* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Measurements of Photoionization Cross Sections of Positive and
Survey
Document related concepts
Cross section (physics) wikipedia , lookup
Hydrogen atom wikipedia , lookup
Quantum electrodynamics wikipedia , lookup
Atomic orbital wikipedia , lookup
Ferromagnetism wikipedia , lookup
Chemical bond wikipedia , lookup
Tight binding wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
Atomic theory wikipedia , lookup
Auger electron spectroscopy wikipedia , lookup
X-ray photoelectron spectroscopy wikipedia , lookup
Electron scattering wikipedia , lookup
X-ray fluorescence wikipedia , lookup
Transcript
Measurements of Photoionization Cross Sections of Positive and Negative Ions Using Synchrotron Radiation Tommy Steindorff Jacobsen University of Aarhus Denmark February 2004 ii Preface This thesis has been submitted to the Department of Physics and Astronomy at the University of Aarhus in order to obtain the MSc degree (Cand. Scient). It is the result of one year of experimental work at the undulator beamline at the ASTRID storage ring in Aarhus where beamtime has been available during the weeks 8, (10+12 lost because of defect dipole supply at the ring) 17, 18, 33, 36, 40 and 42 in 2003. The report will concentrate on photodetachment measurements of negative tellurium and on photoionization of members of the beryllium, boron and carbon isoelectronic sequences, but I have also contributed to the measurements of V− , Cr− , Co− and Ni− as well as Ba2+ , La3+ and Ce4+ Throughout the period I have had the great privilege of working with a number of kind and skilled people. To begin with, I am indepted to my supervisor Henrik Kjeldsen for teaching me about experimental details and physics in general in a patient and inspiring way and I wish him the best of luck in his future activities within the area of AMS. Also, I wish to thank the other group member Finn Folkmann for a very pleasant co-operation and for taking the time to help with the proof-reading. Being a part of their team, sharing both successes and distresses, has been an enjoyable as well as instructive experience and in many ways it can be regarded as the highlight of my time as a student at the University. Of foreign collaborators I have had the pleasure of working with people such as John West, Jean-Marc Bizau, Francis Penent, Denis Cubaynes, Nacer Adrouche and Jørgen Hansen. Finally, I have benefitted from the expertise iii Preface of members of the IFA and ISA staff - in particular Egon Jans, Kaare Iversen and Jens Vestergaard - and their help is gratefully acknowledged. Randers February 2004 Tommy Steindorff Jacobsen iv Contents Preface iii Introduction 1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1 Theory 1.1 5 Atomic theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.1.1 The Schrödinger equation . . . . . . . . . . . . . . . . 5 1.1.2 Many-electron atomic systems . . . . . . . . . . . . . . 6 1.1.3 The independent particle model . . . . . . . . . . . . . 7 1.1.4 Configuration and shell structure . . . . . . . . . . . . 9 1.1.5 LS-coupling and spin-orbit splitting . . . . . . . . . . . 10 1.2 Photoionization cross section . . . . . . . . . . . . . . . . . . . 11 1.3 Interpreting experimental spectra . . . . . . . . . . . . . . . . 13 1.4 1.3.1 Positive ions . . . . . . . . . . . . . . . . . . . . . . . . 13 1.3.2 Negative ions . . . . . . . . . . . . . . . . . . . . . . . 16 “Hardcore” theoretical approaches . . . . . . . . . . . . . . . . 18 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2 The experimental setup 2.1 21 The light source . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.1.1 The undulator . . . . . . . . . . . . . . . . . . . . . . . 22 v CONTENTS 2.1.2 2.2 2.3 CONTENTS The Miyake monochromator . . . . . . . . . . . . . . . 25 The absolute cross section . . . . . . . . . . . . . . . . . . . . 28 2.2.1 Form factors . . . . . . . . . . . . . . . . . . . . . . . . 29 2.2.2 Detector calibration . . . . . . . . . . . . . . . . . . . . 30 2.2.3 Photodiode calibration . . . . . . . . . . . . . . . . . . 31 Recording spectra - an overview . . . . . . . . . . . . . . . . . 33 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3 Ion production 35 3.1 General remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.2 The Middleton sputter source . . . . . . . . . . . . . . . . . . 36 3.3 3.2.1 Surface ionization . . . . . . . . . . . . . . . . . . . . . 36 3.2.2 Sputtering . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.2.3 Description of the Middleton source . . . . . . . . . . . 37 The ECRIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.3.1 The ECRIS principle . . . . . . . . . . . . . . . . . . . 41 3.3.2 Description of the ECR source . . . . . . . . . . . . . . 43 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4 Photodetachment of Te− 47 4.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.2 Results and analysis . . . . . . . . . . . . . . . . . . . . . . . 48 4.2.1 The continuum . . . . . . . . . . . . . . . . . . . . . . 49 4.2.2 Resonances . . . . . . . . . . . . . . . . . . . . . . . . 50 4.2.3 Oscillator strengths . . . . . . . . . . . . . . . . . . . . 52 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 5 Photoionization results 5.1 55 B-like . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 5.1.1 Results and analysis . . . . . . . . . . . . . . . . . . . 56 5.1.2 Metastable fractions . . . . . . . . . . . . . . . . . . . 63 5.1.3 Comparison with theory . . . . . . . . . . . . . . . . . 63 vi CONTENTS 5.2 Be-like . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 5.2.1 5.3 5.4 CONTENTS Results and analysis . . . . . . . . . . . . . . . . . . . 66 C-like . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 5.3.1 Results and analysis . . . . . . . . . . . . . . . . . . . 68 5.3.2 Comparison with theory . . . . . . . . . . . . . . . . . 73 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 Concluding remarks 77 A Useful material 79 A.1 Selection rules in LS-coupling . . . . . . . . . . . . . . . . . . 79 A.2 Coupling of equivalent electrons . . . . . . . . . . . . . . . . . 80 A.3 Hund’s rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 A.4 The periodic table . . . . . . . . . . . . . . . . . . . . . . . . 81 vii viii Introduction Motivation Photoionization of neutral atoms and positive ions and photodetachment of negative ions are important examples of fundamental processes where light and matter interact. Such processes occur in Nature all the time e.g. in stellar and planetary atmospheres, interstellar nebulae, plasmas etc. Schematically photoionization or detachment can be written: Photoionization: Aq+ + hν → A(q+1)+ + e− (1) A− + hν → A + e− . (2) Photodetachment: In the case of inner-shell detachment the neutral atom in (2) will be in an excited state that most likely will decay further by Auger decay, resulting in a positively charged end-product. As should be apparent, absolute photoionization cross sections are extremely important to know especially in astrophysics, since such data are needed to interpret the observations made by space and Earth observatories and for modelling stellar objects. Another important issue is the study of electron-correlation effects which are more dominant compared to the Coulomb interaction with the nucleus in negative ions, due to the more effective screening. Despite these obvious demands of experimental results, by far the largest amount of data has been brought by theorists. For instance, large international collaborations such as The OPAL Project (Roger 1 Motivation Introduction & Iglesias 1994), The OPACITY Project (1995) and The Iron Project (Nahar & Pradhan 1994, Hummer et al 1993) have put considerable effort into the calculations of photoionization cross sections for all elements up to iron, and substantial theoretical work has also been performed for negative ions (Ivanov 1999). The different theoretical approaches (some will be mentioned in the next chapter) all use state-of-the-art techniques, but it is crucial that these models are tested experimentally. The lack of absolute experimental contributions in this field stems mainly from technical difficulties, like the determination of the target-atom density for neutral atoms other than the noble gases. The same problem does not occur with ions since the density of an ion beam can be determined very accurately, but the fact that ions are charged particles prevents the density from getting very high1 . The first absolute measurements for ions using the so-called merged-beam method was obtained in the mid-eighties at Daresbury Laboratory, UK, in the pioneering work of Lyon et al (see Wuilleumier et al 1994 for a review), who used synchrotron radiation from a bending magnet as the source of light. The photon flux allowed only for the measurement of rather large cross sections (& 5M b = 5 · 10−18 cm2 ) and results were obtained for Ba+ , Ca+ , Sr+ , K+ , Zn+ and Ga+ . With the advent of insertion devices such as undulators it has been made possible to achieve intensities that are orders of magnitude higher than from ordinary bending magnets thus switching the main limitations to the production of stable ion beams. Undulator-based beam lines at synchrotron radiation facilities in Denmark (ASTRID), France (Super ACO), USA (ALS) and Japan (SPring8) are using the merged-beam method to measure absolute photoionization cross sections, and results have been obtained for many singly-charged positive ions. Some multiply-charged ions, produced mainly in Electron Cyclotron Resonance Ion Sources (ECRIS), have also been investigated thereby allowing for the study of interesting trends and features along isonuclear and isoelectronic se1 typical densities are in the range of 105 − 106 cm−3 which for comparison corresponds to a pressure of about 10−11 mBar, i.e. it is as low as a good UHV. 2 Introduction Outline quences. So far, inner-shell photodetachment measurements have only been performed for a very small number of negative ions and the need for experimental data is therefore especially profound in this field. Outline In my opinion, a report of this kind should be more or less self-explanatory, which, of course, is a rather ambitious point of view that could probably fill several books if truly carried out. It is my aim and hope, however, that what I have selected will give the reader a basic understanding of the experiment and not leave too many unanswered questions. The content of this thesis is divided into three major parts covering theoretical and experimental details as well as selected experimental results. PART ONE Chapter 1: Here some of the basic theoretical concepts from quantum mechanics and atomic theory are refreshed and more sophisticated models for treating many-electron atomic systems are mentioned briefly. PART TWO Chapter 2: Describes the experimental setup. To a large extent this will follow the work done by former students (see references in chapter 2) and will therefore not be so detailed. Chapter 3: The subject of producing an appropriate ion beam is adressed with emphasis on the description of the two types of ion sources used; namely Middleton’s high-intensity sputter source and the ECRIS. 3 Outline Introduction PART THREE Chapter 4: Presents the results obtained for photodetachment of 4d electrons in Te− Chapter 5: The photoionization cross sections of some ions of astrophysical importance will be presented. More precisely the chapter will concentrate on studies of N2+ , O3+ , O4+ , F3+ , F4+ and Ne4+ . References Hummer D. G., Berrington K. A., Eissner W., Pradhan A. K., Saraph H. E. and Tully J. A. (1993) Astron. Astrophys. 279 298 Ivanov V. K. (1999) J. Phys. B: At. Mol. Opt. Phys. 32 R67 Nahar S. A. and Pradhan A. K. (1994) Phys. Rev. A 49 1816 Roger F. J. and Iglesias C. A. (1994) Science 263 50 The OPACITY Project (1995) (Institute of physics: Bristol, vol. 1) Wuilleumier F. J., Bizau J.-M., Cubaynes D., Rouvelou B. and Journel L. (1994) Nucl. Instr. Meth. B 87 190 4 Chapter 1 Theory As one of the early motivations for performing the photoionization experiments was to provide data for testing different theoretical approaches, a little time should be spent on this subject. On the other hand, a deep analysis is way beyond the scope of this thesis and so only some of the basic things of relevance for the understanding of the obtained results are treated. The chapter is built up of sections with the first one reminding us of some key aspects in atomic theory. Moreover the chapter will briefly deal with the fundamental equations of photoionization (detachment), the interpretation of cross-section spectra and “hardcore” theoretical approaches. 1.1 1.1.1 Atomic theory The Schrödinger equation In modern quantum mechanics a particle is not pictured as a little “glass globe” but is described by a wave function, Ψ. If the particle is moving in a potential, V (r, t), the wave function will satisfy the differential equation n o ∂ ~2 2 i~ Ψ(r, t) = − ∇ + V (r, t) Ψ(r, t). (1.1) ∂t 2m (1.1) is the so-called non-relativistic time-dependent Schrödinger equation, and recalling the relation p = i~∇ for the momentum operator, the term in 5 1.1 Atomic theory Theory the curly brackets is just an expression of the total energy of the particle and is called the Hamiltonian operator or simply the Hamiltonian, H=T +V =− ~2 2 ∇ + V. 2m (1.2) If the potential is static, V (r, t) = V (r), eq. (1.1) admits stationary state solutions of the form Ψ(r, t) = ψE (r)e−iEt/~ , (1.3) where ψE (r) is a solution to the time-independent Schrödinger equation HψE (r) = EψE (r). (1.4) That is, E is an eigenvalue and ψE (r) an eigenfunction of the Hamiltonian, H. Determining the dynamical evolution of the wave function then boils down to solving eq. (1.4) with the set of eigenvalues making up the atom’s energy spectrum. 1.1.2 Many-electron atomic systems If the system is hydrogen-like, the single electron will only feel a Coulomb attraction V (r) = − Ze from the nucleus of charge Ze. (1.4) can then be r solved analytically, and the energy eigenvalues can be written En (eV ) = −Ry · Z2 , n2 (1.5) where Ry = 13.6057 eV is the Rydberg energy and n is the main quantum number describing the energy states of the ion. This result in the special case of hydrogen (Z = 1) was already obtained by Bohr in his famous article from 1913 (Bohr 1913) but without using the full machinery of wavemechanics which was developed only later. In the more general case of a many-electron atom an electron will also feel the presence of the other electrons thereby adding an extra term to the full Hamiltonian which now reads H= N X i=1 ~2 2 Ze2 X e2 − ∇ − − , 2m i ri r i<j ij 6 (1.6) Theory 1.1 Atomic theory (a) (b) Figure 1.1: a) Niels Bohr (1885-1962). b) Bohr’s description of the atom. The electrons can exist in special orbits and jump between orbits by absorbing or emitting photons of energy hνa→b = Eb − Ea . where N is the number of electrons and rij ≡ |ri − rj | is the distance between the ith and the jth electron. Because of the fact that changes for one electron will affect the others, i.e. through the electron correlation, an analytical solution to (1.4) can no longer be found. This is true even in the simplest case of helium-like systems with only two electrons which implies the use of different approximative methods. 