Second-rank tensor operator

In contrast, the second term in (4.6) comprises the full orientation dependence of the nuclear charge distribution in 2nd power. Interestingly, the expression has the appearance of an irreducible (3 x 3) second-rank tensor. Such tensors are particularly convenient for rotational transformations (as will be used later when nuclear spin operators are considered). The term here is called the nuclear quadrupole moment Q. Because of its inherent symmetry and the specific cylindrical charge distribution of nuclei, the quadrupole moment can be represented by a single scalar, Q (vide infra). 

The problem above can also be solved analytically using tensor methods—the preferred technique when higher accuracy is required. In general, any homogeneous deformation can be represented by a second-rank tensor that operates on any vector in the initial material and transforms it into a corresponding vector in the deformed material. For example, in the lattice deformation, each vector, Ffcc, in the initial f.c.c. structure is transformed into a corresponding vector in the b.c.t. structure, Vbct, by... 

It is worth mentioning that expressions (15.75) and (15.76), apart from terms that are scalar with respect to quasispin, also include terms that contain a second-rank tensor part in quasispin space. Substituting (15.75) and (15.76) into the expressions for the Casimir operators of appropriate groups (5.34), (5.29) and (5.33) yields... 

A general Cartesian second-rank tensor operator is represented by a 3><3 matrix. 

Symmetric, but reducible, second-rank tensor operators = ST 1) and... 

The operators so and ss are compound tensor operators of rank zero (scalars) composed of vector (first-rank tensor) operators and matrix (second-rank tensor) operators. We will make use of this tensorial structure when it comes to selection rules for the magnetic interaction Hamiltonians and symmetry relations between their matrix elements. Similar considerations apply to the molecular rotation and hyperfine splitting interaction... 

The first term is a tensor of rank zero involving only spin variables. It does not contribute to the multiplet splitting of an electronic state but yields only a (small) overall shift of the energy and is, henceforth, neglected. The operator 7 is a traceless (irreducible) second-rank tensor operator, the form of which in Cartesian components is... 

Table 11 Irreps of Singlet (S) and Triplet (T) Spin Functions, the Angular Momentum Operators (A and S), an Irreducible Second-Rank Tensor Operator 2, and the Position Operators St, 9,3C in C2v Symmetry...

If they are independent of space, they are constant and the magnetic field varies linearly with space. Because the magnetic field B is a vector with components Bx, By, and B, the magnetic-field gradient is a second-rank tensor with nine components. It can be written as the dyadic product of the gradient operator V and the magnetic field. 

At first order, it can be shown that only the symmetrized part of the interaction tensor contributes to the frequency shift. The majority of second-order contributions arise from large EFG, the EFG tensor being symmetric by definition. Thus, only symmetric second-rank tensors T can be considered, which can be decomposed into two contributions T = isol3 + AT with D3 the identity matrix. The first term is the isotropic part so = l/3Tr(T) that is invariant by any local symmetry operation. The second term is the anisotropic contribution AT, a symmetric second-rank traceless tensor, which depends then on five parameters the anisotropy 8 and the asymmetry parameter rj that measures the deviation from axial symmetry, and three angles to orient the principal axes system (PAS) in the crystal frame. The most common convention orders the eigenvalues of AT such that IAzzL and defines 8 — Xzz and rj — fzvv i )... 

The evaluation of the SSC matrix elements is analogous to the treatment of the SOC matrix elements with exception that the SSC Hamiltonian of (49) is a product of two irreducible second-rank tensors operators. Because SSC has nonvanishing contribution in the first order, it is usually sufficient to neglect contributions from states of different multiplicities. Application of the WET theorem to SSC Hamiltonian of (49) reads (S = S ) ... 

Making use of the properties of nuclear spin (angular momentum), a new spherical second-rank tensor operator can be constructed with the components [5]... 

Add the tensorial rank of each cofactor and subtract 2 for the dot operation. Scalars are zeroth-rank tensors and vectors are first-rank tensors. Since the del operator is a vector and convective momentum flux is a second-rank tensor, V pvv is a vector. The fcth component of V pvv is... 

Now any of the three types of deformation destroys the centre of symmetry of the liquid crystal. The strain tensor is therefore an axial second rank tensor which vanishes identically under a centro-symmetric operation. Since the free energy is a scalar, the components of also form an axial second rank tensor ... 

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. 

With Oq, we denote cartesian components of a vector operator > as well as components of a second-rank tensor operator Oa g, depending on the situation. Similarly, Oq, stands for second- Oap third- Oap-y and fourth-rank tensor operators Oap-ys-The superscripts T and TT are labels attached to the operators in order to associate them with their corresponding fields. In later chapters it will be convenient to express the perturbation operators as the sum over all electrons... 

The deformed charge distribution is generally axially symmetrical and this fact has an important consequence. It permits to characterize the charge distribution asymmetry by means of only one quantity, Q (called the quadrupolar moment), even if the quadrupolar operator, is a 3x3 matrix (in classical physics, a quadrupole is a second rank tensor). The definition of Q and the explicit form of are given in references 2 and 3. 

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, X is the. lA -dimensional vector consisting of K 3-dimensional vectors = AM s(Rs — Rs), where s = 1,2, 3,. .. K numbers the nuclei in the system of the local vibrations, Ms is the mass of 5th nucleus, (Rs — Rs) is the vector of deviation from the equilibrium position Rs, Ol f are the frequency tensors of second rank in the initial and final states. Let us introduce such unitary operators Sy that transformations 2 — Sy Y reduce the quadratic... 

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