Introduction

While atomic magnetometers can measure magnetic fields with exceptional sensitivity and without cryogenics, spin-altering collisions limit the sensitivity of sub-millimeter-scale sensors [1]. In order to probe magnetic fields with nanometer spatial resolution, magnetic measurements using superconducting quantum interference devices (SQUIDs) [2–4] as well as magnetic resonance force microscopes (MRFMs) [5–8] have been performed. However, the spatial resolution of the best SQUID sensors is still not better than a few hundred nanometers [9] and both sensors require cryogenic cooling to achieve high sensitivity, which limits the range of possible applications. A related challenge that cannot be met with existing technology is imaging weak magnetic fields over a wide field of view (millimeter scale and beyond) combined with sub-micron resolution and proximity to the signal source under ambient conditions.

Recently, a new technique has emerged for measuring magnetic fields at the nanometer scale, as well as for wide-field-of-view magnetic field imaging, based on optical detection of electron spin resonances of nitrogen-vacancy (NV) centers in diamond [10–12]. This system offers the possibility to detect magnetic fields with an unprecedented combination of spatial resolution and magnetic sensitivity [8, 12–15] in a wide range of temperatures (from 0 K to well above 300 K), opening up new frontiers in biological [10, 16, 17] and condensed-matter [10, 18, 19] research. Over the last few years, researchers have developed techniques for nanoscale magnetic imaging in bulk diamond [11, 12, 20] and in nanodiamonds [21–23] along with scanning probe techniques [10, 24].

Introduction

Soon after the development of optical magnetometers based on the radio-optical double resonance method (see Chapter 4), it was realized by Bell and Bloom [1] that an alternative method for optical magnetometry was to modulate the light used for optical pumping at a frequency resonant with the Larmor precession of atomic spins. In a Bell-Bloom optical magnetometer, circularly polarized light resonant with an atomic transition propagates through an atomic vapor along a direction transverse to a magnetic field B. Atomic spins immersed in B precess at the Larmor frequency ΩL, and when the light intensity is modulated at Ωm = ΩL, a resonance in the transmitted light intensity is observed. The essential ideas of the Bell–Bloom optical magnetometer are reviewed in Chapter 1 (Section 1.2), and can be summarized in terms of what Bell and Bloom termed optically driven spin precession: in analogy with a driven harmonic oscillator, in a magnetic field B atomic spins precess at a natural frequency equal to ΩL and the light acts as a driving force oscillating at the modulation frequency Ωm. From another point of view, the Bell-Bloom optical magnetometer can be described in terms of synchronous optical pumping: when Ωm = ΩL, there is a “stroboscopic” resonance in which atoms are optically pumped into a spin state stationary in the frame rotating with ΩL. Depending on the details of the atomic structure, the spin state stationary in the rotating frame can be either a dark state that does not interact with the modulated light or a bright state for which the strength of the light–atom interaction is increased.

(e,2e) processes – an overview

An (e,2e) process is one where an electron, of well-defined energy and momentum, is fired at a target, ionizes it and the two exiting electrons are detected in coincidence. The energies and positions in space of these electrons are determined by the experiment, so in effect all but the spin quantum numbers are then known. We can, therefore, describe it as a kinematically complete experiment; if we could also measure all the spins we would have all the information from a scattering experiment that quantum mechanics will allow. The technique offers both the possibility of a direct determination of the target wave function and a profound insights into the nature of few-body interactions. What information you extract from such an experiment really depends on the kinematics you chose and the target you use. Integrated cross sections can be crude things and you need the full power of a highly differential measurement to tease out the delicacies of the interactions. Indeed, often the most intriguing effects turn up in peculiar geometries where the cross sections are small and where a number of relatively subtle few-body interactions are present.

In recent years, attempts to give a complete numerical treatment of electron impact ionization have made considerable progress. In particular, one should mention the pioneering close coupling work of Curran and Walters [1–3], the convergent close coupling approach, [4], the complex exterior scaling calculations, [5], and the propagating exterior complex scaling method, [6].

Introduction

The accurate solution of the Schrödinger equation (SE) for electron-impact collisions leading to discrete elastic and inelastic scattering progressed rapidly with the increase in computing power from the 1970s. A review of the principal methods, including second Born, distorted wave, R-matrix, intermediate-energy R-matrix, pseudo-state close coupling and optical model is given in [1]. However, electron impact collisions leading to ionization on even the simplest atom, hydrogen, were by comparison poorly described; significant progress dates only from the early 1990s when Bray and Stelbovics [2] developed a technique called convergent close coupling (CCC). In this approach they used an in-principle complete set of functions to approximate the hydrogenic target states, both bound and continuous, and used the coupled channels formalism to expand the scattering wave function in these discretized states, reducing the solution of the SE to a set of coupled linear equations in a single co-ordinate. The method was tested in a non-trivial model [3] and shown to provide convergent cross sections not only for discrete elastic and inelastic processes but also for the total ionization cross section. Shortly thereafter the method was applied to the full collision problem from atomic hydrogen and one of the major achievements of the method was that it yielded essentially complete agreement with the (then) recent experiment for total ionization cross section [4]. In the following years, the method was applied to other atoms with considerable success; the range of applications of CCC are covered in the review of Bray et al. [5].

