Interaction quadrupole

The levels m= 1 and m=-l are affected in the same direction by the quadrupolar interaction whereas the m= 0 level is affected in the opposite direction. If the electric 

For a single deuteron in a crystalline material the spectrum consists of a doublet with peak separation equal to  

The peak separation is then dependent on the orientation, expressed by the 0 angle, of the electric field gradient with respect to the magnetic field (Fig. 3.2.2). 

In a powder spectrum, where all the orientations are present, the superimposition of two symmetric powder distributions reversed in sign forms the well known Pake doublet (Fig. 3.2.3) [6]. 

When molecules undergo reorientation on a timescale of the order of the quadrupole interaction or faster (rapid exchange limit, correlation time of the motion 10-7 s), the observed frequencies will be averaged and the resulting lineshape can be analysed to determine the type and the rate of the motion [7]. 

In the volume surrounding the nucleus V(tf) can be expanded as a Taylor series as 

Vij is a second rank symmetric tensor which is diagonal in its PAS so that 

The zero-order term in Eq. 2.100 represents the electrostatic energy which is the same for all orientations. This term will have no influence on the spectrum and can be ignored. The first-order term represents the electric dipole moment and from the fact that the nuclear wavefunction is symmetric (i.e. r(r) 5 r (2 r)), the product rr(r) is antisymmetric and this term will be identically zero for all nuclei. The second-order term is the electric quadrupole moment and this is the most important in giving an 

For the potential Laplace s equation states = 0, from which it follows that for the electric gradient tensor, 

This property means that to fully define V only two of its three components need to be known. It is usually chosen to define these as the largest component 

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Quadrupole interaction energy tensor X Residual resistivity (solid state) Pr... 

A reahstic modeling of N2 monolayers on graphite was done in Refs. 95,96. Concerning the N2-N2 interactions in Hamiltonian (40) and Eq. (42), the X model [146-148,318], consisting of site-site Lennard-Jones and quadrupole interactions, was shown to yield a realistic representation ... 

The van der Waals interaction energy of two hydrogen atoms at large intemuclear distances is discussed by the use of a linear variation function. By including in the variation function, in addition to the unperturbed wave function, 26 terms for the dipole-dipole interaction, 17 for the dipole-quadrupole interaction, and 26 for the quadrupole-quadrupole interaction, the... 

A rough treatment of the dipole-quadrupole and quadrupole-quadrupole interactions of two hydrogen atoms has been published by Mar-... 

The Dipole-Quadrupole Interaction The secular equation for the dipole-quadrupole interaction is... 

If the nucleus feels both a magnetic field and an electric field gradient, and the electric quadrupole interaction is small, then the excited levels shift further and make the sextet asymmetrical, as observed in the spectrum of Fe203. 

Deuterium NMR has recently been used to study molecular motion of organic adsorbates on alumina (1.) and in framework aluminosilicates (2). The advantage of NMR is that the quadrupole interaction dominates the spectrum. This intramolecular interaction depends on the average ordering and dynamics of the individual molecules. In the present work we describe NMR measurements of deuterated benzene in (Na)X and (Cs,Na)X zeolite. 

The quadrupole interaction operator in a Mossbauer experiment may be expressed as ... 

Equation (4.15) would be extremely onerous to evaluate by explicit treatment of the nucleons as a many-particle system. However, in Mossbauer spectroscopy, we are dealing with eigenstates of the nucleus that are characterized by the total angular momentum with quantum number 7. Fortunately, the electric quadrupole interaction can be readily expressed in terms of this momentum 7, which is called the nuclear spin other properties of the nucleus need not to be considered. This is possible because the transformational properties of the quadrupole moment, which is an irreducible 2nd rank tensor, make it possible to use Clebsch-Gordon coefficients and the Wigner-Eckart theorem to replace the awkward operators 3x,xy—(5,yr (in spatial coordinates) by angular momentum operators of the total... 

C. P. Slichter s textbook on magnetic resonance [4] may be recommended for further reading. It presents a very educational introduction into this issue of operator equivalence. A comprehensive, elaborate article on quadrupole interaction in Mossbauer spectroscopy is provided by H. Spiering in [5],... 

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 ... 

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...

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... 

The eigenfunctions for nonaxial nuclear quadrupole interaction are mixtures of the 7, mj) basis functions and thus do not possess well-defined magnetic quantum numbers. Strictly speaking, the states should not be labeled with pure quantum numbers m/. ... 

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... 

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