Quantum theory of ion-atom interactions

Transcription

1 The University of Toledo The University of Toledo Digital Repository Theses and Dissertations 2014 Quantum theory of ion-atom interactions Ming Li University of Toledo Follow this and additional works at: Recommended Citation Li, Ming, "Quantum theory of ion-atom interactions" (2014). Theses and Dissertations This Dissertation is brought to you for free and open access by The University of Toledo Digital Repository. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of The University of Toledo Digital Repository. For more information, please see the repository's About page.

2 A Dissertation entitled Quantum Theory of Ion-Atom Interactions by Ming Li Submitted to the Graduate Faculty as partial fulfillment of the requirements for the Doctor of Philosophy Degree in Physics Dr. Bo Gao, Committee Chair Dr. Song Cheng, Committee Member Dr. Steven R. Federman, Committee Member Dr. Thomas J. Kvale, Committee Member Dr. Biao Ou, Committee Member Dr. Patricia R. Komuniecki, Dean College of Graduate Studies The University of Toledo August 2014

3 Copyright 2014, Ming Li This document is copyrighted material. Under copyright law, no parts of this document may be reproduced without the expressed permission of the author.

4 An Abstract of Quantum Theory of Ion-Atom Interactions by Ming Li Submitted to the Graduate Faculty as partial fulfillment of the requirements for the Doctor of Philosophy Degree in Physics The University of Toledo August 2014 This thesis consists of a series of theoretical efforts aimed at reformulating the quantum theory of ion-atom interactions using quantum-defect theory that is based on the analytic solutions for the long-range, 1/R 4, polarization potential. Ion-atom interactions, especially at cold temperatures of a few kelvin or lower, are complicated by the rapid energy variations induced by the long-range polarization potential, by the generally large number of contributing partial waves, and by the sensitive dependence of the interactions on the short-range potential. The standard numerical method is not only inefficient in addressing these issues, but can also miss important physics such as extremely narrow resonances. Ion-atom interaction at cold temperatures is further complicated by what is normally considered as weak interactions, such as the hyperfine interaction. While they may not be important at high temperatures, they become exceedingly important at 1 K or lower temperatures. The hyperfine effects, and the related effects of identical nuclei, have not been properly treated in existing theories. This thesis contains works that establish the quantum-defect theory for ion-atom interactions, including both its single-channel version, and its multichannel version. Through detailed comparison with numerical calculations, carried out for Na + +Na and proton-hydrogen systems, we show how quantum-defect theories provide a systematic and an efficient understanding of ion-atom interactions. Such an efficient iii

5 description is not only important for two-body systems, but also the key to a systematic understanding of quantum few-body systems, chemical reactions, and many-body systems involving ions. Proper treatments of hyperfine structure and identical nuclei are also developed as a part of these studies. iv

6 Acknowledgments I am deeply indebted to my advisor, Prof. Bo Gao, for his continuous guidance and support. He led me into the world of atomic physics and guided me with wisdom, intelligent insights, rich knowledge, and a deep understanding of physics. He has always been encouraging and patient to me for which I am very grateful. His enthusiasm and rigorous attitude towards physics have been a great influence on me. Constantinos Makrides has always been a great friend and colleague to me, with whom I enjoyed many discussions, collaborations, and more. Many helpful discussions are due to the participants of the AMO seminar, among which are Prof. Robert Deck, Prof. Steven Federman, and Prof. Thomas Kvale. I am also indebted to Prof. Federman for his careful editing of this thesis. Special thanks also go to Prof. Song Cheng and Prof. Scott Lee for their guidance on my role as a teaching assistant. I am also especially thankful to my friends and collegues Thomas Allen, Brad Hubartt, Sam Ibdah, and Carl Starkey. I would like to show great gratitude to Prof. Li You for his guidance and support. I would also like to express my appreciation to my colleagues from Tsinghua University, with special thanks to Prof. Mengkhoon Tey and Dr. Hao Duan. This work would have been impossible without the invaluable support from my parents, for which I am deeply grateful. My thanks also go to Lia Van Dril, who has given me much support and encouragement. Last but not least, I sincerely thank all my friends, who were not mentioned above due to page limitation, for making my time in Toledo enjoyable and delightful. v

7 Contents Abstract iii Acknowledgments v Contents vi List of Tables ix List of Figures x List of Abbreviations xiv 1 Introduction 1 2 Theory background General consideration for two-body interaction Born-Oppenheimer approximation Channel definitions and frame transformation Physical boundary conditions Brief introduction to the quantum-defect theory Quantum-defect thetory for resonant charge exchange Background and introduction General theory for 1 S+ 2 S type of systems Elastic approximation vi

8 3.2.2 Single channel quantum-defect theory Three-parameter QDT implementation The example of Na + +Na Comparison of QDT results with previous results Comparison of QDT results with current numerical results Potential energy curves adopted Comparison of results Comparison of results of different potentials Discussion Chapter summary Multichannel quantum theory for ion-atom interactions Background and introduction Theoretical framework Channel structure and frame transformation Scattering amplitude and cross sections Potential energy curves and numerical method Multichannel quantum-defect theory General formulation K c matrix and short-range parametrization Resonance structure The example of Na + +Na with hyperfine interaction Baseline results from the simplest MQDT parametrization Total scattering cross sections Resonance structures Chapter summary vii

