Tensor operator electric quadrupole

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... 

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 rs. and 4. are the coordinate vectors of electrons i and / belonging to ions S and A, respectively / is the nuclear separation and if is the dielectric constant. The various multipolar terms appear from a power series expansion of the denominator. This expansion was expressed by Kushida (17) in terms of tensor operators. The leading terms are the electric dipole-dipole (EDD), dipole-quadrupole (EQD) and quadrupole-quadrupole (EQQ) interaction. These have radial dependence of/ -3,/ -4 and/ -5 respectively. 

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 ... 

In the optical activity arising from higher-order cross-terms, the effects are in most cases expected to be orientation-dependent. Pseudoscalar terms are the only ones which survive in random orientation (molecules in solution or liquid phase). At the same order of perturbation as El-Ml there is a product of the electric dipole and electric quadrupole transition operators (E1-E2). Since the latter product involves tensors of unequal rank, the result cannot be a pseudoscalar and this term would not, therefore, contribute in random orientation but can be significant for oriented systems with quadrupole-allowed transitions. The E1-E2 mechanism was developed by Buckingham and Dunn and recognized by Barron" as a potential contribution to the visible CD in oriented crystals containing the [Co(en)3] " ion. 

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... 

The second-rank tensor P,y (m) depends on the velocity dipole operator, while Mif(co) depends on the velocity dipole operator and on the magnetic dipole operator and finally T (co) on the velocity dipole operator and the velocity form of the electric quadrupole operator, respectively. Their mathematical expressions are reported and described in detail in Chapter 2. Once more, like we did for TPA, invoking the BO approximation and integrating over the electronic coordinates, the TPCD intensity between vibronic states can be written in terms of elements of electronic transition tensors Pej/it, rj, co), Me,f x, rj, co), and T yr(x, rf, co) between the vibrational states and Z5(/)) associated with the initial and final electronic states 0,) and 0/), respectively. 

The electric hyperfine structure of rotational levels is observed if at least one of the nuclei in the molecule has a spin quantum number / > 1, because the multipole expansion of the electrostatic interaction between the nuclei and electrons gives the quadrupole term as the next non-vanishing term after the monopole. This part can be written in the eoneept of spherical tensor operators [57Edm]... 

Because the mixed electric dipole-magnetic dipole polarizability involves the magnetic dipole operator, in approximate calculations G carries an origin dependence. Indeed, the individual tensor elements of G are origin dependent. The trace of G must be origin independent, since the optical rotation is an experimental observable. In non-isotropic media, contributions to the optical rotation tensor arise from the mixed electric dipole-electric quadrupole polarizability A,... 

Hamiltonian = matrix element of the Hamiltonian H I = nuclear spin I = nuclear spin operator /r( ), /m( ) = energy distributions of Mossbauer y-rays = Boltzmann constant k = wave vector L(E) = Lorentzian line M = mass of nucleus Ml = magnetic dipole transition m = spin projection onto the quantization axes = 1 — a — i/3 = the complex index of refraction p = vector of electric dipole moment P = probability of a nuclear transition = tensor of the electric quadrupole q = eZ = nuclear charge R = reflectivity = radius-vector of the pth proton = mean-square radi-S = electronic spin T = temperature v =... 

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... 

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. 

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... 

In this case, the perturbation term is e.g. proportional to the tensor component of the electric field gradient operator (43) times the nuclear quadrupole moment or is actually multiplied by a perturbation-strength parameter, Q, which is linearly proportional to the nuclear quadrupole moment. 

Here we have introduced Q, the tensor of second electric moments, which is related to the quadrupole moment of the electron distribution, and the scalar and vector product operators for two tensors, O and 0. The expansion may, of course, be carried to... 

Hyperfine Parameters From a quantum chemical point of view, the quantity required for the determination of the nuclear quadrupole-coupling tensor is the electric field gradient at the quadrupolar nucleus. This is a first-order property which can be computed as either the first derivative of the energy with respect to the nuclear quadrupole moment or the expectation value of the corresponding (one-electron) operator... 

In the length gauge, Eq. (2.122), the operator could be the electric dipole or quadrupole operator, defined in Appendix A. It depends on coordinates and momenta of the electrons but it is independent of time, whereas we assume that the time-dependent field. F. (t) does not depend on any electronic variables. The subscript p - again denotes components of a tensor of appropriate rank. On the other hand, in the velocity gauge, Eq. (2.125), the operator is equal to the total canonical... 

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. 

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