Monopole-quadrupole interactions

Fig. 4.6 Quadrupole splitting of the excited state of Fe with I = 3/2 and the resulting Mossbauer spectrum. Quadrupole interaction splits the spin quartet into two degenerate sublevels 7, OT/) with energy separation A q = 2 q. The ground state with I = 1/2 remains unsplit. The nuclear states are additionally shifted by electric monopole interaction giving rise to the isomer shift 8...

Mossbauer spectroscopy senses the hyperfine interactions, which are present at the nucleus of the Mossbauer isotope. The electrical monopole interaction causes the isomer shift and the electric quadrupole interaction leads to the quadrupole splitting, which in the case of Fe causes a two-line Mossbauer pattern. The magnetic dipole interaction leads to a magnetically split six-line pattern (Figure 4). In the following text, these interactions and their deduction from Mossbauer spectra will be discussed. 

Figure 1. Hyperfine interactions for Fe nuclei, showing the nuclear energy level diagram for (a) an unperturbed nucleus (b) electric monopole interaction (isomer shift) (c) electric quadrupole interaction (quadrupole splitting) and (d) magnetic dipole interaction (hyperfine magnetic splitting). Each interaction is shown individually, accompanied by the resulting Mossbauer spectrum.

R l term would be nonzero only if both partners were charged, with nonzero monopoles (i = /—()). An R 2 term appears in the interaction between the monopole of one O = 0) and the dipole of the other ( / = 1). The first term that occurs in the case of a pair of neutral molecules, as in the water dimer, is R 3 which corresponds to the interaction between the dipole moments of the two molecules (i = j = 1). Any nonzero monopole quadrupole terms would appear in this term as well. Dipole-quadrupole interactions die off as R 4, which would also contain charge-octapole interactions, should they exist. Just as the continuation of the multipole expansion to higher orders progressively improves the approximation of the true charge distribution of each monomer, the continuation of the R n summation yields a progressively better approximation to the true electrostatic interaction energy. 

The common expansion of the reciprocal distance in terms of spherical harmonics and collection of the corresponding orders (monopole, dipole, quadrupole,. .., interactions) leads to... 

The full Coulomb interaction energy term can be expressed by combining O Eqs. (25.42) and (O 25.43) as the sum of a number of terms with multipole order 1. The term for / = 0 is the electric monopole term. The / = 2 term gives rise to electric quadrupole interactions and wiU be discussed later. (All the odd / terms vanish for reasons of symmetry. The very small even terms for Z > 2 can also be ignored.)... 

Fig. 4.5. Quadrupole Splitting in with I = 3/2 in the excited state and / = 1/2 in the ground state. The I = 3/2 level is split into two sub-levels by electric quadrupole interaction while the ground state with 7=1/2 does not split because there is no spectroscopic quadrupole moment in a nucleus with I = 1/2. The levels of 7 = 3/2 and 7 = 1/2 are shifted by electric monopole interaction (giving rise to isomer shift). Inset shows the schematic of resultant Mossbauer spectrum...

Brant et al [11] who used the dipole approximation, stressed the importance of the electrostatic term in the calculation of intramolecular interactions in the case of a dipeptide. They have also stressed that when only the nonbonded and torsional interactions are considered, the minimum in energy does not correspond to the experimentally observed conformation. Earlier studies (Rein et al, [15]) on the conformations of simple molecules, such as hydrogen peroxide and methanol, have shown that the monopole and dipole approximations may be insufficient for the electrostatic calculations. Lassetre and Dean [16] considered the interactions in terms of a series of multipoles localized at bond midpoints and included terms up to quadrupole interactions. Tang Au-Chin [17] included interactions up to octopole terms. However, at the time of their work [16, 17], wave functions sufficiently accurate for the calculation of reliable electrical moments were not available. As a result, only rough orders of magnitude were used in their calculations. 

A possible form of the intermolecular potential for R interactions (such as monopole-quadrupole) is the following ... 

Atomic nuclei can be stretched like cigars (prolate shape) or compressed like discs (oblate shape). The deformation is described by the electric quadrii-pole moment Q (prolate Q > 0 oblate Q < 0). The principal interaction is, of course, the normal electrostatic (Coulomb) force on the charged nucleus monopole interaction). The differential interaction, which depends on the structure of the nucleus and on the valuation of the field across its finite extension, is of course very much smaller quadrupole interaction). It gives rise to an electric hyperfine structure. The energy contribution depends on the direction of the nuclear spin in relation to the electric field gradient. For the electric hyperfine interaction one obtains... 

Fig. 2.7 Nuclear energy level scheme ( Fe) for electric monopole interaction (causing the isomer shift, left), pure magnetic dipole interaction (causing magnetic splitting, middle), and combined magnetic dipole interaction and electric quadrupole interaction right)...

Often all three interactions, i.e. the electric monopole, magnetic dipole and electric quadrupole interactions, occur simultaneously. If the quadrupole interaction is small compared with the magnetic interaction (Eq j[i )> a correction to the interaction energy may be applied using first-order perturbation... 

Monopole-quadrupole. Exemplified by interactions of the type ss pp ), these terms have significant value at only very small distances. As with the dipole-dipole terms, the distance dependence is r. ... 

This last integral is the interaction of a monopole with the sum of a monopole and a linear quadrupole. This gives rise to a total of four terms. However, since the interaction of the monopole with each of two negative charges of the dipole are the same, only three terms remain. The remaining integrals are developed by similar manipulations. 

Mossbauer isomer shift and quadrupole splitting are commonly used to obtain information about the bonding environment around source nuclides. The isomer shift arises from the electric monopole interaction of the nucleus with the electrons and depends on the... 

Fig. 7.4 Top Nuclear energy levels of Fe as shifted by electrical monopole (left), or as split by electrical quadrupole (center) or by magnetic dipole interaction (right), schematized for hematite at room temperature (5 > 0 vs. a-Fe, EQ < 0, Bhf 0). Bottom Schematic Mossbauer spectra corresponding to the energy levels schematized on top (J. FriedI, unpubl.).

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