Introduction

The (e,2e) process for an atom describes an electron-impact-induced ionization event in which the momentum states of the incident and two outgoing electrons are defined, i.e., the reaction kinematics is fully specified. Due to its highly differential nature, the cross section describing this process provides a stringent test of electron-scattering theory. However, a quantum mechanically complete description of the (e,2e) process requires additional variables to be specified, namely the spin projection states of the continuum electrons, as well as angular momentum, and its projection state for the target atom before and the residual ion after the collision, respectively. While the goal of performing such a complete measurement is presently beyond experimental capabilities, (e,2e) experiments for which a subset of the quantum mechanical variables were determined have been performed. All employed beams of polarized electrons, enabling cross sections to be determined individually for the two spin states of the projectile (namely ms = ±½); others additionally resolved the angular momentum state of the target atom prior to the collision. In this chapter we will illustrate how the resolution of angular momentum states can powerfully highlight and provide new insight into specific aspects of the (e,2e) collision dynamics.

Electron spin emerges naturally from the relativistic treatment of quantum mechanics and, as a consequence, spin-resolved experiments are ideally suited to probe aspects of relativity in electron–atom scattering. Less obvious is that in the non-relativistic limit, spin-resolved measurements provide a sensitive probe to the nature of electron exchange processes in the (e,2e) ionization dynamics.

Introduction

Electron–electron correlation plays a crucial role in determining physical and chemical properties in a wide class of materials that exhibit fascinating properties including, for example, high-temperature superconductivity, colossal magnetoresistance, metal insulator or ferromagnetic anti-ferromagnetic phase transitions, self assembly and quantum size effects. Furthermore, electron–electron correlation governs the dynamics of charged bodies via long-range Coulomb interaction, whose proper description constitutes one of the more severe tests of quantum mechanics.

Nevertheless, the effects due to correlation remain rather elusive for almost all of the experimental methods currently used to investigate matter in its various states of aggregation. Indeed, being related to processes with two active electrons, like satellite structures in photoemission (i.e., ionization processes with one ejected and one excited electron), or double ionization events, they influence marginally the spectral responses of the target, that are primarily determined by single and independent particle behaviours. Hence the experimental effort devoted in the last 30 years to develop a new class of experiments, whose spectral response is determined mainly by the correlated behaviour of electron pairs.

The common denominator of this class of experiments is the study of reactions whose final state has two holes in the valence orbitals and two unbound electrons in the continuum. It is exactly through interaction of these holes and electron pairs that correlation shapes the cross section of the double ionization processes.

Introduction

As well as probing matter–antimatter interactions, positrons (as positive electrons) have been employed to highlight charge and mass effects in the dynamics of collisions, including those resulting in the ionization of atoms and molecules (see, e.g., [1]). Positronium (Ps), the hydrogenic atom formed from the binding of a positron and an electron, is readily produced in the scattering of positrons from matter. Ps is quasi-stable with a lifetime against annihilation dependent upon its spin: ground-state para-Ps (1 1S0) has a lifetime τ ≃ 125 ps, whilst ortho-Ps (1 3S1) is considerably longer lived (τ ≃ 142 ns). The beam employed for the scattering work discussed in this chapter consists solely of ortho-Ps atoms. In a dense medium, Ps may undergo several cycles of formation and break-up prior to the annihilation of the positron (see, e.g., [2–6]). A quantitative understanding of this cycle is important also for practical applications such as nanodosimetry relating to positron emission tomography (PET) [e.g., 4].

In this chapter, we consider experimental methods employed to investigate positron and positronium impact ionization and fragmentation in collision with atoms and molecules, and associated results. In the case of positrons, an extensive database now exists of integral cross sections for the inert atoms (see, e.g., [5]), less so for molecules (see, e.g., [6]); differential data remain sparse (e.g., [5]). Our focus will be on the latter two topics as well as studies with positronium projectiles.

Introduction

Understanding the collision processes that lead to ionization of atoms and molecules is of great importance, not only in furthering fundamental quantum physics, but also for many applications in technology, industry and science. Applications include lighting, laser development and plasma etching, and these types of collisions occur in stellar and planetary atmospheres, the upper atmosphere of the earth, in a wide range of biological systems, and in the production of greenhouse gases. Indeed, in any area where ionization by electron impact occurs, it is essential to understand the collision mechanisms to fully describe the system under study.

