MSE Schr¨odinger Equation Results Conclusions
(Motion) Stark Effect models
L. Fern´andez-Menchero
ADAS, University of Strathclyde. United Kingdom. Institut Max Plank f¨ur Plasmaphysik. Garching, Germany.
ADAS Workshop 2012, CEA Cadarache, France. September 25th, 2012
ADAS Workshop 2012, CEA Cadarache, France. L. Fern´andez-Menchero (Univ. Strathclyde) (Motion) Stark Effect models / 28 MSE Schr¨odinger Equation Results Conclusions
Contents
1 The Motion Stark Effect (MSE)
2 Solution of the Schor¨odinger Equation
3 Energies and wave functions of the SHA
4 Conclusions
ADAS Workshop 2012, CEA Cadarache, France. L. Fern´andez-Menchero (Univ. Strathclyde) (Motion) Stark Effect models / 28 MSE Schr¨odinger Equation Results Conclusions
Contents
1 The Motion Stark Effect (MSE) The Motion Stark Effect MSE diagnostic
2 Solution of the Schor¨odinger Equation
3 Energies and wave functions of the SHA
4 Conclusions
ADAS Workshop 2012, CEA Cadarache, France. L. Fern´andez-Menchero (Univ. Strathclyde) (Motion) Stark Effect models / 28 MSE Schr¨odinger Equation Results Conclusions
The Motion Stark Effect The Motion Stark Effect (MSE)
In fusion devices, like tokamaks, Neutral Beam Injectors (NBI) insert high-energy neutral atoms inside the magnetic confined plasma. As the atoms are neutrals, they do not react as a hole system to these magnetic fields, being able to penetrate deeply into the plasma until they are ionised. Internally, the neutrals can fell simultaneous electric and magnetic fields, which disturb their electronic structure. The atom is moving rapidly under an intense magnetic field, what causes a Lorentz electric field. The atom is under the influence of simultaneous electric and magnetic fields.
ADAS Workshop 2012, CEA Cadarache, France. L. Fern´andez-Menchero (Univ. Strathclyde) (Motion) Stark Effect models / 28 MSE Schr¨odinger Equation Results Conclusions
MSE diagnostic MSE spectrum diagnostic
MSE spectroscopy to determine magnetic and electric fields in ASDEX-Upgrade Tokamak.
Figure: MSE spectrum of Dα line
ADAS Workshop 2012, CEA Cadarache, France. L. Fern´andez-Menchero (Univ. Strathclyde) (Motion) Stark Effect models / 28 MSE Schr¨odinger Equation Results Conclusions
MSE diagnostic Diagnostic setup overview
Figure: Schematic overview of MSE diagnostic in ASDEX-Upgrade Tokamak
ADAS Workshop 2012, CEA Cadarache, France. L. Fern´andez-Menchero (Univ. Strathclyde) (Motion) Stark Effect models / 28 MSE Schr¨odinger Equation Results Conclusions
MSE diagnostic MSE lines of sight
Figure: Poloidal overview of MSE sight lines
ADAS Workshop 2012, CEA Cadarache, France. L. Fern´andez-Menchero (Univ. Strathclyde) (Motion) Stark Effect models / 28 MSE Schr¨odinger Equation Results Conclusions
MSE diagnostic MSE lines of sight
Figure: Toroidal overview of MSE sight lines
ADAS Workshop 2012, CEA Cadarache, France. L. Fern´andez-Menchero (Univ. Strathclyde) (Motion) Stark Effect models / 28 MSE Schr¨odinger Equation Results Conclusions
Contents
1 The Motion Stark Effect (MSE)
2 Solution of the Schor¨odinger Equation Hamiltonian Basis set
3 Energies and wave functions of the SHA
4 Conclusions
ADAS Workshop 2012, CEA Cadarache, France. L. Fern´andez-Menchero (Univ. Strathclyde) (Motion) Stark Effect models / 28 MSE Schr¨odinger Equation Results Conclusions
Hamiltonian Hamiltonian
Zero order: Coulomb and external constant electric field. Perturbation: constant magnetic field and fine structure. Hydrogen atom under a constant electric field must be determined by any exact method beyond perturbation theory. RHA Rydberg hydrogen atom: Solution to the Schr¨odinger equation for the unperturbed hydrogen atom. Usual wave functions in spherical coordinates and labeled by quantum numbers n, l and m. SHA Stark hydrogen atom: Solution to the Schr¨odinger equation for the hydrogen atom under a constant electric field, which can tend to zero. Wave functions described in parabolic coordinates and labeled by quantum numbers n, k and m. Bethe, H. A. and Salpeter, E. E., 1957, Quantum Mechanics of One- and Two-Electron Systems, New York: Academic Press ADAS Workshop 2012, CEA Cadarache, France. L. Fern´andez-Menchero (Univ. Strathclyde) (Motion) Stark Effect models / 28 MSE Schr¨odinger Equation Results Conclusions
Hamiltonian SHA Hamiltonian. The complex coordinate method.