9 5 Slow proton-hydrogen collision Background and introduction General considerations and potential energy curves QDT parameters Results and discussion Comparison with elastic approximation Total scattering cross sections Threshold behavior of de-excitation rate Chapter summary Conclusions and outlook 87 References 90 A Quantum-defect theory functions for polarization potentials 100 viii

10 List of Tables 4.1 Channel structure for ion-atom interactions of the type 2 S + 1 S with identical nuclei of spin I 2 = I Positions, widths, and classifications of the 7 resonances labeled in Fig Zero energy QDT parameters for proton-hydrogen interaction ix

11 List of Figures 3-1 Comparison of the total and the partial molecular cross sections for the gerade state of Na + 2 from the QDT calculation using parameters from Ref. [21] (3-1a) and from Ref. [21] (3-1b) Comparison of the total and the partial molecular cross sections for the ungerade state of Na + 2 from the QDT calculation using parameters from Ref. [21] (3-2a) and from Ref. [21] (3-2b) Comparison of the BO potential energy curves adopted in this work (3-3a) and from Ref. [21] (3-3b) for gerade (solid lines) and ungerade (dashed lines) states of Na Charge exchange cross sections of Na + +Na obtained from a three-parameter QDT description (dashed line) and from numerical calculations (solid line) Total cross sections of Na + +Na obtained from a three-parameter QDT description (dashed line) and from numerical calculations (solid line) Charge exchange cross sections of Na + +Na obtained from three-parameter QDT descriptions using parameters corresponding to the potential of Ref. [21] (solid line) and using parameters corresponding to our potential (dashed line) Total cross sections of Na + +Na obtained from three-parameter QDT descriptions using parameters corresponding to the potential of Ref. [21] (solid line) and using parameters corresponding to our potential (dashed line) x

12 4-1 Baseline MQDT results (solid line) and numerical results (dashed line) for the total hyperfine de-excitation cross sections of Na + +Na from channel {F 1 = 2, F 2 = 3/2} to channel {F 1 = 1, F 2 = 3/2}. The vertical dotted line identifies the upper hyperfine threshold at K Baseline MQDT (solid line) results and numerical results (dashed line) for the total elastic cross sections of Na + +Na in channel {F 1 = 1, F 2 = 3/2}. The vertical dotted line identifies the upper hyperfine threshold at K Baseline MQDT (solid line) results and numerical results (dashed line) for the partial elastic cross sections of Na + +Na for l = 5 and F = 5/2 in channel {F 1 = 1, F 2 = 3/2}. The vertical dotted line identifies the upper hyperfine threshold at K Total hyperfine de-excitation cross sections from channel {F 1 = 2, F 2 = 3/2} to channel {F 1 = 1, F 2 = 3/2} from MQDT (solid line) and numerical method (dashed line). The vertical dotted line identifies the upper hyperfine threshold at K Total elastic cross sections in the lower channel {F 1 = 1, F 2 = 3/2} from MQDT (solid line) and numerical method (dashed line). The vertical dotted line identifies the upper hyperfine threshold at K Partial wave contribution to the elastic cross section of Fig. 4-5 from l = 5 and F = 5/2. There are seven resonances within the hyperfine splitting that are labelled with numbers 1 through 7. The vertical dotted line identifies the upper hyperfine threshold at K. Their detailed characteristics are tabulated in Table A magnified version of this figure focusing on the resonances within the region between the hyperfine thresholds is presented in Fig xi

13 4-7 Total hyperfine excitation cross sections from channel {F 1 = 1, F 2 = 3/2} to channel {F 1 = 2, F 2 = 3/2} from MQDT (solid line) and numerical method (dashed line). The vertical dotted line identifies the upper hyperfine threshold at K Total elastic cross sections in the higher channel {F 1 = 2, F 2 = 3/2} from MQDT (solid line) and numerical method (dashed line). The vertical dotted line identifies the upper hyperfine threshold at K Magnified version of Fig. 4-6 focussing on the energy region below the hyperfine threshold (vertical dotted line). Labelled 1 through 7 are seven resonances whose detailed characteristics are tabulated in Table BO potential energy curves of the gerade (solid line) and the ungerade (dashed line) states constructed in our work for proton-hydrogen collision Total hyperfine de-excitation cross section of the proton-hydrogen collision with the present numerical calculation (solid line) and the spinexchange cross section from Ref. [67] multiplied by the proper coefficient, 1/4 that accounts for nuclear statistics, and offset by the center-of-gravity K (stars) Total hyperfine de-excitation cross sections from channel {F 1 = 1, F 2 = 1/2} to channel {F 1 = 0, F 2 = 1/2} from MQDT (solid line) and numerical method (dashed line). The vertical dotted line identifies the upper hyperfine threshold located at ɛ 2 /k B K Total hyperfine excitation cross sections from channel {F 1 = 0, F 2 = 1/2} to channel {F 1 = 1, F 2 = 1/2} from MQDT (solid line) and numerical method (dashed line). The vertical dotted line identifies the upper hyperfine threshold located at ɛ 2 /k B K xii