1.1.3 The independent particle model Although each 1/rij term in (1.6) is usually small compared to Z/ri , the sum over j can still be comparable in size with the Coulomb attraction between P electron i and the nucleus. Therefore, treating i<j 1/rij as a perturbation is not a valid option. It is still possible, however, to use perturbation theory by applying an independent particle model. In such a model, each electron is most often imagined to move independently in an effective centrally symmetric potential which represents the attraction of the nucleus and the average 7 1.1 Atomic theory Theory effect of the repulsion between this and the other (N − 1) electrons. This effective potential can be written Vef f (r) = − Ze2 + S(r), r (1.7) where S(r) is some model potential representing the screening of the nucleus by the remaining electrons. Adding and subtracting Vef f (r) in the Hamiltonian (1.6) yields H= N X i=1 | N N X ~2 2 e2 X Ze2 − ∇ + Vef f (ri ) + − + Vef f (ri ) . 2m i r ri i<j ij i=1 {z } | {z } Hc (1.8) Hper The result of this rewriting is that the second term is now much smaller and can be treated perturbatively. By neglecting this term at first and just writing the Schrödinger equation for Hc , one sees that this just separates into N one-electron Schrödinger equations: hi φi = Eφi , hi ≡ − ~2 2 ∇i + Vef f (ri ) . 2m (1.9) Solutions to this equation can be found and the resulting single-electron wave functions or orbitals, φi , are described by the quantum numbers (n, l, ml ), i.e. they are simultaneous eigenfunctions of the Hamiltonian hi , the angular momentum operator Li and its projection onto the z-direction Lz,i . Since electrons are fermions they have an intrinsic magnetic moment, the spin S, which also must be included in the wave functions. Doing this, see e.g. (Bransden & Joachain 1983), the full wave function can be expressed as a so-called Slater determinant, i.e. an antisymmetric product of the N spinorbitals automatically fullfilling the Pauli exclusion principle stating that two electrons in an atom cannot have the same set of quantum numbers. 8 Theory 1.1.4 1.1 Atomic theory Configuration and shell structure The single-electron spin-orbitals are described by the quantum numbers (n, l, ml , s, ms ) and these can take on the values n = 1, 2, . . . l = 0, 1, . . . , n − 1 ml = −l, . . . , l (1.10) s = 1/2 ms = ±1/2 where the reference to s often is omitted since this is always 1/2 for fermions. Because of the spherical symmetry of the potential (1.7) the energy of each electron will be independent of the magnetic quantum numbers ml and ms , but unlike the hydrogenic case the energies will not be degenerate in l. This is because the screening of the nucleus due to the other electrons will be more pronounced for electrons with large angular momentum, as these are forced out by the centrifugal barrier. The total energy of an atom will just be the sum of energies, Eni ,li , of each electron and therefore will be completely determined by the configuration, by which is meant the distribution of the electrons with respect to n and l. This arranges the electrons into shells (electrons with same n) and subshells (electrons with same n, l) and from (1.10) it is seen that each subshell can hold 2(2l + 1) electrons with different ml , ms . Using the typical nomenclature where l = 0, 1, 2, 3, 4, . . . corresponds to the letters s,p,d,f,g,. . . the configuration of e.g. Si with 14 electrons in all can be written 1s2 2s2 2p6 3s2 3p2 , (1.11) where the superscript denotes the number of electrons in the same subshell, i.e. equivalent electrons. 9 1.1 Atomic theory 1.1.5 Theory LS-coupling and spin-orbit splitting In the central-field approximation, the total energy is simply given by the configuration of the system. A more precise description must include the term, Hper , and another important perturbation, the spin-orbit interaction, must also be considered. This is a relativistic effect arising from the interaction of the spin of the electron with the magnetic field induced by the orbiting nucleus (in the restframe of the electron). This interaction is described by the Hamiltonian HSO = N X i=1 1 1 dV (ri ) Li ·Si , 2m2 c2 ri dri (1.12) and will be a small correction for light atoms (ions). The operator Hper will not commute with neither Li nor Si but it will commute with the total orbital and spin angular momenta L= N X Li and S = i=1 N X Si . (1.13) i=1 Including Hper therefore splits each configuration into terms, symbolized as 2S+1 L, that are (2S + 1)(2L + 1) times degenerate with respect to ML and MS . This is the essence of the LS coupling scheme 1 and the terms may be found by adding the spin and orbital angular momenta of the electrons in accordance with the quantum mechanical rules. However, for equivalent electrons not all possible terms are allowed because of the Pauli principle. In appendix A.2 the allowed terms for coupling equivalent s, p- and d electrons are given, as well as Hund’s rules that tell which term will be lowest in energy. Inclusion of the spin-orbit operator will lead to a further splitting of these terms into fine structure components, which are eigenstates of the total angular momentum operator J = L + S, and each eigenstate will be characterized by the symbol 2S+1 LJ . In figure 1.2 the energy splitting due to 1 This coupling scheme is actually also often used for heavier atoms although these are more precisely described using jl coupling where Li and Si are coupled to give Ji and then summed to give the total J. LS coupling will be used throughout this thesis. 10 Theory 1.2 Photoionization cross section the inclusion of the perturbations is pictured schematically in the case of Si. Figure 1.2: Effects (not to scale) of the perturbations Hper and HSO for Si. 1.2 Photoionization cross section Consider an atom or an ion initially in the state Ψi interacting with an unpolarized electromagnetic field. If a photon of energy ~ω, exceeding the binding energy of one of the electrons, is absorbed, the result may be that of photoionization (or detachment for negative ions) where the electron will escape the atomic system. This will then be characterized by a final continuum wave function, Ψf . In the dipole approximation the cross section for this process is given by (in cgs units) σif = 4π ωif |Df i |2 . 3c (1.14) Here ωif is the energy of the absorbed photon which from energy conservation must equal the binding plus the kinetic energy of the free electron. The matrix element Df i is defined as Df i ≡< Ψf |D|Ψi >, 11 (1.15) 1.2 Photoionization cross section Theory with D being the dipole operator D = −e N X ri . (1.16) i=1 Note that the sum runs independently over all electrons which means that the dipole operator is a single-particle operator and that only one photon can interact with one electron. This implies that only main photoprocesses, i.e. excitation or ionization of one electron are possible in this picture. If the wave functions in (1.15) are described by the LS coupling scheme, the non-zero matrix elements must obey the selection rules ∆S = 0 ∆L = 0, ±1 ∆J = 0, ±1 ∆π = ±1 Li + Lf ≥ 1 Ji + Jf ≥ 1. (1.17) As before, S, L and J are the spin, orbital and total angular momenta respectively and π is the parity describing whether the wave function is odd (-) or even (+) under the operation r → −r. Another important parameter, closely related to the cross section, is the oscillator strength, fi,f of a dipole transition between the initial and final states Ψi and Ψf . 2mωif |Dif |2 . (1.18) 3e2 ~ Since both the initial and final state may be multiplet states, it is often more fi,f = useful to consider the mean oscillator strength defined as 1 XX f¯i,f = fi,f . gi α α i (1.19) f If the states are given as terms, αi and αf will refer to the magnetic quantum numbers of the initial and final multiplets and gi is the statistical weight of the initial state, i.e. gi = (2li + 1). It can be shown for hydrogen-like ions (Bransden & Joachain 1983) that both f and f¯ obey the Thomas-ReicheKuhn sum rule X fi,f = 1, f 12 (1.20) Theory 1.3 Interpreting experimental spectra where the sum is over all possible bound or unbound final states. Finally, the following is a useful equation for an experimental determination of the mean oscillator strength: f¯ = 0.00911 · Z σ(E)dE, (1.21) where σ is in Mb and E is in eV. 1.3 Interpreting experimental spectra Although the equations given in the last section apply to both photoionization and photodetachment, the two types of spectra differ markedly due to the structural differences in positive and negative ions, and the description of these spectra has been split up accordingly. Examples and the analysis of both types are given in chapters 4 and 5 and so the following will just be an overview of the gross details. 1.3.1 Positive ions As shown in figure 1.3, the main features of a photoionization spectrum of neutral atoms and positive ions will typically consist of a continuum on which a number of autoionizing resonances are superimposed. The continuum cross section rises abruptly at the threshold where an incoming photon has just enough energy to send an electron directly into the continuum. The cross section will subsequently fall off with increasing photon energy until the threshold of a more strongly bound electron is reached. The resonances correspond to highly excited discrete states embedded in the continuum. Such a resonance can make a radiative transition back to a bound state, but it is much more probable that it makes a radiationless transition into the continuum by emission of an electron, see figure 1.4. This process is called autoionization and is very fast compared to radiative de-excitation. The selection rules for autoionization depend on the operator coupling the discrete state with the continuum and here Coulomb autoionization is the 13 1.3 Interpreting experimental spectra Theory Figure 1.3: Absolute photoionization cross section for Cs+ (5p6 1 S) (not published). The cross section rises abruptly at the 5p−1 threshold at EIP = 23.16 eV (McIlrath et al 1986). Superimposed on the continuum is a number of resonances corresponding to 5p → ns,nd excitations which subsequently autoionize into the continuum. These resonances are members of Rydberg series converging to the Cs2+ (5p5 2 P3/2 ) series limit. most important. However, states that are not allowed to autoionize through the Coulomb interaction may still autoionize by means of e.g. the spinorbit interaction which will be much slower. A table of selection rules and typical lifetimes of autoionizing states is given in appendix A.1 for the most important interactions. As there is no way to determine whether the final continuum state is reached by direct ionization or by resonant ionization (i.e. excitation followed by autoionization, see figure 1.4), the two ionization paths can interfere and the peaks in the spectrum will have characteristic profiles called Beutler-Fano 14 Theory 1.3 Interpreting experimental spectra Figure 1.4: Example of two possible ioniza- Figure 1.5: Beutler-Fano profiles with differ- tion routes of a B-like ion in the ground state. ent values of the Fano parameter, q. The pro- Either by direct ionization or by photoexcita- file becomes Lorentz-like as q goes to infinity. tion followed by autoionization. profiles (Fano 1961). Such profiles are described by the formula σ() = σ0 (q + )2 , 1 + 2 (1.22) where is the reduced energy or the detuning = E − ER . Γ/2 (1.23) σ0 denotes the direct cross section, ER the energy of the resonance and Γ is the natural linewidth. The generally asymmetric shape of the profile is due to interference between the two ionization routes and this is incorporated into the Fano parameter, q. Figure 1.5 shows profiles for different values of q. Note that the cross section is 0 whenever q = − and the case when q = = 0 gives rise to a so-called window resonance. From eq. (1.22) it is also seen that when q is large, i.e. if one route is dominating, the profile approaches a Lorentzian. Furthermore, when we are far from resonance, i.e. → ∞, the cross section is simply the direct cross section, just as one would expect. 15 1.3 Interpreting experimental spectra Theory Quantum defects Resonances are often members of Rydberg series, i.e. they are excited orbitals with successively higher n-values. An example of such Rydberg members are the 1s2 2s2 2p(2 P) → 1s2 2s2p(3 P)np(2 P) excitations that show up in the spectrum of N2+ → N3+ . These will converge to the N3+ (1s2 2s2p(3 P)) ionization limit corresponding to n = ∞. A very powerful tool when trying to identify the experimental peaks is to use the simple concept of Quantum Defect Theory (QDT). The idea is that the energy levels of a Rydberg series are given by a formula similar to (1.5) with the difference that n is replaced by an effective quantum number, n∗ . Z2 . (1.24) n ∗2 is the ionization limit of the series, Z is the charge state of the En∗ = E∞ − Ry Here E∞ ionized ion and n∗ ≡ (n − δnl ) where δnl is the quantum defect that takes into account the partial screening of the nucleus. In the case of hydrogen-like systems we have E∞ = δnl = 0 and eq. (1.24) simply reduces to (1.5). The true power of this theory lies in the fact that the variations of δnl with n is small meaning that δnl ≈ δl . Thus, if one knows a member of the series and the limit to which it converges, it is possible to calculate the quantum defect and thereby the position of all other members under the assumption that δ is constant. 1.3.2 Negative ions Negative ions occupy a unique place among the atomic species because of the strong correlations of the valence electrons. Whereas correlations to a large extent can be regarded as pertubations to the Coulomb attraction betweem the electron and the positive nucelus in atoms and positive ions, these correlations are essential in the binding of the extra electron in negative ions. This fact makes them extremely difficult to handle theoretically and the need for experimental results is therefore particularly profound in this field (Ivanov 1999). 