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Journal ArticleDOI

Fictitious spin 1/2 operator formalism for multiple quantum NMR

15 Jun 1978-Journal of Chemical Physics (American Institute of Physics)-Vol. 68, Iss: 12, pp 5518-5527
TL;DR: In this article, a formalism is presented that describes the time behavior of the spin density matrix of a nuclear spin system with arbitrary spin in terms of fictitious spin −(1/2) operators.
Abstract: A formalism is presented that describes the time behavior of the spin density matrix of a nuclear spin system with arbitrary spin in terms of fictitious spin −(1/2) operators. This formalism is an extension of that used earlier for nuclei with spin I=1. For a spin system with n eigenstates we define for every pair of eigenstates ‖i〉 and ‖j〉 three operators Ii−jp, with p=x, y, and z, according to the three 2×2 Pauli matrices σx, σy, and σz. These operators together constitute a complete set of n2−1 independent Hermitian operators, and we can write the n×n density matrix and the spin Hamiltonian of the system in terms of the Ii−jp operators. The commutation relations among the operators make it possible in many cases to solve the equation of motion of the density matrix analytically. Three examples of the use of the Ii−jp operators are presented. Firstly a system of noninteracting spins with I=1 is considered. The Ii−jp operators for this case are compared with the Iq,k operator defined earlier. The cw sign...
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TL;DR: In this article, the authors propose a method for analyzing experiences en impulsions: modeles vectoriels classiques and semi-classiques, approches par operateur densite and operateur produit.

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TL;DR: NMR pulse experiments are described in terms of pathways through various orders of coherence through the systematic design of phase cycles that select desirable coherence-transfer pathways.

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TL;DR: In this paper, a new method of performing chemical shift correlation spectroscopy in solids with magic angle spinning (MAS) is presented. But the method is not suitable for the case of alanine.
Abstract: We present a new method of performing chemical shift correlation spectroscopy in solids with magic angle spinning (MAS). Its key feature is a longitudinal mixing period of π pulses that recouples the dipolar interaction. We discuss experimental results for triply‐13C‐labeled alanine and a theory combining MAS and π pulses.

619 citations

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TL;DR: In this paper, the generation of nuclear magnetic double quantum echoes and their detection by two-dimensional Fourier transform spectroscopy was reported for partially aligned one-deuteron systems in a nematic liquid crystal.

393 citations

References
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Journal ArticleDOI
TL;DR: In this article, the possibilities for the extension of spectroscopy to two dimensions are discussed, including the elucidation of energy level diagrams, the observation of multiple quantum transitions, and the recording of high-resolution spectra in inhomogenous magnetic fields.
Abstract: The possibilities for the extension of spectroscopy to two dimensions are discussed. Applications to nuclear magnetic resonance are described. The basic theory of two‐dimensional spectroscopy is developed. Numerous possible applications are mentioned and some of them treated in detail, including the elucidation of energy level diagrams, the observation of multiple quantum transitions, and the recording of high‐resolution spectra in inhomogenous magnetic fields. Experimental results are presented for some simple spin systems.

2,968 citations

Book
01 Jan 1974

848 citations

Journal ArticleDOI
TL;DR: In this article, a discussion is given of the effects of a pair of pulses of rf magnetic field (at the NMR frequency) on the Zeeman and dipolar energies of the spin system in a solid.
Abstract: A discussion is given of the effects of a pair of pulses of rf magnetic field (at the NMR frequency) on the Zeeman and dipolar energies of the spin system in a solid. Zeeman order is partly transformed into dipolar order when the two pulses have different rf phases. A theoretical calculation provides a very simple relation between the efficiency of this transfer of energy (as a function of the characteristics of the pulses and their time separation) and the magnitude and shape of the dipolar component of the free-precession signal. This prediction has been verified quantitatively in the case of the ${\mathrm{F}}^{19}$ spins in a Ca${\mathrm{F}}_{2}$ crystal with the large magnetic field in a [100] direction. A maximum efficiency of transfer of Zeeman energy into dipolar energy of 56% has been obtained with a $\frac{\ensuremath{\pi}}{2}$ phase shift between a first $\frac{\ensuremath{\pi}}{2}$ pulse and a second $\frac{\ensuremath{\pi}}{4}$ pulse, separated by a time of the order of ${T}_{2}$. (An ideal adiabatic demagnetization in the rotating frame would have an efficiency of 100%.) The experimental results also indicate that the ${\mathrm{F}}^{19}$ spin system reaches internal quasiequilibrium in a time of the order of a few ${T}_{2}$ after the pulse pair. It is also shown that the use of a pair of rf pulses (both of the same rf phase this time) leads to a new method for measuring the complete shape of free-precession signals, which avoids the usual difficulties due to the finite recovery time of the observation circuit.

531 citations

Journal ArticleDOI
TL;DR: In this article, an alternative definition of single transition operators is given for the description of selective excitation and detection experiments in multilevel spin systems, which has the virtues of a simple physical interpretation and easy application to arbitrarily complicated systems.
Abstract: An alternative definition of single transition operators is given for the description of selective excitation and detection experiments in multilevel spin systems. This definition has the virtues of a simple physical interpretation and easy application to arbitrarily complicated systems. Some applications to the excitation and detection of multiple quantum transitions in spin 1 and spin 3/2 systems as well as in coupled spin systems are described.

352 citations

Journal ArticleDOI
Abstract: An operator formalism is presented which conveniently treats the interaction of a spin‐1 nucleus with a weak radio frequency field. The Hamiltonian in the rotating frame is H=−Δω Iz−ω1Ix+(1/3) ωQ[3 I2z−I (I+1)], where Δω is the resonance offset (Δω=ω0−ω), ω1 is the intensity of the rf field, and ωQ is the quadrupolar splitting. Nine fictitious spin−1/2 operators, Ip,i where p=x,y,z and i=1,2,3, are defined where p refers to the transition between two of the levels and i the Cartesian component. The operators, which are the generators of the group SU (3), satisfy spin‐1/2 commutation relations [Ip,j, Ip,k]=i Ip,l, where j,k,l=1,2,3 or cyclic permutation. Thus each p defines a three‐dimensional space termed p space. For irradiation near one of the quadrupolar satellites, for example, Δω=ωQ+δω with δω, ω1≪ωQ, it is shown that the effective Hamiltonian can be written H?−δωIx,3−√2 ω1Ix,1, i.e., a fictitious spin‐1/2 Hamiltonian in x space with effective magnetogyric ratio γ along the 3 (resonance offset) axis and √2 γ along the 1 (rf field) axis. For irradiation near the center we can effect double quantum transitions between m=±1. The formalism allows us to write the effective operators for these transitions. For example, if we take Δω=δω again with δω, ω1≪ωQ we find the effective double quantum (DQ) Hamiltonian H?−2 δω Iz,1−(ω21/ωQ) Iz,3. Thus the z space is referred to as the double quantum frame with effective magnetogyric ratio 2γ along the 1 (resonance offset) axis and (ω1/ωQ) γ along the 3 (rf field) axis. The limiting expressions are compared with exact calculations for arbitrary ω1 done by high speed computer. The theory is applied to various cases of irradiation including our previously reported technique of Fourier transform double quantum NMR. Various pulse sequences for preparing, storing, and maintaining the evolution of double quantum coherence are analyzed for single crystal and polycrystalline samples. Finally, the effects of rf phase on the double quantum phase are presented briefly, and the possibility of double quantum spin locking is analyzed.

344 citations