Workshop on: ATOMIC STRUCTURE AND TRANSITIONS: THEORY IN USING SUPERSTRUCTURE PROGRAM

Transcription

1 Workshop on: ATOMIC STRUCTURE AND TRANSITIONS: THEORY IN USING SUPERSTRUCTURE PROGRAM PROF. SULTANA N. NAHAR Astronomy, Ohio State U, Columbus, Ohio, USA nahar Textbook: Atomic Astrophysics and Spectroscopy A.K. Pradhan & S.N. Nahar (Cambridge U Press, 2011) Computation: Ohio Supercomputer Center (OSC) Notes from workshops at universities

2 Study the UNIVERSE through RADIATION: Most Complete 3D Map of the universe (Created: By 2MASS - 2Micron All Sky Survey over 3 decades) Includes 43,000 galaxies extended over 380 million light years y Redshifts, or measurements of galaxy distances, were added Missing black band in the middle because of invisibility behind our Milky Way

3 SPECTRUM of the Wind near Blackhole: GRO J Binary Star System Materials from the large star is sucked into companion blackhole - form wind as they spiral to it. Spectrum of the wind (BLUE): Highly charged Mg, Si, Fe, Ni lines.red: Elements in natural widths Doppler Blue Shift - Wind is blowing toward us Information from analysis of light produced from atomic transitions (Miller et al., 2006)

4 ATOMIC STRUCTURE Atomic structure - i) Organization of electrons in various shells and subshells, ii) Determinations of electron energies and wave functions transition probabilities As Fermions, unlike Bosons, electrons form structured arrangements, known as CONFIGURATION, bound by the attractive nuclear potential Electrons move in quantized orbitals with orbital L and spin S angular momenta. L and S give rise to various atomic states. Transitions among those states involve photons which are seen as lines in observed spectra The combination of orbital angular momentun L and spin angular momentum S follow strict coupling rules, known as selection rules The dynamic state of an atom - described by a Schrodinger equation HYDROGEN ATOM - only atomic system that can be treated exactly Approximation begins from 2-electrons systems

5 HYDROGEN ATOM Schrodinger equation of hydrogen, with KE = P 2 /(2m) and nuclear potential energy V(r), is [ h2 ( 2) ] + V (r) Ψ = E Ψ (1) 2m or, [ h2 ( ) ] 2 r + 2 2m + V (r) Ψ = E Ψ V (r) = Ze2 r In spherical coordinates 2 r = 1 ( r 2 r 2 ) r r 2 = 1 r 2 sin ϑ ϑ = 2Z r/a 0 Ry ( sin ϑ ϑ ) + 1 r 2 sin 2 ϑ ϕ 2 The solution or wavefunction has independent variables r, θ, φ, each will correspond to a quantum number, Ψ(r, ϑ, ϕ) = R(r) Y (ϑ, ϕ) 2 (2)

6 MULTI-ELECTRON ATOM A many-electron system requires to sum over (i) all oneelectron operators, that is KE & attractive nuclear Z/r potential, (ii) two-electron Coulomb repulsion potentials HΨ = [H 0 + H 1 ]Ψ, (3) N [ H 0 = 2 i 2Z ], H r 1 = 2 (4) i=1 i r j<i ij H = i f i + j i g ij F + G (5) H 0 : one-body term, stronger, H 1 : two-body term, weaker, can be treated perturbatively Start with a trial wave function Ψ t in some parametric form, Slater Type Orbitals P STO nl (r) = r l+1 e ar A trial function should satisfy variational principle that through optimization an upper bound of energy eigenvalue is obtained in the Schrödinger equation.

7 HARTREE-FOCK EQUATION (Book AAS) The N-electron wavefunction in the determinantal representation ψ 1 (1) ψ 1 (2)... ψ 1 (N) Ψ = 1 N ψ 2 (1) ψ 2 (2)... ψ 2 (N) ψ N (1) ψ N (2)... ψ N (N) This is called the Slater determinant. Ψ vanishes if coordinates of two electrons are the same. Substitution in Schrodinger equation results in Hartree-Fock equation. Simplification gives set of one-electron radial equations, [ 2 i 2Z r i l δ m k L,m l L ] [ u k (r i ) + [ l u l (r j ) 2 r ij u l (r j )dr j ] u k (r i ) u l (r j ) 2 r ij u k (r j )dr j ] u l (r i ) = E k u k (r i ). 1st term= 1-body term, 2nd term= Direct term, 3rd term= Exchange term. The total energy is given by E[Ψ] = I i + 1 [J ij K ij ]. (7) 2 i i j (6)

