G. Wider/Progress in Nuclear Magnetic Resonance Spectroscopy 32(1998)193-275 compensate for the phase change experienced by the The experiment in Fig. 2 starts at time point a in coherence,k takes on values (0, 1, 2, 3, .. An thermal equilibrium(Fig 3)where the populations on example for the design of a phase cycle using the upper Pu and the lower energy level P, across system is given in Section 4.2.1 with the discussion transition fulfil the Boltzmann distribution(Eq. (6) of the double quantum filter. a detailed discussion of Because AE in Eq (1) is typically much smaller than phase cycling can be found in most textbooks on kT we can approximate this exponential distribution NMR,e.g.[6,21,24,261 by the first term in a Taylor expansion 2. 1.3. Descriptive representations of experimental △E △E Pu/P=exp( h kT An NMr experiment can be graphically described to a limited extent based on a classical physical model where k is Boltzmann's constant and T the absolute using populations and magnetization vectors in the temperature. With Eq (1)the energies E1,E,E3,and rotating frame or based on quantum mechanical E4 of the four different energy levels in the system can principles using the product operator formalism. be calculated Both descriptions find widespread applications for =h(-n-Bs+J/2)/2 the discussion and development of NMR experiments The different representations are discussed on the E2=h(-v1+us -J/2)/2 basis of the scheme shown in Fig. 2. The application of this experiment to a system of two scalar coupled E3=h(n-s-J2)/2 spins I and s is described with four different represen tations in Fig 3 for each of the five time points a to e. E4=h(1+s +J/2)/2 Fig.3 represents energy level diagrams(E), the where vi and vs stand for the resonance frequencies of observable spectra( S), magnetization vector diagrams the I and S nuclei, respectively. The resonance fre V)and the notation in the product operator formalism quencies of nuclei with positive gyromagnetic ratio (O)showing the cartesian and the shift operator y such as protons and carbons are negative(Eq(2), basis. In Fig. 3 spins I and s are assumed to be proton and, hence, E, becomes the highest and E4 the lowest and carbon nuclei, respectively, with the size of energy(Fig 3). For nuclei with a positive y value the representative vectors proportional to the correspond- a state(spin 2) has lower energy than the B state(spin ing populations, but qualitatively the figure applies ). The polarizations Mf and Mi are proportional to all nuclei with spin and positive gyromagnetic to the energy differences(E+- E2) and(E3-ED ratIo respectively, and they determine the intensity of the orresponding transitions 2+ 4(24)and 1+3(13). The consistent use of signs and transformation proper ties as presented in Fig. 3 may seem not to be of great importance and, indeed, has very often no direct experimental consequences. But there are situations where inconsistencies occur and the interpretation of Fig. 2. Sketch of the experimental scheme used for the discussion of data becomes confusing or wrong [31, 411 different representations shown in Fig. 3. The black narrow bars from the consistent illustration of different indicate 90 RF pulses. five time-points on the time axis I are representations for the description of an NMR experi denoted by the letters a, b. c, d and e. The RF pulses applied on- ment, Fig. 3 demonstrates that the scheme shown in sonance to the two species of nuclei I and S are indicated on the ig. 2 transfers polarization from proton to carbon nes marked with the corresponding letters, a particular pulse acts uclear species. The scalar coupling between the two pins. At time point d the proton polarization Mi is separated by the time inverted. As a consequence the populations across the rod(2). The phases of the RF pulses are indicated by x or y at arbon transitions 12 and 34 acquire a larger differ he top of the pulses, where x or y stand for the application of the B nce than at thermal equilibrium at time point a and held in the rotating frame along the positive r or y axis, respectively. hence, the experimental scheme allows measurement