16 Theory 1.3 Interpreting experimental spectra Absolute photodetachment studies have so far mainly dealt with detachment of the loosely bound outer-shell electrons utilizing lasers as the source of photons, but in the ASTRID experiments inner-shell detachment is studied. The creation of an inner-shell hole will almost certainly lead to the formation of a positively charged end-product which is subsequently detected (see figure 2.1). The process of interest can be written − − + A− + ~ω → A∗ + e− photo → A + ephoto + eAuger , (1.25) where the decay of the excited neutral state is known as an Auger decay. A cascade of Auger decays may also occur thus forming a multiply charged end-product. Since it is not the long-range Coulomb attraction that holds the electron attached but instead some short-range potential induced by the polarization of the neutral atom by the extra electron, a photodetachment spectrum of negative ions lacks many of the features found in photoionization spectra. Negative ions, for example, usually only exist in one bound state in contrast to the infinity of Rydberg states found in positive ions and neutral atoms. As a consequence, there are no resonances corresponding to different converging Rydberg series. It is sometimes possible, however, as in the case of 4d detachment in Te− (see chapter 4), that an excited state is bound with respect to its ionization threshold. Such states give rise to relatively narrow resonances known as Feshbach resonances below threshold in the detachment spectrum. A more typical type of resonance is the shape resonance which is a broad structure located above threshold and which corresponds to what is sometimes referred to as a “nearly” bound state. The threshold behaviour will also be different because the outgoing photoelectron will move in a potential that is very much influenced by the centrifugal barrier, Vcen (r) = l(l + 1)~2 /2mr2 , and thus show a dependence of l. It can be shown that the cross section immediately (∼ µeV) above the threshold energy, Eth , is given by the Wigner threshold law (Wigner 1948), σ(E) ∝ (E − Eth )l+1/2 . 17 (1.26) 1.4 “Hardcore” theoretical approaches 1.4 Theory “Hardcore” theoretical approaches Returning to eq. (1.14), the calculation of the cross section is seen to depend entirely on the ability to evaluate the matrix elements (1.15) which again means finding initial and final state wave functions as solutions to the Schrödinger equation. One way to accomplish this, as we have seen, is to apply an independent particle picture of which the Hartree-Fock method (HF) or Self-consistent field (SCF) is the most precise and well-known, see e.g. (Fischer 1977). Such an approach may succeed in giving the overall shape of the spectrum but it fails to reproduce features such as autoionizing resonances that are direct consequences of electron correlations. Therefore, one can only hope to get quantitative agreement by using models that include correlations. A number of such models have consequently been developed that differ mainly in the way in which the electron-electron interaction is included. The models most frequently encountered are the Configuration Interaction method (CI), the Random Phase Approximation (RPA), the Local Density Approximation (LDA), Many-Body Perturbation Theory (MBPT), Multi Configuration Dirac-Fock (MCDF) and the R-Matrix method. Time and space allow only for a very few remarks to be given here but details as well as some of the main achievements of these models can be found in e.g. (Becker 1996, Chang 1993, Schmidt 1997) and references therein. In the CI approach the correlated wave functions Ψi and Ψf are expanded into a complete set of un-correlated basis functions with the same symmetry properties. As an example, one would write the ground state of Li+ as Ψcorr = a1 Ψ(1s2 1 S0e ) + a2 Ψ(1s2s 1 S0e ) + a3 Ψ(2s2 1 S0e ) + a4 Ψ(2p2 1 S0e ) + . . . (1.27) where the absolute square of the mixing coefficients, an , is an expression of the weights of the corresponding states. Such an expansion requires, in principle, an infinite number of basis functions which is not possible for practical purposes, since all expansions have to be truncated into a finite number for the calculations to be carried out. The quality of a CI calculation therefore 18 Theory 1.4 “Hardcore” theoretical approaches depends on the number and type (usually HF one-electron wave functions) of basis function used (Schmidt 1997). This way of incorporating electron correlations is common for many of the different models and in chapter 5 some of the experimental results will be compared to both R-matrix results found in the OPACITY database (TOPbase, Cunto et al 1993) and to new MCDF calculations provided by (Bizau 2003). One should note that although these methods are both based on a CI description of initial and final states, they are somewhat different. The R-matrix method calculates the photoionization cross section whereas the MCDF method calculates the photoexcitation cross section under the assumption that the excited states decay by autoionization. Moreover, any interference between direct and resonant ionization, which would lead to asymmetric profiles of the resonances, is neglected in the latter. References Becker U. and Shirley D. A. (1996) VUV and Soft X-Ray Photoionization (Plenum Press) Bizau J. M. (2003) - Private communication Bohr N. (1913) Phil. Mag. 26 1 Bransden B. H. and Joachain C. J. (1983) Physics of Atoms and Molecules (Longman Group Limited) Chang T.-N. (1993) Many-Body Theory of Atomic Structure and Photoionization (World Scientific Publishing Co. Pte. Ltd.) Cunto W., Mendoza C., Ochsenbein F. and Zeippen C. J. (1993) Astronomy and Astrophysics 275 L5 Fano U. (1961) Phys. Rev. 124 1866 Fischer C. F. (1977) The Hartree-Fock method for atoms (John Wiley & Sons, Inc.) Gasiorowicz S. (1996) 2.ed. Quantum Physics (John Wiley & Sons, Inc.) Ivanov V. K. (1999) J. Phys. B: At. Mol. Opt. Phys. 32 R67 19 1.4 “Hardcore” theoretical approaches Theory McIlrath T. J., Sugar J., Kaufman V., Cooper D., and Hill W. T. III. (1986) J. Opt. Soc. Am. B 3 398 Sakurai J. J. (1994) Modern Quantum Mechanics (Addison-Wesley Publishing Company, Inc.) Schmidt V. (1997) Electron Spectrometry of Atoms using Synchrotron Radiation (Cambridge University Press) TOPbase - www.heasarc.gsfc.nasa.gov/topbase/topbase.html Wigner E. P. (1948) Phys. Rev. 73 1002 20 Chapter 2 The experimental setup Figure 2.1: The experimental setup: IS ion source, EL Einzel lens, M1-2 Bending magnets, ED Electrostatic deflector, RC Reaction chamber, D1-2 Ion detectors, FC1-2 Faraday cups (positive target beams), FC3 Faraday cup (negative target beam), PD Photodiode, UN Undulator and MO Monochromator. Fig. 2.1 above shows schematically the experimental setup. In short, a beam of target ions is produced in the ion source (IS) and accelerated by a high voltage, Vacc (typically 2 kV), to an energy E = qeVacc . The beam is then focused (EL) and mass analyzed (M1) before it is steered into the reaction chamber (RC) by means of an electrostatic deflector (ED), and here the target beam is overlapped colinearly with monochromatized synchrotron radiation from the ASTRID undulator (UN and MO). The absolute cross section can now be determined by measuring the photon current in the pho21 2.1 The light source The experimental setup todiode (PD), the current of the primary beam (FC1-2 or FC3 depending on charge), the ion-photon overlap and the number of ionized ions (D1-2). In this chapter the primary experimental parts will be addressed, and it will be shown exactly how measurements of the above mentioned quantities are used to determine absolute cross sections. Since such a description has already been written (Kristensen 2001, Andersen 2001) (both in danish), (Kjeldsen 1999) and (Schwebs 1999), the present text is not intended to be an in-depth treatment and is mainly included for the sake of completeness. A little more attention (next chapter) has been paid to the production of ions, as I have spent a considerable amount of time working with the ion sources that we have used. 2.1 The light source As already mentioned, it is of immense importance to have an extremely high photon flux in order to carry out absolute photoionization cross-section measurements on ions. Therefore, high-intensity light from the undulator at the storage ring ASTRID is used in the experiment. This section will give a short description of both the undulator and the monochromator which together constitutes the light source. A more thorough treatment can be found in (Kjeldsen 1999). 2.1.1 The undulator An undulator is basically built up by a set of permanent magnets arranged in a periodic structure (period λ0 ), as illustrated in fig. 2.2. The magnetic structure gives rise to a periodically changing magnetic field B = y B0 sin( 2πz ) λ0 (2.1) that will exert a Lorentz force F = −ev × B on an entering electron. Since F and v are perpendicular, the magnetic field does no work on the electron 22 The experimental setup 2.1 The light source Figure 2.2: Sketch of an undulator with magnetic period λ0 . The strength of the magnetic field is varied by changing the undulator gap, g. but instead causes it to oscillate in the xz-plane and thereby to emit light as it traverses the undulator. Electrons circulating in the ASTRID storage ring have energies of E = 580 MeV and are therefore highly relativistic: E 1 γ= =p = 1135 1, 2 m0 c 1 − β2 with β = v c (2.2) and m0 the electron mass. In the electrons’ rest frame x0 y 0 z 0 the magnetic period will be much smaller due to the Lorentz contraction λ0 = λ0 , γ (2.3) and the emitted light will correspond to classical dipole radiation from an oscillating point charge with frequency c cγ ν0 = 0 = . (2.4) λ λ0 In the laboratory frame the emitted ligth will be pushed dramatically in the forward direction and will have a Doppler shifted frequency ν= ν0 c = , γ(1 − βcosθ) λ0 (1 − βcosθ) 23 (2.5) 2.1 The light source The experimental setup where θ is the angle from the z-axis. Using the approximations γ2 = 1 1 ≈ (1 − β)(1 + β) 2(1 − β) (2.6) and θ2 , (2.7) 2 which are valid when β ∼ 1 and θ ∼ 0, we can use equation (2.5) to write cos θ ≈ 1 − λ= c λ0 = 2 (1 + γ 2 θ2 ). ν 2γ (2.8) In the calculations leading to equation (2.8) we have assumed that v = vz , that is we have neglected the x-component of the electron velocity and we have therefore overestimated the effect of Lorentz contraction and the Doppler shift. By introducing the so-called undulator parameter K= eB0 λ0 , 2πm0 c (2.9) which is proportional to the maximum electron deflection angle, K = γα, inside the undulator, it can be shown that (2.8) should be extended to λ0 K2 λ= 2 1+ + γ 2 θ2 . (2.10) 2γ 2 By adjusting the undulator gap, B0 and thereby K can be varied between 0 and 2.3 which by insertion yields energies on the z-axis (i.e. θ = 0) between 15 and 58 eV1 . Since K does not greatly exceed 1 in an undulator the electron oscillations will be small and light emitted at different points inside the undulator can interfere, which will lead to the presence of higher harmonics, n, of the fundamental in the undulator spectrum2 . Therefore, the final undulator equation can be written λn = 1 λ0 K2 2 2 1 + + γ θ 2γ 2 n 2 (2.11) This lies in the UV regime and since air is opaque to photons above 6 eV, the beamline must be evacuated. For this reason the light is known as Vacuum Ultra Violet or VUV radiation. 2 In the case where K 1 the insertion device is called a wiggler and here interference effects are minimal. 24 The experimental setup 2.1 The light source In this way one can at a given wavelenght maximize the flux through the monochromator with the appropriate choiche of n and g (or equivalently K). Figure 2.3: Calculated spectrum of the ASTRID undulator at three different gaps with the numbers designating the harmonic value. Lowering the value of K (increasing g) shifts the spectrum towards higher energies. 2.1.2 The Miyake monochromator Since the spectrum of an undulator includes a wide range of energies (see figure 2.3), a monochromator is a necessary supplement in order to pick a specific energy. The Miyake monochromator was constructed at Daresbury Laboratory in 1974 (West et al 1974). It is shown schematically in figure (2.4) and consists of three optical components: • A grating which can be rotated. • A cylindrical, focusing mirror which can both be rotated and translated. • Exit slit with adjustable width. 25 2.1 The light source The experimental setup When the incoming beam of radiation from the undulator enters the monochromator, the grating will select a wavelength in accordance with the grating equation 1 (sinα − sinβ), (2.12) N with k being the diffractive order (k = −1 is used) and N the number of kλ = lines per mm, N = 1200mm−1 . Because the beam must hit the exit slit, we can write another equation relating the mirror angle, θ and the distance d: h sin(π − 2θ) = sin(2θ) = p h2 + (l − d)2 , (2.13) where h and l are constant since the grating can only be rotated. Figure 2.4: The principle of the Miyake monochromator. The grating angle and the mirror position and angle are controlled by stepping motors. By considering similar expressions for the focusing of the mirror and the grating, see (Kjeldsen 1999), it is possible to determine d, θ and α as a function of the beam energy. However, experience has shown that the monochromator should not be operated in this so-called in-focus mode, where the mirror is continuously being moved as to focus the beam on the exit slit. This is due to mechanical instabilities in the mechanism controlling the mirror position, which are considerable especially at low energies where the mirror has 26 The experimental setup 2.1 The light source to be moved a lot. Instead the fixed-focus mode is used, where the mirror is locked at a specific position (d, θ) and only the grating angle, which is highly accurate and reproducable, is varied. This, of course, means that the exit slit will be in focus only at a specific energy and therefore the energy resolution due to defocus will be worse when we are away from this energy. Another important issue is the fact that a nonvanishing amount of secondorder radiation will be transmitted through the monochromator at energies below ∼ 80 eV despite that it was designed to suppress this. This means that a fraction of the light in the reaction chamber will be of second instead of first order, i.e. it will be twice as energetic as expected. The measured energy-dependent cross section will therefore not be correct, but instead be some mean value of the cross section at the energy E, and 2E. σmeasured (E) = (1 − X2. (E))σ(E) + X2. (E)σ(2E). (2.14) Usually the second order fraction, X2. (E), is small, but its contribution to the measured cross section can still be large if σ(2E) σ(E). A way to circumvent this problem is to let light from the monochromator pass through a foil that will absorb the higher-order radiation, but which is semitransparent at lower energy, before it enters the reaction chamber. Figure 2.5 shows the transmission curves of 4 foils of different materials in the energy range between 20 and 130 eV. From this it is clear that Al for instance, which has an L2,3 absorption edge at 72 eV, is practically fully absorbant above this energy and therefore can be used from 36 to 72 eV. It is also apparent from the figure that the insertion of a foil will lead to a pronounced reduction in the first order flux, which is highly undesirable. For this reason the data are often recorded without a foil and thus with a contaminated photon beam. It is possible, however, to correct for this contamination, and according to eq. (2.14) it requires that one can determine the content of second order radiation as a function of energy as well as σ(2E). The technique for doing so is described in (Kristensen 2001). 