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.