Introduction to the physics of highly charged ions. Lecture 7: Direct and resonant photoionization of HCI

Transcription

1 Introduction to the physics of highly charged ions Lecture 7: Direct and resonant photoionization of HCI Zoltán Harman Universität Heidelberg,

2 So far: radiative decay and excitation of ions Radiative decay: photon is emitted in a bound-bound transition Radiative excitation: photon is absorbed in a bound-bound transition Decay rate in nonrel. dipole approximation: 1 τ e 2 = 4 3 c 3 ω3 fi a f r a i 2 ( Eai + E SE i i0 H int f kσ 2 E af ω ) 2 + Γ 2 i 4 Lorentz profile, intensity distribution of the emitted line

3 Photoionization: bound-free transition of the e Direct photoionization (DPI, or photoelectric effect): ω + A q+ A (q+1)+ + e electron removed from the binding potential of the nucleus by absorption of a photon Resonant (Auger) photoionization (RPI): ω + A q+ A q+ A (q+1) + e resonant excitation of an electron by photoabsorption to an autoionizing (often doubly-excited) level Auger effect or autoionization

4 Direct photoionization One-step process, can be described in 1st order of perturbation theory Transition probability (according to Fermi s Golden Rule): w DPI fi = 2π Ψ f H int Ψ i 2 ρ f (E f E i ) with ρ f : density of final states; H int : Hamiltonian for EM interaction (with the a kσ annihilator); Ψ i, Ψ f : initial and final states Cross section e.g. for PI of a H-like ion originally in its 1s ground state, nonrel. dipole approx.: σ DPI fi wdpi fi = 4π2 e 2 j γ m 2 cω ( q p ɛ kσ 1s 2 2 q 2 ) δ 2m E 1s ω q Here: Ψ q ( r) = q r = ei q r V : plane wave (free electron)

5 Formalism for resonant photoionization Resonant processes: cannot be described by perturbation theory in finite orders, only by means of an all-order method RPI: necessarily a many-electron process Generalized transition operator: H int T: transition- or T-operator; interaction included in all orders Transition probability: w fi = 2π Ψ f T Ψ i 2 ρ f (E f E i )

6 How is T defined? Total Hamiltonian: H = H e + H r + H int, with N ) N H e = (c α i p i + mc 2 β i Ze2 e 2 + r i r i=1 i<j i r j, H r = ( ω a + kσ a kσ + 1 ), 2 kσ N H int = e α i A ( r i, t) i=1 Let s define the Green operator (or resolvent): G(z) (z H) 1, with z being some complex energy; G contains all information. T(z) V + VG(z)V ; σ fi = lim ɛ 0+ 2π Ψ f T(E i + ɛ) Ψ i 2 ρ f (E f E i )

7 How to calculate G and T? The unperturbed Green operator: G 0 (z) (z H 0 ) 1. Spectral representation through the eigenfunctions H 0 n = E n n : G 0 (z) = n n n z E n It is connected to G(z) through the Lippmann-Schwinger (LS) equation: G(z) = G 0 (z) + G 0 (z)vg(z) (it follows from the Schrödinger equation and H = H 0 + V). Iterative solution of the LS equation: perturbation expansion G 1 = G 0 + G 0 VG 0 ; G 2 = G 0 + G 0 VG 1 = G 0 + G 0 VG 0 + G 0 VG 0 VG 0 ; etc. Born series for the T operator: T V + VG 0 V + VG 0 VG 0 V +... So far: abstract calculation in terms of V

8 How to define H 0 and V for the resonant photoionization process? Define the relevant subspaces: i R: bound electronic state + 1 photon d Q: bound electronic state + 0 photon f P: 1 e in continuum + 0 photon Define projection operators which project on these subspaces: R = i, k σ i, k σ ; Q = d d ; P = f, p m s f, p m s. i, k σ d f, p m s Properties of projection operators: P 2 = P, Q 2 = Q, R 2 = R, etc. (idempotence); PQ = PR = QR = 0 (orthogonality) We assume completeness of the above subspaces: P + Q + R = 1.