14 5-5 Total elastic cross sections in the lower channel {F 1 = 0, F 2 = 1/2} from MQDT (solid line) and numerical method (dashed line). The vertical dotted line identifies the upper hyperfine threshold at K Total elastic cross sections in the higher channel {F 1 = 1, F 2 = 1/2} from MQDT (solid line) and numerical method (dashed line). The vertical dotted line identifies the upper hyperfine threshold at K Threshold behavior of the hyperfine de-excitation rate W de just above the upper threshold E 2. The x-axis represents the temperature equivalence of the initial kinetic energy (ɛ E 2 )/k B. The results are produced using our numerical method xiii

15 List of Abbreviations a.u BO CC MQDT PES QDT Atomic unit Born-Oppenheimer Coupled-channel Multichannel quantum-defect theory Potential energy surface Quantum-defect theory xiv

16 Chapter 1 Introduction One of the fundamental contributions of cold-atom physics has been its revelation of universal behaviors in quantum many-body [11, 47, 119] and few-body systems [15, 51, 107]. Excluding scaling, ultracold atomic systems behave the same with their only differences being characterized by a few parameters such as the scattering length. At a more fundamental level, such universal behaviors have their origin in the universal ultracold two-body interaction as described by the effective range theory [113, 10, 9, 101]. Since this theory quickly breaks down at slightly higher energies and at shorter distances, it is natural to ask the question of whether universal behaviors exist beyond the ultracold energy regime and at higher densities, and whether they exist for systems of mixed species of, e.g., atoms, ions, and electrons. These are important questions in physics, the answers to which will determine the degree we can understand the world around us, including phenomena as diverse as reactive processes in atomic collisions, chemical reactions, and high-t c superconductivity. One specific question being asked is whether a two-body theory for the interaction between an ion and an atom can be developed to capture the universal behavior beyond the ultracold regime and to be incorporated into few-body and many body theories. Such a theory would have to be efficient and simple enough to be incorporated, but at the same time capable of addressing rapid energy dependence and 1

17 generally the large number of contributing partial waves accurately, especially to describe complex resonance structures, all of which are attributes of ion-atom interaction in cold temperatures where quantum effects are important [21, 88, 45, 89]. Many recent experimental and theoretical efforts have been devoted or greatly related to the development of such a theory. On the experimental side, these efforts include the study of trapped ions interacting with atomic gas in the milli-kelvin regime [52, 111, 130, 129, 53, 116, 87, 57], ultracold plasmas [78, 22, 20, 76, 77, 108], ultracold chemistry [118, 18, 106, 104, 55], and dissociation spectroscopy of molecular ions [58, 80, 61, 115]. On the theoretical side, before this work, developments have been made with numerical calculation [70, 21, 28, 92, 81, 12, 127, 83, 103, 128, 120, 84, 8, 110], semiclassical theories [33, 32], and quantum-defect theory (QDT) [68, 42, 69]. However, theories of the desired characteristics mentioned earlier have not been fully established, especially for systems with fine or hyperfine structures that exhibit nontrivial multichannel characteristics. In the present thesis, we aim to establish a theoretical framework developed around QDT for ion-atom interactions that captures their universal behaviors beyond the ultracold regime. To serve such an intent, we organise the thesis as described in the following outline. Outline of the thesis Chapter 2 briefly introduces the theoretical framework upon which the present work is developed. We start with the general consideration of the fully quantum mechanical description of a two-body interaction including the total Hamiltonian, conserved observables, time-independent Schrödinger equation [112, 85], and expansion of the stationary wavefunction. The Born-Oppenheimer (BO) approximation [14] is then introduced as an essential building block of our work. To effectively solve 2

18 the Schrödinger equation, symmetry properties of the system need to be fully incorporated, and are reflected in the choice of different sets of channel functions used to expand the wavefunction [35]. They are briefly introduced, as well as the frame transformation between them. Next, we briefly go over the physical boundary condition along with the means of extracting physical observables for scattering problems from the solution of the Schrödinger equation [98]. Last, we look at the general concepts behind QDT, which are shared by QDT for the 1/R 4 potential [42, 45] used in our work and the already established Coulombic QDT [72]. In Chapter 3, we present the first installment of QDT for the 1/R 4 potential [42, 45], to study the resonant charge exchange process of systems of the 1 S+ 2 S type as a prototypical system of ion-atom interaction. With the elastic approximation [23], such a problem can be simplified to effective single-channel problems which presents an ideal testing ground for single-channel QDT. We briefly go over the concept of elastic approximation before elaborating on the formulation of single-channel QDT and the three-parameter implementation of QDT for resonant charge exchange. To demonstrate the predictive power of this QDT implementation, we present the comparison of partial and total cross section results from QDT and numerical calculations for the 23 Na Na resonant charge exchange process. We then further compare the results from slightly different potential energy curves to show the dependence of the scattering results on the short range interaction. The content in this chapter is based on our work in Ref. [88]. In Chapter 4, we look at the more complete picture of ion-atom interaction at low temperatures that includes hyperfine structures and the effect of indistinguishable nuclei. To fully resolve the complication raised by these factors, we take a close look at the channel structures of the interaction as well as the physical boundary conditions. Two different sets of channels representing different symmetry properties at the short-range and long-range of internuclear separations are defined and the 3