27 2.2 The absolute cross section The experimental setup Figure 2.5: The figure shows the transmission of radiation of 4 absorption foils as a function of energy. Only foils made of Mg, Al and Si are installed in front of the reaction chamber, whereas both an Al- and a polyimide foil are used as window foils for the ionization chamber. 2.2 The absolute cross section As already mentioned, to measure the cross section on an absolute scale it is necessary to know precisely the number of ions produced (i.e. the signal), the density of target ions, the photon flux and the amount of overlap between the two beams. The photoionization signal S is given by the Lambert-Beer law as S = F (1 − e−σnl ) ≈ F σ n l, (2.15) with F being the photon flux, n the target density, σ the cross section and l the interaction length. The validity of the expansion is recognized by the fact that typical values of the factors in the exponent are σ ∼ 1 Mb = 10−18 cm2 , n ∼ 106 cm−3 and l ∼ 50 cm. If now we by dSxyz refer to the part of the total signal arising from the small volume element dxdydz and let i and j indicate the ion- and photon 28 The experimental setup 2.2 The absolute cross section current, respectively, passing through the area dxdy we get dSxyz = Ωi j σ dz, eη qevdxdy |{z} | {z } (2.16) n F where e is the elementary charge, Ω and η are the detector- and photodiodeefficiencies and v is the target ion velocity. Integration over the xy-plane yields Ωσ dSz = 2 · e ηv RR ij dxdy Ωσ IJ · dz = 2 · · dz, dxdy q e η v dxdy F (z) where we have defined the 2-dimensional form factor F (z) RR RR i dxdy · j dxdy IJ RR F (z) ≡ = RR . ij dxdy ij dxdy (2.17) (2.18) To get the complete signal, eq. (2.17) must be integrated along the z-direction and from this the final expression for the cross section is found to be σ= S · q · e2 · v · η R . I · J · Ω · l dxdy1F (z) dz (2.19) As before, l is the length over which the photons and ions can interact, but referring to fig. (2.1) this is not a well-defined number. To overcome this, a bias of the order of hundreds of volts is applied to the reaction chamber thereby energetically tagging the ions produced in this region and thus allowing for them to be separated from ions produced outside the chamber. The parameters contained in (2.19) all need to be determined with great care and high precission in order to keep the total error within a desired limit of ∼ 10%. The procedure for calibrating the detectors and measurements of form factors is described in great detail in e.g. (Kristensen 2001) and only a few remarks will be given here. 2.2.1 Form factors Beam profiles or form factors are measured with five sets of horizontal and vertical beam scanners installed along the reaction chamber. Two approximations are made in the evaluation of F (z): 29 2.2 The absolute cross section The experimental setup 1. The beam scanners move in steps of finite size which means that inteR P grals are replaced by sums, i.e. → 2. The two-dimensional overlap is approximated as the product of two one-dimensional overlaps; F (z) → Fx (z) · Fy (z). This has the advantage that the time consumption is reduced by a factor of ten or so, and the error has experimentally been shown to be very small. As a consequence (2.18) is replaced by P P P P RR RR j i dxdy · j dxdy x yi· RR P P x y F (z) = ≈ ij dxdy x y ij P P P P j· i yj xi Px P y ≈ = Fx (z) · Fy (z). x ij y ij (2.20) In this way, five values of the form factor {F (z1 ), . . . , F (z5 )} are measured at the points {z1 , . . . , z5 } in the reaction chamber and by fitting these with R a second-order polynomial, l dxdy1F (z) dz can be evaluated. The polynomial description of the z-dependence is remarkably well and experience has shown that it is even safe to leave out measurements at z2 and z4 which is another timesaving factor. All in all an accurate determination of the overlap can be done in less than ten minutes. 2.2.2 Detector calibration The efficiency of the detectors (Johnston multipliers) changes slowly over time and therefore needs to be calibrated once in a while. The procedure is simply to make a stable ion beam and measure the count rate in FC1 (see figure 2.1), which has a 100% efficiency, and then by changing the magnetic field in M2 send it to one of the detectors D1 or D2 and measure the corresponding count rate here. Because the efficiency also depends on the mass, charge and energy of the ions, it is essential that the ions used for the calibration have preferably the same or at least similar (M,q,E)-values as the ones detected in the photoionization experiments. For example, in the measurement of Te− → Te+ , Te2+ a bias of 200 V was applied to the reaction 30 The experimental setup Ions 2.2 The absolute cross section Energy (keV) ΩD1 (%) ΩD2 (%) Xe+ 2.4 59 —– Xe2+ 2.6 68 58 2+ Ar 2.6 77 70 Ar4+ 5.2 82 75 Ar6+ 7.8 85 76 Table 2.1: Selected measured values of detector efficiencies. chamber so with an initial target energy of 2 keV it can be seen that the Te+ -ions detected in D1 had energies of E = (2 + ∆q · Vbias ) keV = 2.4 keV. Similarly the ions detected in D2 had energies of 2.6 keV and so D1 should in principle be calibrated with Te+ -ions at 2.4 keV and D2 with Te2+ -ions at 2.6 keV, but to avoid the difficulties in producing these ions, Xe+ at 2.4 keV and Xe2+ at 2.6 keV were used instead. 2.2.3 Photodiode calibration Figure 2.6: The ionization chamber used to calibrate the photodiode. The length of the collectors and the distances to the window foils are l = 200 mm, x0 = 266 mm and x1 = 428 mm. The photon flux is measured with a calibrated Al2 O3 photodiode but this is unable to detect all photons (actually it detects only a few in a hundred). For this reason, the signal measured with the diode needs to be compared 31 2.2 The absolute cross section The experimental setup with the true flux in order to determine the exact energy-dependent efficiency of the diode, η(E). For this purpose a so-called double ionization chamber sketched in figure 2.6 has been installed in front of the photodiode. The principle is to let in a noble gas (Neon in our case) and measure the resulting ionization current on the two collectors. Using the Lambert-Beer law the true flux can be shown to be given as (Kristensen 2001) J xli (J )2 /e 1 1 Ftrue = · , (2.21) J2 J1 − J2 where i ∈ {0, 1} and J1 and J2 are the currents on the two collectors. Note that for this equation to make sense J1 6= J2 , meaning that the pressure in the chamber needs to be large enough for the signals on the two collectors to be different. In (2.21) it is explicitly assumed that exactly one ion is produced for every incoming photon, which is approximately true for the noble gases (Samson & Haddad 1974). However, at larger energies multiple ionization starts to play a role resulting in a larger registered current on the collectors. Secondary ionization is another phenomenon with the same consequence, and this will occur whenever the photon energy exceeds twice the ionization potential of Ne the gas (EIP = 21.56 eV) where the emitted photoelectrons will carry enough kinetic energy to ionize other atoms by impact. Furthermore the electrons will gain energy from the repeller bias which therefore must not be too large but still large enough as to prevent the electrons from reaching the collectors. Since both multiple and secondary ionization are responsible for an increase in the collector current (and thereby a value of Ftrue that is too high or equivalently too low an efficiency) these effects must be taken into account. Multiple ionization is easily handled as the relative fraction of single and multiple ionization is known from the litterature (Bartlett et al 1992, Samson et al 1992) but the subject of treating secondary ionization is more subtle, since this will depend on the amount of gas in the chamber (i.e. the pressure) and on the vertical position of the photon beam. How compensations are made will not be mentioned here but the reader is recommended to consult (Kristensen 2001) or (Kjeldsen 1999) for details. 32 The experimental setup 2.3 2.3 Recording spectra - an overview Recording spectra - an overview To summarize the chapter it can be instructive to have an overview of some of the basic steps that must be performed when making an actual cross-section measurement. Practically all operations on the undulator, the monochromator and the variety of power supplies can be done remotely using a PC connected to the ISA Controle System (www.isa.au.dk/consys), with some home-made software written in the graphical programming interface Labview. 1. The first step is to produce a target ion beam and steer it all the way to one of the Faraday cups FC1-3. 2. The mirror in the monochromator is then moved and locked at a position corresponding to a photon energy where the highest resolution is wanted (typically around threshold). 3. At an energy where a signal is expected (i.e. σ(E) 6= 0) the beam profiles are measured with the beam scanners and the two beams are aligned as to ensure maximum overlap. 4. Next, the analyzing magnet M2 is scanned to see which magnetic field will send the primary beam to the Faraday cup and the signal to one of the detectors (or both if doubly ionized ions are produced and can be detected simultaneously). 5. A spectrum over a given energy range can now be recorded. At some point before or after the scan the photodiode needs to be calibrated covering this entire range. This must be done every time the diode is hit differently by the photons (e.g. if the width of the exit slit is changed between scans). The monochromator also needs to be energy calibrated which is done by letting in different gases in the ionization chamber and observe the position of known resonances (e.g. 3d → 5p in Kr known to be positioned at 92.463 eV (NIST 2003)). 33 2.3 Recording spectra - an overview The experimental setup References Andersen P. A. (2001) Master thesis University of Aarhus Bartlett R. J., Walsh P. J., He Z. X., Chung Y., Lee E.-M. and Samson J. A. R. (1992) Phys. Rev. A 46 5574 Kjeldsen H. (1999) PhD thesis University of Aarhus Kristensen B. (2001) Master thesis University of Aarhus NIST (2003) http://physics.nist.gov/cgi-bin/AtData/levels form Samson J. A. R., He Z. X. and Bartlett R. J. (1992) Phys. Rev. A 46 7277 Schwebs M. (1999) Master thesis University of Aarhus West J. B., Codling K. and Marr G. V. (1974) J. Phys. E 7 137 34 Chapter 3 Ion production This chapter briefly deals with the problem of producing a target ion beam and describes the two types of ion sources that I have used, namely Middleton’s high-intensity sputter source for negative ion production and the ECRIS (Electron Cyclotron Resonance Ion Source) for the production of multiply-charged positive ions. 3.1 General remarks The production of ion beams has many applications in several areas of physics and other natural sciences as well as for industrial purposes. Each application may have its own needs (charge state, intensity, purity, divergence, etc.) and unfortunately, no universal ion source exists that is able to comply with all the different demands; hence numerous different types of sources are currently in use worldwide. In our experiments at the ASTRID undulator beamline, three points are of main interest: • The extracted ion beam should be stable, since a high degree of stability is absolutely necessary for making precise measurements. • The beam should preferentially be cold, i.e. it should not contain any metastable components. 35 3.2 The Middleton sputter source Ion production • The intensity should be as high as possible, since the measured photoionization signal is directly proportional to the target ion current, see eq. (2.19). 3.2 The Middleton sputter source The ion source used for the production of negative ions is a Cs sputter source which was originally designed in 1983 by Middleton (Middleton 1983). The ions are produced after sputtering of the sample by Cs+ ions created at a spherical ionizer by surface ionization. A schematic overview of the source is shown in figure 3.1 and a more detailed description is given in section 3.2.3. 3.2.1 Surface ionization For atoms that are very near to or adsorbed on a hot metal surface, it is sometimes possible for electrons to move from the metal to the atom or from the atom to the metal, depending on the electropositive/electronegative character of the atom in relation to the work function of the metal. Thus, atoms may be emitted from the surface in ionic form which is known as surface ionization. If we consider atoms with first ionization potential Ip striking a surface with work function φ that is heated to a temperature T , the probability that they leave the surface as positive ions can be written as (Septier 1967, Alton & Mills 1996) P + ! φ − Ip = A · exp , kT (3.1) where A is a constant including the statistical weigths of the ions and neutral atoms. From this equation it is obvious that the surface-ionization efficiency is large only for metals with high work funcktions (e.g. tungsten (4.55 eV), platinum (5.65 eV), rhodium (4.98 eV), tantalum (4.25 eV)) and atoms with low ionization potentials (especially the alkali metals). 36 Ion production 3.2.2 3.2 The Middleton sputter source Sputtering That negative ions can be formed by sputtering a solid surface with positive ions has been an experimental fact for several decades and different variants of sources based on this principle have been developed (Middleton & Adams 1974, Alton 1993). No satisfactory microscopic theory has yet emerged that agrees quantitatively with a large range of experimental results, but the negative ion yield seems to follow a law similar to eq. (3.1). ! E − φ A P − ∝ exp kT (3.2) Here EA is the electron affinity, φ is the work function of the sputter surface and T is its temperature which problably has only little physical significance. From this equation, one would expect a very low ionization efficiency since EA < φ for most elements, but a dramatic enhancement in the negative ion formation can be achieved if the sputtered surface is covered with a layer of an alkali metal (Cs in particular). The reason for this can be ascribed to a decrease in the surface work function from the presence of the adsorbed alkali element. The main theoretical difficulty is the lack of a method for estimating such changes in the work function as well as a realistic model for negative ion formation, so the mechanism of a sputter-type negative ion source is far from beeing completely understood although some promising proposals have been made (e.g. Rao et al 1992). 