Electric monopole interactions

The integral can be solved by conversion from Cartesian to spherical coordinates. Then, the integration variable takes the convenient form dr = r drsinddddcj), which yields 

Finally, a very simple expression is obtained when is replaced by the mean square radius as given above, and the notation (r ) = is introduced, yielding 

When inserting into (4.5), the term ZeR will be multiplied with the elements of the electric field gradient tensor V. Fortunately, the procedure can be restricted to diagonal elements Vu, because V is symmetric and, consequently, a principal axes system exists in which the nondiagonal elements vanish, = 0. The diagonal elements can be determined by using Poisson s differential equation for the electronic potential at point r = 0 with charge density (0), AV = Anp, which yields 

The term causes a uniform shift of the nuclear energy states which, however, is different for the ground and excited state because the nuclear volume and, therefore, also the entity R ) are different for ground and excited states. This gives rise to the isomer shift 6 of the Mbssbauer spectrum. The notation 6E = E is introduced to emphasize the very small change in energy ( 10 eV), which is only a fraction (about 10 ) of the transition energy. The isomer shift will be discussed in detail in Sect. 4.2. 

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The electric monopole interaction between a nucleus (with mean square radius k) and its environment is a product of the nuclear charge distribution ZeR and the electronic charge density e il/ 0) at the nucleus, SE = const (4.11). However, nuclei of the same mass and charge but different nuclear states isomers) have different charge distributions ZeR eR ), because the nuclear volume and the mean square radius depend on the state of nuclear excitation R R ). Therefore, the energies of a Mossbauer nucleus in the ground state (g) and in the excited state (e) are shifted by different amounts (5 )e and (5 )g relative to those of a bare nucleus. It was recognized very early that this effect, which is schematically shown in Fig. 4.1, is responsible for the occurrence of the Mossbauer isomer shift [7]. 

Fig. 4.1 The electric monopole interaction between the nuclear charge and the electron density at the nucleus shifts the energy of the nuclear states and gives the Mossbauer isomer shift...

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...

Fig. 4.9 Magnetic dipole splitting (nuclear Zeeman effect) in pe and resultant Mossbauer spectrum (schematic). The mean energy of the nuclear states is shifted by the electric monopole interaction which gives rise to the isomer shift 5. Afi. g = Sg/tN and A M,e = refer to the...

To calculate Mossbauer spectra, which consist of a finite number of discrete lines, the nuclear Hamiltonian, and thus also Hsu, has to be set up and solved independently for the nuclear ground and excited states. The electric monopole interaction, that is, the isomer shift, can be omitted here since it is additive and independent of Mj. It can subsequently be added as an increment 5 to the transition energies of each of the obtained Mossbauer lines. 

Asymmetry in the ligand environment, either geometric or in charge distribution (or both), affect the asymmetry parameter, tp An r = 0 value corresponds to complete axial symmetry, whereas r = 1 corresponds to pure rhombic symmetry. Electric monopole interactions between the nuclear charge distributions and the electrons at the nucleus cause a shift of the nuclear ground and excited states. These interactions are known as the isomer shift, 8. Both the Mossbauer source and the absorber (the sample of interest) experience an isomer shift, and it is customary to quote 8 relative to a standard, usually Fe metal or Na2[Fe(CN)5NO] 2H2O at... 

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... 

The electric monopole interaction is a function of the s electron densities at the nucleus. This results in a displacement of the spectrum and is expressed as the velocity of the source (mm s ) necessary to counteract the displacement. This isomer (or chemical) shift, 6, provides information about the coordination number, the valency and spin state of the iron in the compound. 

The nuclear ground and excited levels involved in the Mossbauer transition are shifted or split because of the electrostatic interactions between the nuclear charge and the surrounding electric charge (Fig. 2). The first interaction, sometimes called the electric monopole interaction, shifts only the nuclear levels and is related to the perturbation resulting from the electrons inside the nuclear volume. This shift is... 

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. 

The electric monopole interaction is proportional to the s-electron density at the iron nnclens [Pg.2821]

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.

Electric monopole interaction, detectable as a line shift (isomer shift 8, mm s ) ... 

In Mossbauer spectroscopy, the only thing that matters is not the energy shift of an individual level, but the energy-shift difference between the ground-state and excited-state levels. Let us consider the energy shifts of the nuclear levels due to the electric monopole interaction and the energy of the y ray emitted fi-om the source and absorbed by the absorber (O Pig. 25.8). 

The shifts of nuclear energy levels in the source and the absorber due to the electric monopole interaction between the nucleus and the electrons, and the resulted Mossbauer spectrum. According toO Eq. (25.80), the chemical isomer shift Sc is proportional to the energy shift AE s... 

According to O Eq. (25.56), one obtains that the energy shift due to electric monopole interaction is proportional to the product of the s electronic density and the second moment of the nuclear charge distribution also called the mean square nuclear charge radius (see also O Sect. 2.2.3.1 in Chap. 2, Vol. 1). When the nucleus is considered to be a homogeneously charged sphere with a radius R (often called charge equivalent nuclear radius), then... 

In case of Fe, the levels of I = 3/2 and / = 1/2 are shifted by electric monopole interaction giving rise to the isomer shift. The quadrupole splitting... 

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...

Electric monopole interaction between nucleus and electrons at the nuclear site Isomer shift 6 a. Oxidation state (nominal valency) of the Mossbauer atom b. Bonding properties in coordination compounds (covalency effects between central atom/ion and ligands. Delocalization of d-electrons due to back-bonding effects, shielding of s-electrons by p-and d-electron ) c. Electronegativity of ligands... 

A) Electric monopole interaction in source and absorber shifts the nuclear energy levels without affecting the degeneracy B1 resultant Mossbauer spectrum (schematic)... 

B) resultant Mossbauer spectrum (schematic) with equal intensities as for a powder sample the inevitable shift of the nuclear levels due to electric monopole interaction giving rise to the isomer shift is also shown... 

The isomer shift of the absorption lines in the Mossbauer spectrum, also sometimes known as the chemical shift, the chemical isomer shift or the centre shift, is a result of the electric monopole (Coulomb) interaction between the nuclear charge distribution over the finite nuclear volume and the electronic charge density over this volume. This shift arises because of the difference in the nuclear volume of the ground and excited states, and the difference between the electron densities at the Mossbauer nuclei in different materials. In a system where this electric monopole interaction is the only hyperfine interaction affecting the nuclear energy levels, the nuclear ground and excited states are unsplit, but their separation is different in the source and absorber by an amount given by the isomer shift <5. 

In considering the electric monopole interaction and the resulting isomer shift it is implicitly assumed that the nuclear charge distribution is spherical. However, nuclei in states with a nuclear angular momentum quantum number /> have non-spherical charge distributions which are characterised by a nuclear quadrupole moment. When the nuclear... 

On the other hand the isomer shift in a Mossbauer experiment on an intermediate-valent system is considered to be slow relative to the intrinsic time scale of the valence fluctuations. The hyperfine interaction of the nucleus with mainly the 5s character electronic charge at the nucleus is an electric monopole interaction (only s electrons can interact with the nucleus, but the 5s electrons being outside the 4f shell are quite sensitive to the occupation of the 4f shell because of different screening of the nuclear... 

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