Hyperfine structure theory

Eriksson, L. A., O. L. Malkina, V. G. Malkin, and D. S. Salahub. 1994. The hyperfine structures of small radicals from density functional theory. J. Chem. Phys. 100, 5066. 

The spectra discussed in Chapter 4 were analyzed by neglecting the effects of nuclear quadrupole coupling on the nuclear hyperfine structure. Presented here is the way such effects may be incorporated into the spectra using perturbation theory. 

Abragam, A. and Pryce, M.H.L. 1951. Theory of the nuclear hyperfine structure of paramagnetic resonance spectra in crystals. Proceedings of the Royal Society of London A 205 135-153. 

A breakthrough was achieved a few years ago when it was realized that an anal dic calculation of the deuterium recoil, structure and polarizability corrections is possible in the zero range approximation [76, 77]. An analytic result for the difference in (12.29), obtained as a result of a nice calculation in [77], is numerically equal 44 kHz, and within the accuracy of the zero range approximation perfectly explains the difference between the experimental result and the sum of the nonrecoil corrections. More accurate calculations of the nuclear effects in the deuterium hyperfine structure beyond the zero range approximation are feasible, and the theory of recoil and nuclear corrections was later improved in a number of papers [78, 79, 80, 81, 82]. Comparison of the results of these works with the experimental data on the deuterium hyperfine splitting may be used as a test of the deuteron models and state of the art of the nuclear calculations. 

In the conclusion of this section let us notice that a wealth of data on the applications of the relativistic self-consistent field method to the studies of the hyperfine structure of atomic levels is collected in [149]. Investigations of the hyperfine structure by the methods of perturbation theory are described in monograph [17]. 

The second group of codes is written adopting Maple symbolic procedures and are intended for studying the mathematical expressions of the theory of an atom. For example, computer code [9] considers the spin-angular coefficients for single-shell configurations, whereas [10] deals with hyperfine structure parametrisation in Maple. 

The results of research on the NMR spectra of elements with different isotopic composition have been reported. (95) The particular interest of these investigations lies in the possibility of applying perturbation theory to account for the contribution of vibrational states to the shielding of nuclei. (96, 97) In addition, the measurement of nuclear g factors of isotopic pairs, with great accuracy, is required for evaluating small hyperfine structure anomalies. (39,98)... 

Later, after experiments performed by Rabi, Lamb and Kusch and their colleagues, it was discovered that the actual hydrogen spectrum was in part in contradiction to Dirac theory (see Fig. 1). In particular, the theory predicted a value of hyperfine structure interval in the ground state of the hydrogen atom, different from the actual one by one part in 103, and no splitting between 2si/2 and... 

The hyperfine structure interval in hydrogen is known experimentally on a level of accuracy of one part in 1012, while the theory is of only the 10 ppm level [9]. In contrast to this, the muonium hfs interval [12] is measured and calculated for the ground state with about the same precision and the crucial comparison between theory and experiment is on a level of accuracy of few parts in 107. Recoil effects are more important in muonium (the electron to nucleus mass ratio m/M is about 1/200 in muonium, while it is 1/2000 in hydrogen) and they are clearly seen experimentally. A crucial experimental problem is an accurate determination of the muon mass (magnetic moment) [12], while the theoretical problem is a calculation of fourth order corrections (a(Za)2m/M and (Za)3m/M) [11]. 

The latter presents the largest sources of uncertainty in the theory of the muo-nium hfs interval, positronium energy spectrum and the specific nuclear-structure-independent difference for the hfs in the helium ion. The former are crucially important for the theory of the Lamb shift in hydrogen and medium-Z ions, for the difference in Eq. (2) applied to the Lamb shift and hyperfine structure in hydrogen and helium ion, and for the bound electron (/-factor. In the case of high-Z, the Lamb shift, (/-factor and hyperfine structure require an exact treatment of the two-loop correction. 

Abstract. Muonium is a hydrogen-like system which in many respects may be viewed as an ideal atom. Due to the close confinement of the bound state of the two pointlike leptons it can serve as a test object for Quantum Electrodynamics. The nature of the muon as a heavy copy of the electron can be verified. Furthermore, searches for additional, yet unknown interactions between leptons can be carried out. Recently completed experimental projects cover the ground state hyperfine structure, the ls-2s energy interval, a search for spontaneous conversion of muonium into antimuonium and a test of CPT and Lorentz invariance. Precision experiments allow the extraction of accurate values for the electromagnetic fine structure constant, the muon magnetic moment and the muon mass. Most stringent limits on speculative models beyond the standard theory have been set. 