3.2.3 Description of the Middleton source Figure 3.1 shows the buildup of the Middleton Cs sputter source. The Cs reservoir is heated and the Cs vapour is led to the chamber containing the spherical ionizer and sample through the Cs pipe. The sample rod can slide on an o-ring under vacuum and samples can be by changed sliding the rod back and closing the gate valve. This small compartment can be pumped though the valve, thereby keeping the ionizer under vacuum during the whole loading procedure. The ionizer is heated by sending a 37 3.2 The Middleton sputter source Ion production Figure 3.1: Overview of the spherical ionizer sputter ion source, see text. 1) water cooling, 2) gate valve, 3) sample rod, 4) sample, 5) spherical ionizer, 6) Cs pipe and 7) Cs reservoir. Figure 3.2: Details of the region of the ionizer and sample. large current through it, thus resulting in a surface ionization of some of the Cs atoms. The generated Cs+ ions are accelerated towards the sample 38 Ion production 3.2 The Middleton sputter source which is kept at a negative potential and this leads to a sputtering of the sample, see also figure 3.2. The sample itself is pressed into a cathode made of Cu or Al that can be mounted directly on the sample rod (see figure 3.3). By cooling the sample rod, the cathode and sample are prevented from getting very hot which Figure 3.3: cathode with sample. allows for the Cs vapour to condense and cover the sample, thus lowering the work function as described above. The negative ions formed are then extracted in the form of a beam through a hole in the ionizer towards the extraction electrode (not shown in fig. 3.1). Some typical values of the ion source parameters are listed in table 3.1. Parameter Value Ionizer current 18 A Cathode potential -8 kV Reservoir temp. 160 ◦ C Extraction potential 5 kV Pressure 5 · 10−6 mBar Table 3.1: Some typical values of ion source parameters Another important parameter is the cathode distance, i.e. the distance from the sample to the ionizer, which can be varied by moving the sample rod. This adjustment cannot be done continuously while the source is running, since one must push or pull the rod manually and this is on high voltage. This means that the high voltage must be switched off before it is safe to go and change the distance. From our measurements we have experienced that changes of as little as 1-2 mm have had a large effect on the intensity of the extracted ion beam, but generally the same distance seems to apply well to the different samples and only small changes should be made. The main advantages of this source is that negative ions can be made from 39 3.3 The ECRIS Ion production almost every element in the periodic table (see Middleton’s ”A Negative Ion Cookbook”). Moreover, the design makes it extremely easy to reload the source without breaking the vacuum and the whole procedure can be done within 5 min. A little patience is often required when inserting a new cathode since a sputter crater usually has to be formed before the ion yield starts to rise and one must also play around with the different source parameters to find the right settings. This time, of course, depends on how easily the various elements form negative ions, so while for example C and Cu starts to rise immediately, other elements such as Al or Cr may take a few hours. In table 3.2 some of the obtained ion currents measured in FC3 (figure 2.1) are presented. Ion Current (nA) Cathode Te− 30 CdTe − 15 V Cr− 40 Cr Co− 90 Co − 95 Ni V Ni Table 3.2: Obtained ion currents in FC3. 3.3 The ECRIS An ECR ion source is a so-called hot plasma source where the plasma is maintained by electrons interacting resonantly with electromagnetic waves. Many different variants of such sources are now succesfully being used to produce beams of highly charged positive ions and research is still being done in order to enhance their performance. Different attempts have been made to control and optimize on important plasma parameters such as the electron number density, electron temperature (energy) and the plasma confinement. The following is only intended as a short explanation of how an ECRIS work. 40 Ion production 3.3 The ECRIS For more information the reader is referred to e.g. Ciavola & Gammino (1996), Geller (1998), Girard & Melin (1996) or Alton 1996. 3.3.1 The ECRIS principle Figure 3.4: The principle of an ECR ion source. The principle of an ECRIS is shown in figure 3.4 above. A plasma is created inside a chamber surrounded by strong magnets by the injection of radiofrequency (rf) power. The magnets are arranged in a minimum Bstructure where conventional or superconducting solenoids or permanent ring magnets produce a strong axial field and a multipole magnet produces a radial field in such a way that the magnetic field is at a minimum in the middle of the chamber and increases in all directions, see figures 3.5 and 3.6. This magnetic configuration thus constitutes a magnetic trap whichs tends to confine the electrons in the center. The reason for this is a direct consequence of the invariance of the magnetic moment, µ = 2 mv⊥ 2B since electrons moving in a direction of increasing B must increase their perpendicular velocity, which by energy conservation leads to a decrease in the parallel velocity, 41 3.3 The ECRIS Ion production v|| . Depending on the mirror ratio, Bmax /Bmin , the outgoing motion can be stopped hence leading to a reflection of the electron. Furthermore, the electrons will gyrate around the magnetic field lines and Electron Cyclotron Resonance will occur whenever the frequency of gyration matches the rf frequency: ωrf = ωcyc = Figure 3.5: Axial magnetic field. e B. m (3.3) Figure 3.6: Radial magnetic field. By adapting a proper combination of magnetic field strength and rf frequency, a closed surface is obtained on which the resonance condition (3.3) is fulfilled. This defines the boundary of the ECR zone where electrons can be accelerated to very high energies, which enables the plasma atoms/ions to be stripped to high charge states by electron impact. It is important, though, to realize that the presence of energetic electrons in the magnetic trap is not the only criterion for the production of highly charged ions. Since step-by-step ionization is the dominant stripping process, it is also necessary that the ions stay in the plasma for a time sufficient enough for the desired charge state to be reached. Because of their larger mass, the ions are not affected much by the magnetic fields so the ions are kept in the plasma by the space charge potential of the electrons. On the other hand, the ion confinement time must not be too high since ion losses from the plasma is what leads to the extracted output current and therefore the ECRIS is a compromise between these needs. 42 Ion production 3.3.2 3.3 The ECRIS Description of the ECR source Figure 3.7: Schematic overview of the compact 10 GHz all-permanent ECRIS used at the undulator beamline. The ECRIS used in the experiments at ASTRID is shown scematically in figure 3.7. The axial confining magnetic field is provided by strong NdFeB permanent ring magnets and the radial field by a permanent hexapole and the magnets are surrounded by a protective shield of steel. Furthermore, the plasma chamber walls are watercooled to prevent a de-magnetization of the magnets at high temperatures. The field strength in the chamber corresponds to ECR operation at ∼10 GHz, but since there is no way of adjusting the magnetic field, it is necessary to have a rf supply which is tunable both in power and frequency. We have used different supplies with output frequencies ranging from 9 - 10.5 GHz and with rf power up to 250 W. The electromagnetic waves are led from the supply at ground potential to the plasma chamber by air filled wave guides and before they enter the ion source region they must go through a high-vacuum/high-voltage window because the ion source is evacuated and on a high potential. The source has two gas inlets (only one is shown in figure 3.7) and the gas flow is adjusted by needle valves which can be controlled from outside the high-voltage region 43 3.3 The ECRIS Ion production by long insulated rods. The produced multiply-charged ions are extracted towards a puller electrode that is kept on negative potential. One of the great advantages of the ECRIS is that it is extremely easy to operate, since the only parameters one needs to manipulate with are the gas pressure and rf power (and to some extent also the rf frequency). Moreover, ions of many charge states are generated immediately after adding gas and rf power, see figure 3.8. In general, not all the ions will be in the ground state but will instead be in some metastable excited state(s). With so few knobs to turn there is really no way that the metastable component(s) can be reduced and in this respect the simplicity of the ECRIS may be regarded as a drawback. The production of ions from solids is also much less straightforward than from gaseous elements. By evaporating solids from an oven designed for this particular ECRIS, we have succeded, though, in making stable beams of Ba2+ , La3+ and Ce4+ , but only after days of hard work. Table 3.3 below shows some measured output currents for different ions. Ion Current (nA) Made from N2+ 150 N2 (g) 3+ O 120 O2 (g) F4+ 40 SF6 (g) O4+ 115 O2 (g) F 55 SF6 (g) Ne4+ 45 Ne (g) Ba2+ 3+ 28 Ba (s) 3+ 5 C15 H15 La (s) 4+ 4.5 C15 H15 Ce (s) La Ce Table 3.3: Performance of the ECRIS. (g) gas and (s) solid. An example of a measured output spectrum of the ECRIS is shown in figure 3.8. It was obtained after optimizing on the O3+ peak by scanning the field of magnet M1 (figure 2.1). The main peaks have been labeled and it is 44 Ion production 3.3 The ECRIS seen that all charge states of Oq+ are present up to q = 6 whereas ionization of the rest gas is responsible for the nitrogen and hydrogen peaks. It should also be noted that the larger output velocities of the higher charge states results in a decrease in beam divergence and therefore narrower peaks. Figure 3.8: Output spectrum from the ECRIS loaded with O2 gas obtained by scanning the magnet M1 and measuring the ion current on a shutter inserted before the reaction chamber. References Alton G. D. (1993) Nucl. Instr. and Meth. in Phys. B 73 221 Alton G. D. and Mills G. D. (1996) Nucl. Instr. and Meth. in Phys. Res. A 382 232 Alton G. D. (1996) Nucl. Instr. and Meth. in Phys. Res. A 382 276 Ciavola G. and Gammino S. (1996) Nucl. Instr. and Meth. in Phys. Res. A 45 3.3 The ECRIS Ion production 382 267 Geller R. (1998) Rev. Sci. Instrum. 69 1302 Girard A. and Melin G. (1996) Nucl. Instr. and Meth. in Phys. Res. A 382 252 Middleton R. and Adams C. T. (1974) Nucl. Instr. and Meth. 118 329 Middleton R. (1983) Nucl. Instr. and Meth. 214 139 Middleton R. A Negative Ion Cookbook available at http://tvdg10.phy.bnl.gov/COOKBOOK/ Rao Y., Chen H., Boiling X., Dong B. and Baihua X. (1992) Rev. Sci. Instrum. 63 2643 Septier A. (1967) Focusing of Charged Particles, ed. by A. Septier (New York Academic Press) 46 Chapter 4 Photodetachment of Te− In this chapter, results of the 4d photodetachment cross section of negative tellurium will be presented, revealing for the first time strongly bound inner-shell excited states. Absolute cross sections for processes leading to the formation of both Te+ and Te2+ are measured with an uncertainty of approximately 15 % and 50 %, respectively, where the rather large uncertainty for the latter is because of variations in the efficiency of the second detector (D2 in figure 2.1). 4.1 Motivation (Kr)4d10 5s2 5p5 2 P3/2 Ground state Fine-structure splitting (J1/2−3/2 ) 620 meV Binding energy 1.970876(7) eV 4d threshold 40.32 eV Table 4.1: Some facts about Te− . Values of the ground state splitting is taken from (Thøgersen et al 1996) and the electron affinity from (Haeffler et al 1996). The 4d detachment threshold is deduced from the ground state binding energy and the known energy of the lowest lying 4d9 5s2 5p5 level in neutral Te (38.35 eV above Te ground state) (Murphy et al 1999). 47 Photodetachment of Te− 4.2 Results and analysis Table 4.1 summarizes some facts about the negative tellurium ion. The primary motivation for choosing Te− as the target for an inner-shell detachment measurement was the possibility of finding Feshbach resonances in the 4d detachment spectrum. The reason for such excited states to be expected is that the ground state configuration of Te− lacks one electron to completely fill the 5p subshell. Excitations to this particular subshell will therefore result in a filling and the ion is likely to gain extra stability. 4.2 Results and analysis Figure 4.1: Left: Absolute photodetachment cross section for Te− → Te+ (upper) and Te− → Te2+ (lower) in the energy range from 34-130 eV. Right: Details in the region around the resonances. Figure 4.1 above shows the photodetachment spectra for the processes 48 Photodetachment of Te− 4.2 Results and analysis Te− → Te+ and Te− → Te2+ . In the following, the details of the spectra will be discussed. 4.2.1 The continuum In both spectra the cross section increases slowly at ∼ 40.3 eV. This is in agreement with a value of 38.35 eV + 1.97 eV = 40.32 for the 4d detachment threshold, which can be calculated from the measured energies of the Te(4d9 5s2 5p5 ) levels (Murphy et al 1999) and the Te− binding energy (Haeffler et al ), see also table 4.1. At higher photon energy, the cross section eventually shows a broad structure centered at ∼90 eV which corresponds to direct detachment of an electron followed by single or double Auger decay. According to the dipole selection rules, the emitted photoelectron can be either an p or an f electron ( denotes the kinetic energy of the outgoing electron) and since double and triple detachment is very unlikely, the formation of Te+ and Te2+ probably happens in step-wise processes that may be written schematically as Te− (4d10 5s2 5p5 ) + hν → Te(4d9 5s2 5p5 ) + eph ( Te(4d9 5s2 5p5 ) → (4.1) Te+ (4d10 5s2 5p3 ) + eAug Te+ (4d10 5s5p4 ) + eAug → Te2+ (4d10 5s2 5p2 ) + 2eAug . Note that the detachment may also lead to the production of neutral Te and possibly also Te3+ , but neither of these were detected in the experiment. However, we estimate that these processes contribute only little (≤ 10 % of the total oscillator strength), see (Kjeldsen et al 2002). The small increase in the cross section just above threshold is due solely to 4d → p detachment as there is no strength in the 4d → f channel. At larger energies however, this transition dominates and is responsible for the broad so-called giant resonance which have also been observed in the photoionization of low-charged positive ions in the same region of the periodic table, e.g. the iso-electronic Xe+ (Andersen et al 2001). This behaviour is 49 Photodetachment of Te− 4.2 Results and analysis known as a “delayed onset” and can be explained by the large centrifugal repulsion of the f wave function, which prevents the near-threshold continuum electron of penetrating the core (Andersen 2001). Finally, considering the magnitudes, it is seen that the cross sections for Te+ and Te2+ production reach 7 Mb and 15 Mb, respectively, indicating that (sequential) double-Auger decay is more probable than single-Auger decay. 