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Photoionization DLJSRF/R 1 fh ~ · DARESBURY SYNCHROTRON RADIATION LECTURE NOTE SERIES No.1 PHOTOIONIZATION by C. Bottcher Science Research Council DARESBURY LABORATORY Daresbury, Warrington, WA4 4AD DISCLAIMER This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency Thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof. DISCLAIMER Portions of this document may be illegible in electronic image products. Images are produced from the best available original document. © SCIENCE RESEARCH COUNCIL 1974 Enquiries about copyright and reproduction should be addressed to :­ The Librarian, Daresbury Laboratory, Daresbury, Warrington, WA44AD. IMPORTANT The SRC does not accept any responsibility for loss or damage arising from the use of information contained in any of its reports or in any communication about its tests or investigations. Printed by McCorquodale Printers Ltd., NAwton-IA-Willow,;. M ersey side. DARESBURY SYNCHROTRON RADIATION LECTURE NOTE SERIES No.1 PHOTOIONIZATION C. Bottcher Department of Theoretical Physics University of Manchester Notes on a series of lectures given at Daresbury Laboratory, November 1973 Science Research Council DARESBURY LABORATORY 1974 iWASlER FOREWORD These lectures were given in November 1973 to experimental physicists using the Synchrotron Radiation Facility at Daresbury. The aim was a modest one, to survey the basic ideas of the subject and introduce some current theoretical developments. I must thank Dr. Ian Munro for making all the arrangements connected with the lectures and members of the audience for providin~ stimulating feedback. C. Bottcher (i·ii) . v-· <. 'c THIS PAGE. WAS INTENTIONALLY LEFT BLANK CONTENTS Page Foreword ( ; ; ; ) 1. BASIC PHOTOIONIZATION THEORY 1.1 Introduction 1.2 Expressions for the Cross Section 1.3 Oscillator Strengths and their Applications 5 1.4 Simple Examples 5 1.5 Ionization by Fast Projectiles 7. 2. CALCULATION OF NON-RESONANT PHOTOIONIZATION CROSS SECTIONS 8 2. 1 Introduction 8 2.2 Born Approximation 8 2.3 Hartree-Fock Approximation 9 2.4 The Method of Polarized Orbitals 10 2.5 Quantum Defect Theory 10 2.6 Close Coupling and Variational Methods 12 2.7 Recent Calculations 13 3. THEORY OF RESONANCES 23 3. 1 Introduction 23 3.2 Feshbach Projection Operator Theory 24 3.3 Fano Profiles 28 3.4 Calculations 31 4. PHOTOIONIZATION OF HEAVY ATOMS 33 4. 1 Introduction 33 4.2 Minima in Photoionization Cross Sections 33 4.3 Discrete Levels in the Far Continuum 36 .4.4 Other Topics 37 5. PHOTOIONIZATION OF MOLECULES 39 5.1 Non-resonant Photoionization of Molecules 39 5.2 Vibrational Resonances 42 5.3 Electronic Resonances 42 Appendix: SOME RESULTS IN SCATTERING THEORY 46 (v) BASIC PHOTOIONIZATION THEORY l. 1 Introduction Photoionization is the process + .. hv + A(p) ~ A (q) + e ( 1 ) where A(p) is an atom or molecule in state p; A, A+ are usually in their ground states but they can be in excited states. In photo-detachment the electron is removed from a negative ion hv + A- ~ A + e . (2) Any particle X moving sufficiently fast behaves like a photon; the cross section for X + A ~ A+ + e + X ( 3) is closely related to that for (1) and fast electrons, ions, etc. are increas­ ingly being used to complement traditional spectroscopy (see 1.5). Photo­ ionization is usually seen as a probe of atomic and molecular structure and it is increasingly used for this purpose in chemistry. However, these lectures ernphd~:;ise the relation between photoionization and collision processes. The units used will usually be atomic units e = m = n = 1; atomic unit of length a0 ~ 5 x lQ-9 em, atomic unit of energy~ 27.21 eV. Otherwise CGS units 1 2 are used. Two recent review articles are highly recommended( ' ). 1.2 Expressions for the Cross Section We begin by considering a beam of photons of angular frequency w and energy spread ~E falling on an atom A. If the flux is N~E cm- 2 s-1 the rate of photoionization events is ,!» = a ( w) N~E s- 1 (4) where a is the cross section (cm2 ). The energy density in the photon beam is awN~E, a being the fine structure constant(~ 1/137, 1/c in atomic units). To calculate cr we must consider the Hamiltonian of an atom in an electro­ magnetic fie 1d; 1et~ be the unperturbed atomic Hamil toni an and dthe vector potential of the EM field. The perturbed Hamiltonian is (5) For a plane polarized wave ~ = ~ cos(wt- ~ • !), K = 2~ f wavelength. 