TABLE I.

The toggling frame Hamiltonians for the magic echo pulse sequence depicted in Fig. 1. Here θ₁ = 2τω₁ is the flip angle of the first x RF burst as seen in Fig. 1 (time period C).

Time period	Duration	Toggling frame Hamiltonian
A	τ	$D_{12}(3I_1^z{I}_2^z-{\mathit{I}}_{\mathbf{1}}·{\mathit{I}}_{\mathbf{2}})$
B	tπ/2 ≡ π/(2ω1)	$3D_{12}[I_1^z{I}_2^z{\text{cos}}^2({\omega }_{1}t)+(I_1^z{I}_2^x+I_1^x{I}_2^z)\text{cos}({\omega }_{1}t)\text{sin}({\omega }_{1}t)+I_1^x{I}_2^x{\text{sin}}^2({\omega }_{1}t)]-D_{12}{\mathit{I}}_{\mathbf{1}}·{\mathit{I}}_{\mathbf{2}}$
C	2τ	$3D_{12}[I_1^x{I}_2^x{\text{cos}}^2({\omega }_{1}t)+(I_1^x{I}_2^y+I_1^y{I}_2^x)\text{cos}({\omega }_{1}t)\text{sin}({\omega }_{1}t)+I_1^y{I}_2^y{\text{sin}}^2({\omega }_{1}t)]-D_{12}{\mathit{I}}_{\mathbf{1}}·{\mathit{I}}_{\mathbf{2}}$
D	2τ	$3D_{12}[I_1^x{I}_2^x{\text{cos}}^2(-{\omega }_{1}t+{\theta }_1)+(I_1^x{I}_2^y+I_1^y{I}_2^x)\text{sin}(-{\omega }_{1}t+{\theta }_1)\text{cos}(-{\omega }_{1}t+{\theta }_1)+I_1^y{I}_2^y{\text{sin}}^2(-{\omega }_{1}t+{\theta }_1)]-D_{12}{\mathit{I}}_{\mathbf{1}}·{\mathit{I}}_{\mathbf{2}}$
E	tπ/2 ≡ π/(2ω1)	$3D_{12}[I_1^z{I}_2^z{\text{sin}}^2({\omega }_{1}t)+(I_1^z{I}_2^x+I_1^x{I}_2^z)\text{sin}({\omega }_{1}t)\text{cos}({\omega }_{1}t)+I_1^x{I}_2^x{\text{cos}}^2({\omega }_{1}t)]-D_{12}{\mathit{I}}_{\mathbf{1}}·{\mathit{I}}_{\mathbf{2}}$
F	τ	$D_{12}(3I_1^z{I}_2^z-{\mathit{I}}_{\mathbf{1}}·{\mathit{I}}_{\mathbf{2}})$