Two Dimensional NMR --------COSY

The COSY pulse sequence is very simple. We are going to work out the detailed process step by step since it is the simplest and most useful 2D experiment. The equations are very long, but they quite simple.  If you have not read the previous sections, you may check the terms time to time.  If you completely lost, then you may have to read previous sections to understand the basic theory in NMR.

We consider two coupled spins I1 and I2, at equilibrium (delay), the spin one magnetization is I1Z.
 After the first 90 degree pulse, this magnetization will rotate to - I1Y . During the time of t1, it evolves under its own chemical shift (ω1) and the influence of Spin 2 (I2 ) with a coupling constant J12 :

Note:

Spin One at equilibrium  and Spin Two

Spin One and Spin Two after the first 90 degree pulse:
  .       .

Spin One during the t1 evolution time with chemical shift (ω1 ): ----- (Spin Two is doing the same.)

Spin One coupled with Spin Two with the coupling constant J12 during the t1:

After the t1 evolution time t1, a second 90 degree pulse applied:

After the second 90 degree pulse, the acquisition starts:
The terms in Red are not observable:
First term, it is along with the Z magnetization; and
the second term is a double-quantum coherence, it not observable too.

The term:
To be continued

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