Principal-axes frame

The relationship between the components of the EFG tensor in its principal axes frame, and in the coordinate system defined in Fig. 28 is given by... 

The pertubation approach given in Sect. 3 yields the principal values and the direction cosines of the quadrupole tensor Q which writes in its principal axes frame ... 

Equation (10) allows the determination of the principal values of the diffusion tensor if the orientation of its principal axes frame is known. The latter information is included there in implicit form, via Eq. (11). The problem is that neither the principal axes nor principal values are known a priori and are to be determined simultaneously, as outlined below. 

Suppose we are given Cartesian coordinates a, and mass mi for each atom i. To convert the coordinates to principal axes frame of reference, first translate to center-of-mass coordinates... 

The orientational distribution function P cos of axially symmetric coupling tensors (t] — 0) in the laboratory frame can be read directly from the NMR spectrum S S2). Its expansion coefficients x/ can be transformed to those, of the orientational distribution function P(cos0) of the molecules in the sample fixed frame by using the known orientation angles 0 and 0 of the principal-axes frame in the molecule-fixed coordinate frame, and the orientation angle of ihe sample frame in the laboratory frame. By convention, the preferential axis n of the sample is parallel to the Zs-axis of the sample frame. 

Here A and A denote the orientation angles (a, of the magnetic field Bq in the principal axes frame of the interaction tensor (Fig. 3.1.2) during the evolution time t and the detection time (2, respectively. The quantity which characterizes the reorientation process is the combined probability density... 

Let us consider a monocrystalline sample that contains chemically identical spin-1 nuclei. In our physical model it will be assumed that the quadrupole principal axes for each of the spins have the same orientation. The quadrupole Hamiltonian in the quadrupole principal axes frame is given as... 

Fig. 6.7 Orientation of principal axes frames of the g, A(N), and A(Cu) tensors of the Cu(I)-NO moiety. The A(Cu) frame is rotated by the angle, p, about the common x principal axis, but the AjjCCu) principal axis is not necessarily parallel to the Cu—N bond. The y-z plane of the g tensor is spanned by the N—O bond and the symmetry axis of the 2ptiy orbital of the NO molecule. See Table 6.3 for ESR parameters and structural data. The figure is adapted from [31] with permission from the American Chemical Society...

As is well known, the 3x3 matrix Oy can be diagonalized by an appropriate orthogonal coordinate transformation (rotational transformation), provided it is a symmetric matrix generally it is considered to be symmetric because of its physical meaning. If the principal-axes frame of o, where o is expressed by a diagonal matrix, is transformed to the laboratory frame by a rotational transformation R(o, /3, y) which is defined by three Eulerian angles a, /3 and y, then the representations of o in both frames are related to each other by the equation = (5)... 

Here, Op and a are the representations of o in the principal-axes frame and the laboratory frame, respectively Op is defined by its three principal values o, O22, 0 3 as follows ... 

The rotational transformation R is the so-called Unitary and hence if R denotes the transpose of R, the inverse matrix R (RR = 1) is equal to R R = R, and furthermore detR = detR = 1 from the mathematical requirement of keeping the vector length the same in both frames. R comprises three simple rotations (1) a rotation Rz(a ) around the z-axis by an angle a in the principal-axes frame (2) subsequent rotation Ry(/3) around the y-axis by an angle /3 in the new frame and (3) a final rotation Rz(y) around the z-axis by an angle y in the last frame, as shown in Fig. 1. Then it is represented as the consecutive product of three rotational transformations and explicitly... 

Therefore, the experimentally observable chemical shift 033 becomes a function of the principal values as well as the Eulerian angles a and P that transform the principal-axes frame to the laboratory frame. As has been mentioned already, the chemical shift o ou, O22, 033) is defined for individual carbon atoms in a molecule depending on the molecular conformation and O33 depends on Eulerian angles a and p. Therefore, the lineshape is decided by the spatial alignment of the molecule to Bq during measurement, i.e. by the molecular alignment in the sample as well as by the alignment of the sample with respect to Bo. 

In powdered crystals, a particular molecular conformation may be considered to be stable but oriented at random in space. Let p(Q) dQ be the probability that the principal-axes frame of o is oriented in a solid angle between Q(or, j8, y) and 2 -I- dQ. Then, the absorption intensity I co) at a frequency co can be correlated with p( 2)d 2 as follows. 

When samples are available as single crystals, spectra corresponding to specific orientations of the paramagnetic center with respect to the external field can be measured separately. The orientation dependence of the spectrum can then be studied systematically and the principal axes frames of the A- and g-tensors can be related to the crystal frame. In polymer applications, samples are usually macroscopically isotropic, so that only the principal values of the interactions, and in favorable cases the relative orientations of their principal axes frames, can be obtained from spectral simulations. How these flames are related to the molecular geometry then needs to be... 

Difficulties arise from the determination of the realistic average molecular conformation and the average orientation of the local polarizability tensors with respect to the molecular principal axes frame (PAF of the... 

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