19 frame transformation between them is presented. The scattering amplitude and cross sections for the case of resonant charge exchange that take proper account of the effect of identical nuclei are given. With these building blocks in place, we apply the multichannel quantum-defect theory (MQDT) [46] on ion-atom interactions for the first time. We also present the analytical characterization of the resonances through the MQDT formulation, especially within hyperfine splitting, including their positions, widths, and categorization. We demonstrate the predictive power of MQDT by comparing two different implementations to numerical calculations for 23 Na Na, with hyperfine structure included this time. The first implementation takes the same number of parameters as our single-channel QDT calculation, and the second slightly more advanced MQDT implementation includes two more short-range parameters characterizing the small partial wave dependences of the short-range interaction. The results of the comparison are presented and discussed. Resonances in a particular partial wave are also analysed to demonstrate the capability of MQDT to analytically characterize them. The content in Chapter 4 is based on our work in Ref. [89]. Chapter 5 presents another application of the theory in Chapter 4, this time to the proton-hydrogen collision at low temperatures. We first demonstrate the construction of the potential energy curves as well as the extraction of the short-range parameters for the MQDT implementation for this system. Then we present the cross section results of fully multichannel calculation from zero to five kelvin. MQDT results and numerical results are again compared. Threshold behavior of the de-excitation cross sections is investigated. Chapter 6 summarizes the theory and results from the previous chapters. The prospects of further application of the theory are also briefly discussed. 4

20 Chapter 2 Theory background In this chapter, we briefly overview the general framework of the quantum theory for two-body interactions, especially for low energy collisions with spin-orbit and/or hyperfine interaction involved. The framework is the foundation of our work, and contains important physics that helps the development of our theories and their applications. We focus on outlining a relatively self-contained picture of how the theory works for the systems we are interested in this work, without getting too much into the technical details. If further information is needed, please refer to the references cited. 2.1 General consideration for two-body interaction We consider the interaction between two atoms A and B (we use the term atom here in a broader sense that can also refer to an ion, as in a charged atom) in free space. The energy and dynamics of the system are characterized by its total Hamiltonian H, and the wavefunction of the system is governed by the Schrödinger equation [112]. Since the two atoms are in free space, H is time-independent, thus the stationary wavefunction of the system, which is an eigenfunction of the Hamiltonian that satisfies certain boundary conditions, is what we are interested in. 5

21 The total Hamiltonian H, including all the relativistic effects, can be written as H = 2 2µ 2 + H BO + H f + H hf, (2.1) where µ is the reduced mass of the two particles. H BO is the adiabatic BO Hamiltonian, which is used in the determination of the BO molecular states and electronic potential energy surfaces (PESs) within the BO approximation [14]. This will be briefly overviewed in Section 2.2. H f is the Hamiltonian describing the spin-orbit interactions, and H hf describes the hyperfine interactions. The total Hamiltonian H is rotationally invariant, thus commuting with the total angular momentum T of the system. This guarantees the conservation of the total angular momentum through Noether s theorem [49], and that the eigenfunctions of T are also eigenfunctions of H. The same conclusion stands for the magnetic quantum number M T of T. H is also invariant under coordinate-inversion, thus the total parity P T is a conserved quantum number and any eigenfunction of H bears a fixed parity. The total stationary wavefunction can therefore be identified by the quantum numbers mentioned above and is denoted as ψ T M T P T. The time-independent Schrödinger equation that governs the stationary wavefunction takes the form Hψ T M T P T = Eψ T M T P T. (2.2) The total wavefunction can be expanded in terms of adiabatic channel functions (basis functions) as ψ T M T P T = a Φ T M T P T a (R)G T M T P T a (R)/R. (2.3) R is the vector of internuclear separation, where a (and b that will be used later in this work) denotes a particular set of channel functions Φ T M T P T a (R), and the summation is 6

22 over the complete set of these channel functions. The channel functions contain both the electronic wave functions and the angular part of the motion of the two centers of mass of the ion and atom, relative to the center of mass of the whole system. The construction of channel functions and the transformation between different sets of them are presented in Section 2.3. Substituting Eq. (2.3) into the Schrödinger equation (2.2) with the Hamiltonian from Eq. (2.1), and making use of the orthogonality properties of the channel functions, we obtain a set of coupled-channel (CC) equations: ( 2 d 2 ) E 2µR 2 2µ dr + l a(l a + 1) 2 2 G T M T P T a (R) + b [V BO ab (R) + V f hf ab (R) + Vab (R)]GT M T P T b (R) = 0, (2.4) where V BO ab V f (R) Φ T M T P T a H BO Φ T M T P T b, (2.5a) ab (R) ΦT M T P T a H f Φ T M T P T b, (2.5b) V hf ab (R) ΦT M T P T a H hf Φ T M T P T b. (2.5c) The term l a (l a + 1) 2 /2µR 2, usually referred to as the centrifugal barrier, arises from the spherical harmonics describing the angular part of the motion in channel a acting on the total Hamiltonian, and l a is the corresponding orbital angular momentum quantum number. Notice that the channel functions have to comply with the symmetry properties of the Hamiltonian and conserve the three quantum numbers: T, M T, and P T. The boundary conditions for the CC equations and a brief discussion on how to extract physical information from the wavefunctions for scattering problems will be presented in Section