This value depends strongly on the correctness of the theory both for the muonium hyperfine structure and the electron magnetic anomaly. Alternatively, extracting a value for a from Avhfs instead represents a most valuable stringent consistency test for different branches of physics, which each allow to obtain a precise value of a (see Fig.3). 

Fig. 9. The spectroscopic experiments on the hyperfine structure of muonium and the Is-2s energy interval are closely related to a precise measurement of the muon muon magnetic anomaly. The measurements put a stringent test on the internal consistency of the theory of electroweak interaction and on the set of the involved fundamental constants...

To compare the theory of ae with experiment, it is necessary to know the value of a, which has been measured in diverse branches of physics. Currently best values of a, with relative standard uncertainty of 1 x 10-7 or less, are those based on the quantum Hall effect [32], the ac Josephson effect [25], the neutron de Broglie wavelength [33], the muonium hyperfine structure [34,35], and an absolute optical frequency measurement of the Cesium >1 line [36] ... 

Subsequently, search for unfavoured resonances of Av = 2 (n, l) — (n + 1,1 — 1) transitions was carried out [15] with the following motivations. They were expected to yield qualitatively different type of information on the binding energies of pHe+. As Av = 0 transitions alone do not yield energy differences between bands of differing v, information on interband Av = 2 transitions is vitally important for a stringent test of theory. Later, the Av = 2 interband character was found to be essential in finding a hyperfine structure effect [16]. 

Abstract. CPT invariance is a fundamental property of quantum field theories in flat space-time. Principal consequences include the predictions that particles and their antiparticles have equal masses and lifetimes, and equal and opposite electric charges and magnetic moments. It also follows that the fine structure, hyperfine structure, and Lamb shifts of matter and antimatter bound systems should be identical. 

A further structure effect, the proton polarizability, is only estimated to be < 4 ppm [28], of the same order than the value above. The agreement between theory and experiment is therefore only valid on a level of 4 ppm. Thus, we can say that the uncertainty in the hyperfine structure reflects dominantly the electric and magnetic distribution of the proton, which is related to the origin of the proton anomalous moment, being a current topics of particle-nuclear physics. 

Much of the beauty of high-resolution molecular spectroscopy arises from the patterns formed by the fine and hyperfine structure associated with a given transition. All of this structure involves angular momentum in some sense or other and its interpretation depends heavily on the proper description of such motion. Angular momentum theory is very powerful and general. It applies equally to rotations in spin or vibrational coordinate space as to rotations in ordinary three-dimensional space. 

The high-resolution spectroscopy of OH has been perhaps the most important test bed for the development of the theory of the molecular energy levels, both in zero field and in the presence of applied magnetic fields. In this section, we concentrate on the A-doubling and hyperfine structure, as probed by the molecular beam studies. In chapter 9 we discuss the complex theory of the Zeeman effect, and in chapter 10 deal with rotational transitions. Our discussion therefore follows a pattern similar to that adopted for the NO molecule. 

In this section we have described in considerable detail just one aspect of the spectroscopy of OH, namely, the measurement of zl-doubling frequencies and their nuclear hyperfine structure. This has led us to develop the theory of the fine and hyperfine levels in zero field as well as a brief discussion of the Stark effect. We should note at this point, however, that OH was the first transient gas phase free radical to be studied by pure microwave spectroscopy [121], We will describe these experiments in chapter 10. We note also that magnetic resonance investigations using microwave or far-infrared laser frequencies have also provided much of the most important and accurate information these studies are described in chapter 9, where we are also able to compare OH with the equally important radical, CH, a species which, until very recently, had not been detected and studied by either electric resonance techniques or pure microwave spectroscopy. 

The constant b therefore contains contributions from two quite different magnetic interactions, the Fermi contact and the electron-nuclear dipolar interactions. Interpretation of the magnitudes of these constants in terms of electronic structure theory always involves the separate assessment of these different effects, so that we prefer to use an effective Hamiltonian which separates them at the outset. Consequently the effective magnetic hyperfine Hamiltonian used throughout this book is... 

Recent Developments in Configuration Interaction and Density Functional Theory Calculations of Radical Hyperfine Structure. 

During the past few years, however, density functional theory (DFT) has become a serious alternative to conventional Hartree-Fock based approaches, also in the area of hyperfine structure calculations. The DFT methods have drastically expanded the size of the systems accessible to theoretical hfs studies. We will in this paper review some recent developments both in the field of ab initio configuration interaction techniques and in the field of density functional theory applied to hfs studies. 

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