Magnetic dipole hyperfine

The Hamiltonian describing the magnetic dipole hyperfine interaction is... 

Fig. 3 Magnetic dipole hyperfine constant A for the S /2 ground state of gold as a function of the size of the MCDHF expansion circles - CV, squares - SD, triangles - SDT expansions. The arrow refers to a value from Ref [29] and the horizontal line indicates the experimental value [30]...

The Hamiltonian description of the magnetic dipole hyperfine interactions is given by,... 

It had been shown early on by Breit [17] and Racah [18] that the relativistic corrections to magnetic dipole hyperfine interactions can be substantial. For later references, see, for instance, Refs. [19-21]. For the 6 valence orbital of an element such as mercury this correction is roughly a factor of 3. Hence the J(HgHg) coupling constant is increased by an order of magnitude due to relativistic effects. 

Construct a diagram showing the low-field Zeeman splittings of the ground-state hyperfine levels of an alkali atom with nuclear spin 1= j. Indicate the permitted magnetic dipole hyperfine transitions... 

Figure 2A. Schematic diagram of Mossbauer parameters isomer shift (6), quadrupole splitting (AEq) and magnetic dipole splitting of the nuclear energy states of 57pe leading to various hyperfine splitting in Mossbauer spectra.

Fig. 4.13 Combined magnetic hyperfine interaction for Fe with strong electric quadrupole interaction. Top left, electric quadrupole splitting of the ground (g) and excited state (e). Top right first-order perturbation by magnetic dipole interaction arising from a weak field along the main component > 0 of the EFG fq = 0). Bottom the resultant Mossbauer spectrum is shown for a single-crystal type measurement with B fixed perpendicular to the y-rays and B oriented along...

It is much more difficult to observe the Mossbauer effect with the 130 keV transition than with the 99 keV transition because of the relatively high transition energy and the low transition probability of 130 keV transition, and thus the small cross section for resonance absorption. Therefore, most of the Mossbauer work with Pt, published so far, has been performed using the 99 keV transition. Unfortunately, its line width is about five times larger than that of the 130 keV transition, and hyperfine interactions in most cases are poorly resolved. However, isomer shifts in the order of one-tenth of the line width and magnetic dipole interaction, which manifests itself only in line broadening, may be extracted reliably from Pt (99 keV) spectra. 

We have learned from the preceding chapters that the chemical and physical state of a Mossbauer atom in any kind of solid material can be characterized by way of the hyperfine interactions which manifest themselves in the Mossbauer spectrum by the isomer shift and, where relevant, electric quadrupole and/or magnetic dipole splitting of the resonance lines. On the basis of all the parameters obtainable from a Mossbauer spectrum, it is, in most cases, possible to identify unambiguously one or more chemical species of a given Mossbauer atom occurring in the same material. This - usually called phase analysis by Mossbauer spectroscopy - is nondestructive and widely used in various kinds of physicochemical smdies, for example, the studies of... 

In Equation (6) ge is the electronic g tensor, yn is the nuclear g factor (dimensionless), fln is the nuclear magneton in erg/G (or J/T), In is the nuclear spin angular momentum operator, An is the electron-nuclear hyperfine tensor in Hz, and Qn (non-zero for fn > 1) is the quadrupole interaction tensor in Hz. The first two terms in the Hamiltonian are the electron and nuclear Zeeman interactions, respectively the third term is the electron-nuclear hyperfine interaction and the last term is the nuclear quadrupole interaction. For the usual systems with an odd number of unpaired electrons, the transition moment is finite only for a magnetic dipole moment operator oriented perpendicular to the static magnetic field direction. In an ESR resonator in which the sample is placed, the microwave magnetic field must be therefore perpendicular to the external static magnetic field. The selection rules for the electron spin transitions are given in Equation (7)... 

The magnetic hyperfine splitting, the Zeeman effect, arises from the interaction between the nuclear magnetic dipole moment and the magnetic field H at the nucleus. This interaction gives rise to six transitions the separation between the peaks in the spectrum is proportional to the magnetic field at the nucleus. 

The energies of the electric quadrupole (Wg) and magnetic dipole (W ) interactions, which determine the hyperfine structure, are calculated as follows [11,20] ... 

We have also performed the calculation of hyperfine coupling constants the electric quadrupole constant B and magnetic dipole constant A, with inclusion of nuclear finiteness and the Uehling potential for Li-like ions. Analogous calculations of the constant A for ns states of hydrogen-, lithium- and sodiumlike ions were made in refs [11, 22]. In those papers other bases were used for the relativistic orbitals, another model was adopted for the charge distribution in the nuclei, and another method of numerical calculation was used for the Uehling potential. 

In Tables 5 and 6 there are displayed the results for the hyperfine coupling constants in the lowest excited states of Li-like ions. In Table 5 we compare the results of our calculations with those from papers [11, 22] for magnetic dipole coupling constants in the ground state ls 2s of a few lithium-like ions. 

Unlike the Lamb shift, the hyperfine splitting (see Fig. 8.1) can be readily understood in the framework of nonrelativistic quantum mechanics. It originates from the interaction of the magnetic moments of the electron and the nucleus. The classical interaction energy between two magnetic dipoles is given by the expression (see, e.g., [1, 4])... 

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