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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=−$Δ$$ω$ Iz−$ω$1Ix+(1/3) $ω$Q[3 I2z−I (I+1)], where $Δ$$ω$ is the resonance offset ($Δ$$ω$=$ω$0−$ω$), $ω$1 is the intensity of the rf field, and $ω$Q is the quadrupolar splitting. Nine fictitious spin−1/2 operators, Ip,i where p=x,y,z and i=1,2,3, are defined where p refers to the transition between two of the levels and i the Cartesian component. The operators, which are the generators of the group SU (3), satisfy spin‐1/2 commutation relations [Ip,j, Ip,k]=i Ip,l, where j,k,l=1,2,3 or cyclic permutation. Thus each p defines a three‐dimensional space termed p space. For irradiation near one of the quadrupolar satellites, for example, $Δ$$ω$=$ω$Q+$δ$$ω$ with $δ$$ω$, $ω$1≪$ω$Q, it is shown that the effective Hamiltonian can be written H?−$δ$$ω$Ix,3−√2 $ω$1Ix,1, i.e., a fictitious spin‐1/2 Hamiltonian in x space with effective magnetogyric ratio $γ$ along the 3 (resonance offset) axis and √2 $γ$ along the 1 (rf field) axis. For irradiation near the center we can effect double quantum transitions between m=±1. The formalism allows us to write the effective operators for these transitions. For example, if we take $Δ$$ω$=$δ$$ω$ again with $δ$$ω$, $ω$1≪$ω$Q we find the effective double quantum (DQ) Hamiltonian H?−2 $δ$$ω$ Iz,1−($ω$21/$ω$Q) Iz,3. Thus the z space is referred to as the double quantum frame with effective magnetogyric ratio 2$γ$ along the 1 (resonance offset) axis and ($ω$1/$ω$Q) $γ$ along the 3 (rf field) axis. The limiting expressions are compared with exact calculations for arbitrary $ω$1 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.

@article{Vega1977, title = {Operator formalism for double quantum {NMR}}, volume = {66}, issn = {0021-9606}, url = {https://doi.org/10.1063/1.433884}, doi = {10.1063/1.433884}, 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=−\$Δ\$\$ω\$ Iz−\$ω\$1Ix+(1/3) \$ω\$Q[3 I2z−I (I+1)], where \$Δ\$\$ω\$ is the resonance offset (\$Δ\$\$ω\$=\$ω\$0−\$ω\$), \$ω\$1 is the intensity of the rf field, and \$ω\$Q 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, \$Δ\$\$ω\$=\$ω\$Q+\$δ\$\$ω\$ with \$δ\$\$ω\$, \$ω\$1≪\$ω\$Q, it is shown that the effective Hamiltonian can be written H?−\$δ\$\$ω\$Ix,3−√2 \$ω\$1Ix,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 \$Δ\$\$ω\$=\$δ\$\$ω\$ again with \$δ\$\$ω\$, \$ω\$1≪\$ω\$Q we find the effective double quantum (DQ) Hamiltonian H?−2 \$δ\$\$ω\$ Iz,1−(\$ω\$21/\$ω\$Q) Iz,3. Thus the z space is referred to as the double quantum frame with effective magnetogyric ratio 2\$γ\$ along the 1 (resonance offset) axis and (\$ω\$1/\$ω\$Q) \$γ\$ along the 3 (rf field) axis. The limiting expressions are compared with exact calculations for arbitrary \$ω\$1 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.}, number = {12}, journal = {The Journal of Chemical Physics}, author = {Vega, S and Pines, A}, month = jun, year = {1977}, note = {ISBN: 0021-9606}, pages = {5624--5644}, }

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