Chemistry
Spin Orbit Coupling
Spin-orbit coupling is a phenomenon in which the spin of an electron interacts with its orbital motion and the magnetic field created by the nucleus. This interaction leads to splitting of energy levels and affects the electronic structure of atoms and molecules. In chemistry, spin-orbit coupling is important for understanding fine structure in atomic spectra and for predicting chemical reactivity.
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10 Key excerpts on "Spin Orbit Coupling"
eBook - PDF- Andrei G. Kutateladze(Author)
- 2005(Publication Date)
- CRC Press(Publisher)
In organic molecules, spin-orbit coupling plays only a subor-dinate role, while spin-spin dipolar coupling usually domi-nates. Only when molecular symmetry is high, and especially when it forces the spin-spin dipolar contributions to vanish, is there a good opportunity for the contribution from spin–orbit coupling to play a significant role. This is more likely to occur in inorganic molecules, such as transition metal complexes. Zero-field splitting is particularly important in electron para-magnetic resonance spectroscopy. Because the H SO and the H SS terms in the Hamiltonian play related roles, we will mention both of them in this chapter, but our primary focus is the former. For molecules containing atoms of very heavy elements, the spin-orbit term in the Hamiltonian can no longer be viewed as a small perturbation; the importance of spin-orbit coupling may be comparable with that of electron repulsion, 114 Havlas et al. and it is no longer useful to refer to zero-order states of pure spin multiplicity. Under such circumstances, relativistic terms other than spin-orbit coupling normally also have to be considered, and computational treatment of such molecules needs to be more elaborate than those that are adequate for organic molecules. We will deal only with computational procedures that are normal for molecules encountered in organic photochemistry. These methods depend heavily on the assumption that the spin-orbit coupling term H SO is only a minor perturbation. In such an instance, it is common to not include small relativistic terms such as spin-orbit coupling in the Hamiltonian from the beginning but to include them as an afterthought after an ordinary nonrelativistic calculation. This is usually done using perturbation theory or response theory. The chapter is organized as follows. In Section 3.2, we describe the Breit–Pauli Hamiltonian, which forms the his-torical basis for most spin-orbit calculations for organic mole-cules.
eBook - PDF- C J O'connor(Author)
- 1993(Publication Date)
- World Scientific(Publisher)
385 multiplicity are beginning to yield to analysis and there are prospects for understanding the complexity brought about by the clustering of more than two metals in iron-sulfur clusters and high nuclearity oxo-iron and manganese clusters. In many of these areas the overall spin states and structural classes are still at the stage of phenomenological definition. The sustained activity and continuing discovery in the area of iron sulfur clusters, 14 ' 135 the growing interest in dinickel complexes as models for the active site of urease, 136-138 the intense exploration of multi-manganese chemistry as models for the oxygen-evolution site of Photosystem 116,46,126,139,140 a n ( j the continuing interest in understanding the Fe/Cu site of cytochrome oxidase 141 are just a few areas where the application of spin coupling concepts, and perhaps the discovery of new ones, will be seen. Spin coupling will always be important in defining molecular and electronic structure. It is the inevitable signature of bringing paramagnetic centers into close proximity and often, it is an exceedingly sensitive criterion of such structure. Eventually, however, it may be possible to relate spin coupling to the control of reactivity since the nature of a half occupied orbital is ultimately what influences electron transfer rates in redox proteins. Acknowledgments C. A. R. expresses his sincere appreciation to his students and collaborators for their major contributions. His interest in spin coupling began during a sabbatical leave in the laboratories of Dr. Jean-Claude Marchon in 1979 and has been sustained by a long term magnetostructural collaboration with Prof. W. Robert Scheidt. We thank Dr. Gerald Weunschell for preparing the Figures. This research is supported by the National Institutes of Health and the National Science Foundation. References 1. E. I. Solomon and D. E. Wilcox, in Magneto-Structural Correlations in Exchange Coupled Systems, ed. R. D. Willett, D. Gatteschi and O. Kahn (D. Reidel, Dordrecht, 1985) p.