4.2.2 Resonances In the Te+ spectrum two narrow peaks appear below the threshold at 40.32 eV. These are attributed to 4d → 5p transitions in the parent ion with the large peak corresponding to 4d10 5s2 5p5 2 P3/2 → 4d9 5s2 5p6 2 D5/2 and the small peak to 4d10 5s2 5p5 2 P3/2 → 4d9 5s2 5p6 2 D3/2 . Details of these resonances are put in table 4.2 and some of the relevant levels of Te− as well as Te, Te+ and Te2+ are shown in figure 4.2. Transition 2 2 Note that there is no resonance Energy (eV) Binding (eV) P3/2 → 2 D5/2 2 P3/2 → D3/2 f Γ (meV) 200 ± 10 37.37 2.95 0.032 38.85 1.47 0.0028 171 ± 15 Table 4.2: Details of the resonances. corresponding to 4d10 5s2 5p5 2 P1/2 → 4d9 5s2 5p6 2 D3/2 , which implies that the 2 P1/2 level was not populated in the target beam although it lies only 620 meV above the 2 P3/2 ground state. This is in accordance with previous experiments with a similar ion source, where a population of this level of less than 2 % was found (Kristensen et al 1993). Similar to the decay routes of eq. (4.2), the production of Te+ via the 4d → 5p transitions happens through sequential double-Auger, and the absence of these resonances in the Te2+ spectrum means that triple-Auger to the Te2+ continuum is not energetically possible. The two resonances are bound with 2.95 eV (40.32 eV - 37.37 eV) and 1.47 eV (40.32 eV - 38.85 eV) respectively with respect to the 4d−1 threshold, 50 Photodetachment of Te− 4.2 Results and analysis Figure 4.2: The lowest energy levels (bottom) of Te− , Te, Te+ and Te2+ and the 4d → 5p inner-shell excited states (top) of Te− and Te (Murphy et al 1999). The observed transitions are indicated by arrows. and the former is therefore even more strongly bound than the ground state. This extra gain in stability must be ascribed to the full 5p subshell in the 4d9 5s2 5p6 configuration as could be expected. Similar negative ions, with full or half-full outer subshells but one electron, are also likely candidates to form stable inner-shell excited states, but this is so far the only example of such. For comparison, in the recent experiment on 1s detachment of C− (Gibson et al 2003) it was found that the 1s → 2p transition resulted in a shape resonance, in agreement with the fact that the 1s2s2 2p4 configuration represents a loss of stability compared to the half-full 2p-shell in the ground state of C− . The natural widths are found by fitting Voigt profiles to the resonance peaks and the corresponding values (Γ5/2 = 200±10 meV and Γ3/2 = 171±15 meV) are comparable to the values of the 4d9 5s2 5p5 levels of neutral Te that 51 Photodetachment of Te− 4.2 Results and analysis range from 93 to 195 meV (Murphy et al 1999). Furthermore, values of around 120 meV have been measured for the iso-electronic levels of Xe+ (Andersen et al 2001). In the article the fine-structure splitting of the 2 D3/2−5/2 levels is found to be ∼2 eV, which is somewhat higher than the corresponding splitting in Te− (2 D3/2−5/2 = 2.95 - 1.47 = 1.48 eV). This is not surprising as the strength of the spin-orbit operator increases as Z 4 . 4.2.3 Oscillator strengths The oscillator strengths of both resonances as well as the total oscillator strength has been determined by applying the formula (1.21). The oscillator strengths of the resonances (f5/2 = 0.032 and f3/2 = 0.0028) lead to an intensity ratio of 11:1 which is close to the statistical value of 9:1 expected in LS coupling, see e.g. (Woodgate 1970). Integrating over the whole energy region, the total oscillator strength has also been found as ftot = fT e+ + fT e2+ = 3.4 ± 0.5 + 6.7 ± 3.4 = 10.1 ± 3.9. This value is very close to 10, which is the number of 4d electrons and therefore the correct value according to the sum rule (1.20). References Andersen P. A. (2001) Master thesis University of Aarhus Andersen P., Andersen T., Folkmann F., Ivanov V. K., Kjeldsen H. and West J. B. (2001) J. Phys. B: At. Mol. Opt. Phys. 34 2009 Gibson N. D., Walter C. W., Zatsarinny O., Gorczyca T. W., Ackerman G. D., Bozek J. D., Martins M., McLaughlin B. M. and Berrah N. (2003) Phys. Rev. A 67 030703 (R) Haeffler G., Klinkmüller A. E., Rangell J., Berzinh U. and Hanstorp D. (1996) Z. Phys. D 38 211-214 Ivanov V. K. 1999 J. Phys. B: At. Mol. Opt. Phys. 32 R67 Kjeldsen H., Andersen P., Folkmann F., Hansen J. E., Kitajima M. and Andersen T. (2002) J. Phys. B: At. Mol. Opt. Phys. 35 2845 52 Photodetachment of Te− 4.2 Results and analysis Kristensen P., Stapelfeldt H., Balling P., Andersen T. and Haugen H. K. (1993) Phys. Rev. Lett. 71 3435 Murphy N., Costello J. T., Kennedy E. T., McGuinness C., Mosnier J. P., Weinmann B. and O’Sullivan G. (1999) J. Phys. B: At. Mol. Opt. Phys. 32 3905 Thøgersen J. Steele L. D., Scheer M., Haugen H. K., Kristensen P., Balling P. and Andersen T. (1996) Phys. Rev. A 53 3023 Woodgate G. K. (1970) Elementary atomic structure (McGraw-Hill) 53 54 Chapter 5 Photoionization results This chapter will present results on photoionization of positive ions of astrophysical importance. More precisely, absolute cross sections of N2+ , O3+ and F4+ (B-like), O4+ (Be-like) and finally F3+ and Ne4+ (C-like) will be provided with an uncertainty of ∼20 %. This uncertainty is somewhat larger than would normally be obtainable and it is mainly caused by problems with the detector at the time of the measurements. The analysis of the experimental results will not deal with details of the resonance shapes (Fano parameters), but will instead concentrate on characterizing the different transitions involved utilizing simple QDT. Moreover the experimental spectra will be compared to R-matrix calculations taken from the OPACITY database (TOPbase, Cunto et al 1993) as well as results of the MCDF method. It should be noted that in the TOPbase the continuum threshold is calculated for each particular initial state but the cross-section calculations also include points down to 0.01·Z 2 Ry below threshold, see (Luo & Pradhan 1989). These points represent photoabsorbtion in the discrete region where the density of states is high. Consequently, in each of the following graphs of OPACITY results, I have cut away the data below the stated threshold for a better comparison with experiment. 55 5.1 B-like 5.1 Photoionization results B-like Because of its special role in astrophysics, C+ was one of the first ions to be studied in Aarhus. The cross section showed a rich structure with many peaks corresponding to different Rydberg and doubly-excited states and a comparison with OPACITY results showed a generally good agreement (Kjeldsen 1999, Kjeldsen et al 1999). The next three members of the boron sequence are also important to astrophysics because of their abundance in e.g. stellar and planetary atmospheres. The ions were produced with an ECRIS (see chapter 3) with the majority in the 1s2 2s2 2p 2 P ground state and smaller fractions in the 1s2 2s2p2 4 P metastable state (the 1s2 2s2p2 2 S,2 P,2 D terms are not metastable since they are allowed to decay back to the ground state by fluorescence). Hence, the measured cross sections will contain contributions both to the continuum as well as additional resonances arising from these metastable states. 5.1.1 Results and analysis Figure 5.1 shows the results obtained for the isoelectronic ions N2+ , O3+ and F4+ . Also shown are theoretical spectra which have been folded with 100, 250 and 500 meV Gaussians (FWHM), respectively, which represent the experimental resolutions. MCDF calculations for N2+ and F4+ are provided by (Bizau 2003) whereas O3+ calculations are from (Champeaux et al 2003). The R-matrix results by Fernley et al (2004) are available at (TOPbase) and finally the red lines are more recent R-matrix calculations performed by Nahar (Nahar 2004). Referring to figure 5.1 the experimental spectra show the expected behaviour with a continuum on which different Rydberg states are superimposed. The thresholds can be read off from the figure and are located at 47.43 eV, 77.34 eV and 114.27 eV respectively, which is close to the values reported at NIST (NIST 2003), see table 5.1. 56 Figure 5.1: Caption located at next page. 5.1 B-like Photoionization results Caption to figure 5.1: Experimental and theoretical photoionization spectra of N2+ , O3+ and F4+ with energies (horizontal axes) in eV and cross sections (vertical axes) in Mb. Rmatrix results are from Fernley et al (2004) and are found at (Topbase). The red lines are R-matrix data obtained by Nahar (2004) and the MCDF calculations were performed as part of the present work (N2+ and F4+ ) (Bizau 2003) or by (Champeaux et al 2003) (O3+ ). The theoretical spectra have been convoluted with Gaussians of 100 meV, 250 meV and 500 meV (FWHM), respectively, and are calculated under the assumption of a (2 P,4 P) relative fraction of (0.90,0.10) (N2+ and F4+ ) and (0.84,0.16) (O3+ ). Energy (eV) State N2+ O3+ F4+ 1s2 2s2 2p 2 P 0 0 0 7.10 8.88 10.69 1s2 2s2p(3 P)3p 2 D 39.80 59.86 83.84 EIP (1s2 2s2 1 S) 47.45 77.41 114.24 2 1s 2s2p 2 4 P Table 5.1: Some relevant energy levels. From (NIST 2003). The peaks correspond mainly to 2s → np transitions from the ground state and 2p → nd transitions from the metastable state and most of them can be explained from QDT. An analysis of the dipole allowed transitions in the parent ions will be given below. Photoexcitation of an outer 2p electron will lead to the following 1s2 2s2 2p 2 P + hν → 1s2 2s2 (1 S)ns, n0 d (5.1) but such Rydberg series will converge to the 1s2 2s2 1 S ground state of the ionized ions and will therefore not be present in the photoionization signal as they lie below the continuum limit. If the photons are energetic enough to excite an inner 2s electron, the result58 Photoionization results 5.1 B-like ing allowed transitions will be 1s2 2s2 2p 2 P + hν → 1s2 2s2p 1 P 3 P ! np 2 S, 2 P, 2 D 2 S, 2 P, 2 D ! (5.2) Considering the selection rules for Coulomb autoionization, it is easy to see that the 1s2 2s2p(1,3 P)np 2 P terms cannot autoionize into the 1s2 2s2 1 S + p continuum without breaking the parity selection rule. However, the study of C+ showed that these states were indeed responsible for peaks in the photoionization spectrum. This indicates that the 2 P terms may autoionize either by some mechanism other than the Coulomb interaction (e.g. spin-orbit which is not much slower than fluorescence, see table A.2 in appendix A.1) or by a radiative decay followed by Coulomb autoionization (e.g. 1s2 2s2p(1,3 P)np 2 P → 1s2 2s2p(1,3 P)n0 s 2 P + hν → 1s2 2s2 1 S + p). All in all we can expect the presence of 6 Rydberg series converging to the limits of the 1 P and 3 P atomic cores. As described in chapter 1, the location of these Rydberg states have been calculated by means of QDT using eq. (1.24) and the known energy of the 1 P and 3 P ionization thresholds as well as one member of each series (e.g. the 1s2 2s2p(3 P)3p 2 D) (NIST 2003). In a similar way it is possible to analyze resonances originating from transitions from the metastable component. 1s2 2s2p2 4 P + hν → 1s2 2s2p(3 P) n0 s nd ! 4 4 P P, 4 D ! (5.3) These 3 Rydberg series all converge to the energy limit at E(1s2 2s2p 3 P) E(1s2 2s2p2 4 P) but they are not allowed to decay through Coulomb autoionization because of the spin selection rule. Still, from the calculated Rydberg series shown in the experimental spectra it is evident that these transitions are responsible for the resonances below threshold and so the states most likely autoionize via the spin-orbit interaction which is able to flip the spin. In figure 5.2, the experimental spectrum of N2+ is shown in greater detail. Since this is taken with the best resolution, it reveals more structure and is therefore best suited for an illustration of the different Rydberg series. Also 59 5.1 B-like Photoionization results Figure 5.2: Detailed view of the experimental photoionization spectrum of N2+ . Peak numbers refer to table 5.2. Furthermore, the location of Rydberg states calculated from QDT are indicated with the numbers designating the state with the lowest value of n. shown is the location of the different Rydberg states calculated from QDT and this rather simple analysis is sufficient enough to characterize all major resonances in the spectrum and these are given in table 5.2. A few comments ought to be made about the assignments summarized in table 5.2. The peaks below threshold clearly corresponds to transitions from the metastable component although the limited resolution does not allow us to resolve the 2s2p(3 P)nd (4 D,4 P) excited states (transitions to the 4 D term are statistically favoured since the statistical weight of the terms 60 Photoionization results 5.1 B-like Peak number Energy (eV) Transition 2 2 4 1s 2s2p ( P) → 1s2 2s2p(3 P)5s (4 P) 1 42.62 2 43.64 5d (4 D,4 P) 3 44.64 6s (4 P) 4 45.25 6d (4 D,4 P) 5 46.15 7d (4 D,4 P) 6 46.75 8d (4 D,4 P) 7 47.19 9d (4 D,4 P) 8 47.49 10d (4 D,4 P) 9 49.93 1s2 2s2 2p (2 P) → 1s2 2s2p(3 P)5p (2 P) 10 50.37 5p (2 D) 11 50.61 5p (2 S) 12 51.61 6p (2 P) 13 52.19 6p (2 D,2 S) 14 53.14 7p (2 P,2 D,2 S) 15 53.77 8p (2 P,2 D,2 S) 16 54.18 9p (2 P,2 D,2 S) 17 54.49 18 54.65 19 54.86 20 58.13 1s2 2s2 2p (2 P) → 1s2 2s2p(1 P)5p (2 P,2 D) Table 5.2: Location and designation of the resonances in the photoionization spectrum of N2+ . The peak numbers refer to figure 5.2. are (4 D,4 P)=(20,12)). Similarly, in the case of 2s → np transitions from the ground state, it has only been possible to resolve the 2 P, 2 D and 2 S excited states for n = 5 and partially for n = 6. Note that in these cases the 2 P state lies a little lower than the QDT value, but the assignment is supported by the fact that the peaks are not present in the (nonrelativistic) R-matrix calculation but do occur in the MCDF spectrum, see figure 5.1. The assignment of the peaks labelled 17-19 is not straightforward since there will 61 5.1 B-like Photoionization results be some degree of configuration interaction between the 2s2p(3 P)(n ≥ 10)p (2 D,2 P,2 S) and the corresponding 2s2p(1 P)4p terms so the peaks will be a mixture of these. Moreover, additional weak peaks between number 19 and 20, and also higher in energy, does not seem to belong to any of these Rydberg series and are most likely just experimental artefacts or the result of twoelectron excitations, which were also observed in the spectrum of C+ . A comparison of the experimental spectra of the three isoelectronic ions clearly shows that many of the same transitions are responsible for the resonances in the spectra, although the decreasing resolution tends to smear them out. It is also evident that succesive inner-shell resonances move downward in energy relative to the threshold when Z increases. As an example of this, in the case of N2+ the 2s2p(3 P)5p 2 D resonance is located approximately 3 eV above the 2s2p 3 P threshold, whereas the same resonance is located just at threshold in the O3+ spectrum. For F4+ it has plunged down below the ionization limit and into the discrete region and is therefore not visible. This behaviour can be explained by the fact that as Z → ∞, the ions become more and more hydrogenlike and the 2s2p and 2s2 states consequently become degenerate. A downward movement of inner-shell resonances with respect to outer-shell thresholds can therefore be expected for all sequences where the outermost subshell is different from l = 0. A general decrease of the direct photoionization cross section is also observed when moving up the sequence. This behaviour is in accordance with the work of Msezane et al (1977) who predicted that the direct photoionization cross sections for members of isoelectronic sequences decrease approximately parallel with photon energy. At a given energy their will be only a small increase in the cross section, due to the increase in nuclear attraction which leads to a contraction of the orbitals and therefore a larger overlap between the wave functions. Consequently, the decrease of the cross section can be ascribed to the increasing value of the ionization threshold with increasing nuclear charge. 