8 Central Field Approximation for a Multi-Electron System H 1 consists of non-central forces between electrons which contains a large spherically symmetric component We assume that each electron is acted upon by the averaged charge distribution of all the other electrons and construct a potential energy function V(r i ) with one-electron operator. When summed over all electrons, this charge distribution is spherically symmetric and is a good approximation to actual potenial. Neglecting non-radial part, where H = V(r) = N i=1 N i=1 h 2 2m 2 i + V(r). e 2 Z r i + N i j e 2. (8) r ij V(r) is the central-field potential with boundary conditions V(r) = Z r if r 0, = z r if r (9)

9 THOMAS-FERMI-DIRAC-AMALDI (TFDA) APPROXIMATION One most useful procedure (implemented in program SS): Treats electrons as Fermi sea: Electrons, constrained by Pauli exclusion principle, fill in cells up to a highest Fermi level of momentum p = p F at T=0 As T rises, electrons are excited out of the Fermi sea close to the surface levels & approach a Maxwellian distribution spatial density of electrons: ρ = (4/3)πp3 F h 3 /2 Based on quantum statistics, the TFDA model gives a continuous function φ(x) such that the potential is where V(r) = Z eff(λ nl, r) r = Z r φ(x), φ(x) = e Zr/2 + λ nl (1 e Zr/2 ), x = r µ, µ = ( N N 1 ) 2/3 Z 1/3 = constant.

10 THOMAS-FERMI-DIRAC-AMALDI (TFDA) APPROXIMATION The function φ(x) is a solution of the potential equation d 2 φ(x) dx 2 = 1 x φ(x) 3 2 The boundary conditions on φ(x) are φ(0) = 1, φ( ) = Z N + 1. Z The one-electron orbitals P nl (r) can be obtained by solving the wave equation [ ] d 2 l(l + 1) dr2 r 2 + 2V(r) + ɛ nl P nl (r) = 0. This is similar to the radial equation for the hydrogenic case, with the same boundary conditions on P nl (r) as r 0 and r, and (n l + 1) nodes. The second order radial is solved numerically since, unlike the hydrogenic case, there is no general analytic solution. It may be solved using an exponentially decaying function appropriate for a bound state, e.g. Whittaker function

11 THOMAS-FERMI-DIRAC-AMALDI (TFDA) APPROXIMATION The solution is normalized Whittaker function ( ) W(r) = e zr/ν 2zr a 1 + k ν r k N k=1 where ν = z/ ɛ is the effective quantum number and ɛ is the eigenvalue. The coefficients are a 1 = ν {l(l + 1) ν(ν 1)} 1 2z a k = a k 1 ν {l(l + 1) (ν k)(ν k + 1)} 1 2kz and the normalization factor is { } ν 2 1/2 N = Γ(ν + l + 1) Γ(ν 1) z The one-electron spin orbital functions then assume the familiar hydrogenic form ψ n,l,ml,m s (r, θ, φ, m s ) = φ(r, θ, φ)ζ ms

12 THOMAS-FERMI-DIRAC-AMALDI (TFDA) APPROXIMATION - TFDA orbitals are based on a statistical treatment of the free electron gas, & hence neglect the shell-structure However, in practice configuration interaction accounts for much of the discrepancy that might otherwise result.

13 Relativistic Breit-Pauli Approximation (AAS) Hamiltonian: For a multi-electron system, the relativistic Breit-Pauli Hamiltonian is: 1 2 H BP = H NR + H mass + H Dar + H so + N [g ij (so + so ) + g ij (ss ) + g ij (css ) + g ij (d) + g ij (oo )] i j where the non-relativistic Hamiltonian is N H NR = 2 i 2Z + r i the Breit interaction is i=1 H B = i>j and one-body correction terms are H mass = α2 4 p 4 i, H Dar = α2 4 i N j>i 2 r ij [g ij (so + so ) + g ij (ss )] i 2 (Z r i ), H so = Ze2 h 2 L.S 2m 2 c 2 r 3 Wave functions and energies are obtained solving: HΨ = EΨ

14 CONFIGURATION INTERACTION A multi-electron system is described by its configuration and a defined spectroscopic state. All states of the same SLπ, with different configurations, interact with one another - configuration interaction. Hence the wavefunction of the SLπ may be represented by a linear combination of configurations giving the state. Example, the ground state of Si II is [1s 2 2s 2 2p 6 ]3s 2 3p ( 2 P o ). 2 P o state can also be formed from 3s 2 4p ( 2 P o ), 3s3p3d (..., 2 P o ). 3p 3 ( 2 P o ) and so on. These 4 configurations contribute with different amplitudes or mixing coefficients (a i ) to form the four state vectors 2 P o of a 4 4 Hamiltonian matrix. Hence for the optimized energy and wavefunction for each 2 P o state all 4 configurations should be included, 3 Ψ( 2 P o ) = a i ψ[c i ( 2 P o )] = [ a 1 ψ(3s 2 3p) + a 2 ψ(3s 2 4p) i=1 + a 3 ψ(3p 3 ) + a 3 ψ(3s3p3d) ]