eBook - ePub- Kirk W Madison, Kai Bongs, Lincoln D Carr, Ana Maria Rey, Hui Zhai(Authors)
- 2015(Publication Date)
- WCPC(Publisher)
CHAPTER 3
SPIN-ORBIT COUPLING IN OPTICAL LATTICES
Shizhong Zhang*, William S. Cole† , Arun Paramekanti‡ , and Nandini Trivedi§* Department of Physics and Center of Theoretical and Computational Physics, The University of Hong Kong, Hong Kong, China†,§ Department of Physics, The Ohio State University, Columbus, Ohio, 43210, USA‡ Department of Physics, University of Toronto, Toronto M5S1A7, and Canadian Institute for Advanced Research, Toronto, Ontario, M5G 1Z8, CanadaIn this review, we discuss the physics of spin-orbit coupled quantum gases in optical lattices. After reviewing some relevant experimental techniques, we introduce the basic theoretical model and discuss some of its generic features. In particular, we concentrate on the interplay between spin-orbit coupling and strong interactions and show how it leads to various exotic quantum phases in both the Mott insulating and superfluid regimes. Phase transitions between the Mott and superfluid states are also discussed.1.Introduction
Cold atom experiments are performed with charge-neutral atoms.1 ,2At first sight, this would have precluded the effects of orbital magnetism, as well as spin-orbit coupling to be studied in these cold atomic gases. However, in the past few years, by using atom-light coupling (Raman lasers and shaking optical lattice), it has become possible to simulate these effects in neutral atomic samples. This provides cold atom experimentalists with the exciting opportunity to produce and investigate several paradigmatic quantum states such as the quantum Hall liquids, topological insulators and superfluids, Dirac and Weyl semimetals, as well as many other exotic phenomena that have recently been predicted for electron systems in external magnetic fields or with strong spin-orbit interaction.3 –5What is perhaps more interesting is that this capacity would open an entire new vista for the investigation of bosonic topological states that have so far only been subjected to theoretical studies.6 –8Indeed, the great tunability of cold atom systems provides an avenue to the realization of conceptually important models that may not have a natural correspondence to a condensed matter system,9 ,10as was beautifully demonstrated in the recent implementation of Haldane’s honeycomb model of a Chern insulator11 and measurement of Chern number in a Hofstadter band,12
eBook - PDF- Supriyo Bandyopadhyay, Marc Cahay(Authors)
- 2015(Publication Date)
- CRC Press(Publisher)
6 Spin–Orbit Interaction In the previous chapter, we introduced the concept of the longitudinal and transverse relaxation times T 1 and T 2 . One of the major mechanisms that determine T 1 and T 2 for a single spin – or an ensemble of spins – in a solid is spin-orbit interaction . It is caused by the coupling of a moving electron’s spin with an effective magnetic field due to an electric field in the solid. The electric field could be either microscopic (as in an atom due to the charged nucleus) or macroscopic (due to a global electric field caused by doping in a semiconductor or band structure modulation). Either type will make a moving electron experience an effective magnetic field. The magnetic field will not appear in the laboratory frame, but will appear in the rest frame of the electron due to Lorentz transformation of the electric field. Spin-orbit interaction is important to understand since it not only affects spin relaxation, but is also at the heart of many spin-based devices that are discussed in Chapter 14. These devices operate by modulating the spin po-larizations of charge carriers with an external electric field that controls spin-orbit interaction. A simple way of viewing this is that the electric field causes an effective magnetic field via Lorentz transformation, which, in turn, makes spins precess about it. By controlling the electric field (and hence the result-ing magnetic field), one can alter the angular frequency of Larmor precession of spins and therefore the angle through which they precess in a given time. This affords control over the spin precession through an external electric field, which is the basis of many spintronic devices. There are essentially two types of spin-orbit interaction that we need to discuss. One is microscopic or intrinsic (as in an atom) and the other is macroscopic or extrinsic (as in a solid).
eBook - PDFHigh Resolution NMR
Theory and Chemical Applications
- Edwin D. Becker(Author)
- 2013(Publication Date)
- Academic Press(Publisher)
Chapter S Electron-Coupled Spin-Spin Interactions 5.1 Origin o f Spin-Spin C o u p l i n g From the discussion in Chapters 2 and 4, one might anticipate that an N M R spectrum would be made up of a number of single lines of different areas and widths, each arising from one or more chemically discrete nuclei. Actually, most N M R spectra consist not only of individual lines, but also of groups of lines termed multiplets. The multiplet structure arises from interactions between nuclei which cause splitting of energy levels and hence several transitions in place of the single transition expected otherwise. This type of interaction is commonly called spin-spin coupling. There is another kind of spin-spin coupling that we described in Section 2.6: the magnetic dipole-dipole interaction between two different nuclear moments. We found that the magnitude of this dipole-dipole interaction is proportional to /R 3 , where R is the distance between the nuclei, but that it depends also on the angle between R and H 0 . When the nuclei are in molecules that are in rapid, random motion, as are most small molecules in solution, this interaction averages almost completely to zero. The coupling interaction in which we are now interested is normally manifested in solution; hence it must arise from a mechanism that is independent of the rotation of the molecule. Ramsey and Purcell 1 suggested a mechanism for the coupling interaction that involves the electrons that form chemical bonds. Consider, for example, two nuclei, A and B, each with / = . Suppose nucleus A has its spin oriented parallel to H 0 . An electron near nucleus A will tend to orient its spin anti-parallel to that of A because of the tendency of magnetic moments to pair in 86 5.1 Origin of Spin-Spin Coupling 87 antiparallel fashion.