23 2.2 Born-Oppenheimer approximation The BO Hamiltonian from Eq. (2.1) is the complete non-relativistic Hamiltonian of the entire system except for the kinetic term for the nuclei, which includes the kinetic terms for electrons and Coulomb potential terms between all charged particles (electrons and nuclei). It can be written in atomic units as H BO = i e i e i e,i n Z in r iei n + i e>j e 1 r iej e + Z inz jn, (2.6) R i inj n>j n n where i e and j e are labellings for electrons, and i n and j n are labellings for nuclei. Z in or Z jn is the charge carried by nucleus i n or j n respectively, r iej e is the distance between electrons i e and j e, and R inj n is the separation between nuclei i n and j n. With the internuclear separations R inj n fixed, in other words, with a specific nuclear configuration {R injn }, we can obtain the eigenvalues and eigenstates of H BO. The eigenvalues are points on the PESs at that specific nuclear configuration {R injn }, and the eigenstates are adiabatic BO electronic wavefunctions at {R injn }. The electronic wavefunctions at a nuclear configuration can be uniquely identified by a set of quantum numbers, and the ones that share the same set of quantum numbers (or channel) at different internuclear separations form the adiabatic BO molecular states (shortened to BO states for convenience in the following text). For the case of a diatomic system, the BO states are the electronic wavefunctions calculated with the direction and magnitude of the internuclear axis fixed. Thus the BO states conserve M L ( ˆR) which is the projection of the total electronic orbital angular momentum along the internuclear axis ˆR. Also conserved are the total electronic spin S and its projection onto the internuclear axis M S ( ˆR) when relativistic effects such as spin-orbit couplings are not accounted for. Therefore, the eigenvalue equation 8

24 for the BO Hamiltonian can be written as H BO M L ( ˆR)SM S ( ˆR)Γ; R = ε ML SΓ(R) M L ( ˆR)SM S ( ˆR)Γ; R, (2.7) where Γ is the rest of the quantum numbers that can be used to characterize the BO state (i.e. certain symmetry properties for specific systems and the energy ordering. Please refer to Section for an example). The Wigner-Witmer rule [124] states that the molecular BO states are correlated with the electronic states of individual atoms when the two atoms are far apart. Thus the BO states can be expanded asymptotically with individual atomic states of the same symmetry with internuclear orientation carefully taken care of. This is an essential step in our work because the channel functions can be related to the BO states through the asymptotic expansions at large R. The technical details not presented here can be found in Ref. [35]. The BO approximation [14] ignores all the couplings (which we call nonadiabatic couplings) that arise from the BO states acting on the rest of the total Hamiltonian, namely the kinetic term of the nuclei, the spin-orbit coupling term, and the hyperfine interaction term. Besides neglecting all relativistic terms, the approximation assumes that the electronic motion can be decoupled from the nuclear motion. This is usually physically realistic because the mass of an electron is three orders of magnitude smaller than the mass of a nucleus which makes the electron move much faster than the nucleus in most circumstances. However, there are situations where these nonadiabatic couplings become strong enough that they need to be treated properly. The nonadiabatic couplings due to spin-orbit and/or hyperfine interaction, which are important when the interaction energy is comparable to or smaller than the relativistic effect, can be incorporated fairly easily with careful construction of channel functions. Some details will be discussed in Section 2.3. The nonadiabatic cou- 9

25 pling that arises from the kinetic term of the nuclei can become significant within the proximity of avoided crossings between PESs, where the two BO states become near-degenerate and strongly coupled. It is especially important when the avoided crossings are away from the inner region of the PESs where the nuclei are not likely to be. However, this usually happens when electronically excited BO states are involved, which often requires high collision energy, or involves an excited atom or molecule. These are situations beyond the scope of the present work. Another effect that comes from the kinetic term of the nuclei is the isotope effect for similar atoms (nuclei carrying the same charge but having different masses), which gives a different asymptotic threshold due to the mass difference that is not included in the BO Hamiltonian. This effect can also be incorporated with careful construction of the channel functions. Overall, we will be able to cope with the nonadiabatic couplings except for the unlikely avoided crossings, and the BO approximation will be the building block that our theory is built with. 2.3 Channel definitions and frame transformation Channel functions are used to expand the total stationary wavefunction of the system that reflects certain symmetry properties of the Hamiltonians (other than the shared ones: T, M T, and P T ) of different systems or different regions of one system via the evaluation of the potential terms in Eq. (2.5). The structure of the total potential matrix under a specific basis generally varies with internuclear separation, sometimes diagonalized or block-diagonalized, sometimes with large off-diagonal terms (otherwise it will be an effective single-channel problem, see Section 3). Employing the channels that can most reflect the symmetry in a certain region can dramatically reduce the complexity of solving the CC equations, and more importantly is the essential base for developing analytic theories, in this case the QDT. Note that the 10