0 If the wavelength ? 10 A, ~·r hardly varies through the extent of the atom so we can take .st' = A cos wt . (6) This is the dipole approximation. We are interested in a tran!ition between two unperturbed atom1 c states Fermi's golden rule states that under a periodic perturbatinn Veiwt + V*e-iwt the rate of transitions 1 ~ 2 is (7) If we let K = K~z' A = A~x the energy density is o(w) = awN (8) per unit energy interval. From (5) - (8) 2 !JP = (~ ~r p ( w) 1 < h 1 pX I <P 2 > 1 (9) It is easily shown that whence 2 - 1) < <P 1 X <1>2 > > ( E E I I = i < <PI IP X I <1>2 and ( 10} 2 If the radiation is unpolarized* ( 11) We must look at ~ 1 and ~ 2 more closely. If 1 is the.bound state ~ 1 = ~B is normalized in the usual way, < ~B I ~B > = 1 The final state 2 represents an electron being ejected with momentum ~' energy E = k2/2. In practice we observe electrons in a range of energies oc and solid angles bn. If the whole system is enclosed in a cubical box of side L the normalized states are asymptotically, -3/2 ik·r ~2 'V L e - - The number of such states in the range bEbn is 2 n = [~,]3 k bkbn = [~.]3 k bEbn ( 12) To get the rate of transitions into these states we multiply (11) by (12) and use (8), (2TI) 2 2 bEbn n~ = (awN) kl < hlrl ~2 >1 3 [~,]' (21Tl2 2 = aw I< ~B lrl ~(~) > j NbEbn ( 13) 3 where f(~) is the ener·gy-normalized wavefunction ( 14) e ik·r- - Comparing (13) with (4) we find that * We are writing these formulae for a one-electron system. In a system with many electrons ! is replaced by N I r. j =1 -J 3 do(w} 41T 2 am = --3-- aw I< ~B 1~1~(~) > 12 ( 15) The total cross section is obtained by integrating over all directions of ~' o(w} ( 16) Given the frequency w of the radiation, the wavenumber k of the ejected electron is fixed. ( 17) The set of functions ~(~) have the desirable property that <~(~)I'(~·)> = 6(E- E.) 6(n- n•) ( 18) A commonly encountered case is the photoionization of the 1S ground state. This is only connected to the P wave of the continuum. · ~(kp) = Fkp (r) Y1 m (e, ~), ( 19) sin(kr - ~ + 6R.} FkR. (~]~ "" r 1 1 ~(kp} is energy normalized, < ~(kp)l~(k p) >=6(E-E ). After some algebra (16) becomes o(w} (20) An important feature of (16) is that a involves the fine structure constant a and so photoionization cross sections are small ("-' lQ-18 cm2 as opposed to 10-16 cm2 for electron-atom cross sections). We can use the principle of detailed balance to obtain the cross section for the inverse process, radiative recombination, e +A+ -+ A+ hv . ( 21 ) If w8 is the statistical weight of the bound state, 2w 2 = cr(k ~ B) (22) c2 4 1.3 Oscillator Strengths and Their Applications The cross section (16) can be written in terms of the dimensionless oscil­ lator strength f(£), £ = k2/2, df o(w) a dE (23) 2 ~: = ~w j dn I < ~B Ir I '~' ( ~) > 1 (24) Denote the oscillator strengths for transitions from B to the discrete states s by fs and the energies of the discrete states by £s. The complete set of oscillator strengths, both discrete and continuous, satisfy certain sum rules. If we define the sums 00 fs d£ .df S(k) = z: k Cl£ (25) s (I+I£sl)k + f (I + £) 0 S(O) = N ( 26a) 2 N 1 ) < r.) 21 (26b) s ( = ~B I( Z: -J ~B > 3 j =1 S(2) = p (26c) where N is the number of electrons in the system and P the static polarizability. (26a) is the well-known Thomas-Kuhn sum rule. By combining the sum rules with experimental data, including photoionization cross sections, it is possible to build up consistent oscillator strength distributions for many systems. These distributions have a large number of applications, e.g. in calculating long range forces between atoms and molecules. More information is contained in 4 Dalgarno( 3 ) and Starkschall and Gordon( ). 1.4 Simple Examples The calculation of photoionization cross sections has been reduced to that of finding '!'(~) and evaluating (16). It is a comparatively simple matter in most cases to determine ~ 6 . Only one problem, the photoionization of hydro- genic atoms, is soluble in closed terms.
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