,M,µM

iiifffi

),µ,M;J,µ,Mf(J)J(

DCS

Theoretical Calculation

Final quantum numbers

Initial quantum numbers

22 2 2 4 2 4Ar[18][1s 2 2 2 3 3 ]3s p p p ps

Ground state:

1p0 [J=0]: a1 2 43p 3p

Excited states:

3d12[J=0]: a1   2 3 13p 3p 3d + a2   1 4 13p 3p 5s +a3   1 4 13p 3p 4s        

 3d11[J=1]: a1   2 3 13p 3p 3d + a2   2 3 13p 3p 3d +  a3   2 3 13p 3p 5s + a4   1 4 13p 3p 3d + a5   1 4 13p 3p 5s +  a6   1 4 13p 3p 4s + a7   2 3 13p 3p 4s

3d10[J=2]: a1   2 3 13p 3p 3d +  a2   2 3 13p 3p 5s  + a3   2 3 13p 3p 3d + a4   1 4 13p 3p 3d +a5   1 4 13p 3p 3d + a6   2 3 13p 3p 4s

3d9[J=4]: a1 2 3 13p 3p 3d        

3d8[J=3]: a1 2 3 13p 3p 3d + a2 1 4 13p 3p 3d + a32 3 13p 3p 3d

2s5[J=2]: a1   2 3 13p 3p 5s  + a2   2 3 1 3p 3p 3d + a3   2 3 13p 3p 3d +  a4   1 4 13p 3p 3d + a5   2 3 13p 3p 4s  + a61 4 13p 3p 3d

2s4[J=1]: a1   2 3 13p 3p 5s +  a2   2 3 13p 3p 3d + a3   2 3 13p 3p 3d + a4   1 4 13p 3p 5s + a5   2 3 13p 3p 4s + a6   1 4 13p 3p 3d +  a7   1 4 13p 3p 4s

Wave Functions for 3p53d and 3p55s Transitions

3d7[J=2]: a1   2 3 13p 3p 3d +  a22 3 1 3p 3p 3d + a3   1 4 13p 3p 3d + a4   1 4 13p 3p 3d +  a5   2 3 13p 3p 5s  +  a6   2 3 13p 3p 4s

3d6[J=3]: a12 3 13p 3p 3d + a2 1 4 13p 3p 3d + a3 2 3 13p 3p 3d

3d5[J=1]: a1   2 3 13p 3p 3d + a2   2 3 13p 3p 3d + a3   1 4 13p 3p 3d + a4   1 4 13p 3p 5s + a5   2 3 13p 3p 5s +  a6   1 4 13p 3p 4s + a7   2 3 13p 3p 4s

2s3[J=0]: a1   1 4 13p 3p 5s + a2   2 3 13p 3p 3d +a3   1 4 13p 3p 4s

2s2[J=1]: a1   1 4 13p 3p 5s + a2   2 3 13p 3p 3d +  a3   2 3 13p 3p 5s + a4   1 4 13p 3p 3d + a5   1 4 13p 3p 4s + a6   2 3 13p 3p 4s    + a7   2 3 13p 3p 3d

3d4[J=2]: a1   1 4 13p 3p 3d + a2   2 3 13p 3p 3d + a3   1 4 13p 3p 3d + a42 3 1 3p 3p 3d  +  a5   2 3 13p 3p 5s  +  a6   2 3 13p 3p 4s

3d3[J=3]: a1 1 4 13p 3p 3d + a22 3 13p 3p 3d +a3 2 3 13p 3p 3d

3d2[J=2]: a1   1 4 13p 3p 3d + a22 3 1 3p 3p 3d +  a3   1 4 13p 3p 3d + a4   2 3 13p 3p 3d +  a5   2 3 13p 3p 5s  +  a6   2 3 13p 3p 4s

3d1[J=1]: a1   1 4 13p 3p 3d + a2   2 3 13p 3p 3d + a3   2 3 13p 3p 3d + a4   2 3 13p 3p 4s +  a5   2 3 13p 3p 5s +  a6   1 4 13p 3p 4s +a7   1 4 13p 3p 5s                                                                                                                                                                                                                                    

Wave Functions for 3p53d and 3p55s Transitions

States NIST GRASP

3d5 [J=1] 9.32x10-2 5.45x10-2

3d1 [J=1] 0.106 0.164

2s4 [J=1] 2.70x10-2 2.41x10-2

2s2 [J=1] 1.19x10-2 9.83x10-3

Optical Oscillator Strength for allowed Transitions

Salient features of RDW method

Relativistic effects should be included in the study of electron impact excitation of atoms where spin-orbit effects are important. Produces directly fine-structure target states. Includes spin-polarized electrons in a natural way. Our Relativistic Distorted Wave (RDW) method involves solving the Dirac equations to describe both the bound and continuum electrons. The relativistic effects are thus incorporated to all orders in a natural way.