The probability of ionization depends on several factors, including the incident energy of the electron projectile, and the nature of the target that is being ionized. Since electrons cannot be created or destroyed during these collisions (the energy is almost always too low for such a process to occur), the incident electron will be scattered from the target through some angle with respect to the incident direction. Ionization then results in one or more electrons being ejected from the target, and these electrons will also emerge at different angles, depending upon the collision dynamics.

In the types of processes discussed here, we confine our studies to single ionization (with the subsequent release of only one electron from the target), and only consider the interactions when the incident electron has relatively low energy compared to the ionization potential of the target. We here define this as the region from threshold (where the ejected and scattered electrons have almost no energy after the collision) through to intermediate energies, where the incident electron has energies typically several times that of the ionization potential (IP). In this energy regime, the electrons spend sufficient time in the region of the target that impulsive approximations (as successfully used at high incident energies) are no longer applicable. It is hence necessary to carefully consider different mechanisms that lead to ionization.

Introduction

The ionization of atoms and molecules by electron impact is of considerable technological and theoretical relevance. From the practical perspective it plays a central role in many atmospheric, industrial and environmental processes. Examples include the physics and chemistry of the upper atmosphere, the operation of discharges and lasers, radiation-induced damage in biological material and plasma etching processes [1–3]. The extent to which such processes can be controlled and/or optimized is limited by our ability to describe the underlying physical mechanisms which drive them. To refine our understanding, new experimental and theoretical results are required. From a broader perspective, the process of electron-impact-induced ionization of atoms and molecules provides an ideal testbed to refine models for the few- and many-body behaviour of identical particles whose interaction is mediated through the Coulomb potential. Moreover, as the ionization process is extremely sensitive to the electronic structure of the target, comparison of experimentally derived ionization cross sections with calculation provides a powerful means to refine models for the target electronic structure. Historically, (e,2e) measurements can be divided into two categories, namely, those whose primary aim is the determination of the target electronic structure and those whose focus is revealing underlying ionization mechanisms. For the former case (so-called electron momentum spectroscopy (EMS) studies), measurements are performed at relatively high impact energies and for roughly equal energies for the two scattered electrons. Under such conditions the ionization mechanism is quite well understood, with the primary electron interacting predominantly with a single bound target electron.

In the past few years, revolutionary advances in experimental techniques and spectacular increases in computer power have offered unique opportunities to develop a much more profound understanding of the atomic few-body problem. One area of intense effort is the study of fragmentation processes – break-up processes – which are studied experimentally by detecting in coincidence the collisional fragments with their angles and energies resolved. These experiments offer a unique insight into the delicacies of atomic and molecular interactions, being at the limit of what is quantum mechanically knowable; the fine detail that is revealed would be swamped in a less differential measurement. The challenge for the theorist is to develop mathematical and computational techniques which are of sufficient ingenuity and sophistication that they can elucidate the Physics observed in existing measurements and give direction to the next generation of experiments. Fragmentation processes are studied by those interested in electron and photon impact ionization, heavy particle collisions, collisions involving antimatter, as well as molecular collisions.

Introduction

Recent years have witnessed a remarkable progress in high-power short laser pulse generation. Modern conventional and free-electron laser (FEL) systems provide peak light intensities of the order of 1020 W cm−2 or above in pulses in femtosecond and sub-femtosecond regimes. The field strength at these intensities is a hundred times the Coulomb field, binding the ground-state electron in the hydrogen atom. These extreme photon densities allow highly non-linear multiphoton processes, such as above-threshold ionization (ATI), high harmonic generation (HHG), laser-induced tunneling, multiple ionization and others, where up to a few hundred photons can be absorbed from the laser field. In parallel with these experimental developments, massive efforts have been undertaken to unveil the precise physical mechanisms behind multiphoton ionization (MPI) and other strong-field ionization phenomena. It was shown convincingly that multiple ionization of atoms by an ultrashort intense laser pulse is a process in which the highly non-linear interaction between the electrons and the external field is closely interrelated with the fewbody correlated dynamics [1]. A theoretical description of such processes requires development of new theoretical methods to simultaneously account for the field nonlinearity and the long-ranged Coulomb interaction between the particles.

In this chapter, we review our recent theoretical work in which we develop explicitly time-dependent, non-perturbative methods to treat MPI processes in many-electron atoms. These methods are based on numerical solution of the time-dependent Schrödinger equation (TDSE) for a target atom or molecule in the presence of an electromagnetic and/or static electric field. Projecting this solution onto final field-free target states gives us probabilities and cross sections for various ionization channels.