This can be described by the lowering operator I=B><a, the coherent transition from the B to a state by the raising operator I=a><Bl(coherence order +1) The real Cartesian operators Ix and ly correspond to mixtures of both coherence orders, +l, although they are more useful for directly corresponding to the observable x and y components of the magnetization. Their relationship with the complex I operators is simple I=Ix+ lly sing operator Ix=2(I++n) T=Ix"ly lowering operator ly=-h2(+-n) I=/21+lz polarisation operator(a) h(2-1P pP=1h21-L2 polarisation operator(B) The effect of r f pulses(here: an x pulse with flip angle (p)on single-element operators is as follows +1+/cos2((/2)+1-/+sin2(p/2)(+/-iL, sin()) SIPCOS2{q/2}+ Iasin2{(/2}+(1/2)sin{o}[I+-i][3-16 Iacos2( /2)+ISin(c/2)-(1/2)sin([I+-il-l [3-17] Generally it is easier to calculate the effects of r.f. pulses on Cartesian operators and then use the conversion rules to get the single-element version33 This can be described by the lowering operator I - = |b><a|, the coherent transition from the b to a state by the raising operator I + = |a><b| (coherence order +1). The real Cartesian operators Ix and Iy correspond to mixtures of both coherence orders, ±1, although they are more useful for directly corresponding to the observable x and y components of the magnetization. Their relationship with the complex I ± operators is simple: I + = Ix + iIy raising operator Ix = 1 /2 (I+ + I- ) I - = Ix - iIy lowering operator Iy = - i /2 (I+ - I- ) I a = 1 /2 1 + Iz polarisation operator (a) Iz = 1 /2 (Ia - Ib ) I b = 1 /2 1 - Iz polarisation operator (b) 1 = Ia + Ib The effect of r.f. pulses (here: an x pulse with flip angle j) on single-element operators is as follows: jx I +/- ¾¾¾®I +/-cos2{j/2} + I-/+sin2{j/2} (+/- iIz sin{j}) [3-15] jx I b ¾¾¾®I bcos2{j/2} + Iasin2{j/2} + (1/2)sin{j}[I+ - iI- ] [3-16] jx Ia ¾¾¾®Iacos2{j/2} + Ibsin2{j/2} - (1/2)sin{j}[I+ - iI- ] [3-17] Generally it is easier to calculate the effects of r.f. pulses on Cartesian operators and then use the conversion rules to get the single-element version