eBook - PDFHigh Resolution NMR
Theory and Chemical Applications
- Edwin D. Becker(Author)
- 2012(Publication Date)
- Academic Press(Publisher)
Chapter 5 Electron-Coupled Spin-Spin Interactions 5.1 Origin of Spin-Spin Coupling From the discussion in Chapters 2 and 4, one might anticipate that an NMR spectrum would be made up of a number of single lines of different areas and widths, each arising from one or more chemically discrete nu-clei. Actually, most NMR spectra consist not only of individual lines, but also of groups of lines termed multiplets. The multiplet structure arises from interactions between nuclei which cause splitting of energy levels and hence several transitions in place of the single transition expected otherwise. This type of interaction is commonly called spin-spin coupling. There is another kind of spin-spin coupling that we described in Sec-tion 2.6: the magnetic dipole-dipole interaction between two different nu-clear moments. We found that the magnitude of this dipole-dipole in-teraction is proportional to l/R 3 , where R is the distance between the nu-clei, but that it depends also on the angle between R and H 0 . When the nuclei are in molecules that are in rapid, random motion, as are most small molecules in solution, this interaction averages almost completely to zero. The coupling interaction in which we are now interested is nor-mally manifested in solution; hence it must arise from a mechanism that is independent of the rotation of the molecule. Ramsey and Purcell 80 suggested a mechanism for the coupling interac-tion that involves the electrons that form chemical bonds. Consider, for example, two nuclei, A and B, each with / = i . Suppose nucleus A has its spin oriented parallel to H 0 . An electron near nucleus A will tend to orient its spin antiparallel to that of A because of the tendency of magnetic mo-85 86 5. Electron-Coupled Spin-Spin Interactions i 1 Ί i 1 Ί (a) (b) Fig. 5.1 The origin of electron-coupled spin-spin interaction, (a) Antiparallel orienta-tion of nuclear spins; (b) parallel orientation of nuclear spins.
eBook - PDFHigh Resolution NMR
Theory and Chemical Applications
- Bozzano G Luisa(Author)
- 2013(Publication Date)
- Academic Press(Publisher)
Chapter 5 Electron-Coupled Spin-Spin Interactions 5.1 Origin of Spin-Spin Coupling From the discussion in Chapters 2 and 4, one might anticipate that an NMR spectrum would be made up of a number of single lines of different areas and widths, each arising from one or more chemically discrete nu-clei. Actually, most NMR spectra consist not only of individual lines, but also of groups of lines termed multiplets. The multiplet structure arises from interactions between nuclei which cause splitting of energy levels and hence several transitions in place of the single transition expected otherwise. This type of interaction is commonly called spin-spin coupling. There is another kind of spin-spin coupling that we described in Sec-tion 2.6: the magnetic dipole-dipole interaction between two different nu-clear moments. We found that the magnitude of this dipole-dipole in-teraction is proportional to l//? 3 , where R is the distance between the nu-clei, but that it depends also on the angle between R and H 0 . When the nuclei are in molecules that are in rapid, random motion, as are most small molecules in solution, this interaction averages almost completely to zero. The coupling interaction in which we are now interested is nor-mally manifested in solution; hence it must arise from a mechanism that is independent of the rotation of the molecule. Ramsey and Purcell 80 suggested a mechanism for the coupling interac-tion that involves the electrons that form chemical bonds. Consider, for example, two nuclei, A and B, each with / = i . Suppose nucleus A has its spin oriented parallel to H 0 . An electron near nucleus A will tend to orient its spin antiparallel to that of A because of the tendency of magnetic mo-85 86 5. Electron-Coupled Spin-Spin Interactions i 1 Ί i 1 Ί (a) (M Fig. 5.1 The origin of electron-coupled spin-spin interaction, (a) Antiparallel orienta-tion of nuclear spins; (b) parallel orientation of nuclear spins.