Zero order Hamiltonian:
1 1 H = 2 + Fr cos θ 0 −2µ ∇ − r 2 ∂ ∂ 2 ∂ ∂ 1 ∂2 = ξ η ξ + η ∂ξ ∂ξ − ξ + η ∂η ∂η − 2ξη ∂ϕ2 2 1 + F (ξ η) − ξ + η 2 −
ADAS Workshop 2012, CEA Cadarache, France. L. Fern´andez-Menchero (Univ. Strathclyde) (Motion) Stark Effect models / 28 MSE Schr¨odinger Equation Results Conclusions
Hamiltonian SHA Hamiltonian. The complex coordinate method.
Zero order Hamiltonian:
1 1 H = 2 + Fr cos θ 0 −2µ ∇ − r 2 ∂ ∂ 2 ∂ ∂ 1 ∂2 = ξ η ξ + η ∂ξ ∂ξ − ξ + η ∂η ∂η − 2ξη ∂ϕ2 2 1 + F (ξ η) − ξ + η 2 −
Complex coordinate rotation.
′ i r = r e ϑ
ADAS Workshop 2012, CEA Cadarache, France. L. Fern´andez-Menchero (Univ. Strathclyde) (Motion) Stark Effect models / 28 MSE Schr¨odinger Equation Results Conclusions
Hamiltonian SHA Hamiltonian. The complex coordinate method.
Zero order Hamiltonian:
−2iϑ −iϑ e e i H (ϑ) = 2 + e ϑ Fr cos θ 0 − 2µ ∇ − r 2 e−2iϑ ∂ ∂ 2 e−2iϑ ∂ ∂ e−2iϑ ∂2 = ξ η ξ + η ∂ξ ∂ξ − ξ + η ∂η ∂η − 2ξη ∂ϕ2 2 e−iϑ eiϑ + F (ξ η) − ξ + η 2 − Complex coordinate rotation.
′ i r = r e ϑ
ADAS Workshop 2012, CEA Cadarache, France. L. Fern´andez-Menchero (Univ. Strathclyde) (Motion) Stark Effect models / 28 MSE Schr¨odinger Equation Results Conclusions
Basis set Variational method: basis set
N N m 1 imϕ | | − ξ+η Ψ(ξ,η,ϕ) = e (ξη) 2 e 2 cklm ΛNk (ξ)ΛNl (η) √ π 2 Xk=1 Xl=1 Lagrange-Laguerre-mesh polynomials:
i LN (x) ΛNi (x) = ( 1) √xi − x xi − xi : zeros of the Laguerre polynomial LN (x).
ADAS Workshop 2012, CEA Cadarache, France. L. Fern´andez-Menchero (Univ. Strathclyde) (Motion) Stark Effect models / 28 MSE Schr¨odinger Equation Results Conclusions
Basis set Secular equation
|m| − ξ+η |m| − ξ+η Sklk′l′m = (ξη) 2 e 2 ΛNk ΛNl (ξη) 2 e 2 ΛNk ΛNl D E
|m| − ξ+η |m| − ξ+η Hklk′l′m(ϑ) = (ξη) 2 e 2 ΛNk ΛNl Hˆ (ϑ) (ξη) 2 e 2 ΛNk ΛNl D E
Secular equation
(H ES) C = 0 −
ADAS Workshop 2012, CEA Cadarache, France. L. Fern´andez-Menchero (Univ. Strathclyde) (Motion) Stark Effect models / 28 MSE Schr¨odinger Equation Results Conclusions
Contents
1 The Motion Stark Effect (MSE)
2 Solution of the Schor¨odinger Equation
3 Energies and wave functions of the SHA Resonances Wave functions
4 Conclusions
ADAS Workshop 2012, CEA Cadarache, France. L. Fern´andez-Menchero (Univ. Strathclyde) (Motion) Stark Effect models / 28 MSE Schr¨odinger Equation Results Conclusions
Resonances Calculated eigenvalues
F = 0.005 a.u. m = 0 1e+05
1
1e-05 Imaginary part (a.u.)
