The Rotating Frame of Reference

or CH3) is achieved with a ubiquitous building block of NMR pulse sequences the spin echo. 

To understand these new building blocks, we need to look in detail at the motion of the net magnetization vector during a delay as it precesses in the x-y plane. This motion is called evolution because the net magnetization changes with time or evolves as it 

Note that we use A v to refer to the rotating-frame frequency (sometimes called the resonance offset). This is the difference between the Larmor frequency and the reference frequency v0 - vr. The above equation shows that the same physical law expressed in the equation on the left-hand side (precession rate is proportional to y and to B0) is operating in the equation on the right-hand side (resonance offset is proportional to y and to fires) in the rotating frame of reference, as long as we introduce the pseudofield. In the NMR spectrum, A v is the distance from the center of the spectral window to the NMR peak (Fig. 6.2), also represented as 2 in units of radians per second. If the peak is in the downfield half (left half) of the spectrum, the Larmor frequency is greater than the reference frequency ( v0 vr) and we have a positive resonance offset (A v 0). This corresponds to the motion of the net magnetization 

To aid the visualisation of processes occurring during an NMR experiment, a number of simple conceptual changes are employed. Firstly, the oscillating Bi field is considered to be composed of two counter-rotating magnetic vectors in the x-y plane, the resultant 

Strictly, one should use the different co-ordinate labelling scheme for the laboratory and the rotating frames, such as x, y, z and x , /, z, respectively, as in Fig. 2.6. However, since we shall be dealing almost exclusively with a rotating frame description of events, the simpler x, y, z notations will be used throughout the remainder of the book, and explicit indication provided where the laboratory frame of reference is used. 

The example above made use of a 90 pulse, that is a 90° pulse in which the Bi field was applied along the x axis. It is, however, possible to apply the pulse with arbitrary phase, say along any of the axes x, y, —x, or —y as required, which translates to a different starting phase of the excited magnetisation vector. The spectra provided by these pulses show resonances whose phases similarly differ by 90°. The detection system of the spectrometer designates one axis to represent the positive absorption signal (defined by a reeeiver 

The idea of applying a sequence of pulses of different phase angles is of central importance to all NMR experiments. The process of repeating a 

---

The quantitative formulation of chemical exchange involves modification of the Bloch equations making use of Eq. (4-67). We will merely develop a qualitative view of the result." We adopt a coordinate system that is rotating about the applied field Hq in the same direction as the precessing magnetization vector. Let and Vb be the Larmor precessional frequencies of the nucleus in sites A and B. Eor simplicity we set ta = tb- As the frequency Vq of the rotating frame of reference we choose the average of Va and Vb, thus. 

The first pseudo force, Fi, is called the Coriolis force, and its magnitude is directly proportional to the angular velocity of the rotating frame of reference and the linear velocity of the particle in this frame. By definition, this force is perpendicular to the plane where vectors Vi and o are located, Fig. 2.3a, and depends on the mutual position of these vectors. The second fictitious force, F2, is called the centrifugal force. Its magnitude is directly proportional to the square of the angular velocity and the distance from the particle to the center of rotation. It is directed outward from the center and this explains the name of the force. It is obvious that with an increase of the angular velocity the relative contribution of this force... 

Here v, and a, are the velocity and acceleration of the point p in the rotating frame of reference, respectively. Substitution of Equation (2.55) into Newton s second law gives an equation of motion in the non-inertial frame ... 

A transformation into the rotating frame of reference of the nuclear spin system and integrating over all positions then allows us to rewrite eqn (5) as... 

In the eyes of a distant observer using a fixed coordinate system, a meteorite falling in the gravitational field of the earth describes a parabolic path. An observer standing on earth uses the rotating frame of reference of the earth. For him, the complicated path of the falling meteorite simplifies to a straight vertical line. 

From this relation, the time derivative of M in the rotating frame of reference can be calculated ... 

Since only terms with identical dimensions are allowed to be added, a>/y must have the dimension of a magnetic field. Thus, in the rotating frame of reference, the effective field Berr experienced by M differs from B by a term w/y arising from rotation,... 

In the absence of B, the vector M keeps its equilibrium value and position M0 in the z direction. M is thus time-invariant in the rotating frame of reference, so that... 

This means further that the rotational field cojy opposes B0 k in the rotating frame of reference (Fig. 1.6(a)), finally cancelling B0k when the coordinate system rotates at Larmor frequency co0. 

In the rotating frame of reference, the field vector i of the rf field, rotating at angular... 

The third method involves a three pulse sequence, 90 — r — 180° — x — 90°, with a repetition time of tr s. This pulse sequence refocuses the magnetization vector M0 into its equilibrium position within the repetition time, thus representing a pulse driven relaxation acceleration. This technique, known as DEFT NMR [23, 24] (driven equilibrium Fourier transform NMR) can be understood by following the behavior of the magnetization vector Mq under the influence of the pulse sequence in the rotating frame of reference (Fig. 2.17(a-e)). 

---