Electric-quadrupole field-gradient

JJor chemists interested in modem theories of chemical bonding, the most useful data obtainable by the Mossbauer technique are the magnitude and sign of the electric quadrupole field gradient tensor and the magnitude of the shift, 8, (which we prefer to call the chemical isomeric. Cl, shift), of the center of the Mossbauer spectrum relative to some standard absorber. Although a considerable amount of chemical and structural information is potentially available from quadrupole data on iron compounds, relatively little use has been made of such data in the literature, and we will not discuss this parameter here. We will instead restrict ourselves to two main points review of the explanations put forth to explain Cl shift data in iron compounds, and a survey of some of the correlations and generalizations which have been found. 

If the nuclei in a molecule have angular momenta 7 >1 then complications in the spectra may arise due to the electric quadrupole field of the nuclei. Such additional structure is commonly observed and arises because there is an interaction between the electric field gradient along the internuclear axis (due to the electrons), the electric quadrupole moment of the nucleus, and the angular momentum of the molecule. The energy change due to the quadrupole moment can be written ... 

Just like the electric quadrupole moment, the electric field gradient matrix can be written in diagonal form for a suitable choice of coordinate axes. 

In a molecule, a given nucleus will generally experience an electric field gradient due to the surrounding electrons. The energy of interaction U between the nuclear quadrupole and the electric field gradient E is given by... 

In order to find the correct quantum-mechanical energies for a nuclear quadrupole in an electric field gradient, we need to... 

Terms up to order 1/c are normally sufficient for explaining experimental data. There is one exception, however, namely the interaction of the nuclear quadrupole moment with the electric field gradient, which is of order 1/c. Although nuclei often are modelled as point charges in quantum chemistry, they do in fact have a finite size. The internal structure of the nucleus leads to a quadrupole moment for nuclei with spin larger than 1/2 (the dipole and octopole moments vanish by symmetry). As discussed in section 10.1.1, this leads to an interaction term which is the product of the quadrupole moment with the field gradient (F = VF) created by the electron distribution. 

Here q is the net charge (monopole), p, is the (electric) dipole moment, Q is the quadrupole moment, and F and F are the field and field gradient d /dr), respectively. The dipole moment and electric field are vectors, and the pF term should be interpreted as the dot product (p F = + EyPy + Ez z)- "I e quadrupole moment and field... 

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.)...

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. 

Schwerdtfeger, P., Bast, R., Gerry, M.C.L., Jacob, C.R., Jansen, M., Kelld, V., Mudring, A.V., Sadlej, A.J., Saue, T, Sdhnel, T. and Wagner, F.E. (2005) The quadrupole moment of the 3 /2 nuclear groimd state of Au from electric field gradient relativistic coupled cluster and density functional theory of small molecules and the solid slide. Journal of Chemical Physics, 122,124317-1-124317-9. 

Here, I, I, and I are angular momentum operators, Q is the quadrupole moment of the nucleus, the z component, and r the asymmetry parameter of the electric field gradient (efg) tensor. We wish to construct the Hamiltonian for a nucleus if the efg jumps at random between HS and LS states. For this purpose, a random function of time / (f) is introduced which can assume only the two possible values +1. For convenience of presentation we assume equal... 

Both the ground state and the 67.4 keV nuclear excited state of possess a nonzero electric quadrupole moment. If placed in an inhomogeneous electric field (electric field gradient, EFG 0) the Ni nucleus undergoes electric quadmpole interaction with the EFG at the nucleus, as a result of which the 67.4 keV level will split into three substates /, wi) = 5/2, 5/2), 5/2, 3/2), and 5/2, 1/2) and the ground level will split into two substates 3/2, 3/2) and 3/2, 1/2). 

The electric quadrupole Q 2co) involves both the gradient of the electromagnetic incident electric field E u)) and the gradient of the electric quadrupole susceptibility tensor Xq 2o), CO, co). This problem is nonetheless solved by the mere addition of supplementary terms in the surface susceptibility tensor. As a result, the surface susceptibility tensor becomes an effective tensor instead of a purely surface specific one [27,38] ... 

Other key properties of these complexes relevant for comparison with experiment are the nuclear quadrupole coupling constants. Theoretical NQCC values, as calculated via electric field gradients, are discussed exhaustively in several recent theoretical investigations [26,28,30,33,34] and in experimental work [4-11]. BH HLYP/aug-cc-pVTZ calculated NQCCs [30,34], as evaluated for XY- -NH3 complexes, are listed in Table 10. 

The Rh-Ir vector defines the z-axis, and the NN bridges define the x- and y-axes. The N and P atoms are solely electron donors, but the CO ligand involves a lot of re-back-bonding. Thus the quadrupole matrix, which has large components if there is an electric field gradient at the Ir nucleus, is rotated 45° compared with the g-matrix principal axes. 

As such, nuclear contributions to the heat capacity due to the interaction between germanium crystalline electric field gradients and the quadrupole moments of boron nuclei could account for the observed onset of the Schottky anomaly. 

In the literature [55], typical energies involved in the nuclear quadrupole moments -crystalline electric field gradient interactions range up to A E 2x 10-25 J. The measured AE seems to confirm the hypothesis that the excess specific heat of the metallized wafer is due to boron doping of the Ge lattice. 

The strength of the quadrupolar interaction is proportional to the quadrupole moment Q of a nucleus and the electric field gradient (EFG) [21-23]. The size of Q depends on the effective shape of the ellipsoid of nuclear charge distribution, and a non-zero value indicates that it is not spherically symmetric (Fig. 1). 

Fig. 1 (a) Schematic representation of the spherical and non-spherical charge distribution in a nucleus. The value of electric quadrupole moment Q for the quadrupolar nucleus depends on the isotope under consideration, (b) The quadrupolar interaction arises from the interaction of Q with surrounding electric field gradient (EFG)... 

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