1e-10
1e-15 -0.8 -0.6 -0.4 -0.2 0 0.2 Real part (a.u.)
Figure: Calculated eigenvalues for H atom under a constant electric field for several values of the complex rotation angle ϑ.
ADAS Workshop 2012, CEA Cadarache, France. L. Fern´andez-Menchero (Univ. Strathclyde) (Motion) Stark Effect models / 28 MSE Schr¨odinger Equation Results Conclusions
Resonances Calculated eigenvalues
F = 0.005 a.u. m = 0 1e+05
1
1e-05 2 -1 0
2 1 0 Imaginary part (a.u.) 1 0 0 1e-10
1e-15 -0.8 -0.6 -0.4 -0.2 0 0.2 Real part (a.u.)
Figure: Calculated eigenvalues for H atom under a constant electric field for several values of the complex rotation angle ϑ.
ADAS Workshop 2012, CEA Cadarache, France. L. Fern´andez-Menchero (Univ. Strathclyde) (Motion) Stark Effect models / 28 MSE Schr¨odinger Equation Results Conclusions
Resonances Energies
0 3 2 0 3 1 1 3 0 0 3 0 2 3 -2 0 3 -1 1 -0.1 2 1 0 2 -1 0 2 0 1 -0.2
-0.3 State Energy (a.u.) -0.4
-0.5 1 0 0
0 0.002 0.004 0.006 0.008 0.01 Electric field intensitu (a.u.)
Figure: State energies of the H atom versus electric field intensity for m = 0, 1, 2.
ADAS Workshop 2012, CEA Cadarache, France. L. Fern´andez-Menchero (Univ. Strathclyde) (Motion) Stark Effect models / 28 MSE Schr¨odinger Equation Results Conclusions
Resonances Widths
0.025
0.02
0.015 3 0 2
3 -1 1 State Width (a.u.) 0.01 3 0 0
3 -2 0 3 1 1 0.005
3 2 0 2 0 1 0 1 0 0 2 -1 0 2 1 0 0 0.002 0.004 0.006 0.008 0.01 Electric field intensitu (a.u.)
Figure: State widths of the H atom versus electric field intensity for m = 0, 1, 2.
ADAS Workshop 2012, CEA Cadarache, France. L. Fern´andez-Menchero (Univ. Strathclyde) (Motion) Stark Effect models / 28 MSE Schr¨odinger Equation Results Conclusions
Resonances
Stark splitting of Hα line
3000
2500
π+ (nm) λ 2000
1500
1000 σ Stark Splitting Wavelength
500 π−
0 0 0.002 0.004 0.006 0.008 0.01 Electric Field Intensity (a.u.)
Figure: Stark splitting of Hα line versus electric field intensity.
ADAS Workshop 2012, CEA Cadarache, France. L. Fern´andez-Menchero (Univ. Strathclyde) (Motion) Stark Effect models / 28 MSE Schr¨odinger Equation Results Conclusions
Wave functions Wave functions
Wave function state 100 10 10
1
8 0.1
0.01
6 0.001 2 |
η 0.0001 Ψ |
4 1e-05
1e-06
2 1e-07
1e-08
0 1e-09 0 2 4 6 8 10 ξ
Figure: Wave function of the state 100 of the H atom under a constant electric field of 0.005 a.u..
ADAS Workshop 2012, CEA Cadarache, France. L. Fern´andez-Menchero (Univ. Strathclyde) (Motion) Stark Effect models / 28 MSE Schr¨odinger Equation Results Conclusions
Wave functions Wave functions
8 8
6 6
4 2 4 2 |Ψ| |Ψ|
Wave function 2 Wave function 2 Im Im
0 0 Re Re
-2 -2 0 2 4 6 8 10 0 2 4 6 8 10 ξ η
Figure: Wave function of the state 100 of the H atom under a constant electric field of 0.005 a.u..
