Electric quadrupole hyperfine interaction

The form of the nuclear electric quadrupole interaction in the effective Hamiltonian for a diatomic molecule is given in equations (7.158) and (7.161), with the latter applying only to molecules in n electronic states. The two parameters which can be determined from a fit of the experimental data are eqo Q and et/i Q respectively. Since the electric quadrupole moment eQ is known for most nuclei, an experimental observation gives information on q0 (and perhaps qi), the electric field gradient at the nucleus. This quantity depends on the electronic structure of the molecule according to the expression 

Let us consider a nucleus with I 1 in atom 1 of a diatomic molecule. We shall use a local coordinate system (x, y, z) with its origin at the nucleus and z lying along the molecular bond. An unscreened electric charge q at a distance r from the nucleus gives rise to an electrostatic potential 

The total electric field gradient at the nucleus due to all the electrons in the molecule is therefore 

This discussion shows that the most accurate interpretation of the electric quadrupole coupling constant is obtained by evaluating 

Short of the ab initio calculations, there are several semi-empirical approaches to the calculation and interpretation of electric quadrupole coupling constants. These were developed originally by Townes and Dailey [47, 48] and are well documented in the book by Gordy and Cook [49], They are based on the linear combination of atomic orbitals approximation for molecular orbitals, mentioned earlier in equation (7.266) and described in more detail in chapter 6  

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The most important examples of 2S states to be described in this book are CO+, where there is no nuclear hyperfine coupling in the main isotopomer, CN, which has 14N hyperfine interaction, and the Hj ion. A number of different 3E states are described, with and without hyperfine coupling. A particularly important and interesting example is N2 in its A 3ZU excited state, studied by De Santis, Lurio, Miller and Freund [19] using molecular beam magnetic resonance. The details are described in chapter 8 the only aspect to be mentioned here is that in a homonuclear molecule like N2, the individual nuclear spins (1 = 1 for 14N) are coupled to form a total spin, It, which in this case takes the values 2, 1 and 0. The hyperfine Hamiltonian terms are then written in terms of the appropriate value of h As we have already mentioned, the presence of one or more quadrupolar nuclei will give rise to electric quadrupole hyperfine interaction the theory is essentially the same as that already presented for1 + states. 

The first reported Mossbauer spectrum of a-Fe203 was by Kistner and Sunyar [1], who thereby recorded the first chemical isomer shift and electric quadrupole hyperfine interactions to be observed by this technique. With a single-line source the room-temperature spectrum comprises six lines from a hyperfine field of 515 kG the chemical isomer shift (Table 10.1) is... 

The quadrupole splitting. A, is the Mossbauer parameter, which is related to the electric quadrupole hyperfine interaction between the nucleus and the electrons (see Sect. 25.1.5.2). 

The two absorption peaks, appearing in the measured Mossbauer spectrum as a consequence of electric quadrupole hyperfine interaction, are called a doublet. 

Fig. 1 Comparison of the CAS and uncontracted PCFI convergence patterns of the electric quadrupole hyperfine interaction parameter as a function of the size ( niax) orbital active set...

Mossbauer measurements with determination of the electric quadrupole moments have been reported in [253, 254,259]. Wagner et al. [254] measured the quadrupole hyperfine interaction in OSO2 and OSP2 of the Mossbauer isotopes The ratios of the quadrupole moments of the 4 = 72 states in the even osmium isotopes and of the 4 = 5/2 (69.6 keV) and 4 = 3/2 states in Os were deduced very accurately. In Table 7.8, the experimental results [254] are given, from which the following ratios can be calculated ... 

Pure nuclear magnetic hyperfine interaction without electric quadrupole interaction is rarely encountered in chemical applications of the Mossbauer effect. Metallic iron is an exception. Quite frequently, a nuclear state is perturbed simultaneously by... 

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...

The leading term in T nuc is usually the magnetic hyperfine coupling IAS which connects the electron spin S and the nuclear spin 1. It is parameterized by the hyperfine coupling tensor A. The /-dependent nuclear Zeeman interaction and the electric quadrupole interaction are included as 2nd and 3rd terms. Their detailed description for Fe is provided in Sects. 4.3 and 4.4. The total spin Hamiltonian for electronic and nuclear spin variables is then ... 

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... 

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

In formulas (22.12) and (22.13) k acquires only even values for k = 2 we have the usual electric quadrupole interaction, whereas for k = 4 we have the electric hexadecapole interaction, already observed in [145]. The expressions for the matrix elements of the hyperfine structure operators considered above for the closed shells follow straightforwardly from the... 

Hm describes the hyperfine interaction with the 57Fe nucleus. A is the magnetic hyperfine tensor and Hq describes the interaction of the quadrupole moment Q of the 7=3/2 nuclear excited state with the (traceless) electric field gradient (EFG) tensor V (the nuclear ground state has 7= 1/2 and lacks a spectroscopic quadrupole moment). In the absence of magnetic effects (for instance, for S 0, or S = integer for B = 0), the Mossbauer spectrum consists of a doublet with quadrupole splitting ... 

Clearly, all hyperfine interactions can occur simultaneously. In magnetically ordered compounds with a non-vanishing electric field gradient, the shape of the spectrum depends on the relative strengths of the magnetic and the electric quadrupole interaction. In catalysts, the usual situation is that the quadrupole interac-... 

The magnetic hyperfine interaction terms were given in equation (8.351) and the electric quadrupole interaction in equation (8.352). We extend the basis functions by inclusion of the 7Li nuclear spin I, coupled to J to form F the value of / is 3/2. We deal with each term in turn, first deriving expressions for the matrix elements in the primitive basis set (8.353), and then extending these results to the parity-conserved basis. All matrix elements are diagonal in F, and any elements off-diagonal in S and / can of course be ignored. 

The electric quadrupole interaction is handled in exactly the same way as the magnetic hyperfine interactions, by expanding the scalar product (9.36) first in the space-fixed axis system, and then transforming the electronic part of the interaction into the molecule-fixed system. One obtains the result ... 

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