Isotropic filtering using polyhedral phase cycles: Application to singlet state NMR

Orientational Sampling Schemes Based on Four Dimensional Polytopes

“…We expect that these sets of rotations will be useful for the computation of orientational averages in a range of physical sciences, and in experimental procedures such as spherical tensor analysis in nuclear magnetic resonance [7][8][9]. In addition, we anticipate that where necessary, finer sampling of orientational space may be implemented by interpolating between the vertices of the polytopes, or by four-dimensional tiling and honeycomb schemes, such as those described by Coxeter [17].…”

“…Explicit tables of Euler angles are given, based on the vertices of the regular 4-polytopes. These results should be useful for workers in a wide range of physical sciences, especially magnetic resonance, where one such scheme is already in use [9,24].…”

“…For example, in the class of experiments known as spherical tensor analysis [7][8][9], the orientational space of the nuclear spins is sampled in order to derive the spherical tensor components of the quantum statistical operator describing the state of the nuclear spin ensemble. In all such experimental procedures, it is desirable that the orientation sampling scheme is as efficient as possible.…”

Singlet NMR methodology in two-spin-1/2 systems

“…The details of how this method works are described in the original publication [73]; its application for singlet order filtration has been reported [74] and therefore will not be discussed here any further. The general scheme for filtering using polyhedral phase cycles is shown in Fig.…”

“…An ensemble of spin systems each comprising N coupled spins-1/2, can only support spin order terms for which both the rank λ and the component index µ are less than or equal to N. This paper deals only with two-spin-1/2 systems where N = 2, so λ max = N = 2. Under these conditions the polyhedron of choice is the tetrahedron, with V = 4 and N = 12 [73,74], meaning that singlet order filtering via polyhedral phase cycling requires a minimum of 12 steps in the phase cycling. The four phases in Fig.…”