Splitting, quadrupole

A nucleus possesses a nuelear quadrupole moment if it has a spin I greater than 1/2. In this case, its energy levels will be affected by an electric field gradient (EFG) at the nucleus, which leads to sphtting of the Mossbauer line. To detect this it is sufficient if at least one of the nuclear states involved in y-ray excitation possesses a quadrupole moment eQ, and that the electric field at the nucleus is inhomogeneous. This is usually the case if there is a non-cubic valence electron distribution or non-cubic lattice-site symmetry. 

Interaction of the nuclear quadrupole moment eQ with the electric field gradient and resultant splitting of the degenerate nuclear energy levels and permitted transitions for (a) 7g = 1/2 to — 3/2 and (b) 7g =1/2 to = 5/2. 

The energy levels of the split lines can be derived using Eq. 6.5. 

The equation is derived from perturbation theory, and applies only to the special (and more straightforward) case of axial symmetry. Eq is dependent on the quantum numbers 7 and m/, the nuclear quadrupole moment eQ and the zz component of the electric field gradient tensor. This EFG tensor is expressed as a 

In the simplest approximation, two sources can contribute to the total EFG. First, there could be an anisotropic (non-cubic) distribution of electron density in the valence shell of the Mossbauer nucleus, e.g. an asymmetric substitution pattern due to different ligands in a six-coordinate metal atom or an asymmetric (non-cubic) distribution of electrons in the molecular or atomic orbitals in the model of simple ligand field theory. This contribution is called the valence electron contribution to the EFG. The second contribution comes from the lattice, specifically charges or dipoles or distant ions or other components that surround the Mossbauer nucleus in a non-symmetric arrangement. This is the lattice contribution to the EFG. 

The interactions between the nuclear quadrupole moment eQ, which is a measure of the extent of deviation of the nuclear charge distribution from 

The off-diagonal components of the matrix which represents the electric field gradient 3x3 second-rank tensor can be made zero by performing an appropriate coordinate transformation. This makes it possible to specify the electric field gradient by two independent quantities the diagonal element 

The Hamiltonian operator for the electric quadrupole interaction, 7/q, given in (4.29), coimects the spin of the nucleus with quantum number I with the EFG. In the simplest case, when the EFG is axial (y = Vyy, i.e. rf = 0), the Schrddinger equation can be solved on the basis of the spin functions I,mi), with magnetic quantum numbers m/ = 7, 7—1,. .., —7. The Hamilton matrix is diagonal, because 

The electric quadrupole interaction causes a splitting of the (27 +1) magnetic substates without shifting the mean energy of the nuclear spin manifold substates with the same absolute value of w/ remain degenerate for rj = 0. 

The effect of electric quadrupole interaction for Fe is exemplified in Fig. 4.6. The ground state remains unsplit because of the lack of quadrupole moment for 7 = 1/2. The excited state with 7 = 3/2 splits into two doubly degenerate substates 3/2, 3/2) and 3/2, 1/2) due to the w/ dependence of the quadrupole energies  

In a conventional Fe Mossbauer experiment with a powder sample, one would observe a so-called quadrupole doublet with two resonance lines of equal intensities. The separation of the lines, as given by (4.36), represents the quadrupole splitting The parameter Afg is of immense importance for chemical applications of the Mossbauer effect. It provides information about bond properties and local symmetry of the iron site. Since the quadrupole interaction does not alter the mean energy of the nuclear ground and excited states, the isomer shift S can also be derived from the spectrum it is given by the shift of the center of the quadrupole spectrum from zero velocity. 

The quadrupole interaction becomes more sophisticated when the EFG lacks axial symmetry, p 0, because the shift operators connected to p introduce 

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 

Since the splitting of the spectral lines is directly proportional to the magnetic field experienced by the nucleus, Mossbauer spectroscopy provides a very effective means by which this field may be measured. The transition probabilities between the nuclear substates affect the intensities of the lines in the Mossbauer spectrum which can therefore give information on the relative orientation of the magnetic field at the nucleus and the direction of propagation of the gamma-ray beam. 

The total magnetic field experienced by the nucleus is a vector sum of the magnetic hyperfine field and any external applied magnetic field. The 

In the previous section, it was assumed that the distribution of the nuclear charge is spherical [140], However, the charge distribution of a nucleus is not always spherically symmetric. In fact, this is not the case for a nucleus with nuclear angular momentum I 1/2 in this case, the nucleus shows nonspherical nuclear charge distributions [142,143], The electrostatic potential created by a charge distribution localized inside a radius, Ir I R, can be expressed outside the sphere of radius, R, in rectangular coordinates as follows [144]  

The third expressions are the quadrupole terms, with nine components, that is, a second rank tensor  

The quadrupole moment tensor of the nuclear charge distribution is characterized by one term, that is, Q3 3, since Qu = Q i i = - Q3 3 [144], Then, the quadrupole moment, Q, of a nuclear state, in units of area, that is, barns (1 barn = 10 28 m2), is expressed as follows [143] 

0 is the polar angle, which is measured with respect to the nuclear spin direction 

In quadrupole splitting, the existence of a nonspherical nuclear charge distribution produces an electric quadrupole moment, Q, which indicates that the charge distribution in the nucleus is prolate, when Q 0, or oblate, if Q 0 [137-140], 

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Figure 4.54 The effect of an electric field gradient (EFG) creating asymmetry in the electron distribution round a gold nucleus, leading to a quadrupole splitting in the Mossbauer spectrum. (Reproduced with permission from Gold Bull., 1982,15, 53, published by World Gold Council.)...

