In the preceding four chapters, we have mainly described the fundamentals of formal scattering theory at low energy. In this chapter, we apply this theory to cold atomic collisions. We first discuss the applications to atomic scattering in the absence and then in the presence of an external field. In the millikelvin or submillikelvin temperature regime, atomic hyperfine (HF) interactions—that is, interactions between the nuclear and electronic spins of an atom—become important in atomic collisions, even in the absence of any external field, because the energy scale associated with the HF interaction is typically on the order of 100 mK. Therefore, to understand low-energy atomic collisions, it is instructive to begin with a general introduction to atomic HF interactions. It is also essential to have a basic understanding of the origin and the method of constructing diatomic molecular potentials to understand low-energy atomic scattering from first principles. A detailed discussion of these topics, which would require some knowledge of atomic and molecular structure, is beyond the scope of this book.
In the preceding four chapters, we have mainly described the fundamentals of formal scattering theory at low energy. In this chapter, we apply this theory to cold atomic collisions. We first discuss the applications to atomic scattering in the absence and then in the presence of an external field. In the millikelvin or submillikelvin temperature regime, atomic hyperfine (HF) interactions—that is, interactions between the nuclear and electronic spins of an atom—become important in atomic collisions, even in the absence of any external field, because the energy scale associated with the HF interaction is typically on the order of 100 mK. Therefore, to understand low-energy atomic collisions, it is instructive to begin with a general introduction to atomic HF interactions. It is also essential to have a basic understanding of the origin and the method of constructing diatomic molecular potentials to understand low-energy atomic scattering from first principles. A detailed discussion of these topics, which would require some knowledge of atomic and molecular structure, is beyond the scope of this book.
We present the basic ideas about the broad topic of molecular interactions in the context of low-energy collisions. We start by introducing the concept of the interaction energy and the crucial approximation behind it—the Born–Oppenheimer approximation. Next, we focus on understanding of molecular interactions within the framework of the perturbation theory. We discuss the essentials of symmetry-adapted perturbation theory (SAPT) which provides partitioning of the interaction energy into physically meaningful components, such as electrostatic, dispersion, induction, and exchange interactions. We examine the role of these components and how they can be calculated. We also discuss the supermolecular approach to calculation of the interaction energy with various quantum chemistry methods. We indicate potential pitfalls of the latter approach and give recommendations on how to alleviate them. To illustrate the theory presented in this chapter, we provide a detailed discussion of interactions within several systems of relevance in cold chemistry and cold collisions: He-NH(3Σ−), N-NH, He-OH(2Π), NH-NH, O2(3Σ−g)-O2. The calculations of collision rates and scattering cross-sections impose stringent requirements on the accuracy of the potential energy surfaces. To address the issue, we present the analysis of the scattering length dependence upon the potential scaling. Finally, we indicate trends in calculations of intermolecular forces stimulated by the state-of-the-art cold chemistry experiments.
Over the last two decades cold atoms have emerged as a most useful platform for exploring quantum science and quantum technology. Cooling, trapping and manipulating atomic systems require understanding of low energy collisions physics. The purpose of this book is to bridge the gap between graduate teaching in scattering theory, particularly atomic scattering, and the emergent fields of cold atoms and cold molecules. Although this book presents basics of collisions physics as applicable to different branches of physics and chemistry, it is written with a particular emphasis on the physics of low energy quantum collisions and its applications to cold atoms. Key features • Pedagogic description of the topics will make this book suitable for graduate students taking courses on quantum scattering theory and cold atomic physics. • Each chapter will have exercises and worked examples. • Emphasises applications of theory to current research topics in several inter-related areas. • Incorporates numerical techniques. • Self-contained, with appendices summarising related topics needed by students.
In the 1980s the first successful experiments [312] and theory [98], demonstrating that light could be used to cool and confine atoms to submillikelvin temperatures, opened several exciting new chapters in atomic, molecular, and optical (AMO) physics. Atom interferometry [6, 8], matter–wave holography [294], optical lattices [192], and Bose–Einstein condensation in dilute gases [18, 95] all exemplified significant new physics where collisions between atoms cooled with light play a pivotal role. The nature of these collisions has become the subject of intensive study not only because of their importance to these new areas of AMO physics but also because their investigation has led to new insights into how cold collision spectroscopy can lead to precision measurements of atomic and molecular parameters and how radiation fields can manipulate the outcome of a collision itself. As a general orientation Fig. 1.1 shows how a typical atomic de Broglie wavelength varies with temperature and where various physical phenomena situate along the scale. With de Broglie wavelengths on the order of a few thousandths of a nanometer, conventional gas-phase chemistry can usually be interpreted as the interaction of classical nuclear point particles moving along potential surfaces defined by their associated electronic charge distribution. At one time liquid helium was thought to define a regime of cryogenic physics, but it is clear from Fig. 1.1 that optical and evaporative cooling have created “cryogenic” environments below liquid helium by many orders of magnitude.
