Interaction electric quadrupole

The tensor of the nuclear quadrupole moment Q has nine elements 

Quadmpole interaction lifts the degeneracy of nuclear states with spin quantum numbers I 1/2, and is manifested in the Mossbauer spectmm as quadmpole splitting A q (as will be further discussed in Sect. 4.3). According to (4.5), the classical electric monopole and quadmpole interaction energies Ei and q are additive, that is, = E + Eq. 

In quantum mechanics, the classical charge density of the nucleus is replaced by the operator [4]. 

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 

Remarkably, only one nuclear constant, Q, is needed in (4.17) to describe the quadrupole moment of the nucleus, whereas the full quadrupole tensor Q has five independent invariants. The simplification is possible because the nucleus has a definite angular momentum (7) which, in classical terms, imposes cylindrical symmetry of the charge distribution. Choosing x, = z as symmetry axis, the off-diagonal elements Qij are zero and the energy change caused by nuclear 

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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. 

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 ... 

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 ... 

An instructive description of the first-order perturbation treatment of the quadrupole interaction in Ni has been given by Travis and Spijkerman [3]. These authors also show in graphical form the quadrupole-spectrum line positions and the quadrupole-spectrum as a function of the asymmetry parameter r/ they give eigenvector coefficients and show the orientation dependence of the quadrupole-spectrum line intensities for a single crystal of a Ni compound. The reader is also referred to the article by Dunlap [15] about electric quadrupole interaction, in general. 

The low-temperature Mossbauer spectra of the spinel type oxides, NiCr204 [14,18] (Fig. 7.6b) and NiFe204 [3, 18], have been found to exhibit combined magnetic dipole and electric quadrupole interaction (Fig. 7.7). For the evaluation of these spectra, the authors have assumed a small quadrupolar perturbation and a large magnetic interaction, as depicted in Fig. 7.3 and represented by the Hamiltonian [3]... 

MS2 Study of magnetic dipole and electric quadrupole interactions, two Ni sites differing in angle between H and EFG axis phase transitions in NiS2.oo NiS 1 96... 

Due to unresolved electric quadrupole interaction in the hexagonal Re metal lattice, the emission lines are slightly broadened by a factor of about 1.1 and... 

Dunlap, B.D. Mdssbauer effect data index. In Stevens, J.G., Stevens, V.E. (eds.) An Introduction to Electric Quadrupole Interactions in Mdssbauer Spectroscopy. Adam Hilger, London (1972)... 

For the heteronuclear dipole-dipole interaction, the spin I S whereas for the homonuclear dipolar or electric quadrupole interaction, I=S. For the anisotropic chemical shielding interaction, the spin operators are... 

The main NMR interactions in solution of interest to chemists are the chemical shift relative to some stated standard (6), the indirect coupling constant (7) and the relaxation times T1 (spin-lattice) T2 (spin-spin related to the line width) and T p, the relaxation time in the rotating frame. In the case of solids and oriented samples both the direct dipole-dipole and the electric quadrupole interactions assume greater importance. We shall confine our attention in this chapter to diamagnetic compounds so that we may neglect nuclear interactions with electron spins. 

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