Position k space

Neglecting the frequency spread in (5.4.8) by setting s t, r) = 1, the linear impulse response assumes the form of a scattered wave (cf. eqn (2.2.22)) [Man4, Man5], 

The FID (5.4.11) acquired in the presence of a time-invariant gradient defines the signal along a straight line in k space. For example, if the gradient is applied in x-direction, the FID is given by 

For this FID ky =k = 0, so that t otk. Consequently, yi (t) in (5.4.12) defines a cross-section pikjc) in k space along the kx-axis. This cross-section is the Fourier transform of P(x), and P(x) is a ID projection of the 3D distribution Mo(x, y, z) of longitudinal magnetization onto the x-axis corresponding to the integral of Mo(x, y, z) over the space dimensions y and z (cf. Fig. 1.1.5). In the mathematical sense, a projection is just the integral over a function of many variables, so that fewer variables remain. Fourier transformation over t leads to 

Here the delta function introduces the proportionality between frequency and space, 

Equation (5.4.13) states that the NMR spectrum Ti(o)) acquired in the presence of a magnetic-field gradient G produces a projection P —u / yGx)) of the spin density Mo x,y,z)- In comparison to that, the FID (5.4.12) provides a trace through k space. This relationship is known as the projection-cross-section theorem. 

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