Principal axes system

In the principal axes system, Eq. (130) may be written in component notation as ... 

When inserting into (4.5), the term ZeR will be multiplied with the elements of the electric field gradient tensor V. Fortunately, the procedure can be restricted to diagonal elements Vu, because V is symmetric and, consequently, a principal axes system exists in which the nondiagonal elements vanish, = 0. The diagonal elements can be determined by using Poisson s differential equation for the electronic potential at point r = 0 with charge density (0), AV = Anp, which yields... 

The trace vanishes because only p- and /-electrons contribute to the EFG, which have zero probability of presence at r = 0 (i.e. Laplace s equation applies as opposed to Poisson s equation, because the nucleus is external to the EFG-generating part of the electronic charge distribution). As the EFG tensor is symmetric, it can be diagonalized by rotation to a principal axes system (PAS) for which the off-diagonal elements vanish, = 0. By convention, the principal axes are chosen such that... 

Three more parameters are implicitly included which are the Euler angles that describe the orientation of the principal axes system. 

The off-diagonal elements are zero for this arrangement and Vxx = Vyy = — l/214z- As expected, the adopted coordinate system is a principal axes system (PAS) of the EFG, and the asymmetry parameter rj is zero. 

This relation is only valid for a crystal with isotropic /-factor. The effect of crystal anisotropy will be treated in Sect. 4.6.2. The function h(6) describes the probability of finding an angle 6 between the direction of the z-axis and the y-ray propagation. In a powder sample, there is a random distribution of the principal axes system of the EFG, and with h 6) = 1, we expect the intensity ratio to be I2J li = I, that is, an asymmetric Mossbauer spectrum. In this case, it is not possible to determine the sign of the quadmpole coupling constant eQV. For a single crystal, where h ) = — 6o) 5 delta-function), the intensity ratio takes the form... 

Fig. 7.78 Linear relation of the quadmpole splitting A q = ( jl)eqQ (1 + j /3)l/2 and the isomer shift b for aurous (a) and auric (b) compounds. Also included is a correlation with the relative change in electron density at the gold nucleus, Ali/r(o)P, as derived from Dirac-Fock atomic structure calculations for several electron configurations of gold. An approximate scale of the EFG (in the principal axes system) is given on the right-hand ordinate (from [341])...

In (3.1) not all tensors are necessarily coaxial or diagonal. If the principal axes system of the g tensor is chosen as the molecular coordinate system eM, g has diagonal form. The laboratory frame eL is then related to eM by the rotation matrix R according to... 

Solution. First transform the problem to the principal axes system. The eigenvalue equation for D is... 

Therefore, the diffusivity tensor in the principal axes system is... 

The electric field gradient, EFG, at the site of the nucleus can also be described by a second order tensor. The components of the EFG tensor in the principal axes system (x,y,z) are... 

For application, Eq. (II.4) has to be considerably simplified. The simplest model is the assumption of point charges. We take the crystal lattice as composed from the point charges nte, where e is the elementary charge. The index i, o < i k, distinguishes the different kinds of charged points (particles) within the lattice. Assuming the system of crystal axes already transformed to the principal axes system of the tensor, we calculate the coupling constant from... 

The gradient of the electric field is the second derivative of the electrostatic potential, and as such, it obeys certain symmetries The EFG is a symmetric tensor with zero trace. This mean that it can be represented as a physical object An ellipsoid where off-diagonal elements represents reorientation of the principal axes system (See Figure 10). 

The coupling tensors Q, D, o, and J are tensors P of rank 2 [Hael, Mehl, Spil]. For each of the corresponding interactions, A, the tensor, can be separated into an isotropic part Pf, an antisymmetric part P[ and a traceless symmetric part P . For simplicity of notation, the index A is not carried along in the next six equations. In the principal axes system XYZ of the symmetric part of the coupling tensor, the generic coupling tensor P = Pij, where i,j = X, Y, Z, is written as... 

The two Euler angles and are the polar angles which specify the orientation of the magnetic field Bq in the principal axes system of the coupling tensor (Fig. 3.1.2). 

With the help of (3.2.1), it is readily seen that co2q does not depend on the orientation angles a and of the principal axes system. Therefore, the transition corresponds to a narrow resonance which can be favourably exploited for space encoding in imaging. 

The orientational distribution fimction P (cos 0) enters the shape of the wideline spectrum 5(f2) in a slightly hidden way. The angular dependence of the resonance frequency is given by (3.1.23) via the orientation of the magnetic field in the principal axes system XYZ of the coupling tensor (cf. Fig. 3.1.2), while the orientational distribution function specifies the distribution of the preferential direction n in a molecule-fixed coordinate frame (Fig. 3.2.2(a)). Figure 3.2.3 shows the relationship between the different coordinate frames and the definition of the relative orientation angles. 

Fig. 3.2.3 Relationships between coordinate systems for the description of molecular order. The orientation of the laboratory coordinate system in the principal axes system of the coupling tensor determines the angular dependence of the resonance frequency. The orientations of the preferential sample direction in a molecule-fixed coordinate frame determines the orientational distribution function.

According to Rudolph [19], the coordinates of atom k in the principal axes system of the substituted molecule can be obtained by changing the sign of Am and interchanging the primed and unprimed quantities in equations (14-23). The complete transformation between = (xkyk and r = (x y z/ ) can be described by... 

Equations (24)-(26) can be used to transform any vector between the two principal axes systems. Except for die signs of the components of rjt and rj, R and t are completely defined by the planar moments P and P of the parent and the daughter isotopomers, respectively. 

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 )... 

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