Tensors traceless

If the liquid crystalline molecules are not cylindrically symmetrical, S must be expressed by a tensor (traceless tensor). The nematic liquid crystal state (a uniaxial symmetrical system) is the main concern of the book. For such a phase, the order parameter can be denoted by the scalar in Equation 1.11. 

Dipolar D / 10 -10 Through space spin-spin interaction, axially symmetric traceless tensor... 

The strong variation on B and reflects the fact that the EFG is a traceless tensor with positive and negative components. For > 0 and q = 0, the shift Eq is... 

If D is taken as a traceless tensor, Tr(/)) = Da = 0, there remain only two independent components for D (neglecting the three Euler angles for orientation in a general coordinate system). Usually, these are the parameters D and E, for the axial and rhombic contribution to the ZFS ... 

The values of the rhombicity parameters are conventionally limited to the range 0 < EjD < 1/3 without loss of generality. This corresponds to the choice of a proper coordinate system, for which /)zz (in absolute values) is the largest component of the D tensor, and /) is smaller than Dyy. Any value of rhombicity outside the proper interval, obtained from a simulation for instance, can be projected back to 0 < EID < 1/3 by appropriate 90°-rotations of the reference frame, that is, by permutations of the diagonal elements of D. To this end, the set of nonconventional parameters D and EID has to be converted to the components of a traceless 3x3 tensor D using the relationships... 

However, we have previously seen in Chapter 5 that the number of elements in the zero-field tensor can be reduced to two (Equation 5.25) by making D traceless, and so the spin Hamiltonian can be written as... 

In the response function terminology [47] the i, j component of the frequency-dependent dipole polarizability tensor — w) (or the ij, kl component of the traceless quadrupole polarizability tensor is defined through... 

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... 

Expressions (7.2a) and (7.2b) follow from the following general expression for the /th-rank traceless tensor elements ... 

Expressions for the other elements of the traceless quadrupole tensor are obtained by simple permutation of the indices. 

Electric field gradient Traceless Second-rank tensor -3 0... 

The EFG tensor elements can be obtained by differentiation of the operator in expression (8.5) for Ea to each of the three directions / . This procedure removes the spherical component, which does not affect the electric field, and yields the traceless result... 

Both the components of E and the elements of the electric field gradient as given by Eqs. (8.30) and (8.32) are with respect to the reciprocal-lattice coordinate system. A transformation is required if the values in the direct-space coordinate systems are needed. To obtain the elements of the traceless V tensor, the quantity — (47t/3)pe(r) = — (47r/3K) F(H) exp ( — 27tiHT) must be subtracted from each of the diagonal elements VEU. 

Expressions (8.43) for the elements of the traceless quadrupole tensor can be written in a slightly different form as... 

For atoms in sites of low symmetry, the EFG tensor must be diagonalized to obtain its principal components V y. Since the tensor as defined by Eq. (8.9) is traceless, two values will define all three principal elements. They are commonly chosen as V 33, the principal component with the largest magnitude, and the asymmetry parameter t] = V 22 — V n / V 33, which has the range 0 < q < 1. 

The hyperfine tensor can be used to estimate the spin density on the oxygen atoms. Assuming that the unpaired electron is in an axially symmetrical n g orbital, the axial hyperfine tensor can be separated into an isotropic part (aiso) and an anisotropic traceless tensor in the form... 

When a nucleus with spin I is placed in a static magnetic field, the energy splits into 21 + 1 equally spaced levels, and a 2J-fold degenerate resonance line can be observed in an NMR experiment (16). Nuclei with spin I > 1/2 possess an electric quadrupole moment Q which may interact with the gradient of the electric crystal field at the site of the nucleus. This field gradient is a traceless tensor (17). 

The existence of molecules often creates permanent intramolecular optical anisotropy. The optical anisotropy of the liquid is then due to fluctuations in the orientations of the molecules or molecular subunits. If we assign a symmetric traceless anisotropy tensor a to each molecule or molecular subunit in the scattering volume, then the relaxation function for collective optical anisotropy fluctuations can be expressed as... 

Nuclei with spin I > j possess a quadrupole moment eQ and may interact with electric field gradients (EFG) present in the solid. The EFG is described by the traceless symmetrical tensor (72)... 

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