- Edward I. Solomon, Robert A. Scott, R. Bruce King, Edward I. Solomon, Robert A. Scott, R. Bruce King(Authors)
- 2013(Publication Date)
- Wiley(Publisher)
Spin-Orbit Coupling: Effects in Heavy Element Chemistry Nikolas Kaltsoyannis University College London, London, UK1 INTRODUCTION
1.1 General Considerations
The effects of relativity upon the physical and chemical properties of heavy elements are by now well established and documented,1 –5and space does not permit the present contribution to give anything other than a brief summary of the background material. The principal so-called scalar relativistic effect in atoms is the modification of electronic wavefunctions and energies, an effect that can be split into two parts; namely, direct orbital contraction and indirect orbital expansion. The former applies primarily to all s and, to a lesser extent, p orbitals, and may be explained as follows.1 ,6The inner core electrons move with radial velocities that are appreciable fractions of the speed of light. These high velocities lead to modifications in electron mass and radial extension, producing a contraction of the orbital. The atomic orbitals (AOs) of the same l but higher n value then also contract to ensure orthogonality with the core functions. An alternative explanation7 holds that the orthogonalization of high n (valence) s and p functions on the core AOs actually leads to a small expansion of the valence orbitals. The overall contraction of these valence orbitals is in fact due to the mixing in of orbitals higher in energy (especially continuum orbitals) by the relativistically modified Hamiltonian.Notwithstanding the origin of the direct orbital contraction, the indirect orbital expansion describes the effect of relativity on valence d and f functions. It arises from increased shielding of the nucleus as a result of the direct contraction of the outer core s and p electrons of similar radial distribution to the d and f functions. For very heavy elements such as the 5d, 6p, and 5f elements, the relativistic modification of the valence AOs is very significant. Therefore, computational treatments of these elements and their compounds should really incorporate these AO modifications to the largest possible extent.
eBook - ePubHigh Resolution NMR
Theory and Chemical Applications
- Edwin D. Becker(Author)
- 1999(Publication Date)
- Academic Press(Publisher)
Chapter 6 . We defer further discussion until then and in this chapter concentrate on the nature and magnitude of the coupling interaction itself.Our explanation of the origin of spin—spin coupling does not depend on the molecule being in an external field. Unlike the chemical shift, which is induced by and hence proportional to the applied field, spin—spin coupling is characteristic of the molecule itself. The magnitude of the interaction between nuclei A and X is given by a spin—spin coupling constant J AX , which is always expressed in frequency units (hertz or occasionally radians/second). This is a unit of convenient magnitude, which is directly proportional to energy. Mathematically, as we see in more detail in Chapter 6 , the fact that the coupling depends on the relative orientation of the spins of nuclei A and X can be expressed by using the scalar product, J aX Į a ·I X. As a result, this indirect coupling is often called scalar coupling . This term is precise in describing the spin interaction but is sometimes misconstrued to suggest that J AX is a scalar quantity. It is not. J is actually a tensor, just like the chemical shielding tensor, σ . As with σ , when molecules tumble rapidly, only the average is observed; hence, in liquids it is correct to treat J as a simple scalar quantity or “constant.” We shall look into the tensor aspects of J in Chapter 7
eBook - PDFModern Many-Particle Physics
Atomic Gases, Nanostructures and Quantum Liquids
- Enrico Lipparini(Author)
- 2008(Publication Date)
- WSPC(Publisher)
Light and dark gray tones correspond to up and down spin, respectively. circular dot with N electrons, except for a constant representing the charging energy. These values give the chemical potential of the dot for a varying electron number, as measured for instance in capacitance spectroscopy experiments (Ashoori, 1996; Ciorga et al. , 2000). (Notice that, unlike the results reported in Figs. 5.2–5.4, they do not include the effect of the interaction calculated in the constant interaction model. Consequently the results reported in the top of Fig. 6.12 differ from that of Fig. 5.2 by a constant.) Actually the parabola coefficient has been taken from a fit to the experiments. The spin is indicated in Fig. 6.12 with light and dark gray tones. In the absence of SO coupling each line corresponds to a given spin, except for a very small fluctuation due to the Zeeman energy at some cusps and valleys. In this case the traces arrange themselves in parallel pairs of up and down spin. As shown in the two lower panels of Fig. 6.12, the SO coupling produces sizeable up and down fluctuations of the spin. For z 0 = 100 ˚ A, i.e. weak SO coupling, the fluctuations start at low magnetic fields and they extend up to B ≈ 1 T. An even stronger SO ( z 0 = 60 ˚ A) produces spin inversions up to the last level crossing, which marks the filling factor ν = 2 line. Besides, in the latter case the traces are no longer paired 258 Spin–Orbit Coupling in the Confined 2D Electron Gas but, instead, anticorrelated with a π phase shift, specially in the region just before ν = 2. The results of Fig. 6.12 can help to interpret the experiments of Ashoori (1996) and Ciorga et al. (2000), which observed anticorrelated behavior in the traces and spin alternation with increasing B , respectively. The effects of deformation and Coulomb interaction have been analyzed by Val´ ın-Rodr´ ıguez et al.
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