ADAS Workshop 2012, CEA Cadarache, France. L. Fern´andez-Menchero (Univ. Strathclyde) (Motion) Stark Effect models / 28 MSE Schr¨odinger Equation Results Conclusions
Wave functions Wave functions
Wave function state 2 -1 0 10 0.1
8 0.01
6 2 |
η 0.001 Ψ |
4
0.0001 2
0 1e-05 0 2 4 6 8 10 ξ
Figure: Wave function of the state 2 10 of the H atom under a constant electric field of 0.005 a.u.. −
ADAS Workshop 2012, CEA Cadarache, France. L. Fern´andez-Menchero (Univ. Strathclyde) (Motion) Stark Effect models / 28 MSE Schr¨odinger Equation Results Conclusions
Wave functions Wave functions
2 0.8
0.6 1 2 |Ψ| 0.4
Im 0.2 2 0 |Ψ|
Wave function Wave function 0
Re Im -1 -0.2 Re -0.4
-2 -0.6 0 2 4 6 8 10 0 2 4 6 8 10 ξ η
Figure: Wave function of the state 2 10 of the H atom under a constant electric field of 0.005 a.u.. −
ADAS Workshop 2012, CEA Cadarache, France. L. Fern´andez-Menchero (Univ. Strathclyde) (Motion) Stark Effect models / 28 MSE Schr¨odinger Equation Results Conclusions
Wave functions Wave functions
Wave function state 2 1 0 10 1
8 0.1
6 0.01 2 | η Ψ |
4 0.001
2 0.0001
0 1e-05 0 2 4 6 8 10 ξ
Figure: Wave function of the state 210 of the H atom under a constant electric field of 0.005 a.u..
ADAS Workshop 2012, CEA Cadarache, France. L. Fern´andez-Menchero (Univ. Strathclyde) (Motion) Stark Effect models / 28 MSE Schr¨odinger Equation Results Conclusions
Wave functions Wave functions
3 2
2 1.5
2 1 |Ψ| 1
Re Im
Wave function 0 Wave function 0.5 2 Im |Ψ|
-1 0 Re
-2 0 2 4 6 8 10 0 2 4 6 8 10 ξ η
Figure: Wave function of the state 210 of the H atom under a constant electric field of 0.005 a.u..
ADAS Workshop 2012, CEA Cadarache, France. L. Fern´andez-Menchero (Univ. Strathclyde) (Motion) Stark Effect models / 28 MSE Schr¨odinger Equation Results Conclusions
Contents
1 The Motion Stark Effect (MSE)
2 Solution of the Schor¨odinger Equation
3 Energies and wave functions of the SHA
4 Conclusions
ADAS Workshop 2012, CEA Cadarache, France. L. Fern´andez-Menchero (Univ. Strathclyde) (Motion) Stark Effect models / 28 MSE Schr¨odinger Equation Results Conclusions
Conclusions
Stark wave functions nkm should be determined. Cross sections and Einstein coefficients between Stark states. Include directionality. As “rough” approximation cross sections and Einstein coefficients for Rydberg states can be used.
ADAS Workshop 2012, CEA Cadarache, France. L. Fern´andez-Menchero (Univ. Strathclyde) (Motion) Stark Effect models / 28 MSE Schr¨odinger Equation Results Conclusions
Future work
Use the obtained wave functions to calculate directional cross sections of collision with SHA: electron impact, ion impact, charge exchange. Obtain Einstein coefficients for SHA: directional emission. Include these cross sections and Einstein coefficients in the collision-radiative model of ADAS305.
ADAS Workshop 2012, CEA Cadarache, France. L. Fern´andez-Menchero (Univ. Strathclyde) (Motion) Stark Effect models / 28 MSE Schr¨odinger Equation Results Conclusions
Acknowledgments
Thanks to: ASDEX-Upgrade MSE diagnosis team: M. Reich and J. Hobirk.
ADAS Workshop 2012, CEA Cadarache, France. L. Fern´andez-Menchero (Univ. Strathclyde) (Motion) Stark Effect models / 28