Partial quadrupole splittings in inorganic chemistry. G. M. Bancroft. Coord. Chem. Rev., 1973,11, 247-262 (53). 

On the basis of the point-charge model formalism, applied on the experimental nuclear quadrupole splitting rationalization, I Agxp I, the results obtained were interpreted in terms of strong complex formation by either Me2Sn(OH)2 or Me3Sn(0H)(H20) with (n = 1 or 2, obtained in phosphate buffer) and... 

The various, solid-state stereochemistries just described may often be distinguished fairly readily by " Sn Mossbauer spectroscopy 5-9, 452), particularly from the value of the quadrupole splitting parameter, AEq (see Table II). 

Sn Mossbauer Quadrupole Splittings for Organotin Compounds OF Known Stereochemistry... 

For example, octahedral quadrupole splitting observed for the cis-octahedral analogs 7,8). More recently, temperature-dependent Mossbauer measurements have been used in conjunction with Raman spectroscopy to determine molecular weights 453) and lattice rigidity 460) of various organotin compounds. 

Table 1. The 72-atom model examined by different theoretical methods. The energy differences (AE in kcal/mol) are calculated with respect to the lowest SCF energy. q(Fe) stands for Mulliken population charges on the Fe atoms q(S) and SS(b.i.) are the Mulliken population charges and the bond index for the bridging S atoms, respectively AEq is the calculated Mossbauer quadrupole splitting constant [mm/sec]. The PUHF spin states are those projected from the UHF wavefunction with 5 = 5,. 

PUHF, yield the lowest energies as well as reasonable Mossbauer quadrupole splitting constants, A Jq. 

Summary of EPR -Values, Fg-Hyperfine Coupling Constants, Isomer Shifts, and Quadrupole Splittings for Some Representative [Fg3S4] Clusters... 

Most valuable chemical information can be extracted from Mbssbauer parameters such as the isomer shift (5), the quadrupole splitting (AEq), the magnetic splitting (AEm), and the asymmetry parameter (n). 

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.

The results of the XRD measurement showed that the Fe jAl, jPO catalyst was almost in amorphous state. Only a very broad peak at 29 of ca. 23 degree was observed. The Mossbauer spectroscopic study on this catalyst showed one doublet of iron with the isomeric shift of 0.31 mm s (a-Fe was used as the reference) and the quadrupole splitting of 0.62 mm s. These parameters are very close to those observed for FePO [13, 14], suggesting that the iron cation in the catalyst is tetrahedrally coordinated with oxygen and isolated by four PO tetrahedral units. Such coordination circumstance was suggested to be a key factor for the iron site effective for the oxidation of CH to CHjOH by H -Oj gas mixture [15]. 

Sample Fe state Isomer shift (mm/s) Quadrupol splitting AE(mm/s) Reference... 

Table 2. Isomer shifts (IS) and quadrupole splitting (QS) of Pt309phen 36O30+10, fractionally transferred into isotopes.

A unique situation is encountered if Fe-M6ssbauer spectroscopy is applied for the study of spin-state transitions in iron complexes. The half-life of the excited state of the Fe nucleus involved in the Mossbauer experiment is tj/2 = 0.977 X 10 s which is related to the decay constant k by tj/2 = ln2/fe. The lifetime t = l//c is therefore = 1.410 x 10 s which value is just at the centre of the range estimated for the spin-state lifetime Tl = I/Zclh- Thus both the situations discussed above are expected to appear under suitable conditions in the Mossbauer spectra. The quantity of importance is here the nuclear Larmor precession frequency co . If the spin-state lifetime Tl = 1/feLH is long relative to the nuclear precession time l/co , i.e. Tl > l/o) , individual and sharp resonance lines for the two spin states are observed. On the other hand, if the spin-state lifetime is short and thus < l/o) , averaged spectra with intermediate values of quadrupole splitting A q and isomer shift 5 are found. For the intermediate case where Tl 1/cl , broadened and asymmetric resonance lines are obtained. These may be the subject of a lineshape analysis that will eventually produce values of rate constants for the dynamic spin-state inter-conversion process. The rate constants extracted from the spectra will be necessarily of the order of 10 -10 s"F... 

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