26 BO potential matrix is always diagonalized at infinite internuclear separation in the circumstance considered here regardless of the channels chosen, since the difference between different BO energy curves vanishes when they go to the same threshold (without the splittings due to relativistic effects). The channel functions are defined by the angular momentum coupling scheme which dictates in what order the total angular momentum T is constructed from fundamental angular momenta (including spins). These different schemes, which are the embodiment of corresponding symmetry properties, links the channel functions to the BO states, as well as other channel bases. Practically this can be done through comparing the asymptotic forms of the channel functions (or BO states as mentioned in Section 2.2) constructed from individual atomic states plus other ingredients such as rotational wavefunction and possibly nuclear spin state. To be more specific, we will introduce three angular momentum coupling schemes that define three sets of channel functions. Before going into the detailed discussion, it is necessary to introduce the general notations of different angular momenta used here. We use Ls as the electronic orbital angular momenta, Ss as the spins of electrons, Is as the nuclear spins, and ls as the relative orbital angular momenta of the two centers of mass of the two atoms. The bold letters represent the angular momentum vectors, the normal font letters represent the quantum numbers of these angular momentum vectors, and M s with these letters as subscripts represent the corresponding magnetic quantum numbers along a space-fixed axis (lower case m l in the case of l). We also define J = L + S and F = I + J. The first kind of channels to be introduced is called the fragmentation channels, the F F channels, or the F F coupled basis. It reflects the symmetry when two atoms are far apart, where the electrons and nuclei mainly interact within individual atoms, and the coupling between the internal angular momenta of the two atoms is only 11

27 through the two total angular momenta F 1 and F 2 mediated by the relative orbital angular momentum l. Thus the spin-orbit and hyperfine potential are diagonalized in this basis when the two atoms are far apart. Adding to the diagonalization of the BO potential in the long-range, the total long-range interaction is diagonalized. The angular momentum coupling scheme of the F F basis can be specified by the quantum numbers as (α 1 L 1 S 1 J 1 I 1 F 1 ) A (α 2 L 2 S 2 J 2 I 2 F 2 ) B F lt M T, (2.8) where F = F 1 + F 2, and αs are the rest of the quantum numbers that characterize the atomic states. This coupling scheme starts with the two atomic states, α 1 L 1 S 1 J 1 I 1 F 1 A and α 2 L 2 S 2 J 2 I 2 F 2 B, with full coupling within individual atoms including spin-orbit and hyperfine interactions. It then couples the two states through F = F 1 + F 2, and finishes with adding the relative orbital angular momentum to it, as T = F + l. To show as an example of how this works, the F F channel function, for the case of the two nuclei carrying different charges (Z A Z B ), can be written asymptotically as Φ T M T P T (α 1 L 1 S 1 J 1 I 1 F 1 )(α 2 L 2 S 2 J 2 I 2 F 2 )F l R M 1 M 2 M F m l F 1 M 1, F 2 M 2 F M F F M F, lm l T M T α 1 L 1 S 1 J 1 I 1 F 1 A α 2 L 2 S 2 J 2 I 2 F 2 B Y lml ( ˆR). (2.9) As the equation demonstrates, the channel function is expanded asymptotically with the atomic wavefunctions as well as the rotational motion of the nuclei, expressed as spherical harmonics Y lml ( ˆR), through angular momentum coupling. The expansion coefficients here are the Clebsch-Gordan coefficients. For the case of similar atoms (Z A = Z B ) with different nuclei (different isotopes) and the case of identical nuclei, extra symmetrization treatment of the channel functions has to be applied. Since the 12

28 fragmentation channel functions can be seen as the combination of complete atomic internal wavefunctions with the external motion broken down by the partial wave expansion (the spherical harmonics), the scattering boundary conditions are most easily imposed in this channel. The second kind of channels to be introduced is called the LS channels or the LS coupled basis. The channel representation can be specified by the quantum numbers as (α 1 L 1 S 1 I 1 ) A (α 2 L 2 S 2 I 2 ) B LlNSKIT M T, (2.10) where L = L 1 + L 2, S = S 1 + S 2, I = I 1 + I 2, N = L + l, K = N + S, and finally T = K + I. This coupling scheme starts with the uncoupled angular momenta from individual atoms, which are the same as the BO states since the BO states ignore the couplings due to relativistic effects. Therefore the LS coupled basis is more closely related to the molecular BO states, and as a result, the BO Hamiltonian is block-diagonalized in this basis. However, the spin-orbit and hyperfine interaction couplings are not explicitly reflected in the symmetry of this basis; therefore these two Hamiltonians are not diagonalized at large internuclear separation. The third kind of channels is called the JJ channels or the JJ coupled basis. The channel representation can be expressed by the quantum numbers as (α 1 L 1 S 1 J 1 I 1 ) A (α 2 L 2 S 2 J 2 I 2 ) B JlKIT M T, (2.11) where J = J 1 + J 2 and K = J + l. This coupling scheme starts with atomic basis with spin-orbit couplings while the hyperfine coupling is done in the molecular level. It falls in between the F F coupling where the atomic angular momenta are completely coupled and the LS coupling where they are completely uncoupled within individual atoms. This coupling scheme is especially useful when spin-orbit coupling is strong in the inner region of the internuclear separation. 13

