By Gheorghe D. Mateescu, Adrian Valeriu
Not like so much really good books in this topic that emphasize the mathematical foundations of second NMR, this publication deals an experimentally-based, transparent rationalization of the density-matrix and product operator remedies hence permitting non-experts to determine sequences of various levels of complexity.
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Extra info for 2D Nmr: Density Matrix and Product Operator Treatment
The matrix element B is transferred in the slow (24) "slot" and will evolve from now on with the slow frequency W24. The reverse is happening to the matrix element C. 6b t(4) clearly shows that the second evolution te/2 completes the decoupling of proton from carbon. The fast vector 13 catches up with the slow 24 and at t(4) they coincide. They have both precessed a total angle WHte from their starting position along - y. 26)]. The transverse magnetization, calculated from the matrix elements, is 24 Density Matrix Treatment MTH = - ( MoH / 4 )[ 2 i exp( iW H te ) + 2 i exp( iW H te )] = - iMoH exp( iW H te ) After separating the real and imaginary parts in MTH we obtain MTH = - iMoH (cos W H te + i sin W H te ) = MoH (sin W H te - i cos W H te ) MxH = real part of MTH = MoH sin W H te MyH = coefficient of the imaginary part of MTH = - MoH cos W H te This is in full accordance with the vector representation.
7). The receiver phase is matched with the desired signal. It shall be seen that the different phase behavior of the coupled nuclei is connected with their double-quantum coherence. The beauty of INADEQUATE resides in its basic simplicity: only a two-step cycle is theoretically needed to eliminate the unwanted signal. That the real life sequences may reach 128 or more steps is exclusively due to hardware (pulse) imperfections whose effects must be corrected by additional phase cycling. 7. 7. − 90x t(3) t(4) t(5) t(6) t(7) t(8) The INADEQUATE sequence: 90x − τ − 180x − τ − ∆ − 90Φ − AT throughout the experiment).
2 The Second Pulse The rotation operator for the 90yAX pulse can be obtained by multiplying R90yA by R90yX. These operators are (see Appendix C): R90 yA ⎡1 ⎢ 1 ⎢ −1 = 2⎢0 ⎢ ⎣0 1 1 0 0 0 0 1 −1 0⎤ ⎡1 0 ⎥ ⎢ 0⎥ 1 ⎢0 1 ; R90 yX = 1⎥ 2 ⎢ −1 0 ⎥ ⎢ 1⎦ ⎣ 0 −1 1 0 1 0 0⎤ 1 ⎥⎥ 0⎥ ⎥ 1⎦ The result of the multiplication, R = R90 yAX , is shown below together with its reciprocal, R −1 . 97) shows that c and − is are interchanged in all terms. This will lead to the desired phase modulation.