Introduction Interactions between separated atoms are complicated by the large number of degrees of freedom of a many-atom system, and are very difficult to describe quantitatively. Although the Hamiltonian for such systems can be written down easily, finding the eigenfunctions requires a sophisticated effort and large computer resources. This area of quantum chemistry, with all its intricacies, is a complex subfield of science that cannot be adequately described in a single book chapter. Therefore the student is referred to specialized books in this field for further reading [165, 170]. In the 1990s a new field of chemistry emerged that considered such interactions between atoms at ultra-cold temperatures. The field focuses strongly on atomic interactions at large internuclear distances where the atoms are well separated and their electronic clouds do not strongly overlap. In such circumstances the interactions are dominated by the properties of the states of the atoms and they can be understood from an atomic point of view. In this chapter the basic features of this field will be described. The interaction energy (electronic potential energy) between two colliding partners separated by a large distance is called the dissociation limit. Light interactions with the particles can occur with the total energy above or below this limit. The relative kinetic energy does not contribute because it is so very small at such low temperatures. In the presence of a light field tuned below this limit for a pair that has one of the atoms in an excited state, two unbound atoms can be associated into a bound molecular state (for example, an S–P collision of two alkali atoms). Such photo-association is more likely at a large interatomic separation because there is good radial overlap between the incoming, low-energy scattering state and the outer turning point of the bound excited state, so it usually produces a molecule in a high vibrational state. Studying these states provides important information about interatomic interactions near the dissociation limit. The long-range part of the potential is dominated by atomic parameters and in Sec. 18.2 it is shown how these potentials can be constructed. The LeRoy–Bernstein method, which was developed long before the advent of cold-atom photo-association, is described in Sec. 18.3 and can be used to extract these parameters from measurements of the bound states.
Studying chemical reactions at very low temperatures is of importance for the understanding of fundamental physical and chemical processes. At very low energies, collisions are dominated by only a few partial waves. Thus, studies in this regime allow the characterization of quantum effects which depend on the collisional angular momentum, e.g., reactive scattering resonances and tunneling through centrifugal barriers. Additionally, the dynamics of ultralow-energy collisions is dominated by long-range interactions, i.e., “universal” chemical forces. Thus, experiments in this domain probe the details of intermolecular interactions which is of general relevance for the understanding of chemical processes. Moreover, the increasingly more precise experiments provide valuable data for benchmarking theoretical models and quantum chemical calculations. Studies of cold ion–neutral reactions in the laboratory rely on techniques for the generation of cold ions and cold neutrals. Over the last decades, the technology in this domain has made impressive progress. Many of these techniques allow not only the translational cooling of the molecules, but also a precise preparation of their internal quantum state. Thus, by an accurate preparation of both the energy and state of the reaction partners as well as by the application of external electric, magnetic or optical fields, prospects open up to study and control chemical processes at an unprecedented level of accuracy. In this chapter, we review salient theoretical concepts and recent experimental developments for studies of ion–molecule reactions at low temperatures. We first discuss the theoretical background for the description of low-energy ion–molecule reactions. Next we present key experimental methods for laboratory studies of cold ion–molecule reactions. The chapter concludes with a review of illustrative results and an outlook on future directions.
Studying chemical reactions at very low temperatures is of importance for the understanding of fundamental physical and chemical processes. At very low energies, collisions are dominated by only a few partial waves. Thus, studies in this regime allow the characterisation of quantum effects which depend on the collisional angular momentum, e.g., reactive scattering resonances and tunneling through centrifugal barriers. Additionally, the dynamics of ultralow-energy collisions is dominated by long-range interactions, i.e., "universal" chemical forces. Thus, experiments in this domain probe the details of intermolecular interactions which is of general relevance for the understanding of chemical processes. Moreover, the increasingly more precise experiments provide valuable data for benchmarking theoretical models and quantum-chemical calculations. Studies of cold ion-neutral reactions in the laboratory rely on techniques for the generation of cold ions and cold neutrals. Many of these techniques do not only allow the translational cooling of the molecules, but also a precise reparation of their internal quantum state. Thus, by an accurate preparation of both the energy and state of the reaction partners as well as by the application of external electric, magnetic or optical fields, prospects open up to study and control chemical processes at an unprecedented level of accuracy. In this chapter, we will review salient theoretical concepts and recent experimental developments for studies of ion-molecule reactions at low temperatures. We first discuss the theoretical background for the description of low-energy ion-molecule reactions. Next we present key experimental methods for laboratory studies of cold ion-molecule reactions. The chapter concludes with a review of illustrative results and an outlook on future directions.