Multipole Expansion” of the Electron- Photon Interaction Operator

Electric dipole approximation (E1)

We have shown that for interaction of low-Z ions with visible light we ikr may apply electric dipole (long-wave) approximation e ≈ 1

In this approximation transition amplitude reads as:

( E )1 = ψ * α εψ M ab ∫ a (r ) b (r )dr

Electron is bound to ion Evaluation of this amplitude leads us to ……….. set of selection rules : ……….. 3p 3/2 3d 3/2 3p 3s 1/2 1/2 − = ± (au) (au) Energy Energy ja jb ,0 1 2p 3/2 π = −π 2p a b 2s 1/2 1/2

E1 transitions

1s 1/2 Higher multipoles contributions

Transition matrix element can be evaluated by making “multipole expansion” of the electron- photon interaction operator:

M = ψ + (r ) α ε e ikr ψ (r )dr ab ∫ a b 1 e ikr = 1 + ikr + ()ikr 2 + ... For interaction of heavy ions with 2 energetic photons we may find that: electric dipole term kr ≈1

Which means that electric dipole approximation is not longer valid!

− X-rays: k ≈ 10 8 cm 1 We have to take into account higher (non-dipole) terms! − γ-rays: k > 10 9 cm 1

14 January 2010 Beyound the E1 approximation (first correction) Let us consider the next term in the multipole decomposition of electron-photon interaction: 1 e ikr = 1 + ikr + ()ikr 2 + ... And let us analyze matrix element containing this2 term:

M ′ = ψ + (r ) α ε (ikr )ψ (r )dr ab ∫ a b

Again, to make mathematical analysis more “clear” let us restrict ourselves now to the non - relativistic formulation of this matrix element:

′ ∝ ψ + ε ( )ψ M ab ∫ a (r ) p ikr b (r )dr ↑↑ By assuming photon propagating along z-axis (k O z), we can choose photon polarization in x-direction (remember: k ⋅ ε = 0 ). ′ ∝ ψ + ψ M ab ∫ a (r ) p x z b (r )dr

14 January 2010 Beyound the E1 approximation (magnetic dipole and electric quadrupole transitions)

Let us re-write our matrix element as: ′ ∝ ψ + ψ = ψ + − ψ M ab ∫ a (r ) p x z b (r )dr ∫ a (r ) ( p x z x p z ) b (r )dr + ψ + ψ ∫ a (r ) x p z b (r )dr

ˆ = − By using the fact that: L y zp x xp z i and: z& = [Hˆ , z] h 0 (in a.u.)

′ ∝ ψ + ψ + ω ψ + ψ M ab ∫ a (r ) L y b (r )dr i ab ∫ a (r ) x z b (r )dr

14 January 2010 Higher multipoles contributions (M1 and E2 terms)

We just have found that: ( M ,1 E )2 ∝ ψ + ψ + ω ψ + ψ M ab ∫ a (r ) L y b (r )dr i ab ∫ a (r ) x z b (r )dr

magnetic dipole (M1) term electric quadrupole (E2) term

Proportional to magnetic moment of the ion Proportional to electric quadrupole = −µ ˆ + ˆ h moment of the ion µˆ 0 (L gS) / = ( − 2δ ) Qij ∑ q n 3xi x j x ij n

deviation from spherical shape!

After some algebra, selection rules for these transitions can be found!

14 January 2010 Magnetic dipole and electric quadrupole transitions (Selection rules) We just have found that: ( M ,1 E )2 ∝ ψ + ψ + ω ψ + ψ M ab ∫ a (r ) L y b (r )dr i ab ∫ a (r ) x z b (r )dr

magnetic dipole (M1) term electric quadrupole (E2) term

− = ± − = ± ± ja jb ,0 1 ja jb ,0 ,1 2 π = π π = π a b a b

By making further multipole decomposition of the electron-photon interaction we can obtain even higher terms: 1 e ikr = 1 + ikr + ()ikr 2 + ... = E1 + M 1 + E 2 + M 2 + E 3 + .... 2

14 January 2010 Multipole transitions (selection rules) transition selection rules

Electron is free

Electric dipole j − j = ,0 ±1 Electron is bound to ion a b 4f (E1) ……….. 5/2 π = −π ……….. 4f a b 3/2 3p 3/2 3d 3/2

− = ± 3s 3p 1/2 Magnetic ja jb ,0 1 1/2 dipole (M1) π = π a b 2p 3/2

2s 2p Electric 1/2 1/2 − = ± ± quadrupole (E2) ja jb ,0 ,1 2 π = π a b

1s Magnetic − = ± ± 1/2 ja jb ,0 ,1 2 quadrupole (M2) π = −π a b Higher multipoles become more pronounced for heavy ions (large nuclear charges)! (for all transitions 0 → 0 is forbidden and triangle rule − ≤ ≤ + ja jb L ja jb should be satisfied) (Remind yourself our discussion on value of kr )!

14 January 2010 Z-scaling of multipole decay rates

∝ Z 4 2p 3/2 ∝ Z 8 10 E1+ M2 ∝ Z 2s 1/2 2p 1/2

E1 M1

1s 1/2

For low-Z higher multipole transition are orders of magnitude smaller than the leading electric dipole (E1) term.

However, higher terms are enhanced with Z faster than E1 due to the increasing role of relativistic effects.

14 January 2010 Atomic photoionization (theory background)

For the case of the photoabsorption: laser pulse t ce δ − −hω h c )1( (t) = dω A (ω)ei ω ψ α ε eikr ψ td ′e (Ei b Ea t /') b ih ∫ 0 b a ∫ ∆ω 0 We can get total cross section as: t’=0 t’=t 1 2 dσ =C ψ αεeikr ψ ρ dΩ We assume that laser pulse is ab ω b a b independent on time except for ba being “switched on” at t=0.

18 January 2010 Atomic photoionization (theory background)

Bound-free transition matrix element is: = ψ α ε ikr ψ ≡ ψ + α ε ikr ψ M ab b e a ∫ b (r ) e a (r )dr

continuum wave function bound wave function

18 January 2010 K-shell atomic photoionization (non-relativistic framework) We shall consider evaluation of the bound-free matrix element for the moment in the non- relativistic framework: = ψ + ε ikr ∇ψ M ab ∫ b (r ) e 1s (r )dr And let us make an approximation in which continuum electron will be decsribed by the plane wave! (We neglected the interaction of a nucleus with electron in continuum):

− = ⋅ ik b r ε ikr ∇ψ M ab C ∫ e e 1s (r )dr

ε Angular differential photoionization cross section

dσ 2 = C ⋅ M ≈ C ⋅sin 2 θ cos 2 ϕ dΩ ab

18 January 2010 K-shell atomic photoionization (non-relativistic framework) Simple physical interpretation can be obtained for the derived cross section:

dσ 2 = C ⋅ M ≈ C ⋅sin 2 θ cos 2 ϕ dΩ ab ε

Obviously: in non-relativistic electric-dipole (E1) approximation electron should be emitted along the electric field vector = polarization vector of EM wave.

dσ ≈ C ⋅sin 2 θ For the case of unpolarized light: Ω d unp

18 January 2010 Summary of Effects with light and matter (non-relativistic framework)

Effect Initial wavefunction Final wavefunction Photoabsorption bound state bound state (Excitation) Photoionization bound state continuous state Radiative electron bound bound state capture Radiative continuous state bound state recombination bremsstrahlung continuous state continuous state pair production neg. continuous state bound state

18 January 2010