15 1. PHOTO-EXCITATION Photo-Excitation & De-excitation: X +Z + hν X +Z Atomic quantities B 12 - Photo-excitation, Oscillator Strength (f) A 21 - Spontaneous Decay, - Radiative Decay Rate (A-value) B 21 - Stimulated Decay with a radiation field P ij, transition probability, P ij = 2π c2 h 2 νji 2 < j e i > 2 ρ(ν ji ). (10) mcê.peik.r e ik.r = 1 + ik.r + [ik.r] 2 /2! +..., Various terms in e ik.r various transitions 1st term E1, 2nd term E2 and M1,...

16 ALLOWED & FORBIDDEN TRANSITIONS Determined by angular momentum selection rules i) Allowed: Electric Dipole (E1) transitions - same-spin & intercombination (different spin) transition ( J = 0,±1, L = 0,±1,±2; parity changes) Forbidden: ii) Electric quadrupole (E2) transitions ( J = 0,±1,±2, parity does not change) iii) Magnetic dipole (M1) transitions ( J = 0,±1, parity does not change) iv) Electric octupole (E3) transitions ( J= ±2, ±3, parity changes) v) Magnetic quadrupole (M2) transitions ( J = ±2, parity changes) Allowed transitions are much strongher than Forbidden transitions

17 Transition Matrix elements with a Photon 1st term: Dipole operator: D = i r i: Transition matrix for Photo-excitation & Deexcitation: < Ψ B D Ψ B > Matrix element is reduced to generalized line strength (length form): N+1 2 S = Ψ f r j Ψ i (11) j=1 There are also Velocity & Acceleration forms Allowed electric dipole (E1) transtitions The oscillator strength (f ij ) and radiative decay rate (A ji ) for the bound-bound transition are [ ] Eji f ij = S, 3g i [ ] A ji (sec 1 ) = E3 ji S 3g j

18 FORBIDDEN TRANSITIONS i) Electric quadrupole (E2) transitions ( J = 0,±1,±2, π - same) A E2 ji = E5 ij g j S E2 (i, j) s 1, (12) ii) Magnetic dipole (M1) transitions ( J = 0,±1, π - same) A M1 ji = E3 ij S M1 (i, j) s 1, (13) g j iii) Electric octupole (E3) transitions ( J= ±2, ±3, π changes) A E3 ji = E7 ij g j S E3 (i, j) s 1, (14) iv) Magnetic quadrupole (M2) transitions ( J = ±2, π changes) LIEFTIME: A M2 ji = s 1E5 ij g j S M2 (i, j). (15) τ k (s) = 1 i A ki(s 1 ). (16)

19 EX: ALLOWED & FORBIDDEN TRANSITIONS Diagnostic Lines of He-like Ions: w,x,y,z w(e1) : 1s2p( 1 P o 1) 1s 2 ( 1 S 0 ) (Allowed Resonant) x(m2) : 1s2p( 3 P o 2) 1s 2 ( 1 S 0 ) (Forbidden) y(e1) : 1s2p( 3 P o 1) 1s 2 ( 1 S 0 ) (Intercombination) z(m1) : 1s2s( 3 S 1 ) 1s 2 ( 1 S 0 ) (Forbidden) NOTE: 1s-2p are the K α transitions

20 Atomic transitions in PLASMA OPACITY κ ν (i j) = πe2 mc N if ij φ ν N i = ion density in state i, φ ν = profile factor (Gaussian, Lorentzian, or combination of both) Total monochromatic κ ν is obtained from summed contributions of all possible transitions Opacity is a fundamental quantity for radiation transfer in plasmas. It is caused by repeated absorption and emission of the propagating radiation by the constituent plasma elements. 1. Photoexcitation: Atomic parameter - Oscillator Strength (f ij )

21 Monochromatic Opacities κ ν of Fe II on Sun s Surface Monochromatic opacity (κ ν ) depends on f ij κ ν (i j) = πe2 mc N if ij φ ν Increased opacity over 3000Å explains missing radiation from solar surface TOP: κ ν of Fe II (Nahar & Pradhan 1993). BOTTOM: solar blackbody radiation in Å. 3 F λ (10 6 erg cm 2 s 1 A 1 ) λ (A)