Magnetization in the Rotating Frame

Both the mathematical treatments and pictorial representations of many physical phenomena can be simplified by transforming to a coordinate system that moves in some way, rather than one which is fixed in the laboratory. We shall find it convenient to define a coordinate system, or frame of reference, that rotates about B0 at a rate revolutions/second, or corf radians/second, where the frequencies 

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Fig. 2.8. The components v(t) and (/) of the transverse magnetization in the rotating frame of reference for the off-resonance case...

Here a> denotes the frequency axis of the spectrum, while i2o is the precession frequency of the magnetization in the rotating frame. Clearly, the maximum signal amplitude is obtained for a flip angle ot = 90° of the rf pulse. 

It also turns out that we can detect both the x- and y-components of the magnetization in the rotating frame. From now on we will work exclusively in the rotating frame. 

In summary, the output of the mixer is at the offset frequency in the rotating frame and takes the form of a cosine modulation it is thus equivalent to detecting the x component of the magnetization in the rotating frame. 

Fig. 5b this is represented by a tipping of the magnetization in the rotating frame. The magnitude of M along Hn is equal to zero in Fig. 7b. The system is clearly not in equilibrium with H0, and if the rf field, Hj, is removed at this point, the...

The excitation field is the only field seen by the magnetization in the rotating frame. The magnetization processes about it. Starting Ifom equilibrium (Mq=MqM ), transverse components are created and develop according to... 

We now go into pulse NMR in more detail after learning a few Fourier transform theorems. The second section considers the behavior of the magnetization in the rotating frame under pulsed irradiation and how the spectrum is affected, a very important topic. The third section is a discussion of quadrature detection with an educational as well as a practical aim. 

A hypothetical sample is shown in Fig. 7.16. This sample has 5 spins at x = —3, y = +2 cm and 1 spin at x = 1, y = 0 cm. We assume that all spins will, in the absence of any magnetic field gradients, have exactly the same resonance frequency. The evolution of the magnetization in the rotating frame illustrates how the sample spin density distribution is imaged. 

Fig. 7.17 Magnetization in the rotating frame My —) and The acquired data ( ) is taken...

Neglecting the relaxation terms, the equation of motion of the magnetization in the rotating frame associated with can be expressed as ... 

Similar relationships can be derived for ROESY. In this case, the cross peaks are generated by cross relaxation of transverse magnetization. In the rotating frame, o is given by Equation [4]... 

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