29 The LS channels and the JJ channels are called the condensation channels where the short-range interaction is (approximately) block-diagonalized. They should be used in the short-range in order to simplify the CC equations. At long-range, when the BO potential diminishes, the spin-orbit and hyperfine interaction become important and the fragmentation channels should be used. To use two different channel bases, a frame transformation is needed. This can be done by comparing the expressions of the channel functions as R goes to infinity, like the one shown in Eq. (2.9). Note that like the case described for the F F channels, symmetry properties of the nuclei have to be taken into consideration while constructing the asymptotic channel functions. 2.4 Physical boundary conditions To obtain physically viable solutions from the CC equations and to extract useful physical information from them, we need to enforce physical boundary conditions, namely the total wavefunction has to be finite everywhere. There are 2N independent solutions for the CC equations, assuming there are N coupled channels. When R approaches zero, there is a practically infinite wall at a finite R for the overall potential in the CC equations due to the exchange interaction of electrons that results from the exclusion principle and the Coulomb repulsion of nuclei. The solutions and their derivatives well inside the wall can be obtained by solving the CC equations with the infinite potential wall at an approximate R. To prevent the solutions from diverging in this classically forbidden region, half of the solutions that contain an exponentially growing term with decreasing R have to be removed, which leaves only N linearly independent solutions. These N linearly independent solutions and their derivatives can then serve as initial conditions for the global solutions. At large internuclear separation, the potential matrix in the F F coupled basis becomes diagonalized, and the thresholds depend on the fine and/or hyperfine struc- 14

30 tures of the asymptotic atoms when R goes to infinity. If the total energy E is lower than the asymptotic threshold of a certain channel, that channel is called a closed channel. Otherwise, it is called an open channel. If all the channels are closed, it is a bound state problem. If there is at least one open channel, it is a scattering problem. Regardless of how many channels are open or closed, the total wavefunction satisfies the physical boundary conditions ψ T M T P T i R j Φ T M T P T j [f j δ ji g j K (T ) ji ]/R, (2.12) where i and j denote F F channels, f j and g j are asymptotic reference functions determined by the asymptotic behavior of the potential matrix, K (T ) ji is an element in the physical K matrix, and the summation is over all channels. For interactions between two neutral atoms or between an atom and an ion (or an electron), the potential behaves asymptotically as V tot ij (R) = (E i C ni /R n i )δ ij, (2.13) where E i is the threshold energy associated with a fragmentation channel i and n i > 2. The reference functions are spherical Bessel functions for open channels and are modified spherical Bessel functions for closed channels. For scattering problems, the total wavefunction is normalized to unit energy as ψ T M T P T i (E f ) ψ T M T P T j (E f ) = δ T T δ MT M T δ PT P T δ ij δ(e f E f ), (2.14) and for bound state problems, the total wavefunction is normalized to unity. The physical K matrix K (T ) is defined for all channels including open ones and closed ones. It contains the physical information needed to determine scattering properties which is similar to the concept of phase shift in single-channel problems 15

31 (can be understood as a phase shift between incoming and outgoing waves due to the interaction). With proper ordering of the channels, the K matrix can be written in the blocked submatrices form K = K oo K co K oc K cc, (2.15) where K oo, K oc, K co, and K cc are open-open, open-closed, closed-open, and closedclosed submatrices of K (T ) respectively. The S matrix used in the standard scattering theory [98] is given in terms of K oo by S = (1 + ik oo ) 1 (1 ik oo ), (2.16) where 1 is the unit matrix with the same dimension as K oo. The scattering amplitude can be given in terms of the S matrix in a standard fashion [98] with extra consideration of the nuclear symmetry, and all kinds of cross sections can be further derived. 2.5 Brief introduction to the quantum-defect theory The name quantum defect first appeared to denote the parameter, µ l, in the famous equation for atomic energy levels E nl of a Rydberg series of the excited electron in a hydrogen-like atom [109], R y E nl = (n µ l ), (2.17) 2 16

32 which we will call the Rydberg formula, where n is the principle quantum number of the electron, l is the quantum number of the orbital angular momentum of the electron, and R y is the Rydberg constant. The originally empirical parametrization using the quantum defect in the Rydberg formula contains important physics that would lead to the development of the Coulombic QDT [72] and later the QDTs for 1/R n type of potentials with n > 2 [42, 69, 45, 36, 37, 46, 40, 96, 16]. In the rest of this section, we focus on introducing the physical concepts that connect the original empirical parametrization and the later QDTs. For a full acount of the history of the development of QDT, especially the QDT for Coulombic interactions, please refer to Ref. [114] and [105]. The quantum defect is the difference between the Rydberg formula and its counterpart for a hydrogen atom, the Balmer formula [6]. It arises from the deviation of the interaction between the excited electron and the ionic core from a pure Coulombic interaction, as the electron-proton interaction in a hydrogen atom described by the Balmer formula. A closer examination of the physical picture in a hydrogen-like atom reveals that the ionic core mostly occupies a small volume centered on the nucleus, while the excited electron roams much more freely, spending most of its time outside the ionic core. When the excited electron is far away from the ionic core, the interaction between the two is practically Coulombic, with the interaction potential given in the form of 1/r. As the distance between the electron and the ionic core decreases, the non-coulombic interactions contributing more and more significantly, such as other electrostatic interactions from higher multipole moments of the ionic core, second or higher order purterbation terms, and the exchange interaction between electrons [74]. The Rydberg formula succeeds in combining the Balmer formula that describes the bound state structure of the pure Coulombic interaction in the long range with the quantum defect that encapsulates the effect of the non-coulombic part of the interaction in the short range. This idea of separating the total interaction into different 17

