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Multipole NMR. VII: Bromine NMR quadrupolar echoes in crystalline KBr

Abstract Multipole NMR is used to calculate the responses during various pulse sequences for a solid system of spin - 1 2 nuclei subject to a strong do magnetic field and an inhomogeneous distribution of quadrupole and dipole-dipole interactions. The following sequences were treated: 90° X −τ 1 , 90° X −τ 1 −90° X −τ 2 90° X −τ 1 −90° Y −τ 2 , 90° X −τ 1 −90° X −τ 2 −90° X −τ 3 , 90° X −τ 1 −90° Y −τ 2 −90° Y −τ 3 , and 90° X −τ 1 −45° Y −τ 2 −45° Y −τ 3 . The calculations show that all of the 15 multipole polarizations (3 alignments, 6 single-, 4 double-, and 2 triple-quantum coherences) can be created in significant amounts by appropriate sequences. The calculations for each sequence have been verified by a detailed experimental study of the sequence responses for Br in a deformed single-crystal of KBr. Seven independent relaxation times were determined. The experiments provide the first evidence for octupole alignment in spin systems.

Multipole NMR. X. Multispin, multiquantum, multilinear operator bases

Abstract Reasons are given for the choice of the multipole basis which makes it particularly useful for spin dynamic calculations. A parallel development between spin basis states | IM 〉 and an operator basis | kq 〉〉 is given for pedagogical purposes. The direct and indirect methods in NMR and methods for calculating spin dynamics are compared and contrasted. The development is extended to multispin operator bases and related, in the case of two spin systems, to those of others.

Multipole NMR. IX. Polar graphical representation of nuclear spin polarizations

Abstract A graphical representation of the nuclear spin states created in NMR experiments is described which is also applicable to quadrupolar spins ( I > 1 2 ). The multipole polarizations are formally shown to be components of a polarization tensor with respect to the set of basis vectors formed from spherical harmonics. Linear combinations of polarizations have, like atomic orbitals, characteristic “shapes” which can be visualized by diagrams of polar graphs. These diagrams are used to illustrate the rotations produced by rf pulses and the couplings resulting from quadrupolar interactions, and to trace the polarization pathways of a spin-1 system during the quadrupolar solid-echo and Jeener-Broekaert sequences. Also discussed is the relationship between the thermodynamic and multipole approaches to NMR.

Multipole theory of soft pulses in nmr of quadrupolar solids

Abstract By making use of a multipole operator decomposition of the spin density matrix the spin dynamics of quadrupolar nuclei under the influence of soft RF pulses is studied. In particular, exact density matrix solutions for any arbitrary sequence of two soft pulses on an axially symmetric quadrupolar spin I = 3/2 are obtained analytically in terms of all usual experimental pulse parameters. The theory is shown to give good quantitative agreement with the observed cosine and derivative echoes of the FID for a system of I = 3/2 in a single crystal of NaNO 3. The theory then provides some extension of the multipole theory of multiple-hard-pulse sequences. For systems of arbitrary spin magnitudes, the limiting case in which the RF pulse dominates is analyzed and the spin density matrix is obtained perturbatively in the multipole basis. A simple description follows in which all multipole polarizations are rotated plus terms in which the tensor rank is mixed from κ to k ± 1 to first order in the quadrupole coupling strength. This result can be readily implemented to extend other treatments of higher spins.

Theory of pulses in nuclear quadrupole resonance spectroscopy: I. Physical picture

The effect of radiofrequency pulses in nuclear quadrupole resonance (NQR) spectroscopy is examined. The description is different from that needed in nuclear magnetic resonance (NMR) spectroscopy. In particular, both rotating and counter-rotating radiofrequency field components are retained. Pulses in NQR spectroscopy are selective and cause transitions between two pairs of levels (±M) → ±(M + 1), other transitions are not normally excited. The formulation of pulses is then described for any spin I by two 2 × 2 rotation matrices, one causing an M → M + 1 transition and the other causing a - (M + 1) → -M transition. Calculations on resonance for spins with an axially symmetric nuclear quadrupole for up to three pulses are presented for spins I = 1 and I = 3/2. The results agree with the work of Reddy, R. and Narasimhan, P. T., 1991, Molec. Phys., 72, 491, in the appropriate limits.

Analytical solutions for spin 7/2 line intensities in solid state NMR

Developments in computer algebra have prompted us to reconsider earlier, intractable problems. Here the density matrix of spin 7/2 subject to the first order quadrupolar interaction and excited by a RF pulse is calculated ignoring rapidly oscillating terms. The results are studied for a variety of conditions which scale as the ratio of the quadrupolar coupling constant to the RF pulse amplitude. The line intensities of the central and satellite transition are also investigated.

Symmetry-adapted bases and projection superoperators of multiquantum NMR under the groups S2[x]S2 and S4[x]S2

General techniques are presented for factorizing the multispin form of quantum-Liouville equation for spin evolution under scalar coupling. Projection superoperator formalisms are derived that allow partitioning of the spherical basis, |kgv〉〉, components of Liouville space under the group algebra of the inner direct product, Sn[x]Sn …, for multicluster spin systems. A special feature of weakly coupled spin systems allows labelling with respect to the spherical components, q, qA and qx, in addition to irreducible representations under the permutation group, Sn to provide multiple partitioning, denoted by nμq,qx. The importance of permutation symmetry with its related invariance hierarchy is stressed by presenting the discrete subdomain dimensionality structure associated with the [AMX]2 and [A]2[X]4 spin-12 systems under the groups S2 and S4 = D2f, respectively. The inclusion of the effects of finite pulse widths, applicable to pulse experiments involving weakly coupled heteronuclear [AX]n, spin systems, leads to the realizations of further dimensionality reductions and simplifications. The invariance properties of certain quadrupolar, [Q]n spin clusters under Sn are considered in the context of [AQ]2-type systems with general applications to the “particle” subspectral approach.

Composite pulses in nuclear quadrupole resonance

It is shown in this paper that composite pulses optimized for NMR situations can be used also for NQR experiments. Specifically identical composite pulses can be used in both cases and are most effective for single crystal NQR experiments. A number of composite pulses used for NMR are calculated exactly for the NQR case and shown to be as good as or better than existing pulses designed exclusively for NQR.

Multipole N.M.R.: XII. Theory of spin decoupling

Using a two-spin multipole formulation, spin decoupling is analysed analytically from arbitrary initial spin preparations. The rotating frame which is most convenient is ω ≡ 1/2(ω01 + ω02) rather than the commonly used Larmor frequency of the irradiated spin ω02. The importance of unobserved polarizations (bilinear operators) is stressed by identifying these terms as providing pathways to spin coupling which must be destroyed for decoupling. Appropriate linear combinations of the bilinear polarizations can be rotated by the perturbing field. Some general observations of the effects of periodic pulse sequences and amplitude modulation are included.

Theory of three-pulse sequences for systems of single spins of I = 52

Abstract The NMR response of a spin - 5 2 system to a variety of three-pulse sequences is examined theoretically for later use to interpret experiments of solid Kl. Analytical expressions are obtained for the contributions to the nuclear magnetic signal due to vector and tensor polarizations. Sequences containing up to three pulses are studied and some long-lived echoes identified and classified. Schemes are proposed for the indirect detection of higher-tensor-rank alignments.