62 Photoionization results 5.1.2 5.1 B-like Metastable fractions The presence of the Rydberg states below threshold can be used to estimate the fraction of metastable components in the target ion beams by comparing the oscillator strengths of the resonances with the theoretical values. Experimental and MCDF oscillator strengths for the 2s2p(3 P)nd 4 D, 4 P states have been obtained by fitting the peaks with Gaussians and using eq. (1.21). This procedure is obviously not applicable for the R-matrix results since no peaks are present in the spectra, but instead the corresponding oscillator strengths for photoexcitation, which can also be found at the OPACITY database (TOPbase), may be used under the assumption that the excited states autoionize with 100 % probability. The results are presented in table 5.3 and it is estimated that the metastable fractions are 10, 16 and 10 % respectively for the three ions, which is in agreement with previous results obtained for N2+ (8 % metastable) (Bizau et al 2003) and O3+ (16 % metastable) (Champeaux et al 2003) produced with the same ECR source. 5.1.3 Comparison with theory In figure 5.1 the experimental data are shown together with theoretical results of the R-matrix approach and the MCDF method. To account for the instrumental resolution, the spectra have been convoluted with Gaussians of 100 meV, 250 meV and 500 meV (FWHM) respectively, for the three ions. Moreover, in each case the contribution from the metastable component is incorporated into the spectra by applying the formula σtotal = (1 − X4 P ) · σ(2 P) + X4 P · σ(4 P) (5.4) except for the Nahar data (red lines) which are purely ground state. It seems that both theoretical methods are able to calculate the direct cross section fairly well, whereas the OPACITY results in particular fails to reproduce the spectral structure with regards to both the number and relative intensity in the O3+ and F4+ cases. For N2+ , on the other hand, the result is quite good. 63 5.1 B-like Photoionization results Table 5.3: The measured and calculated oscillator strength (f -values) of the 2s2p2 4P → 2s2pnd 4 P, 4 D transitions for N2+ , O3+ , and F4+ . Transition Oscillator strength Ratio Ion n Exp. R-matrix MCDF Exp. R−matrix N2+ 5 0.004126 0.0740 0.08717 0.0558 0.0473 6 0.003079 0.0238 0.05555 0.1294 0.0554 7 0.002656 0.0205 0.02146 0.1296 0.1238 8 0.001437 0.0138 0.01947 0.1041 0.0738 9 0.001076 0.0096 0.00925 0.1121 0.1163 5 0.002695 0.0704 0.06270 0.0383 0.0430 6 0.002388 0.0374 0.03037 0.0638 0.0786 7 0.002501 0.0223 0.01630 0.1121 0.1534 8 0.002241 0.0145 0.00955 0.1545 0.2346 9 0.001998 0.0100 0.00571 0.1998 0.3499 10 0.001206 0.0070 0.00566 0.1723 0.2130 6 0.004945 0.0724 0.04480 0.0683 0.1104 7 0.002717 0.0379 0.02566 0.0717 0.1059 8 0.002055 0.0227 0.01642 0.0905 0.1252 9 0.002247 0.0151 0.03253 0.1488 0.0691 O3+ F4+ Exp. MCDF Exp.: Present experimental data. R-matrix: R-matrix calculation by Fernley et al (2004) MCDF: Multi-Configuration Dirac-Fock calculations performed as part of the present work (N2+ and F4+ ) or by (O3+ ) (Champeaux et al 2003). In the OPACITY data, the 2s2p(3 P)(n=5,6)p 2 P states are not present as the calculation does not include relativistic effects, and these are not present 64 Photoionization results 5.2 Be-like in the calculation by Nahar (red line) either. In fact, this calculation is not really an improvement of the OPACITY result in contrast to the O3+ case, where better (although not convincing) agreement is found in the Nahar result (see also Champeaux et al (2003) for the most recent relativistic Rmatrix result). The MCDF description of the cross sections is generally good. The primary difference when comparing the experimental spectrum of N2+ with the MCDF result is that most theoretical resonances are shifted towards lower energies (approximately -0.7 eV for the peaks below threshold and -0.8 eV for the peaks above). Two significant peaks at 47.15 eV and 55.5 eV respectively are not found in the experiment and these are attributed to the n = 3, 4 members of the 2s2p(1 P)np Rydberg series. The appearance of the n = 3 member is because the calculation incorrectly places this state above threshold and therefore allows it to autoionize into the continuum, whereas the n = 4 member is indeed present but its strength is largely overestimated in the calculation. The calculation for F4+ also looks very reasonable except for a little shift in energy of the resonances above threshold. The appearance of a theoretical double-peak at around 115 eV may be due to a too low threshold value and/or a calculated oscillator strength for the 2s2p(3 P)5p 2 P term that is too large. The most discrepancies occur in the case of O3+ with regards to both the number and relative intensities of resonances. The energy region from around 77-85 eV is rather difficult to handle theoretically because of CI between 2s2p(3 P,1 P)np states and is further complicated by the presence of a 1s2 2p2 4d 2 D doubly-excited state predicted in the calculation to be located just above 81 eV. 5.2 Be-like This section will present the absolute cross section for O4+ which belongs to the Be isoelectronic sequence. Previously, relative data on O4+ have been obtained with a better resolution (Champeaux et al 2003) and absolute data for N3+ were measured at ASTRID this spring by Bizau, Folkmann and cowork65 5.2 Be-like Photoionization results ers, but will not be discussed here. Furthermore, high-resolution absolute measurements of C2+ (Müller et al 2002) and B+ (Schippers et al 2003) have recently been reported and compared to relativistic R-matrix calculations. 5.2.1 Results and analysis The experimental data are shown in figure 5.3 along with results from the OPACITY project (Tully et al 1990) and the MCDF method (Champeaux et al 2003). The experimental spectrum clearly shows two onsets at 113.63 eV and 103.50 eV which correspond to ionization into the O5+ 1s2 2s 2 S continuum from the 1s2 2s2 1 S ground state and the 1s2 2s2p 3 P metastable state of O4+ respectively. According to NIST (NIST 2003) these should be present at 113.90 and 103.74 eV. By comparing the continuum contribution from the metastable component with theory, it was estimated that 50 % of the parent ion beam was produced in the metastable state, which was also observed by Champeaux et al (2003) with an identical ion source. Two series of resonances are present and QDT calculated members of these series are indicated in the upper graph. It is seen that 1s2 2s2p 3 P → 1s2 2p(2 P)np 3 D, 3 S, 3 P transitions from the metastable state are responsible for the peaks labeled 1-6, whereas numbers 8-11 can be ascribed to 1s2 2s2 1 S → 1s2 2p(2 P)nd 1 P double-excitations from the ground state. Peak no. 7 was not observed in the experiment by Champeaux et al and it is not predicted in any of the theoretical approaches which suggests that this is not a physical resonance but a result of statistical fluctuations. A further comparison between experiment and theory shows an overall agreement in both cases. The OPACITY data correctly describes the continuum and also approximately the positions of the resonances but not their relative intensities. The general spectral structure is reproduced much better in the MCDF calculation although the strength of the transitions from the metastable 3 P term seems to be too small. Moreover, the magnitude of the continuum above the threshold at ∼114 66 Photoionization results 5.2 Be-like Figure 5.3: Top: Experimental cross section of O4+ . Middle: R-matrix results from the TOPbase (Tully et al 1990). Bottom: MCDF results from (Champeaux et al 2003). The theoretical spectra have been convolved with Gaussians of widths (FWHM) 650 meV for the 2s2 1 S ground state and 450 meV for the 2s2p 3 P metastable component. For easier comparison between experiment and theory the experimental values at 107 eV and 124 eV are shown in the theoretical spectra. 67 5.3 C-like Photoionization results eV is somewhat smaller than the experiment by as much as 35 % at 124 eV. For further information of the MCDF calculation as well as the most recent (relativistic) R-matrix result, the reader is referred to (Champeaux et al 2003). 5.3 C-like The results for F3+ and Ne4+ are shown in figure 5.4 and recent reports of photoionization measurements of members of the carbon isoelectronic sequence also includes absolute cross sections of N+ (Kjeldsen et al 2002) and O2+ (Champeaux et al 2003). An analysis of the data is given below but this is complicated by the population of several metastable states in the target ion beam. 5.3.1 Results and analysis The ground state configuration of a C-like ion is 1s2 2s2 2p2 which gives rise to a 3 P ground state term and the 1 D and 1 S metastable terms, see table A.3 in appendix A.2. An additional small fraction of the ion beam was also produced in the 1s2 2s2p3 5 S excited state and all these consequently lead to a wealth of possible Rydberg series in the spectra. The following dipole allowed 2s → np transitions from the 3 P ground state and 1 D,1 S metastable states and 2p → nd,n0 s excitations from the 5 S term may show up in the spectra: 1s2 2s2 2p2 3 P + hν → 1s2 2s2p2 4 P 2 P 2 P ! np 3 S, 3 P, 3 D 3 S, 3 P, 3 D 1 P, 1 D ! (5.5) 2 1 1 1 1s2 2s2 2p2 1 D + hν → 1s2 2s2p2 D np P, D, F 2 1 S P 68 (5.6) Photoionization results 5.3 C-like 2 P 1 P 2 np 1 P 1s2 2s2 2p2 1 S + hν → 1s2 2s2p2 D 2 1 S P 1s2 2s2p3 5 S + hν → 1s2 2s2p2 (4 P)nd, n0 s (5 P) (5.7) (5.8) The 3 S terms in eq. (5.5) cannot Coulomb autoionize into the 2s2 2p 2 P continuum because of the parity selection rule and the excited states in (5.8) are spin-forbidden; however, as in the case of the B-like ions, such resonances may still be present. It has not been possible to use QDT to calculate the location of all these Rydberg states since for some of the series no members are reported at NIST, but a few QDT results are shown in the upper panels of figure 5.4. For this reason, a complete assignment of the resonances is not feasible but it still seems reasonable to make educated guesses regarding the spectral structure, especially in the case of F3+ . In the experimental spectrum of F3+ , which is shown in greater detail in figure 5.5, two peaks (1 and 2) corresponding to 2p → (n = 6, 7)d transitions from the 2s2p3 5 S metastable term are clearly visible at 79.58 eV and 80.38 eV, respectively. By comparing the strength of these with the MCDF calculation, a target beam contamination of 2.5 % of the 5 S term was estimated. Direct ionization from the 2p2 1 D excited state is responsible for the sharp onset at 84.07 eV (NIST 84.01 eV, see table 5.4) where the cross section rises to a small plateau with a magnitude of ∼ 0.9 Mb. This plateau has been used to estimate the 1 D population of the ion beam to 30 %. Note that there is no clear evidence of an onset at 80.50 eV because of direct ionization from the 1 S initial excited state. Furthermore, although it has only been possible to calculate one Rydberg series from QDT (figure 5.4), there does not seem to be any trace of transitions belonging to this. For these two reasons it was estimated that the 2p2 1 S state was not populated. The exact experimental location of the ionization threshold for the 3 P ground state is hard to see because of the presence of three prominent peaks in this 69 5.3 C-like Photoionization results Figure 5.4: Top: Experimental photoionization cross sections for F3+ and Ne4+ . Some possible Rydberg series are also indicated. Middle: R-matrix results (Luo & Pradhan 1989) from the OPACITY database (TOPbase). The red line is a newer calculation (100 % 3 P ground state) by Nahar (Nahar 2004). Bottom: MCDF calculations performed as part of this work (Bizau 2003). The theoretical data have been convoluted with Gaussians of 300 meV (F3+ ) and 550 meV (Ne4+ ) (FWHM) to account for the average instrumental resolution. For easier comparison some experimental values indicated by open circles are also shown. 70 Photoionization results 5.3 C-like Figure 5.5: Detailed view of the experimental spectrum of F3+ where the resonances have been labeled. Also shown are relevant Rydberg series calculated from QDT. The dotted series is calculated under the assumption that peak no. 14 corresponds to the 3 P → (2 P)5p 3 L transitions with L = S,P,D. energy region. The NIST threshold is indicated by a thick vertical line and it seems plausible that this marks the onset for ionization from the ground state. It is likely that the (1 D) → (2 D)(n = 4 − 9)p transitions are responsible for the unresolved peaks 4,7,10-13 although 7 and 11 are probably overlapping with other states. Similarly, (3 P) → (4 P)(n = 5 − 9)p can be assigned to the peaks 5,6,7(overlapping),8,9 with some certainty. Number 3 does not fit with any of the QDT levels but as it lies below the ground state threshold, it is undoubtedly due to a (2s → (2 P or 2 S)np) transition from the 1 D metastable component. Moreover, if one carefully assumes that peak no. 14 is the member of the 2 P → (2 P)np 3 L (L=S,P,D) series with n = 5, then peak no. 71 5.3 C-like Photoionization results 11 (overlapping) and 15 also belongs to this series. F3+ Ne4+ State Energy (eV) Population (%) Energy (eV) Population (%) 2s2 2p2 3 P0 0 67.5 0 56.5 2s2 2p2 1 D 3.13 30 3.76 36 S 6.64 0 7.92 6 2s2p3 5 S 9.20 2.5 10.96 1.5 EIP (2s2 2p 2 P) 87.14 — 126.22 — 2 2 1 2s 2p Table 5.4: Estimated population of target beam states as well as their energies according to NIST (NIST 2003). A similar ”guessing procedure” is not possible in the case of Ne4+ since even fewer QDT levels can be calculated and the resolution is worse. The fractions of metastable components in the parent ion beam was estimated in much the same way as for F3+ . Even though it is far from being obvious whether or not there are one or two small peaks corresponding to transitions from the 5 S metastable state it was estimated that this state constituted 1.5 % of the target beam by comparing with the MCDF data. From the small plateau starting at 122.49 eV the 1 D contamination was determined to be 36 %. On top of this plateau a clear resonance is seen which is reproduced by both theoretical approaches. By considering the theoretical cross sections from the 1 S initial state (not shown), it is clear that a transition from this state is responsible for this peak and by comparison the 1 S term was set to 6 %. The estimated metastable fractions and their energy positions are put in table 5.4 for both C-like ions. Finally, as in the case of the B-like ions, a small decrease of the magnitude of the continuum is seen when moving from F3+ to Ne expected downward movement of Rydberg states. 