Electric quadrupole momentum

Changes of the angular correlation between particles or y rays emitted in immediate succession has been observed for nuclides with 1 = 0. These nuclides have a magnetic momentum and an electric quadrupole momentum which interact with the surrounding field. If this field is influenced by variations in chemical bonding, the angular correlation may also be affected. 

We often say that an electron is a spin-1/2 particle. Many nuclei also have a corresponding internal angular momentum which we refer to as nuclear spin, and we use the symbol I to represent the vector. The nuclear spin quantum number I is not restricted to the value of 1/2 it can have both integral and halfintegral values depending on the particular isotope of a particular element. All nuclei for which 7 1 also posses a nuclear quadrupole moment. It is usually given the symbol Qn and it is related to the nuclear charge density Pn(t) in much the same way as the electric quadrupole discussed earlier ... 

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... 

What about parity in electric-quadrupole and magnetic-dipole transitions The quantities (3.58) are even functions. Hence for electric-quadrupole transitions, parity remains the same. Magnetic-dipole transitions involve angular momentum operators. For example, consider Lz = -ih(xd/dy — yd/dx). Inversion of coordinates leaves this operator unchanged. Hence for magnetic-dipole transitions, parity remains the same. 

The interaction between the nuclear electric quadrupole moment and the electrons of the molecule couples the nuclear spin I to the rotational angular momentum J, giving a resultant total angular momentum F, of... 

Yi and Ys - gyromagnetic ratio of spin 1 and spin S nuclear spin, rJS = intemuclear distance, tr= rotational correlation time, x< = reorientation correlation time, xj = angular momentum correlation time, Cs = concentration of spin S, Cq = e2qzzQ/h = quadrupole coupling constant, qzz = the electric field gradient, Q = nuclear electric quadrupole moment in 10 24 cm2, Ceff = effective spin-rotational coupling constant, a = closest distance of appropriate of spin 1 and spin S, D = (DA+DB)/2 = mutual translational self diffusion coefficient of the molecules containing I and S, Ij = moment of inertia of the molecule, Ao = a// - ol-... 

The muon g — 2 value has been determined in a series of experiments at CERN [45,46]. The primary purpose of the new muon g — 2 experiment at Brookhaven National Laboratory is to improve the precision of the experiment by about a factor 20 and verify the presence of the electroweak effect which has been evaluated to two loop orders in the Standard Model. In this experiment, polarized muons from pion decays are captured in a storage ring with a uniform magnetic field and a weak-focusing electric quadrupole field. For a muon momentum of 3.09 GeV/c and 7 = 29.3 the muon spin motion is unaffected by the electric quadrupole field and the difference frequency uia is given by... 

The magnetic dipole interaction and the electric quadrupole interaction, in an atom with nuclear spin I, will cause a level with angular momentum J to split into different energy levels denoted by the total angular momentum F (F=I+J), according to the Casimir formula [11],... 

Mosch et al. [345] have measured the rotational state distribution of RbBr from Rb + HBr. Using an electrical quadrupole state selector, the RbBr has a rotational temperature of 750 100° K this would correspond to the conversion of the orbital angular momentum of the reactants entirely into RbBr rotational angular momentum. Herschbach and coworkers [346] found the rotational angular momentum of the products of Cs + HBr, K + HBr, Cs + HI and CS + CH3I to be polarized perpendicular to the initial relative velocity vector. For Cs + Br2 and Cs + CCI4 there is no discernible polarization [346]. 

The term in equation (4.54) represents the nuclear electric quadrupole interaction. We note that the nuclear spin vibration interaction, anticipated in equation (4.15), is actually identically zero. This is because the only vibrational mode for a diatomic molecule is that associated with bond stretching. Such motion does not generate any angular momentum and so does not produce a magnetic field with which the nuclear spin can interact. 

In part the interest arises from the fact that the spectroscopic state 87/2 is the same for the atom, 4f 6s as for the divalent ion, 4f . The former has been studied by atomic beam triple resonance (Sandars and Woodgate 1960, Evans et al. 1965), and the latter by ENDOR (Baker and Williams 1962) in Cap2, an environment with cubic symmetry. For a half-filled shell, with no orbital momentum and a spherical distribution of electron spin moment with zero density at the nucleus, both the magnetic dipole and electric quadrupole interactions should be zero. Experimentally, they are small compared with the values for other odd-proton lanthanide isotopes with comparable nuclear moments, for which the hyperfine... 

The definition of the moments becomes unambiguous by specifying the orientation of the nucleus. By convention, in the definition of the electric multipole moments the nuclear angular momentum is assumed to be aligned along the z axis. The intrinsic electric quadrupole moment Qo is then defined as... 

Besides the magnetic dipole moment, nuclei with spin higher than 1/2 also possess an electric quadrupole moment. In a semiclassical picture, the nuclear electric quadrupole moment informs about the deviation of nuclear charge distribution from spherical symmetry. Nuclei with spin 0 or 1/2 are therefore said to be spherical, with zero electric quadrupole moment. On the other hand, if the nuclear spin is higher than 1/2, the nuclei are not spherical, assuming cylindri-cally symmetrical shapes around the symmetry axis defined by the nuclear total angular momentum [17]. Within the subspace [/,m), the nuclear electric quadrupole moment operator is a traceless tensor operator of second rank, with Cartesian components written is terms of the nuclear spin [2] ... 

However, /-/ transitions are formally electric dipole forbidden by the Laporte selection rule, (a change in orbital angular momentum of 1 is required to accommodate the loss of photon spin upon absorption), although they are allowed by electric quadrupole, magnetic dipole and forced electric dipole mechanisms to some extent. Direct excitation of the Ln " ion is therefore not easily achieved, due to the low molar absorption coefficients associated with these transitions... 

That will be good enough for us, but there s much more to the story. The photon picture of radiation has its uses, but our version is not sufficient to explain all the properties of the radiation field Photons can have both spin and orbital angular momentum, and single-photon electric quadrupole transitions take place where the angular momentum changes by two units instead of one. 

This expression contains the vector cross-product, written in Cartesian notation, of the position of the electron r and its momentum p. Here is the antisymmetric tensor which equals — 1 for an odd permutations of the Cartesian axes x, y, and z, and +1 for an even permutation. Similarly, one must consider the electric quadrupole moment operator given in Cartesian tensor notation by... 

Principles and Characteristics An interaction that is never directly seen in liquid spectra but that, if present, always dominates solid-state spectra is quadrupole interaction. Nuclei with I > Vi have an electric quadrupole moment Q that is a measure of the deviation of the nuclear charge distribution from spherical symmetry. Nuclei with 7 = 0, Vi do not care about electric field gradients their charge distribution is spherical. Some 74% of all NMR-active nuclei have I > V2, as listed elsewhere [763]. The nuclear electric quadrupole moment, 2, of an 7 > 1 nucleus can interact with the electronic environment near that nucleus to affect the nuclear spin angular momentum energy levels, even in zero magnetic field. Quadrupole interactions can... 

Fig. 13 Definition of labels for transitions in a rotational band. On the left we show the case for a single rotational band, e.g., the ground state rotational band in an even-even nucleus. Two stretched electric quadrupole ( 2) transitions populate and depopulate the band member with angular momentum J. The transition energies Ey and the level energies E J) are indicated. On the right hand side we show the more frequent case of two strongly coupled bands. Stretched 2 transitions connect the members of each band, while the bands are connected by interband transitions of mixed Ml / 2 character. The branching ratios between the mixed interband transitions and the stretched intraband 2 transitions are sensitive to the g-factor of the band-head configuration...

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 ... 

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