Zero-rank tensor

For cubic crystals, which iaclude sUicon, properties described by other than a zero- or a second-rank tensor are anisotropic (17). Thus, ia principle, whether or not a particular property is anisotropic can be predicted. There are some properties, however, for which the tensor rank is not known. In addition, ia very thin crystal sections, the crystal may have two-dimensional characteristics and exhibit a different symmetry from the bulk, three-dimensional crystal (18). Table 4 is a listing of various isotropic and anisotropic sUicon properties. Table 5 gives values for the more common physical properties and for some of the thermodynamic properties. Figure 5 shows some thermal properties. 

An important purpose of tensor analysis is to describe any physical or geometrical quantity in a form that remains invariant under a change of coordinate system. The simplest type of invariant is a scalar. The square of the line element ds of a space is an example of a scalar, or a tensor of rank zero. 

S= kX, provided they point in the same direction. If not, the quotient S/X is of more complicated form. Like the quotient of two integers, which is not always another integer, the quotient of two non-aligned vectors is not necessarily a vector itself, but something else, called a tensor. The scalar k for co-aligned vectors is a tensor of rank zero. A tensor of the first rank has three components, e.g. T = ayTj and is equivalent to a vector. A second rank tensor has the form of a square matrix with nine elements in three dimensions e.g. Tij = Yjk,iaikajiTu, and so forth. 

We are mainly interested in compound tensor operators of rank zero (i.e., scalar operators such as the Hamiltonian). To form a scalar from two tensor operators 0 and /l, their ranks k and j have to be equal. Further, the +q component of lk> has to be combined with the -q component of 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... 

Equation (3.141) consists of three sums of the products of tensors that are scalars with rank zero 2... 

Scalars are specified by a single numerical value. Vectors come with directions as well as numerical values. In a three-dimensional space, a tensor of rank n is determined by 3 elements. A scalar is a tensor with the rank zero, hence 3° = 1. A vector is tensor with the rank 1, hence 31 = 3. For a tensor with n = 2, we have 32 = 9 elements. 

In the very same way as the Bom-Oppenheimer approximation allows the definition of a potential energy surface for a Van der Waals molecule, it enables, too, the conceit of an interaction tensor field. This is a field dependent on the relative coordinates of the monomers and transforming as a tensor under rotation of the complex as a ole. (The potential energy surface is an example of a rank zero interaction tensor field). In the case of tensor fields it is also convenient to base the theory on irreducible tensors and to use an e7q>ansion in terms of a complete set of functions of the five angular coordinates describing a Van der Waals dimer. 

Some operators, such as the interelectronic electrostatic interaction e2/ nj, are obviously scalar quantities. Others are scalar products of two tensorial operators. A tensor of rank zero is a scalar. A tensor of rank one is a vector. There are several ways of combining two vector operators the scalar product... 

Owing to one zero element, all 9/-symbols collapse into 6/-symbols. The tensor rank k = 0 implies a factor s so that the isotropic exchange cannot mix states of different total spin. The simplified reduced matrix elements are collected in Table 11.17. These formulae can be simplified further by substituting the relevant 6/-symbols (Appendix 3). 

The quantities Oy are the components of a second-rank tensor (two subscripts) [second-rank tensor relate the components of two vectors. Vectors may be considered to be first-rank tensors (one subscript) and scalars to be tensors of rank zero. 

The mixed terms in ((Vi Vi))e=o vanish for this case, because jo is a tensor of rank zero in spin space, while m is of rank one. Note that (Vi)o = 0. 

All atoms and molecules can be polarized by an electric field. The polarization (induced dipole of a unit volume) is P = aE where a is molecular polarizability. For spherically symmetric atoms or molecules (like C60 fullerenes) the polarizability is a scalar quantity (tensor of zero rank) and P E. In general case of lath-like molecules, QLij is a second rank tensor (9 components) and Py = a,yE/. By a proper choice of the reference frame the tensor can be diagonalized... 

The essential feature of Ohm s law is that J is directly proportional to the applied field E and (J, being a property of the material, is independent of the field. Note that J and E are vector quantities while cr is a scalar (tensor of rank zero) for an isotropic media however, it will be a tensor of rank 2 for an anisotropic material such as a single crystal. 

The fact that the same function may be represented equally well in two different ways, as in (10.2.8) and (10.2.9), is an example of invariance under a group of transformations. An invariant is a tensor of rank zero. Thus, using F to denote the function constructed from the barred quantities in (10.2.6) and (10.2.10),... 

The rank-zero portion of the dipolar interaction is the PCS 6. Its full expression can be obtained expanding back the ITo.o term as a product of two second-rank tensors according to the inverse of (31) ... 

Consider the following examples for illustration A vector a in n-dimensional space is described completely by its n components a,. It may therefore be seen as a one-index quantity or a tensor of rank one. A matrix A has components A,/ (two indices) and is a rank two tensor. A tensor of rank three has n components, and its components have three indices, T, and so on. As a special case, scalars have only = 1 component and are tensors of rank zero. 

Determination of the fourth-rank tensor term F. 2 remains. Basically, F.,2 cannot be found from any uniaxial test in the principal material directions. Instead, a biaxial test must be used. This fact should not be surprising because F-,2 is the coefficient of the product of a. and 02 in the failure criterion. Equation (2.140). Thus, for example, we can impose a state of biaxial tension described by a, = C2 = c and all other stresses are zero. Accordingly, from Equation (2.140),... 

Second rank (bilinear after pseudospin) zero-field splitting tensor Dap (i.e. the conventional D tensor), its main values and the main anisotropy axes (Xa,Ya,Za). 

D is the zero-field splitting tensor, a traceless, rank-two tensorial quantity. The ZFS tensor is a property of a molecule or a paramagnetic complex, with its origin in the mixing of the electrostatic and spin-orbit interactions (80). In addition, the dipole dipole interaction between individual electron spins can contribute to the ZFS (81), but this contribution is believed to be unimportant... 

We have considered scalar, vector, and matrix molecular properties. A scalar is a zero-dimensional array a vector is a one-dimensional array a matrix is a two-dimensional array. In general, an 5-dimensional array is called a tensor of rank (or order) s a tensor of order s has ns components, where n is the number of dimensions of the coordinate system (usually 3). Thus the dipole moment is a first-order tensor with 31 = 3 components the polarizability is a second-order tensor with 32 = 9 components. The molecular first hyperpolarizability (which we will not define) is a third-order tensor. 

In practice, the transformation of any operator to irreducible form means in atomic spectroscopy that we employ the spherical coordinate system (Fig. 5.1), present all quantities in the form of tensors of corresponding ranks (scalar is a zero rank tensor, vector is a tensor of the first rank, etc.) and further on express them, depending on the particular form of the operator, in terms of various functions of radial variable, the angular momentum operator L(1), spherical functions (2.13), as well as the Clebsch-Gordan and 3n -coefficients. Below we shall illustrate this procedure by the examples of operators (1.16) and (2.1). Formulas (1.15), (1.18)—(1.22) present concrete expressions for each term of Eq. (1.16). It is convenient to divide all operators (1.15), (1.18)—(1.22) into two groups. The first group is composed of one-electron operators (1.18), the first two... 

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