72 4+ and also the Photoionization results 5.3.2 5.4 Conclusion Comparison with theory The theoretical spectra in figure 5.4 are obtained by applying the formula σtot = X(3 P )σ(3 P ) + X(1 D)σ(1 D) + X(1 S)σ(1 S) + X(5 S)σ(5 S) (5.9) where X(2S+1 L) are the fractions of each target beam state, see table 5.4. The best agreement is found in the case of F3+ . The OPACITY data naturally fails to reproduce the peaks from the 5 S state since these occur as the result of relativistic effects, but they are present in the MCDF result. Considering the overall spectral structure, the OPACITY result is generally better both with regards to positions and relative intensities of the resonances. The calculated magnitude of the cross section is within the experimental uncertainty; however, the continuum falls off too rapidly in the MCDF case. For Ne4+ none of the theoretical approaches succesfully reproduces the experimental spectrum. The agreement is only good below the ground state threshold where the 1 D plateau with the prominent peak is reproduced in both the OPACITY and the MCDF data. Above threshold, several discrepancies are seen with respect to both the observed resonances and also the magnitude of the continuum, which seems to be too low in both cases but worst in the MCDF calculation 5.4 Conclusion Absolute cross sections for photoionization of important members of the beryllium, boron and carbon isoelectronic sequences have been presented. In most cases it has been possible to characterize the spectral features from a simple QDT approach although the population of metastable components of the parent ion beams makes the analysis more complicated. A comparison with theoretical results of the R-matrix and MCDF method has also been performed and both of these are generally able to reproduce the experimental data, with some differences, though, especially regarding the relative intensities of the resonances. 73 5.4 Conclusion Photoionization results References Bizau J.-M. (2003) - Private communication Bizau J.-M., Champeaux J.-P., Cubaynes D., Blancard C., Hitz D., Bruneau J., Lemaire J.-L, Compant La Fontaine A., Girard A. and Wuilleumier F. J. (2003) Proceedings of the XXIII’th International Conference on Photonic Electronic and Atomic Collisions (ICPEAC) Champeaux J.-P., Bizau J.-M., Cubaynes D., Blancard C., Nahar S. N., Hitz D., Bruneau J. and Wuilleumier F. J. (2003) Astrophysical Journal Supplement Series, ApJS 148 583 Cunto W., Mendoza C., Ochsenbein F. and Zeippen C. J. (1993) Astronomy and Astrophysics 275 L5 Fernley J. A., Hibbert A., Kingston A. E. and Seaton M. J. to be published Kjeldsen H. (1999) PhD thesis University of Aarhus Kjeldsen H., Folkmann F., Hansen J. E., Knudsen H., Rasmussen M. S., West J. B. and Andersen T. (1999) The Astrophysical Journal 524 L143 Kjeldsen H., Kristensen B., Brooks R. L., Folkmann F., Knudsen H. and Andersen T. (2002) Astrophysical Journal Supplement Series, ApJS 138 219 Luo D. and Pradhan A. K. (1989) J. Phys. B: At. Mol. Opt. Phys. 22 3377 Msezane A., Reilman R. F., Manson S. T., Swanson J. R. and Armstrong Jr. L. (1977) Phys. Rev. A 13 668 Müller A., Phaneuf R. A., Aguilar A., Gharaibeh M. F., Schlachter A. S., Alvarez I., Cisneros C., Hinojosa G. and McLaughlin B. M. (2002) J. Phys. B: At. Mol. Opt. Phys. 35 L137 Nahar (2004) - data for N2+ , O3+ and F3+ downloaded from ftp.astronomy.ohiostate.edu NIST (2003) - http://physics.nist.gov/cgi-bin/AtData/levels form Schippers S., Müller A., McLaughlin B. M., Aguilar A., Cisneros C. Emmons E. D., Gharaibeh M. F. and Phaneuf R. A. (2003) J. Phys. B: At. Mol. Opt. Phys. 36 3371 TOPbase - www.heasarc.gsfc.nasa.gov/topbase/topbase.html Tully J. A., Seaton M. J. and Berrington K. A. (1990) J. Phys. B: At. Mol. 74 Photoionization results 5.4 Conclusion Opt. Phys. 23 3811 75 76 Concluding remarks Photoionization and photodetachment are physical processes that are important in areas such as astrophysics and plasma physics and a substantial amount of theoretical work has been done over the years. By now, absolute photoionization cross-section measurements of a wide range of singly-charged ions have been performed whereas results for multiply-charged and especially negative ions are fewer. In this thesis, results for photoionization of N2+ , O3+ , O4+ , F3+ , F4+ and Ne4+ have been presented and compared to theoretical calculations within the R-matrix approach and the MCDF method. Also presented is the photodetachment cross section of Te− which reveals strongly bound inner-shell excited states. The experimental work has been carried out at the undulator beamline at the storage ring ASTRID in Aarhus using the so-called merged-beam method. In this technique, an ion beam is generated and overlapped over a known distance with an intense photon beam from an undulator. By counting the number of ionized ions in a detector the absolute cross section can be determined if we also know the target ion density, the number of photons and the ion-photon overlap. In the future, studies of trends along isoelectronic and isonuclear sequences will probably be of main interest and the production of multiplycharged ion beams in ECR ion sources makes such experiments feasible. Finally, much more work on inner-shell detachment of negative ions is needed in order to understand effects of correlated electron motion. 77 5.4 Conclusion Photoionization results Figure 5.6 shows absolute results for 3p photodetacment of the V− , Cr− , Co− and Ni− negative ions obtained at ASTRID in the autumn 2003. The results are preliminary and have not yet been analyzed. Figure 5.6: Preliminary results for detachment of V− , Cr− , Co− and Ni− 78 Appendix A Useful material A.1 Selection rules in LS-coupling Interaction Dipole (E1) ∆S ∆L 0 ∆J 0,±1 ∆π 0,±1 Comment τ (s) −10 ±1 La +Lb >1 10 - 10−8 Ja +Jb >1 Table A.1: Selection rules for dipole transitions Interaction ∆S ∆L ∆J ∆π Relative τ (s) rate Coulomb 0 0 0 0 1 10−15 - 10−13 Spin-(other)-orbit 0,±1 0,±1 0 0 α4 10−7 - 10−5 Spin-spin 0,±1,±2 0,±1,±2 0 0 α4 10−7 - 10−5 Hyperfine 0,±1,±2 0,±1 0 me ) α4 ( M p 0,±1 Table A.2: Selection rules for autoionization 79 A.2 Coupling of equivalent electrons A.2 Useful material Coupling of equivalent electrons configuration 2S ns ns2 1S np np5 np2 np4 2P 1 S, 1 D np3 3P 2 P, 2 D np6 4S 1S nd nd9 nd2 nd8 nd3 nd7 nd4 nd6 2D 1 S, 1 D, 1 G 3 P, 3 F 2 P, 2 D, 2 F, 2 G, 2 H 1 S, 1 D, 1 F, 1 G, 1 I nd5 3 P, 3 D, 3 F, 3 G, 3 H 2 S, 2 P, 2 D, 2 F, 2 G, 2 H, 2 I nd10 4 P, 4 F 5D 4 P, 4 D, 4 F, 4 G 1S Table A.3: Allowed terms for electron configuration (nl)k , with l = 0,1,2 A.3 Hund’s rules 1. For a given configuration, the term with the largest value of S has the lowest energy. 2. For a given value of S, the term with the maximum possible L has the lowest energy. 3. For a given value of L and S, the term with J = L + S has the lowest energy if the subshell is more than half-filled. If the subshell is less than half-filled, the term with J = |L − S| has the lowest energy. 80 6S 81 2 S1/2 2 S1/2 2 S1/2 2 S1/2 [Rn] 7s 4.0727 (223) 1 S0 1 S0 1 S0 1 S0 (226) 2 [Rn] 7s 5.2784 Radium Ra 88 137.327 2 [Xe] 6s 5.2117 Barium Ba 56 87.62 2 [Kr] 5s 5.6949 Strontium Sr 38 40.078 2 [Ar] 4s 6.1132 Calcium Ca 20 24.3050 2 [Ne] 3s 7.6462 Magnesium 1 G° 4 T A B L E Sc Ni Cu Solids Liquids Gases Artificially Prepared Zn P° 1/2 B Aluminum Al 2 P1/2 ° 10.811 2 2 1s 2s 2p 8.2980 Boron 13 5 2 13 IIIA P0 Carbon C Silicon Si 14 P0 3 12.0107 2 2 2 1s 2s 2p 11.2603 6 3 14 IVA physics.nist.gov Physics Laboratory ° S3/2 Nitrogen N ° S3/2 P Phosphorus 15 4 14.0067 2 2 3 1s 2s 2p 14.5341 7 4 15 VA P2 Oxygen O S Sulfur 16 P2 3 15.9994 2 2 4 1s 2s 2p 13.6181 8 3 16 VIA P3/2 ° Fluorine F 2 17 VIIA Chlorine ° P3/2 Cl 17 2 18.9984032 2 2 5 1s 2s 2p 17.4228 9 www.nist.gov/srd Standard Reference Data Group Helium He S0 1 18 VIIIA S0 Argon S0 1 Ar 18 20.1797 2 2 6 1s 2s 2p 21.5645 Neon 1 Ne 10 4.002602 2 1s 24.5874 2 Y 2 D3/2 88.90585 2 [Kr]4d 5s 6.2173 Yttrium 39 44.955910 2 [Ar]3d 4s 6.5615 Scandium F2 F2 V 6 D1/2 F3/2 Tantalum 4 Ta 73 92.90638 4 [Kr]4d 5s 6.7589 Niobium Nb 41 50.9415 3 2 [Ar]3d 4s 6.7462 Vanadium Cr S3 D0 Tungsten 5 W 74 95.94 5 [Kr]4d 5s 7.0924 Molybdenum 7 Mo 42 51.9961 5 [Ar]3d 4s 6.7665 Chromium Mn 6 2 S5/2 S5/2 Rhenium 6 Re 75 (98) 5 2 [Kr]4d 5s 7.28 Technetium Tc 43 [Ar]3d 4s 7.4340 5 54.938049 Manganese F5 D4 Osmium 5 Os 76 [Kr]4d 5s 7.3605 7 101.07 Ruthenium 5 Ru 44 55.845 6 2 [Ar]3d 4s 7.9024 Iron Fe Cobalt Co 4 F9/2 4 F9/2 Ir Iridium 77 [Kr]4d 5s 7.4589 8 102.90550 Rhodium Rh 45 58.933200 7 2 [Ar]3d 4s 7.8810 Nickel S0 D3 Platinum 3 Pt 78 106.42 10 [Kr]4d 8.3369 Palladium 1 Pd 46 58.6934 8 2 [Ar]3d 4s 7.6398 F2 ? 2 D3/2 2 D3/2 (227) 2 [Rn] 6d7s 5.17 Actinium Ac 89 138.9055 2 [Xe]5d 6s 5.5769 Lanthanum La 57 (261) 14 2 2 [Rn]5f 6d 7s ? 6.0 ? Rutherfordium 3 Rf 104 G° 4 F2 232.0381 2 2 [Rn]6d 7s 6.3067 Thorium 3 Th 90 140.116 2 [Xe]4f5d 6s 5.5387 Cerium 1 Ce 58 (262) Dubnium Db 105 ° I9/2 5 Nd 60 (264) Bohrium Bh 107 I4 K11/2 231.03588 2 2 [Rn]5f 6d7s 5.89 Protactinium 4 Pa 91 140.90765 3 2 [Xe]4f 6s 5.473 5 L° 6 H° 5/2 L11/2 (237) 4 2 [Rn]5f 6d7s 6.2657 Neptunium 6 Np 93 (145) 5 2 [Xe]4f 6s 5.582 Promethium 6 Pm 61 (277) Hassium Hs 108 F0 F0 (244) 6 2 [Rn]5f 7s 6.0260 Plutonium 7 Pu 94 150.36 6 2 [Xe]4f 6s 5.6437 Samarium 7 Sm 62 (268) Meitnerium Mt 109 S1/2 Cadmium Mercury S0 1 Hg 80 112.411 10 2 [Kr]4d 5s 8.9938 111 112 196.96655 200.59 14 10 2 14 10 [Xe]4f 5d 6s [Xe]4f 5d 6s 9.2255 10.4375 Gold 2 Au 79 107.8682 10 [Kr]4d 5s 7.5762 S0 1 Cd 48 Zinc 65.409 10 2 [Ar]3d 4s 9.3942 S° 7/2 ° S7/2 (243) 7 2 [Rn]5f 7s 5.9738 Americium 8 D° 2 96 D°2 (247) 7 2 [Rn]5f 6d7s 5.9914 Curium 9 157.25 7 2 [Xe]4f 5d6s 6.1498 Gadolinium 9 Gd 64 (272) Am Cm 95 151.964 7 2 [Xe]4f 6s 5.6704 Europium 8 Eu 63 (281) H°15/2 H°15/2 (247) 9 2 [Rn]5f 7s 6.1979 Berkelium 6 Bk 97 158.92534 9 2 [Xe]4f 6s 5.8638 Terbium 6 Tb 65 (285) Ununbium Uun Uuu Uub Ununnilium Unununium 110 S1/2 Silver 2 Ag 47 63.546 10 [Ar]3d 4s 7.7264 Copper Ge Arsenic As Se Selenium Br Bromine Kr Krypton I8 5 I8 (251) 10 2 [Rn]5f 7s 6.2817 Californium Cf 98 162.500 10 2 [Xe]4f 6s 5.9389 Dysprosium 5 Dy 66 204.3833 [Hg] 6p 6.1082 Thallium P1/2 ° 2 Tl 81 Lead ° I15/2 ° I15/2 (252) 11 2 [Rn]5f 7s 6.42 Einsteinium 4 Es 99 164.93032 11 2 [Xe]4f 6s 6.0215 Holmium 4 Ho 67 (289) Ununquadium Uuq 114 207.2 2 [Hg]6p 7.4167 P0 3 Pb 82 Tin P0 3 Sn 50 Antimony ° S3/2 4 Sb 51 Tellurium P2 3 Te 52 ° P3/2 2 I Iodine 53 Xenon S0 1 Xe 54 72.64 74.92160 78.96 79.904 83.798 10 2 2 10 2 3 10 2 4 10 2 5 10 2 6 [Ar]3d 4s 4p [Ar]3d 4s 4p [Ar]3d 4s 4p [Ar]3d 4s 4p [Ar]3d 4s 4p 7.8994 9.7886 9.7524 11.8138 13.9996 Germanium H6 H6 F° 7/2 F° 7/2 (258) 13 2 [Rn]5f 7s 6.58 Mendelevium 2 Md 101 168.93421 13 2 [Xe]4f 6s 6.1843 Thulium 2 Tm 69 (292) Ununhexium Uuh 116 (209) 4 [Hg] 6p 8.414 Polonium P2 3 Po 84 S0 S0 (259) 14 2 [Rn]5f 7s 6.65 Nobelium 1 No 102 173.04 14 2 [Xe]4f 6s 6.2542 Ytterbium 1 Yb 70 (210) 5 [Hg] 6p Astatine ° P3/2 2 At 85 Radon D3/2 P° 1/2? (262) 14 2 [Rn]5f 7s 7p? 4.9 ? Lawrencium 2 Lr 103 174.967 14 2 [Xe]4f 5d6s 5.4259 Lutetium 2 Lu 71 (222) 6 [Hg] 6p 10.7485 S0 1 Rn 86 NIST SP 966 (September 2003) (257) 12 2 [Rn]5f 7s 6.50 Fermium 3 Fm 100 167.259 12 2 [Xe]4f 6s 6.1077 Erbium 3 Er 68 208.98038 3 [Hg]6p 7.2855 Bismuth ° S3/2 4 Bi 83 114.818 118.710 121.760 127.60 126.90447 131.293 10 2 6 10 2 4 10 2 5 10 2 3 10 2 10 2 2 [Kr]4d 5s 5p [Kr]4d 5s 5p [Kr]4d 5s 5p [Kr]4d 5s 5p [Kr]4d 5s 5p [Kr]4d 5s 5p 5.7864 7.3439 8.6084 9.0096 10.4513 12.1298 Indium P1/2 ° 2 In 49 69.723 10 2 [Ar]3d 4s 4p 5.9993 Gallium Ga For a description of the data, visit physics.nist.gov/data 238.02891 3 2 [Rn]5f 6d7s 6.1941 Uranium U 92 144.24 4 2 [Xe]4f 6s 5.5250 Praseodymium Neodymium 4 Pr 59 (266) Seaborgium Sg 106 178.49 180.9479 183.84 186.207 190.23 192.217 195.078 14 2 2 14 6 2 14 5 2 14 3 2 14 7 2 14 9 14 4 2 [Xe]4f 5d 6s [Xe]4f 5d 6s [Xe]4f 5d 6s [Xe]4f 5d 6s [Xe]4f 5d 6s [Xe]4f 5d 6s [Xe]4f 5d 6s 6.8251 7.5496 7.8640 7.8335 8.4382 8.9670 8.9588 Hafnium 3 Hf 72 91.224 2 2 [Kr]4d 5s 6.6339 Zirconium 3 Zr 40 47.867 2 2 [Ar]3d 4s 6.8281 Titanium Ti 26.981538 28.0855 30.973761 32.065 35.453 39.948 3 4 5 6 7 8 9 10 11 12 2 5 2 6 2 4 2 2 2 2 3 [Ne]3s 3p [Ne]3s 3p [Ne]3s 3p [Ne]3s 3p [Ne]3s 3p [Ne]3s 3p IIIB IVB VB VIB VIIB VIII IB IIB 5.9858 8.1517 10.4867 10.3600 12.9676 15.7596 1 3 6 5 2 3 4 7 4 1 3 3 S0 31 2P1/2 F4 29 2S1/2 30 ° 32 ° 34 ° 36 S5/2 26 D4 27 P3/2 P0 33 4S3/2 F3/2 24 S3 25 F9/2 28 S0 P2 35 F2 23 21 2D3/2 22 For the most accurate values of these and other constants, visit physics.nist.gov/constants 1 second = 9 192 631 770 periods of radiation corresponding to the transition between the two hyperfine levels of the ground state of 133Cs -1 speed of light in vacuum c 299 792 458 m s (exact) -34 Planck constant h 6.6261 × 10 J s ( /2 ) -19 elementary charge e 1.6022 × 10 C -31 electron mass me 9.1094 × 10 kg 2 0.5110 MeV me c -27 proton mass mp 1.6726 × 10 kg fine-structure constant 1/137.036 -1 Rydberg constant R 10 973 732 m 15 R c 3.289 842 × 10 Hz R hc 13.6057 eV -23 -1 Boltzmann constant k 1.3807 × 10 J K Frequently used fundamental physical constants C. () indicates the mass number of the most stable isotope. Ionization Energy (eV) 140.116 2 [Xe]4f5d6s 5.5387 Cerium Ce 58 12 S0 Mg 12 1 9.012182 2 2 1s 2s 9.3227 P E R I O D I C Atomic Properties of the Elements S0 Be 1 Beryllium 4 2 IIA Ground-state Level Francium Fr 87 [Xe] 6s 3.8939 132.90545 Cesium Cs 55 85.4678 [Kr] 5s 4.1771 Rubidium Rb 37 39.0983 [Ar] 4s 4.3407 Potassium K 19 22.989770 [Ne] 3s 5.1391 Sodium Based upon † S1/2 Na 2 6.941 2 1s 2s 5.3917 Lithium Ground-state Configuration Atomic † Weight Name S1/2 Li 11 3 2 1.00794 1s 13.5984 Hydrogen H Atomic Number 7 6 5 4 3 2 1 Symbol Period Group 1 IA 2 S1/2 1 Lanthanides A.4 Actinides Useful material A.4 The periodic table The periodic table