Electric quadrupole moment operator

The operator Zj zj xj + Ea Za zaxa that appears above is the z,x element of the electric quadrupole moment operator Qz x it is for this reason that this particular component is labeled E2 and denoted the electric quadrupole contribution. 

Show that (a rfc (-iV ) + r -iVk) >) = ii ab a rkr b) for two energy eigenfunctions o) and b), where Uab = Ea Eb in the units with Planck s constant = 27t, and electron mass=l. Note the relation to the electric quadrupole moment operator for an electronic system f dfrrq(f). Note the dyadic notation ff for the tensor quantity ... 

Our final remark about transitions in atoms concerns the matrix elements for quadrupole transitions. The foregoing selection rules are deduced on the assumption that the electric dipole operator belongs to the representation The electric quadrupole moment operator... 

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] ... 

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... 

Cartesian electric quadrupole moment operator. The electric dipole - magnetic dipole and electric dipole - electric quadrupole polarizabilities govern the optical rotation of chiral molecules. For samples of randomly oriented molecules, the contribution from the electric dipole - electric quadrupole polarizability averages to zero, and only the trace (the sum of the diagonal elements) of the electric dipole - magnetic dipole polarizability contributes to the optical rotation. The... 

Using the fact that the electric quadrupole moment operator is a symmetric second rank tensor and t.he magnetic dipole moment operator transforms as an axial vector, derive the selection rules for magnetic dipole and electric quadrupole radiation given in Table 7.1. 

The probability of a transition being induced by interaction with electromagnetic radiation is proportional to the square of the modulus of a matrix element of the form where the state function that describes the initial state transforms as F, that describing the final state transforms as Tk, and the operator (which depends on the type of transition being considered) transforms as F. The strongest transitions are the El transitions, which occur when Q is the electric dipole moment operator, — er. These transitions are therefore often called electric dipole transitions. The components of the electric dipole operator transform like x, y, and z. Next in importance are the Ml transitions, for which Q is the magnetic dipole operator, which transforms like Rx, Ry, Rz. The weakest transitions are the E2 transitions, which occur when Q is the electric quadrupole operator which, transforms like binary products of x, v, and z. 

The spin selection rule is a consequence of the fact that the electric dipole and quadrupole moment operators do not operate on spin. Integration over the spin variables then always yields zero if the spin functions of the two states 0 and are different, and an electronic transition is spin allowed only if the multiplicities of the two states involved are identical. As a result, singlet-triplet absorptions are practically inobservable in the absorption spectra of hydrocarbons, or for that matter, other organic compounds without heavy atoms. Singlet-triplet excitations are readily observed in electron energy loss spectroscopy (EELS), which obeys different selection rules (Kuppermann et al., 1979). 

Obviously, the 4f - 5 p crossing does not correspond to an electric dipole transition, but to an electric quadrupole moment. It has no clear-cut effect in photo-electron spectra of tungsten, rhenium and osmium, suggesting that the non-diagonal elements of the effective one-electron operator are smaller than 0.3 eV. It would be worthwhile to study volatile molecules such as WF6 or 0s04 under conditions of high resolution using either the continuous spectrum emitted by a synchrotron, or 132.3 eV photons from an yttrium anti-cathode (20, 21). 

The second term in Eq. 8.15 can be shown (3) to reduce to the operator ZjgjXj, which is one component of the electric quadrupole moment of the system of particles. The selection rules are different from those for magnetic dipoles because of the - - sign in the second term of Eq. 8.13. Clearly, the terms ZjXj, yjZj, and Xjyj transform like the direct product of two vector representations. These functions are usually given in the character tables and the deduction of the selection rules in a particular case is straightforward. 

In the previous section we have defined the tensor components aap, - a,p-y and Cap -ys of the electric dipole, dipole uadrupole and quadrupole-quadrupole polarizability tensors as derivatives of the energy E , ) in the presence of a field and field gradient, Eqs. (4.65) to (4.67), or alternatively as derivatives of the perturbation dependent electric dipole p , ) and quadrupole moment 0(5,f), Eqs. (4.46) to (4.48), see also Table B.l. Furthermore, we have seen in Sections 3.3 and 4.3 that the electronic contributions to the electric dipole and quadrupole moments can be expressed as expectation values of the electric dipole and quadrupole moment operators, j2 Ro) and Ro) for the electrons, respectively. Both definitions can be used to derive quantum mechanical expressions for the polarizabilities. 

Here Q / are the elements of the nuclear electric quadrupole moment tensor. Equation [4] is of course a classical expression. A quantum-mechanical expression for the quadrupole Hamiltonian Hq is obtained from Equation [4] by replacing the tensor elements Q / by the operators... 

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... 

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... 

The operators for the potential, the electric field, and the electric field gradient have the same symmetry, respectively, as those for the atomic charge, the dipole moment, and the quadrupole moment discussed in chapter 7. In analogy with the moments, only the spherical components on the density give a central contribution to the electrostatic potential, while the dipolar components are the sole central contributors to the electric field, and only quadrupolar components contribute to the electric field gradient in its traceless definition. 

A) with respect to the operation of inversion about the origin of the system. The electric dipole operator is antisymmetric (A) with respect to inversion at a point of symmetry. The electric quadrupole operator is inversion symmetric (S). A transition is allowed if the product function in the expression for transition moment is symmetric for electric dipole radiation and antisymmetric for electric quadrupole radiation. 

The origin with respect to which the electric quadrupole and magnetic dipole operators are defined is indicated by the superscript. jiPp is the /3 component of the velocity operator. The connection between the quadrupole moment referred to or - for example the centre of nuclear masses - and the EQC is... 

In a rotating molecule containing one quadrupolar nucleus there is an interaction between the angular momentum J of the molecule and the nuclear spin momentum I. The operator of this interaction can be written as a scalar product of two irreducible tensor operators of second rank. The first tensor operator describes the nuclear quadrupole moment and the second describes the electrical field gradient at the position of the nucleus under investigation. 

Here (a)/ are the components of the operator of the quadrupole moment of the molecule in the LSC. Usually in the measurement of the difference of the refraction indices (195) the contribution to the Kerr effect can be separated (Buckingham and Disch, 1968). Therefore in Eq. (196) one can neglect the contribution of the interaction of the dipole moment with the electric field and use the following Hamiltonian instead of H(%) ... 

---