9 Relevant matrix element for our process: f, pm s T i, kσ = f, pm s PTR i, kσ We assume the states of the type P, Q, R to be known, i.e. the unperturbed states are eigenstates of the Hamiltonian in the subspaces P, Q, R: H 0 = PHP + QHQ + RHR. Therefore, the perturbing operator V is by definition: V H H 0 = 1H1 H 0 = (P + Q + R)H(P + Q + R) PHP QHQ RHR = PHQ + PHR + QHP + QHR + RHP + RHQ. PHQ PH e Q: Auger decay; time-reverse: QHP QH e P: dielectronic capture (DC) PHR PH int R: direct photoionization (DPI); RHP RH int P: radiative recombination (RR) QHR QH int R: photoexcitation; RHQ RH int Q: rad. decay

10 So: H 0 is diagonal in P, Q and R; V is non-diagonal We want to use the Born series for the projection PTR: PTR = PVR + PVG 0 VR + PVG 0 VG 0 VR +... = PH int R + PV(P + Q + R)G 0 (P + Q + R)VR +... For simplification, we exploit that just like H 0, its Green operator is also diagonal; e.g. QG 0 (z)q = d d d z E d to thus, the 2nd-order T operator is: PT 2 R = PV(Q+R)G 0 (Q+R)VR = PV(QG 0 Q)VR = PH e (QG 0 Q)H int R Matrix element up to 2nd order: f, pm s P(T 1 + T 2 )R i, kσ = f, pm s H int i, kσ + d f, pm s H e d d H int i, kσ z E d looks good, but...

11 Let s go on 3rd-order term: PT 3 R = PVG 0 VG 0 VR = PV(Q + R)G 0 (Q + R)V(P + Q + R)G 0 (P + Q + R)VR = PV(QG 0 Q)V(QG 0 Q)VR = 0...and so is every odd-order term (except for the 1st order) Let s go on 4th-order term: PT 4 R = PVG 0 VG 0 VG 0 VR (1) = PV(QG 0 Q)V(P + R)G 0 (P + R)V(QG 0 Q)VR = PV(QG 0 Q)V(PG 0 P)V(QG 0 Q)VR + PV(QG 0 Q)V(RG 0 R)V(QG 0 Q)VR

12 What is this? Last row: matrix element of the intermediate-state insertion d V(RG 0 R)V d = i k σ d H int i k σ i k σ H int d E d E i ω + iɛ A somewhat familiar formula; let us use the identity ( ) 1 1 lim ɛ 0+ x + iɛ = P iπδ(x) ; x lim d V(RG 0R)V d ɛ 0+ = P d H int i k σ E d E i ω i k σ = E SE d 2 iπ d H int i k σ i k σ i 2 Γrad d : self-energy and radiative decay width 2

13 Other term in Eq. (1): lim d V(PG 0P)V d ɛ 0+ = P f p m s d H e f p m s 2 E d E f ɛ p m s iπ f p m s d H e f p m s 2 = E A d i 2 ΓA d : Auger shift and Auger decay width Up to 4th order and using the resonance approximation Q = d d d d d : f, pm s T 2 + T 4 i, kσ = f, pm s H e d d H int i, kσ z E ( d + f, pm s H e d d H int i, kσ Ed i 2 Γ ) d z E d z E d Now let s notice that higher even-order terms follow a geometrical series with the quotient x = (... ); and l=0 xl = 1 1 x.

14 Thus, the matrix element for the resonant two-step PI process is: f, pm s T RPI i, kσ = f, pm s H e d d H int i, kσ z E d E d + i 2 Γ d Cross section: ME squared; Lorentzian shape!

15 Photoionization of HCI in the x-ray regime Application: astrophysical plasmas half of the baryonic matter in the universe is in the form of HCIs comprised in the Warm-Hot Intergalactic Medium (WHIM); PI may be the dominant ionization process analysis of ionic absorption spectra (C, N, O, Ne, Ar, Fe ions) temperature, density, mass and velocity PI data are important constituents of astrophysical models: reliable theories and measurements are called for interpretation of data from grating spectrometers on board of the Chandra and XMM Newton observatories

16 Motivation Further motivation: ITER Applications: magnetically confined fusion plasmas and technical plasmas sensitive tests of relativity, relativistic electron correlation, strong-field quantum electrodynamics

17 Photoionization of HCI: novel light sources with high energies and brilliance (synchrotron radiation, XFEL lasers) extending to higher ionic charge states high detection efficiency of the photoions: accurate measurements possible established methods: merged beam experiments, dual laser-produced plasmas, Penning traps new method: PI of HCI in an electron beam ion trap (EBIT)

18 Example: Be-like ions, excitation from the ground state 1s 2 2s 2 Excitation: two-electron one-photon transition Photoionization via 2p5s, 2p5d, 2p6s, 2p6d... states

19 Another possible channel: Excitation from the metastable state 1s 2 2s2p 1/2 3 P 0 state J = 0: long lived state, no single-photon transition to the J = 0 ground state possible, only E1M1 two-photon decay Excitation: normal (one-electron one-photon) transition Photoionization via 2p5p, 2p6p, 2p7p, 2p8p... states

20 No lecture on Dec. 16 (Monday) because of the Christmas party of the MPIK theory division