33 zones that dominated by different effects leads to the eventual development of the Coulombic QDT. To examine the structure of the Coulombic QDT in more detail, we can look at the radial Schrödinger equation describing the combined system of the excited electron and the ionic core in a hydrogen-like atom, which takes a similar form to Eq. (2.4). Applying the idea from above, the radial Schrödinger equation is treated in the short range and in the long range respectively, and then combined. The potential energy term in the long range should take the form of a Coulombic potential, i.e. 1/r. The radial Schrödinger equation can then be solved analytically, and the solution is a combination of Coulomb functions [114, 72]. The short-range interaction is much more complicated to describe using ab initio method and the accurate wavefunction is difficult to obtain even with advanced numerical techniques. However, the effect from the short-range interaction on the long-range wavefunction can be viewed as an initial condition which only determines the coefficients for the Coulomb functions. Thus, physical properties such as bound state energies and scattering cross sections that only require the phase of the wavefunction at infinity 1 can be characterized by the corresponding analytic form derived from the analytic long-range wavefunction, such as the Balmer formula for energy levels, in combination with short-range parameters that encapsulate the short-range interaction, such as the quantum defect. The same idea also leads to the QDTs for 1/R n type of potentials with n > 2, including the case when n = 4 developed in this thesis. The main difference here is that the long-range interactions take different functional forms, which results in different mathematical properties of the long-range wavefunctions. This gives rise to some interesting characteristics unique to the QDTs for 1/R n type of potentials with 1 Calculations of physical processes that require detailed knowledge of wavefunctions, such as transitions, ionizations, detachments, and dissociations, can also make use of QDT in many situations when long-range wavefunctions dominate the integral of the matrix element that corresponds to that physical process. 18

34 n > 2, such as the concept of quantum reflection which does not exist in the Coulombic QDT [40]. Technically, the quantum defect, or any equivalent short-range parameters, depend on the specific quantum state and energy. However, insights into the shortrange physics can help greatly simplify the parametrization and reduce the number of short-range parameters required, which is crucial to the development of any QDT for practical applications. One example of such physical insights in the Coulombic QDT is the insensitivity of the energy dependence of the quantum defect around the ionization threshold, which allows us the use the quantum defect extracted from the Rydberg states, of which the energies are negative (taking the ionization threshold as zero energy), to predict the scattering properties of an electron with the ionic core, of which the energies are positive. We will show in the following chapters that similar patterns of short-range physics exist and are very important for the simplification and optimization of the QDT for ion-atom interactions. 19

35 Chapter 3 Quantum-defect thetory for resonant charge exchange 3.1 Background and introduction Despite being one of the simplest reactive processes that has been a subject of study for a long time [98, 21, 12, 127], quantitative understanding of resonant charge exchange, such as Na + + Na Na + Na +, remains difficult, especially at cold temperatures. This difficulty stems from the sensitive dependence on the PESs when solving the radial Schrödinger equations. It is a common difficulty shared by all heavy particle interactions at cold temperatures (see, e.g., [35]), including not only ion-atom interactions, but also atom-atom interactions [19], and chemical reactions (whenever the Langevin assumption breaks down [41, 43]). In the case of atom-atom interactions, this difficulty has only been overcome by incorporating a substantial amount of spectroscopic data, especially data close to the dissociation limit, to fine tune the PES (see, e.g., Ref. [31, 79]). Without such fine tuning, no ab initio PES for alkali-metal systems has been sufficiently accurate to 20

36 predict the scattering length and other scattering characteristics around the threshold. The availability of such data, however, is limited mostly to alkali-metal atoms and a few other species that can be cooled. For ion-atom systems, with a few exceptions that came close [59], no such data are yet available, though recent efforts on the trapping and cooling of molecular ions (see, e.g., Refs. [64, 100, 99, 126]) show considerable promise. This status on the ion-atom PES is such that at the moment, with the possible exception of H + +H and its isotopic variations [12, 70, 28], no other predictions for cold or ultracold ion-atom processes, including resonant charge exchange, can yet be trusted before experimental verification. We present here a QDT, not only as a general approach to ion-atom interactions, but also as one specific method of dealing with this difficulty of sensitive dependence on PES. It is an initial application of the QDT for a 1/R 4 potential [121, 29, 42, 45], as formulated in Ref. [42], to the resonant charge exchange process. We show that by taking advantage of the partial-wave-insensitive nature of the QDT formulation [37, 40, 42], resonant charge exchange of the type of 1 S+ 2 S, applicable to Group IA (alkali), Group IIA (alkaline earth), and helium atoms in their ground states, can be accurately described over a wide range of energies using only three parameters even at energies where many partial waves contribute to the cross sections. The theory further relates ultracold ion-atom interactions to interactions at much higher temperatures. We adopt the widely used elastic approximation (see, e.g., Refs. [23, 21, 127]), which ignores the hyperfine and isotope effects. The radial Schrödinger equations are effectively single-channel which allows us to examine the range of applicability of QDT in the simplest and purest setting. The more complicated multichannel formulation, which would not be possible without the work in this chapter, will be presented in later chapters. 21