State a1 a2 a3 a4 a5 a6 a7

1p0[J=0] 1.0

3d12[J=0]            0.9924 -0.1118 0.0513

 3d11[J=1] 0.7976 -0.5128 0.2130 0.2053 0.1065 0.0434 -0.0117

3d10[J=2] 0.6659 0.6265 -0.2892 0.2778 -0.0568 -0.0051

3d9[J=4] 1.0

3d8[J=3] 0.9896 0.1364 0.0449

2s5[J=2] 0.7761 -0.5556 0.2401 -0.1604 0.0675 0.0332

2s4[J=1] 0.9674 -0.1789 0.1314 0.0895 0.0787 -0.0233 -0.0034

3d7[J=2] 0.8819 0.3825 -0.2697 -0.0561 0.0009 0.0002

3d6[J=3] 0.9842 -0.1760 -0.0204

3d5[J=1] 0.7123 0.5711 -0.3947 -0.1021 0.0094 0.0087 -0.0063

2s3[J=0] 0.9925 0.1089 0.0561

2s2[J=1] 0.9827 0.1100 -0.1073 - 0.0709 0.0699 -0.0236 -0.0101

3d4[J=2] 0.9411 0.2532 0.1962 0.1086 -0.0003 -0.0001

3d3[J=3] 0.9749 0.1714 -0.1421

3d2[J=2] 0.9243 -0.2976 -0.1930 0.1289 -0.0516 -0.0230

3d1[J=1] 0.8921 0.4455 0.0626 -0.0278 -0.0261 0.0116 0.0108  

Mixing coefficients ai’s for 3p53d and 3p55s transition

; RDW results

; RM-SS calculation, Madison et al

; Unitarized distorted wave results of Madison et al

; Theoretical calculations of Bubelev et al

; Experimental measurements of Chilton et al

; Experimental measurements of Chutjian et al

ICS for the excitation of 3p53d levels from ground state

For allowed transitions the cross section takes on the Bethe-Born form at larger energies

where E (Energy) and ICS are in atomic units, b0 and b1 are the fitting parameters.

( ) 2

010]ln[1 aEbb

EICS +=

Analytic Fits to ICS

2

00

1aEbICS b=

For Forbidden transitions are fitted by the following expression

Resonance levelg J = 1

Metstable Level J= 2, 0

Inter -excitation

Energy level diagram of argon

Population for 1s levels

OES Measurements : Zhu and Pu, J. Phys. D: Appl. Phys. 43 015204 (2010).

Population for 2p levels

OES Measurements : Zhu and Pu, J. Phys. D: Appl. Phys. 43 015204 (2010).

Two Emission lines I750 and I696

750.388 nm

696.54 nm

1s5 1s2

2p2 2p1

Calculation of intensities I750 and I696

The intensities are directly proportional to population densities

750750 2p1

g I = C X P

696696 2p2

g I = C X P

X2p is the relative population density given as X2p = n2p / n0

n0 denotes the total atom density and

Pg denotes the gas pressure.

C750 and C696 are the calibration constants.

Intensity of the 750.38 and 696.54 nm lines

Experimental Measurements : Palmero et al J. Phys. D: Appl. Phys., 101 053306 (2007).

Submitted in JAP-AIP

Conclusion Future Interest

Thus, using detailed fine structure and reliable cross sections one can also improve the CR model to get better results. Presently I am extending my work for hot dense plasma particularly tungsten ions.

Contributors

Prof. Rajesh Srivastava (Ph. D Supervisor), I. I. T. Roorkee

Dr. Lalita Sharma (Collaborator), I. I. T. Roorkee

Prof. A. D. Stauffer (Collaborator),

York University, Toronto

THANK YOU