Page 11 out of 81 total pages , Page 8 out of 12 pages in this chapter
There are several interaction of rank 2 that are important in the treatment of magnetic resonance. These are interactions which modify system energy levels depeding upon their orientation in 3-dimensional space. The concern herein are the irreducible rank 2 spatial tensor components of such interactions.
We will shortly concern ourselves with the mathematical representation of rank 2 interactions, in particular their description in terms of spatial and spin tensors. The spatial tensors will be cast in both Cartesian and spherical coordinates and we will switch between the two when convenient. The figure below relates the orientation angles theta and phi to the standard right handed coordinate system in all GAMMA treatments.
The internal structure of class IntRank2 contains the quantities listed in the following table (names shown are also internal).
The two values DELZZ and ETA are all that is required to specify the rank 2 interaction strength and may be used to represent the rank 2 spatial tensor. However, in GAMMA the value of DELZZ is factored out of the spatial tensor such that all rank two interactions (such as the rank 2 interaction) have the same spatial tensor scaling.
The two angles THETA and PHI indicate how the rank 2 interaction is aligned relative to the interaction principal axes (PAS). These are one in the same as the angles shown in Figure 19-2 when the Cartesian axes are those of the PAS with the origin vaguely being the center of the nucleus. These are intrinsically tied into the values in the array Asph.
There are five values in the complex vector Asph and these are irreducible spherical components of the rank 2 spatial tensor oriented at angle THETA down from the PAS z-axis and over angle PHI from the PAS x-axis. Note that these 5 values are not only orientation dependent, they are also ETA dependent. If either of the three the interaction values {ETA, THETA, PHI} are altered these components will all be reconstructed. The values in Asph will be scaled such that they are consistent with other rank 2 spatial tensors in GAMMA which are independent of the interaction type.
The vector of complex numbers relate to the spherical spatial tensor components via
We begin by examining the 3x3 array which represents a general (reducible) rank 2 Cartesian tensor. The form on the left (below) is a generic representation whereas the tensor is written as a sum over tensors of ranks 0 - 2 on the right.
If the rank 2 spatial tensor is symmetric, the anti-symmetric rank 1 terms are zero. If the rank 2 spatial tensor is traceless then the isotropic rank 1 term will be zero. If both are true then the spatial tensor is equivalent to the irreducible rank 2 tensor which is also both traceless and symmetric.
Any rank 2 spatial tensor can be specified in its principal axis system, the set of axes in which the irreducible rank 2 component is diagonal1.
Symmetric rank 2 spatial tensors are commonly specified in their principal axis system by the three components: the isotropic value 
, the anisotropy 
, and the asymmetry 
. These are given by
, 
In addition, a set of Euler angles 
is used to relate the interaction at an arbitrary orientation to the spatial tensor principle axes.
The 9 irreducible spherical components of a rank 2 spatial tensor, 
, are related to the Cartesian spatial components by the following general formulas2.
The subscript l spans the rank as l = [0, 2], and the subscript m = [-l, l]. The irreducible rank 2 spherical elements of the tensor, 
, in the principal axis system are obtained by placement of (0-5) into (0-6).
These will be the only terms if the tensor is symmetric and has no isotropic component (e.g. a dipolar interaction). Most interactions ignore the antisymmetric terms (l=1) so at most the above terms with 
suffice to describe everything. Herein we are concerned only with the irreducible rank 2 terms.
We can express the irreducible rank 2 spatial tensor components 
relative to any arbitrary axis system (AAS) by a rotation from the principal axes to the new axes via the formula3
where 
are the rank 
Wigner rotation matrix elements and 
the set of three Euler angles which relate the principal axes of the rank 2 spatial tensor to the arbitrary axes. For the treatment liquid isotropic systems these Euler angles are time dependent and averaged. For the treatment of solids they may be static (powder) or time dependent (MAS). Often only two angles suffice to orient the interaction relative to its PAS.
We can expand the components of the oriented spatial tensor in terms of the Wigner rotation elements as well as the reduced Wigner rotation elements 
which are given by
The reduced (rank 2) Wigner rotation matrix elements supplied by this function are given in the following figure4.
is the standard Euler angle which is equal to the spherical phi angle.
Other useful relationships concerning these elements are
Putting these into the formula for rotating the spatial tensor
Of the five components, two may be generated by symmetry. The remaining three are listed in the next equation.
For the m=0 component we can use the relationship 
To summarize the results of these rotations:
Since scaling is arbitrary on these tensors (i.e. the value of 
does not affect the tensor mathematics), we have choosen a standard scaling based on normalized speherical harmonics. In GAMMA we set
Comparison with the spatial tensors defined in GAMMA
Substitution into the previous table of equations produces the spatial tensor components that will be used in GAMMA.
For the sake of completeness we shall rewrite the Cartesian tensor components for the oriented rank 2 spatial tensor. These are used in the literature and may form components of equations found therein. We begin with the generalized relations between the Cartesian and irreducible spherical rank 2 tensor components6.
Generation of the Cartesian components thus involves the substitution of the previous formulae for the oriented spherical components into the above equations.
When the rank 2 interaction has alignment along its principal axes system virtually all of the rank 2 equations simplify. The following figure collects all of these for convenience.
Included are the general relationships between the (GAMMA scaled) Cartesian tensor components to the irreducible spherical components. They are valid when h is defined accordingly! If h is defined by the other common convention (|Azz| 
|Axx| |Ayy|) then the sign on the 
will change as will the sign on the Hamiltonian term multiplied by h.
It is often the case that multiple rotations on a particular tensor must be performed.We can express the irreducible rank 2 spatial tensor components 
relative to any arbitrary axis system (AAS) by a rotation from the principal axes to the new axes via the formula7
where 
are the rank 
Wigner rotation matrix elements and 
the set of three Euler angles which relate the principal axes of the rank 2 spatial tensor to the arbitrary axes. For the treatment liquid isotropic systems these Euler angles are time dependent and averaged. For the treatment of solids they may be static (powder) or time dependent (MAS). Often only two angles suffice to orient the interaction relative to its PAS.
We can expand the components of the oriented spatial tensor in terms of the Wigner rotation elements as well as the reduced Wigner rotation elements 
which are given by
Putting these into the formula for rotating the spatial tensor
The rank 2 principal axis system is set such that |dzz|2
See GAMMA Class Documentation on Spatial Tensors.3
This rotation takes the PAS into the oriented coordinate axes.4
See Brink and Satchler, page 24, TABLE 1.5
The scaling on both A2m is arbitrary, so GAMMA uses a scaling which independent of whatever the tensor is applied to (the interaction type). The {A2m} are scaled so that rotations by angles q & f produce spherical harmonics for a symmetric interaction (h = 0). For specific treatments, a scaling factor (interaction constant) will likely be required.6
See the GAMMA documentation on spatial tensors, class space_T.7
This rotation takes the PAS into the oriented coordinate axes.
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© 1996 Scott A. Smith, The NHMFL, and The Florida State University. All Rights Reserved. |
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